Polarimetric characterization of light and media

• The coherency matrix provides a spectrum-independent mathematical basis for describing electromagnetic-wave polarization, including light beyond the optical range.Source: §1 · p. 2
• Genuine three-dimensional polarization states occur when both the polarization plane and polarization-ellipse shape fluctuate, requiring treatment beyond fixed-plane polarization.Source: §1 · p. 2
• Linear passive media produce refracted, reflected, diffracted, or scattered radiation and require Jones or Stokes-Mueller models depending on coherence and spectral conditions.Source: §1 · pp. 2–3
• Although Mueller matrices can contain up to sixteen measurable elements, their physical information is not straightforwardly organized into quantities with clear meanings.Source: §1 · pp. 3–4
• Polarimetry supports industrial, medical, scientific, and in-situ nondestructive analysis, while ellipsometry measures selected changes in totally polarized light.Source: §1 · p. 3
• The unified framework defines measurable coherency-matrix coefficients and physical parameters, analyzes polarimetric purity, and characterizes media with sixteen meaningful parameters, ten physically invariant.Source: §1 · p. 4

2 Polarized light

• Quasimonochromatic light is represented by an instantaneous Jones vector whose amplitudes and relative phase describe polarization, with longer-time variation producing partial polarization.For stable amplitudes and relative phase, the electric field traces a fixed polarization ellipse; intensity fluctuations remain compatible with total polarization.Source: §2 · pp. 5–6

2.1 The coherency matrix

• The coherency matrix is a Hermitian 2x2 covariance matrix containing all measurable polarization information, and nonnegative eigenvalues are necessary and sufficient for physical realizability.Its elements are time or, under stationarity and ergodicity, ensemble averages of outer products of instantaneous Jones vectors.Source: §2.1 · pp. 6–7

2.2 The Stokes parameters

• Expanding the coherency matrix in the identity-plus-Pauli basis yields four measurable Stokes parameters that form the Stokes vector and obey the coherency matrix’s nonnegativity restrictions.The Pauli matrices provide a linearly independent, Hermitian, trace-orthogonal basis for 2x2 Hermitian matrices.Source: §2.2 · pp. 7–8

2.3 The purity criterion

• The norm relations provide an objective purity criterion, while the limiting case 2^2Φ_0^2Φ=0 identifies unpolarized light as an equiprobable mixture of states.The supplied equations define the purity limits through norms of the coherency matrix and its components.Source: §2.3 · p. 8

2.4 Physical quantities characterizing the polarized light

• The Stokes vector separates intensity, degree of polarization, and the Pauli vector determining the polarization ellipse’s azimuth and ellipticity; the Poincaré sphere distinguishes pure surface states from mixed interior states.The sphere’s origin represents second-order unpolarized light, while all represented states have normalized intensity equal to 1.Source: §2.4 · pp. 8–9

2.5 Decomposition of mixed states

• Mixed Stokes states admit both a pure-plus-unpolarized decomposition and a spectral decomposition into orthogonal pure states, but infinitely many two-pure-state decompositions are also possible.Once one pure component is chosen, the coefficient and remaining pure component are uniquely determined; this nonuniqueness matters for radar-polarimetric target decompositions.Source: §2.5 · pp. 9–10

2.6 Physical invariants and degree of polarization

• The coherency-matrix formulation expresses polarization through physical parameters including intensity, degree of polarization, azimuth, and ellipticity, while supporting later generalization to 3D polarization and 4D media effects.This parameterization reproduces the characteristic decomposition and motivates returning to the coherency-matrix model beyond Stokes-vector representations.Source: §2.6 · p. 10
• Intensity I and degree of polarization P remain invariant under unitary coherency-matrix transformations and are directly related to its eigenvalues.Pure states have rank-one polarization matrices with P = 1, whereas mixed states have rank two with P < 1.Source: §2.6 · p. 11

2.7 Polarization Entropy

• Polarization entropy S is uniquely determined by P, decreasing from log 2 at P = 0 to 0 at P = 1.Thus, entropy distinguishes mixed from totally polarized light regardless of spectral profile.Source: §2.7 · pp. 11–12

