  This paper is concerned with uniqueness, phase retrieval and shape
reconstruction methods for solving inverse electromagnetic source scattering
problems with multi-frequency sparse phased or phaseless far field data. With
the phased data, we show that the smallest strip containing the source support
with the observation direction as the normal can be uniquely determined by the
multi-frequency far field pattern at a single observation direction. A phased
direct sampling method is also proposed to reconstruct the strip. The phaseless
far field data is closely related to the outward energy flux, which can be
measured more easily in practice. We show that the phaseless far field data is
invariant under the translation of the sources, which implies that the location
of the sources can not be uniquely recovered by the data. To solve this
problem, we consider simultaneously the scattering of magnetic dipoles with one
fixed source point and at most three scattering strengths. With this technique,
a fast and stable phase retrieval approach is proposed based on a simple
geometrical result which provides a stable reconstruction of a point in the
plane from three distances to the given points. A novel phaseless direct
sampling method is also proposed to reconstruct the strip. The phase retrieval
approach can also be combined with the phased direct sampling method to
reconstruct the strip. Finally, to obtain a source support, we just need to
superpose the indicators with respect to the sparse observation directions.
Extended numerical examples in three dimensions are conducted with noisy data,
and the results further verify the effectiveness and robustness of the proposed
phase retrieval approach and direct sampling methods.
