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.

In the previous chapter, explicit formulae for the nonlinear susceptibilities were derived. The susceptibilities exhibit various types of symmetry which are of fundamental importance in nonlinear optics: permutation symmetry, time-reversal symmetry, and symmetry in space. (Another kind of symmetry–the relationship between the real and imaginary parts of the susceptibilities – is described in Appendix 8.) The time-reversal and permutation symmetries are fundamental properties of the susceptibilities themselves, whereas the spatial symmetry of the susceptibility tensors reflects the structural properties of the nonlinear medium. All of these have important practical implications. In this chapter we outline the essential features and some practical consequences.

Permutation symmetry

The permutation-symmetry properties of the nonlinear susceptibilities have already been encountered in earlier chapters. Intrinsic permutation symmetry, first described in §§2.1 and 2.2, implies that the nth-order susceptibility is invariant under all n! permutations of the pairs (α1, ω1), (α2, ω2), …, (αn, ωn). Intrinsic permutation symmetry is a fundamental property of the nonlinear susceptibilities which arises from the principles of time invariance and causality, and which applies universally. In some circumstances a susceptibility may also possess a more general property, overall permutation symmetry, in which the susceptibility is invariant when the permutation includes the additional pair (µ, -ωσ); i.e., the nth-order susceptibility is invariant under all (n + 1)! permutations of the pairs (µ, ωσ), (α1, ω1), …, (αn, ωn).

In earlier chapters we have considered in detail the susceptibility formalism of nonlinear optics, which is perhaps the most familiar approach and has a wide range of application. Starting from the constitutive relations of Chapter 2, the susceptibility formalism is quite general. In many practical applications, a particular phenomenon can be described accurately by a single order of nonlinearity, and the susceptibility then provides a useful and convenient description. However, this is not always the case. Some of the most interesting phenomena in nonlinear optics involve close resonance with the transition frequencies of the medium, and perhaps also the use of very intense optical fields. As remarked in §4.5, in these circumstances the resonant susceptibilities display mathematical divergences which are clearly unphysical. Strictly, these divergences occur only because higher-order nonlinearities have been neglected. Successive orders of nonlinearity take into account such effects as saturation, power broadening and level shifts (optical Stark effect). For intense fields or very close resonance, the contributions from several orders of nonlinearity may be comparable in magnitude. Therefore, despite its generality, the susceptibility formalism does not necessarily provide the most practical approach for the description of resonant processes.

This chapter is concerned with the problem of deriving alternative and more manageable descriptions of resonant nonlinear processes, illustrated by examples. For much of the chapter, the nonlinear medium is treated as a two-level system (described in §6.2).

Throughout this book we consider the optical response of materials in the electric-dipole approximation. As mentioned in §2.5, this is a perfectly satisfactory approximation for the majority of practical cases in nonlinear optics because other interactions, such as electric quadrupole and magnetic dipole, are almost always very weak in comparison. However, there are a few cases in which these electric-multipole and magnetic effects need to be considered. For example, second-harmonic generation via electric-dipole interaction is forbidden in centrosymmetric media from symmetry considerations (see §5.3). Yet a weak effect is sometimes observed in centrosymmetric solids-such as the crystal calcite (Terhune et al, 1962)-which can be ascribed to an electric-quadrupole interaction. This is one of the contributory mechanisms being considered currently in an attempt to explain the observation that glass optical fibres can, in certain circumstances, perform efficient second-harmonic generation (Terhune and Weinberger, 1987). Also it is found that certain atomic gases are good systems for the observation of multipole and magnetic nonlinear-optical effects (Hanna et al, 1979). It is possible to tune the optical frequencies in the vicinity of selected electronic transitions that are forbidden in the electric-dipole approximation, but which are allowed via electric-quadrupole and magnetic-dipole interactions; with resonance enhancement, these nonlinear effects can therefore be significant. It is sometimes necessary to take account of such effects in the analysis of very sensitive spectroscopic measurements. As a further example, the electric-dipole approximation may be invalid for highly-extended charge distributions, such as long conjugated-chain molecules.

‘Nonlinear’ optical phenomena are not part of our everyday experience. Their discovery and development were possible only after the invention of the laser.

In optics we are concerned with the interaction of light with matter. At the relatively low light intensities that normally occur in nature, the optical properties of materials are quite independent of the intensity of illumination. If light waves are able to penetrate and pass through a medium, this occurs without any interaction between the waves. These are the optical properties of matter that are familiar to us through our visual sense. However, if the illumination is made sufficiently intense, the optical properties begin to depend on the intensity and other characteristics of the light. The light waves may then interact with each other as well as with the medium. This is the realm of nonlinear optics. The intensities necessary to observe these effects can be obtained by using the output from a coherent light source such as a laser. Such behaviour provides insight into the structure and properties of matter. It is also utilised to great effect in nonlinear-optical devices and techniques which have important applications in many branches of science and engineering.

Another effect of light on matter can sometimes be to induce changes in the chemical composition; such ‘photochemical’ processes lie outside the subject of this book.

Origins of optical nonlinearity

We now consider in a simple way how nonlinear-optical behaviour might arise.

Introduction

LASER is an acronym for Light Amplification by Stimulated Emission of Radiation. As the name implies, in a laser, the process of stimulated emission is used for amplifying light waves. It was as early as 1917 that Einstein first predicted the existence of two different kinds of processes by which an atom can emit radiation; these are called spontaneous and stimulated emissions. The fact that the stimulated emission process could be used in the construction of coherent optical sources was first put forward by Townes and Schawlow in the USA and Basov and Prochorov in the USSR. And finally in 1960 Maiman demonstrated the first laser. Since then the development of lasers has been extremely rapid and laser action has been demonstrated with gases, solids, liquids, free electrons, semiconductors etc.

The three main components of any laser are the amplifying medium, the pump and the optical resonator. The amplifying medium consists of a collection of atoms, molecules or ions which act as an amplifier for light waves. Under normal conditions, the number of atoms in the lower energy state is always larger than the number in the excited energy state; as such, a light wave passing through such a collection of atoms would cause more absorptions than emissions and therefore the wave will be attenuated. Thus in order to have amplification, it is necessary to have population inversion (between two atomic states) in which there is a large number of atoms in the higher energy state as compared to that in the lower energy state.