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REGULAR PAPER
Scattering of electromagnetic waves from random rough metallicsurfaces with one-dimensional structure calculated by the surfaceimpedance boundary condition method (SIBC)
Mehdi Khemiri1 • Imed Sassi1 • Mohamed Oumezzine1
Received: 12 February 2015 / Accepted: 22 June 2015
� The Optical Society of Japan 2015
Abstract A comparison between exact solutions and
surface impedance boundary condition method, for the
phenomenon of light scattering from one-dimensional
random rough metallic surfaces in both cases of polariza-
tion (s- and p-) is presented. The reflective properties of
random rough metallic surfaces at a large angle of inci-
dence have been reported on due to their potential appli-
cations in some of the radiative heat transfer research areas.
The influence of surface roughness, shadowing effects, the
nature of polarization, and the nature of the material on the
behavior of reflectivity and emissivity has been quantified.
1 Introduction
The evaluation of the electromagnetic waves scattering on
randomly rough surfaces has been used in many applica-
tions such as remote sensing, telecommunications, optics,
medical imagery, photorealistic image generation in com-
puter graphics, etc. The calculation of the distribution of
the intensity of light scattered from randomly rough sur-
faces has been reported using several exact and approxi-
mate models. Exact models based on the extinction
theorem [1–3] have been used to resolve Maxwell’s
equations at the surface materials for a wide range of
surface roughness parameters. These models have been
successfully applied to dielectric, metallic or perfectly
conducting surfaces [2–12].
Among these methods, we have, rigorous coupled wave
analysis (RCWA) which has been used to calculate spectral
properties of metallic grating [13–16]. Recently, Qiu et al.
have reported on the reflective properties of randomly
rough metallic surfaces at large incidence angle [17]. Sai
et al. in [18] used the finite-difference time domain method
(FDTD) to investigate the spectral properties of one-di-
mensional tungsten grating.
Exact methods present many disadvantages. They are
very expensive. However, the interpretation of the physical
mechanisms involved in the scattering process is difficult
because different effects, e.g., shadowing or multiple
scattering, are included implicitly in the calculation, and
there is no way of separating them out [19]. Moreover,
these methods are difficult to implement for surfaces in two
dimensions because of requirements on computer memory
and calculation time.
Approximate models, based on simplified assumptions,
have been widely used to treat the problem of the scattering
phenomena for different reasons. These models, explicitly,
enable to take into account certain surface characteristics
(root mean square of height, correlation length, and curva-
ture) and allow to break down the local and non-local inter-
actions involved in the process of electromagnetic waves
scattering. Moreover, these approximate models are very
easy to implement and require short time calculations.
Among these approximate models, we have the small per-
turbation method with first order (SPM1) which requires, in
the case of deterministic surfaces, the knowledge of Fourier
Transform of the surface height. Kirchhoff’s approximation
(KA), Geometric Optics Approximation (GOA), Small Slope
Approximation (SSA), are computationally less expensive
than exact methods. However, many authors have used these
methods, and studied the domains of validity of these meth-
ods by comparing them with exact methods [4, 20–28].
& Mehdi Khemiri
khemirifsm@gmail.com
1 Laboratoire physic-chimie des Materiaux-Faculte des
sciences de Monastir, Universite de Monastir, 5019 Monastir,
Tunisia
123
Opt Rev
DOI 10.1007/s10043-015-0116-3
Electromagnetic waves scattering by random rough
metallic surfaces has been studied by several authors. J. A.
Sanchez and M. Nieto-Vesperinas have reported on the
resonance effects in multiple light scattering from statisti-
cally rough metallic surfaces using a Monte Carlo simu-
lation method. They studied intensity, as well as the effects
associated with the excitation of surface electromagnetic
waves, and, in particular with surface polaritons (SP) [29].
M. Saillard and D. Maystre in [3] reported on the phe-
nomenon of scattering from metallic and dielectric rough
surfaces. Maradudin et al. in [6] used Green’s second
integral theorem to investigate the scattered electromag-
netic field from random metallic grating. For the finite
conductivity metal, it is shown that exact methods are
computationally very intensive and not suitable to metallic
surfaces because of the complex structure of metallic sur-
face. Therefore, an alternative approach called surface
impedance boundary condition (SIBC) was proposed [30–
33]. This method has been used for over 74 years as a
means of analyzing the problems of reflection from con-
ducting surfaces. It is used to calculate the efficiency of a
metallic diffraction grating [34, 35] and grating of small
curvature [36].
