## Abstract

In this review paper I discuss electrically thin composite layers, designed to perform desired operations on applied electromagnetic fields. Starting from a historical overview and based on a general classification of metasurfaces, I give an overview of possible functionalities of the most general linear metasurfaces. The review is concluded with a short research outlook discussion.

## 1. Introduction

A metasurface is a composite material layer, designed and optimized in order to control and transform electromagnetic fields. The layer thickness is negligible with respect to the wavelength in the surrounding space. The composite structure forming the metasurface is assumed to behave as a *material* in the electromagnetic (optical) sense, meaning that it can be homogenized on the wavelength scale, and the metasurface can be adequately characterized by its effective, surface-averaged properties. Similar to volumetric materials, where the notions of permittivity and permeability result from volumetric averaging of microscopic currents over volumes that are small compared with the wavelength, metasurface parameters result from two-dimensional, surface averaging of microscopic currents on the same wavelength scale. This implies that the unit cell sizes of composite metasurfaces are reasonably small as compared with the wavelength. In terms of the optical response, this means that metasurfaces reflect and transmit plane waves like sheets of homogeneous materials, in contrast to diffraction gratings, which produce multiple diffraction lobes. Metasurfaces can be considered as effective two-dimensional structures, which can be designed to ensure desired relations between the electromagnetic field values on the two sides of the engineered sheet. Possibilities for realizing full control over reflected and transmitted waves using thin sheets make this area of research exciting and topical, from both the theoretical and practical points of view.

At first glance it may appear that the possible functionalities of metasurfaces are expected to be rather limited as compared with volumetric metamaterials in general. Indeed, metasurfaces form a very narrow and specific subset of *metamaterials*, which can be defined as general, three-dimensional arrangements of artificial structural elements, designed to achieve advantageous and unusual electromagnetic properties. The shape of a metamaterial sample can be arbitrary, and the special case when one of the dimensions is negligibly small compared with the other dimensions and with the wavelength is what is called a *metasurface*. Sometimes, metasurfaces are defined as two-dimensional analogues of metamaterials. However, according to Huygens' principle, the electromagnetic fields created by arbitrary sources in an arbitrary volume *V* can be found as the fields created by equivalent currents on the volume surface *S*. The conventional paradigm of using metamaterials for control and transformations of electromagnetic fields implies that we engineer artificial materials in such a way that the polarization and conduction currents induced in the material, acting as secondary sources, create the desired fields outside (e.g. [1]) or inside (e.g. [2]) of the metamaterial sample. According to Huygens' principle, however, the same control of the fields outside of the medium volume can be achieved by engineering only surface currents, and there is no reason why the volume enclosed by such an ‘engineered surface’ could not be made negligibly small. For example, scattering-cancellation cloaking devices were first proposed and realized as volumetric metamaterial shells on scattering objects [3], but later it was realized that thin metasurface covers can produce the same effect [4]. However, if the goal is to realize an arbitrary field distribution inside a certain volume, some volumetric current distribution inside the volume is in general required [2,5]. In recent years, metasurfaces have been actively studied for various microwave and optical applications; see review papers [6–8].

## 2. Notes on history

Although the term *metasurface* was introduced less than 10 years ago, thin composite layers have been studied and used for a very long time. One of the earliest examples is a surface filled with a dense array or a mesh of thin conducting wires [9]. The thickness of this metasurface equals the wire diameter, which is usually negligibly small compared with the wavelength and the separations between the wires. If the array period is also electrically small, the array can reflect electromagnetic waves nearly as strongly as a continuous metal sheet, although only a very small part of the surface is filled by metal wires (e.g. [10–12,13]). On the other hand, at resonance (when the array period is equal to the wavelength), the mesh is nearly fully transparent even if the wire diameter is not small compared with the array period (e.g. [14]). Metal wire meshes have been used as various screens and as antenna reflectors for a long time.

