Gap Sensitivity Reveals Universal Behaviors in Optimized Photonic Crystal and Disordered Networks

Through an extensive series of high-precision numerical computations of the optimal complete photonic band gap (PBG) as a function of dielectric contrast $\alpha$ for a variety of crystal and disordered heterostructures, we reveal striking universal behaviors of the gap sensitivity $\mathcal{S}(\alpha )\equiv d\mathrm{\Delta }(\alpha )/d\alpha$, the first derivative of the optimal gap-to-midgap ratio $\mathrm{\Delta }(\alpha )$. In particular, for all our crystal networks, $\mathcal{S}(\alpha )$ takes a universal form that is well approximated by the analytic formula for a 1D quarter-wave stack, ${\mathcal{S}}_{\mathrm{QWS}}(\alpha )$. Even more surprisingly, the values of $\mathcal{S}(\alpha )$ for our disordered networks converge to ${\mathcal{S}}_{\mathrm{QWS}}(\alpha )$ for sufficiently large $\alpha$. A deeper understanding of the simplicity of this universal behavior may provide fundamental insights about PBG formation and guidance in the design of novel photonic heterostructures.

Figure 1

Photonic networks: (a) the crystal diamond network, where cylindrical rods connect nearest neighbors of a diamond lattice, and (b) the disordered nearly hyperuniform network model; see also video S1 [17].

Figure 2

The gap plot shows the optimal gap-to-midgap ratio $\mathrm{\Delta }(\alpha )$ as a function of the dielectric contrast $\alpha$. The plot compares 3D photonic crystals (open symbols) to 3D disordered networks (solid symbols). The dashed-dotted line shows ${\mathrm{\Delta }}_{\mathrm{QWS}}(\alpha )$ for the 1D quarter-wave stack. The inset zooms in on small values of $\alpha$ where the PBGs first open up for the NHN, MD quench, and Laves networks.

Figure 3

The gap-sensitivity plot shows the slope $\mathcal{S}(\alpha )≔d\mathrm{\Delta }(\alpha )/d\alpha$ for each of the curves in Fig. 2. For our 3D crystal networks, $\mathcal{S}(\alpha )$ is well approximated by ${\mathcal{S}}_{\mathrm{QWS}}(\alpha )$ for all $\alpha$. For our disordered networks, $\mathcal{S}(\alpha )$ has a maximum at low $\alpha =\mathcal{O}(1)$ and only converges (approximately) to ${\mathcal{S}}_{\mathrm{QWS}}(\alpha )$ at larger $\alpha \gg 1$.