Simple Design of Polarization-Selective Tunable Triple Terahertz Absorber Based on Graphene Rectangular Ring Resonator

Abstract

In this paper, a simple design of a polarization-selective tunable triple terahertz absorber based on a graphene rectangular ring resonator was proposed and studied. The absorber structure consists of a graphene rectangular ring resonant array on the top, SiO₂ dielectric layer in the middle and gold at the bottom. The calculated results show that the absorber can achieve high-efficiency triple-band absorption under both x and y polarization incident light. When x-polarized light is incident, three distinctive absorption peaks at 2.73, 5.70 and 11.19 THz with absorption rates of 96.7%, 98.5% and 96.5% are achieved. When y-polarized light is incident, three additional absorption peaks at 2.29, 7.55 and 9.98 THz can be obtained with absorption rates of 96.3%, 90.3% and 97.4%, respectively. Moreover, the absorption wavelength of the absorber can be tuned by adjusting the chemical potential of the graphene. Owing to the high efficiency of triple-band absorption in different polarization states, the absorber has broad application prospects in terahertz polarization imaging, sensing and detection.

1. Introduction

Metasurfaces, as artificial sub-wavelength structures, have wide application prospects in the fields of perfect absorption, hyperlenses, and sensing [1,2,3,4]. Metasurface-based absorbers generally consist of a top layer with a periodic pattern, a dielectric middle layer and a bottom metal layer [5]. However, because of the fixed structural parameters and resonant structure, the resonant absorption frequency remains fixed, which limits their application range [6,7]. Therefore, dynamic tuning of metasurface absorbers has become an important research topic.

In recent years, graphene has attracted considerable attention because of its excellent optical properties and ability to adjust its Fermi level under external conditions (such as voltage, optical pump or chemical doping) [8,9,10,11,12]. In 2014, Zhang et al. combined the metasurface of a metal resonator unit with a double layer of graphene to achieve an absorber independent of light polarization [13]. In 2015, Zhang et al. proposed a polarization-insensitive dual-band metasurface absorber composed of a gold resonator, a graphene layer, a dielectric spacer layer and a gold junction plane [14]. In 2016, Guo et al. proposed a bimodal tunable terahertz absorber based on a metallic reflector graphene patch array and surface SRR array [15]. Currently, metasurface absorbers are mostly focused on improving the absorption rate, so they are designed to be insensitive to the polarization of incident light. Therefore, metasurfaces cannot meet the application requirements in the fields of polarization imaging, selective spectral detection and polarization multiplexing [16,17,18]. Therefore, it is of great significance and necessity to design an absorber that can adjust the polarization of light in both orthogonal directions and efficiently absorb it.

In 2016, Deng et al. reported a tunable metasurface absorber comprising a metal substrate, an SiO₂ dielectric spacer layer, a graphene layer and a patterned metal layer. This absorber achieved absorption rates of approximately 0.985 and 0.991 for x-polarized waves at 6.42 THz and 8.37 THz, respectively, and an absorption rate of around 0.99 for y-polarized waves at 7.22 THz, demonstrating significant polarization sensitivity [19]. However, their design did not achieve multi-peak responses for different polarization states. In 2018, Cai et al. utilized stacked multilayer graphene bands to realize polarization-sensitive broadband absorption in the terahertz range. Their design selectively enabled broadband absorption (5.8–6.5 THz) for incident light with TE polarization [20]. However, the multilayer configuration increased the structural complexity. In 2020, Cao et al. proposed a polarization-sensitive multi-band absorber consisting of stacked graphene layers separated by dielectric layers. This design exhibited dual absorption bands for both x and y polarizations. For x-polarized light, two absorption peaks at wavelengths of 27.3 µm and 31.9 µm had absorption rates exceeding 90%. Similarly, for y-polarized light, two absorption peaks at wavelengths of 16.3 µm and 21.2 µm also achieved absorption rates above 90% [21]. In 2023, Zhan et al. introduced a simple and tunable metamaterial absorber composed of a top layer of graphene strips, a middle dielectric layer and a gold substrate. For x-polarized waves, the absorber achieved three nearly perfect absorption peaks of 98.6%, 87.8%, and 98.5% at 2.30 THz, 3.41 THz, and 4.20 THz, respectively [22]. In 2024, Asgari et al. proposed a graphene-based split square ring resonator array metamaterial absorber. For x-polarized waves, the absorber exhibited three nearly perfect absorption peaks at 1.04, 1.58, and 3.63 THz. For y-polarized waves, it achieved 80% absorption at 2.31 THz [23]. The triple-band absorber can precisely control absorption at three frequency bands, improving performance in optical filtering and sensing applications, enhancing signal clarity and detection accuracy. However, tunable intensity of three resonant peaks with high modulation depth in two directions simultaneously has been barely investigated, and the design structure of existing polarization-selective absorbers is mostly based on multilayer designs or complex configurations, which limits their application.

