Seeing Trajectory Without Imaging by Exploring Both Radial Momentum and Orbital Angular Momentum of Light

We report an experiment of visualizing the transversely moving object's trajectory in real time, regardless of not using any imaging components. Our strategy explores both the degrees of freedom of orbital angular momentum and radial momentum of light. We not only resolve the problem regarding the canonical definition of radial Doppler effect associated with radial momentum, but also develop the simultaneous radial-angular-momentum-enabled Doppler metrology. Based on the polarization-decoupled Doppler metrology and the Hilbert–Huang transform, we can reveal the instantaneous radial and angular velocities of the moving object, and thus tracing its transverse motion. In our experiment, we succeed in measuring the radial and angular frequency shifts for a nutation-like motion and reconstructing the real-time trajectory for a spiral motion, respectively. Our method of “seeing” the motion trajectory, even without any imaging component, will trigger new optical sensing applications when other degrees of freedom of light are fully explored.