Electromagnetic characterization of millimetre-scale replicas of the gyroid photonic crystal found in the butterfly Parides sesostris

Abstract

We have used three-dimensional stereolithography to synthetically replicate the gyroid photonic crystal (PC) structure that occurs naturally in the butterfly Parides sesostris. We have experimentally characterized the transmission response of this structure in the microwave regime at two azimuthal angles (ϕ) over a comprehensive range of polar angles (θ). We have modelled its electromagnetic response using the finite-element method (FEM) and found excellent agreement with experimental data. Both theory and experiment show a single relatively broad transmission minimum at normal incidence (θ = 0°) that comprises several narrow band resonances which separate into clearly identifiable stop-bands at higher polar angles. We have identified the specific effective geometric planes within the crystal, and their associated periodicities that give rise to each of these stop-bands. Through extensive theoretical FEM modelling of the gyroid PC structure, using varying filling fractions of material and air, we have shown that a gyroid PC with material volume fraction of 40 per cent is appropriate for optimizing the reflected bandwidth at normal incidence (for a refractive index contrast of 1.56). This is the same gyroid PC material volume fraction used by the butterfly P. sesostris itself to produce its green structurally coloured appearance. This infers further optimization of this biological PC beyond that of its lattice constant alone.

Keywords: Parides sesostris; gyroid; photonic crystal; self-assembly.

Figure 1.

(a) Parides sesostris: the green regions are produced by scales that contain a gyroid photonic crystal structure. (b) The fabricated gyroid sample with 40% material (n = 1.656) and 60% air. Lattice constant is 7 mm.

Figure 2.

Experimental transmission data taken from the synthetically fabricated gyroid structure in the microwave regime, presented in reduced frequency units taken at an azimuthal angle (ϕ) of (a) 0° and (b) 45°. The sample was rotated over a comprehensive polar angular (θ) range. The negative values of polar angle represent experimental data taken with TM-polarized radiation, and the positive values represent experimental data taken with TE-polarized radiation. Miller indices indicate the directions through the crystal for which there exist periodicities of effective planes of the gyroid.

Figure 3.

Theoretical transmission data taken from using finite-element method modelling in the microwave regime, presented in units of reduced frequency. Polar angular data taken at an azimuthal angle of (a) 0° and (b) 45°. The theoretical model replicated the physical model. The data are presented in the same format as the experimental data in figure 2.

Figure 4.

Theoretical modelling of (a) the reflection and (b) transmission response of the gyroid structure with varying filling fraction (FF; shown as a function of t—a parameter taken from the surface equation of the gyroid and also ϕ_s—the material volume FF). The red lines indicate the parameter of t, and the FF, of the gyroid photonic crystal structure found in the butterfly P. sesostris. (c) Band-gap width as a function of FF.