Файл: Doicu A., Wriedt T., Eremin Y.A. Light scattering by systems of particles (OS 124, Springer, 2006.pdf
ВУЗ: Не указан
Категория: Не указан
Дисциплина: Не указана
Добавлен: 28.06.2024
Просмотров: 857
Скачиваний: 0
116 2 Null-Field Method
−iN−1 + QN11(ki,N−1, N − 1, ki,N , N)iN = 0 , |
(2.98) |
−Q31N−1(ki,N−1, N, ki,N−1, N − 1)iN−1 + Q31N (ki,N−1, N, ki,N , N)iN = 0
(2.99) and the matrix equation corresponding to the scattered field representation
s = Q113 (ks, 1, ki,1, 1) i1 + Q111 (ks, 1, ki,1, 1) i1 . |
(2.100) |
For two consecutive layers, the surface fields il−1 and il−1 are related to the surface fields il and il by the matrix equation
|
i |
|
= |
|
|
|
(ki,l |
|
1, ki,l) |
i |
|
|
(2.101) |
||||
|
l−1 |
Q |
l |
− |
l |
, |
|||||||||||
|
il−1 |
|
|
|
|
|
|
|
|
|
il |
|
|
||||
where |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Ql (ki,l−1, ki,l) = |
I |
|
|
|
|
|
|
|
|
0 |
|
|
|
|
|
(2.102) |
|
0 |
Q31 |
|
(ki,l |
− |
1, l, ki,l |
− |
1, l |
− |
1) −1 |
||||||||
|
|
l−1 |
|
|
|
|
|
|
|
|
|
|
|||||
|
Ql13(ki,l−1, l − 1, ki,l, l) Ql11(ki,l−1, l − 1, ki,l, l) |
||||||||||||||||
|
× |
Ql33(ki,l−1, l, ki,l, l) |
|
|
Ql31(ki,l−1, l, ki,l, l) |
|
|||||||||||
and l = 2, 3, . . . , N − 1. For the surface fields iN−1and iN−1, that is, for l = N, we have
with |
|
|
|
|
|
iN−1 = T N iN−1 |
|
|
|
|
|
|
|
(2.103) |
|||||||||
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
T |
= Q11(ki, |
N− |
1 |
, |
N − |
1, ki, |
N |
, |
N |
) Q31(ki, |
1, |
N |
, ki, |
N |
, |
N |
) −1 |
||||||
N |
N |
|
|
|
|
|
|
|
|
N |
|
N− |
|
|
|
||||||||
|
×QN31−1(ki,N−1, N, ki,N−1, N − 1) . |
|
|
|
|
|
|
(2.104) |
|||||||||||||||
Then, using the matrix equation |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
||||||||
|
|
|
|
s |
|
= |
|
1 (ks, ki,1) |
i |
|
|
|
|
|
|
|
|
||||||
|
|
|
|
|
e |
|
Q |
1 |
|
|
|
|
|
|
|
|
|||||||
|
|
|
|
|
|
|
|
|
|
|
|
|
i1 |
|
|
|
|
|
|
|
|
||
with Q1 being given by (2.93), we see that |
|
|
|
|
|
|
|
|
|
||||||||||||||
|
|
|
|
|
s |
|
= |
|
|
i |
|
|
|
|
|
|
|
|
|
|
|||
|
|
|
|
|
|
|
|
|
|
|
N−1 |
, |
|
|
|
|
|
|
|
|
|||
|
|
|
|
|
|
e |
|
|
|
Q iN−1 |
|
|
|
|
|
|
|
|
|
||||
where
Q = Q1 (ks, ki,1) Q2 (ki,1, ki,2) . . . QN−1 (ki,N−2, ki,N−1) .
Finally, using (2.103) and denoting by (Q)ij , i, j = 1, 2, the block-matrix components of Q, we obtain the expression of the transition matrix in terms of Q and T N :
2.5 Layered Particles |
117 |
−1
T = (Q)12 + (Q)11 T N (Q)22 + (Q)21 T N .
