Hybrid Dielectric-loaded Nanoridge Plasmonic Waveguide for Low-Loss Light Transmission at the Subwavelength Scale

The emerging development of the hybrid plasmonic waveguide has recently received significant attention owing to its remarkable capability of enabling subwavelength field confinement and great transmission distance. Here we report a guiding approach that integrates hybrid plasmon polariton with dielectric-loaded plasmonic waveguiding. By introducing a deep-subwavelength dielectric ridge between a dielectric slab and a metallic substrate, a hybrid dielectric-loaded nanoridge plasmonic waveguide is formed. The waveguide features lower propagation loss than its conventional hybrid waveguiding counterpart, while maintaining strong optical confinement at telecommunication wavelengths. Through systematic structural parameter tuning, we realize an efficient balance between confinement and attenuation of the fundamental hybrid mode, and we demonstrate the tolerance of its properties despite fabrication imperfections. Furthermore, we show that the waveguide concept can be extended to other metal/dielectric composites as well, including metal-insulator-metal and insulator-metal-insulator configurations. Our hybrid dielectric-loaded nanoridge plasmonic platform may serve as a fundamental building block for various functional photonic components and be used in applications such as sensing, nanofocusing, and nanolasing.

conventional hybridized waveguide configurations, extensive efforts have also been devoted to the exploration of modified hybrid structures 32,[46][47][48][49][50][51][52][53][54][55][56][57] . Though some of these novel configurations exhibit improved optical performance as opposed to their conventional hybrid counterparts, most of them still suffer from the tradeoff between modal attenuation and field localization. Moreover, due to additional fabrication complexities, many of these modified waveguides face great challenges when leveraged for practical applications. Therefore, there is a need for a simple but feasible way to reduce the propagation loss of traditional hybrid waveguide while maintaining its tight-field localization property.
Here in this article, we propose a new type of HPW by combining dielectric-loaded waveguiding with a traditional hybrid structure, which we refer to as a hybrid dielectric-loaded nanoridge plasmonic waveguide (HDLNRPW). In contrast to the previous hybrid wedge/ridge structures that incorporate metallic nanostructures 52,53 , the hybrid waveguides presented here take full advantage of dielectric nanoridges, which are beneficial for reducing the propagation loss and maintaining the tight field confinement. Based on systematic numerical simulations, we will show in detail the capability of the hybrid dielectric-loaded nanoridge waveguide in balancing the tradeoff between confinement and loss, and we will reveal its tolerance against fabrication errors. Moreover, we will discuss the possibility of applying the waveguide concept to other metal/dielectric structures, which will lay the foundation for future designs and investigations. Figure 1(a) and (b) show schematically three-dimensional (3D) and two-dimensional (2D) geometries of the studied hybrid waveguide, which consists of a silicon slab separated from a silver substrate by a thin silica layer, along with an additional silicon nanoridge sitting on top of the substrate. The unique hybrid gap region facilitates efficient light confinement and transport with moderate attenuation within the nanoscale low-index layers. To reveal the potential of the structure in offering both good confinement and low transmission loss, we used the finite element method (FEM)-based software COMSOL TM to investigate its guiding properties at a telecommunication wavelength of 1550 nm. In our calculations, the refractive indices of SiO 2 , Si, and Ag were chosen to be n l = 1.444, n h = 3.476, and n m = 0.1453 + 14.3587i 30 , respectively. Without loss of generality, semi-circular-shaped dielectric nanostructures were chosen as a proof-of-concept in the following studies. In our later discussions, we will show that our waveguide concept can also be applied to many other configurations with similar nanostructures, including those incorporating rectangular, semi-elliptical and triangular-shaped dielectric nanoridges.

