Nonlinear Fiber Optics - 4 ed. Agrawal

38 Chapter 2. Pulse Propagation in Fibers
The integral in this equation accounts for the energy transfer resulting from intrapulse
Raman scattering. Equation (2.3.36) can be used for pulses as short as a few optical
cycles if enough higher-order dispersive terms are included [22]–[26]. For example,
dispersive effects up to 12th order are sometimes included when dealing with supercontinuum
generation in optical fibers, a phenomenon discussed in Chapter 12.
It is important to note that the use of γ 1 in Eq. (2.3.36) includes automatically the
frequency dependence of both n 2 and A eff . Noting that γ 1 =(dγ/dω) ω=ω0 , the ratio
γ 1 /γ consists of the following three terms:
γ 1 (ω 0 )
γ(ω 0 ) = 1 + 1 ( ) dn2
− 1 ( )
dAeff
. (2.3.37)
ω 0 n 2 dω
ω=ω 0
A eff dω
ω=ω 0
The first term provides the dominant contribution, but the second and third terms become
important in the case of a supercontinuum that may extend over 100 THz or more
[26]. If spectral broadening is limited to 20 THz or so, one can employ γ 1 ≈ γ/ω 0 . This
approximation is often used in practice. If we combine the terms containing the derivative
∂A/∂t, we find that the γ 1 term forces the group velocity to depend on the optical
intensity and leads to the phenomenon of self-steepening [28]–[32].
The nonlinear response function R(t) should include both the electronic and nuclear
contributions. Assuming that the electronic contribution is nearly instantaneous, the
functional form of R(t) can be written as [15]–[20]
R(t)=(1 − f R )δ(t −t e )+ f R h R (t), (2.3.38)
where t e accounts for a negligibly short delay in electronic response (t e < 1 fs) and
f R represents the fractional contribution of the delayed Raman response to nonlinear
polarization P NL . The form of the Raman response function h R (t) is set by vibrations
of silica molecules induced by the optical field.
It is not easy to calculate h R (t) because of the amorphous nature of silica fibers.
An indirect experimental approach is used in practice by noting that the Raman gain
spectrum is related to the imaginary part of the Fourier transform of h R (t) as [15]
g R (Δω)= ω 0
cn(ω 0 ) f Rχ (3)
xxxxIm[˜h R (Δω)], (2.3.39)
where Δω = ω − ω 0 and Im stands for the imaginary part. The real part of ˜h R (Δω)
can be obtained from the imaginary part through the Kramers–Kronig relation [3].
The inverse Fourier transform of ˜h R (Δω) then provides the Raman response function
h R (t). Figure 2.2 shows the temporal form of h R (t) deduced from the experimentally
measured spectrum (see Figure 8.2) of the Raman gain in silica fibers [15].
Attempts have been made to determine an approximate analytic form of the Raman
response function. In view of the damped oscillations seen in Figure 2.2, a useful form
is [16]
h R (t)= τ2 1 + τ2 2
τ 1 τ2
2 exp(−t/τ 2 )sin(t/τ 1 ). (2.3.40)
The parameters τ 1 and τ 2 are two adjustable parameters and are chosen to provide a
good fit to the actual Raman-gain spectrum. In a 1989 study, values of τ 1 = 12.2 fs