Towards a Unified Formalism of Multivariate Coefficients of Variation -- Application to the Analysis of Polarimetric Speckle Time Series

This article primarily aims to unify the various formalisms of multivariate coefficients of variation, leveraging advanced concepts of generalized means, whether weighted or not, applied to the eigenvalues of covariance matrices. We highlight the existence of an infinite number of these coefficients and demonstrate that they are bounded. Moreover, we link the various coefficients of variation identified in the literature to specific instances within our unified formalism. We illustrate the utility of our method by applying it to a time series of polarimetric radar imagery. In this context, the coefficient of variation emerges as a key tool for detecting changes or identifying permanent scatterers, which are characterized by their remarkable temporal stability. The multidimensionality arises from the diversity of polarizations. The introduction of the various possible coefficients demonstrates how their selection impacts the detection of samples exhibiting specific temporal behaviors and underscores the contribution of polarimetry to dynamic speckle analysis

Stern-Gerlach experiment with light: separating photons by spin with the method of A. Fresnel

In 1822 A. Fresnel described an experiment to separate a beam of light into its right- and left-circular polarization components using chiral interfaces. Fresnel's experiment combined three crystalline quartz prisms of alternating handedness to achieve a visible macroscopic separation between the two circular components. Such quartz polyprisms were rather popular optical components in XIXth century but today remain as very little known optical devices. This work shows the analogy between Fresnel's experiment and Stern-Gerlach experiment from quantum mechanics since both experiments produce selective deflection of particles (photons in case of Fresnel's method) according to their spin angular momentum. We have studied a historical quartz polyprism with eight chiral interfaces producing a large spatial separation of light by spin. We have also constructed a modified Fresnel biprism to produce smaller separations and we have examined the analogy with Stern-Gerlach apparatus for both strong and weak measurements. The polarimetric analysis of a Fresnel polyprism reveals that it acts as a spin angular momentum analyzer

Physical significance of the determinant of a Mueller matrix

The determinant of a Mueller matrix M plays an important role in both polarization algebra and the interpretation of polarimetric measurements. While certain physical quantities encoded in M admit a direct interpretation, the understanding of the physical and geometric significance of the determinant of M (detM) requires a specific analysis, performed in this work by using the normal form of M, as well as the indices of polarimetric purity (IPP) of the canonical depolarizer associated with M. We derive an expression for detM in terms of the diattenuation, polarizance and a parameter proportional to the volume of the intrinsic ellipsoid of M. We likewise establish a relation existing between the determinant of M and the rank of the covariance matrix H associated with M, and determine the lower and upper bounds of detM for the two types of Mueller matrices by taking advantage of their geometric representation in the IPP space

Snapshot circular dichroism measurements

Two coherent waves carrying orthogonal polarizations do not interfere when they superpose, but an interference pattern is generated when the two waves share a common polarization. This well-known principle of coherence and polarization is exploited for the experimental demonstration of a novel method for performing circular dichroism measurements whereby the visibility of the interference fringes is proportional to the circular dichroism of the sample. Our proof-of-concept experiment is based upon an analog of Young's double-slit experiment that continuously modulates the polarization of the probing beam in space, unlike the time modulation used in common circular dichroism measurement techniques. The method demonstrates an accurate and sensitive circular dichroism measurement from a single camera snapshot, making it compatible with real-time spectroscopy

Algorithm for the numerical calculation of the serial components of the normal form of depolarizing Mueller matrices

The normal form of a depolarizing Mueller matrix constitutes an important tool for the phenomenological interpretation of experimental polarimetric data. Due to its structure as a serial combination of three Mueller matrices, namely a canonical depolarizing Mueller matrix sandwiched between two pure (nondepolarizing) Mueller matrices, it overcomes the necessity of making a priori choices on the order of the polarimetric components, as this occurs in other serial decompositions. Because Mueller polarimetry addresses more and more applications in a wide range of areas in science, engineering, medicine, etc., the normal form decomposition has an enormous potential for the analysis of experimentally determined Mueller matrices. However, its systematic use has been limited to some extent because of the lack of numerical procedure for the calculation of each polarimetric component, in particular in the case of Type II Mueller matrices. In this work, an efficient algorithm applicable to the decomposition of both Type II and Type I Mueller matrices is presented

Anisotropic integral decomposition of depolarizing Mueller matrices

We propose a novel, computationally efficient integral decomposition of depolarizing Mueller matrices allowing for the obtainment of a nondepolarizing estimate, as well as for the determination of the elementary polarization properties, in terms of mean values and variancescovariances of their fluctuations, of a weakly anisotropic depolarizing medium. We illustrate the decomposition on experimental examples and compare its performance to those of alternative decomposition

Mueller matrix differential decomposition

We present a Mueller matrix decomposition based on the differential formulation of the Mueller calculus. The differential Mueller matrix is obtained from the macroscopic matrix through an eigenanalysis. It is subsequently resolved into the complete set of 16 differential matrices that correspond to the basic types of optical behavior for depolarizing anisotropic media. The method is successfully applied to the polarimetric analysis of several samples. The differential parameters enable one to perform an exhaustive characterization of anisotropy and depolarization. This decomposition is particularly appropriate for studying media in which several polarization effects take place simultaneously