Toroidal resonance based optical modulator employing hybrid graphene-dielectric metasurface

Gui-Dong Liu; Xiang Zhai; Sheng-Xuan Xia; Qi Lin; Chu-Jun Zhao; Ling-Ling Wang

doi:10.1364/OE.25.026045

1. Introduction

Toroidal dipole is a localized electromagnetic excitation produced by currents flowing on the surface of a torus along its meridians, which is distinct from the magnetic and electric dipoles resulting from circulating currents and a pair of charges, respectively. Static toroidal dipole was first considered by Zel’dovich in 1958 and has already been acknowledged in nuclear and particle physics [1]. Unfortunately, the dynamic excitations of toroidal dipole and higher toroidal multipoles were overlooked for some time in classical electrodynamics as their manifestations are often masked by much stronger electric and magnetic multipoles. Until 2010, T. Kaelberer et al. firstly experimentally observed the toroidal-dominated response by deliberately suppressing at the microwave regime in metamaterials consisting of a three-dimensionals (3D) array of four asymmetric split-ring resonators (SRRs) [2], and later the metamaterials were theoretically scaled down by Y. Huang et al. to push the toroidal response to the optical frequency [3]. The 3D arrangements of four SRRs are difficult to fabricate especially at optical wavelengths with toroidal dipolar moments, and many simplified or two-dimensional (2D) planar scheme were demonstrated experimentally or theoretically [4–13]. The toroidal resonances based metamaterials have been used to realize many interesting applications, electromagnetically induced transparency (EIT) [8], sensor with high sensitivity [10], resonant forward scattering of light [12], and so on. Unfortunately, the performances of these metamaterials based devices are usually limited by nonradiative and radiative losses. Fortunately, nonradiative losses can by reduced by employing materials of low loss as elements in metamolecules, such as dielectric [14–18]. Radiative losses can be controlled by optimizing the geometry of the structure to excite sharp asymmetric resonant response widely known as Fano resonances which arise due to destructive interference between broad bright mode and narrow dark mode. At the Fano resonance, the radiative damping of bright mode can be efficiently suppressed by dark mode and thus lead to the reduction of radiative losses [19]. Thus, it will be intriguing if a Fano resonators-based planar structure is designed to achieve toroidal resonance for reason that it is not only easy to fabrication, but also can reduce the nonradiative and radiative losses at the same time.

The modulation of the toroidal resonance is difficult because the geometric parameters of the structures have to be carefully re-optimized. Fortunately, the resonances can be modulated by varying the E_F when the graphene layer is integrated with the proposed structure. Graphene, a monolayer carbon material, has attracted enormous interest due to its unique electronic and optical properties. The optical response of graphene is characterized by its surface conductivity σ [20,21] that greatly relates to its Fermi energy E_F which can be dynamically tuned by applying bias voltage [22]. Actively tunable graphene-based plasmonic devices have been widely studied, such as filter [23], absorber [24–26] and sensor [27], etc. Graphene is also used as active medium to tune and modulate optical response of the metallic plasmonic structures in the mid-infrared and THz regime where the intraband transition of electron dominates and graphene behaves like a metal [28–31]. C. Argyropoulos presented a hybrid graphene-dielectric metasurface to tune and modulate the Fano resonance at near-infrared by making use of the strong interband absorption of the graphene layer [32]. However, to the best of our knowledge, the modulation of toroidal dipolar resonance based on dielectric metasurface by graphene at near-infrared has not been discussed so far.

In this paper, we demonstrate the combination of a dielectric metasurface with a graphene layer to realize a high performance toroidal resonance based optical modulator. For a metasurface with its unit cell being only one ASSRR, it was shown that a high Q-factor magnetic Fano resonance could be excited even under normal electromagnetic incidence conditions in a recent study [27]. The simulations using a finite difference time domain (FDTD) method reveal that ultrasharp toroidal resonance can be obtained in the transmission spectrum by utilizing two mirror-symmetric Fano resonators, namely two mirror-symmetric ASSRRs. The transmission amplitude of the toroidal dipolar resonance can be efficiently modulated by varying the E_F when the graphene layer is integrated with the dielectric metasurface, and the max transmission coefficient difference up to 78% is achieved. We also investigate the effects of the asymmetry degree of the ASSRRs on the toroidal dipolar resonance and the modulation efficiency of transmission amplitude of graphene.

