Exactly solvable inhomogeneous Fibonacci-class quasi-periodic structures (optical filtering)

Optics Communications 247 (2005) 247–256

www.elsevier.com/locate/optcom

Exactly solvable inhomogeneous Fibonacci-classquasi-periodic structures (optical filtering)

Ali Rostami *, Samiyeh Matloub

OIC Research Lab., Faculty of Electrical Engineering, Tabriz University, Tabriz 51664, Iran

Received 18 June 2004; received in revised form 18 October 2004; accepted 23 November 2004

Abstract

In this paper, we will investigate the optical properties of Fibonacci-class quasi-periodic multilayer stacks with inho-

mogeneous index of refraction profile. In this work, the exact solution for inhomogeneous media and the transfer

matrix method for evaluation of quasi-periodic multilayer stack are examined. Also, the inhomogeneous index of

refraction effects on optical filtering properties of Fibonacci-class quasi-periodic multilayer is considered. We show that

using suitable inhomogeneous index of refraction profile we can obtain a narrowband and broadband optical filters,

which is very hard problem in the homogeneous multiplayer structures. In this work, we present semi-exact approach

for optical filter characteristic interpretation.

� 2004 Elsevier B.V. All rights reserved.

Keywords: Fibonacci-class; Inhomogeneous media; Quasi-periodic structures

1. Introduction

The propagation of optical waves in complex

dielectric systems is an intriguing research topic.

Complex dielectrics are dielectric structures in

which the refractive index varies over length scales

comparable to the wavelength of light. In disor-

dered materials light waves undergo a multiple

0030-4018/$ - see front matter � 2004 Elsevier B.V. All rights reserv

doi:10.1016/j.optcom.2004.11.105

* Corresponding author. Tel.: +98 411 3393724; fax: +98 411

3300819.

E-mail address: [email protected] (A. Rostami).

scattering process and are subject to unexpectedinterference effects. Multiple light scattering in dis-

ordered dielectrics shows many similarities with

the propagation of electrons in semiconductors

and various phenomena that are known for elec-

tron transport also appear to have their counter

part in optics [1,2]. Important examples are the

optical Hall effect and optical magneto resistance,

universal conductance fluctuations of light waves,optical negative temperature coefficient resistance

and light localization. On the other extreme, peri-

odic dielectric structures behave as a crystal for

ed.

mailto:[email protected]

248 A. Rostami, S. Matloub / Optics Communications 247 (2005) 247–256

light waves. In periodic structure the interference is

constructive in well-defined propagation direc-

tions, which leads to Bragg scattering and refrac-

tion. At high enough refractive index contrast,

propagation is prohibited in any direction with ina characteristics range of frequencies. This phe-

nomenon is referred to as a photonic band-gap

in analogy with the electronic band-gap in a semi-

conductor. Where as the knowledge on the propa-

gation of light waves in completely ordered and

disordered structures is now rapidly improving, lit-

tle is known about the behavior of optical waves in

the huge intermediate regime between total orderand disorder [3,4]. Quasi-crystals are non-periodic

structures that are constructed following a simple

deterministic generation rule. If made dielectric

material, the resulting structure has fascinating

optical properties. All of previous studies dis-

cussed the homogeneous quasi-crystal behaviors

[5]. In this paper, we will examine the inhomoge-

neous index of refraction effect on optical proper-ties (Bandwidth and stop band damping ratio) of

Fibonacci-class quasi-periodic layers. The control-

ling of optical filter characteristics such as band-

width is very important. In homogeneous

Fibonacci-class quasi-periodic optical filters, the

bandwidth closely depends on the index of refrac-

tion difference between layers. The large band-

width needs to large index of refractiondifference, which is very hard for implementation.

So, for improving this problem, we try to propose

the inhomogeneous index of refraction based

Fibonacci-class quasi-periodic structures as optical

filters.

The organization of this paper is as follows:

In Section 2, the mathematical model for inter-

pretation of inhomogeneous isotropic Fibonacci-class quasi-periodic multilayer stack is presented.

