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Abstract
Graphene-based and metal-based surface plasmon polariton (SPP) waveguides have attracted intense research interest because they can be used as basic components to propagate electromagnetic (EM) waves in future optical integrated systems. We propose a directional coupler, which can couple EM energy from a multilayer-graphene-based cylindrical long-range SPP waveguide to a metal-based cylindrical hybrid SPP waveguide in the mid-infrared range. This coupler exhibits relatively low coupling length, high coupling efficiency, low insertion loss, and high extinction ratio after adjustment of the wave vector mismatch of the two waveguides. Moreover, this coupler is tolerant to practical fabrication errors like misalignment of graphene layres, and can effectively work in the range of Fermi energy Ef > 0.6 eV when the mobility of graphene varies from 10000 to 800 cm2/Vs. Hence, the coupler offers potential applications in signal routing and information exchange between graphene-based and metal-based SPP waveguides in photonic integrated circuits.
© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
To establish future photonic integrated circuits, it is highly desirable to develop an optical waveguide with a long propagation distance and deep sub-wavelength confinement, because such a waveguide can be used as a basic component to propagate electromagnetic (EM) waves in many applications, such as, sensors [1], splitters [2], modulators [3], filters [4], and devices of induced transparency [5]. Dielectric-based nanowire waveguide [6,7] is popular because it is a low-loss optical waveguide with an ultra-long propagation distance. However, the application range of this waveguide is limited because of its poor EM confinement. Surface plasmon polaritons (SPPs) [8,9], which are a collective excitation of electrons propagating along a conductor-dielectric interface, exhibit strong EM confinement and relatively low propagation loss. SPP-based waveguide is a promising candidate to guide EM waves in future photonic integrated circuits. Noble metals and graphene are regarded as important plasmonic materials [10,11]. Various metal-based [12–16] and graphene-based [17–22] SPP waveguides have been previously proposed and investigated. These waveguides exhibit different transmission characteristics and application range and correspond to different working wavebands. Metal-based SPP waveguides are suitable for the mid-infrared to visible range and especially focus on the wavebands in the near-infrared to visible range. Graphene-based SPP waveguides are suitable only for the mid-infrared range. The dielectric waveguide exhibits a broader range than the two types of SPP waveguides.
Hence, there are at least three types of optical waveguides for future photonic integrated circuits. This arises a question of how to achieve signal routing and information exchange between them. This problem can be addressed by coupling EM fields between different waveguides when their efficient indices have similar real parts for achieving wave factor matching. The efficient index can be expressed as neff = nr + ini, where nr and ni are the real and imaginary parts of neff, respectively. EM field coupling has been effectively realized between dielectric waveguide and metal SPP waveguide [23–28] because these two types of waveguides exhibit similar nr values from the near-infrared to visible range. However, the coupling can hardly be excited between metal-based and graphene-based SPP waveguides because the (nr)g value (~101–102) is usually much larger than the (nr)m value (~100) in the mid-infrared range, where (nr)g and (nr)m are the nr of graphene-based and metal-based SPP waveguides, respectively. Thus, to achieve wave vector matching and EM coupling for the two types of waveguides, the (nr)g value should be significantly reduced to near the (nr)m value.
Recently, we have proposed a long-range SPP (LRSPP) waveguide with double layer of graphene [29,30]. The results show that the symmetric coupling mode (SCM) and asymmetric coupling mode (ASCM) can be excited as the two graphene layers couple with each other. The two coupling modes are also defined as the short-range SPP (SRSPP) and LRSPP modes, respectively. Among them, the LRSPP fundamental mode exhibits ultra-long propagation length and ultra-small effective index (~10–30) when the separation distance between the two graphene layers is relatively small, which significantly narrows the nr gap between the graphene-based and metal-based SPP waveguides. Further studies [31–33] show that the nr value of the LRSPP fundamental mode decreases with increasing number of graphene layers. This offers an effective way to reduce (nr)g to near (nr)m. Then, the EM coupling and signal routing can be achieved between the two types of SPP waveguides because of wave vector matching.
