Acoustic and electromagnetic scattering analysis using discrete sources IX discrete sources method in electromagnetic theory

Acoustic and electromagnetic scattering analysis using discrete sources IX discrete sources method in electromagnetic theory Acoustic and electromagnetic scattering analysis using discrete sources IX discrete sources method in electromagnetic theory Acoustic and electromagnetic scattering analysis using discrete sources IX discrete sources method in electromagnetic theory Acoustic and electromagnetic scattering analysis using discrete sources IX discrete sources method in electromagnetic theory Acoustic and electromagnetic scattering analysis using discrete sources IX discrete sources method in electromagnetic theory

of the incident field in the L^-norm We do not repeat the technical pects of the method which are fully presented for the acoustic case Our analysis is mainly concentrated on the construction of convergent approx-imations using the fundamental theorem of discrete approximation For the impedance boundary-value problem we will present a somewhat dif-ferent approach which we call the D-matrix method since the matrix of the corresponding linear system of equations is dissipative We will estab-lish the dissipativity, the convergence of the approximate solution and the solvability of the linear system of equations using the conservation law of energy Special attention is paid to the discrete sources method with dis-tributed vector multipoles The mathematical foundation of the method

as-is accompanied by results of computer simulations The numerical iments include comparison with other methods and scattering analysis of concave particles and clusters of particles

exper-203

1 PROJECTION METHODS

In this section we will use theorems 1.1 and 2.2 of the precedent chapter

to construct approximate solutions to the boundary-value problems for the Maxwell equations The analysis is valid for any linear independent and complete system of functions

To this end let us construct the approximate solution to the exterior Maxwell boundary-value problem as a finite linear combination of fields of

elementary sources If the sources are located in the domain Di the

ap-proximate field E ^ , H ^ will be regular everywhere in the domain £>«, will satisfy the Maxwell equations and the radiation condition at the infinity

In this context the representation formula for the exact solution given by (6.79) is also valid for the approximate solution Combining these relations

we find the estimate

in any closed subset G, of Dg According to (9.1) we see that the exterior

Maxwell boundary-value problem simplifies to the approximation problem

of the tangential components of the incident electric field on the particle surface

In the following, we will consider the system of radiating solutions

to Maxwell equations {*J^,$J[} with the following properties: the set {n X $J[, n X $^} is linearly independent and complete in £tan('S'), V x

^3 -, ;,$3 g^jjj V X $^ = k'if^ Specifically, the system {*^, <^^} stands for

the systems of localized and distributed spherical vector wave functions and vector Mie potentials For the system of magnetic and electric dipoles, we

take {*^i,*^2} ov {$^i,$^2} instead of {*^,*^} , where ^l^ = M^^,

subsets of i?5

Before we proceed to analyze the impedance boundary-value problem

we note some observations:

1 The solution to the exterior Maxwell boundary-value problem can

be represented by

E,(x) = f ; a^M^ (x) + 6^N3 (x), (9.6)

the convergence being uniform in any closed subset which belongs to the

exterior of the circumscribed sphere 5^ Here // is a complex index

con-taining m and n, that is /i = (m, n) and ^ = 1,2, , when n = 1,2, , and

m = —n, , n The coefficients a^ and 6^ are given by

a^ = ^ Z ? ^ J [(n X E,) • N i + j (n x H,) • Mi] dS,

(9.7)

b^ = ^ Z ) ^ y [ ( n x E , ) M 3 + j ( „ x H , ) N | ] d 5 ,

where /i = (—m^n) Let E ^ be an approximate solution to the exterior

Maxwell problem Then, JlEso-" Efo||2n —> 0 as iV —> oo, where E50 and E ^ are the far-field patterns of the exact and approximate solutions,

respectively, and ft is the unit sphere If the system {^^,<l>^} represents

the system of localized spherical vector wave functions we see that

E K - ^I'lKoIlL +1^ - ^riKolL

/ x = l

(9.8)

+ E M' W^Ullu + l^'^i' l|N^o|l2,n - 0 a^ JV - 00

Since, for x € fi,

n-f-l

•i-j)

N3o(x) = N ^ „ o ( x )

jm Pi"! (cos g) sin^

2 The system of equations (9.4) is uniquely solvable even for A = 0

Indeed, assume A: > 0 and set TAT = [a^, 6 ; ^ ] S / i = 1,2, ,iV Multiply each of the first N equations of the homogeneous system by a^*, each of the second N equations by b^*^ and sum the final expressions We obtain

(D^Tyv, TN)I2 = j ( n X E f , H f )^ ^ , (9.11) where D^r is the matrix of the resulting system of equations From Gauss' theorem we known that

