Tailoring light intensity along caustic trajectories

A current challenge in a caustic beam design is to tailor the intensity distribution along the curved trajectory. To address this matter, we present a robust theoretical framework that relates the propagated complex wave field amplitude with the input spectral signal encoded onto a spatial light modulator which is suitable for fold-type monotonic trajectories as well as for cusp-type nonmonotonic trajectories. Specifically, we derive a general closed-form expression that relates the field amplitude along the beam trajectory with the spectral amplitude and the third derivative of the spectral phase for both monotonic and nonmonotonic curved trajectories. This proposal is suitable for direct experimental implementation in a Fourier transform scheme around the focal region, allowing straightforward beam intensity design by selecting the proper spectral amplitude and phase while preserving the beam trajectory. Experimental results from the famous cubic spectral phase support the theoretical predictions. This research lays the foundation for engineering the intensity of curved beams, which can be useful in applications where a specific modulation of the intensity is required over specific regions of the trajectory such as in optical trapping and laser micromachining.