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Polarimetric characterization of light and media

José J. Gil

arXiv:2002.04105v2physics.optics

TL;DR

The paper addresses how to give physical meaning to matrix descriptions of polarized light and material media, including cases involving three-dimensional states and depolarization. It develops a unified coherency-matrix framework, characterizes purity and Mueller matrices, and decomposes media into physically interpretable components. The resulting framework identifies measurable and invariant quantities for analyzing experimental and industrial polarimetry.

  • Problem

    The paper seeks a complete physical interpretation of polarization states and the sixteen elements of Mueller transformation matrices, including depolarizing interactions.

  • Method

    The paper uses coherency matrices to represent light and media, and models linear media through physically realizable passive, deterministic, nondepolarizing components and their ensemble behavior.

  • Results

    The framework defines polarimetric-purity quantities, characterizes Mueller matrices through coherency-matrix eigenvalues and passivity conditions, and identifies physically invariant medium parameters.

  • Takeaways & Limitations

    The unified quantities and decompositions support physical interpretation and analysis of experimental and industrial polarimetry.

  • Takeaways & Limitations

    Jones modeling does not cover depolarizing interactions such as long fibers whose modal shift exceeds the light beam’s coherence length.

Abstract

from arXiv · show

An analysis of the matrix models representing the polarimetric properties of light and material media is carried out by using the concept of the coherency matrix, which leads to the identification and definition of their corresponding physical quantities. For light, cases of homogeneous and inhomogeneous wavefront are analyzed, and a model for 3D polarimetric purity is formulated. For linear passive material media, a general model is developed on the basis that any physically realizable linear transformation of Stokes vectors is equivalent to an ensemble average of passive, deterministic, nondepolarizing transformations. Through this framework, the relevant physical quantities, including the indices of polarimetric purity, are identified and decoupled. Some decompositions of the whole system into a set of well-defined components are considered, as well as techniques for isolating the unknown components by means of new procedures for subtracting coherency matrices. These results and methods constitute a powerful tool for analyzing and exploiting experimental and industrial polarimetry. Some particular application examples are indicated.

1 Introduction

The paper unifies polarimetric descriptions of light and media through coherency matrices, addressing polarization states, material interactions, and physically meaningful parameters. It emphasizes that complete characterization requires models beyond deterministic Jones transformations.

  • Motivation: Coherency matrices provide a general mathematical description of electromagnetic-wave polarization across different spectral regions.The framework applies to light and, with contextual substitutions, microwaves, X-rays, and gamma rays.
  • Polarization states: Polarization states differ according to whether the polarization plane and ellipse shape remain stable or fluctuate.Fixed plane and ellipse correspond to fully polarized light; fluctuations produce partially polarized or genuine three-dimensional states.
  • Polarization states: Three-dimensional polarization states require treatment distinct from light with a fixed polarization plane.Both the polarization plane and ellipse shape may fluctuate in these states.
  • Material media: Linear passive media transform electric-field variables without amplifying light intensity and can produce refracted, reflected, diffracted, or scattered radiation.These effects arise from oscillatory molecular charges and secondary radiation.
  • Material media: Mueller matrices characterize up to sixteen measurable polarimetric parameters of material samples under specified interaction and illumination conditions.Complete Mueller-matrix measurement uses a polarization-state generator and analyzer with multiple independent configurations.
  • Paper framework: The review defines physical parameters through a unified coherency-matrix framework and analyzes polarimetric purity of three-dimensional light and material media.It also characterizes retarding, polarizing, and depolarizing properties using sixteen independent parameters, ten of which are physically invariant.

2 Polarized light

The paper models polarized light with instantaneous Jones vectors and coherency matrices, extending the description from fixed-plane states to partially polarized and three-dimensional states. It relates eigenstructure and invariant quantities to state purity and mixed-state decompositions.

