Nonlinear Fiber Optics - 4 ed. Agrawal

7.6. Polarization Effects 261
4
Pump Intensity
3
2
1
L = 50 L NL
β 2
< 0
Probe Intensity
2
1.5
1
0.5
L D
= 5 L NL
L W
= 10 L NL
0
−10 −5 0 5 10
Time, T/T 0
0
−10 −5 0 5 10
Time. T/T 0
Figure 7.15: Temporal shapes of (a) pump and (b) probe pulses at ξ = 0 (dotted curves) and
ξ = 50 for the x (solid curves) and y (dashed curve) components. The pump pulse is linearly
polarized along the x axis (φ = 0) while the probe is oriented at 45 ◦ with respect to it.
L dj = T0 2/|β
2 j| ( j = 1,2). In the following discussion, we assume for simplicity that
L d1 = L d2 ≡ L d . This may the case for a dispersion-flattened fiber, or when the pump
and probe wavelengths do not differ by more than a few nanometers. The two vector
NLS equations can be solved numerically with the split-step Fourier method. Note that
this amounts to solving four coupled NLS equations.
Similar to the scalar case studied in Section 7.4, three length scales—nonlinear
length L NL , walk-off length L W , and dispersion length L D —govern the interplay between
the dispersive and nonlinear effects. The temporal evolution of the probe pulse
depends considerably on whether the GVD is normal or anomalous. Consider first the
anomalous-dispersion case. Figure 7.14 shows an example of probe evolution over a
length L = 50L NL when both the pump and probe pulses are initially Gaussian, have
the same width, and are linearly polarized at a 45 ◦ angle. The dispersion and walkoff
lengths are chosen such that L D = 5L NL and L W = 10L NL , respectively. Since
N 2 = L D /L NL = 5, the pump pulse evolves toward a second-order soliton. As seen in
Figure 7.15, the pump pulse is compressed considerably at ξ = 50 and has a shape
expected for a second-order soliton. It also shifts toward right because of the groupvelocity
mismatch between the pump and probe.
The probe pulse would disperse completely in the absence of the XPM effects because
of its low peak power. However, as seen in Figure 7.14, its x component, which is
copolarized with the pump pulse and experiences XPM-induced chirp, shifts with the
pump pulse and travels at the same speed as the pump pulse. This feature is similar to
the phenomenon of soliton trapping discussed in Section 6.5.2 in the context of highbirefringence
fibers. The pump pulse traps the copolarized component and the two
move together because of XPM coupling between the two. The XPM also compresses
the probe pulse. This feature is seen more clearly in Figure 7.15, where the copolarized
component of the probe appears to be compressed even more than the pump pulse,