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Photo-induced enhanced negative absorption in the graphene-dielectric hybrid meta-structure

Lan Wang, Sen Gong, Yaxin Zhang, Zezhao He, Cui Yu, Xilin Zhang, Ting Zhang, Hongxin Zeng, Wei Kou, Yuncheng Zhao, Qiye Wen, Lan Feng, Yubing Gong, and Ziqiang Yang

Open Access Open Access

Abstract

Recently, the negative absorption in graphene-based metamaterials became a very attractive direction of THz electronic devices. Here we propose a graphene-dielectric hybrid meta-structure to realize photo-induced enhanced negative absorption in the THz regime, which results from strong graphene-light interaction. The negative absorption is derived from the degradation of the conductivity of graphene under optical pump. Meanwhile, the graphene-dielectric hybrid meta-structure introduces dispersion relation and resonance mode, which can couple with the incident wave to construct a strong resonance. In this case, both the dispersion of the propagating waves and resonance are contributed to the graphene-light interaction and enhance the negative absorption, in which the resonance coupling determines the distribution of negative absorption, and the maximum is dominated by dispersion. More importantly, compared with the previous work, the negative absorption is increased by nearly 100 times by adopting this meta-structure.

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

1. Introduction

Graphene is a unique two-dimensional (2D) material which exhibits extraordinary electrical, optical and mechanical properties, and diverse novel phenomena is demonstrated and reported [1–4]. In the terahertz regime, the graphene has been employed in different kinds of devices, including ultrafast photodetectors [5–7], transparent electrodes [8,9], optical modulators [10–12], lens [13,14] and so on. These devices have shown enhanced performance compared with those in conventional semiconductors because of its high carrier concentration and electron mobility.

Recently, some published results present that under external illumination the linear dispersion of 2D Dirac fermions of graphene leads to negative photo-induced conductivity, which could be connected with generation of hot electrons, enhanced carrier-carrier scattering, and enhanced carrier scattering with lattice vibration [15]. T. Otsuji et al. reported that stimulated emission of THz photons came from the fs-IR laser pumped the single-layer graphene within a picosecond time scale after pumping, which was induced by fast relaxation and relatively slow recombination dynamics of photo-generated electrons and holes [16]. Many other works also provide evidence of the stimulated emission from graphene and degradation to negative about graphene THz conductivity under optical or electrical excitation [16–28]. The remarkable property of graphene has led to growing interest in applying this material in optical gain and negative absorption, which is significant for metal loss compensation in THz regime. It was reported in 2012 that the incident terahertz waves are amplified by the pumped graphene sheet located in sub-wavelength metal hole array [21]. In 2017, Pai-Yen Chen proposed PT-symmetric metasurfaces that simultaneously offer terahertz optical gain and bio-sensing functions [29,30]. After that, they predicted a resonant optical gain in population inverted bilayer graphene [4]. In 2019, Costas M. Soukoulis demonstrated an ultrafast graphene-based absorption modulator through negative photoinduced conductivity in experiment [15]. The unique band structure of graphene provides the possibility of negative absorption. However, the ability to further improve the performance of devices is restricted by the ultra-thin nature of the monolayer graphene with limited light-graphene interaction length making the negative absorption is unobtrusive.

Multilayer graphene-based meta-structure may be an effective way to settle this problem [31–36]. Due to the distinctive dielectric permittivity tensors, the atomic architecture and the controlled excitation, the multilayer meta-structure which consists of multiple layers of patterned or unpatterned subwavelength metal/two-dimensional materials have been proposed to apply in broadband absorber [37], on-chip photonic devices [38], and directional light emitter [39]. Early in 2013, graphene-based tunable hyperbolic metamaterial was used to enhanced near-field absorption [32].

In this work, the graphene-dielectric hybrid meta-structure has been proposed to achieve the photo-induced enhanced negative absorption. Compared with typical graphene metasurfaces, the effect of dispersion of the propagating waves and resonance mode introduced by such multilayer structure have been investigated on the light-matter interaction. The incident THz wave can be coupled with the resonant mode to strengthen the interaction with the graphene and enhance the negative absorption. More importantly, when the dispersion matches the resonance condition, it enables the coupling between propagating and resonant modes, leading to a remarkable negative absorption. This mechanism may provide a promising way to develop THz amplifier, modulator and other electronics devices.

2. Multilayer meta-structure and effective permittivity theory

The proposed graphene-dielectric hybrid meta-structure is shown in Fig. 1, in which the graphene layer is separated by the dielectric layer. In this structure, the region of interaction between the THz wave and the graphene layers is extended to take into account not only the microscale effect but also the macroscopic electromagnetic effect, such as the electromagnetic mode and the boundary condition, which is quite different from the previous multilayer graphene [40,41]. Therefore, the structure of monolayer graphene and dielectric layer are stacked alternately to serve as a cavity by coupling the incident light into similar waveguide modes [37,42,43]. In addition, the component of incident wave can propagate along the Z direction when meeting dispersion relation, as shown in Fig. 1(b). In both cases, the interaction between graphene and terahertz wave is greatly promoted by increasing the interaction length.

 figure: Fig. 1.

Fig. 1. Schematic of the graphene-based metamaterial. The metamaterial consists of alternative pumped graphene sheet and dielectric layer. The dielectric layer is PMMA with ${\varepsilon _d}$ = 4 and a thickness ${t_d}$ = 2 um, and total height of the structure is = 300 um.

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The thickness of the dielectric layer and graphene is ${t_d}$ and ${t_g}$, the period of the metamaterial is ${t_d} + {t_g} \approx {t_d}$, and total height of the structure is $d = N\ast {t_d}$ ($N$ represents the number of periodicity). Since the dielectric layer of the arrangement is much thinner than the wavelength of incident wave, the meta-structure can be simplified to equal to a uniform medium with effective parameters [44,45]:

$${\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\leftrightarrow$}} \over \varepsilon } _{stack}}\textrm{ = }\left[ {\begin{array}{ccc} {{\varepsilon _x}}&0&0\\ 0&{{\varepsilon _y}}&0\\ 0&0&{{\varepsilon _z}} \end{array}} \right],$$
where ${\varepsilon _x} = {\varepsilon _\textrm{z}} \approx {\varepsilon _g}({{{{t_g}} \mathord{\left/ {\vphantom {{{t_g}} {{t_d}}}} \right.} {{t_d}}}} )+ {\varepsilon _d}$ and ${\varepsilon _\textrm{y}} \approx {\varepsilon _d}$. ${\varepsilon _g} = 1 - \frac{{i\sigma }}{{\omega {\varepsilon _0}{t_g}}}$ is the in-plane permittivity of graphene while ${\varepsilon _d}$ is the permittivity of the dielectric and $\sigma $ is the conductivity of monolayer graphene.

In this mechanism, the pump laser is applied to induce the change of the Fermi energy, and the incident THz wave with an angle $\theta$ will interact with graphene layers. Thus, the procedure includes the microscopic stimulated emission and the macroscopic electromagnetic wave transmission. Considering the transmission of incident wave in this structure, which can be expressed as:

$${\nabla ^2}\left[ {\begin{array}{c} {{{\vec{E}}_x}}\\ {{{\vec{E}}_y}}\\ {{{\vec{E}}_z}} \end{array}} \right] - \nabla \left( {\nabla \cdot \left[ {\begin{array}{c} {{{\vec{E}}_x}}\\ {{{\vec{E}}_y}}\\ {{{\vec{E}}_z}} \end{array}} \right]} \right) - {\omega ^2}{\mu _0}\left[ {\begin{array}{ccc} {{\varepsilon_x}}&{}&{}\\ {}&{{\varepsilon_y}}&{}\\ {}&{}&{{\varepsilon_z}} \end{array}} \right] \cdot \left[ {\begin{array}{c} {{{\vec{E}}_x}}\\ {{{\vec{E}}_y}}\\ {{{\vec{E}}_z}} \end{array}} \right]\textrm{ = }0,$$
where ${\vec{E}_i}$ denotes the electric field component.