The result and discussion is presented in Section

3. Finally, the paper ends with a conclusion.

Fig. 1. Fibonacci-class quasi-p

2. Inhomogeneous isotropic Fibonacci-class

quasi-periodic multilayer stack

Fig. 1 illustrates the Fibonacci-class quasi-peri-

odic multi-layer stack. In this case, we assume thatthe index of refraction coefficient in layer A and B

are position dependent. The field distribution is

necessary for the calculation of the optical system

performance. Generally, obtaining the field distri-

bution exactly is very hard and the numerical ap-

proach is usually used. In this case, we will adopt

some special index of refraction coefficient profiles

having exact solution for the electromagneticfields.

By considering Fig. 2, the following equation is

used for light transmission in inhomogeneous iso-

tropic media [6].

d2

dX 2� 1

eðX ÞdeðX ÞdX

d

dXþ k20n1f ðX Þ

� �HY ðX Þ

¼ k2 � k20n20

� �HY ðX Þ; ð2:1Þ

where f(X) is position dependent part of the index

of refraction. Now, we define a new dimensionless

variable x = k0X, where k0 is free space wave vec-tor. According to [6], applying above defined new

variable and adopting the spatial function for the

index of refraction distribution ðf ðxÞ / � 14x2Þ,

we obtain the Schrodinger-like harmonic oscillator

differential equation and we can propose the fol-

lowing equation as a solution for the electromag-

netic fields as

HY ðx; z; tÞ ¼ HY ðxÞeiðxt�kzÞ;

where

HY ðxÞ ¼ CP ðxÞ þ DQðxÞ; � db

2< x <

db

2: ð2:2Þ

In Eq. (2.2), P(x) and Q(x) are corresponds to

the first and the second type of Hermit polynomi-

als and function, respectively, and are given as:

eriodic multilayer stack.

Fig. 2. One layer of Fibonacci-class quasi-periodic multilayer

stack.

A. Rostami, S. Matloub / Optics Communications 247 (2005) 247–256 249

PðxÞ ¼ffiffiffiffiffiffiffiffieðxÞ

pe�

14x2Hn

xffiffi2

p� �

;

QðxÞ ¼ffiffiffiffiffiffiffiffieðxÞ

pe�

14x2Qn

xffiffi2

p� �

:ð2:3Þ

Now, the field distribution for whole structure

using above obtained relations is given as:

HY ðxÞ ¼

A0e�ik0nbðxþdbÞ þ B0e

ik0nbðxþdbÞ; x < �db;

..

.

AiP ðx� ðxi � daÞÞ þ BiQðx� ðxi � daÞÞ; xi�1 < x < xi;

Aiþ1P ðx� ðxi � dbÞÞ þ Biþ1Qðx� ðxi � dbÞÞ; xi < x < xiþ1;

..

.

AF je�ik0nbðx�xF j Þ þ BF je

ik0nbðx�xF j Þx > xF j ;

8>>>>>>>>>>><>>>>>>>>>>>:

ð2:4Þ

where da and db are layer widths of A and B,

respectively. Also, na and nb are the index of

refraction coefficients for layer A and B, respec-

tively. If, we apply the boundary conditions fortangential components of the electric and the

magnetic fields in boundaries (xi), we obtain

the following relations between constants in

Eq. (2.4). For general starting at boundary

x = x0,

A0 þ B0 ¼ A1P ðx� ðx1 � dxÞÞjx¼x0

þ B1Qðx� ðx1 � dxÞÞjx¼x0;

� ik0nbA0 þ ik0nbB0

¼ A1P 0ðx� ðx1 � dxÞÞjx¼x0

þ B1Q0ðx� ðx1 � dxÞÞjx¼x0

; ð2:5Þ

where dx = da for layer A and dx = db for layer B. If

Eq. (2.5) organized in matrix form, we obtain the

following relation:

