Actively controlling the topological transition of dispersion based on electrically controllable metamaterials

Topological transition of the iso-frequency contour (IFC) from a closed ellipsoid to an open hyperboloid, will provide unique capabilities for controlling the propagation of light. However, the ability to actively tune these effects remains elusive and the related experimental observations are highly desirable. Here, tunable electric IFC in periodic structure which is composed of graphene/dielectric multilayers is investigated by tuning the chemical potential of graphene layer. Specially, we present the actively controlled transportation in two kinds of anisotropic zero-index media containing PEC/PMC impurities. At last, by adding variable capacitance diodes into two-dimensional transmission-line system, we present the experimental demonstration of the actively controlled magnetic topological transition of dispersion based on electrically controllable metamaterials. With the increase of voltage, we measure the different emission patterns from a point source inside the structure and observe the phase-transition process of IFCs. The realization of actively tuned topological transition will opens up a new avenue in the dynamical control of metamaterials.


I. INTRODUCTION
Recently, hyperbolic metamaterials (HMMs) have greatly attracted people's attention for its open iso-frequency contour (IFC) in the wave vector space. Once the topological transition have been realized from a closed ellipsoid to an open hyperboloid, interaction between light and matter will be significant changed, such as enhanced spontaneous emission [1][2][3][4], all-angle negative refraction [5][6][7][8][9], super-resolution imaging [10][11][12][13][14], long-range interaction [15][16][17], Cherenkov emission with low energy electrons [18], etc. Up to now, by varying the frequency, the sign of the real part of permittivity (  ) or permeability (  ) in dispersive metamaterials is changed and thereby the topological transition of dispersion is realized [1,8,19,20]. In addition, by maintaining the real parts and tuning the imaginary part of  or  at a fixed frequency, topological transition of IFC have been theoretically analyzed [21] and experimentally demonstrated in metamaterials with added losses [22,23]. In contrast to the passive metamaterials, the study of actively tuned metamaterials and meta-devices is also an outstanding research topic [24][25][26][27]. Actively controlled metamaterial systems have been predicted to be able to yield new applications ranging from electrically switchable devices [26] to the tunable coupling devices [25,27]. The realization of the actively tunable topological transition of IFC is also very useful in the design of new active optical devices. To realize the externally tunable behaviors, two-dimensional (2D) semiconductors such as graphene are usually utilized. Graphene, a 2D honeycomb lattice of carbon atoms, is electrically characterized by its surface conductivity ( , ) c    , where c  is the chemical potential that can be tuned by the gate voltage [24], and  denotes the angular frequency. Tunable absorption [28,29], giant Purcell effect [30] and hyperbolic plasmonics [31,32] have been theoretically proposed in graphene-based structures associated with the hyperbolic property at the suitable condition. In this paper, by tuning the chemical potential of graphene layer, we theoretical analyze and numerically simulate the tunable emission patterns in periodic structure composed of graphene/dielectric multilayers. The emission patterns clearly show the topological transition of IFC from a closed ellipsoid to an open hyperboloid [19,23]. Remarkably, by actively changing the chemical potential of graphene layer, two special cases corresponding to two kinds of anisotropic  -near-zero (ENZ) media can be realized in the proposed structure at a fixed frequency. To see the influence of the topological transition on the wave propagations, we study the transmission property of the two kinds of ENZ media embedded with a PEC or PMC defect. Furthermore, we propose a microwave experimental platform to demonstrate the actively controlled topological transition of IFC. By adding electrically controlled diodes into 2D transmission-line (TLs) system, we experimentally observe the actively tuned topological transition from the emission patterns by modulating the external voltages applied in the diodes.
The paper is organized as follows. In Sec. II, by tuning the chemical potential of graphene layer, we show an actively adjustable topological transition of dispersion in the periodic structure composed of graphene and dielectric layers. Moreover, the influences of the topological transition of dispersion on the scattering properties of structures with embedded PEC or PMC defect have been fully studied. Then, in Sec. III, based on a 2D TL system, we experimentally demonstrate the actively controlled topological transition of dispersion based on electrically controllable metamaterials. Finally, we conclude in Sec. V.

