Image Processing Reference
In-Depth Information
ε
0
is the vacuum permittivity and
x
(
1
)
Where
is the first order susceptibility.
To consider
non-linearities in the system, the Eq.
2 can be re-written as illustrated in Eq.
3 (Agrawal,
2001).
P
(
r
,
t
)=
P
L
(
r
,
t
)+
P
NL
(
r
,
t
)
(3)
(
)
Whereas,
P
NL
is the non-linear part of polarization. Eq. 3 can be used to solve Eq.
1 to derive the propagation equation in non-linear dispersive fibers with few simplifying
assumptions. First,
P
NL
is treated as a small perturbation of
P
L
and the polarization field
is maintained throughout the whole propagation path. Another assumption is that the index
difference between the core and cladding is very small and the center frequency of the wave
is assumed to be much greater than the spectral width of the wave which is also called as
quasi-monochromatic assumption. The quasi-monochromatic assumption is the analogous to
low-pass equivalent modelling of bandpass electrical systems and is equivalent to the slowly
varying envelope approximation in the time domain. Finally, the propagation constant,
r
,
t
,
is approximated by a few first terms of Taylor series expansion about the carrier frequency,
ω
0
,thatcanbegivenas;
β
(
ω
)
1
2
(
ω
−
ω
0
1
6
(
ω
−
ω
0
2
3
β
(
ω
)=
β
0
+(
ω
−
ω
0
)
β
1
+
)
β
2
+
)
β
3
+
.......
(4)
Whereas;
d
n
β
β
n
=
(5)
n
d
ω
ω
=
ω
0
ps
2
/
km
[
]
The second order propagation constant
, accounts for the dispersion effects in the
optical fibers communication systems. Depending on the sign of the
β
2
β
2
, the dispersion region
can be classified into two parts as, normal(
0). Qualitatively,
in the normal-dispersion region, the higher frequency components of an optical signal travel
slower than the lower frequency components. In the anomalous dispersion region it occurs
vice-versa. Fiber dispersion is often expressed by another parameter,
D
β
2
>
0) and anomalous (
β
2
<
[
(
)]
ps
/
nm
.
km
,whichis
1
υ
g
and the mathematical relationship
d
d
called as dispersion parameter.
D
is defined as
D
=
λ
between
β
2
and
D
is given in (Agrawal, 2001), as;
2
β
2
=
−
λ
c
D
(6)
2
π
Where
υ
g
is the group velocity. The cubic and
the higher order terms in Eq. 4 are generally negligible as long as the quasi-monochromatic
assumption remains valid. However, when the center wavelength of an optical signal is near
the zero-dispersion wavelength, as for broad spectrum of the signals, (that is
λ
is the wavelength of the propagating wave and
β
≈
0) then the
β
3
terms should be included.
If the input electric field is assumed to propagate in the
+
z
direction and is polarized in the
x
direction Eq. 1 can be re-written as;
∂
∂
)=
−
2
E
z
E
(
z
,
t
(
z
,
t
)
(linear attenuation)
2
j
β
2
2
∂
+
2
t
E
(
z
,
t
)
(second order dispersion)
∂
