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We present an almost fully analytical technique for computing Casimirinteractions between periodic lamellar gratings based on a modal approach. Ourmethod improves on previous work on Casimir modal approaches for nanostructuresby using the exact form of the eigenvectors of such structures, and computingeigenvalues by solving numerically a simple transcendental equation. In somecases eigenvalues can be solved for exactly, such as the zero frequency limitof gratings modeled by a Drude permittivity. Our technique also allows us topredict analytically the behavior of the Casimir interaction in limiting cases,such as the large separation asymptotics. The method can be generalized to morecomplex grating structures, and may provide a deeper understanding of thegeometry-composition-temperature interplay in Casimir forces betweennanostructures.

A quasi-analytical modal approach for computing Casimir interactions in periodic nanostructures