Macroscopic phase-matching mechanism for orbital angular momentum spectra of high-order harmonics by mixing two Laguerre-Gaussian vortex modes

The high-order harmonic generation (HHG) by the interaction of an intense vortex infrared laser and a gas medium has been employed to produce the extreme ultraviolet (XUV) light beam carrying a high value orbital angular momentum (OAM). Here we extend this approach to generate the XUV light with superimposed OAM modes by mixing two different Laguerre-Gaussian (LG) modes as the driving beam. By solving the three-dimensional Maxwell's wave equations of the high-harmonic field and calculating the coherence lengths of HHG, we reveal the complex nature of phase-matching conditions in the gas medium, especially their dependence on the azimuthal angle. We demonstrate that the OAM spectra of the HHG can be tailored by varying the harmonic order, the position of gas medium, and the energy ratio of two LG beams. We also uncover that only including short-trajectory contribution in the single-atom response could broaden the OAM spectrum of the macroscopic HHG because of the featured phase-matching characteristics caused by the atomic intrinsic phase. We anticipate this work to stimulate some interests in generating an XUV vortex beam with the tunable OAM spectrum through the gaseous HHG process as well as in separating different OAM modes of an XUV vortex beam by advanced techniques.