Acoustic and electromagnetic scattering analysis using discrete sources VII systems of functions in electromagnetic theory

VII
SYSTEMS OF FUNCTIONS IN
ELECTROMAGNETIC THEORY

In this chapter we will analyze complete and linear independent systems of
functions for the Maxwell equations. We will construct complete systems
in £tan('S') and in the product spacefi^anC'^)-Complete systems in ^^anl-^)
will be used to solve the exterior Maxwell and the impedance boundaryvalue problems, while complete systems in Xl^an('S') will be employed to
solve the transmission boundary-value problem.
We begin our analysis by presenting some fundamental results on the
completeness of the localized spherical vector wave functions. To preserve
the completeness at irregular frequencies, linear combinations of these functions will be considered. We then pay attention to the systems of localized
vector multipoles. In Chapter 9 we will apply these results to axisymmetric geometries by taking into account the polarization of the external
excitation. We will then proceed to analyze the completeness properties
of the systems of distributed sources. We start with the spherical vector
wave functions and vector multipoles distributed on a straight line. Our
analysis is based on the addition theorem for spherical wave and vector
wave functions. The next sections concern the completeness of the system
of magnetic and electric dipoles and the system of vector Mie potentials
with singularities distributed on auxiliary closed and open surfaces. These

137


138

CHAPTER VII SYSTEMS OF FUNCTIONS IN ELECTROMAGNETICS

functions are suitable for analyzing the scattering by particles without rotational symmetry. The last section of this chapter deals with the linear
independence of these systems.

1

COMPLETE SYSTEMS OF FUNCTIONS

The completeness properties of the systems of discrete sources are of primary interest since they provide a means for approximating the exact solutions to the scattering problems. For instance, the set of radiating spherical
vector wave functions is known to be complete in /^tanC*^)- Consequently,
any radiating solution to Maxwell equation can be approximated uniformly
in closed subsets of Dg and in the mean square sense on 5 by a sequence
of linear combinations of spherical vector wave functions. In this section
we will present these basic results for localized and distributed sources.
1.1

Localized spherical vector wave functions and vector multipoles

We begin our analysis by defining our notations. The independent solutions
to the vector wave equations
VxVxX-fe2x = o

(7.1)

Mii?.(x) = V«ii?,(x) X X, Nii^Jx) = i v X M^i^Jx),

(7.2)

can be constructed as

where n = 1,2,..., m = —n, ...,n, and in spherical coordinates the uj^^
are the spherical wave functions. The specific forms of the spherical vector
wave functions are
'.

PJr^icose)

dPi^'(cose)

sm 6

do

N M (X) = | n ( n + ifJ^Mp^r^cosd)

^TrTM^'ikr)]

AP]r\cose)
aO

^

pjrmp

e.
.

P^^'(cos0)
sm 9
^

(7.3)

pjm(p

where {er^eg^e^) are the unit vectors in spherical coordinates. The superscript *r stands for the regular spherical vector wave functions while the
superscript '3' stands for the radiating spherical vector wave functions. It is
useful to note that for n = m = 0 we have MJJ? = NQJ? = 0. Mj„^, N^

IS

an entire solution to the Maxwell equations and M^^, N^^ is a radiating
solution to the Maxwell equations in R^ - {0}


139

1. COMPLETE SYSTEMS OF FUNCTIONS

The spherical vector wave expansion of the dyadic ^I is of basic importance in our investigation. It is
5(x,y,fc)I=^5; £ D„
n—\ m^^n

\

[M3__(y)Mi,„(x) + Ni_(y)<„(x)]
-f Irrotational terms, |y| > |x|

(7.4)

X <

[ML^„(y)ML(x) + N!._(y)NL(x)]
[ -f Irrotational terms, |y( < |x|

where the normalization constant Dmn is given by
2n + l (n-|m|)!
(7.5)
•^mn —
4n(n + l) (n+|m|)!"
Using the calculation rules for dyadic functions and the identity ag = a-^I
we find the following simple but useful expansions
VxX[a(y)g(y,x,fc)] = : i ^ 5 ; J2 A
71=1171=—n

{[a(y).Mi„„(y)lNU(x)
+ [a(y) • Ni„„(y)j MlJx)} , |y| > |x|

(7.6)

X <

{[a(y)-ML„„(y)lN^„(x)
+ [a(y) • Nl^Jy)] M3.„(X)} , |y| < |x|
and
Vx X Vx X (a(y)p(y,x,/^)] = ^ E

E

^'

n=rl Tn=—n

[

{[a(y).Mi„„(y)lMj„„(x)
+ [a(y) • Ni„„(yj] Nj„„(x)} . |y| > |x|
(7.7)

{[a(y).Mi^„(y)lM^„(x)
[ +[a(y).Ni^„(y)]NUx)},|y|<|x|


140

CHAPTER VII SYSTEMS OF FUNCTIONS IN ELECTROMAGNETICS

We are now in the position to establish the completeness results for
the systems of spherical vector wave functions.
THEOREM 1.1: Let S be a closed surface of class C^ with unit outward
normal n. Then the following systems of vector functions

(a)

{n X M ^ „ , n x N ^ „ , n = 1,2,..., m = - n , ...,n} ,

(b)

{n X Mj^^,n x N J ^ ^ , n = 1,2,..., m = - n , ...,n/ k i {n X M J ^ ^ , n X N J ^ ^ , n = 1,2,..., m = - n , . . . , n / fc € C T ( A ) }

U{nxa-^, j = l , 2 , . . . , m M } ,
(c)

