Dual plasmon-induced transparency (PIT) and plasmon-induced absorption (PIA) are simultaneously achieved in an integrated metamaterial composed of single layer of graphene. Electric field distribution and coupled mode theory (CMT) are used to demonstrate the physical mechanism of dual PIT and PIA, and the theoretical result of CMT is highly consistent with the finite-difference time-domain (FDTD) method simulation result. Further research shows that both the dual PIT and PIA phenomenon can be effectively modulated by the Fermi level, the carrier mobility of the graphene and the refractive index of the surrounding environment. It is meaningful that the absorption of the dual PIA spectrum can be abruptly increased to 93.5% when the carrier mobility of graphene is 0.8m2/Vs. In addition, the group index can be as high as 328. Thus, our work can pave new way for developing excellent slow-light and light absorption functional devices.
© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
Electromagnetically induced transparency (EIT) is a quantum interference effect in the atomic system, which is produced by the interaction between the atomic layer and the external light field . In general, the EIT effect can significantly slow the light velocity through the medium and thus has significant applications in the slow-light field . Of course, it also has great application value in nonlinear optics  and biochemical sensing . However, EIT is hindered by difficult experimental conditions in practical applications, such as very low experimental temperatures and stable gas lasers . Therefore, plasmon-induced transparent (PIT) and plasmon-induced absorption (PIA) emerge as the times require, which are a phenomenon that is superimposed by interference between a broad-band bright mode and a narrow-band dark mode in metamaterials [6,7]. The bright mode and the excitation light field have a strong coupling, yet the dark mode can only be excited by the localized field which is formed by the coupling of the bright mode and the excitation light field. The PIT and PIA phenomena have potential value in many fields due to anomalous dispersion and narrower transparent peaks, such as light absorbers , light switch , optical filters  and optical sensors .
So far, many plasmonic devices have been proposed, and metal-dielectric-metal (MDM) plasmonic waveguides stand out due to support the acceptable length and high order wavelengths for surface plasmon polaritions (SPPs). Moreover, the coupling of MDM waveguides with double U-cavity , double-circular resonators and multi-ring resonators  have potential applications in the field of slow-light. However, the group index of the slow-light effect is only a few tens at most of these plasmonic waveguides. Moreover, it is hard to modulate the slow-light effect at a specific wavelength. Later, graphene composed of a single layer of carbon atoms as a novel two-dimensional material has been widely concerned by researchers because it can also support the propagation of SPPs in the terahertz and infrared band . In addition, graphene has the advantages of dynamic adjustability, strong locality and low loss. Most importantly, the conductivity of graphene can be modulated by Fermi level or chemical potential, which has unparalleled advantages over other metamaterials. Therefore, it has important application in slow-light device [15,16], modulator , absorber , photon detector  and sensor . However, many devices of light-graphene coupling can only realize one function, and multi-function graphene-based devices are rarely reported.
In our paper, a dynamically tunable plasmon-induced slow-light and light absorption multi-functional graphene-based metamaterial is proposed because the dual PIT and PIA phenomena, which are explained in detail by electric field distribution, electromagnetically induced transparency (EIT) and coupled mode theory (CMT). The slow-light and absorption effects can be effectively modulated by changing the Fermi level, the carrier mobility of the graphene and the surrounding medium environment. Generally, the proposed structure also has such merits. Firstly, the proposed structure base on a single-layer patterned graphene array, which is simpler than other two-layer , three-layer , and four-layer  graphene structures. Secondly, its slow-light characteristics are outstanding, and the group index can be as high as 328. Finally, the absorption rate can be as high as 93.5%, which shows excellent absorption property comparing with other single-layer graphene-based absorbers [21,23]. Based on the above advantages, our proposed structure opens a new door for achieving excellent slow-light and absorption multi-function metamaterial.
2. Structural design and theoretical analysis
Slow-light and light absorption integrated metamaterial is shown in Fig. 1(a). Each patterned graphene unit composed of four graphene blocks and three graphene ribbons is on the top of silicon substrate, and its top view is shown in Fig. 1(b). The specific structural parameters are given in the caption of Fig. 1, where l1, l2, l3, l4, l5 are no longer changed in this paper. A linear polarized plane wave is incident perpendicularly along the positive direction of the z-axis.
