27 Stokes Polarimetry 27.1 INTRODUCTION In this chapter, we discuss methods of measuring (or creating) the Stokes vector, the real four-element entity that describes the state of polarization of a beam of light The measurement process can be represented as I ¼ AS ð27-1Þ where I is the vector of flux measurements as made by the detector, A is a matrix whose dimensions depend on the number of measurements and whose elements depend on the optical system, and S is the incident Stokes vector Since we want to determine the incident Stokes vector, we must invert Eq (27-1) so that S is given by S ¼ AÀ1 I ð27-2Þ This system of equations is generated through a set of measurements and can be solved through Fourier or nonFourier techniques Both solution methods will be discussed in this chapter A set of elements that analyzes a polarization state of incoming light is a polarization state analyzer (PSA) A set of elements that generates a polarization state is a polarization state generator (PSG) The PSA and PSG are functionally depicted in Fig 27-1 All of the polarimeter types described in this chapter can be or have to be used with electronics and computers in order to automate the data collection process A Stokes polarimeter is complete if it measures all four elements of the Stokes vector, and incomplete if it measures less than four We will describe several types of Stokes polarimeters in the remainder of the chapter Rotating element polarimetry, oscillating element polarimetry, and phase modulation polarimetry are all methods that make a series of measurements over time to obtain the Stokes vector [1] Other techniques, division of amplitude and division of wavefront polarimetry, described in the last section of the chapter, are designed to measure all four elements of the Stokes vector simultaneously Copyright © 2003 by Marcel Dekker, Inc All Rights Reserved Figure 27-1 27.2 Functional diagrams of Stokes polarimetry ROTATING ELEMENT POLARIMETRY Stokes polarimeters that use rotating elements are shown in Fig 27-2 The elements shown are all linear retarders and polarizers (analyzers) The measured Stokes elements are shown in the box to the right of each diagram, where the large black dots indicate the Stokes components that are measured 27.2.1 Rotating Analyzer Polarimeter Shown in Fig 27-2a, the polarizer (analyzer) in this polarimeter rotates and produces a modulating signal at the detector, which is given by I¼ a0 a b þ cos 2 þ sin 2 2 ð27-3Þ where is the azimuthal angle of the polarizer The coefficients a0, a2, and b2 are the first three elements of the Stokes vector At least three measurements must be made to determine the three measurable elements of the Stokes vector Equation (27-3) and subsequent expressions for the modulated signal in this chapter on Stokes polarimetry and in Chapter 28 on Mueller matrix polarimetry are all derived from algebraic equations representing these polarimetric systems For example, for the rotating analyzer polarimeter, we have the equation: 01 10 1 cos 2 sin 2 S0 S0 B S01 C B cos 2 CB S1 C cos 2 sin 2 cos 2 B C¼ B CB C ð27-4Þ @ S2 A @ sin 2 sin 2 cos 2 sin2 2 A@ S2 A 0 0 S3 S3 Copyright © 2003 by Marcel Dekker, Inc All Rights Reserved Figure 27-2 Rotating element polarimeters (After Ref 1.) where the input Stokes vector is multiplied by the Mueller matrix for a rotated ideal linear polarizer to obtain the (primed) output Stokes vector We only need carry out the multiplication of the first row of the Mueller matrix with the input Stokes vector because we will be measuring the output signal I ¼ S00 Thus, I¼ S0 S1 S þ cos 2 þ sin 2 2 Copyright © 2003 by Marcel Dekker, Inc All Rights Reserved ð27-5Þ Comparing this equation with (27-3), we have the correspondence: S ¼ a0 S ¼ a2 ð27-6Þ S ¼ b2 The coefficients have been purposely written as a’s and b’s to represent the modulated signal as a Fourier series where the fundamental frequency of modulation and its harmonics are the angle and its multiples We will continue to this for the polarimeters described in this chapter and the next 27.2.2 Rotating Analyzer and Fixed Analyzer Polarimeter A fixed analyzer in front of the detector in this configuration shown in Fig 27-2b means that the detector observes only one polarization, and any detector polarization sensitivity is made superfluous A modulated signal composed of two frequencies is measured, and can be expressed as the Fourier series: I¼ a0 X þ ða cos 2n þ b2n sin 2nÞ 4 n¼1 2n ð27-7Þ The first three elements