Electromagnetic metamaterials have experienced rapid developments due to the unique, fine adjustable properties that may not exist in nature. Metamaterials are artificial structures that have been widely used to manipulate electromagnetic waves and led to many novel devices [1–5]. Metasurfaces (MSs) are the two-dimensional (2D) equivalence of metamaterials, which can provide phase, amplitude, and polarization manipulation of waves [6–10]. They have advantages of thin thickness, low cost, and easy fabrication and have wide application prospects such as filtering, wave absorbance, polarization control, and beamforming [11–15]. Due to the 2D character, conformal metasurfaces meet the requirement of mounting on a curved surface and attract a steadily growing interest from researchers. They can be applied on an antenna array, airborne radome, and wearable devices to reduce RCS, decrease size, or improve integration [16–19].

To design conformal metasurfaces with good performance, theoretical analysis is important. For plane metasurfaces under normally incident waves, there are many methods to analyze the electromagnetic character such as characteristic mode [20,21], filtering theory [22,23], and equivalent circuit model [24,25]. For plane metasurfaces under obliquely incident waves, several researches explain the angular dispersion of the metasurface element [26–29]. The conformal metasurface is in a more complex situation because the incident angle of each element of the metasurface is different. Thus, Floquet model analysis cannot be used to simplify the simulation of the conformal MS. For non-planar quasi-periodic conformal MS structures, it is necessary to simulate the entire conformal MS to obtain the electromagnetic performance. But the full wave simulation is not convenient and quick for large scale or many designs. Although a wide angle conformal metasurface with a high angle stability element is proposed recently [30–33], the electromagnetic performance analysis of conformal metasurfaces is inadequate. In addition, the various couplings between each element make the analysis of conformal metasurfaces more difficult. Therefore, the analysis of conformal MS under different radii of curvature and polarization states is needed. Also, to distinguish between plane metasurfaces under various oblique incidence angles and conformal metasurfaces is demanded.

Because a conformal metasurface is required to assemble on curved surfaces in many cases, the mechanical performance is important. References [34,35] designed metasurfaces with stretchable electronic functionality. A geometric mapping is proposed in Ref. [36] relating mechanically relevant origami substrate properties and electromagnetically relevant conductive element properties. But there is still a lack of mechanical performance results to evaluate the principal strain of conformal metasurfaces. In addition, the metasurface element that can withstand greater stress is demanded to avoid cracking.

Therefore, a novel design of the circular-protrusion-jigsaw-shaped (CPJS) MS is presented inspired by the self-locking structure in the jigsaw puzzle game. In addition, a square-protrusion-jigsaw-shaped (SPJS) MS is proposed for stronger mechanical performance, coming from the idea of ancient Chinese wooden architecture’s mortise and tenon joint. First, a simple equivalent circuit model of planar jigsaw-shaped (JS) MS is proposed and validated by comparing the results with simulation results. The angular and polarization stability of two kinds of JS-MSs is compared with that of the conventional square-grid MS (SG-MS) to analyze the protrusion structure’s effect on the pass band. Second, the electromagnetic and mechanical properties of two conformal JS-MSs to SG-MSs with different curvature radii are investigated.

The electromagnetic property of the conformal MS mounted on a foam cylinder is simulated by HFSS. The conformal stability of the three MSs is compared with the angular stability of the infinite planar one. The reason of electromagnetic character difference under various incident angles, curvature radii and TE/TM mode is given in detail. The mechanical property is obtained by mechanical simulation software called Abaqus, so that the principal strain distribution of MSs can be generated and the mechanical character can be discussed and compared clearly. In addition, the effect of the curvature radii of conformal MS on strain distribution is investigated. Finally, three kinds of S-band MS samples are fabricated and measured to demonstrate the accuracy and effectiveness of the proposed MS design. The analysis method is also suitable for other metasurfaces and valuable to get more comprehension of electromagnetic and mechanical properties under conformal conditions.

In this section, CPJS-MS and SPJS-MS are introduced. The MS structure comprises two metal layers separated by a dielectric substrate. Similar to the self-locking jigsaw puzzle, as shown in Fig. 1(a), the wire grid with CP is designed on the upper layer, as shown in Fig. 1(b). Further, the square patches are placed on the bottom layer. For each unit element, one square patch corresponds to four wire grids, while patches mainly contribute to the capacitance and grids contribute to the inductance.

The mortise and tenon joint of Chinese wooden architecture in Fig. 1(c) is a clever combination of wood components that can effectively limit the twisting of wood pieces. In Fig. 1(d), the square grid with square-protrusion is applied to enhance the mechanical property of MS to mount the curved surface.

