Moiré metasurfaces for dynamic beamforming

Abstract

Recent advances in digitally programmable metamaterials have accelerated the development of reconfigurable intelligent surfaces (RIS). However, the excessive use of active components (e.g., pin diodes and varactor diodes) leads to high costs, especially for those operating at millimeter-wave frequencies, impeding their large-scale deployments in RIS. Here, we introduce an entirely different approach—moiré metasurfaces—to implement dynamic beamforming through mutual twists of two closely stacked metasurfaces. The superposition of two high-spatial-frequency patterns produces a low-spatial-frequency moiré pattern through the moiré effect, which provides the surface impedance profiles to generate desired radiation patterns. We demonstrate experimentally that the direction of the radiated beams can continuously sweep over the entire reflection space along predesigned trajectories by simply adjusting the twist angle and the overall orientation. Our work opens previously unexplored directions for synthesizing far-field scattering through the direct contact of mutually twisted metallic patterns with different plane symmetry groups.

A low-cost approach is proposed for realizing dynamic beamforming through the mutual twist of two stacked Moiré metasurfaces.

INTRODUCTION

In recent years, there has been renewed interest in the rich physics that emerges in twisted bilayer two-dimensional (2D) materials (1–4). The mutual rotation in bilayer graphene induces a wide range of electronic effects (5–11) (7–13), for example, the “magic angle” at which the band structure becomes completely flat leads to superconductivity (12–16). Depending on the twist angle, the two lattices can be commensurate to form a larger unit cell, or incommensurate, which leads to a quasi-periodic lattice (17–20). Bloch’s theorem fails to describe the electronic properties of such twisted bilayer graphene systems because of the absence of crystallinity at most twist angles. Many attempts have been made for theoretical characterization of the quasi-periodic systems, including the use of simplified tight-binding models (21–24), and ab initio or tight-binding calculations (17, 23, 24).

Programmable metasurface is a special type of 2D material composed of an array of digitally controlled unit cells, which can be dynamically programmed to control the wavefront of an incident wave (25–29). The independent control of phase, amplitude, and polarization for each individual unit cell (30–33) has led to many intriguing findings and engineering applications, such as wireless communication (34–38), holographic imaging (39), microwave imager (40, 41), programmable space-time modulation (42–43), programmable nonreciprocity (44), harmonic manipulation (45), and so on. However, because of the large usage of pin diodes (or varactor diodes) and the associate external control circuit, programmable metasurfaces are facing the issue of excessive manufacturing costs, especially at millimeter-wave band, which inevitably affects their large-scale deployment in wireless communications.

In this work, we propose a new mechanism for reaching dynamic beamforming through two mutually twisted metasurfaces. Instead of controlling the voltage across each individual unit cell, we stack two mutually twisted metasurfaces together to produce a moiré pattern, which interacts with the metallic back plate to reconfigure the surface current distribution for engineering the far-field radiation. The superposed moiré pattern varies as a function of the twist angle, enabling continuously beam scanning in the reflection side, as schematically illustrated in Fig. 1A. Moiré effect has been recently introduced to metasurfaces for the purpose of investigating unusual dispersion properties and band structure (46, 47) and interpreting the beam steering mechanism for the variant of leaky-wave antenna designs (48). However, we should remark that our moiré metasurface is very different from the multiple stacked metasurfaces (49–51) and the variforcal metalens doublet designs (52–56). In those design, the phase profiles, or functions, of each individual layer must be designed separately to allow the overall function as the simple combination of the function of individual layers. In addition, the interlayer spacing in those designs must be kept large enough to minimize interlayer coupling, and all layers must be perfectly aligned to guarantee the combined phase profiles. Our moiré metasurface do not have such restrictions, making it more compact in size and more robust in practical applications.

Fig. 1. Moiré metasurface in real space.

