  Using a Keldysh formalism, we study supercurrent (SC) transport in several
types of diffusive ferromagnet(F)/superconductor(S) configurations. We separate
out the even- and odd-frequency components and identify the relative
contributions from the singlet and triplet channels. We first consider 1D
Josephson structures consisting of a uniform SFFS structure and a trilayer
SFFFS configuration. Our results demonstrate that for SCs flowing perpendicular
to the FF interfaces, incorporating a conical texture yields the most effective
way to observe the signatures of the long-ranged SC. We also consider three
different types of finite-sized 2D structures subjected to an applied magnetic
field: a uniform SFS junction, and two SFFS configurations with differing FF
bilayer arrangements. In one case, the FF interface is parallel with the SF
interfaces while in the other case, the FF junction is oriented perpendicular
to the SF interfaces. For the uniform SFS junction, we analytically derive the
magnetic field induced SC and pair potential, providing insight into the
variations of the critical SCs and proximity vortices. Our extensive
computations demonstrate that the induced long-range SCs can penetrate deeply
into uniform FF bilayers when singlet SCs flow parallel to the FF interfaces
which is in stark contrast to configurations where SC flows perpendicular to
the FF interfaces. We pinpoint the origin of the induced triplet and singlet
correlations through spatial profiles of the decomposed total SCs. We find that
the penetration of the long-range SCs associated with SCs flowing parallel to
the FF interfaces, are more pronounced when the thickness of the F strips are
unequal. Lastly, we show that the corresponding experimentally accessible SFFN
spin valve presents an effective platform in which the predicted long-range
effects can be effectively generated and probed.
