Nonlinear Fiber Optics - 4 ed. Agrawal

8.3. SRS with Short Pump Pulses 297
remains undetermined. Numerical solutions of Eqs. (8.3.1) and (8.3.2) show that the
average pulse shapes and spectra at the fiber output are not dramatically affected by
different choices of the shape of the seed pulse. A simple approximation consists of
assuming
A s (0,T )=(Ps0 eff )1/2 , (8.3.12)
where Ps0
eff is given by Eq. (8.1.10). Alternatively, one may take the seed pulse to be
Gaussian with a peak power Ps0 eff.
As a simple application of the analytic solution (8.3.7), consider the Raman threshold
for SRS induced by short pump pulses of width T 0 and peak power P 0 [119]. The
peak power of the Raman pulse at the fiber output (z = L) is given by
P s (L)=|A s (L,0)| 2 = P eff
s0 exp(√ πg s P 0 L W ), (8.3.13)
where Eq. (8.3.10) was used with τ = 0 and L/L W ≫ 1. If we define the Raman
threshold in the same way as for the CW case, the threshold is achieved when P s (L)=
P 0 . The comparison of Eqs. (8.1.12) and (8.3.13) shows that one can use the CW
criterion provided the effective length is taken to be
L eff = √ π L W ≈ T FWHM /|d|. (8.3.14)
In particular, Eq. (8.1.13) can be used to obtain the critical peak power of the pump
pulse if L eff is obtained from Eq. (8.3.14). This change is expected because the effective
interaction length between the pump and Raman pulses is determined by the length
L W —SRS ceases to occur when the two pulses move apart enough that they stop overlapping
significantly. Equations (8.1.13) and (8.3.14) show that the Raman threshold
depends on the width of the pump pulse and increases inversely with T FWHM . For pulse
widths ∼10 ps (L W ∼ 1 m), the threshold pump powers are ∼100 W.
The analytic solution (8.3.7) can be used to obtain both the shape and the spectrum
of the Raman pulse during the initial stages of SRS [124]. The spectral evolution is
governed by the XPM-induced frequency chirp. The chirp behavior has been discussed
in Section 7.4 in the context of XPM-induced asymmetric spectral broadening (see
Figure 7.3). The qualitative features of the XPM-induced chirp are identical to those
shown there as long as the pump remains undepleted. Note, however, that the Raman
pulse travels faster than the pump pulse in the normal-GVD regime. As a result, chirp
is induced mainly near the trailing edge. It should be stressed that both pulse shapes
and spectra are considerably modified when pump depletion is included [121]. As the
energy of the Raman pulse grows, it affects itself through SPM and the pump pulse
through XPM.
8.3.3 Effects of GVD
When the fiber length is comparable to the dispersion length L D , it is important to include
the GVD effects. Such effects cannot be described analytically, and a numerical
solution of Eqs. (8.3.1) and (8.3.2) is necessary to understand the SRS evolution. A
generalization of the split-step Fourier method of Section 2.4 can be used for this purpose.
The method requires specification of the Raman pulse at the fiber input by using
Eq. (8.1.10).