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Abstract
Here, we design a metal-graphene metamaterial to selectively control dual EIT modes. The metallic metamaterial consists of bright, dark, and quasi-dark meta-atoms, leading to two EIT modes. Meanwhile, monolayer graphene ribbons inserted under the dark meta-atoms and quasi-dark meta-atoms are separately connected to different electric sources. In simulation, both the two EIT modes and the time delays can be selectively controlled. Moreover, the number of the EIT modes can be tuned from two to one, and even to zero. Our work provides a strategy to selectively control the two EIT modes and the slow light compacted in a terahertz metamaterial, which may achieve potential applications in actively tunable integrated terahertz devices.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Electromagnetically induced transparency (EIT) is a coherence phenomenon in three-level atomic systems, which can slow down the group velocity of light and hence has extensive application prospects in information storage and processing [1–3]. However, the practical applications are greatly limited by the strict experimental conditions, determined by the short coherence time in the atomic system. Recently, besides classical analogue of EIT [4], the development of metamaterials provides a good solution to this problem [5,6]. Thereafter, a variety of EIT metamaterials have been reported in microwave [7,8], terahertz (THz) [9,10], near-infrared [11–14], and optical regions [15,16], almost covering all bands.
Due to the application demands of miniaturized and versatile optical signal processing devices, more and more attention has been paid to multimode EIT metamaterials [17–21] and actively tunable EIT metamaterials [22–37]. Multimode EIT can be realized via some different designs, such as exciting magnetic dipoles by breaking the symmetry [17], employing inductive coupling among dipoles in similar structures [18–20], or introducing quasi-dark meta-atom into EIT system consisting bright and dark meta-atoms [21]. On the other hand, there have also been a variety of means to actively manipulate the EIT mode, for example, embedding photosensitive silicon [23–26] and electrically sensitive graphene [27–36] into metamaterials, utilizing microelectromechanical systems [37], and so on.
As both the above aspects have been extensively explored, some interest is gradually attracted to combine them, to study actively tunable multimode EIT metamaterials. Till now, some achievements have been made that the wavelength of the multiple EIT modes can be dynamically controlled. However, most of these work modulate the multiple EIT modes as a whole [38–44], and there is little research realizing selective regulation of every mode [45], which is more practical in optical signal processing.
In this work, we design a metal-graphene metamaterial to selectively control the two integrated EIT modes. By simultaneously employing the metallic bright, dark and quasi-dark meta-atoms, the dual-mode EIT metamaterial is constructed. Then, two separate monolayer graphene ribbons connected to different electric sources are inserted between the metallic nanostructure and the SiO2/Si substrate, contacting the dark and quasi-dark meta-atoms, respectively. Applying voltages, the recombination effect of the conductive graphene can dynamically tune the damping rates of the meta-atoms in touch, which, as a result, can regulate the corresponding EIT modes. Moreover, the calculation results show that the delay time accompanying EIT phenomenon can also be selectively tuned, which is beneficial in applications of slow light devices. Consequently, this work provides a strategy to selectively control the EIT modes and the related slow light compacted in a terahertz metamaterial, which may achieve potential applications in dynamically tunable integration devices used in THz communications.
2. Structure design and theoretical model
The dual-mode EIT metamaterial is constructed by three kinds of meta-atoms and the interference process follows the equation that [21]
Fig. 1. (a) The schematic of the selectively tunable EIT metamaterial on the SiO2/Si substrate, with monolayer graphene ribbons inserted between the nanostructure and the substrate. The voltages V1 and V2 are respectively applied to the graphene ribbons through the discrete electrodes, which can be controlled independently. (b) The top view of the unit cell of the metamaterial, with the periods of 120µm in both directions. The geometric parameters are L1 = 38µm, W1 = 36µm, D1 = 10µm, D2 = 20µm, M1 = 20µm, M2 = 18µm, L2 = 98µm, W2 = 10µm, L3 = 49µm, W3 = 28µm, D3 = 8µm, N1=33µm, N2=12µm, f1 = 6µm and f2 = 7µm, respectively. The thickness of the metamaterial is t = 0.2µm.
In simulation, the permittivity of Au in THz band follows the Drude model as [32]
Between the nanostructure and the substrate, two separate monolayer graphene ribbons are inserted under the dark and the quasi-dark meta-atoms, respectively. As shown in Fig. 1(a), discrete golden electrodes are deposited on different ribbons while the substrate is connected to the ground. Separately wire-bonding the electrodes touching the dark meta-atoms and the quasi-dark meta-atoms, we can then connect the corresponding graphene ribbons to two different electric sources. The applied voltages are labelled as V1 and V2, respectively. Changing the voltages, the recombination effect of the conductive graphene can actively tune the damping rates of the metallic meta-atoms in touch.