In this system, the temperature is T = 300K, and the conductivity of the monolayer graphene can be obtained by the Kubo formula. Since the direct interband electronic transitions are negligible in the terahertz frequency range, the main contribution comes from intraband electron photon scattering, whose expression can be expressed as :25].The carrier relaxation time is τ = μEf / (evF2) , where vF = 106m/s and μ = 1m2/(V·s) are Fermi level of graphene and the carrier mobility of graphene, respectively. In addition, since the medium environment around the graphene is silicon, the propagation constant of the SPPs on the surface of the graphene is β = k0(εsi – (2εsi / σgŋ0)2)1/2 [27,28]. Where k0, εsi, ŋ0 are the wave vector of the incident light, the dielectric constant of silicon and the inherent impedance, respectively. At the same time, effective refractive index can also be obtained as: neff = β / k0.
To demonstrate the physical mechanism of dual PIT and PIA clearly, we firstly investigate four metamaterials, as shown in Figs. 2(c)-2(f).The transmission and absorption spectra of the four metamaterials are shown in Figs. 2(a) and 2(b). For convenience of explanation, we have named the dips and peaks in the transmission spectrum of the whole structure from left to right as “dip1”, “peak1”, “dip2”, “peak2”, “dip3”. The black dual PIT line is generated by the metamaterial array of the whole structure in Fig. 2(c), which is composed of the monopole antenna, the quadrupole antenna and the octopole antenna, as shown in Figs. 2(d)-2(f). In general, patterned periodic graphene arrays can well support localized surface plasmons (LSPs). When a linearly polarized plane wave is incident, the monopole antenna is not directly excited due to the absence of electric fields component along the direction of the central axis of the monopole antenna, as indicated by the light blue line. However, the quadrupole antenna and the octopole antenna undergo LSPs resonance by the incident light field, and the classic Lorentz lines are formed at the position of f1 = 2.354THz and f2 = 4.038THz, respectively, as shown by the blue and red lines. In our proposed structure, the monopole antenna can’t be directly excited as a dark mode, while the quadrupole antenna and the octapole antenna are directly excited as two bright modes, and the destructive interference between the three modes forms a strong dual PIT effect and PIA effect, resulting in transparent windows at f3 = 2.093THz, f4 = 3.105THz, f5 = 4.188THz, respectively, wherein the coefficient of absorption transparent window is up to 48%.
In order to better analyze the dual PIT (PIA) phenomenon, we plot the electric field distributions at the resonance frequencies of the quadrupole antenna, the octapole antenna and the whole structure in the y direction, as shown in Fig. 3. Figure 3(a) shows very clearly that there are a pair of dipoles on the upper and lower sides of the graphene ribbon, and the charge polarity is opposite, which is the reason that we call it a quadrupole antenna (the same as the octapole antenna). In Fig. 3(b), the charge intensity of the dipole pair on the adjacent side is smaller than the outer side due to the coupling between the graphene pair. In the field distribution of the whole structure, Figs. 3(c)-3(e) clearly shows the monopole antenna is indirectly excited as a dark mode by two bright modes. The formation mechanism of dip1 is mainly the interaction between monopole antenna and quadrupole antenna, the formation mechanism of dip3 is mainly the interaction between monopole antenna and octupole antenna, and dip2 is caused by the interaction of three modes. Because the resonance dips of the two bright modes are basically the same, the resonance dips of the two single PITs obtained by their interaction with the dark mode are also basically the same. However, after the two single PITs are formed, the adjacent dips are superimposed to form dip2, and the two outer dips are keep the same as dip1 and dip3. The electric field distributions of dip1 and dip3 are basically the same, and the electric field distribution of dip2 also shows the interaction between the three modes, which is in perfect agreement with the above discussion.
Then, the dual PIT (PIA) phenomenon can also be explained from the perspective of EIT. The three modes are assumed to be three photon states, and schematic diagram of their relationship to the Λ-shaped PIT  is shown in Fig. 4(a). The light source corresponds to the ground state, and the quadropole antenna and the octupole antenna can be directly excited by the light source to serve as the first excited state and . However, the monopole antenna cannot be directly excited by light source but can be indirectly excited by the first excited state as the second excited state .There is a coupling field ΩC between and , there is a detection field ΩP between and , and the interference effect between the two fields produces a PIT (PIA) window.
Finally, CMT [30,31] are used to analyze and calculate the dual PIT and PIA phenomena. A diagram of the CMT is shown in Fig. 4(b), where A, B, and C are three imaginary resonators respectively corresponding to one dark mode and two bright modes, whose composite amplitudes are denoted by a, b, c, respectively. represent input or output waves propagating toward the positive or negative direction of the resonator, respectively. Moreover, μnm, γin, γon (n ≠ m, n = 1, 2, 3, m = 1, 2, 3) represent the coupling coefficients between the three resonators and their internal and external loss coefficients. The relationship between the three resonators is easily obtained :
Finally, we can use the CMT to obtain the transmittance of the dual PIT: T = t2, and the absorbance of the dual PIA: A = 1 − t2 − r2.
