We derived the solution in the general case of non perfect phase matching, 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). We understood how to realize the phase matching in the collinear and non collinear case, dealing with walk off. In the non collinear case it is possible to amplify a wide bandwith with a wise choide of the incidence geometry and cristal orientation.
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