A0

B0

� �¼

1 1

�ik0nb ik0nb

� ��1

�P ðx0 � ðx1 � dxÞÞ Qðx0 � ðx1 � dxÞÞP 0ðx0 � ðx1 � dxÞÞ Q0ðx0 � ðx1 � dxÞÞ

� �A1

B1

� �:

ð2:6Þ

Using Eq. (2.4), light transmission from layer A

to B at xi, can be modeled and expressed as:

Ai

Bi

� �¼

P ðdaÞ QðdaÞP 0ðdaÞ Q0ðdaÞ

� ��1

�P ð�dbÞ Qð�dbÞP 0ð�dbÞ Q0ð�dbÞ

� �Aiþ1

Biþ1

� �: ð2:7Þ

So, we define TAB as transmission transfer func-

tion from layer A to B as

T AB ¼P ðdaÞ QðdaÞP 0ðdaÞ Q0ðdaÞ

� ��1 P ð�dbÞ Qð�dbÞP 0ð�dbÞ Q0ð�dbÞ

� �:

ð2:8ÞSimilarly TBA (transmission from layer B to

layer A), can be obtained using transposing prop-

erty. Using similar, above mentioned conclusion

TBB, TAA are obtained as

T BB;AA ¼P ðdb;aÞ Qðdb;aÞP 0ðdb;aÞ Q0ðdb;aÞ

� ��1

�P ð�db;aÞ Qð�db;aÞP 0ð�db;aÞ Q0ð�db;aÞ

� �: ð2:10Þ

1.35 1.4 1.45 1.5 1.55 1.6 1.65 1.7 1.75

x 10-6

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8Reflection

λ

BW=29.2nm

Lg=13.95µm

Fig. 3. Reflection coefficient vs. wavelength for homogeneous

Fibonacci-class multilayer stack (koB = 1.55 lm, n0 = 3, S10,

N = 55, M = 0.5).

1.51 1.52 1.53 1.54 1.55 1.56 1.57 1.58

x 10-6

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9Reflection

λ

BW=3.6nm

Lg=156.37µm

Fig. 4. Reflection coefficient vs. wavelength for homogeneous

Fibonacci-class quasi-periodic multilayer stack (koB = 1.55 lm,


Finally, the transmission Matrix from latest

layer is

AF j�1

BF j�1

" #¼

PðdxÞ QðdxÞP 0ðdxÞ Q0ðdxÞ

� ��11 1

�ik0nb ik0nb

� �AF j

BF j

" #;

ð2:11Þ

where dx = da if we have A! B and dx = db if we

have B ! B. After introducing the transmission

matrix from layer i to j, we multiply all of the ob-

tained matrices and the input–output transfer ma-

trix can be given as

M ¼ T ox . . . T AAT ABT BBT BA . . . T xs; ð2:12Þ

where Tox and Txs are input and output transfer

matrix with x which is determined depends on

the Fibonacci structure how to arranged. If the

first layer is B, then Tox = ToB and similar situation

are given for Txs. The introduced matrix in Eq.(2.12) can be expanded and the closed form is ob-

tained as

M ¼m11 m12

m21 m22

� �: ð2:13Þ

According to [5], the reflection and the trans-

mission coefficients can be obtained as:

t ¼ 1m11

;

r ¼ m21

m11:

ð2:14Þ

Now, in the next section we try to investigate

five different cases for inhomogeneous index of

refraction in multilayer stack and obtain the reflec-

tion coefficient and compare the obtained result

with the homogeneous cases.
n0 = 3, S15, N = 610, M = 0.2).
3. Result and discussion

In this section, we will consider five inhomoge-

neous index of refraction profiles for Fibonacci-

class quasi-periodic multilayer stack and obtain

the field distribution and optical filtering charac-

teristics. For first example, we consider the follow-ing distribution for permittivity.

Case (a). eðxÞ ¼ e0e�ax2 .

