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Optics Communications 281 (2008) 5504–5510

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The curvilinear coordinate method associated with the short-coupling-range approximation for the study of scattering from one-dimensional random rough surfaces K. Aït Braham a,*, R. Dusséaux b a Université Paris-Est Marne-La-Vallée, Laboratoire d’Electronique, Systèmes de Communications et Microsystèmes (ESYCOM), Cité Descartes 5, Boulevard Descartes Champs sur Marne, 77454 Marne-La-Vallée Cedex 2, France b Université de Versailles Saint-Quentin en Yvelines, Centre d’Etude des Environnements Terrestre et Planétaires (CETP), 10/12 avenue de l’Europe, 78140 Vélizy, France

a r t i c l e

i n f o

Article history: Received 7 April 2008 Received in revised form 18 July 2008 Accepted 25 July 2008

Keywords: Scattering from rough surfaces Short-coupling-range approximation C method Huygens principle

a b s t r a c t This paper deals with the scattering of an incident plane wave from a mono-dimensional rough surface. The analysis of the scattering phenomenon is performed using the curvilinear coordinate method (C method) associated with the short-coupling-range approximation (SCRA) and the Huygens principle. The C method is based on the solving of Maxwell’s equations under their covariant form written in a nonorthogonal coordinate system which fits the scattering surface profile. This leads to eigenvalue problems. The scattered surface fields are expressed as linear combinations of eigensolutions and the combination coefficients are determined using the boundary conditions. Electromagnetic fields are obtained using the surface fields and the Huygens principle. The short-coupling-range approximation (SCRA) applied with the C method allows a significant saving in computation time. We confirm the efficiency and the validity of this new approach with respect to the C method. Ó 2008 Elsevier B.V. All rights reserved.

1. Introduction The simulation of electromagnetic scattering from random rough surfaces has aroused physicists and engineers for many years because of its applications both in military and civilian domains, such as radio wave propagation, material sciences, oceanography and remote sensing. The three classical analytical methods commonly used in random rough-surface scattering are the small-perturbation method (SPM), the Kirchhoff method (KM) and the small-slope approximation (SSA). The SPM and the KM are performed for slightly rough surfaces and surfaces with small curvature radii, respectively [1,2]. The SSA can be applied to an arbitrary wavelength, provided the tangent of grazing angles of incident/scattered radiation sufficiently exceeds RMS slopes of roughness [3,4]. If the surface parameters are of order of the incident wavelength, the analysis of the scattering phenomenon with these analytical models becomes inaccurate and should be replaced by numerical methods based on an exact formalism [5–7]. The curvilinear coordinate method, commonly called the C method, is an efficient theoretical tool for analyzing rough surfaces in the resonance domain [8–13]. It consists on solving Maxwell’s

* Corresponding author. Tel.: +33 1 39 25 48 27. E-mail addresses: [email protected] (K. Aït Braham), [email protected] (R. Dusséaux). 0030-4018/$ - see front matter Ó 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2008.07.062

equations under their covariant form written in a nonorthogonal coordinate system which fits the surface profile. The C method leads to eigenvalue systems [10–12]. Then, the scattered fields can be expanded as a linear combination of eigensolutions. The boundary conditions allow the combination coefficients to be determined. The dominant computational cost for the C method is the eigenvalue problem solution which is of order of ð2M s Þ3 where 2M s is the size of the eigenvalue systems. In this paper, we associate the C method with the Huygens principle [14] and with the short-coupling-range approximation (SCRA) [15,16] in order to reduce the computational time. The surfaces under consideration are characterized by a Gaussian height probability distribution and by a Gaussian correlation function [12,13], so that random rough surfaces are completely specified by the rms height (ra ) and the correlation length (‘c ). We suggest working out the bi-static scattering coefficient. The adopted procedure consists of two stages. First, the surface fields are obtained by the C method associated with the SCRA. The Short-Coupling-Range Approximation shows that the surface fields at a given point of a rough surface only depend on the shape of the profile inside an interval centered at this point and that has a width of one or two wavelengths of the incident light [15,16]. According to the SCRA, the whole surface is represented by several elementary ones. For each elementary surface, the surface current densities are derived from the C method. The total surface field is deduced from a concatenation of elementary surface current densities. Second, the

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far-field and the bi-static scattering coefficients are derived from the Huygens principle applied to the total surface fields. The scattering problem is presented in Section 2 and the curvilinear coordinate method and the short-coupling-range approximation are described in Sections 3 and 4, respectively. In Section 5, we validate this approach both for perfect conductors and penetrable media and show that the C method associated with the SCRA permits us the analysis of large rough surface within a reasonable computation time.