{n X M L -f JAn x (n x N ^ ^ ) , n x N ^ „ + j A n x (n x M ^ ) ,

n = 1,2,..., m = —n, ...,n/ Re A > 0} ,

(d)

{n X M i , „ ~ j A n x (n x N J „ ^ ) , n x N^^^ ~ j A n x (n x M^^^),
n = 1,2,..., m = —n, ...,n/ Re A > 0}

are complete in Cl^^{S). Here { a j } ^ ^ is a basis ofN ( ^ J 4- M) andmM =
dim7V(il + Al).
Proof: We consider (a). Let

ya*.(nxM^„)d5 = 0,
s
(7.8)

ya*(nxN^„)d5 = 0,
for n = 1,2,..., m = - n , ...,n, and a € C^^ni'^)- Consider the vector field
£ = {j/k)V X V X Aa' with density a' = n x a*. For x € £>[, where D[ is
the interior of a spherical surface S^ enclosed in i?i, we have
fc2

^w = ~v^ S ^^

/a'(y)-Mi^„(y)d5(y) Mj„„(x)

n = l m——n

+ y"a'(y)-Ni„„(y)d5(y) Ni,„(x)
Ls

(7.9)
The closure relations show that £ = 0 m Di, Application of theorem 2.2
given in Chapter 6 finishes the proof of (a).


1. COMPLETE SYSTEMS OF FUNCTIONS

141

The second part of the theorem corresponding to the casefc^ follows from theorem 2.3 of Chapter 6 and the above arguments. Let us
consider the case k G cr{Di). It has to shown that for a 6 C^g^ni^)^ the
closeness relations

s
(7.10)

s
n = 1,2,..., m = —n,..., n, together with
/ a*, (n X aj)dS

= 0, j = 1,...,TUM,

(7.11)

s
gives a ~ 0 on 5. Before we present the proof we note some basic results. The nuUspace of the operator ^I -{- M corresponds to solutions of
the homogeneous interior Maxwell problem, that means N (^ J -f M) = 3W,
where 9Jl stands for the linear space

m={nxn\s /E,necHDi)nc(Di),
(7.12)
V X E = jfcH, V X H = -jkE in D^, n x E = 0 on 5 } .
In addition
dimN(^I'\-M) =dimN(^I-^M'^ =0

(7.13)

if k is not an interior Maxwell eigenvalue, and
dimN (^I-hM\

= dimN (^I + M'^ = rriM

(7.14)

if k is an eigenvalue. If {hj}^Jl is a basis for N ( ^ J - f A1') and E^ =
VxAfe^xn and Hj = {l/jk)V
x E^, then b^ = n x (n x Ej^) on 5 ,
and the tangential fields a^ = n x H*^., j = l,...,mM, form a basis for
N ( i j + A t ) . Furthermore, the matrix T M = [T^] , T^ = (afe,bj), kj
= 1,..., TUM, is nonsingular. Coming to the proof we define the vector fields
£ and Hhy £ = ij/k) V x V x Aa' and W = V x A a s where a' = n x a*.
Prom (7.10) we obtain £ = H ^ 0 in Ds^ Going to the boundary and
proceeding as in the proof of theorem 2.2 given in the precedent chapter we find that a'^ai^ e Cf^n,d('S'). where ( i j - f A < ) a ( ) = 0. Therefore
ag = X^fe!^ cKfeafc. Finally, the closure relations (7.11) yield a '^ 0 on 5.


142

CHAPTER VII SYSTEMS OF FUNCTIONS.IN ELECTROMAGNETICS

If instead of the system {n x a ^ } ! ^ we consider the system {n x hj}^J[,
from /ao'bjdS = 0, j = l,...,mM, we obtain X)r=^ ^fc (^fc»*^j> = ^^J
s
= 1,..., niM'j whence, by the fact that the matrix T M is nonsingular, a ~ 0
on S follows.
Let us now consider (c). We debut with the closeness relations
y'a*.[nxML-fJAnx(nxND]ci5

=

0,
(7.15)

/ a*.[nxN^„+jAnx(nxM^J]d5
5

=

0,

72 = 1,2,..., m = —n,..., n, written for convenience as
|[a'.M3„„-jA(nxa')N3„„]d5
s

=

0,

|[a'.NL-jA(nxaO-ML]d5

= 0,

(7.16)

5

n = 1,2,..., m = - n , ...,n, where a' = n x a*. Then, from (7.6), (7.7) and
(7.16) we see that
f = V X Aa' - A ^ V X V X Anxa'

(7.17)

vanishes in Di. Theorem 2.4 of Chapter 6 concludes the proof of (c).
To prove (d) we use essentially the same arguments and the result of
theorem 2.5 given in the precedent chapter.
In addition to the completeness results stated in theorem 1.1 we mention that the system
{ {¥ + • ^ ) ('^ ^ M-n). (^2: -f > l ) (n X N L ) ,
n = 1,2,..., m = - n , ...,n/ k ^ (T{Di)} ,

(7 18)

is also complete in ^?an('^)- "^^ prove this assertion we debut with
ya*.(ij + M)(nxMDdS

