Tunable multiple phase-coupled plasmoninduced transparencies in graphene metamaterials Chao Zeng, Yudong Cui, and Xueming Liu* State Key Laboratory of Transient Optics and Photonics, Xi’an Institute of Optics and Precision Mechanics, Chinese Academy of Sciences, Xi’an 710119, China * [email protected]
Abstract: We demonstrate the existence of multiple electromagnetically induced transparencies (EIT)-like spectral responses in graphene metamaterials consisting of a series of self-assembled graphene Fabry-Pérot (FP) cavities. By exploiting the graphene plasmon resonances and phasecoupling effects, the transfer matrix model is established to theoretically predict the EIT-like responses, and the calculated results coincide well with numerical simulations. It is found that high-contrast (~90%) multiple EITlike windows are observed over a broad range of mid-infrared. Additionally, these optical responses can be efficiently tuned by altering the Fermi level in graphene and the separations of FP cavities. The proposed scheme paves the way toward control of the multiple EIT-like responses, enabling exploration of the on-chip multifunctional electro-optic devices including multi-channel-selective filters, sensors, and modulators. © 2014 Optical Society of America OCIS codes: (240.6680) Surface plasmons; (160.3918) Metamaterials; (130.3120) Integrated optics devices; (050.2230) Fabry-Pérot; (050.6624) Subwavelength structures.
References and links 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14.
K. J. Boller, A. Imamolu, and S. E. Harris, “Observation of electromagnetically induced transparency,” Phys. Rev. Lett. 66(20), 2593–2596 (1991). Y. Wu, J. Saldana, and Y. Zhu, “Large enhancement of four-wave mixing by suppression of photon absorption from electromagnetically induced transparency,” Phys. Rev. A 67(1), 013811 (2003). T. F. Krauss, “Why do we need slow light,” Nat. Photonics 2(8), 448–450 (2008). C. Liu, Z. Dutton, C. H. Behroozi, and L. V. Hau, “Observation of coherent optical information storage in an atomic medium using halted light pulses,” Nature 409(6819), 490–493 (2001). M. F. Yanik, W. Suh, Z. Wang, and S. Fan, “Stopping light in a waveguide with an all-optical analog of electromagnetically induced transparency,” Phys. Rev. Lett. 93(23), 233903 (2004). Q. Xu, S. Sandhu, M. L. Povinelli, J. Shakya, S. Fan, and M. Lipson, “Experimental realization of an on-chip alloptical analogue to electromagnetically induced transparency,” Phys. Rev. Lett. 96(12), 123901 (2006). S. Zhang, D. A. Genov, Y. Wang, M. Liu, and X. Zhang, “Plasmon-induced transparency in metamaterials,” Phys. Rev. Lett. 101(4), 047401 (2008). N. Liu, L. Langguth, T. Weiss, J. Kästel, M. Fleischhauer, T. Pfau, and H. Giessen, “Plasmonic analogue of electromagnetically induced transparency at the Drude damping limit,” Nat. Mater. 8(9), 758–762 (2009). R. D. Kekatpure, E. S. Barnard, W. Cai, and M. L. Brongersma, “Phase-coupled plasmon-induced transparency,” Phys. Rev. Lett. 104(24), 243902 (2010). H. Lu, X. M. Liu, and D. Mao, “Plasmonic analog of electromagnetically induced transparency in multinanoresonator-coupled waveguide systems,” Phys. Rev. A 85(5), 053803 (2012). J. Chen, C. Wang, R. Zhang, and J. Xiao, “Multiple plasmon-induced transparencies in coupled-resonator systems,” Opt. Lett. 37(24), 5133–5135 (2012). A. Artar, A. A. Yanik, and H. Altug, “Multispectral plasmon induced transparency in coupled meta-atoms,” Nano Lett. 11(4), 1685–1689 (2011). H. Lu, X. Liu, G. Wang, and D. Mao, “Tunable high-channel-count bandpass plasmonic filters based on an analogue of electromagnetically induced transparency,” Nanotechnology 23(44), 444003 (2012). H. Cheng, S. Q. Chen, P. Yu, X. Y. Duan, B. Y. Xie, and J. G. Tian, “Dynamically tunable plasmonically induced transparency in periodically patterned graphene nanostrips,” Appl. Phys. Lett. 103(20), 203112 (2013).
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Received 22 Oct 2014; revised 25 Dec 2014; accepted 26 Dec 2014; published 9 Jan 2015 12 Jan 2015 | Vol. 23, No. 1 | DOI:10.1364/OE.23.000545 | OPTICS EXPRESS 545
15. X. Shi, D. Z. Han, Y. Y. Dai, Z. F. Yu, Y. Sun, H. Chen, X. H. Liu, and J. Zi, “Plasmonic analog of electromagnetically induced transparency in nanostructure graphene,” Opt. Express 21(23), 28438–28443 (2013). 16. C. Zeng, J. Guo, and X. M. Liu, “High-contrast electro-optic modulation of spatial light induced by grapheneintegrated Fabry-Pérot microcavity,” Appl. Phys. Lett. 105(12), 121103 (2014). 17. L. Ju, B. Geng, J. Horng, C. Girit, M. Martin, Z. Hao, H. A. Bechtel, X. Liang, A. Zettl, Y. R. Shen, and F. Wang, “Graphene plasmonics for tunable terahertz metamaterials,” Nat. Nanotechnol. 6(10), 630–634 (2011). 18. H. Yan, X. Li, B. Chandra, G. Tulevski, Y. Wu, M. Freitag, W. Zhu, P. Avouris, and F. Xia, “Tunable infrared plasmonic devices using graphene/insulator stacks,” Nat. Nanotechnol. 7(5), 330–334 (2012). 19. M. Amin, M. Farhat, and H. Bağcı, “An ultra-broadband multilayered graphene absorber,” Opt. Express 21(24), 29938–29948 (2013). 20. M. A. K. Othman, C. Guclu, and F. Capolino, “Graphene-based tunable hyperbolic metamaterials and enhanced near-field absorption,” Opt. Express 21(6), 7614–7632 (2013). 21. W. Gao, J. Shu, C. Qiu, and Q. Xu, “Excitation of plasmonic waves in graphene by guided-mode resonances,” ACS Nano 6(9), 7806–7813 (2012). 22. Z. Fang, S. Thongrattanasiri, A. Schlather, Z. Liu, L. Ma, Y. Wang, P. M. Ajayan, P. Nordlander, N. J. Halas, and F. J. García de Abajo, “Gated tunability and hybridization of localized plasmons in nanostructured graphene,” ACS Nano 7(3), 2388–2395 (2013). 23. B. Sensale-Rodriguez, T. Fang, R. Yan, M. M. Kelly, D. Jena, L. Liu, and H. G. Xing, “Unique prospects for graphene-based terahertz modulators,” Appl. Phys. Lett. 99, 113104 (2011). 24. A. Andryieuski and A. V. Lavrinenko, “Graphene metamaterials based tunable terahertz absorber: effective surface conductivity approach,” Opt. Express 21(7), 9144–9155 (2013). 25. T. Stauber, N. M. R. Peres, and A. K. Geim, “Optical conductivity of graphene in the visible region of the spectrum,” Phys. Rev. B 78(8), 085432 (2008). 26. K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y. Zhang, S. V. Dubonos, I. V. Grigorieva, and A. A. Firsov, “Electric field effect in atomically thin carbon films,” Science 306(5696), 666–669 (2004). 27. H. Lu, X. Liu, Y. Gong, D. Mao, and L. Wang, “Enhancement of transmission efficiency of nanoplasmonic wavelength demultiplexer based on channel drop filters and reflection nanocavities,” Opt. Express 19(14), 12885–12890 (2011).
