Static Metasurface Reflectors With Independent Magnitude and Phase Control Using Coupled Resonator Configuration

A static metasurface reflector based on a novel coupled resonator configuration is proposed to independently control the reflection phase and magnitude of linearly polarized incident fields and is demonstrated experimentally in the millimeter-wave Ka-band around 30 GHz. The proposed concept is illustrated using a unit cell design consisting of a rectangular ring coupled with a rectangular slot resonator backed by a grounded dielectric slab. By geometrically tuning various dimensions of the two resonators, a near-perfect amplitude–phase coverage is achieved at a fixed design frequency of 30 GHz. To demonstrate the flexible beam-forming capability of the proposed metasurface reflectors, illustrative examples of fixed beam steering with varying reflection magnitudes and asymmetric dual-beam patterns with specified reflection magnitude, reflection angles, and beamwidths are successfully shown. Compared to the standard method based on polarization rotation and resistive loadings with discrete values, the proposed technique does not generate undesired cross-polarization field reflection and provides continuous magnitude tuning including full absorption, along with wide phase coverage.


Static Metasurface Reflectors With Independent Magnitude and Phase Control Using Coupled
Resonator Configuration Joel S. Demetre, Student Member, IEEE, Tom J. Smy , and Shulabh Gupta , Senior Member, IEEE Abstract-A static metasurface reflector based on a novel coupled resonator configuration is proposed to independently control the reflection phase and magnitude of linearly polarized incident fields and is demonstrated experimentally in the millimeter-wave Ka-band around 30 GHz. The proposed concept is illustrated using a unit cell design consisting of a rectangular ring coupled with a rectangular slot resonator backed by a grounded dielectric slab. By geometrically tuning various dimensions of the two resonators, a near-perfect amplitude-phase coverage is achieved at a fixed design frequency of 30 GHz. To demonstrate the flexible beam-forming capability of the proposed metasurface reflectors, illustrative examples of fixed beam steering with varying reflection magnitudes and asymmetric dual-beam patterns with specified reflection magnitude, reflection angles, and beamwidths are successfully shown. Compared to the standard method based on polarization rotation and resistive loadings with discrete values, the proposed technique does not generate undesired cross-polarization field reflection and provides continuous magnitude tuning including full absorption, along with wide phase coverage.

I. INTRODUCTION
E LECTROMAGNETIC metasurfaces are 2-D arrays of subwavelength resonating particles, which are engineered at the microscopic scale to tailor the macroscopic scattered fields in response to the specified incident fields. By manipulating their sizes, materials, and geometrical shapes, a wide variety of wave transformations in transmission and reflection, along with polarization control, can be achieved [1]. Consequently, they have found a myriad of applications ranging across the electromagnetic spectrum, from radio frequencies (RFs) to optical [2], [3], [4], [5], [6], [7], [8], [9].
To achieve complete control over the scattered fields from a metasurface, the subwavelength resonating particles must be Manuscript  engineered to provide all possible combinations of transmission/reflection phase and magnitude. If the surface is reflection type, only the reflection magnitude and phase must be controlled, i.e., the complex reflectance. However, if the surface is transmission type, the problem is more involved, as both complex transmittance and reflectance must be simultaneously manipulated. Once such a resonating particle is conceptualized and designed, the resulting complex reflectance (and/or transmittance) can then form the building blocks to realize an arbitrary spatial distribution of reflection field profile to obtain the desired scattered fields. A few solutions to achieve an independent reflection magnitude and phase have been attempted in the literature. These existing solutions can fundamentally be divided into two categories: 1) polarization rotation and 2) resistive/lumped element loading. In the first approach, part of the incoming polarization is converted into a cross-polarized field component, which acts as an effective loss or amplitude modulation of the specified fields. The geometrical shape of the particle is typically used to independently tune the phase [10], [11], [12], [13], [14], [15]. This strategy can be applied in both reflection and transmission operation [16], [17], [18], [19], [20], [21]. While this is an effective method to control the amplitude and phase, it naturally generates spurious cross-polarized fields, which either must be ignored or be selectively absorbed. Another related method is to utilize anisotropic surfaces, where the complex reflectance of the two incoming polarizations may be independently controlled [22]. While this technique is attractive for generating circular polarization and general simultaneous beam forming of two orthogonal polarizations, it cannot provide an arbitrary complex reflectance if only a single linear polarized wave is assumed incident on the surface without generating any cross polarization.
