  Topological superconducting phases with time-reversal (TR) symmetry have been
widely explored in recent years. However the involved unconventional pairings
are generally implausible in realistic materials. Here we demonstrate via
detailed self-consistent calculation that these topological phases with TR
symmetry in DIII and BDI classes can be realized in a spin-orbit coupled
bilayer system with only $s$-wave interaction. The bilayer freedom enables the
definition of TR symmetry between the layers even in the presence of local
Zeeman fields, which we propose to be realized using four laser beams. The
gapped phase in DIII class is characterized by $\mathbb{Z}_2$, while all the
gapless phases in these two classes are characterized by nontrivial winding
numbers and are also manifested from the Majorana flat bands. We also unveil
the intimate relation between TR symmetry and mirror symmetry due to phase
locking effect between the two layers, which harbors the mirror symmetry
protected topological phases. We finally demonstrate that these phases will not
be spoiled by interlayer pairings.
