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Abstract
To develop on-chip photonic devices capable of transmitting terahertz signals beyond the propagation distance of millimeter while keeping deep subwavelength field confinement has been a challenging task. Herein, we propose a novel multilayer graphene-based hybrid plasmonic waveguide (MLGHPW) consisting of a cylindrical dielectric waveguide and hyperbolic metamaterials. The device is based on alternating graphene and dielectric layers on a rib substrate, operating in the terahertz range (f = 3 THz). We couple the fundamental dielectric waveguide mode with the fundamental volume plasmon polarition modes originated from the coupling of plasmon polaritons at individual graphene sheets. The resulting hybrid mode shows ultra-low loss compared with the conventional GHPW modes at the comparable mode sizes. The present MLGHPW demonstrated a few millimeters of propagation length while keeping the mode area of 10−3 A0, where A0 is the diffraction-limited area, thus possessing a thirty times larger figure of merit (FoM) compared to other GHPWs. The additional degree of freedom (the number of graphene layers) makes the proposed MLGHPW more flexible to control the mode properties. We investigated the geometry and physical parameters of the device and identified optimal FoM. Moreover, we analyzed the crosstalk between waveguides and confirmed the potential to construct compact on-chip terahertz devices. The present design might have the possible extensibility to other graphene-like materials, like silicene, germanen, stanene etc.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Terahertz (THz) waves span in a range of 0.1–10 THz (0.03–3 mm wavelengths) [1]. Guiding such waves with low-energy attenuation has gained researchers’ interest since the development of the first the THz quantum-cascade laser with the working frequency of 70 THz in 1994 [2], followed by 4.4-THz counterpart in 2002 [3]. Terahertz lasers found applications in chemical and biological sensing [4,5], medical imaging [6–8], near-field imaging [9], and spectroscopy [10]. Moreover, some applications required on-chip wave guidance, including noninvasive and label-free molecular detection, and gas and liquid spectroscopy. To tackle this problem, a variety of waveguiding structures, such as dielectric fibers [11,12], hollow-core fibers [13], bare metal wires [14], and hybrid terahertz waveguides [15], have been recently developed. For the dielectric fibers [11–13], the required cross-section diameters to support guided modes are in the range of 200–300 μm. These are comparable with the working wavelengths of the terahertz lasers, and thus the mode sizes of these waveguide structures [11–13] are hindered by the diffraction limit of light. Light propagates as weakly-guided radial surface waves and can be considered as the transverse electromagnetic mode of a conventional coaxial waveguide with a metal wire [14]. Such waveguides feature a remarkably low field attenuation coefficient of about 0.02 cm−1 but have a relatively large diameter of the metal wire, i.e., 900 μm. To break the diffraction limit of light, plasmonic waveguides [16] have been recently proposed. These waveguides support surface plasmon polariton (SPP) modes and operate in visible and near-infrared ranges. However, conventional plasmonic waveguides had high ohmic losses; thus, hybrid plasmonic waveguides (HPW) [17] were suggested to leverage the mode size and propagation length. But hybrid gold terahertz HPWs [15] also had some disadvantages. In particular, they only allowed achieving the normalized mode area Am [17] of 10−1. The latter was attributed to the weak confinement of SPP modes using noble metals while operating in the THz ranges.
To achieve deep subwavelength mode sizes for the THz waves, selecting alternative materials with metallic properties was necessary. Graphene was considered as one of the promising candidates. It has atomically thin sheet and exhibits promising electric, thermal, and optical characteristics [18,19] for building nanophotonic devices. In particular, the surface conductivity of graphene is nearly pure imaginary in the mid-IR and THz ranges [20], and its optical properties can be actively tuned by an external gate voltage or chemical doping. As a result, many graphene-based HPWs (GHPWs) operating in the near-IR [21,22], mid-IR [23,24] and THz [25–27] ranges have been recently reported to exhibit a higher figure of merit (FoM) [17] indicating the ratio between propagation length (Lp) and mode diameter. In Ref. [25], a GHPW operating at frequency f = 3 THz was designed using rectangular dielectric waveguide and high-density polyethylene (HDPE) substrate covered by a five-layer graphene. The obtained propagation length was about 100 μm and Am was about 10−2. With the same propagation length as in Ref. [25], He et al. [26] replaced the rectangular waveguide with circular dielectric waveguide, shrinking the Am to about 8.0 × 10−3. After that, Am was further reduced to 3.0 × 10−3 by replacing the circular dielectric waveguide with hollow tube waveguide [27] with ratio of 0.6 between the inner and outer diameter. However, in this case, the propagation length was moderately reduced to be 80 μm. Notably, the graphene sheet covering HDPE [25] (silicon dioxide, SiO2 [26,27]) substrate was assumed to have a five-layer (three-layer) graphene and thus featured five (three) times higher surface conductivity compared to a monolayer graphene. However, this assumption needs to be moderately modified because of strong overlapping of the π orbitals of the carbon atoms in the individual graphene planes, leading to the variation of the electron band structure. Moreover, as was revealed experimentally with optical spectroscopy [28,29], stacking N layers of graphene was equivalent to N times of surface conductivity of a monolayer while the photon energy was greater than 0.7 eV (λ = 1.77 μm). From the present reports [25–27], although the normalized mode areas were acceptable, the propagation losses were still high.
