Page 438 - Fiber Optic Communications Fund

10

Nonlinear Effects in Fibers

10.1 Introduction

So far, we have treated the fiber optic system as a linear system, but it is actually a nonlinear system because
the refractive index of the fiber changes with the intensity of signal due to the Kerr and Raman effects. In
Section 10.2, the origin of linear and nonlinear refractive indices and the Kerr effect are discussed. Since the
change in refractive index due to the Kerr effect translates into a phase shift, the signal phase is modulated
by its power distribution, which is known as self-phase modulation (SPM). SPM leads to spectral broaden-
ing and the exact balance between dispersion and SPM leads to soliton formation. A soliton is a pulse that
propagates without any change in shape over long distances. Sections 10.3–10.6 present the effects of dis-
persion, SPM, and soliton formation. In WDM systems, several channels co-propagate down the fiber. The
phase of a signal in a channel is modulated not only by its channel power, but also by other channels, which
is known as cross-phase modulation (XPM). In addition, nonlinear interaction among two or more chan-
nels leads to four-wave mixing (FWM), which acts as noise on channels. The impact of XPM and FWM on
the system performance of a WDM system is discussed in Section 10.7. In a high-bit-rate highly dispersive
single-channel system, signal pulses overlap strongly in the time domain, leading to intra-channel four-wave
mixing (IFWM) and intra-channel cross-phase modulation (IXPM). These intrachannel nonlinear effects are
discussed in Sections 10.8–10.10. The propagation of a high-intensity optical pulse leads to an instanta-
neous as well as a delayed change in refractive index. The instantaneous response is responsible for the Kerr
effect, while the delayed response is associated with the Raman effect. Section 10.11 is devoted to the stimu-
lated Raman effect, which is responsible for the amplification of a low-frequency signal by a high-frequency
intense pump.

10.2 Origin of Linear and Nonlinear Refractive Indices
In a dielectric medium, light travels at a speed lower than that in free space. This can be understood
qualitatively as follows. The electric field of the light wave acts on an electron, making it oscillate in
accordance with Coulomb’s law. An oscillating charge acts as a tiny antenna which radiates electromagnetic
radiation at a frequency the same as that of the incident wave in a linear approximation. The newly generated
electromagnetic field is the same as the incident field, except for a phase shift. In other words, absorption of
the incident field by a molecule and re-radiation delays the propagation of light compared with free-space
propagation. The exact determination of the displacement of electrons due to the electric field of a light
wave is a complicated problem of quantum mechanics. Instead, we use a classical electron oscillator model
in which the electron is modeled as a charged cloud surrounding the nucleus, as shown in Fig. 10.1(a) [1].

Fiber Optic Communications: Fundamentals and Applications, First Edition. Shiva Kumar and M. Jamal Deen.
© 2014 John Wiley & Sons, Ltd. Published 2014 by John Wiley & Sons, Ltd.