## Abstract

Metasurfaces (MTSs) constitute a class of thin metamaterials used for controlling plane waves and surface waves (SWs). At microwave frequencies, they are constituted by a metallic texture with elements of sub-wavelength size printed on thin grounded dielectric substrates. These structures support the propagation of SWs. By averaging the tangential fields, the MTSs can be characterized through homogenized isotropic or anisotropic boundary conditions, which can be described through a homogeneous equivalent impedance. This impedance can be spatially modulated by locally changing the size/orientation of the texture elements. This allows for a deformation of the SW wavefront which addresses the local wavevector along not-rectilinear paths. The effect of the MTS modulation can be analysed in the framework of transformation optics. This article reviews theory and implementation of this MTS transformation and shows some examples at microwave frequencies.

## 1. Introduction

Metasurfaces (MTSs) are thin metamaterials constituted by an arrangement of printed elements whose dimensions are smaller than the operational wavelength (figure 1). MTSs may be distinguished as penetrable and impenetrable. A penetrable MTS (also called metascreen or metafilm) consists of a planar distribution of small periodic scatterers in a very thin host medium or of small holes in a conducting screen. Its effective properties can be studied for instance by applying Generalized Sheet Transition Conditions (GSTCs) [1,2], which allow one to characterize an MTS in terms of unambiguous anisotropic sheet impedance. Impenetrable MTSs, which are those treated in this paper, are realized at microwave frequencies through a dense periodic texture of small elements printed on a grounded slab with or without shorting vias to the ground plane. Alternatively, an impenetrable MTS can be realized by etching small holes on the upper wall of a thin parallel plate waveguide. In the following, however, we will focus on patch-type MTSs. By averaging the tangential fields, an MTS can be described macroscopically through impedance boundary conditions. This leads to the definition of a surface impedance tensor, which links the average tangential electric field to the average tangential magnetic field (electric currents). This boundary condition supports the propagation of surface waves (SWs). When the shape of the elements is regular enough (figure 1*a*), the impedance tensor becomes a scalar and therefore the average effect of the boundary conditions is isotropic with respect to the direction of propagation of the SW. When the element shape contains additional features, like slots, grooves or cuts, the effect is anisotropic and the anisotropy can be easily controlled for elements with two orthogonal symmetry axes (figure 1*b*).

In [1] and in many recent papers ([2,3] and references therein), emphasis is given to the effect of reflection/transmission and only in minor part to SW propagation and local impedance modulation, which is instead the main focus of this article. In particular, here it is shown how a proper impedance modulation allows one to address SWs along desired curvilinear paths. In fact, the impedance modulation, obtained by locally changing the sizes of the patches, imposes a local modification of the dispersion equation and hence, at constant operating frequency, of the local wavevector. The general effects of MTS modulation may be therefore treated in the framework of transformation optics (TO) [4,5]. TO establishes a rigorous analytical rule for obtaining a control on wave propagation within inhomogeneous anisotropic metamaterials. This control is achieved by designing the equivalent macroscopic constitutive tensors of the metamaterial on the basis of the differential parameters of a spatial coordinate transformation. The most famous use of TO is concerned with invisibility cloaks [6,7]. However, the technological difficulties in controlling the constitutive parameters in a volume complicate the engineering implementation of TO in practical devices. On the other hand, moulding boundary conditions appears technologically simpler in a microwave regime and can be used for controlling SW propagation and shaping the relevant wavefronts, with the objective of constructing engineering devices [8,9]. Luneburg [10,11] and Maxwell fish-eye lenses [9], beam shifters [12,13] and beam splitters [13] have been recently presented. TO quasi-flat devices can be also constructed by using non-uniform thickness grounded dielectric [14,15]. A solution at terahertz and infrared frequencies is obtained in [16] by using surface plasmon polaritons on a graphene monoatomic layer, where the graphene conductivity is tuned by using static electric fields. The wavenumber modulation can also produce leaky-wave radiation. In [17–21], this is used for realizing leaky-wave antennas. In [22], the MTS modulation has been linked to a generic coordinate transformation like in TO, with the final output of an anisotropic impedance which is able to support a curved-wavefront SW, whose phase is directly related to the transformation. This approach can be conveniently exploited to design beam forming networks for planar antennas. For instance, a planar Luneburg lens allows one to obtain a planar wavefront by using a single pin excitation, and proper geometrical transformations can be used to obtain a squinted beam or a power divider.

