Nonlinear Fiber Optics - 4 ed. Agrawal

4.4. Higher-Order Nonlinear Effects 113
Frequency Shift (THz)
1.5
1
0.5
(a)
−1
−0.5
C 0
= 0
0
0 0.5 1 1.5 2
Distance (m)
Pulse Width (fs)
300
200
100
(b)
−0.5
−1
C 0
= 0
0
0 0.5 1 1.5 2
Distance (m)
Figure 4.24: Evolution of (a) Raman-induced frequency shift and (b) pulse width when an
unchirped Gaussian pulse with T 0 = 50 fs propagates in the normal-dispersion region of a fiber.
Its use requires the introduction of two new moments, representing a temporal shift
q p (z) and a spectral shift Ω p (z), and defined as
q p (z) = 1 ∫ ∞
T |U(z,T )| 2 dT, (4.4.12)
E p −∞
Ω p (z) =
i ∫ ∞
(
U ∗ ∂U )
∂U
∗
−U dT. (4.4.13)
2E p −∞ ∂T ∂T
Following the technique described in Section 4.3.1, these two moments are found to
satisfy
dq p
dz = β dΩ p
2Ω p ,
dz = −T RγP 0 e −αz T
√ 0
. (4.4.14)
2T 3 p
Physically speaking, the Raman term shifts the carrier frequency at which pulse spectrum
is centered. This spectral shift Ω p in turn shifts the pulse position q p in the time
domain because of changes in the group velocity through fiber dispersion.
Equation (4.4.14) should be solved together with Eqs. (4.3.7) and (4.3.8) to study
how Ω p evolves along the fiber length. Figure 4.24 shows the evolution of the RIFS,
Δν R ≡ Ω p /2π, when a chirped Gaussian pulse with T 0 = 50 fs is launched into a fiber
exhibiting normal dispersion [D = −4 ps/(km-nm)]. In the case of unchirped pulses,
Δν R saturates at a value of about 0.5 THz. This saturation is related to pulse broadening.
Indeed, the spectral shift can be increased by chirping Gaussian pulses such that β 2 C <
0. The reason is related to the discussion in Section 3.2.2, where it was shown that such
chirped Gaussian pulses go through an initial compression phase before they broaden
rapidly.
Figure 4.25(a) shows the experimentally recorded pulse spectrum when 109-fs
pulses (T 0 ≈ 60 fs) with 7.4 kW peak power were sent through a 6-m-long fiber [107].
The fiber had β 2 ≈ 4ps 2 /km and β 3 ≈ 0.06 ps 3 /km at the 1260-nm wavelength used
in this experiment. The traces (b) to (d) show the prediction of Eq. (4.4.1) under three
different conditions. Both self-steepening and intrapulse Raman scattering were neglected
in the trace (b) and included in the trace (d), while only the latter was included