2.8 Scope of the coherency matrix model and some extensions

• Because Stokes parameters use second-order moments, non-Gaussian light requires higher-order moments and specialized measurements involving interferometers and fast detectors.Extensions include wavelet-scale, two-point space-frequency, and spatial-angular Stokes parameters, with corresponding Mueller representations.Source: §2.8 · pp. 12–13
• Classical Stokes descriptions do not fully capture quantum polarization, where uncertainty prevents definite ellipses and some states with P = 0 are not unpolarized.Quantum formulations support the study of coherent, squeezed, number, and phase states, including applications involving entanglement and information processing.Source: §2.8 · p. 13
• Quantum degree of polarization can be defined as the distance between a polarization distribution and the uniform unpolarized distribution, or as a trace distance between coherency and identity matrices.This remains an important open subject for describing quantum field states.Source: §2.8 · p. 13

3. 3D Polarized light

• The 3D framework describes waves with time-varying polarization planes using a generalized Jones vector, 3D polarization matrix, and 3D Stokes parameters.It identifies physical quantities characterizing 3D polarization and emphasizes two invariant parameters for polarimetric purity.Source: §3 · p. 13

3.1. 3D Jones vector

• For totally polarized waves, relative phases remain constant, yielding a fixed polarization plane and an elliptical electric-field trajectory; the 2D model follows when the third component vanishes.The 3D Jones vector represents the three electric-field components while separating physically meaningful phase differences from a global phase.Source: §3.1 · pp. 13–14

3.2. 3D coherency matrix

• The 3D coherency matrix is a positive-semidefinite Hermitian covariance matrix expanded in the identity and eight Gell-Mann matrices, whose nine real coefficients are 3D Stokes parameters.Its nonnegative eigenvalues or nested principal minors provide necessary and sufficient physicality conditions.Source: §3.2 · pp. 15–16

3.3. 3D Stokes parameters

• Unlike the 2D case, 3D Stokes parameters obey constraints involving tr R, tr R^2, and tr R^3, while encoding component, projection, handedness, and plane-orientation information.The Poincaré sphere does not represent general 3D polarization states, and quantum-model interpretations of polarization ellipses no longer apply.Source: §3.3 · pp. 17–18

3.4. Purity criterion for 3x3 coherency matrices

• For 3x3 coherency matrices, total polarimetric purity means complete correlation between field variables, with every mutual degree of coherence satisfying |μ_ij| = 1.The symmetric norm criterion is ||R||_0^2 = ||R||^2 for pure states, while ||R||_0^2 = 3||R||^2 characterizes unpolarized light.Source: §3.4 · pp. 18–19

3.5. Decompositions of the coherency matrix

• 3D coherency matrices are decomposed into pure components through spectral, characteristic, or arbitrary convex decompositions.The spectral decomposition yields pure components, the characteristic decomposition combines pure, discriminating, and unpolarized states, and the arbitrary decomposition generalizes pure-state representations.Source: §3.5 · p. 19, §3.5.2 · p. 20, §3.5.3 · p. 20

3.5.1. The spectral decomposition of the coherency matrix

• The spectral decomposition represents every 3D polarization state as an incoherent superposition of three pure states weighted by the eigenvalues of its polarization density matrix.Degenerate eigenvalues make the eigenvectors within the corresponding invariant subspace non-unique.Source: §3.5.1 · p. 20

3.5.2. The characteristic decomposition of the coherency matrix

• The generalized characteristic decomposition requires a pure state, a discriminating state, and a 3D-unpolarized state to account for all nine independent parameters of R.A pure 3x3 coherency matrix contributes five parameters, a discriminating matrix three, and a 3D-unpolarized matrix one.Source: §3.5.2 · p. 20

3.5.3. The arbitrary decomposition of the coherency matrix

• The arbitrary decomposition expresses any mixed 3D polarized-light state as an incoherent convex superposition of up to three independent pure states.For rank 3, any pure state may be chosen first, the second is constrained, and the third is determined; rank 2 fixes the polarization plane, while spectral decomposition is a special case.Source: §3.5.3 · pp. 20, 24

3.6. Polarimetric purity of 3x3 coherency matrices

• The 3D purity formulation uses two relative differences of coherency-matrix eigenvalues as indices of purity, alongside competing invariant parameterizations in prior work.The section situates this formulation among approaches based on scalar invariants, isotropy, three invariant quantities, and related degree-of-polarization measures.Source: §3.6 · p. 25
• Two indices of polarimetric purity, P_1 and P_2, provide complete invariant information about 3D polarization states, unlike the overall degree P_3D alone.The indices lie in the range 0 to 1 and satisfy the physically feasible-region constraint P_1 ≤ P_2.Source: §3.6 · pp. 26, 29
• The purity components P_l, P_c, and P_d describe linear polarization, circular polarization, and polarization-plane stability, respectively, while IPP provide quantitative information on incoherent-component weights.These components provide additional qualitative information about the state’s physical properties.Source: §3.6 · p. 27
• Every 3D polarization state decomposes incoherently into a pure state, a discriminating state, and a 3D unpolarized state.This characteristic decomposition distinguishes pure radiation from random or noise components and supports treatment of measurement errors for known pure states.Source: §3.6 · pp. 29–30

