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Abstract
In this paper, we theoretically and numerically demonstrate a polarization-controlled and symmetry-dependent multiple plasmon-induced transparency (PIT) in a graphene-based metasurface. The unit cell of metasurface is composed of two reversely placed U-shaped graphene nanostructures and a rectangular graphene ring stacking on a dielectric substrate. By adjusting the polarization of incident light, the number of transparency windows can be actively modulated between 1 and 2 when the nanostructure keeps a geometrical symmetry with respect to the x-axis. Especially, when the rectangular graphene ring has a displacement along the y-direction, the number of transparency windows can be arbitrarily switched between 2 and 3. The operation mechanism behind the phenomena can be attributed to the near-field coupling and electromagnetic interaction between the bright modes excited in the unit of graphene resonators. Moreover, the electromagnetic simulations obtained by finite-difference time-domain (FDTD) method agree well with the theoretical results based on the coupled modes theory (CMT). In addition, as applications of the designed nanostructure, we also study the modulation degrees of amplitude, insertion loss and group index of transmission spectra for different Fermi energies, which demonstrates an excellent synchronous switch functionality and slow light effect at multiple frequencies. Our designed metasurface may have potential applications in mid-infrared optoelectronic devices, such as optical switches, modulators, and slow-light devices, etc.
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1. Introduction
Electromagnetically induced transparency (EIT) refers to the quantum destructive interference phenomenon, which can be observed in a three-level atomic system and give rise to a narrow transparent window in a wide opaque spectrum [1,2]. In particular, there exists a strong dispersion near the induced transparent spectral region which can result in slow light effect [3] and enhancement of nonlinear susceptibility [4]. Therefore, it has drawn enormous attention owing to its broad applications including information processing, optical storage, and nonlinear effects, etc. However, the traditional EIT effect usually require harsh experimental conditions, such as low temperature or high-intensity lasers, which limits the practical application of EIT. To overcome these shortcomings, some schemes have been proposed to achieve the classical analogy of EIT phenomenon in various model systems, such as circuit models [5], photonic crystals [6,7], optical microcavities [8,9], and metamaterials [10,11]. The EIT-like phenomena based on metasurfaces have provoked considerable interest due to the ability of confining light to subwavelength scales, which keeps an ongoing interest in the design of nanophotonic devices [12,13]. In general, the EIT-like effect in metamaterials composed of plasmonic molecules, known as plasmon-induced transparency (PIT) [14], can usually be realized by two different ways. The first way is based on the direct destructive interference between a radiative (bright) mode and a subradiative (dark) mode [15,16], and the second way is based on detuning of two bright modes [17,18]. In recent years, the PIT effect has been demonstrated in a wide electromagnetic spectra ranging from microwaves to visible frequency bands [19–22]. However, the performance and working frequency of PIT in metamaterials cannot be flexibly adjusted once the nanostructure was fabricated.
In practical applications, the nanophotonic devices with tunable characteristics are highly desirable. Aiming to improve the flexibility and to enhance the electromagnetic responses of the metamaterials, some strategies have been suggested by integrating metamaterials with active materials, such as semiconductor material [23], micro-/nano-electro-mechanical systems [24], phase-change materials [25,26], two-dimensional (2D) materials [27–29], and so on. Among these materials, graphene as a novel 2D material, which is arranged by an atomic-scale honeycombed hexagonal lattice structure composed of carbon atoms, has been paid a great deal of attention in view of its remarkable optoelectronic characteristics [30]. Compared with the large loss of traditional metal materials, the conductivity of graphene can be actively tailored by applying electrostatic gating or chemical doping. Therefore, one can conclude that the combination of graphene with metamaterials or metasurfaces can be utilized to achieve dynamically tunable PIT effect [31–35]. To date, a variety of graphene-based metamaterials have been presented theoretically and experimentally [36–42]. And the research results have manifested that the graphene metamaterials are excellent platforms for achieving the active manipulation of PIT [43–46]. However, to our knowledge, most of the PIT phenomena occurred in graphene metamaterials are concerned about the illumination with a fixed polarization direction, few attentions have been paid on the influences of polarization angles. And the main modulation method for the transmission spectra and resonance frequency of transparency window is only limited to the electrically tuning the Fermi energy level of graphene.
