Journal of Modern Optics
ISSN: 0950-0340 (Print) 1362-3044 (Online) Journal homepage: http://www.tandfonline.com/loi/tmop20
Tunable multiple mode-splitting in coupled graphene resonators system Jicheng Wang, Xiushan Xia, Xiaosai Wang & Shutian Liu To cite this article: Jicheng Wang, Xiushan Xia, Xiaosai Wang & Shutian Liu (2016) Tunable multiple mode-splitting in coupled graphene resonators system, Journal of Modern Optics, 63:9, 868-873, DOI: 10.1080/09500340.2015.1107649 To link to this article: http://dx.doi.org/10.1080/09500340.2015.1107649
Published online: 04 Nov 2015.
Submit your article to this journal
Article views: 27
View related articles
View Crossmark data
Full Terms & Conditions of access and use can be found at http://www.tandfonline.com/action/journalInformation?journalCode=tmop20 Download by: [Purdue University Libraries]
Date: 18 February 2016, At: 12:04
Journal of Modern Optics, 2016 Vol. 63, No. 9, 868–873, http://dx.doi.org/10.1080/09500340.2015.1107649
Tunable multiple mode-splitting in coupled graphene resonators system Jicheng Wanga,b,†*
, Xiushan Xiab,†, Xiaosai Wangb and Shutian Liua*
a
Department of Physics, Harbin Institute of Technology, Harbin, China; bSchool of Science, Jiangnan University, Wuxi, China
Downloaded by [Purdue University Libraries] at 12:04 18 February 2016
(Received 13 July 2015; accepted 8 October 2015) We investigate a coupled graphene resonator system which exhibits multiple mode-splitting effects and electromagnetically-induced-absorption-like transmission. The finite element method has been employed to study the transmission and electromagnetic responses of our designs at mid-infrared frequency. According to simulation results, the mode-splitting effects are mainly dependent on the destructive interference between two graphene resonators. By varying the chemical potential of graphene or the coupling gap, we are accessible to achieve a dynamically controllable mode-splitting system serving as a sensing application. Keywords: plasmonics; coupled resonator; graphene; mode-splitting; FEM
1. Introduction Surface plasmon polaritons (SPPs), electromagnetic waves coupled to free electron oscillations in a metal [1,2], offers an intriguing approach to localize and guide light in subwavelength metallic structures. A great variety of applications based on SPPs have been vigorously pursued in recent years [3–7], providing techniques to miniaturize the sizes of photonic devices into subwavelength scale. However, it’s hard to tune permittivity functions of noble metals, which limits their ability in some plasmonic devices. Graphene, with only one atomic thin, has already been identified to possess highly tunable optical properties, emerging as the most promising candidate for newgeneration plasmonic material [8–11]. Particularly, the surface conductivity of graphene can be modified by voltage gating or chemical doping [12], which is the most important advantage of graphene over noble metals. F. Javier et al. have studied the propagation properties of graphene plasmonic waveguide based on individual and paired nanoribbons [13]. Liu et al. [14] firstly reported the graphene-based broadband optical modulator. A wide range of researches further promote the development of active plasmonic devices including metamaterial [15–18], optical modulators [19–22], sensors [23,24], graphene waveguides [25–28] and transformation optical devices [29]. In our paper, a model has been developed based on a plasmonic waveguide coupled with two resonator rings formed by graphene. The finite element method [30] has been utilized to study the transmission and electromagnetic responses of our designs at mid-infrared frequency.