It is very instructive to consider the homogenized model of wire meshes and compare that with contemporary volumetric metamaterials. A thin planar wire mesh with the period smaller than the wavelength is a typical metasurface, and it can be characterized in terms of an averaged boundary condition, connecting the averaged electric surface current density *J* and the surface-averaged tangential electric field *E* on the mesh surface. For an array of ideally conducting parallel round metal wires stretched along *x* axis, this boundary condition can be found from the analytical solution for the currents induced in the wires by a given incident plane wave. Knowing the currents in each wire and the electric fields scattered by them, it is possible to write a relation between the surface-averaged current and the surface-averaged tangential electric field in the array plane. The result reads [12,14]
2.1where j is the imaginary unit (the harmonic time dependence is in the form ), is the free-space impedance, *d* is the array period, *k*=2*π*/λ is the free-space wavenumber, and *r*_{0} is the wire radius. In the assumption that the wires are thin, the induced currents in the directions orthogonal to the wire axis (*x*) can be neglected. Detailed derivations of equation (2.1) can be found in [14], §4.2. Parameter *Z*_{g} (Ω) is called *grid impedance* or *sheet impedance*. In the limit it tends to zero, indicating that the response of the array tends to that of a homogeneous perfectly conducting surface.

Note that the sheet impedance *Z*_{g} is not just a complex number, but it is an operator containing spatial derivatives of the current density. Physically, this means that the array response not only is determined by the geometrical parameters and the frequency, but also depends on the spatial distribution of the surface-averaged currents and fields. This effect is called *spatial dispersion*. In contrast to nearly all natural materials with deeply subwavelength distances between atoms and molecules, where the spatial dispersion effects are weak, in metamaterials they are extremely strong, and remain strong even at extremely low frequencies.

Now let us recall the effective permittivity model of one of the most interesting volumetric metamaterials, the wire medium [15]. For a volumetric, three-dimensional periodical arrangement of thin ideally conducting wires stretched along *x*, the permittivity component along *x* reads [16]
2.2where *k*_{p} is the effective plasma wavenumber, related to the plasma frequency *ω*_{p} as *k*_{p}=*ω*_{p}/*c*, and *k*_{x} is the *x* component of the wavenumber of plane waves propagating inside the medium sample. Using the definition of permittivity, **D**=*ϵ*_{0}**E**+**P**=*ϵ***E**, the same can be expressed in the form of a relation between the macroscopic (volume-averaged) electric field *E*_{x} along *x* and the electric polarization density *P*_{x} as
2.3This relation has the same form as the averaged boundary condition for a single-layer wire grid (2.1). The phenomenon of strong spatial dispersion in wire arrays results from the fact that electromagnetic waves can propagate along the wires at arbitrarily long distances (neglecting dissipation) as in a multi-conductor transmission line. Interestingly, this property of volumetric wire media as epsilon-negative and spatially dispersive metamaterials was realized only about 10 years ago [16], while the spatially dispersive averaged boundary condition for single-layer wire meshes (metasurfaces, according to the current terminology) have been known from the mid-twentieth century [12].

Also the notion of an infinitely thin sheet as a material slab with negligible thickness was used in electromagnetics for a long time. The idea of using infinitely thin current sheets to model layers of conventional volumetric materials probably stems from the impedance boundary conditions modelling interfaces with good conductors, introduced in 1940 [17–19]. Owing to a very high value of the refractive index |*n*| in metals at microwave frequencies, the induced electric currents in metal bodies decay very fast from their surfaces, and it is possible to approximate the volumetric current distribution by an equivalent surface current sheet. The corresponding approximate boundary condition has the form
2.4where **E**_{t} is the tangential electric field on the surface, and **J**_{s} is the surface current density. The *surface impedance* *Z*_{s} for very good conductors approximately equals the complex wave impedance in metal. More accurate approximations lead to correction terms which contain spatial derivatives along the surface [14]. Note that this boundary condition has the same form as the above-discussed averaged boundary condition for dense wire meshes (2.1), although the spatial dispersion effects are weak. Equivalent sheet conditions for thin material layers as relations between the tangential fields at their surfaces were used from the mid-twentieth century, mainly for thin coatings of metal bodies, but sometimes also for free-standing layers [20]. At the end of the twentieth century, sheet models for thin layers of anisotropic materials and artificial chiral materials (precursors of modern metamaterials) were developed (e.g. [21–24]).

