  We investigate, in three spatial dimensions, the transition from the normal
state to the Fulde-Ferrel-Larkin-Ovchinnikov superfluid phases. We make use of
a Fourier expansion for the order parameter and the Green's functions to handle
the quasiclassical equations in the vicinity of the transition. We show that,
below the tricritical point, the transition is always first order. We find
that, at the transition, the higher Fourier components in the order parameter
are always essentially negligible. Below the tricritical point we have the
already known result that the order parameter has a spatial dependence which is
essentially $\cos({\bf q}.{\bf r})$. However when the temperature is lowered,
the order parameter switches to a sum of two cosines, with equal weigths and
wavevector with the same length, but orthogonal directions. Finally by further
lowering the temperature, and down to T=0, one finds a another transition
toward an order parameter which is the sum of three cosines with again equal
weigths and orthogonal directions. Hence the structure of the order parameter
gets more complex as the temperature is lowered. On the other hand the
resulting critical temperatures are found to be only slightly higher than the
ones corresponding to the standard second order FFLO transition. We apply our
results to the specific case of ultracold Fermi gases and show that the
differences in atomic populations of the two hyperfine states involved in the
BCS condensation display sizeable variations when one goes from the normal
state to the superfluid FFLO phases, or one FFLO phase to another.
Experimentally this should allow to identify clearly the various phase
transitions.
