Self-locking non-volatile coding metasurfaces via origami-based mechanical bits

Abstract

Digital coding metasurfaces have revolutionized electromagnetic (EM) manipulation, yet typical tunable approaches based on active components suffer from “volatility”. While mechanical modulation provides a potential solution, current implementations generally lack inherent state-locking capabilities, rendering them vulnerable to environmental disturbances and actuation errors. Inspired by the concept of mechanical bits (MBs), this paper presents a self-locking non-volatile coding metasurface platform enabled by Kresling origami-based MBs. Here, the mechanical bi-stability of individual meta-atoms enables robust binary geometric reconfiguration, with distinct states protected by intrinsic energy barriers. Building upon this concept, equivalent current models accounting for the compression-torsion coupling motion are established and analyzed to design efficient and broadband 1-bit EM units. By integrating tailored metallic patterns onto a multi-material 3D-printed truss-variant array, both transmission- and reflection-type prototypes are proposed and experimentally demonstrated, exhibiting reliable wavefront manipulation capabilities through near-field holographic imaging and far-field beam steering. These results bridge mechanical logic and EM information processing, establishing a universal physical paradigm for constructing low-power, highly robust coding metasurfaces resilient to complex environments.

Maintaining states without continuous power remains challenging for tunable metasurfaces. The authors present self-locking non-volatile coding metasurfaces based on origami mechanical bits, enabling holographic imaging and beam steering.

Introduction

The paradigm shift from static structures to digital coding metasurfaces has revolutionized EM wave manipulation, driving breakthroughs in next-generation wireless communications^1,2. By switching on/off states or modulating the capacitance of active components (e.g., PIN diodes, varactors) integrated into independent meta-atoms, electrically tunable coding metasurfaces exhibit fast response speeds and sophisticated wavefront tailoring, facilitating various innovative functionalities, including adaptive beamforming³, wireless information mapping⁴, reconfigurable meta-holograms⁵, and programmable diffractive neural networks^6,7. Recently, the temporal domain has been explored as an additional degree of freedom (DOF) for coding metasurfaces⁸, which has further broadened the scope of applications, such as Doppler cloaks⁹, nonreciprocal antennas¹⁰, and sideband-free radiation^11,12. Despite the success of electrically tunable coding metasurfaces in digitizing EM responses, the requirement for complex biasing networks and external continuous power sources to maintain the coding patterns introduces a “volatility”^13–16 and restricts their deployment in energy-constrained scenarios^17,18.

Mechanical modulation has recently emerged as a powerful framework for the dynamic control of EM waves^19–21, which leverage physical structural changes to tune EM properties, offering a promising avenue for applications requiring non-volatile memory and environmental robustness. Approaches based on stretchable substrates utilize the elastic deformation of elastomers (e.g., PDMS) to continuously modulate the lattice periodicity or inter-particle spacing of meta-atoms^22–24. More sophisticated geometric transformations, such as origami- or kirigami-inspired designs, exploit geometric folding and twisting to reconfigure structural symmetries, enabling the dynamic manipulation of chirality and EM wavefronts^25–29. Furthermore, recent advances have introduced motorized meta-atoms, where the independent rotation of local elements facilitates effective phase shifting, demonstrating remarkable tunability^30–33. However, most mechanically reconfigurable metasurfaces rely on continuous deformation and lack a state-locking capability, rendering EM responses highly susceptible to environmental disturbances and actuation errors, with an increasing risk that slight structural damage could lead to a severe decline in signal fidelity³⁴. Therefore, the shift from continuous physical deformation to discrete state switching with intrinsic stability is crucial for reliable EM performance under long-term operation^35,36.

Multistable mechanical structures, characterized by their discrete stable states and inherent energy barriers, provide a potential method for addressing the volatility challenge in reconfigurable systems^37,38. These energy barriers act as passive mechanical thresholds that can efficiently filter out minor fluctuations and stabilize predefined geometric configurations without continuous energy input. This intrinsic state-retention capability facilitate the construction of mechanical bits (MBs)^39,40. For example, triangulated cylindrical origami (Kresling origami) cells have been demonstrated as non-volatile bit elements for information storage by exploiting their bistable energy landscape⁴¹. Recent developments have shown the potential of such architectures across diverse applications, including mechanical logic gates⁴², data encryption⁴³, and mechanical neural networks^44,45. In the realm of EM metasurfaces, a double-broadband absorbing grid enabled by bistable inflatable origami actuators has been proposed, allowing complementary absorption bands to be switched through structural reconfiguration⁴⁶. A separate study developed a thin multifunctional metamaterial based on bistable curved beams which supports programmable broadband absorption by binary-state encoding of unit cells⁴⁷. More recently, a kirigami-origami hybrid metamaterial has been proposed to realize multiband and ultra-wideband switchable microwave absorption under large 3D deformations⁴⁸. However, current multistable EM designs focused primarily on demonstrating macroscopic impedance matching (e.g., switchable absorbers), lacking independent, pixel-level phase manipulation required for information coding, thereby constraining their versatility for sophisticated spatial wavefront engineering.

