Electromagnetic Polarization Matrix

• Electromagnetic polarization matrix is a Hermitian tensor that generalizes classical coherency matrices by incorporating full electric, magnetic, and cross-correlations.
• It utilizes 3×3 submatrices to quantify field intensities, directional modes, and energy flux with applications in wave scattering and depolarization.
• The framework underpins resource theories through polarization monotones and informs experimental techniques like Mueller matrix analysis in both structured and disordered media.

The electromagnetic polarization matrix is a Hermitian tensor encapsulating all second-order correlations between electromagnetic field components, including electric–electric, magnetic–magnetic, and electric–magnetic cross-terms. This formalism generalizes the classical electric coherency (polarization) matrix to incorporate the full structure of field correlations in arbitrary, possibly random or nonparaxial, electromagnetic environments. Its spectral and geometric invariants underpin the definition of degrees of polarization, resource-theoretic monotones, chiral densities, Mueller–Jones relations, and depolarization kinetics in structured and disordered media.

1. Fundamental Definitions and Structures

For a statistically stationary electromagnetic field, the most general polarization matrix is defined by ensemble averaging the outer product of the analytic signals of the electric $\mathbf E(t) \in \mathbb{C}^3$ and (scaled) magnetic field $\mathbf B(t) = Z_0 \mathbf H(t) \in \mathbb{C}^3$: $$\Psi(t) = \begin{pmatrix} \mathbf E(t) \ \mathbf B(t) \end{pmatrix} \in \mathbb{C}^6, \quad P_{\text{EM}} = \langle \Psi \Psi^\dagger \rangle = \begin{pmatrix} P_{EE} & P_{EB} \ P_{BE} & P_{BB} \end{pmatrix}$$ with $$P_{EE} = \langle \mathbf E \mathbf E^\dagger \rangle$$, $$P_{BB} = \langle \mathbf B \mathbf B^\dagger \rangle$$, $$P_{EB} = \langle \mathbf E \mathbf B^\dagger \rangle$$, $$P_{BE} = \langle \mathbf B \mathbf E^\dagger \rangle = P_{EB}^\dagger$$ (Gil et al., 2 Dec 2025); see also (Forbes et al., 22 May 2025), which separates the blocks into pure electric, pure magnetic, and cross polarization matrices. The spectrally-resolved version, $$\Phi_{ij}(\omega) = \langle E^*_i(\omega) E_j(\omega) \rangle$$, is widely used for stationary fields (Bosyk et al., 2017).

Such matrices are Hermitian and positive semidefinite. Their trace encodes the total field energy (intensity): $I = \mathrm{Tr}[P]$. For paraxial beams, the theory reduces to the 2×2 coherency/Jones matrix familiar from Stokes calculus (Castillo et al., 2013), but general coherent and partially polarized fields require at least a 3×3 description (Alonso, 2020, Bosyk et al., 2017, Santarsiero et al., 22 Jan 2025).

2. Electric, Magnetic, and Mixed Polarization Submatrices

Each 3×3 block $P_{EE}$ (resp. $P_{BB}$) serves as the covariance matrix for the Cartesian components of the electric (resp. magnetic) field, encoding intensities, cross-correlations, and spin properties: $$P_{EE, ij} = \langle E_i E_j^* \rangle; \quad P_{BB, ij} = \langle B_i B_j^* \rangle.$$ Their eigenvalues reveal the polarization purity and correspond to physically meaningful directional modes. The cross-correlation blocks $P_{EB}$, $P_{BE}$ are fundamental for capturing flow and reactive/chiral coupling, as they combine electric–magnetic coherences that cannot be accessed by $P_{EE}, P_{BB}$ alone—essential for accurate description of energy transport, optical helicity, and chirality in complex or nonparaxial fields (Gil et al., 2 Dec 2025, Forbes et al., 22 May 2025).

Complete field characterizations in cylindrical or longitudinally structured media require expansion over vector mode bases (e.g., Hankel functions for cylindrical sources), as in (Santarsiero et al., 22 Jan 2025), yielding cross-spectral density matrices $W_{ij}(\mathbf{r}_1, \mathbf{r}_2)$ in coordinate space or over basis coefficients.

3. Degrees, Measures, and Resource Theory of Polarization

The spectral polarization matrix gives rise to important polarization monotones. For $d$ field components, the normalized density matrix $\rho = \Phi/\mathrm{Tr}[\Phi]$ admits basis-independent eigenvalues $\lambda_1 \geq \dots \geq \lambda_d$. The majorization (or convex-mixing) resource theory of polarization (Bosyk et al., 2017) defines monotones such as:

• 2D degree of polarization: $$P^{2D}(\rho) = \lambda_1 - \lambda_2 = \sqrt{1 - 4\det\rho}$$.
• 3D purity (Samson–Barakat): $$P_\text{SB}(\rho) = \sqrt{(3/2)\mathrm{Tr}\rho^2 - 1/2}$$.
• Linear gap: $P^{\text{lin}}(\rho) = \lambda_1 - \lambda_3$.
• Von Neumann entropy-based: $P^{\text{vN}}(\rho) = 1 - S(\rho)/\ln 3$; $S(\rho) = -\mathrm{Tr}[\rho\ln\rho]$.

Resource theories rigorously define allowed polarization degradation via physically meaningful “free” operations (unital random unitaries, convex mixing), establishing the partial ordering and convex geometry of polarization space (Bosyk et al., 2017). For 3D fields, the structure becomes a simplex with extreme points for fully polarized, partially unpolarized, and fully unpolarized configurations.

