tative results, several methods have been devised to numerically solve Equation 4.39 and their study can start from bibliographies [18,20]. 4.2.7.1 Kerr-Induced Self-Phase Modulation If an amplitude-modulated signal is injected into the fiber, the presence of nonlinear refraction index N2 causes the intensity modulation to be reflected into a phase modulation in such a way that the signal phase, in the approximation of very weak effect, writes (4.40) where J(t) is the field intensity that is time dependent due to modulation. In the simplest case, the presence of a sinusoidal modulation, the sidebands of the signal spectrum are unbalanced for the superposition of the amplitude modulation and phase modulation. If a signal with a more complex amplitude modulation is injected into the fiber, SPM causes a chirp, which is a dependence of the signal phase on time, and causes a nonlinear pulse broadening. Telecommunication Networks 10 mW FIGURE 4.6 20 mW Broadening of a Gaussian pulse propagating in a DS fiber at 1.55 um with zero chromatic dispersion ver- sus the pulse power after the propagation through a variable length of fiber. (After Chraplyvy, A.R., J. Lightwave Technol., 8, 1548, 1990.) Fiber length (km) The broadening of a Gaussian pulse propagating in a DS fiber at 1.55 um with zero chro- matic dispersion is shown in Figure 4.6 [21] versus the pulse power after the propagation through a variable length of fiber. It is interesting to observe that, even if for long distances the pulse is always broadened, at high transmitted power, there is a distance range where the pulse width decreases. This is due to the fact that SPM creates a phase modulation frequency distribution that on one side of the central carrier is out-of-phase with respect to the original frequency distribu- tion that is due to the amplitude modulation. In the long run and at high powers, the non- linear phase distribution prevails and determines the spectrum width, but a range of fiber lengths and field powers exist where the left modulation band