A Hamiltonian structure-preserving discretization of Maxwell's equations in nonlinear media

TL;DR

This paper introduces a Hamiltonian-preserving finite element discretization of Maxwell's equations in nonlinear media, ensuring energy stability and conservation laws, facilitating accurate simulations of nonlinear optical phenomena.

Contribution

It presents a novel, energy-stable finite element discretization of Maxwell's equations that preserves Hamiltonian structure and Gauss's laws in nonlinear media.

Findings

Discretization is energy-stable and conserves Gauss's laws.

Framework is adaptable to various nonlinear media.

Enables accurate time-domain simulations of nonlinear optics.

Abstract

A simple Hamiltonian modeling framework for general models in nonlinear optics is given. This framework is specialized to describe the Hamiltonian structure of electromagnetic phenomena in cubicly nonlinear optical media. The model has a simple Poisson bracket structure with the Hamiltonian encoding all of the nonlinear coupling of the fields. The field-independence of the Poisson bracket facilitates a straightforward Hamiltonian structure-preserving discretization using finite element exterior calculus. The generality and relative simplicity of this Hamiltonian framework makes it amenable for simulating a broad class of time-domain nonlinear optical problems. The main contribution of this work is a finite element discretization of Maxwell's equations in cubicly nonlinear media which is energy-stable and exactly conserves Gauss's laws. Moreover, this approach may be readily adapted to consider more general nonlinear media in subsequent work.