Absorbing electromagnetic (EM) waves in arbitrary directions and polarization has been desired for energy conversion [1–3], EM shielding [4,5], and stealth [6,7] devices in both academia and engineering. By matching the impedance between materials and free space, EM waves can enter materials and then be transformed into other energy [8,9]. There are two typical approaches to design EM absorbers: loss material-based absorbers and structural-based absorbers. Commonly, the loss materials are divided into three types: resistance loss absorption, dielectric loss absorption, and magnetic dielectric loss absorption, which have been utilized to serve the effective absorption in optical, infrared, terahertz and microwave areas [10–22]. While suffering from bulky size, high density, narrowband, or weak resonance, the further development of absorber design is restricted by materials’ inherent drawbacks, especially in broadband and wide angles. As an alternative, the structural resonance-based method is introduced, which achieves the electric or magnetic resonance by meticulous arrangement of the dielectric and metal structure. Particularly, the appearance of metamaterials [23–25], consisting of periodic or quasi-periodic functional motifs, solves the bulky size of the traditional devices and achieves stronger resonance in designed frequency simultaneously, pushing the rapid development EM manipulation as anomalous reflection/deflection [26–28], beam splitting [29–34], RCS reduction [35–38], and holography and imaging [39–44]. Hence, metamaterials have been widely integrated into the study of EM absorbers.

In 2008, Landy et al. proposed an I-shaped based method to realize perfect absorption at a single frequency point [45], arousing interest in the metamaterials absorber. This novel resonant mechanism has been promoted to realize the dual-frequency, multiple frequency, and even wideband absorbers [46–52]. Meanwhile, the composite design methods, frequency selective surface (FSS), and plasmonic structure, have been utilized in metamaterials absorbers to expand the operating bandwidth [53–57]. Heretofore, plenty of existing schemes use broadband effective absorption for normal and small incident angles. However, with an increase of the incident angle, the performance of absorbers will decline, especially for TE polarization waves. This kind of impedance mismatching means that ultrabroadband and wide-angle absorbers rarely coexist, i.e., the so-called property trade-off paradox. As a universal law of nature, such a phenomenon exists widely in the important properties of structural and functional materials. Given that the properties of materials are determined by their structures, the key to realizing the rational design of materials with the desired performance is to dissect the detailed information about the material structures. Recently, achievement of the function-oriented design of new functional materials was developed under the design of functional motifs [58,59], and metamaterials are one of the classical EM functional materials. Thus, it is very necessary to initiate multidimensional design of functional motifs for ultrabroadband and wide-angle absorbers.

In recent years, intelligent optimization algorithms unveiled the curtain of inverse structure’s design. By endowing codes with corresponding structural parameters, the dimension, size, distribution, and shape of the functional motifs can be encoded into a mathematical expression or computer language. Following the guidance of the preset goal orientation, the performance of designed functional motifs will exceed that of empirical intuition. Till now, the optimization method has achieved perfect performance in broadband absorption [60–64], polarization conversion [65–67], precise wavefront manipulation [68–70], and so on. However, these reported performances would decline sharply with an increase of the incident angle of EM waves. And the mechanism in these works cannot surmount the impedance mismatch problem in broadband and wide angle. Recently, some spatially combined structures have been proposed to improve the angular tolerance of absorption [71,72], yet the research on this issue is still in the nascent stage.

In this work, a general and straightforward strategy is proposed for ultrabroadband and wide-angle absorbers. Theoretically, a spatial combined manner is derived from the analysis of EM wave vectors, reacting to the large and small incident angle by the top and bottom absorbing structure, respectively. Then, a vertical atom sticking on $\mathit{N}\times \mathit{N}$ horizontal meta-atoms with conductive film serves as the functional motif. Based on centrosymmetric characteristics, the proposed functional motifs were polarization-independent. Meanwhile, the structural parameters were optimized by the intelligent algorithm to satisfy the wide-angle effective absorption. As a proof of concept, a wide-range broadband microwave absorber was fabricated by resistive sheet and dielectric means. This absorber achieves over 10 dB absorption from 5 to 34 GHz in the range of 0° to 70° for both TE and TM waves and 3 to 40 GHz from 60° to 70° for the TE wave. The measured result is highly coincident with the simulated result, verifying the feasibility of our proposed method. Moreover, the proposed functional motifs can be attached with optical transparency function at will. When the resistive sheet was the transparent material indium tin oxide (ITO) and the dielectric was polymethyl methacrylate (PMMA), the transmittance of the ultrabroadband and wide-angle absorbers is maintained over 70% in optical regimes. Thus, it is believed that this multidimensional design manner of functional motifs can be expanded to optical, infrared, and terahertz absorption, and even to multispectral compatible applications.

