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Metasurfaces enable the shaping of wavefronts in arbitrary ways, by dispersion engineering of the meta-atoms. Here I will review my group research on singularity engineering, including arrays of equally spaced 0D singularities, 2D singular sheets of arbitrary shapes and topoiogically protected singularities in the 4D space encompassing real space and wavelength. In the latter when a perturbation is added to the metasurface the singularity in the focal spot is preserved but shifts in wavelength, signaling topological protection.
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Federico Capasso, Soon Wei Daniel Lim, Christina M. Spaegele, "Singularity engineering with metasurfaces: from 0D to 4D," Proc. SPIE PC12897, High Contrast Metastructures XIII, PC1289701 (13 March 2024); https://doi.org/10.1117/12.3004234