In the previous chapter we examined nonlinear propagation in a two-level medium, which leads to amplification or absorption of the radiation field. As we will see further on in this volume, when the medium is placed in an optical cavity one realizes a laser in the case of amplification or an optically bistable system in the absorptive configuration. On the other hand, the nonlinear response of the atomic polarization to the electric field leads to further fascinating scenarios [29, 51–60]. For example, a field with central ω0 can generate harmonic components with central frequencies 2ω0, 3ω0 etc., processes that are called second-harmonic generation, third-harmonic generation etc. Furthermore, the interaction of two fields with central frequencies ω1 and ω2, with ω1 > ω2, can generate components with central frequency ω1 + ω2 (sum-frequency generation) and ω1 − ω2 (difference-frequency generation). Such phenomena have not yet appeared in this volume because we have assumed that the electric field corresponds to a narrow frequency band around a central frequency ω0 only, whereas to describe them it is necessary to assume that the electric field is the superposition of a number of frequency bands. In this chapter we will provide a description of the basic elements which are necessary to describe these phenomena and we will discuss a small number of them for the case of materials with a quadratic nonlinearity, referring to textbooks in nonlinear optics for the description of other phenomena. The following chapter is devoted, instead, to the case of materials with cubic nonlinearities.

The starting point of nonlinear optics is the expansion of the atomic polarization in powers of the electric field, when the electric field is not too intense. In our treatment in Chapter 5 we saw that the expansion of the nonlinear susceptibility contains only even powers of the electric field (see Eq. (5.27)), and therefore the expansion of the polarization includes only odd powers.