1 ðE Þ ~ Em;sk ðx; yÞ ¼ 2p

We consider the scattering of a harmonic incident plane wave from a one-dimensional local perturbation. The surface is illuminated under incidence angle hi . The time dependence factor varies as expðjx tÞ, where x is the angular frequency. Fig. 1 illustrates the geometry of the problem. The surface separates two homogenous media, the air (er1 ¼ 1) and a lower medium (er2 ) which can be a dielectric material or perfect conductor. The deformation is given by the following equation:

y¼



aðxÞ 0

if x 2 ½L=2; L=2

ð1Þ

elsewhere

L is the length of the perturbation zone and defines, in a first approach, the zone illuminated by a transmitter. aðxÞ is a random function whose values obey to a Gaussian probability density with zero mean value and a root mean square height ra . Its correlation function is also Gaussian with a correlation length ‘c . Both fundamental cases of Ek and Hk polarizations are considered. In case of Ek polarization, the electric vector is parallel to the Oz axis and for Hk polarization, this is the case of the magnetic vector. Let wm ðx; yÞ be one of the electromagnetic field components. Subscript denotes quantities relative to the upper medium (m = 1) and the lower medium (m = 2). With the presence of the roughness, in addition to the specular reflected plane wave w1;r ðx; yÞ and the transmitted plane wave w2;r ðx; yÞ (in case of dielectric lower medium), a scattered field wm;s ðx; yÞ appears. In the upper medium we note the total field

w1 ðx; yÞ ¼ wi ðx; yÞ þ w1;r ðx; yÞ þ w1;s ðx; yÞ;

ð2Þ

1

ð4Þ

 expðjbm ðaÞjyjÞ expðjaxÞda with

~ kd;1 ¼ a~ ux þ b1~ uy ;

a þ

2.1. Presentation of the problem

þ1

ðEk Þ ^m R uz ðaÞ expðjbm ðaÞjyjÞ expðja xÞda ~ ! Z þ1 ~ kd;m 1 ðEk Þ kÞ ~m;s ^ ðE R Zm H ðx; yÞ ¼ ^~ uz m ðaÞ 2p 1 km

2

2. Scattering by random rough surfaces

Z

~ kd;2 ¼ a~ ux  b2~ uy

ð5Þ

2 km ;

b2m

¼ Imbm < 0 2p pffiffiffiffiffiffi ; k2 ¼ er2 k1 k1 ¼ k Z1 Z 1 ¼ 120p; Z 2 ¼ pffiffiffiffiffiffi er2

ð6Þ ð7Þ ð8Þ

Z1 is the intrinsic impedance of free space and Z 2 , the dielectric ^ m ðaÞ is the scattering amplitude associated medium impedance. R with the elementary wave function expðja xÞ expðjbm jyjÞ. The propagation constant bm gives the type of the wave. If ða 6 km Þ then bm ðaÞ is a positive real. The associated wave is an outgoing propagating wave. For a lossless medium and in the far-field zone, the Rayleigh integral is reduced to the only contribution of these propagating waves. At an observation point Mðr; hÞ and using the stationary phase method, we obtain the asymptotic field [18,19]:

rffiffiffiffiffi  p 1 ^ ðEk Þ ~ Rm ðkm sin hÞ cos h expðjkm rÞ exp j uZ kr 4 rffiffiffiffiffi 1 ^ ðEk Þ ðEk Þ ~m;s Rm ðkm sin hÞ cos h Zm H ðr; hÞ  ðerm Þ1=4 kr  p ~  expðjkm rÞ exp j uh ; jhj < 90 4 ð9Þ

