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Solving the diffraction problem of electromagnetic waves on objects with a complex geometry by the pattern equations method
ARTICLE IN PRESS
Journal of Quantitative Spectroscopy & Radiative Transfer 109 (2008) 1417–1429 www.elsevier.com/locate/jqsrt
Solving the diffraction problem of electromagnetic waves on objects with a complex geometry by the pattern equations method Alexander Kyurkchan, Elena A. Skorodumova Moscow Technical University of Communication and Informatics, Aviamotornaya Street 8A, 111024 Moscow, Russian Federation Received 30 September 2007; received in revised form 18 January 2008; accepted 25 January 2008
Abstract A new efficient method for solving the problems of waves diffraction on complex-shaped objects with the use of their replacement by a group of bodies with more simple form (fragments of complex objects) is offered. By the expansion of the scattering patterns of separate bodies in the series of vector spherical harmonics, the problem is reduced to solving the algebraic system of equations. It is shown that the method possesses a high convergence rate. Examples of modeling the scattering patterns of various complex objects are considered. Reliability of the results obtained is validated using the Optical theorem. r 2008 Elsevier Ltd. All rights reserved. Keywords: Diffraction; Scattering pattern; Scatterer; Group of bodies; Optical theorem; Ufimtsev theorem
1. Introduction The problem of effective modeling of the scattering characteristics of electromagnetic waves by objects with a complex geometry is very important in many ranges of science and techniques; however, it is poorly investigated. In many aspects, the solution of this problem presents a severe difficulty. For example, one of the circumstances making the solution more difficult are the perturbations of currents arising in localities of sites of a sharp modification of the geometry of the boundary. Therefore, with numerical realization of the methods, which use these characteristics as required, one should take account of the ‘‘harmonics’’ of high numbers for approximation of the fast oscillating components of the field. It, in turn, essentially affects the consumptions of the computer resources. Notice, that the current distribution, as a rule, is an intermediate quantity, which is used for determination of the other, integral, scattering characteristics. The pattern equations method (PEM) allows building the algorithms of the diffraction problems solution, weakly depending on local perturbations of the currents. In particular, the high efficiency of this technique is demonstrated with solving of the problem of wave diffraction on groups of bodies and on objects with a Corresponding author. Tel.: +7 495 236 2267.
E-mail address: [email protected] (A. Kyurkchan). 0022-4073/$ - see front matter r 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.jqsrt.2008.01.019
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complex structure in acoustical case [1]. In solving the last problem, it is offered to replace the complex-shaped body with a group of simpler bodies. The basis of this idea is the fact that the convergence rate of the computational algorithm weakly depends on the distance between the scatterers. The PEM has also shown its high efficiency for solutions of the diffraction problem on single bodies in a vector case [2]. In this paper, it is offered to extend this approach to electromagnetic problems of the wave scattering by the complex-shaped bodies, using the same idea as in acoustical case. The basic characteristic, for which the diffraction problem solution is founded, is directly the scattering pattern. Being the integral value, it smoothes the current distortions, arising when approaching studied bodies up to their contact. It essentially allows reducing the consumptions on evaluation of the pattern. Therefore, the usage of the scattering pattern as the characteristic required in solving the problem of diffraction on a group of bodies and on complex-shaped bodies is the most expedient. It is also necessary to note that the PEM is the strictly proven method. It distinguishes the PEM from most of the known methods, which as a rule have no explicit rigorous foundation. 2. Statement of the problem and its solution ~0 , H ~0 on the Let us consider the problem of scattering of the primary monochromatic electromagnetic field E object with a complex geometry. We present it as a combination of bodies with more simple form. Here is the detailed study for the case of two bodies with the geometry shown in Fig. 1. This approach can be extended for any set of objects. Let the impedance boundary conditions be set on the surfaces Sj, j ¼ 1,2: h i ~ ¼ Z j ~ ~ , ~ nj H nj E nj ~ Sj
Sj
~¼E ~0 þ E ~1 þ E ~1 , nj is the unit external normal vector to surface Sj, E where Zj is the surface impedance [3], ~ 1 2 ~¼H ~0 þ H ~1 þ H ~ 1 are the total fields and E ~1 , H ~ 1 are the secondary (diffraction) fields, which are generated H 1 2 j j by currents on the jth body and obey the system of homogeneous Maxwell’s equations elsewhere outside Sj: ~1 ; k ¼ opffiffiffiffiffi ~1 ; r H ~ 1 ¼ ik E ~1 ¼ ikzH m, rE j j j z j pffiffiffiffiffiffiffi where z ¼ m= is the wave impedance of a medium, and also to the Sommerfeld’s condition on infinity: ~ ~ rj rj 1 1 ~1 1 1 1 1 ~ ~ ~ Ej rj ! 1. þ zH j ¼ o ; Hj Ej ¼ o ; rj ~ rj z rj rj rj As mentioned above, the main characteristics for which one finds the solution of the stated problem are the scattering patterns of bodies, i.e. the functions, determining the dependence of the diffraction field from the angles (yj, jj) in the spherical coordinates (rj, yj, jj) connected with the jth body for the far zone (for krj b1), z2
n2
r2 r1 n1
r
z1
y1 x1
S1
r12
y2 x2
r02
z θ0
r01
S2
q y
x
ϕ0
Fig. 1. Geometry of the problem.
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where the following asymptotic relations are valid [4]: ! ! exp ikr exp ikr 1 1 1 E 1 H j j ~ ¼ ~ ¼ E F~j yj ; jj þ O 2 ; H F~j yj ; jj þ O 2 . j j rj rj krj krj E
H
Here F~j ; F~j are the scattering patterns for electrical and magnetic fields, respectively. The main idea of the PEM consists in the reduction of the initial boundary-value problem to the system of the integro-operator equations of the second kind with respect to the scattering patterns of separate bodies [1]. However it is necessary to note that this scheme appears too unwieldy for the vector problem. In order to derive the PEM algebraic system in the electromagnetic case, it is most convenient to expand the scattering patterns of the bodies into a series of vector spherical harmonics [5]: ~E ðyj ; jj Þ ¼ F j
1 X n X
1 X n X ~j ðyj ; jj Þ ~j ðyj ; jj Þ, i rj F ajnm in ~ bjnm in z F nm nm
n¼1 m¼n
n¼1 m¼n
