The Non-linear Constitutive Relation in Solids at Optical Frequencies

The constitutive relation between the current density and the electric field in a solid is extended to include terms in all powers of the field. By calculating the density matrix of the charged particles in the solid, a general expression is obtained for the nth-order conductivity tensor which connects the current density with the nth power of the field. The second- and third-order tensors produce the most significant non-linear effects for the radiation fields currently available. They are discussed in more detail and the restrictions placed on them by the symmetry requirements of the thirty-two crystal classes are evaluated. The electronic contribution to the conductivity tensors is treated in the one-electron approximation; for low frequencies and odd values of n, the nth-order conductivity tensor may be expressed in terms of the (n+1)th derivative of the one-electron energy with respect to the wave vector.