- Email: [email protected]
Pattern equation method for solving problems of diffraction of electromagnetic waves by axially symmetric dielectric scatterers
ARTICLE IN PRESS
Journal of Quantitative Spectroscopy & Radiative Transfer 89 (2004) 237–255 www.elsevier.com/locate/jqsrt
Pattern equation method for solving problems of diffraction of electromagnetic waves by axially symmetric dielectric scatterers Alexander Kyurkchan, Dmitrii Demin Moscow Technical University of Communication and Informatics, Aviamotornaya Street 8A, 111024 Moscow, Russia
Abstract A new efficient method is proposed for solving 3D problems of diffraction of electromagnetic waves by a single dielectric body of arbitrary shape. The method offers a high rate of convergence. The method is applied to the problem of scattering by axially symmetrical bodies, and the data are presented that illustrate the convergence rate of the computation algorithm for bodies of different shapes. A possibility is studied to model the scattering characteristics for perfectly absorbing (black) bodies. The Ufimtsev theorem on the ratio of the cross-sections of perfectly reflecting and black bodies is numerically tested. r 2004 Elsevier Ltd. All rights reserved. Keywords: Diffraction; Scattering pattern; Scatterer; Integral-operator equation; Spheroid; Cylinder; Black body
1. Introduction The problem of diffraction of electromagnetic waves by 3D dielectric objects of arbitrary shapes is rather relevant and remains insufficiently studied because of its complexity. In recent years, the method of volume integral [1], the T-matrix method [2], and the method of discrete sources [3] have been commonly used for solving this problem. However, a successful implementation of the methods strongly depends on the geometrical properties of the scatterer. With all these methods, among which the first and third ones are quite universal, the starting boundary-value problem is reduced to a system of linear algebraic equations, with the order of about 50C200ðd=lÞ2 or more Corresponding author. Tel.: +7-95-236-2267.
E-mail address: [email protected], [email protected] (A. Kyurkchan). 0022-4073/$ - see front matter r 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.jqsrt.2004.05.025
ARTICLE IN PRESS 238
A. Kyurkchan, D. Demin / Journal of Quantitative Spectroscopy & Radiative Transfer 89 (2004) 237–255
(d is the characteristic size of the scatterer). Such number of the equations to be solved means great calculation expenses. So, a high urgency exists in developing efficient methods to solve the scattering problems for objects with complicated geometrical characteristics. Here, we consider a new universal method, the pattern equation method (PEM), for solving the diffraction and scattering problems. This method was first presented in Refs. [4,5] and was later applied to a number of diffraction and scattering problems, such as wave scattering by a single object and a group of objects located in homogeneous and planar-stratified media [5–8], wave diffraction by multi-row discrete periodic arrays near plane interface of the media [9], wave propagation in optic fibers with noncircular cross-sections [10], etc. The PEM was also successfully applied to the problems of diffraction of electromagnetic waves by 3D impedance objects [11,12]. The PEM is based on reducing the initial boundary problem for the Helmholtz equation to an integral-operator equation of the second kind with respect to the scattering pattern (spectral function of the wave field). In this paper, we use the generalized Sommerfeld–Weil representation for the diffracted field [13] in the form of integral of plane waves. Such representation converges elsewhere outside the convex envelope of singularities in the continuation of the wave field. Then, the desired function is represented as an expansion onto some basis, and this expansion is substituted into the integral-operator equation. So, the obtained equality is projected onto some basis, generally another one. With certain restrictions on the geometry of the problem, which can be strictly established, the resultant infinite system of algebraic equations is solved by the reduction method, i.e. the method of truncation. The PEM has been shown to offer a number of important advantages in comparison with other methods used in solving the diffraction and scattering problems. Thus, in solving the problem of scattering by 3D impedance bodies with piecewise-smooth boundaries, it has been established that the rate of the algorithm convergence is mainly governed by the scatterer size and weakly depends on its geometry [11,12]. The reason is that, when a wave scattering problem is solved using the PEM, the desired quantity is the scattering pattern itself, i.e., a functional of the field distribution over the scatterer’s surface. Therefore, the integration smoothes rapidly oscillating field components whose approximation implies high-order harmonics to be accounted for. As a result, the body of computations required for a given accuracy to be attained proves to be entirely reduced. Fig. 1 shows the PEM-calculated monostatic radar cross-section of a circular cone with the base radius ka ¼ 3:08 and the vertical half-angle a ¼ 15 . The cross-section is presented in the plane of incidence ðj0 ¼ 0 Þ, versus the incidence angle y of a plane wave (y ¼ 0 corresponds to a case when the plane wave is incident along the z-axis, from the cone vertex). For the sake of comparison, the data of measurements [14] are also presented. It can be seen that our results agree well with these data. The PEM is one of few methods that have a strict foundation, and the foundation technique is universal enough to have been recently used by other authors [15]. Also, the paper of Ochmann [16] appeared that proposed a new method for solving the scattering problems, the full-field method (FFM). The equations of this method are similar to those of the PEM for acoustic scattering. In Ref. [16], the author demonstrates the advantages of the FFM (and, hence, the PEM), as compared to the method of integral equations and the T-matrix method.
ARTICLE IN PRESS A. Kyurkchan, D. Demin / Journal of Quantitative Spectroscopy & Radiative Transfer 89 (2004) 237–255
239
Fig. 1. Cone monostatic RCS in the x–z plane: ka ¼ 3:08; a ¼ 15 .
Fig. 2. Problem layout.
In this paper, we present an extension of the PEM to 3D problems of scattering of electromagnetic waves by a single dielectric body. The limitations on the scatterer’s geometry, which were stated in Ref. [11], are still valid for dielectric bodies.
2. Problem statement Consider an arbitrarily shaped 3D compact object (Fig. 2) bounded by surface S, at which the ~0 impinges. Let the following boundary ~0 ; H primary monochromatic electromagnetic field E condition be met at S: ~0 þ H ~1 ÞjS ¼ ~ ~ i jS ; ~ n ðH nH
0
1
i
~ þE ~ Þ~ ~ ~ ðE njS ¼ E njS ; ð1Þ 0 1 0 1 ~ ;H ~¼H ~ þH ~ are the total external fields, ~¼E ~ þE where ~ n is the outward unit normal to S; E 1 1 ~ ~1 ; H ~ 1 satisfy the ~ and E ; H are the secondary (diffracted) fields. The outer diffracted fields E system of homogeneous Maxwell equations elsewhere outside S (within the domain V e ): 1
1
~ ¼ ik0 zmr H ~ ; rE
1
~ ¼ rH
ik0 ~1 r E : z
ð2Þ
ARTICLE IN PRESS 240
A. Kyurkchan, D. Demin / Journal of Quantitative Spectroscopy & Radiative Transfer 89 (2004) 237–255
~i ; H ~i satisfy the system In the domain V i , the fields E ~ i ¼ ik E ~i ; rH z
~i ¼ ikzH ~i ; rE
pffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffi where k0 ¼ o 0 m0 is the free-space wave number, k ¼ k0 r mr , r and mr are the relative electric pffiffiffiffiffiffiffiffiffiffiffi permittivity and magnetic permeability of volume V i , and z ¼ m0 =0 is the characteristic impedance of the outer medium. Let the outer ðV e Þ and inner ðV i Þ media be homogeneous, linear, and isotropic.
3. Derivation of integral-operator equation According to the PEM standard scheme [4,5], we search for the scattering pattern function, that is, the function that defines the dependence of the diffracted field on angles ðy; jÞ in spherical coordinates ðr; y; jÞ for the far zone (for k0 rb1), where the following asymptotic relations are valid: expð ik0 rÞ ~E 1 expð ik0 rÞ ~H 1 1 1 ~ ~ ; H ¼ : F ðy; jÞ þ O F ðy; jÞ þ O E ¼ r r ðk0 rÞ2 ðk0 rÞ2 ~E ; F ~H are the patterns for electrical and magnetic fields, respectively. Here F Let us briefly consider the idea of the derivation of the PEM integral-operator equation. For the sake of simplicity, we restrict ourselves to the case of a perfectly conducting scatterer. As we have yet mentioned in Introduction, the starting point for further consideration is the following integral representation of the wave field (magnetic one, for instance) [13]: ~1 ¼ 1 H 2pi
Z
Z
2p
p=2þi1
db 0
^H ~ expð ik0 r cos aÞF ðy; j; a; bÞ sin a da;
ð3Þ
0
^H ~ where F is the scattering pattern in the coordinate system rotated so that the OZ axis is directed towards the observation point 2Z
ik ^H ~ ðy; j; a; bÞ ¼ 0 F 4p
~ ~ ðð~ n0 HÞ p^ Þ expðik0~ p^ ~ r0 Þ ds0 : S
In this equation, ~ p ¼ fsin a cos b; sin a sin b; cos ag, ~ p^ ¼ AT~ p, where A is the matrix of the coordinate rotation 1 0 sin j cos j 0 C B A ¼ @ cos j cos y sin j cos y sin y A: cos j sin y sin j sin y cos y
ARTICLE IN PRESS A. Kyurkchan, D. Demin / Journal of Quantitative Spectroscopy & Radiative Transfer 89 (2004) 237–255
241
Integral (3) converges absolutely and uniformly in R3 nB [17] where B is the convex envelope of singularities of the scattered field [5]. Since Z ik20 H ~ ~ q ~ r0 Þ ds0 ; ½ð~ n0 HÞ q expðik0~ F ðx; ZÞ ¼ 4p S ~ q ¼ fsin x cos Z; sin x sin Z; cos xg;
ð4Þ 0
1
~¼H ~ þH ~ , leads to the desirable substituting Eq. (3) into Eq. (4), in view of the fact that H integral-operator equation Z 2p 2 Z 1 ~H ðx; ZÞ ¼ F ~H ðx; ZÞ þ k0 ~ ~ q n F 0 2pi 0 4pi S Z p=2þi1 ^H ~ q ~ rðy; jÞÞ sin a da db ds; (5) expð ik0 r cos aÞF ðy; j; a; bÞ expðik0~ 0
where 2
~H ðx; ZÞ ¼ ik F 0 4p
Z
~0 Þ ~ ½ð~ n0 H q expðik~ q ~ r0 Þ ds0 ; S
r ¼ rðy; jÞ is the equation of surface S in the spherical coordinate system. Further, by using the expansion of the scattering pattern into series in terms of vector angular spherical harmonics that compose the orthogonal basis on the unit sphere and projecting the leftand right-hand parts of Eq. (5) onto the same basis, we reduce the problem to an infinite system of algebraic equations. This method is correct and mathematically justified if B is completely contained inside the scatterer [4,5,18]. For a dielectric scatterer, this scheme is too awkward. Therefore, we use a simpler (though requiring more tricky argumentation) approach for reducing the initial problem to the PEM system of algebraic equations, passing over the stage of integral-operator equation.