  • Jones representation: The Jones vector represents the time-dependent amplitudes and relative phase of the electric-field components.For quasimonochromatic waves, its slow time dependence allows the polarization ellipse to remain effectively fixed over intervals shorter than the coherence time.
  • Jones representation: Totally polarized light is modeled by a Jones vector, while partially polarized light requires a model capturing variation of ellipse azimuth or ellipticity.Coherent superpositions of totally polarized beams are represented by sums of their Jones vectors.
  • Coherency matrix: The coherency matrix contains all measurable polarization information, including intensity, and is formed by time-averaging the outer-product construction of the instantaneous Jones vector.Under stationarity and ergodicity, time averaging can be interpreted as ensemble averaging.
  • Coherency matrix: A coherency matrix is a Hermitian covariance matrix whose nonnegative eigenvalues provide necessary and sufficient physicality conditions.Its normalized form contains information about populations and coherences of polarization states.
  • Coherency matrix: The Pauli matrices plus the identity form a trace-orthogonal Hermitian basis for expanding the 2×2 coherency matrix with real coefficients.This basis connects measurable coefficients to the matrix representation of polarization.
  • Mixed-state decompositions: Mixed states admit infinitely many convex decompositions into two pure states, while three-dimensional mixed states can be represented as incoherent superpositions of up to three pure states.For rank 2, the polarization plane is fixed and the state reduces to a two-dimensional partially polarized state.
  • Physical quantities: Intensity and degree of polarization are invariant under unitary changes of the reference system and are directly related to coherency-matrix eigenvalues.Polarization entropy decreases monotonically as degree of polarization increases, reaching its minimum for totally polarized light.

3. 3D Polarized light

The 3D coherency-matrix model extends polarization analysis to unstable polarization planes, defining invariant purity measures and decompositions for mixed states. It distinguishes pure, discriminating, unpolarized, and intermediate states through eigenvalues and purity indices.

  • 3D polarization model: The 3D polarization matrix and 3D Stokes parameters describe waves whose polarization plane is not stable in time.The model defines and interprets the principal physical quantities of these states, including three-dimensional polarimetric purity.
  • 3D polarization model: A fixed polarization plane reduces the 3D model to the conventional 2D description by setting the third Jones-vector component to zero.In this case, the electric-field endpoint traces a stable ellipse in the polarization plane.
  • Purity criteria: Pure 3D states satisfy R^2−(tr R)R=0, whereas 3D-unpolarized light satisfies R^2−(tr R)R/3=0.The first criterion identifies a single nonzero eigenvalue; the second corresponds to three equal eigenvalues.
  • Spectral and arbitrary decompositions: Every 3D polarization state is an incoherent superposition of three pure states weighted by the eigenvalues of its polarization density matrix.The coherency matrix is the convex combination of the coherency matrices generated by the corresponding eigenvectors.
  • Spectral and arbitrary decompositions: A generalized characteristic decomposition combines a pure state, a discriminating state, and a 3D-unpolarized state, accounting for all nine independent parameters of R.Their parameter counts are five, three, and one, respectively.
  • Spectral and arbitrary decompositions: Any mixed 3D state can be represented as an incoherent superposition of up to three independent pure states, with the number of components equal to rank R.For rank R=3, the first pure component may be chosen arbitrarily, while later choices are progressively constrained; rank R=2 yields a fixed polarization plane.
  • Purity indices: The invariant 3D purity index P_3^D ranges from 0 to 1, spanning total polarimetric purity and a fully unpolarized equiprobable mixture.P_3^D=1 corresponds to one nonzero eigenvalue, while P_3^D=0 corresponds to three equal eigenvalues and zero field-component correlation.
  • Purity indices: Because three eigenvalues exist, the overall index P_3^D does not fully characterize 3D purity; two invariant purity indices are therefore required.These indices provide complete information and separate quantitative contributions to the state’s polarimetric purity.

4. Mathematical representation of the polarimetric effects of material media

The section represents polarimetric transformations of material media with Jones and Mueller formalisms, unified through coherency matrices. It characterizes pure systems, their decompositions, and the conditions governing physical Mueller matrices.

  • General representation: A 4×4 coherency matrix provides the mathematical characterization needed to interpret the sixteen elements of a Mueller transformation matrix.Its properties are analogous to lower-dimensional coherency matrices used for light polarization states.
  • Pure and depolarizing systems: Pure systems transform pure states into pure states and exhibit birefringence and diattenuation, whereas incoherent superposition of emerging pencils produces depolarization.The Jones formalism represents deterministic nondepolarizing interactions, while depolarizing cases require the Mueller formalism.
  • General representation: Physical Mueller matrices can be characterized by explicit necessary and sufficient conditions combining covariance constraints with passivity conditions.The passivity conditions enforce that the constituent elements do not amplify light intensity.
  • Pure-system decompositions: Any polarimetrically pure system can be represented by a Jones matrix, and any pure system is polarimetrically equivalent to serial retarders and diattenuators.A serial combination of three linear retarders also provides a method for designing tunable compensators by adjusting the intermediate retarder.
  • Diattenuators: A diattenuator is completely characterized by its unpolarized-light transmittance and its polarizance/diattenuation vector.Its eigenvalues include the intensity transmittances associated with the two physical eigenstates.
  • Pure-system decompositions: Polar decomposition expresses any passive Jones matrix as a product of a Hermitian diattenuator and a unitary retarder.The equivalent retarder–diattenuator model corresponds to the singular value decomposition of the Jones matrix.

