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

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