Current-driven homogenization and effective medium parameters for finite samples

Abstract

Reflection and refraction of electromagnetic waves by artificial periodic composites (metamaterials) can be accurately modeled by an effective medium theory only if the boundary of the medium is explicitly taken into account and the two effective parameters of the medium -the index of refraction and the impedance -are correctly determined. Theories that consider infinite periodic composites do not satisfy the above condition. As a result, they cannot model reflection and transmission by finite samples with the desired accuracy and are not useful for design of metamaterial-based devices. As an instructive case in point, we consider the "current-driven" homogenization theory, which has recently gained popularity. We apply this theory to the case of one-dimensional periodic medium wherein both exact and homogenization results can be obtained analytically in closed form. We show that, beyond the well-understood zero-cell limit, the current-driven homogenization result is inconsistent with the exact reflection and transmission characteristics of the slab.

Applicability of effective medium description to photonic crystals in higher bands: Theory and numerical analysis

“…For Bloch waves with a nonzero fundamental harmonic (as defined in the Appendices) propagating in threedimensional, intrinsically non-magnetic PCs, the dispersion equation can be written in the following general form 20,26 …”

“…We have shown 20 that the knowledge of Σ(ω, q) for all values of its arguments is insufficient for solving boundary value problems in finite samples. However, in this paper, we restrict attention to dispersion relations, and to this end the knowledge of Σ(ω, q) is sufficient.…”

“…In the other description, the medium is assigned two local parameters (ω) and µ(ω). We have previously argued that the two descriptions are not physically equivalent in general 20 , but it is true that, in the so-called weak nonlocality regime 21 , such an equivalence exists for the refractive index of the medium (but not for the impedance).…”

“…On the other hand, if such local parameters do not exist, then the medium can not be reasonably homogenized and the introduction of the nonlocal permittivity does not solve the problem. Indeed, the knowledge of eff (ω, q) for all values of its arguments is not sufficient to solve any boundary value problems in a finite sample 20 , unlike the knowledge of the local parameters eff (ω) and µ eff (ω). Besides, the typical applications discussed in the literature such as subwavelength imaging require local eff (ω) and µ eff (ω).…”

A non-asymptotic homogenization theory for periodic electromagnetic structures

“…We note that, in the case of homogeneous slabs, a single incident plane wave gives rise to one reflected and one transmitted plane wave; the projections of the wavevectors of all three waves onto the interface are in this case equal. For composite slabs, this is not generally so because of the presence of complicated surface waves [11]. If the lattice periods (in the X-and Y-directions) are smaller than λ, the surface waves are exponentially localized near the physical interface.…”