We demonstrate a type of singular beam that accelerates along a parabolic trajectory and has a cross-section intensity pattern exhibiting a dark central region surrounded by multiple rings with the innermost (main) ring resembling an equilateral triangle. The key to creating such beams is to replace the standard triangle with a rounded one, made up of six circular arcs connected end to end. The individual input phase mask for each arc can be analytically computed, and the whole input phase mask for the beam is thus obtained by piecing together these individual phases. Furthermore, the continuity of field forces of these triangle-like modes is discrete; that is, an index similar to the topological charge of vortex beams arises. Numerical results show that the energy flow in the beam’s cross section circulates around the dark center along the triangle-like main ring, suggesting a possible application in orbiting particles along an irregular path. 

Rays model of a self-bending triangle-like singular beam. (a) Light rays starting from a closed loop in the input plane pass tangentially through a triangle on the surface of a hollow tube. (b) Geometric configuration of the tube’s cross section.