General Mueller Matrix Research Articles

Incoherent phenomena in anisotropic periodic structures: from modeling to experimental demonstration

The periodic structures are widely studied in numerous optical applications and there is a number of good tools for numerical modeling of such a structures (for example rigorous coupled-wave analysis, finite-difference time-domain, finite element method etc.). However, when it comes to the modeling of incoherent effects in many cases of practical interest, the current methods are not rigorous enough or depend on computationally demanding averaging of coherent response. In this paper, we present a novel approach to modeling of incoherent effects in structures with lateral periodicity based on scattering matrix formalism, as a way to describe optical response of a structure, and on application of incoherent wave summation in the form of infinite geometric series and generalized Mueller matrix calculus. This method can be combined with any of the existing coherent methods of modeling periodic structures and it offers significantly faster computational performance than partially coherent/incoherent methods based on averaging. It is compared with other methods for modeling of incoherent effects and also with experimental spectroscopic data. This method is then used to explain phenomena emerging from the complex interaction between diffraction grating and thick substrate.

Interpreting and quantifying depolarization properties of tissue with differential polarization parameters

Calculating the polarization properties and revealing the mechanism or the sources of depolarization properties of tissue are all essential for their successful applications. In this work we report a method of calculating differential polarization parameters directly from measured macroscopic Mueller matrices and revealing the mechanism or the source of depolarization properties of tissue. First a measured Mueller matrix is expressed as a sum of up to four Mueller-Jones matrices. The derivatives of both sides of the equation are carried out to find both the non-depolarizing and depolarizing differential polarization parameters by using the differential Mueller matrix equation. The explicit formulas for each Mueller-Jones matrix are then obtained in terms of the polarization parameters of differential Mueller matrix of the general Mueller matrix. Our method has the advantages of being capable of identifying measurement errors contained in the measured Mueller matrices, separating non-depolarizing and depolarizing differential polarization parameters, and revealing contributions of depolarizing differential polarization parameters to depolarization behaviors of the medium. The experimental results are presented to demonstrate the potential applications of our method.

Features of light scattering on mosaic layers composed of structurally similar birefringent domains

ABSTRACTThe two-point generalized Mueller matrix method and phase screen approximation are applied to determine features of light scattering on mosaic birefringent layers composed of structurally similar domains. Two types of mosaic birefringent layers are considered: mosaic birefringent layers composed of identical non-chiral domains and statistically non-chiral layers composed of structurally similar (but generally not identical) domains. Analytical expressions for matrix operators that connect state vectors of the scattered and non-scattered components with the state vector of the incident beam are derived in terms of statistical structural characteristics of the layer. Based on the theoretical approach developed, some optical phenomena characteristic of mosaic birefringent layers, such as selective light scattering and inversion of circular polarization upon scattering, are explained. The effect of the degree of similarity of domains on the scattering properties of the layer is analyzed. Illustrative experimental examples are given.Abbreviations: LC: liquid crystal; NRPA layer: nematic random planar alignment layer; NPMB layer: non-planar mosaic birefringent layer; RIJM theory: rotationally invariant Jones matrix theory; SMB layer: simple mosaic birefringent layer; SNCM layer: statistically non-chiral mosaic layer

Parameterization of the Mueller matrix.

A deterministic Mueller matrix (Mueller-Jones matrix) contains seven independent parameters. By writing the so-called coherence vector in parametric form, the Mueller matrix can also be written in parametric form, where the matrix elements automatically satisfy the known relationships between each other. Three of these parameters are also related to the so-called anisotropy coefficients. The approach is generalized to express all 16 elements of a general Mueller matrix in terms of a scalar and five three-dimensional vectors. Many properties of a Mueller matrix can be written simply in terms of these vectors. Published experimental matrices are considered by this procedure.

Structure of polarimetric purity of a Mueller matrix and sources of depolarization

The depolarization properties of a medium with associated Mueller matrix M are characterized through two complementary sets of parameters, namely 1) the three indices of polarimetric purity (IPP), which are directly linked to the relative weights of the spectral components of M and provide complete information on the structure of polarimetric randomness, but are insensitive to the specific polarimetric behaviors that introduce the lack of randomness, and 2) the set of three components of purity (CP), constituted by the polarizance, the diattenuation and the degree of spherical purity. The relations between these sets of physical invariant quantities are studied by means of their representation into a common purity figure. Furthermore, the polarimetric properties of a general Mueller matrix M are parameterized in terms of sixteen meaningful quantities, three of them being the IPP, which together with the CP provide complete information on the integral depolarization properties of the medium.

Scattering of partially coherent electromagnetic beams by water droplets and ice crystals

The conventional Lorenz–Mie theory is generalized for a case when the light source is partially spatially coherent. The influence of the degree of coherence of the incident field on the generalized Mueller matrix and the spectral degree of coherence of the scattered light is analytically studied by using the vector field instead of the scalar field to extend previous results on the angular intensity distribution. The results are compared with the Mueller matrix obtained from the Discrete Dipole Approximation (DDA) method, which is an average over an ensemble of stochastic incident beams. Special attention is paid to the Mueller matrix elements in the backward direction, and the results show some Mueller matrix elements, such as P22, depend monotonically on the coherence length of the incident beam. Therefore, detecting back scattering Mueller matrix elements may be a promising method to measure the degree of coherence. The new formalism is applied to cases of large spherical droplets in water clouds and hexagonal ice crystals in cirrus clouds. The corona and glory phenomena due to spheres and halos associated with hexagonal ice crystals are found to disappear if the incident light tends to be highly incoherent.

