Graphene-Coated Two-Layer Dielectric Loaded Surface Plasmon Polariton Rib Waveguide With Ultra-Long Propagation Length and Ultra-High Electro-Optic Wavelength Tuning

A graphene-coated two-layer dielectric loaded surface plasmon polariton (GTDLSPP) rib waveguide is designed. The mode characteristics and electro-optic (EO) modulation performances of the four hybrid plasmonic modes (HPMs) in the designed waveguide are simulated by using the finite element method. The simulation results show that a 103 mm-scale propagation length and an effective mode field area of $\sim \lambda ^{2}$ /1333 are obtained by adjusting the bias voltage. The EO wavelength tunings are −68.6, −42.0, −49.7, and −11.1 nm/V for the HPM 1, HPM 2, HPM 3, and the peak 2 of HPM 4, which are two orders of magnitude larger than those of other EO modulation structures. For the peak 1 of the HPM 4, the EO wavelength tuning is the piecewise linear. For a 150- $\mu \text{m}$ long waveguide, the modulation depths of ~98.7, ~87.9, and 99.5%, and FWHMs of ~450, ~100, and ~42 nm can be achieved for the HPM 1, HPM 2, and HPM 3. For the HPM 4, there are two peaks in the transmission spectrum. The modulation depths are ~97.3 and 75.2%, and FWHMs are ~92 and ~34 nm for the peaks 1 and 2. There is a tradeoff between the modulation depth and FWHM for different waveguide lengths. The GTDLSPP rib waveguide designed has small size, high modulation depth, broad bandwidth, and compatibility with the CMOS technology, so it has potential applications in the EO tunable devices, optical interconnects, and optical switches.

saturable absorption [7]. Moreover, the Fermi energy of graphene can be adjusted by the electrical gating [8], which can convert the electric data to the optical signals [9]. Therefore, the graphene-based EO modulators have been attracting great research interests [10]. However, the interaction between the graphene and vertical incident light becomes weak due to the atomic thickness of the graphene. In order to enhance the interaction, the modulators based on Mach-Zehnder interferometer [11], integrated optical cavities [12], [13], patterned graphene structures [14], [15], hybrid plasmonic structures [16], [17] have been demonstrated. Among them, the graphene-on-silicon (GoS) waveguides show excellent characteristics owing to the advantages of graphene [18], [19].
For the optical devices based on silicon-on-insulator (SOI) waveguides, high integration density and miniaturization remain major challenges in micro and nanotechnology due to the diffraction limit [20]. In contrast, the plasmonic waveguides can break the diffraction limit and achieve the subwavelength optical confinement at the interface between the dielectric and metal [21]. Thus, the hybrid plasmonic waveguides (HPWs), which combine the two or more types of waveguides into a single structure, are widely used as the compact modulators. The HPW can generate the hybrid modes to improve the compromise losses and confinement [22], and provide a scheme for the enhanced EO modulation at a small length scale [23]. Recently, the graphene plasmonic waveguide, which is also called as graphene HPW (GHPW), has been demonstrated to guide the surface plasmon polariton (SPP) wave [24]. The GHPW-based modulators combine the advantages of the graphene and plasmonic waveguides. However, the combination of the graphene and plasmonic waveguides presents a challenge due to the anisotropy permittivity of the graphene [25], which could be improved by optimizing the design of an engineered device. A GHPW composed of a Si-graphene-SiO 2 sandwich and a silver cylindrical nanowire is designed for the EO modulation, whose modulation depth is ∼0.3 dB·µm −1 at low gating voltages [26]. Moreover, the GHPW is combined with a metal-insulator-metal (MIM) structure, which can increase the modulation depth to ∼0.6 dB·µm −1 . To manipulate the angular momentum of photons at the THz frequency, a GHPW based on a gallium arsenide (GaAs) waveguide embedded in high-density polyethylene (HDPE) with five-layer graphene is proposed [27]. The relative phase of π/2 is achieved when the graphene length is 145 µm. A chain of three metal disks on a GHPW is used to realize a submicron EO modulator, whose extinction ratio (ER) is 5.5 dB, waveguide length is 740 nm, and 3-dB bandwidth is 83.4 GHz at wavelength 1.55 µm [28]. To enlarge the bandwidth, a graphene-based plasmonic valley-slot waveguide modulator is presented [29]. A GHPW modulator based on graphene-hexagonal-boron-nitride-graphene sandwich is analyzed [30], where the modulator has a high ER of 39.75 dB and a large bandwidth of 190.5 GHz. By optimizing the parameters, a modulation depth of 3 dB is achieved within a broad wavelength range from 1400 to 1600 nm. A graphene sheet as an absorber tunable layer is embedded inside a sub-wavelength-thick multilayer structure, which is compose of Ag, Si 3 N 4 , and Ta 2 O 5 layers [31]. The modulation depth is 7.5 % at wavelength 1.55 µm. A graphene intensity/phase modulator based on ultra-thin (<80 nm) silicon strip waveguide is designed to improve the modulation efficiency [32]. The modulation depth of 0.280 dB·µm −1 with a bandwidth of 216 GHz at a waveguide thickness of 30 nm is achieved. Silicon nitride (Si 3 N 4 ) waveguide has low propagation loss, relatively small refractive index (RI) contrast, large bandgap, and compatibility with the CMOS process [33]. Thus, it is considered as an ideal candidate for building the optical modulators [34].
