A hybrid plasmonic waveguide design is proposed that incorporates a two-dimensional transition metal dichalcogenide monolayer covered slot-rib in between a cylindrical waveguide and a metal surface. A deep optical energy confinement (mode area ranging from λ2/1000000-λ2/100000) along with a reasonable propagation length (5μm-25μm) can be realized at the working wavelength of 1550 nm. In comparison with a traditional hybrid plasmonic waveguide, the proposed waveguide structure exhibits a smaller mode area as well as a higher figure of merit. Investigation on the influence of various two-dimensional materials on modal properties reveals that a larger permittivity provides a stronger field confinement. Owing to its excellent energy field confinement with low transmission loss, the proposed waveguide can be utilized in a variety of plasmonic devices such as compact plasmonic chips, high-integration plasmonic nano-lasers and high-sensitivity plasmonic detectors.
© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
Hybrid plasmonic waveguides  (HSPWG) incorporate a low-permittivity dielectric gap region embedded between a high-permittivity dielectric nanowire and a metal surface, facilitating light guiding by the hybrid plasmonic mode beyond the diffraction limit. This results in a lower transmission loss than that of surface plasmon polariton (SPP) mode . Due to strong coupling between a SPP and photonic mode, there is a large mode confinement along with a reasonable propagation distance. This transforms the gap region into an optical capacitor holding large electromagnetic energy. Therefore, HSPWG can potentially bridge the gap between integrated electronic circuits and high speed optical circuits, paving way for next generation high-efficiency optoelectronic communication systems. However, there exists a trade-off between mode confinement and propagation distance , which results in a conflict between device miniaturization and low transmission loss. Various types of plasmonic structures based on HSPWG including ribs , wedges , grooves , bow-ties , long-range , slots [9,10] and hybrid structures with graphene [11–13] have been proposed during recent years. However, the key design consideration in these structures is to keep the mode energy away from the metal region, which results in either only a high energy concentration or small propagation loss. Hence, these constraints limit the capability of HSPWG to be used as a small foot-print chip-scale optical communication device.
Recently, two-dimensional layered transition metal dichalcogenides (TMDCs) have gathered considerable interest due to their intriguing physical characteristics such as spatially squeezed optical energy and have led to shrinkage in the device dimensions . In contrast to monolayer graphene , monolayer TMDCs exhibit remarkable optical features in direct band-gap semiconductors, making them suitable for nano-lasing applications , solar cell elements  and field effect-transistors . Therefore, by introducing monolayer TMDCs in the HSPWG structure, the hybrid plasmonic structure has potential applications in ultra-high integrated optical components that not only exhibit strong field confinement but also offer reasonable propagation distance meeting the requirements of optical chip-design.
In this paper, a novel HSPWG with a high integration capability and figure of merit has been proposed. This plasmonic waveguide consists of a mixed structure in which a low-permittivity dielectric slot with a nano-scale metal rib covered by monolayer TMDCs inside the gap region is embedded between the high-permittivity cylindrical dielectric nanowire and metal surface. Most of the mode energy is distributed near the slot-rib region because of the SPP mode emerging from the surface between the rib metal and the surrounding dielectric. This results in a large energy storage inside the dielectric region with the nano-scale metal rib covered by high-permittivity dielectric materials . These factors result in a relatively small propagation loss due to the loss-less slot dielectric area. On the other hand, owing to high-permittivity and ultra-small thickness, monolayer covered TMDCs have the capacity to improve mode energy confinement as well as sustain coupling between the cylinder dielectric mode and SPP mode. All these factors indicate that the proposed waveguide is a promising candidate for compact optical circuits or chips with high performance.
2. Structures and mode properties
A schematic of the three-dimensional geometry and the cross-section of our proposed waveguide are shown in Fig. 1(a). A hybrid low-permittivity slot-rib structure that combines a low-permittivity material slot, a metal Ag rib and a monolayer TMDCs, is placed between the high-permittivity Si cylinder dielectric waveguide and the Ag surface. Height of the SiO2 gap-region (h) is determined by the distance between the bottom of the cylinder waveguide (diameter d) to the top of the hybrid structure where the Ag rib is at the center of the slot covered by monolayer TMDCs with thickness around 0.7nm . Furthermore, according to our previous work , the radius of curvature of the Ag rib corners should be as large as possible, in order to yielding a smooth structure that minimizes scattering loss. In our proposed structure, the Ag rib has two sharp corners and the maximum curvature radius r is equal to the half value of rib width, as shown in Fig. 1(a). In addition, the rib has a height hs that is fixed at 50 nm and ws is defined as the width of the slot. In our simulation, the working wavelength is chosen as λ = 1550nm, and the slot is filled with a porous SiO2 film that has the permittivity as low as 1.1025 . The permittivity of Si, SiO2, and Ag are taken as 12.25, 2.25, and −129 + 3.3i , respectively.
Our proposed waveguide can be used in communication system which has a similar structure as that of our previous work , where couplers are used to connect input/output silicon-on-insulator (SOI) waveguides and the designed waveguides, as shown in Fig. 1(b). Experimentally demonstrated metallic tapered couplers  are capable of converting photonic modes from Si waveguides to hybrid plasmonic modes from our designed waveguides with high coupling efficiency, on the same SOI platform.
