9 The Mathematics of the Mueller Matrix

9 The Mathematics of the Mueller Matrix 9.1 INTRODUCTION Mathematical development to better understand and describe the information contained in the Mueller matrix is given in this chapter The experimental Mueller matrix can be a complicated function of polarization, depolarization, and noise How we separate the specific information we are interested in, e.g., depolarization or retardance, from the measured Mueller matrix? When does an experimental matrix represent a physically realizable polarization element and when does it not? If it does not represent a physically realizable polarization element, how we extract that information which will give us information about the equivalent physically realizable element? These are the questions we attempt to answer in this chapter Two algebraic systems have been developed for the solution of polarization problems in optics, the Jones formalism and the Mueller formalism The Jones formalism is a natural consequence of the mathematical phase and amplitude description of light The Mueller formalism comes from experimental considerations of the intensity measurements of polarized light R.C Jones developed the Jones formalism in a series of papers published in the 1940s [1–3] and reprinted in a collection of historically significant papers on polarization [4] The Jones formalism uses Jones vectors, two element vectors that describe the polarization state of light, and Jones matrices,  matrices that describe optical elements The vectors are complex and describe the amplitude and phase of the light, i.e., ! * * Ex ðtÞ JðtÞ ¼ * ð9-1Þ Ey ðtÞ * * is a time-dependent Jones vector where Ex , Ey are the x and y components of the electric field of light traveling along the z axis The matrices are also complex and describe the action in both amplitude and phase of optical elements on a light beam Copyright © 2003 by Marcel Dekker, Inc All Rights Reserved The Jones matrix is of the form:   j11 j12 J¼ j21 j22 ð9-2Þ where the elements jij ¼ aij þ ibij are complex The two elements of the Jones vector are orthogonal and typically represent the horizontal and vertical polarization states The four elements of the Jones matrix make up the transfer function from the input to the output Jones vector Since these elements are complex, the Jones matrix contains eight constants and has eight degrees of freedom corresponding to the eight kinds of polarization behavior A physically realizable polarization element results from any Jones matrix, i.e., there are no physical restrictions on the values of the Jones matrix elements The Jones formalism is discussed in more detail in Chapter 11 The Mueller formalism, already discussed in previous chapters but reviewed here, owes its name to Hans Mueller, who built on the work of Stokes [5], Soleillet [6], and Perrin [7] to formalize polarization calculations based on intensity This work, as Jones’, was also done during the 1940s but originally appeared in a now declassified report [8] and in a course of lectures at M.I.T in 1945–1946 As we have learned, the Mueller formalism uses the Stokes vector to represent the polarization state The Mueller matrix is a  matrix of real numbers There is redundancy built into the Mueller matrix, since only seven of its elements are independent if there is no depolarization in the optical system In the most general case, the Mueller matrix can have 16 independent elements; however, not every  Mueller matrix is a physically realizable polarizing element For each Jones matrix, there is a corresponding Mueller matrix On conversion to a Mueller matrix, the Jones matrix phase information is discarded A matrix with eight pieces of information is transformed to a matrix with seven pieces of information Transformation equations for converting Jones matrices to Mueller matrices are given in Appendix C The Mueller matrices can also be generated from equations The Jones matrix is related to the Mueller matrix by M ¼ AðJ  Jà ÞAÀ1 ð9-3Þ where  denotes the Kronecker product and A is 0 0 À1 7 A¼6 40 1 i Ài ð9-4Þ The elements of the Mueller matrix can also be obtained from the relation: mij ¼ TrðJi Jy j Þ ð9-5Þ where Jy is the Hermitian conjugate of J and the  are the set of four  matrices that comprise the identity matrix and the Pauli matrices (see Section 9.3) The Jones matrix cannot represent a depolarizer or scatterer The Mueller matrix can represent depolarizers and scatterers (see, e.g., [9]) Since the Mueller matrix contains information on depolarization, the conversion of Mueller matrices Copyright © 2003 by Marcel Dekker, Inc All Rights Reserved to Jones matrices must discard depolarization information There is no phase information in a Mueller matrix, and the conversion conserves seven degrees of freedom The Mueller formalism has two advantages for experimental work over the Jones formalism The intensity is represented explicitly in the Mueller formalism, and scattering can be included in the calculations The Jones formalism is easier to use and more elegant for theoretical studies 9.2 CONSTRAINTS ON THE MUELLER MATRIX The issue of constraints on the Mueller matrix has been investigated by a number of researchers, e.g., [10–15] The fundamental requirement that Mueller matrices must meet in order to be physically realizable is that they map physical incident Stokes