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

Скачать файл
Заказать решение

ВУЗ: Не указан

Категория: Не указан

Дисциплина: Не указана

Добавлен: 28.06.2024

Просмотров: 851

Скачиваний: 0

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

98 2 Null-Field Method

1

I − Ms

ei −

j

Pshi = ee ,

(2.26)

2

k0εs

1

I − Ms

hi +

j

Psei = he ,

(2.27)

2

k0µs

and

1

I + Mi

ei +

j

Pihi = 0 ,

(2.28)

2

k0εi

1

I + Mi

hi −

j

Piei = 0 ,

(2.29)

2

k0µi

respectively. These are four boundary integral equations for the unknowns ei and hi, and we consider two linear combinations of equations, i.e.,

α1 (2.26) + α2 (2.27) + α3 (2.29) and β1 (2.26) + β2(2.27) + β3(2.28) ,

where αi and βi, i = 1, 2, 3, are constants to be chosen. Harrington [97] describes several possible choices, as shown in Table 2.1, and for all these choices we always have existence and unique solvability [155].

The above approach for deriving a pair of boundary integral equations is known as the direct method. In contrast to the null-field method, the direct method considers the null-field equations in both domains Di and Ds, and treats the surface fields as independent unknowns. The indirect method for deriving a pair of boundary integral equations relies on the representation of the electromagnetic fields in terms of four surface fields. Passing to the boundary, using the boundary conditions on the particle surface and imposing two constraints on the surface fields, yields the desired pair of boundary integral equations [97]. It should be emphasized that single integral equations for the transmission boundary-value problem have also been derived by Marx [156], Mautz [157] and Martin and Ola [155]. Another major di erence to the nullfield method is the discretization of the boundary integral equations, which is achieved by using the method of moments [96]. The boundary surface is discretized into a set of surface elements and on each element, the surface fields are approximated by finite expansions of basis functions. Next, a set of

Table 2.1. Choice of constants for the boundary integral equations

Formulation

α1

α2

α3

β1

β2

β3

 

 

 

 

 

 

 

E-field

1

0

0

0

0

1

H-field

0

0

1

0

1

0

Combined field

0

1

−1

1

0

−1

Mautz–Harrington

0

1

−β

1

0

−α

M¨uller

0

µs

µi

εs

0

εi

2.1 Homogeneous and Isotropic Particles

99

testing functions are defined and the scalar product of each testing function is formed with both sides of the equation being solved. This results in a system of equations which is referred to as the element matrix equations. The element matrices are assembled into the global matrix of the entire “structure” and the resulting system of equations is solved for the unknown expansion coe cients.

2.1.6 Spherical Particles

An interesting feature of the null-field method is that all matrix equations become considerably simpler and reduce to the corresponding equations of the Lorenz–Mie theory when the particle is spherically. For a spherical particle of radius R, the orthogonality relations of the vector spherical harmonics show that the Qpq matrices are diagonal

 

 

 

 

 

(Qpq )12

 

 

= (Qpq )21

 

 

= 0

 

 

 

 

 

 

 

 

 

 

 

mn,m1n1

 

 

 

 

mn,m1n1

 

 

 

 

 

 

for all values of m, n, m1 and n1, and

 

 

 

 

 

 

 

 

 

 

Q

31

11

!

 

 

 

 

 

(1)

 

 

 

 

 

 

 

 

 

mn,m1n1 = jx

jn (mrx)

xhn

(x)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

− hn(1) (x) [mrxjn (mrx)] δmm1 δnn1 ,

 

 

31

 

22

 

 

jx

 

 

(1)

 

 

 

 

 

 

 

 

 

 

Q

 

 

 

mn,m1n1

=

 

 

 

 

 

−hn (x) [mrxjn (mrx)]

 

 

 

 

 

 

mr

 

 

 

 

 

 

 

 

 

 

2

j

 

(m

 

 

(1)

"

δ

 

δ

 

,

 

 

 

 

 

 

 

+ m

n

x) xh (x)

mm1

nn1

 

 

 

 

 

 

 

 

 

 

r

 

 

r

 

 

n

 

 

 

 

33

 

11

!

(1)

 

 

 

 

(1)

 

 

 

 

 

 

Q

 

 

mn,m1n1

= jx hn

 

(mrx) xhn

(x)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

(1)

 

 

 

 

 

(1)

 

"

 

 

 

 

 

 

 

 

 

 

− hn

 

(x) mrxhn

 

(mrx)

δmm1 δnn1 ,

33

 

22

 

jx !

 

 

 

(1)

 

 

 

(1)

 

 

 

 

 

 

Q

 

 

mn,m1n1

=

 

 

−hn (x)

mrxhn

(mrx)

 

 

 

 

 

mr

 

 

 

 

 

 

 

 

 

2

 

(1)

 

 

 

 

(1)

"

 

 

 

 

 

 

 

 

 

 

+ mr hn

 

(mrx) xhn (x)

δmm1 δnn1 ,

Q11 11

= jx jn (mrx) [xjn (x)]

 

 

 

 

 

 

 

 

 

 

 

mn,m1n1

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

− jn (x) [mrxjn (mrx)] δmm1 δnn1 ,

(2.30)

(2.31)

(2.32)

(2.33)

(2.34)

100 2 Null-Field Method

Q11 22

=

 

jx

 

−

jn (x) [mrxjn (mrx)]

 

 

 

 

mr

 

 

 

 

 

mn,m1n1

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

+ mr2jn (mrx) [xjn (x)] δmm1 δnn1 ,

 

and

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Q13 11

= jx hn(1) (mrx) [xjn (x)]

 

 

 

 

 

 

 

mn,m1n1

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

(1)

 

"

 

 

 

 

 

 

 

− jn (x) mrxhn (mrx)

δmm1 δnn1 ,

 

13

22

 

jx !

