## Abstract

We present a wave absorption design consisting of periodical arrays of dielectric bricks on the dielectric substrate, which is coated with single-layered and nonstructured graphene, supported by a thick piece of metal. The design is demonstrated to broadband near-perfect absorption with 0.82 terahertz (THz) bandwidth of over 90% absorption and with central frequency of 1.68 THz. The broadband absorption mechanism originates from two contributions. Firstly, the periodical arrays of dielectric bricks on the nonstructured graphene can provide both a set of graphene plasmon resonances with large relative frequency interval and relative radiation rate γ/ω in the THz range. Secondly, the linewidth of each resonance can be broadened by the far-field interaction between neighboring resonators to overlap and spread over a wide frequency region in the THz range. The design in this paper is simple, and consequently facilitates the fabrication and promotes the application of broadband graphene absorbers.

© 2017 Optical Society of America

## 1. Introduction

Wave absorption is crucial for various types of application, including detection, imaging and modulators [1, 2]. Graphene, a single-layered carbon atom arranged in a honeycomb lattice, has attracted considerable attention as a result of its range of potential applications from optical to THz frequencies [1–10]. Recently, graphene has become one of the most promising materials in the design of wave absorbers that can operate in both THz and infrared spectral ranges due to its tunability of carrier mobility and conductivity [11–30]. The first demonstration of complete optical absorption based on graphene takes place in the arrays of doped graphene nanodisks lying on a dielectric layer coating a metal [11]. Then, graphene absorbers have been extensively studied and a variety of structures have been proposed such as graphene ribbons [12–14], anti-dots [15], fishnets [16, 17] or nonstructured graphene with comprising sub-wavelength resonant structures [18–22]. However, these absorbers operate at narrowband spectral ranges because their complete wave absorptions rely on the resonances of surface plasmon polaritons (SPPs) [23], which limits their potential applications. One mean of improving the bandwidth is to use multilayered graphene structures, where several patterned graphene layers biased at different voltages and backed with dielectric substrates are stacked on top of each other [24]. However, it may be difficult to put multilayered graphene structures with different biasing voltages into application. In addition, these structures are based on structured graphene which unavoidably introduce a great many truncation edges of the graphene. These graphene edges from the tailored fabrication haven't been considered in the theory but really exist in experiment, which might lead to edge effects, such as unordered diffuse scattering losses [25, 26], thus rendering it difficult to achieve the unity absorption in practice. Another method is to make use of nanostructured graphene with gradually changed geometric sizes [27]. However, the structures is still based upon the structured graphene to produce varying continuous plasmonic resonances when excited by an incident wave.

In this paper, based on single-layered and nonstructured graphene, we present a broadband absorber with 0.82 THz bandwidth of over 90% absorption and with a central frequency of 1.68 THz. Furthermore, the absorption's dependence on incident angle is also weak and the broadband absorption remains high for wide range of incident angles. Here, we supply the physical insight of broadband absorption of the proposed model system firstly. Then, we run a numerical simulation at terahertz frequency. Finally, we investigate the effects of some relative parameters.

## 2. Structure and principle

The representation of the proposed structure is shown in Fig. 1, which consists of periodical arrays of dielectric bricks on the substrate coated with single-layered graphene supported by a thick piece of metal. The structure is characterized by the periodic interval *P*, the width *W* and thickness *H _{1}* of the brick, the Fermi energy

*E*of graphene and the thickness

_{f}*H*of substrate. In the above configuration, the metal film and arrays of dielectric bricks form two mirrors of the asymmetric Fabry-Perot cavity [12]. The transmission channel could be suppressed as long as the bottom metallic film is thick enough. The reflection path is the sum of all multiple reflection, and the sum would minimize to zero due to the destructive interference if the characters of the structure are well designed. When both the transmission channel and the reflection path close, the only possible outcome will be absorption, which leads to the near-perfect absorption. In this structure, the near-perfect absorption is closely related to the graphene plasmon resonance. So, in order to achieve the broadband near-perfect absorption, the key is to obtain broadband graphene plasmon resonance.