4. Reduction of boundary-value problem to system of algebraic equations ~H into series of vector spherical angular harmonics with ~E and F Let us expand functions F unknown factors anm ; bnm (the details can be found in Ref. [19], for instance) ~E ðy; jÞ ¼ F
1 X n X
~m ðy; jÞÞ anm in ð~ ir F n
n¼1 m¼ n
~H ðy; jÞ ¼ F
1 X n X n¼1 m¼ n
1 X n X
~m ðy; jÞ; bnm in zF n
ð6Þ
n¼1 m¼ n
1 X n X 1 ~m ~m ðy; jÞÞ; anm in F ðy; jÞ bnm in ð~ ir F n n z n¼1 m¼ n
ð7Þ
where ~m ðy; jÞ ¼ ~ r rPm F n ðcos yÞ expðimjÞ: n
ð8Þ
ARTICLE IN PRESS 242
A. Kyurkchan, D. Demin / Journal of Quantitative Spectroscopy & Radiative Transfer 89 (2004) 237–255
Here, Pm r is the radius-vector of the observation point, n are the associated Legendre functions, ~ ~ and ir is the unit vector in the spherical coordinate system. Thus, our purpose is to obtain an algebraic system for factors anm ; bnm . ~ 1 and E ~i ; H ~i can be expanded in terms of the vector spherical wave ~1 ; H The wave fields E functions: 1 X n X
~1 ¼ E
fanm ½r r ð~ rcm rcm n Þ ik 0 zbnm ½r ð~ n Þg;
ð9Þ
n¼1 m¼ n
~1 ¼ H
1 X n X ik0 n¼1 m¼ n
~i ¼ E
1 X n X
z
anm ½r
ð~ rcm n Þ
þ bnm ½r r
ð~ rcm n Þ
;
ð10Þ
i fainm ½r r ð~ rwm rwm in Þ ik0 zmr bnm ½r ð~ in Þg;
ð11Þ
n¼1 m¼ n
~i ¼ H
1 X n X ik0 r n¼1 m¼ n
z
ainm ½r
ð~ rwm in Þ
þ
binm ½r
r
ð~ rwm in Þ
;
ð12Þ
where ð2Þ m cm n ¼ hn ðk0 rÞPn ðcos yÞ expðimjÞ;
m wm in ¼ j n ðkrÞPn ðcos yÞ expðimjÞ:
ð13Þ
Here, ainm and binm are the unknown expansion coefficients for the inner field, j n are the spherical Bessel functions, and hð2Þ n are the spherical Hankel functions of the second kind. The additional subscript i (as in Eq. (11), for instance) designates the functions in which the Bessel function appears whose argument depends on the wave number k for the dielectric material occupying volume V i . The starting point of the PEM consists in representing coefficients anm ; bnm and ainm ; binm in terms of the boundary values of the wave field. The expressions required for the scattered and inner fields to be represented in terms of the boundary values at the surface S of the dielectric can be obtained from the Maxwell equations (2) in the following form: Z z 1 e m ~ ~ ~ ½r r ðJ G 0 Þ ½r ðJ G 0 Þ ds0 ; ð14Þ E ¼ S ik0 ~1 ¼ H
Z S
~i ¼ E
1 m e ~ ~ ½r r ðJ G0 Þ þ ½r ðJ G0 Þ ds0 ; ik0 z
Z S
~i ¼ H
z e i m i ~ ~ ½r r ðJ G 0 Þ ½r ðJ G 0 Þ ds0 ; ik0 r
ð15Þ
Z S
1 m i e i ~ ~ ½r r ðJ G 0 Þ þ ½r ðJ G 0 Þ ds0 ; ik0 zmr
ð16Þ
ð17Þ
ARTICLE IN PRESS A. Kyurkchan, D. Demin / Journal of Quantitative Spectroscopy & Radiative Transfer 89 (2004) 237–255
243
where e ~0 þ H ~ 1 ÞjS ¼ ~ ~ i jS ; n ðH nH J~ ¼ ~
m ~0 þ E ~1 Þ ~ ~i ~ J~ ¼ ðE njS ¼ E njS ;
ð18Þ
G0 ð~ r;~ r0 Þ ¼
1 X n expð ik0 j~ r ~ r0 jÞ k0 X ðn mÞ! m c ð~ ¼ ð2n þ 1Þ rÞwm r0 Þ; 0 n ð~ ðn þ mÞ! n 4pi n¼0 m¼ n 4pj~ r ~ rj
ð19Þ
Gi0 ð~ r;~ r0 Þ ¼
1 X n expð ikj~ r ~ r0 jÞ k X ðn mÞ! m 0 0 0 ¼ ð2n þ 1Þ rÞcm w ð~ 0 in ðr ; y ; j Þ: 4pi n¼0 m¼ n ðn þ mÞ! in 4pj~ r ~ rj
ð20Þ
e m In these expressions, J~ ; J~ are the equivalent surface densities of the electric and magnetic r;~ r0 Þ and G i0 ð~ r;~ r0 Þ are the free-space Green functions for a free-space with currents at S; G 0 ð~ r is the position vector of the observation point; ~ r0 parameters of domains V e and V i , respectively; ~ m m is the position vector of a point at S; cin and wn are determined by Eq. (13) after interchanging k and k0 . Now, by using Eqs. (9)–(20), we obtain Z N nm e 0 0 ~m r0 Þ ½r0 ð~ fzJ~ ð~ r Þ ½r0 r0 ð~ r0 w m r0 w m ð21Þ anm ¼ n Þ ik 0 J ð~ n Þg ds ; 4p S
bnm ¼
ainm
N nm 4p
Z
e
S
kN nm ¼ 4pi
m
0 ~ r0 Þ ½r0 r0 ð~ fik0 J~ ð~ r0 Þ ½r0 ð~ r0 w m r0 w m n Þ þ 1=zJ ð~ n Þg ds ;
Z ( pffiffiffiffiffiffiffiffiffiffi z mr =r ~e 0 0 0 0 r Þ ½r0 r0 ~ r 0 cm J ð~ in ðr ; y ; j Þ ik S ) m 0 0 m 0 0 0 0 r Þ ½r ~ r c ðr ; y ; j Þ ds0 ; J~ ð~ in
binm
Z ( kN nm e 0 0 0 0 ¼ r Þ ½r0 ~ r0 cm J~ ð~ in ðr ; y ; j Þ 4pi S ) pffiffiffiffiffiffiffiffiffiffi r =mr ~m 0 0 0 0 0 J ð~ r Þ ½r0 r0 ~ r 0 cm þ in ðr ; y ; j Þ ds ; ikz
N nm ¼
2n þ 1 ðn mÞ! : nðn þ 1Þ ðn þ mÞ!
ð22Þ
(23)
(24)
ARTICLE IN PRESS 244
A. Kyurkchan, D. Demin / Journal of Quantitative Spectroscopy & Radiative Transfer 89 (2004) 237–255
Let us introduce the following notations: ~h ~e ¼ r r ð~ r cm E nm n Þ ¼ H nm ;
~e ¼ r r ð~ E rcm inm in Þ;
1 ~h ~ e ¼ ik0 r ð~ rcm H nm n Þ ¼ 2 E nm ; z z
~e ¼ ik r ð~ H rcm inm in Þ; z
h
h
~ ~ eenm ¼ r r ð~ rwm n Þ ¼ hnm ; e h~nm
~ ~ eeinm ¼ r r ð~ rwm in Þ ¼ hinm ; rffiffiffiffiffi e ik0 1 h ik r 1 r h m ~ ~ enm ; hinm ¼ einm : ¼ r ð~ rwm r ð~ rwn Þ ¼ 2 ~ in Þ ¼ 2 z mr z z z mr
ð25Þ
It follows from Eqs. (9)–(12) and (25) that q 1 X X ~¼E ~0 þ ~e þ bqp E ~h Þ; E ðaqp E qp qp q¼1 p¼ q
~¼H ~0 þ H
q 1 X X
~e þ bqp H ~h Þ; ðaqp H qp qp
q¼1 p¼ q
~i ¼ E
q 1 X X
eeiqp þ bqp~ ehiqp Þ; ðaiqp~
q¼1 p¼ q
~i ¼ H
q 1 X X
e h ðaiqp h~iqp þ bqp h~iqp Þ:
ð26Þ
q¼1 p¼ q
By using Eqs. (21)–(24) and (26), we arrive at the PEM system of the algebraic equations: q 1 P P 32 ainm ¼ a0nm þ ðG31 nm;qp aqp þ G nm;qp bqp Þ; q¼1 p¼ q
binm ¼ b0nm þ anm ¼
q 1 P P
q¼1 p¼ q
q 1 P P
i 14 i ðG13 nm;qp aqp þ G nm;qp bqp Þ;
q¼1 p¼ q
bnm ¼
n ¼ 1; 2; . . . ; jmjpn;
42 ðG41 nm;qp aqp þ G nm;qp bqp Þ;
q 1 P P
n ¼ 1; 2; . . . ; jmjpn;
ð27Þ
i 24 i ðG23 nm;qp aqp þ G nm;qp bqp Þ;
q¼1 p¼ q
where
Z rffiffiffiffiffi mr ~0 Þ E ~e ðr; y; jÞ ðE ~0 ~ ~e ðr; y; jÞ dS; ð~ nH n Þ
H inm inm r S rffiffiffiffiffi Z zN nm 1 r ~0 0 e e ~ ~ ~ ðE ~ ¼ ð~ n H Þ H inm ðr; y; jÞ þ 2 nÞ E inm ðr; y; jÞ dS 4p S mr z
a0nm ¼ b0nm
zN nm 4p
ð28Þ
ð29Þ
ARTICLE IN PRESS A. Kyurkchan, D. Demin / Journal of Quantitative Spectroscopy & Radiative Transfer 89 (2004) 237–255
245
are the right-hand side terms whose forms are determined by the incident wave, Z rffiffiffiffiffi kN nm z mr 31 ~ e Þ r r ð~ ð~ nH r cm G nm;qp ¼ qp in ðr; y; jÞÞ 4pi S ik r e m ~ ðE qp ~ nÞ r ð~ rcin ðr; y; jÞÞ dS;
G32 nm;qp
G 41 nm;qp
G 42 nm;qp
Z rffiffiffiffiffi z mr ~h Þ r r ð~ ð~ nH rcm qp in ðr; y; jÞÞ r S ik h m ~ ðE qp ~ nÞ r ð~ rcin ðr; y; jÞÞ dS;
kN nm ¼ 4pi
(30)
Z kN nm ~e Þ r ð~ ¼ ð~ nH r cm qp in ðr; y; jÞÞ 4pi S rffiffiffiffiffi 1 r ~e m þ nÞ r r ð~ rcin ðr; y; jÞÞ dS; ðE ~ ikz mr qp Z kN nm ~h Þ r ð~ ¼ ð~ nH r cm qp in ðr; y; jÞÞ 4pi S rffiffiffiffiffi 1 mr ~h m þ ðE ~ nÞ r r ð~ rcin ðr; y; jÞÞ dS; ikz r qp
G13 nm;qp
k0 N nm ¼ 4pi
G14 nm;qp
k0 N nm ¼ 4pi
G23 nm;qp
k0 N nm ¼ 4pi
G24 nm;qp
k0 N nm ¼ 4pi
Z S
Z S
Z S
Z S
e z e m m ~ ð~ n hiqp Þ r r ð~ rw n Þ ð~ eiqp ~ nÞ r ð~ rw n Þ dS; ik0 h z h m m ~ ð~ n hiqp Þ r r ð~ rw n Þ ð~ eiqp ~ nÞ r ð~ rw n Þ dS; ik0 e 1 e m m ~ ð~ n hiqp Þ r ð~ rw n Þ þ nÞ r r ð~ rw n Þ dS; ð~ e ~ ik0 z iqp h 1 h m m ~ ð~ n hiqp Þ r ð~ rw n Þ þ nÞ r r ð~ rw n Þ dS ð~ e ~ ik0 z iqp
are the matrix elements of system (27).
