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

Скачать файл (3,30Мб)

Заказать решение

ВУЗ: Не указан

Категория: Не указан

Дисциплина: Не указана

Добавлен: 28.06.2024

Просмотров: 833

Скачиваний: 0

ВНИМАНИЕ! Если данный файл нарушает Ваши авторские права, то обязательно сообщите нам.

264 B Wave Functions

where er and ek are the unit vectors in the directions of r and k, respectively, and obtain

ejk·r =	2π	δ (er	−	ek ) ejkr	−	δ (er + ek ) e−jkr	, kr	→ ∞	.

	jkr

The vector spherical harmonics are deﬁned as [175, 228, 229]

(θ, ϕ) =

jmπ|m|(θ)e

θ −

τ |m|(θ)e

ejmϕ,

2n(n + 1)

(θ, ϕ) =

τ |m|(θ)e

+ jmπ|m|e

ejmϕ,

2n(n + 1)

(B.7)

(B.8)

(B.9)

where (er , eθ , eϕ) are the spherical unit vectors of the position vector r, and we have er × mmn = nmn and er × nmn = −mmn. We omit the third vector spherical harmonics

pmn(θ, ϕ) = 2n(n + 1) Ymn(θ, ϕ)er

since it will be not encountered in our analysis. The orthogonality relations for vector spherical harmonics are

2π π

0	0	mmn(θ, ϕ) · mm n (θ, ϕ) sin θ dθ dϕ
=	2π π		nmn(θ, ϕ) · nm n (θ, ϕ) sin θ dθ dϕ = πδm,−m δnn ,	(B.10)
=	0	0	nmn(θ, ϕ) · nm n (θ, ϕ) sin θ dθ dϕ = πδm,−m δnn ,	(B.10)
and	2π π
	2π π
			mmn(θ, ϕ) · nm n (θ, ϕ) sin θdθ dϕ = 0 .	(B.11)
		0	0

The system of vector spherical harmonics is orthogonal and complete in L2tan(Ω) (the space of square integrable tangential ﬁelds deﬁned on the unit sphere Ω) and in terms of the scalar product in L2tan(Ω) we have

2π π

0	0	mmn(θ, ϕ) · mm n (θ, ϕ) sin θ dθ dϕ
2π π
=		nmn(θ, ϕ)	·	nm	n	(θ, ϕ) sin θ dθ dϕ = πδmm δnn	(B.12)
0		0	·
and	2π π
	2π π
		mmn(θ, ϕ) · nm n (θ, ϕ) sin θ dθ dϕ = 0.					(B.13)

B.2 Vector Wave Functions

265

We deﬁne the vector spherical harmonics of leftand right-handed type as

lmn(θ, ϕ) = √ [mmn(θ, ϕ) + jnmn(θ, ϕ)] 2

		j		m		m
=			mπ\|		\|(θ) + τ \|		\|(θ)
=			mπ\|		\|(θ) + τ \|		\|(θ)
	2 n(n + 1)		n		n

and

rmn(θ, ϕ) = √ [mmn(θ, ϕ) − jnmn(θ, ϕ)] 2

		j		m				m
=			mπ\|		\|(θ)	−	τ \|		\|(θ)

	2 n(n + 1)		n				n

(B.14)

(eθ + jeϕ)ejmϕ

(B.15)

(eθ − jeϕ)ejmϕ,

respectively, and it is straightforward to verify the orthogonality relations

2π π

0	0	lmn(θ, ϕ) · lm n (θ, ϕ) sin θ dθ dϕ
2π π
=		rmn(θ, ϕ)	·	rm	n	(θ, ϕ) sin θ dθ dϕ = πδmm δnn	(B.16)
0		0	·
and	2π π
	2π π
		lmn(θ, ϕ) · rm n (θ, ϕ) sin θ dθ dϕ = 0.					(B.17)

Since lmn and rmn are linear combinations of mmn and nmn, we deduce that the system of vector spherical harmonics of leftand right-handed type is also orthogonal and complete in L2tan(Ω).

B.2 Vector Wave Functions

The independent solutions to the vector wave equations can be constructed as [215]


M mn1,3 (kr) =		1		umn1,3 (kr) × r,

			2n(n + 1)
N mn1,3 (kr) =	1	× M mn1,3 (kr),

	k

where n = 1, 2, ..., and m = −n, ..., n. The explicit expressions of the vector spherical wave functions are given by

266 B Wave Functions

M 1,3

(kr) =

z1,3(kr) jmπ|m| (θ) e

θ −

τ |m| (θ) e

ejmϕ,

2n(n + 1) n

N 1,3

(kr) =

!n(n + 1)

zn1,3(kr)

P |m|(cos θ)e

2n(n + 1)

krz1,3(kr)

τ |m|(θ)e

+ jmπ|m|(θ)e

ejmϕ,

where

krzn1,3(kr)

krzn1,3(kr) .

d (kr)

The superscript ‘1’ stands for the regular vector spherical wave functions while the superscript ‘3’ stands for the radiating vector spherical wave functions. It

is useful to note that for n = m = 0, we have M 100,3 = N 100,3 = 0. M 1mn, N 1mn is an entire solution to the Maxwell equations and M 3mn, N 3mn is a radiating

solution to the Maxwell equations in R3 − {0}.