of the Stokes vector are S ¼ a0 À a4 S1 ¼ ða2 À a0 þ 2a4 Þ S2 ¼ 0:4ð2b2 þ b4 Þ: 27.2.3 ð27-8Þ Rotating Retarder and Fixed Analyzer Polarimeter This is the basic complete Stokes polarimeter and is illustrated in Fig 27-2c The detector observes only a single polarization, and the modulated signal is again composed of two frequencies The signal is again expressed as a Fourier series: I¼ a0 X þ ða cos 2n þ b2n sin 2nÞ 2 n¼1 2n ð27-9Þ where now the angle is the azimuthal angle of the retarder If the retarder is quarter wave, the Stokes vector is given in terms of the Fourier coefficients as S ¼ a0 À a4 S1 ¼ 2a4 S2 ¼ 2b4 S ¼ b2 Copyright © 2003 by Marcel Dekker, Inc All Rights Reserved ð27-10Þ 27.2.4 Rotating Retarder and Analyzer Polarimeter Both elements rotate in this polarimeter of Fig 27-2d When the analyzer is rotated at three times the retarder angle and the retarder is quarter wave, the detected signal is given by I¼ a0 X þ ða cos 2n þ b2n sin 2nÞ 2 n¼1 2n ð27-11Þ where is the rotation angle of the retarder The Stokes vector is S ¼ a0 S ¼ a2 þ a6 S ¼ b6 À b2 ð27-12Þ S ¼ b4 27.2.5 Rotating Retarder and Analyzer Plus Fixed Analyzer Polarimeter This case, combining the previous two cases and shown in Fig 27-2e, produces as many as nine harmonics in the detected signal when the analyzer is rotated by the factors 5/2, 5/3, or À3/2 times the retarder angle so that I¼ 10 a0 X þ ða cos n þ bn sin nÞ 4 n¼1 n ð27-13Þ n6¼9 The Stokes vector is given in terms of the Fourier coefficients, when the rotation factor is 5/2 and the fixed analyzer is at 0 , as S ¼ a0 À a4 S1 ¼ 2a1 S2 ¼ 2b1 ð27-14Þ S ¼ b3 27.3 OSCILLATING ELEMENT POLARIMETRY Oscillating element polarimeters rotate the polarization of light using some electroor magneto-optical device such as a Faraday cell or a liquid crystal cell (see Chapter 24) If, for example, the plane of polarization is rotated by an angle in a Faraday cell, this has the effect of having mechanically rotated all subsequent elements by an angle À The modulation is typically sinusoidal, which simulates an oscillating element, although a saw-tooth signal could be used to drive the modulation to result in an equivalent to a synchronous rotation of the element The advantages of oscillating element polarimeters include operation at high frequencies, and the absence of moving parts to disturb alignment A disadvantage, when the modulation is sinusoidal, is the additional complication in the signal content The azimuthal angles are sinusoids, and the detected intensity now contains an infinite number of harmonics whose amplitudes depend on Bessel functions of the modulation Copyright © 2003 by Marcel Dekker, Inc All Rights Reserved Figure 27-3 Oscillating element polarimeters (After Ref 1.) amplitude Oscillating element polarimeters derive harmonic content from the relationships (Bessel function expansions): sinð sin !tÞ ¼ X J2nþ1 ðÞ sin½ð2n þ 1Þ!t ð27-15Þ n¼0 cosð sin !tÞ ¼ J0 ðÞ þ X J2n ðÞ cos 2n!t ð27-16Þ n¼0 Experimentally, a lock-in amplifier is required for each detected frequency Three oscillating element polarimeters are shown in Fig 27-3 and we describe these polarimeters in the following subsections 27.3.1 Oscillating Analyzer Polarimeter The oscillating analyzer polarimeter (see Ref 2) is shown in Fig 27-3a This polarimeter, like the rotating analyzer polarimeter, measures the first three components of the Stokes vector and hence is an incomplete polarimeter The oscillating element produces an effective analyzer azimuth of ¼ 0 þ 1 sin !t ð27-7Þ where the azimuth 0 is determined by the mechanical azimuth of the fixed analyzer and/or a d.c bias current in the Faraday cell, and 1 is the amplitude of Copyright © 2003 by Marcel Dekker, Inc All Rights Reserved the sinusoidal optical rotation produced by the Faraday cell Substituting (27-17) into (27-3) we have I¼ a0 a2 cos 20 þ b2 sin 20 þ cosð21 sin !tÞ 2 Àa sin 20 þ b2 cos 20 sinð21 sin !tÞ þ 2 ð27-18Þ If we now use Eqs (27-15) and (27-16) to replace cosð21 sin !tÞ and sinð21 sin !tÞ, we have a0 a2 b2 cos 20 þ sin 20 ½J0 ð21 Þ þ 2J2 ð21 Þ cos 2!t I¼ þ 2 Àa2 b2 sin 20 þ cos 20 ½2J1 ð21 Þ sin !t þ 2 ð27-19Þ where we have neglected terms in frequency higher than 2! The zero frequency (d.c.), fundamental, and second harmonic of the detected signal are then ! a2 b2 cos 20 þ sin 20 Ið0Þ ¼ þ J0 ð21 Þ 2 ! Àa2 b Ið!Þ ¼ 2J1 ð21 Þ sin 20 þ cos 20 sin !t 2 ! a b Ið2!Þ ¼ 2J2 ð21 Þ cos 20 þ sin 20 cos 2!t 2 ð27-20Þ The d.c., fundamental, and second harmonic of the signal are detected synchronously, and the amplitude ratios are ! ¼ Ið!