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.
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GENERAL SCIENTIFIC SUMMARY Introduction and background. The possibility of creating materials with a negative refractive index or permittivity near zero has generated new interest in the fundamental electromagnetic properties of materials. To realise possible applications such as the perfect lens or cloaking, material properties such as permittivity should be constant close to a target value such as −1 or 0 over a range of frequencies, which isn't characteristic of the typical frequency dependence of real materials. This paper demonstrates restrictions on bandwidth imposed by the simple requirements of linearity, causality, time translational invariance, and passivity.
Main results. Using Herglotz functions as our principal modeling tool, we show that the high- and low-frequency asymptotics of a material property such as the permittivity restricts the bandwidth for achieving a given target value. If the target value is between the asymptotics, there are essentially no bandwidth limitations. If it is outside the asymptotics, the fractional bandwidth must be smaller than the maximum deviation from the target value, multiplied by a factor given by the asymptotic values.
Wider implications. The bounds presented in this paper can be used to check whether a hypothetical metamaterial application can be realised using passive materials. The general methodology used in this paper to construct identities can also be used to derive bounds on other linear, causal, time translational invariant, and passive physical systems.