Nonlinear Fiber Optics - 4 ed. Agrawal

158 Chapter 5. Optical Solitons
the effects of third-order dispersion (TOD), self-steepening, and intrapulse Raman scattering.
Their explicit expressions are
δ 3 = β 3
6|β 2 |T 0
, s = 1
ω 0 T 0
,
τ R = T R
T 0
. (5.5.9)
All three parameters vary inversely with pulse width and are negligible for T 0 ≫ 1 ps.
They become appreciable for femtosecond pulses. As an example, δ 3 ≈ 0.03, s ≈ 0.03,
and τ R ≈ 0.1 for a 50-fs pulse (T 0 ≈ 30 fs) propagating at 1.55 μm in a standard silica
fiber if we take T R = 3 fs.
5.5.2 Third-Order Dispersion
When optical pulses propagate relatively far from the zero-dispersion wavelength of an
optical fiber, the TOD effects on solitons are small and can be treated perturbatively. To
study such effects as simply as possible, let us set s = 0 and τ R = 0 in Eq. (5.5.8) and
treat the δ 3 term as a small perturbation. It follows from Eqs. (5.5.3)–(5.5.6) that, since
Ω p = 0 under such conditions, C p = 0, and T p = T 0 . However, the temporal position of
the pulse changes linearly with z as
q p (z)=(β 3 /6T 2
0 )z ≡ δ 3 (z/L D ). (5.5.10)
Thus the main effect of TOD is to shift the soliton peak linearly with distance z.
Whether the pulse is delayed or advanced depends on the sign of β 3 . When β 3 is positive,
the TOD slows down the soliton, and the soliton peak is delayed by an amount
that increases linearly with distance. This TOD-induced delay is negligible in most
fibers for picosecond pulses. If we use a typical value of β 3 = 0.1ps 3 /km, the temporal
shift is only 0.1 ps for T 0 = 10 ps even after a distance of 100 km. However, the shift is
becomes relatively large for femtosecond pulses. For example, when T 0 = 100 fs, the
shift becomes 1 ps after 1 km.
What happens if an optical pulse propagates at or near the zero-dispersion wavelength
of an optical fiber such that β 2 is nearly zero. Considerable work has been done
to understand propagation behavior in this regime [168]–[177]. The case β 2 = 0 has
been discussed in Section 4.2.5 for Gaussian pulses by solving Eq. (4.2.7) numerically.
We can use the same equation for a soliton by using U(0,ξ ′ )=sech(τ) as the input at
z = 0. Figure 5.17 shows the temporal and spectral evolution of a “sech” pulse with
Ñ = 2 for z/L D ′ in the range of 0–4.
The most striking feature of Figure 5.17 is the splitting of the spectrum into two
well-resolved spectral peaks [168]. These peaks correspond to the outermost peaks
of the SPM-broadened spectrum (see Figure 4.2). As the red-shifted peak lies in the
anomalous-GVD regime, pulse energy in that spectral band can form a soliton. The
energy in the other spectral band disperses away simply because that part of the pulse
experiences normal GVD. It is the trailing part of the pulse that disperses away with
propagation because SPM generates blue-shifted components near the trailing edge.
The pulse shapes in Figure 5.17 show a long trailing edge with oscillations that continues
to separate away from the leading part with increasing ξ ′ . The important point
to note is that, because of SPM-induced spectral broadening, the input pulse does not