ð31Þ
ARTICLE IN PRESS 246
A. Kyurkchan, D. Demin / Journal of Quantitative Spectroscopy & Radiative Transfer 89 (2004) 237–255
When the scatterer is an axially symmetric body, i.e., rðy; jÞ ¼ rðyÞ, the algebraic system (27)–(28) can be significantly simplified and reduced to the form 1 X
anm ¼
i 14 i ðG13 nm;qm aqm þ G nm;qm bqm Þ;
q¼jmj 1 X
bnm ¼
i i 24 ðG23 nm;qm aqm þ G nm;qm bqm Þ;
n ¼ 1; 2; . . . ; jmjpn;
q¼jmj
ainm
¼
a0nm
þ
1 X
32 ðG 31 nm;qm aqm þ G nm;qm bqm Þ;
q¼jmj
binm
¼
b0nm
þ
1 X
42 ðG 41 nm;qm aqm þ G nm;qm bqm Þ;
n ¼ 1; 2; . . . ; jmjpn;
(32)
q¼jmj
where the matrix elements Gjlnm;qm ðj; l ¼ 1; 4Þ are expressed in terms of single integrals.
5. Justification of the reduction method In the numerical implementation of the PEM, a problem emerges to truncate the matrix in system of Eq. (27) (or Eq. (32)). In order to justify the algorithm obtained, it is necessary to asymptotically estimate the matrix elements and right-hand sides of system (27) at large indexes n and q. To do so, the technique similar to that of Ref. [5] can be used. To perform the aforementioned estimations, the asymptotic formulas were used for the Bessel functions with n; qbkr and the Legendre functions with n; qb1. For instance, it can be shown that, with nbq, jG jlnm;qp jpconst
sn1 ; n n!
jG jlnm;qp jpconst
n! ; nðjk=k0 js2 Þn
j ¼ 1; 2; l ¼ 3; 4; j ¼ 3; 4; l ¼ 1; 2;
ð33Þ
ð34Þ
where k0 rðys0 ; j0 Þ s expðisy0 Þ; s1 ¼ max s 2 y ;j ;s
ð35Þ
k0 rðys0 ; j0 Þ s : s2 ¼ min Þ expðisy 0 2 ys0 ;j0 ;s
ð36Þ
0
0
The maximum in Eq. (35) should be searched among the roots of the system r0j ðy; jÞ r0y ðy; jÞ ¼ is; s ¼ 1; ¼ 0; exp½isyjys0 ¼ 0: rðy; jÞ ys0 ;j0 rðy; jÞ ys0 ;j0
ð37Þ
ARTICLE IN PRESS A. Kyurkchan, D. Demin / Journal of Quantitative Spectroscopy & Radiative Transfer 89 (2004) 237–255
247
Upon substituting x ¼ rðy; jÞ expðiyÞ, the roots prove to lie inside contours C j . These contours are the mappings of the section of surface S by the ðj; j þ pÞ plane on the complex plane z ¼ r expðiaÞ [5]. The roots of system (37) are the principal singularities of the diffracted field continued inward the scatterer [17]. If function rðy; jÞ has nonanalytic points, these points must be taken into account in calculating the maximum in Eq. (35). The minimum in Eq. (36) is searched among the roots of system (37), which, after substituting x ¼ rðy; jÞ expðiyÞ, correspond to the points lying outside the aforementioned contours C j on the plane z ¼ r expðiaÞ. Similarly, with qbn for Gjlnm;qp , we obtain jG jlnm;qp jpconst
ðjk=k0 js1 Þq ; q!
jG jlnm;qp jpconst
q! ; sq2
j ¼ 1; 2; l ¼ 3; 4:
ð38Þ
j ¼ 3; 4; l ¼ 1; 2:
ð39Þ
In a similar manner, one can show that, at nb1, ja0nm j; jb0nm jpconst
n! ; nðjk=k0 jsÞn
s ¼ maxðs1 ; s0 Þ;
ð40Þ
where s0 ¼ kr0 =2, and r0 is the distance to the point inside S, which is the farthest one from the ~0 ~ 0 ÞjS ðy; jÞ and ðE coordinate origin and corresponds to the singularities of functions ð~ nH ~ nÞjS ðy; jÞ continued into the domain of complex angles y. When the incident field is a plane wave, s0 ¼ 0, i.e., s ¼ s1 . It follows from estimates (33)–(34) and (38)–(40) that one should use the following substitution for the unknown coefficients in the infinite system (27): anm ¼
sn xnm ; n!
bnm ¼
sn y ; n! nm
ainm ¼
n! xi ; ðjk=k0 jsÞn nm
binm ¼
n! yi : ðjk=k0 jsÞn nm
ð41Þ
As a result, the initial system (27) is transformed to the form xnm ¼
q 1 P P
i 14 i ðg13 nm;qp xqp þ gnm;qp yqp Þ;
q¼1 p¼ q
ynm ¼
n ¼ 1; 2; . . . ; jmjpn;
q 1 P P
i ðg23 nm;qp xqp q¼1 p¼ q
xinm ¼ x0nm þ yinm ¼ y0nm þ
q 1 P P
þ
i g24 nm;qp yqp Þ;
32 ðg31 nm;qp xqp þ gnm;qp yqp Þ;
q¼1 p¼ q q 1 P P
42 ðg41 nm;qp xqp þ gnm;qp yqp Þ;
q¼1 p¼ q
n ¼ 1; 2; . . . ; jmjpn
ð42Þ
ARTICLE IN PRESS 248
A. Kyurkchan, D. Demin / Journal of Quantitative Spectroscopy & Radiative Transfer 89 (2004) 237–255
where ðjk=k0 jsÞn 0 ðjk=k0 jsÞn 0 anm ; y0nm ¼ bnm ; n! n! n!q! ðj ¼ 1; 2; l ¼ 3; 4Þ; gjlnm;qp ¼ G jlnm;qp jk=k0 jq sqþn jk=k0 jn sqþn qþn gjlnm;qm ¼ Gjlnm;qm s ðj ¼ 3; 4; l ¼ 1; 2Þ: q!n! x0nm ¼
(43)
System (42) derived from Eq. (27) can be solved with the use of the reduction method if the condition s2 4s
ð44Þ
is met. This statement is also true for the analogous system for axially symmetric bodies. When the incident field is a plane wave, condition (44) imposes a restriction only on the geometry of the scatterer that must belong to a class of weakly nonconvex bodies [5]. In particular, this class contains all convex bodies.
6. Results of numerical studies Let us consider several examples of solving the scattering problems for the following axially symmetric bodies: sphere (Fig. 3a), spheroid (Fig. 3b), and finite circular cylinder (Fig. 3c). The numerical calculations were carried out on the basis of solving the finite system of form (42), in which the upper limit of summation (the maximal ordinal number of the harmonic) was equal
Fig. 3. Shapes of scatterer: (a) sphere; (b) spheroid; (c) finite circular cylinder.
ARTICLE IN PRESS A. Kyurkchan, D. Demin / Journal of Quantitative Spectroscopy & Radiative Transfer 89 (2004) 237–255
249
to N: xnm ¼
N P
i 14 i ðg13 nm;qm xqm þ gnm;qm yqm Þ;
q¼jmj
ynm ¼
i ðg23 nm;qm xqm q¼jmj
xinm ¼ x0nm þ yinm
¼
n ¼ 1; 2; . . . ; jmjpn;
N P
y0nm
þ
N P
þ
i g24 nm;qm yqm Þ;
32 ðg31 nm;qm xqm þ gnm;qm yqm Þ;
q¼jmj N P
n ¼ 1; 2; . . . ; jmjpn:
ðg41 nm;qm xqm q¼jmj
þ
ð45Þ
g42 nm;qm yqm Þ;
The finite matrix of system (45) can be inverted by using any known method for solving linear systems, the numerical Gaussian algorithm, for instance. Because of the specificity of system (45), it can be solved separately for each fixed m ð NpmpNÞ. Therefore, 2N þ 1 final systems of lower order should be solved. In all examples presented, the incident wave is linearly polarized plane one. The first example illustrates scattering by a homogeneous dielectric sphere with radius a; k0 a ¼ 10, and relative permittivity r ¼ 2:25 (Fig. 3a). Fig. 4 shows the bistatic radar cross-section (RCS) s=pa2 in the ~E j2 . Our half-plane j ¼ 0 (curve 1) and in the half-plane j ¼ p=2 (curve 2), where s ¼ 4pjF numerical result completely coincides with that obtained using the Mie series. At N ¼ 20, i.e. N ¼ k0 d (d is the diameter of the scatterer), nine meaningful characters proved to be correct in the scattering pattern. In our computations, a 2.4-GHz Intel Pentiums 4 PC was used. The computation time was 10–15 s. In Fig. 5, the scattering pattern is shown for a magnetic–dielectric sphere with radius k0 a ¼ 5, relative permittivity r ¼ 1, and relative permeability mr ¼ 104 . Such relative permeability corresponds to ferromagnetics, for instance. For the sake of comparison, the scattering pattern PEM-calculated for perfectly magnetically conducting sphere is also shown in the figure. Curves 1 and 2 in Fig. 5 correspond to the aforementioned magnetic–dielectric sphere at N ¼ 22, curves 3 and 4 correspond to the perfectly magnetically conducting sphere at N ¼ 10. These results well agree with the exact solution yielded by the Mie series.
Fig. 4. Bistatic RCS of a sphere with k0 a ¼ 10 and r ¼ 2:25.
ARTICLE IN PRESS 250
A. Kyurkchan, D. Demin / Journal of Quantitative Spectroscopy & Radiative Transfer 89 (2004) 237–255
Fig. 5. The scattering pattern for a magnetic–dielectric sphere with radius k0 a ¼ 5, relative permittivity r ¼ 1, relative permeability mr ¼ 104 and perfectly magnetically conducting sphere.
Fig. 6. Bistatic RCS of a spheroid with k0 a ¼ 5, k0 c ¼ 7, and r ¼ 2:25.
