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Abstract
We present a novel and effective approach for designing and analyzing graphene metasurface-based terahertz absorbers with the desired central frequency and fractional bandwidth. Narrowband and broadband absorbers are designed using the same configuration with a single-layer of graphene ribbons deposited on a metal-backed dielectric film. An analytical circuit model derived for the graphene array applies the impedance matching concept to realize the desired terahertz absorber. Absorbers with a fractional bandwidth ranging from 10-100% are realized at the 1-THz central frequency. The results show excellent agreement with those calculated using full-wave numerical simulations. The proposed method is promising for terahertz imaging, sensing, and filtering applications.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Terahertz radiation (ranging from approximately 0.1 to 10 THz) lies between the infrared and microwave regions of the electromagnetic spectrum and shares some properties with each of these types of radiations [1,2]. The non-ionizing and line of sight nature of terahertz radiation makes it promising for many applications, such as biomedical imaging, non-destructive evaluation, and communication technology [3–7]. However, the lack of natural materials that respond to terahertz waves has continually hindered the development of terahertz devices [8,9]. Nowadays, the use of metamaterials, which are artificial materials composed of sub-wavelength structures, has facilitated the realization of terahertz devices [10–12]. Moreover, Graphene with a monolayer of carbon atoms arranged in a honeycomb lattice has exhibited high carrier mobility and impressive tunability in electrical conductivity [13]. It was reported that continuous and structured graphene supports plasmon polaritons in the terahertz and infrared ranges owing to its exceptional physical and electromagnetic properties [14]. Graphene metamaterials based terahertz devices have attracted considerable research interest [15–20].
One of the outstanding applications of graphene metamaterials is terahertz absorption. Various groups have realized tunable absorbers through graphene structures of different geometries. For example, periodic graphene arrays [21–23], hybrid patterned graphene metasurfaces [24,25], and stacked structures composed of multi-layer graphene or graphene-metal microstructures [26–29] have been proposed. However, all these graphene designs were based on full-wave simulations. In most cases, they usually present complex structures and complicate the fabrication process. Furthermore, these methods can hardly provide accurate analyses for designing the material and geometrical parameters of the absorbers. Recently, some researchers have proposed the equivalent circuit model approach for the analysis and design of metasurface [30–36]. Using these analytical circuit models, multiband or broadband absorption was achieved with simple structures. However, the absorption spectra of the designed absorbers, which were obtained using the simple analytical circuit model reported in Ref. [30], still have considerable discrepancies compared with the full-wave simulations. In addition, the parameters of the accurate analytical model proposed by Khavasi et al. in Ref. [31–33] must be determined numerically. For practical applications, an accurate and simple method for designing simple structures with a step-by-step procedure is required.
In this work, a simple and efficient method for designing terahertz absorbers with a customized central frequency and the desired fractional bandwidth is presented. An analytical circuit model for graphene metamaterials is proposed. Based on the transmission line theory, conditions for both narrowband and broadband absorption are derived analytically, and closed-form equations for the material and geometrical parameters of the structure are formulated. These two different bandwidths for the absorption response can be achieved using the same configuration, which consists of a single-layer graphene array placed on a lossless quarter-wavelength dielectric spacer terminated by a metallic reflector. Then, the proposed method is used to design both narrowband and broadband absorbers with a central frequency of 1 THz. The results are The results are verified with full-wave simulations. In addition, such absorption bands can be tuned by altering the Fermi energy of graphene. The proposed absorbers can function for a wide range of incident angles, which is useful for real applications.
2. Structure and circuit model
The schematic diagram of the proposed absorber is presented in Fig. 1(a). The absorber consists of a graphene micron-ribbon array as the top layer, a dielectric spacing layer with quarter-wavelength thickness and a reflective gold film placed at the bottom. The structure is illuminated by a transverse magnetic (TM) polarized wave at normal incidence. It should be noted that the conductivity and thickness of the Au reflector are assumed to be $4.561 \times 10^7$ S/m and 1 $\mathrm{\mu}$m respectively, where its thickness is sufficient to block the incident terahertz waves [37]. Moreover, it is important to select a suitable dielectric material because the substrate has a direct impact on the fabrication process and lifetime of the device. In this study, a TOPAS polymer ($n_d = 1.53$) was selected as the dielectric material. This polymer has an excellent processing performance, high corrosion resistance, and low losses in the THz range, which make it an excellent material for THz applications [38]. It should be noted that $D$ and $W$ are the period and width of the graphene ribbons. This sub-wavelength size can be processed by photolithography. The Fermi level of graphene can be tuned by changing the gate voltage, which is driven by an external bias circuit [39]. The controllability of the Fermi energy can help achieve the tunability of the graphene absorption characteristics.
