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Abstract
In this paper, a novel graphene-based dual-band perfect electromagnetic absorber operating in the mid-infrared regime has been proposed. The absorber has a periodic structure which its unit cell consists of a sliver substrate and two graphene nanoribbons (GNRs) of equal width separated with a dielectric spacer. Two distinct absorption peaks at 10 and 11.33 µm with absorption of 99.68% and 99.31%, respectively have been achieved due to a lateral displacement of the GNRs. Since graphene surface conductivity is tunable, the absorption performance can be tuned independently for each resonance by adjusting the chemical potential of GNRs. Also, it has been proved that performance of the proposed absorber is independent of the incident angle and its operation is satisfactory when the incident angle varies from normal to ±75°. To simulate and analyze the spectral behavior of the designed absorber, the semi-analytical method of lines (MoL) has been extended. Also, the finite element method (FEM) has been applied in order to validate and confirm the results.
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1. Introduction
Metamaterial absorbers have recently attracted considerable research interest due to their ability to perfectly absorb and trap incident electromagnetic waves in sub-wavelength scale structures [1,2]. These kinds of absorbers have potential applications at microwave [3,4], THz [5,6], near- or mid-infrared [7,8] and visible light [9,10] frequencies. Based on the desired applications, they can be designed to operate as single-band [11–13], dual-band [14,15], multi-band [16,17] or wide-band absorbers [18,19]. Their field of application includes optical absorption and sensing [20], antennas [21], solar energy harvesting [22,23], microwave detection [24] and etc.
Since the first practical implementation of metamaterial absorbers in 2008 by H. Tao et al [25], extensive research has been done to develop and improve their performance; so that some absorbers have been designed to be insensitive to the polarization and angle of incident light while maintaining their high absorption rate [26–28]. According to the type of the absorption band, there are different mechanisms to realize an absorber. For example, to design a narrow band absorber, each of the Fabry-Perot, localized surface plasmon, magnetic and Fano resonances can be utilized [29–31]. Some of these methods result in complex designs, like nanodisk-based and hybrid plasmon structures, which complicate the fabrication process and significantly increase their cost. In addition, to adjust the absorption wavelength and peak, the geometric dimensions should be manipulated, which is very challenging. To overcome these challenges, active tuning methods have been proposed, which include applying an external voltage, electric or magnetic fields, chemical doping, and etc. [32,33]. In order to take advantage of these methods, it is necessary to use suitable materials that can be adjusted using the above-mentioned techniques. Graphene as a 2D material with astonishing electrical, optical, mechanical and thermal properties would be a good candidate with a tunable surface conductivity via applying an external voltage [34–38].
In this paper, a periodic multilayer graphene-based structure has been designed as a dual-band tunable wide-angle perfect absorber in the mid-infrared regime. Its unit cell consists of a silver film as the back reflector to eliminate transmission, a dielectric layer and two GNRs of equal width spaced apart. The presence and effect of these two GNRs can be viewed from two perspectives. First, the dual-band characteristics of the proposed absorber has been obtained through a relative displacement of these GNRs in the lateral direction unlike other designs that are based on GNRs of different width in different layers of the structure. Second, each of the GNRs is responsible to excite surface plasmons in a specific wavelength to achieve perfect absorption with independent tunability. To simulate and analyze the performance of the proposed dual-band absorber, the semi-analytical MoL has been extended. We have presented the detailed analysis of plane wave scattering from periodic multilayer graphene-based structures using MoL in our pervious works [39,40]. To verify the efficiency and accuracy of MoL, the finite element method (FEM) has been applied. It has been shown that their results are in excellent agreement.
The rest of the paper has been organized as follows: In section 2, the geometrical design of the proposed absorber is presented. A brief introduction of MoL as well as simulation results are presented in section 3. Finally, section 4 concludes the paper.
