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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 μm, which is larger than that of the graphene-coated circular nanowire plasmon mode. Meanwhile, the effective mode area is only 10−4A0, 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 (TM0) 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 ε2. The two dielectric nanowire with permittivity of ε1 are coated with monolayer graphene. The radii of the two nanowires are R1 and R2, 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 εg = 1 + iσg/(ε0ωd) [28,31,53], where d is the thickness of monolayer graphene, and ω is the angular frequency of the incident light. ε0 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 σg = σintra + σinter. 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
where τ is the relaxation time, T is the temperature, uc is the chemical potential, ћ is the reduced plank constant, kB is the Boltzmann constant, and e = 1.6 × 10−19C. In what follows, we chose T = 300 K, τ = 0.5 ps, and ε2 = 1.
Fig. 1 Schematic of the cross section of the GCNWD. The radii of the two wires are R1 and R2, and the spacing between two nanowires is D.
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 = k0neff is the propagation constant, k0 = 2π/λ0, and U stands for electric or magnetic fields. neff is the effective mode index and calculated by the finite element method (FEM). The real part Re(neff) of the effective mode index is directly related to the dispersion and the imaginary part Im(neff) is related to the attenuation. Then the propagation length is defined as Lp = 1/(2α) with α = k0Im(neff) and calculated by Lp = λ0/[4πIm(neff)]. The normalized mode area is defined as Aeff/A0 with A0 = λ02/4 being the diffraction-limited mode area. The effective mode area Aeff is evaluated by [58]
where W(r) is the electromagnetic energy density and is given byFigure 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].
3. Results and discussion
3.1 Plasmon modes in GCNWDs
Figure 2 shows the mode properties of the GCNWD. Here, R1 = R2 = 100 nm, D = 50 nm, ε1 = 2.25, and ε2 = 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 (TM0 mode). For increasing the gap distance D, the coupling strength becomes weaker, and finally these two TM0 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)/k0 = Re(neff) 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 LP 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 μm). Figure 2(d) shows the normalized mode area (Aeff/A0) and FoM of the GPM with respect to frequency. For f0 = 20 THz, the propagation length is about 4.5 μm, and a very small effective mode area about 1.6 × 10−4A0 could be obtained. For higher frequencies, the higher absorption leads to the reduction of the performance (decease of FoM) of the GCNWD.
Fig. 2 (a) Field distributions of the fundamental GPM of the GCNWD at 30 THz. The white arrows indicate the polarization directions. (b) Amplitude of Poynting vector(Sz) of the GPM at frequencies of 20 THz, 30 THz, 40 THz, 50 THz and 60 THz. (c) Effective mode index and propagation length, and (d) Normalized mode area and FoM of the plasmon mode as a function of frequency. The parameters are uc = 0.5 eV, T = 300 K, τ = 0.5 ps, R1 = R2 = 100 nm, D = 50 nm, ε1 = 2.25, and ε2 = 1.
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 R1 and R2 could have different values. Due to the commutability of the geometry, one can either hold R1 or R2 constant while changing another. Here we set R1 = R2, D = 20 nm, and R1 varies from 50 nm to 200 nm. Figure 3(a) shows the effective mode index and propagation length with respect to R1 at the frequency of 30 THz. The effective mode indices increase with the enlargement of R1. For the fundamental GPM, decreasing R1 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 R1 reduces, the GCNWD has better performances, i.e., larger FoM.
Fig. 3 (a) Effective mode index and propagation length, and (b) Normalized mode area and FoM as a function of R1 at f0 = 30 THz. The other parameters are uc = 0.5 eV, T = 300 K, τ = 0.5 ps, D = 20 nm, ε1 = 2.25, and ε2 = 1.
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. R1 = R2 = 100 nm, and D ranges from 2 nm to 100 nm. For increasing D, the effective mode indices (Re(neff)) first rapidly decrease (solid blue line in Fig. 4(a)) for D<30 nm, and then moderately decrease. We can also see that Lp 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 (TM0 mode). For increasing the gap distance D, the coupling strength decreases, and finally these two TM0 modes decouple for D exceeding 400 nm here. The blue dashed line in Fig. 4(a) indicates the Re(neff) of the graphene-coated circular nanowire plasmon mode, which is 22.0893 at 30 THz for uc = 0.5 eV, T = 300 K, τ = 0.5 ps, R = 100 nm, ε1 = 2.25, and ε2 = 1. As D exceeding 400 nm, the Re(neff) 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.
