## Abstract

A tunable multifunctional modulator of the stacked graphene-based hyperbolic metamaterial (HM) cells is proposed. The dielectric constant and group index of HM are theoretically investigated. The calculated results show that, for the cell structure, a transmission window in the reflection zone (TWRZ) can be obtained at the normal incidence, but all reflections are converted to the transmission when the incident angle is near 82°. Concurrently, a single frequency absorption in the transmission zone (SFATZ) is realized, which can be adjusted by the chemical potential of graphene. For the whole structure composed of cell structures with different chemical potentials, the ultra-wideband absorption and transmission window in the absorption zone (TWAZ) can be achieved, which can work in different frequency bands if the given structural parameters can be tailored. Those computed results can apply for switchable frequency-dependent and angle-dependent reflection-transmission modulations, single frequency and ultra-wideband absorbers, and a logic switch based on the TWAZ.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

Graphene [1–3], like a honeycomb lattice, is a two-dimensional carbon nanomaterial with a single-layer carbon atom thickness. The graphene was first separated from graphite in 2004 [4], whose excellent electrical and optical properties applied to light modulators [5,6], photodetectors [7,8], and saturated absorbent [9,10] have attracted the interest of researchers.

Unfortunately, the single-pass absorptivity of the monolayer undoped graphene is about 2.3% in the wider band, which is related to the fine structure constant [11,12]. Therefore, the enhanced graphene absorption [13–15] has become a critical research topic. The absorption enhancement of monolayer graphene in the near-infrared was demonstrated numerically by Chen et al. [16]. Their work indicates that the absorption of monolayer graphene can be modulated substantially by bias voltage to make the interband transition of monolayer graphene close to magnetic dipole resonance, and the electrical switching effect can be realized. Different from the single-layer graphene, graphene films with large area possess unusual low-energy electronic structure, whose optical properties vary with their thickness. The band gap of graphene can be adjusted from 0 to 0.25 eV when a voltage is applied to the field-effect transistor of dual gate double-layer graphene at room temperature [17]. Lin et al. [18] experimentally proved a 12.5 cm^{2}, 90-nm-thick graphene metamaterial with an unpolarized absorption rate of about 85% and a spectral range of 300–2500 nm. These results have opened up a new way for the application of photonic devices with large-area strong absorption based on two-dimensional materials.

With the development of nanofabrication technology [19,20], optical metamaterials [21–25] provide more possibilities for controlling the transmission of electromagnetic waves. Hyperbolic metamaterials (HMs) [26–28] highly anisotropic artificial composite electromagnetic materials can be obtained by mixing and superimposing metal [29,30] and graphene [31,32] with a dielectric substrate, which shows a hyperbolic dispersion relation for specific electromagnetic waves. In the terahertz (THz) and near-infrared frequency regions, the unique near-field manipulation characteristics of electromagnetic wave for HMs enable a great application prospect in the image of sub-wavelength [33,34], improvement of photon density [35–37], and enhancement of spontaneous radiation [38,39]. A broadband THz absorption material composed of a graphene-dielectric multilayer frustum pyramids array was designed by He et al. [40]. The multilayered graphene can be regarded as a homogeneous metamaterial with hyperbolic dispersion and anisotropic dielectric constant. They explained that the absorption of electromagnetic waves at different frequencies is due to the compression effect of slow waves [41,42] in the tapered waveguide, and the surface plasmonic waves are excited. Kieliszczyk et al. [43] proposed the possibility of forming the reflectivity characteristics of tunable HMs. The existence of spectral and spatial functions of narrow-band filters and edge dynamically controlled in the spectral range of 3–5 µm is proved.