2. Absorber design
As shown in Fig. 1(a), the proposed absorber consists of a periodic array of GNRs separated by a dielectric layer and a dielectric spacer backed with a silver film. $SiO_2$ with the refractive index of 1.45 is assumed to be the non-dispersive dielectric. Also, a bias circuit is considered to actively tune the GNRs conductivity and consequently absorption behavior of the proposed absorber. Figure 1(b) illustrates the unit cell of the structure with $P=300$ nm, $H=600$ nm, $h=50$ and $w=150$ nm. Note that the silver layer should be thick enough to prevent power transmission. The frequency dependent permittivity of silver is expressed with the Drude model as [12]
in which $\varepsilon _{\infty }=3.7$, the plasma frequency $\omega _p=9.1$ eV and the collision frequency $\gamma =0.018$ eV. Moreover, the frequency dependent effective permittivity of graphene can be calculated as [32]: where $\varepsilon _0$ is the vacuum permittivity and $t_g = 0.34$ nm is the graphene thickness. The graphene’s surface conductivity $\sigma _g$ is calculated through the Kubo formula as follows [12,32] :3. Simulation results and discussion
MoL is an efficient semi-analytical method for solving partial differential equations with which physical phenomena are described. For the case of electromagnetic absorbers, Maxwell’s curl equations are used to analyze the problem. To solve these equations using MoL, all but one of the independent spatial variables have been discretized with finite differences to obtain ordinary differential equations (ODEs) system. Then, the resulting system of ODEs has been solved analytically. For this purpose, the reflection coefficient transformation approach has been introduced [39,40].
In this paper, we have extended MoL for the analysis of our proposed dual-band perfect absorber. To start the analysis, Maxwell’s curl equations are used to obtain the generalized transmission line (GTL) equations as [41]
According to Fig. 1, this absorber is infinitely long in the $y$ direction which translate into $D_{\bar {y}}=0$. Besides, the structure is periodic in the $x$ direction. Hence, the fields and their derivatives need to be discretized using finite differences considering periodic boundary conditions (PBCs) in this direction. For this purpose, consider the unit cell of the absorber as shown in Fig. 2(a) with $N_x$ discretization lines within the unit cell and PBCs on both sides. The discretized GTL equations are obtained as follows
Fig. 2. (a) The discretized scheme of the proposed absorber unit cell with PBCs, (b) Reflection coefficient transformation through layers and interfaces of th proposed absorber.
The general solutions of Eqs. (6) and (7) in each layer of the absorber are
Having obtained the electromagnetic fields in different layers of the proposed absorber (air and silica), the structure is divided into homogeneous layers in the vertical direction $z$ as shown in Fig. 2(b) and transform the known reflection coefficient at the last layer of the absorber ($\textbf {R}_{Ag}=-\textbf {I}$) through layers and interfaces to derive the overall reflection coefficient $\textbf {R}$ at the input of the proposed absorber. According to Eq. (8), we have
andSince $\bar {\textbf {E}}_b(0)=\textbf {R}_{Ag}\bar {\textbf {E}}_f(0)=-\bar {\textbf {E}}_f(0)$ and $\bar {\textbf {E}}_b(H^-)=\textbf {R}_{H^-}\bar {\textbf {E}}_f(H^-)$, we obtain
Next, we should transform $\textbf {R}_{H^-}$ through the interface at $z=H$ to obtain $\textbf {R}_{H^+}$. Note that, there exists GNRs at this interface. To transform reflection coefficient at the graphene-contained interfaces, appropriate boundary conditions should be applied. Since graphene is a single atomic layer with near zero thickness and is characterized with a surface conductivity, it can be electromagnetically modeled as an special impedance boundary condition as
After some mathematical manipulations $\textbf {R}_{H^+}$ is obtained as
To obtain $\textbf {R}_{(h+H)^-}$ and hence $\textbf {R}$, we should transform $\textbf {R}_{H^+}$ through the second silica layer of length $h$ and the graphene-contained interface at $z=h+H$ in the same manner, respectively.
Having obtained $\textbf {R}$, the input impedance of the proposed absorber calculates as
Also, the absorption coefficient calculates as
According to Eq. (17), zero reflection is needed to realize perfect absorption. On the other hand, zero reflection means perfect impedance matching between the absorber and free space based on Eq. (16) which means ${\textbf {Z}_{in}}/\textbf {Z}_0=\textbf {I}$.
In Fig. 3(a), the reflection and normalized input impedance spectra of the proposed absorber have been plotted for $g=75$ nm considering different chemical potentials. In this case, both GNRs have been placed in the middle of the unit cell and there is no offset between them in the horizontal direction ($x$). The corresponding absorption spectra are shown in Fig. 3(b). As it is seen, for ${\mu _c=0.4}$ eV the reflection coefficient is zero at 11.62 $\mu$m which corresponds to perfect impedance matching $(Z_{in}=Z_0)$ at this wavelength. Accordingly, a single perfect absorption occurs due to the localized surface plasmon excitation as a result of asymmetric electric dipole formation at this wavelength. Spatial distribution of the $y$ component of the electric field intensity has been shown in the inset. As the chemical potential increases, the impedance mismatch between the absorber and free space increases the reflection and hence decreases the absorption. Besides, the resonant peak shifts to shorter wavelengths. The reason behind these phenomena can be attributed to the changes of graphene’s optical properties and reduced field confinement around the GNRs as well as in the area between them.
