On the determination of the phases of electromagnetic scattering amplitudes from experimental data

The scattering of electromagnetic waves by systems of unknown charge distribution is considered with the aim of establishing that the phase of the scattered radiation (relative to the incident one) can be computed theoretically from experimentally obtainable data, such as the intensity and the polarization of the scattered wave. By means of the unitarity relation of classical electromagnetic scattering theory, it is demonstrated that, subject to certain restrictions on the magnitude of the scattering data: (i) For an unpolarized incident beam and a spherically symmetric target, knowledge of the differential cross section and the polarization of the scattered radiation is sufficient to determine uniquely the phases of the scattering amplitude tensor, (ii) For a parity-invariant scatterer, and fully circularly or linearly polarized incident beams, knowledge of the differential cross section and the ellipticity of the outgoing radiation is sufficient to completely determine the phases of the scattering amplitude tensor. The scattering data must be available for both the left and the right circularly polarized incoming wave (or for two linearly polarized beams pertaining to the two orthogonal polarizations), (iii) In the above cases it is shown that the phases can be calculated systematically by converging iterative procedures. Conditions for convergence are established, and bounds are provided on the errors associated with the approximants obtained after an arbitrary (but finite) number of iterations. On the basis of result (ii) with the same data assumed to be known, a formal solution to the phase determination problem is given for a general scatterer. However, for this case rigorous results on convergence conditions,and error bounds have not been obtained. An outline of the corresponding treatment valid within the framework of the quantum-electrodynamical theory of light scattering is given. To illustrate the proposed methods for phase determination, we apply them to the scattering of light by very simple systems. These include the free electron and an oscillating charge. We conclude by discussing the possibility of applying the above results to obtain solutions of the phase problem in systems of practical interest.