On the sensitivity of a laser heterodyne polarimeter for vacuum birefringence detection

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On the sensitivity of a laser heterodyne polarimeter for vacuum birefringence detection
Alberts, Gabriel Rene
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Detecting vacuum magnetic birefringence (VMB) requires an immensely accurate, precisely calibrated experiment. We are working on a new design to detect the birefringence (BF) of individual optical components without a cavity that can also be used to measure VMB in reflection off a cavity. Our design uses two overlapping, orthogonally polarized laser beams to measure the relative phase difference between a reference path and one with rotating polarizations, which may experience oscillating phase shifts in vacuum in the presence of a magnetic field. To test the design, we developed a small-scale setup without cavities that can analyze different birefringent sources with the same principle. Our early results from testing mirrors show spatial variations in BF due to imperfect mirror coatings and show no correlation between the strength of a magnetic field applied parallel to the mirror's surface and BF amplitude. In addition to assisting in the selection of suitable components for the final design, our laser heterodyne polarimeter (LHP) promises more sensitive results than previous experiments and may very well be the basis for the very first detection of VMB. ( en )
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Awarded Bachelor of Music, cum laude, on May 8, 2018. Major: Physics. Emphasis/Concentration: Concentration Outside-Instrumental
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College or School: College of Liberal Arts and Sciences
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Advisor: Guido Mueller. Advisor Department or School: Physics

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University of Florida
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! !"#$%&#'&"'($()($*#+,# -#.-'&/# % &$&/+0*"&# 1+.-/(2&$&/ # ,+/ # )-3442#5 (/&,/("6&"3& # 0 &$&3$(+" # 7-5/(&.#8.5&/$' 9 :#;-.#;+..(' 9 :#;-//('+"#<-=+..($> :#74(0+#?4&..&/ 9 # "#$%&'(#)'*+,*-./01203* 4)15#&01'/ +,*67+&18%3*9%1)#05177#3*6:3*;<=!! < "#$%&'(#)'*+,*>%'.#(%'120*%)8*-./01203*-1#8(+)'*?+77#@#3*"#(+�'3*9A3*;BC;C Detecting vacuum magnetic birefringence (VMB) requires an immensely accurate precisely calibrated experiment W e are working on a n ew design to detect the birefringence (BF) of individual optical components without a cavity that can also be used to measure VMB in reflection off a cavity Our design uses two overlapping, orthogonally polarized laser beams to measure the relative phase difference between a reference path and one with rotating polarizations, which may experience oscillating phase shifts in vac uum in the presence of a magnet ic field. To test the design, we developed a small scale setup without cavities that can analyze different birefringent sources with the same principle. Our early results from testing mirrors show spat ial variations in BF due to imperfect mirror coatings a nd show no correlation between the strength of a magnetic field applied parallel to the mirror's surface and BF amplitude. In addition to assisting in the selection of suitable components for the final design, o ur laser heterodyne polarimeter (LHP) promise s more sensitive results than previous experiments and may very well be the basis for the very first detection of VMB @ABC!DEFB@!A # he classical vacuum is a region of space devoid of matter and any physical fields that carry energy and momentum. This "free space" is therefore the lowest possible energy state of a classical system and has zero energy. It also acts the reference state for the permittivity of a material making its relative permittiv ity identically one. This vacuum does not influence matter in any way and therefore acts only as a medium for gravitational and electromagnetic waves to propagate. The quantum vacuum, on the other hand, is the quantum state with the lowest possible energy, which is nonzero due to vacuum fluctuations. These fluctuations can be attributed to the creation and annihilation of virtual particle antiparticle pairs that temporarily disturb the vacuum's energy. In quantum electrodynamics ( QED ) a more particular QED vacuum is needed to fit the theory. This vacuum again describes a lowest possible energy state but now it is of the electromagnetic field when both the electric and magnetic fields are quantized. The quantization of these fields th en gives the QED vacuum a relative permittivity that is not one, as is the case for the classical vacuum Since the propagation of an electromagnetic wave is influenced by this value the QED vacuum is capable of exhibiting bire fringent effects. Birefringe nce is an optical property of a material with a polarization dependent index of refr action VMB was first mentioned formally by Dirac in 1934 during the early deve lopment of QED In a paper on the theory of the positron, or antielectron, Dirac states: Furt her work that remains to be done is to examine the physical consequences of the foregoing assumptions and to see whether it leads to any phenomena of the nature of a polarization of a vacuum by an electromagnetic field. 