Digital Signal Processing Reference
In-Depth Information
(a)
(b)
P
C
Silicon
wave
guide
BPF
Pump
BPF
Receiver
ED
FA
Optical
coupler
P
C
Signal
BPF (band-pass filter)
Fig. 8.6 a
A typical wavelength conversion by FWM.
b
The schematic configuration of the
wavelength converter in silicon waveguide [
20
]
By assuming that the reflections are very weak so that backward propagating
waves can be safely discarded, it is supposed that the optical waves propagate only
in the forward,
+
z
, direction. The total electrical field inside silicon waveguide
will be in the form
E
(
r
,
t
) =
s
p
(
z
)
F
p
(
r
)
A
p
(
z
,
t
)
exp
[
i
(β
0
p
z
− ω
p
t
)]
+
s
s
(
z
)
F
s
(
r
)
A
s
(
z
,
t
)
exp
[
i
(β
0
s
z
− ω
s
t
)]+
c.c.
where
ω
m
=
2
π
c
/
λ
m
(
M
=
P
or
s
for pump and signal waves) is the carrier fre-
quency,
β
0
m
=
n
0
m
k
m
is the propagation constant,
k
m
= ω
m
/
c
is the free-space
wave number, and
n
0
m
is the effective refractive index. To account for the wave-
guide tapering, here introduced
s
m
(
z
)
as the shape function [
22
].
8.6.2 Wavelength Conversion
By means of a standard procedure [
23
], the wave Eq. (
8.2
) can be reduced to the
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