Dielectric loaded graphene plasmon waveguide W. Xu,1 Z. H. Zhu,1,2,* K. Liu,1 J. F. Zhang,1 X. D. Yuan,1 Q. S. Lu 1 and S. Q. Qin1,2 1
College of Optoelectronic Science and Engineering, National University of Defense Technology, Changsha 410073, China 2 State Key Laboratory of High Performance Computing, National University of Defense Technology, Changsha 410073, China * [email protected]
Abstract: Dielectric loaded graphene plasmon waveguide (DLGPW) is proposed and investigated. An analytical model based on effective-index method is presented and verified by the finite element method simulations. The mode effective index, propagation loss, cutoff wavelength of higher order modes and single-mode operation region were derived at mid-infrared spectral region. By changing Fermi energy level, the propagation properties of fundamental mode could be tuned flexibly. The structure of the DLGPW is simple and easy for fabrication. It provided a new freedom to manipulate the graphene surface plasmons, which may led to new applications in actively tunable integrated optical devices. ©2015 Optical Society of America OCIS codes: (240.6680) Surface plasmons; (250.5403) Plasmonics; (230.7370) Waveguides.
References and links 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16.
D. K. Gramotnev and S. I. Bozhevolnyi, “Plasmonics beyond the diffraction limit,” Nat. Photonics 4(2), 83–91 (2010). S. I. Bozhevolnyi, V. S. Volkov, E. Devaux, and T. W. Ebbesen, “Channel plasmon-polariton guiding by subwavelength metal grooves,” Phys. Rev. Lett. 95(4), 046802 (2005). E. Moreno, F. J. Garcia-Vidal, S. G. Rodrigo, L. Martin-Moreno, and S. I. Bozhevolnyi, “Channel plasmonpolaritons: modal shape, dispersion, and losses,” Opt. Lett. 31(23), 3447–3449 (2006). C. L. Zou, F. W. Sun, Y. F. Xiao, C. H. Dong, X. D. Chen, J. M. Cui, Q. Gong, Z. F. Han, and G. C. Guo, “Plasmon modes of silver nanowire on a silica substrate,” Appl. Phys. Lett. 97(18), 183102 (2010). J. A. Dionne, L. A. Sweatlock, H. A. Atwater, and A. Polman, “Plasmon slot waveguides: Towards chip-scale propagation with subwavelength-sacle localization,” Phys. Rev. B 73(3), 035407 (2006). I. Avrutsky, R. Soref, and W. Buchwald, “Sub-wavelength plasmonic modes in a conductor-gap-dielectric system with a nanoscale gap,” Opt. Express 18(1), 348–363 (2010). R. F. Oulton, V. J. Sorger, D. A. Genov, D. F. P. Pile, and X. Zhang, “A hybrid plasmonic waveguide for subwavelength confinement and long-rang propagation,” Nat. Photonics 2(8), 496–500 (2008). B. Steinberger, A. Hohenau, H. Ditlbacher, A. L. Stepanov, A. Drezet, F. Aussenegg, A. Leitner, and J. Krenn, “Dielectric stipes on gold as surface plasmon waveguides,” Appl. Phys. Lett. 88(9), 094104 (2006). T. Holmgaard and S. I. Bozhevolnyi, “Theoretical analysis of dielectric-loaded surface plasmon-polariton waveguides,” Phys. Rev. B 75(24), 245405 (2007). S. Randhawa, A. V. Krasavin, T. Holmgaard, J. Renger, S. I. Bozhevolnyi, A. V. Zayats, and R. Quidant, “Experimental demonstration of dielectric-loaded plasmonic waveguide disk resonators at telecom wavelengths,” Appl. Phys. Lett. 98(16), 161102 (2011). Z. H. Zhu, C. E. Garcia-Ortiz, Z. H. Han, I. P. Radko, and S. I. Bozhevolnyi, “Compact and broadband directional coupling and demultiplexing in dielectric-loaded surface plasmon polariton waveguides based on the multimode interference effect,” Appl. Phys. Lett. 103(6), 061108 (2013). M. Jablan, H. Buljan, and M. Soljačić, “Plasmonics in graphene at infrafred frequencies,” Phys. Rev. B 80(24), 245435 (2009). F. H. L. Koppens, D. E. Chang, and F. J. García de Abajo, “Graphene plasmonics: A platform for strong lightmatter interactions,” Nano Lett. 11(8), 3370–3377 (2011). Z. Fei, A. S. Rodin, G. O. Andreev, W. Bao, A. S. McLeod, M. Wagner, L. M. Zhang, Z. Zhao, M. Thiemens, G. Dominguez, M. M. Fogler, A. H. Castro Neto, C. N. Lau, F. Keilmann, and D. N. Basov, “Gate-tuning of graphene plasmons revealed by infrared nano-imaging,” Nature 487(7405), 82–85 (2012). J. Chen, M. Badioli, P. Alonso-González, S. Thongrattanasiri, F. Huth, J. Osmond, M. Spasenović, A. Centeno, A. Pesquera, P. Godignon, A. Z. Elorza, N. Camara, F. J. García de Abajo, R. Hillenbrand, and F. H. L. Koppens, “Optical nano-imaging of gate-tunable graphene plasmons,” Nature 487(7405), 77–81 (2012). A. Vakil and N. Engheta, “Transformation optics using graphene,” Science 332(6035), 1291–1294 (2011).
