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On the relativistic constraint dynamics of electromagnetic beams

Suppose that for a continuous wave electromagnetic beam the detector is moving relative to the beam waist. For a relativistic treatment of this problem, a Bateman-Hillion solution to Maxwell's equations that takes account of the 4-position of the detector relative to the beam waist is used to represent Gaussian beams. The exact forms of the electric and magnetic fields in the beam are calculated alongside the Poynting vector. This method is shown to produce results that preserve their forms under Lorentz transformations. These results also correspond to solutions of the paraxial wave equation for nearly parallel beams providing the relative time coordinate is set it equal to the axial distance between beam and detector divided by the speed of light. The form of the constraint on the relative coordinates beyond the paraxial limit is uncertain but it is concluded more can be learned by comparing the predictions of the model to experimental data.