Fig. 1. (a) Schematic diagram of an absorber based on an array of graphene ribbons placed above a dielectric layer terminated by a metallic film. (b) Equivalent circuit model of the proposed absorber.
Based on the transmission line theory, the equivalent circuit model of the proposed structure is illustrated in Fig. 1(b). The homogeneous mediums, namely free space and the dielectric spacer, are modeled as transmission lines, with the characteristic propagation constant $\beta = \omega n/c$ and impedance $Z_d = Z_0 / n$, where $n$ is the refractive index of the corresponding region, $c$ is the speed of light, $\omega$ is the angular frequency of free space, and $Z_0 = 120 \pi$ is the free space intrinsic impedance. The Au bottom layer that suppresses the transmittance can be considered as a short circuit, and simulation results show that it is sufficiently accurate enough for the terahertz regime. Because the thickness of the graphene layer is considerately smaller than the typical terahertz wavelengths, the graphene layer can be considered as a shunt impedance at the junction between the two transmission lines [40].
For the numerical simulations, graphene is usually considered as a layer of material with a small thickness $\Delta = 1$nm and an in-plane effective permittivity $\varepsilon _g = \varepsilon _0 - j \sigma _g /(\omega \Delta )$, where $\varepsilon _0$ is the permeability of vacuum and $\sigma _g$ is the graphene surface conductivity. It is known that the surface conductivity of graphene can be derived using the Kubo formula, which is described with interband and intraband contributions [41].
In the circuit model analysis, it is assumed that the graphene array surface acts as an equivalent impedance $Z_g$ in response to the incident wave, which relates the external electric field to the surface current. For the TM-polarized case: $E_x^{ext}=Z_gJ_x$, where $E_x^{ext}$ is the $x$-component of the external electric field and $J_x$ is the surface current density induced on the ribbon in the $x$-direction [43]. Applying the Floquet theorem, an integral equation can be obtained for the surface current on the $i$th ribbon [44]:
Based on the perturbation theory, we treat the second term on the left-hand side of Eq. (6) as an interaction-induced perturbation to the eigenvalue problem for a single ribbon, which is described by the eigenequation without the perturbation term [45]. A method for calculating the eigenvalue problem for a single ribbon using the Fourier expansion of the eigenfunctions is presented in Ref. [45]. The first three eigenfunctions and the corresponding eigenvalues are listed in Table 1. The higher order eigenvalues and eigenfunctions ($n>3$) can be approximately determined by $k_n=(n-0.25)\pi /W$, $\psi _n(x)= \sqrt {2/W}cos(n\pi x / W)$ for odd orders and $\sqrt {2/W}sin(n\pi x / W + (-1)^n \pi /2)$ for even orders. Finally, it can be assumed the eigenfunctions $\phi _n$ are not significantly affected by the perturbation, and set $\phi _n \sim \psi _n$. The eigenvalues $q_n$ can be calculated based on the first-order perturbation theory using the "unperturbed" eigenvalues $k_n$ [46]. According to the expression of the induced surface currents, $Z_g$ can be written as
Table 1. First three eigenvalues and eigenfunctions for the problem of a single graphene ribbon
In most applications, the device is designed near the first resonance frequency of the graphene array. Then, the effect of the higher order modes can be neglected. The impedance of the graphene array can be modeled with a series R-L-C circuit corresponding to the first mode. Because the value of the capacitive impedance is not related to the material properties of the surface, instead of attempting to directly solve the modified first-order eigenvalue, we obtain the expression of the capacitive impedance of the parallel ribbons by analyzing the perfectly conducting strips. First, we consider a periodic array of ideally conducting strips at normal incidence for TE polarized wave (electric field along the $y$-direction), as shown in Fig. 2. When the electric field has a non-zero component parallel to the ideally conducting strips, the response of the array is purely inductive at low frequencies [47]. The averaged boundary conditions for such an array can be found in Ref. [47,48]:
where $Z_{e}=\sqrt {\mu _0 / \varepsilon _0 \varepsilon _{e}}$ is the wave impedance of the uniform host medium and $\alpha$ is the grid parameter. The grid parameter for the first-order Floquet modes can be approximated as $k_{e} D ln \{ csc( \pi W / 2D ) \}/ \pi$, where $k_{e}=k_0 \sqrt {\varepsilon _{e}}$ is the wave number of the incident wave vector in the effective host medium. $\mu _0$ and $k_0$ are the permeability and wave number in free space, respectively. From the Babinet principle, we can derive the capacitive impedance of the complementary structure for TM polarized incident fields (see Fig. 1(a)) as [49]:Fig. 2. The ideally conducting strips with the complementary structure of the graphene array in a homogeneous host medium
Therefore, the surface of the graphene-based ribbon array can be characterized by an equivalent impedance $Z_g = R + j \omega L + 1/j \omega C$. The values of $R$, $L$, and $C$ can be calculated as follows:
