## Abstract

Designing an ultra-wideband metasurface absorber is challenging because of its lossy characteristic that hinders the understanding of its resonance behavior. In this study, a framework was formulated to extend the application of the theory of characteristic modes to the analysis and design of metabsorbers. The metabsorber and its lossless counterpart exhibited similar modal behaviors, hence revealing the absorption mechanism of metabsorber. By introducing absorption modes, a dual-band metabsorber was converted into an ultra-wideband metabsorber. This proposed metabsorber with a thickness of 1.99 times the Rozanov’s limit could measure a bandwidth of 5.51–36.56 GHz (or 6 octaves) with 90% absorptance.

© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

Absorbers convert the incident electromagnetic energy into heat to reduce scattering [1–5]. Thin and optically transparent microwave metabsorbers are attractive for applications that require optical transparency, microwave absorption and a low profile simultaneously, such as radio-frequency identification systems, anechoic chambers and industrial manufacturing [6–10]. However, it is challenging to meet the requirements of low profile and large bandwidth simultaneously.

Although optimization algorithms are effective, optimization-based methods such as genetic algorithm and particle swarm optimization are usually both high time and power consuming processes [11–15]. Moreover, it may be difficult to attain the fabrication tolerance required by machine-optimized designs [16–18]. Most importantly, the difficulties associated with the understanding of the underlying physics of such optimized structures impede the development of general rules or ideas for further optimizations of the bandwidth and thickness. On the other hand, the equivalent circuit method, in which a metasurface is modeled by an equivalent circuit containing L–R–C resonators, can effectively improve the bandwidth of metabsorbers [19–23]. However, a variety of equivalent circuit models are required owing to the large range in electrical sizes of the absorber under wideband conditions; further, its design is troublesome. For irregular shapes, it is difficult to extract equivalent circuit parameters. Therefore, new methods are needed for designing low-profile wideband absorbers.

Metasurfaces are widely used to improve the absorption performance of absorbers. The theory of characteristic modes (TCM), originally proposed for perfect electrical conductors (PECs) and wire antennas, has been recently demonstrated to be a powerful tool for designing metasurface antennas [24–28]. Although some valuable attempts have been made in the TCM formulations for lossy dielectric bodies [29–32], the formulations are not applicable for lossy sheets of infinitesimal thicknesses. As a result, the application of TCM to metabsorbers is limited to the analysis of the lossless counterpart of the metabsorber without including the loss in lossy sheet [33–35], where excitations are not considered; hence, the connection between the modes introduced by the lossless resonator and those by the lossy absorber is missing. For a meta-resonator with a significant number of modes, it is crucial to obtain information on the excited modes in addition to the existing modes to understand the underlying absorption physics and perform effective optimization.

In this study, an alternate method based on the characteristic mode formulation for lossless sheets was used to analyze the absorption performance of metabsorber using lossy sheets under planewave illumination. This developed approximation method was formulated to extend the application of TCM to the analysis and design of metabsorbers. The electric field (E-field) distribution as well as the absorption and scattering powers of the metabsorber and its lossless counterpart were found to be similar, and thus, both of them exhibited similar modal behavior. The approximation method was used to analyze a patch-type metabsorber using the characteristic mode, and it was found that the excitation of absorption modes caused the absorption phenomenon. For proof-of-concept, the patch-type metabsorber exhibiting dual-band absorption was converted into an ultra-wideband metabsorber by introducing additional modes. All the simulations were performed using CST Studio Suite 2020 [36]. Finally, the simulated results were verified through experiments to support the feasibility of the proposed method.

## 2. Designs and analysis

#### 2.1 Theory of characteristic mode

The TCM, first proposed in 1965 [37], can be applied to PECs, dielectrics and magnetic materials [24,38–40]. The characteristics mode formulation can be applied to different surfaces and interfaces using different integral equations, which can be solved by converting them into a matrix equation using the method of moments.

