October 15, 2014 / Vol. 39, No. 20 / OPTICS LETTERS
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Single-mode graphene-coated nanowire plasmonic waveguide Yixiao Gao,1,2 Guobin Ren,1,2,* Bofeng Zhu,1,2 Jing Wang,3 and Shuisheng Jian1,2 1
Key Laboratory of All Optical Network & Advanced Telecommunication Network of EMC, Beijing Jiaotong University, Beijing 100044, China 2 Institute of Lightwave Technology, Beijing Jiaotong University, Beijing 100044, China 3
Science and Technology on Optical Radiation Laboratory, Beijing 100854, China *Corresponding author: [email protected]
Received June 26, 2014; revised September 9, 2014; accepted September 10, 2014; posted September 10, 2014 (Doc. ID 214812); published October 10, 2014 We propose in this Letter a single-mode graphene-coated nanowire surface plasmon waveguide. The single-mode condition and modal cutoff wavelength of high order modes are derived from an analytic model and confirmed by numerical simulation. The mode number diagram of the proposed waveguide in the wavelength-radius space is also demonstrated. By changing the Fermi level of graphene, the performance of the proposed waveguide could be tuned flexibly, offering potential application in tunable nanophotonic devices. © 2014 Optical Society of America OCIS codes: (240.6680) Surface plasmons; (250.5403) Plasmonics; (230.7370) Waveguides. http://dx.doi.org/10.1364/OL.39.005909
Surface plasmons (SPs) are thought to be the only viable way to confine and control light in the subwavelength scale required by the development of nanophotonic technology [1]. The investigations of SP waveguiding focus on how to achieve extreme localization of the electromagnetic field as well as long propagation length. Various types of SP waveguides were proposed, such as thin metal film [2], metal groove/wedge [3], metal nanowire [4], and hybrid plasmonic waveguides [5]. However, noble metals used in these waveguides only support weakly confined SP waves in the mid-infrared to THz frequency band, though this frequency range has important applications in biology, communication, and spectroscopy. Graphene, a newly emerged 2D material consisting of carbon atoms arranged in a hexagonal lattice, is a promising candidate for SP waveguiding in the mid-infrared to THz frequency range [6]. Compared with noble metals, graphene has three major advantages [7]—low loss, extreme mode confinement, and tunable carrier density by electrical gating or chemical doping—which further lead to tunable electromagnetic properties. Graphene surface plasmon (GSP) waveguiding has been investigated intensively, such as GSP mode on a single graphene sheet (GS) [6,8], graphene nanoribbon (GNR) [7,9–11], and graphene groove/wedge waveguide [12]. Graphene-microfiber/ nanowire combined systems have been proposed for various applications [13–17], but GSP modes in the graphene-coated nanowire (GNW) waveguide, which can be seen as an analogy for metal nanowire with azimuthal symmetry, have not been characterized yet. Single-mode operation is highly preferred for most applications due to its high beam quality, lowest transmission loss, and low group velocity dispersion. For GNR, the fundamental mode is even-parity edge mode, whose field is tightly confined near the ribbon edge [9], and single-mode operation occurs with ribbon width shrinking [11]. Transmission properties of GNR are strongly influenced by the morphology of the GNR edge. However, it is still a challenge to control the edge shape with the desired atomic arrangement [18]. Therefore, single-mode propagation in GNR might be strongly influenced by edge non-uniformity. For the graphene 0146-9592/14/205909-04$15.00/0
groove/wedge waveguide, the fundamental mode has a perfect electric conductor symmetry [12], but how to achieve single-mode operation is not specified. It is still significant to realize the single-mode operation in graphene-based SP waveguides. In this Letter, we investigate the SP modes in a GNW. First, the dispersion relation and propagation length are demonstrated. Next, the number of supported modes in the GNW and single-mode condition are derived from an analytic model, and a modal cutoff wavelength formula is presented. In order to verify the proposed formula, the mode number in wavelength-radius space is calculated by commercial software (COMSOL) based on the finite element method (FEM). Finally, we analyze the tunable performance of GNW by varying the graphene Fermi level. The structure of GNW is depicted in Fig. 1(a). A dielectric nanowire with radius R is coated by a graphene layer, which could be regarded as rolling a graphene ribbon perfectly around a dielectric nanowire. Several experiments [14–16] showed that the graphene layer can tightly coat the nanowire due to van der Waals force. Graphene is modeled as a 0.5-nm-thick anisotropic layer [19] with in-plane effective permittivity 2.5 iσ∕ε0 ωt and surfacenormal effective permittivity 2.5. Conductivity σ could be obtained by the Kubo formula [10], which depends on temperature T, Fermi level E f , and scattering time τ