=

0,
(7.19)

|a'.(iJ + Al)(nxNDd5

=

0,


143

1. COMPLETE SYSTEMS OF FUNCTIONS

for n = 1,2,..., m = —n,...,n, and a G£tan('^)- Using the definition of
the adjoint operator with respect to the L^ bilinear form we rewrite the
closeness relations as
(7.20)

n = 1,2,..., m = - n , ...,n, where a' = a*. The completeness of the radiating spherical vector wave functions in £tan('S') yields {^I-\- M^) a' = 0
almost everywhere on S. Therefore ( | l ~ M) (n x a') = 0 almost everywhere on 5 and employing the same arguments as in theorem 2.2 of Chapter 6 we receive n x a ' ' ^ ^ n x a o € C^^ndi"^)- Using this and the fact that
k ^ (T{Di) we deduce that V x Anxa;, vanishes in A - Theorem 2.2 of the
precedent chapter may now be used to conclude.
Next we will analyze complete systems of vector functions in the product space £?an(5') = C^a,n{S) X >C?an(5'). We recall that £'^{S) is a Hilbert
space endowed with the scalar product

and obviously £tan('S') is a subspace of £^(5).
THEOREM 1.2: Let S be a closed surface of class C^ with unit outward
normal n. Then the system of vector functions

n X M^i,
-7\/^nxN^L

\

/ n X N^i

n = 1,2,..., m = —n, ...,n}
is complete in S't&ni'^)' ^^^ radiating spherical vector wave functions are
defined with respect to the wave number kg, while the regular spherical vector
wave functions are defined with respect to the wave number ki.
Proof: It has to shown that for
tions
/

aJ.(nxM^J,)+a|.

(iO € £tan('S') *^he closure rela-

-j./-^nxN^J,

d5 = 0,

5

/

(7.21)
al- (n X N^i„) + a2- - j . ^ n x M^J,

dS

= 0,

144

CHAPTER VII SYSTEMS OF FUNCTIONS IN ELECTROMAGNETICS

n = 1,2,..., m = —n,...,n, gives Si '-^ 0 and a2 ^ 0 on 5. Setting ai =
n X aj and ig = n x £2 we rewrite (7.21) as

/(5^N!:L+jY^5i.M^jJ„„)d5 = 0,
(7.22)

|(ai.Mii„+jYgai.Nii„)d5 = 0,
s
n = 1,2,..., m = —n, ...,n. Prom (7.6) and (7.7) we find that
'^''•^t

+^ ^ ' ' ' ^ ' ' - ^ \

=

O^"^'--

(7.23)

Application of theorem 2.6 given in Chapter 6 finishes the proof of the
theorem.
The same technique can be used to prove the completeness of the system
n X MrtiL

J ,^nxN'^\

\

/ n X N^i

M

jJtl^nxM^

n = 1,2,..., m = —n,..., n} .
We now turn our attention to the general null-field equations for the
exterior Maxwell boundary-value problem. The following theorem states
the equivalence between the null-field equations formulated in terms of
spherical vector wave functions and the general formulation (6.103).
THEOREM

Jinx

1.3: Let hg solve the set of null-field equations

hs)'{n

X M ^ , ) d 5 = - j / ( n x eo) • (n x N ^ ) d5,
(7.24)

Jinx
s

h , ) . (n X N ^ ^ ) d 5 = -J J (^ x eo) • (n x M ^ d S ,

s

for n = 1,2,..., m = —n,..., n. Then hg solve the general null-field equation
(6.103) and conversely.
Proof: The proof follows from the vector spherical waves expansion
of the electric field
5 = - V X Aeo + ^ V X V X A,,,

(7.25)


1. COMPLETE SYSTEMS OF FUNCTIONS

145

inside a sphere enclosed in Dj.
The unique solvabihty of the null-field equations (7.24) follows from
the completeness of the system of vector functions given by theorem 1.1.
Let us consider the general null-field equations for the transmission
boundary-value problem. We have the following result.
THEOREM

1.4: Let e and h solve the set of null-field equations:

J{[nx{e-eo)]-{nxMl„)
s

+j y / ^ [ n x ( h - h o ) ] ( n x N D } d 5
/

= 0,

{[nx(e-eo)]-(nxN^„)

+ j y j [ n x ( h - h o ) ] ( n x M L ) } d 5 = 0,
I [(n X e) • (n X M ^ ) + j ^ (n x h) • (n x N ^ ) ] d5 = 0,

y " [ ( n x e ) - ( n x N j „ „ ) + i y ^ ( n x h ) ( n x M U ] d 5 = 0,
(7.26)
n = 1,2,..., m = —n, ...,n, where the radiating spherical vector wave functions are defined with respect to the wave number kg, while the regular
spherical vector waveJunctions are defined with respect to the wave number ki. Then e and h solve the general null-field equations (6.113) and
conversely.
Proof: The proof of the theorem is provided by the spherical vector
wave expansions of the electric fields
^ . = V X A|_,„ + ^ V

X V X A|_-,^

(7.27)

and
Si = VxAi-\-

7 ^ V X V X Ai

(7.28)

inside a sphere enclosed in Di and outside a sphere enclosing Dj, respectively.

146

CHAPTER VII SYSTEMS OF FUNCTIONS IN ELECTROMAGNETICS

The completeness of the system of vector functions given by theorem
1.2 impUes that the particular null-field equations are uniquely solvable in
£?an(5).
Let us now investigate the completeness properties of the systems of
magnetic and electric vector multipoles. In Cartesian coordinates they are
defined by
Mj^3„p(x) = i v X («ii?.(x)ep) , Nklpi^)

= ^ V x Ml^„^{x),

(7.29)

where p = 1,2,3, n = 0,1,..., m = —n,..., n, and Cp denote the Cartesian
unit vectors. Note that Mj^^^p, ^Innp ^^ ^^ entire solution to the Maxwell
equations and Mj^^p, 'N^^p ^s a radiating solution to the Maxwell equations in R 3 - {0} .
Accounting of the series representation of the Green function and the
identity
(Vx X (a(y)y(x,y,fc))] • Cp = (a(y) x Vyp(x,y,fc)) • Cp
= - a ( y ) - [ep x Vyp(x,y,fc)] = [Vy x (epp(x,y,fc))] • a(y), x 7^ y,
(7.30)
we find the expansion
[Vxx(a(y)^(x,y,fc))].ep = ^ 5 ]