In this paper, a tunable polarization selective triple-band terahertz absorber based on a rectangular ring resonator is proposed. The absorber can achieve high-efficiency triple-band absorption in two polarization directions. The calculated results show that the absorber presents three absorption peaks under different polarized light incident conditions. Among them, the absorptivity of x-polarized and y-polarized light incident is higher than 90%. The physical mechanism of the absorption peaks was analyzed using the impedance matching theory, an equivalent circuit model and electric field distribution. In addition, the absorption operating wavelength of the absorber can be dynamically tuned using the Fermi level of graphene.

2. Materials and Methods

The basic structure of a terahertz absorber is illustrated in Figure 1. Figure 1a,b show the schematic and top view of the absorber structure unit, respectively. The absorber consists of a top graphene ring, middle SiO₂ dielectric layer, and bottom gold reflector.

The dimensions of graphene were negligible. Therefore, it can be modeled as a two-dimensional plane with no thickness in the simulation [24]. According to Kubo’s formula [25], at terahertz frequencies and room temperature, the conductivity of graphene can be divided into intra- and inter-band conductivities:

$${\sigma }_{\mathrm{intra} }\approx {\displaystyle \frac{-{ie}^2{k}_{B}T}{\pi {\hslash }^2\left(\omega -j2\mathsf{\Gamma }\right)}}\left[{\displaystyle \frac{E_f}{k_{B}T}}+2\ln \left(e^{{\displaystyle \frac{E_f}{k_{B}T}}}+1\right)\right]$$

(1)

$${\sigma }_{\mathrm{inter} }\approx {\displaystyle \frac{-{ie}^2}{4\pi \hslash }}\mathrm{l}\mathrm{n}\left({\displaystyle \frac{2\left|E_f \right|-(\omega -i2\mathsf{\Gamma })\hslash }{2\left|E_f\right|+(\omega -i2\mathsf{\Gamma })\hslash }}\right)$$

(2)

where e is the electron charge, k_B is the Boltzmann constant, T is the Kelvin temperature, ħ = h/2π is the reduced Planck’s constant, τ is the relaxation time, and E_f is the Fermi level. In the terahertz band, when ħω << 2E_f, the inter-band conductivity is negligible, and the surface conductivity of graphene depends primarily on the intra-band conductivity. The intra-band electron–photon scattering process produces a Drude-like dispersion, so the conductivity of graphene can be approximated by the Drude model [26]:

$$\sigma (\omega )={\displaystyle \frac{e^2{E}_f}{\pi {\hslash }^2}}{\displaystyle \frac{i}{\omega +i{\tau }^{-1}}}$$

(3)

Equation (3) shows that the conductivity of graphene is related to its Fermi energy level (E_f) in addition to the angular frequency (ω) and relaxation time (τ) of the incident electromagnetic wave.

The thickness d₂ of the SiO₂ layer is 6.55 μm and the refractive index is 1.5, and the base is gold with a thickness of d₁ = 200 nm and an electrical conductivity of 4.56 × 10⁷ S/m. The geometric parameters of the absorber are as follows: P_x = 5.5 μm, P_y = 7.6 μm, a₁ = 4.1 μm, b₁ = 6.2 μm, a₂ = 0.8 μm, b₂ = 1.2 μm. To evaluate the performance of the proposed metamaterial device, numerical simulations were conducted using the finite difference time domain (FDTD) method. In the numerical calculation, the incident light was set to propagate in the z direction throughout the structure. Periodic boundary conditions were used for the x and y directions and perfectly matched boundary conditions were used in the z direction. The gold layer primarily enhances reflection and reduces the transmission of electromagnetic waves. Due to its thickness exceeding the skin depth of the incident light, the transmission coefficient T(ω) = |S₂₁|² is effectively zero, meaning no energy passes through the gold layer. Consequently, all incident energy is either reflected or absorbed. The absorption rate A(ω) is related to the reflection and transmission coefficients, expressed as A(ω) = 1 − T(ω) − R(ω). Since T(ω) = 0, the formula simplifies to A(ω) = 1 − |S₁₁|², where |S₁₁|² represents the reflected energy, and the absorption is the incident energy minus the reflected energy.

