## Abstract

A graphene-based long-range surface plasmon polariton (LRSPP) hybrid waveguide, which is composed of two identical outer graphene nanoribbons and two identical inner silica layers symmetrically placed on both sides of a silicon layer, is investigated using the finite-difference time-domain method. By combining the simulated results with the coupled mode perturbation theory, we demonstrate that the LRSPP and short-range SPP (SRSPP) modes originate from the coupling of the same modes of the two graphene nanoribbons. For the LRSPP mode, an ultra-long propagation length (~10 μm) and an ultra-small mode area (~10^{−7} *A*_{0}, where *A*_{0} is the diffraction-limited mode area) can be simultaneously achieved. This waveguide can be used for future photonic integrated circuits functional in the mid-infrared range.

© 2016 Optical Society of America

## 1. Introduction

Mid-infrared electromagnetic (EM) waves, with wavelengths typically ranging from 3 to 30 μm, have attracted intensive research interest because of their wide application in the fields of communication, biomedicine, spectroscopy, homeland security [1, 2], etc. It would be highly desired to design an optical waveguide with ultra-long propagation length *L _{prop}* and ultra-deep sub-wavelength confinement, because such a device can be used for a basic component to transmit mid-infrared EM waves in certain practical applications, such as optical modulation [3], sensing [4], filtering [5], and near-field imaging [6].

Surface plasmon polartions (SPPs), which are a collective excitation of electrons along a metal-dielectric interface, have relatively low propagation loss as well as deep sub-wavelength confinement from the visible to infrared range [7]. Thus, the SPP-based waveguide is a promising candidate for future application to guide EM waves in this region. Noble metals are typically regarded as the most suitable SPP materials currently available [8], and various noble-metal-based SPP waveguides have been proposed [9–16]. Among them, the long-range SPP (LRSPP) waveguide [13–16], which is composed of a thin metal stripes embedded in infinite homogeneous background dielectric material, can be employed to guide an EM field over a long propagation distance (~10^{4}–10^{6} μm). However, these types of LRSPP waveguide suffer from poor field confinement (~10^{−1}–10 *A*_{0}, where *A*_{0} is the diffraction limited mode area), which renders them unsuitable for compact integration. Recently, a type of metal-based LRSPP hybrid waveguide [17–21], which is composed of two identical low-index inner dielectric layers and two identical high-index outer dielectric layers symmetrically placed on both sides of a thin metal film, has attracted strong research interest. This attention is because of this waveguide’s ability to achieve deep sub-wavelength confinement (~10^{−2}–10^{−1} *A*_{0}) with comparable *L _{prop}* (~10

^{4}–10

^{5}μm), as opposed to the conventional LRSPP waveguide. However, these waveguides are only suitable for application from the near-infrared to visible waveband, and are especially focused on the telecommunication wavelength of

*λ*= 1.55 µm. In the mid-infrared range, SPPs have very weak confinement on the metal surface [22,23], rendering this device unsuitable for use as an SPP waveguide in this waveband.

Graphene, a single-layer combined carbon atom sheet [24], is a promising candidate for SPP wave guiding in the mid-infrared range [25,26]^{, and} various graphene-based SPP waveguides have been investigated [22,23,25–28]· Especially, the monolayer and multilayer graphene nanoribbons waveguides have been widely researched and been applied to many fields in recent year [29–35]. Compared with the metal-based SPP waveguide design, the graphene-based SPP waveguide exhibits extremely strong mode confinement, low propagation loss, and tunable electromagnetic properties [36]. Nonetheless, in the mid-infrared waveband, the graphene plasmon also suffers from high absorption loss because the majority of the light energy is located in the graphene. Inevitably, the *L _{prop}* of this type of waveguide remains relatively small (~10

^{0}μm). Thus, the key challenge to the graphene-based SPP waveguide is determining a means of dramatically improving the

*L*while maintaining the existing (or a similar) degree of confinement.

