Lyapunov exponents for one-dimensional aperiodic photonic bandgap structures

Glen J. Kissel

doi:10.1117/12.893816

9 September 2011 Lyapunov exponents for one-dimensional aperiodic photonic bandgap structures

Glen J. Kissel

Proceedings Volume 8093, Metamaterials: Fundamentals and Applications IV; 80931R (2011) https://doi.org/10.1117/12.893816
Event: SPIE NanoScience + Engineering, 2011, San Diego, California, United States

Abstract

Existing in the "gray area" between perfectly periodic and purely randomized photonic bandgap structures are the socalled aperoidic structures whose layers are chosen according to some deterministic rule. We consider here a onedimensional photonic bandgap structure, a quarter-wave stack, with the layer thickness of one of the bilayers subject to being either thin or thick according to five deterministic sequence rules and binary random selection. To produce these aperiodic structures we examine the following sequences: Fibonacci, Thue-Morse, Period doubling, Rudin-Shapiro, as well as the triadic Cantor sequence. We model these structures numerically with a long chain (approximately 5,000,000) of transfer matrices, and then use the reliable algorithm of Wolf to calculate the (upper) Lyapunov exponent for the long product of matrices. The Lyapunov exponent is the statistically well-behaved variable used to characterize the Anderson localization effect (exponential confinement) when the layers are randomized, so its calculation allows us to more precisely compare the purely randomized structure with its aperiodic counterparts. It is found that the aperiodic photonic systems show much fine structure in their Lyapunov exponents as a function of frequency, and, in a number of cases, the exponents are quite obviously fractal.

Citation Download Citation

Glen J. Kissel "Lyapunov exponents for one-dimensional aperiodic photonic bandgap structures", Proc. SPIE 8093, Metamaterials: Fundamentals and Applications IV, 80931R (9 September 2011); https://doi.org/10.1117/12.893816

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KEYWORDS

Binary data

Matrices

Fourier transforms

Fractal analysis

Nanophotonics

Systems modeling

Algorithm development

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