Revista de Matema´tica: Teor´ıa y Aplicaciones 2013 20(1) : 1–20
cimpa – ucr issn: 1409-2433
pattern equation method for the
solution of electromagnetic
scattering by axially-symmetric
particles with complex anisotropic
surface impedance
me´todo de ecuacio´n por patrones para
la solucio´n de scattering
electromagne´tico por part´ıculas
axialmente sime´tricas con impedancia
de superficie anisotro´pica compleja
Alexander G. Kyurkchan∗ Dmitry B. Demin†
Received: 16/Oct/2011; Revised: 10/Nov/2012;
Accepted: 27/Nov/2012
∗Moscow Technical University of Communication and Informatics, Aviamo-
tornaya Street 8A, Moscow, 111024, Russia. E-Mail: kyurkchan@yandex.ru,
kyurkchan@mtuci2.ru
†Same address as/Misma direccio´n que A.G. Kyurkchan. E-Mail:
mbdeminsds@mail.ru
1
2 a.g. kyurkchan – d.b. demin
Abstract
The pattern equation method (PEM) has been extended to solve
the scattering problems of electromagnetic waves by particles with
mixed anisotropic surface impedance. Thus, the anisotropic impedance
boundary conditions are imposed on lateral surface of the particle,
and isotropic impedance boundary conditions are imposed on end
faces of the particle. The method is formulated for axially-symmetric
bodies. The scattering characteristics of the bodies with artificially
soft and hard lateral surfaces are presented. The comparison of the
results with those obtained by other methods is carried out. The
analysis of convergence’s rate of numerical algorithm of the PEM
and accuracy of numerical calculations are presented. Comparison
of our data with numerical results obtained earlier by the PEM in
absence of an anisotropic impedance is carried out.
Keywords: Scattering problems, pattern equation method, anisotropic
impedance, artificial soft and hard surfaces, bodies with the mixed aniso-
tropic surface impedance.
Resumen
El me´todo de la ecuaciones de patro´n (PEM) han sido extendidos
para resolver problemas de scattering en ondas electromagne´ticas por
medio de part´ıculas con superficie de impedancia mista anisotro´pica.
Luego, las condiciones de frontera de impedancia anisotro´pica se
imponen en la superficie lateral de la part´ıcula, y condiciones de
frontera de impedancia isotro´pica se imponen en caras finales de la
part´ıcula. El me´todo se formula para cuerpos axialmente sime´tricos.
Se presentan las caracter´ısticas del scattering de los cuerpos con su-
perficies laterales artificiales suaves y duras. Se lleva a cabo la com-
paracio´n de los resultados con los que se obtienen con otros me´todos.
Tambie´n se presenta el ana´lisis de la tasa de convergencia del algo-
ritmo nume´rico PEM y la precisio´n de los ca´lculos nume´ricos. Final-
mente se hace la comparacio´n de nuestros datos con los resultados
nume´ricos obtenidos antes con el PEM en ausencia de la impedancia
anisotro´pica.
Palabras clave: Problemas de scattering, me´todo de ecuaciones de pa-
trones, impedancia anisotro´pica, superficies artificiales suaves y duras,
cuerpos con superficie de impedancia anisotro´pica.
Mathematics Subject Classification: 76B15.
Rev.Mate.Teor.Aplic. (ISSN 1409-2433) Vol. 20(1): 1–20, January 2013
pattern equation method for the solution of... 3
1 Introduction
The problem of scattering by impedance bodies is one of the most studied
classical problems of electromagnetism.
In the present paper, the diffraction problem for 3D bodies with mixed
anisotropic impedance is considered. In this case, on the lateral surface
of the scatterer, the full electromagnetic field satisfies to a generalized
anisotropic impedance boundary condition in which the surface impedance
is represented as a tensor with the components corresponding to appropri-
ate directions of anisotropy. On the remained surface isotropic impedance
conditions are imposed. In the case of cylindrical body and the bodies
close to them the remained surface represents its top and bottom bases.
The solution of those problems can be used for simulating the scatter-
ing characteristics of corrugated and chiral structures. In practice, the
simplest situation takes place when the directions of anisotropy are spec-
ified for bodies of revolution. Therefore, only such bodies will be further
considered.
