The shaping of electromagnetic waves by dielectric lenses and metallic reflectors such as parabolic mirrors is a well-established discipline in optics and microwave antenna engineering , , , . However, both dielectric lenses and reflectors lead to considerable volume and weight, especially at microwave and even millimeter-wave frequencies. For these reasons, in the past decades, there has been considerable research to make flat versions of these fundamental wave-shaping devices. At microwaves, there is the well-established field of transmitarrays (to replace lenses) and reflectarrays (to replace reflectors) , , , , . Likewise, mostly in the optical regime, planar gratings and holographic surfaces based on diffraction effects have been developed for various applications , , , , , , . Transmit and reflectarrays typically comprise arrays of elementary antennas (e.g. dipoles or patches) printed on a flat dielectric substrate. The required phasing to achieve a certain wavefront transformation (e.g. to focus an incident plane wave) is provided by spatially modulating the size or the shape of these elementary antennas. Typically, this is accomplished by working close to the resonance of these elementary antennas to obtain the required phase range. Hence, these constituent antenna elements are usually half wavelength in size. Moreover, in the case of transmitarrays, several layers are needed to achieve good matching characteristics and avoid reflections. While traditionally, transmitarrays have been confined to the microwave regime, optical transmitarrays have been reported in recent years due to the advent of precise nanofabrication techniques. Prominent examples on this front include the pioneering work reported in Ref. , . On the other hand, diffraction gratings and holographic surfaces have been traditionally more popular in the optical regime (due to their size, which needs to be several wavelengths long). Primarily, these are periodic structures that manipulate fields in a scalar fashion. Because conventional gratings have unit-cell dimensions comparable to the wavelength, these are diffractive structures. In recent years, gratings and related holographic structures have also been developed for antenna applications at microwaves. In fact, such structures can now handle full tensor-field transformations and hence can manipulate polarization as well , .
In this review article, we report recent advances in Huygens’ metasurfaces (HMSs), which are engineered surfaces also designed for the manipulation of incident electromagnetic waves. These surfaces are related to transmit- and reflectarrays but with some key differences , , . Some of these include the fact that these HMSs are typically subwavelength thin and their constituent unit cells are also subwavelength. Because of the subwavelength scales involved, these surfaces can be homogenized and characterized by spatially varying surface impedances and admittances. This is what justifies the name “metasurface,” which can be thought of as two dimensional (2D) “metamaterials” and also described in terms of electric and magnetic susceptibilities . The conceptual realization of these metasurfaces is based on the Schelkunoff’s equivalence principle, which is the generalization of Huygens’ principle: a known incident electromagnetic wave is transformed to a desired one, resulting to field discontinuities that induce electric and magnetic currents on the metasurface (interface), as depicted in Figure 1 , . These electric and magnetic currents are physically realized using co-located orthogonal electric and magnetic dipoles that correspond to spatially varying admittance and impedance sheet distributions , , . It should be pointed out that these equivalent electric and magnetic currents can be made impressed instead of induced, which leads to the concept of active HMSs, which in fact preceded the passive ones , .
Because of the subwavelength nature of the constituent unit cells and the fact that both electric and magnetic currents are induced according to Maxwell’s equations (rigorous boundary conditions), these HMSs have some interesting intrinsic features: they are naturally well matched even though they are subwavelength thin, and they do not excite spurious radiation modes (such as higher Floquet modes in the case of simple refraction) . In principle, this approach can be used to deterministically design such engineered surfaces to manipulate all attributes of the incident field: its magnitude, phase, and polarization. Some notable successes of HMSs include subwavelength thin “perfect refracting” surfaces that suffer no reflections whatsoever and all incident power is coupled to the refracted beam. Likewise, “perfect reflecting” metasurfaces can be designed (akin to reflectarrays), which can transfer 100% of the incident power to an arbitrary reflection angle. Moreover, because of this total field control that can be enabled by such metasurfaces, aperture antennas can be designed with prescribed characteristics such as beamwidth, beam-pointing direction, and sidelobe-level distribution. Other applications such as absorbing metasurfaces can also be achieved , , , . In the remainder of this article, we elaborate on some of these HMS topics summarizing the basic principles involved and highlighting the latest developments.
2 Basic theory of HMSs
As previously mentioned, HMSs are thin electromagnetic devices that can perform desired field transformations, specifically through the utilization of the Schelkunoff’s equivalence principle , , , . As seen in Figure 1 and Eq. (1), when fields
In this fashion, to produce an HMS, the desired structure simply needs to realize the necessary surface current densities for any given field transformation , . As mentioned, one such approach of realizing these currents is to use impressed sources such as in Ref. ,  and further elaborated in Section 3.5. In this method, the electric and magnetic current densities are directly introduced via external sources. The radiated fields of these impressed sources will then interact with the incident fields to produce the desired output fields. However, the utilization of impressed current sources is not always trivial. Some challenges include the generation and distribution of microwave/optical power, power consumption, and the precise realization of the required current weights . Alternatively, these surface currents can be induced by the incident fields by synthesizing effective electric and magnetic properties. In this case, the metasurface is designed in terms of these intrinsic characteristics instead of arrays of impressed sources. By synthesizing these electric and magnetic properties, the desired boundary surface current densities are excited by the incident fields, thus leading in principle to passive and lossless metasurfaces . The scattered fields from these excited currents are then capable of producing the output fields by interacting with the incident wave. In general, there are three perspectives for designing these metasurface properties, which are the susceptibility , , , polarizability , , , , and impedance/admittance approaches , . It should be noted that regardless of the chosen perspective, all these methods can achieve the same range of field manipulations . In this review article, we will be mainly using the impedance/admittance perspective for demonstrative purposes. To introduce the electric and magnetic properties of the metasurface, the equivalence principle can be combined with another set of equations, which relate the average tangential fields at the desired boundary to the required current densities as seen in Eq. (2) , , .
