[SOLVED] Scheme Time-domain and frequency domain mode solver

Time-domain and frequency domain mode solver
Time-domain method
Finite-difference time-domain (FDTD) method
Frequency-domain method
Plane-wave expansion (PWE) method Beam propagation method
Finite element method (FEM)
Finite difference method (FDM)
Photonic Crystals
C. P. Yu
1
Central difference scheme
Applying Taylor expansion
x (x)2
f(x x) f f f(1) 0 10
f(2)
f(k) f(k)
(x)k 1! 2! k!
f(x x) f f 010
x (x)2 (x)k f(1) f(2)
1! 2! k!
where f ( k ) denotes k th order derivative of function f at point x0
f (1) f1 f1 O[(x)2 ] 2x
f(2) f1 2f0 f1 O[(x)2] (x)2
Photonic Crystals
C. P. Yu
2

Consider a slab waveguide. To find out its TE guided modes.
x n0
n2 2H k2H 0 zz
2 H 2H k2H 0 xyz z z
d2Hz k2H 2H dx2 z z
z
Photonic Crystals
C. P. Yu
3
n1
d2Hz k2H 2H dx2 z z
for f(2) f1 2f0 f1 (x)2
d2Hz Hz,L2Hz,CHz,R dx2 (x)2
Hz,L2Hz,CHz,Rk2H 2H (x)2 z,C z,C
Hz,L Hz,C Hz,R
xx x
C. P. Yu
Photonic Crystals
4

Hz,L 2Hz,C Hz,R k2H 2H (x)2 z,C
z,C
We can devide the slab waveguide in the x-direction by 9 points.
Hz,1
x
Photonic Crystals
Hz,2 Hz,3 Hz,4 Hz,5 Hz,6 Hz,7
n0 n1 n2
Hz,8
Hz,9
C. P. Yu
5
x
x
Hz,12Hz,2Hz,3n2k2H 2H
(x)2 Hz,2 2Hz,3
(x)2 Hz,4 2Hz,5
(x)2 Hz,7 2Hz,8
(x)2 Hz,2 Hz,3
0 0 z,2 z,2 Hz,4 n2k2H 2H
0 0 z,3 z,3

Hz,6 n2k2H 2H
1 0 z,5 z,5

Hz,9 n2k2H 2H 2 0 z,8
z,8
Hz,1
x
Hz,4
Hz,5
Hz,6
Hz,7
Hz,8
Hz,9
C. P. Yu
Photonic Crystals
n0 n1 n2
x
x
6

?2Hz,1 Hz,2 n2k2H 2H (x)2 0 0 z,1
Hz,8 2Hz,9 ?n2k2H
z,1
2H
Hz,6
Hz,1
x
Photonic Crystals
(x)2
Hz,2 Hz,3
Hz,4
2 0 z,9
Hz,5
z,9
Hz,7
Hz,8
Hz,9
C. P. Yu
7
n0 n1 n2
x
x
?2Hz,1 Hz,2 n2k2H 2H (x)2 0 0 z,1
Hz,8 2Hz,9 ?n2k2H (x)2 2 0 z,9
z,1
2H
use proper boundary condition for simulation, for example
02Hz,1 Hz,2 n2k2H 2H (x)2 0 0 z,1 z,1
z,9
Hz,8 2Hz,9 0n2k2H 2H
Hz,1
x
Hz,3
Hz,6
Hz,7
Hz,8
Hz,9
C. P. Yu
(x)2 Hz,2
2 0 z,9
Hz,4 Hz,5
z,9
Photonic Crystals
n0 n1 n2
x
x
8

02Hz,1 Hz,2 n2k2H 2H (x)2 0 0 z,1 z,1
Hz,12Hz,2Hz,3n2k2H 2H
(x)2 Hz,2 2Hz,3
(x)2 Hz,4 2Hz,5
(x)2 Hz,7 2Hz,8
(x)2 Hz,8 2Hz,9
(x)2
0 0 z,2 z,2 Hz,4 n2k2H 2H
0 0 z,3 z,3

Hz,6 n2k2H 2H
1 0 z,5 z,5

Hz,9 n2k2H 2H 2 0 z,8
0n2k2H 2H 2 0 z,9
z,8
z,9
Photonic Crystals
C. P. Yu
9
2/(x)2n2k2 1/(x)2 0 0 0 0 0 0 0 H 00 z,1
1/(x)2 2/(x)2 n2k2 1/(x)2 0 0 0 0 0 00
0 Hz,2
0 1/(x)2 2/(x)2n2k2 1/(x)2 0 0 0 0
0 0 1/(x)2 2/(x)2 n2k2 1/(x)2 0 0 0 0 Hz,4
0 H 00 z,3
10
0 0 0 1/(x)2 2/(x)2 n2k2 1/(x)2 0 0 0 Hz,5 10
2/(x)2 n2k2 1/(x)2 10
0 1/(x)2
0 0 0 0 0 1/(x)2 2/(x)2 n2k2 1/(x)2 0 Hz,7
0 0
H z ,1 H z,2
Hz,3 H z,4
2H z,5
Hz,6
Hz,8 0 0 0 0 0 0 1/(x)2 2/(x)2n2k2Hz,9

20
Hz,7
02Hz,1 Hz,2 n2k2H 2H (x)2 0 0 z,1 z,1
Hz,12Hz,2Hz,3n2k2H 2H
H
0 0 0
0 0
Hz,6
20
22222 0 0 0 0 0 1/(x) 2/(x) n2k0 1/(x)
z,8
H z,9
AH 2H Photonic Crystals
(x)2 Hz,2 2Hz,3
(x)2 Hz,4 2Hz,5
(x)2 Hz,7 2Hz,8
(x)2 Hz,8 2Hz,9
(x)2
0 0 z,2 z,2 Hz,4 n2k2H 2H
0 0 z,3 z,3

Hz,6 n2k2H 2H
1 0 z,5 z,5

Hz,9 n2k2H 2H 2 0 z,8
0n2k2H 2H 2 0 z,9
z,8
z,9
C. P. Yu
10