In the second-order approach, the polarization effects of a material medium on the electromagnetic wave that interacts with it are characterized by means of the corresponding Mueller matrix. Beyond its mere role as the 4×4 matrix that transforms the Stokes vectors of the incoming polarization states into the Stokes vectors of the outgoing states, the Mueller matrix contains, in a intricate manner, rich information on the physical nature and properties of the medium.
This work is devoted to identify and interpret the main physical quantities involved in a Mueller matrix. After an analysis of each significant quantity, the notions of components of purity, indices of polarimetric purity as well as the sets of quantities that remain invariant under certain transformations are revisited and interpreted on the basis of a common framework.
For the sake of self-consistency of this work, it is worth considering briefly the concept of Mueller matrix. The transformation of the state of polarization of an electromagnetic beam caused by its linear interaction with a material medium can be formulated mathematically as Ms = s′, where s and s′ are the respective Stokes vectors of the input and output states of polarization and M is the Mueller matrix associated with the medium (for the given particular conditions of interaction). In general, the medium can exhibit dispersive effects or certain degree of heterogeneity over the area illuminated by the incident electromagnetic beam, producing depolarization. Therefore, the emerging electromagnetic wave is composed of a number of incoherent contributions (mutually coherent or incoherent), in such a manner that the polarimetric effect of the medium not always can be represented by means of a Mueller-Jones matrix (deterministic case).
In other words, while some polarimetric interactions have a deterministic nature and can be formulated either by the Jones calculus or by the Stokes-Mueller calculus (in which case the Mueller matrix M is pure, also called nondepolarizing or Mueller-Jones matrix, that is, M is derived from a unique Jones matrix), in the most general case, M is a convex sum, or ensemble average, of pure Mueller matrices (i.e. a linear combination of pure Mueller matrices with positive coefficients that sum to one). This is a consequence of the fact that, essentially, any macroscopic polarimetric effect is due to a sort of scattering by a myriad of elementary electronic excitations, each one with a well-defined Jones matrix, and the whole action of the medium is a statistical composition of such pure interactions. This physical requisite is called the ensemble criterion  (also covariance criterion or Cloude’s criterion ).
Note that the ensemble criterion imposes additional restrictions beyond the fact that a Mueller matrix is a Stokes matrix transforming Stokes vectors into Stokes vectors . A simple example of a Stokes matrix that is not a Mueller matrix is the Minkowski metric G ≡ diag (1, −1, −1, −1), which plays a key role in the Stokes-Mueller algebra [5-12] but has no associated Jones matrix.
Furthermore, leaving aside certain artificial arrangements , the linear polarimetric effects are passive [3,5,14,15], in the sense that the intensity (power density flux) of the output electromagnetic wave never exceeds the intensity of the input wave.
Thus, by considering jointly the ensemble and passivity criteria, we can state that a 4×4 real matrix M is a Mueller matrix if and only if it can be expressed as
where MJi are pure and passive Mueller matrices (i.e. MJi are associated with respective passive Jones matrices). For a more detailed analysis of this subject we refer the reader to Ref. .
Before identifying the relevant physical parameters involved in a general Mueller matrix M, let us first recall that the structure of the information contained in M is closely related to its block expression ,
where M̂ is the normalized version of M.
For some analyses it is also useful to parameterize the submatrix m as follows 
so that M is expressed in terms of m00 and the five constitutive vectors
It is also worth to consider briefly the reciprocity properties of Mueller matrices, that is, the relation between Mand the Mueller matrix Mr that corresponds to the same medium as M, but interchanging input and output directions of the probe wave interacting with it. It has been proved that (leaving aside systems involving magnetooptic effects, whose reciprocity property Mr = M differs from the usual rule) the reciprocal Mueller matrix Mris given by [18,19]
where the superscript T indicates transposed matrix. Therefore, the knowledge of a depolarizing Mueller matrix M fully determines the reciprocal Mueller matrix Mr. In fact, it has been proved that the constraining inequalities that characterize the Mueller matrices are invariant with respect to the change of Mr by MT [5,1], and consequently all the physical conditions for a matrix M to be a Mueller matrix must also be satisfied by Mr and MT. In other words, M is a Mueller matrix if and only if MT (or M r) is a Mueller matrix.
