# Extended Representation of Mueller Matrices

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## Abstract

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## 1. Introduction

**M**is measured, the analysis of the information provided by the sixteen real elements of

**M**constitutes a critical stage, in which different procedures are applied, including the identification of descriptors of diattenuation, polarizance, retardance, and depolarization, as well as methods for the serial and parallel decompositions of

**M**into simpler components. In particular, so-called arbitrary and characteristic decompositions [1,2,3] constitute powerful tools that provide sets of peculiar parallel constituents that are susceptible to specific interpretations.

**C**[4], a positive semidefinite Hermitian matrix that is biunivocally associated with the given

**M**(that is,

**M**determines

**C**unambiguously and vice versa, see Equation (4)), does not always satisfy the property $\mathrm{rank}C=4$, but its rank can take integer values in the interval $1\le \mathrm{rank}C\le 4$ [5,6]. Recall that $\mathrm{rank}C$ equals the number of independent parallel components of

**M**[1,3].

**M**is combined with a portion of a perfect depolarizer, of which the associated normalized Mueller matrix has the diagonal form ${\widehat{M}}_{\Delta 0}=\mathrm{diag}\hspace{0.17em}(1,0,0,0)$ [6]. Due to the simple structure of ${\widehat{M}}_{\Delta 0}$, which corresponds to polarimetric white noise [7], the composed matrix inherits all the anisotropies exhibited by

**M**. Analogously to what occurs in some image diagnostic techniques, where certain contrast agents or colorants are added to the sample in order to improve the images, in the present case, a fully isotropic element is added with the aim of extending the scope of decompositions, which allows certain featured representations to be obtained.

**M**and a perfect depolarizer; Section 4 deals with the introduction of the homogeneous extended form of

**M**and its decomposition, where the constituents exhibit equal mean intensity coefficients and are analyzed from the mathematical, geometric, and physical points of view; Section 5 describes an alternative extended decomposition of

**M**where the constituents, being of similar nature to those of the homogeneous extended decomposition, exhibit different mean intensity coefficients, and have even simpler forms; Section 6 is devoted to the analysis of the extended decompositions for the particular case in which

**M**lacks either polarizance or diattenuation; and the obtained results are discussed in Section 7.

## 2. Theoretical Background

**s**and ${s}^{\prime}$ are the Stokes vectors that represent the states of polarization of the incident and emerging light beams, respectively, whereas

**M**is the Mueller matrix associated with this kind of interaction, which can always be expressed as [8,9,10]:

**M**; the superscript T indicates transpose; ${m}_{00}$ is the mean intensity coefficient (MIC), i.e., the ratio between the intensity of the emerging light and the intensity of incident unpolarized light;

**D**and

**P**are the diattenuation and polarizance vectors, with absolute values D (diattenuation) and P (polarizance); and

**m**is the normalized 3 × 3 submatrix associated with

**M**, which provides the complementary information on retardance and depolarization properties.

**M**to preserve the degree of polarization (DOP) of totally polarized incident light, a proper measure is given by the degree of polarimetric purity of

**M**(also called the depolarization index) [11], ${P}_{\Delta}$, which can be expressed as

**m**.

**M**, its associated Hermitian coherency matrix $C\hspace{0.17em}(M)$ is positive and semidefinite. The explicit expression of $C\hspace{0.17em}(M)$, in terms of the elements ${m}_{ij}$ of

**M**, is [4]:

**X**is a Mueller matrix if and only if it can be expressed as convex sum of pure and passive Mueller matrices, which is equivalent to saying that $C\hspace{0.17em}(X)$ (which, by construction, is a Hermitian matrix) is positive and semidefinite (i.e., the four eigenvalues of $C\hspace{0.17em}(X)$ are nonnegative), and, in addition,

