A self-calibrating polarimeter to measure Stokes parameters
AA self-calibrating polarimeter to measure Stokes parameters
V. Andreev,
1, 2
C. D. Panda, P. W. Hess, B. Spaun, and G. Gabrielse Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA Technische Universit¨at M¨unchen, Physik-Department, D-85748 Garching, Germany (Dated: February 28, 2017)An easily constructed and operated polarimeter precisely determines the relative Stokes parame-ters that characterize the polarization of laser light. The polarimeter is calibrated in situ withoutremoving or realigning its optical elements, and it is largely immune to fluctuations in the laser beamintensity. The polarimeter’s usefulness is illustrated by measuring thermally-induced birefringencein the indium-tin-oxide coated glass field plates used to produce a static electric field in the ACMEcollaboration’s measurement of the electron electric dipole moment.
I. INTRODUCTION
Light polarimetry remains extremely important inmany fields of physics [1]. Measurements of the polar-ization of light reveal information about interactions be-tween excited states of atoms [2]. Among many appli-cations in astronomy [3], the polarization of light frominterstellar dust reveals the magnetic field that aligns thedust [4, 5]. Light polarization also probes the magneticfield in the plasmas used for nuclear fusion studies, in-sofar as magnetic fields cause the Faraday rotation oflinear light polarization and Cotton-Mouton changes inlight ellipticity [6]. For the most precise measurement ofthe electron’s electric dipole moment, polarimetry of thethermally-induced circular polarization gradient withinglass electric field plates was crucial for understandingthe mechanisms that dominantly contributed to the sys-tematic uncertainty [7].Drawing upon the early work of G. G. Stokes [8, 9],and a later experimental realization [10], we investigatethe limits of a rotating waveplate polarimeter for deter-mining the polarization state of partially polarized laserlight. The design is easy to realize and is robust in its op-eration. With a calibration procedure introduced here, itis straightforward to internally calibrate the polarimeterwithout the need to remove or realign optical elements.The polarimeter is designed to be largely immune to fluc-tuations in light intensity, and it has been used at inten-sities up to a 100 mW / mm . The relative fractions ofcircularly polarized and linearly polarized light can typ-ically be measured with uncertainties below 0 . . / mm [12, 13]and attain uncertainties less than ± .
9% in the Stokesparameters; they have even been recommended for stu-dent labs [14]. Lower precision is also typically attainedusing other measurement methods. Light is sometimessplit to travel along optical paths with differing opticalelements, the polarization state being deduced from therelative intensities transmitted along the paths [15–18].Alternatively, the light can be analyzed using optical ele- ments whose properties vary spatially, with the polariza-tion revealed by the spatially varying intensity [19–21].The usefulness of our internally calibrated polarime-ter is demonstrated by characterizing a circular polariza-tion gradient across a nominally linearly polarized laserbeam. This gradient is produced by thermally-inducedbirefringence caused by the high intensity of the laserlight traveling through glass electric field plates coatedwith an electrically conducting layer of indium tin oxide.Such spatial polarization gradient contributes substan-tially to the systematic uncertainty in the first-generationACME measurement of the electron’s electric dipole mo-ment [7]. The small and well-characterized uncertaintiesof the polarimeter make it possible to characterize newglass electric field plates that were designed to producemuch smaller spatial polarization gradients in the second-generation ACME apparatus.This paper is structured as follows: After reviewingthe Stokes parameters in Section II and introducing thebasics of a rotating waveplate polarimeter in Section III,we describe its laboratory realization together with theintensity normalization scheme in Section IV. Section Vsummarizes how to extract the Stokes parameters froma polarimeter measurement. The calibration techniquewe developed and our analysis of the uncertainties ispresented in Section VI. Finally, we illustrate the per-formance of the polarimeter with an ellipticity gradientmeasurement in Section VII.
