## Abstract

In this paper, we demonstrate a multimode and broadband absorber that is fabricated directly on PET substrate using a commercial direct-to-garment (DTG) inkjet printer. A design procedure of this kind of absorber is presented. Based on the theory of characteristic mode, the underlying modal behaviors of the absorber structure are firstly analyzed to guide the design of multimode absorber. Two modes on the absorber structure are designed to resonate around 1.83 GHz and 4.28 GHz to cover the working frequency range. Simulation and measurement results show that the multimode absorber with a total thickness of 0.0883*λ*_{L} at the lowest operating frequency can achieve broadband microwave absorption with efficiency over 90% in the frequency band of 1.0 ∼ 4.5 GHz (127.3% in fractional bandwidth) through deliberate design. Both the simulated and experimental results demonstrate the validity of the proposed method and indicate that the method can be applied to other microwave and millimeter-wave regions.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

Radar absorbing materials can dissipate the incident electromagnetic energy and have been widely used to minimize the radar signature of targets over limited aspect angles [1]. With the rapid development of radar technology, an absorber with versatile features and enhanced performances, such as ultra-bandwidth [2,3], angular robustness [4], multi-physical manipulation [5,6], low-profile and simple in configurations [7] is desired. Salisbury screen, which is simply comprised of an impedance sheet and a ground plane, is the simplest type of absorber. Jaumann absorber is similar to the Salisbury screen except that it is made of more impedance sheets. By arranging these impedance sheets in multiple layers, broadband characteristics can be obtained. However, Jaumann absorber also has some weakness in thickness and weight. To overcome these disadvantages and limitations, Circuit Analog (CA) absorbers are introduced [8]. A method often used in the design and analysis of CA absorber is equivalent circuit method, which is according to the fact that the frequency selective surfaces (FSS) used in the design are band-stop surface and can be modeled by series RLC resonators [9]. With this method, Debidas Kundu [10] have been proposed an ultra-wideband low-profile absorber (which has the bandwidth of $3 \sim 12$ GHz and the thickness of $0.080\lambda _{L}$ at the lowest operating frequency) based on the printed lossy capacitive surface using the impedance analysis. In [11], a single-layer circuit analog absorber consisted of double-square-loop array with three resonances has been studied. The authors use equivalent circuit method to explain how these three resonances can be produced and its absorption bandwidth can be widened. In these works, the equivalent circuit method can accurately predict the behaviors of CA absorber with simple structure and guide the design of CA absorber. However, it failed for an absorber with complex structures. In addition to equivalent circuit method, many design methods have been proposed to obtain a perfect absorber, including capacitive surface method [7,12], using metasurface groud plane [3], equivalent transmission line method [13] and so on. In recent years, a novel method based on antenna reciprocity is proposed in [14]. The authors firstly design an antenna array with the best transmission performance and then configure it as an absorber with this method. This work reminds us that there are also many mature technologies for antenna design; In the absorber design and analysis, we can use for reference of antenna society, introduce into some advanced design methods that have deep physical meaning.

The three most popular numerical techniques in the computational electromagnetic are the method of moments (MoM), the finite difference time domain method (FDTD), and the finite element method (FEM), respectively. With these techniques, many excellent absorber designs with low-profile and broadband properties are reported [15,16]. To investigate the mechanisms of the absorber, one usually examine the electric field or the surface current distribution on the FSS structure for different frequency points for normal or oblique incidence of angle. Because incident plane wave is considered in these full-wave simulations, an improper selection of the frequency point and the angle of incidence will produce misleading current distributions on the FSS structure. However, Source-free Characteristic Mode Analysis (CMA) [17] allows obtaining the resonant frequencies of the fundamental mode and higher-order modes without needing any sources. These valuable modal analysis results indicate how to excite the electromagnetic structure or introduce new current modes on the structure. Compared with full-wave simulation that is time-consuming and incomplete for modal analysis, CMA is simple and provides all mode information at the same time. Based on CMA, many high-performance antennas with multimode and low-profile are proposed for wideband operation [18–20]. To the authors’ knowledge, the CMA is not already applied to the design and analysis of absorber or FSS.

