## 2 2D Luneburg Lens and Stepping GRIN Profile  
The refractive index distribution of the conventional Luneburg lens in a background medium with refractive index \(n_{0}\) satisfies the equation  
\[n(r)=n_{0}\sqrt{2-\left(\frac{r}{R}\right)^{2}} \tag{1}\]  
where \(R\) denotes the radius of the lens and \(r\) the transverse distance to the center. For a conventional Luneburg lens in air, i.e., \(n_{0}=1\), Figure 1a shows the gradient index profile calculated by Equation (1) with \(R=7.5\)\(\upmu\)m. Here, the size of the lens is benchmarked from the SPP Luneburg lens in Reference [13]. Note that a small variation of the lens size will not cause a distinctive response, as long as the lens is larger than the operating wavelength. Figure 1b shows the schematic of 2D Luneburg lenses for surface wave systems. The surface topology of the lens corresponds to the continuous and stepwise index profiles, respectively, where the surface wave propagates along the \(z\)-axis towards the lenses. Note that the index profiles are acquired by thickness variation of the element along the \(y\)-axis.  
For discretizing a continuous gradient index profile, we propose two schemes. The first is the equilateral step (ELS) profile that consists of the same step width. The second is the equi-index step (EIS) profile that consists of the same index value (i.e., the step height). Graphical examples are shown in Figure 2 with the corresponding surface topology of the devices for each scheme.  
In the EIS scheme, the step width at the rim of the element gradually becomes narrower, which reflects the steeper slope at the rim of the index profile. Although this approach resembles the multi-level phase encoding of diffractive microlenses [16], the operation mechanism is different. Surface wave illuminates a 2D Luneburg lens from the side, i.e., along the in-plane direction (see Figure 1b), whereas conventional diffractive lenses are orthogonally illuminated (i.e., along the out-of-plane direction). The large number of steps will create too narrow steps at the rim, which are difficult to fabricate. Therefore, the smallest number of steps for an optimal performance is of great interest. The 2D FDTD simulations are applied to verify the performance by assessing the field distributions near the Luneburg lens. The upper limit for the number of steps (\(N\)) is set to be 20 for both EIS and EIS schemes, and \(N\) is varied down to 2 with an interval of 1. When the number of steps is sufficiently large, e.g., \(N>6\), the response of the lens shows no difference form the case of the continuous index profile. For instance, EIS \(N=10\) case in Figure 3d leads to an identical result as the continuous-profile device in Figure 3e. For the qualitative comparison, the simulated intensity distributions of \(N=4\), 5, 6 and 10 for the EIS scheme are presented in Figure 3. The focal spot starts to show noticeable degradation from \(N=4\), see Figure 3a. Therefore, the case of \(N=5\) is determined to be the optimal number of steps (\(N_{opt}\)) for the EIS scheme. Figure 4 shows the simulation results for \(N=3\), 4, 5 and 10 of the EIS scheme.  
Figure 1: **(a)** Gradient index profile for a standard Luneburg lens in air with \(n_{0}=1\) and \(R=7.5\) μm. **(b)** Schematic of 2D Luneburg lenses for continuous and stepwise index profiles. Surface wave propagates along the \(z\)-axis.  
Figure 2: **(a)** Equilateral step (ELS) scheme of the index profile and **(b)** the corresponding surface topology of the device; **(c)** Equi-index step (EIS) scheme and **(d)** the corresponding surface topology of the device.  
scheme. Here, the case of \(N=3\) still shows a relatively good focal spot. However, stronger side lobes start to be prominent. The case of \(N=4\) produces a focal spot as good as that of larger \(N\)s. Therefore, we determine \(N_{opt}=4\) for the EIS scheme. In the following section, we will investigate the Luneburg lens on the BSW platform by directly applying the optimal number of steps, \(N_{opt}=5\) and 4 for the ELS and EIS schemes, respectively.