Fig. 1. (a) Schematic of GNW. (b) Mode patterns of the first four order modes of GNW with a radius of 200 nm at the wavelength of 5 μm. © 2014 Optical Society of America
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(τ μE f ∕eV 2f ); additional loss caused by optical phonon is neglected. Carrier mobility μ 10; 000 cm2 ∕V · s, and V f 106 m∕s). In this Letter, GNW is surrounded by air, and we set T 300 K, E f 0.5 eV, and relative permittivity of nanowire εNW 2.1 unless otherwise stated. Due to the azimuthal symmetry of GNW as shown in Fig. 1(a), the electric field of eigenmode could be ex⃗ ⃗r; t Er ⃗ pressed as E expimφ expikz z exp−iωt in cylindrical coordinates. Here, z represents the propa⃗ gation direction. Er is in the form of modified Bessel function I m r when r < R and K m r when r > R. expimφ denotes the standing-wave-like azimuthal field distribution m 0; 1; 2…. Figure 1(b) shows the mode patterns of GNW at the wavelength λ 5 μm. We noticed that mode order is characterized by the mode number of the electromagnetic field along the circular graphene layer. The mth order mode has 2m field maxima, and every high order mode m > 0 is double degenerate. The effective mode index is N eff kz ∕k0 , and k0 ω∕c is the free space wavevector. Figure 2 shows the dispersion relation and propagation length of GNW with a radius of 100 nm. Effective mode indices decrease monotonically as wavelength increases. The fundamental mode m p 0 is cutoff-free, and N eff is asymptotically approaching εNW λ → ∞. (Note: This region is not included in Fig.p 2.) The high order modes m > 0 cutoff when N eff < εNW . Propagation length is defined as 1∕2 Imβ, and the high order mode shows a maximum at a certain wavelength, which originates from the balancing between mode confinement and low group velocity [9]. In the short wavelength region, energy is tightly confined to the graphene layer, leading to a large loss of GSP wave and a consequent short propagation length, but in the long wavelength region near the modal cutoff the group velocity decreases, resulting in a large accumulated loss. Multi-mode propagation always leads to signal fading and unwanted mode conversion, and the fundamental mode of GNW has nearly the largest propagation length among all modes, as can be seen in Fig. 2. Therefore, it is preferred to realize single-mode propagation in GNW.
The mode number can be evaluated as follows, similar to [10]. We first consider GSP mode on an infinite flat GS placed on the XOY plane (the GS system is discussed in Cartesian coordinates, and the z axis is normal to the GS) between two dielectric half spaces with relative permittivities of εr1 z > 0 and εr2 z < 0, respectively. The ˆ GSP wavevector could be written as k⃗ kx xˆ ky y kz zˆ , and at a given frequency the three wavevector components have the relation ω∕c2 εri k20 εri k2x k2y k2z . The GS-normal wavevector kz is an imaginary number for the GSP mode denoting the field decaying as expikz z in the away-from-graphene direction. The in-plane wavevector reads [6] kspp
q ε εr2 2iω : k2x k2y ε0 r1 σ 2
For GSP modes in GNW, we assume that rolling up graphene will not affect the GS-normal wavevector because of the tight confinement of the electromagnetic field as shown in Fig. 1(b); therefore, the radial wavevector kr of the GSP mode in GNW equals kz of that on the GS system under the same graphene parameters and electromagnetic frequency. Therefore, in GNW we have k2z jGNW k2φ ω∕c2 εri − k2r ω∕c2 εri − k2z jGS k2spp . εr1 and εr2 are the relative permittivities of nanowire and its sur2 2 2 rounding material (such p as air). Hence, kφ kspp − kz < k2spp − k20 εr1 (kz > k0 εr1 for the GSP mode). kφ is the azimuthal wave number and for mth order mode kφ m∕R. If the highest order mode GNW supports is mth, then m satisfies m
0 is λc;m Re
Fig. 2. Dispersion diagram (blue curves) and propagation length (red dashed curves) of GSP modes in GNW (R 100 nm). The inset shows mode indices ReN eff of the first four order modes as a function of the GNW radius at λ 10 μm.
