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

We present a detailed theoretical and numerical description of the second-harmonic generation process under phase-mismatched conditions, carefully analyzing different regimes of focusing, beam dimensions, positive or negative phase-mismatch, with presence or absence of group velocity mismatch and group velocity dispersion. It has been shown that in a sufficiently long medium the phase-mismatched SHG becomes self-phase-matched at a shifted frequency.