JOURNAL OF MATHEMATICAL PHYSICS
VOLUME 43, NUMBER 11
NOVEMBER 2002
Scattering relations for point sources:
Acoustic and electromagnetic waves
C. Athanasiadis
Department of Mathematics, University of Athens,
Panepistimiopolis, GR 157 84 Athens, Greece
P. A. Martina)
Department of Mathematical and Computer Sciences, Colorado School of Mines,
Golden, Colorado 80401-1887
A. Spyropoulos and I. G. Stratis
Department of Mathematics, University of Athens,
Panepistimiopolis, GR 157 84 Athens, Greece
~Received 25 March 2002; accepted 20 May 2002!
The problem of scattering of spherical waves by a bounded obstacle is considered.
General scattering theorems are proved. These relate the far-field patterns due to
scattering of waves from a point source put in any two different locations. The
scatterer can have any of the usual properties, penetrable or impenetrable. The
optical theorem is recovered as a corollary. Mixed scattering relations are also
established, relating the scattered fields due to a point source and a plane wave.
© 2002 American Institute of Physics. @DOI: 10.1063/1.1509089#
I. INTRODUCTION
In classical scattering theory, there are some general results that connect the solutions of two
related problems. The most familiar of these is reciprocity: the scattered field at A due to a source
at B is simply related to the scattered field at B due to a source at A.
There are also internal relations within a single problem. A well-known example is the optical
theorem for scattering of plane waves: it relates the far-field pattern in the forward direction to a
certain integral of the far-field pattern over all directions.
In this article, we derive some general relations for scattering of waves emanating from point
sources. Thus, we relate one problem with a point source at A to a similar problem with a point
source at B. By setting A5B and then letting A recede to infinity, we recover the optical theorem.
If we keep A fixed and let B recede to infinity, we obtain so-called mixed scattering theorems,
relating plane-wave incidence to point-source incidence. An example of these is the mixed reciprocity theorem, which has found much use recently in methods for solving inverse scattering
problems.1
As Logan2 points out, Clebsch considered the scattering of elastic waves from a point source
by a rigid sphere 140 years ago, a decade before Lord Rayleigh published his solution for the
scattering of a plane sound wave by a sphere. Collected results for scattering of point-source fields
by simple shapes are given in Ref. 3. More recently, Dassios and his co-workers have studied
incident waves generated by a point source in the vicinity of a scatterer; see, for example, Refs.
4 – 6. There is also some recent work on near-field inverse problems: in addition to the previous
papers, we note the work by Coyle7 and Potthast,1 as well as recent work by three of the present
authors.8,9 The revival of interest in problems related to point-generated wave fields has several
reasons. One is due to the variety of applications coming from the theory of composite materials
and of acoustic emission, from the theoretical analysis of biological studies at the cell level, from
nondestructive testing and evaluation, from geophysics, from modeling in medicine and health
a!
Author to whom correspondence should be addressed. Electronic mail: pamartin@mines.edu
0022-2488/2002/43(11)/5683/15/$19.00
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© 2002 American Institute of Physics
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J. Math. Phys., Vol. 43, No. 11, November 2002
sciences, and from scattering problems connected to environmental data analysis. Another reason
is due to the fact that a point-source field is more easily realizable in a laboratory.
We give results for both acoustic and electromagnetic waves, and we permit all the usual
kinds of scattering obstacles, penetrable and impenetrable. Extensions to elastic-wave problems
are expected.
II. FORMULATION: ACOUSTICS
Let V 2 be a bounded three-dimensional obstacle with a smooth closed boundary S ~the
scatterer!. The exterior R3 \V 2 5V of the scatterer is an infinite homogeneous isotropic lossless
acoustic medium, that is the compressional viscosity d is zero, and with mass density r, phase
velocity c, mean compressibility g and real wave number k5 v Ag r , v being the angular frequency. The interior of the scatterer V 2 is filled with a lossy medium, in general, with corresponding physical parameters d 2 , r 2 , c 2 and g 2 . We consider an incident spherical acoustic
wave due to a source located at a point with position vector a. Suppressing the harmonic time
dependence exp$2iv t % , and following the normalization introduced by Dassios and Kamvyssas,4
we assume the following form for the incident field:
u ia ~ r! 5ae2ika
eik u r2au
,
u r2au
~1!
rÞa,
where a5 u au . We note that when a→`, the spherical wave reduces to a plane wave with direction of propagation 2â, where a5aâ. 4 The total field u ta in the exterior of the scatterer is given
by
u ta ~ r! 5u ia ~ r! 1u sa ~ r! ,
rPV \ $ a% ,
~2!
where u sa is the scattered acoustic field, and solves the Helmholtz equation
¹ 2 u ta ~ r! 1k 2 u ta ~ r! 50,
~3!
rPV.
The scattered as well as the incident fields are solutions of Helmholtz’s equation that satisfy
the Sommerfeld radiation condition
r̂•¹u ~ r! 2iku ~ r! 5o ~ r 21 ! ,
~4!
r→`,
uniformly in all directions r̂PS 2 , where S 2 is the unit sphere. We note that u ta also satisfies the
Sommerfeld radiation condition.
