Issue 
EPJ Applied Metamaterials
Volume 2, 2015
Advanced Metamaterials in Microwaves, Optics and Mechanics



Article Number  12  
Number of page(s)  11  
DOI  https://doi.org/10.1051/epjam/2015016  
Published online  26 January 2016 
https://doi.org/10.1051/epjam/2015016
Research Article
Synthesis of electromagnetic metasurfaces: principles and illustrations
^{1}
Dept. of Electrical Engineering, Polytechnique Montréal, Montréal, QC
H2T 1J3, Canada
^{2}
Electrical and Computer Engineering Dept. King Abdulaziz University, P.O. Box 80204, Jeddah
21589, Saudi Arabia
^{*} email: karim.achouri@polymtl.ca
Received:
1
September
2015
Accepted:
19
November
2015
Published online: 26 January 2016
The paper presents partial overview of the mathematical synthesis and the physical realization of metasurfaces, and related illustrative examples. The synthesis consists in determining the exact tensorial surface susceptibility functions of the metasurface, based on generalized sheet transition conditions, while the realization deals with both metallic and dielectric scattering particle structures. The examples demonstrate the capabilities of the synthesis and realization techniques, thereby showing the plethora of possible metasurface field transmission and subsequent applications. The first example is the design of two diffraction engineering birefringent metasurfaces performing polarization beam splitting and orbital angular momentum multiplexing, respectively. Next, we discuss the concept of the electromagnetic remotely controlled metasurface spatial processor, which is an electromagnetic linear switch based on destructive interferences. Then, we introduce a nonreciprocal nongyrotropic metasurface using a pickup circuit radiator (PCR) architecture. Finally, the implementation of alldielectric metasurfaces for frequency dispersion engineering is discussed.
Key words: Metasurface synthesis / Birefringence / Refraction / Orbital angular momentum / Frequency dispersion engineering / Metamaterial switch / Nonreciprocal metadevice
© K. Achouri et al., Published by EDP Sciences, 2016
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
1 Introduction
Metasurfaces, which are the twodimensional counterparts of volume metamaterials [1–3], have attracted much attention over the past years. Due to their low profile, small losses and rich electromagnetic field manipulation capabilities, they are excellent spatial processors able to manipulate electromagnetic waves with ever more complex possible applications, such as generalized refraction, polarization transformation, signal multiplexing, and nonreciprocal field control.
Metasurfaces are usually made of uniform or nonuniform arrangements of specifically engineered subwavelength scattering particles to produce a desired scattered field. Many such metasurfaces have been reported. However, these metasurface were most often designed using heavy optimization techniques. Fortunately, a few efficient synthesis techniques have been recently introduced, offering powerful and versatile tools for metasurface design [4–8].
In this paper, we briefly recall the synthesis technique which is detailed in reference [4]. Next, we illustrate the capabilities and the advantages of this synthesis technique. The various metasurfaces addressed here will be described and analysed more thoroughly in other future publications.
2 Metasurface design
2.1 Susceptibility synthesis
Let us consider that a metasurface lying in the xy plane at z = 0. Given its inherent subwavelength thickness, a metasurface may be modeled as a zerothickness spatial discontinuity. Rigorous boundary conditions, pertaining to such a discontinuity, were first derived by Idemen [9] and later applied to metasurfaces by Kuester et al. [10]. Conventionally called the Generalized Sheet Transitions Conditions (GSTCs), they read(1a) (1b) (1c) (1d)where the time dependence e^{jωt} is dropped by convenience. The permittivity ε and the permeability μ correspond to the medium surrounding the metasurface. The sign Δ corresponds to the difference of the electromagnetic fields (E, H, D and B) between both sides of the metasurface, P and M are the electric and magnetic polarization densities, respectively. The subscript  refers to the vectorial components that are in the plane of the metasurface. The synthesis technique [4] is based on the relations (1a)–(1d) and determines the susceptibilities required to perform the specified wave transformations.
