  Extended inverse Compton halos are generally anticipated around extragalactic
sources of gamma rays with energies above 100 GeV. These result from inverse
Compton scattered cosmic microwave background photons by a population of
high-energy electron/positron pairs produced by the annihilation of the
high-energy gamma rays on the infrared background. Despite the observed
attenuation of the high-energy gamma rays, the halo emission has yet to be
directly detected. Here, we demonstrate that in most cases these halos are
expected to be highly anisotropic, distributing the up-scattered gamma rays
along axes defined either by the radio jets of the sources or oriented
perpendicular to a global magnetic field. We present a pedagogical derivation
of the angular structure in the inverse Compton halo and provide an analytic
formalism that facilitates the generation of mock images. We discuss exploiting
this fact for the purpose of detecting gamma-ray halos in a set of companion
papers.
