Bounds on Propagation Constants in Dielectric Gratings

We establish analytical bounds on the normalized longitudinal propagation constants $β=k_z/k_0$ of modes in lossless dielectric diffraction gratings. If $ε_{\max}$ denotes the maximum relative permittivity, we show that regular modes ($β^2\in\mathbb{R}$) satisfy $β^2\leε_{\max}$ despite the lack of