In the microwave and antenna engineering community, significant effort has been devoted to development of electrically thin layers as frequency and spatial filters for electromagnetic waves. These layers, called *frequency selective surfaces* (FSS), are usually formed by periodically arranged metal patches of various forms, most commonly squares, loops or crosses. Close to the resonant frequency of the array elements, the reflection and transmission coefficients for the array change rapidly as functions of the frequency, allowing filtering of reflecting and propagating plane waves. FSS were actively studied and practically developed mainly in the last quarter of the last century [25,26]. However, in the vast majority of known designs, the array period is comparable with the wavelength, which makes these FSS distinct from metasurfaces based on subwavelength structures, although application areas can overlap. Another known concept, similar to that of the metasurface, is the concept of reflectarrays and transmitarrays [27,28]. These devices can be considered as non-uniform FSS: tuning the resonance frequency of individual elements of a frequency-selective surface, it is possible to realize the desired variations of phase of reflected and transmitted fields over the array plane, allowing wavefront control. It is interesting to note that the generalized laws of reflection and transmission formulated in a recent paper [29] for surfaces with non-uniform distribution of reflection and transmission phase have the same meaning as the relations used in the design of reflectarrays and transmitarrays, and, more generally, phased-array antennas. The most common application of reflectarrays and transmitarrays is for focusing reflected or transmitted waves. Also in reflect- and transmitarray designs, the array period is usually comparable with the wavelength, although some exceptions are known [30].

## 3. Homogenization models and basic classification

Metasurfaces are homogenizable in the surface plane (they behave as two-dimensional material sheets), which means that illumination by a plane wave results, in the far zone, just in one reflected and one transmitted plane wave. Possible inhomogeneities on the wavelength scale result in some diffuse scattering [31], but no higher-order propagating plane waves can exist. The amplitudes and phases of the reflected and transmitted plane waves depend only on the surface-averaged electric and magnetic current densities **J**_{e} and **J**_{m}, flowing on the metasurface. Thus, the plane-wave response from all metasurfaces is known as soon as we know only two two-dimensional vectors: the electric and magnetic surface current densities induced on them.

Homogenization models describe relations of the induced surface current densities and the averaged electric and magnetic fields on the metasurface. Because the surface current densities are equal to jumps of the tangential components of the surface-averaged fields, these relations can be expressed as relations between the surface-averaged tangential electric and magnetic fields on the two sides of the metasurface. Denoting the averaged tangential fields as **E**_{t±} and **H**_{t±}, where the ± signs refer to the two sides of the surface, we can express the surface current densities in terms of the tangential field components:
3.1where the unit vector **n** is orthogonal to the surface and points to the side marked by the + sign. Equations (3.1) are, actually, just the Maxwellian boundary conditions at an interface between two media (in the absence of surface currents they reduce to the conventional continuity conditions). For simple metasurfaces whose response is the same for illuminations of their two sides and when there is no magnetoelectric coupling, it is convenient to write the matrix relations between the four tangential field vectors expressing the jumps of the fields as functions of the sums of the fields on the two sides:
3.2The double overbar denotes two-dimensional dyadic quantities. Relations (3.2) simply state that induced currents are related to the corresponding fields by linear relations, reflecting the assumption that the metasurface is a linear system. The averaged tangential fields *right at the metasurface* are defined as the averaged tangential field components on its two surfaces:
3.3These boundary conditions for various structures are discussed, for example, in [6,32].

For the most general linear metasurfaces (with magnetoelectric coupling), it is appropriate to model the electromagnetic properties of metasurfaces by the general matrix relation between the tangential fields on its two sides:
3.4In the general case, the matrix elements in (3.2) have no clear physical meaning as effective surface parameters. This dyadic impedance model (3.4) can be used, in fact, for arbitrary linear planar layers which are homogeneous in the layer plane, for example [14]. Sometimes, it is more convenient to rewrite these relations expressing the fields on one side as linear functions of the fields on the other side of the surface. The corresponding matrix is called the *transmission matrix*.

