## Abstract

In this paper, we show that the graphene-coated nanowire dimers could enable outstanding waveguiding performance in the mid-infrared range. The propagating properties of the fundamental graphene plasmon mode and their dependence on the nanowire radius, gap distance, nanowire permittivity and chemical potential of graphene are revealed in detail and compared with the graphene-coated circular nanowire. By improving the geometric parameters and the surface conductivity of graphene, the propagation length could reach about 9 *μ*m, which is larger than that of the graphene-coated circular nanowire plasmon mode. Meanwhile, the effective mode area is only 10^{−4}*A*_{0}, which is one order of magnitude smaller than that of the graphene-coated circular nanowire plasmon mode. Theoretically, the propagation length could be further enhanced by increasing the chemical potential. Besides, the proposed graphene-coated nanowire dimers show quite good fabrication tolerance. The manipulation of mid-infrared waves at the deep subwavelength scale using graphene plasmons may offer potential applications in tunable integrated nanophotonic devices and infrared sensing.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

Surface plasmons (SPs), coupled oscillations of electromagnetic field and free electrons that propagates along metal-dielectric interfaces, are capable of squeezing light into regions much smaller than the diffraction limit, thus becoming a promising candidate for manipulating light on the nanometer scale [1–4]. Noble metals, such as gold and silver are usually chosen to support SPs in the visible to near-infrared frequencies. So far, various plasmonic waveguiding structures have been proposed and investigated, including metal slot waveguide [5], dielectric loaded plasmon waveguide [6], long-range plasmon waveguide [7], metal groove/wedge plasmon waveguide [8,9], metallic nanowire waveguide [10–12], hybrid plasmonic waveguide [13–24], to mention a few. However, these above mentioned waveguides are only suitable for nanoscale applications in the near-infrared and visible frequencies [25]. In the mid-infrared and terahertz (THz) range, SPs have relatively weak confinement on the metal surface [26,27], thus hindering the applications in nanoscale [28].

Recently, experiments show that graphene [29], one of the two-dimensional materials, can support SPs [30–32], thus offering a new approach for subwavelength waveguiding in the mid-infrared range [33]. Up to now, graphene plasmon waveguides such as graphene sheet and ribbons [34,35], graphene slot waveguide [36], graphene groove/wedge [27], dielectric loaded graphene waveguide [37], graphene hybrid waveguide [38–44], graphene-coated nanowire [25,26,45–50], have been proposed and investigated. Compared with the metal-based plasmon waveguides, graphene plasmon waveguides exhibit extremely strong mode confinement, huge field enhancement. Further, the surface conductivity of graphene could be tuned, which makes it more popular [31].

Among the graphene plasmon waveguides, graphene-coated nanowire, which is an analogy of metal nanowire, has attracted lots of research interests for cutoff-free of the fundamental mode (TM_{0}) and simple structure. Gao et. al [45] presented an analytical model for plasmon modes in graphene-coated dielectric nanowire, Hajati et. al [46] investigated the influence of buffer on graphene-coated nanowire. Huang et. al [47] proposed a graphene-coated nanowire with a drop-shaped cross section, which can achieve a propagation length of 1 mm at 2 THz. Davoyan et. al [48] presented a comparison of performance between graphene-coated nanowire and other available THz waveguiding structures. Also, second harmonic generation in graphene-coated nanowire is studied [25]. Recent report shows that through integration with graphene, the ohmic loss of metal nanowires can be reduced [51]. Nevertheless, the mode field in graphene-coated nanowire is less confined as the field decays away from the surface, and the radially polarized mode field makes it difficult to be coupled from the common used linearly polarized sources. Fortunately, we can add another nanowire to form a graphene-coated nanowire dimer (GCNWD) structure, which is an analogy of metal two-wire waveguide [52].

Here in this paper, we explore the GCNWDs in detail including the propagating and confinement properties and their dependence on the nanowire radius, gap, nanowire permittivity, and chemical potential of graphene. We will show that the modal field is mostly restricted in the gap, and also the field in the gap is approximately linearly polarized. And the GCNWD is superior to the graphene-coated circular nanowire in waveguiding performances. We also briefly study four kinds of deformed GCNWDs.