Fig. 3. (a) Reflection spectra along with the real part of normalized input impedance for $g = 75$ nm and different chemical potentials (b) corresponding absorption spectra. $E_y$ distribution at $\lambda =11.62$ $\mu$m for $\mu _c=0.4$ eV has been shown in the inset.
Next, as our main design idea, a situation in which the two GNRs have been displaced horizontally from the middle of the unit cell in opposite directions has been examined. The absorption profile as a function of wavelength and horizontal offset $g$ has been computed using MoL and plotted in Fig. 4(a). In Fig. 4(b), the absorption spectra for $g$ = 30, 40 and 50 nm have been shown to better visualize the offset influence on the absorption behavior of the proposed absorber. As it is seen, for $g = 40$ nm (or equivalently $g=110$ nm) two perfect absorption peaks exist. Each of the GNRs is responsible for localized surface plasmon excitation at a specific wavelength. On the other hand, the resonance wavelength of the plasmons depends on the surrounding medium of the GNRs. The top GNR has been sandwiched between air and silica, while the over- and under-claddings of the bottom GNR are silica. So, the resonance wavelengths of these GNRs are different.
Fig. 4. (a) Absorption profile of the proposed absorber as a function of wavelength and $g$. (b) Absorption spectra for $g = 30, 40, 50$ nm. The graphene’s chemical potential is $\mu _c=0.68$ eV.
The absorption spectrum and normalized input impedance of the proposed absorber for $g=40$ nm have been shown in Fig. 5(a). To further analyze the absorption behavior of the proposed absorber, spatial distributions of $E_y$ at the resonance wavelengths of $\lambda _1=10$ $\mu$m and $\lambda _2=11.33$ $\mu$m have been depicted in the inset. As it is seen, local fields have been enhanced near the GNRs due to excitation of the localized surface plasmons and electric dipoles formation. The upper GNR is responsible for exciting plasmons at the shorter resonance wavelength, while the lower GNR excites plasmons at the longer resonance wavelength. Since the upper GNR has been surrounded by air and silica unlike the lower one which has been sandwiched between two silica layers, it resonates at a different wavelength compared with the lower one. To further explain this phenomenon, the absorption behavior of the proposed absorber for three different cases has been investigated according to Fig. 5(b), considering only the upper GNR, only the lower GNR, and finally both GNRs. As it is seen, each of the GNRs resonates separately at different wavelengths and forms a single band perfect absorption peak. However, using both GNRs, a dual-band absorption forms with a red shift in the resonance wavelengths, which is due to the coupling of the fields generated around the GNRs.
Fig. 5. Absorption spectra of the proposed absorber for $g = 40$ nm and $\mu _c=0.68$ eV, (a) obtained from MoL and compared with FEM results along with the real part of the normalized input impedance and $E_y$ distributions at $\lambda _1=10$ and $\lambda _2=11.33$ $\mu$m, (b) considering only the upper GNR, only the lower GNR, and both GNRs.
From another point of view, the absorption mechanism can be related to the concept of impedance matching. As shown in Fig. 5(a), the normalized input impedance of the absorber equals to unity at the resonance wavelengths, which means the perfect impedance matching $(Z_{in}=Z_0)$ at these wavelengths. Therefore, the reflection coefficient is zero and accordingly, perfect absorption realizes.
