Bounds on Propagation Constants in Dielectric Gratings

Abstract

We establish analytical bounds on the normalized longitudinal propagation constants $\beta =k_z/k_0$ of modes in lossless dielectric diffraction gratings. If ${\epsilon }_{\max }$ denotes the maximum relative permittivity, we show that regular modes (${\beta }^2\in \mathbb{R}$) satisfy ${\beta }^2\le {\epsilon }_{\max }$ despite the lack of a scalar dispersion relation. We further show that the so-called ghost modes (${\beta }^2\notin \mathbb{R}$), admitted by the non-Hermitian nature of the associated eigenproblem, satisfy $|\mathrm{Re}(\beta )|\le \sqrt{{\epsilon }_{\max }}/2$. Both bounds are obtained directly from the Fourier-series formulation of the fully coupled vectorial eigenproblem.

4 pages, no figures