a method for obtaining the solutions of optical rib waveguides Çağatay uluiŞik
TRANSCRIPT
A METHOD FOR OBTAINING
THE SOLUTIONS OF OPTICAL
RIB WAVEGUIDES
Çağatay ULUIŞIK
With a proper configuration of waveguides, one can perform a wide range of operations like :
Modulation Switching Multiplexing Filtering Generation of optical waves
The field of integrated optics is primarily based on the fact that light can be guided and confined in very thin films (with dimensions wavelength of light) of transparentmaterials on suitable substrates.
Wavelength of light = 0.8 m 1.7 m Optical frequencies = 176.5 TeraHz 375 TeraHz
The basic component of an integrated optic device is the optical waveguide.
Optical waveguides are also used as the connecting wires of optical circuits
n0n1
n2
(a)
n0
n1n2
(b)
n0
n1
n2
n3
(c)
n0
n2
n1
(d)
a) Raised Strip waveguideb) Embedded Strip waveguidec) Strip loaded waveguided) Rib waveguide
THE WAVEGUIDES MOSTLY USED IN INTEGRATED OPTICS
n1
n2
n3
D2
D1
2h I
II
III
y
x
a) 1. Reference Problem
n1
n2
n3
D2
D1
2h I
II
III
y
x
L
b) 2. Reference Problem
REAL PROBLEM, INVESTIGATED PROBLEM AND REFERENCE PROBLEMS
c) Investigated Problem
n1
n2
n3
D2
D1
2h I
II
III
y
x
D
Wn1
SS
L
n2 n2
d) Real Problem
n1
n3
2h I
II
III
y
x
DW
n1n2
SOLUTIONS OF DIELECTRIC SLAB LOADED PLANAR WAVEGUIDES
n1
n2
n3
D2
D1
2h I
II
III
y
x
n1> n2 , n3
)(sin)coth()(cos 12221111 yhKDKKyhKKAE x
)sinh(
)(sinh
22
22112 DK
yhDKKAE x
)sinh(
)(sinh)coth(2sin2cos
13
1322121113 DK
yhDKDKhKKhKKAE x
z
E
jH x
y
1
y
E
jH x
z
1
1332212111222112
1 coth)()coth(2sin)(2cos)(2cos)coth())((2sin)( DKhKDKhKhKhKhKhKDKhKhKhKhK
2211 2 nthK
22
22 2 nthK
23
23 2 nthK
22
2222 2 ndDK
23
2113 2 ndDK
hK
DKDK
DK
DKhKhKhKhK
DK
DKhK
hK
DKhK
hKDKhK
hKhK
DKhKhKDKhKhKhN j
3
13132
13
221211
222
211
2
222
1
1222
11
2222
22
122
222
21
2
4
22sinh
sinh
coth2sin2cos
)(sinh2
)()(
2
coth)(
4cos12
)coth(4sin
4
)(coth)()()(coth)()(
EIGENVALUE EQUATION
NORMALIZATION
GRAPHIC USER INTERFACE(GUI)
1 3.39239284511917
2 3.36952676519382
3 3.33128084318311
4 3.27751079657134
5 3.20820984151176
6 3.12407507913493
7 3.02985579154173
1N 10.5461
2N 10.5617
3N 10.5911
4N 10.6420
5N 10.7337
6N 10.9324
7N 11.7887
The roots of the eigenvalue equation (the propogation coefficients) , the normalization coefficients and the normalized modal field functions for the input parameters 4.31 n ,
32 n , 33 n , 1t , 41 d and 42 d
1. Mod
2. Mod 5. Mod
15. Mod – Reel Input Function – The Input Function Computed via Excitation Coefficients
SOLUTIONS OF DIELECTRIC SLAB LOADED and CLOSED (SCREENED) WAVEGUIDES
n1
n2
n3
D2
D1
2h I
II
III
y
x
n1> n2 , n3
L
EIGENVALUE EQUATION
1332212111222112
1 coth)()coth(2sin)(2cos)(2cos)coth())((2sin)( DKhKDKhKhKhKhKhKDKhKhKhKhK
NORMALIZATION :
21
22
21