Figure 1 illustrates the performance of a wide-angle absorber under different EM waves. The proposed functional motifs consist of a vertical atom on $\mathit{N}\times \mathit{N}$ horizontal meta-atoms, which endows the absorber with wide-angle microwave absorption and optical transparency. Under the illumination of microwave, the proposed absorber expresses over 90% absorption for both TE and TM waves in wideband and wide angle, while under the illumination of light waves, the proposed device expresses over 70% transmission. In terms of equivalent circuit parameters, $Z_0$ is the characteristic impedance of air. $Z_R(\theta )$ and $Z_d$ denote the equivalent impedance of the resistive sheet under different incidences and the dielectric substrate, respectively. In sharp contrast to the traditional absorbing metamaterials, the proposed functional motifs can provide equivalent impedance in both vertical and horizontal directions, which will surmount the impedance-matching problem under wide-angle incident waves.

Figure 2 illustrates the vector decomposition of incident waves. First, we define the direction of normally incident wave as the $z$ axis. For the TE wave, the electric (E)-field is on the $y$ axis for both components $k_z$ and $k_x$. Perpendicular to the direction of E-field and $k$, the magnetic (H)-field is along the $x$ axis for $k_z$ and $z$ axis for $k_x$, respectively. Similarly, for the TM wave, the H-field is on the $y$ axis for both components $k_z$ and $k_x$. Perpendicular to the direction of H-field and $k$, the E-field is along the $x$ axis for $k_z$ and along the $z$ axis for $k_x$, respectively. According to the impedance calculation formula, the impedance under different polarizations can be calculated as $$\begin{gathered} Z_{R,\mathrm{TE}}(\theta )=\frac{Z_R}{\cos \theta }\sqrt{{\mu }_r/{\epsilon }_r}, \\ {Z}_{R,\mathrm{TM}}(\theta )=Z_R\cos \theta \sqrt{{\mu }_r/{\epsilon }_r}, \end{gathered}$$(1)where $\theta$ is the incident angle, $Z_R$ is the normal impedance of the structure, and ${\mu }_r$ and ${\epsilon }_r$ denote the relative permeability and permittivity of the material, respectively.

To further deduce the reflection coefficient $R$, the input impedance $Z_{\mathrm{in}}$ of functional motifs is calculated in Eq. (2), which is brought into Eq. (3) to obtain $R$, $$Z_{\mathrm{in}}=\frac{Z_R(\theta )·Z_d}{Z_R(\theta )+Z_d},$$(2)$$R=\frac{\mathrm{Re}\{Z_{\mathrm{in}}\}-Z_0}{\mathrm{Re}\{Z_{\mathrm{in}}\}+Z_0}.$$(3)

Figure 3 depicts the mechanism of wide-angle absorption. For small incident angles, the bottom layer serves as the main absorbing layer, which contributes to the strong absorption under the vertical component [Fig. 3(a)]. For large incident angles, the top layer serves as the main absorbing layer, which contributes to the strong absorption under the horizontal component [Fig. 3(b)]. In order to decrease and even eliminate the mutual interference between two orthogonal structures, the intelligent optimization algorithm is introduced to the inverse functional motifs design. To prove the proposed paradigm, the materials with a light transparent characteristic are selected to realize the microwave absorption. As shown in Fig. 4(a), the cubic PMMA substrate (${\epsilon }_r=2.25-\mathrm{j}0.001$, ${\mu }_r=1$), surrounded by ITO films, is stacked on the $\mathit{N}\times \mathit{N}$ bottom structure. Similarly, the bottom structure consists of ITO, PMMA, and ITO layer, where the top ITO layer exhibits the characteristic as resistive film, the middle PMMA layer as dielectric substrate, and the bottom ITO layer as the microwave reflective back plane with 6 Ω/sq. Figures 4(b) and 4(c) show the surface design strategy of the top structure and the bottom structure, respectively. The surface is divided into $20\mathrm{pixels}\times 20\mathrm{pixels}$, which decide the material in each block. Due to the vector decomposition (Fig. 2) and polarization-independent requirement, symmetry-breaking is utilized to design the same response for TE and TM waves in both vertical and horizontal planes.