(A) Schematic illustration of a moiré metasurface. The mutual twist of two closely attached metasurfaces produces a varying moiré pattern, which reflects the incident wave to a wide range of directions depending on the twist angle. (B) Composition of a moiré metasurface from bottom to top: a metallic back plate, a spacer, and two closely stacked metasurface layers with mutual twist. (C and D) Examples of a chessboard-type and a triangular-type moiré pattern, respectively, with pattern 2 being counterclockwise rotated by 10° relative to pattern 1.

RESULTS

Theoretical prediction of moiré impulse and beam angle

Figure 1B shows the zoom-in view of a typical moiré metasurface, which consists of, from bottom to top, a metallic back plate, a spacer of thickness d, and two closely stacked metasurfaces with a mutual twist angle ψ. Two moiré metasurfaces with the chessboard and triangular patterns at a twist angle ψ = 10° are shown in Fig. 1(C and D, respectively), where the moiré patterns are clearly visible from the overlapped regions. In contrast to the design of conventional metasurfaces, with moiré metasurfaces, the unit cell with locally periodic boundary condition is no longer applicable, and the super unit cell approximation is also not accessible (25–30).

As the superposition of two binary images involves multiplication operation, it is instructive to investigate the formation of the moiré pattern in the reciprocal space (Materials and Methods). Figure 2A illustrates the frequency spectra and their convolution for the two chessboard patterns (p4m plane symmetry group) in the k-space. The four second-order diffraction components marked by red square dots (pattern 1) and green triangular dots (pattern 2) lying on the square grid of the reciprocal lattice (Fig. 2A) are dominant in the frequency spectrum. The convolution between diffraction components of the two patterns, which is manifested geometrically as their vectorial summation, produces two groups of moiré impulses (yellow and blue stars), indexed by the diffraction orders of the two constituent frequency spectra. Each group has equal distance to the origin and respects C₄ rotational symmetry to the k_z axis. At ψ = 10°, the four yellow moiré impulses fall inside the light cone (red circle) and contribute to the beam radiation. Although there are an infinite number of moiré impulses for the binary moiré pattern, we remark that the higher-order moiré impulses take negligibly small values. For this reason, we need only to consider the dominant moiré impulses in the design of moiré metasurface. Further increasing the twist angle ψ, the yellow (blue) moiré impulses gradually leave (approach) the origin and nearly exchange their locations at ψ = 80° (fig. S4). Crucially, the locations of the primary moiré impulses determine the shape and size of the moiré periodicity in the real space, while their intensities determine how clear these moiré periodicities can be visualized. Once we determine the periodicity of both patterns and the twist angle, the location of the moiré impulses will be fixed in the k-space, while the pattern details only affect their amplitude. For the triangular patterns, two groups of moiré impulses (yellow and blue stars) are created by the convolution process, as illustrated in fig. S1A.

Fig. 2. Moiré metasurface in reciprocal space.

(A) Spectrum convolution process for the chessboard-type (p4m plane symmetry group) moiré patterns at ψ = 10°. The dominant frequency components for pattern 1 (red square dots) and pattern 2 (green triangular dots) are labeled by two integers, which represent the diffraction orders along the primary reciprocal lattice vectors in the square and triangular lattices. For illustrative purpose, the amplitudes of the primary reciprocal lattice vectors b₁ and b₂ are set as 3 and 2.5. The spectrum convolution process produces two groups of moiré impulses (yellow and blue stars), labeled with the frequency component indies of patterns 1 and 2. The moiré impulses inside the light cone (red circle) may contribute to real radiations. (B and C) Global view and zoomed view of the moiré impulse trajectories plotted in the polar coordinate for the chessboard-type moiré metasurface (p₁ = 12 mm, p₂ = 11.4 mm) as pattern 2 rotates from 0° to 90° (counterclockwise). As ψ increases from 0° to 90°, the first (second) group of the moiré impulses leaves (approaches) the origin. (D) Beam trajectories of the chessboard-type moiré metasurface obtained from the coordinate transformation of the two groups of moiré impulses in (B) and (C). The red (blue) trajectories correspond to the beam trajectories of the first (second) group of the moiré impulses, which vary by the elevation angle from 11° to 90° (90° to 11°) as ψ increases from 0° to 15.55° (74.45° to 90°). The arrow indicates the moving direction of the moiré impulses with the increasing of twist angle.

