Page 468 - Fiber Optic Communications Fund

Nonlinear Effects in Fibers 449

Let us first consider a single triplet {jkl} corresponding to channels at frequencies Ω , Ω , and Ω satisfying
k
l
j
Ω +Ω −Ω =Ω . (10.209)
j k l n
Let the FWM field generated at Ω be .Now, q may be written as
n
n
n
(0)
q = q n + , (10.210)
n
n
(0)
where q n is the signal field in the absence of nonlinearity. We assume that | | ≪ |q |. Considering only the
n
n
triplet {jkl}, Eq. (10.208) becomes
( (0) ) ( )
dq n + (0) + d n + = iq q q e , (10.211)
(0) (0) ∗(0) iΔ jkln Z
q
dZ 2 n dZ 2 n j k l
where
Δ = + − − (10.212)
jkln j k l n
(0) (0) ∗(0)
∗
is the phase mismatch. Note that q q q of Eq. (10.208) is replaced by q q q in Eq. (10.211), which is
j k l j k l
known as the undepleted pump approximation. When the FWM power is much smaller than the signal power,
(0)
depletion of signal terms (i.e., FWM pumps) appearing in Eq. (10.211) may be ignored. Since q n is the signal
field in the absence of nonlinearity, it can be written as
(0) N N N
q n = A e − Z+i n , n =− , − + 1, … , − 1, (10.213)
2
n
2 2 2
where A and are amplitude and phase at Z = 0, respectively. It can easily be seen that
n n
(0)
dq
n (0)
+ q n = 0. (10.214)
dZ 2
So, Eq. (10.211) becomes
d n (0) (0) ∗(0) iΔ jkln Z
+ = iq q q e . (10.215)
n
dZ 2 j k l
When the third-order dispersion is ignored, the propagation constant is given by Eq. (10.133). Using
Eqs. (10.133), and Eq. (10.209) in Eq. (10.212), we find
2 2 2 2 2
Δ = (Ω +Ω −Ω −Ω )+ [Ω +Ω −Ω −(Ω +Ω −Ω ) ]
jkln 1 j k l n j k l j k l
2
= [Ω Ω −Ω Ω ]. (10.216)
k
l
n
j
2
When the bandwidth of the WDM signal and/or the dispersion slope is large, the third-order dispersion
coefficient can not be ignored. In this case, Eq. (10.216) is modified as (see Example 10.9)
[ ]
( )
3
Δ jkln =(Ω Ω −Ω Ω ) + Ω +Ω k . (10.217)
j
2
j
k
l
n
2
Substituting Eq. (10.213) in Eq. (10.215), we find
d
n −(3∕2−iΔ jkln )Z+iΔ jkl
+ = iA A A e , (10.218)
n
j k l
dZ 2
where
Δ jkl = + − . (10.219)
j
l
k