In the equivalent circuit model for the proposed absorber, the conductance of the Au layer is represented by a short circuit, that is, $Z_{Au} = 0$. Therefore, the input impedance of the dielectric spacer can be calculated as
where $d$ represents the thickness of the dielectric layer. As a result, the total input impedance of the absorber can be calculated as $Z_{in} = Z_g \parallel Z_{tr}$. Based on the above analysis, the absorption of the absorber can be calculated as $A = 1 - |\Gamma |^2$, where $\Gamma = (Z_{in}-Z_0)/(Z_{in}+Z_0)$ is the amplitude reflection coefficient of the absorber.3. Design procedures
In this section, we outline the design procedure of the proposed absorber based on the circuit model derived in the previous section. The impedance matching concept is used to extract the closed-form expressions for the device parameters. To maximize the absorption of the absorber, the total input impedance of the absorber should be matched with that of free space. First, this can be achieved for a given central frequency, $\omega _0$, by the conditions $Re(Z_{in}) \approx Z_0$ and $Im(Z_{in})=0$. Because the transmission line equivalent to the dielectric spacing layer acts as a quarter wavelength line by setting $\beta d = \pi / 2$ at $\omega _0$, the above two conditions can be simplified as:
However, these conditions are not sufficient to achieve high absorption with the desired bandwidth. It is known that the sharpness of a resonance circuit is defined by the Q-factor. The Q-factor of a series RLC circuit is $Q = \sqrt {L / C}/R$. Through simple mathematical processes, the Q factor of the equivalent circuit of this absorber can be calculated as $Q = \tau \omega _0$. It can be concluded that higher values of the relaxation time lead to a narrower bandwidth, whereas lower values of the relaxation time result in a broader bandwidth. To obtain the relations for different parameters of the structure that achieve high absorption with the desired bandwidth, the evaluation criteria for bandwidth are provided. We select $A \geqslant 0.9$ as an acceptable absorption criterion in the continuous spectral range. Because the total input impedance exhibits a certain symmetry near the resonance frequency, an absorption of more than 90% occurs approximately in the interval $[ \omega _0- \Delta \omega / 2, \omega _0+ \Delta \omega / 2]$. The fractional bandwidth of the absorption spectra can be defined as $B = \Delta \omega / \omega _0 \times 100\%$. Considering $\omega _1=(1+B/2)\omega _0$, $Z_{in}$ at this frequency is calculated as:
This results in $Z_{in}|_{\omega =\omega _1}=(R_g +jX_g)||jX_{tr}$, where
Thus, with the given central frequency and desired fractional bandwidth, the relaxation time $\tau$ of graphene can be caculated using Eq. (21). In practice, $\tau =\mu E_F/e v_F^2$ indicates the dependency of the relaxation time on the Fermi energy and electron mobility of graphene. Here, $\mu$ is the electron mobility and $v_F = 10^6$ m/s is the Fermi velocity of graphene. It should be noted that the electron mobility of graphene on a substrate ranges from approximately 0.1 to 6 $\textrm {m}^2/\textrm {Vs}$, depending on the fabrication process [39,50,51]. These facts indicate that the value of the relaxation time should fall in the most practical range of $\tau$ = 0.1 to 2 ps [52,53].
In summary, the material and geometrical parameters of the proposed absorber can be calculated using Eq. (14), Eq. (15), Eq. (16) and Eq. (21). As a result, the ultimate goal, which is the design of a terahertz absorber with the desired central frequency and fractional bandwidth, can be achieved.
4. Results and discussion
According to the design method described above, two graphene-based perfect terahertz absorbers with relatively narrow bandwidths and two high absorption absorbers with broad bandwidths were designed, where the design parameters are presented in Table 2. To investigate the performance of the proposed method in designing both narrowband and broadband absorption, the central frequencies of all the absorbers were uniformly set to 1 THz and the fractional bandwidths were customized, increasing from 10% to 100%. The absorption spectra of the designed narrowband and broadband absorbers are depicted in Figs. 3(a) and 3(b), respectively. To verify the validity of the proposed method, the absorption spectra calculated with the equivalent circuit model approach (solid lines) were compared with those obtained using the full-wave simulation based on the finite element method with COMSOL Multiphysics (dashed lines). For full-wave simulations conducted by COMSOL, graphene is modeled by a transition boundary condition whose permittivity is $\varepsilon _g = \varepsilon _0- j \sigma _g / (\Delta \omega )$, where $\Delta =1$nm. It can be seen that the results obtained using the circuit model and numerical simulations are in good agreement, demonstrating the accuracy and efficacy of the proposed circuit model approach. It should be noted that these absorbers can be achieved using the same configuration composed of only one layer of the graphene array.
