Mode analysis of 2D photonic quasicrystals based on an

approximate analytic model

Ali Rostami a,b,*, Samiye Matloub a, Mohammad Kazem Moravvej-Farshi c

a Photonics and Nano Crystal Research Lab. (PNRL), Faculty of Electrical and Computer Engineering,

University of Tabriz, Tabriz 51666, Iranb School of Engineering-Emerging Technologies, University of Tabriz, Tabriz 51666, Iran

c Department of Electrical and Computer Engineering, Advance Device Simulation Lab. (ADSL),

Tarbiat Modares University, 14115-143, Tehran 1411713116, Iran

Received 2 January 2010; received in revised form 16 July 2010; accepted 21 July 2010

Available online 29 July 2010

Abstract

In this work, we have developed a semi-analytical model based on perturbation theory for evaluating optical properties of

photonic quasicrystals. Although numerical techniques are being used for determining characteristics of photonic quasicrystals, it is

believed that analytical studies based on approximations help us to understand the physical properties and treatment of the structure

more easily. Towards that end, the use of perturbation method can be beneficial in approximating the photonic band structure and

mode profile in photonic quasicrystals made of low dielectric contrast materials with high accuracy. It is shown that error incurred

by the approximation through the perturbation method can be estimated to be a few percent, in comparison to the more accurate

finite difference time domain method. In addition, the effect of variation in the dielectric contrast on the photonic band structure of

2D 12-fold photonic quasicrystal has been studied and it is presented that by decreasing dielectric contrast a smaller photonic band

gap is obtained, which is interesting for optical communication devices.

# 2010 Elsevier B.V. All rights reserved.

Keywords: Photonic quasicrystals; Perturbation theory; Pseudo-Brillouin zone; Band structure; Mode profile

www.elsevier.com/locate/photonics

Available online at www.sciencedirect.com

Photonics and Nanostructures – Fundamentals and Applications 9 (2011) 22–30

1. Introduction

During the past two decades, photonic quasicrystals

(PQCs) have attracted a great deal of attention due to their

ability to localize and control the flow of light within their

structure [1,2]. Quasicrystals are like crystals in that they

have long-range translational and orientational orders.

* Corresponding author at: Photonics and Nano Crystal Research

Lab. (PNRL), Faculty of Electrical and Computer Engineering, Uni-

versity of Tabriz, Tabriz 51666, Iran.

E-mail addresses: rostami@tabrizu.ac.ir (A. Rostami),

matloub@tabrizu.ac.ir (S. Matloub).

1569-4410/$ – see front matter # 2010 Elsevier B.V. All rights reserved.

doi:10.1016/j.photonics.2010.07.006

However, the translational order is not periodic and the

structure does not necessarily have crystallographic

rotational point symmetry [3]. These structures present

more significant features than conventional photonic

crystals (PCs), due to the degree of freedom that has been

concealed in aperiodic structures [3]. In addition, in the

low dielectric contrast regime PQC’s show an isotropic

and complete photonic band gap (PBG) due to their high

level of rotational symmetry [1,4,5]. Although PQCs and

PCs share many common features, some important

concepts such as Brillouin zone and Bloch’s theorem are

invalid in PQCs [6]. However, in a recently published

paper, a method has been presented in which the energy

A. Rostami et al. / Photonics and Nanostructures – Fundamentals and Applications 9 (2011) 22–30 23

spectrum and the eigenstates of a PQC can be obtained by

solving Maxwell’s equations in higher dimensions for

any PQC, to which a generalization of Bloch’s theorem

applies [7]. In addition, for analyzing PQCs, the concept

of pseudo-Brillouin zone (P-Bz) has already been

developed [8].

The exact theoretical prediction of the band

structures and mode profile of PQCs is currently a

numerical challenge and various research groups have

used different numerical techniques, such as finite

difference time domain (FDTD) analysis [2,9–12],

finite element method (FEM) [13], multiple scattering

method (MSM) [14], transfer matrix method (TMM)

[15] and plane wave expansion (PWE) method [4,16].

On the other hand, it is generally believed that analytical

studies based on some approximations can be helpful in

understanding physical properties. And last but not

least, existence of the complete PBG in PQC made of

low dielectric contrast materials makes the use of

perturbation technique beneficial in developing an

approximate analytic model for evaluating their band

structures with high precision and accuracy.

