Critical Phase-Matching

The reason for not achieving the optimum second harmonic output from a particular set-up is in many cases the divergence of the pump beam. For secondharmonic generation in a negative crystal we find that Δk = 0 if n^o₁= n^e₂(Θ_m), which is exactly true only at Θ = Θ_m. Shown that there is a linear relationship between small deviations δθ from the phase-matching angle and Δk. The change of Δk as a function of δθ can be large enough to limit the conversion efficiency in real devices. Consider as an example second-harmonic generation in KDP at 1.064 μm. The linear change in Δk with Θ is sufficiently great to restrict the deviation from the phase-matched direction to approximately 1.2 mrad if the coherence length is to be greater than 1 cm. Phasematching under these unfavorable conditions is termed “critical phase-matching.” Also if phase-matching is accomplished at an angle Θ_m other than 90^◦ with respect to the optic axis of a uniaxial crystal, there will be double refraction.
Therefore, the direction of power flow (Poynting vector) of the fundamental and second harmonics will not be completely collinear but occur at a small angle. For a negative uniaxial crystal and type-I phase-matching, this angle is given by

........(1)

The angle has the effect of limiting the effective crystal length over which harmonic generation can take place. The beams completely separate at a distance of order
........(2)
called the aperture length where D is the beam diameter. Of course, at only a fraction of this distance the reduction of conversion efficiency because walk-off becomes noticeable and has to be taken into account. For weakly focused Gaussian beams, the aperture length can be expressed as
...... (3)
where w₀ is the fundamental beam radius.