Abstract

We have studied a switchable hyperbolic metamaterial composed of a graphene–dielectric periodic structure. By tuning the chemical potential of all graphene sheets simultaneously, the isofrequency curve can switch between an ellipse and a hyperbola conveniently. In particular, a special hyperbolic isofrequency curve with its asymptote perpendicular to the interface is obtained and used to realize the zero reflection effect. Furthermore, a zero-reflection-based optical switch working in the terahertz spectrum is demonstrated. Its bandwidth can be efficiently adjusted by geometric parameters such as permittivity and period. Such an optical switch possesses the merits of low loss, high transmittance contrast, high response speed, compact size, high tolerance of chemical potential, and having all incident angles (0°–90°) simultaneously. Such an optical switch holds great potential in many fields, such as data storage, beam steering, and integrated photonic circuits.

© 2019 Chinese Laser Press

1. INTRODUCTION

Metamaterials, which are artificial subwavelength structured materials, have been a hot topic of central importance due to their properties never observed in nature [13]. Among them, hyperbolic metamaterials are one of the most unusual classes of metamaterials [4,5]. Hyperbolic metamaterials have a diagonal permittivity tensor (or permeability tensor) whose principal components possess opposite signs. This special feature leads to its unusual hyperbolic dispersion and many unique properties. Unlike conventional double-negative metamaterials realized by a resonant-mediated mechanism, hyperbolic metamaterials not only simplify structures, but also make tunability possible. Hyperbolic metamaterials have attracted growing attention recently, whose applications involve negative refraction [68], nano-scale waveguides [9,10], sub-wavelength imaging [11], and spontaneous emission [12,13].

Metal is the most common material used to fabricate hyperbolic metamaterials, and the most common structures are nanowire-array structures [14] and layered metal–dielectric structures [15]. For example, the zero reflection effect in a metal–dielectric metamaterial was studied [16]. However, an inherent problem of such a metal-structured composite is material loss. Moreover, it is difficult to control the permittivity of metals after fabrication. Overcoming these two drawbacks is highly desired. Magnetic metamaterial based on YIG–dielectric layered structure is another kind of hyperbolic metamaterial, and it is used for optical switch design [17]; however, its modulation speed is low, and it works only in the microwave spectrum, since it does not have a magnetic response in infrared and terahertz spectra. What is more, it requires a bulky magnetic field unit for control, which is not convenient for practical applications and is unfavorable for miniaturization. In contrast to metal, graphene, a one-atom-thick allotrope of carbon first discovered in 2004 [18], is lossless and has excellent electronic and optical properties in infrared and terahertz spectra. Furthermore, its dielectric permittivity can be tuned by chemical potential, Fermi energy, external electrostatic field biasing, gate voltage, magnetic field, chemical doping, or the nonlinear effect [1924], so graphene is a good candidate for overcoming the aforementioned drawbacks of metal and can work in infrared and terahertz spectra, which is favorable for on-chip integration. Recently, multilayer structures consisting of graphene and dielectric have been studied as tunable hyperbolic metamaterials [2529]. The topological transitions and associated ultrafast optical properties in graphene-based hyperbolic metamaterials were studied [3032]. Later, it was revealed that a surface wave exists at the interface between two different hyperbolic metamaterials, and the corresponding frequency ranges of the surface wave can be tuned by changing the Fermi energy of graphene sheets [24]. Graphene-based metamaterials have been used to realize negative refraction [68], modulators [33], meta-lenses [34], absorbers [3537], vortex waves [38], waveguides [9,10,25], and optical switches [39,40]. Very recently, a structure consisting of Kerr metallic nanolayers and graphene sheets was proposed to design an optical switch [41], but this metal–graphene structure has a complicated five-layer unit cell and inevitably introduces absorption loss, requiring high fabrication techniques and limiting the performance of the optical switch. From the view of applications, it is highly desired to develop a new mechanism to realize an optical switch that can overcome all the above-mentioned disadvantages, such as material loss, low response speed, and a bulky magnetic control unit, simultaneously.

In this work, we study a graphene–dielectric switchable hyperbolic metamaterial (SHM) and tune its effective permittivity by graphene’s chemical potential. In this way, the isofrequency curve (IFC) of the SHM can switch between ellipse and hyperbola conveniently. When the asymptote of the hyperbolic IFC is perpendicular to the SHM–air interface, there is no reflected wave for any incident angles from the SHM, i.e., the zero reflection effect occurs. Furthermore, by using the peculiar and excellent opto-electronic properties of graphene, an all-angle (0°–90°) optical switch based on the zero reflection effect is demonstrated. Such an optical switch possesses low loss, high transmittance contrast, high response speed, compact size, and high tolerance of chemical potential simultaneously. Additionally, its working frequency range can be adjusted by geometric parameters.

2. ALL-ANGLE ZERO REFLECTION OPTICAL SWITCH

A. Analysis of Graphene–Dielectric Stacking Metamaterials

The schematic of the optical switch based on SHM is depicted in Fig. 1(a). The SHM is constructed by stacking monolayer graphene sheets with surface conductivity σ(f) and dielectric layers with permittivity εd alternatively along the z axis with a period d=0.1μm. εd is adopted to be 2.2, which was used in Ref. [25]. The SHM and air are above and below the interface, respectively. The SHM is truncated at a slanted angle α with respect to the x axis.

 figure: Fig. 1.

Fig. 1. (a) Schematic of the zero-reflection-based optical switch; the period of the SHM is d. (b), (c) Re(ε) and Im(ε) as functions of frequency and chemical potential.

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Graphene is a very thin material layer with a thickness as small as one atom, which is very small compared with the period d of the SHM. Moreover, it has extremely high carrier mobility of 15,000cm2/(V·s), which theoretically makes an optical switch with high response speed on the order of ps possible [42]. In our study, graphene is modeled as a two-dimensional surface and characterized by using a surface conductivity rather than a volumetric permittivity. The surface conductivity of graphene can be calculated by the Kubo formula [19,43]:

σ(f,μc,Γ,T)=ie2kBT2π2(πf+iΓ)[μckBT+2ln(eμc/kBT+1)]+i2e2(πf+iΓ)π20dϵfd(ϵ)fd(ϵ)4(πf+iΓ)24(ϵ/)2,
where e is the charge of an electron, kB is the Boltzmann’s constant, is the reduced Planck constant, fd(ϵ)={1+exp[(ϵμc)/(kBT)]}1 is the Fermi–Dirac distribution, ϵ is the energy, f is the frequency, and μc is the chemical potential determined by the carrier density ns. Γ=0.1meV is the phenomenological scattering rate. T=300K is the temperature. The above surface conductivity material function is available in the material database in finite-difference time-domain (FDTD) software.

To realize the optical switch to be discussed later, the chemical potential of all graphene sheets needs to be changed simultaneously. The chemical potential depends on the carrier density and can be controlled by its chemical potential, external electrostatic field biasing, gate voltage, magnetic field, chemical doping, or nonlinear effect [1922]. For example, one effective way to do this is by using external electrostatic field biasing Ebias. The relation between Ebias and μc is deduced as [18,19]

Ebias=2eπ2vF2ε0εd[(kBT)2μc/kBTμc/kBTxex+1dx+kBTμcln(eμc/kBT+1)+kBTμcln(eμc/kBT+1)],
where vF9.5×105m/s is the Fermi velocity, and ε0 is the permittivity of free space. The other parameters are the same as those in Eq. (1).

In the THz spectrum, the period d of the structure is far less than the working wavelength. So the designed SHM can be modeled as a homogenous uniaxial anisotropic medium by the effective medium approximation method. The diagonal elements of effective relative permittivity tensor εeff of the SHM is described as {ε,ε,ε} and given as follows [44]:

ε=εd,ε=εdiσ(ω,μc,Γ,T)ωε0d,
where and denote the components parallel and perpendicular to the graphene sheets.

From Eq. (3), one knows that ε is a constant equal to the permittivity of dielectric layers, while ε is complex, depending on frequency f and chemical potential μc. By combining Eqs. (1) and (3), Re(ε) and Im(ε) are calculated and shown in Figs. 1(b) and 1(c). In Fig. 1(b), one can see that Re(ε) decreases as μc increases or f decreases. Moreover, the value of Re(ε) can cross the threshold of zero by tuning μc in a wide frequency range. In Fig. 1(c), one can see that Im(ε) is small enough to be neglected as compared with Re(ε) when μc0.2eV.

