Effective Medium Theories Backward in Time: From the 21st to the 19th Century Non-Asymptotic and Nonlocal Approximations, Finite Samples, Symmetry Breaking
ABSTRACT
Most mathematical theories of homogenization are asymptotic – i.e. valid in the limit of the lattice cell size vanishingly small relative to some characteristic scale (free-space wavelength in the case of wave problems). It is now understood, however, that in this asymptotic limit all nontrivial properties of electromagnetic metamaterials – including, notably, magnetic response and negative refraction – vanish. We argue therefore that non-asymptotic and possibly nonlocal homogenization need to be studied and used. Our homogenization methodology applies to periodic electromagnetic structures (photonic crystals and metamaterials), treated on two main scales in the frequency domain. Fields on the fine scale (smaller than the lattice cell size) are approximated by a basis set of Bloch waves traveling in different directions. Fields on the coarse scale are approximated by a respective set of generalized plane waves, constructed in such a way that (i) all interface boundary conditions are satisfied as accurately as possible; (ii) Maxwell’s equations within the cell are also satisfied as accurately as possible. The physical essence of this methodology is in seeking effective material parameters that provide, in some sense, the best approximation of Bloch impedances as well as dispersion relations over an ensemble of physical modes in the structure. Classical theories (Clausius-Mossotti, Lorenz-Lorentz, Maxwell Garnett), while originally derived from very different physical considerations, fit well into this framework.
Several counter-intuitive results in homogenization of periodic electromagnetic structures are worth noting. First, the perception that effective parameters gradually converge to their bulk values as the number of layers increases is incorrect. Second, there exists a peculiar case where the physical nature of the problem and the type of governing mathematical equations change upon homogenization. While fields in laminated magnetic cores of electric machines are governed by Maxwell’s equations in the low frequency limit, this eddy current problem turns into a quasi-magnetostatic one upon homogenization. Third, seemingly incompatible symmetries between a periodic sample and its homogenized version can in fact be reconciled.