Circular Dichroism in Biological Photonic Crystals and Cubic Chiral Nets

Nature provides impressive examples of chiral photonic crystals, with the notable example of the cubic so-called srs network (the label for the chiral degree-three network modeled on $\mathrm{SrS}{\mathrm{i}}_2$) or gyroid structure realized in wing scales of several butterfly species. By a circular polarization analysis of the band structure of such networks, we demonstrate strong circular dichroism effects: The butterfly srs microstructure, of cubic $I{4}_{1}32$ symmetry, shows significant circular dichroism for blue to ultraviolet light, that warrants a search for biological receptors sensitive to circular polarization. A derived synthetic structure based on four like-handed silicon srs nets exhibits a large circular polarization stop band of a width exceeding 30%. These findings offer design principles for chiral photonic devices.

Figure 1

Gyroid structure (see supplemental material [39] for animation). (Left) 1srs network and CMC surface with $\Phi =0.18$ clipped to a truncated octahedron. The chitin domain has a LH three-screw along [111] and a RH four-screw along [100], perpendicular to the truncated octahedron’s faces. (Right) Callophrys rubi and SEM images of its wing scale. The scale bars are $1\mu \mathrm{m}$ (middle) and 100 nm (bottom).

Figure 2

Polarization analysis of the band structure of the 1srs structure with $\Phi =20.8\%$ of the chitin domain ($ϵ=2.4$) in vacuum ($ϵ=1$). (a) Circular dichroism index $C$; (b) Dispersion relation $\omega (k)$ in [100], from $\Gamma$ to $H$ point; (c) Coupling index $\beta$. The point color represents the value of $C$ and the point radius the value of $\beta$. Vertical dashed lines give the thresholds above which bands are considered LCP (blue [dark gray]), RCP (red [medium gray]) or low coupling. (d) Reflectance rate $R$. The color bar represents the color of light if $a=310\mathrm{nm}$ (as in C. rubi).

Figure 3

Partial gap map $\omega (\Phi )$ for the fourfold screw axis of the 1srs chitin structure. The color bar shows the color of light for a lattice constant of $a=310\mathrm{nm}$. The vertical dashed line marks the volume fraction of Fig. 2.

Figure 4

4srs CMC structure with two distinct four-screw axes along [100]. $2^3$ unit cells of $P{4}_{2}32$ [17] are shown (individual nets are LH). As a RH turn, the $4_2(23)$ screw leaves the two nets that coil around it (blue [dark gray] & red [medium gray]) unchanged but exchanges the other pair (green [medium light gray] & orange [light gray]); a LH turn does the opposite.

Figure 5

Band structure of the 4srs geometry with $\Phi =24\%$ for $\mathbf{k}$ along [100]. The relative width of the dichroic band gap is $\Delta =30.2\%$. Open black circles mark the modes for which electric field energies are calculated (page 4). The oscillations in the reflectance $R$ above ${\omega }_2$ are likely due to the finite number of 6 unit cells in the scattering matrix calculation. The color bar represents the color of light if $2a=310\mathrm{nm}$.