B. Realization of the All-Angle Zero-Reflection-Based Optical Switch

As depicted in Fig. 1(a), a Gaussian beam with waist width of 10 μm is incident from the SHM to air at an incident angle θi. Due to the symmetry of the 1D periodic structure, only the xz section needs to be considered. The simplified dispersion relation of extraordinary waves (TM waves) is given as

kx2ε+kz2ε=k02,
where k0=2πf(μ0ε0)1/2 is the wavenumber in vacuum. ε is positive and cannot be tuned after fabrication. ε is a complex number depending on f and μc. However, Figs. 1(b) and 1(c) imply that ε is approximate to a real number due to its tiny Im(ε) when μc0.2eV. The IFC is an ellipse with ε>0, while it will become a hyperbola with ε<0.

To reveal the principle of the zero-reflection-based optical switch, we consider the conditions of f=25THz and α=45°. For the OFF-state, an elliptic IFC is required, i.e., ε>0. Figures 1(b) and 1(c) suggest that Re(ε) decreases from positive to negative as μc increases, and Im(ε) is sufficiently small to be neglected when μc0.2eV. Therefore, we take μc=0.2eV for the OFF-state. For simplicity, we will use ε instead of Re(ε) in the following study unless there are additional specifications. As for the ON-state, the zero-reflection-based optical switch requires a hyperbolic IFC with its asymptote perpendicular to the SHM–air interface. This means that the slope of the asymptote should be kasy=±1 and ε=εd=2.2. According to Fig. 1(b), one can get μc=0.82eV at f=25THz for the ON-state.

IFC analysis is used to study the propagation behaviors on the SHM–air interface, as shown in Figs. 2(a) and 2(b). For simplicity, kx and kz have been normalized by k0. In Fig. 2(a), the optical switch is switched off when μc=0.2eV. Obviously, there exist reflected wave vector kr and energy flow Sr, while there is no refracted wave, since the k conservation line [i.e., the red dashed line in Fig. 2(a)] does not intersect with the air IFC. In Fig. 2(b), the optical switch is switched on when μc=0.82eV. Since the normal of the SHM–air interface is parallel to the asymptote of the hyperbolic IFC, one cannot get the reflected wave vector on the IFC of the SHM. Meanwhile, there is no refracted wave either. In other words, one can realize zero reflection and zero refraction simultaneously at the ON-state.

 figure: Fig. 2.

Fig. 2. IFC analysis and simulation results for the optical switch at θi=60° and f=25THz. (a), (c) Switch-off state (μc=0.2eV). (b), (d) Switch-on state (μc=0.82eV). The inset shows the enlargement of the energy distribution in the denoted square area.

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To verify the above predictions, FDTD simulations for a graphene–dielectric structure with actual parameters are shown in Figs. 2(c) and 2(d), which match well with the analysis in Figs. 2(a) and 2(b). For the ON-state in Fig. 2(d), it is hard to see where the energy goes, since most of it is localized near the incident area at the interface. In order to show the propagation behaviors more clearly, a monitor is introduced to record the evolution of the electromagnetic (EM) wave as time elapses. Figure 3 shows snapshots of the evolution movie of the electric field for a Gaussian beam with a finite length of 4850 time steps recorded by the monitor. Obviously, in the case of no reflection and no refraction, the EM wave changes into a surface wave to propagate in the upper-right direction along the air–SHM interface and escape finally. The enlargement inset in Fig. 2(d) also verifies this point. It should be noticed that in order to efficiently couple the EM wave into the structure, the optical switch is designed to be an isosceles right triangle SHM, as shown in Fig. 1(a). The left SHM–air interface with α=90° is used for coupling light into the device. The transmission for the EM wave incident from the left side is as high as 97% at normal incidence, which is high enough for the application of the device.

 figure: Fig. 3.

Fig. 3. Snapshots at different time steps for a Gaussian beam with a finite length of time steps. (a) ts=100 time steps (before touching the interface), (b) ts=200 time steps (arriving at the interface), and (c) ts=2000 time steps (stably propagating along the interface). (d)–(f) Three snapshots after the Gaussian beam stops emitting from the source.

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It is necessary to discuss the influence of the loss of the dielectric on the optical switch. The loss of the dielectric is characterized by the imaginary part of its permittivity. Figure 4 shows the influence of absorption loss of dielectric layers on the OFF-state (μc=0.2eV) and ON-state (μc=0.82eV). Obviously, the larger the loss, the faster it decays with increasing distance. However, the loss has almost no essential influence on the propagation characteristics of the EM wave. For example, when εd=2.2+0.01i, it results in a lower reflection only at the OFF-state, but keeps near-zero reflection at the ON-state. Therefore, for simplicity, we assume that the loss of the dielectric is zero for our study.

 figure: Fig. 4.

Fig. 4. Influence of absorption loss of dielectric layers on the optical switch. (a) OFF-state (μc=0.2eV). (b) ON-state (μc=0.82eV).

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Further IFC analysis suggests that the optical switch exhibits zero reflection for arbitrary incident angle θi from 0° to 90° as long as the frequency and chemical potential are maintained. However, the phenomenon of zero refraction is limited by θi. If θi decreases continually, a refracted wave vector kr satisfying the k conservation condition will appear on the air IFC. So there exists a critical incident angle θc to decide whether zero refraction appears or not. For f=25THz and μc=0.82eV, θc is calculated to be 42.3°. However, the refracted light would not disappear completely until θi=57° in actual FDTD simulations. This is attributed to the width of the waist of the Gaussian light beam. As we know, the smaller the width of the waist, the larger divergence of the Gaussian beam, and thus it results in a wider incident angle range of the Gaussian beam. As for the Gaussian beam with waist width of 10 μm used here, its divergence angle is about 14.4°. Therefore, the refracted light still exists within 42.3°<θi<57° (i.e., 42.3+14.457°), but its intensity weakens as θi increases. It should be mentioned that similar phenomena also occur for other sources, including a plane wave, which has an ideal incident angle.

C. Influence of Dispersion on the Designed Optical Switch

The influence of dispersion can be studied by calculating the reflection as a function of μc for frequencies around f=25THz. The incident power in FDTD simulations is set to be unity. Reflection is defined as the energy flowing through the top and left sides of the structure in Fig. 1(a). For θi=60°, the curves of reflection versus μc are shown in Fig. 5. For f=25THz, the reflection reaches zero exactly at μc=0.82eV, which agrees well with the IFC analysis in Fig. 2(b). Moreover, the reflection curve shifts rightward as f increases; however, the reflection for the OFF-state (or ON-state) at μc=0.2eV (or 0.82 eV) remains high (>96%) (or near-zero) for f=24.5THz, 25 THz, and 25.5 THz, meaning that the dispersion of the SHM has almost no influence on the optical switch. What is more, from the viewpoint of applications, such a strict condition for the switch, i.e., single value of chemical potential (0.2/0.82 eV), can be relaxed to a range of chemical potentials, provided that the ratio of reflections at the OFF- and ON-states is high enough. Figure 5 shows that the reflection remains high (>96%) and stable within 0.2μc0.32eV for the OFF-state, and it becomes near zero (<1.5%) within 0.77μc0.9eV for the ON-state from 24.5 THz to 25.5 THz. The ratio of reflections at the OFF- and ON-states is as high as 96%:1.5%=64, which is high enough for application. This implies that the optical switch can work in a frequency range by using combinations of μc within these two ranges as control parameters, rather than just 0.2/0.82 eV. Such a property provides high tolerance of chemical potential for the optical switch and makes it very convenient in practical applications.

 figure: Fig. 5.

Fig. 5. Variation of reflection with chemical potential for f=24.5THz, 25 THz, and 25.5 THz.

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3. DISCUSSION

A. Influence of the Permittivity of Dielectric Sheets and the Period of the Structure

In previous studies, we made default assumptions of εd=2.2 and d=0.1μm. Actually, these two parameters can efficiently adjust the working frequency range (fmin,fmax) of the optical switch. From the discussion in Section 2.B, one can get that for α=45°, the condition of the optical switch is that μc should be tuned to guarantee ε>0 (for the OFF-state) or ε=εd (for the ON-state). Actually, ε is affected not only by μc but also by f. In other words, the working frequency range (fmin,fmax) can be determined by ε and μc together.