3. Results

Figure 2 shows the absorption spectra of two different polarizations of incident light when E_f = 0.9 eV. For the x polarization, there are three absorption peaks (P1, P2, and P3) at 2.73, 5.70, and 11.19 THz, with absorption rates of 96.7%, 98.5%, and 96.5%, respectively. For the y polarization, three additional absorption peaks (Q1, Q2, and Q3) are achieved at 2.29, 7.55, and 9.98 THz, with absorption rates of 96.3%, 90.3%, and 97.4%, respectively. Figure 2b illustrates the variation of the Q factor [27] and full width at half maximum (FWHM) as a function of frequency for E_f = 0.9 eV. Q = f/FWHM, f is the center resonant frequency, and FWHM is the full width at half maximum. The blue curve in Figure 2b shows the “Q factor of peaks”, which is relatively low at low frequencies at 2.29 and 2.73 THz. As the frequency increases, the Q factor peaks at 7.55 THz, indicating a high-quality factor and better resonance characteristics in this frequency range. Afterward, the Q factor decreases but rises again at 11.19 THz, suggesting improved resonance once more. The red curve represents the “FWHM of peaks”, showing a similar trend. At 2.29 and 2.73 THz, the FWHM is low, implying a narrower spectral width and higher selectivity. However, as the frequency approaches 7.55 THz, the FWHM increases, indicating a broader spectral width and lower selectivity. Finally, as the frequency reaches 11.19 THz, the FWHM decreases again, implying a higher selectivity in this region.

To better demonstrate the internal mechanism of perfect absorption, the absorption mechanism was analyzed according to impedance matching theory. The relative impedance Z of the absorber can be derived from the S parameter in the simulation results, and its calculation formula is as follows [28]:

$$Z=\sqrt{{\displaystyle \frac{\mu }{\epsilon }}}=\sqrt{{\displaystyle \frac{{\left(1+S_{11}\right)}^2-S_{21}^2}{{\left(1-S_{11}\right)}^2-S_{21}^2}}}$$

(4)

where μ and ε are the effective permeability and dielectric constant of the absorber, respectively, which determine the equivalent impedance. the equivalent conductivity and permittivity can be changed by adjusting the size, shape or material of the absorber. Because the Z-min boundary is ideal, electromagnetic waves are barely transmitted, and S₂₁ is approximately equal to 0. Equation (4) can be simplified to

$$Z={\displaystyle \frac{1+S_{11}}{1-S_{11}}}$$

(5)

Perfect absorption was achieved when the equivalent impedance of the absorber matched the free-space impedance. Under this condition, the real part of the equivalent impedance Re(Z) is 1 and the imaginary part Im(Z) is 0. According to Equation (5), the equivalent impedance of the absorber is calculated, as shown in Figure 3a,b. For x polarization incidence, the equivalent impedances are Z = 0.74 − 0.11i, Z = 0.98 + 0.29i, and Z = 1.14 + 0.54i at resonant frequencies of 2.73, 5.70, and 11.19 THz, respectively. For y polarization incidence, the equivalent impedances are Z = 0.71 − 0.09i, Z = 1.82 − 0.42i, and Z = 0.79 − 0.04i, respectively. All the Re(Z) values approach 1, and Im(Z) approaches 0. The impedance matches the free space; thus, the maximum absorption is obtained.

