418 CHAPTER 9. SOLITON SYSTEMS
used to produce a train of ultrashort pulses. The basic idea consists of injecting a CW
beam, with weak sinusoidal modulation imposed on it, into such a fiber. The combi-
nation of GVD, SPM, and decreasing dispersion converts the sinusoidally modulated
signal into a train of ultrashort solitons [50]. The repetition rate of pulses is governed
by the frequency of initial sinusoidal modulation, often produced by beating two opti-
cal signals. Two distributed feedback (DFB) semiconductor lasers or a two-mode fiber
laser can be used for this purpose. By 1993, this technique led to the development of
an integrated fiber source capable of producing a soliton pulse train at high repetition
rates by using acomb-likedispersion profile, created by splicing pieces of low- and
high-dispersion fibers [50]. Adual-frequencyfiber laser was used to generate the beat
signal and to produce a 2.2-ps soliton train at the 59-GHz repetition rate. In another
experiment, a 40-GHz soliton train of 3-ps pulses was generated using a single DFB
laser whose output was modulated with a Mach–Zehnder modulator before launching
it into a dispersion-tailored fiber with a comb-like GVD profile [51].
A simple method of pulse-train generation modulates the phase of the CW output
obtained from a DFB semiconductor laser, followed by an optical bandpass filter [52].
Phase modulation generates frequency modulation (FM) sidebands on both sides of the
carrier frequency, and the optical filter selects the sidebands on one side of the carrier.
Such a device generates a stable pulse train of widths∼20 ps at a repetition rate that
is controlled by the phase modulator. It can also be used as a dual-wavelength source
by filtering sidebands on both sides of the carrier frequency, with a typical channel
spacing of about 0.8 nm at the 1.55-μm wavelength. Another simple technique uses
a single Mach–Zehnder modulator, driven by an electrical data stream in the NRZ
format, to convert the CW output of a DFB laser into an optical bit stream in the RZ
format [53]. Although optical pulses launched from such transmitters typically do not
have the “sech” shape of a soliton, they can be used for soliton systems because of the
soliton-formation capability of the fiber discussed earlier.
9.3 Loss-Managed Solitons.........................
As discussed in Section 9.1, solitons use the nonlinear phenomenon of SPM to main-
tain their width even in the presence of fiber dispersion. However, this property holds
only if fiber losses were negligible. It is not difficult to see that a decrease in soli-
ton energy because of fiber losses would produce soliton broadening simply because a
reduced peak power weakens the SPM effect necessary to counteract the GVD. Opti-
cal amplifiers can be used for compensating fiber losses. This section focuses on the
management of losses through amplification of solitons.
9.3.1 Loss-Induced Soliton Broadening
Fiber losses are included through the last term in Eq. (9.1.1). In normalized units, the
NLS equation becomes [see Eq. (9.1.5)]
i∂u
∂ξ+
1
2
∂^2 u
∂τ^2+|u|^2 u=−i
2Γu, (9.3.1)