In recent years, the use of nonlinear materials (BBO [44,45], LiNO3 [46,47]), and physical effects (Kerr [48,49], Pockels [50,51]) for switch and modulation [52–54] has become a new research hotspot. Du et al. [55] focused on the nonlinear saturation absorption of ReSe2. The few-layer ReSe2 was prepared by mechanical exfoliation method, and its nonlinear optical response with ambient stability and all-optical adjustable potential were verified at 1.55 µm. A switchable reflection modulator based on HMs with graphene was presented by Pianelli et al. [56]. Theoretical analysis showed that adjusting the chemical potential of graphene from 0.2 eV to 0.8 eV resulted in a blue shift of reflectivity up to 2.3 µm. In the transverse electric (TE) / transverse magnetic (TM) modes, there is a blue/red-shift and reflectivity switch at different incident angles. The proposed reversible tunable graphene-based HMs can be used in the next generation of nonlinear switchable devices in the mid-infrared range. Employing an all-fiber Michelson interferometer, Wang et al. [57] constructed the few-layer bismuthene structure and verified its all-optical phase and intensity modulation with the excellent photothermal effect firstly. There are many interesting reports on the phase, high-order harmonic, and transmission modulation. However, the coexistence of absorption and transmission has always been a bottleneck that caused by the properties of materials. In this paper, a novel method to realize controllable multi-function absorption-transmission modulation is proposed.

Here, the characteristics of the stacked HMs with graphene cells are studied and a fresh multifunctional modulator [58] is demonstrated. The frequency-dependent and angle-dependent modulations and SFATZ can be realized in the unit cell structure, and the ultra-wideband absorption and TWAZ can be achieved in the whole stacked structure. Besides, the effects of the incident angle on the reflection-transmission modulation and the chemical potential on the SFATZ and TWAZ are also discussed systematically. The proposed multifunctional modulator can be used as a THz waveguide with reflection-transmission and absorption-transmission switch, optical shield.

## 2. Theoretical model and method

The cell structure of the proposed HMs with graphene is schematically displayed in Fig. 1. The setting of TM wave and coordinate axis can be found in Fig. 1. *θ* is the incident angle and the background medium is air. A and B represent layers with refractive indices of *n*_{A} = $\sqrt {\textrm{51}} $ and *n*_{B }= 1, whose thicknesses are written as *d*_{A }= 2 µm and *d*_{B }= 8 µm, respectively. HG represents HM consisting of the common dielectric C (*ɛ*_{C }= 11.1) and graphene, and their thicknesses are written as *d*_{C} = 60 nm, *d*_{G }= 1 nm. The proposed structure is arranged in the order of ‘A-HG-B’, and it's period constant is *N *= 3. The thicknesses of dielectrics A, B, HG are optimal values.

For the cell structure of the proposed HMs with graphene, the reflectance and transmittance can be calculated by the transfer matrix method [59]. The whole derivation process is as follows. Graphene is a frequency-dependent medium, whose surface conductivity *σ* can be expressed as the sum of the interband *σ*_{inter} and the intraband *σ*_{intra} [56]:

*e*denotes the electron charge,

*k*

_{B}represents the Boltzmann constant,

*T*describes the temperature that we set the value as

*T*= 300 K, $\hbar $ indicates the reduced Plank constant,

*ω*refers to the angular frequency of the incident electromagnetic wave, and

*τ*signifies the phenomenological scattering rate.

*µ*is the chemical potential, which is a correlation with the applied voltage

*V*and can be written as [56]: where

_{g}*v*corresponds to the Fermi velocity of graphene and its value is set as 10

_{F}^{6}m/s.

*a*

_{0}is an empirical constant and

*a*

_{0 }= 9·10

^{16}m

^{−1}V

^{−1}.

*V*belongs to the offset bias voltage and it is assumed to be 0 V. Assuming that the electronic band structure of the graphene sheet is not affected by adjacent materials, the effective dielectric constant of the graphene can be expressed as [56]: where

_{D}*ɛ*

_{0}and

*d*

_{G}represent the dielectric constant of the vacuum and the thickness of the graphene, respectively.