3.7. Entropy associated with R

• The 3D polarization entropy S_3D is uniquely determined by P_1 and P_2, ranging from S_3D = log 3 = 1 at P_1 = P_2 = 0 to S_3D = 0 at P_1 = P_2 = 1.The maximum corresponds to the fully unpolarized state, while the minimum corresponds to the fully pure state.Source: §3.7 · p. 30
• Two partial entropies, S_2D(P_2) and S_2D(P_1), mirror the purity-index ordering through 0 ≤ S_2D(P_2) ≤ S_2D(P_1) ≤ 1.The parameters I, P_1, and P_2 also form a characteristic set for the polarimetric contrast of a 3 × 3 coherency matrix.Source: §3.7 · p. 30

4. Mathematical representation of the polarimetric effects of material media

• Physical Mueller matrices are completely characterized by covariance conditions and passivity conditions within a parallel-combination model of passive pure elements.The covariance conditions derive from the associated covariance matrix, while passivity requires that the constituent elements do not amplify light intensity.Source: §4 · p. 31

4.1. Basic polarimetric interaction. The Jones matrix

• Pure systems map Jones vectors and coherency matrices to Stokes vectors through Mueller-Jones matrices, with passivity restricting their intensity gain to 0 ≤ g ≤ 1.The Jones model applies when totally polarized incident light remains totally polarized; partially polarized or polychromatic light can nevertheless depolarize in some deterministic systems.Source: §4.1 · pp. 31–33
• Reciprocal Jones systems remain physically realizable under reversal and transposition, whereas magneto-optic systems require special treatment; forward and reverse gains are equal for Mueller-Jones matrices but not generally for Mueller matrices.For a Faraday cell, forward- and counter-propagating Jones matrices are equal, and the maximum reverse transmittance need not equal the forward value for general Mueller matrices.Source: §4.1 · pp. 33–34

4.2. Retarders

• Retarders are unitary birefringent systems, but Jones modeling fails when modal shifts exceed coherence length; integrated depolarization then requires the Mueller formalism.This limitation occurs, for example, in long optical fibers, while Jones methods remain useful for spectral calculations in retarder-based devices.Source: §4.2 · pp. 34–35
• A serial combination of three linear retarders yields a tunable compensator by adjusting the intermediate retarder’s angle, while retarder combinations admit equivalent elliptical or linear forms.The section states that any retarder can be represented using linear retarders and that an elliptical retarder is equivalent to two serial linear retarders.Source: §4.2 · p. 36
• Retarders correspond to unitary Jones or orthogonal Mueller matrices whose action is a Poincaré-sphere rotation about the axis defined by two orthogonal physical eigenstates.Serial retarder combinations remain equivalent to an elliptical retarder, and their Mueller representation uses a 3x3 rotation acting on the vectorial Stokes components.Source: §4.2 · pp. 37–38
• A four-dimensional spherical parameterization represents retarders as solid cylinders whose base projections encode the corresponding Jones eigenvectors.The representation was introduced as an original graphical method for retarders.Source: §4.2 · p. 38

4.3. Diattenuators

• Diattenuators are characterized by eigenvalue structure and by the transmittance and polarizance/diattenuation vector of their Mueller matrix.The eigenvalues are k_1^2, k_2^2, and duplicated k_1k_2; the physical eigenstates have intensity transmittances k_1^2 and k_2^2, while the Mueller matrix is fully determined by m_00 and D.Source: §4.3 · p. 40

