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We present a formulation for deriving effective medium properties ofinfinitely periodic two-dimensional metamaterial lattice structures beyond thestatic and quasi-static limits. We utilize the multipole expansions, where thepolarization currents associated with the supported Bloch modes are expressedvia the electric dipole, magnetic dipole, and electric quadrupole moments perunit length. We then propose a method to calculate the Bloch modes based on thelattice geometry and individual unit element structure. The results revert towell-known formulas in the quasistatic limit and are useful for thehomogenization of nanorod-type metamaterials which are frequently used inoptical applications.

Dynamic Effective Medium Theory for Two-Dimensional Non-Magnetic Metamaterial Lattices using Multipole Expansion