An Electromagnetic Plane Wave in the Spacetime of a Plane Gravitational Wave

I find nearly plane-wave solutions for the Gauss-Ampere law for the 4-vector potential, subject to the Lorenz gauge condition, in the spacetime of a plane gravitational plane wave. I assume that the gravitational wave is weak, in the sense that the dimensionless strain amplitude h is much less than 1. I find a solution for the homogeneous scalar wave equation in this spacetime, and then find a 4-vector potential that solves the Gauss-Ampere law and Lorenz gauge condition in the absence of sources. The solutions are plane waves in Minkowski spacetime, plus additional scattered waves of order h. The problem is analogous to diffraction from a transmission grating, or Brillouin scattering from sound waves in matter. The corrections solve the inhomogeneous wave equation in Minkowski spacetime, with a "distributed source" of order h comprised of terms arising from the non-Minkowski metric and the zero-order solution. The scalar wave solution requires two scattered waves, which can be combined to form a phase correction h phi that varies at the gravitational-wave frequency. This phase correction yields the same time delay and deflection at the observer as for propagation along null geodesics, in the ray approximation. The electromagnetic-wave solution requires four scattered waves. Two correspond to the phase correction h phi found for the scalar field. The other two scattered waves introduce amplitude and polarization changes of order h to the electromagnetic wave. The time delay and deflection match those for the scalar waves. The solution predicts variations of the intensity of the electromagnetic wave of first order in h, at the wavenumber of the gravitational wave. These arise from interference of the first-order scattered waves and the zero-order solution. I briefly discuss possible observations of this effect.