#232262 - $15.00 USD (C) 2015 OSA
Received 12 Jan 2015; revised 10 Feb 2015; accepted 16 Feb 2015; published 19 Feb 2015 23 Feb 2015 | Vol. 23, No. 4 | DOI:10.1364/OE.23.005147 | OPTICS EXPRESS 5147
17. J. Christensen, A. Manjavacas, S. Thongrattanasiri, F. H. Koppens, and F. J. García de Abajo, “Graphene plasmon waveguiding and hybridization in individual and paired nanoribbons,” ACS Nano 6(1), 431–440 (2012). 18. F. J. García de Abajo, “Graphene plasmonics: challenges and opportunities,” ACS Photon. 1(3), 135–152 (2014). 19. E. Forati and G. W. Hanson, “Surface plasmon polaritons on soft-boundary graphene nanoribbons and their application in switching/demultiplexing,” Appl. Phys. Lett. 103(13), 133104 (2013). 20. Y. Gao, G. Ren, B. Zhu, H. Liu, Y. Lian, and S. Jian, “Analytical model for plasmon modes in graphene-coated nanowire,” Opt. Express 22(20), 24322–24331 (2014). 21. P. Liu, X. Zhang, Z. Ma, W. Cai, L. Wang, and J. Xu, “Surface plasmon modes in graphene wedge and groove waveguides,” Opt. Express 21(26), 32432–32440 (2013). 22. J. Kotakoski, D. Santos-Cottin, and A. V. Krasheninnikov, “Stability of graphene edges under electron beam: equilibrium energetics versus dynamic effects,” ACS Nano 6(1), 671–676 (2012). 23. A. B. Buckman, Guided-Wave Photonics, 1 st ed. (Saunders College Publishing, 1992). 24. A. Boltasseva, T. Nikolajsen, K. Leosson, K. Kjaer, M. S. Larsen, and S. I. Bozhevolnyi, “Integrated optical components utilizing long-rang surface plasmon polaritons,” J. Lightwave Technol. 23(1), 413–422 (2005). 25. X. Zhou, T. Zhang, L. Chen, W. Hong, and X. Li, “A graphene-based hybrid plasmonic waveguide with ultradeep subwavelength confinement,” J. Lightwave Technol. 32(21), 4199–4203 (2014). 26. L. A. Falkovsky and S. S. Pershoguba, “Optical far-infrared properties of a graphene monolayer and multilayer,” Phys. Rev. B 76(15), 153410 (2007). 27. D. K. Efetov and P. Kim, “Controlling electron-phonon interactions in graphene at ultrahigh carrier densities,” Phys. Rev. Lett. 105(25), 256805 (2010). 28. L. Ju, B. S. Geng, J. Horng, C. Girit, M. Martin, Z. Hao, H. A. Bechtel, X. G. Liang, A. Zettl, Y. R. Shen, and F. Wang, “Graphene plasmonics for tunable terahertz metamaterials,” Nat. Nanotechnol. 6(10), 630–634 (2011). 29. K. I. Bolotin, K. J. Sikes, Z. Jiang, M. Klima, G. Fudenberg, J. Hone, P. Kim, and H. L. Stormer, “Ultrahigh electron mobility in suspended graphene,” Solid State Commun. 146(9-10), 351–355 (2008).
1. Introduction Surface plasmons (SPs) offer a promising way to confine and control electromagnetic (EM) waves at subwavelength scale which is quite required to realize highly compact optical circuits in nanotechnology [1]. Noble metals are usually used to support SPs in the visible to near-infrared frequencies and various SPs waveguide structures have been studied intensively [2–11]. However in the mid-infrared to terahertz frequency, only loosely bound surfaced waves could be supported by noble meals. Graphene, a newly emerged two dimensional (2D) atomically thin material, is believed as noval plasmonic material from the terahertz to the infrared spectral region [12,13]. Recently, the excitation, propagation and tunability of graphene surface plasmons (GSPs) in mid-infrared frequencies have been experimentally demonstrated [14,15]. Compared with noble metals, GSP could confine EM field at an extremely subwavelength scale in the mid-infrared spectral region. Moreover, GSP could be actively tuned by electrostatically gating or chemical doping which may lead to dynamically tunable plasmonic devices. These extraordinary properties make graphene a promising candidate for mid-infrared to terahertz SP waveguide. GSP modes on graphene sheets [16], graphene nanoribbons [17–19], graphene-coated nanowire [20] and graphene groove/wedge [21] were investigated intensively. These GSP waveguides could be classified into two classes. One is based on graphene patterning. However, the edge shape of graphene could strongly influence the propagation properties of GSP modes, and it is still a challenge to control the edge shape with desired atomic arrangement [22]. The other is based on substrate engineering, which may bring fabrication difficulties. In this paper, we investigate the GSP modes of dielectric loaded graphene plasmon waveguide (DLGPW) in mid-infrared spectral region. This concept originated from dielectric loaded metal plasmon waveguides in visible to near-infrared spectral region [8–11]. The DLGPW is not influenced by the edge shape of graphene and could be produced in a straightforward way by, for example, standard processes of lithography. Moreover the DLGPW could be combined with substrate engineering structures, providing more freedom to manipulate GSPs. So, it is of great significance to characterize the properties of GSP modes in DLGPW. Here, we first present an analytical model based on effective-index method [23,24] to solve eigen guided modes in DLGPW. Then dispersion relation and propagation loss is derived. The single-mode operation region is also illustrated. In the last part, the tunablity of the fundamental mode is dicussed. #232262 - $15.00 USD (C) 2015 OSA
Received 12 Jan 2015; revised 10 Feb 2015; accepted 16 Feb 2015; published 19 Feb 2015 23 Feb 2015 | Vol. 23, No. 4 | DOI:10.1364/OE.23.005147 | OPTICS EXPRESS 5148
2. Analytical model The geometry of the DLGPW is shown in Fig. 1(a). A dielectric strip with width of W and height of h is deposited onto a graphene sheet. The relative permittivity of the strip is εr2. For simplicity, the substrate is supposed to be a half space dielectric with relative permittivity of εr1, and the cladding is air. The dielectric strip results in a higher refractive index for GSP modes on the graphene-dielectric interface compared to graphene-air interface, giving rise to GSP modes bound by the dielectric strip. This is similar to guided modes in dielectric planar waveguides. The effective-index method (EIM) is one of the simple methods for analyze photonic and SP waveguide [9,23,24]. Here, we use this method to present an analytical model for DLGPW.
