Enhancement and redshift of vortex harmonic radiation in epsilon-near-zero materials

The harmonic radiation from a vortex laser field interacting with an epsilon-near-zero (ENZ) material is numerically investigated via solving the Maxwell-paradigmatic-Kerr equations. For a laser field of long duration, the harmonics up to the seventh-order can be generated with a low laser intensity (-109 W/cm2). Moreover, the intensities of high order vortex harmonics at the ENZ frequency are higher than at other frequency points due to the ENZ field enhancement effects. Interestingly, for a laser field of short duration, the obvious frequency redshift occurs beyond enhancement in high order vortex harmonic radiation. The reason is that the strong change of the laser waveform propagating in the ENZ material and the non-constant field enhancement factor around the ENZ frequency. Because the topological number of harmonic radiation is linearly proportional to its harmonic order, the high order vortex harmonics with redshift still possess the exact harmonic orders indicated by the transverse electric field distribution of each harmonic.