Starting from Maxwell equation in presence of a dielectric, non magnetic we re-derived the differential equation for the pulse envelope in presence of linear polarization in a slightly different way.
We then extended to a non linear term, considering the specific case of second order. The most general case is here a three waves mixing, approximating one beam (the pump) much more intense then the others, the signal (initially weak) and the idler (initially empty).We derived the gain expression in the case of perfect phase matching and in the general case, providing the acceptance bandwidth of the signal for given thickness, non linearity and group velocity mismatch. I commented on the need of having large bandwidth to preserve short pulse durations, but also on the opportunity to use long crystals to obtain efficient spectral filtering for example via second harmonic generation (in which w1=w2 and it acts as a pump, while w3 is the generated signal).
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