Electromagnetic pulse propagation in passive media by path integral methods

TL;DR

This paper introduces a new path integral-based time domain solver for Maxwell's equations in passive media, offering higher accuracy and no artificial dispersion compared to traditional methods.

Contribution

The paper presents a novel path integral formalism combined with pseudospectral methods for electromagnetic simulations in passive media, improving accuracy and stability.

Findings

No artificial numerical dispersion in the method

Operates at the Nyquist limit with exponential convergence

Exact Gauss law enforcement without extra cost

Abstract

A novel time domain solver of Maxwell's equations in passive (dispersive and absorbing) media is proposed. The method is based on the path integral formalism of quantum theory and entails the use of ({\it i}) the Hamiltonian formalism and ({\it ii}) pseudospectral methods (the fast Fourier transform, in particular) of solving differential equations. In contrast to finite differencing schemes, the path integral based algorithm has no artificial numerical dispersion (dispersive errors), operates at the Nyquist limit (two grid points per shortest wavelength in the wavepacket) and exhibits an exponential convergence as the grid size increases, which, in turn, should lead to a higher accuracy. The Gauss law holds exactly with no extra computational cost. Each time step requires $O(N\log_2 N)$ elementary operations where $N$ is the grid size. It can also be applied to simulations of electromagnetic waves in passive media whose properties are time dependent when conventional stationary (scattering matrix) methods are inapplicable. The stability and accuracy of the algorithm are investigated in detail.