In this study, a directional coupler, composed of a multilayer-graphene-based cylindrical LRSPP waveguide and a metal-based cylindrical hybrid SPP waveguide, is investigated using the finite-difference time-domain (FDTD) method by using the commercial software Lumerical FDTD Solutions (Version: 8.6.0). We find that EM coupling can be achieved between the LRSPP fundamental mode of the graphene waveguide and the hybrid SPP fundamental mode of the metal waveguide because the two modes exhibit similar nr values and match the wave vectors in the mid-infrared range. Furthermore, we obtain relatively low coupling length, high coupling efficiency, low insertion loss, high extinction ratio, and good tolerance of fabrication errors by selecting appropriate design parameters for this coupler.
2. Multilayer-graphene-based cylindrical LRSPP waveguide
A schematic diagram of the multilayer-graphene-based LRSPP cylindrical waveguide is shown in Fig. 1, in which a cylindrical silicon nanowire is laid through the center of the waveguide and surrounded by several graphene layers uniformly embedded in a silica layer. The radius of the silicon nanowire is labeled as R. The separation distance between two adjacent cylindrical graphene layers is labeled as t. The number of graphene layers is denoted by N. Recently, several groups [34,35] have reported in detail on how to wound a graphene layer onto a nanowire. Several experiments [36–38] showed that the dielectric nanowire can be tightly surrounded by the graphene layer because of the van der Waals force. Thus, several graphene layers can be formed by rolling a graphene flake around the dielectric nanowire from the inner to the outer layer, step by step. The silica layer can be formed using the plasma-enhanced chemical vapor deposition technology and its thickness t can be controlled by tuning the deposition conditions [39].
Fig. 1 Schematic diagram of the proposed multilayer-graphene-based LRSPP waveguide. (a) Cross-section of the waveguide. (b) 3D structure of the waveguide.
The permittivity of the silicon nanowire and the silica layer is set at ε1 = 11.9 and ε2 = 2.09, respectively. The permittivity of graphene εg can be described as a complex permittivity with in-plane effective permittivity, 1 + iσg/(ε0ωΔ), and surface-normal effective permittivity, 2.5 [40], where σg, ω, and Δ are the conductivity of graphene, angular frequency of the incident light, and thickness of the graphene layer, respectively. Note that σg can be calculated from the Kubo formula [41,42], which depends on the Fermi energy Ef, ambient temperature T, ω, and electronic relaxation time τ. The latter is related to electron mobility µ and Fermi velocity vF by τ = µEf /(evF2). We select µ = 10000 m2/(Vs) in this work, and set the other parameters as T = 300 K and Δ = 1 nm. The incident wavelength in the vacuum is λ0 = 10 µm.
First, we set the parameters of the proposed waveguide as N = 6, Ef = 0.8 eV, R = 10 nm, and t = 5 nm. The SCMs and ASCMs can be obtained from constructive and destructive interference, respectively, when several graphene layers couple to each other. For simplicity, we only consider the fundamental SCM and ASCM. The constructive interference results in identical Ey phase distributions on both sides of the outermost and innermost graphene layers for the fundamental SCM [Fig. 2(c)]. On the contrary, the destructive interference results in opposite Ey phase distributions for the fundamental ASCM [Fig. 2(f)]. Meanwhile, according to the transverse electric field Ey distribution and the EM field boundary condition of , we obtain the charge distribution of the two fundamental modes, as shown in Figs. 2(b), 2(e), 2(h) and 2(j). Based on these charge distributions, the EM energy of the fundamental SCM mode is confined between the graphene layers because of the attraction of the opposite charges [Figs. 2(a) and 2(g)]. This increases the interaction between the graphene layers and the EM field, increases the propagation loss and decreases the propagation length. Conversely, the EM field of the fundamental ASCM mode is located on the external surface of the waveguide because of the repulsion of like charges [Figs. 2(d) and 2(i)]. This decreases the propagation loss and increases the propagation length. Thus, we define the fundamental SCM and ASCM as SRSPP and LRSPP fundamental modes, respectively. For simplicity, we further define the SRSPP and LRSPP fundamental modes as S0 and L0 modes, respectively.