In our analysis, we can permit impenetrable or penetrable scatterers. In the former case, we
could have Dirichlet (u ta 50), Neumann ( ] u ta / ] n50) or Robin conditions on S; the Robin ~or
impedance! condition is
S
D
]
1ikl u ta ~ r! 50,
]n
~5!
rPS,
where l is a dimensionless real parameter. In the penetrable case, the incident wave is transmitted
2
into the scatterer; let u 2
a be the total acoustic field in V . Then the following transmission
conditions must hold on the scatterer’s surface
u ta ~ r! 5u 2
a ~ r ! and
] u ta ~ r!
]u2
a ~ r!
5b
,
]n
]n
rPS,
~6!
where the constant b 5( r / r 2 )(12ikc d 2 g 2 ) is complex for a lossy scatterer and real for a
lossless scatterer. More details on the physical parameters of the above problems can be found in
2
Ref. 6. The field u 2
a solves the Helmholtz equation in V ,
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J. Math. Phys., Vol. 43, No. 11, November 2002
Scattering relations for point sources
2 2 2
¹ 2u 2
a ~ r! 1 h k u a ~ r! ,
rPV 2 ,
5685
~7!
where h 5(c/c 2 )(12ikc g 2 d 2 ) 21/2 is the ~complex! index of refraction between the two media
in V and V 2 . The choice of the branch of the square root is such that Im(h)>0 and hence
Im(hk)>0.
In addition, in order to cover various cases that arise in applications, we can easily extend our
analysis to include an impenetrable core in the interior of the scatterer.
As it is well known, all the above problems are well posed.
The behavior of the scattered wave in the far field is given by
u sa ~ r! 5g a ~ r̂! h 0 ~ kr ! 1O ~ r 22 ! ,
r→`,
~8!
where h 0 (x)5eix /(ix) is the spherical Hankel function of the first kind and order zero. Moreover,
g a ~ r̂! 52
ik
4p
E F] ]
u sa ~ r8 !
S
n
G
1ik ~ r̂"n̂! u sa ~ r8 ! e2ikr̂"r8 ds ~ r8 !
~9!
is the far-field pattern.5
III. GENERAL SCATTERING THEOREM
In what follows, we consider two locations for the point source, a and b, from which the
time-harmonic incident spherical waves emanate. Each source generates a corresponding scattered
field, u sa and u sb , respectively. We are interested in relations between these fields.
Let S r denote a large sphere of radius r, surrounding the points a and b, and let
S a,« 5 $ rPR3 : u a2ru 5« % ,
~10!
a small sphere of radius «, surrounding the point a. Then, we introduce the following notation,
@ u, v# S5
ES
S
ū
D
]v
] ū
2v
ds,
]n
]n
where the overbar denotes complex conjugation; in particular, we write @ u, v# [ @ u, v# S .
Lemma 1: Let u ia be a point source at a. Let u ib be a point source at b, with corresponding
scattered field u sb and far-field pattern g b . Then
lim @ u ia ,u sb # S r 52aeika
r→`
E
S2
g b ~ r̂! eikr̂"ads ~ r̂!
~11!
and
lim @ u ia ,u sb # S a,« 54 p aeika u sb ~ a! ,
~12!
«→0
where S r is a large sphere of radius r surrounding a and b, and S a,« is the small sphere defined
by Eq. (10).
Proof: For Eq. ~11!, we use the asymptotic relations
u r2au 5r2r̂"a1O ~ r 21 !
and
u r2au 21 5r 21 1O ~ r 22 ! ,
~13!
as r→`, and obtain
u ia ~ r! 5h 0 ~ kr ! g ia ~ r̂! 1O ~ r 22 ! ,
r→`,
~14!
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5686
Athanasiadis et al.
J. Math. Phys., Vol. 43, No. 11, November 2002
where g ia (r̂)5ika exp$2ika(11â"r̂) % is the far-field pattern of the point-source incident wave.
Using Eqs. ~8! and ~14! gives Eq. ~11!.
For Eq. ~12!, we evaluate u ia and its normal derivative on S a,« . There, we have u ia
5(a/«)eik(a2«) and ( ] / ] n)u ia 52(ik1« 21 )u ia (r), where ] / ] n denotes normal differentiation in
the outward direction. Using the mean value theorem and letting «→0, Eq. ~12! is proved. h
Now, for two point sources, with position vectors a and b, we define a spherical far-field
pattern generator for spherical acoustic waves by
F
G b ~ a! 5ikaeika u sb ~ a! 2
1
2p
E
S2
G
g b ~ r̂! eika"r̂ds ~ r̂! .
~15!
This terminology and definition are appropriate for the following reason @see Eq. ~50! later in this
work#. When both the point source and the observation point recede to infinity, G b (a) reduces to
the far-field pattern for an incident plane wave propagating in the direction 2b̂. Using this
notation, the general scattering theorem for spherical waves is formulated as follows.
Theorem 2: For any two point-source locations in V, a and b, we have
G b ~ a! 1G a ~ b! 1
1
2p
E
S2
g b ~ r̂! g a ~ r̂! ds ~ r̂! 5Ea,b ,
~16!
where
Ea,b 52
ik
@ u t ,u t # .
4p a b
~17!
The value of Ea,b depends on the scatterer:
Ea,b 50 for Dirichlet or Neumann conditions on S;
Ea,b 52
k 2l
2p
E
S
u tb ~ r! u ta ~ r! ds for the Robin condition (5) on S;
~18!
~19!
or
Ea,b 52
k
Im~ b !
2p
E
V2
2
¹u 2
a ~ r ! •¹u b ~ r ! d v~ r ! for a penetrable scatterer,
~20!
where b is the constant in the transmission conditions (6).
Proof: Let us first evaluate Ea,b directly. For Dirichlet (u t50) or Neumann ( ] u t/ ] n50)
conditions, we immediately obtain Eq. ~18!. Similarly, the Robin condition ~5! gives Eq. ~19!. For
a penetrable scatterer, we use the transmission conditions ~6!, apply Green’s first theorem and take
into account that Im(bh2)50 ~see p. 9 of Ref. 6!; this gives Eq. ~20!.
Next, we give an alternative evaluation of Ea,b . Thus, by the relations u at 5u ai 1u as for a
5a,b we have
@ u ta ,u tb # 5 @ u ia ,u ib # 1 @ u sa ,u ib # 1 @ u ia ,u sb # 1 @ u sa ,u sb # .
~21!
Since u ia and u ib are regular solutions of the Helmholtz equation in V 2 , Green’s second theorem
gives
@ u ia ,u ib # 50.
~22!
For the other terms in Eq. ~21!, we consider two small spheres, S a,« 1 and S b,« 2 , centered at a and
b with radii « 1 and « 2 , respectively, with S a,« 1 ùS b,« 2 5B, as well as a large sphere S R centered
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J. Math. Phys., Vol. 43, No. 11, November 2002
Scattering relations for point sources
5687
at the origin surrounding the whole system of the scatterer and the two small spheres. Since u ia (r)
and u sb (r) are solutions of the Helmholtz equation, for rÞa,b, Green’s second theorem gives
@ u ia ,u sb # 5 @ u ia ,u sb # S R 2 @ u ia ,u sb # S a,« 2 @ u ia ,u sb # S b,« .
1
2
The third term is zero because u ia and u sb are regular solutions of the Helmholtz equation in the
interior of S b,« 2 . Then, letting R→` and « 1 →0, using Lemma 1, we obtain
@ u ia ,u sb # 524 p aeika u sb ~ a! 12aeika
E
g b ~ r̂! eikr̂"ads ~ r̂! .
~23!
E
g a ~ r̂! e2ikr̂"bds ~ r̂! .
~24!
S2
As @ u sa ,u ib # 52 @ u ib ,u sa # , we easily deduce that
@ u sa ,u ib # 54 p be2ikb u sa ~ b! 22be2ikb
S2
Finally, in view of the regularity of u sa and u sb in the region exterior to S, we have
@ u sa ,u sb # 5 @ u sa ,u sb # S R .
Then, letting R→`, we pass to the radiation zone and thus we can use the asymptotic form ~8!,
giving
@ u sa ,u sb # 5
2i
k
E
S2
g a ~ r̂! g b ~ r̂! ds ~ r̂! .
~25!
Substituting Eqs. ~22!–~25! in Eq. ~21!, making use of Eq. ~17! and its evaluation, gives Eq. ~16!,
and the theorem is proved.
h
Let us make two remarks. First, if the scatterer is a lossless penetrable obstacle, that is, the
physical parameter b is real, then we obtain Ea,b 50 in Eq. ~16!, just as for soft or hard scatterers.
Second, suppose that we have a penetrable scatterer with a core S 2 on which u 2 50 or
2
] u / ] n50. Then, the relation ~20! still holds, where, now, V 2 denotes the part of the scatterer
between the surfaces S and S 2 .
IV. RECIPROCITY RELATIONS
The proof of Theorem 2 uses two evaluations of @ u ta ,u tb # . If, instead, we start from @ u ta ,u tb # ,
we obtain a reciprocity theorem. This is not surprising, because u ta can be regarded as the exact
Green’s function for the scattering problem. The reciprocity theorem can be found on p. 48 of Ref.
6, for example. We quote it here, using our normalizations.
Theorem 3: For any two point-source locations in V, a and b, and for any scatterer, we have
h 0 ~ ka ! u sa ~ b! 5h 0 ~ kb ! u sb ~ a! ,
~26!
where h 0 (x)5eix /(ix).
From Eq. ~1!, we see that the same reciprocity relation holds for the incident fields, as well:
h 0 (ka)u ia (b)5h 0 (kb)u ib (a). Hence, by Eq. ~2! we conclude that the total exterior fields satisfy
h 0 (ka)u ta (b)5h 0 (kb)u tb (a).
We note that the presence of the multiplicative constant in the above reciprocity relation is due
to the form of the modified spherical wave, Eq. ~1!. If we consider the point sources lying on the
same sphere, that is, a5b, then we obtain the following results:
u sa ~ b! 5u sb ~ a! ,
u ia ~ b! 5u ib ~ a! ,
u ta ~ b! 5u tb ~ a! .
~27!
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5688
Athanasiadis et al.
J. Math. Phys., Vol. 43, No. 11, November 2002
These relations express the fact that interchanging the excitation with the observation point, the
scattered, incident and total fields remain unchanged. From Eq. ~26! we cannot have a reciprocity
relation for the corresponding spherical far-field patterns, because when the point sources go to
infinity the spherical waves reduce to plane waves.
V. THE OPTICAL THEOREM
In Ref. 8, an optical theorem for spherical waves incident upon a soft scatterer has been
proved. Now, this theorem as well as optical theorems for scatterers of other types can be derived
as corollaries of the general scattering theorem. First we define the scattering cross-section due to
a point source at a ~Ref. 6! as
s sa 5
1
k2
E
S2
u g a ~ r̂! u 2 ds ~ r̂! ,
~28!