In this case, the tensors
By stipulating the desired incident fields
In summary, HMSs are defined based on Figure 1 and Eq. (1), with the additional requirement that the surfaces can be homogenized using, for example, equivalent sheet impedances and/or admittances, as implied in Eq. (3). In this way, the excited electric and magnetic surface currents in Figure 1 can produce the desired fields according to Schelkunoff’s equivalence principle, which is the generalization of the Huygens’ principle , , , . Specifically, the equivalent currents are
Applying similar derivations for designing HMSs as shown above, different wave transformations have been demonstrated, which include polarization control, wave refraction, and wave focusing , , , , , , , , , , . However, in certain applications, such as wide-angle refraction, it was found that the above derivations were unable to produce perfect or reflectionless transformations , . Even in cases where no reflected fields were desired, the relations above were still not able to produce unity transformation of the desired input to the transmitted fields. These refraction surfaces often suffered from increased reflections as the refracted wave angles become more oblique relative to the incident wave angle , , . This issue was studied in Ref. , , , , , and the problem was identified to arise due to
In the new bianisotropic formulation, the introduction of the magnetoelectric coupling coefficient
3 HMS applications
3.1 HMS refraction
One of the earliest demonstrations of HMSs for low-reflection refraction was reported in Ref. , , . Through theory, simulations, and experimental verifications, low-reflection refraction at microwave frequencies was observed , . Examples of few of the fabricated refraction surfaces can be seen in Figure 2, displaying two examples of refracting surfaces with different design topologies. While in both cases, wire/loop unit cells were used to represent a Huygens’ source, by realizing the electric response with metallic wires and the magnetic response with loops, the orientations of the two designs is quite different. In Ref. , the wire/loop unit cells were stacked in a transverse fashion, with the structures placed parallel to the direction of propagation. On the other hand, the unit cells in Ref.  were positioned in a single-plane configuration, with the unit cells perpendicular to the direction of propagation. Although physically orientated differently, both of these structures demonstrated good refractive properties and were able to successfully represent Huygens’ sources. However, other designs have also been used to realize HMSs. One such structure is the multi-layer cascaded unit cell , , , . Instead of an intuitive design afforded by the wire/loop unit cells presented in Ref.  and , the cascaded topology utilizes capacitive and inductive properties of stacked planar metallic patterns to obtain Huygens’ unit cell performance. While these stacked planar structures do not contain physical wires and loops to realize Huygens’ sources, the cascaded effect of these capacitive and inductive geometries is still able to synthesize the required Zse and Ysm reactances that are needed by the boundary conditions in (3). Schematics of the two unit cell structures can be seen in Figure 3, and an example of a physical realization of the three-layer structure can be seen in Figure 9. Regardless of the difference in physical realization, all three topologies have demonstrated the capability of realizing Huygens’ responses, specifically in the application of low-reflection refraction , , , .
It should be noted that even though the two unit cells in Figure 3 can generate the Zse and Ysm reactances that are required by Eq. (3), they are actually not equivalent. The wire/loop unit cell naturally synthesizes a Huygens’ source, as it directly realizes the electric and magnetic currents stipulated by the equivalence principle. As was shown in Ref. , the equivalent circuit of the wire/loop unit cell is of the lattice topology, as shown in Figure 4A. This arises from the fact that there is a one-to-one correspondence between the
As previously mentioned, it was shown in Ref. , , , , , , ,  that the refraction efficiency of the first-generation refraction-surface designs deteriorated as the refraction angle became more oblique relative to the incident beam angle. Originally, these refraction surfaces were non-bianisotropic and thus were physically represented with symmetric structures , , , , , . However, due to their physical symmetry, these metasurfaces were unable to simultaneously match both the incident and refracted wave impedances, which are angle-dependent , , , , , . Therefore, as the desired refraction angle was increased, the impedance mismatch and the reflections correspondingly increased, which resulted in a reduction of the refraction efficiency, as shown in Figure 5 .
This issue fundamentally arose from the lack of degrees of freedom. As these surfaces were all designed to be passive devices, they were unable to produce the necessary gain and loss mechanisms required by the boundary conditions. To demonstrate this issue, we will examine a 1D wide-angle refraction of a normally incident transverse electric (TE) plane wave towards 71.8° at 20 GHz, as described in Ref. , . Using the described desired TE refraction scenario, the obtained non-bianisotropic electric impedance and magnetic admittance can be seen in Figure 6.
As shown, for this wide-angle refraction, the obtained Zse and Ysm values contain non-zero real components, which translate to control over loss and gain regions , , , . Therefore, without realizing the required gain and loss mechanisms, the refraction efficiency was reduced. As previously discussed, this is due to the lack of degrees of freedom to choose the desired lossless and passive solution. To introduce an extra degree of freedom, the use of bianisotropy can be included in the boundary condition formulations , , , . Using the same refraction scenario with the bianisotropic boundary conditions and stipulating the lossless, passive, and local power conservation conditions , the corresponding Zse, Ysm, and Kem values can be seen in Figure 7.
Examination of the bianisotropic boundary conditions showed that the obtained Zse and Ysm values now are purely imaginary and Kem is purely real, which corresponds to a true lossless and passive design , . In addition, translating these boundary conditions to generalized scattering or G parameters, as described in Ref. , , , , , results in perfect refraction with no reflection, as can be seen in Figure 8.