Here, we demonstrate a multimode and broadband absorber that are fabricated directly on PET substrate using a commercial Direct To Garment (DTG) inkjet printer. A design procedure of this kind of absorber is presented. Based on the theory of characteristic mode, the underlying modal behaviors of the absorber structure are analyzed firstly to guide the design of multimode absorber. Two modes on the absorber structure are selected to resonate around 1.83 GHz and 4.28 GHz to cover the band of $1.0\sim 4.5$ GHz. Simulation and measurement results show that the multimode absorber with a total thickness of $0.0883\lambda _{\textrm {L}}$ at the lowest operating frequency can achieve broadband microwave absorption with efficiency over 90% in the frequency band of $1.0\sim 4.5$ GHz (127.3% in fractional bandwidth) through deliberate design. All simulations are carried out by the Computer Simulation Technology (CST) Microwave Studio.

## 2. Designs and simulations

#### 2.1 Theory of characteristic mode

When an incident wave incident upon a Perfect Electric Conductor (PEC) structure and will induce surface currents $\boldsymbol {J}$ on the structure. According to the theory of characteristic mode (CM) [17], the current $\boldsymbol {J}$ can be written as a superposition of the characteristic currents:

where $\boldsymbol {J_{n}}$ are a complete orthogonal set of modal currents. $a_{n}$ denote the modal weighting coefficients and are given by where the $\lambda _{n}$ denotes the eigenvalue of mode $\boldsymbol {J_{n}}$, $\boldsymbol {E_{i}}$ denotes the external electric field and $S$ denotes the surface of PEC structure. Modal Significance (MS) is defined as The MS is the intrinsic property of each mode and is independent of any specific external source. It is an important tool to predict the resonant frequency and potential contribution to the radiation of a mode. The MS value ranges from 0 to 1. If $\textrm {MS} = 1$ at a certain frequency, the mode resonates at that frequency and radiates the most efficiently. It states the coupling capability of each mode with incident wave. The second term in the Eq. (2) is named modal excitation coefficient, which is defined as The modal excitation coefficient is usually used to determine which modes are going to be excited by an incident plane wave. However, it is at present not available in CST microwave studio. And the reflection coefficient obtained from FDTD provides an alternative approach. With this method, one can judge whether a mode is excited for a given incident plane wave by combing total current distribution obtained at resonant frequency.#### 2.2 Absorber with one mode

As shown in Fig. 1(a), the proposed structure with one mode consists of dielectric layer (PET), FSS layer (impedance sheet), honeycomb and ground plane. The honeycomb is a dielectric material, which is lightweight, strong and has a low dielectric constant (as close to air). It is used as isolation layer between the ground plane and the FSS layer. The height of honeycomb is typically chosen to be equal to $h=\lambda _\textrm {0}/4=30$ mm at the center frequency $f_\textrm {0}=2.5$ GHz. The surface resistances of FSS is 9 $\Omega /\square$. The PET substrate with a thickness of $t=0.1$ mm has a dielectric constant of 2.8 and a loss tangent of 0.03. The size of the unit cell shown in Fig. 1(b) is $p=34$ mm. The square loop has a length of $a=31$ mm and a line width of $w=1$ mm. In [8], the author uses the equivalent circuit shown in Fig. 1(c) to explain the working principles of CA absorber. The input impedance of the short-circuited transmission line is denoted by $Z_{\textrm {G}}$. The equivalent impedance of FSS is $Z_\textrm {FSS}$, which is given by $Z_\textrm {FSS} = -1/Z_\textrm {0} \cdot (1 + \varGamma )/(2\varGamma )$ [8]. The total input impedance of the absorber is $Z_\textrm {FSS}||Z_\textrm {G}$. The derivations of these impedances as a function of frequency are presented in Fig. 2(a). The Smith chart is normalized to $Z_{\textrm {0}}=377\ \Omega$. The square markers represent the lowest frequency $f_\textrm {L}$ and similarly the triangle markers represent the highest frequency $f_\textrm {H}$. It can be seen from the figure that $Z_{\textrm {G}}$ is located on the rim of the Smith chart. When the frequency is equal to $f_\textrm {0}$ , the input impedance of the short-circuited transmission line is $\infty$. It becomes inductive at a lower frequency $f_\textrm {L}$ and capacitive at the higher frequency $f_\textrm {H}$ as shown in Fig. 2(a). Thus, at the center frequency $f_\textrm {0}$, $Z_\textrm {G}= \infty$ will not change $Z_\textrm {FSS}$ when connected in parallel. At a lower frequency than $f_\textrm {0}$, since $Z_\textrm {FSS}$ is capacitive, it will push $Z_{\textrm {G}}$ downwards toward the capacitive part of the Smith chart. At a higher frequency than $f_\textrm {0}$, $Z_\textrm {FSS}$ is inductive and will consequently push $Z_\textrm {G}$ up toward the inductive part of the Smith chart as indicated by the green curve in Fig. 2(a). That is, the ground plane impedance is partly canceled by the reactive part of FSS impedance at higher and lower frequencies, causing the total input impedance to be located in the -10 dB circle (which is marked with red dash line) as shown in the Smith chart. Thus, we obtain a large of bandwidth of the total input impedance. Here, it should be noted that $Z_\textrm {FSS}$ is not completely symmetric around $f_\textrm {0}$. From the total input impedance we then obtain the reflection coefficient and plot it in Fig. 2(b). In the same figure, the full-ware simulation reflection coefficient is also provided for easy reference. It can be found that the equivalent circuit is coincident with the numerical simulation, showing a powerful ability of the equivalent circuit for the design of broadband absorber. The discrepancy between them can be possibly attributed to the interaction effects between the FSS and the ground plane.

