  We develop a formalism to calculate the fluctuation-induced interactions in
periodic systems. The formalism, which combines the scattering theory with the
C method borrowed from electromagnetic gratings studies, is suitable and
efficient for the calculation of the Casimir forces involving surface relief
gratings. We apply the developed technique to obtain the energy and lateral
force for simple 1-D sinusoidal gratings. Using this formalism we derived known
asymptotic expressions that were previously obtained through perturbative
approximations. At close separation, our numerical results match those obtained
by the proximity force approximation and its first correction using the
derivative expansion.
