April 1, 2015 / Vol. 40, No. 7 / OPTICS LETTERS
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Toward integrated electrically controllable directional coupling based on dielectric loaded graphene plasmonic 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 2
College of Optoelectronic Science and Engineering, National University of Defense Technology, Changsha 410073, China
State Key Laboratory of High Performance Computing, National University of Defense Technology, Changsha 410073, China *Corresponding author: [email protected] Received February 12, 2015; revised March 9, 2015; accepted March 9, 2015; posted March 11, 2015 (Doc. ID 234463); published March 31, 2015 We propose and numerically analyze a mid-infrared electrically controllable plasmonic waveguide directional coupler that is composed of two parallel identical straight dielectric loaded graphene plasmonic waveguide and S-shaped waveguide bends. By varying the Fermi energy level of the graphene sheet, the maximum power coupled from the input waveguide to the cross-waveguide and the corresponding coupling length could be effectively tuned. Under different Fermi energy level, this directional coupler could serve as an electrically controlled optical switch or a 3-dB splitter around the wavelength of 10.5 μm. Moreover, the size of the entire device is really in subwavelength scale making it very facilitative for high density integration. © 2015 Optical Society of America OCIS codes: (130.3120) Integrated optics devices; (250.5403) Plasmonics; (250.6715) Switching. http://dx.doi.org/10.1364/OL.40.001603
Surface plasmon polaritions (SPPs) offer a promising way to realize chip-scale integrated photonic circuits because of their ability of confining and controlling electromagnetic waves at subwavelength scale [1]. Active plasmonic devices such as switches and modulators are essential elements to realize integrated photonic circuits. In the visible to near-infrared frequencies, noble metal combined with active dielectric medium is one of the general methods to realize active plasmonic devices [2–6]. The (SPPs) modes at the metal-dielectric interface can be controlled by electro-, magneto-, and thermo-optic effects (depending on the active dielectric properties and electrode configuration) [7,8]. However in the midinfrared spectral region where SPs also have a broad range of potential applications [9,10], the use of noble metals is limited due to the poor confinement of SPs. Graphene, a newly emerged two-dimensional (2D) atomically thin material, is believed to be a promising plasmonic material from the terahertz to the mid-infrared spectral region [11,12]. Recently, the excitation, propagation and tunability of doped graphene surface plasmon polaritions (GSPPs) in mid-infrared frequencies have been theoretically studied [13–17] and experimentally demonstrated [18,19]. Compared with SPPs on noble metals, GSPPs could confine EM field at an extremely subwavelength scale in the mid-infrared spectral region. Moreover, GSPPs could be actively tuned by electrostatically gating or chemical doping, which makes graphene an excellent platform for active plasmonic devices. Active plasmonic waveguide devices based on vertical coupling between graphene sheets [20] or graphene nanoribbons have been theoretically proposed [21]. However the vertical coupling between graphene sheets is not very suitable for in-plane integration, as the mode on graphene sheet has no in-plane confinement. The edge shape of graphene nanoribbons could strongly influence the propagation properties of GSPP modes, and it is still a challenge to control the edge shape with desired atomic arrangement [22]. Recently, we have proposed the 0146-9592/15/071603-04$15.00/0
