Optics and Photonics Journal, 2012, 2, 270-277
http://dx.doi.org/10.4236/opj.2012.24033 Published Online December 2012 (http://www.SciRP.org/journal/opj)
Studies of Optical Properties of Symmetrical
Quasi-Periodic Photonic Crystals
Jihene Zaghdoudi, Nadia Maaloul, Mounir Kanzari
Photovoltaic and Semiconductor Materials Laboratory, Ecole National d’Ingénieurs de Tunis, Tunis, Tunisia
Email: jihene_zaghdoudi@yahoo.fr
Received August 12, 2012; revised September 14, 2012; accepted September 28, 2012
ABSTRACT
Using the transfer matrix method approach (TMM), the present paper attempts to determine the optical properties of
quasi-periodic symmetric one-dimensional photonic systems. In addition, it studies hybrid hetero-structure systems
constructed by using periodic and quasi-periodic multilayer systems. The effect of symmetry applied to symmetric mul-
tilayer systems results in the appearance of optical windows at the photonic band gaps (PBG) of the system. The use of
hybrid symmetric systems, at normal incidence in the visible range, show that the complete photonic band gap is the
sum of bands from individual systems. The results show also that the width of the PBG depends on the parameters and
nature of the built system.
Keywords: Symmetrical Quasi-Periodic Photonic Crystals; Hybrid Photonic Crystals; Photonic Band Gap
1. Introduction
Unlike electrical materials, today photonic crystals offer
the prospect of controlling the flow of photons in the
dielectric or metallic materials due to their periodic
structure [1].
The speed at which light moves, and the fact that pho-
tons do not tend to interact with transparent matter, is of
enormous benefit to us. It allows us to transmit data over
long distances.
The advent of research into slow light is anticipated to
bring in a wealth of applications, especially in the fields
of telecommunications and optical data processing.
Using light smartly offers many opportunities. Slow
light promotes stronger light-matter interaction, it offers
additional control over the spectral bandwidth of this
interaction and it allows us to delay and temporarily store
light in optical memories [2,3].
Quasi-periodic photonic crystals are dielectric struc-
tures with non-periodic modulations of the refractive
index. These systems can be considered as suitable mod-
els to describe the transition from the perfect periodic
structure to the random structure [4]. In this work, we
have investigated the optical properties of the quasi-pe-
riodic one-dimensional multilayer by using the symmet-
ric structure.
In the first part, we have evaluated the effect of apply-
ing symmetry in quasi-periodic systems.
In the second part, we have studied the reflection prop-
erties of one-dimensional hybrid systems formed by
combinations of periodic and quasi-periodic multilayer.
The quasi-periodic photonic crystals (PCs) used in these
hybrid structures are the symmetric Fibonacci sequence
and the Thue-Morse sequence. The numerical results are
presented in the visible spectral range [0.3, 1] µm for
normal incidence.
2. Problem Formulation and
Quasi-Periodic Models
The method used to calculate the optical response of
symmetric quasiperiodic systems in one-dimensional (1D)
photonic crystals is the transfer matrix method (TMM)
described by Yeh [5]. This method is widely applied for
calculating the transmission and reection spectra of lay-
ered structures because it is quite simple and at the same
time, it is a very powerful tool for simulation of light
propagation through the layered structures and for calcu-
lating the matrix product very quickly.
TMM method consists in the calculation of the back-
ward E and the forward E+ propagating electric field
components. This method shows that the relation be-
tween the amplitudes of the electric fields between two
different planes including a stratified medium is given by
the following matrix product:
01
123 1
01
m
m
m
EE
CCC C
EE


 
 
 
 
where
j
C represent the product of the propagation ma-
C
opyright © 2012 SciRes. OPJ
J. ZAGHDOUDI ET AL. 271
trix
p
r by the interface matrix given respec-
tively by:
Cint
C


1
int
1
exp i01
1
;1
0expi
j
j
pr
j
j
j
r
CC
r
t








;
where
j
t is the Fresnel transmission,
j
r is the reflec-
tion coefficients and 1
j
j
is the change in the phase of
the wave between the and layer [6].
th
1th
j
The values of the change in the phase of the wave are
given by the following equations:
0
1111
for
j
0
2πˆcos 1
jjj
nd j



1
j
d is the thickness layer and

1th
j
is the
wavelength of the incident wave in vacuum.
All the results in this work are given normal incidence,
so the transmittance T for both polarizations is the same.
2.1. Symmetric Fibonacci Structure
The symmetric Fibonacci sequences are multilayer
structures obtained with two different materials H and B,
with refractive index nA and nB respectively [7]. The jth
generation of this sequence can be expressed as

,
nn
SFGn
2
, ,
n
H
1
H

2
H
where Gn and Hn are Fibonacci sequence; they obey by
the following recursion relations,
 
211 for1
nnnn n
GGGH Hn
 

With
00 1
and .GHB GH
Therefore,
21 1n nnnn
SGGH

As an example, the third sequence of symmetric Fibo-
nacci sequences is .

