# Monolitic Hybrid Transmitter-Receiver Lens for Rotary On-Axis Communications
## 2 Materials and Methods  
### Fundamental Concepts  
#### 2.1.1 Channel Model  
OWC use optical emitters like LEDs or LDs at the TX to convert an electrical signal into the optical domain. A PD is used for back-conversion at the RX. Similar to FORJ, both transceivers are placed in front of each other [25]. OWC use lenses instead of optical fibers to enable larger mechanical tolerances. The key goal for the designer is to increase the optical power \(P_{\text{PD}}\) that falls onto the PD in order to improve the RX signal-to-noise ratio (SNR) [26] and to minimize the bit-error-rate (BER). If the SNR is already sufficient, the excess can be converted into a higher data rate by increasing bandwidth or applying a multilevel modulation scheme. There will always be a misalignment between both transceivers due to positioning, assembly tolerances or vibrations. The rotation axis might even exhibit a nutation, i.e., nonideal motion around the ideal rotation axis. The combination of these nonidealities and the communication distance leads to a minimum field of view (FOV) that is required for robust operation. The FOV sizes are denoted by the half-opening angles \(\theta_{\text{TX FOV}}\) and \(\theta_{\text{RX FOV}}\). In order to ensure eye-safe operation, i.e., to classify the system as laser class 1 according to IEC 60825-1:2014 (DIN EN 60825-1:2015-07) [27], the optical transmitter power \(P_{\text{TX}}\) is limited. Therefore, efficient transceiver design is required to meet link-budget requirements, ensure eye-safety and to maximize data rate. The designer tries to maximize the dynamic range by keeping the TX and RX performance constant within a plane perpendicular to the optical axis [28]. In other words, TX has to provide constant irradiance \(E_{\text{TX}}\) and RX has to detect the same signal level within this plane. For a moment we assume the distance \(z\) between both transceivers is large compared to their apertures. Although this does not fully apply for short distances, the assumption is useful to show the fundamental dependencies. As a result, we can assume that parallel rays are incident onto RX. For a homogeneous FOV, the optical power \(P_{\text{PD}}\) can be expressed as product of the irradiance \(E_{\text{TX}}(z)\), the effective receiver input aperture \(A_{\text{RX,eff}}\) and the geometrical coupling coefficient \(\epsilon_{\text{c}}\) as shown in Equation (1). For the ideal optical link, TX and RX have the same FOV size and both FOVs are placed on the optical axis. In practice, both FOVs may differ or they might be misaligned. The geometrical coupling coefficient \(\epsilon_{\text{c}}\) quantifies this misalignment. It becomes crucial for short-distance communication as we know from the challenging assembly tolerances of FORJ. \(\epsilon_{\text{c}}\) is defined as the fraction of the illuminated area of TX, which is overlapping with the FOV of the RX divided by the total illuminated area \(A_{\text{FOV}}\). For the ideal optical channel, i.e., equally sized TX and RX FOVs with no misalignment, \(\epsilon_{\text{c}}=1\) applies. \[P_{\text{PD}}=E_{\text{TX}}(z,\theta_{\text{TX FOV}})\cdot A_{\text{RX,eff}} (\theta_{\text{RX FOV}})\cdot\epsilon_{\text{c}} \tag{1}\]  
The effective receiver input aperture \(A_{\text{RX,eff}}(\theta_{\text{RX FOV}})\) is expressed as product of active PD area \(A_{\text{PD}}\), optical gain \(g(\theta_{\text{RX FOV}})\) and efficiency \(\eta_{\text{RX}}(\theta_{\text{i}})\) at the angle of incidence \(\theta_{\text{i}}\) as it is shown in Equation (2). A large PD area \(A_{\text{PD}}\) is favorable for the link budget but goes along with a large PD capacitance, which limits the modulation bandwidth (BW) [26; 29]. Consequently, a PD with large \(A_{\text{PD}}\) but sufficient BW is chosen.