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Computational Modeling of Quasi-Periodic Rudin-

Shapiro Multilayered Band Gap Structure 
 

Yassine Bouazzi 

Industrial Engineering Department 
University of Hail 
Hail, Saudi Arabia 

yassine.bouazzi@enit.utm.tn 

Naim Ben Ali 

Industrial Engineering Department 
University of Hail 
Hail, Saudi Arabia 
naimgi2@yahoo.fr 

Haitham Alsaif 

Electrical Engineering Department 
University of Hail 
Hail, Saudi Arabia 

hal-saif@hotmail.com 
  

Attia Boudjemline 

Industrial Engineering Department 
University of Hail 

Hail, Saudi Arabia 

a_boudjemline@hotmail.com 

Youssef Trabelsi 

Physics Department 
King Khalid University 

Aseer, Saudi Arabia 

youssef_trabelsi@yahoo.fr 

Ahmed Torchani 

Industrial Engineering Department 
University of Hail 

Hail, Saudi Arabia 
tochahm@yahoo.fr 

 

 

Abstract—The optical transmission spectra proprieties of the one-

dimensional quasi-periodic multilayered photonic structures 

according to the Rudin-Shapiro distribution are studied 

theoretically in this paper by using a theoretical model based on 

the transfer matrix approach for normal incidence geometry. The 

influence of the layer number has been studied, i.e. the iteration 

order of the generating sequence of the quasi-periodic structure 

on the structure spectral behavior and the width of the Photonic 

Band Gap (PBG). It was found that the width of the PBG is 
proportional to the index contrast. 

Keywords-photonic crystal; Rudin-Shapiro distribution; 

transfer-matrix method; transmittance; photonic band gap 

I. INTRODUCTION  

Photonic crystals are new, artificially created materials in 
which the refractive index is periodically modulated in a scale 
compared to the wavelength of operation [1-4]. In fact, the 
periodic modulation of the refractive index of such a structure 
causes the appearance of frequency ranges for which the light 
cannot propagate [5]. Photonic crystals at the end of the last 
century became one of the most active research topics in 
various technological disciplines, e.g. communications, 
optoelectronics, and optics [6, 7]. The simplest form of a 
photonic crystal is the one-dimensional periodic structure, 
known as the Bragg mirror. It consists of a stack of alternating 
layers with low and high refractive index and a thickness in the 
order of ��/4, where �� is the reference wavelength, i.e. the 
thicknesses satisfy the Bragg condition. There is currently a 
need to design photonic materials with broader band gaps and 
multiband. The first manifestation of this goal was the 
realization of a new class of artificial crystals, called quasi-
periodic photonic crystals [8]. These structures are formed by 
the stacking of two or more deposited materials according to a 
predefined recursive inflation rule, so that they can be 

considered as intermediate systems between an ordinary 
periodic crystal and random amorphous solids. These quasi-
periodic structures have become the subject of intense research. 
They have led to many technological achievements in the fields 
of photonics, telecommunications, and microwaves [9]. 

This paper aims to study the optical properties of the 
photonic quasi-periodic unidimensional multilayer structures 
from their spectral response using theoretical modeling and 
numerical simulation based on the Transfer Matrix Method 
(TMM). 

II. MODELING PROCEDURE AND NUMERICAL SIMULATION 

A. Structure Presentation 

The Rudin-Shapiro sequence, also known as the Golay-
Rudin-Shapiro sequence, is an infinite automatic sequence [10, 
11] introduced in 1950 [12]. Rudin-Shapiro's 1D multilayered 
distribution system was exploited for the first time to study 
light confinement in multilayer structures in 2008 [13]. The 
Rudin-Shapiro sequence can be defined by [14-15]: RS(�) = �����
...ε� ...ε�����.    (1) 
with: 

�� = (−1)�(�)    (2) 
where u(i) represents the number “11” in the binary 
development of the integer i. 