Fig. 1. (a) Schematic of the dielectric loaded graphene plasmon waveguide. (b) The equivalent three-layer planar waveguide structure for the derivation of Eq. (1). (b) The equivalent threelayer planar waveguide structure for the derivation of Eq. (3).
In this method, the dielectric strip serves as the core of a three-layer dielectric planar waveguide (Fig. 1(c)). The refractive index of the core ncore is independent on the width of dielectric strip, and is equal to the effective mode index of the highly confined GSP mode (TM mode) in planar graphene sheet sandwiched between dielectrics of relative permittivity εr1 and εr2 (Fig. 1(b)) and is written as [12] ncore = kGSP1 / k0 = ε 0
ε r1 + ε r 2 2ic . σ (ω ) 2
(1)
where σ (ω) is the optical conductivity of graphene, k0 is the vacuum wave number. The refractive index of the cladding nclad is equal to the effective mode index of the GSP mode (TM mode) in planar graphene sheet sandwiched between dielectrics of relative permittivity εr1 and air and is expressed as nclad = kGSP 2 / k0 = ε 0
ε r1 + 1 2ic . 2 σ (ω )
(2)
Then, by simple algebra operation, eigen equation of the equivalent dielectric planar waveguide for the m-th order guided TE mode is given as
μclad T tan(
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Tw mπ ) − μcoreτ = 0. − 2 2
(3)
Received 12 Jan 2015; revised 10 Feb 2015; accepted 16 Feb 2015; published 19 Feb 2015 23 Feb 2015 | Vol. 23, No. 4 | DOI:10.1364/OE.23.005147 | OPTICS EXPRESS 5149
2 2 2 2 Where μclad = μcore = 1 is the relative permeability, T = k0 ncore − neff , τ = k0 neff − nclad , w is
the width of the dielectric strip, neff is the effective mode refractive index of the DLGPW. The cutoff condition of the guided modes is τ = 0. Then, the cutoff wavelength of m-th order guided mode is
λcm = Re(
2w 2 2 ). ncore − nclad m
(4)
By numerical solving Eq. (3), the effective mode index neff of m-th order mode could be derived. The real part of neff corresponds to the GSP wavelength λGSP = λ0/Re(neff), where λ0 is the vacuum wavelength. The imaginary part of neff corresponds to the propagation loss, and determines the propagation length L by L = λ0/[2π⋅Im(neff)]. It should be noted that, in developing the analytical modal for the DLGPW, the influence of the height of the dielectric strip has not been taken into account. This is based on the fact that the GSP mode is highly confined at the interface of graphene sheet (Fig. 2 (c)) and the EM field surpass the top of the dielctric strip could be neglected, if the dielectric strip is not too thin. However, the analytical modal proposed here could be easily expanded to the case of thin dielectric strip or more complicated structure: e. g. the recently proposed graphene based hybrid plasmonic waveguide [25]. In these cases, the eigen equations of planar four-layer or five-layer waveguide structure should be solved first to get the effective refractive index of the core layer. 3. Results and discuss We first numerically solve Eq. (3), the dispersion relation is demonstrated. Then we use commercial software (COMSOL) based on the finite element method (FEM) to verify the proposed analytical model. Next we numerically solve Eq. (4), the single mode operation range is derived. In our calculation, the height of the dielectric strip is 100nm, relative permittivity of the substrate and the dielectric strip are both 3.92. Graphene is modeled as a 0.5nm thick anisotropic layer. The out-of-plane relative permittivity is 2.5. The in-plane relative permittivity is ε / / (ω ) = 2.5 + iσ (ω ) / (ωε 0 t ) , where the optical conductivity of graphene is derived using the random-phase approximation in the local limit [26]: σ (ω ) =
E e2 1 1 ω − 2 EF i i 2 e 2 k BT ( ω + 2 E F ) 2 In 2cos h F + + arc tan In − −1 2 2 2 + + π (ω + iτ ) 2 k T 4 2 π 2 k T 2 π ( ω 2 E ) 4( k T ) B B F B
where T = 300 K is the temperature, k B is the Boltzmann constant, ω is the frequency, EF is the Fermi energy level and τ = μ EF / eVF2 is the carrier relaxation lifetime (μ is the carrier mobility of graphene and VF = 106 m/s is the Fermi velocity). In recent experiments, the Fermi energy level has reached as high as 1.17 eV [27]. The carrier mobility ranges from ~1000 cm2/(V⋅s) [28] in chemical vapor deposition (CVD) grown graphene to 230000 cm2/(V⋅s) [29] in suspended exfoliated graphene. Here, we use moderate Fermi energy level of 0.5eV and carrier mobility of 10000 cm2/(V⋅s), unless otherwise stated.
#232262 - $15.00 USD (C) 2015 OSA
Received 12 Jan 2015; revised 10 Feb 2015; accepted 16 Feb 2015; published 19 Feb 2015 23 Feb 2015 | Vol. 23, No. 4 | DOI:10.1364/OE.23.005147 | OPTICS EXPRESS 5150
Fig. 2. Effective mode indices of the GSP modes in DLGPW with a width of 200 nm: (a) Real part, (b) Imaginary part of the effective mode index. The insets of (b) show the amplitudes of Ey for 1-th mode at the wavelength of 10 μm and 13 μm, respectively. Solid lines are numerical solutions of Eq. (3), symbols are obtained by Comsol simulations, and dashed lines correspond to numerical solutions of Eq. (1) and (2). (c) Mode patterns (the amplitudes of Ey) of the first 4 order modes at the wavelength of 8 μm.