The oblique linear polarized incident THz wave can be regarded as TE or TM electromagnetic wave. The propagating characters of TM waves strongly depend on the dielectric tensor, while that of TE waves are determined by ${\varepsilon _x}\textrm{/}{\varepsilon _z}$. Thus, the absorption coefficient (A = 1 – T – R) for TE or TM polarization wave can be obtained as (the detailed derivation of the absorption coefficient see Appendix A):

$$\begin{array}{l} {A_{TE}} = 1 - \left[ {\left( {1 - \frac{{\sqrt {{\varepsilon _\parallel } - {{\sin }^2}\theta } }}{{\sqrt {1 - {{\sin }^2}\theta } }}} \right)\left( {1 + \frac{{\sqrt {1 - {{\sin }^2}\theta } }}{{\sqrt {{\varepsilon _\parallel } - {{\sin }^2}\theta } }}} \right){e^{j{k_y}d}} + \left( {1 + \frac{{\sqrt {{\varepsilon _\parallel } - {{\sin }^2}\theta } }}{{\sqrt {1 - {{\sin }^2}\theta } }}{\kern 1pt} } \right)\left( {1 - \frac{{\sqrt {1 - {{\sin }^2}\theta } }}{{\sqrt {{\varepsilon _\parallel } - {{\sin }^2}\theta } }}} \right){e^{ - j{k_y}d}}} \right.\\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} + {{\left. {4{e^{j\sqrt {{{\left( {{\omega \mathord{\left/ {\vphantom {\omega c}} \right. } c}} \right)}^2} - {k_z}^2} d}}} \right]} \mathord{\left/ {\vphantom {{\left. {4{e^{j\sqrt {{{\left( {{\omega \mathord{\left/ {\vphantom {\omega c}} \right. } c}} \right)}^2} - {k_z}^2} d}}} \right]} {\left[ {\left( {1 + \frac{{\sqrt {{\varepsilon _\parallel } - {{\sin }^2}\theta } }}{{\sqrt {1 - {{\sin }^2}\theta } }}{\kern 1pt} } \right)\left( {1 + \frac{{\sqrt {1 - {{\sin }^2}\theta } }}{{\sqrt {{\varepsilon _\parallel } - {{\sin }^2}\theta } }}} \right){e^{j{k_y}d}} + \left( {1 - \frac{{\sqrt {{\varepsilon _\parallel } - {{\sin }^2}\theta } }}{{\sqrt {1 - {{\sin }^2}\theta } }}{\kern 1pt} } \right)\left( {1 - \frac{{\sqrt {1 - {{\sin }^2}\theta } }}{{\sqrt {{\varepsilon _\parallel } - {{\sin }^2}\theta } }}} \right){e^{ - j{k_y}d}}} \right]}}} \right. } {\left[ {\left( {1 + \frac{{\sqrt {{\varepsilon _\parallel } - {{\sin }^2}\theta } }}{{\sqrt {1 - {{\sin }^2}\theta } }}{\kern 1pt} } \right)\left( {1 + \frac{{\sqrt {1 - {{\sin }^2}\theta } }}{{\sqrt {{\varepsilon _\parallel } - {{\sin }^2}\theta } }}} \right){e^{j{k_y}d}} + \left( {1 - \frac{{\sqrt {{\varepsilon _\parallel } - {{\sin }^2}\theta } }}{{\sqrt {1 - {{\sin }^2}\theta } }}{\kern 1pt} } \right)\left( {1 - \frac{{\sqrt {1 - {{\sin }^2}\theta } }}{{\sqrt {{\varepsilon _\parallel } - {{\sin }^2}\theta } }}} \right){e^{ - j{k_y}d}}} \right]}},\\ {A_{TM}} = 1 - \left( {\left( {\frac{{ - 2\sqrt {{\varepsilon _ \bot }\left( {1 - {{{{\sin }^2}\theta } \mathord{\left/ {\vphantom {{{{\sin }^2}\theta } {{\varepsilon _\parallel }}}} \right. } {{\varepsilon _\parallel }}}} \right)} }}{{ - \sqrt {{\varepsilon _ \bot }\left( {1 - {{{{\sin }^2}\theta } \mathord{\left/ {\vphantom {{{{\sin }^2}\theta } {{\varepsilon _\parallel }}}} \right. } {{\varepsilon _\parallel }}}} \right)} + {\varepsilon _ \bot }\sqrt {1 - {{\sin }^2}\theta } }}} \right)\left( {\frac{{2{\varepsilon _ \bot }\sqrt {1 - {{\sin }^2}\theta } }}{{{\varepsilon _ \bot }\sqrt {1 - {{\sin }^2}\theta } - \sqrt {{\varepsilon _ \bot }\left( {1 - {{{{\sin }^2}\theta } \mathord{\left/ {\vphantom {{{{\sin }^2}\theta } {{\varepsilon _\parallel }}}} \right. } {{\varepsilon _\parallel }}}} \right)} }}} \right){e^{ - j{k_y}d}} + \left( {\frac{{\sqrt {{\varepsilon _ \bot }\left( {1 - {{{{\sin }^2}\theta } \mathord{\left/ {\vphantom {{{{\sin }^2}\theta } {{\varepsilon _\parallel }}}} \right. } {{\varepsilon _\parallel }}}} \right)} + {\varepsilon _ \bot }\sqrt {1 - {{\sin }^2}\theta } }}{{\sqrt {{\varepsilon _ \bot }\left( {1 - {{{{\sin }^2}\theta } \mathord{\left/ {\vphantom {{{{\sin }^2}\theta } {{\varepsilon _\parallel }}}} \right. } {{\varepsilon _\parallel }}}} \right)} - {\varepsilon _ \bot }\sqrt {1 - {{\sin }^2}\theta } }}} \right)} \right.\\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {{{\kern 1pt} {\kern 1pt} {\kern 1pt} \left. {\left( {\frac{{{\varepsilon _ \bot }\sqrt {1 - {{\sin }^2}\theta } + \sqrt {{\varepsilon _ \bot }\left( {1 - {{{{\sin }^2}\theta } \mathord{\left/ {\vphantom {{{{\sin }^2}\theta } {{\varepsilon _\parallel }}}} \right. } {{\varepsilon _\parallel }}}} \right)} }}{{{\varepsilon _ \bot }\sqrt {1 - {{\sin }^2}\theta } - \sqrt {{\varepsilon _ \bot }\left( {1 - {{{{\sin }^2}\theta } \mathord{\left/ {\vphantom {{{{\sin }^2}\theta } {{\varepsilon _\parallel }}}} \right. } {{\varepsilon _\parallel }}}} \right)} }}} \right){e^{ - 2j{k_y}d}}} \right)} \mathord{\left/ {\vphantom {{{\kern 1pt} {\kern 1pt} {\kern 1pt} \left. {\left( {\frac{{{\varepsilon _ \bot }\sqrt {1 - {{\sin }^2}\theta } + \sqrt {{\varepsilon _ \bot }\left( {1 - {{{{\sin }^2}\theta } \mathord{\left/ {\vphantom {{{{\sin }^2}\theta } {{\varepsilon _\parallel }}}} \right. } {{\varepsilon _\parallel }}}} \right)} }}{{{\varepsilon _ \bot }\sqrt {1 - {{\sin }^2}\theta } - \sqrt {{\varepsilon _ \bot }\left( {1 - {{{{\sin }^2}\theta } \mathord{\left/ {\vphantom {{{{\sin }^2}\theta } {{\varepsilon _\parallel }}}} \right. } {{\varepsilon _\parallel }}}} \right)} }}} \right){e^{ - 2j{k_y}d}}} \right)} {1 + \left( {\frac{{\sqrt {{\varepsilon _ \bot }\left( {1 - {{{{\sin }^2}\theta } \mathord{\left/ {\vphantom {{{{\sin }^2}\theta } {{\varepsilon _\parallel }}}} \right. } {{\varepsilon _\parallel }}}} \right)} + {\varepsilon _ \bot }\sqrt {1 - {{\sin }^2}\theta } }}{{\sqrt {{\varepsilon _ \bot }\left( {1 - {{{{\sin }^2}\theta } \mathord{\left/ {\vphantom {{{{\sin }^2}\theta } {{\varepsilon _\parallel }}}} \right. } {{\varepsilon _\parallel }}}} \right)} - {\varepsilon _ \bot }\sqrt {1 - {{\sin }^2}\theta } }}} \right) + \left( {\frac{{{\varepsilon _ \bot }\sqrt {1 - {{\sin }^2}\theta } + \sqrt {{\varepsilon _ \bot }\left( {1 - {{{{\sin }^2}\theta } \mathord{\left/ {\vphantom {{{{\sin }^2}\theta } {{\varepsilon _\parallel }}}} \right. } {{\varepsilon _\parallel }}}} \right)} }}{{{\varepsilon _ \bot }\sqrt {1 - {{\sin }^2}\theta } - \sqrt {{\varepsilon _ \bot }\left( {1 - {{{{\sin }^2}\theta } \mathord{\left/ {\vphantom {{{{\sin }^2}\theta } {{\varepsilon _\parallel }}}} \right. } {{\varepsilon _\parallel }}}} \right)} }}} \right){e^{ - 2j{k_y}d}}}}} \right. } {1 + \left( {\frac{{\sqrt {{\varepsilon _ \bot }\left( {1 - {{{{\sin }^2}\theta } \mathord{\left/ {\vphantom {{{{\sin }^2}\theta } {{\varepsilon _\parallel }}}} \right. } {{\varepsilon _\parallel }}}} \right)} + {\varepsilon _ \bot }\sqrt {1 - {{\sin }^2}\theta } }}{{\sqrt {{\varepsilon _ \bot }\left( {1 - {{{{\sin }^2}\theta } \mathord{\left/ {\vphantom {{{{\sin }^2}\theta } {{\varepsilon _\parallel }}}} \right. } {{\varepsilon _\parallel }}}} \right)} - {\varepsilon _ \bot }\sqrt {1 - {{\sin }^2}\theta } }}} \right) + \left( {\frac{{{\varepsilon _ \bot }\sqrt {1 - {{\sin }^2}\theta } + \sqrt {{\varepsilon _ \bot }\left( {1 - {{{{\sin }^2}\theta } \mathord{\left/ {\vphantom {{{{\sin }^2}\theta } {{\varepsilon _\parallel }}}} \right. } {{\varepsilon _\parallel }}}} \right)} }}{{{\varepsilon _ \bot }\sqrt {1 - {{\sin }^2}\theta } - \sqrt {{\varepsilon _ \bot }\left( {1 - {{{{\sin }^2}\theta } \mathord{\left/ {\vphantom {{{{\sin }^2}\theta } {{\varepsilon _\parallel }}}} \right. } {{\varepsilon _\parallel }}}} \right)} }}} \right){e^{ - 2j{k_y}d}}}}. \end{array}$$
From Eqs. (1) and (3), it can be found that the absorption coefficient of the EM waves is determined by a key factor - the conductivity of monolayer graphene $\sigma $, which is related to the microcosmic optical parameters of the graphene layer.