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

The section models depolarizing optical systems as parallel combinations or ensemble averages of deterministic, nondepolarizing components, using coherency matrices to characterize their measurable constraints and structure.

  • The resulting Mueller matrix is a convex sum of Mueller-Jones matrices, equivalently an ensemble average of pure Mueller matrices.
  • Physical Mueller matrices form a stricter set than Stokes matrices: every Mueller matrix is a Stokes matrix, but the converse generally fails.
  • Depolarization can arise from spatial inhomogeneity and dispersive effects, preventing representation of the system by a single Jones matrix.
  • The system is modeled as a parallel combination of polarimetrically pure components, each described by a well-defined Jones matrix.
  • Nonnegativity of the nested principal minors of H provides explicit covariance conditions useful for analyzing measurements and physical models.
  • The coherency-matrix formulation connects Mueller-matrix elements with physical quantities through an expansion in Hermitian trace-orthogonal matrices.

5.3. Passivity conditions (transmittance conditions)

The section completes the physical constraints on Mueller matrices by requiring passivity in both forward and reverse directions, in addition to covariance conditions.

  • A physically realizable Mueller matrix must satisfy passivity conditions derived from an ensemble of passive pure Mueller matrices.
  • The transmittance constraints are 0 ≤ r_f^2 ≤ 1 and 0 ≤ r_g^2 ≤ 1.
  • Reverse transmittance is interpreted through r_g, the maximal transmittance for light passing through the system in the reverse direction.
  • The forward condition alone is insufficient: a matrix can satisfy covariance and forward passivity while violating reverse passivity and therefore fail to be physically realizable.

5.4. Characterization theorem for Mueller matrices

The characterization theorem states that a real 4×4 matrix is a Mueller matrix exactly when its coherency matrix has nonnegative eigenvalues and the matrix satisfies both passivity conditions.

  • A real 4×4 matrix M is a Mueller matrix if and only if all four eigenvalues of H(M) are nonnegative and both passivity conditions hold.
  • Equivalently, physical Mueller matrices satisfy six inequalities: four covariance conditions and two passivity conditions.
  • The theorem assumes physically realizable Jones and Mueller matrices, distinguishing them from broader terminology used in the literature.
  • The constraints provide expected ranges for experimental Mueller matrices and a consistency check against real optical systems.
  • Some quantities constructed from Mueller matrices cannot be obtained by direct Stokes-vector transformations, reflecting the specific structure of Mueller matrices.
  • Inequalities established for Stokes matrices are necessary conditions inherited from the broader characterization of Mueller matrices.

5.5. The purity criterion for Mueller matrices and the degree of polarimetric purity of material media

The section links purity and depolarization to the eigenvalue structure of the coherency matrix H, yielding criteria for pure systems and a bounded global purity index.

  • The purity criterion is grounded in the coherency-matrix properties underlying the question of whether a Mueller matrix is a pure Mueller-Jones matrix.
  • A 4×4 coherency matrix corresponds to a Mueller-Jones matrix exactly when it has only one nonzero eigenvalue.
  • The opposite eigenvalue limit corresponds to an equiprobable mixture of pure elements.
  • The degree of polarimetric purity P_Δ lies between 0 and 1, with 0 denoting an ideal total depolarizer and 1 denoting a pure system.
  • P_Δ measures global polarimetric purity and depolarizing power, supporting analysis of measured Mueller matrices.
  • The depolarizance is defined as D_Δ = 1 − P_Δ^2 and can be interpreted as an average depolarization measure over incident pure states.

5.6. Parallel decompositions of matrices representing material media

The section develops physically realizable parallel decompositions of material-media coherency matrices, emphasizing spectral and characteristic forms and arbitrary decompositions for isolating components.

  • A material medium’s coherency matrix H is represented as a convex combination of passive components, corresponding to shared incident light and recombined emerging beams.
  • Spectral decomposition: H’s spectral decomposition expresses any linear system as a parallel combination of up to four pure systems weighted proportionally to H’s eigenvalues.
  • Characteristic decomposition: The characteristic decomposition represents any material system as up to four components with equal mean transmittances: pure, 2D-unpolarized, 3D-unpolarized, and 4D-unpolarized systems.
  • Characteristic decomposition: In general, a nonpure 4D system cannot be decomposed into only a pure component and a 4D unpolarized component.
  • Arbitrary decomposition: Arbitrary decomposition enables subtraction of known pure components, iterative isolation of unknown parts, and improved contrast of target elements in imaging polarimetry.
  • Physical realizability: Not every mathematically possible arbitrary decomposition is physically realizable because its component systems may violate passivity.

