Proceedings Article | 7 February 2008

Proc. SPIE. 6901, Photonic Crystal Materials and Devices VII

KEYWORDS: Visible radiation, Matrices, Ultraviolet radiation, Photonic crystals, Numerical simulations, Wave propagation, Refraction, Analytical research, Disordered photonic crystals, Alternate lighting of surfaces

The transfer matrix formalism has proven to be a powerful tool for analyzing one-dimensional photonic bandgap
structures, whether their multilayers are perfectly periodic or randomized in some fashion. In the randomized structure,
as the number of layers tends to infinity, Furstenberg's formula can be used, at least theoretically, to find the
deterministic Lyapunov exponent (localization factor, sometimes called the inverse localization length) governing the
confinement of energy transmission in the model. The challenge in using Furstenberg's formula is that it requires the
calculation of the invariant probability measure of the direction of the vector propagated by the chain of random
matrices. This invariant measure is usually impossible to find analytically, and so one must resort to numerical
simulation or some other approximating assumption. To aid in the numerical determination of this invariant probability
measure, we consider matrix similarity transformations based on the average plane wave transfer matrix at a given
frequency. These transformations, like the original transfer matrix, are elements of SU(1,1), the special pseudo-unitary
group, and are obtained by moving the fixed points of the bilinear (or Mobius) transformation of the original transfer
matrix to the corresponding fixed points of the canonical forms known from the Iwasawa decomposition. Amazingly, in
some situations, including a quarter-wave stack, such a transformation can cause the invariant probability measure to
become a nearly uniform probability density function, making the Furstenberg formula more readily useable.