Results
In Fig. 2, we show the normalized electric field distribution of the fundamental hybrid mode supported by a typical hybrid nanoridge plasmonic structure, and we compare this distribution with that of the conventional hybrid mode. In the calculations, both the HDLNRPW and the conventional HPW have the same gap size of 5 nm, and the dimensions of the silicon slabs for both structures are fixed at 200 nm × 200 nm. Due to the strong hybridization of the plasmonic and dielectric modes, significant field enhancement was observed inside the gap region for both cases. As illustrated from the 2D panel and 1D cross-sectional field plots, the local field enhancement of the proposed HDLNRPW is even more pronounced than the traditional hybrid structure in both horizontal and vertical directions, which can be attributed to the stronger effect induced by the lower silicon nanoridge. Our calculations also indicate that the HDLNRPW exhibits lower loss than the conventional hybrid waveguide, which is due to the larger distance between the upper silicon slab and the lower silver substrate. In the following section, we will illustrate the characteristics of the plasmonic mode guided by the proposed structure, and demonstrate the possibility of balancing the tradeoff between confinement and loss through tuning structural parameters of the waveguide. In order to reveal the unique potential of the proposed hybrid waveguide in providing tight optical confinement and great propagation length, we calculated the dependence of its modal properties on the gap size and the dimension of the silicon nanoridge. Firstly, we consider the following modal parameters, including the real part of the modal effective index (n eff = Re(N eff )), the propagation length (L), the normalized mode area (A eff /A 0 ), and the figure of merit (FoM) (see methods). As illustrated from the calculated results shown in Fig. 3(a) and (b), both the modal effective index and the propagation distance demonstrate non-monotonic behaviors with the variation of the silicon nanoridge. By contrast, the effective mode area increases slightly as the nanoridge becomes larger, and its value is much less than 1, indicating clearly the subwavelength confinement of the HDLNRPW. For waveguides with small nanoridges (e.g., r < 20 nm), the coupling between the silicon slab and the metal substrate was relatively strong, and it was further enhanced as the dimension of the nanoridge and/or the size of the gap decreased. This strengthened hybridization effect leads to an increased effective index, a shortened propagation distance, and a decreased mode area, as observed in Fig. 3(a)-(c). On the other hand, the overall features of the HDLNRPW were dominated by the silicon nanoridge when the dimension of the nanoridge reached a certain size (e.g., r > 30 nm). With the continuously enlarged nanoridgeas and enhanced modal effect index, a reduced propagation distance and an increased mode size were observed for the considered waveguides and the different gap sizes. Figure 3(b) illustrates that the largest propagation distances are typically obtained when the silicon nanoridge has a moderate radius (e.g., between 20-30 nm). This non-monotonic behavior of the propagation distance, together with the monotonic trend of the mode area, leads to the non-monotonic change of the FoM, as shown in Fig. 3(d). For the considered waveguiding structures, FoMs reach their maxima when the radius of the nanoridge was 15-25 nm. At these conditions, our proposed HDLNRPW features not only a much higher FoM but also a larger propagation distance as compared to its conventional hybrid waveguiding counterpart (see Supplementary Information for details). While compared to modified hybrid structures incorporating inverse metallic nanostructures 58,59 , our proposed waveguide enables much lower loss with subwavelengh field confinement. The propagation distance of HDLNRPW, ranging from tens to hundreds of microns, is more than one orders of magnitude greater than that reported in 58. These features indicate great potential of HDLNRPW for high-performance plasmon waveguiding at the sub-wavelength scale.
In addition to plotting the curves of different mode parameters, we also depict the electric field distributions for typical waveguide configurations, which are shown in Fig. 3(e)-(i). As illustrated from the field profiles, pronounced local field enhancement and tight optical confinement were achieved by waveguides with small gaps (Fig. 3(e)-(g)), due to the strong hybridization of the dielectric-loaded SPP and the dielectric mode supported by the silicon slab. By contrast, less notable field enhancements were achieved for waveguides with relatively large gap distances (Fig. 3(h)-(i)). Under these circumstances, the confinement of the hybrid waveguide is also weaker than the small-gap case, as indicated from the curves of the effective mode area shown in Fig. 3(c).
The field confinement of the HDLNRPW was further revealed by calculating the normalized optical power (NOP) inside the gap region (see methods). Here in Fig. 4 we show the dependence of NOP on the size of the gap for waveguides with different nanoridges. It is seen that the power ratio inside the gap exhibits a non-monotonic trend with the variation of g when r is relatively small (e.g., r < 40 nm), which indicates the existence of an optimal gap size for NOP. Such an optimal g shifts towards a smaller value when the size of the nanoridge increases. As illustrated in Fig. 4, the trend of NOP turns into a monotonic behavior when r becomes greater than 40 nm. Under these circumstances, the power confined inside the silicon nanoridge is significantly greater than the silica layer of the gap region. Further enlarging the gap size will lead to weakened field confinement inside the whole  gap, as illustrated from the decreasing trends of the purple and black curves in Fig. 4. Our calculations show that through choosing appropriate r and g, the confinement of HDLNRPW can be significantly greater than that of the conventional hybrid waveguide inside the gap region. This tight field confinement, along with pronounced local field enhancement, small mode size, and large propagation distance, can potentially enable applications in active waveguides and nanolasers 33,36,40,60 , nonlinear devices 43,45 as well as high-sensitivity optical sensors [61][62][63][64][65][66][67] . On the other hand, combining our HDLNRPW concept with novel materials such as graphene 68,69 and MoS2 70 may lead to other kinds of high-performance waveguides and devices.