The optical properties of the monolayer graphene can be described by the surface conductivity σg in Kubo formula, which can be calculated employing the random-phase approximation (RPA) considering both inter-band and intra-band transition [46,47]. In THz regime, since the conductivity induced by the intra-band process greatly exceeds the one of inter-band at room temperature, the graphene conductivity can be defined by the Drude-like expression [48],
3. Results and discussion
We firstly calculate the transmission spectra by separately arranging every meta-atom periodically. As shown in Fig. 2(a), the bright meta-atom and quasi-dark meta-atom can both be excited by the incident light whose electric component polarized along x-axis. Apart from the difference of the bandwidth, the resonant frequency of the bright meta-atom is close to that of the quasi-dark meta-atom, both around 0.51THz. Although the dark meta-atom cannot be excited by the x-polarized incident wave, it can be excited by the y-polarized wave. Moreover, the resonant frequency of the dark meta-atom is also around 0.51THz. Combining these three kinds of meta-atoms together to construct the periodic array in Fig. 1, the calculated transmission spectra is given in Fig. 2(b), exhibiting two transparency windows around 0.48THz and 0.55THz.
Fig. 2. (a) The calculated transmission spectra of the periodic array consisting only one kind of meta-atom. The purple, blue and orange solid lines are the transmission spectra of the dark, bright and the quasi-dark meta-atom arrays under x-polarization illumination, respectively. The green dashed line are the spectra of dark meta-atom array under y-polarization. (b) The transmission spectra of the metamaterial constructed by the three kinds of meta-atoms. (c) The frequency dependent surface conductivities of the monolayer graphene related to different Fermi levels, where the solid lines and the dashed lines represent the real part and the imaginary part, respectively. (d) The group delay of the metamaterial.
Since EIT phenomenon is always accompanied by the slow light effect, the frequency dependent group delay tg is one of the commonly used parameters to describe it, which is defined as [50]
where φ is the transmission phase shift. As displayed in Fig. 2(d), two largest group delays emerge around the two transparency windows, about 11.5ps and 10.8ps, respectively. Such group delays come from the positive and steepest phase slope, indicating that net energy flows through the metamaterial with slower velocities than that in the free space. Therefore, two EIT modes accompanied by the slow light effect are acquired in this system.Then we calculate the tunable surface conductivity of the monolayer graphene based on Eq. (3). Selecting some Fermi levels that can be tuned by the external electric voltage, the resulted frequency dependent conductivities are displayed in Fig. 2(c). Increasing the Fermi levels, both the real part and the imaginary part of the surface conductivity gradually increase, implying good regulation effects.
Keeping V2 = 0 and tuning the voltage V1 to vary the Fermi level EF1 of the graphene ribbons touching the dark meta-atoms, the variation of the transmission spectra is shown in Fig. 3(a). As EF1 increasing from 0.1 eV to 0.3 eV, the EIT window around higher frequencies gradually attenuates and eventually disappears, while the transparency window around lower frequencies only slightly diminishes with the weakening of the overall transmission intensity. The maximum amplitude modulation depth defined as ${D_T} = \Delta T/{T_0}$ is acquired to be DT1 = 0.73 around 0.55THz, where T0 is the original transmission amplitude before modulation. As a contrast, we tune the voltage V2 to raise EF2 from 0.1 eV to 0.3 eV without applying V1, and then calculate the transmission spectra, as displayed in Fig. 3(b). The transparency window around higher frequencies only slightly weakens accompanying with the decrease of the overall transmission intensity, while the one around lower frequencies gradually diminishes to disappear, with a maximum amplitude modulation depth DT2 = 0.78 around 0.48THz. Hence, the two EIT windows can be selectively controlled via applying voltages to distinct electrodes, with the DT comparable to other techniques [24,37].
Fig. 3. (a) The transmission spectra when tuning the voltage V1 to change the Fermi level EF1 of the graphene in touch with the dark meta-atoms from 0.1 eV to 0.3 eV. (b) The transmission spectra when tuning the voltage V2 to change the Fermi level EF2 of the graphene in touch with the quasi-dark meta-atoms from 0.1 eV to 0.3 eV.
In order to have a better understanding of the modulation process, we calculate the electric field amplitude distribution at the transparency windows. The distribution maps of the metamaterial without graphene, at the EIT windows around 0.48THz and 0.55THz, are shown in Figs. 4(a) and 4(b), respectively. Increasing EF1 up to 0.3 eV via V1, the field distribution map of the remaining EIT window around 0.48THz is drawn in Fig. 4(c). The electric field is mainly localized around the quasi-dark meta-atom, with the dark meta-atom unexcited, which is similar to the one in Fig. 4(a). On the other hand, only the EIT window at 0.55THz exists when EF2 is raised to 0.3 eV by V2, and the corresponding field distribution is displayed in Fig. 4(d). Similar to the results in Fig. 4(b), the electric field is mostly localized around the dark meta-atom, and the quasi-dark meta-atom is hardly excited. In both cases as the Fermi levels are tuned up to 0.3 eV, the damping rates of the meta-atoms in contact with the high conductivity graphene ribbons are so large that they cannot be excited. What’s more, the regulation capacity of the graphene ribbons is localized, which can only affect the contacted meta-atoms. Therefore, different kinds of meta-atoms in this system can be selectively controlled. As a result, the two integrated EIT modes can be switched to single EIT mode.
