## Abstract

In this paper, we theoretically propose a novel graphene-based hybrid plasmonic waveguide (GHPW) consisting of a low-index rectangle waveguide between a high-index cylindrical dielectric waveguide and the substrate with coated graphene on the surface. The geometric dependence of the mode characteristics on the proposed structure is analyzed in detail, showing that the proposed GHPW has a low loss and consequently a relatively long propagation distance. For TM polarization, highly confined modes guided in the low-index gap region between the graphene and the high-index GaAs and the normalized modal area is as small as 0.0018 (λ^{2}/4) at 3 THz. In addition to enabling the building of high-density integration of the proposed structure are examined by analyzing crosstalk in a directional coupler composed of two GHPWs. This structure also exhibits ultra-low crosstalk when a center-to-center separation between adjacent GHPWs is 32μm, which shows great promise for constructing various terahertz integrated devices.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

The control of light on scales much smaller than the wavelength is of vital importance and a critical challenging task in nanophotonics [1]. Surface plasmon-polaritons are among the most promising candidates and its modes are capable of confining light to sizes that are much smaller than the diffracting limit. Due to the strong confinement capability and a relatively long propagation length, various kinds of surface Plasmon polariton (SPPs) have been proposed, such as Cylinder Plasmon polaritons (CyPPs) [2], Wedge plasmon polaritons (WPPs) [3–5], Channel Plasmon polaritons (CPPs) [6–8] and hybrid Plasmon polaritons (HPPs) [9]. Among those SPPs in various architectures, HPPs provide an intriguing approach for linking conventional waveguide technology with subwavelength fields. And those hybrid plasmon-polaritons waveguides can be applied in plasmon laser [10], ultrafast all-optical switching [11] and compact S-bends and Y splitters [12], and so on.

In comparison to SPPs at visible frequencies, terahertz SPPs have very low propagation loss. And a number of architectures have been proposed to guide terahertz SPPs within a deep subwavelength confinement, including metallic grating structures [13], negative permeability metamaterial waveguides [14], and metal-based hybrid plasmonic waveguides [15]. But they are very weakly confined on the metal surface, making them unsuitable for compact integration [16, 17]. Besides, a great many noble-metal-based hybrid SPP waveguides [9, 18] are focused on the telecommunication wavelength (λ = 1.55μm) and they are only suitable for application form the near-infrared to visible waveband. Then it would be highly desired to design an optical waveguide with ultra-long propagation length and ultra-deep sub-wavelength confinement in terahertz domain. Such a device can be used for a basic component to transmit THz electromagnetic waves in certain practical applications, such as optical modulation [19], filtering [20], and near-field imaging [21]. Graphene, as a newly established two dimensional (2D) material, has attracted much attention in nanophotonics. Because its surface conductivity is almost purely imaginary in the THz and far infrared frequencies, and its chemical potential can be actively tuned by a gate voltage with suitable chemical doping [22–24]. Similar to metal plasmon, graphene plasmon also suffers from high absorption loss. However, it is capable of significantly reducing the absorption loss by increasing the chemical potential (Fermi energy). Analogous to the concept of hybrid plasmonic waveguides based on noble metals, a graphene-based hybrid plasmonic terahertz waveguide [25], which is composed of high-index GaAs rectangle waveguide separated from graphene sheets on a substrate by a low-index gap, has been proposed and achieved a much smaller modal size as well as a comparable propagation length, compared with traditional graphene plasmonic waveguides. Reduced effective mode area is realized when the gap between the high-index structure and the graphene sheets is shrunk. However, limited by the mode confinement capability of the corresponding graphene plasmon modes involved, further downscaling of the hybrid mode area seems difficult as the gap width is already minimized for practical implementations.