This article is dedicated to a review of results relevant to SW control through MTS modulation, giving emphasis to new aspects. After illustrating the basic physical phenomenon in §2, the criteria for homogenizing the MTS and retrieving the parameters characterizing its equivalent impedance are presented in §3. Section 4 focuses on the calculation of the dispersion relation for SWs, with particular attention to isofrequency dispersion curves. Section 5 summarizes and generalizes our previous theoretical results for MTS transformation theory. Some numerical examples are presented in §6 and conclusions are drawn in §7.

## 2. Surface waves on a space-variable isotropic impedance

An MTS constituted by subwavelength perfectly electric conducting patches printed on a lossless grounded dielectric substrate imposes at the free-space interface equivalent boundary conditions characterized by a reactive impedance *Z*(** ρ**)=

*jX*(

**), where**

*ρ***is the observation point at the interface plane located at**

*ρ**z*=0. The reactance

*X*(

**) is a scalar quantity if the printed element of the basic cell is regular enough, in practice if it exhibits more than two geometrical symmetry axes. Furthermore,**

*ρ**X*(

**) is positive at low frequency. The variability in space of**

*ρ**X*(

**) is obtained by changing the dimension of the printed elements, while leaving constant the lattice period.**

*ρ*A simple example is illustrated in figure 2. Figure 2*a* shows an MTS consisting of a uniform rectangular lattice of printed square patches modulated in size along *x*, with smaller sizes in the centre and larger sizes at the external side. Assume that this MTS is excited at a certain section *y*=*y*_{0} with a plane wavefront beam, represented in figure 2 by a congruence of parallel equilength wavevectors **k**_{t}. As the wave progresses along *y*, the larger value of reactance at the boundary imposes a local decrease of |**k**_{t}| in the centre and an increase at the sides. Correspondingly, the local phase velocity decreases at the periphery and increases at the centre, thus producing a diverging wavefront. This forces the wavevector to bend towards the region characterized by higher levels of impedance, so as to maintain its amplitude consistent with the dispersion equation for the local reactance value. The opposite occurs when the patches are larger at the centre and smaller at the sides (figure 2*b*). The congruence of rays forms, in this case, a converging wavefront that may focus the field in a point.

Generally speaking, this boundary condition supports an SW of transverse (with respect to *z*) magnetic (TM) type, whose tangential electric field presents an asymptotic behaviour of the type
2.1where the local wavevector **k**_{t}=∇_{t}*Φ*(** ρ**) satisfies the eikonal equation
2.2in which and are the wavenumber and impedance of free space, respectively. In equation (2.1), is the direction of propagation, which is also aligned with the direction of the TM tangential electric field, and

*V*

_{0}(

**) possesses a weak spatial variation determined by the conservation of energy in an elementary strip along the ray. The quantity plays the role of an equivalent refractive index in geometrical optics. To be consistent with the asymptotic form of the wave equation in free space, the condition 2.3should be satisfied. The SW-ray trajectory going from**

*ρ*

*ρ*_{1}to

*ρ*_{2}, namely the curvilinear path of

**k**

_{t}, can be established by minimizing the optical path-length according to the Fermat principle. Equation (2.2) is valid for isotropic elements and becomes more complex for elements like those in figure 1

*b*, which are described by a tensor impedance. This more general class of MTSs is treated in the next section.

## 3. Homogenization of metasurface and impedance retrieval

Several recent papers have been devoted to the homogenization criteria for impenetrable lossless MTS and to the method for retrieving the anisotropic impedance [23,24]. The following overview emphasizes the difference between jump boundary conditions, representing only the printed cladding, and the boundary conditions as seen at the air–MTS interface, which also contain the contribution of the grounded slab. This discussion prepares the ground for the successive §4, devoted to the analysis of SW dispersion.

### (a) Local periodicity and average tangential fields

A modulated MTS is characterized within the assumption that its constituent elements are embedded in a uniform periodic texture. More specifically, we identify the local texture and the relevant value of homogeneous reactance with those of a *periodic* texture which matches the local geometry. We will refer to this concept as ‘local periodicity’ (figure 3). The identification of the local periodic structure is facilitated by the fact that the lattice is often Cartesian and the modulation is obtained not changing the local period, but changing the dimension and orientation of the elements inside a uniform lattice. In the following, we will refer to a lossless substrate with thickness *h* and relative permittivity *ε*_{r}, and to metallic patches arranged on a square lattice of period *d*.