## 2. Theoretical model

Figure 1 is a schematic of the proposed GCNWD embedded in medium with permittivity of *ε*_{2}. The two dielectric nanowire with permittivity of *ε*_{1} are coated with monolayer graphene. The radii of the two nanowires are *R*_{1} and *R*_{2}, respectively. The thickness of monolayer graphene (*d* = 0.33 nm) can be neglected, and the spacing between nanowires is *D* (*D*>>*d*). Further, the permittivity of graphene can be calculated by using *ε*_{g} = 1 + *iσ*_{g}/(*ε*_{0}*ωd*) [28,31,53], where *d* is the thickness of monolayer graphene, and *ω* is the angular frequency of the incident light. *ε*_{0} is the permittivity in free space. Within the random-phase approximation, the dynamic optical response of graphene can be derived from the Kubo’s formula [53–55] consisting of the interband and intraband contributions, that is *σ*_{g} = *σ*_{intra} + *σ*_{inter}. In the terahertz and infrared ranges, the intraband transition of electrons dominates [56,57], and then the surface conductivity of graphene could be approximated as

*τ*is the relaxation time,

*T*is the temperature,

*u*

_{c}is the chemical potential,

*ћ*is the reduced plank constant,

*k*is the Boltzmann constant, and

_{B}*e*= 1.6 × 10

^{−19}C. In what follows, we chose

*T*= 300 K,

*τ*= 0.5 ps, and

*ε*

_{2}= 1.

Assuming that the fundamental graphene plasmon mode (GPM) propagates along *z*-direction and the field of the eigenmode has the form of ** U**(

*x*,

*y*)exp(

*ißz*)exp(

*iωt*), in which

*β*=

*k*

_{0}

*n*

_{eff}is the propagation constant,

*k*

_{0}= 2π/λ

_{0}, and

**stands for electric or magnetic fields.**

*U**n*

_{eff}is the effective mode index and calculated by the finite element method (FEM). The real part Re(

*n*

_{eff}) of the effective mode index is directly related to the dispersion and the imaginary part Im(

*n*

_{eff}) is related to the attenuation. Then the propagation length is defined as

*L*

_{p}= 1/(2

*α*) with

*α*=

*k*

_{0}Im(

*n*

_{eff}) and calculated by

*L*

_{p}=

*λ*

_{0}/[4πIm(

*n*

_{eff})]. The normalized mode area is defined as

*A*

_{eff}/

*A*

_{0}with

*A*

_{0}=

*λ*

_{0}

*/4 being the diffraction-limited mode area. The effective mode area*

^{2}*A*

_{eff}is evaluated by [58]

*W*(r) is the electromagnetic energy density and is given by

Figure of merit (*FoM*) [46,59,60] is an important parameter that provide a proper assessment for the trade-off between the propagation length and the effective mode area. Several different definitions of *FoM* could be found in literature. Here the *FoM* is defined as [59]$FoM=\sqrt{\pi /{A}_{\text{eff}}}\cdot 1/\alpha $.

## 3. Results and discussion

#### 3.1 Plasmon modes in GCNWDs

Figure 2 shows the mode properties of the GCNWD. Here, *R*_{1} *= R*_{2} = 100 nm, *D* = 50 nm, *ε*_{1} = 2.25, and *ε*_{2} = 1. We focus on the fundamental mode for its low transmission loss. Figure 2(a) presents the electric field distribution, and the optical energy is mainly confined in the gap, and the GPM is approximately linearly polarized. The Poynting vector in *z* direction has the maximum value at the graphene surface, seen in Fig. 2(b). Clearly, we can see that the GPM originates from the coupling of the two graphene-coated circular nanowire plasmon modes (TM_{0} mode). For increasing the gap distance *D*, the coupling strength becomes weaker, and finally these two TM_{0} modes decouple for *D* exceeding a certain value. We will briefly illustrate this later. Figure 2(c) demonstrates the dispersion relations of the GPM, and the effective mode indices Re(*β*)/*k*_{0} = Re(*n*_{eff}) increase monotonically with frequency increasing. At higher frequencies, more mode energy penetrates into the nanowire, which can be easily seen in Fig. 2(b). Also we can see from Fig. 2(c), the propagation length *L*_{P} decreases with frequency increasing. Actually, in the mid-infrared range, the GPM suffers from high absorption loss because the majority of the light energy is located at the graphene layer. Inevitably, the propagation length of this type of waveguide remains relatively small (typically about 10 *μ*m). Figure 2(d) shows the normalized mode area (*A*_{eff}/*A*_{0}) and *FoM* of the GPM with respect to frequency. For *f*_{0} = 20 THz, the propagation length is about 4.5 *μ*m, and a very small effective mode area about 1.6 × 10^{−4}*A*_{0} could be obtained. For higher frequencies, the higher absorption leads to the reduction of the performance (decease of *FoM*) of the GCNWD.

