Nonlinear Optics: SHG and OPO I am new to nonlinear optics, and recently I started studying about the second harmonic generation (SHG) and the optical parametric oscillation (OPO) where these two nonlinear interactions are achieved when a beam propagates through a nonlinear crystal.
My question is can the same type of nonlinear crystal be used to achieve both of these interactions (i.e. SHG and OPO)? If yes, how to select which interaction to occur?
 A: I suppose your question is about parametric  up-conversion (such as SHG, or, more general, SFG, sum frequency generation, $\omega_1+\omega_2\rightarrow\omega$) and down-conversion $\omega\rightarrow\omega_1+\omega_2$. For both of these processes, phase matching is crucial, that is, conservation of the total photon momentum. This imposes restrictions on refractive indices of a nonlinear crystal at those frequencies $\omega_i$. In order to enable phase matching, nonlinear crystals have to be birefringent (I intentionally do not mention periodic poling here, that's another, more complicated approach). So basically, as experimental parameters you have 1) incident wavelength, and 2-3) angles of polarization of light with respect to the crystallographic axes of the crystal (because they define the refractive index for this particular beam). Often, these crystals are cut for normal incidence, so only one of these angles remains as an experimental parameter. The question is then, for which of the processes phase matching condition holds at a given incident polarization and wavelength. For instance, same BBO crystals can be used for generating SHG at 800 nm incident wavelength, and down-conversion into a signal and idler beams such that $\omega_s+\omega_i=\omega_{800}$, you just have to choose the angles properly.