The state of polarization of an electromagnetic beam with a well defined direction of propagation can be represented by means of the corresponding Stokes vector
where I is the intensity, P is the degree of polarization (a measure of how close is the state represented by s to a totally polarized one), while ϕ and χ are respectively the azimuth and ellipticity angle of the average polarization ellipse (called the characteristic polarization ellipse ).
The whole set of polarization states ŝ ≡ s/ I (i.e., states with unit intensity and arbitrary values of P, ϕ and χ) constitute the Poincaré sphere. States with P = 1 (said to be pure, or totally polarized) constitute the surface of the sphere, while states satisfying 0 ≤ P < 1 lie inside the sphere (P = 0 corresponding to the origin).
To inspect some polarimetric properties of M, it will be useful to keep in mind the following transformations, through M and through MT, of an input state ŝ into the respective output states sf and sr
MEAN INTENSITY COEFFICIENT
Let us consider the pair of normalized pure Stokes vectors
The effect of a medium represented by a Mueller matrix M on the following equiprobable incoherent mixture of the states ŝp+ and ŝp−
is given by the outgoing Stokes vector
whose intensity is m00. Since each pure state ŝ p+ has its respective orthogonal state ŝ p−, it turns out that an equiprobable incoherent mixture of all pure states (thus covering the entire surface of the Poincaré sphere) is given by a unpolarized state
and therefore, from (11), the intensity of the corresponding output state is m00. Furthermore, if we consider a unpolarized input state with intensity I, the intensity of the transformed state is m00 I. This fact provides the physical meaning of m00 and justifies calling it the mean intensity coefficient of M (also termed transmittance -or reflectance- or gain).
POLARIZANCE AND DIATTENUATION
A given Mueller matrix transforms an input unpolarized state as shown in Eq. (11), so that the ability of M to polarize input unpolarized states is characterized by the polarizance vector P, whose absolute value P is the degree of polarization of the transformed state and is called the polarizance of M [21,22]
The upper limit P = 1 of P (which is restricted to the interval 0 ≤ P ≤ 1) is achieved for media that fully polarize the input unpolarized states. Thus, a medium satisfying P = 1 is called a polarizer and his associated Mueller matrix MP̂D has the form [23,24]
Moreover, the polarizance of a Mueller matrix M can be expressed as 
where the degree of linear polarizance PL is a measure of the ability of M to transform a unpolarized state into a linearly polarized state (0 ≤ PL ≤ 1), while the degree of circular polarizance PC is a measure of the ability of M to C;transform a unpolarized state into a circularly polarized one (|PC| ≤ 1, with PC = −1 for a left-handed circular polarizer and PC = 1 for a right-handed circular polarizer).
To analyze the physical information involved in the diattenuation vector D, let us observe that, from Eq. (7), the and minimum intensities for the output state sf = Mŝ are given by
which correspond respectively to the following mutually-orthogonal input states
showing that the diattenuation D is a measure of the relative difference between the maximum and minimum intensities of the output states with respect to all possible input polarization states. D is limited by 0 ≤ D ≤ 1, where D = 1 corresponds to media for which there exists an input state (s−D̂) whose corresponding output state has zero intensity. When D = 1 the medium is called an analyzer and has the general form [23,24]
As for polarizance, diattenuation can be expressed as a quadratic average of linear diattenuation DL and circular diattenuation DC 
Let us now consider the reverse Mr and transpose MT Mueller matrices associated with M, and observe that, from the mere definition of P, D, DL, DC, PL, PC, these quantities satisfy
In other words, the polarizance of M is also the diattenuation (or reciprocal polarizance) of Mr and MT, while the diattenuation of M is also the polarizance (or reciprocal diattenuation) of Mr and MT .
Vectors P and D, whose respective physical meanings have been analyzed in Section 3, can be represented in the Poincaré sphere, and the polarizance angle η formed by them and defined as
is an interesting quantity involved in M that remains invariant under certain transformations of M (this subject will be considered in further sections on retarder and rotation transformations of M). In particular, the role played by η in the classification of nondepolarizing Mueller matrices is studied in Ref. .