By gradually rotating pattern 2 in the counterclockwise direction, the moiré impulses and the corresponding radiation beams move in the reflection space, as shown in Fig. 2(B to D). The periodicities of the chessboard patterns 1 and 2 are set as: p₁ = 12 mm and p₂ = 11.4 mm. Figure 2 [B (global view) and C (zoomed view)] shows that the first group (red trajectories) of the moiré impulses move away from the origin as ψ increases from 0° to 90°, while the second group (blue trajectories) of the moiré impulses move toward the origin. The trajectories of the two groups exhibit reflection symmetry with respect to ±45° symmetrical lines, agreeing with the pattern symmetry.

The beam trajectories can be obtained directly from the moiré impulse trajectories by performing coordinate transformation from the k-space to the angular space, mathematically expressed as (k_x/π, k_y/π) = ( sin θ cos φ, sin θ sin φ), in which k_x and k_y are the normalized wave vectors along the x and y axes, and θ and φ are the elevation and azimuthal angles, respectively. For example, by assuming a normally incident plane wave of wavelength λ₀, the direction of the beam (+1,+1,−1,−1) can be obtained as

$$\mathrm{\theta }={\text{sin}}^{-1}\left[{\mathrm{\lambda }}_0\frac{\sqrt{2p_1^2+2p_2^2-4p_1{p}_2\text{cos}\mathrm{\psi }}}{p_1{p}_2}\right]$$ (1)

$$\mathrm{\phi }={\text{tan}}^{-1}\left[\frac{p_1\text{cos}\mathrm{\psi }+p_1\text{sin}\mathrm{\psi }-p_2}{p_1\text{cos}\mathrm{\psi }-p_1\text{sin}\mathrm{\psi }-p_2}\right]$$ (2)

Figure 2D shows the trajectories of beam directions for the chessboard pattern in the polar coordinate as ψ increases from 0° to 90°. The elevation angle of the beam sweeps continuously from 11° to the grazing angle as ψ increases from 0° to 15.55° (red traces) and decreases from the grazing angle to 11° as ψ increases from 74.45° to 90° (blue traces). As apparent from Eq. 1, the lower limit of the elevation angle obtained at ψ = 0° can be reduced to zero when p₁ = p₂. We can also derive the maximum rotation angle for reaching grazing reflection ${\mathrm{\psi }}_{\text{max}}={\text{cos}}^{-1}[1-p^2/(4{\mathrm{\lambda }}_0^2)]$, which implies that we can achieve fast beam scanning by reducing the pattern periodicity. Note that the beams can cover the entire upper half space with the in-plane rotation of the whole moiré metasurface. Figure S1 (B to D) shows the moiré impulse and radiation beam trajectories for the triangular case, in which six beams scan across the reflection space in a trend similar to the chessboard case.

Because of the pattern symmetry of the two constituent layers, the moiré pattern repeats itself during the mutual rotation. Mathematically, the chessboard (triangular) moiré metasurface with both layers respecting the p4m (p3m1) plane symmetry group is translational invariant by a minimum rotation period ψ_p of 45°(60°) (section S2). The symmetry of pattern determines the symmetry of the beam trajectories, as verified by the case of the moiré metasurface with regular-square-patch pattern (fig. S5), in which the trajectories of moiré impulses and beams are similar to those of the chessboard case. If each pattern is designed with a different symmetry from the plane symmetry group (57), one expects a wider variety of moiré patterns, with each featuring a distinct way of pattern evolution. This offers us great flexibility in engineering the radiation pattern as well as beam tracing during mutual rotation. For example, the combination of two patterns with p4m (chessboard) and p3m1 (triangular) symmetries creates a pair of beams with C₂ rotation symmetry (fig. S6).