The local periodicity assumption allows for the definition of average tangential fields in a simple way using a local Floquet-wave (FW) expansion. To this end, let us define the observation point on the lattice and impose across the periodic cell a phasing *k*_{x}*d* in the *x* direction and *k*_{y}*d* in the *y*-direction (figure 3). The transverse wavevector is associated with this phasing. Denote as **e**_{t}(** ρ**,

**k**

_{t}) and

**h**

_{t}(

**,**

*ρ***k**

_{t}) the surface tangential components of the

*total*electric and magnetic field on the surface, in the limit for

*z*approaching zero from positive values. Throughout this article, time harmonic fields are considered, with suppressed time dependence . The Floquet theorem allows for expanding these fields in a discrete set of orthogonal

*p*,

*q*indexed FWs with wavevectors at the nodes of the reciprocal (spectral) lattice, i.e. at , (

*p*,

*q*=0,±1,±2,…). We define the

*average*tangential (transverse to

*z*) field on the MTS as 3.1According to this definition, the average field in equation (3.1) coincides with the coefficient of the dominant, zero-indexed FW. In fact, the conjugate phase factor in the integrand makes a projection of the total field on the dominant FW. The electric and magnetic average fields at

*z*=0

^{+}are related through the tensor impedance as 3.2The impedance accounts for the contribution of both the patch cladding and the grounded slab. The impedance representing only the patch cladding can be written in terms of average magnetic field ‘jump’ through the following boundary condition: 3.3In the following, we will refer to the impedance in equation (3.2) as ‘opaque’ MTS impedance and to the one in equation (3.3) as ‘transparent’ MTS impedance. Both and depend on the wavevector

**k**

_{t}and on the angular frequency

*ω*. Furthermore, both of them satisfy, in absence of losses, the anti-Hermitian condition [25], where the superscript † denotes a transpose-conjugate operation. As such, they admit purely imaginary principal values (eigenvalues) and orthogonal principal vectors (eigenvectors). When the constituent elements possess two orthogonal symmetry axes, the impedance is of the form , , where () is real and symmetric. In the following, we will limit the analysis to this latter case.

The boundary conditions (3.2) and (3.3) can be incorporated in a two-port network model [26], where the MTS is represented as a shunt load connected to pieces of transmission lines associated with the transverse magnetic and transverse electric (TE) modes. When the opaque impedance model (3.2) is used, the shunt load terminates two branches of transmission line characterized by *z* propagation constant given by and equivalent impedances and (figure 4*a*), where
3.4Since we are interested in studying SW dispersion, we assume , which implies that the characteristic impedance of the two aforementioned transmission lines is purely imaginary. We also note that although both the two quantities in equation (3.4) are positive, owing to the different sign in the definition of the transmission line impedances, is capacitive while is inductive.

On the other hand, in the network model based on the transparent equivalent impedance, the shunt load is followed by pieces of transmission line of length *h* terminated in a short circuit and associated with TM/TE FWs. In principle, the model must also include transmission lines associated with higher order FWs. The characteristic impedances of these lines are defined as and . The number of interconnected ports depends on the distance between the patch layer and the ground plane. Indeed, higher order FWs decay exponentially away from the periodic surface, and only a few of them provide a non-negligible contribution at a certain distance. This provides a simple criterion for truncating the number of interconnected ports, as the modal ports relevant to non-interacting modes can be considered as not ‘accessible’ to the ground plane [26]. In the homogenization regime, only the dominant mode is accessible to the ground plane, and the multiport network can be reduced to a two-port network like the one represented in figure 4*b*. There, the transmission line branches terminated in a short circuit are characterized by *z*-propagation constant and equivalent impedance given by
3.5The tensor reactances and can be conveniently represented in the orthogonal basis , , where ,
3.6and
3.7where the superscript ‘*e*’ and ‘*h*’ denote TM and TE, respectively. As the unit vectors and are directed along the electric field of the TM and TE modes, respectively, equations (3.6) and (3.7) are denoted as representation ‘in the TE–TM basis’. The ground plane and the free-space loading, reported at the patch level, in the TE–TM basis are represented by two tensors and given by
3.8and
3.9with . Being and in parallel, the relationship between opaque and transparent MTS reactance is
3.10We observe that, since both opaque and transparent metareactances are symmetric, they can be diagonalized by a rotation of the coordinate system that aligns the axes with the eigenvectors. The two tensors can be therefore written in the form
3.11and
3.12where *X*_{n}(*X*_{nS}) and are the eigenvalues and the unit eigenvectors of .

