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

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.