In this work, we employ code optimization strategy to design one-eighth of the surface unit, obtaining all the functional motifs by mirror symmetry and central symmetry, where code 0 denotes nothing on the PMMA and code 1 denotes ITO on the PMMA. To get better surface distribution, we set the period gap width of the bottom atom as $w_1$ and the top atom as $w_2$, which will be optimized by the intelligent algorithm in the joint simulation (specific information in Appendix A). More importantly, another purpose of gap $w_2$ is to reduce or even eliminate the impedance interference between two orthogonal structures. To optimize the dimension of cubic side length $h$, bottom structure period $p$ and thickness $d$, value of N, gap width $w_1$ and $w_2$, surface square resistance values of the bottom meta-atom ${\mathrm{Ohm}}_1$, and the bottom meta-atom ${\mathrm{Ohm}}_2$, these parameters are encoded in binary code and are decoded as follows: $$\{\begin{array}{l}p=(16\times p_1+8\times p_2+4\times p_3+2\times p_4+p_5)\times 0.2+4\\ N=(2\times n_1+n_2)+1\\ h=[(8\times h_1+4\times h_2+2\times h_3+h_4)\times 0.05+0.2]\times Np\\ d=(16\times d_1+8\times d_2+4\times d_3+2\times d_4+d_5)\times 0.1+0.5\\ w_1=(4\times a_1+2\times a_2+a_3)\times 0.2\\ w_2=(4\times b_1+2\times b_2+b_3)\times 0.2+0.4\\ {\mathrm{Ohm}}_1=(32\times r_1+16\times r_2+8\times r_3+4\times r_4+2\times r_5+r_6)\times 5+5\\ {\mathrm{Ohm}}_2=(32\times R_1+16\times R_2+8\times R_3+4\times R_4+2\times R_5+R_6)\times 5+5\end{array},$$(4)where, $p_i$, $n_i$, $h_i$, $d_i$, $a_i$, $b_i$, $r_i$, and $R_i$ are binary codes—in total, 34.

The absorption rate is calculated as follows: $$A(w,\theta )=1-R^2(w,\theta )-T^2(w,\theta ),$$(5)where $w$ is the frequency and $T(w,\theta )$ is the transmission coefficient of the structure. Here, the microwave reflective plane is designed at the bottom of the whole structure. Thus, $T(w,\theta )=0$ at the gigahertz regime, and Eq. (5) can be simplified to Eq. (6), $$A(w,\theta )=1-R^2(w,\theta ).$$(6)

The wide-angle absorber optimization design is a multi-objective optimization problem, so the absorbing efficiency, operating frequency band, and angular stability are considered in the fitness calculation function simultaneously. Here, the optimization goal is to achieve maximum operating bandwidth with an absorption rate more than 90% in the frequency range of 2 to 40 GHz and an incidence angle from 0° to 70° concurrently. The fitness function is set as $$\begin{gathered} \text{fitness}=\sum _{i=1}^4\text{fitness}({\theta }_i), \\ \text{fitness}({\theta }_i)=1-k·\frac{\sum _{j=1}^n\mathrm{\Delta }F(j,{\theta }_i)}{f_{\max }-f_{\min }}, \end{gathered}$$(7)where $\mathrm{\Delta }F(j,{\theta }_i)$ is each continuous bandwidth that meets the absorbing requirement under incident angle ${\theta }_i$ (${\theta }_1$ to ${\theta }_4$ are 0°, 40°, 60°, and 70°, respectively), and the $[f_{\min },f_{\max }]=[2\mathrm{GHz},40\mathrm{GHz}]$. $k$ is the penalty coefficient, which is defined as follows: $$k=\{\begin{array}{cc}1,& 0\le {\text{fitness}}_{\max }({\theta }_i)-{\text{fitness}}_{\min }({\theta }_i)<0.2\\ 0.8,& 0.2\le {\text{fitness}}_{\max }({\theta }_i)-{\text{fitness}}_{\min }({\theta }_i)<0.4\\ 0.5,& 0.4\le {\text{fitness}}_{\max }({\theta }_i)-{\text{fitness}}_{\min }({\theta }_i)<1\end{array},$$(8)where the extreme difference of fitness value under different incident angles will determine the value of $k$. To attain a balanced wide-angle absorber, the $\text{fitness}({\theta }_i)$ should be maintained close to each other. Meanwhile, Eq. (7) reveals that the wider the valid bandwidth, the smaller the fitness value will be, which means the better absorbing effect should be.