Magneto-optical coupling in ferromagnetic thin films investigated by vector-magneto-optical generalized ellipsometry

We performed generalized Mueller matrix ellipsometry measurements in a magnetic field of arbitrary orientation and magnitude up to 400 mT at room temperature and probed the magneto-optical response of capped, ferromagnetic Fe, Ni${}_{20}$Fe${}_{80}$, Co, Ni${}_{80}$Fe${}_{20}$, and Ni thin films on ZnO substrates in the spectral range from 300 to 1100 nm. We determined the off-diagonal elements in the magneto-optical dielectric tensor under saturated magnetization conditions in the sample surface plane via a model analysis. The off-diagonal elements depend on the net spin polarization and the electronic band structure of the ferromagnetic thin films. For the pure ferromagnetic metals Fe, Co, and Ni, the converted off-diagonal elements agree well with the reported experimental optical conductivity data. As a result we use the extracted wavelength-dependent magneto-optical coupling constant to predict the wavelength-dependent magneto-optical response of different Ni/Fe multilayer structures.

Mueller-matrix model of an inhomogeneous, linear, birefringent medium: Single scattering case

Using the anisotropic phase-screen method, we derive the generalized Mueller matrix of an inhomogeneous, linear, birefringent medium in the single-scattering case. We show that the Mueller matrix include eight nonzero elements with relations among elements given by m 11= m 22, m 12= m 21, m 33= m 44, and m 34= m 43. Using the derived Mueller-matrix model, we simulate polarization optics for calcite, quartz, and sodium nitrate crystals. At λ=0.63 μm and with small inhomogeneities, the matrix elements are a function of the standard deviation of the crystal thickness, while m 12 shows pronounced sensitivity to the difference of refractive indices. At an observation angle of 0°, inhomogeneous birefringent media behaves as if they are partial polarizers. Matrix elements most sensitive to the inhomogeneity in the UV to near IR wavelengths have been determined.

Virtual Generalized Mueller Matrix Method for Measurement of Complex Polarization-Mode Dispersion Vector in Optical Fibers

A virtual generalized Mueller matrix method (VGMMM) is proposed to measure the complex polarization-mode dispersion (PMD) vector in a fiber system with polarization-dependent loss or gain. VGMMM can attain the low-noise high-resolution PMD data using a relatively large frequency step, without the knowledge of input polarization states. VGMMM combines the advantages of both matrix-based methods and differentiation-based methods and overcomes their shortcomings. Experimental results on a fiber system confirm the validity and accuracy of VGMMM

Generalized Mueller matrix method for polarization mode dispersion measurement in a system with polarization-dependent loss or gain

A generalized Mueller matrix method (GMMM) is proposed to measure the polarization mode dispersion (PMD) in an optical fiber system with polarization-dependent loss or gain (PDL/G). This algorithm is based on the polar decomposition of a 4X4 matrix which corresponds to a Lorentz transformation. Compared to the generalized Poincaré sphere method, the GMMM can measure PMD accurately with a relatively larger frequency step, and the obtained PMD data has very low noise level.

Anisotropic media with orthogonal eigenpolarizations

For partially polarized light propagating through anisotropic media, we introduce a non-singular eigenvector matrix that diagonalizes the most general differential Mueller matrix of a transmission medium. It is shown that through the similarity matrix it is possible to recover the differential Mueller matrix from the Mueller matrix and vice versa. For non-depolarizing media the similarity matrix is analytically evaluated and the necessary and sufficient condition for such a matrix to be orthogonal is reported. The effects on the degree of polarization for incident natural light are also discussed. The incident light is expressed in terms of the Stokes vector. The transmission medium is assumed to be linear and homogeneous with any simultaneous effects of absorption and dispersion, and is characterized by a differential Mueller matrix that can or cannot exhibit depolarization effects.

Mueller matrices and the degree of polarization

This article shows that only polarizers, retarders, and improper rotation Mueller matrices do not decrease the degree of polarization for any input Stokes vector. On the other hand, a Mueller matrix does not increase the degree of polarization for any input Stokes vector if and only if it has an unpolarized Stokes vector as eigenvector. Furthermore, for any Mueller matrix an unpolarized incident Stokes vector always has the maximum gain in the degree of polarization. It is pointed out that a general Mueller matrix deforms the Poincaré sphere (for input Stokes vectors) into an ellipsoid (for output Stokes vectors). Finally, several inequalities involving the input and output degrees of polarization are derived.

Polarization entropy transfer and relative polarization entropy

The concept of polarization entropy is extended to include its transfer by an optical system (or scattering medium) characterized by a Jones matrix (or equivalently, a Mueller-Jones matrix) and also by a general Mueller matrix. It is shown that physical realizability for a deterministic, passive medium governed by a Jones matrix implies enpolarization (i.e., output degree of polarization is greater than input degree of polarization) and that the output entropy is less than the input entropy. Thus enpolarization leads to a decrease in the uncertainty of the output relative to the input as characterized by the respective degree of polarization. General Mueller matrices can depolarize (i.e., output degree of polarization is less than input degree of polarization), leading to an output entropy that is greater than the input entropy. Thus depolarization leads to an increase in the uncertainty of the output relative to the input. An entropic measure of the “distance” between input and output degrees of polarization, the relative entropy, is introduced and studied.

Bilinear constraints between elements of the 4 x 4 Mueller-Jones transfer matrix of polarization theory

When the 4 x 4 Mueller matrix of an optical system represents a situation in which the polarization of the light is unchanged, then the elements of the Mueller matrix are derivable from a 2 x 2 Jones matrix. The general Mueller matrix contains 16 independent parameters, whereas the Jones matrix contains 7 independent parameters. Consequently, 9 identities exist between the 16 matrix elements. The purpose of this note is to display explicitly these 9 nonlinear (bilinear) equations. These equations are then used to study the number of independent parameters of a recent experimental determination by Howell of the Mueller matrix of a collimator-radiometer system.