In Chapter 2, the nonlinear susceptibility tensors were introduced and their general properties were found by considering some fundamental physical principles. We have now set up, in Chapter 3, all the formal apparatus required to derive explicit formulae for the susceptibility tensors of a medium. This is done in this chapter by considering the dynamical behaviour of the charged particles in the medium under the influence of an electric field. The formulae that we derive are fundamental and quite general; they provide the basis for the treatment of the nonlinear optical properties of any medium in the electric-dipole approximation. We apply the formulae to a simple case – an idealised molecular gas – which is conceptually straightforward and provides a quick route to an understanding of the formulae. We also consider the important, and often difficult, problem of passing from microscopic formulae (which apply to individual molecules or groups of molecules) to the macroscopic formulae which are required later when we consider the resulting wave propagation.

The approach taken in the early part of the chapter is to consider the energy associated with the electric-dipole moment in an electric field; this is probably the most readily understood picture, and leads to formulae which can be directly applied in a quantitative way to atoms and simple molecules. However, in a later section we cast the results into an alternative form in terms of the particle momenta.