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

(er , ek ) =

( j)n+1 jn1 T 11

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58 1 Basic Theory of Electromagnetic Scattering

relying on the rotation transformation rule for vector spherical wave functions are of general use because an explicit expression of the transition matrix is not required. In order to simplify our analysis we consider a vector plane wave of unit amplitude

Ee(r) = epolejke·r , He(r) = εs ek × epolejke·r ,

µs

where epol · ek = 0 and |epol| = 1.

1.5.1 Deﬁnition

Everywhere outside the (smallest) sphere circumscribing the particle it is appropriate to expand the scattered ﬁeld in terms of radiating vector spherical wave functions

∞ n
Es(r) = fmnM mn3 (ksr) + gmnN mn3 (ksr)	(1.93)
n=1 m=−n

and the incident ﬁeld in terms of regular vector spherical wave functions

∞ n
Ee(r) = amnM mn1 (ksr) + bmnN mn1 (ksr) .	(1.94)
n=1 m=−n

Within the vector spherical wave formalism, the scattering problem is solved by determining fmn and gmn as functions of amn and bmn. Due to the linearity relations of the Maxwell equations and the constitutive relations, the relation between the scattered and incident ﬁeld coe cients must be linear. This relation is given by the so-called transition matrix T as follows [256]

fmn		amn	T 11		T 12 amn		(1.95)
gmn	= T	bmn	=	T 21	T 22 bmn	.

Essentially, the transition matrix depends on the physical and geometrical characteristics of the particle and is independent on the propagation direction and polarization states of the incident and scattered ﬁeld.

If the transition matrix is known, the scattering characteristics (introduced in Sect. 1.4) can be readily computed. Taking into account the asymptotic behavior of the vector spherical wave functions we see that the far-ﬁeld pattern can be expressed in terms of the elements of the transition matrix by the relation

Es∞ (er ) = 1 (−j)n+1 [fmnmmn(er ) + jgmnnmn(er )]

ks n,m

= 1 (−j)n+1

ks n,m n1,m1

× Tmn,m11 1n1 am1n1

+ j Tmn,m21 1n1 am1n1

+Tmn,m12 1n1 bm1n1 mmn(er )

+Tmn,m22 1n1 bm1n1 nmn(er ) . (1.96)

1.5 Transition Matrix

To derive the expressions of the tensor scattering amplitude and amplitude matrix, we consider the scattering and incident directions er and ek , and express the vector spherical harmonics as

xmn(er ) = xmn,θ (er ) eθ + xmn,ϕ (er ) eϕ ,

xmn(ek ) = xmn,β (ek ) eβ + xmn,α (ek ) eα ,

where xmn stands for mmn and nmn. Recalling the expressions of the incident ﬁeld coe cients for a plane wave excitation (cf. (1.26))

amn = 4jnepol · mmn (ek ) ,

bmn = −4jn+1epol · nmn (ek ) ,

and using the deﬁnition of the tensor scattering amplitude (cf. (1.71)), we obtain

+ jTmn,m21 1n1 nmn (er ) mm1n1 (ek ) + −jTmn,m12 1n1 mmn (er )

+ Tmn,m22 1n1 nmn (er ) nm1n1 (ek ) .

In view of (1.75), the elements of the amplitude matrix are given by

Spq (er , ek ) =		4	(		−	j)n+1 jn1 T 11			m	(ek )

	ks n,m n1,m1							mn,m1n1	m1n1,q
	−		jT 12	n				(ek ) mmn,p (er )
			mn,m1n1		m1n1,q
	+j Tmn,m21			1n1 mm			1n1,q (ek ) ,
	−		jT 22		n			(ek ) nmn,p (er )		(1.97)
			mn,m1n1		m1n1,q

for p = θ, ϕ and q = β, α. For a vector plane wave linearly polarized in the

β-direction, amn = 4jnmmn,β and bmn = −4jn+1nmn,β , and Sθβ = Es∞,θ and Sϕβ = Es∞,ϕ. Analogously, for a vector plane wave linearly polarized in the

α-direction, amn = 4jnmmn,α and bmn = −4jn+1nmn,α, and Sθα = Es∞,θ and Sϕα = Es∞,ϕ. In practical computer calculations, this technique, relying

on the computation of the far-ﬁeld patterns for parallel and perpendicular polarizations, can be used to determine the elements of the amplitude matrix.

