Generation and Free-Space Transmission Characterization of Tunable Focusing Vortex Beams

In this paper, we introduce a new type of optical beams capable that can auto-focusing multiple times in free space, namely, a tunable focusing vortex beams (TFVBs). The TFVBs are generated by controlling the radial phase distribution to generate multiple vortex beams with pre-determined convex trajectories. After tailoring and overlapping of multiple vortex beams with different predetermined convex trajectories, thus enabling an on-axis multi-focus property. At the same time, we investigate the effect of different OAM states (<tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\boldsymbol{l}=\pm \mathbf{1},\ \pm \mathbf{2}$</tex> and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\pm\mathbf{3}$</tex>) on the focusing. Realize the controllable position of the focusing point and the controllable number of focusing points in the process of dissemination. we expect that this work may bring auto-focusing optical beams further potential applications.