Casimir-Lifshitz force for graphene-covered gratings

We study the Casimir-Lifshitz force (CLF) between a gold plate and a graphene-covered dielectric grating. Using a scattering matrix (S-matrix) approach derived from the Fourier Modal Method (FMM), we find a significant enhancement in the CLF as compared to a mere dielectric slab coated with graphene, over a wide range of temperatures. Additionally, we demonstrate that the CLF depends strongly on the chemical potential of graphene, with maximal effects observed at lower filling fractions. Finally, we analyse the Casimir force gradient between a gold sphere and a graphene-coated dielectric grating, highlighting potential avenues for experimental measurements.