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Abstract
It is well established that the topological transition of the iso-frequency surface (IFS) of hyperbolic metamaterials from the ellipsoid to hyperboloid provides unique capabilities for controlling the propagation of the wave. Here, we present a graphene-based hyperprism (GHP) structure that uses an electronically controlled modulation strategy to achieve optical switching and wide-angle beam steering functions. Numerical simulation results show that, by regulating the chemical potential of GHP, the optical switching system can achieve high transmission (97%) and zero transmission, as well as the beam steering system can reach a maximum adjustable angle of 52.94°. Furthermore, the effects of Fermi energy and relaxation time on transmittance are also investigated. These works may provide new opportunities for applications such as optical data storage, modulators, and integrated photonic circuits.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
In recent years, hyperbolic metamaterials [1–3] (HMMs) have been a critical hot topic due to their unique properties. Since the different diagonal component symbols of the permittivity tensor of HMMs, leads it to produce unusual hyperbolic dispersion [4–7]. By adjusting the dispersion of HMMs, various applications can be achieved, such as imaging [3,8,9], focusing [10], and negative refraction [11–13]. More impressively, when an optically tunable material is combined with the HMM structure, this tunable HMMs can produce a greater variety of beam control capabilities. Graphene, as a new type of two-dimensional tunable nanomaterial, has the advantages of robustness, low loss, and easy integration compared with traditional materials. Another significant feature of graphene is that its optical properties (permittivity) are largely dependent on its surface conductivity, which can be adjusted by Fermi energy, chemical potential, gate voltage or external electrostatic field bias [14]. These unique properties of graphene open up unprecedented possibilities for developing electronic and optoelectronic devices with ideal physical properties [15]. One type of structure is a graphene-based metamaterial, which consists of multilayers of graphene sheets separated by dielectric layers [16–18]. These have been used to realize waveguides [19–21], tunable absorption [22,23], beam steering [24].
In the current work, we analyze the special dispersion relationship in the graphene-based hyperprism structure and study the propagation behavior of the electromagnetic TM polarization wave at the interface between the GHP and the isotropic medium. The GHP structure is formed by cutting a multilayer graphene-based metamaterial into a prism structure with the tilted optical axis. Only by tuning the Fermi level under hyperbolic conditions, zero and high transmission of waves can be achieved. This effect can be applied as the optical switch function. Unlike the transmissive optical switches studied in this paper, Liang proposed a structure composed of graphene sheets and dielectric to design the optical switch [25], which has realized a zero-reflection effect. Maziar et al. reported that a metal / Al2O3 / graphene composite structure implements a negative/total reflection optical switch [26]. The introduction of metal will inevitably increase the loss of beam energy. In addition, when the optical switch is turned on, we find the changes in the beam propagation path is also influenced by Fermi energy. From this, we investigated the ability to control the transmitted beam steering precisely. Forouzmand’s group reported a beam steering device for metal-insulator-metal (MIM) units made of indium tin oxide (ITO) material, which is an electrically tunable reflective array meta-surface [27]. By adjusting the voltage biasing distribution, the scanning angle can be feasibly controlled effectively. Hiroshi Abe designed a non-mechanical on-chip optical beam steering device that uses a photonic crystal waveguide with a dual period structure to regulate the beam [28]. By scanning the wavelength and switching the waveguide, the radiation beam angle is controlled in the longitudinal and transverse directions, respectively. However, they are all controlling the reflected beam, but when a reflected wave signal is received, the incident wave may interfere with the reflected signal. We can implement double functions of a transmissive optical switch and beam steering through the GHP structure. This structure provides a new idea for the realization of photonic devices in the mid-infrared spectrum.
2. Model design and theory
As shown in Fig. 1, a TM-polarization wave with an incident angle of θ is incident from the air to the GHP, then the wave is refracted from the GHP to the medium A. Medium A is considered to be a non-absorptive isotropic medium. The GHP is a prism structure made of graphene-based HMMs. The single cycle of GHP consists of a dielectric layer with a thickness of dd and relative permittivity ɛd, and a graphene monolayer with a thickness of dg = 1 nm and permittivity, ɛg. The optical axis of the GHP is on the x-z plane and makes an angle β with the z′ -axis. For simplicity, we set the medium A to be air and the apex angle of the prism to 90°.
Fig. 1. Diagram of the incident TM-polarization wave with the angle of incidence θ on the interface of air and the GHP. A periodic structure consists of graphene/dielectric. The dielectric layer is marked in yellow, and the graphene layer is marked in black. Here, α is the refraction angle.
