  We analyze the finite temperature phase diagram of fermion mixtures in
one-dimensional optical lattices as a function of interaction strength. At low
temperatures, the system evolves from an anisotropic three-dimensional
Bardeen-Cooper-Schrieffer (BCS) superfluid to an effectively two-dimensional
Berezinskii-Kosterlitz-Thouless (BKT) superfluid as the interaction strength
increases. We calculate the critical temperature as a function of interaction
strength, and identify the region where the dimensional crossover occurs for a
specified optical lattice potential. Finally, we show that the dominant vortex
excitations near the critical temperature evolve from multiplane elliptical
vortex loops in the three-dimensional regime to planar vortex-antivortex pairs
in the two-dimensional regime, and we propose a detection scheme for these
excitations.
