Self-phase matching and X pulse formation in phase-mismatched second harmonic generation due to dimensionality transition

Since 1961, when Franken and his collaborators experimentally discovered second-harmonic generation (SHG) [1], the process of SHG has been widely investigated leading first to important contributions by many authors, who derived the now well-known theory described in nonlinear optics books. It was realized long time ago that the use of short light pulses for SHG gives rise to special effects, like the broadening of the generated light pulse due to the group velocity mismatch between the fundamental (FH) and second-harmonic (SH) pulses, predicted by Comly and coworkers [2] and observed for the first time by Shapiro [3]. Typically SHG is investigated or used at or near the phase-matching condition, in such a way to maximize the conversion efficiency. More recently the study of propagation phenomena in phase-mismatched second-harmonic processes has attracted considerable interest, in particular for the investigation of the effect on the generated pulse shape. However, typically mentioned studies were done in the case when the plane wave approximation is valid and SHG could be recognized as one-dimensional problem.