E
~29!
the absorption cross-section as
s aa 5
1
Im
k
S
u ta
] u ta
ds,
]n
and the extinction cross-section as
s ea 5 s sa 1 s aa .
~30!
If we put a5b in Theorem 2, we obtain
2 Re$ G a ~ a! % 1
1
2p
E
S2
u g a ~ r̂! u 2 ds ~ r̂! 5Ea,a .
We can rewrite this equation using Eq. ~28! to give
s sa 524 p k 22 Re $ G a ~ a! % 12 p k 22 Ea,a .
~31!
From the definitions ~17! and ~29!, we have
s aa 5 12 ik 21 @ u ta ,u ta # 522 p k 22 Ea,a .
~32!
Hence, adding Eqs. ~31! and ~32!, Eq. ~30! gives
s ea 524 p k 22 Re$ G a ~ a! % .
~33!
The value of Ea,a is given in Theorem 2. In particular, for a penetrable scatterer, we obtain
1
s aa 52 Im~ b !
k
E
V2
2
u ¹u 2
a ~ r ! u d v~ r ! ,
~34!
whereas for an impedance surface, we have
s aa 5lk
E
S
u u ta u 2 ds.
We remark that the absorption cross-section s aa provides a measure of the total energy taken
from the incident spherical wave and absorbed by the surface of the scatterer in the impedance
boundary case, or by the lossy medium occupying V 2 in the penetrable case. It is clear that s aa
50 for the soft and hard scatterers and s aa >0 for the other cases.
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J. Math. Phys., Vol. 43, No. 11, November 2002
Scattering relations for point sources
5689
VI. MIXED SCATTERING RELATIONS
For inverse problems, one effective reconstruction method is the point-source method.1 One of
the main steps of this method is the derivation of mixed reciprocity relations. These relations were
introduced in Ref. 10 for sound-soft scatterers, and in Ref. 1 for sound-hard scatterers.
In this section, we allow one of the two point sources considered previously to recede to
infinity, so that we have one spherical incident wave and one plane incident wave. We let both
sources recede to infinity at the end of this section, and recover known results for plane-wave
incidence.
An incident plane wave propagating in the direction of the unit vector d̂ is given by
u i~ r;d̂! 5exp$ ikd̂"r% .
~35!
We have already noted that u ia (r)→u i(r;2â) as a→`.
For an incident plane wave, Eq. ~35!, we denote the total field in V, the scattered field and the
far-field pattern by u t(r;d̂), u s(r;d̂) and g(r̂;d̂), respectively, indicating the dependence on the
incident direction d̂. We have4
u sa ~ r! →u s~ r;2â! ,
as a→`
~36a!
g a ~ r̂! →g ~ r̂;2â! ,
as a→`.
~36b!
and
Now, consider our previous results, involving a and b, and let b→`. Lemma 1 gives the
following results.
Lemma 4: Let u ia (r) be an incident spherical wave and let u i(r;2b̂) be an incident plane
wave. Then
lim @ u ia ,u s~ •,2b̂!# S r 52aeika
r→`
E
S2
g ~ r̂;2b̂! eikr̂"ads ~ r̂!
~37!
and
lim @ u ia ,u s~ •,2b̂!# S a,« 54 p aeika u s~ a;2b̂! .
~38!
«→0
Next, we define a plane far-field pattern generator by
~39!
G ~ a;2b̂! 5 lim G b ~ a!
b→`
F
5ikaeika u s~ a;2b̂! 2
1
2p
E
S2
G
g ~ r̂;2b̂! eika"r̂ds ~ r̂! ,
~40!
where the spherical far-field pattern generator G b (a) is defined by Eq. ~15!. We will also require
lima→` G b (a); this limit is contained in the following theorem.
Theorem 5: For two incident spherical waves, u ia and u ib , we have
lim G b ~ a! 5g b ~ 2â!
~41!
a→`
and
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5690
Athanasiadis et al.
J. Math. Phys., Vol. 43, No. 11, November 2002
~42!
lim G ~ a;2b̂! 5g ~ 2â;2b̂! .
a→`
Proof: We prove Eq. ~41!; a very similar argument gives Eq. ~42!.
We choose coordinates so that the point source at a is on the z-axis. Then, take spherical polar
coordinates ~u, w! on S 2 , so that â"r̂5cos u. Hence for u 50 we have r̂5â, while for u 5 p we
have r̂52â. If we define
F b~ u ! 5
E
2p
0
g b ~ r̂! d w ,
~43!
then we have F b (0)52 p g b (â) and F b ( p )52 p g b (2â). Hence
E
S
g ~ r̂! eikr̂"ads ~ r̂! 5
2 b
E
p
0
i
ka
5
F b ~ u ! eika cos u sin u d u
E
p
0
F b~ u !
d ika cos u
!du
~e
du
2pi
i
@ g b ~ 2â! e2ika 2g b ~ â! eika # 2
ka
ka
5
E
p
eika cos u
dF b ~ u !
du.
du
eika cos u
dF b ~ g v !
du.
du
0
From this equation and Eq. ~15!, we find that
G b ~ a! 5ikaeika u sb ~ a! 1eika @ g b ~ 2â! e2ika 2g b ~ â! eika # 1
eika
2p
E
p
0
~44!
In view of Eq. ~8! and taking into account that the integral in Eq. ~44! tends to zero as a→`, by
the Riemann-Lebesgue lemma, we get Eq. ~41!.
We can now let b→` in the general scattering theorem, Theorem 2; this gives the following
results.