_{2}First, we consider the excitation of the graphene surface plasmons with a free-space wave. It is well known that high doped graphene (*E _{f}* >$\hslash $ω) in which optical loss from interband transition is suppressed can sustain strongly surface plasmons propagating along the sheet with wave vector [9, 10]

*e*is the charge of electron, $\hslash $is the reduced Planck's constant,

*ω*is the angular frequency of graphene surface plasmons,

*E*is the Fermi energy of graphene,

_{f}*τ*is the carrier relaxation time in graphene,

*ε*is permittivity of vacuum,

_{0}*ε*is the effective relative permittivities of periodical arrays of dielectric bricks above the graphene film, and

_{r1}*ε*is the relative permittivities of the dielectric substrate. The wave vector

_{r2}*k*in graphene is much larger than that in the free space. In order to excite the surface plasmons in graphene using free-space wave, the large difference between their wave vectors should be overcome. In the structure, one of the roles of periodical dielectric brick arrays is compensating the wavevector mismatches. Considering the C

_{GSP}_{4v}symmetry of the structures, for simplicity, we assume the incident wave to be in the y-z plane and the polarization to be along the x-direction (TE polarization). The resonant angular frequency

*ω*is determined by the wave-vector matching equationwhere Re(

*k*) is the real part of wave vector

_{GSP}*k*,

_{GSP}*c*is the speed of light in vacuum,

*P*is the array period,

*N*is positive integer,

*θ*is the incident angle defined as the angle between the incident wave and the Z axis.

Then, by simple algebra operation, graphene plasmon resonant frequency *ω* for the N-th order mode can be obtained as

*θ*= 0), the relative frequency interval, is expressed as $\frac{\left|{\omega}_{N}-{\omega}_{N+1}\right|}{\left({\omega}_{N}+{\omega}_{N+1}\right)/2}=2\frac{\sqrt{N+1}-\sqrt{N}}{\sqrt{N+1}+\sqrt{N}}$, between the first and second modes is 34.3%, and between the second and third modes is 20.2%.

Exciting a set of graphene plasmon resonances with large relative frequency interval is the first step to obtain broadband near-perfect absorption. When the excited graphene plasmon modes propagate along the y direction, at the interface between the dielectric bricks and air gap, they partly reflect back into the bricks, and partly transmit into the air gap to become an outgoing wave to re-excite the graphene plasmon resonances supported by the neighboring dielectric bricks. So when the separation between neighboring dielectric bricks is smaller than wavelength, except for the near-field coupling effect, the far-field interaction must be considered [31, 32]. From the temporal coupled mode theory (CMT), the corresponding dynamics are described as follows:

*ω*is the center frequency of the N-th graphene plasmon resonance,

_{Ν}*a*

_{N}_{1}= |

*a*

_{N}_{1}|${e}^{-j{\omega}_{N}t}$and

*a*

_{N}_{2}= |

*a*

_{N}_{2}|${e}^{-j{\omega}_{N}t}$ are the corresponding normalized amplitude in two nearest neighbor resonators,

*s*

_{+}is the amplitude of the incident light,

*Γ*is the absorption rate, and γ

_{N}*is the radiation rate, φ*

_{Ν}*is the phase of the re-radiated wave of one resonator that reaches the nearest neighbor resonator. In the equations, the coefficient κ*

_{Ν}*describes the near-field coupling between the two resonators, while $\sqrt{{\gamma}_{N}}{e}^{-i{\phi}_{N}}$contributes to the far-field interaction between them (this approximation is valid when the separation between neighboring resonators is much smaller than wavelength). For the case*

_{Ν}*Γ*0 and

_{N}=*s*

_{+}

*=*0 in Eqs. (4) and (5), we obtain the quasi-eigen frequencyEquation (6) can be also written as

*= 0 (critical condition), which can be achieved by varying the distance between two neighbor resonators, the subradiant dark mode has zero linewidth and thus cannot be excited by the incident plane wave. On the contrary, the superradiant bright mode reaches the maximal linewidth, which is two times as much as that of original graphene plasmon resonance, and it has strongly interaction with the incident plane wave and then gains imperium on role of wave absorption.*

