THE CASIMIR FORCE BETWEEN PLATES WITH SMALL DEVIATIONS FROM PLANE PARALLEL GEOMETRY

We present a general calculation of the corrections to the Casimir force between plates with small deviations from plane parallel geometry of the arbitrary type. These corrections are considered up to the fourth order with respect to the relative amplitudes of the deviations. The developed formalism is applied to the cases of nonparallel plates and of large scale as well as short scale deformations of the surfaces of different types. A number of particular examples are explicitly calculated. The method used is the previously developed method of additive summation of the Casimir force potentials with a subsequent renormalization. The results obtained by this method are tested by comparison with the results from the Green function method. A special investigation is performed into the role of mixed terms in the expansion with respect to the deviation amplitudes resulting from different plates. It is shown that such contributions are present already in the first nonvanishing correction to the Casimir force, so that this correction is nonadditive.