## Abstract

Optical metasurfaces, typically referred to as two-dimensional metamaterials, are arrays of engineered subwavelength inclusions suitably designed to tailor the light properties, including amplitude, phase and polarization state, over deeply subwavelength scales. By exploiting anomalous localized interactions of surface elements with optical waves, metasurfaces can go beyond the functionalities offered by conventional diffractive optical gratings. The innate simplicity of implementation and the distinct underlying physics of their wave–matter interaction distinguish metasurfaces from three-dimensional metamaterials and provide a valuable means of moulding optical waves in the desired manner. Here, we introduce a general approach based on the electromagnetic equivalence principle to develop and synthesize graded, non-periodic metasurfaces to generate arbitrarily prescribed distributions of electromagnetic waves. Graded metasurfaces are realized with a single layer of spatially modulated, electrically polarizable nanoparticles, tailoring the scattering response of the surface with nanoscale resolutions. We discuss promising applications based on the proposed local wave management technique, including the design of ultrathin optical carpet cloaks, alignment-free polarization beam splitters and a novel approach to enable broadband light absorption enhancement in thin-film solar cells. This concept opens up a practical route towards efficient planarized optical structures with potential impact on the integrated nanophotonic technology.

## 1. Introduction

Natural materials provide the simplest ground to control and tailor electromagnetic fields, widely used by mankind since ancient times. The intrinsic electromagnetic properties of materials are based on the shape, orientation and lattice profile of their constitutive molecules, providing a wide range of refractive indices, chirality, nonlinearity, optical activity and dichroism effects. This variety in bulk constitutive parameters is at the basis of many fundamental applications in electromagnetics and optics [1–5]. However, at the same time, components based on natural materials are not often adequate to ultimately control the wave in the desired way and fulfil the growing demand for efficiency, compactness, speed and cost-effectiveness, particularly for integrated optical devices. The need for a broader space of degrees of freedom to arbitrarily manipulate the flow of light has stimulated large interest in metamaterials during the past years [6–8], with which the desired functionalities may be attained through engineering the shape and composition of *meta-atoms* supporting subwavelength resonances. This field has made substantial progress thanks to the recent advances in micro- and nano-fabrication methods [9–12], enabling giant light–matter interactions at the nanoscale. Several new functionalities have been accomplished or greatly enhanced by incorporating metamaterials in the design of new devices, ranging from negative-refraction and super-resolved imaging [13] to cloaking [14–16], extreme nonlinearity [17–20] and enhanced absorption [21].

Metamaterials offer a valuable potential in terms of design and optimization. However, wave manipulation in three-dimensional metamaterials is typically achieved by relying on the continuous propagation through these media, which is necessarily related to relatively high insertion loss and strong frequency and often spatial dispersions. Complexity of fabrication and associated challenges, moreover, demand alternative approaches to minimize undesired effects while offering similar prospects for wave management and control.

Reducing dimensionality of the structured media offers a possible solution to circumvent these drawbacks. Two-dimensional patterned surfaces, i.e. metasurfaces [22–24], eliminate the seemingly natural dependence on propagation effects inside the structured media by introducing abrupt, surface-confined shifts to the properties of electromagnetic waves in the form of generalized boundary conditions [25]. In general, a metasurface comprises an engineered array of spatially varying scatterers distributed in the lateral direction, with thickness much smaller than the operating wavelength. Exploiting strong wave–matter interactions at the planar interface, a metasurface can modify the properties of the incident light and transform it into the desired distribution of the scattered waves.

