Second Harmonic Generation

A few issues here:

1. How to explain or understand phase-matching easily.
Well, there is dispersion in all materials. For fundamental and 2nd harmonic waves, different wavelength causes different refractive index and then causes different traveling velocity. Consequently, along the propagation direction, the previously-generated 2nd-harmonic-wave and the newly-generated 2nd-harmonic-wave will have different phase. Thus destructive interference will happen and this is phase-mismatching. To fulfill phase-matching, we use birefringence property of some crystals. At certain direction, the cross-polarized fundamental and 2nd-harmonic waves can have the same traveling speed.

2. Type-I and type-II phase-matching.
Type-I: the signal and idler beams have the same polarization. In SHG, signal and idler are equal, which are both the fundamental beam. So in SHG, type-I phase-matching is for polarized fundamental laser.
Type-II: the signal and idler beams have perpendicular polarization. In SHG, type-II phase-matching is for unpolarized fundamental laser.

3. Impedance matched cavity.
For cavity-enhanced SHG, we want the transmission of the input mirror at the pump wavelength equals all intracavity pump losses, including the loss from nonlinear frequency conversion. This way the initial pump reflection and the cavity leakage field will cancel (they always have opposite phase), resulting in zero total pump reflection. Thus all pump light will be coupled into cavity and maximize the conversion efficiency.

4. Major drawback of the intracavity SHG.
It is called "green noise", the intensity instability caused by interaction of multiple longitudinal modes of the fundamental laser.