PC5214 wiki - User contributions [en]

Resonance frequency measurement using a interferometric method

2022-05-01T08:33:21Z

Aucca: /* Discussion */

== Team Members ==
* [[User:Nakarin|Nakarin Jayjong]]
* [[User:Aucca|Joel Auccapuclla]]
* [[User:Xiaoyu|Xiaoyu Nie]]
* [[User:Haotian|Haotian Song]]

== Introduction ==
Interferometers since their invention have been a great tool for experimental physics. Applications cover a wide spectrum of fields such as astronomy, fiber optics, seismology, seismology, velocimetry to say the least. The precision of measurement of small displacements is the key factor, take LIGO as an example. Another important factor is the convenience to set up an interferometer and the low complexity in terms of equipment. This study study aims to measure the resonance frequency of a system, a transducer attached to a mirror, using a Michelson interferometer.

== Theory ==
Our Michelson interferometer setup used a laser, a beam splitter, two mirrors, and a photometer, illustrated in Figure 3. The light propagates from the source to the beam splitter where approximately 50% of the incident light is transmitted and approximately 50% of it is reflected at a right angle. The transmitted light continues to propagate as the reference plane wave to the stationary mirror where it is fully reflected back towards the beam splitter. The reflected light propagates as the target plane wave to the mirror attached to the oscillating loudspeaker. The two waves then recombine at the beam splitter and interfere based on their relative phases based on the idea that the phase shift of the target plane wave varies with the instantaneous position of the speaker mirror combination. Consider a situation in which we setup our interferometer such that no phase shift corresponds to the equilibrium position of the speaker-mirror combination. At this position, we expect constructive interference at the photometer. Likewise, for every λ/2 change in displacement, we expect constructive interference at the photometer because the path length travelled by the target plane wave varies by some integral multiple of λ. Thus, the phase shift is an even multiple of π and the two waves remain in phase. Similarly, when the path length traveled is some odd multiple of λ/4, the phase shift is an odd multiple of π and the two waves interfere destructively.

Consider two coherent waves with the same amplitude A and same frequency <math>\omega</math> that have paths <math>x_1</math> and <math>x_2</math> emitted from different sources that meet together at a point. The net wave at that point can then be given as the real part of the equation

<math>
\psi=Ae^{i(kx_1-\omega t}+Ae^{i(kx_2-\omega t)}=Ae^{-i\omega t}e^{ik\frac{(x_1+x_2)}{2}}[e^{ik\frac{(x_1-x_2)}{2}}e^{-ik\frac{(x_1-x_2)}{2}}]=2Acos[k(\frac{x_1-x_2}{2})]e^{i[k\frac{(x_1+x_2)}{2}-\omega t]}
</math>

where the wave number <math>k=2\pi/\lambda</math>, <math>\lambda</math> is the wavelength of the waves, and c is the phase velocity. The time-averaged intensity of the combined wave is then proportional to the square of the its amplitude. That is,

<math>
I\propto cos^2[\pi(\frac{p_2-p_1}{\lambda})]=cos^2[2\pi(\frac{p_2}{\lambda}-\theta)]
</math>

where <math>\theta</math> is a constant. Now suppose the mirror that the <math>p_2</math> beam is incident to oscillates harmonically back and forth with a frequency <math>f</math> and a displacement <math>B</math>. Then, <math>p_2 =p_2(0)+Bsin(2\pi ft)</math> and the intensity of the detected light varies with time as

<math>
I=A^2cos^2[2\pi(\frac{Bsin(2\pi ft)}{\lambda}-\theta')]
</math>

Note that the frequency of oscillations is itself a sinusoid, meaning that there will be turnaround points in the oscillations.

== Experimental setup ==
As our main source for this experiment, we used a laser diode of wavelength 658.9 nm (as we can see in Figure 6 in the following section). Most red laser diodes are based on either GaInP or AlGaInP, this means we have two semiconductor materials. For one particular configuration, it will allow current flow and produce light amplification. The diode laser is powered by a power supply, see the left top size of Figure 3. Also, the beam coming out of the diode is divergent so an aspheric lens (C110TMD-A) is used to collimate it, while the collimation tube of laser diode is mounted on a homemade aluminum holder as shown in Figure 1.
[[File:Homemade_laser_diode.jpeg|center|thumb|400px|Figure 1: Homemade laser diode.]]

In order to prevent the reverse current through the laser diode, and protect against turn-on transients or spikes in current, we use a protection circuit that includes a resistor, an inductor, and a Schottky diode as shown in Figure 2.

[[File:protection_circuit_v2.jpeg|center|thumb|800px|Figure 2: Protection circuit for laser diode.]]

Our Michelson interferometer consists of a beam splitter BS, which is used to split the laser beam into two beams, and two mirrors M1 and M2. The reflected beam goes to a fixed mirror M1 while the transmitted beam goes to the movable mirror M2. A transducer is attached, using tape, to the mirror M2. The transducer is powered by a signal generator (Aglient 33220A) that is used to vary the vibrating frequency of the mirror. A photodiode PD is used to detect the signal of interference fringes. The schematic Michelson interferometer is shown in Figure 3.
[[File:Schematic Michelson Interferometer.jpeg|center|thumb|600px|Figure 3: Schematic figure of Michelson interferometer. LD is a laser diode, BS is a beam splitter, M1 and M2 are mirrors, PD is a photodiode, and vibration transducer which is driven by a signal generator.]]

The experimental setup Michelson interferometer is shown in figure 4.
[[File:Experimental setup Michelson Interferometer.jpeg|center|thumb|600px|Figure 4: Experimental setup of Michelson interferometer.]]

The interference fringes of the Michelson interferometer are observed when the transducer is turned off in Figure 5.
[[File:Michelson interference fringes.jpeg|center|thumb|600px|Figure 5: Interference fringes which are observed when the transducer is turned off]]

==Experiment Results==
Before we started setting up our interferometer, we measured the wavelength of the laser by an Anstrom WS-7 wavemeter, the wavelength is about 658.9 nm, see Figure 6. Even more, this graph tells us that the Mitsubishi LD is in a single-mode regime, so we proceeded to build the interferometer.
[[File:Wavelength.jpeg|center|thumb|600px|Figure 6: The Wavelength measurement of the Mitsubishi Laser Diode.]]

After the alignment of the optical setup, we applied sinusoidal voltage to the transducer attached to the mirror tightly. intensity fluctuation is recorded by the homemade photodiode. Data from the photodiode is collected by the oscilloscope. The typical figure from the oscilloscope is shown in Figure 7. According to the equation mentioned before, the number of peaks during each period corresponds to the radio of displacement and laser wavelength. In this case, we can measure the displacement for various frequencies. The frequency range is from 86 Hz to 130 Hz.

[[File:ScopeScreen.jpeg|center|thumb|600px|Figure 7A: The display on the oscilloscope at <math>f = 103Hz </math>. After applying the voltage to the transducer, the voltage output from photodiode changes with time.]]

[[File:ult.png|center|thumb|600px|Figure 7B: The display on the oscilloscope at <math>f = 116Hz </math>. After applying the voltage to the transducer, the voltage output from photodiode changes with time.]]

We plot the displacement for different frequencies with the errorbar in Figure 6. Uncertainty is the square root uncertainty of the count number. For a driven, damped, and harmonic oscillator, the relation between displacement and frequency is

<math>|A|=\frac{f_{0}}{\sqrt{\left(\omega_{0}^{2}-\omega^{2}\right)^{2}+4 \beta^{2} \omega^{2}}} </math>

Where <math>f_{0}</math> is the amplitude, <math>\omega_{0}</math> is the resonance frequency, and <math>\beta_{0}</math> is a constant. With this equation, we can fit the frequency response curve. Because there is only one peak or valley for a frequency lower than 92 Hz or higher than 120 Hz, and the error is significant and non-negligible. We exclude those in the fitting process. Finally, the fitting root mean squared error (RMSE) is 130.38. Fitted <math>\beta</math> is <math>1.3 \pm 0.2</math>. The fitting resonance frequency <math>\omega_{0} </math> is <math>103.1 \pm 0.1Hz </math>.

[[File:FitPeakNum.jpeg|center|thumb|600px|Figure 8: Frequency response curve for the transducer system with errorbar. The blue dots represent the displacement measured with different frequencies. Error bars are the square root uncertainty. Red line is the fitting curve for a driven, damped, harmonic oscillator.]]

==Discussion==
In figure 7A, we could see that the nearest you are to the natural resonance frequency of the system, the response of the system to the driven force will coupled and the number of oscillation increase. Meanwhile, in figure 7B the number of oscillations is lower. These phenomena is in concordance with the relation for <math>|A|</math>, in the past section.

Our final result fits our data to the best Lorentzian function. When we say best, we talk about the function with the lowest Root Mean Square Error (RMSE).

RMSE is defined as follow:

<math>\operatorname{RMSE}=\sqrt{\frac{1}{n}\sum_{i=1}^n(A_i-\hat{A_i})^2}</math>

where <math>A_i</math> is the measured displacement and <math>\hat{A_i}</math> is the predicted displacement by the function model. The core of this method is to calculate the distance between the measured and predicted values and take the root mean of them.

The displacements (<math>A_i</math>) we have measured are in the regimen of thousands and the error is ~100. This is attributed to the response of the system to lower and higher frequencies, from the resonance frequency, were we could see they stick together. For near resonance, the model fits pretty well and the fitting method provides a resonance frequency <math>103.1Hz</math>.

Finally, we believe that most of the errors in this experiment came from the fact that the laser diode is a homemade laser diode so it does not have a good temperature control. Moreover, the homemade laser diode holder is not tightly fit to the laser diode collimation tube which can cause displacement of the laser beam.

== Conclusion ==
We have shown indeed that we can characterize the resonance frequency of a system. For this particular application, we could see a resonance frequency of <math>103Hz</math> for the system transducer plus mirror. This could be improve by temperature stabilisation of the laser diode. Finally, this technique has a great potential to be used in initial experimental laboratories due his simplicity to illustrates concepts such as interference and driven oscillator, which are important in advance physics modules.

==References==
[1] Freschi, A. A., et al. "Laser interferometric characterization of a vibrating speaker system." American Journal of Physics 71.11 (2003): 1121-1126.
https://doi.org/10.1119/1.1586262

[2] Skarha, Matthew. "Laser interferometric characterization for a vibrating speaker system."
http://mattskarha.com/assets/docs/Laser%20interferometric%20characterization%20for%20a%20vibrating%20speaker%20system%20good.pdf

[3] Pathare, Shirish, and Vikrant Kurmude. "Low cost Michelson–Morley interferometer." Physics Education 51.6 (2016): 063001.
https://doi.org/10.1088/0031-9120/51/6/063001

Resonance frequency measurement using a interferometric method

2022-05-01T08:31:19Z

Aucca: /* Experiment Results */

== Team Members ==
* [[User:Nakarin|Nakarin Jayjong]]
* [[User:Aucca|Joel Auccapuclla]]
* [[User:Xiaoyu|Xiaoyu Nie]]
* [[User:Haotian|Haotian Song]]

== Introduction ==
Interferometers since their invention have been a great tool for experimental physics. Applications cover a wide spectrum of fields such as astronomy, fiber optics, seismology, seismology, velocimetry to say the least. The precision of measurement of small displacements is the key factor, take LIGO as an example. Another important factor is the convenience to set up an interferometer and the low complexity in terms of equipment. This study study aims to measure the resonance frequency of a system, a transducer attached to a mirror, using a Michelson interferometer.

== Theory ==
Our Michelson interferometer setup used a laser, a beam splitter, two mirrors, and a photometer, illustrated in Figure 3. The light propagates from the source to the beam splitter where approximately 50% of the incident light is transmitted and approximately 50% of it is reflected at a right angle. The transmitted light continues to propagate as the reference plane wave to the stationary mirror where it is fully reflected back towards the beam splitter. The reflected light propagates as the target plane wave to the mirror attached to the oscillating loudspeaker. The two waves then recombine at the beam splitter and interfere based on their relative phases based on the idea that the phase shift of the target plane wave varies with the instantaneous position of the speaker mirror combination. Consider a situation in which we setup our interferometer such that no phase shift corresponds to the equilibrium position of the speaker-mirror combination. At this position, we expect constructive interference at the photometer. Likewise, for every λ/2 change in displacement, we expect constructive interference at the photometer because the path length travelled by the target plane wave varies by some integral multiple of λ. Thus, the phase shift is an even multiple of π and the two waves remain in phase. Similarly, when the path length traveled is some odd multiple of λ/4, the phase shift is an odd multiple of π and the two waves interfere destructively.

Consider two coherent waves with the same amplitude A and same frequency <math>\omega</math> that have paths <math>x_1</math> and <math>x_2</math> emitted from different sources that meet together at a point. The net wave at that point can then be given as the real part of the equation

<math>
\psi=Ae^{i(kx_1-\omega t}+Ae^{i(kx_2-\omega t)}=Ae^{-i\omega t}e^{ik\frac{(x_1+x_2)}{2}}[e^{ik\frac{(x_1-x_2)}{2}}e^{-ik\frac{(x_1-x_2)}{2}}]=2Acos[k(\frac{x_1-x_2}{2})]e^{i[k\frac{(x_1+x_2)}{2}-\omega t]}
</math>

where the wave number <math>k=2\pi/\lambda</math>, <math>\lambda</math> is the wavelength of the waves, and c is the phase velocity. The time-averaged intensity of the combined wave is then proportional to the square of the its amplitude. That is,

<math>
I\propto cos^2[\pi(\frac{p_2-p_1}{\lambda})]=cos^2[2\pi(\frac{p_2}{\lambda}-\theta)]
</math>

where <math>\theta</math> is a constant. Now suppose the mirror that the <math>p_2</math> beam is incident to oscillates harmonically back and forth with a frequency <math>f</math> and a displacement <math>B</math>. Then, <math>p_2 =p_2(0)+Bsin(2\pi ft)</math> and the intensity of the detected light varies with time as

<math>
I=A^2cos^2[2\pi(\frac{Bsin(2\pi ft)}{\lambda}-\theta')]
</math>

Note that the frequency of oscillations is itself a sinusoid, meaning that there will be turnaround points in the oscillations.

== Experimental setup ==
As our main source for this experiment, we used a laser diode of wavelength 658.9 nm (as we can see in Figure 6 in the following section). Most red laser diodes are based on either GaInP or AlGaInP, this means we have two semiconductor materials. For one particular configuration, it will allow current flow and produce light amplification. The diode laser is powered by a power supply, see the left top size of Figure 3. Also, the beam coming out of the diode is divergent so an aspheric lens (C110TMD-A) is used to collimate it, while the collimation tube of laser diode is mounted on a homemade aluminum holder as shown in Figure 1.
[[File:Homemade_laser_diode.jpeg|center|thumb|400px|Figure 1: Homemade laser diode.]]

In order to prevent the reverse current through the laser diode, and protect against turn-on transients or spikes in current, we use a protection circuit that includes a resistor, an inductor, and a Schottky diode as shown in Figure 2.

[[File:protection_circuit_v2.jpeg|center|thumb|800px|Figure 2: Protection circuit for laser diode.]]

Our Michelson interferometer consists of a beam splitter BS, which is used to split the laser beam into two beams, and two mirrors M1 and M2. The reflected beam goes to a fixed mirror M1 while the transmitted beam goes to the movable mirror M2. A transducer is attached, using tape, to the mirror M2. The transducer is powered by a signal generator (Aglient 33220A) that is used to vary the vibrating frequency of the mirror. A photodiode PD is used to detect the signal of interference fringes. The schematic Michelson interferometer is shown in Figure 3.
[[File:Schematic Michelson Interferometer.jpeg|center|thumb|600px|Figure 3: Schematic figure of Michelson interferometer. LD is a laser diode, BS is a beam splitter, M1 and M2 are mirrors, PD is a photodiode, and vibration transducer which is driven by a signal generator.]]

The experimental setup Michelson interferometer is shown in figure 4.
[[File:Experimental setup Michelson Interferometer.jpeg|center|thumb|600px|Figure 4: Experimental setup of Michelson interferometer.]]

The interference fringes of the Michelson interferometer are observed when the transducer is turned off in Figure 5.
[[File:Michelson interference fringes.jpeg|center|thumb|600px|Figure 5: Interference fringes which are observed when the transducer is turned off]]

==Experiment Results==
Before we started setting up our interferometer, we measured the wavelength of the laser by an Anstrom WS-7 wavemeter, the wavelength is about 658.9 nm, see Figure 6. Even more, this graph tells us that the Mitsubishi LD is in a single-mode regime, so we proceeded to build the interferometer.
[[File:Wavelength.jpeg|center|thumb|600px|Figure 6: The Wavelength measurement of the Mitsubishi Laser Diode.]]

After the alignment of the optical setup, we applied sinusoidal voltage to the transducer attached to the mirror tightly. intensity fluctuation is recorded by the homemade photodiode. Data from the photodiode is collected by the oscilloscope. The typical figure from the oscilloscope is shown in Figure 7. According to the equation mentioned before, the number of peaks during each period corresponds to the radio of displacement and laser wavelength. In this case, we can measure the displacement for various frequencies. The frequency range is from 86 Hz to 130 Hz.

[[File:ScopeScreen.jpeg|center|thumb|600px|Figure 7A: The display on the oscilloscope at <math>f = 103Hz </math>. After applying the voltage to the transducer, the voltage output from photodiode changes with time.]]

[[File:ult.png|center|thumb|600px|Figure 7B: The display on the oscilloscope at <math>f = 116Hz </math>. After applying the voltage to the transducer, the voltage output from photodiode changes with time.]]

We plot the displacement for different frequencies with the errorbar in Figure 6. Uncertainty is the square root uncertainty of the count number. For a driven, damped, and harmonic oscillator, the relation between displacement and frequency is

<math>|A|=\frac{f_{0}}{\sqrt{\left(\omega_{0}^{2}-\omega^{2}\right)^{2}+4 \beta^{2} \omega^{2}}} </math>

Where <math>f_{0}</math> is the amplitude, <math>\omega_{0}</math> is the resonance frequency, and <math>\beta_{0}</math> is a constant. With this equation, we can fit the frequency response curve. Because there is only one peak or valley for a frequency lower than 92 Hz or higher than 120 Hz, and the error is significant and non-negligible. We exclude those in the fitting process. Finally, the fitting root mean squared error (RMSE) is 130.38. Fitted <math>\beta</math> is <math>1.3 \pm 0.2</math>. The fitting resonance frequency <math>\omega_{0} </math> is <math>103.1 \pm 0.1Hz </math>.

[[File:FitPeakNum.jpeg|center|thumb|600px|Figure 8: Frequency response curve for the transducer system with errorbar. The blue dots represent the displacement measured with different frequencies. Error bars are the square root uncertainty. Red line is the fitting curve for a driven, damped, harmonic oscillator.]]

==Discussion==
In figure 7A, we could see that the nearest you are to the natural resonance frequency of the system, the response of the system to the driven force will coupled and the number of oscillation increase. Meanwhile, in figure 7B the number of oscillations is lower. These phenomena is in concordance with the relation for <math>|A|</math>, in the past section.

Our final result fits our data to the best Lorentzian function. When we say best, we talk about the function with the lowest Root Mean Square Error (RMSE).

RMSE is defined as follow:

<math>\operatorname{RMSE}=\sqrt{\frac{1}{n}\sum_{i=1}^n(A_i-\hat{A_i})^2}</math>

where <math>A_i</math> is the measured displacement and <math>\hat{A_i}</math> is the predicted displacement by the function model. The core of this method is to calculate the distance between the measured and predicted values and take the root mean of them.

The displacements (<math>A_i</math>) we have measured are in the regimen of thousands and the error is ~100. This is attributed to the response of the system to lower and higher frequencies, from the resonance frequency, were we could see they stick together. However, for near resonance, the model fits pretty well but the fitting method provides a different resonance frequency <math>113.1Hz</math> than the one we can see on Fig. 8, around <math>103Hz</math>.

Finally, we believe that most of the errors in this experiment came from the fact that the laser diode is a homemade laser diode so it does not have a good temperature control. Moreover, the homemade laser diode holder is not tightly fit to the laser diode collimation tube which can cause displacement of the laser beam.

== Conclusion ==
We have shown indeed that we can characterize the resonance frequency of a system. For this particular application, we could see a resonance frequency of <math>103Hz</math> for the system transducer plus mirror. This could be improve by temperature stabilisation of the laser diode. Finally, this technique has a great potential to be used in initial experimental laboratories due his simplicity to illustrates concepts such as interference and driven oscillator, which are important in advance physics modules.

==References==
[1] Freschi, A. A., et al. "Laser interferometric characterization of a vibrating speaker system." American Journal of Physics 71.11 (2003): 1121-1126.
https://doi.org/10.1119/1.1586262

[2] Skarha, Matthew. "Laser interferometric characterization for a vibrating speaker system."
http://mattskarha.com/assets/docs/Laser%20interferometric%20characterization%20for%20a%20vibrating%20speaker%20system%20good.pdf

[3] Pathare, Shirish, and Vikrant Kurmude. "Low cost Michelson–Morley interferometer." Physics Education 51.6 (2016): 063001.
https://doi.org/10.1088/0031-9120/51/6/063001

An interferometric method for measuring the resonance frequency of vibrating system

2022-05-01T06:54:47Z

Aucca: /* Conclusion */

== Team Members ==
* [[User:Nakarin|Nakarin Jayjong]]
* [[User:Aucca|Joel Auccapuclla]]
* [[User:Xiaoyu|Xiaoyu Nie]]
* [[User:Haotian|Haotian Song]]

== Introduction ==
Interferometers since their invention have been a great tool for experimental physics. Applications cover a wide spectrum of fields such as astronomy, fiber optics, seismology, seismology, velocimetry to say the least. The precision of measurement of small displacements is the key factor, take LIGO as an example. Another important factor is the convenience to set up an interferometer and the low complexity in terms of equipment. In this experiment, we will measure the resonance frequency of a vibrating transducer using a Michelson interferometer. Basically, a transducer is attached to one optical path, to one mirror, of the Michelson interferometer and it is expected to see a peak in the response curve for the resonance frequency of the transducer plus mirror.

== Theory ==
Our Michelson interferometer setup used a laser, a beam splitter, two mirrors, and a photometer, illustrated in Fig. 1. Original models used sodium flame as the coherent light source. The light propagates from the source to the beam splitter where approximately 50% of the incident light is transmitted and approximately 50% of it is reflected at a right angle. The transmitted light continues to propagate as the reference plane wave to the stationary mirror where it is fully reflected back towards the beam splitter. The reflected light propagates as the target plane wave to the mirror attached to the oscillating loudspeaker. The two waves then recombine at the beam splitter and interfere based on their relative phases based on the idea that the phase shift of the target plane wave varies with the instantaneous position of the speaker mirror combination. Consider a situation in which we setup our interferometer such that no phase shift corresponds to the equilibrium position of the speaker-mirror combination. At this position, we expect constructive interference at the photometer. Likewise, for every λ/2 change in displacement, we expect constructive interference at the photometer because the path length travelled by the target plane wave varies by some integral multiple of λ. Thus, the phase shift is an even multiple of π and the two waves remain in phase. Similarly, when the path length traveled is some odd multiple of λ/4, the phase shift is an odd multiple of π and the two waves interfere destructively.

Consider two coherent waves with the same amplitude A and same frequency <math>\omega</math> that have paths <math>x_1</math> and <math>x_2</math> emitted from different sources that meet together at a point. The net wave at that point can then be given as the real part of the equation

<math>
\psi=Ae^{i(kx_1-\omega t}+Ae^{i(kx_2-\omega t)}=Ae^{-i\omega t}e^{ik\frac{(x_1+x_2)}{2}}[e^{ik\frac{(x_1-x_2)}{2}}e^{-ik\frac{(x_1-x_2)}{2}}]=2Acos[k(\frac{x_1-x_2}{2})]e^{i[k\frac{(x_1+x_2)}{2}-\omega t]}
</math>

where the wave number <math>k=2\pi/\lambda</math>, <math>\lambda</math> is the wavelength of the waves, and c is the phase velocity. The time-averaged intensity of the combined wave is then proportional to the square of the its amplitude. That is,

<math>
I\propto cos^2[\pi(\frac{p_2-p_1}{\lambda})]=cos^2[2\pi(\frac{p_2}{\lambda}-\theta)]
</math>

where <math>\theta</math> is a constant. Now suppose the mirror that the <math>p_2</math> beam is incident to oscillates harmonically back and forth with a frequency <math>f</math> and a displacement <math>B</math>. Then, <math>p_2 =p_2(0)+Bsin(2\pi ft)</math> and the intensity of the detected light varies with time as

<math>
I=A^2cos^2[2\pi(\frac{Bsin(2\pi ft)}{\lambda}-\theta')]
</math>

Note that the frequency of oscillations is itself a sinusoid, meaning that there will be turnaround points in the oscillations.

== Experimental setup ==
The Michelson interferometer consists of a beam splitter BS which is used to split the laser beam into two beams and two mirrors M1 and M2. The reflected beam goes to a fixed mirror M1 while the transmitted beam goes to the movable mirror M2. A transducer is attached to the mirror M2. The transducer is powered by a signal generator that is used to vary the vibrating frequency of the mirror. A photodiode PD is used to detect the signal of interference fringes. The schematic Michelson interferometer is shown in Figure 1.
[[File:Schematic Michelson Interferometer.jpeg|center|thumb|600px|Figure 1: Schematic figure of Michelson interferometer. LD is a laser diode, BS is a beam splitter, M1 and M2 are mirrors, PD is a photodiode, and vibration transducer which is driven by a signal generator.]]

The experimental setup Michelson interferometer is shown in figure 2.
[[File:Experimental setup Michelson Interferometer.jpeg|center|thumb|600px|Figure 2: Experimental setup of Michelson interferometer.]]

The interference fringes of the Michelson interferometer are observed when the transducer is turned off in Figure 3.
[[File:Michelson interference fringes.jpeg|center|thumb|600px|Figure 3: Interference fringes which are observed when the transducer is turned off]]

==Experiment Results==
After the alignment of the optical setup, we applied sinusoidal voltage to the transducer attached to the mirror tightly. intensity fluctuation is recorded by the homemade photodiode. Data from the photodiode is collected by the oscilloscope. The typical figure from the oscilloscope is shown in Figure 4. According to the equation mentioned before, the number of peaks during each period corresponds to the radio of displacement and laser wavelength. In this case, we can measure the displacement for various frequencies. The frequency range is from 86 Hz to 130 Hz.

[[File:ScopeScreen.jpeg|center|thumb|600px|Figure 4: The display on the oscilloscope at <math>f = 103Hz </math>. After applying the voltage to the transducer, the voltage output from photodiode changes with time.]]

In our setup, we measured the wavelength of Mitsubishi Laser Diode which is 658.9 nm. The wavemeter measurement results are shown here.
[[File:Wavelength.jpeg|center|thumb|600px|Figure 5: The Wavelength measurement of the Mitsubishi Laser Diode.]]

We plot the displacement for different frequencies with the errorbar in Figure 6. Uncertainty is the square root uncertainty of the count number. For a driven, damped and harmonic oscillator, the relation between displacement and frequency is

<math>|A|=\frac{f_{0}}{\sqrt{\left(\omega_{0}^{2}-\omega^{2}\right)^{2}+4 \beta^{2} \omega^{2}}} </math>

Where <math>f_{0}</math> is the amplitude, <math>\omega_{0}</math> is the resonance frequency, and <math>\beta_{0}</math> is a constant. With this equation, we can fit the frequency response curve. Because there is only one peak or valley for a frequency lower than 92 Hz or higher than 120 Hz, and the error is significant and non-negligible. We exclude those in the fitting process. Finally, the fitting root mean squared error is 130.38. Fitted <math>\beta</math> is <math>1.3 \pm 0.2</math>. The fitting resonance frequency <math>\omega_{0} </math> is <math>103.1 \pm 0.1Hz </math>.

[[File:FitPeakNum.jpeg|center|thumb|600px|Figure 6: Frequency response curve for the transducer system with errorbar. The blue dots represent the displacement measured with different frequencies. Error bars are the square root uncertainty. Red line is the fitting curve for a driven, damped, harmonic oscillator.]]

== Conclusion ==
We have shown indeed that we can characterize the resonance frequency of a system. For this particular application, we found a frequency of 103.12Hz for the system transducer plus mirror. Finally, with appropriate handling, this technique could be implemented to characterize more sofisticated systems.

==References==
[1] Freschi, A. A., et al. "Laser interferometric characterization of a vibrating speaker system." American Journal of Physics 71.11 (2003): 1121-1126.
https://doi.org/10.1119/1.1586262

[2] Skarha, Matthew. "Laser interferometric characterization for a vibrating speaker system."
http://mattskarha.com/assets/docs/Laser%20interferometric%20characterization%20for%20a%20vibrating%20speaker%20system%20good.pdf

[3] Pathare, Shirish, and Vikrant Kurmude. "Low cost Michelson–Morley interferometer." Physics Education 51.6 (2016): 063001.
https://doi.org/10.1088/0031-9120/51/6/063001

An interferometric method for measuring the resonance frequency of vibrating system

2022-05-01T06:53:27Z

Aucca: /* Experiment Results */

== Team Members ==
* [[User:Nakarin|Nakarin Jayjong]]
* [[User:Aucca|Joel Auccapuclla]]
* [[User:Xiaoyu|Xiaoyu Nie]]
* [[User:Haotian|Haotian Song]]

== Introduction ==
Interferometers since their invention have been a great tool for experimental physics. Applications cover a wide spectrum of fields such as astronomy, fiber optics, seismology, seismology, velocimetry to say the least. The precision of measurement of small displacements is the key factor, take LIGO as an example. Another important factor is the convenience to set up an interferometer and the low complexity in terms of equipment. In this experiment, we will measure the resonance frequency of a vibrating transducer using a Michelson interferometer. Basically, a transducer is attached to one optical path, to one mirror, of the Michelson interferometer and it is expected to see a peak in the response curve for the resonance frequency of the transducer plus mirror.

== Theory ==
Our Michelson interferometer setup used a laser, a beam splitter, two mirrors, and a photometer, illustrated in Fig. 1. Original models used sodium flame as the coherent light source. The light propagates from the source to the beam splitter where approximately 50% of the incident light is transmitted and approximately 50% of it is reflected at a right angle. The transmitted light continues to propagate as the reference plane wave to the stationary mirror where it is fully reflected back towards the beam splitter. The reflected light propagates as the target plane wave to the mirror attached to the oscillating loudspeaker. The two waves then recombine at the beam splitter and interfere based on their relative phases based on the idea that the phase shift of the target plane wave varies with the instantaneous position of the speaker mirror combination. Consider a situation in which we setup our interferometer such that no phase shift corresponds to the equilibrium position of the speaker-mirror combination. At this position, we expect constructive interference at the photometer. Likewise, for every λ/2 change in displacement, we expect constructive interference at the photometer because the path length travelled by the target plane wave varies by some integral multiple of λ. Thus, the phase shift is an even multiple of π and the two waves remain in phase. Similarly, when the path length traveled is some odd multiple of λ/4, the phase shift is an odd multiple of π and the two waves interfere destructively.

Consider two coherent waves with the same amplitude A and same frequency <math>\omega</math> that have paths <math>x_1</math> and <math>x_2</math> emitted from different sources that meet together at a point. The net wave at that point can then be given as the real part of the equation

<math>
\psi=Ae^{i(kx_1-\omega t}+Ae^{i(kx_2-\omega t)}=Ae^{-i\omega t}e^{ik\frac{(x_1+x_2)}{2}}[e^{ik\frac{(x_1-x_2)}{2}}e^{-ik\frac{(x_1-x_2)}{2}}]=2Acos[k(\frac{x_1-x_2}{2})]e^{i[k\frac{(x_1+x_2)}{2}-\omega t]}
</math>

where the wave number <math>k=2\pi/\lambda</math>, <math>\lambda</math> is the wavelength of the waves, and c is the phase velocity. The time-averaged intensity of the combined wave is then proportional to the square of the its amplitude. That is,

<math>
I\propto cos^2[\pi(\frac{p_2-p_1}{\lambda})]=cos^2[2\pi(\frac{p_2}{\lambda}-\theta)]
</math>

where <math>\theta</math> is a constant. Now suppose the mirror that the <math>p_2</math> beam is incident to oscillates harmonically back and forth with a frequency <math>f</math> and a displacement <math>B</math>. Then, <math>p_2 =p_2(0)+Bsin(2\pi ft)</math> and the intensity of the detected light varies with time as

<math>
I=A^2cos^2[2\pi(\frac{Bsin(2\pi ft)}{\lambda}-\theta')]
</math>

Note that the frequency of oscillations is itself a sinusoid, meaning that there will be turnaround points in the oscillations.

== Experimental setup ==
The Michelson interferometer consists of a beam splitter BS which is used to split the laser beam into two beams and two mirrors M1 and M2. The reflected beam goes to a fixed mirror M1 while the transmitted beam goes to the movable mirror M2. A transducer is attached to the mirror M2. The transducer is powered by a signal generator that is used to vary the vibrating frequency of the mirror. A photodiode PD is used to detect the signal of interference fringes. The schematic Michelson interferometer is shown in Figure 1.
[[File:Schematic Michelson Interferometer.jpeg|center|thumb|600px|Figure 1: Schematic figure of Michelson interferometer. LD is a laser diode, BS is a beam splitter, M1 and M2 are mirrors, PD is a photodiode, and vibration transducer which is driven by a signal generator.]]

The experimental setup Michelson interferometer is shown in figure 2.
[[File:Experimental setup Michelson Interferometer.jpeg|center|thumb|600px|Figure 2: Experimental setup of Michelson interferometer.]]

The interference fringes of the Michelson interferometer are observed when the transducer is turned off in Figure 3.
[[File:Michelson interference fringes.jpeg|center|thumb|600px|Figure 3: Interference fringes which are observed when the transducer is turned off]]

==Experiment Results==
After the alignment of the optical setup, we applied sinusoidal voltage to the transducer attached to the mirror tightly. intensity fluctuation is recorded by the homemade photodiode. Data from the photodiode is collected by the oscilloscope. The typical figure from the oscilloscope is shown in Figure 4. According to the equation mentioned before, the number of peaks during each period corresponds to the radio of displacement and laser wavelength. In this case, we can measure the displacement for various frequencies. The frequency range is from 86 Hz to 130 Hz.

[[File:ScopeScreen.jpeg|center|thumb|600px|Figure 4: The display on the oscilloscope at <math>f = 103Hz </math>. After applying the voltage to the transducer, the voltage output from photodiode changes with time.]]

In our setup, we measured the wavelength of Mitsubishi Laser Diode which is 658.9 nm. The wavemeter measurement results are shown here.
[[File:Wavelength.jpeg|center|thumb|600px|Figure 5: The Wavelength measurement of the Mitsubishi Laser Diode.]]

We plot the displacement for different frequencies with the errorbar in Figure 6. Uncertainty is the square root uncertainty of the count number. For a driven, damped and harmonic oscillator, the relation between displacement and frequency is

<math>|A|=\frac{f_{0}}{\sqrt{\left(\omega_{0}^{2}-\omega^{2}\right)^{2}+4 \beta^{2} \omega^{2}}} </math>

Where <math>f_{0}</math> is the amplitude, <math>\omega_{0}</math> is the resonance frequency, and <math>\beta_{0}</math> is a constant. With this equation, we can fit the frequency response curve. Because there is only one peak or valley for a frequency lower than 92 Hz or higher than 120 Hz, and the error is significant and non-negligible. We exclude those in the fitting process. Finally, the fitting root mean squared error is 130.38. Fitted <math>\beta</math> is <math>1.3 \pm 0.2</math>. The fitting resonance frequency <math>\omega_{0} </math> is <math>103.1 \pm 0.1Hz </math>.

[[File:FitPeakNum.jpeg|center|thumb|600px|Figure 6: Frequency response curve for the transducer system with errorbar. The blue dots represent the displacement measured with different frequencies. Error bars are the square root uncertainty. Red line is the fitting curve for a driven, damped, harmonic oscillator.]]

== Conclusion ==
We have shown indeed that we can characterize the resonance frequency of a system. For this particular application, we found a frequency of 113.12Hz for the system transducer plus mirror. Finally, with appropriate handling, this technique could be implemented to characterize more sofisticated systems.

==References==
[1] Freschi, A. A., et al. "Laser interferometric characterization of a vibrating speaker system." American Journal of Physics 71.11 (2003): 1121-1126.
https://doi.org/10.1119/1.1586262

[2] Skarha, Matthew. "Laser interferometric characterization for a vibrating speaker system."
http://mattskarha.com/assets/docs/Laser%20interferometric%20characterization%20for%20a%20vibrating%20speaker%20system%20good.pdf

[3] Pathare, Shirish, and Vikrant Kurmude. "Low cost Michelson–Morley interferometer." Physics Education 51.6 (2016): 063001.
https://doi.org/10.1088/0031-9120/51/6/063001

Resonance frequency measurement using a interferometric method

2022-04-30T15:55:22Z

Aucca: /* Discussion */

== Team Members ==
* [[User:Nakarin|Nakarin Jayjong]]
* [[User:Aucca|Joel Auccapuclla]]
* [[User:Xiaoyu|Xiaoyu Nie]]
* [[User:Haotian|Haotian Song]]

== Introduction ==
Interferometers since their invention have been a great tool for experimental physics. Applications cover a wide spectrum of fields such as astronomy, fiber optics, seismology, seismology, velocimetry to say the least. The precision of measurement of small displacements is the key factor, take LIGO as an example. Another important factor is the convenience to set up an interferometer and the low complexity in terms of equipment. This study study aims to measure the resonance frequency of a system, a transducer attached to a mirror, using a Michelson interferometer.

== Theory ==
Our Michelson interferometer setup used a laser, a beam splitter, two mirrors, and a photometer, illustrated in Figure 3. The light propagates from the source to the beam splitter where approximately 50% of the incident light is transmitted and approximately 50% of it is reflected at a right angle. The transmitted light continues to propagate as the reference plane wave to the stationary mirror where it is fully reflected back towards the beam splitter. The reflected light propagates as the target plane wave to the mirror attached to the oscillating loudspeaker. The two waves then recombine at the beam splitter and interfere based on their relative phases based on the idea that the phase shift of the target plane wave varies with the instantaneous position of the speaker mirror combination. Consider a situation in which we setup our interferometer such that no phase shift corresponds to the equilibrium position of the speaker-mirror combination. At this position, we expect constructive interference at the photometer. Likewise, for every λ/2 change in displacement, we expect constructive interference at the photometer because the path length travelled by the target plane wave varies by some integral multiple of λ. Thus, the phase shift is an even multiple of π and the two waves remain in phase. Similarly, when the path length traveled is some odd multiple of λ/4, the phase shift is an odd multiple of π and the two waves interfere destructively.

Consider two coherent waves with the same amplitude A and same frequency <math>\omega</math> that have paths <math>x_1</math> and <math>x_2</math> emitted from different sources that meet together at a point. The net wave at that point can then be given as the real part of the equation

<math>
\psi=Ae^{i(kx_1-\omega t}+Ae^{i(kx_2-\omega t)}=Ae^{-i\omega t}e^{ik\frac{(x_1+x_2)}{2}}[e^{ik\frac{(x_1-x_2)}{2}}e^{-ik\frac{(x_1-x_2)}{2}}]=2Acos[k(\frac{x_1-x_2}{2})]e^{i[k\frac{(x_1+x_2)}{2}-\omega t]}
</math>

where the wave number <math>k=2\pi/\lambda</math>, <math>\lambda</math> is the wavelength of the waves, and c is the phase velocity. The time-averaged intensity of the combined wave is then proportional to the square of the its amplitude. That is,

<math>
I\propto cos^2[\pi(\frac{p_2-p_1}{\lambda})]=cos^2[2\pi(\frac{p_2}{\lambda}-\theta)]
</math>

where <math>\theta</math> is a constant. Now suppose the mirror that the <math>p_2</math> beam is incident to oscillates harmonically back and forth with a frequency <math>f</math> and a displacement <math>B</math>. Then, <math>p_2 =p_2(0)+Bsin(2\pi ft)</math> and the intensity of the detected light varies with time as

<math>
I=A^2cos^2[2\pi(\frac{Bsin(2\pi ft)}{\lambda}-\theta')]
</math>

Note that the frequency of oscillations is itself a sinusoid, meaning that there will be turnaround points in the oscillations.

== Experimental setup ==
As our main source for this experiment, we used a laser diode of wavelength 658.9 nm (as we can see in Figure 6 in the following section). Most red laser diodes are based on either GaInP or AlGaInP, this means we have two semiconductor materials. For one particular configuration, it will allow current flow and produce light amplification. The diode laser is powered by a power supply, see the left top size of Figure 3. Also, the beam coming out of the diode is divergent so an aspheric lens (C110TMD-A) is used to collimate it, while the collimation tube of laser diode is mounted on a homemade aluminum holder as shown in Figure 1.
[[File:Homemade_laser_diode.jpeg|center|thumb|400px|Figure 1: Homemade laser diode.]]

In order to prevent the reverse current through the laser diode, and protect against turn-on transients or spikes in current, we use a protection circuit that includes a resistor, an inductor, and a Schottky diode as shown in Figure 2.

[[File:protection_circuit_v2.jpeg|center|thumb|800px|Figure 2: Protection circuit for laser diode.]]

Our Michelson interferometer consists of a beam splitter BS, which is used to split the laser beam into two beams, and two mirrors M1 and M2. The reflected beam goes to a fixed mirror M1 while the transmitted beam goes to the movable mirror M2. A transducer is attached, using tape, to the mirror M2. The transducer is powered by a signal generator (Aglient 33220A) that is used to vary the vibrating frequency of the mirror. A photodiode PD is used to detect the signal of interference fringes. The schematic Michelson interferometer is shown in Figure 3.
[[File:Schematic Michelson Interferometer.jpeg|center|thumb|600px|Figure 3: Schematic figure of Michelson interferometer. LD is a laser diode, BS is a beam splitter, M1 and M2 are mirrors, PD is a photodiode, and vibration transducer which is driven by a signal generator.]]

The experimental setup Michelson interferometer is shown in figure 4.
[[File:Experimental setup Michelson Interferometer.jpeg|center|thumb|600px|Figure 4: Experimental setup of Michelson interferometer.]]

The interference fringes of the Michelson interferometer are observed when the transducer is turned off in Figure 5.
[[File:Michelson interference fringes.jpeg|center|thumb|600px|Figure 5: Interference fringes which are observed when the transducer is turned off]]

==Experiment Results==
Before we started setting up our interferometer, we measured the wavelength of the laser by an Anstrom WS-7 wavemeter, the wavelength is about 658.9 nm, see Figure 6. Even more, this graph tells us that the Mitsubishi LD is in a single-mode regime, so we proceeded to build the interferometer.
[[File:Wavelength.jpeg|center|thumb|600px|Figure 6: The Wavelength measurement of the Mitsubishi Laser Diode.]]

After the alignment of the optical setup, we applied sinusoidal voltage to the transducer attached to the mirror tightly. intensity fluctuation is recorded by the homemade photodiode. Data from the photodiode is collected by the oscilloscope. The typical figure from the oscilloscope is shown in Figure 7. According to the equation mentioned before, the number of peaks during each period corresponds to the radio of displacement and laser wavelength. In this case, we can measure the displacement for various frequencies. The frequency range is from 86 Hz to 130 Hz.

[[File:ScopeScreen.jpeg|center|thumb|600px|Figure 7A: The display on the oscilloscope at <math>f = 103Hz </math>. After applying the voltage to the transducer, the voltage output from photodiode changes with time.]]

[[File:ult.png|center|thumb|600px|Figure 7B: The display on the oscilloscope at <math>f = 116Hz </math>. After applying the voltage to the transducer, the voltage output from photodiode changes with time.]]

We plot the displacement for different frequencies with the errorbar in Figure 6. Uncertainty is the square root uncertainty of the count number. For a driven, damped, and harmonic oscillator, the relation between displacement and frequency is

<math>|A|=\frac{f_{0}}{\sqrt{\left(\omega_{0}^{2}-\omega^{2}\right)^{2}+4 \beta^{2} \omega^{2}}} </math>

Where <math>f_{0}</math> is the amplitude, <math>\omega_{0}</math> is the resonance frequency, and <math>\beta_{0}</math> is a constant. With this equation, we can fit the frequency response curve. Because there is only one peak or valley for a frequency lower than 92 Hz or higher than 120 Hz, and the error is significant and non-negligible. We exclude those in the fitting process. Finally, the fitting root mean squared error (RMSE) is 130.38. Fitted <math>\beta</math> is <math>1.3 \pm 0.2</math>. The fitting resonance frequency <math>\omega_{0} </math> is <math>113.1 \pm 0.1Hz </math>.

[[File:FitPeakNum.jpeg|center|thumb|600px|Figure 8: Frequency response curve for the transducer system with errorbar. The blue dots represent the displacement measured with different frequencies. Error bars are the square root uncertainty. Red line is the fitting curve for a driven, damped, harmonic oscillator.]]

==Discussion==
In figure 7A, we could see that the nearest you are to the natural resonance frequency of the system, the response of the system to the driven force will coupled and the number of oscillation increase. Meanwhile, in figure 7B the number of oscillations is lower. These phenomena is in concordance with the relation for <math>|A|</math>, in the past section.

Our final result fits our data to the best Lorentzian function. When we say best, we talk about the function with the lowest Root Mean Square Error (RMSE).

RMSE is defined as follow:

<math>\operatorname{RMSE}=\sqrt{\frac{1}{n}\sum_{i=1}^n(A_i-\hat{A_i})^2}</math>

where <math>A_i</math> is the measured displacement and <math>\hat{A_i}</math> is the predicted displacement by the function model. The core of this method is to calculate the distance between the measured and predicted values and take the root mean of them.

The displacements (<math>A_i</math>) we have measured are in the regimen of thousands and the error is ~100. This is attributed to the response of the system to lower and higher frequencies, from the resonance frequency, were we could see they stick together. However, for near resonance, the model fits pretty well but the fitting method provides a different resonance frequency <math>113.1Hz</math> than the one we can see on Fig. 8, around <math>103Hz</math>.

Finally, we believe that most of the errors in this experiment came from the fact that the laser diode is a homemade laser diode so it does not have a good temperature control. Moreover, the homemade laser diode holder is not tightly fit to the laser diode collimation tube which can cause displacement of the laser beam.

== Conclusion ==
We have shown indeed that we can characterize the resonance frequency of a system. For this particular application, we could see a resonance frequency of <math>103Hz</math> for the system transducer plus mirror. This could be improve by temperature stabilisation of the laser diode. Finally, this technique has a great potential to be used in initial experimental laboratories due his simplicity to illustrates concepts such as interference and driven oscillator, which are important in advance physics modules.

==References==
[1] Freschi, A. A., et al. "Laser interferometric characterization of a vibrating speaker system." American Journal of Physics 71.11 (2003): 1121-1126.
https://doi.org/10.1119/1.1586262

[2] Skarha, Matthew. "Laser interferometric characterization for a vibrating speaker system."
http://mattskarha.com/assets/docs/Laser%20interferometric%20characterization%20for%20a%20vibrating%20speaker%20system%20good.pdf

[3] Pathare, Shirish, and Vikrant Kurmude. "Low cost Michelson–Morley interferometer." Physics Education 51.6 (2016): 063001.
https://doi.org/10.1088/0031-9120/51/6/063001

Resonance frequency measurement using a interferometric method

2022-04-30T15:54:25Z

Aucca: /* Discussion */

== Team Members ==
* [[User:Nakarin|Nakarin Jayjong]]
* [[User:Aucca|Joel Auccapuclla]]
* [[User:Xiaoyu|Xiaoyu Nie]]
* [[User:Haotian|Haotian Song]]

== Introduction ==
Interferometers since their invention have been a great tool for experimental physics. Applications cover a wide spectrum of fields such as astronomy, fiber optics, seismology, seismology, velocimetry to say the least. The precision of measurement of small displacements is the key factor, take LIGO as an example. Another important factor is the convenience to set up an interferometer and the low complexity in terms of equipment. This study study aims to measure the resonance frequency of a system, a transducer attached to a mirror, using a Michelson interferometer.

== Theory ==
Our Michelson interferometer setup used a laser, a beam splitter, two mirrors, and a photometer, illustrated in Figure 3. The light propagates from the source to the beam splitter where approximately 50% of the incident light is transmitted and approximately 50% of it is reflected at a right angle. The transmitted light continues to propagate as the reference plane wave to the stationary mirror where it is fully reflected back towards the beam splitter. The reflected light propagates as the target plane wave to the mirror attached to the oscillating loudspeaker. The two waves then recombine at the beam splitter and interfere based on their relative phases based on the idea that the phase shift of the target plane wave varies with the instantaneous position of the speaker mirror combination. Consider a situation in which we setup our interferometer such that no phase shift corresponds to the equilibrium position of the speaker-mirror combination. At this position, we expect constructive interference at the photometer. Likewise, for every λ/2 change in displacement, we expect constructive interference at the photometer because the path length travelled by the target plane wave varies by some integral multiple of λ. Thus, the phase shift is an even multiple of π and the two waves remain in phase. Similarly, when the path length traveled is some odd multiple of λ/4, the phase shift is an odd multiple of π and the two waves interfere destructively.

Consider two coherent waves with the same amplitude A and same frequency <math>\omega</math> that have paths <math>x_1</math> and <math>x_2</math> emitted from different sources that meet together at a point. The net wave at that point can then be given as the real part of the equation

<math>
\psi=Ae^{i(kx_1-\omega t}+Ae^{i(kx_2-\omega t)}=Ae^{-i\omega t}e^{ik\frac{(x_1+x_2)}{2}}[e^{ik\frac{(x_1-x_2)}{2}}e^{-ik\frac{(x_1-x_2)}{2}}]=2Acos[k(\frac{x_1-x_2}{2})]e^{i[k\frac{(x_1+x_2)}{2}-\omega t]}
</math>

where the wave number <math>k=2\pi/\lambda</math>, <math>\lambda</math> is the wavelength of the waves, and c is the phase velocity. The time-averaged intensity of the combined wave is then proportional to the square of the its amplitude. That is,

<math>
I\propto cos^2[\pi(\frac{p_2-p_1}{\lambda})]=cos^2[2\pi(\frac{p_2}{\lambda}-\theta)]
</math>

where <math>\theta</math> is a constant. Now suppose the mirror that the <math>p_2</math> beam is incident to oscillates harmonically back and forth with a frequency <math>f</math> and a displacement <math>B</math>. Then, <math>p_2 =p_2(0)+Bsin(2\pi ft)</math> and the intensity of the detected light varies with time as

<math>
I=A^2cos^2[2\pi(\frac{Bsin(2\pi ft)}{\lambda}-\theta')]
</math>

Note that the frequency of oscillations is itself a sinusoid, meaning that there will be turnaround points in the oscillations.

== Experimental setup ==
As our main source for this experiment, we used a laser diode of wavelength 658.9 nm (as we can see in Figure 6 in the following section). Most red laser diodes are based on either GaInP or AlGaInP, this means we have two semiconductor materials. For one particular configuration, it will allow current flow and produce light amplification. The diode laser is powered by a power supply, see the left top size of Figure 3. Also, the beam coming out of the diode is divergent so an aspheric lens (C110TMD-A) is used to collimate it, while the collimation tube of laser diode is mounted on a homemade aluminum holder as shown in Figure 1.
[[File:Homemade_laser_diode.jpeg|center|thumb|400px|Figure 1: Homemade laser diode.]]

In order to prevent the reverse current through the laser diode, and protect against turn-on transients or spikes in current, we use a protection circuit that includes a resistor, an inductor, and a Schottky diode as shown in Figure 2.

[[File:protection_circuit_v2.jpeg|center|thumb|800px|Figure 2: Protection circuit for laser diode.]]

Our Michelson interferometer consists of a beam splitter BS, which is used to split the laser beam into two beams, and two mirrors M1 and M2. The reflected beam goes to a fixed mirror M1 while the transmitted beam goes to the movable mirror M2. A transducer is attached, using tape, to the mirror M2. The transducer is powered by a signal generator (Aglient 33220A) that is used to vary the vibrating frequency of the mirror. A photodiode PD is used to detect the signal of interference fringes. The schematic Michelson interferometer is shown in Figure 3.
[[File:Schematic Michelson Interferometer.jpeg|center|thumb|600px|Figure 3: Schematic figure of Michelson interferometer. LD is a laser diode, BS is a beam splitter, M1 and M2 are mirrors, PD is a photodiode, and vibration transducer which is driven by a signal generator.]]

The experimental setup Michelson interferometer is shown in figure 4.
[[File:Experimental setup Michelson Interferometer.jpeg|center|thumb|600px|Figure 4: Experimental setup of Michelson interferometer.]]

The interference fringes of the Michelson interferometer are observed when the transducer is turned off in Figure 5.
[[File:Michelson interference fringes.jpeg|center|thumb|600px|Figure 5: Interference fringes which are observed when the transducer is turned off]]

==Experiment Results==
Before we started setting up our interferometer, we measured the wavelength of the laser by an Anstrom WS-7 wavemeter, the wavelength is about 658.9 nm, see Figure 6. Even more, this graph tells us that the Mitsubishi LD is in a single-mode regime, so we proceeded to build the interferometer.
[[File:Wavelength.jpeg|center|thumb|600px|Figure 6: The Wavelength measurement of the Mitsubishi Laser Diode.]]

After the alignment of the optical setup, we applied sinusoidal voltage to the transducer attached to the mirror tightly. intensity fluctuation is recorded by the homemade photodiode. Data from the photodiode is collected by the oscilloscope. The typical figure from the oscilloscope is shown in Figure 7. According to the equation mentioned before, the number of peaks during each period corresponds to the radio of displacement and laser wavelength. In this case, we can measure the displacement for various frequencies. The frequency range is from 86 Hz to 130 Hz.

[[File:ScopeScreen.jpeg|center|thumb|600px|Figure 7A: The display on the oscilloscope at <math>f = 103Hz </math>. After applying the voltage to the transducer, the voltage output from photodiode changes with time.]]

[[File:ult.png|center|thumb|600px|Figure 7B: The display on the oscilloscope at <math>f = 116Hz </math>. After applying the voltage to the transducer, the voltage output from photodiode changes with time.]]

We plot the displacement for different frequencies with the errorbar in Figure 6. Uncertainty is the square root uncertainty of the count number. For a driven, damped, and harmonic oscillator, the relation between displacement and frequency is

<math>|A|=\frac{f_{0}}{\sqrt{\left(\omega_{0}^{2}-\omega^{2}\right)^{2}+4 \beta^{2} \omega^{2}}} </math>

Where <math>f_{0}</math> is the amplitude, <math>\omega_{0}</math> is the resonance frequency, and <math>\beta_{0}</math> is a constant. With this equation, we can fit the frequency response curve. Because there is only one peak or valley for a frequency lower than 92 Hz or higher than 120 Hz, and the error is significant and non-negligible. We exclude those in the fitting process. Finally, the fitting root mean squared error (RMSE) is 130.38. Fitted <math>\beta</math> is <math>1.3 \pm 0.2</math>. The fitting resonance frequency <math>\omega_{0} </math> is <math>113.1 \pm 0.1Hz </math>.

[[File:FitPeakNum.jpeg|center|thumb|600px|Figure 8: Frequency response curve for the transducer system with errorbar. The blue dots represent the displacement measured with different frequencies. Error bars are the square root uncertainty. Red line is the fitting curve for a driven, damped, harmonic oscillator.]]

