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2.4 Inhomogeneous Particles

107

on S1 and

n2 × Ei,1 = n2 × Ei,2 ,
n2 × Hi,1 = n2 × Hi,2 ,	(2.67)

onS2, and the Silver–M¨uller radiation condition for the scattered ﬁeld (2.3).

The Stratton–Chu representation theorem for the scattered ﬁeld Es in Di,

where Di = Di,1

Di,2 S2 = Di,1

i,2 = R3 −

s, together with the

boundary conditions (2.66), yield the general null-ﬁeld equation

) +

(r ) g (k

, r

, r ) dS (r )

(2.68)

× S1

i,1

(r ) g (k

, r

) dS (r

) = 0 , r

k0εs 1 × 1 × S1

i,1

while the Stratton–Chu representation theorem for the internal ﬁeld Ei,1 in Ds and Di,2 together with the boundary conditions (2.67), give the general null-ﬁeld equation

(r ) g (k

, r

) dS (r )

− 1 ×

i,1

) g (k

, r

, r ) dS (r )

−k0εi,1

1 × 1 ×

i,1

(r ) g (k

, r

, r ) dS (r )

(2.69)

1 × S2

i,2

i,1

(r ) g (k

, r

, r ) dS (r ) = 0 , r

εi,2

1 × 1 ×

i,2

i,1

i,2

In (2.68) and (2.69), the surface ﬁelds are the tangential components of the electromagnetic ﬁelds in Di,1 and Di,2

ei,1 = n1 × Ei,1 , hi,1 = n1 × Hi,1	on S1
and
ei,2 = n2 × Ei,2, hi,2 = n2 × Hi,2	on S2 ,

respectively.

Considering the general null-ﬁeld equation (2.68), we use the vector spherical wave expansions of the incident ﬁeld and of the dyad gI on a sphere enclosed in Di, to obtain

jks2

$ N

(ksr1) %

(2.70)

ei,1 (r1) ·

(ksr )

π S

$ M

(ksr1) %

$ aν %

µs

+ j

i,1

)

dS (r

) =

−

, ν = 1, 2, . . .

εs

(ksr )

bν

108 2 Null-Field Method

For the general null-ﬁeld equation (2.69) in Ds, we proceed analogously but restrict r1 to lie on a sphere enclosing Di. We then have

jki2,1

$ N

(ki,1r1) %

−

ei,1 (r1) ·

(ki,1r )

µi,1

(ki,1r1)

dS (r )

+ j

i,1

)

εi,1

(ki,1r )

jki2,1

$ N

(ki,1r1 ) %

ei,2 (r2 ) ·

(ki,1r )

$ M

(ki,1r1 ) %

µi,1

+ j

i,2

(r )

dS (r ) = 0 ,

εi,1

(ki,1r )

(2.71)

ν = 1, 2, . . . ,

where the identities ei,2(r2 ) = ei,2(r1 ) and hi,2(r2 ) = hi,2(r1 ) have been used. Finally, for the general null-ﬁeld equation (2.69) in Di,2, we pass from

the origin O1 to the origin O2 by taking into account that the gradient is invariant to a translation of the coordinate system, use the relations

g (ki,1, r1, r1) = g (ki,1, r2, r2) , g (ki,1, r1, r1 ) = g (ki,1, r2, r2 ) ,

and restrict r2 to lie on a sphere enclosed in Di,2. We obtain

jki2,1

$ N

(ki,1r2) %

−

ei,1 (r1) ·

(ki,1r )

µi,1

(ki,1r2)

+ j

i,1

)

dS (r )

εi,1

(ki,1r )

jki2,1

$ N

(ki,1r2 ) %

ei,2 (r2 ) ·

(ki,1r )

$ M

(ki,1r2 ) %

µi,1

+ j

i,2

(r )

dS (r ) = 0 ,

εi,1

(ki,1r )

(2.72)

ν = 1, 2, . . . ,

where, as before, the identities ei,1(r1) = ei,1(r2) and hi,1(r1) = hi,1(r2) have been employed. The set of integral equations (2.70)–(2.72) represent the null-ﬁeld equations for the scattering problem under examination.

