## Abstract

We theoretically study the topological transition of dispersion types and propose a tunable planar lens based on graphene hyperbolic metamaterials (HMMs). By tuning the chemical potential (*μ*_{c}) of graphene, the dispersion relation of the HMM is topologically switchable between ellipse (*μ*_{c}<0.6 eV) and hyperbola (*μ*_{c}>0.6 eV) where positive and negative refractions occur respectively. Especially, for *μ*_{c}>0.6 eV, a Gaussian light beam is negatively refracted twice and focuses at a far-field point finally, acting well as a planar lens. Furthermore, its focal length *l* can be sensitively tuned by controlling *μ*_{c}, and Δ*l* reaches 260 μm (from 528 to 268 μm) while *μ*_{c} varies with only 0.05 eV (from 0.65 to 0.7 eV). The physical reason is attributed to the different anisotropy degrees of EFCs for different *μ*_{c}. Such a compact, high-speed, and sensitively tunable planar lens holds great promise in photonic integration, photonic imaging, and directional coupling applications.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

Metamaterials have attracted great attention due to their abundant properties and potential applications in both science and technology. Many interesting phenomena, such as slow light [1], negative refraction [2], super-collimation [3,4], and superprism effect [5–7], have been explored in theoretical and experimental works. Hyperbolic metamaterials (HMMs) are highly anisotropic metamaterials whose effective electric and/or magnetic tensors have opposite-sign principal components. This special feature results in its unusual hyperbolic dispersion and lots of unique properties. HMMs are usually constructed with metal nanowire-array [8] or layered metal-dielectric structures [9]. Recently, graphene has attracted great attention due to its excellent optoelectronic properties. Its conductivity is very sensitive to external field, chemical potential, Fermi energy, gate voltage, magnetic field, chemical doping, or optical pump [10–19], so that its optoelectronic properties can be effectively and precisely tuned [20]. Graphene has been suggested as an alternative of metal to construct HMMs to confine light [21], guide surface plasmon polaritons [22], and manipulate wavefronts [23,24]. Obviously, graphene-based metamaterials show advantages over metal-based metamaterials which have unavoidable material loss and cannot be tuned after fabrication [25]. Recently, graphene metamaterials have been experimentally realized in the mid-infrared range [15]. A diversity of unusual applications, such as subwavelength imaging, super-lenses, hyperbolic waveguides, and slow light devices [21,26–28], have been proposed.

In recent years, tunable photonic devices are highly desired because they play more and more important roles in integrated photonic/optical circuits. Among them, tunable metamaterial lens is an elementary device and has attracted great attention. Unlike conventional lens design dealing with the shape of a uniform material to satisfy a special performance requirement, a metamaterial lens focuses on structure design and arranging the refractive index profiles. So far, two typical methods have been proposed to realize such lens. The first method is to change the geometrical structure mechanically such as the epsilon-near-zero lens [29] and the MEMS-based metasurface lens [30,31]. This method has the weaknesses of complex structure, slow speed, as well as low stability [32]. The second method is to modify the refractive index by using electro-optic [33] or thermo-optic effect [34]. It seems that this method is more promising because its response is much quicker. However, in order to obtain index change large enough for generating available difference of dispersion properties, it requires some extreme conditions (e.g., very strong external electric field or very high temperature), which significantly reduce the application promise of such metamaterial lens. What is more, even under such extreme conditions, it is still difficult to obtain a large focal-length tuning range. Therefore, it is highly desirable to develop new mechanisms to achieve a large focal-length tuning range of lens without requirements of extreme conditions.

In this paper, we propose to construct a tunable graphene-based HMM and study its switchable topological dispersion properties as well as its lens application. It is found that when the chemical potential (*μ*_{c}) of graphene sheets varies across the critical value of 0.6 eV, the dispersion relation of the HMM switches between ellipse and hyperbola, leading to transition between positive and negative refractions accordingly. Especially, by using the negative refraction effect when *μ*_{c}>0.6 eV, we design a compact planar HMM lens which can make a Gaussian beam redirect twice and focus at a far-field point in air finally. Moreover, its focal length can be sensitively tuned by controlling *μ*_{c}, and its focal-length tuning range as large as 260 μm can be easily achieved which requires only 0.05 eV change of *μ*_{c} from 0.65 to 0.7 eV. Additionally, the resolution of this lens is also discussed.