Fig. 2 (a), (b), and (c) show the |E|, Ey, and Ey phase distribution of the S0 mode on the cross section of the waveguide, respectively. (d), (e), and (f) show the |E|, Ey, and the Ey phase distribution of the L0 mode on the cross section of the waveguide, respectively. (g) and (h) show the |E|, Ey and charge distribution of the S0 mode on the x-z plane of the waveguide, respectively. (i) and (j) show the |E|, Ey and charge distribution of the L0 mode on the x-z plane of the waveguide, respectively. The black curves are the electric field lines of the L0 mode in Fig. 2(j). The “+” and “-” symbols denote the positive and negative charges in Figs. 2(h) and 2(j), respectively.
Next, we investigate the transmission characteristics of the waveguide and focus on the dependence of (nr)L0 and (nr)S0 on the parameters of the waveguide, where (nr)L0 and (nr)S0 are the nr values of the L0 and S0 modes, respectively. First, we set the parameters of this waveguide as E f = 0.8 eV, R = 10 nm, and t = 5nm and vary the number of graphene layers N. We find that (nr)L0 decreases and (nr)S0 increases with increasing N [Fig. 3(a)]. For instance, (nr)L0 = 13.5 and (nr)S0 = 97.3 at N = 2. However, (nr)L0 = 6.2 and (nr)S0 = 120.9 at N = 6. The propagation length Lprop can be defined as Lprop = λ0/4πni [12,19]. Figure 3(b) shows that (Lprop)L0 increases significantly, while (Lprop)S0 has no obvious changes with increasing N, where (Lprop)L0 and (Lprop)S0 are the propagation length of the L0 and S0 modes, respectively. Similarly, increasing Ef can effectively decrease (nr)L0 and improve (Lprop)L0 [Figs. 3(c) and 3(d)]. Figures 3(e) and 3(f) show that increasing R and t also decreases (nr)L0. Thus, (nr)L0 can be effectively decreased by increasing N, Ef, R, and t of the waveguide.
Fig. 3 (a), (b) nr and Lprop as functions of N for both L0 and S0 modes (Ef = 0.8 eV, R = 10 nm, and t = 5 nm). (c), (d) nr and Lprop as functions of Ef for L0 mode (N = 6, R = 10 nm, and t = 5 nm). (e) nr as a function of R for L0 mode (N = 4, t = 5 nm, and Ef = 0.8 eV). (f) nr as a function of t for L0 mode (N = 4, R = 10 nm, and Ef = 0.8 eV).
3. Metal-based cylindrical hybrid SPP waveguide
Figure 4(a) is a schematic diagram of the metal-based cylindrical hybrid waveguide, in which a cylindrical silver nanowire is laid through the center of the waveguide and surrounded by silica and silicon layers from the inner to the outer layer. The radius of the silver nanowire is labeled as RAg. The thicknesses of the silica and silicon layers are labeled as tSiO2 and tSi, respectively. This waveguide has been studied in the telecommunication wavelength of 1550 nm [43]. We will further study the transmission characteristics of this waveguide in the mid-infrared range, because the EM coupling between the graphene-based and metal-based waveguides can only be excited in this waveband.
Fig. 4 (a) Schematic diagram of the proposed metal-based hybrid waveguide. (b) |E| distribution of the H0 mode on the cross section of the waveguide. (c) nr and Lprop as functions of RAg for the H0 mode (tSiO2 = 6 nm, tSi = 25 nm). (d) nr as a function of tSi for the H0 mode (RAg = 10 nm, tSiO2 = 6 nm). (e) nr as a function of tSiO2 for the H0 mode (RAg = 10 nm, tSi = 25 nm).
