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-class quasi-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 inhomogeneous 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. Ó 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 disordered materials light waves undergo a multiple *

Corresponding author. Tel.: +98 411 3393724; fax: +98 411 3300819. E-mail address: [email protected] (A. Rostami).

scattering process and are subject to unexpected interference effects. Multiple light scattering in disordered dielectrics shows many similarities with the propagation of electrons in semiconductors and various phenomena that are known for electron 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, periodic dielectric structures behave as a crystal for

0030-4018/$ - see front matter Ó 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2004.11.105

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

light waves. In periodic structure the interference is constructive in well-defined propagation directions, which leads to Bragg scattering and refraction. At high enough refractive index contrast, propagation is prohibited in any direction with in a characteristics range of frequencies. This phenomenon is referred to as a photonic band-gap in analogy with the electronic band-gap in a semiconductor. Where as the knowledge on the propagation of light waves in completely ordered and disordered structures is now rapidly improving, little is known about the behavior of optical waves in the huge intermediate regime between total order and 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 discussed the homogeneous quasi-crystal behaviors [5]. In this paper, we will examine the inhomogeneous index of refraction effect on optical properties (Bandwidth and stop band damping ratio) of Fibonacci-class quasi-periodic layers. The controlling of optical filter characteristics such as bandwidth is very important. In homogeneous Fibonacci-class quasi-periodic optical filters, the bandwidth closely depends on the index of refraction difference between layers. The large bandwidth needs to large index of refraction difference, 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 interpretation of inhomogeneous isotropic Fibonacciclass quasi-periodic multilayer stack is presented. The result and discussion is presented in Section 3. Finally, the paper ends with a conclusion.

2. Inhomogeneous isotropic Fibonacci-class quasi-periodic multilayer stack Fig. 1 illustrates the Fibonacci-class quasi-periodic multi-layer stack. In this case, we assume that the 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 distribution exactly is very hard and the numerical approach is usually used. In this case, we will adopt some special index of refraction coefficient profiles having exact solution for the electromagnetic fields. By considering Fig. 2, the following equation is used for light transmission in inhomogeneous isotropic media [6].
2  d 1 deðX Þ d 2 þ k
n f ðX Þ H Y ðX Þ 0 1 dX 2 eðX Þ dX dX   ¼ k 2
k 20 n20 H Y 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 vector. According to [6], applying above defined new variable and adopting the spatial function for the index of refraction distribution ðf ðxÞ /
14 x2 Þ, we obtain the Schro¨dinger-like harmonic oscillator differential equation and we can propose the following equation as a solution for the electromagnetic fields as H Y ðx; z; tÞ ¼ H Y ðxÞeiðxt
kzÞ ; where db db
H Y ðxÞ ¼ CP ðxÞ þ DQðxÞ;

Fig. 1. Fibonacci-class quasi-periodic multilayer stack.




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

249

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 1 1
1 ¼ B0
ik 0 nb ik 0 nb

 P ðx0
ðx1
d x ÞÞ Qðx0
ðx1
d x ÞÞ A1 :  0 P ðx0
ðx1
d x ÞÞ Q0 ðx0
ðx1
d x ÞÞ B1

ð2:6Þ Fig. 2. One layer of Fibonacci-class quasi-periodic multilayer stack.

pffiffiffiffiffiffiffiffi
1x2  x  eðxÞe 4 H n pffiffi2 ; pffiffiffiffiffiffiffiffi 1 2   QðxÞ ¼ eðxÞe
4x Qn pxffiffi2 :

P ðxÞ ¼

ð2:3Þ

Using Eq. (2.4), light transmission from layer A to B at xi, can be modeled and expressed as:


Ai



Bi

Now, the field distribution for whole structure using above obtained relations is given as:


¼

P ðd a Þ

Qðd a Þ


1

P 0 ðd a Þ Q0 ðd a Þ 

P ð
d b Þ Qð
d b Þ Aiþ1  : P 0 ð
d b Þ Q0 ð
d b Þ Biþ1

8 x <
d b ; A0 e
ik0 nb ðxþd b Þ þ B0 eik0 nb ðxþd b Þ ; > > > > > . > .. > > > > > < A P ðx
ðx
d ÞÞ þ B Qðx
ðx
d ÞÞ; xi
1 < x < xi ; i i a i i a H Y ðxÞ ¼ > A P ðx
ðx
d ÞÞ þ B Qðx
ðx
d ÞÞ; xi < x < xiþ1 ; iþ1 i b iþ1 i b > > > > > > ... > > > > : AF j e
ik0 nb ðx
xF j Þ þ BF j eik0 nb ðx
xF j Þ x > xF j ;

