Environmental Engineering Reference
In-Depth Information
L
D
;
c
ii
J
i
¼
i
0
xl
0
g
D
J
i
ð
7
:
25
Þ
where
denotes the discrete convolution
g
D
J
i
¼
X
k
0
;
m
0
;
n
0
g
D
ð
k
k
0
;
m
m
0
;
n
n
0
Þ
J
i
ð
k
0
;
m
0
;
n
0
Þ
ð
7
:
26
Þ
and
g
D
ð
k
;
m
;
n
Þ¼
Z
Z
Z
Du
1
Du
2
Du
3
g
ð
u
1
;
k
u
0
1
;
u
2
;
m
u
0
2
;
u
3
;
n
u
0
3
Þ
du
0
1
du
0
2
du
0
3
ð
7
:
27
Þ
0
0
0
Likewise, the operator L
12
in (
7.21
) can be discretized as
i
0
xe
0
Du
1
Du
2
L
D
;
q
12
J
2
¼
g
D
ð
k
þ
1
;
m
;
n
Þ
g
D
ð
k
;
m
;
n
Þ
J
2
ð
k
;
m
;
n
Þ
J
2
ð
k
;
m
1
;
n
Þ
ð
7
:
28
Þ
i
0
xe
0
Du
1
Du
2
g
D
ð
k
þ
1
;
m
;
n
Þ
g
D
ð
k
;
m
;
n
Þ
¼
J
2
ð
k
;
m
;
n
Þ
g
D
ð
k
þ
1
;
m
1
;
n
Þ
g
D
ð
k
;
m
1
;
n
Þ
where the finite-difference method is used for the smooth approximation of the
dyadic Green's function.
The computations of the discrete convolutions can be performed efficiently
by means of cyclic convolutions and FFT [
76
], which is similar to the DDA
method. As a traditional iterative solver of the resulting VIE matrix equation, the
conjugate-gradient method [
77
] converges very slowly and will produce the
nonphysical random errors in the calculation of optical absorption. To tackle
the problem, we employ the fast and smoothly converging biconjugate gradient
stabilized (BI-CGSTAB) method [
78
] (See
Appendix
). The FFT is adopted to
accelerate the matrix-vector multiplications encountered in the BI-CGSTAB
solver with computational complexity of O
ð
N log N
Þ
and memory of O
ð
N
Þ
.
7.3.4 Physical Quantities Extraction
Through the rigorous solutions to Maxwell's equations, we can access some
important physical quantities to reveal the physical mechanism of plasmonic
effects in OSCs and optimize device performances.
The absorption spectrum of OSCs is calculated by
S
A
ð
k
Þ¼
Z
v
n
r
ð
k
Þ
k
i
ð
k
Þ
2pc
0
k
e
0
j
E
j
2
dV
ð
7
:
29
Þ
Search WWH ::
Custom Search