On the algebraic characterization of a Mueller matrix in polarization optics. II. Necessary and sufficient conditions for Jones-derived Mueller matrices

Abstract

We show that every Mueller matrix, that is a real 4 × 4 matrix M which transforms Stokes vectors into Stokes vectors, may be factored as M = L ₂ KL ₁ where L ₁ and L ₂ are orthochronous proper Lorentz matrices and K is a canonical Mueller matrix having only two different forms, namely a diagonal form for type-I Mueller matrices and a non-diagonal form (with only one non-zero off-diagonal element) for type-II Mueller matrices. Using the general forms of Mueller matrices so derived, we then obtain the necessary and sufficient conditions for a Mueller matrix M to be Jones derived. These conditions for Jones derivability, unlike the Cloude conditions which are expressed in terms of the eigenvalues of the Hermitian coherency matrix T associated with M, characterize a Jones-derived matrix M through the G eigenvalues and G eigenvectors of the real symmetric N matrix N = [Mtilde]GM associated with M. Appending the passivity conditions for a Mueller matrix onto these Jones-derivability conditions, we then arrive at an algebraic identification of the physically important class of passive Jones-derived Mueller matrices.     

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.     

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.     

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.     

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.     

State of Polarization in Open- and Closed-loop Helices of Single-mode Fibres

Abstract

The state of polarization in helically wound single-mode fibres is described in terms of coupled-mode equations and the Mueller matrix for an elliptically birefringent single-mode fibre in the quasi-monochromatic case. Possible depolarization has been accounted for by means of the mutual correlation function |γ| between eigenpolarization modes. The polarization state in closed-loop fibre-optic helices has been studied experimentally under single- and dual-mode operation. It has been shown that the closed-loop set-up can be used for the development of compact fibre-optic sensors.     

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.     

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.     

Optimization of a dual-rotating-retarder Mueller matrix polarimeter

The dual-rotating-retarder configuration is one of the most common forms of the Mueller matrix polarimeter. I perform an optimization of this polarimeter configuration by minimizing the condition number of the system data reduction matrix. I find the optimum retardance for the rotating retarders to be 127 degrees. If exactly 16 intensity measurements are used for a Mueller matrix calculation, a complex relationship exists between the condition number and the sizes of the angular increments of the two retarders. If many intensity measurements are made, thus overspecifying the calculation, I find broad optimal ranges of angular increments of the two retarders that yield essentially equal performance. Experimental results are given.     

Birefringence evaluation of helically wound optical fibres

Abstract

A simple, non-destructive method is presented for the characterization of the birefringence properties of a helically wound mono-mode optical fibre. It is shown that the Jones matrix model developed by J.N. Ross can be applied to describe them, if the polarization evolution of light is measured with respect to a local reference frame (Serret-Frenet frame). In this case the Ross model is equal to the matrix model developed by R.C. Jones for a twisted crystal. In order to use a fixed reference frame (laboratory reference frame) it is necessary to consider the rotation of the reference frame at the fibre input face produced by the fibre topology. The orientation of the helical fibre axes at the input is defined following the geometrical criteria proposed by Ross. The use of Poincaré's method and Mueller calculus to rewrite the Ross model simplifies the physical interpretation of the results. The method presented provides an easy way to measure the total linear retardation induced by the fibre curvature, and to separate the circular retardation associated with the photo-elastic response of the fibre, from the topological rotation of the reference frame defined only by the geometry of the helically wound fibre.     

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.     

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.     

Sensitivity of the backscattering Mueller matrix to particle shape and thermodynamic phase

The Mueller matrix (M) corresponding to the phase matrix in the backscattering region (scattering angles ranging from 175 degrees to 180 degrees) is investigated for light scattering at a 0.532-microm wavelength by hexagonal ice crystals, ice spheres, and water droplets. For hexagonal ice crystals we assume three aspect ratios (plates, compact columns, and columns). It is shown that the contour patterns of the backscattering Mueller matrix elements other than M11, M44, M14, and M41 depend on particle geometry; M22 and M33 are particularly sensitive to the aspect ratio of ice crystals. The Mueller matrix for spherical ice particles is different from those for nonspherical ice particles. In addition to discriminating between spherical and nonspherical particles, the Mueller matrix may offer some insight as to cloud thermodynamic phase. The contour patterns for large ice spheres with an effective size of 100 microm are substantially different from those associated with small water droplets with an effective size of 4 microm.     