To circumvent the above problem, we investigated a novel graphene-based hybrid plasmonic waveguide in terahertz domain, with a rectangle waveguide material between a dielectric cylindrical waveguide and substrate with coated graphene on the surface, to achieve ultralow loss and deep subwavelength confinement. Owing to the coupling between the dielectric waveguide and graphene plasmonic modes, a much smaller modal size as well as a comparable propagation length can be achievable, as opposed to a traditional graphene plasmonic waveguide. Moreover, by tuning the geometrical properties of this structure, we can balance the modal size and the propagation distance. In addition to enabling the building of high-density integration of the proposed structure is examined by analyzing crosstalk in a directional coupler composed of two GHPWs. This structure also exhibits ultra-low crosstalk when a center-to-center separation between adjacent GHPWs is 32μm, which shows great promise for constructing various functional devices in future terahertz integrated circuits.

## 2. Waveguide structure and analysis

Figure 1 shows the geometry of the proposed GHPW, where a low-index rectangle waveguide is between a high-index cylindrical dielectric waveguide and the substrate with coated graphene on the surface. They form a hybrid plasmonic waveguide as the strong coupling between the cylinder dielectric waveguide mode and the plasmonic mode supported at the interface results in an extremely confined hybrid plasmonic mode [9]. Here, the high-index cylinder waveguide is GaAs with the refractive index 3.5 and diameter *d*. GaAs has been widely used in the THz waveguides [25, 26] and absorption loss of GaAs can be neglected as opposed to the absorption loss from graphene [25]. Experimentally, GaAs micro-wires can be fabricated in nanochannel glass [27], and GaAs crystalline nanorods can also be grown by using the molecular-beam epitaxy technique (MBE) [28]. The low-index rectangle waveguide is SiO_{2} with relative permittivities 2.25, width *d*_{gap} and thickness g, respectively. Here, the so called ‘gap’ includes air and SiO_{2}, which is different from those hybrid plasmonic waveguide were reported [9, 25]. In addition, the thickness of the SiO_{2} substrate can be treated as large enough as the depth of the waveguide mode. The graphene’s electromagnetic properties are characterized by a surface conductivity *σ*_{g} [25, 29–35], which can be calculated by the Kubo formula under the local random phase approximation (RPA)

*T*, angular frequency

*ω*, relaxation time $\tau $and chemical potential

*μ*

_{c}, and

*k*

_{B}is the Boltzmann's constant, $\hslash =h/2\pi $is the reduced Planck's constant, and -

*e*is the charge of the electron. In our proposed waveguide, the chemical potential

*μ*

_{c}is controlled by the application of a gate voltage with suitable chemical doping [22–24]. In this paper, we let the relaxation time of graphene$\tau =0.5\text{ps}$, which is determined by the carrier mobility

*μ*in graphene. The choice of here is rather conservation to reflect the practical transport loss of graphene [25]. The temperature is

*T*= 300K. At the same time, the frequency

*f*= 3THz and the chemical potential

*μ*

_{c}= 0.5eV, which fulfill the approximation condition${k}_{B}T<<\left|{\mu}_{c}\right|,\hslash \omega $ [34]. Here, we consider the case of trilayer graphene as it can supports plasmonic modes with higher quality factors compared with those supported by monolayer graphene [36]. Since the effective thickness of one graphene sheet is 0.5nm [24] and the numbers of graphene layers are less than 6, i.e., few-layer graphene, graphene sheets can be taken as a non-interacting monolayer so that the conductivity of graphene sheets is approximate to 3

*σ*

_{g}[36, 37], then the total thickness of graphene layers is$\Delta =1.5\text{nm}$. The formula of the equivalent permittivity of graphene can be written aswhere

*η*

_{0}is impedance of air, and

*k*

_{0}is the wave number [24].

For the proposed GHPW shown in Fig. 1, a couple of graphene plasmon and high-index contrast effect at GaAs/air/SiO_{2} multiple interfaces result in a deep-subwavelength mode confinement [9, 38, 39]. Additionally, the gap between GaAs and graphene layers ensure relatively lower propagation loss because the electromagnetic field intensity within the graphene layers would be much weaker. It is to say that our proposed structure is expected to guide optical field with deep-subwavelength as well as low propagation loss. In the following work, the finite-element analysis method is used to calculate the eigenmode of the coupled waveguide system. The mode effective index and propagation length are determined by the real and imaginary parts of the eigenvalue, respectively. In the convergence analysis, the calculation region in the *x*- and *y*-direction is assumed to be sufficiently long to ensure an accurate eigenvalue.