Hyperbolic metamaterials (HMMs) [30] with principal permittivity components of the opposite signs based on an alternating graphene and dielectric layers (AGDLs) have been also extensively studied. They featured isofrequency surfaces in the shape of a hyperboloid, in which the hyperbolic dispersions can support propagating high-k waves. The high-k waves, also referred to as the volume plasmon polariton (VPP) modes, were obtained by coupling the surface plasmon polaritons at individual graphene sheets [31–33]. In contrast to the HMMs formed by the metal-dielectric structures [34–39], the optical properties of HMMs based on AGDLs could be flexibly modulated. Modulation was achieved by the external electric gating or chemical doping [40] and allowed to realize tunable HMMs. In this paper, we report a novel multilayer GHPW (MLGHPW) operating in the THz ranges to achieve both the ultra-low loss and deep subwavelength mode localization by coupling the fundamental dielectric waveguide mode with the fundamental VPP mode. The resulting hybrid fundamental mode exhibits thirty times larger propagation length compared to previously reported structures [25–27] while maintaining the same order of magnitude for normalized mode area.
2. MLGHPW mode properties
A three-dimensional (3D) schematic and front view of MLGHPW is shown in Fig. 1. It consists of AGDLs structure sandwiched between a cylindrical dielectric waveguide and a rib SiO2 substrate. 3D and front views are shown in Figs. 1(a) and 1(b), respectively. Here the cylindrical waveguide and the dielectrics within the AGDLs were made of high-index silicon (Si). MLGHPW operates as a conventional HPW structure [17] with hybrid modes formed by the mode coupling between the dielectric waveguide modes and VPP modes. Physically, the VPP modes originated from the coupling of SPP modes in individual graphene layers separated by dielectric layers. The number of the VPP modes was determined by the number of graphene sheets [33]. The ohmic losses significantly increase for the high-order VPP modes. Therefore, we mainly focus on the fundamental VPP mode (VPP0) with the lowest ohmic loss. The geometry parameters are as follows: the diameter of the cylindrical waveguide is d and the width and height of the AGDLs are w0 and Nt (where Nt ≤ g), respectively. Here N is the number of Si layers with thickness t between graphene sheets. The relative permittivities of Si and SiO2 at a working frequency of f = 3 THz were εSi = 12.25 [41] and εSiO2 = 2.25 [41], respectively.
Fig. 1. (a) A 3D schematic of MLGHPW; (b) The front view of (a) along with the zoomed-in view of the AGDLs. The structure consists of AGDLs sandwiched between a cylindrical dielectric waveguide and a rib SiO2 substrate.
