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

Acoustic and electromagnetic scattering analysis using discrete sources VII systems of functions in electromagnetic theory Acoustic and electromagnetic scattering analysis using discrete sources VII systems of functions in electromagnetic theory Acoustic and electromagnetic scattering analysis using discrete sources VII systems of functions in electromagnetic theory Acoustic and electromagnetic scattering analysis using discrete sources VII systems of functions in electromagnetic theory 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 space fi^anC'^)- Complete systems in ^^anl-^) will be used to solve the exterior Maxwell and the impedance boundary-value 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 func-tions will be considered We then pay attention to the systems of localized vector multipoles In Chapter 9 we will apply these results to axisym-metric 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

functions are suitable for analyzing the scattering by particles without

ro-tational 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

pri-mary interest since they provide a means for approximating the exact

solu-tions 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

V x V x X - f e 2 x = o (7.1) can be constructed as

Mii?.(x) = V«ii?,(x) X X, Nii^Jx) = i v X M^i^Jx), (7.2)

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

where {er^eg^e^) are the unit vectors in spherical coordinates The

super-script *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}

The spherical vector wave expansion of the dyadic ^I is of basic portance in our investigation It is

[ -f Irrotational terms, |y( < |x|

where the normalization constant Dmn is given by

2n + l (n-|m|)!

•^mn —

4n(n + l) (n+|m|)!" (7.5)

Using the calculation rules for dyadic functions and the identity ag = a-^I

we find the following simple but useful expansions

We are now in the position to establish the completeness results for the systems of spherical vector wave functions

T H E O R E M 1.1: Let S be a closed surface of class C^ with unit outward

normal n Then the following systems of vector functions

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

The closure relations show that £ = 0 m Di, Application of theorem 2.2

given in Chapter 6 finishes the proof of (a)

The second part of the theorem corresponding to the case fc ^ <j(A)

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

gives a ~ 0 on 5 Before we present the proof we note some basic

re-sults 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

if k is not an interior Maxwell eigenvalue, and

d i m N (^I-hM\ = d i m N (^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

chap-ter 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

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

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

men-tion that the system

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

radi-ating spherical vector wave functions in £tan('S') yields {^I-\- M^) a' = 0 almost everywhere on S Therefore ( | l ~ M) (n x a') = 0 almost every-

where on 5 and employing the same arguments as in theorem 2.2 of

Chap-ter 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

prod-uct 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

Proof: It has to shown that for

tions (iO € £tan('S') *^he closure

n = 1,2, , m = —n, ,n, gives Si '-^ 0 and a2 ^ 0 on 5 Setting a i =

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)

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)

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

n = 1,2, , m = —n, ,n, where the radiating spherical vector wave

func-tions are defined with respect to the wave number kg, while the regular

spherical vector wave Junctions are defined with respect to the wave

num-ber 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 A i (7.28) inside a sphere enclosed in Di and outside a sphere enclosing Dj, respec-

tively

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

equa-tions 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,

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)

to derive the expansion

The following theorem establishes the completeness of the localized

electric and magnetic vector multipole systems in £?an(*5)*

normal n Then the following systems of vector functions

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

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

normal n Then the following systems of vector functions

for p = 1,2,3, n = 0,1, , m = - n , ,n Setting a'^ = n x aj and

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

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

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 A i , Wi(x) = T T ^ V x ^^(x) (7.46)

1.2 Distributed spherical vector wave functions and vector multipoles

We begin our analysis by presenting the ađition 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 ađition theorems are of vital importance in the multiple

scattering theory of waves (see, ẹg Bruning and Lo [18] and Wang and

Chew [154]) At the same timẹ Chew et al [27] used translation

formu-las 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

com-pute 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 Ọ Let us introduce a second coordinate system with the

origin at the point Ó, whose coordinates are (rc^ôV^o) ^^^^ respect to

Ọ 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 ađition theorem for spherical wave

X y j j''ắ"> —'m'\p, n, n')

p

(7.48)

FIGURE 7.1 Coordinate translation

and the normalization constant

m > 0

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

(7.49)