To reveal the physical mechanisms of the absorber, we will analyze the modal behaviors of unloaded FSS in the following contents. This is because that the elements of a lossy material can be represented by the equivalent loss resistance [8]. Thus, the FSS made up of lossy material can be considered as an FSS consisted of lossless elements loaded with load impedance. The simulation results of an individual element are obtained using CMA tool, where the open boundary is applied and the dielectric layer is infinitely extended in x- and y-directions using layer stackup. In Fig. 3(a), the MSs of the first four modes from 1.0 to 7.0 GHz are calculated and sorted at 4.0 GHz. It can be seen from the figure that the fundamental modes (mode 1/mode 2), which are a pair of orthogonal modes, resonant at 2.67 GHz. There are only fundamental modes that may be excited by an incident plane wave over the working frequency range of $1.0\sim 4.0$ GHz. The other two modes, mode 3 and mode 4 resonant at 4.98 and 5.25 GHz, respectively. n the CM calculation, we can only obtain the modal behaviors of an individual element. However, when placed in an array, the mutual coupling between the individual elements is significant. In order to obtain an accurate resonant frequency, it is necessary to make numerical simulations. The reflection coefficients of the unloaded FSS for different incidence angles are calculated using FDTD method as shown in Fig. 3(b). The inserts of Fig. 3(b) show the current distributions of the fundamental mode and the higher-order mode obtained from FDTD simulations, where the black arrow indicates the current direction and the red arrow indicates the direction of electric field. As can be seen from the figure, only the fundamental mode resonating at 2.21 GHz can be strongly excited for normal incidence. When phi=$0^{\circ }$ and theta=$10^{\circ }$, mode 3 can be excited and mode 4 is still not excited. If we change the incident angle to phi=$45^{\circ }$ and theta=$10^{\circ }$, mode 4 is expectedly excited, as shown in Fig. 3(b). As described in the introduction, the higher-order modes are not successfully excited for normal incidence due to an improper selection of the angle of incidence. Therefore, the full-wave simulation will be not able to give the current distributions of the higher-order modes. In most cases, it is usually not easy to acquire the resonant behavior of a complicated resonator structure in all circumstances. At the same time, we calculate the modal currents of the FSS element at the resonant frequencies as shown in Fig. 4, where the black arrow indicates the current direction. As can be seen, mode 1 is in phase across the entire structure and mode 2 has the same current distributions but orthogonal current polarization. Thus, they both generate a broadside radiation pattern, as shown in Figs. 5(a) and (b). Since the current distributions of mode 3 and mode 4 are out-of-phase, there appears at broadside a radiation null, as shown in Figs. 5(c) and (d). Obviously, mode 1/mode 2 can be excited by any incidence plane wave according to reciprocity theorem. Mode 3 and mode 4, however, are excited only for oblique angles. This is expectedly consistent with observations from Fig. 3(b). A closer look at the current distributions obtained from CMA and FDTD reveals that there is a significant difference between them and the current distribution of fundamental mode resonated at 2.21 GHz obtained from FDTD as shown in the insert of Fig. 3(b) left can be understood as a linear combination of mode 1 and mode 2 as shown in Fig. 4. The limitation of acquiring the resonant behavior of a structure using full-wave simulation is further confirmed continually. From what has been discussed above, we may safely conclude that only one mode within the working frequency range ($1.0\sim 4.0$ GHz) is excited by a plane wave but the higher-order modes are suppressed.