==Discussion==
In figure 7A, we could see that the nearest you are to the natural resonance frequency of the system, the response of the system to the driven force will coupled and the number of oscillation increase. Meanwhile, in figure 7B the number of oscillations is lower. These phenomena is in concordance with the relation for <math>|A|</math>, in the past section.

Our final result fits our data to the best Lorentzian function. When we say best, we talk about the function with the lowest Root Mean Square Error (RMSE).

RMSE is defined as follow:

<math>\operatorname{RMSE}=\sqrt{\frac{1}{n}\sum_{i=1}^n(A_i-\hat{A_i})^2}</math>

where <math>A_i</math> is the measured displacement and <math>\hat{A_i}</math> is the predicted displacement by the function model. The core of this method is to calculate the distance between the measured and predicted values and take the root mean of them.

The displacements (<math>A_i</math>) we have measured are in the regimen of thousands and the error is ~100. This is attributed to the response of the system to lower and higher frequencies were we could see they stick together. However, for near resonance, the model fits pretty well but the fitting method provides a different resonance frequency <math>113.1Hz</math> than the one we can see on Fig. 8, around <math>103Hz</math>.

Finally, we believe that most of the errors in this experiment came from the fact that the laser diode is a homemade laser diode so it does not have a good temperature control. Moreover, the homemade laser diode holder is not tightly fit to the laser diode collimation tube which can cause displacement of the laser beam.

== Conclusion ==
We have shown indeed that we can characterize the resonance frequency of a system. For this particular application, we could see a resonance frequency of <math>103Hz</math> for the system transducer plus mirror. This could be improve by temperature stabilisation of the laser diode. Finally, this technique has a great potential to be used in initial experimental laboratories due his simplicity to illustrates concepts such as interference and driven oscillator, which are important in advance physics modules.

==References==
[1] Freschi, A. A., et al. "Laser interferometric characterization of a vibrating speaker system." American Journal of Physics 71.11 (2003): 1121-1126.
https://doi.org/10.1119/1.1586262

[2] Skarha, Matthew. "Laser interferometric characterization for a vibrating speaker system."
http://mattskarha.com/assets/docs/Laser%20interferometric%20characterization%20for%20a%20vibrating%20speaker%20system%20good.pdf

[3] Pathare, Shirish, and Vikrant Kurmude. "Low cost Michelson–Morley interferometer." Physics Education 51.6 (2016): 063001.
https://doi.org/10.1088/0031-9120/51/6/063001

Resonance frequency measurement using a interferometric method

2022-04-30T15:49:57Z

Aucca: /* Conclusion */

== Team Members ==
* [[User:Nakarin|Nakarin Jayjong]]
* [[User:Aucca|Joel Auccapuclla]]
* [[User:Xiaoyu|Xiaoyu Nie]]
* [[User:Haotian|Haotian Song]]

== Introduction ==
Interferometers since their invention have been a great tool for experimental physics. Applications cover a wide spectrum of fields such as astronomy, fiber optics, seismology, seismology, velocimetry to say the least. The precision of measurement of small displacements is the key factor, take LIGO as an example. Another important factor is the convenience to set up an interferometer and the low complexity in terms of equipment. This study study aims to measure the resonance frequency of a system, a transducer attached to a mirror, using a Michelson interferometer.

== Theory ==
Our Michelson interferometer setup used a laser, a beam splitter, two mirrors, and a photometer, illustrated in Figure 3. The light propagates from the source to the beam splitter where approximately 50% of the incident light is transmitted and approximately 50% of it is reflected at a right angle. The transmitted light continues to propagate as the reference plane wave to the stationary mirror where it is fully reflected back towards the beam splitter. The reflected light propagates as the target plane wave to the mirror attached to the oscillating loudspeaker. The two waves then recombine at the beam splitter and interfere based on their relative phases based on the idea that the phase shift of the target plane wave varies with the instantaneous position of the speaker mirror combination. Consider a situation in which we setup our interferometer such that no phase shift corresponds to the equilibrium position of the speaker-mirror combination. At this position, we expect constructive interference at the photometer. Likewise, for every λ/2 change in displacement, we expect constructive interference at the photometer because the path length travelled by the target plane wave varies by some integral multiple of λ. Thus, the phase shift is an even multiple of π and the two waves remain in phase. Similarly, when the path length traveled is some odd multiple of λ/4, the phase shift is an odd multiple of π and the two waves interfere destructively.

Consider two coherent waves with the same amplitude A and same frequency <math>\omega</math> that have paths <math>x_1</math> and <math>x_2</math> emitted from different sources that meet together at a point. The net wave at that point can then be given as the real part of the equation

<math>
\psi=Ae^{i(kx_1-\omega t}+Ae^{i(kx_2-\omega t)}=Ae^{-i\omega t}e^{ik\frac{(x_1+x_2)}{2}}[e^{ik\frac{(x_1-x_2)}{2}}e^{-ik\frac{(x_1-x_2)}{2}}]=2Acos[k(\frac{x_1-x_2}{2})]e^{i[k\frac{(x_1+x_2)}{2}-\omega t]}
</math>

where the wave number <math>k=2\pi/\lambda</math>, <math>\lambda</math> is the wavelength of the waves, and c is the phase velocity. The time-averaged intensity of the combined wave is then proportional to the square of the its amplitude. That is,

<math>
I\propto cos^2[\pi(\frac{p_2-p_1}{\lambda})]=cos^2[2\pi(\frac{p_2}{\lambda}-\theta)]
</math>

where <math>\theta</math> is a constant. Now suppose the mirror that the <math>p_2</math> beam is incident to oscillates harmonically back and forth with a frequency <math>f</math> and a displacement <math>B</math>. Then, <math>p_2 =p_2(0)+Bsin(2\pi ft)</math> and the intensity of the detected light varies with time as

<math>
I=A^2cos^2[2\pi(\frac{Bsin(2\pi ft)}{\lambda}-\theta')]
</math>

Note that the frequency of oscillations is itself a sinusoid, meaning that there will be turnaround points in the oscillations.

== Experimental setup ==
As our main source for this experiment, we used a laser diode of wavelength 658.9 nm (as we can see in Figure 6 in the following section). Most red laser diodes are based on either GaInP or AlGaInP, this means we have two semiconductor materials. For one particular configuration, it will allow current flow and produce light amplification. The diode laser is powered by a power supply, see the left top size of Figure 3. Also, the beam coming out of the diode is divergent so an aspheric lens (C110TMD-A) is used to collimate it, while the collimation tube of laser diode is mounted on a homemade aluminum holder as shown in Figure 1.
[[File:Homemade_laser_diode.jpeg|center|thumb|400px|Figure 1: Homemade laser diode.]]

In order to prevent the reverse current through the laser diode, and protect against turn-on transients or spikes in current, we use a protection circuit that includes a resistor, an inductor, and a Schottky diode as shown in Figure 2.

[[File:protection_circuit_v2.jpeg|center|thumb|800px|Figure 2: Protection circuit for laser diode.]]

Our Michelson interferometer consists of a beam splitter BS, which is used to split the laser beam into two beams, and two mirrors M1 and M2. The reflected beam goes to a fixed mirror M1 while the transmitted beam goes to the movable mirror M2. A transducer is attached, using tape, to the mirror M2. The transducer is powered by a signal generator (Aglient 33220A) that is used to vary the vibrating frequency of the mirror. A photodiode PD is used to detect the signal of interference fringes. The schematic Michelson interferometer is shown in Figure 3.
[[File:Schematic Michelson Interferometer.jpeg|center|thumb|600px|Figure 3: Schematic figure of Michelson interferometer. LD is a laser diode, BS is a beam splitter, M1 and M2 are mirrors, PD is a photodiode, and vibration transducer which is driven by a signal generator.]]

The experimental setup Michelson interferometer is shown in figure 4.
[[File:Experimental setup Michelson Interferometer.jpeg|center|thumb|600px|Figure 4: Experimental setup of Michelson interferometer.]]

The interference fringes of the Michelson interferometer are observed when the transducer is turned off in Figure 5.
[[File:Michelson interference fringes.jpeg|center|thumb|600px|Figure 5: Interference fringes which are observed when the transducer is turned off]]

==Experiment Results==
Before we started setting up our interferometer, we measured the wavelength of the laser by an Anstrom WS-7 wavemeter, the wavelength is about 658.9 nm, see Figure 6. Even more, this graph tells us that the Mitsubishi LD is in a single-mode regime, so we proceeded to build the interferometer.
[[File:Wavelength.jpeg|center|thumb|600px|Figure 6: The Wavelength measurement of the Mitsubishi Laser Diode.]]

After the alignment of the optical setup, we applied sinusoidal voltage to the transducer attached to the mirror tightly. intensity fluctuation is recorded by the homemade photodiode. Data from the photodiode is collected by the oscilloscope. The typical figure from the oscilloscope is shown in Figure 7. According to the equation mentioned before, the number of peaks during each period corresponds to the radio of displacement and laser wavelength. In this case, we can measure the displacement for various frequencies. The frequency range is from 86 Hz to 130 Hz.

[[File:ScopeScreen.jpeg|center|thumb|600px|Figure 7A: The display on the oscilloscope at <math>f = 103Hz </math>. After applying the voltage to the transducer, the voltage output from photodiode changes with time.]]

[[File:ult.png|center|thumb|600px|Figure 7B: The display on the oscilloscope at <math>f = 116Hz </math>. After applying the voltage to the transducer, the voltage output from photodiode changes with time.]]

We plot the displacement for different frequencies with the errorbar in Figure 6. Uncertainty is the square root uncertainty of the count number. For a driven, damped, and harmonic oscillator, the relation between displacement and frequency is

<math>|A|=\frac{f_{0}}{\sqrt{\left(\omega_{0}^{2}-\omega^{2}\right)^{2}+4 \beta^{2} \omega^{2}}} </math>

Where <math>f_{0}</math> is the amplitude, <math>\omega_{0}</math> is the resonance frequency, and <math>\beta_{0}</math> is a constant. With this equation, we can fit the frequency response curve. Because there is only one peak or valley for a frequency lower than 92 Hz or higher than 120 Hz, and the error is significant and non-negligible. We exclude those in the fitting process. Finally, the fitting root mean squared error (RMSE) is 130.38. Fitted <math>\beta</math> is <math>1.3 \pm 0.2</math>. The fitting resonance frequency <math>\omega_{0} </math> is <math>113.1 \pm 0.1Hz </math>.

[[File:FitPeakNum.jpeg|center|thumb|600px|Figure 8: Frequency response curve for the transducer system with errorbar. The blue dots represent the displacement measured with different frequencies. Error bars are the square root uncertainty. Red line is the fitting curve for a driven, damped, harmonic oscillator.]]

==Discussion==
In figure 7A, we could see that the nearest you are to the natural resonance frequency of the system, the response of the system to the driven force will coupled and the number of oscillation increase. Meanwhile, in figure 7B the number of oscillations is lower. These phenomena is in concordance with the relation for <math>|A|</math>, in the past section.

Our final result fits our data to the best Lorentzian function. When we say best, we talk about the function with the lowest Root Mean Square Error (RMSE).

RMSE is defined as follow:

<math>\operatorname{RMSE}=\sqrt{\frac{1}{n}\sum_{i=1}^n(A_i-\hat{A_i})^2}</math>

where <math>A_i</math> is the measured displacement and <math>\hat{A_i}</math> is the predicted displacement by the function model. The core of this method is to calculate the distance between the measured and predicted values and take the root mean of them.

The displacements (<math>A_i</math>) we have measured are in the regimen of thousands and the error is ~100. This is attributed to the response of the system to lower frequencies. However, for near resonance, the model fits pretty well but the fitting method provides a different resonance frequency <math>113.1Hz</math> than the one we can see on Fig. 8, around <math>103Hz</math>.

Finally, we believe that most of the errors in this experiment came from the fact that the laser diode is a homemade laser diode so it does not have a good temperature control. Moreover, the homemade laser diode holder is not tightly fit to the laser diode collimation tube which can cause displacement of the laser beam.

== Conclusion ==
We have shown indeed that we can characterize the resonance frequency of a system. For this particular application, we could see a resonance frequency of <math>103Hz</math> for the system transducer plus mirror. This could be improve by temperature stabilisation of the laser diode. Finally, this technique has a great potential to be used in initial experimental laboratories due his simplicity to illustrates concepts such as interference and driven oscillator, which are important in advance physics modules.

==References==
[1] Freschi, A. A., et al. "Laser interferometric characterization of a vibrating speaker system." American Journal of Physics 71.11 (2003): 1121-1126.
https://doi.org/10.1119/1.1586262

[2] Skarha, Matthew. "Laser interferometric characterization for a vibrating speaker system."
http://mattskarha.com/assets/docs/Laser%20interferometric%20characterization%20for%20a%20vibrating%20speaker%20system%20good.pdf

[3] Pathare, Shirish, and Vikrant Kurmude. "Low cost Michelson–Morley interferometer." Physics Education 51.6 (2016): 063001.
https://doi.org/10.1088/0031-9120/51/6/063001

Resonance frequency measurement using a interferometric method

2022-04-30T15:49:10Z

Aucca: /* Discussion */

== Team Members ==
* [[User:Nakarin|Nakarin Jayjong]]
* [[User:Aucca|Joel Auccapuclla]]
* [[User:Xiaoyu|Xiaoyu Nie]]
* [[User:Haotian|Haotian Song]]

== Introduction ==
Interferometers since their invention have been a great tool for experimental physics. Applications cover a wide spectrum of fields such as astronomy, fiber optics, seismology, seismology, velocimetry to say the least. The precision of measurement of small displacements is the key factor, take LIGO as an example. Another important factor is the convenience to set up an interferometer and the low complexity in terms of equipment. This study study aims to measure the resonance frequency of a system, a transducer attached to a mirror, using a Michelson interferometer.

== Theory ==
Our Michelson interferometer setup used a laser, a beam splitter, two mirrors, and a photometer, illustrated in Figure 3. The light propagates from the source to the beam splitter where approximately 50% of the incident light is transmitted and approximately 50% of it is reflected at a right angle. The transmitted light continues to propagate as the reference plane wave to the stationary mirror where it is fully reflected back towards the beam splitter. The reflected light propagates as the target plane wave to the mirror attached to the oscillating loudspeaker. The two waves then recombine at the beam splitter and interfere based on their relative phases based on the idea that the phase shift of the target plane wave varies with the instantaneous position of the speaker mirror combination. Consider a situation in which we setup our interferometer such that no phase shift corresponds to the equilibrium position of the speaker-mirror combination. At this position, we expect constructive interference at the photometer. Likewise, for every λ/2 change in displacement, we expect constructive interference at the photometer because the path length travelled by the target plane wave varies by some integral multiple of λ. Thus, the phase shift is an even multiple of π and the two waves remain in phase. Similarly, when the path length traveled is some odd multiple of λ/4, the phase shift is an odd multiple of π and the two waves interfere destructively.

Consider two coherent waves with the same amplitude A and same frequency <math>\omega</math> that have paths <math>x_1</math> and <math>x_2</math> emitted from different sources that meet together at a point. The net wave at that point can then be given as the real part of the equation

<math>
\psi=Ae^{i(kx_1-\omega t}+Ae^{i(kx_2-\omega t)}=Ae^{-i\omega t}e^{ik\frac{(x_1+x_2)}{2}}[e^{ik\frac{(x_1-x_2)}{2}}e^{-ik\frac{(x_1-x_2)}{2}}]=2Acos[k(\frac{x_1-x_2}{2})]e^{i[k\frac{(x_1+x_2)}{2}-\omega t]}
</math>

where the wave number <math>k=2\pi/\lambda</math>, <math>\lambda</math> is the wavelength of the waves, and c is the phase velocity. The time-averaged intensity of the combined wave is then proportional to the square of the its amplitude. That is,

<math>
I\propto cos^2[\pi(\frac{p_2-p_1}{\lambda})]=cos^2[2\pi(\frac{p_2}{\lambda}-\theta)]
</math>

where <math>\theta</math> is a constant. Now suppose the mirror that the <math>p_2</math> beam is incident to oscillates harmonically back and forth with a frequency <math>f</math> and a displacement <math>B</math>. Then, <math>p_2 =p_2(0)+Bsin(2\pi ft)</math> and the intensity of the detected light varies with time as

<math>
I=A^2cos^2[2\pi(\frac{Bsin(2\pi ft)}{\lambda}-\theta')]
</math>

Note that the frequency of oscillations is itself a sinusoid, meaning that there will be turnaround points in the oscillations.

== Experimental setup ==
As our main source for this experiment, we used a laser diode of wavelength 658.9 nm (as we can see in Figure 6 in the following section). Most red laser diodes are based on either GaInP or AlGaInP, this means we have two semiconductor materials. For one particular configuration, it will allow current flow and produce light amplification. The diode laser is powered by a power supply, see the left top size of Figure 3. Also, the beam coming out of the diode is divergent so an aspheric lens (C110TMD-A) is used to collimate it, while the collimation tube of laser diode is mounted on a homemade aluminum holder as shown in Figure 1.
[[File:Homemade_laser_diode.jpeg|center|thumb|400px|Figure 1: Homemade laser diode.]]

In order to prevent the reverse current through the laser diode, and protect against turn-on transients or spikes in current, we use a protection circuit that includes a resistor, an inductor, and a Schottky diode as shown in Figure 2.

[[File:protection_circuit_v2.jpeg|center|thumb|800px|Figure 2: Protection circuit for laser diode.]]

Our Michelson interferometer consists of a beam splitter BS, which is used to split the laser beam into two beams, and two mirrors M1 and M2. The reflected beam goes to a fixed mirror M1 while the transmitted beam goes to the movable mirror M2. A transducer is attached, using tape, to the mirror M2. The transducer is powered by a signal generator (Aglient 33220A) that is used to vary the vibrating frequency of the mirror. A photodiode PD is used to detect the signal of interference fringes. The schematic Michelson interferometer is shown in Figure 3.
[[File:Schematic Michelson Interferometer.jpeg|center|thumb|600px|Figure 3: Schematic figure of Michelson interferometer. LD is a laser diode, BS is a beam splitter, M1 and M2 are mirrors, PD is a photodiode, and vibration transducer which is driven by a signal generator.]]

The experimental setup Michelson interferometer is shown in figure 4.
[[File:Experimental setup Michelson Interferometer.jpeg|center|thumb|600px|Figure 4: Experimental setup of Michelson interferometer.]]

The interference fringes of the Michelson interferometer are observed when the transducer is turned off in Figure 5.
[[File:Michelson interference fringes.jpeg|center|thumb|600px|Figure 5: Interference fringes which are observed when the transducer is turned off]]

==Experiment Results==
Before we started setting up our interferometer, we measured the wavelength of the laser by an Anstrom WS-7 wavemeter, the wavelength is about 658.9 nm, see Figure 6. Even more, this graph tells us that the Mitsubishi LD is in a single-mode regime, so we proceeded to build the interferometer.
[[File:Wavelength.jpeg|center|thumb|600px|Figure 6: The Wavelength measurement of the Mitsubishi Laser Diode.]]

After the alignment of the optical setup, we applied sinusoidal voltage to the transducer attached to the mirror tightly. intensity fluctuation is recorded by the homemade photodiode. Data from the photodiode is collected by the oscilloscope. The typical figure from the oscilloscope is shown in Figure 7. According to the equation mentioned before, the number of peaks during each period corresponds to the radio of displacement and laser wavelength. In this case, we can measure the displacement for various frequencies. The frequency range is from 86 Hz to 130 Hz.

[[File:ScopeScreen.jpeg|center|thumb|600px|Figure 7A: The display on the oscilloscope at <math>f = 103Hz </math>. After applying the voltage to the transducer, the voltage output from photodiode changes with time.]]

[[File:ult.png|center|thumb|600px|Figure 7B: The display on the oscilloscope at <math>f = 116Hz </math>. After applying the voltage to the transducer, the voltage output from photodiode changes with time.]]

We plot the displacement for different frequencies with the errorbar in Figure 6. Uncertainty is the square root uncertainty of the count number. For a driven, damped, and harmonic oscillator, the relation between displacement and frequency is

<math>|A|=\frac{f_{0}}{\sqrt{\left(\omega_{0}^{2}-\omega^{2}\right)^{2}+4 \beta^{2} \omega^{2}}} </math>

Where <math>f_{0}</math> is the amplitude, <math>\omega_{0}</math> is the resonance frequency, and <math>\beta_{0}</math> is a constant. With this equation, we can fit the frequency response curve. Because there is only one peak or valley for a frequency lower than 92 Hz or higher than 120 Hz, and the error is significant and non-negligible. We exclude those in the fitting process. Finally, the fitting root mean squared error (RMSE) is 130.38. Fitted <math>\beta</math> is <math>1.3 \pm 0.2</math>. The fitting resonance frequency <math>\omega_{0} </math> is <math>113.1 \pm 0.1Hz </math>.

[[File:FitPeakNum.jpeg|center|thumb|600px|Figure 8: Frequency response curve for the transducer system with errorbar. The blue dots represent the displacement measured with different frequencies. Error bars are the square root uncertainty. Red line is the fitting curve for a driven, damped, harmonic oscillator.]]

==Discussion==
In figure 7A, we could see that the nearest you are to the natural resonance frequency of the system, the response of the system to the driven force will coupled and the number of oscillation increase. Meanwhile, in figure 7B the number of oscillations is lower. These phenomena is in concordance with the relation for <math>|A|</math>, in the past section.

Our final result fits our data to the best Lorentzian function. When we say best, we talk about the function with the lowest Root Mean Square Error (RMSE).

RMSE is defined as follow:

<math>\operatorname{RMSE}=\sqrt{\frac{1}{n}\sum_{i=1}^n(A_i-\hat{A_i})^2}</math>

where <math>A_i</math> is the measured displacement and <math>\hat{A_i}</math> is the predicted displacement by the function model. The core of this method is to calculate the distance between the measured and predicted values and take the root mean of them.

The displacements (<math>A_i</math>) we have measured are in the regimen of thousands and the error is ~100. This is attributed to the response of the system to lower frequencies. However, for near resonance, the model fits pretty well but the fitting method provides a different resonance frequency <math>113.1Hz</math> than the one we can see on Fig. 8, around <math>103Hz</math>.

Finally, we believe that most of the errors in this experiment came from the fact that the laser diode is a homemade laser diode so it does not have a good temperature control. Moreover, the homemade laser diode holder is not tightly fit to the laser diode collimation tube which can cause displacement of the laser beam.

== Conclusion ==
We have shown indeed that we can characterize the resonance frequency of a system. For this particular application, we found a frequency of <math>113.1 \pm 0.1Hz </math> for the system transducer plus mirror. Finally, this technique has a great potential to be used in initial experimental laboratories due his simplicity to illustrates concepts such as interference and driven oscillator, which are important in advance physics modules.

==References==
[1] Freschi, A. A., et al. "Laser interferometric characterization of a vibrating speaker system." American Journal of Physics 71.11 (2003): 1121-1126.
https://doi.org/10.1119/1.1586262

[2] Skarha, Matthew. "Laser interferometric characterization for a vibrating speaker system."
http://mattskarha.com/assets/docs/Laser%20interferometric%20characterization%20for%20a%20vibrating%20speaker%20system%20good.pdf

[3] Pathare, Shirish, and Vikrant Kurmude. "Low cost Michelson–Morley interferometer." Physics Education 51.6 (2016): 063001.
https://doi.org/10.1088/0031-9120/51/6/063001

Resonance frequency measurement using a interferometric method

2022-04-30T15:47:00Z

Aucca: /* Discussion */

== Team Members ==
* [[User:Nakarin|Nakarin Jayjong]]
* [[User:Aucca|Joel Auccapuclla]]
* [[User:Xiaoyu|Xiaoyu Nie]]
* [[User:Haotian|Haotian Song]]

== Introduction ==
Interferometers since their invention have been a great tool for experimental physics. Applications cover a wide spectrum of fields such as astronomy, fiber optics, seismology, seismology, velocimetry to say the least. The precision of measurement of small displacements is the key factor, take LIGO as an example. Another important factor is the convenience to set up an interferometer and the low complexity in terms of equipment. This study study aims to measure the resonance frequency of a system, a transducer attached to a mirror, using a Michelson interferometer.

== Theory ==
Our Michelson interferometer setup used a laser, a beam splitter, two mirrors, and a photometer, illustrated in Figure 3. The light propagates from the source to the beam splitter where approximately 50% of the incident light is transmitted and approximately 50% of it is reflected at a right angle. The transmitted light continues to propagate as the reference plane wave to the stationary mirror where it is fully reflected back towards the beam splitter. The reflected light propagates as the target plane wave to the mirror attached to the oscillating loudspeaker. The two waves then recombine at the beam splitter and interfere based on their relative phases based on the idea that the phase shift of the target plane wave varies with the instantaneous position of the speaker mirror combination. Consider a situation in which we setup our interferometer such that no phase shift corresponds to the equilibrium position of the speaker-mirror combination. At this position, we expect constructive interference at the photometer. Likewise, for every λ/2 change in displacement, we expect constructive interference at the photometer because the path length travelled by the target plane wave varies by some integral multiple of λ. Thus, the phase shift is an even multiple of π and the two waves remain in phase. Similarly, when the path length traveled is some odd multiple of λ/4, the phase shift is an odd multiple of π and the two waves interfere destructively.

Consider two coherent waves with the same amplitude A and same frequency <math>\omega</math> that have paths <math>x_1</math> and <math>x_2</math> emitted from different sources that meet together at a point. The net wave at that point can then be given as the real part of the equation

<math>
\psi=Ae^{i(kx_1-\omega t}+Ae^{i(kx_2-\omega t)}=Ae^{-i\omega t}e^{ik\frac{(x_1+x_2)}{2}}[e^{ik\frac{(x_1-x_2)}{2}}e^{-ik\frac{(x_1-x_2)}{2}}]=2Acos[k(\frac{x_1-x_2}{2})]e^{i[k\frac{(x_1+x_2)}{2}-\omega t]}
</math>

where the wave number <math>k=2\pi/\lambda</math>, <math>\lambda</math> is the wavelength of the waves, and c is the phase velocity. The time-averaged intensity of the combined wave is then proportional to the square of the its amplitude. That is,

<math>
I\propto cos^2[\pi(\frac{p_2-p_1}{\lambda})]=cos^2[2\pi(\frac{p_2}{\lambda}-\theta)]
</math>

where <math>\theta</math> is a constant. Now suppose the mirror that the <math>p_2</math> beam is incident to oscillates harmonically back and forth with a frequency <math>f</math> and a displacement <math>B</math>. Then, <math>p_2 =p_2(0)+Bsin(2\pi ft)</math> and the intensity of the detected light varies with time as

<math>
I=A^2cos^2[2\pi(\frac{Bsin(2\pi ft)}{\lambda}-\theta')]
</math>

Note that the frequency of oscillations is itself a sinusoid, meaning that there will be turnaround points in the oscillations.

== Experimental setup ==
As our main source for this experiment, we used a laser diode of wavelength 658.9 nm (as we can see in Figure 6 in the following section). Most red laser diodes are based on either GaInP or AlGaInP, this means we have two semiconductor materials. For one particular configuration, it will allow current flow and produce light amplification. The diode laser is powered by a power supply, see the left top size of Figure 3. Also, the beam coming out of the diode is divergent so an aspheric lens (C110TMD-A) is used to collimate it, while the collimation tube of laser diode is mounted on a homemade aluminum holder as shown in Figure 1.
[[File:Homemade_laser_diode.jpeg|center|thumb|400px|Figure 1: Homemade laser diode.]]

In order to prevent the reverse current through the laser diode, and protect against turn-on transients or spikes in current, we use a protection circuit that includes a resistor, an inductor, and a Schottky diode as shown in Figure 2.

[[File:protection_circuit_v2.jpeg|center|thumb|800px|Figure 2: Protection circuit for laser diode.]]

Our Michelson interferometer consists of a beam splitter BS, which is used to split the laser beam into two beams, and two mirrors M1 and M2. The reflected beam goes to a fixed mirror M1 while the transmitted beam goes to the movable mirror M2. A transducer is attached, using tape, to the mirror M2. The transducer is powered by a signal generator (Aglient 33220A) that is used to vary the vibrating frequency of the mirror. A photodiode PD is used to detect the signal of interference fringes. The schematic Michelson interferometer is shown in Figure 3.
[[File:Schematic Michelson Interferometer.jpeg|center|thumb|600px|Figure 3: Schematic figure of Michelson interferometer. LD is a laser diode, BS is a beam splitter, M1 and M2 are mirrors, PD is a photodiode, and vibration transducer which is driven by a signal generator.]]

The experimental setup Michelson interferometer is shown in figure 4.
[[File:Experimental setup Michelson Interferometer.jpeg|center|thumb|600px|Figure 4: Experimental setup of Michelson interferometer.]]

The interference fringes of the Michelson interferometer are observed when the transducer is turned off in Figure 5.
[[File:Michelson interference fringes.jpeg|center|thumb|600px|Figure 5: Interference fringes which are observed when the transducer is turned off]]

==Experiment Results==
Before we started setting up our interferometer, we measured the wavelength of the laser by an Anstrom WS-7 wavemeter, the wavelength is about 658.9 nm, see Figure 6. Even more, this graph tells us that the Mitsubishi LD is in a single-mode regime, so we proceeded to build the interferometer.
[[File:Wavelength.jpeg|center|thumb|600px|Figure 6: The Wavelength measurement of the Mitsubishi Laser Diode.]]

After the alignment of the optical setup, we applied sinusoidal voltage to the transducer attached to the mirror tightly. intensity fluctuation is recorded by the homemade photodiode. Data from the photodiode is collected by the oscilloscope. The typical figure from the oscilloscope is shown in Figure 7. According to the equation mentioned before, the number of peaks during each period corresponds to the radio of displacement and laser wavelength. In this case, we can measure the displacement for various frequencies. The frequency range is from 86 Hz to 130 Hz.

[[File:ScopeScreen.jpeg|center|thumb|600px|Figure 7A: The display on the oscilloscope at <math>f = 103Hz </math>. After applying the voltage to the transducer, the voltage output from photodiode changes with time.]]

[[File:ult.png|center|thumb|600px|Figure 7B: The display on the oscilloscope at <math>f = 116Hz </math>. After applying the voltage to the transducer, the voltage output from photodiode changes with time.]]

We plot the displacement for different frequencies with the errorbar in Figure 6. Uncertainty is the square root uncertainty of the count number. For a driven, damped, and harmonic oscillator, the relation between displacement and frequency is

<math>|A|=\frac{f_{0}}{\sqrt{\left(\omega_{0}^{2}-\omega^{2}\right)^{2}+4 \beta^{2} \omega^{2}}} </math>

Where <math>f_{0}</math> is the amplitude, <math>\omega_{0}</math> is the resonance frequency, and <math>\beta_{0}</math> is a constant. With this equation, we can fit the frequency response curve. Because there is only one peak or valley for a frequency lower than 92 Hz or higher than 120 Hz, and the error is significant and non-negligible. We exclude those in the fitting process. Finally, the fitting root mean squared error (RMSE) is 130.38. Fitted <math>\beta</math> is <math>1.3 \pm 0.2</math>. The fitting resonance frequency <math>\omega_{0} </math> is <math>113.1 \pm 0.1Hz </math>.

[[File:FitPeakNum.jpeg|center|thumb|600px|Figure 8: Frequency response curve for the transducer system with errorbar. The blue dots represent the displacement measured with different frequencies. Error bars are the square root uncertainty. Red line is the fitting curve for a driven, damped, harmonic oscillator.]]

==Discussion==
In figure 7A, we could see that the nearest you are to the natural resonance frequency of the system, the response of the system to the driven force will coupled and the number of oscillation increase. Meanwhile, in figure 7B the number of oscillations is lower. These phenomena is in concordance with the relation for <math>|A|, in the past section.</math>

Our final result fits our data to the best Lorentzian function. When we say best, we talk about the function with the lowest Root Mean Square Error (RMSE).

RMSE is defined as follow:

<math>\operatorname{RMSE}=\sqrt{\frac{1}{n}\sum_{i=1}^n(A_i-\hat{A_i})^2}</math>

where <math>A_i</math> is the measured displacement and <math>\hat{A_i}</math> is the predicted displacement by the function model. The core of this method is to calculate the distance between the measured and predicted values and take the root mean of them.

The displacements (<math>A_i</math>) we have measured are in the regimen of thousands and the error is ~100. This is attributed to the response of the system to lower frequencies. However, for near resonance, the model fits pretty well but the fitting method provides a different resonance frequency <math>113.1Hz</math> than the one we can see on Fig. 8, around <math>103Hz</math>.

Finally, we believe that most of the errors in this experiment came from the fact that the laser diode is a homemade laser diode so it does not have a good temperature control. Moreover, the homemade laser diode holder is not tightly fit to the laser diode collimation tube which can cause displacement of the laser beam.

== Conclusion ==
We have shown indeed that we can characterize the resonance frequency of a system. For this particular application, we found a frequency of <math>113.1 \pm 0.1Hz </math> for the system transducer plus mirror. Finally, this technique has a great potential to be used in initial experimental laboratories due his simplicity to illustrates concepts such as interference and driven oscillator, which are important in advance physics modules.

==References==
[1] Freschi, A. A., et al. "Laser interferometric characterization of a vibrating speaker system." American Journal of Physics 71.11 (2003): 1121-1126.
https://doi.org/10.1119/1.1586262

[2] Skarha, Matthew. "Laser interferometric characterization for a vibrating speaker system."
http://mattskarha.com/assets/docs/Laser%20interferometric%20characterization%20for%20a%20vibrating%20speaker%20system%20good.pdf

[3] Pathare, Shirish, and Vikrant Kurmude. "Low cost Michelson–Morley interferometer." Physics Education 51.6 (2016): 063001.
https://doi.org/10.1088/0031-9120/51/6/063001

Resonance frequency measurement using a interferometric method

2022-04-30T15:46:30Z

Aucca: /* Discussion */

== Team Members ==
* [[User:Nakarin|Nakarin Jayjong]]
* [[User:Aucca|Joel Auccapuclla]]
* [[User:Xiaoyu|Xiaoyu Nie]]
* [[User:Haotian|Haotian Song]]

== Introduction ==
Interferometers since their invention have been a great tool for experimental physics. Applications cover a wide spectrum of fields such as astronomy, fiber optics, seismology, seismology, velocimetry to say the least. The precision of measurement of small displacements is the key factor, take LIGO as an example. Another important factor is the convenience to set up an interferometer and the low complexity in terms of equipment. This study study aims to measure the resonance frequency of a system, a transducer attached to a mirror, using a Michelson interferometer.

== Theory ==
Our Michelson interferometer setup used a laser, a beam splitter, two mirrors, and a photometer, illustrated in Figure 3. The light propagates from the source to the beam splitter where approximately 50% of the incident light is transmitted and approximately 50% of it is reflected at a right angle. The transmitted light continues to propagate as the reference plane wave to the stationary mirror where it is fully reflected back towards the beam splitter. The reflected light propagates as the target plane wave to the mirror attached to the oscillating loudspeaker. The two waves then recombine at the beam splitter and interfere based on their relative phases based on the idea that the phase shift of the target plane wave varies with the instantaneous position of the speaker mirror combination. Consider a situation in which we setup our interferometer such that no phase shift corresponds to the equilibrium position of the speaker-mirror combination. At this position, we expect constructive interference at the photometer. Likewise, for every λ/2 change in displacement, we expect constructive interference at the photometer because the path length travelled by the target plane wave varies by some integral multiple of λ. Thus, the phase shift is an even multiple of π and the two waves remain in phase. Similarly, when the path length traveled is some odd multiple of λ/4, the phase shift is an odd multiple of π and the two waves interfere destructively.

Consider two coherent waves with the same amplitude A and same frequency <math>\omega</math> that have paths <math>x_1</math> and <math>x_2</math> emitted from different sources that meet together at a point. The net wave at that point can then be given as the real part of the equation

<math>
\psi=Ae^{i(kx_1-\omega t}+Ae^{i(kx_2-\omega t)}=Ae^{-i\omega t}e^{ik\frac{(x_1+x_2)}{2}}[e^{ik\frac{(x_1-x_2)}{2}}e^{-ik\frac{(x_1-x_2)}{2}}]=2Acos[k(\frac{x_1-x_2}{2})]e^{i[k\frac{(x_1+x_2)}{2}-\omega t]}
</math>

where the wave number <math>k=2\pi/\lambda</math>, <math>\lambda</math> is the wavelength of the waves, and c is the phase velocity. The time-averaged intensity of the combined wave is then proportional to the square of the its amplitude. That is,

<math>
I\propto cos^2[\pi(\frac{p_2-p_1}{\lambda})]=cos^2[2\pi(\frac{p_2}{\lambda}-\theta)]
</math>

where <math>\theta</math> is a constant. Now suppose the mirror that the <math>p_2</math> beam is incident to oscillates harmonically back and forth with a frequency <math>f</math> and a displacement <math>B</math>. Then, <math>p_2 =p_2(0)+Bsin(2\pi ft)</math> and the intensity of the detected light varies with time as

<math>
I=A^2cos^2[2\pi(\frac{Bsin(2\pi ft)}{\lambda}-\theta')]
</math>

Note that the frequency of oscillations is itself a sinusoid, meaning that there will be turnaround points in the oscillations.

== Experimental setup ==
As our main source for this experiment, we used a laser diode of wavelength 658.9 nm (as we can see in Figure 6 in the following section). Most red laser diodes are based on either GaInP or AlGaInP, this means we have two semiconductor materials. For one particular configuration, it will allow current flow and produce light amplification. The diode laser is powered by a power supply, see the left top size of Figure 3. Also, the beam coming out of the diode is divergent so an aspheric lens (C110TMD-A) is used to collimate it, while the collimation tube of laser diode is mounted on a homemade aluminum holder as shown in Figure 1.
[[File:Homemade_laser_diode.jpeg|center|thumb|400px|Figure 1: Homemade laser diode.]]

In order to prevent the reverse current through the laser diode, and protect against turn-on transients or spikes in current, we use a protection circuit that includes a resistor, an inductor, and a Schottky diode as shown in Figure 2.

[[File:protection_circuit_v2.jpeg|center|thumb|800px|Figure 2: Protection circuit for laser diode.]]

Our Michelson interferometer consists of a beam splitter BS, which is used to split the laser beam into two beams, and two mirrors M1 and M2. The reflected beam goes to a fixed mirror M1 while the transmitted beam goes to the movable mirror M2. A transducer is attached, using tape, to the mirror M2. The transducer is powered by a signal generator (Aglient 33220A) that is used to vary the vibrating frequency of the mirror. A photodiode PD is used to detect the signal of interference fringes. The schematic Michelson interferometer is shown in Figure 3.
[[File:Schematic Michelson Interferometer.jpeg|center|thumb|600px|Figure 3: Schematic figure of Michelson interferometer. LD is a laser diode, BS is a beam splitter, M1 and M2 are mirrors, PD is a photodiode, and vibration transducer which is driven by a signal generator.]]

The experimental setup Michelson interferometer is shown in figure 4.
[[File:Experimental setup Michelson Interferometer.jpeg|center|thumb|600px|Figure 4: Experimental setup of Michelson interferometer.]]

The interference fringes of the Michelson interferometer are observed when the transducer is turned off in Figure 5.
[[File:Michelson interference fringes.jpeg|center|thumb|600px|Figure 5: Interference fringes which are observed when the transducer is turned off]]

==Experiment Results==
Before we started setting up our interferometer, we measured the wavelength of the laser by an Anstrom WS-7 wavemeter, the wavelength is about 658.9 nm, see Figure 6. Even more, this graph tells us that the Mitsubishi LD is in a single-mode regime, so we proceeded to build the interferometer.
[[File:Wavelength.jpeg|center|thumb|600px|Figure 6: The Wavelength measurement of the Mitsubishi Laser Diode.]]

After the alignment of the optical setup, we applied sinusoidal voltage to the transducer attached to the mirror tightly. intensity fluctuation is recorded by the homemade photodiode. Data from the photodiode is collected by the oscilloscope. The typical figure from the oscilloscope is shown in Figure 7. According to the equation mentioned before, the number of peaks during each period corresponds to the radio of displacement and laser wavelength. In this case, we can measure the displacement for various frequencies. The frequency range is from 86 Hz to 130 Hz.

[[File:ScopeScreen.jpeg|center|thumb|600px|Figure 7A: The display on the oscilloscope at <math>f = 103Hz </math>. After applying the voltage to the transducer, the voltage output from photodiode changes with time.]]

[[File:ult.png|center|thumb|600px|Figure 7B: The display on the oscilloscope at <math>f = 116Hz </math>. After applying the voltage to the transducer, the voltage output from photodiode changes with time.]]

We plot the displacement for different frequencies with the errorbar in Figure 6. Uncertainty is the square root uncertainty of the count number. For a driven, damped, and harmonic oscillator, the relation between displacement and frequency is

<math>|A|=\frac{f_{0}}{\sqrt{\left(\omega_{0}^{2}-\omega^{2}\right)^{2}+4 \beta^{2} \omega^{2}}} </math>

Where <math>f_{0}</math> is the amplitude, <math>\omega_{0}</math> is the resonance frequency, and <math>\beta_{0}</math> is a constant. With this equation, we can fit the frequency response curve. Because there is only one peak or valley for a frequency lower than 92 Hz or higher than 120 Hz, and the error is significant and non-negligible. We exclude those in the fitting process. Finally, the fitting root mean squared error (RMSE) is 130.38. Fitted <math>\beta</math> is <math>1.3 \pm 0.2</math>. The fitting resonance frequency <math>\omega_{0} </math> is <math>113.1 \pm 0.1Hz </math>.

[[File:FitPeakNum.jpeg|center|thumb|600px|Figure 8: Frequency response curve for the transducer system with errorbar. The blue dots represent the displacement measured with different frequencies. Error bars are the square root uncertainty. Red line is the fitting curve for a driven, damped, harmonic oscillator.]]

==Discussion==
In figure 7A, we could see that the nearest you are to the natural resonance frequency of the system, the response of the system to the driven force will coupled and the number of oscillation increase. Meanwhile, in figure 7B the number of oscillations is lower. These phenomena is in concordance with the relation for <math>|A|, in the past section.</math>

Our final result fits our data to the best Lorentzian function. When we say best, we talk about the function with the lowest Root Mean Square Error (RMSE).

RMSE is defined as follow:

<math>\operatorname{RMSE}=\sqrt{\frac{1}{n}\sum_{i=1}^n(A_i-\hat{A_i})^2}</math>

where <math>A_i</math> is the measured displacement and <math>\hat{A_i}</math> is the predicted displacement by the function model. The core of this method is to calculate the distance between the measured and predicted values and take the root mean of them.

The displacements (<math>A_i</math>) we have measured are in the regimen of thousands and the error is ~100. This is attributed to the response of the system to lower frequencies. However, for near resonance, the model fits pretty well but the fitting method provides a different resonance frequency (<math>113.1Hz</math>) than the one we can see on Fig. 8, around <math>103Hz</math>.

Finally, we believe that most of the errors in this experiment came from the fact that the laser diode is a homemade laser diode so it does not have a good temperature control. Moreover, the homemade laser diode holder is not tightly fit to the laser diode collimation tube which can cause displacement of the laser beam.

== Conclusion ==
We have shown indeed that we can characterize the resonance frequency of a system. For this particular application, we found a frequency of <math>113.1 \pm 0.1Hz </math> for the system transducer plus mirror. Finally, this technique has a great potential to be used in initial experimental laboratories due his simplicity to illustrates concepts such as interference and driven oscillator, which are important in advance physics modules.

==References==
[1] Freschi, A. A., et al. "Laser interferometric characterization of a vibrating speaker system." American Journal of Physics 71.11 (2003): 1121-1126.
https://doi.org/10.1119/1.1586262

[2] Skarha, Matthew. "Laser interferometric characterization for a vibrating speaker system."
http://mattskarha.com/assets/docs/Laser%20interferometric%20characterization%20for%20a%20vibrating%20speaker%20system%20good.pdf

[3] Pathare, Shirish, and Vikrant Kurmude. "Low cost Michelson–Morley interferometer." Physics Education 51.6 (2016): 063001.
https://doi.org/10.1088/0031-9120/51/6/063001

Resonance frequency measurement using a interferometric method

2022-04-30T15:44:52Z

Aucca: /* Discussion */

== Team Members ==
* [[User:Nakarin|Nakarin Jayjong]]
* [[User:Aucca|Joel Auccapuclla]]
* [[User:Xiaoyu|Xiaoyu Nie]]
* [[User:Haotian|Haotian Song]]

== Introduction ==
Interferometers since their invention have been a great tool for experimental physics. Applications cover a wide spectrum of fields such as astronomy, fiber optics, seismology, seismology, velocimetry to say the least. The precision of measurement of small displacements is the key factor, take LIGO as an example. Another important factor is the convenience to set up an interferometer and the low complexity in terms of equipment. This study study aims to measure the resonance frequency of a system, a transducer attached to a mirror, using a Michelson interferometer.

== Theory ==
Our Michelson interferometer setup used a laser, a beam splitter, two mirrors, and a photometer, illustrated in Figure 3. The light propagates from the source to the beam splitter where approximately 50% of the incident light is transmitted and approximately 50% of it is reflected at a right angle. The transmitted light continues to propagate as the reference plane wave to the stationary mirror where it is fully reflected back towards the beam splitter. The reflected light propagates as the target plane wave to the mirror attached to the oscillating loudspeaker. The two waves then recombine at the beam splitter and interfere based on their relative phases based on the idea that the phase shift of the target plane wave varies with the instantaneous position of the speaker mirror combination. Consider a situation in which we setup our interferometer such that no phase shift corresponds to the equilibrium position of the speaker-mirror combination. At this position, we expect constructive interference at the photometer. Likewise, for every λ/2 change in displacement, we expect constructive interference at the photometer because the path length travelled by the target plane wave varies by some integral multiple of λ. Thus, the phase shift is an even multiple of π and the two waves remain in phase. Similarly, when the path length traveled is some odd multiple of λ/4, the phase shift is an odd multiple of π and the two waves interfere destructively.

Consider two coherent waves with the same amplitude A and same frequency <math>\omega</math> that have paths <math>x_1</math> and <math>x_2</math> emitted from different sources that meet together at a point. The net wave at that point can then be given as the real part of the equation

<math>
\psi=Ae^{i(kx_1-\omega t}+Ae^{i(kx_2-\omega t)}=Ae^{-i\omega t}e^{ik\frac{(x_1+x_2)}{2}}[e^{ik\frac{(x_1-x_2)}{2}}e^{-ik\frac{(x_1-x_2)}{2}}]=2Acos[k(\frac{x_1-x_2}{2})]e^{i[k\frac{(x_1+x_2)}{2}-\omega t]}
</math>

where the wave number <math>k=2\pi/\lambda</math>, <math>\lambda</math> is the wavelength of the waves, and c is the phase velocity. The time-averaged intensity of the combined wave is then proportional to the square of the its amplitude. That is,

<math>
I\propto cos^2[\pi(\frac{p_2-p_1}{\lambda})]=cos^2[2\pi(\frac{p_2}{\lambda}-\theta)]
</math>

where <math>\theta</math> is a constant. Now suppose the mirror that the <math>p_2</math> beam is incident to oscillates harmonically back and forth with a frequency <math>f</math> and a displacement <math>B</math>. Then, <math>p_2 =p_2(0)+Bsin(2\pi ft)</math> and the intensity of the detected light varies with time as

<math>
I=A^2cos^2[2\pi(\frac{Bsin(2\pi ft)}{\lambda}-\theta')]
</math>

Note that the frequency of oscillations is itself a sinusoid, meaning that there will be turnaround points in the oscillations.

== Experimental setup ==
As our main source for this experiment, we used a laser diode of wavelength 658.9 nm (as we can see in Figure 6 in the following section). Most red laser diodes are based on either GaInP or AlGaInP, this means we have two semiconductor materials. For one particular configuration, it will allow current flow and produce light amplification. The diode laser is powered by a power supply, see the left top size of Figure 3. Also, the beam coming out of the diode is divergent so an aspheric lens (C110TMD-A) is used to collimate it, while the collimation tube of laser diode is mounted on a homemade aluminum holder as shown in Figure 1.
[[File:Homemade_laser_diode.jpeg|center|thumb|400px|Figure 1: Homemade laser diode.]]

In order to prevent the reverse current through the laser diode, and protect against turn-on transients or spikes in current, we use a protection circuit that includes a resistor, an inductor, and a Schottky diode as shown in Figure 2.

[[File:protection_circuit_v2.jpeg|center|thumb|800px|Figure 2: Protection circuit for laser diode.]]

Our Michelson interferometer consists of a beam splitter BS, which is used to split the laser beam into two beams, and two mirrors M1 and M2. The reflected beam goes to a fixed mirror M1 while the transmitted beam goes to the movable mirror M2. A transducer is attached, using tape, to the mirror M2. The transducer is powered by a signal generator (Aglient 33220A) that is used to vary the vibrating frequency of the mirror. A photodiode PD is used to detect the signal of interference fringes. The schematic Michelson interferometer is shown in Figure 3.
[[File:Schematic Michelson Interferometer.jpeg|center|thumb|600px|Figure 3: Schematic figure of Michelson interferometer. LD is a laser diode, BS is a beam splitter, M1 and M2 are mirrors, PD is a photodiode, and vibration transducer which is driven by a signal generator.]]

The experimental setup Michelson interferometer is shown in figure 4.
[[File:Experimental setup Michelson Interferometer.jpeg|center|thumb|600px|Figure 4: Experimental setup of Michelson interferometer.]]

The interference fringes of the Michelson interferometer are observed when the transducer is turned off in Figure 5.
[[File:Michelson interference fringes.jpeg|center|thumb|600px|Figure 5: Interference fringes which are observed when the transducer is turned off]]

==Experiment Results==
Before we started setting up our interferometer, we measured the wavelength of the laser by an Anstrom WS-7 wavemeter, the wavelength is about 658.9 nm, see Figure 6. Even more, this graph tells us that the Mitsubishi LD is in a single-mode regime, so we proceeded to build the interferometer.
[[File:Wavelength.jpeg|center|thumb|600px|Figure 6: The Wavelength measurement of the Mitsubishi Laser Diode.]]

After the alignment of the optical setup, we applied sinusoidal voltage to the transducer attached to the mirror tightly. intensity fluctuation is recorded by the homemade photodiode. Data from the photodiode is collected by the oscilloscope. The typical figure from the oscilloscope is shown in Figure 7. According to the equation mentioned before, the number of peaks during each period corresponds to the radio of displacement and laser wavelength. In this case, we can measure the displacement for various frequencies. The frequency range is from 86 Hz to 130 Hz.

[[File:ScopeScreen.jpeg|center|thumb|600px|Figure 7A: The display on the oscilloscope at <math>f = 103Hz </math>. After applying the voltage to the transducer, the voltage output from photodiode changes with time.]]

[[File:ult.png|center|thumb|600px|Figure 7B: The display on the oscilloscope at <math>f = 116Hz </math>. After applying the voltage to the transducer, the voltage output from photodiode changes with time.]]

We plot the displacement for different frequencies with the errorbar in Figure 6. Uncertainty is the square root uncertainty of the count number. For a driven, damped, and harmonic oscillator, the relation between displacement and frequency is

<math>|A|=\frac{f_{0}}{\sqrt{\left(\omega_{0}^{2}-\omega^{2}\right)^{2}+4 \beta^{2} \omega^{2}}} </math>

Where <math>f_{0}</math> is the amplitude, <math>\omega_{0}</math> is the resonance frequency, and <math>\beta_{0}</math> is a constant. With this equation, we can fit the frequency response curve. Because there is only one peak or valley for a frequency lower than 92 Hz or higher than 120 Hz, and the error is significant and non-negligible. We exclude those in the fitting process. Finally, the fitting root mean squared error (RMSE) is 130.38. Fitted <math>\beta</math> is <math>1.3 \pm 0.2</math>. The fitting resonance frequency <math>\omega_{0} </math> is <math>113.1 \pm 0.1Hz </math>.

[[File:FitPeakNum.jpeg|center|thumb|600px|Figure 8: Frequency response curve for the transducer system with errorbar. The blue dots represent the displacement measured with different frequencies. Error bars are the square root uncertainty. Red line is the fitting curve for a driven, damped, harmonic oscillator.]]

==Discussion==
In figure 7A, we could see that the nearest you are to the natural resonance frequency of the system, the response of the system to the driven force will coupled and the number of oscillation increase. Meanwhile, in figure 7B the number of oscillations is lower. These phenomena is in concordance with the relation for <math>|A|, in the past section.</math>

Our final result fits our data to the best Lorentzian function. When we say best, we talk about the function with the lowest Root Mean Square Error (RMSE).

RMSE is defined as follow:

<math>\operatorname{RMSE}=\sqrt{\frac{1}{n}\sum_{i=1}^n(A_i-\hat{A_i})^2}</math>

where <math>A_i</math> is the measured displacement and <math>\hat{A_i}</math> is the predicted displacement by the function model. The core of this method is to calculate the distance between the measured and predicted values and take the root mean of them.

The displacements (<math>A_i</math>) we have measured are in the regimen of thousands and the error is ~100. This is attributed to the response of the system to lower frequencies. However, for near resonance, the model fits pretty well but the fitting method doesn't match gives us a different resonance frequency (<math>113.1Hz</math>).

Finally, we believe that most of the errors in this experiment came from the fact that the laser diode is a homemade laser diode so it does not have a good temperature control. Moreover, the homemade laser diode holder is not tightly fit to the laser diode collimation tube which can cause displacement of the laser beam.

== Conclusion ==
We have shown indeed that we can characterize the resonance frequency of a system. For this particular application, we found a frequency of <math>113.1 \pm 0.1Hz </math> for the system transducer plus mirror. Finally, this technique has a great potential to be used in initial experimental laboratories due his simplicity to illustrates concepts such as interference and driven oscillator, which are important in advance physics modules.

==References==
[1] Freschi, A. A., et al. "Laser interferometric characterization of a vibrating speaker system." American Journal of Physics 71.11 (2003): 1121-1126.
https://doi.org/10.1119/1.1586262

[2] Skarha, Matthew. "Laser interferometric characterization for a vibrating speaker system."
http://mattskarha.com/assets/docs/Laser%20interferometric%20characterization%20for%20a%20vibrating%20speaker%20system%20good.pdf

[3] Pathare, Shirish, and Vikrant Kurmude. "Low cost Michelson–Morley interferometer." Physics Education 51.6 (2016): 063001.
https://doi.org/10.1088/0031-9120/51/6/063001

Resonance frequency measurement using a interferometric method

2022-04-30T15:41:53Z

Aucca: /* Discussion */

== Team Members ==
* [[User:Nakarin|Nakarin Jayjong]]
* [[User:Aucca|Joel Auccapuclla]]
* [[User:Xiaoyu|Xiaoyu Nie]]
* [[User:Haotian|Haotian Song]]

== Introduction ==
Interferometers since their invention have been a great tool for experimental physics. Applications cover a wide spectrum of fields such as astronomy, fiber optics, seismology, seismology, velocimetry to say the least. The precision of measurement of small displacements is the key factor, take LIGO as an example. Another important factor is the convenience to set up an interferometer and the low complexity in terms of equipment. This study study aims to measure the resonance frequency of a system, a transducer attached to a mirror, using a Michelson interferometer.