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 ặ) are defined by the spherical harmonics ex-pansion theorem

we obtain

(2n' -h l)ăm, -m'\p, n, n') = ( - l ) " ' ( 2 p -f l)ăm, 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

ađi-tion coefficients The departure point is the plane wave integral

represen-tation of the regular spherical wave functions, ịẹ

t^mn(x) = jn{kr)PJr^ {cos 0)ê^^

1 ^}^ (7.54)

= ^ _ / / p}r^{cosP)ế^''ê^"'smPdPda,

0 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 ' = O m ' = — n '

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

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 — <Po) + kzo cos (3 (7.58)

and the identity

As mentioned before, the representation (7.62) is valid for the

transla-tion coefficients of the radiating spherical wave functransla-tions 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>

<ni^)=^J J P}rHcos/3)ế^'^ế' sm0dl3dạ (7.63)

0 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 n/2-joo

Q{x,k)=f f Ău;,a;')ế^'"sin/3'd/3'dá, (7.64)

Ăa;,a;') = 6{a - á)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)ẽ^^'"ul,M)- (7.67)

n ' = O m ' = — n '

The function Ău;,a;') allows to formally analytically continue a function

/ ( a , /3), defined for real a and /? onto the complex values of/? In particular

written as

ul^ix) = ^ y y*/^-l(cos/3)ê ế^-^"Q(x',k)sin/?d/?dạ (7.70)

0 0

Inserting (7.67) into (7.70), and assuming that we can interchange the order

of summation and integration we get

oo n'

3

where A!^?^, is given by (7.57) Thus, the proof of our assertion is

com-pleted

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

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

orthogonal-ity properties of the spherical vector wave functions and making some

ap-propriate changes in certain of the summation indices, we find after lengthy

calculations that the addition theorem under coordinate translation has the

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 (7.74)

+ J>k"''(/?)ea] e^*'''e^"'°'sm(3d0da

and

2n TT

•" 0 0 (7.75)

+ jm7rlr'(/3)e„] ế^^'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

polariza-tion vector ê„ by

ê„ (Q, /3) = mTrJTl (/?) ê + jriTl (/3) e , (7.76) Clearly, ê„ • k = 0, and making use of the spherical wave expansion of the

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

M L ( x ) = f ; J2 ^rn'(xo)Mj„,„,(x') + Br„'(xo)Nj„'„-(x'),

(7.79) with

^ r n ' ( x o ) = ^ £ » w / / [ m m ' 7 r L ' " l ( / ? ) 7 r J , 7 ' l ( / 9 )

0 0 (7.80)

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

mVir'(/?)7ri7''(/?)] ế^^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

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 ge-ometrically 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

{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^)-

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 ap- proximated 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

col-lection 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

re-place 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 , re- spectively Then, the resulting systems of vector functions are complete in

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

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')-

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

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)] d 5 ( y )

an-n = 0,1, , where the superscript an-n an-now dean-notes the derivative of order an-n with respect to z Explicitly we have

/i")(0) = Yl Ann' / a ' ( y ) - [n(y) x M ^ „ , ( y ) ] d5(y)

rm

TT

= / cos^ 0 sin'^^l /? P^r' (cos /?) sin /?d^, (7.98)

with n = 0,1, , n' > max(l, |m|) Note, that in (7.96) and (7.97) we set

by convention /!!\^„/ = 0 The integral / ^ / can be computed as follows

Integration by parts and the recurrence relation

for n = 0,1, , n' > max(l, |m|) Therefore, using the series expansion of

the spherical Bessel functions we arrive at

rm w ^ i , n - ^ , n , - n ( ^ ^ + H ) ! ( - l ) ^ ^ ' - ' ( n - 4 - A : ) ! n !

(7.105) The coefficients / ^ , are nonzero if, for a fixed azimuthal mode m and a

given pair {n,n^), there exists an integer k, k = 0,1, , such that 2k =

n - n' -f |m| Consequently, A"^)^, = 0 for n' > n -f 1 and m = 0, while

Ấ^, = 0 forn' > n -f \m\ and m ^ 0 Similarly, B;;;,, = 0 for m = 0, and

B;^^, = 0 forn' > n + |m| - 1 and m^Ọ Thus, from (7.88) we obtain

for n' > max(l, |m|) The completeness of the localized spherical vector

wave functions now concludes the proof of (a) In an analogous manner we

can prove the rest of the theorem

Complete systems in the product space £?an('^) ^^^ given by the

fol-lowing theorem