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Astrophysics, Vol. 50, No. 1, 2007

0571-7256/07/5001-0087 ©2007 Springer Science+Business Media, Inc.

DISSIPATION OF ELECTROMAGNETIC WAVES IN ONE-DIMENSIONALQUASIPERIODIC MEDIA

D. M. Sedrakian,1 A. A. Gevorgyan,1 A. Zh. Khachatrian,2 and V. D. Badalyan1

The transport of electromagnetic radiation through an isotropic quasiperiodic medium of finite size withlinearly or exponentially varying parameters of the dielectric permittivity in one direction is studied. It isshown that quasiperiodicity of the dielectric properties leads to the formation of a forbidden band that isconsiderably wider than the corresponding band for ideally periodic media. Simple approximate formulasare given for the width and central wavelength of the forbidden band.

Keywords: radiation: electromagnetic: dissipation

1. Introduction

The determination of the transmitted and reflected amplitudes of electromagnetic waves incident on inhomoge-

neous media of finite thickness remains one of the most interesting and practically important problems in the theory of

wave propagation [1,2].

This problem has recently acquired a particular urgency in connection with the development of technologies for

creating artificial periodic structures. Of these, the most interesting is photonic crystals (PC)— as special class of artificial

media with periodically varying dielectric properties over a spatial scale on the order of an optical wavelength [3]. These

systems are extensively used in millimeter and submillimeter wavelength technologies, in laser technology, and in optical

communications. Photonic crystals make full control of the propagation of light possible: the localization of light waves

has been observed in them [3-6].

Besides periodic media, quasiperiodic media (QM) are also of great interest in astrophysics and optics. Periodicity

and disorder are the two extremes of a rich spectrum of dielectric media. Although they are nonperiodic structures,

quasiperiodic media are still considered to be deterministically generated and can be regarded as convenient models for

Original article submitted March 31, 2006. Translated from Astrofizika, Vol. 50, No. 1, pp. 113-119 (February2007).

(1) Erevan State University, Armenia, e-mail: dsedrak@server.physdep.r.am, agevorgyan@ysu.am, vbadal@server.physdep.r.am(2) State Engineering University, Armenia, e-mail: akhachat@www.physdep.r.am

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describing the general picture of electromagnetic wave propagation through partially ordered layers of matter. The

forbidden energy band which develops in quasiperiodic media under these conditions makes them unique objects for

physical research and practical applications.

Many theoretical papers are devoted to the passage of fluxes of microscopic particles or the propagation of waves

through layers of inhomogeneous media. Beginning in 1938 Ambartsumyan developed a variant of the method of

superimposed layers [7,8] based on the famous “principle of invariance.” A number of other methods for describing wave

propagation in inhomogeneous media are well known: perturbation theory, immersion, transfer matrices, phase functions,

Green functions, and so on [9-15]. Although these methods do, in principle, make it possible to describe almost all the

effects which arise during the propagation of electromagnetic waves in inhomogeneous media, they still have a number

of disadvantages which make them difficult to use in specific applications.

Recently, a new method has been proposed [16-18] which provides an exact description of the dissipation of light

in a one-dimensional (1D) dielectric slab with an arbitrary refractive index. The difference between this method [16-18]

and the conventional methods is that the problem of the dissipation of an electromagnetic in an inhomogeneous 1D

dielectric is reduced to solving a Cauchy problem for a system of two first order linear differential equations with specified

initial conditions. Formulating the wave problem as a Cauchy problem makes it possible to obtain final results in cases

where an analytic solution of the equations is known or to carry out efficient calculations in cases where the ordinary

methods encounter insurmountable difficulties. This method is applied to the dissipation of a light wave in a 1D

quasiperiodic medium in this paper.

2. Basic equations

Let us consider a 1D quasiperiodic medium in which the dielectric permittivity depends on the coordinates in the

following way:

( ) ( )( )

, 2

sin1 20

σπ+ε=ε xx

xax (1)

where a(x) and ( )xσ are continuous functions (the modulation parameters) which depend on x and ( )00 ε=ε . We first

consider a linear variation in a(x) and ( )xσ : ( ) minminmax ax

d

aaxa +

−

= , ( ) minminmax x

dx σ+

σ−σ

=σ , where d is the

thickness of the layer, while amin

, amax

, minσ , and maxσ are constants. A modulated medium of this sort can be created,

for example, by a powerful optical or ultraviolet field in the presence of a temperature gradient along the x axis. We

consider the case of nonmagnetic media, i.e., we set 1=µ . We shall also neglect absorption.

