Low loss hybrid plasmonic waveguide with graphene multilayers Low loss hybrid plasmonic waveguide with graphene multilayers

A proposed design of a graphene-based hybrid plasmonic waveguide is presented to improve the propagation length with good con ﬁ nement. The suggested design has multilayers of SiO 2 -graphene/SiO 2 e GaAs e graphene/SiO 2 . The full vectorial ﬁ nite element method (FVFEM) is used to study the effective index (n eff ), propagation length (L p ), and normalized effective mode area (A eff ) of the supported hybrid plasmonic modes. In this investigation, the structure and the geometrical parameters are studied to achieve an ultra-small effective mode area with low propagation loss. The numerical results show that a long propagation length of 138 m m at a frequency of 3 THz is achieved with a normalized mode area of ~10 ¡ 5 l 2 and propagation loss of 0.0315 dB/ m m. Therefore, the reported waveguide has advantages in terms of low propagation loss with good ﬁ eld con ﬁ nement, which can be effectively used in integrated photonic devices


Introduction
S urface plasmons (SPs), combined with free electrons and electromagnetic field oscillation at the metal/dielectric interface, can confine the light in a very small area beyond the diffraction limit. Therefore, SPs have sparked a lot of research in different applications including waveguiding (Teng et al., 2020a;Mabrouki and Latrach, 2015), sensing (Azzam et al., 2016), optical antennas (Obayya et al., 2015), data storage (El-Rabiaey et al., 2016;Areed and Obayya, 2014), optical filters (Almewafy et al., 2019), and solar cells (Hussein et al., 2014;Areed et al., 2018). The SPs in the visible to near-infrared frequency range are typically supported by noble metals such as gold and silver. As a result, a variety of plasmonic waveguide structures have been examined and proposed such as the long-range plasmonic waveguide (Oulton et al., 2008), dielectric-loaded plasmon waveguide (Weeber et al., 2017), metallic nanowire waveguide (Bian et al., 2018), metallic slot waveguide , and hybrid plasmonic waveguide (Chen et al., 2012). Sub-wavelength confinement has been reported using surface plasmon polariton modes in different plasmonic waveguides (Oulton et al., 2008). In this context, dielectric-loaded plasmonic waveguides have been introduced using polymer ridges with CMOS-compatible metallic thin films (Weeber et al., 2017). Further, a silicon nano-rib loaded with a metallic nanowire at 1.55 mm was designed (Bian et al., 2018) with a propagation length of 2.2e60.2 mm and a mode area of (l 2 ∕4.5 Â 10 5 l 2 ∕7 Â 10 3 ). Hybrid plasmonic slot-rib waveguide has been presented at a wavelength of 1.55 mm  with mode effective area in the range of l 2 /1000000-l 2 / 100000) along with a reasonable propagation length (5 mme25 mm). A thin metallic film has been placed between two identical dielectric nanowires (Chen et al., 2012) with a propagation length of 434 mm and a mode area of 0.0096 mm 2 at 1.55 mm. However, there is a tradeoff between low modal loss and device miniaturization with modal field containment.
In the terahertz (THz) range, the plasmonic waves are extremely low at the metal surface, limiting their use at the nanoscale.
