Theory of chirped photonic crystals in biological broadband reflectors

TL;DR

This paper develops a quantum scattering model for chirped photonic crystals in biological reflectors, deriving design principles and optimality measures that explain natural phenomena and guide future optical technology design.

Contribution

It introduces a WKB-based analytical approach to model light reflection in adiabatically chirped photonic crystals, providing new insights into their design and biological optimization.

Findings

Derived a closed-form reflectance expression for ACPCs.

Established a differential equation for chirp patterns to achieve target reflectance.

Quantified the minimal bilayer number for desired reflectance and compared with biological structures.

Abstract

One-dimensional photonic crystals with slowly varying, i.e. "chirped", lattice period are responsible for broadband light reflectance in many diverse biological contexts, ranging from the shiny coatings of various beetles to the eyes of certain butterflies. We present a quantum scattering analogy for light reflection from these adiabatically chirped photonic crystals (ACPCs) and apply a WKB-type approximation to obtain a closed-form expression for the reflectance. From this expression we infer several design principles, including a differential equation for the chirp pattern required to elicit a given reflectance spectrum and the minimal number of bilayers required to exceed a desired reflectance threshold. Comparison of the number of bilayers found in ACPCs throughout nature and our predicted minimal required number also gives a quantitative measure of the optimality of chirped biological reflectors. Together these results elucidate the design principles of chirped reflectors in nature and their possible application to future optical technologies.