2. Design and simulations

The geometric parameters of the dielectric metasurface are denoted by black letters in Fig. 1(a). The silicon rings with two equal splits dividing them into pairs of arcs of different length corresponding to β₁ and β₂. Pairs of ASSRRs in a unit cell are of mirror symmetry about yz and xz plane which are placed over silica substrate. We note that throughout our study the structures are illuminated by y-polarized (electric field E is along the y axis) plane waves, as illustrated in Fig. 1(a). The ASSRRs have thick t = 200 nm, inner radius r = 150 nm, outer radius R = 260 nm, and the period P_x = 1200 nm, P_y = 600 nm, respectively. The thickness of the silica substrate is chosen to be: H = 500 nm, but it may be smaller without affecting the performance of the proposed hybrid device. In the hybrid geometry reported in Fig. 1(b), graphene is placed over the ASSRRs and modeled as a 2D flat plane. The optical conductivity of graphene is related to the Fermi energy E_F and carrier mobility μ, through random phase approximation (PRA). In the local limit, it follows [20,21]

σ (ω) = \frac{2 e^{2} T}{π ℏ} \frac{i}{ω + i τ^{- 1}} \log [2 \cos h (\frac{E_{F}}{2 K_{B} T})] + \frac{e^{2}}{4 ℏ} [H (ω / 2) + \frac{4 i ω}{π} \int_{0}^{\propto} d ε \frac{H (ϵ) - H (ω / 2)}{ω^{2} - 4 ε^{2}}],

where

H (ε) = \sin h (ℏ ε / K_{B} T) / [\cos h (E_{F} / K_{B} T) + \cos h (ℏ ε / K_{B} T)]

. Here,

τ = μ E_{F} / e υ_{F}^{2}

is the intrinsic relaxation time, where

υ_{F} \approx c / 300

is the Fermi velocity. Temperature T is set as 300 K and the carrier mobility μ = 10000 cm²∕(V•s). In the proposed hybrid graphene-dielectric metasurface, the Fermi energy of the graphene layer can be modulated by using transparent electrodes which are placed between either the silica substrate or the silicon spilt-rings and the graphene layer [32,33]. The dielectric constant of Si and SiO₂ can be referred from Ref [34]. Our results are obtained by using FDTD method, where periodic boundary conditions are applied along the x and y directions, and perfectly matched layers (PML) are considered in the z directions.

Fig. 1 Schematics of the unit cell for (a) the dielectric metasurface and (b) the hybrid graphene-dielectric metasurface. The proposed structures are composed of a pair of asymmetric Si split-ring resonators (ASSRRs) which are of mirror symmetry about yz and xz plane. Corresponding electromagnetic excitation configuration (with polarization direction along the y axis) and geometric parameters are denoted by black letters in Fig. 1 (a).

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3. Results and discussion

First, we simulate numerically the optical response of the dielectric metasurface shown in Fig. 1(a) without the presence of graphene, in order to study the toroidal response of the dielectric metasurface. As shown in Fig. 2(a), the resonant response appears to have Fano line shape which has shown its immense potential in bio-sensing applications, and one can clearly observe a sharp resonance dip at about 1310.86 nm in the transmission spectrum which is one of the technologically interesting telecommunication wavelengths. The quality factor Q, defined as the ratio of the resonance wavelength λ₀ and the line width Δλ between the peak and the antipeak of the transmission (Q = λ₀ / Δλ), reaches about 1702 while the Δλ is about 0.77 nm. The spectral contrast ratio, defined as [(T_peak – T_antipeak)/ (T_peak + T_antipeak)] × 100%, reaches about 100% [27], where T_peak and T_antipeak are the transmission coefficient at the peak and dip of the resonance, respectively. Thanks to these characteristics of the toroidal dipolar resonance, the proposed metasurface is a promising candidate in many applications, such as optical filters [35] and biochemical sensors [10, 27]. The magnetic field H_z components and the displacement current density J_z components at the resonant wavelength are also computed numerically to qualitatively investigate the underlying physics of the resonance. The z- components of magnetic field H_z, as shown in Fig. 2(d), is distributed with opposite directions in two ASSRRs in the unit cell. The spatial localization of the magnetic field indicates the excitation of toroidal dipolar moment T, which is along the y-direction. The surface currents circulate along the arcs of the split-ring with inverse directions as shown in Fig. 2(e). With electric field of incident plane wave being parallel to the arcs of the split-ring, the displacement current in the two ASSRRs of the unit cell circulates with the opposite direction, resulting in the induced magnetic moments wreathed around the central part of the unit cell, forming a magnetic vortex with head-to-tail configuration. Such a head-to-tail configuration of the individual magnetic moments m can induce the toroidal moments oscillating and thus generate the toroidal dipolar resonant response.

