Journal of Materials

Volume 2016, Article ID 9471312, 7 pages

http://dx.doi.org/10.1155/2016/9471312

## Enhancement of Light Localization in Hybrid Thue-Morse/Periodic Photonic Crystals

^{1}Photovoltaic and Semiconductor Materials Laboratory, El-Manar University-ENIT, P.O. Box 37, Le Belvedere, 1002 Tunis, Tunisia^{2}College of Engineering, Industrial Engineering Department, Haïl University, Haïl 2440, Saudi Arabia

Received 26 April 2016; Accepted 25 July 2016

Academic Editor: Francois M. Peeters

Copyright © 2016 Rihab Asmi et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

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.

#### 1. Introduction

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 [1, 2]. These crystals are structures whose dielectric index varies periodically on the scale of the wavelength on one, two, or three directions in space.

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: , where is the reference wavelength. It is known as Bragg Mirror [3]. This Mirror produces light, which is propagating in the photonic crystal, a similar effect to that of the periodic potential of the electrons in a solid crystal. Just as in the case of a semiconductor, there appear forbidden bands for photons (PBGs) in photonic crystals prohibiting propagation of light in certain directions and for certain energies. The existence of PBGs will lead to many interesting phenomena.

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 [4].

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 [5–7].

The concept of photonic crystal is not limited to periodic order. Since the discovery of quasiperiodic structures in 1984 by Shechtman et al. [8], most of the studies have concentrated on the localization properties of photons.

There is a wide variety of examples of one-dimensional quasicrystal [9]. The most common ones are Fibonacci, Thue-Morse (TM), Cantor, and Rudin-Shapiro structures. These structures have been widely studied, thus opening a way to a wide range of technological applications to many fields.

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. [10].

Sibilia et al. [11] studied the self-similarity effects illustrated in the transmission spectrum of quasiperiodic structures arranged in a triadic Cantor.

Gellermann et al. [12] demonstrated experimentally the existence of band gaps in spectral response of these quasiperiodic structures which can be considered as evidence for the localization of the light waves.

Huang et al. [13] studied numerically the light localization of these structures by means of Transfer Matrix Method. They compare the localization length as a function of the chain length for the random, Thue-Morse, and Fibonacci sequences.

Vasconcelos et al. have studied, for the normal-incidence case, the electric field intensity in quasiperiodic structures: Fibonacci, Thue-Morse, and Period-Doubling [14]. Although Vasconcelos et al. have found that the electrical field distribution for each sequence follows its own structure, their study was not followed by an improvement of the intensity values (i.e., the improvement of the light localization). This is the concept of our study.

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 and stacked alternately following the Fibonacci, Thue-Morse, Cantor, Rudin-Shapiro, Period-Doubling, Paper-Folding, and Baum-Sweet sequences.

#### 2. Quasiperiodic Models

##### 2.1. The Fibonacci Sequence Case

The Fibonacci dielectric multilayer consists of two building blocks and with refraction indices (higher refractive index) and (lower refractive index) and thicknesses and , respectively. The number of layers of the structure depends on the order of the Fibonacci sequence.

The general expression of this sequence is arranged according to the concatenation rule for , where is the th generation of Fibonacci structure. For example, , , and . Here, and [14].

##### 2.2. The Thue-Morse Sequence Case

The Thue-Morse sequence can be also grown by juxtaposing the two building blocks and and can be produced by repeating application of the substitution rules and . For example, the first few generations of Thue-Morse sequence are as follows: , , , , and so on [15].

##### 2.3. The Cantor Sequence Case

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: and . The resulting set is the set of triadic Cantor .

##### 2.4. The Rudin-Shapiro Sequence Case

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: , , , and .

##### 2.5. The Period-Doubling Sequence Case

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

##### 2.6. The Paper-Folding Sequence Case

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

#### 3. Problem Formulation

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 [16]. This method is an excellent tool for accurately analyzing the electromagnetic wave propagation in stratified media which permits particularly extracting and solving the standard problem of the photonic band structures (transmission, reflection, and absorption) spectra [17]. In this method, the electromagnetic field is calculated gradually during the wave propagation in the structure.

To calculate the electric field intensity in a one-dimensional multilayer system, it suffices to calculate the total field at the input of system and add the field inside the system which varies depending on* z*:

is the reflected wave by the whole system, is the incident wave.

The Transfer Matrix Method related the amplitudes of the incident wave , reflected wave , and transmitted wave (Figure 1). This is expressed by the following matrix relation: