LAUSR

By solving the (2 + 1) dimensional Schrödinger equation in free space, we find that the autofocusing off-axis hollow vortex Gaussian beams (HVGBs) are analytically derived for the first time. The off-axis HVGBs can be adjusted by changing the off-axis coordinate (x0,y0) and topological charge n2. In particular, by increasing the off-axis coordinate (x0,y0), the self-focusing intensity can be increased. Besides, the self-focusing property can be more obvious. Furthermore, by increasing the hollow order n1, we can deepen the depth of focus, make the focus position further away, and increase the self-focusing intensity too. We also discuss other propagation properties that are used to enrich the autofocusing off-axis HVGBs, such as Poynting vector, angular momentum, gradient force, and maximum scattering force during propagation.