dielectric-loaded graphene plasmonic waveguide (DLGPW) [23], which is similar to dielectric-loaded metal plasmon waveguide in near-infrared frequencies [24,25]. The DLGPW does not need to pattern the graphene and is easy to fabricated. One peculiarity of the DLGPW is that the mode confinement and transmission loss could be tuned by varying the Fermi energy level of graphene sheet. Inspired by this finding, we think that the coupling between two parallel straight DLGPWs could be dynamically tuned, leading to electrically tunable directional coupler (DC). Based on this DC, active mid-infrared plasmonic devices such as switches could be realized. The schematic of our proposed electrically tunable DC is shown in Fig. 1(a). Two parallel identical straight DLGPWs with edge-to-edge separation d constitute the coupling region. The coupling region with a length of L is connected to the output waveguides (Ps and Pc) by two S-shaped waveguide bends. SiO2 dielectric layer with a thickness of hs is used to separate the graphene sheet and the back-gated metal film. By applying a voltage between the graphene sheet and the back-gated metal film, the carrier concentration and thus the Fermi energy level of garaphene can be electrically tuned. Then the coupling between the two DLGPWs could be tuned. Transmission characteristics of the DC could be analyzed by the well-established coupled-mode theory (CMT) [26], in which electromagnetic field distribution in the DC can be expressed as the superposition of the symmetrical and anti-symmetrical supermodes. For a fixed coupling length, the power coupled from one waveguide to the other is determined by the effective mode index of symmetrical (ns n0s − in00s ) and anti-symmetrical (na n0a − in00a ) supermodes. Here, we use finite element method-based software (COMSOL Multiphysics) to solve the supermodes. In the two dimension (2D) simulations, the width and thickness of the DLGPW are both 100 nm, which ensure singlemode operation from the wavelength of 9 to 11 μm. The edge-to-edge separation is 150 nm, unless otherwise © 2015 Optical Society of America
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curves for hs 200 nm coincide with the ones for halfspace SiO2 layer (without metal), indicating that the back-gated metal film does not affect the modes of the DLGPW, when the separation between the graphene sheet and the metal film is larger than 200 nm. This originates from the vertically tight confinement of the guide mode in the DLGPW. As the mode confinement increases when the Fermi energy level decreases [as shown in the insets of Fig. 2(a)], this conclusion is also valid for Fermi energy level below 0.9 eV. So we fixed the thickness of SiO2 layer to be 200 nm. In general, complete transfer of power from one waveguide to the other is not possible for coupled lossy waveguides, since loss is always present in the transfer process. When jn00s − n00a j∕2 ≪ jn0s − n0a j∕2, which is true for the proposed DC, the maximal power coupled from one waveguide to the other is approximately equal to [31] Fig. 1. (a) Schematic of the electrically tunable directional coupler based on dielectric-loaded graphene plasmonic waveguide. The inset shows the xy-plane cross-section of the directional coupler. (b) Real part of the effective mode index of symmetrical and anti-symmetrical supermodes for different thickness of SiO2 dielectric layer. The insets show the ydirection electric filed component of the supermode. The width and thickness of the DLGPW are both 100 nm. The edge-to-edge separation d is 150 nm, and the Fermi energy of graphene is 0.9 eV.
state. The refractive index of SiO2 was assumed to be 1.96. The graphene sheet was modeled as an anisotropic layer with a thickness of t 0.5 nm, and is uniform meshed with a resolution of 0.125 nm along the thickness direction and 5 nm along the surface direction. The outof-plane permittivity was 2.5, and in-plane permittivity was ε∥ ω 2.5 iσω∕ωε0 t, where the optical conductivity of graphene (σω) was derived using the random-phase approximation in the local limit [27]: i2e2 kB T EF σω 2 In 2 cosh 2kB T πℏ ω iτ−1 2 e 1 1 ℏω − 2E F arctan 4ℏ 2 π 2kB T i ℏω 2E F 2 In ; − 2π ℏω − 2E F 2 4kB T2
(1)
where T 300 K is the temperature, kB is the Boltzmann constant, ω is the frequency, E F is the Fermi energy, and τ μE F ∕eV 2F is the carrier relaxation lifetime (μ is the carrier mobility in graphene and V F 106 m∕s is the Fermi velocity). In recent experiments, the Fermi energy level has reached as high as 1.17 eV [28]. The carrier mobility ranges from ∼1000 cm2 ∕V · s in chemical vapor deposition (CVD) grown graphene [29] to 230000 cm2 ∕V · s in suspended exfoliated graphene [30]. Here, we use a Fermi energy level of 0.9 eV and carrier mobility of 100000 cm2 ∕V · s, unless otherwise stated. The mode dispersion relations (real part of the effective mode index) of the supermodes for different thicknesses of SiO2 layers are shown in Fig. 1(b). From the wavelength of 9 to 11 μm, the mode dispersion
Pcmax ≃ exp−2r arctanr −1 ∕1 r 2 .
