Homodyne detection - Revision history

Johnkhootf at 15:58, 30 April 2022

2022-04-30T15:58:44Z

← Older revision		Revision as of 15:58, 30 April 2022
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	At higher signal frequencies, GR noise is the dominant noise source, and 1/f noise becomes negligible. The noise current can be written as		At higher signal frequencies, GR noise is the dominant noise source, and 1/f noise becomes negligible. The noise current can be written as
	<math display='block'>\left\langle i^2_N \right\rangle=\frac{4e\left\langle i \right\rangle (\tau_0/\tau_d)}{1+4\pi^2f^2\tau^2_0},</math>		<math display='block'>\left\langle i^2_N \right\rangle=\frac{4q_e\left\langle i \right\rangle (\tau_0/\tau_d)}{1+4\pi^2f^2\tau^2_0},</math>
	where <math>\tau_0</math> is the characteristic lifetime random combination occurs, <math>\tau_d</math> is the time taken for a charge carrier to move into external current. So, if random recombination occurs less frequently, less noise is generated.		where <math>\tau_0</math> is the characteristic lifetime random combination occurs, <math>\tau_d</math> is the time taken for a charge carrier to move into external current. So, if random recombination occurs less frequently, less noise is generated.

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	[[File:HomodyneDetection_AutoVariation.mp4\|right\|thumb\|250px\|The voltage varies inexplicably over time without any interference. The variation is not sinusoidal, with long pauses between the extrema. We are unsure what causes this phenomenon.]]		[[File:HomodyneDetection_AutoVariation.mp4\|right\|thumb\|250px\|The voltage varies inexplicably over time without any interference. The variation is not sinusoidal, with long pauses between the extrema. We are unsure what causes this phenomenon.]]

	We now estimate the noise in our detector system. The effective input impedance, from the resistive load of <math>1~\mathrm{M\Omega}</math> and the input impedance of <math>1~\mathrm{M\Omega}</math> being connected in parallel, is <math>0.5~\mathrm{M\Omega}</math>. First, we consider the shot noise. As previously discussed, the rms current of the shot noise is given by <math>\left\langle i^2_N \right\rangle=2e\bar{i}\Delta f</math>, where <math>\bar{i}</math> is the average photocurrent that is given by <math>\bar{i} = \left\langle I \right\rangle A \eta q_e/h\nu</math>. The maximum voltage (with the zero reference being the voltage reading when no light is incident) is 10V, meaning the detected power is <math>2.0 \times 10^{-4}~\mathrm{W}</math>, since by Ohm's law <math>P = V^2/R</math>. This gives us <math>\bar{i}=1.046 \times 10^{-5}</math> and therefore, <math display>\left\langle i^2_N \right\rangle=3.352 \times 10^{-15}</math>, giving a noise voltage of <math>4.09 \times 10^{-2}~\mathrm{V}</math>.		We now estimate the noise in our detector system. The effective input impedance, from the resistive load of <math>1~\mathrm{M\Omega}</math> and the input impedance of <math>1~\mathrm{M\Omega}</math> being connected in parallel, is <math>0.5~\mathrm{M\Omega}</math>. First, we consider the shot noise. As previously discussed, the rms current of the shot noise is given by <math>\left\langle i^2_N \right\rangle=2q_e\bar{i}\Delta f</math>, where <math>\bar{i}</math> is the average photocurrent that is given by <math>\bar{i} = \left\langle I \right\rangle A \eta q_e/h\nu</math>. The maximum voltage (with the zero reference being the voltage reading when no light is incident) is 10V, meaning the detected power is <math>2.0 \times 10^{-4}~\mathrm{W}</math>, since by Ohm's law <math>P = V^2/R</math>. This gives us <math>\bar{i}=1.046 \times 10^{-5}</math> and therefore, <math display>\left\langle i^2_N \right\rangle=3.352 \times 10^{-15}</math>, giving a noise voltage of <math>4.09 \times 10^{-2}~\mathrm{V}</math>.

	We next consider Johnson noise. Through the equation <math>\left\langle i^2_N \right\rangle=4k_BT\Delta f/R</math>, we find a current of <math display>\left\langle i^2_N \right\rangle=1.646 \times 10^{-18}</math> at a room temperature of <math>T = 298~\mathrm{K}</math>. This gives an rms voltage of <math>9.07 \times 10^{-4}~\mathrm{V}</math>.		We next consider Johnson noise. Through the equation <math>\left\langle i^2_N \right\rangle=4k_BT\Delta f/R</math>, we find a current of <math display>\left\langle i^2_N \right\rangle=1.646 \times 10^{-18}</math> at a room temperature of <math>T = 298~\mathrm{K}</math>. This gives an rms voltage of <math>9.07 \times 10^{-4}~\mathrm{V}</math>.

Johnkhootf: /* Noise */

2022-04-30T15:55:56Z

Noise

← Older revision		Revision as of 15:55, 30 April 2022
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	From this, we obtain		From this, we obtain
	<math display='block'>\left\langle i^2_N(f) \right\rangle=2e\bar{i}\mathrm{d}f,</math>		<math display='block'>\left\langle i^2_N(f) \right\rangle=2q_e\bar{i}\mathrm{d}f,</math>
	which must be multiplied against the sampling bandwidth <math>\Delta f</math> to obtain the shot noise current.		which must be multiplied against the sampling bandwidth <math>\Delta f</math> to obtain the shot noise current.

Johnkhootf at 15:52, 30 April 2022

2022-04-30T15:52:43Z

← Older revision		Revision as of 15:52, 30 April 2022
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	[[Image:HomodyneDetection_Michelson.png\|center\|thumb\|300px\|Schematic representation of the Michelson interferometer setup.]]		[[Image:HomodyneDetection_Michelson.png\|center\|thumb\|300px\|Schematic representation of the Michelson interferometer setup.]]

	In order to understand the image formation better, we can consider the virtual images of <math>M_2</math> and the source in the plane of reflection defined by the beam splitter~~. The interference pattern formed can be determined by considering only the light from <math>S_1</math> and <math>S_2</math> incident on the screen~~.		In order to understand the image formation better, we can consider the virtual images of <math>M_2</math> and the source in the plane of reflection defined by the beam splitter.

	[[Image:HomodyneDetection_MichelsonReflections.png\|center\|thumb\|350px\|The images formed by the reflections in a Michelson interferometer.]]		[[Image:HomodyneDetection_MichelsonReflections.png\|center\|thumb\|350px\|The images formed by the reflections in a Michelson interferometer.]]

	~~The interference pattern formed can be determined by considering only the light from <math>S_1</math> and <math>S_2</math> incident on the screen.~~
	[[Image:HomodyneDetection_MichelsonPaths.png\|right\|thumb\|300px\|The apparent paths travelled by the virtual sources to the screen.]]		[[Image:HomodyneDetection_MichelsonPaths.png\|right\|thumb\|300px\|The apparent paths travelled by the virtual sources to the screen.]]

