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Abstract
In general, there is a fundamental trade-off between the operational bandwidth and the attainable absorption. So, obtaining broadband wave absorption of a low reference standard such as 90% is not very difficult. However, when trying to obtain higher absorption such as 99%, the bandwidth will drop dramatically. Here, we demonstrate that broadband near-perfect absorption of over 99% absorption with a 60% relative bandwidth can be obtained utilizing single-layered and nonstructured graphene loaded with periodical dielectric wires. The absorption mechanism originates from the coupling of Mie resonances in dielectric wires excited by the incident wave to the graphene plasmon resonances, which introduces two absorption contributions: direct near-field absorption in the graphene and radiative emission into the graphene.
© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Metamaterial perfect absorbers (MPAs) have attracted increasing attention due to their potential applications, including cloaking [1,2], sensing [3], detection and imaging [4]. The first MPA, which is based on the metallic metamaterial, was demonstrated by Landy et al. at microwave frequencies [5]. Following this work, a great number of MPAs utilizing metallic metamaterials have been proposed and designed with the absorption band expanded from microwave to terahertz (THz) [6–9], infrared and visible region [10–15]. Recently, graphene, a monolayer carbon atom arranged in a honeycomb lattice, has provoked extensive interests in designing frequency-tunable MPAs due to its tunability, broadband response and high carrier mobility. However, many of these graphene-based MPAs’ absorption bands are band-limited or narrowband as a result of the reliance on the resonances of surface plasmon polaritons (SPPs) [16–20], which restricts their applications in some circumstances.
To realize broadband absorption, various methods have been adopted. Typical approaches include integrating multiple graphene resonators in a unit structure [21,22], using multi-layered graphene structures [23], and utilizing the structured graphene with gradually changed geometric sizes [24,25]. Besides, based on impedance matching concept, Khavasi et al. have achieved ultra-broadband absorption at THz frequencies [26,27]. However, the above broadband absorbers are based on structured graphene which unavoidably introduce some edge effects [28], thus leading to the difficult to realize the perfect absorption in practice. To avoid this problem, the absorbers using nonstructured graphene are put forward, such as the millimeter wave absorber which has multilayer graphene sheets on quartz substrates [29], the dual-gated tunable absorber based on graphene hyperbolic metamaterial [30], the absorber using periodical arrays of dielectric bricks loaded with single-layered graphene [31], the terahertz absorber based on multi-band continuous plasmon resonances in geometrically gradient dielectric-loaded graphene plasmon structure [32]. However, all the above mentioned broadband absorbers are based on 90% absorption, which is a low reference standard. In fact, the corresponding bandwidths of 99% absorption for these structures are fairly narrow. This is because, in general, there is a fundamental trade-off between the operational bandwidth and the attainable absorption [33]. When trying to obtain higher absorption such as 99%, the bandwidth will drop dramatically. Therefore, the broadband absorber of higher absorption still remain to be further investigated.
In this letter, utilizing single-layered and nonstructured graphene loaded with periodical dielectric wires, we propose a broadband near-perfect absorber with a 60% relative bandwidth of over 99% absorption based on the coupling of Mie resonances to graphene plasmon resonances. Here, we elucidate the physical mechanism firstly. Then, we present numerical simulations and results. Finally, we investigate the effects of some important structural parameters.
2. Structure and principle
The proposed absorber is presented in Fig. 1, which consists of monolayer graphene loaded with periodical dielectric wires supported by a piece of dielectric substrate on a metallic film. The structure is characterized by the periodic interval P along x-axis, height H1, bottom width W and tilt angle β of the isosceles trapezoid cross section of the dielectric wire, Fermi level EF of the graphene, thickness H2 of the dielectric substrate. The material of the dielectric wire and the substrate are dielectric 1 and dielectric 2, respectively. Here, considering that the transmission is blocked by the thick metallic film, the conditions for perfect absorption are
andwhere Zin is the input impedance of the absorber, Z0 is the impedance of the free space. However, the above conditions do not lead to broadband absorption. According to the preceding analysis, it is hard to achieve the perfect absorption over a wide frequency range. But it is possible to obtain near-perfect absorption (such as 90%) in a wide frequency range. To achieve broadband near-perfect absorption, the Eq. (1) and Eq. (2) can be respectively relaxed toandwhere f is the frequency in the absorption band, and α is a numerical parameter larger than unity and is determined by the minimum allowed absorption Am [26,34]. The relation between α and Am can be expressed as
Fig. 1 Schematic representation of the broadband near-perfect absorber, consisting of monolayer graphene loaded with periodical dielectric wires supported by a piece of dielectric substrate on a metallic film.