(1)
2πR p εGSP − εNW ; m
(4)
in which εGSP iε0 1 εNW c∕σ2 . In order to verify the mode number and modal cutoff wavelength formula derived above, numerical simulation is performed to evaluate the mode number of GNW based on FEM. Figure 3(a) shows mode number as a function of GNW radius and wavelength. The boundary between m − 1th and mth mode regions represents the cutoff wavelength of mth order mode under different GNW radii. The cutoff wavelengths as a function of GNW radius obtained by Eq. (4) are also plotted in Fig. 3(a) as white-dashed curves. We noticed that the analytic result shows very good agreement with the FEM result. The single-mode region (labeled 0) lies in the lowerright of Fig. 3(a). With the radius decreasing, the cutoff wavelengths of all modes decrease, and single-mode operation starts at a shorter wavelength. Figure 3(b) illustrates ReN eff of the fundamental mode as a function of wavelength and radius. ReN eff is nearly
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Fig. 4. (a) Permittivity-dependent cutoff wavelengths of high order modes (m 1; 2; 3, R 100 nm). Dotted lines are obtained by FEM, and solid lines are calculated by Eq. (4). (b) Normalized electric field of fundamental mode along the yellow cut-line in the upper figure. Relative permittivity of nanowire varies from 2 to 5. λ 7 μm, and R 100 nm.
Fig. 3. (a) Regions with single-mode and multi-mode operation. White dashed curves are calculated by Eq. (4). The highest mode order supported in GNW is labeled. (b) Mode index ReN eff of fundamental mode as a function of wavelength and radius. The light blue region on the bottom shows the single-mode region.
120 at λ 3 μm, and drops quickly as the wavelength increases. In the single-mode region in Fig. 3(b), the fundamental mode has a ReN eff ∼ 20, which is a large value compared with conventional silver nanowire SP waveguide [20]. The relative permittivity of nanowire εNW influences the GNW performance. We present the permittivitydependent cutoff wavelength of high order modes and the normalized electric field of the fundamental mode in Fig. 4. The solid lines and dotted lines in Fig. 4(a) show the cutoff wavelengths under different εNW obtained by Eq. (4) and FEM, respectively. Good agreement between the two methods further confirms the accuracy of the analytic model. Cutoff wavelength of high order mode increases as εNW increases. Figure 4(b) depicts the normalized electric field of the fundamental mode under different permittivities of nanowire along the yellow cutline. We noticed that nanowire with higher permittivity shows better mode confinement to the graphene layer. According to Eq. (4), cutoff wavelength λc;m is also related to GNW radius R and graphene conductivity σ. Graphene conductivity σ could beptuned by changing E f . In monolayer graphene E f ℏvf nc π [21], nc is carrier density. In recent experiments, carrier density reaches as high as 1014 cm−2 [22,23], equaling E f 1.17 eV. Fermi
level-dependent performance of GNW is presented in Fig. 5. Figure 5(a) shows the cutoff wavelength as a function of radius ranging from 20 to 150 nm. Each curve stands for a specific Fermi level, varying from 0.3 to 1.1 eV by a 0.2 eV step. To the right of each line is singlemode region under the corresponding Fermi level. An increasing Fermi level results in a blue-shift of cutoff wavelength and a larger single-mode operation region. Propagation properties of the GSP modes in GNW could be adjusted by tuning the graphene Fermi level as well. We investigate the mode behavior of an R 100 nm GNW at λ 10 μm, with the Fermi level tuned from 0.3 to 1.1 eV. As depicted in Fig. 5(b), ReN eff decreases as E f increases. m 1 and m 2 order modes cut off at 0.83 and 0.42 eV, respectively. Modal cutoff could be also observed in Figs. 5(c) and 5(d). Singlemode operation is achieved when E f is larger than 0.83 eV. This observation reveals the possibility of realizing single-mode operation by simply tuning the Fermi level. As shown in Fig. 5(c), the propagation length of the fundamental mode is longer than those of high order modes and could reach as high as 15.9 μm in the singlemode region. The m 1 order mode and m 2 order
Fig. 5. (a) Dependence of cutoff wavelength of m 1 order mode on radius and Ef . (b) Mode indices ReN eff , (c) propagation length, and (d) normalized effective mode areas as a function of Fermi level λ 10 μm and R 100 nm.