The use the Rudin-Shapiro distribution in the photonics 
field remains very limited due to the construction peculiarity of 
this structure. It is represented by quaternary elementary layers 
(A, B, C and D) with substitution rules given by:  

� As={A,	B,	C,	D} → AB,	B → AC,	C → DB,	D → DC    (3) 
Corresponding author: Yassine Bouazzi



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The generation starts with the element labeled by the letter 
A. For example, a multilayer structure of iteration order 2 is 
given by: S2 ACAB, order 3 by S3 ACABACDC… (Figure 1). 
Table I presents the generation of the structure of iteration 
order k=1,2,3. 

TABLE I.  THE FIRST GENERATIONS OF THE 1D MULTILAYER 
STRUCTURE ACCORDING TO THE RUDIN-SHAPIRO DISTRIBUTION 

Iteration order Alphabetic sequence 

RS(0) A 

RS(1) AC 

RS(2) ACAB 

RS(3) ACABACDC 

 
In this work, we chose a multilayer photonic system 

composed of four dielectric layers A, B, C and D, distributed 
according to the Rudin-Shapiro sequence, having refraction 
indices �" , �# , �$  and �% , and optical thicknesses ��/n" , ��/n# , ��/n$  and ��/n%  respectively, where ��  is the 
reference wavelength. Furthermore, the optical and geometrical 
parameters of the structure obey the quarter wavelength 
condition or Bragg condition as indicated by (4): 

�"'" 
 �#'# 
 �$'$ 
 �%'% 
 ()*     (4) 

 
Fig. 1.  One-dimensional Rudin-Shapiro multilayer dielectric structure for 

the 3
rd
 iteration RS(3): ACABACDC 

B. Simulation Method 

For the calculation of reflection and transmission, we used 
TMM, which is a technique well suited to the study of PBG 
materials. This technique can solve the standard problem of the 
photonic band structures to find transmission, reflection, and 
absorption spectra. This method consists in expressing the 

reflected field +,-�� from a multilayer structure as a function of 
incident fields +,-�.  in a matrix form. The studied multilayer 
structure consists of a stack of m layers with different 
thicknesses '� and refractive indices	��, as shown in Figure 2. 
The amplitudes of the electric fields of incident and transmitted 
wave are expressed by the following matrix for stratified films 
within m layers: 

/+�.+�� 0 = ∏ 234.�� /+4.�
.+4.�� 0			    (5) 

where 23 is the propagation matrix for the jth layer, given by: 
23 = 5

�67 exp	��;<3��	 =767 exp	�;<3��	=767 exp	��;<3��	 �67 exp	��;<3��	>			    (6) 

where <j-1  denotes the change in the phase of the wave 
between �A � 1	6B and A6B layers, and can be obtained by: 

C D� 
 0D3�� 
 
F( �G3��'3�� HIJK3��    (7) L3 and M3are the Fresnel transmission and reflection coefficients 
at the interface between �A � 1	6B and A6B  layer. The Fresnel 
coefficients are expressed using the complex refractive index �3 and the complex refractive angle K3: 

For the TM mode, the coefficients are: 

M3NO 
 �G7PQ RST U7��G7 RSTU7PQ�G7PQ RST U7.�G7 RSTU7PQ    (8) 
L3NO 
 
�G7PQ RST U7PQ�G7PQ RSTU7.�G7 RSTU7PQ	    (9) 

and for the TE mode: 

M3NV 
 �G7PQ RST U7PQ��G7 RST U7�G7PQ RST U7PQ��G7 RST U7     (10) 
L3NV 
 �G7PQ RST U7PQ�G7PQ RST U7PQ.�G7 RST U7     (11) 

 
Fig. 2.  Multilayer system with m components 

Taking +4.�� = 0, there will be no reflections from the 
interface of the final structure. Convenient formulas have been 
obtained for the total reflection and transmission coefficients, 
which correspond to the amplitude reflectance r and 
transmittance t respectively, as follows [16-19]: 

