Normal form for Mueller Matrices in Polarization Optics

Abstract

The normal (canonical) form for Mueller matrices in polarization optics is derived: it is shown that a non-singular real 4 × 4 matrix M qualifies to be the bona fide Mueller matrix of some physical system if and only if it has the canonical form M = L′ ΛL, where L and L′ are elements of the proper orthochronous Lorentz group L ^↑ ₊, and where Λ = diag (λ₀, λ₁, λ₂, λ₃) with λ₀ ≥ ¦λ_j¦ > 0. It is further shown that λ₁ and λ₂ can be taken to be positive so that the signature of λ₃ is the same as that of det M. Several experimentally measured Mueller matrices are analysed in the light of the normal form. The case of singular Mueller matrices is briefly discussed as a limiting case.     

A Simple Necessary and Sufficient Condition on Physically Realizable Mueller Matrices

Abstract

Development of simple tools to test physical realizability of measured or computed Mueller matrices is the subject of this paper. In particular, the overpolarization problem, i.e., the problem of ensuring that the output degree of polarization does not exceed unity is solved by finding an easily implementable necessary and sufficient condition. With G being the Lorentz metric, it states that a given matrix M is not overpolarizing if and only if the spectrum of GM ^T GM is real and an eigenvector associated with the largest eigenvalue is a physical Stokes vector. This result is used to characterize some M classes of special interest, and is used to test several examples from recent literature.     

Conditions for the Physical Realizability of Polarization Matrices Characterizing Passive Systems

Abstract

We derive conditions for the physical realizability of polarization matrices characterizing passive systems or scattering media. By physically realizable, we mean that 0  g  1 where g ≡ (output intensity/input intensity). Using the singular-value decomposition of an arbitrary 2 × 2 complex-valued matrix, we prove that a Jones matrix T _J is physically realizable if 0  det T _J ⁺ T _J  1. Consequently singular Jones matrices (i.e. det T _J = 0) completely extinguish the output intensity irrespective of the input intensity because g ≡ 0. Corresponding results are obtained for Mueller-Jones matrices (the 4 × 4 real-valued matrices which are the four-dimensional representations of the two-dimensional 2 × 2 complex-valued Jones matrices). We also study the problem for general Mueller matrices; however because of their phenomenological character they do not admit of such criteria as do the Jones and Mueller-Jones matrices. This is because g now depends upon the matrix elements of the Mueller matrix and the input Stokes parameters; whereas for the Jones and Mueller-Jones matrices, g only depends upon the matrix elements. Finally we study the problem of relating the input and output mean randomness.     

Structure of a general pure Mueller matrix

Changes in the radiance and state of polarization of a beam of radiation can often be described by means of a pure Mueller matrix. Such a 4 × 4 matrix transforms Stokes parameters and can be expressed in terms of the elements of a 2 × 2 Jones matrix. Relations between the two types of matrix are discussed. Explicit expressions are given for changes of a pure Mueller matrix that are caused by certain elementary changes of its Jones matrix, such as when its transpose, complex conjugate, or Hermitian conjugate are taken. It is shown that every pure Mueller matrix has a simple and elegant structure, which is embodied by interrelations that involve either only squares of the elements or only products of different elements. All possible interrelations for the elements of a general pure Mueller matrix are derived from this simple structure.     

Characteristic properties of Mueller matrices

A complete and minimum set of necessary and sufficient conditions for a real 4 x 4 matrix to be a physical Mueller matrix is obtained. An additional condition is presented to complete the set of known conditions, namely, the four conditions obtained from the nonnegativity of the eigenvalues of the Hermitian matrix H associated with a Mueller matrix M and the transmittance condition. Using the properties of H, a demonstration is also presented of Tr(M(T)M) = 4m(2)00 as being a necessary and sufficient condition for a physical Mueller matrix to be a pure Mueller matrix.     

Reduced form of a Mueller matrix

Abstract

Through a simple procedure based on the Lu–Chipman decomposition [S.-Y. Lu and R.C. Chipman, J. Opt. Soc. Am. A 13, 1106 (1996)], any depolarizing Mueller matrix can be transformed into a reduced form which accumulates the depolarization and polarizance properties into a set of six parameters. The simple structure of this reduced form provides straightforward ways for the general characterization of Mueller matrices as well as for the analysis of singular Mueller matrices.     

Errors of Mueller matrix measurements with a partially polarized light source

The linear errors of Mueller matrix measurements, using a partially polarized light source, have been formulated for imperfections of misalignment, depolarization, and nonideal ellipsometric parameters of the polarimetric components. The error matrices for a source-polarizer system and a source-polarizer-compensator system are derived. A polarized light source, when used with an imperfect polarizer, generates extra errors in addition to those for an unpolarized source. The compensator redistributes these errors to different elements of the error matrix. The errors of the Mueller matrices for the polarizer-sample-analyzer and the polarizer-compensator-sample-analyzer systems are evaluated for a straight through case. This error analysis is applied to a Stokes method and an experiment was performed to show the errors by a polarized light source. This general analysis can be used to evaluate errors for ellipsometry and polarimetry.     