Þ=Ið0Þ 2! ¼ Ið2!Þ=Ið0Þ ð27-21Þ and these are, using (28-20), ! ¼ 2! 2J2 ð21 ÞððÀa2 =2Þ sin 20 þ ðb2 =2Þ cos 20 Þ þ J0 ð21 Þðða2 =2Þ cos 20 þ ðb2 =2Þ sin 20 Þ ð27-22Þ 2J2 ð21 Þðða2 =2Þ cos 20 þ ðb2 =2Þ sin 20 Þ ¼ þ J0 ð21 Þðða2 =2Þ cos 20 þ ðb2 =2Þ sin 20 Þ These last equations can be inverted to give the coefficients a2 ¼ ! J2 ð21 Þ sin 20 À 2! J1 ð21 Þ cos 20 J1 ð21 Þ½2! J0 ð21 Þ À 2J2 ð21 Þ À! J2 ð21 Þ cos 20 À 2! J1 ð21 Þ sin 20 b2 ¼ J1 ð21 Þ½2! J0 ð21 Þ À 2J2 ð21 Þ Copyright © 2003 by Marcel Dekker, Inc All Rights Reserved ð27-23Þ If ¼ 0 and 21 ¼ 137:8 , J0 ð21 Þ ¼ 0, and the Stokes vector is given by S00 ¼ I0 2! 2J2 ð21 Þ ! S02 ¼ 2J1 ð21 Þ S01 ¼ ð27-24Þ where the primes indicate the output Stokes parameters 27.3.2 Oscillating Retarder with Fixed Analyzer Polarimeter This polarimeter, the equivalent of the rotating retarder polarimeter, is shown in Fig 27-3b As indicated in the figure, this is a complete Stokes polarimeter A retarder is surrounded by two optical rotators with equal and opposite rotations For example, a quarter-wave retarder might have a Faraday cell on one side and an identical Faraday cell on the other side but connected to an electrical signal source of opposite polarity A light beam passing through a linear retarder of retardance with fast axis azimuth R and a linear polarizer (analyzer) of azimuth A results in an output intensity corresponding to the first Stokes parameter of the emergent light: S00 ¼ S0 S1 þ ½cos 2R cosð2A À 2R Þ À sin 2R sinð2A À 2R Þ cos 2 S2 þ ½sin 2R cosð2A À 2R Þ þ cos 2R sinð2A À 2R Þ cos S3 þ ½sinð2A À 2R Þ sin ð27-25Þ If we assume that ¼ =2 and A ¼ 0, I is the detected signal, and k is a proportionality constant, then we have 1 kI ¼ S0 þ S1 þ S1 cos 4R þ S2 sin 4R À S3 sin 2R ð27-26Þ 2 or kI ¼ þ cos 4R þ sin 4R À sin 2R ð27-27Þ where ¼ S0 þ S1 ð27-28aÞ 1 ¼ S ð27-28bÞ ¼ S 2 ð27-28cÞ ¼ ÀS3 ð27-28dÞ and Copyright © 2003 by Marcel Dekker, Inc All Rights Reserved The two optical rotators on either side of the retarder effectively oscillate the retarder azimuth and we have R ¼ R0 þ R1 sin !t ð27-29Þ where R0 is the bias azimuth, and R1 is the rotation amplitude Using (27-29) in (27-27) and again making use of the Bessel function expansions, we can obtain the Fourier amplitudes of the detected signal as kIdc ¼ þ ½cos 4R0 J0 ð4R1 Þ þ ½sin 4R0 J0 ð4R1 Þ þ ½sin 2R0 J0 ð2R1 Þ ð27-30aÞ kI! ¼ ½À2 sin 4R0 J1 ð4R1 Þ þ ½2 cos 4R0 J1 ð4R1 Þ þ ½2 cos 2R0 J1 ð2R1 Þ ð27-30bÞ kI2! ¼ ½2 cos 4R0 J2 ð4R1 Þ þ ½2 sin 4R0 J2 ð4R1 Þ þ ½2 sin 2R0 J2 ð2R1 Þ ð27-30cÞ kI3! ¼ ½À2 sin 4R0 J3 ð4R1 Þ þ ½2 cos 4R0 J3 ð4R1 Þ þ ½2 cos 2R0 J3 ð2R1 Þ ð27-30dÞ In vector-matrix form, the last three equations are 10 1 À2 sin 4R0 J1 ð4R1 Þ cos 4R0 J1 ð4R1 Þ cos 2R0 J1 ð2R1 Þ I! B C B cos 4 J ð4 Þ sin 4 J ð4 Þ sin 2 J ð2 Þ CB C k@ I2! A ¼ @ R0 R1 R0 R1 R0 R1 A@ A I3! À2 sin 4R0 J3 ð4R1 Þ cos 4R0 J3 ð4R1 Þ cos 2R0 J3 ð2R1 Þ ð27-31Þ This equation can be solved for 1, 2, and by inverting the  matrix Equation (27-30a) can then be used to find 0, and (27-28) used to find the Stokes vector elements 27.3.3 Oscillating Retarder and Analyzer Polarimeter The oscillating retarder and analyzer polarimeter is the generalization of oscillating element designs [3] This polarimeter is shown in Fig 27-3c A retarder is surrounded by two optical rotators as in the oscillating retarder and fixed analyzer polarimeter, but now the rotators produce rotations r1 and r2 The retarder is oriented at some angle R and the linear polarizer is oriented at some angle P With no optical rotators, the detected signal is given by kI ¼ S0 þ ðS1 cos 2R þ S2 sin 2R Þ cosð2P À 2R Þ þ S3 sinð2P À 2R Þ ð27-32Þ Consider that the rotator R2 in Fig 27-3c is replaced by two equivalent rotators in series that have rotations Àr1 and r1 þ r2 The sum of these is r2 and we have not changed the resultant net rotation The retarder is now surrounded by rotators with rotations r1 and Àr1 and this is equivalent to the retarder in the new azimuth R þ r1 The rotator with rotation r1 þ r2 rotates the polarizer azimuth to P þ r1 þ r2 If we Copyright © 2003 by Marcel Dekker, Inc All Rights Reserved replace the angles in (27-32) with the azimuthal angles resulting from the addition of the rotators, we have kI ¼ S0 þ S1 cosð2R þ 2r1 Þ cosð2P À 2R þ 2r2 Þ þ S2 sinð2R þ 2r1 Þ cosð2P À 2R þ 2r2 Þ þ S3 sinð2P À 2R þ 2r2 Þ ð27-33Þ If we reference the angular coordinates to the azimuth of the polarizer, we can set P ¼ and rewrite (27-33) as kI ¼ S0 þ S1 ½cos 4R cosð2r1 À 2r2 Þ À sin 4R sinð2r1 À 2r2 Þ þ cosð2r1 þ 2r2 Þ þ S2 ½sin 4R cosð2r1 À 2r2 Þ þ cos 4R sinð2r1 À 2r2 Þ þ sinð2r1 þ 2r2 Þ ð27-34Þ À S3 ðsin 2R cos 2r2 À cos 2R sin 2r2 Þ Now consider that the rotators are oscillated at the same frequency and are either in phase or out of phase by , then the rotations produced are given by r1 ¼ r1 sin !t ð27-35aÞ r2 ¼ r2 sin !t ð27-35bÞ and We can now substitute the expressions of (27-35) into (27-34) and again use the Bessel function expansions of (27-15) and (27-16) to obtain the equation: kI ¼ MSn ð27-36Þ where I! I ¼ @ I2! A I3! S1 Sn ¼ @ S2 A S3 ð27-37Þ and À sin 4R J1 ð2r1 À 2r2 Þ B M ¼ @ cos 4R J2 ð2r1 À 2r2 Þ þ J2 ð2r1 þ 2r2 Þ À sin 4R J3 ð2r1 À 2r2 Þ cos 4R J2 ð2r1 À 2r2 Þ þ J2 ð2r1 þ 2r2 Þ sin 4R J2 ð2r1 À 2r2 Þ cos 4R J2 ð2r1 À 2r2 Þ þ J2 ð2r1 þ 2r2 Þ cos 2R J1 ð2r2 Þ À2 sin 2R J2 ð2r2 Þ C A cos 2R J3 ð2r2 Þ (27-38) The zero frequency term is given by Idc ¼ S0 þ S1 ½cos 4R J0 ð2r1 À 2r2 Þ þ J0 ð2r1 þ 2r2 Þ þ S2 ½sin 4R J0 ð2r1 À 2r2 Þ À S3 ½sin 2R J0 ð2r2 Þ ð27-39Þ Sn is found by multiplying the signal vector I by the inverse of M and then S0 is obtained from (27-39) Copyright © 2003 by Marcel Dekker, Inc All Rights Reserved Figure 27-4 Phase modulation polarimeters (After Ref 1.) 27.4 PHASE MODULATION POLARIMETRY Phase modulation polarimeters are shown in Fig 27-4 These polarimeters use devices that vary in retardance in response to an electrical signal A common type of phase modulator is the photoelastic modulator (see Chapter 24) 27.4.1 Phase Modulator and Fixed Analyzer Polarimeter This polarimeter, shown in Fig 27-4a, uses a single modulator with a fixed linear analyzer The axes of the modulator and analyzer are inclined at 45 to each other The detected signal is given by I¼ S0 þ ðS cos 2A þ S2 sin 2A Þ cos Á þ S3 sin Á 2 ð27-40Þ where A is the azimuthal angle of the analyzer and Á is the retardance of the modulator The modulator retardance is Á ¼ sin !t ð27-41Þ where ! is the frequency of modulation and is the magnitude of the modulation The detected intensity is given by I¼ I0 I1 I þ sin !t þ cos 2!t 2 Copyright © 2003 by Marcel Dekker, Inc All Rights Reserved ð27-42Þ If ¼ 137:8 [J0() ¼ and ¼ 0 ] the Stokes vector is given by S ¼ I0 I2 2J2 ðÞ I1 S3 ¼ 2J1 ðÞ S1 ¼ ð27-43Þ If the polarimeter elements are both rotated by 45 (see Fig 27-4b), we will measure the Stokes vector: S ¼ I0 I2 2J2 ðÞ I1 S3 ¼ 2J1 ðÞ S2 ¼ 27.4.2 ð27-44Þ Dual-Phase Modulator and Fixed Analyzer Polarimeter The dual-phase modulator and fixed analyzer polarimeter is shown in Fig 27-4c The first modulator (closest to the analyzer) is aligned 45 to the analyzer and has timevarying retardation: Á1 ¼ 1 sin !1 t ð27-45Þ The second modulator, aligned to the analyzer axis, has time-varying retardation: Á2 ¼ 2 sin !2 t ð27-46Þ All four Stokes parameters can be measured with this system The signal is I¼ S0 S1 cos Á2 S2 sin Á2 sin Á1 S3 sin Á2 cos Á1 þ þ À 2 2 ð27-47Þ and if we demand that 1 ¼ 2 ¼ 137.8 then I¼ I0 I1 cos 2!2 t I2 cosð!2 Æ !1 Þ I3 sinð!2 Æ 2!1 Þt Æ þ þ 2 2 ð27-48Þ and higher frequency terms The Stokes vector is then given by S ¼ I0 I1 2J2 ð2 Þ I2 S2 ¼ 2J1 ð1 ÞJ1 ð2 Þ ÀI3 S3 ¼ 2J2 ð1 ÞJ1 ð2 Þ S1 ¼ Copyright © 2003 by Marcel Dekker, Inc All Rights Reserved ð27-49Þ 27.5 TECHNIQUES IN SIMULTANEOUS MEASUREMENT OF STOKES VECTOR ELEMENTS In the polarimetry techniques we have described in this chapter up to this point, all depend on a time sequential activity That is, in rotating element polarimetry, polarizers and retarders are rotated and measurements are made at various angular positions of the elements; in oscillating element polarimetry, rotators are oscillated, and measurements are made at various points in the oscillation; in phase modulation polarimetry, measurements are made at various phase values in the modulation We would like to be able to make all required measurements at the same time to ensure that time is not a factor in the result In order to this we can divide the wavefront spatially and make simultaneous measurements of different quantities at different points in space, or we can separate polarizations by dividing the amplitude of the wavefront Polarimeters of these types generally have no moving parts 27.5.1 Division of Wavefront Polarimetry Wavefront division relies on analyzing different parts of the wavefront with separate polarization elements This has been done using a pair of boresighted cameras that were flown on the space shuttle [4,5] A linear