ðE Þ ~ Em;sk ðr; hÞ  ðerm Þ1=4

k is the wavelength of the incident wave and ~ uh is the unit vector in the polar coordinate system (see Fig. 1). In order to obtain the ðE Þ expression of fields in case of Hk polarization, we substitute ~ Em;sk ðH Þ ðE Þ ðH Þ k k k ~m;s and Z m H ~m;s by ~ by Z m H Em;s in Eqs. (4) and (9). For an incident wave in (a) polarization (a = Ek or a = Hk), the ðaÞ normalized bi-static scattering coefficient rm ðhÞ is defined as follows:

rðaÞ m ðhÞ ¼

and in the lower dielectric medium,

ðaÞ ðaÞ ^m 1 dPm;s jR ðkm sin hÞ cos hj2 ¼ e r m ðaÞ dh kL cos hi P

ð10Þ

i

dP

w2 ðx; yÞ ¼ w2;t ðx; yÞ þ w2;s ðx; yÞ

ð3Þ

The problem consists on working out the field wm;s ðx; yÞ scattered within the two media. 2.2. The Rayleigh representation of the scattered fields The scattered field can be expressed by a Rayleigh expansion [3,17] in the region out of the deformation ðy > max aðxÞÞ or ðy < min aðxÞÞ. In case of Ek incident wave we have

ðaÞ

ðaÞ

where dhm;s is the scattered angular power density and Pi , the total ðaÞ incident power flux through the modulated zone. rm ðhÞ is an angular random process. So, for a set of N R surface profiles, we define an averðaÞ aged bi-static scattering coefficient hrm ðhÞi. The angular brackets h i stand for ensemble averaging. The coherent intensity IðaÞ m;c ðhÞ corresponds to the bi-static scattering coefficient associated with the averðaÞ aged electromagnetic field. The incoherent intensity Im;f ðhÞ is defined as the difference between the two preceding physical quantities: ðaÞ

ðaÞ Im;f ðhÞ ¼ hrðaÞ m ðhÞi  Im;c

¼

erm cos2 h ^ ðaÞ ^ ðaÞ ðkm sin hÞij2 Þ ðhjRm ðkm sin hÞj2 i  jhR m kL cos hi

ð11Þ

u

θi Upper medium

ε1 = 1

Lower medium ε 2

y θ

ki −

L 2

3. The curvilinear coordinate method and the Huygens principle

uθ

0

+

L 2

x

Fig. 1. Local deformation illuminated by a plane wave under incidence hi .

The scattered field cannot be expressed by the Rayleigh expansion inside the modulated zone [3,17]. In order to overcome this problem, we can obtain an expression of fields that is valid over the surface by solving Maxwell’s equations under their covariant form in the translation coordinate system. The curvilinear coordinate method gives the surface current densities and the Huygens–Fresnel principle, the scattering amplitudes.

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3.1. Coordinate system – covariant components of field The translation system is obtained from the Cartesian one ðx; y; zÞ [8]:

8 0 > : 0 z ¼z

requires, for each medium, the solutions of 2M s -dimensional eigenvalue system with Ms ¼ ð2M þ 1Þ. M is the truncation order and determines the eigenvalue system size. 3.2. Surface fields with the C method

ð12Þ

The limit surface a(x) is of equation y0 = 0 in the translation system. The change from Cartesian components (Vx; Vy; Vz) of vector ~ V to covariant components (Vx0 ; Vy’ ; Vz0 ) is given by [8,20]:

8 0 0 _ > < V x0 ðx ; y Þ ¼ V x ðx ; yÞ þ aðxÞV y ðx ; yÞ 0 0 _ V y0 ðx ; y Þ ¼ V y ðx ; y Þ with aðxÞ ¼ da dx > : 0 0 V z0 ðx ; y Þ ¼ V z ðx ; y Þ

ð13Þ

Covariant components V y0 ðx0 ; y0 Þ and V z0 ðx0 ; y0 Þ become identified with Cartesian components V y ðx; yÞ and V z ðx; yÞ where y0 ¼ y aðxÞ. V x0 and V z0 are parallel to coordinate surfaces y0 ¼ y0 , so the covariant components Ex0 Hx0 are tangential to the rough surface. In a source-free medium, from Maxwell’s equations and the constitutive equations written in the translation system, it can be shown that first, covariant components Em;sx0 , Em;sz0 , Hm;sx0 and Hm;sz0 can be expressed only in terms of components Em;sy0 and Hm;sy0 [8,12,13]:

As shown in Refs. [12,13], the Fourier transform of Oy-component is defined as a linear combination of M s eigensolutions satisfying the outgoing wave condition

^ m;s ða; y0 Þ ¼ w

o2 Hm;sx0 o2 Hm;sy0 0 0 2 2 þ km Hm;sx0 ¼  km g x y Hm;sy0 2 0 ox0 oy0 oy   o2 Em;sz0 oZ m Hm;sy0 2 x0 y0 oZ m H m;sy0 þ k E þ m;sz0 ¼ jkm g m 2 0 0 oy ox oy0