_{prop}Combining the unique optical characteristics of graphene with the analogous principles of the metal-based LRSPP hybrid waveguide, in this paper, we propose a graphene-based LRSPP hybrid waveguide, which is composed of two identical outer graphene nanoribbons and two identical inner silica layers symmetrically placed on both sides of a silicon layer. Based on the coupled-mode perturbation theory [37], we demonstrate that the LRSPP and SRSPP modes originate from the coupling of the same modes of the two graphene nanoribbons. Moreover, the LRSPP mode of this waveguide is capable of achieving an ultra-long *L _{prop}* (~10 μm), which is the largest

*L*of all the reported graphene-based waveguides to date (to the best of our knowledge). Further, for this waveguide, an ultra-deep sub-wavelength confinement (~10

_{prop}^{−7}

*A*

_{0}) similar to that of other conventional graphene-based hybrid waveguides is maintained [28, 29].

## 2. Theoretical model

Figure 1 is a schematic of the proposed graphene-based LRSPP hybrid waveguide, where, as noted above, two identical outer graphene nanoribbons and two identical inner silica layers are symmetrically placed on both sides of a thin silicon layer. The thicknesses of the silica and silicon layers are labeled as *t* and *d*, respectively, and the width of the waveguide is labeled as *W*. The permittivities of the silica and silicon are *ε*_{1} and *ε*_{2}, respectively. Further, the permittivity of graphene can be calculated using *ε _{g}* = 2.5 + i

*σ*/(

_{g}*ε*

_{0}

*ω*Δ), where

*σ*is the conductivity of graphene, Δ is the thickness of the given graphene specimen, and

_{g}*ω*is the angular frequency of the incident light. Note that

*σ*can be calculated from the Kubo formula [34, 38, 39], which depends on the Fermi energy

_{g}*E*, the ambient temperature

_{f}*T*,

*ω*, and the electron relaxation time

*τ*. The latter is related to the electron mobility

*µ*and the Fermi velocity

*v*by

_{F}*τ*=

*µE*(

_{f}/*ev*

_{F}^{2}). In our calculations, we set the parameters as

*T*= 300 K,

*µ*= 1.0 m

^{2}/(Vs), Δ = 0.5 nm,

*ε*

_{1}= 2.09, and

*ε*

_{2}= 11.9, and the incident wavelength in vacuum is

*λ*

_{0}= 10 µm.

It is well known that two types of EM field modes can be simultaneously propagated along this waveguide, including the graphene SPP mode (GSPPM) and the dielectric waveguide mode (DWM). Mutual coupling of the EM fields can occur between any two modes, including GSPPM-GSPPM, DWM-DWM, and GSPPM-DWM. As the coupling strength of the DWM-DWM configuration is significantly smaller than that of GSPPM-GSPPM, DWM-DWM coupling can be ignored. Further, the GSPPM-DWM coupling can also be ignored, because of the effective index mismatching [18]. Thus, the operating pattern of the proposed waveguide is equivalent to GSPPM-GSPPM coupling, and we can study the physical mechanism of the waveguide by investigating the GSPPM-GSPPM coupling characteristics.

Normally, both the graphene nanoribbons can support several SPP modes. To avoid the effective index mismatching mentioned above, we consider the coupling of the same GSPPMs only, and focus on the coupling of fundamental GSPPMs in particular, because they exhibit the maximum coupling strength. According to the coupled mode perturbation theory [37], in the perturbation approximation, the EM-field arbitrary component of this waveguide, ${\Phi}_{i}(x,y,z)$, can be expressed as a linear superposition of two GSPPMs

*β*is the GSPPM propagation constant. Further, ${\Phi}_{i}^{(1)}(x,y)\mathrm{exp}(i\beta z)$and ${\Phi}_{i}^{(2)}(x,y)\mathrm{exp}(i\beta z)$represent the two SPP modes originating from the two graphene nanoribbons, respectively, with

*A*

_{1}(

*z*) and

*A*

_{2}(

*z*) being the respective superposition coefficients of the two modes. We can derive

*A*

_{1}(

*z*) =

*i*sin(

*K*) and

_{c}z*A*

_{2}(

*z*) = cos(

*K*) using the coupled mode perturbation theory, where

_{c}z*K*is the coupling coefficient of the two modes.

_{c}*Kc*represents the capabilities of exchanging energy of the two graphene nanoribbons. In general, when two identical modes of the two graphene nanoribbons coupled each other, the energy exchanging becomes easier with decreasing

*d*, and

*K*increases. Substituting

_{c}*A*

_{1}(

*z*) and

*A*

_{2}(

*z*) into Eq. (1) and utilizing the Euler formula, we obtain

*Φ*exp(

_{s}*iβ*) and

_{s}z*Φ*exp(

_{a}*iβ*) represent the symmetric coupling mode (SCM) due to the constructive interference of the two GSPPMs and the anti-symmetric coupling mode (ASCM) due to the destructive interference of the two GSPPMs, respectively.

