Flexible terahertz metasurfaces with ultra-large incident angle tolerance and mechanical adaptability

Abstract

Metasurfaces operating under oblique incidence conditions have attracted considerable interest in modern photonic engineering, owing to their capability to enhance angular coverage for next-generation devices including wide-field detectors, multi-angle communication arrays, and advanced radar architectures. However, conventional metasurfaces frequently suffer from limitations in preserving functional stability at high incidence angles, where performance deterioration typically stems from reduced coupling efficiency and phase mismatch. This work proposes flexible terahertz metasurfaces with an ultra-large incident angle, addressing the limitations of conventional metasurfaces that suffer from severe performance degradation under high angles and lack mechanical adaptability. The proposed metasurfaces are designed through the synergistic integration of spoof localized surface plasmons (S-LSPs) and electromagnetically induced transparency (EIT). Experimental results demonstrate that the metasurfaces exhibit polarization-insensitive behavior while maintaining stable transmission spectra under oblique incidence up to 80°. Furthermore, bending tests confirm that the transmission spectrum of the metasurfaces remains stable even under significant mechanical deformation (with a maximum central angle of 111.7°). The proposed flexible metasurfaces provide a novel design paradigm for oblique-incident metasurfaces, holding great potential for emerging applications in conformal terahertz systems and adaptive wearable photonics.

Graphical abstract

Keywords

metasurfaces / flexible material / oblique incidence / plasmons / terahertz

1 Introduction

Metasurfaces, as two-dimensional counterparts of metamaterials, have emerged as a revolutionary platform for manipulating electromagnetic waves at subwavelength scales [1, 2]. Composed of periodic or aperiodic arrays of nanostructured elements, metasurfaces enable precise control over the amplitude [3−7], phase [8−13], and polarization of incident light [14−17], offering unprecedented opportunities in fields such as imaging [18, 19], sensing [20, 21], and communication [22, 23]. Their compact design, low loss, and high integration capabilities make them superior to traditional three-dimensional metamaterials, paving the way for advanced photonic devices with enhanced functionality and miniaturization.

The application of metasurfaces under oblique incidence has garnered significant attention due to its potential in expanding the angular range of operation for practical devices such as wide-angle sensors [24], communication systems [25], and radar technologies [26]. Traditional metasurfaces design often face challenges in maintaining performance at large incident angles, primarily due to reduced coupling efficiency and phase mismatch [27]. Overcoming these limitations is crucial for realizing robust and versatile optical systems capable of operating in dynamic environments.

Current methods to achieve oblique-incidence metasurfaces include the use of multilayer structures [28, 29], gradient phase designs [30, 31], and resonant unit cells tailored for wide-angle performance [32, 33]. For instance, multilayer frameworks have been employed to achieve achromatic and angle-insensitive responses, enabling consistent performance across a broad range of incident angles [34]. Additionally, gradient phase metasurfaces have been designed to manipulate wavefronts dynamically, ensuring efficient light control even at oblique angles [35]. However, these approaches often require complex fabrication processes and face limitations in achieving high efficiency and large angular ranges simultaneously.

Spoof localized surface plasmons (S-LSPs), which mimic the behavior of traditional localized surface plasmons at lower frequencies, exhibit strong field localization and enhancement, making them ideal for improving light-matter interactions [36−38]. Electromagnetically induced transparency (EIT), is a quantum interference phenomenon that creates a narrow transparency window within an otherwise absorptive medium, leading to enhanced dispersion and slow light effect [39−42]. EIT can be created by bright-dark mode and bright-bright mode. A bright mode is a mode that can be directly excited by incident electromagnetic waves, while a dark mode is a mode that cannot be directly excited by incident electromagnetic waves and requires coupling with a bright mode for excitation. By integrating S-LSPs with EIT-like effects, we intend to demonstrated the potential to achieve large-angle oblique-incidence performance through localized field enhancement and tailored dispersion properties.