Graphene has unique photonics, electronics, and plasmonic features with a single layer of a combined atomic carbon sheet (Grigorenko et al., 2012). Furthermore, SP polariton (SPP) transmission can be stimulated and promoted by graphene layers in the THz to mid-infrared (mid-IR) spectrum, which typically ranges from 10 to 4000 cm À1 (Polini, 2016). Also, graphene-based plasmons have advantages compared with conventional plasmonic materials in terms of sub-wavelength mode containment and strong lightematter interactions. Further, a gate voltage can actively adjust the chemical potential for graphene to improve the wave guidance performance. Several graphene plasmonic waveguides for guiding SPP with deep sub-wavelength were proposed using graphene films (Robinson et al., 2010), graphene nanoribbons (Tran et al., 2017), graphene wedge/groove (Liu et al., 2013), graphene-coated nano-spheres (Christensen et al., 2015), and graphene-coated nanowires (Teng et al., 2020b). Dielectric-loaded  graphene plasmonic waveguide has been also introduced based on a dielectric strip with a graphene sheet (Xu et al., 2015). Terahertz dielectric-loaded graphene waveguide has been studied where the cutoff wavelength of higher order modes has been calculated (Teng and Wang, 2021). In the study by Zhou et al., 2014, high-density polyethylene (HDPE) and GaAs have been used as high-index and low-index dielectric layers in graphene-based hybrid plasmonic waveguides. Small mode area and propagation length of 232.6 mm 2 and 127 mm, respectively, have been obtained at 3 THz. In the study by Rezaei et al., 2015, a strip-assisted hybrid plasmonic waveguide has been suggested with a propagation length of 820 mm, power confinement ratio of 0.65, and normalized mode area (A eff ) of 0.03 at 3 THz. Further, a two-layered MgF 2 eSi structure covered by a graphene-coated nano-wire is reported in Ref (Hajati and Hajati, 2016) with a long propagation length of 13.91 mm. In addition, a longrange SPP hybrid waveguide has been proposed using a graphene material with a long propagation length of 10 mm at 30 THz and normalized A eff of 10 À7 (Liu et al., 2016). A distributed Bragg reflector-based hybrid plasmonic terahertz waveguide is proposed at 3 THz in the study by Mahankali et al., 2022. The normalized mode area was 3.733 Â 10 À3 mm 2 with a propagation length of 16.8 mm. Multilayer graphene-based hybrid plasmonic waveguide has been introduced using the cylindrical dielectric waveguide and hyperbolic metamaterials with a few millimeters of propagation length with a mode area of 10 À3 mm 2 at a wavelength of 100 mm (Huang and Huang, 2021). Further, a compact ultralow-loss graphene-based multilayer hybrid plasmonic waveguide has been suggested with a propagation length of 45.28 mm and a mode area of 0.0547 mm 2 at l ¼ 1550 nm (Bahrami-Chenaghlou et al., 2023). However, the graphene plasmon is affected by high absorption loss in the mid-infrared waveband, as graphene contains the majority of the light energy. In addition, graphene plasmonic wave propagation is still relatively small. The main challenge for the graphene-based SPP waveguide is therefore to dramatically improve the propagation length while retaining the existing degree of containment. Graphene-based hybrid plasmonic waveguides (GHPW) have the advantages of graphene and traditional hybrid plasmonic waveguides. They combine deep sub-wavelength and high-performance characteristics.
In this study, a modified hybrid plasmonic terahertz waveguide is reported and optimized with an ultra-small mode area (~10 À5 l 2 ), which is smaller than those of the traditional graphene plasmonic waveguides (Zhou et al., 2014;He et al., 2018). This is achieved by improving the coupling between the fundamental dielectric waveguide mode and the surface plasmon polaritons mode (Iorsh et al., 2013). The suggested structure consists of an SiO 2 egraphene/SiO 2 eGaAsegraphene/SiO 2 multilayer. The reported structure has the advantage of using multilayers of graphene (Liu et al., 2018;Qin et al., 2014) to improve the propagation properties. The impact of the structural geometrical parameters on the modal characteristics of the reported waveguide is carried out using full vectorial finiteelement methods (Obayya et al., 2000). In this context, the effective index (n eff ), propagation length (L p ), and normalized mode area (A eff ) are studied thoroughly. A small mode area of 10 À5 l 2 is achieved with a propagation loss of 0.0315 dB/mm at f ¼ 3 THz, which is smaller than those reported in Refs (He et al., 2018(He et al., , 2019a(He et al., , 2019b(He et al., , 2021. The reported waveguide has a good potential for highly efficient light transmission in integrated photonic circuits.

Design considerations and numerical results
Fig . 1 shows the schematic diagram of the SiO 2 e graphene/SiO 2 eGaAsegraphene/SiO 2 -multilayer structure. Two identical outer silica layers and two identical inner graphene multilayers are symmetrically placed on both sides of the GaAs layer. The GaAs layer consists of two rectangular waveguides with thickness g and width w. The separation between the rectangular waveguides is labeled as h. Further, a micro semicircular rib with radius r is used, with a high-index material of GaAs (n ¼ 3.5). In addition, 10 graphene layers are used with a thickness of t, which are separated by silicon layers with a thickness of b. The FVFEM (Obayya et al., 2000) is used via COMSOL Multiphysics software [https://www. comsol. com] to analyze the waveguide characteristics. The graphene permittivity can be determined by applying (Hanson, 2008) the following equation: where t ¼ 0.5 nm is the monolayer graphene thickness; 3 0 is the permittivity of free space; and u is the angular frequency. According to Kubo's formulation (Ziegler, 2007), the interband and intraband contributions to the graphene surface conductivity are as follows: where T is the temperature; k B is Boltzmann's constant; ħ is the reduced plank constant; m c is graphene's chemical potential; t is the time of electron relaxation; and e is the electron's charge. Here, T ¼ 300 K, t ¼ 0.5 ps, and m c ¼ 0.5 eV.