For the graphene layers, its electromagnetic properties are characterized by the surface conductivity σ(ω, µc). The σ(ω, µc) can be calculated using the Kubo formula [29,30]
To implement the optical switch and beam steering discussed later, the Fermi Energy of all graphene layers needs to be changed simultaneously. The Fermi Energy can be controlled by graphene’s external electrostatic field biasing, Ebias. The relationship between Ebias and µc is derived as [31,32]
Note that graphene is an anisotropic material, we use an approximation method, assuming that the electronic band structure of the graphene layer is not affected by the adjacent slabs. The effective permittivity ɛg of graphene can be calculated as follows [33,34]
where dg is the thickness of graphene, dg = 1 nm. The real part of the graphene permittivity in the optical range is shown in Fig. 2(a). It is clear from the figure that graphene has a strong dispersibility in the frequency range considered. It can be seen that Re(ɛg) decreases as µc increases or ω decreases.Fig. 2. The relation between the working frequency and (a) Re(ɛg), (b) ɛ⊥, respectively. (c) In the case of fixed frequency ω = 400 THz, the effective parameters ɛ‖, and ɛ⊥ can be tuned according to different values of µc in the multilayer structure. When the value is increases from 0.1 eV to 0.6 eV, the sign of ɛ⊥ is reversed, and ɛ‖ remains positive. (d) Dispersions of GHP. When µc = 0.1 eV, IFS is an ellipse. when µc = 0.6 eV, IFS is a hyperbola.
The GHP with the tilted optical axis has optical characteristics which are different from those of a conventional periodic metamaterial structure. The tilted optical axis gives the dispersion relationship of the GHP with respect to the interface rotation, and perfect absorption can be achieved at a particular angle [35]. In our study, considering that the incident wavelength is much larger than the thickness of the periodic structure, so the effective medium theory method can be used to equivalent the permittivity of GHP. By effective medium theory (EMT) approximation, the permittivity of the GHP can be written as follows
Considering the surface of GHP terminates at a slanted angle β, and the electromagnetic waves are refracted from the GHP into the air. We choose a new Cartesian coordinate system x'-z’ with the same origin as the original coordinate system x-z, and the new coordinate system is obtained by inverting the original coordinate system by β angle. After some algebra, the IFS of the GHP in the new coordinate can be obtained as [36]
To illustrate the tunable character of graphene, Re(ɛ⊥) is calculated and shown in Fig. 2(b). In Fig. 2(b), we observe Re(ɛ⊥) of the graphene/dielectric structure, which changes from positive to negative as the frequency changes. Therefore, in Fig. 2(c), it can be seen that as the µc increases, the values Re(ɛ⊥) and Re(ɛ‖) also change from positive to negative. By combining Eq. (6) and Eq. (7), we can flexibly adjust the value and symbol of Re(ɛ‖) by changing the value of µc, as is shown in Fig. 2(c). When ω = 400 THz, µc changes from 0.1 eV to 0.6 eV, and the sign of ɛ⊥ is inverted at the critical point of 0.478 eV. Figure 2(d) is a dispersive kx-kz diagram of graphene/dielectric metamaterial with Fermi Energy of 0.1 eV and 0.6 eV. The IFS of multilayer HMMs in the original coordinate system x-z is shown in Fig. 2(d), when the Fermi energy is 0.1 eV, the real part of the IFS of the multilayer structure is elliptical. While the Fermi energy is 0.6 eV, the IFS is hyperbolic. As can be seen from the above results, an elliptical or hyperbolic dispersion relation can be generated by regulating the µc of graphene. Different dispersion relations produce a variety of interesting physical phenomena. It should be noticed that although the relationship between Re(ɛ) and µc is only obtained at 400 THz, the above conclusions also apply to other frequencies with appropriate parameters.
Different conditions of the interface between the free space and the anisotropic medium will lead to different propagation behavior of the wave at the interface. In Ref. [37], their interface between metamaterial and air is perpendicular and parallel to the graphene sheets respectively. Therefore, they realized two types of optical switches, namely a positive refraction/total reflection optical switch and a negative refraction/positive refraction optical switch. However, when the interface between metamaterial and air is not simply parallel or perpendicular to the graphene sheets, it will exhibit a unique electromagnetic wave propagation behavior. According to this principle, a transmissive optical switch is designed.