	The path from <math>S_2</math> has length		The interference pattern formed can be determined by considering only the light from <math>S_1</math> and <math>S_2</math> incident on the screen. The path from <math>S_2</math> has length
	<math display='block'>		<math display='block'>
	\begin{align}		\begin{align}
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	<math display='block'>		<math display='block'>
	\begin{align}		\begin{align}
	r &= ~~\ell\cos\theta~~\sin(\theta + \theta_1) = \frac{\ell}{2}\left[ ~~1 -~~ \cos(2\theta + \theta_1)\right] \\		r &= d\sin(\theta + \theta_1) = \frac{\ell}{\cos(\theta)}\left[ \sin\theta\cos\theta_1 + \cos\theta\sin\theta_1 \right] = \ell\left[ \sin\theta_1 + \tan\theta \cos\theta_1 \right] \\
	z &= ~~\ell\cos\theta~~\cos(\theta + \theta_1) = \frac{\ell}{2}\left[~~1 +~~ \cos(2\theta + \theta_1)\right],		z &= d\cos(\theta + \theta_1) = \frac{\ell}{\cos(\theta)}\left[ \cos\theta\cos\theta_1 - \sin\theta\sin\theta_1 \right] = \ell\left[ \cos\theta_1 - \tan\theta \sin\theta_1 \right]
	\end{align}		\end{align}
	</math>		</math>
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	\frac{1}{R(z)} = \frac{1}{z} \left( \frac{1}{1+\left(\frac{z_R}{z}\right)^2} \right) \leq \frac{1}{z},		\frac{1}{R(z)} = \frac{1}{z} \left( \frac{1}{1+\left(\frac{z_R}{z}\right)^2} \right) \leq \frac{1}{z},
	</math>		</math>
	~~and~~ the factor only depends on <math>\theta</math> through <math>z = \ell\left[~~1 +~~ \cos(2\theta + \theta_1)\right]/2</math>, what we might expect to see is the fringes ~~spreading out more~~ as ~~the magnitude angle~~ <math>2\theta + \theta_1</math>~~, which has range~~ <math>[-2\theta_b-\theta_1, 2\theta_b+\theta_1]</math>~~, increases. When this magnitude~~ is small, however, ~~we would expect~~ the ~~approximation to hold well~~. This corresponds to a narrow beam divergence (<math>\theta_b</math> small), which a typical laser would have, and small misalignments (<math>\theta_1</math> small).		it seems the Gaussian beam pattern has a higher spatial frequency than the plane wave. Since the factor only depends on <math>\theta</math> through <math>z = \ell\left[ \cos\theta_1 - \tan\theta \sin\theta_1 \right]</math>, what we might expect to see is the fringes spatial frequency decreasing as <math>\tan\theta\sin\theta_1</math> becomes more positive. If <math>\theta \in [-\theta_b -\theta_1, \theta_b -\theta_1]</math> is small, however, the spatial frequency is approximately constant. This corresponds to a narrow beam divergence (<math>\theta_b</math> small), which a typical laser would have, and small misalignments (<math>\theta_1</math> small). <math>\|\sin\theta_1\| \ll 1</math> by itself would also lead to a good approximation, since it reduces the dependence of <math>z</math> on <math>\theta</math>.

	Based on the equations above, if we can find the size and position of the beam waist, that is, <math>w_0</math> and the physical location where <math>z=0</math>, we can completely characterise the Gaussian beam. If the total power of the beam is given by <math>P_0</math>, and the beam at a particular <math>z</math> is partially obscured by a solid surface (e.g.\ a knife edge) that is modelled as blocking the half plane <math>x \in (-\infty, x_0], y \in (-\infty, \infty)</math>, the measured power will be given by		Based on the equations above, if we can find the size and position of the beam waist, that is, <math>w_0</math> and the physical location where <math>z=0</math>, we can completely characterise the Gaussian beam. If the total power of the beam is given by <math>P_0</math>, and the beam at a particular <math>z</math> is partially obscured by a solid surface (e.g.\ a knife edge) that is modelled as blocking the half plane <math>x \in (-\infty, x_0], y \in (-\infty, \infty)</math>, the measured power will be given by

Johnkhootf: /* Conclusion */

2022-04-30T07:03:45Z

Conclusion

← Older revision		Revision as of 07:03, 30 April 2022
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	== Conclusion ==		== Conclusion ==

	I'm really glad I'm a theorist.		In the course of this project, we analysed and built a Michelson interferometer from an assortment of optomechanical components and a laser pointer. We have learnt various practical skills relating to the improvised construction of mounts and electrical systems, the assembly of components using specialised materials, and the alignment of optical components. We have also performed a theoretical analysis and qualitatively confirmed some of our predictions, and surprisingly managed to construct a shot-noise limited detector.

			Nonetheless, despite extensive effort, we were unable to repair the EOM, due to an electrical fault in the socket. While we had thought to check the electrical connectivity of the rest of the circuit, we did not expect that there would be issues with the socket. This is a learning point to check all parts of each component carefully, especially if they are of dubious provenance.

			In conclusion, I'm really glad I'm a theorist.

Johnkhootf: /* Voltage Readings */ Touch up

2022-04-30T06:47:18Z

Voltage Readings: Touch up

← Older revision		Revision as of 06:47, 30 April 2022
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	[[File:HomodyneDetection_AutoVariation.mp4\|right\|thumb\|250px\|The voltage varies inexplicably over time without any interference. The variation is not sinusoidal, with long pauses between the extrema. We are unsure what causes this phenomenon.]]		[[File:HomodyneDetection_AutoVariation.mp4\|right\|thumb\|250px\|The voltage varies inexplicably over time without any interference. The variation is not sinusoidal, with long pauses between the extrema. We are unsure what causes this phenomenon.]]

	We now estimate the noise in our detector system. The effective input impedance, from the resistive load of <math>1~\mathrm{M\Omega}</math> and the input impedance of <math>1~\mathrm{M\Omega}</math> being connected in parallel, is <math>0.5~\mathrm{M\Omega}</math>. First, we consider the shot noise. As previously discussed, the rms current of the shot noise is given by <math>\left\langle i^2_N~~(f)~~ \right\rangle=2e\bar{i}\Delta f</math>, where <math>\bar{i}</math> is the average photocurrent that is given by <math> \bar{i}=~~\eta q_e/h\nu~~\left\langle I \right\rangle A ~~</math>. <math>P =~~ \~~left\langle I \right\rangle A</math> is the average power of the light incident on the photodiode, <math>~~q_e</~~math> is the elementary charge, and <math>~~\~~eta~~</math> ~~is the quantum efficiency~~. The maximum voltage (with the zero reference being the voltage reading when no light is incident) is 10V, meaning the detected power is <math>2.0 \times 10^{-4}~\mathrm{W}</math>, since by Ohm's law <math>P = V^2/R</math>. This gives us <math>\bar{i}=1.046 \times 10^{-5}</math> and therefore, <math display>\left\langle i^2_N~~(f)~~ \right\rangle=3.352 \times 10^{-15}</math>, giving a noise voltage of <math>4.09 \times 10^{-2}~\mathrm{V}</math>.		We now estimate the noise in our detector system. The effective input impedance, from the resistive load of <math>1~\mathrm{M\Omega}</math> and the input impedance of <math>1~\mathrm{M\Omega}</math> being connected in parallel, is <math>0.5~\mathrm{M\Omega}</math>. First, we consider the shot noise. As previously discussed, the rms current of the shot noise is given by <math>\left\langle i^2_N \right\rangle=2e\bar{i}\Delta f</math>, where <math>\bar{i}</math> is the average photocurrent that is given by <math>\bar{i} = \left\langle I \right\rangle A \eta q_e/h\nu</math>. The maximum voltage (with the zero reference being the voltage reading when no light is incident) is 10V, meaning the detected power is <math>2.0 \times 10^{-4}~\mathrm{W}</math>, since by Ohm's law <math>P = V^2/R</math>. This gives us <math>\bar{i}=1.046 \times 10^{-5}</math> and therefore, <math display>\left\langle i^2_N \right\rangle=3.352 \times 10^{-15}</math>, giving a noise voltage of <math>4.09 \times 10^{-2}~\mathrm{V}</math>.

