## 4 Stepwise Luneburg Lens on the BSW Platform  
To adapt the gradient index profile of Figure 1 to the effective index range of the BSW, we linearly scale the index profile with \(n_{0}\) = 1.14, which is the effective index of the surface mode in the surrounding background (i.e., the BML). This adapted index profile is applied to both the ELS and EIS schemes shown in Figure 6 as solid curves. We apply the optimal number of steps to discretize the continuous profile, \(N_{\textit{opt}}\) = 5 and 4 for the ELS and EIS schemes, respectively. Such stepping schemes of the gradient  
Figure 5: (**a**) Band-gap diagram and dispersion curves, obtained by TMM. The clear area is the localized photonic band-gap. The dispersion curve of the BML indicates the lowest effect index, \(n\) = 1.141 at \(\lambda\) = 1555 nm. Adding the defect layer and the top device layer moves the dispersion curve towards the right-side band edge, where the larger wavenumber indicates the larger effective index; (**b**) The relation between the thickness of the top layer and the effective refractive index at \(\lambda\) = 1555 nm. The maximum index contrast equals approximately 0.3.  
index profile for the BSW are depicted in Figure 6. For the simplicity in the profile stepping, we set the radius of the lens \(R=5\)\(\upmu\)m, which leads to the lateral width of the step for the ELS scheme equal to 1 \(\upmu\)m. As shown in Figure 5b, designed BSW devices allow almost linear conversion between the top layer thickness and the effective index. Therefore, the stepwise index profile can be obtained by patterning the top layer similar to the surface topologies shown in Figure 2b,d. The intrinsic limitation of BSW systems is the maximum achievable index contrast, which is rather lower than that of conventional \"glass/air\" optical systems, where typical \(\Delta n\) equals 0.5. In this study, the designed multilayer yields \(\Delta n_{\text{max}}\) equal to approximately 0.3. Therefore, the lower refractive power loosens the focusing effect, and eventually pushes the focal spot position away from the shadow-side curvature of the lens, as shown in Figure 7, where the conventional circular geometry Luneburg lenses with the ELS \(N=5\) and EIS \(N=4\) are simulated. This lower \(\Delta n\) can be compensated for by increasing the optical path length of the component [14]. To do this, we elongate the circular geometry to a prolate ellipse. Figure 8 shows the prolate elliptical geometries with the aspect ratio (\(A=h/w\)) varying from 1.1 to 1.5, where the minor axis diameter \(w\) is fixed to 10 \(\upmu\)m and the major axis diameter \(h\) is varied from 11 to 15 \(\upmu\)m with an interval of 1 \(\upmu\)m.  
Figure 6: **(a)** Equilateral step (ELS) scheme of the refractive index profile with \(N=5\), where the step width = 1 \(\upmu\)m and \(R=5\)\(\upmu\)m. **(b)** Equi-index step (EIS) scheme with \(N=4\). The solid curve is the adapted gradient index profile for the effective index range of the designed BSW system, where the \(\Delta n_{\text{max}}=0.3\) and \(n_{0}=1.141\). Equivalent height stepping of the top layer will resemble the surface topologies shown in Figure 2b,d. Figure 7: Intensity distributions of stepwise Luneburg lens with circular geometry for **(a)** the ELS and **(b)** EIS profiles shown in Figure 5. The intensities are normalized. Concentric circles depict the step boundary of the index profile. When the aspect ratio rises, the focal spot comes closer to the lens and the confinement becomes tighter. When the aspect ratio is too large, e.g., \(A=1.5\), the focal spot is totally buried in the lens body. For the ELS case, \(A=1.4\) shows the best confinement of the focal spot just on the curvature of the Luneburg lens. The same elongation scheme is applied to the EIS profile (see Figure 9). Similar behavior of the focal spot is observed, the increase of \(A\) brings the focal spot back to the lens and the confinement gets better.