The AGDLs can be considered as HMMs based on the effective medium approximation (EMA) [42,43], and the effective in-plane and out-of-plane permittivities can be approximately by ε‖ = (tgεg + tdεd)/(tg + td) and $\varepsilon_{\perp}$= εgεd (tg + td)/(tgεd + tdεg), respectively, where εg (tg) and εd (td) are permittivities (thicknesses) of graphene and dielectric, respectively. Note that εg denotes the in-plane permittivity of graphene sheet, and it can be written as εg = εSi–iσ/(ωε0tg), where i is imaginary unit, ω is angular frequency, ε0 is vacuum permittivity, and σ is the surface conductivity of monolayer graphene. Here, graphene is modeled as an infinitely thin layer with the surface current density of J = σE on in-plane sheet due to its negligible thickness, where E is electric field vector. The surface conductivity of graphene σ can be calculated using the Kubo formulas [44]
The carrier mobility ranging from >0.1 m2/V·s in graphene grown by chemical vapor deposition [45] to >20 m2/V·s in suspended exfoliated graphene [46] has been demonstrated in recent experiments. By tuning the Fermi energy using electrical gating or chemical doping, the optical properties of graphene can be modulated. To achieve practical feasibility, we set μ = 1 m2/V·s and Ef = 0.5 eV (obtaining τ = 0.5 ps) at T = 300 K. Note that within the considered frequency (f from 1 to 30 THz) and Fermi energy (Ef from 0.2 to 1.2 eV) ranges, the positive imaginary part of σ guarantees the excitation of the graphene plasmon modes.
Before conducting the analyses of the mode properties, we schematically show the fabrication steps of the proposed structure in Fig. 2 and describe them as follows: (1) Depositing Si and positive photoresist (PR) films with thickness gSiO2 on a SiO2 substrate; (2) applying a mask and following by PR exposure and PR development; (3) depositing a SiO2 layer and conducting PR lift-off; (4) forming multilayer graphene-SiO2 stacking by a chemical vapor deposition (CVD), a layer transfer method [40] or the solution-phase method [47,48]; (5) repeating (1) and (2), then depositing Si layer and lifting PR off; (6) Adopting e-beam writer to shape the rectangular Si strip to be a cylinder. Compared to the conventional CVD, transfer method [40], the solution-phase method [47,48] reveals transfer-free, quality independent of the number of layers.
To assess the performances of a plasmonic waveguide, we consider three indexes including Am = Ae/A0 [17] (the degree of light confinement of plasmonic waveguides), Lp = λ/[4πIm(ne)] (the distance where the modal energy attenuates to 1/e of initial value), and the FoM = ${L_p}/2\sqrt {{A_m}/\pi } $[17] (the ratio between Lp and Am). Let us also introduce the effective mode area, Ae, the diffraction-limit mode area, A0 = λ2/4 (λ is the working wavelength in free space), and the imaginary part of effective refractive index, Im(ne). The effective mode area, Ae is given by
Fig. 3. The profiles of electric field |E| of the (a) VPP0-$\textrm{HE}_{\textrm{11}}^{{\kern 1pt} y}$ and (b) VPP0-$\textrm{HE}_{\textrm{11}}^{{\kern 1pt} \textrm{x}}$ HPMs formed by coupling the cylindrical waveguide modes ($\textrm{HE}_{\textrm{11}}^{{\kern 1pt} y}$ and $\textrm{HE}_{\textrm{11}}^{{\kern 1pt} \textrm{x}}$) and VPP0.
We observed that the field profile of the VPP0-$\textrm{HE}_{\textrm{11}}^{{\kern 1pt} y}$ HPM [Fig. 3(a)] was concentrated mainly in the gap between the cylindrical waveguide and AGDLs. The field profile of the VPP0-$\textrm{HE}_{\textrm{11}}^{{\kern 1pt} x}$ HPM [Fig. 3(b)] spread out considerably to the edges of the AGDLs displaying weaker confinement. This is because the major electric field along y-direction of $\textrm{HE}_{\textrm{11}}^{{\kern 1pt} y}$ mode results in stronger coupling with the VPP0 than that of the $\textrm{HE}_{\textrm{11}}^{{\kern 1pt} x}$ mode. The 1D field plots for the two HPMs along y = -d/2 (dashed line in inset) and x = 0+ (dashed line in inset) are shown in Figs. 4(a) and 4(b), respectively.
Fig. 4. Electric fields, |E|, along the (a) y = -d/2 (red dashed line in inset) and (b) x = 0+ (red dashed line in inset) for the two HPMs.
Here, x = 0+ indicates that the position of the peak of |E| is not at x = 0 but slightly deviates (about 10 nm) from the origin with respect to the peak of |E| along y = -d/2. Clearly, the VPP0-$\textrm{HE}_{\textrm{11}}^{{\kern 1pt} y}$ HPM shows the tightest mode confinement and strongest field amplitude. To further analyze the dependence of mode characteristics on N, Re(ne), Lp, Am, and FoM versus N are shown in Figs. 5(a) to 5(d), respectively.