	We next consider Johnson noise. Through the equation <math>\left\langle i^2_N \right\rangle=4k_BT\Delta f/R</math>, we find a current of <math display>\left\langle i^2_N \right\rangle=1.646*10^{-18}</math> at a room temperature of <math>T = 298~\mathrm{K}</math>. This gives an rms voltage of <math>9.07 \times 10^{-4}~\mathrm{V}</math>.		We next consider Johnson noise. Through the equation <math>\left\langle i^2_N \right\rangle=4k_BT\Delta f/R</math>, we find a current of <math display>\left\langle i^2_N \right\rangle=1.646 \times 10^{-18}</math> at a room temperature of <math>T = 298~\mathrm{K}</math>. This gives an rms voltage of <math>9.07 \times 10^{-4}~\mathrm{V}</math>.

	Lastly, while the definition of the GR noise requires detailed knowledge of the electron-hole recombination process, it has estimated by the manufacturer as the ''noise-equivalent power'' or NEP in the datasheet. This is given as <math>1.6 \times 10^{-14}~\mathrm{W}~(\mathrm{Hz})^{-1/2}</math>, giving us an rms noise voltage of <math>1.3 \times 10^{-2}~\mathrm{V}</math>.		Lastly, while the definition of the GR noise requires detailed knowledge of the electron-hole recombination process, it has estimated by the manufacturer as the ''noise-equivalent power'' or NEP in the datasheet. This is given as <math>1.6 \times 10^{-14}~\mathrm{W}~(\mathrm{Hz})^{-1/2}</math>, giving us an rms noise voltage of <math>1.3 \times 10^{-2}~\mathrm{V}</math>.

	Remarkably, despite the simplicity of our setup, the dominant source of noise in this setup is shot ~~noise---we~~ have managed to build a shot-noise limited detector. All sources of noise (aside from the inexplicable low-frequency noise) are several orders of magnitude lower than the variation in voltage from phase changes, implying that this system can be used effectively for modulating and transmitting information.		Remarkably, despite the simplicity of our setup, the dominant source of noise in this setup is shot noise—we have managed to build a shot-noise limited detector. All sources of noise (aside from the inexplicable low-frequency noise) are several orders of magnitude lower than the variation in voltage from phase changes, implying that this system can be used effectively for modulating and transmitting information.

	=== Laser Beam ===		=== Laser Beam ===

Johnkhootf: /* Noise */ Touch up

2022-04-30T06:43:00Z

Noise: Touch up

Johnkhootf: /* Noise */ Fix noise section

2022-04-30T05:59:21Z

Noise: Fix noise section

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 Finally, we have to control for the effect of noise in our experiment. There are four primary sources of noise in semiconductor photodetectors:
 * ''Shot noise'', or ''quantum noise'', arising from the quantisation of light: because the light must arrive at the detector one photon at a time, there will be some fluctuation in the signal read at any point in time.
 * ''Nyquist-Johnson noise'' or ''thermal noise'', arising from thermal fluctuations in the electronics.
 * ''<math>1/f</math> noise'', which is apparently a very mysterious and universal phenomenon caused by various different factors, but has a power spectral density proportional to <math>1/f</math>, hence the name.
 * ''Generation-recombination (GR) noise'', caused by holes and electrons randomly being generated and recombining in the semiconductor.
-<math>\left\langle i^2_N \right\rangle=\frac{4e\left\langle i \right\rangle (\tau_0/\tau_d)}{1+4\pi^2f^2\tau^2_0} </math>
+The noise received depends on both the frequency of the light signal <math>\nu</math> and the signal frequency <math>f</math>, also expressed as <math>\omega = 2\pi{f}</math>. These are not to be confused with each other. <math>\Delta f</math> is the ''sampling bandwidth'', determined by how often take electrical measurements. By Nyquist's theorem, we must have a sampling period <math>T = 1/\Delta f \leq 1/2f</math> to reconstruct a modulated signal of maximum frequency component <math>f</math>: that is, we must have <math>\Delta f \geq 2f</math>.
-<math>\tau_0</math> is the characteristic lifetime random combination occurs, <math>\tau_d</math> is the time charge carrier goes to external current. So, if random recombination occurs less frequently, less noise is generated.
+The Nyquist-Johnson noise current is given by
-<math>\Delta f</math> is the ''sampling'' bandwidth, related to how often we take electrical measurements. By Nyquist's theorem, we must have a sampling period <math>T = 1/\Delta f \leq 1/2f</math> to reconstruct a modulated signal of maximum frequency component <math>f</math>: that is, we must have <math>\Delta f \geq 2f</math>.
+At higher signal frequencies, GR noise is the dominant noise source, and 1/f noise becomes negligible. The noise current can be written as
 == Setup & Methodology ==

Johnkhootf: /* Results & Discussion */

2022-04-30T05:30:37Z

Results & Discussion

← Older revision		Revision as of 05:30, 30 April 2022
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	=== Apparatus Parameters ===		=== Apparatus Parameters ===
	We are using a Hamamatsu [https://www.hamamatsu.com/content/dam/hamamatsu-photonics/sites/documents/99_SALES_LIBRARY/ssd/s5106_etc_kpin1033e.pdf S5106 Si PIN photodiode] as our detector, operated at 10V, with a resistive load of <math>1~\mathrm{M\Omega}</math>. Measurements are taken with a Kenwood CS-5270 oscilloscope with <math>\Delta f = 100~\mathrm{MHz}</math>.		We are using a Hamamatsu [https://www.hamamatsu.com/content/dam/hamamatsu-photonics/sites/documents/99_SALES_LIBRARY/ssd/s5106_etc_kpin1033e.pdf S5106 Si PIN photodiode] as our detector, operated at 10V, with a resistive load of <math>1~\mathrm{M\Omega}</math>. Measurements are taken with a Kenwood CS-5270 oscilloscope with <math>\Delta f = 100~\mathrm{MHz}</math> and an input impedance of <math>1~\mathrm{M\Omega}</math>

	The <chem>LiNbO3</chem> crystal is 3.2mm wide and tall (square cross-section) and 2cm long. The laser pointer lens has a diameter of 3mm, and the wavelength is claimed to be 650nm ± 10nm. We also had access to a pinhole of diameter 1mm.		The <chem>LiNbO3</chem> crystal is 3.2mm wide and tall (square cross-section) and 2cm long. The laser pointer lens has a diameter of 3mm, and the wavelength is claimed to be 650nm ± 10nm. We also had access to a pinhole of diameter 1mm.

	~~=== Laser Beam ===~~

	Despite the process of measuring the laser spot size being rather straightforward to describe, we did not have the equipment to perform a sufficiently precise characterisation. An attempt was made by sliding a box along the edge of a folded piece of paper, with the intention to mark the position of the box on the piece of paper and record the corresponding intensity. However, the lack of rigidity of the apparatus, the lack of precision of our measurement instruments (a pen and a ruler), a lack of manual dexterity and a lack of time to work around these problems made this infeasible.

	~~[[Image:HomodyneDetection_AiryDisk.jpg\|right\|thumb\|250px\|An Airy disk formed by transmitting the beam through a pinhole, and expanding the beam using a lens.]]~~

	Further, as is visible from our images, our beam profile is nowhere close to Gaussian, and instead rather jagged and irregular. One possibility for cleaning up the signal was to send it through a small circular pinhole first. Indeed, by doing so, we were able to observe an Airy disk. However, due to the aperture being much smaller than the beam, a significant fraction of the light was lost. The resulting photocurrent only induced a maximum of a little over 2V, as compared to 10V with the full beam being used. The interference patterns formed were also less visible, so we did not primarily rely on this method to perform our observations.

