## 3 Numerical Results
Under the multiple-scattering mode, the curve in Figure 9b did not change much, but the detector captured trace photons outside the original time range. For application in short-range detection instead of long-distance optical communications, we simulated the received signal a short distance from the transceivers. In Figures 7 and 8, compared to Figures 4-6 and 9, the peaks of the curves were increased by two to three orders of magnitude. The pulse width was reduced by an order of magnitude (in Figure 7a, the pulse time ranged from 0.03 us to 0.16 us.) In the case of short-distance transmission, the off-axis angle \(\alpha\) had little effect on the irradiance. As can be seen from Figure 7a, even when \(\alpha=50\), the peak value dropped from only \(2.25W/m^{2}\) to \(1.75W/m^{2}\). was assumed to be 1 mJ and the detector had a diameter \(d\) of 1.5 cm. During the simulation, \(N=10^{6}\) photons were simulated and the received energy density of the detector can be expressed as  
\[E_{R}=\frac{E_{0}}{N\pi(d/2)^{2}}\sum_{i=1}^{N}P_{ni}, \tag{15}\]  
where \(E_{0}\) represents the total initial energy of the light source. \(P_{ni}\) refers to the total received probability of the \(i\)th photon. The test results are shown in Figure 10. Overall, no matter what kind of light source, the energy density of the received signal \(E_{\text{R}}\) decreased sharply with the increase of the sending and receiving distance. As shown in Figure 10, light sources with different illuminating properties had different rates of acceptance energy reduction. Comparing the two curves of Laser beams 1 and 2 in Figure 10a,b, for received energy density \(E_{R}\), within a certain distance of \(r\) in Figure 1, we can see that a smaller transmitting light convergence \(\phi_{T}\) was more favorable to improve the receiving energy when the laser transmitting energy \(E_{0}\) and the elevation angle \(\theta_{T}\), \(\theta_{R}\) remained unchanged. To verify the correctness of the program above, we compared the results with data obtained using the [PERSON] classic model [6]. The relationships between received energy density \(E_{J}\) and distance \(r\) are shown in Figure 11a on the left Y axis. The numerical differences \(\left(E_{J,MC}-E_{J,Luetgen}\right)\) are shown in Table 2. For the sake of simplicity, we only tested ordinary light sources under the single-scattering model. Since only the molecular scattering coefficient was considered in [PERSON]'s paper, we ignored the effects of aerosols in the following test. The specific parameter settings are shown in Figure 11, and the calculation errors \(\left(E_{J,MC}-E_{J,Luetgen}\right)/E_{J,Luetgen}\) are shown in Figure 11 on the right Y axis. Figure 10: Curves of the received energy density \(E_{R}\) vs. distance \(r\) under different light sources by single scattering. Parameters of the light sources are the same as in Figures 4–9. (**a**) The linear scale in Y axis. (**b**)The log10 scale in Y axis. It can be seen that the results obtained by the two models were similar. The trends of the curves were roughly the same. The differences between the two curves were chiefly caused by the randomness of the Monte Carlo point probability model and the two different ways of curve fitting.