5.7. Geometric maps of the degree of polarization

The section uses Poincaré-sphere mappings to visualize how material media transform output intensity and degree of polarization, distinguishing characteristic behaviors of pure and mixed systems.

  • The P-image maps input Stokes vectors to output states, with each point’s distance from the origin giving its degree of polarization.
  • The DoP surface is the image of totally polarized input states, while the full solid Poincaré sphere includes all input states.
  • Retarders: Retarders rotate the P- and I-images without deforming them, around the axis defined by the retarder’s antipodal eigenstates.
  • Diattenuator mixtures: Statistical mixtures of diattenuators deform the P-image and displace its origin, while also deforming the I-image.
  • Retarder mixtures: Statistical mixtures of retarders deform the P-image while leaving its origin unchanged and preserving the I-image shape.
  • Combined mixtures: Mixtures of diattenuators and retarders deform the P-image, displace its origin, and deform the I-image.

5.8. Indices of polarimetric purity

The section defines three invariant indices of polarimetric purity from the eigenvalues of the 4×4 coherency matrix, providing structural information beyond global purity measures.

  • Three invariant indices P1, P2, and P3 capture all information about the polarimetric purity structure of a material medium.
  • P1, P2, and P3 quantify relative differences between the dominant components, dominant and weaker pairs, and weaker components, respectively.
  • Particular cases: Pure systems satisfy PΔ = P1 = P2 = 1 and P3 = 0, while an ideal depolarizer has PΔ = P1 = P2 = P3 = 0.
  • The IPP provide complete purity information, whereas the degree of polarimetric purity gives only a global measure.
  • Feasible region: The feasible purity region includes lower-dimensional cases corresponding to 3D and 2D coherency matrices through eigenvalue equalities and restrictions such as P3 = 0.
  • Complementary quantities: The components of purity—diattenuation, polarizance, and degree of spherical purity—complement IPP with qualitative information about the medium.

5.9. Polarization entropy

The section extends entropy analysis to 4×4 coherency matrices and relates polarization entropy and global purity measures to the indices of polarimetric purity.

  • Experimental and industrial polarimetry measures up to 16 parameters constrained by complicated nonlinear relations, motivating entropy-based analysis.
  • Entropy and IPP: The indices of polarimetric purity contain broader depolarization information than entropy and global purity, from which both can be calculated.
  • Scattering constraints: Universal constraints on polarization entropy and global purity apply to classical and quantum scattering processes and relate depolarization to decoherence.
  • Partial entropies: Partial entropies are defined for 4×4 coherency matrices as functions associated with P1, P2, and P3, with the latter exclusive to dimensions n ≥ 4.
  • Entropy and IPP: The eigenvalues of the coherency density matrix yield an explicit expression for polarization entropy in terms of the indices of purity.

5.10. Unified polarization algebra

The coherency-matrix framework unifies the description of polarimetric states and systems, providing measurable Stokes quantities and invariant indices of polarimetric purity.

  • Coherency matrices contain the physical measurable information describing polarization states and support a unified polarimetric representation.
  • Expanding an n×n coherency matrix in trace-orthogonal Hermitian matrices produces nD Stokes parameters that completely characterize the system.
  • The degree of purity provides a global invariant measure ranging from 0 for fully random systems to 1 for pure systems.
  • For an n×n coherency matrix, n−1 indices of purity are necessary for complete characterization of polarimetric purity.
  • Spectral, characteristic, and arbitrary coherency-matrix decompositions provide alternative ways to represent polarimetric systems, with the latter two emphasized as underused applications.

5.11. Macroscopic polarimetric behaviors of material media.

The section models material-media behavior through spectral and integrated Stokes–Mueller descriptions, distinguishing polarizance, diattenuation, depolarization, and purity across representative systems.

  • An incoherent mixture of pure elements provides a model for overall material systems, while spectral Mueller–Jones matrices describe frequency-dependent transformations.
  • Integrating the spectral output over frequency yields the resulting Stokes vector and an integrated Mueller matrix.
  • The first row of a Mueller matrix contains diattenuation information, whereas the first column contains polarizance information.
  • Pure Mueller–Jones systems do not depolarize totally polarized input, while incoherent combinations of pure systems produce depolarization.
  • Equal polarizances do not determine whether a system depolarizes; the global purity is given by PΔ.
  • An ideal depolarizer between two total polarizers can behave as pure, with PΔ=1, despite the intermediate depolarizer affecting total transmittance.