	~~[[File:HomodyneDetection_PinholeIncident.mp4\|right\|thumb\|250px\|The voltage variation when blocking and unblocking the laser, after filtering it through a pinhole.]]~~

	~~However, we were able to induce a larger variation in the voltage, consisting of almost the full 2V, by placing pressure on the mirror and thereby inducing phase variations.~~

	~~[[File:HomodyneDetection_PinholePressure.mp4\|right\|thumb\|250px\|The voltage variation when placing pressure on the mirrors, after filtering it through a pinhole.]]~~

	=== Michelson Interferometer Fringes ===		=== Michelson Interferometer Fringes ===
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	=== Voltage Readings ===		=== Voltage Readings ===

	~~We are~~ able to ~~induce~~ a variation in ~~voltage~~ by ~~shifting~~ the ~~phase~~ of the oscilloscope		Even before fixing the EOM, we were able to observe a quantifiable change in the voltage produced with variations in the phase of the interferometer after it was aligned to produce large or circular fringes. The video below shows the variation in intensity of the interference pattern magnified by a lens, while one of the mirrors was being tapped.
			[[File:HomodyneDetection_IntensityTapping.mp4\|center\|thumb\|250px\|Tapping one of the mirrors results in small changes in path length, leading in turn to a change in the interference pattern and therefore the overall intensity.]]

			There is a 10V difference in measured voltage between points in time where the beam is blocked, and when it is unblocked, as shown in the video below.
			[[File:HomodyneDetection_FullIncident.mp4\|center\|thumb\|250px\|The voltage on the oscilloscope drops significantly when the light is incident on the detector.]]

	[[File:~~HomodyneDetection_BlockedVIncident~~.mp4\|~~right~~\|thumb\|250px\|The voltage on the oscilloscope drops significantly when the light is incident on the detector.]]		Placing pressure on and tapping one of the mirrors shifts the phase of the beam, changing the intensity of the interference pattern. The variation in intensity only spanned 7V, providing quantitative evidence of an incoherent component in the light contributing the remaining 3V, which we observed visually in the interference.
			[[File:HomodyneDetection_FullTapping.mp4\|center\|thumb\|250px\|The voltage on the oscilloscope drops significantly when the light is incident on the detector.]]
			[[File:HomodyneDetection_FullPressure.mp4\|center\|thumb\|250px\|The voltage on the oscilloscope drops significantly when the light is incident on the detector.]]

	~~[[File:HomodyneDetection_IntensityTapping~~.~~mp4\|right\|thumb\|250px\|Tapping one of~~ the ~~mirrors results in small changes in path length~~, ~~leading in turn to~~ a ~~change in the~~ interference pattern ~~and therefore~~ the ~~overall intensity~~.]]		However, we also observed an unusual variation in the signal at some points. This behaviour would come and go inexplicably, and we are unsure what causes it. We also observed that, upon being switched on, the laser would sometimes vary its phase at a constant rate, creating a shifting interference pattern. This unusual behaviour resolved after restarting the power supply, but we are also unsure what causes it.

	[[File:HomodyneDetection_AutoVariation.mp4\|right\|thumb\|250px\|The voltage varies inexplicably over time without any interference. The variation is not sinusoidal, with long pauses between the extrema. We are unsure what causes this phenomenon.]]		[[File:HomodyneDetection_AutoVariation.mp4\|right\|thumb\|250px\|The voltage varies inexplicably over time without any interference. The variation is not sinusoidal, with long pauses between the extrema. We are unsure what causes this phenomenon.]]

	~~TODO talk about change~~ in ~~voltage observed with~~ and ~~without~~ the ~~laser and as~~ the ~~phase~~ is ~~changed~~, ~~estimate~~ power being ~~received using~~ voltage and ~~resistance~~, ~~estimate the various sources~~ of ~~noise~~.		We now estimate the noise in our detector system. The effective input impedance, from the resistive load of <math>1~\mathrm{M\Omega}</math> and the input impedance of <math>1~\mathrm{M\Omega}</math> being connected in parallel, is <math>0.5~\mathrm{M\Omega}</math>. First, we consider the shot noise. As previously discussed, the rms current of the shot noise is given by <math>\left\langle i^2_N(f) \right\rangle=2e\bar{i}\Delta f</math>, where <math>\bar{i}</math> is the average photocurrent that is given by <math> \bar{i}=\eta q_e/h\nu\left\langle I \right\rangle A </math>. <math>P = \left\langle I \right\rangle A</math> is the average power of the light incident on the photodiode, <math>q_e</math> is the elementary charge, and <math>\eta</math> is the quantum efficiency. The maximum voltage (with the zero reference being the voltage reading when no light is incident) is 10V, meaning the detected power is <math>2.0 \times 10^{-4}~\mathrm{W}</math>, since by Ohm's law <math>P = V^2/R</math>. This gives us <math>\bar{i}=1.046 \times 10^{-5}</math> and therefore, <math display>\left\langle i^2_N(f) \right\rangle=3.352 \times 10^{-15}</math>, giving a noise voltage of <math>4.09 \times 10^{-2}~\mathrm{V}</math>.

	Finally, we estimate the noise in the detector in order to estimate our SNR. First, we consider the shot noise. As previously discussed, the rms current of the shot noise is given by <math>\left\langle i^2_N(f) \right\rangle=2e\bar{i}\Delta f</math>, where <math>\bar{i}</math> is the average photocurrent that is given by <math> \bar{i}=\frac{\eta q_e}{h\nu}\left\langle I \right\rangle A </math>. <math>\left\langle I \right\rangle A</math> is the power of the light incident on the photodiode, <math>q_e</math> is the elementary charge, and <math>\eta</math> is the quantum efficiency. The maximum voltage (with the zero reference being the voltage reading when no light is incident) is 10 V, meaning the detected power is <math>1.0 \times 10^{-4}~\mathrm{W}</math>, since by Ohm's law <math>P = V^2/R</math>, where load resistance ''R'' is the resistance across which the photocurrent flows. This gives us <math>\bar{i}=5.2310^{-5}</math> and therefore, <math display>\left\langle i^2_N(f) \right\rangle=1.67610^{-15}</math>, giving a noise voltage of <math>4.09 \times 10^{-2}~\mathrm{V}</math>.		We next consider Johnson noise. Through the equation <math>\left\langle i^2_N \right\rangle=4k_BT\Delta f/R</math>, we find a current of <math display>\left\langle i^2_N \right\rangle=1.646*10^{-18}</math> at a room temperature of <math>T = 298~\mathrm{K}</math>. This gives an rms voltage of <math>9.07 \times 10^{-4}~\mathrm{V}</math>.

	We next consider Johnson noise. Through the equation <math>\left\langle i^2_N \right\rangle=~~\frac{~~4k_BT~~}{R}~~\Delta f</math>, we find a current of <math display>\left\langle i^2_N \right\rangle=1.646*10^{-18}</math> at a room temperature of <math>T = 298~\mathrm{K}</math>. This gives an rms voltage of <math>1.29 \times 10^{-3}~\mathrm{V}</math>.

	Lastly, while the definition of the GR noise requires detailed knowledge of the electron-hole recombination process, it has estimated by the manufacturer as the ''noise-equivalent power'' or NEP in the datasheet. This is given as <math>1.6 \times 10^{-14}~\mathrm{W}~(\mathrm{Hz})^{-1/2}</math>, giving us an rms noise voltage of <math>1.3 \times 10^{-2}~\mathrm{V}</math>.		Lastly, while the definition of the GR noise requires detailed knowledge of the electron-hole recombination process, it has estimated by the manufacturer as the ''noise-equivalent power'' or NEP in the datasheet. This is given as <math>1.6 \times 10^{-14}~\mathrm{W}~(\mathrm{Hz})^{-1/2}</math>, giving us an rms noise voltage of <math>1.3 \times 10^{-2}~\mathrm{V}</math>.