II. STOKES PARAMETERS
At any instant point in space and time, the electricfield of a light wave points in a particular direction. Ifthe electric field follows a repeatable path during its oscil-lations, the light wave is said to be polarized. Averagedover some time that is long compared to the oscillationperiod of the light, however, the light may be only par-tially polarized or even completely unpolarized if the di-rection of the electric field varies in a non-periodic way.Stokes showed that fully polarized light and partially po-larized light can be characterized, in principle, by inten-sities transmitted after the light passes through each offour simple configurations of optical elements: a r X i v : . [ phy s i c s . i n s - d e t ] F e b I = I (0 ◦ ) + I (90 ◦ ) = I (45 ◦ ) + I ( − ◦ )= I RHC + I LHC , (1a) M = I (0 ◦ ) − I (90 ◦ ) , (1b) C = I (45 ◦ ) − I ( − ◦ ) , (1c) S = I RHC − I LHC . (1d) The total intensity I , and the two linear polarizations M and C , are given in terms of intensities I ( α ) measuredafter the light passes through a perfect linear polarizerwhose transmission axis is oriented at an angle α with re-spect to the polarization of the incoming light. The circu-lar polarization S is the difference between the intensityof right- and left-handed circularly polarized light, I RHC and I LHC , that, as we shall see, can be deduced using aquarter-waveplate followed by a linear polarizer [11].
A. Fully polarized light
Elliptical polarization is the most general state of afully polarized plane wave traveling in the z directionwith frequency ω and wavenumber k . In cartesian coor-dinates, the electric field is (cid:126) E = ˆ x E x cos( ωt − kz + φ ) + ˆ y E y cos( ωt − kz ) , (2)where E x and E y are the absolute values of orthogonalelectric field components. φ represents the phase dif-ference between the two orthogonal components. TheStokes vector, defined with respect to the polarizationmeasured in the plane perpendicular to the propagationdirection ˆ k , is (cid:126)S = IMCS = E x + E y E x − E y E x E y cos φ E x E y sin φ (3)whereupon it follows that I = M + C + S (4) relates the four Stokes parameters in this case.While I describes the total light intensity, the dimen-sionless quantities M/I , C/I and
S/I determine the po-larization state of light. The linear polarization fractionis
L/I = (cid:112) ( M/I ) + ( C/I ) and the circular polariza-tion fraction is S/I , with(
L/I ) + ( S/I ) = 1 . (5)Because the relative intensities are summed in quadra-ture, nearly complete linear polarization (e.g. L/I =99%) corresponds to a circular polarization that is stillsubstantial (e.g.
S/I = 14%).Points on the Poincar´e sphere (Fig. 1) represent theelliptical polarization state with a relative Stokes vector (cid:126)s = M/IC/IS/I = cos 2 χ cos 2 ψ cos 2 χ sin 2 ψ sin 2 χ . (6) ~s C/IS/IM/I FIG. 1. The three relative Stokes parameters on three orthog-onal axes trace out the Poincar´e sphere, with each point onthe surface a possible state of fully polarized light.
