1. to increase the efficiency of laser and to reduce spontaneous emission. As Bandgap increases their is less chance for spontaneous emission or absorption because it will need to energy to pump up an electron in conduction band. :) 

2. http://pubs.rsc.org/en/content/articlelanding/2017/dt/c6dt04196e#!divAbstract 

I hope this will help you and all the best with your research :) 

Hi

If you are using the NLO crystal for second harmonic generation like KTP, then a large band gap is desirable other wise it will lead to gray tracking problem leading to damage of the crystal. The mechanism is as follows: during SHG some amount of THG signal also generated which may have higer energy than the band gap and creates a color center. These color centers have their own absorption band at the fundamental or at SHG wavelength. So absorption at the incident laser beam or at the generated SHG signal increases rapidly leading to the damage of the crystal.