(2)
r 2Lc ∕πLp , the corresponding coupling length is equal to Lm 2Lc arctanπLp ∕2Lc ∕π;
(3)
where Lc λ0 ∕2n0s − n0a is the beating length of the two supermodes and also the coupling length for coupled lossless waveguides. Lp λ0 ∕πn00s n00a is the mean propagation length. Suppose the input power Pin 1, then the maximum power coupled to Pc at different Fermi energy level is shown in Fig. 2(a), and the corresponding coupling length Lm is shown in Fig. 2(b). As the wavelength decreases, the coupling strength and the maximum power transfer decrease, which originate from the increment of confinement and propagation loss of the mode in the DLGPW. When the coupling is strong and the loss is slight (EF 0.9 eV), that is Lc is much smaller than the mean propagation length Lp , the maximum power transfer approaches 1, and the corresponding coupling length Lm approaches Lc , which is similar to the lossless case. In the opposite limit, when the couple is weak and the loss is huge (E F 0.5 eV for the wavelength under 10.5 μm), that is Lp is much smaller than Lc [the inset of Fig. 1(b)], the maximum power transfer approaches 0, and the corresponding coupling length Lm approaches Lp . So, there is a reverse tendency of the coupling length at E F 0.5 eV when the wavelength is under
Fig. 2. Maximum power transferred from the input DLGPW to the coupled DLGPW (a) and the corresponding coupling length (b) at different Fermi energy level. The geometrical parameters of the DC are the same as Fig. 1.
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Fig. 3. Dependence of the maximum power transfer on the edge-to-edge separation d between the two parallel DLGPWs. The geometrical parameters of the DLGPW are the same as Fig. 1, and the Fermi energy level is 0.9 eV.
10.5 μm, and at the same time, the maximum power transfer approaches 0. Figure 2 clearly shows that by varying the Fermi energy level of graphene sheet, the coupling between the two parallel DLGPWs could be effectively tuned, which enables electrically controllable functional devices, such as optical switch. Dependence of the maximum power transfer on the edge-to-edge separation between the two parallel DLGPW is shown in Fig. 3. When the waveguides are separated by d 900 nm, the maximum power transfer approaches 0. This means at this separation, the coupling between the waveguides is so weak that propagation loss dominates the coupling process. In other words, at this separation, the waveguides are practically not coupled. It should be noted that though this conclusion is drawn at the Fermi energy level of 0.9 eV, it is still true for Fermi energy level lower than 0.9 eV due to higher Fermi energy level leads to higher maximum power transfer [Fig. 2(a)]. So the edge-to-edge separation of the two output waveguides should be larger than 900 nm to ensure zero crosstalk between the output waveguides. In our design, the S-shaped bend waveguide is constructed by two connected circular arcs waveguide with opposite curvature. The radius and radian of the circular arcs is 6 μm and 0.3, respectively. The corresponding edge-to edge separation of the output waveguides is around 1.1 μm, which is large enough. Then we use 3D simulation to study the transmission of the proposed DC with a coupling length L 2.6 μm [Fig. 4(a)]. The fundamental mode of the DLGPW is launched into the input port. The power transferred to the cross-waveguide (Pc) is wavelength dependent and could be tuned by varying Fermi energy level. When E F 0.7 eV, power effectively coupled to the cross-waveguide around the wavelength of 10.5 μm. However, the transmission rate is lower, and the coupling length is shorter than the results calculated by CMT (Fig. 2) due to the additional mode coupling and transmission loss in the S-shaped bends. When E F 0.5 eV, Pc approaches 0 around the wavelength of 10.5 μm. These results enable an electrically controllable optical switch, with E F 0.7 eV as switch-on state and E F 0.5 eV as switch-off state. The switch on-off extinction ratio is larger than 16 dB. If lower Fermi energy level is used as the switch-off state, higher extinction ratio could be realized, as lower Fremi energy level leads to smaller maximum power transfer [Fig. 2(a)]. Moreover,
Fig. 4. (a) Transmission rate of the proposed DC at different Fermi energy level with a coupling length of 2.6 μm. (b) power flow distributions at different Fermi energy level for the wavelength of 10.5 μm.
