Wave propagation through Cantor-set media: Chaos, scaling, and fractal structures

Esaki, Kenta; Sato, Masatoshi; Kohmoto, Mahito

doi:10.1103/physreve.79.056226

Cited by 24 publications

(26 citation statements)

References 26 publications

(83 reference statements)

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“…the Fibonacci sequence, the Thue-Morse sequence, or Cantor sequences [8][9][10][11]. Such one-dimensional systems can be relatively easily produced in reality and a comparison of the theoretical and experimental results shows good agreement [12,13].…”

Section: Introductionmentioning

confidence: 84%

Light transmission through metallic-mean quasiperiodic stacks with oblique incidence

Thiem

Schreiber

Grimm

2010

Philosophical Magazine

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The propagation of s-and p-polarized light through quasiperiodic multilayers, consisting of layers with different refractive indices, is studied by the transfer matrix method. In particular, we focus on the transmission coefficients of the systems and their dependence on the incidence angle and on the ratio of the refractive indices. We obtain additional bands with almost complete transmission in quasiperiodic systems at frequencies in the range of the photonic band-gap of a system with a periodic alignment of the two materials for both types of light polarization. With increasing incidence angle these bands bend towards higher frequencies, where the curvature of the transmission bands in the quasiperiodic stack depends on the metallic mean of the construction rule. Additionally, in the quasiperiodic systems for p-polarized light the bands show almost complete transmission near Brewster's angle, in contrast to the results for s-polarized light. Further, we present results for the influence of the refractive indices at the mid-gap frequency of the periodic stack, where the quasiperiodicity was found to be most effective.

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Section: Introductionmentioning

confidence: 84%

Light transmission through metallic-mean quasiperiodic stacks with oblique incidence

Thiem

Schreiber

Grimm

2010

Philosophical Magazine

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“…Deterministic aperiodic spatial patterns, dubbed photonic quasicrystals [14][15][16][17][18][19][20][21], display optical responses that cannot be found in either periodic or random systems. Among other issues, they include a self-similar energy spectrum [22,23], a pseudo-bandgap of forbidden frequencies [24][25][26], and critically localized states [27,28] whose wave functions are distinguished by power law asymptotes and self-similarity [29][30][31]. In fact, many of these traits have been found to be useful [32,33].…”

Section: Introductionmentioning

confidence: 99%

Lempel-Ziv Complexity of Photonic Quasicrystals

2017

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The properties of one-dimensional photonic quasicrystals ultimately rely on their nontrivial long-range order, a hallmark that can be quantified in many ways depending on the specific aspects to be studied. Here, we assess the quasicrystal structural features in terms of the Lempel-Ziv complexity. This is an easily calculable quantity that has proven to be useful for describing patterns in a variety of systems. One feature of great practical relevance is that it provides a reliable measure of how hard it is to create the structure. Using the generalized Fibonacci quasicrystals as our thread, we give analytical fitting formulas for the dependence of the optical response with the complexity.

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“…The resulting set has a non-integer H-B dimension d f = ln 2/ ln 3 with numerical value less than that of the space d = 1 where the set is embedded [6]. Besides its pedagogical importance, the Cantor set problem has also been of theoretical and practical interest [8][9][10][11]. However, as far as natural fractals are concerned, the Cantor set lacks at least in two ways.…”

Section: Introductionmentioning

confidence: 99%

Dyadic Cantor set and its kinetic and stochastic counterpart

Hassan

Pavel

Pandit

et al. 2014

Chaos, Solitons & Fractals

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Firstly, we propose and investigate a dyadic Cantor set (DCS) and its kinetic counterpart where a generator divides an interval into two equal parts and removes one with probability $(1-p)$. The generator is then applied at each step to all the existing intervals in the case of DCS and to only one interval, picked with probability according to interval size, in the case of kinetic DCS. Secondly, we propose a stochastic DCS in which, unlike the kinetic DCS, the generator divides an interval randomly instead of equally into two parts. Finally, the models are solved analytically; an exact expression for fractal dimension in each case is presented and the relationship between fractal dimension and the corresponding conserved quantity is pointed out. Besides, we show that the interval size distribution function in both variants of DCS exhibits dynamic scaling and we verify it numerically using the idea of data-collapse.Comment: 8 pages, 6 figures, To appear in Chaos, Solitons & Fractal

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Wave propagation through Cantor-set media: Chaos, scaling, and fractal structures

Cited by 24 publications

References 26 publications

Light transmission through metallic-mean quasiperiodic stacks with oblique incidence

Light transmission through metallic-mean quasiperiodic stacks with oblique incidence

Lempel-Ziv Complexity of Photonic Quasicrystals

Dyadic Cantor set and its kinetic and stochastic counterpart

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