Perturbation theory for anisotropic dielectric interfaces, and application to sub-pixel smoothing of discretized numerical methods

TL;DR

This paper develops a first-order perturbation theory for anisotropic dielectric interfaces, providing a surface integral expression that improves accuracy in boundary perturbation analysis and enhances numerical simulation methods.

Contribution

It introduces a correct perturbation approach for anisotropic interfaces and applies it to improve sub-pixel smoothing in numerical electromagnetism simulations.

Findings

Derived a surface integral formula for boundary perturbations in anisotropic media

Reduced numerical errors in discretized simulations using the new smoothing scheme

Applicable to modeling fabrication imperfections in metamaterials

Abstract

We derive a correct first-order perturbation theory in electromagnetism for cases where an interface between two anisotropic dielectric materials is slightly shifted. Most previous perturbative methods give incorrect results for this case, even to lowest order, because of the complicated discontinuous boundary conditions on the electric field at such an interface. Our final expression is simply a surface integral, over the material interface, of the continuous field components from the unperturbed structure. The derivation is based on a "localized" coordinate-transformation technique, which avoids both the problem of field discontinuities and the challenge of constructing an explicit coordinate transformation by taking a limit in which a coordinate perturbation is infinitesimally localized around the boundary. Not only is our result potentially useful in evaluating boundary perturbations, e.g. from fabrication imperfections, in highly anisotropic media such as many metamaterials, but it also has a direct application in numerical electromagnetism. In particular, we show how it leads to a sub-pixel smoothing scheme to ameliorate staircasing effects in discretized simulations of anisotropic media, in such a way as to greatly reduce the numerical errors compared to other proposed smoothing schemes.