Abstract
Frequency dependence of the permittivity and permeability is inevitable in metamaterial applications such as cloaking and perfect lenses. In this paper, Herglotz functions are used as a tool to construct sum rules from which we derive physical bounds suited for metamaterial applications, where the material parameters are often designed to be negative or near zero in the frequency band of interest. Several sum rules are presented that relate the temporal dispersion of the material parameters with the difference between the static and instantaneous parameter values, which are used to give upper bounds on the bandwidth of the application. This substantially advances the understanding of the behavior of metamaterials with extraordinary material parameters, and reveals a beautiful connection between properties in the design band (finite frequencies) and the low- and high-frequency limits.
Highlights
The intriguing physics based on negative index of refraction [28] and -near-zero materials [26] with applications such as the perfect lens [33] and cloaking [2, 24] has created a renewed interest in the fundamental properties of the interaction between electromagnetic elds and materials [14, 20, 32]
The static permittivity and permeability are well dened and can be determined by homogenization techniques for heterogeneous materials; it is well known that the eective material parameter is bounded by the parameters of the included materials [19]
This demonstrates that for instance a high static permeability cannot be created by a composite material unless one of its component materials already has a high static permeability [1]
Summary
The intriguing physics based on negative index of refraction [28] and -near-zero materials [26] with applications such as the perfect lens [33] and cloaking [2, 24] has created a renewed interest in the fundamental properties of the interaction between electromagnetic elds and materials [14, 20, 32]. The classical Kramers-Kronig relations [10, 13, 16] relate the frequency dependence of the real- and imaginary parts of the permittivity and the permeability for causal material models. Since engineered materials with very complex frequency dependence are often considered, it is not sucient to analyze only Drude and Lorentz models if all potentials and restrictions of future metamaterials applications are to be evaluated. Approaches based on the Hilbert transform or equivalently the Kramers-Kronig relations [10] are commonly used to derive bounds for causal material models. The bounds can be tightened if a priori knowledge of the plasma frequency is added, see Sec. 3.4
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