The linear rotation angle is defined by tan 2 ψ = C/M and the ellipticity angle is defined by
S/I = sin 2 χ . B. Partially Polarized Light
The light is partially polarized if the amplitudes E x and E y and the phase φ fluctuate enough so that anaverage over time reduces the size of the average correla-tions between electric field components. The Stokes pa-rameters are then defined by the time averages of Eq. 3,with the averaging interval being long compared to boththe oscillation period and the inverse bandwidth of theFourier components that describe the light. The unpo-larized part of the light contributes only to the first ofthe four Stokes parameters, I , and not to M , C or S .The polarization fraction that survives the averaging, P ,is given by P = ( M/I ) + ( C/I ) + ( S/I ) ≤ . (7)When P = 1, the light is completely polarized and thepolarization vector describes a point on the Poincar´esphere. For partially polarized light, the length of thepolarization vector is shortened such that it will now de-scribe a point inside the sphere. When P = 0, the lightis fully unpolarized. III. ROTATING WAVEPLATE POLARIMETER
The general scheme of a rotating waveplate polarime-ter is shown in Fig. 2. Light travels first through aquarter-waveplate which can be rotated to determine thepolarization state. The light then travels through a lin-ear polarizer which can be rotated to internally calibratethe angular location of the fast axis of the waveplate andthe orientation angle of the linear polarizer transmission
Fast axis of the waveplate Zero axis of 1st rotation stage Polarizer transmission axisIncoming laser beam Outgoing intensityReference plane
Zero axis of 2nd rotation stage ˜ β β ˜ αα I out ( ˜ β ) FIG. 2. A rotating waveplate polarimeter comprised of a ro-tatable waveplate followed by a linear polarizer and a detec-tor. The axes of the optical elements are specified with respectto a reference plane. axis. All optical elements are aligned such that they arein planes perpendicular to the optical axis.Although the measurement axis of the polarimeter isnormal to its optics, the choice of the reference plane(shaded in Fig. 2) is arbitrary. We typically choosethis plane to be aligned with the transmission axis ofa calibration polarizer, through which we can pass lightbefore it enters the polarimeter. With respect to this ref-erence plane, the fast axis of the waveplate has an angle β = ˜ β + β , where ˜ β is the angle of the fast axis of thequarter-waveplate with respect to an initially unknownoffset angle β . Analogously, the transmission axis ofthe polarizer is α = ˜ α + α , where ˜ α is the angle of the polarizer transmission axis with respect to an initiallyunknown offset angle α . The linear polarizer transmis-sion angle α is left fixed during a determination of thefour Stokes parameters for the incident light, and we typ-ically set ˜ α = 0 so that α = α . For the internal cali-bration procedure the linear polarizer is rotated by anangle ˜ α = 45 ◦ . This calibration procedure (see SectionVI A for more details) determines β , α and the phasedelay between the components of the light aligned withthe slow and fast axes of the waveplate, δ ≈ π/ M transforms an input Stokes vector into the Stokes vec-tor for the light leaving the optical elements [11], (cid:126)S out = ˆ M (cid:126)S in . (8)A succession of two such matrices describes the rotatingwaveplate polarimeter of Fig. 2: (cid:126)S out = ˆ P ( α ) ˆΓ( β ) (cid:126)S in . (9)The Mueller matrix for a waveplate whose fast axis isoriented at an angle β with respect to a reference plane,and whose orthogonal slow axis delays the light trans-mission by an angle δ is [11, 22–25]ˆΓ( β ) = β + cos δ sin β cos 2 β sin 2 β (1 − cos δ ) − sin 2 β sin δ β sin 2 β (1 − cos δ ) cos δ cos β + sin β cos 2 β sin δ β sin δ − cos 2 β sin δ cos δ . (10)The Mueller matrix for a linear polarizer [11, 22–25] with transmission axis oriented at an angle α with respect to thereference plane is ˆ P ( α ) = 12 α sin 2 α α cos α cos 2 α sin 2 α α cos 2 α sin 2 α sin α
00 0 0 0 . (11)In order to extract the Stokes parameters of the incoming light, (cid:126)S in , we use a photodetector to measure the intensityof the output light, I out , which is given by I out ( ˜ β ) = I + S sin δ sin(2 α + 2˜ α − β − β )+ C (cid:104) cos δ sin(2 α + 2˜ α − β − β ) cos(2 β + 2 ˜ β ) + cos(2 α + 2˜ α − β − β ) sin(2 β + 2 ˜ β ) (cid:105) (12)+ M (cid:104) cos δ sin(2 α + 2˜ α − β − β ) sin(2 β + 2 ˜ β ) + cos(2 α + 2˜ α − β − β ) cos(2 β + 2 ˜ β ) (cid:105) . The I , M , C and S , upon which this measured intensitydepends, are the Stokes parameters of the incident beam which we wish to determine. This result is in agreementwith Stokes [9] for ˜ β = β = 0 and with [10, 26]. Iris 1 Iris 2QWP Glan-laserpolarizer PD1PD2 mm
190 mm
Rot.stage 1 Rot.stage 2 (a)
Photodetector
Glan-laser polarizer U out I out I U (b) FIG. 3. (a) Scale representation of polarimeter with 1 mmapertures, a waveplate on a rotating stage, and a linear po-larizer and two detectors that rotate on a second stage. (b) AGlan-laser polarizer splits the analyzed light into transmittedand refracted beams which can be used to monitor the totalintensity and correct for amplitude fluctuations in the lightsource.
