Файл: Doicu A., Wriedt T., Eremin Y.A. Light scattering by systems of particles (OS 124, Springer, 2006.pdf

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2.6 Multiple Particles

125

and Ei,2, Hi,2 satisfying the Maxwell equations
× Es = jk0µsHs ,	× Hs = −jk0εsEs in	Ds ,		(2.124)
× Ei,1 = jk0µi,1Hi,1 ,	× Hi,1 = −jk0εi,1Ei,1	in	Di,1 ,	(2.125)
and
× Ei,2 = jk0µi,2Hi,2 , quad × Hi,2 = −jk0εi,2Ei,2		in	Di,2 ,	(2.126)
the boundary conditions
n1 × Ei,1 − n1 × Es = n1 × Ee ,
n1 × Hi,1 − n1 × Hs = n1 × He				(2.127)
on S1 and
n2 × Ei,2 − n2 × Es = n2 × Ee ,
n2 × Hi,2 − n2 × Hs = n2 × He				(2.128)

on S2, 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,1 and Di,2 together with the boundary conditions (2.127) and (2.128) yield the general null-ﬁeld equation


Ee(r) + × S1 ei,1 (r ) g (ks, r, r ) dS (r )
+	j
		× × S1 hi,1 (r ) g (ks, r, r ) dS (r )
	k0εs

+ × S2 ei,2 (r ) g (ks, r, r ) dS (r )
+	j
		× × S2 hi,2 (r ) g (ks, r, r ) dS (r ) = 0 , r Di,1 Di,2 .
	k0εs

Before we derive the null-ﬁeld equations, we seek to ﬁnd a relation between the expansion coe cients of the incident ﬁeld in the global coordinate system

Oxyz,

Ee(r) = aν M 1ν (ksr) + bν N 1ν (ksr)

and the expansion coe cients of the incident ﬁeld in the particle coordinate system O1x1y1z1,

Ee (r1) = a1,ν M 1ν (ksr1) + b1,ν N 1ν (ksr1) .

126 2 Null-Field Method

r1		r
		S
O1	r01	Ds
O1	r01	O

Fig. 2.6. Auxiliary surface S

For this purpose we choose a su ciently large auxiliary surface S enclosing O and O1 (Fig. 2.6) and in each coordinate system we use the Stratton–Chu representation theorem for the incident ﬁeld in the interior of S. We obtain

$ %

aν bν

and

$ %

a1,ν b1,ν

jks2

$ N

(ksr ) %

= −

ee (r )

(ksr )

µs

(ksr )

dS (r ) ,

+ j

he (r )

(ksr )

εs

jks2

) $ N

(ksr1) %

−

(ksr

)


				M		3	(ksr1)
	µs
						ν
+ j			he (r1)		3			dS (r1) ,

		εs		N			(ksr1	)
					ν

respectively. Using the addition theorem for radiating vector spherical wave functions

(k r

)

r )

(ksr )

S10

(ksr )

where

S10rt = R (−γ1, −β1, −α1) T 33

(−ksr01) ,

for

r > r01

and taking into account that S10rt is a block-symmetric matrix, yields

a1,ν

= S10rt

aµ

(2.129)

b1,ν

bµ

2.6 Multiple Particles

127

The condition r > r01 can always be satisﬁed in practice by an appropriate choice of the auxiliary surface S, whence, using the identity T 33(−ksr01) = T 11(−ksr01), we see that

S10rt = R (−γ1, −β1, −α1) T 11 (−ksr01) .

We proceed now to derive the set of null-ﬁeld equations. Passing from the origin O to the origin O1, using the relations

g (ks, r, r ) = g (ks, r1, r1) , g (ks, r, r ) = g (ks, r1, r1 ) ,

and restricting r1 to lie on a sphere enclosed in Di,1, gives

jks2				$ N	3		(ksr1) %
			ei,1 (r1) ·		ν
π	S			M		3		(ksr	)
		1				ν
							1

(ksr1)

µs

dS (r

+ j

)

εs

i,1

(ksr )

jks2

(r )

$ N

(ksr1 ) %

(2.130)

i,2

(ksr )

$ M

(ksr1 ) %

$ a1,ν

µs

+ j

(r )

dS (r ) =

−

, ν = 1, 2, ... ,

εs

i,2

(ksr )

b1,ν

where the identities ei,2(r2 ) = ei,2(r1 ) and hi,2(r2 ) = hi,2(r1 ) have been used. For the general null-ﬁeld equation in Di,2 we proceed analogously but restrict r2 to lie on a sphere enclosed in Di,2. We obtain

jks2

$ N

(ksr2) %

ei,1 (r1) ·

(ksr )

π S

$ M

(ksr2) %

µs

+ j

hi,1

(r1) ·

(ksr )

εs

jks2

(r )

$ N

(ksr2 ) %

i,2

(ksr )

$ M

(ksr2 ) %

µs

+ j

hi,2

(r2 ) ·

(ksr )

εs

(r1)

(2.131)

(r ) =	−	a2,ν	, ν = 1, 2, ... ,
2	−	b2,ν

where, as before, we have taken into account that ei,1(r1) = ei,1(r2) and

hi,1(r1) = hi,1(r2).

µi,1

128 2 Null-Field Method

The surface ﬁelds ei,1, hi,1 and ei,2, hi,2 are the tangential components of the electric and magnetic ﬁelds in the domains Di,1 and Di,2, respectively, and the surface ﬁelds approximations can be expressed as linear combinations of regular vector spherical wave functions,

eNi,1(r1) hNi,1(r1)


% N
= c1N,µ	j
µ=1	−

+dN

1,µ −j

n1(r1) × M 1µ(ki,1r1)

εi,1 n1(r1) × N 1µ(ki,1r1)

n1(r1) × N 1µ(ki,1r1)

		(r	)		M 1 (k		r	) (2.132)
εi,1 n				×
µi,1	1	1			µ	i,1	1

and

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

% N
% N
N
= µ=1 c2,µ −j
+d2N,µ
+d2N,µ		j
	−

n2(r2 ) × M µ1 (ki,2r2 )

εi,2

(r )

1 (k

i,2

r )

µi,2

n2(r2 ) × N µ1 (ki,2r2 )

(r )

M 1

r ) . (2.133)

εi,2 n

µi,2

i,2

Inserting (2.132) and (2.133) into (2.130) and (2.131), using the addition theorem for vector spherical wave functions

(ksr1 )

rtr

(k r

)

(ksr )

with

rtr =

(

β ,

α )

31 (k

)

(α

, β , γ ) ,

for

< r

R −

−

2 2

and

)

(ksr2)

rtr

(ksr

)

(ksr

)

with

rtr =

(

γ ,

−

β ,

−

α )

31 (

−

k r

)

(α , β , γ )

for

< r

R −

s 12

1 1 1

and taking into account the transformation rule for the incident ﬁeld coe - cients (2.129), yields the system of matrix equations

	31		S	rtr	11	(ks, ki,2)i2 =	−S	rt
					Q2			10e ,
Q1 (ks, ki,1)i1 + 12
S	rtr	11			31		−S	rt
		Q1	(ks, ki,1)i1 + Q2			(ks, ki,2)i2 =		20e ,	(2.134)
21

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