Complete pump depletion by autoresonant second harmonic generation in a nonuniform medium

In this paper, we develop for the first time to our knowledge an analytical theory of second harmonic generation (SHG) in a generic nonuniform χ2 medium. It is shown that by varying the properties of the medium gradually enough, the system can enter an autoresonant state in which the phases of the fundamental pump and of the generated second harmonic wave are locked. The effect of autoresonance allows efficient transfer of energy between the waves and, due to the continuous phase-locking in the system, all the energy of the pump could be converted to the second harmonic. Simple closed-form expressions for the waves amplitudes as a function of the longitudinal coordinate are derived, and an explicit criterion for the stability of the autoresonant state is obtained. Our analytical theory is compared to the numerical solution of the coupled mode equations, which are found to be in excellent agreement with each other. The analytical closed-form expressions that we derive could be very useful for practical design of SHG devices with increased performances, such as highly efficient, wideband frequency converters