Transform relationship between Kerr-effect optical phase shift and nonuniform electric field distributions

Electric field distributions measured using the Kerr effect cause a phase shift between light components polarized parallel and perpendicular to the electric field, proportional to the magnitude squared of the electric field components in the plane perpendicular to light propagation integrated over the light path length. One wishes to recover the electric field distribution from measurements of light phase shifts. For axisymmetric geometries where the electric field depends only on the radial coordinate and whose direction is constant along the light path, as is the case along a planar electrode, the total phase shift for light propagating at a constant distance from the center of symmetry and the electric field distribution are related by an Abel transform pair, a special case of Radon transforms typically used in image reconstructions with medical tomography and holography. The more general Radon transform relates the optical phase shift to non-axisymmetric electric field distributions but is restricted to cases where the applied electric field is perpendicular to the plane of light propagation. If the applied electric field direction changes along the light path, it becomes necessary to account for the change in direction of the light components parallel and perpendicular to the applied electric field and the light polarization equations are generalized