The characteristic mode analysis (CMA) is a simple and easy method to understand the physics behind many key factors of radiation and scattering problems. By solving a generalized eigenvalue equation: *X J*_{n}** **= λ

_{n}

*R J*_{n}, the eigenvectors

*J*_{n}and eigenvalues λ

_{n}are obtained, where n is the index of each mode order,

**and**

*R***are the real and imaginary Hermitian parts of the impedance matrix**

*X***=**

*Z***+ j**

*R***[41]. The eigenvalues λ**

*X*_{n}can be transformed to the modal significance given by: MS = |1 / (1 + jλ

_{n})| with a smaller range of [0,1]; the MS is a more convenient parameter for investigating the resonance behavior in wideband. The current

**induced on a PEC under an impressed E-field can be decomposed as:**

*J***= ∑ α**

*J*_{n}

*J*_{n}, where |α

_{n}| or the magnitude of the modal weighting coefficients (MWC) represent the contribution of each

*J*_{n}to the total radiated power

*P*

_{rad}as

*P*

_{rad}= Σ|MWC|

^{2}.

#### 2.2 Material and boundary approximations

Because the TCM is not applicable for lossy sheets of infinitesimal thickness, it cannot be applied to metabsorbers that use lossy sheets. However, the similarity of energy conversion between the lossless and lossy structures makes it possible to analyze the metabsorber using TCM. Figure 1 illustrates the energy conversion by ideal lossless and lossy structures under planewave illumination. In a lossless structure, the induced current scatters the incident energy into space, whereas in a lossy structure, the induced current converts the incident energy into resistive heat. Notably, the energy conversion by a structure with or without loss occurs through scattering and absorption caused by the induced current. According to the energy conservation law, the energy absorbed by a lossy structure is equal to that scattered by its lossless counterpart. In contrast, in an actual absorber, along with the absorption, a small fraction of the incident energy is scattered as well; however, this scattered wave has a negligible effect on the equivalence relationship. Therefore, the TCM can be applied to evaluate a metabsorber by analyzing its lossless counterpart.

Based on the aforementioned analysis, we propose the method of material and boundary approximation to analyze the modal behavior of a metabsorber through its lossless counterpart. A conventional patch-type metabsorber, whose configuration is shown in Fig. 2, was used for the comparing the results obtained by the proposed approximation method. The metabsorber consisted of two layers: a patch layer of periodic square patch resonators each with a width of L_{1} = 12 mm and a ground layer of a continuous film; both the layers were made of an indium tin oxide (ITO), which was sputtered on polyethylene terephthalate (PET) substrates (thickness, d_{pet} = 0.125 mm, relative permittivity: 2.65 and loss tangent: 0.015) to form a ground layer with a sheet resistance of 6Ω/sq and a patch layer with a sheet resistance of 160 Ω/sq. For the unit cell, the width of the lattice was P = 18 mm, and the thickness of the air gap was d_{air} = 6.125 mm.

The approximation method is valid when the metabsorber and its counterpart have similar E-field responses under planewave illumination. Hence, the E-field distribution comparisons are made below and above the patch, where the peak and null of the E-field are of interest. The comparisons of the E-field distributions for the approximations are shown in Table 1. In both the approximations, normal planewave irradiation is considered. The lossy sheet and its periodic boundary are approximated by the lossless sheet and absorbing boundary, respectively. Notably, the approximation of the material is for the patch and ground layers made of ITO. In addition, a coordinate system with origin at the center of the unit cell is used throughout the calculations.

The material approximation is built by approximating the lossy sheet with a lossless sheet. The E-field distributions exhibited by the structures with lossy and lossless sheets are similar when a periodic boundary is used. Figure 3(a) and 3(b) show the E-field distributions below the patch. Evidently, the tangential E-field reaches a maximum at both ends of the patch, whereas a low amplitude appears at y = 0 (below the patch). The “hips-like” structure of the E-field distribution indicates the presence of TM^{x}_{001} mode below the patch, implying that the mode behavior below the patch layer does not show any significant change when the material is changed from lossy to lossless sheet. Figure 3(d) and 3(e) depict the E-field distribution above the patch. Evidently, the maximum field is in the x-direction in this case, and the field at x = 6 is selected for a detailed comparison as demonstrated in Fig. 4(a). The two E-field-amplitude curves corresponding to the lossy and lossless sheets exhibit good fitting at the extrema and show a relative trend of the E-field. Thus, the mode behavior above the patch layer does not show any significant change upon changing the material. These results imply that replacing the lossy sheet of the metabsorber with a lossless sheet does not affect the spatial E-field distribution.

