"Introduction". In: Fiber-Optic Communication Systems

9.6. HIGH-SPEED SOLITON SYSTEMS 453

amplifier can be written as

Ω(zn)=Ω(zn− 1 )+bRE(zn− 1 )+δΩn, (9.6.6)

q(zn)=q(zn− 1 )+b 2 Ω(zn− 1 )+b 2 RE(zn− 1 )+δqn, (9.6.7)

where the parametersb 2 ,bR, andb 2 Rare defined as

b 2 =

∫LA

0

β 2 (z)dz, bR=−

TR

2

√

π

∫LA

0

dzγ(z)

T^3 (z)

[∫z

0

p(z 1 )dz 1

]

, (9.6.8)

b 2 R=−

TR

2

√

π

∫LA

0

dzβ 2 (z)γ(z)

∫z

0

dz 1

T^3 (z 1 )

[∫z

1

0

p(z 2 )dz 2

]

, (9.6.9)

wherep(z)takes into account variations in the pulse energy. In the case of lumped
amplification,p(z)=exp(−αz).
The physical meaning of Eqs. (9.6.6) and (9.6.7) is quite clear. Between any two
amplifiers, the Raman-induced frequency shift within the fiber should be added to the
ASE-induced frequency shift. The former, however, depends on the pulse energy and
would change randomly in response to energy fluctuations. The position of the pulse
changes because of the cumulative frequency shifts induced by the Raman effect and
amplifier noise. The two recurrence relations in Eqs. (9.6.6) and (9.6.7) should be
solved to find the final pulse position at the end of the last amplifier. The timing jitter
depends not only on the variances ofδEδΩ, andδqbut also on their cross-correlations
given in Eqs. (9.5.13)–(9.5.15). It can be written as

σt^2 =σGH^2 +R 1 〈(δE)^2 〉+R 2 〈δEδΩ〉 (9.6.10)

whereσGHis the Gordon–Haus jitter obtained earlier in Eq. (9.5.29).
The Raman contribution to the timing jitter adds two new terms whose magnitude
is governed by

R 1 =b 22 b^2 RNA(NA− 1 )(NA^3 − 10 NA^2 + 29 NA− 9 )/ 120 +b 2 RNA(NA− 1 )

×[b 2 bR( 19 NA^2 − 65 NA+ 48 )/ 96 +b 2 R( 2 NA− 1 )/ 6 ], (9.6.11)

R 2 =b 2 NA(NA− 1 )[b 2 bR(NA− 2 )( 3 NA− 1 )/ 12 +b 2 R( 2 NA− 1 )/ 3 ]. (9.6.12)

TheR 1 term depends on the variance of energy fluctuations and scales asN^5 AforNA

1. TheR 2 term has its origin in the cross-correlation between energy and frequency
   fluctuations and scales asN^4 Afor largeNA. TheR 1 term will generally dominate at long
   distances for high bit rates requiring pulses shorter than 5 ps.
   Figure 9.23 shows the growth of timing jitter with distance for a 160-Gb/s DM soli-
   ton system. The dispersion map consists of alternating 1-km fiber sections with GVD
   of−3.18 and 3.08 ps^2 /km, resulting in an average dispersion of−0.05 ps^2 /km and
   Tmap≈ 1 .25 ps. Fiber losses of 0.2 dB/km are compensated using lumped amplifiers
   every 50 km. The nonlinear parameters have valuesγ= 2 .25 W−^1 /km andTR=3 fs.
   The input pulse parameters are found by solving the variational equations of Section
   9.4.2 and have valuesT 0 = 1 .25 ps,C 0 =1, andE 0 = 0 .12 pJ. The Raman and Gordon–
   Haus contributions to the jitter are also shown separately to indicate when the Raman
   effects begin to dominate. Because of theNA^5 dependence of the Raman contribution,