Recent advances in nanofabrication techniques, such as direct writing [1,2], nanoimprinting [3], lithography [4–6], and assembly approaches [7–11], have promoted the development of moiré metamaterials and metasurfaces, which have enabled numerous distinctive phenomena and unique light–matter interactions. Moiré effects in reciprocal space, existing in bilayer graphene [12–17], $\mathrm{\alpha }\text{-}{\mathrm{MoO}}_3$ [18,19], van der Waals materials [20–23], nanosphere superlattice [24,25], and other dielectric nanophotonic materials [26,27], result in abundant exotic electronic properties and optical features—for instance, superconductivity [28–30], ferromagnetism [31–35], extreme dispersion engineering [36–39]—which allow unprecedented capabilities in the applications of optical chirality [40,41], full-wave manipulation [36], quantum responses [42–44], phonon polaritons [45–47], etc. Interestingly, new symmetries emerge from stacking and twisting, causing hybridization of dispersions, which enables drastic surface state manipulation [48,49]. However, previous works are mostly concentrated on moiré patterns in variable frequency regimes ranging from UV to visible and to infrared (IR) [50–52], and moiré topological transitions of surface states based on twisted hyperbolic metasurfaces (HMSs) at microwave and terahertz frequencies remain unexplored. Since there is no efficient terahertz dipole source to experimentally verify the twist-induced dispersion engineering, we designed a moiré hyperbolic metasurface (MHMS) in the microwave regime to emulate the moiré phenomena. One prominent advantage is the much simpler fabrication process of microwave HMSs than the complicated nanolithography techniques in the UV-IR range [53,54]; another advantage is that the rotation angle between the two relatively independent periodic metasurfaces can be easily and precisely controlled.

Classical plasmonics aims at exploring the fundamental properties of surface plasmon polaritons (SPPs), which describe the confined electromagnetic wave propagation along the interface between a metal and dielectric [55]. However, because metals can be regarded as perfect electric conductors in the microwave region, the SPPs become weakly confined Zenneck waves and the evanescent fields can extend over several wavelengths [56]. Therefore, structured metals are utilized to overcome the limits, and microwave plasmonics composed of subwavelength metal patterns and dielectric slabs can exhibit unique topological transition and have been widely used to manipulate surface waves [57]. For example, planarized sandwich and dielectric-metal structures can be used to obtain the hyperbolic dispersion in the momentum space by varying the permittivity or the structural parameters [58–62]. Topological transitions only occur at a specific excitation frequency, which greatly limits their applications. Nevertheless, moiré effects can arise when mutual rotation is introduced to a stacked bilayer HMS, inducing hyperbolic-to-elliptic topological transitions in the whole frequency range due to the twist-induced coupling between two adjacent layers. Such a photonic phase transition in the equal-frequency contours (EFCs) is analogous to the Lifshitz transition in electronics [63], and the nondiffraction transmission in photonic systems is similar to the dissipation-free flow of electrons (superconductivity). Therefore, the MHMS experiences a topology change after stacking and twisting, and, at the critical angle where the photon density of states reaches a maximum, the dispersion behaves as a flat line, and the angle is thus referred to as the magic angle [64,65]. The flexibility in designing moiré patterns and the sensitivity to rotation-induced dispersion responses allow for on-demand surface wave modifications. The proposed regime offers a new avenue to break the diffraction limits, applicable to nondivergent diffraction [66], negative diffraction [67], and anomalous wave propagation [68,69].

In this paper, we demonstrate theoretically, numerically, and experimentally moiré topological transitions in a sandwiched metal metasurface consisting of two closely stacked periodic hyperbolic microstructures rotated relative to each other. The photonic responses can be actively controlled by rotating the angle between the two stacked layers, leading to hybridization of the EFCs and moiré effects. To fully investigate the underlying mechanism of the numerous wave features, a Fourier transform is employed to map the corresponding iso-frequency lines controlled by the rotation angles. These unusual moiré phenomena emerge as a result of eigenmode coupling and the MHMS formation, presenting an evolution of dispersion from being hyperbolic to elliptical, which facilitates active engineering of surface waves. Besides, at the magic angles, the number of anti-intersection points $N$ between the coupled system and the individual layers can be applied to separate the EFCs with different shapes, where $N$ has a discontinuous change. Finally, we fit the function between the magic angles and frequencies, observing that the magic angle changes linearly with frequency, proving that the MHMS can realize topological transition at each frequency. The proposed MHMS offers a new platform to manipulate light-propagation properties and is therefore promising for nonlinear optics [70,71], near-field focusing [72,73], imaging [74,75], etc.; it also shows exceptional capabilities in designing plasmonic devices, multifunctional devices, and sensors.