1 Just months after, Heisenberg published his "Remarks on Dirac's theory of the positron," which included a discussion on this postulate. 2 With these two papers at hand, many theorists began contributing to the theory with Serber 3 and Uehling 4 expanding Maxwell's field equations and Coul omb's law respectively to account for the effect. In 1936 Heisenberg and Euler derived the Euler Heisenberg Lagrangian 5 which established a foundation to analytically solve for the effects of VMB Consi dering one loop contributions this can be presented in the weak field limit and for slowly varying electromagnetic fields as # $ % & ( ) ) + ) / 0 % & 1 2 ) ) + ) ) 3 4 5 6 7 8 ) 9 : # ; where / < = ) > < ? #@& % A B ? C < ) # D E$ F #& G )H I J G ) : $ ; with > < K : C < ; L being the Compton wavelength of the electron, = 0 ) : M B N A O ; L the fine structure constant, and C < the mass of the electron. Karplus and Neuman showed 6 that for complex indices of refraction, VMB can be expressed P Q Q R + Q S E / < ) I I I D : E ; While multiple astronomical observations are currently b eing pursued to account for these effects we are interested in the direct observation of VMB Several T


GABRIEL ALBERTS HAL HOLLIS HARRISON LABOLLITA GUIDO MUELLER # projects in the past have shared this objective but none have been able to detect the effect with eno ugh sensitivity and confidence. E xperiments conducted at CERN 7 and Brookhaven National Laboratory 8 as well as Legnaro and Ferrara as part of the PVLAS collaboration 9 utilized high finesse cavities and found either a signal was not detected at the expected sensitivity using a given magnetic field str ength, or that the time needed to reach a particular heightened sensitivity was unreasonably long. We believe that perform ing a modified experiment to the one originally published by Hall, Ye, and Ma 10 using advanced gravitational wave detection technology from LIGO and LISA in the ALPS IIc design will give us the sensitivity necessary to measure VMB. While t he ALPS (Any Light Particle Search) experiment was founded to search for WISPs (very Weakly Interacting Sub eV Particles) its facility can serve multiple purposes. The general design, depicted in Fig. 1, is for the "Light Shining through a Wall" (LSW) experiment meant to search for photon oscillations to and from axions a particle theorized to exist beyond the Standard Model. A modified version of the ALPS IIc design, shown in Fig. 2, could be used for a VMB experiment This design has an optical path of approximately 176 m that will be surrounded by 5.3 T HERA dipole magnets making the expected VMB P Q # D # F #& G )) and expected path length difference P T U P Q $ F #& G )A I V Using infrared lasers with a wavelength of 1064 nm, this translates to a phase difference of P W # D #X F #& G Y? I Z[\ Detecting such a miniscule effect requires an immensely accurate experiment that is precisely calibrated One aspect of this calibration is to account for the BF of external sources that cou ld interfere with the final measurement. T he illustration in Fig. 2 is simple but it can be seen that the mirrors are one of these sources so their BF must be known Our experiment tests a new laser heterodyne polarimetry design that will be similar to the final sensing schematic by measuring the BF of mirrors. 8GG8C8BEH # Our laser heterodyn e polarimeter, as seen in Fig. 3 consists of two primary components: a path for laser locking, and a path for signal detection ! "#$%&!"'()*+, Heterodyne interferometry uses the offset frequency ( beat note ) between two independent lasers as a tool for various measurements. While each of our lasers operate at 1064 nm, sub picometer fluctuations create a beat note in the megahertz. In order to obtain accurate results, this beat note must stabilized so any meas urement deviations can I(64/&# > J "#$%&%'$!()*+!,,-!$'.%/0!!23-445!6%1'&1%0/'0-'!$'7'-7%#08 9:! ;<'!=1#$4-7%#0!30$!1'/'0'137%#0!-32%7%'.!&1#5!7<'!)+>!'?='1%5'07!31'! -#56%0'$!7#!!3!.%0/@'!-32%7A!7<37!B#4@$!=1#$ 4-'!7<'!@31/'.7!.%0/@'! =3..!=<3.'!.<%&7!#&!30A!C"D!'?='1%5'078 I(64/&# K J D+$E* )E*!.-<'537%! 4.'$!7#!5'3.41'!5%11#1!DF!GDF!.#41-'H8! F1#5!7<'%1!'5%..%#0I!7<'!@3.'1.!31'!=1'=31'$!7#!6'!=#@31%J'$I!B<%-