Using the bianisotropic boundary conditions, the demonstration of perfect or reflectionless refraction was theoretically demonstrated in Ref. , . Later in Ref. , a physical realization of a printed circuit board (PCB) reflectionless refraction HMS was demonstrated. The fabricated prototype utilized a three-layer asymmetric structure, which exploited the physical asymmetry to synthesize the bianisotropy , . A close-up of a section of the fabricated metasurface can be seen in Figure 9. The realized metasurface was able to achieve good refraction characteristics while causing minimal reflections for wide-angle refraction from normal incidence to 71.8° at 20 GHz . The reflection and refraction of the metasurface were measured using a combination of quasi-optical (setup shown in Figure 10) and far-field measurements and can be seen in Figures 11 and 12.
The fabricated prototype achieved minimal reflections with higher than 80% refraction efficiency of the total scattered power . While this demonstration showed higher than expected losses due to imperfections in fabrication, the obtained refraction efficiency remained clearly higher than what was possible with non-bianisotropic refraction surfaces , . A related experimental demonstration of bianisotropic refraction was also presented in Ref. , which showed good efficiency for two different cases of refraction (0° to −70° and 20° to −28°) at 10 GHz. The designs, which also utilized the asymmetric three-layer PCB structure, were measured using a near-field setup to obtain the transmission characteristics of the metasurface. While the reflections of the prototype were not characterized, the work in Ref.  showed good refraction properties with the use of bianisotropy. The physical realizations hereby described along with previous theoretical designs demonstrate the benefit of utilizing bianisotropic HMSs for applications in wide-angle refraction.
3.2 Antenna beam forming
In the previous examples of refracting surfaces, the structures were all designed to interact with plane waves. However, metasurface applications have also been extended to beam forming from arbitrary sources such as in the case of metasurface lenses and antennas , , , , . These metasurfaces are often sought after for their applications in antenna systems for directive radiation. Metasurfaces used for these antenna applications can be designed in many ways. In this review article, we will briefly discuss two previously demonstrated methods, namely, the impedance/admittance and the bianisotropic boundary condition (BC) methods. In both methods, a key enabler for allowing an elementary feed antenna to interact with a close-by metasurface is to use the Fourier plane-wave expansion to represent the fields of the feed antenna .
The impedance/admittance method utilizes the previously shown non-bianisotropic boundary condition formulation in Eq. (3) to account for any arbitrary excitation source for directive radiation , , . The fields of the incident and desired output waves can be stipulated and the corresponding surface electric impedance Zse and magnetic admittance Ysm can be found, similar to the refracting and reflecting scenarios , , . A detailed exposition of this method is presented in Ref. . In this formulation, the excitation source may be chosen to be any arbitrary field, which may model real antenna excitations, while the output radiated field is chosen to be a modulated plane wave propagating at a desired angle. By choosing the angle of the output plane wave, the beam of the metasurface antenna can be steered, and by choosing a modulated plane-wave profile, directive radiation is specified. To obtain the required boundary conditions, the incident, reflected, and transmitted fields associated with the source and the desired output beam are first decomposed to their respective plane-wave spectra via the Fourier transform . These spectra at first represent the generalized expressions of the desired waves. To determine the exact spectrum decomposition, however, the reflected, transmitted, and incident fields are then related via the local-power-conservation and the local-impedance-equalization conditions at the metasurface boundary . These conditions are applied to stipulate reactive boundary conditions, meaning that the metasurface would be lossless and passive. Once the full decompositions of the waves are obtained, the boundary conditions necessary for the field transformation can be calculated. Utilizing this method, a cavity excited antenna was designed and demonstrated via fullwave simulations and experimental measurements .
The fabricated antenna, which can be seen in Figure 13, displayed high aperture illumination efficiency, low side-lobe level (SLL), and high realized gain. The presence of the cavity helped to illuminate effectively the metasurface from a small feeding antenna in its close proximity. Apart from the radiation direction and high-aperture efficiency, another desired feature of metasurface antennas is the capability to shape the radiated beam. To enable the synthesis of arbitrary antenna patterns, there is a need to precisely control the local reflection coefficients. In turn, this would enable the metasurface modulation to be arbitrarily designed based on any desired antenna pattern. However, originally, the impedance/admittance method utilized the non-bianisotropic boundary condition formulation. As it was desired to define a purely reactive surface enabled by a local-impedance equalization condition, reflected fields that obeyed Fresnel-type reflection coefficients were required. Thus, the modulation of the metasurface could not be arbitrarily set, which limited the control of the output beam shape . Later, when bianisotropic boundary conditions were introduced through the study of the perfect or reflectionless refraction surfaces, it was shown that this new formulation allows arbitrary control of the local reflection coefficients , , . Therefore, by applying the bianisotropic boundary conditions to beam-forming metasurfaces, this new formulation would not be restricted by the Fresnel reflections and should allow arbitrary control over the output beam. In this pursuit, the bianisotropic BC method was introduced.
The bianisotropic BC method allowed the tailoring of the metasurface to produce output wave fronts, which are associated with well-known antenna radiation patterns. Applying this method, antenna radiation patterns, which correspond to a Taylor distribution, were demonstrated using fullwave simulations and can be seen in Figure 14 . By varying the desired distribution, different radiation patterns with varying SLLs were presented using the same design methodology. In addition, the same design approach has been used for demonstration of binomial and Dolph-Chebyshev distributions, which exhibited good agreement with the expected antenna radiation profiles . Utilizing the bianisotropic BC method, various examples of arbitrary beam shaping have been demonstrated, which validates the capability for HMSs to arbitrarily shape beam patterns for antenna applications. In addition, this method further emphasized the advantage of using bianisotropy in metasurface designs.
