Aperiodic photonics of elliptic curves

TL;DR

This paper introduces a new method for designing aperiodic optical media using the complex structure of elliptic curves, enabling advanced control of light scattering and localization for nanophotonics.

Contribution

It links elliptic curve mathematics with optical media design, offering a novel approach to engineer aperiodic structures with unique light transport properties.

Findings

Demonstrates the use of elliptic curves to create aperiodic scattering systems.

Provides a spectral analysis of the scattering properties of these structures.

Shows potential for enhanced light-matter interactions in nanophotonics.

Abstract

In this paper we propose a novel approach to aperiodic order in optical science and technology that leverages the intrinsic structural complexity of certain non-polynomial (hard) problems in number theory and cryptography for the engineering of optical media with novel transport and wave localization properties. In particular, we address structure-property relationships in a large number (900) of light scattering systems that physically manifest the distinctive aperiodic order of elliptic curves and the associated discrete logarithm problem over finite fields. Besides defining an extremely rich subject with profound connections to diverse mathematical areas, elliptic curves offer unprecedented opportunities to engineer light scattering phenomena in aperiodic environments beyond the limitations of traditional random media. Our theoretical analysis combines the interdisciplinary methods of point patterns spatial statistics with the rigorous Green's matrix solution of the multiple wave scattering problem for electric and magnetic dipoles and provides access to the spectral and light scattering properties of novel deterministic aperiodic structures with enhanced light-matter coupling for nanophotonics and metamaterials applications to imaging and spectroscopy.