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, respectively. If, we apply the boundary conditions for tangential 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 = x 0, A0 þ B0 ¼ A1 P ðx
ðx1
d x ÞÞjx¼x0

ð2:4Þ

So, we define TAB as transmission transfer function from layer A to B as

 P ðd a Þ Qðd a Þ
1 P ð
d b Þ Qð
d b Þ T AB ¼ : P 0 ðd a Þ Q0 ðd a Þ P 0 ð
d b Þ Q0 ð
d b Þ ð2:8Þ Similarly TBA (transmission from layer B to layer A), can be obtained using transposing property. Using similar, above mentioned conclusion TBB, TAA are obtained as


þ B1 Qðx
ðx1
d x ÞÞjx¼x0 ; T BB;AA


ik 0 nb A0 þ ik 0 nb B0 ¼ A1 P 0 ðx
ðx1
d x ÞÞjx¼x0 þ B1 Q0 ðx
ðx1
d x ÞÞjx¼x0 ;

ð2:7Þ

ð2:5Þ


1 P ðd b;a Þ Qðd b;a Þ ¼ P 0 ðd b;a Þ Q0 ðd b;a Þ
 P ð
d b;a Þ Qð
d b;a Þ  : P 0 ð
d b;a Þ Q0 ð
d b;a Þ

ð2:10Þ

250

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

Finally, the transmission Matrix from latest layer is

0.8

"

0.7

AF j
1

#

BF j
1


¼

P ðd x Þ 0

Qðd x Þ 0

P ðd x Þ Q ðd x Þ


1


1

1


ik 0 nb ik 0 nb

"

AF j

# ;

0.6

ð2:11Þ

0.5

BF j

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 obtained matrices and the input–output transfer matrix can be given as M ¼ T ox . . . T AA T AB T BB T 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 obtained as
 m11 m12 M¼ : ð2:13Þ m21 m22 According to [5], the reflection and the transmission coefficients can be obtained as: t ¼ m111 ; r ¼ mm2111 :

Reflection

Lg=13.95µ m BW=29.2nm

0.4 0.3 0.2 0.1 0 1.35

1.4

1.45

1.5

1.55

1.6

1.65

1.7

1.75 -6

λ

x 10

Fig. 3. Reflection coefficient vs. wavelength for homogeneous Fibonacci-class multilayer stack (koB = 1.55 lm, n0 = 3, S10, N = 55, M = 0.5).

Reflection

0.9 0.8 0.7

Lg=156.37µ m BW=3.6nm

0.6 0.5 0.4 0.3

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 reflection coefficient and compare the obtained result with the homogeneous cases.

0.2 0.1 0 1.51

1.52

1.53

1.54

1.55

1.56

1.57

λ

1.58 -6

x 10

Fig. 4. Reflection coefficient vs. wavelength for homogeneous Fibonacci-class quasi-periodic multilayer stack (koB = 1.55 lm, n0 = 3, S15, N = 610, M = 0.2).

3. Result and discussion

H Y ðxÞ ¼ Cn e
2ðaþ2Þx H n ðpxffiffi2Þ þ Dn e
2ðaþ2Þx Qn ðpxffiffi2Þ;

In this section, we will consider five inhomogeneous index of refraction profiles for Fibonacciclass quasi-periodic multilayer stack and obtain the field distribution and optical filtering characteristics. For first example, we consider the following distribution for permittivity. 2 Case (a). eðxÞ ¼ e0 e
ax . Using our obtained result in Section 2, the field characteristics for this case are obtained as:

K 2n ¼ K 20 ½n20
ðn þ 12Þ;   n2 ðxÞ ¼ n20 þ a2
14 x2 þ a;

ðnÞ

1

1

2

1

1

2

ð3:1Þ where n0, Cn and Dn are integer number and arbitrary constants, respectively. Also, a is the index of refraction distribution factor. Now, we demonstrate the simulation result in the following figures. Figs. 3 and 4 shows the reflection profiles of homo-

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

0.08 0.07 0.06

n(x)

3.3

Lg=13.95 µm

3.2

BW=23 nm

3.1

0.05

3

0.04

2.9

0.03

2.8

0.02

2.7

0.01

2.6

251

2.5

0 1.4

1.45

1.5

1.55

1.6

1.65

-1

1.7

λ

x 10

-6

0

1

2

3

4

5

6

7

8 -6

x 10

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).

