Tunable multispectral plasmon induced transparency based on graphene metamaterials

Chen Sun; Jiangnan Si; Zhewei Dong; Xiaoxu Deng

doi:10.1364/OE.24.011466

1. Introduction

Electromagnetically induced transparency (EIT) is a quantum interference effect, which is first studied in a laser-driven atomic system and realized in a three-level system [1], causing a narrow transparency window within a broad absorption spectrum. The EIT effect dramatically modifies the effective refractive index of the medium, which can slow down photons and enhance nonlinear characteristics. In order to overcome the scathing experimental requirements of realizing the EIT effect [2], analogues of EIT-like behaviour in classical systems, such as coupled optical resonators [3], electric circuits [4], and metallic plasmonic structures [5–9], have attracted tremendous interest. In particular, plasmonic analogues of EIT based on metamaterial structures, including cut wires [5, 10], split-ring resonators (SRR) [6, 11–15], and coupled waveguide resonators [16], highlight the realization of EIT analogues because of the effective media characteristics [17]. Due to the giant effective refractive index and enhanced nonlinear properties in metamaterials, the EIT-like systems provide great possibilities in developing novel devices, such as slow light photonic components, integrated chip scale buffers and highly sensitive sensors [18–20]. Although active controlling the EIT resonance of the metamaterials is highly desirable for practical applications, it is difficult to be realized in classical EIT-like systems of which EIT resonance can be tuned only by carefully changing geometric parameters of the structures [7, 21].

Graphene has attracted worldwide interests as a promising platform for plasmonics [22–24] due to its unique properties such as electrical tunability [25], strong light confinement [26], and relatively low plasmonic losses [27, 28]. Graphene metamaterials is a promising candidate to design active tunable EIT-like systems [29]. The EIT resonance of the graphene metamaterials structure, such as graphene nanostrips [30, 31] and graphene nanoribbons [32], can be easily tuned by varying the Fermi energy of the graphene through altering the bias voltage, which makes the EIT-like effect in the graphene metamaterials more active than that in metallic systems, indicating great potential applications in tunable sensors, switchers, and slow light devices.

In this paper, the wavelength-tunable multispectral EIT-like responces are investigated for the terahertz frequency range in periodically patterned graphene double layers separated by a dielectric layer. The coupled Lorentz oscillator model is introduced to explain the multispectral PIT phenomena resulting from the near-field coupling in different graphene layers and the dark-bright mode coupling in graphene single layer. The transmission and electric field distributions of the graphene-based multispectral PIT device with different geometrical parameters and Fermi energies are simulated by the the finite-difference time-domain (FDTD) solutions. By varying the Fermi energy of graphene, the tunability of multispectral EIT-like responces can be realized without changing the structure, which offers a promising approach to designing compact elements, such as tunable sensors, switchers, and filters.

2. Theory

The graphene-based multispectral PIT device is composed of periodically patterned graphene double layers deposited on both sides of the dielectric (SiO₂) substrate [22]. The unit cell of the proposed device is shown in Fig. 1(a)–(c). Periodically patterned graphene double layers have the same structural parameters, and each layer has a horizontal cut-out (the slot dipole antenna) and a vertical cut-out pair (the slot quadrupole antenna). The periods of the unit cell are 800nm on x direction and 600nm on y direction. The horizontal cut-out has 540nm length and 40nm width, and the vertical cut-out pair has 360nm length and 40nm width. The in-plane separation between the dipole and quadrupole antennas is 10nm on both sides. The offset in y direction of the dipole antenna from the geometrical center of the structure is denoted as parameter s. The gap between the periodically patterned graphene double layers (that is, the dielectric layer thicknesses) is denoted as parameter d. The carrier concentration in patterned graphene layers is controlled using an ion-gel top gate [22], which is shown in Fig. 1(d).

Fig. 1 (a) Schematic of the unit cell of the graphene-based multispectral PIT device and the incident light polarization configuration. (b) Top view of the unit cell. The geometrical parameters are: L1 = 540nm, w1 = 40nm, L2 = 360nm, w2 = 40nm, Px = 800nm and Py = 600nm, respectively. The small in-plane separation between the horizontal cut-out and the vertical cut-out pair is 10nm on both sides. Parameter s is defined as the offset in y direction of the horizontal cut-out from the geometrical center of the structure. (c) Side view of the unit cell. Parameter d is defined as the gap size between the graphene double layers. (d) Side view of the graphene-based multispectral PIT device.