Figure 2 shows the effective mode indices of GSP modes in DLGPW with a width of 200 nm. Numerical solutions of the analytical model show good agreement with the FEM simulation results, when the wavelength is away from the cutoff wavelength. The guided GSP modes are confined between ncore and nclad, which is the same as dielectric planar waveguide. However, when the modes are approaching cutoff, the results of analytical model show slight deviations from the FEM results. This is because when approaching cutoff, the mode confinement become weak (see the insets of Fig. 2(b)), and the EM field in the corner regions is no longer negligible. However, the effective-index method doesn’t account for the EM field in the corner regions. Effective mode indices decrease monotonically as wavelength increases. The fundamental mode (m = 0) has higher real part of effective mode index than higher order modes, which indicates the shorter GSP wavelength. Moreover the fundamental mode is cutoff-free. The higher order modes cutoff when the real parts of neff approache that of nclad. The propagation loss of higher order mode decreases sharply as the wavelength approaches the cutoff wavelength. However, this is at the expense of poor confinement of the GSP modes. Away from the cutoff wavelength, the fundamental mode has relatively small propagation loss.
Fig. 3. (a) Single-mode and multi-modes operation regions calculated by Eq. (4). The numbers of modes supported by the DLGPW are labeled. (b) The cutoff wavelength of 1-th order mode at different Fermi energy levels.
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Received 12 Jan 2015; revised 10 Feb 2015; accepted 16 Feb 2015; published 19 Feb 2015 23 Feb 2015 | Vol. 23, No. 4 | DOI:10.1364/OE.23.005147 | OPTICS EXPRESS 5151
Single-mode operation is highly preferred for many applications, as multi-mode propagation may lead to signal fading and unwanted mode conversion. By solving Eq. (4), single-mode and multi-modes operation region as a function of the width of the dielectric strip is shown in Fig. 3(a). The white dashed curves are numerical solutions of Eq. (4). At a fixed wavelength, the numbers of guided modes decrease as the width of the dielectric strip decrease. The single-mode operation region (labeled 1) lies in the right of the 1-th order mode cutoff wavelength. As the width of the dielectric strip increase, the single-mode operation region starts at a longer wavelength. Propagation properties of the GSP modes in DLGPW could be manipulated by tuning the Fermi energy level. Figure 3(b) shows the cutoff wavelength of 1-th order mode at different Fermi energy level. The single-mode operation region moves to longer wavelength as the Fermi energy level decrease. When the Fermi energy level changes from 0.3 eV to 0.9 eV and the width is fixed as 50 nm, only fundamental mode is supported for wavelength longer than 10 μm. Then we fix the width as 50nm, and study the dispersion relation of the fundamental mode.
Fig. 4. Real part of the effective mode indices (a) and the propagation length (b) of the fundamental mode at different Fermi energy levels. The carrier mobility is fixed as 10000 cm2/(V⋅s). Real part of the effective mode indices (c) and the propagation length (d) of the fundamental mode at different carrier mobility. The Fermi energy level is fixed as 0.5 eV, and the width of the dielectric strip is 50 nm.
Figure 4(a) shows the real part of the effective mode indices of the fundamental mode in DLGPW with a width of 50 nm. By varying the Fermi energy level, effective mode indices of the fundamental mode could be effectively manipulated. Effective mode indices increase as the Fermi energy level decrease. At a Fermi energy level of 0.3 eV, real part of the effective mode index is larger than 70 for a wavelength of 12 μm, indicating that the GSP mode is tightly confined. However there is a well-known tradeoff between mode confinement and propagation length. The propagation length increases conspicuously as the Fermi energy level increases (Fig. 4(b)). This is due to poor confinement of the GSP mode and increment of the carrier relaxation lifetime (indicating reduction of the intrinsic loss of graphene), as the Fermi energy level increases. The carrier mobility has a great influence on the propagation length (Fig. 4(d)). Higher carrier mobility leads to longer propagation length. But, the real part of the
#232262 - $15.00 USD (C) 2015 OSA
Received 12 Jan 2015; revised 10 Feb 2015; accepted 16 Feb 2015; published 19 Feb 2015 23 Feb 2015 | Vol. 23, No. 4 | DOI:10.1364/OE.23.005147 | OPTICS EXPRESS 5152
effective mode indices is insensitive to the carrier mobility (Fig. 4(c)). So graphene with lower Fermi energy level and higher carrier mobility is better for long distance propagation of highly confined GSP mode. 4. Conclusions In summary, we proposed the concept of dielectric loaded graphene plasmon waveguide. This waveguide is not influenced by the edge shape of graphene and easy to fabricate. An analytical model based on effective-index method was presented and verified by FEM simulations, which provided a simple way to analysis the mode properties of DLGPW. The mode effective index, propagation loss, cutoff wavelength of higher order modes and singlemode operation region were derived. The number of GSP modes supported by the DLGPW increases as either the wavelength decrease or the dielectric strip width increase. By changing Fermi energy level, the propagation properties of fundamental mode could be tuned flexibly. Higher Fermi engery level led to smaller propagation loss at the expense of poorer confinement. The DLGPW provided a new freedom to manipulate the GSP, which may led to new applications in actively tunable integrated optical devices. Acknowledgments This work is supported by the State Key Program for Basic Research of China (No. 2012CB933501) and the National Natural Science Foundation of China (Grant Nos. 61177051, 11304389, 61404174, and 61205087).
