Classical two-component coherency matrix

Fix a named, positively oriented SI laboratory chart with coordinates $(x,y,z,t)$ and angular frequency $\omega >0$. Adopt the SI constants and coordinate units of the laboratory convention: $c>0$ is in m s${}^{-1}$, $k_0:=\omega \mathrm{/}c$ is in m${}^{-1}$, and electric-field components are in V m${}^{-1}$. Let $\mathrm{\Pi }=\{(x,y,z_0):(x,y)\in D\}$ be a specified reference plane, where $z_0\in \mathbb{R}$ and $D\subseteq {\mathbb{R}}^2$ is a specified nonempty region. Let $(e_x,e_y,e_z)$ be the positively oriented orthonormal coordinate basis of the chart. The ordered transverse basis of $\mathrm{\Pi }$ is $(e_x,e_y)$ and its normal is $e_z$.

For the paraxial context, adopt $c>0$, $k_0=\omega \mathrm{/}c$, and $0<\delta <1$. The angular-spectrum variable $\xi \in {\mathbb{R}}^2$ is measured in cycles per metre, $q=2\pi \xi$ is in m${}^{-1}$, and the forward propagating cone is $D_{\delta }=\{\xi :\mathrm{\mid }q\mathrm{\mid }\le \delta k_0\}$ with $k_z(\xi )=\sqrt{k_0^2-\mathrm{\mid }q{\mathrm{\mid }}^2}>0$. Here $\mathrm{\mid }q\mathrm{\mid }:=\sqrt{q_x^2+q_y^2}$. The matrix below is defined pointwise for the two-component projection; it does not assert that an arbitrary random superposition is an exact Maxwell field or that the individual-mode longitudinal bound extends to an ensemble without additional hypotheses.