A useful classification of metasurfaces can be established considering their fundamental response to excitation by normally incident plane electromagnetic waves. The presence of one or both surface currents and the relations between them and the fields define the basic types of metasurfaces.

## 4. Electrically *or* magnetically polarizable metasurfaces

In the limit of zero thickness of a non-magnetic material layer or an array of particles without any natural magnetic properties, we come to the situation when only electric surface current can exist on the sheet. This is because induced non-zero magnetic current in the sheet plane means that there are loops of electric current inside the body of the layer. Obviously, zero-thickness sheets do not allow the existence of such loop currents. Metasurfaces that support only electric surface currents form the simplest class of metasurfaces, often called single-layer FSS [25,26]. In microwave and antenna engineering, such surfaces are single-layer arrays of thin planar and variously shaped metal patches or strips. The thickness of the patches is negligible compared with the free-space wavelength, and at microwaves thin metal sheets can be considered as nearly perfect conductors.

The electromagnetic properties of electric current sheets can be modelled by a single two-dimensional dyadic parameter: surface or grid admittance (or its inverse, surface or grid impedance, as in (2.1)). This parameter defines the relation between the electric surface current density **J**_{e} and the surface-averaged value of the tangential electric field at the sheet plane **E**_{t}:
4.1The surface impedance and admittance depend on the metasurface structure, frequency and, for plane-wave excitation, the incidence angle. For general excitation, the angular dependence is equivalent to dependence on spatial derivatives along the surface. The above relation is sometimes called the *averaged boundary condition* or *transition condition*. The tangential electric field is continuous across the sheet of only electric current, so there is no need to distinguish between the values of **E**_{t±}=**E**_{t} on the two sides of the sheet. This continuity of electric field results in severe limitations on possible functionalities of such metasurfaces, because the plane-wave reflection and transmission coefficients are related to each other:
4.2where is the planar unit dyadic. Analytical modelling of various thin layers maintaining only electric surface current has a long history. For example, various approaches to estimation of for arrays of thin metal wires or strips can be found in [9–13], and in many more recent papers. Arrays of disconnected electrically polarizable particles are treated, for example, in [14], ch. 4.

Metasurfaces that support only magnetic surface currents can be considered using the duality principle, introducing a linear relation between the induced magnetic surface current density and the surface-averaged magnetic field: 4.3The impossibility of controlling transmission independently from reflection limits possible functionalities to FSS and some polarizers.

## 5. Electrically *and* magnetically polarizable metasurfaces

Allowing non-zero (but still negligibly small with respect to the wavelength in surrounding space) thickness, we allow induction of electric current loops inside the metasurface structure, which are manifested as some induced magnetic surface currents. If the electric current loops are excited only in non-uniform (across the layer thickness) external electric fields, it is possible to relate the induced magnetic current density **J**_{m} with the surface-averaged magnetic field **H**_{t} in the sheet plane, as in (3.2).

The plane-wave reflected and transmitted fields, expressed in terms of the two induced surface current densities, read
5.1and
5.2where *η*_{0} is the wave impedance of the surrounding isotropic space, and the upper and lower signs correspond to the normal plane-wave illumination of the metasurface sides marked by the ± signs, respectively. The sheet generates different secondary fields at its two sides, and we can now control reflection independently of transmission. In particular, it is possible to make the metasurface non-reflecting (matched), if the following condition is satisfied:
5.3This is the same relation as between the electric and magnetic fields in the incident plane wave, and such layers are called *Huygens' surfaces*.

Magnetoelectric current sheets offer a number of additional functionalities as compared with purely electric current sheets or purely magnetic current sheets, such as various zero-reflection devices (absorbers, matched FSS, phase-shifting sheets, some polarizers), and various zero-transmission devices (absorbers, mirrors with controlled reflection phase), and others. However, full control over reflection and transmission is still not possible. It is easy to see that the reflection and transmission coefficients are always the same for illuminations of the two opposite sides of the sheet: these metasurfaces always look the same from both sides. Furthermore, these structures are symmetric with respect to space inversion, meaning that effects such as optical activity in transmission are forbidden as long as the metasurface is reciprocal.

