Phase and transport velocities in particle and electromagnetic beams
In a coherent monoenergetic beam of non-interacting particles, the phase velocity and the particle transport velocity are functions of position, with the strongest variation being in the focal region. These velocities are everywhere parallel to each other, and their product is constant in space. For a coherent monochromatic electromagnetic beam, the energy transport velocity is never greater than the speed of light, and can even be zero. The phase velocities (one each for the non-zero components of the electric and magnetic fields, in general) can be different from each other and from the energy transport velocity, both in direction and in magnitude. The phase velocities at a given point are independent of time, for both particle and electromagnetic beams. The energy velocity is independent of time for the particle beam, but in general oscillates (with angular frequency 2w) in magnitude and direction about its mean value at a given point in the electromagnetic beam. However, there exist electromagnetic steady beams, within which the energy flux, energy density and energy velocity are all independent of time.