In this study, we propose the flexible terahertz metasurfaces with polarization-insensitive and oblique incidence characteristics. By integrating S-LSPs and EIT effect, we designed the metasurfaces composed of a spiral metal structure (SMS) and a ring structure (RS). The surface electric field distributions, the surface magnetic field distributions, and the surface current distributions are employed to explore the interaction between the SMS and RS. The multipole scattering theory further provides physics insights into the EIT effect and the two-particle model is used to describe the spectral response of the proposed metasurfaces. Furthermore, we tested the metasurfaces sample under different polarization angles, incident angles and bending degrees. Our work offers a pioneering design strategy for flexible and oblique incident metasurfaces research, holding significant application value in conformal terahertz imaging systems and adaptive wearable electronics where wide-angle stability and mechanical flexibility are paramount.

2 Structure design and numerical model

The schematic of the proposed terahertz flexible metasurfaces under the irradiation of terahertz wave is illustrated in Fig. 1(a). The terahertz pulse is incident to the metasurfaces, forming an angle θ with the surface normal. The directions of electric field and magnetic field are along the y-direction and x-direction respectively. Figure 1(b) shows the unit cell of the terahertz metasurfaces, which is composed of metallic layer and substrate layer. The metallic layer is an EIT configuration and consists of a SMS and a RS. Aluminum (Al) is selected as the material for the metallic structures, which is described by the Drude model in the terahertz band [43]:

(1){\epsilon }_{\mathrm{A}\mathrm{l}}={\epsilon }_{\mathrm{\infty }}-{\displaystyle \frac{{\omega }_p^2}{{\omega }^2+\mathrm{i}\omega \gamma }},

where the dielectric constant at the infinite frequency {\epsilon }_{\mathrm{\infty }} is 3.7. In addition, the plasmon frequency {\omega }_p is 2.24 × 10¹⁶ rad/s, and the damping constant \gamma is 1.22 × 10¹⁴ rad/s. The thickness of metallic layer is 200 nm.

The substrate layer is made of polyimide (PI) and leads to the flexible characteristics of the proposed terahertz metasurfaces. The dielectric constant and loss tangent of PI are 2.93 and 0.05, respectively. The thickness of PI layer is 50 μm. The length of unit cell is P = 160 μm. The outer and inner radii of the RS are 75 μm and 65 μm. The SMS is separated into four arms wrapped in 0.5 turns. The four spiral metal arms have the same width (s) of 10 μm and are separated by a gap (g) of 10 μm. Figure 1(c) shows the optical microscopy image of the fabricated metasurfaces. The proposed mteasurfaces were fabricated using the laser direct writing process, thermal evaporation method and lift-off processes, as shown in S1 in the Supporting Information.

3 Results and discussion

Figure 1(d) illustrates the simulated transmission spectra of the isolated SMS, the isolated RS, and the proposed metasurfaces. The simulated results show that SMS and RS both demonstrate the typical radiation response. All simulations of the proposed terahertz metasurfaces in this work are calculated by a commercial solver (CST Microwave Studio) based on the finite integration technique (FIT). The SMS creates a narrow linewidth (red dotted line), which indicates the weak radiation of SMS under the excitation of the incident electromagnetic wave. The surface electric, magnetic field, and surface current distributions of SMS at 0.63 THz are shown in Figs. S3(a)−(c) (Fig. S3 in the Supporting Information). The near-field patterns indicate that the resonant response of SMS at 0.63 THz is created by the spoof electric localized surface plasmons (S-ELSPs) and has strong light confinement [44]. Due to the existence of the light confinement, the weak radiation and narrow linewidth of SMS can be obtained. Additionally, when the diameter of RS is close to the half of the resonance wavelength, it shows strong radiation with broad linewidth (blue dotted line). Therefore, an electric dipole resonance at 0.5 THz can be observed in the surface electric, magnetic field, and surface current distributions of RS, as shown in Figs. S3(d)−(f) (Fig. S3 in the Supporting Information). When two resonators have similar resonance frequencies but different transmission linewidths, an analogue of EIT effect can be achieved by combining these two types of resonators (SMS and RS).

Merging the SMS and RS into a unit cell, as depicted in Fig. 1(d), produces a prominent transparent window at 0.55 THz (green solid line), flanked by transmission dips at 0.46 THz and 0.68 THz. The transparent window in the spectral response of the proposed metasurfaces closely resembles the spectral characteristics of classical EIT effect, which arises from the hybridization of two bright mode resonators [45]. SMS and RS can be regarded as two different bright modes. The two transmission dips result from the resonances of each independent resonator, while the transparent window forms through the interaction between the SMS and RS.