In this paper, we theoretically and numerically present a graphene-based metasurface, which can achieve a polarization-controlled and symmetry-dependent multiple PIT effect by adjusting the polarization of incident light and geometrical asymmetry. The unit cell of metasurface consists of two reversely placed U-shaped graphene nanostructures and a rectangular graphene ring stacking on a dielectric substrate. By adjusting the polarization of incident light, the number of PIT windows can be arbitrarily switched from 1 to 2. Especially, when the rectangular graphene ring has a displacement along the y-direction, i.e., s = 40 nm, the number of transparency windows can be switched between 2 and 3. The physical mechanism behind the phenomena can be attributed to the near-field coupling and electromagnetic interactions between the radiative modes excited in the unit of graphene resonators, respectively. Moreover, the electromagnetic simulations results agree well with the theoretical calculations based on the coupled modes theory (CMT). At last, as applications of the designed nanostructure, we also study the modulation degrees of amplitude, insertion loss and group index of transmission spectra by modulating the Fermi energies, which demonstrate an excellent synchronous switching function and slow light effect at multiple frequencies.
2. Structural model and method
Figure 1(a) illustrates the schematic of the designed metasurface, unit cell of which is composed of two reversely placed U-shaped graphene nanostructures and a rectangular graphene ring stacking on a substrate. Figure 1(b) illustrates the top view of the periodic unit cell structure. The offset displacement between the graphene rectangular ring and the center of the designed unit cell is labelled by parameter s. The other geometric parameters of the proposed meta-structure presented in Fig. 1(b) are given as follows: px = 360 nm, py = 240 nm, l1 = 90 nm, l2 = 210 nm, l3 = 90 nm, l4 = 95 nm, w1= 35 nm, g = 30 nm, d = 30 nm, s = 40 nm. The substrate can be chosen as the SiNx material with refractive index of 2.05 and the thickness of substrate is set as 150 nm. Figure 1(c) shows the side view of the unit cell, in which the electrodes are applied with a gate voltage for indirect modulation of the graphene Fermi level.
Fig. 1. (a) The schematic diagram of a periodic terahertz three-dimensional structure diagram based on a single layer of graphene. (b) Top view of the unit structure. (c) Side view of the unit cell of the proposed metamaterial.
In this paper, we perform the full-field electromagnetic simulations by the finite-difference time-domain (FDTD) method. The incident light is normally illuminated on the metasurfaces. The periodic boundary conditions are applied to the cells in the x- and y-directions, and perfect matched layer absorption conditions are applied in the z-direction. Based on the random-phase-approximation (RPA), the surface conductivity of graphene can be described by the Drude model as follows [47]
As for the experiment implementation, the proposed graphene-base metasurfaces are feasible at the current level of nanofabrication techniques [48]. First, the graphene layer grown by chemical vapor deposition is transferred onto the aimed substrate via wetting transfer method using PMMA, and then patterned into nanostructure array using electron beam lithography followed by oxygen plasma etching. Subsequently, the ion-gel layer is spin-coated on top of the patterned graphene metasurfaces, while the Au electrodes manufactured with conductive ionic liquid are solidified on the top of the ion-gel layer.