In this work, we propose a self-locking non-volatile coding metasurface platform, consisting of an array of Kresling origami units, where the bistable states of individual units are strictly mapped to 1-bit EM coding phases. This platform introduces a general design framework that exploits the dual DOFs (rotation and height) to realize EM coding in either transmission or reflection modes, thereby extending the functionality of mechanically reconfigurable structures beyond conventional single-DOF implementations. Based on the proposed general theoretical models, the two types of meta-atoms are designed, featuring intrinsic state retention with zero static power consumption while simultaneously achieving highly efficient 1-bit phase reconfiguration over wide frequency ranges (3.5–5.0 GHz and 12.2–13.0 GHz, respectively), overcoming the inherent bandwidth limitations of conventional active metasurfaces. Dynamic near-field holographic imaging for the MB-enabled transmission coding metasurface (MB-TCM) and far-field beam steering for the reflection counterpart (MB-RCM) further validate the wavefront manipulation capability of this system. In addition, the lightweight origami unit (1.5 g) demonstrates strong environmental tolerance, with its bistable states remaining well preserved under varying conditions. The proposed architecture and method bridge the gap between mechanical logic and EM information, opening pathways for low-power, high-robustness, and non-volatile intelligent communication systems, laying the foundation for future realistic applications tailored for complex environments.

Results

Conceptual and design framework of self-locking non-volatile coding metasurfaces via Kresling origami-based mechanical bits

The conceptual framework of the self-locking non-volatile coding metasurface platform via Kresling origami-based mechanical bits (MBs) is illustrated in Fig. 1a. Each MB integrates two key components to achieve mechanical reconfiguration and EM phase switching: Kresling origami units and metallic patterns. The bistability of the Kresling structure establishes an MB mechanism with state-locking capabilities, which can digitize continuous external actuation forces into two stable geometric states (state “0” and state “1”) protected by a physical energy barrier, thereby endowing the system with inherent robustness against environmental perturbations. Based on this mechanical platform, a general design framework leveraging distinct geometric DOFs to achieve EM coding is proposed by establishing two classes of equivalent current models, thereby respectively satisfying the demands of transmissive and reflective operating modes under linearly polarized wave incidence.

Fig. 1. Conceptual and design framework of the proposed mechanically coding metasurfaces.

a The bistable states of individual Kresling meta-atoms are strictly mapped to 1-bit EM coding phases. The mechanical-bit-enabled transmission coding metasurface (MB-TCM) and its reflection counterpart (MB-RCM) are realized by integrating tailored metallic patterns onto multi-material 3D printed Kresling units, demonstrating dynamic holographic imaging and beam steering, respectively. b Equivalent current model schematic for the mechanically driven transmissive 1-bit EM coding meta-atom, with the calculated ideal cross-polarized transmission coefficient as a function of rotation angle (c) Equivalent current model schematic for the mechanically driven reflective 1-bit EM coding meta-atom, with the calculated ideal co-polarized reflection coefficient as a function of height variation.

Figure 1b illustrates the schematic of the equivalent current model for the mechanically driven transmissive 1-bit EM coding meta-atom, alongside the calculated ideal cross-polarized transmission coefficient as a function of the rotation angle (the detailed derivation and comprehensive analysis are provided in Text S1, Supplementary Information). For transmissive coding, the primary bottleneck lies in achieving high radiation efficiency, as a simple single-layer structure is fundamentally limited to a maximum transmission efficiency of 25%⁴⁹. Therefore, to satisfy the requirement for high radiation efficiency, a three-layer sandwich structure is adopted herein, where the top and bottom layers consist of mutually orthogonal metallic gratings functioning as polarization filters, and the intermediate layer is represented by an equivalent surface conductivity tensor ($\overline{\overline{\sigma }}$). By taking into account both the rotational and height variations of the intermediate layer, the analytical solution for the cross-polarized transmission coefficient of the meta-atom has been derived, revealing that the rotational variation serves as a more dominant modulation variable compared to the height variation (see Text S1). Specifically, when the rotation angle is near 45° or 135°, the transmission magnitude can theoretically reach its maximum value close to 1, while the phase undergoes an abrupt π jump at a rotation angle of 90°, thereby achieving a complete phase flip. Consequently, under this model, a bistable carrier capable of providing a relative twist of approximately 90° establishes the ideal conditions for highly efficient 1-bit transmissive modulation.

Figure 1c illustrates the schematic of the equivalent current model for the mechanically driven reflective 1-bit EM coding meta-atom, alongside the calculated ideal co-polarized reflection coefficient as a function of the height variation (the detailed derivation and comprehensive analysis are provided in Text S2). For the co-polarized reflection coding of an arbitrary linearly polarized wave, high radiation efficiency can be readily achieved by adding a metallic mirror. Thus, the model herein consists of a top surface conductivity ($\overline{\overline{\sigma }}$) and a bottom perfect electric conductor (PEC). By considering both the rotational and height variations of the top layer, the analytical solution for the co-polarized reflection coefficient of the meta-atom has been derived, revealing that the co-polarized reflection coefficient is entirely dictated by the height variation when the top surface conductivity is isotropic (see Text S2). Specifically, the reflection magnitude can theoretically reach unity, achieving total reflection. Furthermore, the phase varies approximately linearly with the height, meaning that the phase reconfiguration depends solely on the physical separation between the surface conductivity and the mirror. Mechanically compressing or expanding this distance directly tunes the equivalent electrical length of the reflecting cavity. Consequently, under this model, a bistable carrier capable of producing an axial height variation on the order of the operating wavelength establishes the ideal conditions for highly efficient 1-bit reflective modulation.