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The transmissions of the horizontal cut-out only structure and the vertical cut-out pair only structure for the Fermi energy of the graphene E_F = 0.15eV based on the FDTD simulation are shown in Fig. 2. The black curve shows that the horizontal cut-out only structure can support a typical localized surface plasmon (LSP) resonance, which acts as the bright mode, at 13.66THz with the incident electric field E along the y axis. An inductivecapacitive (LC) resonance is supported, shown by the red curve in Fig. 2, at 15.91THz with the incident electric field E along the x axis in the vertical cut-out pair only structure, which cannot be excited directly by the incident electric field E along x axis, demonstrating that the vertical cut-out pair only structure is considered as the dark mode. The structural asymmetry (s ≠ 0) causes near-field coupling between the dark and bright modes in the same graphene layer and leads to the indirect excitation of dark mode with E along the x axis, which results in a EIT-like response in the periodically patterned graphene single layer(s = 30nm) shown by the blue curve in Fig. 2. The EIT-like response caused by the phonon damping is omitted in the terahertz frequency range [28].

Fig. 2 Simulated transmission spectra of the horizontal cut-out only structure, the vertical cut-out pair only structure and the periodically patterned graphene single layer(s = 30nm) when Fermi energy E_F = 0.15eV. Different geometric structures with the direction of incident electrical field are shown in the insets from top to bottom, respectively.

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For periodically patterned graphene double layers, strong near-field coupling between cutouts in different layers causes splitting of the resonance notches, leading to multiple PIT windows. The bright and dark modes in the graphene-based multispectral PIT device can be expressed as |D^t/b〉 and |Q^t/b〉. The superscript t and b represent the top and bottom layers, respectively. The strong near-field coupling of the bright modes in different layers causes the hybrid bright modes in the in-phase and out-of-phase hybridized states, which are expressed as $| D_{i} 〉 = \frac{1}{\sqrt{2}} [| D^{t} 〉 + | D^{b} 〉]$ and $| D_{o} 〉 = \frac{1}{\sqrt{2}} [| D^{t} 〉 - | D^{b} 〉]$ , respectively. The subscript i and o represent the in-phase and out-of-phase hybridized states, respectively. In the same way, the strong near-field coupling of the dark modes in different layers causes the hybrid dark modes of the in-phase and out-of-phase hybridized states, which are expressed as $| Q_{i} 〉 = \frac{1}{\sqrt{2}} [| Q^{t} 〉 + | Q^{b} 〉]$ and $| Q_{o} 〉 = \frac{1}{\sqrt{2}} [| Q^{t} 〉 - | Q^{b} 〉]$ , respectively. The cross couplings among the bright and dark modes of different graphene layers are assumed to be weak, which can be neglected.

The coupled Lorentz oscillator model is adapted to explain multispectral EIT-like phenomena in the proposed device. The external driving field is denoted as E₀e^iωt. The hybrid bright and dark modes are expressed as |D_i/o〉 = D̃_i/oe^iωt and |Q_i/o〉 = Q̃_i/oe^iωt, respectively. The field amplitude of (|D_i,o〉, |Q_i,o〉) is obtained [21]

[\begin{matrix} ω - ω_{D, i} + i γ_{D, i} & κ & 0 & 0 \\ κ_{i} & ω - ω_{Q, i} + i γ_{D, i} & 0 & 0 \\ 0 & 0 & ω - ω_{D, o} + i γ_{D, o} & κ_{o} \\ 0 & 0 & κ_{o} & ω - ω_{Q, o} + i γ_{D, o} \end{matrix}] [\begin{matrix} {\tilde{D}}_{i} \\ {\tilde{Q}}_{i} \\ {\tilde{D}}_{o} \\ {\tilde{Q}}_{o} \end{matrix}] = [\begin{matrix} g_{i} {\tilde{E}}_{0} \\ 0 \\ g_{o} {\tilde{E}}_{0} \\ 0 \end{matrix}]

where ω_D,i/o, ω_Q,i/o, κ_i/o, g_i/o, γ_D,i/o, and γ_Q,i/o are the resonance frequencies of the hybrid bright modes, the resonance frequencies of the hybrid dark modes, the coupling parameters, the geometrical parameters, the damping rates of the hybrid bright modes, and the damping rates of the hybrid bright modes in the in-phase and out-of-phase hybridized states, respectively. The amplitudes of the hybrid bright mode in both hybridized states are derived as

{\tilde{D}}_{i / o} = \frac{- g_{i / o} {\tilde{E}}_{0} (ω - ω_{Q, i / o} + i γ_{Q, i / o})}{(ω - ω_{D, i / o} + i γ_{D, i / o}) (ω - ω_{Q, i / o} + i γ_{Q, i / o}) - {(κ_{i / o})}^{2}}

The transmission of the graphene-based multispectral PIT device is obtained [30, 33]

T (ω) = 1 - {| \frac{{\tilde{D}}_{i}}{{\tilde{E}}_{0}} |}^{2} - {| \frac{{\tilde{D}}_{0}}{{\tilde{E}}_{0}} |}^{2} .

