Multifunctional graphene metamaterials based on polarization-insensitive plasmon-induced transparency

Liangyu Tao; Xin-Hua Deng; Xin-Hua Deng; Xin-Hua Deng; Pingsheng Zhang; Ming Lu; Jiren Yuan

doi:10.1364/OE.512302

1. Introduction

The electromagnetically induced transparency (EIT) effect was originally generated in a three-level atomic system, which is caused by the coherent process generated by the mutual transition of its electrons on different paths. It will make the originally opaque medium transparent in a certain frequency band and form a resonant window [1,2]. Due to the harsh experimental conditions, including the need for large-scale high-power laser systems and low-temperature environments, the application of the EIT effect in the fields of slow light, filtering and optical switching is hindered [3]. Plasmon-induced transparency (PIT) is an EIT-like effect based on plasmonic structure, which has attracted much attention due to its EIT characteristics at room temperature, small size and easy on-chip integration [4–6]. In 2008, Zhang et al. first proposed the PIT theory of metallic silver metamaterials in the terahertz band [4]. In 2011, Huang et al. realized the PIT effect using a metal-dielectric-metal (MDM) plasmonic waveguide structure [7]. So far, the structures that realize the PIT effect mainly include plasmonic waveguide structure [7,8], metamaterial structure [4,5], metal grating and dielectric waveguide layer coupling (GCDWL) structure [9], etc. Among them, the metamaterial structure has received extensive attention from researchers due to its good slow light characteristics and dynamic tunability.

Surface plasmons (SPs) are evanescent waves localized at the metal-dielectric interface. They have significant near-field enhancement and can break through the optical diffraction limit, laying the foundation for optical manipulation on the sub-wavelength scale [10,11]. Graphene is a two-dimensional material with a hexagonal honeycomb lattice structure formed by the hybridization of single-layer carbon atoms $s{p^2}$, which can support SPs [12,13]. Compared with traditional metal metamaterials, the dielectric properties of graphene can be dynamically adjusted by doping or adding gate voltage [14]. In recent years, there have been many reports on the realization of PIT effect in graphene metamaterials, and researchers have designed related applications, including photoelectric switches, multi-channel filters, slow light, sensors, and absorbers [15–17].

In this paper, a graphene metamaterial is proposed, which can achieve dynamically adjustable and polarization-insensitive PIT effect in terahertz band. In previous polarization-insensitive PIT devices, the PIT effect is mostly generated by bright-bright modes coupling. However, in this paper, the PIT effect can be generated by bright-dark modes coupling. In the application performance, the photoelectric switch based on this structure not only supports linearly polarized light, but also supports left-handed and right-handed circularly polarized light. This structure is applied to the field of sensors to achieve ultra-high sensitivity. In addition, this structure can achieve the slow light effect of high group refractive index. Therefore, the designed graphene metamaterials have great application potential in many fields.

2. Structure and theory

The designed periodic graphene metamaterial structural unit is shown in Fig. 1(a). The structural period of the element is ${P_x} = {P_y} = 11\mu m$. The single-layer graphene is sandwiched between two layers of silica, and the graphene pattern is composed of four L-shaped graphene strips. The specific parameters of the structural unit are as follows: ${L_1} = 0.8\mu \textrm{m}$, ${L_2} = 1.4\mu \textrm{m}$, ${L_3} = 2.9\mu m$, ${L_4} = 4.3\mu \textrm{m}$. The thickness of the upper and lower layers is ${d_1} = 0.05\mu \textrm{m}$ and ${d_2} = 0.15\mu \textrm{m}$ respectively. The relative dielectric constant of silica is equal to 3.9 [18].

Fig. 1. (a) Three-dimensional structure diagram of graphene metamaterial structural unit. (b) Top view and side view of graphene metamaterial structural unit.

