Fast estimation of propagation constants in crossed gratings

TL;DR

This paper introduces simple, efficient techniques for rapidly estimating propagation constants in crossed gratings, reducing computational costs by avoiding eigenvalue equation solutions, especially for regular dielectric structures.

Contribution

It proposes novel methods to estimate propagation constants directly from the modal matrix, bypassing eigenvalue calculations in Fourier-based modal methods.

Findings

Estimation methods are significantly faster than traditional eigenvalue solutions.

Applicable primarily to regular, lossless dielectric gratings.

Improves computational efficiency in grating analysis.

Abstract

Fourier-based modal methods are among the most effective numerical tools for the accurate analysis of crossed gratings. However, leading to computationally expensive eigenvalue equations significantly restricts their applicability, particularly when large truncation orders are required. The resultant eigenvalues are the longitudinal propagation constants of the grating and play a key role in applying the boundary conditions, as well as in the convergence and stability analyses. This paper aims to propose simple techniques for the fast estimation of propagation constants in crossed gratings, predominantly with no need to solve an eigenvalue equation. In particular, we show that for regular optical gratings comprised of lossless dielectrics, nearly every propagation constant appears on the main diagonal of the modal matrix.