Metasurfaces have been originally introduced as periodic arrays, but their interaction with light is different from that of conventional gratings. The strong surface field localization ensures that the response of metasurfaces can be controlled by the resonance of their subwavelength inclusions, and not just by the lattice periodicity. Suitably placed components over different portions of the surface can modify the response in their vicinity, while a periodic arrangement of inclusions or large electrical apertures are not inherently required. The ultrathin, planarized features of metasurfaces are particularly appealing in the prospect of direct integration into nanophotonic systems. Metasurfaces relax the complicated requirements on the fabrication of bulk metamaterials, compatible with current nano-lithographic techniques. Important results have been recently attained in this context, and properly designed metasurfaces have been introduced to efficiently control and manipulate phase, amplitude, polarization and momentum of the optical beams [26–31]. Single and stacked metasurfaces have been exploited to implement several optical elements, such as polarizers, compact lenses, metareflect and transmit arrays, optical vortex plates, optical holograms and quarter wave plates over ultrathin volumes [26,27,31–34]. In this paper, we explore the concept of wavefront manipulation exploiting graded metasurfaces, and we show that the unique properties of metasurfaces, derived from their subwavelength exotic interactions with electromagnetic waves, allow interesting point-by-point scattering management in the near- and far-field, otherwise unattainable with conventional approaches. We use the equivalence principle to design metasurfaces that can control the impinging wave, yet merely comprising surface electric responses to suit practical implementation at shorter wavelengths. We demonstrate a number of advanced functionalities enabled by metasurfaces and provide a systematic design approach to achieve interesting optical effects, while avoiding many inherent limitations associated with currently available techniques. The proposed designs comprise single metareflectors to impart a locally controlled phase signature on the impinging optical beam, operating as ultrathin spatial phase modulators. We demonstrate that, by locally tailoring their surface impedance profile, a single, ultrathin, patterned surface can generate any arbitrary prescribed scattering response, with direct implications for ultrathin carpet cloaks, compact polarization beam splitters and improved solar cell technology. The metasurfaces are realized by employing two commonly available optical materials, in the form of conjoined optical nanoparticles with broadband, low-loss and angularly stable response [27]. The proposed metasurface-based devices eliminate the inevitable complexity associated with manipulating the electromagnetic wave flow in three-dimensional and inhomogeneous metamaterials [14,35,36], and may open new possibilities towards the design of new optical elements with significantly reduced profile, high performance and potential tunability.

## 2. Concept and design of graded metasurfaces

Based on the equivalence principle, it is well established that a proper arrangement of surface currents can independently control the distribution of electromagnetic fields on either side of a mathematical boundary [25]. In general, the solution of Maxwell's equations exhibits abrupt variations in all field components at such a virtual boundary, generated by the presence of both magnetic and electric currents on the surface (figure 1*a*). The discontinuity may be attained through applied currents in the form of locally arranged driven dipole sources or, alternatively, by means of induced polarization currents in electromagnetically polarizable particles [37]. The latter is more appealing, not only because it may eliminate the need for actively controlled sources, but also because it allows using a passive surface for an arbitrary amplitude and phase of the incident wave. In particular, the surface electric admittance and magnetic impedance can be locally tailored to grant a prescribed scattering response, independent from the excitation beam. Clearly, such a polarizable surface may require the control of both electric polarization and magnetization.

Unfortunately, owing to the lack of availability of magnetic effects, and the strict requirements on the magnetic properties of such a surface, the equivalence principle is not directly applicable at high frequencies. At microwaves, an immediate candidate to realize an electromagnetically polarizable surface is to employ loaded wires and loop antennas [37] or metallic patterns [38], designed to follow the desired effective sheet impedances, at least at closely spaced discrete points on the surface. Magnetic effects, however, saturate towards visible wavelengths [39], and an electrically thin metasurface exhibits negligible tangential magnetic response for normal incidence [31]. Therefore, in order to apply the equivalence principle at these frequency bands, it is crucial to suitably adapt a solution based on a polarizable surface with purely non-magnetic elements. There is a possibility to evade the constraints on magnetic properties of the surface by imitating their role with solely electric metasurfaces. Following simple symmetry restrictions, a single electric interface scatters the impinging wave equally into both directions, and thus it is impossible to independently control the electromagnetic wave distribution on both sides of the interface. The principal consequence of introducing mutual electric and magnetic currents on the surface is to break such scattering symmetry, to some extent analogous to asymmetric radiation from a Huygens antenna [40]. In this context, recently we demonstrated the possibility of breaking the scattering symmetry of a non-magnetic metasurface by using stacked or grounded metasurface configurations [27,34], and efficiently controlling the local transmission or reflection phase at will. Following these proposals, here we provide a general design methodology and surface engineering to transform any incident wave form to an arbitrary scattering profile. To avoid the practical challenges associated with experimental implementation of stacked inhomogeneous metasurfaces, which may require precise alignments between layers, this work is focused on fully reflecting configurations, as depicted in figure 1*b*.

