Asymmetric Spatial Power Dividers Using Phase-Amplitude Metasurfaces Driven by Huygens Principle.

Recent years have witnessed an extraordinary spurt in attention toward the wave-manipulating strategies revealed by phase-amplitude metasurfaces. Recently, it has been shown that, when two different phase-encoded metasurfaces responsible for doing separate missions are added together based on the superposition theorem, the mixed digital phase distribution will realize both missions at the same time. In this paper, via a semi-analytical procedure, we demonstrate that such a theorem is not necessarily valid when using phase-only metasurfaces or ignoring the element pattern functions. We introduce the concept of asymmetric spatial power divider (ASPD) with arbitrary power ratio levels in which modulating both amplitude and phase of the meta-atoms is inevitable to fully control the power intensity pattern of a reflective metasurface. Numerical simulations illustrate that the proposed ASPD designed by proper phase and amplitude distribution over the surface can directly generate a desired number of beams with predetermined orientations and power budgets. The C-shaped Pancharatnam-Berry meta-atoms locally realize the optimal phase and amplitude distribution in each case, and the good conformity between simulations and theoretical predictions verifies the presented formalism. A prototype of our ASPD designs is also fabricated and measured, and the experimental results corroborate well our numerical and semi-analytical predictions. Our findings not only offer possibilities to realize arbitrary spatial power dividers over subwavelength scale but also reveal an economical and simple alternative for a beamforming array antenna.

Introduction

One of the favorites of humankind is guiding the electromagnetic (EM) waves in their desired direction. According to the inability of natural materials to provide exotic wave–matter interactions, metamaterials have paved the way to control the EM waves in an unprecedented manner.[1,2] Metamaterials are artificial subwavelength metal/dielectric composites that arm a platform to expose various rich applications including, but not limited to, invisibility cloaks,[3−5] negative refraction,[6,7] illusion,[8,9] beam deflection,[10,11] or epsilon-near-zero behaviors.[12,13] To overcome the metamaterials’ disadvantages arising from fabrication complexities, high inherent losses, strong dispersion, and bulky profiles, metasurfaces have emerged as their 2D version to offer a promising groundwork for tailoring diverse wave signatures such as amplitude,[14,15] phase,[16,17] polarization,[18−20] wave vector,[21] and transverse field profiles.[22,23] The initial overwhelming interest in metasurfaces lies in abrupt and controllable change of wavefronts through a spatial inhomogeneity made by subwavelength scatterers, called meta-atoms, over an infinitesimally thin interface whereby we are able to mold wavefronts into shapes that can be designed at will.

Along the direction of evolution in metasurfaces, it is time to fully manipulate the power intensity pattern of the metasurfaces, an appealing functionality that has not been reported yet. We will illustrate that the conventional beam splitters[24−26] generate two/multiple beams with compulsive power ratios dictated by their tilt angles. Access to asymmetric spatial power dividers (ASPD) with full/independent control over the power level carried by each beam/channel, however, can give us a fabulous flexibility and privilege and is highy demanded in different practical applications, such as satellite communications, multiple-target radar systems, and multiple-input multiple-output (MIMO) communication.[27,28] Formerly, a few studies have contributed to addressing multibeam reflectarrays producing multiple beams with variable power ratio levels. For instance, Nayeri et al.[25] proposed a single-feed reflectarray with asymmetric beam directions and power levels. However, this work has been accompanied with a brute-force and illposed optimization procedure, resulting in a high computational cost that must be repeated afresh if the design specifications change. More recently, Zhang et al.[24] demonstrated that, by changing the angle of incidence, the amount of power carried by each scattered beam can be controlled. Nevertheless, once the power ratios are determined, the direction of the reflected beams is forced to the user. Meanwhile, the presented design, which is restricted to only two beams, is based on numerical predictions, and no semi-analytical framework supports their study. To the best of the authors’ knowledge, the precise control of power levels of multiple beams with arbitrary orientations has not been reported yet and quite remains as a challenging task.

In an effort to manipulate the EM waves with more degrees of freedom, a simple but yet powerful concept had emerged as “Coding metasurface”.[29,30] By purposefully distributing the coding particles ordered by a certain coding pattern over a 2D plane, the coding metasurfaces open the door to many novel and programmable functional devices such as abnormal mirrors,[31−36] low scattering surfaces,[17,37−39] and information encryption interfaces.[40] Recently, it was shown that, when two different coding patterns are added together via the superposition theorem, the mixed coding pattern will elaborately perform both functionalities of primary metasurfaces at the same time.[41] More recently, it has been also demonstrated that involving the complex codes aids in reaching multifunctional meta-reflectors, where several missions of coding metasurfaces could be flexibly superposed together by means of the addition theorem.[42] Nevertheless, in these versions, the addition operation of two digital codes just captured the phase information of the coding particles, and the amplitude data of the final coding pattern were deliberately ignored. Based on the Huygens principle, we will describe that such an assumption would bring irreparable consequences when dealing with the power intensity pattern of the metasurfaces.

