An extensive review of theoretical and experimental investigations conducted over the last two decades since the advent of the laser and relating to the simultaneous absorption of one or more photons by the valence electrons of a crystalline solid is given. The following topics are addressed, with greatest emphasis on the most recent results: the pioneering models of Braunstein and Ockman and that of Basov for two-photon absorption in direct-gap crystals, based on second-order time-dependent perturbation theory and parabolic and isotropic energy bands; extensions and modifications of the above models by various authors to take into account the effects of excitons, crystal anisotropy, laser polarization, and nonparabolicity as well as degeneracy of the electronic energy bands; rigorous band-structure calculations that employ realistic energy bands and momentum matrix elements that include many intermediate states to obtain good convergence in the perturbation calculation; the semiclassical theory of Keldysh that takes into account electric-field effects on the electronic energies and wave functions and that employs first-order perturbation theory to obtain multiphoton transitions of all orders; the fully quantized treatment of the multiphoton absorption (MPA) process along the above lines by Kovarskii and Perlin; the Volkov approximation method of Jones and Reiss; descriptions of the various experimental techniques that are usually employed to study nonlinear phenomena in solids; critical comparison between different theoretical predictions and experimental data; and finally, theoretical and experimental work relating to phonon-assisted two-photon transitions, three-photon absorption, four-photon absorption, and higher-order MPA processes in crystalline solids.
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