Obtaining effective metamaterial parameters through modal analysis

We develop a general method for analyzing the behavior of MTMs using modal decomposition. We replace the inclusions by equivalent lumped circuit elements and use transmission-line theory to solve the propagation problem in the longitudinal direction. This approach calculates physically observable quantities, such as the input impedance, reflection coefficient, dispersion relation, etc., directly without relying on homogenization first. Unlike previous methods, our method can handle complex configurations such as multiple graduated or random inclusion layers within an MTM unit cell. We calculate equivalent constitutive parameters that take into account the entire unit cell and any interactions among the shunt and series elements of the unit cell. We illustrate the utility of our model by evaluating the performance of a MTM composed of unit cells that consist of several layers of SRRs with graduated resonant frequency.