### (b) Retrieval of metasurface reactance from method of moment analysis

The local periodicity assumption allows one to reduce the analysis to a single unit cell imposing periodic boundary conditions and/or using periodic grounded slab Green's function (GF). The analysis may be formulated by the method of moments (MoM) in the spectral domain, imposing a phasing across the periodic unit cell [26]. In principle, the analysis should be repeated for any couple of parameters (*ω*,**k**_{t}); however, it is possible to significantly reduce the computational burden by resorting to some kind of physics-based parametric modelling of the equivalent impedance, as shown in the following.

#### (i) Retrieval via pole-zero matching

The pole-zero matching method [26,27] is based on the two-port transmission line network in figure 4*b* and makes use of the Foster properties of the equivalent impedance of the printed layer. The first step is the derivation from the MoM matrix of the ‘transparent’ MTS reactance (figure 4*b*) in its diagonal form. Afterwards, the frequency dependency of the eigenvalues is approximated through a pole-zero (Foster-type) expansion. As an alternative to the pole-zero-type expansion, one can use an approach like the ones used for approximating the GF in multilayered media [28], e.g. the VECT-FIT algorithm [29]. This allows one to expand the eigenvalues in a summation of residue-pole terms of the following type:
3.13The poles and residues at the right-hand side are found by matching a few samples in the **k**_{t} plane. It is noted that it is preferable to approximate the transparent reactance in place of the opaque one, as its eigenvalues exhibit a smoother variation with **k**_{t}, thus allowing for a simpler interpolation of and . Each term of the expansion in equation (3.13) is associated with an LC series circuit, whose resonant frequency is determined by the pole . However, in the frequency range of interest typically only one pole for each eigenvalue is contained in the irreducible Brillouin region and the other poles are only used to improve the accuracy of the reactance representation.

#### (ii) Retrieval via asymptotic Green's function spectrum

An approximation of the previous method is possible when the ground plane is accessible to only one Floquet mode. The entries of the transparent MTS reactance expressed in the TE–TM basis (3.7) can be represented as [30]
3.14where the capacitances *C*_{n,m} and the inductances *L*_{n,m} can be found in analytical form as a function of the Fourier spectrum of the low-frequency irrotational and solenoidal components of the currents after using the asymptotic form of the spectral GF. The simplified representation in (3.14) is valid under the ‘single accessible mode’ condition, which implies [26]
3.15where λ=2*π*/*k* is the free-space wavelength.

## 4. Surface wave dispersion relation

The SW supported by an anisotropic MTS is in general hybrid; it propagates on the surface with transverse wavevector **k**_{sw} and exponential attenuation *α*_{z} in the *z*-direction. The dispersion equation *ω*=*ω*(**k**_{sw}), which relates **k**_{sw} to the frequency is defined by the resonance condition of the network in figure 4*a*
4.1with *k*_{ρsw}=|**k**_{sw}|.

### (a) Isofrequency dispersion curves

In practical engineering applications involving MTS, the focus is on *isofrequency dispersion curves* (IDCs). At a fixed angular frequency *ω*_{0}, this curve is the locus described by the tip of the vector as a function of the propagation angle (figure 5*a*). For any given angular frequency, the IDC represents the cut at *ω*=*ω*_{0} of the dispersion surface *ω*=*ω*(**k**_{sw}) (figure 5*b*).

The determination of the IDCs all over the irreducible Brillouin region by brute force full-wave analysis is not a trivial task, as it requires the construction of a three-dimensional database in the (*ω*,**k**_{t}) space, and the numerical solution of (4.1). A computationally cheaper approach is based on the retrieval methods described in §3.

After expressing the equivalent reactance in the eigenvector system, the dispersion equation in (4.1) can be factorized as follows:
4.2where
4.3and
4.4The eigenvector in (3.11) forms an angle *δ* with the *x*-axis (figure 6). Both *δ* and *X*_{n} depend in general on the angle of incidence *α*. The TM–TE components of the tensors in (3.2) can be written in terms of the eigenvalues in (3.11) as
4.5and
4.6where *ψ*=*α*−*δ*(*α*) is the angle between the direction of propagation and the eigenvector (. The dispersion equation written in the form (4.2) can be interpreted in terms of decoupled transmission lines with characteristic impedance of TM or TE type and specific (*k*_{ρsw}, *α*)-dependent reactive loads *X*^{QTM} and *X*_{QTE} (figure 7). It is worth noting that in general the modes supported by the two transmission lines are hybrid, although the TM or TE nature prevails when the first or the second factor in (4.2) vanishes, respectively.

