Fig. 2: Sculpting streamlines for adjustable self-similarity and customized self-acceleration.

a, b Bessel beams with adjustable self-similarity, exhibiting self-shrinking a and self-stretching b radii; red curves indicate the energy streamlines along the vortex mainlobe. c, d Bessel beams with customizable self-accelerating dynamics, propagating along a parabolic c and a spiral d trajectory; the yellow dashed lines denote the optical axis. e–g Intensity maps of (a–c) in the y–z plane; red dashed curves denote the vortex mainlobes. h Three-dimensional intensity iso-surface corresponding to (d). The streamlines herein are directly drawn from the streamline function with the distributions of Poynting vector calculated in MATLAB, which are consistent with the analytical solutions from the hydrodynamic differential equations in this work. i–l The corresponding momentum-space angular-spectrum distributions of (a–d) calculated by four-step streamline-engineering approach in Methods. Amp., amplitude; Pha., phase; arb. u., arbitrary units.