Now, dependence of the absorption behavior of the proposed absorber on its geometrical parameters should be investigated. Figure 6 illustrates the effect of dielectric layers thickness ($H,h$) on the absorption. It has been assumed that $g=40$ nm and $\mu _c=0.68$ eV. In Fig. 6(a) the absorption spectra for $h=40, 50$ and 60 nm have been plotted. As it is seen, increasing $h$ from 40 nm to 60 nm results in a blue shift in the resonances’ wavelength. As it is obvious, the first resonance which can be attributed to the upper GNR experiences a larger blue shift in compare to the second one. Based on the presented analyzes in [42,43], this phenomenon is due to the change in capacitance between two GNRs; as $h$ increases, the capacitance decreases and, as a result, the corresponding resonance wavelength decreases. However, due to the fact that $H$ is constant and the corresponding capacitance does not change, the change in the higher resonance wavelength which can be attributed to the lower GNR is negligible. In Fig. 6(b), the absorption spectra for different values of $H$ ranging from 500 to 700 nm have been plotted. Increasing $H$ has negligible effect on the resonance wavelengths and the absorption level as the cavity resonance modes resonate completely and the incident wave is trapped within the dielectric region. On the other hand, the resonance wavelength shift is also very small as $H$ increases, because the capacitance formed between the lower GNR and the back reflector layer does not change significantly.
Fig. 6. Absorption spectra of the proposed absorber for $g = 40$ nm and $\mu _c=0.68$ eV, (a) for $h = 40$, 50 and 60 nm, (b) for $H = 500$, 600 and 700 nm.
The effect of GNRs width on the performance of the proposed absorber has also investigated. For this purpose, according to Fig. 7, $\pm \%$20 changes in $\frac {w}{P}$ have been examined for $P=300$ nm. It is seen that changing the width of GNRs causes a shift in the resonance wavelength and a noticeable decrease in the absorption peak relative to the base case of $\frac {w}{P} =0.5$. The difference is that as $\frac {w}{P}$ decreases, the resonant wavelength experiences a red-shift, while as it increases, a blue-shift occurs. Since the resonance wavelength of the surface plasmons is a function of the surrounding medium as well as the dimensions of the plasmonic material, changing GNRs’ width causes a change in the resonance wavelength. On the other hand, as the GNRs’ width changes the capacitance formed between the upper and lower GNRs varies. As $w$ increases (decreases), the capacitance and hence the resonance wavelength increases (decreases).
Fig. 7. Absorption spectra of the proposed absorber for $g = 40$ nm, $\mu _c=0.68$ eV, and $w/P=0.4$, 0.5 and 0.6.
The next important phenomenon is the change in the absorption rate with the change in $\frac {w}{P}$. So that at $\frac {w}{P} = 0.4$ there is a decrease in the absorption level in both resonance wavelengths compared to the other cases. This happens because the GNRs can not trap the incident wave in the dielectric region. Besides, as discussed in [42,43], it can be concluded that in this case, the input impedance of the absorber does not completely match with the characteristic impedance of free space and hence the reflectance increases which results in a decrease in the absorption level. However, at $\frac {w}{P} = 0.6$, we see a perfect absorption at the higher resonance wavelengths (corresponding to the lower GNR) and $50\%$ absorption at the shorter resonance wavelength (corresponding to the upper GNR). Because in this case, the input impedance of the absorber changes due to the change in the characteristic impedance of the GNRs, and the reflectance increases; as a result, the absorption level decreases. In fact, increasing the width of GNRs reduces the entry of shorter wavelengths to the structure which leads to a decrease in the trapping of these wavelengths inside the absorber.
Note that, the electromagnetic simulations performed with the commercial finite-element electromagnetic solver COMSOL Multiphysics 5.2 have been used to verify the accuracy and efficiency of the extended MoL in Figs. 3 to 7. In COMSOL, graphene has been considered as a conductive layer with a thickness of 0.34 nm; where its conductivity modeled with the Kubo formula. On the contrary, in the extended MoL, graphene has been electromagnetically modeled as an impedance boundary condition. Although different methods have been used to model graphene, the results are in excellent agreement.
Till here, optimum values for the geometrical parameters of the proposed absorber have been obtained to realize the best dual-band performance considering the chemical potential of $\mu _c=0.68$ eV. In the following, tunability of the absorber has been investigated by adjusting the graphene’s chemical potential. Note that, the graphene’s surface conductivity is a function of its chemical potential and a change in $\mu _c$ results in a change in the optical properties of graphene and hence its surface plasmons resonate at different wavelength. Figure 8 shows the absorption profiles as a function of wavelength and chemical potential for different scenarios. In Fig. 8(a), $\mu _{c2}=0.68$ eV has been considered for the lower GNR’s chemical potential and the upper GNR’s chemical potential has been swept from 0.5 to 1 eV. At $\mu _{c1}=0.5$ eV, there exist a single band absorption, while the blue-shift occurs at the resonance wavelength of the upper GNR by increasing its chemical potential. In Fig. 8(b), the lower GNR’s chemical potential has been changed from 0.5 to 1 eV, while the upper GNR has a fixed chemical potential $\mu _{c1}=0.68$ eV. In this case, the resonance wavelength of the second band has been shifted to lower wavelengths as $\mu _{c2}$ increases and at $\mu _{c2}=0.85$ eV the two absorption bands have been coincide with one another and form a single band absorption peak. At last, the assumption of $\mu _{c1}=\mu _{c2}=\mu _c$ has been considered and the impact of sweeping $\mu _c$ from 0.5 to 1 eV has been studied according to Fig. 8(c). In this case, both resonances have been experienced blue-shift with increasing the chemical potential. Hence, a tunable dual-band absorber can be realized by applying a gate voltage to actively control the NRs’ chemical potential. Moreover, it is possible to tune each of the resonances independently by designing an appropriate bias circuit.