Since graphene is involved here, we cannot only tune the geometric parameters, but also tune the chemical potential to improve the waveguiding performance. At the meantime, we need to maintain the existing degree of confinement.

#### 3.2 Propagation properties analysis

Radius of nanowire has a strong impact on the modal behavior in GCNWD. It is worth mentioning that *R*_{1} and *R*_{2} could have different values. Due to the commutability of the geometry, one can either hold *R*_{1} or *R*_{2} constant while changing another. Here we set *R*_{1} = *R*_{2}, *D* = 20 nm, and *R*_{1} varies from 50 nm to 200 nm. Figure 3(a) shows the effective mode index and propagation length with respect to *R*_{1} at the frequency of 30 THz. The effective mode indices increase with the enlargement of *R*_{1}. For the fundamental GPM, decreasing *R*_{1} means that the surface area of graphene decreases, leading to the reduction of loss and increasing of the propagation length. Meanwhile, the normalized mode area decreases with nanowire radius decreasing, shown in Fig. 3(b). Therefore, in order to attain relatively long propagation length and ultra-small mode area, one need to choose nanowire with smaller radius. Apparently, as *R*_{1} reduces, the GCNWD has better performances, i.e., larger *FoM*.

Unlike the graphene-coated circular nanowire, here the separation between two wires can also be adjusted. Figure 4(a) presents the effective mode index and propagation length with respect to *D* at the frequency of 30 THz. *R*_{1} = *R*_{2} = 100 nm, and *D* ranges from 2 nm to 100 nm. For increasing *D*, the effective mode indices (Re(*n*_{eff})) first rapidly decrease (solid blue line in Fig. 4(a)) for *D*<30 nm, and then moderately decrease. We can also see that *L*_{p} increases with increasing *D*. However, increasing *D* also leads to two main issues. The first one is the approximately linearly increased normalized mode area, shown in Fig. 4(b). Another issue is that for *D* exceeding a certain value, the GPM mode does not exist. As we stated before, the fundamental GPM originates from the coupling of the two graphene-coated circular nanowire plasmon modes (TM_{0} mode). For increasing the gap distance *D*, the coupling strength decreases, and finally these two TM_{0} modes decouple for *D* exceeding 400 nm here. The blue dashed line in Fig. 4(a) indicates the Re(*n*_{eff}) of the graphene-coated circular nanowire plasmon mode, which is 22.0893 at 30 THz for *u*_{c} = 0.5 eV, *T* = 300 K, *τ* = 0.5 ps, *R* = 100 nm, *ε*_{1} = 2.25, and *ε*_{2} = 1. As *D* exceeding 400 nm, the Re(*n*_{eff}) of the GPM approaches that of the graphene-coated circular nanowire. Thus, in order to achieve long propagation length and maintain the coupled GPM, a moderate gap distance (50 nm-100 nm) is highly recommended for practical use.

Besides, the figure of merit decreases as *D* increases. For the symmetric case, the highest *FoM* is about 580 for *D* = 2 nm. For the asymmetric case, we set *D* = 2 nm and *R*_{1} = 100 nm, and calculate the *FoM* by varying *R*_{2} from 50 nm to 150 nm. Results show that the *FoM* has the highest value about 623 for *R*_{2} = 50 nm, and then decreases with increasing *R*_{2}. Finally, the *FoM* decreases to 557 for *R*_{2} = 150 nm.