For solving the aforementioned problem, the generalization of the pat-
tern equation method (PEM) has been developed. This method has been
earlier applied to solving the problems of electromagnetic waves scatter-
ing on the perfectly conducting, impedance, and dielectric scatterers as
well as the scatterers coated with several dielectric layers [1-7]. According
to [3] (see also [6]), the impedance approach is suitable for modeling the
problems of diffraction on bodies with the dielectric absorbing covering,
the sizes of which have some lengths of the incident field wave. Recently
the algorithm of the PEM has been developed for the solution of scat-
tering problems on scatterers with the anisotropic impedance boundary
conditions that are imposed everywhere at the scatterers’s surface [8].
The example of bodies with surface anisotropic impedance is a periodic
ridge (or corrugated structure) with the grooves filled with a dielectric
material. The consideration of such a structure has been made in [9] for
the strict electromagnetic statement of the problem.
The numerical algorithm of the PEM is based on the reduction of the
original boundary-value problem for Maxwell’s equations to an infinite
system of linear algebraic equations with respect to the unknown coeffi-
cients of the expansion of the scattering pattern (spectral characteristic
of the wave field) in terms of vector angular spherical harmonics. The
obtained infinite linear system of the algebraic equations is solved by the
method of a reduction with certain restrictions on the geometry of the
problem, which have been strictly established in [1-4].
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4 a.g. kyurkchan – d.b. demin
The PEM is one of the most effective and universal methods for solv-
ing the scattering problems of electromagnetic waves. It has been earlier
shown [1-3] that the rate of convergence of the PEM’s numerical algo-
rithm is mainly governed by the scatterer size and weekly depends on its
geometry. If the scatterer is sphere, this method leads to the obvious an-
alytical solution in the form of infinite Fourier’s series over wave spherical
harmonics, which coincides with the corresponding solution in the theory
of Mie series (see [1, 6], for instance).
In the frames of the boundary-value problem, we carried out a simula-
tion of the scattering problem for a plane wave being incident on axially-
symmetric bodies with artificial soft and hard lateral surfaces. The def-
inition of such surfaces in the electromagnetic case has been first intro-
duced by P-S Kildal [10-11]. The description of such surfaces in terms
of anisotropic impedance independently from polarization of an incident
plane wave was presented more in detail in [8]. Further we study ac-
curacy of the obtained numerical calculations by means of values of the
scattering pattern, and also where it was possible, under the optical the-
orem. We have calculated the some scattering characteristics for finite
circular cylinders and superellipsoids. On lateral surface of these scat-
terers anisotropic impedance boundary conditions were imposed, and the
top and bottom bases were either perfectly conducting or caused by some
isotropic impedance. Further, we have compared these characteristics
with those have been obtained by the PEM for scatterers which surface
has been entirely caused by an anisotropic impedance.
2 Problem statement
Consider an electromagnetic wave scattering problem of a primarymonochro-
matic (eiωt) field ~E0 , ~H0 that is incident on an arbitrarily shaped 3D
compact body bounded by closed surface S as shown in Figure 1. Let
us denote the lateral surface of the body as Sl , and the basis of the
cylindrical body (that is the top and bottom faces) as Sb. Then we have:
S = Sl ∪ Sb.
Let the following anisotropic impedance boundary condition be met
at surface S:
(~n× ~E)|Sl = Z[~n× (~n× ~H)]
∣∣∣
Sl
(1a)
(~n× ~E)|Sb = Z[~n× (~n× ~H)]
∣∣∣
Sb
(1b)
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pattern equation method for the solution of... 5
Figure 1: Layout of the problem.
where ~n is the outward unit normal to S; Z in (1a) is the anisotropic
surface impedance that is represented as a tensor:
Z =
[
Zl Zlϕ
Zϕl Zϕ
]
, (2)
Zb in (1b) is the scalar impedance of the basis of the cylindrical body;
~E = ~E0 + ~E1, ~H = ~H0 + ~H1 is the total field; ~E1, ~H1 is the secondary
(diffracted) field, which satisfies the system of homogeneous Maxwell’s
equations:
5× ~E1 = −ikζ ~H1,5× ~H1 = ik
ζ
~E1 (3)
elsewhere outside S and the Sommerfeld radiation condition at infinity.