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mode have nearly zero propagation lengths near the modal R cutoff. REffective mode area Aeff is defined as jEj2 ds2 ∕ jEj4 ds and normalized by λ2 . In Fig. 5(d), Aeff monotonically increases with the increase of Ef . It is worth noting that the fundamental mode has the largest Aeff , and the m 1 order mode has the smallest. Aeff of any other order mode increases and approaches that of the fundamental mode as the mode order increases, as can be seen in Fig. 5(d). In conclusion, we proposed a single-mode GNW SP waveguide. An analytic modal cutoff wavelength formula was presented and could be applied to calculate cutoff wavelength of each order mode easily and accurately, which could offer valuable references for applications based on GNW. A single-mode operation region has been identified in wavelength-radius space. The single-mode region, transmission property, and mode confinement could be adjusted by tuning the Fermi level of the graphene layer, which renders GNW potential applications in integrated optical devices and plasmon sensors. This work is supported in part by the Major State Basic Research Development Program of China (Grant No. 2010CB328206), the National Natural Science Foundation of China (NSFC) (Grant Nos. 61178008 and 61275092), and the Fundamental Research Funds for the Central Universities, China. References 1. D. K. Gramotnev and S. I. Bozhevolnyi, Nat. Photonics 4, 83 (2010). 2. J. Dionne, L. Sweatlock, H. Atwater, and A. Polman, Phys. Rev. B 72, 075405 (2005). 3. E. Moreno, F. J. Garcia-Vidal, S. G. Rodrigo, L. MartinMoreno, and S. I. Bozhevolnyi, Opt. Lett. 31, 3447 (2006).
4. D. Chang, A. Sørensen, P. Hemmer, and M. Lukin, Phys. Rev. B 76, 035420 (2007). 5. R. F. Oulton, V. J. Sorger, D. A. Genov, D. F. P. Pile, and X. Zhang, Nat. Photonics 2, 496 (2008). 6. M. Jablan, H. Buljan, and M. Soljačić, Phys. Rev. B 80, 245435 (2009). 7. J. Christensen, A. Manjavacas, S. Thongrattanasiri, F. H. Koppens, and F. J. de Abajo, ACS Nano 6, 431 (2012). 8. A. Vakil and N. Engheta, Science 332, 1291 (2011). 9. A. Y. Nikitin, F. Guinea, F. J. García-Vidal, and L. MartínMoreno, Phys. Rev. B 84, 161407 (2011). 10. Y. Francescato, V. Giannini, and S. A. Maier, New J. Phys. 15, 063020 (2013). 11. S. He, X. Zhang, and Y. He, Opt. Express 21, 30664 (2013). 12. P. Liu, X. Zhang, Z. Ma, W. Cai, L. Wang, and J. Xu, Opt. Express 21, 32432 (2013). 13. Q. Bao, H. Zhang, B. Wang, Z. Ni, C. H. Y. X. Lim, Y. Wang, D. Y. Tang, and K. P. Loh, Nat. Photonics 5, 411 (2011). 14. Y. Wu, B. Yao, A. Zhang, Y. Rao, Z. Wang, Y. Cheng, Y. Gong, W. Zhang, Y. Chen, and K. S. Chiang, Opt. Lett. 39, 1235 (2014). 15. H. Xiaoying, Z. Xiangchao, Z. Hao, and X. Min, IEEE J. Sel. Top. Quantum Electron. 20, 55 (2014). 16. X. He, Z.-b. Liu, D. N. Wang, M. Yang, T. Y. Hu, and J.-G. Tian, IEEE Photon. Technol. Lett. 25, 1392 (2013). 17. J. Zhao, X. Liu, W. Qiu, Y. Ma, Y. Huang, J.-X. Wang, K. Qiang, and J.-Q. Pan, Opt. Express 22, 5754 (2014). 18. J. Kotakoski, D. Santos-Cottin, and A. V. Krasheninnikov, ACS Nano 6, 671 (2012). 19. W. Gao, J. Shu, C. Qiu, and Q. Xu, ACS Nano 6, 7806 (2012). 20. Q. Li and M. Qiu, Opt. Express 21, 8587 (2013). 21. C. H. Gan, H. S. Chu, and E. P. Li, Phys. Rev. B 85, 125431 (2012). 22. D. K. Efetov and P. Kim, Phys. Rev. Lett. 105, 256805 (2010). 23. J. Ye, M. F. Craciun, M. Koshino, S. Russo, S. Inoue, H. Yuan, H. Shimotani, A. F. Morpurgo, and Y. Iwasa, Proc. Natl. Acad. Sci. USA 108, 13002 (2011).
5909
Single-mode graphene-coated nanowire plasmonic waveguide Yixiao Gao,1,2 Guobin Ren,1,2,* Bofeng Zhu,1,2 Jing Wang,3 and Shuisheng Jian1,2 1