3. Results and discussion
We first calculate the transmission spectra for different structural elements of the unit cell under the illuminance of x-polarization as shown in Fig. 2. It can be clearly found from Fig. 2(a) and Fig. 2(b) that for the two reversely placed U-shaped graphene rings, there exist coincident transmission spectra with same resonant dips, which can be regarded as the mode degeneracy analogous to that in quantum optics. Both the resonant dips are located at the frequency of 15.4 THz that are labelled by point A and point B, respectively. And we give the corresponding z-components of electric fields (Ez) in Fig. 2(g) and Fig. 2(h), around which the strong dipole moments in the gaps of U-shaped graphene rings are generated, respectively. Meanwhile, when the single rectangular graphene ring is illuminated by x-polarization lights, one can see from Fig. 2(c) that there appears a distinct electromagnetic response at the frequency of 17.7 THz (labelled by point C in the red line) due to the strong excitation of the graphene plasmon polaritions as shown by the distribution of Ez in Fig. 2(i). Therefore, this resonant mode can be defined as a bright mode which can be directly excited by the linearly polarized lights. In addition, we also calculate the transmission spectra of the structural elements combined in pairs. One can see from Fig. 2(c) that when the two reversely placed U-shaped graphene rings are combined together, the transmission spectrum (blue line) takes on a resonant dip at the frequency of 16.2 THz acting as a radiative mode. By contrast, when the rectangular graphene ring is combined with one of the two reversely placed U-shaped graphene rings as shown in Fig. 2(d) and Fig. 2(e), respectively, the optical responses exhibit an analogy of EIT effect, that is the graphene plasmon-induced transparency, which can be attributed to the hybridization coupling between the degeneracy mode and the bright mode. Meanwhile, the transmission spectra keep coincident due to the existence of mode degeneracy. Figure 2(f) calculates the transmission spectrum of the whole integrated unit cell as shown by the inset. One can observe from Fig. 2(f) that the optical response also exhibits an obvious PIT effect, in which a broad transparency window appears in the given frequency region. And the corresponding distributions of Ez at the transmission dips labelled by points D and E are given in Fig. 2(j) and Fig. 2(k), respectively. Comparing with the PIT phenomena appeared in Fig. 2(d) and Fig. 2(e), the transmission spectrum exhibits a broadened transparency window in Fig. 2(f), resulting from the smaller frequency detuning and hybridization coupling between the radiative mode and bright mode.
Fig. 2. (a)-(f) Transmission spectra of the designed metasurfaces with different unit cells. (g)-(k) The corresponding z-components of electric fields (Ez) at the different transmission dips labelled by letters A, B, C, D and E.
Here, it should be noted that the longitudinal offset displacement s between the graphene rectangular ring and the center of the designed unit cell is chosen to be zero in Fig. 2. In fact, the change of the offset displacement parameter s will exert huge influences on the PIT effect. Figure 3(a)-(f) show the transmission spectra of the proposed metasurfaces with different offset displacements s under the forward illumination of x-polarized light. When s = 0 nm, one can see from Fig. 3(a) that there exists only one transparency window between two transmission dips just as the case in Fig. 2(f). When s > 0, one can clearly find that the initial transmission dip at the frequency of 14.6 THz gradually begins tosplit and gives rise to a new transparent window. That is to say, the double PIT effect is induced due to the broken structural symmetry. And the transmittance efficiency of the new transparent window increases correspondingly with further increasing of the offset displacement. To understand the physical mechanism behind this phenomenon as the geometric symmetry is broken, i.e., s = 40 nm, the distributions of electric fields Ez at the newly appeared transmission peaks and dips are depicted as shown in Fig. 3(g)-(j). From the distributions of electric fields, one can conclude that when the geometric symmetry is broken, the coupling lengths between the two reversely placed U-shaped graphene rings and the rectangular graphene ring becomes unequal, thus resulting in the nondegenerate hybridization coupling of bright modes.
Fig. 3. (a)-(f) Transmission spectra at different relative position s. The black lines represent the simulation results, and the red dashed lines represent the CMT theoretical fitting results. (g)-(j) The distributions of electric fields Ez at the dip points and transparency windows for x-polarization (θ = 0°) in Fig. 3(e).
