Some problems of optimum focusing in the process of second harmonic generation in nonlinear crystals. Part 1. Mathematical apparatus

The problem of the second harmonic generation (SHG) of monochromatic spatially coherent paraxial radiation in homogeneous uniaxial quadratically nonlinear crystals upon focusing of a beam into a crystal by two crossed cylindrical lenses with different focal lengths is considered theoretically. The aim of the investigations was to determine optimum focusing parameters as functions of the main radiation power. The first part of this paper presents a mathematical apparatus necessary for carrying out the planned investigations. A variant of transforming a system of nonlinear wave equations to a form more convenient for further use is proposed. Some important particular cases admitting simpler solutions are considered. An asymptotically exact analytical solution of the SHG problem is presented. © 2016, Pleiades Publishing, Ltd.