Figure 2 depicts the structure details of CPJS MS with the dimensions as follows: Lg1=35, Lp1=22.4, Dg1=4.15, R=3.5, w1=0.5, Dr=2.75 (unit: mm). The dielectric substrate, Rogers5880, thickness h is 0.508 mm with a relative dielectric constant (εr) of 2.2 and loss tangent of 0.0009. Figure 3 depicts the SPJS MS structure of the dimensions as follows: Lg2=35, Lp2=22.4, Dg2=5, R=6.5, w2=0.5, Dr=6 (unit: mm).

To efficiently evaluate the transmission performance of JS-MS, a simple equivalent circuit model A is proposed, as shown in Fig. 4(a). The wire grid on the upper layer is modeled as a shunt inductor, L1, whereas the square patches on the bottom layer can be modeled as a shunt combination of a capacitor, C2, in series with an inductor, L2. The dielectric substrate parameters εr and h can be represented by a transmission line of length h and characteristic impedance, Z1=Z0/εr. The half-space outside the substrate can be considered as a semi-infinite transmission line with wave impedance of free space, Z0=377Ω.

According to the telegrapher equation [37], the short transmission line of length h in model A can be represented by a π-type LC network composed of a series inductor, LT, and two parallel capacitors CT1 and CT2 denoted as equivalent circuit model B, as shown in Fig. 4(b). To simplify the equivalent circuit further, the capacitors CT1 and CT2 can be ignored because the effect of the capacitive reactance provided by the capacitors CT1 and CT2 is much smaller than that of the inductive reactance of the inductor LT [38]. Considering the coupling effect between different layers, the equivalent circuit model C is given in Fig. 4(c). The shunt capacitor C1 represents the coupling effect between the relatively prominent parts of the grid, while Cs is a series capacitor representing coupling between the grid and the patch. The LC values of JS-MS are extracted by Q3D [39,40] and listed in Table 1.Table 1.

Parameters of Equivalent Circuit

The transmission matrix, ABCD, cascade method is used to calculate the center frequency of the passband. The ABCD matrices of L1C1, LTCs, and L2C2 are cascaded in this order to get the total matrix. The ABCD parameters of each part can be expressed as follows: [A1B1C1D1]=[10(1−ω2L1C1)/jωL11],(1)[A2B2C2D2]=[1jωLT/(1−ω2LTCs)01],(2)[A3B3C3D3]=[10jωC2/(1−ω2L2C2)1].(3)

By a Laplace transform, the transmission coefficient S21 of model C can be obtained as follows: |S21|=|2A+B/Z0+CZ0+D|,(4)where $$\begin{gathered} [ABCD]=[A1B1C1D1][A2B2C2D2][A3B3C3D3]=[1+Y2Y31Y3Y1+Y2+Y1Y2Y31+Y1Y3], \\ Y1=1+s2L1C1sL1,Y2=sC21+s2L2C2,Y3=1+s2LTCssLT, \end{gathered}$$where s=jω.

The center frequency of the passband can be obtained as |S21| reaches its maximum, which is calculated from Table 1 and found to be f0=2.40GHz. To evaluate the accuracy of the proposed model C equivalent circuit, the transmission coefficient is simulated by HFSS and ADS, as shown in Fig. 5, and it can be seen that the simulated results obtained by HFSS agree quite well with those from ADS. Therefore, the equivalent circuit model C can initially be used to design JS-MS.

The angular stability of the planar JS-MS for TE/TM mode is investigated. The sketch of the normal/oblique incidence to planar MS in Figs. 6 and 7 shows the transmission performance of the infinite planar CPJS-MS.

For the TE mode, it can be seen from Fig. 7(a) that the passband bandwidth decreases by 23.8% from 1.51 to 1.15 GHz as the incidence angle increases. As the wave impedance of free space varies as Z0/cos(θ) for TE mode, the difference between character impedance of MS and wave impedance becomes larger. So the passband bandwidth narrows by increasing the incidence angle θ. The passband bandwidth is inversely proportional to the incidence angle. For the TM mode in Fig. 7(b), the bandwidth increased from 1.47 to 1.88 GHz by 27.9% as the incidence angle increased. The wave impedance of free space varies as Z0cos(θ), the quality factor decreases, and the passband bandwidth becomes wider as the incidence angle increases.