In this paper, the key step in obtaining the proposed ASPD is to design a set of subwavelength meta-atoms with arbitrary reflective phase and amplitude responses to fully control the power intensity pattern of the metasurface. Based on the Huygens principle, a general and straightforward semi-analytical method is presented to predict the exact power level of each radiated beam without any optimization procedure. It is demonstrated that the ASPD design with phase-only meta-atoms fails to achieve satisfactory results. Moreover, the quantization/discretization and the element factor effects are incorporated in our design. The numerical simulations depict that, benefited from reflective C-shaped particles with arbitrary local phase and amplitude response, we can accurately manipulate the number and power ratio levels of the emitting multiple beams without sacrificing the meta-atom amplitude and distorting the tilt angles. As a proof of concept, several illustrative examples numerically demonstrated which one of them is experimentally validated. Eventually, the simulated and measured results have a very good agreement with our theoretical predictions. The authors believe that the proposed ASPD exposes a new opportunity to implement spatial power dividers for various applications such as multiple-target radar systems, beamforming networks, and MIMO communication.

Results and Discussion

Unlike the traditional metamaterials specified with continuous effective permittivity and permeability parameters, the metasurfaces drastically facilitate the wave–matter interaction where the macroscopic designs rely on introducing field discontinuity along the surface by spatially engineering the scattering meta-atoms in an array.[43] By introducing abrupt phase shifts covering the range of [0 – 2π], metasurfaces with spatially varying geometries,[44] DC biases,[40,45] orientations,[46,47] light controlling,[48] and time modulation[49] can imprint specified phase discontinuities on the propagating fields over the subwavelength scale based on the generalized Snell’s law.[50] Referring to this law, the in-plane component of the incident wave vector is artificially mapped to that of the desired reflected one as k∥ = (k0 sin θ cos φ + ∇ φ)x̂ + (k0 sin θ sin φ + ∇ φ)ŷ wherein the gradient symbols denote the slopes of phase variations along the x and y directions. For k∥ > k, the anomalous reflection behavior appears in which the direction of the scattered beam can be determined by[51]

Theoretical Framework

Figure a demonstrates the conceptual illustration of the ASPD design generating multiple beams of different power levels that consists of N × N equal-sized meta-atoms with the periodicity of D along both vertical and horizontal directions. From the antenna array theory and upon illuminating by a normal plane wave, the far-field pattern function of the whole metasurface can be rigorously calculated as the superposition of the fields scattered by each contributing meta-atom[52]

Figure 1

(a) Conceptual illustration of the proposed ASPD architecture and (b) front view of the employed meta-atom. The geometrical parameters are D =10 mm, r1 = 4 mm, r2 = 3 mm, and w = 3.2 mm.

In the above equations, Eelement(θ, φ) and a are, respectively, the pattern function and complex reflection coefficient of the meta-atoms, F(θ, φ) refers to the array factor, θ and φ are the elevation and azimuth observation angles, respectively, and k = 2π/λ indicates the wavenumber, where λ is the working wavelength. The directivity function can be subsequently computed by[31,53]

Regarding the Fourier connection between the complex phase–amplitude distribution over the surface and its radiated beams,[54] many EM theorems can be applied to design the ASPD metasurface. For instance, based on the superposition of the aperture fields and the Huygens principle,[25,55] the additive combination of M different phase–amplitude distributions, a, belonging to distinct single-mission metasurfaces yields a mixed phase–amplitude pattern, b, whereby all M individual/primary missions will appear at the same time. The mixed phase–amplitude pattern determines the map of the local phase and amplitude of the reflection coefficient of the meta-atoms in the superimposed metasurface. To generate M multiple functionalities governed by M different phase–amplitude patterns, simultaneously, the local EM response of the constituent meta-atoms in the combined metasurface must read the following addition principle[42]

Here, |a| and φare the reflection amplitude and the reflection phase information belonging to the (m, n)th particle in the jth phase–amplitude pattern, respectively, and b carries the phase–amplitude information of the (m, n)th particle in the superimposed pattern. Without loss of generality, one can logically assume that the individual functionalities made by the primary phase–amplitude distributions (such as focusing, abnormal deflection, and so on) do not necessarily require the amplitude variation of meta-atoms, that is, |a| = 1.[56] However, it will be shown that ignoring the amplitude information of b coefficients may encounter serious problems when we deal with the power intensity pattern of the superimposed metasurface. Hence, the combined phase–amplitude pattern must possess both phase and amplitude information of the final array. By taking 2D IFFT from eq and assuming that |a| = 1

