9.3. LOSS-MANAGED SOLITONS 423
The advantage of distributed amplification can be seen from Eq. (9.3.7), which can
be written in physical units as
dp
dz=[g(z)−α]p. (9.3.11)Ifg(z)is constant and equal toαfor allz, the peak power or energy of a soliton remains
constant along the fiber link. This is the ideal situation in which the fiber is effectively
lossless. In practice, distributed gain is realized by injecting pump power periodically
into the fiber link. Since pump power does not remain constant because of fiber losses
and pump depletion (e.g., absorption by dopants),g(z)cannot be kept constant along
the fiber. However, even though fiber losses cannot be compensated everywhere locally,
they can be compensated fully over a distanceLAprovided that
∫LA0g(z)dz=αLA. (9.3.12)A distributed-amplification scheme is designed to satisfy Eq. (9.3.12). The distanceLA
is referred to as thepump-station spacing.
The important question is how much soliton energy varies during each gain–loss
cycle. The extent of peak-power variations depends onLAand on the pumping scheme
adopted. Backward pumping is commonly used for distributed Raman amplification
because such a configuration provides high gain where the signal is relatively weak.
The gain coefficientg(z)can be obtained following the discussion in Section 6.3.
If we ignore pump depletion, the gain coefficient in Eq. (9.3.11) is given byg(z)=
g 0 exp[−αp(LA−z)], whereαpaccounts for fiber losses at the pump wavelength. The
resulting equation can be integrated analytically to obtain
p(z)=exp{
αLA[
exp(αpz)− 1
exp(αpLA)− 1]
−αz}
, (9.3.13)
whereg 0 was chosen to ensure thatp(LA)=1. Figure 9.12 shows howp(z)varies
along the fiber forLA=50 km usingα= 0 .2 dB/km andαp= 0 .25 dB/km. The case
of lumped amplification is also shown for comparison. Whereas soliton energy varies
by a factor of 10 in the lumped case, it varies by less than a factor of 2 in the case of
distributed amplification.
The range of energy variations can be reduced further using a bidirectional pumping
scheme. The gain coefficientg(z)in this case can be approximated (neglecting pump
depletion) as
g(z)=g 1 exp(−αpz)+g 2 exp[−αp(LA−z)]. (9.3.14)
The constantsg 1 andg 2 are related to the pump powers injected at both ends. Assuming
equal pump powers and integrating Eq. (9.3.11), the soliton energy is found to vary as
p(z)=exp[
αLA(
sinh[αp(z−LA/ 2 )]+sinh(αpLA/ 2 )
2 sinh(αpLA/ 2 ))
−αz]
. (9.3.15)
This case is shown in Fig. 9.12 by a dashed line. Clearly, a bidirectional pumping
scheme is the best as it reduces energy variations to below 15%. The range over which