	Remarkably, despite the simplicity of our setup, the dominant source of noise in this setup is shot noise---we have managed to build a shot-noise limited detector. All sources of noise (aside from the inexplicable low-frequency noise) are several orders of magnitude lower than the variation in voltage from phase changes, implying that this system can be used effectively for modulating and transmitting information.		Remarkably, despite the simplicity of our setup, the dominant source of noise in this setup is shot noise---we have managed to build a shot-noise limited detector. All sources of noise (aside from the inexplicable low-frequency noise) are several orders of magnitude lower than the variation in voltage from phase changes, implying that this system can be used effectively for modulating and transmitting information.

			=== Laser Beam ===

			Despite the process of measuring the laser spot size being rather straightforward to describe, we did not have the equipment to perform a sufficiently precise characterisation. An attempt was made by sliding a box along the edge of a folded piece of paper, with the intention to mark the position of the box on the piece of paper and record the corresponding intensity. However, the lack of rigidity of the apparatus, the lack of precision of our measurement instruments (a pen and a ruler), a lack of manual dexterity and a lack of time to work around these problems made this infeasible.

			[[Image:HomodyneDetection_AiryDisk.jpg\|right\|thumb\|250px\|An Airy disk formed by transmitting the beam through a pinhole, and expanding the beam using a lens.]]

			Further, as is visible from our images, our beam profile is nowhere close to Gaussian, and instead rather jagged and irregular. One possibility for cleaning up the signal was to send it through a small circular pinhole first. Indeed, by doing so, we were able to observe an Airy disk. However, due to the aperture being much smaller than the beam, a significant fraction of the light was lost. The resulting photocurrent only induced a maximum of a little over 2V, as compared to 10V with the full beam being used. The interference patterns formed were also less visible, so we did not primarily rely on this method to perform our observations.

			[[File:HomodyneDetection_PinholeIncident.mp4\|right\|thumb\|250px\|The voltage variation when blocking and unblocking the laser, after filtering it through a pinhole.]]

			However, we were able to induce a larger variation in the voltage, consisting of almost the full 2V, by placing pressure on the mirror and thereby inducing phase variations.

			[[File:HomodyneDetection_PinholePressure.mp4\|right\|thumb\|250px\|The voltage variation when placing pressure on the mirrors, after filtering it through a pinhole.]]

	=== Phase Modulation ===		=== Phase Modulation ===

	A major issue we faced is that the <chem>LiNbO3</chem> crystal we obtained, and the EOM board it came with, was not characterised. In fact, while the crystal was cuboidal in shape, we were unsure if the long axis along which light would generally be transmitted was the optic axis or not. A further complication is that we do not have an extra crystal to compensate for birefringence, so if the long axis is not the optic axis, we would instead have to resort to trying to ensure the crystal and laser are aligned, to ensure the coherence is minimally disrupted by birefringence.		Having thus established our interferometer, we now proceed to the second major aspect, which is to control the phase, and therefore the received photocurrent and measured voltage. A major issue we faced is that the <chem>LiNbO3</chem> crystal we obtained, and the EOM board it came with, was not characterised. In fact, while the crystal was cuboidal in shape, we were unsure if the long axis along which light would generally be transmitted was the optic axis or not. A further complication is that we do not have an extra crystal to compensate for birefringence, so if the long axis is not the optic axis, we would instead have to resort to trying to ensure the crystal and laser are aligned, to ensure the coherence is minimally disrupted by birefringence.

	[[Image:HomodyneDetection_PhaseMod.jpg\|right\|thumb\|250px\|A closeup view of the phase modulation setup. While some fine details are not visible, a secondary reflection on the mirror is visible.]]		[[Image:HomodyneDetection_PhaseMod.jpg\|right\|thumb\|250px\|A closeup view of the phase modulation setup. While some fine details are not visible, a secondary reflection on the mirror is visible.]]

Johnkhootf: /* Results & Discussion */ Complete discussion of noise

2022-04-30T05:03:43Z

Results & Discussion: Complete discussion of noise

← Older revision		Revision as of 05:03, 30 April 2022
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	=== Apparatus Parameters ===		=== Apparatus Parameters ===
	We are using a Hamamatsu [https://www.hamamatsu.com/content/dam/hamamatsu-photonics/sites/documents/99_SALES_LIBRARY/ssd/s5106_etc_kpin1033e.pdf S5106 Si PIN photodiode] as our detector, with a resistive load of <math>1~\mathrm{M\Omega}</math>. Measurements are taken with a Kenwood CS-5270 oscilloscope with <math>\Delta f = 100~\mathrm{MHz}</math>.		We are using a Hamamatsu [https://www.hamamatsu.com/content/dam/hamamatsu-photonics/sites/documents/99_SALES_LIBRARY/ssd/s5106_etc_kpin1033e.pdf S5106 Si PIN photodiode] as our detector, operated at 10V, with a resistive load of <math>1~\mathrm{M\Omega}</math>. Measurements are taken with a Kenwood CS-5270 oscilloscope with <math>\Delta f = 100~\mathrm{MHz}</math>.

	The <chem>LiNbO3</chem> crystal is 3.2mm wide and tall (square cross-section) and 2cm long. The laser pointer lens has a diameter of 3mm, and the wavelength is 650nm ± 10nm. We also had access to a pinhole of diameter 1mm.		The <chem>LiNbO3</chem> crystal is 3.2mm wide and tall (square cross-section) and 2cm long. The laser pointer lens has a diameter of 3mm, and the wavelength is claimed to be 650nm ± 10nm. We also had access to a pinhole of diameter 1mm.

	=== Laser Beam ===		=== Laser Beam ===
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	TODO talk about change in voltage observed with and without the laser and as the phase is changed, estimate power being received using voltage and resistance, estimate the various sources of noise.		TODO talk about change in voltage observed with and without the laser and as the phase is changed, estimate power being received using voltage and resistance, estimate the various sources of noise.

			Finally, we estimate the noise in the detector in order to estimate our SNR. First, we consider the shot noise. As previously discussed, the rms current of the shot noise is given by <math>\left\langle i^2_N(f) \right\rangle=2e\bar{i}\Delta f</math>, where <math>\bar{i}</math> is the average photocurrent that is given by <math> \bar{i}=\frac{\eta q_e}{h\nu}\left\langle I \right\rangle A </math>. <math>\left\langle I \right\rangle A</math> is the power of the light incident on the photodiode, <math>q_e</math> is the elementary charge, and <math>\eta</math> is the quantum efficiency. The maximum voltage (with the zero reference being the voltage reading when no light is incident) is 10 V, meaning the detected power is <math>1.0 \times 10^{-4}~\mathrm{W}</math>, since by Ohm's law <math>P = V^2/R</math>, where load resistance ''R'' is the resistance across which the photocurrent flows. This gives us <math>\bar{i}=5.2310^{-5}</math> and therefore, <math display>\left\langle i^2_N(f) \right\rangle=1.67610^{-15}</math>, giving a noise voltage of <math>4.09 \times 10^{-2}~\mathrm{V}</math>.

			We next consider Johnson noise. Through the equation <math>\left\langle i^2_N \right\rangle=\frac{4k_BT}{R}\Delta f</math>, we find a current of <math display>\left\langle i^2_N \right\rangle=1.646*10^{-18}</math> at a room temperature of <math>T = 298~\mathrm{K}</math>. This gives an rms voltage of <math>1.29 \times 10^{-3}~\mathrm{V}</math>.