#232262 - $15.00 USD (C) 2015 OSA
Received 12 Jan 2015; revised 10 Feb 2015; accepted 16 Feb 2015; published 19 Feb 2015 23 Feb 2015 | Vol. 23, No. 4 | DOI:10.1364/OE.23.005147 | OPTICS EXPRESS 5153
College of Optoelectronic Science and Engineering, National University of Defense Technology, Changsha 410073, China 2 State Key Laboratory of High Performance Computing, National University of Defense Technology, Changsha 410073, China * [email protected]
Abstract: Dielectric loaded graphene plasmon waveguide (DLGPW) is proposed and investigated. An analytical model based on effective-index method is presented and verified by the finite element method simulations. The mode effective index, propagation loss, cutoff wavelength of higher order modes and single-mode operation region were derived at mid-infrared spectral region. By changing Fermi energy level, the propagation properties of fundamental mode could be tuned flexibly. The structure of the DLGPW is simple and easy for fabrication. It provided a new freedom to manipulate the graphene surface plasmons, which may led to new applications in actively tunable integrated optical devices. ©2015 Optical Society of America OCIS codes: (240.6680) Surface plasmons; (250.5403) Plasmonics; (230.7370) Waveguides.
References and links 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16.
D. K. Gramotnev and S. I. Bozhevolnyi, “Plasmonics beyond the diffraction limit,” Nat. Photonics 4(2), 83–91 (2010). S. I. Bozhevolnyi, V. S. Volkov, E. Devaux, and T. W. Ebbesen, “Channel plasmon-polariton guiding by subwavelength metal grooves,” Phys. Rev. Lett. 95(4), 046802 (2005). E. Moreno, F. J. Garcia-Vidal, S. G. Rodrigo, L. Martin-Moreno, and S. I. Bozhevolnyi, “Channel plasmonpolaritons: modal shape, dispersion, and losses,” Opt. Lett. 31(23), 3447–3449 (2006). C. L. Zou, F. W. Sun, Y. F. Xiao, C. H. Dong, X. D. Chen, J. M. Cui, Q. Gong, Z. F. Han, and G. C. Guo, “Plasmon modes of silver nanowire on a silica substrate,” Appl. Phys. Lett. 97(18), 183102 (2010). J. A. Dionne, L. A. Sweatlock, H. A. Atwater, and A. Polman, “Plasmon slot waveguides: Towards chip-scale propagation with subwavelength-sacle localization,” Phys. Rev. B 73(3), 035407 (2006). I. Avrutsky, R. Soref, and W. Buchwald, “Sub-wavelength plasmonic modes in a conductor-gap-dielectric system with a nanoscale gap,” Opt. Express 18(1), 348–363 (2010). R. F. Oulton, V. J. Sorger, D. A. Genov, D. F. P. Pile, and X. Zhang, “A hybrid plasmonic waveguide for subwavelength confinement and long-rang propagation,” Nat. Photonics 2(8), 496–500 (2008). B. Steinberger, A. Hohenau, H. Ditlbacher, A. L. Stepanov, A. Drezet, F. Aussenegg, A. Leitner, and J. Krenn, “Dielectric stipes on gold as surface plasmon waveguides,” Appl. Phys. Lett. 88(9), 094104 (2006). T. Holmgaard and S. I. Bozhevolnyi, “Theoretical analysis of dielectric-loaded surface plasmon-polariton waveguides,” Phys. Rev. B 75(24), 245405 (2007). S. Randhawa, A. V. Krasavin, T. Holmgaard, J. Renger, S. I. Bozhevolnyi, A. V. Zayats, and R. Quidant, “Experimental demonstration of dielectric-loaded plasmonic waveguide disk resonators at telecom wavelengths,” Appl. Phys. Lett. 98(16), 161102 (2011). Z. H. Zhu, C. E. Garcia-Ortiz, Z. H. Han, I. P. Radko, and S. I. Bozhevolnyi, “Compact and broadband directional coupling and demultiplexing in dielectric-loaded surface plasmon polariton waveguides based on the multimode interference effect,” Appl. Phys. Lett. 103(6), 061108 (2013). M. Jablan, H. Buljan, and M. Soljačić, “Plasmonics in graphene at infrafred frequencies,” Phys. Rev. B 80(24), 245435 (2009). F. H. L. Koppens, D. E. Chang, and F. J. García de Abajo, “Graphene plasmonics: A platform for strong lightmatter interactions,” Nano Lett. 11(8), 3370–3377 (2011). Z. Fei, A. S. Rodin, G. O. Andreev, W. Bao, A. S. McLeod, M. Wagner, L. M. Zhang, Z. Zhao, M. Thiemens, G. Dominguez, M. M. Fogler, A. H. Castro Neto, C. N. Lau, F. Keilmann, and D. N. Basov, “Gate-tuning of graphene plasmons revealed by infrared nano-imaging,” Nature 487(7405), 82–85 (2012). J. Chen, M. Badioli, P. Alonso-González, S. Thongrattanasiri, F. Huth, J. Osmond, M. Spasenović, A. Centeno, A. Pesquera, P. Godignon, A. Z. Elorza, N. Camara, F. J. García de Abajo, R. Hillenbrand, and F. H. L. Koppens, “Optical nano-imaging of gate-tunable graphene plasmons,” Nature 487(7405), 77–81 (2012). A. Vakil and N. Engheta, “Transformation optics using graphene,” Science 332(6035), 1291–1294 (2011).
