Exploring the ellipticity dependency on vector helical Ince-Gaussian beams and their focusing properties

We present a numerical study of the intensity and polarization structure of vector helical Ince-Gaussian (VHIG) modes, which present a distinct subclass of vector Ince-Gaussian modes with defined parameter settings. The intensity profile of VHIG beams has an elliptic hollow structure, while the polarization distribution shows multiple single-charge polarization vortices arranged along a line. By selecting the mode order, phase factor and ellipticity of the VHIG beams, we can control the number of elliptic rings, the number of polarization vortices, and the topology of the vector singularity. Furthermore, we simulate the focusing properties of VHIG beams based on vector diffraction theory. Our results indicate that the ellipticity parameter of VHIG beams could be a valuable degree of freedom to generate attractive transverse profiles and longitudinal distributions under focusing, which may have implications for lithography, material processing, optical communication, and even optical trapping and manipulation.