Nonlinear dynamics and extreme events were explored in a spatially distributed polarization laser based on an azimuthally polarized Nd:GdVO₄ laser with pump modulation. When the modulation frequency approached the relaxation oscillation frequency, a period-doubling route to chaos was observed with an increasing modulation depth, and spatiotemporal extreme events occurred with the generation of the chaos. An efficient azimuthal polarization was kept, even though the intensity fluctuated in the chaotic region.

Differing from the standing-wave mode, which is usually observed in a linear cavity configuration, a type of geometric ring mode can self-sustain in a multi-element linear cavity with axial pumping. When the configurations are adjusted in a way that the marginal boundary of the stable region is associated with the self-imaging condition, the off-axis aberration and symmetric breaking with respect to the optical axis results in the existence of a ring trajectory within the cavity. Trigonometrical ray tracing based on numerical simulations showed good agreement with the experimental results.

A period-doubling route to chaos was observed in a passively Q-switched Nd:GdVO4 laser beam with azimuthal polarization. At a period of two, the peak intensities gradually varied with the azimuthal angle. When the pump power was increased, the chaotic behavior was spatiotemporal, but the polarization was still azimuthal. To the best of our knowledge, this is the first report of spatiotemporal chaos under a stationary spatial distribution of the polarization.

Considering the birefringence of the gain medium, we propose a selection rule for designing cavity configurations that enable the generation of radially and azimuthally polarized laser beams with a low beam quality factor, i.e., high beam quality. Using this rule, all stable regions can support cylindrical vector (CV) beams by using the position of the end mirror as the tuning parameter to vary the cavity configuration. Such cavity configurations tend to sustain the fundamental or lowest-order CV beam, and radially or azimuthally polarized beams can be obtained simply by varying the tuning parameter. Based on experimental measurements of the beam quality factor and polarization characteristics, we verified our analyses and simulations for a four-element laser system.