Let us start by formulating the reflection equivalence principle in its most general form. In figure 1*a*, an inhomogeneous electromagnetic metasurface is excited by an optical beam with field components (*E*_{i}(** r**),

*H*_{i}(

**)), described at each point**

*r***in space, producing a scattered wave profile (**

*r*

*E*_{s}(

**),**

*r*

*H*_{s}(

**)) in the upper region. The total fields are enforced to be zero in the region below the metasurface. Both incident and scattered waves can be arbitrary, as long as they are proper solutions of the homogeneous Maxwell's equations in free space. To account for field discontinuities over the boundary**

*r***=**

*r*

*r*_{s}, the sheet electric admittance and magnetic impedance are imposed over the surface and the primary boundary conditions read [25,41] 2.1where the subscript

*t*denotes the surface tangential component of the fields, and is the local normal unit vector on the surface, which, without loss of generality, we set towards at the point of interest. An e

^{jωt}time harmonic variation is also assumed, where

*ω*is the angular frequency of operation.

The surface tensors and required to produce the desired scattered fields are in general 2×2 matrices directly retrievable from (2.1). For simplicity of notation, we assume that the scattered fields are chosen such that the surface impedance and admittance tensors are diagonal and that the metasurface does not require any polarization coupling effect, i.e. *Y*_{e,xy}=*Y*_{e,yx}=*Z*_{m,xy}=*Z*_{m,yx}=0. Readily available from (2.1), the local required surface admittances and impedances read *Y*_{e,xx}=4/*Z*_{m,yy}=−2*H*_{y}/*E*_{x} and *Y*_{e,yy}=4/*Z*_{m,xx}=2*H*_{x}/*E*_{y}, where (*E*,*H*)_{x,y} denote the total field components on the surface. In general, the derived surface admittances may be complex, as a function of the distribution of impinging and scattered fields; in addition, the necessity of operation in reflection mode imposes an essential constraint on the relationship between electric and magnetic properties of the surface, i.e. and are not independent. The local surface impedance values can also be described through scattering parameters [41], i.e. reflection and transmission matrices, which are defined under normal plane wave illumination in the electronic supplementary material.

We now consider the asymmetric configuration illustrated in figure 1*b*, which models a single *electric* metasurface interface [41,42], and it can be analogously described by local reflection coefficients
2.2where *Y*_{eg,xx} and *Y*_{eg,yy} are the diagonal elements of the non-magnetic admittance tensor describing the surface, *η*_{0}=120*π*Ω is the free-space characteristic impedance and *k*_{s} is the angular wavenumber in the substrate. Subscripts TE/TM are defined as orthogonal polarization states with electric/magnetic fields along the *y*-axis. The two systems in figure 1*a*,*b* have a similar response, provided that
2.3Employing an electrically polarizable metasurface to emulate the response of magneto-electric surfaces is a critical step to apply the equivalence principle in a simplified form, amenable to work at optical frequencies, and it enables nanoscale scattering wave management. Indeed, our goal is to control the scattering signature of the metasurface in order to locally tailor the outgoing wave, as desired by design. By engineering the metasurface texture, appropriate secondary sources (induced surface currents) are produced on the metasurface which perform as equivalent source distributions for the scattered wave. The metasurface design can be summarized as the synthesis of an inhomogeneous surface admittance profile in (2.3) that transforms the impinging wavefront into the desired scattered one. As noted above, the required admittance is generally complex, and therefore loss and gain elements may be required to synthesize the required real portion of . As intuitively expected, this condition becomes extreme if the incident and scattered wavefronts do not sufficiently overlap on the surface, and the surface properties are desired to be maintained local. In this scenario, in fact, the metasurface would necessarily be required to absorb energy in one location, and re-radiate it from somewhere else to generate the desired scattering profile. Fortunately, in many practical applications impinging and scattered beams overlap, and passive metasurfaces may be sufficient. In order to facilitate the physical implementation of the structure and avoid incorporation of active elements [43], the required surface profile in these cases may be approximated as purely imaginary (i.e. lossless), while allowing a high conversion efficiency [44] to the desired scattering profile. The potential performance degradation in doing this approximation from equation (2.3) may also be minimized through a round of optimization of the obtained reactive surface profile.

