Non-linear eigenvalue problems and applications to photonic crystals

TL;DR

This paper develops comprehensive spectral analysis tools for non-linear eigenvalue problems in photonic crystals, including variational principles, bounds, and numerical methods, applicable to complex dielectric models.

Contribution

It introduces new analytic and numerical techniques for spectral analysis of rational operator Nevanlinna functions in photonic crystal modeling, including multi-pole Lorentz models.

Findings

Established complete spectral analysis with variational principles and bounds.

Derived explicit bounds on band gaps near Lorentz poles.

Implemented high-order finite element methods for eigenvalue estimation.

Abstract

We establish new analytic and numerical results on a general class of rational operator Nevanlinna functions that arise e.g. in modelling photonic crystals. The capability of these dielectric nano-structured materials to control the flow of light depends on specific features of their eigenvalues. Our results provide a complete spectral analysis including variational principles and two-sided estimates for all eigenvalues along with numerical implementations. They even apply to multi-pole Lorentz models of permittivity functions and to the eigenvalues between the poles where classical min-max variational principles fail completely. In particular, we show that our abstract two-sided eigenvalue estimates are optimal and we derive explicit bounds on the band gap above a Lorentz pole. A high order finite element method is used to compute the two-sided estimates of a selection of eigenvalues for several concrete Lorentz models, e.g. polaritonic materials and multi-pole models.