Analytical prediction for the optical matrix

TL;DR

This paper derives an analytical expression for the average scattering matrix of a locally periodic structure, linking it to the transport properties of a single cell and confirming the distribution with numerical simulations.

Contribution

It introduces an analytical method to predict the average optical S matrix for periodic structures, bypassing experimental measurements.

Findings

Analytical expression for the average S matrix derived.

Distribution of S matrix matches numerical simulations.

Average S depends only on single-cell transport properties.

Abstract

Contrary to praxis, we provide an analytical expression, for a physical locally periodic structure, of the average $\langle S\rangle$ of the scattering matrix, called optical $S$ matrix in the nuclear physics jargon, and fundamentally present in all scattering processes. This is done with the help of a strictly analogous nonlinear dynamical mapping where iteration time is the number $N$ of scatterers. The ergodic property of chaotic attractors implies the existence and analyticity of $\langle S\rangle$. We find that the optical $S$ matrix depends only on the transport properties of a single cell, and that the Poisson kernel is the distribution of the scattering matrix $S_N$ in the large size limit $N\rightarrow \infty$. The theoretical distribution shows perfect agreement with numerical results for a chain of delta potentials. A consequence of our findings is the a priori knowledge of $\langle S\rangle$ without resort to experimental data.