Theorem 6: Let u ia (r) be an incident spherical wave and let u i(r;2b̂) be an incident plane
wave. Then
1
2p
E
g ~ r̂;2b̂! g a ~ r̂! ds ~ r̂! 5Ma ~ 2b̂! ,
~45!
Ma ~ 2b̂! 50 for Dirichlet or Neumann conditions on S;
~46!
G ~ a;b̂! 1g a ~ 2b̂! 1
S2
where Ma (2b̂)5limb→` Ea,b :
Ma ~ 2b̂! 52
k 2l
2p
E
S
u t ~ r;2b̂! u ta ~ r! ds ~ r̂! for the Robin condition (5) on S;
~47!
or
Ma ~ 2b̂! 52
k
Im~ b !
2p
E
V2
2
¹u 2
a ~ r ! •¹u ~ r;2b̂ ! d v~ r ! for a penetrable scatterer. ~48!
The mixed reciprocity principle is contained in the next theorem.
Theorem 7: Let u ia (r) be an incident spherical wave and let u i(r;2b̂) be an incident plane
wave. Then, we have the following reciprocity relation:
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J. Math. Phys., Vol. 43, No. 11, November 2002
Scattering relations for point sources
g a ~ b̂! 5ikae2ika u s ~ a;2b̂! .
5691
~49!
Proof: Let b→` in Theorem 3, using Eqs. ~8! and ~36a!.
h
This result means that the spherical far-field pattern of the point source at a in the direction b̂
is proportional to the scattered field at a due to the incident plane wave with direction of propagation 2b̂.
Finally, if we combine Eqs. ~36b!, ~39!, ~41! and ~42!, we obtain
lim lim G b ~ a! 5 lim lim G b ~ a! 5g ~ 2â;2b̂! .
~50!
b→` a→`
a→` b→`
We can then check that letting both point sources recede to infinity, a→` and b→`, yields the
known scattering and optical theorems for plane-wave scattering; see Refs. 11–13, 6, and 8.
VII. FORMULATION: ELECTROMAGNETICS
In the remainder of the article, we consider electromagnetic problems. The exterior V is an
infinite homogeneous medium with electric permittivity «, magnetic permeability m, phase velocity c and conductivity s 50. The scatterer V 2 is filled with a homogeneous medium with corresponding physical parameters « 2 , m 2 , c 2 and s 2 Þ0.
We consider an incident spherical electromagnetic wave due to a source located at a point with
position vector a, with respect to the origin O. This incident wave (Eia ,Hia ) has the form9
S
D
ae2ika
eik u r2a u
â3p̂ ,
¹3
ik
u r2au
~51!
Hia ~ r;p̂! 5 ~ ik ! 21 ~ «/ m ! 1/2¹3Eia ~ r;p̂! ,
~52!
Eia ~ r;p̂! 5
where p̂ is a constant unit vector with p̂•â50, k5 v A« m .0 is the free-space wave number, and
a5 u au . Physically, (Eia ,Hia ) represents the field generated by a magnetic dipole with dipole
moment â3p̂; see p. 163 of Ref. 14 or p. 23 of Ref. 6. The coefficient ae2ika /(ik) in Eq. ~51!
assures that when the point source tends to infinity the spherical wave reduces to a plane electric
wave with direction of propagation 2â and polarization p̂. The total electric exterior field Eta is
given by
Eta ~ r;p̂! 5Eia ~ r;p̂! 1Esa ~ r;p̂! ,
rPV\ $ a% ,
~53!
where Esa (r;p̂) is the scattered electric field, which satisfies the Silver-Müller radiation condition
lim ~ r3¹3Esa 1ikrEsa ! 50
~54!
r→`
uniformly in all directions r̂PS 2 , where S 2 is the unit sphere. Eta solves the equation
¹3¹3Eta 5k 2 Eta in V.
~55!
We note that the incident electric field satisfies the radiation condition ~54!, and hence the total
electric field also satisfies Eq. ~54!.
The surface of the scatterer may be perfectly conducting, in which case
n̂3Eta 50 on S,
~56!
or it may be an impedance surface, in which case
n̂3¹3Eta 52 ~ ik/Z s ! n̂3 ~ n̂3Eta ! on S,
~57!
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5692
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J. Math. Phys., Vol. 43, No. 11, November 2002
where the dimensionless parameter Z s denotes the surface impedance relative to the characteristic
impedance of the medium and may vary on S.
If the scatterer is a dielectric, the incident electromagnetic waves are transmitted into the
2
scatterer. Let E2
a be the total electric field in the interior. Then Ea satisfies
2 2 2
2
¹3¹3E2
a 5 h k Ea in V ,
~58!
where the complex constant h is the relative index of refraction; on the surface of the scatterer we
have the following transmission conditions:
2
t
2
n̂3Eta 5n̂3E2
a and n̂3¹3Ea 5 ~ m / m ! n̂3¹3Ea on S.
The behavior of the scattered electric field in the radiation zone is given by
Esa ~ r! 5h 0 ~ kr ! ga ~ r̂! 1O ~ r 22 ! , r→`,
~59!
where ga (r̂) is the far-field pattern.