M = NWQNQQ    (12) 
L = �NQQ    (13) 

where X��  and X
�  are the matrix elements of the product 
matrix: 

∏ 234.�� = /X�� X�
X
� X�
0      (14) 
Starting from (12) and (13), the reflection R and the 

transmission T are given by [6]: 

Y = |M|
 = [NWQNQQ [
	    (15) 
X 
 |L|
 
 [ �NQQ[
		    (16) 

High index layer nA 

Low index layer nC 

ZZZZ    

θθθθ    Incident wave 
XXXX    

YYYY    

High index layer nB 

Low index layer nD 



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III. NUMERICAL RESULTS AND DISCUSSION 

Figure 2 shows the reflection spectra with normal incidence 
for the quasi-periodic multilayer system distributed according 
to the generalized Rudin-Shapiro model. We choose the 
following four materials as elementary layers of the stack: TiO2 
(nA=2.3), HfO2 (nB=2), SiO2 (nC=1.45) and TaO5 (nD=2.2). 

 

(a) 

 

(b) 

 

(c) 

 

(d) 

 

Fig. 3.  The reflection spectra of Rudin-Shapiro distribution multilayer 

photonic structures for: (a) 4
th
, (b) 5

th
, (c) 6

th
, and (d) 7

th
 iteration 

We notice from the spectra in Figure 3 that the width of the 
photonic band gap of the studied system increases with the 
order of generation k. The width of the PBG (Figure 4) 
observed near λ0=0.5µm increases when progressing from one 
iteration to another. 

 
Fig. 4.  Variation of the PBG width as a function of the number of 

iterations (k) 

TABLE II.  WIDTH AND LIMITATIONS OF PBGS 

Iteration k ∆∆∆∆λλλλPBG (µµµµm) 

6 0.0898 

7 0.0972 

8 0.1002 

9 0.1025 

10 0.1039 

11 0.1039 

12 0.1039 
 

Figure 5 and Table III show the evolution of the maximum 
reflection Rmax(%) as a function of the iteration k. We can 
consider that the first PBG appears for the 6

th
 iteration with 

PBG width ∆λPBG=0.1039µm. 

TABLE III.  VARIATION OF Rmax (%) ACCORDING TO k 

Iteration k Rmax (%) 

2 0 

3 29.7051 

4 81.2423 

5 96.2897 

6 99.9874 

7 99.9999 

8 100 

9 100 

10 100 

 
Fig. 5.  Variation of Rmax (%) as a function of the iteration value k 

We notice that starting from the 7
th
 iteration two transparent 

windows (Figure 6) located in the central PBG and almost 

symmetrical to λ0=0.5µm with T=100% appear. These optical 
windows indicate a strong light confinement in the 
microcavities of low refractive indices formed by the quasi-



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periodic stack for the corresponding wavelengths. This last 
characteristic makes the system a good candidate for the design 
of an optical filter. 

 

 
Fig. 6.  Number and position of optical windows according to the number 

of iterations k 

TABLE IV.  WIDTHS OF PBGS, NUMBER OF OPTICAL WINDOWS AND 
THEIR POSITIONS, ACCORDING TO THE NUMBER OF ITERATIONS k 

k ∆λPBG 
Optical windows 

Number Positions 

6 0.0898 - - 

7 0.0972 2 0.4636-0.5421 

8 0.1002 2 0.4568-0.5515 

9 0.1025 2 0.4548-0.5550 

10 0.1039 2 0.4544-0.5564 

11 0.1039 2 0.4542-0.5564 

12 0.1039 2 0.4542-0.5564 

 

The objective of the remaining part of this work is to check 
the correlation between the width of the PBG and the indices 
ratio nH/nL, with nH and nL representing the high and the low 
refractive indices respectively. We studied the reflection of the 
one-dimensional Rudin-Shapiro photonic structure as a 
function of the index contrast. The index contrast or the optical 
contrast is defined as the ratio between the high and low 
refractive indices of the layers constituting the structure. For a 
one-dimensional structure, the index contrast is given by [14]:  