The permittivity of nanowire also has a large impact on GPM in GCNWD. Figure 5(a) depicts the permittivity dependent effective mode index and propagation length at the frequency of 30 THz. The parameters are *u*_{c} = 0.5 eV, *T* = 300 K, *τ* = 0.5 ps, *R*_{1} = *R*_{2} = 100 nm, *D* = 20 nm, and *ε*_{2} = 1. With decreasing nanowire permittivity, effective mode indices almost linearly decrease and the loss also reduces. Figure 5(b) presents the normalized mode area and *FoM* with respect to *ε*_{1}. The irregular changes of *A*_{eff}/*A*_{0} with different permittivity could be ignored, since *A*_{eff}/*A*_{0} is around 1 × 10^{−4} for all permittivity values considered and the permittivity seems to have a little influence on the normalized mode area. This is probably because that the dielectric permittivity is much smaller than the equivalent permittivity of graphene. From Fig. 5, we can deduce that smaller permittivity shows a much better performance of the GPM.

The surface conductivity *σ*_{g} of graphene could be tuned by changing chemical potential *u*_{c}. Figure 6 shows the chemical potential dependent characteristics of the GPM at *f*_{0} = 30 THz. The other parameters are *T* = 300 K, *τ* = 0.5 ps, *R*_{1} = *R*_{2} = 100 nm, *D* = 20 nm, *ε*_{1} = 2.25, and *ε*_{2} = 1. We change *u*_{c} from 0.2 eV to 1 eV. The increase of chemical potential provides two large benefits. First, Re(*n*_{eff}) and Im(*n*_{eff}) decrease with increasing *u*_{c}, implying an increase in propagation length as depicted in Fig. 6(a). Second, the normalized mode area enlarges only about 10% when *u*_{c} ranging from 0.2 eV to 1 eV, thus the chemical potential seems to have a slight influence on the normalized mode area. Finally, increasing *u*_{c} leads to the almost linearly increase of *FoM*, seen in Fig. 6(b). These results indicate the possibility of realizing higher performance of the GCNWD by simply enlarging the chemical potential.

So far, we have investigated the waveguiding performance of the GCNWD by changing geometric parameters and surface conductivity of graphene. We have also shown that the smaller nanowire radius, moderate gap distance (*D* = 50 nm-100 nm), smaller nanowire permittivity, and larger chemical potential *u*_{c} could offer better performance of the GCNWDs. Figure 7 shows the improved mode properties of the GPM with *T* = 300 K, *τ* = 0.5 ps, *ε*_{1} = 2, *ε*_{2} = 1, *u*_{c} = 1 eV, *R*_{1} = *R*_{2} = 50 nm, and *f*_{0} = 30 THz. With the increase of gap distance, the propagation length could reach about 9 *μ*m, which is larger than that of the graphene-coated circular nanowire plasmon mode [45,46]. Meanwhile, the effective mode area is only 10^{−4}*A*_{0}, which is one order of magnitude smaller than that of the graphene-coated circular nanowire plasmon mode [26]. Further, the propagation length could be enhanced by increasing *u*_{c}. The trade-off between the propagation length and mode area still exist, thus one should choose to realize either better mode confinement or longer propagation length.

#### 3.3 Comparison with graphene-coated nanowire

In this Section, we compare the plasmon modal behaviors of three kinds of graphene-coated nanowire based waveguides in the frequency range of 40 THz-60 THz. These waveguides are the symmetric GCNWD (Type **A**), graphene-coated nanowire on a graphene layer (Type **B**), and graphene-coated circular nanowire (Type **C**), shown in Fig. 8(a).

Here, the radii of all nanowires are set to be 100 nm, and gap distance is 30 nm. As shown in Fig. 8(b), the effective mode indices of GPM in GCNWD (ranges from 24.3 to 34.2) are much larger than those of the other two structures, implying the GPM has a much shorter effective wavelength (*λ*_{eff} = *λ*_{0}/Re(*n*_{eff})) and better mode confinement. Figure 8(c) shows the comparison of propagation length, the GPM have a propagation length about a few micrometers, which is due to the high absorption in this band mentioned above. One can see that the GPM in GCNWD has larger *L*_{p} compare with that of graphene-coated circular nanowire. At lower frequencies, Type **B** waveguide has larger *L*_{p} than other two structures. The normalized mode area is depicted in Fig. 8(d). Clearly, the plasmon modes in Type **A** and Type **B** waveguides have comparable mode confinement, which is one order of magnitude smaller than that of the plasmon mode in Type **C** waveguide.