The permittivity of silver is εAg = –2726 + i1410 [44] at the incident wavelength λ0 = 10 µm. We set the parameters of this waveguide as tSiO2 = 6 nm and tSi = 25 nm and vary RAg from 6 nm to 22 nm. In this case, this waveguide can support a hybrid SPP fundamental mode [Fig. 4(b)]. For simplicity, we define this mode as the H0 mode. Figure 4(c) shows that (nr)H0 increases and (Lprop)H0 decreases with decreasing RAg, where (nr)H0 and (Lprop)H0 are the nr and Lprop values of the H0 mode. (nr)H0 can also be effectively increased by increasing tSi [Fig. 4(d)]. Figure 4(e) shows that (nr)H0 is not sensitive to varying tSiO2 values. Thus, (nr)H0 can be effectively increased by decreasing RAg and increasing tSi.
4. Coupling between graphene-based and metal-based waveguides
Because the L0 and H0 modes have similar nr values, EM field coupling can be achieved between them because of wave vector matching. However, the nr values exhibit obvious differences between the other modes of the two waveguides [29,30,43]. Thus, the EM field coupling can only be effectively excited between the L0 and H0 modes of the directional coupler. According to the analysis in Sections 2 and 3, (nr)L0 should be decreased and (nr)H0 should be increased to achieve wave vector matching of the two modes. For the L0 mode, (nr)L0 can be decreased by increasing N, Ef, R, and t. However, a larger R and t will result in a significant increase in computation time. Thus, we mainly adopt the method of increasing N and Ef in this work. Considering that the decrease of (nr)L0 becomes less distinct significant for N > 6, we chose the number of graphene layers as 6 in this work. (neff)L0 = (nr)L0 + (ni)L0 = 5.0528 + i0.0171 at R = 16 nm, t = 4 nm, N = 6, and Ef = 0.9 eV. This Ef value is relatively high but was still allowed in a previously reported experiment [45,46]. On the other hand, the low (nr)L0 value, caused by the high Ef value, can also be obtained by increasing R and t while employing a low Ef value in practice. For the H0 mode, (neff)H0 = (nr)H0 + (ni)H0 = 5.0579 + i1.3916 at RAg = 6 nm, tSiO2 = 6 nm, and tSi = 30 nm. In this case, coupling can be achieved between the L0 and H0 modes because their wave vectors match.
The fundamental SCM and ASCM can be excited when L0 and H0 modes couple to each other. For convenience, we abbreviate the fundamental SCM and ASCM as s0 and a0 modes, respectively. For the s0 mode, the strong EM field is located between the two waveguides owing to their constructive interference [Fig. 5(a)]. The transverse electric field Ey and the Ey phase distribution of the s0 mode exhibit an approximate symmetry with respect to the xz-plane [Figs. 5(b) and 5(c)]. On the contrary, |E|, Ey, and the Ey phase distribution of the a0 mode exhibit opposite characteristics owing to the destructive interference [Figs. 5(d)–6(f)].
Fig. 5 (a), (b), and (c) show the |E|, Ey, and Ey phase distribution of the s0 mode on the cross section of the coupler, respectively. (d), (e), and (f) show the |E|, Ey, and Ey phase distribution of the a0 mode on the cross section of the waveguide, respectively. (g) nr as a function of d for the s0 and a0 modes. (h) The normalized coupling length Lc /(Lprop)L0 versus d.
Fig. 6 (a) Schematic diagram of the directional coupler and the |E| distribution of the excited coupling mode on the x-z plane. (b) Normalized output power P/PG0 from the two output arms versus the normalized propagation length z/Lc at d = 16 nm when the EM field is launched from the multilayer-graphene-based LRSPP waveguide at z = –0.9 μm (RAg = 10 nm, tSiO2 = 6 nm, and tSi = 26 nm). (c–e) Coupling efficient (CE), insertion loss (IL), and excited ratio (ER) versus d for different RAg, respectively.
