A metasurface absorber (MA) is an electromagnetic (EM) wave absorber built upon an artificially engineered conductive metapattern that not only enables impedance matching of the total structure with free space but also converts incident EM energy into heat [1]. This energy conversion is triggered by the ohmic loss induced by the effects of the electric current on the conductive patterns or dielectric loss from a dielectric substrate. Owing to its design freedom pertaining to the use of conductive patterns, an MA can enhance absorption levels as well as improve the bandwidth (BW) using relatively thin dielectric substrates, in contrast to a conventional EM wave absorber. However, one of the limitations of MA is that its absorption capacities deteriorate significantly with an increase in the incident angle.

A narrow single [2] and multi-band [3] MAs have been reported, however, the absorptions larger than 90% that indicate 10% reflectance, i.e., −10 dB reflectance, were not achieved for the oblique incidences. By focusing on extending BW that satisfies the −10 dB reflectance, many types of broadband MAs have been reported successfully [4–10]. However, satisfactory performance could not be maintained for the extreme incident angle of 60°. To overcome this bottleneck, super cells composed of four split-ring resonators sequentially rotated by 90° [11] and a form of split ring [12] were reported, which achieved −10 dB reflectance for the incident angle of 60°. However, this reflectance could not be simultaneously achieved for the transverse electric (TE) and transverse magnetic (TM) polarizations.

To attain this extremely complicated absorption performance, previous studies have utilized a version of a split electric resonator [13], an eight-circular sector [14], and a pinwheel-shaped structure [15]. Nonetheless, the overlapped BWs for both polarizations were still quite narrow at an incident angle of 60°. Another approach to achieve this aim involved inserting vertical structures into MA, such as via holes [16, 17] or an I-shaped loop [18]. Even though this approach allowed for favorable performances for the extreme oblique incident condition, certain problems persisted, be it the reflectance for the TE polarization not decreasing below −10 dB [16] or the presence of unstable electrical connectivity between the resistive pattern and the metal cylinder [17]. Moreover, the bulky structure was not attractive from the perspective of the applicability of the MA [18].

In this paper, an MA composed of a single-layer metapattern that can simultaneously achieve wide-angle absorption for both TE and TM polarizations up to an incident angle of 60° is proposed. First, a pair of two broadband MAs [19] was employed as the initial set, consisting of four chip resistors and square copper tiles as the initial condition. Second, a transformative genetic algorithm (GA) was applied to the initial set after updating the figure of merit (FOM) based on the full-wave simulation results obtained for the initial set. To suppress the reflectance of the worst case, the TM polarization with the largest incident angle 60°, a scaling factor is multiplied to the reflectance of that one. The accuracy of the proposed design was verified by comparing the full-wave simulated and measured results.

To design the wide-angle MA, a configuration that allowed broadband absorption for a limited range of incident angles [19] was chosen as the initial state. Fig. 1(a) presents the simulation setting constructed using COMSOL Multiphysics. The MA consisted of an FR4 substrate, with its top and bottom surfaces covered by a metapattern and a perfect electric conductor (PEC), respectively. The thickness and relative permittivity of the FR4 substrate were 3.6 mm and 4.46–i0.13, respectively. Notably, this thickness represents a quarter of the wavelength inside the substrate λ_g at 10 GHz, thus supporting coherent absorption at this frequency [7] and satisfying the impedance matching condition when the impedance of the pattern is 377 Ω [11]. Therefore, by designing an MA with an impedance close to 377 Ω at the target band around the center frequency of 10 GHz, the desired absorption properties could be achieved. Meanwhile, the simulation domain outside the MA was filled with air and truncated by a perfectly matched layer (PML). Furthermore, the periodic boundary condition (PBC) was set to the side boundary of the entire setting, while the wave port responsible for irradiating the EM wave was placed at the surface of contact between the air and the PML.