#### 3.4 Higher order modes and deformed GCNWDs

Here we briefly compare the fundamental mode (*m* = 0) with the higher order modes (*m* = 1,2,3,4), as shown in Fig. 9. Figure 9(b) demonstrates the dispersion relations of GPM modes in GCNWD with *R*_{1} = *R*_{2} = 100 nm and *D* = 30 nm. We noticed that *m* = 0 mode is cutoff-free and the effective mode indices of all modes increase monotonically with frequency increasing. At high frequencies, the modes are strongly confined and result in an increasing absorption loss, which leads to shorter propagation length, seen in Fig. 9(c). Also we can see that the propagation lengths of higher order modes are less than 4 *μ*m in the frequency range of 25 THz to 60 THz. This means that the fundamental mode dominates over the higher order modes after propagating about 4 *μ*m at low frequencies (*f*_{0}<35 THz).

In addition to the waveguiding configurations considered here, the radius of the nanowires could be different, which provides another freedom to be tuned. Further, the shape of cross section of the nanowire could be changed, such as elliptical, triangular or other shapes, to achieve deep subavelength waveguiding and long propagation. Figure 10 shows four typical structures along with the energy distrbutions. We find that despite different cross sections, the proposed GCNWDs maintain outstanding optical performances. And all these deformed structures can achieve propagation length about 6 *μ*m-8 *μ*m and very small mode area about 10^{−4}*A*_{0}. In other words, the above results clearly indicate that the proposed GCNWDs are quite tolerant to fabrication errors, which is beneficial for its implementations.

The numerical simulation analysis is based on the finite element method (FEM). Also, a convergence analysis was conducted to ensure that the effective mode indices varied by less than 1%. In the simulation, the graphene could be either treated as a thin layer or surface current density. As for the former, one should carefully deal with mesh and the geometry. While the use of surface current density instead of thin layer would make the simulation easier. The relative error between two approaches is less than 1%.

## 4. Summary

We propose GCNWDs for deep subwavelength waveguiding of mid-infrared waves. The results show that the properties of the fundamental GPM are dependent on the geometric parameter of the GCNWDs as well as the chemical potential. By carefully tuning the geometric parameters and surface conductivity of graphene, long propagation length about 9 *μ*m and ultra-small modal area about 10^{−4}*A*_{0} could be obtained. The concept reported in this work could be easily applied to other GCNWDs with different cross sections, and could also be extended to terahertz band. The using of graphene plasmon for deep-subwavelength waveguiding of middle infrared waves opens up a new horizon for applications that would be otherwise difficult to realize with traditional material systems, offering potential application in miniaturized integrated photonic devices and a variety of intriguing applications at the sub-diffraction-limited scale.

## Funding

Open Research Fund of Zhengzhou Normal University; Aid program for Science and Technology Innovative Research Team of Zhengzhou Normal University.

## Acknowledgments

We are indebted to the reviewers and Minning Zhu from Rutgers, the State University of New Jersey for their comments and suggestions for improving the paper.

## References

**1. **S. A. Maier, *Plasmonics: Fundamentals and Applications* (Springer, 2007).

**2. **S. A. Maier and H. A. Atwater, “Plasmonics: Localization and guiding of electromagnetic energy in metal/dielectric structures,” J. Appl. Phys. **98**(1), 011101 (2005). [CrossRef]

**3. **S. A. Maier, P. G. Kik, H. A. Atwater, S. Meltzer, E. Harel, B. E. Koel, and A. A. G. Requicha, “Local detection of electromagnetic energy transport below the diffraction limit in metal nanoparticle plasmon waveguides,” Nat. Mater. **2**(4), 229–232 (2003). [CrossRef] [PubMed]

**4. **D. K. Gramotnev and S. I. Bozhevolnyi, “Plasmonics beyond the diffraction limit,” Nat. Photonics **4**(2), 83–91 (2010). [CrossRef]

**5. **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 Condens. Matter Mater. Phys. **73**(3), 035407 (2006). [CrossRef]

**6. **B. Steinberger, A. Hohenau, H. Ditlbacher, A. L. Stepanov, A. Drezet, F. Aussenegg, A. Leitner, and J. Krenn, “Dielectric stripes on gold as surface plasmon waveguides,” Appl. Phys. Lett. **88**(9), 094104 (2006). [CrossRef]