4.4. Polar decomposition of pure systems

• Any passive pure Jones system admits two equivalent polar decompositions into a Hermitian diattenuator and a unitary retarder, directly transferring to Mueller–Jones matrices.The allowed parameters provide all information obtainable through polarimetric measurements, while an alternative primed parameter set can characterize the pure system.Source: §4.4 · p. 41
• When the Jones matrix is singular, the polar decomposition is not unique and arbitrary retardance parameters can be selected.For the singular case, one may choose δ = δ′ = 0, yielding a linear equivalent retarder; when nonsingular, the retarder parameters are obtained after applying inverse equivalent-diattenuator matrices.Source: §4.4 · pp. 41–43
• The maximum and minimum transmittances of a pure system are determined by the singular values of J = M M^T and can be expressed through traces and determinants.The maximum transmittance is independent of the mathematical formalism used, and the minimum transmittance has an analogous invariant expression.Source: §4.4 · p. 42
• Every pure component in the polar decomposition is normal or homogeneous, with orthogonal eigenstates; the equivalent diattenuator’s orthogonal eigenstates are a peculiarity of the decomposition.Normal systems have coincident eigenvectors for the equivalent retarder and diattenuator, whereas inhomogeneity is characterized by η, with η = 0 for normal systems and η = 1 at maximum inhomogeneity.Source: §4.4 · p. 43
• The retarder–diattenuator polar decomposition of a pure system is equivalent to the singular value decomposition of its Mueller matrix.This representation distinguishes normal systems, whose eigenvalues coincide with singular values, from nonnormal systems, whose eigenvalues differ from them.Source: §4.4 · p. 43
• For pure systems, the magnitudes of the polarizance and diattenuation vectors are equal.The vector P contains all information about polarizing power for incident unpolarized radiation, while D completely describes the system’s diattenuation power.Source: §4.4 · p. 44

5.1. Construction of a Mueller matrix

• The resulting Mueller matrix is a convex sum of Mueller-Jones matrices, equivalently an ensemble average of pure Mueller matrices with nonnegative probabilities summing to one.The same result follows from either the parallel-element description or the ensemble interpretation.Source: §5 · p. 45, §5.1 · p. 46

5. Interaction of polarized light with non‐deterministic passive optical systems

• Physically realizable Mueller matrices are distinguished from general Stokes matrices by the ensemble criterion, with the former representing physically relevant passive optical systems.Every physical Mueller matrix is a Stokes matrix, but the converse generally does not hold.Source: §5 · p. 45

5.1. Construction of a Mueller matrix

• A depolarizing optical system is modeled as a parallel combination of deterministic nondepolarizing elements, each described by a Jones matrix, whose incoherent outputs form the emerging beam.This construction applies to systems with spatial inhomogeneity or dispersive effects that cannot be represented by one Jones matrix.Source: §5.1 · pp. 45–46

5.2. Coherency matrix associated with a Mueller matrix and covariance conditions

• The Hermitian coherency matrix H provides the central physicality test: its four eigenvalues must be nonnegative, together with the passivity condition that maximum transmittance does not exceed unity.The eigenvalue criterion derives from the ensemble model, while passivity constrains the gain for all incident polarization states.Source: §5.2 · p. 47
• Four nested principal minors of H yield simpler necessary and sufficient covariance conditions for physical Mueller matrices, supporting analysis of Mueller-matrix structure, measurements, and physical models.These conditions are equivalent to requiring H to be positive semidefinite.Source: §5.2 · pp. 47–48
• The coherency matrix expands in a Hermitian, trace-orthogonal basis whose sixteen real coefficients are the Mueller-matrix elements, giving them the mathematical status of 4D Stokes parameters.This establishes an analogy with the relation between the 2×2 coherency matrix and ordinary Stokes parameters.Source: §5.2 · pp. 48–50
• The 4×4 coherency matrix H also contains the light models: eliminating extra variables recovers the 2D coherency matrix and permits restriction to the 3D model.The 2D model appears as the upper-left 2×2 matrix, while the 3D restriction has more complicated expressions because of basis asymmetry.Source: §5.2 · pp. 49–50

5.3. Passivity conditions (transmittance conditions)

• For an ensemble of passive pure Mueller matrices, the derived transmittance criteria impose both forward and reverse passivity conditions.The derivation assumes the Mueller matrix is an ensemble average and every realization is passive.Source: §5.3 · p. 50
• The conditions include the bounds f_r^2 ≤ 1 and g_r^2 ≤ 1, with f_r and g_r defined from sums of Mueller-matrix elements.These are the demonstrated pair of passivity inequalities for the forward and reverse directions.Source: §5.3 · p. 50
• Reverse passivity corresponds to the maximal transmittance for light passing through the system in the reverse direction.Direct application of the integral passivity condition requires transmittance never to exceed 1.Source: §5.3 · pp. 50–51
• A matrix satisfying covariance and forward passivity but violating reverse passivity cannot be obtained as an average of pure Mueller matrices and is therefore unrealizable.The example shows that covariance and forward passivity alone are insufficient for physical realizability.Source: §5.3 · pp. 50–51