First, we discuss the lower limit fmin. Figures 1(b) and 1(c) show that Re(ε) increases with decreasing μc; however, Im(ε) increases rapidly when μc decreases from 0.2 eV to 0 eV, so μc=0.2eV is the minimum available value. On the other hand, Re(ε) decreases from positive to negative as f decreases. Obviously, there exists a frequency threshold where Re(ε)=0 [or ε=0 for simplicity due to negligible Im(ε)]. Since the OFF-state requires ε>0, this frequency threshold is fmin. In a word, fmin is determined by μc=0.2eV and ε=0 together. As for fmax, Fig. 1(b) shows that with increasing f,Re(ε) increases, and a larger μc is required to maintain ε=εd=2.2 for the ON-state. However, μc cannot increase indefinitely. So we assume μc=0.9eV to be the maximum available value, since in this case, the reflection is still near zero as discussed in Section 2.C. Therefore, fmax is determined by μc=0.9eV and ε=εd.

It is convenient to obtain fmin and fmax from the μc-f curves to study the influence of εd and d. We first study the influence of εd while keeping d=0.1μm. The relations of μc versus f satisfying ε=0 and ε=εd for different εd are presented in Figs. 6(a) and 6(b), respectively. fmin and fmax are determined by the intersections of μc-f curves and the two dashed horizontal lines located at μc=0.2eV and 0.9 eV, respectively. Obviously, both fmin and fmax decrease as εd increases, and the absolute bandwidth Δf=fmaxfmin decreases, too. In other words, the smaller the εd, the wider the bandwidth. Specifically, the working frequency ranges, i.e., (fmin,fmax), are (24.8, 38.7) THz, (17.1, 26.1) THz, (14.8, 22.4) THz, (12.7, 19.3) THz, and (11.5, 17.3) THz for εd=1, 2.2, 3, 4, and 5, respectively. For the designed optical switch with εd=2.2 and d=0.1μm in Section 2, it can work in a broad frequency range from 17.1 THz to 26.2 THz.

 figure: Fig. 6.

Fig. 6. μc-f curves with different parameter conditions: (a) ε=0, d=0.1μm; (b) ε=εd, d=0.1μm; (c) ε=0, εd=1; (d) ε=1, εd=1. The dashed lines indicate μc=0.2eV and 0.9 eV, respectively.

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Then we study the influence of d while keeping εd as a constant. Without loss of generality, we keep εd=1, though it is impossible to select air (εd=1) to construct a real SHM. Figures 6(c) and 6(d) show the μc-f curves satisfying conditions of ε=0 and ε=1, respectively, for different d. Through similar analysis, one can find that no matter whether ε=0 or 1, all f-μc curves shift rightward as d decreases, meaning that both fmin and fmax increase with decreasing d. Specifically, the working frequency ranges are (48.2, 85.4) THz, (31.2, 49.7) THz, (24.8, 38.7) THz, (21.6, 32.7) THz, and (19.1, 28.9) THz for d=0.02μm, 0.06 μm, 0.1 μm, 0.14 μm, and 0.18 μm, respectively. Apparently, the absolute bandwidth Δf=fmaxfmin decreases significantly as d increases (from 37.2 THz to 9.8 THz when d increases from 0.02 μm to 0.18 μm).

B. Influence of the Slanted Angle α on the Zero Reflection Effect

So far, we have assumed α=45°. Actually, no matter what α is, it is always possible to realize zero reflection provided that the asymptote of the hyperbolic IFC is perpendicular to the interface whose slope is kα=tanα. For the hyperbola described by Eq. (4), the slope of one asymptote is kasy=(ε/ε)1/2(ε<0). So we get the required ε described by α from kα·kasy=1 as

ε=εtan2α.
For parameters of εd=2.2 and d=0.1μm, the required ε for zero reflection are 16.6, 2.2, 0.73, and 0.07 when α=20°,45°,60°, and 80°, respectively. For these special values of ε, the relations of μc versus f are shown in Fig. 7(a), which can be used to determine the upper limit fmax. When α increases, the corresponding μc-f curve shifts rightward, leading to a larger fmax. The upper limits fmax for α=20°,45°,60°, and 80° are 12.6 THz, 26.2 THz, 32 THz, and 36.3 THz, respectively, as denoted by the dashed line at μc=0.9eV in Fig. 7(a). On the other hand, since the required μc for OFF-state still remains at 0.2 eV, the lower limit fmin is still 17.1 THz. Therefore, it is an effective way to enlarge the working bandwidth by increasing the slanted angle α.

 figure: Fig. 7.

Fig. 7. (a) μc-f curves for α=20°, 45°, 60°, and 80°. (b) IFC at f=25THz for α=80° and ε=0.07. (c) Dependence of maximal incident angle θimax on the slanted angle α.

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We further discuss the effective incident angle range (θimin,θimax). One asymptote of the hyperbolic IFC perpendicular to the SHM–air interface has a slope of kasy=1/kα=1/tanα. The SHM supports wave vectors ki only within these two asymptotes, as shown in Fig. 7(b) where the hyperbolic IFC at f=25THz for the case of α=80° and ε=0.07 is drawn. Obviously, θimin is 0°. As for θimax, if the angle between the two asymptotes is less than 90°, θimax=2arctan(1/tanα)=2(90°α); otherwise, θimax=90° because an incident angle cannot be larger than 90°. The curve of θimax versus α is shown in Fig. 7(c). Obviously, θimax decreases linearly when α>45°. Usually, fmax is desired to be as large as possible, and at the same time the SHM should support arbitrary θi between 0° and 90°. Therefore, α=45° is the optimal selection for the SHM–air interface.

4. CONCLUSION

In conclusion, we have realized the zero reflection effect in a graphene–dielectric SHM and demonstrated an all-angle compact optical switch. The IFC of this SHM can be conveniently switched between an ellipse and a hyperbola by tuning the chemical potential. In particular, when the asymptote of the hyperbolic IFC is perpendicular to the SHM–air interface, zero reflection can be realized. Based on these unique properties, an all-angle optical switch is proposed. It can work at high reflection (>96%) (OFF-state) over 0.2μc0.32eV and near-zero reflection (ON-state) over 0.77μc0.9eV, providing high tolerance of chemical potential in practical applications. Moreover, its working bandwidth can be effectively adjusted by permittivity of the dielectric, period of the structure, and the slanted angle of the SHM–air interface. These results hold great potential in many fields, such as data storage, beam steering, and THz wave sensing, as well as integrated photonic circuits.

Funding

National Natural Science Foundation of China (NSFC) (11504114, 11434017); National Key R&D Program of China (2018YFA 0306200); Science and Technology Program of Guangzhou, China; SRP of South China University of Technology.

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24. T. Gric and O. Hess, “Tunable surface waves at the interface separating different graphene-dielectric composite hyperbolic metamaterials,” Opt. Express 25, 11466–11476 (2017). [CrossRef]  

25. B. Zhu, G. Ren, S. Zheng, Z. Lin, and S. Jian, “Nanoscale dielectric-graphene-dielectric tunable infrared waveguide with ultrahigh refractive indices,” Opt. Express 21, 17089–17096 (2013). [CrossRef]  

26. I. V. Iorsh, I. S. Mukhin, I. V. Shadrivov, P. A. Belov, and Y. S. Kivshar, “Hyperbolic metamaterials based on multilayer graphene structures,” Phys. Rev. B 87, 075416 (2013). [CrossRef]  

27. H. G. Liu, P. G. Liu, L. A. Bian, C. X. Liu, Q. H. Zhou, and Y. W. Chen, “Electrically tunable terahertz metamaterials based on graphene stacks array,” Superlattices Microstruct. 112, 470–479 (2017). [CrossRef]  

28. B. Janaszek, A. Tyszka-Zawadzka, and P. Szczepański, “Tunable graphene-based hyperbolic metamaterial operating in SCLU telecom bands,” Opt. Express 24, 24129–24136 (2016). [CrossRef]  

29. M. Shoaei, M. K. Moravvej-Farshi, and L. Yousefi, “Nanostructured graphene-based hyperbolic metamaterial performing as a wide-angle near infrared electro-optical switch,” Appl. Opt. 54, 1206–1211 (2015). [CrossRef]  

