A rigorous homogenization theory of metamaterials—artificial periodic structures judiciously designed to control the propagation of electromagnetic (EM) waves—is developed. The theory is an amalgamation of two concepts: Smith and Pendry’s physical insight into field averaging and the mathematical framework of Whitney-like interpolation. All coarse-grained fields are unambiguously defined and satisfy Maxwell’s equations exactly. Fields with tangential and normal continuity across boundaries are associated with two different kinds of interpolation, which reveals the physical and mathematical origin of “artificial magnetism.” The new approach is illustrated with several examples and agrees well with the established results (e.g., the Maxwell–Garnett formula and the zero cell-size limit) within the range of applicability of the latter. The sources of approximation error and the respective suitable error indicators are clearly identified, along with systematic routes for improving the accuracy further. The proposed methodology should be applicable in areas beyond metamaterials and EM waves (e.g., in acoustics and elasticity).
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