Page 477 - Fiber Optic Communications Fund

458 Fiber Optic Communications

Central pulse

Ch. 1
t

Ch. 2
t

Ch. 3
t

Ch. 4
t

Ch. 5
t
Intrachannel effects only
Figure 10.25 Illustration of the difference between intrachannel and inter channel nonlinear effects.

where (Z) is the loss/gain profile which includes fiber loss as well as amplifier gain, is the second-order
2
dispersion profile, and is the fiber nonlinear coefficient. Fig. 10.26(a) shows a typical fiber-optic transmission
system. We assume that the amplifier compensates for the fiber loss. To separate the fast variation of optical
power due to fiber loss/gain, we use the following transformation [50]:
q(T, Z)= a(Z)u(T, Z), (10.243)
where a(Z) is real. Differentiating Eq. (10.243), we find
̇q u
= ̇au + a , (10.244)
Z Z
where ⋅ denotes differentiation with regard to Z.Let
(Z)
̇ a =− a. (10.245)
2
Substituting Eqs. (10.244) and (10.245) in Eq. (10.242), we obtain the NLS equation in the lossless form as
2
u (Z) u 2 2
2
i − =−a (Z)|u| u. (10.246)
Z 2 T 2
Solving Eq. (10.245) with the initial condition a(0)= 1, we obtain
[ ]
Z (s)
a(Z)= exp − ds . (10.247)
∫
0 2
The choice of this initial condition is arbitrary. The sole purpose of introducing a(Z) is to separate the varia-
tions of the optical field due to loss/gain from that due to dispersion and nonlinear effects. Between amplifiers,
when the fiber loss is constant, (Z)= , it becomes
0
′
a(Z)= exp (− Z ∕2), (10.248)
0