  Optical singularities play a major role in modern optics and are frequently
deployed in structured light, super-resolution microscopy, and holography.
While phase singularities are uniquely defined as locations of undefined phase,
polarization singularities studied thus far are either partial, i.e., bright
points of well-defined polarization, or unstable for small field perturbations.
We demonstrate for the first time a complete, topologically protected
polarization singularity; it is located in the 4D space spanned by the three
spatial dimensions and the wavelength and is created in the focus of a cascaded
metasurface-lens system. The field Jacobian plays a key role in the design of
such higher-dimensional singularities, which can be extended to
multidimensional wave phenomena, and pave the way to novel applications in
topological photonics and precision sensing.
