Introduction

A key weakness of the simple approach to nonlinear optics adopted in Chapter 1 was that the physical origin of nonlinearity in the interaction of light and matter was hidden inside the χ(n) coefficients of the polarisation expansion. This is such a fundamental issue that it is difficult to avoid some mention of how nonlinearity arises within a quantum mechanical framework, even in an introductory text. Unfortunately, the standard technique for calculating the nonlinear coefficients is based on time-dependent perturbation theory, and the expressions that emerge begin to get large and unwieldy even at second order. While every effort has been made in this chapter to provide a gentle lead-in to this aspect of the subject, this is almost impossible to achieve given the inherent complexity of the mathematical machinery.

From a mathematical point of view, Schrödinger's equation is linear in the wave function, but nonlinear in its response to perturbations. At a fundamental level, this is where nonlinear optics comes from. The perturbations of atoms and molecules referred to here arise from external electromagnetic fields. When the fields are relatively weak, the perturbations are relatively small, and the theoretical machinery of time-dependent perturbation theory can be deployed to quantify the effects. This is the regime where the traditional polarisation expansion of Eq. (1.24) applies, indeed the terms in the expansion correspond to successive orders of perturbation theory.