To analyze the formation mechanism of EIT, we use the two-particle model to describe the spectral response of the proposed metasurfaces [46, 47]. SMS and RS represent particles 1 and 2 respectively, which are excited by the electric field E=E_0{\mathrm{e}}^{\mathrm{i}\omega t}. Two particles can be analytically expressed by the following equations [48]:

(2){\ddot{x}}_1\left(t\right)+{\gamma }_1{\dot{x}}_1\left(t\right)+{{\omega }_{01}}^2{x}_1\left(t\right)+{\kappa }^2{x}_2\left(t\right)={\displaystyle \frac{g_1{E}_0}{m_1}},

(3){\ddot{x}}_2\left(t\right)+{\gamma }_2{\dot{x}}_2\left(t\right)+{{\omega }_{02}}^2{x}_2\left(t\right)+{\kappa }^2{x}_1\left(t\right)={\displaystyle \frac{g_2{E}_0}{m_2}}.

(x_1,x_2), ({\omega }_{01},{\omega }_{02}), (m_1,m_2), and ({\gamma }_1,{\gamma }_2) represent the electric field amplitudes, resonance frequencies, masses, resistances of SMS and RS, respectively. (g_1, g_2) represent the coupling strength coefficients of SMS and RS under the irradiation of incident wave. \kappa refers to the coupling strength between SMS and RS. By expressing x_1 and x_2 under harmonic excitation E(t)=E(\omega ){\mathrm{e}}^{-\mathrm{i}\omega t}, we get the expressions of x_1 and x_2:

(4)x_1={\displaystyle \frac{\left({\displaystyle \frac{B}{A}}\right){\kappa }^2+\left({\omega }^2-{{\omega }_{02}}^2+\mathrm{i}\omega {\gamma }_2\right)\left({\displaystyle \frac{g_{1}E}{m_1}}\right)}{{\kappa }^4-({\omega }^2-{{\omega }_{01}}^2+\mathrm{i}\omega {\gamma }_1)({\omega }^2-{{\omega }_{02}}^2+\mathrm{i}\omega {\gamma }_2)}},

(5)x_2={\displaystyle \frac{\left[{\kappa }^2+\left({\displaystyle \frac{B}{A}}\right)\left({\omega }^2-{{\omega }_{01}}^2+\mathrm{i}\omega {\gamma }_1\right)\right]\left({\displaystyle \frac{g_{1}E}{m_1}}\right)}{{\kappa }^4-({\omega }^2-{{\omega }_{01}}^2+\mathrm{i}\omega {\gamma }_1)({\omega }^2-{{\omega }_{02}}^2+\mathrm{i}\omega {\gamma }_2)}}.

To simplify the calculation, g_1/g_2 and m_1/m_2 are defined as A and B, respectively. Then, the expression of effective polarization in the metasurfaces is P=g_1{x}_1+g_2{x}_2. Then, we can obtain the effective electric susceptibility:

(6)\begin{array}{rl}{x}_{eff}=& x_{ef{f}_{\mathrm{\_}r}}+i{x}_{ef{f}_{\mathrm{\_}i}}=\frac{P}{{\epsilon }_{0}E}\\ =& \frac{K}{A^{2}B}(\frac{A\left(B+1\right){\kappa }^2+A^2\left({\omega }^2-{{\omega }_{02}}^2\right)+B({\omega }^2-{{\omega }_{01}}^2)}{{\kappa }^4-({\omega }^2-{{\omega }_{01}}^2+\mathrm{i}\omega {\gamma }_1)({\omega }^2-{{\omega }_{02}}^2+\mathrm{i}\omega {\gamma }_2)}\\ & +\mathrm{i}\omega \frac{A^2{\gamma }_2+B{\gamma }_1}{{\kappa }^4-({\omega }^2-{{\omega }_{01}}^2+\mathrm{i}\omega {\gamma }_1)({\omega }^2-{{\omega }_{02}}^2+\mathrm{i}\omega {\gamma }_2)}).\end{array}

Here, {\chi }_{eff\mathrm{\_}r} and {\chi }_{eff\mathrm{\_}i} are the dispersion and the absorption of the metasurfaces. K is a proportionality factor. Finally, the transmission can be obtained:

(7)T=1-\mathrm{I}\mathrm{m}[{\chi }_{eff}].