Next, we analyze the dependence of the coupling characteristics on the coupling distance d. According to the perturbation theory of the coupled mode [47], the propagation constants of the s0 and a0 modes, (kz)s0 and (kz)a0, can be expressed as (kz)s0 = (kz)0 + Kc and (kz)a0 = (kz)0 – Kc, respectively, where (kz)0 is the propagation constant of the decoupled L0 mode or the decoupled H0 mode, and Kc is the coupling coefficient of the two modes. nr and kz have similar trends because kz = nrk0, where k0 is the propagation constant of the incident light in vacuum. With increasing d, the energy exchange between the two modes becomes more difficult and Kc decreases. Then, (kz)s0 and (nr)s0 decrease, while (kz)a0 and (nr)a0 increase, where (nr)s0 and (nr)a0 represent the nr value of the s0 and a0 modes, respectively. When d increases to a very large value, Kc approaches to 0 and (kz)s0 and (kz)a0 converge to (kz)0. Similarly, (nr)s0 and (nr)a0 converge to (nr)0, which is the nr value for the uncoupled L0 mode or the uncoupled H0 mode [Fig. 5(g)].
The phase factor of the s0 and a0 modes can be expressed as exp{i[(kz)0 + Kc]} and exp{i[(kz)0 – Kc]} [20], respectively. Thus, we can express the phase difference of the two modes as Δφ = 2Kcz. The EM energy can be transferred from one waveguide to the other as Δφ varies from 0 to π. The coupling length Lc can be defined as Lc = Δφ/2Kc = π/[(kz)s0 – (kz)a0] and corresponds to the propagation distance at Δφ = π. The EM energy exchang becomes more difficult with increasing coupling distance d. A longer propagation length is needed to transfer the EM energy between the two waveguides. Thus, the coupling length increases exponentially with d [Fig. 5(h)]. Note that Lc of this coupler lies between 0.028 (Lprop)L0 and 0.136 (Lprop)L0 as d increases from 2 nm to 74 nm, where (Lprop)L0 is the propagation length of the uncoupled L0 mode. This Lc is rather low compared to that of the other types of couplers [21–26].
To further understand the power transfer between the two waveguides, we assume that the incident light launches into the graphene waveguide at z = – 0.9 μm to excite the L0 mode [Fig. 6(a)]. Strong coupling occurs as the EM wave is propagated along the graphene waveguide at z = 0, which is the left endpoint of the metal waveguide. The power of the incident light is P0 at z = – 0.9 μm. The transmission powers of graphene and metal waveguides are PG0 and PM0 at z = 0, respectively. Figure 6(b) shows the normalized powers of the two waveguides, P/PG0, versus the normalized propagation length, z/Lc, which is similar to a sinusoidal oscillation with strong attenuation along z. With increasing z, the EM field energy is transferred from the graphene waveguide to the metal hybrid waveguide, owing to the interference of the s0 and the a0 modes. Note that PM0 0 (point B) as PG0 = 1 (point A). The power of the metal waveguide rises to the maximum value (point C) as z approaches to Lc, while the power of the graphene waveguide decreases to the minimum value (point D) as z diverges from Lc. The positions of points C and D are not in agreement. Moreover, the maximum power of the metal waveguide is only 0.792 PG0 at RAg = 10 nm, which is less than that of the coupling between dielectric and metal SPP waveguides [21–26]. This is because the metal hybrid SPP waveguide has relatively high propagation losses in the mid-infrared range.
