The focus in this chapter is on intensity-dependent changes in the refractive index of a GRIN medium, responsible for the Kerr effect. In Section 5.1, we consider self-focusing of an optical beam inside a GRIN medium. Pulsed beams are considered in Section 5.2, where we derive a nonlinear propagation equation and discuss the phenomena of self- and cross-phase modulations. Section 5.3 is devoted to modulation instability and the formation of multimode solitons. Intermodal nonlinear effects are considered in Section 5.4 with emphasis on four-wave mixing and stimulated Raman scattering. Nonlinear applications discussed in Section 5.5 include supercontinuuum generation, spatial beam cleanup, and second harmonic generation.

Optical interactions can generally be categorized into parametric processes and nonparametric processes. A parametric process does not cause any change in the quantum-mechanical state of the material, whereas a nonparametric process causes some changes in the quantum-mechanical state of the material. Phase matching among interacting optical fields is not automatically satisfied in a parametric process but is always automatically satisfied in a nonparametric process. All second-order nonlinear optical processes are parametric in nature. The nonlinear polarization and phase-matching condition of each second-order process are discussed in the second section. Some third-order nonlinear optical processes are parametric, and others are nonparametric. The nonlinear polarization and phase-matching condition of each third-order process are discussed in the third section.

All-optical modulation of an optical wave is accomplished through a nonlinear optical process that involves one or multiple optical waves. A nonlinear optical modulator can be based on either self-modulation or cross-modulation. Such nonlinear optical modulators and switches are also known as all-optical modulators and all-optical switches, respectively. Most all-optical modulators and switches are based on third-order nonlinear optical processes, but some rely on the high-order process of optical saturation, either absorption saturation or gain saturation. There are two fundamentally different types of all-optical modulators and switches: the dispersive type and the absorptive type. All-optical modulation of the dispersive type, which is based on the optical Kerr effect, is discussed in this chapter. In the first four sections, the physics, phenomena, and measurement of the optical Kerr effect are discussed. The last two sections cover all-optical modulators and switches in the bulk form and those in the waveguide form.