In this paper, by combining the advantages of the GHPW, Si 3 N 4 , and two-layer dielectric loaded surface plasmon polariton (TDLSPP) waveguide, we propose a graphenecoated two-layer dielectric loaded surface plasmon polariton (GTDLSPP) rib waveguide. Two dielectric layers of the GTDLSPP rib waveguide are silicon and Si 3 N 4 , respectively. The mode characteristics and EO tunability of the GTDLSPP rib waveguide are analyzed and optimized. The ultra-long propagation length and ultra-high electro-optic tuning can be achieved. The GTDLSPP rib waveguide can be used as the EO modulator by applying the low bias voltage, which has potential applications in the ultra-compact on-chip micro-nano photonic system. The rest of this paper is organized as follows. In Section II, the structure of the GTDLSPP rib waveguide is designed. In Section III, the mode characteristics of the GTDLSPP rib waveguide are analyzed and optimized. In Section IV, the EO wavelength tuning and modulation performances of the GTDLSPP rib waveguide with the different bias voltage are investigated. Conclusions are drawn in Section V.

II. STRUCTURE DESIGN OF THE GTDLSPP RIB WAVEGUIDE
The schematic of the designed GTDLSPP rib waveguide is shown in Fig. 1. The GTDLSPP rib waveguide consists of a TDLSPP rib waveguide and a monolayer graphene. The GTDLSPP rib waveguide is made of two dielectric layers with different refractive indices (RIs) deposited on a 70-nm thick gold film. The upper layer is the silicon with a high RI of 3.45, and the middle layer is the silicon nitride with a low RI of 2. The cross-section of the GTDLSPP rib waveguide is shown in Fig. 1(b). The thicknesses of the rectangular and rib waveguides are h 1 , h 2 , and h 3 , the length of the waveguide coated with graphene is L G , and the ridge width of the GTDLSPP rib waveguide is w = 500 nm. h Au and g 0 are the thicknesses of the gold film and monolayer graphene, respectively. The substrate and upper cladding of the GTDLSPP waveguide are the silica and air, respectively.
The frequency-dependent relative permittivity of Au can be described by the Drude model [35] where ε ∞ is the dielectric constant at the infinite frequency, γ is the electron collision frequency, ω p is the bulk plasma frequency, ω = 2π/λ is the angular frequency of the incident light in the vacuum, and λ is the wavelength. For Au, ε ∞ = 9.75, ω p = 1.36×10 16 rad/s, and γ = 1.45×10 14 rad/s [36]. The relative permittivity of the graphene can be equivalently expressed as ε g = 1 + iσ g /(ωε 0 g 0 ) [37], where ε 0 is the permittivity of the vacuum. For the monolayer graphene, g 0 is set as 1 nm. σ g is the complex conductivity of the monolayer graphene, and can be described as σ g = σ intra + σ inter . σ intra and σ inter correspond to the intraband electron-photon scattering and interband transition contribution, which can be characterized by the Kubo formula [35], [38] where e 0 ,h, and k B are the electron charge, reduced Plank's constant, and Boltzman's constant, respectively. T = 300 K is the temperature in Kelvin. τ −1 is the free carrier scattering rate, which is chosen as 2 × 10 12 s −1 [35]. E F is the Fermi energy (chemical potential), which can be electrically controlled by the bias voltage supplied to the monolayer graphene. In Eq. (2), at room temperature, E F is an experimentally tunable parameter, and can be modified by the applied bias voltage as the following equation [39] where v F = 10 6 m/s is the Fermi velocity, a is the capacitance per unit area per charge, and V g is the applied bias voltage between the electrodes. To demonstrate the EO tuning, V g changes from 0 to 16 V, and the corresponding E F changes from 0 to 0.75 eV.