Re J|(nxE,^)Hf*d5Uy Efoi'da; (9.12)

provided that k > 0 Taking the imaginary part of (9.11) we are led to

Q

The above equation shows the dissipativity of the matrix D;v? i.e

Im(DivTAr,TAr),2 >Ofor any T^r ^0

Now, let {DNTN,TN)I2 = 0 From (9.13) we see that E ^ = 0 on ft,

and further that E ^ = 0 in D^ The linear independence of the system

{n X ^ ^ , n X $^} finishes the proof of our assertion

The systems of discrete sources can also be used in the

boundary-integral equations treatment of the exterior problem For instance, let the

scattered electric field be represented as a surface distribution of magnetic

dipoles

Es{x) = V X y a(y)^(x,y,A:)d5(y), (9.14)

where a solves the integral equation

f ^ r + - M ) a = eo (9.15)

Fork^a {D^) the operator \l-\-M : Cl^^^{S) -> C^t^^^^{S) is bijective and

has a bounded inverse An approximate solution to the integral equation

(9.15) can be sought in the form

N

by using the projection relations given in theorem 3.4 of Chapter 1 Note

that the operator M : £?an('S') —^ ^tan('5) is compact and for fc ^ cr [Di),

by the Fredholm alternative applied in different dual systems, the

opera-tor ^T-\- M : >C^an(«S') -^ ^tan('^) ^^ bijcctive and has a bounded inverse

Consequently, the approximate solution

E f (x) = V X y a;v(y)5(x,y,fe)d5(y) (9.17)

converges to the exact solution in closed subsets of Dg- Furthermore, we

have n x E^^ — n x Ef*^ -^ 0 as AT -^ oo, and this conclusion agrees

with the completeness result establish for the system of functions

{ Q l + A^) (n X ^ 3 ) Q j ^ ^ - J ( „ ^ $3^^

M = l , 2 , / f c ^ < 7 ( A ) }

Similar arguments can be employed for the combined integral equation

(Ix-^M-^jXAfPisi) a = eo, (9.18)

with a real coupling parameter A ^ 0 Here, the operator Pr stands for the

projection of a vector field defined on S onto the tangent plane

P r a = ( n x a ) x n (9.19)

and So is the single layer operator in the potential Umit case fc = 0 In this

case the scattered electric field is represented as a combined magnetic and

electric dipole distribution

E,(x) = V X Ja{y)g{x,y,k)dS{y)

s -h jXV X V X jn{y) x {SSeL){y)g{x,y,k)dS{y)

(9.20)

s

Note that the operator i J -h A^ -h j W P r 5g : ^ ^ , ^ ( 5 ) -^ Cj*;^,rf(5) is

bi-jective and has a bounded inverse Furthermore, we have compacteness of

M -f j W P r 5 g from £tan('S) into £tan('5), and by the Predholm

alterna-tive applied in different dual systems, the operator ^X •\- M •\- JXMVXSQ :

^tan('S') —• ^tan('S') is bijcctivc and has a bounded inverse

Approximate solutions to the impedance boundary-value problem can

be constructed in an analogous manner The representation formulas for

the exact and approximate solutions E«,Hs, and E ^ , H ^ yield the

esti-mate

i | E - E n U a + l | H - H ^ ' I U

< C ||(n X E, - 7n X (n X H,)) - [n x E ^ - i n x (n x H^)] ||

(9.21)

in any closed subset Gs of Dg Evidently, the impedance boundary-value

problem simplifies to the minimization problem of the residual fields on the

It is readily seen that we may use part (a) of theorem 1.1 given in Chapter 8

to determine the ampUtudes of discrete sources from the normal equations

/ n X E f - 7n X (n X H f ) - f,n x ^l -hnn x (n x ^l)\ = 0,

/ n X E f ~ 7n X (nx H f ) - f, n x $^ -h j'yn x (n x * ^ ) \ = 0,

(9.23) where i/ = 1,2, , AT, Re7 > 0 and f = n x E ^ - i n x (n x H^) The linear

independence and the completeness of the system

{n X *^ H-J7n x (n x $^) ,n x $^ -f J7n x (n x * ^ ) ,

i/ = l,2, / R e 7 > 0 }

guarantee the convergence of the approximate solution in closed subsets of

We will now present an alternative approach which we call the

D-matrix method since the D-matrix of the resulting system of equations is

dissipative The dissipativity is established as a consequence of the

conser-vation law of energy

Let us determine the coefficients a^ and b^ from the following

projec-tion relaprojec-tions

^ n x E f - 7 n x ( n x H f ) - f , $ ^ ) = 0,

(9.24)