To further support the above thesis, we can use the calculated result of a three-level plasma system to quantitatively analyze the above discussion. The coupled mode theory (CMT) [49] is used to study the theoretical mechanism in the proposed radiation pattern structure. As shown in Fig. 4, A1, A2, and A3 represent three different hypothetical resonator modes, where the superscripts ‘in’ and ‘out’ separately indicate the inflow and outflow radiation waves, the subscripts ‘+’ and ‘−’ respectively imply the propagation direction of the radiation waves. ${\gamma _{in}} = {\omega _n}/2{Q_{in}}$ and ${\gamma _{on}} = {\omega _n}/2{Q_{on}}$, (n = 1, 2, 3) are the inherent loss coefficient and external loss coefficient of the nth mode. Where ${\omega _n}$ is the resonant angular frequency in the optical bright mode, ${Q_{in}} = Re({{n_{\textrm{eff}}}} )/Im({{n_{\textrm{eff}}}} )$ and ${Q_{on}}$, respectively, are the internal loss quality factor and external loss quality factor of the nth mode, and ${n_{\textrm{eff}}}$ is the effective refractive index. The relational expression of ${Q_{in}}$ and ${Q_{on}}$ is $1/{Q_{tn}} = 1/{Q_{on}} + 1/{Q_{in}}$. Here, ${Q_{tn}}$ can be obtained by the ratio of resonance frequency: ${Q_{tn}} = f/\triangle f$. In addition, the mutual coupling coefficient among the three radiation modes can be defined as ${\mu _{nm}}\; ({m,n = 1,\; 2,\; 3,\; \; m \ne n} )$.
Therefore, the complex amplitudes of the time-dependent harmonics (A1, A2, A3) can be provided from the following equations [17]
Since the designed metasurface structure is located on the same layer, φ1, φ2 (the phase difference between 1st mode and 2nd mode, 2nd mode and 3rd mode) can be treated as zero. After straightforward calculations, the transmission coefficient and reflection coefficient can be obtained as
Thus, the transmission and reflection can be obtained by $T = {|t |^2}$ and $R = {|r |^2}$. The relative absorptivity can be obtained by $A = 1 - T - R$.
Based on the above CMT model, the theoretical transmittances for the designed graphene metasurface with different displacement parameter values are calculated as shown by red dashed lines in Fig. 3. One can see from Fig. 3 that the theoretical calculations obtained from CMT are in good agreement with the simulation results as indicated by the black solid line.
Figure 5 shows the transmission curves for polarization angle of the incident light varying from 0° to 90° at s = 0 nm and s = 40 nm, respectively. One can clearly see that the transmission spectra are highly sensitive to the polarization direction of incident light. When θ = 0°, the polarization direction is actually oriented to the x-axis, there exists only a wide transparent window in the PIT spectrum for the metasurface with longitudinal offset displacement s = 0 nm. As the polarization angle θ is gradually increased, i.e., θ = 30°, there appears an obvious resonance dip at the frequency of 16.5 THz as shown in Fig. 5(a), thus leading to the splitting of transparency window. As a result, the number of PIT window is switched from 1 to 2. In contrast, for the case of s = 40 nm, as shown in Fig. 5(b), the number of PIT window can be switched from 2 to 3 due to the polarization splitting and the broken structural symmetry when θ = 30°. With further increasing of polarization angle, the transparency windows gradually take place a change due to the enhancement of resonance dip for both cases. In particular, when θ = 90°, that is the y-polarization, all the transparent windows completely disappear, and the transmission spectrum takes on a Lorentz-like line shape. Based on Jones matrix, the linearly polarized light can be decomposed into two orthogonal directions of linearly polarized light with a fixed phase difference. When light is incident on the metasurfaces with a polarization angle $\theta $, the total transmission can be calculated as the sum of two polarization components with different ratios, i.e., $T = {T_x}\textrm{co}{\textrm{s}^2}(\theta )+ {T_y}\textrm{si}{\textrm{n}^2}(\theta )$, and their transmission spectra would also be superposed by the x- and y-polarizations. Therefore, the polarization-controlled and symmetry-dependent multiple PIT effect can be realized in the designed graphene metasurfaces, in which the number of transparency window can be flexibly and dynamically switched from 1 to 2, and even 2 to 3. Moreover, the corresponding absorption spectra are calculated as indicated by the red line in Fig. 5, which manifest the multi-peak absorption characteristics.