The expression E_f of graphene Fermi level is as follows [29]:

$$E_f=\hslash v_F\sqrt{n_g\pi }=\hslash v_F\sqrt{{\displaystyle \frac{\pi {\epsilon }_d{\epsilon }_0{V}_g}{e{t}_d}}}$$

(6)

In Equation (6), n_g = ε_dε₀V_g/et_d is the carrier concentration of graphene, ε_d is the dielectric constant of the dielectric layer, ε₀ is the dielectric constant of the vacuum, t_d is the thickness of the dielectric layer between the graphene layer and the electrode, and V_g is the bias voltage applied to the graphene layer. Equation (6) shows that the Fermi level mainly depends on the carrier concentration of graphene. The carrier concentration n_g of graphene can be varied by adjusting V_g. Thus, the Fermi level E_f can be tuned using V_g. Since the Fermi level can change the conductivity of graphene, it can also change the equivalent impedance of the entire absorber, thereby changing the absorption wavelength.

Figure 4a,b illustrate the influence of changes in the Fermi level on the absorption performance under two polarization states. For both x- and y-polarized states, with an increase in the Fermi level of graphene, the frequency of the absorption peaks shifted blue. For x polarization, the absorption at P1 increased, while that at P2 and P3 decreased. For y polarization, the absorptions of all three peaks increased. When the Fermi level of graphene changes, the surface conductivity also changes. The impedance match between the absorber and free space was disrupted. This demonstrates that the absorption characteristics can be effectively tuned by manipulating the Fermi level of the graphene metasurface, eliminating the need for redesigning the device structure.

The absorption spectra at different polarization angles are shown in Figure 5. The polarization angle for x polarization was set to 0°, and 90° for y polarization. As shown in Figure 5a, when the polarization angle increased, the absorption of peaks P1, P2, and P3 weakened. Meanwhile, the absorption of peaks Q1, Q2, and Q3 became stronger. At a polarization angle of 45°, the electric field response in the x and y directions reached dynamic equilibrium, which produced six modes of equal intensity. In the case of x-polarized light, the three high absorption peaks correspond to zero absorption in the y direction, and vice versa for y-polarized light. When the polarization angle is 45°, the energy is evenly distributed between the x and y directions, causing the absorption intensity in each direction to be halved. Therefore, the absorption intensities of P1, P2, and P3 for x-polarized light, and Q1, Q2, and Q3 for y-polarized light, are each half of their original values. The maximum absorption of the six peaks at different polarization angles were obtained. The results are shown in Figure 5b. The maximum absorption of the resonant peaks varied with the polarization angle. Therefore, the polarization angle can actively control the peak number and intensity of the multiband absorber.

To further study the physical mechanism of the absorption, the electric field |E| and magnetic field |H| of the resonant absorption peaks for x and y polarizations were analyzed, as shown in Figure 6 and Figure 7, respectively. Figure 6a–c illustrate the electric and magnetic fields in the x-y plane, as well as the magnetic field in the x-z plane at the resonant absorption frequency of 2.73 THz. From Figure 6a,b, it can be seen that when x-polarized light is incident, the electric field is mainly distributed on the left and right sides of the graphene ring, and the magnetic field is mainly inside the upper and lower sides of the graphene ring. When the frequency was 5.70 THz, the electric field was mainly concentrated at the outer four corners of the graphene ring, and the magnetic field was mainly outside the left and right sides of the graphene ring. The resonance is mainly generated by the electromagnetic dipole of the gap between adjacent cells between the left and right adjacent units, as shown in Figure 6d,e. Figure 6g,h show the electric and magnetic fields at 11.19 THz. The electric field was mainly concentrated in the inner and outer sides of the graphene ring, whereas the magnetic field was mainly concentrated in the inner corners of the graphene ring.

Figure 7 shows the distribution of the electric field |E| and magnetic field intensity |H| at the resonant absorption peak in y polarization mode. Figure 7a–c show the electric and magnetic fields in the x-y plane, as well as the magnetic field in the y-z plane at the resonant absorption frequency 2.29 THz. From Figure 7a,b, it can be observed that the electric field is mainly on the upper and lower sides of the graphene ring, and the magnetic field is mainly inside the left and right sides of the graphene ring. Resonance is mainly generated by the inter-unit electromagnetic dipoles of adjacent units. At a frequency is 7.55 THz, the electric field is mainly concentrated at the outer four corners of the graphene ring, the magnetic field is mainly concentrated on the upper and lower outer sides of the graphene ring, as shown in Figure 7d,e. Resonance is mainly generated by the electromagnetic dipole between the adjacent units above and below. Figure 7g,h show the electric and magnetic fields of 9.98 THz. The electric field was mainly concentrated on the inner and outer sides of the upper and lower parts of the graphene ring, whereas the magnetic field was mainly concentrated at the inner corners of the graphene ring.

