Nonlinear Fiber Optics - 4 ed. Agrawal

5.2. Fiber Solitons 135
Figure 5.6: (a) Temporal and (b) spectral evolution over one soliton period of the third-order
soliton. Note pulse splitting near z/L D = 0.5 and soliton recovery beyond that.
from Eqs. (5.2.8)–(5.2.11). Using ζ 1 = i/2 and ζ 2 = 3i/2 for the two eigenvalues, the
field associated with the second-order soliton is given by [80]
u(ξ ,τ)=
4[cosh(3τ)+3exp(4iξ )cosh(τ)]exp(iξ /2)
. (5.2.23)
[cosh(4τ)+4cosh(2τ)+3cos(4ξ )]
An interesting property of the solution (5.2.23) is that |u(ξ ,τ)| 2 is periodic in ξ
with the period ξ 0 = π/2. In fact, this periodicity occurs for all higher-order solitons.
Using the definition ξ = z/L D from Eq. (5.2.1), the soliton period z 0 in real units becomes
z 0 = π 2 L D = π T0
2
2 |β 2 | ≈ T FWHM
2
2|β 2 | . (5.2.24)
Periodic evolution of a third-order soliton over one soliton period is shown in Figure
5.6(a). As the pulse propagates along the fiber, it first contracts to a fraction of its
initial width, splits into two distinct pulses at z 0 /2, and then merges again to recover
the original shape at the end of the soliton period at z = z 0 . This pattern is repeated
over each section of length z 0 .
To understand the origin of periodic evolution for higher-order solitons, it is helpful
to look at changes in the pulse spectra shown in Figure 5.6(b) for the N = 3 soliton.
The temporal and spectral changes result from an interplay between the SPM and GVD
effects. The SPM generates a frequency chirp such that the leading edge of the pulse
is red-shifted, while its trailing-edge is blue-shifted from the central frequency. The