## 2 Monte Carlo Point Probability Model and the Modeling of Light Sources  
Based on these studies, this paper explored the pulses acquired at the receiver under different initial photon distributions, both coplanar and non-coplanar. Simulations of different light sources (mainly including LEDs, collimated lasers, and expanded laser beams) were conducted. The effect of the pulse broadening and the received energy density were evaluated.  
In the case of multiple scattering and single scattering, respectively, we simulated the atmospheric transmission of UV light based on different kinds of light source. By changing the light intensity distribution of the light source and the geometric features of the transceiver (\(\alpha\), \(\theta_{T}\), \(\theta_{R}\), \(\phi_{T}\), \(\phi_{R}\)), we found that the light energy density of different light sources affected the received signal under the single-scattering mode. Compared with long-distance light transmission, the received energy increased more significantly through multiple scattering in short-range light propagation. Under the same divergence angle, the more concentrated the light energy density is, the higher the received energy efficiency. However, this led to greater sensitivity of the received energy to changes of the off-axis angle.  
Figure 1 illustrates the process of ultraviolet light atmospheric transmission. \(T\) is the position of the emitting light source. A photon eventually reaches the receiver \(R\) (such as a photomultiplier tube, PMT) after it is scattered by aerosols and air molecules multiple times [29]. \(TF\) and \(RG\) indicate the central axes of the emergent light and the field of view (FOV), respectively. \(\phi_{T}\) and \(\phi_{R}\) are defined as the half-beam angle of the source and half-FOV angle, respectively. \(\theta_{T}\) and \(\theta_{R}\) represent the elevation angle of light source and receiver. \(\theta_{sn}\) is the scattering angle after the \(n\)th scattering, and \(\alpha\) refers to  
the off-axis angle. When \(\alpha=0\), \(TF\) and \(RG\) are coplanar. \(\theta_{s0}\) is the angle between the initial photon emission direction and \(TF\). And \(r\) is the horizontal distance between \(T\) and \(R\). \(TF\)' is the projection of \(TF\) in the \(XTZ\) plane.  
Because of the randomness of photons' direction distribution and the limited receiving area of the detector, a large number of photons need to be traced for more accuracy. This greatly increases the running time of the program, as large numbers of photons are absorbed and have little contribution to the received signals. In the direct MC method, it is necessary to determine whether a photon reaches the detector at each scattering point, and calculate the weight of the photon after the last scattering. For the point probability method, we did not need to consider whether the scattered photons reached the detector; we just needed to calculate the point probability that the photon reached the detector at each scattering point [21].  
The Monte Carlo point probability method simulates the propagation process of a large number of photons, and calculates the reception probability of the received photons. In this way, every scattering of each photon is valuable. Compared with the direct Monte Carlo method, it improves the photon utilization and reduces the program operation time, meaning that we can obtain the received light intensity as a function of time. The entire simulation process can be divided into three steps: the simulation of light, the propagation of the photons, and the reception of photons.