Relations (1a)–(1d) form a set of coupled partial differential equations. To simplify the analysis, we consider that the metasurface is composed of only surface polarization densities, which reduces relations (1a) and (1b) to a set of coupled linear equations that can be solved for closedform expressions of the metasurface susceptibilities. Upon substitution of the general definitions of P and M, in terms of the bianisotropic susceptibility tensors and [11], (1a)–(1d) reduces to(2a) (2b)where the polarizations densities P and M are expressed in terms of the arithmetic average of the fields (as denoted by the subscript “av”) instead of the acting or total fields at the position of each scattering particles [10].
The mathematical synthesis of the metasurface essentially consists in inserting the electromagnetic fields of the specified transformations into (2a) and (2b) and solving for the susceptibilities. Depending on the desired transformation, the system of equations (2a) and (2b) may be underdetermined, in which case some of the tensor elements must be set to zero for a definite solution [4]. Once the susceptibilities of the metasurface are determined, the metasurface can be physically realized, which is the topic of the next section.
2.2 Scattering particle synthesis and implementation
The physical realization of metasurfaces is, as of today, not a trivial routine. Here, two possible approaches are briefly discussed, while many more exist. In both approaches, the susceptibility functions, obtained by the aforementioned synthesis procedure, are spatially discretized in the xy plane where each discrete point corresponds to a unit cell to be implemented. The unit cells are simulated (assuming periodic boundary conditions), one by one or group by group when the structure is fully or partly nonuniform, using commercial softwares that compute their scattering parameters. The required physical parameters for the scattering particles are obtained by mapping the scattering parameters into the susceptibility function [4]. Finally, an enhanced design is achieved by tuning the parameters of the scattering particles via parametric analysis or standard optimization techniques.
2.2.1 Metallic scatterers
Metasurfaces consisting of metallic scatterers, typically on dielectric substrates, have been the most commonly reported types of metasurfaces, as well as their predecessors, the frequency selective surfaces (FSS) [12]. In all cases, using more than one layer is an effective way to increase the available number of degrees of freedom, and hence achieving enhanced properties, including higher bandwidth and larger phase coverage range of the structure’s unit cell. It was recently shown that three cascaded layers, where the two outer layers are identical, represents the minimum configuration to achieve full transmission and a 2π phase coverage [13, 14]. With three layers, the overall thickness of the metasurface generally remains subwavelength (usually in the order of t ≈ λ/10) with negligible loss increase. Further increasing the number of layers may naturally be an approach for even broader bandwidth, at the expense of loss and weight.
A typical shape for the scattering particles forming the metallic layers is the Jerusalem cross, as shown in Figure 1. The Jerusalem cross has the advantage of featuring fairly well decoupled responses for x and y polarizations, consequently simplifying the implementation. In the structure of Figure 1, relatively strong capacitive coupling in the transverse (xy) plane offers the benefit of lowering the resonance frequencies [12, 15] or, alternatively, of reducing the freespace electrical size of the unit cell (d ≈ λ/5), while introducing more complexity in terms of coupling.
Figure 1. Generic unit cell with three metallic (PEC) Jerusalem crosses separated by dielectric slabs, the outer layers are identical. 
The realization of the unit cells is greatly simplified if the longitudinal evanescent mode coupling between the three metallic layers can be minimized. This occurs when the dielectric spacer relative permittivity is decreased or/and its thickness is increased. In that case, each layer can be designed separately and the overall response of the multilayer unit cell can then be found using simple transmission matrix approaches [14].
2.2.2 Dielectric scatterers
The second implementation method is based on alldielectric scattering particles. It has been known for a long time that dielectric resonators exhibit both electric and magnetic resonances [16, 17]. More recently, dielectric resonators have been used to realize alldielectric metasurfaces [18, 19]. A typical unit cell is shown in Figure 2a where the particles are dielectric cylinders of circular cross section with permittivity ε_{r,1} embedded in a substrate with permittivity ε_{r,2}. Other types of particle shapes are naturally also possible. It is possible, by tuning the physical dimensions of the resonator as well as the permittivities ratio ε_{r,1}/ε_{r,2}, to merge the electric and magnetic resonances. In this scenario, if the two resonances have the same strength and are associated with orthogonal dipole moments in the transverse