We also evaluate the coupling efficiency (CE) and extinction ratio (ER) to characterize the performance of the transfer power of the directional coupler. CE is defined as the ratio of the maximum power on the metal waveguide to the launched power on the graphene waveguide at z = 0. ER is defined as the ratio of the maximum power on the metal waveguide to the minimum power on the graphene waveguide for the first cycle of the energy exchange. In Fig. 6(b), CE = PC/PA = PC/PG0 and ER = PC/PD. Considering that the power transfer is very sensitive to RAg and coupling distance d, we analyze the dependence of CE and ER on d for different RAg values. Figure 6(c) shows that CE decreases monotonously with increasing d and decreasing RAg. It is understandable that the corresponding insertion loss (IL = 10log(CE)) increases with increasing d and decreasing RAg [Fig. 6(d)]. Therefore, the CE exhibits very small value as RAg is quite small because of the large insertion loss in this case. For example, when d = 16 nm, IL = –2.60 dB and CE = 54.9% at RAg = 6 nm, while IL = –1.07 dB and CE = 78.1% at RAg = 12 nm. Thus, a high CE value can be obtained by increasing RAg and decreasing d.
The dependence of ER on d is determined by two factors [26,27]. First, the difference in the amplitudes of the s0 and a0 modes decreases at the input end with increasing d, which results in an improved ER. Second, the coupling length and the propagation loss increase considerably with increase in d, which leads to a smaller ER. Therefore, with the increase of d, ER arises from the competition of the two factors. The simulated results show that, with increasing d, the ER increases at RAg = 6nm and decreases at RAg = 9nm, 12nm, and 18 nm, respectively, as shown in Fig. 6(e). Figure 6(e) also shows that ER decreases with increasing RAg. This can be attributed to the fact that (nr)H0 decreases with increasing RAg, which exacerbates the wave vector mismatch between the L0 and H0 modes and weakens the ER. Therefore, the maximum values of ER occur at RAg = 6 nm because of the matching of the wave vectors of the two modes. In this case, ERs lie between 8 and 12 dB as d increases from 2 nm to 40 nm.
5. Discussion
5.1 Fabrication error tolerance
Considering that fabrication of such a coaxial cylindrical waveguide with multilayer graphene is very difficult, the directional coupler may have a misalignment of graphene layers in practice. This will have a possible impact on the performance of the coupler. We analyze the fabrication error tolerance resulting from the misalignment of graphene layers.
We assume that the cylindrical graphene layers have different axes, with coordinates of (0,0), (0, δx), (0, –δx), (δx, 0), (–δx, 0), and (δx, δx) from inside to outside, respectively. To avoid the contact of adjacent graphene layers, δx should be less than t/2. We define the rate of error tolerance of misalignment η = 2δx/t. From Figs. 7(c) and 7(d), we find that the coupling performance of the directional coupler, including Lc, CE, IL, and ER, does not show obvious change when η varies from 0 to 0.75. This can be explained as follows. Because the multilayer-graphene-based cylindrical waveguide is a closed system, the excellent coupling characteristics between the graphene layers can be maintained even if η becomes very large. For instance, when η = 0.5, the EM distribution of L0 mode still exhibits an almost perfect axial symmetry [Fig. 7(a)], which is similar to that for η = 0. In this case, strong coupling can be excited between the graphene waveguide and metal waveguide [Fig. 7(b)]. Therefore, the misalignment of graphene layers resulting from fabrication errors has only slight impact on the coupling performance.
Fig. 7 (a) shows the |E| distribution of L0 mode on the cross section of the graphene waveguide for the rate of error tolerance of misalignment η = 0.5. (b) shows the |E| distribution of the excited coupling mode on the xy-plane (η = 0.5). (c) Lc and CE as functions of η for different RAg. (d) IL and ER as functions of η for different RAg.