Fig. 3. Absorption spectra of (a) Absorber 1 in Table 2 with a central frequency of $f_0$ = 1 THz and fractional bandwidth of 10% and (b) Absorber 4 in Table 2 with a central frequency of $f_0$ = 1 THz and fractional bandwidth of 100%.
Table 2. Designed parameters of the absorbers with a given center frequency of 1 THz and customed relative bandwidth ranging from 10% to 100%
The normalized input impedances of the two absorbers with a narrow bandwidth of 10% and broadband bandwidth of 100% are plotted in Figs. 4(a) and 4(b), respectively. As seen, the sharp changes in both the real and imaginary parts of the normalized input impedance of Absorber 1 near the center frequency leads to a rapid mismatch off the center frequency thus resulting in narrowband absorption. On the other hand, a near-zero imaginary part and a near-unity real part of the normalized input admittance of Absorber 4 around the central frequency indicate wide impedance matching and broadband absorption. Figures 4(c) and 4(d) illustrate the loss distributions on a graphene ribbon of Absorber 1 and Absorber 4 at their resonance frequencies, respectively. As depicted, the power loss density distributions are similar at the resonance frequencies of both absorbers, where the fundamental resonant mode of the graphene ribbon is responsible for both the resonant frequencies. This demonstrates the accuracy and efficacy of the proposed circuit model for the design of both narrowband and broadband absorbers.
Fig. 4. Normalized input impedances of (a) the narrowband Absorber 1 and (b) the broadband Absorber 4 presented in Table 2. The loss density distributions on the graphene ribbon of (c) Absorber 1 and (d) Absorber 4 at their resonance frequencies.
Figure 5 shows the tunability of the designed narrowband and broadband absorbers observed by changing the Fermi energy of graphene. As depicted in the Fig. 5(a), at different Fermi energies, the peak absorption frequency of Absorber 1 varies from 0.9 to 1.4 THz, whereas its absorption peak remains nearly constant with a near-unity value. Note that further lower down the bandwidth, the calculated value of the Fermi level is extremely small which cannot be accurately modulated in practical. Figure 5(b) illustrates the Fermi energy variations of Absorber 4 from 0.3 eV to 0.6 eV. This indicates that the performance of the device in the lower frequency band is not sensitive to variations in the Fermi energy of graphene, while the absorption spectrum in the higher frequency band shows a blueshift, leading to a change in the bandwidth. In general, any change in the Fermi energy of graphene results in a change in the surface conductivity of graphene. As defined in Eq. (12), the variation in the Fermi energy leads to a change in the $R$ and $L$ values in the circuit model. Due to the different relaxation times of the narrowband and broadband absorbers, the change in the Fermi energy has different effects on the absorption spectra of absorbers with different bandwidths. This indicates that the narrowband absorber has peak frequency tunability and the broadband absorber has fraction bandwidth tunability.
Fig. 5. Absorption spectra of (a) Absorber 1 and (b) Absorber 4 presented in Table 2 for different Fermi energies, calculated using the circuit model (solid lines) and full-wave simulations (dashed lines).
Next, the angular dependence of the designed absorbers was investigated. Figures 6(a) and 6(b) show the absorption spectra of Absorber 1 and Absorber 4 as a function of frequency and incident angle for TM polarization, respectively. The absorption peak of the narrowband absorber can be observed for large incident angles in Fig. 6(a). As shown in Fig. 6(b), a broadband absorption of approximately 80% can still be achieved at an incident angle of $70^0$. Hence, this verifies that the proposed absorbers are angular insensitive within this range.
Fig. 6. Absorption spectra of (a) Absorber 1 and (b) Absorber 4 presented in Table 2 as a function of incident angle and frequency for TM polarizations.
5. Conclusion
In summary, a novel and efficient approach for designing terahertz absorbers based on a single layer of graphene ribbons was proposed. Based on the proposed circuit model and closed-formed relations of different design parameters of the structure, both narrowband and broadband absorbers can be designed with a customized central frequency and the desired fractional bandwidth. For a given central frequency at 1 THz, graphene-based absorbers with a fractional bandwidth ranging from 10% to 100% were designed using the proposed approach. The results were verified via full-wave numerical simulations. In addition, it was demonstrated that the narrowband absorber has peak frequency tunability, while the broadband absorber has fraction bandwidth tunability. Futhermore, these absorbers are insensitive to the incident angle. The proposed approach significantly simplifies the design and analysis of graphene-based terahertz absorbers.
Funding
Key-Area Research and Development Program of Guangdong Province (2020B090922006); South China Normal University (2019LKXM010); Guangzhou Municipal Science and Technology Project (202002030165).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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