if E F 0.82 eV, without changing the structure, Pc is equal to Ps around 10.5 μm indicating that this DC could serve as a 3-dB splitter. Figure 4(b) shows the power flow distributions at different Fermi energy level for the wavelength of 10.5 μm, which clearly shows the function of switch and splitter. It should be noted that the results in Fig. 4 are acquired under the carrier mobility of 100000 cm2 ∕V · s. The decrease of carrier mobility will bring more insertion loss; however, the function could still be maintained. So this proposed DC could serve as electrically controllable multifunctional devices with a sub-wavelength size of 6 μm × 3 μm. As for practical applications, the required bias-voltage to raise the graphene Fermi energy level from 0 to 0.82 eV is estimated up to about 500 V when the SiO2 insulator layer is 200 nm thick. However, the bias-voltage can be reduced by using transparency electrode [32] in place of metal electrode or using insulator with higher static permittivity (for example Al2 O3 ) in place of SiO2 , or chemical-doping [33] combined with electric-doping. In addition, optical resist can be used as the top dielectric ridge to simplify the fabrication procedure by standard processes of electron beam lithography and lift-off. In summary, we have studied the coupling characters of a directional coupler (DC) consisting of two parallel identical straight DLGPWs in the mid-infrared regime. The maximal power coupled from the input waveguide to the cross-waveguide and the corresponding coupling length could be manipulated by changing the Fermi energy level of the graphene sheet, which enables electrically controlled functional devices. Based on this DC, electrically controlled optical switch operating around the wavelength of 10.5 μm is proposed and simulated. The switch on-off extinction ratio is larger than 16 dB. Moreover, without changing the structure, by varying
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the Fermi energy level, the function of the DC could be changed from an optical switch to a 3-dB splitter, making this DC a multifunction device. At the same time, the size of the entire device is only around 6 μm × 3 μm. This DCbased electrically controlled device may find potential applications in high-density integrated active plasmonic circuits. 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). References 1. W. L. Barnes, A. Dereux, and T. W. Ebbesen, Nature 424, 824 (2003). 2. T. Nikolajsen, K. Leosson, and S. I. Bozhevolnyi, Appl. Phys. Lett. 85, 5833 (2004). 3. D. Kalavrouziotis, S. Papaioannou, K. Vyrsokinos, L. Markey, A. Dereux, G. Giannoulis, D. Apostolopoulos, H. Avramopoulos, and N. Pleros, IEEE Photon. Technol. Lett. 24, 1819 (2012). 4. A. Pitilakis and E. E. Kriezis, J. Lightwave Technol. 29, 2636 (2011). 5. K. Hassan, J. C. Weeber, L. Markey, A. Dereux, A. Pitilakis, O. Tsilipakos, and E. E. Kriezis, Appl. Phys. Lett. 99, 241110 (2011). 6. J. C. Weeber, K. Hassan, A. Bouhelier, G. Golas-des-Francs, J. Arocas, L. Markey, and A. Dereux, Appl. Phys. Lett. 99, 031113 (2011). 7. R. G. Hunsperger, Integrated Optics: Theory and Technology (Springer, 1995). 8. A. V. Krasavin and A. V. Zayats, Appl. Phys. Lett. 90, 211101 (2007). 9. R. Stanley, Nat. Photonics 6, 409 (2012). 10. P. Chen, Q. Gan, F. J. Bartoli, and L. Zhu, J. Opt. Soc. Am. B 27, 685 (2010). 11. M. Jablan, H. Buljan, and M. Soljačić, Phys. Rev. B 80, 245435 (2009). 12. F. H. L. Koppens, D. E. Chang, and F. J. García de Abajo, Nano Lett. 11, 3370 (2011). 13. A. Vakil and N. Engheta, Science 332, 1291 (2011). 14. J. Christensen, A. Manjavacas, S. Thongrattanasiri, F. H. Koppens, and F. J. García de Abajo, ACS Nano 6, 431 (2012).