== Theory ==
Our Michelson interferometer setup used a laser, a beam splitter, two mirrors, and a photometer, illustrated in Figure 3. The light propagates from the source to the beam splitter where approximately 50% of the incident light is transmitted and approximately 50% of it is reflected at a right angle. The transmitted light continues to propagate as the reference plane wave to the stationary mirror where it is fully reflected back towards the beam splitter. The reflected light propagates as the target plane wave to the mirror attached to the oscillating loudspeaker. The two waves then recombine at the beam splitter and interfere based on their relative phases based on the idea that the phase shift of the target plane wave varies with the instantaneous position of the speaker mirror combination. Consider a situation in which we setup our interferometer such that no phase shift corresponds to the equilibrium position of the speaker-mirror combination. At this position, we expect constructive interference at the photometer. Likewise, for every λ/2 change in displacement, we expect constructive interference at the photometer because the path length travelled by the target plane wave varies by some integral multiple of λ. Thus, the phase shift is an even multiple of π and the two waves remain in phase. Similarly, when the path length traveled is some odd multiple of λ/4, the phase shift is an odd multiple of π and the two waves interfere destructively.

Consider two coherent waves with the same amplitude A and same frequency <math>\omega</math> that have paths <math>x_1</math> and <math>x_2</math> emitted from different sources that meet together at a point. The net wave at that point can then be given as the real part of the equation

<math>
\psi=Ae^{i(kx_1-\omega t}+Ae^{i(kx_2-\omega t)}=Ae^{-i\omega t}e^{ik\frac{(x_1+x_2)}{2}}[e^{ik\frac{(x_1-x_2)}{2}}e^{-ik\frac{(x_1-x_2)}{2}}]=2Acos[k(\frac{x_1-x_2}{2})]e^{i[k\frac{(x_1+x_2)}{2}-\omega t]}
</math>

where the wave number <math>k=2\pi/\lambda</math>, <math>\lambda</math> is the wavelength of the waves, and c is the phase velocity. The time-averaged intensity of the combined wave is then proportional to the square of the its amplitude. That is,

<math>
I\propto cos^2[\pi(\frac{p_2-p_1}{\lambda})]=cos^2[2\pi(\frac{p_2}{\lambda}-\theta)]
</math>

where <math>\theta</math> is a constant. Now suppose the mirror that the <math>p_2</math> beam is incident to oscillates harmonically back and forth with a frequency <math>f</math> and a displacement <math>B</math>. Then, <math>p_2 =p_2(0)+Bsin(2\pi ft)</math> and the intensity of the detected light varies with time as

<math>
I=A^2cos^2[2\pi(\frac{Bsin(2\pi ft)}{\lambda}-\theta')]
</math>

Note that the frequency of oscillations is itself a sinusoid, meaning that there will be turnaround points in the oscillations.

== Experimental setup ==
As our main source for this experiment, we used a laser diode of wavelength 658.9 nm (as we can see in Figure 6 in the following section). Most red laser diodes are based on either GaInP or AlGaInP, this means we have two semiconductor materials. For one particular configuration, it will allow current flow and produce light amplification. The diode laser is powered by a power supply, see the left top size of Figure 3. Also, the beam coming out of the diode is divergent so an aspheric lens (C110TMD-A) is used to collimate it, while the collimation tube of laser diode is mounted on a homemade aluminum holder as shown in Figure 1.
[[File:Homemade_laser_diode.jpeg|center|thumb|400px|Figure 1: Homemade laser diode.]]

In order to prevent the reverse current through the laser diode, and protect against turn-on transients or spikes in current, we use a protection circuit that includes a resistor, an inductor, and a Schottky diode as shown in Figure 2.

[[File:protection_circuit_v2.jpeg|center|thumb|800px|Figure 2: Protection circuit for laser diode.]]

Our Michelson interferometer consists of a beam splitter BS, which is used to split the laser beam into two beams, and two mirrors M1 and M2. The reflected beam goes to a fixed mirror M1 while the transmitted beam goes to the movable mirror M2. A transducer is attached, using tape, to the mirror M2. The transducer is powered by a signal generator (Aglient 33220A) that is used to vary the vibrating frequency of the mirror. A photodiode PD is used to detect the signal of interference fringes. The schematic Michelson interferometer is shown in Figure 3.
[[File:Schematic Michelson Interferometer.jpeg|center|thumb|600px|Figure 3: Schematic figure of Michelson interferometer. LD is a laser diode, BS is a beam splitter, M1 and M2 are mirrors, PD is a photodiode, and vibration transducer which is driven by a signal generator.]]

The experimental setup Michelson interferometer is shown in figure 4.
[[File:Experimental setup Michelson Interferometer.jpeg|center|thumb|600px|Figure 4: Experimental setup of Michelson interferometer.]]

The interference fringes of the Michelson interferometer are observed when the transducer is turned off in Figure 5.
[[File:Michelson interference fringes.jpeg|center|thumb|600px|Figure 5: Interference fringes which are observed when the transducer is turned off]]

==Experiment Results==
Before we started setting up our interferometer, we measured the wavelength of the laser by an Anstrom WS-7 wavemeter, the wavelength is about 658.9 nm, see Figure 6. Even more, this graph tells us that the Mitsubishi LD is in a single-mode regime, so we proceeded to build the interferometer.
[[File:Wavelength.jpeg|center|thumb|600px|Figure 6: The Wavelength measurement of the Mitsubishi Laser Diode.]]

After the alignment of the optical setup, we applied sinusoidal voltage to the transducer attached to the mirror tightly. intensity fluctuation is recorded by the homemade photodiode. Data from the photodiode is collected by the oscilloscope. The typical figure from the oscilloscope is shown in Figure 7. According to the equation mentioned before, the number of peaks during each period corresponds to the radio of displacement and laser wavelength. In this case, we can measure the displacement for various frequencies. The frequency range is from 86 Hz to 130 Hz.

[[File:ScopeScreen.jpeg|center|thumb|600px|Figure 7A: The display on the oscilloscope at <math>f = 103Hz </math>. After applying the voltage to the transducer, the voltage output from photodiode changes with time.]]

[[File:ult.png|center|thumb|600px|Figure 7B: The display on the oscilloscope at <math>f = 116Hz </math>. After applying the voltage to the transducer, the voltage output from photodiode changes with time.]]

We plot the displacement for different frequencies with the errorbar in Figure 6. Uncertainty is the square root uncertainty of the count number. For a driven, damped, and harmonic oscillator, the relation between displacement and frequency is

<math>|A|=\frac{f_{0}}{\sqrt{\left(\omega_{0}^{2}-\omega^{2}\right)^{2}+4 \beta^{2} \omega^{2}}} </math>

Where <math>f_{0}</math> is the amplitude, <math>\omega_{0}</math> is the resonance frequency, and <math>\beta_{0}</math> is a constant. With this equation, we can fit the frequency response curve. Because there is only one peak or valley for a frequency lower than 92 Hz or higher than 120 Hz, and the error is significant and non-negligible. We exclude those in the fitting process. Finally, the fitting root mean squared error (RMSE) is 130.38. Fitted <math>\beta</math> is <math>1.3 \pm 0.2</math>. The fitting resonance frequency <math>\omega_{0} </math> is <math>113.1 \pm 0.1Hz </math>.

[[File:FitPeakNum.jpeg|center|thumb|600px|Figure 8: Frequency response curve for the transducer system with errorbar. The blue dots represent the displacement measured with different frequencies. Error bars are the square root uncertainty. Red line is the fitting curve for a driven, damped, harmonic oscillator.]]

==Discussion==
In figure 7A, we could see that the nearest you are to the natural resonance frequency of the system, the response of the system to the driven force will coupled and the number of oscillation increase. Meanwhile, in figure 7B the number of oscillations is lower. These phenomena is in concordance with the relation we found for <math>|A|.</math>

Our final result fits our data to the best Lorentzian Function. When we say best we talk about the function with the lowest Root Mean Square Error (RMSE).

RMSE is defined as follow:

<math>\operatorname{RMSE}=\sqrt{\frac{1}{n}\sum_{i=1}^n(A_i-\hat{A_i})^2}</math>

where <math>A_i</math> is the measured displacement and <math>\hat{A_i}</math> is the predicted displacement by the function model. The core of this method is to calculate the distance between the measured and predicted values and take the root mean of them.

The displacements (<math>A_i</math>) we have measured are in the regimen of thousands and the error is ~100. This is attributed to the response of the system to lower frequencies. However, for near resonance, the model fits pretty well.

Finally, we believe that most of the errors in this experiment came from the fact that the laser diode is a homemade laser diode so it does not have a good temperature control. Moreover, the homemade laser diode holder is not tightly fit to the laser diode collimation tube which can cause displacement of the laser beam.

== Conclusion ==
We have shown indeed that we can characterize the resonance frequency of a system. For this particular application, we found a frequency of <math>113.1 \pm 0.1Hz </math> for the system transducer plus mirror. Finally, this technique has a great potential to be used in initial experimental laboratories due his simplicity to illustrates concepts such as interference and driven oscillator, which are important in advance physics modules.

==References==
[1] Freschi, A. A., et al. "Laser interferometric characterization of a vibrating speaker system." American Journal of Physics 71.11 (2003): 1121-1126.
https://doi.org/10.1119/1.1586262

[2] Skarha, Matthew. "Laser interferometric characterization for a vibrating speaker system."
http://mattskarha.com/assets/docs/Laser%20interferometric%20characterization%20for%20a%20vibrating%20speaker%20system%20good.pdf

[3] Pathare, Shirish, and Vikrant Kurmude. "Low cost Michelson–Morley interferometer." Physics Education 51.6 (2016): 063001.
https://doi.org/10.1088/0031-9120/51/6/063001

Resonance frequency measurement using a interferometric method

2022-04-30T15:27:38Z

Aucca: /* Experiment Results */

== Team Members ==
* [[User:Nakarin|Nakarin Jayjong]]
* [[User:Aucca|Joel Auccapuclla]]
* [[User:Xiaoyu|Xiaoyu Nie]]
* [[User:Haotian|Haotian Song]]

== Introduction ==
Interferometers since their invention have been a great tool for experimental physics. Applications cover a wide spectrum of fields such as astronomy, fiber optics, seismology, seismology, velocimetry to say the least. The precision of measurement of small displacements is the key factor, take LIGO as an example. Another important factor is the convenience to set up an interferometer and the low complexity in terms of equipment. This study study aims to measure the resonance frequency of a system, a transducer attached to a mirror, using a Michelson interferometer.

== Theory ==
Our Michelson interferometer setup used a laser, a beam splitter, two mirrors, and a photometer, illustrated in Figure 3. The light propagates from the source to the beam splitter where approximately 50% of the incident light is transmitted and approximately 50% of it is reflected at a right angle. The transmitted light continues to propagate as the reference plane wave to the stationary mirror where it is fully reflected back towards the beam splitter. The reflected light propagates as the target plane wave to the mirror attached to the oscillating loudspeaker. The two waves then recombine at the beam splitter and interfere based on their relative phases based on the idea that the phase shift of the target plane wave varies with the instantaneous position of the speaker mirror combination. Consider a situation in which we setup our interferometer such that no phase shift corresponds to the equilibrium position of the speaker-mirror combination. At this position, we expect constructive interference at the photometer. Likewise, for every λ/2 change in displacement, we expect constructive interference at the photometer because the path length travelled by the target plane wave varies by some integral multiple of λ. Thus, the phase shift is an even multiple of π and the two waves remain in phase. Similarly, when the path length traveled is some odd multiple of λ/4, the phase shift is an odd multiple of π and the two waves interfere destructively.

Consider two coherent waves with the same amplitude A and same frequency <math>\omega</math> that have paths <math>x_1</math> and <math>x_2</math> emitted from different sources that meet together at a point. The net wave at that point can then be given as the real part of the equation

<math>
\psi=Ae^{i(kx_1-\omega t}+Ae^{i(kx_2-\omega t)}=Ae^{-i\omega t}e^{ik\frac{(x_1+x_2)}{2}}[e^{ik\frac{(x_1-x_2)}{2}}e^{-ik\frac{(x_1-x_2)}{2}}]=2Acos[k(\frac{x_1-x_2}{2})]e^{i[k\frac{(x_1+x_2)}{2}-\omega t]}
</math>

where the wave number <math>k=2\pi/\lambda</math>, <math>\lambda</math> is the wavelength of the waves, and c is the phase velocity. The time-averaged intensity of the combined wave is then proportional to the square of the its amplitude. That is,

<math>
I\propto cos^2[\pi(\frac{p_2-p_1}{\lambda})]=cos^2[2\pi(\frac{p_2}{\lambda}-\theta)]
</math>

where <math>\theta</math> is a constant. Now suppose the mirror that the <math>p_2</math> beam is incident to oscillates harmonically back and forth with a frequency <math>f</math> and a displacement <math>B</math>. Then, <math>p_2 =p_2(0)+Bsin(2\pi ft)</math> and the intensity of the detected light varies with time as

<math>
I=A^2cos^2[2\pi(\frac{Bsin(2\pi ft)}{\lambda}-\theta')]
</math>

Note that the frequency of oscillations is itself a sinusoid, meaning that there will be turnaround points in the oscillations.

== Experimental setup ==
As our main source for this experiment, we used a laser diode of wavelength 658.9 nm (as we can see in Figure 6 in the following section). Most red laser diodes are based on either GaInP or AlGaInP, this means we have two semiconductor materials. For one particular configuration, it will allow current flow and produce light amplification. The diode laser is powered by a power supply, see the left top size of Figure 3. Also, the beam coming out of the diode is divergent so an aspheric lens (C110TMD-A) is used to collimate it, while the collimation tube of laser diode is mounted on a homemade aluminum holder as shown in Figure 1.
[[File:Homemade_laser_diode.jpeg|center|thumb|400px|Figure 1: Homemade laser diode.]]

In order to prevent the reverse current through the laser diode, and protect against turn-on transients or spikes in current, we use a protection circuit that includes a resistor, an inductor, and a Schottky diode as shown in Figure 2.

[[File:protection_circuit_v2.jpeg|center|thumb|800px|Figure 2: Protection circuit for laser diode.]]

Our Michelson interferometer consists of a beam splitter BS, which is used to split the laser beam into two beams, and two mirrors M1 and M2. The reflected beam goes to a fixed mirror M1 while the transmitted beam goes to the movable mirror M2. A transducer is attached, using tape, to the mirror M2. The transducer is powered by a signal generator (Aglient 33220A) that is used to vary the vibrating frequency of the mirror. A photodiode PD is used to detect the signal of interference fringes. The schematic Michelson interferometer is shown in Figure 3.
[[File:Schematic Michelson Interferometer.jpeg|center|thumb|600px|Figure 3: Schematic figure of Michelson interferometer. LD is a laser diode, BS is a beam splitter, M1 and M2 are mirrors, PD is a photodiode, and vibration transducer which is driven by a signal generator.]]

The experimental setup Michelson interferometer is shown in figure 4.
[[File:Experimental setup Michelson Interferometer.jpeg|center|thumb|600px|Figure 4: Experimental setup of Michelson interferometer.]]

The interference fringes of the Michelson interferometer are observed when the transducer is turned off in Figure 5.
[[File:Michelson interference fringes.jpeg|center|thumb|600px|Figure 5: Interference fringes which are observed when the transducer is turned off]]

==Experiment Results==
Before we started setting up our interferometer, we measured the wavelength of the laser by an Anstrom WS-7 wavemeter, the wavelength is about 658.9 nm, see Figure 6. Even more, this graph tells us that the Mitsubishi LD is in a single-mode regime, so we proceeded to build the interferometer.
[[File:Wavelength.jpeg|center|thumb|600px|Figure 6: The Wavelength measurement of the Mitsubishi Laser Diode.]]

After the alignment of the optical setup, we applied sinusoidal voltage to the transducer attached to the mirror tightly. intensity fluctuation is recorded by the homemade photodiode. Data from the photodiode is collected by the oscilloscope. The typical figure from the oscilloscope is shown in Figure 7. According to the equation mentioned before, the number of peaks during each period corresponds to the radio of displacement and laser wavelength. In this case, we can measure the displacement for various frequencies. The frequency range is from 86 Hz to 130 Hz.

[[File:ScopeScreen.jpeg|center|thumb|600px|Figure 7A: The display on the oscilloscope at <math>f = 103Hz </math>. After applying the voltage to the transducer, the voltage output from photodiode changes with time.]]

[[File:ult.png|center|thumb|600px|Figure 7B: The display on the oscilloscope at <math>f = 116Hz </math>. After applying the voltage to the transducer, the voltage output from photodiode changes with time.]]

We plot the displacement for different frequencies with the errorbar in Figure 6. Uncertainty is the square root uncertainty of the count number. For a driven, damped, and harmonic oscillator, the relation between displacement and frequency is

<math>|A|=\frac{f_{0}}{\sqrt{\left(\omega_{0}^{2}-\omega^{2}\right)^{2}+4 \beta^{2} \omega^{2}}} </math>

Where <math>f_{0}</math> is the amplitude, <math>\omega_{0}</math> is the resonance frequency, and <math>\beta_{0}</math> is a constant. With this equation, we can fit the frequency response curve. Because there is only one peak or valley for a frequency lower than 92 Hz or higher than 120 Hz, and the error is significant and non-negligible. We exclude those in the fitting process. Finally, the fitting root mean squared error (RMSE) is 130.38. Fitted <math>\beta</math> is <math>1.3 \pm 0.2</math>. The fitting resonance frequency <math>\omega_{0} </math> is <math>113.1 \pm 0.1Hz </math>.

[[File:FitPeakNum.jpeg|center|thumb|600px|Figure 8: Frequency response curve for the transducer system with errorbar. The blue dots represent the displacement measured with different frequencies. Error bars are the square root uncertainty. Red line is the fitting curve for a driven, damped, harmonic oscillator.]]

==Discussion==
Our final result fits our data to the best Lorentzian Function. When we say best we talk about the function with the lowest Root Mean Square Error (RMSE).

RMSE is defined as follow:

<math>\operatorname{RMSE}=\sqrt{\frac{1}{n}\sum_{i=1}^n(A_i-\hat{A_i})^2}</math>

where <math>A_i</math> is the measured displacement and <math>\hat{A_i}</math> is the predicted displacement by the function model. The core of this method is to calculate the distance between the measured and predicted values and take the root mean of them.

The displacements (<math>A_i</math>) we have measured are in the regimen of thousands and the error is ~100. This is attributed to the response of the system to lower frequencies. However, for near resonance, the model fits pretty well.

Finally, we believe that most of the errors in this experiment came from the fact that the laser diode is a homemade laser diode so it does not have a good temperature control. Moreover, the homemade laser diode holder is not tightly fit to the laser diode collimation tube which can cause displacement of the laser beam.

== Conclusion ==
We have shown indeed that we can characterize the resonance frequency of a system. For this particular application, we found a frequency of <math>113.1 \pm 0.1Hz </math> for the system transducer plus mirror. Finally, this technique has a great potential to be used in initial experimental laboratories due his simplicity to illustrates concepts such as interference and driven oscillator, which are important in advance physics modules.

==References==
[1] Freschi, A. A., et al. "Laser interferometric characterization of a vibrating speaker system." American Journal of Physics 71.11 (2003): 1121-1126.
https://doi.org/10.1119/1.1586262

[2] Skarha, Matthew. "Laser interferometric characterization for a vibrating speaker system."
http://mattskarha.com/assets/docs/Laser%20interferometric%20characterization%20for%20a%20vibrating%20speaker%20system%20good.pdf

[3] Pathare, Shirish, and Vikrant Kurmude. "Low cost Michelson–Morley interferometer." Physics Education 51.6 (2016): 063001.
https://doi.org/10.1088/0031-9120/51/6/063001

File:Ult.png

2022-04-30T15:25:03Z

Aucca:

Resonance frequency measurement using a interferometric method

2022-04-30T14:31:50Z

Aucca: /* Conclusion */

== Team Members ==
* [[User:Nakarin|Nakarin Jayjong]]
* [[User:Aucca|Joel Auccapuclla]]
* [[User:Xiaoyu|Xiaoyu Nie]]
* [[User:Haotian|Haotian Song]]

== Introduction ==
Interferometers since their invention have been a great tool for experimental physics. Applications cover a wide spectrum of fields such as astronomy, fiber optics, seismology, seismology, velocimetry to say the least. The precision of measurement of small displacements is the key factor, take LIGO as an example. Another important factor is the convenience to set up an interferometer and the low complexity in terms of equipment. This study study aims to measure the resonance frequency of a system, a transducer attached to a mirror, using a Michelson interferometer.

== Theory ==
Our Michelson interferometer setup used a laser, a beam splitter, two mirrors, and a photometer, illustrated in Figure 3. The light propagates from the source to the beam splitter where approximately 50% of the incident light is transmitted and approximately 50% of it is reflected at a right angle. The transmitted light continues to propagate as the reference plane wave to the stationary mirror where it is fully reflected back towards the beam splitter. The reflected light propagates as the target plane wave to the mirror attached to the oscillating loudspeaker. The two waves then recombine at the beam splitter and interfere based on their relative phases based on the idea that the phase shift of the target plane wave varies with the instantaneous position of the speaker mirror combination. Consider a situation in which we setup our interferometer such that no phase shift corresponds to the equilibrium position of the speaker-mirror combination. At this position, we expect constructive interference at the photometer. Likewise, for every λ/2 change in displacement, we expect constructive interference at the photometer because the path length travelled by the target plane wave varies by some integral multiple of λ. Thus, the phase shift is an even multiple of π and the two waves remain in phase. Similarly, when the path length traveled is some odd multiple of λ/4, the phase shift is an odd multiple of π and the two waves interfere destructively.

Consider two coherent waves with the same amplitude A and same frequency <math>\omega</math> that have paths <math>x_1</math> and <math>x_2</math> emitted from different sources that meet together at a point. The net wave at that point can then be given as the real part of the equation

<math>
\psi=Ae^{i(kx_1-\omega t}+Ae^{i(kx_2-\omega t)}=Ae^{-i\omega t}e^{ik\frac{(x_1+x_2)}{2}}[e^{ik\frac{(x_1-x_2)}{2}}e^{-ik\frac{(x_1-x_2)}{2}}]=2Acos[k(\frac{x_1-x_2}{2})]e^{i[k\frac{(x_1+x_2)}{2}-\omega t]}
</math>

where the wave number <math>k=2\pi/\lambda</math>, <math>\lambda</math> is the wavelength of the waves, and c is the phase velocity. The time-averaged intensity of the combined wave is then proportional to the square of the its amplitude. That is,

<math>
I\propto cos^2[\pi(\frac{p_2-p_1}{\lambda})]=cos^2[2\pi(\frac{p_2}{\lambda}-\theta)]
</math>

where <math>\theta</math> is a constant. Now suppose the mirror that the <math>p_2</math> beam is incident to oscillates harmonically back and forth with a frequency <math>f</math> and a displacement <math>B</math>. Then, <math>p_2 =p_2(0)+Bsin(2\pi ft)</math> and the intensity of the detected light varies with time as

<math>
I=A^2cos^2[2\pi(\frac{Bsin(2\pi ft)}{\lambda}-\theta')]
</math>

Note that the frequency of oscillations is itself a sinusoid, meaning that there will be turnaround points in the oscillations.

== Experimental setup ==
As our main source for this experiment, we used a laser diode of wavelength 658.9 nm (as we can see in Figure 6 in the following section). Most red laser diodes are based on either GaInP or AlGaInP, this means we have two semiconductor materials. For one particular configuration, it will allow current flow and produce light amplification. The diode laser is powered by a power supply, see the left top size of Figure 3. Also, the beam coming out of the diode is divergent so an aspheric lens (C110TMD-A) is used to collimate it, while the collimation tube of laser diode is mounted on a homemade aluminum holder as shown in Figure 1.
[[File:Homemade_laser_diode.jpeg|center|thumb|400px|Figure 1: Homemade laser diode.]]

In order to prevent the reverse current through the laser diode, and protect against turn-on transients or spikes in current, we use a protection circuit that includes a resistor, an inductor, and a Schottky diode as shown in Figure 2.

[[File:protection_circuit_v2.jpeg|center|thumb|800px|Figure 2: Protection circuit for laser diode.]]

Our Michelson interferometer consists of a beam splitter BS, which is used to split the laser beam into two beams, and two mirrors M1 and M2. The reflected beam goes to a fixed mirror M1 while the transmitted beam goes to the movable mirror M2. A transducer is attached, using tape, to the mirror M2. The transducer is powered by a signal generator (Aglient 33220A) that is used to vary the vibrating frequency of the mirror. A photodiode PD is used to detect the signal of interference fringes. The schematic Michelson interferometer is shown in Figure 3.
[[File:Schematic Michelson Interferometer.jpeg|center|thumb|600px|Figure 3: Schematic figure of Michelson interferometer. LD is a laser diode, BS is a beam splitter, M1 and M2 are mirrors, PD is a photodiode, and vibration transducer which is driven by a signal generator.]]

The experimental setup Michelson interferometer is shown in figure 4.
[[File:Experimental setup Michelson Interferometer.jpeg|center|thumb|600px|Figure 4: Experimental setup of Michelson interferometer.]]

The interference fringes of the Michelson interferometer are observed when the transducer is turned off in Figure 5.
[[File:Michelson interference fringes.jpeg|center|thumb|600px|Figure 5: Interference fringes which are observed when the transducer is turned off]]

==Experiment Results==
Before we started setting up our interferometer, we measured the wavelength of the laser by an Anstrom WS-7 wavemeter, the wavelength is about 658.9 nm, see Figure 6. Even more, this graph tells us that the Mitsubishi LD is in a single-mode regime, so we proceeded to build the interferometer.
[[File:Wavelength.jpeg|center|thumb|600px|Figure 6: The Wavelength measurement of the Mitsubishi Laser Diode.]]

After the alignment of the optical setup, we applied sinusoidal voltage to the transducer attached to the mirror tightly. intensity fluctuation is recorded by the homemade photodiode. Data from the photodiode is collected by the oscilloscope. The typical figure from the oscilloscope is shown in Figure 7. According to the equation mentioned before, the number of peaks during each period corresponds to the radio of displacement and laser wavelength. In this case, we can measure the displacement for various frequencies. The frequency range is from 86 Hz to 130 Hz.

[[File:ScopeScreen.jpeg|center|thumb|600px|Figure 7: The display on the oscilloscope at <math>f = 103Hz </math>. After applying the voltage to the transducer, the voltage output from photodiode changes with time.]]

We plot the displacement for different frequencies with the errorbar in Figure 6. Uncertainty is the square root uncertainty of the count number. For a driven, damped, and harmonic oscillator, the relation between displacement and frequency is

<math>|A|=\frac{f_{0}}{\sqrt{\left(\omega_{0}^{2}-\omega^{2}\right)^{2}+4 \beta^{2} \omega^{2}}} </math>

Where <math>f_{0}</math> is the amplitude, <math>\omega_{0}</math> is the resonance frequency, and <math>\beta_{0}</math> is a constant. With this equation, we can fit the frequency response curve. Because there is only one peak or valley for a frequency lower than 92 Hz or higher than 120 Hz, and the error is significant and non-negligible. We exclude those in the fitting process. Finally, the fitting root mean squared error (RMSE) is 130.38. Fitted <math>\beta</math> is <math>1.3 \pm 0.2</math>. The fitting resonance frequency <math>\omega_{0} </math> is <math>113.1 \pm 0.1Hz </math>.

[[File:FitPeakNum.jpeg|center|thumb|600px|Figure 8: Frequency response curve for the transducer system with errorbar. The blue dots represent the displacement measured with different frequencies. Error bars are the square root uncertainty. Red line is the fitting curve for a driven, damped, harmonic oscillator.]]

==Discussion==
Our final result fits our data to the best Lorentzian Function. When we say best we talk about the function with the lowest Root Mean Square Error (RMSE).

RMSE is defined as follow:

<math>\operatorname{RMSE}=\sqrt{\frac{1}{n}\sum_{i=1}^n(A_i-\hat{A_i})^2}</math>

where <math>A_i</math> is the measured displacement and <math>\hat{A_i}</math> is the predicted displacement by the function model. The core of this method is to calculate the distance between the measured and predicted values and take the root mean of them.

The displacements (<math>A_i</math>) we have measured are in the regimen of thousands and the error is ~100. This is attributed to the response of the system to lower frequencies. However, for near resonance, the model fits pretty well.

Finally, we believe that most of the errors in this experiment came from the fact that the laser diode is a homemade laser diode so it does not have a good temperature control. Moreover, the homemade laser diode holder is not tightly fit to the laser diode collimation tube which can cause displacement of the laser beam.

== Conclusion ==
We have shown indeed that we can characterize the resonance frequency of a system. For this particular application, we found a frequency of <math>113.1 \pm 0.1Hz </math> for the system transducer plus mirror. Finally, this technique has a great potential to be used in initial experimental laboratories due his simplicity to illustrates concepts such as interference and driven oscillator, which are important in advance physics modules.

==References==
[1] Freschi, A. A., et al. "Laser interferometric characterization of a vibrating speaker system." American Journal of Physics 71.11 (2003): 1121-1126.
https://doi.org/10.1119/1.1586262

[2] Skarha, Matthew. "Laser interferometric characterization for a vibrating speaker system."
http://mattskarha.com/assets/docs/Laser%20interferometric%20characterization%20for%20a%20vibrating%20speaker%20system%20good.pdf

[3] Pathare, Shirish, and Vikrant Kurmude. "Low cost Michelson–Morley interferometer." Physics Education 51.6 (2016): 063001.
https://doi.org/10.1088/0031-9120/51/6/063001

Resonance frequency measurement using a interferometric method

2022-04-30T14:17:49Z

Aucca: /* Conclusion */

== Team Members ==
* [[User:Nakarin|Nakarin Jayjong]]
* [[User:Aucca|Joel Auccapuclla]]
* [[User:Xiaoyu|Xiaoyu Nie]]
* [[User:Haotian|Haotian Song]]

== Introduction ==
Interferometers since their invention have been a great tool for experimental physics. Applications cover a wide spectrum of fields such as astronomy, fiber optics, seismology, seismology, velocimetry to say the least. The precision of measurement of small displacements is the key factor, take LIGO as an example. Another important factor is the convenience to set up an interferometer and the low complexity in terms of equipment. This study study aims to measure the resonance frequency of a system, a transducer attached to a mirror, using a Michelson interferometer.

== Theory ==
Our Michelson interferometer setup used a laser, a beam splitter, two mirrors, and a photometer, illustrated in Figure 3. The light propagates from the source to the beam splitter where approximately 50% of the incident light is transmitted and approximately 50% of it is reflected at a right angle. The transmitted light continues to propagate as the reference plane wave to the stationary mirror where it is fully reflected back towards the beam splitter. The reflected light propagates as the target plane wave to the mirror attached to the oscillating loudspeaker. The two waves then recombine at the beam splitter and interfere based on their relative phases based on the idea that the phase shift of the target plane wave varies with the instantaneous position of the speaker mirror combination. Consider a situation in which we setup our interferometer such that no phase shift corresponds to the equilibrium position of the speaker-mirror combination. At this position, we expect constructive interference at the photometer. Likewise, for every λ/2 change in displacement, we expect constructive interference at the photometer because the path length travelled by the target plane wave varies by some integral multiple of λ. Thus, the phase shift is an even multiple of π and the two waves remain in phase. Similarly, when the path length traveled is some odd multiple of λ/4, the phase shift is an odd multiple of π and the two waves interfere destructively.

Consider two coherent waves with the same amplitude A and same frequency <math>\omega</math> that have paths <math>x_1</math> and <math>x_2</math> emitted from different sources that meet together at a point. The net wave at that point can then be given as the real part of the equation

<math>
\psi=Ae^{i(kx_1-\omega t}+Ae^{i(kx_2-\omega t)}=Ae^{-i\omega t}e^{ik\frac{(x_1+x_2)}{2}}[e^{ik\frac{(x_1-x_2)}{2}}e^{-ik\frac{(x_1-x_2)}{2}}]=2Acos[k(\frac{x_1-x_2}{2})]e^{i[k\frac{(x_1+x_2)}{2}-\omega t]}
</math>

where the wave number <math>k=2\pi/\lambda</math>, <math>\lambda</math> is the wavelength of the waves, and c is the phase velocity. The time-averaged intensity of the combined wave is then proportional to the square of the its amplitude. That is,

<math>
I\propto cos^2[\pi(\frac{p_2-p_1}{\lambda})]=cos^2[2\pi(\frac{p_2}{\lambda}-\theta)]
</math>

where <math>\theta</math> is a constant. Now suppose the mirror that the <math>p_2</math> beam is incident to oscillates harmonically back and forth with a frequency <math>f</math> and a displacement <math>B</math>. Then, <math>p_2 =p_2(0)+Bsin(2\pi ft)</math> and the intensity of the detected light varies with time as

<math>
I=A^2cos^2[2\pi(\frac{Bsin(2\pi ft)}{\lambda}-\theta')]
</math>

Note that the frequency of oscillations is itself a sinusoid, meaning that there will be turnaround points in the oscillations.

== Experimental setup ==
As our main source for this experiment, we used a laser diode of wavelength 658.9 nm (as we can see in Figure 6 in the following section). Most red laser diodes are based on either GaInP or AlGaInP, this means we have two semiconductor materials. For one particular configuration, it will allow current flow and produce light amplification. The diode laser is powered by a power supply, see the left top size of Figure 3. Also, the beam coming out of the diode is divergent so an aspheric lens (C110TMD-A) is used to collimate it, while the collimation tube of laser diode is mounted on a homemade aluminum holder as shown in Figure 1.
[[File:Homemade_laser_diode.jpeg|center|thumb|400px|Figure 1: Homemade laser diode.]]

In order to prevent the reverse current through the laser diode, and protect against turn-on transients or spikes in current, we use a protection circuit that includes a resistor, an inductor, and a Schottky diode as shown in Figure 2.

[[File:protection_circuit_v2.jpeg|center|thumb|800px|Figure 2: Protection circuit for laser diode.]]

Our Michelson interferometer consists of a beam splitter BS, which is used to split the laser beam into two beams, and two mirrors M1 and M2. The reflected beam goes to a fixed mirror M1 while the transmitted beam goes to the movable mirror M2. A transducer is attached, using tape, to the mirror M2. The transducer is powered by a signal generator (Aglient 33220A) that is used to vary the vibrating frequency of the mirror. A photodiode PD is used to detect the signal of interference fringes. The schematic Michelson interferometer is shown in Figure 3.
[[File:Schematic Michelson Interferometer.jpeg|center|thumb|600px|Figure 3: Schematic figure of Michelson interferometer. LD is a laser diode, BS is a beam splitter, M1 and M2 are mirrors, PD is a photodiode, and vibration transducer which is driven by a signal generator.]]

The experimental setup Michelson interferometer is shown in figure 4.
[[File:Experimental setup Michelson Interferometer.jpeg|center|thumb|600px|Figure 4: Experimental setup of Michelson interferometer.]]

The interference fringes of the Michelson interferometer are observed when the transducer is turned off in Figure 5.
[[File:Michelson interference fringes.jpeg|center|thumb|600px|Figure 5: Interference fringes which are observed when the transducer is turned off]]

==Experiment Results==
Before we started setting up our interferometer, we measured the wavelength of the laser by an Anstrom WS-7 wavemeter, the wavelength is about 658.9 nm, see Figure 6. Even more, this graph tells us that the Mitsubishi LD is in a single-mode regime, so we proceeded to build the interferometer.
[[File:Wavelength.jpeg|center|thumb|600px|Figure 6: The Wavelength measurement of the Mitsubishi Laser Diode.]]

After the alignment of the optical setup, we applied sinusoidal voltage to the transducer attached to the mirror tightly. intensity fluctuation is recorded by the homemade photodiode. Data from the photodiode is collected by the oscilloscope. The typical figure from the oscilloscope is shown in Figure 7. According to the equation mentioned before, the number of peaks during each period corresponds to the radio of displacement and laser wavelength. In this case, we can measure the displacement for various frequencies. The frequency range is from 86 Hz to 130 Hz.

[[File:ScopeScreen.jpeg|center|thumb|600px|Figure 7: The display on the oscilloscope at <math>f = 103Hz </math>. After applying the voltage to the transducer, the voltage output from photodiode changes with time.]]

We plot the displacement for different frequencies with the errorbar in Figure 6. Uncertainty is the square root uncertainty of the count number. For a driven, damped, and harmonic oscillator, the relation between displacement and frequency is

<math>|A|=\frac{f_{0}}{\sqrt{\left(\omega_{0}^{2}-\omega^{2}\right)^{2}+4 \beta^{2} \omega^{2}}} </math>

Where <math>f_{0}</math> is the amplitude, <math>\omega_{0}</math> is the resonance frequency, and <math>\beta_{0}</math> is a constant. With this equation, we can fit the frequency response curve. Because there is only one peak or valley for a frequency lower than 92 Hz or higher than 120 Hz, and the error is significant and non-negligible. We exclude those in the fitting process. Finally, the fitting root mean squared error (RMSE) is 130.38. Fitted <math>\beta</math> is <math>1.3 \pm 0.2</math>. The fitting resonance frequency <math>\omega_{0} </math> is <math>113.1 \pm 0.1Hz </math>.

[[File:FitPeakNum.jpeg|center|thumb|600px|Figure 8: Frequency response curve for the transducer system with errorbar. The blue dots represent the displacement measured with different frequencies. Error bars are the square root uncertainty. Red line is the fitting curve for a driven, damped, harmonic oscillator.]]

==Discussion==
Our final result fits our data to the best Lorentzian Function. When we say best we talk about the function with the lowest Root Mean Square Error (RMSE).

RMSE is defined as follow:

<math>\operatorname{RMSE}=\sqrt{\frac{1}{n}\sum_{i=1}^n(A_i-\hat{A_i})^2}</math>

where <math>A_i</math> is the measured displacement and <math>\hat{A_i}</math> is the predicted displacement by the function model. The core of this method is to calculate the distance between the measured and predicted values and take the root mean of them.

The displacements (<math>A_i</math>) we have measured are in the regimen of thousands and the error is ~100. This is attributed to the response of the system to lower frequencies. However, for near resonance, the model fits pretty well.

Finally, we believe that most of the errors in this experiment came from the fact that the laser diode is a homemade laser diode so it does not have a good temperature control. Moreover, the homemade laser diode holder is not tightly fit to the laser diode collimation tube which can cause displacement of the laser beam.

== Conclusion ==
We have shown indeed that we can characterize the resonance frequency of a system. For this particular application, we found a frequency of <math>113.1 \pm 0.1Hz </math> for the system transducer plus mirror. Finally, this technique has a great potential to be used in initial experimental laboratories due his simplicity to illustrates concepts such as interference and driven oscillator, which will be covered later on in advance physics modules.

==References==
[1] Freschi, A. A., et al. "Laser interferometric characterization of a vibrating speaker system." American Journal of Physics 71.11 (2003): 1121-1126.
https://doi.org/10.1119/1.1586262

[2] Skarha, Matthew. "Laser interferometric characterization for a vibrating speaker system."
http://mattskarha.com/assets/docs/Laser%20interferometric%20characterization%20for%20a%20vibrating%20speaker%20system%20good.pdf

[3] Pathare, Shirish, and Vikrant Kurmude. "Low cost Michelson–Morley interferometer." Physics Education 51.6 (2016): 063001.
https://doi.org/10.1088/0031-9120/51/6/063001

Resonance frequency measurement using a interferometric method

2022-04-30T14:11:31Z

Aucca: /* Introduction */

== Team Members ==
* [[User:Nakarin|Nakarin Jayjong]]
* [[User:Aucca|Joel Auccapuclla]]
* [[User:Xiaoyu|Xiaoyu Nie]]
* [[User:Haotian|Haotian Song]]

== Introduction ==
Interferometers since their invention have been a great tool for experimental physics. Applications cover a wide spectrum of fields such as astronomy, fiber optics, seismology, seismology, velocimetry to say the least. The precision of measurement of small displacements is the key factor, take LIGO as an example. Another important factor is the convenience to set up an interferometer and the low complexity in terms of equipment. This study study aims to measure the resonance frequency of a system, a transducer attached to a mirror, using a Michelson interferometer.

== Theory ==
Our Michelson interferometer setup used a laser, a beam splitter, two mirrors, and a photometer, illustrated in Figure 3. The light propagates from the source to the beam splitter where approximately 50% of the incident light is transmitted and approximately 50% of it is reflected at a right angle. The transmitted light continues to propagate as the reference plane wave to the stationary mirror where it is fully reflected back towards the beam splitter. The reflected light propagates as the target plane wave to the mirror attached to the oscillating loudspeaker. The two waves then recombine at the beam splitter and interfere based on their relative phases based on the idea that the phase shift of the target plane wave varies with the instantaneous position of the speaker mirror combination. Consider a situation in which we setup our interferometer such that no phase shift corresponds to the equilibrium position of the speaker-mirror combination. At this position, we expect constructive interference at the photometer. Likewise, for every λ/2 change in displacement, we expect constructive interference at the photometer because the path length travelled by the target plane wave varies by some integral multiple of λ. Thus, the phase shift is an even multiple of π and the two waves remain in phase. Similarly, when the path length traveled is some odd multiple of λ/4, the phase shift is an odd multiple of π and the two waves interfere destructively.

Consider two coherent waves with the same amplitude A and same frequency <math>\omega</math> that have paths <math>x_1</math> and <math>x_2</math> emitted from different sources that meet together at a point. The net wave at that point can then be given as the real part of the equation

<math>
\psi=Ae^{i(kx_1-\omega t}+Ae^{i(kx_2-\omega t)}=Ae^{-i\omega t}e^{ik\frac{(x_1+x_2)}{2}}[e^{ik\frac{(x_1-x_2)}{2}}e^{-ik\frac{(x_1-x_2)}{2}}]=2Acos[k(\frac{x_1-x_2}{2})]e^{i[k\frac{(x_1+x_2)}{2}-\omega t]}
</math>

where the wave number <math>k=2\pi/\lambda</math>, <math>\lambda</math> is the wavelength of the waves, and c is the phase velocity. The time-averaged intensity of the combined wave is then proportional to the square of the its amplitude. That is,

<math>
I\propto cos^2[\pi(\frac{p_2-p_1}{\lambda})]=cos^2[2\pi(\frac{p_2}{\lambda}-\theta)]
</math>

where <math>\theta</math> is a constant. Now suppose the mirror that the <math>p_2</math> beam is incident to oscillates harmonically back and forth with a frequency <math>f</math> and a displacement <math>B</math>. Then, <math>p_2 =p_2(0)+Bsin(2\pi ft)</math> and the intensity of the detected light varies with time as

<math>
I=A^2cos^2[2\pi(\frac{Bsin(2\pi ft)}{\lambda}-\theta')]
</math>

Note that the frequency of oscillations is itself a sinusoid, meaning that there will be turnaround points in the oscillations.

== Experimental setup ==
As our main source for this experiment, we used a laser diode of wavelength 658.9 nm (as we can see in Figure 6 in the following section). Most red laser diodes are based on either GaInP or AlGaInP, this means we have two semiconductor materials. For one particular configuration, it will allow current flow and produce light amplification. The diode laser is powered by a power supply, see the left top size of Figure 3. Also, the beam coming out of the diode is divergent so an aspheric lens (C110TMD-A) is used to collimate it, while the collimation tube of laser diode is mounted on a homemade aluminum holder as shown in Figure 1.
[[File:Homemade_laser_diode.jpeg|center|thumb|400px|Figure 1: Homemade laser diode.]]

In order to prevent the reverse current through the laser diode, and protect against turn-on transients or spikes in current, we use a protection circuit that includes a resistor, an inductor, and a Schottky diode as shown in Figure 2.

[[File:protection_circuit_v2.jpeg|center|thumb|800px|Figure 2: Protection circuit for laser diode.]]

Our Michelson interferometer consists of a beam splitter BS, which is used to split the laser beam into two beams, and two mirrors M1 and M2. The reflected beam goes to a fixed mirror M1 while the transmitted beam goes to the movable mirror M2. A transducer is attached, using tape, to the mirror M2. The transducer is powered by a signal generator (Aglient 33220A) that is used to vary the vibrating frequency of the mirror. A photodiode PD is used to detect the signal of interference fringes. The schematic Michelson interferometer is shown in Figure 3.
[[File:Schematic Michelson Interferometer.jpeg|center|thumb|600px|Figure 3: Schematic figure of Michelson interferometer. LD is a laser diode, BS is a beam splitter, M1 and M2 are mirrors, PD is a photodiode, and vibration transducer which is driven by a signal generator.]]

The experimental setup Michelson interferometer is shown in figure 4.
[[File:Experimental setup Michelson Interferometer.jpeg|center|thumb|600px|Figure 4: Experimental setup of Michelson interferometer.]]

The interference fringes of the Michelson interferometer are observed when the transducer is turned off in Figure 5.
[[File:Michelson interference fringes.jpeg|center|thumb|600px|Figure 5: Interference fringes which are observed when the transducer is turned off]]

==Experiment Results==
Before we started setting up our interferometer, we measured the wavelength of the laser by an Anstrom WS-7 wavemeter, the wavelength is about 658.9 nm, see Figure 6. Even more, this graph tells us that the Mitsubishi LD is in a single-mode regime, so we proceeded to build the interferometer.
[[File:Wavelength.jpeg|center|thumb|600px|Figure 6: The Wavelength measurement of the Mitsubishi Laser Diode.]]

After the alignment of the optical setup, we applied sinusoidal voltage to the transducer attached to the mirror tightly. intensity fluctuation is recorded by the homemade photodiode. Data from the photodiode is collected by the oscilloscope. The typical figure from the oscilloscope is shown in Figure 7. According to the equation mentioned before, the number of peaks during each period corresponds to the radio of displacement and laser wavelength. In this case, we can measure the displacement for various frequencies. The frequency range is from 86 Hz to 130 Hz.

[[File:ScopeScreen.jpeg|center|thumb|600px|Figure 7: The display on the oscilloscope at <math>f = 103Hz </math>. After applying the voltage to the transducer, the voltage output from photodiode changes with time.]]

We plot the displacement for different frequencies with the errorbar in Figure 6. Uncertainty is the square root uncertainty of the count number. For a driven, damped, and harmonic oscillator, the relation between displacement and frequency is

<math>|A|=\frac{f_{0}}{\sqrt{\left(\omega_{0}^{2}-\omega^{2}\right)^{2}+4 \beta^{2} \omega^{2}}} </math>

Where <math>f_{0}</math> is the amplitude, <math>\omega_{0}</math> is the resonance frequency, and <math>\beta_{0}</math> is a constant. With this equation, we can fit the frequency response curve. Because there is only one peak or valley for a frequency lower than 92 Hz or higher than 120 Hz, and the error is significant and non-negligible. We exclude those in the fitting process. Finally, the fitting root mean squared error (RMSE) is 130.38. Fitted <math>\beta</math> is <math>1.3 \pm 0.2</math>. The fitting resonance frequency <math>\omega_{0} </math> is <math>113.1 \pm 0.1Hz </math>.

[[File:FitPeakNum.jpeg|center|thumb|600px|Figure 8: Frequency response curve for the transducer system with errorbar. The blue dots represent the displacement measured with different frequencies. Error bars are the square root uncertainty. Red line is the fitting curve for a driven, damped, harmonic oscillator.]]

==Discussion==
Our final result fits our data to the best Lorentzian Function. When we say best we talk about the function with the lowest Root Mean Square Error (RMSE).

RMSE is defined as follow:

<math>\operatorname{RMSE}=\sqrt{\frac{1}{n}\sum_{i=1}^n(A_i-\hat{A_i})^2}</math>

where <math>A_i</math> is the measured displacement and <math>\hat{A_i}</math> is the predicted displacement by the function model. The core of this method is to calculate the distance between the measured and predicted values and take the root mean of them.

The displacements (<math>A_i</math>) we have measured are in the regimen of thousands and the error is ~100. This is attributed to the response of the system to lower frequencies. However, for near resonance, the model fits pretty well.

Finally, we believe that most of the errors in this experiment came from the fact that the laser diode is a homemade laser diode so it does not have a good temperature control. Moreover, the homemade laser diode holder is not tightly fit to the laser diode collimation tube which can cause displacement of the laser beam.

== Conclusion ==
We have shown indeed that we can characterize the resonance frequency of a system. For this particular application, we found a frequency of <math>113.1 \pm 0.1Hz </math> for the system transducer plus mirror. Finally, this technique has the great potential to be used in initial experimental laboratories due his simplicity to illustrates concepts such as interference and driven oscillator, which will be covered later on in advance physics modules.

==References==
[1] Freschi, A. A., et al. "Laser interferometric characterization of a vibrating speaker system." American Journal of Physics 71.11 (2003): 1121-1126.
https://doi.org/10.1119/1.1586262

[2] Skarha, Matthew. "Laser interferometric characterization for a vibrating speaker system."
http://mattskarha.com/assets/docs/Laser%20interferometric%20characterization%20for%20a%20vibrating%20speaker%20system%20good.pdf

[3] Pathare, Shirish, and Vikrant Kurmude. "Low cost Michelson–Morley interferometer." Physics Education 51.6 (2016): 063001.
https://doi.org/10.1088/0031-9120/51/6/063001

Resonance frequency measurement using a interferometric method

2022-04-30T09:19:03Z

Aucca: /* Conclusion */

== Team Members ==
* [[User:Nakarin|Nakarin Jayjong]]
* [[User:Aucca|Joel Auccapuclla]]
* [[User:Xiaoyu|Xiaoyu Nie]]
* [[User:Haotian|Haotian Song]]

== Introduction ==
Interferometers since their invention have been a great tool for experimental physics. Applications cover a wide spectrum of fields such as astronomy, fiber optics, seismology, seismology, velocimetry to say the least. The precision of measurement of small displacements is the key factor, take LIGO as an example. Another important factor is the convenience to set up an interferometer and the low complexity in terms of equipment. In this experiment, we will measure the resonance frequency of a system, a transducer attached to a mirror, using a Michelson interferometer.

== Theory ==
Our Michelson interferometer setup used a laser, a beam splitter, two mirrors, and a photometer, illustrated in Fig. 1. The light propagates from the source to the beam splitter where approximately 50% of the incident light is transmitted and approximately 50% of it is reflected at a right angle. The transmitted light continues to propagate as the reference plane wave to the stationary mirror where it is fully reflected back towards the beam splitter. The reflected light propagates as the target plane wave to the mirror attached to the oscillating loudspeaker. The two waves then recombine at the beam splitter and interfere based on their relative phases based on the idea that the phase shift of the target plane wave varies with the instantaneous position of the speaker mirror combination. Consider a situation in which we setup our interferometer such that no phase shift corresponds to the equilibrium position of the speaker-mirror combination. At this position, we expect constructive interference at the photometer. Likewise, for every λ/2 change in displacement, we expect constructive interference at the photometer because the path length travelled by the target plane wave varies by some integral multiple of λ. Thus, the phase shift is an even multiple of π and the two waves remain in phase. Similarly, when the path length traveled is some odd multiple of λ/4, the phase shift is an odd multiple of π and the two waves interfere destructively.

Consider two coherent waves with the same amplitude A and same frequency <math>\omega</math> that have paths <math>x_1</math> and <math>x_2</math> emitted from different sources that meet together at a point. The net wave at that point can then be given as the real part of the equation

<math>
\psi=Ae^{i(kx_1-\omega t}+Ae^{i(kx_2-\omega t)}=Ae^{-i\omega t}e^{ik\frac{(x_1+x_2)}{2}}[e^{ik\frac{(x_1-x_2)}{2}}e^{-ik\frac{(x_1-x_2)}{2}}]=2Acos[k(\frac{x_1-x_2}{2})]e^{i[k\frac{(x_1+x_2)}{2}-\omega t]}
</math>

where the wave number <math>k=2\pi/\lambda</math>, <math>\lambda</math> is the wavelength of the waves, and c is the phase velocity. The time-averaged intensity of the combined wave is then proportional to the square of the its amplitude. That is,

<math>
I\propto cos^2[\pi(\frac{p_2-p_1}{\lambda})]=cos^2[2\pi(\frac{p_2}{\lambda}-\theta)]
</math>

where <math>\theta</math> is a constant. Now suppose the mirror that the <math>p_2</math> beam is incident to oscillates harmonically back and forth with a frequency <math>f</math> and a displacement <math>B</math>. Then, <math>p_2 =p_2(0)+Bsin(2\pi ft)</math> and the intensity of the detected light varies with time as

<math>
I=A^2cos^2[2\pi(\frac{Bsin(2\pi ft)}{\lambda}-\theta')]
</math>

Note that the frequency of oscillations is itself a sinusoid, meaning that there will be turnaround points in the oscillations.

== Experimental setup ==
As our main source for this experiment, we used a laser diode of wavelength 658.9 nm (as we can see in fig. 4 in the following section). Most red laser diodes are based on either GaInP or AlGaInP, this means we have two semiconductor materials. For one particular configuration, it will allow current flow and produce light amplification. The diode laser is powered by a power supply, see the left top size of Figure 3. Also, the beam coming out of the diode is divergent so an aspheric lens (C110TMD-A) is used to collimate it, while the collimation tube of laser diode is mounted on a homemade aluminum holder as shown in Figure 1.
[[File:Homemade_laser_diode.jpeg|center|thumb|400px|Figure 1: Homemade laser diode.]]

In order to prevent the reverse current through the laser diode, and protect against turn-on transients or spikes in current, we use a protection circuit that includes a resistor, an inductor, and a Schottky diode as shown in Figure 2.

[[File:Protection circuit.jpeg|center|thumb|800px|Figure 2: Protection circuit for laser diode.]]

Our Michelson interferometer consists of a beam splitter BS, which is used to split the laser beam into two beams, and two mirrors M1 and M2. The reflected beam goes to a fixed mirror M1 while the transmitted beam goes to the movable mirror M2. A transducer is attached, using tape, to the mirror M2. The transducer is powered by a signal generator (Aglient 33220A) that is used to vary the vibrating frequency of the mirror. A photodiode PD is used to detect the signal of interference fringes. The schematic Michelson interferometer is shown in Figure 3.
[[File:Schematic Michelson Interferometer.jpeg|center|thumb|600px|Figure 3: Schematic figure of Michelson interferometer. LD is a laser diode, BS is a beam splitter, M1 and M2 are mirrors, PD is a photodiode, and vibration transducer which is driven by a signal generator.]]

The experimental setup Michelson interferometer is shown in figure 4.
[[File:Experimental setup Michelson Interferometer.jpeg|center|thumb|600px|Figure 4: Experimental setup of Michelson interferometer.]]

The interference fringes of the Michelson interferometer are observed when the transducer is turned off in Figure 5.
[[File:Michelson interference fringes.jpeg|center|thumb|600px|Figure 5: Interference fringes which are observed when the transducer is turned off]]

==Experiment Results==
Before we started setting up our interferometer, we measured the wavelength of the laser by an Anstrom WS-7 wavemeter, the wavelength is about 658.9 nm, see Figure 6. Even more, this graph tells us that the Mitsubishi LD is in a single-mode regimen, so we proceeded to build the interferometer.
[[File:Wavelength.jpeg|center|thumb|600px|Figure 6: The Wavelength measurement of the Mitsubishi Laser Diode.]]

After the alignment of the optical setup, we applied sinusoidal voltage to the transducer attached to the mirror tightly. intensity fluctuation is recorded by the homemade photodiode. Data from the photodiode is collected by the oscilloscope. The typical figure from the oscilloscope is shown in Figure 7. According to the equation mentioned before, the number of peaks during each period corresponds to the radio of displacement and laser wavelength. In this case, we can measure the displacement for various frequencies. The frequency range is from 86 Hz to 130 Hz.