As we know, the optical response of monolayer graphene is described as [20,46]:

$$\begin{array}{l} \sigma (\omega )= {\sigma _{{\mathop{\rm int}} ra}}(\omega )+ {\sigma _{{\mathop{\rm int}} er}}(\omega )\\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} = \frac{{{e^2}}}{{4\hbar }}\frac{{8{k_B}T\tau }}{{\pi \hbar ({1 + j\omega \tau } )}}\ln \left( {1 + \exp \left( {\frac{{Ef}}{{{k_B}T}}} \right)} \right)\\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} + \frac{{{e^2}}}{{4\hbar }}\left( {\tanh \left( {\frac{{\hbar \omega - 2Ef}}{{4{k_B}T}}} \right) + \frac{{4\hbar \omega }}{{j\pi }}\int\limits_0^\infty {\frac{{G({\varepsilon ,Ef} )- G({{{\hbar \omega } \mathord{\left/ {\vphantom {{\hbar \omega } 2}} \right.} 2},Ef} )}}{{{{({\hbar \omega } )}^2} - 4{\varepsilon^2}}}d\varepsilon } } \right), \end{array}$$
where $G({\varepsilon ,Ef} )= \frac{{\sinh ({{\varepsilon \mathord{\left/ {\vphantom {\varepsilon {{k_B}T}}} \right.} {{k_B}T}}} )}}{{\cosh ({{\varepsilon \mathord{\left/ {\vphantom {\varepsilon {{k_B}T}}} \right.} {{k_B}T}}} )+ \cosh ({{{Ef} \mathord{\left/ {\vphantom {{Ef} {{k_B}T}}} \right.} {{k_B}T}}} )}}$ and $Ef = 12\frac{{{e^2}}}{\hbar }{\left( {\frac{{\hbar {u_F}}}{{{k_B}T}}} \right)^2}\frac{{{\tau _R}{I_\varOmega }}}{{\hbar \varOmega }}{k_B}T$.$\omega$ is the angular frequency of incident wave, e is the charge quantity of an electron, ${k_B}$ is the Boltzmann constant, T is the temperature, and $\hbar$ is the reduced Planck constant. $\tau$ = 10 ps is the momentum relaxation time related to carrier scattering, and ${\tau _R}$ = 0.5 ps is the inter-band recombination time. $Ef$ is the non-equilibrium Fermi energy of the pumped monolayer graphene, which is proportional to the light intensity ${I_\varOmega }$.

Numerical calculations of the conductivity of graphene with different pump power and temperature are shown in Fig. 2. The variety of pump power leads to the change of the Fermi energy, achieving tunability of graphene conductivity. More importantly, the calculated results demonstrate that the increase of the power causes a decrease in the photoinduced THz conductivity and even a distribution of the negative real part. The negative real part conductivity of graphene and the EM mode characteristics of the hybrid structure make enhanced negative absorption possible.

 figure: Fig. 2.

Fig. 2. The distribution of negative real-part of graphene complex conductivity. Calculated surface conductivity versus frequency and pump intensity at the temperature of 77 K (a) and 300 K (b).

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3. Results and discussion

We performed numerical calculation to study the TE waves response of the metamaterial structure. First, the dependence of the absorption of the structure on the frequency and incident angle with 4.8 meV of graphene is illustrated in Fig. 3. In a wide-angle range and broad frequency band, negative absorption of the meta-structure is induced by the negative conductivity of graphene. The notable negative absorption presents a regular zonal distribution, corresponding to the resonance condition ${\mathop{\rm Re}\nolimits} ({{k_y}({\omega ,\theta } )} )\cdot d = n\pi$ ($n$ is integer), as shown the dashed line in Fig. 3. When the angle and frequency of the incident TE wave are just at the resonance point, the wave can be coupled into the structure to form a similar dielectric waveguide mode, oscillating back and forth in the metamaterial and extending the interaction distance of the wave with graphene. Therefore, the negative absorption at different incident angles can be enhanced by the resonance, as shown the black dashed line in Fig. 3.

 figure: Fig. 3.

Fig. 3. Calculated negative absorption versus frequency and angle with 4.8 meV of graphene, corresponding to 3.2 W/cm2 of pump power at 77 K. The dashed line and yellow hollow circle represent resonance modes and dispersion, respectively. The calculating steps of the frequency and the incident angle are 0.01 THz and 0.1 °.

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In addition, another noteworthy feature is that the maximum is distributed where dispersion matches the resonance condition as shown the overlap between the dashed line and the hollow circle in Fig. 3. The dispersion relation in this periodic structure represents the relationship between the wave number in the Z direction and the frequency (${k_z}\sim \omega$), which can be obtained as (the detailed derivation see Appendix B):

$$({{k_1} + {k_z}} )({{k_3} + {k_z}} ){e^{j{k_z}d}} = ({{k_1} - {k_z}} )({{k_3} - {k_z}} ){e^{ - j{k_z}d}}.$$
To further explain the effect of dispersion and resonance on negative absorption, absorption spectra with fixed incident angle is presented in Fig. 4. It can be found that the absorption spectrums display multiple peaks, corresponding to red solid circle in Fig. 3. But not all of peaks are caused by resonance since the transmission of wave in the meta-structure is also affected by dispersion. In Fig. 4(a), point P and M1 meet the dispersion and resonance condition respectively, and their frequency are very close, which enables the coupling between propagating and resonant modes. The interaction with graphene is promoted to the greatest extent, including the extension of the interaction distance and the enhancement of the intensity. Therefore, this remarkable negative absorption scenario is induced by the enough interaction of graphene and light, which is determined by the structure characteristics of hybrid meta-structure as well as negative conductivity of graphene. As illustrated in Figs. 4(c) and 4(e), the value of point P increases with the increase of incident angle, that means there are more parallel components of incident wave propagating in the structure, thereby enhancing the coupling between the resonant and propagating mode and obtaining the stronger negative absorption.

 figure: Fig. 4.

Fig. 4. (a), (c) and (e) are the absorption spectra with incident angles 30°, 45° and 60°, respectively. (b), (d) and (e) are the distribution of magnetic field (under the same display settings) at P∼M4, respectively.

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The magnetic field distribution of M2∼M4 is also shown in Fig. 4, which demonstrates that the fields are concentered in the metamaterial structure and form similar dielectric waveguide modes, corresponding to the TE2-like ∼ TE4-like modes. The active medium acts as a two-dimensional anisotropic dielectric cavity, and the co-phase stacking of the incident waves at the resonance points makes resonance mode. Thus, it is appreciated that both the dispersion relation and resonance can enhance the interaction between terahertz wave and graphene.

We also study the effect of graphene Fermi energy on negative absorption and compare with previous work to demonstrate that the structure can enhance negative absorption. With different Fermi energy of graphene, the transmission (T), reflection (R) and absorption (A) spectra are shown in Fig. 5. Firstly, multiple weak resonance points appear in the spectrum without pump light. As the Fermi energy increases from 0 to 2.8 meV, the resonance of the meta-structure is continuously enhanced and the corresponding absorption decreases. Further increasing the Fermi energy of graphene, the absorption changes from positive to negative and shows an increment reversely, indicating a negative real part of the conductivity of graphene. Meanwhile, the enhancement of negative absorption occurs not only at the resonance point but also at dispersion point and is closely related to the intensity of optical pump.

 figure: Fig. 5.

