Transverse and longitudinal structure of bessel vortex beam solutions to Maxwell´s equations

Solutions to Maxwell´s equations show that localized waves such as vortex waves, knotted waves, and linked waves, may exist. Such waves exhibit orbital angular momentum (OAM) and can be derived from Maxwell´s equations in the frequency or time domains by classical techniques. Here we consider vortex waves using the frequency-domain solution of the vector wave equation in cylindrical coordinates, absent circular waveguide boundary conditions. Bessel waves are doubly indexed, nondenumerable, and overcomplete, i.e. contain denumerable orthonormal subsets of modes. Axial phase velocity is superluminal, i.e. exceeds c, while energy velocity is subluminal, i.e. less than c. We conclude that photons having OAM need not travel on straight line paths, and consequently Einstein´s c is merely an upper bound on the speed of wave propagation in free space.