O N THE SENSITIVITY OF A LASER HETERODYNE POLARIMET ER FOR VACUUM BIREFR ING ENCE DETECTION $%&'()*&+,!-.!/0-)&12!3!4-5)%20!-.!$%1()6)2152 +(!7(*(2)89!3!:-05;(!#<=!>**5(!" 3 ?@)&%6!#<"A B confidently be attributed t o the source being measured. One method of stabilization is by locking the two lasers so their offset frequency and phase remain constant. =&-2#G-$%J# Each beam in o ur apparatus traveled through an identical set of components before being superimposed beginning with a F araday isolator to prevent backscattered light from damaging the laser. The beams then travel ed throu gh a half wave plate (HWP), which rotat ed the polarization of the in cident light. One field then had p polarized light, represented by the red line, while the other had s polarized light, represented by the blue line. The glass plates reduce d the power of each laser by one order of magnitude leaving the beams with a power of approximately 10 mW. The y were then guided by steering mirrors into the first polarizing beam splitter ( PBS) where they overlap ped and propagate d coherently to the rest of the phase lock loop (PLL) path and the signal path In order for the photodetector to detect a beat between the two orthogonally polarized beams, they must first be projected into the same plane. A HWP set to 22.5 ¡ followed the combining PBS, which rotated the beams by 45 ¡ With the use of another PBS, the s polarization of each beam was selected giving the PLL PD a linearly polarized field containing equal power contributions from each laser. G %-'&#< +3L J# After the opt ical signal has been acquired to the PLL PD, several electronic components are used stabilize the beat note with specification s detailed in Table 1 The PD first cove rts the optical signal to an electrical one that mixes wi th a signal of the same frequency from the func tion generator. Our chosen carrier frequency was 5 MHz because it was well within the bandwidth of all the components. The mixer then outputs the sum an d difference frequencies of the two inputs before the low pass filter (LPF) selects the difference frequency, 0 Hz, with minimal loss. This signal is then fed into an analog PLL made in house that returns two outputs to minimize the drift of the beat between the two lasers. One channel outputs a "slow" correction signal to the temperat ure control of Laser 2 in Fig. 4 This signal is used to offset the slight drift in room temperatur e o ver hours The second channel outputs a "fast" correction signal that can quickly adjust the voltage across the piezo crystal in the laser to offset random phase fluctuations. With these controls in place, the beat between the two lasers remains in phase w ith the local oscillator. From the stability of the function generator, we know the frequency of the beat is also locked. -*,+#.!/%0%(0*'+ =&-2#G-$%J# Once the two beams combine d at the first PBS, one path was used for locking while the other path was for signal detection. The subsequent procedure follow s that of Hollis et. al. 13 The two orthogonally polarized, phase locked beam s at this stage, shown in Fig. 3 can be represented through the Jones matrices as 5 ] 4 # & 8 / ^ 0 _`a 4 & # 8 / b 0 : ` c d ; a : M ; where / ^ and / b are real amplitudes and e is the carrier frequency. The reflected field from a beam splitter (BS) that further divides the experiment into reference and signal paths is then 5 ] f # g $ 4 # & 8 / ^ 4 & # 8 / b 0 d a 0 _`a I I I D : h ; The reference path simply projected the field into a single p lane, as with the locking path, so a PBS selected the s polarization after the field traveled through a HWP rotated at 22.5 ¡ The field on the reference detector, RPD after this co mbination of components is given by f LO /f RF (MHz) f IF (MHz) LO Power Mini Circuits ZAD 6+ Mixer 0.003 100 DC 100 +7 dBm Passband (MHz) Impedance Mini Circuits BLP 5+ L PF DC 22 50 Frequency Range Amplitude Range (Vpp) Impedance SRS DS345 Synthesized Function Generator 1 Hz 30.2 MHz 0.01 10 ( @ 50 ) 50 / 1 M Bandwidth Gain NEP EOT 3000A Amplified PD 30 kHz 1.5 GHz 770 V/W ( @ 1064 nm) ~39 nW/ Hz I(64/&# M J )3.'1!@#-M%0/!'@'-71#0%-.!.-<'537%-8!+%/03@.!&1#5!7<'!*))!*L! 30$!&40-7%#0!/'0'137#1!31'!-#56%0'$!%0!3!5%?'18!(!@#B!=3..!&%@7'1!.'@'-7.! 7<'!$%&&'1'0-'!&1'N4'0-A!7#!%0=47!%07#!7<'!*))8!O0'!&''$63-M!@##=! -#00'-7.!$%1'-7@A!7#!7<'!=%'J#!-1A.73@!%0!7< '!@3.'1!B<%@'!7<'!#7<'1!3$P4.7.! 7<'!@3.'1Q.!7'5='13741'!-#071#@8 B -5. J )3.'1!@#-M%0/!'@'-71%-3@!-#5=#0'07!.='-%&%-37%#0. !


