Page 500 - Fiber Optic Communications Fund

Nonlinear Effects in Fibers 481

P (0)= 10 P s (0) (dBm)∕10 mW
s
= 0.1 mW, (10.419)
1 − exp (− L)
p
L eff,p =
p
1 − exp (−0.08 × 2)
=
0.08 × 10 −3
= 1.848 km, (10.420)
g P (0)L eff,p 1 × 10 −13 × 100 × 10 −3 × 1.848 × 10 3
R p
=
A 40 × 10 −12
eff
= 0.462, (10.421)
L = 0.092. (10.422)
s
So, the gain dominates the loss. From Eq. (10.413), we find
P (L)= P (0)e −0.092+0.462
s s
= 0.1447 mW. (10.423)

10.11.1 Time Domain Description
In the time domain, stimulated Raman scattering can be explained as follows [81]. When a high-intensity
optical pulse interacts with a molecule, it perturbs the electronic structure of the molecule and results in
intensity-dependent polarizability of the molecule (see Section 10.2). This electronic effect occurs on a time
scale shorter than the pulse width of the optical pulse, and it can be considered instantaneous. However,
perturbation of the electronic structure by the optical pulse also perturbs the Coloumb interaction between the
nuclei and the electronic structure, which can excite molecular vibrations. These vibrations, in turn, perturb the
electronic structure, leading to a delayed change in polarizability. The time- and intensity-dependent change
in polarizability (or equivalent refractive index) associated with the excitation of a molecular vibration is the
Raman effect, whereas the instantaneous intensity-dependent change in polarizability is the Kerr effect.
2
As mentioned in Section 10.4, the Kerr effect is taken into account by the term |q| q in the nonlinear
Schrödinger equation. In the presence of the Raman effect, it can be modified as [82]
∞
2 2 2
|q(T, Z)| → (1 − )|q(T, Z)| + h(s)|q(T − s, Z)| ds, (10.424)
∫
−∞
where is the fraction of the nonlinearity resulting from the Raman contributions and h(T) is the normalized
Raman response function, with
∞
h(T)dT = 1 (10.425)
∫
−∞
and h(−|t|)= 0 to ensure causality. The response function h(T) is specific to the medium and the imaginary
part of its Fourier transform is related to the Raman gain coefficient g(Ω) [81, 83].Let
2
P(T, Z)= |q(T, Z)| , (10.426)
2
P(T − s, Z)= |q(T − s, Z)| . (10.427)