Graphene-based HMs are anisotropic mediums and possess uniaxial dielectric tensor components, which can be approximately written as [56]:

*θ*and can be expressed as [60]:

The transfer matrix can be written as [59]:

Where *η _{j }*=

*n*cos

_{j}*θ*,

_{j}*δ*=

_{j}*ωn*cos

_{j}d_{j}*θ*/c,

_{j}*n*indicates the refractive index of the layer

_{j}*j*.

*d*and

_{i}*θ*signify the thickness and the incident angle of the layer

_{i}*j*, respectively.

The transmission matrix of the designed HM structure based on graphene can be written as [59]:

Finally, the reflection coefficient *r* and transmission coefficient *t* can be obtained as [59]:

Where *η*_{0 }= *n*_{0}cos*θ*_{0} (TE wave), *η*_{0} = *n*_{0} /cos*θ*_{0} (TM wave), *n*_{0} describes the refractive index of air. The absorptance *A*(*ω*) can be written as

*R*(

*ω*)=|

*r|*

^{2}and transmittance

*T*(

*ω*)=|

*t|*

^{2}.

## 3. Results and discussion

#### 3.1 Properties of the dielectric constant and group index for the proposed HMs with graphene

The real parts ${\varepsilon _ \bot }$ and ${\varepsilon _{||}}$ of the proposed HMs based on the graphene with different frequencies are presented in Fig. 2. The designed structure parameters are *d*_{C} = 60 nm, *τ *= 5·10^{−12} s, *µ *= 0.1 eV. It can be seen from Fig. 2 that the real of ${\varepsilon _ \bot }$ beyonds 0 in the whole frequency domain and the real of ${\varepsilon _{||}}$ changes from negative to positive at 7.07 THz. It is well known that the material exhibits hyperbolic properties at $Re({\varepsilon _ \bot })Re({\varepsilon _{\textrm{||}}}) < 0$ and c3onventional medium properties at $Re({\varepsilon _ \bot })Re({\varepsilon _{\textrm{||}}}) > 0$. One can observe from Fig. 2 that material properties alter from hyperbolic medium to ordinary medium at 7.07 THz.

The group index *n*_{g} of the HMs with graphene is shown in Fig. 3. It can be found that both the real part and imaginary part of *n _{g}* have extreme value points, the former is a maximum value at 7.08 THz, and the latter is a minimum value at 7.06 THz. These noted frequency points play a critical role in controlling the transmission modulation of electromagnetic waves.

To investigate the effects of the phenomenological scattering rates *τ*, chemical potential *µ*, thickness of dielectric C *d*_{C} on the performance of HMs properties, the relationships between three parameters and *n*_{g} are given in Figs. 4–6, respectively.

The real part and imaginary part of *n _{g}* with different phenomenological scattering rates of

*τ*= 10

^{−12}s,

*τ*= 10

^{−13}s, and

*τ*= 10

^{−14}s are plotted in Fig. 4. It can be observed from Figs. 4(a) and (b) that the maximum value of the real part of

*n*becomes smaller and shifts to the higher frequencies when

_{g}*τ*is decreased, while the minimum value of the imaginary part of

*n*becomes larger and moves to the lower frequencies, which indicates that the phenomenological scattering rate can markedly affect the properties of HMs.

_{g}The real part and imaginary part of *n _{g}* with different chemical potentials and thicknesses of dielectric C are plotted in Fig. 5 and Fig. 6, respectively. It can be noticed that increasing the value of

*µ*and

*d*

_{C}can significantly affect

*n*, and all the extreme points of

_{g}*n*are blue-shifted, which shows that

_{g}*µ*and

*d*

_{C}can adjust the properties of HMs as well as

*τ*and should be optimized in structural design.