Fig. 2 (a) Transmission spectrum of the dielectric metasurface without a graphene layer. (b) Scattered power and (c) phases of the five major multipoles of the dielectric metasurface, including electric dipoles (P), magnetic dipoles (M), toroidal dipoles (T), electric quadrupoles (Q)_e, and magnetic quadrupoles (Q)_m. The log scale in the y axis of Fig. 1(b) is chosen so as to reveal more clearly the contribution of the M and Qe as well. The magnetic field H_z components (d) and the displacement current density J_z components (e) at the resonant wavelength of 1310.86 nm are also simulated. Arrows indicate instantaneous directions of the displacement current flow.

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In order to study the toroidal dipole resonance quantitatively, the radiated powers of the magnitude of electric and magnetic multipoles and toroidal dipole were calculated by the induced volume displacement current density J [2]:

electric dipole moment : P = \frac{1}{i ω} \int^{​} J d^{3} r

magnetic dipole moment : M = \frac{1}{2 c} \int^{​} (r \times J) d^{3} r

toroidal dipole moment : T = \frac{1}{10 c} \int^{​} [(r \cdot J) r - 2 r^{2} J] d^{3} r

electric quadrupole moment : Q_{α β} = \frac{1}{i ω} \int^{​} [r_{α} J_{β} + r_{β} J_{α} - \frac{2}{3} (r \cdot J)] d^{3} r

and magnetic quadrupole moment : M_{α β} = \frac{1}{3 c} \int^{​} [{(r \times J)}_{α} r_{β} + {(r \times J)}_{β} r_{α}] d^{3} r

where c is the speed of light in the vacuum, r is the distance vector from the origin to point (x, y, z) in a Cartesian coordinate system, and α, β = x, y, z. Therefore, the decomposed far-field scatter power by these multipole moments can be calculated by using the following equations:

I_{P} = \frac{2 ω^{4}}{3 c^{3}} | P |^{2}

,

I_{M} = \frac{2 ω^{4}}{3 c^{3}} | M |^{2}

,

I_{T} = \frac{2 ω^{6}}{3 c^{5}} | T |^{2}

,

I_{Q}^{e} = \frac{ω^{6}}{5 c^{5}} \sum^{​} | Q_{α β} |^{2}

, and

I_{Q}^{m} = \frac{ω^{6}}{20 c^{5}} \sum^{​} | M_{α β} |^{2}

. The scatted powers and phase of the five major multipoles of the proposed metasurface as functions of wavelength are shown in Figs. 2(b) and 2(c), respectively. To study the toroidal dipolar resonance in two mirrored asymmetric silicon split-ring resonators more clearly, the displacement currents in the substrate are ignored. We notice that in the vicinity of the resonant wavelength the toroidal dipole T is the strongest contribution which is about 1.1 times stronger than the magnetic quadrupole Q_m and much stronger than P, M, Q_e. The electric dipole moment of the proposed metasurface is suppressed because the polarization of the incident plane wave is orthogonal to the split of the Si split-rings. The presence of relative strong quadrupole Qm prevents us from achieving the even higher Q-factor.

To investigate the performance of the toroidal dipolar resonance based optical modulator, the transmission spectra of the dielectric metasurface without and with graphene layer (Fermi energies: 0.6 eV, 0.5 eV, 0.4 eV) are obtained numerically and showed in Figs. 3(a) and Figs. 3(b)-3(d), respectively. Compared with the optical response of the dielectric metasurface, the pronounced dip in the transmission spectrum is nearly no change when the graphene layer with E_F = 0.6 eV is introduced, while the resonance is broadened and becomes more flatted with the decrease of the E_F. Furthermore, it is obviously seen that the spectral contrast ratio and the Q-factor are rapidly decreased. Therefore, the proposed hybrid graphene/dielectric metasurface can be utilized as the active optical modulator by using a graphene layer. To explore the underlying physics mechanism, we plot the distributions of the displacement current density |J| at the resonant wavelength without and with graphene layer while E_F varied from 0.6 eV to 0.4 eV in Figs. 3(e) and Figs. 3(f)-3(h), respectively. The magnitude of the displacement current density |J| in the dielectric metasurface becomes weaker when the graphene layer with E_F = 0.6 eV placed over the dielectric metasurface, although it have little effect on the transmission spectrum. Decreasing of the E_F yielding a decrease in the displacement current density strength and it nearly disappear when the E_F decrease to 0.4 eV, as shown in Fig. 3(e)-3(h).

