Calculating Casimir interactions for Periodic Surface Relief Gratings using the C-Method

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.