As shown in Fig. 1, a moiré structure is formed by stacking two mutually twisted H-shaped periodic HMSs, which consist of a lossless substrate (${\mathrm{F}}_4\mathrm{BM}300$) with a permittivity of 3.0 and an anisotropic copper pattern. Compared with other microwave HMSs, the H-shaped ultrathin meta-array exhibits an excellent ability to manipulate surface waves. The detailed structural parameters of the unit cell are designed as $p=8\mathrm{mm}$, $a=6\mathrm{mm}$, $b=5\mathrm{mm}$, and $w=0.5\mathrm{mm}$; the thicknesses of the dielectric substrate and the copper structure are 1 and 0.035 mm, respectively. The relative rotation angle $\mathrm{\Delta }\theta$ is defined as the anticlockwise rotation of the top layer with respect to the bottom layer. A microwave dipole resonating in the $z$ direction is placed at the center of the designed MHMS to excite surface waves. When the superimposed top layer rotates relative to the bottom layer, a drastic plasmon hybridization effect between the two HMSs occurs due to the strong anisotropy of the monolayers. Therefore, the coupling of eigenmodes between the two layers will change the shape and intensity of the surface waves when a misalignment of the optical axes between the two layers appears, thereby enabling the manipulation of surface plasmons. As also depicted at the top right in Fig. 1, topological transitions of the surface states appear as the rotation angle increases (from a V-shaped pattern to a flattened line and finally to a spherical trajectory), which will provide new degrees of freedom for the engineering of light–matter interactions and present great potential in designing active plasmonic devices.

Previous studies are mostly concerned with the tight-binding approximation to express the $k$-space properties [39]; here, the effective medium theory (EMT) is utilized to describe the moiré configurations in the microwave region. Therefore, in order to analyze the SPP redirection effect of the proposed MHMS, we start with the electromagnetic characteristics of the designed individual monolayer HMS using the time-domain solver of the commercial software Computer Simulation Technology (CST). As depicted in Fig. 2(a), the transmission spectra have Lorentzian profiles, which are symmetrically shaped lines when the incident waves are polarized in the $x$ and $y$ directions. Figures 2(b) and 2(c) exhibit greatly different electric distributions when the plane waves propagate along the $x$ and $y$ axes, respectively. These results provide insight into the electric dipoles in the orthogonal directions induced by the incident polarization states; they also indicate that the model has a strong anisotropic response. Furthermore, we built equivalent circuit models to describe the electric dipoles with a dipole moment along the $x$ and $y$ directions, respectively. As presented in Figs. 2(d) and 2(e), the resonance frequencies are $f_x=\frac{1}{2\pi \sqrt{C_1{L}_1}}$ and $f_y=\frac{1}{2\pi \sqrt{\frac{L_2{L}_2^{\prime }(C_2+C_2^{\prime })}{L_2+L_2^{\prime }}}}$. Besides, we employed the EMT to simplify the designed monolayer HMS to a homogeneous anisotropic effective-medium slab. For the nonmagnetic uniaxial material whose permeability mainly comes from the metal patterns, it is reasonable to set $\mu =1$ as an approximation; the permittivity can also be reduced to a diagonal tensor, $\widehat{\mathit{\epsilon }}=\left(\begin{array}{cc}{\epsilon }_{xx}& 0\\ 0& {\epsilon }_{yy}\end{array}\right)$, where ${\epsilon }_{xx}$ and ${\epsilon }_{yy}$ can be obtained by the $S$-parameter retrieval process [76–78]. According to Snell’s law, $S_{21}$ (transmittance) and $S_{11}$ (reflectance) are related to the effective refractive index $n$ and the impedance $z$ of the dielectric slab by $$S_{21}^{-1}=\cos (nkh)-\frac{i}{2}(z+\frac{1}{z})\sin (nkh),$$(1)$$S_{11}=\frac{i}{2}(\frac{1}{z}-z)\sin (nkh),$$(2)where $h$ is the thickness of the substrate and for the nonmagnetic metasurface $z=1/n$. In this case, the permittivity can be derived as $$\epsilon =n^2=\frac{1}{k^2{h}^2{\cos }^2[\frac{1}{2S_{21}}(1-S_{11}^2+S_{21}^2)]}.$$(3)