3.3 Perfect reflection
Somewhat surprisingly, recent developments have shown that it is highly non-trivial to arbitrarily reflect an incident wave with perfect efficiency. The following perspective helps us understand why: in Section 3.1, we showed that perfect transmission in a metasurface is achieved by using a multilayer surface to match the impedances of the input and output waves. However, for a reflective metasurface, the incident and reflected waves share a half-space and are hence superimposed atop each other. Hence, when their wave impedances differ, simple impedance matching techniques cannot be applied to reflect the wave with perfect efficiency. Consider a perfect TE anomalous reflection metasurface shown schematically in Figure 15. Recent works have found that above a perfect anomalous reflection surface, the normal Poynting vector, which represents power flow into or out of the metasurface, fluctuates in a sinusoidal fashion , , :
Here the surface runs along the xy-plane and anomalous reflection happens in the y-direction. Ex,i is the incident electric field, η0 is the characteristic impedance of free space, kg=2π/λg is the metasurface’s spatial frequency, λg is its period of variation, θi and θr are the incidence and reflection angles, and ϕ is a phase shift upon reflection from the metasurface. We see that, aside from specular reflection (θr=θi) and retroreflection (θr=−θi), (6) assumes a sinusoidal variation along the y-direction. Hence, half the area of the surface must be lossy (reradiates less power than is incident upon it) while the other half must be active (reradiates more power than is incident upon it). Lossy metasurfaces can be implemented with relative ease, but they reduce the metasurface’s power efficiency; active metasurfaces represent a level of complication best to be avoided. This dilemma hence hindered, until very recently, the demonstration of perfect anomalous reflection with a metasurface.
The earliest reflection metasurfaces redirected an electromagnetic wave by applying a linear phase shift upon reflection. It has been shown that, besides obtaining a reflected beam in the desired direction, such surfaces also diverted some power to spurious reflected directions. It was shown ,  that for such surfaces, the maximum power efficiency is as follows:
The spurious reflection components can be eliminated if one can tailor both the amplitude (from 0 to 1) and phase (from 0 to 2π) of the reflection coefficient. Reference  demonstrated an HMS that achieves the necessary tuning; Ref.  and  calculated the necessary (lossy) impedance profile that allows one to perform anomalous reflection without reradiating spurious reflection components. However, in this case, the maximum achievable power efficiency worsens, to the following:
Most recently, study on this topic has led to a class of “non-local” metasurfaces, which can redistribute power along the metasurface to satisfy (6). Reference  first showed this functionality by exciting auxiliary evanescent standing waves along the metasurface, which facilitated a rapid power fluctuation that satisfied (6), hence leading to perfect power transfer from the incident to the reflected wave. Reference  achieved power redistribution through a leaky-wave based metasurface design – the wave incident to the “lossy” region of the metasurface is somewhat absorbed by the leaky wave; this absorbed power then propagated briefly along the metasurface and was re-radiated in the “active” region of the metasurface. Reference  also showed that this redistribution could be achieved without resorting to non-local effects if one physically curved the metasurface along a trajectory where the Poynting vector remained constant. The operations of these perfect anomalous reflectors have been verified experimentally and/or numerically. However, the design and fabrication of these surfaces are rather complicated due to the sensitivity of leaky and evanescent waves, the metasurfaces’ requirement for deeply subwavelength and/or multilayer elements and for  the curvature of the metasurface.
On a parallel front, another group of researchers demonstrated perfect anomalous reflection through a strikingly different paradigm. Whereas most works heretofore on metasurfaces focused on tailoring a continuous impedance profile, a few recent works examined the metasurface as a discrete structure that engineers the diffraction of waves. Prior to the advent of metasurfaces, researchers studying retroreflection from gratings found that some thick gratings could reflect waves at anomalous, or “off-Bragg”, angles, with very high efficiency , , . A theoretical investigation from the perspective of plane-wave diffraction showed that perfect anomalous reflection was possible from such structures when two angular parameters can be achieved, for example, by tuning the depth of the grooves in a rectangular groove grating . A recent work from our group, discussed later in this section, explains why a conceptually simple device such as a grating may achieve the power redistribution described in (6). More recently, Ra’di et al.  proposed the concept of “meta-gratings”. This work, through diffraction calculations and numerical investigation, demonstrated that when electric and magnetic current pairs were properly spaced from each other and from the ground plane below them, the interference characteristics canceled re-radiation from the specular direction and thereby allowed one to transfer all the power into the anomalous direction. Reference  presented an analytical investigation on a meta-grating built from an array of dipoles and demonstrated their use toward the design of perfect reflection beam splitters, which also required a redistribution of power along a metasurface, in a similar fashion to (6). These works provide exciting insights to constructing anomalous reflection metasurfaces; they show the potential of achieving wave manipulation properties using discrete meta-scatterers.
We have experimentally demonstrated perfect anomalous reflection using a discrete HMS without a tunable thickness. Our concept of the discrete HMS is to take the discrete nature of the metasurface into account from the initial design phase. Specifically, when one considers a reflection metasurface as a surface that is spatially discrete and periodic, the reflection profile in spectral (spatial frequency) domain is that of a train of delta functions, as shown in Figure 16. Here diffraction orders are separated by kg=2π/λg, where λg is the grating period. In general, each diffraction order has a unique reflection amplitude and phase. Let N denote the number of diffraction orders within the propagation range (i.e. k∈[−k0, k0]). This number can be upperbounded, i.e. N≤2k0/kg. We have shown that, in order to tune arbitrarily the amplitudes and phases of each propagating diffraction order, one needs to discretize each period of the metasurface into at least N elements. In this paradigm, specular reflection, which obeys the reflection law θr=θi, is represented by the zeroth diffraction order. We achieve anomalous reflection by (i) tuning the metasurface period to steer the first (or minus first) diffraction order into the desired direction and thereafter (ii) maximizing the amplitude of that diffraction order. We find that in many cases, kg is sufficiently large, such that N=2. In these cases, metasurface design and fabrication are dramatically simplified to a structure featuring two cells per grating period. We call this structure a binary HMS.
