Second-Harmonic Generation of Light with Double Refraction

Experimental and theoretical investigations are reported on the second-harmonic generation (SHG) of light in crystals long enough so that SHG is limited by double refraction. This effect has been called the aperture effect. SHG has been observed in ammonium dihydrogen phosphate (ADP) under matching conditions using the highly parallel beam from a gas laser. Crystals of lengths 5 and 10 cm were used. It was observed that the peak of the SHG is displaced from the peak of the fundamental by a few millimeters. This is the order of magnitude to be expected, since the fundamental is an ordinary ray and the SHG is an extraordinary ray. Since the position of the SHG peak should depend upon absorption, the absorption coefficients of ADP for the fundamental (${\lambda }_1=1.1526\mathrm{}$ μ) and the second harmonic (${\lambda }_2=0.5763\mathrm{}$ μ) were measured and found to be ${\alpha }_1=0.151\mathrm{}$ ${\mathrm{cm}}^{-1\mathrm{}}$, ${\alpha }_2=0.024\mathrm{}$ ${\mathrm{cm}}^{-1\mathrm{}}$; the latter value may represent small-angle scattering rather than true absorption. The matching angle, the angle between the beam and the optic axis for index matching, was measured and found to be ${\theta }_m=42.7\mathrm{}°$. Two kinds of theory are presented. The experiments are first discussed in terms of a heuristic theory which relies upon intuition and plausible physical arguments. The experiments are shown to be in quantitative agreement with this theory. A formal theory is also given for SHG under matching conditions by beams of finite aperture taking into account double refraction and absorption. For parallel beams this theory reduces to the heuristic theory.