Quantum and classical tomography with equidistant reference angles

Abstract

Concepts of quantum tomography are useful for classical image reconstructions as well. The overall error caused by a finite number of equidistant reference phases is quantified in terms of the density matrix in Fock representation. Moreover, an exact condition for perfect reconstruction is derived.     

Improved results for neutral systems with nonlinear perturbations

Abstract

In this paper, a simple robust stability problem for uncertain neutral systems is considered. The uncertainty under consideration is nonlinear parameter perturbations and does not require a matching condition. Based on the constructive use of the Lyapunov functional approach and linear matrix inequality (LMI) technique, a new delay‐dependent criterion is proposed to guarantee robust stability for such systems. Finally, less conservative than previous results are illustrated by the five examples.     

Neural network pattern recognition by means of differential absorption Mueller matrix spectroscopy

Artificial neural network systems were built for detecting amino acids, sugars, and other solid organic matter by pattern recognition of their polarized light scattering signatures in the form of a Mueller matrix. Backward-error propagation and adaptive gradient descent methods perform network training. The product of the training is a weight matrix that, when applied as a filter, discerns the presence of the analytes on the basis of their cued susceptive Mueller matrix difference elements. This filter function can be implemented as a software or a hardware module to a future differential absorption Mueller matrix spectrometer.     

Definition of a parametric form of nonsingular Mueller matrices

The goal of this paper is to propose a mathematical framework to define and analyze a general parametric form of an arbitrary nonsingular Mueller matrix. Starting from previous results about nondepolarizing matrices, we generalize the method to any nonsingular Mueller matrix. We address this problem in a six-dimensional space in order to introduce a transformation group with the same number of degrees of freedom and explain why subsets of O(5,1), the orthogonal group associated with six-dimensional Minkowski space, is a physically admissible solution to this question. Generators of this group are used to define possible expressions of an arbitrary nonsingular Mueller matrix. Ultimately, the problem of decomposition of these matrices is addressed, and we point out that the "reverse" and "forward" decomposition concepts recently introduced may be inferred from the formalism we propose.     

Pole‐assignment of dynamic interval systems

Abstract

A simple sufficient condition which guarantees that all the eigenvalues of an interval matrix will lie inside a circle centered at α with radius r is formulated. A design procedure of robust state feedback controllers for dynamic interval systems is demonstrated by an example.     

Photopolarimetric measurement of single, intact pulp fibers by mueller matrix imaging polarimetry

A method based on Mueller matrix polarimetry is developed and demonstrated for determining the fibril angle and relative phase retardation of single, intact pulp fibers. The method permits quantitative and nondestructive determination of these parameters from measurements at one wavelength without any fiber alignment. The Mueller matrix of a pulp fiber and its relationship with the fibril angle and phase retardation are described. A nonmodulation method for determining the Mueller matrix is then proposed that is based on a set of intensity data registered by a single detector. Measurements were carried out with single pulp fibers as samples to test the theoretical prediction. The test measurements and results are described and presented.     

Electro-optic coefficient spectrum of cadmium telluride

A spectrum for the electro-optic coefficient of cadmium telluride measured from 3 to 14 μm is reported. The spectrum shows that the quantity n(3)r(41) has a nearly constant value of 1.09 × 10(-10) m/V over this spectral band, with a slight (5%) dip at the weak absorption band centered at 6 μm. Measurements were performed with an infrared Mueller matrix spectropolarimeter. Transmission spectra of the Mueller matrix were acquired at a set of applied voltages. Retardance spectra were calculated from Mueller matrix spectra, and then the electro-optic coefficient was calculated at each wavelength by a least-squares fit to the resulting retardance as a function of voltage.