Graphene-based extremely wide-angle tunable metamaterial absorber

We investigate the absorption properties of graphene-based anisotropic metamaterial structures where the metamaterial layer possesses an electromagnetic response corresponding to a near-zero permittivity. We find that through analytical and numerical studies, near perfect absorption arises over an unusually broad range of beam incidence angles. Due to the presence of graphene, the absorption is tunable via a gate voltage, providing dynamic control of the energy transmission. We show that this strongly enhanced absorption arises due to a coupling between light and a fast wave-mode propagating along the graphene/metamaterial hybrid.

solved using a transfer matrix approach that accounts for the presence of graphene through boundary conditions imposed on the magnetic field. We discuss the behavior of the absorption in terms of the energy flow throughout the entire structure.
Our main result is that by tuning E F , we are able to determine experimentally relevant regimes where near perfect absorption can occur over an extremely wide range of incident angles for an ENZ metamaterial layer that has a small component of the permittivity tensor. We show that the enhanced absorption occurs due to coherent perfect absorption which couples light to a fast wave propagating along the graphene/ENZ hybrid.

Theory
The principal coordinate axes are chosen so that both the permittivity and permeability tensor in the anisotropic medium are both diagonal. A diagram of our setup is shown in Fig. 1. For concreteness, we focus on p-polarized incident light (E x , E z , B y ) ≠ 0. In the anisotropic medium (region 2), the nonzero components to the tensors are (ε x,2 , ε z,2 , μ y,2 ). It is backed by a perfect electric conductor (PEC) in region 1, where no electric or magnetic fields exist. The finite surface conductivity σ of graphene will be effectively taken into account in the boundary condition for the magnetic field H at the interface between air (region 3) and the anisotropic medium, where 26  Thus, at a temperature T of 300 K, and THz frequencies, we have k B T = 25 meV, and ħω ~ 1 meV, meaning doping levels should exceed E F = 100 meV when using Eq. (2). Here, E F is the Fermi energy, ω is the frequency of the EM wave, and t rel is the relaxation time. For frequencies in the THz region and lower, Eq. (2) reflects that the interband part to the conductivity is negligible compared its the intraband part, and consequently to the Drude-like behavior for σ .
We now consider Maxwell's equation (in SI units) in the incident air region and in the anisotropic medium, both having no free charges or currents. For p-polarized incident light and diagonal tensors, we obtain the following set of equations determining the components of the fields when the time-dependence is harmonic (e −iωt ): These equations are solved with a plane-wave ansatz for the field components where ∝ E e k x i x and k x is conserved due to translational invariance. The Methods section contains additional details of the numerical and analytical solutions of the above equations.
Once the equations have been solved, one may compute the absorption  from the scattering coefficients: A 2 and B 2 are associated with the transmitted and reflected electric field wave in region 2 whereas B 3 ≡ r is the coefficient for the reflected wave in region 3. Since there is no transmission into the PEC layer, the absorbance  can be computed from the incident wave after subtracting the reflected part: = − r 1 2  . The reflection coefficient r is obtained as where we have defined The coefficients for the other waves are given as: , and B 2 = − A 2 . We are now interested to see how the presence of the graphene layer modifies the absorption properties of the metamaterial, and in particular determine the influence of the gate-controlled Fermi level which is tunable in situ. The analytical expression for the reflection coefficient r allows us to estimate which order of magnitude E F should have in order to have any influence on the results. If the terms proportional to the conductivity σ are to be comparable in magnitude with the other terms, we see that e.g. ε x,3 ω ~ k z,3 σ needs to be satisfied. Since we are considering an EM wave incident from air, ε x,3 = ε z,3 = ε 0 . Inserting the expression for σ , one finds that the above equation corresponds to a Fermi level in graphene of For grazing incidence θ ~ π /2, the graphene layer thus has no influence for any realistic values of E F . We also note that when σ → ∞ , r → 1 such that the absorption vanishes,  → 0. In order to gain information about the spatial distribution of the power flow in the structure, we also compute the Poynting vector S = E × H. We write E = E 0 e −iωt and H = H 0 e −iωt . For future use, we define the normalization constant for the power flow as the time-averaged incident power at z = τ : S 0 = ε 0 c/(2 cos θ ).
By appropriately tuning E F , graphene-dielectric stacks can exhibit a wavevector dispersion that traces out a hyperbolic contour at a given frequency 21,23 . Such hyperbolic metamaterial (HMM) systems are described by a permittivity tensor whose principle components are opposite in sign 49 , and allow for the propagation of evanescent modes in subwavelength imaging 29 , and can be designed to exhibit substantial near-field absorption 22 . Such structures are often nonresonant and possess low loss 50 . There have been a variety of HMM systems that have been fabricated [52][53][54] , including semiconductor hybrids, metallic layers 52 , and silver nanowires 54 . Gate tunable HMM systems have also been made that allow control of their optical response 53 . Here, we will determine how the absorption properties of HMM structures are modified by the presence of a graphene layer over a broad range of angles θ and Fermi energies E F . Specifically, we focus on gate-tunable HMM structures in the ENZ limit, whereby the tensor components ε x,2 and ε z,2 are opposite in sign. We set μ y,2 = μ 0 , and the relaxation time in graphene to τ = 10 −13 s. For incident light in the THz regime, with E F = 100 meV and λ = 1.6 mm, this leads to a surface con- We first consider the standard Drude-like frequency response in the metamaterial, such that Here, α = λ /λ z and f = 0.02, where λ z is the characteristic wavelength, with λ z = 1.6 mm. Designing a metamaterial within effective medium theory to possess the prescribed dielectric response in the THz regime can be challenging with current fabrication and materials capabilities. It may be possible however to design a multilayer dielectric structure or rod lattice 55 with polaritonic components to achieve the desired HMM response since the permittivity of polaritonic materials mimic metals in the optical regime 56,57 . Therefore construction of an anisotropic HMM that has a longitudinal ENZ response should be possible at THz frequencies.
It will also be of interest to see how the absorption properties depend on the sign of ε z,2 , determined by the frequency of the incident wave, and also its magnitude. We set λ λ  z , so that ε  0 z ,2 . If we take the limit ε x,2 → 0, the reflection coefficient simplifies to In the absence of graphene (σ = 0), we have zero absorption since |r| 2 = 1. However, when graphene is present (σ ≠ 0) an interesting opportunity arises. It is seen from the above equation that r = 0, meaning perfect absorption =1 when the following condition is satisfied: In the low THz regime and below (ω ≪ 10 THz), we have ω ≪ τ and full absorption takes place for angles  θ satisfying θ = . .
F  Although this equation represents the criterium for 100% absorption, we show in the next section that the absorption in fact is close to 100% for an extremely wide range of incident angles.