1. Introduction Electromagnetically induced transparency (EIT) occurs in laser-activated atomic systems as a result of quantum interference between the atomic level and the excitation pathways [1]. Due to the strong dispersion in transparency windows, the EIT effects have shown a plethora of intriguing potentials in nonlinear optical enhancement [2] and all-optical data processing [3]. However, complex experimental requirements have significantly constrained the chip-scale implements of conventional EIT in atomic systems [4]. Fortunately, the EIT-like effects were later demonstrated in classical systems as diverse as high-Q resonators [5,6] and plasmonic structures [7–9]. For instance, Xu et al. reported the first experimental observations of EITlike spectra in silicon resonator systems, which brought the original quantum phenomena into the realm of classical optics [6]. Zhang et al. proposed a coupled radiative-dark plasmonic mechanism to mimic the functionality of EIT by metamaterials [7]. Because of the capabilities to operate at subwavelength scales, especially, metal-based plasmonic EIT-like configurations have been extensively exploited [7–13]. Recently, plasmonic analogs of EIT have been demonstrated in periodically patterned graphene nanostructures [14–16]. Due to its unprecedented advantages including flexible tunability, extremely tight field confinement, and low losses at terahertz and mid-infrared frequencies, graphene has been proved to be a promising alternative material to metals for chip-integrated nanophotonics [17–26]. However, as Brongersma et al. analyzed in [9], the majority of previous EIT-like effects in both metallic and graphene structures arise from the near-field coupling of the bright- and dark-mode resonators [7,8,12–15]. These mechanisms strongly depend on the coupling strength of resonators, significantly restricting the access to the coupling medium between resonators [9]. In addition, being similar to the multi-EIT-like effects in metallic plasmonic systems for complicated multi-channel applications, the exploration of graphene-based multiEIT-like effects is appealing for the on-chip, active, and multifunctional devices [10–13,27]. In this paper, multiple EIT-like optical responses are demonstrated in graphene metamaterials comprising a series of self-assembled graphene Fabry-Pérot (FP) cavities. In contrast to the classical near-field coupling structures, the phase-coupling scheme is #225474 - $15.00 USD © 2015 OSA
Received 22 Oct 2014; revised 25 Dec 2014; accepted 26 Dec 2014; published 9 Jan 2015 12 Jan 2015 | Vol. 23, No. 1 | DOI:10.1364/OE.23.000545 | OPTICS EXPRESS 546
employed to conceive the proposed graphene metamaterials for the realization of EIT-like spectral features. Furthermore, the observed multiple EIT-like windows can be efficiently tuned in the mid-infrared regime by adjusting the Fermi level in graphene and the separations of FP cavities. The transfer matrix model incorporating the graphene plasmon (GP) resonances and phase-coupling effects is established to explain the observed EIT-like responses, and the theoretical results agree well with the finite-difference time-domain (FDTD) simulations. 2. Structure model and analytical theory
Fig. 1. (a) Schematic configuration of the graphene metamaterials on a substrate consisting of a series of spatially separated GNR layers with different ribbon widths. The dielectric A isolates the adjacent GNR layers and dielectric B is the surrounding environment. A linearly polarized (along x-axis direction) mid-infrared wave incident along + z-axis direction is employed to excite the plasmon resonances in GNRs. (b) Side-view illustration of one unit cell with N-layer GNRs in the metamaterials. The adjacent two GNR layers construct a cavity and the inset zooms in the jth cavity. σj and Wj are the optical conductivity and ribbon width of the jth-layer GNR, respectively. hj is the separation of the jth cavity. The refractive indices of the media above and below the jth-layer GNR are denoted by nj and nj+1, respectively. P is the period. The ribbon width difference δWj = |Wj+1–Wj| is termed detuning in the jth cavity.
