## 3 Bloch Surface Wave and Multilayer Design  
BSWs are supported at the edge of a truncated 1D periodic dielectric media, i.e., a dielectric multilayer serving as 1D photonic crystal [17]. BSW are sustained in the frequency region of the localized photonic band-gap. The band-gap denies propagation into the half space containing the multilayer structure. Since the propagation constant of the mode is outside the light cone of the medium in the other half space, a surface mode is supported. For designing the multilayer for desired BSWs, the band-gap diagram and dispersion curves can be calculated by the transfer matrix method (TMM) [17] or numerical methods, e.g., eigenmode solver CAMFR [18]. In this study, the designed multilayer stack consists of six periods of silicon nitride (Si\({}_{3}\)N\({}_{4}\), \(n_{\text{Si3N4}}=1.94\), \(t=283\) nm) and silicon dioxide (SiO\({}_{2}\), \(n_{\text{SiO2}}=1.47\), \(t=472\) nm), where \(t\) being thickness. We call this stage of the multilayer  
Figure 4: Simulated intensity distributions of Luneburg lenses with the EIS index profile: (**a**) \(N=3\), (**b**) 4, (**c**) 5 and (**d**) 10, respectively. Larger \(N\)s (>6) show almost the same response as the Luneburg lens with a continuous GRIN profile. The intensities are normalized. Concentric circles depict the step boundary of the index profile in (**a**). Here, the smallest step number for the optimal performance (\(N_{opt}\)) is determined to be 4. Figure 3: Simulated intensity distributions of Luneburg lenses with the ELS index profile: (**a**) \(N=4\), (**b**) 5, (**c**) 6 and (**d**) 10, respectively. Larger \(N\)s (>6) show almost the same response as a Luneburg lens with a continuous GRIN profile. The intensities are normalized. Concentric circles depict the step boundary of the index profile in (**a**); Here, the smallest step number for an optimal performance (\(N_{opt}\)) is determined to be 5; (**e**) The result of the continuous-profile device shows an identical result as that of \(N=10\) shown in (**d**). \"bare multilayer (BML)\". An additional 20-nm Si\({}_{3}\)N\({}_{4}\) is added on the top of the periodic layers as a defect layer. Finally, the device layer, 140-nm titanium dioxide layer (TiO\({}_{2}\), \(n_{\text{TiO2}}\) = 2.23), is added on the defect layer. The dispersion curves with corresponding band-gap diagram of the current chip design is shown in Figure 5a, where \(\omega\) (=2\(\pi\cdot\)\(c/\lambda\)) being the angular frequency with \(c\) the speed of light and \(\beta\) (=\(k\)-sin\(\theta\)) being the wave number of the BSW with \(k\) the wave number in the glass substrate and \(\theta\) the incident angle in the Kretschmann configuration. Accordingly, an effective index (\(n_{\textit{eff}}\)) of a BSW is defined as  
\[n_{\textit{eff}}=\frac{\beta_{c}}{\omega} \tag{2}\]  
which can be simplified to be  
\[n_{\textit{eff}}=n_{\textit{sub}}\cdot\sin\theta \tag{3}\]  
with \(n_{\textit{sub}}\) the refractive index of the substrate. The dispersion curves of the BML indicates the lowest effective index, \(n_{\textit{eff}}\) = 1.141 for \(\lambda\) = 1555 nm with \(n_{\textit{sub}}\) = 1.444 and \(\theta\) = 52.2\({}^{\circ}\). Adding the defect layer and the top device layer augments the effective index and increasing the thickness of the top layer moves the dispersion curve towards the right-side band edge, where the larger wavenumber imposes the larger index of the surface mode. Like this, varying the thickness of the top element allows adjusting the effective index contrast. Therefore, the thickness stepping of this device layer can produce the stepwise index profile. The relation between the top layer thickness and the effective index are obtained by TMM for the target \(\lambda\) = 1555 nm, see Figure 5b, where the empty circle markers are the calculated data points and the dashed line is the result of interpolation. The interpolation shows a good linear behavior. For higher thickness of the top layer, the increase of the effective index will be saturated.