[[File:ScopeScreen.jpeg|center|thumb|600px|Figure 7: The display on the oscilloscope at <math>f = 103Hz </math>. After applying the voltage to the transducer, the voltage output from photodiode changes with time.]]

We plot the displacement for different frequencies with the errorbar in Figure 6. Uncertainty is the square root uncertainty of the count number. For a driven, damped, and harmonic oscillator, the relation between displacement and frequency is

<math>|A|=\frac{f_{0}}{\sqrt{\left(\omega_{0}^{2}-\omega^{2}\right)^{2}+4 \beta^{2} \omega^{2}}} </math>

Where <math>f_{0}</math> is the amplitude, <math>\omega_{0}</math> is the resonance frequency, and <math>\beta_{0}</math> is a constant. With this equation, we can fit the frequency response curve. Because there is only one peak or valley for a frequency lower than 92 Hz or higher than 120 Hz, and the error is significant and non-negligible. We exclude those in the fitting process. Finally, the fitting root mean squared error (RMSE) is 130.38. Fitted <math>\beta</math> is <math>1.3 \pm 0.2</math>. The fitting resonance frequency <math>\omega_{0} </math> is <math>113.1 \pm 0.1Hz </math>.

[[File:FitPeakNum.jpeg|center|thumb|600px|Figure 8: Frequency response curve for the transducer system with errorbar. The blue dots represent the displacement measured with different frequencies. Error bars are the square root uncertainty. Red line is the fitting curve for a driven, damped, harmonic oscillator.]]

==Discussion==
Our final result fits our data to the best Lorentzian Function. When we say best we talk about the function with the lowest Root Mean Square Error (RMSE).

RMSE is defined as follow:

<math>\operatorname{RMSE}=\sqrt{\frac{1}{n}\sum_{i=1}^n(A_i-\hat{A_i})^2}</math>

where <math>A_i</math> is the measured displacement and <math>\hat{A_i}</math> is the predicted displacement by the function model. The core of this method is to calculate the distance between the measured and predicted values and take the root mean of them.

The displacements (<math>A_i</math>) we have measured are in the regimen of thousands and the error is ~100. This is attributed to the response of the system to lower frequencies. However, for near resonance, the model fits pretty well.

Finally, we believe that most of the errors in this experiment came from the fact that the laser diode is a homemade laser diode so it does not have a good temperature control. Moreover, the homemade laser diode holder is not tightly fit to the laser diode collimation tube which can cause displacement of the laser beam.

== Conclusion ==
We have shown indeed that we can characterize the resonance frequency of a system. For this particular application, we found a frequency of <math>113.1 \pm 0.1Hz </math> for the system transducer plus mirror. Finally, this technique has the great potential to be used in initial experimental laboratories due his simplicity to illustrates concepts such as interference and driven oscillator, which will be covered later on in advance physics modules.

==References==
[1] Freschi, A. A., et al. "Laser interferometric characterization of a vibrating speaker system." American Journal of Physics 71.11 (2003): 1121-1126.
https://doi.org/10.1119/1.1586262

[2] Skarha, Matthew. "Laser interferometric characterization for a vibrating speaker system."
http://mattskarha.com/assets/docs/Laser%20interferometric%20characterization%20for%20a%20vibrating%20speaker%20system%20good.pdf

[3] Pathare, Shirish, and Vikrant Kurmude. "Low cost Michelson–Morley interferometer." Physics Education 51.6 (2016): 063001.
https://doi.org/10.1088/0031-9120/51/6/063001

Resonance frequency measurement using a interferometric method

2022-04-30T09:18:03Z

Aucca: /* Conclusion */

== Team Members ==
* [[User:Nakarin|Nakarin Jayjong]]
* [[User:Aucca|Joel Auccapuclla]]
* [[User:Xiaoyu|Xiaoyu Nie]]
* [[User:Haotian|Haotian Song]]

== Introduction ==
Interferometers since their invention have been a great tool for experimental physics. Applications cover a wide spectrum of fields such as astronomy, fiber optics, seismology, seismology, velocimetry to say the least. The precision of measurement of small displacements is the key factor, take LIGO as an example. Another important factor is the convenience to set up an interferometer and the low complexity in terms of equipment. In this experiment, we will measure the resonance frequency of a system, a transducer attached to a mirror, using a Michelson interferometer.

== Theory ==
Our Michelson interferometer setup used a laser, a beam splitter, two mirrors, and a photometer, illustrated in Fig. 1. The light propagates from the source to the beam splitter where approximately 50% of the incident light is transmitted and approximately 50% of it is reflected at a right angle. The transmitted light continues to propagate as the reference plane wave to the stationary mirror where it is fully reflected back towards the beam splitter. The reflected light propagates as the target plane wave to the mirror attached to the oscillating loudspeaker. The two waves then recombine at the beam splitter and interfere based on their relative phases based on the idea that the phase shift of the target plane wave varies with the instantaneous position of the speaker mirror combination. Consider a situation in which we setup our interferometer such that no phase shift corresponds to the equilibrium position of the speaker-mirror combination. At this position, we expect constructive interference at the photometer. Likewise, for every λ/2 change in displacement, we expect constructive interference at the photometer because the path length travelled by the target plane wave varies by some integral multiple of λ. Thus, the phase shift is an even multiple of π and the two waves remain in phase. Similarly, when the path length traveled is some odd multiple of λ/4, the phase shift is an odd multiple of π and the two waves interfere destructively.

Consider two coherent waves with the same amplitude A and same frequency <math>\omega</math> that have paths <math>x_1</math> and <math>x_2</math> emitted from different sources that meet together at a point. The net wave at that point can then be given as the real part of the equation

<math>
\psi=Ae^{i(kx_1-\omega t}+Ae^{i(kx_2-\omega t)}=Ae^{-i\omega t}e^{ik\frac{(x_1+x_2)}{2}}[e^{ik\frac{(x_1-x_2)}{2}}e^{-ik\frac{(x_1-x_2)}{2}}]=2Acos[k(\frac{x_1-x_2}{2})]e^{i[k\frac{(x_1+x_2)}{2}-\omega t]}
</math>

where the wave number <math>k=2\pi/\lambda</math>, <math>\lambda</math> is the wavelength of the waves, and c is the phase velocity. The time-averaged intensity of the combined wave is then proportional to the square of the its amplitude. That is,

<math>
I\propto cos^2[\pi(\frac{p_2-p_1}{\lambda})]=cos^2[2\pi(\frac{p_2}{\lambda}-\theta)]
</math>

where <math>\theta</math> is a constant. Now suppose the mirror that the <math>p_2</math> beam is incident to oscillates harmonically back and forth with a frequency <math>f</math> and a displacement <math>B</math>. Then, <math>p_2 =p_2(0)+Bsin(2\pi ft)</math> and the intensity of the detected light varies with time as

<math>
I=A^2cos^2[2\pi(\frac{Bsin(2\pi ft)}{\lambda}-\theta')]
</math>

Note that the frequency of oscillations is itself a sinusoid, meaning that there will be turnaround points in the oscillations.

== Experimental setup ==
As our main source for this experiment, we used a laser diode of wavelength 658.9 nm (as we can see in fig. 4 in the following section). Most red laser diodes are based on either GaInP or AlGaInP, this means we have two semiconductor materials. For one particular configuration, it will allow current flow and produce light amplification. The diode laser is powered by a power supply, see the left top size of Figure 3. Also, the beam coming out of the diode is divergent so an aspheric lens (C110TMD-A) is used to collimate it, while the collimation tube of laser diode is mounted on a homemade aluminum holder as shown in Figure 1.
[[File:Homemade_laser_diode.jpeg|center|thumb|400px|Figure 1: Homemade laser diode.]]

In order to prevent the reverse current through the laser diode, and protect against turn-on transients or spikes in current, we use a protection circuit that includes a resistor, an inductor, and a Schottky diode as shown in Figure 2.

[[File:Protection circuit.jpeg|center|thumb|800px|Figure 2: Protection circuit for laser diode.]]

Our Michelson interferometer consists of a beam splitter BS, which is used to split the laser beam into two beams, and two mirrors M1 and M2. The reflected beam goes to a fixed mirror M1 while the transmitted beam goes to the movable mirror M2. A transducer is attached, using tape, to the mirror M2. The transducer is powered by a signal generator (Aglient 33220A) that is used to vary the vibrating frequency of the mirror. A photodiode PD is used to detect the signal of interference fringes. The schematic Michelson interferometer is shown in Figure 3.
[[File:Schematic Michelson Interferometer.jpeg|center|thumb|600px|Figure 3: Schematic figure of Michelson interferometer. LD is a laser diode, BS is a beam splitter, M1 and M2 are mirrors, PD is a photodiode, and vibration transducer which is driven by a signal generator.]]

The experimental setup Michelson interferometer is shown in figure 4.
[[File:Experimental setup Michelson Interferometer.jpeg|center|thumb|600px|Figure 4: Experimental setup of Michelson interferometer.]]

The interference fringes of the Michelson interferometer are observed when the transducer is turned off in Figure 5.
[[File:Michelson interference fringes.jpeg|center|thumb|600px|Figure 5: Interference fringes which are observed when the transducer is turned off]]

==Experiment Results==
Before we started setting up our interferometer we measured the wavelength of the Mitsubishi LD in an Anstrom WS-7 wavemeter, the wavelength of the LD is about 658.9 nm, see Figure 6. Even more, this graph tells us that the Mitsubishi LD is in a single-mode regimen, so we proceeded to build the interferometer.
[[File:Wavelength.jpeg|center|thumb|600px|Figure 6: The Wavelength measurement of the Mitsubishi Laser Diode.]]

After the alignment of the optical setup, we applied sinusoidal voltage to the transducer attached to the mirror tightly. intensity fluctuation is recorded by the homemade photodiode. Data from the photodiode is collected by the oscilloscope. The typical figure from the oscilloscope is shown in Figure 7. According to the equation mentioned before, the number of peaks during each period corresponds to the radio of displacement and laser wavelength. In this case, we can measure the displacement for various frequencies. The frequency range is from 86 Hz to 130 Hz.

[[File:ScopeScreen.jpeg|center|thumb|600px|Figure 7: The display on the oscilloscope at <math>f = 103Hz </math>. After applying the voltage to the transducer, the voltage output from photodiode changes with time.]]

We plot the displacement for different frequencies with the errorbar in Figure 6. Uncertainty is the square root uncertainty of the count number. For a driven, damped, and harmonic oscillator, the relation between displacement and frequency is

<math>|A|=\frac{f_{0}}{\sqrt{\left(\omega_{0}^{2}-\omega^{2}\right)^{2}+4 \beta^{2} \omega^{2}}} </math>

Where <math>f_{0}</math> is the amplitude, <math>\omega_{0}</math> is the resonance frequency, and <math>\beta_{0}</math> is a constant. With this equation, we can fit the frequency response curve. Because there is only one peak or valley for a frequency lower than 92 Hz or higher than 120 Hz, and the error is significant and non-negligible. We exclude those in the fitting process. Finally, the fitting root mean squared error (RMSE) is 130.38. Fitted <math>\beta</math> is <math>1.3 \pm 0.2</math>. The fitting resonance frequency <math>\omega_{0} </math> is <math>113.1 \pm 0.1Hz </math>.

[[File:FitPeakNum.jpeg|center|thumb|600px|Figure 8: Frequency response curve for the transducer system with errorbar. The blue dots represent the displacement measured with different frequencies. Error bars are the square root uncertainty. Red line is the fitting curve for a driven, damped, harmonic oscillator.]]

==Discussion==
Our final result fits our data to the best Lorentzian Function. When we say best we talk about the function with the lowest Root Mean Square Error (RMSE).

RMSE is defined as follow:

<math>\operatorname{RMSE}=\sqrt{\frac{1}{n}\sum_{i=1}^n(A_i-\hat{A_i})^2}</math>

where <math>A_i</math> is the measured displacement and <math>\hat{A_i}</math> is the predicted displacement by the function model. The core of this method is to calculate the distance between the measured and predicted values and take the root mean of them.

The displacements (<math>A_i</math>) we have measured are in the regimen of thousands and the error is ~100. This is attributed to the response of the system to lower frequencies. However, for near resonance, the model fits pretty well.

Finally, we believe that most of the errors in this experiment came from the fact that the laser diode is a homemade laser diode so it does not have a good temperature control. Moreover, the homemade laser diode holder is not tightly fit to the laser diode collimation tube which can cause displacement of the laser beam.

== Conclusion ==
We have shown indeed that we can characterize the resonance frequency of a system. For this particular application, we found a frequency of <math>113.1 \pm 0.1Hz </math> for the system transducer plus mirror. Finally, this technique has the great potential to be used in initial experimental laboratories due his simplicity to illustrates concepts such as interference and driven oscillator, which will be cover later on in advance physics modules.

==References==
[1] Freschi, A. A., et al. "Laser interferometric characterization of a vibrating speaker system." American Journal of Physics 71.11 (2003): 1121-1126.
https://doi.org/10.1119/1.1586262

[2] Skarha, Matthew. "Laser interferometric characterization for a vibrating speaker system."
http://mattskarha.com/assets/docs/Laser%20interferometric%20characterization%20for%20a%20vibrating%20speaker%20system%20good.pdf

[3] Pathare, Shirish, and Vikrant Kurmude. "Low cost Michelson–Morley interferometer." Physics Education 51.6 (2016): 063001.
https://doi.org/10.1088/0031-9120/51/6/063001

Resonance frequency measurement using a interferometric method

2022-04-30T08:22:58Z

Aucca: /* Experimental setup */

== Team Members ==
* [[User:Nakarin|Nakarin Jayjong]]
* [[User:Aucca|Joel Auccapuclla]]
* [[User:Xiaoyu|Xiaoyu Nie]]
* [[User:Haotian|Haotian Song]]

== Introduction ==
Interferometers since their invention have been a great tool for experimental physics. Applications cover a wide spectrum of fields such as astronomy, fiber optics, seismology, seismology, velocimetry to say the least. The precision of measurement of small displacements is the key factor, take LIGO as an example. Another important factor is the convenience to set up an interferometer and the low complexity in terms of equipment. In this experiment, we will measure the resonance frequency of a system, a transducer attached to a mirror, using a Michelson interferometer.

== Theory ==
Our Michelson interferometer setup used a laser, a beam splitter, two mirrors, and a photometer, illustrated in Fig. 1. The light propagates from the source to the beam splitter where approximately 50% of the incident light is transmitted and approximately 50% of it is reflected at a right angle. The transmitted light continues to propagate as the reference plane wave to the stationary mirror where it is fully reflected back towards the beam splitter. The reflected light propagates as the target plane wave to the mirror attached to the oscillating loudspeaker. The two waves then recombine at the beam splitter and interfere based on their relative phases based on the idea that the phase shift of the target plane wave varies with the instantaneous position of the speaker mirror combination. Consider a situation in which we setup our interferometer such that no phase shift corresponds to the equilibrium position of the speaker-mirror combination. At this position, we expect constructive interference at the photometer. Likewise, for every λ/2 change in displacement, we expect constructive interference at the photometer because the path length travelled by the target plane wave varies by some integral multiple of λ. Thus, the phase shift is an even multiple of π and the two waves remain in phase. Similarly, when the path length traveled is some odd multiple of λ/4, the phase shift is an odd multiple of π and the two waves interfere destructively.

Consider two coherent waves with the same amplitude A and same frequency <math>\omega</math> that have paths <math>x_1</math> and <math>x_2</math> emitted from different sources that meet together at a point. The net wave at that point can then be given as the real part of the equation

<math>
\psi=Ae^{i(kx_1-\omega t}+Ae^{i(kx_2-\omega t)}=Ae^{-i\omega t}e^{ik\frac{(x_1+x_2)}{2}}[e^{ik\frac{(x_1-x_2)}{2}}e^{-ik\frac{(x_1-x_2)}{2}}]=2Acos[k(\frac{x_1-x_2}{2})]e^{i[k\frac{(x_1+x_2)}{2}-\omega t]}
</math>

where the wave number <math>k=2\pi/\lambda</math>, <math>\lambda</math> is the wavelength of the waves, and c is the phase velocity. The time-averaged intensity of the combined wave is then proportional to the square of the its amplitude. That is,

<math>
I\propto cos^2[\pi(\frac{p_2-p_1}{\lambda})]=cos^2[2\pi(\frac{p_2}{\lambda}-\theta)]
</math>

where <math>\theta</math> is a constant. Now suppose the mirror that the <math>p_2</math> beam is incident to oscillates harmonically back and forth with a frequency <math>f</math> and a displacement <math>B</math>. Then, <math>p_2 =p_2(0)+Bsin(2\pi ft)</math> and the intensity of the detected light varies with time as

<math>
I=A^2cos^2[2\pi(\frac{Bsin(2\pi ft)}{\lambda}-\theta')]
</math>

Note that the frequency of oscillations is itself a sinusoid, meaning that there will be turnaround points in the oscillations.

== Experimental setup ==
As our main source for this experiment, we used a laser diode of wavelength 658.9 nm (as we can see in fig. 4 in the following section). Most red laser diodes are based on either GaInP or AlGaInP, this means we have 2 semiconductor materials. The laser diode is collimated by an aspheric lens (C110TMD-A). For one particular configuration, the data sheet tells us this, it will allow current flow and produce light amplification, this is what we called laser. This is why the diode laser is connected to a power supply, see left top size of fig. 2. Also, the beam coming out of the diode is divergent so Aspheric lens is used to collimate it.

Our Michelson interferometer consists of a beam splitter BS, which is used to split the laser beam into two beams, and two mirrors M1 and M2. The reflected beam goes to a fixed mirror M1 while the transmitted beam goes to the movable mirror M2 (see fig. 1). A transducer is attached, using tape, to the mirror M2. The transducer is powered by a signal generator that is used to vary the vibrating frequency of the mirror. A photodiode PD is used to detect the signal of interference fringes. The schematic Michelson interferometer is shown in Figure 1.
[[File:Schematic Michelson Interferometer.jpeg|center|thumb|600px|Figure 1: Schematic figure of Michelson interferometer. LD is a laser diode, BS is a beam splitter, M1 and M2 are mirrors, PD is a photodiode, and vibration transducer which is driven by a signal generator.]]

The experimental setup Michelson interferometer is shown in figure 2.
[[File:Experimental setup Michelson Interferometer.jpeg|center|thumb|600px|Figure 2: Experimental setup of Michelson interferometer.]]

The interference fringes of the Michelson interferometer are observed when the transducer is turned off in Figure 3.
[[File:Michelson interference fringes.jpeg|center|thumb|600px|Figure 3: Interference fringes which are observed when the transducer is turned off]]

==Experiment Results==
Before we started setting up our interferometer we measured the wavelength of the Mitsubishi LD in an Anstrom WS-7 wavemeter, the wavelength of the LD is about 658.9 nm, see fig. 4. Even more, this graph tells us that the Mitsubishi LD is in a single mode regimen, so we proceeded to build the interferometer.
[[File:Wavelength.jpeg|center|thumb|600px|Figure 4: The Wavelength measurement of the Mitsubishi Laser Diode.]]

After the alignment of the optical setup, we applied sinusoidal voltage to the transducer attached to the mirror tightly. intensity fluctuation is recorded by the homemade photodiode. Data from the photodiode is collected by the oscilloscope. The typical figure from the oscilloscope is shown in Figure 5. According to the equation mentioned before, the number of peaks during each period corresponds to the radio of displacement and laser wavelength. In this case, we can measure the displacement for various frequencies. The frequency range is from 86 Hz to 130 Hz.

[[File:ScopeScreen.jpeg|center|thumb|600px|Figure 5: The display on the oscilloscope at <math>f = 103Hz </math>. After applying the voltage to the transducer, the voltage output from photodiode changes with time.]]

We plot the displacement for different frequencies with the errorbar in Figure 6. Uncertainty is the square root uncertainty of the count number. For a driven, damped and harmonic oscillator, the relation between displacement and frequency is

<math>|A|=\frac{f_{0}}{\sqrt{\left(\omega_{0}^{2}-\omega^{2}\right)^{2}+4 \beta^{2} \omega^{2}}} </math>

Where <math>f_{0}</math> is the amplitude, <math>\omega_{0}</math> is the resonance frequency, and <math>\beta_{0}</math> is a constant. With this equation, we can fit the frequency response curve. Because there is only one peak or valley for a frequency lower than 92 Hz or higher than 120 Hz, and the error is significant and non-negligible. We exclude those in the fitting process. Finally, the fitting root mean squared error (RMSE) is 130.38. Fitted <math>\beta</math> is <math>1.3 \pm 0.2</math>. The fitting resonance frequency <math>\omega_{0} </math> is <math>113.1 \pm 0.1Hz </math>.

[[File:FitPeakNum.jpeg|center|thumb|600px|Figure 6: Frequency response curve for the transducer system with errorbar. The blue dots represent the displacement measured with different frequencies. Error bars are the square root uncertainty. Red line is the fitting curve for a driven, damped, harmonic oscillator.]]

==Discussion==
Our final result fits our data to the best Lorentzian Function. When we say best we talk about the function with the lowest Root Mean Square Error (RMSE).

RMSE is defined as follow:

<math>\operatorname{RMSE}=\sqrt{\frac{1}{n}\sum_{i=1}^n(A_i-\hat{A_i})^2}</math>

where <math>A_i</math> is the measured displacement and <math>\hat{A_i}</math> is the predicted displacement by the function model. The core of this method is to calculate the distance between the measured and predicted values and take the root mean of them.

The displacements (<math>A_i</math>) we have measured are in the regimen of thousands and the error is ~100. This is attributed to the response of the system to lower frequencies. However, for near resonance, the model fits pretty well.

== Conclusion ==
We have shown indeed that we can characterize the resonance frequency of a system. For this particular application, we found a frequency of <math>113.1 \pm 0.1Hz </math> for the system transducer plus mirror. Finally, this technique has the great potential to be used in initial experimental laboratories due his simplicity and it illustrates several physics concepts such as optics and mechanical resonance.

==References==
[1] Freschi, A. A., et al. "Laser interferometric characterization of a vibrating speaker system." American Journal of Physics 71.11 (2003): 1121-1126.
https://doi.org/10.1119/1.1586262

[2] Skarha, Matthew. "Laser interferometric characterization for a vibrating speaker system."
http://mattskarha.com/assets/docs/Laser%20interferometric%20characterization%20for%20a%20vibrating%20speaker%20system%20good.pdf

[3] Pathare, Shirish, and Vikrant Kurmude. "Low cost Michelson–Morley interferometer." Physics Education 51.6 (2016): 063001.
https://doi.org/10.1088/0031-9120/51/6/063001

Resonance frequency measurement using a interferometric method

2022-04-30T08:20:15Z

Aucca: /* Conclusion */

== Team Members ==
* [[User:Nakarin|Nakarin Jayjong]]
* [[User:Aucca|Joel Auccapuclla]]
* [[User:Xiaoyu|Xiaoyu Nie]]
* [[User:Haotian|Haotian Song]]

== Introduction ==
Interferometers since their invention have been a great tool for experimental physics. Applications cover a wide spectrum of fields such as astronomy, fiber optics, seismology, seismology, velocimetry to say the least. The precision of measurement of small displacements is the key factor, take LIGO as an example. Another important factor is the convenience to set up an interferometer and the low complexity in terms of equipment. In this experiment, we will measure the resonance frequency of a system, a transducer attached to a mirror, using a Michelson interferometer.

== Theory ==
Our Michelson interferometer setup used a laser, a beam splitter, two mirrors, and a photometer, illustrated in Fig. 1. The light propagates from the source to the beam splitter where approximately 50% of the incident light is transmitted and approximately 50% of it is reflected at a right angle. The transmitted light continues to propagate as the reference plane wave to the stationary mirror where it is fully reflected back towards the beam splitter. The reflected light propagates as the target plane wave to the mirror attached to the oscillating loudspeaker. The two waves then recombine at the beam splitter and interfere based on their relative phases based on the idea that the phase shift of the target plane wave varies with the instantaneous position of the speaker mirror combination. Consider a situation in which we setup our interferometer such that no phase shift corresponds to the equilibrium position of the speaker-mirror combination. At this position, we expect constructive interference at the photometer. Likewise, for every λ/2 change in displacement, we expect constructive interference at the photometer because the path length travelled by the target plane wave varies by some integral multiple of λ. Thus, the phase shift is an even multiple of π and the two waves remain in phase. Similarly, when the path length traveled is some odd multiple of λ/4, the phase shift is an odd multiple of π and the two waves interfere destructively.

Consider two coherent waves with the same amplitude A and same frequency <math>\omega</math> that have paths <math>x_1</math> and <math>x_2</math> emitted from different sources that meet together at a point. The net wave at that point can then be given as the real part of the equation

<math>
\psi=Ae^{i(kx_1-\omega t}+Ae^{i(kx_2-\omega t)}=Ae^{-i\omega t}e^{ik\frac{(x_1+x_2)}{2}}[e^{ik\frac{(x_1-x_2)}{2}}e^{-ik\frac{(x_1-x_2)}{2}}]=2Acos[k(\frac{x_1-x_2}{2})]e^{i[k\frac{(x_1+x_2)}{2}-\omega t]}
</math>

where the wave number <math>k=2\pi/\lambda</math>, <math>\lambda</math> is the wavelength of the waves, and c is the phase velocity. The time-averaged intensity of the combined wave is then proportional to the square of the its amplitude. That is,

<math>
I\propto cos^2[\pi(\frac{p_2-p_1}{\lambda})]=cos^2[2\pi(\frac{p_2}{\lambda}-\theta)]
</math>

where <math>\theta</math> is a constant. Now suppose the mirror that the <math>p_2</math> beam is incident to oscillates harmonically back and forth with a frequency <math>f</math> and a displacement <math>B</math>. Then, <math>p_2 =p_2(0)+Bsin(2\pi ft)</math> and the intensity of the detected light varies with time as

<math>
I=A^2cos^2[2\pi(\frac{Bsin(2\pi ft)}{\lambda}-\theta')]
</math>

Note that the frequency of oscillations is itself a sinusoid, meaning that there will be turnaround points in the oscillations.

== Experimental setup ==
As our main source for this experiment, we used a laser diode of wavelength 658.9 nm (as we can see in fig. 4 in the following section). Most red laser diodes are based on either GaInP or AlGaInP, this means we have 2 semiconductor materials. The laser diode is collimated by an aspheric lens (C110TMD-A). For one particular configuration, the data sheet tells us this, it will allow current flow and produce light amplification, this is what we called laser. This is why the diode laser is connected to a power supply, see left top size of fig. 2.

Our Michelson interferometer consists of a beam splitter BS, which is used to split the laser beam into two beams, and two mirrors M1 and M2. The reflected beam goes to a fixed mirror M1 while the transmitted beam goes to the movable mirror M2 (see fig. 1). A transducer is attached, using tape, to the mirror M2. The transducer is powered by a signal generator that is used to vary the vibrating frequency of the mirror. A photodiode PD is used to detect the signal of interference fringes. The schematic Michelson interferometer is shown in Figure 1.
[[File:Schematic Michelson Interferometer.jpeg|center|thumb|600px|Figure 1: Schematic figure of Michelson interferometer. LD is a laser diode, BS is a beam splitter, M1 and M2 are mirrors, PD is a photodiode, and vibration transducer which is driven by a signal generator.]]

The experimental setup Michelson interferometer is shown in figure 2.
[[File:Experimental setup Michelson Interferometer.jpeg|center|thumb|600px|Figure 2: Experimental setup of Michelson interferometer.]]

The interference fringes of the Michelson interferometer are observed when the transducer is turned off in Figure 3.
[[File:Michelson interference fringes.jpeg|center|thumb|600px|Figure 3: Interference fringes which are observed when the transducer is turned off]]

==Experiment Results==
Before we started setting up our interferometer we measured the wavelength of the Mitsubishi LD in an Anstrom WS-7 wavemeter, the wavelength of the LD is about 658.9 nm, see fig. 4. Even more, this graph tells us that the Mitsubishi LD is in a single mode regimen, so we proceeded to build the interferometer.
[[File:Wavelength.jpeg|center|thumb|600px|Figure 4: The Wavelength measurement of the Mitsubishi Laser Diode.]]

After the alignment of the optical setup, we applied sinusoidal voltage to the transducer attached to the mirror tightly. intensity fluctuation is recorded by the homemade photodiode. Data from the photodiode is collected by the oscilloscope. The typical figure from the oscilloscope is shown in Figure 5. According to the equation mentioned before, the number of peaks during each period corresponds to the radio of displacement and laser wavelength. In this case, we can measure the displacement for various frequencies. The frequency range is from 86 Hz to 130 Hz.

[[File:ScopeScreen.jpeg|center|thumb|600px|Figure 5: The display on the oscilloscope at <math>f = 103Hz </math>. After applying the voltage to the transducer, the voltage output from photodiode changes with time.]]

We plot the displacement for different frequencies with the errorbar in Figure 6. Uncertainty is the square root uncertainty of the count number. For a driven, damped and harmonic oscillator, the relation between displacement and frequency is

<math>|A|=\frac{f_{0}}{\sqrt{\left(\omega_{0}^{2}-\omega^{2}\right)^{2}+4 \beta^{2} \omega^{2}}} </math>

Where <math>f_{0}</math> is the amplitude, <math>\omega_{0}</math> is the resonance frequency, and <math>\beta_{0}</math> is a constant. With this equation, we can fit the frequency response curve. Because there is only one peak or valley for a frequency lower than 92 Hz or higher than 120 Hz, and the error is significant and non-negligible. We exclude those in the fitting process. Finally, the fitting root mean squared error (RMSE) is 130.38. Fitted <math>\beta</math> is <math>1.3 \pm 0.2</math>. The fitting resonance frequency <math>\omega_{0} </math> is <math>113.1 \pm 0.1Hz </math>.

[[File:FitPeakNum.jpeg|center|thumb|600px|Figure 6: Frequency response curve for the transducer system with errorbar. The blue dots represent the displacement measured with different frequencies. Error bars are the square root uncertainty. Red line is the fitting curve for a driven, damped, harmonic oscillator.]]

==Discussion==
Our final result fits our data to the best Lorentzian Function. When we say best we talk about the function with the lowest Root Mean Square Error (RMSE).

RMSE is defined as follow:

<math>\operatorname{RMSE}=\sqrt{\frac{1}{n}\sum_{i=1}^n(A_i-\hat{A_i})^2}</math>

where <math>A_i</math> is the measured displacement and <math>\hat{A_i}</math> is the predicted displacement by the function model. The core of this method is to calculate the distance between the measured and predicted values and take the root mean of them.

The displacements (<math>A_i</math>) we have measured are in the regimen of thousands and the error is ~100. This is attributed to the response of the system to lower frequencies. However, for near resonance, the model fits pretty well.

== Conclusion ==
We have shown indeed that we can characterize the resonance frequency of a system. For this particular application, we found a frequency of <math>113.1 \pm 0.1Hz </math> for the system transducer plus mirror. Finally, this technique has the great potential to be used in initial experimental laboratories due his simplicity and it illustrates several physics concepts such as optics and mechanical resonance.

==References==
[1] Freschi, A. A., et al. "Laser interferometric characterization of a vibrating speaker system." American Journal of Physics 71.11 (2003): 1121-1126.
https://doi.org/10.1119/1.1586262

[2] Skarha, Matthew. "Laser interferometric characterization for a vibrating speaker system."
http://mattskarha.com/assets/docs/Laser%20interferometric%20characterization%20for%20a%20vibrating%20speaker%20system%20good.pdf

[3] Pathare, Shirish, and Vikrant Kurmude. "Low cost Michelson–Morley interferometer." Physics Education 51.6 (2016): 063001.
https://doi.org/10.1088/0031-9120/51/6/063001

Resonance frequency measurement using a interferometric method

2022-04-30T08:18:43Z

Aucca: /* Conclusion */

== Team Members ==
* [[User:Nakarin|Nakarin Jayjong]]
* [[User:Aucca|Joel Auccapuclla]]
* [[User:Xiaoyu|Xiaoyu Nie]]
* [[User:Haotian|Haotian Song]]

== Introduction ==
Interferometers since their invention have been a great tool for experimental physics. Applications cover a wide spectrum of fields such as astronomy, fiber optics, seismology, seismology, velocimetry to say the least. The precision of measurement of small displacements is the key factor, take LIGO as an example. Another important factor is the convenience to set up an interferometer and the low complexity in terms of equipment. In this experiment, we will measure the resonance frequency of a system, a transducer attached to a mirror, using a Michelson interferometer.

== Theory ==
Our Michelson interferometer setup used a laser, a beam splitter, two mirrors, and a photometer, illustrated in Fig. 1. The light propagates from the source to the beam splitter where approximately 50% of the incident light is transmitted and approximately 50% of it is reflected at a right angle. The transmitted light continues to propagate as the reference plane wave to the stationary mirror where it is fully reflected back towards the beam splitter. The reflected light propagates as the target plane wave to the mirror attached to the oscillating loudspeaker. The two waves then recombine at the beam splitter and interfere based on their relative phases based on the idea that the phase shift of the target plane wave varies with the instantaneous position of the speaker mirror combination. Consider a situation in which we setup our interferometer such that no phase shift corresponds to the equilibrium position of the speaker-mirror combination. At this position, we expect constructive interference at the photometer. Likewise, for every λ/2 change in displacement, we expect constructive interference at the photometer because the path length travelled by the target plane wave varies by some integral multiple of λ. Thus, the phase shift is an even multiple of π and the two waves remain in phase. Similarly, when the path length traveled is some odd multiple of λ/4, the phase shift is an odd multiple of π and the two waves interfere destructively.

Consider two coherent waves with the same amplitude A and same frequency <math>\omega</math> that have paths <math>x_1</math> and <math>x_2</math> emitted from different sources that meet together at a point. The net wave at that point can then be given as the real part of the equation

<math>
\psi=Ae^{i(kx_1-\omega t}+Ae^{i(kx_2-\omega t)}=Ae^{-i\omega t}e^{ik\frac{(x_1+x_2)}{2}}[e^{ik\frac{(x_1-x_2)}{2}}e^{-ik\frac{(x_1-x_2)}{2}}]=2Acos[k(\frac{x_1-x_2}{2})]e^{i[k\frac{(x_1+x_2)}{2}-\omega t]}
</math>

where the wave number <math>k=2\pi/\lambda</math>, <math>\lambda</math> is the wavelength of the waves, and c is the phase velocity. The time-averaged intensity of the combined wave is then proportional to the square of the its amplitude. That is,

<math>
I\propto cos^2[\pi(\frac{p_2-p_1}{\lambda})]=cos^2[2\pi(\frac{p_2}{\lambda}-\theta)]
</math>

where <math>\theta</math> is a constant. Now suppose the mirror that the <math>p_2</math> beam is incident to oscillates harmonically back and forth with a frequency <math>f</math> and a displacement <math>B</math>. Then, <math>p_2 =p_2(0)+Bsin(2\pi ft)</math> and the intensity of the detected light varies with time as

<math>
I=A^2cos^2[2\pi(\frac{Bsin(2\pi ft)}{\lambda}-\theta')]
</math>

Note that the frequency of oscillations is itself a sinusoid, meaning that there will be turnaround points in the oscillations.

== Experimental setup ==
As our main source for this experiment, we used a laser diode of wavelength 658.9 nm (as we can see in fig. 4 in the following section). Most red laser diodes are based on either GaInP or AlGaInP, this means we have 2 semiconductor materials. The laser diode is collimated by an aspheric lens (C110TMD-A). For one particular configuration, the data sheet tells us this, it will allow current flow and produce light amplification, this is what we called laser. This is why the diode laser is connected to a power supply, see left top size of fig. 2.

Our Michelson interferometer consists of a beam splitter BS, which is used to split the laser beam into two beams, and two mirrors M1 and M2. The reflected beam goes to a fixed mirror M1 while the transmitted beam goes to the movable mirror M2 (see fig. 1). A transducer is attached, using tape, to the mirror M2. The transducer is powered by a signal generator that is used to vary the vibrating frequency of the mirror. A photodiode PD is used to detect the signal of interference fringes. The schematic Michelson interferometer is shown in Figure 1.
[[File:Schematic Michelson Interferometer.jpeg|center|thumb|600px|Figure 1: Schematic figure of Michelson interferometer. LD is a laser diode, BS is a beam splitter, M1 and M2 are mirrors, PD is a photodiode, and vibration transducer which is driven by a signal generator.]]

The experimental setup Michelson interferometer is shown in figure 2.
[[File:Experimental setup Michelson Interferometer.jpeg|center|thumb|600px|Figure 2: Experimental setup of Michelson interferometer.]]

The interference fringes of the Michelson interferometer are observed when the transducer is turned off in Figure 3.
[[File:Michelson interference fringes.jpeg|center|thumb|600px|Figure 3: Interference fringes which are observed when the transducer is turned off]]

==Experiment Results==
Before we started setting up our interferometer we measured the wavelength of the Mitsubishi LD in an Anstrom WS-7 wavemeter, the wavelength of the LD is about 658.9 nm, see fig. 4. Even more, this graph tells us that the Mitsubishi LD is in a single mode regimen, so we proceeded to build the interferometer.
[[File:Wavelength.jpeg|center|thumb|600px|Figure 4: The Wavelength measurement of the Mitsubishi Laser Diode.]]

After the alignment of the optical setup, we applied sinusoidal voltage to the transducer attached to the mirror tightly. intensity fluctuation is recorded by the homemade photodiode. Data from the photodiode is collected by the oscilloscope. The typical figure from the oscilloscope is shown in Figure 5. According to the equation mentioned before, the number of peaks during each period corresponds to the radio of displacement and laser wavelength. In this case, we can measure the displacement for various frequencies. The frequency range is from 86 Hz to 130 Hz.

[[File:ScopeScreen.jpeg|center|thumb|600px|Figure 5: The display on the oscilloscope at <math>f = 103Hz </math>. After applying the voltage to the transducer, the voltage output from photodiode changes with time.]]

We plot the displacement for different frequencies with the errorbar in Figure 6. Uncertainty is the square root uncertainty of the count number. For a driven, damped and harmonic oscillator, the relation between displacement and frequency is

<math>|A|=\frac{f_{0}}{\sqrt{\left(\omega_{0}^{2}-\omega^{2}\right)^{2}+4 \beta^{2} \omega^{2}}} </math>

Where <math>f_{0}</math> is the amplitude, <math>\omega_{0}</math> is the resonance frequency, and <math>\beta_{0}</math> is a constant. With this equation, we can fit the frequency response curve. Because there is only one peak or valley for a frequency lower than 92 Hz or higher than 120 Hz, and the error is significant and non-negligible. We exclude those in the fitting process. Finally, the fitting root mean squared error (RMSE) is 130.38. Fitted <math>\beta</math> is <math>1.3 \pm 0.2</math>. The fitting resonance frequency <math>\omega_{0} </math> is <math>113.1 \pm 0.1Hz </math>.

[[File:FitPeakNum.jpeg|center|thumb|600px|Figure 6: Frequency response curve for the transducer system with errorbar. The blue dots represent the displacement measured with different frequencies. Error bars are the square root uncertainty. Red line is the fitting curve for a driven, damped, harmonic oscillator.]]

==Discussion==
Our final result fits our data to the best Lorentzian Function. When we say best we talk about the function with the lowest Root Mean Square Error (RMSE).

RMSE is defined as follow:

<math>\operatorname{RMSE}=\sqrt{\frac{1}{n}\sum_{i=1}^n(A_i-\hat{A_i})^2}</math>

where <math>A_i</math> is the measured displacement and <math>\hat{A_i}</math> is the predicted displacement by the function model. The core of this method is to calculate the distance between the measured and predicted values and take the root mean of them.

The displacements (<math>A_i</math>) we have measured are in the regimen of thousands and the error is ~100. This is attributed to the response of the system to lower frequencies. However, for near resonance, the model fits pretty well.

== Conclusion ==
We have shown indeed that we can characterize the resonance frequency of a system. For this particular application, we found a frequency of <math>113.1 \pm 0.1Hz </math> for the system transducer plus mirror. Finally, this technique has the great potential to be used in initial experimental laboratories due his simplicity and mix of physics concept such as optics and mechanical resonance.

==References==
[1] Freschi, A. A., et al. "Laser interferometric characterization of a vibrating speaker system." American Journal of Physics 71.11 (2003): 1121-1126.
https://doi.org/10.1119/1.1586262

[2] Skarha, Matthew. "Laser interferometric characterization for a vibrating speaker system."
http://mattskarha.com/assets/docs/Laser%20interferometric%20characterization%20for%20a%20vibrating%20speaker%20system%20good.pdf

[3] Pathare, Shirish, and Vikrant Kurmude. "Low cost Michelson–Morley interferometer." Physics Education 51.6 (2016): 063001.
https://doi.org/10.1088/0031-9120/51/6/063001

Resonance frequency measurement using a interferometric method

2022-04-30T08:15:02Z

Aucca: /* Conclusion */

== Team Members ==
* [[User:Nakarin|Nakarin Jayjong]]
* [[User:Aucca|Joel Auccapuclla]]
* [[User:Xiaoyu|Xiaoyu Nie]]
* [[User:Haotian|Haotian Song]]

== Introduction ==
Interferometers since their invention have been a great tool for experimental physics. Applications cover a wide spectrum of fields such as astronomy, fiber optics, seismology, seismology, velocimetry to say the least. The precision of measurement of small displacements is the key factor, take LIGO as an example. Another important factor is the convenience to set up an interferometer and the low complexity in terms of equipment. In this experiment, we will measure the resonance frequency of a system, a transducer attached to a mirror, using a Michelson interferometer.

== Theory ==
Our Michelson interferometer setup used a laser, a beam splitter, two mirrors, and a photometer, illustrated in Fig. 1. The light propagates from the source to the beam splitter where approximately 50% of the incident light is transmitted and approximately 50% of it is reflected at a right angle. The transmitted light continues to propagate as the reference plane wave to the stationary mirror where it is fully reflected back towards the beam splitter. The reflected light propagates as the target plane wave to the mirror attached to the oscillating loudspeaker. The two waves then recombine at the beam splitter and interfere based on their relative phases based on the idea that the phase shift of the target plane wave varies with the instantaneous position of the speaker mirror combination. Consider a situation in which we setup our interferometer such that no phase shift corresponds to the equilibrium position of the speaker-mirror combination. At this position, we expect constructive interference at the photometer. Likewise, for every λ/2 change in displacement, we expect constructive interference at the photometer because the path length travelled by the target plane wave varies by some integral multiple of λ. Thus, the phase shift is an even multiple of π and the two waves remain in phase. Similarly, when the path length traveled is some odd multiple of λ/4, the phase shift is an odd multiple of π and the two waves interfere destructively.

Consider two coherent waves with the same amplitude A and same frequency <math>\omega</math> that have paths <math>x_1</math> and <math>x_2</math> emitted from different sources that meet together at a point. The net wave at that point can then be given as the real part of the equation

<math>
\psi=Ae^{i(kx_1-\omega t}+Ae^{i(kx_2-\omega t)}=Ae^{-i\omega t}e^{ik\frac{(x_1+x_2)}{2}}[e^{ik\frac{(x_1-x_2)}{2}}e^{-ik\frac{(x_1-x_2)}{2}}]=2Acos[k(\frac{x_1-x_2}{2})]e^{i[k\frac{(x_1+x_2)}{2}-\omega t]}
</math>

where the wave number <math>k=2\pi/\lambda</math>, <math>\lambda</math> is the wavelength of the waves, and c is the phase velocity. The time-averaged intensity of the combined wave is then proportional to the square of the its amplitude. That is,

<math>
I\propto cos^2[\pi(\frac{p_2-p_1}{\lambda})]=cos^2[2\pi(\frac{p_2}{\lambda}-\theta)]
</math>

where <math>\theta</math> is a constant. Now suppose the mirror that the <math>p_2</math> beam is incident to oscillates harmonically back and forth with a frequency <math>f</math> and a displacement <math>B</math>. Then, <math>p_2 =p_2(0)+Bsin(2\pi ft)</math> and the intensity of the detected light varies with time as

<math>
I=A^2cos^2[2\pi(\frac{Bsin(2\pi ft)}{\lambda}-\theta')]
</math>

Note that the frequency of oscillations is itself a sinusoid, meaning that there will be turnaround points in the oscillations.

== Experimental setup ==
As our main source for this experiment, we used a laser diode of wavelength 658.9 nm (as we can see in fig. 4 in the following section). Most red laser diodes are based on either GaInP or AlGaInP, this means we have 2 semiconductor materials. The laser diode is collimated by an aspheric lens (C110TMD-A). For one particular configuration, the data sheet tells us this, it will allow current flow and produce light amplification, this is what we called laser. This is why the diode laser is connected to a power supply, see left top size of fig. 2.

Our Michelson interferometer consists of a beam splitter BS, which is used to split the laser beam into two beams, and two mirrors M1 and M2. The reflected beam goes to a fixed mirror M1 while the transmitted beam goes to the movable mirror M2 (see fig. 1). A transducer is attached, using tape, to the mirror M2. The transducer is powered by a signal generator that is used to vary the vibrating frequency of the mirror. A photodiode PD is used to detect the signal of interference fringes. The schematic Michelson interferometer is shown in Figure 1.
[[File:Schematic Michelson Interferometer.jpeg|center|thumb|600px|Figure 1: Schematic figure of Michelson interferometer. LD is a laser diode, BS is a beam splitter, M1 and M2 are mirrors, PD is a photodiode, and vibration transducer which is driven by a signal generator.]]

The experimental setup Michelson interferometer is shown in figure 2.
[[File:Experimental setup Michelson Interferometer.jpeg|center|thumb|600px|Figure 2: Experimental setup of Michelson interferometer.]]

The interference fringes of the Michelson interferometer are observed when the transducer is turned off in Figure 3.
[[File:Michelson interference fringes.jpeg|center|thumb|600px|Figure 3: Interference fringes which are observed when the transducer is turned off]]

==Experiment Results==
Before we started setting up our interferometer we measured the wavelength of the Mitsubishi LD in an Anstrom WS-7 wavemeter, the wavelength of the LD is about 658.9 nm, see fig. 4. Even more, this graph tells us that the Mitsubishi LD is in a single mode regimen, so we proceeded to build the interferometer.
[[File:Wavelength.jpeg|center|thumb|600px|Figure 4: The Wavelength measurement of the Mitsubishi Laser Diode.]]

After the alignment of the optical setup, we applied sinusoidal voltage to the transducer attached to the mirror tightly. intensity fluctuation is recorded by the homemade photodiode. Data from the photodiode is collected by the oscilloscope. The typical figure from the oscilloscope is shown in Figure 5. According to the equation mentioned before, the number of peaks during each period corresponds to the radio of displacement and laser wavelength. In this case, we can measure the displacement for various frequencies. The frequency range is from 86 Hz to 130 Hz.

[[File:ScopeScreen.jpeg|center|thumb|600px|Figure 5: The display on the oscilloscope at <math>f = 103Hz </math>. After applying the voltage to the transducer, the voltage output from photodiode changes with time.]]

We plot the displacement for different frequencies with the errorbar in Figure 6. Uncertainty is the square root uncertainty of the count number. For a driven, damped and harmonic oscillator, the relation between displacement and frequency is

<math>|A|=\frac{f_{0}}{\sqrt{\left(\omega_{0}^{2}-\omega^{2}\right)^{2}+4 \beta^{2} \omega^{2}}} </math>

Where <math>f_{0}</math> is the amplitude, <math>\omega_{0}</math> is the resonance frequency, and <math>\beta_{0}</math> is a constant. With this equation, we can fit the frequency response curve. Because there is only one peak or valley for a frequency lower than 92 Hz or higher than 120 Hz, and the error is significant and non-negligible. We exclude those in the fitting process. Finally, the fitting root mean squared error (RMSE) is 130.38. Fitted <math>\beta</math> is <math>1.3 \pm 0.2</math>. The fitting resonance frequency <math>\omega_{0} </math> is <math>113.1 \pm 0.1Hz </math>.

[[File:FitPeakNum.jpeg|center|thumb|600px|Figure 6: Frequency response curve for the transducer system with errorbar. The blue dots represent the displacement measured with different frequencies. Error bars are the square root uncertainty. Red line is the fitting curve for a driven, damped, harmonic oscillator.]]

==Discussion==
Our final result fits our data to the best Lorentzian Function. When we say best we talk about the function with the lowest Root Mean Square Error (RMSE).

RMSE is defined as follow:

<math>\operatorname{RMSE}=\sqrt{\frac{1}{n}\sum_{i=1}^n(A_i-\hat{A_i})^2}</math>

where <math>A_i</math> is the measured displacement and <math>\hat{A_i}</math> is the predicted displacement by the function model. The core of this method is calculate the distant between the measured and predicted values and take the root mean of them.

The displacements (<math>A_i</math>) we have measured are in the regimen of thousands and the error is ~100. This is attributed to the response of the system to lower frequencies. However, for near resonance the model fits pretty well.

== Conclusion ==
We have shown indeed that we can characterize the resonance frequency of a system. For this particular application, we found a frequency of <math>113.1 \pm 0.1Hz </math> for the system transducer plus mirror. Finally, this technique has proved that is easy to implement and allows to visualise concepts of optics and mechanical resonance.

==References==
[1] Freschi, A. A., et al. "Laser interferometric characterization of a vibrating speaker system." American Journal of Physics 71.11 (2003): 1121-1126.
https://doi.org/10.1119/1.1586262

[2] Skarha, Matthew. "Laser interferometric characterization for a vibrating speaker system."
http://mattskarha.com/assets/docs/Laser%20interferometric%20characterization%20for%20a%20vibrating%20speaker%20system%20good.pdf

[3] Pathare, Shirish, and Vikrant Kurmude. "Low cost Michelson–Morley interferometer." Physics Education 51.6 (2016): 063001.
https://doi.org/10.1088/0031-9120/51/6/063001

Resonance frequency measurement using a interferometric method

2022-04-30T08:07:35Z

Aucca: /* Introduction */

== Team Members ==
* [[User:Nakarin|Nakarin Jayjong]]
* [[User:Aucca|Joel Auccapuclla]]
* [[User:Xiaoyu|Xiaoyu Nie]]
* [[User:Haotian|Haotian Song]]

== Introduction ==
Interferometers since their invention have been a great tool for experimental physics. Applications cover a wide spectrum of fields such as astronomy, fiber optics, seismology, seismology, velocimetry to say the least. The precision of measurement of small displacements is the key factor, take LIGO as an example. Another important factor is the convenience to set up an interferometer and the low complexity in terms of equipment. In this experiment, we will measure the resonance frequency of a system, a transducer attached to a mirror, using a Michelson interferometer.

== Theory ==
Our Michelson interferometer setup used a laser, a beam splitter, two mirrors, and a photometer, illustrated in Fig. 1. The light propagates from the source to the beam splitter where approximately 50% of the incident light is transmitted and approximately 50% of it is reflected at a right angle. The transmitted light continues to propagate as the reference plane wave to the stationary mirror where it is fully reflected back towards the beam splitter. The reflected light propagates as the target plane wave to the mirror attached to the oscillating loudspeaker. The two waves then recombine at the beam splitter and interfere based on their relative phases based on the idea that the phase shift of the target plane wave varies with the instantaneous position of the speaker mirror combination. Consider a situation in which we setup our interferometer such that no phase shift corresponds to the equilibrium position of the speaker-mirror combination. At this position, we expect constructive interference at the photometer. Likewise, for every λ/2 change in displacement, we expect constructive interference at the photometer because the path length travelled by the target plane wave varies by some integral multiple of λ. Thus, the phase shift is an even multiple of π and the two waves remain in phase. Similarly, when the path length traveled is some odd multiple of λ/4, the phase shift is an odd multiple of π and the two waves interfere destructively.

Consider two coherent waves with the same amplitude A and same frequency <math>\omega</math> that have paths <math>x_1</math> and <math>x_2</math> emitted from different sources that meet together at a point. The net wave at that point can then be given as the real part of the equation

<math>
\psi=Ae^{i(kx_1-\omega t}+Ae^{i(kx_2-\omega t)}=Ae^{-i\omega t}e^{ik\frac{(x_1+x_2)}{2}}[e^{ik\frac{(x_1-x_2)}{2}}e^{-ik\frac{(x_1-x_2)}{2}}]=2Acos[k(\frac{x_1-x_2}{2})]e^{i[k\frac{(x_1+x_2)}{2}-\omega t]}
</math>

where the wave number <math>k=2\pi/\lambda</math>, <math>\lambda</math> is the wavelength of the waves, and c is the phase velocity. The time-averaged intensity of the combined wave is then proportional to the square of the its amplitude. That is,

<math>
I\propto cos^2[\pi(\frac{p_2-p_1}{\lambda})]=cos^2[2\pi(\frac{p_2}{\lambda}-\theta)]
</math>

where <math>\theta</math> is a constant. Now suppose the mirror that the <math>p_2</math> beam is incident to oscillates harmonically back and forth with a frequency <math>f</math> and a displacement <math>B</math>. Then, <math>p_2 =p_2(0)+Bsin(2\pi ft)</math> and the intensity of the detected light varies with time as

<math>
I=A^2cos^2[2\pi(\frac{Bsin(2\pi ft)}{\lambda}-\theta')]
</math>

Note that the frequency of oscillations is itself a sinusoid, meaning that there will be turnaround points in the oscillations.

== Experimental setup ==
As our main source for this experiment we used a Laser Diode of wavelength 658,85 nm (as we can see in fig. 4 in the following section). Most red laser diodes are based on either GaInP or AlGaInP, this means we have 2 semiconductor materials. For one particular configuration, the data sheet tells us this, it will allow current flow and produce light amplification, this is what we called laser. This is why the diode laser is connected to a power supply, see left top size of fig. 2.

Our Michelson interferometer consists of a beam splitter BS, which is used to split the laser beam into two beams, and two mirrors M1 and M2. The reflected beam goes to a fixed mirror M1 while the transmitted beam goes to the movable mirror M2 (see fig. 1). A transducer is attached, using tape, to the mirror M2. The transducer is powered by a signal generator that is used to vary the vibrating frequency of the mirror. A photodiode PD is used to detect the signal of interference fringes. The schematic Michelson interferometer is shown in Figure 1.
[[File:Schematic Michelson Interferometer.jpeg|center|thumb|600px|Figure 1: Schematic figure of Michelson interferometer. LD is a laser diode, BS is a beam splitter, M1 and M2 are mirrors, PD is a photodiode, and vibration transducer which is driven by a signal generator.]]

The experimental setup Michelson interferometer is shown in figure 2.
[[File:Experimental setup Michelson Interferometer.jpeg|center|thumb|600px|Figure 2: Experimental setup of Michelson interferometer.]]

The interference fringes of the Michelson interferometer are observed when the transducer is turned off in Figure 3.
[[File:Michelson interference fringes.jpeg|center|thumb|600px|Figure 3: Interference fringes which are observed when the transducer is turned off]]

==Experiment Results==
Before we started setting up our interferometer we measured the wavelength of the Mitsubishi LD in an Anstrom WS-7 wavemeter, the wavelength of the LD is about 658.9 nm, see fig. 4. Even more, this graph tells us that the Mitsubishi LD is in a single mode regimen, so we proceeded to build the interferometer.
[[File:Wavelength.jpeg|center|thumb|600px|Figure 4: The Wavelength measurement of the Mitsubishi Laser Diode.]]