Reflection

0.12

0.45 1)α=-1 2)α=-1.1 3)α=-1.2 4)α=-1.3 5)α=-1.4

0.1 0.08

0.4

0.04

BW=3 nm

0.3 0.25 0.2 0.15 0.1

0.02

0.05 0

0 1.45

Lg=156. µ m

0.35

Lg=13.95 µm BW1=23 nm BW2=23.8 nm BW3=24.4 nm BW4=25 nm BW5=25.2 nm

0.06

Reflection

1.5

1.55

1.6

λ

1.65

1.7 x 10

-6

Fig. 6. Reflection coefficient vs. wavelength for inhomogeneous Fibonacci-class quasi-periodic multilayer stack (koB = 1.55 lm, n0 = 3, S10, N = 55, M = 0.5).

geneous quasi-periodic structures for different N (number of layers), M (the index of refraction difference 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 index 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 simulation is low and the index of refraction is varied slowly, so, main incident light should be transmit-

1.53 1.535 1.54 1.545 1.55 1.555 1.56 1.565 1.57 1.575

λ

x 10

-6

Fig. 7. Reflection coefficient vs. wavelength for Inhomogeneous Fibonacci-class quasi-periodic multilayer stack (koB = 1.55 lm, n0 = 3, S15, N = 610, M = 0.2, a =
1).

ted. By increasing the number of layer, we can increase 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, using above 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 obtain narrowband filter. Also, the side wall ringing is displaced. The ringing magnitude is increased and the damping ratio is decreased.

252

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

0.12

Reflection

0.7 1)m=0.6 2)m=0.5 3)m=0.3 4)m=0.2 5)m=0.1

0.1

0.6

Lg=154.3 µ m BW=3.24 nm

0.5

0.08

0.4 0.06

0.3 0.04

0.2

0.02

0.1

0 1.45

1.5

1.55

1.6

1.65

1.7

λ

λ

x 10

Fig. 8. Reflection coefficient vs. wavelength for inhomogeneous Fibonacci-class quasi-periodic multilayer stack (koB = 1.55 lm, n0 = 3, S10, N = 55, 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 quasiperiodic structure is Gaussian, then Figs. 6 and 8 demonstrate the system behavior and it effect on system performance. As a second example, we consider the power law distribution for permittivity as follows. Case (b). e(x) = e0(1 + ax)b.

3

0.25

2.8

2.4 -2

1 2

b

1 2

þDn ð1 þ axÞ2 e
4x Qn ðpxffiffi2Þ; K 2n ¼ K 20 ½n20
ðn þ 12Þ; h2 i bðbþ2Þ 2 n2 ðxÞ ¼ n20 þ 14 að1þax ; 2
x Þ ð3:2Þ

Reflection 1)β=-1.5 2)β=-1.6 3)β=-1.8

0.2 0

2 L

0.15

4

6 x 10

-6

0.15 0.1

0.1

0.05

0.05 0

b

ðnÞ

H Y ðxÞ ¼ C n ð1 þ axÞ2 e
4x H n ðpxffiffi2Þ

0.25

2.6

0.2

In this case similar to the previous case we derive exactly the field distribution, wave vector and the index of refraction coefficient, which is shown in Eq. (3.2) as

0.3

3.2

0.3

-6

x 10

Fig. 10. Reflection coefficient vs. wavelength for inhomogeneous Fibonacci-class quasi-periodic multilayer stack (koB = 1.55 lm, n0 = 3, S15, N = 610, M = 0.2, a = 1).