Figure 1 schematically shows the configuration of the graphene metamaterials, which consists of a series of graphene nanoribbon (GNR) layers spatially separated by dielectric spacers. The ribbon widths of each adjacent GNR layer are slightly detuned, resulting in the detuning of GP resonant wavelengths in GNRs [17,18]. To model the dynamic transmission of the proposed metamaterials, the transfer matrix method is employed [16,23]. As illustrated in Fig. 1(b), the jth-layer (j = 1, 2, …, N) GNR is surrounded by dielectrics with the refractive indices nj and nj+1. The transfer matrix of the N-layer system can be defined as H H = M N S N −1 M N −1 S N − 2 M 2 S1 M 1 = 11 H 21
H12 , H 22
(1)
where Mj and Sj matrices represent the zero-thickness graphene interface and hj-thickness dielectric interlayer, respectively. The matrices are governed by [16,23] eiϕ j Sj = 0
0 1 t j , j +1t j +1, j − rj , j +1rj +1, j , M j = − iϕ j −rj , j +1 t j +1, j e
rj +1, j . 1
(2)
Here φj = nj+1hjω/c denotes the phase difference between the adjacent GNR layers j and j + 1, hj is the corresponding spatial separation, ω is the angular frequency of incident wave, and c is the speed of light in vacuum. The Fresnel coefficients in the Mj matrix for the jth-layer GNR are expressed as tj,j+1 = 2nj/Δ, tj+1,j = 2nj+1/Δ, rj,j+1 = (nj–nj+1–Z0σj)/Δ, and rj+1,j = (nj+1–nj– Z0σj)/Δ, wherein Δ = (nj + nj+1 + Z0σj) [24,25]. Z0 = 376.73 Ω is the vacuum impedance. The average sheet optical conductivity of the jth-layer GNR is σj = ifjDjω/{π(ω2–ω2p,j + iωГp,j)}
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Received 22 Oct 2014; revised 25 Dec 2014; accepted 26 Dec 2014; published 9 Jan 2015 12 Jan 2015 | Vol. 23, No. 1 | DOI:10.1364/OE.23.000545 | OPTICS EXPRESS 547
[16–18]. fj = Wj/P is the filling factor of the periodically patterned GNRs. Dj = e2EF,j/ћ2 is the Drude weight, e is the electron charge, EF,j is the Fermi level in graphene, and ħ is the reduced Planck’s constant. The plasmon resonance width Гp,j of GNRs is usually 10% larger than the Drude scattering width Гj = evF2/(μEF,j) in the unpatterned graphene [18]. vF≈c/300 is the Fermi velocity and μ = 10000 cm2V−1s−1 is the measured DC mobility, which is rather conservative compared to experimental observations and a moderate value to improve the performance of graphene-based devices [14,16,26]. The plasmon resonance frequency in GNRs is given by ω p , j = D j (η j ε j ε 0W j ) [18]. εj = (nj2 + nj+12)/2 is the effective dielectric
constant of media surrounding the jth-layer GNR, ε0 is the vacuum permittivity, and Wj is the ribbon width of the jth-layer GNR. ηj is a fitting parameter deduced from the simulated results. Therefore, the output transmission efficiency of the graphene metamaterials is derived as N
H H − H12 H 21 T = 11 22 H 22
2
=
∏n j =1
2
nj j +1
H 22
(3)
.
The above equation is the typical transmission spectrum form of a FP resonator with two frequency-dependent mirrors [9–11]. Consequently, the underlying principle of the spectral responses in the proposed system can be attributed to the FP oscillation, which means that the designed metamaterials can be regarded as the multiple graphene-integrated FP-cavity system with tunable operating wavelengths and Fresnel coefficients. For the sake of simplicity and clarity, in the following discussions, the refractive indices of media surrounding GNRs are set the same as n = 1.48 and the period P = 200 nm. Thus, Eq. (3) is further simplified to T = 1/|H22|2. 3. Results and discussions Theoretical
0.5
Transmission
0.0 1.0 0.5 0.0 1.0
δW=1 nm
(a)
δW=3 nm
(b) λ12
0.5 0.0 1.0 0.5
Numerical
λ1
(c)
λ2
δW=5 nm
δW=7 nm
(d)
0.0 7.5
z (μm)
1.0
8.0
8.5
9.0
Wavelength (μm)
9.5
2 (e) 1 0 -1 -2 2 (f) 1 0 -1 -2 2 (g) 1 0 -1 -2 -100
1.0
λ1=8.18 μm
0.8 0.6 0.4
λ12=8.31 μm
0.2 0.0
-50
0
x (nm)
λ2=8.44 μm
50
100
Fig. 2. Transmission spectra of the two-layer GNRs system with various ribbon width detunings (a) δW = 1 nm, (b) δW = 3 nm, (c) δW = 5 nm, and (d) δW = 7 nm. The solid curves are the theoretical calculations and the circles are the simulation results. The ribbon widths (W1 and W2) of the two GNRs are set as 92 and 93 nm (a), 91 and 94 nm (b), 90 and 95 nm (c), 89 and 96 nm (d). h1 = 2.8 μm and EF = 0.60 eV. Numerical field distributions (|Ez|2) of one unit cell with the incident wavelengths at (e) λ1 = 8.18 μm, (f) λ12 = 8.31 μm, and (g) λ2 = 8.44 μm. The other parameters are the same as that in (c). The inset (white dash square) zooms in the field distribution around the 2nd-layer GNR (red dash square).
To demonstrate the phase-coupled plasmon-induced transparency scheme in our system, the transmission characteristics of two-layer GNRs system with different ribbon widths are investigated firstly. The FDTD method is utilized to numerically simulate the performances
#225474 - $15.00 USD © 2015 OSA
Received 22 Oct 2014; revised 25 Dec 2014; accepted 26 Dec 2014; published 9 Jan 2015 12 Jan 2015 | Vol. 23, No. 1 | DOI:10.1364/OE.23.000545 | OPTICS EXPRESS 548
1.0
λ12
0.0
λ23
10
0.5 Theoretical Numerical
(a)
0.0 7.5
8.0
z (μm)
8.5
9.0
9.5
Wavelength (μm)
0.0
0.2
0.4
0.6
0.8
1.0
(c)
2 (b) 0 λ12=8.31 μm
-2 -50
0
50
λ23=8.60 μm
-50
x (nm)
0
50
Cavity Separation (μm)
Transmission
of the system [16,21]. The perfectly matched absorbing boundary in z-axis direction and periodic boundary conditions in x- and y-axis directions are employed. The thickness of monolayer graphene sheet is set as 0.5 nm, and the maximum mesh size is Δx = Δy = 0.5 nm and Δz = 0.05 nm. The theoretical and numerical results are presented in Fig. 2. It is found that a narrow transparency peak emerges in the transmission spectrum when there is a ribbon width detuning δW = 1 nm, and the peak transmission can be enhanced by increasing the detuning from 1 to 7 nm, as shown in Figs. 2(a)-2(d). By optimizing the geometric parameters, the contrast ratio can exceed 90%, which is much higher than that in [14] and [15] (using near-field coupling mechanism). Meanwhile, the spectral responses reveal that a smaller detuning induces a narrower spectral bandwidth but a reduced peak transmission, indicating a tradeoff between peak transmission and quality factor [10,16]. These results exhibit the typical EIT-like optical responses [7–13]. The transmission dips in Figs. 2(a)-2(d) originate from the localized plasmon resonances in GNRs [17,18]. Because of exciting the GP resonances, the incident waves are partially absorbed and mostly reflected by GNRs at the resonant wavelengths. As illustrated in Figs. 2(e)-2(g), the field distributions show the evidence of plasmon resonances in GNRs. It can be seen that when the incident wavelength is 8.18 or 8.44 μm, the local GP resonance is excited in the individual GNR and the incident wave is reflected, as depicted in Figs. 2(e) and 2(g). At the central transparency wavelength, as shown in Fig. 2(f), both GNRs are partially resonant, resulting in the FP oscillation between the two GNRs. The incident waves can thus pass through the system, exhibiting a transparency effect [9–11]. In accordance with the FP model, the pronounced EIT-like transparency peak should nearly locate at λ12 = (λ1 + λ2)/2 when the FP resonant condition is satisfied, namely, the phase difference φ1 = mπ (m = 1, 2, …) [10]. In our investigation, as shown in Fig. 2(c), the wavelengths of transmission dips are λ1 = 8.18 μm and λ2 = 8.44 μm, and the transparency peak locates at the central wavelength λ12 of 8.31 μm. When the cavity separation h1 between the two GNRs is λ12/(2n), φ1 equals π at λ12 = (λ1 + λ2)/2, which is consistent with the parameter setting h1 = 2.80 μm. The theoretical results are in good agreement with the FDTD simulations. 0.2
0.4
0.6
0.8
(d)
1.0
ϕ ∼ 3π
8 6
ϕ ∼ 2π
4
ϕ∼π 2
Window 2
Window 1
7.5
8.0
8.5
9.0
Wavelength (μm)
9.5
Fig. 3. (a) Transmission spectrum of the three-layer GNRs system with W3 = 100 nm and h2 = 2.90 μm. (b) and (c) Field distributions (|Ez|2) of one unit cell with the incident wavelengths at λ12 = 8.31 μm and λ23 = 8.60 μm, respectively. (d) Evolution of transmission spectrum as a function of cavity separation. The other parameters are the same as that in Fig. 2(c).