Under linearly polarized wave incidence, to design coding metasurfaces on a unified mechanical platform capable of satisfying diverse functional requirements (both transmissive and reflective operating modes), the Kresling-based MB, which inherently provides coupled compression-torsion DOFs, naturally emerges as a highly compatible hardware carrier. Furthermore, compared with other typical bistable architectures, Kresling origami exhibits distinct engineering advantages. For instance, in contrast to a spring-latch structure or a cam-based push-button mechanism, Kresling origami can be monolithically fabricated via multi-material 3D printing, offering a lightweight design and eliminating the need for complex assembly. Crucially, since the polymer rods are insulating and electrically small relative to the operating wavelengths, their scattering contributions are negligible. This renders the Kresling-based mechanical bit highly transparent to EM waves, thereby leaving ample design space for the functional metallic patterns. Additionally, compared to a conventional bistable beam, Kresling origami inherently features flat top and bottom surfaces, which facilitate integration with the designed metallic patterns.

Geometry and mechanical properties of the Kresling structure

The bistability of the proposed meta-atom stems from the Kresling origami topology^50,51. In the deployed state, the structure adopts a cuboid configuration (Fig. 2a) with square top and bottom surfaces of side length a = 17.5 mm. As a typical non-rigid origami, the Kresling pattern exhibits intrinsic panel deformation during folding, leading to its nonlinear mechanical response. Despite this complexity, the truss model, which simplifies panel deformation to the crease lines, serves as a qualitative yet sufficient analytical tool to capture the bistable energy landscape⁵². Guided by the model (see Text S3, Supplementary Information), the initial height (h₀) of the Kresling unit was calculated to be 20 mm to satisfy the “critical design”⁵³. Under this configuration, the structure transitions to the folded state with a 90° relative rotation between the top and bottom surfaces and an approximately 50% reduction in axial height, creating the appropriate conditions for 1-bit transmissive and reflective coding, respectively. Following the principle of minimum potential energy (see Text S3), the normalized energy profile during free-rotation compression is shown in Fig. 2b, exhibiting two local minima that confirm the bistability of the structure. The inset further illustrates the energy map as a function of normalized displacement and rotation angle, where the minimum-energy valley (dark region) and the corresponding path (dashed curve) indicate a kinematically coupled single-DOF transition path. Notably, in the truss model, the bistable behavior of Kresling origami primarily originates from the geometric topology, while the material stiffness mainly affects the energy scale rather than the existence of bistability (see Text S4). This characteristic enables the fabrication of truss-variant Kresling structures with robust bistability using multi-material 3D printing.

Fig. 2. Geometry and mechanical properties of the Kresling structure.

a Geometric illustration of the Kresling unit that satisfies critical design conditions. b The calculated normalized energy profile at a = 17.5 mm and h₀ = 20 mm. The inset plot shows the normalized energy map as a function of normalized displacement Δh/h₀ and rotational angle α, in which the white dashed curve traces the minimum potential energy path. c Photographs of the fabricated Kresling unit in the state “0” and state “1”. d Experimentally measured force-displacement response curve of the Kresling unit. The solid line represents the mean values, and the shaded region denotes the standard deviation. e The multi-material 3D printed model, where white and black regions correspond to stiff (Vero) and flexible (Agilus) materials, respectively. f Simulated von Mises equivalent stress distribution at different compression stages. g–i The stability and reliability of the Kresling unit under complex environments, supported by force-displacement curves measured (g) at various temperatures ( − 40–60 °C), (h) under different relative humidity levels (30–90% RH), and (i) after varying holding durations (0–100 h) in the folded state.

Figure 2c presents the fabricated Kresling unit, showing a height difference of 6 mm between the two stable states and a corresponding relative rotation of 77°. To ensure reliable EM performance, all subsequent EM simulations are conducted based on these measured geometric variations. The discrepancy from theoretical prediction is attributed to the finite thickness of the printed members, which deviates from the idealized zero-thickness assumption in the truss model. The bistable behavior is experimentally validated by the force–displacement response in Fig. 2d, which can be divided into three regimes. In the first segment (Δh < 3 mm), the structure undergoes elastic compression, with the force increasing to an average peak value of 1.84 N, corresponding to a maximum stable load of 125 times its own weight (1.5 g), demonstrating excellent load-bearing capability. In the second segment (Δh ≈ 3–6 mm), the Kresling structure undergoes snap-through buckling, transitioning from state “0” to state “1”. During this process, the structural motion exceeds the speed of the compression platen, resulting in a transient loss of contact with the force sensor and a near-zero force reading. In the third segment (Δh > 6 mm), contact is re-established and the force increases again under further compression.

