Three-dimensional coherent optical waves localized on trochoidal parametric surfaces

We theoretically demonstrate that the three-dimensional (3D) coherent laser waves formed by the degenerate Laguerre-Gaussian modes with different longitudinal indices are well localized on rotating trochoidal parametric surfaces. We further use a large-Fresnel-number laser system to realize the existence of the laser modes related to trochoidal coherent states. Experimental results reveal that the exotic laser modes generally originate from a superposition of two degenerate standing-wave trochoidal coherent states.