30. M. A. K. Othman, C. Guclu, and F. Capolino, “Graphene-dielectric composite metamaterials: evolution from elliptic to hyperbolic wavevector dispersion and the transverse epsilon-near-zero condition,” J. Nanophoton. 7, 073089 (2013). [CrossRef]  

31. H. N. S. Krishnamoorthy, Z. Jacob, E. Narimanov, I. Kretzschmar, and V. M. Menon, “Topological transitions in metamaterials,” Science 336, 205–209 (2012). [CrossRef]  

32. S. Campione, T. S. Luk, S. Liu, and M. B. Sinclair, “Optical properties of transiently-excited semiconductor hyperbolic metamaterials,” Opt. Mater. Express 5, 2385–2394 (2015). [CrossRef]  

33. F. H. Shi, Y. H. Chen, P. Han, and P. Tassin, “Broadband, spectrally flat, graphene-based terahertz modulators,” Small 11, 6044–6050 (2015). [CrossRef]  

34. W. G. Liu, B. Hu, Z. D. Huang, H. Y. Guan, H. T. Li, X. K. Wang, Y. Zhang, H. X. Yin, X. L. Xiong, J. Liu, and Y. T. Wang, “Graphene-enabled electrically controlled terahertz meta-lens,” Photon. Res. 6, 703–708 (2018). [CrossRef]  

35. C. Guclu, S. Campione, and F. Capolino, “Hyperbolic metamaterial as super absorber for scattered fields generated at its surface,” Phys. Rev. B 86, 205130 (2012). [CrossRef]  

36. Y. Zhang, Y. Shi, and C. H. Liang, “Broadband tunable graphene-based metamaterial absorber,” Opt. Mater. Express 6, 3036–3044 (2016). [CrossRef]  

37. Y. T. Zhao, B. A. Wu, B. J. Huang, and Q. A. Cheng, “Switchable broadband terahertz absorber/reflector enabled by hybrid graphene-gold metasurface,” Opt. Express 25, 7161–7169 (2017). [CrossRef]  

38. Y. Shi and Y. Zhang, “Generation of wideband tunable orbital angular momentum vortex waves using graphene metamaterial reflectarray,” IEEE Access 6, 5341–5347 (2018). [CrossRef]  

39. Z. Li, W. Y. Liang, and W. H. Chen, “Switchable hyperbolic metamaterials based on the graphene-dielectric stacking structure and optical switches design,” Europhys. Lett. 120, 37001 (2017). [CrossRef]  

40. H. N. S. Krishnamoorthy, B. Gholipour, N. I. Zheludev, and C. Soci, “A non-volatile chalcogenide switchable hyperbolic metamaterial,” Adv. Opt. Mater. 6, 1800332 (2018). [CrossRef]  

41. M. Shoaei, M. K. Moravvej-Farshi, and L. Yousefi, “All-optical switching of nonlinear hyperbolic metamaterials in visible and near-infrared regions,” J. Opt. Soc. Am. B 32, 2358–2365 (2015). [CrossRef]  

42. J. Qin, H. M. Dong, K. Han, and X. F. Wang, “Ultrafast dynamic optical properties of graphene,” Acta Phys. Sin. 64, 237801 (2015). [CrossRef]  

43. V. P. Gusynin, S. G. Sharapov, and J. P. Carbotte, “Magneto-optical conductivity in graphene,” J. Phys. Condens. Matter 19, 026222 (2007). [CrossRef]  

44. M. A. K. Othman, C. Guclu, and F. Capolino, “Graphene-based tunable hyperbolic metamaterials and enhanced near-field absorption,” Opt. Express 21, 7614–7632 (2013). [CrossRef]  

References

  • View by:

  1. D. R. Smith, J. B. Pendry, and M. C. K. Wiltshire, “Metamaterials and negative refractive index,” Science 305, 788–792 (2004).
    [Crossref]
  2. J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312, 1780–1782 (2006).
    [Crossref]
  3. Q. Zhao, T. Zhou, T. Wang, W. Liu, J. Liu, T. Yu, Q. Liao, and N. Liu, “Active control of near-field radiative heat transfer between graphene-covered metamaterials,” J. Phys. D 50, 145101 (2017).
    [Crossref]
  4. D. R. Smith and D. Schurig, “Electromagnetic wave propagation in media with indefinite permittivity and permeability tensors,” Phys. Rev. Lett. 90, 077405 (2003).
    [Crossref]
  5. Y. Guo, W. Newman, C. L. Cortes, and Z. Jacob, “Applications of hyperbolic metamaterial substrates,” Adv. OptoElectron. 2012, 452502 (2012).
    [Crossref]
  6. K. V. Sreekanth, A. De Luca, and G. Strangi, “Negative refraction in graphene-based hyperbolic metamaterials,” Appl. Phys. Lett. 103, 023107 (2013).
    [Crossref]
  7. J. Hoffman, L. Alekseyev, S. S. Howard, K. J. Franz, D. Wasserman, V. A. Podolskiy, E. E. Narimanov, D. L. Sivco, and C. Gmachl, “Negative refraction in semiconductor metamaterials,” Nat. Mater. 6, 946–950 (2007).
    [Crossref]
  8. C. Argyropoulos, N. M. Estakhri, F. Monticone, and A. Alù, “Negative refraction, gain and nonlinear effects in hyperbolic metamaterials,” Opt. Express 21, 15037–15047 (2013).
    [Crossref]
  9. A. D. Neira, G. A. Wurtz, and A. V. Zayats, “Superluminal and stopped light due to mode coupling in confined hyperbolic metamaterial waveguides,” Sci. Rep. 5, 17678 (2015).
    [Crossref]
  10. S. H. Liang, C. H. Jiang, Z. Q. Yang, D. C. Li, W. D. Zhang, T. Mei, and D. W. Zhang, “Plasmonic slow light waveguide with hyperbolic metamaterials claddings,” J. Opt. 20, 065001 (2018).
    [Crossref]
  11. T. F. Li, V. Nagai, D. H. Gracias, and J. B. Khurgin, “Limits of imaging with multilayer hyperbolic metamaterials,” Opt. Express 25, 13588–13601 (2017).
    [Crossref]
  12. D. Lu, J. J. Kan, E. E. Fullerton, and Z. W. Liu, “Enhancing spontaneous emission rates of molecules using nanopatterned multilayer hyperbolic metamaterials,” Nat. Nanotechnol. 9, 48–53 (2014).
    [Crossref]
  13. K. J. Lee, Y. U. Lee, S. J. Kim, and P. André, “Hyperbolic dispersion dominant regime identified through spontaneous emission variations near metamaterial interfaces,” Adv. Mater. Interfaces 5, 1701629 (2018).
    [Crossref]
  14. T. A. Morgado, S. I. Maslovski, and M. G. Silveirinha, “Ultrahigh Casimir interaction torque in nanowire systems,” Opt. Express 21, 14943–14955 (2013).
    [Crossref]
  15. M. Kim, S. Kim, and S. Kim, “Optical bistability based on hyperbolic metamaterials,” Opt. Express 26, 11620–11632 (2018).
    [Crossref]
  16. X. Li, Z. X. Liang, X. H. Liu, X. Y. Jiang, and J. Zi, “All-angle zero reflection at metamaterial surfaces,” Appl. Phys. Lett. 93, 171111 (2008).
    [Crossref]
  17. W. Li, Z. Liu, X. Zhang, and X. Y. Jiang, “Switchable hyperbolic metamaterials with magnetic control,” Appl. Phys. Lett. 100, 161108 (2012).
    [Crossref]
  18. K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y. Zhang, S. V. Dubonos, I. V. Grigorieva, and A. A. Firsov, “Electric field effect in atomically thin carbon films,” Science 306, 666–669 (2004).
    [Crossref]
  19. G. W. Hanson, “Dyadic Green’s functions and guided surface waves for a surface conductivity model of graphene,” J. Appl. Phys. 103, 064302 (2008).
    [Crossref]
  20. A. Vakil and N. Engheta, “Transformation optics using graphene,” Science 332, 1291–1294 (2011).
    [Crossref]
  21. H. Deng, F. Ye, B. A. Malomed, X. Chen, and N. C. Panoiu, “Optically and electrically tunable Dirac points and Zitterbewegung in graphene-based photonic superlattices,” Phys. Rev. B 91, 201402 (2015).
    [Crossref]
  22. H. Deng, X. Chen, B. A. Malomed, N. C. Panoiu, and F. Ye, “Tunability and robustness of Dirac points of photonic nanostructures,” IEEE J. Sel. Top. Quantum Electron. 22, 98–106 (2016).
    [Crossref]
  23. Y. C. Chang, C. H. Liu, C. H. Liu, S. Zhang, S. R. Marder, E. E. Narimanov, Z. Zhong, and T. B. Norris, “Realization of mid-infrared graphene hyperbolic metamaterials,” Nat. Commun. 7, 10568 (2016).
    [Crossref]
  24. T. Gric and O. Hess, “Tunable surface waves at the interface separating different graphene-dielectric composite hyperbolic metamaterials,” Opt. Express 25, 11466–11476 (2017).
    [Crossref]
  25. B. Zhu, G. Ren, S. Zheng, Z. Lin, and S. Jian, “Nanoscale dielectric-graphene-dielectric tunable infrared waveguide with ultrahigh refractive indices,” Opt. Express 21, 17089–17096 (2013).
    [Crossref]
  26. I. V. Iorsh, I. S. Mukhin, I. V. Shadrivov, P. A. Belov, and Y. S. Kivshar, “Hyperbolic metamaterials based on multilayer graphene structures,” Phys. Rev. B 87, 075416 (2013).
    [Crossref]
  27. H. G. Liu, P. G. Liu, L. A. Bian, C. X. Liu, Q. H. Zhou, and Y. W. Chen, “Electrically tunable terahertz metamaterials based on graphene stacks array,” Superlattices Microstruct. 112, 470–479 (2017).
    [Crossref]
  28. B. Janaszek, A. Tyszka-Zawadzka, and P. Szczepański, “Tunable graphene-based hyperbolic metamaterial operating in SCLU telecom bands,” Opt. Express 24, 24129–24136 (2016).
    [Crossref]
  29. M. Shoaei, M. K. Moravvej-Farshi, and L. Yousefi, “Nanostructured graphene-based hyperbolic metamaterial performing as a wide-angle near infrared electro-optical switch,” Appl. Opt. 54, 1206–1211 (2015).
    [Crossref]
  30. M. A. K. Othman, C. Guclu, and F. Capolino, “Graphene-dielectric composite metamaterials: evolution from elliptic to hyperbolic wavevector dispersion and the transverse epsilon-near-zero condition,” J. Nanophoton. 7, 073089 (2013).
    [Crossref]
  31. H. N. S. Krishnamoorthy, Z. Jacob, E. Narimanov, I. Kretzschmar, and V. M. Menon, “Topological transitions in metamaterials,” Science 336, 205–209 (2012).
    [Crossref]
  32. S. Campione, T. S. Luk, S. Liu, and M. B. Sinclair, “Optical properties of transiently-excited semiconductor hyperbolic metamaterials,” Opt. Mater. Express 5, 2385–2394 (2015).
    [Crossref]
  33. F. H. Shi, Y. H. Chen, P. Han, and P. Tassin, “Broadband, spectrally flat, graphene-based terahertz modulators,” Small 11, 6044–6050 (2015).
    [Crossref]
  34. W. G. Liu, B. Hu, Z. D. Huang, H. Y. Guan, H. T. Li, X. K. Wang, Y. Zhang, H. X. Yin, X. L. Xiong, J. Liu, and Y. T. Wang, “Graphene-enabled electrically controlled terahertz meta-lens,” Photon. Res. 6, 703–708 (2018).
    [Crossref]
  35. C. Guclu, S. Campione, and F. Capolino, “Hyperbolic metamaterial as super absorber for scattered fields generated at its surface,” Phys. Rev. B 86, 205130 (2012).
    [Crossref]
  36. Y. Zhang, Y. Shi, and C. H. Liang, “Broadband tunable graphene-based metamaterial absorber,” Opt. Mater. Express 6, 3036–3044 (2016).
    [Crossref]
  37. Y. T. Zhao, B. A. Wu, B. J. Huang, and Q. A. Cheng, “Switchable broadband terahertz absorber/reflector enabled by hybrid graphene-gold metasurface,” Opt. Express 25, 7161–7169 (2017).
    [Crossref]
  38. Y. Shi and Y. Zhang, “Generation of wideband tunable orbital angular momentum vortex waves using graphene metamaterial reflectarray,” IEEE Access 6, 5341–5347 (2018).
    [Crossref]
  39. Z. Li, W. Y. Liang, and W. H. Chen, “Switchable hyperbolic metamaterials based on the graphene-dielectric stacking structure and optical switches design,” Europhys. Lett. 120, 37001 (2017).
    [Crossref]
  40. H. N. S. Krishnamoorthy, B. Gholipour, N. I. Zheludev, and C. Soci, “A non-volatile chalcogenide switchable hyperbolic metamaterial,” Adv. Opt. Mater. 6, 1800332 (2018).
    [Crossref]
  41. M. Shoaei, M. K. Moravvej-Farshi, and L. Yousefi, “All-optical switching of nonlinear hyperbolic metamaterials in visible and near-infrared regions,” J. Opt. Soc. Am. B 32, 2358–2365 (2015).
    [Crossref]
  42. J. Qin, H. M. Dong, K. Han, and X. F. Wang, “Ultrafast dynamic optical properties of graphene,” Acta Phys. Sin. 64, 237801 (2015).
    [Crossref]
  43. V. P. Gusynin, S. G. Sharapov, and J. P. Carbotte, “Magneto-optical conductivity in graphene,” J. Phys. Condens. Matter 19, 026222 (2007).
    [Crossref]
  44. M. A. K. Othman, C. Guclu, and F. Capolino, “Graphene-based tunable hyperbolic metamaterials and enhanced near-field absorption,” Opt. Express 21, 7614–7632 (2013).
    [Crossref]

2018 (6)

S. H. Liang, C. H. Jiang, Z. Q. Yang, D. C. Li, W. D. Zhang, T. Mei, and D. W. Zhang, “Plasmonic slow light waveguide with hyperbolic metamaterials claddings,” J. Opt. 20, 065001 (2018).
[Crossref]

K. J. Lee, Y. U. Lee, S. J. Kim, and P. André, “Hyperbolic dispersion dominant regime identified through spontaneous emission variations near metamaterial interfaces,” Adv. Mater. Interfaces 5, 1701629 (2018).
[Crossref]

M. Kim, S. Kim, and S. Kim, “Optical bistability based on hyperbolic metamaterials,” Opt. Express 26, 11620–11632 (2018).
[Crossref]

W. G. Liu, B. Hu, Z. D. Huang, H. Y. Guan, H. T. Li, X. K. Wang, Y. Zhang, H. X. Yin, X. L. Xiong, J. Liu, and Y. T. Wang, “Graphene-enabled electrically controlled terahertz meta-lens,” Photon. Res. 6, 703–708 (2018).
[Crossref]

Y. Shi and Y. Zhang, “Generation of wideband tunable orbital angular momentum vortex waves using graphene metamaterial reflectarray,” IEEE Access 6, 5341–5347 (2018).
[Crossref]

H. N. S. Krishnamoorthy, B. Gholipour, N. I. Zheludev, and C. Soci, “A non-volatile chalcogenide switchable hyperbolic metamaterial,” Adv. Opt. Mater. 6, 1800332 (2018).
[Crossref]

2017 (6)

Z. Li, W. Y. Liang, and W. H. Chen, “Switchable hyperbolic metamaterials based on the graphene-dielectric stacking structure and optical switches design,” Europhys. Lett. 120, 37001 (2017).
[Crossref]

Y. T. Zhao, B. A. Wu, B. J. Huang, and Q. A. Cheng, “Switchable broadband terahertz absorber/reflector enabled by hybrid graphene-gold metasurface,” Opt. Express 25, 7161–7169 (2017).
[Crossref]

H. G. Liu, P. G. Liu, L. A. Bian, C. X. Liu, Q. H. Zhou, and Y. W. Chen, “Electrically tunable terahertz metamaterials based on graphene stacks array,” Superlattices Microstruct. 112, 470–479 (2017).
[Crossref]

T. Gric and O. Hess, “Tunable surface waves at the interface separating different graphene-dielectric composite hyperbolic metamaterials,” Opt. Express 25, 11466–11476 (2017).
[Crossref]

T. F. Li, V. Nagai, D. H. Gracias, and J. B. Khurgin, “Limits of imaging with multilayer hyperbolic metamaterials,” Opt. Express 25, 13588–13601 (2017).
[Crossref]