ð14Þ ð15Þ ð16Þ ð17Þ

and secondly, longitudinal components Em;y0 and Hm;y0 obey to the same propagation Eq. (18) [8,12,13]

o2 wm;s ox0 2

þ

o2 wm;s oy0 2

2 þ km wm;s

þg

y0 y0

o2 wm;s oy0 2

" # 0 0 owm;s og x y wm;s o x0 y0 þ 0 g þ ¼0 oy ox0 ox0

^ m;n ða; y0 Þ: Am;n w

ð20Þ

n¼1

^ m;s ða; y0 Þ represents the Fourier Transform of Function w 0 0 wm;s ðx ; y Þ. According to the sampling theorem [21], the elementary ^ m;n ða; y0 Þ can be constructed from samples wave functions w /m;n ðaq Þ by the following interpolations:

^ m;n ða; y0 Þ ¼ w

p  /m;n ðaq Þsinc ða  aq Þ expðjk1 r m;n jy0 jÞ; Da q¼M þM X

ð21Þ where

aq ¼ k1 sin hi þ qDa:

ð22Þ

Da is the spectral resolution that is fixed by the cut-off integer MC . Da is less than or equal to 2p=L

Da ¼ o2 Em;sx0 o2 Em;sy0 0 0 2 2 þ km Em;sx0 ¼  km g x y Em;sy0 2 0 ox0 oy0 oy   o2 Z m Hm;sz0 oEm;sy0 2 x0 y0 oEm;sy0 0 þ k Z H ¼ þjk g þ m m m m;sz oy0 ox0 oy0 2

2Mþ1 X

2k1 2MC þ 1

and Da 

2p : L

ð23Þ

^ m;n ða; y0 Þ is constructed from the samples The eigenfunction w /m;n ðaq Þ (eigenvector components) and the eigenvalue rm;n . The ^ m;n ða; y0 Þ represents an outgoing wave propagating eigenfunction w with no attenuation if Reðr m;n Þ > 0 and Imðr m;n Þ ¼ 0. For an evanescent wave, Imðr m;n Þ < 0. For each medium, it is observed numerically that among the 2Ms eigenfunctions, M s of them correspond to outgoing waves and as many to incoming waves. Substituting Em;sy0 ¼ 0 and wm;s ¼ Z m Hm;sy0 into Eqs. (14)–(17) and applying the numerical procedure reported in Refs. [12,13], we obtain the Fourier transforms of Ek-polarized transverse components as follows:

0 @

^ ðEk Þ 0 ða; y0 Þ Zm H m;sx ^ðEk Þ 0 ða; y0 Þ E m;sz

1 A¼

2Mþ1 X n¼1

0

^ ðEk Þ 0 ð Zm H ðEk Þ m;sx ;n @ Am;n ðE Þ

aÞ

^ k 0 ðaÞ E m;sz ;n

1 A expðjkrm;n jy0 jÞ ð24Þ

ð18Þ ðE Þ

where

wm;s ðx0 ; y0 Þ ¼ Em;sy0 ðx0 ; y0 Þ for the polarization Ek

0

wm;s ðx0 ; y0 Þ ¼ Z m Hm;sy0 ðx0 ; y0 Þ for the polarization Hk 0 0

@

0 0

g x y and g y y are elements of matrix tensor which depend on the derivatives of function aðx0 Þ with respect to x0 [8]: 0 0

da dx0  2 da ¼1þ dx0

gx y ¼  0 0

gy y

ðE Þ

^ k 0 ða; y0 Þ ¼ 0; ^ k 0 ða; y0 Þ ¼ E Zm H m;sz m;sy

with

ð19Þ

In the previous work [13], we have proposed a procedure for solving the propagation equation in the spectral domain. The Oycomponents are expanded as a linear combination of eigensolutions satisfying the outgoing wave condition. We deduce from (14)–(17) the Ek-polarized components of electric and magnetic fields by taking Em;sy0 ¼ 0 and the Hk-polarized components by taking Hm;sy0 ¼ 0. The expansion coefficients are found by solving the boundary conditions. As shown in reference [12], the method