Investigating equation (2.2), it is clear that, under the lossless condition of , the local amplitude of reflection is unitary, and the reflection phase can span the entire 360° range by varying the magnitude of surface reactance. In view of recent advances in metasurface technology, such a metasurface may be realized with several approaches, ranging from improved reflectarray and frequency-selective surface technology [45,46] at microwaves and radio-frequencies to V-shaped nano-rod antennas, gap-plasmon resonators and hybrid plasmonic–dielectric nanoparticles at infrared and optical frequencies [26,27,47]. The proper choice of metasurface building blocks can offer additional design flexibilities in terms of attainable resolution, chromatic dispersion and intrinsic loss. Figure 1*c* illustrates two suitable candidates to realize an inhomogeneous metasurface at optical frequencies, as originally introduced in [27]. The surface elements consist of only two materials, selected to have opposite signs of permittivity, i.e. the combination of a dielectric material and a plasmonic metal. In the limit of small size of these *meta-atoms*, i.e. *l*≪λ_{0}, *d*≪λ_{0}, the dielectric element operates as a quasi-static nanocapacitor and the plasmonic portion forms a confined nanoinductor whose effective values depend on the filling ratio as well as on the optical properties of the constitutive materials. Their combination offers a size-independent and surface-confined impedance block, directly and effectively tunable through varying the filling ratio of the metallic portion [27]. Specifically, the local surface impedance of such nanoblocks can be tailored to acquire any desired value by employing readily available optical materials, such as silver (Ag), gold, silicon (Si), silicon dioxide (SiO_{2}) or optical semiconductors [48]. Figure 1*c*(ii) shows two examples of inhomogeneous metareflectors realized based on these elements, deposited on a thin layer of dielectric substrate and backed by a ground plane, physically implementing the configuration in figure 1*b*. The two examples show a polarization-dependent surface (left), in which the local surface admittance strongly changes for *x*-polarized and *y*-polarized illumination, and a polarization-insensitive surface (right), for which the surface is essentially invariant with polarization, i.e. *Y*_{eg,xx}=*Y*_{eg,yy}. The optical nanoblocks introduced in figure 1*c*, moreover, exhibit robust performance to intrinsic material losses and allow reactance control at deeply subwavelength scales. As the local reactance of each surface element is solely determined by the filling ratio between the two constitutive materials, the total size of each block can be deeply subwavelength (*l*≈λ_{0}/10), providing high spatial resolution and strong angular stability [27].

Either when working at low frequencies (i.e. with magneto-electric metasurfaces) or at visible wavelengths (i.e. purely electric metasurfaces), equations (2.1) and (2.3) must be cautiously handled while physically realizing the structure. Once the required values of surface admittance and/or impedance are known, a typical approach to implement the surface is to discretize the continuous required surface impedance profile into subwavelength portions. Each segment is then realized using the associated meta-atom as in figure 1*c*, based on the phase that would be obtained from a periodic configuration [49]. In reality, however, coupling between neighbouring elements cannot be neglected [50], and it may affect their properties when assembled together to realize the inhomogeneous surface profile in (2.1) or (2.3). A more consistent approach to ensure the desired performance is to fine-tune the properties of the surface through optimization or direct measurement of the local fields. The details of the proposed procedure to design these graded metasurfaces can be found in the recent literature [27,34,51] and in the electronic supplementary material.