\ = �]/�^    (17) 
Figures 7(a) and (c) show the optical reflection of a 

multilayer structure based on the Rudin-Shapiro model for the 
5
th
 and 6

th
 iterations respectively, as a function of the index 

contrast and wavelength. Figures 7(b) and (d) illustrate the 
maps of the PBGs for the 5

th
 and 6

th
 iterations, respectively, 

where the blue zone represents the forbidden bands and the 
white zone the permissible ones. We observe that for the 5

th
 

iteration, the first PBG appears with index contrast δ≈1.774 and 
the width and number of PBGs increase considerably with the 
index contrast. Moreover, from Table V, we notice that the 

difference between the middle of the PGB (λm) and the 
reference wavelength (λ0=0.5) 	increases with the increase in 
the index contrast. 

 

(a) 

 

(b) 

 

(c) 

 

(d) 

 

Fig. 7.  Spectral aspect as a function of the contrast variation for the 

Rudin-Shapiro distribution. (a), (c) Variation of reflection spectra as a 

function of contrast and wavelength for the 5
th
 and 6

th
 iterations respectively. 

(b), (d) PBGs maps as a function of contrast and wavelength for the 5
th
 and 6

th
 

iterations respectively 

TABLE V.  VARIATION OF THE PBGS WIDTH AND POSITIONS ACCORDING TO THE INDEX CONTRAST 

δδδδ=nH/nL 
5th Iteration 6th Iteration 

λmax λmin λm ∆λBPG λmax λmin λm ∆λBPG 

1,6 - - - - 0,4625 0,5437 0,5031 0,0812 

1,9 0,4581 0,5506 0,5044 0,0925 0,4452 0,5718 0,5085 0,1266 

2,2 0,4420 0,5746 0,5083 0,1326 0,4344 0,5898 0,5121 0,1554 

2,5 0,4332 0,5922 0,5127 0,1590 0,4280 0,5998 0,5139 0,1718 

2,8 0,4260 0,6038 0,5149 0,1778 0,4044 0,6310 0,5177 0,2266 

 
IV. CONCLUSION 

Based on the simulation of the photometric response of the 
Rudin-Shapiro multilayer structures as a function of the 
iteration k we found that the number of PGBs and their widths 
increase with k due to the increase of the number of layers from 
one iteration to another, which results in the generation of 

multiple internal reflections in the structure giving rise to more 
constructive interferences. The latter prevents certain 
wavelengths from propagating in the system in order to 
increase the width of the PBG. Additionally, the influence of 
the index contrast on the photonic band gap was investigated. It 
was found that the width of the PBG is proportional to the 



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index contrast. Moreover, the number of PGBs increases 
considerably and their position shifts according to the index 
ratio so that the medium becomes more and more inhibiting for 
the propagation of light. Indeed, the increase of the index 
contrast generates more internal reflections on the interfaces of 
the high refractive indices’ layers. 

ACKNOWLEDGMENT 

We would like to acknowledge the research deanship of 
University of Hail for the funding under the project RG-
191250. 

REFERENCES 

[1] K. Yasumoto, Electromagnetic theory and applications for photonic 

crystals, CRC Press, 2006 

[2] M. Johri, Y. A. Ahmed, T. Bezboruah, “Photonic band gap materials: 
Technology, applications and challenges”, Current Science, Vol. 92, No. 