where ~j ðyj ; jj Þ ¼ ~ rj rPm F nm n ðcos yj Þ expðimjj Þ, Pm n ðcos yj Þ are the associated Legendre functions [6]. Thus, to solve the initial problem, we need to derive the algebraic system for unknown coefficients ajnm ; bjnm . The first step of PEM is the expression of the desired coefficients via the boundary values of the wave field. The expressions required can be derived (by analogy with [7]) on a basis of the following relations: Z h e i h m i z ~ ~ ~1 ¼ ~ r r I E G r I j G 0j (1) þ Z ds0j , 0j j j j S j ik ~1 ¼ H j
Z
Sj
m i h e i Zj h ~ ~ r r I j G 0j þ r ~ I j G 0j ds0j , ikz
(2)
where e ~ ~ ; Ij ¼ ~ nj H
~m ~ ~ , Ij ¼ ~ nj ~ nj H
Sj
Sj
and G 0j ¼ G0 ð~ rj ; ~ r0 j Þ ¼
expðikj~ rj ~ r0j jÞ r0j j kj~ rj ~
(3)
is the Green’s function of the free space (the fundamental solution of the scalar Helmholtz equation). ~ and H ~ can be presented as the following expansions: The wave fields E ~¼E ~0 þ E ~¼H ~0 þ H
1 X n X
2 2 ~ e2 ~ h1 ðr1 ; y1 ; j1 Þ b1 z2 H ~e1 ðr1 ; y1 ; j1 Þ þ a2 H ~h2 a1nm H nm nm nm nm nm ðr2 ; y2 ; j2 Þbnm z H nm ðr2 ; y2 ; j2 Þ ,
n¼1 m¼n 1 X n X
2 ~ h2 2 ~ e2 ~ e1 ðr1 ; y1 ; j1 Þ þ b1 H ~ h1 a1nm H nm nm nm ðr1 ; y1 ; j1 Þ þ anm H nm ðr2 ; y2 ; j2 Þþbnm H nm ðr2 ; y2 ; j2 Þ ,
(4)
n¼1 m¼n
where the wave spherical functions are expressed using the formulas: ~hj ¼ r r ð~ H rj cm nm n ðyj ; jj ÞÞ;
~ej ¼ ik r ð~ rj c m H nm n ðyj ; jj ÞÞ, z
(5)
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ð2Þ m cm n ðyj ; jj Þ ¼ hn ðkrj ÞPn ðcos yj Þ expðimjj Þ;
j ¼ 1; 2,
(6)
hð2Þ n ðkrj Þ are the Hankel spherical functions of the second kind [6]. Furthermore, using the formulas (1)–(5) the coefficients ajnm ; bjnm can be presented as the integrals [5]: Z e z ¯ 0 j ~ ~ anm ¼ N nm (7) e ðr0 ; y0 ; j0 Þ ds0j , nj H ~ nm j j j 4p Sj Sj
bjnm
z N nm ¼ 4p
Z e ¯ 0 0 0 0 ~ h~ ~ n0j H nm ðrj ; yj ; jj Þ dsj , Sj
(8)
Sj
where the following designations are introduced: ð2n þ 1Þ ðn mÞ! N nm ¼ , nðn þ 1Þ ðn þ mÞ! e ~ e¯ nm rj ; yj ; jj ¼ r r ~ rj w¯ jnm rj ; yj ; jj , ik ¯e rj w¯ jnm rj ; yj ; jj , h~nm rj ; yj ; jj ¼ r ~ z w¯ jnm rj ; yj ; jj ¼ j n krj Pm n cos yj exp imjj , j n ðkrj Þ are the Bessel spherical functions of the first kind [6], and the line means a sign of complex conjugation. As a result, using the relations (4), (7) and (8), we obtain the PEM system of algebraic equations: 8 1 P n P jl;aa l > ajnm ¼ aj0 þ Gjj;aa ajnm þ G jj;ab bjnm þ G nm;nm anm þ G jl;ab blnm ; > nm;nm nm;nm nm;nm nm > > n¼1 m¼n > < 1 P n P jj;bb j jl;ba l jl;bb l j Gjj;ba bjnm ¼ bj0 > nm þ nm;nm anm þ G nm;nm bnm þ G nm;nm anm þ G nm;nm bnm ; > > n¼1 m¼n > > : n ¼ 1; 2; . . . ; jmjon; j; l ¼ 1; 2; jal:
(9)
All the coefficients in this system represent the sum of two components: j00 jz0 aj0 nm ¼ anm þ Z j anm ;
j00 jz0 bj0 nm ¼ bnm þ Z j bnm ;
zjl G jlnm;nm ¼ G 0jl nm;nm þ Z j G nm;nm ;
j; l ¼ 1; 2,
where the index ‘‘0’’ corresponds to the case of zero impedance (Zj ¼ 0), and the index ‘‘z’’ conforms to additional addends, caused by the difference of the value of impedance from zero. E ~E ðy; jÞ for each of the scatterers, we obtain the general scattering Defining the patterns F~1 ðy; jÞ and F 2 pattern for two bodies using the following formula: ~E ðy; jÞ ¼ R ~E ðy; jÞ, ¯ 1 F~E1 ðy; jÞ þ R ¯ 2F F 2 where Rj ¼ eik~q~r0j ; ~ q ¼ f sin y0 cos j0 ; ~ r0j ¼ r0j f sin y0j cos j0j ;
sin y0 sin j0 ;
sin y0j sin j0j ;
cos y0 g,
cos y0j g,
(r0j,y0j,j0j) is the origin of coordinates of the jth body in the general system of coordinates.
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j0 Thus the coefficients aj0 nm ; bnm can be obtained by the following integrals: Z e z ¯ ~0 ~ ~ N ¼ H r ; y ; j aj00 n e nm j j j j dsj , nm nm 4p Sj Z e z ¯ ~ 0 h~ ~ N ajz0 ¼ nj H nj ~ nm nm rj ; yj ; jj dsj , nm 4p Sj Z e z ¯ j00 ~0 h~ ~ N nm bnm ¼ nj H nm rj ; yj ; jj dsj , 4p Sj Z e 1 ~0 ~ ~ N nm bjz0 nj H nj ~ e¯ nm rj ; yj ; jj dsj . nm ¼ 4pz Sj
The matrix elements of the PEM system at Zj ¼ 0 are expressed in the form of relations: Z e z 0 jl;aa ~ e ðrl ; yl ; jl Þ ~ ~ ¼ N nm e¯ nm ðrj ; yj ; jj Þ dsj ; G nm;nm nj H nm 4p Sj Z e z 0 jl;ab ~ h ðrl ; yl ; jl Þ ~ ~ G nm;nm ¼ N nm e¯ nm ðrj ; yj ; jj Þ dsj ; nj H nm 4p Sj Z e z ¯ 0 jl;ba ~e ðrl ; yl ; jl Þ h~ ~ N nm G nm;nm ¼ nj H nm nm ðrj ; yj ; jj Þ dsj , 4p Sj Z e z ¯ 0 jl;bb ~h ðrl ; yl ; jl Þ h~ ~ N nm G nm;nm ¼ nj H nm nm ðrj ; yj ; jj Þ dsj . 4p Sj
(10)
When the value of impedance is distinguished from zero, for the additional addends of the matrix elements, we have Z e z ¯ zjl;aa ~e ðrl ; yl ; jl Þ h~ ~ N nm nj H G nm;nm ¼ nj ~ nm nm ðrj ; yj ; jj Þ dsj , 4p Sj Z e z ¯ ~h ðrl ; yl ; jl Þ h~ ~ ~ N G zjl;ab ¼ n H n nm j j nm nm;nm nm ðrj ; yj ; jj Þ dsj ; 4p Sj Z e 1 ~e ðrl ; yl ; jl Þ ~ ~ ~ N G zjl;ba ¼ n H e¯ nm ðrj ; yj ; jj Þ dsj ; n nm j j nm nm;nm 4pz Sj Z e 1 h ~ ~ ~ ~ N G zjl;bb ¼ n H ðr ; y ; j Þ e¯ nm ðrj ; yj ; jj Þ dsj ; j; l ¼ 1; 2. (11) n nm j j l nm l l nm;nm 4pz Sj In the spherical coordinate system: 1 ~ irj rj sin yj ~ iyj r0jy sin yj ~ ijj r0jj ; j j kj rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi kj ¼ ðr2j þ ðr0j y Þ2 Þsin2 yj þ ðr0j j Þ2 , ~ nj ¼
j
dsj ¼ kj rj dyj djj ,
j
and rj ¼ rj ðyj ; jj Þ are the equations of surfaces Sj. When evaluating the coefficients of the PEM algebraic system, we face the problem that lies in the fact that at j6¼l the integrands in the expressions (10) and (11) contain the vectors, designated in various coordinate systems (jth and lth, respectively). Therefore, before calculating the values of these integrals, it is necessary to reduce all the vectors in a uniform coordinate system. So, for determination of the coefficients jlab jlba jlbb G jlaa nm;nm , G nm;nm , G nm;nm and G nm;nm at j6¼l, it is expedient to take advantage of the addition theorems for vector
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spherical functions [8–10]: ~ e ðrl ; yl ; jl Þ ¼ H nm ~ h ðrl ; yl ; jl Þ ¼ H nm
eð1Þ i hð1Þ ~ ~ Amn rj ; yj ; jj þ Bmn rj ; yj ; jj , pq rlj ; ylj ; jlj H qp pq rlj ; ylj ; jlj H qp z q¼0 p¼q
q 1 X X
hð1Þ eð1Þ mn ~ ~ H H Amn r ; y ; j r ; y ; j r ; y ; j r ; y ; j izB lj lj j j lj lj j j lj j lj j . qp qp pq pq
q 1 X X q¼0 p¼q
~ hð1Þ are calculated by the formulas (5) and (6), in which the ~eð1Þ and H Here, the wave spherical functions H qp qp Hankel spherical functions of the second kind hð2Þ n ðkrj Þ should be replaced by the Bessel spherical functions of the first kind j n ðkrj Þ. mn Coefficients Amn pq ðrlj ; ylj ; jlj Þ and Bpq ðrlj ; ylj ; jlj Þ can be obtained from the following relations: p Amn pq ðrlj ; ylj ; jlj Þ ¼ ð1Þ
nþq X
mq aðm; nj p; qjsÞaðn; q; sÞhð2Þ ðcos ylj Þ exp½iðm qÞjlj , s ðkrlj ÞPs
s¼jnqj pþ1 Bmn pq ðrlj ; ylj ; jlj Þ ¼ ð1Þ
nþq X
mq aðm; nj p; qjs; s 1Þbðn; q; sÞhð2Þ ðcos ylj Þ exp½iðm qÞjlj , s ðkrlj ÞPs
s¼jnqj
where 2q þ 1 ðqðq þ 1Þ þ nðn þ 1Þ sðs þ 1ÞÞ, 2qðq þ 1Þ 2q þ 1 ððn þ q þ s þ 1Þðq n þ sÞðn q þ sÞðn þ q s þ 1ÞÞ1=2 , bðn; q; sÞ ¼ iqþsn 2qðq þ 1Þ
aðn; q; sÞ ¼ iqþsn
and
ðn þ mÞ!ðp þ qÞ!ðs m pÞ! 1=2 n aðm; njp; qjsÞ ¼ ð1Þ ð2s þ 1Þ ðn mÞ!ðp qÞ!ðs þ m þ pÞ! 0
1=2 n ðn þ mÞ!ðp þ qÞ!ðs m pÞ! aðm; njp; qjs; tÞ ¼ ð1Þmþp ð2s þ 1Þ ðn mÞ!ðp qÞ!ðs þ m þ pÞ! 0 mþp
Multipliers j1
j2
j
m1
m2
ðm1 þ m2 Þ
q
s
0 0 q
t
0 0
! !
n
q
s
m
p
m p
n
q
s
m
p
m p
! , ! .
