Spiral beams shaped like a planar curve with self-crossings

Spiral beams are structurally stable laser beams, which intensity remains the same within scale and rotation as they propagate. Last years, a family of spiral beams with intensity structure shaped like a planar curve without self-crossings has been constructed theoretically and realized in experiment. For closed curves and corresponding beams, a specific quantization condition has been found: the beam intensity is similar to the closed curve only for discrete values of square of domain bounding by the curve. In this work, spiral beams for planar curves with self-crossings are investigated. A condition of the intensity distribution to be homogeneous along the curve is found. Some results about the number and motion of phase singularities of a spiral beam for the curve with self-crossings are presented.