After the alignment of the optical setup, we applied sinusoidal voltage to the transducer attached to the mirror tightly. intensity fluctuation is recorded by the homemade photodiode. Data from the photodiode is collected by the oscilloscope. The typical figure from the oscilloscope is shown in Figure 5. According to the equation mentioned before, the number of peaks during each period corresponds to the radio of displacement and laser wavelength. In this case, we can measure the displacement for various frequencies. The frequency range is from 86 Hz to 130 Hz.

[[File:ScopeScreen.jpeg|center|thumb|600px|Figure 5: The display on the oscilloscope at <math>f = 103Hz </math>. After applying the voltage to the transducer, the voltage output from photodiode changes with time.]]

We plot the displacement for different frequencies with the errorbar in Figure 6. Uncertainty is the square root uncertainty of the count number. For a driven, damped and harmonic oscillator, the relation between displacement and frequency is

<math>|A|=\frac{f_{0}}{\sqrt{\left(\omega_{0}^{2}-\omega^{2}\right)^{2}+4 \beta^{2} \omega^{2}}} </math>

Where <math>f_{0}</math> is the amplitude, <math>\omega_{0}</math> is the resonance frequency, and <math>\beta_{0}</math> is a constant. With this equation, we can fit the frequency response curve. Because there is only one peak or valley for a frequency lower than 92 Hz or higher than 120 Hz, and the error is significant and non-negligible. We exclude those in the fitting process. Finally, the fitting root mean squared error (RMSE) is 130.38. Fitted <math>\beta</math> is <math>1.3 \pm 0.2</math>. The fitting resonance frequency <math>\omega_{0} </math> is <math>113.1 \pm 0.1Hz </math>.

[[File:FitPeakNum.jpeg|center|thumb|600px|Figure 6: Frequency response curve for the transducer system with errorbar. The blue dots represent the displacement measured with different frequencies. Error bars are the square root uncertainty. Red line is the fitting curve for a driven, damped, harmonic oscillator.]]

==Discussion==
Our final result fits our data to the best Lorentzian Function. When we say best we talk about the function with the lowest Root Mean Square Error (RMSE).

RMSE is defined as follow:

<math>\operatorname{RMSE}=\sqrt{\frac{1}{n}\sum_{i=1}^n(A_i-\hat{A_i})^2}</math>

where <math>A_i</math> is the measured displacement and <math>\hat{A_i}</math> is the predicted displacement by the function model. The core of this method is calculate the distant between the measured and predicted values and take the root mean of them.

The displacements (<math>A_i</math>) we have measured are in the regimen of thousands and the error is ~100. This is attributed to the response of the system to lower frequencies. However, for near resonance the model fits pretty well.

== Conclusion ==
We have shown indeed that we can characterize the resonance frequency of a system. For this particular application, we found a frequency of <math>113.1 \pm 0.1Hz </math> for the system transducer plus mirror. Finally, with appropriate handling, this technique could be implemented to characterize more sofisticated systems.

==References==
[1] Freschi, A. A., et al. "Laser interferometric characterization of a vibrating speaker system." American Journal of Physics 71.11 (2003): 1121-1126.
https://doi.org/10.1119/1.1586262

[2] Skarha, Matthew. "Laser interferometric characterization for a vibrating speaker system."
http://mattskarha.com/assets/docs/Laser%20interferometric%20characterization%20for%20a%20vibrating%20speaker%20system%20good.pdf

[3] Pathare, Shirish, and Vikrant Kurmude. "Low cost Michelson–Morley interferometer." Physics Education 51.6 (2016): 063001.
https://doi.org/10.1088/0031-9120/51/6/063001

Resonance frequency measurement using a interferometric method

2022-04-30T08:06:55Z

Aucca: /* Introduction */

== Team Members ==
* [[User:Nakarin|Nakarin Jayjong]]
* [[User:Aucca|Joel Auccapuclla]]
* [[User:Xiaoyu|Xiaoyu Nie]]
* [[User:Haotian|Haotian Song]]

== Introduction ==
Interferometers since their invention have been a great tool for experimental physics. Applications cover a wide spectrum of fields such as astronomy, fiber optics, seismology, seismology, velocimetry to say the least. The precision of measurement of small displacements is the key factor, take LIGO as an example. Another important factor is the convenience to set up an interferometer and the low complexity in terms of equipment. In this experiment, we will measure the resonance frequency of a system, a transducer attach to a mirror, using a Michelson interferometer.

== Theory ==
Our Michelson interferometer setup used a laser, a beam splitter, two mirrors, and a photometer, illustrated in Fig. 1. The light propagates from the source to the beam splitter where approximately 50% of the incident light is transmitted and approximately 50% of it is reflected at a right angle. The transmitted light continues to propagate as the reference plane wave to the stationary mirror where it is fully reflected back towards the beam splitter. The reflected light propagates as the target plane wave to the mirror attached to the oscillating loudspeaker. The two waves then recombine at the beam splitter and interfere based on their relative phases based on the idea that the phase shift of the target plane wave varies with the instantaneous position of the speaker mirror combination. Consider a situation in which we setup our interferometer such that no phase shift corresponds to the equilibrium position of the speaker-mirror combination. At this position, we expect constructive interference at the photometer. Likewise, for every λ/2 change in displacement, we expect constructive interference at the photometer because the path length travelled by the target plane wave varies by some integral multiple of λ. Thus, the phase shift is an even multiple of π and the two waves remain in phase. Similarly, when the path length traveled is some odd multiple of λ/4, the phase shift is an odd multiple of π and the two waves interfere destructively.

Consider two coherent waves with the same amplitude A and same frequency <math>\omega</math> that have paths <math>x_1</math> and <math>x_2</math> emitted from different sources that meet together at a point. The net wave at that point can then be given as the real part of the equation

<math>
\psi=Ae^{i(kx_1-\omega t}+Ae^{i(kx_2-\omega t)}=Ae^{-i\omega t}e^{ik\frac{(x_1+x_2)}{2}}[e^{ik\frac{(x_1-x_2)}{2}}e^{-ik\frac{(x_1-x_2)}{2}}]=2Acos[k(\frac{x_1-x_2}{2})]e^{i[k\frac{(x_1+x_2)}{2}-\omega t]}
</math>

where the wave number <math>k=2\pi/\lambda</math>, <math>\lambda</math> is the wavelength of the waves, and c is the phase velocity. The time-averaged intensity of the combined wave is then proportional to the square of the its amplitude. That is,

<math>
I\propto cos^2[\pi(\frac{p_2-p_1}{\lambda})]=cos^2[2\pi(\frac{p_2}{\lambda}-\theta)]
</math>

where <math>\theta</math> is a constant. Now suppose the mirror that the <math>p_2</math> beam is incident to oscillates harmonically back and forth with a frequency <math>f</math> and a displacement <math>B</math>. Then, <math>p_2 =p_2(0)+Bsin(2\pi ft)</math> and the intensity of the detected light varies with time as

<math>
I=A^2cos^2[2\pi(\frac{Bsin(2\pi ft)}{\lambda}-\theta')]
</math>

Note that the frequency of oscillations is itself a sinusoid, meaning that there will be turnaround points in the oscillations.

== Experimental setup ==
As our main source for this experiment we used a Laser Diode of wavelength 658,85 nm (as we can see in fig. 4 in the following section). Most red laser diodes are based on either GaInP or AlGaInP, this means we have 2 semiconductor materials. For one particular configuration, the data sheet tells us this, it will allow current flow and produce light amplification, this is what we called laser. This is why the diode laser is connected to a power supply, see left top size of fig. 2.

Our Michelson interferometer consists of a beam splitter BS, which is used to split the laser beam into two beams, and two mirrors M1 and M2. The reflected beam goes to a fixed mirror M1 while the transmitted beam goes to the movable mirror M2 (see fig. 1). A transducer is attached, using tape, to the mirror M2. The transducer is powered by a signal generator that is used to vary the vibrating frequency of the mirror. A photodiode PD is used to detect the signal of interference fringes. The schematic Michelson interferometer is shown in Figure 1.
[[File:Schematic Michelson Interferometer.jpeg|center|thumb|600px|Figure 1: Schematic figure of Michelson interferometer. LD is a laser diode, BS is a beam splitter, M1 and M2 are mirrors, PD is a photodiode, and vibration transducer which is driven by a signal generator.]]

The experimental setup Michelson interferometer is shown in figure 2.
[[File:Experimental setup Michelson Interferometer.jpeg|center|thumb|600px|Figure 2: Experimental setup of Michelson interferometer.]]

The interference fringes of the Michelson interferometer are observed when the transducer is turned off in Figure 3.
[[File:Michelson interference fringes.jpeg|center|thumb|600px|Figure 3: Interference fringes which are observed when the transducer is turned off]]

==Experiment Results==
Before we started setting up our interferometer we measured the wavelength of the Mitsubishi LD in an Anstrom WS-7 wavemeter, the wavelength of the LD is about 658.9 nm, see fig. 4. Even more, this graph tells us that the Mitsubishi LD is in a single mode regimen, so we proceeded to build the interferometer.
[[File:Wavelength.jpeg|center|thumb|600px|Figure 4: The Wavelength measurement of the Mitsubishi Laser Diode.]]

After the alignment of the optical setup, we applied sinusoidal voltage to the transducer attached to the mirror tightly. intensity fluctuation is recorded by the homemade photodiode. Data from the photodiode is collected by the oscilloscope. The typical figure from the oscilloscope is shown in Figure 5. According to the equation mentioned before, the number of peaks during each period corresponds to the radio of displacement and laser wavelength. In this case, we can measure the displacement for various frequencies. The frequency range is from 86 Hz to 130 Hz.

[[File:ScopeScreen.jpeg|center|thumb|600px|Figure 5: The display on the oscilloscope at <math>f = 103Hz </math>. After applying the voltage to the transducer, the voltage output from photodiode changes with time.]]

We plot the displacement for different frequencies with the errorbar in Figure 6. Uncertainty is the square root uncertainty of the count number. For a driven, damped and harmonic oscillator, the relation between displacement and frequency is

<math>|A|=\frac{f_{0}}{\sqrt{\left(\omega_{0}^{2}-\omega^{2}\right)^{2}+4 \beta^{2} \omega^{2}}} </math>

Where <math>f_{0}</math> is the amplitude, <math>\omega_{0}</math> is the resonance frequency, and <math>\beta_{0}</math> is a constant. With this equation, we can fit the frequency response curve. Because there is only one peak or valley for a frequency lower than 92 Hz or higher than 120 Hz, and the error is significant and non-negligible. We exclude those in the fitting process. Finally, the fitting root mean squared error (RMSE) is 130.38. Fitted <math>\beta</math> is <math>1.3 \pm 0.2</math>. The fitting resonance frequency <math>\omega_{0} </math> is <math>113.1 \pm 0.1Hz </math>.

[[File:FitPeakNum.jpeg|center|thumb|600px|Figure 6: Frequency response curve for the transducer system with errorbar. The blue dots represent the displacement measured with different frequencies. Error bars are the square root uncertainty. Red line is the fitting curve for a driven, damped, harmonic oscillator.]]

==Discussion==
Our final result fits our data to the best Lorentzian Function. When we say best we talk about the function with the lowest Root Mean Square Error (RMSE).

RMSE is defined as follow:

<math>\operatorname{RMSE}=\sqrt{\frac{1}{n}\sum_{i=1}^n(A_i-\hat{A_i})^2}</math>

where <math>A_i</math> is the measured displacement and <math>\hat{A_i}</math> is the predicted displacement by the function model. The core of this method is calculate the distant between the measured and predicted values and take the root mean of them.

The displacements (<math>A_i</math>) we have measured are in the regimen of thousands and the error is ~100. This is attributed to the response of the system to lower frequencies. However, for near resonance the model fits pretty well.

== Conclusion ==
We have shown indeed that we can characterize the resonance frequency of a system. For this particular application, we found a frequency of <math>113.1 \pm 0.1Hz </math> for the system transducer plus mirror. Finally, with appropriate handling, this technique could be implemented to characterize more sofisticated systems.

==References==
[1] Freschi, A. A., et al. "Laser interferometric characterization of a vibrating speaker system." American Journal of Physics 71.11 (2003): 1121-1126.
https://doi.org/10.1119/1.1586262

[2] Skarha, Matthew. "Laser interferometric characterization for a vibrating speaker system."
http://mattskarha.com/assets/docs/Laser%20interferometric%20characterization%20for%20a%20vibrating%20speaker%20system%20good.pdf

[3] Pathare, Shirish, and Vikrant Kurmude. "Low cost Michelson–Morley interferometer." Physics Education 51.6 (2016): 063001.
https://doi.org/10.1088/0031-9120/51/6/063001

Resonance frequency measurement using a interferometric method

2022-04-30T08:03:28Z

Aucca: /* Conclusion */

== Team Members ==
* [[User:Nakarin|Nakarin Jayjong]]
* [[User:Aucca|Joel Auccapuclla]]
* [[User:Xiaoyu|Xiaoyu Nie]]
* [[User:Haotian|Haotian Song]]

== Introduction ==
Interferometers since their invention have been a great tool for experimental physics. Applications cover a wide spectrum of fields such as astronomy, fiber optics, seismology, seismology, velocimetry to say the least. The precision of measurement of small displacements is the key factor, take LIGO as an example. Another important factor is the convenience to set up an interferometer and the low complexity in terms of equipment. In this experiment, we will measure the resonance frequency of a vibrating transducer using a Michelson interferometer. Basically, a transducer is attached to one optical path, to one mirror, of the Michelson interferometer and it is expected to see a peak in the response curve for the resonance frequency of the transducer plus mirror.

== Theory ==
Our Michelson interferometer setup used a laser, a beam splitter, two mirrors, and a photometer, illustrated in Fig. 1. The light propagates from the source to the beam splitter where approximately 50% of the incident light is transmitted and approximately 50% of it is reflected at a right angle. The transmitted light continues to propagate as the reference plane wave to the stationary mirror where it is fully reflected back towards the beam splitter. The reflected light propagates as the target plane wave to the mirror attached to the oscillating loudspeaker. The two waves then recombine at the beam splitter and interfere based on their relative phases based on the idea that the phase shift of the target plane wave varies with the instantaneous position of the speaker mirror combination. Consider a situation in which we setup our interferometer such that no phase shift corresponds to the equilibrium position of the speaker-mirror combination. At this position, we expect constructive interference at the photometer. Likewise, for every λ/2 change in displacement, we expect constructive interference at the photometer because the path length travelled by the target plane wave varies by some integral multiple of λ. Thus, the phase shift is an even multiple of π and the two waves remain in phase. Similarly, when the path length traveled is some odd multiple of λ/4, the phase shift is an odd multiple of π and the two waves interfere destructively.

Consider two coherent waves with the same amplitude A and same frequency <math>\omega</math> that have paths <math>x_1</math> and <math>x_2</math> emitted from different sources that meet together at a point. The net wave at that point can then be given as the real part of the equation

<math>
\psi=Ae^{i(kx_1-\omega t}+Ae^{i(kx_2-\omega t)}=Ae^{-i\omega t}e^{ik\frac{(x_1+x_2)}{2}}[e^{ik\frac{(x_1-x_2)}{2}}e^{-ik\frac{(x_1-x_2)}{2}}]=2Acos[k(\frac{x_1-x_2}{2})]e^{i[k\frac{(x_1+x_2)}{2}-\omega t]}
</math>

where the wave number <math>k=2\pi/\lambda</math>, <math>\lambda</math> is the wavelength of the waves, and c is the phase velocity. The time-averaged intensity of the combined wave is then proportional to the square of the its amplitude. That is,

<math>
I\propto cos^2[\pi(\frac{p_2-p_1}{\lambda})]=cos^2[2\pi(\frac{p_2}{\lambda}-\theta)]
</math>

where <math>\theta</math> is a constant. Now suppose the mirror that the <math>p_2</math> beam is incident to oscillates harmonically back and forth with a frequency <math>f</math> and a displacement <math>B</math>. Then, <math>p_2 =p_2(0)+Bsin(2\pi ft)</math> and the intensity of the detected light varies with time as

<math>
I=A^2cos^2[2\pi(\frac{Bsin(2\pi ft)}{\lambda}-\theta')]
</math>

Note that the frequency of oscillations is itself a sinusoid, meaning that there will be turnaround points in the oscillations.

== Experimental setup ==
As our main source for this experiment we used a Laser Diode of wavelength 658,85 nm (as we can see in fig. 4 in the following section). Most red laser diodes are based on either GaInP or AlGaInP, this means we have 2 semiconductor materials. For one particular configuration, the data sheet tells us this, it will allow current flow and produce light amplification, this is what we called laser. This is why the diode laser is connected to a power supply, see left top size of fig. 2.

Our Michelson interferometer consists of a beam splitter BS, which is used to split the laser beam into two beams, and two mirrors M1 and M2. The reflected beam goes to a fixed mirror M1 while the transmitted beam goes to the movable mirror M2 (see fig. 1). A transducer is attached, using tape, to the mirror M2. The transducer is powered by a signal generator that is used to vary the vibrating frequency of the mirror. A photodiode PD is used to detect the signal of interference fringes. The schematic Michelson interferometer is shown in Figure 1.
[[File:Schematic Michelson Interferometer.jpeg|center|thumb|600px|Figure 1: Schematic figure of Michelson interferometer. LD is a laser diode, BS is a beam splitter, M1 and M2 are mirrors, PD is a photodiode, and vibration transducer which is driven by a signal generator.]]

The experimental setup Michelson interferometer is shown in figure 2.
[[File:Experimental setup Michelson Interferometer.jpeg|center|thumb|600px|Figure 2: Experimental setup of Michelson interferometer.]]

The interference fringes of the Michelson interferometer are observed when the transducer is turned off in Figure 3.
[[File:Michelson interference fringes.jpeg|center|thumb|600px|Figure 3: Interference fringes which are observed when the transducer is turned off]]

==Experiment Results==
Before we started setting up our interferometer we measured the wavelength of the Mitsubishi LD in an Anstrom WS-7 wavemeter, the wavelength of the LD is about 658.9 nm, see fig. 4. Even more, this graph tells us that the Mitsubishi LD is in a single mode regimen, so we proceeded to build the interferometer.
[[File:Wavelength.jpeg|center|thumb|600px|Figure 4: The Wavelength measurement of the Mitsubishi Laser Diode.]]

After the alignment of the optical setup, we applied sinusoidal voltage to the transducer attached to the mirror tightly. intensity fluctuation is recorded by the homemade photodiode. Data from the photodiode is collected by the oscilloscope. The typical figure from the oscilloscope is shown in Figure 5. According to the equation mentioned before, the number of peaks during each period corresponds to the radio of displacement and laser wavelength. In this case, we can measure the displacement for various frequencies. The frequency range is from 86 Hz to 130 Hz.

[[File:ScopeScreen.jpeg|center|thumb|600px|Figure 5: The display on the oscilloscope at <math>f = 103Hz </math>. After applying the voltage to the transducer, the voltage output from photodiode changes with time.]]

We plot the displacement for different frequencies with the errorbar in Figure 6. Uncertainty is the square root uncertainty of the count number. For a driven, damped and harmonic oscillator, the relation between displacement and frequency is

<math>|A|=\frac{f_{0}}{\sqrt{\left(\omega_{0}^{2}-\omega^{2}\right)^{2}+4 \beta^{2} \omega^{2}}} </math>

Where <math>f_{0}</math> is the amplitude, <math>\omega_{0}</math> is the resonance frequency, and <math>\beta_{0}</math> is a constant. With this equation, we can fit the frequency response curve. Because there is only one peak or valley for a frequency lower than 92 Hz or higher than 120 Hz, and the error is significant and non-negligible. We exclude those in the fitting process. Finally, the fitting root mean squared error (RMSE) is 130.38. Fitted <math>\beta</math> is <math>1.3 \pm 0.2</math>. The fitting resonance frequency <math>\omega_{0} </math> is <math>113.1 \pm 0.1Hz </math>.

[[File:FitPeakNum.jpeg|center|thumb|600px|Figure 6: Frequency response curve for the transducer system with errorbar. The blue dots represent the displacement measured with different frequencies. Error bars are the square root uncertainty. Red line is the fitting curve for a driven, damped, harmonic oscillator.]]

==Discussion==
Our final result fits our data to the best Lorentzian Function. When we say best we talk about the function with the lowest Root Mean Square Error (RMSE).

RMSE is defined as follow:

<math>\operatorname{RMSE}=\sqrt{\frac{1}{n}\sum_{i=1}^n(A_i-\hat{A_i})^2}</math>

where <math>A_i</math> is the measured displacement and <math>\hat{A_i}</math> is the predicted displacement by the function model. The core of this method is calculate the distant between the measured and predicted values and take the root mean of them.

The displacements (<math>A_i</math>) we have measured are in the regimen of thousands and the error is ~100. This is attributed to the response of the system to lower frequencies. However, for near resonance the model fits pretty well.

== Conclusion ==
We have shown indeed that we can characterize the resonance frequency of a system. For this particular application, we found a frequency of <math>113.1 \pm 0.1Hz </math> for the system transducer plus mirror. Finally, with appropriate handling, this technique could be implemented to characterize more sofisticated systems.

==References==
[1] Freschi, A. A., et al. "Laser interferometric characterization of a vibrating speaker system." American Journal of Physics 71.11 (2003): 1121-1126.
https://doi.org/10.1119/1.1586262

[2] Skarha, Matthew. "Laser interferometric characterization for a vibrating speaker system."
http://mattskarha.com/assets/docs/Laser%20interferometric%20characterization%20for%20a%20vibrating%20speaker%20system%20good.pdf

[3] Pathare, Shirish, and Vikrant Kurmude. "Low cost Michelson–Morley interferometer." Physics Education 51.6 (2016): 063001.
https://doi.org/10.1088/0031-9120/51/6/063001

Resonance frequency measurement using a interferometric method

2022-04-30T08:00:23Z

Aucca: /* Experiment Results */

== Team Members ==
* [[User:Nakarin|Nakarin Jayjong]]
* [[User:Aucca|Joel Auccapuclla]]
* [[User:Xiaoyu|Xiaoyu Nie]]
* [[User:Haotian|Haotian Song]]

== Introduction ==
Interferometers since their invention have been a great tool for experimental physics. Applications cover a wide spectrum of fields such as astronomy, fiber optics, seismology, seismology, velocimetry to say the least. The precision of measurement of small displacements is the key factor, take LIGO as an example. Another important factor is the convenience to set up an interferometer and the low complexity in terms of equipment. In this experiment, we will measure the resonance frequency of a vibrating transducer using a Michelson interferometer. Basically, a transducer is attached to one optical path, to one mirror, of the Michelson interferometer and it is expected to see a peak in the response curve for the resonance frequency of the transducer plus mirror.

== Theory ==
Our Michelson interferometer setup used a laser, a beam splitter, two mirrors, and a photometer, illustrated in Fig. 1. The light propagates from the source to the beam splitter where approximately 50% of the incident light is transmitted and approximately 50% of it is reflected at a right angle. The transmitted light continues to propagate as the reference plane wave to the stationary mirror where it is fully reflected back towards the beam splitter. The reflected light propagates as the target plane wave to the mirror attached to the oscillating loudspeaker. The two waves then recombine at the beam splitter and interfere based on their relative phases based on the idea that the phase shift of the target plane wave varies with the instantaneous position of the speaker mirror combination. Consider a situation in which we setup our interferometer such that no phase shift corresponds to the equilibrium position of the speaker-mirror combination. At this position, we expect constructive interference at the photometer. Likewise, for every λ/2 change in displacement, we expect constructive interference at the photometer because the path length travelled by the target plane wave varies by some integral multiple of λ. Thus, the phase shift is an even multiple of π and the two waves remain in phase. Similarly, when the path length traveled is some odd multiple of λ/4, the phase shift is an odd multiple of π and the two waves interfere destructively.

Consider two coherent waves with the same amplitude A and same frequency <math>\omega</math> that have paths <math>x_1</math> and <math>x_2</math> emitted from different sources that meet together at a point. The net wave at that point can then be given as the real part of the equation

<math>
\psi=Ae^{i(kx_1-\omega t}+Ae^{i(kx_2-\omega t)}=Ae^{-i\omega t}e^{ik\frac{(x_1+x_2)}{2}}[e^{ik\frac{(x_1-x_2)}{2}}e^{-ik\frac{(x_1-x_2)}{2}}]=2Acos[k(\frac{x_1-x_2}{2})]e^{i[k\frac{(x_1+x_2)}{2}-\omega t]}
</math>

where the wave number <math>k=2\pi/\lambda</math>, <math>\lambda</math> is the wavelength of the waves, and c is the phase velocity. The time-averaged intensity of the combined wave is then proportional to the square of the its amplitude. That is,

<math>
I\propto cos^2[\pi(\frac{p_2-p_1}{\lambda})]=cos^2[2\pi(\frac{p_2}{\lambda}-\theta)]
</math>

where <math>\theta</math> is a constant. Now suppose the mirror that the <math>p_2</math> beam is incident to oscillates harmonically back and forth with a frequency <math>f</math> and a displacement <math>B</math>. Then, <math>p_2 =p_2(0)+Bsin(2\pi ft)</math> and the intensity of the detected light varies with time as

<math>
I=A^2cos^2[2\pi(\frac{Bsin(2\pi ft)}{\lambda}-\theta')]
</math>

Note that the frequency of oscillations is itself a sinusoid, meaning that there will be turnaround points in the oscillations.

== Experimental setup ==
As our main source for this experiment we used a Laser Diode of wavelength 658,85 nm (as we can see in fig. 4 in the following section). Most red laser diodes are based on either GaInP or AlGaInP, this means we have 2 semiconductor materials. For one particular configuration, the data sheet tells us this, it will allow current flow and produce light amplification, this is what we called laser. This is why the diode laser is connected to a power supply, see left top size of fig. 2.

Our Michelson interferometer consists of a beam splitter BS, which is used to split the laser beam into two beams, and two mirrors M1 and M2. The reflected beam goes to a fixed mirror M1 while the transmitted beam goes to the movable mirror M2 (see fig. 1). A transducer is attached, using tape, to the mirror M2. The transducer is powered by a signal generator that is used to vary the vibrating frequency of the mirror. A photodiode PD is used to detect the signal of interference fringes. The schematic Michelson interferometer is shown in Figure 1.
[[File:Schematic Michelson Interferometer.jpeg|center|thumb|600px|Figure 1: Schematic figure of Michelson interferometer. LD is a laser diode, BS is a beam splitter, M1 and M2 are mirrors, PD is a photodiode, and vibration transducer which is driven by a signal generator.]]

The experimental setup Michelson interferometer is shown in figure 2.
[[File:Experimental setup Michelson Interferometer.jpeg|center|thumb|600px|Figure 2: Experimental setup of Michelson interferometer.]]

The interference fringes of the Michelson interferometer are observed when the transducer is turned off in Figure 3.
[[File:Michelson interference fringes.jpeg|center|thumb|600px|Figure 3: Interference fringes which are observed when the transducer is turned off]]

==Experiment Results==
Before we started setting up our interferometer we measured the wavelength of the Mitsubishi LD in a Thorlabs wavemeter,the wavelength of the LD which is 658.9 nm, see fig. 4. Even more, this graph tells us that the Mitsubishi LD is in a single mode regimen, so we proceeded to build the interferometer.
[[File:Wavelength.jpeg|center|thumb|600px|Figure 4: The Wavelength measurement of the Mitsubishi Laser Diode.]]

After the alignment of the optical setup, we applied sinusoidal voltage to the transducer attached to the mirror tightly. intensity fluctuation is recorded by the homemade photodiode. Data from the photodiode is collected by the oscilloscope. The typical figure from the oscilloscope is shown in Figure 5. According to the equation mentioned before, the number of peaks during each period corresponds to the radio of displacement and laser wavelength. In this case, we can measure the displacement for various frequencies. The frequency range is from 86 Hz to 130 Hz.

[[File:ScopeScreen.jpeg|center|thumb|600px|Figure 5: The display on the oscilloscope at <math>f = 103Hz </math>. After applying the voltage to the transducer, the voltage output from photodiode changes with time.]]

We plot the displacement for different frequencies with the errorbar in Figure 6. Uncertainty is the square root uncertainty of the count number. For a driven, damped and harmonic oscillator, the relation between displacement and frequency is

<math>|A|=\frac{f_{0}}{\sqrt{\left(\omega_{0}^{2}-\omega^{2}\right)^{2}+4 \beta^{2} \omega^{2}}} </math>

Where <math>f_{0}</math> is the amplitude, <math>\omega_{0}</math> is the resonance frequency, and <math>\beta_{0}</math> is a constant. With this equation, we can fit the frequency response curve. Because there is only one peak or valley for a frequency lower than 92 Hz or higher than 120 Hz, and the error is significant and non-negligible. We exclude those in the fitting process. Finally, the fitting root mean squared error (RMSE) is 130.38. Fitted <math>\beta</math> is <math>1.3 \pm 0.2</math>. The fitting resonance frequency <math>\omega_{0} </math> is <math>113.1 \pm 0.1Hz </math>.

[[File:FitPeakNum.jpeg|center|thumb|600px|Figure 6: Frequency response curve for the transducer system with errorbar. The blue dots represent the displacement measured with different frequencies. Error bars are the square root uncertainty. Red line is the fitting curve for a driven, damped, harmonic oscillator.]]

==Discussion==
Our final result fits our data to the best Lorentzian Function. When we say best we talk about the function with the lowest Root Mean Square Error (RMSE).

RMSE is defined as follow:

<math>\operatorname{RMSE}=\sqrt{\frac{1}{n}\sum_{i=1}^n(A_i-\hat{A_i})^2}</math>

where <math>A_i</math> is the measured displacement and <math>\hat{A_i}</math> is the predicted displacement by the function model. The core of this method is calculate the distant between the measured and predicted values and take the root mean of them.

The displacements (<math>A_i</math>) we have measured are in the regimen of thousands and the error is ~100. This is attributed to the response of the system to lower frequencies. However, for near resonance the model fits pretty well.

== Conclusion ==
We have shown indeed that we can characterize the resonance frequency of a system. For this particular application, we found a frequency of 113.12Hz for the system transducer plus mirror. Finally, with appropriate handling, this technique could be implemented to characterize more sofisticated systems.

==References==
[1] Freschi, A. A., et al. "Laser interferometric characterization of a vibrating speaker system." American Journal of Physics 71.11 (2003): 1121-1126.
https://doi.org/10.1119/1.1586262

[2] Skarha, Matthew. "Laser interferometric characterization for a vibrating speaker system."
http://mattskarha.com/assets/docs/Laser%20interferometric%20characterization%20for%20a%20vibrating%20speaker%20system%20good.pdf

[3] Pathare, Shirish, and Vikrant Kurmude. "Low cost Michelson–Morley interferometer." Physics Education 51.6 (2016): 063001.
https://doi.org/10.1088/0031-9120/51/6/063001

Resonance frequency measurement using a interferometric method

2022-04-30T07:57:32Z

Aucca: /* Experiment Results */

== Team Members ==
* [[User:Nakarin|Nakarin Jayjong]]
* [[User:Aucca|Joel Auccapuclla]]
* [[User:Xiaoyu|Xiaoyu Nie]]
* [[User:Haotian|Haotian Song]]

== Introduction ==
Interferometers since their invention have been a great tool for experimental physics. Applications cover a wide spectrum of fields such as astronomy, fiber optics, seismology, seismology, velocimetry to say the least. The precision of measurement of small displacements is the key factor, take LIGO as an example. Another important factor is the convenience to set up an interferometer and the low complexity in terms of equipment. In this experiment, we will measure the resonance frequency of a vibrating transducer using a Michelson interferometer. Basically, a transducer is attached to one optical path, to one mirror, of the Michelson interferometer and it is expected to see a peak in the response curve for the resonance frequency of the transducer plus mirror.

== Theory ==
Our Michelson interferometer setup used a laser, a beam splitter, two mirrors, and a photometer, illustrated in Fig. 1. The light propagates from the source to the beam splitter where approximately 50% of the incident light is transmitted and approximately 50% of it is reflected at a right angle. The transmitted light continues to propagate as the reference plane wave to the stationary mirror where it is fully reflected back towards the beam splitter. The reflected light propagates as the target plane wave to the mirror attached to the oscillating loudspeaker. The two waves then recombine at the beam splitter and interfere based on their relative phases based on the idea that the phase shift of the target plane wave varies with the instantaneous position of the speaker mirror combination. Consider a situation in which we setup our interferometer such that no phase shift corresponds to the equilibrium position of the speaker-mirror combination. At this position, we expect constructive interference at the photometer. Likewise, for every λ/2 change in displacement, we expect constructive interference at the photometer because the path length travelled by the target plane wave varies by some integral multiple of λ. Thus, the phase shift is an even multiple of π and the two waves remain in phase. Similarly, when the path length traveled is some odd multiple of λ/4, the phase shift is an odd multiple of π and the two waves interfere destructively.

Consider two coherent waves with the same amplitude A and same frequency <math>\omega</math> that have paths <math>x_1</math> and <math>x_2</math> emitted from different sources that meet together at a point. The net wave at that point can then be given as the real part of the equation

<math>
\psi=Ae^{i(kx_1-\omega t}+Ae^{i(kx_2-\omega t)}=Ae^{-i\omega t}e^{ik\frac{(x_1+x_2)}{2}}[e^{ik\frac{(x_1-x_2)}{2}}e^{-ik\frac{(x_1-x_2)}{2}}]=2Acos[k(\frac{x_1-x_2}{2})]e^{i[k\frac{(x_1+x_2)}{2}-\omega t]}
</math>

where the wave number <math>k=2\pi/\lambda</math>, <math>\lambda</math> is the wavelength of the waves, and c is the phase velocity. The time-averaged intensity of the combined wave is then proportional to the square of the its amplitude. That is,

<math>
I\propto cos^2[\pi(\frac{p_2-p_1}{\lambda})]=cos^2[2\pi(\frac{p_2}{\lambda}-\theta)]
</math>

where <math>\theta</math> is a constant. Now suppose the mirror that the <math>p_2</math> beam is incident to oscillates harmonically back and forth with a frequency <math>f</math> and a displacement <math>B</math>. Then, <math>p_2 =p_2(0)+Bsin(2\pi ft)</math> and the intensity of the detected light varies with time as

<math>
I=A^2cos^2[2\pi(\frac{Bsin(2\pi ft)}{\lambda}-\theta')]
</math>

Note that the frequency of oscillations is itself a sinusoid, meaning that there will be turnaround points in the oscillations.

== Experimental setup ==
As our main source for this experiment we used a Laser Diode of wavelength 658,85 nm (as we can see in fig. 4 in the following section). Most red laser diodes are based on either GaInP or AlGaInP, this means we have 2 semiconductor materials. For one particular configuration, the data sheet tells us this, it will allow current flow and produce light amplification, this is what we called laser. This is why the diode laser is connected to a power supply, see left top size of fig. 2.

Our Michelson interferometer consists of a beam splitter BS, which is used to split the laser beam into two beams, and two mirrors M1 and M2. The reflected beam goes to a fixed mirror M1 while the transmitted beam goes to the movable mirror M2 (see fig. 1). A transducer is attached, using tape, to the mirror M2. The transducer is powered by a signal generator that is used to vary the vibrating frequency of the mirror. A photodiode PD is used to detect the signal of interference fringes. The schematic Michelson interferometer is shown in Figure 1.
[[File:Schematic Michelson Interferometer.jpeg|center|thumb|600px|Figure 1: Schematic figure of Michelson interferometer. LD is a laser diode, BS is a beam splitter, M1 and M2 are mirrors, PD is a photodiode, and vibration transducer which is driven by a signal generator.]]

The experimental setup Michelson interferometer is shown in figure 2.
[[File:Experimental setup Michelson Interferometer.jpeg|center|thumb|600px|Figure 2: Experimental setup of Michelson interferometer.]]

The interference fringes of the Michelson interferometer are observed when the transducer is turned off in Figure 3.
[[File:Michelson interference fringes.jpeg|center|thumb|600px|Figure 3: Interference fringes which are observed when the transducer is turned off]]

==Experiment Results==
Before we started setting up our interferometer we measured the wavelength of the LD in a Thorlabs wavemeter,the wavelength of the LD which is 658.9 nm, see fig. 4. Even more, this graph
[[File:Wavelength.jpeg|center|thumb|600px|Figure 4: The Wavelength measurement of the Mitsubishi Laser Diode.]]

After the alignment of the optical setup, we applied sinusoidal voltage to the transducer attached to the mirror tightly. intensity fluctuation is recorded by the homemade photodiode. Data from the photodiode is collected by the oscilloscope. The typical figure from the oscilloscope is shown in Figure 5. According to the equation mentioned before, the number of peaks during each period corresponds to the radio of displacement and laser wavelength. In this case, we can measure the displacement for various frequencies. The frequency range is from 86 Hz to 130 Hz.

[[File:ScopeScreen.jpeg|center|thumb|600px|Figure 5: The display on the oscilloscope at <math>f = 103Hz </math>. After applying the voltage to the transducer, the voltage output from photodiode changes with time.]]

We plot the displacement for different frequencies with the errorbar in Figure 6. Uncertainty is the square root uncertainty of the count number. For a driven, damped and harmonic oscillator, the relation between displacement and frequency is

<math>|A|=\frac{f_{0}}{\sqrt{\left(\omega_{0}^{2}-\omega^{2}\right)^{2}+4 \beta^{2} \omega^{2}}} </math>

Where <math>f_{0}</math> is the amplitude, <math>\omega_{0}</math> is the resonance frequency, and <math>\beta_{0}</math> is a constant. With this equation, we can fit the frequency response curve. Because there is only one peak or valley for a frequency lower than 92 Hz or higher than 120 Hz, and the error is significant and non-negligible. We exclude those in the fitting process. Finally, the fitting root mean squared error (RMSE) is 130.38. Fitted <math>\beta</math> is <math>1.3 \pm 0.2</math>. The fitting resonance frequency <math>\omega_{0} </math> is <math>113.1 \pm 0.1Hz </math>.

[[File:FitPeakNum.jpeg|center|thumb|600px|Figure 6: Frequency response curve for the transducer system with errorbar. The blue dots represent the displacement measured with different frequencies. Error bars are the square root uncertainty. Red line is the fitting curve for a driven, damped, harmonic oscillator.]]

==Discussion==
Our final result fits our data to the best Lorentzian Function. When we say best we talk about the function with the lowest Root Mean Square Error (RMSE).

RMSE is defined as follow:

<math>\operatorname{RMSE}=\sqrt{\frac{1}{n}\sum_{i=1}^n(A_i-\hat{A_i})^2}</math>

where <math>A_i</math> is the measured displacement and <math>\hat{A_i}</math> is the predicted displacement by the function model. The core of this method is calculate the distant between the measured and predicted values and take the root mean of them.

The displacements (<math>A_i</math>) we have measured are in the regimen of thousands and the error is ~100. This is attributed to the response of the system to lower frequencies. However, for near resonance the model fits pretty well.

== Conclusion ==
We have shown indeed that we can characterize the resonance frequency of a system. For this particular application, we found a frequency of 113.12Hz for the system transducer plus mirror. Finally, with appropriate handling, this technique could be implemented to characterize more sofisticated systems.

==References==
[1] Freschi, A. A., et al. "Laser interferometric characterization of a vibrating speaker system." American Journal of Physics 71.11 (2003): 1121-1126.
https://doi.org/10.1119/1.1586262

[2] Skarha, Matthew. "Laser interferometric characterization for a vibrating speaker system."
http://mattskarha.com/assets/docs/Laser%20interferometric%20characterization%20for%20a%20vibrating%20speaker%20system%20good.pdf

[3] Pathare, Shirish, and Vikrant Kurmude. "Low cost Michelson–Morley interferometer." Physics Education 51.6 (2016): 063001.
https://doi.org/10.1088/0031-9120/51/6/063001

Resonance frequency measurement using a interferometric method

2022-04-30T07:46:47Z

Aucca: /* Experiment Results */

== Team Members ==
* [[User:Nakarin|Nakarin Jayjong]]
* [[User:Aucca|Joel Auccapuclla]]
* [[User:Xiaoyu|Xiaoyu Nie]]
* [[User:Haotian|Haotian Song]]

== Introduction ==
Interferometers since their invention have been a great tool for experimental physics. Applications cover a wide spectrum of fields such as astronomy, fiber optics, seismology, seismology, velocimetry to say the least. The precision of measurement of small displacements is the key factor, take LIGO as an example. Another important factor is the convenience to set up an interferometer and the low complexity in terms of equipment. In this experiment, we will measure the resonance frequency of a vibrating transducer using a Michelson interferometer. Basically, a transducer is attached to one optical path, to one mirror, of the Michelson interferometer and it is expected to see a peak in the response curve for the resonance frequency of the transducer plus mirror.

== Theory ==
Our Michelson interferometer setup used a laser, a beam splitter, two mirrors, and a photometer, illustrated in Fig. 1. The light propagates from the source to the beam splitter where approximately 50% of the incident light is transmitted and approximately 50% of it is reflected at a right angle. The transmitted light continues to propagate as the reference plane wave to the stationary mirror where it is fully reflected back towards the beam splitter. The reflected light propagates as the target plane wave to the mirror attached to the oscillating loudspeaker. The two waves then recombine at the beam splitter and interfere based on their relative phases based on the idea that the phase shift of the target plane wave varies with the instantaneous position of the speaker mirror combination. Consider a situation in which we setup our interferometer such that no phase shift corresponds to the equilibrium position of the speaker-mirror combination. At this position, we expect constructive interference at the photometer. Likewise, for every λ/2 change in displacement, we expect constructive interference at the photometer because the path length travelled by the target plane wave varies by some integral multiple of λ. Thus, the phase shift is an even multiple of π and the two waves remain in phase. Similarly, when the path length traveled is some odd multiple of λ/4, the phase shift is an odd multiple of π and the two waves interfere destructively.

Consider two coherent waves with the same amplitude A and same frequency <math>\omega</math> that have paths <math>x_1</math> and <math>x_2</math> emitted from different sources that meet together at a point. The net wave at that point can then be given as the real part of the equation

<math>
\psi=Ae^{i(kx_1-\omega t}+Ae^{i(kx_2-\omega t)}=Ae^{-i\omega t}e^{ik\frac{(x_1+x_2)}{2}}[e^{ik\frac{(x_1-x_2)}{2}}e^{-ik\frac{(x_1-x_2)}{2}}]=2Acos[k(\frac{x_1-x_2}{2})]e^{i[k\frac{(x_1+x_2)}{2}-\omega t]}
</math>

where the wave number <math>k=2\pi/\lambda</math>, <math>\lambda</math> is the wavelength of the waves, and c is the phase velocity. The time-averaged intensity of the combined wave is then proportional to the square of the its amplitude. That is,

<math>
I\propto cos^2[\pi(\frac{p_2-p_1}{\lambda})]=cos^2[2\pi(\frac{p_2}{\lambda}-\theta)]
</math>

where <math>\theta</math> is a constant. Now suppose the mirror that the <math>p_2</math> beam is incident to oscillates harmonically back and forth with a frequency <math>f</math> and a displacement <math>B</math>. Then, <math>p_2 =p_2(0)+Bsin(2\pi ft)</math> and the intensity of the detected light varies with time as

<math>
I=A^2cos^2[2\pi(\frac{Bsin(2\pi ft)}{\lambda}-\theta')]
</math>

Note that the frequency of oscillations is itself a sinusoid, meaning that there will be turnaround points in the oscillations.

== Experimental setup ==
As our main source for this experiment we used a Laser Diode of wavelength 658,85 nm (as we can see in fig. 4 in the following section). Most red laser diodes are based on either GaInP or AlGaInP, this means we have 2 semiconductor materials. For one particular configuration, the data sheet tells us this, it will allow current flow and produce light amplification, this is what we called laser. This is why the diode laser is connected to a power supply, see left top size of fig. 2.

Our Michelson interferometer consists of a beam splitter BS, which is used to split the laser beam into two beams, and two mirrors M1 and M2. The reflected beam goes to a fixed mirror M1 while the transmitted beam goes to the movable mirror M2 (see fig. 1). A transducer is attached, using tape, to the mirror M2. The transducer is powered by a signal generator that is used to vary the vibrating frequency of the mirror. A photodiode PD is used to detect the signal of interference fringes. The schematic Michelson interferometer is shown in Figure 1.
[[File:Schematic Michelson Interferometer.jpeg|center|thumb|600px|Figure 1: Schematic figure of Michelson interferometer. LD is a laser diode, BS is a beam splitter, M1 and M2 are mirrors, PD is a photodiode, and vibration transducer which is driven by a signal generator.]]

The experimental setup Michelson interferometer is shown in figure 2.
[[File:Experimental setup Michelson Interferometer.jpeg|center|thumb|600px|Figure 2: Experimental setup of Michelson interferometer.]]

The interference fringes of the Michelson interferometer are observed when the transducer is turned off in Figure 3.
[[File:Michelson interference fringes.jpeg|center|thumb|600px|Figure 3: Interference fringes which are observed when the transducer is turned off]]

==Experiment Results==
In our setup, we measured the wavelength of Mitsubishi Laser Diode which is 658.9 nm. The wavemeter measurement results are shown here.
[[File:Wavelength.jpeg|center|thumb|600px|Figure 5: The Wavelength measurement of the Mitsubishi Laser Diode.]]

After the alignment of the optical setup, we applied sinusoidal voltage to the transducer attached to the mirror tightly. intensity fluctuation is recorded by the homemade photodiode. Data from the photodiode is collected by the oscilloscope. The typical figure from the oscilloscope is shown in Figure 4. According to the equation mentioned before, the number of peaks during each period corresponds to the radio of displacement and laser wavelength. In this case, we can measure the displacement for various frequencies. The frequency range is from 86 Hz to 130 Hz.

[[File:ScopeScreen.jpeg|center|thumb|600px|Figure 4: The display on the oscilloscope at <math>f = 103Hz </math>. After applying the voltage to the transducer, the voltage output from photodiode changes with time.]]

We plot the displacement for different frequencies with the errorbar in Figure 6. Uncertainty is the square root uncertainty of the count number. For a driven, damped and harmonic oscillator, the relation between displacement and frequency is

<math>|A|=\frac{f_{0}}{\sqrt{\left(\omega_{0}^{2}-\omega^{2}\right)^{2}+4 \beta^{2} \omega^{2}}} </math>

Where <math>f_{0}</math> is the amplitude, <math>\omega_{0}</math> is the resonance frequency, and <math>\beta_{0}</math> is a constant. With this equation, we can fit the frequency response curve. Because there is only one peak or valley for a frequency lower than 92 Hz or higher than 120 Hz, and the error is significant and non-negligible. We exclude those in the fitting process. Finally, the fitting root mean squared error (RMSE) is 130.38. Fitted <math>\beta</math> is <math>1.3 \pm 0.2</math>. The fitting resonance frequency <math>\omega_{0} </math> is <math>113.1 \pm 0.1Hz </math>.

[[File:FitPeakNum.jpeg|center|thumb|600px|Figure 6: Frequency response curve for the transducer system with errorbar. The blue dots represent the displacement measured with different frequencies. Error bars are the square root uncertainty. Red line is the fitting curve for a driven, damped, harmonic oscillator.]]

==Discussion==
Our final result fits our data to the best Lorentzian Function. When we say best we talk about the function with the lowest Root Mean Square Error (RMSE).

RMSE is defined as follow:

<math>\operatorname{RMSE}=\sqrt{\frac{1}{n}\sum_{i=1}^n(A_i-\hat{A_i})^2}</math>

where <math>A_i</math> is the measured displacement and <math>\hat{A_i}</math> is the predicted displacement by the function model. The core of this method is calculate the distant between the measured and predicted values and take the root mean of them.

The displacements (<math>A_i</math>) we have measured are in the regimen of thousands and the error is ~100. This is attributed to the response of the system to lower frequencies. However, for near resonance the model fits pretty well.

== Conclusion ==
We have shown indeed that we can characterize the resonance frequency of a system. For this particular application, we found a frequency of 113.12Hz for the system transducer plus mirror. Finally, with appropriate handling, this technique could be implemented to characterize more sofisticated systems.

==References==
[1] Freschi, A. A., et al. "Laser interferometric characterization of a vibrating speaker system." American Journal of Physics 71.11 (2003): 1121-1126.
https://doi.org/10.1119/1.1586262

[2] Skarha, Matthew. "Laser interferometric characterization for a vibrating speaker system."
http://mattskarha.com/assets/docs/Laser%20interferometric%20characterization%20for%20a%20vibrating%20speaker%20system%20good.pdf

[3] Pathare, Shirish, and Vikrant Kurmude. "Low cost Michelson–Morley interferometer." Physics Education 51.6 (2016): 063001.
https://doi.org/10.1088/0031-9120/51/6/063001

Resonance frequency measurement using a interferometric method

2022-04-30T07:45:19Z

Aucca: /* Theory */

== Team Members ==
* [[User:Nakarin|Nakarin Jayjong]]
* [[User:Aucca|Joel Auccapuclla]]
* [[User:Xiaoyu|Xiaoyu Nie]]
* [[User:Haotian|Haotian Song]]

== Introduction ==
Interferometers since their invention have been a great tool for experimental physics. Applications cover a wide spectrum of fields such as astronomy, fiber optics, seismology, seismology, velocimetry to say the least. The precision of measurement of small displacements is the key factor, take LIGO as an example. Another important factor is the convenience to set up an interferometer and the low complexity in terms of equipment. In this experiment, we will measure the resonance frequency of a vibrating transducer using a Michelson interferometer. Basically, a transducer is attached to one optical path, to one mirror, of the Michelson interferometer and it is expected to see a peak in the response curve for the resonance frequency of the transducer plus mirror.

== Theory ==
Our Michelson interferometer setup used a laser, a beam splitter, two mirrors, and a photometer, illustrated in Fig. 1. The light propagates from the source to the beam splitter where approximately 50% of the incident light is transmitted and approximately 50% of it is reflected at a right angle. The transmitted light continues to propagate as the reference plane wave to the stationary mirror where it is fully reflected back towards the beam splitter. The reflected light propagates as the target plane wave to the mirror attached to the oscillating loudspeaker. The two waves then recombine at the beam splitter and interfere based on their relative phases based on the idea that the phase shift of the target plane wave varies with the instantaneous position of the speaker mirror combination. Consider a situation in which we setup our interferometer such that no phase shift corresponds to the equilibrium position of the speaker-mirror combination. At this position, we expect constructive interference at the photometer. Likewise, for every λ/2 change in displacement, we expect constructive interference at the photometer because the path length travelled by the target plane wave varies by some integral multiple of λ. Thus, the phase shift is an even multiple of π and the two waves remain in phase. Similarly, when the path length traveled is some odd multiple of λ/4, the phase shift is an odd multiple of π and the two waves interfere destructively.

Consider two coherent waves with the same amplitude A and same frequency <math>\omega</math> that have paths <math>x_1</math> and <math>x_2</math> emitted from different sources that meet together at a point. The net wave at that point can then be given as the real part of the equation

<math>
\psi=Ae^{i(kx_1-\omega t}+Ae^{i(kx_2-\omega t)}=Ae^{-i\omega t}e^{ik\frac{(x_1+x_2)}{2}}[e^{ik\frac{(x_1-x_2)}{2}}e^{-ik\frac{(x_1-x_2)}{2}}]=2Acos[k(\frac{x_1-x_2}{2})]e^{i[k\frac{(x_1+x_2)}{2}-\omega t]}
</math>

where the wave number <math>k=2\pi/\lambda</math>, <math>\lambda</math> is the wavelength of the waves, and c is the phase velocity. The time-averaged intensity of the combined wave is then proportional to the square of the its amplitude. That is,

<math>
I\propto cos^2[\pi(\frac{p_2-p_1}{\lambda})]=cos^2[2\pi(\frac{p_2}{\lambda}-\theta)]
</math>

where <math>\theta</math> is a constant. Now suppose the mirror that the <math>p_2</math> beam is incident to oscillates harmonically back and forth with a frequency <math>f</math> and a displacement <math>B</math>. Then, <math>p_2 =p_2(0)+Bsin(2\pi ft)</math> and the intensity of the detected light varies with time as

<math>
I=A^2cos^2[2\pi(\frac{Bsin(2\pi ft)}{\lambda}-\theta')]
</math>

Note that the frequency of oscillations is itself a sinusoid, meaning that there will be turnaround points in the oscillations.

== Experimental setup ==
As our main source for this experiment we used a Laser Diode of wavelength 658,85 nm (as we can see in fig. 4 in the following section). Most red laser diodes are based on either GaInP or AlGaInP, this means we have 2 semiconductor materials. For one particular configuration, the data sheet tells us this, it will allow current flow and produce light amplification, this is what we called laser. This is why the diode laser is connected to a power supply, see left top size of fig. 2.

Our Michelson interferometer consists of a beam splitter BS, which is used to split the laser beam into two beams, and two mirrors M1 and M2. The reflected beam goes to a fixed mirror M1 while the transmitted beam goes to the movable mirror M2 (see fig. 1). A transducer is attached, using tape, to the mirror M2. The transducer is powered by a signal generator that is used to vary the vibrating frequency of the mirror. A photodiode PD is used to detect the signal of interference fringes. The schematic Michelson interferometer is shown in Figure 1.
[[File:Schematic Michelson Interferometer.jpeg|center|thumb|600px|Figure 1: Schematic figure of Michelson interferometer. LD is a laser diode, BS is a beam splitter, M1 and M2 are mirrors, PD is a photodiode, and vibration transducer which is driven by a signal generator.]]

The experimental setup Michelson interferometer is shown in figure 2.
[[File:Experimental setup Michelson Interferometer.jpeg|center|thumb|600px|Figure 2: Experimental setup of Michelson interferometer.]]

The interference fringes of the Michelson interferometer are observed when the transducer is turned off in Figure 3.
[[File:Michelson interference fringes.jpeg|center|thumb|600px|Figure 3: Interference fringes which are observed when the transducer is turned off]]

==Experiment Results==
After the alignment of the optical setup, we applied sinusoidal voltage to the transducer attached to the mirror tightly. intensity fluctuation is recorded by the homemade photodiode. Data from the photodiode is collected by the oscilloscope. The typical figure from the oscilloscope is shown in Figure 4. According to the equation mentioned before, the number of peaks during each period corresponds to the radio of displacement and laser wavelength. In this case, we can measure the displacement for various frequencies. The frequency range is from 86 Hz to 130 Hz.

[[File:ScopeScreen.jpeg|center|thumb|600px|Figure 4: The display on the oscilloscope at <math>f = 103Hz </math>. After applying the voltage to the transducer, the voltage output from photodiode changes with time.]]

In our setup, we measured the wavelength of Mitsubishi Laser Diode which is 658.9 nm. The wavemeter measurement results are shown here.
[[File:Wavelength.jpeg|center|thumb|600px|Figure 5: The Wavelength measurement of the Mitsubishi Laser Diode.]]

We plot the displacement for different frequencies with the errorbar in Figure 6. Uncertainty is the square root uncertainty of the count number. For a driven, damped and harmonic oscillator, the relation between displacement and frequency is

<math>|A|=\frac{f_{0}}{\sqrt{\left(\omega_{0}^{2}-\omega^{2}\right)^{2}+4 \beta^{2} \omega^{2}}} </math>

Where <math>f_{0}</math> is the amplitude, <math>\omega_{0}</math> is the resonance frequency, and <math>\beta_{0}</math> is a constant. With this equation, we can fit the frequency response curve. Because there is only one peak or valley for a frequency lower than 92 Hz or higher than 120 Hz, and the error is significant and non-negligible. We exclude those in the fitting process. Finally, the fitting root mean squared error (RMSE) is 130.38. Fitted <math>\beta</math> is <math>1.3 \pm 0.2</math>. The fitting resonance frequency <math>\omega_{0} </math> is <math>113.1 \pm 0.1Hz </math>.