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The surface conductivity of graphene can be expressed as [16]:

(1)$$\sigma (\omega ) = {\sigma _{i \textrm{ntra}}}(\omega ) + {\sigma _{\textrm{inter}}}(\omega )$$

where ${\sigma _{\textrm{inter}}}$ is the inter-band conductivity, ${\sigma _{\textrm{intra}}}$ is the intra-band conductivity, $\omega $ is the angular frequency. ${\sigma _{\textrm{intra}}}$ and ${\sigma _{\textrm{inter}}}$ are expressed as [16]:

(2)$${\sigma _{\textrm{intra}}} = j\frac{{{e^2}{k_B}T}}{{\pi {\hbar ^2}({w + j{\tau^{ - 1}}} )}}\left\{ {\frac{{{E_F}}}{{{k_B}T}} + 2\ln \left[ {\exp \left( { - \frac{{{E_F}}}{{{k_B}T}}} \right) + 1} \right]} \right\}$$

(3)$${\sigma _{\textrm{inter }}}(\omega ) = \textrm{j}\frac{{{e^2}}}{{4\pi \hbar }}\ln \left[ {\frac{{2{E_\textrm{F}} - ({\omega + \textrm{j}{\tau^{ - 1}}} )\hbar }}{{2{E_\textrm{F}} + ({\omega + \textrm{j}{\tau^{ - 1}}} )\hbar }}} \right]$$

where j is the imaginary unit, e represents the elementary charge, $\hbar $ is the reduced Planck constant, T represents the temperature (set to $300K$), ${k_B}$ represents the Boltzmann constant, $\tau $ is the relaxation time, and ${E_F}$ represents the Fermi level of graphene. Because the conductivity of graphene is mainly affected by ${\sigma _{{\mathop{\rm int}} ra}}$ in the terahertz band, ${\sigma _{{\mathop{\rm int}} er}}$ can be ignored. In addition, under the simulation conditions, $T = 300K$, so the Fermi level ${E_F}$ of graphene is much larger than ${k_B}T$, and the conductivity of graphene is commonly approximated by the Drude model as follows [17]:

(4)$$\sigma (\omega ) \approx {\sigma _{{\mathop{\rm int}} \textrm{ra}}}(\omega ) = j\frac{{{e^2}{E_F}}}{{\pi {\hbar ^2}({\omega + j{\tau^{ - 1}}} )}}$$

where $\tau $ can be expressed as $\tau = \mu {E_F}/({eV_F^2} )$, ${V_F} = \textrm{1} \times \textrm{1}{\textrm{0}^\textrm{6}}\textrm{m/s}$ denotes Fermi velocity and $\mu = \textrm{4}{\textrm{m}^\textrm{2}}\textrm{/Vs}$ is the carrier mobility. The graphene pattern shown in Fig. 1 can be obtained by the following process: first, the sing le-layer graphene is generated by chemical vapor deposition (CVD) method and transferred to the $\textrm{Si}{\textrm{O}_\textrm{2}}$ dielectric, and finally the graphene pattern in the structure is obtained by electron beam lithography (EBL) [19,20]. In this paper, it is assumed that the terahertz wave irradiates vertically on the surface of the structure., the periodic boundary conditions are set in the x and y directions of the structure, and the perfectly matched layer (PML) is set in the z direction. The COMSOL Multiphysics 6.0 frequency domain solver based on the finite element method (FEM) is used to simulate the structure. The Fermi level of graphene and the polarization angle $\theta $ of the incident wave are set to 0.8 eV and ${0^ \circ }$, respectively, unless otherwise stated.

In addition, in order to reveal the physical mechanism of PIT generation, we use the widely used Lorentz coupling model to analyze the interaction between the bright mode and the dark mode. The bright-dark mode is equivalent to two harmonic oscillator models to describe, then the coupling differential equation between the two modes is [4]

(5)$$\left\{ {\begin{array}{l} {{{\ddot{x}}_1}(t) + {\gamma_1}{{\dot{x}}_1}(t) + \omega_0^2{x_1}(t) - {\kappa^2}{x_2}(t) = gE(t)}\\ {{{\ddot{x}}_2}(t) + {\gamma_2}{{\dot{x}}_2}(t) + {{({{\omega_0} + \delta } )}^2}{x_2}(t) - {\kappa^2}{x_1}(t) = 0} \end{array}} \right.$$

