# Monolitic Hybrid Transmitter-Receiver Lens for Rotary On-Axis Communications
## 2 Materials and Methods
### Fundamental Concepts
#### 2.1.1 Channel Model
\[P_{\text{PD}}=E_{\text{TX}}(z,\theta_{\text{TX FOV}})\cdot A_{\text{PD}} \cdot g(\theta_{\text{RX FOV}})\cdot\eta_{\text{RX}}(\theta_{\text{i}}) \cdot\epsilon_{\text{c}} \tag{2}\]  
The RX FOV should always be chosen as small as possible, since the optical gain \(g(\theta_{\text{RX FOV}})\) decreases with increasing FOV due to conservation of Etendue [30]. Moreover, a restricted RX FOV improves the robustness against noise and interchannel interference. Next, \(g(\theta_{\text{RX FOV}})\) is substituted by the maximum theoretical optical gain [30] as shown in Equation (3). Now \(\eta_{\text{RX}}(\theta_{\text{i}})\) is used as a figure of merit for the optical RX efficiency. The ideal lossless receiver achieves \(\eta_{\text{RX}}(\theta_{\text{i}})=1\) for all \(\theta_{\text{i}}\in[0,\theta_{\text{RX FOV}}]\). The angle \(\theta_{\text{RX,PD,max}}\) denotes the maximum coupling angle from the RX optics to thePD surface normal. In the next step, the irradiance \(E_{\text{TX}}(z,\theta_{\text{TX FOV}})\) is substituted by the product of the optical TX power \(P_{\text{TX}}\) and the TX efficiency \(\eta_{\text{TX}}\) divided by the illuminated spot area \(A_{\text{FOV TX}}\). \[P_{\text{PD}}=\frac{P_{\text{TX}}\cdot\eta_{\text{TX}}}{A_{\text{FOV TX}}} \cdot\left(\frac{n_{1}\cdot\sin\theta_{\text{RX,PD}}}{n_{air}\cdot\sin\theta_ {\text{RX FOV}}}\right)^{2}\cdot A_{\text{PD}}\cdot\eta_{\text{RX}}(\theta_{ \text{i}})\cdot\epsilon_{\text{c}} \tag{3}\]  
\(A_{\text{FOV TX}}\) is replaced by the corresponding triangular relationship, which includes the communication distance \(z\) and the tangent of the TX FOV \(\theta_{\text{TX FOV}}\). Finally, Equation (4) is a useful expression for the most important geometric dependencies of \(P_{\text{PD}}\). \[P_{\text{PD}}=\frac{P_{\text{TX}}\cdot\eta_{\text{TX}}}{\pi\cdot(z\cdot\tan \theta_{\text{TX FOV}})^{2}}\cdot\left(\frac{n_{1}}{\sin\theta_{\text{RX FOV}}} \right)^{2}\cdot A_{\text{PD}}\cdot\eta_{\text{RX}}\cdot\epsilon_{\text{c}} \tag{4}\]  
The performance merits from Equation (4) are \(\eta_{\text{TX}}\), \(\eta_{\text{RX}}\) and \(\epsilon_{\text{c}}\). The transmitter efficiency \(\eta_{\text{TX}}\) specifies how much of the emitted power \(P_{\text{TX}}\) is concentrated into the FOV. Its loss mechanisms are \(\zeta_{\text{TX M}}\) and \(\zeta_{\text{TX F}}\). \(\zeta_{\text{TX M}}\) describes rays that strike the target plane outside the FOV. \(\zeta_{\text{TX F}}\) describes back-reflected rays due to Fresnel-reflections. Since \(\eta_{\text{TX}}\) contains no information concerning irradiance homogeneity, we additionally introduce the effective transmitter efficiency \(\eta_{\text{TX eff}}\). It is defined according to Equation (5) by using the minimum irradiance within the FOV \(E_{\text{min}}\)[28; 31]. The difference between \(\eta_{\text{TX}}\) and \(\eta_{\text{TX eff}}\) is called the inhomogeneity factor \(\zeta_{\text{TX H}}\). \[\eta_{\text{TX eff}}=\frac{E_{\text{min}}}{E_{\text{min,ideal}}}=\frac{E_{ \text{min}}A_{\text{FOV TX}}}{P_{\text{TX}}} \tag{5}\]  
The sensitivity of RX is typically limited by the interaction of several internal noise mechanisms that exhibit a Gaussian probability function [26]. The signal can additionally be corrupted by optical crosstalk, i.e., adjacent communication channels. This nonGaussian noise has a limited range of variation. This effect introduces a power penalty \(PP\)[26] that reduces the usable peak-to-peak amplitude of the signal. The optical power that falls onto the PD consists of a signal part \(P_{\text{PD}}\) and another part arising from crosstalk \(P_{\text{PD cross}}\). The corrected power value \(P_{\text{PD PP}}\) takes the eye-closure effect into account by subtracting the \(P_{\text{PD cross}}\) from \(P_{\text{PD}}\)[26]. It holds \(P_{\text{PD PP}}<P_{\text{PD}}\) as soon as crosstalk is present. In this work, all optical power values are understood as average values to ensure comparability with literature. When dealing with the power penalty the signal strength is considered in a peak-to-peak manner.