A self-calibrating polarimeter to measure Stokes parameters V. Andreev,1, 2 C. D. Panda,1 P. W. Hess,1 B. Spaun,1 and G. Gabrielse1 1Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA 2Technische Universit¨atM¨unchen,Physik-Department, D-85748 Garching, Germany (Dated: February 19, 2021) 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 without removing or realigning its optical elements, and it is largely immune to fluctuations in the laser beam intensity. The polarimeter's usefulness is illustrated by measuring thermally-induced birefringence in the indium-tin-oxide coated glass field plates used to produce a static electric field in the ACME collaboration's measurement of the electron electric dipole moment. I. INTRODUCTION ments whose properties vary spatially, with the polariza- tion revealed by the spatially varying intensity [19{21]. Light polarimetry remains extremely important in The usefulness of our internally calibrated polarime- many fields of physics [1]. Measurements of the polar- ter is demonstrated by characterizing a circular polariza- ization of light reveal information about interactions be- tion gradient across a nominally linearly polarized laser tween excited states of atoms [2]. Among many appli- beam. This gradient is produced by thermally-induced cations in astronomy [3], the polarization of light from birefringence caused by the high intensity of the laser interstellar dust reveals the magnetic field that aligns the light traveling through glass electric field plates coated dust [4, 5]. Light polarization also probes the magnetic with an electrically conducting layer of indium tin oxide. field in the plasmas used for nuclear fusion studies, in- Such spatial polarization gradient contributes substan- sofar as magnetic fields cause the Faraday rotation of tially to the systematic uncertainty in the first-generation linear light polarization and Cotton-Mouton changes in ACME measurement of the electron's electric dipole mo- light ellipticity [6]. For the most precise measurement of ment [7]. The small and well-characterized uncertainties the electron's electric dipole moment, polarimetry of the of the polarimeter make it possible to characterize new thermally-induced circular polarization gradient within glass electric field plates that were designed to produce glass electric field plates was crucial for understanding much smaller spatial polarization gradients in the second- the mechanisms that dominantly contributed to the sys- generation ACME apparatus. tematic uncertainty [7]. This paper is structured as follows: After reviewing the Stokes parameters in Section II and introducing the Drawing upon the early work of G. G. Stokes [8, 9], basics of a rotating waveplate polarimeter in Section III, and a later experimental realization [10], we investigate we describe its laboratory realization together with the the limits of a rotating waveplate polarimeter for deter- intensity normalization scheme in Section IV. Section V mining the polarization state of partially polarized laser summarizes how to extract the Stokes parameters from light. The design is easy to realize and is robust in its op- a polarimeter measurement. The calibration technique eration. With a calibration procedure introduced here, it we developed and our analysis of the uncertainties is is straightforward to internally calibrate the polarimeter presented in Section VI. Finally, we illustrate the per- without the need to remove or realign optical elements. formance of the polarimeter with an ellipticity gradient The polarimeter is designed to be largely immune to fluc- measurement in Section VII. tuations in light intensity, and it has been used at inten- sities up to a 100 mW=mm2. The relative fractions of circularly polarized and linearly polarized light can typ- ically be measured with uncertainties below 0:1 % and II. STOKES PARAMETERS 0:4 %, respectively. Light polarization can be measured in various ways At any instant point in space and time, the electric arXiv:1703.00963v3 [physics.ins-det] 19 Feb 2021 [11]. Polarimeters similar to ours, but lacking the in- field of a light wave points in a particular direction. If ternal calibration mechanism and immunity to intensity the electric field follows a repeatable path during its oscil- fluctuations, can handle up to several mW=mm2 [12, 13] lations, the light wave is said to be polarized. Averaged and attain uncertainties less than ±0:9% in the Stokes over some time that is long compared to the oscillation parameters; they have even been recommended for stu- period of the light, however, the light may be only par- dent labs [14]. Lower precision is also typically attained tially polarized or even completely unpolarized if