The surface ﬁelds ei,1 and hi,1 are the tangential components of the electric and magnetic ﬁelds in the domain Di,1 bounded by the closed surfaces S1 and S2. Taking into account the completeness property of the system of regular and radiating vector spherical wave functions on two enclosing surfaces

2.4 Inhomogeneous Particles

109

(Appendix D), we approximate the surface ﬁelds ei,1 and hi,1 by the ﬁnite expansions

$ eN

) %

n1(r1)

M µ1 (ki,1r1)

i,1

= c1,µ

)

−

εi,1

(r )

1 (k

i,1

r )

i,1

µ=1

µi,1

)

N 1

)

i,1

+d1,µ

εi,1

)

M 1

i,1

)

−

µi,1

(r )

M 3

)

i,1

+c1,µ

εi,1

)

N 3 (k

)

−

µi,1

i,1

)

N 3

)

i,1

+d1,µ

)

M 3

) . (2.73)

εi,1 n

−

µi,1

i,1

The surface ﬁelds ei,2 and hi,2 are the tangential components of the electric and magnetic ﬁelds in the domain Di,2 and the surface ﬁelds approximations can be expressed as linear combinations of regular vector spherical wave functions:

eNi,2(r2 ) hNi,2(r2 )

% N

n2(r2 ) × M µ1 (ki,2r2 )

= c2,µ

−

εi,2

(r )

1 (k

i,2

r )

µ=1

µi,2

(r )

N 1

r )

i,2

+d2,µ

(r )

M 1

r ) . (2.74)

εi,2 n

−

µi,2

i,2

To express the resulting system of equations in matrix form we introduce the Qpqt (k1, k2) matrix as the Qpq (k1, k2) matrix of the surface St. The elements of the Qpqt matrix are given by (2.11)–(2.14) but with St in place of S. For example, the elements (Qpqt )11νµ read as

jk12

n (r )

)

(k2r )

(k1r )

νµ

(k1r )" dS (r ) .

ε2

n (r )

N q

(k2r )

ε1

Inserting (2.73) and (2.74) into (2.70), (2.71) and (2.72), using the identities (cf. (B.23), (B.24) and (B.25)),

Qpp	(k, k) = 0 ,	for p = 1 or p = 3 ,
1
Q13	(k, k) = I ,
1

Q311 (k, k) = −I ,

110 2 Null-Field Method

and taking into account the transformation rules

i,1

)

r )

i,1

(ki,1r )

S12

(ki,1r )

and

i,1

)

r )

i,1

(ki,1r

)

S21

(ki,1r )

we obtain the system of matrix equations

Q131(ks, ki,1)i1 + Q133(ks, ki,1)i1 = −e ,

−i1 + S12tr Q211(ki,1, ki,2)i2 = 0 ,

S21rt i1 + Q231(ki,1, ki,2)i2 = 0 ,

(2.75)

, i2

]

, and as before,

where i1 = [c1,µ

, d1,µ] , i1

= [c1,µ, d1,µ

]

= [c2,µ, d2,µ

e = [aν , bν ]

is the vector containing the expansion coe cients of the incident

ﬁeld. The S12tr matrix relates the vector spherical wave functions deﬁned with respect to the coordinate system O1x1y1z1 to those deﬁned with respect to the coordinate system O2x2y2z2, and can be expressed as the product of a translation and a rotation matrix:

S12tr

= T

11 (ki,1r12) R (α, β, γ) ,

(2.76)

where T and R are deﬁned in Appendix B. The S21rt

matrix describes the

inverse transformation and is given by

rt =

(

−

γ, β,

α)

33 ( k

)

for r

> r

(2.77)

S21

− −

− i,1

Since R is a block-diagonal matrix and T is a block-symmetric matrix it follows that S12tr and S21rt are also block-symmetric matrices.

Solving the system of matrix equations gives

i1	=	−	Q31	(ks, ki,1) + Q33		(ks, ki,1)	tr T 2	rt −1 e ,
		−	1		1		S12	S21
i1 = S12tr T 2S21rt i1 ,
i2	=	−	Q31	(ki,1, ki,2) −1		rt i1 ,		(2.78)
		−	2		S21

where

T 2 = −Q112 (ki,1, ki,2) Q312 (ki,1, ki,2) −1

is the transition matrix of the inhomogeneity imbedded in a medium with relative media constants εi,1 and µi,1.

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