roughness scattering

8/2/2019 Roughness Scattering

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ROUGHNESS-INDUCED SCATTERING AND ABSORPTION OF

ELECTROMAGNETIC RADIATION PROPAGATING IN COMPOSITE MATERIALS

AND THEIR DIELECTRIC PROPERTIES

Nedialko M. Nedeltchev

Institute of Electronics, Bulg. Acad. of Sciences, 72 Tzarigradsko chosse, 1784 Sofia, Bulgaria

TEL: +359(2) 9795919, FAX: +359(2) 9753201, e-mail: [email protected]

Abstract: For the purposes of nanoscience and nanotechnology, the roughness- inducedscattering and absorption of electromagnetic radiation, propagating in composite materials, is

studied. The medium of investigation is stratified one. It consists of one overlayer and a semi-

infinite material underneath. The common boundary surface between the two media is

considered as rough. The electromagnetic wave scattering model of studied composite materials

is composed of three layers- the original overlayer and semi-infinite material, and one new

intermediate anisotropic layer having definite thickness, which embrace the rough boundary

surface. The considered rough boundary surface is characterized and presented in the model by

the probability density function of its elevations and the nth-order probability density functions

of surface heights. The expressions of roughness-induced electromagnetic field in the overlayer

are derived. For this purpose the Greens functions of stratified medium having anisotropiclayers are as well drawn, namely in the case of medium with anisotropic regions characterized by

dielectric dyadics being function of the vertical distance z. The derived Greens functions can

be also used in the field of environmental remote sensing, e.g. radar cross-sections (RCS) study

of soil covered by snow and/or vegetation. The effective dielectric constant of studied composite

materials is investigated and calculated.

Keywords: nanoscience, electromagnetic scattering and absorption, composite materials,

Greens functions.

1. Introduction

The composite materials, namely their dielectric properties and their behavior from

electromagnetic point of view, are in the focus of intensive study during the last two decades.

This growing interest has been promoted by the prominent progress of nanoscience and

nanotechnology in the recent time and the needs of new raw products, having well defined

properties and features as a consequence. One important task in this domain is to find out the

electric field, expressed in a form convenient for further analysis, propagating in composite

materials and inhomogeneous media in general [1, 2]. One largely spread out parameter in

practice is their effective dielectric constant eff, (the effective permittivity may be an operator in

general), being in the scope of active investigations by many authors [2, 3, 4, 5, 6]. Theelectromagnetic properties of composite materials embedded with randomly oriented and

positioned discrete scatterers have been recently studied by the aid of strong permittivity

fluctuation theory [2]. Commonly encountered practice in the field of nanoscience and

nanotechnology is the employment of composite materials consisted of isotropic homogeneous

thin layers having, in general, rough boundary surfaces between any two of these regions. This

kind of inhomogeneous media is a subject of present analysis. This study is a generalization of

[5, 6]. In present analysis the singularities of Greens functions in the source region [7] have

been avoided. Compact form of expressions of electric field, which is propagating in such

composite materials are obtained, as well as their effective permittivity is analyzed. Elsewhere in

this analysis the boundary rough surfaces are considered as random ones. Their stochastic

properties, i.e. the joint probability density function of surface altitudes is explicitly presented inderived roughness- induced electric fields expressions.


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2. Roughness-induced electric field

The composite material, subject of this analysis, is depicted on fig. 1. It consists of two

semi-infinite isotropic homogeneous layers (Region 1, Region 2), characterized by the

permeability 0 of vacuum, and complex, in general, permittity 1, 2 respectively, 1 = 1 + i1,1 = 2 + i2, Im(1) 0, Im(2) 0. Here the altitude of boundary rough surface, which

separates the two regions, i.e. its position referenced to basic coordinate system, defined by theunit vectors [ zyx ,, ], is h( pr ), yyxxrp += , d2 < h( pr ) < d1 (fig.2). In this study the time-

harmonic case is considered, e. g. the electric field )]exp()(Re[),( tirEtrE =

. It can be

shown that the electric field propagating in the medium (fig.1) is

0)()()( 20 = rErkrE

, (1)

where k0 = (00)1/2 is the wavenumber of vacuum, and

)]([)()( 212 prhzr += . (2)

Here = () is the Heaviside step function,

>


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)(),(].)([ 2020 rrIkrrGzk =

. (8)

Its regular and singular parts are ),(),,( rrGrrG

respectively, i.e.

),(),(),( rrGrrGrrG +=

,

zzrrz

rrGrrG )()(

1),(),(

||

= . (9)

Here and everywhere in the text )( rr stands for Dirac delta function as usual. Full analysisand corresponding Greens function expressions will be presented in a coming soon paper.

Finally the solution of )(rE

can be derived from (6 8),

)()()( 00 rETGrErE

+= , (10)

where the operatorT is defined by

)(. TGIT += . (11)

Having as a main goal to avoid the singularities of Greens function in the source region

[7], the following expression can be derived from (9, 10, 11) for operatorT,

)(. TGIPT += , (12)

where ),( rrG is the regular part of ),( rrG (9), and the dyadic P ,

||

||

2|| ]

)(

1

)(

1[)]()([ P

rzPzrP

+= . (13)

The explicit expression of operator T is derived and presented in the next section.

2. Delectric properties

As it was stated above, one important electric characteristic of composite materials is the

effective permittivity eff ,

>=< )()()( rErEr eff

, (14)

where as usual < > represents an ensemble averaging. To evaluate eff its necessary to

retrieve the operatorT. The solution of (12) gives

= =

0).(

n

nPGPT , (15)


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where the dyadic P is presented by (13), and ),( rrG

is the regular part of Greens function

(9). On the other hand it can be shown (10, 11), that

>>= = 0. As a consequence, the

following values of )(),( || zz stem from this condition (2, 13),

)()()()( 1212 zFrz +=>=

===< z

d

dhhprhzzF

2

)()]([)( 11

, (20)

and p1(h) stands for probability density function of rough interface random height h (fig.1). It

worth to be noted as well that IdId 2211 )(,)( == (18, 19, 20). In the same spirit, P takesthe following form applying (18, 19, 20) on (13),

)()( 210 rPP

= , (21)where

zzz

IP )1)(

(21

2||

0

+= , (22)

and the random function >


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For example, the second-order solution >< 2T of the mean value of operator T, retrieved from(15, 21, 22), can be represented as

)().,().(),()( 0022

212 zPrrGzPrrT >=

roughness scattering

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