In this work, the values of {\omega }_{01}, and {\omega }_{02} are 2π × 0.5 × 10¹² rad/s and 2π × 0.63 × 10¹² rad/s, respectively. The values of {\gamma }_1 and {\gamma }_2 can be obtained by calculating the linewidths of the simulated transmission spectrum of SMS and SS ({\gamma }_1=2\text{π}\times 0.1\times {10}^{12} rad/s, {\gamma }_2=2\text{π}\times 0.3\times {10}^{12} rad/s). The value of \kappa is calculated based on the \kappa =\sqrt{{\omega }_{02}^2-{\omega }^2} (or \kappa =\sqrt{{\omega }^2-{\omega }_{01}^2}). Substituting the transmission values for the metamaterial at{\omega }_{01}, {\omega }_{02}, and {\omega }_0 into Eq. (6) can lead to three equations concerning T({\omega }_0), T({\omega }_{01}), and T({\omega }_{02}), solving these equations relative to T({\omega }_0), T({\omega }_{01}), and T({\omega }_{02}), we can obtain the corresponding values of A=g_1/g_2 and B=m_1/m_2. By substituting the above values into Eq. (7), we can obtain the calculated transmission spectrum. According to the two-particle model, we draw the calculated transmission spectrum in Fig. 1(e). The simulated transmission spectrum matches well with the theoretical transmission spectrum based on the two-particle model.

Figure 1(f) shows the photo of fabricated metasurfaces sample and the experimental transmission spectrum of the sample. The experimental setup of the continuous-wave THz spectroscopy system (TeraScan 1550, TOPTICA Photonics AG) is used to measure the metasurfaces sample, as shown in Fig. S2 in the Supporting Information. The lower transmission amplitude at the EIT transparent window can be attributed to factors such as measurement instrument accuracy, sampling resolution, and manufacturing process errors. However, the most significant cause is the losses in the dielectric substrate and the metal structures. Although the amplitude of the experimental transmission spectrum is lower than that of the simulated spectrum, the results remain within an acceptable range and can still validate the correctness of the simulation results.

Figure 1(g) shows the transmission spectra of the metasurfaces within the polarization angle range from 0° to 90°. Due to the four-fold rotational symmetry of the metasurfaces, we only need to obtain the transmission spectra from 0° to 90°. The change of polarization angle has no influence on the transmission spectrum and the proposed metasurfaces exhibit polarization-insensitive characteristics. When the electric field direction is along the y-polarized direction, the transmission amplitude and bandwidth of the transparent window gradually decrease with the increase of incident angle, as shown in Fig. 1(h). When the incident angle is greater than 60°, EIT phenomenon disappears. However, the proposed metasurfaces show extremely robust to the incident angle under TM polarization. Although the change of incident angle has a great influence on the transmission amplitudes and bandwidths of two transmission dips, it has little influence on the formation of transparent window. When the incident angle increases from 0° to 80°, the transmission amplitude at transparent window (0.55 THz) hardly changes, as shown in Fig. 1(i).

To further explore the interaction between the SMS and RS in metasurfaces, the surface electric field, magnetic field, and the current distributions at two transmission dips and the transparent peak are discussed. For the first transmission dip at 0.46 THz, the typical electric dipole resonance can be observed on the surface electric field distributions and the surface current distributions, as shown in Figs. 2(a) and (c). The strong surface magnetic field distributions are mainly concentrated on the RS, as shown in Fig. 2(b). The RS is not excited at 0.46 THz, and therefore cannot couple with the SMS. As a result, the electric dipole resonance is solely excited in the RS, leading to radiation loss in the proposed metasurfaces. In contrast, contrary behavior can be observed at second transmission dip. The surface electric field, the surface magnetic field, and the surface current distributions are primarily localized on the surface of SMS, as shown in Figs. 2(g)−(i). This phenomenon indicates that a typical S-ELSPs mode is excited within the SMS, becoming the dominant excitation mode at 0.68 THz. At the transparent peak, the strong surface electric field, the surface magnetic field, and the surface current distributions are concentrated on the surfaces of SMS and RS, as shown in Figs. 2(d)−(f). Apparently, SMS and RS are excited by incident electromagnetic waves at the same time and the destructive interference between SMS and RS leads to the transparent window. Additionally, the two magnetic dipoles with opposite directions can be observed in Fig. 2(e) and the out-of-phase magnetic field is created by the near-field coupling between SMS and RS. The reversed magnetic moments suppress the localized magnetic field and weaken the coupling between the proposed metasurfaces and the incident electromagnetic wave, resulting in the formation of EIT effect.