As the next example, a prolate spheroid is considered with k0 a ¼ 5, k0 c ¼ 7, and r ¼ 2:25 (Fig. 3b). Fig. 6 shows the bistatic RCS s=l2 of the spheroid in the half-plane j ¼ 0 (curve 1) and in the half-plane j ¼ p=2 (curve 2). Our result completely coincides with that of Ref. [20]. Note that four correct meaningful decimal digits were established in the scattering pattern at N k0 d. Now, let us consider modeling the scattering characteristics of absorbing or black bodies [21]. In Ref. [22], this problem was analyzed in the impedance approximation. As a physical model of the black body, a metallic body was used, with the dielectric coating of thickness t that is small relative to the body’s size. The permittivity and permeability of the body were [21] ¼ ia0 ;
m ¼ iam0 ;
where a ! 1 (the model of Zommerfeld) [23]. In the impedance approximation, the scattering characteristics of the dielectric-coated body are modeled by solving the diffraction problem for a body of the same geometrical properties with the impedance boundary condition met at its surface
ARTICLE IN PRESS A. Kyurkchan, D. Demin / Journal of Quantitative Spectroscopy & Radiative Transfer 89 (2004) 237–255
251
[11,22]. The impedance is given by the formula rffiffiffi m Z¼i tg kt: For the Zommerfeld model, Z ffi z. Here, we consider the problem of diffraction by an absorbing body in its strict statement. In the framework of such approach, the boundary-value problem is to be solved for a magnetic–dielectric body whose material parameters, and m, are complex and provide a substantial absorption of the energy of the incident electromagnetic wave. Figs. 7–9 show the scattering patterns for a sphere with ka ¼ 4, a prolate (along the z-axis) spheroid with the semiaxes kc ¼ 4, and ka ¼ 2, and a circular cylinder with the height kh ¼ 8 (along the z-axis) and the base radius ka ¼ 2, respectively. The plane electromagnetic wave is incident along the x-axis, with the vector of the electric field oriented along the z-axis. The relative permittivity and permeability of these bodies were specified to be equal: r ¼ mr ¼ 1 4i. For the sake of comparison, the figures also show the scattering patterns for the corresponding black bodies (according to Zommerfeld). These patterns were calculated in the impedance approximation at Z ¼ z. Both curves well coincide for the three bodies, such coincidence confirming the validity of the impedance approach for calculating the scattering characteristics of well-absorbing bodies. For black bodies, the Ufimtsev [24] theorem exists that argues that the integral scattering crosssection of a black body is strictly two times lower than that of a perfectly conducting body having the same shadow contour, that is, the boundary between the illuminated and shadowed parts of the body. This statement is valid for all convex bodies whose linear dimensions and minimal curvature radius are much larger than the wavelength. The integral scattering cross-section can be calculated as follows: Z Z 1 2p p ~E jF ðy; jÞj2 sin y dy dj: PS ¼ 2z 0 0 In Table 1, the values of the integral scattering cross-section are summarized that were obtained for the aforementioned bodies, namely: the sphere ðka ¼ 4Þ, the spheroid ðkc ¼ 4; ka ¼ 2Þ, and the cylinder ðkh ¼ 8; ka ¼ 2Þ with different physical parameters. In the table, PS2 is the value of the sphere ka = 4 12
Ζ =ζ
10
ε r = µ r = 1 − 4i
| FθE |
8 6 4 2 0 0
30
60
90
120
150
180
, degrees Fig. 7. The scattering patterns for a sphere with ka ¼ 4; r ¼ mr ¼ 1 4i.
ARTICLE IN PRESS A. Kyurkchan, D. Demin / Journal of Quantitative Spectroscopy & Radiative Transfer 89 (2004) 237–255
252
spheroid: kc = 4, ka = 2 6 Ζ = ζ 5
ε r = µ r = 1 − 4i
| F θE |
4 3 2 1 0 0
60
120
180
240
30 0
360
, degrees Fig. 8. The scattering patterns for a spheroid with the semiaxes kc ¼ 4, and ka ¼ 2; r ¼ mr ¼ 1 4i.
cylinder: kh = 8, ka = 2
8
Ζ = ζ
7
ε r = µ r = 1 − 4i
6
| FθE |
5 4 3 2 1 0 0
60
120
180
240
30 0
360
, degrees Fig. 9. The scattering patterns for a circular cylinder with the height kh ¼ 8 and the base radius ka ¼ 2; r ¼ mr ¼ 1 4i.
Table 1 The values of the integral scattering cross-section for the sphere ðka ¼ 4Þ, the spheroid ðkc ¼ 4; ka ¼ 2Þ, and the cylinder ðkh ¼ 8; ka ¼ 2Þ Parameters of the body
Sphere
Spheroid
Cylinder
Z¼0
PS ¼ 0:14264 PS2 ¼ 0:14263 PS ¼ 0:07877 PS ¼ 0:07809 PS ¼ 0:05827
PS ¼ 0:09319 PS2 ¼ 0:09319 PS ¼ 0:03530 PS ¼ 0:03412 PS ¼ 0:03057
PS ¼ 0:10537 PS2 ¼ 0:10845 PS ¼ 0:04981 PS ¼ 0:04423 PS ¼ 0:04141
Z¼z r ¼ mr ¼ 1 4i Black body (the McDonald model)
integral scattering cross-section calculated according to the optical theorem that states PS ¼ PS2 ; where PS2 is a quantity proportional to the imaginary part of the scattering pattern in the direction of incidence of the initial plane wave. For the perpendicular incidence of the plane wave
ARTICLE IN PRESS A. Kyurkchan, D. Demin / Journal of Quantitative Spectroscopy & Radiative Transfer 89 (2004) 237–255
253
Fig. 10. The ratios of the integral scattering cross-section of the perfectly conducting body to that of the black body.
and the polarization specified [25], l PS2 ¼ ImfF Ey ðy ¼ 90 ; j ¼ 0 Þg: z The last row of the table contains the values of PS for the black body of the McDonald model [21] for which the scattering pattern is equal to the sum of the scattering patterns of perfectly electrically and magnetically conducting bodies with the same geometrical properties. According to the table, the ratio of the integral cross-section of the perfectly conducting body to that of the absorbing body only approximately follows the Ufimtsev theorem for the bodies of sizes at hand. It is advantageous to consider the dependence of this ratio on the body’s size in more detail. Fig. 10 shows the ratios of the integral scattering cross-section of the perfectly conducting body to that of the black body for the prolate spheroid (curve 1) and cylinder (curve ~0 vector oriented 2). The plane wave is normally incident at the bodies along the x-axis, with the E along the z-axis. The minor semiaxis of the prolate spheroids, ka, varies from 0.05 to 12, with a ratio of 1:2 for its two semiaxes. The sizes of the cylinders were specified so that the spheroids could be inscribed into the cylinders. The plots show that the agreement with the Ufimtsev theorem is good for large sizes of the bodies ðka45Þ.
7. Conclusions From the performed calculations of the scattering characteristics of dielectric bodies, a conclusion can be drawn that the PEM allows one to effectively solve the diffraction problems. In all examples considered, high accuracy and convergence rate of the PEM were demonstrated for both dielectric scatterers with analytical boundary and those having a piecewisesmooth boundary. This conclusion completely coincides with that of Refs. [11,12] for perfectly conducting and impedance scatterers. Thus, the PEM can be used to solve various scattering problems for bodies of complicated shapes and different physical nature, with an acceptable computational accuracy and expanses. A possibility is studied to model the scattering characteristics of perfectly absorbing (black) bodies. The Ufimtsev theorem on the ratio of the
ARTICLE IN PRESS 254
A. Kyurkchan, D. Demin / Journal of Quantitative Spectroscopy & Radiative Transfer 89 (2004) 237–255
scattering cross-sections of perfectly reflecting and black bodies is numerically tested, and its validity limits are established.
Acknowledgements This work was supported by Russian Foundation for Basic Research, Project no. 03-02-16336.
References [1] Colton D, Kress R. Integral equation methods in scattering theory. New York: Wiley-Interscience; 1983. [2] Mishchenko MI, Travis LD, Mackowski DW. T-matrix computations of light scattering by nonspherical particles: a review. JQSRT 1996;55:16–21. [3] Wriedt T, editor. Generalized multipole techniques for electromagnetic and light scattering. Amsterdam: Elsevier; 1999. [4] Kyurkchan AG. A new integral equation in the diffraction theory. Soviet Phys Dokl 1992;37:338–40. [5] Kyurkchan AG. On a method of solution to the problem of wave diffraction by finite-size scatterers. Phys Dokl 1994;39:546–9. [6] Kyurkchan AG. On solving the problem of waves scattering by several bodies. Dokl Math 1996;53:453–61. [7] Kyurkchan AG, Manenkov SA. A new method for solving the problem of diffraction by a compact obstacle in a layered medium. Radiophys Quantum Electron 1998;41:589–99. [8] Kyurkchan AG, Manenkov SA. Wave diffraction by the group of bodies located in a layered medium. In: . Vigo, Spain Proceedings of the Fourth Conference on Electromagnetic and Light Scattering by Nonspherical Part. Theory and Applications; 1999. p. 111–9. [9] Kyurkchan AG, Soloveichik AL. Scattering of waves by a periodic grating situated close to a planar interface. J Com Tech Electron 2000;45:357–64. [10] Kyurkchan AG, Manenkov SA, Kleev AI, Sukov AI. Pattern equation method for solution of electromagnetic scattering problems. J Appl Electromagn 1999;2:17–31. [11] Kyurkchan AG, Demin DB. Electromagnetic wave diffraction from impedance scatterers with piecewise-smooth boundaries. J Com Tech Electron 2002;47:856–63. [12] Kyurkchan AG, Demin DB. On the application of a pattern equation method to solve the 3-d electromagnetic scattering problem by an impedance body with a piecewise smooth surface. In: . Gainesville, USA Proceedings of the Sixth Conference on Electromagnetic and Light Scattering by Nonspherical Part. Theory, Measurements and Applications; 2002. p. 183–6. [13] Kyurkchan AG. Obtaining of solutions of the mathematical physics equations by their reduction to the ordinary differential equations with operator coefficients. Math Dokl 1992;45:54–9. [14] Bechtel ME. Application of geometric diffraction theory to scattering from cones and disks. Proc IEEE 1965;53:877–82. [15] Farafonov VG, Il’in VB, Prokopjeva MS. Light scattering by multilayered nonspherical particles: a set of methods. JQSRT 2003;79–90:599–626. [16] Ochmann M. The full-field equations for acoustic radiation and scattering. J Acoust Soc Am 1999;105:2574–84. [17] Kyurkchan AG, Sternin BYu, Shatalov VE. Singularities of continuation of wave fields. Phys Uspekhi 1996;39:1221–42. [18] Do Dyk Tkhang, Kyurkchan AG. Efficient method for solving the problems of wave diffraction by scatterers with broken boundaries. Acoust Phys 2003;49:51–8. [19] Papas CH. Theory of electromagnetic wave propagation. New York: McGraw-Hill; 1965. [20] Doicu A, Wriedt T. Extended boundary condition method with multipole sources located in the complex plane. Optics Comm 1997;139:85–91. [21] Zaharyev LN, Lemansky AA. Wave scattering by ‘‘black’’ bodies. Moscow: Sovetskoye Radio; 1972.
ARTICLE IN PRESS A. Kyurkchan, D. Demin / Journal of Quantitative Spectroscopy & Radiative Transfer 89 (2004) 237–255
255
[22] Kyurkchan AG, Demin DB. Modeling of wave scattering characteristics from bodies with absorbing cover and ‘‘black’’ bodies (Russian). J Tech Phys 2004;74:24–31. [23] Sommerfeld A. Optik. New York: Academic Press; 1950. [24] Ufimtsev PYa. Three lectures on physical diffraction theory. Leningrad: Leningrad University; 1972. [25] Bohren CF, Huffman DR. Absorption and scattering of light by small particles. New York: Wiley-Interscience; 1983.