IV. LABORATORY REALIZATION ANDINTENSITY FLUCTUATIONS
Our realization of a rotating waveplate polarimeter (toscale in Fig. 3 (a)) utilizes a quarter-waveplate (Thor-labs WPQ05M-1064), a polarizer (Thorlabs GL10-B) andtwo detectors (Thorlabs PDA100A). The waveplate ismounted on a first rotation stage, while the linear po-larizer and the two detectors are mounted on a secondrotation stage (both Newport URS50BCC).The aperture before the polarimeter constrains the col-limation of the beam inside the device. If the aperture istoo large, the imperfections of optical elements (e.g. spa-tial inhomogeneity of the waveplate retardance) reducethe accuracy of the measurement. An aperture that is toosmall results in errors due to diffraction. We found thatan aperture with a diameter of 1 ± .
25 mm minimizedthe uncertainties for our measurements at the wavelength of 1090 nm. The second aperture is used to align the po-larimeter by maximizing the light admitted by the pairof apertures.For measuring the polarization we typically vary theangle ˜ β over time with 120 discretized values ˜ β = { ◦ , ◦ , . . . , ◦ } covering one full rotation of the wave-plate. We are able to similarly rotate the linear polar-izer angle for the internal calibration, though a more re-stricted calibration rotation turns out to be optimal forreducing the uncertainties (see Section VI A).Fluctuations in light intensity contribute noise in themeasured polarization since it is deduced from the inten-sity of light transmitted through the polarimeter. Forthe sample measurements to be discussed, intensity fluc-tuations on the scale of few percent over the time of onepolarimetry measurement contributed to fluctuations inmeasurements of S/I of up to ∼ V. EXTRACTING THE STOKESPARAMETERS
A measured signal on the polarimeter detector in Fig. 4illustrates the variation of the transmitted intensity withthe waveplate angle ˜ β that is described in Eq. 12. Interms of its Fourier components, Eq. 12 can be writtenas I out ( ˜ β ) = C + C cos(2 ˜ β ) + S sin(2 ˜ β )+ C cos(4 ˜ β ) + S sin(4 ˜ β ) , (13)with the Fourier coefficients C = I + 1 + cos( δ )1 − cos( δ ) · [ C cos(4 α + 4˜ α − β ) + S sin(4 α + 4˜ α − β ) , (14a) C = S sin δ sin(2 α + 2˜ α − β ) , (14b) S = − S sin δ cos(2 α + 2˜ α − β ) , (14c) C = 1 − cos( δ )2 [ M cos(2 α + 2˜ α − β ) − C sin(2 α + 2˜ α − β )] , (14d) S = 1 − cos( δ )2 [ M sin(2 α + 2˜ α − β ) + C sin(2 α + 2˜ α − β )] . (14e)These Fourier components can be extracted from measurements like the one in Fig. 4. The circular polarization S isrelated to S and C . The linear polarization intensities, M and C , are related to C , S and C . Inverting Eqs. 14determines the Stokes parameters of the incoming light in terms of the Fourier coefficients: I = C − δ )1 − cos( δ ) · [ C cos(4 α + 4˜ α − β ) + S sin(4 α + 4˜ α − β ) , (15a) M = 21 − cos( δ ) [ C cos(2 α + 2˜ α − β ) + S sin(2 α + 2˜ α − β )] , (15b) C = 21 − cos( δ ) [ S cos(2 α + 2˜ α − β ) − C sin(2 α + 2˜ α − β )] , (15c) S = C sin( δ ) sin(2 α + 2˜ α − β ) = − S sin( δ ) cos(2 α + 2˜ α − β ) . (15d)The Stokes parameters are thus determined by theFourier coefficients that are extracted from measure-ments like the one in Fig. 4, along with the values of theangles α , β and δ from the calibration to be described.The angle ˜ α is 0 during a polarization measurement andis stepped away from 0 only during a calibration, as weshall see.Both of the two expressions for S must be used cau-tiously given the possibility that a denominator couldvanish. Combining them gives a more robust expressionthat is independent of the two calibration angles, α and ⇥ ! ° ⇥ M ea s u r e dpho t od i od e vo lt a g e V ⇥ FIG. 4. Illustration of how the light transmitted through thepolarimeter varies with the angle of the waveplate axis aspredicted. β [10], S = − sign(S ) (cid:112) C + S sin( δ ) . (16)We choose to make the angle 2( α − β ) small, whereupon | S | > | C | for nonvanishing S and the sign of S is thatof − S . Similarly, combining equations (15b) and (15c)gives a more robust expression for the magnitude of thelinear polarization L = √ M + C , independent of α and β , L = (cid:112) C + S sin (cid:0) δ (cid:1) . (17)Of course, both S/I and
