Biomedical Engineering Reference
In-Depth Information
1
2
−
⎡
2
⎤
3
⎛
⎜
pA
a
3
⎞
⎟
A
1
5
3
5
⎢
⎢
⎥
⎥
e
=
1
−
(5.11)
5
π
1
+
A A
+
/
π
a
3
5
⎣
⎦
1
2
−
⎡
2
⎤
2
⎛
⎜
qA
b
2
⎞
⎟
A
1
4
2
4
⎢
⎢
⎥
⎥
n
=
1
−
(5.12)
4
2
2
2
π
1
+
n A n A
+
/
π
n b
2
2
4
4
1
⎣
⎦
π
λ
0
(5.13)
A
=
=
2 3 4 5
,
,
,
1
2
1
2
(
)
(
)
2
2
2
2
k
−
k
2
n
−
n
1
2 3 4 5
,
,
,
1
2 3 4 5
,
,
,
The
E
p
y
modes are polarized such that
E
y
is the only significant component
of electric field with
E
x
and
E
z
negligibly small. In the case of the
E
p
x
modes,
E
x
is the only significant electric field component.
A computer code was written to calculate the modes of channel waveguides
using Marcatili's equations. The program provides the
e
's,
n
's, and
n
eff
's =
k
z
's
for any particular channel guide design. The goal of the design process was
to determine the fabrication parameters that would produce a guide with the
least amount of propagation loss possible and with only single mode propa-
gation. The key factor in the determination of these optimum parameters
is the minimization of the optical power guided in the mode tails residing
in the substrate, that is,
n
3
,
n
4
,
n
5
region. Calculations of waveguide cutoff
curves and losses for any aspect ratio can be performed for any propagation
wavelength. Figures 5.7 through 5.9 are the propagation cutoff curves at 1.3,
3.4, and 10.6 μm wavelengths for an aspect ratio of 1. These curves, along
with the accompanying loss versus
N
D
curves, provide the basis for the selec-
tion of
N
D
,
t
, and aspect ratio at each wavelength for channel waveguides. In
addition, Figures 5.10 through 5.12 indicate the specific range or distance on
the cutoff curve
x
-axis, over which the
E
11
and
E
11
modes are guided but the
E
21
and
E
21
modes are not guided, versus
N
D
and aspect ratio. This thickness
tolerance is important since it decreases, for example, with increasing aspect
ratio. Since a thickness is generally chosen somewhere near the center of this
range to prevent any fabrication error causing either multimode or no mode
guiding, this range is an important design consideration.
5.4 DirectionalCouplers
The dual channel directional coupler consists of parallel channel waveguides
spaced so that synchronous coherent coupling between the overlapping
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