Knowing that F(θ, φ) = IFFT(e),[40] thenin which Fsup(θ, φ) stands for the superimposed array factor. Thus, the scattered fields of the superimposed metasurface can be written according to eq

It should be noted that the multiple scattered beams created by the combined phase–amplitude pattern obey the above equation. Henceforth, in line with our outlined intentions, we develop a generalized format for the addition principle whereby the power level of each radiating beam can be independently controlled by incorporating different multiplicative power constants, , to eqs and 6. Here, j is just the same superscript representing the index of each contributing phase–amplitude pattern. A mathematical manipulation similar to that performed for eqs –9yields

The above formalism represents a more general form of the superposition theorem in which Esupgen(θ, φ) caused by the superimposed phase–amplitude distribution, bgen, contains M independent multiple beams pointing at (θ, φ) directions (see Figure ). It should be remembered that the jth contributing phase–amplitude pattern is designed in a manner that generates a pencil beam along the (θ, φ) direction. Subsequently, for each couple of pencil beams, the power ratio level readswhere Psupgen(θ, φ) and Psupgen(θ, φ) illustrate the peak power intensity of two arbitrarily selected beams oriented along (θ, φ) and (θ, φ) orientations, respectively. It should be noted that, in deducing eq , we assume that the angular position of the maximum in the scattering pattern of each individual metasurface is located in the vicinity of the null of the other scattering patterns, that is, Escatt(θ, φ) ≃ 0. Obviously, eq does not remain further valid if we ignore either the amplitude information of the mixed phase–amplitude pattern, |b|, or the far-field pattern function of the meta-atoms, Eelement(θ, φ). Nevertheless, these important factors were neglected in most of the previous studies.[42]

Figure 2

Schematic illustration of the general form of the superposition theorem by incorporating the power coefficients to generate multiple asymmetric beams with arbitrary power ratio levels.

Concept Verification

In particular, we assume the element pattern function asNevertheless, the mere cosine function of eq closely resembles the element factor of the metasurface particles.[24,37,40] By substituting eq into eq , one can readily infer that, to design an ASPD with a desired set of power ratio levels for the emitting beams, the power coefficients should be chosen as

Once the power coefficients are determined, the phase and amplitude information of the superimposed phase–amplitude pattern can be immediately obtained from eq . After that, the phase/amplitude-adjustable meta-atoms are aimed at realizing the required local EM reflections dictated by the superimposed phase–amplitude pattern. For the sake of simplicity, we put the focus of our design into the case of metasurfaces with two and three asymmetric multiple beams to inspect the performance of the proposed ASPD.