To demonstrate the binary HMS paradigm, we first used it to design a near-perfect retroreflector for an electromagnetic wave at near-grazing incidence , . This was a special case of anomalous reflection where, as obtainable from (8), power reflection with perfect efficiency can be achieved by conventional blazed gratings . Nonetheless, the binary HMS made a robust, low-profile, and efficient retroreflector for this application. For this work, we chose as the metasurface element an electric dipole backed by a dielectric spacer and a metallic ground plane, as depicted in Figure 17A. In a prior work , we showed how this simple element functioned as a passive Huygens’ source and that its operation extended from microwave to optical frequencies. Figure 17B shows that by varying the length of the dipole element, we realized a phase tuning range of nearly 360°. We chose two element lengths, which gave reflection phase shifts that differ by 180°, and placed them adjacent to one another to form the elements of the binary HMS. Figure 17C shows a period of the resultant metasurface. This arrangement should provide total suppression of the specular reflection mode and hence redirect the incident wave into retroreflection with perfect efficiency. After slight length adjustments to account for coupling effects between these elements, the resultant metasurface achieved a highly efficient retroreflection for an incident wave at f=24 GHz and θi=82.87°. Figure 17D shows the metasurface mounted on a rotation stage in a far-field antenna chamber. A horn antenna to the right of the photo illuminates the metasurface and receives the retroreflected wave. As the metasurface rotates with its stage, the retroreflected signal strength is measured as a function of the incidence angle. This hence forms a monostatic radar cross-section (RCS) measurement for the metasurface, which we show in Figure 17E. As expected, we measured strong retroreflection signals at normal incidence as well as ±82°, where we obtained a retroreflection power efficiency of 94% in simulation and 93% in experiment. While this surface retroreflects a TE-polarized wave, we designed an analogous surface with slots replacing dipoles to retroreflect a TM-polarized wave, with similar power efficiency (99% in simulation, 93% in experiment).
Following the same methodology, we designed a binary HMS that achieved perfect anomalous reflection at 24 GHz , . This metasurface was designed to accept an incoming plane wave at θi=50° and reflect it to θr=−22.5°. The unit cell and single period architecture for this metasurface is the same as that for the retroreflection metasurface, shown in Figure 17A and C. Table 1 tabulates the material and feature dimensions for this binary HMS. A (periodic) Floquet simulation showed that this metasurface performed the desired anomalous reflection with 99.98% efficiency, which was perfect within the tolerance of the simulation. Figure 18A plots the incident and reflected (scattered) waves as calculated by the simulation software. We then truncated the metasurface to 78 cells (413.4 mm) and simulated its scattering pattern across a range of frequencies. Figure 18B (top panel) shows the simulated scattering pattern for the binary HMS at 20 GHz, compared with that of a perfect conductor of the same size. Clearly, power from the incident wave was completely redirected into the anomalous direction of θr=−22.5°. Figure 18B (bottom panel) shows the same comparison over a range of frequencies, from 22.0 GHz to 28.5 GHz. We observed beam squinting (the variation of beam angle with respect to frequency), as was common with various antenna arrays and reflect arrays. Notwithstanding this, the non-resonant nature of the design allows the BHM to perform high-efficiency anomalous reflection. An efficiency surpassing 90% was achieved over this 25.7% bandwidth. Figure 19A shows a close-up of the fabricated metasurface; Figure 19B shows the experimental apparatus we used for its characterization. From the right, a lens antenna launched a 24-GHz wave toward the metasurface at θi=50° from a distance of 1.5 m. A second lens antenna equidistant from the metasurface measured the scattering. This receive antenna was rotated to measure the bistatic RCS of the metasurface upon the prescribed incidence. Figure 19C compares the bistatic RCS of the BHM with that of a metal plate of the same size and shape. The experimental results agree well with simulation: specular reflection was suppressed by 20 dB compared to the metal plate, and near-perfect anomalous reflection was achieved.
Specifications and key dimensions for the perfect anomalous reflection binary HMS.
|Operation frequency||24 GHz|
|Operation angles||θi=50°; θr=−22.5°|
|Printed circuit board|
|Substrate||Rogers RT/Duroid 5880|
|Key design dimensions (see Figure 17)|
|Ux=Uy=5.44; Px1=2.0; Px2=3.6; Py=1.5 (all units in mm).|
To investigate how this apparently simple metasurface achieved the power redistribution required in (6), we examined the electromagnetic field and power flow normal to selected planes parallel and in close proximity to the metasurface. We observed strong evanescent electromagnetic fields existed at subwavelength distances from the metasurface. Most likely, this strong evanescent component performed similarly to auxiliary waves proposed in Ref. . However, their implicit existence in a much simpler structure boded well for the broadband operation of the perfect anomalous reflection BHM and its practicality for scaling up to millimter-wave and THz frequencies. This discovery also provided one plausible explanation on how some grating structures demonstrated near-perfect anomalous reflection notwithstanding the recently discovered requirement (6).