plane of the metasurface, as shown in Figure 2b, reflection may be totally suppressed. This is due to perfect destructive interference of the waves scattered by the electric and magnetic scattering particles in the incident side of the metasurface, and their constructive interference at the transmission side of it [17]. In this case, the transmission is theoretically 100% and flat over a wide bandwidth. Moreover, the transmission phase covers a full 2π range around the resonance frequency ω. This powerful concept may therefore apply to allpass metasurfaces with controllable phase over large bandwidth. It must be noted that, in general, the electric and magnetic resonances have different dispersion [20] which leads to a transmission amplitude that is not perfectly flat over the bandwidth of interest. However, such small variations in transmission may be in general neglected since the transmission coefficient S_{21} does not drop below −2 dB (assuming no dielectric loss), as will be shown in Section 3.4.
Figure 2. (a) Representation of an alldielectric allpass metasurface unit cell consisting of a dielectric resonator (ε_{r,1}) embedded in a host layer of permittivity ε_{r,2}. (b) Operation principle for full transmission (zero reflection). 
Structurally symmetric shapes like cylinders or squares present the same behavior for x and ypolarized waves. However, using asymmetric shapes, such as ellipses and rectangles, allow for a complete and independent control of the two orthogonal polarizations, as recently demonstrated in reference [19]. As an additional advantage, dielectric unit cells have a greatly reduced number of physical parameters to adjust compared to the three layer Jerusalem crosses of Figure 1, effectively simplifying the optimization procedure to achieve the specified response.
3 Illustrative examples
3.1 Birefringent metasurface
The first example we present is a birefringent metasurface. Birefringence is the property, available in certain crystal, according to which different refractive indices are presented to different wave polarizations. This property requires anisotropy [21]. A birefringent crystal has, in the plane transverse to propagation, a fast axis and slow axis corresponding to low and high refractive indices, respectively. Thus, a wave polarized along the direction of the fast axis undergoes a smaller phase shift than a wave polarized along the slow axis. Based on this principle, several interesting devices can be realized. For example, a halfwave plate, which induces a π phase shift between the two orthogonal polarizations so as to change the polarization angle of linearly polarized waves (rotation of polarization without chirality) or change the handedness of circularly polarized waves. Similarly, a quarterwave plate induces a π/2 phase shift between the two polarizations, which allows transforming a linearly polarized wave into a circularly polarized wave or vice versa.
Birefringent metasurfaces can be mathematically described using the synthesis technique of Section 2.1. Since bianisotropy (or chirality) are not required to achieve birefringence, (2a) and (2a) reduces to(3a) (3b)
Following the procedure described in reference [4], the system is straightforwardly solved and yields two sets of orthogonal susceptibilities, given for xpolarized waves by(4a) (4b)and for ypolarized waves by(5a) (5b)
We shall next consider two specific types of birefringences, generalized refraction and orbital angular momentum birefringences, that are both achieved by using proper nonuniform susceptibilities in (4a)–(5b).
3.1.1 Generalized refraction birefringence
The concept of generalized refraction birefringence is proposed here as a direct application of the general synthesis technique described in reference [4] and simplified for the case of birefringence in (4a)–(5b). Generalized refraction birefringence consists in independently and simultaneously controlling the reflection and the transmission coefficients of xpolarized and ypolarized plane waves incident on the metasurface. The principle is illustrated in Figure 3, where the reflected waves are specified to be zero.
Figure 3. Example of a birefringent “generalized refraction”. Two orthogonally polarized plane waves incident on the metasurface get refracted at different and independent angles. 