Here, we consider the case where the absorption is larger than 99%, which implies Am equals to 99%. Then, according Eq. (5), α should be set to 1.22. To obtain the absorption with broad bandwidth of over 99% absorption, the real part of Zin/Z0 should be between 1/1.22 and 1.22, and the imaginary part of Zin/Z0 should be small in a wide frequency range. In general, these two conditions can be fulfilled simultaneously by introducing broadband resonances.
In this configuration, the periodical dielectric wires with high refractive index can be viewed as Mie resonators that can capture and trap the wave by multiple internal reflections from the boundaries [35–38]. Under the incident excitation with the electric field parallel to x-axis, the Mie modes which behave as magnetic resonances could be developed in the dielectric wires. Here, the formation of the Mie modes can be explained through the following analysis. Basically, the wave front of an incident plane wave undergoes a strong distortion close to the dielectric wires to satisfy simultaneously the continuity and discontinuity conditions of tangential and normal electric field components at the wire-air interfaces, respectively. The electric field, developed inside the dielectric wires, is then predominantly tangential close to the boundaries. This leads to the creation of displacive eddy currents in the cross section of the dielectric wire, which enhance the magnetic field confined along the y-axis [38]. Owning to the small size of the subwavelength dielectric wires, the internal Mie resonance becomes leaky, which implies it is broadband and interacts more effectively with the outside world [33]. In this structure, Mie resonances in dielectric wires can be coupled to graphene plasmon resonances, which introduce two absorption contributions: direct near-field absorption in the graphene and radiative emission into the graphene. Both the two absorption contributions introduce a leaky channel into the graphene for the field of Mie resonances, leading to the broadening of Mie resonances. Hence, as previously described, Eq. (3) and Eq. (4) can be satisfied as a result of the broadband Mie resonances, yielding the broadband near-perfect absorption.
3. Simulation and discussion
Here, we use CST Microwave Studio to explore the absorption mechanism and the performance of the proposed absorber. In the simulation, relative permittivities of dielectric 1 and dielectric 2 are set to 12 and 4, which can be approximatively provided by silicon and silicon dioxide respectively in the THz range [39]. The metallic material is gold with thickness 2 μm and is described by Drude model. The relative permittivity is εgold (ω) = ε∞ - ωP2/(ω2 + iωγ). ε∞, ωP and γ are 1.0, 1.38 × 1016 rad·s−1, and 1.23 × 1013 s−1, respectively [40]. Graphene is modeled as an anisotropic effective media of thickness t = 1 nm with the in-plane relative permittivity εin = 2.5 + iσ(ω)/(ωε0t) and the out-of-plane relative permittivity εout = 2.5, where the conductivity σ(ω) can be expressed in the THz range as
where ω is the angular frequency of the incident wave, e is the charge of an electron, kB is the Boltzmann constant, T = 300 K is the temperature, is the reduced Planck constant, τ is the carrier relaxation time, and EF is the Fermi level. Here, the relaxation time is given as τ = μEF/evF2 [41], where the mobility μ is 10000 cm2/(V·s) and the Fermi velocity vF is 106 m/s. In the simulation, the open boundary condition is adopted in z direction and the unit cell boundary condition is adopted in x and y directions. The absorption is calculated as A = 1-R-T, where R and T are reflection and transmission respectively. Here, the reflection R is given by |S11|2, where S11 is the reflection coefficient of the incident wave and is extracted directly from the CST. The transmission channel is blocked by the thick metallic film, leading to T ≈0. Therefore, the absorption is obtained asFirstly, we consider the case where EF = 0.3 eV, W = 61 μm, β = 72°, H1 = 61 μm, H2 = 34 μm and P = 80 μm, respectively. Figure 2(a) shows the calculated absorption spectrum for the incident polarization with the electric vector that is parallel to x-axis under normal incidence excitation. From Fig. 2(a), we can see the above 99% absorption covers the frequency range of 0.66-1.21 THz and the relative bandwidth reaches about 60%. To confirm the preceding theoretical analysis, we calculate the ratio of the input impedance of the absorber to the impedance of the free space in the 99% absorption band using
where F is the reflection efficient, which is given by S11. By simple algebra operation, Zin/Z0 can be obtained asUtilizing Eq. (9), Zin/Z0 is obtained and shown in Fig. 2(b). It can be seen from Fig. 2(b) that the real part of Zin/Z0 is between 1/1.22 and 1.22, and the imaginary part of Zin/Z0 is small in the 99% absorption band, which is in accordance with the preceding analysis. To elucidate the broadband mechanism, the magnetic field amplitude patterns on the x-z plane for six representational frequencies in the 99% absorption band are calculated and shown in Fig. 2(c). As can be seen from Fig. 2(c), the field is mainly concentrated on the graphene-dielectric interface in the low-frequency band while confined inside the dielectric wire in the high-frequency band. Besides, we find the first and second order graphene plasmon resonances are produced in the low-frequency band. Hence, for the low-frequency band, the broadband mechanism can be explained through the following analysis. Firstly, the periodical dielectric wires on the graphene can provide graphene plasmon resonances of different orders with large relative frequency interval and relative radiation rate 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 [31]. For the high-frequency band, we can see that Mie resonances are excited by the incident wave in the dielectric wire. Hence, the corresponding broadband absorption results from the coupling of the Mie resonances to graphene plasmon resonances (see Visualization 1). Therefore, these two different mechanisms contribute jointly to the whole absorption band.Fig. 2 (a) The absorption spectrum under normal incident wave with the electric field parallel to x-axis. (b) The ratio of the input impedance of the absorber to the impedance of the free space in the 99% absorption band. (c) Magnetic field amplitude patterns on the x-z plane for six representational frequencies in the 99% absorption band.