DEGREE OF POLARIZANCE
As we have seen, despite the fact that both polarizance P and diattenuation D have respective specific physical meanings, they rely on a common dual physical nature, so that, for some purposes, it is useful to define the degree of polarizance PP 
which provides a measure of the combined contribution of P and D to polarimetric purity. PP is restricted to the range 0 ≤ PP ≤ 1, PP = 1 corresponding to a pure polarizer and PP = 0 corresponding to a nonpolarizing Mueller matrix M (with zero diattenuation and zero polarizance).
Concerning the notion of the degree of polarizance, it is worth to recall that, contrary to that it would seem at first sight, PP is related to the ability of the medium represented by M to reduce the degree of polarization of certain input states of polarization . To analyze this feature, let us consider the example of a normal diattenuator  (which is a kind of pure Mueller matrix ), with diattenuation 0 < D < 1, whose Mueller matrix has the form 
The action of MD on the partially polarized state (1,DT)T, whose degree of polarization D equals the diattenuation of MD, produces an output partially polarized state represented by the following Stokes vector
whose degree of polarization is D (1 + D2) < DD and, consequently, the fact that D > 0 implies that MD reduces the degree of polarization of some input partially polarized states.
DEGREE OF SPHERICAL PURITY
so that it is limited by 0 ≤ PS ≤ 1 and constitutes a measure of how close is M to the Mueller matrix of a retarder (in general elliptic, and regardless of the effective value of its retardance) . That is, PS = 1 corresponds to a Mueller matrix whose normalized version is the Mueller matrix MR of a retarder (i.e. and det MR = +1). The lower limit and PS = 0 corresponds to a depolarizing medium satisfying m = 0, which is fully characterized by m00, P and D.
DEGREE OF POLARIMETRIC PURITY
Polarimetric purity refers to the closeness of a Mueller matrix to that of a deterministic nondepolarizing medium (i.e., a pure medium, hence characterized by the fact that it preserves the degree of polarization of any input totally polarized state in both forward and reverse directions). An overall measure of the degree of polarimetric purity of M is given by the depolarization index PΔ defined as 
This quantity is limited by 0 ≤ PΔ ≤ 1, the upper limit PΔ = 1 corresponding to a pure medium (regardless of its particular retardance and diattenuation properties) and the lower limit PΔ = 0 corresponding to an ideal depolarizer with associated Mueller matrix MΔ0 = m00 diag (1, 0, 0, 0).
In accordance with the physical meaning of PΔ, an overall measure of the depolarizing power (or polarimetric randomness) of the medium is given by the depolarizance 
Pure Mueller matrices are characterized by PΔ = 1 (DΔ = 0), while Mueller matrices satisfying PΔ < 1 (DΔ > 0) are called nonpure or depolarizing.
Despite the fact that PΔ constitutes a well-defined overall measure of the polarimetric purity, it has been shown that it does not provide enough information for a complete parameterization of the polarimetric purity of M [4,31]. Such parameterization can be obtained from two complementary sets of quantities called the components of purity and the indices of polarimetric purity which will be described in respective sections.
COMPONENTS OF PURITY
The block expression of a Mueller matrix in Eq. (2) is particularly significant because it reflects relevant properties of M concerning the structure of the physical information contained in it. Accordingly, and from the inspection of the right term of Eq. (26) the quantities P, D and PS, taken as a set, are called the components of purity (hereafter CP) of M. The analysis of the particular values of the CP allows for a classification of Mueller matrices (see Refs. [20,26]).
The parameters PP and PS, taken as a set, are called the sources of purity of M, which represent complementary contributions to the overall polarimetric purity given by PΔ 
INDICES OF POLARIMETRIC PURITY
While the components of purity provide information of how the polarimetric purity is shared among polarizance, diattenuation and spherical purity, a complementary view of how the polarimetric purity (or, conversely, its randomness) is quantitatively structured, is achieved from the eigenvalues of the covariance matrix H associated with M and defined as
where ⊗ stands for the Kronecker product and σk are the Pauli matrices (ordered as usual in Optics)
Since H is a positive semidefinite Hermitian matrix, its eigenvalues are nonnegative, and, taken in decreasing order, are denoted by λi (λ0 ≥ λ1 ≥ λ2 ≥ λ3). Thus, H can be expressed as
where λ̂ ≡ λ (tr H) and HJi are the covariance matrices associated with respective pure Mueller matrices MJi (the subscript J is used to indicate that these matrices are associated with pure Mueller matrices, that is, each HJi has only one nonzero eigenvalue λi). The spectral decomposition of M is formulated as follows in terms of the pure Mueller matrices MJi ≡ m00 M̂Ji with the same mean intensity coefficients  equal to m00 = tr H
The above decomposition means that, from a polarimetric point of view, the medium represented by M is equivalent to a parallel composition [32,33] of the pure media represented by MJi, that is, the medium behaves like if the input electromagnetic beam was shared among a set of pencils interacting separately with the pure components represented by MJi, in such a manner that the emerging pencils recombine incoherently into a whole output beam [3,4]. Consequently, the normalized eigenvalues λ̂i of H represent the relative weights of the pure components MJi in the spectral decomposition.