By introducing the 3rd-layer GNR (W3 = 100 nm) into the aforementioned two-layer system, a three-layer GNRs system is constructed and the corresponding results are shown in Fig. 3. It is worth noting that two transparency peaks are observed in the transmission spectrum and the maximal transmission of the second peak can be achieved by setting the separation h2 at the optimum point 2.90 μm, as shown in Fig. 3(a), revealing the double EITlike optical responses [10,11]. Being similar to the two-layer system, there exist two FP cavities in the three-layer system: the first one between the 1st- and 2nd-layer GNRs, and the second one between the 2nd- and 3rd-layer GNRs. The field distributions at the two transparency peaks plotted in Figs. 3(b) and 3(c) visually demonstrate that the transparency
#225474 - $15.00 USD © 2015 OSA
Received 22 Oct 2014; revised 25 Dec 2014; accepted 26 Dec 2014; published 9 Jan 2015 12 Jan 2015 | Vol. 23, No. 1 | DOI:10.1364/OE.23.000545 | OPTICS EXPRESS 549
window 1 (λ12 = 8.31 μm) is due to the optical oscillation of the first FP cavity and the window 2 (λ23 = 8.60 μm) originates from that of the second cavity [27]. To further illustrate the phase-coupling effect in our system, the transmission evolution as a function of cavity separation is theoretically calculated and presented in Fig. 3(d). Obviously, the phasecoupling conditions (φ = π, 2π, 3π, …) of the two windows are periodically satisfied with the increase of cavity separation, which indicates the nature of phase-induced spectral characteristics [10,16]. The above principle can be generalized to the multi-layer GNRs system for the generation of multiple EIT-like optical responses. Here, the four- and five-layer GNRs systems are implemented to validate the tunability of the graphene-based multiple EIT-like systems. The four-layer system is achieved by adding the 4th-layer GNR (W4 = 105 nm) into the above three-layer system with the optimal separation h3 = 3.0 μm. Figures 4(a)-4(c) show the observed triple EIT-like responses in this four-layer system. In particular, the transparency windows can be shifted by altering the Fermi level in GNRs. It can be seen that when the Fermi level is changed the original transparency windows are no longer transparent while the original dips become transparent, indicating a transition between passband and stopband. Successively, the quadruple EIT-like responses are achieved in five-layer system, as shown in Figs. 4(d)-4(h). It is found that each transparency window can be independently closed and opened by adjusting the corresponding cavity separations. These features prove that by reasonably tuning the parameters, multi-channel-selective switches and filters over a broad range of mid-infrared can be realized in the desired manner. Theoretical
1.0
Numerical
0.0 (a)
EF=0.60 eV
1.0 0.5
0.0 (b) 1.0
EF=0.58 eV
Transmission
Transmission
0.5
0.5 0.0 (c) 7.5 8.0
EF=0.56 eV
8.5
9.0
9.5
Wavelength (μm)
10.0
Theoretical 1.0 0.5 (d) 0.0 1.0 0.5 0.0 (e) 1.0 0.5 0.0 (f) 1.0 0.5 0.0 (g) 1.0 0.5 0.0 (h) 7.5 8.0 8.5
Numerical
h1=2.8 μm h2=2.9 μm h3=3.0 μm h4=3.1 μm
h1=1.4 μm
h2=1.6 μm
h3=1.8 μm
h4=2.0 μm
9.0
9.5
Wavelength (μm)
10.0
Fig. 4. Transmission spectra of the four-layer GNRs system with different Fermi levels (a) Ef = 0.60 eV, (b) Ef = 0.58 eV, and (c) Ef = 0.56 eV. W4 = 105 nm and h3 = 3.0 μm. The other parameters are the same as that in Fig. 3(a). Transmission spectra of the five-layer GNRs system with different cavity separations (d) h1 = 2.8 μm, h2 = 2.9 μm, h3 = 3.0 μm, h4 = 3.1 μm, (e) h1 = 1.4 μm, (f) h2 = 1.6 μm, (g) h3 = 1.8 μm, and (h) h4 = 2.0 μm. The unspecified parameters in (e)-(h) are the same as that in (d). W5 = 110 nm. The other parameters are the same as that in (a).
4. Conclusions
In summary, multiple EIT-like effects in graphene metamaterials have been demonstrated theoretically and numerically. Comparing with classical structures, the proposed scheme is based on the phase coupling instead of the near-field coupling, and exhibits a noteworthy advantage of the unimpeded access to the coupling medium in the self-assembled graphene FP cavities. This type of design facilitates the multifunctional sensing applications [9]. Moreover, due to the intrinsic tunability and two-dimensional character of graphene, the proposed graphene-based multiple EIT-like metamaterials promise the active multi-channel filters, sensors, and modulators in the next generation of chip-integrated photonic circuits [10–13,27].
#225474 - $15.00 USD © 2015 OSA
Received 22 Oct 2014; revised 25 Dec 2014; accepted 26 Dec 2014; published 9 Jan 2015 12 Jan 2015 | Vol. 23, No. 1 | DOI:10.1364/OE.23.000545 | OPTICS EXPRESS 550
Acknowledgments
The authors would like to acknowledge Jing Guo for assistance and fruitful discussions. This work was supported by the National Natural Science Foundation of China under Grants Nos. 10874239, 10604066, and 61223007. Corresponding author (X. Liu). Tel.: + 862988881560; fax: + 862988887603; electronic mail: [email protected] and [email protected].