^ n x E f - 7 n x ( n x H f ) - f , ^ ^ ^ = 0,

for u = 1,2, Substituting the expressions of the approximate electric

and magnetic fields into (9.24) we obtain the following system of equations

Dll = J[nx^l-i-nnx{nx^l)]-^l*dS,

s

(9.26)

For simplicity, let us assume fe > 0 and R e 7 > 0 Then, the following

theorems are valid

THEOREM 1.1: The system (9.25) is uniquely solvable for any fixed

N

Proof: We will establish the dissipativity of the I^N matrix, i.e we

will establish that the inequahty Im (DTVTTV, TAr)^2 < 0 holds for any T^v =

[a^, 6^] , /x = 1,2, , AT, and in addition that {DNTI^.TN)^ = 0 if and

only if TN = 0 We choose an arbitrary T^^ and construct the vector fields

E ^ and H ^ according to (9.22) It is readily seen that

(DyvT^v, Tw), = -j ( n X E,^ - 7 n x ( n x H,^) , H , ^ ) ^ ^ (9.27)

Taking the imaginary part of this equation and using (9.12) we obtain

Im (D;vT,v, T;v),2 = - J |E,^ol'du; - J Re7|n x H^\^dS, (9 28)

Since R e 7 > 0, (9.28) shows the dissipativity of the matrix D^^ The claim

(DTVTJV, Tiv)/2 = 0 if and only if T^^ = 0 follows as in observation 2

T H E O R E M 1.2: Let the approximate solution to the impedance

boundary-value problem be given by (9.22), where the vector of amplitudes

Tjsj solve the system (9.25) Then

lim ||E«o-E,^o|l2n = 0, (9-29)

where E50 is the far-field pattern of the exact solution E^, Hg

Proof: We debut by showing that the sequence H ^ is bounded for

any N In (9.24) we multiply the first set of N equations by ja* , the second

one by jbl and sum the resulting expressions We get as before

( n X E,^ - 7 n X (n X H ^ ) - f, H^)^^^ = 0 (9.30)

2

Taking the real part of (9.30), using (9.12), and the identity | a ~ b | =

|a|^ 4- |b|^ - 2 Re (a • b*), we find that

+ / | v / R i T n x H f - ^ n x f

(9.31)

The above relation shows that the sequence H ^ is bounded for any N

Since the system

Here, cjv = n x £^ — jnx (n x H^) — f represents the discrepancies of

the tangential fields on the surface 5, and £^ is the far-field pattern of the

approximate solution £^, H^ Then, we use (9.24) and

(9.35) ( n x f f - 7 n x ( n x K f ) - f , * ^ > 2 , 5 = (€7V,*^>2,5

to derive the following set of equations

(9.36)

( n x 5 E f - 7 n x ( n x 6 H f ) , * 3 ) ^ ^ = {eN,^l\s

for the residual fields SE^ = f f - E f and 6H^ = W f - H ^ Since 6 E f and

5 H ^ are expressed as linear combinations of ^ ^ and ^ ^ , fi = 1,2, ,A/^,

gives

The triangle inequality and (9.34) may now be used to conclude

From (9.26) we see that in the case of axisymmetric scatterers the

surface integrals simplify to one-dimensional integrals along the particle

generator The problem decouples over the azimuthal modes and therefore

the amount of computer storage required to solve the scattering problem is

not excessive high In contrast, for particles without rotational symmetry

it is not possible to obtain a separate solution for each azimuthal mode

Consequently, the dimensions of the linear systems of equations

consider-ably increases This leads to increased difficulties that are associated with

the stability of the solutions Effective solutions of these systems appear to

be possible only by means of iterative schemes It is therefore reasonable to

analyze the correct solvability of the system (9.25) in order to investigate

the applicability of iterative schemes For the system of localized spherical

vector wave functions we can prove the following theorem

T H E O R E M 1.3: The system (9.25) is correctly solvable for any fixed

N

Proof: We have

/

^ 1 (9 40)

This implies the correct solvability of the system (9.25), i.e ||DivTAr||/2 >

c\\TM\y

In this context it is possible to use iterative schemes such as GMRES

(see, e.g Saad and Schultz, [132]) for any fixed value of the truncation

index N

We will now present some computer simulations in order to give a clear

picture on the convergence of the D-matrix method As reference we choose

the conventional null-field method The conventional null-field method for

the impedance boundary-value problem consists in the projection relations:

imation of the tangential component of the total magnetic field may be

sought in the form

N

n x H ^ = - j ^ a ; y n x $ i - f 6 ^ n x * i , (9.43)

where the amplitudes a ^ and b^, ^1= 1,2, , iV, solve the truncated system

of equations (9.41) The approximate scattered field is

E f = f ; / , ^ M ^ + 5 ^ N ^ , (9.44) where

fc2 / i ^ = - ^ y ( n x H ^ ) ( M H i 7 n x N ^ ) d 5 ,

s 9^ = ~ y ' ( n x H ^ ) ( N ^ + J 7 n x M i ) d 5

(9.45)