The calculated permittivity is plotted in Fig. 2(f), where the gray areas represent the hyperbolic EFCs $({\epsilon }_{xx}\times {\epsilon }_{yy}<0)$ and topological transition frequencies (8.63 and 19.49 GHz). The structure corresponds to a type II HMS for $8.63<f<14.8\mathrm{GHz}$, where ${\epsilon }_{xx}<0$ and ${\epsilon }_{yy}>0$; when $f>19.49\mathrm{GHz}$, the structure has the characteristics of a type I HMS with ${\epsilon }_{xx}>0$ and ${\epsilon }_{yy}<0$. As the surface wave transmits in the direction perpendicular to the EFC, nondiffraction transmission occurs at the transition frequencies in two perpendicular directions, and the detailed dispersion properties of the H-shaped single-layer structure are investigated through CST eigenvalue analysis in the following.

Assuming that the electromagnetic waves propagate in the $x-y$ plane and decay along the $z$ direction, the dispersion relation can be calculated by substituting the permittivity tensor into Maxwell’s equations as follows: $$\frac{k_x^2}{{\epsilon }_{yy}}+\frac{k_y^2}{{\epsilon }_{xx}}=k_0^2,$$(4)where $k_0$ describes the wave number in vacuum at an angular frequency $\omega$. As depicted in Figs. 2(g)–2(i), the first three energy bands in the first Brillouin zone and the first two EFCs of the energy bands were calculated with the CST eigenmode solver to understand the dispersion characteristic of the HMS. It can be seen that, for the first three energy bands, the directions of topological transitions are along the $x$, $y$, and $x$ axes, respectively. The hyperbolic and transition phenomena show great consistency with the permittivity values calculated above, and the deviation in the transition frequencies comes mainly from the simplified 2D homogeneous model, as the realistic 3D structure is too complicated to be analyzed simply, and the retrieval process will also introduce some error when the $S$ parameter values are too small [79]. In order to get a clearer perspective, we investigated the eigenmode EFCs based on the equation $k=\pi \times \theta /180p$. As shown in Fig. 2(h), with the frequency increasing, a topological transition appears when the dispersion curves of the first mode gradually change from elliptical to flat and finally to hyperbolic, showing an unprecedented capability to control the SPPs. As the direction of the surface plasmon group velocity is perpendicular to the EFC ($\stackrel{\rightharpoonup }{{\mathit{V}}_g}={\nabla }_{\stackrel{\rightharpoonup }{\mathit{k}}}\omega$), various wavefront geometries can be observed at different frequencies. However, when the geometric parameters are fixed, the corresponding dispersion curve at a specific frequency will not change any further. Therefore, in order to modify the dispersion relation, the only way is to change the permittivity, permeability, shape, or structural parameters of the HMS, which will incur considerable time and manufacturing costs. Inspired by the moiré configurations in the visible range, we have developed an MHMS to verify whether the twist-induced photonic dispersion can achieve the engineering of surface waves in the microwave region.