Fig. 5. Mode characterization: (a) real part of effective refractive index Re(ne), (b) propagation length Lp, (c) normalized mode area Am, and (d) FoM versus the number of Si layers N (N + 1 graphene layers) for the two HPMs at t = 50 nm, g = 0.5 μm, d = 30 μm, and w0 = 30 μm.
As N increases, Re(ne) and Am of the VPP0-$\textrm{HE}_{\textrm{11}}^{{\kern 1pt} x}$ HPM decrease moderately to the constant, but the Lp decrease significantly. In contrast, Re(ne) of the VPP0-$\textrm{HE}_{\textrm{11}}^{{\kern 1pt} y}$ HPM decreases significantly, but Lp and Am increase significantly as N increases. For the VPP0-$\textrm{HE}_{\textrm{11}}^{{\kern 1pt} x}$ HPM, although it can achieve subwavelength mode size of Am = 3.37 × 10−3 (while N = 10), the Lp is below 250 μm. In contrast, the VPP0-$\textrm{HE}_{\textrm{11}}^{{\kern 1pt} y}$ HPM achieves the much longer Lp = 590.34 μm while still exhibiting better Am = 2.92 × 10−3 compared to that in the VPP0-$\textrm{HE}_{\textrm{11}}^{{\kern 1pt} x}$ HPM. For the VPP0-$\textrm{HE}_{\textrm{11}}^{{\kern 1pt} y}$ HPM, Am moderately increased with N. At the same time, significant increase in propagation length (from Lp = 19.06 μm for N = 1 to Lp = 590.34 μm for N = 10) was observed. A thirty-fold increase in propagation length resulted in FoM improvement from 2394 to 19135. According to the analytical formula [33] of a multilayer graphene waveguide, the effective conductivity of the VPP0 with a major electric field component Ey is approximately by Nσ, and the mode field profile expands moderately as N increases. Therefore, the strong coupling between the VPP0 and $\textrm{HE}_{\textrm{11}}^{{\kern 1pt} y}$ modes leads to larger Lp and Am for the VPP0-$\textrm{HE}_{\textrm{11}}^{{\kern 1pt} y}$ HPM as N increases. In contrast, the $\textrm{HE}_{\textrm{11}}^{{\kern 1pt} x}$ mode with a major electric field component Ex interacts weakly with the VPP0 mode, leading to the moderate variation of the VPP0-$\textrm{HE}_{\textrm{11}}^{{\kern 1pt} x}$ HPM properties for smaller N and slight variation as N exceeds 4. Therefore, the mode characteristics of the VPP0-$\textrm{HE}_{\textrm{11}}^{{\kern 1pt} x}$ HPM can be controlled by varying the number of graphene layers. To observe the variations of mode profiles of the VPP0-$\textrm{HE}_{\textrm{11}}^{{\kern 1pt} y}$ HPM, the |E|’s with N = 1, 4, 7, and 10, are shown in Figs. 6(a) to 6(d), respectively.
Fig. 6. The mode profiles of |E| of the VPP0-$\textrm{HE}_{\textrm{11}}^{{\kern 1pt} y}$ HPM at N = (a) 1, (b) 4, (c) 7, and (d) 10.
As can be seen from this figure, the mode field leaks gradually to the cylindrical waveguide as N increases. This results in moderate increase in Am [Fig. 5(c)] but significant reduction of ohmic losses. One-dimensional field plots of the FHPM with several N’s along y = -d/2 and x = 0+ are shown in Figs. 7(a) and 7(b), respectively.
As N increases, the full width at half maximum along y = -d/2 is compensated by the enhancement of field peak [Fig. 7(a)]. The serrated mode distribution in the left inset of Fig. 6(b) clearly demonstrates that the VPP0-$\textrm{HE}_{\textrm{11}}^{{\kern 1pt} y}$ HPM of the present MLGHPW is formed by the coupling between the VPP0 and cylindrical waveguide mode.