15. E. Forati and G. W. Hanson, Appl. Phys. Lett. 103, 133104 (2013). 16. P. Liu, X. Zhang, Z. Ma, W. Cai, L. Wang, and J. Xu, Opt. Express 21, 32432 (2013). 17. Y. Gao, G. Ren, B. Zhu, H. Liu, Y. Lian, and S. Jian, Opt. Express 22, 24322 (2014). 18. 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, Nature 487, 82 (2012). 19. J. Chen, M. Badioli, P. A. 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, Nature 487, 77 (2012). 20. B. Wang, X. Zhang, X. Yuan, and J. Teng, Appl. Phys. Lett. 100, 131111 (2012). 21. K. J. A. Ooi, H. S. Chu, L. K. Ang, and P. Bai, J. Opt. Soc. Am. B 30, 3111 (2013). 22. J. Kotakoski, D. Santos-Cottin, and A. V. Krasheninnikov, ACS Nano 6, 671 (2012). 23. W. Xu, Z. H. Zhu, K. Liu, J. F. Zhang, X. D. Yuan, Q. S. Lu, and S. Q. Qin, Opt. Express 23, 5147 (2015). 24. B. Steinberger, A. Hohenau, H. Ditlbacher, A. L. Stepanov, A. Drezet, F. Aussenegg, A. Leitner, and J. Krenn, Appl. Phys. Lett. 88, 094104 (2006). 25. T. Holmgaard and S. I. Bozhevolnyi, Phys. Rev. B 75, 245405 (2007). 26. C. L. Chen, Foundations for Guided-Wave Optics (Wiley, 2007) pp. 121–126. 27. L. A. Falkovsky and S. S. Pershoguba, Phys. Rev. B. 76, 153410 (2007). 28. D. K. Efetov and P. Kim, Phys. Rev. Lett. 105, 256805 (2010). 29. 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, Nat. Nanotechnol. 6, 630 (2011). 30. K. I. Bolotin, K. J. Sikes, Z. Jiang, M. Klima, G. Fudenberg, J. Hone, P. Kim, and H. L. Stormer, Solid State Commun. 146, 351 (2008). 31. G. Veronis and S. Fan, Opt. Express 16, 2129 (2008). 32. Z. Fang, S. Thongrattanasiri, A. Schlather, Z. Liu, L. Ma, Y. Wang, P. M. Ajayan, P. Nordlander, N. J. Halas, and F. J. García de Abajo, ACS Nano 7, 2388 (2013). 33. I. Khrapach, F. Withers, T. H. Bointon, D. K. Polyushkin, W. L. Barnes, S. Russo, M. F. Craciun, and F. Monica, Adv. Mater. 24, 2844 (2012).
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Toward integrated electrically controllable directional coupling based on dielectric loaded graphene plasmonic 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 2
College of Optoelectronic Science and Engineering, National University of Defense Technology, Changsha 410073, China
State Key Laboratory of High Performance Computing, National University of Defense Technology, Changsha 410073, China *Corresponding author: [email protected] Received February 12, 2015; revised March 9, 2015; accepted March 9, 2015; posted March 11, 2015 (Doc. ID 234463); published March 31, 2015 We propose and numerically analyze a mid-infrared electrically controllable plasmonic waveguide directional coupler that is composed of two parallel identical straight dielectric loaded graphene plasmonic waveguide and S-shaped waveguide bends. By varying the Fermi energy level of the graphene sheet, the maximum power coupled from the input waveguide to the cross-waveguide and the corresponding coupling length could be effectively tuned. Under different Fermi energy level, this directional coupler could serve as an electrically controlled optical switch or a 3-dB splitter around the wavelength of 10.5 μm. Moreover, the size of the entire device is really in subwavelength scale making it very facilitative for high density integration. © 2015 Optical Society of America OCIS codes: (130.3120) Integrated optics devices; (250.5403) Plasmonics; (250.6715) Switching. http://dx.doi.org/10.1364/OL.40.001603
Surface plasmon polaritions (SPPs) offer a promising way to realize chip-scale integrated photonic circuits because of their ability of confining and controlling electromagnetic waves at subwavelength scale [1]. Active plasmonic devices such as switches and modulators are essential elements to realize integrated photonic circuits. In the visible to near-infrared frequencies, noble metal combined with active dielectric medium is one of the general methods to realize active plasmonic devices [2–6]. The (SPPs) modes at the metal-dielectric interface can be controlled by electro-, magneto-, and thermo-optic effects (depending on the active dielectric properties and electrode configuration) [7,8]. However in the midinfrared spectral region where SPs also have a broad range of potential applications [9,10], the use of noble metals is limited due to the poor confinement of SPs. Graphene, a newly emerged two-dimensional (2D) atomically thin material, is believed to be a promising plasmonic material from the terahertz to the mid-infrared spectral region [11,12]. Recently, the excitation, propagation and tunability of doped graphene surface plasmon polaritions (GSPPs) in mid-infrared