Characterization theorem

• A real 4×4 matrix is a Mueller matrix if and only if H(M) has four nonnegative eigenvalues and M satisfies the two passivity conditions.This theorem provides necessary and sufficient conditions for physical realizability.Source: §5.4 · p. 51
• Equivalently, Mueller-matrix realizability is specified by six inequalities: four covariance conditions and two passivity conditions.The explicit formulation is given by the covariance conditions in Eq. (167) together with passivity conditions in Eqs. (176) and (177).Source: §5.4 · p. 51

Scope

• The theorem applies to physically realizable Jones and Mueller matrices, a qualification emphasized because terminology and definitions differ across the literature.The physical-realizability qualification prevents confusion when interpreting matrix classes.Source: §5.4 · p. 51

Experimental interpretation

• The characterization constrains experimental Mueller matrices, supports consistency checks despite measurement errors, and enables interpretation through parameters with specific physical meaning.General constraints define the range of possible expectations for measured optical systems.Source: §5.4 · p. 51

Mueller and Stokes matrices

• Every Mueller matrix is a Stokes matrix, but the converse fails because some Stokes matrices generate vectors that are not Stokes vectors; therefore, Stokes-matrix inequalities are necessary consequences of Mueller characterization.The counterexamples include D_X(1,0,1,−1) and D_Y(1,1,1,0), whose corresponding s_03 and s_12 vectors are not Stokes vectors.Source: §5.4 · p. 53

Purity criterion

• A Mueller matrix is Mueller–Jones when its coherency matrix has only one nonzero eigenvalue; under convex-sum and passivity conditions, Eq. (187) is also sufficient.Contrary claims arise from real 4x4 matrices that fail the required eigenvalue or passivity conditions.Source: §5.5 · pp. 53–54

Degree of polarimetric purity

• The degree of polarimetric purity P_Δ is an invariant parameter ranging from 0 for an ideal total depolarizer to 1 for a pure system.P_Δ = 0 corresponds to equal coherency-matrix eigenvalues and no polarimetric preference, whereas P_Δ = 1 corresponds to one positive eigenvalue and three zero eigenvalues.Source: §5.5 · p. 54

Depolarizance

• P_Δ objectively measures global polarimetric purity and depolarizing power, while the depolarizance is defined as D_Δ = 1 − P_Δ^2.D_Δ represents the average depolarization produced over all incident pure states and supports analysis of measured Mueller matrices.Source: §5.5 · p. 54

Comparison with alternative depolarization measures

• Unlike the direct-propagation average degree of polarization A, invariant P_Δ treats direct and reciprocal Mueller matrices symmetrically and gives D_Δ a geometric interpretation as distance from an ideal depolarizer.The distinction is especially significant for asymmetric serial arrangements of diattenuators and depolarizers.Source: §5.5 · p. 54

5.6. Parallel decompositions of matrices representing material media

• Physically realizable parallel decompositions require convex combinations of passive components whose incident-light portions recombine into the complete emerging beam.The convex combination preserves the passive behavior of the whole material system.Source: §5.6 · p. 55

5.6.1. The spectral decomposition of the Mueller matrix of a material medium

• Any linear material system decomposes spectrally into up to four pure systems, with weights proportional to the nonnegative eigenvalues of its coherency matrix.Repeated eigenvalues make the associated orthonormal eigenvectors nonunique, while the eigenvalues admit probabilistic interpretations relevant to entropy and polarimetric-purity indices.Source: §5.6.1 · p. 55

5.6.2. The characteristic decomposition of the Mueller matrix of a material medium

• The characteristic decomposition represents any material state as up to four equal-mean-transmittance components: pure, 2D unpolarized, 3D unpolarized, and 4D unpolarized systems.The number of pure components equals rank(H), and a nonpure system generally cannot be written as only a pure plus fully random component.Source: §5.6.2 · pp. 56–57
• Characteristic and arbitrary decompositions distinguish pure from random components, support experimental-error treatment, and enable iterative subtraction of known pure components to isolate unknown sample contributions.In imaging polarimetry, subtraction can improve contrast between target elements and a homogeneous substrate.Source: §5.6.2 · pp. 56, 58
• Only decompositions whose components satisfy passivity are physically realizable; mathematically valid convex decompositions can instead contain nonpassive components.The ideal-depolarizer example demonstrates that one arbitrary decomposition violates passivity, whereas an alternative decomposition satisfies it.Source: §5.6.2 · pp. 58–59