The propagation length (L p ), normalized mode field area (A), and figure of merit (FoM) are used to evaluate the performances of the GTDLSPP rib waveguide. L p is defined as the distance where the initial electromagnetic energy of the guided mode decreases to 1/e [40] where n eff is the mode effective RI of the GTDLSPP rib waveguide, Im (n eff ) is the imaginary part of n eff . The effective mode field area A eff is used to describe the energy confinement, which is defined as the ratio of the total electromagnetic energy to the peak energy density of the guided mode [40] A eff = W (x, y)dxdy max(W (x, y)) .
In Eq. (6), W (x, y) and max (W (x, y)) represent the total electromagnetic energy density distribution of the guided mode and the corresponding peak energy, respectively. In order to evaluate the waveguide performance for guiding light beyond the diffraction limit, the normalized mode field area (A = A eff /A 0 ) is used. Here, A 0 = λ 2 /4 is the diffraction limit mode field area.
To compare the performances of the different waveguides, FoM is defined as the ratio of the propagation distance to the mode field diameter [41] The propagation loss of the GTDLSPP rib waveguide is defined as following [30] α loss = 40π λ ln 10 Im(n eff ) (dB µm).
To analyze the mode characteristics of the GTDLSPP rib waveguide, the commercial COMSOL Multiphysics software based on the finite element method (FEM) is employed. The scattering bound condition is used. To ensure good convergence of the calculated results, the maximum grid sizes for the coated graphene layer and TDLSPP rib waveguide are set as 0.1 nm and 0.08 µm, respectively.

III. WAVEGUIDE CHARACTERISTICS
In this work, λ = 1.55 µm, h 3 = 50 nm, h 2 , h 1 , and V g are selected as the sweep parameters. The complex effective RIs of the four hybrid plasmonic modes (HPMs) in the GTDLSPP rib waveguide are calculated. The RI real parts Re (n eff ) of the four HPMs with different h 2 and h 1 at a fixed V G of 3.182 V are shown in Fig. 2(a). Both the mode number and Re (n eff ) increase with h 1 when h 2 remains unchanged. For the HPM 1 and HPM 4, Re (n eff ) increases with h 2 when h 1 remains unchanged. However, Re (n eff ) increases and then decreases with h 2 when h 1 changes from 50 to 250 nm. In order to ensure that the four HPMs are propagated inside the GTDLSPP rib waveguide, h 1 is chosen as 250 nm. To understand the influences of λ, the relationships between Re (n eff ) and λ are shown in Fig. 2(b). From Fig. 2   as a red dot. The reason for the maximum peak is that Re (n eff ) of the monolayer graphene has a maximum value at V G =∼3.2 V. However, Re (n eff ) of the monolayer graphene permittivity is nearly zero at V G =∼7.8V, which causes the smaller peak at V G =∼8.2 V.
As shown in the inset of Fig. 3(a), the HPM 1 is a photonic-like mode. The electric field direction of the HPM 1 is along the x-axis, and the corresponding electric field energy is mainly concentrated in the upper Si layer. In the inset of Fig. 3(b), the HPM 2 is a conventional HPM, which is generated by a TM mode. Hence, the electric field direction of the HPM 2 is along the y-axis, and the electric field energy is mainly concentrated in the middle silicon nitride layer. In the insets of Figs. 3(c) and 3(d), the HPM 3 and HPM 4 are the higher-order HPMs excited by TM modes.
