Theory of Second Harmonic Generation of Light

A theory is given for the generation of second harmonic light in crystals analogous to the classical theory of crystal optics. Optical activity and absorption are neglected, but the anisotropy is treated rigorously. An approximate treatment is also given, valid for most crystals, which retains the essential effects of the anisotropy and neglects the minor effects. All coherence effects are treated exactly, and special attention is given to the case of index matching. An exact solution is given for the second harmonic waves produced in and outside the medium when the fundamental beam enters the medium at a plane surface. A brief discussion is given of the effects of the finite aperture of the beam. The theory is applied to recent experiments on KDP and quartz, and values are given for the second-order polarization coefficients of these materials. Finally, the theory is given for second harmonic generation by focused beams based upon the diffraction theory of an ideal focus.