7.8. LONG-HAUL LIGHTWAVE SYSTEMS 307
lightwave system. Figure 7.15 shows such a recirculating fiber loop schematically.
It was used to demonstrate transmission of a 10-Gb/s signal over a distance of up to
10,000 km over standard fibers with periodic loss and dispersion management [135].
Two optical switches determine how long a pseudorandom bit stream circulates in-
side the loop before it reaches the receiver. The loop length and the number of round
trips determine the total transmission distance. The loop length is typically 300–
500 km. The length of DCF is chosen in accordance with Eq. (7.8.1) and is set to
L 2 =−D 1 L 1 /D 2 for complete compensation (D ̄=0). An optical bandpass filter is also
inserted inside the loop to reduce the effects of amplifier noise.
7.8.2 Simple Theory
The major nonlinear phenomenon affecting the performance of a single-channel sys-
tem is SPM. As before, the propagation of an optical bit stream inside a dispersion-
managed system is governed by the nonlinear Schr ̈odinger (NLS) equation [Eq. (7.7.4)]:
i∂A
∂z−
β 2
2∂^2 A
∂t^2+γ|A|^2 A=−iα
2A, (7.8.2)
with the main difference thatβ 2 ,γ, andαare now periodic functions ofzbecause of
their different values in two or more fiber sections used to form the dispersion map.
Loss compensation at lumped amplifiers can be included by changing the loss param-
eter suitably at the amplifier locations.
In general, Eq. (7.8.2) is solved numerically to study the performance of dispersion-
managed systems [120]–[129]. It is useful to eliminate the last term in this equation
with the transformation [see Eq. (7.7.5)]
A(z,t)=B(z,t)exp[
−
1
2
∫z0α(z)dz]
. (7.8.3)
Equation (7.8.2) then takes the form
i∂B
∂z−
β 2 (z)
2∂^2 B
∂t^2
+γ ̄(z)|B|^2 B= 0 , (7.8.4)where power variations along the dispersion-managed fiber link are included through a
periodically varying nonlinear parameterγ ̄(z)=γexp[−
∫z
0 α(z)dz].
Considerable insight into the design of a dispersion-managed system can be gained
by solving Eq. (7.8.4) with a variational approach [123]. Its use is based on the ob-
servation that a chirped Gaussian pulse maintains its functional form in the linear case
(γ=0) although its amplitude, width, and chirp change with propagation (see Section
2.4). Since the nonlinear effects are relatively weak locally in each fiber section com-
pared with the dispersive effects, the pulse shape is likely to retain its Gaussian shape.
One can thus assume that the pulse evolves along the fiber in the form of a chirped
Gaussian pulse such that
B(z,t)=aexp[−( 1 +iC)t^2 / 2 T^2 +iφ], (7.8.5)