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}$(\alpha)\equiv d\Delta(\alpha)/d\alpha$, 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 one-dimensional quarter-wave stack, ${\mathcal{S}}_{\text{QWS}}(\alpha )$. Even more surprisingly, the values of $\mathcal{S}(\alpha )$ for our disordered networks converge to ${\mathcal{S}}_{\text{QWS}}(\alpha )$ 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.