The numerical simulations are first carried out in the MATLAB software using the well-known antenna array theory. We begin the study with semi-continuous designs in which the ASPD structures are discretized in an imperceptible manner and no phase/amplitude quantization has been adopted. The configuration of the semi-continuous demonstrations is the same as depicted in Figure in which all metasurfaces are composed of N = 200 particles separated periodically at the distance of D = D = λ/20 together (to approximately mimic a laterally-infinite surface with continuous phase/amplitude modulation). In the following, we will present three different illustrative examples in which the ASPD structure symmetrically/asymmetrically divides the power between two multiple beams oriented along (θ1 = 10 ° , φ1 = 180 ° ) and (θ2 = 30 ° , φ2 = 270 ° ) directions. In the first illustration, we demonstrate how to achieve two asymmetrically oriented scattering beams with a specific power ratio level by applying the general form of the superposition theorem on two contributing phase–amplitude distributions. The first metasurface driven by the gradient phase-only map of a1 individually reflects the incident waves into the direction of (θ1 = 10 ° , φ1 = 180 ° ), while the second one is designed with the gradient phase-only map of a2 to scatter a single pencil beam pointing at the direction of (θ2 = 30 ° , φ2 = 270 ° ). All individual single-beam metasurfaces in the following examples have been designed based on the gradient phase-only patterns achieved with the same method given in refs[16] and.[17] Finally, the superimposed metasurface whose b phase–amplitude pattern is obtained from eq with plays the role of an ASPD architecture that elaborately splits the normal incidence into two asymmetric reflected beams with the power ratio of Psupgen(θ2, φ2)/Psupgen(θ1, φ1) = 0.77 (see the simulated 2D scattering patterns in Figure a,b). Outstandingly, the semi-analytical predictions based on the Huygens principle and the general form of the superposition theorem estimates well the power ratio level as 0.773. This is the best place to explain that ignoring the element pattern function of eq , like what is done in previous studies,[31,52,53,57,58] imports noticeable error in predicting the power ratio levels of differently oriented pencil beams. For instance, without considering the element pattern function, the superimposed metasurface of our example should generate two pencil beams carrying equal power intensity, which essentially contradict with the results of Figure a,b. To investigate the flexibility of the design, we intend to demonstrate another ASPD dividing the scattered power equally (Psupgen(θ2, φ2)/Psupgen(θ1, φ1) = 1) into the same two asymmetric pencil beams, a special functionality that the conventional wave-splitting platforms fail to achieve.[24] Referring to the Huygens principle and the general form of the superposition theorem in eq , the ASPD structure must be endowed by the superimposed phase–amplitude pattern, b, obtained by assuming to expose two differently oriented beams with identical power budgets. As can be seen in Figure c,d, the power ratio level of two scattered beams satisfactorily approaches to unity (about 0.99) with the same desired tilt angles. We continue to study another peculiar performance that cannot be easily realized via the traditional phase-only metasurfaces, that is, producing the same two independent asymmetric beams on different planes with the power ratio level of (Psupgen(θ2, φ2)/Psupgen(θ1, φ1) = 1.85). In this case, we apply the general form of the superposition theorem on two phase-only patterns by setting . As the inset of Figure e,f demonstrates, 65% of the reflected power approximately propagates toward the higher elevation angle, and the remaining power is arrested in the lower elevation angle, as theoretically expected. We remark that the very little discrepancy between our theoretical predictions and MATLAB simulations can be attributed to the unwanted side lobes (information losses) originating from our initial assumptions: continuous and laterally-infinite phase/amplitude modulating that are not ideally satisfied during the simulations. Anyway, the correctness and robustness of the proposed design strategy are confirmed theoretically and numerically. The presented concept can be further surveyed by illustrating two different examples in which the superimposed phase–amplitude metasurfaces act as an ASPD with three individual scattered beams. In these scenarios, three equations are postulated to disclose the power ratio level between each couple of beams

Figure 3

Numerical simulations of (a, c, e) 1D and (b, d, f) 2D far-field scattering patterns for two-beam ASPD metasurfaces having different power ratio levels.

In these examples, we wish the ASPD structure to generate three independent/asymmetric pencil beams with (θ1 = 10 ° , φ1 = 90 ° ), (θ2 = 20 ° , φ2 = 270 ° ), and (θ3 = 35 ° , φ3 = 180 ° ). Figure a–c represents three different power intensity patterns in each of which three reflected beams have been successfully acquired by proper sets of predetermined power ratio levels , and , , and , as well as , , , respectively. As can be observed from Figure a–c, the tilt angles and power ratios excellently corroborate our theoretical predictions (for further information, see Table ).

Figure 4

Numerical simulations of (a, c, e) 1D and (b, d, f) 2D far-field scattering patterns for three-beam ASPD metasurfaces having different power ratio levels.

Table 1

Quantitative Comparison between the Full-Wave Simulations and Theoretical Predictions for the Power Ratio Levels of the Semi-Continuous ASPD Designs

power ratio level	example #1	example #2	example #3
Psupgen(θ2, φ2)/Psupgen(θ1, φ1) (theory)	0.91	1.1	2
Psupgen(θ2, φ2)/Psupgen(θ1, φ1) (MATLAB simul.)	0.905	1.095	2.02
Psupgen(θ3, φ3)/Psupgen(θ1, φ1) (theory)	0.691	1.556	1
Psupgen(θ3, φ3)/Psupgen(θ1, φ1) (MATLAB simul.)	0.705	1.58	1.03
Psupgen(θ3, φ3)/Psupgen(θ2, φ2) (theory)	0.76	1.413	0.5
Psupgen(θ3, φ3)/Psupgen(θ2, φ2) (MATLAB simul.)	0.779	1.443	0.509

