Optical ﬁltering properties of inhomogeneous isotropic...

340 Laser Phys. Lett. 1, No. 7 , 340–346 (2004) / DOI 10.1002/lapl.200410078

Abstract: Inhomogeneous optical slab or channel wave-guidesare basic alternative for realization of the passive and active de-vices and systems in the full-optical engineering domain. Usu-ally, the exact treatment for obtaining the reflected and transmit-ted light is very hard and the numerical methods are used. In thispaper, we will investigate the optical filtering properties of theexactly solvable large set of the inhomogeneous index of refrac-tion profiles. We show that tuning of the inhomogeneous medialength easily can control the bandwidth of our proposed filter.

X = - L / 2

X = L / 2

Y

X

Z

Slab wave-guide structure

c© 2004 by Astro Ltd.Published exclusively by WILEY-VCH Verlag GmbH & Co. KGaA

Optical filtering properties of inhomogeneous isotropicslab waveguidesA. Rostami ∗ and S. Matloub

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

Received: 26 February 2004, Accepted: 8 March 2004Published online: 9 June 2004

Key words: optical filters; inhomogeneous media; exact solution

PACS: 42.79.Ci

1. Introduction

Nowadays optical integrated circuits have key role for in-creasing the speed of signal processing in electronic andcommunication engineering. Usually in this domain thelarge band-gap material such as photonic crystals or di-electric materials are used. According to similarity be-tween electronic and optical engineering in realization ofsame engineering purposes, optical layered media has im-portant role as a substrate or device structures [1]. Periodiclayered media are a special class of layered media in whichlayers of dielectric material are stacked in a periodic fash-ion. The simplest example of a periodic medium consistsof alternating layers of two different materials with equalthickness. Wave propagation in these media exhibits manyinteresting and potentially useful phenomenon. These in-clude Bragg reflectors, holography and optical filtering.The diffraction of X-rays in crystals is a good example ofwave propagation in periodic layered media [1]. In addi-tion to the existence of these naturally occurring periodicmaterials, periodic layered media can also be grown arti-ficially by using various techniques, including molecularbeam epitaxy (MBE), metal organic chemical vapor phasedeposition (MOCVD) and atomic layer epitaxy (ALE).

The necessary calculation for investigation of these struc-tures can be performed numerically [2,3] and analytically[4–7]. The optical layered media can be divided into twogroups and are homogeneous and inhomogeneous media.In homogeneous media the index of refraction is constantversus position in each segment. But in inhomogeneousmedia the index of refraction distribution is non-uniformversus position. In this paper, we will investigate the filter-ing properties of inhomogeneous-layered media for three-layer structure. We try to give the exact treatment for re-flection and transmission coefficients [8,9].

The organization of this paper is as follows.In section 2, the optical filtering properties of TM-

mode electromagnetic waves travelling in inhomogeneousslab waveguides is investigated. In this section the theo-retical background is presented. Result and discussion ispresented in section 3. Finally, the paper ends with a con-clusion.

2. Inhomogeneous media and optical filteringproperties for TM-modeFig. 1 is used as optical filter for TM-mode electromag-netic waves.


∗ Corresponding author: e-mail: [email protected]

Laser Phys. Lett. 1, No. 7 (2004) / www.lphys.org 341

X = -L /2

X = L /2

Y

X

Z

Figure 1 Slab wave-guide structure

According to [9], the following equation is used forlight transmission in inhomogeneous isotropic media.[

d2

dX2− 1

ε(X)dε(X)dX

d

dX+ k2

0n1f(X)]

HY (X) = (1)

= (k2 − k20n

20)HY (X),

where f(X) is position dependent part of the index of re-fraction. Now, we define a new dimensionless variable (x)as

x = k0X. (2)

With applying this new variable and using HY (x) =ε

12 (x)hY (x), Eq. (1) is converted to[− d2

dx2+ U(x)

]hY (x) = λhY (x), (3)

where

U(x) =3ε′(x)2 − 2ε(x)ε′′(x)

4ε2(x)− n1f(x), (4)

λ = n20 − k2/k2

0. (5)

Now, we try to solve the Eq. (3) for U(x) = x2/4whereas there are many other possible alternatives. In thiscase, Eq. (3) has the following solution [9]

h(n)Y (x) = Nne−x2/4Hn

(x/

√2)

(6)

with eigenvalue λn = n + 1/2.In Eq. (6) n is nonnegative integer, Hn(x) is Hermite

polynomial and Nn is the normalization constant. UsingEq. (6), we obtain the following position dependent indexof refraction and wave-vector respectively

n2(x) = n20 +

14

[3ε′(x)2 − 2ε(x)ε′′(x)

ε2(x)− x2

], (7)

k2n = k2

0

[n2

0 − (n + 1/2)].