The bistable transition is enabled by multi-material 3D printing (Fig. 2e), utilizing rigid Vero resin for the rods and flexible Agilus resin for the joints. The geometric parameters are: a = 17.5, h₀ = 20, t_v = 1.2, t_a = 3, w = 1.75, t_r = 1.5, and d = 1.5, in millimeter, in which the rod thickness (w) is determined through a trade-off between stiffness and deformability (As detailed in Text S5). To further qualitatively elucidate the deformation mechanism, the finite element simulation based on a 3D solid model (Fig. S5) was performed. As shown in Fig. 2f, the von Mises stress is predominantly concentrated at the flexible joints, and mechanical interference between rods occurs at a displacement of 6 mm. This stress localization allows the structure to approximate an ideal rigid-foldable system, where the rods remain largely undeformed. In contrast, single-material designs often suffer from parasitic bending, which may degrade structural stability and switching reliability.

Furthermore, the robustness of the mechanical performance under varying environmental conditions is systematically evaluated. As shown in Fig. 2g–i, the force-displacement curves measured under different temperatures, humidity levels, and state-retention durations exhibit consistent overall profiles, indicating the preservation of bistable switching behavior. Notably, the peak force shows a pronounced dependence on temperature (Fig. 2g), increasing significantly at lower temperatures due to the enhanced stiffness of the constituent materials, while decreasing at elevated temperatures as a result of material softening. Despite this variation in force magnitude, the overall curve shape and critical buckling displacement remain largely unchanged, confirming that the underlying bistable mechanism is insensitive to temperature. In contrast, the responses under varying humidity conditions (30%–90% RH) show only marginal variations (Fig. 2h), suggesting negligible influence of moisture absorption on the mechanical performance. Similarly, under long-term retention in the fold state, the peak force exhibits a moderate reduction due to stress relaxation but stabilizes after prolonged holding (Fig. 2i), with no degradation in bistability. Additionally, the cyclic testing (Fig. S7) demonstrates stable operation within 1000 compression-torsion cycles, confirming good fatigue resistance. Overall, these results verify that the bistable behavior of the Kresling unit is largely preserved under practical environmental perturbations, supporting its potential for reliable and long-term operation resilient to complex environments.

Electromagnetic reconfigurability of the meta-atoms

The inherent bistability of the Kresling structure serves as a robust physical basis for constructing a reconfigurable EM platform. To verify the versatility of this platform, full wave simulations in commercial software CST Microwave Studio are performed. The MB-TCM and MB-RCM meta-atoms are illustrated in Fig. 3a, b, respectively, both with a lattice period of 22 mm, and the geometric dimensions for these meta-atoms are shown in Fig. S8 (Supplementary Information). Because the resin rods are electrically small relative to the operating wavelength, their influence on EM wave scattering is minimal. Consequently, the rods were omitted from the simulation models to optimize computational efficiency. This simplification is validated in Fig. S9, which demonstrates a negligible discrepancy for the meta-atoms between models with and without strut modeling. Based on the simplified model, the optimization process for the metallic pattern is illustrated in Figs. S10, S11. Additionally, the potential impact of the 3D printed material on the EM response is discussed in Text S7 (Supplementary Information), which confirms that the proposed designs possess excellent tolerance to fluctuations in dielectric properties of the resin materials.

Fig. 3. Reconfigurable characteristics of the MB-TCM and MB-RCM meta-atoms.

a Schematic of the MB-TCM unit, featuring a grating-SRR-grating sandwich architecture designed for polarization-selective transmission. b Schematic of the MB-RCM unit, consisting of a Jerusalem cross and a metallic backplane. c Simulated EM response of the MB-TCM unit, demonstrating a high cross-polarized transmission coefficient |T_xy | , and a stable phase difference between the two states. d Simulated EM response of the MB-RCM unit, showing a high co-polarized reflection coefficient |R_xx | , and a stable phase difference between the two states.

The MB-TCM meta-atom adopts a grating‑SRR‑grating sandwich configuration, in which the top and bottom surfaces consist of two mutually orthogonal metallic gratings, while the middle layer is an SRR attached to the rotating surface of the Kresling structure. The orthogonal gratings act as a polarization filter effectively blocking direct transmission of co-polarized waves while forming a high-quality resonant cavity that reinforces the interaction between incident waves and the middle-layer SRR⁵⁴. This EM reconfiguration here stems directly from harnessing the rotational DOF of Kresling origami, as detailed in Text S8. Notably, in contrast to the Pancharatnam-Berry (PB) phase principle, which typically leverages the linear relationship between phase and rotation⁵⁵, the coherent interference within this multilayer cavity induces a non-linear phase response. Consequently, the actual 77° rotation is sufficient to yield a phase flip of approximately 180° (as shown in Fig. S14), ensuring effective 1-bit coding despite the 13° deviation from the theoretical 90° case. The EM response of the MB-TCM meta-atom is plotted in Fig. 3c. The unit maintains a high cross‑polarized transmission coefficient ( | T_xy | > 0.75) in both stable states across the broad frequency range of 3.5–5 GHz, ensuring high radiation efficiency and immunity to scattering noise. Additionally, the minor magnitude difference ensures balanced scattering contributions from individual meta-atoms, which is critical for achieving complete destructive interference, thereby maximizing the signal-to-noise ratio (SNR) in EM information processing. Furthermore, once the unit transitions from state “0” to state “1”, the SRR undergoes a precise geometric rotation, triggering a near 180° transmission phase flip for the cross-polarized component. The resulting phase difference remains remarkably stable within the range of 160°–200°, effectively fulfilling the requirement for robust 1‑bit phase coding.

