Second-Order Optical Processes and Harmonic Fields in Solids

A consistent method has been developed to calculate induced electromagnetic fields and optical transitions of electrons in a solid, in response to an incident laser beam of (circular) frequency $\omega$. The analysis is based upon the independent-particle Schrödinger equation for electrons and Maxwell's equations for the electromagnetic fields. General expressions for linear and bilinear currents as well as second-order optical transition probabilities have been derived. It is shown that the second-order transition probability, which is proportional to the fourth power in the incident field, contains two different types of terms, describing double-photon transitions of the incident frequency $\omega$ and single-photon transitions of the harmonic frequency $2\omega$. An estimate has been made to show that in the case of centrosymmetric solids like metals, the relative contribution due to the single second-harmonic photon transition is of the order ${\left(\frac{e^2}{\hslash c}\right)}^2\ll 1$ in the optical region, compared with the double-fundamental-photon transition. However, in the case of solids lacking inversion symmetry, the contributions due to these two processes are estimated to be of the same order in magnitude.