## 2. Model of Graphene HMM

Figure 1(a) shows the designed HMM structure consisting of alternative graphene sheets and dielectric layers with *ε _{d}* = 2.2 along the

*z*-axis. The thickness of the dielectric layer is

*d*= 100 nm. Graphene is a very thin material layer with a thickness as small as one atom, and it possesses extremely high carrier mobility of 15000 cm

^{2}/(V·s) which theoretically makes graphene-based devices with high response speed in the order of

*picosecond*possible. Owing to negligible thickness of graphene sheet as compared with the dielectric layer, the period of HMM can be regarded the same as

*d*. In this study, the working frequency is

*f =*30 THz (i.e.,

*λ*= 10 μm). Because of

*λ*>>

*d*= 100 nm, the designed structure can be modeled as an anisotropic effective medium by effective medium theory [35]. A Gaussian light beam is normally incident on the structure and propagates along the

*x*-axis. The

*x-z*view of the structure is presented in Fig. 1(b).

In our model, graphene is characterized by the following surface conductivity *σ*(*f*,*μ _{c}*,Γ,Τ) model according to the well-known Kubo formula [10],

*f*(

_{d}*E*) = {exp[(

*E*-

*μ*)/(

_{c}*k*T)] + 1}

_{B}^{−1}is the Fermi-Dirac distribution with the energy

*E*.

*f*is the working frequency and

*e*is the charge of the electron. Τ = 300 K is the temperature.

*k*and

_{B}*ћ*are Boltzmann’s constant and reduced Planck’s constant, respectively. Γ is the phenomenological scattering rate (which is assumed to be 0.1 meV) and independent of the energy

*E*.

*μ*is the chemical potential of graphene sheets.

_{c}To realize the tunable planar lens to be discussed later, the chemical potentials ${\mu}_{c}$ of all graphene sheets needed to be changed simultaneously. One effective way to do this is using external electrostatic field biasing ${E}_{bias}$ by electrical gating. The relation between ${E}_{bias}$ and ${\mu}_{c}$is deduced as [10],

According to effective medium theory [35,36], the effective relative permittivity tensor *ε _{eff}* of the HMM is described as,

*ε*

_{||}and

*ε*

_{⊥}are shown as follows,

Here *ε*_{⊥} is a positive real number because of *ε _{d}* = 2.2>0. As for

*ε*

_{||}, Fig. 2(a) clearly shows that Re(

*ε*

_{||}) varies from positive to negative values as

*μ*

_{c}increases from 0 to 1 eV within a wide frequency range; while Fig. 2(b) indicates that Im(

*ε*

_{||}) is near zero under conditions of

*μ*≥0.2 eV and

_{c}*f*>27 THz. Specifically, for the working frequency

*f*= 30 THz, the curves of Re(

*ε*

_{||})-

*μ*

_{c}and Im(

*ε*

_{||})-

*μ*

_{c}are shown in Fig. 2(c). We can see that for 0<

*μ*

_{c}<0.2 eV, both of Re(

*ε*

_{||}) and Im(

*ε*

_{||}) decrease nonlinearly as

*μ*

_{c}increases. Obviously, the existence of non-zero Im(

*ε*

_{||}) originated from losses of graphene layers is unfavorable for the design of tunable planar lens to be discussed later. But for

*μ*

_{c}≥0.2 eV, the Re(

*ε*

_{||})-

*μ*

_{c}curve is quite linear and it changes from positive to negative values at the critical value of

*μ*

_{c}= 0.6 eV where Re(

*ε*

_{||}) = 0, while Im(

*ε*

_{||}) keeps to be zero. This property is quite useful for the design of super-sensitive planar HMM lens later. Therefore, we will choose

*μ*

_{c}≥0.2eV in our later study unless there are additional specifications, and use

*ε*

_{||}instead of Re(

*ε*

_{||}) because of the near-zero Im(

*ε*

_{||}) for simplicity.