_{a}z*β*and

_{s}*β*are the propagation constants of the SCM and the ASCM, respectively, with

_{a}*β*=

_{s}*β*+

*K*and

_{c}*β*=

_{a}*β*–

*K*. Thus, the SCM and ASCM of this waveguide originate from the coupling of the same modes of the two graphene nanoribbons. In other words, the fundamental SCM and the fundamental ASCM originate from the coupling of the fundamental modes of the two graphene nanoribbons, for example.

_{c}We also employ the finite-difference time-domain (FDTD) method by the commercial software of the Lumerical FDTD Solutions to investigate the propagation properties of the proposed waveguide. Due to FDTD is a time-domain technique, the EM fields are solved as a function of time. Based on the time-domain function of the EM fields, the Lumerical FDTD Solutions is used to calculate the EM fields as a frequency-domain function by performing Fourier transforms. The perfectly matched absorbing boundary condition and the non-uniform mesh are applied at the simulation space. The mesh sizes inside the two graphene nanoribbons along the *x*, *y*, and *z* axes are set as 0.05, 0.05, and 5 nm, respectively, and the mesh size gradually increases outside the two graphene nanoribbons. The “mode source” of the Lumerical FDTD Solution, which can solve all EM field modes along the waveguide by mode solver, is used to analyze the characteristics of waveguide modes, including the EM field patterns, the graphene charge distributions, the effective index, and the propagation length. For our proposed waveguide, the TM polarization modes can be obtained by the mode solver of the “mode source”.

## 3. LRSPP hybrid mode in proposed waveguide

Combining Eq. (2) with the numerical simulation by FDTD, we investigate the mode characteristics of the proposed waveguide. First, we study the mode features of the fundamental SCM and ASCM by varying *d*. Figure 2(a) shows the effective index *n _{eff}* as a function of

*d*for the two modes, which can be expressed as

*n*=

_{eff}*n*+ i

_{r}*n*, where

_{i}*n*and

_{r}*n*correspond to the real and imaginary parts of

_{i}*n*, respectively.

_{eff}*β*and

*n*exhibit similar trend variations for

_{r}*β*=

*n*

_{r}β_{0}, where

*β*

_{0}is the propagation constant of incident light in vacuum. With increasing

*d*, energy exchange between the two graphene nanoribbons becomes more difficult. Thus,

*K*decreases. As

_{c}*β*=

_{s}*β*+

*K*and

_{c}*β*=

_{a}*β*–

*K*,

_{c}*β*and (

_{s}*n*)

_{r}*decrease while*

_{s}*β*and (

_{a}*n*)

_{r}*increase, where (*

_{a}*n*)

_{r}*and (*

_{s}*n*)

_{r}*correspond to the*

_{a}*n*value of the SCM and ASCM, respectively. When

_{r}*d*becomes quite large,

*K*tends to 0. Note that

_{c}*β*≈

_{s}*β*≈(

_{a}*β*)

_{∞}and (

*n*)

_{r}*≈(*

_{s}*n*)

_{r}*≈(*

_{a}*n*)

_{r}_{∞}, where (

*β*)

_{∞}and (

*n*)

_{r}_{∞}represent the values of

*β*and

*n*as

_{r}*d*→ ∞, respectively. However, when

*d*is very small, (

*n*)

_{r}*> (*

_{s}*n*)

_{r}*, and (*

_{a}*λ*)

_{SPP}_{s}< (

*λ*)

_{SPP}*, where (*

_{a}*λ*)

_{SPP}_{s}and (

*λ*)

_{SPP}*represent the SPP wavelengths of the SCM and ASCM, respectively. For the fundamental SCM and ASCM, when*

_{a}*t*= 4 nm,

*W*= 100 nm,

*E*= 0.8 eV, and

_{f}*d*= 8 nm, we can obtain (

*n*)

_{r}*= 79.6, and (*

_{s}*n*)

_{r}*= 21.6. Thus, (*

_{a}*λ*)

_{SPP}*= 125.6 nm, and (*

_{s}*λ*)