The two transmission lines provide individual solutions for different modes that may or may not exist simultaneously at a given frequency. In particular, there is always a certain frequency under which the only mode that can propagate is dominantly TM. The evolution of the IDCs with the frequency identifies three frequency bands, which are illustrated in figure 7 and discussed next.

#### (i) Low-frequency band (figure 6a)

At very low frequency (quasi-static regime), the eigenvalues of are approximately equal (*X*_{1}≈*X*_{2}≈*X*). Their frequency dependence is almost linear and can be approximated as *X*≈*ωL*_{eq}, with *L*_{eq}=*μ*_{0}*h*(*ε*_{r}−1)/*ε*_{r}. This implies isotropic impedance and purely TM polarization. In metamaterial theory, this region is sometimes called ‘effective medium’ region. The IDCs are described by
4.7In this band, the shape of the printed element is weakly relevant, since the patches are too small to significantly affect the wavenumber with their shape.

#### (ii) Transition band (figure 6b)

For increasing frequencies, one has *X*_{1}≠*X*_{2} and the geometrical shape of the periodic element plays a role. The dominant polarization is still TM, with a very small TE contribution. If the element possesses at least one symmetry axis, an eigenvector of the reactance tensor is aligned with it. In this band, it is assumed that both *X*^{QTM} and *X*^{QTE} are positive and smaller than *ζ*. Therefore, the second factor in (4.2) never vanishes, that is, one cannot find a QTE-type solution. The dispersion equation becomes simply . When the direction of incidence coincides with one of the eigenvectors, , one has *X*^{QTM}=*X*^{ee}(*ψ*=0)=*X*_{1}(*ψ*=0). Furthermore, when *X*^{QTM}=*X*^{ee}(*ψ*=*π*/2)=*X*_{2}(*ψ*=*π*/2). Also, when the element has two orthogonal symmetry axes, the values obtained for these propagation directions are the maximum and minimum values of *X*^{QTM}, for instance and . Along these directions, the dispersion equation becomes therefore
4.8and
4.9The IDC can be approximated through the following elliptical-shape representation:
4.10The maximum and minimum TM eigenvalues can be approximated through a **k**_{t}-independent single-pole expression
4.11where *L*_{eq}=*μ*_{0}*h*(*ε*_{r}−1)/*ε*_{r}, *C*_{eq}=*ε*_{0}*ε*_{r}*h*/3 and *C*^{min,max} are the minimum and maximum quasi-static capacitance of the metallic cladding in the two principal axes of the cell. Within the validity limit in (3.15) and for some simple geometries, the latter can be approximated by
4.12where and *g*_{eff} is the equivalent gap in the two principal directions [31,32]. Note that *ν* is a quantity that can range from 0.55 to 1.85 for *g*_{eff}/*d*∈(0.1,0.4). The limit of validity of the approximation (4.11) with (4.12) is given by (3.15) and
4.13The above limit is identified as the limit of the transition bandwidth. Equations (4.8)–(4.10), although approximated, can be employed for a preliminary design of TO MTS. It is worth noting that in MTS-TO one should always operate close to the upper limit of this band. In fact, in this frequency region it is possible to exploit the capability of the elements to shape the IDCs without uncontrollable effects like onset of band-gap or excitation of other modes, which are described in the following section.

#### (iii) Dynamic band (figure 6c)

Increasing the frequency beyond the transition band implies a significant deformation of the IDCs. In fact, the dependence on frequency of the eigenvalues is very fast close to the resonance of the opaque impedance. In this region, one cannot use simple approximations; furthermore, the IDCs are also influenced by the periodicity lattice. The IDCs can be found using equation (4.2), which leads to 4.14and 4.15It may happen that one can jump from (4.14) to (4.15) when resonances are encountered. It may also occur that both (4.14) and (4.15) can be satisfied at the same frequency, thus leading to a dual mode regime.

The dispersion relations above require the numerical estimate of the eigenvalues *X*_{n} of the opaque impedance. These eigenvalues cannot be assumed to be independent of the wavenumber as in (4.8) and (4.9). In other words, (4.14) and (4.15) are implicit dispersion equations in which both sides of the equality depend on *k*_{ρsw}. *X*_{n} can be found by using the methods described in §3b.