For the MB-RCM meta-atom, in contrast to the rotation‑based phase modulation used in the transmission unit, the reconfiguration strategy here shifts to exploiting the axial height DOF of the Kresling structure, effectively decoupling the EM response from its inherent twisting motion. This decoupling is enabled by the Jerusalem Cross resonator with C₄ rotational symmetry, which ensures its EM response remains invariant to in‑plane orientation⁵⁶. Numerical sensitivity analysis (Text S8) confirms this robustness, as the phase difference at 12.5 GHz for the actual 77° state deviates by only 5.6° relative to the theoretical 90° case (Fig. S14). Consequently, the phase switching depends solely on the vertical displacement between the resonator and the metallic ground plane. When the Kresling unit transitions from the deployed to the folded state, the cavity spacing undergoes an abrupt reduction, which directly alters the effective electrical length, shifting the resonant condition and producing the observed near 180° phase shift (as detailed in Text S9). The EM response of the MB-RCM meta-atom is plotted in Fig. 3d, clearly showing a high co-polarized reflection coefficient ( | R_xx | > 0.85) across 12.2–13 GHz in both stable states, accompanied by a stable phase difference of 160°–200°, successfully validating its effectiveness as a reflective 1-bit coding unit. Meanwhile, the minimal magnitude difference ensures the suppression of the parasitic 0th-order beam, thereby maximizing the diffraction efficiency of wavefront manipulation such as beam steering.

Crucially, throughout EM reconfiguration, the Kresling unit functions as an MB with intrinsic memory. During the compression process, for instance, the structure exhibits a well-defined mechanical threshold, which means that any external force below this threshold triggers elastic recovery to state “0” and only forces exceeding it induce snap‑through instability, locking the structure into state “1”. This bistable snap‑through behavior delivers two key advantages: zero static power consumption (energy is solely expended during switching) and inherent disturbance immunity (sub‑threshold vibrations or contacts are mechanically filtered out). These features make the proposed approach significantly superior to conventional electrically tunable metasurfaces based on PIN diodes or varactors in terms of non‑volatility and environmental robustness.

Near‑field holographic

As a revolutionary imaging technique capable of recording and reconstructing EM amplitude and phase information, metasurface holography has demonstrated immense potential in information storage and image display. Analogous to artificial neural networks, metasurface holography can be conceptualized as a diffractive neural network (DNN) comprising an input layer, a single hidden layer, and an output layer⁵⁷. This architecture enables efficient wavefront control and ultrafast information processing through light propagation. The densely packed unit cells within the metasurface act as neurons in the hidden layer. According to Huygens’ principle and the Rayleigh–Sommerfeld diffraction equation, the field intensity at any position ${\mathbf{r}}^{l+1}=\left(x^{l+1},y^{l+1},z^{l+1}\right)$ in the subsequent layer is determined by the collective contribution of all superatoms in the preceding layer, which can be expressed as:

$$E\left({\mathbf{r}}^{l+1}\right)={\iint }_{-\infty }^{\infty }\frac{Z-Z_i}{r^2}\left(\frac{1}{2\pi r}+\frac{1}{j\lambda }\right)\exp \left(\frac{j2\pi r}{\lambda }\right)\cdot E\left({\mathbf{r}}_i^l\right)\cdot T\left({\mathbf{r}}_i^l\right)dxdy$$ 1

where ${\mathbf{r}}_i^l=\left(x_i^l,y_i^l,z_i^l\right)$ denotes the position of the i-th neuron in the l-th layer, λ is the incident wavelength, and $r={\left({\left(x^{l+1}-x_i^l\right)}^2+{\left(y^{l+1}-y_i^l\right)}^2+{\left(z^{l+1}-z_i^l\right)}^2\right)}^{1/2}$ represents the Euclidean distance between two neurons in adjacent layers. The complex transmission coefficient of each unit is denoted as $T\left({\mathbf{r}}_i^l\right)=a_i^l\left(x_i^l,y_i^l,z_i^l\right)\cdot \exp \left(j{\phi }_i^l\left(x_i^l,y_i^l,z_i^l\right)\right)$, where $a_i^l\left(x_i^l,y_i^l,z_i^l\right)$ and ${\phi }_i^l\left(x_i^l,y_i^l,z_i^l\right)$ represent the amplitude and phase, respectively. To simplify the calculation, the amplitude term can be treated as constant. Assuming the DNN consists of M hidden layers, the energy intensity at the output plane is $s_k^{M+1}={∣E_k^{M+1}∣}^2$, and the loss function is defined as the weighted mean square error (MSE) between the target field $g_k^{M+1}$ and the actual predicted field $s_k^{M+1}$:

$$f\left({\varphi }_i^l\right)=\frac{1}{K}{{\sum }_k{\omega }_k\left(s_k^{M+1}-g_k^{M+1}\right)}^2$$ 2

In the DNN framework for our MB-TCM holography, the number of hidden layers is M = 1. K represents the total number of sampling points in the output layer, and ω_k denotes the weight assigned to the k-th sampling point. By assigning appropriate weights to regions requiring field energy concentration, noisy areas can be effectively suppressed, thereby significantly enhancing the overall image quality (as shown in Figure S16, Supplementary Information).