#225474 - $15.00 USD © 2015 OSA
Received 22 Oct 2014; revised 25 Dec 2014; accepted 26 Dec 2014; published 9 Jan 2015 12 Jan 2015 | Vol. 23, No. 1 | DOI:10.1364/OE.23.000545 | OPTICS EXPRESS 551
Abstract: We demonstrate the existence of multiple electromagnetically induced transparencies (EIT)-like spectral responses in graphene metamaterials consisting of a series of self-assembled graphene Fabry-Pérot (FP) cavities. By exploiting the graphene plasmon resonances and phasecoupling effects, the transfer matrix model is established to theoretically predict the EIT-like responses, and the calculated results coincide well with numerical simulations. It is found that high-contrast (~90%) multiple EITlike windows are observed over a broad range of mid-infrared. Additionally, these optical responses can be efficiently tuned by altering the Fermi level in graphene and the separations of FP cavities. The proposed scheme paves the way toward control of the multiple EIT-like responses, enabling exploration of the on-chip multifunctional electro-optic devices including multi-channel-selective filters, sensors, and modulators. © 2014 Optical Society of America OCIS codes: (240.6680) Surface plasmons; (160.3918) Metamaterials; (130.3120) Integrated optics devices; (050.2230) Fabry-Pérot; (050.6624) Subwavelength structures.
References and links 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14.
K. J. Boller, A. Imamolu, and S. E. Harris, “Observation of electromagnetically induced transparency,” Phys. Rev. Lett. 66(20), 2593–2596 (1991). Y. Wu, J. Saldana, and Y. Zhu, “Large enhancement of four-wave mixing by suppression of photon absorption from electromagnetically induced transparency,” Phys. Rev. A 67(1), 013811 (2003). T. F. Krauss, “Why do we need slow light,” Nat. Photonics 2(8), 448–450 (2008). C. Liu, Z. Dutton, C. H. Behroozi, and L. V. Hau, “Observation of coherent optical information storage in an atomic medium using halted light pulses,” Nature 409(6819), 490–493 (2001). M. F. Yanik, W. Suh, Z. Wang, and S. Fan, “Stopping light in a waveguide with an all-optical analog of electromagnetically induced transparency,” Phys. Rev. Lett. 93(23), 233903 (2004). Q. Xu, S. Sandhu, M. L. Povinelli, J. Shakya, S. Fan, and M. Lipson, “Experimental realization of an on-chip alloptical analogue to electromagnetically induced transparency,” Phys. Rev. Lett. 96(12), 123901 (2006). S. Zhang, D. A. Genov, Y. Wang, M. Liu, and X. Zhang, “Plasmon-induced transparency in metamaterials,” Phys. Rev. Lett. 101(4), 047401 (2008). N. Liu, L. Langguth, T. Weiss, J. Kästel, M. Fleischhauer, T. Pfau, and H. Giessen, “Plasmonic analogue of electromagnetically induced transparency at the Drude damping limit,” Nat. Mater. 8(9), 758–762 (2009). R. D. Kekatpure, E. S. Barnard, W. Cai, and M. L. Brongersma, “Phase-coupled plasmon-induced transparency,” Phys. Rev. Lett. 104(24), 243902 (2010). H. Lu, X. M. Liu, and D. Mao, “Plasmonic analog of electromagnetically induced transparency in multinanoresonator-coupled waveguide systems,” Phys. Rev. A 85(5), 053803 (2012). J. Chen, C. Wang, R. Zhang, and J. Xiao, “Multiple plasmon-induced transparencies in coupled-resonator systems,” Opt. Lett. 37(24), 5133–5135 (2012). A. Artar, A. A. Yanik, and H. Altug, “Multispectral plasmon induced transparency in coupled meta-atoms,” Nano Lett. 11(4), 1685–1689 (2011). H. Lu, X. Liu, G. Wang, and D. Mao, “Tunable high-channel-count bandpass plasmonic filters based on an analogue of electromagnetically induced transparency,” Nanotechnology 23(44), 444003 (2012). H. Cheng, S. Q. Chen, P. Yu, X. Y. Duan, B. Y. Xie, and J. G. Tian, “Dynamically tunable plasmonically induced transparency in periodically patterned graphene nanostrips,” Appl. Phys. Lett. 103(20), 203112 (2013).
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15. X. Shi, D. Z. Han, Y. Y. Dai, Z. F. Yu, Y. Sun, H. Chen, X. H. Liu, and J. Zi, “Plasmonic analog of electromagnetically induced transparency in nanostructure graphene,” Opt. Express 21(23), 28438–28443 (2013). 16. C. Zeng, J. Guo, and X. M. Liu, “High-contrast electro-optic modulation of spatial light induced by grapheneintegrated Fabry-Pérot microcavity,” Appl. Phys. Lett. 105(12), 121103 (2014). 17. L. Ju, B. Geng, J. Horng, C. Girit, M. Martin, Z. Hao, H. A. Bechtel, X. Liang, A. Zettl, Y. R. Shen, and F. Wang, “Graphene plasmonics for tunable terahertz metamaterials,” Nat. Nanotechnol. 6(10), 630–634 (2011). 18. H. Yan, X. Li, B. Chandra, G. Tulevski, Y. Wu, M. Freitag, W. Zhu, P. Avouris, and F. Xia, “Tunable infrared plasmonic devices using graphene/insulator stacks,” Nat. Nanotechnol. 7(5), 330–334 (2012). 19. M. Amin, M. Farhat, and H. Bağcı, “An ultra-broadband multilayered graphene absorber,” Opt. Express 21(24), 29938–29948 (2013). 20. M. A. K. Othman, C. Guclu, and F. Capolino, “Graphene-based tunable hyperbolic metamaterials and enhanced near-field absorption,” Opt. Express 21(6), 7614–7632 (2013). 21. W. Gao, J. Shu, C. Qiu, and Q. Xu, “Excitation of plasmonic waves in graphene by guided-mode resonances,” ACS Nano 6(9), 7806–7813 (2012). 22. Z. Fang, S. Thongrattanasiri, A. Schlather, Z. Liu, L. Ma, Y. Wang, P. M. Ajayan, P. Nordlander, N. J. Halas, and F. J. García de Abajo, “Gated tunability and hybridization of localized plasmons in nanostructured graphene,” ACS Nano 7(3), 2388–2395 (2013). 23. B. Sensale-Rodriguez, T. Fang, R. Yan, M. M. Kelly, D. Jena, L. Liu, and H. G. Xing, “Unique prospects for graphene-based terahertz modulators,” Appl. Phys. Lett. 99, 113104 (2011). 24. A. Andryieuski and A. V. Lavrinenko, “Graphene metamaterials based tunable terahertz absorber: effective surface conductivity approach,” Opt. Express 21(7), 9144–9155 (2013). 25. T. Stauber, N. M. R. Peres, and A. K. Geim, “Optical conductivity of graphene in the visible region of the spectrum,” Phys. Rev. B 78(8), 085432 (2008). 26. K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y. Zhang, S. V. Dubonos, I. V. Grigorieva, and A. A. Firsov, “Electric field effect in atomically thin carbon films,” Science 306(5696), 666–669 (2004). 27. H. Lu, X. Liu, Y. Gong, D. Mao, and L. Wang, “Enhancement of transmission efficiency of nanoplasmonic wavelength demultiplexer based on channel drop filters and reflection nanocavities,” Opt. Express 19(14), 12885–12890 (2011).