0.35

n(x)

Reflection 3.4

0.35

0 1.53 1.535 1.54 1.545 1.55 1.555 1.56 1.565 1.57 1.575

-6

0 1.45

1.5

1.55

λ

1.6

1.65

1.7 -6

x 10

Fig. 9. Reflection coefficient vs. wavelength for inhomogeneous Fibonacci-class quasi-periodic multilayer stack (koB = 1.55 lm, n0 = 3, S10, b =
1.5, N = 55, M = 0.5, a = 1).

1.45

1.5

1.55

1.6

λ

1.65

1.7 -6

x 10

Fig. 11. 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).

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

The index of refraction difference between layers is changed and the result is shown in Fig. 12. Fig. 11 is demonstrated that the increasing of b can 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).

Reflection

0.35

1)m=1.1 2)m=0.8 3)m=0.5 4)m=0.2 5)m=0.05

0.3 0.25 0.2 0.15 0.1 0.05 0

1.45

1.5

1.55

1.6

1.65

1.7 -6

λ

x 10

Fig. 12. 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, b =
1.5).

Reflection

0.4 0.35

where Cn and Dn are arbitrary constants. Also, a and 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 different parameters. Fig. 10 shows the similar filtering operation corresponds to our previous simulated result, which is shown in Fig. 7. In this case the amplitude, bandwidth 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.

BW=2.8 nm

0.25 0.2 0.15 0.1 0.05 0

1.53 1.535 1.54 1.545 1.55 1.555 1.56 1.565 1.57 1.575 1.58

λ

-6

x 10

Fig. 14. Reflection coefficient vs. wavelength for inhomogeneous Fibonacci-class quasi-periodic multilayer stack (koB = 1.55 lm, n0 = 3, S15, N = 610, M = 0.2, a =
1.5).

n(x)

0.3 0.25

Lg=153.2 µm

0.3

Reflection

0.35

253

3.2 Lg=13.95 ∝m BW=22.8 nm

3.1

0.2 3 0.15 2.9

0.1

2.8

0.05 0 1.45

1.5

1.55

λ

1.6

1.65

1.7 -6

x 10

2.7 -1

0

1

2

3

4

5

6 -6

L

x 10

Fig. 13. 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 =
2.7).

254

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

According to our result reported in Section 2, the field characteristics for this case are obtained exactly as pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 2 ðnÞ H Y ðxÞ ¼ C n coshðaxÞe
4x H n ðpxffiffi2Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 2 þDn coshðaxÞe
4x Qn ðpxffiffi2Þ;

Reflection

0.35

1) 2) 3) 4)

0.3

=-2.4 =-2.5 =-2.7 =-2.9

0.25 BW1=23.8 nm BW2=23.2 nm BW3=23 nm BW4=21.2 nm

0.2 0.15

K 2n ¼ K 20 ½n20
ðn þ 12Þ;

0.1

2

ðaxÞ þ x2 ; n2 ðxÞ ¼ n20
14 ½a2 3
cosh cosh2 ðaxÞ

0.05 0

1.45

ð3:3Þ 1.5

1.55

1.6

1.65

λ

1.7 -6

x 10

Fig. 15. Reflection coefficient vs. wavelength for inhomogeneous Fibonacci-class quasi-periodic multilayer stack (koB = 1.55 lm, n0 = 3, S10, N = 55, M = 0.5).

0.9

1)m=0.05 2)m=0.5 3)m=1

0.3

0.8 0.7

BW1=22.6 nm BW2=22.2 nm BW3=21.2 nm

0.25 0.2

0.5 0.4 0.3

0.1

0.2 0.1

0.05

0 1.4

1.45

1)α=-1 2)α=-1.5 3)α=-1.8

Lg=13.95µm BW1=85 nm BW2=95 nm BW3=100nm

0.6

0.15

0

Reflection

1

Reflection

0.35

where Cn and Dn are arbitrary constants. Also, a is the index of refraction distribution parameter. Now, we demonstrate the simulation result.

1.5

1.55

1.6

1.65

λ

1.5

1.55

1.6

1.65

1.7

1.75

1.8

λ

1.85 -6

x 10

-6

x 10

Fig. 16. Reflection coefficient vs. wavelength for inhomogeneous Fibonacci-class quasi-periodic multilayer stack (koB = 1.55 lm, n0 = 3, S10, N = 55, a =
2.7).

