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Casimir force between two substrates within three different layers using the scattering approach

We compute the Casimir force for a system composed of two layers as substrates within three different homogenous layers. We use the scattering approach along with the Matsubara formalism in order to calculate the Casimir force at finite temperature. We impose the appropriate boundary condition on the tangential components of the electric and magnetic fields to construct the reflection matrices. Our calculation is simple such that one can extend it to systems containing inhomogeneous layers as good candidates for designing nanomachines. We find that the Casimir force is proportional to L^(-5) where L is the thickness of the inner layer.