With the above discussions, we have made two observations in the general form of the superposition principle when considering a laterally-infinite continuously modulated metasurface: (i) deflecting the incident wave into multiple arbitrarily oriented reflected beams and (ii) dividing the power asymmetrically between the reflected beams regardless of their tilt angles. Such illustrative examples divulge that the weighted combination of individual phase-only patterns by incorporating the amplitude information of the superimposed metasurface will drastically boost the current wave-manipulating abilities in the framework of the general superposition theorem and furnish a robust and flexible design approach for such power-based complicated manipulations. Theoretically speaking, any controllable power ratio levels for multibeam emissions can be obtained by using the proposed ASPD structures without resorting to any brute-force optimization schemes. Prior to providing the realization detail, we stress that, in this paper, the meta-atoms occupying the ASPD will be characterized with both amplitude and phase information. To highlight the necessity of this revisiting, let us consider the same design approach given in ref,[42] which deliberately neglects the amplitude information of the superimposed metasurface. The results of Figures c and 4c have been reproduced by aggressively sacrificing the amplitude data of the combined metasurface. As can be noticed in Figure , the power ratio level of the scattered beams does not further match with our theoretical predictions, thereby highlighting the significant role of the amplitude information in designing a perfect ASPD structure. Unlike the previous strategies that are applicable only for predicting the direction of tilted beams, the regulations presented in this study are more general and applicable to study the power intensity patterns of the phase–amplitude metasurfaces. It sounds effective in various antenna applications and simultaneous multitarget detection scenarios where the power ratio levels of differently oriented scattered beams are required to be manipulated independently.

Figure 5

Highlighting the effects of the amplitude information in the general form of the superposition theorem by comparing the 2D scattering patterns of the ASPD designs with/without amplitude data for the results of (a) Figure c and (b) Figure c.

Numerical Simulations: Incorporating Discretization and Quantization Effects

Up to now, we have studied semi-continuous phase/amplitude modulation over a laterally-infinite surface, an optimistic hypothesis that cannot be realized in practice. Nevertheless, aggressive discretization and quantization may bring up great advantages such as lower quality factors caused by relatively large element separations, less susceptibility to mutual coupling effects, immunity against fabrication tolerance, low-cost manufacturing, and robust implementation.[37] While most theoretical proposals on metasurfaces deal with continuously varying phase–amplitude or impedance distributions,[40] we will demonstrate how aggressive discretization/quantization deteriorates the ASPD performance. Particularly, the ASPD structure is realized using spatially inhomogeneous distribution of an agile meta-atom whose size and phase/amplitude levels specify the grade of discretization and quantization, respectively. Since the phase–amplitude patterns of the proposed ASPD structures in eq were assumed to be ideally continuous in level and spatial position; henceforth, the discretization and quantization effects should be involved in our study as the constituent meta-atoms occupy a certain size and expose a quantized reflection phase/amplitude.

To investigate the discretization and quantization impacts, the antecedent ASPD demonstrations are reaccomplished this time for structures consisting of N = 30 meta-atoms with the inter-element space of D = D = λ/3 (aggressive discretization), where the phase–amplitude profile describing the superimposed metasurfaces is quantized into two or three levels. With four-level quantization (2-bit) for both amplitude and phase responses, the ASPDs are constructed by 16 elements manifesting phase/amplitude states of “0°/0”, “90°/0.33”, “180°/0.66”, and “270°/1”, while for eight-level quantization (3-bit), the building units of ASPDs are characterized with 64 distinct phase/amplitude responses of “0°/0”, “45°/0.14”, “90°/0.28”, “135°/0.42”, “180°/0.57”, “225°/0.71”, “270°/0.85”, and “325°/1”. Regarding the above-mentioned criteria, the phase–amplitude patterns obtained for the ASPDs presented in the previous section have been discretized and quantized. The numerical simulations have been reaccomplished using MATLAB. A fair comparison between the power intensity patterns generated by the semi-continuous and quantized/discretized phase–amplitude distributions has been carried out in Figure a–d. In the first illustration, the ASPD structures with two-level (Figure a) and three-level quantization (Figure b) serve to divide the incident power equally, Psupgen(θ2, φ2)/Psupgen(θ1, φ1) = 1, between two scattered beams oriented along (θ1 = 15 ° , φ1 = 180°) and (θ2 = 30 ° , φ2 = 270°) directions. In the other example, the ASPD meta-devices with two-level (Figure c) and three-level quantization (Figure d) sets for Psupgen(θ2, φ2)/Psupgen(θ1, φ1) = 1.44 are responsible for asymmetrically scattering two pencil beams with the tilt angles of (θ1 = 30 ° , φ1 = 180°) and (θ2 = 30 ° , φ2 = 270°). As the inset of these figures displays, although the architectures with two-level quantization fail to achieve satisfactory results in comparison to those of continuously modulated designs, the ASPDs with three-level quantization efficiently operate, even in the presence of aggressive discretization. The quantitative summary of the above-mentioned results is tabulated in Table . Consequently, one can conclude that the general form of the superposition theorem based on the Huygens principle does not remain valid under aggressive quantization. As a deduction, conventional 1-bit and 2-bit coding metasurfaces are not able to act as a perfect ASPD. Meanwhile, to have a full control over the power pattern intensity of the ASPD designs, it is required to modulate both phase and amplitude states of the meta-atoms in >3 quantization levels.[44,59−63]

Figure 6

(a, b) Power intensity patterns for ASPDs with (θ1 = 15 ° , φ1 = 90 ° ) and (θ2 = 30 ° , φ2 = 270 ° ) assuming two- and three-level quantization, respectively. (c, d) Power intensity patterns for ASPDs with two reflected beams in the same elevation and different azimuth angles. The blue lines are the continuous results, while the red lines are the discretized results.

