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 for doing 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)].

Introduction

The distinguishing feature of crystalline solids is their symmetry, manifest microscopically in their X-ray diffraction patterns and macroscopically in crystal morphology. An ideal crystal is an infinite regular repetition in space of identical structural units. The symmetry of a particular ideal crystal is specified by the set of symmetry elements, comprising rotations, reflections and translations, which leave it invariant. Real crystals are not only of finite extent, but also contain a variety of imperfections such as inclusions of minority phases, grain boundaries, dislocations, impurities and point defects; the latter two are especially relevant in the present context. An isolated impurity or point defect in an otherwise ideal crystal obviously removes translational symmetry. It may also reduce the residual point symmetry in some circumstances, exemplified by ions with degenerate electronic states (the Jahn–Teller effect), impurities in the form of small molecules, small substitutional cations which move off-centre, vacancy pairs, bipolarons, etc. A principal theme of the present monograph is exploration of the ways in which the properties of a laser-active centre are controlled or affected by its crystalline surroundings, including their residual point symmetry. We shall find that it is often useful to distinguish a dominant site symmetry, determined by coordination alone, which is somewhat higher than the actual point symmetry.

The formal description of symmetry exploits a branch of mathematics called ‘Group Theory’, which is not a physical theory in the sense of ‘Quantum Theory’, but is rather a collection of principles deduced from a chosen set of axioms.

Natural minerals and gemstones

There are several thousand naturally occurring minerals, inorganic compounds of fixed chemical composition and regular sub-microscopic structure. Not more than 100 or so of these are recognized as gemstones although the number may vary as fashions change and new sources are found. Gemstones are hard and durable to withstand regular use without damage. But most particularly, they are beautiful, in their colour and lustre, especially when cut, faceted or polished for personal adornment. Beauty notwithstanding, the most important quality of a gemstone is scarcity. To be rare is to be greatly valued.

Minerals that are the basis of gems are found in rocks which form over aeons in time: where they are found reflects the process of continuous formation. There are three main classes of gem-bearing rocks. Sedimentary rocks are formed by the accumulation of eroded rock fragments, which settle over time, are compressed and again harden into rock. They are set down in layers which eventually emerge from below the earth's surface. Gypsum is a typical sedimentary rock with important variations in alabaster and selenite. Opal and tourmaline occur as veins in such sedimentary rocks as shale. Igneous rocks solidify from molten rock deep beneath the earth's surface, sometimes escaping in lava flows from erupting volcanoes. The slower the rate at which the molten rocks cool, the larger are the gems that grow within them. In consequence, gems grow at high temperature, under huge hydrostatic pressures.

Physical principles

The present chapter is concerned with transitions between optical levels of point imperfections in solids in which energy is conserved by emission of phonons rather than photons. Such transitions can introduce an undesired loss mechanism in laser materials which competes with stimulated emission, but they also play an essential role in the performance of laser materials, since excited states reached in electric-dipole-allowed pumping transitions subsequently relax to metastable lasing states by a sequence of radiationless processes.

Radiationless transitions are, paradoxically, both ubiquitous and elusive. They are the rule, rather than the exception, following optical excitation; yet, they are inaccessible to direct observation. Radiationless relaxation is generally inferred from its consequences, including such familiar phenomena as radiant heating and the absence of luminescence. In favourable circumstances, the observation of luminescence with diminished intensity and duration provides more detailed information about radiationless processes. Quantitative understanding of some aspects of radiationless relaxation has also proved elusive, in that theory has been more successful in elucidating trends than in the prediction of absolute transition rates. Radiationless transitions generally belong to one of two categories: static processes which are thermally activated from a metastable state, and dynamic processes which occur during rapid relaxation immediately following excitation [Bartram (1990)]. Point imperfections in solids provide examples of both categories.

Prepared state

Radiationless transitions can only occur between non-stationary states of a system; thus the radiationless transition rate depends critically on the sort of non-stationary state which is prepared in a given experiment.

Historical notes

Laser is an acronym for Light Amplification by Stimulated Emission of Radiation. The operating principles of the laser were originally elucidated for devices operating at microwave frequencies (masers). The maser was invented by Gordon, Zeiger and Townes (1955), who used an inverted population between the excited vibrational levels of ammonia. Extension of the wavelength range of the maser into the optical regime was proposed by Schawlow and Townes (1958), whose various laser schemes all comprised a gain medium, an excitation source to pump atoms or ions in the gain medium into higher energy levels and a mirror feedback system to enable one or multiple passes of the emitted radiation through the laser medium. The special qualities that distinguish laser light from other optical sources include extreme brightness, monochromaticity, coherence and directionality. Also the laser output is linearly polarized and very frequency stable. After development over forty years modern lasers operate over wavelength ranges from the mid-infrared, through the visible and beyond into the ultraviolet and vacuum ultraviolet ranges.