where ${x_1}(t)$ and ${x_2}(t)$ denote the bright-mode and dark-mode resonance strengths, respectively. ${\omega _0}$ represents the resonant frequency of the bright mode, ${\gamma _1}$ represents the damping factor of the bright mode, ${\gamma _2}$ represents the damping factor of the dark mode, $\delta $ represents the resonant detuning frequency of the bright mode and the dark mode, k represents the coupling coefficient of the bright mode and the dark mode, g is the coupling factor of the incident electromagnetic wave electric field and the bright mode. Let ${x_1}(t) = {x_1}\exp (j\omega t)$, ${x_2}(t) = {x_2}\exp (j\omega t)$ and $E(t) = {E_0}\exp( j\omega \textrm{t}) $ represent the time domain changes. By solving Eq. (5) with the approximation ${\omega ^2} - \omega _0^2 \approx 2({\omega - {\omega_0}} ){\omega _0}$, the relationship between the transmission coefficient and the frequency is as follows [3]:

(6)$$T = 1 - \frac{{g({\omega - {\omega_0} - \delta } )+ i\frac{{{\gamma _2}}}{2}}}{{\left( {\omega - {\omega_0} + i\frac{{{\gamma_1}}}{2}} \right)\left( {\omega - {\omega_0} - \delta + i\frac{{{\gamma_2}}}{2}} \right) - \frac{{{\kappa ^2}}}{4}}}$$

3. Results and discussion

The transmission spectrum of the complete graphene metamaterial is obtained by using the FEM method. As shown in the blue curve in Fig. 2(a), a PIT window is formed at 4.45 THz of the transmission spectrum. In order to fully understand the physical mechanism of the PIT effect, we decompose the complete graphene structure into four transverse graphene strips and four longitudinal graphene strips. In order to facilitate the representation, four transverse graphene strips and four longitudinal graphene strips are called ${G_1}$ and ${G_2}$ respectively.

Fig. 2. (a) The transmission spectra of ${G_1}$, ${G_2}$ and the complete structure. (b) The electric field distribution of ${G_2}$ at “b” point. (c) The electric field distribution map of ${G_1}$, at the transmission valley. (d) electric field distribution of entire structure at transmission peak. (e) The electric field distribution map at dip1 (3.61 THz) of the overall structure. (f) The electric field distribution at dip2 (5.31 THz) of the whole structure.

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The transmission spectra of ${G_1}$ and ${G_2}$ correspond to the red curve and green line in Fig. 2(a), respectively. Obviously, ${G_1}$ can directly interact with the incident light to form a transmission valley, and the transmission valley frequency is 4.51 THz. This part of the structure is regarded as a bright mode. The transmission spectrum of ${G_2}$ corresponds to the green line in Fig. 2(a), and the transmittance is close to 100%, indicating that it cannot directly interact with the incident light, and this part of the structure is regarded as a dark mode. In the complete structure, the dark mode is indirectly excited by the bright mode, and the transparent window is generated due to the destructive interference between the bright and dark modes, as shown in the blue line in Fig. 2(a). Next, the electric field distribution analysis is used to further explain the generation of PIT phenomenon. As shown in Fig. 2(c), when the incident wave is irradiated on ${G_1}$, many electrons gather at both ends of the graphene strip to form a strong local light field, which indicates that the mode is directly excited by the incident wave. As shown in Fig. 2(b), when the incident wave is irradiated on ${G_2}$, no local light field is formed, which indicates that it can not be directly excited by the incident wave. As shown in Fig. 2(d), when the incident wave is illuminated on the complete structure, due to the near-field coupling between the bright mode and the dark mode, ${G_2}$ is excited by the ${G_1}$ near-field, which makes the strong local field excited by ${G_1}$ disappear, thus forming a transparent window. At the same time, the field distributions of two transmission dips with frequencies of dip1 (1.98 THz) and dip2 (3.69 THz) are also given, showing strong local field distribution on the complete structure, as shown in Figs. 2(e) and 2(f).

The wide application of graphene in metamaterial structure benefits from its tuning characteristics. The resonance characteristics of the structure can be regulated by electric field, magnetic field, optical field and chemical doping, among which the applied gate voltage is the most common [21,22]. The Fermi level of graphene can be expressed as [23]:

(7)$${E_F} = \hbar {v_F}\sqrt {\pi {n_s}}$$

where ${v_F}$ is the Fermi velocity, and $ {n_s}$ is the doping concentration of carriers. The Fermi level of graphene is related to the applied bias, that is, adjusting the applied bias voltage can control the Fermi level of graphene, thus affecting the surface conductivity of graphene, and ultimately affecting the resonance characteristics of the structure without changing the geometric parameters. The transmission spectra of metamaterials at different Fermi levels of graphene are plotted, as shown in Fig. 3(a) red curve. With the increase of the Fermi level of graphene, the transparent window has a blue shift. In addition, we also introduce the Lorentz resonance model to fit the transmission spectrum of the FEM numerical simulation, and the results are shown in Fig. 3(a) with a black imaginary curve. The observation results show that the simulation results are highly consistent with the theoretical results.