Using our obtained result in Section 2, the field

characteristics for this case are obtained as:

H ðnÞY ðxÞ ¼ Cne

�12ðaþ1

2Þx2Hnð xffiffi

2p Þ þ Dne

�12ðaþ1

2Þx2Qnð xffiffi

2p Þ;

K2n ¼ K2

0½n20 � ðnþ 12Þ�;

n2ðxÞ ¼ n20 þ a2 � 14

� �x2 þ a;

ð3:1Þwhere n0, Cn and Dn are integer number and arbi-

trary constants, respectively. Also, a is the index ofrefraction distribution factor. Now, we demon-

strate the simulation result in the following figures.

Figs. 3 and 4 shows the reflection profiles of homo-

1.4 1.45 1.5 1.55 1.6 1.65 1.7x 10

-6

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08Reflection

λ

Lg=13.95 µm

BW=23 nm

-1 0 1 2 3 4 5 6 7 8

x 10-6

2.5

2.6

2.7

2.8

2.9

3

3.1

3.2

3.3n(x)

L

Fig. 5. Reflection coefficient vs. wavelength for inhomogeneous Fibonacci-class quasi-periodic multilayer stack (koB = 1.55 lm, n0 = 3,

S10, N = 55, M = 0.5, a = �1).

1.45 1.5 1.55 1.6 1.65 1.7

x 10-6

0

0.02

0.04

0.06

0.08

0.1

0.12Reflection

λ

1)α=-12)α=-1.13)α=-1.24)α=-1.35)α=-1.4

Lg=13.95 µmBW1=23 nmBW2=23.8 nmBW3=24.4 nmBW4=25 nmBW5=25.2 nm

Fig. 6. Reflection coefficient vs. wavelength for inhomogeneous


n0 = 3, S10, N = 55, M = 0.5).

1.53 1.535 1.54 1.545 1.55 1.555 1.56 1.565 1.57 1.575

x 10-6

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45Reflection

λ

Lg=156. µm

BW=3 nm

Fig. 7. Reflection coefficient vs. wavelength for Inhomogeneous


n0 = 3, S15, N = 610, M = 0.2, a = �1).


geneous quasi-periodic structures for different N

(number of layers), M (the index of refraction dif-

ference between two layers) and Bragg wavelength(k0B) values. Figs. 5–7 show the inhomogeneous

index of refraction profiles for Fibonacci-class

quasi-periodic systems.

As you see, for the case of inhomogeneous in-

dex of refraction coefficient and the simulation

parameters the reflection coefficient is low and all

of incident light in transmitted. This subject is

acceptable. Since the number of layers in this sim-ulation is low and the index of refraction is varied

slowly, so, main incident light should be transmit-

ted. By increasing the number of layer, we can in-

crease the reflection coefficient. Also, the effect of

the index of refraction distribution (a) variation

on filtering characteristics is shown in Fig. 6. As

it is seen, by decreasing a the bandwidth and the

reflection coefficient is increased. Generally, usingabove simulated result, we can obtain the narrow

bandwidth filters.

Fig. 7 shows our simulation for large number of

layers. As it is shown, in this case which is similar

to our homogeneous case shown in Fig. 4, we ob-

tain narrowband filter. Also, the side wall ringing

is displaced. The ringing magnitude is increased

and the damping ratio is decreased.

1.45 1.5 1.55 1.6 1.65 1.7

x 10-6

0

0.02

0.04

0.06

0.08

0.1

0.12Reflection

λ

1)m=0.62)m=0.53)m=0.34)m=0.25)m=0.1



n0 = 3, S10, N = 55, a = �1).

1.53 1.535 1.54 1.545 1.55 1.555 1.56 1.565 1.57 1.575

x 10-6

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7Reflection

λ

Lg=154.3µ m

BW=3.24 nm

Fig. 10. Reflection coefficient vs. wavelength for inhomoge-

neous Fibonacci-class quasi-periodic multilayer stack (koB =

1.55 lm, n0 = 3, S15, N = 610, M = 0.2, a = 1).