GABRIEL ALBERTS HAL HOLLIS HARRISON LABOLLITA GUIDO MUELLER C 5 ] fij k & & & # l # g $ k # # # + # l 5 ] f # $ k & & # + # l 4 # & 8 / ^ 4 & # 8 / b 0 d a 0 _`a I I I # $ 4 & # 8 m / ^ + / b 0 d a n 0 _`a I I I D : X ; It follows that the intensity is o fij p 5 ] fij p ) # M m / ^ ) / b ) n + # $ / ^ / b qrs e t I I I D : 3 ; The RPD signal was used as a standard measurement for noise inherent to the setup. Since we only desired to detect the BF from the source mirror a BB1 E03 Broadband Dielectric mirror from Thorlabs any extraneous effects could be eliminated by subtracting the phase inf ormation on RPD. The signal path began with the BS transmiss ion field: 5 ] u v g $ 4 # & 8 / ^ 4 & # 8 / b 0 d a 0 _`a I I I D : @ ; The field propagated through another HWP but now one rotated at an angle w x before reflecting off the source mirror rotated at w y The phase of each polarization after this reflection is denoted z { and z | The field after propagating back through the rotated HWP is then represented: 5 ] u v $ g $ } $ 0 ~ qrs $€ m 0 ~ + 0 ~  n s‚ƒ $€ / ^ I I I I I I I I I I I I I I I I I m 0 ~ + 0 ~  n s‚ƒ $€ $ 0 ~  qrs $€ / b 0 d a „ 0 _`a : … ; where † $ w x + w y Finally, the field is reflected at the BS and enters an equivalent HWP PBS combination as bef ore all PDs. This field represented by 5 ] ‡ij v M g $ ˆ m 0 ~ + 0 ~  n m / ^ / b 0 d a n qrs$ € + m 0 ~ + 0 ~  n m / ^ + / b 0 d a n s‚ƒ$ € m 0 ~ 0 ~  n m / ^ + / b 0 d a n ‰ 0 _`a I I : #& ; now contains a phase shifted beat note compared to the RPD signal because of the mirror's BF In the case when this BF, Š ‹ z { + z | is small, we can approximate the intensity on the signal P D to be o ‡ij Œ # @ m / ^ ) / b ) n + # M / ^ / b : Š qrs$ † s‚ƒ e t qrs e t ; Œ # @ m / ^ ) / b ) n + # M / ^ / b qrs : e t z ; I I I I : ## ; where z $ Š qrs$ € $ qrs : M w x + $ w y ; By rotating the HWP at a constant angular velocity such that w x Ž x t we show that z  qrs : M Ž x t + $ w y ; I I I D : #$ ; Thus, with ideal components the mirror's BF produce s a signal at four times the rotation rate of the HWP with amplitude Š We were able to rotate the HWP consistently using a set of Thorlabs components: a DDR05 rotation mount, a KBD101 K Cube Brushless DC Servo controller and an APT system software. The physical rotation of the mirror was possible due to the mount, which was controlled by a brushless motor capable of steps, jogs, and continuous rotation. The motor's actions were then controlled by the A PT software where the rotation rate was set to 5 Hz With imperfect components, Eq. 12 deviates slightly. An error in the rotating HWP can be modeled by adjusting its retardation to B  Taking only first order terms in Š and  an extra factor of  $ / ^ / b qrs$ w x s‚ƒ e  : #E ; arises in the SPD intensity when again considering Š to be small Letting w x Ž x t in Eq. 13, Eq. 12 then becomes z  qrs : M Ž x t + $ w y ; + $  qrs$ Ž x t : #M ; showing a signal from the HWP error appears at twice its rotation rate with an amplitude of $  In order to ensure the mirror's BF signal does not appear at a frequency laced with other spurious signals, we extended the SPD intensity to include the next higher order term in  : +  ) M / ^ / b : # qr sM w x ; qrs e  I I I D : #h ; This leads to a beat note phase shift of z Œ  qrs : M Ž x t + $ w y ; + $  qrs$ Ž x t +  ? E qrsX Ž x t I ‘ : #X ; which identifies an additional peak at six times the HWP rotation due to its retardation error, but no additional signal at M Ž x t the frequency of the mirror BF signal. G%-'&#G/+3&''("6J# To gather phase information from the RPD and SPD, we used a Moku:Lab Phasemeter 1 4 from Liquid Instruments. This instrument tracked the phase, frequency, and amplitude of each PD through two indepe ndent channels The interface was provided through an iPad application, seen in Fig. 5 and provided optimal control of v arious parameters.