vectors into physical resultant Stokes vectors This recalls our requirement on Stokes vectors that the degree of polarization must always be less than or equal to one, i.e., P¼ ðS21 þ S22 þ S23 Þ1=2 S0 ð9-6Þ A well-known constraint on the Mueller matrix is the inequality [16]: TrðMMT Þ ¼ X m2ij 4m200 ð9-7Þ i, j¼0 The equals sign applies for nondepolarizing systems and the inequality otherwise Many more constraints on Mueller matrix elements have been recorded However, we shall not attempt to list or even to discuss them further here The reason for this is that they may be largely irrelevant when one is making measurements with real optical systems The measured Mueller matrices are a mixture of pure (nondepolarizing) states, depolarization, and certainly noise (optical and electronic) Is the magnitude of a particular Mueller matrix element due to diattenuation or retardance or is it really noise, or is it a mixture? If it is a mixture, what are the proportions? It is the responsibility of the experimenter to reduce noise sources as much as possible, determine the physical realizability of his Mueller matrices, and if they are not physically realizable, find the closest physically realizable Mueller matrices A method of finding the closest physically realizable Mueller matrix and a method of decomposing nondepolarizing and depolarizing Mueller matrices are discussed in the remaining sections of this chapter These are very important and provide useful results; however, only so much can be done to reduce noise intrusion A study was done [17] to follow error propagation in the process of finding the best estimates, and it was found that the noise was reduced by one-third in nondepolarizing systems and reduced by one-tenth in depolarizing systems in going from the nonphysical matrix to the closest physically realizable matrix The reduction is significant and worth doing, but no method can completely eliminate measurement noise We will give examples in Section 9.4 Copyright © 2003 by Marcel Dekker, Inc All Rights Reserved 9.3 EIGENVECTOR AND EIGENVALUE ANALYSIS Cloude [18,19] has formulated a method to obtain polarization characteristics and answer the question of physical realizability Any  matrix J (in particular, a Jones matrix) can be expressed as X J¼ ki i ð9-8Þ i where the  i are the Pauli matrices:     0 1 ¼ 2 ¼ À1  3 ¼ i Ài  with the addition of the identity matrix:   0 ¼ ð9-9Þ ð9-10Þ and the ki are complex coefficients given by ki ¼ TrðJ Á i Þ ð9-11Þ The components of this vector can also be written: k0 ¼ ð j11 þ j22 Þ k1 ¼ ð j11 À j22 Þ k2 ¼ ð j12 þ j21 Þ i k3 ¼ ð j12 À j21 Þ ð9-12Þ ð9-13Þ ð9-14Þ ð9-15Þ Cloude introduces a  Hermitian ‘‘target coherency matrix’’ obtained from the tensor product of the k’s, i.e., Tc ¼ k  kÃT ð9-16Þ The elements of the Mueller matrix are given in terms of the Jones matrix as mij ¼ TrðJi Jy j Þ ð9-17Þ and Cloude shows that this can also be written as mij ¼ TrðTc 4iþj Þ ð9-18Þ where the  are the 16 Dirac matrices, a set of matrices which form a basis for  matrices The Dirac matrices are shown in Table 9-1 The matrix Tc can be expressed as Tc ¼ mij i  j Copyright © 2003 by Marcel Dekker, Inc All Rights Reserved ð9-19Þ Table 9-1 0 B0 B @0 4 0 B1 B @0 Dirac Matrices 1 0 0 1 0 0C C 0A B1 B @0 2 1 0 0 0C C 0 iA À1 0 B0 0 B @1 0 i 5 0 0 0 i 0 C C Ài A B0 B @0 0 B0 B @1 0 Ài 1 0 iC C 0A 0 12 0 0 B 0 Ài B @0 i 0 B B @ Ài 0 0 Ài 0 0 0 C C A À1 B0 B @0 i B0 B @i 0 Ài C C A 0 B0 B @ Ài 0 0 i 0C C 0A 0 0 Ài C C A 0 B À1 0 C C B @0 A 0 À1 B B @ Ài 0 Ài 0 0 0 1C C 0A 0 B Ài B @ 0 i 0 i 0 1C C 0 0A 0 0 Ài Bi B @0 0 14 1 0C C 0A 11 1 i 0 7 10 13 0C C 0A 0 6 9 8 3 0 1 0C C 1A 15 0 1 0C C 1A 0 1 0 B À1 0 C B C @ 0 À1 A 0 where i  j ð9-20Þ are the Dirac matrices Tc can be written in the parametric form: A0 þ A C À iD H þ iG I À iJ B C B C þ iD B0 þ B E þ iF K À iL C B C B H À iG E À iF B À B M þ iN C @ A I þ iJ K þ iL M À iN ð9-21Þ A0 À A where A through N are real numbers If these real numbers are arranged into a  matrix where the ijth element is the expansion coefficient of the Dirac matrix 4iþj then the matrix: A0 þ B0 C þ N H þ L FþI B C GþK C B CÀN AþB EþJ B C ð9-22Þ B HÀL EÀJ AÀB DþM C @ A IÀF KÀG M À D A0 À B0 is just the Mueller matrix when Tc is expressed in the Pauli base The coherency matrix is then obtained from the experimental Mueller matrix by solving for the real Copyright © 2003 by Marcel Dekker, Inc All Rights Reserved elements A through N When this is done the elements of the coherency matrix are found to be m þ m22 þ m33 þ m44 ð9-23Þ t11 ¼ 11 m þ m21 À iðm34 À m43 Þ ð9-24Þ t12 ¼ 12 m þ m31 þ iðm24 À m42 Þ t13 ¼ 13 ð9-25Þ m þ m41 À iðm23 À m32 Þ t14 ¼ 14 ð9-26Þ m þ m21 þ iðm34 À m43 Þ t21 ¼ 12 ð9-27Þ m þ m22 À m33 À m44 ð9-28Þ t22 ¼ 11 m þ m32 þ iðm14 À m41 Þ t23 ¼ 23 ð9-29Þ m þ m42 À iðm13 À m31 Þ t24 ¼ 24 ð9-30Þ m þ m31 À iðm24 À m42 Þ t31 ¼ 13 ð9-31Þ m þ m32 À iðm14 À m41 Þ t32 ¼ 23 ð9-32Þ m À m22 þ m33 À m44 ð9-33Þ t33 ¼ 11 m þ m43 þ iðm12 À m21 Þ t34 ¼ 34 ð9-34Þ m þ m41 þ iðm23 À m32 Þ ð9-35Þ t41 ¼ 14 m þ m42 þ iðm13 À m31 Þ t42 ¼ 24 ð9-36Þ m þ m43 À iðm12 À m21 Þ t43 ¼ 34 ð9-37Þ m À m22 À m33 þ m44 ð9-38Þ t44 ¼ 11 The eigensystem for the coherency matrix Tc can be found and provides the decomposition of Tc into four components i.e., Tc ¼ 1 Tc1 þ 2 Tc2 þ 3 Tc3 þ 4 Tc4 ð9-39Þ where the  are the eigenvalues of Tc and Tci ¼ ki  kÃT i ð9-40Þ are the eigenvectors