! ,
entered in the given expressions, are the Wigner 3j-symbols [6]: ! j1 j2 j ¼ ð1Þj 1 j2 þm ð2j þ 1Þ1=2 ðj 1 j 2 m1 m2 jjmÞ, m1 m2 m where ðj 1 j 2 m1 m2 jjmÞ are the Clebsch–Gordan coefficients. 3. Results of calculations 3.1. Examination of the convergence of calculation algorithm Table 1 illustrates the convergence rate of the calculation algorithm for various objects—bodies of revolution with the common axis (the axis z)—under the perpendicular incidence of the plane wave. The value of impedance is chosen to be equal to zero (Zj ¼ 0). The values of the scattering pattern module jF Ey ðy; jÞj are calculated for the angles y ¼ p/2 and j ¼ 0. As one can see from the table, for the scatterers with an analytic boundary (spheres), four or five valid significant figures are already established when the reduction parameter
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Table 1 The values jF Ey ðy; jÞj at y ¼ p/2 and j ¼ 0 Two spheres, ka1,2 ¼ 10
Two superellipsoids, ka1,2 ¼ 5, kc1,2 ¼ 10, m ¼ 20
N
kr ¼ 20.02
kr ¼ 40
N
kr ¼ 20.02
kr ¼ 40
19 20 21 22 23
99.048619814 99.048550775 99.048546408 99.048546069 99.048546097
104.2595164167 104.2595164144 104.2595164142 104.2595164141 104.2595164142
21 22 23 24 25
75.72585373 75.66363669 75.67027629 75.64396840 75.67981606
74.62930903 74.63440138 74.60449618 74.61549045 74.61731606
z c
a x
Fig. 2. An axial section of a superellipsoid.
of the system (9) N ¼ kd 1;2 ¼ 20 even at the minimum distance between the objects (kr ¼ 20.02) (where d is the maximum size of the scatterer). However in the case of superellipsoids, the axial cross-section of which is given by the following equation: x2m z2m þ ¼1 a c (see Fig. 2), three or four valid significant figures are established only at N ¼ 1:5 kd 1;2 . It is caused because the singularities of the analytic continuation of the secondary field for such scatterers are located close enough to their boundaries. These investigations show that the convergence rate of the numerical algorithm remains almost the same when scattering bodies are arranged close together up to their contact, like in the previously described acoustical cases [1]. This fact allows us to extend the PEM to solving the diffraction problem for the scatterers with a complex geometry replacing them by the combination of the objects of more simple forms with minimum consumption of the computer resources. 3.2. Examination of mutual influence of objects The difficulty of the problem of wave diffraction on a group of bodies consists in the mutual interaction of objects, caused by multiple reflections between them. Fig. 3a and b illustrates the dependence of the integral scattering cross-sections: Z Z 1 2p p ~E s¼ jF ðy; jÞj2 sin y dy dj 4p 0 0 for two superellipsoids on the distance between them. The bodies have the parameters ka1;2 ¼ 1; kc1;2 ¼ ~E ðy; jÞ is the scattering pattern for 2; m ¼ 20 and the impedances Z1,2 ¼ 0 and Z1,2 ¼ z, respectively. Here F two bodies with (curves 1 and 2) and without taking into account the mutual influence. With increase in the distance between the objects, the value of the general integral scattering cross-section comes close to the sum of the cross-sections of separate bodies (curve 3). Furthermore, these figures show that when the bodies come together, there is a diminution of the aggregate cross-section relative to the aggregate cross-section, calculated without taking into account the mutual influence. This can be explained by the ‘‘partial accumulation’’
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Fig. 3. (a) Dependence of the integral scattering cross sections of the group of two superellipsoids with parameters ka1,2 ¼ 1, kc1,2 ¼ 2, m ¼ 20 from the distance between them at Z1,2 ¼ 0, the perpendicular incidence of the wave. (b) Dependence of the integral scattering cross sections of the group of two superellipsoids with parameters ka1,2 ¼ 1, kc1,2 ¼ 2, m ¼ 20 from the distance between them at Z1,2 ¼ z, the perpendicular incidence of the wave.
Fig. 4. Geometries of the bodies researched: (a) a superellipsoid; (b) a group of two closely located superellipsoids; (c) a ‘‘mushroom’’; (d) an inverted ‘‘mushroom’’ and (e) a ‘‘grenade’’
of power of the incident wave in the area between the reflectors during the average period. Curve 4 corresponds to the integral cross section of the superellipsoid of double size, to which the aggregate cross section of two bodies with their contact is coming close. 3.3. Powers estimation of simulation of the scattering characteristics for bodies with a complex geometry Let us carry out an examination of the method proposed, which will be based on comparison of the scattering pattern of a single body (Fig. 4a) with the scattering pattern of an object, composed of halves of these bodies (Fig. 4b). We locate these elements one over the other and set the minimum distance kD between them equal to 0.01. Fig. 5a and b shows the examples of such comparison for superellipsoids with the parameters ka1;2 ¼ 4; kc1;2 ¼ 8; m ¼ 20 for various values of the impedance. Fig. 2a corresponds to zero impedance (Z1,2 ¼ 0) and Fig. 2b to Z1,2 ¼ z for the perpendicular incidence of the plane wave in both cases. The figures show that the differences of the corresponding patterns are very small. Thus, as one can see from the investigations given above, the PEM algorithm is good for converging with no respect to the distance between the scattering objects and gives valid results in the vicinity of the interacting bodies. It allows us to apply the present method for modeling of the scattering characteristics of the bodies with a complex geometry.
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Fig. 5. (a) Comparison of the scattering patterns of two superellipsoids with parameters ka1,2 ¼ 4, kc1,2 ¼ 8, m ¼ 20 with the scattering pattern of a single superellipsoid with ka1,2 ¼ 4, kc1,2 ¼ 16, m ¼ 20 at Z1,2 ¼ 0, the perpendicular incidence of the wave. (b) Comparison of the scattering patterns of two superellipsoids with parameters ka1,2 ¼ 4, kc1,2 ¼ 8, m ¼ 20 with the scattering pattern of a single superellipsoid with ka1,2 ¼ 4, kc1,2 ¼ 16, m ¼ 20 at Z1,2 ¼ z, the perpendicular incidence of the wave.
Fig. 6. The scattering pattern of a ‘‘mushroom’’—a hemisphere with radius kahs ¼ 4 and a superellipsoid with parameters kase ¼ kcse ¼ 2, kD ¼ 0.01l, Z ¼ 0, the longitudinal (along the rotation axis) incidence of the wave.
Consider, as an example, the scattering problem on the object, having the shape of a ‘‘mushroom’’ (Fig. 4c), i.e. the body, composed of a hemisphere (ka ¼ 4) and a superellipsoid (ka ¼ kc ¼ 2; m ¼ 20). Fig. 6 shows the scattering pattern of such a body for Z1,2 ¼ 0. Calculations are carried out for the case of the longitudinal (along the rotation axis) incidence of the primary wave. Let us consider a similar example for other values of the impedance. We ‘‘protect’’ a perfect conducting superellipsoid (Z ¼ 0) with the parameters ka ¼ kc ¼ 1:5 by an absorbing hemisphere with radius ka ¼ 5 having located it from the incidence of the wave (Fig. 4d). Compare the pattern obtained with the analogous pattern for the case, when both components of the ‘‘mushroom’’ are perfect conducting (Fig. 7a and b,
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Fig. 7. (a) The scattering pattern of an ‘‘inverted mushroom’’—a hemisphere with radius kahs ¼ 5 and a superellipsoid with parameters kase ¼ kcse ¼ 1.5, located at the minimal distance kD ¼ 0.01l, Z1 ¼ 0, Z2 ¼ z, the longitudinal (along the rotation axis) incidence of the wave. (b) The scattering pattern of an ‘‘inverted mushroom’’—a hemisphere with radius kahs ¼ 5 and a superellipsoid with parameters kase ¼ kcse ¼ 1.5, located at the minimal distance kD ¼ 0.01l, Z1,2 ¼ 0, the longitudinal (along the rotation axis) incidence of the wave.