The transmission performance of the infinite planar SPJS-MS is provided in Fig. 8. It can be seen that as the incidence angle increases, the passband bandwidth decreases by 21.2% from 1.51 to 1.19 GHz for TE mode while the passband bandwidth increases from 1.50 to 1.90 GHz by 26.7% for TM mode; the passband bandwidth of CPJS-MS and SPJS-MS varies obviously while the center frequency does not change much; and SPJS-MS has better angular stability.

For comparison purposes, the square-grid MS (SG-MS) with similar bandpass performance is designed, as shown in Fig. 9 with dimensions as follows: Lg3=54, Lp3=34.6, w3=0.5 (unit: mm).

The length of the SG-MS unit element (Lg3) is 54 mm, which is 1.5 times longer than that of CPJS-MS (Lg1). We compare the unit element of CPJS-MS, SPJS-MS, and SG-MS in Fig. 10. The reflection and transmission performance of three MS structures under normal incidence angles is shown in Fig. 11.

It can be seen that these three MSs have similar center frequencies and bandwidths of passbands under normal incidence. In addition, we compare the transmission coefficient of these three MS structures under different oblique incidence angles, as shown in Fig. 12.

Figure 12(a) shows that in the lower frequency range (0.3–2.4 GHz), the transmission performance of SG-MS is in good agreement with that of CPJS-MS under different incidence angles for TE mode. In the higher frequency range (2.4–3.6 GHz), the bandwidth of SG-MS becomes narrower than that of CPJS-MS as the incidence angle increases. Figure 12(b) shows that in the lower frequency range (0.3–2.4 GHz), the transmission performance of SG-MS is in good agreement with that of CPJS-MS under different incidence angles for TM mode. In the higher frequency range (2.4–3.6 GHz), the passband bandwidth of SG-MS increases, but the increase is smaller than that of CPJS-MS. Figure 12(c) shows that the passband bandwidths of SG-MS and SPJS-MS become narrower as the incidence angle increases, and the passband bandwidth of SPJS-MS is narrower than that of CPJS-MS. Figure 12(d) shows that the passband bandwidths of SG-MS and SPJS-MS become wider as the incidence angle increases, and the passband bandwidth of SPJS-MS is wider than that of CPJS-MS.

The length of the SG-MS unit element is considerably longer than that of CPJS-MS and SPJS-MS in Fig. 10; the angular stability of SG-MS is the worst, especially for the high frequency range. Detailed simulated results for TE and TM modes are listed in Table 2. It can be observed that the passband bandwidth of CPJS-MS, SPJS-MS, and SG-MS decreases by 23.8%, 21.2%, and 21.6% as incidence angle increases from 0° to 40° for TE mode, respectively. The increase in passband bandwidth of these three MSs is 27.9%, 26.7%, and 27.6% for TM mode. The bandpass performance of the CPJS-MS and SPJS-MS is more stable than that of SG-MS. SPJS-MS has the best angular stability for TE and TM modes because of the smallest bandwidth variation.Table 2.

Comparison of Transmission Coefficient of Infinite Planar MS

This section investigates the bandpass performance of the finite planar MS and conformal MS for CPJS-MS, SPJS-MS, and SG-MS using HFSS. We simulate the cylindrical conformal finite MS (10×14 array) with curvature radii of 100, 150, and 200 mm, as shown in Fig. 13. Both TE and TM modes are considered for the transmission performance of the conformal MS. The sketch of the conformal MS for normal incidence is shown in Fig. 14.

The simulated transmission coefficients of the finite planar CPJS-MS and conformal CPJS-MS are presented in Fig. 15. For comparison purposes, the simulated results of the infinite planar CPJS-MS are shown in the same figure.

It can be observed from Fig. 15 that the TE and TM mode transmission performances of conformal CPJS-MS are significantly different. For TE mode in Fig. 15(a), the transmission performance of conformal CPJS-MS varies greatly in the higher frequency range, and the passband bandwidth becomes narrower as the curvature radius decreases. On the contrary, the transmission performance for the TM mode in Fig. 15(b) varies greatly at lower frequencies, and the passband bandwidth becomes wider as the curvature radius decreases. Detailed information on the transmission performance of the conformal CPJS-MS is listed in Table 3.Table 3.

Simulated Results of Planar and Conformal CPJS-MSs

Compared with the infinite planar case, the center frequency of the passband for the finite planar CPJS-MS moves toward lower frequency (from 2.35 to 2.34 GHz), and the bandwidth becomes narrower (from 1.51 to 1.40 GHz) for TE mode. For TM mode, the center frequency of the passband increased from 2.33 to 2.42 GHz, and the bandwidth decreased from 1.47 to 1.27 GHz. It can be explained in that the current distribution on the edge element of the finite planar CPJS-MS is different from that on the unit element of the infinite planar CPJS-MS because the edge element does not satisfy the periodic boundary condition—the different current distributions cause a wave impedance mismatch between finite planar CPJS-MS and air.

