410 CHAPTER 9. SOLITON SYSTEMS
Figure 9.4: (a) Intensity and (b) phase profiles of dark solitons for several values of the internal
phaseφ. The intensity drops to zero at the center for black solitons.
interesting feature of dark solitons is related to their phase. In contrast with bright
solitons which have a constant phase, the phase of a dark soliton changes across its
width. Figure 9.4 shows the intensity and phase profiles for several values ofφ.For
a black soliton (φ=0), a phase shift ofπoccurs exactly at the center of the dip. For
other values ofφ, the phase changes by an amountπ− 2 φin a more gradual fashion.
Dark solitons were observed during the 1980s in several experiments using broad
optical pulses with a narrow dip at the pulse center. It is important to incorporate a
πphase shift at the pulse center. Numerical simulations show that the central dip can
propagate as a dark soliton despite the nonuniform background as long as the back-
ground intensity is uniform in the vicinity of the dip [18]. Higher-order dark solitons
do not follow a periodic evolution pattern similar to that shown in Fig. 9.1 for the third-
order bright soliton. The numerical results show that whenN>1, the input pulse forms
a fundamental dark soliton by narrowing its width while ejecting several dark-soliton
pairs in the process. In a 1993 experiment [19], 5.3-ps dark solitons, formed on a 36-ps
wide pulse from a 850-nm Ti:sapphire laser, were propagated over 1 km of fiber. The
same technique was later extended to transmit dark-soliton pulse trains over 2 km of
fiber at a repetition rate of up to 60 GHz. These results show that dark solitons can be
generated and maintained over considerable fiber lengths.
Several practical techniques were introduced during the 1990s for generating dark
solitons. In one method, a Mach–Zehnder modulator driven by nearly rectangular elec-
trical pulses, modulates the CW output of a semiconductor laser [20]. In an extension of
this method, electric modulation is performed in one of the arms of a Mach–Zehnder in-
terferometer. A simple all-optical technique consists of propagating two optical pulses,
with a relative time delay between them, in the normal-GVD region of the fiber [21].
The two pulses broaden, become chirped, and acquire a nearly rectangular shape as
they propagate inside the fiber. As these chirped pulses merge into each other, they
interfere. The result at the fiber output is a train of isolated dark solitons. In another
all-optical technique, nonlinear conversion of a beat signal in a dispersion-decreasing