Here k = ω
√
εµ and ζ =
√
µ/ε are the wave number and the free-space
wave impedance, respectively.
The component Zi of tensor Z corresponds to the direction of unit
vector ~iϕ, which is tangential to Sland perpendicular to unit vectors ~iϕ
(unit vector of a spherical coordinate system (r, θ, ϕ)) and to ~n. Let
vector ~il be equal to (~iϕ × ~n). Thus, vectors ~il, ~iϕ, and ~n form a right-
hand orthogonal system. It is clear that components Zl and Zϕ are the
surface impedances along the main directions of anisotropy corresponding
to vectors ~il and ~iϕ.
Using the expansion of the vector ~E (and similar for ~H) in the form
~E = Eϕ~iϕ +El~il +En~n,
and substituting it into boundary condition (1a), we obtain
Eϕ~il = (ZlHl + ZlϕHϕ)~il + (ZϕlHl + ZϕHϕ)~iϕ. (4)
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6 a.g. kyurkchan – d.b. demin
If Zlϕ = Zϕl = 0, we obtain the following impedance condition:
Eϕ~il −El~iϕ = ZlHl~il + ZϕHϕ~iϕ,
which is similar that is resulted in paper [11].
For solving the scattering problem (1)–(3) in the framework of the
PEM we used the spherical basis for decomposition the scattering pattern
and wave fields. It reduces directly the initial problem to the algebraic
system with respect to the unknown coefficients of the pattern expansion
in terms of spherical harmonics.
3 Reduction of boundary-value problem to sys-
tem of algebraic equations
Further, we describe the standard scheme of deriving the numerical algo-
rithm of PEM.
According to papers [1-4], we are going to find the scattering pattern
function, that is, the function that defines the dependence of the diffracted
field on angles (θ, ϕ) in spherical coordinates (r, θ, ϕ) for the far zone
(kr 1) where the following asymptotic relations are valid:
~E1 =
exp(−ikr)
r
~FE(θ, ϕ) + O
(
1
(kr)2
)
,
~H1 =
exp(−ikr)
r
~FH(θ, ϕ) + O
(
1
(kr)2
)
.
Here ~FE and ~FH are the scattering patterns for electrical and magnetic
fields, respectively.
The basic point of the PEM consists in obtaining the infinite system
of algebraic equations with respect to unknown expansion coefficients of
the scattering patterns into series in terms of vector angular spherical har-
monics [12], which form the orthogonal basis in the spherical coordinates.
The series expansions of the scattering patterns have the following form
(the details can be found in [1], for instance):
~FE(θ, ϕ) = −
∞∑
n=1
n∑
m=−n
anmi
n(~ir × ~Φmn (θ, ϕ))−
∞∑
n=1
n∑
m=−n
bnmi
nζ~Φmn (θ, ϕ),
(5)
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pattern equation method for the solution of... 7
~FH(θ, ϕ) =
∞∑
n=1
n∑
m=−n
anmi
n1
ζ
~Φmn (θ, ϕ)−
∞∑
n=1
n∑
m=−n
bnmi
n(~ir × ~Φmn (θ, ϕ)),
(6)
where
~Φmn (θ, ϕ) = ~r ×5Pmn (cos θ) · exp(imϕ), (7)
and anm, bnm are the unknown expansion coefficients of the scattering
patterns that are to be determined. In formulas (5)-(7),i =
√−1 is imag-
inary unit, ~ir is the unit vector in the spherical coordinate system, and
Pmn (cosθ) are the associated Legendre functions.
Moreover, the wave field ~E1, ~H1 also can be expanded into series of
the vector spherical wave functions with respect to unknown coefficients
anm, bnm :
~E1 =
∞∑
n=1
n∑
m=−n
{anm ~Eenm + bnm ~Ehnm}, (8)
~H1 =
∞∑
n=1
n∑
m=−n
{anm ~Henm + bnm ~Hhnm}, (9)
where
~Eenm = 5×5× (~rΨnm) = ~Hhnm,
~Ehnm = −ikζ 5×(~rΨnm) = −ζ2 ~Henm
Ψnm = h
(2)
n (kr)P
m
n (cos θ) exp(imϕ), (10)
h
(2)
n are the spherical Hankel functions of the second kind.