#### 3.2 Features of the cell structure for the proposed HMs with graphene

The reflectance, transmittance, and absorptance diagrams of TM wave for the proposed cell structure at *θ *= 0° and *θ *= 82° are illustrated in Fig. 7. The designed structure parameters are *d*_{C} = 60 nm, *τ *= 5·10^{−12} s, *µ *= 0.1 eV, *N *= 3. In Fig. 7, the blue dash-dot line, red dash line, and dark solid line denote the curves of transmittance, reflectance, and absorptance, respectively. In Fig. 7(a), the TWRZ can be found that a transmission band (the transmittance beyond 0.9) is located at 9.74 - 10.78 THz, and the reflection bands (the reflectance beyond 0.9) are situated at 4 - 9.40 THz and 11.16 - 16 THz, except for an absorption peak at 7.10 THz. The phenomenon of absorption peak can be explained by the strong absorbing properties of the graphene. Combined with the results in Figs. 4–6, it can also be seen that the absorption peak appears near the frequency point of the extreme value of *n _{g}*, so the position of absorption frequency point can be controlled by adjusting

*τ*,

*µ*, and

*d*

_{C}. These simulated results show that the designed structure can be used as a frequency-dependent switchable reflection-transmission modulator and a tunable single-frequency absorber. In order to discuss the angle characteristics of the designed structure, the reflection, transmission, and absorption diagrams of TM wave at

*θ*= 82° are described in Fig. 7(b). It can be noticed from Fig. 7(b) that the designed structure exhibits transmission characteristics in the whole frequency domain, except for the absorption peak at 7.05 THz and the reflection peak at 7.09 THz when

*θ*= 82°, which means that the designed structure can be applied as an angle-dependent switchable reflection-transmission modulator with the function of reflected at a small angle and transmitted at a large angle.

The reflectance diagrams of TM wave for the designed structure under different angles *θ* are placed in detail in Fig. 8. It can be found that there are obvious angular stable reflection region (*RR*) and absorption region (*AR*) in the range of 0° - 50°, and the transmission region (*TR*) becomes wider slightly with the increase of *θ*, which indicates that the frequency-dependent reflection-transmission modulator possesses angular stability within a certain range. In Fig. 8(b), it can be observed that when *θ* is 80° - 85°, there is a *TR* with frequency-independent, except the reflection and absorption region (*RAAR*) around 7 THz, and we can state that the angle-dependent reflection-transmission modulator can be achieved within a specific angle.

The phenomenon of broadband transmission can be explained by Brewster’s angle *θ*_{B}. Derived from Brewster’s law, we can understand that the TM wave will be transmitted by the structure at *θ*_{B}, and the TE wave will be reflected. *θ*_{B} can be written as

*n*

_{1}and

*n*

_{2}represent the refractive indices of dielectrics. The incident direction is set from medium 1 to medium 2. In this work, the background medium and dielectric B are air. Because the thickness of HMs layer is thin, the dielectric constants of dielectric A and the HMs can be equivalent to an effective dielectric constant [60].

*ɛ*

_{A }= 51,

*ɛ*

_{HG}and

*d*

_{HG}represent the relative permittivity and thickness of HMs, respectively and

*d*

_{HG}= 61 nm. It can be viewed from Figs. 5 and 6 that the real part of

*n*is around 0 - 5 except the maximum value. Suppose

_{g}*ɛ*

_{HG}= 6.25, we can get

*ɛ*≈ 49.68 and

_{eff}*θ*= arctan ($\sqrt {49.68} $/ 1) ≈ 82°. Since

_{B}*ɛ*is equivalent and

_{eff}*n*of HMs is frequency-dependent, there is a broadband transmission around 82°. To sum up, the broadband transmission can be realized reasonably for the designed cell structure.

_{g}#### 3.3 Analysis of the SFATZ

The SFATZ can be found in Fig. 7, while a reflection defect is undesirable. After parameter optimization, a perfect filter can be obtained in Fig. 9. The parameters are *d*_{C} = 60 nm, *τ *= 5·10^{−12} s, *µ *= 0.04 eV, *N *= 3.

