  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.