Key Laboratory of All Optical Network & Advanced Telecommunication Network of EMC, Beijing Jiaotong University, Beijing 100044, China 2 Institute of Lightwave Technology, Beijing Jiaotong University, Beijing 100044, China 3
Science and Technology on Optical Radiation Laboratory, Beijing 100854, China *Corresponding author: [email protected]
Received June 26, 2014; revised September 9, 2014; accepted September 10, 2014; posted September 10, 2014 (Doc. ID 214812); published October 10, 2014 We propose in this Letter a single-mode graphene-coated nanowire surface plasmon waveguide. The single-mode condition and modal cutoff wavelength of high order modes are derived from an analytic model and confirmed by numerical simulation. The mode number diagram of the proposed waveguide in the wavelength-radius space is also demonstrated. By changing the Fermi level of graphene, the performance of the proposed waveguide could be tuned flexibly, offering potential application in tunable nanophotonic devices. © 2014 Optical Society of America OCIS codes: (240.6680) Surface plasmons; (250.5403) Plasmonics; (230.7370) Waveguides. http://dx.doi.org/10.1364/OL.39.005909
Surface plasmons (SPs) are thought to be the only viable way to confine and control light in the subwavelength scale required by the development of nanophotonic technology [1]. The investigations of SP waveguiding focus on how to achieve extreme localization of the electromagnetic field as well as long propagation length. Various types of SP waveguides were proposed, such as thin metal film [2], metal groove/wedge [3], metal nanowire [4], and hybrid plasmonic waveguides [5]. However, noble metals used in these waveguides only support weakly confined SP waves in the mid-infrared to THz frequency band, though this frequency range has important applications in biology, communication, and spectroscopy. Graphene, a newly emerged 2D material consisting of carbon atoms arranged in a hexagonal lattice, is a promising candidate for SP waveguiding in the mid-infrared to THz frequency range [6]. Compared with noble metals, graphene has three major advantages [7]—low loss, extreme mode confinement, and tunable carrier density by electrical gating or chemical doping—which further lead to tunable electromagnetic properties. Graphene surface plasmon (GSP) waveguiding has been investigated intensively, such as GSP mode on a single graphene sheet (GS) [6,8], graphene nanoribbon (GNR) [7,9–11], and graphene groove/wedge waveguide [12]. Graphene-microfiber/ nanowire combined systems have been proposed for various applications [13–17], but GSP modes in the graphene-coated nanowire (GNW) waveguide, which can be seen as an analogy for metal nanowire with azimuthal symmetry, have not been characterized yet. Single-mode operation is highly preferred for most applications due to its high beam quality, lowest transmission loss, and low group velocity dispersion. For GNR, the fundamental mode is even-parity edge mode, whose field is tightly confined near the ribbon edge [9], and single-mode operation occurs with ribbon width shrinking [11]. Transmission properties of GNR are strongly influenced by the morphology of the GNR edge. However, it is still a challenge to control the edge shape with the desired atomic arrangement [18]. Therefore, single-mode propagation in GNR might be strongly influenced by edge non-uniformity. For the graphene 0146-9592/14/205909-04$15.00/0
