Support: Princeton University/US Army Research Office

Period of Performance: May 1, 2024 — April 30, 2026

Budget: $1,000,000 (CRSC share $300,000)

Summary: We plan to address a number of mathematical and computational questions related to the analysis of three types of electromagnetic fields: two different types of topological solutions known as electromagnetic knots and vortex knots, and also the so-called accelerating beams. All these solutions exhibit some unusual features that may prove useful for various applications if harnessed properly. Topological solutions to Maxwell’s equations are characterized by a twisted or knotted structure. They may be different on the substance, but still share a common terminology. Topological solutions of the first type are the so-called electromagnetic knots (e.g., Hopfions) in free space. They can be interpreted as standing waves. Solutions of the second type are (optical) vortex knots, which can be thought of as traveling waves. As concerns the accelerating waves (beams), these are solutions with curved caustics that allow for an interpretation as an accelerating propagation trajectory. A key objective of the proposed project is to investigate analytically and numerically the fundamental properties of topological solutions of both types and understand how and to what degree they may be related to one another. Another important objective is to build a more formal mathematical framework for accelerating waves and extend their analysis from optical to radio frequencies and microwaves. Of considerable interest is the capability to synthesize and manipulate the near-field topological solutions and accelerating beams, as the currently available approaches do not amount to a systematic methodology. An important potential application is in the communications theory where the various twisted modes and/or knotted electromagnetic configurations may present an additional degree of freedom allowing to distinguish between different signals transmitted via the same channel. Moreover, the curved trajectories of accelerating waves have the potential of circumventing the obstacles. To allow for the analysis of topological and accelerating solutions, the existing numerical capacity for the simulation of Maxwell’s equations will need to be enhanced. The corresponding effort is also a part of the proposed project. The theoretical and computational approach that we propose to pursue in this work employs Calderon’s operators. It will offer new advances in understanding the topology of optical vortices and their relation to knotted electromagnetic fields, mathematical representation of topological solutions as standing vs.\ traveling waves, efficient generation of the various knotted solutions and accelerating waves, in particular, non-paraxial accelerating waves, and potential benefits they may offer for applications in telecommunications and remote sensing.