Edge Corrections to Electromagnetic Casimir Energies From General-Purpose Mathieu Function Routines

TL;DR

This paper develops a new computational package for Mathieu functions to accurately calculate edge effects in Casimir energies of elliptic cylinders and strips, advancing precision in quantum electromagnetic force predictions.

Contribution

It introduces a general-purpose Mathieu function package capable of complex argument computations, enabling precise edge correction calculations in Casimir energy models.

Findings

Successfully computed edge corrections to Casimir energies.

Enhanced Mathieu function computations for complex arguments.

Validated edge effects against proximity force approximation.

Abstract

Scattering theory methods make it possible to calculate the Casimir energy of a perfectly conducting elliptic cylinder opposite a perfectly conducting plane in terms of Mathieu functions. In the limit of zero radius, the elliptic cylinder becomes a finite-width strip, which allows for the study of edge effects. However, existing packages for computing Mathieu functions are insufficient for this calculation, because none can compute Mathieu functions of both the first and second kind for complex arguments. To address this shortcoming, we have written a general purpose Mathieu function package, based on algorithms developed by Alhargan [1,2]. We use these routines to find edge corrections to the proximity force approximation for the Casimir energy of a perfectly conducting strip opposite a perfectly conducting plane.