The eigenvalues of Tc are real since Tc is Hermitian The eigenvectors are in general complex Each eigenvalue/eigenvector corresponds to a Jones matrix (and every Jones matrix corresponds to a physically realizable Copyright © 2003 by Marcel Dekker, Inc All Rights Reserved Table 9-2 Matrix 0 0 1 1 2 2 3 3 Meaning of the C-vector Components Coefficient 0 0 1 1 2 2 3 3 Meaning Amplitude Phase Amplitude Phase Amplitude Phase Amplitude Phase Absorption Phase Linear diattenuation along axes Linear retardance along axes Linear diattenuation 45 Linear retardance 45 Circular diattenuation Circular retardance polarization element) The Jones matrix corresponding to the dominant eigenvalue is the matrix that describes the dominant polarizing action of the element Extraction of this Jones matrix may be of interest for some applications: however, here the properties of the sample are most important These properties may be found with the realization that the eigenvector corresponding to the dominant eigenvalue is the quantity known as the C-vector [20] The eigenvector components are the coefficients of the Pauli matrices in the decomposition of the Jones matrix: this is identical to the definition of the C-vector The components of the C-vector give the information shown in Table 9.2 Cloude has shown that for an experimental Mueller matrix to be physically realizable, the eigenvalues of the corresponding coherency matrix must be nonnegative The ratio of negative to positive eigenvalues is a quantitative measure of the realizability of the measured matrix Further, a matrix that is not physically realizable can be ‘‘filtered,’’ or made realizable by subtracting the component corresponding to the negative eigenvalue from the coherency matrix Calculation of a new Mueller matrix then yields one that may include errors and scattering, but one that can be constructed from real polarization components 9.4 EXAMPLE OF EIGENVECTOR ANALYSIS In this section, a simple example of the calculations described in Section 9.3 is given We will also give examples of the calculations to derive the closest physically realizable Mueller matrix from experimentally measured matrices The Mueller matrix for a partial linear polarizer with principal intensity transmission coefficients k1 ¼ 0.64 and k2 ¼ 0.36 along the principal axes and having an orientation  ¼ is given by 0:50 0:14 0:0 0:0 0:14 0:50 0:0 0:0 7 ð9-41Þ 0:0 0:0 0:48 0:0 0:0 0:0 0:0 0:48 The equivalent Jones matrix is ! 0:8 0:0 0:0 0:6 Copyright © 2003 by Marcel Dekker, Inc All Rights Reserved ð9-42Þ The Cloude coherency matrix is 0:98 0:14 0:0 0:0 0:14 0:02 0:0 0:0 7 0:0 0:0 0:0 0:0 0:0 0:0 0:0 0:0 ð9-43Þ There is only one nonzero eigenvalue of this matrix and it has a value of one The eigenvector corresponding to this eigenvalue is 0:9899 0:1414 7 ð9-44Þ 0:000 0:000 where the second element of this vector is the measure of the linear diattenuation Note that the terms corresponding to diattenuation at 45 and circular diattenuation are zero Now suppose that the polarizer with the same principal transmission coefficients is rotated 40 The Mueller matrix is 0:500000 0:024311 0:137873 0:000000 0:024360 0:480725 0:003578 0:000000 7 ð9-45Þ 0:137900 0:003270 0:499521 0:000000 0:000000 0:000000 0:000000 0:480000 The dominant eigenvalue is approximately one, and the corresponding eigenvector is 0:9899 0:0246 7 ð9-46Þ 0:1393 0:0002i With the rotation, the original linear polarization has coupled with polarization at 45 and circular polarization, and, in fact, the polarization at 45 is now the largest The linear diattenuation can now be calculated from (1) the original Mueller matrix, (2) the Jones matrix as found by Gerrard and Burch, and (3) the Cloude coherency matrix eigenvector The linear diattenuation is given by k1 À k2 0:64 À 0:36 ¼ 0:28 ¼ k1 þ k2 0:64 þ 0:36 ð9-47Þ Calculation of the linear diattenuation from the Jones matrix derived directly from the Mueller matrix gives r21 À r22 0:82 À 0:62 ¼ ¼ 0:28 r21 þ r22 0:82 þ 0:62 Copyright © 2003 by Marcel Dekker, Inc All Rights Reserved ð9-48Þ In the method of Cloude, the components of the eigenvector corresponding to the dominant eigenvalue (i.e., the components of the C-vector) are given by k0 ¼ ðr1 þ r2 Þ ð9-49Þ k1 ¼ ðr1 À r2 Þ ð9-50Þ and so that, solving for r1 and r2, and calculating diattenuation, a value of 0.28 is again obtained Let us now examine experimental Mueller matrices that have noise and are not likely to be physically realizable, and convert these into the closest possible physically realizable Mueller matrix We will follow a slightly different prescription (D.M Hayes, Pers Commun., 1996) from that given above [19] First, create the covariance matrix n for the experimental Mueller matrix m from the following equations: n11 ¼ m11 þ m22 þ m12 þ m21 ð9-51Þ n12 ¼ n21 ð9-52Þ n13 ¼ n31 ð9-53Þ n14 ¼ n41 ð9-54Þ n21 ¼ m13 þ m23 À iðm14 þ m24 Þ ð9-55Þ n22 ¼ m11 À m22 À m12 þ m21 ð9-56Þ n23 ¼ n32 ð9-57Þ n24 ¼ n42 ð9-58Þ n31 ¼ m31 þ m32 þ iðm41 þ m42 Þ ð9-59Þ n32 ¼ m33 À m44 þ iðm34 þ m43 Þ ð9-60Þ n33 ¼ m11 À m22 þ m12 À m21 ð9-61Þ n34 ¼ n43 ð9-62Þ n41 ¼ m33 þ m44 À iðm34 À m43 Þ ð9-63Þ n42 ¼ m31 À m32 þ iðm41 À m42 Þ ð9-64Þ n43 ¼ m13 À m23 À iðm14 À m24 Þ ð9-65Þ n44 ¼ m11 þ m22 À m12 À m21 ð9-66Þ Since this results in a Hermitian matrix, the eigenvalues will be real and the eigenvectors orthogonal Now find the eigenvalues