To demonstrate the imaging capabilities of the MB-TCM experimentally, a prototype was fabricated and measured using the setup shown in Fig. 4a. The prototype adopts the polarization-selective sandwich architecture designed for cross-polarization manipulation, with its top and bottom layers consisting of orthogonal metallic gratings, as presented in Fig. 4b, c. The internal region of the sample consists of a 16 × 16 array of unit cells, as depicted in Fig. 4d, e, with a total aperture size of 352 × 352 mm² (5.6λ × 5.6λ at 4.8 GHz). Each unit can independently switch states through mechanical folding and expansion, enabling pixel-level programming of the aperture phase. To validate the reconfigurability of the MB-TCM, the letters “Z”, “J”, and “U” are selected as target images shown in Fig. 4f, corresponding to three distinct coding configurations. Figure 4g displays the 1-bit phase coding matrices optimized for an operating frequency of 4.8 GHz and a focal plane distance of 100 mm, using the modified simulated annealing (SA) algorithm, as detailed in Text S10 (Supplementary Information). The optimization efficiency is further evidenced by the iterative decay of the loss function, as illustrated in Fig. S17. The theoretical holograms calculated based on different phase distributions and Eq. (3) are shown in Fig. 4h, which clearly reconstruct the target characters. Figure 4i presents the simulation results obtained with the commercial software CST Microwave Studio. It is worth noting that the optimal imaging plane observed in the full-wave simulation is located approximately 65 mm from the lower surface of the MB-TCM, which is smaller than the theoretically preset distance of 100 mm, as shown in Fig. S18. This discrepancy arises because the theoretical model approximates the metasurface as an ideal zero-thickness phase screen, neglecting the physical height of the Kresling units, as detailed in Text S11. The experimental results, as shown in Fig. 4j, clearly reconstruct the contours of the three target letters. Furthermore, the broadband robustness of the imaging system is confirmed by the normalized holographic images at different frequencies, as shown in Fig. S20. The high consistency among the theoretical, simulated, and experimental results strongly demonstrates the effectiveness of the proposed MB-TCM in near-field wavefront engineering.

Fig. 4. Near-field holographic imaging demonstration of the MB-TCM.

a Photograph of the near‑field scanning experimental setup. b, c Photographs of the fabricated MB-TCM prototype, displaying the top (b) and bottom (c) layers with orthogonal metallic grating structures. d Internal view of the array with the top grating layer removed to reveal the Kresling units. e Close-up view of individual unit cells in different coding states. f The three target holographic images (letters “Z”, “J”, “U”). g Corresponding optimized 1-bit phase coding matrices, where black and white pixels represent 0 and π phase responses, respectively. h–j Comparison of the holographic imaging results at 4.8 GHz. Rows represent distinct letter patterns, while columns display (h) theoretical calculations based on the Rayleigh-Sommerfeld diffraction integral, (i) full-wave numerical simulations, and (j) experimental near-field scanning measurements.

Far‑field beam steering

Based on the proposed mechanical reconfiguration methodology, the MB-RCM is further employed for beam steering. According to antenna array theory⁵⁸, the far-field scattering pattern of the metasurface can be derived by coherently superimposing the contributions from all individual unit cells. Under normal plane wave incidence, the far-field scattering function F(θ, φ) can be expressed as:

$$F\left(\theta ,\phi \right)=f_e\left(\theta ,\phi \right){\sum }_{m=1}^N{\sum }_{n=1}^N\exp \left\{j\left[\phi \left(m,n\right)+kD\sin \theta \left(m\cos \varphi +n\sin \varphi \right)\right]\right)$$ 3

where θ and φ denote the elevation and azimuth angles in the spherical coordinate system, respectively. f_e(θ, φ) is the element factor, (m, n) denotes the indices of the unit cell in the 2D array, corresponding to the x and y axis directions, k is the free-space wavenumber, D is the period of the metasurface unit, and φ(m, n) denotes the coding phase distribution. For 1-bit coding, the value of φ(m, n) is discretized to either 0 or π.

By digitally programming the spatial phase distribution φ(m, n), the wavefront propagation direction of the scattered beam can be flexibly controlled. To validate this capability, four typical coding matrices are designed based on an 8 × 8 MB-RCM array as shown in Fig. 5a: two uniform coding patterns (“0000…” and “1111…”) and two periodic coding sequences (“0101…” and “0110…”). The corresponding simulated far-field radiation patterns are presented in Fig. 5b. For the uniform coding configurations, all units exhibit in-phase reflection responses, rendering the MB-RCM electromagnetically equivalent to a perfect metallic plate. The incident EM wave undergoes specular reflection, with its energy highly concentrated in the normal direction. For the “01010101” coding sequence, a π phase shift between adjacent rows is introduced, leading to beam splitting and generating a symmetric dual-beam scattering pattern. For the more complex “01100110” sequence, a quad-beam scattering pattern is further achieved. The fabricated MB-RCM prototype configured in the four distinct coding states is presented in Fig. 5c. Benefiting from the independent mechanical bistability of the Kresling units, the array can be precisely configured and mechanically locked into any predefined geometric pattern without continuous power input.