## 3. Switchable topological dispersion properties of HMM

In this work, the dispersion of the HMM is computed after the homogenization of the HMM has been performed by using effective medium theory. Considering the symmetry of the 1D periodic structure shown in Fig. 1(a), the dispersion relation after homogenization for extraordinary waves (i.e. TM waves) can be written as [37],

*k*and

_{x}*k*are

_{z}*x*and

*z*components of the wave vector respectively, and

*k*

_{0}is the wave number in vacuum determined by ${k}_{0}=2\pi f\sqrt{{\mu}_{0}{\epsilon}_{0}}$ with the vacuum permeability

*μ*

_{0}and the vacuum permittivity

*ε*

_{0}. For

*f*= 30 THz, Fig. 2(c) shows that

*ε*

_{||}is positive (or negative) when

*μ*

_{c}is smaller (or larger) than the critical value of 0.6 eV, resulting in an elliptic (or hyperbolic) topological type of dispersion relation. This property provides an effective way to realize the topological transition of dispersion types and makes a tunable graphene-HMM-based planar lens possible.

For better understanding the transmission behaviors of electromagnetic (EM) waves, equi-frequency contour (EFC) analysis is employed to predict the refraction behaviors at an interface. A EFC is a curve consisting of all the points with the same frequency in *k*-space. As is well known, the general Snell’s law *n*_{1}sin*θ*_{1} = *n*_{2}sin*θ*_{2} (*n*_{1} and *n*_{2} are refractive indices, while *θ*_{1} and *θ*_{2} are the incident and refractive angles) is only applicable to the case of isotropic medium. As for a periodic structure with anisotropic dispersion relation, it is necessary to develop it to be the extended Snell’s law by the following three steps: (1) The tangential components of the incident and refractive wave vectors (**k*** _{i}* and

**k**) parallel to the interface should be conserved (i.e.,

_{r}**k**=

_{i}_{||}**k**), because of |

_{r}_{||}**k**

_{0}|

*n*

_{1}sin

*θ*

_{1}= |

*k*_{0}|

*n*

_{2}sin

*θ*

_{2}(

**k**

_{0}is the wave vector in vacuum), i.e.,

**k**sin

_{i}*θ*

_{1}=

**k**sin

_{r}*θ*

_{2}. (2) The propagation direction of EM wave at certain point of a EFC is along its gradient direction, because the energy velocity in a lossless periodic structure is the same as the group velocity vector defined as

**v**

*= ▽*

_{g}

_{k}*ω*(

**k**). (3) According to the energy conservation law, the direction of the group velocity should point to the direction away from the source, so that the angle between the direction of the incident wave and that of the group velocity is acute. As a result, we are able to qualitatively determine the propagation direction of the refraction light at an interface.

The transmission behaviors for two topological types of EFCs of the graphene HMM are further studied. Equation (5) shows that as *ε*_{||} varies from positive to negative values, the EFC diagram changes from an ellipse to a hyperbola, which are clearly shown by the yellow EFC curves in Figs. 3(a1) and 3(b1). For the case of *μ*_{c} = 0.2 eV in Fig. 3(a1), *ε*_{||} is 1.58, which together with *ε*_{⊥} = 2.2 makes the EFC to be an ellipse according to Eq. (5). Such an elliptic EFC shows slightly anisotropy. Based on the extended Snell’s law, the direction of energy velocity in HMM deviates a little from that of phase velocity, i.e., the energy direction **S _{2}** is non-parallel to the wave vector

**k**inside the HMM. The FDTD simulations in Fig. 3(a2) clearly shows that a Gaussian light beam incident from region 1 (i.e. air) undergoes two times of positive refractions at interfaces 1/2 and 2/3, and finally diverges to propagate in air region 3, which agrees well with the EFC analysis in Fig. 3(a1). While for the case of

_{2}*μ*

_{c}= 0.65 eV (

*ε*

_{||}= −0.15), the results are very different. In Fig. 3(b1), the EFC of HMM becomes a hyperbola with strong anisotropy, so that