Such an approximation technique has already been

employed by Ochiai and Sakoda [17] for obtaining an

analytic model for extraction of the band structure of a

2D-hexagonal PC, nevertheless to the best of our

knowledge this work is the first instance of reporting

the use of perturbation technique for developing an

approximate analytic model for evaluating the band

structure and mode profile of 2D 12-fold PQC structures.

In implementing this technique, we have expanded the

refractive index of the PQC structure in terms of periodic

functions and have also developed a P-Bz for the PQCs

and finally using analytical perturbation theory, band

structure and mode profile of these structures have been

obtained. Furthermore, we have demonstrated how the

photonic band structure and its corresponding gap can be

affected by variations in the dielectric contrast.

Accordingly, the rest of the paper is organized as

follows. Section 2 is dedicated to the modeling of 2D

PQC. In Section 3, an approximate analytic model for

evaluating of the band structure and mode profile of

PQCs is developed in a perturbation technique frame-

work. The simulation results and a discussion of 2D 12-

fold PQC are presented in Section 4. Finally, the paper

ends with a conclusion in Section 5.

2. Modeling of 2D photonic quasicrystals

Before developing our analytic model by use of the

perturbation technique, we first construct the refractive

index for the PQC structure in the low contrast regime.

Then, a P-Bz for the structures is defined. At last, we use

the perturbation technique to solve the wave equations.

As mentioned earlier, the refractive index of

quasicrystals can be developed as a superposition of

periodic crystals refractive index [3]. A function is

defined as quasiperiodic if it is expressed as a sum of

periodic functions with periods, where at least some of

these periods are incommensurate (i.e. their ratio is

irrational). In other words, we can express the refractive

index of 2D PQCs in the same way provided that the

ratio of periods of periodic terms must be irrational.

Therefore, the refractive index of PQCs can be given as

nðrÞ ¼ n0 þ mX

K

nðKÞcosðKrÞ; (1)

where n0, n, r, K and m are average refractive index,

amplitude of periodic terms, the projection of r in the x–

y plane, the reciprocal lattice vector of quasicrystals and

the small perturbation parameter, respectively. In our

approximation we limit the summation to N vectors Ki.

The number (N = 5, 8, 10 or 12) determines the order of

the rotational symmetry of the quasicrystal pattern. We

make the simplification that the only reciprocal lattice

vectors in Eq. (1) are given by

Ki ¼2p

acos

2pi

N

� �; sin

2pi

N

� �� �; (2)

where a is the pitch of rods. For N even, the sum in

Eq. (1) is restricted to N/2 terms (that is because then

Ki = � Ki + N/2). If the amplitude of periodic terms, ni,

are much smaller in comparison to the constant part, n0,

the perturbation theory can be applied for determining

the eigenfrequency and eigenmodes of quasicrystal. It is

good to mention that there is some confusion about the

notion of quasicrystal. For some people every quasipe-

riodic structure with sharp diffraction peaks is a quasi-

crystal, while others make a distinction between

incommensurate modulated structures with periodic

basic structure [18–21] and quasicrystals where not

such a basic structure exists [22,23]. The non-crystal-

lographic symmetry is not crucial for a quasicrystal, but

in our case a structure with N-fold (N = 5, 8, 10 or 12)

symmetry makes it clear that it is a quasicrystal [24]. In

Eq. (1) the function n(r) is smooth and not localized.

But that is acceptable for a dielectric constant.