			Lastly, while the definition of the GR noise requires detailed knowledge of the electron-hole recombination process, it has estimated by the manufacturer as the ''noise-equivalent power'' or NEP in the datasheet. This is given as <math>1.6 \times 10^{-14}~\mathrm{W}~(\mathrm{Hz})^{-1/2}</math>, giving us an rms noise voltage of <math>1.3 \times 10^{-2}~\mathrm{V}</math>.

			Remarkably, despite the simplicity of our setup, the dominant source of noise in this setup is shot noise---we have managed to build a shot-noise limited detector. All sources of noise (aside from the inexplicable low-frequency noise) are several orders of magnitude lower than the variation in voltage from phase changes, implying that this system can be used effectively for modulating and transmitting information.

	=== Phase Modulation ===		=== Phase Modulation ===
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	Despite our best efforts, we were not able to perfectly align the EOM board. The <chem>LiNbO3</chem> crystal's optical surfaces seem to be somewhat reflective, producing an additional red spot on the screen. The spot was confirmed to be produced by the crystal by blocking the two mirrors and observing that the spot is still there, whereas it disappears if light from <math>B_1</math> is not allowed to reach the crystal. If the setup was perfectly aligned, this additional spot would have formed exactly at the position of the beam that reflected from <math>M_1</math>, but this was not the case. While this additional spot is not visible in the image, a similar effect produces the additional misaligned spot on <math>M_1</math>, this time caused by the beam, reflected by <math>M_1</math>, hitting the crystal. While some light is transmitted onwards to the screen, some is reflected, forming the additional spot on <math>M_1</math>.		Despite our best efforts, we were not able to perfectly align the EOM board. The <chem>LiNbO3</chem> crystal's optical surfaces seem to be somewhat reflective, producing an additional red spot on the screen. The spot was confirmed to be produced by the crystal by blocking the two mirrors and observing that the spot is still there, whereas it disappears if light from <math>B_1</math> is not allowed to reach the crystal. If the setup was perfectly aligned, this additional spot would have formed exactly at the position of the beam that reflected from <math>M_1</math>, but this was not the case. While this additional spot is not visible in the image, a similar effect produces the additional misaligned spot on <math>M_1</math>, this time caused by the beam, reflected by <math>M_1</math>, hitting the crystal. While some light is transmitted onwards to the screen, some is reflected, forming the additional spot on <math>M_1</math>.

	While the additional spot sometimes displays visible fringes, it does not interfere with the beam from <math>M_2</math> when said beam is overlapped onto it. That the fringes arise from interference can be verified by applying a slight pressure to the EOM board, which produces a shift in the pattern characteristic of interference. ~~This tells us~~ that the path difference between this path reflecting from the EOM, and the path reflecting from <math>M_2</math>, exceeds the coherence length of the laser. However, it is unclear what this reflected beam is interfering with.		While the additional spot sometimes displays visible fringes, it does not interfere with the beam from <math>M_2</math> when said beam is overlapped onto it. That the fringes arise from interference can be verified by applying a slight pressure to the EOM board, which produces a shift in the pattern characteristic of interference. One possibility is that the path difference between this path reflecting from the EOM, and the path reflecting from <math>M_2</math>, exceeds the coherence length of the low-qualtiy laser. However, the laser claims to have a wavelength of 650nm ± 10nm. If that was truly the case, then the coherence length would be would be
			<math display='block'>
			\ell_c \approx \frac{c}{\Delta \nu} = \frac{1}{\lambda_{\min}} - \frac{1}{\lambda_{\max}} = 4.7 \times 10^4~\mathrm{m},
			</math>
			far greater than the size of our experiment. It may be possible that the bandwidth stated on the laser is not truly the bandwidth, but the uncertainty in the average frequency, or it may be that the light is somehow being scattered incoherently before landing on the screen. Additionally, it is unclear what this reflected beam is interfering with.

	A far worse problem, however, was discovered very close to the end of the project. Despite all the effort put into repairing the EOM board, the coaxial socket does not seem properly connected to the top plate, and even worse, there seems to be a short circuit within the coaxial socket. This was first suspected when it was observed that driving the EOM with a function generator through a high voltage amplifier did not produce any change whatsoever in the observed voltage when the signal was switched on and off, and strongly suspected when it was observed that a high voltage power supply would have its output drop to 0V when connected to the EOM. These faults were confirmed by direct measurement of resistance using a multimeter, which discovered these faults.		A far worse problem, however, was discovered very close to the end of the project. Despite all the effort put into repairing the EOM board, the coaxial socket does not seem properly connected to the top plate, and even worse, there seems to be a short circuit within the coaxial socket. This was first suspected when it was observed that driving the EOM with a function generator through a high voltage amplifier did not produce any change whatsoever in the observed voltage when the signal was switched on and off, and strongly suspected when it was observed that a high voltage power supply would have its output drop to 0V when connected to the EOM. These faults were confirmed by direct measurement of resistance using a multimeter, which discovered these faults.
Line 532:		Line 544:

	In the course of trying to repair the EOM board, we had attempted to desolder the coaxial socket and transfer it to a new board, or else use a new coaxial socket with a new board. Comically, but very unfortunately for our experiment, despite there being a vast number of similar boards and coaxial sockets in the drawers at the CQT Workshop, none of the coaxial sockets can fit in any of the new boards, and the socket that we have, which definitely fits, seems resistant to desoldering (the solder on its legs did not melt even after being exposed to a soldering iron at 450 °C). Having exhausted these options (and the crystal breaking off from the board again shortly after this discovery, for good measure), it was decided to admit defeat and limit ourselves to presenting these results and observations, instead of trying again to get our phase modulation to work.		In the course of trying to repair the EOM board, we had attempted to desolder the coaxial socket and transfer it to a new board, or else use a new coaxial socket with a new board. Comically, but very unfortunately for our experiment, despite there being a vast number of similar boards and coaxial sockets in the drawers at the CQT Workshop, none of the coaxial sockets can fit in any of the new boards, and the socket that we have, which definitely fits, seems resistant to desoldering (the solder on its legs did not melt even after being exposed to a soldering iron at 450 °C). Having exhausted these options (and the crystal breaking off from the board again shortly after this discovery, for good measure), it was decided to admit defeat and limit ourselves to presenting these results and observations, instead of trying again to get our phase modulation to work.

	~~=== TBD ===~~
	We consider the shot noise. As previously discussed, the shot noise spectrum is given by equation <math display> \left\langle i^2_N(f) \right\rangle=2e\bar{i}\Delta f </math>. <math display> \Delta f </math> is the modulation bandwidth, which is <math display> 100MHz</math> for our oscilloscope. <math display> \bar{i}</math> is the photocurrent that is determined by <math display> \bar{i}=\frac{\eta e}{h\nu}\left\langle I \right\rangle A </math>.
	<math display>\left\langle I \right\rangle A </math> term gives how much light power shoots on the photodiode. Multiplying it with quantum efficiency, we have how much power are detected. The detected voltage is 10 V, meaning the detected power is 0.0001 W, given by <math display> \frac{V^2}{R_{diode}} </math>. <math display>R_{diode}=1M\Omega </math> in our case.
	~~<math display>\bar{i}=5.2310^{-5} </math> and therefore, <math display>\left\langle i^2_N(f) \right\rangle=1.67610^{-15}</math>~~

	We then consider Johnson noise. Through the equation <math display>\left\langle i^2_N \right\rangle=\frac{4k_BT}{R}\Delta f</math>, we find <math display>\left\langle i^2_N \right\rangle=1.656*10^{-18}</math>.