The starting point for the subsequent analysis is representing coeffi-
cients anm, bnm in terms of the boundary values of the wave field (1). By
analogy with [2, 8], we use the following integral relations for the field ~E1,
~H1, which can be obtained from the Maxwell’s equations (3):
~E1 =
∫
S
ζ
ik
[5×5× (~IeG0)]ds′ +
∫
Sl
Z[5× (~ImG0)]ds′
+Zb
∫
Sb
[5× (~ImG0)]ds′, (11)
~H1 =
∫
Sb
[5× (~IeG0)]ds′ −
∫
Sl
Z
ikζ
[5×5× (~ImG0)]ds′
− Zb
∫
Sb
1
ikζ
[5×5× (~ImG0)]ds′, (12) (12)
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8 a.g. kyurkchan – d.b. demin
where
~Ie = (~n× ~H)
∣∣∣
S
; ~Im = ~n × (~n× ~H)
∣∣∣
S
= (~nHn − ~H)
∣∣∣
S
= − ~Hτ
∣∣∣
S
,
Hn = ~n · ~H, (13)
G0 =
exp(−ik|~r − ~r′|)
4pi|~r− ~r′| is the fundamental solution (the free-space Green
function) for the scalar Helmholtz equation in free space, which is decom-
posed in the following series at r > r′:
G0(~r − ~r′) = k
4pii
∞∑
n=1
n∑
m=−n
(2n+ 1)
(n−m)!
(n+m)!
ψnm(~r)χ¯(~r
′), (14)
χnm = jn(kr)P
m
n (cos θ) exp(imϕ). (15)
Here jn are the spherical Bessel functions and ~r
′ is the position vector of
a point on S.
By comparing Eqs.(8)-(9) with (11)-(12), it was established [2] that
anm = −Nnmζ
4pi
{∫
S
~Ie(r′) · ~eenm(r′)ds′
−
∫
Sl
Z · ~Im(r′) · ~henm(r′)ds′
−Zb
∫
Sb
~Im(r′) · ~henm(r′)ds′
}
, (16)
bnm = −Nnm
4piζ
{∫
S
~Ie(r′) · ~ehnm(r′)ds′
−
∫
Sl
Z · ~Im(r′) · ~hhnm(r′)ds′
−Zb
∫
Sb
~Im(r′) · ~hhnm(r′)ds′
}
, (17)
where
~eenm = 5×5× (~rχnm)
= ~hhnm; ~e
h
nm?− ikζ 5×(~rχnm)
= −ζ2~henm; Nnm =
2n+ 1
n(n+ 1)
(n−m)!
(n−m)! . (18)
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pattern equation method for the solution of... 9
Now, using relations (16)–(17) and replacing quantities ~Ie and ~Im by
corresponding expansions of wave fields in view of Eq. (13), we obtain the
following infinite system of the linear algebraic equations of PEM


anm = a
0
nm +
∑∞
q=1
∑q
p=−q(G
11
nm,qpaqp +G
12
nm,qpbqp),
bnm = b
0
nm +
∑∞
q=1
∑q
p=−q(G
21
nm,qpaqp +G
22
nm,qpbqp)
n = 1, 2, ..., |m| ≤ n,
(19)
where
{
a0nm = a
00
nm + a
z˜0
nm; b
0
nm = b
00
nm + b
z˜0
nm; G
ij
nm,qp = G
0ij
nm,qp +G
z˜ij
nm,qp;
i, j = 1, 2. (20)
Here, the additional superscript marked by “0” corresponds to the per-
fect conductor (Z = 0), and those marked by “z˜” designates the additional
terms caused by the anisotropic impedance Z. Coefficients a0nm, b
0
nmare
determined by the incident wave. These coefficients and the matrix ele-
ments Gijnm,qp, i, j = 1, 2 in Eq. (20) are represented by surface integrals
on S as follows:
a00nm = −
Nnmζ
4pi
∫
S
(~n× ~H0) · ~eenmds, b00nm
= −Nnm
4piζ
∫
S
(~n × ~H0) · ~ehnmds,
az˜0nm = −
Nnmζ
4pi
{∫
Sl
(Z ~H0τ ) ·~henmds+ Zb
∫
Sb
~H0τ · ~henmds
}
, (21)
bz˜0nm = −
Nnm
4piζ
{∫
Sl
(Z ~H0τ ) · ~eenmds+ Zb
∫
Sb
~H0τ · ~eenmds
}
;
G011nm,qp = −
Nnmζ
4pi
∫
S
(~n× ~Heqp) · ~eenmds, G012nm,qp
= −Nnmζ
4pi
∫
S
(~n× ~Hhqp) · ~eenmds,
G021nm,qp = −
Nnmζ
4pi
∫
S
(~n× ~Heqp) ·~henmds, G022nm,qp
= −Nnmζ
4pi
∫
S
(~n× ~Hhqp) ·~henmds,