4. Geometric, Spinor, and Tensor Representations

For paraxial (2D) beams, the Jones vector represents the polarization state as an SU(2) spinor, mapped onto the Poincaré sphere via Pauli matrix expansion. The 2×2 coherency matrix $J$: $$J = \frac12(S_0 \sigma_0 + S_1 \sigma_1 + S_2 \sigma_2 + S_3 \sigma_3)$$ yields diametrically opposite principal polarization points on the sphere, with degree of polarization $P = \|\mathbf{S}\|/S_0$ (Castillo et al., 2013). Lossless optical elements act by SU(2) rotations, transforming $J$ via congruence and rotating the Stokes vector on the sphere.

Nonparaxial fields require a 3×3 polarization (coherency) matrix $$\Gamma = \langle \mathbf{E} \mathbf{E}^\dagger \rangle$$ (Alonso, 2020). Its geometric invariants include:

• eigenvalues $\Lambda_n$ (yield barycentric and center-of-mass coordinates for DOP measures),
• spin density $$\mathbf{S} = \operatorname{Im} \langle \mathbf{E}^* \times \mathbf{E} \rangle$$,
• inertia ellipsoid from $\operatorname{Re}\Gamma$, with directors encoding principal axes and strengths,
• two-point Majorana/Hannay or Poincarana representations for fully polarized modes,
• three-point and director constructs for partial polarization.

Stokes–Gell-Mann expansions of $\Gamma$ extend this framework to the full 8D space of nonparaxial polarization parameters, with intricate constraints and visualizations.

5. Physical Interpretation and Applications of the 6×6 Matrix

The electromagnetic polarization matrix $P_\text{EM}$ (Gil et al., 2 Dec 2025) enables extraction of physically distinct quantities: | Matrix | Description | Physical significance | |--------|-----------------------------------|----------------------------| | $CRA$ | Active (Poynting) flux matrix | Time-averaged energy flux | | $CIA$ | Reactive flux matrix | Oscillatory energy exchange| | $CRS$ | In-phase alignment matrix | Synchronous E–B correlations| | $CIS$ | Quadrature alignment matrix | $90^\circ$-phase E–B coupling|

Global indices $p, q, p_u, p_i$ quantify the norms of these matrices, and $c$ summarizes total E–B coupling through the Frobenius metric. The purity of the normalized $P_\text{EM}$ reflects quantum-like separability, with “mixed” and “pure” electromagnetic states characterized analogously to quantum density matrices.

6. Scattering, Mueller Matrices, and Depolarization in Anisotropic Media

Mueller matrix formalism (Ray et al., 2016, Byrnes et al., 2022) provides the operational mechanism by which electromagnetic polarization states transform upon anisotropic scattering. The 4×4 Mueller matrix $M$ relates input and output Stokes vectors: $$M_{ij} = \frac12 \operatorname{Tr}[\sigma_i J \sigma_j J^\dagger].$$ The spectral mapping of Fano resonance asymmetry is encapsulated by key Mueller elements, diattenuation, and retardance, with experimental pre- and post-selection of polarization states enabling control over Fano phase shifts and amplitude ratios. Random-matrix theory models full vector scattering, unitarity, reciprocity, and the statistics of depolarization in multiple-scattering environments (Byrnes et al., 2022).

Disordered media and multiple scattering admit a multi-channel decomposition of the polarization matrix, with diffusion equations governing depolarization, spatial coherence anisotropy, and relaxation rates determined by polarization eigenchannel lifetimes and ladder-operator spectra (Vynck et al., 2013).

7. Chirality, Nonparaxiality, and Extensions

Generalized polarization matrices are required for faithful representation of optical chirality (helicity) in near- and far-field regimes. The mixed electric–magnetic blocks, $$\Phi_{ij}^{M(3D)} = \epsilon_0 c \langle E^*_i B_j \rangle$$, are necessary to compute the helicity density $h$: $$h = \frac{1}{2\omega} \operatorname{Im} \left\{ \delta_{ij} [\Phi_{ij}^{N(3D)} - \Phi_{ij}^{M(3D)}] \right\}$$ enabling analysis of chiral dipole emission and tightly focused beams (Forbes et al., 22 May 2025). Paraxial reduction delivers the standard Stokes description only when longitudinal E/B correlations vanish. In nonparaxial (3D) fields, the need for tensorial extensions is critical for predicting and interpreting phenomena including local helicity, reactive coupling, and chiral light–matter interactions.

8. Methodologies and Experimental Realization

Computation and measurement of polarization matrices require:

• Spectroscopic polarimetry over multiple input/output states (e.g., 16-state Mueller matrix analysis (Ray et al., 2016)).
• Basis expansion (Hankel/cylindrical harmonics) and statistical inference for structured sources (Santarsiero et al., 22 Jan 2025).
• Random-matrix and transfer-matrix cascades for light propagation in disordered media or composite slabs (Byrnes et al., 2022).
• Diagonalization, trace normalization, and invariant extraction for resource-theoretic polarization monotone computation (Bosyk et al., 2017).

Experimental extraction of DOP, Fano parameters, and energy flux indices proceeds via fit to analytic forms and comparison with directional, spin, or coherence observables.

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Electromagnetic polarization matrices thus provide a unified, rigorous, and physically interpretable representation for electromagnetic field correlations, generalizing classical polarization descriptions and exposing the full manifold of polarization, coherence, chirality, scattering, and energy transport in complex media (Gil et al., 2 Dec 2025Forbes et al., 22 May 2025Alonso, 2020Santarsiero et al., 22 Jan 2025Ray et al., 2016Byrnes et al., 20221309.34561709.073071303.44961303.1836).