  In this work we theoretically consider the Casimir force between two periodic
arrays of nanowires (both in vacuum, and on a substrate separated by a fluid)
at separations comparable to the period. Specifically, we compute the
dependence of the exact Casimir force between the arrays under both lateral
translations and rotations. Although typically the force between such
structures is well-characterized by the Proximity Force Approximation (PFA), we
find that in the present case the microstructure modulates the force in a way
qualitatively inconsistent with PFA. We find instead that effective-medium
theory, in which the slabs are treated as homogeneous, anisotropic dielectrics,
gives a surprisingly accurate picture of the force, down to separations of half
the period. This includes a situation for identical, fluid-separated slabs in
which the exact force changes sign with the orientation of the wire arrays,
whereas PFA predicts attraction. We discuss the possibility of detecting these
effects in experiments, concluding that this effect is strong enough to make
detection possible in the near future.