Fig. 8. The scattering pattern of an object, composed of two superellipsoids with ka1 ¼ kc1 ¼ 2:5, ka2 ¼ kc2 ¼ 5, located at the minimal distance kD ¼ 0.01l, Z1,2 ¼ 0, the perpendicular incidence of the wave.
respectively). The figures reveal that, in the case when the superellipsoid is ‘‘protected’’, scattering in the direction of a radiant of the incident wave is noticeably weaker. Thus, it is advantageous to use the presentation of the complex-shaped object as a combination of more simple ones since it allows modeling, especially, when the components of the body have various conductances as in our example. Fig. 8 illustrates another example of a complex body with the shape of a ‘‘grenade’’. This figure shows the scattering pattern calculated for two superellipsoids that are located one over the other (Fig. 4e). The parameters of the upper superellipsoid (ka1 ¼ kc1 ¼ 2:5; m ¼ 20) being half as the lower one (ka2 ¼ kc2 ¼ 5). In all the examples of modeling the scattering characteristics of the objects with a complex geometry given above, the distances between surfaces of their constituent elements were smaller than 0.01% of average
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sizes of the objects. Thus the minimal distance between the objects practically did not influence on the operational time of the computation algorithm. It allows one to speak about a high efficiency of the method proposed. 3.4. Verification of the validity of the optical theorem One of the estimation methods of the confirmation of the diffraction problem solution is the verification of the fulfillment of the optical theorem, according to which [11] Z Z n E o 1 2p p ~E ~ Im F ðy ¼ y0 ; j ¼ j0 Þ ~ p ¼ jF ðy; jÞj2 sin y dy dj, 2l 0 0 where (y0, j0) are the angles of incidence of the primary wave, ~ p is a polarization vector, l is a wave length. The data illustrating the optical theorem verification for two objects with various distances between them are given in Table 2. Calculations were carried out for parameters of reduction of the system (9) N ¼ 20 (spheres) and N ¼ 15 (superellipsoids). The accuracy of fulfillment of the optical theorem for two spheres was practically equivalent to the calculation-based precision. However for superellipsoids, the accuracy is noticeably lower. It can be explained by the fact that for these objects singularities of the analytical continuation of the wave field inside of the bodies are located close enough to their boundaries. It is necessary to note that the accuracy of the optical theorem validity is quite acceptable. Table 3 illustrates the verification of the validity of the optical theorem for perfect conducting bodies with a complex shape. Despite the complexity of the boundaries of explored objects, the theorem is fulfilled with the acceptable accuracy: the differences do not exceed 0.5%. Table 2 Verification of the validity of the optical theorem for two objects at the perpendicular incidence of the wave Two superellipsoids, ka1,2 ¼ 5, m ¼ 20
Two spheres, ka1,2 ¼ 10 kr
s
ImfF Ey ðy0 ; j0 Þg
Kr
s
ImfF Ey ðy0 ; j0 Þg
20.02 21 22 25 30
99.0200083585383 101.446125623356 102.394301062974 104.786289469724 102.728098089288
99.0200083585355 101.446125623356 102.394301062975 104.786289469723 102.728098089289
10.02 11 12 15 20
36.72808 35.40544 36.58867 37.35219 37.09743
36.79654 35.41377 36.66545 37.30462 37.17219
Table 3 Verification of the validity of the optical theorem for various objects with a complex geometry n E o p Im F~ ðy ¼ y0 ; j ¼ j0 Þ ~
s
ka1 ¼ 2.5 kc1 ¼ 2.5 ka2 ¼ 5 kc2 ¼ 5
22.7213
22.6195
Two superellipsoids (a ‘‘grenade’’), the incidence of the wave is along the rotation axis
ka1 ¼ 2.5 kc1 ¼ 2.5 ka2 ¼ 5 kc2 ¼ 5
17.4079
17.4142
A superellipsoid and a hemisphere (a ‘‘mushroom’’)
kase ¼ 2 kcse ¼ 2 kahemisph ¼ 4
5.9363
5.9387
Objects
Parameters
Two superellipsoids (a ‘‘grenade’’), the incidence of the wave is perpendicular to the rotation axis
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Fig. 9. Verification of the validity of the Ufimtsev theorem for two spheroids with parameters: (1) ka1;2 ¼ 3; kc1;2 ¼ 6; (2) ka1;2 ¼ 4; kc1;2 ¼ 8; (3) ka1;2 ¼ 5, kc1,2 ¼ 10; (4) ka1;2 ¼ 6; kc1;2 ¼ 12; (5)ka1;2 ¼ 7; kc1;2 ¼ 14, with various distances between them, the perpendicular incidence of the wave.
3.5. Verification of the Ufimtsev theorem Let us carry out the verification of the Ufimtsev theorem [12] for ‘‘black’’ (or absorbing) bodies. According to this theorem, the integral scattering cross section of a perfect conducting body is exactly two times bigger than the integral cross section of a ‘‘black’’ body with the same shadow contour (i.e. the boundary between the illuminated and shadowed parts of the body’s surface). Calculations for two spheroids of various sizes with the perpendicular incidence of the wave (perpendicular to the rotation axis) in the case of the Sommerfeld’s model (i.e. Z ¼ z) are carried out. The dependences of the ratio of the integral cross sections for two bodies on the distances between them are shown in Fig. 9 by the solid line. The dashed line designates the ratio of the integral cross sections of single objects with corresponding sizes. This figure shows that, at values of ka1,245, the Ufimtsev theorem is fulfilled with good enough accuracy. Notice that with increase in the distance between the researched bodies, the ratio of their cross sections comes closer to the analogous ratio of single scatterers. It is also visible from the graphs that the obtained results correlate well with the analogous data for single bodies [2]. It is necessary to note that the calculations of the Ufimtsev theorem for a group of bodies were not carried out earlier. 4. Conclusions Numerical calculations of the scattering characteristics of groups of bodies showed the high convergence rate of the computational algorithm weakly depending on the distance between the researched objects. Comparison of the scattering patterns of a single body and a combination of ‘‘components’’ of this body showed the possibility of application of this method for solving the problem of diffraction on the objects with a complex geometry by representing them as a combination of bodies with more simple shapes. Examples for objects in which the constituent elements have different conductances are given. The verification of the validity of the optical theorem showed the good accuracy of similar researches. For the first time, verification of the Ufimtsev theorem for two bodies was carried out. It was shown that the numerical results obtained correspond well with the analogous investigations for single objects. Acknowledgment This work was supported by Russian Foundation for Basic Research, Project no. 06-02-16483.
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References [1] Kyurkchan A, Skorodoumova EA. Modeling the characteristics of the waves scattering by a group of scatterers. JQSRT 2006; 100(1–3):207–19. [2] Kyurkchan AG, Demin DB. Pattern equation method for solving problems of diffraction of electromagnetic waves by axially symmetric dielectric scatterers. JQSRT 2004;89(1–4):237–55. [3] Markov GT, Chaplin AF. Excitation of electromagnetic waves. Moscow: Energiya; 1967 (in Russian). [4] Colton D, Kress R. Integral equation methods in scattering theory. New York: Wiley-Interscience; 1983. [5] Papas CH. Theory of electromagnetic wave propagation. New York, London, Sydney: McGraw-Hill Book Comp; 1965. [6] Vilenkin NYa. Special functions and representations group theory. Moscow: Nauka; 1965 (in Russian). [7] Kyurkchan AG. Solution of vector scattering problem by the pattern equations method. J Commun Technol Electron 2000;45(9):970–5. [8] Stein S. Addition theorems for spherical wave functions. Q Appl Math 1961;19(1):15–24. [9] Cruzan OR. Translational addition theorems for spherical vector wave functions. Q Appl Math 1962;20(1):33–40. [10] Felderhof BU, Jones RB. Addition theorems for spherical wave solutions of the vector Helmholtz equation. J Math Phys 1987;26(4):836–9. [11] Bohren CF, Huffman DR. Absorption and scattering of light by small particles. New York: Wiley-Interscience; 1983. [12] Ufimtsev PYa. Theory of edge diffraction in electromagnetics. Encino, CA: Tech Science Press; 2003.