More details on the physical parameters of the above problems can be found in Ref. 6. Each
problem has a unique and stable classical solution.14,15
VIII. GENERAL SCATTERING THEOREM: ELECTROMAGNETICS
In the sequel, for an incident time-harmonic spherical wave Eia (r;p̂) due to a point source
located at a, we will denote the total field in V, the scattered field and the far-field pattern by
writing Eta (r;p̂), Esa (r;p̂) and ga (r̂;p̂), respectively, indicating the dependence on the position a of
the point source and the polarization p̂. Also, the total electric field in V 2 will be denoted by
E2
a (r;p̂). We consider a point source at a with polarization p̂1 and another point source at b with
polarization p̂2 .
For a shorthand notation, we use
$ E,E8 % S 5
E
S
@~ n̂3Ē! • ~ ¹3E8 ! 2 ~ n̂3E8 ! • ~ ¹3Ē!# ds;
in particular, we write $ E,E8% [ $ E,E8% S .
Lemma 8: Let Eia (r;p̂1 ) be a point source at a. Let Eib (r;p̂2 ) be a point source at b, with
corresponding scattered field Esb (r;p̂2 ) and far-field pattern gb (r̂;p̂2 ). Then
lim $ Eia ~ •;p̂1 ! ,Esb ~ •;p̂2 ! % S r 52aeika
r→`
E
S2
gb ~ r̂;p̂2 ! • ~ r̂3 ~ â3p̂1 !! eikr̂"ads ~ r̂!
~60!
and
lim $ Eia ~ •;p̂1 ! ,Esb ~ •,p̂2 ! % S a,« 54 p i~ a/k ! eika ~ ¹3Esb ~ a;p̂2 !! • ~ â3p̂1 ! ,
~61!
«→0
where S r is a large sphere of radius r enclosing a and b, and S a,« is a small sphere surrounding
a, defined by Eq. (10).
Proof: For Eq. ~60!, we use the asymptotic forms ~13!. These show that the incident electric
wave takes the form
Eia ~ r;p̂1 ! 5h 0 ~ kr ! gia ~ r̂;p̂1 ! 1O ~ r 22 ! ,
r→`,
~62!
where gia (r̂;p̂1 )5ika exp$2ika(11r̂"â) % (r̂3(â3p̂1 )) is the far-field pattern of the point source
incident wave. Using Eqs. ~59! and ~62! we establish Eq. ~60!. Note that r̂"gia (r̂;p̂1 )50.
For Eq. ~61!, some calculations show that
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J. Math. Phys., Vol. 43, No. 11, November 2002
$ Eia ~ •,p̂1 ! ,Esb ~ •,p̂2 ! % S a,« 5aeika
1
HE
S a,«
E
S a,«
2k 2
Scattering relations for point sources
5693
n̂•¹3 @~~ â3p̂1 ! •¹h 0 ~ k u r2au !! Esb # ds
~ ik1 u r2au 21 ! h 0 ~ k u r2au !~ ¹3Esb ! • ~ â3p̂1 ! ds
E
S a,«
J
n̂• @ Esb 3 ~ â3p̂1 !# h 0 ~ k u r2au ! ds .
The first integral on the right-hand side vanishes by Stokes’s theorem. Applying the mean value
theorem on the remaining integrals and letting «→0, we obtain Eq. ~61!.
h
For two incident spherical electric waves, Eia (r;p̂1 ) and Eib (r;p̂2 ), we define a spherical
far-field pattern generator, as follows:
F
Gb ~ a;p̂2 ! 5eika a3 ¹3Esb ~ a;p̂2 ! 2
ik
2p
E
S2
G
r̂3gb ~ r̂;p̂2 ! eikr̂"ads ~ r̂! .
~63!
As we shall see later, when the point sources recede to infinity, Gb (a;p̂2 ) is reduced to the far-field
pattern for an incident plane electric wave propagating in the direction 2â and of polarization p̂2 .
Using this notation, the general scattering theorem for spherical electric waves is formulated as
follows.
Theorem 9: For any two point-source locations in V, a and b, and for any polarizations, p̂1
and p̂2 , we have
p̂1 •G1 ~ a;p̂2 ! 1p̂2 •Ga ~ b;p̂1 ! 1
1
2p
E
S2
gb ~ r̂;p̂2 ! •ga ~ r̂;p̂1 ! ds ~ r̂! 5Ea,b ~ p̂1 ;p̂2 ! ,
~64!
where
Ea,b ~ p̂1 ;p̂2 ! 52
ik
$ Et ~ •,p̂1 ! ,Etb ~ •,p̂2 ! % .
4p a
~65!
The value of Ea,b depends on the scatterer:
Ea,b 50 for a perfectly conducting surface;
Ea,b ~ p̂1 ;p̂2 ! 52
k2
2p
E
Re~ Z S !
t
t
2 ~ n̂3Ea ~ r̂;p̂1 !! • ~ n̂3Eb ~ r̂;p̂2 !! ds ~ r !
S u Z Su
~66!
~67!
for the impedance boundary condition (57); or
Ea,b ~ p̂1 ;p̂2 ! 52
k 3m
Im~ h 2 !
2pm2
E
V2
2
E2
a ~ r̂;p̂1 ! •Eb ~ r̂;p̂2 ! d v
~68!
for a dielectric scatterer.
Proof: We proceed exactly as in the proof of Theorem 2. First, we evaluate Ea,b directly, using
the boundary or transmission conditions on S; for the dielectric scatterer, we have to apply the
divergence theorem in V 2 . This gives the stated expressions for Ea,b .
Next, we give an alternative evaluation of Ea,b , using the relations Eat 5Eai 1Eas , for a 5a, b.