Fig. 5. (a), (b) and (c) are transmission, reflection and absorption spectra under various Fermi energy of graphene with d=300 um and θ = 0°, respectively; (d) The maximum absolute absorption values are plotted as a function of the pump intensity.

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 figure: Fig. 6.

Fig. 6. Negative absorption versus frequency and angle with carrier relaxation time is (a)10ps, (b) 5ps and (c) 1ps, respectively. The calculating steps of the frequency and the incident angle are 0.1 THz and 0.1 °.

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By recording the maximum absolute absorption values of multilayer metamaterial, the evolution of negative absorption under different pump intensity is investigated, as shown in Fig. 5(d). The positive absorption and negative absorption correspond to the pink and blue regions shown in the inset, respectively. With the increase of the pump intensity, the absorption changes from positive to negative, and continues to be enhanced until the threshold of the optical pump (4.8 W/cm2). This indicates that the negative absorption will not be enhanced indefinitely under the action of optical pump, but will decrease after reaching the threshold. More importantly, compared with the previous work [47], the negative absorption is increased by nearly 100 times by adopting this meta-structure.

In addition, negative absorption versus frequency and angle with various carrier relaxation time is illustrated in Fig. 6. The carrier relaxation time is 10  ps, 5 ps and 1 ps, while the corresponding optical pump power is 3.2 W/cm2, 5.2 W/cm2, 6.1 W/cm2, respectively. As carrier relaxation time decreases, negative absorption requires a stronger optical pump to be excited. Although the whole distribution of negative absorption moves to higher frequency, the overall mechanism remains unchanged. Negative absorption is still mainly distributed along the resonance line, and the maximum is distributed where dispersion matches the resonance. Therefore, smaller relaxation time values can still be applied to the mechanism proposed in this paper, but the parameters of optical pump or structure might require a thorough optimization.

4. Conclusions

The present work introduces a graphene-dielectric hybrid meta-structure to boost light-matter interaction. This method relies on negative conductivity of graphene and macroscopic electromagnetic effect of meta-structure. Taking the TE-polarized wave as an example for the structure, we demonstrate a significant enhancement in negative absorption, which results from resonance modes and dispersion relation. When the dispersion of the propagating waves matches the resonance condition, it enables the coupling between propagating and resonant modes, leading to a remarkable negative absorption. The mechanism we propose may also be applied to other polarized waves and 2D materials. Here we only discuss the negative conductivity of graphene induced by ultrafast optical excitation, but population inversion might be induced by external voltage. Therefore, our work provides theoretical foundations to conduct experiment of multilayer graphene metamaterial, which might help to further develop the graphene-based THz lasers and other modulation devices.

Appendix A. The detailed derivation of the absorption coefficient

First, the propagation of the wave in the multilayer stack should meet the wave equations, as:

$${\nabla ^2}\left[ {\begin{array}{c} {{{\vec{E}}_x}}\\ {{{\vec{E}}_y}}\\ {{{\vec{E}}_z}} \end{array}} \right] - \nabla \left( {\nabla \cdot \left[ {\begin{array}{c} {{{\vec{E}}_x}}\\ {{{\vec{E}}_y}}\\ {{{\vec{E}}_z}} \end{array}} \right]} \right) - {\omega ^2}{\mu _0}\left[ {\begin{array}{ccc} {{\varepsilon_x}}&{}&{}\\ {}&{{\varepsilon_y}}&{}\\ {}&{}&{{\varepsilon_z}} \end{array}} \right] \cdot \left[ {\begin{array}{c} {{{\vec{E}}_x}}\\ {{{\vec{E}}_y}}\\ {{{\vec{E}}_z}} \end{array}} \right]\textrm{ = }0.$$
Making use of the expression of the electric field
$$\vec{E}\textrm{ = }({{{\vec{e}}_x}{E_x} + {{\vec{e}}_y}{E_y} + {{\vec{e}}_z}{E_z}} ){e^{ - j{k_x}x}}{e^{ - j{k_y}y}}{e^{ - j{k_z}z}},$$
the wave vector relations can be obtained by making the vector wave Eq. (6) have a nontrivial solution. there should be:
$$\left|{\begin{array}{ccc} {{k_y}^2 + {k_z}^2 - {\omega^2}\mu {\varepsilon_x}}&{ - {k_x}{k_y}}&{ - {k_x}{k_z}}\\ { - {k_x}{k_y}}&{{k_x}^2 + {k_z}^2 - {\omega^2}\mu {\varepsilon_y}}&{ - {k_z}{k_y}}\\ { - {k_x}{k_z}}&{ - {k_y}{k_z}}&{{k_x}^2 + {k_y}^2 - {\omega^2}\mu {\varepsilon_z}} \end{array}} \right|= 0.$$
Considering the case that the width of the stack is much larger than the wavelength of the incident waves, ${k_x}$ can be set to 0 to simplify this question. For the perpendicular polarized (TE) waves $({{{\vec{H}}_x},{{\vec{E}}_y},{{\vec{E}}_z}} )$, Eq. (8) can be expressed as:
$${\omega ^2}{\mu _0}{\varepsilon _y} - {k_z}^2 - {k_y}^2 = 0.$$
In the stack, the magnetic field along the Z direction can be written as:
$${H_z}^{stack} = {B_{stack}}^a{e^{ - j{k_z}z - j{k_y}y}} + {B_{stack}}^b{e^{ - j{k_z}z + j{k_y}y}}.$$
According to the Maxwell equation, the electric field expression can be obtained as:
$$\begin{array}{l} \nabla \times \overrightarrow H = j\omega {\varepsilon _x}{E_x}\\ {E_x} = \frac{1}{{j\omega {\varepsilon _x}}}\nabla \times \overrightarrow H = \frac{1}{{j\omega {\varepsilon _x}}}\left( {\frac{{\partial {H_z}}}{{\partial y}} - \frac{{\partial {H_y}}}{{\partial z}}} \right). \end{array}$$
Accordingly, there is:
$${E_x} ={-} {B_{stack}}^a\frac{{{k_y}^2 + {k_z}^2}}{{\omega {\varepsilon _x}{k_y}}}{e^{ - j{k_z}z - j{k_y}y}} + {B_{stack}}^b\frac{{{k_y}^2 + {k_z}^2}}{{\omega {\varepsilon _x}{k_y}}}{e^{ - j{k_z}z + j{k_y}y}}.$$

Thus, the expression of the fields in the incident and emergence region can be obtained as:

$$\begin{array}{l} \left\{ \begin{array}{l} {H_z}^{stack} = {B_{stack}}^a{e^{ - j{k_z}z - j{k_y}y}} + {B_{stack}}^b{e^{ - j{k_z}z + j{k_y}y}}\\ {E_x}^{stack} ={-} {B_{stack}}^a\frac{{{k_y}^2 + {k_z}^2}}{{\omega {\varepsilon_x}{k_y}}}{e^{ - j{k_z}z - j{k_y}y}} + {B_{stack}}^b\frac{{{k_y}^2 + {k_z}^2}}{{\omega {\varepsilon_x}{k_y}}}{e^{ - j{k_z}z + j{k_y}y}} \end{array} \right.,\\ \left\{ \begin{array}{l} {H_z}^{inc} = {B_{inc}}^a{e^{ - j{k_z}z - j{k_1}y}} + {B_{inc}}^b{e^{ - j{k_z}z + j{k_1}y}}\\ {E_x}^{inc} ={-} {B_{inc}}^a\frac{{{k_0}^2}}{{\omega {k_1}}}{e^{ - j{k_z}z - j{k_1}y}} + {B_{inc}}^b\frac{{{k_0}^2}}{{\omega {k_1}}}{e^{ - j{k_z}z + j{{({{k_y}} )}_I}y}}{k_1} \end{array} \right.,\\ \left\{ \begin{array}{l} {H_z}^{eme} = {B_{eme}}^a{e^{ - j{k_z}z - j{k_3}y}}\\ {E_x}^{eme} ={-} {B_{eme}}^a\frac{{{k_0}^2}}{{\omega {k_3}}}{e^{ - j{k_z}z - j{k_3}y}} \end{array} \right.. \end{array}$$