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10 a.g. kyurkchan – d.b. demin
Gz˜11nm,qp = −
Nnmζ
4pi
{∫
Sl
Z[~n× (~n× ~Heqp)] · ~henmds
+Zb
∫
Sb
[~n× (~n× ~Heqp)] ·~henmds
}
, (22)
Gz˜12nm,qp = −
Nnmζ
4pi
{∫
Sl
Z[~n× (~n× ~Hhqp)] · ~henmds
+Zb
∫
Sb
[~n× (~n× ~Hhqp)] ·~henmds
}
,
Gz˜21nm,qp = −
Nnm
4piζ
{∫
Sl
Z[~n× (~n× ~Heqp)] · ~eenmds
+Zb
∫
Sb
[~n× (~n× ~Heqp)] · ~eenmds
}
,
Gz˜22nm,qp = −
Nnm
4piζ
{∫
Sl
Z[~n× (~n× ~Hhqp)] · ~eenmds
+Zb
∫
Sb
[~n× (~n× ~Hhqp)] · ~eenmds
}
.
The system of the linear algebraic equations of PEM (19) can be used
for calculating the scattering characteristics of arbitrarily shaped scatter-
ers, which are not axially symmetric. When the scatterer is an axially
symmetric object (body of revolution), i.e. the surface equation takes the
form ρ(θ, ϕ = ρ(θ), the algebraic system (19) considerably simplifies and
can be written as

anm = a
0
nm +
∑N
q=|m|(G
11
nm,qmaqm +G
12
nm,qmbqm),
n = 1, 2, ..., N ; |m| ≤ n,
bnm = b
0
nm +
∑N
q=|m|(G
21
nm,qmaqm +G
22
nm,qmbqm)
(23)
where N is the upper limit of summation in Eq. (16), that is, the maximal
number of harmonic functions in series (5)-(6). Matrix elements in (23)
are expressed in terms of single integrals.
To justify the applicability of the method of reducing to the obtained
infinite system (19), the matrix elements and the right-hand part of the
system can be estimated for large values of n and q. Such an estimation
is similar to that earlier performed for the numerical algorithm of PEM
(see, for example, [1-4]). That approach allows us to specify rigorous
limitations on the geometry of the scatterers. If the incident field is a
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pattern equation method for the solution of... 11
plane wave, the method of reducing is valid if the scatterer belongs to a
class of weakly non-convex bodies [1-4]. In particular, this class contains
all convex bodies.
Note that, for perfectly conducting bodies of revolution, the algebraic
system similar to (23), seems to be first obtained in [13] for the case
when the scatterer belongs to a class of so-called Rayleigh bodies (see
[14], for instance). However, the PEM has been independently developed
[15] on the basis of the strict integral-operator equation with respect to
the scattering pattern of a body, and, as it was mentioned above, the
PEM is applicable to considerably wider class of scatterers rather than
the Rayleigh bodies.
4 Numerical results
In this section, we present some results of calculating of the scatter-
ing characteristics for different axially symmetric cylindrical scatterers
at whose surfaces the mixed impedance boundary conditions are fulfilled.
The z-axis was chosen as the symmetry axis of the scatterers. In all ex-
amples, the incident field is a plane unit wave.
The aim of our study is to compute the scattering characteristics for
scatterers with artificially soft and hard lateral surfaces.