Journal of Quantitative Spectroscopy & Radiative Transfer 89 (2004) 237–255 www.elsevier.com/locate/jqsrt
Pattern equation method for solving problems of diffraction of electromagnetic waves by axially symmetric dielectric scatterers Alexander Kyurkchan, Dmitrii Demin Moscow Technical University of Communication and Informatics, Aviamotornaya Street 8A, 111024 Moscow, Russia
Abstract A new efficient method is proposed for solving 3D problems of diffraction of electromagnetic waves by a single dielectric body of arbitrary shape. The method offers a high rate of convergence. The method is applied to the problem of scattering by axially symmetrical bodies, and the data are presented that illustrate the convergence rate of the computation algorithm for bodies of different shapes. A possibility is studied to model the scattering characteristics for perfectly absorbing (black) bodies. The Ufimtsev theorem on the ratio of the cross-sections of perfectly reflecting and black bodies is numerically tested. r 2004 Elsevier Ltd. All rights reserved. Keywords: Diffraction; Scattering pattern; Scatterer; Integral-operator equation; Spheroid; Cylinder; Black body
1. Introduction The problem of diffraction of electromagnetic waves by 3D dielectric objects of arbitrary shapes is rather relevant and remains insufficiently studied because of its complexity. In recent years, the method of volume integral [1], the T-matrix method [2], and the method of discrete sources [3] have been commonly used for solving this problem. However, a successful implementation of the methods strongly depends on the geometrical properties of the scatterer. With all these methods, among which the first and third ones are quite universal, the starting boundary-value problem is reduced to a system of linear algebraic equations, with the order of about 50C200ðd=lÞ2 or more Corresponding author. Tel.: +7-95-236-2267.
E-mail address: [email protected], [email protected] (A. Kyurkchan). 0022-4073/$ - see front matter r 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.jqsrt.2004.05.025
ARTICLE IN PRESS 238
A. Kyurkchan, D. Demin / Journal of Quantitative Spectroscopy & Radiative Transfer 89 (2004) 237–255
(d is the characteristic size of the scatterer). Such number of the equations to be solved means great calculation expenses. So, a high urgency exists in developing efficient methods to solve the scattering problems for objects with complicated geometrical characteristics. Here, we consider a new universal method, the pattern equation method (PEM), for solving the diffraction and scattering problems. This method was first presented in Refs. [4,5] and was later applied to a number of diffraction and scattering problems, such as wave scattering by a single object and a group of objects located in homogeneous and planar-stratified media [5–8], wave diffraction by multi-row discrete periodic arrays near plane interface of the media [9], wave propagation in optic fibers with noncircular cross-sections [10], etc. The PEM was also successfully applied to the problems of diffraction of electromagnetic waves by 3D impedance objects [11,12]. The PEM is based on reducing the initial boundary problem for the Helmholtz equation to an integral-operator equation of the second kind with respect to the scattering pattern (spectral function of the wave field). In this paper, we use the generalized Sommerfeld–Weil representation for the diffracted field [13] in the form of integral of plane waves. Such representation converges elsewhere outside the convex envelope of singularities in the continuation of the wave field. Then, the desired function is represented as an expansion onto some basis, and this expansion is substituted into the integral-operator equation. So, the obtained equality is projected onto some basis, generally another one. With certain restrictions on the geometry of the problem, which can be strictly established, the resultant infinite system of algebraic equations is solved by the reduction method, i.e. the method of truncation. The PEM has been shown to offer a number of important advantages in comparison with other methods used in solving the diffraction and scattering problems. Thus, in solving the problem of scattering by 3D impedance bodies with piecewise-smooth boundaries, it has been established that the rate of the algorithm convergence is mainly governed by the scatterer size and weakly depends on its geometry [11,12]. The reason is that, when a wave scattering problem is solved using the PEM, the desired quantity is the scattering pattern itself, i.e., a functional of the field distribution over the scatterer’s surface. Therefore, the integration smoothes rapidly oscillating field components whose approximation implies high-order harmonics to be accounted for. As a result, the body of computations required for a given accuracy to be attained proves to be entirely reduced. Fig. 1 shows the PEM-calculated monostatic radar cross-section of a circular cone with the base radius ka ¼ 3:08 and the vertical half-angle a ¼ 15 . The cross-section is presented in the plane of incidence ðj0 ¼ 0 Þ, versus the incidence angle y of a plane wave (y ¼ 0 corresponds to a case when the plane wave is incident along the z-axis, from the cone vertex). For the sake of comparison, the data of measurements [14] are also presented. It can be seen that our results agree well with these data. The PEM is one of few methods that have a strict foundation, and the foundation technique is universal enough to have been recently used by other authors [15]. Also, the paper of Ochmann [16] appeared that proposed a new method for solving the scattering problems, the full-field method (FFM). The equations of this method are similar to those of the PEM for acoustic scattering. In Ref. [16], the author demonstrates the advantages of the FFM (and, hence, the PEM), as compared to the method of integral equations and the T-matrix method.
ARTICLE IN PRESS A. Kyurkchan, D. Demin / Journal of Quantitative Spectroscopy & Radiative Transfer 89 (2004) 237–255
239
Fig. 1. Cone monostatic RCS in the x–z plane: ka ¼ 3:08; a ¼ 15 .
Fig. 2. Problem layout.
In this paper, we present an extension of the PEM to 3D problems of scattering of electromagnetic waves by a single dielectric body. The limitations on the scatterer’s geometry, which were stated in Ref. [11], are still valid for dielectric bodies.
2. Problem statement Consider an arbitrarily shaped 3D compact object (Fig. 2) bounded by surface S, at which the ~0 impinges. Let the following boundary ~0 ; H primary monochromatic electromagnetic field E condition be met at S: ~0 þ H ~1 ÞjS ¼ ~ ~ i jS ; ~ n ðH nH
0
1
i
~ þE ~ Þ~ ~ ~ ðE njS ¼ E njS ; ð1Þ 0 1 0 1 ~ ;H ~¼H ~ þH ~ are the total external fields, ~¼E ~ þE where ~ n is the outward unit normal to S; E 1 1 ~ ~1 ; H ~ 1 satisfy the ~ and E ; H are the secondary (diffracted) fields. The outer diffracted fields E system of homogeneous Maxwell equations elsewhere outside S (within the domain V e ): 1
1
~ ¼ ik0 zmr H ~ ; rE
1
~ ¼ rH
ik0 ~1 r E : z
ð2Þ
ARTICLE IN PRESS 240
A. Kyurkchan, D. Demin / Journal of Quantitative Spectroscopy & Radiative Transfer 89 (2004) 237–255
~i ; H ~i satisfy the system In the domain V i , the fields E ~ i ¼ ik E ~i ; rH z
~i ¼ ikzH ~i ; rE
pffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffi where k0 ¼ o 0 m0 is the free-space wave number, k ¼ k0 r mr , r and mr are the relative electric pffiffiffiffiffiffiffiffiffiffiffi permittivity and magnetic permeability of volume V i , and z ¼ m0 =0 is the characteristic impedance of the outer medium. Let the outer ðV e Þ and inner ðV i Þ media be homogeneous, linear, and isotropic.
3. Derivation of integral-operator equation According to the PEM standard scheme [4,5], we search for the scattering pattern function, that is, the function that defines the dependence of the diffracted field on angles ðy; jÞ in spherical coordinates ðr; y; jÞ for the far zone (for k0 rb1), where the following asymptotic relations are valid: expð ik0 rÞ ~E 1 expð ik0 rÞ ~H 1 1 1 ~ ~ ; H ¼ : F ðy; jÞ þ O F ðy; jÞ þ O E ¼ r r ðk0 rÞ2 ðk0 rÞ2 ~E ; F ~H are the patterns for electrical and magnetic fields, respectively. Here F Let us briefly consider the idea of the derivation of the PEM integral-operator equation. For the sake of simplicity, we restrict ourselves to the case of a perfectly conducting scatterer. As we have yet mentioned in Introduction, the starting point for further consideration is the following integral representation of the wave field (magnetic one, for instance) [13]: ~1 ¼ 1 H 2pi
Z
Z
2p
p=2þi1
db 0
^H ~ expð ik0 r cos aÞF ðy; j; a; bÞ sin a da;
ð3Þ
0
^H ~ where F is the scattering pattern in the coordinate system rotated so that the OZ axis is directed towards the observation point 2Z
ik ^H ~ ðy; j; a; bÞ ¼ 0 F 4p
~ ~ ðð~ n0 HÞ p^ Þ expðik0~ p^ ~ r0 Þ ds0 : S
In this equation, ~ p ¼ fsin a cos b; sin a sin b; cos ag, ~ p^ ¼ AT~ p, where A is the matrix of the coordinate rotation 1 0 sin j cos j 0 C B A ¼ @ cos j cos y sin j cos y sin y A: cos j sin y sin j sin y cos y
ARTICLE IN PRESS A. Kyurkchan, D. Demin / Journal of Quantitative Spectroscopy & Radiative Transfer 89 (2004) 237–255
241
Integral (3) converges absolutely and uniformly in R3 nB [17] where B is the convex envelope of singularities of the scattered field [5]. Since Z ik20 H ~ ~ q ~ r0 Þ ds0 ; ½ð~ n0 HÞ q expðik0~ F ðx; ZÞ ¼ 4p S ~ q ¼ fsin x cos Z; sin x sin Z; cos xg;
ð4Þ 0
1
~¼H ~ þH ~ , leads to the desirable substituting Eq. (3) into Eq. (4), in view of the fact that H integral-operator equation Z 2p 2 Z 1 ~H ðx; ZÞ ¼ F ~H ðx; ZÞ þ k0 ~ ~ q n F 0 2pi 0 4pi S Z p=2þi1 ^H ~ q ~ rðy; jÞÞ sin a da db ds; (5) expð ik0 r cos aÞF ðy; j; a; bÞ expðik0~ 0
where 2
~H ðx; ZÞ ¼ ik F 0 4p
Z
~0 Þ ~ ½ð~ n0 H q expðik~ q ~ r0 Þ ds0 ; S
r ¼ rðy; jÞ is the equation of surface S in the spherical coordinate system. Further, by using the expansion of the scattering pattern into series in terms of vector angular spherical harmonics that compose the orthogonal basis on the unit sphere and projecting the leftand right-hand parts of Eq. (5) onto the same basis, we reduce the problem to an infinite system of algebraic equations. This method is correct and mathematically justified if B is completely contained inside the scatterer [4,5,18]. For a dielectric scatterer, this scheme is too awkward. Therefore, we use a simpler (though requiring more tricky argumentation) approach for reducing the initial problem to the PEM system of algebraic equations, passing over the stage of integral-operator equation.