3. Regulation and discussion
Numerical simulations of dual PIT and PIA phenomena are shown in the blue solid line of Fig. 5, both the transmission and absorption spectra exhibit obvious blue-shift as the Fermi level increases from 0.9eV to 1.2eV. We also get results of the CMT theoretical calculation by coupling model formulas introduced earlier, as shown by the red dotted line in Fig. 5. As a result, the theoretical calculation of CMT is highly consistent with the finite-difference time-domain (FDTD) numerical simulation. The relationships between the resonant dips and peaks versus frequency at different Fermi levels are plotted in Fig. 6(a), which show a good linear relationship. Figure 6(b) shows the relationship between the internal loss quality factor and frequencies at different Fermi levels in the whole system, which can be obtained by: Qin = Re(neff) / Im(neff) . Then, the quality factors of the transmission dips in the dual PIT line and the quality factors of the absorption peaks in the dual PIA line at different Fermi levels are shown in Figs. 6(c) and 6(d). In addition, the transmission and absorption evolution between Fermi levels and frequency are plotted in Figs. 6(e) and 6(f), and we can see the blue-shift of dual PIT and PIA lines and the approximate range of transmission and absorption coefficients as the increases of Fermi level.
The evolution of dual PIT and PIA are further explored by increasing the length l6 of monopole antenna from 0.20μm to 0.50μm with increments 0.06μm. Figures 7(a) and 7(b) show how the transmission (absorption) spectra evolve from a single PIT (PIA) to a dual PIT (PIA) as increases of l6. Interestingly, the first resonant dip (peak) of the dual PIT (PIA) is almost unchanged; however, the third resonant dip (peak) is changed rapidly with the decrease (increase). The above results highlight the importance of dark mode from the side-coupled, which has the greatest influence on the third resonance dip (peak) of dual PIT (PIA).
The refractive index of the surrounding media and the carrier mobility of graphene are changed in order to achieve the purpose of effectively regulating the dual PIT and PIA phenomenon, the results are shown in Fig. 8. When only the refractive index of the medium around graphene increases from 2.92 to 3.52, the dual PIT (PIA) spectra are monotonically red-shifted. However, both the transmittance and reflectance of the proposed metamaterial are very low when only the carrier mobility of graphene is increased from 0.6m2/Vs to 0.8m2/Vs, so the absorption of the dual PIA line can suddenly increase to 93.5%. Therefore, we can fix the carrier mobility of graphene as μ = 0.8m2/Vs, and then change the surrounding medium environment to effectively modulate the absorption with excellent performance.
Slow-light effect is an important area in the application of PIT, so we analyzed the slow-light characteristics of the proposed metamaterial. In general, the slow-light effect is determined by the group index, the larger the group index, the better the slow-light characteristics. It can be obtained by: ng = c · dk / dω = c · dθ / (l · dω) , Where c, k, l are the light velocity, the wave vector in vacuum and the thickness of the substrate silicon, respectively, and transmission phase shift is θ = arg(t). Therefore, we plot the group index and phase shift at transparent windows from 0.9eV to 1.2eV in Fig. 9, the max group index of the metamaterials increases with increasing of the Fermi level. When the Fermi level is 1.2eV, the group index can be as high as 328, which is better than most of the slow-light equipment. Since the sharp change in the phase shift at the transmission dip (absorption peak), the group index drops sharply. Thence, large group index is produced around the transmission dip (absorption peak), which is caused by strong dispersion. Therefore, we can effectively modulate the dual PIT (PIA) phenomenon to seek the desired slow-light effect.
In summary, the dual PIT and PIA phenomena are numerically simulated and theoretically studied based on monolayer of periodically patterned graphene-based metamaterials, and the results of FDTD numerical simulation is highly consistent with CMT theoretical calculations. Dual PIT and PIA phenomena can be effectively modulated by the Fermi level, the carrier mobility of the graphene and the refractive index of the surrounding environment, which have potential applications in modulators. Interestingly, the absorption can be as high as 93.5% by changing the carrier mobility of graphene, which opens a new door for monolayer graphene to achieve excellent absorbers. In addition, the slow-light characteristics of the proposed structure are very outstanding, and the group index is as high as 328. Therefore, our work can pave new way for developing excellent slow-light and light absorption functional devices.
The National Natural Science Foundation of China (No. 11847026, 61764005 and 11804093); The Scientific Project of Jiangxi Education Department of China (No. GJJ160532); The Graduate Education Reform Project of Jiangxi Province of China (No. JXYJG-2017-080); The authors thank Professor Jicheng Zhao and Mary Anna Wildermuth of Ohio State University for discussions and revision of this paper.
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