ðE Þ

^ k 0 ðaÞ Zm H m;sx ;n ^ðEk Þ 0 ð E m;sz ;n

aÞ

1 A¼

þM X q¼M

0 @

ðE Þ

^ k 0 ðaq Þ Zm H m;sx ;n ^ðEk Þ 0 ð q Þ E m;sz ;n

a

1

  A sinc p ða  aq Þ : Da ð25Þ

^ ðEk Þ 0 ðaq Þ and E ^ðEk Þ 0 ðaq Þ are obAs shown in Refs. [12,13], samples H m;sx ;n m;sz ;n tained from /m;n ðaq Þ. Substituting Hm;sy0 ¼ 0 and wm;s ¼ Em;sy0 into Eqs. (14)–(17), we obtain the Fourier transforms of Hk-polarized transverse components. The combination coefficients AðaÞ m;n are found by solving the boundary conditions according to the type of the lower medium (perfect conductor or dielectric medium) and the polarization of the impinging wave. ðE Þ ðE Þ In case of Ek polarization, we notice F m;k z0 ðx0 ; y0 Þ ¼ Em;k sz0 ðx0 ; y0 Þ ðE Þ

ðE Þ

ðH Þ

and F m;kx0 ðx0 ;y0 Þ¼ZHm;ksx0 ðx0 ;y0 Þ, and for Hk polarization F m;kz0 ðx0 ; y0 Þ ¼ ðH Þ Z m Hm;ksz0 ðx0 ; y0 Þ

ðH Þ ðH Þ and F m;kx0 ðx0 ; y0 Þ ¼ Em;ksx0 ðx0 ; y0 Þ. The surface ðaÞ ðaÞ functions F m; z0 ðx0 ; y0 Þ and F m; x0 ðx0 ; y0 Þ where y0 ¼

fields

are given by

0.

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The Cartesian component F ðaÞ m; z ðx; yÞ of the scattered field can be calculated from surface fields and Huygens principle [14,15]. In the upper medium, we obtain

¼

j 4

1 þ 4

Z Z

þp=Da

p=Da þp=Da

p=Da

ðaÞ

F 1; z0 ðx0 ; y0 ¼ 0Þ ðaÞ k1 F 1; x0 ðx0 ; y0

¼

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð2Þ oH0 ðk1 j~ r ~ r 0 jÞ _2 0 on

p

Z

þ1

1

0

-5

1 expðjaxÞ expðjb1 jyjÞ da b1

ð27Þ

and the Rayleigh expansion (4) into (26), we obtain the far field angular dependence for y > maxðaðxÞÞ as follows [15]:

^ ðaÞ ðk1 sin hÞ cos h R 1 Z 1 þp=Da ðaÞ 0 0 0 0 _ 0 Þ sin hÞF ðaÞ ¼ ½ðcos h  aðx 1;z0 ðx ; y ¼ 0Þ þ F 1;x0 ðx ; y ¼ 0Þ 2 p=Da  expðjk1 sin h x0 Þ expðjk1 cos h að x0 ÞÞ dx0 :

0

5

10

-0.5 0

5

10

15

20

x

Fig. 3. Two elementary profiles bq ðxÞ of length L=N b þ 2d þ 2‘t . Surface parameters: L ¼ 50k, N b ¼ 5, d ¼ 2k, ‘t ¼ k.

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ a_ 2 ðx0 Þ dx0 ;

ð2Þ 0ÞH0 ðk1 j~ r ~ r 0 jÞ

ð26Þ

1

0

x

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð2Þ where j~ r  r j ¼ ðx  x0 Þ2 þ ðy  aðx0 ÞÞ2 , H0 ðk1 j~ r ~ r 0 jÞ is the zeroth ~ order Hankel function of the second kind and n is the outward normal to the surface. Formula (26) associates the surface fields ðaÞ ðaÞ F m; z0 ðx0 ; y0 ¼ 0Þ and F m; x0 ðx0 ; y0 ¼ 0Þ given by the C method with the Huygens principle. Above the highest point of the surfaceðy > max aðxÞÞ; the CarteðaÞ sian component F 1; z ðx; yÞ can be expressed by a Rayleigh expansion. Using the Weyl representation of the Hankel function [15] ð2Þ