#### 2.3 Absorber with multiple modes

In the previous section, an absorber with the bandwidth of $1.13\sim 3.92$ GHz is obtained after continuously optimized. However it still does not meet our desired bandwidth. This is because that the equivalent impedance $Z_{\textrm {FSS}}$ is highly capacitive at the lower frequencies and highly inductive at the higher frequencies. The equivalent impedance $Z_{\textrm {FSS}}$ and the ground plane impedance $Z_{\textrm {G}}$ can not partly cancel each other and thus yield a larger reflection from the back at the lower and higher frequencies. From the above analysis of the modal behaviors on the single square loop, it can be found that a single square loop can support four modes in the frequency range of $1.0\sim 7.0$ GHz. Only the fundamental mode is excited for normal of incidence and other modes may be excited only for oblique angles of incidence. It should note that, when the fundamental mode and higher modes are excited at the same time, the optimization and design method will become extremely complicated and hard to operate. Therefore, we have considered only the fundamental modes in the follow-up study. In this section, an absorber with multiple modes is proposed to broaden bandwidth and reduce thickness. The basic design is to introduce a fundamental mode resonating at the lower frequency and a higher mode resonating at the higher frequency to the structure for improving impedance characteristic of the FSS layer. The detailed configuration of the multimode absorber element composed of triple square loops is shown in Fig. 6(a), where $a_{\textrm {1}} = 32.4$ mm, $a_{\textrm {2}} = 21.1$ mm, $a_{\textrm {3}} = 8.0$ mm, $p = 34$ mm, and $w = 1.0$ mm. The geometry of the absorber with multiple modes is the same as the absorber with one mode. But the height of honeycomb is equal to $h=26.5$ mm. The MSs of the four modes from 1.0 to 7.0 GHz are shown in Fig. 6(b). Mode 1#/mode 2# and mode 3# /mode 4# resonate at 2.49 GHz and 3.99 GHz, respectively. In order to obtain the accurate resonant frequency, the reflection coefficient of the FSS layer as a function of frequency for normal of incidence is illustrated in Fig. 6(b). It can be seen from the figure that there are two modes which are excited at 1.83 GHz and 4.28 GHz, respectively. The corresponding modal currents and radiation patterns of the FSS element at resonant frequencies are shown in Fig. 7 and Fig. 8, respectively. As observed, the current distributions of mode 1# and mode 2# are localized and mainly distributed on the outer circle. The current distributions of mode 3# and mode 4# are also localized and mainly distributed on the inner circle. The current mode on the innermost circle is a special non-resonant mode within the working frequency range and can be used to adjust the impedance characteristic of FSS at the higher frequencies. The resonant frequencies for mode 1# and mode 2# are determined only by the outer circle size. Similarly, the resonant frequencies for mode 3# and mode 4# are determined only by the inner circle size. As discussed previously, the current distributions obtained from FDTD as shown in the insert of Fig. 6(c) are the linear combinations of current modes as shown in Fig. 7. The modal radiation patterns shown in Fig. 8 prove that the four modes can be excited by plane wave for normal or oblique angles of incidence.

