Pr. Villeneuve et M. Piche, PHOTONIC BANDGAPS - WHAT IS THE BEST NUMERICAL REPRESENTATION OF PERIODIC STRUCTURES, J. mod. opt., 41(2), 1994, pp. 241-256
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.