Far-Field Topological Structure of the Second Harmonic from Higher-Order Poincaré Sphere Beam

In this paper, the far-field topological structures (FFTSs) of the second harmonic (SH) from higher-order Poincaré sphere (HOPS) beams, including circularly polarized vortex beams (VBs), cylindrically vector beams (CVBs) and elliptically polarized CVBs (EPCVBs), were demonstrated and reported. To begin with, the hidden FFTSs of the SH after propagating the twice Rayleigh range were simulated based on the vectorial coupled wave equations and the Collins formula. Then, the experimental setup was established to achieve the SH from the HOPS by applying two orthogonal 5% MgO: PPLN crystals, the FFTSs of which were demonstrated. The theoretical and experimental results indicate that for the circularly polarized VBs, the FFTSs of the SH still exhibit the 135°-linearly polarized VBs, which is similar to that of the SH in-source plane, because the SH is the eigen-mode of free space, while for the CVBs, the FFTSs of the SH generally show the disappearance of the central dark core, replaced by the maximum light intensity at the center due to the topological phase transition during propagation. Especially of note, for the EPCVBs, the FFTSs of the SH display the maximum light intensity at the center, but the FFTSs in the horizontal and vertical directions reveal rotational symmetry related to the chirality of the EPCVBs. The results reveal the evolution mechanisms of the SH from the HOPS in the far field, which may facilitate the applications of the SH from HOPS beam.