Gap Sensitivity Reveals Universal Behaviors in Optimized Photonic Crystal and Disordered Networks
Michael A. Klatt, Paul J. Steinhardt, Salvatore Torquato
Abstract
Through an extensive series of high-precision numerical computations of the optimal complete photonic band gap (PBG) as a function of dielectric contrast $\ensuremath{\alpha}$ for a variety of crystal and disordered heterostructures, we reveal striking universal behaviors of the gap sensitivity $\mathcal{S}(\ensuremath{\alpha})\ensuremath{\equiv}d\mathrm{\ensuremath{\Delta}}(\ensuremath{\alpha})/d\ensuremath{\alpha}$, the first derivative of the optimal gap-to-midgap ratio $\mathrm{\ensuremath{\Delta}}(\ensuremath{\alpha})$. In particular, for all our crystal networks, $\mathcal{S}(\ensuremath{\alpha})$ takes a universal form that is well approximated by the analytic formula for a 1D quarter-wave stack, ${\mathcal{S}}_{\mathrm{QWS}}(\ensuremath{\alpha})$. Even more surprisingly, the values of $\mathcal{S}(\ensuremath{\alpha})$ for our disordered networks converge to ${\mathcal{S}}_{\mathrm{QWS}}(\ensuremath{\alpha})$ for sufficiently large $\ensuremath{\alpha}$. 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.