and eigenvectors of this matrix, and form a diagonal matrix from the eigenvalues, i.e., 1 0 2 0 7 ü6 ð9-67Þ 0 3 0 4 Copyright © 2003 by Marcel Dekker, Inc All Rights Reserved We now set any negative eigenvalues in à equal to zero because negative eigenvalues correspond to nonphysical components Construct a matrix V composed of the eigenvectors of n and perform the similarity transform: N ¼ VÃVÀ1 ð9-68Þ where N is the covariance matrix corresponding to the closest physical Mueller matrix to m Finally, construct the physical Mueller matrix using the linear transformation: M21 ¼ N11 þ N22 À N33 À N44 ð9-69Þ M12 ¼ M21 þ N33 À N22 ð9-70Þ M22 ¼ N11 À N22 À M12 ð9-71Þ M11 ¼ 2N11 À M22 À M12 À M21 ð9-72Þ M13 ¼ ReðN21 þ N43 Þ ð9-73Þ M23 ¼ Reð2N21 Þ À M13 ð9-74Þ M31 ¼ ReðN31 þ N42 Þ ð9-75Þ M32 ¼ Reð2N31 Þ À M31 ð9-76Þ M33 ¼ ReðN41 þ N32 Þ ð9-77Þ M44 ¼ Reð2N41 Þ À M33 ð9-78Þ M14 ¼ ÀImðN21 þ N43 Þ ð9-79Þ M24 ¼ Imð2N43 Þ þ M14 ð9-80Þ M41 ¼ ImðN31 þ N42 Þ ð9-81Þ M42 ¼ Imð2N31 Þ À M41 ð9-82Þ M43 ¼ ImðN41 þ N32 Þ ð9-83Þ M34 ¼ Imð2N32 Þ À M43 ð9-84Þ Let us now show numerical examples The first example is an experimental calibration matrix for a rotating retarder polarimeter The (normalized) matrix, which should ideally be the identity matrix, is 0:978 0:003 0:005 1:000 À0:007 0:006 7 ð9-85Þ 0:007 0:999 À0:007 0:005 À0:003 À0:002 0:994 The eigenvalues of the corresponding coherency matrix are, written in vector form,  à 1:986 À0:016 À0:007 À0:005 ð9-86Þ Copyright © 2003 by Marcel Dekker, Inc All Rights Reserved Three of these eigenvalues are negative so that the three corresponding eigenvalues must be removed (substracted) from the diagonal matrix formed by the set of four eigenvalues In this case, the filtered matrix is 0:993 0:002 0:005 0:993 0 7 ð9-87Þ 0:002 0:993 0:005 0 0:993 The eigenvalue ratio, the ratio of the negative eigenvalue to the dominant eigenvalue in decibels, is a measure of the closeness to realizability For this example the ratio of the largest negative eigenvalue to the dominant eigenvalue is approximately À21 dB The original matrix was quite close to being physically realizable In a second example we have the case of a quartz plate that has its optic axis misaligned from the optical axis, inducing a small birefringence The measured matrix was 1:000 0:019 0:021 À0:130 À0:024 À0:731 À0:726 0:005 7 ð9-88Þ 0:008 0:673 À0:688 À0:351 À0:009 0:259 À0:247 0:965 The eigenvalues of the corresponding coherency matrix are  à 2:045 À0:073 0:046 À0:017 ð9-89Þ and the eigenvalue ratio is approximately À14.5 dB In this case there are two negative eigenvalues that must be subtracted The filtered matrix becomes 0:737 À0:005 0:006 À0:067 À0:005 À0:987 À0:024 0:131 7 ð9-90Þ 0:006 À0:024 À0:989 À0:304 À0:067 0:131 À0:304 0:674 9.5 THE LU–CHIPMAN DECOMPOSITION Given an experimental Mueller matrix, we would like to be able to separate the diattenuation, retardance, and depolarization A number of researchers had addressed this issue e.g., [21, 22] for nondepolarizing matrices A general decomposition, a significant and extremely useful development, was only derived with the work of Lu and Chipman This polar decomposition, which we call the Lu–Chipman decomposition [23, 24], allows a Mueller matrix to be decomposed into the product of the three factors Let us first review the nondepolarizing factors of diattenuation and retardance in this context Diattenuation changes the intensity transmittances of the incident polarization states The diattenuation is defined as D Tmax À Tmin Tmax þ Tmin Copyright © 2003 by Marcel Dekker, Inc All Rights Reserved ð9-91Þ and takes values from to Eigenpolarizations are polarization states that are transmitted unchanged by an optical element except for a change in phase and intensity A diattenuator has two eigenpolarizations For example, a horizontal polarizer has the eigenpolarizations of horizontal polarization and vertical polarization If the eigenpolarizations are orthogonal, the element is a homogeneous polarization element, and is inhomogeneous otherwise The axis of diattenuation is along the direction of the eigenpolarization with the larger transmittance Let this diattenuation axis be along the eigenpolarization described by the Stokes vector: ð1 d1 d3 ÞT ¼ ð1, D^ T ÞT d2 ð9-92Þ where qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi d12 þ d22 þ d32 ¼ jD^ j ¼ ð9-93Þ Let us define a diattenuation vector: 1 DH Dd1 * B C B C D  DD^ ¼ @ Dd2 A ¼ @ D45 A ð9-94Þ Dd3 DC where DH is the horizontal diattenuation, D45 is the 45 linear diattenuation, and DC is the circular diattenuation The linear diattenuation is defined as qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð9-95Þ DL  D2H þ D245 and the total diattenuation is qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi * D ¼ D2H þ D245 þ D2C ¼ D2L þ D2C ¼ jDj ð9-96Þ The diattenuation vector provides a complete description of the diattenuation properties of a diattenuator The intensity transmittance can be written as the ratio of energies in the exiting to incident Stokes vector: T¼ s00 m00 s0 þ m01 s1 þ m02 s2 þ m03 s3 ¼ s0 s0 ð9-97Þ where there is an intervening element with Mueller matrix M The first row of the Mueller matrix completely determines the intensity transmittance Equation (9-97) can be rewritten as * * T ¼ m00 þ mÁs s0 ð9-98Þ *  ðm01 , m02 , m03 Þ and * s  ðs1 , s2 , s3 Þ The where the vectors