Optimum angles for a polarimeterpart II

Optimum angles for a polarimeter: part II Amrit Ambirajan Dwight C. Look, Jr., MEMBER SPIE University of Missouri-Rolla Thermal Radiative Transfer Group Mechanical and Aerospace Engineering and Engineering Mechanics Department Rolla, Missouri 65401 1 Introduction In most polarimeters13 involving a quarter-wave plate and a linear polarizer, the polarizer is fixed and the quarter-wave plate is free to rotate.'3 It was shown3 that, for this system, a set of angles exists such that a quarter-wave plate rotated to these positions would lead to a polarimeter with minimal sensitivity to errors in the azimuthal alignment of the optical components and fluctuations in the incident light field. In this paper, we apply the procedure introduced in Ref. 3 to find sets of angles that will ensure an accurate estimation of an incident Stokes vector when a polarimeter consists of a ro- tatable quarter-wave plate and a rotatable linear polarizer. 2 Optimum Angles To determine the optimum angles for both the components of the polarimeter, consider the optical system illustrated in Fig. 1. The first element in this optical system is the quarter- wave plate and the second is the linear polarizer. The azi- muthal position of the former is given by and the azimuthal position of the latter by 0. Note that the incident beam has 1656/OPTICAL ENGINEERING / June 1995 / Vol. 34 No.6 Abstract. The four sets of two optimum rotation angles for a polarimeter consisting of a quarter-wave plate in conjunction with a linear polarizer, both of which are free to rotate, are determined. These angles are ob- tained by maximizing the determinant of the system measurement ma- trix. The determinant for this system is approximately twice the value of the determinant when only the quarter-wave plate is free to rotate. In addition, the condition numbers of the measurement matrix decrease with the additional consideration of a rotatable linear polarizer. 2. —I' = {I + Q, cos21 cos[2(O —e + U1 sin241 cos[2(0 — + V, sin[2(0 — (1) where 1e 5 the j'th total intensity measurement and 4 and 01 are the conesponding rotation angles of the quarter-wave plate and linear polarizer respectively. Also t1 and t2 are the isotropic transmittances of the quarter-wave plate and linear polarizer, respectively. Estimation of the components of the vector S, requires at least four total intensity measurements be taken at four corresponding sets of angles (,0). Thus Subject terms: polarization analysis and measurement; Mueller matrix; polarime- ter; optimum angles. Optical Engineering 34(6), 1656- 1658 (June 1995). a Stokes vector of S, ={I,Q,U, V}T and the exit beam has a Stokes vector of Se ={'eQe Ue Ve}T. Using the Mueller matrices for the quarter-wave plate and linear polarizer,4 the intensity detected by a polarization in- dependent detector is derived to be5 Paper PAM-I 1 received Nov. 8, 1994; revised manuscript received Jan. 3, 1995; accepted for publication Jan. 26, 1995. This paper is a revision of a paper presented at the SPIE conference on Polarization Analysis and Measurement II, July 1994. San Diego, CA. The paper presented there appears (unrefereed) in SPIE Proceedings Vol. 2265. 1995 Society of Photo-Optical Instrumentation Engineers. 0091-3286/95/$6.OO. [l1 I'! IiiI i I — 2 1 I j4 ILei [1 cosa1cosl31 sina1cos1 sinI31 cosa2cos2 sina2cosf32 sinI32 cosa3cosf33 sina3cosl33 sint33 cosii4cosl34 sina4cos4 sinI34 Ii Qi (2) Downloaded from SPIE Digital Library on 23 Feb 2011 to 222.201.64.27. Terms of Use: http://spiedl.org/terms OPTIMUM ANGLES FOR A POLARIMETER IeaMPSi , (4) where 'e and ' are the measured intensity vector and mea— surement matrix, respectively. Thus, given four suitable sets of data of (a1, ,I), Eq. (4) can be inverted to estimate the incident Stokes vector. It is the aim of this paper to determine four sets of angles (a,3) that minimize the sensitivity of this polarimetric system to noise in the measured flux vector and errors in the optical components constituting the optical system. For a matrix to be well posed, the condition number must be as small as possible and the determinant of the matrix as large as possible.6 A well—posed measurement matrix is de- fined to have a lower sensitivity to noise in the measured intensity vector and errors in the optical components com- posing the polarimeter.3 The smallest possible value that a condition number can have is one.6 Attempts to use the con- dition numbers as objective functions in a minimization pro- cedure as reported in Ref. 3 was not possible using any of the IMSL subroutines. In this paper, the condition numbers are only used for comparison purposes. The angles that are obtained are based solely on maximization of the determinant of M'. Consider a unit vector defined by (cosa, cos3, sina, cos3, sin 13k), where the angles are as shown in Fig. 2 and i is an integer that can take values from 1 to 4. This sphere is analagous to the Poincaré sphere.4 The tips of these unit vectors will lie on this unit sphere. It can be shown7 by vector algebra that the volume of the tetrahedron formed by the tips of four such unit vectors is proportional to the determinant of M'. Further it can be shown by geometric considerations that the circumscribed tetrahedron with the largest volume is equal sided (each side equal to 1. 