Propagation of interacting first and second optical harmonics in a nonlinear active medium

A theoretical study is made of the propagation of coupled plane monochromatic first (ω₁) and second (ω₂) harmonics in a nonlinear active medium combining the functions of generation of laser radiation of frequency ω₁ and of doubling of this frequency. The wave mismatch is not restricted. The cases of homogeneous and inhomogeneous broadening of a luminescence line are discussed. Equilibrium (asymptotic) solutions are found and their stability is investigated. One or two stable equilibrium solutions can be obtained, depending on the wave mismatch and on the pumping rate. Transient conditions and establishment of asymptotic solutions are studied. It is shown that wave mismatch can be used to optimize the generation or amplification of a second harmonic in a nonlinear active medium.