  We derive an exact solution for the Casimir force between two arbitrary
periodic dielectric gratings and illustrate our method by applying it to two
nanostructured silicon gratings. We also reproduce the Casimir force gradient
measured recently [1] between a silicon grating and a gold sphere taking into
account the material dependence of the force. We find good agreement between
our theoretical results and the measured values both in absolute force values
and the ratios between the exact force and PFA predictions.
