# Monolitic Hybrid Transmitter-Receiver Lens for Rotary On-Axis Communications
## 2 Materials and Methods
### Hybrid Lens
#### 2.2.2 Optic Design Methods  
The hybrid lens is a nonimaging optical system. It can be designed by two fundamental approaches [38]: numerical optimization and direct calculation. Numerical optimization is a straight-forward approach for designing complex optic modules. Modern optic simulation tools like Optic Studio Zemax enable forming and optically simulating arbitrarily shaped optics by overlapping parametric objects. Optimization algorithms like the Levenberg-Marquardt algorithm are used for adjusting parameters of those objects until a sufficiently good result is achieved. Due to the large amount of variables, the optimization is typically inefficient, because of many local minima in the merit function [38]. As a result, it is easy to find a solution, but its performance is very limited, especially if the systems become more complex. In contrast, direct calculation algorithms follow well-defined design procedures and yield deterministic outcomes. Thereby, they provide better results than numerical optimization methods [38], especially if the systems are complex. A great variety of design methods are known, for example ray mapping [39, 40, 41, 42, 43, 44, 45, 46], forming surfaces using Cartesian ovals [47], the simultaneous multiple surface method in 2D [48] and 3D [38, 49] or the tailored freeform design method proposed by [PERSON] and [PERSON] [50]. Nowadays ray mapping approach, i.e., the combination of energy mapping in  
Figure 4: Ray path in the hybrid lens system. (**a**) Center TX (blue) and off-axis RX (green). (**b**) Center RX and off-axis TX. conjunction with geometrical surface construction, is in the focus of illumination research [39; 40; 41; 42; 43; 44; 45; 46]. Here, a mass-transfer problem is solved by transforming the source power irradiance \(E_{\text{s}}\) into the target power irradiance \(E_{\text{t}}\). This transformation is represented by a mapping \(\phi:\Omega_{\text{s}}\rightarrow\Omega_{\text{t}}\) from the source domain \(\Omega_{\text{s}}\) to the target domain \(\Omega_{\text{t}}\). Then, the laws of refraction and reflection are applied for \(k\times l\) points to calculate a corresponding vector field \(\mathbf{N}\) containing the surface normals \(\mathbf{n}_{i,j}\) with \(i\in[1,l]\), \(j\in[1,k]\). The challenge is to find a mapping \(\phi\) which yields a vector field \(\mathbf{N}\) that satisfies the integrability condition for a continuous surface. This condition is shown in Equation (6) [41; 42; 50]. It states that \(\mathbf{N}\) has to be curl-free or exhibit at least minimum curl. \[\mathbf{N}\cdot(\
abla\times\mathbf{N})=0 \tag{6}\]  
Nowadays, parametrization and consecutive optimization are widely employed for generating a mapping \(\phi\)[28; 31; 39; 40]. Circular shaped FOV are formed with the mapping shown in Figure 5a. An equi-flux grid around the source in spherical coordinates \(\theta_{\text{s}},\phi\) is mapped onto a target grid in polar coordinates \(\beta,r\). In 3D, mappings like these typically lead to a normal vector field \(\mathbf{N}\) with substantial curl. Therefore, an optimization procedure purposely distorts the target grid, for instance by varying \(r\), to improve the performance. It was shown that this approach works well for on-axis scenarios [28; 39]. In case of an off-axis placed emitter and two optical surfaces, the scenario is more challenging due to the nonparaxial nature. Hence, the result will deviate from the anticipated irradiance pattern. An additional variable \(\beta\) could be used in lateral direction as illustrated in Figure 5a. The downside is a slower optimization process. For the TX lens part of the hybrid lens, inefficient optimization can be avoided by taking the curl of \(\phi\) directly into account. An initial curl-free mapping \(\phi_{0}\) is generated, assuming that the resulting vector field \(\mathbf{N}\) exhibits minimum curl [42]. Figure 5b illustrates the irradiance in front of the source \(E_{\text{s}}\) and the anticipated irradiance at the target plane \(E_{\text{t}}\).