Differences in crystals used for SHG and OPO SHG: second harmonic generation
OPO: optical parametric oscillator
As we know these are non linear cavities (optical cavities with non linear crystals in them along a part of light beam's path) used for frequency up-conversion and down conversion respectively. The latter is also called a squeezer. Apart from minor differences in the coating of the crystal surfaces, what is the main difference between the crystals used in both these cavities, given that I use same crystals i.e. PPKTP? Since they are used of two complementary processes, I surmise the difference's gotta do with crystal cut direction or something along that line of thought, but I am not sure! 
 A: So three-wave mixing is the umbrella term for the nonlinear effects in $\chi^{\left(2\right)}$ media. For SHG we have two input photons at the same frequency are converted to one photon of twice the frequency, $\omega_1+\omega_1 \rightarrow \omega_2$ with $\omega_2 = 2\omega_1$.
This is essentially a case of degenerate sum frequency generation (SFG), which is  $\omega_1+\omega_2 \rightarrow \omega_3$. At the same time the reverse process of difference frequency generation can take place $\omega_3\rightarrow \omega_2+\omega_1$.
This difference frequency generation process is sometimes called optical parametric amplification, because you can send frequency $\omega_3$ and $\omega_1$ into a crystal and $\omega_1$ amplified as $\omega_3$ is depleted.
The only difference between an OPA and an OPO is the optical cavity built around the crystal.
The difference in these processes is purely down to the frequencies of the three fields involved. The crystal used will therefore need to be periodically poled with a different period, assuming the process is quasi-phasematched.
Please feel free to ask questions if anything isn't clear.
