  We investigate the proximity effect in diffusive superconducting hybrid
structures with a spin-orbit (SO) coupling. Our study is focused on the
singlet-triplet conversion and the generation of long-range superconducting
correlations in ferromagnetic elements. We derive the quasiclassical equations
for the Green's functions including the SO coupling terms in form of a
background SU(2) field. With the help of these equations, we first present a
complete analogy between the spin diffusion process in normal metals and the
generation of the triplet components of the condensate in a diffusive
superconducting structure in the presence of SO coupling. From this analogy it
turns out naturally that the SO coupling is an additional source of the
long-range triplet component (LRTC) besides the magnetic inhomogeneities
studied in the past. We demonstrate an explicit connection between an
inhomogeneous exchange field and SO coupling mechanisms for the generation of
the LRTC and establish the conditions for the appearance of the LRTC in
different geometries. We also consider a S/F bilayer in contact with normal
metal with SO coupling and show that the latter can be used as a source for the
LRTC. Our work gives a global description of the singlet-triplet conversion in
hybrids structures in terms of generic spin-fields and our results are
particularly important for the understanding of the physics underlying
spintronics devices with superconductor elements.
