  We study the lateral dependence of the Casimir energy for different
corrugated gratings of arbitrary periodic profile. To this end we model the
profiles as stacks of horizontal rectangular slices following the profiles'
shape and evaluate numerically the Casimir energy between them for different
relative lateral displacements of the two corrugated plates. We compare our
results with predictions obtained within the proximity force approximation
(PFA). At comparable separation of the corrugated plates and geometric
parameters, we find a strong dependence of the Casimir energy on the shape of
the corrugation profiles.