X^ P,mn

n=Om=—n

a(y) • Mi„„p(y)uJ„„(x), |y| > |x|
(

^^'^^^

a(y)-ML„„JyR„(x),
|y|<|x|
-mnpy

Analogously, we use the identity
[Vx X Vx X (a(y)5(x,y,A;))] • ep = {Vx x [a(y) x Vyg(x,y,/i;)]} • ep
= [- (a(y)-Vx) Vyg{x,y,k)

+ a(y)Vx • (Vy^Cx,y,A))] • ep

= [(a(y)-Vy) Vy5(x,y,fc) - a(y)Ay5(x, y,A;)] • Bp
= [(ep-Vy) Vyff(x,y,A;) - epAy^(x, y,A;)] • a(y)
= - {Vy X (ep X Vy5(x,y,fc)]} • a(y)
= [Vy X Vy X (ep5(x,y,A;))) • a(y), x ^ y,
(7.32)


1. COMPLETE SYSTEMS OF FUNCTIONS

147

to derive the expansion
oo

[V^xVxX(a(y)g(x,y,/j))]-ep = ^ 2 J

n

z J Tl
^ 'mn

n=OTn=—n

a(y)-Ni_^(yKJx),|y|>|x|

(7.33)

x<
a(y)-Ni^,p(y)tx^,(x), | y | < | x |
The following theorem establishes the completeness of the localized
electric and magnetic vector multipole systems in £?an(*5)*
T H E O R E M 1.5: Let S be a closed surface of class C^ with unit outward
normal n. Then the following systems of vector functions

(a)

{n X N ^ ^ p , p = 1,2,3, n = 0,1,..., m = - n , . . . , n} ,
{n X M^^p, p = 1,2,3, n = 0,1,..., m = - n , . . . , n} ,

(b)

{n X Nj„np» P = 1,2,3, n = 0,1,..., m = - n , . . . , n / fc ^ ^ ( A ) } ,
{n X Mj^^p, p = 1,2,3, n = 0,1,...,m = ~n,..., n / fc ^ (T{D^)} ,
{n X Nj^np. p = 1,2,3, n = 0,1, ...,m = - n , ...,n/ k e ( T ( A ) }

U { n x a j , J = l,2,...,mM},
{n X Mj^^p, p = 1,2,3, n = 0,1,..., m = ~n,..., n/ k e (T(A)}
U { n x a j , j = l,2,...,mA/},

(c)

{n X M^^p 4- jA n X (n X N ^ ^ p ) ,
p = 1,2,3, n = 0, l,...,m = —n, ...,n/ ReA > 0} ,
{n X N^^p + J A n X (n X M ^ ^ p ) ,
p = 1,2,3, n = 0, l,...,m = —n,...,n/ ReA > 0 } ,

(d)

{n X Mj^^p ~ jA n X (n X Nj^„p),
p = 1,2,3, n = 0, l,...,m = ~n, ...,n/ ReA > 0} ,
{n X Nj^^p - j A n X (n X Mj„„p),
p = 1,2,3, n = 0, l,...,m = - n , ...,n/ ReA > 0}

are complete in C^Q,^{S). The significance of the system of functions
is as in theorem 1.1.

{dijj^Jl


148

CHAPTER VII SYSTEMS OF FUNCTIONS IN ELECTROMAGNETICS

Proof: We consider (a). Let a € Cl^^{S) satisfy
/ a - . ( n x N ^ „ , ) d 5 = 0 . p = 1 . 2 , 3 , „ = 0,l,...,™ = - „ , . . . , n . (T.34)

Define the electric field £ hy S = {j/k)V x V x A a s where a' = n x a*,
and let S^ be a sphere enclosed in Di. For x € -D[, where £)[ is the interior
of S^^ we use (7.33) to obtain

fc2

^w-p = -v5:
E^'
n=OTn=—n

(y)-N!.„„Jy)d5(y)
-mnp

«mn(x)

Ls
(7.35)

for p = 1,2,3. The closeness relations show that E-ep = 0 in Z)[,p = 1,2,3.
Thus, £^ = 0 in D[, whence, by the analyticity of £^, £^ = 0 in Di follows.
Finally, theorem 2.2 of Chapter 6 may be used to conclude the proof of the
first part of (a). The second part can be proved by employing the same
arguments for the magnetic field W = V x Aa'.
Going further, we see that the closeness relations for the regular electric
and magnetic vector multipoles imply £ = 0 in Dg and W = 0 in D^,
respectively. Hence, theorem 2.3 of Chapter 6 accounts for the first part of
(b). The proof of the second part proceeds as in theorem 1.1.
To prove (c) and (d) we use essentially the same arguments and the
results of theorems 2.4 and 2.5 given in the precedent chapter.
The following theorem is the analog of theorem 1.2.

T H E O R E M 1.6: Let S be a closed surface of class C^ with unit outward
normal n. Then the following systems of vector functions

,p = 1,2,3, n = 0, l,...,m = - n , .