groove/wedge waveguide, the fundamental mode has a perfect electric conductor symmetry [12], but how to achieve single-mode operation is not specified. It is still significant to realize the single-mode operation in graphene-based SP waveguides. In this Letter, we investigate the SP modes in a GNW. First, the dispersion relation and propagation length are demonstrated. Next, the number of supported modes in the GNW and single-mode condition are derived from an analytic model, and a modal cutoff wavelength formula is presented. In order to verify the proposed formula, the mode number in wavelength-radius space is calculated by commercial software (COMSOL) based on the finite element method (FEM). Finally, we analyze the tunable performance of GNW by varying the graphene Fermi level. The structure of GNW is depicted in Fig. 1(a). A dielectric nanowire with radius R is coated by a graphene layer, which could be regarded as rolling a graphene ribbon perfectly around a dielectric nanowire. Several experiments [14–16] showed that the graphene layer can tightly coat the nanowire due to van der Waals force. Graphene is modeled as a 0.5-nm-thick anisotropic layer [19] with in-plane effective permittivity 2.5 iσ∕ε0 ωt and surfacenormal effective permittivity 2.5. Conductivity σ could be obtained by the Kubo formula [10], which depends on temperature T, Fermi level E f , and scattering time τ
Fig. 1. (a) Schematic of GNW. (b) Mode patterns of the first four order modes of GNW with a radius of 200 nm at the wavelength of 5 μm. © 2014 Optical Society of America
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(τ μE f ∕eV 2f ); additional loss caused by optical phonon is neglected. Carrier mobility μ 10; 000 cm2 ∕V · s, and V f 106 m∕s). In this Letter, GNW is surrounded by air, and we set T 300 K, E f 0.5 eV, and relative permittivity of nanowire εNW 2.1 unless otherwise stated. Due to the azimuthal symmetry of GNW as shown in Fig. 1(a), the electric field of eigenmode could be ex⃗ ⃗r; t Er ⃗ pressed as E expimφ expikz z exp−iωt in cylindrical coordinates. Here, z represents the propa⃗ gation direction. Er is in the form of modified Bessel function I m r when r < R and K m r when r > R. expimφ denotes the standing-wave-like azimuthal field distribution m 0; 1; 2…. Figure 1(b) shows the mode patterns of GNW at the wavelength λ 5 μm. We noticed that mode order is characterized by the mode number of the electromagnetic field along the circular graphene layer. The mth order mode has 2m field maxima, and every high order mode m > 0 is double degenerate. The effective mode index is N eff kz ∕k0 , and k0 ω∕c is the free space wavevector. Figure 2 shows the dispersion relation and propagation length of GNW with a radius of 100 nm. Effective mode indices decrease monotonically as wavelength increases. The fundamental mode m p 0 is cutoff-free, and N eff is asymptotically approaching εNW λ → ∞. (Note: This region is not included in Fig.p 2.) The high order modes m > 0 cutoff when N eff < εNW . Propagation length is defined as 1∕2 Imβ, and the high order mode shows a maximum at a certain wavelength, which originates from the balancing between mode confinement and low group velocity [9]. In the short wavelength region, energy is tightly confined to the graphene layer, leading to a large loss of GSP wave and a consequent short propagation length, but in the long wavelength region near the modal cutoff the group velocity decreases, resulting in a large accumulated loss. Multi-mode propagation always leads to signal fading and unwanted mode conversion, and the fundamental mode of GNW has nearly the largest propagation length among all modes, as can be seen in Fig. 2. Therefore, it is preferred to realize single-mode propagation in GNW.
