  This paper concerns time-harmonic inverse source problems with a single
far-field pattern in two dimensions, where the source term is compactly
supported in an a priori given inhomogeneous background medium. For
convex-polygonal source terms, we prove that the source support together with
the zeroth and first order derivatives of the source function at corner points
can be uniquely determined. Further, we prove that an admissible set of source
functions (including harmonic functions) having a convex-polygonal support can
be uniquely identified by a single far-field pattern. A class of radiating
sources is characterized and the extension of the radiated field across a
corner point is proven impossible. The corner scattering theory leads to a
data-driven inversion scheme for imaging an arbitrarily convex-polygonal source
support.