According to the theory [16-18], the complex transmission pst , and reflection psr , amplitudes (where s and p

denote the polarized electromagnetic waves) in a dielectric layer with a continuous refractive index and bounded on both

sides by a homogeneous medium (with dielectric constant ε1) can be expressed in terms of the real functions ( )xH ps,

,21

and ( )xN ps,,21 at the point x = d using the formulas

89

{ } ( ) ( )( ) ( ) ( )( )[ ], exp2

11 ,2

,1

,2

,10,

dHdNidNdHdikt

pspspspsxps

−−+= (2)

{ } ( ) ( )( ) ( ) ( )( )[ ]. exp2

1 ,2

,1

,2

,10,

,

dHdNidNdHdikt

r pspspspsxps

ps

+−−−= (3)

The functions ( )xH ps,,21 and ( )xN ps,

,21 are solutions of the system of differential equations

( ), and

sin2,10

2,12,1

0

21

2

22,1 s

x

ss

x

s

Nkdx

dHH

k

x

cdx

dN−=

ε−αεω= (4)

and

( )( ) , and

sin1

12,10

2,12,1

21

02

22,1 p

x

pp

x

p

Nkxdx

dHH

xkcdx

dNε−=

ε

αε−ω= (5)

with the initial conditions

. 1 0and0 1 2121 =, ==, = ,,,, pspspsps NNHH (6)

In Eqs. (4) and (5) αω= cos0 ck x , where α is the angle of incidence of the light, ω is the angular frequency, and c is the

speed of light. The reflection psR , and transmission psT , coefficients are defined as 2,, psps rR = and

2,, psps tT = .

3. Results and discussion

First of all we examine the case a(x) = const and ( ) minminmax x

dx σ+

σ−σ

=σ , i.e., a quasiperiodic medium with

a constant periodicity amplitude and a linear profile for the variation in the identity period.

Figure 1a is a plot of the reflection coefficient R as a function of wavelength λ. The dashed curve corresponds

to the case a(x) = const and ( ) const=σ x , i.e. to an ideally periodic medium. This figure shows that there is a limited

range of wavelengths for the incident light where the reflection coefficient is maximal, i.e., 1≈R . It is referred to as the

diffractive reflection region and is analogous to the Bragg reflection region for x rays off crystal planes (it is also referred

to as the “forbidden band” from the band theory of solids). Curve 1 of Fig. 1a corresponds to a structure with

a(x) = const and ( ) x~xσ . This figure shows that a forbidden band appears in a quasiperiodic system, as in a periodic

one. Furthermore, the gradient in the identity period also leads to a shift in the Bragg reflection region, as well as to

its significant broadening.

We now introduce the formulas for the width and central wavelength of the forbidden band for ideally periodic

and quasiperiodic one dimensional media. For a(x) = a = const and ( ) const=σ=σ x the dielectric constant ( )xε can be

expanded in the Fourier series

90

( ) ( ) , 2

2

2

∑−=

σπ−

ε=εl

lxil ex (7)

where

( ) ( ) ( ) ( ) ( ) . 4

and0, 2

1 022110

0 ε−=ε=ε=ε=ε

+ε=ε −− a

,a

The width of the forbidden band of an ideally periodic medium in the two-wave approximation of the dynamic

theory for the scattering of electromagnetic waves [14] is given in terms of the Fourier coefficients of the expansion (7)

Fig. 1. The reflection coefficient R as a function of wavelength λ. (a) dashedcurve: a(x) = const = 0.5, m0.42const)( µ==σ x ; curve 1: a(x) = const = 0.5,

minminmax x

dx σ+

σ−σ

σ =)( ; curve 2: a(x) = const = 0.5, xex βγ=σ )( . (b)

dashed curve: a(x) = const = 0.5, m0.42const)( µ==σ x ; curve 1:

m0.42const)( µ==σ x , minminmax ax

d

aaxa +

−

=)( ; curve 2:

m0.42const)( µ==σ x , xexa νχ=)( . The remaining parameters are:

m44µ=d ; 2520 .=ε ; m380 µσ = .min ; m460 µ=σ .max ; amin

= 0.25; amax

=

0.75; m380 µ=γ . ; -1m004340 µ=β . ; 0.25=χ ; -1m024970 µ=ν . .