Fig. 4. (a) and (b) show the electric field amplitude distributions at the transparency windows of 0.48THz and 0.55THz, respectively, without external voltages. (c) The electric field amplitude distribution at the remained transparency window, when EF1 is tuned to 0.3 eV by V1. (d) The electric field amplitude distribution at the remained transparency window, when EF2 is tuned to 0.3 eV by V2. The white dashed lines outline the meta-atoms.
As mentioned before, the group delay tg is an important parameter to value the slow light effect along with the EIT phenomenon. We then calculate the group delay related to different Fermi levels and show the results in Fig. 5. Only applying V1 to change EF1 from 0.1 eV to 0.3 eV, the group delay at higher frequency gradually diminishes to zero, while the one at lower frequency drops to 3.1ps, compared to the value of 11.5ps without graphene. Similar variations occur when merely applying V2 to increase EF2 from 0.1 eV to 0.3 eV. The slow light effect at lower frequency vanishes in the end, and the group delay at higher frequency finally falls from 10.8ps to 3.5ps. Therefore, increasing the Fermi levels, the group delay falls as the phase slop becomes slower, accompanied with the transmission decrease of the EIT window. Although the graphene can only manipulate the damping rate of the meta-atom in touch, the slow light effect at the two EIT windows are both affected. That is because both the two EIT modes come from the destructive interference among the three meta-atoms, even though there is a distinct primary role in different modes. Therefore, changing each voltage from 0 to 0.3 eV, one slow light effect would be turned from on to off, accompanied by the other group delay varying within a limited range.
Fig. 5. (a) and (b) display the frequency dependent group delays when the Fermi levels of the graphene ribbons are tuned from 0.1 eV to 0.3 eV by V1 and V2, respectively.
As seen from the above results, each EIT mode can be selectively modulated to swith on or off by individually changing the corresponding voltage. Thus, dual EIT modes can be turned to single EIT mode. Besides, it is worth mentioning that the remained EIT mode can be turned on or off as well. Fixing voltage V2 to keep EF2 as 0.3 eV, we then change the voltage V1 to raise the Fermi level EF1 of the rest graphene ribbons from 0.1 eV to 0.3 eV, the transmission spectra are shown in Fig. 6(a). The transmission amplitude of the EIT window decrease and eventually diappear as EF1 increasing to 0.3 eV, accompanied by the group delay in Fig. 6(b) diminishing to near zero. It offers more freedom to manipulate the integrated dual EIT modes, which may broaden the applications of the actively tunable multimode system.
Fig. 6. (a) The transmission spectra when voltage V2 is applied to fix EF2 as 0.3 eV and then tune V1 to increase EF1 from 0.1 eV to 0.3 eV. (b) The corresponding group delay.
Apart from the Fermi level, the carrier mobility μ is also an important parameter which can affect the surface conductivity. Independently fixing the Fermi levels as 0.2 eV and changing μ of each graphene ribbons from 2000cm2/V·s to 4000cm2/V·s, we calculate the variation of the transmission spectra and show the results in Fig. 7. In both cases, the modulated EIT mode gradually diminishes as the carrier mobility increasing. Therefore, the carrier mobility could affect the threshold of the Fermi level to switch off the EIT mode. The higher the carrier mobility, the lower the Fermi level required to turn the EIT mode off.
Fig. 7. The transmission spectra as μ is changed from 2000cm2/V·s to 4000cm2/V·s, when (a) only apply voltage V1 to fix EF1 as 0.2 eV and (b) only apply voltage V2 to fix EF2 as 0.2 eV.
4. Conclusions
In summary, we have designed a metal-graphene metamaterial in which the integrated dual EIT modes can be selectively controlled by the independently applied voltages. In this system, both the transmission amplitudes of the EIT windows and the related slow light effects can be controlled. The simulation results show that the two EIT windows can be separately tuned from on to off when increasing the Fermi levels of the corresponding graphene ribbons up to 0.3 eV, and the threshold of the Fermi level is affected by the carrier mobility of graphene. Only applying V1 or V2, one of the group delay gradually vanishes, while the other one becomes smaller. Therefore, one of the slow light effect can be turned on or off, with the other one varies simultaneously. Moreover, keeping the Fermi level of the graphene ribbons in contact with the quasi-dark meta-atoms as 0.3 eV and increasing the Fermi level of the rest graphene ribbons to 0.3 eV, the remained EIT window can also be switched from on to off. Therefore, we offer a distinct strategy to actively tune the integrated EIT modes. One can selectively turn the dual-mode EIT to single-mode EIT, and even then switch the single EIT mode from on to off. Consequently, the number of the EIT modes in this system can be tuned from two to one, and even to zero. Our work may inspire related studies about tunable multimode EIT metamaterials, and achieve potential applications on integrated optical circuits in terahertz band.
Funding
National Natural Science Foundation of China (11804178, 11274188, 11847002, 11904008); Natural Science Foundation of Shandong Province (ZR2018BA027).
Disclosures
The authors declare no conflicts of interest.
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