L/I still depend on all of thecalibration angles since I does, but the use of the morerobust expression can reduce the uncertainties in S/I and
L/I . VI. CALIBRATION AND UNCERTAINTIES
The in situ calibration procedure is centrally responsi-ble for the low uncertainties realized with this polarime-ter. The basic idea is to rotate the linear polarizer inorder to calibrate the angles α , β and the waveplate de-lay δ (which varies with wavelength). This has been donebefore [10]. The advance here is to determine the opti-mal rotation needed to minimize the calibration time anduncertainties. We avoid realignments of optical elementscaused by either temporarily removing or by flipping op-tical elements with respect to the light transmission axis[26].The uncertainties arising from the calibration causeuncertainties in the measured Stokes parameters, as dothe uncertainties with which the Fourier coefficients C , C , C , S , and S are determined. Statistical uncertain-ties that can be averaged down with more measurementsare typically on the order of 0 .
01% for
S/I and 0 .
05% for
L/I . We also discuss the systematic uncertainties thatarise in addition to calibration uncertainties, specificallydiscussing those that arise from waveplate imperfections,misalignment of the incident light pointing relative to the measurement axis, and the finite extinction ratio of thepolarizer.
A. Calibration Method
The calibration procedure starts with a high extinctionratio polarizer placed in the light beam before it entersthe polarimeter. The light analyzed by the polarimeteris in this case known to be almost fully polarized. Thetransmission axis of this external calibration polarizerthen defines the reference plane in terms of which thelinear polarizations M and C are determined. For therelative Stokes vector (1,0,0), Eqs. (15) simplify to C − δ )1 − cos( δ ) [ C cos(4 α − β ) + S sin(4 α − β )] = 21 − cos( δ ) [ C cos(2 α − β ) + S sin(2 α − β )] , (18a)arctan S C = 2 α − β . (18b)For the third linearly independent equation needed to determine the three calibration parameters we rotate the linearpolarizer from ˜ α = 0 to ˜ α = 45 ◦ , a choice shortly to be justified, whereupon˜ C − δ )1 − cos( δ ) · [ ˜ C cos(4 α + 4˜ α − β ) + ˜ S sin(4 α + 4˜ α − β )]= 21 − cos( δ ) [ ˜ C cos(2 α + 2˜ α − β ) + ˜ S sin(2 α + 2˜ α − β )] (19)These cumbersome equations simplify when the wave-plate is reasonably close to being a quarter-waveplate,whereupon cos δ (cid:39) π − δ . They simplify further when wedeliberately choose to make α small, which means thatthe zero-axis of the internal polarizer is approximatelyaligned with the external polarizer that defines the refer-ence plane for the Stokes parameters. The greatly sim-plified calibration equations are then δ = π (cid:112) C + S − C (cid:112) C + S + C , (20) α = cot ˜ α − α ) (cid:112) S + C − ˜ C (cid:112) S + C − C , (21) β = 14 (cid:18) arctan S C − α (cid:19) . (22)The uncertainty in α is determined by the standard errorpropagation: σ α = (cid:18) ∂α ∂ ˜ α (cid:19) σ α + (cid:18) ∂α ∂ ˜ C (cid:19) σ C + (cid:18) ∂α ∂C (cid:19) σ C + (cid:18) ∂α ∂C (cid:19) σ C + (cid:18) ∂α ∂S (cid:19) σ S , (23)with α from Eq. 21 given our simplifying choice to make α small.To illustrate that the choice ˜ α ≈ ◦ minimizes uncer-tainties, Fig. 5 shows the uncertainty in α for a typicalchoice of uncertainties in the normalized Fourier coeffi-cients of 0 .