In this study, the n eff , the normalized A eff, L p , and figure of merit (FoM) (Chowdhury, 2011) of the supported hybrid modes are studied and analyzed. The FoM is calculated using the equation The propagation length is given by The effective mode area A m can be found by the formula (Chowdhury, 2011) The W(r) is the energy density of the supported mode with electric field E(r) and magnetic field H(r), which is defined by the equation where m 0 is the free space magnetic permeability, and 3(r) is the dielectric permittivity. It is required to improve the field confinement of the supported modes with long propagation lengths and minimum propagation losses.

Numerical results of plasmonic
To achieve long propagation length with strong field confinement, the geometrical parameters of the reported structure are studied. First, the effect of the number of graphene layers is investigated. Fig. 2 shows the mode properties' dependence on the numbers of graphene layers N. Here, N varies from 3 to 10 layers. However, the other geometrical parameters are fixed at g ¼ 5 mm, h ¼ 1 mm, r ¼ 0.5 mm, w ¼ 20 mm, and b ¼ 14 nm. Fig. 2a shows that the effective index n eff increases and then decreases by increasing the number of graphene layers N. As N increases, the propagation length L p and figure of merit FOM increase as shown in Fig. 2b, d. It is also evident that the normalized mode area A eff increases slightly from 1.5 Â 10 À4 to 3.17 Â 10 À4 as shown in Fig. 2c. It can be noticed that the propagation length L p increases from 25.4 mm to 98.2 mm, and the FoM increases from 36 to 97 by increasing N from 3 layers to 10 layers. Therefore, N ¼ 10 is taken with high propagation length and sub-wavelength mode confinement. The effective conductivity of the fundamental volume plasmon polariton modes with a significant electric field is approximately Ns, and the mode field profile widens moderately as N increases, according to the analytical formula (Smirnova et al., 2014) of a multilayer graphene waveguide. Hence, when N grows, larger L p and A eff are due to the strong coupling between the volume plasmon polariton and major electric fields. Fig. 3 shows the 2D energy density of the supported modes at N ¼ 3, 6, and 10 layers, while the 1D plots are shown on the right at y ¼ 0. The optical energy is confined around a micro semicircular rib as N increases. Therefore, the A eff is increased [ Fig. 2c] with a significant reduction of ohmic losses. When N increases, the enhancement of the field peak, as shown in Fig. 3, compensates for the full width at half maximum along y ¼ 0. Further, the number of graphene layers can improve the achieved optical energy. Next, the effect of radius r is studied. Fig. 4 shows the variation of the n eff , L p , A eff , and FoM, depending on thickness g at various micro semicircular rib r values (0.5 mm, 1 mm, and 2 mm) within the g range of 0.5e10 mm. It can be seen that n eff , increases, while the L p and FoM decrease as the distance g increases. Therefore, at increased micro semicircular rib r, smaller A eff , L p high, and high FOM can be obtained with strong optical confinement as shown in Fig. 4bed.
Especially when r ¼ 2 mm, a minimum of A eff of [~9 Â 10 À4 (l 2 /4)] is achieved at g ¼ 0.5 mm with L p of 103.5 mm and FOM of 61 as illustrated in Fig. 4aec. It can be noticed that to obtain better performance using the micro semicircular rib, r equals h/2. Fig. 5 shows the 2D energy density distribution of the supported hybrid mode at different micro semicircular rib radii (r). It may be seen that at r ¼ 0.5 mmas shown in Fig. 5a; low-energy flow density distribution exists in graphene multilayers due to the weak coupling between plasmonic and optical modes. It may be seen that when the distance r increases, the optical energy increases owing to the strong coupling with the SPP mode. Due to the increase in the area of a high-index region, the mode field is more concentrated toward the region between two micro semicircular ribs (Fig. 5d and e) showing 1D energy density distributions along the vertical (at y ¼ 0) and horizontal (at x ¼ 0) directions as r changes from 0.5 mm to 2 mm. It is evident from Fig. 5e that the energy density in the gap area is sharply increased as the distance r increases to 2 mm. Therefore, the optical energy of the supported mode can be confined in the gap, when a large r is used.