The metapattern, a combination of square copper tiles connected to four 100 Ω chip resistors, was optimally designed using a GA according to the configuration shown in Fig. 1(b). The GA is a global optimization method used to find an optimum solution that satisfies the target performance [7, 17, 19–21]. This tool enables optimal designing of MAs without having to model it as an analytical equivalent circuit [5, 6, 16]. Here, the 100 Ω resistance is chosen because the reflectance for the normal incidence at each resonance frequency is lowest among those of the designs utilizing resistances from 80.6 to 120 Ω [19]. The material properties of the square tiles were determined in terms of those of the air or the copper, depending on the state of the bits shuffled by GA being 0 or 1, respectively.

To achieve absorption functionality independent of the horizontal angle of rotation of the incident EM wave, chip resistors were placed at the boundaries of the four quadrants and square tiles, as shown in Fig. 1(b), are grouped to satisfy the axial symmetries for the x and y axes. They were grouped starting from the origin of the coordinates, as shown in the inset in Fig. 1(b), with the process concluding on reaching the sixty-second pair, meaning that the total number of tile groups was determined to be 62. Furthermore, by attaching T-shaped copper pads to the chip resistors, as shown in Fig. 1(b), the copper tiles could be connected to the chip resistors in various directions. For instance, the copper-tile pattern could diverge toward the boundary or converge into the center of the unit cell depending on the design criterion determined by the FOM of GA. Furthermore, to save computational resources and reduce simulation time, the material properties of the square tiles were set on the two-dimensional (2D) structure using the transition boundary condition (TBC) [22]. The effective thickness and conductivity of the tiles were set to 34.8 μm and 5.8 × 10⁷ S/m, respectively. A detailed explanation of the GA is provided later in this section. The size of the chip resistor was maintained at 1 mm × 0.5 mm. Although increasing the tile resolution could possibly improve the BW [23], the size was fixed at 0.5 mm × 0.5 mm accounting for fabrication tolerance and the size of the chip resistor. Finally, to separate the unit cells, the outer boundary of the metapattern was surrounded by an air gap of 0.25 mm width.

Fig. 2(a) presents the simulation results for the initial state, considered by [19] as the best metapattern. It shows that the reflectance of MA increased significantly for the TM polarization at the incident angle of 60°. The inset in Fig. 2(a) shows that the metapattern matched with the simulation results. Furthermore, the overlapped band with reflectance below −10 dB for both TE and TM polarizations at incident angles of 0° and 60° shifted upward, achieving a narrow BW. More significantly, the overlapped BW could not be maintained for other oblique incident angles [19]. Since the bottom of the MA was blocked by the PEC, and the scattered power leaking through the PBC was negligible [19], absorption was estimated in terms of the reflectance, which was calculated as the ratio of the reflected power captured by the wave port to the incident power.

To suppress the reflectance in the worst case, which was TM polarization at the incident angle of 60°, a design strategy of updating the FOM of the GA applied to the initial state was adopted. The transformative GA, responsible for changing an optimal state to another state by updating the FOM, is markedly distinct from an adaptive GA [21], which updates FOM in a single loop of iteration to achieve fixed target functionality. To enhance the absorption in the worst case, a scaling factor of 1.5 was multiplied with the reflectance of the TM polarization at the incident angle of 60°, which can be expressed as follows:

where f₁, f₂, and f₃ are 9.5, 10, and 10.5 GHz, respectively. The frequencies are selected to achieve the wide-angle absorption within a BW as broad as that confirmed in Fig. 2(a) near 12 GHz. By setting the center frequency at 10 GHz, the absorption band was adjusted to be located near 10 GHz. Meanwhile, the scaling factor was determined through a trial-and-error process to check for the overlapped −10 dB BW for both the TE and TM polarizations at incident angles of 0° and 60°. Since the square tiles were grouped to maintain symmetries for the x and y axes, the reflectance for TM polarization at the incident angle of 0° was omitted from Eq. (1).

