Digital Signal Processing Reference
In-Depth Information
numerical modeling of SC, we can use Eq. (
6.1
) by adding higher order dispersion
terms. Then this will yield the following equation:
M
∂
M
A
∂
T
M
∂
A
∂
Z
1
2
(α(ω
O
) +
I
α
1
∂
∂
T
)
A
+
I
M
−
1
β
M
M
!
+
M
=
2
(6.2)
∞
∂
∂
T
T
′
Z
,
T
−
T
′
|
2
D
T
′
=
I
γ (ω
O
) +
I
γ
1
A
(
Z
,
T
)
R
|
A
0
where
M
is order representing the limit of dispersive effect,
β
m
is coefficient of
Taylor's series expansion of propagation constant (
t
) is the nonlinear response that
includes Raman contribution.
F
R
H
R
(
T
) |
A
(
Z
,
T
−
T
′
)|
2
R
(
T
) = (
1
−
F
R
)δ(
T
−
T
E
) +
where
t
e
represents short delay which is often negligible, fractional contribution of
delayed Raman response, and Raman response function are represented by
f
r
and
h
r
(
t
).
Equation (
6.2
) gave successful result in modeling some features of SC by using
ultra short pulses in nonlinear fibers. The split step Fourier transform method can be
used to solve the equation. Here the choice of
M
is not clear. It can
M
=
6 for numer-
ical simulation or values as large as 12 for some experiments. In fact, to all orders
dispersion can be included numerically because in split step method dispersive effect
is carried out in spectral domain of pulse by ignoring all nonlinear terms [
20
].
The term on right-hand side of equation is due to several nonlinear effects like
SPM, SRS, and self steepening.
In Fourier domain, Eq. (
6.2
) can be written as
∞
β
M
M
!
(ω − ω
O
)
M
(6.3)
M
=
2
= β(ω) − β(ω
O
) − β
1
(ω
O
)(ω − ω
O
)
where
β
1
=
1/
v
g
,
v
g
represents group velocity and is centered frequency [
21
]. This
approach needs knowledge about
β
(
ω
) or propagation constant over entire frequency
range over SC can spread. In case of tapered fiber or fiber with constant core cladding
refractive indices,
β
(
ω
) can be obtained by solving the Eigen value equation. This
approach is inappropriate for highly nonlinear fibers like photonic crystal fibers [
20
].
6.5 SC Generation with Picosecond Pulses
In 1993, SC generation acted as an ideal source for WDM optical systems. It was
used to generate pulse-train at multiple wavelengths by using picoseconds pulses
at 1.55
μ
m [
20
]. Research has been done for high average power SC sources; a
picoseconds laser generates the pulses with duration of 14 ps at repetition rate of
480 MHz and that exhibit 20 W average output power. This generates the SC over
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