As an example, we present the implementation of a polarization beam splitter (or, reciprocally, a polarization combiner). The corresponding metasurface, that is shown in Figure 4, is first described in terms of its surface susceptibilities, by inserting the specified incident, reflected and transmitted electric and magnetic fields into (4a)–(5b). Here, the reflected waves are specified to be zero and the transmitted waves are specified to be fully refracted at ϕ_{⊥} = +60° and ϕ_{ } = −60°, respectively, in the diagonal of the square metasurface. The computed susceptibilities, which are periodic in the xy plane, are then discretized into eight fundamental unit cells of sides d ≈ λ/5 and thickness t ≈ λ/10. Following the procedure described in Section 2.2.1, each unit cell is designed separately to realize the required equivalent surface susceptibilities. The fabricated metasurface, shown in Figure 4b, consists in 24 × 24 unit cells. From periodicity, the unit cells can be grouped into periodic supercells composed of 8 × 8 unit cells.
Figure 4. (a) Representation of the polarization beam splitter/combiner. (b) Fabricated structure with 24 × 24 unit cells. The two polarizations are refracted in the diagonal direction where a macroscopic stripelike pattern is apparent. 
Because the metasurface in Figure 4b is electrically too large to be simulated, a simplified version of the metasurface is considered for the fullwave simulations. Only the first row of the 8 × 8 supercell is selected. Numerical simulations (CST) are performed for x and y polarizations. The results are shown in Figures 5a and 5b, respectively. It has to be noted that, because the periodicity is now only in the x direction, instead of being in the diagonal of the supercell, the refraction angle is reduced. This can be understood by considering that the projected phase of the refracted wave on the metasurface undergoes one full cycle along the x direction but two full cycles along the diagonal of the supercell.
Figure 5. Fullwave simulations, assuming periodic boundary conditions, of the first row of the supercell shown in Figure 4b. (a) An xpolarized normally incident plane wave is impinging from left to right on the metasurface that refracts it upward. (b) A ypolarized normally incident plane wave is impinging from left to right on the metasurface that refracts it downward. 
As may be seen in Figure 5a, the simulation is in good agreement with the expected result. However, the simulation in Figure 5b shows a refracted wave perturbed by the presence of a zeroth diffraction order contribution. The reason for the mediocre results of Figure 5b is the altered coupling between the unit cells due to truncation, which yields the observed zeroth order transmission.
3.1.2 Orbital angular momentum birefringence
The second example for a birefringent metasurface is an orbital angular momentum (OAM), or vortex wave, multiplexing metasurface [22]. The generation of OAM waves using metasurfaces has been presented in several recent reports [23–25]. Bessel waves are a typical choice for vortex waves. However, Bessel waves are either radially or azimuthally polarized, demanding for either inhomogeneous rotation of polarization if the incident plane wave is linearly polarized, or that the incident plane wave be circularly polarized. To avoid this constraint, HypergeometricGaussian (HyG) waves are considered instead. The electric field of a HyG wave is given by [26](6)where _{1} F_{1} (a, b; x) is the confluent hypergeometric function, Γ(x) is the gamma function, m is the OAM order, p ≥ −m is a real parameter, and where , ξ = z/z_{r } w_{0}, with w_{0} being the beam waist and z_{r } the Rayleigh range given by .
Compared to Bessel waves, HyG waves have the advantage of being linearly polarized, thus allowing direct transformation (i.e. no rotation of polarization) from a linearly polarized plane wave, as well as the multiplexing of orthogonal plane waves into orthogonal vortex HyG of different order m.
Consider the following transformation, where two normally incident plane waves, polarized along x and y, respectively, are transformed into HyG waves of order m = +1 and m = −1, respectively. Figure 6 shows the typical amplitude and phase distributions of a vortex HyG wave of order m = −1. Because of its complex structure, a complete discretization of the HyG wave would result in a very large number of scattering particles. In order to simplify the problem, the HyG wave phase is coarsely sampled with only four phase samples, as shown in Figure 6d. The four phase samples are sufficient, according to the Nyquist criterion, to describe the full phase evolution of the wave. As for the magnitude of the HyG wave, it is transformed into a flat uniform plane of magnitude E = 1, as shown in Figure 6c. Although the implemented magnitude is flat, the radiated field magnitude variations follow naturally from selfinterference.
Figure 6. Magnitude (a) and phase (b) of the HypergeometricGaussian (HyG) wave with parameters p = 1, m = −1, w_{0} = λ, ξ = 1. Simplified version of the specified fields with flat magnitude (c) and four phase samples (d). 