As the curvature radius of conformal CPJS-MS decreased, the neighbor square patches become closer, and the mutual coupling increased, which causes the corresponding equivalent capacitance of parallel capacitor C2 to increase (see Fig. 4 for the equivalent circuit), so the center frequency and passband bandwidth decreased.

For the TE mode in Fig. 15(a), the higher frequency range of the passband of conformal CPJS-MS is significantly different from that of finite planar CPJS-MS. For the case of r=100mm, the passband bandwidth decreased from 1.40 to 1.26 GHz by 10.0%, which is more serious than that for the larger curvature radius case (3.6% at 150 mm and 2.9% at 200 mm). For conformal CPJS-MS, the central element and the edge element are not on the same plane, so the normal incidence for the central element becomes the oblique incidence for the edge element. Hence, the transmission performance of conformal CPJS-MS is similar to that of the planar CPJS-MS under oblique incidence. Comparing Table 2 with Table 3, it can be demonstrated that the passband bandwidth of conformal CPJS-MS with different curvature radii (1.26–1.36 GHz), shown in Table 3, is between that of infinite planar CPJS-MS under incidence angles of 20° and 40° (1.43 GHz at 20°, and 1.15 GHz at 40°), as shown in Table 2.

For the TM mode in Fig. 15(b), the passband bandwidth of conformal CPJS-MS increased from 1.29 to 1.45 GHz by 12.4%, and the center frequency moves toward lower frequency (from 2.45 to 2.43 GHz) as the curvature radius decreased from 200 mm to 100 mm. The increase of passband bandwidth of conformal CPJS-MS also can be explained by the fact that the bandwidth of the infinite planar CPJS-MS increases as the incidence angle increased (1.47 GHz at 0°, and 1.88 GHz at 40°), as shown in Table 2. For a smaller curvature radius, the transmission performance of conformal CPJS-MS is more similar to that of infinite CPJS-MS under oblique incidence.

It can be seen from Table 3 that the center frequency of the passband increases first and then decreases for TE mode as the curvature radius decreases. It can be explained that for the larger curvature radius, the center frequency increases because of the effect of oblique incidence on the passband. However, the center frequency decreases for the smaller curvature radius due to the enhanced mutual coupling between patches. For the conformal CPJS-MS with a larger curvature radius, the effect of mutual coupling between conformal CPJS-MS patches is similar to that of the planar CPJS-MS. However, the oblique incidence angle effect becomes dominant at the center frequency of the passband of conformal CPJS-MS. The center frequency increased for TE and TM modes as the curvature radius decreased and the oblique incidence angle increased. For the conformal CPJS-MS with a smaller curvature radius, the effect of the mutual coupling between patches becomes dominant, which causes the center frequency to decrease. The analysis of SPJS-MS and SG-MS is also given in Appendix A. In order to illustrate the proposed MS performance in realistic use cases, the RCS is also simulated as shown in Appendix B.

Figures 15–17 can be summarized as follows. (1)For TE mode and TM mode, the center frequency of the passband of JS-MS increases first and then decreases as the curvature radius decreases. However, the passband bandwidth decreases for TE mode, while the change of passband bandwidth is the opposite for TM mode.(2)Since most conformal MS unit elements are under oblique incidence, the transmission performance of conformal MS is similar to that of the infinite plane MS under oblique incidence. However, the center frequency and bandwidth of the passband of the conformal MS show more complex change characteristics because of the coupling between conformal MS elements and the aperiodic boundary of edge elements.(3)The protrusion structure provides the MS with better angular stability by reducing the size of the MS unit element and increasing the passband bandwidth for conformal cases. From the comparison of SPJS-MS and CPJS-MS, it can be seen that different shapes of protrusion structures correspond to different center frequencies and bandwidths of passbands by affecting the equivalent inductance of MS. Section 3.B discusses the mechanical properties of the protrusion structure’s different shapes.

It can be seen from Section 3.A that the transmission performance of planar MS could no longer be maintained as the MS structure is deformed, such as cylindrical conformal MS. Hence, in the design process of conformal MS, the strain analysis should be carried out, and the resulting structure deformation should be considered to evaluate the change in transmission performance. This section uses Abaqus to simulate the strain distribution of the conformal MS with different curvature radii, and the maximum principal strain value is obtained to evaluate the deformation degree of conformal MS structures.