_{SPP}*= 462.9 nm. Figure 2(e) shows that the fundamental SCM completes 10 harmonic oscillations with a propagation distance of 1256 nm, while Fig. 2(f) shows that the fundamental ASCM completes two harmonic oscillations with a propagation distance of 926 nm.*

_{a}On the other hand, (*n _{i}*)

*< (*

_{s}*n*)

_{i}*as*

_{a}*d*> 20 nm. With decreasing

*d*, (

*n*)

_{i}*first increases gradually and then decreases sharply as*

_{a}*d*< 30 nm. In particular, when

*d*< 20nm, (

*n*)

_{i}

_{s}_{>}(

*n*)

_{i}*. For instance, for*

_{a}*d*= 8nm, (

*n*)

_{i}*= 0.362 and (*

_{s}*n*)

_{i}*= 0.146. In that case, the loss of the fundamental SCM is significantly larger than that of the fundamental ASCM, owing to the fact that the loss of an optical waveguide is given by*

_{a}*l*[dB/µm] = –8.86

_{m}*n*

_{i}k_{0}[40]. Therefore, from Figs. 2(e) and 2(f), we can determine that the electric |

*E*| field distribution of the SCM is attenuated significantly along the propagation direction of the SPP wave, while that of the ASCM exhibits no obvious attenuation.

The propagation length can be expressed as *L _{prop}* = 1/[Im(

*n*)

_{eff}*k*

_{0}] =

*λ*

_{0}/2π

*n*[23, 29]. Figure 2(b) shows (

_{i}*L*)

_{prop}*> (*

_{s}*L*)

_{prop}*as*

_{a}*d*> 20nm, where (

*L*)

_{prop}*and (*

_{s}*L*)

_{prop}*represent the propagation lengths of the fundamental SCM and ASCM, respectively. However, (*

_{a}*L*)

_{prop}*increases dramatically as*

_{a}*d*< 20 nm, and is significantly larger than (

*L*)

_{prop}*. Therefore, we can define SCM and ASCM as the SRSPP and LRSPP modes, respectively. These definitions are exactly opposite to those of the metal-based LRSPP hybrid waveguide [17–21]. To compare the propagation performance of our proposed waveguide, we also investigate the propagation properties of a normal graphene-based hybrid waveguide, which is composed of a silica layer sandwiched between a graphene layer and a silicon layer, as shown in the insert of Fig. 2(b). The parameters of this waveguide are same with that of the LRSPP hybrid waveguide. Figure 2(b) show that, with decreasing*

_{s}*d*, the propagation length of the normal hybrid SPP (NHSPP) fundamental mode (

*L*)

_{prop}*first decrease gradually and then increase rapidly as*

_{n}*d*< 12 nm. This attributes the fact that the SPP confinement decrease with

*d*. Nonetheless, (

*L*)

_{prop}*is significantly smaller than (*

_{n}*L*)

_{prop}*as*

_{a}*d*< 20 nm. This indicates that the increase of (

*L*)

_{prop}*not only comes from the decrease of the SPP confinement, but also mostly comes from the increase of the coupling of the two graphene nanoribbons.*

_{a}Next, we study the physical mechanism behind the difference between the LRSPP and SRSPP modes. For simplicity, we further define the SRSPP and LRSPP fundamental mode as modes 1' and 1, respectively. It is possible to differentiate between these modes by analyzing their respective *E _{y}* field distributions [29]. Figures 3(a), 3(b), and 3(e) show that the

*E*field distribution of mode 1 is anti-symmetric with respect to the

_{y}*x*-

*z*plane. Conversely, Figs. 3(c), 3(d), and 3(f) show that the

*E*field distribution of mode 1' is symmetric with respect to the

_{y}*x*-

*z*plane. Further, using the EM-field boundary condition, $\overrightarrow{n}\cdot ({\overrightarrow{D}}_{2}-{\overrightarrow{D}}_{1})=\sigma $, we can determine the charge distributions of modes 1 and 1', as shown in Figs. 3(a)–3(f). Here, mode 1 has like charges while mode 1' has opposite charges in the opposite position of the two graphene nanoribbons. Based on such charge distributions, by the way, the mode 1 and 1' can also be excited by a

*z*-oriented and

*y*-oriented polarized electric dipole source placed in the middle end of the proposed waveguide [9], respectively. When