Q. Zhao, T. Zhou, T. Wang, W. Liu, J. Liu, T. Yu, Q. Liao, and N. Liu, “Active control of near-field radiative heat transfer between graphene-covered metamaterials,” J. Phys. D 50, 145101 (2017).
[Crossref]

2016 (4)

H. Deng, X. Chen, B. A. Malomed, N. C. Panoiu, and F. Ye, “Tunability and robustness of Dirac points of photonic nanostructures,” IEEE J. Sel. Top. Quantum Electron. 22, 98–106 (2016).
[Crossref]

Y. C. Chang, C. H. Liu, C. H. Liu, S. Zhang, S. R. Marder, E. E. Narimanov, Z. Zhong, and T. B. Norris, “Realization of mid-infrared graphene hyperbolic metamaterials,” Nat. Commun. 7, 10568 (2016).
[Crossref]

B. Janaszek, A. Tyszka-Zawadzka, and P. Szczepański, “Tunable graphene-based hyperbolic metamaterial operating in SCLU telecom bands,” Opt. Express 24, 24129–24136 (2016).
[Crossref]

Y. Zhang, Y. Shi, and C. H. Liang, “Broadband tunable graphene-based metamaterial absorber,” Opt. Mater. Express 6, 3036–3044 (2016).
[Crossref]

2015 (7)

M. Shoaei, M. K. Moravvej-Farshi, and L. Yousefi, “All-optical switching of nonlinear hyperbolic metamaterials in visible and near-infrared regions,” J. Opt. Soc. Am. B 32, 2358–2365 (2015).
[Crossref]

J. Qin, H. M. Dong, K. Han, and X. F. Wang, “Ultrafast dynamic optical properties of graphene,” Acta Phys. Sin. 64, 237801 (2015).
[Crossref]

M. Shoaei, M. K. Moravvej-Farshi, and L. Yousefi, “Nanostructured graphene-based hyperbolic metamaterial performing as a wide-angle near infrared electro-optical switch,” Appl. Opt. 54, 1206–1211 (2015).
[Crossref]

S. Campione, T. S. Luk, S. Liu, and M. B. Sinclair, “Optical properties of transiently-excited semiconductor hyperbolic metamaterials,” Opt. Mater. Express 5, 2385–2394 (2015).
[Crossref]

F. H. Shi, Y. H. Chen, P. Han, and P. Tassin, “Broadband, spectrally flat, graphene-based terahertz modulators,” Small 11, 6044–6050 (2015).
[Crossref]

A. D. Neira, G. A. Wurtz, and A. V. Zayats, “Superluminal and stopped light due to mode coupling in confined hyperbolic metamaterial waveguides,” Sci. Rep. 5, 17678 (2015).
[Crossref]

H. Deng, F. Ye, B. A. Malomed, X. Chen, and N. C. Panoiu, “Optically and electrically tunable Dirac points and Zitterbewegung in graphene-based photonic superlattices,” Phys. Rev. B 91, 201402 (2015).
[Crossref]

2014 (1)

D. Lu, J. J. Kan, E. E. Fullerton, and Z. W. Liu, “Enhancing spontaneous emission rates of molecules using nanopatterned multilayer hyperbolic metamaterials,” Nat. Nanotechnol. 9, 48–53 (2014).
[Crossref]

2013 (7)

T. A. Morgado, S. I. Maslovski, and M. G. Silveirinha, “Ultrahigh Casimir interaction torque in nanowire systems,” Opt. Express 21, 14943–14955 (2013).
[Crossref]

K. V. Sreekanth, A. De Luca, and G. Strangi, “Negative refraction in graphene-based hyperbolic metamaterials,” Appl. Phys. Lett. 103, 023107 (2013).
[Crossref]

C. Argyropoulos, N. M. Estakhri, F. Monticone, and A. Alù, “Negative refraction, gain and nonlinear effects in hyperbolic metamaterials,” Opt. Express 21, 15037–15047 (2013).
[Crossref]

B. Zhu, G. Ren, S. Zheng, Z. Lin, and S. Jian, “Nanoscale dielectric-graphene-dielectric tunable infrared waveguide with ultrahigh refractive indices,” Opt. Express 21, 17089–17096 (2013).
[Crossref]

I. V. Iorsh, I. S. Mukhin, I. V. Shadrivov, P. A. Belov, and Y. S. Kivshar, “Hyperbolic metamaterials based on multilayer graphene structures,” Phys. Rev. B 87, 075416 (2013).
[Crossref]

M. A. K. Othman, C. Guclu, and F. Capolino, “Graphene-dielectric composite metamaterials: evolution from elliptic to hyperbolic wavevector dispersion and the transverse epsilon-near-zero condition,” J. Nanophoton. 7, 073089 (2013).
[Crossref]

M. A. K. Othman, C. Guclu, and F. Capolino, “Graphene-based tunable hyperbolic metamaterials and enhanced near-field absorption,” Opt. Express 21, 7614–7632 (2013).
[Crossref]

2012 (4)

H. N. S. Krishnamoorthy, Z. Jacob, E. Narimanov, I. Kretzschmar, and V. M. Menon, “Topological transitions in metamaterials,” Science 336, 205–209 (2012).
[Crossref]

C. Guclu, S. Campione, and F. Capolino, “Hyperbolic metamaterial as super absorber for scattered fields generated at its surface,” Phys. Rev. B 86, 205130 (2012).
[Crossref]

Y. Guo, W. Newman, C. L. Cortes, and Z. Jacob, “Applications of hyperbolic metamaterial substrates,” Adv. OptoElectron. 2012, 452502 (2012).
[Crossref]

W. Li, Z. Liu, X. Zhang, and X. Y. Jiang, “Switchable hyperbolic metamaterials with magnetic control,” Appl. Phys. Lett. 100, 161108 (2012).
[Crossref]

2011 (1)

A. Vakil and N. Engheta, “Transformation optics using graphene,” Science 332, 1291–1294 (2011).
[Crossref]

2008 (2)

G. W. Hanson, “Dyadic Green’s functions and guided surface waves for a surface conductivity model of graphene,” J. Appl. Phys. 103, 064302 (2008).
[Crossref]

X. Li, Z. X. Liang, X. H. Liu, X. Y. Jiang, and J. Zi, “All-angle zero reflection at metamaterial surfaces,” Appl. Phys. Lett. 93, 171111 (2008).
[Crossref]

2007 (2)

J. Hoffman, L. Alekseyev, S. S. Howard, K. J. Franz, D. Wasserman, V. A. Podolskiy, E. E. Narimanov, D. L. Sivco, and C. Gmachl, “Negative refraction in semiconductor metamaterials,” Nat. Mater. 6, 946–950 (2007).
[Crossref]

V. P. Gusynin, S. G. Sharapov, and J. P. Carbotte, “Magneto-optical conductivity in graphene,” J. Phys. Condens. Matter 19, 026222 (2007).
[Crossref]

2006 (1)

J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312, 1780–1782 (2006).
[Crossref]

2004 (2)

D. R. Smith, J. B. Pendry, and M. C. K. Wiltshire, “Metamaterials and negative refractive index,” Science 305, 788–792 (2004).
[Crossref]

K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y. Zhang, S. V. Dubonos, I. V. Grigorieva, and A. A. Firsov, “Electric field effect in atomically thin carbon films,” Science 306, 666–669 (2004).
[Crossref]

2003 (1)

D. R. Smith and D. Schurig, “Electromagnetic wave propagation in media with indefinite permittivity and permeability tensors,” Phys. Rev. Lett. 90, 077405 (2003).
[Crossref]

Alekseyev, L.

J. Hoffman, L. Alekseyev, S. S. Howard, K. J. Franz, D. Wasserman, V. A. Podolskiy, E. E. Narimanov, D. L. Sivco, and C. Gmachl, “Negative refraction in semiconductor metamaterials,” Nat. Mater. 6, 946–950 (2007).
[Crossref]

Alù, A.

André, P.

K. J. Lee, Y. U. Lee, S. J. Kim, and P. André, “Hyperbolic dispersion dominant regime identified through spontaneous emission variations near metamaterial interfaces,” Adv. Mater. Interfaces 5, 1701629 (2018).
[Crossref]

Argyropoulos, C.

Belov, P. A.