polarizer was placed in front of each camera where the polarizers were orthogonal to each other Chun et al [6] have performed wavefront division polarimetry using a single infrared camera Metal wire-grid polarizers were formed on a substrate using microlithography in the pattern shown in Fig 27-5 This wire-grid array was placed in front of the detector array so that light from different parts of the object space pass through different polarization elements and on to different detectors Each detector element of the infrared focal plane array has its own polarizer These polarizers are linear polarizers at four different orientations, as shown in Fig 27-5, and the pattern is repeated up to the size of the array There are no circular components measured and thus this is an incomplete polarimeter Figure 27-5 Pattern of micropolarizers in a wavefront division polarimeter Copyright © 2003 by Marcel Dekker, Inc All Rights Reserved The advantage of this polarimetric measurement method is the simultaneous measurement of the Stokes vector elements available from the polarization element array The reduction in resolution of the detector by the number of different polarization elements and the spatial displacement of information within the polarization element pattern are disadvantages 27.5.2 Division of Amplitude Polarimetry In amplitude division polarimetry, the energy in the entire wavefront of the incident beam is split and analyzed before passing to detectors The detectors should be spatially registered so that any detector element is looking at the same point in space as all other detector elements This method can employ as few as two detectors with analysis of two orthogonally polarized components of light, or it can measure the complete Stokes vector using four detectors There are a number of variations of division of amplitude polarimetry and we will describe several Four-Channel Polarimeter Using Polarizing Beam Splitters A diagram of a four-channel polarimeter [7] is shown in Fig 27-6 This polarimeter uses three polarizing beam splitters and two retarders Readings are made at four detectors The input Stokes vector is determined from the four detector measurements and from use of a transfer Mueller matrix found during the calibration procedure The polarizing beam splitters have transmissions of 80% and 20% for the Figure 27-6 A four-channel polarimeter PBS is a beam splitter, QWR is a quarter-wave retarder, and HWR is a half-wave retarder Copyright © 2003 by Marcel Dekker, Inc All Rights Reserved parallel and perpendicular components A quarter-wave retarder before detectors and is oriented at 45 and the half-wave retarder before detectors and is oriented at 22.5 The advantage of this system is the simultaneous measurement of all four Stokes components for each point in object space Care must be taken to ensure spatial registration of the detectors and equalization of detector response Twochannel polarimeters [8] are substantially easier to construct Azzam’s Four-Detector Photopolarimeter Another type of amplitude division complete Stokes polarimeter is the four-detector photopolarimeter of Azzam [9,10] A diagram of this polarimeter is shown in Fig 27-7, and a photograph of a commercial version of this instrument is given in Fig 27-8 In this four-detector polarimeter, a light beam strikes four detectors in sequence, as shown in Fig 27-7 Part of the light striking the first three is specularly reflected to the remaining detectors in the sequence, while the last detector absorbs substantially all the remaining light The signal measured by each detector is proportional to the fraction of the light that it absorbs, and that fraction is a linear combination of the Stokes parameters The light intensity measured by the detector is then linearly related to the input Stokes vector The four detected signals are related to the input Stokes vector by 0 1 i0 S0 B i1 C B S1 C B C C I¼B @ i2 A ¼ A@ S2 A ¼ AS i3 S3 ð27-50Þ Figure 27-7 Optical diagram of the four-detector photopolarimeter (From Ref 9.) Copyright © 2003 by Marcel Dekker, Inc All Rights Reserved Figure 27-8 Photograph of a commercial four-detector photopolarimeter (Courtesy of Gaertner Scientific Corp., Skokie, IL.) where A is a Mueller matrix of the instrument The input Stokes vector is then obtained from S ¼ AÀ1 I ð27-51Þ In order to determine the Stokes vector uniquely, the instrument matrix must be nonsingular We now derive this instrument matrix The Stokes vectors of the light