3.4 Review of optical HMSs
At optical frequencies, the difficulty in obtaining magnetic response and the inevitable high Ohmic loss have been the main bottlenecks in early optical HMS research. As such, various “Huygens’-like” metasurfaces that utilize multiple layers of admittance sheets have been initially demonstrated for which some of these metasurfaces are shown in Figure 20A–C , , . Here a key Huygens’ property (i.e. properly weighted orthogonal electric and magnetic induced currents) can be conceptually envisioned by considering the effect of an incident electric and magnetic field separately. Specifically, an incident electric field would induce electric currents in all layers while an incident magnetic field would excite anti-parallel currents in the outermost layers that result in magnetic current (Figure 20B). The directions of these induced electric and magnetic currents are orthogonal to each other; hence, they mimic the Huygens’ property. Despite such a conceptual Huygens’ interpretation, however, they closely resemble transmitarrays and their method of analysis indeed follows that of transmitarrays at microwaves. For example, Monticone et al.  have analyzed their Huygens’-like metasurfaces by modeling each layer as a shunt reactive network separated by a transmission-line segment, thereby determining the required surface reactances in each layer. Their analysis has also established that electrical responses alone from each layer sufficed to synthesize arbitrary reflection and transmission properties. See also related discussion surrounding Figure 4. These Huygens’-like metasurfaces operate in the transmission mode, and their transmission efficiencies are mainly limited due to plasmonic Ohmic loss (e.g. 30% power transmission efficiency was reported in ).
On the other hand, high-efficiency optical Huygens’ metasurfaces (OHMs) have been demonstrated, which operate in the reflection mode , , , . Figure 20D shows the so-called gap-surface plasmon (GSP) resonators, which have been used as unit cells in these OHMs. They take advantage of a metallic back plate for efficient light reflection, while the metallic nanorods spaced by a dielectric layer provide necessary electric and magnetic currents for synthesizing the surface currents in Schelkunoff’s equivalence principle . Here, an incident electric field is polarized along the nanorods, which would induce electric currents in the nanorods and the substrate. On the other hand, an incident magnetic field would induce circulating electric currents in the nanorods and the substrate, thereby generating a magnetic current. This is illustrated in Figure 20E and F. In particular, Figure 20E shows strongly enhanced magnetic field between the nanorods and the substrate, as the field would be in a typical Fabry-Perot resonator. In such a case, unequal and opposite electric currents are induced in the nanorods and the substrate (inset in Figure 20E), which indicates simultaneous existence of both electric and magnetic currents. On the contrary, when the nanorods are not properly tuned, the field is no longer confined and the induced current is mostly confined within the nanorods, as shown in Figure 20F. As a further illustration, the radiation patterns of single isolated unit cells that are electromagnetically similar to the GSP resonators are shown in Figure 20H–I. In this isolated cell analysis, the dielectric layer and the ground plane are truncated to match the size of the nanorod, as the purpose is to simulate a single unit cell that is not coupled. The geometrical parameters are set similar to the GSP resonators at an operational wavelength of 800 nm (see the caption of Figure 20H–I for details). It is seen that the radiation patterns are unidirectional for certain resonator lengths, which is a unique property of a Huygens’ unit cell and demonstrates the simultaneous existence of orthogonal electric and magnetic dipoles. Specifically, when a magnetic current is absent, they scatter in both the forward and backward directions, while they scatter either in the forward or the backward direction in the presence of both electric and magnetic currents.
Based on the GSP resonators, anomalous reflection, asymmetric beam splitting, far-field pattern engineering, focusing, birefringence polarization control, and holograms have been demonstrated , , , , . In particular, the demonstrations in Ref.  have established that the surface needs a user-designed spatially varying reflection loss in addition to surface reactance for complete reflected wavefront control (in particular, for controlling the amplitude of the reflection coefficient). In this respect, the authors have deliberately introduced polarization losses into their surfaces by rotating the gold nanorods with respect to the incident polarization vector, thereby designing passive and lossy OHMs. However, as pointed out by Asadchy et al.  and Epstein and Eleftheriades  the efficiency of such a lossy and reactive surface is not unity. In this regard, the topic of perfect reflection/refraction has gained much attention recently, as was previously discussed. Following the introduction of passive OHMs, their reconfigurable version has also been demonstrated , . Instead of utilizing non-tunable materials (e.g. silica) that are placed between the gold nanowires and the substrate, these reconfigurable metasurfaces employ vanadium dioxide (VO2) or indium tin oxide (ITO) as the phase changing materials. For the VO2-based structure, a DC current is applied to the gold nanowire, which in turn increases the temperature of VO2 and induces its insulator-to-metal phase transition. This enables the unit cell’s reflection coefficient to be dynamically tuned based on the state of VO2. On the other hand, a DC voltage is applied between the gold nanowire and the metallic back plate for the ITO-based surfaces. This changes the refractive index of ITO in a unit cell, thereby altering its reflection coefficient.
