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The electric field intensity in one-dimensional (1D) quasiperiodic and hybrid photonics band-gap structures is studied in the present paper. The photonic structures are ordered according to Fibonacci, Thue-Morse, Cantor, Rudin-Shapiro, Period-Doubling, Paper-Folding, and Baum-Sweet sequences. The study shows that the electric field intensity is higher for the Thue-Morse multilayer systems. After that the Thue-Morse structure will be combined with a periodic structure to form a hybrid photonic structure. It is shown that this hybrid system is the best for a strong localization of light. The proposed structures have been modeled using the Transfer Matrix Method.

During the last decades, a great deal of attention has been devoted to photonic crystals (PC) as a new type of materials, whose optical properties are used to manipulate the light on the scale of the wavelength [

Although the two-dimensional (2D) and three-dimensional (3D) photonic crystals have attracted and still attract many research efforts, the one-dimensional structure (1D) is the simplest in both geometry and handling, allowing mastering the properties of these structures and studying the influence of various physical parameters on these properties.

It consists of a stack of alternating layers having low and high refractive indices, whose thicknesses satisfy the Bragg condition:

During the last ten years, researchers have tried to fabricate several configurations of crystals to photonic band gap (PBG). Within this noteworthy structuring, light (or more generally an electromagnetic field) cannot propagate freely. It can be blocked (reflected), allowed only in certain directions, or even localized in certain areas [

Photonic crystals have paved the way for a new field of research and application possibilities. These crystals could for example improve the performance of lasers and light-emitting diodes or allow the manufacture of new types of antennas and amplifiers [

The concept of photonic crystal is not limited to periodic order. Since the discovery of quasiperiodic structures in 1984 by Shechtman et al. [

There is a wide variety of examples of one-dimensional quasicrystal [

The electronic properties of a one-dimensional quasicrystal arranged in a Fibonacci sequence, the wave functions at the center and at the edge of the band, the fractal nature (or self-similarity), and other critical properties of these wave functions are investigated by Kohmoto et al. [

Sibilia et al. [

Gellermann et al. [

Huang et al. [

Vasconcelos et al. have studied, for the normal-incidence case, the electric field intensity in quasiperiodic structures: Fibonacci, Thue-Morse, and Period-Doubling [

The aim of this work is to study and ameliorate the variation of the electric field intensity as a function of the thickness of quasiperiodic and hybrid photonics band-gap structures. These structures are composed of dielectric multilayer

The Fibonacci dielectric multilayer consists of two building blocks

The general expression of this sequence is arranged according to the concatenation rule

The Thue-Morse sequence can be also grown by juxtaposing the two building blocks

The distribution of Cantor consists in cutting each segment by three and the middle segment is removed. And the algorithm is repeated an infinite number of times.

Indeed, the Cantor sequence is created by the inflation rule:

The inflation rule used to generate the Rudin-Shapiro (RS) arrays can simply be obtained by the iteration of the two-letter inflation as follows:

In a two-letter alphabet, the Period-Doubling sequence can simply be obtained by the substitution

This substitution rule is also simple, and like the other chains, if we start with the correct tile, the tiling is fixed:

The method that we introduce here for calculating the electric field intensity spectra in one-dimensional photonic crystal is the Transfer Matrix Method (TMM) introduced by Yeh and Yariv [

To calculate the electric field intensity in a one-dimensional multilayer system, it suffices to calculate the total field at the input of system

The Transfer Matrix Method related the amplitudes of the incident wave

1D periodic photonic crystal composed of alternating layers of dielectric permittivities

The values

where

with

These equations allow deducing the expression of the total electric field according to the thickness

Finally, the intensity of the total electric field will be

In this numerical investigation, silicon dioxide

We propose here showing the influence of optogeometrical parameters of 1D photonic structure on the localization of the electromagnetic field. We compare the distributions of Cantor, Fibonacci, Thue-Morse, Rudin-Shapiro, Paper-Folding, Period-Doubling, and Baum-Sweet for an almost equal number of layers.

The strong localization of the electromagnetic field in the material requires structures in which the electromagnetic wave is slowed sharply to certain frequencies. From Figure ^{5} u.a) with a number of layers equal to 128. Here and in contrast to other structures, the wave is located at the outlet of the structure.

Electric field intensity of quasiperiodic photonic crystals according to the thickness (

In this second part, we will study the variation of the electric field intensity as a function of the thickness of one-dimensional hybrid systems. The hybrid assembly design implies putting the periodic and the quasiperiodic multilayer together in a sandwich configuration. The quasiperiodic photonic crystals (PCs) used in these hybrid structures are the Thue-Morse sequence, the best performing structure that we have found in the first part. The geometrical configurations considered are of the types Bragg/Thue-Morse/Bragg and Thue-Morse/Bragg/Thue-Morse.

In the first section, we are interested in the multilayer system composed of Thue-Morse sequence

Geometry example of the hybrid system formed by a Thue-Morse sequence sandwiched between two periodic structures.

Variation of the maximal electric field intensity according to the parameter

^{6} u.a but the total number of layers becomes 2150 (Figure

Variation of the maximal electric field intensity according to the parameter

In the second section, the spectra of intensity variation were obtained at normal incidence for a hybrid structure composed of a periodic photonic crystal interposed between two Thue-Morse sequences

Geometry example of the hybrid system formed by a periodic sequence sandwiched between two Thue-Morse sequences.

Variation of the maximal electric field intensity according to the parameter

Variation of the maximal electric field intensity according to the parameter

Figures

In this paper we have studied firstly the localization of light in one-dimensional quasiperiodic photonic crystals, according to Fibonacci, Thue-Morse, Cantor, Rudin-Shapiro, Period-Doubling, Paper-Folding, and Baum-Sweet sequences.

By comparing these structures, we try to highlight the importance of light localization in quasiperiodic photonic crystals. We found that the Thue-Morse distribution is the best structure to localize light; it permits getting the highest electric field intensity.

Secondly, to enhance the importance of this structure and to emphasize the importance of hybrid system, we have made an optimization by studying the combination of Thue-Morse and periodic structures.

We found that the hybrid photonic system (Bragg/Thue-Morse/Bragg) is performing well in localizing light, as well as optimizing the number of layers and the thickness of the photonic system.

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