Time-dependent 3-D caustic beams over arbitrary trajectories

In this study, we introduce a method for synthesizing a time-dependent caustic beam along a generic beam-axis trajectory with arbitrary curvature and torsion. Our approach evaluates the pulsed aperture field that radiates the beam along a predefined trajectory by constructing a time-dependent caustic surface around the beam-axis skeleton. Initially, we derive the aperture field delay to form a caustic of rays along the beam-axis, extending this to other points over the aperture to construct a smooth caustic surface for points near the beam-axis. The amplitude is selected to confine the transverse (off-axis) beam. We also provide a theoretical analysis of the pulsed caustic beam propagation speed along the curved beam-axis. Finally, we present several numerical examples demonstrating the method's ability to synthesize aperture field distributions that generate pulsed beams propagating along trajectories with various curvatures and torsions.