Fig. 3 (a) – (d) Simulated transmission spectra without and with the graphene layer, showing the active modulation of toroidal dipolar resonance, respectively. (e) – (h) Corresponding distributions of the displacement current density |(J)| at resonant wavelength for (a) – (d), respectively.

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The underlying physics mechanism is further investigated and explained by calculating the wavelength dependent graphene conductivity for different E_F according to the Eq. (1), and the results shown in Fig. 4. Interestingly, the real conductivity of graphene Re (σ) is always greater than zero with the decrease of the E_F from 0.6 eV to 0.4 eV, while the imaginary conductivity of graphene Im (σ) is greater than zero when E_F = 0.6 eV, in which case graphene features a metallic behavior, and the Im (σ) reduce to less than zero when E_F = 0.5 eV meaning that graphene behaves like a lossy dielectric [36]. A doping charge density n_g raises the Fermi level to $E_{F} = ℏ υ_{F} \sqrt{π | n_{g} |}$ , so the lower charge density n_g indicates the lower E_F, which yields to higher loss and absorption. With the decrease of the E_F from 0.6 eV to 0.4 eV, the Re (σ) of graphene increases rapidly at the toroidal dipolar resonant wavelength which indicates the increase of the absorption of the graphene [37]. Note that the electromagnetic waves reemitted by the induced multipoles of the dielectric metasurface are strongly coupled to the absorption of the graphene layer [32], therefore the shrunk outline of the toroidal dipolar resonance can attribute to the increase of the absorption of the graphene. It is worth noting that Xiao et al [29] and Chen et al [30] recently investigated the modulations of EIT effect and toroidal resonance with monolayer graphene in THz hybrid meta-graphene metamaterials by making use of the high and tunable conductivity of graphene, respectively, while the on-to-off modulation of the transmission amplitude is realized with the increase of the Fermi energy of graphene which is in the opposite trend with our proposed hybrid graphene-dielectric metasurface. That’s because the Re (σ) of graphene increase with the increase of the E_F of graphene at the THz frequency while it decrease with the increase of the E_F of graphene in the near-infrared regime for a fixed frequency or wavelength.

Fig. 4 The wavelength dependent (a) the real part (b) the imaginary part of graphene conductivity. The Fermi energy is varied from 0.4 eV to 0.6 eV, as shown in the insets.

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Next, we will investigate the effect of the asymmetry degree of the ASSRRs, defined as Δβ = β₂ – β₁, on the toroidal dipolar resonance and further demonstrate the modulating potential of graphene. To present the transmission amplitude modulation more quantitative and clear, the different in transmission amplitude between without and with graphene layer (E_F = 0.4 eV) are calculated. The absolute value of the transmission difference, defined as ΔT = |T(E_F = 0.4 eV) – T(without the graphene layer)| × 100%, as a function of the wavelength of the incident plane wave where the Δβ is varied from 0° to 40° are plotted in Figs. 5(e)-5(h), respectively. As Δβ = 0° and in the absence of the graphene layer, there is no resonance in the transmission spectrum which keep almost unchanged when the graphene layer with E_F = 0.4 eV is introduced as shown in Fig. 5(a). Moreover, the transmission coefficient difference is nearly 0 demonstrate that the graphene layer almost do not have modulating ability in this case as shown in Fig. 5(e). When Δβ = 20° is introduced, a resonance possessing high Q and pronounced contrast ratio appear in the transmission spectrum, and the max transmission coefficient difference is up to 78% demonstrating that the graphene layer have strong ability to modulate the resonance as shown in Figs. 5(b) and 5(f). As the Δβ is increased from 20° to 40° at a step of 10°, we can see from the Figs. 5(c) and 5(d) that the contrast ratio nearly unchange while the resonant wavelength increases and the Q of the resonances decrease slowly. Furthermore, the modulating ability of graphene also decreases as the max transmission coefficient difference decrease with the increase of the Δβ as shown in Figs. 5(g) and 5(h).