In [11], the definition of the artificially soft and hard surfaces in an
electromagnetic case has been introduced by using the special values of
anisotropic impedance. This impedance corresponds to some corrugated
structure of surface S with grooves, the edges of which are parallel to
either vector ~iϕ or vector ~il. In the same paper, the values of anisotropic
impedance are obtained for surfaces that are polarization-independently
soft and hard. Generally, those surfaces are artificially soft and hard. It
was established (see Eq. (15) in [11]) that, independently of the field
polarization, the components of Z corresponding to the artificially soft
surface take the following values
|Zϕ| =∞, Zl = 0, Zlϕ = Zϕl = 0. (24)
Moreover, the direction of vector ~il corresponds to the direction of wave
propagation along surface S, and vector ~iϕ (transverse direction) is per-
pendicular to the plane of incidence of the wave and to the direction of
wave propagation (the direction of vector ~iϕ corresponds to that of the
grooves).
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12 a.g. kyurkchan – d.b. demin
Similarly, the artificially hard surface can be defined by the following
values of the components of anisotropic impedance Z (see Eq.(17) in [11]):
|Zl| =∞, Zϕ = 0, Zlϕ = Zϕl = 0. (25)
In [8] it was already noticed that the artificially soft surface is a surface
with ideal mixed conductivity along the direction~iϕ, and, in the same way,
the artificially hard surface is a surface with ideal mixed conductivity along
the direction~il. Thus, the artificially soft scatterer, in general case, should
not support of a surface wave. In [9], the question is in detail analyzed
about what properties of hard surfaces, in terms of the sizes of grooves,
and what properties of materials of a covering of corrugated structures
should be.
The scattering problem was considered for the plane wave with circular
polarization in the form
~E0 = (~ixcosθ0+~iz sin θ0+~iy(±i)) exp(−ikr(− sinθ sin θ0 cosϕ+cos θ cos θ0)),
~H0 =
∓i
ζ
· ~E0. (26)
The scatterers were cylindrical bodies, such as a finite circular cylinder and
a superellipsoid. In Eq. (26), ~ix and ~iy are the unit vectors in Cartesian
coordinate system, and the super- and subscripts mean the right and left
polarization, respectively. The surface of a superellipsoid in Cartesian
coordinate system is defined by equation
x2m + y2m
a2m
+
z2m
c2m
= 1 (27)
where m is the coefficient of roundedness. The sizes of circular cylinder
are as follows: ka = 1(a is a radius of the basis of the cylinder), kh = 10
(h is the height of the cylinder), and the parameters of the superellipsoid
are specified as ka = 1, kc = 5, and m = 8 (the sizes of the superellipsoid
correspond to the sizes of the cylinder). Figures 2-7 show the scattering
patterns of the electric field (quantities FEθ (θ, ϕ) and F
E
ϕ (θ, ϕ)) in two
half-planes: ϕ = 0 (E-plane), and ϕ = pi/2 (H-plane), respectively. In
these planes angle θ changes from 0 to 180 degrees. In Figs. 2-3, curves
1 and 2 correspond to the cylinder and superellipsoid with Z = 0, Zb = ζ
(matched impedance), and curves 3 correspond to the perfectly conducting
cylinder (Z and Zb = 0). In Figs. 4-5, curves 1 and 2 correspond to the
cylinder and superellipsoid with Zl = 1000ζ, Zϕ = 0 at Sl (hard surface)
Rev.Mate.Teor.Aplic. (ISSN 1409-2433) Vol. 20(1): 1–20, January 2013
pattern equation method for the solution of... 13
and Zb = ζ at Sb, and curves 3 correspond to the superellipsoid with
the artificially hard lateral surface (Zl = 1000ζ, Zϕ = 0, Zb = 0). By
analogy, the curves 1 and 2 in Figs. 6-7 correspond to the cylinder and
superellipsoid with Zl = 0, Zϕ = 1000ζ (soft surface) at Sl and Zb = ζ
at Sb, and curves 3 correspond to the superellipsoid with the artificially
soft lateral surface (Zl = 0, Zϕ = 1000ζ, Zb = 0), respectively. In our
calculations, the number 1000ζ replaces infinity in Eqs. (24) and (25),
and N is equal to 17-20 (that is, N equals to approximately the height
of cylinder). In all calculations, 3-4 correct significant digits in the values
of the scattering pattern are obtained. It can be seen From Figs. 6-7
that the scattering patterns for all the scatterers are almost identical in
both half-planes but it is not true for artificially hard particles (Figs.4-
5). In addition, Fig.6-7 clearly show that the scattering pattern of the
superellipsoid with mixed impedance weakly differs from the pattern of
the soft superellipsoid in both half-planes. The similar situation is also
exhibited by Figs.4-5.