' ^ n X N^'^

^ n x M^'^

mnp

I ' ^ " ^ •^'^'^' n = 0,l,...,m =

-n,...,n

are complete in £?an('5')
Proof: Let

/

it)

a j . (n X M'^\^)

G £?an('5') satisfy the closure relations

+i^
V

- j j ^ n x N^^J,

mnp
V /^s,i

dS = 0

(7.36)


1. COMPLETE SYSTEMS OF FUNCTIONS

149

for p = 1,2,3, n = 0,1,..., m = - n , ...,n. Setting a'^ = n x aj and
a2 = n X a2 we rewrite (7.36) as
^ a l . M l i „ p - j / l I a ^ . N i i „ p ) cI5 = 0,
V ^«.^
/

(7.37)

p = 1,2,3, n = 0,1,..., m = - n , . . . , n . Then, from (7.31) and (7.33) we
have
V X Af'/ - 7 - ^ V X V X A^; = 0 in A,5.

(7.38)

Theorem 2.6 of Chapter 6 can now be used to conclude. Turning to the
second part of the theorem we remark that the closure relations lead to
V X At)' 4- r ^ V

X V X At)' = 0 in A , , .

(7.39)

Once again application of theorem 2.6 given in Chapter 6 finishes the proof
of the theorem.
Next, we will formulate the null-field equations in terms of magnetic
and electric vector multipoles. We begin with the exterior Maxwell boundaryvalue problem.
THEOREM

1.7: Let hg solve one of the following sets of null-field equa-

tions
J (n X h,) • (n X N ^ p ) dS = -j j{nx
s
s

eo) • (n x M ^ , ^ ) dS

(7.40)

for p = 1,2,3, n = 0,1,..., m = —n,..., n, and
Jinx
s

hs)' (n X M ^ , ^ ) dS = -j J {nx eo) • (n x N ^ p ) dS
s

(7.41)

for p = 1,2,3, n = 0,1,..., m = —n, ...,n. Then hg solve the general
null-field equation (6.103) and conversely.
Proof: To prove the theorem we define the electric and magnetic fields
S and H by
5 = - V X Aeo + | v X V X A,,,, W = - i v X f

(7.42)

and use the spherical wave expansions of the projections E-^p and Wcp,
p = 1,2,3, inside a sphere enclosed in Dj.


150

CHAPTER VII

SYSTEMS OF FUNCTIONS IN ELECTROMAGNETICS

The following theorem is the analog of theorem 1.4 for the system of
magnetic and electric vector multipoles.
THEOREM

1.8: Let e and h solve one of the following sets of null-field

equations
(a)

y*{[nx(e-eo)]-(nxMLp)
s
+ j

^ [n X (ii - ho)] . (n X N L P ) } d5 = 0,
(7.43)

J [(n X e). (n X M^^^^)
s
+ jJ^

(n X h) . (n X N ^ p ) ] dS = 0,

for p = 1,2,3, n = 0,1,..., m = —n,..., n,
(b)

y{[nx(5-eo)]-(nxNLp)
s
+ jJ^

[n X (h - ho)] • (n X M L P ) | d5 = 0,
(7.44)

I [(n X e) • (n X N^„p)
+ j y ^ ( n x h ) . ( n x M i „ „ p ) dS = 0,
for p = 1,2,3, n — 0,1,..., m = —n, ...,n. Then e anrf h 5o/w the general
null-field equations (6.113) and conversely.
Proof: _The proof follows from the spherical wave expansion of the
projections Es,i • Cp and W«,t • ep, p = 1,2,3, where

(7.45)
and
5, = V X Ai + - j ^ V X V X Ai, Wi(x) = T T ^ V x ^^(x).

(7.46)


1. COMPLETE SYSTEMS OF FUNCTIONS

1.2

151

Distributed spherical vector wave functions and vector multipoles

We begin our analysis by presenting the addition theorems for spherical
scalar and vector wave functions. These theorems were formulated by
Friedman and Russek [63], Stein [136] and Cruzan [36]. We note here that
the translational addition theorems are of vital importance in the multiple
scattering theory of waves (see, e.g. Bruning and Lo [18] and Wang and
Chew [154]). At the same time. Chew et al. [27] used translation formulas in fast-scattering algorithms to reduce the computational complexities
of volume integral equations. These scattering algorithms, have in turn,
brought about renewed interest in the development of efficient ways to compute the translation coefficients of scalar (see Chew [23]) and vector (see
Chew and Wang [25]) wave functions.
To this end let us consider a point x 6 R^ which has the spherical
coordinates (r, 0, v?) with respect to a coordinate system having the origin
at the point O. Let us introduce a second coordinate system with the
origin at the point O', whose coordinates are (rc^o^V^o) ^^^^ respect to
O. The set of spherical coordinates [r\0\(p') is introduced with respect
to the second coordinates system, such that the polar axis, ^' = 0, and
the azimuthal axis cp' = 0, are respectively parallel to the corresponding
axes, 0 = 0 and (f = 0. This is then a rigid translation of the coordinate
system as it shown in Figure 7.1. The addition theorem for spherical wave

functions is:

oo

n j

^^Jt^'(xo)^i;'(A:rO/^r'l(cos0O^^•^'^^r'>ro

(7.47)

n'=Om'=-n' [ B;;j/;^,(xo)^^(A:r')Flr''(cos0')^^^^'^', ^ ' < ^0
where the translation coefficients A^V^, and B!^Jl^, are given by

X }J''{2p+

l)a{m,m'

-

m\n',n,p)

V

X 2i(A;ro)Pp'"-'"'(cos6»o)e-'('"-'"')'^o,
(7.48)
B-,"„,(xo)

=

j"'-"(2n' + l)(-l)'"'(Cw/C^n)

X y j j''a('"> —'m'\p, n, n')
p


152

CHAPTER VII SYSTEMS OF FUNCTIONS IN ELECTROMAGNETICS

FIGURE 7.1

Coordinate translation.

and the normalization constant
m>0
(7.49)

^

^

(n-f|m|)!'
5|m|

has been introduced to assure the equivalence P^ = CmnPn • Here, P^
stand for the Legendre functions with positive and negative values of the
index m. The coefficients a(.) are defined by the spherical harmonics expansion theorem
P^{cos0)P;p'{cos0)

= ^a(m,m'|p,n,n')Pp^+^'(cos0).

