  We calculate the zero temperature phase diagram of a polarized two-component
Fermi gas in an array of weakly-coupled parallel one-dimensional (1D) 'tubes'
produced by a two-dimensional optical lattice. Increasing the lattice strength
drives a crossover from three-dimensional (3D) to 1D behavior, stabilizing the
Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) modulated superfluid phase. We argue
that the most promising regime for observing the FFLO phase is in the quasi-1D
regime, where the atomic motion is largely 1D but there is weak tunneling in
the other directions that stabilizes long range order. In the FFLO phase, we
describe a phase transition where the quasiparticle spectrum changes from
gapless near the 3D regime to gapped in quasi-1D.
