**Abstract.**
The measured Mueller matrices contain up to 16 independent parameters for each measurement configuration (spectral profile of the wave probe of the polarimeter, angle of incidence, observation direction, etc.) and for each spatially resolved element of the sample (imaging polarimetry). Thus, the polarimetric techniques are widely used for the study of a great variety of material samples in optics and remote sensing. Nevertheless, the relevant physical information does not appear explicitly in the measured parameters, and thus, the best knowledge of the structure of the physical information contained in a Mueller matrix is required in order to develop appropriate procedures for the polarimetric analysis. The main approaches for serial and parallel decompositions as well as for the geometric representation of measured Mueller matrices are reviewed. Furthermore, the physically invariant polarimetric quantities are identified and decoupled.

# Review on Mueller matrix algebra for the analysis of polarimetric measurements

**José J. Gil**

Universidad de Zaragoza, Facultad de Educación, Pedro Cerbuna 12, 50009 Zaragoza, Spain

*J. Appl. Remote Sens*. 8(1), 081599 (Mar 17, 2014). doi:10.1117/1.JRS.8.081599

#### Open Access

## Abstract

## Introduction

The polarimetric techniques are powerful measurement tools that are commonly used today for the characterization, study, and analysis of great variety of material samples, with important applications to several fields of science and industry and, particularly, to remote sensing.^{1} Therefore, the knowledge of the mathematical structure and properties of the Mueller matrices plays a key role in the interpretation and exploitation of polarimetric measurements. In particular, it is greatly desirable to have at hand general mathematical models for the decomposition of a given Mueller matrix into equivalent optical systems constituted by simple components. From an experimental point of view, it is of great interest to be able to analyze the various polarimetric properties that can be identified from each specific decomposition.

In the recent years, a number of algebraic methods for such decompositions of a Mueller matrix have been developed. Several issues concerning polarimetric decomposition are reviewed along next sections, including parallel, serial, and differential decompositions. Moreover, the relevant physical quantities are identified and analyzed.

Parallel decompositions consist of representing a Mueller matrix as convex sum of Mueller matrices. The physical meaning of parallel decompositions is that the incoming light beam is shared among a set of pencils that interact with a number of optical components spatially distributed in the irradiated area without overlapping, in such a manner that the emerging pencils are incoherently recombined into a single output light beam (Fig. 1).

From the concept of arbitrary parallel decompositions, the polarimetric subtraction of Mueller matrices consist of identifying a parallel component $M0$ that is subtractable from the whole system $M$, in such a manner that the cross-section $p0$ of $M0$ as well as the Mueller matrix of the remainder system (or difference system) $MX$ are calculated (Fig. 2). The subtraction procedure and the conditions for a subtraction to be physically realizable are dealt with in Sec. 7.

**F2 :**

Polarimetric subtraction of a Mueller matrix $M0$ from the whole Mueller matrix $M$. The subtraction procedure provides the test for checking if the subtraction is physically realizable or not and, in the affirmative case, gives the values of the cross-section $p0$ of $M0$ and the Mueller matrix $MX$ of the remainder system $MX=(M-p0M0)/(1\u2212p0)$.

Serial decompositions consist of representing a general Mueller matrix as a product of particular Mueller matrices. The physical meaning of serial decompositions is that the whole system is considered as a cascade of polarization components so that the incoming light interacts sequentially with them. This arrangement of the components constitutes the serial equivalent system (Fig. 3).

Serial and parallel decompositions can be combined in order to generate different sets of equivalent systems constituted by simple components.^{2}

Differential decompositions consist of identifying an elementary representative of a given integral Mueller matrix and then separating the mean values of the elementary properties from the depolarizing sources given by their uncertainties and by the anisotropic absorptions. When the depolarization occurs uniformly and continuously along the optical path, a differential depolarizing Mueller matrix can be properly defined as $m\u2261(ln\u2009M)/L$, where $L$ is the optical path length (Fig. 4). Otherwise, an equivalent differential Mueller matrix can mathematically be defined, but without a direct interpretation in terms of the nature of the sample.^{3}^{,}^{4}

Geometric representations of a Mueller matrix $M$ are generally based on different kinds of Poincaré sphere mappings by $M$. These approaches are useful to provide visual and geometric images of the physical properties of the medium represented by $M$, and hence, they promise improvements in applied polarimetry for diagnostic in remote sensing, medicine, and other important fields of science and industry. Among the several geometric approaches for the representation of Mueller matrices presented in the literature, we emphasize the recent representation of a given depolarizing Mueller matrix through a set of three complementary ellipsoids^{5} (Fig. 5), which is dealt with in Sec. 13.

Wherever appropriate, we use along this paper the common conventions and notations used in works dealing with the algebraic properties of Mueller matrices. Particularly convenient for some purposes is to express a Mueller matrix $M$ in the following partitioned form:^{6}