Generally, the smaller the maximum principal strain value obtained by simulation, the smaller the deformation degree of the MS structure. Furthermore, the strain-curvature relations of three kinds of MS are discussed. The simulated model of the conformal SG-MS is shown in Fig. 16.

As shown in Fig. 16, the left edge of the MS structure is fixed while the right edge is bent with a certain curvature radius (i.e., r=100, 150, and 200 mm). The effect of curvature radius on the maximum principal strain value is investigated. The simulated strain distributions of MSs and comparison are given in Appendix C with different curvature radii shown in Fig. 19. The maximum strain values of three MSs with different curvature radii are listed in Table 4.Table 4.

Simulated Results of Conformal MS

For SG-MS and CPJS-MS, the maximum principal strain value position appears at the protrusion. SPJS-MS has the smallest maximum principal strain value for a curvature radius of 100 mm. Therefore, SPJS-MS has the best mechanical properties because the maximum principal strain value increased slowly as the curvature radius decreased.

In order to clarify the effect of various substrates, mechanical simulations of CPJS-MS under four kinds of substrates are completed, as shown in Table 5. It can be observed that the maximum principal strain increases with the decrease of radius for all substrates. The maximum principal strain between different substrates is different because of the different modulus of elasticity and Poisson’s ratio. As for the effect of substrate thickness, the simulation results indicate that the maximum principal strain is about linearly correlated with thickness. It should be mentioned that the larger thickness substrate is more difficult to bend. Another method such as a thin metal layer adhered to the already formed curved surface could be more suitable.Table 5.

Mechanically Simulated Maximum Principal Strain of CPJS-MS with Different Radii and Substrates

The transmission performance of the proposed JS-MS is experimentally demonstrated using a free-space measurement system in Fig. 17. Three types of MS samples (CPJS-MS, SPJS-MS, and SG-MS) are fabricated and mounted on the foam cylinder. The foam cylinder with three curvature radii (r=100mm, 150 mm, and 200 mm) is in the middle of two standard horn antennas (0.4–6 GHz). The measured transmission coefficients of MS for TE/TM mode are shown in Fig. 24 (Appendix D), showing good agreement between the simulated and measured results. Also, the angular stability of the two JS-MSs is much better than that of SG-MS.

An electrical-resistance strain gauge is used to measure the principal strain of the conformal MS sample mounted on the foam cylinder, a device that measures the normal strain on the surface of a stressed object. By bonding the strain gage to the surface of planar MS firmly, the length of the strain gage could be proportional to the strain of conformal MS with different curvature radii. Since each gage could measure the normal strain along only one direction, we chose a 45° strain rosette consisting of three strain gages arranged to measure the strains along two perpendicular directions and at a 45° angle between them.

A 45° strain rosette is bonded on the surface of the three MS samples for the strain measurement. These MSs are mounted on the foam cylinder with different curvature radii, as shown in Fig. 18.

The maximum principal strain values of the three MSs with different curvature radii are listed in Table 6.Table 6.

Measurement Results of Principal Strain

Table 5 shows that the agreement between the simulated results by Abaqus and the measured data by a 45° strain rosette is reasonably good. Also, the results show that the maximum principal strain value of the SG-MS for each curvature radius is larger than that of two JS-MSs, and the SPJS-MS sample has the smallest strain among the three MSs.

To realize high mechanical performance for conformal application and further understand electromagnetic characteristics under conformal conditions, two novel designs of JS-MS, based on the idea of a jigsaw and mortise-tenon joint, are introduced in this paper. Firstly, the ECM of the planar JS-MS is proposed to evaluate the passband with different incidence angles efficiently. Secondly, the cylindrical conformal JS-MS and SG-MS with different radii are simulated by HFSS and Abaqus, respectively. The effect of curvature radius on electromagnetic and mechanical properties of three MSs is investigated in detail. In addition, the distinction of passband width and center frequency of the three MSs with infinite planar form under various incident angles and polarization states is discussed based on ECM to get more understanding of the conformal condition. Finally, three MS samples are measured by a free-space measurement system and a 45° strain rosette. It can be found that the proposed SPJS-MS exhibits the best electromagnetic and mechanical performance for the conformal applications. The equivalent circuit and simulation are both adopted to analyze the electromagnetic character of conformal MS. The summarized rules and analysis method are helpful to get deeper comprehension of conformal MSs and can be expanded to other MSs. The mechanical character is also studied and both electromagnetic and mechanical performances are considered for the first time.