1547). Thus any set of angles (a,13) that leads to a circumscribed uniform terahedron is a valid angle set for optimum operating conditions of the polarimeter. Four such unit, vectors producing optimum an- gles are shown in Fig. 3 and are numbered from one to four. A corresponding set of (a,13) can be found to be (0 deg, 90 deg), [0 deg, —arcsin(1/3)j, [120 deg, —arcsin(1/3)], and [240 deg, —arcsin(1/3)]. Other orientations of the enclosed tetrahedron would lead to other settings for optimum operation. For the angles obtained using the definitions in Ref. 3, the L1 condition number (K1) ofthemeasurement matrix is 5.864, the L condition number (Kj is 3.932, and the determinant is 3.079. It was found3 that for a polarimeter consisting of a fixed polarizer and a rotating quarter-wave plate, the smallest possible K1 ofthe measurementmatrix was 7.95 1, the smallest possible K was 5.872, and the largest absolute value possible for the determinant was 1 .487. Thus with the additional con- sideration of a rotatable polarizer, the magnitude of the de- terminant more than doubles and the condition numbers de- crease significantly. 3 Conclusion In this paper, a polarimetric system consisting of a rotatable quarter-wave plate and a rotatable linear polarizer was ana- lyzed to find rotation angles that lower the sensitivity of the OPTICAL ENGINEERING / June 1995 / Vol.34 No.6/1657 Si M1 M2 Linear Plate Polarizer Se Detector Fig. 1 Schematic of the polarimeter under consideration. where (3) In matrix notation, Eq. (2) can be written in vector form as Fig. 2 Angles used in representing the rows of the measurementmatrix. ,,'—_—-Tetrahedron Fig. 3 Rows of the measurement matrix portrayed as numbered vectors comprising a terahedron in the unit sphere. Downloaded from SPIE Digital Library on 23 Feb 2011 to 222.201.64.27. Terms of Use: http://spiedl.org/terms AMB!RAJAN and LOOK system to noise in the measured flux and errors in the optical components. A class of angles was found that are the unit vectors pointing to the corners of uniform tetrahedron lying inside the unit sphere. It was seen that for this system, the condition numbers of the measurement matrix dropped sig- nificantly, and the magnitude of the determinant more than doubled. With the introduction of rotatable linear polarizer as an additional degree of freedom, the polarimetric system becomes less sensitive to noise in the measured flux and errors in the optical components comprising the polarimeter. Acknowledgment The authors wish to acknowledge the partial support of the National Science Foundation (NSF) through Grant CTS- 9103971. The authors would also like to thank the reviewers of this paper for some very helpful suggestions. 1658 / OPTICAL ENGINEERING / June 1995 /VoI. 34 No. 6 References 1. J.L. Pezzaniti and R. A. Chipman, ' 'Imaging polarimeters for optical metrology,' ' in Polarimetry: Radar, Infrared, Visible, Ultraviolet, and X-Ray (1990), Proc. SPIE 1317, 280—294 (1990). 2. F. M. Morgan, R. A. Chipman, and D. G. Torr, ' 'An ultraviolet polarim- eter for characterization of an imaging spectrometer,' ' in Polarimetry: Radar, Infrared, Visible, Ultraviolet, and X-Ray (1990), Proc. SPIE, 1317, 384—394 (1990). 3. A. Ambirajan and D. C. Look, Jr., ' 'Optimum angles for a Mueller matrix polarimeter," inPolarization Analysis and Measurement II, Proc. SPIE 2265, 314-326 (1994). Also appears in this issue as "Optimum angles for a polarimeter—I." 4. D. S. Kliger, J. W. Lewis, and C. E. Randall, Polarized Light in Optics and Spectroscopy, Academic Press, Boston (1990). 5. E. Collett, Polarized Light: Fundamentals and Applications, Marcel Dekker, New York (1993). 6. E. Issacson and H. B. Keller, Analysis of Numerical Methods, John Wiley & Sons, New York (1966). 7. M. R. Spiegel, Theory and Problems of Vector Analysis, Schaum's Out- line Series, New York (1959). Biographies and photographs of the authors appear with the paper "Optimum angles for a polarimeter: part I" in this issue. Downloaded from SPIE Digital Library on 23 Feb 2011 to 222.201.64.27. Terms of Use: http://spiedl.org/terms