**7. **P. Berini, “Long-range surface plasmon polaritons,” Adv. Opt. Photonics **1**(3), 484–588 (2009). [CrossRef]

**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. **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]

**10. **M. I. Stockman, “Nanofocusing of optical energy in tapered plasmonic waveguides,” Phys. Rev. Lett. **93**(13), 137404 (2004). [CrossRef] [PubMed]

**11. **X. Guo, Y. Ma, Y. Wang, and L. Tong, “Nanowire plasmonic waveguides, circuits and devices,” Laser Photonics Rev. **7**(6), 855–881 (2013). [CrossRef]

**12. **L. Gao, L. Chen, H. Wei, and H. Xu, “Lithographically fabricated gold nanowire waveguides for plasmonic routers and logic gates,” Nanoscale **10**(25), 11923–11929 (2018). [CrossRef] [PubMed]

**13. **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]

**14. **D. Dai and S. He, “A silicon-based hybrid plasmonic waveguide with a metal cap for a nano-scale light confinement,” Opt. Express **17**(19), 16646–16653 (2009). [CrossRef] [PubMed]

**15. **M. Z. Alam, J. Meier, J. S. Aitchison, and M. Mojahedi, “Propagation characteristics of hybrid modes supported by metal-low-high index waveguides and bends,” Opt. Express **18**(12), 12971–12979 (2010). [CrossRef] [PubMed]

**16. **Y. Bian and Q. Gong, “Metallic-nanowire-loaded silicon-on-insulator structures: a route to low-loss plasmon waveguiding on the nanoscale,” Nanoscale **7**(10), 4415–4422 (2015). [CrossRef] [PubMed]

**17. **Y. Bian, Z. Zheng, X. Zhao, J. Zhu, and T. Zhou, “Symmetric hybrid surface plasmon polariton waveguides for 3D photonic integration,” Opt. Express **17**(23), 21320–21325 (2009). [CrossRef] [PubMed]

**18. **H. Liang, S. Ruan, M. Zhang, H. Su, and I. L. Li, “Modified surface plasmon polaritons for the nanoconcentration and long-range propagation of optical energy,” Sci. Rep. **4**(1), 5015 (2015). [CrossRef]

**19. **H. S. Chu, E. P. Li, P. Bai, and R. Hegde, “Optical performance of single-mode hybrid dielectric-loaded plasmonic waveguide-based components,” Appl. Phys. Lett. **96**(22), 221103 (2010). [CrossRef]

**20. **X. Y. He, Q. J. Wang, and S. F. Yu, “Numerical study of gain-assisted terahertz hybrid plasmonic waveguide,” Plasmonics **7**(3), 571–577 (2012). [CrossRef]

**21. **D. Chen, “Cylindrical hybrid plasmonic waveguide for subwavelength confinement of light,” Appl. Opt. **49**(36), 6868–6871 (2010). [CrossRef] [PubMed]

**22. **L. Chen, T. Zhang, X. Li, and W. Huang, “Novel hybrid plasmonic waveguide consisting of two identical dielectric nanowires symmetrically placed on each side of a thin metal film,” Opt. Express **20**(18), 20535–20544 (2012). [CrossRef] [PubMed]

**23. **M. Z. Alam, J. S. Aitchison, and M. Mojahedi, “A marriage of convenience: Hybridization of surface plasmon and dielectric waveguide modes,” Laser Photonics Rev. **8**(3), 394–408 (2014). [CrossRef]

**24. **D. Teng, Q. Cao, and K. Wang, “An extension of the generalized nonlocal theory for the mode analysis of plasmonic waveguides at telecommunication frequency,” J. Opt. **19**(5), 055003 (2017). [CrossRef]

**25. **Y. Gao and I. V. Shadrivov, “Second harmonic generation in graphene-coated nanowires,” Opt. Lett. **41**(15), 3623–3626 (2016). [CrossRef] [PubMed]

**26. **Y. Gao, G. Ren, B. Zhu, J. Wang, and S. Jian, “Single-mode graphene-coated nanowire plasmonic waveguide,” Opt. Lett. **39**(20), 5909–5912 (2014). [CrossRef] [PubMed]

**27. **P. Liu, X. Zhang, Z. Ma, W. Cai, L. Wang, and J. Xu, “Surface plasmon modes in graphene wedge and groove waveguides,” Opt. Express **21**(26), 32432–32440 (2013). [CrossRef] [PubMed]