Next, we formulate the boundary approximation by approximating an infinite periodic boundary with a finite absorbing boundary. Similar E-field distributions are obtained under the two boundaries when a lossless sheet is used in both the structures. Figure 3(b) and 3(c) show the E-field distributions below the patch. The polarization of the field and the location where the extreme value occurs remain similar. Figure 3(e) and 3(f) show the similarity of the E-field distributions above the patch, and a detailed comparison is presented in Fig. 4(b), which further establishes the similarity in the field distribution. Based on these results, it can be considered that a periodic boundary can be approximated with a finite absorbing boundary. Therefore, similar E-field responses are obtained under the material and boundary approximations, and hence, the CMA can be used to analyze a metabsorber through the approximation method.

#### 2.3 Analysis and design based on CMA

The CMA results of the patch-type metabsorber were calculated and analyzed using the CMA tools and the proposed approximation method. Because the loss of substrates produces a negligible effect on the total absorption, the loss tangent of PET substrate was not considered in the calculations. The ground layer and substrates were extended infinitely; the metabsorber unit cells were extended infinitely in the x-y direction to simulate the absorption, which can be calculated by: A = 1 – R– T, where R = |S_{11}|^{2} and T = |S_{21}|^{2} are the reflectance and transmittance derived from the S-parameter, respectively.

Figure 5 shows a comparison between the scattering power of the lossless counterpart, represented by Σ|MWC|^{2}, and the absorbing power of the patch-type metabsorber. The metabsorber based on the ITO patch resonator (sheet resistance: 160 Ω/sq) achieves dual-band absorption in the ranges of 6.8–15.9 and 30.6–37.1 GHz (absorption > 0.9). The absorbing power of the metabsorber and the scattering power of the reactive counterpart showed similar trends. In addition, the minimum was observed at 23.5 GHz, and the two observed wideband peaks resemble each other. Furthermore, when we adjust the surface resistance of the patch, the magnitude of the absorption changes; however, the position of the absorption peak band is always roughly consistent with that of the scattering peak band. Thus, the absorption power of the patch-type metabsorber is similar to the scattering power (Σ|MWC|^{2}) of the lossless counterpart.

To reveal the dual-band absorption mechanism of the patch-type metabsorber, the CMA results are analyzed at the absorption peaks and absorption minimums at 9, 34, and 23.5 GHz. Since |MWC|^{2} measures the contribution of each mode to the total power, the excited modes are arranged in descending order of |MWC|^{2}, shown in Fig. 6. Normalization is performed by defining the |MWC|^{2} maximum as 1 at each frequency. Although the structure of metabsorber has a significant number of modes, only a few of them are excited. As can be seen, the excited single-dominant mode, mode 12 and mode 29 appear at 9 GHz and 23.5 GHz, respectively. In this scenario, the normalized |MWC|^{2} of mode 12 and mode 29 are much higher than that of other modes, hence the modal behavior of the single-dominate mode is consistent with that of the metabsorber. Unlike other frequencies, three dominant modes are excited at 34 GHz, as shown in Fig. 6(c).

The consistency between the radiation and absorption patterns becomes evident with the extended application of the reciprocity theorem [42]. Therefore, the mode radiation pattern can be used to analyze the absorption performance. Accordingly, the modal radiation patterns of the dominant modes are calculated and plotted in Fig. 7. As shown in Fig. 7(a), the dominant mode—mode 12—generates broadside radiation at 9 GHz, whereas the other modal patterns generate a lower magnitude of the E-field in the + z direction with a lower value of |MWC|^{2}, as shown in Fig. 7(b) and 7(c). In addition, at the other absorption peak, the three excited dominant modes generate similar broadside radiation, as shown in Fig. 7(h)–7(j). Conversely, a radiation null appears at the broadside of the frequency of the absorption minimum, as shown in Fig. 7(d). Consequently, when the planewave is normally incident on the counterpart of the metabsorber, only the modal currents that generate broadside radiation are excited. According to Joule’s law, the dominant modal currents flowing on the lossy sheet of the metabsorber cause absorption. Thus, under normal planewave irradiation, the excited dominant modes can be called as normal absorption modes (hereafter, referred to as absorption modes). The CMA results indicated that the absorption was caused by the occurrence of absorption modes.