O N THE SENSITIVITY OF A LASER HETERODYNE POLARIMET ER FOR VACUUM BIREFR ING ENCE DETECTION $%&'()*&+,!-.!/0-)&12!3!4-5)%20!-.!$%1()6)2152 +(!7(*(2)89!3!:-05;(!#<=!>**5(!" 3 ?@)&%6!#<"A D Each channel locked onto a reference frequency equal to that of our beat note with a 1 Vpp range since the maximum RPD and SPD voltages were approximately 0.6 V and 0.3 V respectively. The sampling rate was set to the 120 samples per second (S/s) setting although the true value was closer to 122.07 S/s. Data was saved as a CSV file which was then use d for analysis in MATLAB. To convert the raw phase data in to plots containing the amplitudes of various BF signal s we took the discrete Fourier transform (DFT) of the phase difference between the two channels HG8B@8<#N8C@8AFO # One test performed on the mirror involved determining how its BF varied when probed at different locations on its surface. 1&'(%23&% In order to obtain a BF surface map, we needed to be able to adjust the mirror's position in the 2 D plane perpendicular to the incident light. We used a combination of a Thorlabs 6XS mirror mount and a Newport 460 XYZ mount to accomplish this with enough translational freedom to cover a significant portion of the mirror 's surface We then constructed a 4 x 4 grid of points shown in Fig. 6, to effectively probe 144 mm 2 which maximized our area covered while maintaining that the beam was full y reflected. A two minute measurement was taken at each point. 4%$3.0$ #+2!5'+(.3$*'+$ It is shown f rom Fig. 7 that the birefringence across the mirror 's surface varied approximately 2 mrad from 22 .4 to 24.4 m rad In comparison, repeated measurements on a single point showed a standard deviation of approximately 0.2 mrad. Both plo ts showed the maximum BF at the point second from the bottom and left while the minimum BF w as at the point directly above it. These values were consistent over multiple runs covering all 16 points before measuring another point again. The surface fluctuations can therefore be attributed to imperfections over random error. Since this mirror, a BB1 E03 Broadband Dielectric mirro r from Thorlabs, is composed of a polished glass substrate under alternating I(64/&# P J "#M4K)36!*<3.'5'7'1!%*3$!%07'1&3-'8!,0=47!9!30$!:!<3$!7<'! .35'!.'77%0/.K!R!"EJ!1'&'1'0-'!&1'N4'0-AI!9S MEJ!630$B%$7

GABRIEL ALBERTS HAL HOLLIS HARRISON LABOLLITA GUIDO MUELLER E layers of reflective coatings, we can conclude the imperfections are due to errors in the coating layers. ?87AOB@F#OIIOFBH # T he 5 T magnets present in the modified ALPS IIc design from Fig. 2 will produce strong fields outside of their intended target area. The mirrors will therefore be subject to some amount of these stray fields. In order to precisely characterize all BF effects and eliminate the possibility of a false positive wh en attempting to detect VMB, we examined how a mirror's BF varied when placed in a magnetic field parallel to its surface. 677#$!#+2! 1&'(%23&% First, we wanted to ensure all measurement s contained contributions strictly from the magnetic effect so we i n serted a telescope between the rotating HWP and source mirror. This expanded the beam so its cross sectional area at the mirror was nearly as large as the mirror's surface itself. With this beam expander in place, the minor surface variations were average d out giving us a more stable measurement for the mirror's BF on its own. To replicate the stray fields present on the mirror's surface, we obtained strong magnets and created a device to hold them in place over a range of distances. The four magnets we us ed were 2" x 1" x 3/8" N52 Neodynium magnets (BY0X06 N52) from K&J Magnets, Inc. that each produced a surface field strength near the 3400 Gauss advertised We were able to measure this with a Gauss meter