Page 489 - Fiber Optic Communications Fund

470 Fiber Optic Communications

The nonlinear phase shift is

2
(T)= |u(T, 0)| Z , (10.330)
NL
eff
= P(T, 0)Z . (10.331)
eff
At T = 0, we have
3
(0)= 1.1 × 10 −3 × 6 × 10 −3 × 18.27 × 10 = 0.1206 rad. (10.332)
NL
The Taylor expansion of Eq. (10.326) yields
{ 2 }
4 2
2
u(T, Z)= u(T, 0) 1 + i|u(T, 0)| Z eff − |u(T, 0)| Z eff +··· . (10.333)
2!
Here, the second and third terms on the right-hand sied of Eq. (10.333) represent the first-order and
second-order corrections due to nonlinear effects. First, let us consider only the first-order term
2
u(T, Z)= u(T, 0){1 + i|u(T, 0)| Z } (10.334)
eff
= u(T, 0)B(T)e i(T) , (10.335)
where
√
4 2
2
B(T)= 1 + |u(T, 0)| Z , (10.336)
eff
2
−1
(T)= tan (|u(T, 0)| Z ). (10.337)
eff
At T = 0, we have
B(0)= 1.007, (10.338)
(0)= 0.12002 rad. (10.339)

B and represent the amplitude shift and nonlinear phase shift using the first-order theory. Note that from the
exact solution given by Eq. (10.326), we see that there is no change in amplitude due to fiber nonlinearity;
the first-order approximation shows that the amplitude is shifted by a factor of 1.007.
Next, consider the terms up to second order in Eq. (10.333),

u(T, Z)= u(T, 0){x + iy}, (10.340)
2 4 2
x = 1 − |u(T, 0)| Z , (10.341)
eff
2!
2
y = |u(T, 0)| Z . (10.342)
eff
i
Let x + iy = Be . Proceeding as before, we find
B(0)= 1.00002, (10.343)
(0)= 0.12089 rad. (10.344)

Comparing Eqs. (10.338) and (10.339) with Eqs. (10.343) and (10.344), we see that the second-order theory
is closer to the exact result.