The coupling parameters κ_i/o indicating the coupling between the hybrid bright and dark modes, increase with the structural asymmetry s [30]. When the structure is symmetrical s = 0, the resonance notches arise at the resonance frequencies ω_D,i/o in the transmission. As the structural asymmetry s increases, the transmission at the resonance frequencies ω_Q,i/o increases due to the excitation of the hybrid dark mode, resulting in the emerging of the EIT peaks in the resonance notches. The resonance frequencies ω_D,i/o and ω_Q,i/o can be tuned by the Fermi energy of the graphene double layers [30]. Though the proposed device are based on graphene double layers, the physical mechanism of the multispectral EIT-like responses can be easily extended to a larger number of graphene layers.

3. Simulation and results

The transmission spectra of the graphene-based multispectral PIT device is simulated by the FDTD solutions (Lumerical Solutions, Inc.). In the three-dimensional simulations, a y-polarized TEM beam normally incidents on the unit cell in z direction with periodic boundary conditions in x–z plane and symmetric conditions in y–z plane. The index of the dielectric (SiO₂) layer is considered to be 1.45. The surface conductivity of graphene σ is computed from the Kubo formula [34]

σ (ω) = \frac{i e^{2} (ω - 2 i Γ)}{π {\bar{h}}^{2}} [\frac{1}{{(ω - 2 i Γ)}^{2}} \int_{0}^{\infty} ε (\frac{\partial f_{d} (ε)}{\partial ε} - \frac{\partial f_{d} (- ε)}{\partial ε}) d ε - \int_{0}^{\infty} \frac{f_{d} (- ε) - f_{d} (ε)}{{(ω - 2 i Γ)}^{2} - 4 {(ε / \bar{h})}^{2}} d ε]

where, f_d(ε) = (e^{(ε − E_F)/(k_BT) +1)−1}, E_F is the Fermi energy of graphene, ω is the angular frequency, Γ = 1.98THz is a phenomenological scattering rate [35], and T = 300K is temperature. The quantum finite-size effects associated with the structure edges can be ignored due to the width of the graphene nanostrips above 10nm [36].

The transmission spectra of the graphene-based multispectral PIT device with different gap sizes by employing the FDTD solutions are demonstrated in Fig. 3(a) (s = 0nm and E_F = 0.15eV for graphene double layers). For the periodically patterned graphene double layer, two resonance notches are supported owing to the strong near-field coupling between the different graphene layers, which is denoted as the out-of-phase(OP) and in-phase(IP) hybridized states respectively, shown in Fig. 3(a), instead of only one resonance notch for the periodically patterned graphene single layer shown by the black curve in Fig. 3(a). As the gap size d decreases, the resonance notch of the in-phase hybridized state is gradually enhanced and blue shifted, and the resonance notch of the out-of-phase hybridized state is gradually weakened and red shifted, for the increase of coupling between different graphene layers. The cross-sectional electrical field distributions of the coupled graphene double layers at the resonance frequencies in the out-of-phase and in-phase hybridized states are shown in Fig. 3(b) at the position marked with the black dashed line in the inset of Fig. 3(a) for d = 130nm. The hybrid bright modes in the out-of-phase and in-phase hybridized state are radiant as a result of the symmetric and antisymmetric combination of electrical field distributions in graphene double layers, respectively, shown in Fig. 3(b) A and B. The resonance notch in the in-phase hybridized state is easier to excite due to the overall superposition of the electrical field and blue-shifted with the decrease of gap d, and that of the out-of-phase hybridized state is harder to excite due to the partial detraction and red-shifted with the decrease of gap d [37].

Fig. 3 (a) Simulated transmission spectra of the graphene-based multispectral PIT device with different gap sizes d for s = 0nm and E_F = 0.15eV. The out-of-phase(OP) and in-phase(IP) hybridized states are demonstrated. (b) The cross-sectional electrical field distributions of the graphene double layers (d = 130nm) at the resonance notches in the out-of-phase and in-phase hybridized states are shown in A and B for d = 130nm, respectively, which are observed at a position marked with the black dashed line in the inset of (a).

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The simulated transmission spectra of the graphene-based multispectral PIT device with different Fermi energies for graphene double layers are demonstrated in Fig. 4 (s = 0nm and d = 130nm). As the Fermi energy E_F increases, the resonance notches of the out-of-phase and in-phase hybridized states are both enhanced and blue shifted. The difference between the resonance wavelengths in the out-of-phase and in-phase hybridized states remains about the same. The resonance wavelength is written as λ_res ∝ [2π²h̄²c²L/(e²E_F)]^1/2 (where L represents the length of the graphene nanostrips) [30], which can be controlled by the Fermi energy of graphene.