GRAPHENE/DIELECTRIC MULTILAYERS
We consider the structure, shown in Fig. 1, which is composed of graphene layers separated by dielectric slabs. For the graphene sheet, electromagnetic properties are characterized by its surface conductivity ( , ) c    . The ( , ) c    can be calculated as the Kubo formula [33,34]:   [24].
At high frequencies, the contribution from inter-band can be neglected. For the Fermi-Dirac statistics, conductivity can be simplified as [33]: When electronic band structure of a graphene sheet is unaffected by the neighboring, the effective permittivity of graphene can be calculated as follow [24]:  ,when the value of c  increase from 0.1e to 0.5e, the sign of //  reversal at the critical point 0.439e while the sign of   remain positive, as displayed in Fig. 3(a).
Considering the light with TM polarization propagating in the x-y plane of the 2D uniaxial media, the iso-frequency surface in such a strongly anisotropic metamaterial is given by [12]: , the IFC of the graphene/dielectric multilayers will be a hyperbola, which is marked by the solid blue line in Fig. 3(b). The change of IFC will strongly modify the propagation and emission properties of electromagnetic waves. The emission pattern is an open line because of the density of states is maximum along the hyperbolic asymptote ( Fig. 3(d)) [37]. The chemical potential of graphene is modulated by an applied electric field and the topological transition of dispersion is realized with the changed chemical potential [24][25][26][27][28][29][30][31][32]. This actively controlled topological transition will greatly change the emission pattern of a source in the medium. From the simulated emission patterns, we can find that the emission patterns as shown in Fig. 3(c) and 3(d) coincide with the IFCs based on the effective medium theory ( Fig.   3(b)). So, the actively tunable IFCs in graphene/dielectric multilayers are realized by tuning the chemical potential of graphene layer. 6 We further study the influences of the topological transition of dispersion on the scattering properties of structures if PEC or PMC defect is added. According to the Eq. (5), the sign of   also can be controlled with the varied c  , which is shown in Fig. 4(a).  Fig. 4(b)), which can be regarded as one kind of anisotropic ENZ medium. As another kind of anisotropic ENZ medium for 0.439 Fig. 3(a), the IFC is a very flat elliptic curve along vertical direction (marked by purple line in Fig. 4(b)). So far, based on passive system, unusual transportation properties of the light in anisotropic ENZ metamaterials have been demonstrated, including collimation [38], flux manipulation [39] and total transmission (reflection) [40]. Now, with the aid of active control, we study the actively tuned transmissions of two kinds of anisotropic ENZ media based on the graphene/dielectric multilayers. For the type Ⅰ of anisotropic ENZ medium ( 18 c e   ), when a plane wave is incident on this structure embedded with a tiny PEC defect, the wave can perfectly pass through this defect without any influence, as is shown in Fig.4 (c). The transmission behavior will change once the PEC defect is replaced by a PMC defect and the incident wave will be scattered by the defect (Fig. 4(d)). Similar to the type Ⅰ of anisotropic ENZ medium, for the type Ⅱ of anisotropic ENZ medium ( 0.439 c e   ), Fig. 4(e) displays that the incident wave also will not be affected by the PEC defect just as in the case of Fig.   4(c). However, the transmission will dramatically change for the structure containing PMC defect.
In this case, the incident wave will be completely blocked by the tiny defect ( Fig.4 (f)), which is just as the total reflection effect realized in double near-zero-index material [41]. So, inspired by the demonstration of the topological transition of dispersion, we present the novel transportation behavior that can be controlled in an active manner. 7 In visible range, the experimental realization of actively tunable topological transition of dispersion is still a great challenge. In this section, we introduce a microwave platform based on 2D

TOPOLOGICAL TRANSITION BASED ON TLS
TLs to experimentally demonstrate the actively controlled topological transition. The experimental schematic of the TLs-based metamaterial is shown in Fig. 5(a) and the corresponding circuit model is shown in Fig. 5(b). The insets below Fig. 5(b) show the enlarged lumped variable capacitance diodes and the protected elements, respectively. Here, our sample is composed of 20  C , the effective permittivity and permeability of 2D TLs in the long-wavelength limit can be written as [22,42]: where 0  is permeability of vacuum. 0  Fig. 7.
The 2D TLs can be described by the effective medium theory and the dispersion relation is derived as [17]: Based on Eq. (8), we calculate different IFCs by varying the value of C at 0.8 GHz, as is shown in Fig. 8(a). One can see that the topological transition of IFC will happen once the value of C changes from 10 pF to 4 pF. Moreover, the IFC will become more and more flat with the decrease of C .
We further use the CST (computer simulation technology) microwave studio software to perform the simulation. A linearly polarized source is loaded near the center of the sample marked by the blue point. In Fig. 8 The near-field probe is vertically placed 1 mm above the TLs to measure the signals of electric fields of the TLs. In order to accurately probe the field distributions in the near-field scanning measurement, our experimental sample is placed on an automatic translation device. The spatial step of scanning the near field is set to be 1 mm in the x and y directions, respectively. With the increase of the voltage, we measure the different emission patterns at 0.8 GHz to observe the changing process of IFCs. The measured E y patterns in Fig. 8(c) agree well with the numerical simulated one in Fig. 8(b). So, based on 2D TLs, we have experimentally demonstrated the actively controlled magnetic topological transition of dispersion by changing the external voltages.

IV. CONCLUTION
In conclusion, we demonstrate the actively tuned topological transition of dispersion in graphene/dielectric multilayers and the influence on the scattering properties when a PEC or PMC defect is embedded in the structure. Moreover, based on 2D TLs loaded with lumped variable capacitance diodes, we experimentally realize the actively controlled topological transition of dispersion. Our results represent a step towards the active control of wave propagations based on metamaterials.