Notes on Deterministic and Stochastic Approaches in Electromagnetic Information Theory

TL;DR

The paper addresses why deterministic approaches remain highly effective in Electromagnetic Information Theory (EIT) by establishing a fundamental link to stochastic models. It shows that, for spatially uncorrelated, homogeneous sources, the stochastic field’s coherent modes and eigenvalues coincide with the deterministic radiation operator’s left singular functions and singular values, i.e., the eigenpairs satisfy $\lambda_n = \sigma_n^2$ and $\psi_n(\mathbf r) = e^{i\phi_n} u_n(\mathbf r)$. The analysis leverages Hilbert–Schmidt expansions, Karhunen–Loève decomposition, and Wigner-function phase-space methods to connect $N_{\rm DoF}$ across frameworks and to relate it to radiometric metrics like étendue. The results rationalize the success of classical EM methods in EIT and provide a unified perspective on the trade-offs between spatial information and energy transfer in complex propagation environments. This work thereby bridges deterministic field representations and stochastic coherence theory, offering practical guidance for design and analysis in ESIT/modern EIT contexts.

Abstract

This paper investigates the relationship between the Number of Degrees of Freedom ($N_{\rm DoF}$) of the field in deterministic and stochastic source models within Electromagnetic Information Theory (EIT). Our findings demonstrate a fundamental connection between these two approaches. Specifically, we show that a deterministic model and a stochastic model with a spatially incoherent and homogeneous source yield not only the same $N_{\rm DoF}$ but also identical eigenvalues and basis functions for field representation. This key equivalence not only explains the effectiveness of deterministic approaches in EIT but also corroborates the use of classical electromagnetic methods within this new discipline.

Figures (6)

• Figure 1: Geometry of the problem in case of non stochastic source; $\mathbf{r}' \in D$, where $D$ is the the source domain while the $\mathbf{r}$ lies on the observation manifold $S_{obs}$, that in this paper will be a curve or a surface.
• Figure 2: The red curves show the $\xi$ hyperbolas for a linear source of length $\ell = 8 \lambda$ (drawn as a green line). The gray rectangles are "receiving domains".
• Figure 3: From a 'spatial information' point of view, linear source sees the scene in Fig. \ref{['fig:urbanov2a']} in a distorted space.
• Figure 4: Real part of the Wigner distribution along the hyperbola passing through $\bar{\mathbf{r}}=(20 \lambda, 16 \lambda)$; $s$ is the abscissa along the curve normalize to the wavelength, and $k_s$ the component tangent to the curve of the $\mathbf{k}$ vector; close to the source we have a large range of $\mathbf{k}$ vector pointing in different direction, giving a complicated 'power flow' pattern; for large $s$ the component the vector $\mathbf{k}$ tangent to the curve tends to $\beta=2 \pi/\lambda$ and $\mathbf{k}$ tends to be aligned to the curve.
• Figure 5: Spectral degree of coherence amplitude, $|\mu(\mathbf{r}_1, \mathbf{r}_2)|$ (left), and mutual information, $I_{MI}(\mathbf{r}_1, \mathbf{r}_2)$ (right), for a linear source of length $\ell=8 \lambda$. The point $\mathbf{r}_1$, indicated as $\bar{\mathbf{r}}$ in the main text, is marked with a white circle, while $\mathbf{r}_2$ ranges across the entire domain of the figure. The subset $Q_{\xi}$ of hyperbolas of constant $\xi$ is drawn as dashed curves.