2 - Frequency mixing

Summary

Introduction and preview

In this chapter, we will consider several of the basic frequency-mixing processes of nonlinear optics. The simplest is second harmonic generation (SHG), and we will take this as our basic example. In SHG, a second harmonic wave at 2ω grows at the expense of the fundamental wave at ω. As we will discover, whether energy flows from ω to 2ω or vice versa depends on the phase relationship between the second harmonic field and the nonlinear polarisation at 2ω. Maintaining the optimal phase relationship is therefore of crucial importance if efficient frequency conversion is to be achieved.

The SHG process is governed by a pair of coupled differential equations, and their derivation will be our first goal. The analysis in Section 2.2 is somewhat laborious, although the material is standard, and can be found in many other books, as well as in innumerable PhD theses. The field definitions of Eqs (2.4)–(2.5) are used repeatedly throughout the book, and are worth studying carefully.

In Section 2.3, the coupled-wave equations are solved for SHG in the simplest approximation. The results are readily extended to the slightly more complicated cases of sum and difference frequency generation, and optical parametric amplification, which we move on to in Section 2.4. The important case of Gaussian beams is treated in Section 2.5, where the effect of the Gouy phase shift in the waist region of a focused beam is highlighted.