Figure 2: Scattering patterns for the cylinder and superellipsoid (E-plane).
Aaxial incidence of a plane wave, perfectly conducting lateral sur-
face.
For comparison, Figs. 8-10 show the scattering patterns for the cylin-
der and superellipsoid of the same sizes as in Figs. 2-7 but the plane wave
is incident perpendicularly to a symmetry axis. Each pattern corresponds
Rev.Mate.Teor.Aplic. (ISSN 1409-2433) Vol. 20(1): 1–20, January 2013
14 a.g. kyurkchan – d.b. demin
Figure 3: Scattering patterns for the cylinder and superellipsoid (H-plane),
Aaxial incidence of a plane wave, perfectly conducting lateral sur-
face.
to plane ϕ = [0, pi], so as angle θ varies from 0 to 360 degrees. Curves 1 and
2 correspond to the cylinder and superellipsoid, and curves 3 correspond
to the superellipsoid with Zb = 0. The following values are specified:
Z = 0 and Zb = ζ at Sl in Fig.8, Zl = 1000ζ and Zϕ = 0 in Fig.9, with
Zb = ζ, and Zl = 0, Zϕ = 1000ζ at Sl in Fig.10, also with Zb = ζ. In that
case, it can be seen that the scattering patterns shown in all the figures
are almost close to each other. It means that the value of impedance Zb
at face surfaces weakly influences the changes in the patterns.
Rev.Mate.Teor.Aplic. (ISSN 1409-2433) Vol. 20(1): 1–20, January 2013
pattern equation method for the solution of... 15
Figure 4: Scattering patterns for the cylinder and superellipsoid (E-plane). Ax-
ial incidence of a plane wave, artificially hard lateral surface.
Figure 5: Scattering patterns for the cylinder and superellipsoid (H-plane). Ax-
ial incidence of a plane wave, artificially hard lateral surface.
Rev.Mate.Teor.Aplic. (ISSN 1409-2433) Vol. 20(1): 1–20, January 2013
16 a.g. kyurkchan – d.b. demin
Figure 6: Scattering patterns for the cylinder and superellipsoid (E-plane). Ax-
ial incidence of a plane wave, artificially soft lateral surface.
Figure 7: Scattering patterns for the cylinder and superellipsoid (H-plane). Ax-
ial incidence of a plane wave, artificially soft lateral surface.
Rev.Mate.Teor.Aplic. (ISSN 1409-2433) Vol. 20(1): 1–20, January 2013
pattern equation method for the solution of... 17
Figure 8: Scattering patterns for the cylinder and superellipsoid (E-plane). Per-
pendicular incidence of a plane wave, perfectly conducting lateral sur-
face.
Figure 9: Scattering patterns for the cylinder and superellipsoid (E-plane). Per-
pendicular incidence of a plane wave, artificially hard lateral surface.
Rev.Mate.Teor.Aplic. (ISSN 1409-2433) Vol. 20(1): 1–20, January 2013
18 a.g. kyurkchan – d.b. demin
Figure 10: Scattering patterns for the cylinder and superellipsoid (E-plane).
Perpendicular incidence of a plane wave, artificially soft lateral sur-
face.
