Page 508 - Fiber Optic Communications Fund

Nonlinear Effects in Fibers 489

a(z)
1

G

z
L a 2L a nL a
Figure 10.33 Plot of a(Z) vs. distance Z.

Example 10.13
2
Show that the Fourier transformation of |u | u is
0 0
P 3∕2 3 √ ∑
T
0
0
2
2
{|u | u }= b b b exp [g +(i − d) ∕4C],
0
0
l m n
|T | T C
2
1 1 lmn
where
√ ∞ [ ]
P T ∑ (T − nT ) 2
0 0
s
u = b exp − ,
0
n
T 1 n=−∞ 2T 1 2
2
3T + iS
0
C = 4 ,
2
2(T + S )
0
2
[(l + m + n)T + i(l + m − n)S]T s
0
d = ,
4
T + S 2
0
2
2
2
2
2
2
2
[(l + m + n )T +(l + m − n )iS]T 2 s
0
g = .
4
2(T + S )
2
0
Solution:
Let [ ]
(T − lT ) 2
s
r = exp − , (10.491)
l
2T 2
1
∞
∞
∞
T
P 3∕2 3 ∑ ∑ ∑
0
0
∗
2
|u | u = u u u = b r b r b r ∗
n n
l l
m m
0
0
0 0 0
2
|T | T
1 1 l=−∞ m=−∞ n=−∞
P 3∕2 3
T ∑
0
0
∗
= b b b r r r , (10.492)
l m n l m n
2
|T | T
1 1 lmn