The first operational laser used synthetic rubies, corundum crystals containing ∼0.1 wt.% of Cr2O3, as the gain medium, pumped with white light from a helical flashlamp to oscillate on the sharp R-line at a wavelength of 693.4 nm [Maiman (1960)]. Soon afterwards the He–Ne laser was operated on the 3s → 2p (632.8 nm), 2s → 2p (1.15 μm) and 3s → 3p (3.39 μm) transitions of atomic Ne.

The successful launch of the alexandrite laser by Allied Chemicals Inc. and the elucidation of essential design parameters [Walling et al. (1979), (1980)] spawned rapid growth of research into Cr3+-based lasers [Caird and Payne (1991)]. Possible alternative gain media to alexandrite included the Cr3+-doped garnets [Struve and Huber (1985)]. However, the development of Cr3+ : colquiriite lasers at Lawrence Livermore National Laboratory [Payne et al. (1988a), (1989a)] deflected attention away from the Cr3+ garnets, these mixed fluoride gain media being as efficient as alexandrite and almost as broadband as Ti-sapphire. Two distractions from the dominance of Cr3+ ion broadband tunable lasers were the inventions of the Ti-sapphire (Ti3+ : Al2O3) [Moulton (1982a,b)] and the Cr4+ : forsterite [Petricevic et al. (1988)] lasers. At present Ti3+ : Al2O3 and Nd3+ : YAG are the market leaders in solid state laser production against which new developments are assessed. Their pre-eminence derives in part from the quality and quantity of laser rods that can be produced at modest cost. Both materials have excellent photothermal and thermomechanical properties, and are robust components under laser operating conditions. However, despite much spectroscopic research Ti-sapphire is still the only usable Ti3+-activated solid state laser. In contrast, operation of several 3d2-ion doped lasers have been reported giving broadband tunability at near-infrared wavelengths (1.0–1.7 μm) which have potential applications in optical communications, medical sciences and on remote sensing LIDAR platforms.

Almost four decades have passed since Maiman invented the ruby (Cr3+:Al2O3) laser. The intervening years have witnessed many major achievements and much innovation, culminating in a plethora of devices that impinge in many walks of life. Fundamental studies and technology have been pursued with vigour, separately and in tandem, with resulting applications in materials and device processing, optical, communication and information sciences, and medical and paramedical sciences. Such applications have the potential to materially transform our everyday living at home, at work and at leisure. Optical devices that were once regarded as revolutionary but which are taken now for granted include the fibre laser, responsible for dramatically improving the quality and cost of intercontinental telephone calls, and the compact disc, now the ubiquitous storage medium for optical information. The scope of optical materials discussed in this monograph is very broad, indicating the potential sweep of exciting and novel developments in the short and long term futures.

Ruby lasers of the kind invented by Maiman, and discussed in this monograph, operate on the Cr3+ R-line emission near 694 nm: they are pumped by the ruby's absorption of some of the visible light from a xenon flashlamp. The He–Ne laser was reported soon thereafter, since when laser action has been observed on many thousands of transitions in the gas phase, including 200 or so on neutral Ne alone.

Ligand-field theory

Limitations of crystal-field theory

The limitations of crystal-field theory became apparent soon after its formulation by Bethe (1929). Van Vleck, who was primarily responsible for its early applications, recognized that the point-ion model on which it is based is quantitatively unreliable, and proposed an alternative formulation based on covalent bonding [Van Vleck (1935), Van Vleck and Sherman (1935)], now called ligand-field theory [Ballhausen (1962)]. Nevertheless, the popularity of crystal-field theory with adjustable parameters remains undiminished, contrary to expectation [Jørgensen (1971)]. By virtue of its elegance and relative conceptual simplicity, it continues to provide a useful framework for summarizing and interpolating empirical spectral information [Morrison (1992), Kaminskii (1996)].

The essential similarity of ligand-field theory and crystal-field theory is attributable to the underlying symmetry of the complex, and its implications for the wave functions and energy levels involved. However, ligand-field theory has the capacity to explain phenomena not contemplated in crystal-field theory, such as the nephelauxetic effect discussed in §4.4.8 and §9.6.3 and transferred hyperfine interactions [Spaeth et al. (1992)]. In addition, models based on covalency provide a deeper understanding of optical properties addressed in preceding chapters, and may ultimately yield quantitatively reliable predictions of crystal-field parameters.

Molecular orbitals

In the molecular-orbital theory of covalency [Hund (1927b), Mulliken (1928)], electrons occupy orbital wave functions which are delocalized over the entire complex consisting, for example, of a transition-metal ion and its immediate ligands. An approximate molecular orbital can be constructed as a linear combination of atomic orbitals (LCAO).