Mueller matrix roots depolarization parameters

The Mueller matrix roots decomposition recently proposed by Chipman in [1] and its three associated families of depolarization (amplitude depolarization, phase depolarization, and diagonal depolarization) are explored. Degree of polarization maps are used to differentiate among the three families and demonstrate the unity between phase and diagonal depolarization, while amplitude depolarization remains a distinct class. Three families of depolarization are generated via the averaging of different forms of two nondepolarizing Mueller matrices. The orientation of the resulting depolarization follows the cyclic permutations of the Pauli spin matrices. The depolarization forms of Mueller matrices from two scattering measurements are analyzed with the matrix roots decomposition-a sample of ground glass and a graphite and wood pencil tip.     

Mueller Matrices and Depolarization Criteria

Abstract

The question of whether a given Mueller matrix represents a deterministic or a non-deterministic system is analysed by means of a matrix condition. The possibility of replacing this matrix condition by a scalar condition is examined. It is shown that this is permissible only for those cases where a Hermitian matrix constructed from the Mueller matrix is positive semidefinite.     

Error analysis for Mueller matrix measurement

The linear errors of Mueller matrix measurements are formulated for misalignment, depolarization, and incorrect retardation of the polarimetric components. The measured errors of a Mueller matrix depend not only on the imperfections of the measuring system but also on the Mueller matrix itself. The error matrices for different polarimetric systems are derived and also evaluated for the straight-through case. The error matrix for a polarizer-sample-analyzer system is much simpler than those for more complicated systems. The general error matrix is applied to null ellipsometry, and the obtained errors in ellipsometric parameters psi and delta are identical to the errors specifically derived for null ellipsometry with depolarization.     

Mueller matrices and information derived from linear polarization lidar measurements: theory

A Mueller matrix M is developed for a single-scattering process such that G(theta, phi) = T (phi(a))M T (phi(p))u, where u is the incident irradiance Stokes vector transmitted through a linear polarizer at azimuthal angle phi(p), with transmission Mueller matrix T (phi(p)), and G(theta, phi) is the polarized irradiance Stokes vector measured by a detector with a field of view F, placed after an analyzer with transmission Mueller matrix T (phi(a)) at angle phi(a). The Mueller matrix M is a function of the Mueller matrix S (theta) of the scattering medium, the scattering angle (theta, phi), and the detector field of view F. The Mueller matrixM is derived for backscattering and forward scattering, along with equations for the detector polarized irradiance measurements (e.g., cross polarization and copolarization) and the depolarization ratio. The information that can be derived from the Mueller matrix M on the scattering Mueller matrixS (theta) is limited because the detector integrates the cone of incoming radiance over a range of azimuths of 2pi for forward scattering and backscattering. However, all nine Mueller matrix elements that affect linearly polarized radiation can be derived if a spatial filter in the form of a pie-slice slit is placed in the focal plane of the detector and azimuthally dependent polarized measurements and azimuthally integrated polarized measurements are combined.     

Dual-modulator broadband infrared Mueller matrix ellipsometry

A broadband mid-infrared Mueller matrix ellipsometer is described based on two photoelastic modulators and a step-scan interferometer. The data are analyzed using a combination hardware-software double Fourier transformation. Obtaining spectra of the Mueller matrix elements requires that the infrared wavelength-dependent retardation amplitude of the modulators be known through calibration and subsequently incorporated into the data processing. The spectroscopic capability of the instrument is demonstrated in transmission and reflection geometries by the measured Mueller matrices of air, an anisotropic quartz crystal, and the ZnSe-water interface, each from 2500-4000 cm^-1.     

Investigation of Mueller matrices of anisotropic nonhomogeneous layers in application to an optical model of the cornea

We consider a system of anisotropic layers as an optical model of the eye cornea. Effective refraction indices for normally incident light are calculated with the assumption that each layer consists of closely packed uniform cylinders (fibrils). Jones matrix formalism is used to describe light propagation through the cornea. We calculate the Jones matrices from the experimentally measured Mueller matrices. Two algorithms are used for this purpose. The experiments have shown that ~20% of cornea area studied had the structure well described by the helical model proposed.     

Mueller calculus of polarization change in the cube-corner retroreflector

The optical ray properties of the cube-corner retroreflector (CCR) are first recalled. The change of polarization of the radiation due to CCR reflection is then derived by use of the Mueller matrix calculus. It is found that, in general, when the faces are not ideal reflectors, the useful cross section of the CCR consists of six zones, each of which produces a different change of polarization, i.e., it gives a different Mueller matrix. All the Mueller matrices depend on wavelength. The results are quite general and can be used directly also for partially polarized radiation.     