	Therefore, we conclude both Shot noise and Johnson noise are in a reasonable amount. We still need to address on GR noise. However, the characteristic time for random electron – hole recombination and for electrons to move from diode to the circuit is beyond our ability to decide.

	== Conclusion ==		== Conclusion ==

	I'm really glad I'm a theorist.		I'm really glad I'm a theorist.

Johnkhootf: /* Results & Discussion */ Complete fringe discussion

2022-04-30T04:24:16Z

Results & Discussion: Complete fringe discussion

← Older revision		Revision as of 04:24, 30 April 2022
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	We are using a Hamamatsu [https://www.hamamatsu.com/content/dam/hamamatsu-photonics/sites/documents/99_SALES_LIBRARY/ssd/s5106_etc_kpin1033e.pdf S5106 Si PIN photodiode] as our detector, with a resistive load of <math>1~\mathrm{M\Omega}</math>. Measurements are taken with a Kenwood CS-5270 oscilloscope with <math>\Delta f = 100~\mathrm{MHz}</math>.		We are using a Hamamatsu [https://www.hamamatsu.com/content/dam/hamamatsu-photonics/sites/documents/99_SALES_LIBRARY/ssd/s5106_etc_kpin1033e.pdf S5106 Si PIN photodiode] as our detector, with a resistive load of <math>1~\mathrm{M\Omega}</math>. Measurements are taken with a Kenwood CS-5270 oscilloscope with <math>\Delta f = 100~\mathrm{MHz}</math>.

	The <chem>LiNbO3</chem> crystal is 3.2mm wide and tall (square cross-section) and 2cm long. The laser pointer lens has a diameter of 3mm, and the wavelength is 650nm ± 10nm.		The <chem>LiNbO3</chem> crystal is 3.2mm wide and tall (square cross-section) and 2cm long. The laser pointer lens has a diameter of 3mm, and the wavelength is 650nm ± 10nm. We also had access to a pinhole of diameter 1mm.

	=== Laser Beam ===		=== Laser Beam ===

	~~TODO estimate~~ beam ~~waist? talk about~~ pinhole~~/non-~~Gaussian?		Despite the process of measuring the laser spot size being rather straightforward to describe, we did not have the equipment to perform a sufficiently precise characterisation. An attempt was made by sliding a box along the edge of a folded piece of paper, with the intention to mark the position of the box on the piece of paper and record the corresponding intensity. However, the lack of rigidity of the apparatus, the lack of precision of our measurement instruments (a pen and a ruler), a lack of manual dexterity and a lack of time to work around these problems made this infeasible.

			[[Image:HomodyneDetection_AiryDisk.jpg\|right\|thumb\|250px\|An Airy disk formed by transmitting the beam through a pinhole, and expanding the beam using a lens.]]

			Further, as is visible from our images, our beam profile is nowhere close to Gaussian, and instead rather jagged and irregular. One possibility for cleaning up the signal was to send it through a small circular pinhole first. Indeed, by doing so, we were able to observe an Airy disk. However, due to the aperture being much smaller than the beam, a significant fraction of the light was lost. The resulting photocurrent only induced a maximum of a little over 2V, as compared to 10V with the full beam being used. The interference patterns formed were also less visible, so we did not primarily rely on this method to perform our observations.

			[[File:HomodyneDetection_PinholeIncident.mp4\|right\|thumb\|250px\|The voltage variation when blocking and unblocking the laser, after filtering it through a pinhole.]]

			However, we were able to induce a larger variation in the voltage, consisting of almost the full 2V, by placing pressure on the mirror and thereby inducing phase variations.

			[[File:HomodyneDetection_PinholePressure.mp4\|right\|thumb\|250px\|The voltage variation when placing pressure on the mirrors, after filtering it through a pinhole.]]

	=== Michelson Interferometer Fringes ===		=== Michelson Interferometer Fringes ===

	The fine adjustment of the pitch and yaw of the mirrors using the knobs on their optomechanical mounts allowed us to explore the variation in fringe patterns as the mirrors rotate through their angular range of motion. Although our cameras were not able to resolve the details of the interference fringes, we were able to make a few qualitative observations of the fringes formed, using a piece of white paper as a screen:		The fine adjustment of the pitch and yaw of the mirrors using the knobs on their optomechanical mounts allowed us to explore the variation in fringe patterns as the mirrors rotate through their angular range of motion. Although our cameras were not able to resolve the details of the interference fringes, we were able to make a few qualitative observations of the fringes formed, using a piece of white paper as a screen:
			# It is difficult to tell if the fringes are truly circular, since there may be only one fringe when the mirrors are highly aligned, and it is not possible to tell if the fringe is truly circular or simply the result of viewing very widely separated fringes through a circular aperture.
			# Leaving one mirror stationary, say <math>M_1</math>, and aligning <math>M_2</math> to it would produce a larger number of fringes. Adjusting <math>M_1</math> slightly, then adjusting again <math>M_2</math> to coincide with it, would be required to produce fringes that were closer to being (what we believe to be) circular, which have only around one fringe and experience large variation in intensity over the variation in <math>\delta</math>.
	# As the beams became more and more coincident, the fringes seemed to rotate, while increasing in thickness and decreasing in number. After crossing each other, and as they became less coincident, the process was reversed: the fringes continued rotating, but decreased in thickness and increased in number, until they were no longer visible.		# As the beams became more and more coincident, the fringes seemed to rotate, while increasing in thickness and decreasing in number. After crossing each other, and as they became less coincident, the process was reversed: the fringes continued rotating, but decreased in thickness and increased in number, until they were no longer visible.
	# Leaving one mirror stationary, say <math>M_1</math>, and adjusting <math>M_2</math> often would not produce optimal fringes. Adjusting <math>M_1</math> slightly, then adjusting <math>M_2</math> to coincide with it, would be required to produce fringes that were closer to being (what we believe to be) circular and experiencing a large variation in intensity over the variation in <math>\delta</math>.
	# It is difficult to tell if the fringes are truly circular, since there is only a single fringe when the mirrors are highly aligned, and it is not possible to tell if the fringe is truly circular or simply the result of viewing very widely separated fringes through a circular aperture.

	~~This agrees~~ with our theoretical ~~models~~ for ~~Michelson interferometer~~. ~~TODO~~		Our observations agree with our theoretical derivations. We first consider the case where the mirrors are aligned to each other, although possibly not to the screen, which can be visually verified from the overlap of the beams. In such a case, <math>\theta_1 = \theta_2</math> and we have
			<math display='block'>
			\begin{align}
			s &= 2\|\delta\cos\theta_1\| \\
			\psi &= \arctan\left(\frac{1}{\cos\theta_1}\right) = \frac{\pi}{2} - \theta_1 \\
			\therefore \Delta d &= 2\|\delta\cos\theta_1\|\left( \cos\theta_1 \cos\theta + \sin\theta_1 \sin\theta \cos\phi \right) + O(s^2/d^2),
			\end{align}
			</math>
			which tells us that increasing the misalignment <math>\theta_1</math> from the screen makes the fringes straighter and less circular, which means it is not enough to align the beams to each other: they must also be aligned to the screen. However, such a misalignment also reduces the value of <math>s</math>, decreasing the spatial frequency and therefore increasing the size of the fringes. This makes it more difficult to judge if the fringes are truly circular when the mirrors are aligned to each other.

