Factorization of the coherency matrix of polarization optics

A Table of Some Coherency Matrices, Coherency Matrix Factors, and Their Respective Mueller Matrices

Many books on polarization give tables of Mueller matrices. The coherency matrix has been found useful for interpretetion of the Mueller matrix. Here we give a table of Mueller matrices M, coherency matrices C, and coherency matrix factors F for different polarization components and systems. F is not given for some complicated nondeterministic cases. In many cases, though, F has a very simple form. In particular, we give expressions for F for the general case of an homogeneous elliptic diattenuating retarder. Different coordinate systems for describing diattenuating retarders are compared, on a generalized retardation sphere, analogous to the Poincaré sphere. For the general homogeneous deterministic case, expressions for the Mueller matrix have particularly simple forms for Cartesian or stereographic coordinates in generalized retardation space.

Characterization of the Mueller Matrix: Purity Space and Reflectance Imaging

Depolarization has been found to be a useful contrast mechanism in biological and medical imaging. The Mueller matrix can be used to describe polarization effects of a depolarizing material. An historical review of relevant polarization algebra, measures of depolarization, and purity spaces is presented, and the connections with the eigenvalues of the coherency matrix are discussed. The advantages of a barycentric eigenvalue space are outlined. A new parameter, the diattenuation-corrected purity, is introduced. We propose the use of a combination of the eigenvalues of coherency matrices associated with both a Mueller matrix and its canonical Mueller matrix to specify the depolarization condition. The relationships between the optical and polarimetric radar formalisms are reviewed. We show that use of a beam splitter in a reflectance polarization imaging system gives a Mueller matrix similar to the Sinclair–Mueller matrix for exact backscattering. The effect of the reflectance is canceled by the action of the beam splitter, so that the remaining features represent polarization effects in addition to the reflection process. For exact backscattering, the Mueller matrix is at most Rank 3, so only three independent complex-valued measurements are obtained, and there is insufficient information to extract polarization properties in the general case. However, if some prior information is known, a reconstruction of the sample properties is possible. Some experimental Mueller matrices are considered as examples.

Phasor approach of Mueller matrix optical scanning microscopy for biological tissue imaging

Mueller matrix microscopy is an advanced imaging technique providing a full characterization of the optical polarization fingerprint of a sample. The Lu-Chipman (LC) decomposition, a method based on the modeling of elementary polarimetric arrangements and matrix inversions, is the gold standard to extract each polarimetric component separately. However, this models the optical system as a small number of discrete optical elements and requires a priori knowledge of the order in which these elements occur. In stratified media or when the ordering is not known, the interpretation of the LC decomposition becomes difficult. In this work, we propose a new, to our knowledge, representation dedicated to the study of biological tissues that combines Mueller matrix microscopy with a phasor approach. We demonstrate that this method provides an easier and direct interpretation of the retardance images in any birefringent material without the use of mathematical assumptions regarding the structure of the sample and yields comparable contrast to the LC decomposition. By validating this approach through numerical simulations, we demonstrate that it is able to give access to localized structural information, resulting in a simple determination of the birefringent parameters at the microscopic level. We apply our novel, to our knowledge, method to typical biological tissues that are of interest in the field of biomedical diagnosis.

Polarimetric optical scanning microscopy of zebrafish embryonic development using the coherency matrix

Many of the most important resolution improvements in optical microscopy techniques are based on the reduction of scattering effects. The main benefit of polarimetry-based imaging to this end is the discrimination between scattering phenomena originating from complex systems and the experimental noise. The determination of the coherency matrix elements from the experimental Mueller matrix can take advantage of scattering measurements to obtain additional information on the structural organization of a sample. We analyze the contrast mechanisms extracted from (a) the coherency matrix elements, (b) its eigenvalues and (c) the indices of polarimetric purity at different stages of zebrafish embryos, based on previous work using Mueller matrix optical scanning microscopy. We show that the use of the coherency matrix and related decompositions leads to an improvement in the imaging contrast, without requiring any complicated algebraic operations or any a priori knowledge of the sample, in contrast to standard polarimetric methods.