22 2 hKnnthK
21
23
21
23 2 hKnnthK
21
22
21
2222 2 hKnnt
t
dDK
21
23
21
2113 2 hKnnt
t
dDK
21
22
1 2
t
hK
Le
nn
220)(
240
LenhI
LetE
y
hK
DKDK
DK
DKhKhKhKhK
DK
DKhK
hK
DKhK
hKDKhK
hKhK
DKhKhKDKhKhKhI y
3
13132
13
221211
222
211
2
222
1
1222
11
2222
22
122
222
21
4
22sinh
sinh
coth2sin2cos
)(sinh2
)()(
2
coth)(
4cos12
)coth(4sin
4
)(coth)()()(coth)()(
The normalized modal field functions for the input parameters 4.31 n , 32 n , 33 n , 1t , 41 d 42 d and Le=10
n=3 ; 3. mod in xm=1; 1. mod in y
n=1 ; 1. mod in xm=3; 3. mod in y
SOLUTIONS OF CLOSED RIB WAVEGUIDES
ztttztt uH
njuHj
z
E
2
11
0
tztttzt Eu
jEuj
z
H
1
VQV
22
CBBCQ
n1
n2
n3
D2
D1
2h I
II
III
y
x
D
Wn1 SS
L
n2 n2
xixi uee
yiyxixi uhuhh
yx
eh ix
iix
2
1
ixi
ixiy ek
h
22
xixiy
yixiy
zzitt hy
h
yx
hh
yx
h
x
huhuh
2
222
2
2
)(sin)coth()(cos)cos( ,12,2,2,1,1 yhKDKKyhKKxke iiiiixix
h
Dh
iL
W
xx
h
Dh
i
L
W
xxxi
xi dy
y
yg
y
ygdxxkkdyygygdxxkkk
nkk
nnB
)()()(sin2)()()(cos2
111 2/2
2/222
22
21
20
22
210
,
t
dt
it
dt
i
i
i dYY
hYg
Y
hYg
Le
wn
Le
wnn
tdYhYghYg
Le
wn
Le
wnn
Le
nt
tLen
nnnB
2222
322
21
22
,
)()(4sin
42
1)()(
4sin
422
2120
30
)(sin)coth()(cos)( ,12,2,2,1,1 yhKDKKyhKKyg iiiiii
S
zizi dSeueunnC 21
220,
t
dt
ii dYhYghYgLe
wn
Le
wnn
tn
LennC )()(4sin42
240 2
21
22
,
xixi uee
xx uee
S
xixi dSeennC 21
220,
)()cos()(sin)coth()(cos)cos( ,12,2,2,1,1 ygxkyhKDKKyhKKxke ixiiiixix
)()cos()(sin)coth()(cos)cos( ,12,2,2,1,1 ygxkyhKDKKyhKKxke xxx
S
ixi dSygygxknnC )()()(cos221
220,
h
Dh
ixxx
i dyygygWkWLkk
nnC )()(2sin2
2
21
220
,
The field function of the rib waveguide for the input parameters 3.4n1 , 3n2 , 3n3 , 1t , 4d1 , 4d2 , Le=10, w=2.5; d=0.5 corresponding to the 5th mode
in x and 1st mode in y.
The field function of the rib waveguide for the input parameters 3.4n1 , 3n2,3n3, 1t, 4d1, 4d2 , Le=10, w=2.5; d=0.5 corresponding to the 5th mode
in x and 2nd mode in y.
The field function of the rib waveguide for the input parameters 3.4n1 , 3n2 , 3n3 , 1t , 4d1 , 4d2 , Le=10, w=2.5; d=0.5 corresponding to the 10th mode
in x and 1st mode in y.
The field function of the rib waveguide for the input parameters 3.4n1 , 3n2 , 3n3 , 1t , 4d1 , 4d2 , Le=10, w=2.5; d=0.5 corresponding to the 20th mode
in x and 1st mode in y.
COMPUTATIONAL TIMES AND COMPARISON
Our solution is a general way and can easily be applied to other waveguides used in integrated optics.
n=5 n=10 n=20PII 333 MHz, 64 Mbayt Ram 284 s 512 s 962 sPIII 733 MHz, 384 Mbayt Ram 107 s 187 s 354 s
The required computational times for different values of n and on different platforms
1. dominant mod find via 3 different methods for the inputparameters 4.31n , 32n , 33n , 1t , 41d , 42d ,Le=10, w=2.5, d=0.5
Method of Moments 1= 3.3863382
Effective Index Method 1= 3.3904312
FDTD 1=3.3893602
Error % 0.4