are defined as m maximum and minimum values of the dot product can be taken to be * * * s Á m ¼ s0 jmj Copyright © 2003 by Marcel Dekker, Inc All Rights Reserved ð9-99Þ and * * * s Á m ¼ Às0 jmj so that the maximum and minimum transmittances Tmax and Tmin are qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Tmax ¼ m00 þ m201 þ m202 þ m203 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Tmin ¼ m00 À m201 þ m202 þ m203 The normalized Stokes vectors associated with Tmax and Tmin are 1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi C B m01 B m201 þ m202 þ m203 C C B B qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi C ^ C B Smax ¼ B m02 m201 þ m202 þ m203 C C B B qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi C A @ m03 m201 þ m202 þ m203 ð9-100Þ ð9-101Þ ð9-102Þ ð9-103Þ and 1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi C B C B Àm01 m201 þ m202 þ m203 C B C B C B ¼ B Àm02 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi C þ m2 þ m2 C B m 01 02 03 C B C B @ Àm qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi A 03 2 m01 þ m02 þ m03 S^min The diattenuation of the Mueller matrix is qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi T À Tmin D ¼ max ¼ m201 þ m202 þ m203 Tmax þ Tmin m00 ð9-104Þ ð9-105Þ and the axis of diattenuation is along the maximum transmittance and thus the direction of S^max The axis of diattenuation is along the state S^max and the diattenuation vector of the Mueller matrix is then given by 1 m D H 01 * @ ð9-106Þ m02 A D ¼ @ D45 A ¼ m00 DC m03 so that the first row of a Mueller matrix gives its diattenuation vector The expressions for S^max and S^min can be written as   S^max ¼ ^ ð9-107Þ D and S^min ¼  ÀD^  Copyright © 2003 by Marcel Dekker, Inc All Rights Reserved ð9-108Þ Operational definitions for the components of the diattenuation vector are given by TH À TV m01 ¼ ¼ DH TH þ TV m00 ð9-109Þ T45 À T135 m02 ¼ ¼ D45 T45 þ T135 m00 ð9-110Þ TR À TL m03 ¼ ¼ DC TR þ TL m00 ð9-111Þ where TH is the transmittance for horizontally polarized light, TV is the transmittance for vertically polarized light, T45 is the transmittance for linear 45 polarized light, T135 is the transmittance for linear 135 polarized light, TR is the transmittance for right circularly polarized light, and TL is the transmittance for left circularly polarized light Now consider that we have incident unpolarized light, i.e., only one element of the incident Stokes vector is nonzero The exiting state is determined completely by the first column of the Mueller matrix The polarization resulting from changing completely unpolarized light to polarized light is called polarizance The polarizance is given by qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi P¼ m210 þ m220 þ m230 ð9-112Þ m00 and can take values from to A normalized polarizance vector is given by 1 m10 PH * @ m20 A ð9-113Þ P  @ P45 A ¼ m00 PR m30 The components of the polarizance vector are equal to the horizontal degree of polarization, 45 linear degree of polarization, and circular degree of polarization resulting from incident unpolarized light Retarders are phase-changing devices and have constant intensity transmittance for any incident polarization state Eigenpolarizations are defined for retarders according to the phase changes they produce The component of light with leading phase has its eigenpolarization along the fast axis (see Chaps 24 and 26) of the retarder Let us define a vector along this direction: ð 1, a1 , a2 , a3 ÞT ¼ ð 1, R^ T ÞT where qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a21 þ a22 þ a23 ¼ jR^ j ¼ The retardance vector and the fast axis are described by 1 RH Ra1 * R  RR^ ¼ @ Ra2 A  @ R45 A Ra3 RC Copyright © 2003 by Marcel Dekker, Inc All Rights Reserved ð9-114Þ ð9-115Þ ð9-116Þ * where the components of R give the horizontal, 45 linear, and circular retardance components The net linear retardance is qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð9-117Þ RL ¼ R2H þ R245 and the total retardance is qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi * R ¼ R2H þ R245 þ R2C ¼ R2L þ R2C ¼ jRj ð9-118Þ Now that we have laid the groundwork for nondepolarizing Mueller matrices, let us consider the decomposition of these matrices Nondepolarizing Mueller matrices can be written as the product of a retarder and diattenuator, i.e., M ¼ MR M D ð9-119Þ where MR is the Mueller matrix of a pure retarder and MD is the Mueller matrix of a pure diattenuator A normalized Mueller matrix M can be written: 1 m01 m02 m03 ! *T B m10 m11 m12 m13 C D C M¼B ð9-120Þ @ m20 m21 m22 m23 A ¼ * P m m30 m31 m32 m33 where the submatrix m m11 m12 m ¼ @ m21 m22 m31 m32 * is m13 m23 A m33 ð9-121Þ * and D and P are the diattenuation and polarizance vectors as given in (9.106) and (9.113) The diattenuator MD is calculated from the first row of M, and MÀ1 D can then be multiplied by M to obtain the retarder matrix MR ¼ MMÀ1 The diattenuator D matrix is given by ! *T D MD ¼ * ð9-122Þ D mD where * *T  mD ¼ aI3 þ b D Á D ð9-123Þ and where I3 is the  identity matrix, and a and b are scalars derived from the norm of the diattenuation vector, i.e., * D ¼ jDj pffiffiffiffiffiffiffiffiffiffiffiffiffiffi a ¼ À D2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffi À À D2 b¼ D2 Copyright © 2003 by Marcel Dekker, Inc All Rights Reserved ð9-124Þ ð9-125Þ ð9-126Þ Writing the diattenuator matrix out, we have m02 m03 m01 B C B m01 a þ bm201 bm01 m02 bm01 m03 C C MD ¼ B Bm C @ 02 bm02 m01 a þ bm02 bm02 m03 A m03 bm03 m01 bm03 m02 a þ bm203 ð9-127Þ where a¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi À ðm201 þ m202 þ m203 Þ ð9-128Þ and 1À b¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi À ðm201 þ m202 þ m203 Þ ð9-129Þ ðm201 þ m202 þ m203 Þ MÀ1 D is then given by MÀ1 D ¼ a *T ! 