Curved platforms with arbitrary packaging and shape are widely used in practice, especially in fields in which aerodynamics and aesthetics are a major concern, such as in high-speed aircraft, missiles, and vehicles. Moreover, multifaced or curved structures are also highly preferable and utilized in stealth applications due to their low bistatic scattering. Therefore, in the past two decades, many studies have investigated conformal techniques and array antennas ^[1,2]. Among them, studies pertaining to conformal design for improving the scattering of a curved object are rarely seen because most studies have focused on radar cross section (RCS) reduction and isotropic scattering ^[3]. Moreover, to some extent, a superscatterer with a flexible beam number and beam steering on a curved structure may be crucial to enhance the survivability, rescue, or navigation safety of a friendly high-speed target in remote areas because of their strong return signal intensity ^[4]. Therein, we also hope that the detected signal can be safely obtained for emitting waves of any polarization.

Metasurfaces, comprising a set of planar subwavelength meta-atoms ^[5–44], have recently attracted considerable interest among science and engineering researchers for manipulating electromagnetic (EM) waves. With the use of abrupt phase discontinuities, thick materials are not required for accumulating the required phase. Thus, super-thin, flexible, low-loss, easily fabricated metasurfaces can be utilized. These features greatly aid in the real-life applications of metasurfaces. Despite valuable progress toward planar design, some studies have focused on an arbitrary platform with a large curvature ^[37–40]. To circumvent this problem, an ultra-thin acoustic cloak for a complex random structure was realized using a large phase gradient ^[41]. Nevertheless, the available conformal designs have a narrow bandwidth and rigorous polarization restriction, which hinders their real-life applicability.

Herein, to address the aforementioned open issues in the literature, we introduce a sophisticated strategy inspired by the ray-tracing technique. In contrast to the strategy used in Refs. ^[37–40], the Pancharatnam–Berry (PB) phase ^[29–33] was utilized for the first time to realize the desired broadband and polarization-independent high-efficiency superscattering behavior. Such engineered functionality differs from the ultra-thin invisibility cloak by restoring normal reflections ^[37,41], backscatter ^[38], diffusion ^[39], and highly directive emissions ^[40]. In our scheme, the measurface’s functionality is polarization insensitive because any linearly polarized (LP) wave can be decoupled as a linear combination of two circularly polarized (CP) waves of different handedness. For demonstration, the metasurface is designed to wrap over a metallic cylinder to realize a simple design. Two inverse linear-phase gradients are symmetrically formed about the axes of the metallic cylinder to yield a pair of symmetric highly directive beams in two opposite directions. The key to achieving high-efficiency superscattering is that the phase error induced by the curvature should be compensated and minimized point by point so that planar uniform wavefronts are restored in the predicted directions.

The basic principle behind the dual-beam superscatterer in planar geometry lies in generalized Snell’s law ^[5]. However, for a curved platform in this study [see the conceptual topology and function illustrated in Fig. 1(a)], the situation is much complex, and most efforts should be focused on the nonlinear-phase design, which completely extends the scope of anomalous reflections. Two highly directive scattering beams are obviously inspected across the horizontal plane when our designed metasurface is wrapped over a cylinder. However, as displayed in Fig. 1(b), a doughnut-like scattering pattern with almost homogeneous intensity is expected to lie across half the circumference of the azimuth ϕ (illumination region) for the bare cylinder. The super-thin metasurface modulates the wavefronts of the metallic cylinder, thus facilitating the recollection of the homogeneous scattering energy and the concentration of the energy into two arbitrary angles. To modulate the EM wave without distortion on arbitrary platforms in a reflection scheme, we first project the flexible surface on a common reference plane by calculating the optical path difference φC(i) at each point i through ray optics approximation. Subsequently, we obtain the required phase distribution φ0(i) on the basis of the desired plane functionality, as shown in Fig. 1(c). The physics can be also understood as that φ0(i) determines the functionality, whereas φC(i) modifies the phase error induced in nonplanar packaging.