and calculate it using the following equation provi ded by K&J Magnets, Inc.: I I I I I I I I I ’ “ x [ƒ G Y k ”• ) – g H – — c ” — c • — l + [ƒ G Y 4 ”• ) : c – ; — ™ H : c – ; — c ” — c • — 8 I I I D : #3 ; In Eq. 17, š is the remanence field, U › and œ are the length, width, and thickness of the magnet respectively, and  is the distance away from a pole assuming the poles are oriented with the axis of the thickness The units of U › œ and  are arbitrary so long as they are all the same. The theoretical and measured fields for a single magnet are plotted in Fig. 8 The holder s shown in Fig. 9, had spaces to fit each magnet independently, which would produce a combined field that is parallel to the surface of the mirror. The holder s also had holes for the two poles on either side that had distance markings on them. This, in conjunction with the side screws, allowed us to adjust the distance of the magnets from the mirror in eighth inch increments and thus alter the field strength present at the mirror's surface. In addition, the casing s were made out of phenolic making neither them nor the mirror mount magnetic. With all four magnets in place, we took two minute measurements at eighth inch increments. The nearest position is depicted in Fig. 9 and is when the closest magnet is one inch from the cente r of the mirror. The furthest position is when the bottom casing is sitting on the table, which makes the closest magnet 2.5 inches from the mirror's center. 4%$3.0$!#+2!5'+(.3$*'+$ As seen in Fig. 10, the BF of a New Focus 5104 mirror remains essentially unchanged over a range of nearly 1000 Gauss. The uncertainty in field strength arises from the sixteenth inch uncertainty in our distance measurements while t he uncertainty in our BF amplitude was determined after repeated measurements The mean BF amplit ude was 18,200 M&& I % rad which was near the initial value and on the final value even in a magnetic field great than 1 kG. I(64/&# S J )36!=<#7#/13=

O N THE SENSITIVITY OF A LASER HETERODYNE POLARIMET ER FOR VACUUM BIREFR ING ENCE DETECTION $%&'()*&+,!-.!/0-)&12!3!4-5)%20!-.!$%1()6)2152 +(!7(*(2)89!3!:-05;(!#<=!>**5(!" 3 ?@)&%6!#<"A F H@7A8<#@H!<8B@!A # From Eq. 14, we expect BF signals at twice and four time the rotation rate of the HWP and from Eq. 16, we can even expect a small signal at six times this rate. When observing the DFT of the phase difference between our two channels, however, we spot these peaks along with others at integer multiples of the rotation rate, as seen in Fig. 11. In an attempt to confirm the 20 Hz amplitude as only our mirror's birefringence, we utilize the notion that under constant rotation, we can set w y Ž y t so Eq. 14 beco mes z  qrs : M Ž x t + $ Ž y t ; + $  qrs$ Ž x t : #@ ; meaning the mirror's BF signal can be moved off 20 Hz We used a function generator connected to a stepper motor that controlled the gears on a mirror mount to set the mirror's rotatio n rate to 3.5 Hz in the opposite direction of the HWP rotation 4%$3.0$!#+2!5 '+(.3$*'+$ Accor ding to Eq. 18, the mirror's birefringence should have moved to 27 Hz with an amplitude of Š and that is exactly what we see, shown in Fig. 12. The BB1 E03 mirror's BF was determined to be 1.3 4 mrad with phase noise of 2 rad/ Hz. From this, we can reach a sensitivity of 0.1 rad in 400 s averaging time, matching the precision of previous experiments 15 without the complications of a cavity. Unfortunately, including the mirror's own rotation rat e introduced peaks at its fundamental frequency and higher harmonics, as seen in Fig. 13. We know that a signal will appear at the mirror's rotation rate if the rotation axis is not perfectly aligned with the incident beam but are unclear about the other p eaks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