5.2 Dependence of coupling performance on graphene parameters
To obtain a good coupling performance for this directional coupler, we have selected a relatively high mobility μ and Fermi energy Ef. However, it may not be practical to assume that graphene has such high μ and Ef at the same time. Therefore, in order to ensure that our proposed coupler is physically realizable, we would first select an experimentally feasible range of μ for SiO2/Si-supported graphene, and then employ a lower Ef value, which is easier to be realized in experiment. Novoselov et al. [48] observed μ up to 15000 cm2/Vs at 300 K for SiO2/Si-supported multilayer graphene, and between 3000 and 10000 cm2/Vs for few-layer graphene. Dean et al. [49] have obtained experimentally the electron mobility of μ = 60000 cm2/Vs for boron nitride-supported monolayer graphene. Fei et al. [50] reported that μ as high as 8000 cm2/Vs has been achieved for SiO2/Si-supported monolayer graphene. Considering that our proposed directional coupler is a device based on SiO2/Si substrate, we would investigate the dependence of coupling performance on graphene parameters by selecting μ as 10000, 6000, 2000, and 800 cm2/Vs, respectively, and varying Ef from 0.9 eV to 0.5 eV.
From Fig. 8(a), we can find that (ni)L0 increases substantially with decreasing μ and Ef, which would influence the directional coupler in two ways. First, because (ni)H0 (point A) is considerably larger than (ni)L0, increasing (ni)L0 narrows the gap between (ni)H0 and (ni)L0. This will help achieve a more perfect wave vector matching for the two waveguides, and hence improve the coupling performance. Second, a larger (ni)L0 would result in higher losses of the graphene waveguide because the loss of an optical waveguide is given by lm[dB/μm] = –8.86ni(kz)0 [51]. This will degrade the coupling performance. Under these combined effects, from Figs. 8(b)–8(d), we find that the coupling performances are improved instead of deteriorated when Ef is varied from 0.9 eV to 0.6 eV. With constantly decreasing Ef, the losses increase rapidly, and the coupling performance drops significantly when Ef < 0.6 eV. In particular, the EM energy of two waveguides almost attenuate to zero after an energy exchange at Ef < 0.5 eV. Obviously, periodic energy exchange is ceased in this case. Based on the above observation, one can draw a conclusion that the optimal performance is achieved around Ef > 0.6 eV for this directional coupler [Fig. 8(e)]. On the other hand, the coupling performance gradually worsens with decreasing μ, which can be attributed to the continuous increase in the transmission loss of the graphene waveguide under this condition.
Fig. 8 (a) (nr)L0 versus Ef for different mobilities μ of graphene (R = 16 nm, t = 4 nm, and N = 6). Point A presents the (nr)H0 value as RAg = 6 nm, tSi = 30 nm, and tSiO2 = 6 nm for the metal-based hybrid waveguide. (b) Coupling efficient (CE), (c) insertion loss (IL) and coupling length Lc, and (d) extinction ratio (ER) versus Ef for different μ, respectively. (e) show the |E| distribution on the x-z plane of the directional coupler (Ef = 0.45 eV and μ = 2000 cm2/Vs).
6. Conclusion
In summary, we proposed a directional coupler, composed of a multilayer-graphene-based LRSPP waveguide and a metal-based hybrid SPP waveguide. The transmission characteristics of the two waveguides have been investigated. The L0 mode of the graphene waveguide exhibited the similar nr value as the H0 mode of the metal waveguide. Thus, coupling can be achieved between them because of wave vector matching. Moreover, the coupler features a relatively low coupling length, high coupling efficiency, low insertion loss, high extinction ratio, and good fabrication error tolerance. Our simulation results also show that this coupler can effectively work in the range of Ef > 0.6 eV when μ varies from 10000 to 800 cm2/Vs. Hence, this directional coupler can be used for signal routing, information changing, and power splitting or combining between graphene SPP waveguide and metal SPP waveguide in future photonic integrated circuits.
Funding
National Natural Science Foundation of China (Grant Nos. 61741515, 61701431); Scientific Research Fund of Jiangxi Provincial Education Department of China (Grant No. GJJ161017); Educational Reform Research Fund of Jiangxi Provincial Education Department of China (Grant No. JXJG171516).
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