The multipole scattering theory further provides valuable physics insights into the EIT effect of the proposed metasurfaces. By solving the current density of the proposed metasurfaces through CST software, scattered powers of the multipole moments can be calculated. The formulas of electric dipole \mathit{P}, magnetic dipole \mathit{M}, toroidal dipole \mathit{T}, and electric quadrupole Q_e can be calculated [49]. The contributions of multipole moments are calculated from the conducting current density \mathit{J}, which is extracted from the structural calculations. The formulas of electric dipole, magnetic dipole, toroidal dipole, and electric quadrupole are expressed as \mathit{P}=\frac{1}{\mathrm{i}\omega }\int \mathit{J}{\mathrm{d}}^{3}r, \mathit{M}=\frac{1}{2c}\int (\mathit{r}\times \mathit{J}){\mathrm{d}}^{3}r, \mathit{T}=\frac{1}{10c}\int [\left(\mathit{r}\cdot \mathit{J}\right)\mathit{r}-2r^2\mathit{J}]{\mathrm{d}}^{3}r, and Q_e=\frac{1}{\mathrm{i}2\omega }\int \left[r_{\alpha }{J}_{\beta }+r_{\beta }{J}_{\alpha }-\frac{2}{3}\left(\mathit{r}\cdot \mathit{J}\right){\delta }_{\alpha \beta }\right]{\mathrm{d}}^{3}r. Here, c is the speed of light in the vacuum, \mathit{J} is the conducting current density, and \mathit{r} is the displacement vector from the origin to point (x,y,z) in a Cartesian coordinate system (\alpha ,\beta =x,y,z). The far field scattered power for \mathit{P}, \mathit{M}, \mathit{T}, and Q_e are as I_P=\frac{2{\omega }^4{\left|\mathit{P}\right|}^2}{3c^3}, I_M=\frac{2{\omega }^4{\left|\mathit{M}\right|}^2}{3c^3}, I_T=\frac{2{\omega }^6{\left|\mathit{T}\right|}^2}{3c^5}, and I_Q^{(e)}=\frac{{\omega }^6\sum {\left|Q_{\alpha \beta }^{(e)}\right|}^2}{5c^5}.

Then, we can calculate the scattered powers of the multipole moments. We calculate the scattered powers of the multipole moments at 0.46 THz, 0.55 THz, and 0.68 THz. At the first transmission dip (0.46 THz), scattered powers of the multipole moments are shown in Fig. 2(j). The incident electromagnetic wave mainly excites the RS and produces the electric dipole moment, so the proportion of electric dipole moment is the largest. The distributions of electric dipole moment and magnetic dipole moment at 0.46 THz are shown in Fig. S4(a) (Fig. S4 in the Supporting Information). The counter-rotating circular currents formed on both sides of the RS create an electric dipole moment, while the circular currents on the left and right sides of the RS generate magnetic dipole moments oriented perpendicular to the metasurfaces, downward and upward respectively. The surface currents on both sides of the RS form two electric quadrupole moments at the head and tail positions, as shown in Fig. S4(d) (Fig. S4 in the Supporting Information). Although the two opposite magnetic dipole moments induce a toroidal dipole moment, the scattered power of the generated toroidal dipole is relatively weak. Subsequently, at the transparent window (0.55 THz), electric dipole moment increases rapidly, which is caused by the electric dipole generated by RS and the S-ELSPs mode generated by SMS. The proportion of electric dipole moment is much larger than that of the other three moments, as shown in Fig. 2(k). Compared to Fig. 2(j), scattered powers of magnetic dipole moment, toroidal dipole moment, and toroidal dipole moment are enhanced. The high scattered power of the electric dipole moment arises from the simultaneous generation of electric dipole moments by both the RS and SMS, as shown in Fig. S4(b) (Fig. S4 in the Supporting Information). The opposite surface currents induced on both sides of the RS generate two opposite magnetic dipole moments, which collectively give rise to a toroidal dipole moment. Furthermore, besides the electric quadrupole moments generated at the head and tail regions of the surface currents along the RS, opposing surface currents emerging between the metallic arms of the SMS likewise produce electric quadrupole moments, as shown in Fig. S4(e) (Fig. S4 in the Supporting Information). At the second transmission dip (0.68 THz), surface currents exist predominantly on the SMS with negligible intensity on the RS. This leads to a diminished scattering power of the electric dipole moment and a weakening of the toroidal dipole moment due to the absence of surface currents on the RS, as illustrated in Fig. 2(l). Meanwhile, the circular surface currents on the SMS generate an electric dipole moment while simultaneously forming a magnetic dipole moment, as shown in Fig. S4(c) (Fig. S4 in the Supporting Information). Similarly, electric quadrupole moments are generated between the counter-rotating circular currents on the metallic arms of the SMS, as illustrated in Fig. S4(f) (Fig. S4 in the Supporting Information). The distributions of these multipoles in Fig. S4 are consistent with the field distributions in Fig. 2, and the difference is that Fig. S4 is displayed in a quantitative method and three-dimensional representation.

