Multi-band terahertz anisotropic metamaterial absorber composed of graphene-based split square ring resonator array featuring two gaps and a connecting bar

Abstract

A multi-band anisotropic metamaterial absorber operating in the terahertz (THz) range is constructed using a graphene-based split square ring resonator array featuring two gaps and a connecting bar. The design is meticulously simulated through the finite element method (FEM) using CST Software. Subsequently, an equivalent circuit model (ECM) is introduced, leveraging impedance and transmission lines, and implemented with a rapid MATLAB code to evaluate the absorber’s behavior in the THz spectrum. The proposed absorber, dynamically adjustable through a one-layered resonator array, exhibits a strong linear dichroism response of 99% within a frequency range of 0.3–4 THz. The metamaterial has an absorption rate of 81% for one absorption band in transverse magnetic mode and its three absorption bands in transverse electric mode have an average of 99.3% in each absorption band with absorption over 99%. This absorber holds potential applications in polarization-sensitive devices and THz systems. The ECM model was established to provide an efficient analytical tool for assessing the absorber’s performance, and the FEM simulation results align well with those derived from the ECM.

Introduction

Anisotropic metamaterials exhibit non-equal responses to the incident waves along some different directions. Metamaterials fall into the category of anisotropic metamaterials if they possess at least one of the following features: anisotropic geometries, anisotropic material within the metamaterial structure, or applying an external magnetic field to the metamaterial. Metamaterials featuring anisotropic geometries do not coincide with their mirror image at some rotational angles¹.

Graphene, which is a two-dimensional layer derived from graphite, possesses exceptional qualities that render it a favorable option for optical devices and systems. Recently, there have been proposals, analyses, and investigations into anisotropic metamaterials based on graphene. These metamaterials demonstrate tunable anisotropic responses like linear dichroism (LD)^2,3,4,5. Tunable graphene-based terahertz (THz) anisotropic metamaterials can find potential applications in polarization-sensitive devices and systems.

Through the adjustment of the graphene-based metamaterial absorber’s spectral response by the alternation of the applied bias voltage dynamically, graphene-based multi-band absorbers can be employed in a variety of applications. They can be used to reduce electromagnetic interference for next-generation (6G) wireless communication multi-channel systems by absorbing undesired electromagnetic radiation⁶. Moreover, they have the potential to play a crucial role in advancing telecommunication multi-channel systems in the future. In addition to multi-channel telecommunication systems⁷ these multi-band metamaterial absorbers can be used in highly sensitive and selective sensing and imaging, spectroscopy applications^{8,9,10,11,12,13}, filtering¹⁰, switching¹², modulation¹³, energy absorbing and energy converting¹⁴, power detecting¹⁵, power receiving¹⁶, and more. However, most of the earlier reported structures have limited number of absorption peaks thus hindering their usability in some applications.

Creating multi-channel systems involves constructing components with multiple absorption bands. Metamaterials commonly achieve this by integrating either multiple layers of resonators or super unit cells to encompass a multi-band spectrum of absorption bands^17,18,19. Graphene-based anisotropic metamaterials reported in^2,5 are dual-band THz absorbers. The absorber in² has two layers of resonators each layer containing one resonator and the absorber in⁵ has two resonators in one layer of resonator. To increase the number of absorption peaks without compromising with simplicity of the structure, the aim of this work was to design a multi-band THz graphene-based anisotropic metamaterial absorber containing simple geometry with few resonator layers and few resonators per unit cell.

There is one paper reporting four THz graphene-based multi-band metamaterial absorbers composed of square split ring resonator arrays with different number of gaps²⁰. The absorbers were designed in 0.1–5.5 THz with limited maximum absorption and limited maximum LD response. In this paper, we have designed and developed a multi-band THz graphene-based anisotropic metamaterial absorber which is composed of a square split ring resonator array featuring two gaps and a connecting bar, with increased absorption, LD, and number of absorption bands. In addition, the ECM in this work differs from that reported in²⁰.

