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Structured light with more controllable degrees-of-freedom (DoFs) is an exciting topic with versatile applications. In contrast to conventional vector vortex beams (VVBs) with two DoFs of orbital angular momentum (OAM) and polarization, a hybrid ray-wave structure was recently proposed [Optica 7(7), 820-831 (2020)], which simultaneously manifests multiple DoFs such as ray trajectory, coherent state phase, trajectory combination, besides OAM and polarization. Here we further generalize this exotic structure as the astigmatic hybrid VVB by hatching a new DoF of astigmatic degree. Importantly, the transverse topology varies with propagation, e.g. a linearly distributed hybrid trajectory pattern can topologically evolve to a circularly polygonal star shape, where the number of singularity changes from zero to multiple in a single beam. The propagation-dependent evolution can be easily controlled by the astigmatic degree, including as a vector vortex state such that different astigmatic trajectories have different polarizations. We experimentally generate such beams from a simple laser with a special astigmatic conversion by combined spherical and cylindrical lenses, and the results agree well with our theoretical simulation. With our new structured light, the propagation-multiplexing multi-DoF patterns can be controlled in a single beam, which can largely extend related applications such as high-dimensional large-capacity optical communication, laser machining, and particle trapping.

We propose a new method to determine the wavefront of a laser beam based on modal decomposition by computer-generated holograms. The hologram is encoded with a transmission function suitable for measuring the amplitudes and phases of the modes in real-time. This yields the complete information about the optical field, from which the Poynting vector and the wavefront are deduced. Two different wavefront reconstruction options are outlined: reconstruction from the phase for scalar beams, and reconstruction from the Poynting vector for inhomogeneously polarized beams. Results are compared to Shack-Hartmann measurements that serve as a reference and are shown to reproduce the wavefront and phase with very high fidelity.