It can be easily shown that, even with such a drastic simplification, the fields scattered by the metasurface do exhibit the smooth shape of the continuous fields of Figures 6a and 6b. This can be verified by Fourier propagating the fields of Figures 6c and 6d, as shown in Figure 7. Similar considerations apply to the HyG wave of order m = +1 whose phase spirals in the opposite direction than the phase of the HyG wave of order m = −1.
Figure 7. Fourier propagated (a) magnitude and (b) phase, at a distance of z = 100λ, generated by the simplified fields of Figures 6c and 6d. Fourier propagated (c) magnitude and (d) phase, at a distance of z = 100λ, for the HyG wave of order m = +1. The four stronger regions in (a) and (c) are due to the square shape of the aperture formed by the metasurface (size 5λ × 5λ). Using a circular metasurface would result in a better approximation of the ideal doughnut shaped magnitude of Figure 6a. 
The final metasurface is implemented by combining both m = +1 and m = −1 HyG waves, which results in a total number of 16 different unit cells distributed over the area of the metasurface. The final fabricated structure is shown in Figure 8.
Figure 8. Fabricated OAM multiplexing metasurface. 
3.2 Electromagnetic remotely controlled spatial processor
In this section, we propose to synthesize and implement a metasurface spatial processor whose transmitted wave (or reflected wave) can be remotely modified by the presence of a control incident wave. The metasurface has two main operation states, with or without control wave. In the most general case, all waves can be arbitrarily specified and the control wave can therefore dramatically modify the initial transmitted wave. Moreover, changing the amplitude, phase or polarization of the control wave separately can be used as additional degrees of freedom to tune the transmitted wave. The metasurface itself is not necessarily tunable, in contrast to certain structures with tunable scattering particles [27–30]. Likewise, the electromagnetic properties of the metasurface are not modified by the presence of the control wave in contrast to, for instance [31, 32].
Consider for instance the metasurface spatial processor illustrated in Figure 9. In this example, the reflected waves are specified to be zero. When no control wave is present, the metasurface transforms an incident wave (or signal) into a transmitted wave refracted at a 45° angle with respect to the normal of the metasurface, as shown in Figure 9a. When the control wave is impinging on the metasurface, as in Figure 9b, the transmitted wave refraction angle becomes −45°.
Figure 9. Electromagnetic remotely controlled metasurface concept. The metasurface is represented as an array of Jerusalem crosses. In this example, zero reflection is specified. (a) An incident wave (or signal) is transformed into a transmitted wave refracted at 45°. (b) When a control wave (in red) is impinging on the metasurface, the transmitted wave refraction angle becomes −45°. Note that both signal and control waves have the same frequency. 
We now present a simpler operation for the metasurface spatial processor concept: switching. Consider two plane waves, one xpolarized (the signal wave) and one ypolarized (the control wave), both normally incident on the metasurface. The signal wave is normally transmitted by the metasurface, whereas the control wave is also normally transmitted but its polarization is rotated to match the polarization of the transmitted signal wave. The control is achieved by destructive interference between the signal and the control waves, as shown in Figure 10a. In the configuration of Figure 10a, the signal wave and the control wave have the same point of incidence and their respective sources would therefore overlap. To overcome this difficulty, we propose the configuration of Figure 10b, where the metasurface presented in Section 3.1.1 is used as a polarization combiner instead of a polarization beam splitter. The physical implementation of the metasurface presented in Figure 10 inherently requires chirality and could therefore not be directly realized with the concept of birefringence. To synthesize the metasurface of Figure 10a, the system (2a) and (2b) can be solved by specifying appropriate scattering parameters, as discussed in references [4, 6]. One possible set of scattering parameters is given here(7a) (7b)where port 1 corresponds to the incident side of Figure 10a (lefthand side) and port 2 corresponds to the transmit side (righthand side). The reflection scattering tensors and are here left as free parameters. Due to the chiral nature of the metasurface, the structure is bianisotropic irrespectively to the values of and . An important point, apparent in relations (7a) and (7b), is that the efficiency of the metasurface switch described here is limited to 50%. This can be understood by considering a ypolarized normally incident plane wave from port 2. As given by (7b), the ypolarized incident wave splits equally into two waves respectively polarized along x and y. Since the metasurface is reciprocal (i.e. , as given by the Jones matrix calculus [33]), the transmission of either the signal or the control wave is limited to 50%. Higher efficiency could be achieved with a nonreciprocal metasurface which is the topic of the next section.