1. Introduction Electromagnetically induced transparency (EIT) occurs in laser-activated atomic systems as a result of quantum interference between the atomic level and the excitation pathways [1]. Due to the strong dispersion in transparency windows, the EIT effects have shown a plethora of intriguing potentials in nonlinear optical enhancement [2] and all-optical data processing [3]. However, complex experimental requirements have significantly constrained the chip-scale implements of conventional EIT in atomic systems [4]. Fortunately, the EIT-like effects were later demonstrated in classical systems as diverse as high-Q resonators [5,6] and plasmonic structures [7–9]. For instance, Xu et al. reported the first experimental observations of EITlike spectra in silicon resonator systems, which brought the original quantum phenomena into the realm of classical optics [6]. Zhang et al. proposed a coupled radiative-dark plasmonic mechanism to mimic the functionality of EIT by metamaterials [7]. Because of the capabilities to operate at subwavelength scales, especially, metal-based plasmonic EIT-like configurations have been extensively exploited [7–13]. Recently, plasmonic analogs of EIT have been demonstrated in periodically patterned graphene nanostructures [14–16]. Due to its unprecedented advantages including flexible tunability, extremely tight field confinement, and low losses at terahertz and mid-infrared frequencies, graphene has been proved to be a promising alternative material to metals for chip-integrated nanophotonics [17–26]. However, as Brongersma et al. analyzed in [9], the majority of previous EIT-like effects in both metallic and graphene structures arise from the near-field coupling of the bright- and dark-mode resonators [7,8,12–15]. These mechanisms strongly depend on the coupling strength of resonators, significantly restricting the access to the coupling medium between resonators [9]. In addition, being similar to the multi-EIT-like effects in metallic plasmonic systems for complicated multi-channel applications, the exploration of graphene-based multiEIT-like effects is appealing for the on-chip, active, and multifunctional devices [10–13,27]. In this paper, multiple EIT-like optical responses are demonstrated in graphene metamaterials comprising a series of self-assembled graphene Fabry-Pérot (FP) cavities. In contrast to the classical near-field coupling structures, the phase-coupling scheme is #225474 - $15.00 USD © 2015 OSA
Received 22 Oct 2014; revised 25 Dec 2014; accepted 26 Dec 2014; published 9 Jan 2015 12 Jan 2015 | Vol. 23, No. 1 | DOI:10.1364/OE.23.000545 | OPTICS EXPRESS 546
employed to conceive the proposed graphene metamaterials for the realization of EIT-like spectral features. Furthermore, the observed multiple EIT-like windows can be efficiently tuned in the mid-infrared regime by adjusting the Fermi level in graphene and the separations of FP cavities. The transfer matrix model incorporating the graphene plasmon (GP) resonances and phase-coupling effects is established to explain the observed EIT-like responses, and the theoretical results agree well with the finite-difference time-domain (FDTD) simulations. 2. Structure model and analytical theory
Fig. 1. (a) Schematic configuration of the graphene metamaterials on a substrate consisting of a series of spatially separated GNR layers with different ribbon widths. The dielectric A isolates the adjacent GNR layers and dielectric B is the surrounding environment. A linearly polarized (along x-axis direction) mid-infrared wave incident along + z-axis direction is employed to excite the plasmon resonances in GNRs. (b) Side-view illustration of one unit cell with N-layer GNRs in the metamaterials. The adjacent two GNR layers construct a cavity and the inset zooms in the jth cavity. σj and Wj are the optical conductivity and ribbon width of the jth-layer GNR, respectively. hj is the separation of the jth cavity. The refractive indices of the media above and below the jth-layer GNR are denoted by nj and nj+1, respectively. P is the period. The ribbon width difference δWj = |Wj+1–Wj| is termed detuning in the jth cavity.
Figure 1 schematically shows the configuration of the graphene metamaterials, which consists of a series of graphene nanoribbon (GNR) layers spatially separated by dielectric spacers. The ribbon widths of each adjacent GNR layer are slightly detuned, resulting in the detuning of GP resonant wavelengths in GNRs [17,18]. To model the dynamic transmission of the proposed metamaterials, the transfer matrix method is employed [16,23]. As illustrated in Fig. 1(b), the jth-layer (j = 1, 2, …, N) GNR is surrounded by dielectrics with the refractive indices nj and nj+1. The transfer matrix of the N-layer system can be defined as H H = M N S N −1 M N −1 S N − 2 M 2 S1 M 1 = 11 H 21
H12 , H 22
(1)
where Mj and Sj matrices represent the zero-thickness graphene interface and hj-thickness dielectric interlayer, respectively. The matrices are governed by [16,23] eiϕ j Sj = 0
0 1 t j , j +1t j +1, j − rj , j +1rj +1, j , M j = − iϕ j −rj , j +1 t j +1, j e
rj +1, j . 1
(2)
Here φj = nj+1hjω/c denotes the phase difference between the adjacent GNR layers j and j + 1, hj is the corresponding spatial separation, ω is the angular frequency of incident wave, and c is the speed of light in vacuum. The Fresnel coefficients in the Mj matrix for the jth-layer GNR are expressed as tj,j+1 = 2nj/Δ, tj+1,j = 2nj+1/Δ, rj,j+1 = (nj–nj+1–Z0σj)/Δ, and rj+1,j = (nj+1–nj– Z0σj)/Δ, wherein Δ = (nj + nj+1 + Z0σj) [24,25]. Z0 = 376.73 Ω is the vacuum impedance. The average sheet optical conductivity of the jth-layer GNR is σj = ifjDjω/{π(ω2–ω2p,j + iωГp,j)}
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Received 22 Oct 2014; revised 25 Dec 2014; accepted 26 Dec 2014; published 9 Jan 2015 12 Jan 2015 | Vol. 23, No. 1 | DOI:10.1364/OE.23.000545 | OPTICS EXPRESS 547
[16–18]. fj = Wj/P is the filling factor of the periodically patterned GNRs. Dj = e2EF,j/ћ2 is the Drude weight, e is the electron charge, EF,j is the Fermi level in graphene, and ħ is the reduced Planck’s constant. The plasmon resonance width Гp,j of GNRs is usually 10% larger than the Drude scattering width Гj = evF2/(μEF,j) in the unpatterned graphene [18]. vF≈c/300 is the Fermi velocity and μ = 10000 cm2V−1s−1 is the measured DC mobility, which is rather conservative compared to experimental observations and a moderate value to improve the performance of graphene-based devices [14,16,26]. The plasmon resonance frequency in GNRs is given by ω p , j = D j (η j ε j ε 0W j ) [18]. εj = (nj2 + nj+12)/2 is the effective dielectric
constant of media surrounding the jth-layer GNR, ε0 is the vacuum permittivity, and Wj is the ribbon width of the jth-layer GNR. ηj is a fitting parameter deduced from the simulated results. Therefore, the output transmission efficiency of the graphene metamaterials is derived as N
H H − H12 H 21 T = 11 22 H 22
2
=
∏n j =1
2
nj j +1
H 22
(3)
.