So, with suitable selection of ε(x), we can obtain largeset of inhomogeneous media with exact solution. Now,the filtering properties of the inhomogeneous media intro-duced by Eq. (7) are investigated. The Field distribution inour proposed structure is

HY (x, z, t) = HY (x) exp {i(ωt − kz)}, (8)

where

HY (x) =

Ae−ik0x + Beik0x, x < −L2

CP (x) + DQ(x), −L2 < x < L

2Fe−ik0x, L

2 < x. (9)

The functions P (x) and Q(x) in Eq. (9) corresponds tothe first and the second type of Hermite polynomials andfunctions respectively and are given as

P (x) =√

ε(x)e−x2/4Hn

(x/

√2)

, (10)

Q(x) =√

ε(x)e−x2/4Qn

(x/

√2)

.

Using the boundary condition for x = ±L/2, we ob-tain the reflection and transmission coefficients as

r = B/A, t = F/A, where (11)

A = [Q′(−L) − ik0Q(−L)][P ′(L) + iksP (L)]−−[Q′(L) + iksQ(L)][P ′(−L) − ik0P (−L)],

B = [P ′(−L) + ik0P (−L)][Q′(L) + iksQ(L)]−−[Q′(−L) + ik0Q(−L)][P ′(L) + iksP (L)],

F = [P ′(−L) + ik0P (−L)][Q′(−L) − ik0Q(−L)]−−[Q′(−L) + ik0Q(−L)][P ′(−L) − ik0P (−L)].

Table 1 shows the P (x) and Q(x) for n = 2, 3, 4.Using Eq. (11), we can obtain the reflection and trans-

mission coefficients of the large set of exactly solvableinhomogeneous media given in Eqs. (2–6,7). Discussionabout our proposed structures are given in section 3.

3. Results and discussion

In this section, we will concentrate on four special casesfor given U(x) = x2/2, which was assumed in section 2.

As a first example, we consider the ε(x) = ε0.

Case a) ε(x) = ε0

Using Eq. (11) and ε(x) = ε0, we obtain the followingfunction for index of refraction, P (x) and Q(x) as

n2(x) = n20 − x2/4, (12)

P (x) = e−x2/4Hn

(x/

√2)

,

Q(x) = e−x2/4Qn

(x/

√2)

.

Our simulated results for this case are shown in Fig. 2.Fig. 2a shows the reflection and transmission coefficientsfor n=2,3,4 versus incident light wavelength. As we see,with increasing n the reflection and transmission coeffi-cients will change very fast and one feel that the degree ofthe designed filter is increased. The reflection and trans-mission coefficients versus wavelength and incoming lightmedium index of refraction are demonstrated in Fig. 2b.


342 A. Rostami and S. Matloub: Optical filtering properties

Re

fle

ctio

n a

nd

Tra

nsm

issio

n (

R, T

)

Wavelength

n0

1

2

3

4

1

0.8

0.6

0.4

0.2

0

00.5

11.5

2

+10-6

Ba

nd

wid

th

L

0 0.5 1.0 1.5 2.0 2.5

+10-6

3.0

2.5

2.0

1.5

1.0

0.5

0

+10-6

+10-6

Ba

nd

wid

th

n0

1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0

1.11

1.10

1.09

1.08

1.07

1.06

1.05

1.04

Ba

nd

wid

th

n s

+10-6

1.12

1.11

1.10

1.09

1.08

1.07

1.06

1.05

1.04

1.03

1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0

1.0

0.9

0

0 0.2 1.0 2.0

+10-6

c b a

Wavelength (m)

Re

fle

ctio

n a

nd

Tra

nsm

issio

n (

R, T

)

Re

fle

ctio

n a

nd

Tra

nsm

issio

n (

R, T

)

Wavelength

n0

1

2

3

4

1

0.8

0.6

0.4

0.2

0

00.5

11.5

2

+10-6

Wavelength0

1

2

3

1

0.8

0.6

0.4

0.2

0

00.5

11.5

2

+10-6

L

+10-6

Re

fle

ctio

n a

nd

Tra

nsm

issio

n (

R, T

)

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0.4 0.6 0.8 1.2 1.4 1.6 1.8

a) n=2b) n=3c) n=4

a) b)

c) d)

e) f)

g)

Figure 2 (online color at www.interscience.wiley.com) Case a)



Reflection a

nd T

ransm

issio

n (

R, T

)

