Bounds on Propagation Constants in Dielectric Gratings

RESUMO

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.