What is a reconfigurable intelligent surface? What is a smart radio environment? What is a metasurface? How do metasurfaces work and how to model them? How to reconcile the mathematical theories of communication and electromagnetism? What are the most suitable uses and applications of reconfigurable intelligent surfaces in wireless networks? What are the most promising smart radio environments for wireless applications? What is the current state of research? What are the most important and challenging research issues to tackle? These are a few of the many questions that we investigate in this short opus, which has the threefold objective of introducing the emerging research field of smart radio environments empowered by reconfigurable intelligent surfaces, putting forth the need of reconciling and reuniting C. E. Shannon's mathematical theory of communication with G. Green's and J. C. Maxwell's mathematical theories of electromagnetism, and reporting pragmatic guidelines and recipes for employing appropriate physics-based models of metasurfaces in wireless communications.

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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.

Non-uniform metasurfaces (electrically thin composite layers) can be used for shaping refracted and reflected electromagnetic waves. However, known design approaches based on the generalized refraction and reflection laws do not allow realization of perfectly performing devices: there are always some parasitic reflections into undesired directions. In this paper we introduce and discuss a general approach to the synthesis of metasurfaces for full control of transmitted and reflected plane waves and show that perfect performance can be realized. The method is based on the use of an equivalent impedance matrix model which connects the tangential field components at the two sides on the metasurface. With this approach we are able to understand what physical properties of the metasurface are needed in order to perfectly realize the desired response. Furthermore, we determine the required polarizabilities of the metasurface unit cells and discuss suitable cell structures. It appears that only spatially dispersive metasurfaces allow realization of perfect refraction and reflection of incident plane waves into arbitrary directions. In particular, ideal refraction is possible only if the metasurface is bianisotropic (weak spatial dispersion), and ideal reflection without polarization transformation requires spatial dispersion with a specific, strongly non-local response to the fields.