There are two ways out when we encounter this

situation. The suitable method that one usually used for

electrons and phonons is to study a periodic approx-

imant. In this case, a quasiperiodic structure in 2D is the

intersection of a periodic structure in 4D with the 2D

plane for our case. Then, the refractive index becomes a

A. Rostami et al. / Photonics and Nanostructures – Fundamentals and Applications 9 (2011) 22–3024

function in 4D and it may be written with a Bloch

expression. By changing slightly the axes, the structure

in 2D becomes periodic. In the components of the basic

vectors appears, for the case of N = 12, the irrational

numberffiffiffi3p

=2. This can be approximated in a continued

fraction way: the successive approximations are 1, 6/7,

13/16, 78/91, etc. In doing so the intersection in 2D

becomes periodic, but with increasing size of the unit

cell. However, in the examples studied, already for

small approximants (e.g. 6/7), the results for the

phonons and electrons in the approximant are very

similar to those in the quasicrystal. For an approximant

with 6/7 instead offfiffiffi3p

=2 the unit cell is still small, and

the Brillouin zone becomes finite, but the agreement

with the quasicrystal will be better. In this case, we have

a periodic structure that is similar to our quasicrystal,

and we can use the standard procedure [7,24,25]. The

other way is to consider a P-Bz. The notion of P-Bz is

not unique. One may define it as a region of points

closer to a diffraction point than to other diffraction

points above the certain intensity [8]. In the following

section, we have developed our model using the concept

of P-Bz.

3. Perturbation technique framework

For the purpose of evaluating photonic band

structure and mode profile of electromagnetic field,

Maxwell’s equations must be solved. Assuming lossless

and linear materials, the well known Maxwell’s

equations are written [26]

r� 1

eðrÞr �HðrÞ� �

¼ v2

c2eðrÞHðrÞ; (3a)

r �HðrÞ ¼ 0; (3b)

EðrÞ ¼ 1

ve0eðrÞr �HðrÞ (3c)

where H(r), E(r), e(r), e0, v, c, and r are the magnetic

field, electric field, dielectric constant, free space per-

mittivity, the light radian frequency, speed of light in free

space, and the position vector in a 3D coordinate system.

We identify the left side of Eq. (3a) as operator Q acting

on H(r) similar to eigenvalue equation. It should be

mentioned that the operator Q is a Hermitian operator

and thus the Maxwell’s equation is arranged as an

eigenvalue problem for the magnetic field H(r) and then

the E(r) can be calculated via Eq. (3c). For 2D case, the

eigenvalue equations are much simplified if the wave-

vector is parallel to the 2D plane. Therefore, the modes

must be oscillatory in the z-direction due to system’s

homogeneity in this direction, with no restrictions on the

wavevector along that direction (kz). By solving eigen-

value equation described by Eq. (3a), the eigenfrequen-

cies and eigenmodes of system are obtained.

In the case of 2D PCs, the system has discrete

translational symmetry in the x–y plane. So by applying

Bloch’s theorem, one can focus on the values of in-plane

wavevector kjj that are in the Brillouin zone.

Unfortunately, for quasiperiodic systems, Brillouin

zone in the usual sense does not exist. In these systems,

the reciprocal lattice vectors of a quasiperiodic structure

densely fill all reciprocal space. However, it is often

useful to choose a subset of basic reciprocal lattice

vectors that correspond to the relatively intense spots in

the diffraction pattern. In other words, quasicrystals do

not possess a Brillouin zone; nevertheless it is possible

to construct an analogue called the P-Bz which is

defined by lines bisecting the basic reciprocal lattice

vectors [4,8]. By defining P-Bz for quasicrystal, one

pays attention to the values of kjj in to this zone. The

label n (band number) is used to label the modes in order

of increasing frequency. Indexing the modes of

quasicrystal by kz and kjj and n, they take the familiar

form of Bloch-like states.

Hðn;kz;kjjÞðrÞ ¼ eikjjreikzzuðn;kz;kjjÞðrÞ (4)

In this equation, r is the projection of r in the x–y

plane and u(r) is a profile of modes and it is a

quasiperiodic function with Fourier components

belonging to the spots in the diffraction pattern. Also,

in this case kz is unrestricted and we mainly restrict

ourselves to in-plane (kz = 0) propagation. It should be

mentioned that we know it is an approximation, that is

supposed to work for k vectors inside the mentioned P-

Bz around the origin, and that the neglected components

are supposed to be small.

Right now, perturbation method is considered to

solve the Maxwell’s equation. Perturbation theory is an

efficient method to modal analysis of this system due to

small variation of dielectric constant which is con-

sidered as a small perturbation to homogeneous media.

The idea is to begin with the modes of idealized

homogeneous medium (as an unperturbed system), and

use analytical tools to approximately evaluate the effect

of small changes in the dielectric function (as a profile

of quasiperiodic function) on the modes and their

frequencies. For many realistic problems, the error in

this approximation is negligible.