Fig. 8. Absorption profile (a) for $\mu _{c2}=0.68$ eV and $\mu _{c1}$ ranging from 0.5 to 1 eV, (b) for $\mu _{c1}=0.68$ eV and $\mu _{c2}$ ranging from 0.5 to 1 eV and (c) for $\mu _{c1}=\mu _{c2}=\mu _c$ ranging from 0.5 to 1 eV.
Last, effect of the incident angle on the absorption has been investigated. Figure 9 shows the absorption profile as a function of the incident angle. The absorption peaks slightly shift to lower wavelengths; however, the absorption level is insensitive to the angle of incidence while it varies from 0 to $\pm 75^\circ$. So, performance of the designed tunable dual-band absorber is independent of the incident angle to a great extent.
As an application-type functional device, the selection of actual fabrication steps is particularly important. The fabrication process consists of three steps as shown in Fig. 10 [44]. First, the silver back reflector and silica spacer are deposited on a substrate by physical vapour deposition technique. Next, the synthesized graphene is deposited on the silica layer. The GNRs of desired width are then created using a tightly focused femtosecond laser beam. Finally, the absorber fabrication is completed by repeating these three steps.
However, we may encounter some fabrication tolerances specially for the horizontal displacement of GNRs $g$. To gain an insight into the sensitivity of our proposed absorber to small changes in $g$, a fabrication tolerance analysis has been done. For this purpose, the absorption spectra of the absorber for $\pm 1$ nm to $\pm 5$ nm tolerances in the relative displacement $g$ calculated. The results are shown in Fig. 11(a) and 11(b) for the positive and negative tolerances, respectively.
To quantitatively investigate the fabrication tolerance, the wavelength shift of the resonance peaks and their corresponding absorption difference relative to the ideal case of $g=40$ or 110 nm have been calculated. For $g=115$ nm, the first resonance peak experiences $1.72\%$ blue shift with $5.2\%$ absorption reduction while the second one shifts $0.17\%$ to longer wavelengths and its absorption reduces $0.05 \%$. On the contrary, for $g=105$ nm, the red shift and absorption reduction of the first resonance are $0.76\%$ and $4.8\%$, respectively while the second resonance experiences $0.52\%$ red shift with $16\%$ absorption reduction.
Compared to the mid-infrared graphene-based dual-band absorbers designed in [14,15], our proposed absorber has a smaller thickness with a desirable performance in a wider range of the incident angle. Also, approximately perfect absorption in its both bands were realized. As well, unlike [14,15], the fabrication tolerance analysis was performed for our proposed absorber and it was shown that $\pm 5$ nm tolerance in the relative displacement $g$ has no significant effect on its performance. These differences are tabulated in Table 1.
Table 1. Comparison of the proposed absorber with some mid-infrared graphene-based dual-band absorbers
Fig. 12. Sensing analysis for (a) and (b) the single-band absorber, (c) and (d) the dual-band absorber.
Moreover, as a practical example, the proposed absorber has a potential to be used as an optical sensor. In what follows, the sensing ability of the designed absorber has been examined for different refractive indices. For this purpose, the air layer has been replaced with various gases and liquids of different refractive indices to be sensed. In Fig. 12, the absorption spectra of the single- and dual-band absorbers for different refractive indices have been shown. As it is seen, red shifts have been occurred in the resonance of the single-band absorber and in the first resonance of the dual-band absorber with variation of refractive index. Variations of the resonance wavelengths in each of the absorbers for gas and water with different refractive indices are approximately 30 nm which gives an approximate sensitivity of 3000 $(nm / RIU)$. Note that, the sensing operation of the dual-band absorber in its second resonance can also be studied.