0.5

1 þ a ðx Þ dx0

~0

H0 ðk1 rÞ ¼

0.5

-0.5 -10

ðaÞ

F 1; z ðx; yÞ

y

3.3. Scattering amplitudes derived from the C method and the Huygens principle

ð28Þ

The bi-scattering coefficient is given by Eq. (10), the C method gives the surface fields and the scattering amplitudes ^ ðaÞ ðk1 cosðhÞÞ are calculated using Eq. (28). R 1

ization, respectively. The SCRA does not hold when the height of asperities are greater than their width [15,16]. In this paper, we use this approximation with the C method and we apply it to obtain the electromagnetic fields scattered from perfect conductor and dielectric surfaces. The surface with a total length L is represented by N b elementary profiles aq ðxÞ having a shorter length L=N b (see Fig. 2). As shown in Fig. 3, the elementary profile bq ðxÞ analyzed with the C method is of length lb ¼ L=N b þ 2d þ 2‘t . Two consecutive elementary profiles present a common zone having a length 2d þ 2‘t (see Fig. 3). This reduces the discontinuities over the surface fields at the truncation places. Elementary profiles are defined as follows:

bq ðxÞ ¼ V q ðxÞaðxÞ:

ð29Þ

Vq(x) is a weighting window 8 xþlb =2 xþlb =2 1 > > > ‘t  2p sinð2p ‘t Þ > 8 > > > x 2 ½lb =2 þ qlb ;lb =2 þ qlb þ ‘t  > > > > > > < x 2 ½l =2 þ ql þ ‘ ;l =2 þ ql  ‘  > > t b t b b b < 1 if > x 2 ½lb =2 þ qlb  ‘t ; lb =2 þ qlb  V q ðxÞ ¼ > > > : > > > elswhere > > > > xþl =2 xþl =2 > b  1 sinð2p b Þ > > ‘ ‘t 2p > : t 0 ð30Þ

4. The short-coupling-range approximation (SCRA) This approximation assumes that for a perfect conducting random rough surface, the surface field at a given point only depends on the shape of the profile inside an interval centered at this point. The width d of this interval is close to one wavelength for an Ek polarized incident wave and to two wavelengths for the Hk polar-

0.5

Function V q ðxÞ has continuous first and second derivatives. The length ‘t defines a transition zone where V q ðxÞ varies continuously from 1 to 0. This weighting window is used to limit the edge effects [12,13]. The C method is used to perform the surface fields associated with each elementary profile bq ðxÞ. Using elementary profiles with a perturbation length L=N b þ 2d, we obtain surface fields ða;qÞ ða;qÞ F m; z0 ðx0 ; y0 ¼ 0Þ and F m; x0 ðx0 ; y0 ¼ 0Þ associated with profiles aq ðxÞ ðaÞ of length L=N b . The total surface fields F m; z0 ðx0 ; y0 ¼ 0Þ and ðaÞ 0 0 F m; x0 ðx ; y ¼ 0Þ are obtained by concatenations of elementary surface fields

y

h i ðaÞ ða;1Þ ða;2 ða;3Þ ða;N Þ F m;z0 ðx0 ; 0Þ ¼ F m;z0 ðx0 ; 0Þ; F m;z0 ðx0 ; 0Þ; F m;z0 ðx0 ; 0Þ . . . ; F m;z0 b ðx0 ; 0Þ h i: ða;N Þ ðaÞ ða;1Þ ða;2 ða;3Þ F m;x0 ðx0 ; 0Þ ¼ F m;x0 ðx0 ; 0Þ; F m;x0 ðx0 ; 0Þ; F m;x0 ðx0 ; 0Þ . . . ; F m;x0 b ðx0 ; 0Þ ð31Þ

0

The scattered amplitude is calculated using the overall surface fields (31) into Eq. (28).

5. Numerical results 5.1. Saving in the computation time

-0.5 -25

-20

-15

-10

-5

0

x

5

10

15

20

Fig. 2. Total surface profile of length L divided into N b sub-domains.

25

MC and M are the two numerical parameters of the C method. For a given perturbation length L, the cut-off order MC gives the spectral resolution Da (23). The truncation order M gives the high-

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est spatial frequency amax . The Mth-order truncation removes the highest spatial frequencies of the field components and of the associated eigenwaves. Indeed, integration variable a varies within interval ½amax ; þamax  where amax depends on ratio M=MC

amax ¼ aM ¼ MDa ¼

2k1 M M  ki 2M C þ 1 MC

if M C >> 1:

