Theory of chirped photonic crystals in biological broadband reflectors

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.

Convergent Evolution of Broadband Reflectors Underlies Metallic Coloration in Butterflies

“…These examples of insect metallic colors rely on broadband reflectance from multiple layers of chitin with differing thicknesses, often termed chirped stacks (Kinoshita and Yoshioka, 2005;Biro and Vigneron, 2011). As in the chaotic reflector, those chirped mirrors are often hundreds of microns thick because they rely on the stacking of bilayers of dielectric materials with distinctive refractive indices (Deparis et al, 2006;Cook and Amir, 2016;Chiadini et al, 2017), typically from air or cytoplasm (low-index 1 < n L < 1.33), and chitin (high-index 1.53 < n H < 1.56), melanized chitin (high-index 1.56 < n H < 1.8), or guanine platelets (1.46 < n H < 1.85) (Leertouwer et al, 2011;Jordan et al, 2012;Stavenga et al, 2015a). Reducing the thickness of a chirped mirror theoretically requires increasing the number and refractive index of the dielectric materials.…”

“…In particular, the difference in refractive index between layers determines how much light is reflected at each interface. The smaller this difference, the less light is reflected, implying that chirped stacks relying on natural materials can only achieve reasonable reflectances using multilayers of 10 thin films or more (Land, 1972;Cook and Amir, 2016).…”

Circularly polarized reflection from the scarab beetleChalcothea smaragdina: light scattering by a dual photonic structure

“…A number of previous studies concerning the reflection of CP light by coleopteran species have focused upon rutelinid scarabs belonging to the Chrysina genus [20,21,[72][73][74]. Several of these beetles display silver and gold hues owing the chirped distribution of pitch lengths within their helicoids [62,72,73,75]. Additionally, the intensity of LCP light reflected by these brilliantly coloured specimens, in response to illumination at normal incidence, exceeds that observed herein for C. smaragdina [72].…”

The Chirp Function Revisited: A Uniqueness Conjecture for Chirplet Modulation

“…an integral whose kernel is characterised by a quadratic phase function. Chirped mirrors incorporate 'chirped spaces' in a dielectric designed to reflect wavelengths of light and compensate for dispersion effects that can be created by some optical elements [6]. More recently, in the field of cryptology, the chirp function has been used for key exchange [7], for the self-authentication of digital signals [8] and high resilience watermarking [9].…”