Materials and methods

Figure 1 illustrates the metamaterial absorber configuration from various perspectives, including periodic, unit cell, front, and side views. The layers making the metamaterial from the top to the bottom are respectively ion gel, graphene, Rogers RT5880LZ, and gold. Atop the metamaterial structure, there is a layer of ion gel with a permittivity value of ε_ig = 2.0164 and a thickness of 150 nm. The primary function of this ion gel layer is to apply bias to the underlying graphene resonator layer²¹.

Figure 1

(a) Periodic, (b) unit cell, (c) front, and (d) side views of the proposed multi-band terahertz anisotropic metamaterial absorber which is comprised of a graphene-based split square ring resonator array with two gaps and a bar between the gaps. The layers making the metamaterial from the top to the bottom are respectively ion gel, graphene, Rogers RT5880LZ, and gold. We have used a layer of ion gel to bias the resonator array and a gold layer beneath the metamaterial to avoid the transmission of electromagnetic waves.

The metamaterial is composed of a gold layer with high conductivity (σ = 4.56 × 10⁷ S/m)²². The maximum penetration depth ($\delta_{\max } = \sqrt {\frac{1}{{\sigma \pi f_{\min } \mu_{r} \mu_{0} }}}$) for the gold layer happens in 0.3 THz (minimum frequency f_min of the frequency range) where the relative permeability μ_r is 1 and vacuum permeability μ₀ is 4π × 10^–7. So, the maximum penetration depth is 0.1361 μm. The gold layer with a thickness of 0.5 μm which is much larger than the penetration depth (⁓ 3.7 × maximum penetration depth) of the THz waves is selected to ensure that the incident electromagnetic wave cannot transmit to the other side of the absorber and the transmission is zero because the electromagnetic waves can penetrate inside the gold layer up to its penetration depth²³. The gold layer acts as a perfect reflector for the considered THz design range. Moreover, it doesn’t produce any loss since it has a big conductivity and energy cannot go through it²⁴. Positioned on this gold layer is a dielectric layer, on which graphene patterns are arranged. In this configuration, graphene is depicted as an extremely thin layer (Δ = 0.335 nm) embedded within Rogers RT5880LZ. Rogers RT5880LZ has a permittivity value of ɛ_d = 2. The design also incorporates a graphene-based split square ring resonator array, featuring two gaps and a connecting bar, situated on the dielectric layer.

Fabrication of the metamaterial is not in the scope of this work, but we briefly summarize the fabrication process here for future readers of the work who may want to fabricate this metamaterial: Begin by applying a 13 µm thick dielectric with gold coating on one side through chemical vapor deposition (CVD)²⁵. Subsequently, use a standard lithography process to create the graphene patterns on the dielectric²⁶. Transfer the ion gel dielectric onto the graphene resonator array using thermal evaporation²⁷.

Dimensional and material parameters of the metamaterial of Fig. 1 are optimized. d_ion-gel is the thickness of the ion gel layer = 150 nm, d_substrate is the thickness of the dielectric substrate = 13 μm, d_gold is the thickness of the gold layer = 0.5 μm, P_x is the unit cell dimension in x direction = 16 μm, P_y is the unit cell dimension in y direction = 16 μm, a is the width of connecting bar = 1.2 μm, L is the length of connecting bar and outer length of split square ring = 15 μm, g is the gap distance = 0.6 μm, b is the width of split square ring = 1.5 μm, E_f is the Fermi energy level of graphene = 1 eV, and τ is the relaxation time of graphene = 2 ps.

Numerical simulations were conducted using the finite element method (FEM) with the frequency domain solver in CST Microwave Studio^{5,20,28,29,30}. Periodic (unit cell) boundary conditions were applied in the x and y directions, while an absorbing (open (add space)) boundary condition was implemented in the z direction. The metamaterial was meshed using a tetrahedral mesh^5,20,29,30. Optimization of the metamaterial was performed using the genetic algorithm (the parameter values were randomly changed to find the optimal values) within the CST software^{5,20,30,31,32}. The unit cell dimensions were set as P_x = P_y = 16 μm, a size smaller than λ_min = 75 μm, which corresponds to f_max = 4 THz, the upper limit of the simulated frequency range. This sizing decision was made to prevent the propagation of higher-order Floquet modes^{5,20,30,33,34}.