The second approach is to use resistive loadings in the resonators to modulate its Q-factor. This is typically achieved by adding lumped resistive components in strategic locations in the unit cell to change the magnitude, while the phase is tuned using the geometrical dimensions or by lumped inductors and capacitors [23], [24], [25], [26]. For static cases, these components are typically off-the-shelf commercially available RF resistors, for magnitude control. For dynamic cases, electronically tunable elements, such as varactor diodes and p-i-n diodes, are naturally required to modulate the phase and amplitude independently, as recently been shown for metasurface reflectors [27], [28], [29]. Similar ideas have been proposed in the past such as the concept of digital coding metasurfaces [30], [31]. They are dynamic surfaces and naturally based on voltage/circuit biasing of active lumped circuit elements on the metasurface to achieve the desired complex reflectance states. Due to their natural reliance on tunable resistors (e.g., p-i-n diodes), there are no static counterparts of these surfaces. However, in all these approaches, no cross polarization is generated, as desired.
For static surfaces, since the amplitude control is through RF resistors, which are typically available in discrete values only, there is no simple way to achieve a continuous range of resistance values (and thus the magnitude values). Moreover, from the fabrication perspective, it requires an extra lumped element assembly step, which increases its prototyping costs. The use of lumped elements also presents inherent difficulties when scaling to higher frequencies or when moving to a more subwavelength unit cell, as these elements have a fixed size in comparison to the unit cell.
In this work, we are proposing a third method of achieving an independent reflection magnitude and phase response at a given design frequency using a static metasurface reflector, when operated with linearly polarized incident waves, without generating any cross polarization and without any discrete static lumped resistors. This proposed technique is based on a concept inspired by [33], [27], and [28], where the metasurface cell consists of two tightly coupled resonators. Dimensional changes in one resonator affect the reflection response of the second resonator, via electromagnetic coupling, enabling an efficient way to independently, and continuously, tune the reflection amplitude and phase. The proposed concept is illustrated using transmission line models, a surface susceptibility-based coupled Lorentz oscillator model and fullwave simulations, and is experimentally confirmed using a variety of reflector prototypes to produce steered beams with varying amplitude and asymmetric dual-beam profiles.
There are other alternative techniques proposed in the literature to provide independent amplitude/phase responses. One technique is to use two metasurfaces separated by a distance, which by properly designing the metasurfaces provide an independent control of the transmission amplitude and phase. This technique has been referred to as compound metaoptics [34], [35]. While the compound metaoptics technique can provide independent amplitude and phase control, they are well known to require electrically large thicknesses (typically more than a wavelength), as sufficient propagation distances are needed between two metasurfaces. The proposed unit cell on the other hand is electrically thin and can be homogenized with effective surface parameters (e.g., angle-independent effective surface susceptibilities). They consequently qualify as electromagnetic Metasurfaces, as opposed to the compound metaoptic systems. Another technique to achieve independent amplitude and phase control of wavefronts is through embedding sources inside the metasurface structures [36]. The overall structure in fact may be considered closer to a metasurface antenna. This is a fundamentally different field scattering problem compared to what is being addressed in the proposed work here, where generic incident fields of arbitrary distributions are assumed to be propagating waves, presumably coming from a source(s) located in the far-field regions. Consequently, the proposed surfaces that are passive field reflectors should be considered as scatterers with a very different operating principle compared to those in [36].