**S**inside HMM deviates dramatically from

_{2}**k**. According to the extended Snell’s law, negative refraction will occur at the interface between HMM and air. Since a Gaussian light beam always contains a range of wave vectors but not a single one, negative refraction occurs for each wave vector at the air-HMM interface, leading to a redirected light beam to focus at a focal point. As shown in Fig. 3(b2), due to the strong anisotropy of the HMM, the Gaussian beam first focuses at a focal point quite near the left air-HMM interface inside the HMM, then propagates with quite large divergent angle, and finally passes through the right HMM-air interface to focus again at a far-field point in air region 3. These results agree quite well with the predictions in Fig. 3(b1). In other words, a planar lens can be realized provided that the HMM possesses hyperbolic EFCs under the condition of

_{2}*μ*

_{c}>0.6 eV. It should be noticed that from the viewpoint of applications, only the second focal point in air region 3 is the key performance parameter to characterize the planar HMM lens, so that we will further investigate the properties of such focal point in the following text.

## 4. Discussions of tunable sensitive HMM lens

From the aforementioned analyses, it is found that the shape of the hyperbolic EFC described by Eq. (5) sensitively depends on *μ*_{c}. Here we take four representative *μ*_{c} to quantitatively study the influence of the chemical potential on the HMM lens. The EFC diagrams and FDTD simulation results are shown in Fig. 4. It is noticed that all of the considered *μ*_{c} here are larger than 0.6 eV where *ε*_{||} = 0, leading to negative *ε*_{||}, i.e., *ε*_{||} = −0.15, −0.53, −0.9, and −1.28 for *μ*_{c} = 0.65, 0.75, 0.85, and 0.95 eV, respectively. Hence all of their corresponding EFCs are hyperbolic according to Eq. (5). Due to the negative refraction behaviors occurring at 1/2 and 2/3 interfaces, a Gaussian light beam finally focuses in region 3 to form a focal point, as shown in Fig. 4. Though the incident angle *θ*_{1} keeps the same, the energy direction of EM wave inside the HMM (i.e. **S _{2}**) varies sensitively as

*μ*

_{c}increases, resulting in a sensitive change of focal lengths of the planar HMM lens in region 3. These theoretical analyses are well verified by the FDTD simulations shown in Figs. 4(a2)-4(d2) whose focal lengths are 528, 170, 91, and 52 μm, respectively.

Next, we will discuss the sensitivity of the focal length of the HMM lens. Figure 4 shows that **S _{1}** and

**S**always keep unchanged whatever

_{3}*μ*

_{c}is, whereas

**S**is sensitive to

_{2}*μ*

_{c}. Therefore, the variation of focal length mainly depends on the direction of

**S**. In a word, one can tune the focal length of the HMM lens by changing

_{2}*μ*

_{c}conveniently. Figure 5(a) presents more details about the relationship between the focal length

*l*and the chemical potential

*μ*

_{c}. With increasing

*μ*

_{c}, the focal length

*l*decreases dramatically at first (when

*μ*

_{c}<0.75 eV) and then slowly (when

*μ*

_{c}>0.75 eV). For examples, for the case that

*μ*

_{c}is near the critical point

*μ*= 0.6 eV, the decrement of the focal length is as large as Δ

_{c}*l*= 260 μm (from 528 to 268 μm, about 26 times of the incident wavelength of 10 μm) when

*μ*

_{c}increases with only 0.05 eV (from 0.65 to 0.7 eV); while for the case that

*μ*

_{c}is much large than 0.6 eV, Δ

*l*is only 8 μm (from 52 to 44 μm) when

*μ*

_{c}increases with 0.05 eV (from 0.95 to 1 eV). The physical reason is attributed to the different anisotropy degrees of EFCs for different

*μ*

_{c}. The variation of anisotropy degree for 0.65<

*μ*

_{c}<0.7 eV is much larger than that for 0.95<

*μ*

_{c}<1 eV. This sensitive dependence characteristics of focal length on chemical potential provides an effective means to realize tunable compact planar HMM lens.

