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

The numerical analysis of photonic bandgaps in periodic dielectric str uctures leads to an infinite-dimensional eigenvalue problem which must be truncated to be solved. The truncation alters the numerical repres entation of the dielectric structure. By changing the formulation of t he problem, the representation of the periodic structure and the conve rgence of the solutions can be improved. Our calculations have shown t hat convergence can be reached for one- and two-dimensional structures but insufficient computer memory has prevented us from reaching conve rgence for three-dimensional structures. High-order super-Gaussians ca n be used to estimate photonic bandgaps accurately and to overcome the problems caused by the poor numerical representation of step function s. However, results obtained with super-Gaussians in one- and two-dime nsional 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 pro vided that certain conditions are satisfied.