This article is structured as follows. Section II presents the comparison of a single resonator surface with a double resonator using a transmission line model and introduces the concept described in terms of effective surface susceptibilities. Section III shows the proposed unit cell architecture and demonstrates independent amplitude and phase tuning using Ansys FEM-HFSS, along with the proposed practical metasurface implementation. Section IV presents the fabricated metasurface reflector prototypes for various wave transformations in the Ka-band around 30 GHz, and the custom build metasurface measurement system used to characterize the surfaces. Conclusions are finally provided in Section V.

II. PROPOSED METASURFACE PRINCIPLE A. Conventional Single Resonator Unit Cell
A conventional reflective metasurface unit cell typically consists of a subwavelength resonator on top of a host dielectric backed by a ground plane, as shown in Fig. 1(a). To understand its reflection characteristics, an equivalent transmission line model may be built [37], [38], [39], where a generic resonator is represented by a shunt resonator, Z 1 , which could be a series LC R resonator, for instance, i.e., It is followed by a transmission line of length ℓ, corresponding to a dielectric slab of permittivity ϵ r , backed by a short circuit representing the ground plane. A resistance is added to account for the inherent conductor losses. Fig. 1(a) shows a typical reflection magnitude and phase response, as the resistance and the capacitance of the resonator are changed. While the capacitance change may be brought about by changing the dimension of the resonator geometry, to tune its resonant frequency, a precise change of the resistance is only possible by adding external lumped elements. The amplitude and phase plots reveal that an effective tuning of the phase is achieved as capacitance is changed while keeping a constant resistance, R 0 . However, this is accompanied by an uncontrolled change in the magnitude. Fig. 1(a) further shows various contours of constant magnitudes, which offers full 2π phase; however, it confirms that an independent magnitude and phase control in this single resonator configuration necessarily requires simultaneous tuning of capacitance and resistance [23].

B. Proposed Coupled Resonator Unit Cell
It is clear from the example of a single resonator of Fig. 1(a) that two control elements are needed to independently tune the reflection magnitude and phase. To avoid resistance tuning, an alternative way is to add another resonator structure, resulting in a dual resonator configuration, as shown in Fig. 1(b). The resonance control of each of the two resonators can thus be seen as two independent tuning mechanisms to manipulate its complex reflectance. This is more clearly seen in an equivalent transmission line model, where the two resonators are chosen to be separated by a dielectric slab, which are both placed on a short-circuited host dielectric. Each of the resonators is considered here to be a series LC R resonator, where their capacitances C 1 and C 2 may be varied to tune their resonance frequencies. Such resonance shifts can be practically brought about by structural changes in the two resonators. While the two resonators in this equivalent model are shown as uncoupled for simplicity, in physical practical implementations, they will naturally be electromagnetically coupled.
The typical reflection amplitude and phase response is further shown in Fig. 1(b) for varying values of the two capacitances, C 1 and C 2 , and for a fixed value of the resistance. The reflection features are distinctly different from the case of a single resonator, while still containing regions of full reflection and absorption, along full phase coverage. To evaluate whether an independent tuning of reflection phase and magnitude is possible, contours of constant magnitude are shown in the phase plots for various values. It is clearly visible that each of these constant magnitude contours passes through the full range of reflection phase. In other words, specific combinations of C 1 and C 2 are possible to achieve a specified arbitrary combination of reflection magnitude and phase at the desired frequency, without changing the resistance. This thus provides an opportunity for complete control over the complex reflectance by tuning the resonant frequencies of each resonator, which can be easily accomplished by changing their geometrical dimensions and without adding any extra lumped circuit elements.