[[File:FitPeakNum.jpeg|center|thumb|600px|Figure 6: Frequency response curve for the transducer system with errorbar. The blue dots represent the displacement measured with different frequencies. Error bars are the square root uncertainty. Red line is the fitting curve for a driven, damped, harmonic oscillator.]]

==Discussion==
Our final result fits our data to the best Lorentzian Function. When we say best we talk about the function with the lowest Root Mean Square Error (RMSE).

RMSE is defined as follow:

<math>\operatorname{RMSE}=\sqrt{\frac{1}{n}\sum_{i=1}^n(A_i-\hat{A_i})^2}</math>

where <math>A_i</math> is the measured displacement and <math>\hat{A_i}</math> is the predicted displacement by the function model. The core of this method is calculate the distant between the measured and predicted values and take the root mean of them.

The displacements (<math>A_i</math>) we have measured are in the regimen of thousands and the error is ~100. This is attributed to the response of the system to lower frequencies. However, for near resonance the model fits pretty well.

== Conclusion ==
We have shown indeed that we can characterize the resonance frequency of a system. For this particular application, we found a frequency of 113.12Hz for the system transducer plus mirror. Finally, with appropriate handling, this technique could be implemented to characterize more sofisticated systems.

==References==
[1] Freschi, A. A., et al. "Laser interferometric characterization of a vibrating speaker system." American Journal of Physics 71.11 (2003): 1121-1126.
https://doi.org/10.1119/1.1586262

[2] Skarha, Matthew. "Laser interferometric characterization for a vibrating speaker system."
http://mattskarha.com/assets/docs/Laser%20interferometric%20characterization%20for%20a%20vibrating%20speaker%20system%20good.pdf

[3] Pathare, Shirish, and Vikrant Kurmude. "Low cost Michelson–Morley interferometer." Physics Education 51.6 (2016): 063001.
https://doi.org/10.1088/0031-9120/51/6/063001

Resonance frequency measurement using a interferometric method

2022-04-30T07:40:12Z

Aucca: /* Experimental setup */

== Team Members ==
* [[User:Nakarin|Nakarin Jayjong]]
* [[User:Aucca|Joel Auccapuclla]]
* [[User:Xiaoyu|Xiaoyu Nie]]
* [[User:Haotian|Haotian Song]]

== Introduction ==
Interferometers since their invention have been a great tool for experimental physics. Applications cover a wide spectrum of fields such as astronomy, fiber optics, seismology, seismology, velocimetry to say the least. The precision of measurement of small displacements is the key factor, take LIGO as an example. Another important factor is the convenience to set up an interferometer and the low complexity in terms of equipment. In this experiment, we will measure the resonance frequency of a vibrating transducer using a Michelson interferometer. Basically, a transducer is attached to one optical path, to one mirror, of the Michelson interferometer and it is expected to see a peak in the response curve for the resonance frequency of the transducer plus mirror.

== Theory ==
Our Michelson interferometer setup used a laser, a beam splitter, two mirrors, and a photometer, illustrated in Fig. 1. Original models used sodium flame as the coherent light source. The light propagates from the source to the beam splitter where approximately 50% of the incident light is transmitted and approximately 50% of it is reflected at a right angle. The transmitted light continues to propagate as the reference plane wave to the stationary mirror where it is fully reflected back towards the beam splitter. The reflected light propagates as the target plane wave to the mirror attached to the oscillating loudspeaker. The two waves then recombine at the beam splitter and interfere based on their relative phases based on the idea that the phase shift of the target plane wave varies with the instantaneous position of the speaker mirror combination. Consider a situation in which we setup our interferometer such that no phase shift corresponds to the equilibrium position of the speaker-mirror combination. At this position, we expect constructive interference at the photometer. Likewise, for every λ/2 change in displacement, we expect constructive interference at the photometer because the path length travelled by the target plane wave varies by some integral multiple of λ. Thus, the phase shift is an even multiple of π and the two waves remain in phase. Similarly, when the path length traveled is some odd multiple of λ/4, the phase shift is an odd multiple of π and the two waves interfere destructively.

Consider two coherent waves with the same amplitude A and same frequency <math>\omega</math> that have paths <math>x_1</math> and <math>x_2</math> emitted from different sources that meet together at a point. The net wave at that point can then be given as the real part of the equation

<math>
\psi=Ae^{i(kx_1-\omega t}+Ae^{i(kx_2-\omega t)}=Ae^{-i\omega t}e^{ik\frac{(x_1+x_2)}{2}}[e^{ik\frac{(x_1-x_2)}{2}}e^{-ik\frac{(x_1-x_2)}{2}}]=2Acos[k(\frac{x_1-x_2}{2})]e^{i[k\frac{(x_1+x_2)}{2}-\omega t]}
</math>

where the wave number <math>k=2\pi/\lambda</math>, <math>\lambda</math> is the wavelength of the waves, and c is the phase velocity. The time-averaged intensity of the combined wave is then proportional to the square of the its amplitude. That is,

<math>
I\propto cos^2[\pi(\frac{p_2-p_1}{\lambda})]=cos^2[2\pi(\frac{p_2}{\lambda}-\theta)]
</math>

where <math>\theta</math> is a constant. Now suppose the mirror that the <math>p_2</math> beam is incident to oscillates harmonically back and forth with a frequency <math>f</math> and a displacement <math>B</math>. Then, <math>p_2 =p_2(0)+Bsin(2\pi ft)</math> and the intensity of the detected light varies with time as

<math>
I=A^2cos^2[2\pi(\frac{Bsin(2\pi ft)}{\lambda}-\theta')]
</math>

Note that the frequency of oscillations is itself a sinusoid, meaning that there will be turnaround points in the oscillations.

== Experimental setup ==
As our main source for this experiment we used a Laser Diode of wavelength 658,85 nm (as we can see in fig. 4 in the following section). Most red laser diodes are based on either GaInP or AlGaInP, this means we have 2 semiconductor materials. For one particular configuration, the data sheet tells us this, it will allow current flow and produce light amplification, this is what we called laser. This is why the diode laser is connected to a power supply, see left top size of fig. 2.

Our Michelson interferometer consists of a beam splitter BS, which is used to split the laser beam into two beams, and two mirrors M1 and M2. The reflected beam goes to a fixed mirror M1 while the transmitted beam goes to the movable mirror M2 (see fig. 1). A transducer is attached, using tape, to the mirror M2. The transducer is powered by a signal generator that is used to vary the vibrating frequency of the mirror. A photodiode PD is used to detect the signal of interference fringes. The schematic Michelson interferometer is shown in Figure 1.
[[File:Schematic Michelson Interferometer.jpeg|center|thumb|600px|Figure 1: Schematic figure of Michelson interferometer. LD is a laser diode, BS is a beam splitter, M1 and M2 are mirrors, PD is a photodiode, and vibration transducer which is driven by a signal generator.]]

The experimental setup Michelson interferometer is shown in figure 2.
[[File:Experimental setup Michelson Interferometer.jpeg|center|thumb|600px|Figure 2: Experimental setup of Michelson interferometer.]]

The interference fringes of the Michelson interferometer are observed when the transducer is turned off in Figure 3.
[[File:Michelson interference fringes.jpeg|center|thumb|600px|Figure 3: Interference fringes which are observed when the transducer is turned off]]

==Experiment Results==
After the alignment of the optical setup, we applied sinusoidal voltage to the transducer attached to the mirror tightly. intensity fluctuation is recorded by the homemade photodiode. Data from the photodiode is collected by the oscilloscope. The typical figure from the oscilloscope is shown in Figure 4. According to the equation mentioned before, the number of peaks during each period corresponds to the radio of displacement and laser wavelength. In this case, we can measure the displacement for various frequencies. The frequency range is from 86 Hz to 130 Hz.

[[File:ScopeScreen.jpeg|center|thumb|600px|Figure 4: The display on the oscilloscope at <math>f = 103Hz </math>. After applying the voltage to the transducer, the voltage output from photodiode changes with time.]]

In our setup, we measured the wavelength of Mitsubishi Laser Diode which is 658.9 nm. The wavemeter measurement results are shown here.
[[File:Wavelength.jpeg|center|thumb|600px|Figure 5: The Wavelength measurement of the Mitsubishi Laser Diode.]]

We plot the displacement for different frequencies with the errorbar in Figure 6. Uncertainty is the square root uncertainty of the count number. For a driven, damped and harmonic oscillator, the relation between displacement and frequency is

<math>|A|=\frac{f_{0}}{\sqrt{\left(\omega_{0}^{2}-\omega^{2}\right)^{2}+4 \beta^{2} \omega^{2}}} </math>

Where <math>f_{0}</math> is the amplitude, <math>\omega_{0}</math> is the resonance frequency, and <math>\beta_{0}</math> is a constant. With this equation, we can fit the frequency response curve. Because there is only one peak or valley for a frequency lower than 92 Hz or higher than 120 Hz, and the error is significant and non-negligible. We exclude those in the fitting process. Finally, the fitting root mean squared error (RMSE) is 130.38. Fitted <math>\beta</math> is <math>1.3 \pm 0.2</math>. The fitting resonance frequency <math>\omega_{0} </math> is <math>113.1 \pm 0.1Hz </math>.

[[File:FitPeakNum.jpeg|center|thumb|600px|Figure 6: Frequency response curve for the transducer system with errorbar. The blue dots represent the displacement measured with different frequencies. Error bars are the square root uncertainty. Red line is the fitting curve for a driven, damped, harmonic oscillator.]]

==Discussion==
Our final result fits our data to the best Lorentzian Function. When we say best we talk about the function with the lowest Root Mean Square Error (RMSE).

RMSE is defined as follow:

<math>\operatorname{RMSE}=\sqrt{\frac{1}{n}\sum_{i=1}^n(A_i-\hat{A_i})^2}</math>

where <math>A_i</math> is the measured displacement and <math>\hat{A_i}</math> is the predicted displacement by the function model. The core of this method is calculate the distant between the measured and predicted values and take the root mean of them.

The displacements (<math>A_i</math>) we have measured are in the regimen of thousands and the error is ~100. This is attributed to the response of the system to lower frequencies. However, for near resonance the model fits pretty well.

== Conclusion ==
We have shown indeed that we can characterize the resonance frequency of a system. For this particular application, we found a frequency of 113.12Hz for the system transducer plus mirror. Finally, with appropriate handling, this technique could be implemented to characterize more sofisticated systems.

==References==
[1] Freschi, A. A., et al. "Laser interferometric characterization of a vibrating speaker system." American Journal of Physics 71.11 (2003): 1121-1126.
https://doi.org/10.1119/1.1586262

[2] Skarha, Matthew. "Laser interferometric characterization for a vibrating speaker system."
http://mattskarha.com/assets/docs/Laser%20interferometric%20characterization%20for%20a%20vibrating%20speaker%20system%20good.pdf

[3] Pathare, Shirish, and Vikrant Kurmude. "Low cost Michelson–Morley interferometer." Physics Education 51.6 (2016): 063001.
https://doi.org/10.1088/0031-9120/51/6/063001

Resonance frequency measurement using a interferometric method

2022-04-30T07:37:42Z

Aucca: /* Experimental setup */

== Team Members ==
* [[User:Nakarin|Nakarin Jayjong]]
* [[User:Aucca|Joel Auccapuclla]]
* [[User:Xiaoyu|Xiaoyu Nie]]
* [[User:Haotian|Haotian Song]]

== Introduction ==
Interferometers since their invention have been a great tool for experimental physics. Applications cover a wide spectrum of fields such as astronomy, fiber optics, seismology, seismology, velocimetry to say the least. The precision of measurement of small displacements is the key factor, take LIGO as an example. Another important factor is the convenience to set up an interferometer and the low complexity in terms of equipment. In this experiment, we will measure the resonance frequency of a vibrating transducer using a Michelson interferometer. Basically, a transducer is attached to one optical path, to one mirror, of the Michelson interferometer and it is expected to see a peak in the response curve for the resonance frequency of the transducer plus mirror.

== Theory ==
Our Michelson interferometer setup used a laser, a beam splitter, two mirrors, and a photometer, illustrated in Fig. 1. Original models used sodium flame as the coherent light source. The light propagates from the source to the beam splitter where approximately 50% of the incident light is transmitted and approximately 50% of it is reflected at a right angle. The transmitted light continues to propagate as the reference plane wave to the stationary mirror where it is fully reflected back towards the beam splitter. The reflected light propagates as the target plane wave to the mirror attached to the oscillating loudspeaker. The two waves then recombine at the beam splitter and interfere based on their relative phases based on the idea that the phase shift of the target plane wave varies with the instantaneous position of the speaker mirror combination. Consider a situation in which we setup our interferometer such that no phase shift corresponds to the equilibrium position of the speaker-mirror combination. At this position, we expect constructive interference at the photometer. Likewise, for every λ/2 change in displacement, we expect constructive interference at the photometer because the path length travelled by the target plane wave varies by some integral multiple of λ. Thus, the phase shift is an even multiple of π and the two waves remain in phase. Similarly, when the path length traveled is some odd multiple of λ/4, the phase shift is an odd multiple of π and the two waves interfere destructively.

Consider two coherent waves with the same amplitude A and same frequency <math>\omega</math> that have paths <math>x_1</math> and <math>x_2</math> emitted from different sources that meet together at a point. The net wave at that point can then be given as the real part of the equation

<math>
\psi=Ae^{i(kx_1-\omega t}+Ae^{i(kx_2-\omega t)}=Ae^{-i\omega t}e^{ik\frac{(x_1+x_2)}{2}}[e^{ik\frac{(x_1-x_2)}{2}}e^{-ik\frac{(x_1-x_2)}{2}}]=2Acos[k(\frac{x_1-x_2}{2})]e^{i[k\frac{(x_1+x_2)}{2}-\omega t]}
</math>

where the wave number <math>k=2\pi/\lambda</math>, <math>\lambda</math> is the wavelength of the waves, and c is the phase velocity. The time-averaged intensity of the combined wave is then proportional to the square of the its amplitude. That is,

<math>
I\propto cos^2[\pi(\frac{p_2-p_1}{\lambda})]=cos^2[2\pi(\frac{p_2}{\lambda}-\theta)]
</math>

where <math>\theta</math> is a constant. Now suppose the mirror that the <math>p_2</math> beam is incident to oscillates harmonically back and forth with a frequency <math>f</math> and a displacement <math>B</math>. Then, <math>p_2 =p_2(0)+Bsin(2\pi ft)</math> and the intensity of the detected light varies with time as

<math>
I=A^2cos^2[2\pi(\frac{Bsin(2\pi ft)}{\lambda}-\theta')]
</math>

Note that the frequency of oscillations is itself a sinusoid, meaning that there will be turnaround points in the oscillations.

== Experimental setup ==
As our main source for this experiment we used a Laser Diode of wavelength 658,85 nm (as we can see in fig. 4 in the following section). Most red laser diodes are based on either GaInP or AlGaInP, this means we have 2 semiconductor materials. For one particular configuration, the data sheet tells us this, it will allow current flow and produce light amplification, this is what we called laser. This is why the diode laser is connected to a power supply, see left top size of fig. 2.

Our Michelson interferometer consists of a beam splitter BS, which is used to split the laser beam into two beams and two mirrors M1 and M2. The reflected beam goes to a fixed mirror M1 while the transmitted beam goes to the movable mirror M2 (see fig. 1). A transducer is attached, using tape, to the mirror M2. The transducer is powered by a signal generator that is used to vary the vibrating frequency of the mirror. A photodiode PD is used to detect the signal of interference fringes. The schematic Michelson interferometer is shown in Figure 1.
[[File:Schematic Michelson Interferometer.jpeg|center|thumb|600px|Figure 1: Schematic figure of Michelson interferometer. LD is a laser diode, BS is a beam splitter, M1 and M2 are mirrors, PD is a photodiode, and vibration transducer which is driven by a signal generator.]]

The experimental setup Michelson interferometer is shown in figure 2.
[[File:Experimental setup Michelson Interferometer.jpeg|center|thumb|600px|Figure 2: Experimental setup of Michelson interferometer.]]

The interference fringes of the Michelson interferometer are observed when the transducer is turned off in Figure 3.
[[File:Michelson interference fringes.jpeg|center|thumb|600px|Figure 3: Interference fringes which are observed when the transducer is turned off]]

==Experiment Results==
After the alignment of the optical setup, we applied sinusoidal voltage to the transducer attached to the mirror tightly. intensity fluctuation is recorded by the homemade photodiode. Data from the photodiode is collected by the oscilloscope. The typical figure from the oscilloscope is shown in Figure 4. According to the equation mentioned before, the number of peaks during each period corresponds to the radio of displacement and laser wavelength. In this case, we can measure the displacement for various frequencies. The frequency range is from 86 Hz to 130 Hz.

[[File:ScopeScreen.jpeg|center|thumb|600px|Figure 4: The display on the oscilloscope at <math>f = 103Hz </math>. After applying the voltage to the transducer, the voltage output from photodiode changes with time.]]

In our setup, we measured the wavelength of Mitsubishi Laser Diode which is 658.9 nm. The wavemeter measurement results are shown here.
[[File:Wavelength.jpeg|center|thumb|600px|Figure 5: The Wavelength measurement of the Mitsubishi Laser Diode.]]

We plot the displacement for different frequencies with the errorbar in Figure 6. Uncertainty is the square root uncertainty of the count number. For a driven, damped and harmonic oscillator, the relation between displacement and frequency is

<math>|A|=\frac{f_{0}}{\sqrt{\left(\omega_{0}^{2}-\omega^{2}\right)^{2}+4 \beta^{2} \omega^{2}}} </math>

Where <math>f_{0}</math> is the amplitude, <math>\omega_{0}</math> is the resonance frequency, and <math>\beta_{0}</math> is a constant. With this equation, we can fit the frequency response curve. Because there is only one peak or valley for a frequency lower than 92 Hz or higher than 120 Hz, and the error is significant and non-negligible. We exclude those in the fitting process. Finally, the fitting root mean squared error (RMSE) is 130.38. Fitted <math>\beta</math> is <math>1.3 \pm 0.2</math>. The fitting resonance frequency <math>\omega_{0} </math> is <math>113.1 \pm 0.1Hz </math>.

[[File:FitPeakNum.jpeg|center|thumb|600px|Figure 6: Frequency response curve for the transducer system with errorbar. The blue dots represent the displacement measured with different frequencies. Error bars are the square root uncertainty. Red line is the fitting curve for a driven, damped, harmonic oscillator.]]

==Discussion==
Our final result fits our data to the best Lorentzian Function. When we say best we talk about the function with the lowest Root Mean Square Error (RMSE).

RMSE is defined as follow:

<math>\operatorname{RMSE}=\sqrt{\frac{1}{n}\sum_{i=1}^n(A_i-\hat{A_i})^2}</math>

where <math>A_i</math> is the measured displacement and <math>\hat{A_i}</math> is the predicted displacement by the function model. The core of this method is calculate the distant between the measured and predicted values and take the root mean of them.

The displacements (<math>A_i</math>) we have measured are in the regimen of thousands and the error is ~100. This is attributed to the response of the system to lower frequencies. However, for near resonance the model fits pretty well.

== Conclusion ==
We have shown indeed that we can characterize the resonance frequency of a system. For this particular application, we found a frequency of 113.12Hz for the system transducer plus mirror. Finally, with appropriate handling, this technique could be implemented to characterize more sofisticated systems.

==References==
[1] Freschi, A. A., et al. "Laser interferometric characterization of a vibrating speaker system." American Journal of Physics 71.11 (2003): 1121-1126.
https://doi.org/10.1119/1.1586262

[2] Skarha, Matthew. "Laser interferometric characterization for a vibrating speaker system."
http://mattskarha.com/assets/docs/Laser%20interferometric%20characterization%20for%20a%20vibrating%20speaker%20system%20good.pdf

[3] Pathare, Shirish, and Vikrant Kurmude. "Low cost Michelson–Morley interferometer." Physics Education 51.6 (2016): 063001.
https://doi.org/10.1088/0031-9120/51/6/063001

Resonance frequency measurement using a interferometric method

2022-04-30T07:37:09Z

Aucca: /* Experimental setup */

== Team Members ==
* [[User:Nakarin|Nakarin Jayjong]]
* [[User:Aucca|Joel Auccapuclla]]
* [[User:Xiaoyu|Xiaoyu Nie]]
* [[User:Haotian|Haotian Song]]

== Introduction ==
Interferometers since their invention have been a great tool for experimental physics. Applications cover a wide spectrum of fields such as astronomy, fiber optics, seismology, seismology, velocimetry to say the least. The precision of measurement of small displacements is the key factor, take LIGO as an example. Another important factor is the convenience to set up an interferometer and the low complexity in terms of equipment. In this experiment, we will measure the resonance frequency of a vibrating transducer using a Michelson interferometer. Basically, a transducer is attached to one optical path, to one mirror, of the Michelson interferometer and it is expected to see a peak in the response curve for the resonance frequency of the transducer plus mirror.

== Theory ==
Our Michelson interferometer setup used a laser, a beam splitter, two mirrors, and a photometer, illustrated in Fig. 1. Original models used sodium flame as the coherent light source. The light propagates from the source to the beam splitter where approximately 50% of the incident light is transmitted and approximately 50% of it is reflected at a right angle. The transmitted light continues to propagate as the reference plane wave to the stationary mirror where it is fully reflected back towards the beam splitter. The reflected light propagates as the target plane wave to the mirror attached to the oscillating loudspeaker. The two waves then recombine at the beam splitter and interfere based on their relative phases based on the idea that the phase shift of the target plane wave varies with the instantaneous position of the speaker mirror combination. Consider a situation in which we setup our interferometer such that no phase shift corresponds to the equilibrium position of the speaker-mirror combination. At this position, we expect constructive interference at the photometer. Likewise, for every λ/2 change in displacement, we expect constructive interference at the photometer because the path length travelled by the target plane wave varies by some integral multiple of λ. Thus, the phase shift is an even multiple of π and the two waves remain in phase. Similarly, when the path length traveled is some odd multiple of λ/4, the phase shift is an odd multiple of π and the two waves interfere destructively.

Consider two coherent waves with the same amplitude A and same frequency <math>\omega</math> that have paths <math>x_1</math> and <math>x_2</math> emitted from different sources that meet together at a point. The net wave at that point can then be given as the real part of the equation

<math>
\psi=Ae^{i(kx_1-\omega t}+Ae^{i(kx_2-\omega t)}=Ae^{-i\omega t}e^{ik\frac{(x_1+x_2)}{2}}[e^{ik\frac{(x_1-x_2)}{2}}e^{-ik\frac{(x_1-x_2)}{2}}]=2Acos[k(\frac{x_1-x_2}{2})]e^{i[k\frac{(x_1+x_2)}{2}-\omega t]}
</math>

where the wave number <math>k=2\pi/\lambda</math>, <math>\lambda</math> is the wavelength of the waves, and c is the phase velocity. The time-averaged intensity of the combined wave is then proportional to the square of the its amplitude. That is,

<math>
I\propto cos^2[\pi(\frac{p_2-p_1}{\lambda})]=cos^2[2\pi(\frac{p_2}{\lambda}-\theta)]
</math>

where <math>\theta</math> is a constant. Now suppose the mirror that the <math>p_2</math> beam is incident to oscillates harmonically back and forth with a frequency <math>f</math> and a displacement <math>B</math>. Then, <math>p_2 =p_2(0)+Bsin(2\pi ft)</math> and the intensity of the detected light varies with time as

<math>
I=A^2cos^2[2\pi(\frac{Bsin(2\pi ft)}{\lambda}-\theta')]
</math>

Note that the frequency of oscillations is itself a sinusoid, meaning that there will be turnaround points in the oscillations.

== Experimental setup ==
As our main source for this experiment we used a Laser Diode of wavelength 658,85 nm (as we can see in fig. 4 in the following section). Most red laser diodes are based on either GaInP or AlGaInP, this means we have 2 semiconductor materials. For one particular configuration, the data sheet tells us this, it will allow current flow and produce light amplification, this is what we called laser. This is why the diode laser is connected to a power supply, see left top size of fig. 2.
Our Michelson interferometer consists of a beam splitter BS, which is used to split the laser beam into two beams and two mirrors M1 and M2. The reflected beam goes to a fixed mirror M1 while the transmitted beam goes to the movable mirror M2 (see fig. 1). A transducer is attached, using tape, to the mirror M2. The transducer is powered by a signal generator that is used to vary the vibrating frequency of the mirror. A photodiode PD is used to detect the signal of interference fringes. The schematic Michelson interferometer is shown in Figure 1.
[[File:Schematic Michelson Interferometer.jpeg|center|thumb|600px|Figure 1: Schematic figure of Michelson interferometer. LD is a laser diode, BS is a beam splitter, M1 and M2 are mirrors, PD is a photodiode, and vibration transducer which is driven by a signal generator.]]

The experimental setup Michelson interferometer is shown in figure 2.
[[File:Experimental setup Michelson Interferometer.jpeg|center|thumb|600px|Figure 2: Experimental setup of Michelson interferometer.]]

The interference fringes of the Michelson interferometer are observed when the transducer is turned off in Figure 3.
[[File:Michelson interference fringes.jpeg|center|thumb|600px|Figure 3: Interference fringes which are observed when the transducer is turned off]]

==Experiment Results==
After the alignment of the optical setup, we applied sinusoidal voltage to the transducer attached to the mirror tightly. intensity fluctuation is recorded by the homemade photodiode. Data from the photodiode is collected by the oscilloscope. The typical figure from the oscilloscope is shown in Figure 4. According to the equation mentioned before, the number of peaks during each period corresponds to the radio of displacement and laser wavelength. In this case, we can measure the displacement for various frequencies. The frequency range is from 86 Hz to 130 Hz.

[[File:ScopeScreen.jpeg|center|thumb|600px|Figure 4: The display on the oscilloscope at <math>f = 103Hz </math>. After applying the voltage to the transducer, the voltage output from photodiode changes with time.]]

In our setup, we measured the wavelength of Mitsubishi Laser Diode which is 658.9 nm. The wavemeter measurement results are shown here.
[[File:Wavelength.jpeg|center|thumb|600px|Figure 5: The Wavelength measurement of the Mitsubishi Laser Diode.]]

We plot the displacement for different frequencies with the errorbar in Figure 6. Uncertainty is the square root uncertainty of the count number. For a driven, damped and harmonic oscillator, the relation between displacement and frequency is

<math>|A|=\frac{f_{0}}{\sqrt{\left(\omega_{0}^{2}-\omega^{2}\right)^{2}+4 \beta^{2} \omega^{2}}} </math>

Where <math>f_{0}</math> is the amplitude, <math>\omega_{0}</math> is the resonance frequency, and <math>\beta_{0}</math> is a constant. With this equation, we can fit the frequency response curve. Because there is only one peak or valley for a frequency lower than 92 Hz or higher than 120 Hz, and the error is significant and non-negligible. We exclude those in the fitting process. Finally, the fitting root mean squared error (RMSE) is 130.38. Fitted <math>\beta</math> is <math>1.3 \pm 0.2</math>. The fitting resonance frequency <math>\omega_{0} </math> is <math>113.1 \pm 0.1Hz </math>.

[[File:FitPeakNum.jpeg|center|thumb|600px|Figure 6: Frequency response curve for the transducer system with errorbar. The blue dots represent the displacement measured with different frequencies. Error bars are the square root uncertainty. Red line is the fitting curve for a driven, damped, harmonic oscillator.]]

==Discussion==
Our final result fits our data to the best Lorentzian Function. When we say best we talk about the function with the lowest Root Mean Square Error (RMSE).

RMSE is defined as follow:

<math>\operatorname{RMSE}=\sqrt{\frac{1}{n}\sum_{i=1}^n(A_i-\hat{A_i})^2}</math>

where <math>A_i</math> is the measured displacement and <math>\hat{A_i}</math> is the predicted displacement by the function model. The core of this method is calculate the distant between the measured and predicted values and take the root mean of them.

The displacements (<math>A_i</math>) we have measured are in the regimen of thousands and the error is ~100. This is attributed to the response of the system to lower frequencies. However, for near resonance the model fits pretty well.

== Conclusion ==
We have shown indeed that we can characterize the resonance frequency of a system. For this particular application, we found a frequency of 113.12Hz for the system transducer plus mirror. Finally, with appropriate handling, this technique could be implemented to characterize more sofisticated systems.

==References==
[1] Freschi, A. A., et al. "Laser interferometric characterization of a vibrating speaker system." American Journal of Physics 71.11 (2003): 1121-1126.
https://doi.org/10.1119/1.1586262

[2] Skarha, Matthew. "Laser interferometric characterization for a vibrating speaker system."
http://mattskarha.com/assets/docs/Laser%20interferometric%20characterization%20for%20a%20vibrating%20speaker%20system%20good.pdf

[3] Pathare, Shirish, and Vikrant Kurmude. "Low cost Michelson–Morley interferometer." Physics Education 51.6 (2016): 063001.
https://doi.org/10.1088/0031-9120/51/6/063001

Resonance frequency measurement using a interferometric method

2022-04-30T06:33:54Z

Aucca: /* Experimental setup */

== Team Members ==
* [[User:Nakarin|Nakarin Jayjong]]
* [[User:Aucca|Joel Auccapuclla]]
* [[User:Xiaoyu|Xiaoyu Nie]]
* [[User:Haotian|Haotian Song]]

== Introduction ==
Interferometers since their invention have been a great tool for experimental physics. Applications cover a wide spectrum of fields such as astronomy, fiber optics, seismology, seismology, velocimetry to say the least. The precision of measurement of small displacements is the key factor, take LIGO as an example. Another important factor is the convenience to set up an interferometer and the low complexity in terms of equipment. In this experiment, we will measure the resonance frequency of a vibrating transducer using a Michelson interferometer. Basically, a transducer is attached to one optical path, to one mirror, of the Michelson interferometer and it is expected to see a peak in the response curve for the resonance frequency of the transducer plus mirror.

== Theory ==
Our Michelson interferometer setup used a laser, a beam splitter, two mirrors, and a photometer, illustrated in Fig. 1. Original models used sodium flame as the coherent light source. The light propagates from the source to the beam splitter where approximately 50% of the incident light is transmitted and approximately 50% of it is reflected at a right angle. The transmitted light continues to propagate as the reference plane wave to the stationary mirror where it is fully reflected back towards the beam splitter. The reflected light propagates as the target plane wave to the mirror attached to the oscillating loudspeaker. The two waves then recombine at the beam splitter and interfere based on their relative phases based on the idea that the phase shift of the target plane wave varies with the instantaneous position of the speaker mirror combination. Consider a situation in which we setup our interferometer such that no phase shift corresponds to the equilibrium position of the speaker-mirror combination. At this position, we expect constructive interference at the photometer. Likewise, for every λ/2 change in displacement, we expect constructive interference at the photometer because the path length travelled by the target plane wave varies by some integral multiple of λ. Thus, the phase shift is an even multiple of π and the two waves remain in phase. Similarly, when the path length traveled is some odd multiple of λ/4, the phase shift is an odd multiple of π and the two waves interfere destructively.

Consider two coherent waves with the same amplitude A and same frequency <math>\omega</math> that have paths <math>x_1</math> and <math>x_2</math> emitted from different sources that meet together at a point. The net wave at that point can then be given as the real part of the equation

<math>
\psi=Ae^{i(kx_1-\omega t}+Ae^{i(kx_2-\omega t)}=Ae^{-i\omega t}e^{ik\frac{(x_1+x_2)}{2}}[e^{ik\frac{(x_1-x_2)}{2}}e^{-ik\frac{(x_1-x_2)}{2}}]=2Acos[k(\frac{x_1-x_2}{2})]e^{i[k\frac{(x_1+x_2)}{2}-\omega t]}
</math>

where the wave number <math>k=2\pi/\lambda</math>, <math>\lambda</math> is the wavelength of the waves, and c is the phase velocity. The time-averaged intensity of the combined wave is then proportional to the square of the its amplitude. That is,

<math>
I\propto cos^2[\pi(\frac{p_2-p_1}{\lambda})]=cos^2[2\pi(\frac{p_2}{\lambda}-\theta)]
</math>

where <math>\theta</math> is a constant. Now suppose the mirror that the <math>p_2</math> beam is incident to oscillates harmonically back and forth with a frequency <math>f</math> and a displacement <math>B</math>. Then, <math>p_2 =p_2(0)+Bsin(2\pi ft)</math> and the intensity of the detected light varies with time as

<math>
I=A^2cos^2[2\pi(\frac{Bsin(2\pi ft)}{\lambda}-\theta')]
</math>

Note that the frequency of oscillations is itself a sinusoid, meaning that there will be turnaround points in the oscillations.

== Experimental setup ==
As our main source for this experiment we used a Laser Diode of wavelength 658,85 nm (as we can see in fig. 4 in the following section). Our Michelson interferometer consists of a beam splitter BS, which is used to split the laser beam into two beams and two mirrors M1 and M2. The reflected beam goes to a fixed mirror M1 while the transmitted beam goes to the movable mirror M2 (see fig. 1). A transducer is attached, using tape, to the mirror M2. The transducer is powered by a signal generator that is used to vary the vibrating frequency of the mirror. A photodiode PD is used to detect the signal of interference fringes. The schematic Michelson interferometer is shown in Figure 1.
[[File:Schematic Michelson Interferometer.jpeg|center|thumb|600px|Figure 1: Schematic figure of Michelson interferometer. LD is a laser diode, BS is a beam splitter, M1 and M2 are mirrors, PD is a photodiode, and vibration transducer which is driven by a signal generator.]]

The experimental setup Michelson interferometer is shown in figure 2.
[[File:Experimental setup Michelson Interferometer.jpeg|center|thumb|600px|Figure 2: Experimental setup of Michelson interferometer.]]

The interference fringes of the Michelson interferometer are observed when the transducer is turned off in Figure 3.
[[File:Michelson interference fringes.jpeg|center|thumb|600px|Figure 3: Interference fringes which are observed when the transducer is turned off]]

==Experiment Results==
After the alignment of the optical setup, we applied sinusoidal voltage to the transducer attached to the mirror tightly. intensity fluctuation is recorded by the homemade photodiode. Data from the photodiode is collected by the oscilloscope. The typical figure from the oscilloscope is shown in Figure 4. According to the equation mentioned before, the number of peaks during each period corresponds to the radio of displacement and laser wavelength. In this case, we can measure the displacement for various frequencies. The frequency range is from 86 Hz to 130 Hz.

[[File:ScopeScreen.jpeg|center|thumb|600px|Figure 4: The display on the oscilloscope at <math>f = 103Hz </math>. After applying the voltage to the transducer, the voltage output from photodiode changes with time.]]

In our setup, we measured the wavelength of Mitsubishi Laser Diode which is 658.9 nm. The wavemeter measurement results are shown here.
[[File:Wavelength.jpeg|center|thumb|600px|Figure 5: The Wavelength measurement of the Mitsubishi Laser Diode.]]

We plot the displacement for different frequencies with the errorbar in Figure 6. Uncertainty is the square root uncertainty of the count number. For a driven, damped and harmonic oscillator, the relation between displacement and frequency is

<math>|A|=\frac{f_{0}}{\sqrt{\left(\omega_{0}^{2}-\omega^{2}\right)^{2}+4 \beta^{2} \omega^{2}}} </math>

Where <math>f_{0}</math> is the amplitude, <math>\omega_{0}</math> is the resonance frequency, and <math>\beta_{0}</math> is a constant. With this equation, we can fit the frequency response curve. Because there is only one peak or valley for a frequency lower than 92 Hz or higher than 120 Hz, and the error is significant and non-negligible. We exclude those in the fitting process. Finally, the fitting root mean squared error (RMSE) is 130.38. Fitted <math>\beta</math> is <math>1.3 \pm 0.2</math>. The fitting resonance frequency <math>\omega_{0} </math> is <math>113.1 \pm 0.1Hz </math>.

[[File:FitPeakNum.jpeg|center|thumb|600px|Figure 6: Frequency response curve for the transducer system with errorbar. The blue dots represent the displacement measured with different frequencies. Error bars are the square root uncertainty. Red line is the fitting curve for a driven, damped, harmonic oscillator.]]

==Discussion==
Our final result fits our data to the best Lorentzian Function. When we say best we talk about the function with the lowest Root Mean Square Error (RMSE).

RMSE is defined as follow:

<math>\operatorname{RMSE}=\sqrt{\frac{1}{n}\sum_{i=1}^n(A_i-\hat{A_i})^2}</math>

where <math>A_i</math> is the measured displacement and <math>\hat{A_i}</math> is the predicted displacement by the function model. The core of this method is calculate the distant between the measured and predicted values and take the root mean of them.

The displacements (<math>A_i</math>) we have measured are in the regimen of thousands and the error is ~100. This is attributed to the response of the system to lower frequencies. However, for near resonance the model fits pretty well.

== Conclusion ==
We have shown indeed that we can characterize the resonance frequency of a system. For this particular application, we found a frequency of 113.12Hz for the system transducer plus mirror. Finally, with appropriate handling, this technique could be implemented to characterize more sofisticated systems.

==References==
[1] Freschi, A. A., et al. "Laser interferometric characterization of a vibrating speaker system." American Journal of Physics 71.11 (2003): 1121-1126.
https://doi.org/10.1119/1.1586262

[2] Skarha, Matthew. "Laser interferometric characterization for a vibrating speaker system."
http://mattskarha.com/assets/docs/Laser%20interferometric%20characterization%20for%20a%20vibrating%20speaker%20system%20good.pdf

[3] Pathare, Shirish, and Vikrant Kurmude. "Low cost Michelson–Morley interferometer." Physics Education 51.6 (2016): 063001.
https://doi.org/10.1088/0031-9120/51/6/063001

Resonance frequency measurement using a interferometric method

2022-04-30T05:00:45Z

Aucca: /* Discussion */

== Team Members ==
* [[User:Nakarin|Nakarin Jayjong]]
* [[User:Aucca|Joel Auccapuclla]]
* [[User:Xiaoyu|Xiaoyu Nie]]
* [[User:Haotian|Haotian Song]]

== Introduction ==
Interferometers since their invention have been a great tool for experimental physics. Applications cover a wide spectrum of fields such as astronomy, fiber optics, seismology, seismology, velocimetry to say the least. The precision of measurement of small displacements is the key factor, take LIGO as an example. Another important factor is the convenience to set up an interferometer and the low complexity in terms of equipment. In this experiment, we will measure the resonance frequency of a vibrating transducer using a Michelson interferometer. Basically, a transducer is attached to one optical path, to one mirror, of the Michelson interferometer and it is expected to see a peak in the response curve for the resonance frequency of the transducer plus mirror.

== Theory ==
Our Michelson interferometer setup used a laser, a beam splitter, two mirrors, and a photometer, illustrated in Fig. 1. Original models used sodium flame as the coherent light source. The light propagates from the source to the beam splitter where approximately 50% of the incident light is transmitted and approximately 50% of it is reflected at a right angle. The transmitted light continues to propagate as the reference plane wave to the stationary mirror where it is fully reflected back towards the beam splitter. The reflected light propagates as the target plane wave to the mirror attached to the oscillating loudspeaker. The two waves then recombine at the beam splitter and interfere based on their relative phases based on the idea that the phase shift of the target plane wave varies with the instantaneous position of the speaker mirror combination. Consider a situation in which we setup our interferometer such that no phase shift corresponds to the equilibrium position of the speaker-mirror combination. At this position, we expect constructive interference at the photometer. Likewise, for every λ/2 change in displacement, we expect constructive interference at the photometer because the path length travelled by the target plane wave varies by some integral multiple of λ. Thus, the phase shift is an even multiple of π and the two waves remain in phase. Similarly, when the path length traveled is some odd multiple of λ/4, the phase shift is an odd multiple of π and the two waves interfere destructively.

Consider two coherent waves with the same amplitude A and same frequency <math>\omega</math> that have paths <math>x_1</math> and <math>x_2</math> emitted from different sources that meet together at a point. The net wave at that point can then be given as the real part of the equation

<math>
\psi=Ae^{i(kx_1-\omega t}+Ae^{i(kx_2-\omega t)}=Ae^{-i\omega t}e^{ik\frac{(x_1+x_2)}{2}}[e^{ik\frac{(x_1-x_2)}{2}}e^{-ik\frac{(x_1-x_2)}{2}}]=2Acos[k(\frac{x_1-x_2}{2})]e^{i[k\frac{(x_1+x_2)}{2}-\omega t]}
</math>

where the wave number <math>k=2\pi/\lambda</math>, <math>\lambda</math> is the wavelength of the waves, and c is the phase velocity. The time-averaged intensity of the combined wave is then proportional to the square of the its amplitude. That is,

<math>
I\propto cos^2[\pi(\frac{p_2-p_1}{\lambda})]=cos^2[2\pi(\frac{p_2}{\lambda}-\theta)]
</math>

where <math>\theta</math> is a constant. Now suppose the mirror that the <math>p_2</math> beam is incident to oscillates harmonically back and forth with a frequency <math>f</math> and a displacement <math>B</math>. Then, <math>p_2 =p_2(0)+Bsin(2\pi ft)</math> and the intensity of the detected light varies with time as

<math>
I=A^2cos^2[2\pi(\frac{Bsin(2\pi ft)}{\lambda}-\theta')]
</math>

Note that the frequency of oscillations is itself a sinusoid, meaning that there will be turnaround points in the oscillations.

== Experimental setup ==
The Michelson interferometer consists of a beam splitter BS which is used to split the laser beam into two beams and two mirrors M1 and M2. The reflected beam goes to a fixed mirror M1 while the transmitted beam goes to the movable mirror M2. A transducer is attached to the mirror M2. The transducer is powered by a signal generator that is used to vary the vibrating frequency of the mirror. A photodiode PD is used to detect the signal of interference fringes. The schematic Michelson interferometer is shown in Figure 1.
[[File:Schematic Michelson Interferometer.jpeg|center|thumb|600px|Figure 1: Schematic figure of Michelson interferometer. LD is a laser diode, BS is a beam splitter, M1 and M2 are mirrors, PD is a photodiode, and vibration transducer which is driven by a signal generator.]]

The experimental setup Michelson interferometer is shown in figure 2.
[[File:Experimental setup Michelson Interferometer.jpeg|center|thumb|600px|Figure 2: Experimental setup of Michelson interferometer.]]

The interference fringes of the Michelson interferometer are observed when the transducer is turned off in Figure 3.
[[File:Michelson interference fringes.jpeg|center|thumb|600px|Figure 3: Interference fringes which are observed when the transducer is turned off]]

==Experiment Results==
After the alignment of the optical setup, we applied sinusoidal voltage to the transducer attached to the mirror tightly. intensity fluctuation is recorded by the homemade photodiode. Data from the photodiode is collected by the oscilloscope. The typical figure from the oscilloscope is shown in Figure 4. According to the equation mentioned before, the number of peaks during each period corresponds to the radio of displacement and laser wavelength. In this case, we can measure the displacement for various frequencies. The frequency range is from 86 Hz to 130 Hz.

[[File:ScopeScreen.jpeg|center|thumb|600px|Figure 4: The display on the oscilloscope at <math>f = 103Hz </math>. After applying the voltage to the transducer, the voltage output from photodiode changes with time.]]

In our setup, we measured the wavelength of Mitsubishi Laser Diode which is 658.9 nm. The wavemeter measurement results are shown here.
[[File:Wavelength.jpeg|center|thumb|600px|Figure 5: The Wavelength measurement of the Mitsubishi Laser Diode.]]

We plot the displacement for different frequencies with the errorbar in Figure 6. Uncertainty is the square root uncertainty of the count number. For a driven, damped and harmonic oscillator, the relation between displacement and frequency is

<math>|A|=\frac{f_{0}}{\sqrt{\left(\omega_{0}^{2}-\omega^{2}\right)^{2}+4 \beta^{2} \omega^{2}}} </math>

Where <math>f_{0}</math> is the amplitude, <math>\omega_{0}</math> is the resonance frequency, and <math>\beta_{0}</math> is a constant. With this equation, we can fit the frequency response curve. Because there is only one peak or valley for a frequency lower than 92 Hz or higher than 120 Hz, and the error is significant and non-negligible. We exclude those in the fitting process. Finally, the fitting root mean squared error (RMSE) is 130.38. Fitted <math>\beta</math> is <math>1.3 \pm 0.2</math>. The fitting resonance frequency <math>\omega_{0} </math> is <math>113.1 \pm 0.1Hz </math>.

[[File:FitPeakNum.jpeg|center|thumb|600px|Figure 6: Frequency response curve for the transducer system with errorbar. The blue dots represent the displacement measured with different frequencies. Error bars are the square root uncertainty. Red line is the fitting curve for a driven, damped, harmonic oscillator.]]

==Discussion==
Our final result fits our data to the best Lorentzian Function. When we say best we talk about the function with the lowest Root Mean Square Error (RMSE).

RMSE is defined as follow:

<math>\operatorname{RMSE}=\sqrt{\frac{1}{n}\sum_{i=1}^n(A_i-\hat{A_i})^2}</math>

where <math>A_i</math> is the measured displacement and <math>\hat{A_i}</math> is the predicted displacement by the function model. The core of this method is calculate the distant between the measured and predicted values and take the root mean of them.

The displacements (<math>A_i</math>) we have measured are in the regimen of thousands and the error is ~100. This is attributed to the response of the system to lower frequencies. However, for near resonance the model fits pretty well.

== Conclusion ==
We have shown indeed that we can characterize the resonance frequency of a system. For this particular application, we found a frequency of 113.12Hz for the system transducer plus mirror. Finally, with appropriate handling, this technique could be implemented to characterize more sofisticated systems.

==References==
[1] Freschi, A. A., et al. "Laser interferometric characterization of a vibrating speaker system." American Journal of Physics 71.11 (2003): 1121-1126.
https://doi.org/10.1119/1.1586262

[2] Skarha, Matthew. "Laser interferometric characterization for a vibrating speaker system."
http://mattskarha.com/assets/docs/Laser%20interferometric%20characterization%20for%20a%20vibrating%20speaker%20system%20good.pdf

[3] Pathare, Shirish, and Vikrant Kurmude. "Low cost Michelson–Morley interferometer." Physics Education 51.6 (2016): 063001.
https://doi.org/10.1088/0031-9120/51/6/063001

Resonance frequency measurement using a interferometric method

2022-04-30T04:59:25Z

Aucca: /* Discussion */

== Team Members ==
* [[User:Nakarin|Nakarin Jayjong]]
* [[User:Aucca|Joel Auccapuclla]]
* [[User:Xiaoyu|Xiaoyu Nie]]
* [[User:Haotian|Haotian Song]]

== Introduction ==
Interferometers since their invention have been a great tool for experimental physics. Applications cover a wide spectrum of fields such as astronomy, fiber optics, seismology, seismology, velocimetry to say the least. The precision of measurement of small displacements is the key factor, take LIGO as an example. Another important factor is the convenience to set up an interferometer and the low complexity in terms of equipment. In this experiment, we will measure the resonance frequency of a vibrating transducer using a Michelson interferometer. Basically, a transducer is attached to one optical path, to one mirror, of the Michelson interferometer and it is expected to see a peak in the response curve for the resonance frequency of the transducer plus mirror.

== Theory ==
Our Michelson interferometer setup used a laser, a beam splitter, two mirrors, and a photometer, illustrated in Fig. 1. Original models used sodium flame as the coherent light source. The light propagates from the source to the beam splitter where approximately 50% of the incident light is transmitted and approximately 50% of it is reflected at a right angle. The transmitted light continues to propagate as the reference plane wave to the stationary mirror where it is fully reflected back towards the beam splitter. The reflected light propagates as the target plane wave to the mirror attached to the oscillating loudspeaker. The two waves then recombine at the beam splitter and interfere based on their relative phases based on the idea that the phase shift of the target plane wave varies with the instantaneous position of the speaker mirror combination. Consider a situation in which we setup our interferometer such that no phase shift corresponds to the equilibrium position of the speaker-mirror combination. At this position, we expect constructive interference at the photometer. Likewise, for every λ/2 change in displacement, we expect constructive interference at the photometer because the path length travelled by the target plane wave varies by some integral multiple of λ. Thus, the phase shift is an even multiple of π and the two waves remain in phase. Similarly, when the path length traveled is some odd multiple of λ/4, the phase shift is an odd multiple of π and the two waves interfere destructively.

Consider two coherent waves with the same amplitude A and same frequency <math>\omega</math> that have paths <math>x_1</math> and <math>x_2</math> emitted from different sources that meet together at a point. The net wave at that point can then be given as the real part of the equation

<math>
\psi=Ae^{i(kx_1-\omega t}+Ae^{i(kx_2-\omega t)}=Ae^{-i\omega t}e^{ik\frac{(x_1+x_2)}{2}}[e^{ik\frac{(x_1-x_2)}{2}}e^{-ik\frac{(x_1-x_2)}{2}}]=2Acos[k(\frac{x_1-x_2}{2})]e^{i[k\frac{(x_1+x_2)}{2}-\omega t]}
</math>

where the wave number <math>k=2\pi/\lambda</math>, <math>\lambda</math> is the wavelength of the waves, and c is the phase velocity. The time-averaged intensity of the combined wave is then proportional to the square of the its amplitude. That is,

<math>
I\propto cos^2[\pi(\frac{p_2-p_1}{\lambda})]=cos^2[2\pi(\frac{p_2}{\lambda}-\theta)]
</math>

where <math>\theta</math> is a constant. Now suppose the mirror that the <math>p_2</math> beam is incident to oscillates harmonically back and forth with a frequency <math>f</math> and a displacement <math>B</math>. Then, <math>p_2 =p_2(0)+Bsin(2\pi ft)</math> and the intensity of the detected light varies with time as

<math>
I=A^2cos^2[2\pi(\frac{Bsin(2\pi ft)}{\lambda}-\theta')]
</math>

Note that the frequency of oscillations is itself a sinusoid, meaning that there will be turnaround points in the oscillations.

== Experimental setup ==
The Michelson interferometer consists of a beam splitter BS which is used to split the laser beam into two beams and two mirrors M1 and M2. The reflected beam goes to a fixed mirror M1 while the transmitted beam goes to the movable mirror M2. A transducer is attached to the mirror M2. The transducer is powered by a signal generator that is used to vary the vibrating frequency of the mirror. A photodiode PD is used to detect the signal of interference fringes. The schematic Michelson interferometer is shown in Figure 1.
[[File:Schematic Michelson Interferometer.jpeg|center|thumb|600px|Figure 1: Schematic figure of Michelson interferometer. LD is a laser diode, BS is a beam splitter, M1 and M2 are mirrors, PD is a photodiode, and vibration transducer which is driven by a signal generator.]]

The experimental setup Michelson interferometer is shown in figure 2.
[[File:Experimental setup Michelson Interferometer.jpeg|center|thumb|600px|Figure 2: Experimental setup of Michelson interferometer.]]

The interference fringes of the Michelson interferometer are observed when the transducer is turned off in Figure 3.
[[File:Michelson interference fringes.jpeg|center|thumb|600px|Figure 3: Interference fringes which are observed when the transducer is turned off]]

==Experiment Results==
After the alignment of the optical setup, we applied sinusoidal voltage to the transducer attached to the mirror tightly. intensity fluctuation is recorded by the homemade photodiode. Data from the photodiode is collected by the oscilloscope. The typical figure from the oscilloscope is shown in Figure 4. According to the equation mentioned before, the number of peaks during each period corresponds to the radio of displacement and laser wavelength. In this case, we can measure the displacement for various frequencies. The frequency range is from 86 Hz to 130 Hz.

[[File:ScopeScreen.jpeg|center|thumb|600px|Figure 4: The display on the oscilloscope at <math>f = 103Hz </math>. After applying the voltage to the transducer, the voltage output from photodiode changes with time.]]

In our setup, we measured the wavelength of Mitsubishi Laser Diode which is 658.9 nm. The wavemeter measurement results are shown here.
[[File:Wavelength.jpeg|center|thumb|600px|Figure 5: The Wavelength measurement of the Mitsubishi Laser Diode.]]

We plot the displacement for different frequencies with the errorbar in Figure 6. Uncertainty is the square root uncertainty of the count number. For a driven, damped and harmonic oscillator, the relation between displacement and frequency is

<math>|A|=\frac{f_{0}}{\sqrt{\left(\omega_{0}^{2}-\omega^{2}\right)^{2}+4 \beta^{2} \omega^{2}}} </math>

Where <math>f_{0}</math> is the amplitude, <math>\omega_{0}</math> is the resonance frequency, and <math>\beta_{0}</math> is a constant. With this equation, we can fit the frequency response curve. Because there is only one peak or valley for a frequency lower than 92 Hz or higher than 120 Hz, and the error is significant and non-negligible. We exclude those in the fitting process. Finally, the fitting root mean squared error (RMSE) is 130.38. Fitted <math>\beta</math> is <math>1.3 \pm 0.2</math>. The fitting resonance frequency <math>\omega_{0} </math> is <math>113.1 \pm 0.1Hz </math>.

[[File:FitPeakNum.jpeg|center|thumb|600px|Figure 6: Frequency response curve for the transducer system with errorbar. The blue dots represent the displacement measured with different frequencies. Error bars are the square root uncertainty. Red line is the fitting curve for a driven, damped, harmonic oscillator.]]

==Discussion==
Our final result fits our data to the best Lorentzian Function. When we say best we talk about the function with the lowest Root Mean Square Error (RMSE).

RMSE is defined as follow:

<math>\operatorname{RMSE}=\sqrt{\frac{1}{n}\sum_{i=1}^n(A_i-\hat{A_i})^2}</math>

where <math>A_i</math> is the measured displacement and <math>\hat{A_i}</math> is the predicted displacement by the function model. The core of this method is calculate the distant between the measured and predicted values and take the root mean of them.

The displacements (<math>A_i</math>) we have measured are in the regimen of thousands and the error is ~100. This could be attribute to the response of the system to lower frequencies. However, for near resonance the model fits pretty well.

== Conclusion ==
We have shown indeed that we can characterize the resonance frequency of a system. For this particular application, we found a frequency of 113.12Hz for the system transducer plus mirror. Finally, with appropriate handling, this technique could be implemented to characterize more sofisticated systems.

==References==
[1] Freschi, A. A., et al. "Laser interferometric characterization of a vibrating speaker system." American Journal of Physics 71.11 (2003): 1121-1126.
https://doi.org/10.1119/1.1586262

[2] Skarha, Matthew. "Laser interferometric characterization for a vibrating speaker system."
http://mattskarha.com/assets/docs/Laser%20interferometric%20characterization%20for%20a%20vibrating%20speaker%20system%20good.pdf

[3] Pathare, Shirish, and Vikrant Kurmude. "Low cost Michelson–Morley interferometer." Physics Education 51.6 (2016): 063001.
https://doi.org/10.1088/0031-9120/51/6/063001

Resonance frequency measurement using a interferometric method

2022-04-30T04:28:04Z

Aucca: /* Discussion */

== Team Members ==
* [[User:Nakarin|Nakarin Jayjong]]
* [[User:Aucca|Joel Auccapuclla]]
* [[User:Xiaoyu|Xiaoyu Nie]]
* [[User:Haotian|Haotian Song]]

== Introduction ==
Interferometers since their invention have been a great tool for experimental physics. Applications cover a wide spectrum of fields such as astronomy, fiber optics, seismology, seismology, velocimetry to say the least. The precision of measurement of small displacements is the key factor, take LIGO as an example. Another important factor is the convenience to set up an interferometer and the low complexity in terms of equipment. In this experiment, we will measure the resonance frequency of a vibrating transducer using a Michelson interferometer. Basically, a transducer is attached to one optical path, to one mirror, of the Michelson interferometer and it is expected to see a peak in the response curve for the resonance frequency of the transducer plus mirror.