Journal of Quantitative Spectroscopy & Radiative Transfer 109 (2008) 1417–1429 www.elsevier.com/locate/jqsrt
Solving the diffraction problem of electromagnetic waves on objects with a complex geometry by the pattern equations method Alexander Kyurkchan, Elena A. Skorodumova Moscow Technical University of Communication and Informatics, Aviamotornaya Street 8A, 111024 Moscow, Russian Federation Received 30 September 2007; received in revised form 18 January 2008; accepted 25 January 2008
Abstract A new efficient method for solving the problems of waves diffraction on complex-shaped objects with the use of their replacement by a group of bodies with more simple form (fragments of complex objects) is offered. By the expansion of the scattering patterns of separate bodies in the series of vector spherical harmonics, the problem is reduced to solving the algebraic system of equations. It is shown that the method possesses a high convergence rate. Examples of modeling the scattering patterns of various complex objects are considered. Reliability of the results obtained is validated using the Optical theorem. r 2008 Elsevier Ltd. All rights reserved. Keywords: Diffraction; Scattering pattern; Scatterer; Group of bodies; Optical theorem; Ufimtsev theorem
1. Introduction The problem of effective modeling of the scattering characteristics of electromagnetic waves by objects with a complex geometry is very important in many ranges of science and techniques; however, it is poorly investigated. In many aspects, the solution of this problem presents a severe difficulty. For example, one of the circumstances making the solution more difficult are the perturbations of currents arising in localities of sites of a sharp modification of the geometry of the boundary. Therefore, with numerical realization of the methods, which use these characteristics as required, one should take account of the ‘‘harmonics’’ of high numbers for approximation of the fast oscillating components of the field. It, in turn, essentially affects the consumptions of the computer resources. Notice, that the current distribution, as a rule, is an intermediate quantity, which is used for determination of the other, integral, scattering characteristics. The pattern equations method (PEM) allows building the algorithms of the diffraction problems solution, weakly depending on local perturbations of the currents. In particular, the high efficiency of this technique is demonstrated with solving of the problem of wave diffraction on groups of bodies and on objects with a Corresponding author. Tel.: +7 495 236 2267.
E-mail address: [email protected] (A. Kyurkchan). 0022-4073/$ - see front matter r 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.jqsrt.2008.01.019
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complex structure in acoustical case [1]. In solving the last problem, it is offered to replace the complex-shaped body with a group of simpler bodies. The basis of this idea is the fact that the convergence rate of the computational algorithm weakly depends on the distance between the scatterers. The PEM has also shown its high efficiency for solutions of the diffraction problem on single bodies in a vector case [2]. In this paper, it is offered to extend this approach to electromagnetic problems of the wave scattering by the complex-shaped bodies, using the same idea as in acoustical case. The basic characteristic, for which the diffraction problem solution is founded, is directly the scattering pattern. Being the integral value, it smoothes the current distortions, arising when approaching studied bodies up to their contact. It essentially allows reducing the consumptions on evaluation of the pattern. Therefore, the usage of the scattering pattern as the characteristic required in solving the problem of diffraction on a group of bodies and on complex-shaped bodies is the most expedient. It is also necessary to note that the PEM is the strictly proven method. It distinguishes the PEM from most of the known methods, which as a rule have no explicit rigorous foundation. 2. Statement of the problem and its solution ~0 , H ~0 on the Let us consider the problem of scattering of the primary monochromatic electromagnetic field E object with a complex geometry. We present it as a combination of bodies with more simple form. Here is the detailed study for the case of two bodies with the geometry shown in Fig. 1. This approach can be extended for any set of objects. Let the impedance boundary conditions be set on the surfaces Sj, j ¼ 1,2: h i ~ ¼ Z j ~ ~ , ~ nj H nj E nj ~ Sj
Sj
~¼E ~0 þ E ~1 þ E ~1 , nj is the unit external normal vector to surface Sj, E where Zj is the surface impedance [3], ~ 1 2 ~¼H ~0 þ H ~1 þ H ~ 1 are the total fields and E ~1 , H ~ 1 are the secondary (diffraction) fields, which are generated H 1 2 j j by currents on the jth body and obey the system of homogeneous Maxwell’s equations elsewhere outside Sj: ~1 ; k ¼ opffiffiffiffiffi ~1 ; r H ~ 1 ¼ ik E ~1 ¼ ikzH m, rE j j j z j pffiffiffiffiffiffiffi where z ¼ m= is the wave impedance of a medium, and also to the Sommerfeld’s condition on infinity: ~ ~ rj rj 1 1 ~1 1 1 1 1 ~ ~ ~ Ej rj ! 1. þ zH j ¼ o ; Hj Ej ¼ o ; rj ~ rj z rj rj rj As mentioned above, the main characteristics for which one finds the solution of the stated problem are the scattering patterns of bodies, i.e. the functions, determining the dependence of the diffraction field from the angles (yj, jj) in the spherical coordinates (rj, yj, jj) connected with the jth body for the far zone (for krj b1), z2
n2
r2 r1 n1
r
z1
y1 x1
S1
r12
y2 x2
r02
z θ0
r01
S2
q y
x
ϕ0
Fig. 1. Geometry of the problem.
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where the following asymptotic relations are valid [4]: ! ! exp ikr exp ikr 1 1 1 E 1 H j j ~ ¼ ~ ¼ E F~j yj ; jj þ O 2 ; H F~j yj ; jj þ O 2 . j j rj rj krj krj E
H
Here F~j ; F~j are the scattering patterns for electrical and magnetic fields, respectively. The main idea of the PEM consists in the reduction of the initial boundary-value problem to the system of the integro-operator equations of the second kind with respect to the scattering patterns of separate bodies [1]. However it is necessary to note that this scheme appears too unwieldy for the vector problem. In order to derive the PEM algebraic system in the electromagnetic case, it is most convenient to expand the scattering patterns of the bodies into a series of vector spherical harmonics [5]: ~E ðyj ; jj Þ ¼ F j
1 X n X
1 X n X ~j ðyj ; jj Þ ~j ðyj ; jj Þ, i rj F ajnm in ~ bjnm in z F nm nm
n¼1 m¼n
n¼1 m¼n
where ~j ðyj ; jj Þ ¼ ~ rj rPm F nm n ðcos yj Þ expðimjj Þ, Pm n ðcos yj Þ are the associated Legendre functions [6]. Thus, to solve the initial problem, we need to derive the algebraic system for unknown coefficients ajnm ; bjnm . The first step of PEM is the expression of the desired coefficients via the boundary values of the wave field. The expressions required can be derived (by analogy with [7]) on a basis of the following relations: Z h e i h m i z ~ ~ ~1 ¼ ~ r r I E G r I j G 0j (1) þ Z ds0j , 0j j j j S j ik ~1 ¼ H j