**28. **Y. Y. Dai, X. L. Zhu, N. A. Mortensen, J. Zi, and S. S. Xiao, “Nanofocusing in a tapered graphene plasmonic waveguide,” J. Opt. **17**(6), 065002 (2015). [CrossRef]

**29. **F. Bonaccorso, Z. Sun, T. Hasan, and A. C. Ferrari, “Graphene photonics and optoelectronics,” Nat. Photonics **4**(9), 611–622 (2010). [CrossRef]

**30. **A. Politano and G. Chiarello, “Plasmon modes in graphene: status and prospect,” Nanoscale **6**(19), 10927–10940 (2014). [CrossRef] [PubMed]

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

**32. **X. He, P. Gao, and W. Shi, “A further comparison of graphene and thin metal layers for plasmonics,” Nanoscale **8**(19), 10388–10397 (2016). [CrossRef] [PubMed]

**33. **M. Jablan, H. Buljan, and M. Soljacic, “Plasmonics in graphene at infrared frequencies,” Phys. Rev. B Condens. Matter Mater. Phys. **80**(24), 245435 (2009). [CrossRef]

**34. **B. Wang, X. Zhang, F. J. García-Vidal, X. Yuan, and J. Teng, “Strong coupling of surface plasmon polaritons in monolayer graphene sheet arrays,” Phys. Rev. Lett. **109**(7), 073901 (2012). [CrossRef] [PubMed]

**35. **A. N. Grigorenko, M. Polini, and K. S. Novoselov, “Graphene plasmonics,” Nat. Photonics **6**(11), 749–758 (2012). [CrossRef]

**36. **Y. Ding, X. Guan, X. Zhu, H. Hu, S. I. Bozhevolnyi, L. K. Oxenløwe, K. J. Jin, N. A. Mortensen, and S. Xiao, “Efficient electro-optic modulation in low-loss graphene-plasmonic slot waveguides,” Nanoscale **9**(40), 15576–15581 (2017). [CrossRef] [PubMed]

**37. **W. Xu, Z. H. Zhu, K. Liu, J. F. Zhang, X. D. Yuan, Q. S. Lu, and S. Q. Qin, “Dielectric loaded graphene plasmon waveguide,” Opt. Express **23**(4), 5147–5153 (2015). [CrossRef] [PubMed]

**38. **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).

**39. **Y. Kim and M. S. Kwon, “Mid-infrared subwavelength modulator based on grating-assisted coupling of a hybrid plasmonic waveguide mode to a graphene plasmon,” Nanoscale **9**(44), 17429–17438 (2017). [CrossRef] [PubMed]

**40. **J. P. Liu, X. Zhai, L. L. Wang, H. J. Li, F. Xie, Q. Lin, and S. X. Xia, “Analysis of mid-infrared surface plasmon modes in a graphene-based cylindrical hybrid waveguide,” Plasmonics **11**(3), 703–711 (2016). [CrossRef]

**41. **M. Chen, P. Sheng, W. Sun, and J. Cai, “A symmetric terahertz graphene-based hybrid plasmonic waveguide,” Opt. Commun. **376**, 41–46 (2016). [CrossRef]

**42. **J. P. Liu, X. Zhai, L. L. Wang, H. J. Li, F. Xie, S. X. Xia, X. J. Shang, and X. Luo, “Graphene-based long-range SPP hybrid waveguide with ultra-long propagation length in mid-infrared range,” Opt. Express **24**(5), 5376–5386 (2016). [CrossRef] [PubMed]

**43. **L. Ye, K. Sui, Y. Liu, M. Zhang, and Q. H. Liu, “Graphene-based hybrid plasmonic waveguide for highly efficient broadband mid-infrared propagation and modulation,” Opt. Express **26**(12), 15935–15947 (2018). [CrossRef] [PubMed]

**44. **D. Wu, J. Tian, and R. Yang, “Study of mode performances of graphene-coated nanowire integrated with triangle wedge substrate,” J. Nonlinear Opt. Phys. Mater. **27**(02), 1850013 (2018). [CrossRef]

**45. **Y. Gao, G. Ren, B. Zhu, H. Liu, Y. Lian, and S. Jian, “Analytical model for plasmon modes in graphene-coated nanowire,” Opt. Express **22**(20), 24322–24331 (2014). [CrossRef] [PubMed]