#232262 - $15.00 USD (C) 2015 OSA
Received 12 Jan 2015; revised 10 Feb 2015; accepted 16 Feb 2015; published 19 Feb 2015 23 Feb 2015 | Vol. 23, No. 4 | DOI:10.1364/OE.23.005147 | OPTICS EXPRESS 5147
17. J. Christensen, A. Manjavacas, S. Thongrattanasiri, F. H. Koppens, and F. J. García de Abajo, “Graphene plasmon waveguiding and hybridization in individual and paired nanoribbons,” ACS Nano 6(1), 431–440 (2012). 18. F. J. García de Abajo, “Graphene plasmonics: challenges and opportunities,” ACS Photon. 1(3), 135–152 (2014). 19. E. Forati and G. W. Hanson, “Surface plasmon polaritons on soft-boundary graphene nanoribbons and their application in switching/demultiplexing,” Appl. Phys. Lett. 103(13), 133104 (2013). 20. Y. Gao, G. Ren, B. Zhu, H. Liu, Y. Lian, and S. Jian, “Analytical model for plasmon modes in graphene-coated nanowire,” Opt. Express 22(20), 24322–24331 (2014). 21. P. Liu, X. Zhang, Z. Ma, W. Cai, L. Wang, and J. Xu, “Surface plasmon modes in graphene wedge and groove waveguides,” Opt. Express 21(26), 32432–32440 (2013). 22. J. Kotakoski, D. Santos-Cottin, and A. V. Krasheninnikov, “Stability of graphene edges under electron beam: equilibrium energetics versus dynamic effects,” ACS Nano 6(1), 671–676 (2012). 23. A. B. Buckman, Guided-Wave Photonics, 1 st ed. (Saunders College Publishing, 1992). 24. A. Boltasseva, T. Nikolajsen, K. Leosson, K. Kjaer, M. S. Larsen, and S. I. Bozhevolnyi, “Integrated optical components utilizing long-rang surface plasmon polaritons,” J. Lightwave Technol. 23(1), 413–422 (2005). 25. X. Zhou, T. Zhang, L. Chen, W. Hong, and X. Li, “A graphene-based hybrid plasmonic waveguide with ultradeep subwavelength confinement,” J. Lightwave Technol. 32(21), 4199–4203 (2014). 26. L. A. Falkovsky and S. S. Pershoguba, “Optical far-infrared properties of a graphene monolayer and multilayer,” Phys. Rev. B 76(15), 153410 (2007). 27. D. K. Efetov and P. Kim, “Controlling electron-phonon interactions in graphene at ultrahigh carrier densities,” Phys. Rev. Lett. 105(25), 256805 (2010). 28. L. Ju, B. S. Geng, J. Horng, C. Girit, M. Martin, Z. Hao, H. A. Bechtel, X. G. Liang, A. Zettl, Y. R. Shen, and F. Wang, “Graphene plasmonics for tunable terahertz metamaterials,” Nat. Nanotechnol. 6(10), 630–634 (2011). 29. K. I. Bolotin, K. J. Sikes, Z. Jiang, M. Klima, G. Fudenberg, J. Hone, P. Kim, and H. L. Stormer, “Ultrahigh electron mobility in suspended graphene,” Solid State Commun. 146(9-10), 351–355 (2008).
1. Introduction Surface plasmons (SPs) offer a promising way to confine and control electromagnetic (EM) waves at subwavelength scale which is quite required to realize highly compact optical circuits in nanotechnology [1]. Noble metals are usually used to support SPs in the visible to near-infrared frequencies and various SPs waveguide structures have been studied intensively [2–11]. However in the mid-infrared to terahertz frequency, only loosely bound surfaced waves could be supported by noble meals. Graphene, a newly emerged two dimensional (2D) atomically thin material, is believed as noval plasmonic material from the terahertz to the infrared spectral region [12,13]. Recently, the excitation, propagation and tunability of graphene surface plasmons (GSPs) in mid-infrared frequencies have been experimentally demonstrated [14,15]. Compared with noble metals, GSP could confine EM field at an extremely subwavelength scale in the mid-infrared spectral region. Moreover, GSP could be actively tuned by electrostatically gating or chemical doping which may lead to dynamically tunable plasmonic devices. These extraordinary properties make graphene a promising candidate for mid-infrared to terahertz SP waveguide. GSP modes on graphene sheets [16], graphene nanoribbons [17–19], graphene-coated nanowire [20] and graphene groove/wedge [21] were investigated intensively. These GSP waveguides could be classified into two classes. One is based on graphene patterning. However, the edge shape of graphene could strongly influence the propagation properties of GSP modes, and it is still a challenge to control the edge shape with desired atomic arrangement [22]. The other is based on substrate engineering, which may bring fabrication difficulties. In this paper, we investigate the GSP modes of dielectric loaded graphene plasmon waveguide (DLGPW) in mid-infrared spectral region. This concept originated from dielectric loaded metal plasmon waveguides in visible to near-infrared spectral region [8–11]. The DLGPW is not influenced by the edge shape of graphene and could be produced in a straightforward way by, for example, standard processes of lithography. Moreover the DLGPW could be combined with substrate engineering structures, providing more freedom to manipulate GSPs. So, it is of great significance to characterize the properties of GSP modes in DLGPW. Here, we first present an analytical model based on effective-index method [23,24] to solve eigen guided modes in DLGPW. Then dispersion relation and propagation loss is derived. The single-mode operation region is also illustrated. In the last part, the tunablity of the fundamental mode is dicussed. #232262 - $15.00 USD (C) 2015 OSA
Received 12 Jan 2015; revised 10 Feb 2015; accepted 16 Feb 2015; published 19 Feb 2015 23 Feb 2015 | Vol. 23, No. 4 | DOI:10.1364/OE.23.005147 | OPTICS EXPRESS 5148
2. Analytical model The geometry of the DLGPW is shown in Fig. 1(a). A dielectric strip with width of W and height of h is deposited onto a graphene sheet. The relative permittivity of the strip is εr2. For simplicity, the substrate is supposed to be a half space dielectric with relative permittivity of εr1, and the cladding is air. The dielectric strip results in a higher refractive index for GSP modes on the graphene-dielectric interface compared to graphene-air interface, giving rise to GSP modes bound by the dielectric strip. This is similar to guided modes in dielectric planar waveguides. The effective-index method (EIM) is one of the simple methods for analyze photonic and SP waveguide [9,23,24]. Here, we use this method to present an analytical model for DLGPW.