Z
Sj
m i h e i Zj h ~ ~ r r I j G 0j þ r ~ I j G 0j ds0j , ikz
(2)
where e ~ ~ ; Ij ¼ ~ nj H
~m ~ ~ , Ij ¼ ~ nj ~ nj H
Sj
Sj
and G 0j ¼ G0 ð~ rj ; ~ r0 j Þ ¼
expðikj~ rj ~ r0j jÞ r0j j kj~ rj ~
(3)
is the Green’s function of the free space (the fundamental solution of the scalar Helmholtz equation). ~ and H ~ can be presented as the following expansions: The wave fields E ~¼E ~0 þ E ~¼H ~0 þ H
1 X n X
2 2 ~ e2 ~ h1 ðr1 ; y1 ; j1 Þ b1 z2 H ~e1 ðr1 ; y1 ; j1 Þ þ a2 H ~h2 a1nm H nm nm nm nm nm ðr2 ; y2 ; j2 Þbnm z H nm ðr2 ; y2 ; j2 Þ ,
n¼1 m¼n 1 X n X
2 ~ h2 2 ~ e2 ~ e1 ðr1 ; y1 ; j1 Þ þ b1 H ~ h1 a1nm H nm nm nm ðr1 ; y1 ; j1 Þ þ anm H nm ðr2 ; y2 ; j2 Þþbnm H nm ðr2 ; y2 ; j2 Þ ,
(4)
n¼1 m¼n
where the wave spherical functions are expressed using the formulas: ~hj ¼ r r ð~ H rj cm nm n ðyj ; jj ÞÞ;
~ej ¼ ik r ð~ rj c m H nm n ðyj ; jj ÞÞ, z
(5)
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ð2Þ m cm n ðyj ; jj Þ ¼ hn ðkrj ÞPn ðcos yj Þ expðimjj Þ;
j ¼ 1; 2,
(6)
hð2Þ n ðkrj Þ are the Hankel spherical functions of the second kind [6]. Furthermore, using the formulas (1)–(5) the coefficients ajnm ; bjnm can be presented as the integrals [5]: Z e z ¯ 0 j ~ ~ anm ¼ N nm (7) e ðr0 ; y0 ; j0 Þ ds0j , nj H ~ nm j j j 4p Sj Sj
bjnm
z N nm ¼ 4p
Z e ¯ 0 0 0 0 ~ h~ ~ n0j H nm ðrj ; yj ; jj Þ dsj , Sj
(8)
Sj
where the following designations are introduced: ð2n þ 1Þ ðn mÞ! N nm ¼ , nðn þ 1Þ ðn þ mÞ! e ~ e¯ nm rj ; yj ; jj ¼ r r ~ rj w¯ jnm rj ; yj ; jj , ik ¯e rj w¯ jnm rj ; yj ; jj , h~nm rj ; yj ; jj ¼ r ~ z w¯ jnm rj ; yj ; jj ¼ j n krj Pm n cos yj exp imjj , j n ðkrj Þ are the Bessel spherical functions of the first kind [6], and the line means a sign of complex conjugation. As a result, using the relations (4), (7) and (8), we obtain the PEM system of algebraic equations: 8 1 P n P jl;aa l > ajnm ¼ aj0 þ Gjj;aa ajnm þ G jj;ab bjnm þ G nm;nm anm þ G jl;ab blnm ; > nm;nm nm;nm nm;nm nm > > n¼1 m¼n > < 1 P n P jj;bb j jl;ba l jl;bb l j Gjj;ba bjnm ¼ bj0 > nm þ nm;nm anm þ G nm;nm bnm þ G nm;nm anm þ G nm;nm bnm ; > > n¼1 m¼n > > : n ¼ 1; 2; . . . ; jmjon; j; l ¼ 1; 2; jal:
(9)
All the coefficients in this system represent the sum of two components: j00 jz0 aj0 nm ¼ anm þ Z j anm ;
j00 jz0 bj0 nm ¼ bnm þ Z j bnm ;
zjl G jlnm;nm ¼ G 0jl nm;nm þ Z j G nm;nm ;
j; l ¼ 1; 2,
where the index ‘‘0’’ corresponds to the case of zero impedance (Zj ¼ 0), and the index ‘‘z’’ conforms to additional addends, caused by the difference of the value of impedance from zero. E ~E ðy; jÞ for each of the scatterers, we obtain the general scattering Defining the patterns F~1 ðy; jÞ and F 2 pattern for two bodies using the following formula: ~E ðy; jÞ ¼ R ~E ðy; jÞ, ¯ 1 F~E1 ðy; jÞ þ R ¯ 2F F 2 where Rj ¼ eik~q~r0j ; ~ q ¼ f sin y0 cos j0 ; ~ r0j ¼ r0j f sin y0j cos j0j ;
sin y0 sin j0 ;
sin y0j sin j0j ;
cos y0 g,
cos y0j g,
(r0j,y0j,j0j) is the origin of coordinates of the jth body in the general system of coordinates.
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j0 Thus the coefficients aj0 nm ; bnm can be obtained by the following integrals: Z e z ¯ ~0 ~ ~ N ¼ H r ; y ; j aj00 n e nm j j j j dsj , nm nm 4p Sj Z e z ¯ ~ 0 h~ ~ N ajz0 ¼ nj H nj ~ nm nm rj ; yj ; jj dsj , nm 4p Sj Z e z ¯ j00 ~0 h~ ~ N nm bnm ¼ nj H nm rj ; yj ; jj dsj , 4p Sj Z e 1 ~0 ~ ~ N nm bjz0 nj H nj ~ e¯ nm rj ; yj ; jj dsj . nm ¼ 4pz Sj
The matrix elements of the PEM system at Zj ¼ 0 are expressed in the form of relations: Z e z 0 jl;aa ~ e ðrl ; yl ; jl Þ ~ ~ ¼ N nm e¯ nm ðrj ; yj ; jj Þ dsj ; G nm;nm nj H nm 4p Sj Z e z 0 jl;ab ~ h ðrl ; yl ; jl Þ ~ ~ G nm;nm ¼ N nm e¯ nm ðrj ; yj ; jj Þ dsj ; nj H nm 4p Sj Z e z ¯ 0 jl;ba ~e ðrl ; yl ; jl Þ h~ ~ N nm G nm;nm ¼ nj H nm nm ðrj ; yj ; jj Þ dsj , 4p Sj Z e z ¯ 0 jl;bb ~h ðrl ; yl ; jl Þ h~ ~ N nm G nm;nm ¼ nj H nm nm ðrj ; yj ; jj Þ dsj . 4p Sj
(10)
When the value of impedance is distinguished from zero, for the additional addends of the matrix elements, we have Z e z ¯ zjl;aa ~e ðrl ; yl ; jl Þ h~ ~ N nm nj H G nm;nm ¼ nj ~ nm nm ðrj ; yj ; jj Þ dsj , 4p Sj Z e z ¯ ~h ðrl ; yl ; jl Þ h~ ~ ~ N G zjl;ab ¼ n H n nm j j nm nm;nm nm ðrj ; yj ; jj Þ dsj ; 4p Sj Z e 1 ~e ðrl ; yl ; jl Þ ~ ~ ~ N G zjl;ba ¼ n H e¯ nm ðrj ; yj ; jj Þ dsj ; n nm j j nm nm;nm 4pz Sj Z e 1 h ~ ~ ~ ~ N G zjl;bb ¼ n H ðr ; y ; j Þ e¯ nm ðrj ; yj ; jj Þ dsj ; j; l ¼ 1; 2. (11) n nm j j l nm l l nm;nm 4pz Sj In the spherical coordinate system: 1 ~ irj rj sin yj ~ iyj r0jy sin yj ~ ijj r0jj ; j j kj rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi kj ¼ ðr2j þ ðr0j y Þ2 Þsin2 yj þ ðr0j j Þ2 , ~ nj ¼
j
dsj ¼ kj rj dyj djj ,
j
and rj ¼ rj ðyj ; jj Þ are the equations of surfaces Sj. When evaluating the coefficients of the PEM algebraic system, we face the problem that lies in the fact that at j6¼l the integrands in the expressions (10) and (11) contain the vectors, designated in various coordinate systems (jth and lth, respectively). Therefore, before calculating the values of these integrals, it is necessary to reduce all the vectors in a uniform coordinate system. So, for determination of the coefficients jlab jlba jlbb G jlaa nm;nm , G nm;nm , G nm;nm and G nm;nm at j6¼l, it is expedient to take advantage of the addition theorems for vector
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spherical functions [8–10]: ~ e ðrl ; yl ; jl Þ ¼ H nm ~ h ðrl ; yl ; jl Þ ¼ H nm
eð1Þ i hð1Þ ~ ~ Amn rj ; yj ; jj þ Bmn rj ; yj ; jj , pq rlj ; ylj ; jlj H qp pq rlj ; ylj ; jlj H qp z q¼0 p¼q
q 1 X X
hð1Þ eð1Þ mn ~ ~ H H Amn r ; y ; j r ; y ; j r ; y ; j r ; y ; j izB lj lj j j lj lj j j lj j lj j . qp qp pq pq
q 1 X X q¼0 p¼q
~ hð1Þ are calculated by the formulas (5) and (6), in which the ~eð1Þ and H Here, the wave spherical functions H qp qp Hankel spherical functions of the second kind hð2Þ n ðkrj Þ should be replaced by the Bessel spherical functions of the first kind j n ðkrj Þ. mn Coefficients Amn pq ðrlj ; ylj ; jlj Þ and Bpq ðrlj ; ylj ; jlj Þ can be obtained from the following relations: p Amn pq ðrlj ; ylj ; jlj Þ ¼ ð1Þ
nþq X
mq aðm; nj p; qjsÞaðn; q; sÞhð2Þ ðcos ylj Þ exp½iðm qÞjlj , s ðkrlj ÞPs
s¼jnqj pþ1 Bmn pq ðrlj ; ylj ; jlj Þ ¼ ð1Þ
nþq X
mq aðm; nj p; qjs; s 1Þbðn; q; sÞhð2Þ ðcos ylj Þ exp½iðm qÞjlj , s ðkrlj ÞPs
s¼jnqj
where 2q þ 1 ðqðq þ 1Þ þ nðn þ 1Þ sðs þ 1ÞÞ, 2qðq þ 1Þ 2q þ 1 ððn þ q þ s þ 1Þðq n þ sÞðn q þ sÞðn þ q s þ 1ÞÞ1=2 , bðn; q; sÞ ¼ iqþsn 2qðq þ 1Þ
aðn; q; sÞ ¼ iqþsn
and
ðn þ mÞ!ðp þ qÞ!ðs m pÞ! 1=2 n aðm; njp; qjsÞ ¼ ð1Þ ð2s þ 1Þ ðn mÞ!ðp qÞ!ðs þ m þ pÞ! 0
1=2 n ðn þ mÞ!ðp þ qÞ!ðs m pÞ! aðm; njp; qjs; tÞ ¼ ð1Þmþp ð2s þ 1Þ ðn mÞ!ðp qÞ!ðs þ m þ pÞ! 0 mþp
Multipliers j1
j2
j
m1
m2
ðm1 þ m2 Þ
q
s
0 0 q
t
0 0
! !
n
q
s
m
p
m p
n
q
s
m
p
m p
! , ! .