The derivation of perturbation theory for Hermitian

eigen problem is straightforward and is covered in many

texts on quantum mechanics [27,28]. Suppose a

�;

A. Rostami et al. / Photonics and Nanostructures – Fundamentals and Applications 9 (2011) 22–30 25

Hermitian operator Q is altered by a small amount DQ.

The resulting eigenfrequencies and eigenmodes of the

perturbed operator can be written as series expansions, in

terms that depend on increasing powers of the perturba-

tion strength DQ. The resulting equation can be solved

order-by-order using only the eigenmodes of the

unperturbed operator. The methods of applying perturba-

tion theory according to our system have been discussed

in recently published papers [29,30]. We briefly review

this method, here. Eq. (3a) for the transverse magnetic

(TM) polarization and 2D case is considered.

� @

@x

1

eðrÞ@

@xþ @

@y

1

eðrÞ@

@y

� �HzðrÞ ¼

v2

c2HzðrÞ;

Qð2ÞHzðrÞ�v2

c2HzðrÞ;

(5)

where superscript (2) denotes the 2D operator. Then, by

substituting Eq. (4) in this equation and assuming the in-

plane propagation (kz = 0), the 2D operator can be

written as

Qð2Þuðn;kjjÞðrÞ�v2

c2uðn;kjjÞðrÞ;Q

ð2Þ

¼ � 1

eðrÞ@2

@x2þ @2

@y2

� �þ @

@x

1

eðrÞ@

@xþ @

@y

1

eðrÞ@

@y�kjj

2

�(6)

where in this notation the subscript n and kjj denotes the

band number and corresponding kjj in the P-Bz. After

that, according to Eq. (1), e�1(r) is expanded and

substituted into the 2D eigenvalue problem given by

Eq. (6). The unperturbed Hermitian operator Q̂ð2Þ

and

the perturbed operator DQ̂ð2Þ

are obtained as follows

ðQ̂ð2Þ þ DQ̂ð2ÞÞu0ðn;kjjÞðrÞ�V0ðn;kjjÞu

0ðn;kjjÞðrÞ; (7a)

Q̂ð2Þ � � 1

n20

@2

@x2þ @2

@y2

� �� k2

jj

� �; (7b)

DQ̂ð2Þ � 2n0

XN

i¼1

ni cosðKirÞ@2

@x2þ @2

@y2

� �

þ @

@x2n0

XN

i¼1

ni cosðKirÞ !

@

@x

þ @

@y2n0

XN

i¼1

ni cosðKirÞ !

@

@y; (7c)

where u0ðn;kjjÞðrÞ and V0ðn;kjjÞ ¼ v2=c2 are the eigen-

modes and eigenfrequencies of the perturbed operator

that can be written as series expansion. Now, perturbed

eigenfrequencies and eigenmodes are substituted into

Eq. (6). Eigenmodes and eigenfrequencies of perturbed

system can be expressed as

u0ðn;kjjÞðrÞ ¼ uð0Þðn;kjjÞðrÞ þ u

ð1Þðn;kjjÞðrÞ (8a)

V0ðn;kjjÞ ¼ Vð0Þðn;kjjÞ þV

ð1Þðn;kjjÞ; (8b)

where the superscripts denote the order of eigenmodes

and eigenfrequencies corrections. uð0Þðn;kjjÞðrÞ, V

ð0Þðn;kjjÞ

are

the solution of unperturbed system, which can be

obtained using different methods such as finite differ-

ence method. In addition, uð1Þðn;kjjÞðrÞ and V

ð1Þðn;kjjÞ

are the

correction of eigenmodes and eigenfrequencies, that is

calculated as follows [29,30],

uð1Þðn;kjjÞðrÞ ¼

Xm 6¼ n

Mmn

ðVð0Þðn;kjjÞ �Vð0Þðm;kjjÞÞ

RRuð0Þ�ðm;kjjÞu

ð0Þðm;kjjÞdr

ð0Þðn;kjjÞðrÞ;

