The designed optical transparent MMA based on resistance loss and multi-resonance characteristics is shown in Fig. 1(a), consisting a periodic arrayed ITO-PET on the top layer, PMMA plate and low resistance ITO-PET reflective backplane placed in the middle and bottom layers in sequence. The top and side view of the MMA unit-cell structure are illustrated in Fig. 1(b) and Fig. 1(c), respectively. All the constituent materials have prosperous optical transparency, which ensures the overall structure possesses distinguished optical transmittance properties. Multiple resonance modes are introduced into the periodic arrayed ITO-PET to generate different EM resonances to ensure the broadband microwave absorption. Excellent microwave absorption under different polarization modes can be achieved because of the proposed MMA with four-fold symmetry. In this work, both of the upper and lower surfaces were composed of ITO-PET film with commercial availability. The dielectric constants of PMMA and PET are 2.25(1-j0.001) and 3.0(1-j0.06), respectively.

The numerical simulation for the unit-cell structure of the MMA was performed by the CST Microwave Studio 2016. The boundary conditions of the unit-cell structure were set to periodic boundary condition and add space in x-y plane and z direction respectively. The absorbance of the MMA in the frequency range of 2–18 GHz can be calculated by S-parameter. The relationship among absorbance, reflectance and transmittance is as follows36:

$$R(\omega )=|{S_{11}}{|^2}$$

1

$$T(\omega )=|{S_{21}}{|^2}$$

2

$$A(\omega )=1 - R(\omega ) - T(\omega )$$

3

where *A*(*ω*), *R*(*ω*) and *T*(*ω*) are energy absorbance, reflectance and transmittance, respectively. The bottom ITO film with low resistance can be regarded as a metal backplane with good conductivity, which ensuring the incident EM wave was almost reflected. In consequence, the transmittance can be neglected, and the absorbance is approximately equal to36:

$$A(\omega )=1 - R(\omega )$$

4

To clarify the process of our proposed structure for ideal absorption bandwidth, the intermediate configuration process of the absorber is analyzed, as shown in Fig. 2. An absorption peak appears at 8.4 GHz, and the absorbance is less than 90% in the whole simulation frequency range when there is only the cross structure on the top layer. The bandwidth was enhanced by incorporating the cross ring in the structure, and the resonant frequency moves to lower frequency. Finally,, the windmill structure was integrated in the final design, which further enhances the level of absorbance.

Microwave absorption bandwidth of the MMA is a key factor to evaluate the microwave absorption performance, which can be regulated by changing the geometric structure and composition materials. To obtain excellent microwave absorption performance, the geometric parameters of the proposed structure are optimized and the absorbance change with the structural parameters is shown in Figs. 3(a-d). With the increase of PMMA thickness *t**d*, the absorption performance of the low-frequency part becomes better, while the high-frequency part becomes weaker, and the overall absorption bandwidth becomes narrower. The proposed MMA obtains the best absorbance when the PMMA thickness td equals 3.2 mm. In the upper ITO layer, the absorbance at low frequency becomes worse with the increase of *g*1. The MMA shows the best absorption effect when *g*1 is 0.2 mm. It can be seen from Fig. 3(c) that the change of the resistance *R**s2* value of the reflection backplane has little effect on the absorbance, which means that ITO resistance can have a certain error in the actual production of the bottom. The relationship between the microwave absorption bandwidth and the value of the top layer surface resistance (*R**s1*) is shown in Fig. 3(d). The absorbance shows three sharp and discontinuous absorption peaks when the value of top layer surface resistance is 45Ω/sq. As the *R**s1* value increases to 65Ω/sq, the three absorption peaks become mild and continuous. Therefore, the enhanced effective absorption bandwidth maybe due to the improved impedance matching. In the resonant circuit, the quality factor *Q* of the circuit is closely related to the characteristics of the resonant peak. The higher the *Q* value, the sharper the resonant curve. In the designed structure, with the increase of *R**s1*, the quality factor *Q* gradually decreases, and the three resonant peaks tend to be continuous. The relation formula between quality factor and bandwidth is30:

$$Q={\omega _0} \times \frac{{{P_T}}}{{{P_L}}}=\frac{{{f_0}}}{{\Delta f}}$$

5

where *P**T* and *P**L* are the stored and consumed energy, *ω*0 and *f*0 are the resonant frequencies and Δ*f* is the absorption bandwidth. It can be seen from the absorbance curve that the resonant frequency *f*0 of the structure is constant, and the decrease of *Q* factor will increase the absorption bandwidth of the MMA. The ITO film with high resistance can increase the value of *P**L* and enables the fusion of multiple absorption peaks to achieve broadband absorption. The optimized geometric parameters of the proposed unit-cell structure are as follows: *p* = 16.6 mm, *w*1 = 4 mm, *w*2 = 11.2 mm, *w*3 = 2 mm, *w*4 = 5.4 mm, *g*1 = 0.2 mm, *g*2 = 0.1 mm, *d*1 = 1.3 mm, *d*2 = 4.6 mm, *t*p=0.175 mm and *t*d=3.2 mm. The resistance of the top ITO pattern layer and the bottom ITO reflective backplane are *R**S1*=55 Ω/sq and *R**S2*=6 Ω/sq, respectively.

The microwave absorbance of normal incident transverse electric (TE) wave along the surface of MMA is shown in Fig. 4(a). There are three absorption peaks at about 7 GHz, 11.4 GHz and 16.8 GHz, respectively. The absorption peaks at 7 GHz and 16.8 GHz are moderate and near perfect absorption is realized at 11.4 GHz. The MMA achieves effective absorption (90% absorbance) in the frequency range of 6.0-17.8 GHz, exhibiting a relative bandwidth of 99.2% with a center frequency of 11.9 GHz. To explain the response of the proposed MMA to the incident EM waves, the homogenization algorithm was used to retrieve the relative permittivity (\(\epsilon\)) and relative permeability (\(\mu\)) of the structure. The S-parameters *S**11* and *S**21* are necessary according to the equivalent medium theory to successfully retrieve the EM parameters of the MMA structure. Since the reflective backplane of the proposed transparent MMA is a whole piece of ITO film, in order to obtain *S**21*, we dig four square holes at four corners of the ITO backplane of the unit-cell.37 The calculation formulas of equivalent relative impedance (*z*), equivalent permittivity (\(\epsilon\)) and permeability (\(\mu\)) can be expressed as38:

$$z= \pm \sqrt {\frac{{{{(1+{S_{11}})}^2} - {S_{21}}^{2}}}{{{{(1 - {S_{11}})}^2} - {S_{21}}^{2}}}}$$

6

$$\varepsilon =\frac{n}{z}$$

7

The calculation formula of relative refractive index *n* is38:

$$n= \pm \frac{1}{{kd}}{\cos ^{ - 1}}\left[ {\frac{1}{{2{S_{21}}}}\left( {1 - {S_{11}}^{2}+{S_{21}}^{2}} \right)+2m\pi } \right]$$

9

and

$$k=\frac{{2\pi \lambda }}{c}$$

10

where *d* is the overall thickness of the proposed transparent MMA, *m* is any integer, *c* is the propagation velocity of wave in vacuum, and \(\lambda\) is wavelength. The S-parameter is obtained by EM simulation, and combined with Formulas (6)-(10), the relative EM parameters of the proposed MMA structure are shown in Figs. 4(b-d). Figure 4(b) shows the normalized input impedance (*z*) obtained in the frequency range of 4–18 GHz. The value of the real part of *z* is almost close to 1 besides one peak at about 14 GHz, and the imaginary part of *z* is around 0, indicating that the designed MMA meets the basic principle of impedance matching and the incident EM waves would enter into the absorber. Almost perfect impedance matching is implemented at 11.4 GHz, which is also consistent with the ultra-strong absorption band of incident EM waves at this frequency. Figure 4(c) and Fig. 4(d) show the curves of the real and imaginary parts of the relative permittivity and permeability of the proposed MMA. In the working frequency range (6.0-17.8GHz) of the MMA, the imaginary part of the relative permittivity varies from 0 to 20, and decreases with the increase of frequency, which indicates that the electrical loss decreases with the increase of frequency. The imaginary part of relative permeability varies from 0 to 15 in the working range of the MMA, and the response to EM waves at 6 GHz and 14 GHz is magnetic resonance. The relative EM parameters prove that electric resonance and magnetic resonance coexist in the proposed transparent MMA.