A Padé‐based factorization‐free algorithm for identifying the eigenvalues missed by a generalized symmetric eigensolver

When computing the solution of a generalized symmetric eigenvalue problem of the form Ku =λ Mu , the Sturm sequence check, also known as the inertia check, is the most popular method for reporting the number of missed eigenvalues within a range [σ_L,σ_R]. This method requires the factorization of the matrices K ?σ_L M and K ?σ_R M . When the size of the problem is reasonable and the matrices K and M are assembled, these factorizations are possible. When the eigensolver is equipped with an iterative solver, which is nowadays the preferred choice for large‐scale problems, the factorization of K ?σ M is not desired or feasible and therefore the inertia check cannot be performed. To this effect, the purpose of this paper is to present a factorization‐free algorithm for detecting and identifying the eigenvalues that were missed by an eigensolver equipped with an iterative linear equation solver within an interval of interest [σ_L,σ_R]. This algorithm constructs a scalar, rational, transfer function whose poles are exactly the eigenvalues of the symmetric pencil ( K , M ), approximates it by a Padé expansion, and computes the poles of this approximation to detect and identify the missed eigenvalues. The proposed algorithm is illustrated with an academic numerical example. Its potential for real engineering applications is also demonstrated with a large‐scale structural vibrations problem. Copyright © 2009 John Wiley & Sons, Ltd.     

Data analysis for a rotating quarter-wave, far-infrared Stokes polarimeter

Data analysis techniques are reviewed and extended for the measurement of the Stokes vector of partially or completely polarized radiation by the rotating quarter-wave method. It is shown that the conventional technique, based on the Fourier analysis of the recorded signal, can be efficiently replaced by a weighted least-squares best fit, so that the different accuracy of the measured data can be taken into account to calculate the measurement errors of the Stokes vector elements. Measurement errors for the polarization index P and for the azimuth and ellipticity angles psi and chi of the radiation are also calculated by propagation error theory. For those cases in which the above technique gives a nonphysical Stokes vector (i.e., with a polarization degree of P>1) a constrained least-squares best fit is introduced, and it is shown that in this way a Stokes vector with P = 1 (rather than P相似文献   

Formulation of a Mueller matrix for modeling of depolarization and scattering of nitrobenzene in a Kerr cell

We have studied the depolarization of light from nitrobenzene in a Kerr cell. We observed that absorption in nitrobenzene is electric-field dependent. For modeling a nitrobenzene device we formulated a Mueller matrix for the Kerr-cell assembly, and by operating it on a Stokes vector of the input light we obtained a corresponding Stokes vector for the output light. The first parameter of the output Stokes vector corresponds to the intensity transmittance. It was simulated and compared with the measured intensity transmittance for several orientations of the polarizer-analyzer pair with respect to the applied voltages. The measurement of all unknown coefficients in a Mueller matrix consisting of the superposition of nondepolarizing and depolarizing components predicts the depolarization, scattering, and absorption in the nitrobenzene electro-optic device. The output intensities of the orthogonally polarized and cross-coupled depolarizing coefficients are in good agreement for a semi-isotropic medium. The formulated Mueller matrix agrees with the experimentally measured transmittance.     

Bisection for parallel computing using Ritz and Fiedler vectors

Summary. In this article, an efficient algorithm is developed for the decomposition of large-scale finite element models. A weighted incidence graph with N nodes is used to transform the connectivity properties of finite element meshes into those of graphs. A graph G₀ constructed in this manner is then reduced to a graph G_n of desired size by a sequence of contractions G₀ G₁ G₂ G_n. For G₀, two pseudoperipheral nodes s₀ and t₀ are selected and two shortest route trees are expanded from these nodes. For each starting node, a vector is constructed with N entries, each entry being the shortest distance of a node n_i of G₀ from the corresponding starting node. Hence two vectors v₁ and v₂ are formed as Ritz vectors for G₀. A similar process is repeated for G_i (i=1,2,,n), and the sizes of the vectors obtained are then extended to N. A Ritz matrix consisting of 2(n+1) normalized Ritz vectors each having N entries is constructed. This matrix is then used in the formation of an eigenvalue problem. The first eigenvector is calculated, and an approximate Fiedler vector is constructed for the bisection of G₀. The performance of the method is illustrated by some practical examples.     

Components of purity of a Mueller matrix

The degree of polarimetric purity of a Mueller matrix, also called "depolarization index" [Opt. Acta 33, 185 (1986)] is expressed as a quadratic average of two contributions of different nature. The contribution due to the polarizance and diattenuation properties is given by a unique parameter called "degree of polarizance," and the complementary contribution due to nonpolarizing properties is given by a parameter called "degree of spherical purity." These two intrinsic quantities are useful in order to analyze the sources of the polarimetric purity of a material sample whose Mueller matrix has been measured and provide criteria for the classification of Mueller matrices.