Exact solutions of Maxwell's equations representing light beams are explored. The solutions satisfy all of the physical requirements of causal propagation and of energy, momentum and angular momentum conservation. A set of solutions can be found from a proto-beam by an imaginary translation along the beam direction. The proto-beam is given analytically in terms of the Bessel functions J0, J1 and the Lommel functions U0, U1, or equivalently in terms of products of the spherical Bessel functions and Legendre polynomials. The complex wavefunction has rings of zeros in the focal plane. Localization of the focal region is to within about one half of the vacuum wavelength.
- Received July 4, 2016.
- Accepted November 2, 2016.
- © 2016 The Author(s)
Published by the Royal Society. All rights reserved.