S HLHHLH
3
The number of layers depends on the order of the sym-
metric Fibonacci sequence.
22. Thue-Morse Structure
The Thue-Morse sequence is defined by the recursive
relation as follows:


0
1
1
12
n
n
nn
TH
THL
TTT

where H indicates the high refractive index layer and B is
the low refractive index layer.
The number of the nth iteration is 2n [8].
In this work, we will use only the symmetric Thue-
Morse structure.
3. Result and Discussion
In the following numerical investigation, we chose the
titanium TiO2 (H) and the dioxide of Silicon SiO2 (L) as
two elementary layers, with refractive indices nH = 2.3
and nL = 1.45 respectively. The optical thickness of each
layer has been chosen to satisfy the Bragg condition,
where, λ0 is the reference wavelength which is equal to
0.5 µm.
Here we study the optical properties for normal inci-
dence, in the spectral range
0.3, 1μm which corre-
sponds to
00.5, 1.66

. We notice that the transfer
matrix Method [4] is used to study the optical properties
of a one-dimensional multilayer system.
3.1. Hybrid H(LH)j/SF(n)/H(LH)j Systems
In this part, we are interested in the multilayer system
composed of symmetric Fibonacci sequence (SFn) sand-
wiched between two periodical multilayer systems
H(LH)j (Bragg mirror). Where (n) and (j) are the number
of iterations of the symmetric Fibonacci and the periodic
systems respectively.
Figure 1 shows an example of the geometry of hybrid
H(LH)j/SFn/H(LH)j system.
3.1.1. The Optimization of the Repetition Number j of
the Periodic System
In order to optimize the repetition number j of the peri-
odic multilayer structure H(LH)j we study under normal
incidence the optical proprieties of the system H(LH)j/
SFn/H(LH)j.
So we fixed the number of iteration of the symmetric
Fibonacci n to 4 and let j vary from 3 to 10. Figure 2
show the numerical results for different values of j.
From Figure 2, it is clear that the reflection spectra
present the same number of peaks. This number is fixed
to 1. Moreover, we note that:
The position and the intensity of this peak are the
same for different values of j (Ipeak = 96%).
The increase of j induces a reduction in full width at
half maximum (FWHM) for this peak as shown in
Figure 3.
Transmitted
Wave
Reflected
Wave
Incident
Wave
H(LH)
j
SF
n
H(LH)
j
Figure 1. Example of the geometry of the hybrid system
formed by a symmetric Fibonacci sequence sandwiched
etween two periodic structures. b
Copyright © 2012 SciRes. OPJ
J. ZAGHDOUDI ET AL.
Copyright © 2012 SciRes. OPJ
272
0.6 0.811.2 1.4 1.6
0
0.2
0.4
0.6
0.8
1
R
0/
j = 3
λ0/λ
0.6 0.811.2 1.41.6
0
0.2
0.4
0.6
0.8
1
R
0/
j = 5
λ
0
/
λ
0.6 0.811.2 1.4 1.6
0
0.2
0.4
0.6
0.8
1
R
0/
j = 8
λ
0
/
λ
0.6 0.8 11.21.4 1.6
0
0.2
0.4
0.6
0.8
1
R
0/
j = 10
λ0/λ
Figure 2. Reflection spectrum versus λ0/λ for the H(LH)j/SF4/H(LH)j system for different j values.
345678910
0
0.5
1
1.5
2x 1 0
-3
Repetition number j of the periodic system
FWHM (µm)
Figure 3. Plot of the average of FWHM of the peak versus
the repetition number j of H(LH)j/SF4/H(LH)j.
We notice that when j is equal to 8 the FWHM of the
peak is almost zero. Then we can conclude that the opti-
mal value of j which can be chosen for the configuration
H(LH)j/SF4 /H(LH)j is equal to 8.
3.1.2. Numerical Results of the Study Configuration
H(LH)8/SFn /H(LH)8
Now we move to study the reflection spectrum of the
configuration H(LH )8/SFn/H(LH)8 for the optimal value j
= 8 and for different values of n ranges from 3 to 9.
Here we note the different systems by [H(LH)8/SFn/
H(LH)8]p where p represents the total number of layers.
Figure 4 shows that the number of peaks increases with
n. We notice that the variation of values n affects the