the di- using other measurement methods. Light is sometimes rection of the electric field varies in a non-periodic way. split to travel along optical paths with differing optical Stokes showed that fully polarized light and partially po- elements, the polarization state being deduced from the larized light can be characterized, in principle, by inten- relative intensities transmitted along the paths [15{18]. sities transmitted after the light passes through each of Alternatively, the light can be analyzed using optical ele- four simple configurations of optical elements: 2 S/I I =I(0◦) + I(90◦) = I(45◦) + I(−45◦) =IRHC + ILHC; (1a) ◦ ◦ M =I(0 ) − I(90 ); (1b) ~s C =I(45◦) − I(−45◦); (1c) S =IRHC − ILHC: (1d) 2χ The total intensity I, and the two linear polarizations M and C, are given in terms of intensities I(α) measured C/I after the light passes through a perfect linear polarizer whose transmission axis is oriented at an angle α with re- spect to the polarization of the incoming light. The circu- 2 lar polarization S is the difference between the intensity of right- and left-handed circularly polarized light, IRHC M/I and ILHC, that, as we shall see, can be deduced using a quarter-waveplate followed by a linear polarizer [11]. FIG. 1. The three relative Stokes parameters on three orthog- onal axes trace out the Poincar´esphere, with each point on the surface a possible state of fully polarized light. A. Fully polarized light Elliptical polarization is the most general state of a The linear rotation angle is defined by tan 2 = C=M fully polarized plane wave traveling in the z direction and the ellipticity angle is defined by S=I = sin 2χ. with frequency ! and wavenumber k. In cartesian coor- dinates, the electric field is B. Partially Polarized Light ~ E = x^ E0x cos(!t − kz + φ) + y^ E0y cos(!t − kz); (2) where E0x and E0y are the absolute values of orthogonal The light is partially polarized if the amplitudes E0x electric field components. φ represents the phase dif- and E0y and the phase φ fluctuate enough so that an ference between the two orthogonal components. The average over time reduces the size of the average correla- Stokes vector, defined with respect to the polarization tions between electric field components. The Stokes pa- measured in the plane perpendicular to the propagation rameters are then defined by the time averages of Eq. 3, direction k^, is with the averaging interval being long compared to both the oscillation period and the inverse bandwidth of the 0 1 0 2 2 1 I E0x + E0y Fourier components that describe the light. The unpo- M B E2 − E2 C S~ = B C = B 0x 0y C (3) larized part of the light contributes only to the first of @ C A @2 E0xE0y cos φA the four Stokes parameters, I, and not to M, C or S. S 2 E0xE0y sin φ The polarization fraction that survives the averaging, P , whereupon it follows that is given by 2 2 2 2 I = M + C + S (4) P 2 = (M=I)2 + (C=I)2 + (S=I)2 ≤ 1: (7) relates the four Stokes parameters in this case. When P = 1, the light is completely polarized and the While I describes the total light intensity, the dimen- polarization vector describes a point on the Poincar´e sionless quantities M=I, C=I and S=I determine the po- sphere. For partially polarized light, the length of the larization state of light. The linear polarization fraction p polarization vector is shortened such that it will now de- is L=I = (M=I)2 + (C=I)2 and the circular polariza- scribe a point inside the sphere. When P = 0, the light tion fraction is S=I, with is fully unpolarized. (L=I)2 + (S=I)2 = 1: (5) Because the relative intensities are summed in quadra- III. ROTATING WAVEPLATE POLARIMETER ture, nearly complete linear polarization (e.g. L=I = 99%) corresponds to a circular polarization that is still substantial (e.g. S=I = 14%). The general scheme of a rotating waveplate polarime- Points on the Poincar´esphere (Fig. 1) represent the ter is shown in Fig. 2. Light travels first through a elliptical polarization state with a relative Stokes vector quarter-waveplate which can be rotated to determine the polarization state. The light then travels through a lin- 0 1 0 1 M=I cos 2χ cos 2 ear polarizer which can be rotated to internally calibrate ~s = @ C=I A = @cos 2χ sin 2 A : (6) the angular location of the fast axis of the waveplate and S=I sin 2χ the orientation angle of the linear polarizer transmission 3 Zero axis of Zero axis of polarizer transmission axis with respect to an initially ˜ 1st rotation 2nd rotation unknown offset angle α . The linear polarizer transmis- β stage stage 0 α˜ sion angle α is left fixed during a determination of the Incoming β0 α0 four Stokes parameters for the incident light, and we typ- laser beam ically setα ~ = 0 so that α = α0.