Fig. 1. (a) Schematic of the dielectric loaded graphene plasmon waveguide. (b) The equivalent three-layer planar waveguide structure for the derivation of Eq. (1). (b) The equivalent threelayer planar waveguide structure for the derivation of Eq. (3).
In this method, the dielectric strip serves as the core of a three-layer dielectric planar waveguide (Fig. 1(c)). The refractive index of the core ncore is independent on the width of dielectric strip, and is equal to the effective mode index of the highly confined GSP mode (TM mode) in planar graphene sheet sandwiched between dielectrics of relative permittivity εr1 and εr2 (Fig. 1(b)) and is written as [12] ncore = kGSP1 / k0 = ε 0
ε r1 + ε r 2 2ic . σ (ω ) 2
(1)
where σ (ω) is the optical conductivity of graphene, k0 is the vacuum wave number. The refractive index of the cladding nclad is equal to the effective mode index of the GSP mode (TM mode) in planar graphene sheet sandwiched between dielectrics of relative permittivity εr1 and air and is expressed as nclad = kGSP 2 / k0 = ε 0
ε r1 + 1 2ic . 2 σ (ω )
(2)
Then, by simple algebra operation, eigen equation of the equivalent dielectric planar waveguide for the m-th order guided TE mode is given as
μclad T tan(
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Tw mπ ) − μcoreτ = 0. − 2 2
(3)
Received 12 Jan 2015; revised 10 Feb 2015; accepted 16 Feb 2015; published 19 Feb 2015 23 Feb 2015 | Vol. 23, No. 4 | DOI:10.1364/OE.23.005147 | OPTICS EXPRESS 5149
2 2 2 2 Where μclad = μcore = 1 is the relative permeability, T = k0 ncore − neff , τ = k0 neff − nclad , w is
the width of the dielectric strip, neff is the effective mode refractive index of the DLGPW. The cutoff condition of the guided modes is τ = 0. Then, the cutoff wavelength of m-th order guided mode is
λcm = Re(
2w 2 2 ). ncore − nclad m
(4)
By numerical solving Eq. (3), the effective mode index neff of m-th order mode could be derived. The real part of neff corresponds to the GSP wavelength λGSP = λ0/Re(neff), where λ0 is the vacuum wavelength. The imaginary part of neff corresponds to the propagation loss, and determines the propagation length L by L = λ0/[2π⋅Im(neff)]. It should be noted that, in developing the analytical modal for the DLGPW, the influence of the height of the dielectric strip has not been taken into account. This is based on the fact that the GSP mode is highly confined at the interface of graphene sheet (Fig. 2 (c)) and the EM field surpass the top of the dielctric strip could be neglected, if the dielectric strip is not too thin. However, the analytical modal proposed here could be easily expanded to the case of thin dielectric strip or more complicated structure: e. g. the recently proposed graphene based hybrid plasmonic waveguide [25]. In these cases, the eigen equations of planar four-layer or five-layer waveguide structure should be solved first to get the effective refractive index of the core layer. 3. Results and discuss We first numerically solve Eq. (3), the dispersion relation is demonstrated. Then we use commercial software (COMSOL) based on the finite element method (FEM) to verify the proposed analytical model. Next we numerically solve Eq. (4), the single mode operation range is derived. In our calculation, the height of the dielectric strip is 100nm, relative permittivity of the substrate and the dielectric strip are both 3.92. Graphene is modeled as a 0.5nm thick anisotropic layer. The out-of-plane relative permittivity is 2.5. The in-plane relative permittivity is ε / / (ω ) = 2.5 + iσ (ω ) / (ωε 0 t ) , where the optical conductivity of graphene is derived using the random-phase approximation in the local limit [26]: σ (ω ) =
E e2 1 1 ω − 2 EF i i 2 e 2 k BT ( ω + 2 E F ) 2 In 2cos h F + + arc tan In − −1 2 2 2 + + π (ω + iτ ) 2 k T 4 2 π 2 k T 2 π ( ω 2 E ) 4( k T ) B B F B
where T = 300 K is the temperature, k B is the Boltzmann constant, ω is the frequency, EF is the Fermi energy level and τ = μ EF / eVF2 is the carrier relaxation lifetime (μ is the carrier mobility of graphene and VF = 106 m/s is the Fermi velocity). In recent experiments, the Fermi energy level has reached as high as 1.17 eV [27]. The carrier mobility ranges from ~1000 cm2/(V⋅s) [28] in chemical vapor deposition (CVD) grown graphene to 230000 cm2/(V⋅s) [29] in suspended exfoliated graphene. Here, we use moderate Fermi energy level of 0.5eV and carrier mobility of 10000 cm2/(V⋅s), unless otherwise stated.
#232262 - $15.00 USD (C) 2015 OSA
Received 12 Jan 2015; revised 10 Feb 2015; accepted 16 Feb 2015; published 19 Feb 2015 23 Feb 2015 | Vol. 23, No. 4 | DOI:10.1364/OE.23.005147 | OPTICS EXPRESS 5150
Fig. 2. Effective mode indices of the GSP modes in DLGPW with a width of 200 nm: (a) Real part, (b) Imaginary part of the effective mode index. The insets of (b) show the amplitudes of Ey for 1-th mode at the wavelength of 10 μm and 13 μm, respectively. Solid lines are numerical solutions of Eq. (3), symbols are obtained by Comsol simulations, and dashed lines correspond to numerical solutions of Eq. (1) and (2). (c) Mode patterns (the amplitudes of Ey) of the first 4 order modes at the wavelength of 8 μm.
