Gaussian dispersion analysis in the time domain: efficient conversion with Pad\'e approximants

TL;DR

This paper introduces an efficient method to implement Gaussian dispersion analysis in time-domain simulations using Padé approximants, enabling accurate modeling of optical materials with disorder in FDTD solvers.

Contribution

It develops explicit analytical formulas and a universal FDTD implementation for Gaussian dispersion using Padé approximants, facilitating practical time-domain simulations of dispersive media.

Findings

Derived analytical formulas for Gaussian oscillator implementation

Created prototype FDTD codes with automated approximant generation

Demonstrated compatibility with various numerical schemes and solvers

Abstract

We present an approach for adapting the Gaussian dispersion analysis (GDA) of optical materials to time-domain simulations. Within a GDA model, the imaginary part of a measured dielectric function is presented as a sum of Gaussian absorption terms. Such a simple model is valid for materials where inhomogeneous broadening is substantially larger than the homogeneous linewidth. The GDA model is the essential broadband approximation for the dielectric function of many glasses, polymers, and other natural and artificial materials with disorder. However, efficient implementation of this model in time-domain full-wave electromagnetic solvers has never been fully achieved. We start with a causal form of an isolated oscillator with Gaussian-type absorption - Causal Dawson-Gauss oscillator. Then, we derive explicit analytical formulas to implement the Gaussian oscillator in a finite-difference time-domain (FDTD) solver with minimal use of memory and floating point operations. The derivation and FDTD implementation employ our generalized dispersive material (GDM) model - a universal, modular approach to describing optical dispersion with Pad\'e approximants. We share the FDTD prototype codes that include automated generation of the approximants and a universal FDTD dispersion implementation that employs various second-order accurate numerical schemes. The codes can be used with non-commercial solvers and commercial software for time-domain simulations of light propagation in dispersive media, which are experimentally characterized with GDA models.