frequencies have been theoretically studied [13–17] and experimentally demonstrated [18,19]. Compared with SPPs on noble metals, GSPPs could confine EM field at an extremely subwavelength scale in the mid-infrared spectral region. Moreover, GSPPs could be actively tuned by electrostatically gating or chemical doping, which makes graphene an excellent platform for active plasmonic devices. Active plasmonic waveguide devices based on vertical coupling between graphene sheets [20] or graphene nanoribbons have been theoretically proposed [21]. However the vertical coupling between graphene sheets is not very suitable for in-plane integration, as the mode on graphene sheet has no in-plane confinement. The edge shape of graphene nanoribbons could strongly influence the propagation properties of GSPP modes, and it is still a challenge to control the edge shape with desired atomic arrangement [22]. Recently, we have proposed the 0146-9592/15/071603-04$15.00/0
dielectric-loaded graphene plasmonic waveguide (DLGPW) [23], which is similar to dielectric-loaded metal plasmon waveguide in near-infrared frequencies [24,25]. The DLGPW does not need to pattern the graphene and is easy to fabricated. One peculiarity of the DLGPW is that the mode confinement and transmission loss could be tuned by varying the Fermi energy level of graphene sheet. Inspired by this finding, we think that the coupling between two parallel straight DLGPWs could be dynamically tuned, leading to electrically tunable directional coupler (DC). Based on this DC, active mid-infrared plasmonic devices such as switches could be realized. The schematic of our proposed electrically tunable DC is shown in Fig. 1(a). Two parallel identical straight DLGPWs with edge-to-edge separation d constitute the coupling region. The coupling region with a length of L is connected to the output waveguides (Ps and Pc) by two S-shaped waveguide bends. SiO2 dielectric layer with a thickness of hs is used to separate the graphene sheet and the back-gated metal film. By applying a voltage between the graphene sheet and the back-gated metal film, the carrier concentration and thus the Fermi energy level of garaphene can be electrically tuned. Then the coupling between the two DLGPWs could be tuned. Transmission characteristics of the DC could be analyzed by the well-established coupled-mode theory (CMT) [26], in which electromagnetic field distribution in the DC can be expressed as the superposition of the symmetrical and anti-symmetrical supermodes. For a fixed coupling length, the power coupled from one waveguide to the other is determined by the effective mode index of symmetrical (ns n0s − in00s ) and anti-symmetrical (na n0a − in00a ) supermodes. Here, we use finite element method-based software (COMSOL Multiphysics) to solve the supermodes. In the two dimension (2D) simulations, the width and thickness of the DLGPW are both 100 nm, which ensure singlemode operation from the wavelength of 9 to 11 μm. The edge-to-edge separation is 150 nm, unless otherwise © 2015 Optical Society of America
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curves for hs 200 nm coincide with the ones for halfspace SiO2 layer (without metal), indicating that the back-gated metal film does not affect the modes of the DLGPW, when the separation between the graphene sheet and the metal film is larger than 200 nm. This originates from the vertically tight confinement of the guide mode in the DLGPW. As the mode confinement increases when the Fermi energy level decreases [as shown in the insets of Fig. 2(a)], this conclusion is also valid for Fermi energy level below 0.9 eV. So we fixed the thickness of SiO2 layer to be 200 nm. In general, complete transfer of power from one waveguide to the other is not possible for coupled lossy waveguides, since loss is always present in the transfer process. When jn00s − n00a j∕2 ≪ jn0s − n0a j∕2, which is true for the proposed DC, the maximal power coupled from one waveguide to the other is approximately equal to [31] Fig. 1. (a) Schematic of the electrically tunable directional coupler based on dielectric-loaded graphene plasmonic waveguide. The inset shows the xy-plane cross-section of the directional coupler. (b) Real part of the effective mode index of symmetrical and anti-symmetrical supermodes for different thickness of SiO2 dielectric layer. The insets show the ydirection electric filed component of the supermode. The width and thickness of the DLGPW are both 100 nm. The edge-to-edge separation d is 150 nm, and the Fermi energy of graphene is 0.9 eV.