The total thickness of the metamaterial, including the ion gel, graphene, Rogers RT5880LZ, and gold layers, was 13.65 μm, approximately 0.182 × λ_min which shows that the metamaterial is relatively thin in the simulated frequency range^5,30. The relative permittivity and the surface conductivity of graphene are modeled as reported in^5,20.

As is clear in Fig. 1, the graphene-based resonator array is composed of the graphene-based split ring resonator array and the graphene-based middle connecting bar array. As shown in Fig. 2a,b, the impedance of the split ring resonator array is considered as Z_1, and the impedance of the middle connecting bar array is assumed as Z₂. The split ring resonator array consists of horizontal and vertical arms as shown in Fig. 2a. The ECM of the graphene-based resonator array in TE and TM modes are respectively given in Fig. 2c,d.

Figure 2

Equivalent circuit model (ECM) of the (a) graphene-based split ring resonator array composing of horizontal and vertical arms, (b) graphene-based middle connecting bar array, and graphene-based resonator array layer in (c) TE (E field along the y direction) and (d) TM (E field along the x direction) modes.

The graphene resonator layer has a non-symmetric geometry so the ECMs in TE and TM modes differ. For TE mode (Fig. 2c), the incident E field is along the y direction. There are four gaps between the middle connecting bar and the split ring resonator (shown in Fig. 2c). The E field is parallel to the gaps, so the gaps are not modeled. E field is along the middle connecting bar, so it’s modeled by Z₂. The split ring resonator is modeled by Z₁ and each half of the split ring resonator is modeled by 2Z₁. As depicted in Fig. 2c, the anisotropic metamaterial contains three parallel impedances in TE mode. So, the impedance in TE mode Z_TE is calculated by:

$$Z_{TE} = \frac{{Z_{1} Z_{2} }}{{Z_{1} + Z_{2} }}$$

(1)

Since the E field is along the y direction, the produced impedance by the horizontal arms (shown in Fig. 2a) of the split ring resonator is very small and neglected (the width of the arms is much smaller than their length). Each vertical arm (shown in Fig. 2a) of the split ring resonator produces 2Z₁ and they have equal length and width with the middle connecting bar, so they have equal impedances:

$$2Z_{1} \sim Z_{2}$$

(2)

So:

$$Z_{TE} = \frac{{Z_{2} }}{3}$$

(3)

For TM mode (Fig. 2d), the incident E field is along the x direction. The E field is normal to the gaps, so the gaps are modeled by capacitors and each capacitor is assumed to have a capacitance of 4c. E field is not along the middle connecting bar, so it’s modeled as an open circuit (O.C). The split ring resonator is modeled by Z₁ and each half of the split ring resonator is modeled by Z₁/2. The ECM in TM mode simplifies to a Seri of Z₁ and c.

The effective conductivity of the graphene resonator layer by Fresnel equation in TE mode can be obtained by³⁵:

$$\sigma_{TE} = \frac{{\cos \left( {\theta_{{{\text{in}}}} } \right) - \sqrt {\varepsilon_{d} } \cos \left( {\theta_{{{\text{out}}}} } \right) - r^{TE} \left( {\cos \left( {\theta_{{{\text{in}}}} } \right) + \sqrt {\varepsilon_{d} } \cos \left( {\theta_{{{\text{out}}}} } \right)} \right)}}{{Z_{0} \left( {1 + r^{TE} } \right)}}$$

(4)

in which θ_in, ε_d, θ_out, r^TE, and Z₀ are respectively the angle of the incident illuminated wave, the relative permittivity of the dielectric substrate, angle of the transmitted wave, the reflection coefficient of the graphene resonator layer in TE mode, and the vacuum impedance. The relation between θ_in and θ_out is:

$$\sin \left( {\theta_{{{\text{out}}}} } \right) = \sqrt {\frac{1}{{\varepsilon_{d} }}} \sin \left( {\theta_{{{\text{in}}}} } \right)$$