Figure 10. Representations of the metasurface switch concept. (a) The control and the signal waves are both normally incident on the metasurface. The polarization of the control wave is rotated to match the polarization of the signal wave. A πphase shift is imposed between the two transmitted waves, so that they cancel each other by destructive interference. (b) The control and the signal waves are first combined together by the polarization combiner described in Section 3.1.1. 
To implement the metasurface switch, the three metallic layer approach is used. To obtain the chiral specified behavior, the central metallic layer is rotated 45°, as shown in Figure 11.
Figure 11. (a) Representation of a three layer unit cell of the metasurface switch. (b) Fabricated structure, with 17 × 18 unit cells, on the measurement stage. 
Simulations of the metasurface when illuminated by the signal wave, the control wave and with the two waves simultaneously is shown in Figures 12a–12c, respectively.
Figure 12. Numerical simulations (CST) of the ypolarized fields, the metasurface is at z = 0. (a) Fields when only the control wave is incident on the structure. The control wave is initially xpolarized (not shown here) from left to right. In the positive zdirection, the control wave is transmitted with a y polarization. In the negative zdirection, the control wave is reflected with a y polarization. (b) Fields when only the signal wave is incident on the metasurface. (c) Combination of control and signal waves, the cancellation of the transmission by destructive interference is clearly visible. 
3.3 Nonreciprocal nongyrotropic metasurface
Nonreciprocal devices are usually based on magnetic material nonreciprocal gyrotropy, whose Faraday rotation is a particular effect [34, 35]. Recently, nonreciprocal gyrotropic metasurfaces, having a similar response to magnetic materials but without requiring a magnet, have been reported [36–38].
We will discuss here a onewave transparent metasurface that does not induce polarization rotation and whose principle is shown in Figure 13a. The specified response of the structure is(8)while the other scattering parameters are . Solving the system (2a) and (2b), following the procedure described in references [4, 6], with the specified scattering parameters, yields the susceptibility tensors(9a) (9b)
Figure 13. Spatial nonreciprocal nongyrotropic metasurface. (a) Functional illustration. (b) Proposed pick upcircuitradiator (PCR) implementation. 
The nonreciprocal behavior of the metasurface is evident from the fact that the susceptibility tensors (9b) dot not respect the reciprocity condition, i.e. violate . It can be easily understood why the synthesized metasurface needs to be bianisotropic even though no rotation of polarization is specified: because it is specified to be nongyrotropic, the nondiagonal elements of and , as well as the diagonal elements of and , are necessarily zero since they correspond to gyrotropic behavior. Consequently, the tensors and , being diagonal, can only correspond to reciprocal behavior according to the reciprocity conditions (i.e. and ). The remaining way to simultaneously achieve nonreciprocity and nongyrotropy in such case is by leveraging bianisotropy. The metasurface described by relations (9a) and (9b) is a particular case of a oneway transparent sheet [39]. The magnetoelectric coupling coefficients, given in (9b), have nonzero asymmetric nondiagonal terms, which classifies this kind of structure as a moving medium [40].
Because the structure described by relations (9a) and (9b) is not trivial to implement with the tools given in Section 2.2, we have used an alternative approach based on a similar design to the one used in reference [41]. The metasurface, as shown in Figure 13b, consists in a pickup circuit radiator (PCR) structure. The pickup and radiator faces of the metasurface are composed of patch antennas linked together by an electric circuit present in the center of the structure. When the electric circuit is loaded with an insulating device (like a FET transistor), transmission is allowed only in one direction, effectively realizing the specified scattering parameters. More details about the implementation of this metasurface will be presented elsewhere. Figure 14 shows the simulated field when the metasurface is illuminated from left to right or from right to left, the nonreciprocal behavior can clearly be seen.