In an effort to further minimize the Ohmic loss, all-dielectric OHMs have also been demonstrated. The fundamentals of these all-dielectric Huygens’ unit cells have been established by the early works of Mie, Lewin, and Kerker et al. , , . The former has established the connection between the scattering properties of a dielectric structure to its electric and magnetic multipole coefficients . This allows the backward (Qbs) and forward (Qfs) scattering efficiencies to be expressed in terms of electric (am) and magnetic (bm) multipole coefficients given by ,
On the other hand, Lewin  has formulated the effective permittivity and permeability of a material loaded with spherical particles, while Kerker et al.  have shown that a dielectric sphere with equal electric permittivity and magnetic permeability possesses zero backscattering and no depolarization. Such properties are similar to those of a Huygens’ source, and it is known as the first Kerker condition. For example, Figure 21 shows a scattered radiation pattern of a nanosphere that satisfies this condition. The observed pattern is unidirectional (a cardioid pattern) and radiates either in the forward or in the backward direction similar to the pair of GSP resonators (Figure 20H and G). The first Kerker condition corresponds to the case where the electric and magnetic multipoles have equal coefficients in which the zero backscattering can also been seen from (9) for am=bm. In light of the first Kerker condition, dielectric nanoparticles of various shapes (spheres, cubes, cylindrical disks, and rods) have been investigated to engineer their forward/backward scattering properties for certain functionalities , , , , . In particular, Cheng et al. and Decker et al. have demonstrated zero backscattering with an array of silicon nanodisks that also offer full 2π transmission phase by spectrally overlapping electric and magnetic dipole resonances , , . Figure 22C shows the transmission properties of silicon nanodisks when the electric and magnetic dipole resonances are separated and collocated. Specifically, when they are collocated, the transmission magnitude is unity, while the transmission phase offers full 2π variation. Subsequent works by numerous researchers have utilized such an idea and similar inclusions to demonstrate various wavefront manipulations, ranging from THz to visible wavelengths , , , , , , , , , , , , , , , , , , , , , , , . For example, Figure 22D and E shows arbitrary beam refraction and focusing with an array of silicon nanodisks . Such functionalities have also been demonstrated with similar dielectric inclusions, and regardless of the shape of the unit cells being used, the common goal is to parametrically characterize them and utilize a set that provides full 2π phase variation with maximized transmission magnitude. This is to implement a certain phase profile that is deduced from the generalized Snell’s law. On the other hand, Shanei et al.  have demonstrated beam focusing with desired full width half maximum (FWHM) and SLL based on an array of silicon nanoblocks. To achieve arbitrary FWHM and SLL, however, both phase and magnitude modulation must be simultaneously applied on unit cells . While the proposed silicon nanoblocks do not offer complete/systematic independent control over the phase and magnitude, the work in Ref.  presents parametric characterization results as a function of transmission magnitude and phase. From these results, certain geometries of unit cells are selected, which feature the nearest values to the required ones, as shown in Figure 22F. Figure 22G shows another type of all-dielectric unit cells . These metasurfaces are, strictly speaking, not HMSs, as they rely on the waveguiding effect (i.e. phase accumulation) for achieving 2π phase shift. As such, their height should be tall enough to cover a 2π transmission phase. The variation in local phase accumulation is obtained by having a variable proportion of dielectric within a period (i.e. mixing of materials that results to certain effective local mode index). However, Kruk et al.  have recently demonstrated Huygens’ property as represented by Figure 21 with similar unit cells (nanopillars). Instead of viewing them as local waveguides, the authors have engineered the heights and other geometrical parameters of nanopillars to employ higher order electric and magnetic multipoles (in addition to electric and magnetic dipoles) for better suppression of backward scattering waves and enhanced forward scattering. Another Huygens’-related metasurface is shown in Figure 22H, which employs an array of indium antimonide (InSb) nanoblocks that are directly placed on a metallic backplate . The overall system can be interpreted as an array of reconfigurable magnetic dipoles on a metallic backplate. To obtain the reconfigurability, a metallic contact is deposited on InSb, as shown in the figure. This allows dynamic alteration of magnetic dipole resonance of the InSb blocks depending on the biased voltage such that the local reflection coefficients can be reconfigured.
3.5 Active HMSs
Heretofore in this paper, we have examined various kinds of HMSs, which manipulate incoming waves in a passive manner to achieve a desired output waveform. In this section, we review works on active HMSs and their applications. Ample works exist in the literature on the topic of active-controlled metasurfaces , , , , , , , , , , , . These works modify an inherently passive metasurface by injecting an auxiliary signal thermally, optically, or electrically to change the refractive index, the conductivity, or the material phase at selected locations on the metasurface. At radio frequencies, tunable and/or switchable electrical components, such as varactors and p-i-n diodes, are also used to allow one to tune a metasurface element through voltage bias lines. The aforementioned active tuning schemes change the reflection and transmission phases, loss characteristics, and metallization patterns of the metasurfaces and hence make the metasurfaces highly reconfigurable. Some devices built from such active-controlled metasurfaces include anomalous reflectors, elliptical polarizers, switchable reflectors, absorbers, wave plates, reconfigurable lenses, and frequency-tunable devices.
However, the objective of this section is devoted to HMSs featuring impressed Huygens’ source elements. Figure 1 illustrates the general concept of an HMS: an incident (primary) wave generates electric and magnetic currents along the metasurface, which become Huygens’ (secondary) sources as they re-radiate the energy into the surrounding media, thus superimposing with the incident wave to form a desired electromagnetic field
An extremely intriguing application of an active HMS is that of an active surface invisibility cloak. While the first proposals of electromagnetic cloaks ,  involved a passive metamaterial shell with exotic electromagnetic properties, it soon became apparent that such cloaks came with three disadvantages: (i) they had appreciable losses, which led to imperfect cloaking; (ii) they were inevitably large and heavy (the thickness of the cloak was comparable to the size of the cloaked region); and (iii) their achievable bandwidths were inevitably limited due to the sum rule on the extinction cross-section . In principle, one may overcome all three limitations with an active electromagnetic cloak. It was first proposed in the realm of acoustics  and scalar optics  that, through exciting active sources surrounding (or partially surrounding) an object, one could cloak an object by suppressing its scattering. In similar spirit, one could also disguise the object into a decoy by altering its scattered signal . Such active cloaks were desirable in that they could be arranged in an ultrathin layer and had potential to be a broadband, thus bypassing requirements for achieving stringent material parameters over a broad bandwidth.