Vð1Þðn;kjjÞ ¼

RRu�ð0Þðn;kjjÞðrÞDQ̂

ð2Þuð0Þðn;kjjÞðrÞ drRR

uð0Þðn;kjjÞðrÞ

��� ���2dr

(9b)

where Mmn is defined as

Mmn ¼Z Z

u�ð0Þðm;kjjÞðrÞDQ̂

ð2Þuð0Þðn;kjjÞðrÞdr (10)

It is necessary to notice that if there is degeneracy in

the eigenfrequency of unperturbed system, which it

means any two zeroth order eigenmodes had the same

frequency, the perturbation theory developed before is

no longer valid. Eventually, the degenerate perturbation

theory can be used to identify the eigenmodes of

perturbed system [27,28]. The eigenfrequencies and the

coefficients of g-fold degenerate eigenmodes expansion

aam, are calculated by solving the following equation

Xg

m¼1

ðM pm �Vð1Þðn;kjjÞdm pÞaa

m ¼ 0: (11)

4. Simulation results and discussion

In this section, we present the numerical results

obtained from the analytical model developed in

Section 3. We consider a 2D 12-fold PQC made of

air rods in a dielectric base with pitch size of a. The

refractive indices vary around an average value of n0 in

a quasiperiodic manner defined by Eq. (1). The 2D cross

sectional view of this PQC is illustrated in Fig. 1a, and

values of all geometrical and physical parameters are

given in Table 1. It is good to mention that initial values

A. Rostami et al. / Photonics and Nanostructures – Fundamentals and Applications 9 (2011) 22–3026[(Fig._1)TD$FIG]

Fig. 1. (a) Modeling of refractive index of 2D PQC is composed of air rods arranged on a pitch of 260 nm and the refractive index of back ground

material is chosen 2.04. Modeling parameters can be given by n0, ni, a, and N equal to 1.69,�0.115, 260 nm and 6, respectively. (b) Corresponding

reciprocal lattice of 2D 12-fold PQC, where the P-Bz is indicated by white dodecagon.

of n0 and ni have been chosen as the average refractive

index and contrast, respectively. Then, by fitting the

parameters in Eq. (1), the appropriate parameters for n0

and ni can be found. As mentioned before, the parameter

ni should be much smaller than n0 and choosing the

parameters as given in Table 1 will satisfy this

condition. So, we can use the perturbation method

which was examined in previous section.

In order to start simulation process, first of all P-Bz

must be defined for the present structure. In this way,

diffraction pattern of 2D 12-fold PQC is calculated and

it is depicted in Fig. 1b. In fact, the concept of P-Bz is

used to give a first understanding of the interaction of

plane waves in the system. Two waves in 2D interact if

their wave vectors differ by a wave vector Ki from the

Fourier module (i.e. the projection of the 4D reciprocal

lattice). The strength of the interaction depends on the

intensity of the peak at Ki in the diffraction pattern.

Therefore, the first P-Bz boundary is halfway between

the origin and the strongest diffraction peaks. The P-Bz

is indicated in Fig. 1b by white dodecagon.

The band structure of the unperturbed structure

(empty lattice) is shown in Fig. 2a, for corresponding k-

Table 1

Structural parameters for modeling refractive index of 2D 12-fold

PQC.

Parameter Value

nbackground 2.04

n0 1.69

ni �0.115

a 260 nm

N 6

vector in P-Bz. This figure is evaluated from zeroth

order equation of perturbation method using finite

difference technique with perfectly matched layered

(PML) boundary condition. As indicated in Fig. 2a,

there are some points with high symmetry for which the

eigenfrequency of unperturbed operator is degenerated

(i.e. they have different eigenmodes with equal

eigenfrequency). The correction of eigenfrequency of

perturbed operator in these cases can be obtained using

Eq. (11) [29]. In other points of Fig. 2a where there is no

degeneracy, correction of eigenfrequency can be

calculated using Eq. (9b). Finally, the band structure

of 2D 12-fold PQC is illustrated in Fig. 2b.

To compare the calculation results with more

accurate numerical methods, FDTD method based on

Yee’s algorithm which provides accurate numerical

solution to Maxwell’s equations is used to compute the

band structure of 2D 12-fold PQC. FDTD method is

widely used to characterize PQC structures [2,9–12].