We further study the influence of *μ*_{c} on the transmittance of the HMM lens. Transmittance is calculated by the ratio of the power flowing through the 2/3 interface to the power of incident Gaussian light beam. As shown in Fig. 5(b), for 0.2≤*μ _{c}*≤0.54eV or 0.64≤

*μ*≤1 eV, the HMM lens can act as a concave or convex lens, and the transmittance maintains high (>92%) and stable, meaning that the tunable HMM lens can work well within a wide chemical potential range. However, for 0.54<

_{c}*μ*<0.64 eV, there exists a minimum transmittance close to zero at the critical point

_{c}*μ*= 0.6 eV where

_{c}*ε*

_{||}= 0. In this special case of

*μ*= 0.6 eV, the hyperbolic EFC degenerates to a horizontal line parallel to the

_{c}*k*-axis, causing the corresponding

_{x}**S**to be along the

_{2}*y*-axis. In other words, all of the EM waves inside the HMM propagates upwards and downwards, causing no energy to propagate rightwards, thus the transmittance is almost zero. As

*μ*deviates from 0.6 eV, the transmittance increases rapidly. No matter the EFC is elliptical or hyperbolic, more and more energy transmits rightwards, leading to higher and higher transmittance. It should be noticed that as

_{c}*μ*increases from 0 to 0.2 eV, the transmittance increases quickly from 70% to 92% and then keeps stable, as shown by the inset in Fig. 5(b). By further referring to the Im(

_{c}*ε*

_{||})-

*μ*

_{c}curve in Fig. 2(c) where Im(

*ε*

_{||}) characterizing the losses of graphene decreases to near-zero with increasing

*μ*from 0 to 0.2 eV, we can conclude that the transmittance increases quickly to be stable when the losses of graphene reduces to near-zero. That is to say, the losses of graphene will reduce the transmittance of the structure, which is unfavorable for the design of tunable HMM lens. Thus we choose

_{c}*μ*≥0.2 eV where the losses of graphene can be neglected for our planar lens design.

_{c}We also investigate the resolution of the HMM lens. Figure 5(c) shows the |E| profiles along the *y* direction at focal points in region 3 for four different *μ _{c}*. The resolution is characterized by the full width half maximum (FWHM) of the curve. The FWHMs are 25.6, 20, 19, and 18.8 μm for

*μ*

_{c}= 0.65, 0.75, 0.85, and 0.95 eV, respectively. Obviously, the resolution decreases quickly at first and then slowly as

*μ*

_{c}increases from 0.65 to 0.95 eV. This means that higher

*μ*

_{c}is beneficial for enhancing the resolution of the lens, but the resolution will gradually reach a saturation value after

*μ*

_{c}= 0.85 eV.

## 5. Conclusions

To summarize, we have constructed a tunable graphene-based HMM and studied its switchable topological dispersion properties and lens application. There exists a critical value of chemical potential (*μ*_{c} = 0.6 eV) leading to *ε*_{||} = 0, and the EFC can be topologically switched between ellipse (*μ*_{c}<0.6 eV) and hyperbola (*μ*_{c}>0.6 eV) conveniently. Accordingly, positive or negative refraction can be realized for elliptic or hyperbolic EFCs respectively. Based on these properties, we design a compact tunable planar HMM lens. For *μ*_{c}>0.6 eV, FDTD simulations results show that a Gaussian light beam is negatively refracted two times and focuses at a far-field point in air region finally, which agrees well with the theoretical predictions of EFC analysis. Furthermore, the influence of *μ*_{c} on the focal length *l* is studied and the results show that the focal length varies sensitively for *μ*_{c}<0.75 eV and smoothly for *μ*_{c}>0.75 eV. The physical reason is attributed to the different anisotropy degrees of EFCs for different *μ*_{c}. These results have potential in photonic integration, photonic imaging, and directional coupling applications.

## Funding

National Natural Science Foundation of China (NSFC) (11504114, 11434017); Science and Technology Program of Guangzhou (201904010105); National Key R&D Program of China (2018YFA 0306200); Guangdong Innovative and Entrepreneurial Research Team Program (2016ZT06C594); Natural Science Foundation of Guangdong Province, 2019; Fundamental Research Funds for the Central Universities, (x2wl/D2191420); Teaching and Research Reform Project of SCUT (x2wl/Y1190281); Student Research Program of SCUT (x2wl/C9192092).

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