Description and explanation of electromagnetic behaviors in artificial metamaterials based on effective medium theory

We present a general theory of effective media to set up the relationship between the particle responses and the macroscopic system behaviors for artificial metamaterials composed of periodic resonant structures. By treating the unit cell of the periodic structure as a particle, we define the average permittivity and permeability for different unit structures and derive a general form of discrete Maxwell's equations on the macroscale, from which we obtain different wave modes in metamaterials including the propagation mode, pure plasma mode, and resonant crystal band-gap mode. We explain unfamiliar behaviors of metamaterials from the numerical $S$ parameter retrieval approach. The excellent agreement between theoretical predictions and retrieval results indicates that the defined model and method of analysis fit the physical structures very well. Thereafter, we propose a more advanced form of the fitting formulas for the effective electromagnetic parameters of metamaterials.