Discussion
In addition to demonstrating good optical performance under ideal conditions as discussed above, the proposed HDLNRPW also exhibits good tolerance to possible fabrication imperfections, such as the lateral misalignment between the silicon nanoridge and the upper silicon slab. Our calculations show that less than ~3.5% of the propagation distance and less than ~1.8% of the normalized optical power can be observed when the deviation of the nanoridge was varied between 0 nm and 50 nm. Meanwhile, the subwavelength mode area and high FoM were maintained within the considered geometric parameter range (see Supplementary Information for details). Such optical performance clearly indicates the robust modal behavior of the proposed waveguide against fabrication errors, making the HDLNRPW promising for practical applications.
Besides the semi-circular nanoridge-based HDLNRPW in the above case studies, our hybrid waveguide concept is readily applicable to many other structures as well. Here in Fig. 5, we show the electric field distributions of the fundamental hybrid modes guided by two different HDLNRPWs, which incorporate square and triangular-shaped silicon nanoridges, respectively. As clearly illustrated from the 2D field panels and 1D cross-sectional curves, these hybrid configurations also demonstrate stronger field enhancement inside the gap region as compared to the conventional HPW; simultaneously they feature less mode attenuation and greater figure of merit. These optical performances are similar to those of their semi-circular waveguiding counterparts.
In Fig. 6, we plot the geometries of two other types of modified hybrid nanoridge waveguides, which were obtained by combining the HDLNRPW concept with a metal-insulator-metal (MIM) waveguide or an insulator-metal-insulator (IMI) configuration. Both configurations can be realized using modern micro-nanofabrication technologies (See Supplementary Information for details). The electric field distributions of low-loss hybrid modes guided by typical MIM-HDLNRPW and IMI-HDLNRPW are shown in Fig. 7. Significant local field enhancement was observed inside the gap regions for both waveguides. Our calculations further show that the MIM-type hybrid nanoridge structure is capable of providing ultra-tight confinement of the optical field (e.g., larger NOP), whereas the IMI-type hybrid configuration features extremely low propagation loss, with centimeter-range propagation distance at appropriate geometries.
In summary, we have proposed and investigated a new class of plasmonic waveguiding platform based on the combination of dielectric-loaded configurations and hybrid plasmonic structures. Through optimizations of key structural parameters, high-performance waveguiding with both large propagation length and subwavelength mode size can be achieved at a telecommunication wavelength using our proposed HDLNRPW. Compared with conventional HPWs, our proposed structure demonstrates not only lower propagation loss but also a greater figure of merit under optimal conditions. Furthermore, we show that the HDLNRPW is also highly tolerant to

Methods
The modal characteristics of HDLNRPWs are investigated numerically by solving the Helmholtz equation using the eigenmode solver of the finite element method (FEM) based software COMSOL TM . A scattering boundary condition was applied to mimic the open boundary. Convergence tests ensured that the numerical boundaries and meshing did not interfere with the solutions. The modal properties were characterized by a complex wave vector, whose parallel component defines the propagating constant with β + iα. Here, β and α are the phase and attenuation constants, respectively. The real part of the modal effective index was calculated by n eff = Re(N eff ) = β/k 0 , where k 0 is the vacuum wavevector. The propagation length was defined by L = 1/2α = λ/[4πIm(N eff )], whereas the effective mode area was defined as the ratio of the total mode energy and the maximum electromagnetic energy density 30 : eff In order to accurately account for the energy in the metallic region, the electromagnetic energy density W(r) is defined as 22,30 : In equation (2), E(r) and H(r) are the electric and magnetic fields, ε (r) is the electric permittivity and μ 0 is the vacuum magnetic permeability. The normalized effective mode area is defined as A eff /A 0 where A 0 = λ 2 /4 is the diffraction-limited mode area in free space. The figure of merit was defined as the ratio of the propagation length to the diameter of the effective mode area (2(A eff /π)½). The normalized optical power was defined as the ratio of the power inside the whole gap region to the total power of the waveguide.