The CMA results showed that it is difficult for the conventional patch-type metabsorber to achieve ultra-wideband absorption because of the lack of absorption modes. To the best of our knowledge, additional absorption modes can be excited by inserting additional resonators. Thus, the CMA-based method for wideband optimization can be summarized in the following three steps:

- 1. Replace the lossy sheet and periodic boundary of the metabsorber array with a lossless sheet and absorbing boundary. Hence, the lossless counterpart of the metabsorber can be analyzed using TCM.
- 2. Calculate Σ|MWC|
^{2}of the metabsorber under normally incident planewave, and find the frequency band in which no dominant modes (absorption modes) are excited. - 3. Insert additional resonators to introduce absorption modes in the corresponding frequency band to achieve ultra-wideband absorption optimization.

As a proof-of-concept design, the ring resonator is inserted into the patch-type metabsorber, and an absorption mode is introduced to achieve ultra-wideband optimization. The metabsorber and layer designs are illustrated in Fig. 8. The detailed parameter configurations are: P = 18 mm, L_{1} = 12 mm, L_{2} = 3 mm, d_{up} = 2.5 mm and d_{low} = 3.5 mm. For the ITO sheets on the substrates, the sheet resistance of the ground layer was R_{1} = 6 Ω/sq and that of the remaining layers was R_{2} = 160 Ω/sq. Figure 9 shows Σ|MWC|^{2} of the metabsorbers with patch (denoted by MA-A) and ring resonators (denoted by MA-B). Evidently, the Σ|MWC|^{2} peak of MA-B appears in the range of 18–28 GHz, indicating the appearance of the absorption mode; the absorption null of MA-A also appears in the same band. Therefore, it is possible to introduce additional absorption modes around 20 GHz by inserting the ring resonator of MA-B into MA-A for achieving ultra-wideband absorption. The geometrical model of the proposed full metabsorber (denoted by MA-C) and its absorption value are shown in Fig. 8(c) and Fig. 9, respectively. The proposed metabsorber achieves over 90% absorption in the range of 5.51–36.56 GHz. The three absorption peaks roughly correspond to the peaks of Σ|MWC|^{2}. This indicates that the absorption modes of MA-A and MA-B appear in MA-C, which, in turn, achieves ultra-wideband absorption.

To verify the introduced absorption modes, the modal behavior of the ring resonator with and without the influence of the patch resonator is elaborated here. The |MWC|^{2} values are shown in Fig. 10(a) and 10(d), which show that a similar single dominant mode was excited in both the metabsorbers. In addition, the modal radiation patterns of the dominant modes are plotted for comparison in Fig. 10(b) and 10(e). Similar strong E-fields in the + z direction are obtained in both the metabsorbers, despite the differences in the minor lobes. For further confirmation, the modal currents of the dominant mode in the ring resonator are compared, as shown in Fig. 10(c) and 10(f); the peak and direction of the modal currents are of interest in this case. An upward current is generated on the left and right sides of the ring, whereas a downward current is generated on the upper and lower sides. These results demonstrate that the absorption mode of the ring resonator remains almost unchanged with or without the influence of the patch layer. This proves that ultra-wideband absorption can be achieved by introducing absorption modes into the metabsorber.

The above-mentioned analysis and optimization indicate that Σ|MWC|^{2} can be used to predict the potential absorption performance of a metabsorber because it represents the total contribution of the absorption modes. Furthermore, an additional absorption mode can be introduced by inserting meta-resonators for ultra-wideband optimization. Most importantly, the CMA results provide physical insights into the absorption phenomenon as well as aid in developing insightful guidelines for the ultra-wideband optimization of the metabsorber.

## 3. Performance characterization

#### 3.1 Simulating performance characterization

In addition to the wideband absorption characteristic, it is necessary for the proposed absorbers to be thin and possess angular stability for practical applications. To evaluate the thickness of the proposed absorber, the ratio of the metabsorber’s thickness to Rozanov’s limit was compared. Rozanov’s limit of the wideband absorber can be calculated as:

_{s}is the static permeability, and d is the thickness of the absorber [43]. Because the material of the transparent metabsorber is nonmagnetic (μ

_{s}≡ 1), Eq. (1) can be simplified to

_{0}is expressed as: Γ

_{0 }= 20log (ρ

_{0}), and λ

_{min}and λ

_{max}are the maximum and minimum wavelengths of the frequency range for Γ

_{0}reflectance, respectively. Equation (2) shows that the Rozanov’s limit is (λ

_{max}– λ

_{min})/17.2 for the metabsorber exhibiting 10 dB reflectance (90% absorption). For comparison, the ratio of the metabsorber thickness to Rozanov’s limit is listed in Table 2. As can be seen, the proposed absorber achieves a wider absorption bandwidth with a similar thickness.