## 6. General bi-anisotropic metasurfaces

In the most general case of a linear metasurface, both electric and magnetic polarizations in the metalayer can be induced by both incident electric and magnetic fields. Such layers are called *bi-anisotropic metasurfaces*. For the surfaces of this class it is not appropriate to link the electric and magnetic current densities with the averaged fields in the sheet plane as in (3.2), but it is possible to write the most general relations between the induced currents and the *incident* fields, as in [33]:
6.1Knowing the induced currents from the above relations, we can find the reflected and transmitted fields using (5.1) and (5.2). It can be shown that bi-anisotropic metasurfaces can control reflection and transmission properties in the most general way. The only restrictions are because of possible limitations on the symmetry properties and on the values of the coefficients in (6.1). These limitations follow from the symmetry of the layer structure and from the reciprocity and passivity conditions. In principle, all these limitations can be lifted using layers of the most general symmetry classes, by including non-reciprocal or time-dependent components, and using active elements.

Analytical modelling of bi-anisotropic metasurfaces can be based on equations (6.1). If the metasurface is formed by an array of electrically and magnetically polarizable bi-anisotropic particles, the incident fields can be linked to the local fields exciting every single particle, using so-called *interaction constants* (e.g. [14], ch. 4). Alternatively, the layer properties can be modelled by linear relations between the surface-averaged tangential components of the total electric and magnetic fields at the two sides of the surface (3.4). These can be written in terms of the admittance or impedance matrices with dyadic coefficients, for example [14], ch. 2. These coefficients define the values of the reflection and transmission coefficients (often called *S-parameters*) through the standard formulas of microwave engineering. Some approaches to the synthesis of general metasurfaces (finding the appropriate topologies and dimensions to realize the desired effective parameters) are known (e.g. [33]), but this field needs further developments before we get mature synthesis instruments.

## 7. Metasurfaces for surface-wave control

Metasurfaces can be used to control not only reflection and transmission of waves incident on the sheet but also waves that travel *along* the metasurface: surface waves. As discussed above, metasurfaces are characterized by impedance relations between the surface-averaged currents and tangential electric and magnetic fields (as in (2.4) or (3.1)). It is well known that surface waves can propagate along a reactive (lossless) impedance surface; for details, see, for example, [14], §6.4. The wave properties (in particular, the propagation constant along the surface) depend on the surface impedance. Thus, controlling and modulating the metasurface properties, surface waves can be manipulated in a rather general fashion. Some results on surface-wave control by modulating surface reactance were published as early as 1959 [34]. Modern active research on metasurfaces for surface-wave control resulted in a new concept of *metasurfing* [35]. Metasurfaces with engineering surface impedance can be used to shape propagating waves along the surface (usually forming a parallel-plate waveguide where one of the walls is a metasurface and the other one is a perfect conductor) or to control radiation of leaky waves along the metasurface. Interestingly, it is possible to use the two-dimensional version of the transformation-electromagnetics theory to find the required surface impedance for some classes of desired field transformations [36]. However, most of the known results refer to only impenetrable metasurfaces without bi-anisotropy. In view of the above review of the most general metasurfaces, it appears that there are still many unexplored possibilities in development of metasurfaces for surface-wave manipulations.

## 8. Examples of field-transforming metasurfaces

Here I list a few examples of metasurfaces for various field transformations, based on recent results of our research group. Let us imagine that the space is filled by a plane-wave field and assume that we want to manipulate the field in one half-space without disturbing the field in the other half, inserting a certain metasurface between the two spaces. In other words, we assume that the metasurface produces no reflections and study how we can control the transmission properties of the surface. This problem has been considered in [33] in view of a particular application: transformation of polarization of transmitted waves. Writing the reflected and transmitted fields in terms of the effective parameters (6.1), it is possible to find the necessary conditions for not disturbing fields in one half-space (no reflected waves) in terms of the collective polarizabilities of unit cells. Next, analysing the expressions for the transmitted fields, it is possible to find what transformations are possible and what physical properties of the surface (chirality, non-reciprocity, gain, etc.) are necessary to invoke in order to realize the desired function. Paper [33] presents an example of design and an experimental test of a matched twist-polarizer at microwave frequencies.