An objective view of how the polarimetric purity is structured in M is obtained by rearranging the spectral decomposition in the form of the following trivial, or characteristic, decomposition 
where the quantities P1, P2 and P3 are the so-called indices of polarimetric purity (IPP) of M, defined as follows from the normalized eigenvalues of H 
while M̂J0, M̂J1, M̂J2 and M̂Δ0 are normalized Mueller matrices associated with the covariance matrices ĤJ 0, ĤJ1, ĤJ2, and ĤΔ0 defined as
where I is the 4×4 identity matrix. The above expressions allows for the following interpretation of the characteristic components of M:
• MJ1 ≡ m00 M̂J1 is called the characteristic pure component (note that it coincides with the first component of the spectral decomposition) and its relative weight in the characteristic decomposition (34) is precisely the first index of polarimetric purity, or degree of polarization P1 of M.
• M2 ≡ m00 M̂2 represents a 2D depolarizer, and its relative weight is given by the difference P2 − P1.
• M3 ≡ m00 M̂3 represents a 3D depolarizer, and its relative weight is given by the difference P3 − P2.
• MΔ0 ≡ m00 diag (1, 0, 0, 0) represents an ideal depolarizer (or 4D depolarizer), whose relative weight is given by the difference 1 − P3.
The overall degree of polarimetric purity PΔ can be calculated through the following weighted quadratic average of the IPP 
It is remarkable that, from the role played by the IPP in the characteristic decomposition and from the scaled structure of their values 
the IPP constitute a privileged set of parameters providing quantitative information of how the polarimetric purity (or, conversely, the polarimetric randomness) is organized in the medium represented by M. Note that the IPP are insensitive to the specific features of M relative to its CP.
Although IPP and CP provide independent and complementary information, note that the set (P1, P2, P3, P, D) of purity parameters is sufficient to calculate PΔ and PS. The purity figure (Fig. 1) provides a powerful geometric tool to inspect the intricate relations between the IPP and the CP. A detailed analysis of the physical interpretation of both sets of parameters, can be found in Ref. 
The term retarder is used for pure media with zero diattenuation-polarizance, P = D = 0. The Mueller matrix of an ideal retarder MR has the general form
so that a given Mueller matrix M corresponds to a retarder if and only if MT = M−1, that is, if and only if M is orthogonal.
The action of MR can be represented in the Poincaré sphere as a rotation by the angle Δ about the axis defined by the unit Pauli vector uR of MR, so that any input state is transformed into an output state obtained through such rotation. The two eigenstates of MR are given by the mutually orthogonal and totally polarized Stokes vectors
Thus, a given retarder can also be represented by means of its corresponding straight retardance vector R defined as
where uR is the Pauli vector determining the azimuth and ellipticity of the fast eigenstate of MR and Δ is the retardance. Note that R has been defined so as to satisfy 0 ≤ R ≤ 1 which allows for its representation in the Poincaré sphere.
Unlike Mueller matrices of retarders, Mueller matrices (pure or nonpure) with nonzero diattenuation or polarizance, are necessarily associated with media that exhibit forward or reverse diattenuation, which is a property linked to selective absorption (dichroism) or selective reflection, hence producing power loss, in the sense that such effects cannot be compensated by means of serial combinations with additional passive media. Therefore, unlike the action of diattenuators, the action of a retarder produces lossless effects on the interacting electromagnetic wave. In fact, given MR, always exists a passive pure Mueller matrix MR−1 (which also corresponds to a retarder) whose serial combination with the retarder represented by MR, MR−1 MR = I produces a neutral effect.
