Up to this point our discussion has avoided any detailed reference to the underlying physical mechanisms producing scattered waves. One most often thinks of scattering in terms of particles undergoing elastic collision, either with other particles or with macroscopic objects such as walls. In the microscopic domain this view is even extended to light when the photon picture is appropriate. As we have seen, geometric optics permits a similar interpretation wherein rays mimic the scattering behavior of particles.

When one can no longer neglect the wave nature of light, however, this intuitive view of the scattering process is not entirely adequate and we are compelled to look beyond it for physical origins. On a microscopic level an electron, atom, or molecule will couple to an incident electromagnetic wave in an oscillating fashion, such that it re-radiates in all directions, producing a ‘scattered’ wave. It is tempting to amplify this mechanism to macroscopic targets, since they are certainly composed of these constituents, but to do so in detail would be a forbidding problem in many-body physics. A more appropriate macroscopic approach might be to envision the incident fields as inducing electric and magnetic multipoles that oscillate, and hence radiate, while maintaining definite phase relations with the incident wave. When the wavelength of the incident radiation is long compared with the dimensions of the scatterer, only the lowest-order multipoles will be important, and the re-radiation process can be approximated by invoking electric and magnetic dipoles.

Following the rather lengthy mathematical onslaught in the theoretical development of scattering from a large dielectric sphere in the preceding chapter, it is perhaps time to take a break from these labors and apply the results to prediction of some measurable quantities. As well as judging the largesphere theory against the exact expressions provided by the Mie solution, some pleasure may also be found in using the results to explore various meteorological optical phenomena.

It cannot be emphasized too strongly that the preceding theory is exact, in the same sense that the Mie solution is exact. In principle we can calculate amplitudes and cross sections to all orders in β – it is not an approximate theory. In practice, however, one need only retain a few terms in the asymptotic expansions and in the residue series to achieve reasonable accuracy, so that we can appropriately approximate the theoretical expressions and reduce the calculational labor. However, this is also in contrast with the partial-wave expansions, which must include indices l to at least O(β). The difference between approximate theories and approximations to the exact theory is that in the latter we are able to control the estimates rather precisely. A separate issue relates to how many terms must be retained in the Debye expansion itself, and this will be addressed as the need arises.

Given the way we seem to have continually ignored the ripple, one could be excused for thinking it nothing but a noisy nuisance. It is certainly ubiquitous, as seen in Fig. 3.4 for Qext(β), in Fig. 6.4 for Qabs(β), in Fig. 6.6 for Qpr(β), and in Figs. 3.6 and 6.16 for backscattering. Far from being a nuisance, however, the ripple reflects a great deal of additional physics taking place within the transparent sphere, and at this point requires much closer scrutiny. Unless clearly stated otherwise we shall consider n to be real.

What is it that needs to be explained about the ripple? First and foremost, we should like to understand clearly the origin of the sharp, almost chaoticlooking peaks in these cross sections, not only mathematically, but also physically. In addition, the ripple structure appears to oscillate about the slowly varying background of Eq. (6.3), as seen in Fig. 6.1b. Why is this?

Ultimately the ripple must be linked to the behavior of the electromagnetic fields produced by the encounter of the incident plane wave with the sphere. What processes are taking place in terms of these fields that might lead to the ripple structure? Physics beyond the scattering mechanism emerges here, in the fields internal to the sphere, and a major goal in this chapter will be to explicate these phenomena.

Many of the figures in the text relating to the Mie theory were constructed from computational data generated using the Mathematica® system fordoing mathematics on a computer. Although this is not a necessary choice of software, it was found to be very convenient and efficient. It is a functional-programming-based language that, although it is not as fast as C or Fortran, is quite user friendly and contains very efficient routines for all the special functions in the preceding appendices. Nevertheless, almost all routines used to compute Mie scattering functions ran fairly rapidly on Pentium II and Pentium III processors. All figures were eventually converted to PostScript® for plotting and were labeled in TEX

The Mie partial-wave coefficients are given by Eqs. (3.88) and (3.89) in terms of Ricatti–Bessel functions, and it is the latter that encompass most of the computational effort. Although Mathematica's built-in functions were used frequently, we employ iterated recursion relations here because the former are too slow for large indices and arguments.

Various excited state processes, in addition to radiative decay, are important to laser performance. The quenching of luminescence reduces the excited state lifetime and can cause sample heating, thereby contributing to photothermal effects such as thermal lensing and thermal shock. Luminescence quenching also results in reduced laser slope efficiency, as discussed briefly here. Other excited state processes that require consideration are excited state absorption and energy transfer. In excited state absorption (ESA) a photon excites an electronic centre from the ground state to an excited state, which then relaxes to some lower lying metastable level. A second photon promotes the centre to an even higher energy state. Energy transfer arises when the optical centres are close enough together to interact, and this occurs when the concentration exceeds some lower bound, which need not be large. Although the energy levels of the interacting ions can be unaffected at such concentrations, the interion interaction is strong enough to enable excitation to be transferred between them. Prior to 1966, energy transfer was understood to involve excited states of donors (|D*〉) interacting with the ground states of acceptors (|A〉). Auzel (1966) pointed out that excited acceptors (|A*〉) also receive energy from excited donors (|D*〉), and that energy differences can be exchanged as well as absolute energies. Energy transfer from excited donors to the metastable levels of acceptors can be treated by generalization of the Förster–Dexter theory outlined in §7.1.

Principles and objectives

Crystal-field engineering seeks to use present knowledge to establish appropriate design principles for the development of new laser and nonlinear optical materials. First, the wavelength range of the optical device and its possible application (e.g. CW, ultrashort pulse, single frequency or tunable) are specified. This determines the chemical nature of the optical centre. The host environment is then selected, guided by historical knowledge of gain media or intuition of novel hosts with potentially beneficial properties, and then the theoretical and experimental techniques outlined in earlier chapters are invoked. The numerous objectives of crystal-field engineering include shifting the wavelength ranges of optical transitions, increasing the rates of radiative transitions and minimizing loss by nonradiative decay and excited state absorption. In addition, there may be reason to minimize or maximize energy transfer between centres, to avoid concentration quenching and to enhance laser efficiency respectively. Such objectives may be achieved by manipulating the unit cell containing the optical centre using such external perturbations as hydrostatic pressure, uniaxial stress or electric field. More usually, however, manipulating the unit cell is accomplished by changing its chemical composition.

Manipulating the unit cell

Hydrostatic pressure shortens bond lengths, reducing the unit cell dimensions without changing its symmetry. Such hydrostatic pressures will enhance the crystal field and, in consequence, shift spectra to shorter wavelengths. Studies of the Cr3+-doped elpasolites and garnets under pressure demonstrate the continuous tuning of the crystal field and of the coupling of the 2E and 4T2 states of the Cr3+ ion [Dolan et al. (1986), Hommerich and Bray (1995)].