In this chapter, the characteristics of pulse propagation in an isotropic and spatially homogeneous Kerr medium are discussed. The general optical pulse propagation equation and its form under the rate equation approximation are presented in the first section. The second section addresses the effect of dispersion on the propagation of an optical pulse in a linear optical medium where the nonlinear susceptibility does not exist. The third section addresses the effect of self-phase modulation on the propagation of an optical pulse in a nonlinear optical Kerr medium without the effect of dispersion. The following two sections cover the phenomena and characteristics of spectral stretching, pulse stretching, pulse compression, soliton formation, and soliton evolution that appear under different conditions in the propagation of an optical pulse in a nonlinear optical Kerr medium with the effect of dispersion. The final section addresses the process of modulation instability from the viewpoint of nonlinear wave propagation.

The optical response of a material is described by an electric polarization through an optical susceptibility. In the presence of optical nonlinearity, the total optical susceptibility is generally a function of the optical field. When the electric polarization can be expressed as a perturbation series of linear and nonlinear polarizations, field-independent linear and nonlinear susceptibilities can be defined. The linear susceptibility is a second-order tensor, and the second-order and third-order nonlinear susceptibilities are respectively third-order and four-order tensors. Each tensor element of these susceptibilities satisfies the reality condition. All tensor elements as functions of interacting optical frequencies generally possess intrinsic permutation symmetry. A full permutation symmetry exists when the material causes no loss or gain at all of the optical frequencies, and Kleinman’s symmetry exists when the medium is nondispersive at these frequencies. The spatial symmetry of a linear or nonlinear susceptibility tensor depends on the structure of the material.

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.

Stimulated Raman scattering leads to Raman gain for a Stokes signal at a frequency that is down-shifted at a Raman frequency, and stimulated Brillouin scattering leads to Brillouin gain at a frequency that is down-shifted by a Brillouin frequency. This chapter begins with a general discussion of Raman scattering and Brillouin scattering. After a discussion of the characteristics of the Raman gain, Raman amplification and generation based on stimulated Raman scattering are addressed through their applications as Raman amplifiers, Raman generators, and Raman oscillators. After a discussion of the characteristics of the Brillouin gain, Brillouin amplification and generation based on stimulated Brillouin scattering are addressed through their applications as Brillouin amplifiers, Brillouin generators, and Brillouin oscillators. This chapter ends with a comparison of Raman and Brillouin devices.

The general formulation for optical propagation in a nonlinear medium is given in this chapter. In the first section, the general equation for the propagation in a spatially homogeneous medium is obtained. This equation can be expressed either in the frequency domain or in the time domain. In the second section, the general pulse propagation equation for a waveguide mode is obtained in the time domain. In the third section, the propagation of an optical pulse in an optical Kerr medium is considered for three useful equations: nonlinear equation with spatial diffraction for propagation in a spatially homogeneous medium, nonlinear Schrödinger equation without spatial diffraction for propagation in a spatially homogeneous medium or in a waveguide, and generalized nonlinear Schrödinger equation for the nonlinear propagation of an optical pulse that has a pulsewidth down to a few optical cycles or that undergoes extreme spectral broadening.

This chapter addresses optical wave propagation in isotropic and anisotropic media. This chapter begins with general discussions on the energy flow and power exchange as an optical wave propagates through a medium. The next two sections respectively address the propagation of plane waves in isotropic and anisotropic homogeneous media. The polarization normal modes of propagation are defined for a birefringent crystal, which can be uniaxial with only one optical axis or biaxial with two optical axes. The concepts and characteristics of phase velocity, group velocity, and various types of dispersion are then discussed.