Results and Discussion
To establish how perfect absorption can be achieved, we discuss how the incident beam can couple to the intrinsic EM modes of the system. Our approach involves finding the poles of the reflection coefficient, Eq. Deviations from  θ introduce reflected waves which shift the net energy flow in the air region. Additionally, for r to be single-valued, the appropriate branch cuts at the points ± k 0 need to be made for k z, 3 . When examining the perfect absorption phenomena, we utilize a simple yet useful property of the reflection coefficient: Eq. (8) reveals that interchanging the direction of the incident and reflected waves, transforms r into its reciprocal: with k z,3 = k 0 cos θ . This property simplifies the solution process since finding a complex pole, related to − k z, 3 , is equivalent to finding the perfect absorption modes for k z,3 59,60 . We underline that the absorption occurring in this way is an interference phenomenon and not based on losses.
We now introduce more possibilities for enhanced absorption, and investigate anisotropic systems where the permittivity components perpendicular and parallel to the interfaces are of opposite sign. The corresponding HMM dispersion relations are then identified as either type I if ε x,2 > 0 and ε z,2 < 0, or type II if ε x,2 < 0 and ε z,2 > 0. The operating wavelength of the incident beam is tuned around λ z to control the sign of ε z,2 . The component of ε parallel to the interface always has the same magnitude, but can differ in sign according to ε x,2 = (± 4 + 0.1i)ε 0 .
Extremely wide-angle absorption for type-I and type-II anisotropic structures is demonstrated clearly in Fig. 2, where we provide results for these two cases for different thicknesses of the metamaterial and values of the Fermi level. The difference between Type I and Type II for a fixed thickness primarily occurs around normal incidence, θ = 0, where the absorption is much higher in the Type I case. It is seen that when the thickness is close to the wavelength [τ /λ = 0.85 in panels (e) and (f)], the presence of graphene provides very high absorption for a remarkably wide range of angles. This should be contrasted to the case without graphene (dashed line), where the absorption peaks for an angle θ < 45° and then rapidly declines. For sufficiently small metamaterial thicknesses, and smaller angles, the metamaterial layer becomes insignificant and the EM wave interacts with mainly the graphene sheet and metal backing. Tuning E F is seen to improve the absorption dramatically not only in magnitude, but also in terms of which angles of incidence that become absorbed. Remarkably, the absorption coefficient can be tuned to be nearly perfect, for an unusually broad range of angles. This is distinct from a scenario involving conventional dielectrics where isolated absorption peaks can appear at specific angles of incidence.
To pinpoint the mechanism of the observed perfect absorption, the mode characteristics of the system should be identified. By correlating the longitudinal wavevector component k x of the incident EM wave with the permitted EM modes of the graphene-based ENZ structure, the critical coupling responsible for perfect absorption can be identified. Since k x /k 0 < 1, and ℑ m(k x ) = 0, we search for the allowed fast-wave EM modes (whereby the phase velocities exceed c) that the structure supports 51 . By solving the transcendental equation R + = 0 [Eq. (5)], we find the permitted incident wavevector components that yield  = 0. The obtained k x , are converted to a critical coupling angle via θ = arcsin(k x /k 0 ), giving the necessary incident beam angles for fast-wave mode coupling to occur. To illustrate this, in Fig. 3 the perfect absorption curves (solutions to R + = 0) are shown as a function of E F and θ . The wavelength considered λ = 1.01λ z corresponds to a negative real part of ε z,2 that is also near zero. In (a) ε x,2 = (4 + 0.1i)ε 0 corresponding to a type I HMM, while in (b) both permittivity components are of the same sign, since now ε x,2 = (− 4 + 0.1i)ε 0 . The dispersion curves are shown together with density plots of the absorption , which gives a global view of the high absorption regions spanned by E F and θ . Clearly the fast-wave dispersion curves overlap with the regions of perfect absorption. A broader range of incident angles leading to perfect absorption is found in the graphene/HMM structure (a) compared to and we set the characteristic wavelength of the metamaterial dispersion to λ z = 1.6 mm.
the same sign. It is evident that incident angles corresponding to perfect absorption occur when a fast-wave mode is excited in the structure, due to the coherent superposition of waves with k x below the light line. These modes are distinct from conventional guided wave modes that have k x /k 0 > 1, and consequently the EM fields would be evanescent in the air region near the graphene layer. They also differ from leaky wave modes 58 that dissipate when propagating along x since in that case, ℑ m{k x } ≠ 0. Therefore, by coupling the incident beam to fast wave modes of the structure, the enhanced absorption features provided by the HMM in the ENZ regime 61 , complimented with the tunable E F afforded by graphene, generates a broad range of angles for perfect absorption to arise.
The behavior of the energy flow is exhibited in Fig. 4, where the Poynting vector components S x and S z are shown within the air (z/τ > 1) and metamaterial (z/τ ≤ 1) regions for the structure considered in Fig. 3(a). It is evident that magnitudes of S x and S z are largest near the interfaces, except for near normal angles of incidence, whereby the energy flows along the metal backing. The flow of S x in the metamaterial is opposite to the incident beam, demonstrating negative refraction that arises from the small negative permittivity component ε z,2 . The perfect absorption features seen in Fig. 3(a) are also consistent with Fig. (d) for θ = 41° where the energy flow is maximal in the vacuum region at this angle, indicating purely downward energy flow with no upward contributions from surface reflections.