! ,
entered in the given expressions, are the Wigner 3j-symbols [6]: ! j1 j2 j ¼ ð1Þj 1 j2 þm ð2j þ 1Þ1=2 ðj 1 j 2 m1 m2 jjmÞ, m1 m2 m where ðj 1 j 2 m1 m2 jjmÞ are the Clebsch–Gordan coefficients. 3. Results of calculations 3.1. Examination of the convergence of calculation algorithm Table 1 illustrates the convergence rate of the calculation algorithm for various objects—bodies of revolution with the common axis (the axis z)—under the perpendicular incidence of the plane wave. The value of impedance is chosen to be equal to zero (Zj ¼ 0). The values of the scattering pattern module jF Ey ðy; jÞj are calculated for the angles y ¼ p/2 and j ¼ 0. As one can see from the table, for the scatterers with an analytic boundary (spheres), four or five valid significant figures are already established when the reduction parameter
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Table 1 The values jF Ey ðy; jÞj at y ¼ p/2 and j ¼ 0 Two spheres, ka1,2 ¼ 10
Two superellipsoids, ka1,2 ¼ 5, kc1,2 ¼ 10, m ¼ 20
N
kr ¼ 20.02
kr ¼ 40
N
kr ¼ 20.02
kr ¼ 40
19 20 21 22 23
99.048619814 99.048550775 99.048546408 99.048546069 99.048546097
104.2595164167 104.2595164144 104.2595164142 104.2595164141 104.2595164142
21 22 23 24 25
75.72585373 75.66363669 75.67027629 75.64396840 75.67981606
74.62930903 74.63440138 74.60449618 74.61549045 74.61731606
z c
a x
Fig. 2. An axial section of a superellipsoid.
of the system (9) N ¼ kd 1;2 ¼ 20 even at the minimum distance between the objects (kr ¼ 20.02) (where d is the maximum size of the scatterer). However in the case of superellipsoids, the axial cross-section of which is given by the following equation: x2m z2m þ ¼1 a c (see Fig. 2), three or four valid significant figures are established only at N ¼ 1:5 kd 1;2 . It is caused because the singularities of the analytic continuation of the secondary field for such scatterers are located close enough to their boundaries. These investigations show that the convergence rate of the numerical algorithm remains almost the same when scattering bodies are arranged close together up to their contact, like in the previously described acoustical cases [1]. This fact allows us to extend the PEM to solving the diffraction problem for the scatterers with a complex geometry replacing them by the combination of the objects of more simple forms with minimum consumption of the computer resources. 3.2. Examination of mutual influence of objects The difficulty of the problem of wave diffraction on a group of bodies consists in the mutual interaction of objects, caused by multiple reflections between them. Fig. 3a and b illustrates the dependence of the integral scattering cross-sections: Z Z 1 2p p ~E s¼ jF ðy; jÞj2 sin y dy dj 4p 0 0 for two superellipsoids on the distance between them. The bodies have the parameters ka1;2 ¼ 1; kc1;2 ¼ ~E ðy; jÞ is the scattering pattern for 2; m ¼ 20 and the impedances Z1,2 ¼ 0 and Z1,2 ¼ z, respectively. Here F two bodies with (curves 1 and 2) and without taking into account the mutual influence. With increase in the distance between the objects, the value of the general integral scattering cross-section comes close to the sum of the cross-sections of separate bodies (curve 3). Furthermore, these figures show that when the bodies come together, there is a diminution of the aggregate cross-section relative to the aggregate cross-section, calculated without taking into account the mutual influence. This can be explained by the ‘‘partial accumulation’’
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Fig. 3. (a) Dependence of the integral scattering cross sections of the group of two superellipsoids with parameters ka1,2 ¼ 1, kc1,2 ¼ 2, m ¼ 20 from the distance between them at Z1,2 ¼ 0, the perpendicular incidence of the wave. (b) Dependence of the integral scattering cross sections of the group of two superellipsoids with parameters ka1,2 ¼ 1, kc1,2 ¼ 2, m ¼ 20 from the distance between them at Z1,2 ¼ z, the perpendicular incidence of the wave.
Fig. 4. Geometries of the bodies researched: (a) a superellipsoid; (b) a group of two closely located superellipsoids; (c) a ‘‘mushroom’’; (d) an inverted ‘‘mushroom’’ and (e) a ‘‘grenade’’
of power of the incident wave in the area between the reflectors during the average period. Curve 4 corresponds to the integral cross section of the superellipsoid of double size, to which the aggregate cross section of two bodies with their contact is coming close. 3.3. Powers estimation of simulation of the scattering characteristics for bodies with a complex geometry Let us carry out an examination of the method proposed, which will be based on comparison of the scattering pattern of a single body (Fig. 4a) with the scattering pattern of an object, composed of halves of these bodies (Fig. 4b). We locate these elements one over the other and set the minimum distance kD between them equal to 0.01. Fig. 5a and b shows the examples of such comparison for superellipsoids with the parameters ka1;2 ¼ 4; kc1;2 ¼ 8; m ¼ 20 for various values of the impedance. Fig. 2a corresponds to zero impedance (Z1,2 ¼ 0) and Fig. 2b to Z1,2 ¼ z for the perpendicular incidence of the plane wave in both cases. The figures show that the differences of the corresponding patterns are very small. Thus, as one can see from the investigations given above, the PEM algorithm is good for converging with no respect to the distance between the scattering objects and gives valid results in the vicinity of the interacting bodies. It allows us to apply the present method for modeling of the scattering characteristics of the bodies with a complex geometry.
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Fig. 5. (a) Comparison of the scattering patterns of two superellipsoids with parameters ka1,2 ¼ 4, kc1,2 ¼ 8, m ¼ 20 with the scattering pattern of a single superellipsoid with ka1,2 ¼ 4, kc1,2 ¼ 16, m ¼ 20 at Z1,2 ¼ 0, the perpendicular incidence of the wave. (b) Comparison of the scattering patterns of two superellipsoids with parameters ka1,2 ¼ 4, kc1,2 ¼ 8, m ¼ 20 with the scattering pattern of a single superellipsoid with ka1,2 ¼ 4, kc1,2 ¼ 16, m ¼ 20 at Z1,2 ¼ z, the perpendicular incidence of the wave.
Fig. 6. The scattering pattern of a ‘‘mushroom’’—a hemisphere with radius kahs ¼ 4 and a superellipsoid with parameters kase ¼ kcse ¼ 2, kD ¼ 0.01l, Z ¼ 0, the longitudinal (along the rotation axis) incidence of the wave.
Consider, as an example, the scattering problem on the object, having the shape of a ‘‘mushroom’’ (Fig. 4c), i.e. the body, composed of a hemisphere (ka ¼ 4) and a superellipsoid (ka ¼ kc ¼ 2; m ¼ 20). Fig. 6 shows the scattering pattern of such a body for Z1,2 ¼ 0. Calculations are carried out for the case of the longitudinal (along the rotation axis) incidence of the primary wave. Let us consider a similar example for other values of the impedance. We ‘‘protect’’ a perfect conducting superellipsoid (Z ¼ 0) with the parameters ka ¼ kc ¼ 1:5 by an absorbing hemisphere with radius ka ¼ 5 having located it from the incidence of the wave (Fig. 4d). Compare the pattern obtained with the analogous pattern for the case, when both components of the ‘‘mushroom’’ are perfect conducting (Fig. 7a and b,
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Fig. 7. (a) The scattering pattern of an ‘‘inverted mushroom’’—a hemisphere with radius kahs ¼ 5 and a superellipsoid with parameters kase ¼ kcse ¼ 1.5, located at the minimal distance kD ¼ 0.01l, Z1 ¼ 0, Z2 ¼ z, the longitudinal (along the rotation axis) incidence of the wave. (b) The scattering pattern of an ‘‘inverted mushroom’’—a hemisphere with radius kahs ¼ 5 and a superellipsoid with parameters kase ¼ kcse ¼ 1.5, located at the minimal distance kD ¼ 0.01l, Z1,2 ¼ 0, the longitudinal (along the rotation axis) incidence of the wave.