5.12. The generalized polar decomposition

The generalized polar decomposition separates diattenuation, retardation, and polarizing-depolarizing behavior while imposing physical passivity conditions on the resulting components.

  • The generalized polar decomposition identifies a symmetric polarizing-depolarizer component and a proper rotational retarder component from the polar decomposition of a reduced matrix.
  • The conventional decomposition may assign the overall coefficient so that a polarizer-depolarizer component fails the reverse passivity condition.
  • A nonsingular Mueller matrix can be decomposed into a diattenuator, retarder, and polarizing-depolarizer when the passive-realizability condition is satisfied.
  • Singular cases include systems with maximum polarizance or diattenuation and require separate treatment in the decomposition.
  • Transposition preserves the nature of pure components while replacing polarizing-depolarizers with diattenuating-depolarizers.
  • Any Mueller matrix is polarimetrically equivalent to an output pure system, a canonical depolarizer, and an input pure system in serial combination.

5.13. The Kernel and the arrow forms of a Mueller matrix

Kernel and arrow forms reorganize Mueller-matrix factors to isolate invariant physical information while ensuring that the kernel decomposition uses physically realizable Mueller-matrix components.

  • Direct singular-value decomposition generally yields orthogonal factors that are not Mueller matrices, except for special diagonal cases.
  • Algebraic rearrangement produces a kernel matrix that is a product of Mueller matrices and therefore satisfies passivity and covariance conditions.
  • The kernel form represents the Mueller matrix through physically valid retarder-related components and a kernel component.
  • The arrow form is obtained through singular-value decomposition of the 3×3 submatrix and lacks retardance, retaining diattenuation, polarizance, and depolarization information.
  • Diattenuation, polarizance, and depolarization generally remain coupled, whereas retardation can always be isolated using input and output retarders.

5.14. Physical invariants of a Mueller matrix

The arrow decomposition separates a Mueller matrix into input and output birefringence and ten quantities invariant under dual retarder transformations. These invariants include transmittance, purity indices, and intrinsic polarizance vectors for characterizing material samples.

  • Matrix properties: The decomposed matrix ΔD is depolarizing, except in the degenerate retarder case, while U and V produce overpolarizing effects.For ΔD, the purity index is less than one except when the diagonal values coincide with the retarder case.
  • Decomposition: The arrow decomposition represents M as input and output retarder transformations surrounding its arrow form.This decomposition provides the basis for identifying the physical quantities embedded in M.
  • Decomposition: The decomposition separates six parameters describing overall birefringence into input and output birefringence.The input and output contributions are specified by the respective retarder transformations.
  • Invariants: Ten parameters remain invariant under dual retarder transformations of M.These invariant quantities characterize properties independent of the corresponding input and output retarder transformations.
  • Invariants: The invariant set includes unpolarized-light transmittance, three purity indices, and intrinsic input and output polarizance vectors.The purity indices characterize the system’s polarimetric purity, while the intrinsic polarizances are distinct from the non-invariant polarizance vectors of M.
  • Applications: These physical invariants provide a basis for objectively characterizing material samples in experimental polarimetry.The cited set includes both coherency-matrix invariants and physical invariants derived from the Mueller matrix.

6. Some applications of polarimetry

Polarimetry supports a broad range of scientific, medical, and industrial applications. These applications use polarization measurements for sensing, imaging, characterization, detection, and metrology across diverse media and environments.

  • Sensing and scattering: Light-scattering polarimetry supports remote sensing, lidar, atmospheric studies, particle sizing, contaminant characterization, and surface characterization.Applications include aerosols, hydrosols, surface roughness, and biological microorganisms.
  • Sensing and scattering: SAR polarimetry is applied to airborne and spaceborne remote sensing, imaging, detection, agriculture, forestry, meteorology, and mapping.Listed targets include vegetation, sea ice, oil spills, buried targets, and devastated areas.
  • Medicine and photonics: Polarimetry is used in medicine and biology to study, detect, and image tissues, immunological reactions, the eye, oral precancer, DNA structure, and optical coherence tomography.The application list also includes optical fiber and photonic-device characterization and polarization-mode-dispersion control.
  • Optics and industry: Polarimetric techniques are used in optics research and industry for fabrication, component characterization, optical-system analysis, ray tracing, and spectral-filter design.Imaging polarimetry is also identified as a distinct application area.
  • Other fields: Additional applications include plasma and particle physics, microelectronics metrology, quantum physics, LCD technologies, layered media, microwave transmission, photoelasticity, and astronomy.Astronomical applications include X-ray and solar polarimetry, planetary atmospheres, and black-hole studies.
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