**X**satisfies the passivity condition ${x}_{00}(1+Q)\le 1$ [16].

**M**, but with the incident and emergent directions of the light probe interchanged, is given by [18,19]:

**M**).

**N**is diagonalizable (i.e., there exists an invertible matrix

**A**such that ${A}^{-1}NA$ is diagonal), then

**M**can be written in the type-I normal form [21,22,23,24,25,26,27,28]

**N**is not diagonalizable,

**M**is type-II and it can always be written in the type-II normal form [28]

**M**can be represented geometrically by means of the pair of ellipsoids ${E}_{\Delta P}$ and ${E}_{\Delta D}$ generated by

**M**and ${M}^{r}$, respectively. The canonical depolarizer ${M}_{\Delta}$ (with ${M}_{\Delta}$ representing either ${M}_{\Delta d}$ or ${M}_{\Delta nd}$, depending on whether

**M**is type-I or type-II) is fully characterized by its associated canonical ellipsoid ${E}_{\Delta}$. The use of the three characteristic ellipsoids ${E}_{\Delta P}$, ${E}_{\Delta D}$, and ${E}_{\Delta}$ leads to a complete and significant geometric view of the properties of

**M**[29].

**m**of

**M**,

**m**(taken in decreasing order), so that

**M**is then defined as [30]

**M**is defined as

**M**are recovered from those of ${M}_{A}$ through the respective transformations $D=\hspace{0.17em}{m}_{RI}^{\mathrm{T}}{D}_{A}$ and $P={m}_{RO}{P}_{A}$, which preserve the absolute values of the transformed vectors and are determined by the entrance and exit retarders ${M}_{RI}$ and ${M}_{RO}$ of

**M**.

## 3. Parallel Compositions of a Given Mueller Matrix and That of a Perfect Depolarizer

**M**for which the components exhibit certain essential properties of

**M**in a decoupled and simple manner, the extended form of

**M**is built by adding to

**M**an appropriate proportion of a perfect depolarizer ${M}_{\Delta 0}$.

**M**(depolarizing or not), it is always possible to build the depolarizing Mueller matrix

**I**being the identity matrix),

**M**with respect to the whole composed matrix, the resulting matrix admits certain parallel decompositions which are not realizable for

**M**itself. In particular, as will be shown in the next section, $q=1/3$ constitutes a critical value that ensures the realizability of a parallel decomposition of ${M}^{\prime}$ into three components with very simple structures depending on

**P**,

**D,**and

**m**, respectively, regardless of the value of $\mathrm{rank}C$. Such a decomposition is not possible, in general, when $q>1/3$, whereas $q<1/3$ leads to similar parallel decompositions but with an additional component that is proportional to ${\widehat{M}}_{\Delta 0}$ with the respective coefficient $1-3q$.

## 4. Homogeneous Extended Decomposition of a Mueller Matrix

**M,**where the term homogeneous is used to note that the MICs $({m}_{00})$ of both components are equal to that of

**M,**which is consistent with the name coined for arbitrary decompositions where all components have equal MICs $({m}_{00})$ [3].

**m**,

**P**, and

**D**appear isolated within respective components. The above decomposition will be called the homogeneous extended decomposition of

**M**, which should be interpreted as a parallel composition of media represented by ${M}_{m}$, ${M}_{P}$, and ${M}_{D}$ with equal MICs $({m}_{00})$ and respective portions (or cross sections) equal to 1/3 (i.e., the intensity I of the incident light probe is shared among the three components with equal intensities I/3).

