Periodic-trajectory-controlled, coherent-state-phase-switched, and wavelength-tunable SU(2) geometric modes in a frequency-degenerate resonator

We report a method to realize the periodic trajectory controlling, coherent-state phase switching, and wavelength tuning of SU(2) geometric modes (GMs) in a frequency-degenerate resonator (FDR). FDR is a resonator in which the ratio of transverse and longitude modes frequency spacing is a simple rational number, which would lead to a laser wave-packet in a SU(2) coherent state related to a periodic ray trajectory. We demonstrate that the periodic trajectory can be switched with coherent-state phase changing from 0 to π (or π to 0) by simply controlling the position of gain medium in the same FDR. For the period-of-four state, the geometries are switched between "W" and "M" shapes. For the period-of-three state, those are switched between "IV" and "VI" shapes. Moreover, because a special crystal Yb:CALGO with extremely broad emission band is used as the gain medium, our SU(2) GMs have the wavelength-tunable property in contrast to the conventional single-frequency GMs. The center wavelength can be tuned with the range of about 10 nm by adjusting the pump power. These effects can further enrich the various applications of structured light manipulation.