C. Coupled Lorentz Oscillator Model
While the reflection phase tuning is well understood in terms of geometrical size of the resonators, how the general complex reflectance (and especially the reflection magnitude without any resistive lumped components) can be changed in a proposed coupled resonator unit cell is less obvious. To gain better insight into the complex reflectance control, we can use a coupled Lorentz oscillator model to understand the behavior of the unit cell. Since the unit cell is subwavelength (i.e., period ≪ λ 0 ), it can be described in terms of effective surface susceptibilities, which are the constitutive parameters of the surface [40]. Let us consider, for simplicity, a monoisotropic metasurface reflector (i.e., T (ω) = 0) modeled using electric and magnetic surface susceptibilities, i.e., χ ee (ω) and χ mm (ω), respectively. A typical frequency dispersive susceptibility can be expressed as a sum of Lorentzians (and Drude) as where ω 0 and ω p are the resonant and plasma frequencies, respectively, and γ is the loss coefficient. For a reflector, to ensure zero transmission across frequency, the magnetic surface susceptibility assuming normal plane-wave incidence is related to the electric one as A typical frequency response of a coupled resonator unit cell is shown in Fig. 2(a), with two resonances across frequency modeled using (2) with three expansion terms (with ω 0,2 and ω 0,3 as the two resonant frequencies of the cell). This, for example, can be extracted from a given complex reflectance obtained from a unit cell simulation. This reflectance response is shown in Fig. 2(b) as case #1. The cell exhibits a desired reflection phase of φ at the design frequency f des. with a given reflection magnitude. One may ask, how can we tune this reflection magnitude while maintaining the phase without adding extra material losses?
To achieve this, the second resonance of the coupled resonator can be tuned, in conjunction with the main resonance of the structure, to affect the complex reflectance at f des. . 1 For example, Fig. 2(b) shows a second case #2, where a new set of resonant frequencies { f 0,2 , f 0,3 } is chosen, which maintains the same phase value of φ at f des. , while increasing the corresponding reflection magnitude. The main difference between the two cases is the phase slopes around f des. (and thus the Q-factor of the resonance controlling the reflection magnitude), as evident in the inset of Fig. 2(b). This is a result of an electromagnetic interaction between the coupled resonances of the unit cell only and not adding any extra material losses. This behavior will now be confirmed next using a practical unit cell design.

A. Proposed Unit Cell Design
The proposed metasurface reflector of Fig. 1(b) requires a subwavelength double resonator configuration backed by a grounded dielectric slab. A simple practical implementation of such a configuration is shown in Fig. 3(a) chosen here for a 30 GHz operation in the Ka-band, for later demonstration. It consists of a rectangular ring resonator coupled with a rectangular slot resonator, which is then placed on top of a conductor-backed dielectric slab. Any changes in one resonator affect the response of the second resonator via electromagnetic coupling, which is then used to achieve independent amplitude-phase tuning. The resulting unit cell has a subwavelength periodicity, i.e., ≪ λ 0 (e.g., 2 mm for this example equals λ 0 /5 at 30 GHz). Practically, the proposed unit cell architecture is a multilayer configuration, as shown in Fig. 3(a), where the two dielectric layers are attached using a bonding layer. The unit cell layer stack has a ground plane of thickness 52 µm, a rectangular slot of 35 µm, and a rectangular ring 35 µm, all being made of copper. The base substrate (h 1 = 730 µm) and the top substrate (h 2 = 100 µm) are both composed of Rogers 4350B, which was simulated using ϵ r of 3.62 and δ of 0.0037. The PrePreg bonding layer (t = 110 µm) was made of Rogers RO4450F and simulated using ϵ r of 3.8 with δ of 0.0039; 9 µm-thick resin layers were used adjoining the metal and substrate/PrePreg that have ϵ r of 2.4. These design parameters are used throughout this work for later prototyping and testing.
The unit cell operates on a specific linearly polarized incident wave, as shown in Fig. 3(a). Due to the symmetry of the structure, no cross-polarization is generated, as desired. Moreover, since the structure is not rotationally symmetric, the complex reflectance of the orthogonally polarized mode is different. If the slot resonator has zero width, the proposed cell simply becomes a conventional single resonator structure with a uniform ground plane. The addition of a slot in that ground can thus be seen as introducing an effective defected ground plane structure, with engineerable surface impedance, instead of a fixed short-circuit impedance [41]. Alternatively, it may be viewed as cascade of two impedance layers backed by a ground plane, to control the complex reflectance, similar to multilayer Huygen's metasurfaces [9], [42]. While the proposed unit cell structure with slot resonator may be seen as a simple and intuitive evolution of a single resonator-based metasurface, other architectures are naturally possible with two or more resonance structures.