Fig. 5. (a) The transmission and absorption spectrum of the PIT metamaterial device with different polarization angles θ for s = 0 nm. (b) The transmission and absorption spectrum of the PIT metamaterial device with different polarization angles θ for s = 40 nm.
To illustrate the dynamically tunable characteristics of the designed graphene metasurface, we study the transmission spectra of the PIT metasurface at different Fermi energies Ef as shown in Fig. 6. As we know, the complex surface conductivity of graphene can be varied by controlling its Fermi energy, which can be achieved by chemical doping or electrostatic gating. It is clearly observed from Fig. 6 that the resonance frequencies and PIT windows can be actively tuned by varying the Fermi energy level while fixing the geometrical parameters. Furthermore, when the Fermi energy level increases from 0.8 eV to 1 eV, the PIT windows have an obvious blue-shift as the resonance frequency satisfies f$\propto \sqrt {{E_\textrm{f}}}$ [31]. Thus, it implies a good modulation performance for the application of electro-optical switch. Comparing the transmission spectra of the graphene-based PIT metasurfaces, it can be found from Fig. 6(a) that for different Fermi energies, there are significant differences in transmission amplitudes at the resonant frequencies, i.e., f1 = 14.6 THz, f2 = 16.7 THz, and f3 = 18.7 THz. The modulation degree (MD) of the amplitude and the corresponding insertion loss (IL) can be used to describe the adjustable capability, which are expressed as [34]:
where Ton is the maximum transmission amplitude, and Toff is the minimum transmittance. When the Fermi energy level Ef is set to 0.8 eV, as shown by the black line, the transmission amplitudes at the frequencies of f1 (14.6 THz) and f3 (18.7 THz) are 0.98 and 0.99, which are the ‘on’ states with the insertion loss of 0.09 dB and 0.04 dB, respectively. When Ef = 1 eV, as indicated by the red line, the transmission amplitudes at the same frequencies are 0.05 and 0.16, which can be regarded as the ‘off’ states. Therefore, the MD at 14.6 THz and 18.7 THz are 95% and 84%, respectively. In addition, the transmission amplitude at the frequency of f2 (16.7 THz) is changed from 0.26 to 0.99 when the Fermi level is switched from 0.8 to 1 eV, and the MD is 74%, IL is 0.04 dB. So, a simultaneous switching effect can be realized in our designed nanostructure. In contrast, when s = 40 nm, the on-to-off electro-optical switching at multiple frequencies can be obtained due to the broken structure symmetry. In this case, there appears a synchronous multi-switching effect at different frequencies, i.e., f4, f5, f6 and f7, as shown in Fig. 6(b). And the MD of amplitudes at the frequencies of f4 and f5 are 83% and 90% as the Fermi level is changed from 0.8 to 1 eV. In particular, it is shown that the MD at the frequencies of f6 and f7 keep the same to f3 and f4. And the minimum of IL is 0.04 dB in all the above cases. To demonstrate the benefits of our proposed nanostructure, we compare the on-off characteristics of some graphene-based modulators as shown in Table 1. In Table 1, we list the structure configuration, modulation mode, the maximum of MD and the minimum of IL. One can find from Table 1 that our proposed nanostructure shows an excellent optical switch modulation performance, which can be used to design the optical synchronous switching at multiple frequencies. Therefore, the proposed graphene-based PIT device offers additional degrees of freedom in realizing universal tunable functionalities.Fig. 6. Schematic diagram of the on-to-off electro-optical switching modulation at displacement parameter (a) s = 0 nm and (b) s = 40 nm, and the calculated transmission spectra as a function of graphene Fermi energy level and frequency at (c) s = 0 nm and (d) s = 40 nm.
