Theory of chirped photonic crystals in biological broadband reflectors

Photonic crystals with slowly varying lattice period are responsible for broadband light reflectance in many biological contexts, ranging from the coatings of shiny beetles to the eye shine of butterflies. We utilize a quantum scattering analogy to obtain an approximate but closed-form expression for the reflectance of these adiabatically chirped photonic crystals (ACPCs). This expression allows us to devise a method for designing ACPCs with tailored reflectance spectra and obtain an estimate for the minimal number of layers required to exceed a given reflectance threshold. Comparison against the number of layers found in ACPCs throughout nature gives a quantitative measure of the optimality of chirped biological reflectors. Together, these results elucidate the design of chirped reflectors in nature and their possible application to future optical technologies.