PHOTONIC BANDGAPS - WHAT IS THE BEST NUMERICAL REPRESENTATION OF PERIODIC STRUCTURES

The numerical analysis of photonic bandgaps in periodic dielectric structures leads to an infinite-dimensional eigenvalue problem which must be truncated to be solved. The truncation alters the numerical representation of the dielectric structure. By changing the formulation of the problem, the representation of the periodic structure and the convergence of the solutions can be improved. Our calculations have shown that convergence can be reached for one- and two-dimensional structures but insufficient computer memory has prevented us from reaching convergence for three-dimensional structures. High-order super-Gaussians can be used to estimate photonic bandgaps accurately and to overcome the problems caused by the poor numerical representation of step functions. However, results obtained with super-Gaussians in one- and two-dimensional structures converge to the same values as those obtained with step functions; hence we conclude that the expansion of the dielectric function with step functions yields reliable and accurate results provided that certain conditions are satisfied.