In the following, we explore the significance of our proposed point-by-point scattering management technique for three relevant applications at visible wavelengths. First, we analytically design and physically implement a unidirectional carpet cloak which is capable of reducing unwanted scattering from arbitrarily large objects employing a single inhomogeneous surface admittance, expanding and extending our recent work on the topic [51]. Next, we discuss an ultrathin polarization beam splitter implemented using a polarization-sensitive wave-deflecting metasurface. The configuration is tailored to create large divergence angles between TE and TM incident optical beams and efficiently split a circularly polarized wave into its linear components. Finally, we discuss a way to increase the propagation path length inside thin-film solar cells and improve their absorption properties by replacing the solar cell back-reflector with a suitably designed graded metasurface.

## 3. Applications

### (a) Conformal carpet cloaking

To prove the generality and power of the proposed concept, we demonstrate the possibility of hiding objects from an external observer using an ultrathin unidirectional carpet cloak, which is not based on bulk metamaterials, as in [36,52,53], but rather using a single, ultrathin, non-periodic metasurface. We aim at concealing the object between the metasurface and a ground plane, such that the whole system mimics a flat reflecting surface. For simplicity, we assume a two-dimensional configuration illuminated by a TE-polarized Gaussian beam at λ_{0}=500 nm, as depicted in figure 2*a*. A triangular PEC bump with lateral size *L*=10λ_{0} and centre height *H* = 1.6λ_{0} is placed on the PEC ground and illuminated at 45°. The presence of the bump results in total deformation of the input wave, as expected, and the incident beam experiences a new boundary condition at each point on the surface of the obstacle (figure 2*c*). The metasurface to cloak the object is readily characterized following equations (2.1) and (2.3), by inserting the incident and desired scattered waves. The admittance profile is calculated assuming *d*_{s}=λ_{0}/10 and *n*_{sub} = 1, as plotted in figure 2*b*. The detailed design procedure for this example can be found in the electronic supplementary material, pushing forward the performance and operation limits of our first results on graded metasurfaces operating as carpet cloaks presented in [51], which were based on a more basic ray-optics approach.

For cloaking, the optimal surface admittance is generally complex and, interestingly, the requirement on the surface admittance is to have lossy elements (i.e. Re[*Y*_{eg,yy}]>0) on one edge and active components (i.e. Re[*Y*_{eg,yy}]<0) on the opposite side, while the reactive portions are symmetric. This shares interesting analogies with the recently proposed parity-time symmetric cloaking configurations in which a balanced combination of loss and gain may create unidirectional cloaking in various set-ups [54–56]. The distribution of the electric field around the obstacle is shown in figure 2*d*, when the cloaking layer illustrated in figure 2*b* is applied at the predesignated position around the object. The presence of the cloak successfully enforces the desired specular reflection pattern, with negligible residual scattering associated with the finite thickness of the configuration (*d*_{s}=50 nm). Next, we relax the requirement on spatially distributed loss/gain over the surface and replace the exact complex admittance of figure 2*b* with its reactive portion, i.e. *Y*_{eg,yy}=*j* Im[*Y*_{eg,yy}]. Shown in figure 2*e* is the corresponding electric field distribution of the passive cloak, clearly displaying sub-optimal performance compared with the exact solution. Yet, even with the approximate passive cloak, the unwanted scattering is largely suppressed, which is particularly interesting considering the large dimensions of the obstacle compared with the operation wavelength. The residual scattering can be minimized by further optimizing the passive cloaking layer, which we do in the next example, physically implementing a surface cloak based on the admittance elements shown in figure 1*c*. The results presented in this example have been performed via full-wave simulations in COMSOL Multiphysics (http://www.comsol.com/comsol-multiphysics).