For our further analysis, it is more convenient to express the above equa-

tions in matrix form. Deﬁning the vectors
s =	fmn	, e =	amn
	gmn		bmn

60 1 Basic Theory of Electromagnetic Scattering

and the “augmented” vector of spherical harmonics

(−j)n mmn (er ) v (er ) = j (−j)n nmn (er )

we see that

Es∞ (er ) = −

vT (er ) s =

−

vT (er ) T e =

−

eTT Tv (er ) ,

and, since e = 4epol · v (ek ), we obtain

, e

) =

vT (e

) T v

) =

v†

) T Tv

(e ) .

−ks p

−ks q

The superscript † means complex conjugate transpose, and

vp ( ) =

(

−

j)n mmn,p ( )

j (

−

j) nmn,p ( )

(1.98)

(1.99)

where p = θ, ϕ for the er -dependency and p = β, α for the ek -dependency. The extinction and scattering cross-sections can be expressed in terms

of the expansion coe cients amn, bmn, fmn and gmn. Denoting by Sc the circumscribing sphere of outward unit normal vector er and radius R, and using the deﬁnition of the extinction cross-section (cf. (1.82) and (1.83) with

|Ee0| = 1), yields

µs

Cext = −

Sc er · Re {Ee × Hs + Es × He } dS

εs

−

Re j

fmna

+ gmnb

n,m n1,m1

m1n1

M 3

N 3

m1n1

+ fmnbm1n1 + gmnam1n1

M 3

M 1

+ er

dS .

mn ·

m1n1

mn ·

m1n1

Taking into account the orthogonality relations of the vector spherical wave functions on a spherical surface (cf. (B.18) and (B.19)) we obtain

jπR

∞

= Re

(f a

+ g

)

ext

−

mn mn

n=1 m=−n

(1)

× hn

(ksR) [ksRjn(ksR)] − jn(ksR) ksRhn

(ksR)

whence, using the Wronskian relation

(1)

(ksR) [ksRjn(ksR)] − jn(ksR) ksRhn

(ksR) = −

ksR

1.5 Transition Matrix

we end up with

				π	∞ n
C		=			Re	f	a	+ g	b	.	(1.100)
			−ks2
	ext				n=1 m=−n	{	mn mn		mn mn}

For the scattering cross-section, the expansion of the far-ﬁeld pattern in terms of vector spherical harmonics (cf. (1.96)) and the orthogonality relations of the vector spherical harmonics (cf. (B.12) and (B.13)), yields

	π	∞ n	2		2
Cscat =				+ \|gmn\|			(1.101)
	ks2	n=1 m=−n \|fmn\|				.

Thus, the extinction cross-section is given by the expansion coe cients of the incident and scattered ﬁeld, while the scattering cross-section is determined by the expansion coe cients of the scattered ﬁeld.

1.5.2 Unitarity and Symmetry

It is of interest to investigate general constraints of the transition matrix such as unitarity and symmetry. These properties can be established by applying the principle of conservation of energy to nonabsorbing particles (εi > 0 and µi > 0). We begin our analysis by deﬁning the S matrix in terms of the T matrix by the relation

S = I + 2T ,

where I is the identity matrix. In the literature, the S matrix is also known as the scattering matrix but in our analysis we avoid this term because the scattering matrix will have another signiﬁcance.

First we consider the unitarity property. Application of the divergence theorem to the total ﬁelds E = Es + Ee and H = Hs + He in the domain D bounded by the surface S and a spherical surface Sc situated in the far-ﬁeld region, yields