Since the thickness of the MHMS is much smaller than the wavelength, the bilayer HMS can be treated as a single layer with period $p$ when $\mathrm{\Delta }\theta =0°$ [80]. Based on the EMT, the model can still be regarded as a homogeneous anisotropic effective-medium slab whose permittivity is a diagonal tensor. However, the stacking between different layers will excite magnetic modes manifesting as ring-like surface current distributions in the split rings [see Figs. 3(h)–3(k)]; hence, the polaritons in the MHMS change from TM modes to hybrid TM and TE eigenmodes and the permeability becomes a diagonal tensor as well. In general, conductivity is introduced to describe the dispersion relation: $\widehat{\mathit{\sigma }}=\left(\begin{array}{cc}{\sigma }_{xx}& {\sigma }_{xy}\\ {\sigma }_{yx}& {\sigma }_{yy}\end{array}\right)$, $\widehat{\mathit{\sigma }}=-i\omega d·\widehat{\mathit{\epsilon }}$. The total electric and magnetic fields can be written as the summation of the corresponding components of pure TE and TM modes: $\mathit{H}={\mathit{H}}_{\mathrm{TM}}+{\mathit{H}}_{\mathrm{TE}}$, $\mathit{E}={\mathit{E}}_{\mathrm{TM}}+{\mathit{E}}_{\mathrm{TE}}$. According to the electromagnetic boundary conditions, the dispersion relation of the surface waves is written as [81] $$(\frac{k_1}{{\mu }_1}+\frac{k_2}{{\mu }_2}-i{\sigma }_{yy})(\frac{{\epsilon }_1}{k_1}+\frac{{\epsilon }_2}{k_2}-i{\sigma }_{xx})={\sigma }_{xy}{\sigma }_{yx},$$(5)where $k_j$, ${\epsilon }_j$, and ${\mu }_j$ ($j=1$, 2) describe the wave numbers, permittivity, and permeability in the media above ($j=1$) and below ($j=2$) the single anisotropic metasurface. Here, the effective conductivity tensor can be regarded as a sum of the two metasurfaces: ${\widehat{\mathit{\sigma }}}_{\sum }={\widehat{\mathit{\sigma }}}_{\text{bottom}}+{\widehat{\mathit{\sigma }}}_{\mathrm{top}}$. The detailed description of the conductivity tensors is provided in Appendix A. By retrieving the $S$ parameters, we can obtain the effective permittivity and permeability by the relations $\mu =nz$ and $\epsilon =n/z$. Here, $n$ and $z$ are given by $$\begin{gathered} n=\frac{1}{kh\cos [\frac{1}{2S_{21}}(1-S_{11}^2+S_{21}^2)]}, \\ z=\sqrt{\frac{{(1+S_{11})}^2-S_{21}^2}{{(1-S_{11})}^2-S_{21}^2}}. \end{gathered}$$(6)

As exhibited in Fig. 3, by analyzing the transmittance and reflectance curves under $x$- and $y$-polarized incidence, we obtain the following conclusions from the effective permittivity and permeability data [Figs. 3(b) and 3(c)]. 1) The topological transition frequencies decrease because of the presence of new resonance valleys (6.59 and 8.63 GHz when the incident wave is polarized in the $x$ direction; 14.60 and 19.30 GHz when the electric field is polarized in the $y$ direction) with lower frequencies. 2) The increase of coupling modes causes an increase in the number of topological transition frequencies, and obvious transition occurs at 6.39, 8.57, and 19.09 GHz. 3) The three gray regions (6.39–6.70, 8.57–8.99, and 19.09–22.04 GHz) correspond to type II HMS, type II HMS, and type I HMS, respectively. As the surface plasmon group velocity is perpendicular to the EFC, the directions of the self-collimation frequencies are along the $x$, $x$, and $y$ axes, respectively. 4) Compared with the monolayer structure, the relative permeability tensor changes from a unit matrix to a diagonal one, which means that magnetic responses appear as a result of stacking. Analogously, the resonance dips represent similar electric distributions, which show the same excited modes, as shown in Figs. 3(d)–3(g). 5) Finally, the calculated energy band diagram and the first two EFCs in the first Brillouin zone show the same phenomena as the relative permittivity predicted above [see Figs. 3(l)–3(n)].

The MHMS turns to an aperiodic structure when $\mathrm{\Delta }\theta \ne 0°$, where the relative permittivity and permeability change from diagonal tensors to fully populated tensors owing to the rotation and stacking. We can, however, still use Eq. (5) to characterize the dispersion relations of the MHMS. Appendix B studies the corresponding conductivity tensors of the twisted bilayer hyperbolic metasurface. Then, taking $\mathrm{\Delta }\theta =15°$ as an example, we simulated the co- and cross-polarized transmittance and reflectance spectra [Figs. 4(a) and 4(b)] to obtain the nondiagonal terms of the relative permittivity and permeability. The appearance of the cross-polarization (the orange and blue solid lines) means that the proposed MHMS has the ability for polarization conversion, which has great potential in the active control of metasurfaces and will enrich the multifunctional devices. Based on the $S$-parameter retrieval process, the corresponding calculated results of the real and imaginary parts of the relative permittivity and permeability are plotted in Figs. 4(c) and 4(d), respectively. It is clear that the relative rotation between the two HMSs leads to nondiagonal components, which means that the topological transition frequencies and hyperbolic characteristics cannot be simply obtained based on the relative permittivity. Thus, we obtain the EFCs of the MHMS by analyzing the Fourier transform of the $E_z$ distributions in the following part.