4. Conclusion
A perfect dual-band tunable graphene-based absorber was designed in the mid-infrared spectral region. The unit cell of the proposed periodic absorber consists of two graphene nanoribbons of equal width with a constant distance in the vertical direction and a relative lateral displacement. The whole structure is backed with a silver substrate to eliminate power transmission. It was shown that when the GNRs are placed in the middle of the unit cell on top of each other with no lateral offset ($g=75$ nm), a single-band absorber achieved. The two distinct absorption peaks at 10 $\mu$m and 11.33 $\mu$m with near perfect absorption with efficiencies of $99.68\%$ and $99.31\%$, respectively were achieved through the relative displacement of GNRs when $g=40$ or 110 nm. Besides, each of the absorption peaks can be adjusted independently by changing the Fermi energy of GNRs through applying a variable external voltage source. It is also shown that the proposed absorber is independent of the incident angle and acts as a perfect absorber when the incident angle varies from 0 to $\pm 75^\circ$. Moreover, based on the fabrication tolerance analysis it is shown that $\pm 5$ nm tolerance in the relative displacement $g$ has no significant effect on its performance. The proposed absorber may find application in thermal imaging, filtering, data storage and sensing; that its operation as a refractive-index sensor was examined. Last but not least, we developed the semi-analytical MoL for the analysis of the proposed absorber where its accuracy and efficiency were evaluated and confirmed by FEM. In this method, GNRs modeled with an special impedance boundary conditions and the proposed absorber divided into homogeneous layers in the vertical direction while discretized in the $x$ direction using finite differences regarding periodic boundary conditions. Then, the reflection coefficient transformation approach used to analytically analyze the absorber.
Disclosures
The authors declare no conflicts of interest.
Data availability
The datasets generated and analyzed during the current study are available from the corresponding author upon reasonable request.
References
1. L. Feng, P. Huo, Y. Liang, and T. Xu, “Photonic metamaterial absorbers: morphology engineering and interdisciplinary applications,” Adv. Mater. 32, 1903787 (2019). [CrossRef]
2. C. Zhang, S. Yin, C. Long, B. W. Dong, D. He, and Q. Cheng, “Hybrid metamaterial absorber for ultra-low and dual-broadband absorption,” Opt. Express 29(9), 14078–14086 (2021). [CrossRef]
3. Q. Zhou, X. Yin, F. Ye, R. Mo, Z. Tang, X. Fan, L. Cheng, and L. Zhang, “Optically transparent and flexible broadband microwave metamaterial absorber with sandwich structure,” Appl. Phys. A 125(2), 131 (2019). [CrossRef]
4. H. Zhuang, X. Wang, J. Wang, M. Guo, D. Tang, B. Zhang, X. Chen, P. Chen, and Y. Yang, “Broadband microwave metamaterial absorber based on magnetic periodic elements,” J. Phys. D: Appl. Phys. 53(25), 255502 (2020). [CrossRef]
5. M. Zhang and Z. Song, “Switchable terahertz metamaterial absorber with broadband absorption and multiband absorption,” Opt. Express 29(14), 21551–21561 (2021). [CrossRef]
6. L. Qi, C. Liu, and S. M. A. Shah, “A broad dual-band switchable graphene-based terahertz metamaterial absorber,” Carbon 153, 179–188 (2019). [CrossRef]
7. L. Guo, X. Ma, Y. Zou, R. Zhang, and D. Zhang, “Wide-angle infrared metamaterial absorber with near-unity absorbance,” Opt. Laser Technol. 98, 247–251 (2018). [CrossRef]
