Analytical approximations for the dispersion of electromagnetic modes in slabs of biaxial crystals

TL;DR

This paper derives analytical dispersion relations for electromagnetic modes in biaxial crystal slabs, aiding the understanding and design of anisotropic polaritons in advanced nanostructures.

Contribution

It provides the first general dispersion relation and simplified analytical formulas for electromagnetic modes in biaxial slabs, enhancing theoretical understanding and experimental analysis.

Findings

Derived a general dispersion relation for biaxial slabs.

Provided simplified formulas for short wavelength and thin slabs.

Facilitates analysis of anisotropic polaritons in biaxial materials.

Abstract

Anisotropic crystals have recently attracted considerable attention because of their ability to support polaritons with a variety of unique properties, such as hyperbolic dispersion, negative phase velocity, or extreme confinement. Particularly, the biaxial crystal $\alpha$-MoO$_3$ has been demonstrated to support phonon polaritons, light coupled to lattice vibrations, with in-plane anisotropic propagation and unusually long lifetime. However, the lack of theoretical studies on electromagnetic modes in biaxial crystal slabs impedes a complete interpretation of the experimental data, as well as an efficient design of nanostructures supporting such highly anisotropic polaritons. Here, we derive the dispersion relation of electromagnetic modes in biaxial slabs surrounded by semi-infinite isotropic dielectric half-spaces with arbitrary dielectric permittivities. Apart from a general dispersion relation, we provide very simple analytical expressions in typical experiments in nano-optics: the limits of short polaritonic wavelength and/or very thin slabs. The results of our study will allow for an in-depth analysis of anisotropic polaritons in novel biaxial van der Waals materials.