According to simulation results, the mode-splitting by the destructive interference between resonant modes leads to electromagnetically-induced-absorptionlike (EIA-like) transmission [31]. The destructive interference is strongly dependent on the frequency detuning and coupling gap between them. Therefore, by varying the chemical potential of graphene or the coupling gap, the multiple mode-splitting can be effectively controlled. Based on transmittance characteristics, a tunable sensing application can be realized in our configuration. 2. Model and numerical method Figure 1 depicts the designed coupled graphene resonator system. The input and output graphene bus waveguides and two graphene ring resonators are in shoulder-coupled arrangement. The optical conductivity of graphene is determined by the Kubo formalism considering the interband and intraband transition contributions [32]. The surface conductivity of graphene σg, is governed by the Kubo formula ie2 lc ie2 2lc ðx þ is1 Þh rg ¼ 2 ln þ h 2lc þ ðx þ is1 Þh ph ðx þ is1 Þ 4p ie2 kB T lc ln exp þ 2 þ 1 : (1) kB T ph ðx þ is1 Þ It depends on temperature T, chemical potential μc, momentum relaxation time τ, and photon frequency. Here, momentum relaxation time follows equation: τ = μμc/(evf2) relating to the carrier mobility μ and Fermi velocity vf = 106 m/s in graphene. By employing biased
*Corresponding authors. Email:
[email protected] (S. Liu);
[email protected] (J. Wang) †These authors contributed equally to this work. © 2015 Taylor & Francis
Journal of Modern Optics
Downloaded by [Purdue University Libraries] at 12:04 18 February 2016
Figure 1. Schematic of coupled graphene resonator system consists of the input and output graphene bus waveguides and two graphene ring resonators. R1 and R2 are the radius of the two graphene rings; d is the coupling distance between the graphene waveguide and graphene rings; t is the gap between two graphene rings; VG1, VG2 and VG3 represent the gate voltages on graphene waveguides and the two graphene rings, respectively. (The colour version of this figure is included in the online version of the journal.)
voltage, the chemical potential of graphene can be tuned in a controllable fashion. In our investigation, the employed incident light is in mid-infrared range where the intraband transition contribution dominates in graphene sheets [33]. Under this condition, the optical conductivity is simplified as rg ðxÞ ¼
ie2 lc =ph2 x þ is1
(2)
The equivalent permittivity of graphene follows the equation [29]: εg,eq = 1 + iσgη0/(k0Δ), where η0 ≈ 377 Ω is the intrinsic impedance of air and k0 = 2πf/c is wavenumber in vacuum. In our simulation, the thickness of graphene is modeled as Δ = 0.5 nm, and the carrier mobility is reasonably chosen to be μ = 20,000 cm2 V−1 s−1 from experiment results [8,34]. For simplicity on simulation, the graphene-based system is assumed to be suspended in air. The TM-polarized graphene surface plasmon polaritons (GSPP) supported by single-layer graphene is only in consideration for the investigation. The dispersion relation of this TM GSPP surface wave follows the equation: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi 2 2 bGSPP ¼ k0 1 (3) g0 rg where βGSPP is the propagation constant of GSPP. Another important parameter derived from the above equation is the effective refractive index of GSPP NGSPP = βGSPP/k0, which shows the ability to confine GSPP on graphene. The propagation length is defined as LGSPP = 1/Im(βGSPP) featuring the GSPP propagation loss. Furthermore, it should be noted that the dispersion relation of GSPP on graphene waveguide works on graphene rings as well [35]. The dependence of Re(NGSPP) and LGSPP on the chemical potential μc and incident light frequency are shown in Figure 2. Obviously, from Figure 2(a), the Re(NGSPP) increases as the chemical
869
potential μc decreases for a fixed frequency implying GSPP are better confined at lower chemical potential. Whereas, the trend in Figure 2(b) is apparently opposite to that in Figure 2(a) indicating that lower chemical potential gives shorter propagation distance. Hence, these two important factors should be both taken into consideration for the design of coupled graphene resonator system. Interestingly, the Re(NGSPP) varies greatly when the chemical potential is slightly changed, which gives a way of active control on our configuration. 3. Simulations and results In this section, we mainly discuss the single coupled graphene resonator system depicted in Figure 3(a). First, the influence of coupling distance d on the resonant modes has been investigated via numerical simulation, as shown in Figure 3. The chemical potential μc of graphene waveguides and graphene ring are set as 0.4 and 0.5 eV, respectively. The coupling distance d is varied from 10 to 50 nm in step of 10 nm. In Figure 3(b), the transmission spectra show multiple pronounced transmittance peaks corresponding to each resonant mode. The resonance frequency of GSPP on graphene ring should follow the phase-matching equation [36]: ReðbGSPP Þ 2pR ¼ 2mp;
(4)
where m is a positive integer, resonance mode number. With the increasing of coupling distance d, the transmission peaks get smaller and a little red-shift can be observed. It is because the increment of d weakens the coupling between graphene waveguides and ring, and adds more phase delay. Figure 3(c) depicts the magnetic field Hz(left) and normalized electric field norm. E(Right) distributions for the first-order to third-order resonant modes and a non-resonance mode with d = 20 nm. Apparently, the GSPP will be only enhanced in the ring at resonance frequencies and then efficiently coupled to the output graphene waveguide. For higher frequencies, less energy can be transported to the output due to the weak coupling between the graphene waveguides and ring since less GSPP field extends off them. Next, we study the dependency of resonant modes on chemical potential μc2 of graphene ring. As shown in Figure 4(a), the transmission spectra of each mode show obvious blue-shift with an increasing μc2, which is in well agreement with the phase-matching equation. The Figure 4(b) further illustrates that the numerical results of resonant modes agree well with the theoretical predictions determined by Equation (3). It should be noted that the phase delay from the coupling distance leads to the slight discrepancy which can be compromised by enlarging coupling distance or considering high-order resonant modes. Therefore, we have an access to dynamically