The variation effect of the index of refraction

difference is demonstrated in Fig. 8. As it is shown,

the increasing of the index of refraction difference

will increase the reflection coefficient.

From our simulation in case a, we can conclude

that if the index of refraction distribution in quasi-

periodic structure is Gaussian, then Figs. 6 and 8

demonstrate the system behavior and it effect onsystem performance. As a second example, we con-

sider the power law distribution for permittivity as

follows.

Case (b). e(x) = e0(1 + ax)b.

1.45 1.5 1.55 1.6 1.65 1.7x 10

-6

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35Reflection

λ

-2 0 2 4 6x 10

-6

2.4

2.6

2.8

3

3.2

3.4n(x)

L



n0 = 3, S10, b = �1.5, N = 55, M = 0.5, a = 1).

In this case similar to the previous case we de-

rive exactly the field distribution, wave vector

and the index of refraction coefficient, which is

shown in Eq. (3.2) as

H ðnÞY ðxÞ ¼ Cnð1þ axÞ

b2e�

14x2Hnð xffiffi

2p Þ

þDnð1þ axÞb2e�

14x2Qnð xffiffi

2p Þ;

K2n ¼ K2

0½n20 � ðnþ 12Þ�;

n2ðxÞ ¼ n20 þ 14

a2b bþ2ð Þ1þaxð Þ2 � x2

h i;

ð3:2Þ

1.45 1.5 1.55 1.6 1.65 1.7

x 10-6

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35Reflection

λ

1)β=-1.52)β=-1.63)β=-1.8


neous Fibonacci-class quasi-periodic multilayer stack

(koB = 1.55 lm, n0 = 3, S10, N = 55, M = 0.5, a = 1).

1.45 1.5 1.55 1.6 1.65 1.7

x 10-6

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35Reflection

λ

1)m=1.12)m=0.83)m=0.54)m=0.25)m=0.05



(koB = 1.55 lm, n0 = 3, S10, N = 55, M = 0.5, a = 1, b = �1.5).

1.53 1.535 1.54 1.545 1.55 1.555 1.56 1.565 1.57 1.575 1.58

x 10-6

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4Reflection

λ

Lg=153.2 µm

BW=2.8 nm



(koB = 1.55 lm, n0 = 3, S15, N = 610, M = 0.2, a = �1.5).


where Cn and Dn are arbitrary constants. Also, aand b are the permittivity distribution parameters.

Now, we demonstrate the simulation result in the

following figures. Figs. 9 and 10 are demonstrated

the reflectivity for Fibonacci-class inhomogeneous

multiplayer structure for some differentparameters.

Fig. 10 shows the similar filtering operation cor-

responds to our previous simulated result, which is

shown in Fig. 7. In this case the amplitude, band-

width and ringing amplitude are increased. Also,

the ringing damping is very low. Also, the effect

of the index of refraction distribution parameters

(b) on the reflectivity is shown in the Fig. 11.

1.45 1.5 1.55 1.6 1.65 1.7x 10

-6

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35Reflection

λ

Lg=13.95 ∝m

BW=22.8 nm

Fig. 13. Reflection coefficient vs. wavelength for inhomogeneous F

n0 = 3, S10, N = 55, M = 0.5, a = �2.7).

The index of refraction difference between lay-

ers is changed and the result is shown in Fig. 12.

Fig. 11 is demonstrated that the increasing of bcan shift the filter pass band and central frequency

to the lower frequencies and also, it is broadened.The effect of the index of refraction difference on

the reflectivity is shown in Fig. 12. As it is shown,

the m (the index of refraction difference) factor has

not efficient effect on the reflection coefficient (see

Fig. 13).

As a third example, we consider the following

distribution:

Case (c). e(x) = e0cosh(ax).

