Modal approach to Casimir forces in periodic structures

We present a modal approach to calculate finite-temperature Casimir interactions between two periodically modulated surfaces. The scattering formula is used and the reflection matrices of the patterned surfaces are calculated by decomposing the electromagnetic field into the natural modes of the structures. The Casimir force gradient from a deeply etched silicon grating is evaluated using the modal approach and compared to experiment for validation. The Casimir force involving a two-dimensional periodic structure is computed and deviations from the proximity force approximation are examined.