Basic concepts of electromagnetic theory; Coulomb gauge; intensity of electromagnetic field. Electrons in an electromagnetic field: from the Lagrangian to the Hamiltonian; canonical momentum. Interaction Hamiltonian. Semiclassical approximation; weak-field limit. Electric dipole approximation. Calculation of the optical susceptibility by using the density matrix approach. From optical susceptibility to absorption coefficient. Momentum of an electron in a periodic crystal.

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.