ð32Þ

The proportion of evanescent waves becomes larger when amax increases, leading to a better description of the coupling phenomena and a better accuracy on results. The cut-off order M C linearly increases with the perturbation length and the higher spatial frequency amax increases with the increasing of the root-mean-square slope [10–12]. In each medium, the C method requires the solution of a 4M þ 2-dimensional eigenvalue system and for each polarization, the solution of a linear 4M þ 2-dimensional system when dealing with the boundary conditions. The dominant computational cost for the C method is the eigenvalue problem solution which is of order of ð2M s Þ3 (with M s ¼ 2M þ 1). The accuracy on results depend on the value of the highest spatial frequency amax . In order to compare the C method and the C method associated with the SCRA, we should give the same value of amax for both approaches. So, with respect to the direct C method, the computational time required by the C method and the SCRA is divided by the theoretical factor qth given by

Table 1 CPU time versus truncation order and common zone length d (k) C method

C method + SCRA

L/k

MCM

Tcpu(s)

Nb

L/Nb

d/k

M CMþSCRA

Tcpu (s)

50 60 200

156 372 606

75 670 4000

5 6 25

10 10 8

3 3 4

54 108 48

21 87 90

amax

qreal

qth

18.85 37.70 18.85

3.6 7.7 44

4.8 6.8 56.5

The surface under consideration is a single realization of random spatial process with ra ¼ 0:5k, lC ¼ 1:5k. For all the configurations, ‘t ¼ k.

!3  3 1 MCM 1 DaCMþSCRA s ¼ Nb M CMþSCRA Nb DaCM s  3 1 L þ 2‘t ¼ : Nb L=Nb þ 2d þ 2‘t

qth 

ð33Þ

Table 1 gives computational times, theoretical factors qth and real factors qreal for different lengths L, truncation order M, cutoff order M C and number of elementary profiles N b . qreal and qth are close to each other. This table shows that the SCR approximation allows a significant saving in computation time. Fig. 4 shows the incoherent intensity estimated over 10 realizations. Curves are given for an Hk-polarized incident wave. The rough surface is illuminated under hi ¼ 30 . The surface separates the air from a dielectric medium. The relative permittivity is fixed at er2 ¼ 3:6  0:2j. The simulation parameters are those of Table 1 corresponding to the length L ¼ 200k. As illustrated in Fig. 4, comparison between results derived from both versions of the C method is conclusive. 5.2. Validation of results in the far-field zone Fig. 5 shows the incoherent intensity given from both versions of the C method. The perfectly conducting rough surfaces under study are characterized by a rms height ra ¼ k=2 and a correlation length ‘c ¼ 3k=2. Results are estimated over 100 realizations and given for an Ek-polarized incident wave impinging up on the surface at hi ¼ 30 . The total length L is fixed at 60k. Each surface is represented by six elementary profiles. The common zone takes two values with d = 2k and 4k. As shown in Fig. 5, comparison is good for both lengths of the common zone and confirms the validity of SCR approximation for the rough surface under consideration. A peak in the backscattering direction can be seen. The Monte-Carlo simulations based on the C method associated with the SCRA enable us to predict backscattering enhancement. Let’s consider now a dielectric lower medium with er2 ¼ 6. The surfaces under consideration present relatively large slopes with a rms height ra ¼ 0:8k and a correlation length ‘c ¼ 1k. We apply

Incoherent intensity(H//)

0.2 0.18

C method

0.16

C method + SCRA

0.14 0.12 0.1 0.08 0.06 0.04 0.02 0

-8 0

-60

-40

-2 0

0

20

40

60

80

θ Fig. 4. Incoherent intensity for a dielectric lower medium illuminated by a Hk polarized incident wave. The parameters are those of Table 1 where L ¼ 200k. ‘t ¼ k, N R ¼ 10 and er2 ¼ 3:6  0:2j.

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K. Aït Braham, R. Dusséaux / Optics Communications 281 (2008) 5504–5510

0. 7 C method C method + SCRA : δ = 2λ C me tho d + SCRA : δ= 4λ

Incoherent intensity (E//)

0. 6

0. 5

0. 4

0. 3

0. 2

0. 1

0

-8 0

-60

-40

-2 0

0

20

40

60

80

θ Fig. 5. Incoherent intensity for a perfect conductor surface illuminated by an Ek polarized incident wave. Simulations parameters: L ¼ 60k, ra ¼ k=2, lc ¼ 3k=2, N b ¼ 6, ‘t ¼ k, M CM ¼ 372, amax ¼ 6k1 . MCMþSRCA ¼ 96 for d ¼ 2k, M CMþSRCA ¼ 120 for d ¼ 4k.