(7.50)

It is apparent from (7.50) that a(.) can be determined in terms of an integral
over a special triple product of associated Legendre functions, i.e.
a(m, m'|p,n,n')

=

2p-f 1 (p — m — m')!
2 (p -h m -f m')!
X / P^(cos e)P;^' (cos (9)Pp^+^' (cos 0) sin 0d0.
0

(7.51)
We remark that both representations in (7.47) can be used for z^{kr) =
Jn(fc^) without restriction on the relative size of r' and TQ. Indeed, using
(7.51) and the identity

prn ^ (_i)ml!i±ili}!p^
(n - m) | - n

(7.52)


1. COMPLETE SYSTEMS OF FUNCTIONS

153

we obtain
(2n' -h l)a(m, -m'\p, n, n') = (-l)"'(2p -f l)a(m, m' - m|n', n,p); (7.53)

whence ^J^^i/ = B^Jl^, follows. An inspection of (7.47) reveals that for
r' > ro the translation coefficients are identically for any dependence on
position (z^{kr) or z^{kr)).
We will now derive an integral representation for the translation addition coefficients. The departure point is the plane wave integral representation of the regular spherical wave functions, i.e.
t^mn(x)

=

jn{kr)PJr^ {cos 0)e^^^

=

1
^ _

^}^
/ /
0

(7.54)
p}r^{cosP)e^'^''e^^"'smPdPda,

0

where a and /3 are the angular spherical coordinates of the wave vector k.
Using the spherical wave expansion of the plane wave
oo

n'

n'=Om'=—n'

(7.55)
and the identity exp(jk • x) = exp(jk • XQ) exp(jfk • x'), we obtain
oo

"i.„(x) = l ]

n'

^

^-r„,(xoR,„,(x'),

(7.56)

n ' = O m ' =—n'

with
;

A^^n'i^o)

=

^^^Vm'n^

27r TT

f

f

PJT^(cOS0)PI?'^{cOS0)

{{

(7.57)

X e^"("^-^')^e^^'^osin/?d/?da.
Note that the expansion (7.55) is a uniformly convergent series and hence
when it is substituted into equation (7.54) the order of summation and
integration can be interchanged. We also remark that the expansion (7.56)
is valid without restriction on the relative size of r' and TQ. The integration
with respect to the azimuthal angle a can be analytically performed by
using the representation
k • xo = fcpo sin /? cos(a —
(7.58)


154

CHAPTER VII SYSTEMS OF FUNCTIONS IN ELECTROMAGNETICS

and the identity
27r

f^jxcos{a-<Po)eJ(^-^>da

= 27rj^'-^J^/_n,(a:)e-^'(^'-^)^o,

(7.59)

0

where

(PQ, (PQ^ZQ) are

the cyUndrical coordinates of O'. We arrive at
n


=

2j-'+-'-—P^,n'e-^(^'-^)^o/'j^,_^(fcposin/3)
0

X e^^'^ cos0plm\^^^g^)pl^7^'l(cos/?) sin/?d/3.
(7.60)
If the translation is along the ^-axis the addition theorem involves a single
summation
CX)

txJnnW = Yl ^ m n ' ( ^ 0 ) « ; „ „ . ( x ' )
n'=0

(7.61)

and the translation coefficients simplify to
TT

AX'(^o) = 2 j " ' - " P „ „ , y*e^'=^<'~»^/^'"l(cos/?)/^7'l(cos^)sin^d/3.
0

(7.62)
As mentioned before, the representation (7.62) is valid for the translation coefficients of the radiating spherical wave functions in the case r' > TQ.
This result can also be established by using the concept of quasi-plane
waves. To show this let us consider the plane wave integral representation
of the radiating spherical wave functions
2ir 7r/2-j(x>


J
0

P}rHcos/3)e^'^'^e^''--sm0dl3da.

(7.63)

0

The representation (7.63) is valid for 2 > 0, since only then the integral
converges. Let us recall the definition of the quasi-plane wave Q(x, k). For
^ > 0 it is
27r

Q{x,k)=f

n/2-joo

f
0

0

A(u;,a;')e^'^'"sin/3'd/3'da',

(7.64)


1. COMPLETE SYSTEMS OF FUNCTIONS

155

where
A{w,w') = - f ;

X ; P„.„Pi'"l(cos/3)/^'"l(cos/?')e^'"("'-").