Wavelength

n0

1.0

1.5

2.5

3.0

1

0.8

0.6

0.4

0.2

0

00.5

11.5

2

+10-6

Bandw

idth

L

0 0.5 1.0 1.5 2.0 2.5

+10-6

5

4

3

2

1

0

+10-6

+10-6

Bandw

idth

n0

1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0

1.12

1.10

1.08

1.06

1.04

1.02

1.00

Bandw

idth

n s

+10-6

1.12

1.11

1.10

1.09

1.08

1.07

1.06

1.05

1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0

1.0

0.9

0

0 0.2 1.0 2.0

+10-6

Wavelength (m)

Reflection a

nd T

ransm

issio

n (

R, T

)

Reflection a

nd T

ransm

issio

n (

R, T

)

Wavelength

n0

1

0.8

0.6

0.4

0.2

0

00.5

11.5

2

+10-6

Wavelength0

1

2

3

1.5

1.0

0.5

0

00.5

11.5

2

+10-6

L

+10-6

Reflection a

nd T

ransm

issio

n (

R, T

)

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0.4 0.6 0.8 1.2 1.4 1.6 1.8

a) b)

c) d)

e) f)

g)

a

b

c

a) n=2b) n=3c) n=4

2.0

1.0

1.5

2.5

3.0

2.0

6

7

Figure 3 (online color at www.interscience.wiley.com) Case b)



Re

fle

ctio

n a

nd

Tra

nsm

issio

n (

R, T

)

Wavelength

n0

1.0

1.5

2.5

3.0

1

0.8

0.6

0.4

0.2

0

00.5

1.0

2.02.5

+10-6

Ba

nd

wid

th

L

0 0.5 1.0 1.5 2.0 2.5

+10-6

2.5

2.0

1.5

1.0

0.5

0

+10-6

+10-6

Ba

nd

wid

th

n0

1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0

1.110

1.100

1.085

1.080

1.070

1.065

1.060

Ba

nd

wid

th

n s

+10-6

1.12

1.11

1.10

1.09

1.08

1.07

1.06

1.05

1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0

1.0

0.9

0

0 2.5

+10-6

Wavelength (m)

Re

fle

ctio

n a

nd

Tra

nsm

issio

n (

R, T

)

Re

fle

ctio

n a

nd

Tra

nsm

issio

n (

R, T

)

Wavelength

nS

1

0.8

0.6

0.4

0.2

0

+10-6

Wavelength

+10-6

LRe

fle

ctio

n a

nd

Tra

nsm

issio

n (

R, T

)

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0.5 1.0 1.5 2.0

a) b)

c) d)

e) f)

g)

abc

a) n=2b) n=3c) n=4

2.0

1.0

1.5

2.5

3.0

2.0

3.0

3.5

1.5

00.5

1.0

2.02.5

1.5

0

2.5

1

0.8

0.6

0.4

0.2

0

00.5

1.0

2.02.5

1.5

2.01.5

1.00.5

+10-6

1.075

1.090

1.095

1.105

Figure 4 (online color at www.interscience.wiley.com) Case c)



Re

fle

ctio

n a

nd

Tra

nsm

issio

n (

R, T

)

Wavelength

n0

1.0

1.5

2.5

3.0

1

0.8

0.6

0.4

0.2

0

00.5

1.0

2.02.5

+10-6

Ba

nd

wid

th

L

0 0.5 1.0 1.5 2.0 2.5

+10-6

2.5

2.0

1.5

1.0

0.5

0

+10-6

+10-6

Ba

nd

wid

th

n0

1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0

1.10

1.08

1.04

1.02

1.00

Ba

nd

wid

th

n s

+10-6

1.10

1.12

1.08

1.06

1.04

1.02

1.00

0.98

0.96

1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0

1.0

0.9

0

0 1.2

+10-6

Wavelength (m)

Re

fle

ctio

n a

nd

Tra

nsm

issio

n (

R, T

)

Re

fle

ctio

n a

nd

Tra

nsm

issio

n (

R, T

)

Wavelength

nS

1

0.8

0.6

0.4

0.2

0

+10-6

Wavelength

+10-6

LRe

fle

ctio

n a

nd

Tra

nsm

issio

n (

R, T

)

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0.4 0.5 0.8 1.0

a) b)

c) d)

e) f)

g)

abc

2.0

1.0

1.5

2.5

3.0

2.0

3.0

3.5

1.5

00.5

1.0

2.02.5

1.5

0

2.5

1

0.8

0.6

0.4

0.2

0

00.5

1.0

2.02.5

1.5

2.01.5

1.00.5

+10-6

1.06

1.12

a) n=2b) n=3c) n=4

Figure 5 (online color at www.interscience.wiley.com) Case d)



n P (x) Q(x)

2√

ε(x)e−x2/4[2x2 − 2]√

ε(x)e−x2/4[−2x + 13x3 + 1

60x5 + 1

840x7 + . . .]

