9.4. DISPERSION-MANAGED SOLITONS 433
period when the nonlinear length is much larger than the local dispersion length, we
average it over one map period and obtain the following relation betweenT 0 andC 0 :
T 0 =Tmap√
1 +C 02
|C 0 |
, Tmap=(
|β 2 nβ 2 alnla|
β 2 nln−β 2 ala) 1 / 2
, (9.4.11)
whereTmapis a parameter with dimensions of time involving only the four map param-
eters. It provides a time scale associated with an arbitrary dispersion map in the sense
that the stable periodic solutions supported by it have input pulse widths that are close
toTmap(within a factor of 2 or so). The minimum value ofT 0 occurs for|C 0 |=1 and
is given byT 0 min=
√
2 Tmap.
Equation (9.4.11) can also be used to find the shortest pulse within the map. Recall-
ing from Section 2.4 that the shortest pulse occurs at the point at which the propagating
pulse becomes unchirped,Tm=T 0 /( 1 +C^20 )^1 /^2 =Tmap/
√
|C 0 |. When the input pulse
corresponds to its minimum value (C 0 =1),Tmis exactly equal toTmap. The optimum
value of the pulse stretching factor is equal to
√
2 under such conditions. These conclu-
sions are in agreement with the numerical results shown in Fig. 9.16 for a specific map
for whichTmap≈ 3 .16 ps. If dense dispersion management is not used for this map and
LmapequalsLA=80 km, this value ofTmapincreases to 9 ps. Since the FWHM of in-
put pulses then exceeds 21 ps, such a map is unsuitable for 40-Gb/s soliton systems. In
general, the required map period becomes shorter and shorter as the bit rate increases
as is evident from the definition ofTmapin Eq. (9.4.11).
It is useful to look for other combinations of the four map parameters that may play
an important role in designing a DM soliton system. Two parameters that help for this
purpose are defined as [89]
β ̄ 2 =β^2 nln+β^2 ala
ln+la, Sm=
β 2 nln−β 2 ala
TFWHM^2, (9.4.12)
whereTFWHM≈ 1. 665 Tmis the FWHM at the location where pulse width is minimum
in the anomalous-GVD section. Physically,β ̄ 2 represents the average GVD of the
entire link, while the map strengthSmis a measure of how much GVD varies between
two fibers in each map period. The solutions of Eqs. (9.4.7)–(9.4.9) as a function of
map strengthSfor different values ofβ ̄ 2 reveal the surprising feature that DM solitons
can exist even when the average GVD is normal provided the map strength exceeds a
critical valueScr[97]–[101]. Moreover, whenSm>Scrandβ ̄ 2 >0, a periodic solution
can exist for two different values of the input pulse energy. Numerical solutions of Eqs.
(9.4.1) confirm these predictions but the critical value of the map strength is found to
be only 3.9 instead of 4.8 obtained from the variational equations [89].
The existence of DM solitons in maps with normal average GVD is quite intriguing
as one can envisage dispersion maps in which a soliton propagates in the normal-GVD
regime most of the time. An example is provided by the dispersion map in which
a short section of standard fiber (β 2 a≈−20 ps^2 /km) is used with a long section of
dispersion-shifted fiber (β 2 n≈1ps^2 /km) such thatβ ̄ 2 is close to zero but positive. The
formation of DM solitons under such conditions can be understood by noting that when
Smexceeds 4, input energy of a pulse becomes large enough that its spectral width is