Figure 14. (a) Fabricated nonreciprocal nongyrotropic metasurface with 16 unit cells. Fullwave simulated electric fields of the metasurface loaded with a perfect isolator when: (b) excited from port 1 and (c) excited from port 2. The operating frequency is 9.5 GHz. 
The realization of nonreciprocal metasurfaces may become an important part in the implementation of more complicated metamaterial structures allowing more complex field transformations as, for example, the metasurface switch discussed in the previous section.
3.4 Frequency dispersion engineering
So far, most existing metasurfaces perform spatial transformations for monochromatic waves only. As metasurface applications will become more and more advanced, the demand for nonuniform phase control over large frequency bands will increase. As it stands, the synthesis technique of Section 2.1 is restricted to monochromatic wave transformations. However, (2a) and (2b) may be solved repeatedly for different frequencies, consequently forming frequencydependent, or temporallydispersive, susceptibility components, i.e. χ(ω). The response of the metasurface could therefore be engineered for the specified frequency band in the similar fashion that phasers can achieve realtime analog signal processing (RASP) [42].
One possible approach, to engineer frequencydependent metasurfaces is to use the additional degrees of freedom provided by coupled resonators as demonstrated in reference [43]. Another possibility, that is proposed here, is to cascade metasurfaces made of dielectric scattering particles, as depicted in Figure 15. As shown in Section 2.2.2, cylindrical dielectric resonators can be tuned to achieve near full transmission and 2π phase coverage over a specific bandwidth. The proposed allpass metasurface is shown in Figure 16. In this configuration, each cylinder is mechanically supported by dielectric interconnections, which also suppress spurious FabryPérot resonances that would occur if the dielectric particles were placed on a supporting slab.
Figure 15. Conceptual idea of cascading several dielectric metasurfaces for dispersion engineering. 
Figure 16. Proposed allpass and alldielectric metasurface based on 2D array of interconnected coupled cylinders. (a) Structure illustration. (b) Fabricated prototype, the exciting horn antenna is visible in the background. 
The simulated response of the dielectric metasurface of Figure 16b is presented in Figure 17. A flat and almost full transmission is achieved over a large bandwidth while a full phase cycle occurs around the resonant frequency ω = 11.15 GHz.
Figure 17. Rigorous coupled wave analysis (RCWA) of the metasurface of Figure 16b. The plots show the transmission and reflection coefficients as well as the transmission phase and the transmission group delay, respectively. The field is polarized in the x direction (perpendicular to the dielectric interconnections). 
4 Conclusion
In this paper, we have presented a partial overview of the synthesis and possible applications of electromagnetic metasurfaces and several illustrative examples of metasurface applications, in an increasing order of synthesis complexity. First, the concept of birefringence, which allows an independent control of both orthogonal polarizations, was used to implement two diffraction engineering metasurfaces: one for polarization dependent generalized refraction and the other one for orbital angular momentum multiplexing. Next, the implementation of an electromagnetic remotely controlled metasurface spatial processor was discussed. It turns out that birefringence, which is based on anisotropy, is fundamentally insufficient to realize this remotely controlled metasurface processor, the latter requiring bianisotropic constituents. It was shown that the efficiency of such metasurfaces is limited due to its inherent reciprocal operating principle. However, a nonreciprocal implementation of these metasurfaces could theoretically achieve a much better efficiency. Based on this consideration, a nonreciprocal nongyrotropic metasurface was introduced. This metasurface, which also requires bianisotropic features, could be a possible future solution to realize efficient nonreciprocal remotely controlled metasurface processors. Finally, we have introduced a conceptual idea to realize allpass dispersion engineering metasurfaces able to control the group delay of electromagnetic signals of large bandwidth, which constitute a missing piece in the metasurface arsenal.
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Cite this article as: Achouri K, Khan BA, Gupta S, Lavigne G, Salem MA & Caloz C: Synthesis of electromagnetic metasurfaces: principles and illustrations. EPJ Appl. Metamat. 2015, 2, 12.
All Figures
Figure 1. Generic unit cell with three metallic (PEC) Jerusalem crosses separated by dielectric slabs, the outer layers are identical. 

