## 2 Theoretical Formula Derivation  
### Single-Scattering Channel Model  
The communication system we discuss is established based on a coplanar single-scattering model, which forms the foundation for analyzing the transmission-channel-reception communication link. Figure 1 illustrates the diagram of a UV NLOS scattering communication link in coplanar geometry, where the transmitter (Tx) is located at point T and the receiver (Rx) is at point R. The defined parameters are as follows: \(\theta_{1}\) and \(\theta_{2}\) are the angles between the communication devices and the horizontal axis at Tx and Rx, respectively; \(\phi_{1}\) and \(\phi_{2}\) are the transmitter-side beam divergence angle and receiver-side field of view (FOV) angle; \(\theta_{s}\) represents the scattering angle between the emission direction of the incident light and the receiving direction, satisfying \(\theta_{s}=\theta_{1}+\theta_{2}\); \(r\) denotes the straight-line distance between the transmitter and receiver; \(r_{1}\) and \(r_{2}\), respectively, represent the distances from the scattering center to Tx and Rx.  
The scattered energy at Rx caused by a differential volume source [20] is calculated as follows:  
\[\delta E_{r}=\frac{E_{t}k_{s}P(\mu)A_{r}\delta V\cos\zeta\exp[-k_{e}(r_{1}+r_{ 2})]}{\Omega_{1}r_{1}^{2}r_{2}^{2}}, \tag{1}\]  
In this context, \(E_{t}\) represents the emitted UV light energy; \(\Omega_{1}\) denotes the solid angle of radiation cone from the transmitter (Tx); \(\delta V\) is the common volume intersecting the emission and reception ends; \(k_{e}\) is the atmospheric extinction coefficient, given by \(k_{e}=k_{s}+k_{a}\), where \(k_{s}\) and \(k_{a}\) are the scattering and absorption coefficients, respectively; \(A_{r}\) represents the area of the receiving aperture; \(P(\mu)\) is the scattering phase function, where \(\mu=\cos\theta s\); and \(\zeta\) is the angle between the Rx axis and the vector from Rx to the common volume. Based on the respective scattering coefficients, the scattering phase function is modeled as a weighted sum of Rayleigh (molecular) and Mie (aerosol) scattering phase functions [9] calculated as follows:  
\[P(\mu)=\frac{k_{s}^{Ray}}{k_{s}}p^{Ray}(\mu)+\frac{k_{s}^{Mie}}{k_{s}}p^{Mie}( \mu), \tag{2}\]  
Here, \(k_{s}^{Ray}\) and \(k_{s}^{Mie}\) represent the Rayleigh and Mie scattering coefficients, respectively, with \(k_{s}=k_{s}^{Ray}+k_{s}^{Mie}\). \(p^{Ray}(\mu)\) and \(p^{Mie}(\mu)\) representing the [PERSON] and Mie scattering phase functions, following the generalized Rayleigh model and the generalized Henyey-Greenstein function. Subsequently, we define the path loss as \(PL=E_{t}/E_{r}\). The path loss expression of the truncated cone approximation model is given by the following equation [9]:  
\[PL\approx\frac{96r\sin\theta_{1}\sin^{2}\theta_{2}(1-\cos\frac{\phi_{1}}{2}) \exp[\frac{k_{e}r(\sin\theta_{1}+\sin\theta_{2})}{\sin\theta_{s}}]}{k_{s}P( \mu)A_{r}\phi_{2}^{2}\phi_{2}\sin\theta_{s}(12\sin^{2}\theta_{2}+\phi_{2}^{2} \sin^{2}\theta_{1})}, \tag{3}\]  
Figure 1: The diagram of a UV NLOS scattering communication link in coplanar geometry.  
This closed-form expression allows for the analysis and handling of path loss, as well as the BER performance of communication systems under different geometric conditions. Subsequent simulations and experiments are also based on this expression.