Excitations of electrons in solids in the presence of two electromagnetic beams are discussed. Two excitation processes are considered: (a) transition of an electron from valence band 1 to conduction band 2, absorbing a photon with energy $\hslash \omega$, and subsequent excitation of the electron to conduction band 3 absorbing a photon with energy $\hslash \Omega$; (b) excitation of an electron to band 2, followed by an intraband transition accompanied by the emission of a phonon, and subsequent absorption of a photon $\hslash \Omega$ and excitation of the electron to band 3. It is shown that there is a distinct threshold energy for the process of type (a) and in certain cases there is a distinct threshold energy for processes of type (b). The transition rate is evaluated as a function of $\hslash \omega$ for values close to the threshold value. At threshold, cascade two-photon transition may be allowed in points, along a line, or over a surface in $\overrightarrow{\mathrm{K}}$ space. These different conditions lead to different types of transition-rate spectra. The effect of the intensity of the $\Omega$ beam on the different spectra is discussed. The intensity is allowed to exceed the values for which cascade transitions may be considered. The transition-rate spectrum for process (b) is also evaluated. It is shown that for polar crystals, process (b) will lead to transition rates of the same order of magnitude as those obtained in process (a). Possible use of these effects for the investigation of band structure of solids is discussed.

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