Assuming the incidence of the terahertz waves in Y-Z plane shown in Fig. 1(a) in the manuscript with an angle $\theta$, the boundary condition can be written as:

$$\begin{array}{l} { {{E_x}^{inc}} |_{y = 0}} = { {{E_x}^{stack}} |_{y = 0}}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} { {{H_z}^{inc}} |_{y = 0}} = { {{H_z}^{stack}} |_{y = 0}}\\ { {{E_x}^{stack}} |_{y = d}} = { {{E_x}^{eme}} |_{y = d}}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} { {{H_z}^{stack}} |_{y = d}} = { {{H_z}^{eme}} |_{y = d}}. \end{array}$$
By substituting the field expressions for each region, we can get the transmission and reflection coefficients.
$$\begin{array}{l} R = \frac{{\left( {1 - \frac{{{k_y}}}{{{k_1}}}} \right)\left( {1 + \frac{{{k_3}}}{{{k_y}}}} \right){e^{j{k_y}d}} + \left( {1 + \frac{{{k_y}}}{{{k_1}}}} \right)\left( {1 - \frac{{{k_3}}}{{{k_y}}}} \right){e^{ - j{k_y}d}}}}{{\left( {1 + \frac{{{k_y}}}{{{k_1}}}} \right)\left( {1 + \frac{{{k_3}}}{{{k_y}}}} \right){e^{j{k_y}d}} + \left( {1 - \frac{{{k_y}}}{{{k_1}}}} \right)\left( {1 - \frac{{{k_3}}}{{{k_y}}}} \right){e^{ - j{k_y}d}}}},\\ T = \frac{{4{e^{j{k_3}d}}}}{{\left( {1 + \frac{{{k_y}}}{{{k_1}}}} \right)\left( {1 + \frac{{{k_3}}}{{{k_y}}}} \right){e^{j{k_y}d}} + \left( {1 - \frac{{{k_y}}}{{{k_1}}}} \right)\left( {1 - \frac{{{k_3}}}{{{k_y}}}} \right){e^{ - j{k_y}d}}}}. \end{array}$$
where ${k_1} = \sqrt {k_0^2 - k_z^2}$, ${k_y} = \sqrt {{\varepsilon _\parallel }k_0^2 - k_z^2}$, ${k_3} = \sqrt {k_0^2 - k_z^2}$, ${k_z} = {k_0}\sin \theta$ and ${k_0}$ is the wavenumber in the vacuum. The reflection and transmission coefficients for TM polarization can also be obtained by the same method. Thus, the absorption coefficient (A = 1 – T – R) for TE or TM polarization wave can be obtained as:
$$\begin{array}{l} {A_{TE}} = 1 - \left[ {\left( {1 - \frac{{\sqrt {{\varepsilon _\parallel } - {{\sin }^2}\theta } }}{{\sqrt {1 - {{\sin }^2}\theta } }}} \right)\left( {1 + \frac{{\sqrt {1 - {{\sin }^2}\theta } }}{{\sqrt {{\varepsilon _\parallel } - {{\sin }^2}\theta } }}} \right){e^{j{k_y}d}} + \left( {1 + \frac{{\sqrt {{\varepsilon _\parallel } - {{\sin }^2}\theta } }}{{\sqrt {1 - {{\sin }^2}\theta } }}{\kern 1pt} } \right)\left( {1 - \frac{{\sqrt {1 - {{\sin }^2}\theta } }}{{\sqrt {{\varepsilon _\parallel } - {{\sin }^2}\theta } }}} \right){e^{ - j{k_y}d}}} \right.\\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} + {{\left. {4{e^{j\sqrt {{{\left( {{\omega \mathord{\left/ {\vphantom {\omega c}} \right. } c}} \right)}^2} - {k_z}^2} d}}} \right]} \mathord{\left/ {\vphantom {{\left. {4{e^{j\sqrt {{{\left( {{\omega \mathord{\left/ {\vphantom {\omega c}} \right. } c}} \right)}^2} - {k_z}^2} d}}} \right]} {\left[ {\left( {1 + \frac{{\sqrt {{\varepsilon _\parallel } - {{\sin }^2}\theta } }}{{\sqrt {1 - {{\sin }^2}\theta } }}{\kern 1pt} } \right)\left( {1 + \frac{{\sqrt {1 - {{\sin }^2}\theta } }}{{\sqrt {{\varepsilon _\parallel } - {{\sin }^2}\theta } }}} \right){e^{j{k_y}d}} + \left( {1 - \frac{{\sqrt {{\varepsilon _\parallel } - {{\sin }^2}\theta } }}{{\sqrt {1 - {{\sin }^2}\theta } }}{\kern 1pt} } \right)\left( {1 - \frac{{\sqrt {1 - {{\sin }^2}\theta } }}{{\sqrt {{\varepsilon _\parallel } - {{\sin }^2}\theta } }}} \right){e^{ - j{k_y}d}}} \right]}}} \right. } {\left[ {\left( {1 + \frac{{\sqrt {{\varepsilon _\parallel } - {{\sin }^2}\theta } }}{{\sqrt {1 - {{\sin }^2}\theta } }}{\kern 1pt} } \right)\left( {1 + \frac{{\sqrt {1 - {{\sin }^2}\theta } }}{{\sqrt {{\varepsilon _\parallel } - {{\sin }^2}\theta } }}} \right){e^{j{k_y}d}} + \left( {1 - \frac{{\sqrt {{\varepsilon _\parallel } - {{\sin }^2}\theta } }}{{\sqrt {1 - {{\sin }^2}\theta } }}{\kern 1pt} } \right)\left( {1 - \frac{{\sqrt {1 - {{\sin }^2}\theta } }}{{\sqrt {{\varepsilon _\parallel } - {{\sin }^2}\theta } }}} \right){e^{ - j{k_y}d}}} \right]}},\\ {A_{TM}} = 1 - \left( {\left( {\frac{{ - 2\sqrt {{\varepsilon _ \bot }\left( {1 - {{{{\sin }^2}\theta } \mathord{\left/ {\vphantom {{{{\sin }^2}\theta } {{\varepsilon _\parallel }}}} \right. } {{\varepsilon _\parallel }}}} \right)} }}{{ - \sqrt {{\varepsilon _ \bot }\left( {1 - {{{{\sin }^2}\theta } \mathord{\left/ {\vphantom {{{{\sin }^2}\theta } {{\varepsilon _\parallel }}}} \right. } {{\varepsilon _\parallel }}}} \right)} + {\varepsilon _ \bot }\sqrt {1 - {{\sin }^2}\theta } }}} \right)\left( {\frac{{2{\varepsilon _ \bot }\sqrt {1 - {{\sin }^2}\theta } }}{{{\varepsilon _ \bot }\sqrt {1 - {{\sin }^2}\theta } - \sqrt {{\varepsilon _ \bot }\left( {1 - {{{{\sin }^2}\theta } \mathord{\left/ {\vphantom {{{{\sin }^2}\theta } {{\varepsilon _\parallel }}}} \right. } {{\varepsilon _\parallel }}}} \right)} }}} \right){e^{ - j{k_y}d}} + \left( {\frac{{\sqrt {{\varepsilon _ \bot }\left( {1 - {{{{\sin }^2}\theta } \mathord{\left/ {\vphantom {{{{\sin }^2}\theta } {{\varepsilon _\parallel }}}} \right. } {{\varepsilon _\parallel }}}} \right)} + {\varepsilon _ \bot }\sqrt {1 - {{\sin }^2}\theta } }}{{\sqrt {{\varepsilon _ \bot }\left( {1 - {{{{\sin }^2}\theta } \mathord{\left/ {\vphantom {{{{\sin }^2}\theta } {{\varepsilon _\parallel }}}} \right. } {{\varepsilon _\parallel }}}} \right)} - {\varepsilon _ \bot }\sqrt {1 - {{\sin }^2}\theta } }}} \right)} \right.\\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {{{\kern 1pt} {\kern 1pt} {\kern 1pt} \left. {\left( {\frac{{{\varepsilon _ \bot }\sqrt {1 - {{\sin }^2}\theta } + \sqrt {{\varepsilon _ \bot }\left( {1 - {{{{\sin }^2}\theta } \mathord{\left/ {\vphantom {{{{\sin }^2}\theta } {{\varepsilon _\parallel }}}} \right. } {{\varepsilon _\parallel }}}} \right)} }}{{{\varepsilon _ \bot }\sqrt {1 - {{\sin }^2}\theta } - \sqrt {{\varepsilon _ \bot }\left( {1 - {{{{\sin }^2}\theta } \mathord{\left/ {\vphantom {{{{\sin }^2}\theta } {{\varepsilon _\parallel }}}} \right. } {{\varepsilon _\parallel }}}} \right)} }}} \right){e^{ - 2j{k_y}d}}} \right)} \mathord{\left/ {\vphantom {{{\kern 1pt} {\kern 1pt} {\kern 1pt} \left. {\left( {\frac{{{\varepsilon _ \bot }\sqrt {1 - {{\sin }^2}\theta } + \sqrt {{\varepsilon _ \bot }\left( {1 - {{{{\sin }^2}\theta } \mathord{\left/ {\vphantom {{{{\sin }^2}\theta } {{\varepsilon _\parallel }}}} \right. } {{\varepsilon _\parallel }}}} \right)} }}{{{\varepsilon _ \bot }\sqrt {1 - {{\sin }^2}\theta } - \sqrt {{\varepsilon _ \bot }\left( {1 - {{{{\sin }^2}\theta } \mathord{\left/ {\vphantom {{{{\sin }^2}\theta } {{\varepsilon _\parallel }}}} \right. } {{\varepsilon _\parallel }}}} \right)} }}} \right){e^{ - 2j{k_y}d}}} \right)} {1 + \left( {\frac{{\sqrt {{\varepsilon _ \bot }\left( {1 - {{{{\sin }^2}\theta } \mathord{\left/ {\vphantom {{{{\sin }^2}\theta } {{\varepsilon _\parallel }}}} \right. } {{\varepsilon _\parallel }}}} \right)} + {\varepsilon _ \bot }\sqrt {1 - {{\sin }^2}\theta } }}{{\sqrt {{\varepsilon _ \bot }\left( {1 - {{{{\sin }^2}\theta } \mathord{\left/ {\vphantom {{{{\sin }^2}\theta } {{\varepsilon _\parallel }}}} \right. } {{\varepsilon _\parallel }}}} \right)} - {\varepsilon _ \bot }\sqrt {1 - {{\sin }^2}\theta } }}} \right) + \left( {\frac{{{\varepsilon _ \bot }\sqrt {1 - {{\sin }^2}\theta } + \sqrt {{\varepsilon _ \bot }\left( {1 - {{{{\sin }^2}\theta } \mathord{\left/ {\vphantom {{{{\sin }^2}\theta } {{\varepsilon _\parallel }}}} \right. } {{\varepsilon _\parallel }}}} \right)} }}{{{\varepsilon _ \bot }\sqrt {1 - {{\sin }^2}\theta } - \sqrt {{\varepsilon _ \bot }\left( {1 - {{{{\sin }^2}\theta } \mathord{\left/ {\vphantom {{{{\sin }^2}\theta } {{\varepsilon _\parallel }}}} \right. } {{\varepsilon _\parallel }}}} \right)} }}} \right){e^{ - 2j{k_y}d}}}}} \right. } {1 + \left( {\frac{{\sqrt {{\varepsilon _ \bot }\left( {1 - {{{{\sin }^2}\theta } \mathord{\left/ {\vphantom {{{{\sin }^2}\theta } {{\varepsilon _\parallel }}}} \right. } {{\varepsilon _\parallel }}}} \right)} + {\varepsilon _ \bot }\sqrt {1 - {{\sin }^2}\theta } }}{{\sqrt {{\varepsilon _ \bot }\left( {1 - {{{{\sin }^2}\theta } \mathord{\left/ {\vphantom {{{{\sin }^2}\theta } {{\varepsilon _\parallel }}}} \right. } {{\varepsilon _\parallel }}}} \right)} - {\varepsilon _ \bot }\sqrt {1 - {{\sin }^2}\theta } }}} \right) + \left( {\frac{{{\varepsilon _ \bot }\sqrt {1 - {{\sin }^2}\theta } + \sqrt {{\varepsilon _ \bot }\left( {1 - {{{{\sin }^2}\theta } \mathord{\left/ {\vphantom {{{{\sin }^2}\theta } {{\varepsilon _\parallel }}}} \right. } {{\varepsilon _\parallel }}}} \right)} }}{{{\varepsilon _ \bot }\sqrt {1 - {{\sin }^2}\theta } - \sqrt {{\varepsilon _ \bot }\left( {1 - {{{{\sin }^2}\theta } \mathord{\left/ {\vphantom {{{{\sin }^2}\theta } {{\varepsilon _\parallel }}}} \right. } {{\varepsilon _\parallel }}}} \right)} }}} \right){e^{ - 2j{k_y}d}}}}. \end{array}$$