width of the PBG of the system.
The most outstanding result of this study is the Bragg-
PBGs covering the spectral range existing between the
two PBGs of the symmetric Fibonacci system.
From Figure 5 we can see that for each iteration the
number and the position of the peaks are symmetric about
the central peak (λ0/λ = 1). In Table 1, we present the
position X and the intensities I of the peaks also the width
(
0/
) of the PBG for some iterations.
According to the results obtained, we notice that the
peaks intensities are grater or equal to 96%. In addition
there are two types of peaks:
The peaks which are located at the band gap of the
symmetric Fibonacci system.
The peaks which are concentrated in the band gap
resulting from periodic systems. The peak number on
either sides of the central peak depends on the parity
of the number of iterations. It is odd when the number
of iterations is even and vice versa.
3.2. Hybrid H(LH)j/Tm/H(LH)j Systems
At this juncture, the reflection spectra are extricated for
1D hybrid quasi-periodic multilayer stack at normal in-
cident wave constructed through the use of Thue-Morse
sequence (Tm) intercalated between two periodic systems
H(LH)j. Where m and j are the numbers of iterations of
Thue-Morse structure and periodic systems respectively.
Figure 6 shows an example of the geometry of hybrid
J. ZAGHDOUDI ET AL. 273
0.6 0.811.21.4 1.6
0
0.5
1
R
0/
H(LH)
8
[H(LH)
8
/SF
3
/
H(LH)
8
]
40
λ
0
/λ
0.6 0.811.21.4 1.6
0
0.5
1
R
0/
H(LH)
8
[H(LH)
8
/SF
4
/ H(LH)
8
]
44
λ
0
/λ
0.6 0.8 11.2 1.4 1.6
0
0.5
1
R
0/
H(LH)
8
[H(LH)
8
/SF
5
/ H(LH)
8]
50
λ
0
/λ
0.6 0.811.21.4 1.6
0
0.5
1
R
0/
H(LH)
8
[H(LH)
8
/SF
6
/ H(LH)
8
]
60
λ
0
/λ
0.6 0.811.2 1.41.6
0
0.5
1
R
0/
H(LH)
8
[H(LH)
8
/SF
7
/ H(LH)
8]
76
λ
0
/λ
0.6 0.811.21.4 1.6
0
0.5
1
R
0/
λ
0
/
λ
H(LH)
8
[H(LH)
8
/SF
8
/ H(LH)
8
]
102
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J. ZAGHDOUDI ET AL.
274
0.6 0.811.2 1.4 1.6
0
0.5
1
R
0/
λ
0
/λ
H(LH)
8
[H(LH)
8
/SF
9
/ H(LH)
8
]
144
0.6 0.8 11.2 1.4 1.6
0
0.5
1
R
0/
SF
9
)
110
λ
0
/
λ
Figure 4. Reflection spectra for the H(LH)8/SFn/H(LH)8 structure for different n values.
0.7 0.8 0.911.1 1.2 1.3
5
6
7
8
9
n
0/pic
λ0pic
Figure 5. Peaks position for Hybrid H(LH)8/SFn/H(LH)8 systems.
H(LH)j/Tm/H(LH)j system.
To illustrate the reflection spectra properties in the
spectral range
0.3, 1μm of the hybrid H(LH)j/T(m)/
H(LH)j system, we start by determining the optimal value
of j.
3.2.1. The optimization of Repetition Number j of the
Periodic System
To determine the optimal value of j of the periodic sys-
tem H(LH)j, we study under normal incidence the optical
response of the hybrid H(LH)j/Tm/H(LH)j system for dif-
ferent values of j and by fixing m to 4.
Figure 7 presents the numerical results for different j
values.
We notice that the reflection spectra present the trans-
mission peaks at the photonic band gap. The variation of
the full width at half maximum (FWHM) of the central
peak, as shown in Figure 8, indicate that the optimal
value of j is equal to 8.
3.2.2. Variation Effect of the Repetitive Number m
For the optimal value j = 8, we study the reflection spec-
tra properties of the configuration H(LH)8/Tm/H(LH)8
for
different values of m. It should be noted that we choose
the even number of m because the Thue-Morse with even
iteration values can be builds a symmetric system.
Figure 9 shows the reflection spectra generated by this
hybrid photonic system. It is clear that if m increases the
Copyright © 2012 SciRes. OPJ
J. ZAGHDOUDI ET AL. 275
Table 1. Position and intensities of the peaks and the width of the PBG for different value of n.
n = 5
Structure Δ (λ0/λ) Δλ (µm) Peak
Position Intensities