state. The refractive index of SiO2 was assumed to be 1.96. The graphene sheet was modeled as an anisotropic layer with a thickness of t 0.5 nm, and is uniform meshed with a resolution of 0.125 nm along the thickness direction and 5 nm along the surface direction. The outof-plane permittivity was 2.5, and in-plane permittivity was ε∥ ω 2.5 iσω∕ωε0 t, where the optical conductivity of graphene (σω) was derived using the random-phase approximation in the local limit [27]: i2e2 kB T EF σω 2 In 2 cosh 2kB T πℏ ω iτ−1 2 e 1 1 ℏω − 2E F arctan 4ℏ 2 π 2kB T i ℏω 2E F 2 In ; − 2π ℏω − 2E F 2 4kB T2
(1)
where T 300 K is the temperature, kB is the Boltzmann constant, ω is the frequency, E F is the Fermi energy, and τ μE F ∕eV 2F is the carrier relaxation lifetime (μ is the carrier mobility in graphene and V F 106 m∕s is the Fermi velocity). In recent experiments, the Fermi energy level has reached as high as 1.17 eV [28]. The carrier mobility ranges from ∼1000 cm2 ∕V · s in chemical vapor deposition (CVD) grown graphene [29] to 230000 cm2 ∕V · s in suspended exfoliated graphene [30]. Here, we use a Fermi energy level of 0.9 eV and carrier mobility of 100000 cm2 ∕V · s, unless otherwise stated. The mode dispersion relations (real part of the effective mode index) of the supermodes for different thicknesses of SiO2 layers are shown in Fig. 1(b). From the wavelength of 9 to 11 μm, the mode dispersion
Pcmax ≃ exp−2r arctanr −1 ∕1 r 2 .
(2)
r 2Lc ∕πLp , the corresponding coupling length is equal to Lm 2Lc arctanπLp ∕2Lc ∕π;
(3)
where Lc λ0 ∕2n0s − n0a is the beating length of the two supermodes and also the coupling length for coupled lossless waveguides. Lp λ0 ∕πn00s n00a is the mean propagation length. Suppose the input power Pin 1, then the maximum power coupled to Pc at different Fermi energy level is shown in Fig. 2(a), and the corresponding coupling length Lm is shown in Fig. 2(b). As the wavelength decreases, the coupling strength and the maximum power transfer decrease, which originate from the increment of confinement and propagation loss of the mode in the DLGPW. When the coupling is strong and the loss is slight (EF 0.9 eV), that is Lc is much smaller than the mean propagation length Lp , the maximum power transfer approaches 1, and the corresponding coupling length Lm approaches Lc , which is similar to the lossless case. In the opposite limit, when the couple is weak and the loss is huge (E F 0.5 eV for the wavelength under 10.5 μm), that is Lp is much smaller than Lc [the inset of Fig. 1(b)], the maximum power transfer approaches 0, and the corresponding coupling length Lm approaches Lp . So, there is a reverse tendency of the coupling length at E F 0.5 eV when the wavelength is under
Fig. 2. Maximum power transferred from the input DLGPW to the coupled DLGPW (a) and the corresponding coupling length (b) at different Fermi energy level. The geometrical parameters of the DC are the same as Fig. 1.
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Fig. 3. Dependence of the maximum power transfer on the edge-to-edge separation d between the two parallel DLGPWs. The geometrical parameters of the DLGPW are the same as Fig. 1, and the Fermi energy level is 0.9 eV.
