of wire media to that of conventional plasma [141]. In 1968, Veselago mathematically described the response of materials exhibiting both negative permittivity and permeability and demonstrated that such material parameters would correspond to a negative index of refraction [170]. Later, during the 1980s and 1990s, important theoretical and practical developments were made toward the realization of bianisotropic media and microwave absorbers [165]. However, it is only after the beginning of the twentieth century that the field of metamaterials really started to attract massive attention. It is notably due to the first experimental demonstration by Shelby et al. [150] that a negative index of refraction was indeed feasible and the associated concept of the perfect lens imagined by Pendry [120]. The interest in metamaterials grew even more when, in 2006, Pendry and Smith proposed and realized the concept of electromagnetic cloaking, based on transformation optics [121, 146].
At the beginning of this section, we provided a very general and broad definition of metamaterials. To be more specific, we shall now define more rigorously what we mean by a metamaterial. Throughout this book, we will consider that a metamaterial is an artificial structure consisting of an arrangement, usually periodic, of scattering particles. The effective physical size of these scattering particles, as well as the distance between them, must be smaller than that of the wavelength of the electromagnetic wave interacting with them. This prevents the existence of diffraction orders or Bragg scattering due to the lattice period and allows the metamaterial structure to be homogenized into effective material parameters. Note that this condition implies that photonic crystals [71] and electromagnetic band-gap structures [174], which typically exhibit unit cell size larger than that of the wavelength, cannot be considered to be metamaterials since it is impossible to describe them in terms of effective material parameters. The effective material parameters of a typical metamaterial stem from the combined effects of the chemical composition, the shape, and the orientation of their scattering particles. From a general perspective, the scattering particles may be composed of metal or dielectric materials; they may be resonant or non-resonant structures; they may be active, lossy, nonlinear, and even time-varying or contain electronic circuitry.
1.2 Emergence of Metasurfaces
Although the interest in metamaterials was at its peak in the first decade of the twenty-first century, it is their two-dimensional counterparts – the metasurfaces – that progressively became the dominant source of attention. The main reason for this sudden rise in interest is explained by the fact that metasurfaces are easier to fabricate, less bulky, and less lossy than conventional volume metamaterials [48, 57, 58, 105].
While the interest in metasurfaces only became substantial in recent years, the conceptual idea of controlling the propagation of electromagnetic waves with thin artificial structures is more than 100 years old. One of the earliest examples of this concept was most likely proposed by Lamb, at the end of the twentieth century, who investigated the scattering properties of an array of metallic strips [87]. This was soon followed by the polarization reflectors made by Marconi in the 1910s. However, the most important progresses in this area were made in conjunction with the development of the radar technology during World War II and which continued through the 1950s and the 1960s leading to a rich diversity of new systems and technologies. We illustrate some of these systems in Figure 1.1. For instance, Figure 1.1a depicts a Fresnel zone plate reflector, typically used in radio transmitters [23], whose working principle is based on the concept of the Fresnel lens already pioneered 200 hundred years ago. Figure 1.1b illustrates another very well-known operation, which is that of the frequency-selective surfaces used as microwave frequency filters [110, 139]. Then, Figure 1.1c represents a rather modern version of a reflectarray antenna (or reflectarrays) [61, 98, 107], whose technology, dating back to 1970s, consists of an array of microstrip patches implementing the functionality of conventional parabolic reflectors. Note that the first types of reflectarray antennas were designed using short-ended waveguides instead of microstrip patches [17]. Finally, Figure 1.1d illustrates the working principle of a transmitarray, the transmission counterpart of a reflectarray, which appeared already in the early 1960s [75, 101, 160]. This kind of systems was typically made of an array of receiving and transmitting antennas connected to each other by a delay line, whose phase shift could be tuned to shape the wavefront of the transmitted wave. Later, in the 1990s, microstrip patches were proposed as potential replacement for the interconnected antenna array used to implement transmitarrays [34, 86, 132]. However, the transmission properties of microstrip patches were then plagued with low efficiency as their phase shifting capabilities could not reach the required
Figure 1.1 Examples of two-dimensional waves manipulating structures [2]. (a) Fresnel-zone plate reflector. (b) Reflectarray. (c) Interconnected-array lens. (d) Frequency-selective surface. Due to their structural configuration, (a) and (c) do not qualify as metasurfaces. On the other hand, the structures in (b) and (d) can be homogenized into effective material parameters and hence correspond to metasurfaces.
The structures and systems presented in Figure 1.1 are the precursors of current metasurfaces. However, they are not the only ones. Indeed, there are many other examples of electrically thin structures that would be today considered as metasurfaces. For instance, consider the case of near-field plates [51, 65, 66] and impedance surfaces converting surface waves into space waves [43, 97], which, unlike the structures in Figure 1.1 that are restricted to space-wave transformations, can accommodate both space waves and surface waves.
As of today, there are countless examples of metasurface concepts and applications that have been reported in the literature and it would therefore be impossible to provide a complete description of all of them. Instead, we shall now present a non-exhaustive list of some typical applications of metasurfaces. A very common application is that of polarization conversion [45, 60, 69, 90, 99, 157, 173, 180], where a linearly polarized wave may be transformed either into a circularly polarized one or be rotated using chirality [77]. Metasurfaces have also been commonly used to control the amplitude of the incident wave and, for instance, been used as perfect absorbers by altering the amplitude of the fields so that no reflection or transmission can occur [19, 40, 91, 135, 168, 172, 177]. Nevertheless, one of the most spectacular aspects of metasurfaces is their wavefront manipulating capabilities, which is typically based on the same physical concept used to realize Fresnel lenses [106] and blazed gratings [38, 112], and which have been used for applications such as generalized refraction, collimation, focalization, and holography [1, 12, 46, 47, 55, 62, 63, 95, 96, 136, 153, 154, 175, 178, 179, 181]. Finally, more exotic applications include, for instance, the generation of vortex beams characterized by orbital angular momentum [76, 93, 127, 161, 171, 176], stable tractor beams for nanoparticle manipulation [128], nonreciprocal scattering for electromagnetic isolation [33, 36, 137, 159], nonlinear interactions [89, 169, 182], analog computing [131, 156], and spatial filtering [117, 151,
