Recursion relations for generalized Fresnel coefficients: Casimir force in a planar cavity

TL;DR

This paper introduces a generalized recursion relation approach for calculating Fresnel coefficients in multilayer systems, enabling more versatile and accurate computation of optical properties and Casimir forces in complex media.

Contribution

It presents a novel recursion relation method involving stacks of layers, applicable to diverse multilayer structures including nonlocal and inhomogeneous media.

Findings

Derived algorithms for Bragg mirror reflectivity

Extended Casimir force formula to arbitrary media

Validated approach for complex multilayer systems

Abstract

We emphasize and demonstrate that, besides using the usual recursion relations involving successive layers, generalized Fresnel coefficients of a multilayer can equivalently be calculated using the recursion relations involving stacks of layers, as introduced some time ago [M. S. Tomas, Phys. Rev. A 51, 2545 (1995)]. Moreover, since the definition of the generalized Fresnel coefficients employed does not imply properties of the stacks, these nonstandard recursion relations can be used to calculate Fresnel coefficients not only for local systems but also for a general multilayer consisting of various types (local, nonlocal, inhomogeneous etc.) of layers. Their utility is illustrated by deriving a few simple algorithms for calculating the reflectivity of a Bragg mirror and extending the formula for the Casimir force in a planar cavity to arbitrary media.