Table 2

Quantitative Comparison between the Semi-Continuous and Quantized ASPD Designs for Spatially Dividing the Incident Power

pattern characteristic	(2-bit)	(3-bit)	(theoretical)
(θ1 = 15 °, θ2 = 30 °), (φ1 = 90 °, φ2 = 0 °), and	1.113	1	1
(θ1 = θ2 = 30 °), (φ1 = 180 °, φ2 = 270 °), and	2.754	1.45	1.44

Different methodological attempts have been made to simultaneously modulate the amplitude and phase profiles of a metasurface by tuning the geometry of antennas.[44] We begin the realization of the ASPD designs by discussing an agile phase–amplitude-controlling meta-atom whose topology is depicted in Figure b. The meta-atoms employed in this paper integrate the functionality of a metasurface for phase control and a metasurface for amplitude control, which are adjusted with the geometrical configuration and angular orientation of C-shaped particles, respectively.[44,47] The phase-controlling metasurface is composed of geometrically engineered C-shaped antennas and functions in a linear cross-polarization scheme. Indeed, by changing the arm length and the open angle of the employed meta-atoms, the phase can be robustly and independently controlled in the reflection mode with a polarization orthogonal to that of the incident wave (Figure b). The symmetry line of each particle is oriented along the +45° or −45° angle to maximize the polarization conversion ratio merit.[44] The amplitude-controlling metasurface, however, consists of C-shaped meta-atoms of the same geometry in which the reflection amplitude can be independently modulated by varying the orientations.[44,46,47] Merging the design rules of these two types of metasurfaces yields a C-shaped meta-atom whose cross-polarized phase and amplitude responses can be separately manipulated by changing its geometry and orientation (Figure a–d). The front view of the established meta-atoms is pictured in Figure b in which they are composed of a C-shaped metallic structure etched on a 3.2 mm-thick FR4 substrate (ϵ = 4.3 and tanδ = 0.025). To block the transmitted power, the ASPD structure is terminated with a copper (σ = 5.8 × 107 S/m) ground plane. The periodicity of the meta-atom is equal to λ/3, where λ is the operating wavelength at 10 GHz, and the other geometrical parameters are given in the caption of Figure . By engineering the geometrical parameters of open angle (α) and orientation angle (β), one can independently control the phase and amplitude of the cross-polarized reflection, respectively, a necessary condition to realize the general form of the superposition theorem. The proposed meta-atoms are characterized with the Floquet solver of the commercial software CST Microwave Studio where periodic boundary conditions are applied to x and y directions to simulate a laterally-infinite periodic array, while the Floquet ports are assigned to the z direction. Figure a,b and Figure c,d demonstrate the simulated cross-polarized reflection phase and amplitude coefficients with varying β and α parameters, respectively. When β = ± 45°, the maximum energy from the incident wave will be coupled to the cross-polarization component, and for the case of β = 0° or 90°, the cross-polarization component will vanish. Meanwhile, the cross-polarized reflection phase spans the whole 180° phase range when α varies and β is kept constant. By adding ±90° to β, the amplitude remains constant, while the reflection phase experiences further changes of ±180°.[14] Due to the important role of the element factor in our presentations, the pattern function of the meta-atom employed in this paper is also simulated in CST Microwave Studio, and the corresponding result is displayed in Figure . This figure illustrates that the designed meta-atoms have maximum scattering field intensity along the boresight direction, while it falls down gradually when the observation direction deviates from θ = 0. This element factor, which can be approximated by a cosine function, will be involved in eq to accurately predict the power ratio level of multiple beams scattered by the ASPD designs.

Figure 7

Simulated (a) amplitude and (b) phase of the C-shaped meta-atom with different β values when α = 50°. Simulated (c) amplitude and (d) phase of the C-shaped meta-atom with different α values when β = 13° and β = – 77°.

Figure 8

Simulated scattering pattern function of the employed meta-atoms.

