  We describe an imbalanced superfluid Fermi gas in three dimensions within the
path-integral framework. To allow for the formation of the
Fulde-Ferell-Larkin-Ovchinnikov-state (FFLO-state), a suitable form of the
saddle-point is chosen, in which the pairs have a finite centre-of-mass
momentum. To test the correctness of this path-integral description, the
zero-temperature phase diagram for an imbalanced Fermi gas in three dimensions
is calculated, and compared to recent theoretical results. Subsequently, we
investigate two models that describe the effect of imposing a one-dimensional
optical potential on the 3D imbalanced Fermi gas. We show that this 1D optical
potential can greatly enlarge the stability region of the FFLO-state, relative
to the case of the 3D Fermi gas without 1D periodic modulation. Furthermore it
is show that there exists a direct connection between the centre-of-mass
momentum of the FFLO-pairs and the wavevector of the optical potential. We
propose that this concept can be used experimentally to resonantly enhance the
stability region of the FFLO-state.