The mode number can be evaluated as follows, similar to [10]. We first consider GSP mode on an infinite flat GS placed on the XOY plane (the GS system is discussed in Cartesian coordinates, and the z axis is normal to the GS) between two dielectric half spaces with relative permittivities of εr1 z > 0 and εr2 z < 0, respectively. The ˆ GSP wavevector could be written as k⃗ kx xˆ ky y kz zˆ , and at a given frequency the three wavevector components have the relation ω∕c2 εri k20 εri k2x k2y k2z . The GS-normal wavevector kz is an imaginary number for the GSP mode denoting the field decaying as expikz z in the away-from-graphene direction. The in-plane wavevector reads [6] kspp
q ε εr2 2iω : k2x k2y ε0 r1 σ 2
For GSP modes in GNW, we assume that rolling up graphene will not affect the GS-normal wavevector because of the tight confinement of the electromagnetic field as shown in Fig. 1(b); therefore, the radial wavevector kr of the GSP mode in GNW equals kz of that on the GS system under the same graphene parameters and electromagnetic frequency. Therefore, in GNW we have k2z jGNW k2φ ω∕c2 εri − k2r ω∕c2 εri − k2z jGS k2spp . εr1 and εr2 are the relative permittivities of nanowire and its sur2 2 2 rounding material (such p as air). Hence, kφ kspp − kz < k2spp − k20 εr1 (kz > k0 εr1 for the GSP mode). kφ is the azimuthal wave number and for mth order mode kφ m∕R. If the highest order mode GNW supports is mth, then m satisfies m
0 is λc;m Re
Fig. 2. Dispersion diagram (blue curves) and propagation length (red dashed curves) of GSP modes in GNW (R 100 nm). The inset shows mode indices ReN eff of the first four order modes as a function of the GNW radius at λ 10 μm.
(1)
2πR p εGSP − εNW ; m
(4)
in which εGSP iε0 1 εNW c∕σ2 . In order to verify the mode number and modal cutoff wavelength formula derived above, numerical simulation is performed to evaluate the mode number of GNW based on FEM. Figure 3(a) shows mode number as a function of GNW radius and wavelength. The boundary between m − 1th and mth mode regions represents the cutoff wavelength of mth order mode under different GNW radii. The cutoff wavelengths as a function of GNW radius obtained by Eq. (4) are also plotted in Fig. 3(a) as white-dashed curves. We noticed that the analytic result shows very good agreement with the FEM result. The single-mode region (labeled 0) lies in the lowerright of Fig. 3(a). With the radius decreasing, the cutoff wavelengths of all modes decrease, and single-mode operation starts at a shorter wavelength. Figure 3(b) illustrates ReN eff of the fundamental mode as a function of wavelength and radius. ReN eff is nearly
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Fig. 4. (a) Permittivity-dependent cutoff wavelengths of high order modes (m 1; 2; 3, R 100 nm). Dotted lines are obtained by FEM, and solid lines are calculated by Eq. (4). (b) Normalized electric field of fundamental mode along the yellow cut-line in the upper figure. Relative permittivity of nanowire varies from 2 to 5. λ 7 μm, and R 100 nm.
Fig. 3. (a) Regions with single-mode and multi-mode operation. White dashed curves are calculated by Eq. (4). The highest mode order supported in GNW is labeled. (b) Mode index ReN eff of fundamental mode as a function of wavelength and radius. The light blue region on the bottom shows the single-mode region.