A few years ago, the design and full-wave simulation of the first active electromagnetic cloak have been reported in Ref. . Figure 23 shows a diagram for the operation of this active cloak. Essentially, the active Huygens’ cloak has been designed such that, in the presence of the object (a dielectric cylinder), the cloak’s impressed currents generated a secondary scattered field, which formed perfect destructive interference with the primary scattered field caused by the incident plane wave. Hence, this suppressed scattering from the object and effectively concealed it. The calculated necessary electric and magnetic currents from the desired free-space fields are given by (10) the following:
where the subscript i denotes incidence, the subscript s denotes electric/magnetic surface currents, and
Figure 24A–B displays the fullwave simulation results showing the successful cloaking of a dielectric cylinder. The cloak surrounded a dielectric cylinder with εr=10, and radius a=0.7λ. The incident plane wave, as depicted in Figure 23, has a z-polarized E-field and traveled in the x-direction (left to right in Figures 23 and 24A–B). Whereas a shadow region existed when the cloak is switched off, the plane wave flowed through the object unhindered when the cloak was activated. More recently, it was shown that the cloak does not need to conform to the shape of the object being concealed. Figure 24C shows an example simulation where a square-shaped cloak concealed an arbitrarily shaped object within the region enclosed by the cloak. Essentially, the active HMS generates a zero field region interior to the cloak, such that objects within would not generate electromagnetic scattering and were thus concealed from the region exterior to the cloak. While the examples shown here demonstrate active electromagnetic cloaking in 2D, the principle would extend directly into three-dimensional with the involvement of properly directed electric and magnetic currents that form the solution to (10).
Active electromagnetic cloaking has been demonstrated experimentally using an HMS . Figure 25A shows the parallel-plate waveguide environment in which the experiment was performed. A metallic cylinder of radius a=0.56λ (f=1 GHz) was cloaked by an active HMS. The metallic shell of the cylinder shorted out the electric currents; the magnetic currents, calculable by (10), were implemented by small loops in close proximity to the cylinder. The cylinder was placed in a parallel-plate waveguide environment, with a perforated top plate, which allowed the probe to scan the electric field within the waveguide. A monopole source illuminated the environment from 3λ away. The measured total electric field (real-part) showed the existence of a shadow region when the cloak was removed (Figure 25B) and the restoration of the incident wave when the cloak was activated (Figure 25C). This work inspired the proposal of other active cloaking schemes, which included a parity-time symmetric cloak with both active and loss components .
While the active Huygens’ cloak engineers the electromagnetic field external to the metasurface, a related development, which we call the Huygens’ box, investigated the possibilities for engineering an electromagnetic waveform within a region enclosed by an HMS. Again invoking the equivalence principle, one can synthesize electric and magnetic currents along a closed surface, which result to the generation of an arbitrary waveform inside the enclosed region. We synthesized the required currents using a simple active Huygens’ source element , , comprising a current strip or a dipole in close proximity to a metallic cavity wall, which shorted out the electric current and left only the magnetic dipole response through the image current. Using this device, we designed metasurfaces that generated unconventional modes within a metallic cavity. Figure 26 shows fullwave simulations performed at 1 GHz for a square metallic cavity of size λ×λ. (The third (z-) dimension is made smaller than half-wavelength to render the cavity a 2D environment.) Travelling waves of arbitrary amplitude, phase, and propagation direction can be generated (Figure 26A–B); standing waves can also be generated with strong tangential electric fields very close to the metallic cavity walls (Figure 26C) . The generation of these modes, which are naturally forbidden in a metallic cavity environment, allows one to build an arbitrary waveform by way of Fourier plane-wave superposition. Accordingly, Figure 26D shows a subwavelength focus formed by propagating plane waves, using a theory known as selective superoscillation . Such waveforms can find potential use in medical imaging and hyperthermia – localized heating of the human body as part of a medical treatment procedure . An ongoing effort towards experimental demonstration of arbitrary waveform generation in a Huygens’ box has also been described in Ref. .
4 Conclusion and prospects
From the exposition of the basic principles and the various applications of HMSs described in this review paper, the versatility and effectiveness of HMSs in manipulating electromagnetic waves become apparent. Compared to gratings, including holographic ones, HMSs offer complete control over the excited Floquet modes, allowing the suppression of unwanted spurious radiation. In addition, compared to reflect- and transmitarrays, they can be made subwavelength thin while offering perfect conversion efficiency between the incident and reflected/refracted waves. The HMSs can be realized from microwaves to optics using both metallic and purely dielectric structures.
Prospects for the future look bright. The possibility of inducing auxiliary surface waves in the design of HMSs offers a way for even more complex wave manipulations such as beam splitting and novel antenna functionality . In the optical regime, nanoscale thin lenses can be envisioned to further miniaturize personal communication devices . Equipping the HMSs unit cells with various control elements such as varactors at microwaves and liquid crystals or phase-transition materials in the optical regime can open up new possibilities for antenna and optical beam-forming applications. One can also envision loading the unit cells with non-linear or non-reciprocal elements to synthesize new functionalities such as frequency conversion, optical switching, and wave isolation/duplexing , . Time modulating HMS is another possibility for obtaining various parametric effects, including magnetless non-reciprocity . Finally, the front of active HMSs, as described in Section 3.5, offers many possibilities for future developments such as active cloaking, steerable antennas, and beam focusing for medical applications.
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