As seen from Figure 6 and Figure 7, the magnetic field energy at P1, P2, Q1, and Q2 is concentrated in the dielectric layer below the graphene ring, as shown in Figure 6c,f and Figure 7c,f, which is consistent with the characteristics of local surface plasmon resonance (LSPR) [30]. The concentration of magnetic field energy in the dielectric layer indicates that the LSPR mode is effectively excited, resulting in enhanced local energy and increased absorption at these locations. The magnetic field energies at P3 and Q3 are strongly confined to the region both below and within the graphene ring, as shown in Figure 6i and Figure 7i, which indicates that the propagating surface plasmonic resonance (PSPR) [31] is excited. Unlike LSPR, PSPR causes plasmon waves to propagate along the graphene surface, leading to a broader energy distribution, which results in a wider resonance frequency range and distinct absorption characteristics. Additionally, the energy is localized and dissipated between the surface graphene and the underlying metal layer.

The changes in the structural parameters of the absorber also affect the resonant frequency. According to equivalent circuit theory, the resonant frequencies f, L, and C of the terahertz absorber caused by LSPR are as follows [32]:

$$f={\displaystyle \frac{1}{2\pi \sqrt{LC/2}}}\propto {\displaystyle \frac{1}{l\sqrt{{\epsilon }_d}}}$$

(7)

$$L ={\mu }_0{\displaystyle \frac{hl}{w}}$$

(8)

$$C=C_d+C_a\approx \phi {\epsilon }_d+{\epsilon }_0{\displaystyle \frac{wl}{h}}$$

(9)

Here, L is the effective inductor, C is the capacitor, ε_d is the dielectric constant of the dielectric layer, ε₀ is the dielectric constant of the vacuum, φ is the correlation coefficient, and l and w are the length and width of the graphene structure through which the surface current flows, respectively, l = a₁, w = b₁. h is the height of the microchannel, h = d₂. The resonant frequency f associated with the PSPR can be expressed as follows [32,33,34]:

$$f={\displaystyle \frac{c\sqrt{i^2+j^2}}{P_{eff}}}\sqrt{{\displaystyle \frac{{\epsilon }_m+{\epsilon }_d}{{\epsilon }_m{\epsilon }_d}}}$$

(10)

where ε_m represents the dielectric constant of the metal, i and j are the scattering orders, P_eff = P_x − k₁a₁ − k₂a₂, or P_eff = P_y − k₃b₁ + k₄b₂, depending on the condition, is the equivalent period related to the cavity structure parameters, and k₁, k₂, k₃ and k₄ are the correlation coefficients.

Figure 8 shows the absorption spectra of different thicknesses of SiO₂. When the thickness d₂ increases from 6.15 μm to 6.95 μm, the wavelength of absorption peaks for both x and y polarization are basically unchanged. According to Equations (7)–(9), the resonant frequencies of peaks P1, P2, Q1, and Q2, which are caused by LSPR, are determined by the dielectric constant ε_d. Therefore, the resonant frequency remains unchanged. However, according to Equation (10), the resonant frequency of peaks P3 and Q3 caused by the PSPR remains unchanged because the resonant frequency is independent of the dielectric layer thickness.

Figure 9a,b show the absorption spectra corresponding to x polarization and y polarization as the graphene transverse strip length a₁ increases from 3900 nm to 4300 nm, respectively. For x polarization, with the increase in a₁, the absorption frequencies of peaks P1 and P2 exhibit redshift and the frequency of P3 shows blueshift, as shown in Figure 9a. For y polarization, the absorption frequencies of Q1 and Q2 exhibit redshift and the absorption frequency of Q3 is blueshift, as shown in Figure 9b. This is because when a₁ gradually increases, the length of the surface current flowing through the graphene structure l increases, resulting in increased equivalent inductance L and equivalent capacitance C. According to Equation (7), the resonant frequencies exhibit redshift. On the other hand, due to the increase in a₁, P_eff decreases, resulting in a blueshift of P3 and Q3 in the absorption spectrum.

