Theory of backward second-harmonic generation of short laser pulses in periodically and aperiodically poled nonlinear crystals

We study numerically and analytically the backward second-harmonic generation in periodically and aperiodically poled nonlinear crystals in the quasiphase-matched regime. In our numerical analysis, we apply the optimum control technique based on the shooting method. Due to the fast and efficient numerical technique, we are able to analyze this frequency-conversion process in detail, taking into account various factors – the pump intensity, group-velocity mismatch and dispersion, linear phase mismatch, and “chirp” of nonlinear grating. Also we derive several approximate solutions for the backward harmonic efficiency employed in the undepleted pump approximation.