Localization and the invariant probability measure for photonic band gap structures

Abstract

Optical localization in a randomly disordered infinite length one-dimensional photonic band gap structure is studied using the transfer matrix formalism. Asymptotically, the infinite product of random matrices acting on a nonrandom input vector induces an invariant probability measure on the direction of the propagated vector. This invariant measure is numerically calculated for use in Furstenberg's master formula giving the upper Lyapunov exponent (localization factor) of the infinite random matrix product. A quarter-wave stack model with one of the bilayer thicknesses disordered is used for simulation purposes. In this plane wave model the invariant measure is rarely a uniform probability density function, as is sometimes assumed in the literature. Yet, the assumption of a uniform probability density function for the invariant measure gives surprisingly good results for a highly disordered system in the UV region.

Analyzing light localization using Iwasawa-canonical transfer matrices

“…4, we do see in 4 (b) a more nearly uniform pdf than in 4 (a), and yet the frequency, 2.0, is so near the right hand edge of the first stopband, that the invariant measure still has a strong point mass characteristic, typical of the invariant measures for frequencies inside stopbands 14 . The result at the frequency 3.355, at a vanished even photonic bandgap, is interesting in that the plane wave basis gives a slightly more uniform invariant measure than does the Iwasawa-canonical basis.…”

“…In the table we present the localization factor calculated via the modified Wolf algorithm 14 , and use that as a "truth model." We compare this to the estimation of the localization factor given by Furstenberg's formula, assuming the invariant measure is uniform, first using a plane wave basis, Eq.…”

“…The numerical technique used to calculate the invariant measure was described in a previous paper 14 . As was done in the previous paper, the simulation here used 5,000,000 pairs of layers but threw out the results from the first 50,000 layers to determine the invariant measure's probability density function.…”

“…In addition, the localization factor was numerically calculated using the modified Wolf algorithm discussed in the previous paper 14 . The simulation used 5,000,000 pairs of layers, ignoring the first 50,000, and no ensemble averaging.…”

“…But surprisingly 14 , the assumption of a uniform pdf for the invariant measure gave excellent results for the quarter-wave stack model at high disorder (the layer thickness a disordered at most +/-90%) and high frequency -in the lower UV range -, despite the fact that the invariant measure was in fact not a uniform pdf. This result, along with an observation by Baluni and Willemsen 18 has motivated us in part to examine using similarity transformations to produce transfer matrices in different bases in the expectation that a different basis might render the invariant measure a uniform pdf in a natural way.…”