*d*> 30 nm, a strong

*E*distribution is located between the two graphene nanoribbons for both modes, as shown in Figs. 3(b) and 3(c). This indicates that there is a strong interaction between the EM field and the two graphene nanoribbons, which causes additional propagation loss. Thus, the

_{y}*L*values of modes 1 and 1' are very small (approximately 3–5 μm), and are only comparable to those of other conventional graphene-based hybrid waveguides [28, 29]. However, with decreasing

_{prop}*d*, the

*E*distribution of mode 1 almost disappears between the two graphene nanoribbons because of the repulsion between like charges, as shown in Fig. 3(e). Thus, the interaction between the EM field and the two graphene nanoribbons decreases rapidly and, hence, the propagation loss is decreased and

_{y}*L*is increased significantly. Conversely, the

_{prop}*E*distribution of mode 1' is enhanced owing to the attraction between opposite charges, as shown in Fig. 3(f). Therefore, the interaction between the EM field and the two graphene nanoribbons is increased, which induces a slight increasing in the propagation loss and a decrease in

_{y}*L*.

_{prop}Then, we discuss the dependence of the LRSPP-mode characteristics on *t*. The EM field confinement can be described using the normalized mode area, *A _{eff}* /

*A*

_{0}, where

*A*

_{0}=

*λ*

_{0}

^{2}/4. Note that

*A*is the effective mode area, which is defined as ${A}_{eff}={\displaystyle \int W(r)ds/\mathrm{max}[W(r)]},$ where

_{eff}*W*(

*r*) is the EM field energy density, given by

*n*/

_{r}*n*.

_{i}The waveguide coupling strength can be set to weak or strong based on *d*. However, the silica layer plays a different role in different coupling scenarios. For weak coupling (for *d* > 30 nm, approximately), the waveguide is similar to an ordinary graphene-based hybrid waveguide. The SPP is excited at the two graphene nanoribbons and the electric field decays exponentially at both sides of the two nanoribbons. As the normal boundary conditions of the electric field is *ε*_{1}*E*_{1}* _{y}* =

*ε*

_{2}

*E*

_{2}

*, a strong electric field is induced in the silica layer due to*

_{y}*ε*

_{1}

*< ε*

_{2}, as shown in Fig. 4(e). This is equivalent to transferring the electric field energy from the graphene nanoribbons and silicon layer to the silica layers. Thus, the silica layers play a role of storing energy. With increasing

*t*, more energy can be transferred to these layers. This indicates that the absorption losses of the graphene nanoribbons decrease and the

*L*of the waveguide increases. Simultaneously, the confinement of the EM field increases, whereas (

_{prop}*A*/

_{eff}*A*

_{0}) decreases. Obviously, the FOM of the waveguide is improved with increasing

*t*. However, in the strong coupling case (for

*d*< 20 nm, approximately), the majority of the electric field’s energy shifts from the silica layer to the outer space of the waveguide, as shown in Fig. 4(f). In this scenario, the storing-energy capacity of the silica layer is weakened. Note that the waveguide characteristics are primarily determined by the coupling distance. With decreasing

*t*, the coupling distance decreases, which results in a slight improvement in the waveguide performance. On the other hand, whether in a weak-coupling or strong-coupling scenario, the normalized mode area of this waveguide is extremely small (approximately 10

^{−7}

*A*

_{0}). This is similar to the smallest value reported for another graphene-based hybrid waveguide [29]. The other interesting point to be noted is that the overall performance of the waveguide containing silica layers is clearly superior to that of the waveguide with no silica layers (

*t*= 0 nm), as shown in Figs. 4(a)–4(d).

The *L _{prop}* of the graphene-based waveguide can be effectively improved by enhancing the

*E*of the graphene. The Fermi level depends on the carrier concentration, which can be controlled via electrical gating or chemical doping. Experimental carrier density values of as high as 10

_{f}^{14}cm

^{−2}have been reported [41], which is equivalent to

*E*= 1.17 eV. Therefore, we set the Fermi level tuning range from 0.4 to 1.0 eV. For comparison, two grahphene-based waveguides, including the LRSPP hybrid waveguide and the normal hybrid waveguide, are investigated with identical parameters of

_{f}*t*= 4 nm,

*d*= 6nm, and

*W*= 100 nm. Figure 5(a) shows that (

*L*)

_{prop}*and (*

_{a}*L*)

_{prop}*increase monotonically with*

_{n}*E*. Moreover, with increasing

_{f}*E*, the increase of (

_{f}*L*)

_{prop}*is more sensitive to*

_{a}*E*than that of (

_{f}*L*)

_{prop}*. Therefore, our proposed waveguide has more potential for improving*

_{n}*L*, compared with the graphene-based normal hybrid waveguide.

