Fourier transform of the hyperbola and its role in hyperbolic photonics

TL;DR

This paper derives the Fourier transform of hyperbolic dispersion to understand wave emission in hyperbolic media, revealing new insights into hyperbolic wave behavior, negative refraction, and imaging artifacts across various physical systems.

Contribution

It introduces a generalized Fourier analysis of hyperbolic dispersion, providing analytical tools for modeling hyperbolic wave phenomena in anisotropic materials.

Findings

Derived Fourier transform of hyperbolic dispersion.

Connected hyperbolic wave behavior to a generalized Huygens' principle.

Identified aliasing artifacts in polariton imaging.

Abstract

Motivated by recent breakthrough studies of wave hyperbolicity in extremely anisotropic natural materials and artificial composites, we investigate the radiation pattern of a localized emitter in a hyperbolic medium. Since the emission of a point source is associated with the Fourier transform of the iso-frequency contours of a medium, we derive and analyze the properties of the Fourier transform of hyperbolic dispersion, which sheds light into the emission properties in the presence of hyperbolic bands. Our analysis leads to a generalized form of Huygens' principle for hyperbolic waves, connecting to the emergence of negative refraction and focusing with hyperbolic media. We also highlight the occurrence of aliasing artifacts in polariton imaging. More broadly, our findings provide analytical tools to model polariton propagation in materials with extreme anisotropy, and may be applied to several other physical platforms featuring hyperbolic responses, from astrophysics to seismology.