Fig. 5. Far‑field beam steering demonstration of the MB‑RCM.

a The four reconfigurable phase coding masks designed for the 8 × 8 array, corresponding to the “all-0” state (“0000…”), “all-1” state (“1111…”), dual-beam splitting state (“0101…”), and quad-beam splitting state (“0110…”). Yellow and green pixels represent the “0” and “1” states, respectively. b Simulated far‑field radiation patterns in the normalized wavevector space at 12.5 GHz for the respective coding configurations above. c Photographs of the MB‑RCM prototype under the respective coding states. d Photograph of the far-field measurement setup. e Comparison between simulated (blue lines) and measured (red lines) normalized far-field scattering patterns in the x-z plane at 12.5 GHz. The dashed regions indicate the angular range ( ± 3°) where the receiving antenna physically obstructs the transmitting antenna, resulting in a measurement shadowing effect.

The far-field measurement setup is illustrated in Fig. 5d. A pair of horn antennas operating within the 11.9–18 GHz were connected to a vector network analyzer (VNA) to serve as the transmitter and receiver for co-polarized EM waves. The simulated and measured normalized far-field scattering patterns in the x-z plane at 12.5 GHz are depicted in Fig. 5e, showing high agreement in terms of main-lobe direction and beamwidth. Notably, a pronounced dip in the measured power is observed around the 0° direction across all measured coding states. This phenomenon arises from a shadowing effect within a narrow angular range of ± 3°, where the receiving antenna physically obstructs the EM waves from the transmitting horn under the bistatic measurement configuration. Overall, these results confirm the reliable beam steering capability of the MB-RCM under mechanical coding control. Furthermore, to evaluate the spectral stability of the proposed platform, the far-field scattering patterns were measured across 1 GHz bandwidth. As illustrated in Fig. S21 (Supplementary Information), the radiation patterns for all four coding states maintain high pointing stability from 12.0 to 13.0 GHz, further validating the broadband effectiveness of the MB-RCM in diverse wavefront manipulation scenarios.

Discussion

In conclusion, we propose a robust coding metasurface platform enabled by origami-based MB for non-volatile 1-bit phase modulation. By exploiting mechanical snap-through instability, continuous mechanical deformations are discretized into robust binary geometric states. As demonstrated in Table 1, this mechanism offers advantages including zero static power consumption and intrinsic error-correction capabilities, thereby overcoming the volatility of electrically controlled devices and the limited stability of conventional mechanically reconfigurable metasurfaces. Furthermore, the proposed platform enables versatile transmissive and reflective phase coding across two broad frequency bands, providing both disturbance immunity and flexible wavefront manipulation capabilities.

Table 1.

Comparison of different approaches to dynamic modulation in metasurfaces

Reference	Approach	Operation frequency	Operation type	Coding scheme	State retention mechanism
6	Electronics	5.4 GHz, 5.5 GHz	Transmission	Amplitude & Phase	Continuous bias voltage
3	Electronics	3.5 GHz	Reflection	Phase	Continuous bias voltage
4	Electronics	9.7 GHz	Radiation	Phase	Continuous bias voltage
27	Mechanics	4.96–38.8 GHz	Reflection		External mechanical locking
28	Mechanics	12–17 GHz	Reflection		External mechanical locking
Ours	Mechanics	3.5–5 GHz, 12.2–13 GHz	Transmission & Reflection	Phase	Mechanical self-locking

Leveraging this physical mechanism, two theoretical models accounting for the compression-torsion coupling behavior were proposed to design efficient 1-bit coding meta-atoms for transmission and reflection modulation under linearly polarized incidence, respectively. Based on the ideal conditions derived from the models, the geometry and bistable feature of the specific Kresling topology was systematically analyzed through both theoretical and numerical models. Experimental results further demonstrate that the lightweight sample fabricated by multi-material 3D printing technology possesses structural reliability against environmental disturbances. The MB-TCM and MB-RCM meta-atoms deliver efficient and broadband EM responses, with |T_yx | > 0.75 over 3.5–5.0 GHz and |R_xx | > 0.85 over 12.2–13.0 GHz, respectively, together with a stable phase difference of 160°–200°. Functional prototype arrays were also fabricated and measured, successfully demonstrating reliable wavefront manipulation capabilities at different frequencies by high-fidelity near-field holographic imaging and pointing-stability far-field beam steering. The high consistency observed between theoretical predictions, numerical simulations, and measurements fully validates the reliability of this design. While current prototypes utilize mechanical actuation, future integration of multi-physical stimuli (e.g., magnetic or thermal fields) promises to enable rapid, fully self-adaptive array addressing. The proposed conceptual and design framework for mechanically driven coding metasurfaces, characterized by high efficiency, wideband operation, non-volatility and low-power consumption, lays a solid foundation for future explorations into multi-bit coding, intelligent control systems, and miniaturization.