I. V. Iorsh, I. S. Mukhin, I. V. Shadrivov, P. A. Belov, and Y. S. Kivshar, “Hyperbolic metamaterials based on multilayer graphene structures,” Phys. Rev. B 87, 075416 (2013).
[Crossref]

Bian, L. A.

H. G. Liu, P. G. Liu, L. A. Bian, C. X. Liu, Q. H. Zhou, and Y. W. Chen, “Electrically tunable terahertz metamaterials based on graphene stacks array,” Superlattices Microstruct. 112, 470–479 (2017).
[Crossref]

Campione, S.

S. Campione, T. S. Luk, S. Liu, and M. B. Sinclair, “Optical properties of transiently-excited semiconductor hyperbolic metamaterials,” Opt. Mater. Express 5, 2385–2394 (2015).
[Crossref]

C. Guclu, S. Campione, and F. Capolino, “Hyperbolic metamaterial as super absorber for scattered fields generated at its surface,” Phys. Rev. B 86, 205130 (2012).
[Crossref]

Capolino, F.

M. A. K. Othman, C. Guclu, and F. Capolino, “Graphene-based tunable hyperbolic metamaterials and enhanced near-field absorption,” Opt. Express 21, 7614–7632 (2013).
[Crossref]

M. A. K. Othman, C. Guclu, and F. Capolino, “Graphene-dielectric composite metamaterials: evolution from elliptic to hyperbolic wavevector dispersion and the transverse epsilon-near-zero condition,” J. Nanophoton. 7, 073089 (2013).
[Crossref]

C. Guclu, S. Campione, and F. Capolino, “Hyperbolic metamaterial as super absorber for scattered fields generated at its surface,” Phys. Rev. B 86, 205130 (2012).
[Crossref]

Carbotte, J. P.

V. P. Gusynin, S. G. Sharapov, and J. P. Carbotte, “Magneto-optical conductivity in graphene,” J. Phys. Condens. Matter 19, 026222 (2007).
[Crossref]

Chang, Y. C.

Y. C. Chang, C. H. Liu, C. H. Liu, S. Zhang, S. R. Marder, E. E. Narimanov, Z. Zhong, and T. B. Norris, “Realization of mid-infrared graphene hyperbolic metamaterials,” Nat. Commun. 7, 10568 (2016).
[Crossref]

Chen, W. H.

Z. Li, W. Y. Liang, and W. H. Chen, “Switchable hyperbolic metamaterials based on the graphene-dielectric stacking structure and optical switches design,” Europhys. Lett. 120, 37001 (2017).
[Crossref]

Chen, X.

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J. Qin, H. M. Dong, K. Han, and X. F. Wang, “Ultrafast dynamic optical properties of graphene,” Acta Phys. Sin. 64, 237801 (2015).
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D. Lu, J. J. Kan, E. E. Fullerton, and Z. W. Liu, “Enhancing spontaneous emission rates of molecules using nanopatterned multilayer hyperbolic metamaterials,” Nat. Nanotechnol. 9, 48–53 (2014).
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M. A. K. Othman, C. Guclu, and F. Capolino, “Graphene-dielectric composite metamaterials: evolution from elliptic to hyperbolic wavevector dispersion and the transverse epsilon-near-zero condition,” J. Nanophoton. 7, 073089 (2013).
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J. Hoffman, L. Alekseyev, S. S. Howard, K. J. Franz, D. Wasserman, V. A. Podolskiy, E. E. Narimanov, D. L. Sivco, and C. Gmachl, “Negative refraction in semiconductor metamaterials,” Nat. Mater. 6, 946–950 (2007).
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H. N. S. Krishnamoorthy, Z. Jacob, E. Narimanov, I. Kretzschmar, and V. M. Menon, “Topological transitions in metamaterials,” Science 336, 205–209 (2012).
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S. H. Liang, C. H. Jiang, Z. Q. Yang, D. C. Li, W. D. Zhang, T. Mei, and D. W. Zhang, “Plasmonic slow light waveguide with hyperbolic metamaterials claddings,” J. Opt. 20, 065001 (2018).
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K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y. Zhang, S. V. Dubonos, I. V. Grigorieva, and A. A. Firsov, “Electric field effect in atomically thin carbon films,” Science 306, 666–669 (2004).
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W. Li, Z. Liu, X. Zhang, and X. Y. Jiang, “Switchable hyperbolic metamaterials with magnetic control,” Appl. Phys. Lett. 100, 161108 (2012).
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D. Lu, J. J. Kan, E. E. Fullerton, and Z. W. Liu, “Enhancing spontaneous emission rates of molecules using nanopatterned multilayer hyperbolic metamaterials,” Nat. Nanotechnol. 9, 48–53 (2014).
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H. N. S. Krishnamoorthy, B. Gholipour, N. I. Zheludev, and C. Soci, “A non-volatile chalcogenide switchable hyperbolic metamaterial,” Adv. Opt. Mater. 6, 1800332 (2018).
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K. J. Lee, Y. U. Lee, S. J. Kim, and P. André, “Hyperbolic dispersion dominant regime identified through spontaneous emission variations near metamaterial interfaces,” Adv. Mater. Interfaces 5, 1701629 (2018).
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K. J. Lee, Y. U. Lee, S. J. Kim, and P. André, “Hyperbolic dispersion dominant regime identified through spontaneous emission variations near metamaterial interfaces,” Adv. Mater. Interfaces 5, 1701629 (2018).
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S. H. Liang, C. H. Jiang, Z. Q. Yang, D. C. Li, W. D. Zhang, T. Mei, and D. W. Zhang, “Plasmonic slow light waveguide with hyperbolic metamaterials claddings,” J. Opt. 20, 065001 (2018).
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Li, T. F.

Li, W.

W. Li, Z. Liu, X. Zhang, and X. Y. Jiang, “Switchable hyperbolic metamaterials with magnetic control,” Appl. Phys. Lett. 100, 161108 (2012).
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X. Li, Z. X. Liang, X. H. Liu, X. Y. Jiang, and J. Zi, “All-angle zero reflection at metamaterial surfaces,” Appl. Phys. Lett. 93, 171111 (2008).
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Q. Zhao, T. Zhou, T. Wang, W. Liu, J. Liu, T. Yu, Q. Liao, and N. Liu, “Active control of near-field radiative heat transfer between graphene-covered metamaterials,” J. Phys. D 50, 145101 (2017).
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Liu, W. G.

Liu, X. H.

X. Li, Z. X. Liang, X. H. Liu, X. Y. Jiang, and J. Zi, “All-angle zero reflection at metamaterial surfaces,” Appl. Phys. Lett. 93, 171111 (2008).
[Crossref]

Liu, Z.

W. Li, Z. Liu, X. Zhang, and X. Y. Jiang, “Switchable hyperbolic metamaterials with magnetic control,” Appl. Phys. Lett. 100, 161108 (2012).
[Crossref]

Liu, Z. W.

D. Lu, J. J. Kan, E. E. Fullerton, and Z. W. Liu, “Enhancing spontaneous emission rates of molecules using nanopatterned multilayer hyperbolic metamaterials,” Nat. Nanotechnol. 9, 48–53 (2014).
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D. Lu, J. J. Kan, E. E. Fullerton, and Z. W. Liu, “Enhancing spontaneous emission rates of molecules using nanopatterned multilayer hyperbolic metamaterials,” Nat. Nanotechnol. 9, 48–53 (2014).
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Malomed, B. A.

H. Deng, X. Chen, B. A. Malomed, N. C. Panoiu, and F. Ye, “Tunability and robustness of Dirac points of photonic nanostructures,” IEEE J. Sel. Top. Quantum Electron. 22, 98–106 (2016).
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H. Deng, F. Ye, B. A. Malomed, X. Chen, and N. C. Panoiu, “Optically and electrically tunable Dirac points and Zitterbewegung in graphene-based photonic superlattices,” Phys. Rev. B 91, 201402 (2015).
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Y. C. Chang, C. H. Liu, C. H. Liu, S. Zhang, S. R. Marder, E. E. Narimanov, Z. Zhong, and T. B. Norris, “Realization of mid-infrared graphene hyperbolic metamaterials,” Nat. Commun. 7, 10568 (2016).
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Mei, T.

