Bandgap Simulations in Randomized 3D Photonic Crystal Supercells

Periodic supercell lattice structures with elements of random polydispersity disorder were created to simulate the effect of randomization on photonic crystals using finite-difference time domain (FDTD) methods. As a key exemplar system, a three-dimensional “inverse opal” structure of a face-centered cubic lattice with air spheres in a silicon dielectric was simulated, with sphere radii within supercells following a randomized Gaussian distribution, with characteristic standard deviation and mean. A corresponding ordered lattice with a bandgap with magnitude 3.5% of the normalized frequency range was used as a direct control, with sphere radius 0.34 times the lattice constant a. For a range of standard deviations, up to 5.9% of the 0.34a mean, a Monte Carlo-style approach was adopted, with photonic band properties analyzed over a large number of repeat simulations to ensure statistical significance. The corresponding Gaussian distribution in the resultant photonic bandgap magnitudes is broadened with increasing polydispersity such that an evolving fraction of simulations no longer exhibits a non-zero bandgap. A characteristic pseudo-transition occurs at a standard deviation of approximately 4.1% of the 0.34a mean, above where the frequency of simulations still returning a finite bandgap rapidly diminishes. Some isolated configurations, with a high degree of uniqueness, can exhibit enhanced bandgap properties (greater than the 3.5% benchmark) despite considerable polydisperse disordering; we envisage that these findings point towards the use of engineered randomness in supercell systems to create desired photonic crystal properties and functionality, such as localization and photonic bandgaps.