The preceding analysis examines the impacts of six critical parameters on EIT transmission, as shown in Fig. S5 in the Supporting Information. As demonstrated, modifications to three RS parameters (s, g, r) exhibit the most pronounced influence on EIT characteristics, with parameter s demonstrating particular sensitivity. This phenomenon is fundamentally attributed to the fact that minute adjustments in RS parameters induce localized surface plasmon (LSP) variations, thereby altering the coupling dynamics between the RS and SMS. Consequently, optimizing RS parameters and meticulously analyzing the coupling bewteen RS and SMS constitute crucial considerations in practical device design. Furthermore, the observed phenomena reveal that broadband EIT effect can be effectively achieved through strategic RS parameter tuning, with particularly significant enhancement potential.

Schematic diagram of polarization angle φ rotation is described in Fig. 3(a). The experimental device for testing polarization-insensitive characteristics is shown in Fig. 3(d). By varying the polarization angle, we measured the experimental results of the proposed metasurfaces at polarization angles of 0° and 90°, as shown in Fig. 3(g). The experimental results demonstrate a close match with the simulations, and the transmission spectra of the metasurfaces remains unchanged even when the polarization angle is varied. Figure 3(e) shows the experimental device for testing oblique incidence under TE polarization. We plotted the schematic diagram of the metasurfaces testing setup under TE polarization [Fig. 3(b)] and measured the transmission spectra by rotating the sample through angles from 0 to 90°, as shown in Fig. 3(h). The experimental results demonstrate that the metasurfaces can maintain unchanged transmission spectra under oblique incidence angles ranging from 0 to 60° under TE polarization, showing good agreement with the corresponding simulation results. Figure 3(f) illustrates the experimental setup for the metasurfaces under oblique incidence with TM polarization. We measured the transmission spectra of the metasurfaces at incidence angles ranging from 0° to 90° [schematic diagram in Fig. 3(c)], and the experimental results are shown in Fig. 3(i). The experimental results demonstrate that the metasurfaces can maintain its transmission spectrum unchanged even at oblique incidence angles up to 80°, showcasing exceptional angle-insensitive performance. We simulated the electric and magnetic field distributions of the proposed terahertz metasurface under oblique incidence using simulation software, as shown in Fig. S6 of the Supporting Information. The electric and magnetic field distributions indicate that the SMS can still be effectively excited by the RS under oblique incidence. Compared to other planar EIT terahertz metasurfaces, the distinctive feature of our proposed metasurfaces lies in the use of S-ELSPs as a coupling mode for EIT. Its localized field enhancement and tailored dispersion characteristics enable the metasurfaces to maintain strong coupling even under oblique incidence. This capability is crucial for advancing research and development in applications, such as ultra-high-resolution imaging, next-generation wireless communications, and compact terahertz sensing systems.

