  We study the Fermi polaron problem with dipolar fermions in a bilayer
geometry, where a single dipolar particle in one layer interacts with a Fermi
sea of dipolar fermions in the other layer. By evaluating the polaron spectrum,
we obtain the appearance of a series of attractive branches when the distance
between the layers diminishes. We relate these to the appearance of a series of
bound two-dipole states when the interlayer dipolar interaction strength
increases. By inspecting the orbital angular momentum component of the polaron
branches, we observe an interchange of orbital character when system parameters
such as the gas density or the interlayer distance are varied. Further, we
study the possibility that the lowest energy two-body bound state spontaneously
acquires a finite center of mass momentum when the density of fermions exceeds
a critical value, and we determine the dominating orbital angular momenta that
characterize the pairing. Finally, we propose to use the tunneling rate from
and into an auxiliary layer as an experimental probe of the impurity spectral
function.