05% and an uncertainty in ˜ α of 0 . ◦ . For ˜ α close to 0 ◦ or 90 ◦ the equations (18) and (19) becomedegenerate and therefore the uncertainty in α grows to
50 0 500.050.100.150.200.250.30 ⇥⌅ ° ⇥ ⇤ ⇥ ° ⇥ FIG. 5. The uncertainty with which α is determined as afunction of the change in linear polarizer angle used in thecalibration procedure shows that a 45 ◦ step is close to optimal. infinity. The 45 ◦ choice minimizes the uncertainties incalibration parameters, typically making them less than0 . ◦ , when the uncertainties in the Fourier coefficientsare approximately equal.To summarize, these are the calibration steps:1. Place a linear polarizer before the polarimeter. Itspolarization axis then defines the reference planefor M and C . Set ˜ α = 0.2. Perform at least one full rotation of the waveplaterecording the output intensity I out ( ˜ β ).3. Determine the Fourier coefficients C , C , and S from the scan in step 2.4. Rotate the polarizer inside the polarimeter by ˜ α =45 ◦ and repeat step 2.5. Determine the Fourier coefficients ˜ C , ˜ C , and ˜ S from the scan in step 4.6. Using the Eqs. (18)-(19), calculate the calibrationparameters δ , α , and β from the measured Fouriercoefficients. B. Calibration Uncertainties
Typically our calibration procedure determines thewaveplate delay to about 0 . ◦ out of δ ≈ ◦ . For ourchoice of angles, α = 2 ◦ and β = 20 ◦ , we typically de-termine the differences α − β and α − β to betterthan 0 . ◦ .It is straightforward but tedious to propagate uncer-tainties of this size to resulting uncertainties in the rela-tive Stokes parameters. Fig. 6 shows the contribution ofthe uncertainty in α − β alone to errors in S/I and
L/I with a dashed curve. This is typically much smaller thanthe contribution from the uncertainty in δ alone, shownwith a solid curve.The calibration uncertainties for S/I are always below0.1%, and the calibration uncertainties for
L/I are below0.35% for any analyzed input polarization. For our ex-ample measurement of small
S/I values the calibrationuncertainty is below 0 . C. Waveplate Imperfections
Even after input intensity fluctuations were normal-ized out, there was still a variation in the transmittedlight intensity as the waveplate rotated. The initially ob-served variation, for an achromatic waveplate (ThorlabsAQWP05M-980), is shown by the light gray points inFig. 7. This variation limited the uncertainty in
S/I toabout 0 . .
01% in the normalized Fourier coefficients.This translates into an error in
S/I of smaller than 0 . D. Misalignments
The waveplate and the polarizer that make up the po-larimeter are ideally aligned so that their optical surfacesare exactly perpendicular to the direction of propagationof the laser beam. Fig. 8 shows an example of the sys-tematic uncertainty that arises in measurements of
S/I due to misalignments. We routinely align the polarimeterto better than 0 . ◦ which translates into uncertaintiesof 0 . S/I and 0 .
