Some exact solutions for light beams

We give an infinite class of exact analytical solutions for monochromatic light beams with strong focusing. As the solutions do not contain integrals, they are easy to explore compared with diffraction-theory results for strongly focused light. All monochromatic beams can be decomposed into two standing waves, each proportional to a Hilbert transform of the other. This means a beam can be built from any standing wave and our class is derived using this procedure. We give a visual overview of some of the beams, which reveals many interesting energy and field structures, including vortices in the energy flow, angular momentum in the propagation direction, and knotted field lines. We also show how the method can be used to design beams with an arbitrary focal shape. (paper)