_{Ν}On the other hand, from Eq. (1), we can see a quadratic dependence of Re(*k _{GSP}*) on

*ω*, which implies the ability of light confinement provided by the graphene plasmon is proportional to

_{Ν}*ω*. The quantified degree of confinement is given by the ratio of graphene plasmon to free-space-light wave vectors

_{Ν}*and relative radiation γ*

_{N}*ω*

_{N}/*of graphene plasmon resonance in the terahertz range is much larger than in the mid- and far-infrared range.*

_{N}Since the structure can provide both a set of graphene plasmon resonances with large relative frequency interval and larger relative radiation rate γ* _{N}/* ω

*in the terahertz range, we can expect the broadened linewidth for each resonance caused by the far-field interaction between neighboring resonators to overlap and spread over a wide frequency region in the terahertz range. The achieved broadband graphene plasmon resonances imply that the broadband near-perfect absorption may be obtained under certain conditions.*

_{N}## 3. Simulation and discussion

To verify the theoretical prediction, we conducted the fullwave numerical simulations using frequency domain solver in CST Microwave Studio. In the numerical simulation, the graphene is modeled as an anisotropic effective media of thickness *t _{g}* = 1 nm with the in-plane component of relative complex permittivity as ε

_{rx}(ω) = ε

_{ry}(ω) = 2.5 + i

*σ*(ω)/(ωε

_{0}

*t*) and the surface normal component as ε

_{g}_{rz}(ω) = 2.5, where σ(ω) is the optical conductivity and calculated in the terahertz range as

*=*10

^{6}m/s is the Fermi energy level, and μ = 10000

*cm*(

^{2}/*V*·

*s*) is the DC mobility), and T = 300

*K*is the temperature. In our simulation, the dielectric material and substrate material are considered to be non-dispersive with relative permittivity

*ε*= 12 and

_{dielectric}*ε*= 4, respectively. The metallic material is gold with thickness 1 μm and is treated as a dispersive medium following the Drude model. The relative permittivity is $\epsilon (\omega )={\epsilon}_{\infty}-{\omega}_{p}^{2}/({\omega}^{2}+i\omega \gamma )$. The value of${\epsilon}_{\infty}$,

_{substrate}*ω*and γ are 1.0, 1.38 × 10

_{p}^{16}rad•s

^{−1}and 1.23 × 10

^{13}s

^{−1}[33], respectively. In this work, the absorption A is calculated by A = 1-R-T, where R and T represent reflection and transmission respectively.

First, we consider the case with *P* = 40 μm, *W* = 29 μm, *H _{1}* = 14 μm

*H*= 22 μm,

_{2}*T*= 300

*K*and the graphene with

*E*= 0.5

_{f}*eV*. Figure 2(a) shows the calculated absorption spectra for TE polarization under normal incident excitation. From Fig. 2(a), we find the 90% absorption bandwidth is as high as 0.82 THz with a central frequency of 1.68 THz. The electric field amplitude patterns on the central cutting y-z plane for three representational frequencies 1.3, 1.6, and 2.2 THz, corresponding to three absorption bands labeled by A, B and C, are calculated and shown in Figs. 2(b)–2(d), respectively. Figures 2(b)–2(d) clearly show that the three absorption bands correspond to the first, second and third order graphene plasmon resonances, respectively. In addition, we can also see that, in the air gap between two neighboring dielectric bricks, the electric fields are composed of propagating waves that correspond to the far-field interaction and evanescent waves that correspond to the near-field interaction effect. In order to investigate the absorption sensitivity to the oblique incident wave, we vary the incident angle θ from 0 to 89 degree while maintaining the incidence wave in the y-z plane and the electric field vector along the x-direction with all other parameters kept the same as before. Figure 2(e) shows the absorption as a function of frequency and incident angle. From Fig. 2(e), three absorption peaks are always observed and they are only slightly sensitive to the incident angle when the incident angle varies between 0 and 80 degree. By solving Eq. (3), we can obtain the analytical graphene plasmon resonant frequencies as a function of incident angle for the first, second and third modes, as shown in Fig. 2(f). Comparing Figs. 2(e) and 2(f), we can see that the three absorption peak frequencies are in agreement with the analytical graphene plasmon resonant frequencies, which slightly increase with increased incident angle. As the incident angle increases beyond 80 degree, the absorption decreases rapidly. The reason corresponds to the fact that the near-unity absorption of this type of the structure is closely related to destructive interference. However, under conditions of glancing incidence, the reflection amplitude from graphene plasmon resonances and periodical arrays of dielectric bricks is much larger than that from thick metal, and the destructive interference condition is destroyed.