4. Reduction of boundary-value problem to system of algebraic equations ~H into series of vector spherical angular harmonics with ~E and F Let us expand functions F unknown factors anm ; bnm (the details can be found in Ref. [19], for instance) ~E ðy; jÞ ¼ F
1 X n X
~m ðy; jÞÞ anm in ð~ ir F n
n¼1 m¼ n
~H ðy; jÞ ¼ F
1 X n X n¼1 m¼ n
1 X n X
~m ðy; jÞ; bnm in zF n
ð6Þ
n¼1 m¼ n
1 X n X 1 ~m ~m ðy; jÞÞ; anm in F ðy; jÞ bnm in ð~ ir F n n z n¼1 m¼ n
ð7Þ
where ~m ðy; jÞ ¼ ~ r rPm F n ðcos yÞ expðimjÞ: n
ð8Þ
ARTICLE IN PRESS 242
A. Kyurkchan, D. Demin / Journal of Quantitative Spectroscopy & Radiative Transfer 89 (2004) 237–255
Here, Pm r is the radius-vector of the observation point, n are the associated Legendre functions, ~ ~ and ir is the unit vector in the spherical coordinate system. Thus, our purpose is to obtain an algebraic system for factors anm ; bnm . ~ 1 and E ~i ; H ~i can be expanded in terms of the vector spherical wave ~1 ; H The wave fields E functions: 1 X n X
~1 ¼ E
fanm ½r r ð~ rcm rcm n Þ ik 0 zbnm ½r ð~ n Þg;
ð9Þ
n¼1 m¼ n
~1 ¼ H
1 X n X ik0 n¼1 m¼ n
~i ¼ E
1 X n X
z
anm ½r
ð~ rcm n Þ
þ bnm ½r r
ð~ rcm n Þ
;
ð10Þ
i fainm ½r r ð~ rwm rwm in Þ ik0 zmr bnm ½r ð~ in Þg;
ð11Þ
n¼1 m¼ n
~i ¼ H
1 X n X ik0 r n¼1 m¼ n
z
ainm ½r
ð~ rwm in Þ
þ
binm ½r
r
ð~ rwm in Þ
;
ð12Þ
where ð2Þ m cm n ¼ hn ðk0 rÞPn ðcos yÞ expðimjÞ;
m wm in ¼ j n ðkrÞPn ðcos yÞ expðimjÞ:
ð13Þ
Here, ainm and binm are the unknown expansion coefficients for the inner field, j n are the spherical Bessel functions, and hð2Þ n are the spherical Hankel functions of the second kind. The additional subscript i (as in Eq. (11), for instance) designates the functions in which the Bessel function appears whose argument depends on the wave number k for the dielectric material occupying volume V i . The starting point of the PEM consists in representing coefficients anm ; bnm and ainm ; binm in terms of the boundary values of the wave field. The expressions required for the scattered and inner fields to be represented in terms of the boundary values at the surface S of the dielectric can be obtained from the Maxwell equations (2) in the following form: Z z 1 e m ~ ~ ~ ½r r ðJ G 0 Þ ½r ðJ G 0 Þ ds0 ; ð14Þ E ¼ S ik0 ~1 ¼ H
Z S
~i ¼ E
1 m e ~ ~ ½r r ðJ G0 Þ þ ½r ðJ G0 Þ ds0 ; ik0 z
Z S
~i ¼ H
z e i m i ~ ~ ½r r ðJ G 0 Þ ½r ðJ G 0 Þ ds0 ; ik0 r
ð15Þ
Z S
1 m i e i ~ ~ ½r r ðJ G 0 Þ þ ½r ðJ G 0 Þ ds0 ; ik0 zmr
ð16Þ
ð17Þ
ARTICLE IN PRESS A. Kyurkchan, D. Demin / Journal of Quantitative Spectroscopy & Radiative Transfer 89 (2004) 237–255
243
where e ~0 þ H ~ 1 ÞjS ¼ ~ ~ i jS ; n ðH nH J~ ¼ ~
m ~0 þ E ~1 Þ ~ ~i ~ J~ ¼ ðE njS ¼ E njS ;
ð18Þ
G0 ð~ r;~ r0 Þ ¼
1 X n expð ik0 j~ r ~ r0 jÞ k0 X ðn mÞ! m c ð~ ¼ ð2n þ 1Þ rÞwm r0 Þ; 0 n ð~ ðn þ mÞ! n 4pi n¼0 m¼ n 4pj~ r ~ rj
ð19Þ
Gi0 ð~ r;~ r0 Þ ¼
1 X n expð ikj~ r ~ r0 jÞ k X ðn mÞ! m 0 0 0 ¼ ð2n þ 1Þ rÞcm w ð~ 0 in ðr ; y ; j Þ: 4pi n¼0 m¼ n ðn þ mÞ! in 4pj~ r ~ rj
ð20Þ
e m In these expressions, J~ ; J~ are the equivalent surface densities of the electric and magnetic r;~ r0 Þ and G i0 ð~ r;~ r0 Þ are the free-space Green functions for a free-space with currents at S; G 0 ð~ r is the position vector of the observation point; ~ r0 parameters of domains V e and V i , respectively; ~ m m is the position vector of a point at S; cin and wn are determined by Eq. (13) after interchanging k and k0 . Now, by using Eqs. (9)–(20), we obtain Z N nm e 0 0 ~m r0 Þ ½r0 ð~ fzJ~ ð~ r Þ ½r0 r0 ð~ r0 w m r0 w m ð21Þ anm ¼ n Þ ik 0 J ð~ n Þg ds ; 4p S
bnm ¼
ainm
N nm 4p
Z
e
S
kN nm ¼ 4pi
m
0 ~ r0 Þ ½r0 r0 ð~ fik0 J~ ð~ r0 Þ ½r0 ð~ r0 w m r0 w m n Þ þ 1=zJ ð~ n Þg ds ;
Z ( pffiffiffiffiffiffiffiffiffiffi z mr =r ~e 0 0 0 0 r Þ ½r0 r0 ~ r 0 cm J ð~ in ðr ; y ; j Þ ik S ) m 0 0 m 0 0 0 0 r Þ ½r ~ r c ðr ; y ; j Þ ds0 ; J~ ð~ in
binm
Z ( kN nm e 0 0 0 0 ¼ r Þ ½r0 ~ r0 cm J~ ð~ in ðr ; y ; j Þ 4pi S ) pffiffiffiffiffiffiffiffiffiffi r =mr ~m 0 0 0 0 0 J ð~ r Þ ½r0 r0 ~ r 0 cm þ in ðr ; y ; j Þ ds ; ikz
N nm ¼
2n þ 1 ðn mÞ! : nðn þ 1Þ ðn þ mÞ!
ð22Þ
(23)
(24)
ARTICLE IN PRESS 244
A. Kyurkchan, D. Demin / Journal of Quantitative Spectroscopy & Radiative Transfer 89 (2004) 237–255
Let us introduce the following notations: ~h ~e ¼ r r ð~ r cm E nm n Þ ¼ H nm ;
~e ¼ r r ð~ E rcm inm in Þ;
1 ~h ~ e ¼ ik0 r ð~ rcm H nm n Þ ¼ 2 E nm ; z z
~e ¼ ik r ð~ H rcm inm in Þ; z
h
h
~ ~ eenm ¼ r r ð~ rwm n Þ ¼ hnm ; e h~nm
~ ~ eeinm ¼ r r ð~ rwm in Þ ¼ hinm ; rffiffiffiffiffi e ik0 1 h ik r 1 r h m ~ ~ enm ; hinm ¼ einm : ¼ r ð~ rwm r ð~ rwn Þ ¼ 2 ~ in Þ ¼ 2 z mr z z z mr
ð25Þ
It follows from Eqs. (9)–(12) and (25) that q 1 X X ~¼E ~0 þ ~e þ bqp E ~h Þ; E ðaqp E qp qp q¼1 p¼ q
~¼H ~0 þ H
q 1 X X
~e þ bqp H ~h Þ; ðaqp H qp qp
q¼1 p¼ q
~i ¼ E
q 1 X X
eeiqp þ bqp~ ehiqp Þ; ðaiqp~
q¼1 p¼ q
~i ¼ H
q 1 X X
e h ðaiqp h~iqp þ bqp h~iqp Þ:
ð26Þ
q¼1 p¼ q
By using Eqs. (21)–(24) and (26), we arrive at the PEM system of the algebraic equations: q 1 P P 32 ainm ¼ a0nm þ ðG31 nm;qp aqp þ G nm;qp bqp Þ; q¼1 p¼ q
binm ¼ b0nm þ anm ¼
q 1 P P
q¼1 p¼ q
q 1 P P
i 14 i ðG13 nm;qp aqp þ G nm;qp bqp Þ;
q¼1 p¼ q
bnm ¼
n ¼ 1; 2; . . . ; jmjpn;
42 ðG41 nm;qp aqp þ G nm;qp bqp Þ;
q 1 P P
n ¼ 1; 2; . . . ; jmjpn;
ð27Þ
i 24 i ðG23 nm;qp aqp þ G nm;qp bqp Þ;
q¼1 p¼ q
where
Z rffiffiffiffiffi mr ~0 Þ E ~e ðr; y; jÞ ðE ~0 ~ ~e ðr; y; jÞ dS; ð~ nH n Þ
H inm inm r S rffiffiffiffiffi Z zN nm 1 r ~0 0 e e ~ ~ ~ ðE ~ ¼ ð~ n H Þ H inm ðr; y; jÞ þ 2 nÞ E inm ðr; y; jÞ dS 4p S mr z
a0nm ¼ b0nm
zN nm 4p
ð28Þ
ð29Þ
ARTICLE IN PRESS A. Kyurkchan, D. Demin / Journal of Quantitative Spectroscopy & Radiative Transfer 89 (2004) 237–255
245
are the right-hand side terms whose forms are determined by the incident wave, Z rffiffiffiffiffi kN nm z mr 31 ~ e Þ r r ð~ ð~ nH r cm G nm;qp ¼ qp in ðr; y; jÞÞ 4pi S ik r e m ~ ðE qp ~ nÞ r ð~ rcin ðr; y; jÞÞ dS;
G32 nm;qp
G 41 nm;qp
G 42 nm;qp
Z rffiffiffiffiffi z mr ~h Þ r r ð~ ð~ nH rcm qp in ðr; y; jÞÞ r S ik h m ~ ðE qp ~ nÞ r ð~ rcin ðr; y; jÞÞ dS;
kN nm ¼ 4pi
(30)
Z kN nm ~e Þ r ð~ ¼ ð~ nH r cm qp in ðr; y; jÞÞ 4pi S rffiffiffiffiffi 1 r ~e m þ nÞ r r ð~ rcin ðr; y; jÞÞ dS; ðE ~ ikz mr qp Z kN nm ~h Þ r ð~ ¼ ð~ nH r cm qp in ðr; y; jÞÞ 4pi S rffiffiffiffiffi 1 mr ~h m þ ðE ~ nÞ r r ð~ rcin ðr; y; jÞÞ dS; ikz r qp
G13 nm;qp
k0 N nm ¼ 4pi
G14 nm;qp
k0 N nm ¼ 4pi
G23 nm;qp
k0 N nm ¼ 4pi
G24 nm;qp
k0 N nm ¼ 4pi
Z S
Z S
Z S
Z S
e z e m m ~ ð~ n hiqp Þ r r ð~ rw n Þ ð~ eiqp ~ nÞ r ð~ rw n Þ dS; ik0 h z h m m ~ ð~ n hiqp Þ r r ð~ rw n Þ ð~ eiqp ~ nÞ r ð~ rw n Þ dS; ik0 e 1 e m m ~ ð~ n hiqp Þ r ð~ rw n Þ þ nÞ r r ð~ rw n Þ dS; ð~ e ~ ik0 z iqp h 1 h m m ~ ð~ n hiqp Þ r ð~ rw n Þ þ nÞ r r ð~ rw n Þ dS ð~ e ~ ik0 z iqp
are the matrix elements of system (27).