(5)

Impedance is the inverse of conductivity (admittance). So²⁰:

$$Z_{TE} = \frac{1}{{\sigma_{TE} }}$$

(6)

By substituting Eq. (6) in Eq. (3), we have:

$$Z_{2} = \frac{3}{{\sigma_{TE} }}$$

(7)

which is equal to:

$$Z_{2} = \frac{{3Z_{0} \left( {1 + r^{TE} } \right)}}{{\cos \left( {\theta_{in} } \right) - \sqrt {\varepsilon_{d} } \cos \left( {\theta_{out} } \right) - r^{TE} \left( {\cos \left( {\theta_{in} } \right) + \sqrt {\varepsilon_{d} } \cos \left( {\theta_{out} } \right)} \right)}}$$

(8)

As shown in Fig. 2b, the graphene resonator array in TM mode is modeled by a series circuit containing Z₁ and c. So:

$$Z_{TM} = Z_{c} + Z_{1}$$

(9)

where Z_c is the impedance of the capacitor. Also²⁰:

$$Z_{TM} = \frac{1}{{\sigma_{TM} }}$$

(10)

The effective conductivity of the graphene resonator layer by Fresnel equation in TM mode can be obtained by³⁵:

$$\sigma_{TM} = \frac{{\sec \left( {\theta_{{{\text{in}}}} } \right) - \sqrt {\varepsilon_{d} } \sec \left( {\theta_{{{\text{out}}}} } \right) - r^{TM} \left( {\sec \left( {\theta_{{{\text{in}}}} } \right) + \sqrt {\varepsilon_{d} } \sec \left( {\theta_{{{\text{out}}}} } \right)} \right)}}{{Z_{0} \left( {1 + r^{TM} } \right)}}$$

(11)

where r™ is the reflection coefficient of the graphene resonator layer in TM mode. The impedance of the gap capacitance can be obtained by²⁰:

$$Z_{c} = \frac{1}{j\omega c}$$

(12)

where ω is the angular frequency. The gap capacitance is calculated by:

$$c = \varepsilon_{{{\text{eff}}}} \frac{b}{g}$$

(13)

where ε_eff is the effective permittivity calculated by²⁰:

$$\varepsilon_{{{\text{eff}}}} = \varepsilon_{0} \frac{{1 + \varepsilon_{d} }}{2}$$

(14)

in which ε₀ is the permittivity of vacuum. Substituting Eqs. (13) and (14) in Eq. (12), we will have:

$$Z_{c} = \frac{2g}{{j\omega b\varepsilon_{0} \left( {1 + \varepsilon_{d} } \right)}}$$

(15)

Then, Z₁ can be calculated from:

$$Z_{1} = \frac{{Z_{0} (1 + r^{TM} )j\omega l\varepsilon_{0} (1 + \varepsilon_{d} ) - 2w\left[ {\sec \left( {\theta_{{{\text{in}}}} } \right) - \sqrt {\varepsilon_{d} } \sec \left( {\theta_{{{\text{out}}}} } \right) - r^{TM} \left( {\sec \left( {\theta_{{{\text{in}}}} } \right) + \sqrt {\varepsilon_{d} } \sec \left( {\theta_{{{\text{out}}}} } \right)} \right)} \right]}}{{j\omega l\varepsilon_{0} \left( {1 + \varepsilon_{d} } \right)\left[ {\sec \left( {\theta_{{{\text{in}}}} } \right) - \sqrt {\varepsilon_{d} } \sec \left( {\theta_{{{\text{out}}}} } \right) - r^{TM} \left( {\sec \left( {\theta_{{{\text{in}}}} } \right) + \sqrt {\varepsilon_{d} } \sec \left( {\theta_{{{\text{out}}}} } \right)} \right)} \right]}}$$

(16)

The entire metamaterial is represented by transmission lines, as depicted in Fig. 3. The input impedances for each segment of the metamaterial absorber are illustrated in the same figure. Given that the thickness of the graphene layer is significantly smaller than the minimum wavelength within the simulated wavelength range, the graphene resonator layer is simplified and modeled as a point load^36,37.

