Green function for hyperbolic media

TL;DR

This paper analyzes the electromagnetic Green function in hyperbolic media, revealing highly anisotropic dipole emission patterns, the coexistence of different wave polarizations, and a singular term affecting photonic density of states and Purcell effect.

Contribution

It provides a detailed analysis of the Green function in hyperbolic media, highlighting anisotropic emission patterns and the role of singular terms in photonic properties.

Findings

Emission patterns are highly anisotropic with a cross-like shape.

Coexistence of cone-like and elliptical emission patterns due to different polarizations.

Identification of a singular delta-function term influencing the photonic density of states.

Abstract

We revisit the problem of the electromagnetic Green function for homogeneous hyperbolic media, where longitudinal and transverse components of the dielectric permittivity tensor have different signs. We analyze the dipole emission patterns for both dipole orientations with respect to the symmetry axis and for different signs of dielectric constants, and show that the emission pattern is highly anisotropic and has a characteristic cross-like shape: the waves are propagating within a certain cone and are evanescent outside this cone. We demonstrate the coexistence of the cone-like pattern due to emission of the extraordinary TM-polarized waves and elliptical pattern due to emission of ordinary TE-polarized waves. We find a singular complex term in the Green function, proportional to the $\delta-$function and governing the photonic density of states and Purcell effect in hyperbolic media.