Note that the moiré periodicity that is perceived by human eyes does not generally correspond to true 2D crystallinity for most rotation angles except at certain discrete rotation angles. To investigate this further, we derive from section S3 the commensuration condition for two disorientated lattices with different periodicities and reveal that the lower bound of the rigorous periodicity corresponds exactly to the dominant moiré periodicity.

Numerical validation of radiation control with mutual rotation

As previously mentioned, the k-space representation of the moiré pattern predicts only the directions of the beams that can be possibly radiated by the physical structure, but not their intensity. To evaluate the beamforming performance, we perform full-wave numerical simulations for the chessboard and triangular models in CST Microwave Studio using time-domain solver (Materials and Methods). Note that although modal expansion has been recently reported for calculating the transmission of dielectric moiré metasurfaces (58, 59), it does not converge with our metallic-type moiré metasurfaces. As moiré metasurfaces are locally anisotropic at most twist angles, we use a left-handed circularly polarized (LHCP) plane wave as the excitation. Figure 3A shows the simulated 3D radiation patterns (LHCP component) for the chessboard case at ψ = 0.25°, 2°, 4°, 6°, 8°, 10°, 12°, and 14° (top view in fig. S11), in which uniform intensity can be identified from the four beams at all twist angles. The beam efficiency (normalized amplitude of beam maxima; see Materials and Methods) is indicated by the red line (average value) and color-shaded regions (fluctuation) in Fig. 3A, which decreases with increasing elevation angle but remains above 0.3. The efficiency is greater than that of the conventional digital coding metasurface with regularly shaped patterns (fig. S17) (60).

Fig. 3. Controlling the far-field radiations by the twist angle.

(A) Simulated 3D radiation patterns (LHCP component) for a round-shaped chessboard-type moiré metasurface with 250-mm diameter illuminated by an LHCP plane wave for the twist angles ψ = 0.25°, 2°, 4°, 6°, 8°, 10°, 12°, and 14°. Nearly uniform beam intensity is identified from all radiation patterns, as also observed from the averaged value (solid line) and standard deviation (colored region) of the normalized beam maxima (Materials and Method) of the four beams as ψ increases from 0.25° to 14° with the step of 0.25°. (B) Cutting-plane view of the 3D radiation patterns in (A) extracted from the plane of the maximum radiation of beams #1 and #3, showing the beam scanning capability in the elevation plane. (C) Trajectories of the beam maxima as ψ increases from 0.25° to 14° with the step of 0.25°, which follow closely with the theoretical prediction (red line). Color intensity in (C) represents the normalized amplitudes. The beams are indexed in (C). Note that the 3D radiation patterns in (A) have been self-normalized.

The elevation angle of the beam gradually increases from 10° to 63° as the rotation angle increases from 0.25° to 14°, as observed from the 2D radiation patterns in the linear (Fig. 3B) and dB scaling (fig. S12) on the plane of the maximum radiation. Figure 3C shows the trajectories (color representing the normalized beam intensity) of the beam maxima as the twist angle increases from 0.25° to 14° with a step of 0.25°, further validating the good agreement of the beams in the azimuthal direction between the numerical simulation (colored dots) and theoretical calculations (solid lines). The ellipticity (E_R − E_L)/(E_R + E_L) along the beam maxima direction ranges from −0.42 to −0.18 (fig. S13), indicating that the reflected wave is mostly left polarized. The simulated 3D far-field patterns for the triangular-type moiré metasurface show similar trend to the chessboard case (fig. S2), with beam intensity ~25% lower than the chessboard case due to the increase of the beam number.