In this paper, we report on the numerical study of
scattering from random rough metallic surfaces using SIBC
method. We are interested in studying the variation of the
bidirectional reflectivity (BDR) of two different surface
materials (gold and silver). The effects of roughness, the
nature of polarization, incidence angle, and surface mate-
rials on the behavior of the radiation properties are pre-
sented. The regions of validity of SIBC method are
discussed in comparison with results from literature.
2 Surface geometry
We consider a one-dimensional rough surface. The surface
profile function in the x–z plane Pr[z = h(x)] separates a
vacuum region V1[z[ h(x)] from a metal medium
V2[z\ h(x)]. The surface is only varying in the x direction,
so that scattering is considered in the plane of incidence,
and it has a Gaussian random process which is completely
characterized by the correlation function defined by:
h x1ð Þh x2ð Þh i ¼ r2C x1 � x2ð Þ ¼ r2 exp � x1 � x2ð Þ2
s2
!
ð1Þ
where r is the root mean square (RMS) height and s is thecorrelation length. The Fourier transform of r2C(x) is the
power spectral density W(kx). A surface of size L is to be
generated from the power spectral density. We make
h(x) periodic outside, i.e.,
h(x) = h(x ? L). A Fourier series is used to represent
h(x) is given by
h xð Þ ¼ 1
L
X1n¼�1
bn kxnð Þ exp� ikxnxð Þ ð2Þ
where bn(kxn) = 2p(LW(kxn)) is a Gaussian random vari-
able, and kxn = 2pn/LTo model rough surfaces, we use a Gaussian correlation
function given by:
C xð Þ ¼ r2 exp � x2
s2
� �ð3Þ
The corresponding power spectral density is:
W kxð Þ ¼ r2s2ffiffiffip
p exp � k2xs2
4
� �ð4Þ
Random rough surface scattering were initially modeled by
Gaussian correlation functions in Monte Carlo simulations
[37–39].
Figure 1 shows the scattered electromagnetic waves
from 1-D Gaussian rough surface with Gaussian correlation
function. e is the permittivity of air, and e(x) is the per-
mittivity of material. ki, ks, hi, and hs, designate, respec-tively, an incident field, a scattered field, an angle of
incidence, and scattering angle.
3 Integral formalism for rough surfaces
Let U1 exp (-jxt) represent the y component of the electric
field in the s-polarization case or the magnetic field in the
p-polarization case. The indices inc, 1, and 2 designate,
respectively, an incident field, a total field component
above the surface, and that below the profile of surface.
The incident field is
Uincðx; zÞ ¼ expðj ða0x� b0zÞÞ; ð5Þ
where a0 = k0 sin h0, b0 = (k02 - a0
2)1/2and k0 = 2p/k.The total field y component is a solution of Helmholtz
wave equations in air (or vacuum) region,
Fig. 1 Rough surface scattering geometry
Opt Rev
123
r2U 1ð Þ x; zð Þ þ k20Uð1Þ x; zð Þ ¼ 0; z[ h xð Þ; ð6Þ
and in the metal region,
r2U 2ð Þ x; zð Þ þ k20e xð ÞUð2Þ x; zð Þ ¼ 0; z\h xð Þ; ð7Þ
and it satisfies the boundary conditions
Uð1Þðx; zÞÞ��z¼hþðxÞ¼ Uð2Þðx; zÞÞ
��z¼h�ðxÞ; ð8Þ
oUð1Þðx; zÞon
����z¼hþðxÞ
¼ 1
C
oUð2Þðx; zÞÞon
����z¼h�ðxÞ
; ð9Þ
as well as radiation condition at the infinite z (outgoing
waves at z ? ±?). q/qn indicates the normal derivative,
and C is a constant equal to 1 (s-polarization) or e (p-
polarization). The superscript on h? indicates that the field
and its normal derivative are evaluated in the limit
approached from above, and analogous comments hold for
h-.
