9.5. IMPACT OF AMPLIFIER NOISE 441
[D=±4 ps/(km-nm)]. Optical amplifiers withnsp= 1 .3 (noise figure 4.1 dB) are
placed every 80.8 km (4 map periods) along the fiber link for compensating 0.2-dB/km
losses. Variational equations were used to find the input pulse parameters for which
the soliton recovers periodically after each map period:T 0 = 6 .87 ps,C 0 = 0 .56, and
E 0 = 0 .402 pJ. The nonlinear parameterγwas 1.7 W−^1 /km.
An important question is whether the use of dispersion management is helpful or
harmful from the standpoint of timing jitter. The timing jitter for standard solitons can
be found in a closed form by using Eq. (9.5.17) in Eq. (9.5.28) and is given by
σt^2 =SspT 02
3 Es[NA+^16 NA(NA− 1 )( 2 NA− 1 )d^2 ], (9.5.32)where we have usedEsfor the input soliton energy to emphasize that it is different
from the DM soliton energyE 0 used in Eq. (9.5.29). For a fair comparison of the DM
and standard solitons, we consider an identical soliton system except that the dispersion
map is replaced by a single fiber whose GVD is constant and equal to the average value
β ̄ 2. The soliton energyEscan be found by using Eq. (9.2.3) in Eq. (9.2.5) and is given
by
Es= 2 fLM|β ̄ 2 |/(γT 0 ), (9.5.33)
where the factorfLMis the enhancement factor resulting from loss management (fLM≈
3 .8 for a 16-dB gain). The dashed line in Fig. 9.17 shows the timing jitter using Eqs.
(9.5.32) and (9.5.33). A comparison of the two curves shows that the jitter is consider-
ably smaller for DM solitons. The physical reason behind the jitter reduction is related
to the enhanced energy of the DM solitons. In fact, the energy ratioE 0 /Esequals the
energy enhancement factorfDMintroduced earlier in Eq. (9.4.14). From a practical
standpoint, reduced jitter of DM solitons permits much longer transmission distances
as evident from Fig. 9.17. Note that Eq. (9.5.32) also applies for DDFs because the
GVD variations along the fiber can be included through the parameterdas defined in
Eq. (9.5.30).
For long-haul soliton systems, the number of amplifiers is large enough that theN^3 A
term dominates in Eq. (9.5.32), and the timing jitter for standard solitons is approxi-
mately given by [116]
σt^2
T 02
=
SspL^3 T
9 EsL^2 DLA. (9.5.34)
Comparing Eqs. (9.5.31) and (9.5.34), one can conclude that the timing jitter is reduced
by a factor of(fDM/ 3 )^1 /^2 when DM solitons are used. The factor of 3 has its origin in
the nearly Gaussian shape of the DM solitons.
To find a simple design rule, we can use Eq. (9.5.34) with the conditionσt<bj/B,
wherebjis the fraction of the bit slot by which a soliton can move without affecting
the system performance adversely. UsingB=( 2 q 0 T 0 )−^1 andEsfrom Eq. (9.5.33), the
bit rate–distance productBLTfor standard solitons is found to be limited by
BLT<
(
9 b^2 jfLMLA
Sspq 0 γβ ̄ 2