Appendix B. The detailed derivation of dispersion relationship for TE waves

When the thickness of dielectric layer is much smaller than the wavelength of the incident waves, the graphene-based metamaterial can be considered as an effective anisotropy medium:

$${\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\leftrightarrow$}} \over \varepsilon } _{stack}}\textrm{ = }\left[ {\begin{array}{ccc} {{\varepsilon_x}}&0&0\\ 0&{{\varepsilon_y}}&0\\ 0&0&{{\varepsilon_z}} \end{array}} \right],$$
where ${\varepsilon _x} = {\varepsilon _\textrm{z}} \approx {\varepsilon _g}({{{{t_g}} \mathord{\left/ {\vphantom {{{t_g}} {{t_d}}}} \right.} {{t_d}}}} )+ {\varepsilon _d}$, ${\varepsilon _\textrm{y}} \approx {\varepsilon _d}$ and ${\varepsilon _g} = 1 - \frac{{i\sigma }}{{\omega {\varepsilon _0}{t_g}}}$ . The dynamic conductivity of the pumped monolayer graphene and effective dielectric tensor of the stack is displayed in Fig. 7(a) and 7(b). It can be seen from Fig. 7(b) that the pumped graphene sheets make ${\varepsilon _{x/\textrm{z}}}$ a positive imaginary part, indicating a gain region. While for the real part, it changes from negative to positive with the increasing frequency due to the imaginary part of the conductivity of the pumped graphene, meaning a hyperbolic or elliptical dispersive characteristic depending on the frequency.

 figure: Fig. 7.

Fig. 7. (a) The normalized conductivity by ${{{e^2}} \mathord{\left/ {\vphantom {{{e^2}} {4\hbar }}} \right.} {4\hbar }}$ of the pumped monolayer graphene sheet; (b) the effective dielectric tensor of the structure in the parallel direction of the graphene sheet.

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When a wave propagates along the direction of parallel media in an anisotropic medium, the medium can be considered as a waveguide structure. The longitudinal field component of TE wave can be written as:

$${H_z}^{stack} = {B_{stack}}^a{e^{ - j{k_z}z - j{k_y}y}} + {B_{stack}}^b{e^{ - j{k_z}z + j{k_y}y}}.$$
According to the Maxwell equation, the electric field expression can be obtained as:
$$\begin{array}{l} \nabla \times \overrightarrow H = j\omega {\varepsilon _x}{E_x},\\ {E_x} = \frac{1}{{j\omega {\varepsilon _x}}}\nabla \times \overrightarrow H = \frac{1}{{j\omega {\varepsilon _x}}}\left( {\frac{{\partial {H_z}}}{{\partial y}} - \frac{{\partial {H_y}}}{{\partial z}}} \right). \end{array}$$
The corresponding field expression for each region can be written as:
$$\begin{array}{l} \left\{ \begin{array}{l} {H_z}^{stack} = {B_{stack}}^a{e^{ - j{k_z}z}}{e^{ - j{k_y}y}} + {B_{stack}}^b{e^{ - j{k_z}z}}{e^{j{k_y}y}}\\ {E_x}^{stack} ={-} {B_{stack}}^a\frac{{{k_0}^2}}{{\omega {k_y}}}{e^{ - j{k_z}z}}{e^{ - j{k_y}y}} + {B_{stack}}^b\frac{{{k_0}^2}}{{\omega {k_y}}}{e^{ - j{k_z}z}}{e^{j{k_y}y}} \end{array} \right.,\\ \left\{ \begin{array}{l} {H_z}^{inc} = {B_{inc}}^b{e^{ - j{k_z}z}}{e^{j{k_1}y}}\\ {E_x}^{inc} = {B_{inc}}^b\frac{{{k_0}^2}}{{\omega {k_1}}}{e^{ - j{k_z}z}}{e^{j{k_1}y}} \end{array} \right.,\\ \left\{ \begin{array}{l} {H_z}^{eme} = {B_{eme}}^a{e^{ - j{k_z}z}}{e^{ - j{k_3}y}}\\ {E_x}^{eme} ={-} {B_{eme}}^a\frac{{{k_0}^2}}{{\omega {k_3}}}{e^{ - j{k_z}z}}{e^{ - j{k_3}y}} \end{array} \right.. \end{array}$$
where ${k_1} = \sqrt {k_0^2 - k_z^2}$, ${k_y} = \sqrt {{\varepsilon _\parallel }k_0^2 - k_z^2}$, ${k_3} = \sqrt {k_0^2 - k_z^2}$. The following equations can be obtained by substituting the boundary conditions:
$$\begin{array}{l} {B_{inc}}^b = {B_{stack}}^a + {B_{stack}}^b\\ {B_{inc}}^b\frac{{{k_0}^2}}{{\omega {k_1}}} ={-} {B_{stack}}^a\frac{{{k_0}^2}}{{\omega {k_y}}} + {B_{stack}}^b\frac{{{k_0}^2}}{{\omega {k_y}}}\\ {B_{stack}}^a{e^{ - j{k_y}d}} + {B_{stack}}^b{e^{j{k_y}d}} = {B_{eme}}^a{e^{ - j{k_3}d}}\\ - {B_{stack}}^a\frac{{{k_0}^2}}{{\omega {k_y}}}{e^{ - j{k_y}d}} + {B_{stack}}^b\frac{{{k_0}^2}}{{\omega {k_y}}}{e^{j{k_y}d}} ={-} {B_{eme}}^a\frac{{{k_0}^2}}{{\omega {k_3}}}{e^{ - j{k_3}d}}. \end{array}$$
To obtain a non-zero solution for the above equations, the following conditions must be satisfied:
$${k_y}^2 + {k_z}^2 = {\varepsilon _x}{k_0}^2,$$
$$\left|{\begin{array}{cccc} 1&{ - 1}&{ - 1}&0\\ {\frac{{{k_0}^2}}{{\omega {k_1}}}}&{\frac{{{k_0}^2}}{{\omega {k_y}}}}&{ - \frac{{{k_0}^2}}{{\omega {k_y}}}}&0\\ 0&{{e^{ - j{k_y}d}}}&{{e^{j{k_y}d}}}&{ - {e^{ - j{k_3}d}}}\\ 0&{ - \frac{{{k_0}^2}}{{\omega {k_y}}}{e^{ - j{k_y}d}}}&{\frac{{{k_0}^2}}{{\omega {k_y}}}{e^{j{k_y}d}}}&{\frac{{{k_0}^2}}{{\omega {k_3}}}{e^{ - j{k_3}d}}} \end{array}} \right|= 0.$$
Thus, the dispersion relation for TE waves can be written as:
$$({{k_1} + {k_z}} )({{k_3} + {k_z}} ){e^{j{k_z}d}} = ({{k_1} - {k_z}} )({{k_3} - {k_z}} ){e^{ - j{k_z}d}}.$$

