What is the best beam for second harmonic generation?

Summary form only. Efficiency optimization of second harmonic generation (SHG) with freely propagating beams in bulk nonlinear crystals is of primary importance in a number of applications. In one of the early works on SHG Boyd and Kleinman found the gaussian beam that maximizes the efficiency as a result of the best compromise between the beam focalization, which enhances the efficiency, and the diffractive spread, which reduces it. More recently it has been suggested that "non diffracting" Bessel beams could generate second harmonic more efficiently than Gaussian ones, while other authors disagree with this conclusion. In any case, a question remains open: what is the beam that, for given crystal and beam power, gives the absolute maximum of the SHG efficiency. In this work we answer that question. To this purpose an arbitrary beam of finite power is expanded as a sum of Laguerre-Gauss modes and the SHG efficiency is calculated assuming negligible pump depletion. The values of the expansion coefficients and phase mismatch that give the absolute maximum of the SHG efficiency are then precisely calculated with a numerical procedure. The optimum beam has a plane wave front in the center of the crystal and a bell-shaped profile with a weak intensity ring.