1 0.9 L =13.95 µm g 0.8 BW=100 nm 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 1.4 1.45

1.45

1.7

Reflection

Fig. 18. Reflection coefficient vs. wavelength for inhomogeneous 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).

n(x)

3.15 3.1 3.05 3 2.95 2.9 2.85 2.8

1.5

1.55

λ

1.6

1.65 x 10

1.7 -6

2.75 -1

0

1

2

L

3

4

5 -6

x 10

Fig. 17. Reflection coefficient vs. wavelength for Inhomogeneous 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, ae =
1.8, ap = 1, bp = 0.4).

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

1 0.9 0.8 0.7

0.9 0.8 0.7

Lg=13.95 µ m BW1=90 nm BW2=78 nm BW3=87.5 nm BW4=100 nm

0.6 0.5 0.4

0.6 0.4 0.3

0.2

0.2

0.1

0.1

1.4

1.45

1.5

1.55

1.6

1.65

λ

1.7

0 1.45

1.75 -6

x 10

1.6

1.65

1.7 -6

x 10

consider the exponential and power law distributions. 2 Case (d). e1 ðxÞ ¼ e0 e
ax , 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 selection 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 nm bandwidth. The distribution parameters effect on

Reflection

n(x)

3.15 3.1 3.05

Lg=13.95 µ m BW=39 nm

3 2.95

0.5 0.4

2.9 2.85

0.3

2.8 2.75

0.2 0.1 0 1.45

1.55

Fig. 21. Reflection coefficient vs. wavelength for inhomogeneous Fibonacci-class quasi-periodic multilayer stack layer A (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 narrower than previous two cases. The effects of a and the index of refraction difference between layers are demonstrated in Figs. 15 and 16, respectively. Similar to previous case, increasing the a will 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 combined cases, with different index of refraction profiles for layer A and B. For first example, we

0.6

1.5

λ

Fig. 19. Reflection coefficient vs. wavelength for inhomogeneous 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, ae =
1.8, ap = 1).

0.7

1)α=-1 2)α=-1.1 3)α=-1.2 4)α=-1.3

Lg=13.95 µ m BW1=39.0 nm BW2=48.4 nm BW3=45.2 nm BW4=42.2 nm

0.5

0.3

0

Reflection

1 1)β=-0.5 2)β=-0.2 3)β=0.2 4)β=0.4

255

1.5

1.55

λ

1.6

1.65

1.7

2.7 2.65 -1

0

1

2

3

4

-6

x 10

L

5

6

7

8 x 10

-6

Fig. 20. Reflection coefficient vs. wavelength for inhomogeneous Fibonacci-class quasi-periodic multilayer stack layer A (exponential) and layer B (hyperbolic) (koB = 1.55 lm, n0 = 3, S10, N = 55, M = 0.5, ae =
1, ah =
2.7).

256

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

4. Conclusion

Reflection

1

1)α=-1.8 2)α=-2.1 3)α=-2.4 4)α=-2.7

0.9 0.8 0.7

Lg=13.95 µ m BW1=29 nm BW2=35.2 nm BW3=42 nm BW4=48.2 nm

0.6 0.5 0.4 0.3 0.2 0.1 0 1.35

1.4

1.45

1.5

λ

1.55

1.6

1.65

1.7 x 10

-6

Fig. 22. Reflection coefficient vs. wavelength for inhomogeneous Fibonacci-class quasi-periodic multilayer stack layer A (hyperbolic) and layer B (exponential) (koB = 1.55 lm, n0 = 3, S10, N = 55, M = 0.5, ae =
1).

In this paper, we examined the inhomogeneous Fibonacci-class quasi-periodic multilayer stacks from optical filtering point of views. In this 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 structures in special cases. Using our approach, one can study the practical inhomogeneous effects on the optical filters designed by quasi-periodic structures. These effects are illustrated in our simulations.

References reflectivity is shown in the Figs. 18 and 19. Using our simulations, we obtain the suitable controlling methods for filter characteristics. 2 Case (e). e1 ðxÞ ¼ e0 e
ax , 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.

[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 structures, in: Proceedings of the Asia Pacific Radio Science Conference 2004, China. [6] A. Rostami, S. Matloub, Laser Phys. Lett. 1 (7) (2004).