Figs. 4(a) and 4(b) show the propagation length L p and normalized effective mode field area A of the four HPMs as functions of V G when h 1 = 250 nm, h 2 = 150 nm, and h 3 = 50 nm. In Fig. 4(a), L p of HPM 1, HPM 2, and HPM 3 have the large peaks at V G of 13.8, 1.6, and 3.6 V. Moreover, L p of the HPM 1, HPM 2, and HPM 3 can achieve 10 3 , 10 2 , and 10 mm-scale, respectively, which is longer than that of a conventional TDLSPP waveguide. For the HPM 4, there is only a smaller peak at V G of 8.2 V. L p is ∼10 2 µm, which is comparable to that of a TDLSPP waveguide. A of the four HPMs are shown in Fig. 4(b). It can be seen from Fig. 4(b)  that A of the HPM 1 has little change with V G . As a photoniclike mode, the mode field distribution of the HPM 1 hardly changes with the monolayer graphene. However, A of the HPM 2, HPM 3, and HPM 4 change significantly when V G goes up to a certain value. There is a minimum when V G is ∼13.8 V (E F =∼0.75 eV). To explain the reasons for the changes, the real and imaginary parts Re (n g ) and Im (n g ) of the refractive index n g of the graphene are shown in Fig. 4(c). From Fig. 4(c), Re (n g ) is larger than Im (n g ) when V G is lower than 8 V. The guided mode plays a dominant role in the HPM, which result in a larger A. However, Im (n g ) becomes larger than Re (n g ) when V G is higher than 8 V. The monolayer graphene is close to a very thin ''metal'' layer [37]. The SPP mode plays an important role in the HPM, which can improve the field confinement of the GTDLSPP rib waveguide. When V G increase to ∼13 V, Re (n g ) reduces to less than 0.1, and Im (n g ) increases to larger than 1.5. The ohmic loss increases due to the enhanced metallic property of the graphene, which reduces the field distribution ratio in the core of the GTDLSPP rib waveguide and makes A larger. The normalized A ef of ∼0.010, ∼0.003, and ∼0.012 for the HPM 2, HPM 3, and HPM 4 are achieved. That is, A eff of the HPM 2, HPM 3, and HPM 4 are ∼ λ 2 /400, ∼ λ 2 /1333, and ∼ λ 2 /333, respectively.
To illustrate the effect of graphene, the mode characteristics of the GTDLSPP rib waveguide and TDLSPP rib waveguide are listed in Table 1 for comparison. L p and FoMs of the GTDLSPP rib waveguide are much better than those of the TDLSPP waveguide.
In order to optimize the structure parameters of the GTDLSPP rib waveguide, FoM is investigated when h 1

IV. THE EO MODULATION PERFORMANCES
The Au electrode is deposited directly above the GTDLSPP rib waveguide, as shown in Fig. 1(a), and Au slab under the rib waveguide is used as another electrode. The bias voltage V G can be changed by applying the static electric potential across the Au electrodes. Fig. 6 shows the propagation loss (α loss ) spectra of the four HPMs for the different V G . VOLUME 8, 2020  For comparison, α loss spectra of the TDLSPP rib waveguide (without graphene) are also shown in Figs. 6(a)-6(d). λ changes from 1.4 to 2.4 µm for the HPM 1, and from 1.1 to 1.7 µm for the HPM 2, HPM 3, and HPM 4. There is only one peak for the HPM 1, HPM 2, and HPM 3. However, there are two peaks for the HPM 4. To show the EO tuning of the designed GTDLSPP rib waveguide, the range of V G changes between 0 and 10.4 V. The loss peaks of the HPM 1, HPM 2, and HPM 3 and peak 2 of the HPM 4 are blue-shifted when V G changes from 0 to 10.4 V. However, the peak 1 of the HPM 4 changes slowly.
The wavelength changes of the peaks in the loss spectra with V G and the linear fittings are shown in Fig. 6(e). The EO   in Table 2. The EOWTs of −68.2, −42.0, and −49.7 nm/V are achieved for the HPM 1, HPM 2, and HPM 3. For the peak 2 of the HPM 4, the EOWT is −11.1 nm/V. Moreover, the EOWTs of the HPM 1, HPM 2, HPM 3, and peak 2 of the HPM 4 have good linearity when V G is changed from 0 to 10.4 V. However, the peak 1 of the HPM 4 shows the piecewise linearity. The EOWTs are −5.6 and −3.2 nm/V when V G changes from 0 to 4.8 V and 6.2 to 10.4 V, respectively.
The EOWTs of several other modulators are also listed in Table 2 for comparison. From Table 2, the EOWTs of the GTDLSPP rib waveguide are two orders of magnitude higher than that of other structures.