10.5 μm, and at the same time, the maximum power transfer approaches 0. Figure 2 clearly shows that by varying the Fermi energy level of graphene sheet, the coupling between the two parallel DLGPWs could be effectively tuned, which enables electrically controllable functional devices, such as optical switch. Dependence of the maximum power transfer on the edge-to-edge separation between the two parallel DLGPW is shown in Fig. 3. When the waveguides are separated by d 900 nm, the maximum power transfer approaches 0. This means at this separation, the coupling between the waveguides is so weak that propagation loss dominates the coupling process. In other words, at this separation, the waveguides are practically not coupled. It should be noted that though this conclusion is drawn at the Fermi energy level of 0.9 eV, it is still true for Fermi energy level lower than 0.9 eV due to higher Fermi energy level leads to higher maximum power transfer [Fig. 2(a)]. So the edge-to-edge separation of the two output waveguides should be larger than 900 nm to ensure zero crosstalk between the output waveguides. In our design, the S-shaped bend waveguide is constructed by two connected circular arcs waveguide with opposite curvature. The radius and radian of the circular arcs is 6 μm and 0.3, respectively. The corresponding edge-to edge separation of the output waveguides is around 1.1 μm, which is large enough. Then we use 3D simulation to study the transmission of the proposed DC with a coupling length L 2.6 μm [Fig. 4(a)]. The fundamental mode of the DLGPW is launched into the input port. The power transferred to the cross-waveguide (Pc) is wavelength dependent and could be tuned by varying Fermi energy level. When E F 0.7 eV, power effectively coupled to the cross-waveguide around the wavelength of 10.5 μm. However, the transmission rate is lower, and the coupling length is shorter than the results calculated by CMT (Fig. 2) due to the additional mode coupling and transmission loss in the S-shaped bends. When E F 0.5 eV, Pc approaches 0 around the wavelength of 10.5 μm. These results enable an electrically controllable optical switch, with E F 0.7 eV as switch-on state and E F 0.5 eV as switch-off state. The switch on-off extinction ratio is larger than 16 dB. If lower Fermi energy level is used as the switch-off state, higher extinction ratio could be realized, as lower Fremi energy level leads to smaller maximum power transfer [Fig. 2(a)]. Moreover,
Fig. 4. (a) Transmission rate of the proposed DC at different Fermi energy level with a coupling length of 2.6 μm. (b) power flow distributions at different Fermi energy level for the wavelength of 10.5 μm.
if E F 0.82 eV, without changing the structure, Pc is equal to Ps around 10.5 μm indicating that this DC could serve as a 3-dB splitter. Figure 4(b) shows the power flow distributions at different Fermi energy level for the wavelength of 10.5 μm, which clearly shows the function of switch and splitter. It should be noted that the results in Fig. 4 are acquired under the carrier mobility of 100000 cm2 ∕V · s. The decrease of carrier mobility will bring more insertion loss; however, the function could still be maintained. So this proposed DC could serve as electrically controllable multifunctional devices with a sub-wavelength size of 6 μm × 3 μm. As for practical applications, the required bias-voltage to raise the graphene Fermi energy level from 0 to 0.82 eV is estimated up to about 500 V when the SiO2 insulator layer is 200 nm thick. However, the bias-voltage can be reduced by using transparency electrode [32] in place of metal electrode or using insulator with higher static permittivity (for example Al2 O3 ) in place of SiO2 , or chemical-doping [33] combined with electric-doping. In addition, optical resist can be used as the top dielectric ridge to simplify the fabrication procedure by standard processes of electron beam lithography and lift-off. In summary, we have studied the coupling characters of a directional coupler (DC) consisting of two parallel identical straight DLGPWs in the mid-infrared regime. The maximal power coupled from the input waveguide to the cross-waveguide and the corresponding coupling length could be manipulated by changing the Fermi energy level of the graphene sheet, which enables electrically controlled functional devices. Based on this DC, electrically controlled optical switch operating around the wavelength of 10.5 μm is proposed and simulated. The switch on-off extinction ratio is larger than 16 dB. Moreover, without changing the structure, by varying
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