Multilayer optical calculations

TL;DR

This paper derives comprehensive formulas for multilayer optical calculations, including oblique incidence, absorption profiles, and ellipsometry, addressing gaps in existing literature and supporting the Python 'tmm' software.

Contribution

It provides explicit, consistent derivations of transfer-matrix formulas for multilayer optics, including incoherent layers and complex absorption effects, enhancing theoretical understanding and implementation.

Findings

Derived formulas for multilayer optical properties including oblique angles.

Addressed absorption effects and strange phenomena like T>1 without gain.

Provided implementation details for complex-plane branch cuts.

Abstract

When light hits a multilayer planar stack, it is reflected, refracted, and absorbed in a way that can be derived from the Fresnel equations. The analysis is treated in many textbooks, and implemented in many software programs, but certain aspects of it are difficult to find explicitly and consistently worked out in the literature. Here, we derive the formulas underlying the transfer-matrix method of calculating the optical properties of these stacks, including oblique-angle incidence, absorption-vs-position profiles, and ellipsometry parameters. We discuss and explain some strange consequences of the formulas in the situation where the incident and/or final (semi-infinite) medium are absorptive, such as calculating $T>1$ in the absence of gain. We also discuss some implementation details like complex-plane branch cuts. Finally, we derive modified formulas for including one or more "incoherent" layers, i.e. very thick layers in which interference can be neglected. This document was written in conjunction with the "tmm" Python software package, which implements these calculations.