Methods
We here provide some details for how to obtain the solution of the Maxwell equations in the presence of graphene. The wave-equations for E x and E z take the same form in region j = 2, 3: The dispersion relation then reads: . An electric field propagating in region j is then written: = + . The amplitudes are related via: , Using this relation, we may now write down the solution of the electric field in region j which takes into account both transmitted and reflected waves: where we defined κ j = k x ε x,j /k z,j ε z,j . Since the light is incident from air, we take A 3 = 1. The magnetic field is obtained as =Ĥ H y j yj , where: y j There are three coefficients to be determined: {A 2 , B 2 , B 3 }. To do this, we need to specify the boundary conditions at the air/anisotropic medium interface (z = τ) and the anisotropic medium/PEC interface (z = 0). Since the tangential component of E is always continuous, it follows that ⋅ | = ⋅ | . The presence of the graphene layer now enters in the boundary condition for the tangential component of H by writing the free current as J f = σ E: . Specifically, we find that the boundary conditions may be written in the form  where we defined ζ = ε ± σ ±k / x z ,2 ,2 . Thus, the coefficients are solved by a = M −1 b, which in turn fully determines E and H throughout the entire structure.
We also provide an explicit expression for the time-averaged Poynting vector = × ⁎ S E H Re{ } 1 2 in the metamaterial region 2 is: We also examine the effects of loss on the absorption properties of the graphene-based metamaterial. In Fig. 6, we consider the same system that was studied in Fig. 2 except now substantial losses are included whereby the component of the permittivity orthogonal to the interface, ε z,2 , has a large imaginary component. Although the wide-angle absorption properties are seen to be reduced in the type-II case, the top panels (b) and (c) reveal that remarkably for the type-I HMM, wide-angle absorption remains intact when considerable losses are present.

Figure 6. Absorption vs. angle of incidence for the Type I case (top row) and Type II case (bottom row).
Here we consider the high loss regime where Im{ε z,2 }/ε 0 = 1, while all other parameters are the same as those used in Fig. 2.