Measurement of dynamic beamforming

To experimentally illustrate dynamic beamforming performance of the moiré metasurface, we fabricated two round-shaped samples (250-mm diameter) with the chessboard (Fig. 4A) and triangular (fig. S3A) patterns. Figures S9 and S10 show the photograph of the chessboard-type and triangular-type samples at different twist angles. The sample was manufactured by tightly stacking the following parts into a nylon holder: a metal plate, a foam spacer, two layers of moiré patterns, and a foam cover (Materials and Methods). We measured the 3D far-field pattern in a microwave chamber at ψ = 6°, 8°, 10°, and 12°, as shown in Fig. 4(B to J) (chessboard), which are obtained on four equally spaced (4° interval in the azimuthal direction) elevation planes around the maxima of each beam. For the chessboard case, the 3D radiation pattern presented in the perspective view (Fig. 4, B to E) and top view (Fig. 4, G to J) are in good agreement with numerical simulations (Fig. 3A and fig. S11). The elevation angle of the measured beam (Fig. 4F) also matches precisely with the numerical simulations (Fig. 3B) for ψ ranging from 0.25° to 14°. The excellent agreement is also shown in the comparison between the simulated (fig. S12, A to H) and measured (fig. S12, I to P) 2D far-field patterns in dB scaling. Notably, the experimentally measured beam efficiency is close to that of simulation at smaller elevation angle, with slight decrease at larger elevation angles. For completeness, the experimental results of the triangular case are provided in figs. S3 and S14 to S16.

Fig. 4. Experimental results of far-field radiations.

Photo of the fabricated sample (A) and measured far-field radiation patterns (B to J) for the chessboard-type moiré metasurface. Black and yellow colors in the sample represent metal and polyimide substrate, respectively. The far-field patterns at ψ = 6°, 8°, 10°, and 12° are measured at four equally spaced elevation planes (4° interval in the azimuthal direction) around the direction of the four beam maxima and are shown in the perspective view (B to E) and top view (G to J), which faithfully reproduce the major signatures of the simulated radiations (see Fig. 3A and fig. S11). The measured radiation patterns also show precise beam angles in the elevation plane (F) as compared to the simulation results in Fig. 3B. The measured beam efficiency is close to that of the simulation at smaller elevation angles but is slightly reduced at larger elevation angle.

DISCUSSION

In this work, we have provided a simple yet effective approach to realize dynamic beamforming using two closely stacked metasurfaces. The mutual twist between the high-spatial-frequency periodic patterns produces a low-spatial-frequency moiré pattern, which provides the distribution of surface impedance for reflecting the illuminating wave to the desired directions. By mutually twisting the stacked patterns, we can achieve continuous adjustment on the beam directions. The beam efficiency shows small variation for different radiation angles and is comparable to the conventional programmable metasurfaces. The moiré metasurface requires zero power to maintain a certain configuration, thus offering much lower energy consumption and hardware cost.

The moiré metasurfaces are expected to show great potentials for large-scale deployment in wireless networks for cost-efficient improvement in wireless channel quality, which may facilitate the development of reconfigurable intelligent surfaces (RIS) for enhancing radio propagation for future 6G wireless networks (61, 62). Although the tuning speed of moiré metasurfaces may not compete with actively programmable metasurfaces because of the speed limitation of mechanical rotation, there are many scenarios in wireless communications that require only second-level and minute-level channel refinement based on the statistical channel state information (63). One important advantage of the moiré metasurfaces is that they do not suffer from receiver noise as they are completely passive. In addition to RIS, we also anticipate that the moiré metasurfaces can find applications in radar detections; for example, a flat dish antenna featuring in-plane rotation provides lower profile and better convenience for installation. The moiré metasurface can operate with much higher power input than the digitally programmable metamaterials because of the disposal of pin diodes (or varactor diode).