0.550.0

R

0.65 0.75 0.85 0.95

0.2

0.4

0.6

0.8

1.0 a

1

2

0.650.0

R

0.7 0.75 0.8 0.85

0.2

0.4

0.6

0.8

1.0 b

1

2

m ,µλ

91

by

( )

( ) , 0

20

ε

ελ=λ∆ (8)

where 0λ is the central wavelength of the forbidden band, given by ( )00 εσ=λ . According to the calculations

(Fig. 1a), in the ideal case the forbidden band extends from 0.6681 =λ mm to 0.7392 =λ mm and the width of the band

is 070.≈λ∆ mm. Equation (8) gives the same number on substituting the values of 0ε , σ , and a. Equation (8) can be

written differently by introducing the effective refractive indices

( )( )

( ) . 2

10

20

2,1

ε

ε±ε=effn (9)

Then we obtain the following for the central wavelength and width of the forbidden band: σ=λ n0 and σ∆=λ∆ n , where

n and n∆ are the average refractive index and the birefringence of the medium, respectively.

For a quasiperiodic medium with the parameters a(x) = a and ( ) minminmax x

dx σ+

σ−σ

=σ , the forbidden band

extends from 0.621 =λ mm to 0.952 =λ mm ( 330.=λ∆ mm). It might seem that in this case the width of the forbidden

band could be calculated approximately using the formula mineff

maxeff nn σ−σ≈λ∆ 21 . However, calculations show that

this is not so. Our studies of the dependence of R on l have shown that when there is a gradient in the identity period,

the width of the forbidden band is given by

, 21 mineff

maxeff nx

dx

dnn σ−∆

σ+σ=λ∆ (10)

where dxdσ is the gradient in the identity period and x∆ is the width of the part of the layer where a gradient in the

modulation parameter exists.

We now consider the case of ( ) const=σ x and ( ) minminmax ax

d

aaxa +

−

= , when the amplitude of the periodicity

varies linearly. Figure 1b (curve 1) shows the wavelength dependence of the reflection coefficient for this case. (As in

Fig. 1a, the dashed curve corresponds to the case with a(x) = const and ( ) const=σ x .) This graph shows that the forbidden

band extends from 0.6341 =λ mm to 0.8962 =λ mm ( 2620.=λ∆ mm). when there is a gradient in the periodicity

amplitude, as in the previous case the Bragg reflection region is shifted and broadened. A study of the variation of

( )λ= RR shows that the width of the forbidden band and the average value of the refractive index can be approximated

as follows:

, 4

∆+σ=λ∆ x

dx

dan

a(11)

, 22

1

21

212

0

∆

++ε= x

dx

daaan (12)

where da/dx is the gradient in the periodicity amplitude and ( )[ ] 2122 2minmax aaa += .

92

As another quasiperiodicity model we consider a system with an exponential variation in the modulation param-

eters: ( ) xexa νχ= and ( ) xex βγ=σ (with the constants χ , ν , and β , γ chosen so that the functions a(x) and ( )xσ vary

over the same ranges as in the linear case).

Curve 2 of Fig. 1a shows the dependence of R on λ in the case of a(x) = const, ( ) xex βγ=σ . The figure shows

that for known values of the parameters β and γ the use of an exponential function σ(x) instead of the linear form leads

to an insignificant change in the location and width of the forbidden band.

In Fig. 1b, curve 2 shows the variation with λ for a model quasiperiodic medium with ( ) const=σ x , ( ) xexa νχ= .

Clearly the broadening of the forbidden band with an exponential variation in the periodicity amplitude is less than with

a linear variation.

4. Conclusion

To summarize, we have studied the dissipation of electromagnetic waves in 1D quasiperiodic media with a

specified dielectric permittivity ( )xε . A harmonic function with amplitude and phase modulations was used as a simple

function for ( )xε . These calculations show that a forbidden band appears for both linear and exponential variations in

the modulation parameters. Under certain conditions the forbidden band can be much broader than the corresponding

band for a periodic medium. In the case of a linear dependence of the modulation parameters on x, simple approximate

formulas have been obtained for the width and central wavelength of the forbidden band that are a good approximation

to numerical calculations. It appears that we have done this type of calculation for the first time. A recently published

article [19] examines ideally periodic 1D photonic crystals consisting of alternating identical layers with a gradient and

isotropic profile of the modulation parameters within the confines of an individual layer.

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