To get a deeper knowledge of the information contained in M, it is therefore interesting to analyze its possible lossless transformations through its serial combination with other Mueller matrices. The most general form of a lossless transformation of M is that of a dual retarder transformation 
whose physical interpretation is that the medium represented by M is sandwiched by the retarders represented by MR1 (entrance retarder) and MR2 (exit retarder), so that this serial combination is represented by the resulting Mueller matrix M′. Dual retarder transformations preserve important quantities like m00, D, P, PS, P1, P2, P3 and PΔ, thus including those that fully characterize the depolarizing properties of M; this is the reason why Mueller matriceslinked to M through dual retarder transformations are said to be invariant-equivalent to M .
Among the infinite Mueller matrices that are invariant-equivalent to M, it is worth to pay attention to the so-called arrow form MA of M , which is built from the singular value decomposition of the submatrix m of M
where mRI and mRO are proper orthogonal matrices, and mA is the diagonal matrix whose elements are defined as follows from the singular values (a1, a2, a3) of m,
The arrow form MA of M is defined as 
so that the dual retarder transformation M = MRO MA MRI is called the arrow decomposition of M.
The main feature of the arrow form MA is that the six off-diagonal elements of mA are zero, so that the ten nonzero elements of MA provide all the information of M that is invariant under dual retarder transformations. In fact, MA is built from m00 together with (1) the diattenuation vector DA ≡ mRI D, (2) the polarizance vector PA ≡ mTROP, and (3) the spherical vector
whose absolute value equals the degree of spherical purity
Other interesting class of retarder transformations is the single retarder transformation, defined as an orthogonal similarity transformation of M 
whose physical interpretation is that the medium represented by M is sandwiched by two identical retarders whose fast eigenstates are mutually orthogonal.
This section is devoted to dual retarder transformations that can be physically performed by means of rotations of the respective laboratory reference frames used for the representation of the Stokes vectors of the input and output electromagnetic beam .
The matrix MG that transforms a Mueller matrix M into the Mueller matrix of the same medium but with respect to a rotated laboratory reference frame, is commonly called a rotator and has the form
where θ is the angle rotated. Note that MG (θ) is a Mueller matrix. In fact, it coincides with the Mueller matrix of circular retarder with retardance 2θ .
A dual-rotation transformation  is a kind of dual retarder transformation where MR2 and MR1 have the form of respective rotation matrices MR2 = MG (θ2), MR1 = MG (θ1)
By using the five-vector expression of M in terms of the elements of its constitutive vectors D, P, k, r and q, 
the dual rotation transformations can be expressed as 
An especially interesting subclass of dual rotation transformations is that constituted by single rotation transformations , which correspond to the case that the laboratory reference frames for the representation of input and output polarization states are rotated jointly (and, hence, applicable for the case of direct transmission experiments where input and output reference frames coincide)
When expressed in terms of the elements of the five constitutive vectors of M, the single rotation transformation adopts the form 
PHYSICAL PARAMETERIZATION OF A MUELLER MATRIX
The retarder and rotation transformations provide meaningful ways for describing the physical information contained in M in terms of certain sets of quantities.
From the point of view of dual retarder transformations and, in particular, from the arrow decomposition of M, we get that a complete set of sixteen physical parameters is provided by
that is, by
• The mean intensity coefficient m00
• The entrance straight retardance vector RI, which contains all the information of the entrance equivalent retarder through its Pauli vector uI and its retardance ΔI. Thus, RI fully determines the entrance retarding properties of M.
• The exit straight retardance vector RO, which contains all the information of the exit equivalent retarder through its Pauli vector uO and its retardance ΔO. Thus, RO fully determines the exit retarding properties of M.
• The polarizance vector P = mRO PA, which fully determines the polarizance properties of M.
• The three indices of polarimetric purity (P1, P2, P3), which fully determines the quantitative depolarizing properties of M.