In MMAs, the augment of absorption intensity and absorption bandwidth is usually caused by the excited electric and magnetic resonance. Electric and magnetic resonance are closely related to the current flow direction of the top and bottom layers. To further understand the microwave attenuation mechanisms of the designed MMA, the electric current distributions on the top and bottom layers at the three resonant frequencies were investigated under normal incidence, as shown in Fig. 5. The excited electric current on the top layer induced by the incident EM waves is mainly concentrated at the corner of the internal cross, and the current direction is along the positive direction of x axis tt 7 GHz, as shown in Fig. 5(a). As shown in Fig. 5(d), the induced current on the bottom plate is along the negative direction of x-axis, which is opposite to the top layer, proving the near-field coupling between the top and the reflective backplane led to the magnetic resonance loss. Except for the opposite direction of the excited current, the induced current distribution at 11.4 GHz and 16.8 GHz are similar, which mainly concentrated in the windmill structure along the x-axis direction of the branch, as shown in Fig. 5(b) and Fig. 5(c). As shown in Fig. 5(e) and Fig. 5(f), the current on the backplane is anti-parallel to the current on the top layer at 11.4 GHz and 16.8 GHz which indicating that magnetic resonance loss occur at 11.4 GHz and 16.8 GHz,. As a result, the magnetic resonance makes the absorption performance of the MMA significantly enhanced. It also can be found that the current intensity at the top surface are far stronger than the bottom layer, indicating that the magnetic resonance maybe not the main reason for EM energy attenuation. The strong EM energy maybe caused by the resistance loss. By comparing Figs. 5(a)-(c) and Figs. 5(g)-(h), it is found that the EM waves energy loss is highly consistent with the location of the induced current concentration of ITO layers. This is because the ITO layers has a certain resistance, which will produce significant ohmic loss under the action of the induced current, according to39:

$${P_{loss}}={I^2}{R_{ITO}}$$

11

where *Ι* is the induced current intensity and *R**ITO* is the sheet resistance of the top ITO film.

Microwave absorption performance at different incident angles were discussed for most EM waves are not vertically incident on the absorber surface in practical applications. The absorption performance of the MMA with different incident angles under TE and TM polarization are shown in Fig. 6. For TE polarization and TM polarization, the microwave absorption performance of the MMA gradually deteriorates as the incident angle of EM waves increases from 0° to 60°. Compared with TE polarization, the microwave absorption performance under TM polarization at different incident angles is higher, indicating that the angle stability is better for TM polarization. In the case of TE polarized mode (Fig. 6(a)), when the incident angle is less than 30°, over 90% absorption can be still maintained. When the incident angle is greater than 45°, the absorption performance of the proposed MMA is greatly reduced. The main factor for the decrease of the absorbance is that the magnetic flux through the multilayer structure decreases with the increase of the incident angle, leading to the decrease of the circulating current of the ITO surface and the impedance mismatching. In the case of TM polarized mode (see Fig. 6(b)), when the incident angle is 0° − 45°, over 90% absorbance can still be maintained. Over85% absorbance can still be guaranteed when the incident angle reaches 60° under TM polarized mode, which is due to the magnetic field is parallel to the surface in TM polarization mode and the EM wave absorption is mainly caused by the magnetic resonance. 40