  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.