To apply GA using Eq. (1), a pair of two bit sequences utilized for designing broadband absorption [19] was employed, the best one matched with the simulation results of Fig. 2(a). The number of bits included in each sequence was 62, the same as the total number of tile groups. Notably, the transformative GA process involved four steps. First, to proliferate the bit sequences, a cross operation was applied 16 times to the adopted pair, resulting in an exchange of bits, which were then arrayed after randomly selected locations on the sequence. Second, a mutation operation was applied 5 times to each generated sequence to convert the bits at the randomly selected locations from 1 to 0 or vice versa. The cross and mutation operations were repeated until they produced 62 bit sequences as a new set, thus guaranteeing diversity. Third, the bit sequences were decoded into metapatterns using the configuration presented in Fig. 1(b). Therefore, when a bit was 1 or 0, the material properties of the matched group of square tiles were set to those of copper or air, respectively. Finally, the best pair of two bit sequences was selected by comparing the FOM, calculated using Eq. (1), after simulating the MA using the decoded metapattern. If the FOMs of the best pair did not reach a minimum value, the metapatterns were encoded into bit sequences, and the GA process was repeated.

Fig. 2(b) depicts the FOMs calculated for iterations of the GA process, showing that minimum FOM was achieved at the seventeenth iteration. Between the two bit sequences of the optimal pair, the sequence with the minimum FOM was selected as the final design. Fig. 2(c) illustrates the metapattern decoded from the final bit sequence. The blue, white, and red colors indicate copper, air, and the chip resistors, respectively. The inset in Fig. 2(c) illustrates the incidences of TE and TM polarizations with regard to the vertical angle θ. Fig. 2(d) presents the simulated reflectance for both TE and TM polarizations at incident angles of 0° and 60°, confirming the −10 dB reflectance BWs to be 9.22–11.05 GHz, 9.77–10.77 GHz, and 10.13–11.1 GHz for TE with θ = 0°, TE with θ = 60°, and TM with θ = 0°, respectively. Notably, although the bands for θ = 60° shifted upward due to decomposition of the wave vector k⇀ into the horizontal and vertical ones, it is confirmed that the BW overlapped from 10.13 to 10.77 GHz, with the fractional BW being 6.12%.

To confirm the design accuracy, the 2D metapattern designed using TBC was changed into a 3D one featuring the same material parameters as the original one. Moreover, simulation results calculated using another commercial software, the Ansys High Frequency Structure Simulator (HFSS), based on the 3D metapattern were examined for further verification. Fig. 3(a) depicts the metapattern translated using the Ansys HFSS. Notably, to guarantee electrical connectivity between the copper tiles connected through their corners, 0.1 mm-wide thin copper patches were added at the junctions. Although a comparison of the results, i.e., the reflectance before and after attaching the patches, is not presented here, the effects of the patches were found to be negligible [19].

Fig. 3(b) and 3(c) present the simulation results for both TE and TM polarizations at incident angles ranging from 0° to 60°, increased by intervals of 15°. Fig. 3(b) shows that the overlapped −10 dB reflectance BWs for both TE and TM polarizations at incident angles of 0° and 60° ranged from 10.36 to 11.15 GHz and from 10.36 to 11.36 GHz, as obtained by implementing COMSOL Multiphysics and Ansys HFSS, respectively. The fractional BWs for the former and the latter were 7.35% and 9.21%, respectively. Although an inherent discrepancy is evident between the two values calculated by the two different simulators, the accuracy of the analysis is confirmed by the well-matched trends in Fig. 3(b) and 3(c) for the same simulation conditions. Moreover, the accuracy of the original 2D design was verified based on the negligible error between the fractional BWs of the 2D and 3D metapatterns, calculated using COMSOL Multiphysics.