**46. **M. Hajati and Y. Hajati, “High-performance and low-loss plasmon waveguiding in graphene-coated nanowire with substrate,” J. Opt. Soc. Am. B **33**(12), 2560–2565 (2016). [CrossRef]

**47. **Y. Huang, L. Zhang, H. Yin, M. Zhang, H. Su, I. L. Li, and H. Liang, “Graphene-coated nanowires with a drop-shaped cross section for 10 nm confinement and 1 mm propagation,” Opt. Lett. **42**(11), 2078–2081 (2017). [CrossRef] [PubMed]

**48. **A. R. Davoyan and N. Engheta, “Salient features of deeply subwavelength guiding of terahertz radiation in graphene-coated fibers,” ACS Photonics **3**(5), 737–742 (2016). [CrossRef]

**49. **H. Liang, L. Zhang, S. Zhang, T. Cao, A. Alù, S. Ruan, and C. W. Qiu, “Gate-Programmable Electro-Optical Addressing Array of Graphene-Coated Nanowires with Sub-10 nm Resolution,” ACS Photonics **3**(10), 1847–1853 (2016). [CrossRef]

**50. **D. A. Kuzmin, I. V. Bychkov, V. G. Shavrov, V. V. Temnov, H. I. Lee, and J. Mok, “Plasmonically induced magnetic field in graphene-coated nanowires,” Opt. Lett. **41**(2), 396–399 (2016). [CrossRef] [PubMed]

**51. **W. Wang, W. Zhou, T. Fu, F. Wu, N. Zhang, Q. Li, Z. Xu, and W. Liu, “Reduced propagation loss of surface plasmon polaritons on Ag nanowire-graphene hybrid,” Nano Energy **48**, 197–201 (2018). [CrossRef]

**52. **H. Gao, Q. Cao, D. Teng, M. Zhu, and K. Wang, “Perturbative solution for terahertz two-wire metallic waveguides with different radii,” Opt. Express **23**(21), 27457–27473 (2015). [CrossRef] [PubMed]

**53. **T. Zhang, L. Chen, and X. Li, “Graphene-based tunable broadband hyperlens for far-field subdiffraction imaging at mid-infrared frequencies,” Opt. Express **21**(18), 20888–20899 (2013). [CrossRef] [PubMed]

**54. **A. Yu. Nikitin, F. Guinea, F. J. Garcia-Vidal, and L. Martin-Moreno, “Fields radiated by a nanoemitter in a graphene sheet,” Phys. Rev. B Condens. Matter Mater. Phys. **84**(19), 195446 (2011). [CrossRef]

**55. **Y. Francescato, V. Giannini, and S. A. Maier, “Strongly confined gap plasmon modes in graphene sandwiches and graphene-on-silicon,” New J. Phys. **15**(6), 063020 (2013). [CrossRef]

**56. **Y. Zhou, Y. Y. Zhu, K. Zhang, H. W. Wu, R. W. Peng, R. H. Fan, and M. Wang, “Plasmonic band structures in doped graphene tubes,” Opt. Express **25**(11), 12081–12089 (2017). [CrossRef] [PubMed]

**57. **G. W. Hanson, “Quasi-transverse electromagnetic modes supported by a graphene parallel-plate waveguide,” J. Appl. Phys. **104**(8), 084314 (2008). [CrossRef]

**58. **J. Xu, N. Shi, Y. Chen, X. Lu, H. Wei, Y. Lu, N. Liu, B. Zhang, and J. Wang, “TM_{01} mode in a cylindrical hybrid plasmonic waveguide with large propagation length,” Appl. Opt. **57**(15), 4043–4047 (2018). [CrossRef] [PubMed]

**59. **R. Buckley and P. Berini, “Figures of merit for 2D surface plasmon waveguides and application to metal stripes,” Opt. Express **15**(19), 12174–12182 (2007). [CrossRef] [PubMed]

**60. **J. Grandidier, S. Massenot, G. Colas des Francs, A. Bouhelier, J. C. Weeber, L. Markey, and A. Dereux, “Dielectric-loaded surface plasmon polariton waveguides: figures of merit and mode characterization by image and Fourier plane leakage microscopy,” Phys. Rev. B Condens. Matter Mater. Phys. **78**(24), 245419 (2008). [CrossRef]