• Because material interactions change intensity and degree of polarization, normalized Poincaré-sphere mapping alone cannot directly represent output states.The Poincaré sphere represents unit-intensity states, with radius equal to degree of polarization, but material targets generally alter both quantities.Source: §5.7 · p. 59
• The P-image maps every input point of the solid Poincaré sphere to an output Stokes state, whose radial distance gives the corresponding degree of polarization.Its boundary, the DoP surface, represents totally polarized input states.Source: §5.7 · p. 59
• Together, the P-image and I-image provide geometric representations for classifying distinct polarimetric behaviors of material media.The I-image is defined as a surface associated with a given Mueller matrix.Source: §5.7 · p. 60
• Normal diattenuators preserve a spherical P-image of radius 1 while producing a nonspherical I-image and pole concentration linked to maximum and minimum polarizance vectors.The Mueller matrix is symmetric, and the segment joining the poles intersects the image of the Poincaré-sphere origin.Source: §5.7 · p. 60
• Retarders rotate both objects without morphological or topological deformation, whereas statistical mixtures deform the images with behavior determined by the component types.Mixtures of diattenuators displace the P-image origin and deform both images; mixtures of retarders deform only the P-image while preserving its origin; combined mixtures deform both and displace the origin.Source: §5.7 · p. 60

Definition and interpretation

• Three invariant indices of polarimetric purity, P1, P2, and P3, capture all information about the relative eigenvalue weights, whereas 4D P and polarization entropy provide only global purity measures.The indices are derived from relative differences among the four eigenvalues of H, and the global measures can be calculated from them.Source: §5.8 · pp. 60, 62
• The indices have probabilistic interpretations: P1 compares the two dominant pure components, P2 contrasts dominant and subordinate pairs, and P3 compares the two subordinate components.Together, these relative measures provide complete information about the relative amounts of the equivalent pure components.Source: §5.8 · p. 61

Limiting cases

• Pure systems satisfy P1 = P2 = 1 and P3 = 0, whereas P1 = P2 = P3 = PΔ = 0 describes an equiprobable mixture producing an ideal depolarizing Mueller matrix.In the latter case, the Mueller matrix elements vanish except for m00.Source: §5.8 · p. 62

Feasible purity space

• Fig. 11 defines the feasible purity space, whose P3 = 0 plane reproduces the 3D region and whose 2D degree-of-polarization range is the segment 0 ≤ P1 ≤ 1 with P2 = 1.For fixed P1, the maximum P3 is (1 − P1)/3, reached when P2 = (1 + P1)/2.Source: Figure 11 · p. 62, §5.8 · p. 62

Physical cases

• The feasible faces and edges correspond to mixtures of two, three, or four pure elements, with equal or unequal eigenvalue cross sections determining whether states reproduce lower-dimensional representations.States on the P3 = 0 face are mathematically equivalent to 3D coherency matrices, while several other faces are exclusive to the 4D representation.Source: §5.8 · pp. 63–64

Complementary purity descriptions

• The IPP quantify the purity structure, while diattenuation, polarizance, and spherical purity provide complementary qualitative information about the contributions to Mueller-matrix purity.The two descriptions are linked by a relation using the updated IPP expression.Source: §5.8 · p. 64

Polarization entropy

• Polarization entropy S [57] measures material-sample depolarization and is commonly applied in light scattering, SAR polarimetry, and related problems.S is defined from the eigenvalues of the normalized coherency density matrix.Source: §5.9 · p. 64
• IPP provide more complete information about depolarization and polarimetric purity than S and P_Δ, while S and P_Δ can be obtained directly from IPP.The entropy expression can be written explicitly in terms of IPP by expressing coherency-matrix eigenvalues through the definitive 4D IPP.Source: §5.9 · pp. 64–65
• Relations between P_Δ and S impose universal constraints on all classical and quantum scattering media, with applications including quantum communication where depolarization corresponds to decoherence.Source: §5.9 · p. 64
• Shannon entropy and Kullback relative entropy for optical waves relate to n-dimensional degrees of polarization and can also analyze polarimetric purity in material systems.Source: §5.9 · pp. 64–65
• Partial entropies can be extended from 3×3 to 4×4 coherency matrices through quantities involving P_1, P_2, and P_3, with the third exclusive to n×n matrices for n≥4.Source: §5.9 · p. 65