For convenience and to obtain a simplistic design, the inversely symmetric phase gradients are designed only along the circumference, and those along the axial and radial directions are left constant. To obtain the required phase profile and minimize amplitude distortions (maximum efficiency) of edged meta-atoms at large incidence angles, the following discussion is presented. The position of the ith element starting from a specific point (the original point in this study) should satisfy P(i)=R0arcsin[(i−1)p/R0]. Moreover, the initial phases of the elements were progressively increased with φ0(i+1)−φ0(i)=90°, i=1,2,…,n. In these equations, p is the lattice constant of the meta-atoms, R0 is the radius of the cylinder, and n is the number of elements utilized for shaping each high-efficiency beam. Therein, the meta-atom interval becomes larger and larger as it departs from the center to the edge, so that the performance of the resulting devices is dominated by the centered meta-atoms. However, each meta-atom is still periodically arranged when projected to the reference plane. Then, the imparted phase utilized to correct the distortions that are induced by the curved shape at f0 is determined by considering the optical path difference: Therefore, the phase shift imposed by the metasurface should be φ(i)=φ0(i)+φC(i). By using this design procedure, the arbitrary function can be exactly realized by the flexible conformal metasurface.

Now that the physics and strategy are clear, the next step focuses on the design of meta-atoms to map the determined phase distribution. To achieve a broad operation bandwidth, a rotational metallic Jerusalem cross with a PB phase is utilized, as shown in Fig. 2(a). The meta-atom is composed of a dielectric layer, an air spacer, and a backed metallic ground. The Jerusalem cross is etched on top of the dielectric layer with a dielectric constant of ϵr=3.4 and thickness of 0.06 mm. Moreover, the air layer is utilized to enhance the high-efficiency bandwidth. Note that the air layer is not necessary if a trade-off is considered between the mechanical strength and performance. The criterion φyy−φxx=180° is utilized to guarantee near-unity CP conversion efficiency by suppressing any component that does not carry PB phase information ^[29–31]. Here, we employ the dispersion engineering method ^[42] to enable the slopes of φyy and φxx to be approximately equal within a broad bandwidth by modulating the length of three orthogonal bars of both “H” structures. As shown in Fig. 2(b), after cautious optimizations, we achieved φyy−φxx=180°±45° in the range of 8.3–23.5 GHz, thus corresponding to a fractional bandwidth of 95.6%. Moreover, the reflection magnitude remains almost unity across the entire band, which is necessary for a high-efficiency superscatterer. As displayed in Fig. 2(c), the CP copolarization conversion efficiency is more than 93.5% in the range of 8.17–22.1 GHz, where the cross-polarization component (less than 0.25) is almost suppressed.

Finally, the EM response under oblique illumination is displayed in Fig. 2(d). We can see that the efficiency and bandwidth (high-edge frequency shifted downward) decrease when the incidence increases from 0° to 45° at intervals of 15°. Nevertheless, a desirable efficiency of more than 87.7% is achieved when θ approaches 30°, and the efficiency remains higher than 70.5% even for θ=45°. These features are critical for conformal design on arbitrary platforms with moderate curvature. In the above cases, the effect of contrast for different incidences on performance is negligible. However, the meta-atoms with large curvature are typically illuminated obliquely with a larger incidence of θ>45°. In this case, the curvature affects the final performance of the resulting device through distortion of the magnitude, thus decreasing the efficiency. However, this decrease in efficiency can be rectified through amplitude compensation, which is beyond the scope of this work. Note that PB phases are typically affected by the angle of incidence ^[43]. However, it should be emphasized that the PB phases in this work are not significantly affected by the oblique incidence and are almost the same with those at normal incidence. The condition of broadband φyy−φxx=180° equals that for overcoming the angular limitation of a helicity-preserving mirror ^[43]. Therefore, in theory, our method using phase correction enables us to restore the EM wavefront of any curved platform with arbitrary curvature provided that the utilized meta-atom can work with high efficiency at a large incidence.

By progressively rotating the meta-atoms through an appropriate ϕ, we can achieve a PB phase of φ=2ϕ under CP wave excitation. As a consequence, the imposed correctional phase calculated using Eq. (1) can be realized by rotating the aforementioned meta-atoms with an appropriate ϕ at predicted positions. Then, the superscatterer can be realized by wrapping the flexible conformal metasurface over a conducting cylinder.