Formally, the calculations proceed as before, with $•,•% in place of @•,•#. We also use the vector
version of Green’s second theorem, which gives
$ Eia ,Eib % 50,
~69!
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5694
Athanasiadis et al.
J. Math. Phys., Vol. 43, No. 11, November 2002
$ Eia ,Esb % 524 p i~ a/k ! eika ~ ¹3Esb ~ a;p̂2 !! • ~ â3p̂1 ! 12aeika
E
S2
gb ~ r̂;p̂2 ! • ~ r̂3 ~ â3p̂1 !! eikr̂"ads ~ r̂!
~70!
and
$ Esa ,Esb % 5
2i
k
E
S2
ga ~ r̂;p̂1 ! •gb ~ r̂;p̂2 ! ds ~ r̂! .
~71!
The remaining details are omitted.
h
There is also a reciprocity theorem, as Eta is an exact Green’s function. It can be found on p.
63 of Ref. 6, for example. With our normalizations, it takes the following form.
Theorem 10: For any two point-source locations in V, a and b, for any polarizations, p̂1 and
p̂2 , and for any scatterer, we have
h 0 ~ ka !~ b̂3p̂2 ! • ~ ¹3Esa ~ b̂;p̂1 !! 5h 0 ~ kb !~ â3p̂1! • ~ ¹3Esb ~ â;p̂2 !! .
~72!
IX. OPTICAL THEOREM: ELECTROMAGNETICS
In Ref. 9, an optical theorem for spherical waves incident upon a perfect conductor has been
proved. Here, we generalize this result to other scatterers, using the general scattering theorem.
First we define the scattering cross-section due to a point source at a ~Ref. 6! as
s sa 5
1
k2
E
S2
u ga ~ r̂;p̂! u 2 ds ~ r̂! ,
~73!
n̂"~ Eta 3¹3Eta ! ds
~74!
the absorption cross-section as
1
s aa 5 Im
k
E
S
and the extinction cross-section, s ea , by Eq. ~30!.
If we put a5b and p̂1 5p̂2 5p̂ in Theorem 9, we obtain
2 Re@ p̂"Ga ~ a;p̂!# 1
1
2p
E
S2
u ga ~ r̂;p̂! u 2 ds ~ r̂! 5Ea,a ~ p̂;p̂! ,
which we can rewrite as
s sa 524 p k 22 Re@ p̂"Ga ~ a;p̂!# 12 p k 22 Ea,a ~ p̂;p̂! .
~75!
From the definitions ~65! and ~74!, we have
s aa 5 21 ik 21 $ Eta ~ •,p̂! ,Eta ~ •,p̂! % 522 p k 22 Ea,a ~ p̂;p̂! .
~76!
Hence, adding Eqs. ~75! and ~76!, the definition ~30! gives
s ea 524 p k 22 Re@ p̂•Ga ~ a;p̂!# .
~77!
The value of Ea,a (p̂;p̂) is given in Theorem 9; it depends on the scatterer’s properties.
X. MIXED SCATTERING RELATIONS
Let
Ei~ r;d̂,p̂! 5p̂ exp$ ikd̂"r% .
~78!
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J. Math. Phys., Vol. 43, No. 11, November 2002
Scattering relations for point sources
5695
be an incident time-harmonic plane electric wave, where the unit vector d̂ describes the direction
of propagation and the unit vector p̂ gives the polarization. We will indicate the dependence of the
total field in V, the total field in V 2 , the scattered field and the electric far-field pattern on the
incident direction d̂ and the polarization p̂ by writing Et(r;d̂,p̂), E2 (r;d̂,p̂), Es(r;d̂,p̂) and
g(r̂;d̂,p̂), respectively.
Here, we consider mixed situations, and relate fields due to one spherical electric wave
Eia (r;p̂1 ) and one plane electric wave Ei(r;2b̂,p̂2 ); we do this by letting b→` in our previous
results.
Using the asymptotic forms ~13!, we can easily show that for the spherical electric wave ~51!
we have
lim Eib ~ r;p̂! 5Ei~ r;2b̂,p̂! ,
~79!
b→`
that is the spherical electric wave, when the point source goes to infinity, reduces to a plane
electric wave with direction of propagation 2b̂ and polarization p̂. Similarly, we have Etb (r;p̂)
→Et(r;2b̂,p̂), Esb (r;p̂)→Es(r;2b̂,p̂) and gb (r̂;p̂)→g(r̂;2b̂,p̂) as b→`.
Next, let b→` in Lemma 8 to give the following result.
Lemma 11: Let Eia (r;p̂1 ) be an incident spherical electric wave and let Ei(r;2b̂,p̂2 ) be an
incident plane electric wave. Then
lim $ Eia ~ •;p̂1 ! ,Es~ •;2b̂,p̂2 ! % S r 52aeika
r→`
E
S2
g~ r̂;2b̂,p̂2 ! • ~ r̂3 ~ â3p̂1 !! eikr̂"ads ~ r̂!
and
lim $ Eia ~ •;p̂1 ! ,Es~ •;2b̂,p̂2 ! % S a,« 54 p i~ a/k ! eika ~ ¹3Es~ a;2b̂,p̂2 !! • ~ â3p̂1 ! .
«→0
We define a plane far-field pattern generator by the formula
F
G~ a;2b̂,p̂2 ! 5 lim Gb ~ a;p̂2 ! 5eika a3 ¹3Es~ a;2b̂,p̂2 ! 2
b→`
ik
2p
E
S2
G
r̂3g~ r̂;2b̂,p̂2 ! eikr̂"ads ~ r̂! ,
where Gb (a;p̂2 ) is defined by Eq. ~63!. Other limiting values are given in the next theorem.