To further validate the concept and dive in the performance of our designs in a more realistic configuration, the finite-size ASPD architectures are excited by a normal plane wave in a full-wave simulation host, CST Microwave Studio. The design follows our previously developed analytical formalism, and we characterize the functionality of the ASPDs (with three-level quantization) through a bistatic scattering pattern measurement setup numerically. In the following, we present two different classes of power-dividing examples in which two scattered beams have the same evaluational (Figure ) and azimuthal (Figure ) angles, respectively. The phase and amplitude distributions of the contributing metasurfaces to the superposition principle for achieving the final ASPD designs of Figure a–f are presented in the Supporting Information. A three-beam-generating metasurface (Figure c) is also perused by a full-wave simulation to show the powerful ability of the proposed ASPD meta-devices in generating multibeam scattered beams with asymmetric power ratio levels. Without loss of generality, a y-polarized normal incidence is considered here. As the first demonstration, based on the general form of the superposition theorem and Huygens principle, we have combined two gradient phase-only metasurfaces with considering to produce pencil beams along (θ1 = 15 ° , φ1 = 90 ° ) and ( θ2 = 35 ° , φ2 = 180 ° ) directions, respectively. Therefore, the superimposed phase–amplitude metasurface should emit two main beams whose carried power levels are dictated by both their tilt angles and power coefficients. In this way, as expected from eq , the power ratio of two scattered beams must obey Psupgen(θ2, φ2)/Psupgen(θ1, φ1) = 1.

Figure 9

Simulated (a) 3D and (b) 2D scattering patterns of the ASPD structures responsible for (a, b) equally and (c, d) unequally dividing the incident power between two beams and (e, f) three beams with different angles.

Figure 10

Simulated (a, c) 3D and (b, d) 2D scattering patterns of the ASPD structures responsible for unequally dividing the incident power between two beams with the same elevation angles.

The full-wave simulation results of the 3D and 2D scattering patterns (φ = 0° and 90° planes) depicted in Figure a,b corroborate well our theoretical prediction where they report the power ratio level and the beam angles as 1.03 and θ1 = 15° and θ2 = 35°, respectively. The slight discrepancy (only 0.12 dB difference) between the results can be attributed to the discretization effects, finite size of the metasurface, and the inevitable coupling between the meta-atoms. For the second demonstration, the basic gradient phase-only metasurfaces creating two pencil beams pointing at (θ1 = 15 ° , φ1 = 90 ° ) and ( θ2 = 35 ° , φ2 = 180 ° ) directions are added by choosing the power coefficients of . Furthermore, the difference between the power intensity levels are just commanded by their distinct tilt angles. By applying the power coefficients into the general form of the superposition theorem, the scattered beams are theoretically achieved to have the power ratio level of Psupgen(θ2, φ2)/Psupgen(θ1, φ1) = 0.719. The simulated results of the 2D and 3D bistatic scattering patterns (see Figure c,d) show that the power ratio reaches 0.705 (only 0.08 dB difference), which has an excellent conformity with our analytical predictions. Moreover, the scattered beams are satisfactorily oriented along the predetermined directions. The summary of the quantitative results is tabulated in the first and second rows of Table . Our final example is devoted to a three-beam-generating ASPD whose numerical and theoretical power specifications are listed in Figure e,f and Table . The achievements observed by the comparison between analytical and numerical results depict a perfect concordance. The existing negligible errors are mostly due to the discretization and finite size of the ASPD, which are interestingly less than 3%. By skimming the scattering results of Figure a–d and the detailed comparison given in the third and fourth rows of Table , one can immediately deduce that the fixed-θ radiating ASPD structures also function well. The phase and amplitude distributions of the contributing metasurfaces to the superposition principle for achieving the final ASPD designs of Figure a–d are presented in the Supporting Information. Overall, the proposed ASPD structures successfully perform their missions, that is, dividing the power asymmetrically and arbitrarily between multiple beams pointing at our desired directions, a prominent functionality that was not reported.

Table 3

Quantitative Comparison between the Full-Wave Simulations and Theoretical Predictions for the Power Ratio Levels of Two-Beam-Generating ASPD Designs

pattern characteristic	(theoretical)	(numerical)
(θ1 = 15 °, θ2 = 35 °), (φ1 = 90 °, φ2 = 180 °), and	1	1.03
(θ1 = 15 °, θ2 = 35 °), (φ1 = 90 °, φ2 = 180 °), and	0.719	0.705
(θ1 = 20 ° , θ2 = 20 ° ), (φ1 = 90 ° , φ2 = 180 ° ), and	0.446	0.444
(θ1 = 25 ° , θ2 = 25 ° ), (φ1 = 180 ° , φ2 = 315 ° ), and	1.44	1.48

