A planar metal-dielectric multilayer whose dielectric layer thicknesses act as coordinates of a generalized parameter space, so that stacking additional nickel and dielectric layers raises the dimensionality of that space beyond the two thicknesses of the original gradient-thickness optical cavity. The nickel layers act as partially reflecting mirrors coupling the dielectric layers as resonators, and whether all of them participate sets an effective dimension that determines the topological phase. Zeros of the complex reflection coefficient, computed by the transfer matrix method, form vortical structures in that space: infinitely stretched vortex lines, closed vortex rings in three dimensions, and vortex sheets, cylinders and manifolds in four. Liquid-crystal layers whose optical path length is tuned by an applied bias supply electrically scannable synthetic dimensions, so electro-optic tomography projects two-dimensional slices of the parameter space into real space and the projected optical vortices trace out trajectories set by the geometry of the underlying structure - translational, zig-zag, or the creation and annihilation of vortex-antivortex pairs. A topological phase diagram in the nickel-thickness coordinates classifies trivial, vortex-line and vortex-ring phases, and the transitions between them are confirmed experimentally across ten fabricated samples.
The dimensionality of vortical structures has recently been extended beyond two dimensions, providing higher-order topological characteristics and robustness for high-capacity information processing …