The coupled resonators of the unit cell have several key design parameters: slot resonator length ℓ s , width w s , ring resonator length and width ℓ r and w r , and the ring resonator thickness w, while the slot thicknesses always remain at a thickness of 110 µm, due to fabrication considerations. The left of Fig. 3(b) shows a typical simulated amplitude and phase coverage range when excited with normally incident plane waves, using FEM-HFSS, which are achievable using this unit cell by varying all these key design parameters for a fixed number of simulation runs and for a fixed design frequency (30 GHz, here). Each point indicates that a set of resonator dimensions could be readily found and that a specific amplitude-phase pair is achievable. A near-perfect amplitude coverage from full reflection to full absorption and phase coverage of 2π is observed, as desired. It is further found that there is no simple relationship between the reflection magnitude and the phase, with the five key parameters of the unit cell. Consequently, Fig. 3(b) essentially serves as the main lookup table of this unit cell for 30 GHz operation frequency. A small subset of the full lookup table where all the parameters except w s and ℓ r are kept constant is shown in the right of Fig. 3(b), which indicates   Fig. 4. The two unit cells are chosen from Fig. 3(b) exhibiting a high absorption and reflectance. As expected, strong currents are observed for high reflection magnitudes, whereas weak ones are observed for strong absorption.

B. Metasurface Design
A metasurface can now be designed by cascading unit cells of Fig. 3(a) of varying resonator dimensions to realize a spatial complex reflectance along the metasurface to achieve a specified far-field reflection pattern. A full-wave model of the N unit cell metasurface is shown in Fig. 5. The metasurface is  Table I. excited with a y-polarized uniform plane wave (i.e., transverse magnetic (TM), for instance), and perfect magnetic conductor (PMC) boundary condition is used along the x-direction to enforce uniformity along the x-axis. The unit cell geometry is varied along y, to engineer a desired complex reflectance profile, (y).
A simple way to obtain the space-dependent complex reflection profile to generate a multibeam pattern is using standard antenna array theory (see [43]). The surface is first spatially discretized with N × N unit cells with a unit cell period , i.e., surface size is ℓ = N . For a dual beam, the desired far-field radiation pattern can be explicitly expressed as E(θ ) = N n=1 a n e jn{k 0 sin θ } + b n e jn{k 0 sin θ } resulting in the overall complex reflectance of the cell as n = (a n + b n ). The variable k 0 is the free-space wavenumber at the design frequency ω, N is the number of unit cells in the metasurface, and {a n , b n } is the complex reflectance profile of the nth unit cell, for each beam. To prescribe the desired far-field pattern properties, the beam reflectances can be represented in the form of a n = w(n)e jnβ n e jnβ 0 (5) for instance, where w(n) is the amplitude window function used to control the beamwidth and amplitude of the generated beam, and β n = k sin θ 0 to control the steering angle, with θ 0 being the incident angle of the exciting plane wave. The amplitude function w n is constructed here as a Chebyshev window [43]. In MATLAB, this can easily be implemented using the chebwin(·), as w(n) = b n chebwin(N , r n ) function, where N is the total number of unit cells in the metasurface and r n is the relative sidelobe attenuation, which also controls the beamwidth in the far field. Using an appropriate set of {β n , b n , r n } in (4) and (5), an approximate multibeam pattern can be generated with the two beams at desired angles, peak amplitudes, and beamwidths. A more general synthesis of the surface can, however, be performed using surface susceptibility-based methods [40]. Fig. 6 shows a few examples of the proposed metasurface reflector with various spatial profiles of complex reflectances and corresponding far-field reflection patterns. The first metasurface example is where an oblique incident plane wave at θ = −25 • is used, to generate an output reflection beam at an angle of 40 • at a design frequency of 30 GHz. Since the angle of reflection does not follow conventional Snell's law, a linear phase gradient ̸ (y) is needed to steer the beam. Simultaneously, it is desired to change the reflection magnitude of the output beam to various levels, including almost perfect absorption. Thus, reflection magnitude control is required. Using (4), the complex reflectance is generated, and the lookup table of Fig. 3 is used to generate three different designs, exhibiting three different | |'s independently of the required phase gradient, as shown in the bottom plots of Fig. 6(a)-(c). The top plots of Fig. 6(a)-(c) show the full-wave simulated farfield reflection profile using the model of Fig. 5. As expected, an output reflection beam is successfully generated at 40 • in each case, with varying field magnitudes, as specified. Differences in the ideal reflectance profile and ones obtained from FEM-HFSS may be attributed to interunit cells couplings in a nonuniform structure, which are not accounted for in both the array factor of (4) as well as the lookup table of Fig. 3 obtained using ideal periodic boundary conditions.