reflected from the surfaces of the photodetectors D0, D1, and D2 are Sð0Þ ¼ M0 S Sð1Þ ¼ M1 R1 M0 S S ð2Þ ð27-52Þ ¼ M2 R2 M1 R1 M0 S where S is the input Stokes vector, À cos l 6 À cos l Ml ¼ r l 6 0 0 sin l À sin is the Mueller matrix of the lth detector, and 0 sin 2 l cos 2 l Rl ¼ À sin 2 l cos 2 l 0 0 7 sin l sin Ál cos Ál l sin Ál sin l ð27-53Þ cos Ál ð27-54Þ is the rotation matrix describing the rotation of the plane of incidence between successive reflections; rl is the reflectance of the lth detector for incident unpolarized Copyright © 2003 by Marcel Dekker, Inc All Rights Reserved or circularly polarized light and tan l eiÁl ¼ rpl =rsl is the ratio of the complex reflection coefficients of the surface for polarizations parallel and perpendicular to the local plane of incidence Let us form a vector L composed of the first elements of the Stokes vectors S, Sð0Þ , Sð1Þ , and Sð2Þ , i.e., the elements that are proportional to the intensities This can be accomplished by multiplying each of these Stokes vectors by the row vector:  à G¼ 0 ð27-55Þ so that we have S0 ð0Þ 6S0 7 L ¼ ð1Þ 6S0 S ð02Þ ð27-56Þ This vector L is linearly related to the input Stokes vector by L ¼ FS ð27-57Þ where F is given in terms of its rows F0 G GM0 F1 7 F¼6 F ¼ GM R M 25 1 by 7 7 ð27-58Þ GM2 R2 M1 R1 M0 F3 The last three rows of this matrix are the first three rows of the matrices M0, M1R1M0, and M2R2M1R1M0 If we insert the appropriate forms of Eqs (27-53) and (27-54) into (27-58) we obtain the matrix: 0 7 f10 f11 ð27-59Þ F¼6 f20 f21 f22 f23 f30 f31 f32 f33 where f10 ¼ r0 f11 ¼ Àr0 cos f20 ¼ r0 r1 ð1 þ cos cos cos 2 1 Þ þ cos cos 2 1 Þ f21 ¼ Àr0 r1 ðcos f22 ¼ Àr0 r1 ðsin cos Á0 cos sin 2 1 Þ f23 ¼ Àr0 r1 ðsin sin Á0 cos sin 2 1 Þ Copyright © 2003 by Marcel Dekker, Inc All Rights Reserved f30 ¼ r0 r1 r2 ð1 þ cos cos cos 2 1 þ cos þ cos cos 2 cos 2 1 cos 2 2 À cos sin cos Á1 cos f31 ¼ Àr0 r1 r2 ðcos þ cos 2 þ cos 2 cos Á0 cos cos 2 1 þ cos cos Á0 sin cos Á1 cos À sin sin Á0 sin sin Á1 cos sin Á0 cos cos cos Á1 cos sin 2 1 þ sin þ sin f33 ¼ Àr0 r1 r2 ðsin cos 2 cos 2 2 cos sin 2 1 sin 2 2 Þ cos 2 1 cos 2 2 À sin f32 ¼ Àr0 r1 r2 ðsin 2 cos 2 2 sin 2 1 sin 2 2 Þ cos Á0 cos 2 sin 2 1 cos 2 2 cos 2 1 sin 2 2 sin 2 2 Þ sin 2 1 þ sin sin Á0 cos þ sin cos Á0 sin sin Á1 cos 2 sin 2 2 þ sin sin Á0 sin cos Á1 cos 2 cos 2 1 sin 2 2 Þ sin 2 1 cos 2 2 ð27-60Þ The signal from each of the four detectors is proportional to the light absorbed by it The light absorbed is the difference between the incident flux and the reflected flux; thus, the signal from the first detector is the difference between the first two elements of the vector L (27-56) multiplied by a proportionality constant that is dependent on the detector responsivity; the signal from the second detector is proportional to the difference between the second and third elements of the vector L; the signal from the third detector is proportional to the difference between the third and fourth elements of the vector L; and since the last detector is assumed to absorb the remaining light, the signal from this detector is proportional to the remaining flux The signal from each detector is then expressed as i0 ¼ k0 ðS0 À S ð0Þ Þ ð1Þ i1 ¼ k1 ðS ð0Þ ÀS0 Þ ð27-61Þ ð2Þ i2 ¼ k2 ðS ð1Þ ÀS0 Þ i3 ¼ k3 S ð2Þ In matrix form, (27-61) can be expressed as I ¼ KDL ð27-62Þ where K is the detector responsivity matrix, L is the vector in (27-56), and D is constructed so that it takes the difference between elements of the vector L, i.e., k0 0 60 K¼6 40 k1 0 k2 07 7 05 k3 Copyright © 2003 by Marcel Dekker, Inc All Rights Reserved ð27-63Þ and 60 D¼6 40 À1 À1 0 0 7 À1 ð27-64Þ Substituting (27-57) into (27-62) we obtain I ¼ KDFS ð27-65Þ and we observe in comparing (27-65) and (27-50) that the instrument matrix A is A ¼ KDF ð27-66Þ We know K, D, and F from (27-59), (27-63), and (27-64), and we have found the instrument matrix In order to compute AÀ1, A must be nonsingular and its determinant must be nonzero We find the determinant from det A ¼ ðdet KÞðdet DÞðdet FÞ ð27-67Þ which becomes, when we make substitutions, det A ¼ Àðk0 k1 k2 k3 Þðr30 r21 r2 Þðsin 2 1 sin 2 2 Þ Â ðsin2 cos sin cos cos 2 Þ sin Á1 ð27-68Þ If any factor in this equation is zero, the determinant becomes zero We can now make some observations about the conditions under which this can happen The first term in parentheses is the product of the responsivities of the detectors It is undesirable and unlikely that any of these are zero, but this might happen if a detector