By the use of Green’s second theorem and boundary
conditions we can derive the integral equations for the
scattered electromagnetic field. This field is expressed in
terms of the unknown field and its normal derivative
(source functions); these functions are defined in vacuum
on the boundary (Pr) as follows:
-for p-polarization case
HðxÞ ¼ Hð1Þðx; zÞ��z¼hþðxÞ; ð10Þ
LðxÞ ¼ �h0ðxÞ o
oxþ o
oz
� �Hð1Þðx; zÞ
�����z¼hþðxÞ
; ð11Þ
-for s-polarization case
EðxÞ ¼ Eð1Þðx; zÞ��z¼hþðxÞ; ð12Þ
FðxÞ ¼ ð�h0ðxÞ o
oxþ o
ozÞEð1Þðx; zÞ
����z¼hþðxÞ
; ð13Þ
where h0(x) is the derivative of h(x).
From integral equations giving a field at any point M(x,
z), one can obtain a pair of coupled integral equations for
the source functions for both polarizations. The resolution
of these equations, as described in references [6, 43] con-
sists in converting the infinite systems of integral equations
into two finite systems of linear equations as shown, in
turn, in Refs. [5] and [40].
4 Description of the SIBC method
Somemetals used in optics, such asAl, Ag, andAu has a high
conductivity. Although when the values of the refraction
index of the metals increase, that yields the resolution of
coupled integral equations for the source functions become
very hard because of the appearance of the term |n*| on the
argument of the Hunkel function. This problem can also be
solved if the relation between the tangential components of
the fields E// and H// along the surface is known. In this
second approach, Maxwell’s boundary conditions are
replaced by a boundary condition of the form:
E== ¼ Z n~ x H==
� �; ReðZÞ[ 0; ð14Þ
where n~ is a normal unit vector pointing towards the
exterior of the scatter and Z is a second rank tensor called
the surface impedance tensor. If Z is known the fields
outside the scattered can be found without solving Max-
well’s equation inside it.
Using the boundary conditions (Eqs. 8–9) and the inte-
gral equations for Refs. [5, 38, 39], the surface impedance
is:
Zs�pol ¼ k
j
EðxÞdEðxÞdn
: ð15Þ
By using the definition of F given by the Eq. (13), we get
Zs�pol ¼ k
j
EðxÞc FðxÞ ; c ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ h0 2ðxÞ
p: ð16Þ
In the p-polarization, we get:
Zp�pol ¼ j
k
dHðx;hðxÞÞdn
Hðx; hðxÞÞ : ð17Þ
By using the definition of L given by the Eq. (11), we get
Zp�pol ¼ j
k
c LðxÞHðxÞ : ð18Þ
We define two operators (G and N) associated to the
function of Green and its normal derivative, respectively.
The application of such operators to the function / is
represented as:
G /ð Þ ¼Zþ1
�1
G x; x0ð Þ/ x0ð Þdx0 and
Nð/Þ ¼Zþ1
�1
dGðx; x0Þdn
/ðx0Þ dx0:
ð19Þ
Using the delta family as |n*| ? ? the kernel G(x, x0)tends to a delta function G(/) can be approximated in the
following way:
Gð/Þ � /ðxÞZþ1
�1
Gðx; x0Þ dx0: ð20Þ
In the approximation, it is found that the operators G and N
tend towards multiplicative operators [34–36, 44].
Opt Rev
123
Gð/Þ ¼ /ðxÞ2 j k n � cðxÞ and Nð/Þ � j
4kn�h00ðxÞ /ðxÞ
c3ðxÞ :
ð21Þ
When the above relations are introduced into the integral
equations [44], we get:
1þ jh00
2k n � q3
� �HðxÞ ¼ jn�
kq3LðxÞ; p-polarization ð22Þ
1þ jh00
2k n � q3
� �EðxÞ ¼ j
kn � q3 FðxÞ s-polarization
ð23Þ
q is the radius of curvature of the random rough surface
z = h(x), qðxÞ ¼ Z01þh02ðxÞð Þ
32
h00ðxÞ : By using the Eqs. (16) and
(18) the surface impedance become:
Zp�polðxÞ ¼ Z0 1þ j
2k n � qðxÞ
� ; ð24Þ
Zs�polðxÞ ¼ Z0 1þ j
2k n � qðxÞ
� �1
; ð25Þ
where Z0 = 1/n* is the constant value obtained for flat
boundary [45].
Using the Poynting theorem relation, [46, 47] the bidi-
rectional reflection function and the bidirectional trans-
mission function are stated in Refs [5, 38, 39]. The
hemispherical radiative properties of reflection and emis-
sion are obtained by integrating the bidirectional functions
[5, 38, 39, 42].