A second example of a surface is one that generates two reflection beams of specified characteristics when excited with a uniform plane wave at an angle of −10 • . Two different designs are made. The first design generates a reflection beam in each half of the surface (i.e., one beam with anomalous reflection), where both the reflection angles do not follow conventional Snell's law, as shown in Fig. 5(d). A second design generates one anomalously reflected beam and another following conventional Snell's law, as shown in Fig. 5(e). In these examples, generally, higher background reflection is observed compared to the specifications, which can be attributed to the noticeable differences in the near-field complex reflectance of  FIG. 6 the surface, especially for the phase. Nevertheless, in both cases, two reflection beams of specified beam characteristics are successfully generated. These two examples of asymmetric dual-beam surfaces illustrate the usefulness of the proposed coupled resonator metasurface reflector providing independent reflection amplitude and phase control, which are otherwise not possible to obtain using phase-only metasurfaces. The summary of the various beam characteristics is tabulated in Table I. It should be noted that the choice of these specifications was arbitrarily made to illustrate the concept.

A. Measurement Setup
To measure the scattered field characteristics of various metasurface reflectors, no standard metasurface measurement setup exists. Consequently, an in-house measurement system is built, which operates in the Ka-band , as shown in Fig. 7. It consists of a circular track on which transmitter (Tx) and receiver (Rx) standard gain horn antennas (Eravant/Sage Millimeter SAR-2013-282F-E2, Ka-band horn with 20 dBi gain) are installed and moved using an automated control. The two horn antennas are connected to the two ports of a vector network analyzer (PNA, Agilent N5230A). To enable co-angle measurements, where the Tx and Rx horns are located at the same angle (i.e., monostatic configuration), an angular tilt is added on each horn in the vertical plane in addition to slightly different heights, as shown in Fig. 7(c). The metasurface to be tested is placed at the center of the circular track. The separation between the metasurface and the Tx/Rx horns is 700 mm, to ensure far-field measurement conditions, which essentially limits the size of the surface in the given operating frequency band. Several absorbers were placed to minimize reflections from the background. A typical noise floor of −30 dB is measured.
Next, to calibrate the system, a PEC reflector of the same size as the metasurface to be tested is first used, as a reference reflector. The measured angular field distribution and the peak field amplitude are then used to calibrate the system, which is then used to estimate the reflection characteristics of the surface. The angular response of the PEC reflector at various frequencies was measured and compared with full-wave simulations (not shown here), and a typical amplitude difference is found to be ≈0.5 dB between the two. After the calibration step, the PEC reflector is replaced with a metasurface of interest for actual characterization.