After the joint simulation process, the parameters of the proposed structure are optimized as $p=10\mathrm{mm}$, $h=24\mathrm{mm}$, $d=2.7\mathrm{mm}$, $w_1=0.2\mathrm{mm}$, $w_2=1\mathrm{mm}$, ${\mathrm{Ohm}}_1=35\mathrm{\Omega }/\text{sq}$, ${\mathrm{Ohm}}_2=25\mathrm{\Omega }/\mathrm{sq}$, and $N=4$. The optimal surface distributions are obtained as shown in Figs. 4(b) and 4(c). Figure 5 depicts the simulated results of the absorption under both TE and TM incident waves. For TE polarization [Fig. 5(a)], the proposed absorber achieves over 10 dB reduction, from 4.8 to 34.0 GHz at 0°, 5.0 to 40.0 GHz at 40°, 3.0 to 40 GHz at 60°, and 2.8 to 39.4 GHz at 70°. For TM polarization [Fig. 5(c)], the proposed absorber achieves over 10 dB reduction, from 4.8 to 34.0 GHz at 0°, 5.1 to 40.0 GHz at 40°, 5.2 to 40 GHz at 60°, and 5.2 to 40.0 GHz at 70°. Obviously, the resonant peaks vary with the polarization and incident angles. Figures 5(b) and 5(d) display the absorption distribution with the frequency and incident angles under TE and TM waves, respectively, where the light red, red, and crimson denote the absorption of over 80%, 90%, and 95%. The absorption performance is so stable that this inverse polarization-independent design method achieves more than 148.7% fractional bandwidth (FBW) and exhibits forceful angular absorption stability from 0° to 70° for both TE and TM waves. Especially for TE polarization, this absorber, realizing a better performance than TM polarization at wide angles, achieves more than 172.1% FBW from 60° to 70°, which is a breakthrough in the design of wide-angle dual polarization absorption.

In order to validate the improvement from the multidimensional design of functional motifs, the absorption of functional motifs, only the top structure and only the bottom structure are discussed in Fig. 6. For both TE and TM polarization waves, the absorptions of top and bottom structures are simulated from 0° to 70°, respectively, and compared with the spatial structure. Figures 6(a)–6(d) showcase each absorption curve of these three structures under TE waves from 0° to 70°. Under small incident angles [Figs. 6(a) and 6(b)], the top structure cannot exhibit strong absorption, while the bottom structure can assist the top structure to improve performance. Under large incident angles [Figs. 6(c) and 6(d)], the top structure has performance very close to the spatial structure, while the absorption of the bottom structure obviously decreases. Figures 6(e)–6(h) showcase each absorption curve of these three structures under TM waves from 0° to 70°. Similar to the results for the TE wave, the top structure shows the strong absorption in the range of large angles, and the bottom structure can assist the top structure to increase the absorption in the range of small angles. Importantly, the absorbing phenomenon of spatial structure has verified our design theme, that the bottom and top structures are utilized to mostly respond to vertical and horizontal waves, respectively (shown in Fig. 3).

To further understand the mechanism of the designed absorber in the range of wide angles, we have monitored the power loss density of the proposed metastructure. Figure 7 shows the energy loss distribution of the strongest resonant peaks under TE and TM waves from 0° to 70°. For incident angle 0°, Fig. 7(a) illustrates the loss distribution of both the top and bottom structures at 8.9 GHz for both TE and TM waves, where most of the energy loss distributes in the horizontal area. For incident angle 40°, Fig. 7(b) depicts the loss distribution at 21.1 GHz for the TE wave and 23.1 GHz for the TM wave, where the energy loss in the horizontal area accounts for a little more than the vertical area around the top structure. Figures 7(a) and 7(b) reveal the fact that the small incident angles have mainly been solved by the bottom structure, which is coherent with our design principle for absorption under small incident angles. For incident angle 60°, Fig. 7(c) showcases the power loss density at 3.4 GHz for the TE wave and 31.3 GHz for the TM waves. With the increase of incident angle, the energy loss in the vertical area around the top structure starts to account for more than in the horizontal area, especially for the TE wave. Meanwhile, the absorption of TE expands to low frequency. For incident angle 70°, Fig. 7(d) displays the loss distribution at 3.5 GHz for the TE wave and 7.1 GHz for the TM wave. In contrast to small incident angles, most of the energy loss distributes in the vertical area, which verifies our proposed design method to absorb EM waves in large incident angles. Moreover, the frequency of resonant peaks varies with incident polarization and angles, revealing the operating mechanism that the broadband polarization-independent absorption is implemented by the coupling of electric and magnetic resonances.