**M**is parameterized through the following sixteen parameters: the MIC,${m}_{00}$, of

**M**; the three angular parameters determining the entrance retarder ${M}_{RI}$; the three angular parameters determining the exit retarder ${M}_{RO}$; the polarizance vector

**P**of

**M**; the diattenuation vector

**D**of

**M**; and the three diagonal elements $({a}_{1},{a}_{2},\epsilon {a}_{3})$ of ${m}_{mA}$.

#### 4.1. Nonenpolarizing Component

**m**of the nonenpolarizing component, ${M}_{m}$, coincides with that of

**M**. Thus, by considering the procedure used to define the arrow form ${M}_{A}(M)$ [30],

**m**can be written as in Equation (10), $m={m}_{RO}\hspace{0.17em}{m}_{A}\hspace{0.17em}{m}_{RI}$, with ${m}_{A}=\mathrm{diag}\hspace{0.17em}({a}_{1},{a}_{2},\epsilon \hspace{0.17em}{a}_{3})$, and therefore ${M}_{m}$ can be expressed through the following dual retarder transformation [31] (which coincides with the normal form of ${M}_{m}$):

**M**.

**C**associated with

**M**. Furthermore, since

**M**is a Mueller matrix,

**C**is positive and semidefinite, and therefore its diagonal elements are necessarily nonnegative; consequently, the eigenvalues ${\lambda}_{mi}$ of the Hermitian matrix ${C}_{m}$ are nonnegative, showing that ${C}_{m}$ is a proper coherency matrix and consequently ${M}_{m}$ is a Mueller matrix.

**M**. Thus, from the point of view of the arbitrary decomposition [3], ${M}_{m}$ has a number ${r}_{m}$ of parallel pure components. Moreover, since ${P}_{P}({M}_{m})=0$, the only source of polarimetric purity of ${M}_{m}$ is ${P}_{S}({M}_{m})={P}_{\Delta}({M}_{m})={P}_{S}(M)$.

**M**entails the passivity of ${M}_{m}$, that is, ${m}_{00}(1+Q)\le 1$ $\Rightarrow {m}_{00}\le 1$, where $Q\equiv \mathrm{max}\hspace{0.17em}(D,P)$. Observe also that Equation (19) shows that ${M}_{m}$ is always a type-I Mueller matrix.

#### 4.2. Enpolarizing Components

**M**[11].

**M**implies the passivity of both ${M}_{P}$ and ${M}_{D}$.

**P**and

**D**[10,32,33,34], ${M}_{R}$ represents an arbitrary retarder (which plays no role in the definitive forms of ${M}_{P}$ and ${M}_{D}$), $\otimes $ stands for the Kronecker product, and ${I}_{3}$ is the 3 × 3 identity matrix.

## 5. Extended Decomposition of a Mueller Matrix

**M**, the MIC of the perfect depolarizer added to

**M**can be taken as ${m}_{00}(P+D)/2$ (instead of ${m}_{00}$), so that the extended form of

**M**is defined as

**M**is then defined as

**M**itself.

## 6. Extended Decompositions of Matrices Lacking Polarizance or Diattenuation

**M**for a well-defined extended form of

**M**, without an excess of ${M}_{\Delta 0}$, changes with respect to the case where $P>0$ and $D>0$. Appropriate particular forms of extended decompositions are analyzed below, which, together those dealt with in Section 4 and Section 5, provide a complete case analysis.

**M**take the following forms, where the denominator of the coefficients (polarimetric cross sections) equals the number of components (two):

## 7. Discussion

**M**involves its convex sum with a perfect depolarizer ${M}_{\Delta 0}$, which does not exhibit any anisotropy (or polarimetric preference). Consequently, the anisotropies inherited by the extended representations ${M}_{H}(M)$ and ${M}_{E}(M)$ are precisely those of

**M**. Furthermore, for any given

**M**, both ${M}_{H}(M)$ and ${M}_{E}(M)$ are biunivocally related to

**M**through simple expressions. In fact, the

**m, P**, and

**D**of ${M}_{H}(M)$ are none other than those of

**M,**whereas the structure of ${M}_{E}(M)$ is given by

**m,**$\widehat{P}$, $\widehat{D}$, P, and D (with $P=P\widehat{P}$ and $D=D\hspace{0.17em}\widehat{D}$).

**M**, and makes it possible to obtain the extended decompositions considered in Section 4, Section 5 and Section 6.

**M**are type-I, and are therefore free from the intricate structure exhibited by type-II Mueller matrices [28,36].

**M**can be straightforwardly interpreted regardless of its structural complexity. When the arrow form of

**M**is considered, the analysis becomes even simpler because in that case, once the entrance and exit retarders have been decoupled from