Circular Intensity Differential Scattering for Label-Free Chromatin Characterization: A Review for Optical Microscopy

Circular Intensity Differential Scattering (CIDS) provides a differential measurement of the circular right and left polarized light and has been proven to be a gold standard label-free technique to study the molecular conformation of complex biopolymers, such as chromatin. In early works, it has been shown that the scattering component of the CIDS signal gives information from the long-range chiral organization on a scale down to 1/10th-1/20th of the excitation wavelength, leading to information related to the structure and orientation of biopolymers in situ at the nanoscale. In this paper, we review the typical methods and technologies employed for measuring this signal coming from complex macro-molecules ordering. Additionally, we include a general description of the experimental architectures employed for spectroscopic CIDS measurements, angular or spectral, and of the most recent advances in the field of optical imaging microscopy, allowing a visualization of the chromatin organization in situ.

Eigenvectors of polarization coherency matrices

Calculation of the eigenvectors of two- and three-dimensional coherency matrices, and the four-dimensional coherency matrix associated with a Mueller matrix, is considered, especially for algebraic cases, in the light of recently published algorithms. The preferred approach is based on a combination of an evaluation of the characteristic polynomial and an adjugate matrix. The diagonal terms of the coherency matrix are given in terms of the characteristic polynomial of reduced matrices as functions of the eigenvalues of the coherency matrix. The analogous polynomial form for the off-diagonal elements of the coherency matrix is also presented. Simple expressions are given for the pure component in the characteristic decomposition.

Polarization in reflectance imaging

The Sinclair and Kennaugh matrices are widely used in the remote sensing discipline for signals detected in the backward direction. The connections between the Jones matrix and the Sinclair matrix, and between the Mueller matrix and the Kennaugh matrix, are explored. Different operations on the Jones matrix and their corresponding effects on the Mueller matrix, coherency matrix, and coherence vector are derived. As an example, the Sinclair matrix leads to a Mueller-Sinclair matrix, and a transformed coherence vector. The Kennaugh matrix is not, however, a Mueller matrix, but can be determined from the Mueller or Mueller-Sinclair matrices. We consider backscattering through a medium on a perfect mirror. We propose that backscattering from a uniform medium can be modeled as an effective uniform medium situated on a perfectly reflective substrate, and the elementary polarization properties derived. In this way, the concept of a uniform polarizing medium can be extended to the reflectance geometry. An experimental Mueller matrix from the literature is considered as an example.

Eigenvalues of the coherency matrix for exact backscattering

An important approach to interpretation of the Mueller matrix is based on the eigenvalues of the coherency matrix, given by the roots of a quartic characteristic equation. For the case of backscattering, one eigenvalue is zero from reciprocity arguments, and the characteristic equation reduces to a cubic. These two approaches (quartic and cubic) to calculation of the eigenvalues for exact backscattering are analytically considered and compared. As expected, the cubic approach is usually simpler, but for the special case of two zero eigenvalues, either approach reduces to the predictions of the simple quadratic characteristic equation. Either approach can be used for numerical calculation of the eigenvalues. The variation in different purity measures with the values of the Mueller matrix elements is presented. An experimental Mueller matrix for backscattering from a turbid chiral medium is investigated.

Coherency and differential Mueller matrices for polarizing media

The elements of the coherency matrix give the strength of the components of a Mueller matrix in the coherency basis. The Z-matrix (called the polarization-coupling matrix or state-generating matrix) represents a partial sum of the coherency expansion. For transmission through a deterministic medium, the coherency elements can be used directly as generators to calculate the development of polarization upon propagation. The commutation properties of the coherency elements are investigated. New matrices that we call the W-matrix and the X-matrix, both different representations of the Z-matrix in a Jones basis, are introduced. The W-matrix controls the transformation of the Jones vector and also the covariance matrix. The product of the X-matrix with its complex conjugate gives the matrix representation of the Mueller matrix in the Jones basis. The development of Mueller matrix and coherency matrix elements upon propagation through some examples of a uniform medium is investigated. It is shown that the coherency matrix is more easily interpreted than the Mueller matrix. Analytic expressions are presented to calculate the elementary polarization properties from coherency matrix elements or Mueller matrix parameters.