ð31Þ
ARTICLE IN PRESS 246
A. Kyurkchan, D. Demin / Journal of Quantitative Spectroscopy & Radiative Transfer 89 (2004) 237–255
When the scatterer is an axially symmetric body, i.e., rðy; jÞ ¼ rðyÞ, the algebraic system (27)–(28) can be significantly simplified and reduced to the form 1 X
anm ¼
i 14 i ðG13 nm;qm aqm þ G nm;qm bqm Þ;
q¼jmj 1 X
bnm ¼
i i 24 ðG23 nm;qm aqm þ G nm;qm bqm Þ;
n ¼ 1; 2; . . . ; jmjpn;
q¼jmj
ainm
¼
a0nm
þ
1 X
32 ðG 31 nm;qm aqm þ G nm;qm bqm Þ;
q¼jmj
binm
¼
b0nm
þ
1 X
42 ðG 41 nm;qm aqm þ G nm;qm bqm Þ;
n ¼ 1; 2; . . . ; jmjpn;
(32)
q¼jmj
where the matrix elements Gjlnm;qm ðj; l ¼ 1; 4Þ are expressed in terms of single integrals.
5. Justification of the reduction method In the numerical implementation of the PEM, a problem emerges to truncate the matrix in system of Eq. (27) (or Eq. (32)). In order to justify the algorithm obtained, it is necessary to asymptotically estimate the matrix elements and right-hand sides of system (27) at large indexes n and q. To do so, the technique similar to that of Ref. [5] can be used. To perform the aforementioned estimations, the asymptotic formulas were used for the Bessel functions with n; qbkr and the Legendre functions with n; qb1. For instance, it can be shown that, with nbq, jG jlnm;qp jpconst
sn1 ; n n!
jG jlnm;qp jpconst
n! ; nðjk=k0 js2 Þn
j ¼ 1; 2; l ¼ 3; 4; j ¼ 3; 4; l ¼ 1; 2;
ð33Þ
ð34Þ
where k0 rðys0 ; j0 Þ s expðisy0 Þ; s1 ¼ max s 2 y ;j ;s
ð35Þ
k0 rðys0 ; j0 Þ s : s2 ¼ min Þ expðisy 0 2 ys0 ;j0 ;s
ð36Þ
0
0
The maximum in Eq. (35) should be searched among the roots of the system r0j ðy; jÞ r0y ðy; jÞ ¼ is; s ¼ 1; ¼ 0; exp½isyjys0 ¼ 0: rðy; jÞ ys0 ;j0 rðy; jÞ ys0 ;j0
ð37Þ
ARTICLE IN PRESS A. Kyurkchan, D. Demin / Journal of Quantitative Spectroscopy & Radiative Transfer 89 (2004) 237–255
247
Upon substituting x ¼ rðy; jÞ expðiyÞ, the roots prove to lie inside contours C j . These contours are the mappings of the section of surface S by the ðj; j þ pÞ plane on the complex plane z ¼ r expðiaÞ [5]. The roots of system (37) are the principal singularities of the diffracted field continued inward the scatterer [17]. If function rðy; jÞ has nonanalytic points, these points must be taken into account in calculating the maximum in Eq. (35). The minimum in Eq. (36) is searched among the roots of system (37), which, after substituting x ¼ rðy; jÞ expðiyÞ, correspond to the points lying outside the aforementioned contours C j on the plane z ¼ r expðiaÞ. Similarly, with qbn for Gjlnm;qp , we obtain jG jlnm;qp jpconst
ðjk=k0 js1 Þq ; q!
jG jlnm;qp jpconst
q! ; sq2
j ¼ 1; 2; l ¼ 3; 4:
ð38Þ
j ¼ 3; 4; l ¼ 1; 2:
ð39Þ
In a similar manner, one can show that, at nb1, ja0nm j; jb0nm jpconst
n! ; nðjk=k0 jsÞn
s ¼ maxðs1 ; s0 Þ;
ð40Þ
where s0 ¼ kr0 =2, and r0 is the distance to the point inside S, which is the farthest one from the ~0 ~ 0 ÞjS ðy; jÞ and ðE coordinate origin and corresponds to the singularities of functions ð~ nH ~ nÞjS ðy; jÞ continued into the domain of complex angles y. When the incident field is a plane wave, s0 ¼ 0, i.e., s ¼ s1 . It follows from estimates (33)–(34) and (38)–(40) that one should use the following substitution for the unknown coefficients in the infinite system (27): anm ¼
sn xnm ; n!
bnm ¼
sn y ; n! nm
ainm ¼
n! xi ; ðjk=k0 jsÞn nm
binm ¼
n! yi : ðjk=k0 jsÞn nm
ð41Þ
As a result, the initial system (27) is transformed to the form xnm ¼
q 1 P P
i 14 i ðg13 nm;qp xqp þ gnm;qp yqp Þ;
q¼1 p¼ q
ynm ¼
n ¼ 1; 2; . . . ; jmjpn;
q 1 P P
i ðg23 nm;qp xqp q¼1 p¼ q
xinm ¼ x0nm þ yinm ¼ y0nm þ
q 1 P P
þ
i g24 nm;qp yqp Þ;
32 ðg31 nm;qp xqp þ gnm;qp yqp Þ;
q¼1 p¼ q q 1 P P
42 ðg41 nm;qp xqp þ gnm;qp yqp Þ;
q¼1 p¼ q
n ¼ 1; 2; . . . ; jmjpn
ð42Þ
ARTICLE IN PRESS 248
A. Kyurkchan, D. Demin / Journal of Quantitative Spectroscopy & Radiative Transfer 89 (2004) 237–255
where ðjk=k0 jsÞn 0 ðjk=k0 jsÞn 0 anm ; y0nm ¼ bnm ; n! n! n!q! ðj ¼ 1; 2; l ¼ 3; 4Þ; gjlnm;qp ¼ G jlnm;qp jk=k0 jq sqþn jk=k0 jn sqþn qþn gjlnm;qm ¼ Gjlnm;qm s ðj ¼ 3; 4; l ¼ 1; 2Þ: q!n! x0nm ¼
(43)
System (42) derived from Eq. (27) can be solved with the use of the reduction method if the condition s2 4s
ð44Þ
is met. This statement is also true for the analogous system for axially symmetric bodies. When the incident field is a plane wave, condition (44) imposes a restriction only on the geometry of the scatterer that must belong to a class of weakly nonconvex bodies [5]. In particular, this class contains all convex bodies.
6. Results of numerical studies Let us consider several examples of solving the scattering problems for the following axially symmetric bodies: sphere (Fig. 3a), spheroid (Fig. 3b), and finite circular cylinder (Fig. 3c). The numerical calculations were carried out on the basis of solving the finite system of form (42), in which the upper limit of summation (the maximal ordinal number of the harmonic) was equal
Fig. 3. Shapes of scatterer: (a) sphere; (b) spheroid; (c) finite circular cylinder.
ARTICLE IN PRESS A. Kyurkchan, D. Demin / Journal of Quantitative Spectroscopy & Radiative Transfer 89 (2004) 237–255
249
to N: xnm ¼
N P
i 14 i ðg13 nm;qm xqm þ gnm;qm yqm Þ;
q¼jmj
ynm ¼
i ðg23 nm;qm xqm q¼jmj
xinm ¼ x0nm þ yinm
¼
n ¼ 1; 2; . . . ; jmjpn;
N P
y0nm
þ
N P
þ
i g24 nm;qm yqm Þ;
32 ðg31 nm;qm xqm þ gnm;qm yqm Þ;
q¼jmj N P
n ¼ 1; 2; . . . ; jmjpn:
ðg41 nm;qm xqm q¼jmj
þ
ð45Þ
g42 nm;qm yqm Þ;
The finite matrix of system (45) can be inverted by using any known method for solving linear systems, the numerical Gaussian algorithm, for instance. Because of the specificity of system (45), it can be solved separately for each fixed m ð NpmpNÞ. Therefore, 2N þ 1 final systems of lower order should be solved. In all examples presented, the incident wave is linearly polarized plane one. The first example illustrates scattering by a homogeneous dielectric sphere with radius a; k0 a ¼ 10, and relative permittivity r ¼ 2:25 (Fig. 3a). Fig. 4 shows the bistatic radar cross-section (RCS) s=pa2 in the ~E j2 . Our half-plane j ¼ 0 (curve 1) and in the half-plane j ¼ p=2 (curve 2), where s ¼ 4pjF numerical result completely coincides with that obtained using the Mie series. At N ¼ 20, i.e. N ¼ k0 d (d is the diameter of the scatterer), nine meaningful characters proved to be correct in the scattering pattern. In our computations, a 2.4-GHz Intel Pentiums 4 PC was used. The computation time was 10–15 s. In Fig. 5, the scattering pattern is shown for a magnetic–dielectric sphere with radius k0 a ¼ 5, relative permittivity r ¼ 1, and relative permeability mr ¼ 104 . Such relative permeability corresponds to ferromagnetics, for instance. For the sake of comparison, the scattering pattern PEM-calculated for perfectly magnetically conducting sphere is also shown in the figure. Curves 1 and 2 in Fig. 5 correspond to the aforementioned magnetic–dielectric sphere at N ¼ 22, curves 3 and 4 correspond to the perfectly magnetically conducting sphere at N ¼ 10. These results well agree with the exact solution yielded by the Mie series.
Fig. 4. Bistatic RCS of a sphere with k0 a ¼ 10 and r ¼ 2:25.
ARTICLE IN PRESS 250
A. Kyurkchan, D. Demin / Journal of Quantitative Spectroscopy & Radiative Transfer 89 (2004) 237–255
Fig. 5. The scattering pattern for a magnetic–dielectric sphere with radius k0 a ¼ 5, relative permittivity r ¼ 1, relative permeability mr ¼ 104 and perfectly magnetically conducting sphere.
Fig. 6. Bistatic RCS of a spheroid with k0 a ¼ 5, k0 c ¼ 7, and r ¼ 2:25.