			However, for the variation in the fringes as the beams cross each other, we were not able to find a convincing theoretical explanation. The rotation of the fringes could perhaps be explained by the maxima and minima shifting with variation in <math>\psi</math>, but the expressions for <math>s</math> and <math>\psi</math> in terms of <math>\theta_1</math> and <math>\Delta \theta = \theta_2 - \theta_1</math> are too complicated for us to extract meaningful insight from, even with series expansions and other approximations.

	=== Voltage Readings ===		=== Voltage Readings ===

			We are able to induce a variation in voltage by shifting the phase of the oscilloscope

	[[File:HomodyneDetection_BlockedVIncident.mp4\|right\|thumb\|250px\|The voltage on the oscilloscope drops significantly when the light is incident on the detector.]]		[[File:HomodyneDetection_BlockedVIncident.mp4\|right\|thumb\|250px\|The voltage on the oscilloscope drops significantly when the light is incident on the detector.]]

	[[File:HomodyneDetection_IntensityTapping.mp4\|right\|thumb\|250px\|Tapping one of the mirrors results in small changes in path length, leading in turn to a change in the interference pattern and therefore the overall intensity.]]		[[File:HomodyneDetection_IntensityTapping.mp4\|right\|thumb\|250px\|Tapping one of the mirrors results in small changes in path length, leading in turn to a change in the interference pattern and therefore the overall intensity.]]

	~~[[File:HomodyneDetection_VoltageTapping.mp4\|right\|thumb\|250px\|The phase difference arising from tapping the mirror is registered as a notable variation in the voltage.]]~~

	[[File:HomodyneDetection_AutoVariation.mp4\|right\|thumb\|250px\|The voltage varies inexplicably over time without any interference. The variation is not sinusoidal, with long pauses between the extrema. We are unsure what causes this phenomenon.]]		[[File:HomodyneDetection_AutoVariation.mp4\|right\|thumb\|250px\|The voltage varies inexplicably over time without any interference. The variation is not sinusoidal, with long pauses between the extrema. We are unsure what causes this phenomenon.]]

← Older revision		Revision as of 06:43, 30 April 2022
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	The noise received depends on both the frequency of the light signal <math>\nu</math> and the signal frequency <math>f</math>, also expressed as <math>\omega = 2\pi{f}</math>. These are not to be confused with each other. <math>\Delta f</math> is the ''sampling bandwidth'', determined by how often take electrical measurements. By Nyquist's theorem, we must have a sampling period <math>T = 1/\Delta f \leq 1/2f</math> to reconstruct a modulated signal of maximum frequency component <math>f</math>: that is, we must have <math>\Delta f \geq 2f</math>.		The noise received depends on both the frequency of the light signal <math>\nu</math> and the signal frequency <math>f</math>, also expressed as <math>\omega = 2\pi{f}</math>. These are not to be confused with each other. <math>\Delta f</math> is the ''sampling bandwidth'', determined by how often take electrical measurements. By Nyquist's theorem, we must have a sampling period <math>T = 1/\Delta f \leq 1/2f</math> to reconstruct a modulated signal of maximum frequency component <math>f</math>: that is, we must have <math>\Delta f \geq 2f</math>.

	The shot noise is primarily determined by two parameters: the average current <math>\overline{i}</math> and noise current <math>i_N~~(\omega)~~</math>. The average current is given by		The shot noise is primarily determined by two parameters: the average current <math>\overline{i}</math> and noise current <math>i_N</math>. The average current is given by
	<math display='block'>\overline{i} = \frac{\eta q_e}{h\nu} \left\langle I \right\rangle A,</math>		<math display='block'>\overline{i} = \frac{\eta q_e}{h\nu} \left\langle I \right\rangle A,</math>
	where <math>P = \left\langle I \right\rangle A</math> is the average power of the light incident on the photodiode, <math>q_e</math> is the elementary charge, and <math>\eta</math> is the quantum efficiency, a measure of the proportion of incident photons that are absorbed.		where <math>P = \left\langle I \right\rangle A</math> is the average power of the light incident on the photodiode, <math>q_e</math> is the elementary charge, and <math>\eta</math> is the quantum efficiency, a measure of the proportion of incident photons that are absorbed.
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	The frequency spectrum of the noise current, on the other hand, is given by		The frequency spectrum of the noise current, on the other hand, is given by
	<math display='block'>\left\langle i^2_N(\omega) \right\rangle = 4\pi \eta \left\langle N \right\rangle \left\| F(\omega) \right\|^{2}d\omega</math>		<math display='block'>\left\langle i^2_N(\omega) \right\rangle = 4\pi \eta \left\langle N \right\rangle \left\| F(\omega) \right\|^{2}d\omega</math>
	where <math>F(\omega)</math> is the frequency spectrum (that is, the Fourier transform) of current signal <math>i(t)</math>. From this, we obtain		where <math>F(\omega)</math> is the frequency spectrum (that is, the Fourier transform) of current signal <math>i(t)</math>, and <math>N</math> is the number of photons impinging on the detector.
	<math display='block'>\left\langle i^2_N(~~\omega~~) \right\rangle=2e\bar{i}\mathrm{d}f,</math>
	which must be multiplied ~~agains~~ the sampling bandwidth to obtain the shot noise current.		From this, we obtain
			<math display='block'>\left\langle i^2_N(f) \right\rangle=2e\bar{i}\mathrm{d}f,</math>
			which must be multiplied against the sampling bandwidth <math>\Delta f</math> to obtain the shot noise current.

	The shot noise limit, which is the best possible signal-to-noise ratio (SNR), is given by examining the ratio		The shot noise limit, which is the best possible signal-to-noise ratio (SNR), is given by examining the ratio
	<math display='block'>\frac{\bar{i}^2}{\left\langle i^2_N \right\rangle} = \frac{\eta \left\langle I \right\rangle A}{2h\nu\Delta f} =\frac{\eta P}{2h\nu\Delta f}.</math>		<math display='block'>\frac{\bar{i}^2}{\left\langle i^2_N(f) \right\rangle \Delta f} = \frac{\eta \left\langle I \right\rangle A}{2h\nu\Delta f} =\frac{\eta P}{2h\nu\Delta f}.</math>
	If this ratio falls below 1, then we are unable to recover the signal. Therefore, even if we have perfect detector with <math>\eta = 1</math>, ''P'' should be at least <math>2h\nu \Delta f</math>. Otherwise, the signal is lost in the noise.		If this ratio falls below 1, then we are unable to recover the signal. Therefore, even if we have perfect detector with <math>\eta = 1</math>, ''P'' should be at least <math>2h\nu \Delta f</math>. Otherwise, the signal is lost in the noise.

	The Nyquist-Johnson noise current is given by		The mean-squared Nyquist-Johnson noise current is given by
	<math display='block'>\left\langle i^2_N \right\rangle=\frac{4k_BT}{R}\Delta f,</math>		<math display='block'>\left\langle i^2_N \right\rangle=\frac{4k_BT}{R}\Delta f,</math>
	where <math>k_B</math> is the Boltzmann constant, <math>T</math> is the temperature, and <math>R</math> is the resistance of the component being measured. If our measurement bandwidth is <math>\Delta f</math>, we can treat a noisy component as being equivalent to a noiseless component in parallel with a current source with mean-square current <math>\left\langle i^2_N \right\rangle \Delta f </math>.		where <math>k_B</math> is the Boltzmann constant, <math>T</math> is the temperature, and <math>R</math> is the resistance of the component being measured. If our measurement bandwidth is <math>\Delta f</math>, we can treat a noisy component as being equivalent to a noiseless component in parallel with a current source with mean-square current <math>\left\langle i^2_N \right\rangle \Delta f </math>.