Structured electromagnetic pulses with temporally knotted polarization

By solution of the inverse problem of classical electrodynamics, we propose a construction of nearly arbitrary spatially localized structured electromagnetic pulses, including wave packets with temporal polarization singularities. The latter are based on the topology of electric field hodographs. They encompass topological knots of both finite and infinite complexity and polarization ribbons, including one-sided M"obius stripes of variable width. These electromagnetic wave packets are generated by the localized dynamics of positive and negative electric charges and currents in a zero-net-charge plasma. The admissible electric field distributions are constrained by the requirement that the field of the pulse's electric area, the time integral of the electric field, must be curl-free. Our results open new avenues for the complete spatiotemporal structuring of ultrashort electromagnetic pulses; generalizations for a wide class of vector fields of various natures are possible.