To show the propagation characteristics of the HPMs in the GTDLSPP rib waveguide, L G is set as 150 µm. The transmission spectra of the four HPMs with different V G are shown in Figs. 7(a)-7(d). It can be seen from Figs. 7(a)-7(c) that for the HPM 1, HPM 2, and HPM 3, the large peaks emerge at a fixed V G . The wavelengths of the peaks have distinct blue-shifts and the widths of the peaks enlarge when V G increases. However, there are two peaks at wavelengths ∼1.3 and ∼1.5 µm for the HPM 4, as shown in Fig. 7(d). The peak 1 drops rapidly when V G is larger than 4 V. The width of the peak 2 hardly changes when V G < 4 V, and the peak wavelength slightly occurs to blue-shift when V G >4 V. VOLUME 8, 2020 In order to evaluate the modulation performances, the normalized transmission spectra of the four HPMs are shown in Figs. 8(a)-8(d) when V G = 0.8, 3.6, 4.8, and 10.4 V, respectively, and L G = 150 µm. Moreover, the modulation depth (M ) is defined as = 10 log 10 ( where T max and T min are the maximum and minimum of the transmission spectra, corresponding to the ''ON'' and ''OFF'' states, respectively. Here, the ''OFF'' or ''ON'' state of the modulators means that the light is blocked or not blocked, and the gate voltages at the ''ON'' and ''OFF'' states are represented by V ON and V OFF , respectively. ME (dB) is another representation of the modulation depth. In Fig. 8(a) Fig. 8(e). When L G changes from 0.5 to 300 µm, M increases sharply and then approaches to the saturation, while FWHM decreases gradually. The FWHM of the transmission peak is out of the considered wavelength range when L G < 50 nm. The insertion losses (ILs) for the different L G are calculated by IL(dB) = −10 log 10 (T max ), as shown in Fig. 8(f). ILs for the four HPMs increase when L G increases from 0.5 to 300 µm. When L G is less than 150 µm, ILs are lower than 6 dB. ILs of ∼0.05 and 1.01 dB are achieved for the HPM 1 and HPM 2, respectively. Considering a compromise of M , FWHM, and IL, L G = 150 µm is the optimum value. The normalized electric field distributions of the ''ON'' and ''OFF'' states in the xy plane and the major components in the xz and yz planes for the four HPMs in the GTDLSPP rib waveguide are shown in Figs. 9(a)-9(e). The major component of the electric field is E x for the HPM 1, and E y for the HPM 2, HPM 3, and HPM 4. From Figs. 9(a)-9(e), the normalized electric field distributions of the ''ON'' and ''OFF'' states indicate that most of the light energy can or not pass through the GTDLSPP rib waveguide. Table 3 shows the related parameters, including M , IL, λ on , FWHM of the peaks at λ on , voltage difference between V ON and V OFF ( V G = V ON -V OFF ), and E F in ''on'' and ''OFF'' states (E F (ON) and E F (OFF)). The modulation performances of some different on-chip EO modulators are listed in Table 4 for comparison. From Table 4, M and IL of the EO modulator based on the designed GTDLSPP rib waveguide is comparable or superior to those of other structures, while the bandwidth is larger than those of other structures.

V. CONCLUSIONS
In summary, we propose a GTDLSPP rib waveguide. The simulation results show that the GTDLSPP rib waveguide can achieve a 10 3 mm-scale propagation length and a max (FoM) of ∼10 6 . A high EO wavelength tuning of -68.6 nm/V is also achieved. The proposed GTDLSPP rib waveguide provides a feasible scheme for the EO amplitude modulation in a wide wavelength range from 1.2 to 2.4 µm. Considering the tradeoff between M and FWHM, M of ∼98.7% and FWHM of ∼450 nm are achieved for a 150-µm long GTDLSPP rib waveguide by changing V G .
The GTDLSPP rib waveguide could provide a feasible on-off keying scheme for the on-chip EO modulation, along with compact footprint, high modulation depth, and broad bandwidth. It also has the promising applications in the EO tunable devices, optical interconnects, and optical switches due to the high EO wavelength tuning. Moreover, the silicon nitride has good optical characteristics in the mid-infrared spectral region, so the proposed GTDLSPP waveguide can also find application in the mid-infrared optoelectronic devices. At the same time, the biocompatibility of the graphene makes the GTDLSPP waveguide convenient for the biochemical sensing.