[Paper Review] Multilayer optical calculations

Steven J. Byrnes|arXiv (Cornell University)|Mar 6, 2016

Photonic Crystals and Applications4 references169 citations

TL;DR

This note derives the transfer-matrix method for multilayer stacks, including oblique incidence, absorption profiles, ellipsometry, and incoherent layers; it also discusses branch cuts and implementation details for tmm.

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.

Motivation & Objective

• Derive the transfer-matrix framework for multilayer thin films and generalize it to oblique incidence.
• Provide explicit formulas for reflection, transmission, and ellipsometric parameters.
• Explain absorption distribution and Poynting-vector calculations within layered media.
• Discuss special cases with absorptive starting/final media and incoherent thick layers.
• Address implementation details such as complex-plane branch cuts and sign conventions.

Proposed method

• Develop explicit E, H, and k vectors for s- and p-polarization.
• Derive single-interface Fresnel equations and extend to multilayer stacks via transfer matrices (Eq. 10–15).
• Compute ellipsometric parameters from r_s and r_p (Eq. 16).
• Formulate power flow, transmitted/reflected power, and absorbed energy density (Eq. 17–24).
• Introduce incoherent-thick-layer treatment and derive the corresponding transfer-matrix approach (Eq. 25–30).
• Discuss branch cuts, sign conventions, and implementation considerations for the tmm software.

Experimental results

Research questions

• RQ1 How can the transfer-matrix method be applied to arbitrary multilayer planar stacks with oblique incidence?
• RQ2 How do absorption and complex refractive indices affect R, T, and energy conservation in multilayer interfaces?
• RQ3 How can incoherent (thick) layers be incorporated into the multilayer formalism without resolving Fabry–Pérot fringes?
• RQ4 What are the correct handling rules (branch cuts and sign conventions) when implementing these formulas computationally?
• RQ5 How can ellipsometric parameters be computed from the derived reflection coefficients in general multilayer systems?

Key findings

• A closed-form transfer-matrix formalism yields r and t for arbitrary multilayer stacks (Eq. 15).
• Explicit expressions for transmittance and reflectance are provided for s- and p-polarizations (Eq. 21–23).
• Ellipsometric parameters can be computed from r_p and r_s (ψ and Δ, Eq. 16).
• Absorption profiles and Poynting-vector calculations are derived for both polarizations (Eq. 17–24).
• A coherent/incoherent layer framework is developed to handle thick layers where interference averages out (Eq. 25–30).
• The notes discuss nontrivial effects when the starting or ending medium is absorptive and address branch-cut issues in complex Snell’s law.