Nonlinear Fiber Optics - 4 ed. Agrawal

296 Chapter 8. Stimulated Raman Scattering
where the XPM term has been neglected assuming |A s | 2 ≪|A p | 2 . For the same reason,
the SPM term in Eq. (8.3.2) can be neglected. The solution of Eq. (8.3.2) is then given
by [124]
A s (z,T )=A s (0,T + zd)exp{[g s /2 + iγ s (2 − f R )]ψ(z,T )}, (8.3.7)
where
∫ z
ψ(z,T )= |A p (0,T + zd − z ′ d)| 2 dz ′ . (8.3.8)
0
Equation (8.3.6) shows that the pump pulse of initial amplitude A p (0,T ) propagates
without change in its shape. The SPM-induced phase shift imposes a frequency chirp
on the pump pulse that broadens its spectrum (see Section 4.1). The Raman pulse,
by contrast, changes both its shape and spectrum as it propagates through the fiber.
Temporal changes occur owing to Raman gain while spectral changes have their origin
in XPM. Because of pulse walk-off, both kinds of changes are governed by an overlap
factor ψ(z,T ) that takes into account the relative separation between the two pulses
along the fiber. This factor depends on the pulse shape. For a Gaussian pump pulse
with the input amplitude
A p (0,T )= √ P 0 exp(−T 2 /2T 2
0 ), (8.3.9)
the integral in Eq. (8.3.8) can be performed in terms of error functions with the result
ψ(z,τ)=[erf(τ + δ) − erf(τ)]( √ πP 0 z/δ), (8.3.10)
where τ = T /T 0 and δ is the distance in units of the walk-off length, that is,
δ = zd/T 0 = z/L W . (8.3.11)
An analytic expression for ψ(z,τ) can also be obtained for pump pulses having “sech”
shape [133]. In both cases, the Raman pulse compresses initially, reaches a minimum
width, and then begins to rebroaden as it is amplified through SRS. It also acquires
a frequency chirp through XPM. This qualitative behavior persists even when pump
depletion is included [133]–[135].
Equation (8.3.7) describes Raman amplification when a weak signal pulse is injected
together with the pump pulse. The case in which the Raman pulse builds from
noise is much more involved from a theoretical viewpoint. A general approach should
use a quantum-mechanical treatment, similar to that employed for describing SRS in
molecular gases [51]. It can be considerably simplified for optical fibers if transient
effects are ignored by assuming that pump pulses are much wider than the Raman response
time. In that case, Eqs. (8.3.1) and (8.3.2) can be used, provided a noise term
(often called the Langevin force) is added to the right-hand side of these equations.
The noise term leads to pulse-to-pulse fluctuations in the amplitude, width, and energy
of the Raman pulse, similar to those observed for SRS in molecular gases [22]. Its
inclusion is essential if the objective is to quantify such fluctuations.
The average features of noise-seeded Raman pulses can be described with the theory
of Section 8.1.2, where the effective Stokes power at the fiber input is obtained by
using one photon per mode at all frequencies within the Raman-gain spectrum. Equation
(8.1.10) then provides the input peak power of the Raman pulse, while its shape