The calculated impedances as a function of frequency are plotted in Fig. 9(a). The Smith chart is also normalized to $Z_\textrm {0}=377\ \Omega$. Obviously the ground plane impedance $Z_\textrm {G}$ at the lowest frequency decreases with reducing absorber thickness. Meanwhile the equivalent impedances of the FSS layer, which is represented by the orange line, are located closer to the line going through the zero point and the infinity point in the Smith chart especially at the lowest and highest frequencies. Thus, the problem that we face in the previous can be addressed by introducing multiple modes. Since the FSS equivalent impedance $Z_\textrm {FSS}$ and the ground plane impedance $Z_\textrm {G}$ to a large degree cancel each other over a broad frequency band, the parallel combination $Z_\textrm {FSS} || Z_\textrm {G}$ is entirely located in the -10 dB circle as shown in the Fig. 9(a); it leads to a small reflection. The reflection coefficients of the multimode absorber for normal incidence obtained from equivalent circuit and full-wave simulation are shown in Fig. 9(b). As expected, Simulation results show that the multimode absorber can achieve broadband microwave absorption with efficiency over 90% in the frequency band of $1.0\sim 4.5$ GHz (127.3% in fractional bandwidth).

## 3. Experimental results

The FSS layer is often made by silk-screening commercially available graphite- and carbon-based shielding paint into a pattern on a dielectric substrate. Engineers still need to depend heavily on personal experiences to obtain the proper resistivity and thickness to get the correct resistivity [21,22]. A thin resistive film obtained by vapor deposition is a significant improvement [8]. However, the high cost of the resistive pattern obtained, for example, by lithography etching processes should be considered. Instead, inkjet printing, which is a well-known drop-on-demand technology for depositing tiny droplets at a specific location of a substrate as shown in Fig. 10(a), is an appropriately scalable fabrication approach capable of producing resistive sheets with very close tolerances. It is important that this approach is a low-cost, simple and repeatable means to manufacture the FSS layer. Photograph of the inkjet-printed FSS on the PET substrate with an overall dimension of $30.6\times 30.6$ $\textrm {cm}^{2}$ fabricated using a commercial Direct To Garment (DTG) inkjet printer is shown in Fig. 10(b).

The measurement is carried out in a microwave anechoic chamber. The antenna used in the experiment operates between 0.5 to 6.0 GHz and is well suited for experimentation. The measurement system is based on an Agilent 8720ES vector network analyzer. At last, the measured and simulated reflection coefficients are illustrated in Fig. 10(c). As expected, the good agreement in bandwidth between the simulated and measured results validates that using multi-mode structures in the design of absorber can further enhance its broadband absorption performance. However, comparing the results from them, a significant difference in the absorption peaks especially at lower frequencies exists due to the finite size of the fabricated absorber used in the measurement. And the surface waves only exited in finite array may lead to some variation of the terminal input impedance of the individual elements [23]. Impedance matching becomes better at lower frequencies, causing the significant attenuation of echo signals. Even so, the factor does not change the analysis of the -10 dB absorption property of the proposed multimode absorber.

## 4. Conclusions

In summary, a multimode and broadband absorber that is fabricated directly on PET substrate using a commercial Direct To Garment (DTG) inkjet printer is proposed. Based on CMA, the modal behaviors and the mode excitation on the proposed multimode structure are analyzed in order to guide the absorber design. The results show that the equivalent impedance of the FSS with one mode is highly capacitive at the lower frequencies and highly inductive at the higher frequencies. In order to improve the impedance characteristic of the FSS, a fundamental mode resonating at the lower frequency and a higher mode resonating at the higher frequency are introduced to the structure. For the FSS with multiple modes, two modes on the structure are designed to resonate around 1.83 GHz and 4.28 GHz to cover the frequency band of $1.0\sim 4.5$ GHz. Simulations and measurements show that the multimode absorber with a total thickness of $0.0883\lambda _{\textrm {L}}$ at the lowest operating frequency can achieve broadband microwave absorption with efficiency over 90% in the frequency band of $1.0\sim 4.5$ GHz through deliberate design. Both the simulated and experimental results indicate that the multimode and broadband absorber may find promising applications for broadband stealth and the method can be applied to other microwave and millimeter-wave regions.

## Funding

National Natural Science Foundation of China (61172003).

## Disclosures

The authors declare that there are no conflicts of interest related to this article.

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