Table 1. Comparisons of the designed metamaterial in this paper with previous works.a
Finally, it is noted that the PIT phenomena are accompanied by the strong phase dispersion near the transparency window, which can result in a distinct slow light effect. To elucidate the slow light characteristics of the proposed PIT metasurfaces, Fig. 7 and Fig. 8 calculate the transmission spectra, phases and the corresponding group index of the proposed metasurface for different Fermi energies Ef at s = 0 nm and s = 40 nm, respectively. In general, the slow light effect can be characterized by group index ${n_g}$, which represents the slowdown factor of light and is defined as follows [50]:
where c, k, φ(ω), and h are the velocity of light in vacuum, the wave vector, transmission phase, and the thickness of the substrate, separately. One can clearly see from Fig. 7 and Fig. 8 that strong phase dispersion occurs around the transmission window, thus resulting in a high group index. In this case, the photon has a weak energy dissipation and undergoes a group delay for a long time in the nanophotonic structure. Moreover, the group index gradually increases with the increasing of Fermi level. When s = 0 nm, the maximum group index is greater than 300, indicating light propagating slower near the transparency window. Especially, when s = 40 nm, a larger group index appears around the newly emerged transparency window due to the broken structure symmetry, and the obtained maximum group index reaches up to 542. Therefore, the proposed graphene metasurface can be applied in designing dynamically tunable slow light devices.Fig. 7. (a) Transmission spectra of the graphene-based PIT nanostructure when the incident wave is polarized in the x direction. (b) Transmission phase of graphene-based metasurfaces. (c) Corresponding group index at different Fermi levels Ef, where s = 0 nm.
Fig. 8. (a) Transmission spectra of the graphene-based PIT nanostructure when the incident wave is polarized in the x direction. (b) Transmission phase of graphene-based metasurfaces. (c) Corresponding group index at different Fermi levels Ef, where s = 40 nm.
4. Conclusions
In summary, we numerically and theoretically presented a polarization-controlled and symmetry-dependent multiple PIT effect in a graphene-based metasurface, which is composed of two reversely placed U-shaped graphene nanostructures and a rectangular graphene ring stacking on a substrate. The designed nanostructure can achieve active control of multiple PIT windows by adjusting the polarization angle of the incident light and the relative position of the structure elements. Concretely speaking, by adjusting the polarization of incident light, the number of transparency windows can be actively modulated between 1 and 2 when the nanostructure keeps a geometrical symmetry with respect to the x-axis. Especially, when the rectangular graphene ring has a displacement along the y-direction, i.e., s = 40 nm, the number of transparency windows can be arbitrarily switched between 2 and 3. The operation mechanism behind the phenomena can be attributed to the near-field coupling and electromagnetic interactions between the bright modes excited in the unit of graphene resonators, respectively. The electromagnetic simulations results obtained by FDTD method agree well with the theoretical calculations based on the coupled modes theory. In addition, an on-to-off electro-optical switching modulation and the maximum group index of 542 can be achieved by adjusting the Fermi energy level of graphene, which demonstrates an excellent synchronous switch functionality at multiple frequencies and slow light effect. Therefore, the proposed graphene-based metasurfaces provide a platform for multi-controlled PIT which has potential applications in the development of next-generation integrated optical processing chips, optical modulation and slow light devices, etc.
Funding
Graduate Research and Innovation Projects of Jiangsu Province (KYCX21_2818); Natural Science Foundation of Jiangsu Province (BK20201446); Applied characteristic Disciplines of Electronic Science and Technology of Xiangnan University.
Disclosures
The authors declare that there are no conflicts of interest related to this article.
Data availability
The data that support the findings of this study are available from the corresponding author upon reasonable request.
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