The TM band structure of 2D 12-fold PQC for

corresponding k-vector in the P-Bz and based on

FDTD method is depicted in Fig. 2b, too. A careful

inspection of this comparison shows that the error

incurred by the approximation through the perturbation

technique is an overestimation of about 2–3%, in

comparison to the more accurate FDTD method.

However, the error on the gap width has approximately

been 10%. Knowing this reasonable error, we use our

developed approximate analytic model to evaluate band

structure for the 2D 12-fold PQC.

Moreover, electromagnetic field profile of PQC can

be given based on the method developed in previous

section. In general, magnetic field is given by Eq. (4). In

this equation, in-plane propagation (kz = 0) is consid-

A. Rostami et al. / Photonics and Nanostructures – Fundamentals and Applications 9 (2011) 22–30 27[(Fig._2)TD$FIG]

Fig. 2. (a) Band structure of unperturbed structure (empty lattice) with homogeneous refractive index n0 for TM polarization. (b) Comparison the

TM band structure for a considered 2D 12-fold PQC evaluated by perturbation theory (solid lines marked as blue circles) with those obtained from

FDTD numerical method (indicated as purple square). The inset is zooming out of indicated region to show that the frequencies obtained using the

perturbation approximation match the numerical FDTD results within 2–3%. However, the error on the gap width has approximately been 10%. (For

interpretation of the references to color in this figure legend, the reader is referred to the web version of the article.)

ered and uðn;;kjjÞðrÞ for any points of band structure and

for the determined kjj, can be calculated using Eq. (8a).

In Eq. (8a), the zeroth order of mode profile has been

calculated by solving zeroth order equation based on

finite difference method with PML boundary condition

and the first order correction can be obtained based on

Eqs. (9a) or (11), according to the order of degeneracy

of unperturbed eigenmodes. For instance, at the point

X0, in which the kjj is (1/2, 0) in units of 2p/a, there is 2-

fold degeneracy between first and second bands. The

degeneracy is lifted using Eq. (11) and the coefficients

of eigenmodes correction can be calculated. The profile

of magnetic field intensity is illustrated in Fig. 3. The

frequencies of modes for the first and second band are

0.3219 and 0.3927 in units of (va/2pc), respectively. In

addition, at the point of J0, in which the kjj is (1/2, 1/

(2 tan(p/12)) in units of 2p/a, with 3-fold degeneracy,

the profile of modes is shown in Fig. 4. The frequencies

of modes for the first, second and third band are 0.3607,

0.4307 and 0.4403 in units of (va/2pc), respectively. It

is good to mention that in these figures the intensity of

the modes are illustrated and 12-fold symmetry of mode

profile in the central part is seen.

In the following paragraphs, we have investigated the

effect of variation in the dielectric contrast on the

photonic band structure of 2D 12-fold PQC. Therefore,

the eigenfrequencies of the foregoing structure for

variation of background material refractive index is

sketched in Fig. 5. It should be noticed that the

alteration in the refractive index has been restricted to

A. Rostami et al. / Photonics and Nanostructures – Fundamentals and Applications 9 (2011) 22–3028[(Fig._3)TD$FIG]

Fig. 3. The profile of magnetic field intensity for odd parity at the point X0, in which the kjj is (1/2, 0) in units of 2p/a, for considered 2D 12-fold

PQC. (a) For the first band and (b) for the second band. The frequencies of modes for (a) and (b) are 0.3219 and 0.3927 in units of (va/2pc),

respectively.

the values for which the condition of perturbation theory

(i.e. ni� n0) be satisfied. Furthermore, the variation of

the band structure in Fig. 5 has been calculated for two

first bands in special value of k-vector inside P-Bz and

between the X and J points. The needed parameters for

modeling refractive index of 2D 12-fold PQC according

to Eq. (1) are listed in Table 2.