As presented in Fig. 5(a), a moiré pattern is assembled by two H-shaped bilayer metasurfaces with a subwavelength thickness (total size $248\mathrm{mm}\times 248\mathrm{mm}$), and the individual layers have the same parameters and are placed closely to each other in the $z$ direction, where the top layer is twisted counterclockwise with respect to the bottom layer by $\mathrm{\Delta }\theta$. The CST time-domain solver is used to simulate the $E_z$ distributions at 0.05 mm above the top layer, with the dipole source resonating in the $z$ direction and located at the center of the designed model. Interestingly, it is found that the mode hybridization controlled by rotation enhances the electromagnetic response of the MHMS and achieves effective modulation of the topological transition of the surface waves at different frequencies. To illustrate the underlying mechanism of the moiré properties, we numerically calculated the electric field distribution and the corresponding dispersion, which shows that a distinct topological transition occurs with an increase of the relative twist angle. In addition, we observe an obvious electric field enhancement by about $9\times {10}^4\mathrm{V}/\mathrm{m}$ at 0.5 mm above the metasurface, which will provide a new route to construct detectors with enhanced sensitivity [82].

With the $z$-oriented electric field $E_z$ distributions at 5.90 GHz taken as an example, the Fourier transform is utilized to map the corresponding EFCs [see Figs. 5(b) and 5(c)]. In the reciprocal space, the white solid curves are the fitted dispersion contours, while the red and yellow dotted curves are the EFCs of the top and bottom layers, respectively, and the red arrows denote the directions of the wave vectors. When $\mathrm{\Delta }\theta =0°$, the surface waves propagate along a convergent path, and the resulting EFC is a hyperbola. When $\mathrm{\Delta }\theta =15°$, the excited SPPs transmit along a nondiffraction collimated path, which means that the corresponding EFC is a tilted flattened line. When $\mathrm{\Delta }\theta =30°$, the trajectory of polariton propagation exhibits a V-shape, which originates from the elliptical EFC. When $\mathrm{\Delta }\theta =45°$, the EFC is a closed circle, leading to a spherical expanding wavefront. Besides, it can be found that the anti-intersection points ($N$) can be used to qualitatively characterize the moiré topological transition angle: when $N=4$, the resulting dispersion is hyperbolic, but, when $N=0$, the dispersion turns into an ellipse, where the moiré topological transition occurs.

To further demonstrate the varying pattern of the magic angle, we summarize the change of the topological transition frequencies with respect to the rotation angles, as illustrated in Fig. 5(d). The black dotted line denotes the magic angles that separate the range of hyperbolic (the blue area, $N=4$) and elliptical dispersion (the orange area, $N=0$). Therefore, the designed moiré structure makes a topological transition at an arbitrary frequency possible by merely controlling the twist angle, showing great potential in multifunctional devices and integrated plasmonic devices and promising a new platform to manipulate diverse surface plasmon transmission processes. In addition, the number of anti-intersection points, $N$, in the $k$-space directly determines the shape of the dispersion contour. When $N$ changes discontinuously, the EFCs change simultaneously, from a closed ellipse to an open hyperbola, resulting in a magic angle topological transition due to enhanced light–matter interactions and coupling between the individual isolated layers. Detailed descriptions about the magic angles are given in Appendix C. In addition, the spacer thickness dependence of the moiré structure is discussed in Appendix D to further investigate the electromagnetic interaction. In a word, the proposed MHMS, in which the superimposed layer is rotated with respect to the bottom layer, enables the observation of polariton hybridization, drastic dispersion modifications, and highly tunable optical responses.