To prove the principle, we first employ a total of 21 dipoles that were mounted on the cylindrical surface with R0=99.06mm to mimic a conformal metasurface, as shown in Fig. 3(a). Each dipole is spaced with progressively enlarged distance along the circumference and is fed with the same amplitude but specific phase determined by using the aforementioned procedure at 12 GHz. In such a case, the effect of scattering amplitude loss (distortion) induced by the curvature can be avoided. Because of the shielding effect, a half-circle of all 21 dipoles is necessary. Different from the scattering scenario, we require φ(i)=φ0(i)+φC(i)/2 to shape the radiation wave to the desired directions because no illumination phase error is induced. For a comprehensive study, three dipole arrays are considered on the same curved platform, and the corresponding phases are displayed in Fig. 3(b). The first dipole array (case 1) is designed with two beams directed toward ϕ=±45°. The phase tolerance at f0 is compensated and minimized point by point on the basis of the following equation:

The other two dipole arrays (cases 2 and 3) contain the aforementioned linear gradient phases [φ0(i+1)−φ0(i)=90°]. However, in case 3, the linear phases are superposed with the compensation phase for the curvature.

As expected in Figs. 3(c) and 3(d), highly desirable radiation patterns are clearly observed in the range of 6–18 GHz, i.e., two symmetric radiation beams with enhanced intensity directed toward ϕ=±45° in case 1, whereas two high-efficiency anomalous beams steered at ϕ=±arcsin(λ/Np) with specular radiations almost suppressed in case 2. The highly concentrated beam was steered from 69.6° at 8 GHz to 24.6° at 18 GHz. The slightly enhanced specular radiation and the distorted beam angle at low frequencies are attributable to the reduced electrical array aperture, and the ray-tracing approximation introduces some tolerances. Nevertheless, this does affect the verification of our compensation strategy. This declaration finds strong support from the almost completely deteriorative radiations (emanative radiations with dispersive intensity) in case 3; see Fig. 3(e). A clear beam cannot be observed; instead, the power intensity is discretized toward numerous directions without any phase correction, thus illustrating the effectiveness of our approach.

We now further evaluated the performance of our conformal metasurfaces by replacing the dipoles with real PB meta-atoms. For a comprehensive study, two metallic cylindrical platforms were utilized with R0=95 and R0=156mm, in which a total of 21 and 33 meta-atoms were accommodated, respectively. The inhomogeneously arranged position and rotated angle of each meta-atom along the circumference were determined using the aforementioned strategy, as shown in Figs. 4(a) and 4(b). Figure 4(c) depicts the numerically calculated full-wave scattering intensity. As expected, two symmetric beams with almost equal intensity and completely suppressed specular scattering are clearly observed at all inspected frequencies within a broad bandwidth. This is very desirable for highly directive scattering enhancement. Moreover, perfect high-efficiency scattering occurs near the target frequency f0 and shifts as f0 varies from 18 to 15 GHz. This is quite physical because the required phase corresponding to the optical path difference is dispersive (frequency dependent), whereas the utilized correctional phase is constant. The phase distortion accounts for the enhanced normal scattering at frequencies substantially different from f0. Moreover, the beam angle of our superscatterer decreases with the frequency. This finding deviates from the conclusion drawn from theory and planar metasurface of the same projected size. This deviation can be attributed to the reflection magnitude loss of each meta-atom induced by the curvature. Such a deviation is substantial at low frequencies because the metasurface becomes electrically large, and thus, the reflection magnitude uniformity of each meta-atom is worsened. This finding is especially true for the edged meta-atoms, for which the incidence angle of the impinging wave is relatively large. However, the curvature effect becomes weak as the frequency increases because the curved metasurface can be approximated as a planar plate. This proposal is verified by analyzing the far-field patterns of the dipole arrays and planar metasurface displayed in Fig. 3 and Fig. 4(c), respectively, where the magnitude of each meta-atom or dipole is near unity. Therein, in both cases, the reflection angles are in good agreement with the theoretical ones predicted by ϕ=arcsin(λ/Np).