-1 0 1 2 3 4 5 6

x 10-6

2.7

2.8

2.9

3

3.1

3.2

n(x)

L

ibonacci-class quasi-periodic multilayer stack (koB = 1.55 lm,

1.45 1.5 1.55 1.6 1.65 1.7x 10

-6

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35Reflection

λ

1) =-2.42) =-2.53) =-2.74) =-2.9

BW1=23.8 nmBW2=23.2 nmBW3=23 nmBW4=21.2 nm



(koB = 1.55 lm, n0 = 3, S10, N = 55, M = 0.5).

1.45 1.5 1.55 1.6 1.65 1.7x 10

-6

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35Reflection

λ

1)m=0.052)m=0.53)m=1

BW1=22.6 nmBW2=22.2 nmBW3=21.2 nm



(koB = 1.55 lm, n0 = 3, S10, N = 55, a = �2.7).

1.4 1.45 1.5 1.55 1.6 1.65 1.7x 10

-6

00.10.20.30.40.50.60.70.80.91 Reflection

λ

Lg=13.95 µm

BW=100 nm

2

2

2

2

2

3

3

3

Fig. 17. Reflection coefficient vs. wavelength for Inhomogeneous Fibo

and layer B (polynomial) (koB = 1.55 lm, n0 = 3, S10, N = 55, M = 0.5


According to our result reported in Section 2,

the field characteristics for this case are obtained

exactly as

H ðnÞY ðxÞ ¼ Cn

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffifficoshðaxÞ

pe�

14x2Hnð xffiffi

2p Þ

þDn

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffifficoshðaxÞ

pe�

14x2Qnð xffiffi

2p Þ;

K2n ¼ K2

0½n20 � ðnþ 12Þ�;

n2ðxÞ ¼ n20 � 14½a2 3�cosh2ðaxÞ

cosh2ðaxÞ þ x2�;

ð3:3Þ

where Cn and Dn are arbitrary constants. Also, a is

the index of refraction distribution parameter.Now, we demonstrate the simulation result.

-1 0 1 2 3 4 5x 10

-6

.75

.8

.85

.9

.95

3

.05

.1

.15 n(x)

L

nacci-class quasi-periodic multilayer stack layer A (exponential)

, ae = �1.8, ap = 1, bp = 0.4).

1.4 1.45 1.5 1.55 1.6 1.65 1.7 1.75 1.8 1.85x 10

-6

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1Reflection

λ

1)α=-12)α=-1.53)α=-1.8

Lg=13.95µmBW1=85 nmBW2=95 nmBW3=100nm


neous Fibonacci-class quasi-periodic multilayer stack layer A

(exponential) and layer B (polynomial) (koB = 1.55 lm, n0 = 3,

S10, N = 55, M = 0.5, ap = 1, bp = 0.4).

1.4 1.45 1.5 1.55 1.6 1.65 1.7 1.75x 10

-6

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1Reflection

λ

1)β=-0.52)β=-0.23)β=0.24)β=0.4

Lg=13.95µ m

BW1=90 nmBW2=78 nmBW

3=87.5 nm

BW4=100 nm



(exponential) and layer B (polynomial) (koB = 1.55 lm, n0 = 3,

S10, N = 55, M = 0.5, ae = �1.8, ap = 1).

1.45 1.5 1.55 1.6 1.65 1.7

x 10-6

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1 Reflection

λ

1)α=-12)α=-1.13)α=-1.24)α=-1.3

Lg=13.95 µmBW

1=39.0 nm

BW2=48.4 nm

BW3=45.2 nm

BW4=42.2 nm



(hyperbolic) and layer B (exponential) (koB = 1.55 lm, n0 = 3,

S10, N = 55, M = 0.5, ah = �2.7).


As we see, the bandwidth of this filter is nar-

rower than previous two cases. The effects of aand the index of refraction difference between lay-

ers are demonstrated in Figs. 15 and 16, respec-tively. Similar to previous case, increasing the awill shift the pass band to the lower frequencies

and the band width will increased.

The effect of the index of refraction differences

is demonstrated in Fig. 16 and generally has not

efficient effect on reflectivity.