In the previous example, the reactance profile presented in figure 2*b* is imposed on the metasurface in a continuous manner and with infinite spatial resolution. It would be insightful to physically implement a metasurface-based cloak and study its performance under realistic conditions. For this purpose, here we implement a graded metasurface to effectively conceal a PEC triangle with physical dimensions of *L*=1500 nm and *H*=200 nm at λ_{0}=500 nm and under 45° TE illumination. The metasurface thickness is set at *d*=λ_{0}/10 to ensure accurate sheet approximation, and we choose *d*_{s}=100 nm and *n*_{sub}=1. The metasurface elements are the conjoined dielectric–plasmonic particles shown in figure 1*c* (left) and are made of combinations of a high index dielectric (*n*=3.46) and a plasmonic metal (Ag). Silver dispersion and realistic losses are taken into account considering the Drude-type permittivity model , with , *ω*_{p}=2*π*×2175 THz and *γ*=2*π*×4.35 THz based on experimental data [57]. The required surface profile is then discretized into 26 segments and the entire surface of the object is covered with elements of 97 and 112 nm lateral sizes (approx. λ_{0}/5) to guarantee an acceptable resolution at 500 nm. Detailed design parameters are available in the electronic supplementary material. The filling ratio of each element is theoretically calculated to acquire the required reflection properties according to the theory developed in the previous section, and then fine-tuned through proper optimization to properly consider mutual coupling effects and the approximation of neglecting the resistive portion of the impedance. A three-dimensional sketch of the final set-up is shown in figure 3*c*.

Figure 3*a*,*b* compares the electric field distribution in the incidence plane, when the bare and cloaked objects are illuminated by a Gaussian beam at 45° (the design angle). As desired, the cloaked set-up scatters like a flat ground plane and the near-field around the scatterer is fully restored. This is quantitatively shown in figure 3*d*, in which we plot the total electric field (i.e. ) along a hypothetical line placed 20 nm above the surface. Incorporating a single 50 nm thick graded surface, both amplitude and phase of the electric field are successfully reconstructed at each point to those of a reference flat mirror. Owing to the subwavelength thickness of the metasurface and small lateral foot-print of the elements, the metasurface response is stable with respect to the angle of incidence [27,46]. This is further studied in figure 3*e*, in which we excite the object under different incident angles and record the field magnitude on a half-circle placed at *R*=10λ_{0} (indicated in figure 3*a* by a dashed line). The numerical simulations confirm the angular stability of the designed cloak over more than ±10° range (http://www.cst.com/).

The presented design methodology is completely independent of the shape of the scatterer, and the metasurface may be placed at any virtual boundary to replicate the desired near-field distribution, allowing one not only to conceal but also to create virtual illusions and deceive the observer [58]. In our examples, unidirectional cloaking is achieved by exploiting a single ultrathin surface, providing a viable solution for practical implementation of metamaterial cloaks. The angular stability of the design ensures an acceptable performance for a wide range of applications. In comparison, transformation optics techniques [36,52,53,58,59] typically require an electrically large (several free-space wavelengths) metamaterial block placed over the object and a careful refractive index control inside the entire cloaking volume. At microwave frequencies, real-time tuning and reconfigurability of circuit elements may even further reinforce our cloaking technique. In this regard, tunable carpet cloaks may be conveniently designed with deeply subwavelength thicknesses employing varactor-loaded tunable impedance surfaces [45].

### (b) Ultrathin polarization beam splitter

A polarization beam splitter is an optical device intentionally designed to distinguish different polarization states of an unpolarized optical beam. Conventionally, natural birefringent materials, such as calcite crystals, may grant such functionality through creating beam displacement between orthogonal polarizations or by reflecting one polarization state while fully transmitting the other. These effects are yet naturally weak and entail long-distance propagation of the optical beam inside the birefringent crystal to accumulate the desired levels of extinction ratio. Graded metasurfaces offer a potential solution to these inherent limitations, as the metasurface elements can be designed to exhibit a drastically polarization-dependent scattering signature [27,60,61], as discussed for the plasmonic–dielectric composite particles in figure 1*c*.