10, pp. 1361-1366, 2007 

[3] P. K. Bhattacharya, “Photonic crystal devices”, Journal of Physics D: 
Applied Physics, Vol. 40, No. 9, 2007 

[4] S. Hong-Yi, L. Zhi-Yean, G. Ben-Yuan, “Defect modes in multiple-
constituent one-dimensional photonic crystals examined by an analytic 

Bloch-Mode approach”, Chinese Physics Letters, Vol. 22, No. 2, pp. 
365-368, 2005 

[5] A. H. Aly, “Electromagnetic propagation characteristics in one 

dimensional photonic crystal”, in Wave Propagation in Materials for 
Modern Applications, InTech, pp. 193-200, 2011 

[6] W. A. Goddard, Handbook of nanoscience, engineering, and technology, 

CRC Press, 2003 

[7] I. Iliev, M. Nedelchev, E. Markov, “A novel 2D Z-shaped 
electromagnetic bandgap structure”, Engineering, Technology & 

Applied Science Research, Vol. 5, No. 1, pp. 760-763, 2015 

[8] D. Schechtman, I. Blech, D. Gratias, J. W. Cahn, “Metallic phase with 
long-range orientational order and no translational symmetry”, Physical 

Review Letters, Vol. 53, No. 20, Article ID 1951, 1984 

[9] A. Bousselmi, A. Gharsallah, T. P. Vuong, “A novel high-gain quad-
band antenna with AMC metasurface for satellite positioning systems”, 

Engineering, Technology & Applied Science Research, Vol. 9, No. 5, 
pp. 4581-4585, 2019 

[10] M. Rattier, H. Benisty, C. J. M. Smith, A. Bernaud, D. Cassagne, C. 
Jouanin, T. F. Krauss, C. Weisbuch, “Performance of waveguide-based 

two-dimensional photonic-crystal mirrors studied with Fabry-Perot 
resonators”, IEEE Journal of Quantum Electronics, Vol. 37, No. 2, pp. 

237-243, 2001 

[11] J. H. Kim, L. Chrostowski, “Fabry-Perot cavity design in AlGaAs/GaAs 
using a photonic crystal slab for a resonant cavity detector”, 19th Annual 

Meeting of the IEEE Lasers and Electro-Optics Society, Montreal, 
Canada, October 29-November 2, 2006 

[12] J. P. Allouche, P. Liardet, “Generalized Rudin–Shapiro sequences”, Acta 

Arithmethmetica, Vol. 60, No. 1, pp. 1-27, 1991 

[13] V. Agarwal, M. E. Mora-Ramos, B. Alvarado-Tenorio, “Optical 
properties of multilayered Period-Doubling and Rudin-Shapiro porous 

silicon dielectric heterostructures”, Photonics and Nanostructures–
Fundamentals and Applications, Vol. 7, No. 2, pp. 63-68, 2008 

[14] W. Rudin, “Some theorems on Fourier coefficients”, Proceedings of the 

American Mathematical Society, Vol. 10, No. 6, pp. 855-859, 1959 

[15] H. S. Shapiro, Extremal problems for polynomials and power series, 
MSc Thesis, Massachussetts Institute of Technology, 1951 

[16] Z. Y. Li, “Principles of the plane-wave transfer-matrix method for 
photonic crystals”, Science and Technology of Advanced Materials, Vol. 

6, No. 7, pp. 837–841, 2005 

[17] Y. Zeng, Y. Fu, X. Chen, W. Lu, H. Agren, “Extended planewave-based 
transfer-matrix approach to simulating dispersive photonic crystals”, 

Solid State Communications, Vol. 139, No. 7, pp. 328-333, 2006 

[18] A. Mouldi, M. Kanzari, “Design of microwave devices exploiting 
Fibonacci and hybrid periodic/Fibonacci one dimensional photonic 

crystals”, Progress In Electromagnetics Research B, Vol. 40, pp. 221–
240, 2012 

[19] K. Ben Abdelaziz, J. Zaghdoudi, M. Kanzari, B. Rezig, “A broad 

omnidirectional reflection band obtained from deformed Fibonacci 
quasi-periodic one dimensional photonic crystals”, Journal of Optics A: 

Pure and Applied Optics, Vol. 7, pp. 544–549, 2005