Casimir energy of $N$ magnetodielectric $\delta$-function plates

To investigate Casimir electromagnetic interaction in $N$ bodies, we implement multiple $\delta$-function plates with electric and magnetic properties. We use their optical properties to study the Casimir energy between the plates by implementing multiple scattering formalism. We initially solve Green’s functions for two and three plates configurations to obtain their reflection coefficients. Further, the coefficients are implemented in multiple scattering formalism, and a simple method was obtained to depict energy density distribution in the multiple scattering expansions using diagrammatic loops. The Casimir energy for $N$ bodies depends on multiple scattering parameter $\mathrm{\Delta }$; this parameter was distributed into nearest neighbour scattering and next-to-nearest neighbour scattering terms represented by different loops depending on reflection, transmission and propagation distance. In this manner, the Casimir energy density was generalized to $N$ plates by identifying a systematic pattern in the representation of diagrammatic loops.