0.06

C method C method + PFC : δ = 1λ C method + PFC : δ = 4λ

Incoherent intensity (H//)

0.05

0.04

0.03

0.02

0.01

0

-80

-60

-40

-20

0

20

40

60

80

θ Fig. 6. Incoherent intensity in case of dielectric medium for a Hk polarized incident wave Simulations parameters: L ¼ 60k, amax ¼ 9k1 . MCMþSRCA ¼ 144 for d ¼ 1k, MCMþSRCA ¼ 180 for d ¼ 4k.

the C method direct to each surface of length L ¼ 60k with a truncation order MCM ¼ 558. The SCRA is used with N b ¼ 6. The common zone d takes two values d = 1k and 4k. Results are estimated over 100 realizations and given for an Hk-polarized incident wave. The incidence angle is hi ¼ 30 and the wavelength, k ¼ 5:36 cm. As ðH Þ shown in Fig. 6, comparison on the incoherent intensity Im;fk ðhÞ is good and it illustrates the validity of the SCRA in the case of dielectric medium. A common zone of 1k is sufficient to obtain the averaged power in the far-field zone. The scattering pattern also presents a peak in the backscattering direction that is well described by using the SCR approximation.

ra ¼ k=2, lc ¼ 3k=2, Nb ¼ 6, ‘t ¼ k, MCM ¼ 558,

6. Conclusion In this paper, a new version of the C method based on the shortcoupling-range approximation has been presented and implemented for analyzing the electromagnetic field scattered from a one-dimensional random surface. The Monte Carlo technique has been applied for estimating the averaged scattering pattern from the results over several realizations. The SCRA applied with the C method allows the analysis of large surfaces and an important saving in computation time with respect to the C method alone. We have shown the efficiency and the validity of SCRA for perfect con-

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K. Aït Braham, R. Dusséaux / Optics Communications 281 (2008) 5504–5510

ductors and penetrable media in both fundamental polarizations. The surfaces under study in this paper present relatively large slopes for which the predictions of the standard analytic methods are inaccurate. The Monte-Carlo simulations based on the new version of the C method have been successful in predicting backscattering enhancement. In order to reduce the computational time of the C method, we have successfully implemented the short-coupling-range approximation in the case of one-dimensional rough surfaces. The implementation in the case of two-dimensional surfaces is in progress. References [1] L. Tsang, J.A. Kong, R.T. Shin, Theory of Microwave Remote Sensing, WileyInterscience, New York, 1985. [2] P. Beckmann, A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces, Pergamon, Oxford, 1963. [3] G. Voronovich, Wave Scattering from Rough Surfaces, Springer, Berlin, 1994. [4] G. Voronovich, Waves Random Media 4 (3) (1994) 337.

[5] J.A. Ogilvy, Theory of Waves Scattering from Random Rough Surfaces, Adam Hilger, Bristol, 1991. [6] K.F. Warnick, W.C. Chew, Waves Random Media 11 (2001) 1. [7] M. Saillard, A. Sentenac, Waves Random Media 11 (2001) R103. [8] J. Chandezon, D. Maystre, G. Raoult, J.Opt. (Paris) 11 (1980) 235. [9] L. Li, J. Chandezon, J. Opt. Soc. Am. 13 (1995) 2247. [10] R. Dusséaux, R. de Oliveira, PIER 34 (2001) 63. [11] R. Dusséaux, C. Baudier, PIER 37 (2003) 289. [12] C. Baudier, R. Dusséaux, K.S. Edee, G. Granet, Waves Random Media 14 (2004) 61. [13] K. A Braham, R. Dusséaux, G. Granet, Waves Random Complex Media 18 (2) (2008) 255. [14] A. Ishimaru, Electromagnetic Wave Propagation, Radiation and Scattering, Prentice Hall, New Jersey, 1991. [15] D. Maystre, IEEE Trans, Antennas Propagat. 31 (1983) 885. [16] D. Maystre, P. Rossi, J. Opt. Soc. Am. 3 (1986) 1276. [17] P.M. Van DenBerg, J.T. Fokkema, Radio Sci. 15 (1980) 723. [18] M. Born, E. Wolf, Principal of the Optics – Electromagnetic Theory of Propagation Interference and Diffraction of Light, Pergamon, Oxford,, 1980. Appendix III. [19] C. Baudier, R. Dusséaux, PIER 34 (2001) 1. [20] J.A. Stratton, Electromagnetic Theory, McGraw-Hill, New York, 1941. [21] D. Middleton, Introduction to Statistical Communication Theory, McGraw-Hill, New York, 1960.