8. Z. Li, L. Stan, D. A. Czaplewski, X. Yang, and J. Gao, “Wavelength-selective mid-infrared metamaterial absorbers with multiple tungsten cross resonators,” Opt. Express 26(5), 5616–5631 (2018). [CrossRef]
9. Y. Zhang, Z. Yi, X. Wang, P. Chu, W. Yao, Z. Zhou, S. Cheng, Z. Liu, P. Wu, M. Pan, and Y. Yougen, “Dual band visible metamaterial absorbers based on four identical ring patches,” Phys. E 127, 114526 (2021). [CrossRef]
10. S. Ogawa and M. Kimata, “Metal-insulator-metal-based plasmonic metamaterial absorbers at visible and infrared wavelengths: a review,” Materials 11(3), 458 (2018). [CrossRef]
11. M. Zhong, “Design and measurement of a narrow band metamaterial absorber in terahertz range,” Opt. Mater. 100, 109712 (2020). [CrossRef]
12. H. Li, L. Wang, and X. Zhai, “Tunable graphene-based mid-infrared plasmonic wide-angle narrowband perfect absorber,” Sci. Rep. 6(1), 1–8 (2016). [CrossRef]
13. S. Luo, J. Zhao, D. Zuo, and X. Wang, “Perfect narrow band absorber for sensing applications,” Opt. Express 24(9), 9288–9294 (2016). [CrossRef]
14. C. Cen, Y. Zhang, X. Chen, H. Yang, Z. Yi, W. Yao, Y. Tang, Y. Yi, J. Wang, and P. Wu, “A dual-band metamaterial absorber for graphene surface plasmon resonance at terahertz frequency,” Phys. E 117, 113840 (2020). [CrossRef]
15. Y. Zhao, Q. Huang, H. Cai, X. Lin, H. He, T. Ma, and Y. Lu, “Dual band and tunable perfect absorber based on dual gratings-coupled graphene-dielectric multilayer structures,” Opt. Express 27(4), 5217–5229 (2019). [CrossRef]
16. B. Mulla and C. Sabah, “Multiband metamaterial absorber design based on plasmonic resonances for solar energy harvesting,” Plasmonics 11(5), 1313–1321 (2016). [CrossRef]
17. M. Moniruzzaman, M. T. Islam, G. Muhammad, M. S. J. Singh, and M. Samsuzzaman, “Quad band metamaterial absorber based on asymmetric circular split ring resonator for multiband microwave applications,” Results Phys. 19, 103467 (2020). [CrossRef]
18. Y. Wu, H. Lin, J. Xiong, J. Hou, R. Zhou, F. Deng, and R. Tang, “A broadband metamaterial absorber design using characteristic modes analysis,” J. Appl. Phys. 129(13), 134902 (2021). [CrossRef]
19. M. Bağmancı, O. Akgöl, M. Özaktürk, M. Karaaslan, E. Ünal, and M. Bakır, “Polarization independent broadband metamaterial absorber for microwave applications,” Int. J. RF Microw. Comput. Eng. 29(1), e21630 (2019). [CrossRef]
20. Y. Yao, Z. Liao, Z. Liu, X. Liu, J. Zhou, G. Liu, Z. Yi, and J. Wang, “Recent progresses on metamaterials for optical absorption and sensing: a review,” J. Phys. D: Appl. Phys. 54(11), 113002 (2021). [CrossRef]
21. J. Zhang, J. Li, and J. Chen, “Mutual coupling reduction of a circularly polarized four-element antenna array using metamaterial absorber for unmanned vehicles,” IEEE Access 7, 57469–57475 (2019). [CrossRef]
22. M. Bagmanci, M. Karaaslan, E. Unal, O. Akgol, M. Bakır, and C. Sabah, “Solar energy harvesting with ultra-broadband metamaterial absorber,” Int. J. Mod. Phys. B 33(08), 1950056 (2019). [CrossRef]
23. A. Hoque and M. T. Islam, “Numerical analysis of single negative broadband metamaterial absorber based on tri thin layer material in visible spectrum for solar cell energy harvesting,” Plasmonics 15(4), 1061–1069 (2020). [CrossRef]
24. F. Ö. Alkurt, O. Altıntaş, M. Bakır, M. Karaaslan, E. Ünal, F. Karadag, O. Akgöl, and C. Sabah, “Microwave power imaging detector based on metamaterial absorber,” Opt. Eng. 59(8), 087104 (2020). [CrossRef]
25. H. Tao, N. I. Landy, C. M. Bingham, X. Zhang, R. D. Averitt, and W. J. Padilla, “A metamaterial absorber for the terahertz regime: design, fabrication and characterization,” Opt. Express 16(10), 7181–7188 (2008). [CrossRef]