In Fig. 9, a high absorption peak can be observed at 4.89 THz (the absorptance is greater than 0.9), and the cell structure shows transmission characteristics in the whole frequency band. To verify the correctness of the simulated results, the electric field distributions of the absorption frequency point (4.89 THz) and transmission frequency point (12 THz) are also given in Fig. 9. One can clearly see from the electric field distribution of 4.89 THz that the color of the electric field in the last few layers turns to dark blue, which means that the energy of the electromagnetic wave is basically absorbed. For the electric field distribution of 12 THz, there are different phenomena. The color of the electric field is light blue in the last few layers, which indicates that the electromagnetic wave passes through the cell structure. In addition, the interference pattern can be observed in Fig. 9, which can be interpreted as the incident wave and reflected wave redistribute the light intensity to form interference. Because the incident angle is large and the component of the wave vector on the *x*-axis possesses a large value, there will be obvious an interference pattern in the *x*-axis direction in Fig. 9. According to the analyses of Fig. 9, the obtained results are consistent. Finally, we can conclude that a single frequency absorption in the transmission band (SFATB) can be realized in the proposed cell structure.

To discuss the effect of chemical potential *µ* on the SFATB, the transmittance and absorptance diagrams of TM wave at different *µ* are plotted in Fig. 10.

The parameters are *d*_{C} = 60 nm, *τ *= 5·10^{−12} s, *θ *= 82°, *N *= 3. Figures 10(a)-(d) are the transmittance and absorptance curves of the proposed cell structure with different *µ* of 0.01 eV, 0.03 eV, 0.05 eV and 0.07 eV, respectively. It can be noticed from Fig. 10 that if *µ* increases, the position of the absorption peak will move towards the higher frequencies, and these absorption frequencies are located at 3.79 THz, 4.55 THz, 5.25 THz, and 5.98 THz, respectively. The calculated results show that *µ* plays a vital role in adjusting SFATB, which can greatly change the position of the absorption peak.

#### 3.4 Realization of the ultra-wideband absorption and TWAZ

The position of the absorption peak can be controlled by *µ*. Suppose that we can discretize a segment of *µ* closely, and assign them to different cell structures, and splice cell structures together, so that the continuous absorption peaks can be obtained. In order to verify this hypothesis, the cell structures with different *µ* are stacked. The overall structure is shown in Fig. 11. Here, *µ* is adopted. *µ* can be controlled more conveniently than *τ* and *d*_{C}, and only the applied voltage *V _{g}* needs to be changed.

Two kinds of devices with ultra-wideband absorption in different frequency bands are designed by using the proposed stacked structure. The simulation results are shown in Fig. 12. For Fig. 12(a), the structure parameters are *d*_{A }= 2 µm, *d*_{B }= 8 µm, *d*_{C} = 60 nm, *τ *= 5·10^{−12} s, *θ *= 82°, *N *= 2. *µ _{n}* is treated as the discrete value of

*µ*, which is expressed as

*µ*= (0.001 + 0.001×(

_{n }*n*- 1)) eV (

*n*= 1 - 120, and

*n*is an integer). The value of

*µ*changes from

_{n}*µ*= 0.001 eV to

_{n }*µ*= 0.12 eV, and the step length is 0.001 eV. It can be observed from Fig. 12(a) that the absorption band is situated at 3.95 - 7.72 THz, whose bandwidth is 3.77 THz and its relative bandwidth (RB) is 32.31%, which is an ultra-wideband absorption. For Fig. 12(b), the structure parameters are

_{n }*d*

_{A }= 0.5 µm,

*d*

_{B }= 9 µm,

*d*

_{C}= 60 nm,

*τ*= 5·10

^{−12}s,

*θ*= 82°,

*N*= 2.

*µ*is written as

_{n}*µ*= (0.12 + 0.0019×(

_{n }*n*- 1)) eV (

*n*= 1 - 158, and

*n*is an integer). The value of

*µ*converts from

_{n}*µ*= 0.12 eV to

_{n }*µ*= 0.4183 eV, and the step length is 0.0019 eV. It can be noticed from Fig. 12(b) that the absorption band covers 7.89 - 14.65 THz, and its bandwidth and RB are 6.76 THz and 29.99%, respectively. It is also an ultra-wideband absorption. According to the results of Fig. 12, it can be concluded that an ultra-wideband absorption can be realized in the proposed stacked whole structure, and it can work in different frequency bands by adjusting structural parameters.