In the text 
Figure 2. (a) Representation of an alldielectric allpass metasurface unit cell consisting of a dielectric resonator (ε_{r,1}) embedded in a host layer of permittivity ε_{r,2}. (b) Operation principle for full transmission (zero reflection). 

In the text 
Figure 3. Example of a birefringent “generalized refraction”. Two orthogonally polarized plane waves incident on the metasurface get refracted at different and independent angles. 

In the text 
Figure 4. (a) Representation of the polarization beam splitter/combiner. (b) Fabricated structure with 24 × 24 unit cells. The two polarizations are refracted in the diagonal direction where a macroscopic stripelike pattern is apparent. 

In the text 
Figure 5. Fullwave simulations, assuming periodic boundary conditions, of the first row of the supercell shown in Figure 4b. (a) An xpolarized normally incident plane wave is impinging from left to right on the metasurface that refracts it upward. (b) A ypolarized normally incident plane wave is impinging from left to right on the metasurface that refracts it downward. 

In the text 
Figure 6. Magnitude (a) and phase (b) of the HypergeometricGaussian (HyG) wave with parameters p = 1, m = −1, w_{0} = λ, ξ = 1. Simplified version of the specified fields with flat magnitude (c) and four phase samples (d). 

In the text 
Figure 7. Fourier propagated (a) magnitude and (b) phase, at a distance of z = 100λ, generated by the simplified fields of Figures 6c and 6d. Fourier propagated (c) magnitude and (d) phase, at a distance of z = 100λ, for the HyG wave of order m = +1. The four stronger regions in (a) and (c) are due to the square shape of the aperture formed by the metasurface (size 5λ × 5λ). Using a circular metasurface would result in a better approximation of the ideal doughnut shaped magnitude of Figure 6a. 

In the text 
Figure 8. Fabricated OAM multiplexing metasurface. 

In the text 
Figure 9. Electromagnetic remotely controlled metasurface concept. The metasurface is represented as an array of Jerusalem crosses. In this example, zero reflection is specified. (a) An incident wave (or signal) is transformed into a transmitted wave refracted at 45°. (b) When a control wave (in red) is impinging on the metasurface, the transmitted wave refraction angle becomes −45°. Note that both signal and control waves have the same frequency. 

In the text 
Figure 10. Representations of the metasurface switch concept. (a) The control and the signal waves are both normally incident on the metasurface. The polarization of the control wave is rotated to match the polarization of the signal wave. A πphase shift is imposed between the two transmitted waves, so that they cancel each other by destructive interference. (b) The control and the signal waves are first combined together by the polarization combiner described in Section 3.1.1. 

In the text 
Figure 11. (a) Representation of a three layer unit cell of the metasurface switch. (b) Fabricated structure, with 17 × 18 unit cells, on the measurement stage. 

In the text 
Figure 12. Numerical simulations (CST) of the ypolarized fields, the metasurface is at z = 0. (a) Fields when only the control wave is incident on the structure. The control wave is initially xpolarized (not shown here) from left to right. In the positive zdirection, the control wave is transmitted with a y polarization. In the negative zdirection, the control wave is reflected with a y polarization. (b) Fields when only the signal wave is incident on the metasurface. (c) Combination of control and signal waves, the cancellation of the transmission by destructive interference is clearly visible. 

In the text 
Figure 13. Spatial nonreciprocal nongyrotropic metasurface. (a) Functional illustration. (b) Proposed pick upcircuitradiator (PCR) implementation. 

In the text 
Figure 14. (a) Fabricated nonreciprocal nongyrotropic metasurface with 16 unit cells. Fullwave simulated electric fields of the metasurface loaded with a perfect isolator when: (b) excited from port 1 and (c) excited from port 2. The operating frequency is 9.5 GHz. 

In the text 
Figure 15. Conceptual idea of cascading several dielectric metasurfaces for dispersion engineering. 

In the text 
Figure 16. Proposed allpass and alldielectric metasurface based on 2D array of interconnected coupled cylinders. (a) Structure illustration. (b) Fabricated prototype, the exciting horn antenna is visible in the background. 

In the text 
Figure 17. Rigorous coupled wave analysis (RCWA) of the metasurface of Figure 16b. The plots show the transmission and reflection coefficients as well as the transmission phase and the transmission group delay, respectively. The field is polarized in the x direction (perpendicular to the dielectric interconnections). 

In the text 
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