(7.65)

n = 0 7Ti=—n

Note, that if /3 and ^ ' belongs to [0, TT], the expression of A(a;, a;') simpHfies
to
A(a;,a;') = 6{a - a')5(cos^ - cos^O-

(7-66)

The expansion of quasi-plane waves in terms of radiating spherical wave
functions is

Q(x',k) = f ;

f2

2r'V,n,n.Pl?'\cos0)e~^^'"ul,M)-

(7.67)

n'=Om'=—n'

The function A(u;,a;') allows to formally analytically continue a function
/ ( a , /3), defined for real a and /? onto the complex values of/?. In particular
we see that
/(a,/?) = y y/(a',/?')A(a;,a;Osin/?'d/?'da',
0

(7.68)

0

where f3 can assume all values lying on the contour [0,7r/2 — joo]. Then,
we have
27r n/2-joo

2TT TT

/

/

/(a,/3)e^*^-^sin^d/?da = /

0

0

0

//(a,^)Q(x,k)sin/3d/3da,
0

(7.69)
where we have used the definition of Q(x, k). Consequently, (7.63) can be
written as
ul^ix)

= ^

y y*/^-l(cos/3)e^--e^'^-^"Q(x',k)sin/?d/?da.
0

(7.70)

0

Inserting (7.67) into (7.70), and assuming that we can interchange the order
of summation and integration we get
oo
3
'^mn

W = E

n'

E

n'=OTn' =—n'

^rn'(xo)«^<„'(x'),

(7.71)


156

CHAPTER VII SYSTEMS OF FUNCTIONS IN ELECTROMAGNETICS

where A!^?^, is given by (7.57). Thus, the proof of our assertion is completed.
We now pay attention to the vector case. The standard technique
for deriving the translation formula is to start from the definition of the
spherical vector wave functions and to use the addition theorem for scalar
wave functions. We obtain
Mmn

(x)

=

VUmn (x) X (XQ + x ' )

= 5Z E ^;;:^„'(xo)vu„»-„'(x')x(xo+x')
OD

n'

n'= 0 m' ——n'

(7.72)
and the remainder of the analysis then consists in expressing the vector
quantity Vum'n' x XQ in terms of M ^ ' n ' and Nm'n'- Using the orthogonality properties of the spherical vector wave functions and making some appropriate changes in certain of the summation indices, we find after lengthy
calculations that the addition theorem under coordinate translation has the
form

M„„(x) =

N„„(x)

=

oo

n

E

E

c»

n'

E

E

^;;;J'„'(XO)M„,„-(X') + B;;:J;,(XO)N„.,„,(X'),

^;;j^„'(xo)N„'n'(x') + B r n ' ( x o ) M w ( x ' ) ,
(7.73)

where the explicit expressions for A!^?^, and B^J\^, can be found elsewhere
(see, e.g. Stein [136]).
As in our previous analysis we will now derive integral representations
for the translation coefficients A!^?^, and B^/^/. To this end let us consider
the plane wave integral representation of the regular spherical vector wave
functions

0 0

+ J>k"''(/?)ea]

(7.74)

e^*'''e^"'°'sm(3d0da


1. COMPLETE SYSTEMS OF FUNCTIONS

157

and
2n TT

•"

0 0

(7.75)

+ jm7rlr'(/3)e„] e^'^^'eJ'""sin/3d/3da,
where TTIT'C^) = PI"*'(cos/?)/sin/? and rL'"'(/?) =dPl"''(cos/3)/d/J.
We concentrate on the representation of M^„ and define the polarization vector e^„ by
e^„ (Q, /3) = mTrJTl (/?) e^ + jriTl (/3) e,.

(7.76)

Clearly, e^„ • k = 0, and making use of the spherical wave expansion of the
plane wave
elAa,0)e^^-'

=

oo

n

"£

Yl

^m'n'[a^^^j;,, (a,^)M^,„,(x')

n'=lm'=-n'

(7.77)

+ 6-r„,(a,/3)Nj„,„,(x')],
where
a-i;, (a,;3) = -4i"'+i [mm'Trk'"! (/3)7rl,7'l (^) + rL'"l (/?)rL7'' (/?)] e"^-'-,
6-r„, (a,/?) = -4j"'+i [mTrl'"! (/?)T^T'I (^) + m'rt^ (/?)TrL?'' (/?)] e-^-'",
(7.78)
we obtain
ML(x) = f;

J2

^rn'(xo)Mj„,„,(x') + Br„'(xo)Nj„'„-(x'),
(7.79)

with
^rn'(xo)

=

^£»w//[mm'7rL'"l(/?)7rJ,7'l(/9)
0 0
•f rn

(/?)r|;?''(/?)] e-?'(^-"^')^e^*^-^o sin/3d/?dQ;

(7.80)


158

CHAPTER VII SYSTEMS OF FUNCTIONS IN ELECTROMAGNETICS

and
27r TT

,

B'i^^n'i^o)

=

^D„,„,yy[m7rL'"l(^)rLT'l(/3)
0

0

+ m'TL"''(/3)7rl,7'l(/3)] e>('"-'"')°eJ'''"'sin/3d/3da.
(7.81)

Further integration with respect to the azimuthal angle a gives

TT

xjjm'.rnikposinp)

[mm'7rL"^«(/3)7rl,7''(/3)

(7.32)

[m7rL^'(/?)Tl,?''(^)

. (7, 83)

0

and

TT

X

J Jm'-m{kposin^)
0

mVir'(/?)7ri7''(/?)] e^'^^o ^^«^ sin^d/?.
If the translation is along the ^-axis the double summation reduces to a
single summation over the index n' and we have

(7.84)

0

+ TI'"'(^)^1,'^'(/?)] eJ'=*»^°«^sin/?d/?
and
TT

0

+ rL'"'(/3)7rl:?l(/3)] eJ^^o-^^^^sin^^dyS.

(7.85)


159

1. COMPLETE SYSTEMS OF FUNCTIONS

FIGURE 7.2

The support of multiple spherical vector wave functions.