As the next example, a prolate spheroid is considered with k0 a ¼ 5, k0 c ¼ 7, and r ¼ 2:25 (Fig. 3b). Fig. 6 shows the bistatic RCS s=l2 of the spheroid in the half-plane j ¼ 0 (curve 1) and in the half-plane j ¼ p=2 (curve 2). Our result completely coincides with that of Ref. [20]. Note that four correct meaningful decimal digits were established in the scattering pattern at N k0 d. Now, let us consider modeling the scattering characteristics of absorbing or black bodies [21]. In Ref. [22], this problem was analyzed in the impedance approximation. As a physical model of the black body, a metallic body was used, with the dielectric coating of thickness t that is small relative to the body’s size. The permittivity and permeability of the body were [21] ¼ ia0 ;
m ¼ iam0 ;
where a ! 1 (the model of Zommerfeld) [23]. In the impedance approximation, the scattering characteristics of the dielectric-coated body are modeled by solving the diffraction problem for a body of the same geometrical properties with the impedance boundary condition met at its surface
ARTICLE IN PRESS A. Kyurkchan, D. Demin / Journal of Quantitative Spectroscopy & Radiative Transfer 89 (2004) 237–255
251
[11,22]. The impedance is given by the formula rffiffiffi m Z¼i tg kt: For the Zommerfeld model, Z ffi z. Here, we consider the problem of diffraction by an absorbing body in its strict statement. In the framework of such approach, the boundary-value problem is to be solved for a magnetic–dielectric body whose material parameters, and m, are complex and provide a substantial absorption of the energy of the incident electromagnetic wave. Figs. 7–9 show the scattering patterns for a sphere with ka ¼ 4, a prolate (along the z-axis) spheroid with the semiaxes kc ¼ 4, and ka ¼ 2, and a circular cylinder with the height kh ¼ 8 (along the z-axis) and the base radius ka ¼ 2, respectively. The plane electromagnetic wave is incident along the x-axis, with the vector of the electric field oriented along the z-axis. The relative permittivity and permeability of these bodies were specified to be equal: r ¼ mr ¼ 1 4i. For the sake of comparison, the figures also show the scattering patterns for the corresponding black bodies (according to Zommerfeld). These patterns were calculated in the impedance approximation at Z ¼ z. Both curves well coincide for the three bodies, such coincidence confirming the validity of the impedance approach for calculating the scattering characteristics of well-absorbing bodies. For black bodies, the Ufimtsev [24] theorem exists that argues that the integral scattering crosssection of a black body is strictly two times lower than that of a perfectly conducting body having the same shadow contour, that is, the boundary between the illuminated and shadowed parts of the body. This statement is valid for all convex bodies whose linear dimensions and minimal curvature radius are much larger than the wavelength. The integral scattering cross-section can be calculated as follows: Z Z 1 2p p ~E jF ðy; jÞj2 sin y dy dj: PS ¼ 2z 0 0 In Table 1, the values of the integral scattering cross-section are summarized that were obtained for the aforementioned bodies, namely: the sphere ðka ¼ 4Þ, the spheroid ðkc ¼ 4; ka ¼ 2Þ, and the cylinder ðkh ¼ 8; ka ¼ 2Þ with different physical parameters. In the table, PS2 is the value of the sphere ka = 4 12
Ζ =ζ
10
ε r = µ r = 1 − 4i
| FθE |
8 6 4 2 0 0
30
60
90
120
150
180
, degrees Fig. 7. The scattering patterns for a sphere with ka ¼ 4; r ¼ mr ¼ 1 4i.
ARTICLE IN PRESS A. Kyurkchan, D. Demin / Journal of Quantitative Spectroscopy & Radiative Transfer 89 (2004) 237–255
252
spheroid: kc = 4, ka = 2 6 Ζ = ζ 5
ε r = µ r = 1 − 4i
| F θE |
4 3 2 1 0 0
60
120
180
240
30 0
360
, degrees Fig. 8. The scattering patterns for a spheroid with the semiaxes kc ¼ 4, and ka ¼ 2; r ¼ mr ¼ 1 4i.
cylinder: kh = 8, ka = 2
8
Ζ = ζ
7
ε r = µ r = 1 − 4i
6
| FθE |
5 4 3 2 1 0 0
60
120
180
240
30 0
360
, degrees Fig. 9. The scattering patterns for a circular cylinder with the height kh ¼ 8 and the base radius ka ¼ 2; r ¼ mr ¼ 1 4i.
Table 1 The values of the integral scattering cross-section for the sphere ðka ¼ 4Þ, the spheroid ðkc ¼ 4; ka ¼ 2Þ, and the cylinder ðkh ¼ 8; ka ¼ 2Þ Parameters of the body
Sphere
Spheroid
Cylinder
Z¼0
PS ¼ 0:14264 PS2 ¼ 0:14263 PS ¼ 0:07877 PS ¼ 0:07809 PS ¼ 0:05827
PS ¼ 0:09319 PS2 ¼ 0:09319 PS ¼ 0:03530 PS ¼ 0:03412 PS ¼ 0:03057
PS ¼ 0:10537 PS2 ¼ 0:10845 PS ¼ 0:04981 PS ¼ 0:04423 PS ¼ 0:04141
Z¼z r ¼ mr ¼ 1 4i Black body (the McDonald model)
integral scattering cross-section calculated according to the optical theorem that states PS ¼ PS2 ; where PS2 is a quantity proportional to the imaginary part of the scattering pattern in the direction of incidence of the initial plane wave. For the perpendicular incidence of the plane wave
ARTICLE IN PRESS A. Kyurkchan, D. Demin / Journal of Quantitative Spectroscopy & Radiative Transfer 89 (2004) 237–255
253
Fig. 10. The ratios of the integral scattering cross-section of the perfectly conducting body to that of the black body.
and the polarization specified [25], l PS2 ¼ ImfF Ey ðy ¼ 90 ; j ¼ 0 Þg: z The last row of the table contains the values of PS for the black body of the McDonald model [21] for which the scattering pattern is equal to the sum of the scattering patterns of perfectly electrically and magnetically conducting bodies with the same geometrical properties. According to the table, the ratio of the integral cross-section of the perfectly conducting body to that of the absorbing body only approximately follows the Ufimtsev theorem for the bodies of sizes at hand. It is advantageous to consider the dependence of this ratio on the body’s size in more detail. Fig. 10 shows the ratios of the integral scattering cross-section of the perfectly conducting body to that of the black body for the prolate spheroid (curve 1) and cylinder (curve ~0 vector oriented 2). The plane wave is normally incident at the bodies along the x-axis, with the E along the z-axis. The minor semiaxis of the prolate spheroids, ka, varies from 0.05 to 12, with a ratio of 1:2 for its two semiaxes. The sizes of the cylinders were specified so that the spheroids could be inscribed into the cylinders. The plots show that the agreement with the Ufimtsev theorem is good for large sizes of the bodies ðka45Þ.
7. Conclusions From the performed calculations of the scattering characteristics of dielectric bodies, a conclusion can be drawn that the PEM allows one to effectively solve the diffraction problems. In all examples considered, high accuracy and convergence rate of the PEM were demonstrated for both dielectric scatterers with analytical boundary and those having a piecewisesmooth boundary. This conclusion completely coincides with that of Refs. [11,12] for perfectly conducting and impedance scatterers. Thus, the PEM can be used to solve various scattering problems for bodies of complicated shapes and different physical nature, with an acceptable computational accuracy and expanses. A possibility is studied to model the scattering characteristics of perfectly absorbing (black) bodies. The Ufimtsev theorem on the ratio of the
ARTICLE IN PRESS 254
A. Kyurkchan, D. Demin / Journal of Quantitative Spectroscopy & Radiative Transfer 89 (2004) 237–255
scattering cross-sections of perfectly reflecting and black bodies is numerically tested, and its validity limits are established.
Acknowledgements This work was supported by Russian Foundation for Basic Research, Project no. 03-02-16336.
References [1] Colton D, Kress R. Integral equation methods in scattering theory. New York: Wiley-Interscience; 1983. [2] Mishchenko MI, Travis LD, Mackowski DW. T-matrix computations of light scattering by nonspherical particles: a review. JQSRT 1996;55:16–21. [3] Wriedt T, editor. Generalized multipole techniques for electromagnetic and light scattering. Amsterdam: Elsevier; 1999. [4] Kyurkchan AG. A new integral equation in the diffraction theory. Soviet Phys Dokl 1992;37:338–40. [5] Kyurkchan AG. On a method of solution to the problem of wave diffraction by finite-size scatterers. Phys Dokl 1994;39:546–9. [6] Kyurkchan AG. On solving the problem of waves scattering by several bodies. Dokl Math 1996;53:453–61. [7] Kyurkchan AG, Manenkov SA. A new method for solving the problem of diffraction by a compact obstacle in a layered medium. Radiophys Quantum Electron 1998;41:589–99. [8] Kyurkchan AG, Manenkov SA. Wave diffraction by the group of bodies located in a layered medium. In: . Vigo, Spain Proceedings of the Fourth Conference on Electromagnetic and Light Scattering by Nonspherical Part. Theory and Applications; 1999. p. 111–9. [9] Kyurkchan AG, Soloveichik AL. Scattering of waves by a periodic grating situated close to a planar interface. J Com Tech Electron 2000;45:357–64. [10] Kyurkchan AG, Manenkov SA, Kleev AI, Sukov AI. Pattern equation method for solution of electromagnetic scattering problems. J Appl Electromagn 1999;2:17–31. [11] Kyurkchan AG, Demin DB. Electromagnetic wave diffraction from impedance scatterers with piecewise-smooth boundaries. J Com Tech Electron 2002;47:856–63. [12] Kyurkchan AG, Demin DB. On the application of a pattern equation method to solve the 3-d electromagnetic scattering problem by an impedance body with a piecewise smooth surface. In: . Gainesville, USA Proceedings of the Sixth Conference on Electromagnetic and Light Scattering by Nonspherical Part. Theory, Measurements and Applications; 2002. p. 183–6. [13] Kyurkchan AG. Obtaining of solutions of the mathematical physics equations by their reduction to the ordinary differential equations with operator coefficients. Math Dokl 1992;45:54–9. [14] Bechtel ME. Application of geometric diffraction theory to scattering from cones and disks. Proc IEEE 1965;53:877–82. [15] Farafonov VG, Il’in VB, Prokopjeva MS. Light scattering by multilayered nonspherical particles: a set of methods. JQSRT 2003;79–90:599–626. [16] Ochmann M. The full-field equations for acoustic radiation and scattering. J Acoust Soc Am 1999;105:2574–84. [17] Kyurkchan AG, Sternin BYu, Shatalov VE. Singularities of continuation of wave fields. Phys Uspekhi 1996;39:1221–42. [18] Do Dyk Tkhang, Kyurkchan AG. Efficient method for solving the problems of wave diffraction by scatterers with broken boundaries. Acoust Phys 2003;49:51–8. [19] Papas CH. Theory of electromagnetic wave propagation. New York: McGraw-Hill; 1965. [20] Doicu A, Wriedt T. Extended boundary condition method with multipole sources located in the complex plane. Optics Comm 1997;139:85–91. [21] Zaharyev LN, Lemansky AA. Wave scattering by ‘‘black’’ bodies. Moscow: Sovetskoye Radio; 1972.
ARTICLE IN PRESS A. Kyurkchan, D. Demin / Journal of Quantitative Spectroscopy & Radiative Transfer 89 (2004) 237–255
255
[22] Kyurkchan AG, Demin DB. Modeling of wave scattering characteristics from bodies with absorbing cover and ‘‘black’’ bodies (Russian). J Tech Phys 2004;74:24–31. [23] Sommerfeld A. Optik. New York: Academic Press; 1950. [24] Ufimtsev PYa. Three lectures on physical diffraction theory. Leningrad: Leningrad University; 1972. [25] Bohren CF, Huffman DR. Absorption and scattering of light by small particles. New York: Wiley-Interscience; 1983.