Fig. 5 (a) - (d) Simulated transmission spectra of the dielectric metasurface without and with graphene layer (E_F = 0.4 eV). (e) - (h) Computed percentage of the absolute value of the transmission coefficient difference versus the incident wavelength. The asymmetry degree is varied from 0° to 40°, as shown in the insets.

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To reveal the physical mechanism behind the novel phenomenon, simulated z-component of the displacement current density J_z at the wavelength λ = 1310.86 nm without and with graphene layer when Δβ = 0° are plotted in Figs. 6(a) and 6(b), respectively. As shown in Fig. 6(a), the displacement current circulates along the arcs of the split-ring with same directions and thus the magnetic vortex can’t form the head-to-tail configuration no longer, indicating that the toroidal dipolar resonance isn’t excited when Δβ = 0°. The graphene layer nearly has no effect on the magnitude of the J_z for reason that in the absence of resonance, the electromagnetic waves reemitted by the induced multipoles are too weak to couple to the absorption of the graphene layer.

Fig. 6 Simulated the z-component of the displacement current density Jz at the wavelength λ = 1310.86 nm (a) without and (b) with graphene layer while asymmetry degree is Δβ = 0°.

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The displacement current density |J| distributions at the toroidal dipolar resonant wavelength without and with the graphene layer are also calculated and plotted in Figs. 7(a) - 6(c) and Figs. 7(d)-7(f), respectively, where the Δβ is varied from 20° to 40° at a step of 10°.

Fig. 7 Simulated the displacement current density |(J)| distributions at the resonant wavelength (a) - (c) without and (e) - (f) with the graphene layer, respectively. The asymmetry degree Δβ is varied from 20° to 40° at a step of 10°, as shown in the insets.

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The Q of a resonator commonly indicates the rate of the stored energy and the energy loss in the resonator. With the absence of the graphene layer, increasing the Δβ yields a decrease in the magnitude of the displacement current density indicating the decrease of the stored energy in the metasurface on one hand, On the other hand, increasing the Δβ yields increase of coupling to free space and therefore the energy loss [38]. As a result, the Q of the resonance decreases with the increase of Δβ, as shown in Figs. 5(b)-5(d). With the absence of the graphene layer, as shown in Figs. 7(a)-7(c), increasing the Δβ yields the decrease of the magnitude of the displacement current density indicating the decrease of field enhancement, which will make the degree of graphene absorption enhancement reduce when the graphene layer is introduced [39]. What’s more, the real conductivity of the graphene decreases slowly as the toroidal dipolar resonant wavelength increase with the increase of the Δβ, which lead to the absorption of the graphene layer decrease a little. As a result, the absorption of graphene and the interaction between the field and the graphene layer decrease with the increase of the Δβ leading to the increase of the magnitude of the displacement current density as shown in Figs. 7(d)-7(f), and the decrease of the strength of transmission amplitude modulation by the graphene layer.

4. Summary and conclusion

In this paper, we demonstrated the combination of a dielectric metasurface with a graphene layer to realize a high performance toroidal resonance based optical modulator. The dielectric metasurface consisted of two mirrored asymmetric Fano resonators which could support strong toroidal dipolar resonance with narrow line width (~0.77 nm) and high quality (Q)-factor (~1702), contrast ratio (~100%). Numerical simulation results showed that the transmission amplitude of the toroidal dipolar resonance can be efficiently modulated by varying the Fermi energy E_F when the graphene layer was integrated with the dielectric metasurface, and the max transmission coefficient difference up to 78% was achieved. The toroidal dipolar resonant wavelength red-shifted and the efficiency of transmission amplitude modulation of graphene decreased with the increase of the asymmetry degree of the proposed metasurface. Our results may provide potential applications in optical modulator, optical filter and bio-chemical sensing.

Funding

National Natural Science Foundation of China (Grant Nos. 61505052, 61775055, 61176116, 11074069).

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Toroidal resonance based optical modulator employing hybrid graphene-dielectric metasurface

Abstract

1. Introduction

2. Design and simulations

3. Results and discussion

4. Summary and conclusion

Funding

References and links

Cited By

Figures (7)

Equations (6)

Optics Express