Fig. 4 Simulated transmission spectra of the graphene-based multispectral PIT device with different Fermi energies for graphene double layers (s = 0nm and d = 130nm).

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The difference between the resonance wavelengths in the out-of-phase and in-phase hybridized states are tuned by varying the Fermi energies of the top and bottom periodically patterned graphene layers respectively. The simulated transmission spectra of the graphene-based multispectral PIT device with different Fermi energies of the top and bottom graphene layers (s = 0nm and d = 130nm) are shown in Fig. 5. The Fermi energies of the top and bottom graphene layers are: 0.2/0.1, 0.175/0.125, 0.15/0.15 and 0.125/0.175 for different curves, respectively. As the Fermi energy of top graphene layer increases and bottom one decreases, the resonance notch of the in-phase hybridized state is gradually blue shifted, and the resonance notch of out-of-phase hybridized state is gradually red shifted. Therefore, the frequency difference between the in-phase and out-of-phase hybridized states is tunable.

Fig. 5 Simulated transmission spectra of the graphene-based multispectral PIT device with different Fermi energies of the top and bottom graphene layers (s = 0nm and d = 130nm).

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The transmission spectra of the graphene-based multispectral PIT device with different offsets s of the horizontal cut-out from the geometrical center by employing the FDTD solutions are shown in Fig. 6(a) (d = 130nm and E_F = 0.15eV). As s increases, two EIT peaks emerge in the resonance notches of the in-phase and out-of-phase hybridized states, respectively, resulting from the increase of coupling between the dark and bright modes in the same graphene layer. The top view electrical field distributions at the EIT peaks (s = 30nm) in both out-of-phase and in-phase hybridized states are shown in Fig. 6(b) A and B, respectively, demonstrating strong excitation of the hybrid dark modes in both hybridized states. By introducing the structural asymmetry (s ≠ 0), the electromagnetic field is coupled back and forth between the hybrid bright and dark mode, leading to a destructive interference, resulting in multispectral EIT-like behavior. The analytic fitting of Eq. 3 to the transmission spectrum (s = 30nm) is shown by the blue circles, which traces the numerical results very well. The fitting parameters are ω_D,i/ω_D,o = 6.64/5.08THz, ω_Q,i/ω_Q,o = 6.50/4.96THz, g_i/g_o = 0.36/0.10, γ_D,i/γ_D,o = 0.34/0.20THz, γ_Q,i/γ_Q,o = 0.34/0.44THz and κ_i/κ_o = 0.30/0.38THz.

Fig. 6 (a) Simulated transmission spectra of the graphene-based multispectral PIT device with different offsets s for d = 130nm and E_F = 0.15eV. The analytic fitting based on Lorentzian harmonic oscillators mode to the simulated transmission (s = 30nm) is shown by the blue circles. (b) The top view electrical field distributions of the device at the EIT peaks (s = 30nm) in the out-of-phase and in-phase hybridized states are shown in A and B, respectively.

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The multiple PIT windows are realized in the proposed device by introducing the near-field coupling caused by the periodically patterned graphene double layers. The EIT peaks emerge in the multiple PIT windows through the structural asymmetry (s ≠ 0) which leads to the coupling between the bright and dark modes in the same graphene layer. The multispectral EIT-like responses are actively controlled by varying the Fermi energy via altering the voltage on graphene, showing advantages in the applications of optical components, such as tunable sensors, switches and filters.

4. Summary

In summary, periodically patterned graphene double layers separated by a dielectric layer are adapted to support multispectral EIT-like responses in the terahertz frequency range. The coupled Lorentz oscillator model, incorporating the near-field coupling in different graphene layers and the bright-dark coupling in the same graphene layer, is employed to explain multispectral EIT-like responses. The resonances in multiple PIT windows are controlled by changing the Fermi energy of graphen or the structural parameters, which is demonstrated by the simulated transmission based on the FDTD solutions. The dynamic tunability of multispectral EIT-like responses are realized via varying the Fermi energy of graphene without changing the geometric parameters, which makes the proposed graphene-based multispectral PIT device more active than the metallic ones.

Acknowledgments

This work was supported by National Natural Science Foundation of China (61378067, 61178050).

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Tunable multispectral plasmon induced transparency based on graphene metamaterials

Abstract

1. Introduction

2. Theory

3. Simulation and results

4. Summary

Acknowledgments

References and links

Cited By

Figures (6)

Equations (4)

Optics Express