Figure 2 shows the electromagnetic field distributions of energy flux density for the hybrid plasmonic mode with TM polarization (where the electric field is perpendicular to the surface of the substrate) for different diameter *d* of the GaAs waveguide and thickness g of the SiO_{2} gap. For large *d* and g (e.g., [*d*, g] = [30, 7]μm), the energy of hybrid plasmonic mode with optical energy confined in the dielectric GaAs waveguide core, as shown in Fig. 2(a). When decrease the diameter *d* of the GaAs waveguide or the SiO_{2} gap’s thickness g, some optical energy confined both in the gap region and the GaAs waveguide, as shown in the Figs. 2(b) and 2(c). But for small *d* and g (e.g., [*d*, g] = [20, 0.5]μm), the most optical energy confined in the gap region, as shown in Fig. 2(d).

For more intuitive observation the changes of optical energy, Fig. 3 shows the energy flux density along *x* = 0 and *y* = -(*d* + g)/2 when *d* = *d*_{gap} = 30μm. We can clearly see that the optical energy couple to the gap region from the dielectric waveguide increases as the gap distance decreases from 7μm, 5μm, 3μm, 1μm to 0.5μm. When g = 0.5μm, the cylinder mode is strongly coupled to the SPP mode, and most of the optical energy is concentrated inside the gap region.

In order to gain a deeper understanding of the mode properties for the present GHPW, we analyse the mode effective index, the modal area and propagation length for the hybrid mode by varying the diameter *d* of the cylindrical GaAs waveguide and the gap distance g between the graphene and GaAs waveguide. First, we consider a relatively simple case, e.g., *d* = *d*_{gap}. Figure 4(a) shows the real part of the effective refractive index of the proposed GHPW versus the diameter of the GaAs waveguide for different gap thickness. Fixed the value of g, the *n*_{eff} increases when the diameter *d* increases. While fixed the diameter *d*, the *n*_{eff} decreases when the value of g increases. Figure 4(b) shows the dependences of the normalized modal area *A*_{m} on the cylinder diameter *d* for different gap distance g. Here the normalized modal area *A*_{m} = *A*_{eff}/*A*_{0}, where *A*_{eff} is the effective mode area and *A*_{0} represents the diffraction-limited area in free space, which is equal to *λ*^{2}/4. Here, the effective mode area [9] is defined as

*P*(

*r*) is the energy flux density (Poynting vector). It can be seen that when the gap distance g decreases to a certain extent, the smaller normalized modal area

*A*

_{m}can be obtained by tuning the diameter

*d*of the GaAs cylinder waveguide. Such as g = 1μm and

*d*= 15μm, the smallest normalized modal area

*A*

_{m}can be reduced to 0.0033 (

*λ*

^{2}/4), which is an ultra-deep sub-wavelength confinement compared with the proposed graphene-based hybrid plasmonic terahertz waveguide [25]. It is possible to further reduce the normalized modal area

*A*

_{m}if reducing the thickness of the gap, e.g., fixed g = 0.5μm, the normalized modal area

*A*

_{m}decreases from 0.0032 (

*λ*

^{2}/4) to 0.0018 (

*λ*

^{2}/4) when

*d*decreases from 26μm to 15μm. This is very useful for nonlinear optical applications or optical modulations. Figures 4(c) and 4(d) show the dependences of the propagation length

*L*

_{prop}on the diameter

*d*of GaAs cylinder waveguide for different gap thickness g and the gap thickness g for different diameter

*d*, respectively. Here, the propagation length is defined as the distance that the amplitude of the field attenuates to 1/e, i.e.,

*L*

_{prop}=

*λ*/{4πIm(

*n*

_{eff})}. It can be seen that the propagation distance for a larger GaAs waveguide (e.g.,