We emphasize that, although in other practical applications (i.e. EBG, FSS) periodic surfaces are used within this bandwidth, the use of MTSs in the dynamic band is not recommended in the frame of TO.

### (b) Numerical examples

Examples of IDCs are given in the coloured maps of figure 8, which report the quantity *kd*/*π*=2*d*/λ as a function of *k*_{x}*d*/*π* and *k*_{y}*d*/*π*. These results have been obtained by using (4.14) and (4.15) with different resonance frequencies for the eigenvalues, which leads to two different shapes of the IDCs. Figure 8*a*,*b* is obtained with resonance frequency of *X*^{hh} lower and higher than that of *X*^{ee}, respectively. In the first case, the IDC may exhibit a hyperbolic behaviour starting from 2*d*/λ=0.3.

The results in figure 9 are relevant to the case of elliptical patches periodically printed over a grounded dielectric substrate with *h*=1 mm and relative permittivity *ε*_{r}=9.8. The lattice has period *d*=3 mm, and the semi-axes of the ellipses are *a*=0.45*d* and *b*=0.5*a*. Two cases are presented, where the axes of the printed ellipses are aligned with the lattice axes (figure 9*a*) or rotated of 45° with respect to them (figure 9*b*). The labels denote frequency *f*=*ω*/2*π* in gigahertz. For this example, we have

— quasi-static region (0<

*f*≤9 GHz): the IDCs are circles;— transition region (9 GHz<

*f*≤16 GHz): IDCs are quasi-elliptical and follow the orientation of the elliptical patches in the lattice;— dynamic region (

*f*>16 GHz): IDCs are deformed by the geometry of the lattice and depend on patches' rotation.

## 5. Metasurface transformation

Let us consider two half-spaces: a ‘virtual’ one, identified by a primed position vector , *z*′>0 and a ‘real’ one, whose points are identified by an unprimed position vector , *z*>0. Both the position vectors are expressed in Cartesian coordinates and unit vectors of their respective spaces. Both half-spaces *z*′>0 and *z*>0 are filled by *free space*; however, they have two *different boundary conditions* at *z*′=0 and *z*=0. We assume that the *virtual* space possesses boundary conditions described by a scalar reactive *ω*-dependent opaque reactance *X*_{ref}(** ρ**). In the

*real*space, instead, we define at

*z*=0 two complementary surfaces

*S** and

*S*, separated by a continuous line boundary ∂

*S*, whose summation covers the entire plane (figure 10). In

*S**, we assume the same impedance boundary condition described by the surface reactance

*X*

_{ref}(

**). In**

*ρ**S*, we impose an anisotropic opaque boundary condition characterized by an impedance . We note that, as a difference from previous papers of the authors of this paper [9,12], here we assume a space dependent .

Let us define a coordinate transformation that maps the real space in the virtual space, leaving unchanged the *z* coordinate. In particular, the transformation ** γ** maps univocally any point of the virtual space into a point of the real space. Its inverse transformation

**maps the real space into the virtual space. We also impose that the transformations are identities in the region of space**

*Γ**S**: 5.1and 5.2As the coordinate transformation is the identity in

*S**, the primed space is coincident with the real space there. The Jacobian of the transformation

**and its inverse can be expressed in covariant and contravariant bases as 5.3where**

*Γ***g**

_{x}=∂

**/∂**

*ρ**x*′,

**g**

^{x}=∇

_{t}

*x*′,

*γ*_{x}=∂

**′/∂**

*ρ**x*,

*γ*^{x}=∇′

_{t}

*x*, with analogous definitions for

**g**

_{y},

**g**

^{y},

*γ*_{y}and

*γ*^{y}.

### (a) Local isofrequency dispersion curves in the real space

Consider a TM–SW propagating in the virtual space, supported by the reactive impedance *jX*_{ref}(** ρ**′). Since

*jX*

_{ref}(

**′) is non-uniform in space, this SW propagates, some wavelengths away from the source, with a phase factor and a**

*ρ***′-dependent wavevector locally satisfying the eikonal equation 5.4Owing to the isotropic nature of the reference impedance in the virtual space, the IDCs in this space are circles (figure 10**

*ρ**a*). On the other hand, the SW in the transformed space is obtained by applying to the phase factor the transformation dictated by the coordinate mapping. This implies , or 5.5where ∇

