Designing Optical Fields in Three Dimensions Using Source-Boundary Transformations

Recent advances in optical beam shaping have enabled a wide class of structured lights that follow exotic trajectories, opening many routes for numerous applications. Although the Helmholtz equation sets stringent conditions on the fields that can be physically realized, these constraints can be removed if sources can be introduced inside the medium. However, introducing sources is not always possible or practical. Here, we propose an analytical approach that transforms the effects of the required sources from the inside of the medium to its boundaries. The derived boundary conditions can then be used to generate any desired optical field (amplitude and phase). We demonstrate the feasibility of the approach by experimentally generating two optical fields inside a source-free simple medium, demonstrating control over the trajectories, phases, and intensities of the fields.