S. H. Liang, C. H. Jiang, Z. Q. Yang, D. C. Li, W. D. Zhang, T. Mei, and D. W. Zhang, “Plasmonic slow light waveguide with hyperbolic metamaterials claddings,” J. Opt. 20, 065001 (2018).
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H. N. S. Krishnamoorthy, Z. Jacob, E. Narimanov, I. Kretzschmar, and V. M. Menon, “Topological transitions in metamaterials,” Science 336, 205–209 (2012).
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K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y. Zhang, S. V. Dubonos, I. V. Grigorieva, and A. A. Firsov, “Electric field effect in atomically thin carbon films,” Science 306, 666–669 (2004).
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I. V. Iorsh, I. S. Mukhin, I. V. Shadrivov, P. A. Belov, and Y. S. Kivshar, “Hyperbolic metamaterials based on multilayer graphene structures,” Phys. Rev. B 87, 075416 (2013).
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H. N. S. Krishnamoorthy, Z. Jacob, E. Narimanov, I. Kretzschmar, and V. M. Menon, “Topological transitions in metamaterials,” Science 336, 205–209 (2012).
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Narimanov, E. E.

Y. C. Chang, C. H. Liu, C. H. Liu, S. Zhang, S. R. Marder, E. E. Narimanov, Z. Zhong, and T. B. Norris, “Realization of mid-infrared graphene hyperbolic metamaterials,” Nat. Commun. 7, 10568 (2016).
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Y. C. Chang, C. H. Liu, C. H. Liu, S. Zhang, S. R. Marder, E. E. Narimanov, Z. Zhong, and T. B. Norris, “Realization of mid-infrared graphene hyperbolic metamaterials,” Nat. Commun. 7, 10568 (2016).
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K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y. Zhang, S. V. Dubonos, I. V. Grigorieva, and A. A. Firsov, “Electric field effect in atomically thin carbon films,” Science 306, 666–669 (2004).
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M. A. K. Othman, C. Guclu, and F. Capolino, “Graphene-dielectric composite metamaterials: evolution from elliptic to hyperbolic wavevector dispersion and the transverse epsilon-near-zero condition,” J. Nanophoton. 7, 073089 (2013).
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M. A. K. Othman, C. Guclu, and F. Capolino, “Graphene-based tunable hyperbolic metamaterials and enhanced near-field absorption,” Opt. Express 21, 7614–7632 (2013).
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H. Deng, X. Chen, B. A. Malomed, N. C. Panoiu, and F. Ye, “Tunability and robustness of Dirac points of photonic nanostructures,” IEEE J. Sel. Top. Quantum Electron. 22, 98–106 (2016).
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H. Deng, F. Ye, B. A. Malomed, X. Chen, and N. C. Panoiu, “Optically and electrically tunable Dirac points and Zitterbewegung in graphene-based photonic superlattices,” Phys. Rev. B 91, 201402 (2015).
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J. Hoffman, L. Alekseyev, S. S. Howard, K. J. Franz, D. Wasserman, V. A. Podolskiy, E. E. Narimanov, D. L. Sivco, and C. Gmachl, “Negative refraction in semiconductor metamaterials,” Nat. Mater. 6, 946–950 (2007).
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J. Qin, H. M. Dong, K. Han, and X. F. Wang, “Ultrafast dynamic optical properties of graphene,” Acta Phys. Sin. 64, 237801 (2015).
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Schurig, D.

J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312, 1780–1782 (2006).
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D. R. Smith and D. Schurig, “Electromagnetic wave propagation in media with indefinite permittivity and permeability tensors,” Phys. Rev. Lett. 90, 077405 (2003).
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I. V. Iorsh, I. S. Mukhin, I. V. Shadrivov, P. A. Belov, and Y. S. Kivshar, “Hyperbolic metamaterials based on multilayer graphene structures,” Phys. Rev. B 87, 075416 (2013).
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V. P. Gusynin, S. G. Sharapov, and J. P. Carbotte, “Magneto-optical conductivity in graphene,” J. Phys. Condens. Matter 19, 026222 (2007).
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F. H. Shi, Y. H. Chen, P. Han, and P. Tassin, “Broadband, spectrally flat, graphene-based terahertz modulators,” Small 11, 6044–6050 (2015).
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J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312, 1780–1782 (2006).
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D. R. Smith, J. B. Pendry, and M. C. K. Wiltshire, “Metamaterials and negative refractive index,” Science 305, 788–792 (2004).
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D. R. Smith and D. Schurig, “Electromagnetic wave propagation in media with indefinite permittivity and permeability tensors,” Phys. Rev. Lett. 90, 077405 (2003).
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H. N. S. Krishnamoorthy, B. Gholipour, N. I. Zheludev, and C. Soci, “A non-volatile chalcogenide switchable hyperbolic metamaterial,” Adv. Opt. Mater. 6, 1800332 (2018).
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Tassin, P.

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A. Vakil and N. Engheta, “Transformation optics using graphene,” Science 332, 1291–1294 (2011).
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Q. Zhao, T. Zhou, T. Wang, W. Liu, J. Liu, T. Yu, Q. Liao, and N. Liu, “Active control of near-field radiative heat transfer between graphene-covered metamaterials,” J. Phys. D 50, 145101 (2017).
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J. Qin, H. M. Dong, K. Han, and X. F. Wang, “Ultrafast dynamic optical properties of graphene,” Acta Phys. Sin. 64, 237801 (2015).
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Wang, Y. T.

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J. Hoffman, L. Alekseyev, S. S. Howard, K. J. Franz, D. Wasserman, V. A. Podolskiy, E. E. Narimanov, D. L. Sivco, and C. Gmachl, “Negative refraction in semiconductor metamaterials,” Nat. Mater. 6, 946–950 (2007).
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Figures (7)

Fig. 1.
Fig. 1. (a) Schematic of the zero-reflection-based optical switch; the period of the SHM is d . (b), (c)  Re ( ε ) and Im ( ε ) as functions of frequency and chemical potential.
Fig. 2.
Fig. 2. IFC analysis and simulation results for the optical switch at θ i = 60 ° and f = 25 THz . (a), (c) Switch-off state ( μ c = 0.2 eV ). (b), (d) Switch-on state ( μ c = 0.82 eV ). The inset shows the enlargement of the energy distribution in the denoted square area.
Fig. 3.
Fig. 3. Snapshots at different time steps for a Gaussian beam with a finite length of time steps. (a)  t s = 100 time steps (before touching the interface), (b)  t s = 200 time steps (arriving at the interface), and (c)  t s = 2000 time steps (stably propagating along the interface). (d)–(f) Three snapshots after the Gaussian beam stops emitting from the source.
Fig. 4.
Fig. 4. Influence of absorption loss of dielectric layers on the optical switch. (a) OFF-state ( μ c = 0.2 eV ). (b) ON-state ( μ c = 0.82 eV ).
Fig. 5.
Fig. 5. Variation of reflection with chemical potential for f = 24.5 THz , 25 THz, and 25.5 THz.
Fig. 6.
Fig. 6. μ c - f curves with different parameter conditions: (a)  ε = 0 , d = 0.1 μm ; (b)  ε = ε d , d = 0.1 μm ; (c)  ε = 0 , ε d = 1 ; (d)  ε = 1 , ε d = 1 . The dashed lines indicate μ c = 0.2 eV and 0.9 eV, respectively.
Fig. 7.
Fig. 7. (a)  μ c - f curves for α = 20 ° , 45°, 60°, and 80°. (b) IFC at f = 25 THz for α = 80 ° and ε = 0.07 . (c) Dependence of maximal incident angle θ i max on the slanted angle α .

Equations (5)

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σ ( f , μ c , Γ , T ) = i e 2 k B T 2 π 2 ( π f + i Γ ) [ μ c k B T + 2 ln ( e μ c / k B T + 1 ) ] + i 2 e 2 ( π f + i Γ ) π 2 0 d ϵ f d ( ϵ ) f d ( ϵ ) 4 ( π f + i Γ ) 2 4 ( ϵ / ) 2 ,
E bias = 2 e π 2 v F 2 ε 0 ε d [ ( k B T ) 2 μ c / k B T μ c / k B T x e x + 1 d x + k B T μ c ln ( e μ c / k B T + 1 ) + k B T μ c ln ( e μ c / k B T + 1 ) ] ,
ε = ε d , ε = ε d i σ ( ω , μ c , Γ , T ) ω ε 0 d ,
k x 2 ε + k z 2 ε = k 0 2 ,
ε = ε tan 2 α .

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