Page 506 - Fiber Optic Communications Fund

Nonlinear Effects in Fibers 487

Example 10.12
The evolution of the complex field envelope in a periodically amplified fiber-optic system is governed by
N
2
q (Z) q 2 (Z) ∑
2
i − + (Z)|q| q =−i q + iA (Z − nL )q(T, nL ), (10.471)
a−
a
Z 2 T 2 2
n=1
where
(Z)= for 0 < mod(Z, L ) < L
2 20 a a
= 0 otherwise, (10.472)
(Z)= for 0 < mod(Z, L ) < L
0 a a
= 0 otherwise (10.473)

(Z)= for 0 < mod(Z, L ) < L
0 a a
= 0 otherwise (10.474)
Using the transformation

q(Z, T)= a(Z)u(Z, T), (10.475)
show that
2
u (Z) u 2 2
2
i − + a (Z)|u| u = 0, (10.476)
Z 2 T 2 2
a(Z)= e − 0 Z∕2 for 0 < mod(Z, L ) < L a
a
= 1 otherwise. (10.477)

Assume that the fiber loss is exactly compensated by the amplifier gain.

Solution:
Consider the propagation over a short length from nL a− to nL a− +ΔZ corresponding to the amplifier located
at nL . In this short length, (Z)= (Z)= (Z)= 0. Integrating Eq. (10.471) from nL a− to nL a− +ΔZ,we
a
2
obtain
nL a− +ΔZ nL a− +ΔZ
dq
i dZ = iA (Z − nL )q(T, nL ), (10.478)
∫ dZ ∫
a
a−
nL a− nL a−
q(T, nL +ΔZ)− q(T, nL )= Aq(T, nL ), (10.479)
a− a− a−
q(T, nL +ΔZ)
a−
= A + 1. (10.480)
q(T, nL )
a−
Since q(T, nL a− +ΔZ) and q(T, nL ) represent the amplifier output and input, respectively, we have
a−
√
q(T, nL a− +ΔZ)= Gq(T, nL ), (10.481)
a−