MATERIALS AND METHODS

k-space representation of the moiré pattern

The Fourier expansion of a periodic pattern with in-plane translation vectors a₁ and a₂ is simply the discrete value sampled at the 2D reciprocal lattice F = m₁b₁ + m₂b₂ with translation vector b₁ = 2πRa₂/(a₁ ⋅ Ra₂), b₂ = 2πRa₁/(a₂ ⋅ Ra₁), where R is 90° rotation matrix, and m₁ and m₂ are integers. The superposition of two binary images is equivalent to the logical operation OR (|), which can be decomposed into a series of logic operations: A₁∣A₂ = A₁ + A₂ − A₁A₂ (Materials and Methods). Consequently, the spatial frequency spectrum of a moiré pattern contains the convolution of the frequency spectra of the constituent layers, which is simply equivalent to their vectorial sum: ${\mathit{F}}_{\mathit{\text{moire}}}={{\mathit{F}}_1}^{*}{\mathit{F}}_2={\mathit{m}}_1^{(1)}{\mathit{b}}_1^{(1)}+{\mathit{m}}_2^{(1)}{\mathit{b}}_2^{(1)}+{\mathit{m}}_1^{(2)}{\mathit{b}}_1^{(2)}+{\mathit{m}}_2^{(2)}{\mathit{b}}_2^{(2)}$, in which the superscript represents the layer index. Each moiré impulse is labeled by the lattice vector indices of the two constituent layers as $({\mathit{m}}_1^{(1)},{\mathit{m}}_2^{(1)},{\mathit{m}}_1^{(2)},{\mathit{m}}_2^{(2)})$. Note that some combinations may correspond to the same wave vector when the two layers of lattices commensurate, but it will not cause ambiguity for engineering purposes.

Numerical simulation

The far-field scattering properties of our moiré metasurfaces are computed through full-wave numerical simulations with the time-domain solver in CST Microwave Studio. The model was cropped into round shape with 250-mm diameter, with the geometrical and material parameters following exactly from fabrication. Because of the absence of optical chirality in such lossless reflection-type metasurface, the radiation performance under RHCP and LHCP excitations are almost identical, with only flip of polarizations. All results have been normalized to the back-scattering maxima of a bare perfectly electrical conducting plate with the same diameter.

Sample preparation

The moiré metasurface was manufactured by stacking two layers of moiré patterns on top of a rigid metal plate, separated by a foam spacer (ε_r = 1.07) with thickness d. A judicious choice of the spacer thickness (d = 3.1 mm for chessboard case, d = 2.7 mm for the triangular case, ε_r = 1.07) allows us to ensure the optimized radiation performance at all twist angles. The moiré pattern was fabricated on a flexible polyimide substrate (ε_r = 3.1, δ = 0.03) with 40-μm thickness using standard printed circuit board technology. The periodicity p and side length L for the chessboard and triangular patterns are specified as follows: chessboard: p₁ = 12 mm, p₂ = 11.4 mm, L₁ = 5.25 mm, L₂ = 4.95 mm; triangular: p₁ = 11.4 mm, p₂ = 12 mm, L₁ = 10.1 mm, L₂ = 10.7 mm. All layers are tightly assembled in a nylon holder with the help of a 5-mm-thick foam cover on the top side, which ensures precise and uniform separation between the moiré pattern and back plate. Unlike the multiple stacked metasurfaces (49–51) and the varifocal metalens (52–56), in which all the layers must be perfectly aligned to guarantee the designed phase profiles, the radiation performance of our moiré metasurface will not be affected by in-plane misalignment errors (section S4), making it more robust in practical applications.

Experimental measurement

The radiation pattern of the sample was measured in a microwave chamber using a signal analyzer (Keysight N9040B) and an analog signal generator (Keysight E8257D), which were connected, respectively, to a pair of dual circularly polarized horn antennas with a nominal gain of 20 dB (fig. S19). The sample and the transmitting antenna were coaxially placed on a wooden board at a distance R of 1.79 m, which together rotate in the horizontal plane at the angular precision of 0.1°. To measure the far fields on different elevation planes, the sample was attached to a turntable, which was wirelessly controlled at an angular precision of 0.2°. Note that as the sample was not placed at the center of rotation, the measured far-field patterns are stretched in the elevation angle. All far-field results have been corrected using the formula θ^′ = cos⁻¹[(R + L cos θ₀)/(R² + L² + 2RL cos θ₀)], in which R represents the distance between the sample and transmitting antenna, L = 8.84 m the distance between the center of rotation and receiving antenna, and θ₀ the rotation angle of the rotator in the horizontal plane.

Supplementary Materials

This PDF file includes:

Sections S1 to S5

Figs. S1 to S19