The overlapped BWs for both polarizations at all incident angles are also shown in Fig. 3(b) and 3(c), confirmed by COMSOL Multiphysics and Ansys HFSS to be 10.8–11.21 GHz and 10.93–11.47 GHz, respectively. The fractional BWs for the former and the latter were 3.73% and 4.82%, respectively. Between these two results, the former was selected owing to the measured resonance frequencies of TE and TM polarizations for the normal incidence being closer to those calculated by COMSOL Multiphysics than by Ansys HFSS. The measured results are discussed in more detail in the next section. Since the FOM did not account for the reflectance of oblique incidences at angles of 30° and 45°, their resonance frequencies, for the most part, did not gather around the center frequency, as they did in the case of angles 0° and 60°, as shown in Fig. 3. This might have contributed to the reduction in fractional BW from 7.35% to 3.73% after the inclusion of the results for the incident angles of 15°, 30°, and 45°.

Table 1 presents a comparison of the simulated overlapped −10 dB reflectance BWs at the extreme oblique incident angle of 60° and at all incident angles from 0° to 60°. Although multiple articles have reported on broadband absorption, to the best of the authors’ knowledge, the studies summarized in Table 1 are the references that are most comparable to the proposed work from the perspective of wide-angle absorption. Notably, Table 1 clearly implies that the proposed design offers considerable advantages from the perspective of the overlapped −10 dB BW for all polarizations and incident angles.

The transition from broadband to wide-angle MA can be explained by the quality (Q) factor of a series resistance (R)-inductance (L)-capacitance (C) resonator circuit model [5, 6, 16]. The Q factor can be described as follows:

where fc=1/2πLC is the resonance frequency of the circuit. The Q factor in Eq. (2) is the same as that of a conventional R-L-C circuit because the incident electric field can be matched with the voltage source of the circuit. For the initial design shown in Fig. 2(a), chip resistors were utilized to extend the BW [9, 10, 19] by decreasing the Q factor. However, for wide-angle absorption, the role of the chip resistors was transformed to enhance absorption at the 60° angle in the target band. By comparing the −10 dB reflectance bandwidths of broadband MAs without and with chip resistors in [23] and [19], respectively, it was found that the BW obtained using chip resistors was considerably wider than that achieved without using them. Moreover, since an inverse relationship exists between BW and the Q factor, the absorption attained using chip resistors was expected to improve with a reduction in BW. If copper tiles without chip resistors were recombined to achieve a high Q factor to compensate for the performance degradation at the extreme angle of 60°, reducing the reflectance levels below −10 dB for all incidence scenarios would be extremely difficult or even impossible. This is indirectly evidenced by the very narrow −10 dB reflectance BWs confirmed in [14] and [15], which proposed MAs composed of an eight-circular-sector and a crescent-shape resonator, respectively, without any chip resistors.

To locate the resonance frequency in the target band, as well as to increase the Q factor for the proposed design, the copper tiles connected with the chip resistors were recombined using transformative GA in a specific direction to increase the inductance of the MA. As a result, the resonance frequencies f_c, which were observed to gather around 12 GHz in Fig. 2(a), shifted downward. In addition, the Q factor increased due to the increment in inductance resulting from Eq. (2). Consequently, the −10 dB reflectance BW in Fig. 2(d) declined compared to that in Fig. 2(a) while maintaining it for the wide range of the incident angle. Furthermore, since inductance is proportional to the total length of the pattern, the increased inductance was confirmed by the increased pattern length in Fig. 3(a) compared to that in the inset in Fig. 2(a).

Drawing on Fig. 3(a), which guaranteed electrical connectivity in the metapattern, an MA sample was fabricated, as shown in Fig. 4(a). The inset shows the detailed structure of the metasurface unit cell, consisting of 12 × 12-unit cell arrays of which the size is 150 mm × 150 mm. The metapattern was fabricated by etching copper onto FR4 substrate, while the chip resistors were soldered using surface mount technology. Fig. 4(b) presents the measurement setting, consisting of two lens-horn antennas and a sample holder. For the normal incidence, the reflectance was estimated by dividing the |S₁₁|² measured using the MA by that measured using the reference copper plate, with |S₁₁|² denoting the ratio of the backward reflected power to the incident power. For this case, the lens-horn antenna connected to port 1 of the vector network analyzer (VNA) was employed. The S-parameters were measured using Keysight’s PNA N5227B VNA. For the oblique incidences, the reflectance was evaluated by dividing the |S₂₁|² measured using the MA by that measured using the copper plate, with |S₂₁|² being the ratio of the specular reflected power to the incident power. To measure |S₂₁|², the lens-horn antennas were rotated in accordance with the circular rails on the table. Among the angles used for the simulations, the smallest oblique incident angle for the measurement was set to 45°, since the large size of the lens-horn antenna prohibited the arrangement of the two antennas at an angle less than 45°.