With the proper choice of surface admittance, graded metasurfaces can be designed to enforce distinct functionalities based on the polarization of the excitation field, e.g. to steer incident light into different directions under TE or TM illuminations [60]. To further outline our proposed point-by-point scattering management technique, in this section we design and implement a beam steering surface to redirect an obliquely incident TE plane wave illuminating the surface at *θ*_{i}=10° towards *θ*_{r,TE}=50° while the TM wave experiences a specular reflection and *θ*_{r,TM}=10°. The metasurface creates an abrupt birefringence with 40° divergence angle at the design wavelength of 500 nm, emulating the functionality of a Wollaston prism over an ultrathin profile, with the potential of integrability into nanophotonic systems and polarization control at the subwavelength scale.

The required local reflection coefficient of the surface is analytically calculated and plotted in figure 4*a* (solid lines). Owing to the plane wave nature of both incident and scattered waves, the surface has a superlattice periodicity *L*=844 nm, as also apparent in the plot. For a physical implementation of the structure, the surface profile is divided into eight equally sized segments within each supercell, which we then realize using the conjoined particle illustrated in figure 4*a*(i). In this case, we consider a substrate thickness *d*_{s}=50 nm and *n*_{sub}=1.45. The parallel sheet admittance *Y*_{eg,yy} is effectively controlled by varying the filling ratio of the plasmonic portion. Over this range, *Y*_{eg,xx} is approximately constant and the variations in the TM reflection phase *R*_{g,TM} are less than 30°, ensuring efficient specular reflection for this polarization. The discrete points in figure 4*a* indicate the physically realized local reflection coefficients for the graded metasurface, also schematically shown in figure 4*b*. Numerical simulations (http://www.cst.com/) of the designed surface under plane wave illumination confirm that over 91% of the incident TE component and less than 0.19% of the TM component are redirected towards the *θ*_{r}=50° direction, while 98% of the incident TM wave and less than 1% of the TE polarized wave are reflected towards *θ*_{r}=10°. We further examine the performance of the designed structure when illuminated by a finite size, circularly polarized Gaussian beam, as shown in figure 4*b*. The incident wave clearly splits into two branches, which we verified to be TE and TM in the field plots presented in figure 4*c*,*d*. We discuss more details of the design, bandwidth and angular stability of the structure in the electronic supplementary material.

### (c) Light trapping in thin-film solar cells

Owing to weak light–matter interactions inside its absorbing material, a thin-film solar cell intrinsically absorbs only a small percentage of the incoming wave, proportional to its physical thickness. However, around the natural Fabry–Perot resonances of the film, a strong vertical standing wave is created inside the slab and relatively high absorption levels may be attained from the structure. Here, we present a way to employ graded metasurfaces as ground planes for thin-film photovoltaic cells in order to artificially create large standing waves in the lateral direction and increase the optical path of the impinging beam inside the cell. We show that the generated local hot spots inside the active layer can significantly improve the absorption properties of the film, and this effect may be induced over broad bandwidths.

Figure 5*a* schematically illustrates the proposed technique: a weakly absorbing thin film is coupled to a graded metasurface back-reflector, and the whole configuration is tailored to redirect the impinging wave towards a new direction (i.e. 45° off-normal, as shown in figure 5*a*). Wave deflection, which is equivalent to imposing a constant transverse momentum along the surface [26], enforces multiple partial internal reflections inside the active layer, owing to the refractive index contrast between the dielectric film and free space. This process successfully creates lateral standing waves inside the semiconductor layer that are anticipated to increase carrier collection efficiency due to enhanced light–matter interactions.