For verification, we fabricated a proof-of-concept sample mounted on a cylinder with R0=95 and h=250mm. Because the available CP horns in the laboratory operate within 6–18 GHz, we designed another metasurface target at that frequency band, as shown in Figs. 5 and 6. As can be seen, the scattering behavior was similar to that observed in Fig. 4. Again, two high-efficiency scattering patterns were clearly observed. Such behavior sharply contrasts with that of the bare cylinder, where the scattering is almost homogeneous across the circumference (flat curve in Fig. 6). The total scattering by the metasurface-coated cylinders is very close to that of the flat conducting plate with the same projected size. The operation band with a precise steering angle shifts as f0 shifts for a conformal metasurface. However, almost identical beam direction is achieved in the theoretical and numerical calculations for the planar metasurface. Moreover, the beam is narrowed when R0 increases from 95 to 156 mm due to the contribution of the array factor. For the bare cylinder, the specular scattering enhances progressively as the frequency increases and finally dominates the energy across the azimuth like a planar metallic plate. This occurs because a curved surface can be approximated as a plane at high frequencies.

Figure 7(a) displays a photograph of the fabricated flexible sample with a planar area of L·h=π·95·250mm2. In the fabrication, we first attached a double-faced cardboard with ϵr=1 and thickness of 4 mm on the metallic cylinder. Subsequently, the super-thin film was fixed to the cardboard. The sample was then characterized in a microwave anechoic chamber by using an angle-resolved bistatic scattering experimental setup; also see Fig. 7(a). Two CP horns that exhibited a voltage standing wave ratio of less than 2.5 and axial ratio of less than 3.5 dB in the range of 6–18 GHz were adopted as the emitter and receiver. These horns were placed 0.8 m away from the sample and were connected to an Agilent N5230 C vector network analyzer through two 13-m-long cables. The emitting horn and metasurface were fixed on a large foam that could be freely rotated along its axial center. Moreover, the receiving horn was aligned with the metasurface and recorded the signals that were scattered in the range of −90°<ϕr<90°.

As shown in Fig. 7(b), very well-defined dual beams can be clearly observed in the range of 8–18 GHz. However, the cross-polarized scattering normalized to the copolarized scattering is negligible within 8–17 GHz; see Fig. 7(c). Thus, high conversion efficiency is obtained because φyy−φxx=180° is maintained. Therefore, we obtain a high efficiency of approximately 82% (90.3%) in measurement (simulation) across the range of 8–18 GHz, as shown in Fig. 7(d). In contrast to a bare cylinder, for which the reflection is approximately averaged across the azimuth, the highly concentrated scattering is very desirable for a metasurface-covered cylinder, as shown in Fig. 7(e). The slightly reversed beam-angle variation trend at low- and upper-edge frequencies between measurements in Fig. 7(b) and theory in Fig. 3 is the same as that discussed between simulations in Fig. 4, Fig. 5, and theory. Moreover, the measured scattering beamwidth is slightly larger than that observed in theory and simulations and is due to the nonideal plane wavefront of the transmitting horn.

The high-efficiency strong scattering can be further verified by analyzing Fig. 8. The slightly deteriorative patterns in experiments, in particular the asymmetric scattering intensity of the dual beams and undesirable scattering at other angles, can be attributed to the incoherence and misalignment of the flexible film and cylinder in the assembly. The slight deviation in the anomalous reflection angles obtained theoretically and experimentally is due to the magnitude distortion of edged meta-atoms, as previously discussed. In addition to the phase compensation, the simultaneous independent magnitude compensation of each meta-atom would overcome the distorted reflection angle. It should be emphasized that although an accurate conformal design is target at normal incidence and the wave reflection direction is also not flexible, the versatile complex wavefront control (random incidence and beam direction in practice) can be engineered by using smart sensors and tunable techniques based on the solid fundamentals established in this study.

To sum up, we have proposed and verified a general method to design flexible high-efficiency superscatterers that can exhibit the desired broadband performance as exhibited by their planar counterparts. The geometrical form of the superscatterer can be decoupled from its EM function, enabling the metasurface to be applied to nonplanar platforms and stringent packaging. For verification, a set of conformal metasurfaces were designed at microwave frequency by modulating the nondispersive PB phase. A dual planar wave with minimized wavefront error was recovered at two predicted angles because a perfect phase profile was restored at each point of the curved surfaces by using the ray-tracing technique. Theoretical, numerical, and experimental results coincide well, indicating that the measured (simulated) efficiency of anomalous reflection can approach more than 82% (90.3%) in the range of 8–18 GHz. Our strategy, low profile and flexible, affords a possible avenue to generate EM superscattering for any curved surface with a versatile beam number and beam angle without requiring polarization considerations.