Casimir Energies of Periodic Dielectric Gratings

TL;DR

This paper presents a novel computational method for calculating Casimir energies of periodic dielectric gratings, especially effective for deep corrugations, by modeling electromagnetic reflection as a multichannel scattering problem and using a generalized Helmholtz equation.

Contribution

It introduces a variable phase method for efficient Casimir energy computation of complex gratings, extending applicability beyond Rayleigh expansion limitations.

Findings

Successfully computed Casimir energies for sinusoidal gratings

Method handles deep corrugations effectively

Provides energy variation with separation and displacement

Abstract

Reflection of electromagnetic waves from a periodic grating can be described in terms of a discrete coupled multichannel scattering problem. By modeling the grating as a space- and frequency-dependent dielectric, it is possible to use a variable phase method, applied to a generalized Helmholtz equation incorporating both transverse and longitudinal modes, to efficiently compute the scattering $S$-matrix. The projection onto transverse modes of this result, evaluated for imaginary wave vector, provides the information necessary for a Casimir energy calculation. This approach is of particular interest for gratings with deep corrugations, which can limit the applicability of techniques based on the Rayleigh expansion. We demonstrate the method by calculating the Casimir interaction energy between sinusoidal grating profiles as a function of separation and lateral displacement.