Figure 3

ECM of the whole metamaterial absorber.

The input impedances of the different segments of Fig. 3 are calculated by^20,36:

$$Z_{1}^{TE/TM} = Z_{d}^{TE/TM} \frac{{Z_{Au} + jZ_{d}^{TE/TM} \tan \left( {\beta_{d} d} \right)}}{{Z_{d}^{TE/TM} + jZ_{Au} \tan \left( {\beta_{d} d} \right)}}$$

(17)

in which $Z_{d}^{TE/TM}$ and β_d are respectively the TE/TM impedances of the dielectric substrate and the propagation constant of the THz electromagnetic wave in the dielectric substrate.

Since the conductivity of gold is high and its thickness is much larger than the maximum penetration depth of electromagnetic waves in the designed frequency range, the waves cannot transmit through the gold layer to the other side of the metamaterial. So, it acts as a perfect reflector and its impedance can be considered as zero, Z_Au = 0. Then, Eq. (17) is simplified to^20,36:

$$Z_{1}^{TE/TM} = jZ_{d}^{TE/TM} \tan \left( {\beta_{d} d} \right)$$

(18)

$$Z_{2}^{TE/TM} = Z_{g}^{TE/TM} \left\| {Z_{1}^{TE/TM} } \right.$$

(19)

where $Z_{g}^{TE/TM}$ is the TE/TM impedances of the graphene resonator layer. The input impedance of the metamaterial can be obtained by^20,36:

$$Z_{in}^{TE/TM} = Z_{ig}^{TE/TM} \frac{{Z_{2}^{TE/TM} + jZ_{ig}^{TE/TM} \tan \left( {\beta_{ig} d_{ig} } \right)}}{{Z_{ig}^{TE/TM} + jZ_{2}^{TE/TM} \tan \left( {\beta_{ig} d_{ig} } \right)}}$$

(20)

in which $Z_{ig}^{TE/TM}$ and β_ig are respectively the TE/TM impedances of the ion gel layer and the propagation constant of the THz electromagnetic wave in the ion gel²⁰.

$$Z_{ig/d}^{TE} = \frac{{Z_{0} }}{{\sqrt {\varepsilon_{ig/d} } }}\cos \left( {\theta_{in/d} } \right)$$

(21)

$$Z_{ig/d}^{TM} = \frac{{Z_{0} }}{{\sqrt {\varepsilon_{ig/d} } }}\sec \left( {\theta_{in/d} } \right)$$

(22)

in which θ_d is the electrical length of the substrate²⁰.

$$\theta_{d} = \frac{{d_{{{\text{substrate}}}} \omega \sqrt {\varepsilon_{d} } }}{c}$$

(23)

$$\beta_{ig/d} = \frac{{\omega \sqrt {\varepsilon_{ig/d} } }}{c}$$

(24)

The TE/TM scattering parameters (return losses) of the metamaterial $S_{11}^{TE/TM}$ can be obtained by²⁰:

$$S_{11}^{TE} = \frac{{Z_{{{\text{in}}}}^{TE} - Z_{0} \cos (\theta_{{{\text{in}}}} )}}{{Z_{{{\text{in}}}}^{TE} + Z_{0} \cos (\theta_{{{\text{in}}}} )}}$$

(25)

$$S_{11}^{TM} = \frac{{Z_{{{\text{in}}}}^{TM} - Z_{0} \sec (\theta_{{{\text{in}}}} )}}{{Z_{{{\text{in}}}}^{TM} + Z_{0} \sec (\theta_{{{\text{in}}}} )}}$$

(26)

in which $Z_{in}^{TE/TM}$ is obtained from Eq. (20). The TE/TM reflections of the metamaterial R^TE/™ are obtained by^20,38:

$$R^{TE/TM} = \left| {S_{11}^{TE/TM} } \right|^{2}$$

(27)