*d*= 35μm/30μm) decreases greatly as the thickness g decreases. However, the propagation distance does not change much for a slimly GaAs waveguide (e.g.,

*d*= 20μm/15μm). Fixed the gap thickness g, the propagation length

*L*

_{prop}increases with the increasing diameter

*d*[see Fig. 4(c)]. Fixed the diameter

*d*, the propagation length

*L*

_{prop}increases with the increasing gap thickness g [see Fig. 4(d)]. It is worth noting that when the normalized modal area is reduced to 10

^{−3}(

*λ*

^{2}/4) (e.g., g = 1μm/0.5μm), the propagation length

*L*

_{prop}can be reached as long as a few hundred microns. Therefore, we properly choose the thickness g of gap and diameter

*d*of the GaAs waveguide to obtain the ultra-deep sub-wavelength confinement and comparable propagation length, which is conducive to nanophotonic integration.

To further analyse the effect of geometry parameters on the mode and propagation properties, we fixed the diameter of the GaAs waveguide *d* = 30μm. As shown in Fig. 5(a), one can see that the normalized modal area *A*_{m} decreases firstly, and then increases with the increasing of gap width *d*_{gap} for large gap thickness (e.g., g = 7μm/5μm). But for the small gap thickness (e.g., g = 3μm/1μm/0.5μm), the normalized modal area *A*_{m} just has a few changes. It is worth noting that a smaller normalized modal area *A*_{m} can still be attained by tuning the gap width *d*_{gap} [see the inset in Fig. 5(a)]. The calculated propagation length is shown in Fig. 5(b) as the gap width *d*_{gap} varies. In the Fig. 5(b), one can see that the propagation distance decreases when *d*_{gap} increase. So for a certain GaAs waveguide, we can obtain the ultra-deep sub-wavelength confinement and comparable propagation length by properly choosing the geometry of gap.

## 3. Crosstalk between two adjacent GHPWs

For high photonic integration, one key point is the component density. From this perspective, it is important to examine how closely two adjacent parallel waveguides can be placed on a chip so that the cross talk between them due to coupling is negligible after a propagation distance [40, 41]. In our work, as shown in Fig. 6(a), two GHPWs in parallel are selected for evaluating the coupling length, where *S* is the center-to-center separation distance between the two GHPWs. Based on the coupled-mode theory (CMT), the coupling length can be estimated by a well-known equation *L*_{c} = 0.5*λ*/(*n*_{e}–*n*_{o}), where *n*_{e} and *n*_{o} are effective refractive indices of the even and odd modes of two coupled GHPWs, respectively. Figures 6(b)-6(c) show the *E*_{y} field of even mode (symmetric mode) and odd mode (anti-symmetric mode), respectively. The corresponding parameters are *d* = *d*_{gap} = 30μm, *S* = 35μm.

Fixed the *d* = *d*_{gap} = 30μm as a simple case, the dependences of the calculated effective index of the even mode and odd mode on the separation distance S for different gap thickness g = 7μm, 5μm, 3μm, 1μm and 0.5μm are shown in Fig. 7(a). It can be seen that, as the separation distance increases the values of effective index for even and odd modes approach some certain values, which is close to the effective index of a single GHPW [see Fig. 4(a)]. This phenomenon means that the decoupling (no crosstalk) appears when the two waveguides are sufficiently far apart. Figure 7(b) shows the normalized coupling length *L*_{c}/*L*_{prop} of two parallel GHPWs versus the normalized center-to-center distance between adjacent waveguides (*S*-*d*)/*d* for different gap thickness. Here *L*_{prop} denotes the propagation length for individual GHPW [see Fig. 4(c)]. It is clearly seen that *L*_{c}/*L*_{prop} increases exponentially as the normalized separations (*S*-*d*)/*d* increases, this is also similar to the case of a conventional dielectric optical waveguide. It is interesting to note that the value of the gap thickness g has great effect on *L*_{c}/*L*_{prop}. With the decrease of gap thickness from 7μm to 3μm, the necessary distance decreases (e.g. 50μm, 46μm, 40μm). Even when g = 1, 0.5μm, the *L*_{c}/*L*_{prop} is almost larger than 1, which is much better than those had been reported in hybrid plasmonic waveguides [25]. These results may provide additional way to achieve no coupling, the results verify that the proposed GHPW allows higher integration than these other designs.