Theorem 12: For two incident point source electric waves, Eia (r;p̂1 ) and Eib (r;p̂2 ), we have
lim Gb ~ a;p̂! 5gb ~ 2â;p̂2 !
~80!
a→`
and
lim G~ a;2b̂,p̂2 ! 5g~ 2â;2b̂,p̂2 ! .
~81!
a→`
Proof: For Eq. ~80!, we use spherical polar coordinates ~u, w! as in the proof of Theorem 5,
and define
Fb ~ u ! 5
E
2p
0
r̂3gb ~ r̂;p̂2 ! d w .
In particular, we have Fb (0)52 p â3gb (â;p̂2 ) and Fb ( p )522 p â3gb (2â;p̂2 ). Hence
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5696
Athanasiadis et al.
J. Math. Phys., Vol. 43, No. 11, November 2002
E
S
r̂3ĝb ~ r̂;p̂2 ! eikr̂"ads ~ r̂! 5
2
5
E
p
0
Fb ~ u ! eika cos u sin u d u
i
i
@ Fb ~ p ! e2ika 2Fb ~ 0 ! eika # 2
ka
ka
;2
E
p
eika cos u
0
dFb ~ u !
du
du
2pi
â3 @ gb ~ 2â;p̂2 ! e2ika 1gb ~ â;p̂2 ! eika #
ka
for large a. From this equation and Eq. ~63!, we arrive at Eq. ~80!. The proof of Eq. ~81! is
similar.
h
We can now let b→` in the general scattering theorem, Theorem 9.
Theorem 13: Let Eia (r;p̂1 ) be an incident spherical electric wave and let Ei(r;2b̂,p̂2 ) be an
incident plane electric wave. Then
1
2p
p̂1 "G~ a;2b̂,p̂2 ! 1p̂2 •ga ~ b̂;p̂1 ! 1
E
S2
g~ r̂;2b̂,p̂2 ! •ga ~ r̂;p̂1 ! ds ~ r̂! 5Ma ~ 2b̂;p̂1 ,p̂2 ! ,
where Ma (2b̂;p̂1 ,p̂2 )5limb→` Ea,b (p̂1 ;p̂2 ):
~82!
Ma ~ 2b̂;p̂1 ,p̂2 ! 50
for a perfectly conducting surface;
Ma ~ 2b̂;p̂1 ,p̂2 ! 52
k2
2p
E
Re~ Z S !
t
t
2 ~ n̂3E ~ r̂;2b̂,p̂2 !! • ~ n̂3Ea ~ r̂;p̂1 !! ds ~ r̂ !
S u Z Su
~83!
for the impedance boundary condition; or
Ma ~ 2b̂;p̂1 ,p̂2 ! 52
k 3m
Im~ h 2 !
2pm2
E
V2
2
E2
a ~ r̂;p̂1 ! "E ~ r̂;2b̂,p̂2 ! d v
~84!
for a dielectric scatterer.
Proof: The proof is similar to that of Theorem 9. The only substantial difference appears in the
formula for $ Esa ,Ei% ; cf. Eq. ~70!. Now, using the plane electric wave ~78!, we find that ~see p. 59
of Ref. 6!
$ Esa ~ •;p̂1 ! ,Ei ~ •;2b̂,p̂2 ! % 5p̂2 •
E
S
ˆ
@ n̂3¹3Esa ~ r;p̂1 ! 1ikb̂3 ~ n̂3Esa ~ r;p̂1 !!# e2ikb"rds ~ r!
54 p ik 21 p̂2 •ga ~ 2b̂,p̂1 ! .
h
The mixed reciprocity relation for perfect conductors is Theorem 2.3.4 in Ref. 1. It is valid
more generally, as follows.
Theorem 14: Let Eia (r;p̂1 ) be an incident spherical electric wave and let Ei(r;2b̂,p̂2 ) be an
incident plane electric wave. Then
p̂2 "ga ~ b̂,p̂1 ! 5e2ika p̂1 • @~ ¹3Es~ a;2b̂,p̂2 !! 3â# .
~85!
Proof: Working as in the proof of Theorem 10 and taking into account that
$ Eia ~ •;p̂1 ! ,Es~ •;2b̂,p̂2 ! % 54 p i~ a/k ! e2ika ~ ¹3Es~ a;2b̂,p̂1 !! • ~ â3p̂1 !
and
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J. Math. Phys., Vol. 43, No. 11, November 2002
Scattering relations for point sources
5697
$ Esa ~ •;p̂1 ! ,Ei~ •;2b̂,p̂2 ! % 524 p ik 21 p̂2 •ga ~ b̂,p̂1 ! ,
h
Theorem 14 is proved.
To conclude, we note that we also have
lim lim Gb ~ a;p̂2 ! 5 lim lim Gb ~ a;p̂2 ! 5g~ 2â;2b̂,p̂2 ! .
a→` b→`
b→` a→`
This can be used to verify that the known scattering relations for plane-wave incidence14,6 are
recovered when a→` and b→`. Furthermore, Eq. ~81! and the reciprocity principle for plane
waves6 gives the following limiting property:
lim p̂1 •G~ a;2b̂,p̂2 ! 5 lim p̂2 •G~ 2b;â,p̂1 ! .
a→`
b→`
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2
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