Page 469 - Fiber Optic Communications Fund

450 Fiber Optic Communications

Eq. (10.218) is a first-order ordinary differential equation. The integrating factor is e Z∕2 . So, multiplying Eq.
(10.218) by e Z∕2 , we find
d( e Z∕2 )
n −(−iΔ jkln )Z+iΔ jkl
= iA A A e . (10.220)
j k l
dZ
Integrating Eq. (10.220) from 0 to L with the condition (0)= 0, we obtain
n
L
(L)= iA A A e −L∕2+iΔ jkl e −(−iΔ jkln )Z dZ
n j k l ∫
0
[1 − e − jkln L ]
= K jkl , (10.221)
jkln
where
K jkl = iA A A e iΔ jkl −L∕2 (10.222)
j k l
and
= − iΔ . (10.223)
jkln jkln
The power of the FWM component is [23, 24]

2
|K | |1 − e − jkln L 2
|
jkl
2
P = | | = e −L
FWM,n n 2
| jkln |
2
2
= P P P L e −L , (10.224)
j k l eff jkln
2
2
+ 4e −L sin (Δ jkln L∕2)∕L 2 eff
= , (10.225)
jkln 2 2
+(Δ jkln )
2
P = A , (10.226)
j j
1 − exp (−L)
L = . (10.227)
eff

Here, jkln represents the FWM efficiency. Fig. 10.16 shows the dependence of the efficiency on the dispersion
coefficient when j = 1, k = 2, and l = 3. When = 0, the efficiency is maximum and this is known as
2
2
phase matching.As | | increases, the FWM efficiency decreases and it becomes significantly smaller when
2
2
| | > 6ps /km. When the fiber is sufficiently long, the second term in Eq. (10.225) may be ignored and
2
Eq. (10.225) may be approximated as
2
jkln ≅ . (10.228)
2
+(Δ ) 2
jkln
Let the channel spacing be Δf and Ω = j2Δf, j =−N∕2, −N∕2 + 1, … , N∕2 − 1. Now, Eqs. (10.209) and
j
(10.216) become
j + k − l = n, (10.229)
2
Δ =(2Δf) [nl − jk]. (10.230)
jkln 2
With j = 1, k = 2, and l = 3, we find n = 0 and
2
1230 ≅ 2 . (10.231)
2
+ 4 (2Δf) 4
2