_{n }Furthermore, we can also reprocess the above-stacked structure. Suppose that a part of the discrete *µ* is set as the *µ*_{1} of the first cell structure, and the original absorption band of this part will act on the absorption frequency point of the first cell structure at the same time, and then the original absorption of this part will be converted to transmission, and the transmission window in the absorption zone (TWAZ) will be obtained in the proposed multi-stack structure. The simulation results are shown in Fig. 13.

Two different transmission windows in the ultra-wideband absorption band (3.95 - 7.72 THz) of Fig. 12(a) are designed, as shown in Fig. 13. For Fig. 13(a), the structure parameters are *d*_{A }= 2 µm, *d*_{B }= 8 µm, *d*_{C} = 60 nm, *τ *= 5·10^{−12} s, *θ *= 82°, *N *= 2. The partial *µ _{n}* can be expressed as

*µ*= (0.001 + 0.001×(

_{n }*n*- 1)) eV (

*n*= 1 - 25, 60 - 120). The step length is 0.001. The remaining

*µ*can be expressed as

_{n}*µ*= 0.001 eV (

_{n }*n*= 26 - 59). One can observe from Fig. 13(a) that the transmission window is laid at 4.59 - 5.33 THz, whose bandwidth is 0.74 THz and RB is 7.46%.

For Fig. 13(b), the structure parameters are *d*_{A }= 2 µm, *d*_{B }= 8 µm, *d*_{C} = 60 nm, *τ *= 5·10^{−12} s, *θ *= 82°, *N *= 2. The integrant *µ _{n}* can be written as

*µ*= (0.001 + 0.001×(

_{n }*n*- 1)) eV (

*n*= 1 - 55, 90 - 120). The step length is 0.001. The rest

*µ*can be written as

_{n}*µ*= 0.001 eV (

_{n }*n*= 56 - 89). It can be seen from Fig. 13(b) that the transmission window is seated at 5.70 - 6.36 THz, and its bandwidth is 0.66 THz and the RB is 5.47%. From the above results, it can be known that the TWAZ can be achieved in different frequency bands in the proposed multi-stack HMs with graphene.

## 4. Conclusions

In short, a periodic cell structure of graphene-based HM is constructed and the whole structure is designed by stacking the cell structures with different *µ*. The properties of the frequency-dependent and angle-dependent reflection-transmission switch, ultra-wideband absorption, SFATZ, and TWAZ are theoretically calculated, and the effects of *µ*, *τ*, and *d*_{C} on the *n _{g}* of HM are also investigated. Numerical simulations indicate that the TWRZ can be found when

*θ*is 0° - 50°, and the reflection alters to transmission in the whole frequency domain at

*θ*= 80° - 85°. The SFATB can be regulated by

*µ*, whose absorption peak is blue-shifted as

*µ*is increased. The ultra-wideband absorptions (3.95 - 7.72 THz, 7.89 - 14.65 THz.) and TWABs (4.59 - 5.33 THz, 5.70 - 6.36 THz) can be realized in the stacked unit cells by assigning different

*µ*functions. The proposed stacked graphene-based HM cells can be employed to design the tunable multifunctional modulators including reflection-transmission modulation with frequency-dependent and the angle-dependent, absorption-transmission switches, and filters.

## Funding

Postgraduate Research & Practice Innovation Program of Jiangsu Province (KYCX20_0807); Open Research Program in China’s State Key Laboratory of Millimeter Waves (K201927); Shenzhen Science and Technology Plan Basic Research Project (JCYJ20180305164708625).

## Disclosures

The authors declare no conflicts of interest.

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