Table 4

Quantitative Comparison between the Full-Wave Simulations and Theoretical Predictions for the Power Ratio Levels of Two-Beam-Generating ASPD Designs when (θ1 = 10 ° , θ2 = θ3 = 25 ° ), (φ1 = 180 ° , φ2 = 90 ° , φ3 = 135 ° ) and , , and

power ratio level	example #3
Psupgen(θ2, φ2)/Psupgen(θ3, φ3) (theoretical)	1.69
Psupgen(θ2, φ2)/Psupgen(θ3, φ3) (numerical)	1.7
Psupgen(θ1, φ1)/Psupgen(θ3, φ3) (theoretical)	0.95
Psupgen(θ1, φ1)/Psupgen(θ3, φ3) (numerical)	0.915
Psupgen(θ2, φ2)/Psupgen(θ1, φ1) (theoretical)	1.779
Psupgen(θ2, φ2)/Psupgen(θ1, φ1) (numerical)	1.858

Conclusions

In summary, a general version of the superposition theorem based on the Huygens principle was proposed for the first time to introduce the concept of ASPD architectures whereby the power can be asymmetrically and arbitrarily divided between multiple beams oriented along the predetermined directions. The proposed design scheme is not accompanied with any brute-force optimization, trial-and-error steps, and time-consuming procedures. Particularly, it was theoretically shown that, unlike the previous demonstrations, both phase and amplitude profiles of metasurfaces must be modulated to empower us to flexibly control their power intensity patterns. Benefited from the C-shaped meta-atoms, we were able to independently tailor the local reflection phase and amplitude with three-level quantization. Several illustrative examples were presented in this paper to validate the concept. A prototype of our ASPD designs is also fabricated and measured, and the corresponding experimental results are in good accordance with our numerical and semi-analytical predictions. This work takes a great step forward in designing spatial power dividers for which many promising applications such as beamforming networks and MIMO communication can be envisioned.

Experimental Section

To verify the simulation results, we fabricate one of the previously designed ASPD prototypes with the dimensions of 27 cm × 27 cm (including 27 × 27 meta-atoms) through printed circuit board (PCB) technique on a commercial FR4 substrate (see Figure a). The ASPD has the thickness of 3.2 mm and is terminated by a copper ground plane. In our experiments, we designed, fabricated, and measured the ASPD design of Figure a, whereby the incident power is spatially divided between two differently oriented pencil beams with the power ratio level of 0.444 (−3.5 dB). The experiments are totally performed in an anechoic chamber (Figure a) to avoid any possible interference from the environment, and the schematic photograph of the measurement setup (i.e., the NRL Arc setup[64] for bistatic radar cross-section measurement) is illustrated in Figure b. Two X-band horn antennas covering the frequency bandwidth of 8.2 to 12.4 GHz were established in our tests whose locations and angular orientations could be freely/precisely adjusted on the arc setup. One of the horn antennas is responsible for exciting the ASPD from the normal direction, while the other one moves along the arc curvature while recording the cross-polarized amplitude of the far-field scattering pattern. To provide a quasi-plane wave illumination, the feeding antenna was located at the distance of about 2 m from the measured ASPD sample. Due to the finite size of the sample and the nonplanar wavefront of the feeding antenna, the phase compensation method was accomplished during the ASPD design in the same manner given in ref.[65] The ASPD sample was enveloped by pyramidal absorbing materials to block possible edge diffractions. The vector network analyzer 8720C (50 MHz–20 GHz) was connected to the horn antennas to capture the transmission coefficients between them. A personal computer with GPIB interface is also responsible for gathering the far-field pattern data. According to the restrictions of our experimental setup, the cross-polarized component of the far-field scattering pattern was measured only in two principal planes. The noise effects were removed to some extent by subtracting the results of the background scene. In this manner, the measured far-field patterns were recorded at 10 GHz, and the corresponding results are given in Figure c,d. As can be observed, the measured beam angles in both perpendicular planes corroborate well the simulation results (i.e., about θ = 19°). Most importantly, the amplitude of the power pattern pertaining to the scattered beam #1 is about 3.95 dB larger than that of the beam #2, which is in acceptable accordance with our numerical and semi-analytical predictions illustrating 3.5 dB power difference. The little discrepancy between the simulated and measured results can be attributed to the fabrication tolerances, the dispersion of the substrate, and imperfections caused by our experimental setup. Eventually, the proposed ASPD design operates well in arbitrary, dividing the incident power between multiple spatial channels oriented along our desired directions.

Figure 11

Experimental validation. (a) Photo of the fabricated ASPD prototype generating two asymmetric beams with the power difference of 3.5 dB and the environment of the experimental procedure with the ARC setup. (b) Schematic sketch of the performed bistatic scattering measurements in the anechoic chamber. (c, d) Cross-polarized components of the 2D far-field pattern in the φ = 90° (c) and φ = 180° (d) planes.