It is shown that SIBC method is more suitable than exact
methods. This method greatly reduces the storage and
computational requirements for the problem of scattering
by random rough metallic surfaces.
5 Results and discussion
The results presented in this section are for scattering from a
surface of length L = 25.6k, where k is the wavelength. Thesurface is discretized into at 1024 points in x, depending on
the roughness of the surface. The roughness parameters are
r and s. The incident beam is p- or s-polarized at a given
angle of incidence hi. The result in both figures is an averageof the results for 300 surface realizations.
The influence of the angle of incidence, the nature of
polarization, and surface materials is interesting. As a
check on the validity of the SIBC method, the results were
compared with numerical [6] results published previously.
In Fig. 2 we compare the results of the present method
and numerical results taken from ref [6]. In this figure, we
are interested in the variation of the bidirectional
reflectivity (BDR) as a function of the scattering angle hsfor p-polarized light incident on a random rough metallic
surface characterized by the statistical parameters: RMS
height r = 1.4142 lm and a correlation length s = 2 lm.
The wavelength of the incident light was k = 0.6127 lm.
The surface material is the silver with dielectric constant at
this wavelength e(x) = -17.2 ? i0.498. What is reported
here is the behavior of the enhanced backscattering peak in
the retro-reflection direction (hs = -h0) as the angle of
incidence h0 is increased from 0� to 45�. The results of
SIBC calculation were fitted to exact results at different
angles of incidence. We can say that in the case of p-po-
larization, the agreement between the present results and
the results obtained through exact calculations is very
good, at different angles of incidence.
At normal incidence, a more intense and acute peak in
the retro-reflection direction is seen. This peak is also
observed for the incidence angles equal to 20�, 30� and 45�.It is shown that this peak is less intensive and very wide
when the angle of incidence increases. These observations
are verified by both methods.
In Fig. 2a, b, c, and d, we noted that the bidirectional
reflectivity (BDR) calculated by SIBC method is less
intense than that presented in [6]. This observation can be
explained by the limit of validity of the present method.
In Fig. 3, we compare the results of the present method
and numerical results taken from Ref. [6]. We present the
variation of the BDR as a function of the scattering angle hsfor the scattering of s-polarized light incident normally on a
random rough surface ruled on the surface of silver. In
Fig. 3a, we observe a good agreement between the results
found by SIBC method and those given in Ref. [6]. At
normal incidence, the peak is well defined. This peak is
very intense. Figure 3b is the same as Fig. 3a, only when
the angle of incidence is 20�. Here, the peak of the BDR
calculated by SIBC method is not observed, in the
backscattering direction (hs = -20�). This can be
explained by the limit of validity of the present method and
the nature of polarization. The peak is visible at an angle of
incidence hi equal to 20�Figure 4 shows the variation of the BDR as a function of
the scattering angle hs for the scattering of s- and p-po-
larized light incident normally on a random rough metallic
surface. The surface material is the gold characterized by
its permittivity e(x) = -9.8949 ? 1.0458i. It is estimated
that the curves found by the SIBC method and those given
in ref [41] have the same allure in both cases of polariza-
tion. The peak is observed at normal incidence. This peak
found by the present method is less intense and less closed
than the peak given in Ref. [41]
Figure 5 shows curves displaying the reflectivity versus
the angle of incidence hi in both cases of polarization. The
surface material is the gold. Figure 5a and b shows the
Opt Rev
123
behavior of the reflectivity of the gold for two different
roughnesses. In Fig. 5a, we note that the value of reflec-
tivity is constant for angles of incidence less than 60�. Atlarge angles of incidence (more than 60�), the value of
reflectivity increases and tends to unity. However, in
Fig. 5b, it is seen that the value of reflectivity is constant
for angles of incidence less than 70� and for angles of
incidence more than 70� the value of reflectivity increases
and tends to unity, This difference of variation of the
reflectivity can be explained by the influence of the surface
roughness; the surface in Fig. 5a is rougher than the surface
in Fig. 5b so the value of reflectivity in Fig. 5a is less than
that in Fig. 5b. For the case of p-polarization, in Fig. 5b,
we talk about the limit of validity of the SIBC method
because for large incidence angle, the value of the reflec-
tivity exceeds unity.
