Propagation dynamics and X-pulse formation in phase-mismatched second-harmonic generation

This paper concerns the theoretical, numerical, and experimental study of the second-harmonic-generation (SHG) process under conditions of phase and group-velocity mismatch and aims to demonstrate the dimensionality transition of the SHG process caused by the change of the fundamental wave diameter. We show that SHG from a narrow fundamental beam leads to the spontaneous self-phase-matching process with, in addition, the appearance of angular dispersion for the off-axis frequency components generated. The angular dispersion sustains the formation of the short X pulse in the second harmonic (SH) and is recognized as three-dimensional (3D) dynamics. On the contrary, the large diameter fundamental beam reduces the number of the degrees of freedom, does not allow the generation of the angular dispersion, and maintains the so-called one-dimensional (1D) SHG dynamics, where the self-phase-matching appears just for axial components and is accompanied by the shrinking of the SH temporal bandwidth, and sustains a long SH pulse formation. The transition from long SH pulse generation typical of the 1D dynamics to the short 3D X pulse is illustrated numerically and experimentally by changing the conditions from the self-defocusing to the self-focusing regime by simply tuning the phase mismatch. The numerical and experimental verification of the analytical results are also presented.