26. N. I. Landy, S. Sajuyigbe, J. J. Mock, D. R. Smith, and W. J. Padilla, “Perfect metamaterial absorber,” Phys. Rev. Lett. 100(20), 207402 (2008). [CrossRef]
27. N. Thi Quynh Hoa, P. Huu Lam, and P. Duy Tung, “Wide-angle and polarization-independent broadband microwave metamaterial absorber,” Microw. Opt. Technol. Lett. 59(5), 1157–1161 (2017). [CrossRef]
28. H. Zhang, H. Qin, W. Zhang, L. Huang, and X. Zhang, “Moiré graphene nanoribbons: nearly perfect absorptions and highly efficient reflections with wide angles,” Opt. Express 30(2), 2219–2229 (2022). [CrossRef]
29. K. Zhou, Q. Cheng, L. Lu, B. Li, J. Song, and Z. Luo, “Dual-band tunable narrowband near-infrared light trapping control based on a hybrid grating-based fabry–perot structure,” Opt. Express 28(2), 1647–1656 (2020). [CrossRef]
30. S. Shu, Z. Li, and Y. Y. Li, “Triple-layer fabry-perot absorber with near-perfect absorption in visible and near-infrared regime,” Opt. Express 21(21), 25307–25315 (2013). [CrossRef]
31. H. Lu, B. P. Cumming, and M. Gu, “Highly efficient plasmonic enhancement of graphene absorption at telecommunication wavelengths,” Opt. Lett. 40(15), 3647–3650 (2015). [CrossRef]
32. F. Jabbarzadeh, M. Heydari, and A. Habibzadeh-Sharif, “A comparative analysis of the accuracy of kubo formulations for graphene plasmonics,” Mater. Res. Express 6(8), 086209 (2019). [CrossRef]
33. H. Khalilzadeh, A. Habibzadeh-Sharif, and N. Anvarhaghighi, “Design of a broadband infrared absorber based on multiple layers of black phosphorus nanoribbons,” J. Opt. Soc. Am. B 38(12), 3920–3928 (2021). [CrossRef]
34. P. Ghaderian and A. Habibzadeh-Sharif, “Rainbow trapping and releasing in graded grating graphene plasmonic waveguides,” Opt. Express 29(3), 3996–4009 (2021). [CrossRef]
35. F. Jabbarzadeh and A. Habibzadeh-Sharif, “Double v-groove dielectric loaded plasmonic waveguide for sensing applications,” J. Opt. Soc. Am. B 36(3), 690–696 (2019). [CrossRef]
36. F. Jabbarzadeh and A. Habibzadeh-Sharif, “High performance dielectric loaded graphene plasmonic waveguide for refractive index sensing,” Opt. Commun. 479, 126419 (2021). [CrossRef]
37. W. Zhang, T. Wu, and X. Zhang, “Reconfigurable topological phases in photoexcited graphene nanoribbon arrays,” J. Opt. 20(9), 095005 (2018). [CrossRef]
38. W. Zhang, J. Ren, and X. Zhang, “Tunable superradiance and quantum phase gate based on graphene wrapped nanowire,” Opt. Express 23(17), 22347–22361 (2015). [CrossRef]
39. M. Z. Bideskan, K. Forooraghi, Z. Atlasbaf, A. Mehrdadian, and A. V. Lavrinenko, “Method of lines for the analysis of tunable plasmonic devices composed of graphene-dielectric stack arrays,” Opt. Express 29(18), 28787–28804 (2021). [CrossRef]
40. M. Ziaee Bideskan, K. Forooraghi, and Z. Atlasbaf, “Method of lines for analysis of plane wave scattering by periodic arrays of magnetically-biased graphene strips,” Sci. Rep. 11(1), 7588 (2021). [CrossRef]
41. R. Pregla, Analysis of electromagnetic fields and waves: the method of lines, vol. 21 (John Wiley & Sons, 2008).
42. Z. Xu, D. Wu, Y. Liu, C. Liu, Z. Yu, L. Yu, and H. Ye, “Design of a tunable ultra-broadband terahertz absorber based on multiple layers of graphene ribbons,” Nanoscale Res. Lett. 13(1), 143 (2018). [CrossRef]
43. K. Arik, S. AbdollahRamezani, and A. Khavasi, “Polarization insensitive and broadband terahertz absorber using graphene disks,” Plasmonics 12(2), 393–398 (2017). [CrossRef]
44. H. Lin, B. C. Sturmberg, K.-T. Lin, Y. Yang, X. Zheng, T. K. Chong, C. M. de Sterke, and B. Jia, “A 90-nm-thick graphene metamaterial for strong and extremely broadband absorption of unpolarized light,” Nat. Photonics 13(4), 270–276 (2019). [CrossRef]