_{t}

*Φ*(

**)=**

*ρ***k**

_{t}is the local wavevector of the SW in the real space. Combining (5.5) with (5.4), one can find the local IDC in the real space as 5.6where 5.7The practical implementation of boundary conditions which satisfy equation (5.7) at a particular frequency is based on the identification of (5.5) with the IDC at the same frequency of an anisotropic MTS. In fact, equation (5.5) represents ellipses with axes aligned with the eigenvectors of . By denoting these eigenvectors as one can write the diagonal form representation as where

*σ*

_{i}are the eigenvalues of This allows one to rewrite equation (5.6) in the form 5.8As shown in the previous sections, printing patches on a grounded slab realizes equivalent anisotropic boundary conditions which support SWs with IDCs which can be approximated by ellipses with axes oriented along the symmetry axes of the elements (equation (4.10)). Hence, the synthesis of the required IBC can be performed by matching the local wavenumber along the principal directions with the values given by (5.8). Comparing (5.8) with (4.10), one has 5.9where are the maximum and minimum TM-type principal values of the tensor reactance. Therefore, in order to implement the desired transformation, one should use a variable opaque impedance locally satisfying equation (5.9). To this end, and provided not to be in the dynamic band, one can also use (4.10) as a first approximation.

### (b) Conformal mapping

A particular case of transformation is represented by conformal mappings. These mappings satisfy the condition that angles of intersection between two lines are maintained after mapping. If we assume that the transformations *γ* and *Γ* are conformal, they satisfy the Cauchy–Riemann (CR) conditions. For the transformation *γ*(** ρ**′), these conditions are ∂

*x*/∂

*x*′=∂

*y*/∂

*y*′, ∂

*x*/∂

*y*′=−∂

*y*/∂

*x*′ which imply

*γ*^{x}⋅

*γ*^{y}=0, |

*γ*^{x}|=|

*γ*^{y}| and |∇′

_{t}

*x*|

^{2}=|∇′

_{t}

*y*|

^{2}. Therefore, , where is the identity dyad, and from (5.6) one has 5.10namely the impedance is isotropic, and therefore realizable through geometrically regular printed elements. If the mapping is quasi-conformal, one can have a good approximation by using 5.11

### (c) Non-radiating conditions and smooth variability

As in TO the local IDCs are constructed by using a mathematical transformation, control on physical realizability should be applied. The first condition is that the wave should not radiate during the path. This is obtained when the left-hand member of equation (5.9) is larger than one, which implies
5.12in any point of the synthesized plane. This condition actually limits the derivative of the transformation in any point of the synthesized plane. A second constraint comes from the condition in (2.3), namely ∇⋅**k**_{t}≪**k**_{t}⋅**k**_{t}. This latter and (5.12) pose the following limit on the variability of the transformation:
5.13

### (d) Design procedure for microwave devices based on metasurface transformation

The MTS transformation theory described above can be employed to design beam forming networks for planar antennas, and other flat devices. A flow chart representation of the design process is depicted in figure 11. The first step is the determination of the mathematical transformation, starting from the requirements on phase fronts and power flow direction. When the requirements can be matched by an orthogonal transformation, the design is performed by using (5.10) and resorting to an equivalent refraction index concept, which leads to a scalar inhomogeneous impedance. This impedance can be implemented by using patches with regular shapes, namely circular patches, circular patches with a circular hole, or square patches with square hole. As a general rule, larger patches correspond to higher impedance values, and etching a hole inside a patch allows to further increase the impedance value [9].

If the transformation is not orthogonal, equation (5.9) is used to determine the anisotropic impedance tensor, starting from the parameters of the transformation. The impedance tensor is next implemented in terms of anisotropic elements, like circular patches with a cut or a slot, or elliptical patches.

In any case, the patch associated with a given local value of impedance (either scalar or tensorial) is designed by using a two-step approach [9,19]: first, impedance maps are created representing the values of the impedance versus some geometrical parameters of the patch, then, the points on these maps that reconstruct the needed impedance profile are found. The maps are generated by analysing the single patch in a periodic environment; this approach relies on a local periodicity assumption.