Fig. 4 (a) Effective mode index and propagation length. The blue dashed line indicates the Re(neff) of the graphene-coated circular nanowire plasmon mode, and (b) Normalized mode area and FoM as a function of gap distance at 30 THz. The other parameters are uc = 0.5 eV, T = 300 K, τ = 0.5 ps, R1 = R2 = 100 nm, ε1 = 2.25, and ε2 = 1.
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 R1 = 100 nm, and calculate the FoM by varying R2 from 50 nm to 150 nm. Results show that the FoM has the highest value about 623 for R2 = 50 nm, and then decreases with increasing R2. Finally, the FoM decreases to 557 for R2 = 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 uc = 0.5 eV, T = 300 K, τ = 0.5 ps, R1 = R2 = 100 nm, D = 20 nm, and ε2 = 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 ε1. The irregular changes of Aeff/A0 with different permittivity could be ignored, since Aeff/A0 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.
Fig. 5 (a) Effective mode index and propagation length, (b) Normalized mode area and FoM as a function of nanowire permittivity at 30 THz. The other parameters are uc = 0.5 eV, T = 300 K, τ = 0.5 ps, R1 = R2 = 100 nm, D = 20 nm, and ε2 = 1.
The surface conductivity σg of graphene could be tuned by changing chemical potential uc. Figure 6 shows the chemical potential dependent characteristics of the GPM at f0 = 30 THz. The other parameters are T = 300 K, τ = 0.5 ps, R1 = R2 = 100 nm, D = 20 nm, ε1 = 2.25, and ε2 = 1. We change uc from 0.2 eV to 1 eV. The increase of chemical potential provides two large benefits. First, Re(neff) and Im(neff) decrease with increasing uc, implying an increase in propagation length as depicted in Fig. 6(a). Second, the normalized mode area enlarges only about 10% when uc 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 uc 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.
Fig. 6 (a) Effective mode index and propagation length, and (b) Normalized mode area and FoM as a function of uc at f0 = 30 THz. The other parameters are T = 300 K, τ = 0.5 ps, R1 = R2 = 100 nm, D = 20 nm, ε1 = 2.25, and ε2 = 1.
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 uc 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, ε1 = 2, ε2 = 1, uc = 1 eV, R1 = R2 = 50 nm, and f0 = 30 THz. With the increase of gap distance, the propagation length could reach about 9 μm, which is larger than that of the graphene-coated circular nanowire plasmon mode [45,46]. Meanwhile, the effective mode area is only 10−4A0, 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 uc. 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.
Fig. 7 Improved mode properties of the GPM. The related parameters are T = 300 K, τ = 0.5 ps, ε1 = 2, ε2 = 1, uc = 1 eV, R1 = R2 = 50 nm, and f0 = 30 THz.
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).
Fig. 8 (a) Three kinds of graphene-coated nanowire based waveguides. (b) Effective mode index. (c) Propagation length. (d) Normalized mode area. The parameters are uc = 1 eV, T = 300 K, τ = 0.5 ps, R = 100 nm, D = 30 nm, ε1 = 3, and ε2 = 1.
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 (λeff = λ0/Re(neff)) 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 Lp compare with that of graphene-coated circular nanowire. At lower frequencies, Type B waveguide has larger Lp 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 R1 = R2 = 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 μm 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 μm at low frequencies (f0<35 THz).
Fig. 9 (a) Mode patterns (Ez) of first 5 order modes at 60 THz. (b) Effective mode index. (c) Propagation length. The parameters are uc = 1 eV, T = 300 K, τ = 0.5 ps, R1 = R2 = 100 nm, D = 30 nm, ε1 = 3, and ε2 = 1.
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 μm-8 μm and very small mode area about 10−4A0. In other words, the above results clearly indicate that the proposed GCNWDs are quite tolerant to fabrication errors, which is beneficial for its implementations.
Fig. 10 Energy density distributions of GPMs for four extended GCNWDs. (a) Two circular nanowires with R1 = 100 nm, R2 = 200 nm, LP = 6.7 μm, Aeff = 0.85 × 10−4A0. (b) Circular and elliptical nanowires with R1 = 100 nm, a = 100 nm, b = 200 nm, LP = 6.4 μm, Aeff = 0.87 × 10−4A0. (c) Two elliptical nanowires with a = 100 nm, b = 200 nm, LP = 6 μm, Aeff = 1 × 10−4A0. (d) Two elliptical nanowires with a = 200 nm, b = 100 nm, LP = 8.5 μm, Aeff = 0.57 × 10−4A0. a and b stand for the short axis and long axis, respectively.The other parameters are uc = 1 eV, T = 300 K, τ = 0.5 ps, D = 30 nm, ε1 = 2.25, ε2 = 1, and f0 = 20 THz.
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 μm and ultra-small modal area about 10−4A0 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.
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