Funding

Key Technologies Research and Development Program (2018YFB1801503); National Natural Science Foundation of China (61771327, 61921002, 61931006).

Disclosures

The authors declare no conflicts of interest.

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Figures (7)
Fig. 1.
Fig. 1. Schematic of the graphene-based metamaterial. The metamaterial consists of alternative pumped graphene sheet and dielectric layer. The dielectric layer is PMMA with ${\varepsilon _d}$  = 4 and a thickness ${t_d}$  = 2 um, and total height of the structure is = 300 um.

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Fig. 2.
Fig. 2. The distribution of negative real-part of graphene complex conductivity. Calculated surface conductivity versus frequency and pump intensity at the temperature of 77 K (a) and 300 K (b).

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Fig. 3.
Fig. 3. Calculated negative absorption versus frequency and angle with 4.8 meV of graphene, corresponding to 3.2 W/cm2 of pump power at 77 K. The dashed line and yellow hollow circle represent resonance modes and dispersion, respectively. The calculating steps of the frequency and the incident angle are 0.01 THz and 0.1 °.

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Fig. 4.
Fig. 4. (a), (c) and (e) are the absorption spectra with incident angles 30°, 45° and 60°, respectively. (b), (d) and (e) are the distribution of magnetic field (under the same display settings) at P∼M4, respectively.

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Fig. 5.
Fig. 5. (a), (b) and (c) are transmission, reflection and absorption spectra under various Fermi energy of graphene with d=300 um and θ = 0°, respectively; (d) The maximum absolute absorption values are plotted as a function of the pump intensity.

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Fig. 6.
Fig. 6. Negative absorption versus frequency and angle with carrier relaxation time is (a)10ps, (b) 5ps and (c) 1ps, respectively. The calculating steps of the frequency and the incident angle are 0.1 THz and 0.1 °.

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Fig. 7.
Fig. 7. (a) The normalized conductivity by ${{{e^2}} \mathord{\left/ {\vphantom {{{e^2}} {4\hbar }}} \right.} {4\hbar }}$ of the pumped monolayer graphene sheet; (b) the effective dielectric tensor of the structure in the parallel direction of the graphene sheet.

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Equations (24)