Methods

Electromagnetic simulation

Full-wave simulations were conducted using CST Studio Suite to evaluate the EM performance of the MB-TCM and MB-RCM meta-units under discrete stable states. The dielectric substrates were modeled as F4B high-frequency laminates (relative permittivity ɛ_r = 2.65 for MB-TCM and ɛ_r = 2.2 for MB-RCM). The polyimide film (0.05 mm thickness) was assigned a permittivity of ɛ_r = 3, and the copper layers were defined with an electrical conductivity σ = 5.8 × 10⁷ S/m. The Vero resin was modeled with ɛ_r = 3.05 and a loss tangent of tanδ = 0.06. Unit cell boundaries were adopted along the x- and y-axes while open boundaries were defined in the z-direction. For the holographic imaging and beam steering simulations, the metasurfaces were illuminated by plane waves, and field monitors were utilized to record the near-field distributions and far-field radiation patterns within the operating bands.

Prototype fabrication

The Kresling units were fabricated using a commercial PolyJet multi-material 3D printer (Stratasys J850, Stratasys Ltd.). Each unit comprises rigid rods (VeroWhitePlus, Shore D hardness 85) and flexible joints (Agilus, Shore A hardness 50). The functional EM layers were produced via commercial flexible printed circuit (FPC) techniques on polyimide/copper (0.05 mm/35 μm) films. The SRR and Jerusalem Cross patterns were formed by etching away the excess copper, followed by laser cutting into individual patches. These patches were directly glued on the Kresling units, which were subsequently assembled into F4B dielectric substrates featuring computer numerical control (CNC) drilled blind holes to form the final arrays (as shown in Figure S22, Supplementary Information). The resulting MB-TCM prototype is a 16 × 16 array (352 × 352 mm²), while the MB-RCM prototype is an 8 × 8 array (176 × 176 mm²).

Mechanical experiments

Quasi-static compression tests were conducted using a universal testing machine (Instron 34SC-1) equipped with a high-precision 50 N load cell. Displacement-control was applied at a constant rate of 0.2 mm/s. To ensure stability of the specimen during testing, the four cylindrical feet of the Kresling unit were inserted into a customized 3D printed PLA base fixture, which was secured to the base plate. Supplementary Movie 1 shows the mechanical testing. Temperature-dependent measurements were performed using a cryogenic chamber ( − 40–0 °C) and a climatic chamber (0–70 °C). Samples were conditioned for 2 hours before testing to ensure thermal equilibrium and measured within 1 minute after removal. Humidity-dependent tests were conducted at 20 °C under relative humidity (RH) levels of 30–90% after 24 hours conditioning. For long-term state-retention tests, the Kresling unit was maintained in the folded state at room temperature for 0–100 hours prior to measurement.

Electromagnetic experiments

The actuation of the Kresling-based metasurface is enabled by a motorized 3D translation stage, as illustrated in Supplementary Movie 2, along with the relevant discussions in Text S13 (Supplementary Information). Near-field holographic imaging was performed in an anechoic chamber. A standard horn antenna (3.94–5.99 GHz) served as the transmitter, placed 2 m away from the MB-TCM to generate an x-polarized quasi-plane wave. The imaging plane was located 65 mm behind the MB-TCM. A waveguide probe raster-scanned the 352 × 352 mm² area with a step size of 4.4 mm to map the y-polarized electric field component. Data acquisition was synchronized using a VNA. Suppl. Movie 3 illustrates the evolution of the holographic pattern as the number of coding rows changes. Initially, the MB-TCM is in an unencoded state, which is equivalent to free space, and the probe detects a quasi-uniform plane wave. As the number of encoded rows gradually increases, the target holographic patterns (“Z”, “J”, and “U”) progressively emerge.

For far-field beam steering, a bistatic measurement setup was employed using a pair of horn antennas (11.9–18 GHz) connected to a VNA. The transmitting horn was positioned 1.5 m from the MB-RCM. The prototype and the receiving horn were mounted on a rotary stage with a 1 m separation. The receiving antenna was rotated in the x-z plane from −90° to 90° with a 1° step to record the angular scattering profile. The far-field radiation pattern measurement of MB-RCM under four different coding states, together with the corresponding real-time beam steering results, is demonstrated in Suppl. Movie 4.

Author contributions

D.Z. and P.T. contributed equally to this work. D.Z., P.T. and Z.W. initiated the plan, and L.J., H.C. and Z.W. supervised the entire study. D.Z. and P.T. conceived the idea of this work and designed the simulations and experiments. D.Z., P.T., X.Y., B.Z., E.W., Y.L. and E.F. carried out the measurements and data analyses. D.Z., P.T. and Z.W. prepared the manuscript with input from all authors. All authors discussed the research.

Peer review

Peer review information

Nature Communications thanks Fei Ding, Jiafang Li and the other anonymous reviewer(s) for their contribution to the peer review of this work. A peer review file is available.

Funding

The work at Zhejiang University was sponsored by the National Natural Science Foundation of China (NSFC) (U25A20411, 62222115, 62201500), the Key Research and Development Program of Zhejiang Province under Grant No.2024C01241(SD2), the Natural Science Foundation of Zhejiang Province under Grant No.ZCLY24F0101.

Data availability

All the data supporting the findings of this study are provided in the Supporting Information and Source Data file. Source data are provided with this paper.

Competing interests

The authors declare no competing interests.

Supplementary information

The online version contains supplementary material available at https://doi.org/10.1038/s41467-026-77313-6.