 

 

 

 

(1)

 

 

 

 

 

Q

 

mn,m1n1

=

 

 

−jn (x)

mrxhn (mrx)

 

 

 

 

mr

 

 

 

 

 

 

+ m2h(1)

(m

x) [xj

(x)] δ

mm1

δ

nn1

,

 

 

 

 

 

 

 

r

n

r

 

n

 

 

 

where x = ksR is the size parameter and

mr =

εi

εs

(2.35)

(2.36)

(2.37)

is the relative refractive index of the particle with respect to the ambient medium. The above relations are not suitable for computing the Qpq matrices. Denoting by An and Bn the logarithmic derivatives [2, 17]

An(x) =

d

{ln [xjn (x)]} =

[xjn (x)]

 

 

 

 

 

,

 

 

dx

xjn(x)

 

 

 

 

 

 

(1)

 

 

Bn(x) =

d

ln xhn(1) (x) =

 

xhn (x)

 

,

dx

 

 

 

 

 

 

 

 

xhn(1)(x)

and using the recurrence relation (cf. (A.8))

[xzn (x)] = xzn−1(x) − nzn(x) ,

where zn stands for jn

Q31 11 mn,m1n1

Q31 22 mn,m1n1

or h(1)n , we rewrite (2.30)–(2.37) as

2

 

 

 

 

 

n

(1)

 

= −jx

jn

(mrx) mrAn (mrx) +

 

 

hn

(x)

x

(1)

 

 

 

 

 

 

 

 

 

 

(2.38)

− hn−1 (x)

δmm1 δnn1 ,

 

 

 

 

 

 

 

2

 

 

! An (mrx)

 

 

n

 

(1)

 

= −jmrx

jn (mrx)

 

 

+

 

hn

(x)

 

mr

x

(1)

 

 

 

 

 

 

 

 

 

(2.39)

− hn−1

(x)

δmm1 δnn1 ,

 

 

 

 

 

 

2.1 Homogeneous and Isotropic Particles

33

 

 

11

 

 

 

 

 

2

 

(1)

 

 

 

 

 

 

 

 

n

(1)

 

Q

 

 

mn,m1n1

= −jx

hn

(mrx)

mrBn (mrx) +

 

 

 

 

 

 

 

 

 

hn

(x)

 

 

x

 

 

 

 

 

 

 

 

 

 

 

 

(1)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

− hn−1

(x) δmm1 δnn1 ,

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Q33 22

=

−

jmrx2h(1)

(mrx)

!

Bn (mrx)

 

 

+

 

n

h(1)

(x)

 

 

 

 

 

 

 

 

 

mn,m1n1

 

 

 

 

 

 

 

n

 

 

 

 

mr

 

 

 

 

 

 

x

n

 

 

 

 

 

 

 

 

 

 

 

 

 

(1)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

− hn−1 (x) δmm1 δnn1 ,

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

11

11

 

 

 

 

 

 

2

 

 

 

 

 

 

 

 

 

 

n

 

 

Q

 

 

 

 

mn,m1n1

 

= −jx

jn (mrx)

mrAn (mrx) +

 

 

 

 

 

 

 

 

 

jn (x)

 

 

 

 

 

 

x

 

 

 

 

 

 

 

 

 

 

 

− jn−1 (x)} δmm1 δnn1 ,

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

22

 

 

 

 

 

 

 

 

 

 

 

 

 

! An (mrx)

n

 

Q11

mn,m1n1

= −jmrx2jn

(mrx)

 

 

 

 

+

 

 

 

 

 

 

jn (x)

 

 

mr

 

x

 

 

 

 

 

 

 

 

 

 

− jn−1 (x)} δmm1 δnn1 ,

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

and

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

13

 

 

11

 

 

 

 

 

2

 

(1)

 

 

 

 

 

 

 

 

 

 

n

 

Q

 

 

 

mn,m1n1

= −jx

 

hn (mrx)

mrBn (mrx) +

 

 

 

 

 

 

jn (x)

 

 

 

 

 

 

x

 

 

 

 

 

 

 

 

 

 

− jn−1 (x)} δmm1 δnn1 ,

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Q13 22

=

−

jmrx2h(1) (mrx) !

Bn (mrx)

+

n

jn (x)

 

 

 

 

 

 

 

mn,m1n1

 

 

 

 

 

 

 

n

 

 

 

 

mr

 

 

 

 

 

 

 

x

 

 

 

 

 

 

 

 

 

 

 

 

− jn−1 (x)} δmm1 δnn1 .

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

The functions An and Bn satisfy the recurrence relation

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Ψn(x) =

n + 1

−

 

1

,

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

x

Ψn+1(x) + n+1

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

x

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

101

(2.40)

(2.41)

(2.42)

(2.43)

(2.44)

(2.45)

where Ψn stands for An and Bn, and a stable scheme for computing Ψn relies on a downward recursion. Beginning with an estimate for Ψn, where n is larger that the number of terms required for convergence, successively lower-order logarithmic derivatives can be generated by downward recursion. It should be noted that the downward stability of Ψn is a consequence of the downward stability of the spherical Bessel functions jn (see Appendix A).

The transition matrix of a spherical particle is diagonal with entries

Tmn,m11 1n1 = Tn1δmm1 δnn1 ,

Tmn,m22 1n1 = Tn2δmm1 δnn1 ,

Смотрите также файлы