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ε ↔ s t a c k  =  [ ε x 0 0 0 ε y 0 0 0 ε z ] ,
∇ 2 [ E → x E → y E → z ] − ∇ ( ∇ ⋅ [ E → x E → y E → z ] ) − ω 2 μ 0 [ ε x ε y ε z ] ⋅ [ E → x E → y E → z ]  =  0 ,
A T E = 1 − [ ( 1 − ε ∥ − sin 2 θ 1 − sin 2 θ ) ( 1 + 1 − sin 2 θ ε ∥ − sin 2 θ ) e j k y d + ( 1 + ε ∥ − sin 2 θ 1 − sin 2 θ ) ( 1 − 1 − sin 2 θ ε ∥ − sin 2 θ ) e − j k y d + 4 e j ( ω / ω c c ) 2 − k z 2 d ] / 4 e j ( ω / ω c c ) 2 − k z 2 d ] [ ( 1 + ε ∥ − sin 2 θ 1 − sin 2 θ ) ( 1 + 1 − sin 2 θ ε ∥ − sin 2 θ ) e j k y d + ( 1 − ε ∥ − sin 2 θ 1 − sin 2 θ ) ( 1 − 1 − sin 2 θ ε ∥ − sin 2 θ ) e − j k y d ] [ ( 1 + ε ∥ − sin 2 θ 1 − sin 2 θ ) ( 1 + 1 − sin 2 θ ε ∥ − sin 2 θ ) e j k y d + ( 1 − ε ∥ − sin 2 θ 1 − sin 2 θ ) ( 1 − 1 − sin 2 θ ε ∥ − sin 2 θ ) e − j k y d ] , A T M = 1 − ( ( − 2 ε ⊥ ( 1 − sin 2 θ / sin 2 θ ε ∥ ε ∥ ) − ε ⊥ ( 1 − sin 2 θ / sin 2 θ ε ∥ ε ∥ ) + ε ⊥ 1 − sin 2 θ ) ( 2 ε ⊥ 1 − sin 2 θ ε ⊥ 1 − sin 2 θ − ε ⊥ ( 1 − sin 2 θ / sin 2 θ ε ∥ ε ∥ ) ) e − j k y d + ( ε ⊥ ( 1 − sin 2 θ / sin 2 θ ε ∥ ε ∥ ) + ε ⊥ 1 − sin 2 θ ε ⊥ ( 1 − sin 2 θ / sin 2 θ ε ∥ ε ∥ ) − ε ⊥ 1 − sin 2 θ ) ( ε ⊥ 1 − sin 2 θ + ε ⊥ ( 1 − sin 2 θ / sin 2 θ ε ∥ ε ∥ ) ε ⊥ 1 − sin 2 θ − ε ⊥ ( 1 − sin 2 θ / sin 2 θ ε ∥ ε ∥ ) ) e − 2 j k y d ) / ( ε ⊥ 1 − sin 2 θ + ε ⊥ ( 1 − sin 2 θ / sin 2 θ ε ∥ ε ∥ ) ε ⊥ 1 − sin 2 θ − ε ⊥ ( 1 − sin 2 θ / sin 2 θ ε ∥ ε ∥ ) ) e − 2 j k y d ) 1 + ( ε ⊥ ( 1 − sin 2 θ / sin 2 θ ε ∥ ε ∥ ) + ε ⊥ 1 − sin 2 θ ε ⊥ ( 1 − sin 2 θ / sin 2 θ ε ∥ ε ∥ ) − ε ⊥ 1 − sin 2 θ ) + ( ε ⊥ 1 − sin 2 θ + ε ⊥ ( 1 − sin 2 θ / sin 2 θ ε ∥ ε ∥ ) ε ⊥ 1 − sin 2 θ − ε ⊥ ( 1 − sin 2 θ / sin 2 θ ε ∥ ε ∥ ) ) e − 2 j k y d 1 + ( ε ⊥ ( 1 − sin 2 θ / sin 2 θ ε ∥ ε ∥ ) + ε ⊥ 1 − sin 2 θ ε ⊥ ( 1 − sin 2 θ / sin 2 θ ε ∥ ε ∥ ) − ε ⊥ 1 − sin 2 θ ) + ( ε ⊥ 1 − sin 2 θ + ε ⊥ ( 1 − sin 2 θ / sin 2 θ ε ∥ ε ∥ ) ε ⊥ 1 − sin 2 θ − ε ⊥ ( 1 − sin 2 θ / sin 2 θ ε ∥ ε ∥ ) ) e − 2 j k y d .
σ ( ω ) = σ int r a ( ω ) + σ int e r ( ω ) = e 2 4 ℏ 8 k B T τ π ℏ ( 1 + j ω τ ) ln ⁡ ( 1 + exp ⁡ ( E f k B T ) ) + e 2 4 ℏ ( tanh ⁡ ( ℏ ω − 2 E f 4 k B T ) + 4 ℏ ω j π ∫ 0 ∞ G ( ε , E f ) − G ( ℏ ω / ℏ ω 2 2 , E f ) ( ℏ ω ) 2 − 4 ε 2 d ε ) ,
( k 1 + k z ) ( k 3 + k z ) e j k z d = ( k 1 − k z ) ( k 3 − k z ) e − j k z d .
∇ 2 [ E → x E → y E → z ] − ∇ ( ∇ ⋅ [ E → x E → y E → z ] ) − ω 2 μ 0 [ ε x ε y ε z ] ⋅ [ E → x E → y E → z ]  =  0.
E →  =  ( e → x E x + e → y E y + e → z E z ) e − j k x x e − j k y y e − j k z z ,
| k y 2 + k z 2 − ω 2 μ ε x − k x k y − k x k z − k x k y k x 2 + k z 2 − ω 2 μ ε y − k z k y − k x k z − k y k z k x 2 + k y 2 − ω 2 μ ε z | = 0.
ω 2 μ 0 ε y − k z 2 − k y 2 = 0.
H z s t a c k = B s t a c k a e − j k z z − j k y y + B s t a c k b e − j k z z + j k y y .
∇ × H → = j ω ε x E x E x = 1 j ω ε x ∇ × H → = 1 j ω ε x ( ∂ H z ∂ y − ∂ H y ∂ z ) .
E x = − B s t a c k a k y 2 + k z 2 ω ε x k y e − j k z z − j k y y + B s t a c k b k y 2 + k z 2 ω ε x k y e − j k z z + j k y y .
{ H z s t a c k = B s t a c k a e − j k z z − j k y y + B s t a c k b e − j k z z + j k y y E x s t a c k = − B s t a c k a k y 2 + k z 2 ω ε x k y e − j k z z − j k y y + B s t a c k b k y 2 + k z 2 ω ε x k y e − j k z z + j k y y , { H z i n c = B i n c a e − j k z z − j k 1 y + B i n c b e − j k z z + j k 1 y E x i n c = − B i n c a k 0 2 ω k 1 e − j k z z − j k 1 y + B i n c b k 0 2 ω k 1 e − j k z z + j ( k y ) I y k 1 , { H z e m e = B e m e a e − j k z z − j k 3 y E x e m e = − B e m e a k 0 2 ω k 3 e − j k z z − j k 3 y .
E x i n c | y = 0 = E x s t a c k | y = 0 H z i n c | y = 0 = H z s t a c k | y = 0 E x s t a c k | y = d = E x e m e | y = d H z s t a c k | y = d = H z e m e | y = d .
R = ( 1 − k y k 1 ) ( 1 + k 3 k y ) e j k y d + ( 1 + k y k 1 ) ( 1 − k 3 k y ) e − j k y d ( 1 + k y k 1 ) ( 1 + k 3 k y ) e j k y d + ( 1 − k y k 1 ) ( 1 − k 3 k y ) e − j k y d , T = 4 e j k 3 d ( 1 + k y k 1 ) ( 1 + k 3 k y ) e j k y d + ( 1 − k y k 1 ) ( 1 − k 3 k y ) e − j k y d .
A T E = 1 − [ ( 1 − ε ∥ − sin 2 θ 1 − sin 2 θ ) ( 1 + 1 − sin 2 θ ε ∥ − sin 2 θ ) e j k y d + ( 1 + ε ∥ − sin 2 θ 1 − sin 2 θ ) ( 1 − 1 − sin 2 θ ε ∥ − sin 2 θ ) e − j k y d + 4 e j ( ω / ω c c ) 2 − k z 2 d ] / 4 e j ( ω / ω c c ) 2 − k z 2 d ] [ ( 1 + ε ∥ − sin 2 θ 1 − sin 2 θ ) ( 1 + 1 − sin 2 θ ε ∥ − sin 2 θ ) e j k y d + ( 1 − ε ∥ − sin 2 θ 1 − sin 2 θ ) ( 1 − 1 − sin 2 θ ε ∥ − sin 2 θ ) e − j k y d ] [ ( 1 + ε ∥ − sin 2 θ 1 − sin 2 θ ) ( 1 + 1 − sin 2 θ ε ∥ − sin 2 θ ) e j k y d + ( 1 − ε ∥ − sin 2 θ 1 − sin 2 θ ) ( 1 − 1 − sin 2 θ ε ∥ − sin 2 θ ) e − j k y d ] , A T M = 1 − ( ( − 2 ε ⊥ ( 1 − sin 2 θ / sin 2 θ ε ∥ ε ∥ ) − ε ⊥ ( 1 − sin 2 θ / sin 2 θ ε ∥ ε ∥ ) + ε ⊥ 1 − sin 2 θ ) ( 2 ε ⊥ 1 − sin 2 θ ε ⊥ 1 − sin 2 θ − ε ⊥ ( 1 − sin 2 θ / sin 2 θ ε ∥ ε ∥ ) ) e − j k y d + ( ε ⊥ ( 1 − sin 2 θ / sin 2 θ ε ∥ ε ∥ ) + ε ⊥ 1 − sin 2 θ ε ⊥ ( 1 − sin 2 θ / sin 2 θ ε ∥ ε ∥ ) − ε ⊥ 1 − sin 2 θ ) ( ε ⊥ 1 − sin 2 θ + ε ⊥ ( 1 − sin 2 θ / sin 2 θ ε ∥ ε ∥ ) ε ⊥ 1 − sin 2 θ − ε ⊥ ( 1 − sin 2 θ / sin 2 θ ε ∥ ε ∥ ) ) e − 2 j k y d ) / ( ε ⊥ 1 − sin 2 θ + ε ⊥ ( 1 − sin 2 θ / sin 2 θ ε ∥ ε ∥ ) ε ⊥ 1 − sin 2 θ − ε ⊥ ( 1 − sin 2 θ / sin 2 θ ε ∥ ε ∥ ) ) e − 2 j k y d ) 1 + ( ε ⊥ ( 1 − sin 2 θ / sin 2 θ ε ∥ ε ∥ ) + ε ⊥ 1 − sin 2 θ ε ⊥ ( 1 − sin 2 θ / sin 2 θ ε ∥ ε ∥ ) − ε ⊥ 1 − sin 2 θ ) + ( ε ⊥ 1 − sin 2 θ + ε ⊥ ( 1 − sin 2 θ / sin 2 θ ε ∥ ε ∥ ) ε ⊥ 1 − sin 2 θ − ε ⊥ ( 1 − sin 2 θ / sin 2 θ ε ∥ ε ∥ ) ) e − 2 j k y d 1 + ( ε ⊥ ( 1 − sin 2 θ / sin 2 θ ε ∥ ε ∥ ) + ε ⊥ 1 − sin 2 θ ε ⊥ ( 1 − sin 2 θ / sin 2 θ ε ∥ ε ∥ ) − ε ⊥ 1 − sin 2 θ ) + ( ε ⊥ 1 − sin 2 θ + ε ⊥ ( 1 − sin 2 θ / sin 2 θ ε ∥ ε ∥ ) ε ⊥ 1 − sin 2 θ − ε ⊥ ( 1 − sin 2 θ / sin 2 θ ε ∥ ε ∥ ) ) e − 2 j k y d .
ε ↔ s t a c k  =  [ ε x 0 0 0 ε y 0 0 0 ε z ] ,
H z s t a c k = B s t a c k a e − j k z z − j k y y + B s t a c k b e − j k z z + j k y y .
∇ × H → = j ω ε x E x , E x = 1 j ω ε x ∇ × H → = 1 j ω ε x ( ∂ H z ∂ y − ∂ H y ∂ z ) .
{ H z s t a c k = B s t a c k a e − j k z z e − j k y y + B s t a c k b e − j k z z e j k y y E x s t a c k = − B s t a c k a k 0 2 ω k y e − j k z z e − j k y y + B s t a c k b k 0 2 ω k y e − j k z z e j k y y , { H z i n c = B i n c b e − j k z z e j k 1 y E x i n c = B i n c b k 0 2 ω k 1 e − j k z z e j k 1 y , { H z e m e = B e m e a e − j k z z e − j k 3 y E x e m e = − B e m e a k 0 2 ω k 3 e − j k z z e − j k 3 y .
B i n c b = B s t a c k a + B s t a c k b B i n c b k 0 2 ω k 1 = − B s t a c k a k 0 2 ω k y + B s t a c k b k 0 2 ω k y B s t a c k a e − j k y d + B s t a c k b e j k y d = B e m e a e − j k 3 d − B s t a c k a k 0 2 ω k y e − j k y d + B s t a c k b k 0 2 ω k y e j k y d = − B e m e a k 0 2 ω k 3 e − j k 3 d .
k y 2 + k z 2 = ε x k 0 2 ,
| 1 − 1 − 1 0 k 0 2 ω k 1 k 0 2 ω k y − k 0 2 ω k y 0 0 e − j k y d e j k y d − e − j k 3 d 0 − k 0 2 ω k y e − j k y d k 0 2 ω k y e j k y d k 0 2 ω k 3 e − j k 3 d | = 0.
( k 1 + k z ) ( k 3 + k z ) e j k z d = ( k 1 − k z ) ( k 3 − k z ) e − j k z d .
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