120 at λ 3 μm, and drops quickly as the wavelength increases. In the single-mode region in Fig. 3(b), the fundamental mode has a ReN eff ∼ 20, which is a large value compared with conventional silver nanowire SP waveguide [20]. The relative permittivity of nanowire εNW influences the GNW performance. We present the permittivitydependent cutoff wavelength of high order modes and the normalized electric field of the fundamental mode in Fig. 4. The solid lines and dotted lines in Fig. 4(a) show the cutoff wavelengths under different εNW obtained by Eq. (4) and FEM, respectively. Good agreement between the two methods further confirms the accuracy of the analytic model. Cutoff wavelength of high order mode increases as εNW increases. Figure 4(b) depicts the normalized electric field of the fundamental mode under different permittivities of nanowire along the yellow cutline. We noticed that nanowire with higher permittivity shows better mode confinement to the graphene layer. According to Eq. (4), cutoff wavelength λc;m is also related to GNW radius R and graphene conductivity σ. Graphene conductivity σ could beptuned by changing E f . In monolayer graphene E f ℏvf nc π [21], nc is carrier density. In recent experiments, carrier density reaches as high as 1014 cm−2 [22,23], equaling E f 1.17 eV. Fermi
level-dependent performance of GNW is presented in Fig. 5. Figure 5(a) shows the cutoff wavelength as a function of radius ranging from 20 to 150 nm. Each curve stands for a specific Fermi level, varying from 0.3 to 1.1 eV by a 0.2 eV step. To the right of each line is singlemode region under the corresponding Fermi level. An increasing Fermi level results in a blue-shift of cutoff wavelength and a larger single-mode operation region. Propagation properties of the GSP modes in GNW could be adjusted by tuning the graphene Fermi level as well. We investigate the mode behavior of an R 100 nm GNW at λ 10 μm, with the Fermi level tuned from 0.3 to 1.1 eV. As depicted in Fig. 5(b), ReN eff decreases as E f increases. m 1 and m 2 order modes cut off at 0.83 and 0.42 eV, respectively. Modal cutoff could be also observed in Figs. 5(c) and 5(d). Singlemode operation is achieved when E f is larger than 0.83 eV. This observation reveals the possibility of realizing single-mode operation by simply tuning the Fermi level. As shown in Fig. 5(c), the propagation length of the fundamental mode is longer than those of high order modes and could reach as high as 15.9 μm in the singlemode region. The m 1 order mode and m 2 order
Fig. 5. (a) Dependence of cutoff wavelength of m 1 order mode on radius and Ef . (b) Mode indices ReN eff , (c) propagation length, and (d) normalized effective mode areas as a function of Fermi level λ 10 μm and R 100 nm.
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mode have nearly zero propagation lengths near the modal R cutoff. REffective mode area Aeff is defined as jEj2 ds2 ∕ jEj4 ds and normalized by λ2 . In Fig. 5(d), Aeff monotonically increases with the increase of Ef . It is worth noting that the fundamental mode has the largest Aeff , and the m 1 order mode has the smallest. Aeff of any other order mode increases and approaches that of the fundamental mode as the mode order increases, as can be seen in Fig. 5(d). In conclusion, we proposed a single-mode GNW SP waveguide. An analytic modal cutoff wavelength formula was presented and could be applied to calculate cutoff wavelength of each order mode easily and accurately, which could offer valuable references for applications based on GNW. A single-mode operation region has been identified in wavelength-radius space. The single-mode region, transmission property, and mode confinement could be adjusted by tuning the Fermi level of the graphene layer, which renders GNW potential applications in integrated optical devices and plasmon sensors. This work is supported in part by the Major State Basic Research Development Program of China (Grant No. 2010CB328206), the National Natural Science Foundation of China (NSFC) (Grant Nos. 61178008 and 61275092), and the Fundamental Research Funds for the Central Universities, China. References 1. D. K. Gramotnev and S. I. Bozhevolnyi, Nat. Photonics 4, 83 (2010). 2. J. Dionne, L. Sweatlock, H. Atwater, and A. Polman, Phys. Rev. B 72, 075405 (2005). 3. E. Moreno, F. J. Garcia-Vidal, S. G. Rodrigo, L. MartinMoreno, and S. I. Bozhevolnyi, Opt. Lett. 31, 3447 (2006).
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