## 4. Conclusion

In conclusion, we have proposed and investigated an ultralow loss GHPW for deep sub-wavelength confinement in terahertz domain. The simulation results show that the GHPW could have an ultra-small normalized mode areas (~10^{−3}*λ*^{2}) and long propagation by changing the geometry of the dielectric waveguide and the rectangle gap waveguide. In addition, the crosstalk of a directional coupler composed of two proposed GHPWs were discussed, showing that normalized coupling length can be tuned by changing the separations *S* in different gap thickness g. When g = 1μm/0.5μm, the normalized coupling length is almost larger than 1 and the crosstalk is weak within this distance. Therefore, the separations *S* can be as small as possible, which is smaller than the reported hybrid plasmonic waveguides [25]. Those results may provide additional way to design terahertz integrated devices with ultra-high integration density.

## Funding

This work is jointly supported by the National Natural Science Foundation of China (61471033,61525501), the National Natural Science Foundation of Beijing (4154081) and the fundamental research funds for the central universities(2016JBM001).

## References and links

**1. **R. Kirchain and L. Kimerling, “A roadmap for nanophotonics,” Nat. Photonics **1**(6), 303–304 (2007). [CrossRef]

**2. **C. A. Pfieffer, E. N. Economou, and K. L. Ngai, “Surface polaritons in a circularly cylindrical interface: Surface plasmons,” Phys. Rev. B **10**(8), 3038–3051 (1974). [CrossRef]

**3. **D. F. P. Pile, T. Ogawa, D. K. Gramotnev, T. Okamoto, M. Haraguchi, M. Fukui, and S. Matsuo, “Theoretical and experimental investigation of strongly localized plasmons on triangular metal wedges for subwavelength waveguiding,” Appl. Phys. Lett. **87**(6), 061106 (2005). [CrossRef]

**4. **E. Moreno, S. G. Rodrigo, S. I. Bozhevolnyi, L. Martín-Moreno, and F. J. García-Vidal, “Guiding and Focusing of Electromagnetic Fields with Wedge Plasmon Polaritons,” Phys. Rev. Lett. **100**(2), 023901 (2008). [CrossRef] [PubMed]

**5. **A. Boltasseva, V. S. Volkov, R. B. Nielsen, E. Moreno, S. G. Rodrigo, and S. I. Bozhevolnyi, “Triangular metal wedges for subwavelength plasmon-polariton guiding at telecom wavelengths,” Opt. Express **16**(8), 5252–5260 (2008). [CrossRef] [PubMed]

**6. **D. F. P. Pile and D. K. Gramotnev, “Channel plasmon-polariton in a triangular groove on a metal surface,” Opt. Lett. **29**(10), 1069–1071 (2004). [CrossRef] [PubMed]

**7. **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). [CrossRef] [PubMed]

**8. **S. I. Bozhevolnyi, V. S. Volkov, E. Devaux, J. Y. Laluet, and T. W. Ebbesen, “Channel plasmon subwavelength waveguide components including interferometers and ring resonators,” Nature **440**(7083), 508–511 (2006). [CrossRef] [PubMed]

**9. **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-range propagation,” Nat. Photonics **2**(8), 496–500 (2008). [CrossRef]

**10. **R. F. Oulton, V. J. Sorger, T. Zentgraf, R. M. Ma, C. Gladden, L. Dai, G. Bartal, and X. Zhang, “Plasmon lasers at deep subwavelength scale,” Nature **461**(7264), 629–632 (2009). [CrossRef] [PubMed]

**11. **A. Y. Elezzabi, Z. Han, S. Sederberg, and V. Van, “Ultrafast all-optical modulation in silicon-based nanoplasmonic devices,” Opt. Express **17**(13), 11045–11056 (2009). [CrossRef] [PubMed]