19 2 3 27

T r u n c a t i o n i n d e x

31 3 5

FIGURE 9.1 Normalized differential scattering cross-section at the scattering angle

0 = 180° for different values of the truncation index The scatterer is a prolate spheroid with 7 = 1 and semiaxes ka = 10 and kb = 2

and V = {—m^n)

In our examples we consider the scattering by prolate spheroids The cident field is a p-polarized plane wave propagating along the particle sym-metry axis The surface current density is approximated by linear combi-

in-nations of distributed spherical vector wave functions M^\ and M^n- The

expansion coefficients can be found separately for each azimuthal mode m and note that only two azimuthal modes, m = ± 1 , are required for solution construction Therefore, the convergence of the projections schemes can be analyzed by varying the number of discrete sources rimax- Assuming the incident field to have unit amplitude we evaluate the differential scattering cross-section (DSCS) in the azimuthal plane (p = 0° Figures 9.1 and 9.2 show the normalized differential scattering cross-section at the scattering

angle 0 = 180° for a prolate spheroid with semiaxes ka = 10 and kb = 2

and two values of 7, 7 = 1 and 7 = 8 In addition to the conventional null-field method we used an approach in which the surface current density

is approximated by the tangential components of radiating functions, i.e

n X

N

(9.46)

Q B B B

B-22 26 30 34 38 42 Truncation index n rn^^

FIGURE 9.2 The same as in Figure 9.1 but the curves correspond to 7 = 8

In Figure 9.3 we plot the differential scattering cross-section for a prolate

spheroid with ka = 10 and kb = 2 The curves are computed with the

con-ventional null-field method and the D-matrix method In order to achieve convergence we use 18 discrete sources in the first case, while 38 discrete sources are necessary in the second case The discrepancy concerning the number of sources used is more pronounced if the aspect ratio increases The results plotted in Figure 9.4 correspond to a prolate spheroid with

ka = 10 and kb = 1 As before, 18 discrete sources lead to accurate results

in the conventional method, while 48 discrete sources are required in the D-matrix method Our numerical analysis demonstrates that the D-matrix method leads to convergent results but its efficiency is inferior to that of the conventional null-field method

We pay now attention to the transmission boundary-value problem Let us represent the approximate electric and magnetic fields E ^ and H ^

in the exterior domain D5 as a linear combination of discrete sources with

singularities distributed in A In this case E^ and H ^ are analytic in Ds' Analogously^ let us represent the approximate electric and magnetic fields Ef^ and Hf^ in the interior domain Dt using elementary sources distributed in Dg Consequently, Ef^and Hf^ are analytic in D^ Then, the

representation formulas (6.93) and (6.94) are also valid for the approximate solutions Taking into account the continuity conditions on the particle surface, we obtain the estimate

FIGURE 9.3 Differential scattering cross-section for a prolate spheroid with 7 = 8

and semiaxes ka = 10 and kb — 2

FIGURE 9.4 Differential scattering cross-section for a prolate spheroid with 7 = 8

and semiaxes ka = 10 and kb — 1

in any closed sets Gg C As and G/ C D? The estimate (9.47) reflects the

basic principle of the discrete sources method: an approximate solution to the transmission boundary-value problem minimizes the residual electric and magnetic fields on the particle surfacẹ

Once again we may rely on the result of the preceding chapter to struct the solution The departure point is to represent the approximate solutions as

con-M = i Ms,t

_#3,l(x)

(9.48)

V l^s,i

for X € Ds^i, and to determine the amplitudes ấ*^, 6^'*^, fi = 1,2, , AT,

by using the completeness of the system

for 1/ = 1,2, , N Further application of this theorem leads to projection

schemes involving the control parameter A

The systems of discrete sources can also be used in the integral equations treatment of the transmission problem The transmis-sion problem is usually reduced to a pair of coupled integral equations for a pair of unknown tangential vector fields These fields may be approximated

boundary-by entire-boundary bases consisting in fields of discrete sources (9.49) An excellent review of direct and indirect methods using pairs of coupled inte-gral equations over the particle surface was given by Martin and Ola [100) Their analysis also includes several integral equations for single unknown tangential vector fields