Figure 10a,b show the absorption spectra corresponding to x and y polarizations, respectively, due to the increase in the graphene vertical bar width b₁ from 6000 nm to 6400 nm. It can be seen that with the increase in b₁, for x polarization, the absorption frequencies of the peaks of P1 and P2 remain basically unchanged, and the frequency of P3 is blueshifted, as shown in Figure 10a. For y polarization, the absorption frequency of Q1 exhibits a redshift, the absorption frequency of Q2 remains unchanged and the Q3 absorption peak is blueshifted, as shown in Figure 10b. This is because when b₁ gradually increases, the length of the surface current flowing through the graphene structure w increases, resulting in an increased equivalent capacitance C. According to Equation (7), the resonant frequencies exhibit a redshift. On the other hand, when b₁ increased, P_eff decreased, which resulted in the blueshift of P3 and Q3.

The results are summarized in

Table 1. Parameter sensitivity analysis table.

	Parameter (μm)	Error (μm)	Sensitivity to Peaks (P1, P2, P3)	Sensitivity to Peaks (Q1, Q2, Q3)
Thickness of SiO2 (d2)	6.55	±0.2	Nonsensitive parameter	Nonsensitive parameter
Graphene transverse strip length (a1)	4.1	±0.1	P1 and P2: Nonsensitive P3: Sensitive	Q1 and Q2: Nonsensitive Q3: Sensitive
Graphene vertical bar width (b1)	6.2	±0.1	P1 and P2: Nonsensitive P3: Sensitive	Q1 and Q2: Nonsensitive Q3: Sensitive

based on the analysis above. It can be seen that the thickness of SiO₂ has no significant effect on the absorption peaks, while the graphene transverse strip length and vertical bar width show strong sensitivity to both the P3 and Q3 absorption peaks. The other absorption peaks are not significantly affected.

Table 2. Comparison of absorption characteristics of polarization-dependent metasurface absorbers.

Ref. No.	x Polarization			y Polarization			Year	Structural Composition
	Peak Number	Absorption Wavelength	Absorptivity	Peak Number	Absorption Wavelength	Absorptivity
[19]	2	6.42 and 8.37 THz	>98%	1	7.22 THz	>99%	2016	Au-graphene-SiO2-Au
[20]	broadband	5.8–6.5 THz	>95%	0	-	-	2018	three graphene-Al2O3- gold
[21]	2	27.3 and 31.9 µm	>90%	2	16.3 and 21.2 µm	>90%	2020	double graphene-Si-Au
[35]	2	4.09 and 6.28 THz	>96%	2	3.89 and 5.39 THz	>85%	2021	graphene-SiO2-Au
[22]	3	2.30, 3.41 and 4.27 THz	>87%	0	-	-	2023	graphene-SiO2-Au
[23]	3	1.04, 1.58 and 3.63 THz	>99%	1	2.31 THz	>81%	2024	Ion gel-graphene-Rogers RT5880LZ-gold
[36]	1	17.57 THz	>90%	broadband	8.0–16.0 THz	>90%	2024	VO2-GaA-Ge-SiO2-Metal
This work	3	2.73, 5.70 and 11.19 THz	>96%	3	2.29, 7.55 and 9.98 THz	>90%		graphene-SiO2-Au

presents the publication status of various polarization-sensitive absorbers based on tunable materials. It can be seen that the absorber proposed in this paper has better performance, especially the number of the absorption peaks for both x polarization and y polarization. Our results provide new avenues for a multi-band, high-efficiency, polarization-sensitive absorber with a simple structure design.

4. Conclusions

A simple polarimetric-sensitive, triple-band, tunable terahertz absorber is proposed and studied in this paper. The absorber consists of a graphene rectangular resonant ring, silica dielectric spacer and gold substrate. When the incident light is of x polarization, the absorption rates of the absorptions peaks at 2.73, 5.70, and 11.19 THz are 96.7%, 98.5%, and 96.5%, respectively. When the incident light is of y polarization, the absorption rates of the absorber at 2.29, 7.55, and 9.98 THz are 96.3%, 90.3% and 97.4%, respectively. The impedance matching method and equivalent circuit model were used to analyze the absorption characteristic mechanism. Moreover, the absorber can adjust the wavelengths of the absorption peak by adjusting the Fermi level of the graphene. This absorber can realize three-band absorption in different polarization directions with a simpler structure than that of a traditional absorber, which has broad application prospects in the fields of terahertz polarization imaging, sensing, and detection.