Nonlinear Fiber Optics - 4 ed. Agrawal

458 Chapter 12. Novel Nonlinear Phenomena
Figure 12.4: X-FROG traces (top row) and optical spectra (bottom row) for fiber lengths of
10 m (left column) and 180 m (right column). Dashed curves show the delay time as a function
of wavelength. (After Ref. [23]; c○2001 OSA.)
the two phases at a distance z after a delay t = z/v g are given by [28]
φ(ω d )=β(ω d )z − ω d (z/v g ), (12.1.4)
φ(ω s )=β(ω s )z − ω s (z/v g )+ 1 2 γP sz, (12.1.5)
where ω d and ω s are the frequencies of dispersive waves and the soliton, respectively,
and v g is the group velocity of the soliton. The last term in Eq. (12.1.5) is due to
the nonlinear phase shift occurring only for solitons. Its origin is related to the phase
factor exp(z/2L D ) appearing in Eq. (5.2.16), where L D = L NL =(γP s ) −1 . If we expand
β(ω d ) in a Taylor series around ω s , the two phases are matched when the frequency
shift Ω d = ω d − ω s satisfies
∞
∑
m=2
β m (ω s )
Ω m d
m!
= 1 2 γP s. (12.1.6)
Note that P s is the peak power of the Raman soliton formed after the fission process
(and not that of the input pulse). Similarly, the dispersion parameters β m appearing in
Eq. (12.1.6) are at the soliton central frequency ω s . As this frequency changes because
of a RIFS, the frequency of dispersive waves would also change.