To study the tunable effects of the absorber, we varied the Fermi level EF between 0.3 eV and 0.5 eV, fixing W = 61 μm, β = 72°, H1 = 61 μm, H2 = 34 μm and P = 80 μm. Figure 3 shows the absorption curves with varying Fermi level under normal incident excitation with the electric field parallel to x-axis. From Fig. 3, we find there is a remarkable blueshift of the lower absorption band as EF increases, while the upper absorption band is affected weakly by EF. This is because graphene plasmon resonances dominate the low-frequency absorption and the plasmonic resonance frequency of the graphene increases with increased Fermi level [42], while the high-frequency absorption is mainly dependent on Mie resonances which are less affected by Fermi level, as shown in Fig. 2(c).
Next, we consider the situation where Fermi level EF is fixed at 0.3 eV and structural parameters of the dielectric wire (W, β, H1) are varied, as shown in Fig. 4. We have to mention that in each group of the simulation, all the other parameters and conditions are kept the same as the first simulation. Figure 4(a), Fig. 4(b) and Fig. 4(c) respectively shows the absorption curves under normal incident excitation with the electric field parallel to x-axis for variable bottom width W, tilt angle β and height H1. As can be seen from Fig. 4(a) and Fig. 4(b), a redshift of the absorption band is evident with increased W or β. The reason for this corresponds to the fact that the increased W or β leads to an increased effective refractive index of the resonant mode [43,44]. However, from Fig. 4(c), we find the absorption is insensitive to H1, which results from the structural feature that H1 has little effect on the effective refractive index of the resonant mode when W and β are fixed. In addition, we find a common characteristic from Fig. 4 that the upper absorption band is more susceptible to the structural parameters of the dielectric wire compared with the lower absorption band, which is because Mie resonances are more affected by the geometry of the dielectric wire than graphene plasmon resonances.
Fig. 4 (a) The absorption curves with varying bottom width of the dielectric wire. (b) The absorption curves with varying tilt angle of the dielectric wire. (c) The absorption curves with varying height of the dielectric wire.
Then, in order to investigate the absorption sensitivity to the oblique incident plane wave, we vary the incident angle θ (defined as the angle between the incident plane wave and positive z-direction) with all the structural parameters and Fermi level kept the same as the first simulation, while maintaining the incident plane wave in the x-z plane and the magnetic field parallel to y-axis. Figure 5 shows the absorption as a function of frequency and the incident angle. From Fig. 5, we find the incident angle dependence is relatively weak when the incident angle varies between 0 and 60 degree. As the incident angle increases beyond 80 degree, the absorption decreases rapidly. The phenomenon stems from the fact that the condition of broadband near-perfect absorption is not satisfied in the case of large angle incidence.
4. Conclusion
In summary, we propose a broadband near-perfect absorber utilizing monolayer and nonstructured graphene loaded with periodical dielectric wires. Numerical simulations demonstrate that a 60% relative bandwidth of over 99% absorption can be achieved. The absorption mechanism originates from the coupling of Mie resonances in dielectric wires excited by the incident wave to the graphene plasmon resonances, which introduces two absorption contributions: direct near-field absorption in the graphene and radiative emission into the graphene. This work breaks a fundamental trade-off between the operational bandwidth and the attainable absorption, and provides a new idea for the design of graphene broadband absorber.
Funding
National Natural Science Foundation of China (NSFC) (11674396).
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