Fig. 8. The scattering pattern of an object, composed of two superellipsoids with ka1 ¼ kc1 ¼ 2:5, ka2 ¼ kc2 ¼ 5, located at the minimal distance kD ¼ 0.01l, Z1,2 ¼ 0, the perpendicular incidence of the wave.
respectively). The figures reveal that, in the case when the superellipsoid is ‘‘protected’’, scattering in the direction of a radiant of the incident wave is noticeably weaker. Thus, it is advantageous to use the presentation of the complex-shaped object as a combination of more simple ones since it allows modeling, especially, when the components of the body have various conductances as in our example. Fig. 8 illustrates another example of a complex body with the shape of a ‘‘grenade’’. This figure shows the scattering pattern calculated for two superellipsoids that are located one over the other (Fig. 4e). The parameters of the upper superellipsoid (ka1 ¼ kc1 ¼ 2:5; m ¼ 20) being half as the lower one (ka2 ¼ kc2 ¼ 5). In all the examples of modeling the scattering characteristics of the objects with a complex geometry given above, the distances between surfaces of their constituent elements were smaller than 0.01% of average
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sizes of the objects. Thus the minimal distance between the objects practically did not influence on the operational time of the computation algorithm. It allows one to speak about a high efficiency of the method proposed. 3.4. Verification of the validity of the optical theorem One of the estimation methods of the confirmation of the diffraction problem solution is the verification of the fulfillment of the optical theorem, according to which [11] Z Z n E o 1 2p p ~E ~ Im F ðy ¼ y0 ; j ¼ j0 Þ ~ p ¼ jF ðy; jÞj2 sin y dy dj, 2l 0 0 where (y0, j0) are the angles of incidence of the primary wave, ~ p is a polarization vector, l is a wave length. The data illustrating the optical theorem verification for two objects with various distances between them are given in Table 2. Calculations were carried out for parameters of reduction of the system (9) N ¼ 20 (spheres) and N ¼ 15 (superellipsoids). The accuracy of fulfillment of the optical theorem for two spheres was practically equivalent to the calculation-based precision. However for superellipsoids, the accuracy is noticeably lower. It can be explained by the fact that for these objects singularities of the analytical continuation of the wave field inside of the bodies are located close enough to their boundaries. It is necessary to note that the accuracy of the optical theorem validity is quite acceptable. Table 3 illustrates the verification of the validity of the optical theorem for perfect conducting bodies with a complex shape. Despite the complexity of the boundaries of explored objects, the theorem is fulfilled with the acceptable accuracy: the differences do not exceed 0.5%. Table 2 Verification of the validity of the optical theorem for two objects at the perpendicular incidence of the wave Two superellipsoids, ka1,2 ¼ 5, m ¼ 20
Two spheres, ka1,2 ¼ 10 kr
s
ImfF Ey ðy0 ; j0 Þg
Kr
s
ImfF Ey ðy0 ; j0 Þg
20.02 21 22 25 30
99.0200083585383 101.446125623356 102.394301062974 104.786289469724 102.728098089288
99.0200083585355 101.446125623356 102.394301062975 104.786289469723 102.728098089289
10.02 11 12 15 20
36.72808 35.40544 36.58867 37.35219 37.09743
36.79654 35.41377 36.66545 37.30462 37.17219
Table 3 Verification of the validity of the optical theorem for various objects with a complex geometry n E o p Im F~ ðy ¼ y0 ; j ¼ j0 Þ ~
s
ka1 ¼ 2.5 kc1 ¼ 2.5 ka2 ¼ 5 kc2 ¼ 5
22.7213
22.6195
Two superellipsoids (a ‘‘grenade’’), the incidence of the wave is along the rotation axis
ka1 ¼ 2.5 kc1 ¼ 2.5 ka2 ¼ 5 kc2 ¼ 5
17.4079
17.4142
A superellipsoid and a hemisphere (a ‘‘mushroom’’)
kase ¼ 2 kcse ¼ 2 kahemisph ¼ 4
5.9363
5.9387
Objects
Parameters
Two superellipsoids (a ‘‘grenade’’), the incidence of the wave is perpendicular to the rotation axis
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Fig. 9. Verification of the validity of the Ufimtsev theorem for two spheroids with parameters: (1) ka1;2 ¼ 3; kc1;2 ¼ 6; (2) ka1;2 ¼ 4; kc1;2 ¼ 8; (3) ka1;2 ¼ 5, kc1,2 ¼ 10; (4) ka1;2 ¼ 6; kc1;2 ¼ 12; (5)ka1;2 ¼ 7; kc1;2 ¼ 14, with various distances between them, the perpendicular incidence of the wave.
3.5. Verification of the Ufimtsev theorem Let us carry out the verification of the Ufimtsev theorem [12] for ‘‘black’’ (or absorbing) bodies. According to this theorem, the integral scattering cross section of a perfect conducting body is exactly two times bigger than the integral cross section of a ‘‘black’’ body with the same shadow contour (i.e. the boundary between the illuminated and shadowed parts of the body’s surface). Calculations for two spheroids of various sizes with the perpendicular incidence of the wave (perpendicular to the rotation axis) in the case of the Sommerfeld’s model (i.e. Z ¼ z) are carried out. The dependences of the ratio of the integral cross sections for two bodies on the distances between them are shown in Fig. 9 by the solid line. The dashed line designates the ratio of the integral cross sections of single objects with corresponding sizes. This figure shows that, at values of ka1,245, the Ufimtsev theorem is fulfilled with good enough accuracy. Notice that with increase in the distance between the researched bodies, the ratio of their cross sections comes closer to the analogous ratio of single scatterers. It is also visible from the graphs that the obtained results correlate well with the analogous data for single bodies [2]. It is necessary to note that the calculations of the Ufimtsev theorem for a group of bodies were not carried out earlier. 4. Conclusions Numerical calculations of the scattering characteristics of groups of bodies showed the high convergence rate of the computational algorithm weakly depending on the distance between the researched objects. Comparison of the scattering patterns of a single body and a combination of ‘‘components’’ of this body showed the possibility of application of this method for solving the problem of diffraction on the objects with a complex geometry by representing them as a combination of bodies with more simple shapes. Examples for objects in which the constituent elements have different conductances are given. The verification of the validity of the optical theorem showed the good accuracy of similar researches. For the first time, verification of the Ufimtsev theorem for two bodies was carried out. It was shown that the numerical results obtained correspond well with the analogous investigations for single objects. Acknowledgment This work was supported by Russian Foundation for Basic Research, Project no. 06-02-16483.
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References [1] Kyurkchan A, Skorodoumova EA. Modeling the characteristics of the waves scattering by a group of scatterers. JQSRT 2006; 100(1–3):207–19. [2] Kyurkchan AG, Demin DB. Pattern equation method for solving problems of diffraction of electromagnetic waves by axially symmetric dielectric scatterers. JQSRT 2004;89(1–4):237–55. [3] Markov GT, Chaplin AF. Excitation of electromagnetic waves. Moscow: Energiya; 1967 (in Russian). [4] Colton D, Kress R. Integral equation methods in scattering theory. New York: Wiley-Interscience; 1983. [5] Papas CH. Theory of electromagnetic wave propagation. New York, London, Sydney: McGraw-Hill Book Comp; 1965. [6] Vilenkin NYa. Special functions and representations group theory. Moscow: Nauka; 1965 (in Russian). [7] Kyurkchan AG. Solution of vector scattering problem by the pattern equations method. J Commun Technol Electron 2000;45(9):970–5. [8] Stein S. Addition theorems for spherical wave functions. Q Appl Math 1961;19(1):15–24. [9] Cruzan OR. Translational addition theorems for spherical vector wave functions. Q Appl Math 1962;20(1):33–40. [10] Felderhof BU, Jones RB. Addition theorems for spherical wave solutions of the vector Helmholtz equation. J Math Phys 1987;26(4):836–9. [11] Bohren CF, Huffman DR. Absorption and scattering of light by small particles. New York: Wiley-Interscience; 1983. [12] Ufimtsev PYa. Theory of edge diffraction in electromagnetics. Encino, CA: Tech Science Press; 2003.