Fig. 2. Engineering desired caustics in light, demonstrating their propagation invariance and momentum conservation.

a–f Astroid. a Desired caustic. b 1D phase function. c Phase function mapped onto 2D Fourier ring. d Transverse projection of the rays. e, f Experimentally measured transverse intensity and phase. g–l Deltoid. m–r Cardioid. s Invariant propagation of the astroid from image (e), xz-cross-section through the intensity volume. t, u Invariant astroid with charge ℓ = 2. v–z Self-healing/momentum transfer in astroid: v, w Diffractive astroid, initial field with phase vortex of charge ℓ = 2. x Propagation showing diffraction and transfer to final OAM state. y, z Momentum transfer to final astroid state same as (e, f), which then propagates invariantly.