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Proceedings Paper

Discretely disordered photonic bandgap structures: a more accurate invariant measure calculation
Author(s): Glen J. Kissel
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Paper Abstract

In the one-dimensional optical analog to Anderson localization, a periodically layered medium has one or more parameters randomly disordered. Such a randomized system can be modeled by an infinite product of 2x2 random transfer matrices with the upper Lyapunov exponent of the matrix product identified as the localization factor (inverse localization length) for the model. The theorem of Furstenberg allows us, at least theoretically, to calculate this upper Lyapunov exponent. In Furstenberg's formula we not only integrate with respect to the probability measure of the random matrices, but also with respect to the invariant probability measure of the direction of the vector propagated by the random matrices. This invariant measure is difficult to find analytically, and, as a result, the most successful approach is to determine the invariant measure numerically. A Monte Carlo simulation which uses accumulated bin counts to track the direction of the propagated vector through a long chain of random matrices does a good job of estimating the invariant probability measure, but with a level of uncertainty. A potentially more accurate numerical technique by Froyland and Aihara obtains the invariant measure as a left eigenvector of a large sparse matrix containing probability values determined by the action of the random matrices on input vectors. We first apply these two techniques to a random Fibonacci sequence whose Lyapunov exponent was determined by Viswanath. We then demonstrate these techniques on a quarter-wave stack model with binary discrete disorder in layer thickness, and compare results to the continuously disordered counterpart.

Paper Details

Date Published: 16 February 2009
PDF: 12 pages
Proc. SPIE 7223, Photonic and Phononic Crystal Materials and Devices IX, 722318 (16 February 2009); doi: 10.1117/12.809489
Show Author Affiliations
Glen J. Kissel, Univ. of Southern Indiana (United States)

Published in SPIE Proceedings Vol. 7223:
Photonic and Phononic Crystal Materials and Devices IX
Ali Adibi; Shawn-Yu Lin; Axel Scherer, Editor(s)

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