The coupled-wave theory is used in the analysis of the interactions among optical waves of different frequencies. In the analysis of the coupling of waveguide modes, coupled-mode theory has to be used. In general, both the interaction among different optical frequencies and the characteristics of the waveguide modes have to be considered for a nonlinear optical interaction in an optical waveguide. In the first section, a combination of coupled-wave and coupled-mode theories is formulated for the analysis of nonlinear optical interaction in a waveguide. In the second section, the coupled equations for a parametric nonlinear interaction in a waveguide are formulated by using three-frequency parametric interaction, second-harmonic generation, and the optical Kerr effect as three examples. In the third section, the coupled equations for a nonparametric nonlinear interaction in a waveguide are formulated by using stimulated Raman scattering and two-photon absorption as two examples.

This chapter addresses the physics and applications of optical saturation, including optical absorption saturation and optical gain saturation. Optical saturation is a nonlinear optical process that usually cannot be approximated with a perturbation expansion as a second-order or third-order nonlinear process. Instead, a fully nonlinear analysis is required. Following a discussion on the general physics and characteristics of absorption saturation and gain saturation in the first section, the properties and applications of saturable absorbers and saturated amplifiers are discussed in the second and third sections. The last section covers laser oscillation as a consequence of optical gain saturation.

The coupled-wave theory deals with the coupling of waves of different frequencies in nonlinear optical interactions. In the first section, the general coupled-wave equation is derived. Its form under the slowly varying amplitude approximation is then obtained, followed by a form under the transverse approximation. In the second section, the coupled-wave equations for a parametric process are formulated by using three-frequency parametric interaction and second-harmonic generation as two examples. In the third section, the coupled-wave equations for a nonparametric process are formulated by using stimulated Raman scattering and two-photon absorption as two examples.

Most parametric frequency-conversion processes are not automatically phase matched, thus requiring arrangements to achieve phase matching. If a parametric frequency-conversion process is perfectly phase matched, optical power can be efficiently converted from one frequency to another. Otherwise, the conversion efficiency is reduced. The geometric arrangement and the conditions for collinear phase matching and noncollinear phase matching are discussed in the first section. The second section addresses the concept and techniques of birefringent phase matching, which employs the birefringence of a uniaxial or biaxial crystal to accomplish phase matching of a nonlinear optical process. It is the most commonly used method of obtaining collinear phase matching for a second-order frequency-conversion process. The third section covers the concept and techniques of quasi-phase matching, which uses periodic modulation of the nonlinear susceptibility for phase matching. Phase matching in an optical waveguide is discussed in the fourth section.

Optical nonlinearity emerges from nonlinear interaction of light with matter. In this chapter, the basic concept and formulation of light‒matter interaction are discussed through a semiclassical approach with the behavior of the optical field classically described by Maxwell’s equations and the state of the material quantum mechanically described by a wave function governed by the Hamiltonian of the material. An optical field interacts with a material through its interaction with the electrons in the material. A Schrödinger electron is nonrelativistic with a nonzero mass, and a Dirac electron is relativistic with a zero mass. The interaction Hamiltonian can be expressed in terms of the vector and scalar potentials by using the Coulomb gauge. It can be expressed in terms of the electric and magnetic fields through multipole expansion as a series of electric and magnetic multipole interactions, with the first term being the electric dipole interaction. The electric polarization of a material induced by an optical field is obtained through density matrix analysis. The optical susceptibility of the material is then obtained from the electric polarization.

Bistability is a phenomenon that has two stable states under one condition. A bistable device has two possible stable output values for one input condition. The necessary conditions for optical bistability are optical nonlinearity and positive feedback. Depending on whether the optical nonlinearity that is responsible for the bistable function comes from the real or the imaginary part of a nonlinear susceptibility, a bistable optical device can be classified as either dispersive or absorptive. Depending on the type of feedback, a bistable optical device can also be classified as either intrinsic or hybrid. After a general discussion on the condition for optical bistability, this chapter covers dispersive optical bistability, absorptive optical bistability, and hybrid optical bistability of passive optical systems in three sections. The final chapter covers optical bistability in the active optical system of a laser oscillator.