Image Processing Reference
In-Depth Information
3
+
β
3
6
∂
3
t
E
(
z
,
t
)
(third order dispersion)
∂
2
E
−
γ
|
(
)
|
(
)
j
E
z
,
t
z
,
t
(Kerr effect)
T
R
∂
2
E
+
∂
t
|
(
)
|
(
)
j
γ
E
z
,
t
z
,
t
(SRS)
∂
ω
0
∂
∂
t
|
2
E
−
E
(
z
,
t
)
|
(
z
,
t
)
(self-steeping effect)
(7)
Where
E
(
z
,
t
)
is the varying slowly envelope of the electric field,
z
is the propagation distance,
t
=
t
-
(
t
=physicaltime,
v
g
υ
g
=the group velocity at the center wavelength),
α
is the fiber
β
2
is the second order propagation constant [
ps
2
/
km
],
loss coefficient [1/
km
],
β
3
is the third
n
2
λ
0
A
ef f
2
π
is the non-linear coefficient [
km
−
1
W
−
1
],
n
2
order propagation constant [
ps
3
/
km
],
γ
=
·
is the non-linear index coefficient,
A
eff
is the effective core area of the fiber,
λ
0
is the center
wavelength and
ω
0
is the central angular frequency. When the pulse width is greater than
1ps, Eq. 7 can further be simplified because the Raman effects and self-steepening effects
are negligible compared to the Kerr effect (Agrawal, 2001). Mathematically the generalized
form of non-linear Schrödinger equation suitable to describe the signal propagation in
communication systems can be given as;
E
2
=
N
D
E
∂
E
j
β
2
2
∂
t
2
−
2
2
z
=
j
γ
|
E
|
+
−
+
(8)
∂
∂
Also that
D
and
N
are termed as linear and non-linear operators as in Eq. 9.
2
j
β
2
2
∂
t
2
−
2
N
2
;
D
=
j
γ
|
E
|
=
−
(9)
∂
3.1 Split-step Fourier method (SSFM)
As described in the previous section, it is desirable to solve the non-linear Schrödinger
equation to estimate various fiber impairments occurring during signal transmission with
high precision. The split-step Fourier method (SSFM) is the most popular algorithm because
of its good accuracy and relatively modest computing cost.
As depicted in Eq. 8, the generalized form of NLSE contains the linear operator
D
and
non-linear operators
N
and they can be expressed as in Eq. 9. When the electric field envelope,
E
(
z
,
t
)
, has propagated from
z
to
z
+
h
, the analytical solution of Eq. 8 can be written as;
exp
h
N
D
z
,
t
(10)
In the above equation
h
is the propagation step length also called as step-size, through the
fiber section. In the split-step Fourier method, it is assumed that the two operators commute
with each other as in Eq. 11;
E
(
z
+
h
,
t
)=
+
·
E
(
exp
h
N
exp
h
D
·
z
,
t
(
+
)
≈
(
E
z
h
,
t
E
(11)
Eq.11 suggests that
E
can be estimated by applying the two operators independently.
If
h
is small, Eq.11 can give high accuracy results. The value of
h
is usually chosen such that
the maximum phase shift
(
z
+
h
,
t
)
Ep
2
(
φ
max
=
γ
|
|
h
,
Ep
=peak value of
E
(
z
,
t
)
) due to the non-linear
