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This paper introduces simple analytical formulas for the grid impedance of electrically dense arrays of square patches and for the surface impedance of high-impedance surfaces based on the dense arrays of metal strips or square patches over ground planes. Emphasis is on the oblique-incidence excitation. The approach is based on the known analytical models for strip grids combined with the approximate Babinet principle for planar grids located at a dielectric interface. Analytical expressions for the surface impedance and reflection coefficient resulting from our analysis are thoroughly verified by full-wave simulations and compared with available data in open literature for particular cases. The results can be used in the design of various antennas and microwave or millimeter wave devices which use artificial impedance surfaces and artificial magnetic conductors (reflect-array antennas, tunable phase shifters, etc.), as well as for the derivation of accurate higher-order impedance boundary conditions for artificial (high-) impedance surfaces. As an example, the propagation properties of surface waves along the high-impedance surfaces are studied. I. INTRODUCTIONIn this paper we consider planar periodic arrays of infinitely long metal strips and periodic arrays of square patches, as well as artificial high-impedance surfaces based on such grids. [17]. Capacitive strips and square patches have been studied extensively in the literature (e.g., [18]-[20]). However, to the best of the authors' knowledge, there is no known easily applicable analytical model capable of predicting the plane-wave response of these artificial surfaces for large angles of incidence with good accuracy.Models of planar arrays of metal elements excited by plane waves can be roughly split into two categories: computational and analytical methods. Computational methods as a rule are based on the Floquet expansion of the scattered field (see, e.g., [2], [3], [21], [22]). These methods are electromagnetically strict and general (i.e., not restricted to a particular design geometry). Periodicity of the total field in tangential directions allows one to consider the incidence of a plane wave on a planar grid or on a high-impedance surface as a single unit cell problem. The field in the unit cell of the structure can be solved using,

It is found that there exist composite media that exhibit strong spatial dispersion even in the very large wavelength limit. This follows from the study of lattices of ideally conducting parallel thin wires ͑wire media͒. In fact, our analysis reveals that the description of this medium by means of a local dispersive uniaxial dielectric tensor is not complete, leading to unphysical results for the propagation of electromagnetic waves at any frequencies. Since nonlocal constitutive relations have been usually considered in the past as a secondorder approximation, meaningful in the short-wavelength limit, the aforementioned result presents a relevant theoretical interest. In addition, since such wire media have been recently used as a constituent of some discrete artificial media ͑or metamaterials͒, the reported results open the question of the relevance of the spatial dispersion in the characterization of these artificial media. Causality imposes that all material media must be dispersive. In most cases this behavior results in local dispersive constitutive relations, i.e., in frequency-dependent constitutive permittivity and permeability tensors. Nonlocal dispersive behavior ͑i.e., spatial dispersion͒, which results in constitutive operators depending also on the spatial derivatives of the mean fields ͑or, for plane electromagnetic waves, on the wave-vector components͒, is usually considered as a small effect, meaningful in the short-wavelength limit. Specifically, spatial dispersion will always appear when the higher-order terms in the series expansion of the constitutive parameters in power series of the dimensionless parameter a/ (a is the lattice constant of the crystal and the wavelength inside the medium͒ are not neglected.

A model for a chiral material in which both the permittivity and permeability are equal to zero is discussed. Such a material is referred by us as a "chiral nihility". It is shown that this exotic material can be realized as a mixture of small helical inclusions. Wave solutions and energy in such a medium are analyzed. It is shown that an extraordinary wave in chiral nihility is a backward wave. Wave reflection and refraction on a chiral nihility interface is also considered. It is shown that a linearly polarized wave normally incident onto this interface produces the wave of "standing phase" and the same wave in the case of oblique incidence causes two refracted waves, one of them with an anomalous refraction.