Modal parameters such as mode area Ma, propagation distance L and figure of merit (FoM) , are used to evaluate the modal characteristics of the designed waveguide. The mode area is defined as the ratio of the total electromagnetic energy to the maximum energy density of the hybrid mode :
The propagation distance which is a measure of the propagation loss of designed waveguide is defined as :
In Eq. (2), is the imaginary part of effective refractive index of the hybrid mode.
To evaluate the performance of the waveguide for mode confinement with low-loss, the FoM is employed as the ratio of propagation distance to the diameter of mode area:
In this paper, the investigation of modal properties is performed using finite element method (FEM) based on COMSOL software with scattering boundary conditions. The simulation results are stable and accurate when the mesh element size is one order smaller than the thickness of monolayer TMDCs.
3. Results and discussions
The dependence of modal characteristics of the designed waveguide with monolayer WSe2 on d is shown in Fig. 2 with the following parameters, εWSe2 = 19.8353 + 1.3342i , wr = 10 nm, ws = 200 nm, and h = 5 nm. In order to demonstrate the advantages of the proposed design structure, its modal properties were compared with those of the traditional HSPWG designed by R. F. Oulton, et al . From Fig. 2, it is evident that the modal properties for all of the waveguides exhibited a similar variation as a function of d, which can be attributed to a similar coupling mechanism between cylinder dielectric mode and SPP mode for these waveguides. Additionally, due to the high permittivity of the ultra-thin monolayer WSe2, the proposed waveguide has the capability of confining a large fraction of the energy within the low-index slot-rib region. This makes its mode area approximately four orders of magnitude smaller than that of traditional HSPWG, as shown in Fig. 2. Moreover, the proposed waveguide without the monolayer WSe2 could only support a hybrid mode which a 2 orders of magnitude smaller Meff and larger L than its traditional HSPWG counterpart under an optimal d. On one hand, a smaller Meff is useful in restricting most of the modal energy to the low-index slot-rib region, which leads to the gap region having lower energy and therefore obtaining a hybrid mode with a stronger energy convergence. On the other hand, owing to the slot filled with a loss-less material, the supported hybrid mode can propagate over a longer length as compared to the traditional design. Although the proposed waveguide with monolayer WSe2 has larger propagation loss due to increased loss from WSe2, it exhibits almost 10 times larger FoM over the conventional HSPWG .
Figure 3 shows the variation of modal properties of the proposed waveguide with and without monolayer WSe2 on ws at different h. It is observed that higher h corresponds to a larger L, Meff, and FoM, which can be explained by the increased modal energy located in the enlargement of gap region with low-loss. Additionally, L and FoM gradually increase with increasing ws for the proposed waveguide without WSe2, which is attributed to the enhanced modal energy located in the increased low-loss slot region. Due to the strong energy confinement in slot-rib area introduced by relatively high index covered monolayer WSe2, the effect of slot-size has limited impact on the modal properties of the proposed waveguide, as shown in Fig. 3. In other words, the dependence of different modal properties on ws in the case of the proposed waveguides with and without WSe2 suggest that the monolayer WSe2 is capable of enhancing the energy confinement within the ultra-small size region.
In order to determine the influence of the two-dimensional materials on the modal characteristics, L, Meff and FoM of the proposed waveguide as a function of wr, three types of monolayer TMDCs(thickness ~0.7 nm )–WSe2, WS2, and MoSe2 are studied, as shown in Fig. 4. The permittivity of WSe2, WS2 and MoSe2 are 19.8353 + 1.3342i, 17.3742 + 1.3989i, and 12.4126 + 1.3315i , respectively. It is evident that L and Meff gradually increase with increasing wr, which ultimately leads to a smaller FoM. The reduced propagation loss and increased Meff can be attributed to the smooth curved rib structure and the larger rib metal surface as a result of increasing wr. Furthermore, among three kinds of monolayer TMDCs, WSe2 has the longest L, the smallest Meff, and the largest FoM, which is the result of its Re(ε) being the largest and Im(ε) being the smallest among the monolayers studied. Hence, in order to attain high energy confinement and low transmission loss, the two dimensional material with large Re(ε) and small Im(ε) will be a better choice for the proposed waveguide structure.
To fabricate the proposed waveguiding structure, a highly precise Ag rib could be realized using Helium focused-ion beam technique (HeFIB) , following which a low-index porous SiO2 thin film can be evaporated onto the metal part by oblique deposition technique . By using the micromechanical cleavage (MC) technique , monolayer TMDCs can be further prepared and then wet-transferred onto the top of rib-slot region. After depositing SiO2 layer in the gap region, the Si nanowire will be ultimately formed by nanoparticle-catalyzed vapor-liquid-solid (VLS) method .
In conclusion, by embedding a low-permittivity slot-rib structure covered by monolayer TMDCs into the gap region of HSPWG, we have simulated a high-performance waveguiding configuration, achieving reasonable propagation distance and ultra-high optical energy confinement. In addition, our designed waveguide without monolayer TMDCs is able to support a hybrid mode that has lower transmission loss and a two orders of magnitude smaller mode area over the traditional HSPWG. By adding monolayer TMDCs, the figure of merit of our waveguide is increased by 10 times as compared to a traditional HSPWG. The mode area is further reduced by about four orders of magnitude over its HSPWG counterpart. A waveguide that offers a large mode confinement could be potentially used for ultra-compact nano-lasers, high integration chip-level optical circuits and platforms that study enhanced optical field at nano-scales.
National Basic Research Program of China (2015CB352005) and the National Natural Science Foundation of China(61775145, 61605124, 31771584, 61525503, 61620106016, 81727804, 61605130, 51602201).
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