Second-harmonic-generation rings and refractive-index measurement in uniaxial crystals

Processes of ring formation due to phase-matched noncollinear second-harmonic generation involving an unfocused laser beam in birefringent uniaxial crystals are examined. First-order calculations are presented of the generation of a cone of second-harmonic radiation surrounding the main beam due to off-axis phase matching in a birefringent uniaxial crystal, and it is noted that for a type II second harmonic generation crystal, up to three rings corresponding to second-harmonic cones may be observed. Expressions for the ring position and radius obtained as functions of crystal refractive indices and geometry are shown to be in agreement with experiments performed on type I and type II crystals. It is noted that the values of the ring parameters are very sensitive to index mismatch, implying that measurements of ring centers and radii may be used for precise relative determinations of crystal refractive indexes. A second-order calculation is then performed for use in such calculations. It is noted that the results may also be used in the maintenance of optimized second-harmonic generation without the use of optics in the beam.