5.10. Unified polarization algebra

• Coherency matrices contain all measurable polarization information, while expansion in trace-orthogonal Hermitian matrices yields the corresponding nD Stokes parameters.The basis comprises SU(n) traceless generators plus the identity, and the expansion coefficients are measurable system quantities.Source: §5.10 · p. 65
• The purity criterion is Ω^2_0 = Ω_0^2, which is necessary and sufficient for a coherency matrix to represent a pure system.Here Ω_0^2 is defined from the trace-based matrix norm.Source: §5.10 · pp. 65–66
• The degree of purity provides a global invariant measure, ranging from 0 for a fully random system to 1 for a pure system.The associated indices of purity organize detailed invariant information and offer a simpler alternative to entropy-like characterization.Source: §5.10 · p. 66
• Complete polarimetric-purity characterization requires n−1 indices of purity, constrained by the nested limits 0 ≤ P_1 ≤ ... ≤ P_{n−1} ≤ 1.These invariant quantities apply to any problem modeled by n×n positive semidefinite Hermitian matrices.Source: §5.10 · p. 66
• Spectral, characteristic, and arbitrary coherency-matrix decompositions are generalized, with the latter two highlighted as underused approaches having high potential in polarimetry and other fields.The framework is linked to the broader algebraic relation between coherency matrices and SU(n).Source: §5.10 · pp. 65–66

Macroscopic polarimetric behaviors of material media.

• Polarizance P and diattenuation D are bounded between 0 and 1 and encode forward and reverse polarimetric behavior, respectively.All diattenuation information is contained in the Mueller matrix’s first row, while polarizance information is contained in its first column.Source: §5.11 · p. 67
• Pure Mueller-Jones systems satisfy P_Δ = 1 and P = D, whereas P ≠ D implies nonpurity; however, P = D alone is not sufficient to establish Mueller-Jones purity.Total purity is obtained when P = D = 1.Source: §5.11 · p. 68
• Incoherent mixtures of pure systems can depolarize while combining depolarization with retardation and polarization properties.The section begins by considering systems formed from incoherent parallel combinations of pure elements.Source: §5.11 · p. 68
• A depolarizing polarizer has P = 1 and 0 < D < 1, while a depolarizing analyzer has 0 < P < 1 and D = 1; each can result from placing an ideal depolarizer between polarizing elements.The analyzer is the reciprocal configuration of the depolarizing polarizer, with the ordering of total and partial polarizers reversed.Source: §5.11 · p. 68
• Systems with 0 < P < 1 and 0 < D < 1 exhibit forward polarizance, reverse polarizance, and depolarization, as modeled by a partial polarizer–depolarizer–partial polarizer sequence.When P = D within this range, the equality does not determine whether depolarization occurs; global purity is given by P_Δ.Source: §5.11 · p. 68
• A system with P = D = 1 can behave as pure even when depolarizing elements lie between two total polarizers, because the intermediate depolarizer changes only total transmittance and leaves P_Δ = 1.Thus, an ideal depolarizer between two polarizers is equivalent in this respect to a pure system.Source: §5.11 · p. 68

5.12.1. The Lu‐Chipman decomposition

• The Lu–Chipman construction identifies the polarizing-depolarizer and retarder as the symmetric and orthogonal factors from the polar decomposition of the diattenuation-free matrix.The retarder is a proper rotation, while the polarizing-depolarizer is symmetric; generally, the diattenuator and polarizing-depolarizer do not individually satisfy passivity.Source: §5.12.1 · p. 70
• For a passive realizable decomposition, when Eq. (233) is satisfied, the Mueller matrix M can be decomposed as a product of a diattenuator, retarder, and polarizing-depolarizer.Source: §5.12.1 · p. 70

5.12.2. The forward and reverse generalized polar decompositions

• Forward generalized polar decompositions always yield Stokes-matrix components, whereas reverse orderings may produce unphysical components and require explicit passivity checks.Transposition preserves the nature of pure components but replaces polarizing-depolarizers with diattenuating-depolarizers; the latter can exhibit nonzero forward and reverse polarizance.Source: §5.12.2 · pp. 71–72
• Any Mueller matrix is polarimetrically equivalent to an ordered serial combination of an output pure system, a canonical depolarizer, and an input pure system.Applying polar decompositions to the pure components yields a symmetric representation with two retarders, two normalized diattenuators, and a canonical depolarizer.Source: §5.12.2 · p. 73