is not working The next term in parentheses is a product of the reflectances of the first three detectors If any of these are zero, light will not get to the fourth detector, and the system will not work Again, this is a condition that is undesirable and unlikely The third term in parentheses is a geometrical condition: these factors are nonzero as long as the planes of incidence of two successive reflections are not coincident or orthogonal The detectors can be arranged so that this does not happen The fourth term in parentheses vanishes when ¼ 0, ¼ 0, ð27-69Þ ¼ ¼ ¼ The first two conditions in (27-69) are equivalent to having the first two detectors as perfect linear polarizers The last three conditions would require that the first three detectors reflect p and s polarizations equally or function as retarders Since the detectors are designed to be absorbing elements and typical reflections Copyright © 2003 by Marcel Dekker, Inc All Rights Reserved from absorbing surfaces will not fulfill these conditions, they are unlikely The last factor, sin Á1 , is the sine of the differential reflection phase shift at the second detector A phase shift of or is usually associated with Fresnel reflections from nonabsorbing dielectrics Again, we have absorbing detectors and this condition is not fulfilled Further details of polarimeter optimization, light path choice, spectral performance, and calibration are given in Azzam [10] A fiber-optic implementation of the four-detector polarimeter is described in Bouzid et al [11], and a corner cube configuration version of the polarimeter is discussed in Liu and Azzam [12] Division of Amplitude Polarimeters Using Gratings A number of polarimeters based on division of amplitude using gratings have been proposed [13–16] Diffraction gratings split a single incident light beam into multiple beams and introduce significant polarization [17] Azzam has demonstrated a polarimeter based on conical diffraction [10] This instrument is shown in Fig 27-9 An incident beam strikes a metal diffraction grating at an oblique incidence angle The grating is positioned such that the lines of the gratings are at some arbitrary angle to the plane of incidence, and this is the condition for conical diffraction With this geometry, the diffraction efficiency is dependent on all elements of the Stokes vector, and thus this instrument is a complete polarimeter A linear detector is placed at the location of each diffracted order to be detected When four detectors are used, the same relationships apply to the grating polarimeter as in the four-detector polarimeter; i.e., the signal is linearly related to the incident Stokes vector by I ¼ AS ð27-70Þ and we again invert the instrument matrix A to obtain the Stokes vector as in (27-51), i.e., S ¼ AÀ1 I ð27-51Þ The derivation of the instrument matrix for this polarimeter follows the calibration procedures established for the four-detector polarimeter A polarimeter using a grating in the normal spectroscopic orientation, i.e., in a planar diffraction condition, has been designed and constructed [14] This polarimeter is illustrated in Fig 27-10 Polarizers are placed in front of the detectors in this Figure 27-9 Photopolarimeter using conical diffraction (From Ref 13.) Copyright © 2003 by Marcel Dekker, Inc All Rights Reserved Figure 27-10 Photopolarimeter using planar diffraction (From Ref 14.) design in order to make the instrument sensitive to all Stokes parameters Four detectors are used in four diffracted orders At least two of the diffracted beams must have polarizers in order for this polarimeter to be complete An instrument matrix is determined through a calibration process A 16-beam grating-based polarimeter has also been designed and demonstrated [16] A proposed polarimeter using transmission gratings and four linear detector arrays is designed to measure spectral and polarization information simultaneously [15] Division of Amplitude Polarimeter Using a Parallel Slab A wavefront may be divided in amplitude using the multiple reflections obtained in a planar dielectric slab [18] Figure 27-11 shows a polarimeter based on a parallel plane slab of material of index n1 ðÞ A coating of metal of complex index n2 À ik2 is placed on the bottom surface of the slab A light beam incident on the slab at angle undergoes multiple reflections in the slab, which results in a set of parallel and equally spaced outgoing beams Linear polarizers are arranged in front of detectors in these beams