A closed-form design rule for photonic hopfions of arbitrary order

Abstract

Topological textures of light are robust against perturbation and are emerging as carriers of information and probes of light-matter interaction. The most intricate of them, the photonic hopfion, in which the loops of different polarization states are linked, has so far been assembled from chosen superpositions of laser modes, with weights set by hand or found by numerical optimization. Here we replace this search with a design rule. Two scalar beams, an optical vortex of order $\ell$ and a field with n dark rings, propagate in closed form, and their phases on one plane are the two masks of a metasurface that controls two orthogonal polarizations. Although a beam, unlike the closed spaces of topology, extends to infinity, we prove that it is a hopfion of order $n\ell$ whenever three algebraic conditions on the ring coefficients and the vortex order hold, and that the polarization loops of every state that the far field does not reach then link exactly $n\ell$ times; each condition guards against one way in which the topology can escape. The order is thus chosen on paper and the device follows directly, without optimization. In scalar simulation, one terahertz metasurface geometry produces hopfions up to order nine, and a full-wave simulation of the complete first-order device preserves its topology. The rule applies to any platform that shapes the phases of two orthogonal polarizations.