In order to demonstrate the proposed light-trapping scheme, we consider a thin-film active layer with refractive index and absorption length ranging from 0.9 to 5.7 μm (between 450 and 1100 THz). The optical properties of the thin film are shown in figure 5*b* (inset). The semiconductor layer is assumed to be 180 nm thick, and it is placed on top of a graded metareflector composed of conjoined composite particles shown in figure 1*c* (left). The structure is then designed to impart a transverse momentum able to bend the wave impinging from normal incidence by an angle of 45° at 500 nm. Analogous to the polarization beam-splitting surface, this functionality is realized with a superlattice periodicity, suitable for large-scale solar cell designs. The required profile periodicity is discretized into eight steps, and in each segment the filling ratio of the plasmonic portion (Ag) is tailored to replicate the calculated ideal reflection phase from the structure. We set the metasurface thickness at *d*=50 nm and the entire structure is deposited on a thin layer of *d*_{s}=30 nm silicon dioxide (*n*_{sub}=1.45) backed by a PEC ground plane. In this example, the *meta-reflectarray* is designed for a one-dimensional set-up, tuned for TE polarized impinging waves. A similar concept can be easily extended into a two-dimensional matrix in order to provide an isotropic response for all input polarizations, for instance by employing the concentric graded metasurface shown in figure 1*c* (right).

Figure 5*b* shows the numerically calculated absorption spectra of the solar cell, highlighting the overall absorption enhancement. The shaded area represents the additional absorption attained by patterning the back-reflector, indicating an improvement factor of 2.6 at the design frequency. As predicted, while the total structure aims at coupling the impinging wave to the first Floquet order (i.e. *θ*_{r}=45°), the higher refractive index in the semiconductor allows multiple internal reflections inside the dielectric, ensuring an overall longer propagation distance inside the thin-film active layer (figure 5*c*). Notably, the coupling efficiency to the first scattering order (i.e. wave bending) using graded metasurfaces is very high over a relatively broad range of frequencies [27]. This phenomenon resides in the inherently broadband response of the exploited composite nanoparticles, and accordingly allows us to predict that this light-trapping mechanism can be broadband, as verified in figure 5*c*. The reflections build up to ensure strong light concentration and trapping inside the cell, consequently increasing the total collected photocurrent, except possibly in the vicinity of Fabry–Perot resonances of the original configuration (occurring around 720 THz). In addition, depending on its thickness, the film may support a number of guided optical modes [62–64]. Typically, a substantial light-trapping effect may also be achieved at the resonant frequency of these modes (around 540 and 1000 THz in the current example), over narrow spectral widths of a few nanometres. However, these resonances are considerably broadened in our configuration and, combined with the broadband background absorption enhancement produced by the gradient metareflector, they significantly boost the overall efficiency, which is particularly important in the frequency bands where the original material fails to properly harness the solar radiation.

## 4. Conclusion and discussion

A general approach to apply the equivalence principle for optical metasurfaces has been discussed here, allowing one to locally manipulate the scattering distribution at will. We have applied this concept to a few relevant examples at optical frequencies. We have shown that a single electrical metasurface, with deeply subwavelength thickness, can be used to fully control the scattering signature of optical elements such as thin-film solar cells and polarization beam splitters, and drastically boost their performance, while minimizing the size of the device and improving the robustness to losses. Moreover, we have designed graded metasurfaces to perform more complex functionalities, such as cloaking, by reconstructing the desired field distribution in the near-field. Owing to their conformal profile and the fact that our designs are all based on the alternation of commonly available optical materials, the proposed optical films may be directly integrated into nanophotonic devices. Our technique may also be scaled up to longer wavelengths to realize ultrathin infrared, terahertz or microwave elements, including broadband couplers, selective filters and real-time conformal cloaks. The described control and manipulation of optical scattering through metasurfaces, along with their compatibility with standard lithographic techniques and on-chip fabrication technologies, pave the way to several applications in optics, and open up a new route to design compact, planarized optical devices.

## Data accessibility

Supporting data are available in the electronic supplementary material.

## Authors' contributions

N.M.E. and A.A. conceived and developed the concept. N.M.E. set up and performed the computations and structure optimization; A.A. planned and directed the research; all authors wrote and revised the manuscript.

## Competing interests

The authors declare that they have no competing interests.

## Funding

This work has been supported by ONR MURI grant no. N00014–10–1–0942, NSF CAREER award no. ECCS-0953311, Welch Foundation grant no. F-1802, AFOSR grant no. FA9550–13–1–0204 and ARO grant no. W911NF-11–1–0447.

## Footnotes

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

- Accepted February 11, 2015.

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