== Theory ==
Our Michelson interferometer setup used a laser, a beam splitter, two mirrors, and a photometer, illustrated in Fig. 1. Original models used sodium flame as the coherent light source. The light propagates from the source to the beam splitter where approximately 50% of the incident light is transmitted and approximately 50% of it is reflected at a right angle. The transmitted light continues to propagate as the reference plane wave to the stationary mirror where it is fully reflected back towards the beam splitter. The reflected light propagates as the target plane wave to the mirror attached to the oscillating loudspeaker. The two waves then recombine at the beam splitter and interfere based on their relative phases based on the idea that the phase shift of the target plane wave varies with the instantaneous position of the speaker mirror combination. Consider a situation in which we setup our interferometer such that no phase shift corresponds to the equilibrium position of the speaker-mirror combination. At this position, we expect constructive interference at the photometer. Likewise, for every λ/2 change in displacement, we expect constructive interference at the photometer because the path length travelled by the target plane wave varies by some integral multiple of λ. Thus, the phase shift is an even multiple of π and the two waves remain in phase. Similarly, when the path length traveled is some odd multiple of λ/4, the phase shift is an odd multiple of π and the two waves interfere destructively.

Consider two coherent waves with the same amplitude A and same frequency <math>\omega</math> that have paths <math>x_1</math> and <math>x_2</math> emitted from different sources that meet together at a point. The net wave at that point can then be given as the real part of the equation

<math>
\psi=Ae^{i(kx_1-\omega t}+Ae^{i(kx_2-\omega t)}=Ae^{-i\omega t}e^{ik\frac{(x_1+x_2)}{2}}[e^{ik\frac{(x_1-x_2)}{2}}e^{-ik\frac{(x_1-x_2)}{2}}]=2Acos[k(\frac{x_1-x_2}{2})]e^{i[k\frac{(x_1+x_2)}{2}-\omega t]}
</math>

where the wave number <math>k=2\pi/\lambda</math>, <math>\lambda</math> is the wavelength of the waves, and c is the phase velocity. The time-averaged intensity of the combined wave is then proportional to the square of the its amplitude. That is,

<math>
I\propto cos^2[\pi(\frac{p_2-p_1}{\lambda})]=cos^2[2\pi(\frac{p_2}{\lambda}-\theta)]
</math>

where <math>\theta</math> is a constant. Now suppose the mirror that the <math>p_2</math> beam is incident to oscillates harmonically back and forth with a frequency <math>f</math> and a displacement <math>B</math>. Then, <math>p_2 =p_2(0)+Bsin(2\pi ft)</math> and the intensity of the detected light varies with time as

<math>
I=A^2cos^2[2\pi(\frac{Bsin(2\pi ft)}{\lambda}-\theta')]
</math>

Note that the frequency of oscillations is itself a sinusoid, meaning that there will be turnaround points in the oscillations.

== Experimental setup ==
The Michelson interferometer consists of a beam splitter BS which is used to split the laser beam into two beams and two mirrors M1 and M2. The reflected beam goes to a fixed mirror M1 while the transmitted beam goes to the movable mirror M2. A transducer is attached to the mirror M2. The transducer is powered by a signal generator that is used to vary the vibrating frequency of the mirror. A photodiode PD is used to detect the signal of interference fringes. The schematic Michelson interferometer is shown in Figure 1.
[[File:Schematic Michelson Interferometer.jpeg|center|thumb|600px|Figure 1: Schematic figure of Michelson interferometer. LD is a laser diode, BS is a beam splitter, M1 and M2 are mirrors, PD is a photodiode, and vibration transducer which is driven by a signal generator.]]

The experimental setup Michelson interferometer is shown in figure 2.
[[File:Experimental setup Michelson Interferometer.jpeg|center|thumb|600px|Figure 2: Experimental setup of Michelson interferometer.]]

The interference fringes of the Michelson interferometer are observed when the transducer is turned off in Figure 3.
[[File:Michelson interference fringes.jpeg|center|thumb|600px|Figure 3: Interference fringes which are observed when the transducer is turned off]]

==Experiment Results==
After the alignment of the optical setup, we applied sinusoidal voltage to the transducer attached to the mirror tightly. intensity fluctuation is recorded by the homemade photodiode. Data from the photodiode is collected by the oscilloscope. The typical figure from the oscilloscope is shown in Figure 4. According to the equation mentioned before, the number of peaks during each period corresponds to the radio of displacement and laser wavelength. In this case, we can measure the displacement for various frequencies. The frequency range is from 86 Hz to 130 Hz.

[[File:ScopeScreen.jpeg|center|thumb|600px|Figure 4: The display on the oscilloscope at <math>f = 103Hz </math>. After applying the voltage to the transducer, the voltage output from photodiode changes with time.]]

In our setup, we measured the wavelength of Mitsubishi Laser Diode which is 658.9 nm. The wavemeter measurement results are shown here.
[[File:Wavelength.jpeg|center|thumb|600px|Figure 5: The Wavelength measurement of the Mitsubishi Laser Diode.]]

We plot the displacement for different frequencies with the errorbar in Figure 6. Uncertainty is the square root uncertainty of the count number. For a driven, damped and harmonic oscillator, the relation between displacement and frequency is

<math>|A|=\frac{f_{0}}{\sqrt{\left(\omega_{0}^{2}-\omega^{2}\right)^{2}+4 \beta^{2} \omega^{2}}} </math>

Where <math>f_{0}</math> is the amplitude, <math>\omega_{0}</math> is the resonance frequency, and <math>\beta_{0}</math> is a constant. With this equation, we can fit the frequency response curve. Because there is only one peak or valley for a frequency lower than 92 Hz or higher than 120 Hz, and the error is significant and non-negligible. We exclude those in the fitting process. Finally, the fitting root mean squared error (RMSE) is 130.38. Fitted <math>\beta</math> is <math>1.3 \pm 0.2</math>. The fitting resonance frequency <math>\omega_{0} </math> is <math>113.1 \pm 0.1Hz </math>.

[[File:FitPeakNum.jpeg|center|thumb|600px|Figure 6: Frequency response curve for the transducer system with errorbar. The blue dots represent the displacement measured with different frequencies. Error bars are the square root uncertainty. Red line is the fitting curve for a driven, damped, harmonic oscillator.]]

==Discussion==
Our final result fits our data to the best Lorentzian Function. When we say best we talk about the function with the lowest Root Mean Square Error (RMSE).

RMSE is defined as follow:

<math>\operatorname{RMSE}=\sqrt{\frac{1}{n}\sum_{i=1}^n(A_i-\hat{A_i})^2}</math>

where <math>A_i</math> is the measured displacement and <math>\hat{A_i}</math> is the predicted displacement by the function model. The core of this method is calculate the distant between the measured and predicted values and take the root mean of them.

== Conclusion ==
We have shown indeed that we can characterize the resonance frequency of a system. For this particular application, we found a frequency of 113.12Hz for the system transducer plus mirror. Finally, with appropriate handling, this technique could be implemented to characterize more sofisticated systems.

==References==
[1] Freschi, A. A., et al. "Laser interferometric characterization of a vibrating speaker system." American Journal of Physics 71.11 (2003): 1121-1126.
https://doi.org/10.1119/1.1586262

[2] Skarha, Matthew. "Laser interferometric characterization for a vibrating speaker system."
http://mattskarha.com/assets/docs/Laser%20interferometric%20characterization%20for%20a%20vibrating%20speaker%20system%20good.pdf

[3] Pathare, Shirish, and Vikrant Kurmude. "Low cost Michelson–Morley interferometer." Physics Education 51.6 (2016): 063001.
https://doi.org/10.1088/0031-9120/51/6/063001

Resonance frequency measurement using a interferometric method

2022-04-30T04:27:38Z

Aucca: /* Discussion */

== Team Members ==
* [[User:Nakarin|Nakarin Jayjong]]
* [[User:Aucca|Joel Auccapuclla]]
* [[User:Xiaoyu|Xiaoyu Nie]]
* [[User:Haotian|Haotian Song]]

== Introduction ==
Interferometers since their invention have been a great tool for experimental physics. Applications cover a wide spectrum of fields such as astronomy, fiber optics, seismology, seismology, velocimetry to say the least. The precision of measurement of small displacements is the key factor, take LIGO as an example. Another important factor is the convenience to set up an interferometer and the low complexity in terms of equipment. In this experiment, we will measure the resonance frequency of a vibrating transducer using a Michelson interferometer. Basically, a transducer is attached to one optical path, to one mirror, of the Michelson interferometer and it is expected to see a peak in the response curve for the resonance frequency of the transducer plus mirror.

== Theory ==
Our Michelson interferometer setup used a laser, a beam splitter, two mirrors, and a photometer, illustrated in Fig. 1. Original models used sodium flame as the coherent light source. The light propagates from the source to the beam splitter where approximately 50% of the incident light is transmitted and approximately 50% of it is reflected at a right angle. The transmitted light continues to propagate as the reference plane wave to the stationary mirror where it is fully reflected back towards the beam splitter. The reflected light propagates as the target plane wave to the mirror attached to the oscillating loudspeaker. The two waves then recombine at the beam splitter and interfere based on their relative phases based on the idea that the phase shift of the target plane wave varies with the instantaneous position of the speaker mirror combination. Consider a situation in which we setup our interferometer such that no phase shift corresponds to the equilibrium position of the speaker-mirror combination. At this position, we expect constructive interference at the photometer. Likewise, for every λ/2 change in displacement, we expect constructive interference at the photometer because the path length travelled by the target plane wave varies by some integral multiple of λ. Thus, the phase shift is an even multiple of π and the two waves remain in phase. Similarly, when the path length traveled is some odd multiple of λ/4, the phase shift is an odd multiple of π and the two waves interfere destructively.

Consider two coherent waves with the same amplitude A and same frequency <math>\omega</math> that have paths <math>x_1</math> and <math>x_2</math> emitted from different sources that meet together at a point. The net wave at that point can then be given as the real part of the equation

<math>
\psi=Ae^{i(kx_1-\omega t}+Ae^{i(kx_2-\omega t)}=Ae^{-i\omega t}e^{ik\frac{(x_1+x_2)}{2}}[e^{ik\frac{(x_1-x_2)}{2}}e^{-ik\frac{(x_1-x_2)}{2}}]=2Acos[k(\frac{x_1-x_2}{2})]e^{i[k\frac{(x_1+x_2)}{2}-\omega t]}
</math>

where the wave number <math>k=2\pi/\lambda</math>, <math>\lambda</math> is the wavelength of the waves, and c is the phase velocity. The time-averaged intensity of the combined wave is then proportional to the square of the its amplitude. That is,

<math>
I\propto cos^2[\pi(\frac{p_2-p_1}{\lambda})]=cos^2[2\pi(\frac{p_2}{\lambda}-\theta)]
</math>

where <math>\theta</math> is a constant. Now suppose the mirror that the <math>p_2</math> beam is incident to oscillates harmonically back and forth with a frequency <math>f</math> and a displacement <math>B</math>. Then, <math>p_2 =p_2(0)+Bsin(2\pi ft)</math> and the intensity of the detected light varies with time as

<math>
I=A^2cos^2[2\pi(\frac{Bsin(2\pi ft)}{\lambda}-\theta')]
</math>

Note that the frequency of oscillations is itself a sinusoid, meaning that there will be turnaround points in the oscillations.

== Experimental setup ==
The Michelson interferometer consists of a beam splitter BS which is used to split the laser beam into two beams and two mirrors M1 and M2. The reflected beam goes to a fixed mirror M1 while the transmitted beam goes to the movable mirror M2. A transducer is attached to the mirror M2. The transducer is powered by a signal generator that is used to vary the vibrating frequency of the mirror. A photodiode PD is used to detect the signal of interference fringes. The schematic Michelson interferometer is shown in Figure 1.
[[File:Schematic Michelson Interferometer.jpeg|center|thumb|600px|Figure 1: Schematic figure of Michelson interferometer. LD is a laser diode, BS is a beam splitter, M1 and M2 are mirrors, PD is a photodiode, and vibration transducer which is driven by a signal generator.]]

The experimental setup Michelson interferometer is shown in figure 2.
[[File:Experimental setup Michelson Interferometer.jpeg|center|thumb|600px|Figure 2: Experimental setup of Michelson interferometer.]]

The interference fringes of the Michelson interferometer are observed when the transducer is turned off in Figure 3.
[[File:Michelson interference fringes.jpeg|center|thumb|600px|Figure 3: Interference fringes which are observed when the transducer is turned off]]

==Experiment Results==
After the alignment of the optical setup, we applied sinusoidal voltage to the transducer attached to the mirror tightly. intensity fluctuation is recorded by the homemade photodiode. Data from the photodiode is collected by the oscilloscope. The typical figure from the oscilloscope is shown in Figure 4. According to the equation mentioned before, the number of peaks during each period corresponds to the radio of displacement and laser wavelength. In this case, we can measure the displacement for various frequencies. The frequency range is from 86 Hz to 130 Hz.

[[File:ScopeScreen.jpeg|center|thumb|600px|Figure 4: The display on the oscilloscope at <math>f = 103Hz </math>. After applying the voltage to the transducer, the voltage output from photodiode changes with time.]]

In our setup, we measured the wavelength of Mitsubishi Laser Diode which is 658.9 nm. The wavemeter measurement results are shown here.
[[File:Wavelength.jpeg|center|thumb|600px|Figure 5: The Wavelength measurement of the Mitsubishi Laser Diode.]]

We plot the displacement for different frequencies with the errorbar in Figure 6. Uncertainty is the square root uncertainty of the count number. For a driven, damped and harmonic oscillator, the relation between displacement and frequency is

<math>|A|=\frac{f_{0}}{\sqrt{\left(\omega_{0}^{2}-\omega^{2}\right)^{2}+4 \beta^{2} \omega^{2}}} </math>

Where <math>f_{0}</math> is the amplitude, <math>\omega_{0}</math> is the resonance frequency, and <math>\beta_{0}</math> is a constant. With this equation, we can fit the frequency response curve. Because there is only one peak or valley for a frequency lower than 92 Hz or higher than 120 Hz, and the error is significant and non-negligible. We exclude those in the fitting process. Finally, the fitting root mean squared error (RMSE) is 130.38. Fitted <math>\beta</math> is <math>1.3 \pm 0.2</math>. The fitting resonance frequency <math>\omega_{0} </math> is <math>113.1 \pm 0.1Hz </math>.

[[File:FitPeakNum.jpeg|center|thumb|600px|Figure 6: Frequency response curve for the transducer system with errorbar. The blue dots represent the displacement measured with different frequencies. Error bars are the square root uncertainty. Red line is the fitting curve for a driven, damped, harmonic oscillator.]]

==Discussion==
Our final result fits our data to the best Lorentzian Function. When we say best we talk about the function with the lowest Root Mean Square Error (RMSE).

RMSE is defined as follow:

<math>\operatorname{RMSE}=\sqrt{\frac{1}{n}\sum_{i=1}^n(A_i-\hat{A_i})^2}</math>

where <math>A_i</math> is the measured displacement and <math>\hat{A_i}</math> is the predicted displacement by the function model.

The core of this method is calculate the distant between the measured and predicted values and take the root mean of them.

== Conclusion ==
We have shown indeed that we can characterize the resonance frequency of a system. For this particular application, we found a frequency of 113.12Hz for the system transducer plus mirror. Finally, with appropriate handling, this technique could be implemented to characterize more sofisticated systems.

==References==
[1] Freschi, A. A., et al. "Laser interferometric characterization of a vibrating speaker system." American Journal of Physics 71.11 (2003): 1121-1126.
https://doi.org/10.1119/1.1586262

[2] Skarha, Matthew. "Laser interferometric characterization for a vibrating speaker system."
http://mattskarha.com/assets/docs/Laser%20interferometric%20characterization%20for%20a%20vibrating%20speaker%20system%20good.pdf

[3] Pathare, Shirish, and Vikrant Kurmude. "Low cost Michelson–Morley interferometer." Physics Education 51.6 (2016): 063001.
https://doi.org/10.1088/0031-9120/51/6/063001

Resonance frequency measurement using a interferometric method

2022-04-30T03:46:39Z

Aucca: /* Experiment Results */

== Team Members ==
* [[User:Nakarin|Nakarin Jayjong]]
* [[User:Aucca|Joel Auccapuclla]]
* [[User:Xiaoyu|Xiaoyu Nie]]
* [[User:Haotian|Haotian Song]]

== Introduction ==
Interferometers since their invention have been a great tool for experimental physics. Applications cover a wide spectrum of fields such as astronomy, fiber optics, seismology, seismology, velocimetry to say the least. The precision of measurement of small displacements is the key factor, take LIGO as an example. Another important factor is the convenience to set up an interferometer and the low complexity in terms of equipment. In this experiment, we will measure the resonance frequency of a vibrating transducer using a Michelson interferometer. Basically, a transducer is attached to one optical path, to one mirror, of the Michelson interferometer and it is expected to see a peak in the response curve for the resonance frequency of the transducer plus mirror.

== Theory ==
Our Michelson interferometer setup used a laser, a beam splitter, two mirrors, and a photometer, illustrated in Fig. 1. Original models used sodium flame as the coherent light source. The light propagates from the source to the beam splitter where approximately 50% of the incident light is transmitted and approximately 50% of it is reflected at a right angle. The transmitted light continues to propagate as the reference plane wave to the stationary mirror where it is fully reflected back towards the beam splitter. The reflected light propagates as the target plane wave to the mirror attached to the oscillating loudspeaker. The two waves then recombine at the beam splitter and interfere based on their relative phases based on the idea that the phase shift of the target plane wave varies with the instantaneous position of the speaker mirror combination. Consider a situation in which we setup our interferometer such that no phase shift corresponds to the equilibrium position of the speaker-mirror combination. At this position, we expect constructive interference at the photometer. Likewise, for every λ/2 change in displacement, we expect constructive interference at the photometer because the path length travelled by the target plane wave varies by some integral multiple of λ. Thus, the phase shift is an even multiple of π and the two waves remain in phase. Similarly, when the path length traveled is some odd multiple of λ/4, the phase shift is an odd multiple of π and the two waves interfere destructively.

Consider two coherent waves with the same amplitude A and same frequency <math>\omega</math> that have paths <math>x_1</math> and <math>x_2</math> emitted from different sources that meet together at a point. The net wave at that point can then be given as the real part of the equation

<math>
\psi=Ae^{i(kx_1-\omega t}+Ae^{i(kx_2-\omega t)}=Ae^{-i\omega t}e^{ik\frac{(x_1+x_2)}{2}}[e^{ik\frac{(x_1-x_2)}{2}}e^{-ik\frac{(x_1-x_2)}{2}}]=2Acos[k(\frac{x_1-x_2}{2})]e^{i[k\frac{(x_1+x_2)}{2}-\omega t]}
</math>

where the wave number <math>k=2\pi/\lambda</math>, <math>\lambda</math> is the wavelength of the waves, and c is the phase velocity. The time-averaged intensity of the combined wave is then proportional to the square of the its amplitude. That is,

<math>
I\propto cos^2[\pi(\frac{p_2-p_1}{\lambda})]=cos^2[2\pi(\frac{p_2}{\lambda}-\theta)]
</math>

where <math>\theta</math> is a constant. Now suppose the mirror that the <math>p_2</math> beam is incident to oscillates harmonically back and forth with a frequency <math>f</math> and a displacement <math>B</math>. Then, <math>p_2 =p_2(0)+Bsin(2\pi ft)</math> and the intensity of the detected light varies with time as

<math>
I=A^2cos^2[2\pi(\frac{Bsin(2\pi ft)}{\lambda}-\theta')]
</math>

Note that the frequency of oscillations is itself a sinusoid, meaning that there will be turnaround points in the oscillations.

== Experimental setup ==
The Michelson interferometer consists of a beam splitter BS which is used to split the laser beam into two beams and two mirrors M1 and M2. The reflected beam goes to a fixed mirror M1 while the transmitted beam goes to the movable mirror M2. A transducer is attached to the mirror M2. The transducer is powered by a signal generator that is used to vary the vibrating frequency of the mirror. A photodiode PD is used to detect the signal of interference fringes. The schematic Michelson interferometer is shown in Figure 1.
[[File:Schematic Michelson Interferometer.jpeg|center|thumb|600px|Figure 1: Schematic figure of Michelson interferometer. LD is a laser diode, BS is a beam splitter, M1 and M2 are mirrors, PD is a photodiode, and vibration transducer which is driven by a signal generator.]]

The experimental setup Michelson interferometer is shown in figure 2.
[[File:Experimental setup Michelson Interferometer.jpeg|center|thumb|600px|Figure 2: Experimental setup of Michelson interferometer.]]

The interference fringes of the Michelson interferometer are observed when the transducer is turned off in Figure 3.
[[File:Michelson interference fringes.jpeg|center|thumb|600px|Figure 3: Interference fringes which are observed when the transducer is turned off]]

==Experiment Results==
After the alignment of the optical setup, we applied sinusoidal voltage to the transducer attached to the mirror tightly. intensity fluctuation is recorded by the homemade photodiode. Data from the photodiode is collected by the oscilloscope. The typical figure from the oscilloscope is shown in Figure 4. According to the equation mentioned before, the number of peaks during each period corresponds to the radio of displacement and laser wavelength. In this case, we can measure the displacement for various frequencies. The frequency range is from 86 Hz to 130 Hz.

[[File:ScopeScreen.jpeg|center|thumb|600px|Figure 4: The display on the oscilloscope at <math>f = 103Hz </math>. After applying the voltage to the transducer, the voltage output from photodiode changes with time.]]

In our setup, we measured the wavelength of Mitsubishi Laser Diode which is 658.9 nm. The wavemeter measurement results are shown here.
[[File:Wavelength.jpeg|center|thumb|600px|Figure 5: The Wavelength measurement of the Mitsubishi Laser Diode.]]

We plot the displacement for different frequencies with the errorbar in Figure 6. Uncertainty is the square root uncertainty of the count number. For a driven, damped and harmonic oscillator, the relation between displacement and frequency is

<math>|A|=\frac{f_{0}}{\sqrt{\left(\omega_{0}^{2}-\omega^{2}\right)^{2}+4 \beta^{2} \omega^{2}}} </math>

Where <math>f_{0}</math> is the amplitude, <math>\omega_{0}</math> is the resonance frequency, and <math>\beta_{0}</math> is a constant. With this equation, we can fit the frequency response curve. Because there is only one peak or valley for a frequency lower than 92 Hz or higher than 120 Hz, and the error is significant and non-negligible. We exclude those in the fitting process. Finally, the fitting root mean squared error (RMSE) is 130.38. Fitted <math>\beta</math> is <math>1.3 \pm 0.2</math>. The fitting resonance frequency <math>\omega_{0} </math> is <math>113.1 \pm 0.1Hz </math>.

[[File:FitPeakNum.jpeg|center|thumb|600px|Figure 6: Frequency response curve for the transducer system with errorbar. The blue dots represent the displacement measured with different frequencies. Error bars are the square root uncertainty. Red line is the fitting curve for a driven, damped, harmonic oscillator.]]

==Discussion==

== Conclusion ==
We have shown indeed that we can characterize the resonance frequency of a system. For this particular application, we found a frequency of 113.12Hz for the system transducer plus mirror. Finally, with appropriate handling, this technique could be implemented to characterize more sofisticated systems.

==References==
[1] Freschi, A. A., et al. "Laser interferometric characterization of a vibrating speaker system." American Journal of Physics 71.11 (2003): 1121-1126.
https://doi.org/10.1119/1.1586262

[2] Skarha, Matthew. "Laser interferometric characterization for a vibrating speaker system."
http://mattskarha.com/assets/docs/Laser%20interferometric%20characterization%20for%20a%20vibrating%20speaker%20system%20good.pdf

[3] Pathare, Shirish, and Vikrant Kurmude. "Low cost Michelson–Morley interferometer." Physics Education 51.6 (2016): 063001.
https://doi.org/10.1088/0031-9120/51/6/063001

Resonance frequency measurement using a interferometric method

2022-04-30T03:44:42Z

Aucca:

== Team Members ==
* [[User:Nakarin|Nakarin Jayjong]]
* [[User:Aucca|Joel Auccapuclla]]
* [[User:Xiaoyu|Xiaoyu Nie]]
* [[User:Haotian|Haotian Song]]

== Introduction ==
Interferometers since their invention have been a great tool for experimental physics. Applications cover a wide spectrum of fields such as astronomy, fiber optics, seismology, seismology, velocimetry to say the least. The precision of measurement of small displacements is the key factor, take LIGO as an example. Another important factor is the convenience to set up an interferometer and the low complexity in terms of equipment. In this experiment, we will measure the resonance frequency of a vibrating transducer using a Michelson interferometer. Basically, a transducer is attached to one optical path, to one mirror, of the Michelson interferometer and it is expected to see a peak in the response curve for the resonance frequency of the transducer plus mirror.

== Theory ==
Our Michelson interferometer setup used a laser, a beam splitter, two mirrors, and a photometer, illustrated in Fig. 1. Original models used sodium flame as the coherent light source. The light propagates from the source to the beam splitter where approximately 50% of the incident light is transmitted and approximately 50% of it is reflected at a right angle. The transmitted light continues to propagate as the reference plane wave to the stationary mirror where it is fully reflected back towards the beam splitter. The reflected light propagates as the target plane wave to the mirror attached to the oscillating loudspeaker. The two waves then recombine at the beam splitter and interfere based on their relative phases based on the idea that the phase shift of the target plane wave varies with the instantaneous position of the speaker mirror combination. Consider a situation in which we setup our interferometer such that no phase shift corresponds to the equilibrium position of the speaker-mirror combination. At this position, we expect constructive interference at the photometer. Likewise, for every λ/2 change in displacement, we expect constructive interference at the photometer because the path length travelled by the target plane wave varies by some integral multiple of λ. Thus, the phase shift is an even multiple of π and the two waves remain in phase. Similarly, when the path length traveled is some odd multiple of λ/4, the phase shift is an odd multiple of π and the two waves interfere destructively.

Consider two coherent waves with the same amplitude A and same frequency <math>\omega</math> that have paths <math>x_1</math> and <math>x_2</math> emitted from different sources that meet together at a point. The net wave at that point can then be given as the real part of the equation

<math>
\psi=Ae^{i(kx_1-\omega t}+Ae^{i(kx_2-\omega t)}=Ae^{-i\omega t}e^{ik\frac{(x_1+x_2)}{2}}[e^{ik\frac{(x_1-x_2)}{2}}e^{-ik\frac{(x_1-x_2)}{2}}]=2Acos[k(\frac{x_1-x_2}{2})]e^{i[k\frac{(x_1+x_2)}{2}-\omega t]}
</math>

where the wave number <math>k=2\pi/\lambda</math>, <math>\lambda</math> is the wavelength of the waves, and c is the phase velocity. The time-averaged intensity of the combined wave is then proportional to the square of the its amplitude. That is,

<math>
I\propto cos^2[\pi(\frac{p_2-p_1}{\lambda})]=cos^2[2\pi(\frac{p_2}{\lambda}-\theta)]
</math>

where <math>\theta</math> is a constant. Now suppose the mirror that the <math>p_2</math> beam is incident to oscillates harmonically back and forth with a frequency <math>f</math> and a displacement <math>B</math>. Then, <math>p_2 =p_2(0)+Bsin(2\pi ft)</math> and the intensity of the detected light varies with time as

<math>
I=A^2cos^2[2\pi(\frac{Bsin(2\pi ft)}{\lambda}-\theta')]
</math>

Note that the frequency of oscillations is itself a sinusoid, meaning that there will be turnaround points in the oscillations.

== Experimental setup ==
The Michelson interferometer consists of a beam splitter BS which is used to split the laser beam into two beams and two mirrors M1 and M2. The reflected beam goes to a fixed mirror M1 while the transmitted beam goes to the movable mirror M2. A transducer is attached to the mirror M2. The transducer is powered by a signal generator that is used to vary the vibrating frequency of the mirror. A photodiode PD is used to detect the signal of interference fringes. The schematic Michelson interferometer is shown in Figure 1.
[[File:Schematic Michelson Interferometer.jpeg|center|thumb|600px|Figure 1: Schematic figure of Michelson interferometer. LD is a laser diode, BS is a beam splitter, M1 and M2 are mirrors, PD is a photodiode, and vibration transducer which is driven by a signal generator.]]

The experimental setup Michelson interferometer is shown in figure 2.
[[File:Experimental setup Michelson Interferometer.jpeg|center|thumb|600px|Figure 2: Experimental setup of Michelson interferometer.]]

The interference fringes of the Michelson interferometer are observed when the transducer is turned off in Figure 3.
[[File:Michelson interference fringes.jpeg|center|thumb|600px|Figure 3: Interference fringes which are observed when the transducer is turned off]]

==Experiment Results==
After the alignment of the optical setup, we applied sinusoidal voltage to the transducer attached to the mirror tightly. intensity fluctuation is recorded by the homemade photodiode. Data from the photodiode is collected by the oscilloscope. The typical figure from the oscilloscope is shown in Figure 4. According to the equation mentioned before, the number of peaks during each period corresponds to the radio of displacement and laser wavelength. In this case, we can measure the displacement for various frequencies. The frequency range is from 86 Hz to 130 Hz.

[[File:ScopeScreen.jpeg|center|thumb|600px|Figure 4: The display on the oscilloscope at <math>f = 103Hz </math>. After applying the voltage to the transducer, the voltage output from photodiode changes with time.]]

In our setup, we measured the wavelength of Mitsubishi Laser Diode which is 658.9 nm. The wavemeter measurement results are shown here.
[[File:Wavelength.jpeg|center|thumb|600px|Figure 5: The Wavelength measurement of the Mitsubishi Laser Diode.]]

We plot the displacement for different frequencies with the errorbar in Figure 6. Uncertainty is the square root uncertainty of the count number. For a driven, damped and harmonic oscillator, the relation between displacement and frequency is

<math>|A|=\frac{f_{0}}{\sqrt{\left(\omega_{0}^{2}-\omega^{2}\right)^{2}+4 \beta^{2} \omega^{2}}} </math>

Where <math>f_{0}</math> is the amplitude, <math>\omega_{0}</math> is the resonance frequency, and <math>\beta_{0}</math> is a constant. With this equation, we can fit the frequency response curve. Because there is only one peak or valley for a frequency lower than 92 Hz or higher than 120 Hz, and the error is significant and non-negligible. We exclude those in the fitting process. Finally, the fitting root mean squared error is 130.38. Fitted <math>\beta</math> is <math>1.3 \pm 0.2</math>. The fitting resonance frequency <math>\omega_{0} </math> is <math>113.1 \pm 0.1Hz </math>.

[[File:FitPeakNum.jpeg|center|thumb|600px|Figure 6: Frequency response curve for the transducer system with errorbar. The blue dots represent the displacement measured with different frequencies. Error bars are the square root uncertainty. Red line is the fitting curve for a driven, damped, harmonic oscillator.]]

==Discussion==

== Conclusion ==
We have shown indeed that we can characterize the resonance frequency of a system. For this particular application, we found a frequency of 113.12Hz for the system transducer plus mirror. Finally, with appropriate handling, this technique could be implemented to characterize more sofisticated systems.

==References==
[1] Freschi, A. A., et al. "Laser interferometric characterization of a vibrating speaker system." American Journal of Physics 71.11 (2003): 1121-1126.
https://doi.org/10.1119/1.1586262

[2] Skarha, Matthew. "Laser interferometric characterization for a vibrating speaker system."
http://mattskarha.com/assets/docs/Laser%20interferometric%20characterization%20for%20a%20vibrating%20speaker%20system%20good.pdf

[3] Pathare, Shirish, and Vikrant Kurmude. "Low cost Michelson–Morley interferometer." Physics Education 51.6 (2016): 063001.
https://doi.org/10.1088/0031-9120/51/6/063001

Resonance frequency measurement using a interferometric method

2022-04-30T03:27:38Z

Aucca: Created page with "== Team Members == * Nakarin Jayjong * Joel Auccapuclla * Xiaoyu Nie * Haotian Song == Introduction == Interferometers since their invention have been a great tool for experimental physics. Applications cover a wide spectrum of fields such as astronomy, fiber optics, seismology, seismology, velocimetry to say the least. The precision of measurement of small displacements is the key factor, take LIGO as an..."

== Team Members ==
* [[User:Nakarin|Nakarin Jayjong]]
* [[User:Aucca|Joel Auccapuclla]]
* [[User:Xiaoyu|Xiaoyu Nie]]
* [[User:Haotian|Haotian Song]]

== Introduction ==
Interferometers since their invention have been a great tool for experimental physics. Applications cover a wide spectrum of fields such as astronomy, fiber optics, seismology, seismology, velocimetry to say the least. The precision of measurement of small displacements is the key factor, take LIGO as an example. Another important factor is the convenience to set up an interferometer and the low complexity in terms of equipment. In this experiment, we will measure the resonance frequency of a vibrating transducer using a Michelson interferometer. Basically, a transducer is attached to one optical path, to one mirror, of the Michelson interferometer and it is expected to see a peak in the response curve for the resonance frequency of the transducer plus mirror.

== Theory ==
Our Michelson interferometer setup used a laser, a beam splitter, two mirrors, and a photometer, illustrated in Fig. 1. Original models used sodium flame as the coherent light source. The light propagates from the source to the beam splitter where approximately 50% of the incident light is transmitted and approximately 50% of it is reflected at a right angle. The transmitted light continues to propagate as the reference plane wave to the stationary mirror where it is fully reflected back towards the beam splitter. The reflected light propagates as the target plane wave to the mirror attached to the oscillating loudspeaker. The two waves then recombine at the beam splitter and interfere based on their relative phases based on the idea that the phase shift of the target plane wave varies with the instantaneous position of the speaker mirror combination. Consider a situation in which we setup our interferometer such that no phase shift corresponds to the equilibrium position of the speaker-mirror combination. At this position, we expect constructive interference at the photometer. Likewise, for every λ/2 change in displacement, we expect constructive interference at the photometer because the path length travelled by the target plane wave varies by some integral multiple of λ. Thus, the phase shift is an even multiple of π and the two waves remain in phase. Similarly, when the path length traveled is some odd multiple of λ/4, the phase shift is an odd multiple of π and the two waves interfere destructively.

Consider two coherent waves with the same amplitude A and same frequency <math>\omega</math> that have paths <math>x_1</math> and <math>x_2</math> emitted from different sources that meet together at a point. The net wave at that point can then be given as the real part of the equation

<math>
\psi=Ae^{i(kx_1-\omega t}+Ae^{i(kx_2-\omega t)}=Ae^{-i\omega t}e^{ik\frac{(x_1+x_2)}{2}}[e^{ik\frac{(x_1-x_2)}{2}}e^{-ik\frac{(x_1-x_2)}{2}}]=2Acos[k(\frac{x_1-x_2}{2})]e^{i[k\frac{(x_1+x_2)}{2}-\omega t]}
</math>

where the wave number <math>k=2\pi/\lambda</math>, <math>\lambda</math> is the wavelength of the waves, and c is the phase velocity. The time-averaged intensity of the combined wave is then proportional to the square of the its amplitude. That is,

<math>
I\propto cos^2[\pi(\frac{p_2-p_1}{\lambda})]=cos^2[2\pi(\frac{p_2}{\lambda}-\theta)]
</math>

where <math>\theta</math> is a constant. Now suppose the mirror that the <math>p_2</math> beam is incident to oscillates harmonically back and forth with a frequency <math>f</math> and a displacement <math>B</math>. Then, <math>p_2 =p_2(0)+Bsin(2\pi ft)</math> and the intensity of the detected light varies with time as

<math>
I=A^2cos^2[2\pi(\frac{Bsin(2\pi ft)}{\lambda}-\theta')]
</math>

Note that the frequency of oscillations is itself a sinusoid, meaning that there will be turnaround points in the oscillations.

== Experimental setup ==
The Michelson interferometer consists of a beam splitter BS which is used to split the laser beam into two beams and two mirrors M1 and M2. The reflected beam goes to a fixed mirror M1 while the transmitted beam goes to the movable mirror M2. A transducer is attached to the mirror M2. The transducer is powered by a signal generator that is used to vary the vibrating frequency of the mirror. A photodiode PD is used to detect the signal of interference fringes. The schematic Michelson interferometer is shown in Figure 1.
[[File:Schematic Michelson Interferometer.jpeg|center|thumb|600px|Figure 1: Schematic figure of Michelson interferometer. LD is a laser diode, BS is a beam splitter, M1 and M2 are mirrors, PD is a photodiode, and vibration transducer which is driven by a signal generator.]]

The experimental setup Michelson interferometer is shown in figure 2.
[[File:Experimental setup Michelson Interferometer.jpeg|center|thumb|600px|Figure 2: Experimental setup of Michelson interferometer.]]

The interference fringes of the Michelson interferometer are observed when the transducer is turned off in Figure 3.
[[File:Michelson interference fringes.jpeg|center|thumb|600px|Figure 3: Interference fringes which are observed when the transducer is turned off]]

==Experiment Results==
After the alignment of the optical setup, we applied sinusoidal voltage to the transducer attached to the mirror tightly. intensity fluctuation is recorded by the homemade photodiode. Data from the photodiode is collected by the oscilloscope. The typical figure from the oscilloscope is shown in Figure 4. According to the equation mentioned before, the number of peaks during each period corresponds to the radio of displacement and laser wavelength. In this case, we can measure the displacement for various frequencies. The frequency range is from 86 Hz to 130 Hz.

[[File:ScopeScreen.jpeg|center|thumb|600px|Figure 4: The display on the oscilloscope at <math>f = 103Hz </math>. After applying the voltage to the transducer, the voltage output from photodiode changes with time.]]

In our setup, we measured the wavelength of Mitsubishi Laser Diode which is 658.9 nm. The wavemeter measurement results are shown here.
[[File:Wavelength.jpeg|center|thumb|600px|Figure 5: The Wavelength measurement of the Mitsubishi Laser Diode.]]

We plot the displacement for different frequencies with the errorbar in Figure 6. Uncertainty is the square root uncertainty of the count number. For a driven, damped and harmonic oscillator, the relation between displacement and frequency is

<math>|A|=\frac{f_{0}}{\sqrt{\left(\omega_{0}^{2}-\omega^{2}\right)^{2}+4 \beta^{2} \omega^{2}}} </math>

Where <math>f_{0}</math> is the amplitude, <math>\omega_{0}</math> is the resonance frequency, and <math>\beta_{0}</math> is a constant. With this equation, we can fit the frequency response curve. Because there is only one peak or valley for a frequency lower than 92 Hz or higher than 120 Hz, and the error is significant and non-negligible. We exclude those in the fitting process. Finally, the fitting root mean squared error is 130.38. Fitted <math>\beta</math> is <math>1.3 \pm 0.2</math>. The fitting resonance frequency <math>\omega_{0} </math> is <math>113.1 \pm 0.1Hz </math>.

[[File:FitPeakNum.jpeg|center|thumb|600px|Figure 6: Frequency response curve for the transducer system with errorbar. The blue dots represent the displacement measured with different frequencies. Error bars are the square root uncertainty. Red line is the fitting curve for a driven, damped, harmonic oscillator.]]

== Conclusion ==
We have shown indeed that we can characterize the resonance frequency of a system. For this particular application, we found a frequency of 113.12Hz for the system transducer plus mirror. Finally, with appropriate handling, this technique could be implemented to characterize more sofisticated systems.

==References==
[1] Freschi, A. A., et al. "Laser interferometric characterization of a vibrating speaker system." American Journal of Physics 71.11 (2003): 1121-1126.
https://doi.org/10.1119/1.1586262

[2] Skarha, Matthew. "Laser interferometric characterization for a vibrating speaker system."
http://mattskarha.com/assets/docs/Laser%20interferometric%20characterization%20for%20a%20vibrating%20speaker%20system%20good.pdf

[3] Pathare, Shirish, and Vikrant Kurmude. "Low cost Michelson–Morley interferometer." Physics Education 51.6 (2016): 063001.
https://doi.org/10.1088/0031-9120/51/6/063001

Main Page

2022-04-30T03:26:22Z

Aucca: /* An interferometric method for measuring the resonance frequency of a vibrating system */

<strong>MediaWiki has been installed.</strong>
Welcome to the main page for the PC5214 graduate module AY2122, Sem2</strong>.
Here, we leave project descriptions, literature references, and other collateral information. You will need to create an account in class to obtain write access.

Actual lecture locations will be placed here until we have reached a stable state. If you are interested and have not been able to register, please send me an email (if you have not done so already) to [mailto:phyck@nus.edu.sg phyck@nus.edu.sg].

'''<span style="color:#ff0000">Deadline for all the wiki entries will be 30 April 2022, 23:59! Please ensure that you hare happy with your entries by that time!</span>'''

Cheers, Christian

This page is currently set up.

==Lab spaces==
* '''S11-02-04''' (next to physics dept resource room). This is where most optics-related projects should go.
* '''S12 level 4''', "year 1 teaching lab", back room, "vanderGraff lab". This is perhaps were non-optics related projects would fit.
* '''S13-01-??''' (blue door: former accoustics lab). Not sure yet who could go there, but it is a really really quiet place!
* anything else you have access to

==Projects==
Please leave a a link to your project page (or pages) here, and leave a short description what this is about. Write the '''stuff you need''' under the description too.

===[[Project 1 (example)]]===
Keep a very brief description of a project or even a suggestion here, and perhaps the names of the team members, or who to contact if there is interest to join.

===[[Confocal Microscopy]]===
Team Members: Wang Tingyu, Xue Rui, Yang Hengxing

A Confocal Microscopy or Confocal Laser Scanning Microscopy (CLSM) uses pinhole to block out all out of focus light to enhance optical resolution, very different from traditional wide-field fluorescence microscopes. To offset the block of out of focus lights, the light intensity is detected by a photomultiplier tube or avalanche photodiode, which transforms the light signal into an electrical one. We will try to build a Setup like this to enhance optical resolution and maybe get profile information about the sample.

===[[Resonance frequency measurement using a interferometric method]]===

Members: [[User:Nakarin|Nakarin Jayjong]], [[User:Aucca|Joel Auccapuclla]], [[User:Xiaoyu|Xiaoyu Nie]], [[User:Haotian|Haotian Song]].

In this project, the resonance frequency of the vibrating system namely the vibration transducer is measured using a Michelson interferometer.

===[[Homodyne detection]]===
Proposed By: [[User:Johnkhootf|John Khoo]]

Team Members: [[User:Johnkhootf|John Khoo]], [[User:Xie_Chengkun|Xie Chengkun]]

[https://en.wikipedia.org/wiki/Homodyne_detection ''Optical'' homodyne detection] is a method for detecting messages transmitted in optical signals, where a frequency or phase modulated signal is compared to what is misleadingly called the "local oscillator" (LO) signal, which is generated from the same source but not modulated with the message. In order to probe quantum effects, it is important to bring the noise of the detector down to the [https://en.wikipedia.org/wiki/Shot_noise ''shot-noise limit''], where the only fluctuations observed arise from the discrete nature of photons, which can be theoretically modelled as the vacuum-state fluctuations of the quantised electromagnetic field. This project's first objective is to build a homodyne detector from scratch.

'''Stuff we need''': Acousto-optical modulator, electro-optical modulator, transformer to control EOM, photodiodes, current-to-voltage converter (I'm not sure what this is - can we just use a resistor connected to the ground and measure the voltage?), Raspberry Pi (I hope the ADC is good enough for this), mirrors and beamsplitters

===[[Laser Microphone]]===
Team Members: [[User:Nicholas cjl|Nicholas Chong Jia Le]], [[User:Marcuslow|Marcus Low Zuo Wu]]

A laser spot illuminating a vibrating surface should move along with it, and tracking the motion of the spot should theoretically allow us to retrieve some of the information regarding the vibrations of the surface. If a loud enough sound causes the surface to vibrate, this should theoretically be enough for the transmission of audio information through visual means. The signal obtained will then be put through a few different digital signal processing techniques in an attempt to retrieve a (good enough) copy of the original audio.

===[[Argon gas discharge lamp]]===
Proposed By: Park Kun Hee

Team Members: Park Kun Hee, Yang Jincheng, Qin Jingwen

By applying a sufficiently high DC voltage across a gas, the gas atoms/molecules are ionised by the strong electric field.
In this project, we construct an Argon-based gas discharge lamp, with adjustable pressure and voltage.
The breakdown voltage of Argon gas with respect to pressure changes is observed, and compared with [https://en.wikipedia.org/wiki/Paschen%27s_law Paschen's law].
We also observe changes in the spectroscopic properties of the plasma with varying pressure.

===[[Characterization of Single Photon Counters]]===
Proposed By: Yeo Zhen Yuan

The project is to characterize an Avalanche PhotoDiode (APD) and compare its efficiency with commercial counterparts like [https://www.digikey.com/en/products/detail/excelitas-technologies/SPCM-AQRH-10-FC/6235280 this device]. It works based on the photoelectric effect to turn incident photon into photoelectron. This photoelectron is then accelerated in an electric field to produce cascading electrons and this "electron avalanche" is detected as a spike in the current. Analog signals will need to be processed via custom electronics and ultimately provide a digital readout. Current commercial detectors boast 50% Photon Detector Efficiency (PDE) at room temperature and that will be our goal. They typically cost $2000-$5000 which seems over-priced and ready for disruption. Liquid nitrogen temperatures may be needed to see how large a PDE we can get.

What is SPCM good for? Copied from the datasheet/brochure: LIDAR, Quantum Cryptography, Photon correlation spectroscopy, Astronomical observation, Optical range finding, Adaptive optics, Ultra-sensitive fluorescence, Particle sizing, Microscopy. So maybe this would become a toy/tool for next year's students.

[NEW CAPABILITY]

High throughput Oscilloscope data collection. ~700 "screenshots" per minute. Demonstration on APD, 10K screenshots of 2 Channel Digital Oscilloscope [https://github.com/zhenyuan992/OpenWave-1KB/raw/88a85a7f18741b370563b03d87a53f913b714a4c/src/results03_01_apdvoltage.png].

Semi-seamless data collection [https://www.tek.com/en/support/faqs/can-i-use-my-oscilloscope-do-data-logging].

===[[Kerr Microscope]]===
Proposed By: Sim May Inn (write up by Joel Yeo)

'''Team members: Gan Jun Herng, Joel Yeo, Sim May Inn'''

'''Project Location: S11-02-04'''

Imaging a sample can be done in many ways, depending on the light-matter interaction we are interested in observing. The magneto-optic Kerr effect describes the change in polarization and intensity of incident light when it impinges on the surface of a magnetic material. The resultant reflected light can then form an image through focusing optics which provides high contrast between areas of different magnetization.

In this project, we will be aiming to build a basic Kerr microscope using off-the-shelf polarizers, objectives, detectors and laser source. An example of a magnetic sample is the magnetic tape from an old school cassette tape. To increase the field of view, we also plan to incorporate automatic raster scanning of the sample through means of an Arduino-controlled sample stage.

'''Items needed (as of 28 Feb 2022):'''
* Light source (visibile wavelength): <s> Laser, LED </s>, laser diode
* <s> Linear polarizer (sheet) x 2Camera (CCD/CMOS) </s>
* <s> Non-polarizing beam splitter </s>
* <s> Camera (CCD/CMOS) </s>
* <s> Pinhole/aperture </s>
* <s> Magnetic samples for Kerr microscopy (eg. Magnetic film, magnets, ferromagnetic materials) </s>
* Arduino
* Microscope stage
* Piezoelectrics (?) for moving stage

===[[Electron Gun]]===
Team Members: Aliki Sofia Rotelli, Lai Tian Hao, Lim En Liang Irvin, Tan Chuan Jie

The purpose of this project is to design and build an electron gun from the initial concept in order to create a detectable electron beam through the use of a phosphor-coated screen. Additionally, the beam current will be examined in order to better define the devices' capabilities. Mass spectrometry, x-ray production for linear accelerators, and electron-beam lithography are just a few of the applications for electron gun technology.

===[[Smoke detection in air]]===
Team Members: Cheng De Hao, Huang Hai Tao, Wang Zheng Yu

Using detector to detect the scattering light and amplify the signal by using the lock-in amplifier.

===[[Anti-glare LCD]]===
Team members: Zhang Yuanyuan, Ming Xiaohan, Han Shixin

As s bad lighting phenomenon, glare phonomenon brings inconvenience to all aspects of human life, especially people's access to information on instruments. In order to suppress glare effectively, anti-glare film is put into research. The common anti-glare film in the market is an optical film using the principle of optical scattering, but it can not adapt to the change of light environment in time, which has some limitations in practical application. In this study, a two-dimensional barcode micro-region orientation structure, based on the characteristics of liquid crystal, namely a random grating structure, was designed by simulation in the lab and using MATLAB software, and its optional parameters were searched.

===[[Custom atomic beam source]]===
Team Members: Lu Tiangao, Li Putian

===[[Schlieren Imaging]]===
Team members: Zhang Xingjian, Du Jinyi

We built a Schlieren imaging setup and saw the airflow generated by the lighter, the heat of the hand, and the blow. Other than that, we also make a high-frequency blinking light source to "stop" the 40 kHz ultrasound wave generated by an ultrasonic speaker and captured it by the Schlieren imaging setup.

===[[Contactless Conductivity Measurement]]===

Team members: Chen Guohao, Jiang Luwen

The purpose of this project is to measure the conductivity of materials without having to make electrical contact with them. Specifically, we make use of the eddy-current induced in the materials to calculate the conductivity.

===[[Quantum Random Number Generator]]===

'''Proposed by: Zhang Munan'''

'''Team members: Wang Yang, Xiao Yucan, Zhang Munan'''

'''Venue: S14-03-04'''

Random numbers are a fundamental resource in science and engineering with important applications in simulation and cryptography. The inherent randomness at the core of quantum mechanics makes quantum systems a perfect source of entropy. Quantum random number generation is one of the most mature quantum technologies with many alternative generation methods. The purpose of our project is to build a simple optics-based QRNG. We will also collect the random number generated by our device and use some methods to check the randomness.

===[[Orbits of the Galilean Moons]]===

Team Members: [[User:Matthew|Matthew Wee]]

Galileo Galilei’s discovery of celestial bodies that orbit something other than the Earth marked the beginning of the end of the geocentric model of the universe. In this project, we will perform the same observations on those moons as Galileo did 400 years ago.

===[[A digital oscilloscope Based on MCU]]===

Team Members: Zhang Chengyue, Yang Ningli, Guo Diandian, Chen Jiayu

We design a digital oscilloscope with STC8A8K chip as the control core, which mainly consists of two modules: hardware circuit and software program. The hardware module mainly includes OLED screen, voltage sampling circuit, clock system, power supply and management module and so on. The software module mainly includes A/D sampling, OLED display, interrupt timing and some necessary data processing. its measurable bandwidth is 0-3000Hz, the measured range is 0-30V. After many tests and comparisons, the design achieves the amplification and reduction of waveform and the measurement of different frequency waveform in the experimental process, so as to achieve the desired goal.

===[[Test]]===
Keep a very brief description of a project or even a suggestion here, and perhaps the names of the team members, or who to contact if there is interest to join.

==Resources==
===[[Recorded sessions]]===
Some of the sessions will be recorded and uploaded to youtube. Find a description on the [[Recorded sessions]] page.

===Devices and material===
Apart form all the stuff in the teaching lab, we have a few resources you may want to consider for your project
*...

'''Books:'''
* P.R. Bevington, D.K. Robinson: Data Reduction and Error Analysis for the Physical Sciences, 3rd edition. McGrawHill, ISBN0-07-119926-8. A very good book containing all the questions you never allowed yourself to ask about error treatment, statistics, fitting of data to models etc.
* Horrowitz/Hill: The Art of Electronics
* C.H. Moore, C.C. Davis, M.A. Coplan: Building Scientific Apparatus. 2nd or higher edition. Perseus Books, ISBN0-201-13189-7. A very comprehensive book about many dirty details in experimental physics, and ways to get simple problems solved. Appears a bit dated, but is a good start for many experimental projects up to this day!
* Christopher C. Davis: Laser and Electro-optics. Useful as a general introduction to many contemporary aspects you come across when working with lasers, with a reasonable introduction of the theory. Very practical for optics.

'''Software:'''
Some of the more common data processing tools used in experimental physics:
* [http://www.gnuplot.info/ '''Gnuplot''']: A free and very mature data display tool that works on just about any platform used that produces excellent publication-grade eps and pdf figures. Can be also used in scripts. Open source and completely free.
* Various '''Python''' extensions. [http://www.python.org/ Python] is a very powerful free programming language that runs on just about any computer platform. It is open source and completely free.
* '''Matlab''': Very common, good toolset also for formal mathematics, good graphics. Expensive. We may have a site license, but I am not sure how painful it is for us to get a license for this course. Ask me if interested.
* '''Mathematica''': More common among theroetical physicists, very good in formal maths, now with better numerics. Graphs are ok but can be a pain to make looking good. As with Matlab, we do have a campus license but an increasingly painful licensing ritual. Ask me if interested or follow the instruction to install the software in your desktop.
* '''Origin''': Very widespread data processing software with a complete graphical user interface, integrates well into a Windows environment. Most likely available in your research labs, not sure if NUS has a site license.
* '''Labview''': Many of you may have seen this in your labs, but I am not too familiar with it, and chances are it is too resource-hungry to run on the machines we have there. It keeps its promise of a fast learning curve if you want to do simple things but it can get a REAL pain if you want to do subtle things, or want to do things fast, or want to debug code. Expensive and resource-hungry, but comes with good integration of also expensive hardware. May not be worth it if you know any programming language.
* [https://www.circuitlab.com/ '''Circuit Lab''']: a convenient software to design and simulate electrical circuits directly at your browser. I think Flash is required. It works well in Chrome.

===[[Acronym database]]===
This is an attempt to clarify the countless acronyms we use in our sub-communities (follow headline link)

===[[Gnuplot tricks]]===
Follow the headline link for some of the random questions that came up with gnuplot.

== Previous PC5214 wikis ==
* [http://pc5214.org/AY1819S2 AY2018/19 Sem2]
* [http://pc5214.org/AY1415S1 AY2014/15 Sem1]
* [http://pc5214.org/AY1314S1 AY2013/14 Sem1]
* [http://pc5214.org/AY1213S1 AY2012/13 Sem1]
* [http://pc5214.org/AY1112S1 AY2011/12 Sem1]
* [http://pc5214.org/AY1011S1 AY2010/11 Sem1]

== Some wiki reference materials==
Consult the [//meta.wikimedia.org/wiki/Help:Contents User's Guide] for information on using the wiki software. Other sources:

* [//www.mediawiki.org/wiki/Manual:FAQ MediaWiki FAQ]
* [[Writing mathematical expressions]]
* [[Uploading images]]

An interferometric method for measuring the resonance frequency of a vibrating system

2022-04-30T03:07:59Z

Aucca: /* An interferometric method for measuring the resonance frequency of a vibrating system */

===[[Resonance frequency measurement using a interferometric method]]===

== Team Members ==
* [[User:Nakarin|Nakarin Jayjong]]
* [[User:Aucca|Joel Auccapuclla]]
* [[User:Xiaoyu|Xiaoyu Nie]]
* [[User:Haotian|Haotian Song]]

== Introduction ==
Interferometers since their invention have been a great tool for experimental physics. Applications cover a wide spectrum of fields such as astronomy, fiber optics, seismology, seismology, velocimetry to say the least. The precision of measurement of small displacements is the key factor, take LIGO as an example. Another important factor is the convenience to set up an interferometer and the low complexity in terms of equipment. In this experiment, we will measure the resonance frequency of a vibrating transducer using a Michelson interferometer. Basically, a transducer is attached to one optical path, to one mirror, of the Michelson interferometer and it is expected to see a peak in the response curve for the resonance frequency of the transducer plus mirror.