Let $(\mathrm{\Xi },\mathcal{A},\mathbb{P})$ be a specified probability space. Identify ${\mathbb{C}}^2$ with ${\mathbb{R}}^4$ by real and imaginary coordinates and equip it with the Euclidean Borel $\sigma$-algebra; measurable maps to ${\mathbb{C}}^2$ are understood with respect to this $\sigma$-algebra. At each fixed $\rho \in D$, let $Z_{\rho }:\mathrm{\Xi }\to {\mathbb{C}}^2$ be measurable. Let ${\pi }_x,{\pi }_y:{\mathbb{C}}^2\to \mathbb{C}$ be the first and second continuous coordinate projections, and define the coordinate random variables $Z_x(\rho ):={\pi }_x\circ Z_{\rho }:\mathrm{\Xi }\to \mathbb{C}$ and $Z_y(\rho ):={\pi }_y\circ Z_{\rho }:\mathrm{\Xi }\to \mathbb{C}$. Thus $Z_{\rho }(\sigma )=(Z_x(\rho )(\sigma ),Z_y(\rho )(\sigma ){)}^{\mathsf{T}}$ for $\sigma \in \mathrm{\Xi }$. The map $X_{\rho }:\mathrm{\Xi }\to {\mathbb{R}}^4$ is defined by the four real coordinate maps of $Z_{\rho }$; these maps are continuous, so $X_{\rho }$ is measurable as their composition with $Z_{\rho }$. No joint measurability in $\rho$ is assumed or used. Define $X_{\rho }(\sigma ):=(\mathrm{Re}{Z}_x(\rho )(\sigma ),\mathrm{Im}{Z}_x(\rho )(\sigma ),\mathrm{Re}{Z}_y(\rho )(\sigma ),\mathrm{Im}{Z}_y(\rho )(\sigma ))\mathrm{.}$ Since $\mathrm{\mid }{X}_{\rho }(\sigma ){\mathrm{\mid }}^2=\mathrm{\mid }{Z}_x(\rho )(\sigma ){\mathrm{\mid }}^2+\mathrm{\mid }{Z}_y(\rho )(\sigma ){\mathrm{\mid }}^2$, assume $X_{\rho }$ is square-integrable, equivalently $\mathbb{E}[\mathrm{\mid }{Z}_x(\rho ){\mathrm{\mid }}^2+\mathrm{\mid }{Z}_y(\rho ){\mathrm{\mid }}^2]<\mathrm{\infty }$. For $\rho \in D$, $\sigma \in \mathrm{\Xi }$ and $t\in \mathbb{R}$, define the associated transverse real-field model by $E_{\perp }(\rho ,\sigma ,t):=\mathrm{Re}(Z_{\rho }(\sigma )e^{-i\omega t})$, where $\mathrm{Re}$ is applied componentwise. Expectation is taken over this declared classical ensemble and is applied componentwise to complex integrable variables. Regard vectors in ${\mathbb{C}}^2$ as columns. For a column $w=(w_1,w_2{)}^{\mathsf{T}}$ define $w^{†}:=(\overline{w_1},\overline{w_2})$; for a $2\times 2$ matrix $M$ define $M^{†}:={\bar{M}}^{\mathsf{T}}$, where $\bar{M}$ is obtained by conjugating each entry. Put $Z_1(\rho ):=Z_x(\rho )$ and $Z_2(\rho ):=Z_y(\rho )$. Define $\mathrm{\Gamma }(\rho ):=\mathbb{E}[Z_{\rho }{Z}_{\rho }^{†}],{\mathrm{\Gamma }}_{ij}(\rho ):=\mathbb{E}[Z_i(\rho )\overline{Z_j(\rho )}](i,j\in \{1,2\})\mathrm{.}$ Thus $\mathrm{\Gamma }$ is the uncentered second-moment matrix of the two selected components; no zero-mean assumption is made. At each fixed $\rho$, suppressing the argument $\rho$ in these coordinate names, define the Stokes coordinates $S_0:={\mathrm{\Gamma }}_{11}+{\mathrm{\Gamma }}_{22},S_1:={\mathrm{\Gamma }}_{11}-{\mathrm{\Gamma }}_{22},S_2:=2\mathrm{Re}{\mathrm{\Gamma }}_{12},S_3:=-2\mathrm{Im}{\mathrm{\Gamma }}_{12}\mathrm{.}$ When $S_0>0$, define the degree of polarization by $\mathcal{P}:=\frac{\sqrt{S_1^2+S_2^2+S_3^2}}{S_0}\mathrm{.}$ The minus sign in $S_3$ is fixed by the stated phasor convention and the ordered basis $(e_x,e_y)$. In this two-component model, call the ensemble at $\rho$ unpolarized when $S_0>0$ and $\mathrm{\Gamma }=(S_0\mathrm{/}2)I_2$, where $I_2$ is the $2\times 2$ identity matrix; this label concerns second moments and does not say that any realization is a quantum state. For the transverse electric-field interpretation, the $S_j$ have units (V m${}^{-1}$)${}^2$; $S_0$ is a mean-square field coordinate, not an absolutely calibrated irradiance.

The same algebraic construction applies to any ordered pair of square-integrable classical complex amplitudes of a common physical unit; the second moments then have the square of that unit. When the pair labels transverse electric components, the $S_j$ have the polarization interpretation just defined. When it labels two beam channels, the same matrix entries are channel correlations; they are not polarization components.

Polarization of a transverse plane wave: linear, circular and elliptical traces, and handedness

Framework-local SI units, the vacuum constants and the exact/measured status of their values

Paraxial phase and longitudinal-field bounds

Mean vector and covariance matrix of a square-integrable random vector

Real and imaginary parts, complex conjugation, and modulus

Linearity, monotonicity, and the modulus bound for expectation

The matrix is uncentered: unless the ensemble mean vanishes, it is not the centered covariance matrix. The individual-mode longitudinal bound from Paraxial phase and longitudinal-field bounds does not give an aggregate bound for a random superposition, so none is claimed. A detector model, calibration, noise law, finite-sample uncertainty, or equality between time and ensemble averages must be supplied separately.