A comparison of localization effects in one-dimensional disordered photonic bandgap structures

Kissel, Glen J.; McGill, Cody A.

doi:10.1117/12.860876

Cited by 1 publication

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“…For computational purposes, the above equation can be simplified even further, and the reader is referred to previous papers 1, 3 for the details of this simplification, resulting in what is called the modified Wolf algorithm.…”

Section: Calculating the Lyapunov Exponentsmentioning

confidence: 99%

“…The computation of Lyapunov exponents for randomly disordered one-dimensional photonic bandgap structures has been readily demonstrated using various techniques [1][2][3][4][5] giving insight into Anderson localization as a function of frequency. In this paper we bring to bear one of those techniques 5 to calculate the Lyapunov exponents for onedimensional photonic bandgap structures constructed according to five different deterministic aperiodic sequences.…”

Section: Introductionmentioning

confidence: 99%

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Lyapunov exponents for one-dimensional aperiodic photonic bandgap structures

Kissel

2011

SPIE Proceedings

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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.

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Section: Calculating the Lyapunov Exponentsmentioning

confidence: 99%