H(LH)8/SF5/H(LH)8)50 0.1498 0.0753 x1 = 1 λ1 = 0.5 µm I1 = 96%
n = 6
Structure Δ (λ0/λ) Δλ (µm) Peak
Position Intensities
H(LH)8/SF6/H(LH)8)60 0.2396 0.1215 x1 = 0.9517 λ1 = 0.525 µm I1 = 96.197%
x2 = 1 λ2 = 0.5 µm I2 = 96%
x3 = 1.048 λ3 = 0.477 µm I3 = 96.197%
n = 7
Structure Δ (λ0/ λ) Δλ (µm) Peak
Position Intensities
H(LH)8/SF7/H(LH)8)76 0.6304 0.350 x1 = 0.8647 λ1 = 0.5782 µm I1 = 98.116%
x2 = 0.8768 λ2 = 0.5702 µm I2 = 97.652%
x3 = 0.9296 λ3 = 0.5379 µm I3 = 96.436%
x4 = 0.9748 λ4 = 0.5129 µm I4 = 96.052%
x5 = 1 λ5 = 0.5 µm I5 = 96%
x6 = 1.025 λ6 = 0.4878 µm I6 = 96.052%
x7 = 1.07 λ7 = 0.4673 µm I7 = 96.436%
x8 = 1.123 λ8 = 0.4452 µm I8 = 97.652%
x9 = 1.35 λ9 = 0.44405 µm I9 = 97.116%
Transmitted
Wave
Reflected
Wave
Incident
Wave
Thue-Morse
Figure 6. Example of the geometry of hybrid H(LH)j/Tm/H(LH)j systems.
0.6 0.811.2 1.4 1.6
0
0.2
0.4
0.6
0.8
1
R
0/
j = 1 0
0.6 0.811.2 1.4 1.6
0
0.2
0.4
0.6
0.8
1
R
0/
j = 8
0.6 0.811.2 1.4 1.6
0
0.2
0.4
0.6
0.8
1
R
0/
j = 5
λ
0
λ
0
λ
0
Figure 7. Reflection spectra for the H(LH)j/T4/H(LH)j system for different j values.
Copyright © 2012 SciRes. OPJ
J. ZAGHDOUDI ET AL.
276
345678910
0
0.5
1
1.5
2x 10-3
R epetition n um ber j of the periodic system
FW HM (µm )
Figure 8. Variation of the FWHM of the central peak.
0.6 0.811.2 1.4 1.6
0
0.5
1
R
0/
λ
0
/λ
H(LH)
8
[H(LH)
8
/T
2
/ H(LH)
8]
38
0.6 0.811.21.4 1.6
0
0.5
1
R
0/
λ
0
/λ
H(LH)8
[H(LH)8
/T
4
/
H(LH)8
]
50
0.6 0.811.21.4 1.6
0
0.5
1
R
0/
0.6 0.811.21.4 1.6
0
0.5
1
R
0/
λ
0
/λ
H(LH)
8
[H(LH)
8
/T
6
/ H(LH)
8
]
98
0.6 0.811.21.4 1.6
0
0.5
1
R
0/
T
6
λ
0
/λ
Figure 9. Reflection spectra for the hybrid H(LH)8/Tm/H(LH)8 systems for different m values.
Copyright © 2012 SciRes. OPJ
J. ZAGHDOUDI ET AL.
Copyright © 2012 SciRes. OPJ
277
width of the PBG is larger and the peak number increases.
We can notice that the effect of the Thue-Morse structure
of the system considered is obtained only from m = 4. In
the case where m = 2, it is clear that just the effect of
symmetry that appears (the appearance of a central peak at
the PBG of the system). It follows that when m increases
we have the following results:
The peak positions are symmetricaly about the central
peak.
The width of the complete PBG increases. Indeed, the
width of the band gap of the resulting system Br8/Tm/BR8
is the sum of the bandgaps of periodic systems and quasi-
periodic system (Thue-Morse). We note that the PBG of
the periodic system covers the spectral range between the
two PBGs of the Thue-Morse system. This result is very
important because it is possible to calculate the widh of
the PBG of the considered hybrid system from the spectra
of individual systems (Bragg and Thue-Morse) without
representing the spectrum of the entire structure.
4. Conclusions
The effect of symmetry applied to symmetric multilayer
systems results in the appearance of optical windows at
the PBG of the system which are symmetrical about the
value λ0/λ = 1.
The use of hybrid symmetric systems kinds PS/QPS/
PS and QPS/PS/QPS, at normal incidence in the visible
range show that:
The complete photonic bandage is the sum of individ-
ual bands from systems.
The width of the PBG and the optical windows, which
appear at this band, depend on the parameters and the
nature of the system built.
Also, the numbers of PBGs and transmission peaks are
controlled by the variations of the parameters n and m of
the symmetric Fibonacci and Thue-Morse structures. There-
fore, against the conventional quasi-periodic structures,
the hybrid systems offer the possibility of obtaining poly-
chromatic filters and of controlling the properties of these
filters.
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