Second-Harmonic Generation of Light by Focused Laser Beams

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Abstract
An experimental and theoretical study is reported on second-harmonic generation (SHG) by focused laser beams in nonlinear crystals which are large compared with the extent of the focus. The gas-laser beam is in the lowest order (Gaussian) mode and very close to index-matching conditions in the crystal. The intensity pattern of the SHG observed photographically has roughly the shape of a half moon with a very sharp edge. The distribution on the bright side of the edge has been measured using a traveling slit and photomultiplier detector. Fine structure has been observed consisting of a series of fringes extending from the edge into the bright region. The nature of the fine structure depends upon where the pattern is observed. The pattern at the surface of the crystal has fringes at the positions one would expect for diffraction of a second-harmonic beam by a straight-edged screen placed at the focus. The pattern at great distance from the crystal has uniformly spaced fringes; in this case fringes have also been seen on the dim side of the edge. It has been shown that the position of the edge relative to the axis of the laser beam is sensitive to extremely fine adjustments of the crystal orientation. The power has also been measured as a function of crystal orientation, and the position of the edge has been determined for the orientation giving maximum power. The theoretical treatment of the problem, which occupies most of the paper, is based upon an exact formula for the second-harmonic field for the case in which the laser beam is in a Guassian mode. In a far-field approximation this formula is decomposed into three types of terms having different dependence upon distance. The leading term in the limit of large distance has a sharp edge and describes the gross behavior on the bright side of the edge. The other terms are associated with fine structure of the Fresnel and Fraunhofer types arising from the focus and the crystal surfaces, respectively. The theory contains a parameter to specify the phase-matching conditions which can be simply related to changes in the crystal orientation. The power is obtained as a function of this parameter, and the optimum matching condition corresponding to maximum power is thereby determined. A detailed analysis of the experimental data is presented which shows the theory to be in quantitative as well as qualitative agreement with experiment.