In the case r' > ro , the addition theorem for radiating spherical vector wave
functions contains the same translation coefficients A^, and 6JJJ"/ . This is
obvious, from the derivation leading to (7.73), and the fact that the addition
theorem for the scalar case involves the same translation coefficients for
regular and radiating functions.
We are now well prepared to present the completeness results of this
section. Since the localized spherical vector wave functions are complete
on the particle surface it seems to be possible to approximate surface fields

by a number of sequences of spherical vector wave functions with different
origins. These expansions do indeed provide enough freedom to solve geometrically complicated problems. In this context, let us consider a finite
sequence of poles {xop} ^ ^ distributed inside Di and let us define the set
of multiple spherical vector wave functions by
Mj^3,p(x) = M ^ ^ J x - xop), Ar^'^p(x) = N^^^Jx - xop),

(7.86)

where p = 1,2,...,P, n = l,2,...,m = - n , ...,n. Clearly, A 4 ^ „ P , A/'^J.^p is an
entire solution to the Maxwell equations and A<^„p, A/'^^^p is a radiating
solution to the Maxwell equations in R^ - {xop}. The distribution of the
poles is shown in Figure 7.2.
Then we can state the following theorem.
T H E O R E M 1.9:
Consider Di a bounded domain of class C^ and let
{xop} J be a finite collection of poles inside Di. Replace in theorem 1.1 the
localized spherical vector wave functions M^J^ and N^J^, n = 1,2,..., m =
—n,..., n, by the multiple spherical vector wave functions M^^ip ^'^^-^mnp^
p = 1,2, ...,P, n = 1,2, ...,m = —n^,..n, respectively. Then, the resulting
systems of vector functions are complete in C^ani^)-


160

CHAPTER VII SYSTEMS OF FUNCTIONS IN ELECTROMAGNETICS

Proof: We prove only (a). Let a G >C^an('S') and fix the pole p. Then,
for any e > 0 there exists the integer Np = Np{e) and the coefficients
dmnp and 6mnp, n = 1,2, ...jATp, m = - n , ...,n, such that a / P can be approximated by linear combinations of localized functions n x M^^^p ^^^
n X N^rip with an approximation error smaller than e/P. Thus, for a collection of poles {xop} ^ 1 the triangle inequality may be used to conclude.

Using similar arguments we can prove the following theorem which
establishes the completeness of the multiple spherical vector wave functions
in the product space £tan('S^)T H E O R E M 1.10: Under the same assumption as in theorem 1.9 replace in theorem 1.2 the localized spherical vector wave functions M ^ ^
and N^Jj, n = 1,2,..., m = —n,..., n, by the multiple spherical vector wave
functions M'^r^^p (^^^ J^mlpj P = 1,2,...,P, n = l,2,...,m = - n , . . . , n , respectively. Then, the resulting systems of vector functions are complete in
£?an(5).
Turning now to the null-field equations, we recall that in the case of
the exterior Maxwell problem the null-field equations formulated in terms
of localized spherical vector wave functions guarantee that the total electric
field vanishes inside a sphere enclosed in the particle. Because of its analyticity, the total electric field will vanishes throughout the entire interior
volume. If we now formulate the null-field equations in terms of multiple
spherical vector wave functions, we will guarantee that the total electric
field vanishes simultaneously inside several inscribed spheres.
Let us now extend the system of distributed spherical wave functions
to the vector case. For the time being we consider a set of points {zn}^^i
distributed on the 2;-axis, and define the set of distributed spherical vector
wave functions by

Mkii^)

= M ; ^ 5 „ | + , ( X - znes), ^r^'„{x) = N ^ 5 ^ | + , ( x - z„es),

(7.87)

where n = 1,2,...,m E Z, and / = lif m = 0 and / = Oif m 7^ 0.
M}nni -^mn ^^ ^^ entire solution to the Maxwell equations and M^^^ -^mn
is a radiating solution to the Maxwell equations in R^— {2:ne3}.
Then, the following result holds.
T H E O R E M 1.11: Consider the bounded sequence (zn) C F^, where F
z

is a segment of the z-axis. Assume S is a surface of class C^ enclosing F^.
Replace in theorem 1.1 the localized spherical vector wave functions M^Jj
and N^Jj, n = 1,2, ...,m = —n, ...,n, by the distributed spherical vector
wave functions M^^ andM^^^n = 1,2,..., m G Z, respectively. Then, the
resulting systems of vector functions are complete in /^tan('5')-


161

1. COMPLETE SYSTEMS OF FUNCTIONS

FIGURE 7.3 The support of distributed spherical vector wave functions and the
auxiliary surface S^.

Proof: We consider (a). We have to show that for a € ^tanC'^') the
set of closure relations
Ja^{yy[n{y)xMl,n{y)]dS{y)

=

0,
(7.88)

y^a*(y).[n(y)xAr^,(y)]d5(y)

=

0,

n = 1,2,..., m G Z, gives a ~ 0 on 5. Let S^ be a sphere enclosed in A

and assume F^ c £>[, where Z>[ is the interior of S^. The illustration of
the auxiliary surface S^ and the support of discrete sources is shown in
Figure 7.3. For a fixed azimuthal mode m we use the addition theorem for
spherical vector wave functions to rewrite the closeness relations as
fmizn) = y"a*(y)- [n(y) x M^,|„|+,(y - Zn^s)] d5(y)
s
=

E
n'>max(l,|m|)

-4m;!^'^'(-^n) /a*(y)- [n(y) x M^-Cy)] d5(y)
^

+ C i ™ ' " " ( - ^ n ) y a*(y)- [n(y) x N^-Cy)] d5(y) = 0, n = 1,2,...,
s
(7.89)