Nonlinear Fiber Optics - 4 ed. Agrawal

136 Chapter 5. Optical Solitons
spectral broadening is clearly seen in Figure 5.6(b) for z/L D = 0.32 with its typical
oscillatory structure. In the absence of GVD, the pulse shape would have remained
unchanged. However, anomalous GVD contracts the pulse as the pulse is positively
chirped (see Section 3.2). Only the central portion of the pulse contracts because the
chirp is nearly linear only over that part. However, as a result of a substantial increase in
the pulse intensity near the central part of the pulse, the spectrum changes significantly
as seen in Figure 5.6(b) for z/L D = 0.48. It is this mutual interaction between the GVD
and SPM effects that is responsible for the temporal evolution pattern seen in Figure
5.6(a).
In the case of a fundamental soliton (N = 1), GVD and SPM balance each other in
such a way that neither the pulse shape nor the pulse spectrum changes along the fiber
length. In the case of higher-order solitons, SPM dominates initially but GVD soon
catches up and leads to pulse contraction seen in Figure 5.6. Soliton theory shows that
for pulses with a hyperbolic-secant shape and with peak powers determined from Eq.
(5.2.3), the two effects can cooperate in such a way that the pulse follows a periodic
evolution pattern with original shape recurring at multiples of the soliton period z 0
given by Eq. (5.2.24). Near the 1.55-μm wavelength, typically β 2 = −20 ps 2 /km for
standard silica fibers. The soliton period is ∼80 m for T 0 = 1 ps and scales as T0 2,
becoming 8 km when T 0 = 10 ps. For dispersion-shifted fibers with β 2 ≈−2ps 2 /km,
z 0 increases by one order of magnitude for the same value of T 0 .
5.2.4 Experimental Confirmation
The possibility of soliton formation in optical fibers was suggested as early as 1973 [76].
However, the lack of a suitable source of picosecond optical pulses at wavelengths
>1.3 μm delayed their experimental observation until 1980. Solitons in optical fibers
were first observed in an experiment [81] that used a mode-locked color-center laser
capable of emitting short optical pulses (T FWHM ≈ 7 ps) near 1.55 μm, a wavelength
near which optical fibers exhibit anomalous GVD together with minimum losses. The
pulses were propagated inside a 700-m-long single-mode fiber with a core diameter of
9.3 μm. The fiber parameters for this experiment were estimated to be β 2 ≈−20 ps 2 /
km and γ ≈ 1.3 W −1 /km. Using T 0 = 4 ps in Eq. (5.2.20), the peak power for exciting
a fundamental soliton is ∼ 1W.
In the experiment, the peak power of optical pulses was varied over a range 0.3–
25 W, and their pulse shapes and spectra were monitored at the fiber output. Figure
5.7 shows autocorrelation traces and pulse spectra at several power levels and compares
them with those of the input pulse. The measured spectral width of 25 GHz of
the input pulse is nearly transform limited, indicating that mode-locked pulses used in
the experiment were unchirped. At a low power level of 0.3 W, optical pulses experienced
dispersion-induced broadening inside the fiber, as expected from Section 3.2.
However, as the power was increased, output pulses steadily narrowed, and their width
became the same as the input width at P 0 = 1.2 W. This power level corresponds to
the formation of a fundamental soliton and should be compared with the theoretical
value of 1 W obtained from Eq. (5.2.20). The agreement is quite good in spite of many
uncertainties inherent in the experiment.