Means for the general control of reflected fields using metasurfaces formed by single-layer arrays of subwavelength particles have been studied in [37,38]. Here, the goal is to ensure full power reflection (although a single-layer particle array cannot incorporate a continuous mirror), and find ways to fully control the reflection phase. Arrays of small lossless resonant electric dipoles (only electric polarization, §4) fully reflect plane waves at the particle resonance, although the geometrical cross-section of the particles can be very small as compared with the unit-cell area. However, the reflection phase in this case is fixed to 180° (emulation of perfect electric conductor). Using duality, we can conclude that arrays of small lossless resonant magnetic dipoles fully reflect plane waves with the reflection phase 0°, emulating a magnetic wall. It is easy to see that metasurfaces which have both electric and magnetic dipolar inclusions (§5) can be tuned to fully reflect plane waves with an arbitrary phase of the reflected plane wave. However, practical realizations of such arrays are challenging, because the inclusions of the electric and magnetic dipole types must operate in a metamirror at non-resonant frequencies [37] and be adjusted very precisely taking into account all interactions in the array. For this reason, in paper [38] bi-anisotropic inclusions (§6) were used to realize full control over reflection phase. With the use of bi-anisotropic coupling, the general control over reflection becomes possible using only resonant particles. Practical examples described in [38] demonstrate, both theoretically and experimentally, single-layer metasurfaces for plane-wave deflection and focusing in reflection.

A study of the full range of possibilities offered by the most general bi-anisotropic metasurfaces (§6) has been published in [39]. In that paper, asymmetric response of metasurfaces that are fully transparent when illuminated from one of their sides has been studied, with the emphasis on functionalities offered by non-reciprocal bi-anisotropic effects in the layer.

## 9. Conclusion

The classical Huygens' principle (1690) establishes the equivalence between the electromagnetic fields generated by volumetric distributions of sources and surface-bound equivalent electric and magnetic currents (the electromagnetic formulation for vector fields is attributed to Stratton & Chu [40]). This is the starting point in investigations of engineered metasurfaces for field control and field transformations. The generalized Huygens' principle [41] tells that a layer of a finite thickness (which can be negligibly small compared with the wavelength in the surrounding media) can, for example, ‘glue’ together two field systems created by two independent sets of sources, in addition to offering an infinite number of alternative realizations of Huygens' layers. Historically, metasurfaces designed for a number of applications (polarizers and polarization transformers, frequency filters, absorbers, etc.) have been known for a long time, well before the very name ‘metamaterial’ or ‘metasurface’ appeared. For many applications, the sheet should reflect as little power as possible (e.g. in absorbers or polarization transformers), and all such sheets are Huygens' layers. More recently, some studies on the potentials offered by the most general linear metasurfaces started to appear, for example [37,39], but there is still much work ahead. We need better understanding of what the limits of metasurface performance are. What field transformations of arbitrary incident fields can be done with a theoretically infinitely thin sheet? What tasks cannot be done without creating a volumetric metamaterial? How can we overcome the limitations set by causality and passivity with the use of active or parametric (time-varying) metasurfaces? And, most importantly, we need to find practical means for realizations of most general functional metasurfaces for various applications in various frequency ranges and for waves of different physical nature.

## Competing interests

I declare I have no competing interests.

## Funding

I received no funding for this study.

## Footnotes

One contribution of 14 to a Theo Murphy meeting issue ‘Spatial transformations: from fundamentals to applications’.

- Accepted February 4, 2015.

- © 2015 The Author(s) Published by the Royal Society. All rights reserved.