Since the metamaterial is backed by a gold layer, there is zero transmission (T^TE/™ = 0) for the metamaterial. So, the insertion losses (T^TE/™ or $S_{21}^{TE/TM}$) for TE/TM modes are zero³⁸. The TE/TM absorptions of the metamaterial A^TE/™ are obtained by^20,38:

$$A^{TE/TM} = 1 - R^{TE/TM}$$

(28)

The linear dichroism (LD) is the difference in absorbance for TE and TM polarized waves^5,20,38:

$$LD = A^{TM} - A^{TE}$$

(29)

Results and discussion

Absorption spectra of the anisotropic metamaterial absorber of Fig. 1 for both TE and TM mode are obtained by CST and the results are shown in Fig. 4. The metamaterial has different responses for TE and TM waves, and it works in 0.3–4 THz. There are three resonances in TE mode with average absorption of 99.3% and one resonance in TM mode with absorption rate of 81%.

Figure 4

TE and TM absorption spectra of the anisotropic metamaterial absorber of Fig. 1.

Surface current distributions in the resonance frequencies of the anisotropic metamaterial absorber of Fig. 1 are displayed in Fig. 5. In 1.04 THz, the surface currents make two loops (two magnetic dipoles), one of them anticlockwise and the other one clockwise. In 1.58 THz, the surface currents also make two loops (two magnetic dipoles), one of them clockwise and the other one anticlockwise. In 2.31 THz, the surface currents make two electric dipoles with the directions shown in Fig. 5c. In 3.63 THz, the surface currents make seven electric dipoles with the directions shown in Fig. 5d.

Figure 5

Surface current distributions of the metamaterial absorber of Fig. 1 in (a) 1.04, (b) 1.58, (c) 2.31, and (d) 3.63 THz.

To start the ECM analysis, the graphene resonator layer is placed on the dielectric with thickness of 300 μm which is considered as a half-space to minimize the influence of the substrate on the impedances of the graphene resonator layer. The configuration is simulated in CST and the reflection spectra are obtained. The obtained TE and TM reflection spectra (r^TE in Eq. (4) and r™ in Eq. (11)) are given in Fig. 6a,b. Then, Eqs. (4) and (11) are calculated to obtain the equivalent conductivities of the graphene resonator layer in TE and TM modes, and the results are given in Fig. 6c–f. The real parts are positive (Fig. 6c,d) representing the resistive nature and the imaginary parts contain both positive and negative sections (Fig. 6e,f) indicating the inductive and capacitive natures of the graphene resonator array.

Figure 6

(a) TE and (b) TM reflection spectra of the graphene-based resonator array layer when its located on the half-space substrate without considering the ion gel and gold layers. (c) and (d) real, (e) and (f) imaginary parts of the equivalent conductivity of the graphene resonator array layer in TE and TM modes (Eqs. (10) and (11)).

The resonator array of the metamaterial of Fig. 1 consists of a split ring resonator array Z₁ and the middle connecting bar array Z₂. We have calculated and plotted the real and imaginary parts of the impedances of these arrays separately and the results are given in Fig. 7. Z₁ and Z₂ are shown in the inset figures in Fig. 7. The real parts are positive producing great loss, and the imaginary parts contain both positive and negative sections indicating the inductive and capacitive natures of the split ring resonator array and the middle connecting bar array.

Figure 7

(a) Real and (b) imaginary parts of Z₁ (impedance of the split ring resonator array of the metamaterial of Fig. 1 as shown in the inset figure). (c) Real and (d) imaginary parts of Z₂ (impedance of the middle connecting bar array of the metamaterial of Fig. 1 as shown in the inset figure).

TE and TM absorption spectra of the metamaterial absorber of Fig. 1 for three different values of incident angle θ_in are respectively given in Fig. 8a,b. By change of θ_in, As shown, by change of θ_in, the absorption spectra for both modes alter slightly so the metamaterial is incident angle independent.