The structure of the above equations is such that a recurrence relation for computing the transition matrix can be established. For this purpose, we define the matrix T l+1,l+2,...,N as
|
il = T l+1,l+2,...,N il . |
and use (2.101) to obtain |
|
il−1 |
= (Ql)12 + (Ql)11 T l+1,l+2,...,N il , |
il−1 |
= (Ql)22 + (Ql)21 T l+1,l+2,...,N il , |
where (Ql)ij , i, j = 1, 2, are the block-matrix components of Ql(ki,l−1, ki,l). Hence, the matrix T l,l+1,...,N , satisfying il−1 = T l,l+1,...,N il−1, can be computed by using the downward recurrence relation
T l,l+1,...,N = (Ql)12 + (Ql)11 T l+1,l+2,...,N |
|
× (Ql)22 + (Ql)21 T l+1,l+2,...,N −1 |
(2.105) |
for l = N − 1, N − 2, . . . , 1. For l = N − 1, T N is given by (2.104), while for
l = 1, Ql is the matrix Q1(ks, ki,1) and T l,l+1,...,N is the transition matrix of the layered particle
T = T 1,2,...,N = − Q111 (ks, 1, ki,1, 1) + Q131 (ks, 1, ki,1, 1) T 2,3,...,N
−1
× Q311 (ks, 1, ki,1, 1) + Q331 (ks, 1, ki,1, 1)T 2,3,...,N .
If the origins coincide, the above relations simplify considerably, since
|
|
|
|
|
Ql13(ki,l−1, 1, ki,l, 1) Ql11(ki,l−1, 1, ki,l, 1) |
|
||||||||||
Ql (ki,l−1, ki,l) = |
−Ql33(ki,l−1, |
1, ki,l, 1) −Ql31(ki,l−1, 1, ki,l, 1) |
(2.106) |
|||||||||||||
and |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
T |
= T |
N |
= |
− |
Q11 |
(ki, |
N− |
1, 1, ki, |
N |
, 1) |
Q31(ki, |
N− |
1, 1, ki, |
N |
, 1) −1 . |
|
N |
|
|
N |
|
|
N |
|
|
|
|||||||
We obtain
Tl,l+1,...,N = − Q11l (ki,l−1, 1, ki,l, 1) + Q13l (ki,l−1, 1, ki,l, 1)T l+1,l+2,...,N
×Q31l (ki,l−1, 1, ki,l, 1) + Q33l (ki,l−1, 1, ki,l, 1)T l+1,l+2,...,N −1
(2.107)
118 2 Null-Field Method and further
T l,l+1,...,N = T l − Q13l (ki,l−1, 1, ki,l, 1)T
× Q31l (ki,l−1, 1, ki,l, 1) −1
l+1,l+2,...,N
I + Q33l (ki,l−1, 1, ki,l, 1)
× |
T l+1,l+2,..., |
Q31 |
(ki,l |
− |
1, 1, ki,l, 1) −1 −1 |
(2.108) |
|
|
N |
l |
|
|
|
||
of which (2.82) is the simplest special case. Note that in (2.107) and (2.108), T is the total transition matrix of the layered particle with outer surface Sl.