As we have seen in previous sections, the quantitative characterization of the polarimetric purity of M is given by the IPP: P1, P2, P3 (which, obviously, also determine the quantitative structure of the polarimetric randomness and depolarizing properties of M), while the qualitative characterization of the polarimetric purity (or of the corresponding sources of depolarization) of M is given by the CP: P, D, PS. Moreover, it has been demonstrated  that any depolarizing Mueller matrix M has an associated reference pure Mueller matrix MJ (M) to which M is reduced when the IPP of M are replaced by P1 = P2 = P3 = 1. That is, the IPP act as regulators of the quantitative structure of the polarimetric purity of M, while the CP provide detailed information on the sources and nature of the polarimetric purity associated with M.
where the central horizontal diattenuator has the form
the entrance equivalent retarder MR1 is determined by a sequence of a linear retarder (with retardance ΔI and whose fast linear eigenstate has azimuth ϕI) and a horizontal linear retarder with retardanceΔ/2, while the exit equivalent retarder MR2 is determined by a sequence of a horizontal linear retarder with retardance Δ/2 and a linear retarder (with retardance ΔO and whose fast linear eigenstate has azimuth ϕO).
Therefore, when the IPP of M are replaced by their values P1 = P2 = P3 = 1 corresponding to total purity, the six parameters associated with the set composed of the entrance and exit retarders, MRI and MRO, of the arrow decomposition of M, collapse into the five parameters ϕI, ΔI, Δ, ϕO, ΔO, while the nine parameters of M A collapse into the two parameters of MDL0 (m00, D). Consequently, contrary to the apparent necessity of nine parameters for the complete integral quantitative and qualitative characterization of the depolarization properties of M, they are fully determined by its five purity parameters P1, P2, P3, P, D. Obviously, the complete joint characterization of retarding, polarizing, diattenuating and depolarizing properties of M is given by R I, RO, P, D, P1, P2, P3 (fifteen parameters that, together with the mean intensity coefficient m00, complete the sixteen physical parameters that fully characterize M).
Thus, the above physical parameterization of M has important consequences for the physical interpretation of Mueller matrices. In fact, the dual retarder transformation that links M and its arrow form MA is reversible in the sense that it does not involve diattenuation effects (the CP of MA are equal to those of M) or changes in the structure of polarimetric purity (the IPP of MA are equal to those of M). Conversely, transformations involving diattenuators or depolarizers necessarily affect to the CP or to the IPP.
Obviously, there are many quantities that can be derived from the above minimum set of sixteen independent parameters in Eq. (54), like, for instance, PΔ, DΔ and PS.
It has been demonstrated  that other alternative parameterizations of M, based on dual retarder transformations, are possible, as for instance the one obtained by replacing P1, P2 and P3 in the set (52) by the scalar quantities PS, det M (whose physical meaning is studied in Refs. [24,38]), and PT mD (whose physical interpretation would require further analysis). Other possible choice is that constituted by
OTHER PARAMETERIZATIONS OF A MUELLER MATRIX
Concerning the parameterization of M in terms of the single retarder transformation, let us observe that a complete set is constituted by 
Note that PT D = PD cosη, where η is the polarizance angle. Nevertheless, the physical interpretation of PT m D, PT mT D and trM is not straightforward and require additional study.
where, in analogy to the linear and circular components of vectors P and D, the linear and circular components of vectors r, q and k have been defined as
And finally, from the single rotation transformation, the following parameterization in terms of sixteen independent parameters is obtained 
The relevant physical information contained in a Mueller matrix M has been identified, decoupled and interpreted on the basis of the arrow decomposition of M:
• The mean intensity coefficient of M (mean transmittance or gain) is given by m00 (one parameter).
• The birefringence properties of M are fully determined by the entrance and exit straight retardation vectors, RI and RO (3+3 parameters).
• The polarizance and diattenuation properties are fully determined by the respective polarizance and diattenuation vectors P and D (3+3 parameters).
• The quantitative structure of the polarimetric purity (or, conversely, the polarimetric randomness) is fully determined by the indices of polarimetric purity P1, P2, P3 (3 parameters).
Complete quantitative and qualitative information on the depolarizing properties of M is provided by the set of five purity parameters P1, P2, P3, P, D, from which other relevant physically invariant quantities relative to polarimetric purity, like PS, PΔ or DΔ can be calculated
Funding. This research was supported by Ministerio de Economía y Competitividad, grants FIS2011-22496 and FIS2014-58303-P, and by Gobierno de Aragón (E99).
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