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We investigate, in three spatial dimensions, the transition from the normalstate to the Fulde-Ferrel-Larkin-Ovchinnikov superfluid phases. We make use ofa Fourier expansion for the order parameter and the Green's functions to handlethe quasiclassical equations in the vicinity of the transition. We show that,below the tricritical point, the transition is always first order. We findthat, at the transition, the higher Fourier components in the order parameterare always essentially negligible. Below the tricritical point we have thealready known result that the order parameter has a spatial dependence which isessentially $\cos({\bf q}.{\bf r})$. However when the temperature is lowered,the order parameter switches to a sum of two cosines, with equal weigths andwavevector with the same length, but orthogonal directions. Finally by furtherlowering the temperature, and down to T=0, one finds a another transitiontoward an order parameter which is the sum of three cosines with again equalweigths and orthogonal directions. Hence the structure of the order parametergets more complex as the temperature is lowered. On the other hand theresulting critical temperatures are found to be only slightly higher than theones corresponding to the standard second order FFLO transition. We apply ourresults to the specific case of ultracold Fermi gases and show that thedifferences in atomic populations of the two hyperfine states involved in theBCS condensation display sizeable variations when one goes from the normalstate to the superfluid FFLO phases, or one FFLO phase to another.Experimentally this should allow to identify clearly the various phasetransitions.Comment: 15 pages, 12 figure

Transition to Fulde-Ferrell-Larkin-Ovchinnikov phases in three dimensions: A quasiclassical investigation at low temperature with Fourier expansion