  The imbalanced superfluid state of spin-1/2 fermions with s-wave pairing is
numerically studied by solving the Bogoliubov-de-Gennes equation at zero
temperature in an annular disk geometry with narrow radial width. Two distinct
types of systems are considered. The first case may be relevant to heavy
fermion superconductors, where magnetic field causes spin imbalance via Zeeman
interaction and the system is studied in a grand canonical ensemble. As the
magnetic field increases, the system is transformed from the uniform superfluid
state to the Fulde-Ferrell-Larkin-Ovchinnikov state, and finally to the spin
polarized normal state. The second case may be relevant to cold fermionic
systems, where the numbers of fermions of each species are fixed as in a
canonical ensemble. In this case, the groundstate depends on the pairing
strength. For weak pairing, the order parameter exhibits a periodic domain wall
lattice pattern with a localized spin distribution at low spin imbalance, and a
sinusoidally modulated pattern with extended spin distribution at high spin
imbalance. For strong pairing, the phase separation between superfluid state
and polarized normal state is found to be more preferable, while the increase
of spin imbalance simply changes the ratio between them.
