Phase Matching

ωp - ωs = ωi (or 1/λp -1/λs = 1/λi in wavelength)

In order to obtain high conversion efficiency, the phase vectors of input beams and generated beams have to be matched:
Δk = k3 - k2 - k1 = 2πn3/λ3 - 2πn2/λ2 - 2πn1/λ1 = 0 (for sum frequency generation)

Where:Δk is phase mismatching, k(i) is phase vector at λ(i) and n(i) is refractive index at λ(i). In low power case, the relationship between conversion efficiency and phase mismatching is: η∝[(sinΔkL)/ΔkL]2

It is clear that the conversion efficiency will drop dramatically if Δk increases.

The phase-matching can be obtained by angle tilting, temperature tuning or other methods. The angle tilting is mostly used to obtain phase-matching as shown in the left figure. If the angle between optical axis and beam propagation (θ) isn't equal to 90o or 0o, we call it critical phase-matching (CPM). Otherwise, 90o non-critical phase-matching (NCPM) is for θ=90o and 0o NCPM is for θ=0o.

	For sum frequency generation		Critical Phase Matching
	Type I phase-matching		Type II phase-matching

Two types of phase-matching are classified in consideration of polarization of lasers. If the polarizations of two input beams (for sum frequency) are parallel to each other, it is called type I phase-matching. If the polarizations are perpendicular to each other, it is called type II phase-matching.