Fig. 3. (a) The transmission spectra at different Fermi levels, in which the numerical simulation results are represented by red real curves, and the theoretical results are represented by black lines. (b) Theoretical design of photoelectric switch. (c) Transmission spectrum at different polarization angle.

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When the transmission spectrum moves with the change of Fermi level, there will be two states of high transmittance and low transmittance at some frequencies, which can be used to design photoelectric switches. The transmission spectra of ${E_F} = 0.8eV$ and ${E_F} = 1.2eV$ are taken as an example, as shown in Fig. 3(b). Low transmittance at ${f_1}(3.61THz)$ and ${f_3}(5.31THz)$ is defined as “off” state, while high transmittance at ${f_2}(4.38THz)$ and ${f_4}(6.46THz)$ is defined as “on” state when the Fermi level is 0.8 eV. The states of ${f_1}(3.61THz)$ and ${f_3}(5.31THz)$ become “on” at the same time, and the states of ${f_2}(4.38THz)$ and ${f_4}(6.46THz)$ become “off” at the same time when the Fermi level is switched to $1.2eV$.At the same time, the modulation depth (MD) and insertion loss (IL) are used to evaluate the performance of the photoelectric switch. The modulation depth (MD) is used to represent the degree of transmittance change, which can be expressed as [24]:

(8)$$\textrm{MD} = \frac{{|{{T_{\textrm{on}}} - {T_{\textrm{off}}}} |}}{{{T_{\textrm{on}}}}} \times 100\%$$

where ${T_{on}}$ and ${T_{off}}$ are the transmittance in the “on” and “off” states, respectively. Insertion loss can be calculated using $\textrm{IL} ={-} 10 \times \lg ( {\textrm{T}_{\textrm{on}}}) $ [24]. The modulation depth and insertion loss at four frequencies are 96.69%, 99.24%, 96.86%, 99.01% and 0.33 dB, 0.07 dB, 0.04 dB, 0.06 dB, respectively. Therefore, in the case of linearly polarized light, a multi-frequency switch with high MD and low IL is realized.

The transmission spectrum of the polarization angle of the linearly polarized light from $0^\circ$ to $90^\circ$ is shown in Fig. 3(c). We can find that the transmission spectra of different polarization angles are exactly the same, which is caused by the symmetry of the graphene pattern. The graphene metamaterial achieves a polarization-insensitive PIT effect in the terahertz band, and the photoelectric switch is also insensitive to the polarization direction of linearly polarized light. In addition, Figs. 4(a) and 4(b) are the transmission spectra at different incident angles for TE polarization and TM polarization, respectively. It can be seen from Fig. 4 that no matter the incident wave is TE polarization or TM polarization, when the incident angle is less than 30, the transmission spectrum does not change greatly. It shows that the metamaterial structure has a certain incident angle insensitivity.

Fig. 4. Transmission spectra at different incidence angles. (a) TE polarization; (b) TM polarization.

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When the light source is left-handed circularly polarized light and right-handed polarized light, the transmission curves obtained by vertically incident on the surface of the graphene metamaterial structure completely coincide, as shown in Fig. 5(a), indicating that the designed photoelectric switch can support circularly polarized light. The left-handed circularly polarized light is vertically incident on the surface of the structure, and the transmission spectra when the Fermi level is 0.8 eV and 1.2 eV are shown in Fig. 5(b). The modulation depth and insertion loss at four frequencies are calculated to be 96.69%, 99.24%, 96.86%, 99.01% and 0.33 dB, 0.07 dB, 0.04 dB, 0.06 dB, respectively. Therefore, a multi-frequency switch with excellent performance is also achieved in the case of circularly polarized light. This result is undoubtedly of great significance to the design and application of PIT-based photoelectric switches.