3√

ε(x)e−x2/4[−8.48x + 2.82x3]√

ε(x)e−x2/4[−8.48 + 12.72x2 − 1.06x4 − 0.035x6 + . . .]

4√

ε(x)e−x2/4[12 − 24x2 + 4x4]√

ε(x)e−x2/4[12x − 6x3 + 0.3x5 + 1140

x7 + . . .]

Table 1 P(x) and Q(x) in special cases

Also, the reflection and transmission versus wavelength,outgoing index of refraction and medium length is shownin Figs. 2c,2d). Finally, we give the bandwidth of our de-signed filter as incoming medium, outgoing medium indexof refractions and medium length in Figs. (2e,2f,2g). Aswe see from Eq. (5), the bandwidth of our proposed struc-ture strongly depends on length of the inhomogeneous me-dia.

Case b) ε(x) = ε0e−αx2

As a second example, we consider the ε(x) = ε0e−αx2

and using Eq. (11), we obtain the following function forindex of refraction, P (x) and Q(x) as

n2(x) = n20 +

(α2 − 1

4

)x2 + α, (13)

P (x) = e−12 (α+ 1

2 )x2Hn

(x/

√2)

,

Q(x) = e−12 (α+ 1

2 )x2Qn

(x/

√2)

.

The similar results about this case can be seen in Fig. 3.Also, the description of obtained results is similar to theabove-mentioned explanation for case a).

Case c) ε(x) = ε0(1 + αx)β

Using Eq. (11) and ε(x) = ε0(1+αx)β , we obtain thefollowing function for index of refraction, P (x)and Q(x)as

n2(x) = n20 +

14

[α2β(β + 2)(1 + αx)2

− x2

], (14)

P (x) = (1 + αx)β/2e−x2/4Hn

(x/

√2)

,

Q(x) = (1 + αx)β/2e−x2/4Qn

(x/

√2)

.

The similar results about this case can be seen in Fig. 4.Also, the description of obtained results is similar to theabove-mentioned explanation for case a).

Case d) ε(x) = ε0 cosh(αx)As a final example we consider ε(x) = ε0 cosh(αx)

and using Eq. (11), we obtain the following function forindex of refraction, P (x) and Q(x) as

n2(x) = n20 −

14

[α2 3 − cosh2(αx)

cosh2(αx)+ x2

], (15)

P (x) =√

cosh(αx)e−x2/4Hn

(x/

√2)

,

Q(x) =√

cosh(αx)e−x2/4Qn

(x/

√2)

.

4. Conclusion

In this work, we present a new class of inhomogeneousmedia and its filtering properties were investigated. Weshow that the inhomogeneous medium length can beused for the bandwidth control effectively. The effect ofcladding and substrate index of refractions on reflectionand transmission coefficients and bandwidth of designedfilter were analyzed. According to our obtained result, thelength of inhomogeneous media is best way for bandwidthcontrol.

References

[1] P. Yeh, Optical Waves in Layered Media (John Wiley, 1988);J. M. Senior, Optical Fiber Communications, Principles andPractice, 2nd ed. (Cambridge Univ. Press, Cambridge, 1992);A. Billings, Optics, Optoelectronics and Photonics Engineer-ing principles and Applications (Prentice Hall, 1993); H.Nishihara et al, Optical Integrated Circuits, (Mc-Graw Hill,1989); K. Kim, H. Lim, and D. Lee, J. Korean Phys. Soc. 39,L956 (2001); H. Vainola, J. Sinkkonen, and J. Aho, J. KoreanPhys. Soc. 39, 506 (2001); A. Yariv, Quantum Electronics,3rd Ed. (John Wiley, 1989); A. Ishimaru, ElectromagneticWave Propagation, Radiation and Scattering (Prentice Hall,1991).

[2] Li-Pen Yuan, IEEE J. Quantum Electron. 30, 1 (1994).[3] B. Engguist and G.A. Kriegsmann, Computational Wave

Propagation (Springer-Verlag, New York, 1997).[4] S.M. Chumakov and K.B. Wolf, Phys. Lett. A 193, 51 (1994).[5] A. Rostami and S.K. Moayedi, Indian J. Phys. 75B (4), 357

(2001).[6] A. Rostami and S K. Moayedi, J. Opt. A 5, 380 (2003).[7] A. Rostami, Laser Phys. 12, 1358 (2002).[8] M. Abramowitz and I.A. Stegun, Handbook of Mathematical

Functions (Dover publications, New York, 1970).[9] A. Rostami and S.K. Moayedi, Il Nuovo Cimento B, in press.


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