  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 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 from a two dimensional periodic structure is computed and
deviations from the proximity force approximation examined.
