Boundary conditions for interfaces of electromagnetic (photonic) crystals and generalized Ewald-Oseen extinction principle

TL;DR

This paper develops boundary conditions and a generalized extinction principle for electromagnetic crystal interfaces, enabling analytical and numerical solutions for diffraction and reflection in complex photonic structures.

Contribution

It introduces a comprehensive set of boundary conditions and a generalized extinction principle for semi-infinite electromagnetic crystals, including analytical solutions for point dipolar inclusions.

Findings

Derived boundary conditions relating incident, eigenmodes, and scattered waves.

Formulated a generalized Ewald-Oseen extinction principle for photonic crystal interfaces.

Provided analytical formulas for mode amplitudes in dipolar scatterer crystals.

Abstract

The problem of plane-wave diffraction on semi-infinite orthorhombic electromagnetic (photonic) crystals of general kind is considered. Boundary conditions are obtained in the form of infinite system of equations relating amplitudes of incident wave, eigenmodes excited in the crystal and scattered spatial harmonics. Generalized Ewald-Oseen extinction principle is formulated on the base of deduced boundary conditions. The knowledge of properties of infinite crystal's eigenmodes provides option to solve the diffraction problem for the corresponding semi-infinite crystal numerically. In the case when the crystal is formed by small inclusions which can be treated as point dipolar scatterers with fixed direction the problem admits complete rigorous analytical solution. The amplitudes of excited modes and scattered spatial harmonics are expressed in terms of the wave vectors of the infinite crystal by closed-form analytical formulae. The result is applied for study of reflection properties of metamaterial formed by cubic lattice of split-ring resonators.