In the earlier days of nonlinear optics, the materials used for experiments and devices were mainly inorganic dielectric crystals, vapours, liquids and bulk semiconductors. The search for ‘good’ nonlinear-optics media was made amongst the known materials. However in more recent years, with the growing interest in optical devices and applications, attention has focused increasingly on new artificial solid-state materials which may offer higher nonlinearity; in particular, those that will allow nonlinearoptical devices to operate efficiently at relatively low power levels, such as the outputs from semiconductor-diode lasers. Organic materials offer great scope since modern methods of synthesis allow considerable flexibility in the design of materials at the molecular level. As mentioned in §4.4.3, the macroscopic nonlinear-optical properties of many organic crystals are given by the tensor sum of the properties of the constituent molecules, with due regard to local-field factors and molecular orientation. It is this feature of organic materials that allows a ‘molecular-engineering’ approach to the optimisation of macroscopic properties. Several materials with large second-order nonlinearity have been successfully fabricated (Chemla and Zyss, 1, 1987). Some of these newer organic optical materials also exhibit other desirable properties, such as a greater resistance to optical damage. These have applications in devices such as compact optical-frequency doublers and parametric- amplifiers and -oscillators. However, materials with large third-order nonlinearity are perhaps of greater interest currently, since the nonlinear refractive-index effect can be exploited for switching, optical bistability, phase conjugation and other types of signal processing (Gibbs, 1985).

Semiconductors contain free carriers. That is the characteristic feature which makes them different from the other systems which we have considered hitherto. The optical nonlinearities discussed in previous chapters arose from bound charges. Similar effects arise from bound charges in semiconductors but we do not consider them here. The optical nonlinearities which arise from free carriers in semiconductors are particularly important for applications because of the high degree of control which we have over the free-carrier densities and therefore on the performance of devices which make use of them.

When the free-carrier densities are changed by optical excitation we are concerned with real transitions (see §6.6). The resulting nonlinear processes proceed via a real exchange of energy from the optical field to the medium, and are often referred to as ‘dynamic nonlinearities’ (Miller et al, 1981a; Oudar, 1985); this is the nomenclature used here.

Another feature of semiconductors which has become of particular significance for applications in recent years is the ability to fabricate multiple ‘quantum well’ structures, in which the carriers are confined in one direction in repeated layers of the order of 5 nm wide. Within the layers the carrier motion is two-dimensional, which drastically affects their behaviour. This topic belongs more naturally to the next chapter. In this chapter, we confine our attention to the nonlinear-optical properties of bulk semiconductors.

In §§8.1 and 8.2 we outline the one-electron band structure and the behaviour of phonons in Group IV and III – V semiconductors.

In recent years there has been a rapid expansion of activity in the field of nonlinear optics. Judging by the proliferation of published papers, conferences, international collaborations and enterprises, more people than ever before are now involved in research and applications of nonlinear optics. This intense activity has been stimulated largely by the increasing interest in applying optics and laser technology in tele-communications and information processing, and has been propelled by significant advances in nonlinear-optical materials.

The origins of these recent developments can be traced through three decades of work since the invention of the laser and the first observations of nonlinear-optical phenomena by Franken et al (1961). From the earliest days it was recognised that such phenomena can have useful practical applications; for example, effects such as optical-frequency doubling allow the generation of coherent radiation at wavelengths different from those of the available lasers. In the 1960s many of the most fundamental discoveries and investigations were made. Work was then mainly concerned with the interaction of high-power lasers with inorganic dielectric crystals, gases and liquids. Effects such as parametric wave-mixing, stimulated Raman scattering and self-focusing of laser beams were investigated intensively. The invention of the wavelength-tunable dye laser paved the way for the development during the 1970s of many tunable sources utilising nonlinear effects, such as harmonic generation, sum- and difference-frequency mixing, and stimulated Raman scattering. In this way tunable coherent radiation became available in a wide spectral range from the far infrared to vacuum ultraviolet.

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.