Most prograde metamorphic reactions involve dehydration or decarbonation. The large increase in entropy that accompanies the liberation of a fluid phase from a mineral ensures that rising metamorphic temperatures will favor reactions that produce a separate fluid phase.

Optical nonlinearity emerges from nonlinear interaction of light with matter. In this chapter, the basic concept and formulation of light‒matter interaction are discussed through a semiclassical approach with the behavior of the optical field classically described by Maxwell’s equations and the state of the material quantum mechanically described by a wave function governed by the Hamiltonian of the material. An optical field interacts with a material through its interaction with the electrons in the material. A Schrödinger electron is nonrelativistic with a nonzero mass, and a Dirac electron is relativistic with a zero mass. The interaction Hamiltonian can be expressed in terms of the vector and scalar potentials by using the Coulomb gauge. It can be expressed in terms of the electric and magnetic fields through multipole expansion as a series of electric and magnetic multipole interactions, with the first term being the electric dipole interaction. The electric polarization of a material induced by an optical field is obtained through density matrix analysis. The optical susceptibility of the material is then obtained from the electric polarization.

Metamorphism is the sum of all the changes that take place in a rock as a result of changes in the rock’s environment; that is, changes in temperature, pressure (directed as well as lithostatic), and composition of fluids. The changes in the rock may be textural, mineralogical, chemical, or isotopic. These changes proceed at varying rates, so time is an important factor in metamorphism.

Bistability is a phenomenon that has two stable states under one condition. A bistable device has two possible stable output values for one input condition. The necessary conditions for optical bistability are optical nonlinearity and positive feedback. Depending on whether the optical nonlinearity that is responsible for the bistable function comes from the real or the imaginary part of a nonlinear susceptibility, a bistable optical device can be classified as either dispersive or absorptive. Depending on the type of feedback, a bistable optical device can also be classified as either intrinsic or hybrid. After a general discussion on the condition for optical bistability, this chapter covers dispersive optical bistability, absorptive optical bistability, and hybrid optical bistability of passive optical systems in three sections. The final chapter covers optical bistability in the active optical system of a laser oscillator.

All-optical modulation of an optical wave is accomplished through a nonlinear optical process that involves one or multiple optical waves. A nonlinear optical modulator can be based on either self-modulation or cross-modulation. Such nonlinear optical modulators and switches are also known as all-optical modulators and all-optical switches, respectively. Most all-optical modulators and switches are based on third-order nonlinear optical processes, but some rely on the high-order process of optical saturation, either absorption saturation or gain saturation. There are two fundamentally different types of all-optical modulators and switches: the dispersive type and the absorptive type. All-optical modulation of the dispersive type, which is based on the optical Kerr effect, is discussed in this chapter. In the first four sections, the physics, phenomena, and measurement of the optical Kerr effect are discussed. The last two sections cover all-optical modulators and switches in the bulk form and those in the waveguide form.

In this chapter we discuss how magmas differentiate to produce the wide range of igneous rocks. Many processes have been invoked, but fractional crystallization is undoubtedly the most important of these. After dealing with the chemical evidence for fractional crystallization, we discuss the actual mechanisms by which crystals can be segregated from liquid in magmas. Historically this was thought to be due to gravitative crystal settling.

The determination of metamorphic temperatures and pressures is fundamental for understanding the petrotectonic histories of mountain belts. Quantifying metamorphic fluid compositions, rates, and processes of heat transfer through the crust, rock deformation mechanisms, and rates of tectonic burial and exhumation are just a few examples of applications that require pressure–temperature estimates.

This chapter deals with isotope geochemistry and its role in igneous and metamorphic petrology. Isotopes are of two types: those that are radioactive and decay to other isotopes of a different element, and those that are stable and do not change with time.

Metamorphic rocks have mineral assemblages that crystallized at elevated pressures and temperatures. With certain assemblages, these conditions can be estimated quite closely using thermodynamic, kinetic, and experimental data, as we have seen in Chapters 19–22.

In this chapter we discuss the forms taken by bodies of igneous rock, starting with volcanic forms because they are the best understood and following with intrusive bodies.

In this chapter we deal with two important diffusion processes: the transfer of heat (conduction) and the migration of chemical constituents (diffusion). They typically operate at very different scales; heat moves, for example, through the lithosphere, whereas chemical diffusion operates over distances of microns to a few meters at the most. The laws governing their transport, however, are similar.

Supercontinuum generation is a nonlinear optical process that produces a broad continuous spectrum, often spanning over an octave, when an intense laser beam of an initially narrow bandwidth propagates through a nonlinear medium. Given a sufficiently high laser power, supercontinuum generation can be observed in any material. In general, many nonlinear processes are involved, including self-phase modulation, cross-phase modulation, four-wave mixing, modulation instability, self-focusing, stimulated Raman scattering, soliton dynamics, and dispersive wave generation. The specific nonlinear optical processes that are involved depend on the optical properties of the material and on the wavelength and the temporal characteristics of the laser beam. The usage of optical fibers greatly facilitated the development of supercontinuum generation because an optical fiber provides the favorable combination of both high intensity and long interaction length for efficient supercontinuum generation.

Optical nonlinearity manifests nonlinear interaction of an optical field with a material. The origin of optical nonlinearity is the nonlinear response of electrons in a material to an optical field. Macroscopically, the nonlinear optical response of a material is described by an optical polarization that is a nonlinear function of the optical field. This optical polarization is obtained through density matrix analysis by using the interaction Hamiltonian, which can be approximated with electric dipole interaction in most cases. When the interaction Hamiltonian is small compared to the Hamiltonian of the system, it can be treated as a perturbation to the system by expanding the density matrix in a perturbation series and the total optical polarization in terms of a series of polarizations. In most nonlinear optical processes of interest, the perturbation expansion of the polarization is valid and only the three terms of linear, second-order, and third-order polarizations are significant. The perturbation expansion is not valid in the cases of high-order harmonic generation and optical saturation. Then, a full analysis is required.

So far, our discussion has almost exclusively been about inequality in the United States. Although inequality and stratification in the U.S. are complex enough to warrant the space given them, it is important to remember that patterns of inequality and the processes leading to them are different in other countries. It is also useful to consider where the U.S. falls in the larger picture. There are no contemporary societies in which resources are equally distributed, but the degree of inequality varies dramatically among countries. In particular, the disparity between the very rich and the very poor often differs notably across countries. This is especially true when comparing developed countries (e.g., Australia, the United States, Canada, and the United Kingdom) with those that are still developing (e.g., Brazil, China, Hungary, India, and Mexico). Understanding how the United States compares to other countries puts in perspective the processes we encounter close to home.