870
J. Wang et al.
Downloaded by [Purdue University Libraries] at 12:04 18 February 2016
Figure 2. (a) The real part of effective refractive index for GSPP as a function of incident frequency and chemical potential. (b) The dependence of propagation length LGSPP on incident frequency and chemical potential. (The colour version of this figure is included in the online version of the journal.)
Figure 3. (a) Schematic of single coupled graphene resonator system. (b) Transmission spectra of single coupled graphene resonator system with different coupling distance d. (c) The magnetic field Hz and normalized electric field Norm.E distributions for the firstorder to third-order resonant modes and a non-resonance mode. (The colour version of this figure is included in the online version of the journal.)
Figure 4. (a) Transmission spectra of single coupled graphene resonator system with different chemical potential on graphene resonator for each mode. (b) Dependence of the resonant frequency of theoretical results (solid red line) and numerical results (dashed blue line) on the chemical potential of the graphene resonator for each mode. (The colour version of this figure is included in the online version of the journal.)
Downloaded by [Purdue University Libraries] at 12:04 18 February 2016
Journal of Modern Optics control the transmittance characteristics by biased voltage applied on graphene. Armed with the above analysis, a two directly coupled graphene resonator system has been proposed, as depicted in Figure 1. Here, the focus is on the interference between two graphene ring resonators. At first, the effect of gap t between two coupled graphene resonators on the interference has been researched. The chemical potential μc of graphene waveguides and graphene rings are fixed at 0.4 and 0.5 eV, respectively. The coupling distance d is reasonably chosen as 20 nm to make sure strong coupling between the waveguides and rings. The researched gap t is set as from 30 to 100 nm. From Figure 5, multiple mode-splitting leading to EIA-like transmission can be obviously observed. The original peak has turned to be a dip and split into two new resonant modes. As the gap t grows, an on-to-off effect for each mode can be achieved in the transmission spectra. To get more insight into the interference, the magnetic field distributions at the newborn dip for second-order mode with different gap t are shown in Figure 5(b). It is found that the growing gap t brings about more coupling loss and weakens the destructive interference, which well explains the on-to-off effect. Furthermore, we have plotted the electric field response at the newborn dip and two splitting resonant modes for first-order, second-order and third-order modes, shown in
871
Figure 6. It can be seen that the destructive interference between two graphene resonators causes strong reflection to the input giving rise to the mode-splitting and a sharp transmission dip in each mode. For high-order modes, better confinement of GSPP can be observed agreeing with the definition of NGSPP. Successively, by introducing the resonant frequency detuning between two resonators, we study the dynamical tunability of the mode-splitting transmission characteristics. Based on the phase-matching equation, we know the resonance frequency can be adjusted via biased voltage. The gap t is set as 30 nm. The chemical potential μc2 of left graphene ring is fixed at 0.50 eV while the chemical potential μc3 of right graphene ring is changed from 0.46 to 5.4 eV. The other parameters keep the same with the above discussion. Seeing from Figure 7, the red-shift occurs when μc3 is less than μc2 while the blue-shift occurs when μc3 is more than μc2. Moreover, the increasing absolute value of μc3–μc2 results in a broader transmittance dip in mode-splitting region. It’s found that the dip-broadened effect is more apparent in high-order modes. From the aforesaid analysis, we find dip-broadened effect for high-order modes is more sensitive to the frequency detuning and greater frequency detuning leads to more obvious mode-splitting shift, which provides useful guidelines for the design of a dynamically controllable mode-splitting system.
