ABSTRACT
In Chap. 1, we reviewed the basic concepts of linear electromagnetic wave propagation in isotropic and anisotropic materials. However, optical propagation in many materials is nonlinear, in the sense that the polarization depends nonlinearly on the optical field in themedium. In order to understand the origin of such nonlinearities, we demonstrate, through a simple physical model of an electron around a nucleus, how the displacement of the electron, and hence the induced polarization, depends nonlinearly on the applied timeharmonic optical field. This explanation is, however, far too simplified to be used as a viable mathematical model in a practical nonlinear material. With rigor in mind, we introduce the time domain formulation of optical nonlinearity by relating the polarization to the optical field. A more widely used formulation is, however, found in the frequency domain. Thus, we use the Fourier transform to re-express the nonlinear polarizations in terms of the spectra of the optical fields and the so-called nonlinear susceptibilities. In most practical applications, the applied optical field is time-harmonic and so is the nonlinear polarization. We therefore provide the relations between the nonlinear polarization phasors and the optical field phasors. We also derive a nonlinear extension of the wave equation for the optical field in terms of the induced nonlinear polarization, assuming that both are expressible in terms of phasors. Finally, we illustrate another simple mathematical model for nonlinear propagation which has its foundations in fluid dynamics and compare with our more rigorously derived results.
