Optical Double Resonance in Solids

Nonlinear interaction of two electromagnetic beams with a semiconducting or insulating crystal is considered under the following conditions: The photon energy $\hslash \Omega$ of an intense laser beam is equal to the gap between two conduction bands $\hslash {\Omega }_{32}$ at certain points in the crystal momentum space. The photon energy $\hslash \omega$ of a relatively weak beam is approximately equal to the gap between the valence band and one of the conduction bands $\hslash {\Omega }_{21}$ at the same points in crystal momentum space. These conditions are referred to as double resonance. A theory is developed for two nonlinear effects: (1) the change of the dielectric coefficient of the crystal at frequency $\omega$ caused by the $\Omega$ perturbation; and (2) the generation of a parametric beam at frequency $\omega +\Omega$. An interesting result of the theory is that both effects are very sensitive to the conditions at the point of contact of the surfaces of constant ${\Omega }_{31}$ and ${\Omega }_{21}$ in $K$ space. The effects obtained when $\frac{\partial {\Omega }_{31}}{\partial K_{\perp }}$ and $\frac{\partial {\Omega }_{21}}{\partial K_{\perp }}$ have the same sign are much smaller than those obtained when they have opposite signs. ($K_{\perp }$ is the component of K perpendicular to the surfaces at the point of contact.) This phenomenon is explained in terms of combined electronic and electromagnetic states. Possible applications of these effects to the investigation of the band structure of solids are discussed. In particular, it is shown that measurements of double-resonance effects will yield information about the band structure at noncritical points in the Brillouin zone.