_{prop}The *L _{prop}* of the proposed waveguide can also be effectively improved by changing

*W*. In this experiment, we set

*t*= 4 nm and

*E*= 0.8 eV and vary

_{f}*W*from 20 to 180 nm with different

*d*. Figure 5(b) shows that the

*L*of mode 1 is insensitive to

_{prop}*W*under weak coupling case. However, under strong coupling, the

*L*of mode 1 rapidly increases with

_{prop}*W*at first, and then gradually tends to a constant value as

*W*> 100 nm. This can be explained as follows. For the LRSPP mode, as mentioned above, the EM fields of the two graphene nanoribbons are mutually exclusive, because of the existing like charges in their opposing positions. Obviously, with increasing

*W*, greater EM-field energy is excluded from the two graphene nanoribbons. Thus,

*L*increases with

_{prop}*W*. On the other hand, since the majority of the EM field energy is excluded from the waveguide for

*W*> 100 nm,

*L*is then insensitive to

_{prop}*W*. Hence, we typically select

*W*= 100 nm in our calculations.

Finally, we investigate the dispersion relation of this waveguide. Figure 6(a) and 6(b) show that the *n _{r}* values of the LRSPPs and SRSPPs decrease monotonically with increasing

*λ*

_{0}. The higher-order modes have cutoff wavelengths, at which

*n*tends toward zero and

_{r}*n*tends toward infinity. Note that the higher-order modes correspond to smaller cutoff wavelengths and that the SRSPP modes have larger cutoff wavelengths than the LRSPP modes of same mode order. For example, for

_{i}*t*= 4 nm,

*d*= 4nm,

*W*= 100 nm, and

*E*= 0.8 eV, the cutoff wavelengths of SRSPP modes 2', 3′, and 4' are 16.5, 8.5, and 6.26 μm, respectively. Correspondingly, the cutoff wavelengths of LRSPP modes 2, 3, and 4 are 7, 5.5, and 4.8 μm, respectively. The fundamental modes are cutoff-free. The

_{f}*L*values of the different modes have different trend variations with increasing

_{prop}*λ*

_{0}. For the fundamental modes, the

*L*increases monotonically with

_{prop}*λ*

_{0}, and mode 1 corresponds to a significantly increasing value. For the higher-order modes,

*L*increases at first and then rapidly drops to zero at the cutoff wavelength. Note that the

_{prop}*L*values of the higher-order LRSPP modes are significantly smaller than that of mode 1. This can be primarily attributed to the fact that mode 1 and the higher-order LRSPP modes correspond to the graphene SPP edge mode and the graphene SPP waveguide modes [42], respectively, as shown in Figs. 6(d)–6(g). In comparison to the edge mode, the waveguide modes have larger energy confined on the graphene nanoribbon surface, which corresponds to larger propagation losses and smaller

_{prop}*L*values.

_{prop}## 4. LRSPP hybrid mode in proposed waveguide with substrate

For practical applications, we place the graphene-based LRSPP hybrid waveguide on a buffer layer of thickness *h* on top of a substrate, as shown in Fig. 7(a). In this case, the effective index matching is destroyed because of the asymmetric waveguide profile. Then, the coupling strength of the GSPPMs inevitably decreases. The waveguide EM-field distribution becomes asymmetric, as shown in Fig. 7(b) and its insets. From Fig. 7(c), it is apparent that the *L _{prop}* of the asymmetric waveguide is smaller than that of the symmetric waveguide discussed in previous research, which results in a decrease in performance. Nonetheless, the asymmetric waveguide has similar LRSPP mode properties to the symmetric waveguide. Moreover, the asymmetric waveguide has a significantly increased

*L*and a comparable ultra-small mode area compared to other conventional graphene-based hybrid waveguides [28,29].

