  Due to their collective nature Casimir forces can strongly depend on the
geometrical shape of the interacting objects. We study the effect of strong
periodic shape deformations of two ideal metal plates on their quantum
interaction. A non-perturbative approach which is based on a path integral
quantization of the electromagnetic field is presented in detail. Using this
approach, we compute the force for the specific case of a flat plate and a
plate with a rectangular corrugation. We obtain complementary analytical and
numerical results which allow us to identify two different scaling regimes for
the force as a function of the mean plate distance, corrugation amplitude and
wave length. Qualitative distinctions between transversal electric and magnetic
modes are revealed. Our results demonstrate the importance of a careful
consideration of the non-additivity of Casimir forces, especially in strongly
non-planar geometries. Non-perturbative effects due to surface edges are found.
Strong deviations from the commonly used proximity force approximation emerge
over a wide range of corrugation wave lengths, even though the surface is
composed only of flat segments. We compare our results to that of a
perturbative approach and a classical optics approximation.