We conclude this section with the following remarks

1 The projection schemes (9.3), (9.23) and (9.50) use normal tions for amplitudes determination This technique is employed in the discrete sources method by Cadilhac and Petit [19], and Ikuno

equa-et al [74] There are two special features which are associated

with this method First, the rate of convergence is low For dimensional geometries Yasuura mastered this problem by equip-ping the standard method with a smoothing procedure In this case the convergence of the numerical scheme becomes faster by in-

two-creasing the order p of the iterated kernel In the electromagnetic

case the projection schemes (9.4) and (9.5) for the exterior Maxwell problem, and their analog for the transmission problem have a su-perior rate of convergence than (9.3) and (9.50), respectively We will prove this assertion in the framework of the null-field method The parameter A is a measure of the convergence rate and plays the

same role as the parameter p in the Yasuura smoothing method

The second feature is that this scheme is numerically unstable As

it was shown in the acoustic case the collocation method with square matrices leads to a more stable numerical algorithm

non-2 Actually, any complete system of functions can be used to represent the solution However, the widely used system of functions is the system of multiple spherical vector wave functions In this case one must be careful to choose the location and the order of each pole, the number of multipoles and the distribution of matching points Some useful criteria have been summarized as follows, and a com-puter program for automatically optimizing the expansion functions has been successfully developed Firstly, the matching points on the boundary should be set with enough density, and the intervals be-tween adjacent pairs of matching points must be far less than the wavelength for suppressing errors of expansion on the boundary Secondly, the poles should be neither too near nor too far from the boundary, since the former requirement may avoid alternative errors

corresponding to non-physical rough solution on that interval; and

a well-conditioned matrix will be formulated with the latter

require-ment According to the quasi-local behavior of multipole expansion,

only a restricted area around its origin is influenced by this pole,

so that a reasonable scheme is to let the area of influence of each

pole fit a portion of the boundary, but the whole boundary must

be covered by the areas of influence of all the poles Finally, the

highest order of a pole is limited by the density of matching points

covered by its area of influence On the other hand a term with a

high order n^^^ will result in an oscillatory solution on the

inter-val spanned by the largest view-angle ^p from the nearest poles;

for example, let nj^^x^p <^ ^/2- Pertinent analysis concerning the

location and number of expansion poles was given by Leuchtmann

[91], RegH [129] and Beshir and Richie [12]

2 MODEL WITH DISTRIBUTED VECTOR MULTIPOLES

2.1 Formulations

In this section we will analyze the scattering by axisymmetric particles in

the framework of the discrete sources method with distributed vector

mul-tipoles Specifically, we will construct the approximate solution as a linear

combination of discrete sources by taking into account the polarization of

the external excitation Essentially, our analysis relies on the Bromwich

theorem which states that any entire solution to Maxwell equation can be

represented as a superposition of a TM and a TE wave

We debut with the exterior Maxwell boundary-value problem Let S

be an axisymmetric surface and let (2„) c F^ be a bounded sequence of

points distributed on the symmetry axis Suppose that the approximate

solution is represented as a linear combination of vector multipoles M^^p

and M^np' Straightforward calculations show that the components of these

functions, in cylindrical coordinates, depends explicitly on exp[j(m + 1)<^],

exp{J7rnp) and exp\j{m — 1)<^) This leads to a recursive scheme for

ampli-tude determination with an increased computational complexity In order

to master this problem we consider the set of vector multipoles with positive

values of the index m

k

where r„'"(Tj) = /i^>(A;r„)/^(cos0„), p = 1,2,3, n = 1,2, , and m =

0,1,

Let us assume that the external excitation is a linearly polarized plane

wave with the wave vector lying in the xz-plane and enclosing the angle

j e ^ W = i V x A ^ ^ C x ) , (9.52)

(77 — 7) with the particle symmetry axis In the case of the TM polarization

^ m ^ ( ^ ) = ^ ^ = ^ ( - j ) " ^ + ^ C O S 7 l J m + l ( f e p s i n 7 ) - J m - l ( f c ^

^m^(^) = ( - j ) ^ + ^ c o s 7 [ J ^ + i ( f c p s i n 7 ) + J^_i(fcpsin7)]G(^,7),

^^^ry) = -(2~6„,o)(-J)"^sin7Jm(fcpsin7)G(^,7)

(9.55) and G(z,7) = exp(—jA;>2:cos7) Taking into account that the unit normal

vector to 5 has only p- and 2;-components we have

0 0

n (x) X Eo (x) = ^ e^(T?) sin(mv?)ep + e^^(»7) cos(m(^)e^

m=0

(9.56) + e^^(»7)sin(mv')e, = J ] e ^ ( x )