REFERENCES 469
9.11Derive Eq. (9.3.15) by integrating Eq. (9.3.11) in the case of bidirectional pump-
ing. Plotp(z)forLA=20, 40, 60, and 80 km usingα= 0 .2 dB/km and
αp= 0 .25 dB/km.
9.12Use Eq. (9.3.15) to determine the pump-station spacingLAfor which the soliton
energy deviates at most 20% from its input value.
9.13Consider soliton evolution in a dispersion-decreasing fiber using the NLS equa-
tion and prove that soliton remains unperturbed when the fiber dispersion de-
creases exponentially asβ 2 (z)=β 2 ( 0 )exp(−αz).
9.14Starting from the NLS equation (9.4.5), derive the variational equations for the
pulse widthTand the chirpCusing the Gaussian ansatz given in Eq. (9.4.6).
9.15Solve Eqs. (9.4.7) and (9.4.8) numerically by imposing the periodicity condition
given in Eq. (9.4.9). PlotT 0 andC 0 as a function ofE 0 for a dispersion map
made using 70 km of the standard fiber withD=17 ps/(km-nm) and 10 km of
dispersion-compensating fiber withD=−115 ps/(km-nm). Useγ=2W−^1 /km
andα= 0 .2 dB/km for the standard fiber andγ=6W−^1 /km andα= 0 .5 dB/km
for the other fiber.
9.16Calculate the map strengthSand the map parameterTmapfor the map used in the
preceding problem. Estimate the maximum bit rate that this map can support.
9.17Verify using Eqs. (9.5.8)–(9.5.12) that the variances and correlations of amplifier-
induced fluctuations are indeed given by Eqs. (9.5.13)–(9.5.15).
9.18Prove that the variances ofE,Ω, andqare given by Eq. (9.5.17) for the standard
solitons using Eq. (9.5.16) in Eqs. (9.5.8)–(9.5.11).
9.19Derive Eq. (9.5.29) for the timing jitter starting from the recurrence relation in
Eq. (9.5.26). Show all the steps clearly.
9.20Find the peak value of the collision-induced frequency and temporal shifts by
integrating Eq. (9.7.8) withb=1.
9.21Explain how soliton collisions limit the number of channels in a WDM soliton
system. Find how the maximum number of channels depends on the channel and
amplifier spacings using the conditionLcoll> 2 LA.
References
[1] N. Zabusky and M. D. Kruskal,Phys. Rev. Lett. 15 , 240 (1965).
[2] A. Hasegawa and F. Tappert,Appl. Phys. Lett. 23 , 142 (1973).
[3] L. F. Mollenauer, R. H. Stolen, and J. P. Gordon,Phys. Rev. Lett. 45 , 1095 (1980).
[4] L. F. Mollenauer and K. Smith,Opt. Lett. 13 , 675 (1988).
[5] A. Hasegawa and Y. Kodama,Solitons in Optical Communications, Clarendon Press,
Oxford, 1995.
[6] L. F. Mollenauer, J. P. Gordon, and P. V. Mamychev, inOptical Fiber Telecommunica-
tions IIIA, I. P. Kaminow and T. L. Koch, Eds., Academic Press, San Diego, CA, 1997,
Chap. 12.