_{prop}## 5. Conclusion

In summary, we have proposed and investigated a graphene-based LRSPP hybrid waveguide, which is composed of two identical outer graphene nanoribbons and two identical inner silica layers symmetrically placed on both sides of a silicon layer. The coupled mode perturbation theory shows that the LRSPP and SRSPP modes originate from the coupling of the same modes of the two graphene nanoribbons, which agrees well with the numerical simulation results. Compared with the normal graphene-based hybrid waveguide design, the proposed waveguide with the LRSPP fundamental mode has a distinct advantage that it can simultaneously achieve an ultra-long propagation length (~10 μm) and an ultra-small mode area (~10^{−7}*A*_{0}) through variation of the silica- and silicon-layer thicknesses, the width of the waveguide, and the Fermi energy of the graphene. This waveguide can be used to construct various functional devices for guiding mid-infrared SPP waves in future photonic integrated circuits.

## Acknowledgments

This work was supported by the National Natural Science Foundation of China (NSFC) (Grant Nos. 61505052, 11074069, 61176116).

## References and links

**1. **R. Soref, “Mid-infrared photonics in silicon and germanium,” Nat. Photonics **4**(8), 495–497 (2010). [CrossRef]

**2. **B. Ferguson and X. C. Zhang, “Materials for terahertz science and technology,” Nat. Mater. **1**(1), 26–33 (2002). [CrossRef] [PubMed]

**3. **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]

**4. **O. Limaj, F. D’Apuzzo, A. D. Gaspare, V. Giliberti, F. Domenici, S. Sennato, F. Bordi, S. Lupi, and M. Ortolani, “Mid-Infrared Surface Plasmon Polariton Sensors Resonant with the Vibrational Modes of Phospholipid Layers,” J. Phys. Chem. C **117**(37), 19119–19126 (2013). [CrossRef]

**5. **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]

**6. **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]

**7. **W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature **424**(6950), 824–830 (2003). [CrossRef] [PubMed]

**8. **P. R. West, S. Ishii, G. V. Naik, N. K. Emani, V. M. Shalaev, and A. Boltasseva, “Searching for better plasmonic materials,” Laser Photonics Rev. **4**(6), 795–808 (2010). [CrossRef]

**9. **S. Sun, H. T. Chen, W. J. Zheng, and G. Y. Guo, “Dispersion relation, propagation length and mode conversion of surface plasmon polaritons in silver double-nanowire systems,” Opt. Express **21**(12), 14591–14605 (2013). [CrossRef] [PubMed]

**10. **S. E. Kocabaş, G. Veronis, D. A. B. Miller, and S. Fan, “Modal analysis and coupling in metal-insulator- metal waveguides,” Phys. Rev. B **79**(3), 035120 (2009). [CrossRef]

**11. **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]

**12. **Y. Bian and Q. Gong, “Low-loss light transport at the subwavelength scale in silicon nano-slot based symmetric hybrid plasmonic waveguiding schemes,” Opt. Express **21**(20), 23907–23920 (2013). [CrossRef] [PubMed]

**13. **A. V. Krasavin and A. V. Zayats, “Numerical analysis of long-range surface plasmon polariton modes in nanoscale plasmonic waveguides,” Opt. Lett. **35**(13), 2118–2120 (2010). [CrossRef] [PubMed]

**14. **P. Berini, “Plasmon-polariton modes guided by a metal film of finite width bounded by different dielectrics,” Opt. Express **7**(10), 329–335 (2000). [CrossRef] [PubMed]

**15. **R. Charbonneau, N. Lahoud, G. Mattiussi, and P. Berini, “Demonstration of integrated optics elements based on long-ranging surface plasmon polaritons,” Opt. Express **13**(3), 977–984 (2005). [CrossRef] [PubMed]

**16. **R. Adato and J. P. Guo, “Modification of dispersion, localization, and attenuation of thin metal stripe symmetric surface plasmon-polariton modes by thin dielectric layers,” J. Appl. Phys. **105**(3), 034306 (2009). [CrossRef]

**17. **Z. Zhang and J. Wang, “Long-range hybrid wedge plasmonic waveguide,” Sci. Rep. **4**, 6870 (2014). [CrossRef] [PubMed]

**18. **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]

**19. **J. Chen, Z. Li, S. Yue, and Q. Gong, “Hybrid long-range surface plasmon-polariton modes with tight field confinement guided by asymmetrical waveguides,” Opt. Express **17**(26), 23603–23609 (2009). [CrossRef] [PubMed]