05% in
L/I . E. Finite extinction ratios of the polarizers
The two linear polarizers used within and ahead of thepolarimeter each have a finite extinction ratio r . A simplemodel of a polarizer perfectly transmits light along oneaxis and suppresses light transmission by a factor of r along the orthogonal axis. The corresponding Muellermatrix [24] for such a polarizer isˆ P imp = 12 − r − r (cid:112) r (1 − r ) 00 0 0 2 (cid:112) r (1 − r ) . (24)For polarizers used here, r (cid:46) − rather than beingperfectly r = 0.For calibration, the relative Stokes vector sent into thepolarimeter after circular polarized light passed throughan imperfect calibration polarizer is (cid:126)s = − r (cid:112) r (1 − r ) (cid:39) − r √ r , (25)For an extinction ratio of r ∼ − , there is a residual S/I of up to 2 √ r (cid:39) . . ◦ for α and β , and smaller than 0 . ◦ for δ .The finite extinction ratio of the polarizer in the po-larimeter modifies the Fourier components measured inthe out-going intensity I out . Eq. 12 is re-obtainedwith the transformation of the Stokes parameters M → M (1 − r ), C → C (1 − r ) and S → S (1 − r ). Thatmeans that the measured Stokes parameters differ from � �� �� �� �� ����������������������� � / � [%] � / � � �� � � [ % ] δ ��� = ��� ° � ( α � - β � ) ��� = � ° δ ��� = � ° � ( α � - β � ) ��� = ��� ° � �� �� �� �� ����������������������������������� � / � [%] � / � � �� � � [ % ] FIG. 6. Calculated uncertainties in
S/I (left) and
L/I (right) for typical calibration parameters of δ = 92 ◦ , α = 2 ◦ , β = 20 ◦ ,and calibration uncertainties of 0 . ◦ in δ (solid curve) and also for an uncertainty of 0 . ◦ in α − β (dashed curve) for a rangeof S/I and
L/I values. The curves for
S/I are symmetric around zero.
Achromatic WPMonochromatic WP0 50 100 150 200 250 300 3500.9900.9951.0001.0051.010 Waveplate angle, ⇥ ° ⇥ N o r m a li ze d P D vo lt a g e FIG. 7. An achromatic waveplate makes the detected inten-sity vary much more as a function of waveplate orientationthan does a monochromatic waveplate. ⇥ ⇥ ⇥ ⇥ Misalignment deg ⇥ ⇤ L ⇤ I ⇥ L I S I ⇥ ⇥ ⇥ ⇥ deg ⇥ ⇤ S ⇤ I ⇥ FIG. 8. Uncertainties in
S/I (light gray) and
L/I (dark gray)change with misalignment of the polarimeter with respect tothe light propagation axis, performed with a fixed incomingpolarization of
S/I = 16 .