Also, in the following we will simulate the com-

bined cases, with different index of refraction pro-files for layer A and B. For first example, we

1.45 1.5 1.55 1.6 1.65 1.7x 10

-6

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7 Reflection

λ

Lg=13.95µm

BW=39 nm

Fig. 20. Reflection coefficient vs. wavelength for inhomogeneous Fibo

and layer B (hyperbolic) (koB = 1.55 lm, n0 = 3, S10, N = 55, M = 0.5

consider the exponential and power law

distributions.

Case (d). e1ðxÞ ¼ e0e�ax2 , and e2(x) = e0(1 + ax)b.In this case, we will consider the combination of

different index of refraction profiles for layers A

and B. Fig. 17 shows the reflectivity for this selec-

tion of the index of refractions (see Fig. 14).

In this figure, we demonstrate the reflection

coefficient from inhomogeneous Fibonacci-class

quasi-periodic structure. Using this selection we

obtain a suitable broadband filter with 100 nmbandwidth. The distribution parameters effect on

-1 0 1 2 3 4 5 6 7 8x 10

-6

2.65

2.7

2.75

2.8

2.85

2.9

2.95

3

3.05

3.1

3.15n(x)

L

nacci-class quasi-periodic multilayer stack layer A (exponential)

, ae = �1, ah = �2.7).

1.35 1.4 1.45 1.5 1.55 1.6 1.65 1.7

x 10-6

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1Reflection

λ

1)α=-1.82)α=-2.13)α=-2.44)α=-2.7

Lg=13.95 µmBW

1=29 nm

BW2=35.2 nm

BW3=42 nm

BW4=48.2 nm



(hyperbolic) and layer B (exponential) (koB = 1.55 lm, n0 = 3,

S10, N = 55, M = 0.5, ae = �1).


reflectivity is shown in the Figs. 18 and 19. Using

our simulations, we obtain the suitable controlling

methods for filter characteristics.

Case (e). e1ðxÞ ¼ e0e�ax2 , and e2(x) = e0cosh(ax).In this case the other alternative for index of

refractions in layers A and B is considered. Similar

to previous case, the effects of different parameters

on optical filtering properties are demonstrated in

the following figures. In Fig. 20, we demonstrate

the reflectivity. Also, in the Figs. 21 and 22, we

illustrated the distribution parameters effect on

the reflectivity.

4. Conclusion

In this paper, we examined the inhomoge-

neous Fibonacci-class quasi-periodic multilayer

stacks from optical filtering point of views. Inthis work, we obtained the narrow band filters

with suitable index of refraction selection. Also,

we reported the broadband optical filters with

suitable index of refractions selection for layers

A and B. In this work, we try to present

semi-exact treatment for quasi-periodic struc-

tures in special cases. Using our approach, one

can study the practical inhomogeneous effectson the optical filters designed by quasi-periodic

structures. These effects are illustrated in our

simulations.

References

[1] M.S. Vasconcelos, E.L. Albuquerque, A.M. Mariz, J. Phys.:

Condens. Matter 10 (1998) 5839.

[2] L.D. Negro, C.J. Oton, Z. Gaburro, L. PAvesi, P. Johnson,

A.d.L. Agendijk, R. Righini, M. Colocci, D.S. Wiersma,

Phys. Rev. Lett. 90 (5) (2003).

[3] L. Schachter, J.A. Nation, Beam-wave Interaction in a

Quasi-periodic Structure, IEEE 1993, PAC.

[4] X. Yang, et al., Phys. Rev. B 59 (7) (1999).

[5] A. Rostami, S. Matloub, Multiband optical filter design

using fibonacci based quasi-periodic homogeneous struc-

tures, in: Proceedings of the Asia Pacific Radio Science

Conference 2004, China.

[6] A. Rostami, S. Matloub, Laser Phys. Lett. 1 (7) (2004).

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Exactly solvable inhomogeneous Fibonacci-class quasi-periodic structures (optical filtering)