## 1 Introduction  
Propagating electromagnetic surface waves are a key technological element for many near-field and nano-photonics devices. The most frequently studied surface wave is the surface plasmon polariton (SPP), which propagates along the interface between a dielectric and a metal [1]. Apart from long-range SPPs available at the expense of burying the field in a dielectric layer [2], SPPs are in general limited in their propagation length by dissipation. This constitutes limitations for many existing and potential applications, in particular, in integrated photonic devices. For an alternative solution, we present a different type of surface wave, which is called \"Bloch surface wave\" sustained by dielectric multilayers. The use of lossless dielectric media guarantees a long propagation length. For instance, a BSW with a propagation length of 3.24 mm at \(\lambda\) = 1.558 \(\mu\)m has been reported [3]. This complementary character to the SPP makes the BSW a good candidate for integrated photonic devices, and this has inspired much research in recent years. Devices with basic functionalities have been explored, e.g., a prism demonstrating refraction of the propagating surface wave [4], a grating generating 2D Talbot images on the chip surface [4], a refractive micro lens for focusing [5], and waveguide components [6; 7; 8]. All these elements came into reach by adding a structured device layer on the top of the stratified media. This locally changes the dispersion relation of the BSW and allows for control of the effective refractive index contrast. More advanced functionalities have also been demonstrated, e.g., linear [9] and circular [10] grating couplers that couple the surface waves without a prism coupling setup. Furthermore, the research has been extended to demonstrate applications for enhanced fluorescence detection [10], resonator [11], and bio-sensing [12]. The control of in-plane propagation of surface waves is usually accomplished by modifying the top surface or adding discrete scattering structures on the top. Those patterned structures have a single height or depth, which produces a single-step index contrast. To enrich controlling capabilities of surface waves, [PERSON] et al. have introduced the concept of GRIN optics onto SPP devices [13],which is a kind of transformational optics approach. Instead of directly modifying the refractive index of the dielectric medium, the gradient index profile has been achieved by slowly changing the thickness of the dielectric top layer (i.e., the height profile modification), which gradually varies the local effective index of the surface wave. To spatially vary the top layer's height profile, grey-scale electron beam lithography (EBL) is used, where the electron dose is continuously varied across the sample to modulate the height of the resist layer in Reference [13]. However, obtaining the designed height profile, which is a non-linear and continuous form, with nanometer precision is a challenging task in fabrication. In this paper, in order to facilitate more repeatable and controllable fabrication processes, we propose a stepwise GRIN optics approach, where discrete index steps replace a continuous gradient index profile. Investigations are carried out exclusively using a 2D FDTD simulation (CST microwave studio, at \(\lambda\) = 1555 nm). In our previous work [14], the validity of 2D numerical simulations was confirmed by showing a perfect agreement with the results of 3D simulations and scanning near-field optical microscopy measurements. Since the large number of height steps will be a time-consuming and challenging task, we will first determine the smallest number of index steps for optimal performances. We consider a Luneburg lens for a demonstrator of the GRIN optics devices, which is proven to produce tight focusing in micro-sized elements of the SPP [13] and the photonic nanojet [15]. Afterwards, we apply the optimal number of the index profile step to the BSW-based Luneburg lens design.