Fig. 4(c) and 4(d) show the measured and simulated results for the TE and TM polarizations, respectively, at incident angles of 0°, 45°, and 60°. As described in Section II, between the resonance frequencies of 10.4 and 10.69 GHz calculated for the normal incidence using COMSOL Multiphysics and Ansys HFSS, respectively, the former was found to be closer to the measured value of 10.49 GHz. Therefore, in Fig. 4, the measured results are compared to those obtained using COMSOL Multiphysics. Fig. 4(c) confirms that the trends exhibited by the measured results for TE polarization matched well with the simulated results. In contrast, some discrepancies were observed in the case of TM polarization at incident angles of 45° and 60°. This can be attributed to differences in radiation patterns of the horn antenna for TE and TM polarizations, which may lead to the deterioration of the focused beam at the sample location, especially in the case of the oblique incidences of TM polarization. Nevertheless, the accuracy of the fabricated metasurface absorber was verified by the well-matched results for the normal incidences in the case of both TE and TM polarizations.

Table 2 presents a comparison of the measured −10 dB reflectance BW at normal incidence and at the extreme oblique incident angle of 60°. Since the reflectance for the incident angles of 15° and 30° could not be measured owing to the large sizes of the lens-horn antennas, the results for only two representative cases are summarized in Table 2 instead of those for all angles ranging from 0° to 60°. Table 2 clearly indicates that the proposed design achieved the widest −10 dB BWs for both scenarios, as well as for the overlapped BW. The results obtained for the reference study [19] are omitted from Table 2, since its measurements were limited to the incident angle of 15°. Furthermore, Table 2 shows that the fractional −10 dB reflectance BW at the oblique incidence of 60° for both TE and TM polarizations is 14.48%. Due to inaccuracies in the measurements, especially for the oblique incidences pertaining to TM polarization, a gap was identified between this value and the 7.35% calculated from the simulated results in Table 1. Nonetheless, the accuracy of the simulated results was indirectly verified by the fractional BWs for the normal incidence condition, calculated as 18.06% and 19.56% by the simulation and the measurement, respectively, indicating that they were well matched.

In this study, a novel design strategy is proposed for constructing a wide-angle MA that can optimally recombine copper tiles connected to four 100 Ω chip resistors. For the initial condition, a broadband MA was designed using GA and then modified into a wide-angle MA by applying a transformative GA. For the FOM of the transformative GA, a scaling factor of 1.5 was multiplied by the reflectance of the TM polarization at the incident angle of 60°, which exhibited a large variation in the spectrum for broadband design. The full-wave simulation results revealed that the overlapped −10 dB reflectance BW for both TE and TM polarizations at all incident angles up to 60° was 10.8–11.21 GHz, with the fractional BW being 3.73%. Although the overlapped BW achieved by the proposed MA was confirmed to be the largest when comparing it with those of other MAs reported in the literature, there is still room for extending the BW further by adding via-holes in the substrate [16, 17]. In addition, reducing thickness of MA with maintaining insensitivity for the incident angle is remained as a future work. The accuracy of the proposed design was verified not only through two well-matched sets of simulation results calculated using two different commercial software but also by comparing the simulated and measured results. Overall, the proposed method can be employed to design wide-angle MAs that can be easily integrated into printed circuit boards [24] for microwave circuits or used in antenna systems requiring the prevention of interferences of EM waves.