2.5.2 Practical Formulation
In practical computer calculations it is simpler to solve the system of matrix equations (2.95)–(2.99) for all unknown vectors il and il, l = 1, 2, . . . , N − 1, and iN . For this purpose, we consider the global matrix
|
A1 |
0 |
|
|
0 ... |
0 |
|
|
|
|
0 |
|
|
|
|||||||
|
A12 A21 |
|
0 ... |
0 |
|
|
|
|
0 |
|
|
|
|||||||||
|
|
0 |
|
|
A23 |
A32 ... |
0 |
|
|
|
|
0 |
|
|
|
||||||
|
|
|
|
|
|
|
|
|
|
|
|||||||||||
A = |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
, |
|
|||
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
0 |
|
|
0 ... AN− |
1, |
N− |
2 |
0 |
|
|
|
||||||
|
0 |
|
|
|
|
|
|
|
|
|
|||||||||||
|
|
0 |
|
|
0 |
|
|
0 ... |
|
|
|
|
|
|
|
|
|
|
|||
|
|
|
|
|
|
AN−1,N |
|
|
AN |
|
|
|
|||||||||
where |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
A1 = Q133(ks, 1, ki,1, 1) Q131(ks, 1, ki,1, 1) , |
|
||||||||||||||||||||
Q13(ki,l |
− |
1, l |
− |
1, ki,l, l) Q11(ki,l |
− |
1, l |
− |
1, ki,l, l) |
|||||||||||||
Al,l−1 = |
l |
|
|
|
|
, l) |
l |
|
|
|
|
|
|
, l) |
|||||||
|
Q33 |
(k |
i,l−1 |
, l, k |
i,l |
Q31 |
(k |
|
, l, k |
i,l |
|||||||||||
|
l |
|
|
|
|
|
|
|
l |
|
|
|
i,l−1 |
|
|
|
|||||
|
|
|
|
|
|
|
|
|
|
0 |
|
|
|
|
|
|
|
|
|
||
Al−1,l = |
−I |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
, |
|||||
|
|
|
0 −Ql31−1(ki,l−1, l, ki,l−1, l − 1) |
|
|
||||||||||||||||
and
Q11(ki,N−1, N − 1, ki,N , N)
AN = N .
Q31N (ki,N−1, N, ki,N , N)
Then, denoting by A the inverse of A,
(2.109)
,(2.110)
(2.111)
(2.112)
|
|
|
|
A11 |
A12 ... |
A1,2N−1 |
|
|
|||
A |
= A−1 |
= |
|
A |
21 |
A |
22 ... |
A |
2,2N−1 |
|
, |
|
|
|
|
|
|
|
|
||||
|
|
|
|
|
|
|
|
|
|
|
|
A2N−1,1 A2N−1,2 ... A2N−1,2N−1
2.5 Layered Particles |
119 |
we express i1 and i1 as
i1 = −A11e , i1 = −A21e ,
and use (2.100) to obtain
T = − Q131 (ks, 1, ki,1, 1) A11 + Q111 (ks, 1, ki,1, 1) A21 .
For axisymmetric layers and axial positions of the origins Ol (along the z-axis of rotation), the scattering problem decouples over the azimuthal modes and the transition matrix can be computed separately for each m. Specifically, for each layer l, we compute the Ql matrices and assemble these matrices into the global matrix A. The matrix A is inverted, and the blocks 11 and 21 of the inverse matrix are used for T -matrix calculation. Because A is a sparse matrix, appropriate LU–factorization routines (for sparse systems of equations) can be employed.
An important feature of this solution method is that the expansion orders of the surface field approximations can be di erent. To derive the dimension of the global matrix A, we consider an axisymmetric particle. If Nrank(l) is the maximum expansion order of the layer l and, for a given azimuthal mode m, 2Nmax(l) × 2Nmax(l) is the dimension of the corresponding Q matrices, where
Nmax(l) = |
# Nrank(l) , |
m = 0 |
, |
Nrank(l) − |m| + 1 , m = 0 |
|||
then, the dimension of the global matrix A is given by
dim (A) = 2Nmax × 2Nmax ,
with
N−1
Nmax = Nmax (N) + 2 Nmax(l) .
l=1
The dimension and occupation of the matrix A is shown in Table 2.2 for three layers.
Since
dim A11 = dim A21 = dim Q131 = dim Q111 = 2Nmax(1) × 2Nmax(1) , it follows that
dim (T ) = 2Nmax(1) × 2Nmax(1) .
Thus, the dimension of the transition matrix is given by the maximum expansion order corresponding to the first layer, while the maximum expansion