Nonlinear Fiber Optics - 4 ed. Agrawal

138 Chapter 5. Optical Solitons
Figure 5.8: Temporal evolution over 10 dispersion lengths when N = 1.2 atz = 0. The pulse
evolves into a fundamental soliton of narrower width for which N approaches 1 asymptotically.
section focuses on formation of solitons when the parameters of an input pulse do not
correspond to a soliton (see Appendix B for a sample numerical code).
Consider first the case when the peak power is not exactly matched and the value of
N obtained from Eq. (5.2.3) is not an integer. Figure 5.8 shows the evolution of a “sech”
pulse launched with N = 1.2 by solving the NLS equation numerically. Even though
pulse width and peak power change initially, the pulse eventually evolves toward a
fundamental soliton of narrower width for which N = 1 asymptotically. Perturbation
theory has been used to study this behavior analytically [80]. Because details are cumbersome,
only results are summarized here. In physical terms, the pulse adjusts its
shape and width as it propagates along the fiber and evolves into a soliton. A part of
the pulse energy is dispersed away in the process. This part is known as the continuum
radiation. It separates from the soliton as ξ increases and its amplitude decays
as ξ −1/2 . For ξ ≫ 1, the pulse evolves asymptotically into a soliton whose order is
an integer Ñ closest to the launched value of N. Mathematically, if N = Ñ + ε, where
|ε| < 1/2, the soliton part corresponds to an initial pulse shape of the form
u(0,τ)=(Ñ + 2ε)sech[(1 + 2ε/Ñ)τ]. (5.2.25)
The pulse broadens if ε < 0 and narrows if ε > 0. No soliton is formed when N ≤ 1 2 .
The effect of pulse shape on soliton formation can also be investigated solving
Eq. (5.2.5) numerically. Figure 4.8 of Chapter 4 shows evolution of a Gaussian pulse
using the initial field u(0,τ)=exp(−τ 2 /2). Even though N = 1, pulse shape changes
along the fiber because of deviations from the “sech” shape required for a fundamental
soliton. The interesting feature of Figure 4.8 is that the pulse adjusts its width and
evolves asymptotically into a fundamental soliton. In fact, the evolution appears to
be complete by z/L D = 5, a distance that corresponds to about three soliton periods.
An essentially similar evolution pattern occurs for other pulse shapes such as a super-
Gaussian shape. The final width of the soliton and the distance needed to evolve into a
fundamental soliton depend on the exact shape but the qualitative behavior remains the
same.
As pulses emitted from laser sources are often chirped, we should also consider
the effect of initial frequency chirp on soliton formation [85]–[90]. The chirp can be