with as many inclination angles of the transmission axes as there are detectors The signal from the mth detector is then a linear combination of the elements of the Stokes vector, i.e., im ¼ X amj Sj , m ¼ 0, 1, 2, ð27-71Þ j¼0 where the mth vector am ¼ ½am0 am1 am2 am3 is the first row of the Mueller matrix of the mth light path If we limit the detectors to four, the output signal vector is related to the input Stokes vector by the equation we have seen before for division of amplitude polarimeters: I ¼ AS Copyright © 2003 by Marcel Dekker, Inc All Rights Reserved ð27-70Þ Figure 27-11 Parallel slab polarimeter (From Ref 18.) The matrix A is the instrument matrix determined through calibration, and, as in previous division of amplitude examples, an unknown Stokes vector is found from the equation S ¼ AÀ1 I ð27-51Þ El-Saba et al [18] show that for a slab of fused silica coated with a layer of silver and operated at 633 nm, the preferred angle of incidence for maximum energy in the beams and maximum value of the determinant of the instrument matrix is around 80 27.6 OPTIMIZATION OF POLARIMETERS To this point we have not discussed specific polarization element angular settings We have made reference to the use of quarter-wave retarders, primarily because we can construct a complete Stokes polarimeter using the readily available quarter-wave retarder and linear polarizer We now ask the question, are there measurement angles and values of retardance that will result in a more efficient and/or better polarimeter? This question was first addressed with regard to the angular positions of the quarter-wave retarder and linear polarizer in a rotating retarder and fixed analyzer polarimeter [19] and a rotating retarder, rotating analyzer polarimeter [20] It was found in the first instance that angles of (À45 , 0 , 30 , 60 ) or (À90 , À45 , 30 , 60 ) resulted in the least sensitivity with regard to flux noise and rotation positional errors In the second instance, if we let the rotation angle of the polarizer be and the rotation angle of the retarder be ’ and define an and such that ¼ 2’ ¼ 2ð À ’Þ Copyright © 2003 by Marcel Dekker, Inc All Rights Reserved ð27-72Þ Figure 27-12 Locus of points on the Poincare´ sphere for retardance values 45 , 90 , 132 , and 180 for a rotating retarder polarimeter (From Ref 21.) then an optimal set of and is ½0 , 90 , 0 , À sinÀ1 ! ! ! 1 , 120 , À sinÀ1 , 240 , À sinÀ1 3 ð27-73Þ If we allow both the measurement angles and retardance to take part in the optimization process for a rotating retarder polarimeter, we find that the optimal value of retardance is 0.3661 (%132 ) and the optimal retarder positions are either (Æ15.12 , Æ51.69 ) or (Æ74.88 , Æ38.31 ) where these angle pairs are complements of each other [21,22] These values were found through numerical optimization described in the cited references where the optimal values offer the best signal-to-noise performance and least sensitivity to element misalignment Figure 27-12 shows the locus of points on the Poincare´ sphere for values of retardance of 45 , 90 , 132 , and 180 The figure indicates that better ‘‘global coverage’’ of the sphere is made possible by using the retardance of 132 Figure 27-13 reinforces this intuition where the intersection of the curve for the retardance value 132 with the four retarder positions (Æ15.12 , Æ51.69 ) forms the corners of a regular tetrahedron inscribed in the Poincare´ sphere, points as far apart as possible as one can make them on the surface of the sphere Figure 27-14 shows plots of a figure of merit for the rotating retarder fixed polarizer polarimeter versus number of measurements for the system with a quarterwave retarder and an optimal retarder with both equally spaced angles and the optimal measurement angles The results of this plot indicate that the optimal retarder with repeated optimal angles offers the best performance At this time, 132 retarders are not standard items from optical supply houses, and the improvement in performance gained by using these optimal elements may not be worth the cost and risk of ordering custom elements Copyright © 2003 by Marcel Dekker, Inc All Rights Reserved Curves for retardance values of 90 and 132 intersecting the retarder angles (Æ15.12 , Æ51.69) to form the regular tetrahedron (From Ref 21.) Figure 27-13 Figure 27-14 Plots of a figure of merit versus number of measurements for several measurement methods (From Ref 21.) 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