The above equation is the typical transmission spectrum form of a FP resonator with two frequency-dependent mirrors [9–11]. Consequently, the underlying principle of the spectral responses in the proposed system can be attributed to the FP oscillation, which means that the designed metamaterials can be regarded as the multiple graphene-integrated FP-cavity system with tunable operating wavelengths and Fresnel coefficients. For the sake of simplicity and clarity, in the following discussions, the refractive indices of media surrounding GNRs are set the same as n = 1.48 and the period P = 200 nm. Thus, Eq. (3) is further simplified to T = 1/|H22|2. 3. Results and discussions Theoretical
0.5
Transmission
0.0 1.0 0.5 0.0 1.0
δW=1 nm
(a)
δW=3 nm
(b) λ12
0.5 0.0 1.0 0.5
Numerical
λ1
(c)
λ2
δW=5 nm
δW=7 nm
(d)
0.0 7.5
z (μm)
1.0
8.0
8.5
9.0
Wavelength (μm)
9.5
2 (e) 1 0 -1 -2 2 (f) 1 0 -1 -2 2 (g) 1 0 -1 -2 -100
1.0
λ1=8.18 μm
0.8 0.6 0.4
λ12=8.31 μm
0.2 0.0
-50
0
x (nm)
λ2=8.44 μm
50
100
Fig. 2. Transmission spectra of the two-layer GNRs system with various ribbon width detunings (a) δW = 1 nm, (b) δW = 3 nm, (c) δW = 5 nm, and (d) δW = 7 nm. The solid curves are the theoretical calculations and the circles are the simulation results. The ribbon widths (W1 and W2) of the two GNRs are set as 92 and 93 nm (a), 91 and 94 nm (b), 90 and 95 nm (c), 89 and 96 nm (d). h1 = 2.8 μm and EF = 0.60 eV. Numerical field distributions (|Ez|2) of one unit cell with the incident wavelengths at (e) λ1 = 8.18 μm, (f) λ12 = 8.31 μm, and (g) λ2 = 8.44 μm. The other parameters are the same as that in (c). The inset (white dash square) zooms in the field distribution around the 2nd-layer GNR (red dash square).
To demonstrate the phase-coupled plasmon-induced transparency scheme in our system, the transmission characteristics of two-layer GNRs system with different ribbon widths are investigated firstly. The FDTD method is utilized to numerically simulate the performances
#225474 - $15.00 USD © 2015 OSA
Received 22 Oct 2014; revised 25 Dec 2014; accepted 26 Dec 2014; published 9 Jan 2015 12 Jan 2015 | Vol. 23, No. 1 | DOI:10.1364/OE.23.000545 | OPTICS EXPRESS 548
1.0
λ12
0.0
λ23
10
0.5 Theoretical Numerical
(a)
0.0 7.5
8.0
z (μm)
8.5
9.0
9.5
Wavelength (μm)
0.0
0.2
0.4
0.6
0.8
1.0
(c)
2 (b) 0 λ12=8.31 μm
-2 -50
0
50
λ23=8.60 μm
-50
x (nm)
0
50
Cavity Separation (μm)
Transmission
of the system [16,21]. The perfectly matched absorbing boundary in z-axis direction and periodic boundary conditions in x- and y-axis directions are employed. The thickness of monolayer graphene sheet is set as 0.5 nm, and the maximum mesh size is Δx = Δy = 0.5 nm and Δz = 0.05 nm. The theoretical and numerical results are presented in Fig. 2. It is found that a narrow transparency peak emerges in the transmission spectrum when there is a ribbon width detuning δW = 1 nm, and the peak transmission can be enhanced by increasing the detuning from 1 to 7 nm, as shown in Figs. 2(a)-2(d). By optimizing the geometric parameters, the contrast ratio can exceed 90%, which is much higher than that in [14] and [15] (using near-field coupling mechanism). Meanwhile, the spectral responses reveal that a smaller detuning induces a narrower spectral bandwidth but a reduced peak transmission, indicating a tradeoff between peak transmission and quality factor [10,16]. These results exhibit the typical EIT-like optical responses [7–13]. The transmission dips in Figs. 2(a)-2(d) originate from the localized plasmon resonances in GNRs [17,18]. Because of exciting the GP resonances, the incident waves are partially absorbed and mostly reflected by GNRs at the resonant wavelengths. As illustrated in Figs. 2(e)-2(g), the field distributions show the evidence of plasmon resonances in GNRs. It can be seen that when the incident wavelength is 8.18 or 8.44 μm, the local GP resonance is excited in the individual GNR and the incident wave is reflected, as depicted in Figs. 2(e) and 2(g). At the central transparency wavelength, as shown in Fig. 2(f), both GNRs are partially resonant, resulting in the FP oscillation between the two GNRs. The incident waves can thus pass through the system, exhibiting a transparency effect [9–11]. In accordance with the FP model, the pronounced EIT-like transparency peak should nearly locate at λ12 = (λ1 + λ2)/2 when the FP resonant condition is satisfied, namely, the phase difference φ1 = mπ (m = 1, 2, …) [10]. In our investigation, as shown in Fig. 2(c), the wavelengths of transmission dips are λ1 = 8.18 μm and λ2 = 8.44 μm, and the transparency peak locates at the central wavelength λ12 of 8.31 μm. When the cavity separation h1 between the two GNRs is λ12/(2n), φ1 equals π at λ12 = (λ1 + λ2)/2, which is consistent with the parameter setting h1 = 2.80 μm. The theoretical results are in good agreement with the FDTD simulations. 0.2
0.4
0.6
0.8
(d)
1.0
ϕ ∼ 3π
8 6
ϕ ∼ 2π
4
ϕ∼π 2
Window 2
Window 1
7.5
8.0
8.5
9.0
Wavelength (μm)
9.5
Fig. 3. (a) Transmission spectrum of the three-layer GNRs system with W3 = 100 nm and h2 = 2.90 μm. (b) and (c) Field distributions (|Ez|2) of one unit cell with the incident wavelengths at λ12 = 8.31 μm and λ23 = 8.60 μm, respectively. (d) Evolution of transmission spectrum as a function of cavity separation. The other parameters are the same as that in Fig. 2(c).