To assess the mechanical flexibility of the proposed metasurfaces, we measured the transmission spectra under different bending degrees. Since the proposed metasurfaces feature a flexible structure composed of PI and AI, we performed bending operations on the original metasurfaces. In this work, the bending degree of the metamaterial was quantitatively analyzed by calculating the central angle formed after bending. When no bending operation is performed on the metasurfaces, the central angle is 0. When the square sheet is bent, one edge remains straight (fixed), while the opposite edge curves into an arc. By calculating the central angle after bending, we qualitatively evaluated the bending degree of the metasurface. The detailed calculation process of bending degree is described in Fig. S7 in the Supporting Information.

Therefore, during testing, we only need to measure the chord length of the bent metasurfaces to calculate its central angle. The metasurfaces sample under test is fixed on a metal plate with through-holes, and its schematic is shown in Fig. 4(a). Here, we considered two different bending conditions: vertical and horizontal bending conditions. Figure 4(b) depicts the experimental setup for testing the bending of the metasurfaces sample under vertical and horizontal bending conditions. The relationship between the bending angle (central angle) and chord length of the proposed metasurfaces is described in Fig. 4(c). By controlling the distance (denoted as d) between the two sides of the sample, we induced bending deformation. The value of d is inversely proportional to the bending curvature of the sample, and the central angle formed by bending the sample is θ. Without applied bending force (i.e., in the flat state), d equals 20 mm (θ = 0°). As the bending force is gradually applied, d decreases, resulting in metasurfaces curvature. The experimental results in Fig. 4(d) reveal that the transmission spectrum remains stable as the central angle of the sample bending increases from 0° to 111.7° (i.e., d decreases from 20 mm to 17 mm). Under horizontal bending condition, the same phenomenon can be found. In order to better observe the influence of sample bending condition on the transmission spectrum, we draw the transmission spectra of the sample at angles of 0°, 63°, 90°, and 111.7°. As can be seen from Fig. 4(e), the bending of the sample has no influence on the frequency and transmission amplitude of the transmission spectra. The above phenomenon demonstrates the excellent bending resistance of our proposed metasurfaces. Such performance holds significant implications for research in flexible optoelectronic devices and wearable technology, while showcasing great potential in adaptive aerospace structures and mechanically reconfigurable sensors for real-time strain monitoring.

A comprehensive comparison of polarization-insensitive characteristics and angular stability among state-of-the-art terahertz metasurfaces is systematically summarized in Table 1. Notably, our proposed metasurfaces achieve stable transmission window preservation under oblique incidence up to 80° while maintaining polarization-independent characteristics, surpassing existing designs in angular tolerance. More importantly, the proposed terahertz metasurfaces can maintain the working bandwidth under oblique incidence. This exceptional angular robustness is attributed to the tightly confined LSPs enabled by the SMS, which effectively suppresses phase distortion under extreme illumination conditions. This work opens new avenues for developing ultra-reliable terahertz devices in scenarios demanding both wide-angle operation and polarization diversity. Furthermore, the integration of such angle-immune metasurfaces with flexible substrates could catalyze breakthroughs in conformal wearable terahertz systems for biomedical diagnostics and human-machine interfaces.

4 Conclusion

In conclusion, flexible terahertz metasurfaces with polarization-insensitive and oblique incidence characteristics are proposed. By combing S-LSPs and EIT effect, we designed the metasurfaces composed of SMS and RS. The surface electric field, magnetic field, and current distributions of the proposed metasurfaces collectively show that EIT effect is created by the destructive interference between SMS and RS. The multipole scattering theory further proves that EIT is caused by the electric dipole generated by RS and the S-ELSPs mode generated by SMS. The simulated transmission spectrum is in good agreement with the theoretical transmission spectrum based on the two-particle model. Experimental results show that the transmission spectra of the metasurfaces remain unchanged at polarization angles of 0° and 90°. Furthermore, the experimental results demonstrate that the metasurfaces can keep the transmission spectrum unchanged even at oblique incidence angles up to 80°, showing excellent performance of angle insensitivity. We measured the transmission spectra under different bending degrees and experimental results show the proposed metasurfaces with an excellent bending resistance. Our work offers a novel design strategy for flexible and oblique incident metasurfaces, showing applications in conformal terahertz systems and adaptive wearable photonics.

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