B. Metasurface Prototypes and Results
Next, all the metasurface designs of Table I operating  at 30 GHz were fabricated for testing based on the unit cell geometry of Fig. 3. The detailed fabrication parameters have already been summarized in Section II-A. Fig. 8 shows the results of the single-beam steered metasurface with three different reflection amplitudes, where a picture of one of the prototypes is shown in Fig. 8(a) consisting of a 30 × 30 unit cell array. Two identical sets of each metasurface (#1 and #2) were fabricated to test the repeatability of the prototypes/measurements. The Tx horn is fixed at an angle of θ inc. = −25 • , while the Rx horn is moved around the metasurfaces in the front half-space, i.e., θ ∈ {−90 • , 90 • }. The top row shows the transmission frequency response (S 21 ) of the surfaces at various Rx angular positions, for the cases of high reflection (large S 21 ), medium reflection, and low reflection (small S 21 ). All the S-parameters plots are normalized with respect to the calibrated data measured using the reference PEC reflector. This broadband response helps to identify any frequency shifts that may have occurred due to fabrication and material tolerances. Indeed, a frequency downshift is observed of about 1 GHz, clearly evident from the location of the transmission dips across frequency. Next, angular field patterns were extracted at 29 GHz for each case and are shown in the bottom row of Fig. 8(b)-(d). A clear beam tilting at 40 • is observed, especially for the high and  medium reflection cases, while the transmission level is almost −20 dB for the low reflection case approaching a strong absorption. The responses of two identical surfaces further show good repeatability of the measurements. Despite a clear main reflection beam, in general, the background reflection appears to be higher than expected, which could be due to spurious reflections from the environment, as the setup is not completely shielded in an enclosure. The operational 3 dB bandwidths are found to be 5.5 GHz, 850 MHz, and 150 MHz for each of the cases. Nevertheless, the variation in the reflection amplitude, while the beam is being steered, is clearly demonstrated with an acceptable agreement with the full-wave simulations.
The second set of measurements corresponds to the dual-beam reflection patterns. The picture of one of the prototypes is shown in Fig. 9(a). The incident beam is fixed at θ inc. = −10 • , and two beams are generated to the left and right of the incident beam. Two different surfaces are fabricated with different beam specifications, corresponding to the last two entries of Table I, and their corresponding reflection responses are shown in Fig. 9(b) and (c). In both cases, a narrow but higher reflection beam is observed at θ 2 , as desired with very good agreement with full-wave simulations. The anomalously reflected beam on the left is lower in amplitude and wider in beamwidth. While noticeable ripples are observed on this beam, it has an acceptable agreement with the full-wave response in terms of the reflection amplitude and beam location, in particular. An improved measurement setup with better isolation from the environment is expected to reduce the background reflection and occurrence of local spurious reflections. Overall, both surfaces generated two distinct beams of desired characteristics with an operational bandwidth of about 150 MHz, and the measurements for both single-and dual-beam surfaces validate the proposed metasurface architecture to achieve independent amplitude and phase characteristics.

V. CONCLUSION
A static metasurface reflector based on a novel coupled resonator configuration has been proposed to independently control the reflection phase and magnitude of linearly polarized incident fields and has been demonstrated experimentally in the millimeter-wave Ka-band. The chosen practical design consists of rectangular ring coupled with a rectangular slot resonator backed by a grounded dielectric slab, due to its design simplicity and ease of implementation. By geometrically tuning various dimensions of the two resonators, nearperfect amplitude-phase coverage has been achieved at a fixed design frequency of 30 GHz. Using the generated lookup table, various metasurface reflectors have been designed, simulated, and successfully measured. To demonstrate the flexible beam-forming capability of the proposed metasurface reflectors, illustrative examples of fixed beam steering with varying reflection magnitudes and dual-beam patterns with specified reflection magnitude, reflection angles, and beamwidths have been demonstrated.
The proposed concept of utilizing coupled resonances to achieve independent reflection amplitude and phase represents an attractive technique for achieving arbitrary complex reflection profiles from metasurfaces, without generating spurious cross-polarization. In addition, it does not require discrete resistive elements that are typically available with limited quantized values, thereby eliminating the lumped element assembly step in standard PCB fabrication and simplifying the overall prototyping process, making the proposed techniques ideal for high-frequency implementations, including various millimeter-wave bands. While a multilayer unit cell architecture of the coupled ring and slot resonator has been shown here for demonstration due to its intuitive conceptualization, other resonant shapes are possible, which may provide more optimized reflection response.