Figure 8

(a) TE and (b) TM absorption spectra of the metamaterial absorber of Fig. 1 for three different values of incident angle θ_in.

TE and TM absorption spectra for the metamaterial of Fig. 1 obtained by CST and ECM for three different values of θ_in are given in Fig. 9. Obtained spectra with both methods are in good agreement and the ECM works for any oblique incident angle of the illuminated wave θ_in.

Figure 9

Absorption spectra of the metamaterial absorber of Fig. 1 obtained by CST and ECM in (a) TE mode for θ_in = 0, (b) TM mode for θ_in = 0, (c) TE mode for θ_in = 30, (d) TM mode for θ_in = 30, (e) TE mode for θ_in = 60, and (f) TM mode for θ_in = 60.

The real and imaginary parts of the normalized input impedances in TE and TM modes (${Z}_{in}^{TE}/{Z}_{o}$ and ${Z}_{in}^{TM}/{Z}_{o}$) of the proposed anisotropic metamaterial absorber of Fig. 1 with the properties presented in Fig. 9a,b are shown in Fig. 10a,b, respectively. Z₀ is the vacuum impedance equal to 377 Ω.

As shown in the curve obtained by ECM in Fig. 9a, absorption peaks in the first and second peaks reach 1 while the absorption in the third peak reaches 0.8. As shown in Fig. 10a, the imaginary part of the normalized input impedance is zero at the first and second absorption peaks and 0.27 at the third absorption peak. Also, the real part approaches 1 in the first and second peaks while 0.82 in the third peak. As shown in the curve obtained by ECM in Fig. 9b, absorption peak reach 1. As shown in Fig. 10b, the imaginary part of the normalized input impedance is zero at the absorption peak. Also, the real part approaches 1 in the absorption peak. These characteristics show the impedance matching condition to achieve the perfect absorption at the resonance peaks (Fig. 10).

Figure 10

Normalized input impedance of the anisotropic metamaterial absorber of Fig. 1 in (a) TE and (b) TM modes with properties respectively presented in Fig. 9a,b.

The linear dichroism (LD: the difference in absorbance for TE and TM polarized waves) vs frequency for the metamaterial of Fig. 1 obtained by Eq. (29) for three different values of graphene Fermi energy E_f are shown in Fig. 11. The maximum LD reaches 99% for this metamaterial. By increasing of E_f, the resonance frequencies of the LD spectra increase, showing a blueshift. This is because the real part of the β decreases as the E_f increases (Eq. (4) in²⁰). So, the resonances increase with the increase of E_f.

Figure 11

Linear dichroism (LD: the difference in absorbance for TE and TM polarized waves) of the metamaterial absorber of Fig. 1 obtained by Eq. (29) for three different values of graphene Fermi energy E_f.

The results of this work are compared with some previously published works in Table 1. The performance of the metamaterial in this work is improved compared to the metamaterial containing of two gaps (without the connecting bar) in²⁰.

Table 1 Comparison of this work with some previously published works.

Conclusion

In this study, we propose an Equivalent Circuit Modeling (ECM) approach utilizing impedance and transmission lines, implemented through a straightforward MATLAB code. The focus is on a terahertz (THz) graphene absorber with a graphene-based split square ring resonator array, featuring two gaps and a connecting bar. The design is subjected to numerical simulation using CST Microwave Studio Software, employing the finite element method (FEM). The results from both FEM and ECM exhibit good agreement. The absorber design is dynamically tunable, comprising a one-layer resonator array with two resonators per unit cell. The metamaterial absorber demonstrates a pronounced linear dichroism (LD) response, achieving 99% efficiency. The metamaterial has an absorption rate of 81% for one absorption band in transverse magnetic (TM) mode. Its three absorption bands in transverse electric (TE) mode have an average of 99.3% in each absorption band with absorption over 99%. This delineates a single absorption band for TM mode and three for TE mode. The proposed graphene absorber holds promise as a component in polarization-sensitive systems for controllable absorption and sensing applications.