**20. **Y. S. Bian and Q. H. Gong, “Multilayer metal-dielectric planar waveguides for subwavelength guiding of long-range hybrid plasmon polaritons at 1550nm,” J. Opt. **16**(1), 015001 (2014). [CrossRef]

**21. **L. Chen, X. Li, G. P. Wang, W. Li, S. H. Chen, L. Xiao, and D. S. Gao, “A silicon-based 3-D hybrid long-range plasmonic waveguide for nanophotonic integration,” J. Lightwave Technol. **30**(1), 163–168 (2012). [CrossRef]

**22. **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]

**23. **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]

**24. **K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, M. I. Katsnelson, I. V. Grigorieva, S. V. Dubonos, and A. A. Firsov, “Two-dimensional gas of massless Dirac fermions in graphene,” Nature **438**(7065), 197–200 (2005). [CrossRef] [PubMed]

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

**26. **L. Ju, B. Geng, J. Horng, C. Girit, M. Martin, Z. Hao, H. A. Bechtel, X. Liang, A. Zettl, Y. R. Shen, and F. Wang, “Graphene plasmonics for tunable terahertz metamaterials,” Nat. Nanotechnol. **6**(10), 630–634 (2011). [CrossRef] [PubMed]

**27. **A. Y. Nikitin, F. Guinea, F. J. García-Vidal, and L. Martín-Moreno, “Fields radiated by a nanoemitter in a graphene sheet,” Phys. Rev. B **84**(19), 195446 (2011). [CrossRef]

**28. **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 **95**, 1-9 (2015).

**29. **S. He, X. Zhang, and Y. He, “Graphene nano-ribbon waveguides of record-small mode area and ultra-high effective refractive indices for future VLSI,” Opt. Express **21**(25), 30664–30673 (2013). [CrossRef] [PubMed]

**30. **J. Christensen, A. Manjavacas, S. Thongrattanasiri, F. H. L. Koppens, and F. J. García de Abajo, “Graphene plasmon waveguiding and hybridization in individual and paired nanoribbons,” ACS Nano **6**(1), 431–440 (2012). [CrossRef] [PubMed]

**31. **D. Correas-Serrano, J. S. Gomez-Diaz, J. Perruisseau-Carrier, and A. Álvarez-Melcón, “Graphene-based plasmonic tunable low-pass filters in the terahertz band,” IEEE Trans. NanoTechnol. **13**(6), 1145–1153 (2014). [CrossRef]

**32. **P. Y. Chen, C. Argyropoulos, and A. Alù, “Terahertz antenna phase shifters using integrally-gated graphene transmission-lines,” IEEE Trans. Antenn. Propag. **61**(4), 1528–1537 (2013). [CrossRef]

**33. **P. Y. Chen, H. Huang, D. Akinwande, and A. Alù, “Graphene-based plasmonic platform for reconfigurable terahertz nanodevices,” ACS Photonics **1**(8), 647–654 (2014). [CrossRef]

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

**35. **B. Wang, X. Zhang, X. C. Yuan, and J. H. Teng, “Optical coupling of surface plasmons between graphene sheets,” Appl. Phys. Lett. **100**(13), 131111 (2012). [CrossRef]

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

**37. **S. S. Yu, *Physical basis of guided wave optics*, (Northern Jiaotong University, 2002), Chap. 5.

**38. **L. A. Falkovsky and A. A. Varlamov, “Space-time dispersion of graphene conductivity,” Eur. Phys. J. B **56**(4), 281–284 (2007). [CrossRef]

**39. **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]

**40. **D. Dai, Y. Shi, S. He, L. Wosinski, and L. Thylen, “Gain enhancement in a hybrid plasmonic nano-waveguide with a low-index or high-index gain medium,” Opt. Express **19**(14), 12925–12936 (2011). [CrossRef] [PubMed]

**41. **D. K. Efetov and P. Kim, “Controlling Electron-Phonon Interactions in Graphene at Ultrahigh Carrier Densities,” Phys. Rev. Lett. **105**(25), 256805 (2010). [CrossRef] [PubMed]

**42. **Y. Nikitin, F. Guinea, F. J. García-Vidal, and L. Martín-Moreno, “Edge and waveguide terahertz surface plasmon modes in graphene microribbons,” Phys. Rev. B **84**(16), 161407 (2011). [CrossRef]