The vector spherical wave functions can be expressed in terms of vector spherical harmonics as follows:

M mn1,3 (kr) = zn1,3(kr)mmn(θ, ϕ),
					(kr)
	n(n + 1)	z1,3(kr)		krz1,3
N mn1,3 (kr) =		n	Ymn (θ, ϕ) er +	n		nmn(θ, ϕ),
		kr
2				kr

In the far-ﬁeld region, the asymptotic behavior of the spherical Hankel functions for large value of the argument yields the following representations for the radiating vector spherical wave functions [40]:

ejkr !

n+1

M mn(kr) =

(−j)

mmn(θ, ϕ) + O

, r → ∞,

ejkr !

n+1

N mn(kr) =

j (−j)

nmn(θ, ϕ) + O

, r → ∞.

The orthogonality relations on a sphere Sc, of radius R, are

[er × M mn(kr)] · [er × M m n (kr)] dS(r) = πR2zn2 (kR)δm,−m δnn ,

	[er	×	N mn(kr)]	·	[er	×	N m n (kr)] dS(r) = πR2	!	[kRzn(kR)]	"2
									kR
Sc

×δm,−m δnn ,

B.2 Vector Wave Functions

267

and

[er

M mn(kr)]

[er

N m n (kr)] dS(r) = 0,

and we also have

[er

M mn(kr)]

N m n (kr)dS (r)

[er

N mn(kr)]

M m n (kr)dS (r)

−

= πR2zn(kR)

[kRzn(kR)]

δm,−m δnn

(B.18)

and

M mn(kr)]

M m n (kr)dS (r)

[er

N mn(kr)]

N m n (kr)dS (r) = 0.

(B.19)

The spherical vector wave expansion of the dyad gI is of basic importance in our analysis and is given by [175, 228, 229]

g(k, r, r )

M 3

(kr )M 1

(kr) + N 3

(kr

−

∞

+Irrotational terms ,

r < r

π n=1 m= n

M 1

(kr )M 3

(kr) + N 1

(kr

−

+Irrotational terms ,

r > r

)N 1mn(kr)

(B.20)

)N 3mn(kr)

Using the calculation rules for dyadic functions and the identity ag = a · gI, we ﬁnd the following simple but useful expansions

[a(r )g(k, r, r )]

a(r )

(kr ) N 1

(kr)

−mn

∞

+ a(r ) N mn(kr ) M mn(kr) , r < r

−

(B.21)

π n=1 m= n

a(r )

(kr ) N 3

(kr)

−

+ a(r )

(kr ) M

(kr) , r > r

−mn

268 B Wave Functions

and

× ×

[a(r )g(k, r, r )]

a(r )

(kr ) M 1

(kr)

−mn

+ a(r ) N mn(kr ) N mn(kr) , r < r

∞

−

(B.22)

π n=1 m= n

a(r )

(kr ) M 3

(kr)

−

+ a(r )

(kr ) N

(kr)

, r > r

−mn

Relying on these expansions and using the Stratton–Chu representation theorem, orthogonality relations of vector spherical wave functions on arbitrarily closed surfaces can be derived. Let Di be a bounded domain with boundary S and exterior Ds, and let n be the unit normal vector to S directed into Ds. The wave number in the domain Ds is denoted by ks, while the wave number in the domain Di is denoted by ki. For r Di, application of Stratton–Chu representation theorem to the vector ﬁelds Es(r) = M 3mn(ksr)

and Hs(r) = −j εs/µsN 3mn(ksr) gives

M 3

N 3

−m n

+ n

−m n

dS = 0,

mn ·

N 3

mn ·

M 3

−m n

(B.23)

while for r Ds, yields

jks2

M 1

N 1

−m n

+ n

−m n

N 1

mn ·

M 1

−m n

(B.24)

δmm δnn

Similarly, for r Ds, vector ﬁelds Ei(r) =

n × M 1mn

the Stratton–Chu representation theorem applied to the

M 1	(k r) and H		(r) =		−	j	ε	/µ	N 1	(k	r) leads to
mn	i	i			−		i	i	mn	i
M	1			N 1		N 1
	−m n	+ n	×	N 1				−m n			dS = 0.
· N −1 m n			×	mn		·		M −1 m n

(B.25) The regular and radiating spherical vector wave functions can be expressed as integrals over vector spherical harmonics [26]

1	1	2π π		(−j) mmn (β, α) e		jk(β,α)			r	sin β dβ dα,
M mn(kr) = −								·
M mn(kr) = −	4πjn+1	0	0					·
										(B.26)
1	1	2π π		nmn (β, α) e	jk(β,α)		r	sin β dβ dα,
N mn(kr) = −						·
N mn(kr) = −	4πjn+1	0	0			·
										(B.27)

Смотрите также файлы

Ufimtsev P. Fundamentals of the physical theory of diffraction (Wiley 2007)(348s) PEo .pdf

Antsiferov V.V., Smirnov G.I. Physics of solid-state lasers (ISBN 1898326177) (CISP, 2005)(179s).pdf

Staliunas K., Sanchez-Morcillo V.J. (eds.) Transverse Patterns in Nonlinear Optical Resonators(ST.pdf

Pye D. Polarised light in science and nature (IOP)(133s).pdf

Weber H., Herziger G., Poprawe R. (eds.) Laser Fundamentals. Part 1 (Springer 2005)(263s) PEo .pdf

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

Смотрите также файлы

Информация

Списки файлов

Дополнительно