Fig. 5. (a) Transmission spectra for right rotation and left rotation circularly polarized light. (b) Theoretical design of photoelectric switch for circularly polarized light.

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The slow light effect associated with PIT metamaterials has important application value in many applications, which is caused by the strong dispersion of the PIT window. Usually, the group refractive index is used to evaluate the slow light performance of a material. The group index $ {n_g}$ is defined as [25]:

(9)$${n_g} = c\frac{{d{k_0}}}{{\; d\omega }} = \frac{c}{h}\frac{{d\theta }}{{d\omega }}$$

where $c$, $\theta $, $h$ represents the speed of light in vacuum, phase shift, the thickness of metamaterials, respectively. The phase shift and group index corresponding to different Fermi levels are shown in Figs. 6(a) and 6(b), respectively. The results show that the group index is up to 980.

Fig. 6. (a) The phase shift at different Fermi levels. (b) The group index of different Fermi levels.

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Small changes in the refractive index of the surrounding environment of the PIT device will cause changes in the transmission spectrum, so it can be used to detect biomolecules and toxic gases [26]. Therefore, the sensing performance of PIT devices is evaluated by transmission spectra under different environmental refractive indexes. Sensitivity is an important index to measure the performance of refractive index sensor. Its expression is [27]

(10)$$s = \frac{{\Delta \lambda }}{{\Delta n}}$$

where $\Delta \lambda $ is the offset of the resonance wavelength. In this paper, the resonance wavelength shift of dip1 and dip2 is studied. The figure of merit (FOM) is also an important index to measure the performance of the refractive index sensor, and its expression is [28]

(11)$$FOM(\lambda ) = \frac{{\Delta T}}{{T\Delta n}} = \frac{{T(\lambda ,n + \Delta n) - T(\lambda ,n)}}{{T(\lambda ,n)\Delta n}}$$

where $T(\lambda ,n + \Delta n)$ and $T(\lambda ,n)$ are the transmittances at the same wavelength $\lambda $ under the refractive indices of $n\textrm{ } + \textrm{ }\Delta n$ and n, respectively. As shown in Fig. 7(a), for every 0.05 increase in the ambient refractive index, the transmission spectrum of the structure can move significantly. Then, from Fig. 7(b), it can be seen that the change of the resonance wavelength of the structure is linearly related to the change of the refractive index. The sensitivity of dip1 and dip2 calculated by Eqs. (10) and (11) are ${s_1} = \textrm{49156 nm/RIU}$ and ${s_2}\textrm{ = 29504 nm/RIU}$, respectively, and the maximum $\textrm{FOM1}$ and $\textrm{FOM2}$ are 301 and 259, respectively. A comparison of the performance of this sensor and the reported PIT sensors is shown in Table 1. Based on the above discussion and calculation, it is shown that the PIT sensor we designed has ultra-high sensitivity.

Fig. 7. (a) The transmission spectra of metamaterials under different refractive indexes. (b) The relationship between the change of resonance wavelength of dip1 and dip2 and the change of environmental refractive index. (c) FOM data.

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Table 1. Comparison of sensitivity reported in other terahertz PIT sensors

View Table

4. Conclusion

In this paper, a graphene metamaterial is proposed to realize the PIT effect which is insensitive to the polarization angle. This kind of structure can realize high MD low IL photoelectric switch with multiple light sources, ultra-high sensitivity sensor and high group index slow light effect. This paper provides some ideas for the multifunctional design of polarization-insensitive PIT devices.

Funding

Natural Science Foundation of Jiangxi Province (20224BAB202032); National Natural Science Foundation of China (11664025, 11964018); Natural Science Foundation of Chongqing (CSTB2023NSCQ-MSX0730); State Key Laboratory of Advanced Technology for Materials Synthesis and Processing (2022-KF-15).

Disclosures

The authors declare no conflicts of interest

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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Multifunctional graphene metamaterials based on polarization-insensitive plasmon-induced transparency

Abstract

1. Introduction

2. Structure and theory

3. Results and discussion

4. Conclusion

Funding

Disclosures

Data availability

References

Data availability

Cited By

Figures (7)

Tables (1)

Equations (11)

Optics Express