3. Modulation of the properties of the VPP0-${\mathbf{HE}}_{{\mathbf{11}}}^{{\kern 1pt} {\mathbf{\textit{y}}}}$ HPM
In this section, benefiting from the superior performances of the AGDLs, we aim to study the dependence of modal characteristics of the VPP0-$\textrm{HE}_{\textrm{11}}^{{\kern 1pt} y}$ HPM on the relevant geometry and physical parameters. Following the approach developed in previous section for g = 0.5 μm, we investigate mode properties versus N for different t’s at g = 1 μm (Fig. 8). For t = 10, 20, and 50 nm, N = 100, 50, and 25, respectively, are required to completely fill the rib part of SiO2 substrate (i.e., gSiO2 = 0). We observe that Re(ne)’s and Am’s are nearly independent of t for the same values of N; this is attributed to t being much larger than the thickness of a single graphene sheet. Therefore, the graphene filling ratios of the AGDLs vary very slightly for different t’s, resulting in similar effective permittivities of AGDLs based on the EMA [42,43]. Interestingly, maximum Lp’s occur at specific N’s for different t’s rather than monotonically increasing as N increases. Here, we introduce parameter η = Nt/g as the ratio between thicknesses of AGDLs and g. The optimal N’s are 60 (η = 0.6), 35 (η = 0.7), and 15 (η = 0.75) for t = 10, 20, and 50 nm, respectively [Fig. 8(b)].
Fig. 7. Electric field, |E| along (a) y = -d/2 and (b) x = 0+ for the VPP0-$\textrm{HE}_{\textrm{11}}^{{\kern 1pt} y}$ HPM at N = 1, 4, 7, and 10.
Fig. 8. The (a) Re (ne), (b) Lp, (c) Am, and (d) FoM versus N for different t’s at g = 1 μm, d = 30 μm, and w0 = 30 μm.
Different from the case shown in Fig. 5(b), the optimal Lp for t = 50 nm was achieved for N = 15 (η = 0.75) but not for N = 20 (η = 1). Similar results were obtained for t = 10 and 20 nm. For instance, the optimal FoM = 42,336 (28,457) with Lp = 3439.5 (2274.5) μm and Am = 5.18 × 10−3 (5.01 × 10−3) for t = 10 (20) nm was achieved at N = 55 (35). Thus, we need to choose larger η for larger t to obtain optimal Lp. The optimal Lp for different t and g is attributed to both N and the thickness of gSiO2. For several values of g and t, we list η’s for the optimal Lp’s in Table 1. We find that the optimal η’s are in a range from 0.5 to 0.6, 0.7 to 0.8, and 0.75 to 0.85 for t = 10, 20, and 50 nm, respectively. Here, we also varied g from 1.0 to 2.5 μm.
Table 1. The values of η for different values of g and t.
The optimal η’s for several values of t and g shown in Table 1 were used to study mode properties (Fig. 9). In particular, as g increased, the variations in Re(ne) and Am were minor for t = 10 and 20 nm but moderate for t = 50 nm. However, propagation lengths showed relatively strong variations. The proposed MLGHPW features the largest propagation length Lp = 4,734 μm (∼ 4.7 mm) ever reported in THz range when the subwavelength mode area is kept Am = 5.54 × 10−3 for t = 10 nm and g = 2 μm. The latter results in extremely high FoM of 56,374.
Fig. 9. The (a) Re (ne), (b) Lp, (c) Am, and (d) FoM versus g at the chosen optimal η’s for different t’s at d = 30 μm and w0 = 30 μm.
Next, we analyzed the dependence of mode behaviors from d and w0 (Fig. 10).
Fig. 10. Mode dynamics for (a) Re (ne), (b) Lp, (c) Am, and (d) FoM versus the diameter of the cylindrical waveguide d for different w0’s at t = 10 nm and g = 2 μm.
At t = 10 nm and g = 2 μm, the values of Re(ne), Am, and Lp increased significantly with d. The latter was attributed to the fact that larger d led to generation of the cylinder-like mode [17]. Nevertheless, Re(ne) and Am were almost invariant for different w0’s. In contrast, Lp’s displayed moderate variation for different w0’s. Comparing Lp’s for w0 ranging from 20 to 40 μm, w0 = 30 exhibited the longest Lp [Fig. 10(d)]. Here we chose optimal N = 110 (η = 0.55, see Table 1) at t = 10 nm and g = 2 μm corresponding to w0 = 30 μm. Therefore, the deviation of w0 from 30 μm resulted in shorter Lp. In contrast, when d exceeds 30 μm, cylinder-like mode compensates propagation loss. Thus, even for the values of d deviating from 30 μm, longer Lp can be still obtained. To verify the performance of the proposed MLGHPW, the mode properties for N = 1, 10, and 50 (optimal value) at t = 10 nm, g = 1 μm, d = w0 = 30 μm, and Ef = 0.5 eV were compared with those from the previously reported results [25–27]. For a fair comparison, the parameters in the Refs. [25–27] were similar to the above values. For instance, w = 20 μm, h = 30 μm, and d = 1 μm in Ref. [25]; d = dgap = 30 μm, g = 1 μm in Ref. [26]; and d = w0 = 30 μm, di = 18 μm, and g = 1 μm in Ref. [27]. The comparisons of Am, Lp, and FoM are listed in Table 2.
