Quantitative determination of fractional topological charge based on the rotational Doppler effect.

The utilization of fractional-order vortex beams extends the diversity of optical field manipulation, permits for more flexible control over beam propagation, and provides novel applications in optical communications, edge enhancement imaging, and particle manipulation. However, compared with the integer-order vortex beams, the topological charge measurement techniques for fractional-order vortex beams are not well developed, impeding the further exploration of its applications. In this paper, the frequency signal of rotational Doppler effect and corresponding broadening behavior under the fractional-order vortex beam illumination were analyzed. When the fractional topological charge approaches a half integer, the broadening is minimized. Leveraging this relationship, we designed a phase-compensated scheme coupled with signal-to-noise ratio detection to realize the real-time fractional topological charge measurement. The single pixel photodetector was used and eliminated the need for two-dimensional image acquisition and analysis, ensuring efficient acquisition and quantitative analysis. Both theoretical and experimental results confirm the feasibility of this method, thereby advancing the comprehension of the optical Doppler effect and potentially paving the way for future investigations into fractional vortex beams.