An interferometric method for measuring the resonance frequency of a vibrating system

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 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 $\omega$ that have paths $x_{1}$ and $x_{2}$ 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

${\begin{alignedat}{2}\psi &=Ae^{i(kx_{1}-\omega t}+Ae^{i(kx_{2}-\omega t)}\\&=Ae^{-i\omega t}e^{ik{\frac {(x_{1}+x_{2})}{2}}}[e^{ik{\frac {(x_{1}-x_{2})}{2}}}e^{-ik{\frac {(x_{1}-x_{2})}{2}}}]\\&=2Acos[k({\frac {x_{1}-x_{2}}{2}})]e^{i[k{\frac {(x_{1}+x_{2})}{2}}-\omega t]}\end{alignedat}}$

where the wave number $k=2\pi /\lambda$, $\lambda$ 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,

$I\propto cos^{2}[\pi ({\frac {p_{2}-p_{1}}{\lambda }})]=cos^{2}[2\pi ({\frac {p_{2}}{\lambda }}-\theta )]$

where $p_{1}$ is the distance from beam splitter to fixed mirror $M1$, and $p_{2}$ is the distance from beam splitter to movable mirror $M2$. Now suppose the mirror that the $p_{2}$ beam is incident to oscillates harmonically back and forth with a frequency $f$ and a displacement $B$, while the distance $p_{1}$ is constant, and all of constant term can be rewritten into a phase different $\theta$. Then, $p_{2}=p_{2}(0)+Bsin(2\pi ft)$ and the intensity of the detected light varies with time as

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

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.

Figure 1: Schematic figure of Michelson interferometer. $LD$ is a laser diode, $BS$ is a beam splitter, $M1$ and $M2$ are mirrors, $p_{1}$ is the distance from $BS$ to $M1$, and $p_{2}$ is the distance from $BS$ to $M2$, $PD$ is a photodiode, and vibration transducer is driven by a signal generator.

The experimental setup Michelson interferometer is shown in figure 2.

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.

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.

Figure 4: The display on the oscilloscope at $f=103Hz$. 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

$|A|={\frac {f_{0}}{\sqrt {\left(\omega _{0}^{2}-\omega ^{2}\right)^{2}+4\beta ^{2}\omega ^{2}}}}$

Where $f_{0}$ is the amplitude, $\omega _{0}$ is the resonance frequency, and $\beta _{0}$ 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 $\omega _{0}$ is 113.12Hz.

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