**M**, ${M}_{m}$ becomes diagonal [37].

## 8. Conclusions

**M**and a perfect depolarizer are always susceptible to be submitted to respective kinds of parallel decompositions named the homogeneous extended decomposition and the extended decomposition, the components of which have very simple structures which are directly inherited from the anisotropies exhibited by

**M**.

**M**and a perfect depolarizer, with appropriate convex coefficients, only affects the MIC (mean intensity coefficient) of the resulting composed matrix, but it is the key for

**M**to be interpreted in terms of the properties of the components of the corresponding extended decompositions. In particular, two components can be straightforwardly determined from the polarizance and diattenuation vectors of

**M**, respectively, whereas the third component depends exclusively on the 3 × 3 submatrix

**m**of

**M**, which encompasses the remaining polarimetric information. That is to say, once the information on polarizance and diattenuation has been decoupled and allocated to respective parallel components, the structure of the remaining nonenpolarizing component allows for the recovery of the complete polarimetric information (including the depolarization and retardance properties) held by

**M**.

**M**(depolarizing or nondepolarizing) is susceptible to being represented, uniquely, through the extended representations ${M}_{H}(M)$ and ${M}_{E}(M)$, and admits respective extended decompositions where the structural properties of

**M**appear decoupled in a very simple manner and encoded into separate components.

## Author Contributions

## Funding

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**Characteristic ellipsoids of the nonenpolarizing component ${M}_{m}$ of the extended form of a Mueller matrix

**M**: (

**a**) forward ellipsoid ${E}_{\Delta P}({M}_{m})$; (

**b**) canonical ellipsoid ${E}_{\Delta}({M}_{m})$, of which the semiaxes ${a}_{1},{a}_{2},{a}_{3}$ (with ${a}_{1}\ge {a}_{2}\ge {a}_{3}$) are aligned with the Poincaré axes ${S}_{1},{S}_{2},{S}_{3}$, respectively; and (

**c**) reverse ellipsoid ${E}_{\Delta D}({M}_{m})$.

**Figure 2.**The characteristic ellipsoids of the polarizing component ${M}_{P}$ of the homogeneous extended form of a Mueller matrix

**M**are given by single points (the three semiaxes are zero-valued): (

**a**) forward ellipsoid, ${E}_{\Delta P}({M}_{P})$, which lies on the surface of the Poincaré sphere if and only if $P=1$; (

**b**) canonical ellipsoid ${E}_{\Delta}({M}_{P})$; and (

**c**) reverse ellipsoid ${E}_{\Delta D}({M}_{P})$.

**Figure 3.**The characteristic ellipsoids of the polarizing component ${M}_{D}$ of the homogeneous extended form of a Mueller matrix

**M**are given by single points (the three semiaxes are zero-valued): (

**a**) forward ellipsoid ${E}_{\Delta P}({M}_{D})$; (

**b**) canonical ellipsoid ${E}_{\Delta}({M}_{D})$; and (

**c**) reverse ellipsoid, ${E}_{\Delta P}({M}_{D})$, which lies on the surface of the Poincaré sphere if and only if $D=1$.

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San José, I.; Gil, J.J. Extended Representation of Mueller Matrices. *Photonics* **2023**, *10*, 93.
https://doi.org/10.3390/photonics10010093

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San José I, Gil JJ. Extended Representation of Mueller Matrices. *Photonics*. 2023; 10(1):93.
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**Chicago/Turabian Style**

San José, Ignacio, and José J. Gil. 2023. "Extended Representation of Mueller Matrices" *Photonics* 10, no. 1: 93.
https://doi.org/10.3390/photonics10010093