Figure 5. (a) Transmission spectra of single and dual coupled graphene resonator system with different gap t. (b) The magnetic field distributions for second-order resonant mode with different gap t. (The colour version of this figure is included in the online version of the journal.)
Downloaded by [Purdue University Libraries] at 12:04 18 February 2016
872
J. Wang et al.
Figure 6. Normalized electric field distributions corresponding to transmission dip and peaks in the mode -splitting region for firstorder, second-order and third-order modes. (The colour version of this figure is included in the online version of the journal.)
detuning. All these analysis will provide useful guidelines for the design of a dynamically controllable sensing device based on mode-splitting effect. Disclosure statement No potential conflict of interest was reported by the authors.
Funding This work was supported by the National Basic Research Program of China [grant number 2013CBA01702]; the National Natural Science Foundation of China [grant number 61377016], [grant number 61308017], [grant number 11504139]; the Jiangsu Natural Science Foundation [grant number BK20140167]. Figure 7. Transmisssion spectra of the dual graphene resonator system with various chemical potential on the right-side graphene ring. (a) μc3 = 0.50, 0.48, 0.46 eV. (b) μc3 = 0.50, 0.52, 0.54 eV. (The colour version of this figure is included in the online version of the journal.)
ORCID Jicheng Wang
http://orcid.org/0000-0001-7246-3256
References 4. Conclusions We numerically study the transmission and electromagnetic responses of a coupled graphene resonator structure. The simulation results indicate that the coupling distance and chemical potential of graphene ring both have an effect on the resonant modes providing an access to effectively control the transmittance characteristics. In addition, we investigate two directly coupled graphene resonators system in the same way. The influence of coupling gap and resonant frequency detuning between graphene resonators have been discussed in detail. It’s found that greater frequency detuning leads to more obvious mode-splitting shift. For high-order modes, dip-broadened effects are more sensitive to the frequency
[1] Barnes, W.L.; Dereux, A.; Ebbesen, T.W. Nature 2003, 424, 824–830. [2] Zia, R.; Schuller, J.A.; Chandran, A. Mater. Today 2006, 9, 20–27. [3] Srituravanich, W.; Fang, N.; Sun, C.; Luo, Q.; Zhang, X. Nano Lett. 2004, 4, 1085–1088. [4] Skorobogatiy, M.; Kabashin, A.V. Appl. Phys. Lett. 2006, 89, 143518-1–143518-3. [5] Ebbesen, T.W.; Genet, C.; Bozhevolnyi, S.I. Phys. Today 2008, 61, 44–50. [6] Briggs, R.M.; Grandidier, J.; Burgos, S.P.; Feigenbaum, E.; Atwater, H.A. Nano Lett. 2010, 10, 4851–4857. [7] Fang, Z.Y.; Peng, Q.A.; Song, W.T.; Hao, F.H.; Wang, J.; Nordlander, P.; Zhu, X. Nano Lett. 2011, 11, 893–897. [8] Novoselov, K.S.; Geim, A.K.; Morozov, S.V.; Jiang, D.; Zhang, Y.; Dubonos, S.V.; Grigorieva, I.V.; Firsov, A.A. Science 2004, 306, 666–669.
