Effective permittivity of finite inhomogeneous objects

Abstract

A generalization of the S-parameter retrieval method for finite three-dimensional inhomogeneous objects under arbitrary illumination and observation conditions is presented. The effective permittivity of such objects may be rigorously defined as a solution of a nonlinear inverse scattering problem. We confirm analytically and observe numerically effects that were previously reported in the one-dimensional strongly inhomogeneous slabs: the nonuniqueness of the effective permittivity and its dependence on the illumination and observation conditions, and the geometry of the object. Moreover, we show that, although the S-parameter retrieval of the effective permittivity is scale free at the level of problem statement, the exact solution of this problem either does not exist or is not unique. Using the results from the spectral analysis we describe the set of values of the effective permittivity for which the scattering problem is ill-posed. Unfortunately, real nonpositive values, important for negative refraction and invisibility, belong to this set. We illustrate our conclusions using a numerical reduced-order inverse scattering algorithm specifically designed for the effective permittivity problem.

Macroscopic optical response and photonic bands

“…Its lower part begins at the pole of M (ω), which corresponds to a double zero of µ(ω) [42]. This part of the local dispersion relation is spurious, as the second-order Taylor expansion of the dielectric function that yields equations (33) and (32) starting from equation (27) would be meaningless at a pole. Consequently, this part of the local band does not correspond to any band within the exact calculation.…”

“…Thus, our photonic crystal, made up of holes within a dispersionless dielectric, behaves for some frequencies like a left-handed metamaterial [55,56]. Nevertheless, caution should be exercised when using equations (33) and (32) close to singularities of M T (ω, k).…”

“…4 Effective medium theories [18][19][20][21][22][23][24][25][26] have been proposed for describing inhomogeneous systems, such as diluted colloidal suspensions in the quasistatic or long-wavelength limit, in terms of a homogeneous macroscopic response. Further developments on homogenization of composites have been proposed [27][28][29][30][31][32][33][34][35] and their limits of validity have been discussed [2,36,37].…”

Generalized method for retrieving effective parameters of anisotropic metamaterials

“…For non resonant structures, the parameters are expected to remain relatively unchanged but the idea is to get more accurate parameter values in the frequency range of interest [16,17,18] (more accurate than the values obtained from homogenization). As pointed out in [19], the main virtue of this technique is that it does not require any assumption except that the concept of effective medium is valid, which is not incidental [20,21,22,23,24].…”

Disturbances Analysis Considering an Effective Servo System for Holographic Disc Drive

“…However, in many cases these approximations may be unreliable, or fail completely, leading to the necessity of more elaborate homogenization theories. One of the most commonly used procedures is based on the assumption that a finite slab of the crystal under consideration is equivalent to a homogeneous slab of the same thickness, in the sense that both, for a given incident wave, produce the same transmitted and reflected fields in the far-field zone [10][11][12][13][14]. Then, by inverting Fresnel's equations, one obtains appropriate scalar effective permittivity and permeability functions (S-matrix retrieval method).…”