## 6. Numerical examples

### (a) Maxwell's fish-eye lens

A planar Maxwell's fish-eye lens focuses each point on the lens edge to the opposite point on the same edge and has a refractive index profile that varies radially according to
6.1where , *R* is the lens' radius and *n*_{0} is the refractive index of the surrounding medium. A lens of this type, working at 7.5 GHz, has been implemented by using a modulated MTS providing local IDCs consistent with (6.1). The MTS has been obtained by printing two types of elements, circular patches and slotted circular patches (see insets in figure 12), on a grounded slab with permittivity 9.8 and thickness 1.575 mm. Such elements are arranged in a periodic lattice with square unit cell of side equal to 5 mm. In this implementation, *n*_{0}=*k*_{sw0}/*k*, where *k*_{sw0} is the SW wavenumber supported by the unprinted grounded slab. Full-wave numerical simulations have been performed by using the commercial software ADF (https://www.idscorporation.com/images/Downloads/Space/SPACE_ADF-EMS.pdf) based on the MoM, and numerical results are reported in figure 12. The simulated structure consists of an infinite grounded slab with patches printed inside a circular area with radius *R*=12.5 cm. A printed elementary dipole placed at the edge of the lens is used as excitation. As expected, the SW rays lunched by the dipole are all focused on the opposite side of the circumference.

### (b) Modified Luneburg lens

In the modified Luneburg lens, unlike in the classical Luneburg lens, the focus is not at the periphery, but inside the lens. It is characterized by an equivalent refractive index profile given by
6.2where is the distance from the lens' centre, *R* is the lens' radius, *n*_{0} is the refractive index of the background and *f* is the distance of the focus from the centre. The local refractive index (6.2) has been synthesized at 7.5 GHz by using both circular and square patches printed on a grounded slab with relative permittivity 9.8 and thickness 1.575 mm. Two cases, corresponding to *f*=*R*/2 (figure 13*a*) *f*=2*R*/3 (figure 13*b*), have been implemented with printed elements and analysed with ADF. In both cases, the printed elements are arranged in a periodic lattice with square unit cell of 5 mm size. A horizontal short monopole has been placed at the focal point. In the case *f*=*R*/2, the refractive index in (6.2) requires lower value of reactance, compatible with the use of circular patches. When the focal point is located more inside the lens, the increased level of reactance prevents the use of circular patches and imposes the use of square patches.

### (c) Beam splitter

A beam splitter is a device which allows one to halve an SW beam into two specular directions, while leaving unchanged the wavefront direction. To this end, we consider the coordinate transformation presented in [33] for bending the Poynting vector of an SW beam by an angle *θ* without changing the wavefront direction. A SW beam shifter based on this coordinate transformation was analysed in [8] by using an impedance sheet. In [13], as a further step, the tensor impedance is also synthesized by using ring-mushroom unit cells. Here the synthesis is carried out by using elliptical patches.

The Jacobian of the aforementioned coordinate transformation is a constant matrix given by
6.3It is noted that the linearity of the coordinate transformation yields the independence of the Jacobian matrix from the spatial coordinate; this means that the MTS needed to implement the transformation will be uniform. Therefore, to design a beam-splitting device one may simply use in the region *x*∈[−*A*,*A*], *y*∈[0,*B*] and in the region *x*∈[−*A*,*A*], *y*∈[0,−*B*]. The designed beam-splitting device operates at 9 GHz and spans a rectangular surface with dimensions *A*=30 cm and *B*=35 cm, while the deflection angle is *θ*=15°. In order to synthesize the require IDC, we have used elliptical patches [30] printed in 5×5 mm cells on a grounded slab with a thickness of 1.55 mm and a relative permittivity of 9.8. In accordance with (4.10), the values of the geometrical parameters of the elliptical patch have been set so as to approximate the principal axes of the desired elliptical dispersion curve arising from equation (5.8). The device has been analysed by using ADF and the results are shown in figure 14. As expected, the SW beam is split into two beams shifted by 15° upon encountering the anisotropic MTS.

## 7. Conclusion

A general methodology has been presented for designing TO devices using MTSs composed by patches printed over a grounded substrate. The wavefront of the SW is controlled by designing boundary conditions. The approach leads to the design of devices with novel functionalities, characterized by low losses, reduced thickness and inexpensive fabrication. The validity of the proposed approach has been demonstrated by means of some examples in the microwave range. The performances of the devices have been verified through full-wave simulations. Although the presented designs are focused in the microwave range, the proposed approach is applicable also in the terahertz and infrared regions, as well as at optical frequencies provided to have the right technology for MTS implementation.

## Authors' contributions

All the authors contributed to conceptualizing this manuscript. M.M. carried out simulations. S.M. wrote the first draft, E.M. and M.M. revised the manuscript. All authors gave final approval for publication.

## Competing interests

The authors have no competing interests.

## Funding

We received no funding for this study.

## Footnotes

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

- Accepted March 19, 2015.

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