== Theory ==
Our Michelson interferometer setup used a laser, a beam splitter, two mirrors, and a photometer, illustrated in Fig. 1. Original models used sodium flame as the coherent light source. The light propagates from the source to the beam splitter where approximately 50% of the incident light is transmitted and approximately 50% of it is reflected at a right angle. The transmitted light continues to propagate as the reference plane wave to the stationary mirror where it is fully reflected back towards the beam splitter. The reflected light propagates as the target plane wave to the mirror attached to the oscillating loudspeaker. The two waves then recombine at the beam splitter and interfere based on their relative phases based on the idea that the phase shift of the target plane wave varies with the instantaneous position of the speaker mirror combination. Consider a situation in which we setup our interferometer such that no phase shift corresponds to the equilibrium position of the speaker-mirror combination. At this position, we expect constructive interference at the photometer. Likewise, for every λ/2 change in displacement, we expect constructive interference at the photometer because the path length travelled by the target plane wave varies by some integral multiple of λ. Thus, the phase shift is an even multiple of π and the two waves remain in phase. Similarly, when the path length traveled is some odd multiple of λ/4, the phase shift is an odd multiple of π and the two waves interfere destructively.

Consider two coherent waves with the same amplitude A and same frequency <math>\omega</math> that have paths <math>x_1</math> and <math>x_2</math> emitted from different sources that meet together at a point. The net wave at that point can then be given as the real part of the equation

<div class="center" style="width:auto; margin-left:auto; margin-right:auto;">
<math>
\begin{alignat}{2}
\psi & = Ae^{i(kx_1-\omega t}+Ae^{i(kx_2-\omega t)} \\
& = Ae^{-i\omega t}e^{ik\frac{(x_1+x_2)}{2}}[e^{ik\frac{(x_1-x_2)}{2}}e^{-ik\frac{(x_1-x_2)}{2}}]\\
& = 2Acos[k(\frac{x_1-x_2}{2})]e^{i[k\frac{(x_1+x_2)}{2}-\omega t]}
\end{alignat}
</math>
</div>

where the wave number <math>k=2\pi/\lambda</math>, <math>\lambda</math> is the wavelength of the waves, and c is the phase velocity. The time-averaged intensity of the combined wave is then proportional to the square of the its amplitude. That is,

<div class="center" style="width:auto; margin-left:auto; margin-right:auto;">
<math>
I\propto cos^2[\pi(\frac{p_2-p_1}{\lambda})]=cos^2[2\pi(\frac{p_2}{\lambda}-\theta)]
</math>
</div>

where <math>p_1</math> is the distance from beam splitter to fixed mirror <math>M1</math>, and <math>p_2</math> is the distance from beam splitter to movable mirror <math>M2</math>. Now suppose the mirror that the <math>p_2</math> beam is incident to oscillates harmonically back and forth with a frequency <math>f</math> and a displacement <math>B</math>, while the distance <math>p_1</math> is constant, and all of constant term can be rewritten into a phase different <math>\theta</math>. Then, <math>p_2 =p_2(0)+Bsin(2\pi ft)</math> and the intensity of the detected light varies with time as

<div class="center" style="width:auto; margin-left:auto; margin-right:auto;">
<math>
I=A^2cos^2[2\pi(\frac{Bsin(2\pi ft)}{\lambda}-\theta')]
</math>
</div>

Note that the frequency of oscillations is itself a sinusoid, meaning that there will be turnaround points in the oscillations.

== Experimental setup ==
The Michelson interferometer consists of a beam splitter BS which is used to split the laser beam into two beams and two mirrors <math>M1</math> and <math>M2</math>. The reflected beam goes to a fixed mirror <math>M1</math> while the transmitted beam goes to the movable mirror <math>M2</math>. A transducer is attached to the mirror <math>M2</math>. The transducer is powered by a signal generator that is used to vary the vibrating frequency of the mirror. A photodiode <math>PD</math> is used to detect the signal of interference fringes. The schematic Michelson interferometer is shown in Figure 1.
[[File:Schematic_Michelson_Interferometer_v2.jpeg|center|thumb|600px|Figure 1: Schematic figure of Michelson interferometer. <math>LD</math> is a laser diode, <math>BS</math> is a beam splitter, <math>M1</math> and <math>M2</math> are mirrors, <math>p_1</math> is the distance from <math>BS</math> to <math>M1</math>, and <math>p_2</math> is the distance from <math>BS</math> to <math>M2</math>, <math>PD</math> is a photodiode, and vibration transducer is driven by a signal generator.]]

The experimental setup Michelson interferometer is shown in figure 2.
[[File:Experimental setup Michelson Interferometer.jpeg|center|thumb|600px|Figure 2: Experimental setup of Michelson interferometer.]]

The interference fringes of the Michelson interferometer are observed when the transducer is turned off in Figure 3.
[[File:Michelson interference fringes.jpeg|center|thumb|600px|Figure 3: Interference fringes which are observed when the transducer is turned off]]

==Experiment Results==
After the alignment of the optical setup, we applied sinusoidal voltage to the transducer is attached to the mirror tightly. intensity fluctuation is recorded by the home-made photodiode. Data from the photodiode is collected by the oscilloscope. The typical figure from the oscilloscope is shown in Figure 4. According to the equation ?, the number of peaks during each period corresponds to the radio of displacement and laser wavelength. In this case, we can measure the displacement for various frequencies. The frequency range is from 86 Hz to 130 Hz.

[[File:ScopeScreen.jpeg|center|thumb|600px|Figure 4: The display on the oscilloscope at <math>f = 103Hz </math>. After applying the voltage to the transducer, the voltage output from photodiode changes with time.]]

In our setup, Mitsubishi Laser Diode operated at a wavelength of 660 nm. We plot the displacement for different frequencies with the errorbar in Figure 5. Uncertainty is the square root uncertainty of the count number. For a driven, damped and harmonic oscillator, the relation between displacement and frequency is

<math>|A|=\frac{f_{0}}{\sqrt{\left(\omega_{0}^{2}-\omega^{2}\right)^{2}+4 \beta^{2} \omega^{2}}} </math>

Where <math>f_{0}</math> is the amplitude, <math>\omega_{0}</math> is the resonance frequency, and <math>\beta_{0}</math> is a constant. With this equation, we can fit the frequency response curve. Because there is only one peak or valley for a frequency lower than 92 Hz or higher than 120 Hz, and the error is significant and non-negligible. We exclude those in the fitting process. Finally, the fitting root mean squared error is 130.38, and the fitting resonance frequency <math>\omega_{0} </math> is 113.12Hz.

[[File:FitPeakNum.jpeg|center|thumb|600px|Figure 5: Frequency response curve for the transducer system with errorbar. The blue dots represent the displacement measured with different frequencies. Error bars are the square root uncertainty. Red line is the fitting curve for a driven, damped, harmonic oscillator.]]
== Conclusion ==
We have shown indeed that we can characterize the resonance frequency of a system. For this particular application, we found a frequency of 113.12Hz for the system transducer plus mirror. Finally, with appropriate handling, this technique could be implemented to characterize more sofisticated systems.

==References==
[1] Freschi, A. A., et al. "Laser interferometric characterization of a vibrating speaker system." American Journal of Physics 71.11 (2003): 1121-1126.
https://doi.org/10.1119/1.1586262

[2] Skarha, Matthew. "Laser interferometric characterization for a vibrating speaker system."
http://mattskarha.com/assets/docs/Laser%20interferometric%20characterization%20for%20a%20vibrating%20speaker%20system%20good.pdf

[3] Pathare, Shirish, and Vikrant Kurmude. "Low cost Michelson–Morley interferometer." Physics Education 51.6 (2016): 063001.
https://doi.org/10.1088/0031-9120/51/6/063001

An interferometric method for measuring the resonance frequency of vibrating system

2022-04-29T04:14:06Z

Aucca: /* Experiment Results */

== Team Members ==
* [[User:Nakarin|Nakarin Jayjong]]
* [[User:Aucca|Joel Auccapuclla]]
* [[User:Xiaoyu|Xiaoyu Nie]]
* [[User:Haotian|Haotian Song]]

== Introduction ==
Interferometers since their invention have been a great tool for experimental physics. Applications cover a wide spectrum of fields such as astronomy, fiber optics, seismology, seismology, velocimetry to say the least. The precision of measurement of small displacements is the key factor, take LIGO as an example. Another important factor is the convenience to set up an interferometer and the low complexity in terms of equipment. In this experiment, we will measure the resonance frequency of a vibrating transducer using a Michelson interferometer. Basically, a transducer is attached to one optical path, to one mirror, of the Michelson interferometer and it is expected to see a peak in the response curve for the resonance frequency of the transducer plus mirror.

== Theory ==
Our Michelson interferometer setup used a laser, a beam splitter, two mirrors, and a photometer, illustrated in Fig. 1. Original models used sodium flame as the coherent light source. The light propagates from the source to the beam splitter where approximately 50% of the incident light is transmitted and approximately 50% of it is reflected at a right angle. The transmitted light continues to propagate as the reference plane wave to the stationary mirror where it is fully reflected back towards the beam splitter. The reflected light propagates as the target plane wave to the mirror attached to the oscillating loudspeaker. The two waves then recombine at the beam splitter and interfere based on their relative phases based on the idea that the phase shift of the target plane wave varies with the instantaneous position of the speaker mirror combination. Consider a situation in which we setup our interferometer such that no phase shift corresponds to the equilibrium position of the speaker-mirror combination. At this position, we expect constructive interference at the photometer. Likewise, for every λ/2 change in displacement, we expect constructive interference at the photometer because the path length travelled by the target plane wave varies by some integral multiple of λ. Thus, the phase shift is an even multiple of π and the two waves remain in phase. Similarly, when the path length traveled is some odd multiple of λ/4, the phase shift is an odd multiple of π and the two waves interfere destructively.

Consider two coherent waves with the same amplitude A and same frequency <math>\omega</math> that have paths <math>x_1</math> and <math>x_2</math> emitted from different sources that meet together at a point. The net wave at that point can then be given as the real part of the equation

<math>
\psi=Ae^{i(kx_1-\omega t}+Ae^{i(kx_2-\omega t)}=Ae^{-i\omega t}e^{ik\frac{(x_1+x_2)}{2}}[e^{ik\frac{(x_1-x_2)}{2}}e^{-ik\frac{(x_1-x_2)}{2}}]=2Acos[k(\frac{x_1-x_2}{2})]e^{i[k\frac{(x_1+x_2)}{2}-\omega t]}
</math>

where the wave number <math>k=2\pi/\lambda</math>, <math>\lambda</math> is the wavelength of the waves, and c is the phase velocity. The time-averaged intensity of the combined wave is then proportional to the square of the its amplitude. That is,

<math>
I\propto cos^2[\pi(\frac{p_2-p_1}{\lambda})]=cos^2[2\pi(\frac{p_2}{\lambda}-\theta)]
</math>

where <math>\theta</math> is a constant. Now suppose the mirror that the <math>p_2</math> beam is incident to oscillates harmonically back and forth with a frequency <math>f</math> and a displacement <math>B</math>. Then, <math>p_2 =p_2(0)+Bsin(2\pi ft)</math> and the intensity of the detected light varies with time as

<math>
I=A^2cos^2[2\pi(\frac{Bsin(2\pi ft)}{\lambda}-\theta')]
</math>

Note that the frequency of oscillations is itself a sinusoid, meaning that there will be turnaround points in the oscillations.

== Experimental setup ==
The Michelson interferometer consists of a beam splitter BS which is used to split the laser beam into two beams and two mirrors M1 and M2. The reflected beam goes to a fixed mirror M1 while the transmitted beam goes to the movable mirror M2. A transducer is attached to the mirror M2. The transducer is powered by a signal generator that is used to vary the vibrating frequency of the mirror. A photodiode PD is used to detect the signal of interference fringes. The schematic Michelson interferometer is shown in Figure 1.
[[File:Schematic Michelson Interferometer.jpeg|center|thumb|600px|Figure 1: Schematic figure of Michelson interferometer. LD is a laser diode, BS is a beam splitter, M1 and M2 are mirrors, PD is a photodiode, and vibration transducer which is driven by a signal generator.]]

The experimental setup Michelson interferometer is shown in figure 2.
[[File:Experimental setup Michelson Interferometer.jpeg|center|thumb|600px|Figure 2: Experimental setup of Michelson interferometer.]]

The interference fringes of the Michelson interferometer are observed when the transducer is turned off in Figure 3.
[[File:Michelson interference fringes.jpeg|center|thumb|600px|Figure 3: Interference fringes which are observed when the transducer is turned off]]

==Experiment Results==
After the alignment of the optical setup, we applied sinusoidal voltage to the transducer attached to the mirror tightly. intensity fluctuation is recorded by the homemade photodiode. Data from the photodiode is collected by the oscilloscope. The typical figure from the oscilloscope is shown in Figure 4. According to the equation mentioned before, the number of peaks during each period corresponds to the radio of displacement and laser wavelength. In this case, we can measure the displacement for various frequencies. The frequency range is from 86 Hz to 130 Hz.

[[File:ScopeScreen.jpeg|center|thumb|600px|Figure 4: The display on the oscilloscope at <math>f = 103Hz </math>. After applying the voltage to the transducer, the voltage output from photodiode changes with time.]]

In our setup, we measured the wavelength of Mitsubishi Laser Diode which is 658.9 nm. The wavemeter measurement results are shown here.
[[File:Wavelength.jpeg|center|thumb|600px|Figure 5: The Wavelength measurement of the Mitsubishi Laser Diode.]]

We plot the displacement for different frequencies with the errorbar in Figure 6. Uncertainty is the square root uncertainty of the count number. For a driven, damped and harmonic oscillator, the relation between displacement and frequency is

<math>|A|=\frac{f_{0}}{\sqrt{\left(\omega_{0}^{2}-\omega^{2}\right)^{2}+4 \beta^{2} \omega^{2}}} </math>

Where <math>f_{0}</math> is the amplitude, <math>\omega_{0}</math> is the resonance frequency, and <math>\beta_{0}</math> is a constant. With this equation, we can fit the frequency response curve. Because there is only one peak or valley for a frequency lower than 92 Hz or higher than 120 Hz, and the error is significant and non-negligible. We exclude those in the fitting process. Finally, the fitting root mean squared error is 130.38. Fitted <math>\beta</math> is <math>1.3 \pm 0.2</math>. The fitting resonance frequency <math>\omega_{0} </math> is <math>113.1 \pm 0.1Hz </math>.

[[File:FitPeakNum.jpeg|center|thumb|600px|Figure 6: Frequency response curve for the transducer system with errorbar. The blue dots represent the displacement measured with different frequencies. Error bars are the square root uncertainty. Red line is the fitting curve for a driven, damped, harmonic oscillator.]]

== Conclusion ==
We have shown indeed that we can characterize the resonance frequency of a system. For this particular application, we found a frequency of 113.12Hz for the system transducer plus mirror. Finally, with appropriate handling, this technique could be implemented to characterize more sofisticated systems.

==References==
[1] Freschi, A. A., et al. "Laser interferometric characterization of a vibrating speaker system." American Journal of Physics 71.11 (2003): 1121-1126.
https://doi.org/10.1119/1.1586262

[2] Skarha, Matthew. "Laser interferometric characterization for a vibrating speaker system."
http://mattskarha.com/assets/docs/Laser%20interferometric%20characterization%20for%20a%20vibrating%20speaker%20system%20good.pdf

[3] Pathare, Shirish, and Vikrant Kurmude. "Low cost Michelson–Morley interferometer." Physics Education 51.6 (2016): 063001.
https://doi.org/10.1088/0031-9120/51/6/063001

An interferometric method for measuring the resonance frequency of vibrating system

2022-04-29T04:10:42Z

Aucca: /* Experiment Results */

== Team Members ==
* [[User:Nakarin|Nakarin Jayjong]]
* [[User:Aucca|Joel Auccapuclla]]
* [[User:Xiaoyu|Xiaoyu Nie]]
* [[User:Haotian|Haotian Song]]

== Introduction ==
Interferometers since their invention have been a great tool for experimental physics. Applications cover a wide spectrum of fields such as astronomy, fiber optics, seismology, seismology, velocimetry to say the least. The precision of measurement of small displacements is the key factor, take LIGO as an example. Another important factor is the convenience to set up an interferometer and the low complexity in terms of equipment. In this experiment, we will measure the resonance frequency of a vibrating transducer using a Michelson interferometer. Basically, a transducer is attached to one optical path, to one mirror, of the Michelson interferometer and it is expected to see a peak in the response curve for the resonance frequency of the transducer plus mirror.

== Theory ==
Our Michelson interferometer setup used a laser, a beam splitter, two mirrors, and a photometer, illustrated in Fig. 1. Original models used sodium flame as the coherent light source. The light propagates from the source to the beam splitter where approximately 50% of the incident light is transmitted and approximately 50% of it is reflected at a right angle. The transmitted light continues to propagate as the reference plane wave to the stationary mirror where it is fully reflected back towards the beam splitter. The reflected light propagates as the target plane wave to the mirror attached to the oscillating loudspeaker. The two waves then recombine at the beam splitter and interfere based on their relative phases based on the idea that the phase shift of the target plane wave varies with the instantaneous position of the speaker mirror combination. Consider a situation in which we setup our interferometer such that no phase shift corresponds to the equilibrium position of the speaker-mirror combination. At this position, we expect constructive interference at the photometer. Likewise, for every λ/2 change in displacement, we expect constructive interference at the photometer because the path length travelled by the target plane wave varies by some integral multiple of λ. Thus, the phase shift is an even multiple of π and the two waves remain in phase. Similarly, when the path length traveled is some odd multiple of λ/4, the phase shift is an odd multiple of π and the two waves interfere destructively.

Consider two coherent waves with the same amplitude A and same frequency <math>\omega</math> that have paths <math>x_1</math> and <math>x_2</math> emitted from different sources that meet together at a point. The net wave at that point can then be given as the real part of the equation

<math>
\psi=Ae^{i(kx_1-\omega t}+Ae^{i(kx_2-\omega t)}=Ae^{-i\omega t}e^{ik\frac{(x_1+x_2)}{2}}[e^{ik\frac{(x_1-x_2)}{2}}e^{-ik\frac{(x_1-x_2)}{2}}]=2Acos[k(\frac{x_1-x_2}{2})]e^{i[k\frac{(x_1+x_2)}{2}-\omega t]}
</math>

where the wave number <math>k=2\pi/\lambda</math>, <math>\lambda</math> is the wavelength of the waves, and c is the phase velocity. The time-averaged intensity of the combined wave is then proportional to the square of the its amplitude. That is,

<math>
I\propto cos^2[\pi(\frac{p_2-p_1}{\lambda})]=cos^2[2\pi(\frac{p_2}{\lambda}-\theta)]
</math>

where <math>\theta</math> is a constant. Now suppose the mirror that the <math>p_2</math> beam is incident to oscillates harmonically back and forth with a frequency <math>f</math> and a displacement <math>B</math>. Then, <math>p_2 =p_2(0)+Bsin(2\pi ft)</math> and the intensity of the detected light varies with time as

<math>
I=A^2cos^2[2\pi(\frac{Bsin(2\pi ft)}{\lambda}-\theta')]
</math>

Note that the frequency of oscillations is itself a sinusoid, meaning that there will be turnaround points in the oscillations.

== Experimental setup ==
The Michelson interferometer consists of a beam splitter BS which is used to split the laser beam into two beams and two mirrors M1 and M2. The reflected beam goes to a fixed mirror M1 while the transmitted beam goes to the movable mirror M2. A transducer is attached to the mirror M2. The transducer is powered by a signal generator that is used to vary the vibrating frequency of the mirror. A photodiode PD is used to detect the signal of interference fringes. The schematic Michelson interferometer is shown in Figure 1.
[[File:Schematic Michelson Interferometer.jpeg|center|thumb|600px|Figure 1: Schematic figure of Michelson interferometer. LD is a laser diode, BS is a beam splitter, M1 and M2 are mirrors, PD is a photodiode, and vibration transducer which is driven by a signal generator.]]

The experimental setup Michelson interferometer is shown in figure 2.
[[File:Experimental setup Michelson Interferometer.jpeg|center|thumb|600px|Figure 2: Experimental setup of Michelson interferometer.]]

The interference fringes of the Michelson interferometer are observed when the transducer is turned off in Figure 3.
[[File:Michelson interference fringes.jpeg|center|thumb|600px|Figure 3: Interference fringes which are observed when the transducer is turned off]]

==Experiment Results==
After the alignment of the optical setup, we applied sinusoidal voltage to the transducer attached to the mirror tightly. intensity fluctuation is recorded by the homemade photodiode. Data from the photodiode is collected by the oscilloscope. The typical figure from the oscilloscope is shown in Figure 4. According to the equation mentioned before, the number of peaks during each period corresponds to the radio of displacement and laser wavelength. In this case, we can measure the displacement for various frequencies. The frequency range is from 86 Hz to 130 Hz.

[[File:ScopeScreen.jpeg|center|thumb|600px|Figure 4: The display on the oscilloscope at <math>f = 103Hz </math>. After applying the voltage to the transducer, the voltage output from photodiode changes with time.]]

In our setup, we measured the wavelength of Mitsubishi Laser Diode which is 658.9 nm. The wavemeter measurement results are shown here.
[[File:Wavelength.jpeg|center|thumb|600px|Figure 5: The Wavelength measurement of the Mitsubishi Laser Diode.]]

We plot the displacement for different frequencies with the errorbar in Figure 6. Uncertainty is the square root uncertainty of the count number. For a driven, damped and harmonic oscillator, the relation between displacement and frequency is

<math>|A|=\frac{f_{0}}{\sqrt{\left(\omega_{0}^{2}-\omega^{2}\right)^{2}+4 \beta^{2} \omega^{2}}} </math>

Where <math>f_{0}</math> is the amplitude, <math>\omega_{0}</math> is the resonance frequency, and <math>\beta_{0}</math> is a constant. With this equation, we can fit the frequency response curve. Because there is only one peak or valley for a frequency lower than 92 Hz or higher than 120 Hz, and the error is significant and non-negligible. We exclude those in the fitting process. Finally, the fitting root mean squared error is 130.38, and the fitting resonance frequency <math>\omega_{0} </math> is <math>113.1 \pm 0.1Hz </math>.

[[File:FitPeakNum.jpeg|center|thumb|600px|Figure 6: Frequency response curve for the transducer system with errorbar. The blue dots represent the displacement measured with different frequencies. Error bars are the square root uncertainty. Red line is the fitting curve for a driven, damped, harmonic oscillator.]]

== Conclusion ==
We have shown indeed that we can characterize the resonance frequency of a system. For this particular application, we found a frequency of 113.12Hz for the system transducer plus mirror. Finally, with appropriate handling, this technique could be implemented to characterize more sofisticated systems.

==References==
[1] Freschi, A. A., et al. "Laser interferometric characterization of a vibrating speaker system." American Journal of Physics 71.11 (2003): 1121-1126.
https://doi.org/10.1119/1.1586262

[2] Skarha, Matthew. "Laser interferometric characterization for a vibrating speaker system."
http://mattskarha.com/assets/docs/Laser%20interferometric%20characterization%20for%20a%20vibrating%20speaker%20system%20good.pdf

[3] Pathare, Shirish, and Vikrant Kurmude. "Low cost Michelson–Morley interferometer." Physics Education 51.6 (2016): 063001.
https://doi.org/10.1088/0031-9120/51/6/063001

An interferometric method for measuring the resonance frequency of vibrating system

2022-04-29T03:49:18Z

Aucca: /* Experiment Results */

== Team Members ==
* [[User:Nakarin|Nakarin Jayjong]]
* [[User:Aucca|Joel Auccapuclla]]
* [[User:Xiaoyu|Xiaoyu Nie]]
* [[User:Haotian|Haotian Song]]

== Introduction ==
Interferometers since their invention have been a great tool for experimental physics. Applications cover a wide spectrum of fields such as astronomy, fiber optics, seismology, seismology, velocimetry to say the least. The precision of measurement of small displacements is the key factor, take LIGO as an example. Another important factor is the convenience to set up an interferometer and the low complexity in terms of equipment. In this experiment, we will measure the resonance frequency of a vibrating transducer using a Michelson interferometer. Basically, a transducer is attached to one optical path, to one mirror, of the Michelson interferometer and it is expected to see a peak in the response curve for the resonance frequency of the transducer plus mirror.

== Theory ==
Our Michelson interferometer setup used a laser, a beam splitter, two mirrors, and a photometer, illustrated in Fig. 1. Original models used sodium flame as the coherent light source. The light propagates from the source to the beam splitter where approximately 50% of the incident light is transmitted and approximately 50% of it is reflected at a right angle. The transmitted light continues to propagate as the reference plane wave to the stationary mirror where it is fully reflected back towards the beam splitter. The reflected light propagates as the target plane wave to the mirror attached to the oscillating loudspeaker. The two waves then recombine at the beam splitter and interfere based on their relative phases based on the idea that the phase shift of the target plane wave varies with the instantaneous position of the speaker mirror combination. Consider a situation in which we setup our interferometer such that no phase shift corresponds to the equilibrium position of the speaker-mirror combination. At this position, we expect constructive interference at the photometer. Likewise, for every λ/2 change in displacement, we expect constructive interference at the photometer because the path length travelled by the target plane wave varies by some integral multiple of λ. Thus, the phase shift is an even multiple of π and the two waves remain in phase. Similarly, when the path length traveled is some odd multiple of λ/4, the phase shift is an odd multiple of π and the two waves interfere destructively.

Consider two coherent waves with the same amplitude A and same frequency <math>\omega</math> that have paths <math>x_1</math> and <math>x_2</math> emitted from different sources that meet together at a point. The net wave at that point can then be given as the real part of the equation

<math>
\psi=Ae^{i(kx_1-\omega t}+Ae^{i(kx_2-\omega t)}=Ae^{-i\omega t}e^{ik\frac{(x_1+x_2)}{2}}[e^{ik\frac{(x_1-x_2)}{2}}e^{-ik\frac{(x_1-x_2)}{2}}]=2Acos[k(\frac{x_1-x_2}{2})]e^{i[k\frac{(x_1+x_2)}{2}-\omega t]}
</math>

where the wave number <math>k=2\pi/\lambda</math>, <math>\lambda</math> is the wavelength of the waves, and c is the phase velocity. The time-averaged intensity of the combined wave is then proportional to the square of the its amplitude. That is,

<math>
I\propto cos^2[\pi(\frac{p_2-p_1}{\lambda})]=cos^2[2\pi(\frac{p_2}{\lambda}-\theta)]
</math>

where <math>\theta</math> is a constant. Now suppose the mirror that the <math>p_2</math> beam is incident to oscillates harmonically back and forth with a frequency <math>f</math> and a displacement <math>B</math>. Then, <math>p_2 =p_2(0)+Bsin(2\pi ft)</math> and the intensity of the detected light varies with time as

<math>
I=A^2cos^2[2\pi(\frac{Bsin(2\pi ft)}{\lambda}-\theta')]
</math>

Note that the frequency of oscillations is itself a sinusoid, meaning that there will be turnaround points in the oscillations.

== Experimental setup ==
The Michelson interferometer consists of a beam splitter BS which is used to split the laser beam into two beams and two mirrors M1 and M2. The reflected beam goes to a fixed mirror M1 while the transmitted beam goes to the movable mirror M2. A transducer is attached to the mirror M2. The transducer is powered by a signal generator that is used to vary the vibrating frequency of the mirror. A photodiode PD is used to detect the signal of interference fringes. The schematic Michelson interferometer is shown in Figure 1.
[[File:Schematic Michelson Interferometer.jpeg|center|thumb|600px|Figure 1: Schematic figure of Michelson interferometer. LD is a laser diode, BS is a beam splitter, M1 and M2 are mirrors, PD is a photodiode, and vibration transducer which is driven by a signal generator.]]

The experimental setup Michelson interferometer is shown in figure 2.
[[File:Experimental setup Michelson Interferometer.jpeg|center|thumb|600px|Figure 2: Experimental setup of Michelson interferometer.]]

The interference fringes of the Michelson interferometer are observed when the transducer is turned off in Figure 3.
[[File:Michelson interference fringes.jpeg|center|thumb|600px|Figure 3: Interference fringes which are observed when the transducer is turned off]]

==Experiment Results==
After the alignment of the optical setup, we applied sinusoidal voltage to the transducer attached to the mirror tightly. intensity fluctuation is recorded by the homemade photodiode. Data from the photodiode is collected by the oscilloscope. The typical figure from the oscilloscope is shown in Figure 4. According to the equation mentioned before, the number of peaks during each period corresponds to the radio of displacement and laser wavelength. In this case, we can measure the displacement for various frequencies. The frequency range is from 86 Hz to 130 Hz.

[[File:ScopeScreen.jpeg|center|thumb|600px|Figure 4: The display on the oscilloscope at <math>f = 103Hz </math>. After applying the voltage to the transducer, the voltage output from photodiode changes with time.]]

In our setup, we measured the wavelength of Mitsubishi Laser Diode which is 658.9 nm. The wavemeter measurement results are shown here.
[[File:Wavelength.jpeg|center|thumb|600px|Figure 5: The Wavelength measurement of the Mitsubishi Laser Diode.]]

We plot the displacement for different frequencies with the errorbar in Figure 6. Uncertainty is the square root uncertainty of the count number. For a driven, damped and harmonic oscillator, the relation between displacement and frequency is

<math>|A|=\frac{f_{0}}{\sqrt{\left(\omega_{0}^{2}-\omega^{2}\right)^{2}+4 \beta^{2} \omega^{2}}} </math>

Where <math>f_{0}</math> is the amplitude, <math>\omega_{0}</math> is the resonance frequency, and <math>\beta_{0}</math> is a constant. With this equation, we can fit the frequency response curve. Because there is only one peak or valley for a frequency lower than 92 Hz or higher than 120 Hz, and the error is significant and non-negligible. We exclude those in the fitting process. Finally, the fitting root mean squared error is 130.38, and the fitting resonance frequency <math>\omega_{0} </math> is 113.12Hz.

[[File:FitPeakNum.jpeg|center|thumb|600px|Figure 6: Frequency response curve for the transducer system with errorbar. The blue dots represent the displacement measured with different frequencies. Error bars are the square root uncertainty. Red line is the fitting curve for a driven, damped, harmonic oscillator.]]

== Conclusion ==
We have shown indeed that we can characterize the resonance frequency of a system. For this particular application, we found a frequency of 113.12Hz for the system transducer plus mirror. Finally, with appropriate handling, this technique could be implemented to characterize more sofisticated systems.

==References==
[1] Freschi, A. A., et al. "Laser interferometric characterization of a vibrating speaker system." American Journal of Physics 71.11 (2003): 1121-1126.
https://doi.org/10.1119/1.1586262

[2] Skarha, Matthew. "Laser interferometric characterization for a vibrating speaker system."
http://mattskarha.com/assets/docs/Laser%20interferometric%20characterization%20for%20a%20vibrating%20speaker%20system%20good.pdf

[3] Pathare, Shirish, and Vikrant Kurmude. "Low cost Michelson–Morley interferometer." Physics Education 51.6 (2016): 063001.
https://doi.org/10.1088/0031-9120/51/6/063001

File:Wavelength.jpeg

2022-04-29T03:46:19Z

Aucca:

An interferometric method for measuring the resonance frequency of vibrating system

2022-04-29T03:42:12Z

Aucca: /* Experiment Results */

== Team Members ==
* [[User:Nakarin|Nakarin Jayjong]]
* [[User:Aucca|Joel Auccapuclla]]
* [[User:Xiaoyu|Xiaoyu Nie]]
* [[User:Haotian|Haotian Song]]

== Introduction ==
Interferometers since their invention have been a great tool for experimental physics. Applications cover a wide spectrum of fields such as astronomy, fiber optics, seismology, seismology, velocimetry to say the least. The precision of measurement of small displacements is the key factor, take LIGO as an example. Another important factor is the convenience to set up an interferometer and the low complexity in terms of equipment. In this experiment, we will measure the resonance frequency of a vibrating transducer using a Michelson interferometer. Basically, a transducer is attached to one optical path, to one mirror, of the Michelson interferometer and it is expected to see a peak in the response curve for the resonance frequency of the transducer plus mirror.

== Theory ==
Our Michelson interferometer setup used a laser, a beam splitter, two mirrors, and a photometer, illustrated in Fig. 1. Original models used sodium flame as the coherent light source. The light propagates from the source to the beam splitter where approximately 50% of the incident light is transmitted and approximately 50% of it is reflected at a right angle. The transmitted light continues to propagate as the reference plane wave to the stationary mirror where it is fully reflected back towards the beam splitter. The reflected light propagates as the target plane wave to the mirror attached to the oscillating loudspeaker. The two waves then recombine at the beam splitter and interfere based on their relative phases based on the idea that the phase shift of the target plane wave varies with the instantaneous position of the speaker mirror combination. Consider a situation in which we setup our interferometer such that no phase shift corresponds to the equilibrium position of the speaker-mirror combination. At this position, we expect constructive interference at the photometer. Likewise, for every λ/2 change in displacement, we expect constructive interference at the photometer because the path length travelled by the target plane wave varies by some integral multiple of λ. Thus, the phase shift is an even multiple of π and the two waves remain in phase. Similarly, when the path length traveled is some odd multiple of λ/4, the phase shift is an odd multiple of π and the two waves interfere destructively.

Consider two coherent waves with the same amplitude A and same frequency <math>\omega</math> that have paths <math>x_1</math> and <math>x_2</math> emitted from different sources that meet together at a point. The net wave at that point can then be given as the real part of the equation

<math>
\psi=Ae^{i(kx_1-\omega t}+Ae^{i(kx_2-\omega t)}=Ae^{-i\omega t}e^{ik\frac{(x_1+x_2)}{2}}[e^{ik\frac{(x_1-x_2)}{2}}e^{-ik\frac{(x_1-x_2)}{2}}]=2Acos[k(\frac{x_1-x_2}{2})]e^{i[k\frac{(x_1+x_2)}{2}-\omega t]}
</math>

where the wave number <math>k=2\pi/\lambda</math>, <math>\lambda</math> is the wavelength of the waves, and c is the phase velocity. The time-averaged intensity of the combined wave is then proportional to the square of the its amplitude. That is,

<math>
I\propto cos^2[\pi(\frac{p_2-p_1}{\lambda})]=cos^2[2\pi(\frac{p_2}{\lambda}-\theta)]
</math>

where <math>\theta</math> is a constant. Now suppose the mirror that the <math>p_2</math> beam is incident to oscillates harmonically back and forth with a frequency <math>f</math> and a displacement <math>B</math>. Then, <math>p_2 =p_2(0)+Bsin(2\pi ft)</math> and the intensity of the detected light varies with time as

<math>
I=A^2cos^2[2\pi(\frac{Bsin(2\pi ft)}{\lambda}-\theta')]
</math>

Note that the frequency of oscillations is itself a sinusoid, meaning that there will be turnaround points in the oscillations.

== Experimental setup ==
The Michelson interferometer consists of a beam splitter BS which is used to split the laser beam into two beams and two mirrors M1 and M2. The reflected beam goes to a fixed mirror M1 while the transmitted beam goes to the movable mirror M2. A transducer is attached to the mirror M2. The transducer is powered by a signal generator that is used to vary the vibrating frequency of the mirror. A photodiode PD is used to detect the signal of interference fringes. The schematic Michelson interferometer is shown in Figure 1.
[[File:Schematic Michelson Interferometer.jpeg|center|thumb|600px|Figure 1: Schematic figure of Michelson interferometer. LD is a laser diode, BS is a beam splitter, M1 and M2 are mirrors, PD is a photodiode, and vibration transducer which is driven by a signal generator.]]

The experimental setup Michelson interferometer is shown in figure 2.
[[File:Experimental setup Michelson Interferometer.jpeg|center|thumb|600px|Figure 2: Experimental setup of Michelson interferometer.]]

The interference fringes of the Michelson interferometer are observed when the transducer is turned off in Figure 3.
[[File:Michelson interference fringes.jpeg|center|thumb|600px|Figure 3: Interference fringes which are observed when the transducer is turned off]]

==Experiment Results==
After the alignment of the optical setup, we applied sinusoidal voltage to the transducer attached to the mirror tightly. intensity fluctuation is recorded by the homemade photodiode. Data from the photodiode is collected by the oscilloscope. The typical figure from the oscilloscope is shown in Figure 4. According to the equation mentioned before, the number of peaks during each period corresponds to the radio of displacement and laser wavelength. In this case, we can measure the displacement for various frequencies. The frequency range is from 86 Hz to 130 Hz.

[[File:ScopeScreen.jpeg|center|thumb|600px|Figure 4: The display on the oscilloscope at <math>f = 103Hz </math>. After applying the voltage to the transducer, the voltage output from photodiode changes with time.]]

In our setup, we measured the wavelength of Mitsubishi Laser Diode which is 658.9 nm. The wavemeter measurement results are shown here.
[[File:Wavelength.jpeg|center|thumb|600px|Figure 5: The display on the oscilloscope at <math>f = 103Hz </math>. After applying the voltage to the transducer, the voltage output from photodiode changes with time.]]

We plot the displacement for different frequencies with the errorbar in Figure 6. Uncertainty is the square root uncertainty of the count number. For a driven, damped and harmonic oscillator, the relation between displacement and frequency is

<math>|A|=\frac{f_{0}}{\sqrt{\left(\omega_{0}^{2}-\omega^{2}\right)^{2}+4 \beta^{2} \omega^{2}}} </math>

Where <math>f_{0}</math> is the amplitude, <math>\omega_{0}</math> is the resonance frequency, and <math>\beta_{0}</math> is a constant. With this equation, we can fit the frequency response curve. Because there is only one peak or valley for a frequency lower than 92 Hz or higher than 120 Hz, and the error is significant and non-negligible. We exclude those in the fitting process. Finally, the fitting root mean squared error is 130.38, and the fitting resonance frequency <math>\omega_{0} </math> is 113.12Hz.

[[File:FitPeakNum.jpeg|center|thumb|600px|Figure 6: Frequency response curve for the transducer system with errorbar. The blue dots represent the displacement measured with different frequencies. Error bars are the square root uncertainty. Red line is the fitting curve for a driven, damped, harmonic oscillator.]]

== Conclusion ==
We have shown indeed that we can characterize the resonance frequency of a system. For this particular application, we found a frequency of 113.12Hz for the system transducer plus mirror. Finally, with appropriate handling, this technique could be implemented to characterize more sofisticated systems.

==References==
[1] Freschi, A. A., et al. "Laser interferometric characterization of a vibrating speaker system." American Journal of Physics 71.11 (2003): 1121-1126.
https://doi.org/10.1119/1.1586262

[2] Skarha, Matthew. "Laser interferometric characterization for a vibrating speaker system."
http://mattskarha.com/assets/docs/Laser%20interferometric%20characterization%20for%20a%20vibrating%20speaker%20system%20good.pdf

[3] Pathare, Shirish, and Vikrant Kurmude. "Low cost Michelson–Morley interferometer." Physics Education 51.6 (2016): 063001.
https://doi.org/10.1088/0031-9120/51/6/063001

An interferometric method for measuring the resonance frequency of vibrating system

2022-04-22T02:12:13Z

Aucca: /* Experiment Results */

== Team Members ==
* [[User:Nakarin|Nakarin Jayjong]]
* [[User:Aucca|Joel Auccapuclla]]
* [[User:Xiaoyu|Xiaoyu Nie]]
* [[User:Haotian|Haotian Song]]

== Introduction ==
Interferometers since their invention have been a great tool for experimental physics. Applications cover a wide spectrum of fields such as astronomy, fiber optics, seismology, seismology, velocimetry to say the least. The precision of measurement of small displacements is the key factor, take LIGO as an example. Another important factor is the convenience to set up an interferometer and the low complexity in terms of equipment. In this experiment, we will measure the resonance frequency of a vibrating transducer using a Michelson interferometer. Basically, a transducer is attached to one optical path, to one mirror, of the Michelson interferometer and it is expected to see a peak in the response curve for the resonance frequency of the transducer plus mirror.

== Theory ==
Our Michelson interferometer setup used a laser, a beam splitter, two mirrors, and a photometer, illustrated in Fig. 1. Original models used sodium flame as the coherent light source. The light propagates from the source to the beam splitter where approximately 50% of the incident light is transmitted and approximately 50% of it is reflected at a right angle. The transmitted light continues to propagate as the reference plane wave to the stationary mirror where it is fully reflected back towards the beam splitter. The reflected light propagates as the target plane wave to the mirror attached to the oscillating loudspeaker. The two waves then recombine at the beam splitter and interfere based on their relative phases based on the idea that the phase shift of the target plane wave varies with the instantaneous position of the speaker mirror combination. Consider a situation in which we setup our interferometer such that no phase shift corresponds to the equilibrium position of the speaker-mirror combination. At this position, we expect constructive interference at the photometer. Likewise, for every λ/2 change in displacement, we expect constructive interference at the photometer because the path length travelled by the target plane wave varies by some integral multiple of λ. Thus, the phase shift is an even multiple of π and the two waves remain in phase. Similarly, when the path length traveled is some odd multiple of λ/4, the phase shift is an odd multiple of π and the two waves interfere destructively.

Consider two coherent waves with the same amplitude A and same frequency <math>\omega</math> that have paths <math>x_1</math> and <math>x_2</math> emitted from different sources that meet together at a point. The net wave at that point can then be given as the real part of the equation

<math>
\psi=Ae^{i(kx_1-\omega t}+Ae^{i(kx_2-\omega t)}=Ae^{-i\omega t}e^{ik\frac{(x_1+x_2)}{2}}[e^{ik\frac{(x_1-x_2)}{2}}e^{-ik\frac{(x_1-x_2)}{2}}]=2Acos[k(\frac{x_1-x_2}{2})]e^{i[k\frac{(x_1+x_2)}{2}-\omega t]}
</math>

where the wave number <math>k=2\pi/\lambda</math>, <math>\lambda</math> is the wavelength of the waves, and c is the phase velocity. The time-averaged intensity of the combined wave is then proportional to the square of the its amplitude. That is,

<math>
I\propto cos^2[\pi(\frac{p_2-p_1}{\lambda})]=cos^2[2\pi(\frac{p_2}{\lambda}-\theta)]
</math>

where <math>\theta</math> is a constant. Now suppose the mirror that the <math>p_2</math> beam is incident to oscillates harmonically back and forth with a frequency <math>f</math> and a displacement <math>B</math>. Then, <math>p_2 =p_2(0)+Bsin(2\pi ft)</math> and the intensity of the detected light varies with time as

<math>
I=A^2cos^2[2\pi(\frac{Bsin(2\pi ft)}{\lambda}-\theta')]
</math>

Note that the frequency of oscillations is itself a sinusoid, meaning that there will be turnaround points in the oscillations.

== Experimental setup ==
The Michelson interferometer consists of a beam splitter BS which is used to split the laser beam into two beams and two mirrors M1 and M2. The reflected beam goes to a fixed mirror M1 while the transmitted beam goes to the movable mirror M2. A transducer is attached to the mirror M2. The transducer is powered by a signal generator that is used to vary the vibrating frequency of the mirror. A photodiode PD is used to detect the signal of interference fringes. The schematic Michelson interferometer is shown in Figure 1.
[[File:Schematic Michelson Interferometer.jpeg|center|thumb|600px|Figure 1: Schematic figure of Michelson interferometer. LD is a laser diode, BS is a beam splitter, M1 and M2 are mirrors, PD is a photodiode, and vibration transducer which is driven by a signal generator.]]

The experimental setup Michelson interferometer is shown in figure 2.
[[File:Experimental setup Michelson Interferometer.jpeg|center|thumb|600px|Figure 2: Experimental setup of Michelson interferometer.]]

The interference fringes of the Michelson interferometer are observed when the transducer is turned off in Figure 3.
[[File:Michelson interference fringes.jpeg|center|thumb|600px|Figure 3: Interference fringes which are observed when the transducer is turned off]]

==Experiment Results==
After the alignment of the optical setup, we applied sinusoidal voltage to the transducer attached to the mirror tightly. intensity fluctuation is recorded by the homemade photodiode. Data from the photodiode is collected by the oscilloscope. The typical figure from the oscilloscope is shown in Figure 4. According to the equation mentioned before, the number of peaks during each period corresponds to the radio of displacement and laser wavelength. In this case, we can measure the displacement for various frequencies. The frequency range is from 86 Hz to 130 Hz.

[[File:ScopeScreen.jpeg|center|thumb|600px|Figure 4: The display on the oscilloscope at <math>f = 103Hz </math>. After applying the voltage to the transducer, the voltage output from photodiode changes with time.]]

In our setup, we measured the wavelength of Mitsubishi Laser Diode which is 658.9 nm. We plot the displacement for different frequencies with the errorbar in Figure 5. Uncertainty is the square root uncertainty of the count number. For a driven, damped and harmonic oscillator, the relation between displacement and frequency is

<math>|A|=\frac{f_{0}}{\sqrt{\left(\omega_{0}^{2}-\omega^{2}\right)^{2}+4 \beta^{2} \omega^{2}}} </math>

Where <math>f_{0}</math> is the amplitude, <math>\omega_{0}</math> is the resonance frequency, and <math>\beta_{0}</math> is a constant. With this equation, we can fit the frequency response curve. Because there is only one peak or valley for a frequency lower than 92 Hz or higher than 120 Hz, and the error is significant and non-negligible. We exclude those in the fitting process. Finally, the fitting root mean squared error is 130.38, and the fitting resonance frequency <math>\omega_{0} </math> is 113.12Hz.

[[File:FitPeakNum.jpeg|center|thumb|600px|Figure 5: Frequency response curve for the transducer system with errorbar. The blue dots represent the displacement measured with different frequencies. Error bars are the square root uncertainty. Red line is the fitting curve for a driven, damped, harmonic oscillator.]]

== Conclusion ==
We have shown indeed that we can characterize the resonance frequency of a system. For this particular application, we found a frequency of 113.12Hz for the system transducer plus mirror. Finally, with appropriate handling, this technique could be implemented to characterize more sofisticated systems.

==References==
[1] Freschi, A. A., et al. "Laser interferometric characterization of a vibrating speaker system." American Journal of Physics 71.11 (2003): 1121-1126.
https://doi.org/10.1119/1.1586262

[2] Skarha, Matthew. "Laser interferometric characterization for a vibrating speaker system."
http://mattskarha.com/assets/docs/Laser%20interferometric%20characterization%20for%20a%20vibrating%20speaker%20system%20good.pdf

[3] Pathare, Shirish, and Vikrant Kurmude. "Low cost Michelson–Morley interferometer." Physics Education 51.6 (2016): 063001.
https://doi.org/10.1088/0031-9120/51/6/063001

An interferometric method for measuring the resonance frequency of a vibrating system

2022-04-08T03:48:53Z

Aucca: /* Theory */

===[[An interferometric method for measuring the resonance frequency of a vibrating system]]===

== Team Members ==
* [[User:Nakarin|Nakarin Jayjong]]
* [[User:Aucca|Joel Auccapuclla]]
* [[User:Xiaoyu|Xiaoyu Nie]]
* [[User:Haotian|Haotian Song]]

== Introduction ==
Interferometers since their invention have been a great tool for experimental physics. Applications cover a wide spectrum of fields such as astronomy, fiber optics, seismology, seismology, velocimetry to say the least. The precision of measurement of small displacements is the key factor, take LIGO as an example. Another important factor is the convenience to set up an interferometer and the low complexity in terms of equipment. In this experiment, we will measure the resonance frequency of a vibrating transducer using a Michelson interferometer. Basically, a transducer is attached to one optical path, to one mirror, of the Michelson interferometer and it is expected to see a peak in the response curve for the resonance frequency of the transducer plus mirror.

== Theory ==
Our Michelson interferometer setup used a laser, a beam splitter, two mirrors, and a photometer, illustrated in Fig. 1. Original models used sodium flame as the coherent light source. The light propagates from the source to the beam splitter where approximately 50% of the incident light is transmitted and approximately 50% of it is reflected at a right angle. The transmitted light continues to propagate as the reference plane wave to the stationary mirror where it is fully reflected back towards the beam splitter. The reflected light propagates as the target plane wave to the mirror attached to the oscillating loudspeaker. The two waves then recombine at the beam splitter and interfere based on their relative phases based on the idea that the phase shift of the target plane wave varies with the instantaneous position of the speaker mirror combination. Consider a situation in which we setup our interferometer such that no phase shift corresponds to the equilibrium position of the speaker-mirror combination. At this position, we expect constructive interference at the photometer. Likewise, for every λ/2 change in displacement, we expect constructive interference at the photometer because the path length travelled by the target plane wave varies by some integral multiple of λ. Thus, the phase shift is an even multiple of π and the two waves remain in phase. Similarly, when the path length traveled is some odd multiple of λ/4, the phase shift is an odd multiple of π and the two waves interfere destructively.

Consider two coherent waves with the same amplitude A and same frequency <math>\omega</math> that have paths <math>x_1</math> and <math>x_2</math> emitted from different sources that meet together at a point. The net wave at that point can then be given as the real part of the equation

<div class="center" style="width:auto; margin-left:auto; margin-right:auto;">
<math>
\begin{alignat}{2}
\psi & = Ae^{i(kx_1-\omega t}+Ae^{i(kx_2-\omega t)} \\
& = Ae^{-i\omega t}e^{ik\frac{(x_1+x_2)}{2}}[e^{ik\frac{(x_1-x_2)}{2}}e^{-ik\frac{(x_1-x_2)}{2}}]\\
& = 2Acos[k(\frac{x_1-x_2}{2})]e^{i[k\frac{(x_1+x_2)}{2}-\omega t]}
\end{alignat}
</math>
</div>

where the wave number <math>k=2\pi/\lambda</math>, <math>\lambda</math> is the wavelength of the waves, and c is the phase velocity. The time-averaged intensity of the combined wave is then proportional to the square of the its amplitude. That is,

<div class="center" style="width:auto; margin-left:auto; margin-right:auto;">
<math>
I\propto cos^2[\pi(\frac{p_2-p_1}{\lambda})]=cos^2[2\pi(\frac{p_2}{\lambda}-\theta)]
</math>
</div>

where <math>\theta</math> is a constant. Now suppose the mirror that the <math>p_2</math> beam is incident to oscillates harmonically back and forth with a frequency <math>f</math> and a displacement <math>B</math>. Then, <math>p_2 =p_2(0)+Bsin(2\pi ft)</math> and the intensity of the detected light varies with time as

<div class="center" style="width:auto; margin-left:auto; margin-right:auto;">
<math>
I=A^2cos^2[2\pi(\frac{Bsin(2\pi ft)}{\lambda}-\theta')]
</math>
</div>

Note that the frequency of oscillations is itself a sinusoid, meaning that there will be turnaround points in the oscillations.

== Experimental setup ==
The Michelson interferometer consists of a beam splitter BS which is used to split the laser beam into two beams and two mirrors M1 and M2. The reflected beam goes to a fixed mirror M1 while the transmitted beam goes to the movable mirror M2. A transducer is attached to the mirror M2. The transducer is powered by a signal generator that is used to vary the vibrating frequency of the mirror. A photodiode PD is used to detect the signal of interference fringes. The schematic Michelson interferometer is shown in Figure 1.
[[File:Schematic Michelson Interferometer.jpeg|center|thumb|600px|Figure 1: Schematic figure of Michelson interferometer. LD is a laser diode, BS is a beam splitter, M1 and M2 are mirrors, PD is a photodiode, and vibration transducer which is driven by a signal generator.]]

The experimental setup Michelson interferometer is shown in figure 2.
[[File:Experimental setup Michelson Interferometer.jpeg|center|thumb|600px|Figure 2: Experimental setup of Michelson interferometer.]]

The interference fringes of the Michelson interferometer are observed when the transducer is turned off in Figure 3.
[[File:Michelson interference fringes.jpeg|center|thumb|600px|Figure 3: Interference fringes which are observed when the transducer is turned off]]

==Experiment Results==
After the alignment of the optical setup, we applied sinusoidal voltage to the transducer is attached to the mirror tightly. intensity fluctuation is recorded by the home-made photodiode. Data from the photodiode is collected by the oscilloscope. The typical figure from the oscilloscope is shown in Figure 4. According to the equation ?, the number of peaks during each period corresponds to the radio of displacement and laser wavelength. In this case, we can measure the displacement for various frequencies. The frequency range is from 86 Hz to 130 Hz.

[[File:ScopeScreen.jpeg|center|thumb|600px|Figure 4: The display on the oscilloscope at <math>f = 103Hz </math>. After applying the voltage to the transducer, the voltage output from photodiode changes with time.]]

In our setup, Mitsubishi Laser Diode operated at a wavelength of 660 nm. We plot the displacement for different frequencies with the errorbar in Figure 5. Uncertainty is the square root uncertainty of the count number. For a driven, damped and harmonic oscillator, the relation between displacement and frequency is

<math>|A|=\frac{f_{0}}{\sqrt{\left(\omega_{0}^{2}-\omega^{2}\right)^{2}+4 \beta^{2} \omega^{2}}} </math>

Where <math>f_{0}</math> is the amplitude, <math>\omega_{0}</math> is the resonance frequency, and <math>\beta_{0}</math> is a constant. With this equation, we can fit the frequency response curve. Because there is only one peak or valley for a frequency lower than 92 Hz or higher than 120 Hz, and the error is significant and non-negligible. We exclude those in the fitting process. Finally, the fitting root mean squared error is 130.38, and the fitting resonance frequency <math>\omega_{0} </math> is 113.12Hz.

[[File:FitPeakNum.jpeg|center|thumb|600px|Figure 5: Frequency response curve for the transducer system with errorbar. The blue dots represent the displacement measured with different frequencies. Error bars are the square root uncertainty. Red line is the fitting curve for a driven, damped, harmonic oscillator.]]
== Conclusion ==
We have shown indeed that we can characterize the resonance frequency of a system. For this particular application, we found a frequency of 113.12Hz for the system transducer plus mirror. Finally, with appropriate handling, this technique could be implemented to characterize more sofisticated systems.

==References==
[1] Freschi, A. A., et al. "Laser interferometric characterization of a vibrating speaker system." American Journal of Physics 71.11 (2003): 1121-1126.
https://doi.org/10.1119/1.1586262

[2] Skarha, Matthew. "Laser interferometric characterization for a vibrating speaker system."
http://mattskarha.com/assets/docs/Laser%20interferometric%20characterization%20for%20a%20vibrating%20speaker%20system%20good.pdf

[3] Pathare, Shirish, and Vikrant Kurmude. "Low cost Michelson–Morley interferometer." Physics Education 51.6 (2016): 063001.
https://doi.org/10.1088/0031-9120/51/6/063001