By introducing the 3rd-layer GNR (W3 = 100 nm) into the aforementioned two-layer system, a three-layer GNRs system is constructed and the corresponding results are shown in Fig. 3. It is worth noting that two transparency peaks are observed in the transmission spectrum and the maximal transmission of the second peak can be achieved by setting the separation h2 at the optimum point 2.90 μm, as shown in Fig. 3(a), revealing the double EITlike optical responses [10,11]. Being similar to the two-layer system, there exist two FP cavities in the three-layer system: the first one between the 1st- and 2nd-layer GNRs, and the second one between the 2nd- and 3rd-layer GNRs. The field distributions at the two transparency peaks plotted in Figs. 3(b) and 3(c) visually demonstrate that the transparency
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Received 22 Oct 2014; revised 25 Dec 2014; accepted 26 Dec 2014; published 9 Jan 2015 12 Jan 2015 | Vol. 23, No. 1 | DOI:10.1364/OE.23.000545 | OPTICS EXPRESS 549
window 1 (λ12 = 8.31 μm) is due to the optical oscillation of the first FP cavity and the window 2 (λ23 = 8.60 μm) originates from that of the second cavity [27]. To further illustrate the phase-coupling effect in our system, the transmission evolution as a function of cavity separation is theoretically calculated and presented in Fig. 3(d). Obviously, the phasecoupling conditions (φ = π, 2π, 3π, …) of the two windows are periodically satisfied with the increase of cavity separation, which indicates the nature of phase-induced spectral characteristics [10,16]. The above principle can be generalized to the multi-layer GNRs system for the generation of multiple EIT-like optical responses. Here, the four- and five-layer GNRs systems are implemented to validate the tunability of the graphene-based multiple EIT-like systems. The four-layer system is achieved by adding the 4th-layer GNR (W4 = 105 nm) into the above three-layer system with the optimal separation h3 = 3.0 μm. Figures 4(a)-4(c) show the observed triple EIT-like responses in this four-layer system. In particular, the transparency windows can be shifted by altering the Fermi level in GNRs. It can be seen that when the Fermi level is changed the original transparency windows are no longer transparent while the original dips become transparent, indicating a transition between passband and stopband. Successively, the quadruple EIT-like responses are achieved in five-layer system, as shown in Figs. 4(d)-4(h). It is found that each transparency window can be independently closed and opened by adjusting the corresponding cavity separations. These features prove that by reasonably tuning the parameters, multi-channel-selective switches and filters over a broad range of mid-infrared can be realized in the desired manner. Theoretical
1.0
Numerical
0.0 (a)
EF=0.60 eV
1.0 0.5
0.0 (b) 1.0
EF=0.58 eV
Transmission
Transmission
0.5
0.5 0.0 (c) 7.5 8.0
EF=0.56 eV
8.5
9.0
9.5
Wavelength (μm)
10.0
Theoretical 1.0 0.5 (d) 0.0 1.0 0.5 0.0 (e) 1.0 0.5 0.0 (f) 1.0 0.5 0.0 (g) 1.0 0.5 0.0 (h) 7.5 8.0 8.5
Numerical
h1=2.8 μm h2=2.9 μm h3=3.0 μm h4=3.1 μm
h1=1.4 μm
h2=1.6 μm
h3=1.8 μm
h4=2.0 μm
9.0
9.5
Wavelength (μm)
10.0
Fig. 4. Transmission spectra of the four-layer GNRs system with different Fermi levels (a) Ef = 0.60 eV, (b) Ef = 0.58 eV, and (c) Ef = 0.56 eV. W4 = 105 nm and h3 = 3.0 μm. The other parameters are the same as that in Fig. 3(a). Transmission spectra of the five-layer GNRs system with different cavity separations (d) h1 = 2.8 μm, h2 = 2.9 μm, h3 = 3.0 μm, h4 = 3.1 μm, (e) h1 = 1.4 μm, (f) h2 = 1.6 μm, (g) h3 = 1.8 μm, and (h) h4 = 2.0 μm. The unspecified parameters in (e)-(h) are the same as that in (d). W5 = 110 nm. The other parameters are the same as that in (a).
4. Conclusions
In summary, multiple EIT-like effects in graphene metamaterials have been demonstrated theoretically and numerically. Comparing with classical structures, the proposed scheme is based on the phase coupling instead of the near-field coupling, and exhibits a noteworthy advantage of the unimpeded access to the coupling medium in the self-assembled graphene FP cavities. This type of design facilitates the multifunctional sensing applications [9]. Moreover, due to the intrinsic tunability and two-dimensional character of graphene, the proposed graphene-based multiple EIT-like metamaterials promise the active multi-channel filters, sensors, and modulators in the next generation of chip-integrated photonic circuits [10–13,27].
#225474 - $15.00 USD © 2015 OSA
Received 22 Oct 2014; revised 25 Dec 2014; accepted 26 Dec 2014; published 9 Jan 2015 12 Jan 2015 | Vol. 23, No. 1 | DOI:10.1364/OE.23.000545 | OPTICS EXPRESS 550
Acknowledgments
The authors would like to acknowledge Jing Guo for assistance and fruitful discussions. This work was supported by the National Natural Science Foundation of China under Grants Nos. 10874239, 10604066, and 61223007. Corresponding author (X. Liu). Tel.: + 862988881560; fax: + 862988887603; electronic mail: [email protected] and [email protected].
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