Page 481 - Fiber Optic Communications Fund

462 Fiber Optic Communications

The first-order correction u (T, Z) is obtained by performing the inverse Fourier transformation of ̃u (, Z).
1
1
Typically, in quasi-linear systems, the nonlinear effects are smaller than the dispersive effects and the
first-order correction u (T, Z) is often adequate to describe the nonlinear propagation. However, when
1
the transmission distance is long and/or the launch power is large, a second-order perturbation theory is
needed [20].
A closed-form expression for u (T, Z) and u (T, Z) can be obtained if we assume that the pulse shape f(T)
0
1
is Gaussian, i.e.,
( )
√ T 2
f(T)= P exp − 2 , (10.272)
0
2T
0
where P is the peak power. The linear propagation of this pulse is described by (see Eq. (2.158))
0
√ [ ]
T 0 P 0 T 2
exp − , (10.273)
T (Z) 2T (Z)
2
1
1
2
2
where T = T − iS(Z). When a long bit sequence is launched to the fiber, u (T, 0) is given by Eq. (10.263).
1 0 0
In this case, the linear solution is
√ ∞ [ ]
T 0 P 0 ∑ (T − nT ) 2
s
u (T, Z)= b exp − , (10.274)
n
0
2
T 1 n=−∞ 2T (Z)
1
∞
∞
∞
T 0 3 ∑ ∑ ∑
3∕2
2
F(T, Z)= |u (T, Z)| u (T, Z)= P 0 2 b b b
0
0
l m n
|T | T
1 1 l=−∞ m=−∞ n=−∞
[ ]
(T − lT ) 2 (T − mT ) 2 (T − nT ) 2
s
s
s
× exp − − − . (10.275)
2 ∗
2T 2 2T 2 2(T )
1 1 1
The Fourier transform of F(T, Z) is (Example 10.13)
P 3∕2 3 √ ∑
T
0
0
̃
2
F(, Z)= b b b exp [−g(Z)+[i − d(Z)] ∕4C(Z)], (10.276)
l m n
2
|T | T C(Z)
1 1 lmn
where
2
3T + iS
0
C(Z)= 4 , (10.277)
2
2(T + S )
0
2
T [(l + m + n)T + i(l + m − n)S]
s
0
d(Z)= , (10.278)
4
T + S 2
0
2
2
2
2
2
2
2
2
T [(l + m + n )T + i(l + m − n )S]
s
0
g(Z)= 4 . (10.279)
2
2(T + S )
0
Substituting Eq. (10.276) in Eq. (10.269), and after performing the inverse Fourier transformation, we find
∑
u (T, L )= u lmn (T, L )b b b , (10.280)
tot
l m n
tot
1
lmn
2
2
L tot a (x) exp [−(2CT − d) ∕(4C(1 + i2SC)− C)]
u (T, L )= iP 3∕2 3 dx. (10.281)
T
lmn tot 0 0 ∫ √
0 2 4 2
(1 + i2SC)(T − iS)(T + S )
0 0