Figure 2 shows the effective mode indices of GSP modes in DLGPW with a width of 200 nm. Numerical solutions of the analytical model show good agreement with the FEM simulation results, when the wavelength is away from the cutoff wavelength. The guided GSP modes are confined between ncore and nclad, which is the same as dielectric planar waveguide. However, when the modes are approaching cutoff, the results of analytical model show slight deviations from the FEM results. This is because when approaching cutoff, the mode confinement become weak (see the insets of Fig. 2(b)), and the EM field in the corner regions is no longer negligible. However, the effective-index method doesn’t account for the EM field in the corner regions. Effective mode indices decrease monotonically as wavelength increases. The fundamental mode (m = 0) has higher real part of effective mode index than higher order modes, which indicates the shorter GSP wavelength. Moreover the fundamental mode is cutoff-free. The higher order modes cutoff when the real parts of neff approache that of nclad. The propagation loss of higher order mode decreases sharply as the wavelength approaches the cutoff wavelength. However, this is at the expense of poor confinement of the GSP modes. Away from the cutoff wavelength, the fundamental mode has relatively small propagation loss.
Fig. 3. (a) Single-mode and multi-modes operation regions calculated by Eq. (4). The numbers of modes supported by the DLGPW are labeled. (b) The cutoff wavelength of 1-th order mode at different Fermi energy levels.
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Received 12 Jan 2015; revised 10 Feb 2015; accepted 16 Feb 2015; published 19 Feb 2015 23 Feb 2015 | Vol. 23, No. 4 | DOI:10.1364/OE.23.005147 | OPTICS EXPRESS 5151
Single-mode operation is highly preferred for many applications, as multi-mode propagation may lead to signal fading and unwanted mode conversion. By solving Eq. (4), single-mode and multi-modes operation region as a function of the width of the dielectric strip is shown in Fig. 3(a). The white dashed curves are numerical solutions of Eq. (4). At a fixed wavelength, the numbers of guided modes decrease as the width of the dielectric strip decrease. The single-mode operation region (labeled 1) lies in the right of the 1-th order mode cutoff wavelength. As the width of the dielectric strip increase, the single-mode operation region starts at a longer wavelength. Propagation properties of the GSP modes in DLGPW could be manipulated by tuning the Fermi energy level. Figure 3(b) shows the cutoff wavelength of 1-th order mode at different Fermi energy level. The single-mode operation region moves to longer wavelength as the Fermi energy level decrease. When the Fermi energy level changes from 0.3 eV to 0.9 eV and the width is fixed as 50 nm, only fundamental mode is supported for wavelength longer than 10 μm. Then we fix the width as 50nm, and study the dispersion relation of the fundamental mode.
Fig. 4. Real part of the effective mode indices (a) and the propagation length (b) of the fundamental mode at different Fermi energy levels. The carrier mobility is fixed as 10000 cm2/(V⋅s). Real part of the effective mode indices (c) and the propagation length (d) of the fundamental mode at different carrier mobility. The Fermi energy level is fixed as 0.5 eV, and the width of the dielectric strip is 50 nm.
Figure 4(a) shows the real part of the effective mode indices of the fundamental mode in DLGPW with a width of 50 nm. By varying the Fermi energy level, effective mode indices of the fundamental mode could be effectively manipulated. Effective mode indices increase as the Fermi energy level decrease. At a Fermi energy level of 0.3 eV, real part of the effective mode index is larger than 70 for a wavelength of 12 μm, indicating that the GSP mode is tightly confined. However there is a well-known tradeoff between mode confinement and propagation length. The propagation length increases conspicuously as the Fermi energy level increases (Fig. 4(b)). This is due to poor confinement of the GSP mode and increment of the carrier relaxation lifetime (indicating reduction of the intrinsic loss of graphene), as the Fermi energy level increases. The carrier mobility has a great influence on the propagation length (Fig. 4(d)). Higher carrier mobility leads to longer propagation length. But, the real part of the
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Received 12 Jan 2015; revised 10 Feb 2015; accepted 16 Feb 2015; published 19 Feb 2015 23 Feb 2015 | Vol. 23, No. 4 | DOI:10.1364/OE.23.005147 | OPTICS EXPRESS 5152
effective mode indices is insensitive to the carrier mobility (Fig. 4(c)). So graphene with lower Fermi energy level and higher carrier mobility is better for long distance propagation of highly confined GSP mode. 4. Conclusions In summary, we proposed the concept of dielectric loaded graphene plasmon waveguide. This waveguide is not influenced by the edge shape of graphene and easy to fabricate. An analytical model based on effective-index method was presented and verified by FEM simulations, which provided a simple way to analysis the mode properties of DLGPW. The mode effective index, propagation loss, cutoff wavelength of higher order modes and singlemode operation region were derived. The number of GSP modes supported by the DLGPW increases as either the wavelength decrease or the dielectric strip width increase. By changing Fermi energy level, the propagation properties of fundamental mode could be tuned flexibly. Higher Fermi engery level led to smaller propagation loss at the expense of poorer confinement. The DLGPW provided a new freedom to manipulate the GSP, which may led to new applications in actively tunable integrated optical devices. Acknowledgments This work is supported by the State Key Program for Basic Research of China (No. 2012CB933501) and the National Natural Science Foundation of China (Grant Nos. 61177051, 11304389, 61404174, and 61205087).
#232262 - $15.00 USD (C) 2015 OSA
Received 12 Jan 2015; revised 10 Feb 2015; accepted 16 Feb 2015; published 19 Feb 2015 23 Feb 2015 | Vol. 23, No. 4 | DOI:10.1364/OE.23.005147 | OPTICS EXPRESS 5153