Downloaded by [Purdue University Libraries] at 12:04 18 February 2016
Journal of Modern Optics [9] Wang, F.; Zhang, Y.; Tian, C.; Girit, C.; Zettl, A.; Crommie, M.F.; Shen, Y.R. Science 2008, 320, 206–209. [10] Li, Z.Q.; Henriksen, E.; Jiang, Z.; Hao, Z.; Martin, M.C.; Kim, P.; Stormer, H.L.; Basov, D.N. Nat. Phys. 2008, 4, 532–535. [11] Naik, G.V.; Shalaev, V.M.; Boltasseva, A. Adv. Mater. 2013, 25, 3264–3294. [12] Geim, A.K. Science 2009, 324, 1530–1534. [13] Christensen, J.; Manjavacas, A.; Thongrattanasiri, S.; Koppens, F.H.L. ACS Nano 2012, 6, 431–440. [14] Liu, M.; Yin, X.B.; Ulin-Avila, Erick; Geng, B.S.; Zentgraf, T.; Ju, L.; Wang, F.; Zhang, X. Nature 2011, 474, 64–67. [15] Papasimakis, N.; Luo, Z.Q.; Shen, Z.X.; De Angelis, F.; Di Fabrizio, E.; Nikolaenko, A.E.; Zheludev, N.I. Opt. Express 2010, 18, 8353–8359. [16] Yan, H.G.; Li, X.S.; Chandra, B.; Tulevski, G.; Wu, Y.Q.; Freitag, M.; Zhu, W.J.; Avouris, P.; Xia, F.N. Nat. Nanotechnol. 2012, 7, 330–334. [17] Amin, M.; Farhat, M.; Bagci, H. Sci. Rep. 2013, 3, 21051–2105-8. [18] Ding, J.; Arigong, B.; Ren, H.; Zhou, M.; Shao, J.; Lu, M.; Chai, Y.; Lin, Y.K.; Zhang, H.L. Sci. Rep. 2014, 4, 6128-1–6128-7. [19] Lu, Z.L.; Zhao, W.S. J. Opt. Soc. Am. B 2012, 29, 1490– 1496. [20] Sensale, R.B.; Yan, R.S.; Zhu, M.D.; Jena, D.; Liu, L.; Xing, H.G. Appl. Phys. Lett. 2012, 101, 261115-1– 261115-3.
873
[21] Lao, J.; Tao, J.; Wang, Q.J.; Huang, X.G. Laser Photonics Rev. 2014, 8, 569–574. [22] Das, S.; Salandrino, A.; Wu, J.Z.; Hui, R.Q. Opt. Lett. 2015, 40, 1516–1519. [23] Maharana, P.K.; Srivastava, T.; Jha, R. IEEE Photonics Technol. Lett. 2013, 25, 122–125. [24] Hosseini, T.; Kouklin, N.A. Appl. Phys. Lett. 2014, 105, 043104. [25] Kim, J.T.; Choi, S.Y. Opt. Express 2011, 24, 24557– 24562. [26] Ooi, K.J.A.; Chu, H.S.; Ang, L.K.; Bai, P. J. Opt. Soc. Am. B 2013, 30, 3111–3116. [27] Huang, Z.R.; Wang, L.L.; He, M.D.; Li, H.J.; Sun, B.; Liu, J.Q.; Zhai, X. J. Mod. Opt. 2014, 62, 514–518. [28] Xia, X.; Wang, J.; Zhang, F.; Liu, C.; Hu, Z.D.; Yan, X.; Yuan, L. Plasmonics 2015, doi:10.1007/s11468-015-9955-9. [29] Vakil, A.; Engheta, N. Science 2011, 332, 1291–1294. [30] Jin, J.M. The Finite Element Method in Electromagnetics; Wiley-IEEE Press: New York, 2002. [31] Li, Q.; Wang, T.; Su, Y.K.; Yan, M.; Qiu, M. Opt. Express 2010, 18, 8367–8382. [32] Chen, P.Y.; Alù, A. ACS Nano 2011, 5, 5855–5863. [33] Hanson, G.W. J. Appl. Phys. 2008, 104, 084314. [34] Thongrattanasiri, S.; Koppens, F.H.L.; de Abajo, F.J.G. Phys. Rev. Lett. 2012, 108, 047401. [35] Lu, W.B.; Zhu, W.; Xu, H.J.; Ni, Z.H.; Dong, Z.G.; Cui, T.J. Opt. Express 2013, 21, 10475–10482. [36] Huang, Z.R.; Wang, L.L.; Sun, B.; He, M.D.; Liu, J.Q.; Li, H.J.; Zhai, X. J. Opt. 2014, 16, 105004.