To study how the photonic band structure of

PQC and its corresponding gap is affected by

variations in the dielectric contrast in more detail,

the alteration of central frequency of PBG and

relative PBG has been investigated. In this way,[(Fig._4)TD$FIG]

Fig. 4. The profile of magnetic field intensity for odd parity at the point J0, in

12-fold PQC. (a) For the first band, (b) for the second band and (c) for the thir

and 0.4403 in units of (va/2pc), respectively.

according to Fig. 5, the relative PBG between first

and second band is defined as

gap ¼ Dv

v0

¼ 2minkjj 2 P-Bz : fvhðkjjÞg �maxkjj 2P-Bz : fvlðkjjÞgminkjj 2 P-Bz : fvhðkjjÞg þmaxkjj 2P-Bz : fvlðkjjÞg

(12)

where the minima and maxima are taken over all kjj on

the boundary of the P-Bz, and vl and vh are bands just

below and just above the gap, respectively. The propor-

which the kjj is (1/2, 1/(2 tan(p/12)) in units of 2p/a, for considered 2D

d band. The frequencies of modes for (a), (b) and (c) are 0.3607, 0.4307

A. Rostami et al. / Photonics and Nanostructures – Fundamentals and Applications 9 (2011) 22–30 29[(Fig._5)TD$FIG]

Fig. 5. First and second bands of considered 2D 12-fold PQC’s TM

band structure between X and J points with variation of background

material refractive index as a parameter. The PBG between first and

second band is indicated for one case.

[(Fig._6)TD$FIG]

Fig. 6. Central frequency of PBG between first and second band

(orange circles, left axis) and relative PBG between first and second

band (blue stars, right axis) versus background material index for

considered 2D 12-fold PQC. (For interpretation of the references to

color in this figure legend, the reader is referred to the web version of

the article.)

tional effect of dielectric contrast variation on the

relative PBG and central frequency is depicted in

Fig. 6. As presented in Fig. 6, by increasing dielectric

contrast, PBG has been larger and a little bit reduction in

central frequency has been observed. For description of

the illustrated simulation results, we had better mention

that a large band of frequencies around central frequen-

cy can be considerably diffracted by increasing dielec-

tric contrast in PQC structures. On the other hand, since

the reflection coefficient directly corresponds to dielec-

tric contrast, the relative PBG of the system has been

increased by increment of the dielectric contrast. More-

over, for central frequency the situation is completely

inverted, because by increasing of the dielectric contrast

Table 2

The needed parameters for modeling refractive index of 2D 12-fold

PQC.

nbackground n0 ni

1.7 1.47 �0.079

1.8 1.54 �0.089

1.9 1.6 �0.1

2.04 1.69 �0.115

2.1 1.75 �0.12

2.2 1.8 �0.14

2.3 1.9 �0.16

2.4 1.95 �0.16

2.5 2 �0.169

2.6 2.07 �0.18

2.7 2.15 �0.195

2.8 2.2 �0.2

2.9 2.28 �0.217

optical phase condition for resonance has been obtained

in lower frequencies. Therefore, we have expected to

decrease the central frequency by increasing the dielec-

tric contrast.

Indeed, the main idea of investigation into the

alteration in dielectric constant has been to present how

much the relative PBG and central frequency will

change due to variation of dielectric constant. As we

know, the dielectric constant varies because tempera-

ture dependency. Therefore, the examination of PBG

dependency on the dielectric contrast will be important

for design of optical filters, multiplexers, etc. from OIC

design’s point of view.

5. Conclusion

In summary, optical characteristics in 2D 12-fold

PQCs were successfully examined based on semi-

analytical perturbation method. According to the fact

that a complete PBG can be observed in PQCs, when

low dielectric contrast materials is used, developing

perturbation method in this case is so good and

applicable for modal analysis. By defining P-Bz, the

Bloch theory was applied to the PQCs and conse-

quently, based on perturbation theory, the band structure

and its corresponding mode profile have been calcu-

lated. Additionally, for the points with high symmetry,

degenerate perturbation theory was used in order to

extract the precise band structure. It has been shown that

the error incurred by the perturbation method is

estimated to be a few percent, in comparison to the

A. Rostami et al. / Photonics and Nanostructures – Fundamentals and Applications 9 (2011) 22–3030

more accurate FDTD numerical method. Moreover, it

was pointed out that PBG for 2D 12-fold PQC is

reduced, as a result of decrement in dielectric contrast

and central frequency is increased. In the end, the

proposed semi-analytical method presents a conceptual

view point for interested applied designers.

Acknowledgment

One of the authors, Samiye Matloub, would like to

appreciate Prof. Ted Janssen in the Institute for

Theoretical Physics at the University of Nijmegen for

constructive and useful discussions about the modeling

of PQC structures.

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