1* ÀD ÀD I3 þ a ða þ 1Þ * *T * *T ðD Á D Þ ! The retarder matrix is ! *T MR ¼ * mR ð9-130Þ ð9-131Þ where mR ¼ i * *T 1h m À bðP Á D Þ a The retarder matrix can be written a 0 m À bðm 16 11 10 m01 Þ MR ¼ a m21 À bðm20 m01 Þ m31 À bðm30 m01 Þ ð9-132Þ explicitly as 0 m12 À bðm10 m02 Þ m13 À bðm10 m03 Þ 7 m22 À bðm20 m02 Þ m23 À bðm20 m03 Þ m32 À bðm30 m02 Þ m33 À bðm30 m03 Þ ð9-133Þ The total retardance R and the retardance vector can be found from the equations:   * À1 TrðmR Þ À R  ð9-134Þ R ¼ jRj ¼ cos   * TrðmR Þ À R ¼ jRj ¼ 2 À cosÀ1  R 2 ð9-135Þ 1 ðMR Þ23 À ðMR Þ32 RH * R ð9-136Þ R ¼ @ R45 A ¼ @ ðMR Þ31 À ðMR Þ13 A sinðRÞ RC ðMR Þ12 À ðMR Þ21 Copyright © 2003 by Marcel Dekker, Inc All Rights Reserved The total retardance is then given explicitly as   À1 ½m þ m22 þ m33 À bðm10 m01 þ m20 m02 þ m30 m03 Þ À aŠ R ¼ cos 2a 11 ð9-137Þ and the retardance vector is given by m23 À m32 À bðm20 m03 À m30 m02 Þ * R ¼ m31 À m13 À bðm30 m01 À m10 m03 Þ m12 À m21 À bðm10 m02 À m20 m01 Þ cosÀ1 ð1=2a½m11 þ m22 þ m33 À bðm10 m01 þ m20 m02 þ m30 m03 Þ À aŠÞ ffi ð9-138Þ Â pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4a2 À ½m11 þ m22 þ m33 À bðm10 m01 þ m20 m02 þ m30 m03 Þ À aŠ2 A pure depolarizer can be represented by the matrix: 0 60 a 07 40 b 05 0 c ð9-139Þ where |a|, |b|, |c| The principal depolarization factors are À |a|, À |b|, and À |c|, and these are measures of the depolarization of this depolarizer along its principal axes The parameter Á given by Á1À jaj þ jbj þ jcj , Á ð9-140Þ is the average of the depolarization factors, and this parameter is called the depolarization power of the depolarizer An expression for a depolarizer can be written as " # *T ð9-141Þ , mTÁ ¼ mÁ * mÁ where mÁ is a symmetric  submatrix The eigenvalues of mÁ are the principal depolarization factors, and the eigenvectors are the three orthogonal principal axes This last expression is not the complete description of a depolarizer, because it contains only six degrees of freedom when we require nine The most general expression for a depolarizer can be written as " # *T MÁ ¼ * ð9-142Þ , mTÁ ¼ mÁ PÁ mÁ * where PÁ is the polarizance vector, and with this expression we have the required nine degrees of freedom and no diattenuation or retardance Thus, we see that a depolarizer with a nonzero polarizance may actually have polarizing properties according to our definition here Depolarizing Mueller matrices can be written as the product of the three factors of diattenuation, retardance, and depolarization, i.e., M ¼ MÁ MR MD Copyright © 2003 by Marcel Dekker, Inc All Rights Reserved ð9-143Þ where MÁ is the depolarization, and this equation is the generalized polar decomposition for depolarizing Mueller matrices It is useful for the decomposition of experimental Mueller matrices to allow the depolarizing component to follow the nondepolarizing component As in the nondepolarizing case, we first find the matrix for the diattenuator We then define a matrix M0 such that M0 ¼ MMÀ1 D ¼ MÁ MR This expression can be written out as the product of the  matrices: " #" # " # *T *T *T 1 ¼ * M Á MR ¼ * * PÁ mÁ mR PÁ mÁ mR " # *T ¼ * ¼ M0 PÁ m0 ð9-144Þ ð9-145Þ Let 1, 2, and 3 be the eigenvalues of m0 ðm0 ÞT ¼ mÁ mR ðmÁ mR ÞT ¼ m2Á ð9-146Þ We can obtain the relations: * * PÁ ¼ * P À mD À D2 ð9-147Þ and m0 ¼ mÁ mR ð9-148Þ from (9-144) and (9-145) pffiffiffiffiffi pffiffiffiffiffi pffiffiffiffiffi The eigenvalues of mÁ are then 1 , 2 , and 3 It should be pointed out that there is an ambiguity in the signs of the eigenvalues [17] The retarder submatrix mR is a rotation matrix and has a positive determinant so that the sign of the determinant of m0 indicates the sign of the determinant of mÁ The assumption that the eigenvalues all have the same sign is reasonable, especially since depolarization in measured systems is usually small and the eigenvalues are close to one This assumption simplifies the expression for mÁ An expression for mÁ is given by, from the Cayley–Hamilton theorem (a matrix is a root of its characteristic polynomial), mÁ ¼ ƽm0 ðm0 ÞT þ 2 IŠÀ1 ½1 m0 ðm0 ÞT þ 3 I ð9-149Þ where 1 ¼ pffiffiffiffiffi pffiffiffiffiffi pffiffiffiffiffi 1 þ 2 þ 3 pffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffi 2 ¼ 1 2 þ 2 3 þ 3 1 ð9-150Þ ð9-151Þ and 3 ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 2 3 Copyright © 2003 by Marcel Dekker, Inc All Rights Reserved ð9-152Þ The sign in front of the expression on the right-hand side in Eq (9-149) follows the sign of the determinant of m0 We can now find mR from the application of mÀ1 Á to m0 , i.e., 0 T À1 0 T 0 mr ¼ mÀ1 Á m ¼ ƽ1 m ðm Þ þ 3 IŠ ½m ðm Þ m þ 2 m Š ð9-153Þ The eigenvalues 1, 2, and 3 can be found in terms of the original Mueller matrix elements by solving a cubic equation, but the expressions that result are long and complicated It is more feasible to find the ’s We have qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 ¼ detðmÁ Þ ¼ detðm2Á Þ ¼ det½m0 ðm0 ÞT Š ¼ detðm0 Þ ð9-154Þ Recall that M0 ¼ M(MÁ)À1 has the form: " # *T M ¼ *1 0 PÁ m ð9-155Þ so that 3 ¼ detðm0 Þ ¼ detðM0 Þ ¼ detðMÞ detðMÀ1 Á Þ ¼ detðMÞ detðMÞ ¼ detðMÁ Þ a4 ð9-156Þ Let us define a  and  such that 1 ¼ Tr½m2Á Š ¼ 1 þ 2 þ 3 ð9-157Þ 2 ¼ Tr½23 ðm2Á ÞÀ1 Š ¼ 1 2 þ 1 3 þ 2 3 ð9-158Þ and Then 1 satisfies the recursive equation: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 ¼ 1 þ 2 þ 3 1 ð9-159Þ This can be approximated by rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffi 1 % 1 þ 2 þ 23 1 ð9-160Þ Since 2 ¼ ½21 À 1 Š we can use the approximation for 1 to obtain the approximation for 2: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffi 2 % 2 þ 23 1 ð9-161Þ ð9-162Þ Expressions for  and  are given in terms of the original Mueller matrix elements and the elements of m2Á : " # !2 3 3 X X X X 1 ð9-163Þ 1 ¼ m2i, j À m2i, þ 4 mi, À mi, j m0, j a i, j¼1 a i¼1 i¼1 j¼1 Copyright © 2003 by Marcel Dekker, Inc All Rights Reserved and   2 ¼ mÁ2, mÁ3, þ mÁ1, mÁ3, þ mÁ1, mÁ2, À m2Á2, þ m2Á1, þ m2Á1, where the elements mÁ1, m2Á ¼ mÁ2, mÁ3, ð9-164Þ of m2Á are mÁ1, mÁ2, mÁ3, mÁ1, mÁ2, mÁ3, ð9-165Þ where we note that mÁi, j ¼ mÁj, i and " ! # " #" # 3 X X X 1 mÁi, j ¼ mik mjk À mi0 mj0 þ mi0 À mik m0k mj0 À mjk m0k a a k¼1 k¼1 k¼1 ð9-166Þ We can then write: mÁ2,2 mÁ3,3 À m2Á2,3 23 ðm2Á ÞÀ1 ¼ 4mÁ1,3 mÁ2,3 À mÁ1,2 mÁ3,3 mÁ1,3 mÁ2,3 À mÁ1,2 mÁ3,3 mÁ1,2 mÁ2,3 À mÁ2,2 mÁ2,3 mÁ1,1 mÁ3,3 À m2Á1,3 mÁ1,2 mÁ2,3 À mÁ2,2 mÁ2,3 mÁ1,2 mÁ1,3 À mÁ1,1 mÁ2,3 mÁ1,2 mÁ1,3 À mÁ1,1 mÁ2,37 mÁ1,1 mÁ2,2 À m2Á1,2 ð9-167Þ and the retarder rotation matrix is given by mR ¼ mÀ1 Á m ¼ à 1 I À m2Á þ 23 ðm2Á ÞÀ1 m0 1 If we can find approximations for the depolarizer eigenvalues then we can write an expression for mÀ1 Á as mÀ1 Á ¼ à 1 I À m2Á þ 23 ðm2Á ÞÀ1 1 ð9-168Þ pffiffiffiffiffi pffiffiffiffiffi pffiffiffiffiffi 1 , 2 , and 3 , ð9-169Þ where Àpffiffiffiffiffi pffiffiffiffiffi pffiffiffiffiffiÁ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð1 þ 2 þ 3 Þ 1 þ 2 þ 3 À 1 2 3 Àpffiffiffiffiffi pffiffiffiffiffiÁÀpffiffiffiffiffi pffiffiffiffiffiÁÀpffiffiffiffiffi pffiffiffiffiffiÁ þ1 ¼ 2 þ 3 1 þ 3 1 þ 2 ð9-170Þ ¼ Àpffiffiffiffiffi pffiffiffiffiffiÁÀpffiffiffiffiffi pffiffiffiffiffiÁÀpffiffiffiffiffi pffiffiffiffiffiÁ 1 þ 2 1 þ 3 2 þ 3 ð9-171Þ Àpffiffiffiffiffi pffiffiffiffiffi pffiffiffiffiffiÁ 1 þ 2 þ 3 ¼ ÀpffiffiffiffiffiffiffiffiffiffiffiffiffiffiÁÀpffiffiffiffiffi pffiffiffiffiffiÁÀpffiffiffiffiffi pffiffiffiffiffiÁÀpffiffiffiffiffi pffiffiffiffiffiÁ 1 2 3 1 þ 2 2 þ 3 1 þ 3 ð9-172Þ and Copyright © 2003 by Marcel Dekker, Inc All Rights Reserved 9.6 SUMMARY We have answered the questions posed at the beginning of this chapter With the material presented here, we now have the tools to determine whether or not a Mueller matrix is physically realizable and we have a method to bring it to the closest physically realizable matrix We can then separate the matrix into its constituent components of diattenuation, retardance, and depolarization We must remember, however, that noise, once introduced into the system, is impossible to remove entirely The experimentalist must take prudent precautions to minimize the influence of errors peculiar to the system at hand REFERENCES 10 11 12 13 14 15 16 17 18 19 20 Jones, R C., ‘‘A new calculus for the treatment of optical systems: I Description and discussion of the calculus,’’ J Opt Soc Am., 31, 488–493 (1941) Jones, R C., ‘‘A new calculus for the treatment of optical systems: IV,’’ J Opt Soc Am., 32, 486–493 (1942) Jones, R C., ‘‘A new calculus for the treatment of optical systems: V A more general formulation, and description of another calculus,’’ J Opt Soc Am., 37, 107–110 (1947) Swindell, W., Polarized Light in Optics, Dowden, Hutchinson, and Ross, Stroudsberg, PA, 1975 Stokes, G G., ‘‘On the composition and resolution of streams of polarized light from different sources,’’ Trans Cambridge Phil Soc., 9, 399 (1852) Soleillet, P., ‘‘Sur les parameters caracterisant la polarization partielle de la lumiere dans les phenomenes de fluorescence,’’ Ann Phys., 12, 23 (1929) Perrin, F., ‘‘Polarization of light scattered by isotropic opalescent media,’’ J Chem Phys., 10, 415 (1942) Mueller, H., ‘‘Memorandum on the polarization optics of the photoelastic shutter,’’ Report No of the OSRD project OEMsr-576, Nov 15, 1943 van de Hulst, H C., Light Scattering by Small Particles, Dover, New York, 1981 Anderson, D G M and Barakat, R., ‘‘Necessary and sufficient conditions for a Mueller matrix to be derivable from a Jones matrix,’’ J Opt Soc Am A, 11, 2305–2319 (1994) Brosseau, C., Givens, C R and Kostinski, A B., ‘‘Generalized trace condition on the Mueller–Jones polarization matrix,’’ J Opt Soc Am A, 10, 2248–2251 (1993) Kostinski, A B., Givens, C R and Kwiatkowski, J M., ‘‘Constraints on Mueller matrices of polarization optics,’’ Appl Opt., 32(9), 1646–1651 (1993) Givens, C R and Kostinski, A B., ‘‘A simple necessary and sufficient condition on physically realizable Mueller matrices,’’ J Mod Opt., 40(3), 471–481 (1993) Hovenier, J W., van de Hulst, H C and van der Mee, C V M., ‘‘Conditions for the elements of the scattering matrix,’’ Astron Astrophys, 157, 301–310 (1986) Barakat, R., ‘‘Bilinear constraints between elements of the  Mueller–Jones transfer matrix of polarization theory,’’ Opt Comm., 38(3), 159–161 (1981) Fry, E S and Kattawar, G W., ‘‘Relationships between elements of the Stokes matrix,’’ Appl Opt., 