Nonlinear Fiber Optics - 4 ed. Agrawal

8.3. SRS with Short Pump Pulses 295
the pump pulse, these equations take the form
∂A p
∂z + iβ 2p ∂ 2 A p
2 ∂T 2
∂A s
∂z − d ∂A s
∂T + iβ 2s ∂ 2 A s
2 ∂T 2
= iγ p[|A p | 2 +(2 − f R )|A s | 2 ]A p − g p
2 |A s| 2 A p , (8.3.1)
= iγ s[|A s | 2 +(2 − f R )|A p | 2 ]A p + g s
2 |A p| 2 A s , (8.3.2)
where
T = t − z/v gp , d = vgp −1 − v −1
gs . (8.3.3)
The walk-off parameter d accounts for the group-velocity mismatch between the pump
and Raman pulses and is typically 2–6 ps/m. The GVD parameter β 2 j , the nonlinearity
parameter γ j , and the Raman-gain coefficient g j ( j = p or s) are slightly different for
the pump and Raman pulses because of the Raman shift of about 13 THz between their
carrier frequencies. In terms of the wavelength ratio λ p /λ s , these parameters for pump
and Raman pulses are related as
β 2s = λ p
λ s
β 2p ,
γ s = λ p
λ s
γ p ,
g s = λ p
λ s
g p . (8.3.4)
Four length scales can be introduced to determine the relative importance of various
terms in Eqs. (8.3.1) and (8.3.2). For pump pulses of duration T 0 and peak power P 0 ,
these are defined as
L D = T 0
2
|β 2p | , L W = T 0
|d| , L NL = 1 , L G = 1 . (8.3.5)
γ p P 0 g p P 0
The dispersion length L D , the walk-off length L W , the nonlinear length L NL , and the
Raman-gain length L G provide, respectively, the length scales over which the effects
of GVD, pulse walk-off, nonlinearity (both SPM and XPM), and Raman gain become
important. The shortest length among them plays the dominant role. Typically, L W ∼
1 m (for T 0 < 10 ps) while L NL and L G become smaller or comparable to it for P 0 >
100 W. In contrast, L D ∼ 1 km for T 0 = 10 ps. Thus, the GVD effects are generally
negligible for pulses as short as 10 ps. The situation changes for pulse widths ∼1 ps
or less because L D decreases faster than L W with a decrease in the pulse width. The
GVD effects can then affect SRS evolution significantly, especially in the anomalousdispersion
regime.
8.3.2 Nondispersive Case
When the second-derivative term in Eqs. (8.3.1) and (8.3.2) is neglected, these equations
can be solved analytically [133]–[136]. The analytic solution takes a simple form
if pump depletion during SRS is neglected. As this assumption is justified for the initial
stages of SRS and permits us to gain physical insight, let us consider it in some detail.
The resulting analytic solution includes the effects of both XPM and pulse walk-off.
The XPM effects without walk-off effects were considered relatively early [115]. Both
of them can be included by solving Eqs. (8.3.1) and (8.3.2) with β 2p = β 2s = 0 and
g p = 0. Equation (8.3.1) for the pump pulse then yields the solution
A p (z,T )=A p (0,T )exp[iγ p |A p (0,T )| 2 z], (8.3.6)