Kernel form

• The kernel decomposition rearranges the SVD-based factors so that all three components are Mueller matrices satisfying passivity and covariance conditions.The kernel matrix is a product of Mueller matrices, unlike the original symmetric and orthogonal factors, which are generally not Mueller matrices.Source: §5.13 · pp. 74–75

Arrow form

• The arrow form lacks retardance and contains information on depolarization, diattenuation, polarizance, and mean transmittance.It is obtained through the singular value decomposition of the 3x3 submatrix of the Mueller matrix and belongs to a family generated by dual retarder transformations.Source: §5.13 · p. 75

Physical invariants

• Retardation properties can always be isolated through the input and output retarders, whereas diattenuation, polarizance, and depolarization generally remain coupled.The input and output retarders are characterized by retardance plus eigenstate azimuth and ellipticity.Source: §5.13 · p. 76
• The decomposition describes a Mueller matrix through mean intensity transmittance, input and output retarders, intrinsic diattenuation and polarizance vectors, and three spherical purity components.The purity components equal the semiaxes of the intrinsic ellipsoid, with SP = a_1^2 + a_2^2 + a_3^2.Source: §5.13 · p. 76

Matrix-factor properties

• The serial factors U and V are non-Mueller overpolarizing transformations, whereas their combination with the physical retarder factors always yields a Mueller matrix.The associated depolarizing factor is strictly depolarizing except for the degenerate retarder case, while U and V can produce DoP values exceeding one.Source: §5.14 · pp. 76–77

Invariant quantities

• The arrow form preserves M’s purity indices and unpolarized transmittance, while its polarizance magnitudes remain invariant even though the original polarizance vectors do not.The physical invariants are therefore associated with Z or the arrow form rather than with the coordinate-dependent vectors P and D of M.Source: §5.14 · pp. 77–78

Arrow decomposition

• The arrow decomposition expresses M as output and input retarders surrounding its arrow form, separating six birefringence parameters from ten dual-retarder invariants.The six parameters comprise input and output birefringence; the ten invariants include unpolarized transmittance, three purity indices, and intrinsic polarizance vectors.Source: §5.14 · p. 78

Physical invariants

• The invariant set provides an objective description of material samples through mean transmittance, coherency-matrix purity indices, and intrinsic polarizance vectors.The purity indices characterize the system’s polarimetric purity, while the intrinsic vectors distinguish physical invariants from the non-invariant polarizance vectors of M.Source: §5.14 · pp. 77–78

Structural theorems

• The arrow-form theorems characterize special cases: diagonal M occurs exactly when intrinsic polarizances vanish, and vanishing polarizances yield a retarder parallel combination.These conditions also identify a ten-parameter parallel-retarder representation; an ideal depolarizer can likewise be synthesized from parallel retarders.Source: §5.14 · p. 78

• Polarimetry supports light-scattering applications including remote sensing, lidar, atmospheric and surface studies, aerosol and hydrosol analysis, particle sizing, and characterization of contaminants and microorganisms.Source: §6 · p. 79
• Optical fiber and photonic-device polarimetry enables communication systems, polarization-mode-dispersion control, and fiber-optic sensing [143,146,223‐228].Source: §6 · p. 79
• SAR polarimetry serves airborne and spaceborne remote sensing, imaging, and detection across vegetation, agriculture, forestry, sea ice, terrain, meteorology, disasters, spills, and buried or immersed targets.Source: §6 · p. 79
• Medical and biological polarimetry supports detection and imaging of immunological reactions and tissues, optical coherence tomography, eye studies, oral-precancer research, and DNA-structure investigations [232‐236].Source: §6 · p. 79
• Additional applications extend to microgravity experimentation, photoelasticity, astronomy and astrophysics, plasma and nuclear-fusion research, microelectronics metrology, quantum physics, LCD technologies, layered media, and microwave transmission systems.Source: §6 · p. 79
• Optics-industry and research applications include fabrication, optical-component characterization, optical-system analysis, ray tracing, and spectral-filter design and fabrication [248‐253], alongside imaging polarimetry [31,254‐257,233,239,241].Source: §6 · p. 79