20, 2811–2814 (1981) Hayes, D M., ‘‘Error propagation in decomposition of Mueller matrices,’’ Proc SPIE, 3121, 112–123 (1997) Cloude, S R., ‘‘Group theory and polarisation algebra,’’ Optik, 75(1), 26–36 (1986) Cloude, S R., ‘‘Conditions for the physical realisability of matrix operators in polarimetry,’’ Proc SPIE, 1166, 177–185 (1989) Chipman, R A., ‘‘Polarization aberrations,’’ PhD thesis, University of Arizona (1987) Copyright © 2003 by Marcel Dekker, Inc All Rights Reserved 20a Gerrard, A., and Burch, J M., Introduction to Matrix Methods in Optics, Wiley, London, 1975 22 Gil, J J and Bernabeu, E., ‘‘Obtainment of the polarizing and retardation parameters of a non-depolarizing optical system from the polar decomposition of its Mueller matrix,’’ Optik, 76, 26–36 (1986) 23 Xing, Z -F., ‘‘On the deterministic and non-deterministic Mueller matrix,’’ J Mod Opt., 39, 461–484 (1992) 24 Lu, S -Y., ‘‘An interpretation of polarization matrices,’’ PhD dissertion, Dept of Physics, University of Alabama at Huntsville (1995) 25 Lu, S -Y and Chipman, R A., ‘‘Interpretation of Mueller matrices based on polar decomposition,’’ J Opt Soc Am A, 13, 1106–1113 (1996) Copyright © 2003 by Marcel Dekker, Inc All Rights Reserved [...]...Three of these eigenvalues are negative so that the three corresponding eigenvalues must be removed (substracted) from the diagonal matrix formed by the set of four eigenvalues In this case, the filtered matrix is 2 3 0 :99 3 0 0:002 0:005 6 0 0 :99 3 0 0 7 6 7 9- 87Þ 6 7 4 0:002 0 0 :99 3 0 5 0:005 0 0 0 :99 3 The eigenvalue ratio, the ratio of the negative eigenvalue to the dominant eigenvalue... S^min The diattenuation of the Mueller matrix is qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi T À Tmin 1 D ¼ max ¼ m201 þ m202 þ m203 Tmax þ Tmin m00 9- 104Þ 9- 105Þ and the axis of diattenuation is along the maximum transmittance and thus the direction of S^max The axis of diattenuation is along the state S^max and the diattenuation vector of the Mueller matrix is then given by 0 1 0 1 m D H 01 * 1 @ 9- 106Þ... complete description of the diattenuation properties of a diattenuator The intensity transmittance can be written as the ratio of energies in the exiting to incident Stokes vector: T¼ s00 m00 s0 þ m01 s1 þ m02 s2 þ m03 s3 ¼ s0 s0 9- 97Þ where there is an intervening element with Mueller matrix M The first row of the Mueller matrix completely determines the intensity transmittance Equation (9- 97) can be rewritten... Reserved 9- 152Þ The sign in front of the expression on the right-hand side in Eq (9- 1 49) follows the sign of the determinant of m0 We can now find mR from the application of mÀ1 Á to m0 , i.e., 0 0 0 T À1 0 0 T 0 0 mr ¼ mÀ1 Á m ¼ ƽ1 m ðm Þ þ 3 IŠ ½m ðm Þ m þ 2 m Š 9- 153Þ The eigenvalues 1, 2, and 3 can be found in terms of the original Mueller matrix elements by solving a cubic equation, but the. .. pointed out that there is an ambiguity in the signs of the eigenvalues [17] The retarder submatrix mR is a rotation matrix and has a positive determinant so that the sign of the determinant of m0 indicates the sign of the determinant of mÁ The assumption that the eigenvalues all have the same sign is reasonable, especially since depolarization in measured systems is usually small and the eigenvalues... matrices, let us consider the decomposition of these matrices Nondepolarizing Mueller matrices can be written as the product of a retarder and diattenuator, i.e., M ¼ MR M D 9- 1 19 where MR is the Mueller matrix of a pure retarder and MD is the Mueller matrix of a pure diattenuator A normalized Mueller matrix M can be written: 0 1 1 m01 m02 m03 ! *T B m10 m11 m12 m13 C 1 D C M¼B 9- 120Þ @ m20 m21 m22 m23... 0:005 7 6 7 9- 88Þ 6 7 4 0:008 0:673 À0:688 À0:351 5 À0:0 09 0:2 59 À0:247 0 :96 5 The eigenvalues of the corresponding coherency matrix are  à 2:045 À0:073 0:046 À0:017 9- 89 and the eigenvalue ratio is approximately À14.5 dB In this case there are two negative eigenvalues that must be subtracted The filtered matrix becomes 2 3 0:737 À0:005 0:006 À0:067 6 À0:005 À0 :98 7 À0:024 0:131 7 6 7 9- 90Þ 4 0:006... out as the product of the 2  2 matrices: " #" # " # *T *T *T 1 0 1 0 1 0 ¼ * M Á MR ¼ * * PÁ mÁ 0 mR PÁ mÁ mR " # *T 1 0 ¼ * ¼ M0 PÁ m0 9- 144Þ 9- 145Þ Let 1, 2, and 3 be the eigenvalues of m0 ðm0 ÞT ¼ mÁ mR ðmÁ mR ÞT ¼ m2Á 9- 146Þ We can obtain the relations: * * PÁ ¼ * P À mD 1 À D2 9- 147Þ and m0 ¼ mÁ mR 9- 148Þ from (9- 144) and (9- 145) pffiffiffiffiffi pffiffiffiffiffi pffiffiffiffiffi The eigenvalues of mÁ are then 1... for a Mueller matrix to be derivable from a Jones matrix, ’’ J Opt Soc Am A, 11, 2305–23 19 ( 199 4) Brosseau, C., Givens, C R and Kostinski, A B., ‘‘Generalized trace condition on the Mueller Jones polarization matrix, ’’ J Opt Soc Am A, 10, 2248–2251 ( 199 3) Kostinski, A B., Givens, C R and Kwiatkowski, J M., ‘‘Constraints on Mueller matrices of polarization optics,’’ Appl Opt., 32 (9) , 1646–1651 ( 199 3) Givens,... physically realizable Mueller matrices,’’ J Mod Opt., 40(3), 471–481 ( 199 3) Hovenier, J W., van de Hulst, H C and van der Mee, C V M., ‘‘Conditions for the elements of the scattering matrix, ’’ Astron Astrophys, 157, 301–310 ( 198 6) Barakat, R., ‘‘Bilinear constraints between elements of the 4  4 Mueller Jones transfer matrix of polarization theory,’’ Opt Comm., 38(3), 1 59 161 ( 198 1) Fry, E S and Kattawar,