Interpretation of Mueller matrices based on polar decomposition.pdf
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1、1106J. Opt. Soc. Am. A/Vol. 13, No. 5/May 1996S-Y. Lu and R. A. Chipman Interpretation of Mueller matrices based on polar decomposition Shih-Yau Lu and Russell A. Chipman Department of Physics, University of Alabama in Huntsville, Huntsville, Alabama 35899 Received July 17, 1995; accepted October 30
2、, 1995 We present an algorithm that decomposes a Mueller matrix into a sequence of three matrix factors:a diattenuator, followed by a retarder, then followed by a depolarizer.Those factors are unique except for singular Mueller matrices.Based on this decomposition, the diattenuation and the retardan
3、ce of a Mueller matrix can be defined and computed.Thus this algorithm is useful for performing data reduction upon experimentally determined Mueller matrices. 1996 Optical Society of America 1.INTRODUCTION The objective of this paper is to decompose an arbi- trary Mueller matrix to determine its di
4、attenuation, retardance, and depolarization.For the diattenuation the three diattenuation components xyy, 45y135, and rightyleft provide a complete description.Similarly the xyy, 45y135, and rightyleft components of retardances provide a full specification of retardance. The decomposition of Mueller
5、 matrices has been ad- dressed by many authors.14Cloude1showed that any (physical) Mueller matrix can be expressed as a sum of four nondepolarizing Mueller matrices.This decompo- sition into a sum was also discussed by Anderson and Barakat.5Gil and Bernabeu2applied the polar decom- position to nonde
6、polarizing Mueller matrices.Xing3 also discussed the polar decomposition for nondepo- larizing Mueller matrices.In addition, Xing3pointed out that any Mueller matrix can be expressed as a product of three factors:one nondepolarizing Mueller matrix, followed by a diagonal Mueller matrix, then fol- lo
7、wed by another nondepolarizing Mueller matrix.This decomposition was further elaborated by Sridhar and Simon.4 The theories presented by Cloude and by Sridhar and Simon are useful in discussing the physical realizabil- ity of a Mueller matrix.But these methods have not been applied to determine the
8、diattenuation and the retardance of a Mueller matrix.The polar decomposi- tion has been proven to be a feasible approach for this purpose.2,6This paper extends the methods of the two previous works in Refs. 2 and 6 to the derivation of the diattenuation and the retardance for a Mueller ma- trix usin
9、g the polar decomposition.We show that any Mueller matrix can be decomposed into three factors:a diattenuator, followed by a retarder, then followed by a purely depolarizing factor.Those factors are uniquely defined except for singular Mueller matrices such as polarizers and analyzers.These algorith
10、ms also pro- vide a method to perform data reduction upon a Mueller matrix. 2.DIATTENUATION AND RETARDANCE:A REVIEW Nondepolarizing Mueller matrices describe nondepolariz- ing polarization elements that convert completely polari- zed light into completely polarized light.Those Mueller matrices have
11、equivalent Jones matrices.The relation- ship between nondepolarizing Mueller matrices and Jones matrices has been well presented in the literature; see, for example, Ref. 5.On the other hand, depolarizing Mueller matrices describe depolarizing elements that may convert completely polarized light int
12、o partially polarized light.Those Mueller matrices have no equivalent Jones matrices. We begin by reviewing the diattenuation and the retar- dance of nondepolarizing elements.For a more complete treatment refer to Refs. 6 and 7.The Jones matrix repre- sentation of a polarization element is used in t
13、his section, in addition to the Mueller matrix representation, because of the simplicity of the polarization element forms when expressed as Jones matrices. Physically, a polarization element alters the polariza- tion state of light by changing the amplitudes andyor the phases of the components of i
14、ts electric-field vector. Two types of nondepolarizing elements, called diattenu- ators and retarders, are basic.The diattenuator changes only the amplitudes of the components of the electric- field vector.A diattenuator is described by a Hermitian Jones matrix.The retarder changes only the phases o
15、f components of the electric-field vector.A retarder has a unitary Jones matrix.Polarizers and wave plates are examples of diattenuators and retarders, respectively. A diattenuator has an intensity transmission that de- pends on the incident polarization state.The diattenu- ation of a diattenuator i
16、s defined as7 D ; jTq2 Trj Tq1 Tr kjqj22 jjrj2j jjqj21 jjrj2 , 0 # D # 1.(1) Here jqand jrare the eigenvalues of the diattenuator Jones matrix, and Tq jjqj2and Tr jjrj2are trans- 0740-3232/96/051106-08$10.00 1996 Optical Society of America S-Y. Lu and R. A. ChipmanVol. 13, No. 5/May 1996/J. Opt. Soc
17、. Am. A1107 mittances for (orthogonal) eigenpolarizations.It turns out that Tqand Trare also the maximum and minimum transmittances.The diattenuation characterizes the de- pendence of transmission upon the incident polarization state.Since different diattenuators can have the same diattenuation, add
18、itional degrees of freedom are needed for a complete description. First, the eigenpolarizations of a diattenuator describe its principal axes.The axis (or direction) of diattenu- ation for a diattenuator is defined to be along the eigenpolarization with larger transmittance.Let this diattenuation ax
19、is be along the eigenpolarization de- scribed by the Stokes vector s1, d1, d2, d3dT s1, DTdT, with pd 121 d221 d32 j Dj 1.Define a diattenu- ation vector D ! as D ! ; D D 0 B B Dd1 Dd2 Dd3 1 C C A; 0 B B DH D45 DC 1 C C A. (2) The three components of D ! are called the horizontal, 45-linear, and cir
20、cular diattenuations of this diattenu- ation, respectively.The linear diattenuation is defined as DL; q DH21 D452.(3) Note that D q DH21 D4521 DC2 q DL21 DC2 jD !j. (4) A diattenuator described by a diattenuation vector D ! has a diattenuation jD !j and its axis along s1, DTdT.Hence the diattenuatio
21、n vector describes both the magnitude and the axis of diattenuation, providing a complete de- scription of the diattenuation properties of a diattenuator. The diattenuation vector also characterizes the diattenu- ator Jones matrix up to a complex factor.The diattenu- ator Jones matrix with a diatten
22、uation vector D ! is given by JD/ exp a 2 D ? s ! ! s0cosh a 2 ! 1 sD ? s !dsinh a 2 ! / s01 D ! ? s ! 1 1 p1 2 D2,(5) where a relates to the diattenuation through D tanh a. Here s ! ; ss1, s2, s3dT.Those s matrices are the Pauli spin matrices, and s0is the 2 3 2 identity matrix: s0 “ 10 01 # , s1 “
23、 10 021 # , s2 “ 01 10 # , s3 “ 02i i0 # . (6) A retarder causes different phase changes for its (or- thogonal) eigenpolarizations.A retarder has a constant intensity transmittance independent of the incident po- larization state.The retardance for a retarder is defined as7 R ; jdq2 drj,0 # R # p ,(
24、7) in which dqand drare the phase changes for the eigen- polarizations.The fast axis (or direction) of retardance is defined to be along the eigenpolarization that emerges first from the retarder, i.e., the eigenpolarization with the leading phase.Let this retardance axis be characteri- zed by the S
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