ABSTRACT
Now instead of the nonlinearity originatingwithχ(2), we considerχ(3) effects. In this chapter we assume that the origin of χ(3) is parametric (no energy exchange with the medium), but the techniques we develop also apply to processes where χ(3) is nonparametric. The nonlinear polarization for χ(3) processes is proportional to the total input field cubed (see Equation 3.3), whichgives rise tomanypossible output frequencies.Table 8.1 lists the frequencies present after cubing a field that has three frequency components at the input:ω2,ω3, andω4. Fewer distinct frequenciesmay be present depending on the input frequencies. For example, if the three inputs are 3ω, 2ω, and ω, then only 10 frequencies are present in the nonlinear polarization. Each of the entries listed in Table 8.1 potentially has a different nonlinear susceptibility.
Before writing a general expression for the nonlinear polarization let us consider a few specific examples from Table 8.1 to see how degeneracy is included in the polarization expression. To simplify our discussion we consider a situation where the fields and nonlinear polarization point in the same direction so that we treat them as scalars. As a first example, consider a nonlinear polarization at ω1 = ω2 +ω3 +ω4, which has a complex amplitude proportional to the component of the field cubed at ω1,