Next, we consider the situation where the Fermi level is fixed at 0.50 *eV* and the thickness *H _{2}*, and width

*W*are varied. Figure 3(a) shows the calculated absorption as a function of frequency and thickness

*H*under normal incident excitation with

_{2}*P*,

*W*and

*H*fixed at 40

_{1}*μm*, 29

*μm*and 14

*μm*, respectively. From Fig. 3(a), we can see that the absorption is sensitive to the thickness of the substrate. It results from the fact that the phase-matching condition of destructive interference is dependent on the phase delay provided by the thickness of the substrate material. Figure 3(b) shows the calculated absorption as a function of frequency and width

*W*with

*P*,

*H*and

_{1}*H*fixed at 40

_{2}*μm*, 14

*μm*and 22

*μm*, respectively. The width

*W*varies between 0 and

*P*. From Fig. 3(b), we can find the width

*W*has a great influence on the bandwidth of the absorption and have an optimized value for obtaining maximal broadband absorption. This can be understood through the following qualitative analysis. The φ

*is tightly related with the distance between two neighbor resonators, and the critical condition of φ*

_{Ν}*= 0 can be achieved for the optimized value, where the superradiant bright mode reaches the maximal linewidth while the subradiant dark mode has zero linewidth. When the distance deviates from the critical condition, the linewidth of the superradiant bright mode decreases while that of subradiant dark mode increases and the situation becomes quite different. For the limit case of*

_{Ν}*W*= 0, the structure consists of graphene, dielectric substrate and metallic mirror, and the consequent F-P effect (arising from multiple reflections at the substrate ends) enhances the absorption, resulting in a 55% maximum absorption at 2.4 THz. For

*W*=

*P*, the F-P effect and absorption enhancement is similar, however the effective refraction index is higher than the case of

*W*= 0 due to the higher-index material on the top, and so the maximum absorption occurs at the lower frequency (about 1.0 THz). In the two limit cases (

*W*= 0 and

*W*=

*P*), the field interaction disappears and the absorption is narrow.

Then, we study the tunable effects of this absorption by varying the Fermi level. Figure 3(c) is the calculated absorption spectra for Fermi energy 0.4eV, 0.5eV and 0.6eV. From Fig. 3(c), we can observe a blue shift of the working frequency range at large absorption with the increased Fermi level, which implies that the operation frequency can be tuned by the Fermi level. This is because the broadband near-unity absorption of this type of structure is closely related to the lowest order graphene plasmon resonances, and simultaneously, these resonant frequencies increase with increased Fermi level, which is confirmed by Fig. 3(d) that shows the analytical graphene plasmon resonant frequencies as a function of Fermi level by solving Eq. (3).

## 4. Conclusion

In summary, we design a broadband near-perfect absorber based on single-layered and nonstructured graphene. Numerical simulations demonstrate that the working bandwidth of 90% absorption is as high as 0.82 THz, with a central frequency of 1.68 THz. The broadband mechanism originates from two key contributions. One is the multiple graphene plasmon resonances with large relative frequency interval and radiant rate. The other is the far-field interactions between the neighboring resonators. The absorber is based on nonstructured graphene, which can avoid destroying the unique properties of graphene and be beneficial to the development of many related applications. In additon, it is simple and thus facilitates the application of the broadband graphene absorbers.

## Funding

National Natural Science Foundation of China (NSFC) (11674396, 1304389, 61404174).

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