**12. **M. Wu, Z. Han, and V. Van, “Conductor-gap-silicon plasmonic waveguides and passive components at subwavelength scale,” Opt. Express **18**(11), 11728–11736 (2010). [CrossRef] [PubMed]

**13. **S. A. Maier, S. R. Andrews, L. Martín-Moreno, and F. J. García-Vidal, “Terahertz Surface Plasmon-Polariton Propagation and Focusing on Periodically Corrugated Metal Wires,” Phys. Rev. Lett. **97**(17), 176805 (2006). [CrossRef] [PubMed]

**14. **A. Ishikawa, S. Zhang, D. A. Genov, G. Bartal, and X. Zhang, “Deep Subwavelength Terahertz Waveguides Using Gap Magnetic Plasmon,” Phys. Rev. Lett. **102**(4), 043904 (2009). [CrossRef] [PubMed]

**15. **S. H. Nam, A. J. Taylor, and A. Efimov, “Subwavelength hybrid terahertz waveguides,” Opt. Express **17**(25), 22890–22897 (2009). [CrossRef] [PubMed]

**16. **J. A. Dionne, L. A. Sweatlock, H. A. Atwater, and A. Polman, “Plasmon slot waveguides: Towards chip-scale propagation with subwavelength-scale localization,” Phys. Rev. B **73**(3), 035407 (2006). [CrossRef]

**17. **L. Y. M. Tobing, L. Tjahjana, and D. H. Zhang, “Demonstration of low-loss on-chip integrated plasmonic waveguide based on simple fabrication steps on silicon-on-insulator platform,” Appl. Phys. Lett. **101**(4), 041117 (2012). [CrossRef]

**18. **Y. Bian, Z. Zheng, Y. Liu, J. Liu, J. Zhu, and T. Zhou, “Hybrid wedge plasmon polariton waveguide with good fabrication-error-tolerance for ultra-deep-subwavelength mode confinement,” Opt. Express **19**(23), 22417–22422 (2011). [CrossRef] [PubMed]

**19. **M. Liu, X. Yin, E. Ulin-Avila, B. Geng, T. Zentgraf, L. Ju, F. Wang, and X. Zhang, “A graphene-based broadband optical modulator,” Nature **474**(7349), 64–67 (2011). [CrossRef] [PubMed]

**20. **H. J. Li, L. L. Wang, J. Q. Liu, Z. R. Huang, B. Sun, and X. Zhai, “Investigation of the graphene based planar plasmonic filters,” Appl. Phys. Lett. **103**(21), 211104 (2013). [CrossRef]

**21. **R. L. Olmon, P. M. Krenz, A. C. Jones, G. D. Boreman, and M. B. Raschke, “Near-field imaging of optical antenna modes in the mid-infrared,” Opt. Express **16**(25), 20295–20305 (2008). [CrossRef] [PubMed]

**22. **F. Wang, Y. Zhang, C. Tian, C. Girit, A. Zettl, M. Crommie, and Y. R. Shen, “Gate-Variable Optical Transitions in Graphene,” Science **320**(5873), 206–209 (2008). [CrossRef] [PubMed]

**23. **Y. J. Yu, Y. Zhao, S. Ryu, L. E. Brus, K. S. Kim, and P. Kim, “Tuning the graphene work function by electric field effect,” Nano Lett. **9**(10), 3430–3434 (2009). [CrossRef] [PubMed]

**24. **A. Vakil and N. Engheta, “Transformation optics using graphene,” Science **332**(6035), 1291–1294 (2011). [CrossRef] [PubMed]

**25. **X. Zhou, T. Zhang, L. Chen, W. Hong, and X. Li, “A Graphene-Based Hybrid Plasmonic Waveguide With Ultra-Deep Subwavelength Confinement,” J. Lightwave Technol. **32**(21), 3597–3601 (2014).

