New nonlinear laser effects in $\alpha$-quartz: generation of a two-octave Stokes and anti-Stokes comb and cascaded lasing in the spectral range of the second and third harmonics

Crystals that are simultaneously $\chi^{(2)}$- and $\chi^{(3)}$-active offer a wide range of possibilities for the generation of new coherent wavelengths of light. Frequency conversion processes such as stimulated Raman scattering, second and third harmonic generation, or parametric sum and difference frequency mixing can be combined effectively in the same noncentrosymmetric crystal in cascaded $\chi^{(3)}\leftrightarrow\chi^{(2)}$ lasing processes. We present several new manifestations of these effects under picosecond laser excitation in $\alpha$-quartz ($\mathrm{SiO_2}$), the oldest nonlinear-laser crystal. Among them are 45 Stokes and anti-Stokes wavelength comb generation of more than two octaves (from 0.3692 $\mu$m to 1.5142 $\mu$m) and self-conversion of Raman-generation frequencies into the wavelength region of the second and third harmonics of one-micron pumping via many-step cascaded $\chi^{(3)}\leftrightarrow\chi^{(2)}$ processes.