  We originally demonstrate the vortex beams with patterns of closed polygons
[namely polygonal vortex beams (PVBs)] generated by a
quasi-frequency-degenerate (QFD) Yb:CALGO laser resonator with astigmatic
transformation. The PVBs with peculiar patterns of triangular, square, and
parallelogram shapes carrying large orbital angular momentums (OAMs) are
theoretically investigated and experimentally obtained in the vicinity of the
SU(2) degenerate states of laser resonator. The PVBs in QFD states are compared
with the vortex beams with patterns of isolated spots arrays located on the
triangle-, square-, and parallelogram-shaped routes [namely
polygonalspots-array vortex beams (PSA-VBs)] under normal SU(2) degenerate
states. Beam profile shape of PVB or PSA-VB and OAM can be controlled by
adjusting the cavity length and the position of pump spot. The simulated and
experimental results validate the performance of our method to generate PVB,
which is of great potential for promoting novel technologies in particle
trapping and beam shaping.