In all the considered examples, the validity of the optical theorem
with accuracy not less than 0.001 has been confirmed. According to the
optical theorem, the integral scattering cross-section PS for nonabsorbing
scatterers is proportional to the imaginary part of the quantity of the
scattering pattern for electrical field PS2 in the direction of incidence of
the initial plane wave. According to the shown figures, the following results
were obtained for superellipsoid with the artificially hard lateral surface:
PS ≈ 0, 07472, Ps2 ≈ 0, 07488,
where
PS =
1
2ζ
∫ 2pi
0
∫ pi
0
∣∣∣~FE(θ, ϕ)∣∣∣2 sin θdθdϕ, (28)
PS2 =
2pi
ζ
Im
(
~FE · ~g(θ0, ϕ0)
)
, ~E0 = ~g · e−i~k~r. (29)
5 Conclusions
We demonstrated that impedance conditions with anisotropic impedance
are applicable to simulating the scattering characteristics of particles with
Rev.Mate.Teor.Aplic. (ISSN 1409-2433) Vol. 20(1): 1–20, January 2013
pattern equation method for the solution of... 19
mixed anisotropic surface impedance. The results indicate that the PEM
is efficient to solve these complicated problems. This approach will be
extended to the solution of the electromagnetic wave scattering problems
by group of bodies with anisotropic surface impedance.
Acknowledgments
This work was supported by Russian Foundation for Basic Research,
Project no. 09-02-00126.
References
[1] Kyurkchan, A.G. (2000) “Solution of vector scattering problems by
the pattern equation method”, Journ Comm Tech and Electron 45:
970–975.
[2] Kyurkchan, A.G.; Demin. D.B. (2002) “Electromagnetic wave diffrac-
tion from impedance scatterers with piecewise–smooth boundaries”,
Journ Comm Tech and Electron 47: 856–863.
[3] Kyurkchan, A.G.; Demin. D.B. (2004) “Simulation of wave scattering
by bodies with an absorbing coating and Black Bodies”, Technical
Physics 49: 165–173.
[4] Kyurkchan, A.G.; Demin. D.B. (2004) “Pattern equation method for
solving problems of diffraction of electromagnetic waves by axially
dielectric scatterers”, JQSRT 89: 237–255.
[5] Kyurkchan, A.G.; Demin. D.B.; Orlova, N.I. (2007) “Solution of scat-
tering problems of electromagnetic waves from objects with dielec-
tric covering using pattern equation method”, Journal of Commun.
Techn. and Electron 52: 131–40.
[6] Kyurkchan, A.G.; Demin. D.B.; Orlova, N.I. (2007) “Solution based
on the pattern equation method for scattering of electromagnetic
waves by objects coated with dielectric materials”, JQSRT 106: 192–
202.
[7] Kyurkchan, A.G.; Demin. D.B. (2009) “Solution of problem of elec-
tromagnetic waves scattering on inhomogeneously layered scatterers
using pattern equation method”, JQSRT 110: 1345–1355.
Rev.Mate.Teor.Aplic. (ISSN 1409-2433) Vol. 20(1): 1–20, January 2013
20 a.g. kyurkchan – d.b. demin
[8] Kyurkchan, A.G.; Demin. D.B. (2011) “Solution of problems of elec-
tromagnetic wave scattering by bodies with an anisotropic impedance
with the help of the pattern equation method”, Journal of Commun.
Techn. and Electron 56: 134–141.
[9] Kyurkchan, A.G. (1999) “Excitation of a periodic ribbed structure
possessing the properties of an artificially hard surface by a thread of
current”, Journal of Commun. Techn. and Electron. 44: 731–737.
[10] Kildal, P.-S. (1988) “Definition of artificially soft and hard surfaces
for electromagnetic waves”, Electron. Lett. 24(3): 168–170.
[11] Kildal, P.-S. (1990) “Artificially soft and hard surfaces in electromag-
netics”, IEEE Transactions on Antennas and Propagation 38(10):
1537–1544.
[12] Papas, C.H. (1965) Theory of Electromagnetic Wave Propagation.
McGraw–Hill, New York.
[13] Hizal, A.; Marincic, A. (1970) “New rigorous formulation of elec-
tromagnetic scattering from perfectly conducting bodies of arbitrary
shape”, Proc. IEEE 117(8): 1639–1647.
[14] Millar, R.F.; Burrows, M.L. (1969) “Rayleigh hypothesis in scattering
problems”, Electron. Lett. 5: 416–418.
[15] Kyurkchan, A.G. (1992) “A new integral equation in the diffraction
theory”, Soviet Physics Doklady 37(7): 338–340.
Rev.Mate.Teor.Aplic. (ISSN 1409-2433) Vol. 20(1): 1–20, January 2013