Theory of Two-Photon Spectroscopy in Solids

The coefficient of absorption of two photons is calculated for a two-band model of a solid. The frequency and polarization dependence is evaluated for the case that one photon (laser) frequency is fixed while the second is varied. Wannier exciton states are explicitly included in both the intermediate and final states. Depending upon the symmetry of the two energy bands, the final state may either be an exciton $s$, $p$, or $d$ state. All three cases have different frequency and polarization dependences. The final formulas for the matrix element are complicated, and approximate formulas are given which are useful for many applications. It is shown that some of the two-photon spectroscopy experiments in semiconductors and alkali halides which have been reported so far can be explained by these calculations.