The impact of the rectangular waveguide is next investigated. Fig. 6 illustrates the variation of the real part of the effective index n eff , L p , normalized A eff , and FoM, with the distance h at various thicknesses and g values (0.5 mm, 1 mm, and 3 mm) within the range 0.5e15 mm. It can be seen that n eff , L p , and A eff increase, while the FoM decreases as the distance g increases. Therefore, at a reduced distance h, a smaller A eff and high L p can be obtained with strong optical confinement as shown in Fig. 6bed. At g ¼ 0.5 mm, A eff of [6.6 Â 10 À5~1 4.4 Â 10 À3 (l 2 /4)] can be achieved, where an L p of (64 mm,~150 mm) and an FOM of (139~22) are achieved as illustrated in Fig. 6aed. These results show that the distance g and h can be simultaneously controlled to achieve deep sub-wavelength confinement with the improved overall optical performance. The numerical simulations indicate that by designing the proposed geometric structure, a smaller A eff [~6.6 Â 10 À5 (l 2 /4)] can be achieved at a propagation length of 64 mm. Fig. 7 illustrates the energy density of the 2D and 1D (at y ¼ 0) electric field distributions of the supported modes at different distances h: 1 mm, 7 mm, and 15 mm. The optical energy of the supported mode is improved by decreasing the distance h.
To achieve high efficiency with tunable performance, the effect of the applied potential on the graphene layer is studied. Fig. 8 shows the effect of the chemical potential of the graphene m c on the n eff , A eff, and L p of the fundamental hybrid mode at w ¼ 25 mm and b ¼ 14 nm. It can be seen that the effective index n eff decreases monotonically, while L p  and FOM increase monotonically with the increase of m c . In this circumstance, A eff of [3 Â 10 À5~9 .5 Â 10 À5 ] can be achieved where the chemical potential m c can be tuned to achieve relative lengths of propagation (13.6 mme138 mm). When m c is increased, the conductivity value of graphene (s g ) will be increased, while the imaginary part of the effective index (Im (n eff ) will be reduced. Then, the graphene loss is low at large m c values. The proposed structure has deep sub-wavelength confinement, which is very promising for the integration of compact photonic devices.
The effect of the temperature on the graphene layer used in the proposed design with its optimal geometries (g ¼ 0.5 mm, h ¼ 0.5 mm, w ¼ 25 mm, b ¼ 14 nm) is next studied. Fig. 9 shows the impact of temperature on the L p and A eff of the fundamental hybrid mode. It can be observed that the temperature has a negligible effect on the normalized A eff as well as the propagation length over the temperature ranging from 200 K to 400.
The optical properties of the reported waveguide are compared with those presented in the literature as shown in Table 1. It is revealed from this table that the reported waveguide has a much lower loss than the previously graphene plasmonic waveguides (He et al., 2018(He et al., , 2019a(He et al., , 2019b(He et al., , 2021. Thereby, a mode area of about (~10 À5 l 2 ) with a long propagation length of 138 mm can be achieved. This is a significant improvement over the previous structures (Zhou et al., 2014;He et al., 2018He et al., , 2019aHe et al., , 2019bHe et al., , 2021. Furthermore, the reported waveguide has stronger optical confinement and longer propagation length than the proposed waveguide with GaAs and graphene (He et al., 2018(He et al., , 2019a(He et al., , 2019b(He et al., , 2021. It is evident from Table 1 that the suggested design has a propagation length 3 times longer than that previously designed in the study by He et al., 2018 with the same material. Compared with the recently published designs presented by Huang and Huang (2021) and Bahrami-Chenaghlou et al. (2023), which are operating at l ¼ 100 mm and 1550 nm, respectively, the suggested hybrid plasmonic terahertz waveguide based on multilayers of graphene has a propagation length of 138 mm with a mode normalized area of~10 À5 l 2 at f ¼ 3 THz (l ¼ 100 mm). This improvement is due to using multilayers of graphene that can be used to realize compact hybrid waveguides. Fig. 9. Variation of (a) L p and (b) normalized A eff of the supported hybrid mode with temperature T, while the other geometrical parameters are fixed at g ¼ 0.5 mm, h ¼ 0.5 mm, w ¼ 25 mm, and b ¼ 14 nm.

Conclusion
A hybrid plasmonic multilayer-graphene waveguide is proposed and investigated for deep terahertz sub-wavelength confinement. The modal analysis of the reported waveguide is made using FVFEM. The numerical results show that the suggested normalized mode area could be very small (~10 À5 l 2 ) with a long propagation length of 138 mm. The reported structure has also low propagation loss with good mode confinement. Therefore, the structure has a good potential capacity for developing THz waveguide with compact field confinement for photonic integrated circuits.