Table 2. Comparisons of the modal properties of Am, Lp, and FoM.
Clearly, our design achieves the least Am and about 30 times larger Lp than that in Refs. [25] and [26]. Comparing with that in Ref. [27], although Am = 5.2 × 10−3 in this work was a slightly larger than Am = 3.0 × 10−3 in Ref. [27], the Lp = 3,440 μm was much larger (about 40 times) than Lp = 80 μm in Ref. [27]. Thus, we achieved the extremely high FoM of 42,336. Note that the significant improvement of Lp of the proposed structure cannot be achieved by directly adopting N times the conductivity in the structure of [26]. For instance, the proposed structure and the structure in [26] obtain Lp = 697 (1695) and 318 (730) μm, respectively, for N = 10 (20) at the parameters of t = 10 nm, g = 0.5 μm, d = 30 μm, and w0 = 30 μm, thus demonstrating the effect of hyperbolic metamaterials. For observing the mode field leaking to the cylindrical waveguide as N increases to the optimal condition of N = 50, we show the mode profiles of |E| for N = 10 and 50 for comparison in Figs. 11(a) and 11(b), respectively.
Fig. 11. The mode profiles of |E| for (a) N = 10 and (d) N = 50 at t = 10 nm, g = 1 μm, d = w0 = 30 μm, and Ef = 0.5 eV.
Different from the graphene-based devices with a few (<5) graphene layers adopting a conventional single voltage to a top contact [50,51], an inhomogeneous carrier density distribution occurs due to interlayer screening [52] in a multilayer graphene structure. Therefore, a potential electrical gating scheme [40] is adopted to achieve the required Fermi energy in the proposed MLGHPW. This scheme [40] controls the carrier concentrations of individual graphene layers by different gate voltages, in which each graphene layer could be accessed independently, making the carrier concentrations alter together and thus induce the same surface charge density in all layers. Finally, we investigated the effect of varying Ef for d = w0 = 30 μm, t = 10 nm, and g = 2 μm, and the mode features versus Ef for several values of N (Fig. 12). For Ef = 0.5 eV and N = 110, the longest Lp can be achieved [Fig. 12(b)]. Similarly, N = 115, 95, and 90 were optimal values for Ef = 0.4, 0.6, and 0.7 eV, respectively, showing that the optimal N decreases as Ef increases. This result also reflects that larger Ef with looser mode confinement has an equivalent effect as increasing N. Therefore, to achieve optimal performance, N needs to be reduced while increasing Ef. Theoretically, the hybrid mode of the proposed MLGHPW can be considered as that of a conventional HPW [17] by observing the mode profiles (see Fig. 11) and Re(ne) (<2 for N >10), excepting its ultra-low-loss mode property. Different from exciting the high-k VPP modes (higher order modes) by etched metal grating structures [53,54], the excitation of the resultant mode of the present structure is analogous to that of an HPW or a metal-insulator-metal waveguide structure, lowering the difficulty of the excitation of graphene SPP modes.
Fig. 12. The dynamics of (a) Re (ne), (b) Lp, (c) Am, and (d) FoM versus Ef for different N’s at t = 10 nm, g = 2 μm, d = 30 μm, and w0 = 30 μm.