To evaluate the angular stability, the absorptance of MA-C under the TE and TM modes of obliquely incident planewave was simulated as shown in Fig. 11. Evidently, under TE mode incidence, the absorption is almost 90% at incident angles of 0°–25°, except at 27 GHz under which the absorption shows a decreasing trend. Under the TM modes, the absorption exhibits a slow blue shift as the incident angle increases, and the angular stability improves. Overall, the absorptivity of the metamaterial absorber is above 80% at incident angles of 0°–40° under both the TM and TE mode incidence across the working bandwidth, which spans the entire X-, Ku- and K-bands. These results establish the practical applicability of the proposed absorber.

#### 3.2 Experimental performance characterization

The absorption performance of the proposed MA-C metabsorber was experimentally verified as well. A square array of 16 × 16 unit cells with an overall width of 288 mm was fabricated, as shown in the inset of Fig. 12(b). The patch and ring resonators were developed by laser etching a uniform ITO sheet with a sheet resistance of 155 Ω/sq (the proposed value is 160 Ω/sq). Another ITO sheet with a sheet resistance of 6 Ω/sq was used as the ground layer. Two commercial polymethyl methacrylate (PMMA) frames with average thicknesses of 2.47 and 3.58 mm were used as supports for the upper and lower air gaps, respectively. The three layers were glued to frames with an optically transparent adhesive to fabricate air gaps with specific thicknesses. However, the optical adhesive and frames produced a slight thickness error owing to which the MA-C sample thickness was found to be d_{up} = 2.48 mm and d_{low} = 4.1 mm (the proposed is d_{up} = 2.5 mm and d_{low} = 3.5 mm).

Figure 12(a) shows the setup for the measurement of the reflectance at normal incidence. Owing to the excellent shielding performance of the ground layer, the transmittance was ignored. The measured absorption, which is defined by A = 1 – |S_{11}|^{2}, showed that the absorptance was above 90% over the frequency ranges of 5–9.95 and 19–36.06 GHz. As the fabricated sample had different parameters, the simulation was performed again to compare the resulting values. Evidently, the trend and the frequencies of the simulated and measured maxima were approximately similar to each other, although a difference was observed in the minimum at approximately 15 GHz. The error in the minimum was possibly caused by: (1) the larger size of the normalized metal plate, which had a width of 300 mm; (2) inconsistency in the measured thicknesses caused by slight sagging of the layers; (3) inclusion of PMMA frame; (4) resonance in the actual finite structure; and (5) inhomogeneity of the ITO sheet. In summary, the appearance of the three resonant peaks demonstrated the effectiveness of introducing absorption modes for wideband optimization and provided insights into the physics of the excited modes of the meta-resonators.

## 4. Conclusion

In this study, we extended TCM to the analysis and design of lossy absorbers. By comparing the vector E-field distributions as well as the absorption and scattering powers of the same meta-resonators with and without loss, a connection was built between the resonant modes of the metabsorbers and their counterparts. With the absorption physics clearly revealed, additional absorption modes were introduced to improve the bandwidth of the conventional patch-type metabsorber significantly, thereby forming the proposed optically transparent metabsorber that simultaneously exhibited ultra-wideband absorption characteristic and a low profile. Furthermore, the proposed structure exhibited a wider absorption bandwidth than did the previously reported transparent wideband metabsorbers with similar thicknesses. The proposed analysis and design framework paves the way for the physics-oriented optimization of absorbers with the aid of CMA.

## Funding

National Natural Science Foundation for Distinguished Young Scholars of China (51625201); Key Project of National Natural Science Foundation of China (52032004); Shanghai Pujiang Program (20PJ1411400).

## Disclosures

The authors declare no conflicts of interest.

## Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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