3. Results and discussion
3.1 Transmissive optical switch
In order to reveal the principle of the transmissive optical switch, we choose the conditions of ω = 400 THz and θ = 45°. IFS analysis is adopted to study the propagation behavior of electromagnetic waves at the interface, as shown in Figs. 3(a) and 3(b). In Figs. 3(a) and 3(b), gray solid line and gray dashed line are the conservation lines. The gray dashed line is the optical axis direction of the GHP and the gray solid line is the optical axis direction of the composite structure. The gray solid conservation line intersects the IFS twice which means energy passes through the same interface twice instead of the air when µc = 0.53 eV, in which case only energy flow Sr exists. Here the optical switch is switched off. Of course, according to Ref. [25], this phenomenon also occurs when the IFS of GHP is an elliptical dispersion relationship. Nevertheless, when µc = 0.6 eV, the gray solid conservation line intersects the IFS of the GHP once and passes through the IFS of the medium A (air). Thus, the refracted wave energy flow Sa is generated. In this case, the optical switch is switched on. Therefore, the phenomenon we studied shows that zero transmittance (total reflection) and high transmittance can be achieved under the premise of fixed incident direction and reasonable adjustment of angles θ and β.
To verify these predictions, the propagation behavior of the electromagnetic wave is simulated by the finite element method, as shown in Figs. 3(c) and 3(d). The results of the simulation are in good agreement with the analysis in Figs. 3(a) and 3(b). For the off-state in Fig. 3(c), the energy is reflected at the interface and escape finally at the upper-left direction of the GHP. For the on-state in Fig. 3(d), the energy is transmitted through the interface and comes out from the interface at the below direction of the GHP. In other words, by tuning the Fermi energy, zero and high transmittance can be achieved in our structure.
Fig. 3. Theoretical off state and on state deduced by IFS analysis for (a) µc = 0.53 eV and (b) µc = 0.6 eV at ω = 400 THz and θ = 45°, β = 45°. The black arrows Sr and St represent the direction of the reflected and refracted energy flow in the GHP, respectively. The black arrow Sa corresponds to the refracted beam in output air. (c) and (d) are the corresponding |H| distributions for (a) and (b), respectively.
In the dynamic process of adjusting µc, as shown in Fig. 4(a), we calculate the transmittance of the below medium A layer (air) in the optical switch. Transmittance is defined as the ratio of energy flowing through the interface of air and the GHP to the incident energy. As can be seen from Fig. 4 that the optical switch state changes at µc = 0.539 eV for θ = 45°. Obviously, both the off and on states can be maintained over a wide range. Therefore, transmittance can be adjusted by adjusting the external bias voltage of the structure. In other words, the ability to control transmittance can be viewed as an optical switch of energy. This optical switch can achieve high efficiency in the on and off states. As shown in Fig. 4(a), the transmittance is 0 in the off state, and the transmittance can reach 97% in the on-state. For our structure, it is not limited to use the above two specific values. Such properties are very convenient for practical applications. To continue the discussion of beam control, it is necessary to consider the effect of the loss on the optical switches. The loss is due to the relaxation time of graphene, which comes from the imaginary part of the permittivity. Figure 4(b) shows the relationship between loss and transmittance. Obviously, when the relaxation time is less than 100 ps, the smaller the relaxation time, the lower the transmittance. Therefore, it can be seen from Fig. 4 that the loss has little substantial influence on the propagation characteristics of electromagnetic waves.
Fig. 4. The transmissive (a) as a function of µc with τ = 100 ps, and (b) as a function of τ with µc = 0.6 eV.
3.2 Precise beam steering
From the above studies, it is worth noting that when the switch is on-state, the transmitted beam will change direction with the change of µc. We define the angle between the transmitted beam that varies with µc and the z′-axis as the angle of beam steering, which is α. Therefore, we will further discuss the phenomenon of changing the angle of beam steering.
