## 3 Numerical Results  
For LEDs, from Figure 2, we can see that when \(\phi_{\mathrm{T}}=\pi/12\), \(\theta=0.62 rad\), it has \(I(\theta)/I_{0}\approx 0.015\ll 1\). We assumed that \(I(\theta)/I_{0}=0\) when \(\theta>\pi/5\approx 0.628 rad\). The results of the Monte Carlo simulation under single-scattering (SS) mode are shown in Figures 4a, 5a, 6a, 7a, 8a and 9a, and the results under multiple-scattering (MS) mode are shown in Figures 4b, 5b, 6b, 7b, 8b and 9b. Table 1 illustrates the specific parameter settings. Figure 4: Curves of the UV irradiance vs. time with an ordinary light source under single scattering (SS) (**a**) and multiple scattering (MS) (**b**) modes. Figure 5: Curves of the UV irradiance vs. time with a Lambertian LED light source under SS (**a**) and MS (**b**) modes. Figure 6: Curves of the UV irradiance vs. time with a Gaussian expanded laser beam under SS (**a**) and MS (**b**) modes. Figure 8: Curves of the UV irradiance vs. time with Gaussian laser beam 2 under SS (**a**) and MS (**b**) modes. Figure 7: Curves of the UV irradiance vs. time with Gaussian laser beam 1 under SS (**a**) and MS (**b**) modes. Figure 9: Curves of the UV irradiance vs. time with Gaussian laser beam 3 under SS (**a**) and MS (**b**) modes. In general, when the distance r is a constant, there is a time delay of the received scattering light signal. Different off-axis angles \(\alpha\) will lead to different signal delays in single-scattering mode, as shown in Figures 4a, 5a, 6a, 7a, 8a and 9a. This is because as \(\alpha\) increases, the non-coplanarity of the transceiver increases, and so does the shortest propagation range for a photon. The shortest propagation time of the received photons increases and it will let fewer photons be absorbed and scattered, and the peak of the received irradiance decreases. In Figure 5a, when the light source is an LED with a Lambertian distribution, the received signals are also delayed. The difference is that the peak of the received irradiance has shifted to the right along the time line, and the received irradiance is much higher (nearly one order of magnitude) than that from the ordinary light source. This is because the light intensity of the LED is more concentrated in the paraxial region. It also makes the light source more sensitive to non-coplanar deflection angles \(\alpha\). When \(\alpha=40^{\circ}\), the received irradiance is very weak and close to zero. When \(\alpha=50\), photons simply cannot reach the receiver with only a single scattering in (Figure 5a). However, there are still some weak signals in multiple-scattered signals, as shown in Figure 5b. For the Gaussian expanded laser beam, the curves of UV irradiance shown in Figure 6 behaved similarly to those of the LEDs in Figure 5, but there were differences specifically in the following two aspects. First, in Figure 6, the irradiance curve of the expanded laser beam is always located below the LED. Second, in Figure 6, when \(\alpha=0^{\circ}\), \(10^{\circ}\), \(20^{\circ}\), the peak value of the expanded laser is lower than the LED's in Figure 5, but when \(\alpha=30^{\circ}\), \(40^{\circ}\), \(50^{\circ}\), the situation is reversed. This shows that the irradiance of the detector is not only related to the non-coplanarity of the transceiver, but also to the light intensity distribution of the light source. In Figure 9a,b, the other variables were kept the same and the light source was replaced with an unexpanded Gaussian beam (\(\phi_{\rm T}=0.01\)rad), In Figure 9a, the received signals became more sensitive to \(\alpha\). When \(\alpha\geq 30^{\circ}\), we could not detect the signals in single-scattering mode. We could still detect weak signals in multiple-scattering mode under this circumstance. Under the single-scattering mode, the common scattering volume formed by the thin laser beam and the receiving FOV was very small, which resulted in a very limited time-span of photon propagation.