The experimental sample was composed of two identical H-shaped HMSs with a twisted angle $\mathrm{\Delta }\theta$ between them. A 1 mm thick ${\mathrm{F}}_4\text{BM}300$ was employed as the dielectric substrate with a permittivity of 3.0 and tan $\delta \le 7\times {10}^{-4}$ at 10 GHz, and a 0.035 mm thick H-shaped copper pattern was printed on the substrate slab, as shown in Fig. 6. Besides, the organic solderability preserving anti-oxidation process was carried out on the metal surface to facilitate long-term experimental measurement needs. The constructed monolayer was chosen with the following geometries: $p=8\mathrm{mm}$, $a=6\mathrm{mm}$, $b=5\mathrm{mm}$, and $w=0.5\mathrm{mm}$; the total size was $248\mathrm{mm}\times 248\mathrm{mm}$. Since the coupling between free-space radiation and the HMSs was extremely weak, a probe connected with a vector network analyzer was placed close to the center of the copper pattern to excite a microwave dipole and improve the coupling efficiency. In order to shield the spatial electromagnetic signal radiated by the dipole source during the experiments, we used a metal box with a small round hole on the top to localize the dipole in the box, which could not only reduce the radiation loss of the electromagnetic waves but also enhance the coupling between the copper pattern and the impinging wave. Another probe was used to detect the $E_z$ field distributions located at 0.05 mm above the surface, being moved with a step of 2 mm along the $x$ and $y$ axes by a 3D translation stage.

Taking into account the symmetry of the electric fields on both sides of the source, we selected the right half of the model (red dotted area) as the fine scanning range, as depicted in Fig. 7(a). Similarly, to experimentally observe the topological transition of the twist-induced surface states, we mapped the $E_z$ distributions at 6.53 GHz in Fig. 7(b). Interestingly, when $\mathrm{\Delta }\theta =30°$, we observed a nondiffraction transmission of surface waves at their self-collimation frequencies where the photon density of states is relatively large, and the localization of surface waves is strong. Unlike at other angles, the electric field pattern turns from a straight trajectory into a spherical wavefront when $\mathrm{\Delta }\theta \ge 60°$. That is, with the increase of the mutually twisted angle, the dispersion changes from an open hyperbola to a closed ellipse based on the rule that the transmission direction of surface waves is perpendicular to the EFC. The corresponding Fourier transform of the experimental results is performed in Appendix E, which exhibits the same topological transition process as the simulation results. However, in the measurement process, the unevenly distributed electric fields and deviation in the rotation angles mainly come from the fabrication error and height difference of the MHMS. But our experimental results are also numerically supported by our simulations with excellent agreement. In addition, other structured hyperbolic metasurfaces and magnetic topological transition configurations with double split-rings may provide new methods to reduce the propagation loss.

We then summarized the relationship between the magic angle and the frequency in Fig. 7(c); it can be seen that the magic angle changes linearly with frequency, as shown by the black dotted line, and the dispersion is hyperbolic below the line and elliptic above. The magic angle observed at the topological transition point offers a new avenue for engineering light–matter interactions and optical responses. Different from previous methods based on sweeping the topological transition frequencies in different structures, the proposed regime can realize the control of surface states at arbitrary frequencies.

A new type of sandwiched twisted bilayer metal hyperbolic metasurface is proposed, which enables the steering of electromagnetic responses (the evolution process of the relative permittivity and permeability tensors) and the excitation of magic angle phenomena (drastic dispersion modification) by the relative rotation of the superimposed layer. The designed MHMS can realize topological transitions at arbitrary frequencies rather than a single transition frequency compared with a monolayer counterpart. Besides, Fourier transform of the real space is introduced to study the underlying mechanism of the surface states with the increase of the twist angles. In particular, the fitted curve of the magic angles separating the closed (ellipse) and open (hyperbola) dispersion contours changes linearly with frequency, and the position of the magic angles can be found quantitatively by calculating the number of anti-intersection points between the moiré structure and the distinct layers. Finally, experimental measurements are implemented to verify the interesting twist-induced moiré effects, and the results are in good agreement with the simulations. The proposed MHMS provides a unique way to manipulate light propagation properties and offers new opportunities in designing plasmonic devices, which can be widely used in sensing, imaging, and slow light applications in the microwave region [82].