# Bloch-Surface-Polariton-Based Hybrid Nanowire Structure for Subwavelength, Low-Loss Waveguiding
## 3 Optical Properties and Modal Analysis of the Proposed Hybrid Waveguide  
In order to quantitatively investigate the proposed HBSPP waveguide's potential in low-loss light propagation at the subwavelength scale, the dependence of the hybrid modal properties on the thickness of the buffer layer are studied by varying \(t_{\text{b}}\). The considered key parameters of the guided optical hybrid mode usually include the modal effective refractive index (\(n_{\textit{eff}}\)), the propagation length (\(L\)), the normalized mode area (\(A_{\textit{eff}}\)/\(A_{0}\)), and the confinement factor (\(\Gamma\)). The propagation length is obtained by \(L=\lambda/[4\pi\text{Im}(n_{\textit{eff}})]\), which is determined by the effective propagation loss of the mode. The effective mode area \(A_{\textit{eff}}\) is calculated using [45]:  
\[A_{\textit{eff}}=\iint W(r)dA/max(W(r)) \tag{1}\]  
where, in order to exactly evaluate the energy in the metallic region, the electromagnetic energy density is defined as:  
\[W(r)=\frac{1}{2}\text{Re}\bigg{\{}\frac{d[\omega\varepsilon(r)]}{d\omega} \bigg{\}}|E\ (r)|^{2}+\frac{1}{2}\mu_{0}|H(r)|^{2} \tag{2}\]  
where \(E(r)\) and \(H(r)\) are the electric and magnetic fields, respectively. \(\varepsilon(r)\) is the electric permittivity and \(\mu_{0}\) is the vacuum magnetic permeability. \(A_{0}\) is the diffraction-limited mode area, defined as \(\lambda^{2}/4\). Thus, the mode confinement factor is defined as the ratio of the power confined in the considered gap region to the total power of the hybrid waveguide. Figure 4 illustrates the variations of the above key characteristics for different thicknesses of the buffer layer when \(t_{\text{b}}\) varies within the range of 0-10 nm, where the radius of the Au nanowire is fixed at 100 nm for better optical confinement. To control the modal performance, the thickness of the dielectric ridge is set as 100 nm, 125 nm, 150 nm, and 175 nm, respectively. This illustrates the trends of effective refractive index, propagation length, mode area, and confinement factor with the  
Figure 3: (**a-c**) 2D electric field distributions and the corresponding 1D electric field plots of the excited hybrid modes guided by the hybrid Bloch surface plasmon polariton (HBSPP) waveguide with different buffer layer thicknesses. (**a**) \(t_{\text{b}}\) = 0 nm; (**b**) \(t_{\text{b}}\) = 5 nm; (**c**) \(t_{\text{b}}\) = 10 nm. Other geometric parameters of all these waveguides are chosen as: \(t_{\text{r}}\) = 150 nm, \(r\) = 100 nm. The field profiles are taken at the bottom corner of the Au nanowire for \(t_{\text{b}}\) = 0 nm, and at the center of the buffer layer for \(t_{\text{b}}\) = 5 nm and 10 nm. gradual thickening of the buffer layer. It is shown that the strengthened coupling between the metal nanowire and the dielectric-ridge results in deep-subwavelength mode confinement with orders of magnitude enhancement (_Aeff/A_0 lower than \(10^{-3}\)) (see Figure 4c), while a tens of microns propagation distance can simultaneously be reached within the range of the considered structural parameters. It is worth noting that in spite of the buffer layer being very thin, the optical power confined in the gap region--shown in Figure 4d--is reasonable and could grow larger as the buffer layer thickens further. We further investigate the feasibility of controlling the mode properties when the radius of the Au nanowire varies, whereas the thickness of the dielectric ridge is fixed at 150 nm to ensure relatively strong interaction between the BSP mode and SPP mode with low propagation loss and subwavelength mode confinement. Figure 5 shows the calculated results for different buffer thicknesses when the radius of the metallic nanowire varies from 60 nm to 200 nm. It can be observed that with the increment of the radius of the metallic nanowire (\(r\)), the propagation length (see Figure 5b) can reach up to tens of micrometers and can be further increased under different buffer thicknesses (\(t_{\text{b}}\)).