Figure 6 illustrates the variation of emissivity as a
function of the ratio s/k for two different surface materials
(gold and silver) in both cases of polarization and at normal
incidence (hi = 0�). The RMS height r is fixed to 1k and
(a) (b)
(c) (d)
Fig. 2 The bidirectional
reflectivity (BDR) for the
scattering of p-polarized beam
of light from random rough
metallic surface characterized
by: r = 1.4142 lm, s = 2 lm,
k = 0.6127 lm. The surface
material is the silver with
e(x) = -17.2 ? i0.498, while
L = 25.6 lm, Ns = 300,
Np = 1024
(a) (b)Fig. 3 The bidirectional
reflectivity (BDR) for the
scattering of s-polarized beam
of light from random rough
metallic surface characterized
by: r = 1, 2 lm, s = 2 lm.
k = 0.6127 lm. The surface
material is the silver with
e(x) = -17.2 ? i0.498, while
L = 25.6 lm, Ns = 300,
Np = 1024
Opt Rev
123
the correlation length s is varied from 0 to 28k. In Fig. 6a,
we note that for 0\ s/k\ 4 the surface considered is very
rough, the intensity of emissivity increases and we have a
resonant peak, at s/k equal to 1.5. This peak is due to the
emission mediated by surface plasmon [48, 49]. For s/k[ 4, the value of emissivity is invariant when the ratio s/
k increases and tends to the value of emissivity of a flat
surface given by using Fresnel’s formula [39]. It is seen
that SIBC result show the resonance region of the material
(the gold) in both cases of polarization. In Fig. 6b, for s/k\ 4, we have a slight variation of the value of emissivity.
Unlike gold, silver does not have a resonance region. This
(a) (b)Fig. 4 The bidirectional
reflectivity (BDR) for the
scattering of s-polarized beam
of light from random rough
metallic surface characterized
by: r = 1,9 lm, s = 4.58 lm,
k = 0.633 lm. The surface
material is the gold with
e(x) = -9.8949 ? 1.0458i,
while L = 25.6 lm, Ns = 300,
Np = 1024
(a) (b)Fig. 5 The reflectance from
two different gold one-
dimensional rough surfaces in
both cases of polarization with
different roughness. The length
of each surface is L = 25.6 lm,
Ns = 300, Np = 1024
(a) (b)Fig. 6 Emissivity from two
different random rough metallic
surfaces in both cases of
polarization at normal
incidence. The length of each
surface is L = 25.6 lm,
Ns = 300, Np = 1024
Opt Rev
123
observation can be described on the basis of complex
dielectric constants [50] as real part of the dielectric per-
mittivity of material determines the resonance region
position while imaginary part determines the relative
contribution of absorption. In the case p-polarization,
emissivity decreases whereas in the case of s-polarization
emissivity is slightly increases. For s/k[ 4 the value of
emissivity tends to the value of emissivity of a flat surface.
In addition, when the ratio s/k is increased, the emissivity
tends to the value of a plane surface. Fig. 7, is the same as
Fig. 6 but at angle of incidence hi = 60�. Here, we study
the shadowing effects, and the influence of the angle of
incidence, on the behavior of emissivity. It is shown that
curves in Fig. 7a and b have the same asymptotic behavior
in both cases of polarization. For s/k\ 4, the value of
emissivity decreases at s/k equal to 2. This observation is
due to the shadowing effects however, if s/k[ 4, the value
of the emissivity is invariant and tends to the value of
emissivity of a plane surface.
6 Conclusion
A comparison between exact method and SIBC method of
random rough metallic surfaces is investigated and dis-
cussed. The effects of the angle of incidence, nature of
polarization, and the geometrical and physical parameters
on the validity of SIBC method are discussed. Backscat-
tering phenomenon for p-polarized beam of light is
observed, whereas in the case of s-polarized beam of light
is absent. However, for the surface materials of gold and
silver, the domain of validity of SIBC method is quantified
for reflection and emission in both cases of polarization.
From the result obtained, some important conclusions
are derived concerning the domains of validity of SIBC
method:
• The region of validity of SIBC method depends on the
nature of polarization, incident wavelength, direction of
incidence, and the physical parameters of surface
material.
• The domains of validity of the SIBC model are more
extensive for p-polarization than for s-polarization for
gold and silver.
• The accuracy of SIBC method is better for gold
material than for silver material. The resonance region
for gold is clear but is absent for silver.
• The use of the SIBC method is less expensive and it
requires a very short computation time.
• SIBC is suitable for dielectric and metallic surfaces.
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