So far, we have assumed that β is 45°. In fact, the transmitted beam can be generated at many β values. It stipulates that the conservation line intersects with the IFS of the isotropic medium A (air). The relationship between α, θ, and β is calculated. As shown in Fig. 5, we turn our attention to the behavior of the α and µc at different θ and β, respectively. From Figs. 5(a) and 5(b), one can see that the angle of refraction α is affected by changing in the chemical potential of the graphene. When θ = 45° and β = 45°, the adjustable angle range is Δα = 30.13°. At this time, αmax is 87.53°, and αmin is 57.39°. In order to more clearly show the propagation behavior of electromagnetic waves, we chose six interval points (µc = 0.55 eV, 0.57 eV, 0.59 eV, 0.61 eV, 0.63 eV and 0.65 eV), which are selected within the hyperbolic range of µc. These points are used to record changes in electromagnetic waves with µc. Obviously, the results of the calculations and simulations are consistent. From Fig. 6(a) to Fig. 6(f), we can clearly see the changes of angles with µc, which are 74.98°, 67.58°, 63.75°, 61.30°, 59.57°, and 58.27° respectively. In terms of application, we can choose the appropriate adjustment range by selecting the values of θ and β reasonably. Moreover, when we select the desired adjustment range, we only need to change the applied bias voltage of graphene in GHP. These characteristics make it very convenient for practical applications that control the angle of the beam.
Fig. 5. The angle α (a) as a function of µc with β = 45°, and (b) as a function of µc with θ = 45°. Here, the lines indicate the frequency ω = 400 THz of the incident TM wave.
Fig. 6. Magnetic-field distribution of the chemical potential (a) µc = 0.55 eV (b) µc = 0.57 eV (c) µc = 0.59 eV (d) µc = 0.61 eV (e) µc = 0.63 eV (f) µc = 0.65 eV.
Further analysis and discussion of the IFS will show the principle of beam steering. In Fig. 7, ki is an incident wave vector. Sa and Sa’ are refractive energy flows in the medium A. The angle between the black arrows Sa and Sa’ is the maximum adjustable angle that the GHP can achieve under this condition. The gray solid conservation line is used to determine the direction of the refracted energy flow in the medium A layer. The gray dashed conservation line represents the optical axis of GHP and is used to determine the position of the gray solid conservation line. It can be seen from Fig. 7 that the final angle of refraction is determined by the gray solid conservation line. The maximum refraction angle is obtained by the tangent between the conservation line and the IFS of the isotropic medium A, that is, the angle corresponding to the refracted wave vector represented by Sa’. The minimum adjustable angle is obtained at the maximum value of 0.667 eV in the hyperbolic range. Finally, we can give a relation of α, θ, and β, as shown in Eq. (8).
Fig. 7. IFS analysis. When IFS is hyperbolic, all the µc values result in a hyperbolic change dispersion diagram. The black arrow Sa corresponds to the refracted beam in output air. Sa and Sa’ indicate the maximum and minimum angles that can be adjusted.
The maximum adjustable refraction angle Δα can be achieved by changing different incident angles θ and optical axis rotation angles β. Figure 8 is the Δα diagram corresponding to different incident angle θ. For different rotation angles β of the optical axis, two large adjustable angle areas will be generated. The maximum adjustable angle Δα can reach 52.94 ° under hyperbolic conditions.
Fig. 8. The maximum adjustable refraction angle Δα diagram corresponding to different θ and β in GHP.
It is necessary to review the feasibility of GHP briefly. As far as the preparation technology of GHP is concerned, a periodic layered structure composed of several graphene sheets and dielectric slabs is feasible. The most common approaches for fabrication single and multiple graphene layers are CVD [38] and MBE [39] methods. There are many ways to implement layer transfer, such as PMMA and PMGI. The applicability of these methods has also been demonstrated in many studies [40,41].
4. Conclusion
Based on the variable hyperbolic dispersion of the graphene-based multilayer hyperbolic metamaterial prism structure, the tunable beam manipulation phenomenon is investigated in the mid-infrared spectrum. The theoretical analysis and numerical calculations show that the IFS of GHP can be conveniently switched by adjusting the Fermi energy of the graphene layer. Especially when the conservation line intersects IFS twice, zero transmission can be achieved. Based on this characteristic, a transmissive optical switch is proposed. This optical switch can work at zero transmittance (on-state) over 0.478 - 0.53 eV, and high transmittance (>97%) (off-state) 0.53 - 0.667 eV. The advantage of this type of optical switch is that switching effect can be generated by actively adjusting the Fermi energy without changing the structure and angle of the incident wave. In addition, we have further explored its applications in beam steering. The results clearly show that when the conservation line intersects IFS once, the transmitted beam can achieve a beam steering effect with the increase of the Fermi energy and the maximum adjustable angle can reach 52.94°. This structure has a high working efficiency in practical applications. These results have potential applications in the fields of optical data storage, modulator, and integrated photonic circuit.
Funding
National Natural Science Foundation of China (11474078).
Disclosures
The authors declare no conflicts of interest.
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