**26. **W. Wang, J. He, X. Li, and Z. Hong, “Slow light in the GaAs-rod-loaded metallic waveguide for terahertz wave,” Opt. Express **18**(11), 11132–11137 (2010). [CrossRef] [PubMed]

**27. **A. D. Berry, R. J. Tonucci, and M. Fatemi, “Fabrication of GaAs and InAs wires in nanochannel glass,” Appl. Phys. Lett. **69**(19), 2846–2848 (1996). [CrossRef]

**28. **H. G. Lee, H. C. Jeon, T. W. Kang, and T. W. Kim, “Gallium arsenide crystalline nanorods grown by molecular-beam epitaxy,” Appl. Phys. Lett. **78**(21), 3319–3321 (2001). [CrossRef]

**29. **P. Y. Chen and A. Alù, “Atomically thin surface cloak using graphene monolayers,” ACS Nano **5**(7), 5855–5863 (2011). [CrossRef] [PubMed]

**30. **R. J. Li, X. Lin, S. S. Lin, X. Liu, and H. S. Chen, “Tunable deep-subwavelength superscattering using graphene monolayers,” Opt. Lett. **40**(8), 1651–1654 (2015). [CrossRef] [PubMed]

**31. **R. Li, X. Lin, S. Lin, X. Liu, and H. Chen, “Atomically thin spherical shell-shaped superscatterers based on a Bohr model,” Nanotechnology **26**(50), 505201 (2015). [CrossRef] [PubMed]

**32. **R. Li, B. Zheng, X. Lin, R. Hao, S. Lin, W. Yin, E. Li, and H. Chen, “Design of Ultracompact Graphene-Based Superscatterers,” IEEE J. Sel. Top. Quant. **23**(1), 4600208 (2017). [CrossRef]

**33. **X. He, T. Ning, R. Li, L. Pei, J. Zheng, and J. Li, “Dynamical manipulation of Cosine-Gauss beams in a graphene plasmonic waveguide,” Opt. Express **25**(12), 13923–13932 (2017). [CrossRef] [PubMed]

**34. **G. W. Hanson, “Dyadic Green’s functions and guided surface waves for a surface conductivity model of graphene,” J. Appl. Phys. **103**(6), 064302 (2008). [CrossRef]

**35. **R. Li, X. Lin, S. Lin, X. Zhang, E. Li, and H. Chen, “Graphene induced mode bifurcation at low input power,” Carbon **98**, 463–467 (2016). [CrossRef]

**36. **C. H. Gan, “Analysis of surface plasmon excitation at terahertz frequencies with highly doped graphene sheets via attenuated total reflection,” Appl. Phys. Lett. **101**(11), 111609 (2012). [CrossRef]

**37. **H. Yan, X. Li, B. Chandra, G. Tulevski, Y. Wu, M. Freitag, W. Zhu, P. Avouris, and F. Xia, “Tunable infrared plasmonic devices using graphene/insulator stacks,” Nat. Nanotechnol. **7**(5), 330–334 (2012). [CrossRef] [PubMed]

**38. **X. Y. Zhang, A. Hu, J. Z. Wen, T. Zhang, X. J. Xue, Y. Zhou, and W. W. Duley, “Numerical analysis of deep sub-wavelength integrated plasmonic devices based on Semiconductor-Insulator-Metal strip waveguides,” Opt. Express **18**(18), 18945–18959 (2010). [CrossRef] [PubMed]

**39. **H. Benisty and M. Besbes, “Plasmonic inverse rib waveguiding for tight confinement and smooth interface definition,” J. Appl. Phys. **108**(6), 063108 (2010). [CrossRef]

**40. **D. Dai, Y. Shi, and S. He, “Comparative study of the integration density for passive linear planar light-wave circuits based on three different kinds of nanophotonic waveguide,” Appl. Opt. **46**(7), 1126–1131 (2007). [CrossRef] [PubMed]

**41. **Z. Han and S. I. Bozhevolnyi, “Radiation guiding with surface plasmon polaritons,” Rep. Prog. Phys. **76**(1), 016402 (2013). [CrossRef] [PubMed]