Frequency Conversion
The frequency conversion processes include frequency doubling (which is a special case of sum frequency generation), sum frequency generation (SFG), differentialfrequency generation (DFG) and optical parametric generation (OPG) which are demonstrated in the following equations:
Sum Frequenby Generation (SFG) 
DifferentialFrequency Generation (DFG) 
Optocal Parametric Generatiom (OPG) 
ω1+ω2=ω3 (or 1/λ1+1/λ2=1/λ3 in wavelength) It combines two low energy (or low frequency) photons into a high energy photon. For example:1064nm+1064nm→532nm 
ω1ω2=ω3(or 1/λ1+1/λ2=1/λ3 in wavelength) It combines two high energy photons into a low energy photon. For example:532nm810nm→1550nm 
ωpωs=ωi(or 1/λp=1/λs+1/λi in wavelength) It splits one high energy photon into two low energy photons. For example:532nm→810nm+1550nm 
Second Harmonic Generation(SHG) 
Third Harmonic Generation(THG) 

Frequency Doubling is a special case of sum frequency generation if the two input wavelengths are the same. λ1=λ2=2Xλ3 1064nm+1064nm→532nm 
Frequency Tripling is an example of Sum Frequency Generation where. λ1=2Xλ2=3Xλ3 1064nm+532nm→355nm 
Optical Parametric Oscillation
Optical Parametric Generation (OPG) is an inverse process of sum Frequency Generation. It splits one highfrequency photon (pumping wavelength, lp) into two lowfrequency photons (signal, ls, and idler wavelength, li). If two mirrors are added to from a cavity as shown in fig. 7, an Optical Parametric Oscillator (OPO) is established. For a fixed pump wavelength, an infinite number of sigal and idler wavelengths can be generated by tilting a crystal. Therefore, OPO is an excellent source for generating wide tunable range coherent radiation. KTP and LINbO3 are good crystal for OPO and Optical Parametic Amplifier (OPA)Applications.
Phasematching
IN order to obtain high conversion efficiency, the phase vectors of input beams and generated beams have to be matched:
ΔK=k3k2k1=2p(n3/l3n2/l2n1/l1)=0 (For sum frequency generation)
Where:ΔK is phase mismatching, ki is phase vector at li and ni is refractive index atli.
In low power case, the relationship between conversion efficiency and phase mismatching is:
h∞(SIN(ΔKL)/ΔKL)^2
It is clear that the conversion efficiency will drop dramatically if ΔK increases. The phasematching can be obtained by angle tilting, temperature tuning or other methods. The angle tilting is mostly used to obtain phasematching as shown. If the angle between optical axis and beam propagation (q) is not equal to 90 deg. or 0 deg., we call it Critical phasematching (CPM). Otherwise, 90deg. noncritical phasematching (NCPM) is for q=90deg. and 0 deg. NCPM is for q=0deg.
Two types of phasematching are classsified in consideration of polarization of lasers. If the polarizations of two input beams (for sum frequency) are parallel to each other., it is type I phasematching. If the polarizations are perpendicular to each other, it is called type II phasematching.
Crystal Acceptance If a laser light propagates in the direction with angle Δq to phase matching direction, the conversion efficiency will reduce dramatically. We define the acceptance angle (Δq) as full angle at half maximum (FAHM), where q=0 deg. is phasematching direction. For example, the acceptance angle of BBO for type I frequecny doubling of Nd:YAG at 1064nm is about 1mradcm. therefore, if a Nd:YAG laser has beam dibergence of 3mrad for frequencydoubling, over half of the input power is useless. In this case, LBO may be better because of its larger acceptance angle, about 8 mradcm. For NCMP, the acceptance angle is normally much bigger than that for CPM, for example, 52 mradcm(1/2) for type I NCPM LBO. In addtion, you have to consider the Spectral acceptance (Δl) of crystal and the spectral bandwidth of your laser; crystal temperature acceptance (ΔT)and the temperature change of environment. efficiency. 


Group Velocity Mismatching
For NLO processes of ultrafast lasers such as Ti:Sapphire and Dye lasers with femtosecond (fs) pulse width, the main limitation to conversion efficiency is group velocity mismatching(GVM). The GVM is caused by group velocity dispersion of NLO crystal. For frequency doubling a Ti:sapphire laser at 800nm, for example, the inverse group velocities (1/VG)of BBO are respectively 1/VG=56.09ps/cm at 800nm and 1/VG=58.01ps/cm at 400nm and GVM=1.92ps/cm. That means an 1mm long BBO crystal will make 192fs separation between the pulses at two wavelengths. Therefore, for an 100fs Ti:sapphire laser, we normally recommend a 0.5mm long BBO crystal (with 96 fs separation) in order to obtain high effciency without dramatic pulse broadening.