3% and
L/I = 98 . the true values by a factor 1 / (1 − r ) (cid:39) r , which is0 . r (cid:39) − . TABLE I. Summary of the systematic errors for
S/I <
L/I >
L/I ) err [%] ( S/I ) err [%]( α − β ) calibration to ± . ◦ < . < . δ calibration to ± . ◦ < . < . < . < . < . < . < . < . < . < . < . < . F. Systematic Uncertainty Summary
A summary of the investigated systematic errors for
S/I <
30% and
L/I >
S/I is smaller than 0 . L/I are significantly larger, up to 0 . δ . From Eqs.16 and 17, S ∝ sin − ( δ ) and L ∝ sin − ( δ/ δ = 90 ◦ is a first order effect in L and issecond order for S . VII. APPLICATION: THERMALLY-INDUCEDBIREFRINGENCE
To illustrate the use of our internally calibrated po-larimeter we measure the circular polarization inducedin laser light intense enough to create thermal gradientsin glass electric field plates coated with a conducting layerof indium tin oxide used in the ACME measurement ofthe electric dipole moment of the electron. This effectcontributed to a systematic error mechanism that domi-nated the systematic uncertainty in a measurement thatwas an order of magnitude more sensitive than previ-ous measurements [7]. The polarimeter makes it possi-
Gen I field plateGen II field plate mm ⇥ ⇥ S ⇤ I ⇥ Intensity profile a.u. ⇥ FIG. 9. The self-calibrated polarimeter has uncertainties lowenough to compare circular polarization gradients producedby thermal gradients in first and second-generation glass fieldplates used by the ACME collaboration for the electron elec-tric dipole moment experiment. Measurements were takenwith an elongated Gaussian laser beam at 1090 nm with waists w x = 1 . (cid:28) w y (cid:39)
30 mm and a total power of 2 W. Errorbars represent a quadrature sum of statistical and systematicuncertainties. ble to see whether improved electric plates produced fora second-generation experiment succeed in reducing thethermally-induced birefringence.To measure the polarization induced by the field platebirefringence, we start with a collimated high-power laserbeam with the total power of 2 W, wavelength 1090 nm,and a circular Gaussian beam shape with waists of w x (cid:39) w y (cid:39) . y di-rection using two cylindrical lenses with focal lengths of f = 10 mm and f = 200 mm, so that the beam shapeis elongated with w x = 1 . (cid:28) w y (cid:39)
30 mm. Thelaser beam then passes through the glass plate and en-ters the polarimeter.
S/I is measured as the polarimeteris translated on a linear translation stage in the x di-rection across the narrow illuminated area on the fieldplate.We compare the spatial gradient in S/I for ACME’sfirst- and second-generation plates. The first-generationplate was made of borosilicate glass with a thermal ex-pansion coefficient of 3 . · − . · − S/I are shown in Fig. 9. Theintensity profile of the laser beam in the x direction isthe upper dashed curve. The spatial variation of S/I forthe first-generation plate are the dark gray points, witha smooth curve from a theoretical model [33, 34] thatis beyond the scope of this report. The much smaller spatial gradient of the light gray points was measuredwith the second-generation plate. The substantial reduc-tion, from
S/I = 0 .
6% to
S/I < . .
1% over the diameter of the laser beam, this small vari-ation is superimposed upon a much larger
S/I ≈
8% off-set. This offset can be reduced to be less than 0 .
1% byaligning the intensity profile of the intense laser with thepolarization axis. However, the offset is a reminder thatmechanical stress in optical windows and other opticalelements will typically produce birefringence.The 8% offset in our experiment comes primarily fromstress in the 5.5 x 3.5 inch vacuum windows that are0.75 inch thick, made from the same material as the fieldplates. With atmospheric pressure on both sides of thesewindows, adjusting the tension in screws holding the win-dows to the vacuum chamber changed
S/I from about 8%to 6%. Pumping out the chamber to put a differentialpressure of one atmosphere across such a window typi-cally changed
S/I by up to 3%. Related measurementswith the polarimeter showed that optical elements suchas a zero-order half-waveplate could produce circular po-larization of up to 3%. We did not observe unexpectedlinear polarization changes from the windows larger thanthe systematic uncertainties in the measurement.
VIII. CONCLUSION
A highly sensitive, easy-to-construct polarimeter withhigh power-handling capabilities is demonstrated. Thepolarimeter is calibrated internally and in situ, withoutthe need for removing or realigning any optical elements.The calibration procedure is critical for the low uncer-tainties that have been achieved. A detailed error anal-ysis shows that the
S/I
Stokes parameter that describescircular polarization can be measured to better than 0 . L/I is determined to below 0 . S/I . The usefulness of a polarimeter withlow uncertainty was demonstrated by measuring circularpolarization gradients due to thermally-induced birefrin-gence in a glass field plate that is critical to the mostprecise measurement of the electron electric dipole mo-ment.
ACKNOWLEDGEMENT
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