4. Waveguide crosstalk
To achieve high-density integration of photonic components on chips, the crosstalk between adjacent waveguides should be necessarily addressed. The coupling length of a coupled lossless waveguide Lc = λ/[2(ns – na)] [55] which is the length required to completely transfer power from one waveguide to another, is used to estimate the degree of crosstalk according to the coupled mode theory. Here ns and na are the effective refractive indices of the symmetric and asymmetric modes, respectively. Nevertheless, a more appropriate criterion considers both the power attenuation and maximum power transfer to measure the crosstalk of a coupled lossy waveguide, such as plasmonic waveguides, using the normalized coupling length Lc/Lp [56]. Once the condition of Lc/Lp >1 is reached, the adjacent waveguides can be considered as isolated, because the transferred power from one channel to the other is relatively weak within the distance of Lp. Figure 13(a) shows a coupled waveguide consisting of two MLGHPWs with a center-to-center separation of s. The electric field components Ey of the symmetric and asymmetric modes with their zoomed-in views are shown in Figs. 13(b) and 13(c), respectively. We set t = 50 nm, g = 0.5 μm, N = 10, d = 30 μm, w0 = 30 μm, Ef = 0.5 eV, and s = 35 μm. Clearly, the relatively weak overlaps of field Ey between the adjacent MLGHPWs result in the large value of Lc/Lp = 1.693 (Lc = 999.54 μm and Lp = 590.34 μm) realizing the highly-dense photonic integrated circuits. Note that the optimal values of N were adopted in the following calculations.
Fig. 13. (a) Schematic of a coupled waveguide consisting of two parallel MLGHPWs with a center-to-center separation s. Field profiles of Ey for (b) symmetric and (c) asymmetric modes at t = 50 nm, g = 0.5 μm, d = 30 μm, w0 = 30 μm, Ef = 0.5 eV, and s = 35 μm.
Several geometry parameters of the AGDLs were considered, provided d = 30 μm, w0 = 30 μm, and Ef = 0.5 eV. We first studied the normalized coupling length, Lc/Lp versus the separation s for several t’s at g = 2 μm. Next, the same length was investigated for several g’s at t = 20 nm. The corresponding plots are shown in Figs. 14(a) and 14(b).
Fig. 14. The normalized coupling length, Lc/Lp versus the center-to-center separation, s between the adjacent MLGHPWs with d = 30 μm, w0 = 30 μm for several values of (a) t at g = 2 μm and Ef = 0.5 eV, (b) g at t = 20 nm and Ef = 0.5 eV, and (c) Ef at t = 50 nm and g = 0.5 μm.
For any s, Lc/Lp increased with t [Fig. 14(a)] because of better mode confinement at larger t [Fig. 9(c)]. To reach isolation [48] between waveguides, the values of s should be greater than 37, 42, and 45 μm for t = 50, 20, and 10 nm, respectively. In Fig. 14(b), we observed that Lc/Lp increased with decreasing g. Therefore, larger t and smaller g resulted in weaker crosstalk because of better mode confinement. The latter allowed to significantly enhance the density of integration. For t = 50 nm and g = 0.5 μm, we considered the effect of varying Ef on Lc/Lp [Fig. 14(c)]. In particular, smaller Ef led to tighter mode confinement, thus attaining larger Lc/Lp. Finally, for Ef = 0.5 (0.6) eV, s >32 (35) μm is sufficient to decouple the adjacent MLGHPWs.
5. Summary
We have reported a compact multilayer graphene-based hybrid plasmonic waveguide (MLGHPW) with unprecedentedly low operating losses in the THz range (f = 3 THz). The MLGHPW consisted of an AGDLs sandwiched between a cylindrical high-index dielectric waveguide and a low-index rib substrate. We demonstrated that coupling the dielectric waveguide mode and the fundamental VPP mode of the AGDLs resulted in hybrid mode generation. This mode featured extremely large propagation length of few millimeters while keeping a deep subwavelength mode area of 10−3 below the diffraction limit. The mode characteristics of the present MLGHPW can be controlled by the number of graphene layers. More specifically, it allows to effectively increase the degree of freedom to tailor the mode properties. We have also determined the optimal number of graphene layers for different geometry parameters and Fermi energies. For a comparable order of mode area, the MLGHPW can transmit THz waves forty times further compared to the previously reported GHPWs. Moreover, the negligible crosstalk between the two adjacent MLGHPWs also demonstrates its advantage and possibility to realize highly-integrated on-chip terahertz devices. Our design is expected to pave the way for potential applications in building ultra-low-loss, compact, and tunable THz photonic components and can be extended to other extraordinary two-dimensional materials.
Funding
Ministry of Science and Technology, Taiwan (110-2112-M-005-005).
Acknowledgments
The authors would like to thank Enago (http://www.enago.tw) for the English language review.
Disclosures
The author declares no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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