In addition, the far-field patterns of the designed absorber with $10\times 10$ units and group metal of the same size are simulated. Figure 8(a) illustrates the simulated 3D far-field patterns of both absorber and metal under normal incidence at 8, 14, 22, and 28 GHz. Compared to the total reflection at the bottom side (control group metal), the proposed absorber achieves the wavefront absorption at the top side. To directly show the benefit brought by the designed absorber, Fig. 8(b) gives the monostatic radar cross section (RCS) of the designed absorber and control group metal. The over 10 dB reduction is achieved from 5 to 34 GHz, which is highly coherent with the absorbing operating band under normal incidence. For visually analyzing the EM field distribution, the 3D far-field information is transformed to the 2D patterns shown in Fig. 9, where the plane diagrams consist of the far-field value at each elevation angle $\theta$ and azimuth angle $\phi$. To depict the absorbing effect clearly, the RCS of both the designed absorber and the control group at the direction $\theta ={\theta }_i$ and $\phi =0°$ is marked at 4, 8, 16, 24, and 32 GHz, respectively. Apparently, the marked points are also the maximum value in the spatial areas. Thus, the RCS reduction effect of the absorber can be derived from the marked point comparison between absorber and the control group. For 0° and 40° incidences [Figs. 9(a) and 9(b)], the designed absorber expresses over 10 dB reduction at 8, 16, 24, and 32 GHz within the operating band, while at outband 4 GHz, the reduction is only 6.9 dB at 0° and 8.7 dB at 40°. For 60° and 70° incidences [Figs. 9(c) and 9(d)], owing to the wide operating band under large angles, the designed absorber achieves over 10 dB reduction at 4, 8, 16, 24, and 32 GHz. Above all, the wide-angle broadband absorption has been strongly proved from different aspects such as reflection, power loss density, and far-field distribution, which all verify the good performance of our proposed absorber.

To prove the proposed design paradigm and the simulated performance experimentally, a prototype [Fig. 10(a)] with size $400\mathrm{mm}\times 400\mathrm{mm}\times 26.7\mathrm{mm}$ is fabricated. The 6 Ω/sq, 25 Ω/sq, and 35 Ω/sq ITO films are taken by magnetron sputtering with different tin–oxygen contents. To get the designed PMMA structures, precise matching is utilized to slice and polish each PMMA prototype. For microwave testing, the top structure is placed in a foam mold (${\epsilon }_r\approx 1.01$) and is then stacked on the bottom structure. In the foam mold, the ITO films of cubes are attached to the four faces of the cubes by electrostatic attachment. The whole prototype is placed in the mirror reflection test platform [Fig. 10(b)], which is surrounded by the microwave absorbing wedges. In the measurement process, two pairs of broadband antennas (2–18 GHz and 18–40 GHz, respectively) serve as a transmitter and receiver. Figures 10(c) and 10(d) showcase the measured reflection curves of TE and TM waves, respectively. The experimental results coincide with the simulated results practically from 0° to 70°, verifying the design theory for wide-angle absorption. For the optical frequency, the transmittance of the prototype is tested as being one of two types [Fig. 10(e)]. The red curve shows that the optical wave illuminates as the schematic of the red arrow type, which only penetrates the bottom structure; the optical transmittance is over 71.6%, from 400 to 800 nm. The blue curve denotes that the optical wave illuminates as the schematic of the blue arrow type, which penetrates both the top and bottom structures; the optical transmittance is over 69.5%, from 400 to 800 nm. In this work, the duty ratio of the top structure is calculated as ${(h/\mathit{N}p)}^2=0.36$, and the average optical transmittance is derived at over 70.8%, from 400 to 800 nm, which ensures high transparent performance for the naked eye.

In summary, a general strategy was proposed for ultrabroadband and wide-angle absorbers via multidimensional design of functional motifs. By coupling the orthogonal meta-atoms, functional motifs exhibit the angular tolerance of impedance matching. To broaden the operating frequency band and maintain angular tolerance simultaneously, an intelligent algorithm is utilized to search the optimal structure parameters. As a demonstration, a wide-angle broadband polarization-independent absorber is designed and verified in microwave regimes. What is more, the proposed functional motifs can be attached with optical transparency function at will, which strongly proves the feasibility of the multidimensional design. We firmly believe that the proposed method can be expanded from microwave to infrared and optical regimes, and the proposed design mentality can be extended to be multispectrum-compatible as an optical hologram and infrared (microwave) stealth-compatible, infrared and microwave stealth-compatible, optical visibility and wavefront manipulation-compatible, etc.