## 4 Simulation Results
### Magnitude Response  
In this section, we show how temporal dispersion is closely linked to multiple scattering and the system's geometry. Since each photon has been traced independently, the total propagation distance and received power of all the received photons can be easily calculated for each simulation round. This allows calculating the temporal response of the channel. In our model, we have assumed that every photon has some weight and that the total received power is the sum of the photon weights at each scattering event. Every scattering event reduces the weight of the photon by a factor equal to water albedo \(w_{o}\). To determine the magnitude response, we form a histogram of received powers to calculate channel bandwidth and thereby understand how the channel spreads light pulses in time. A standard method to analyze temporal response is to measure it in the frequency domain.  
The results in Figures 3-5 use the calculations for computing the maximum frequency and impulse response presented in Section 3. The magnitude response curves presented contain the received power of photons. As the power of total received photons drops, the magnitude response curves tend to die out quickly. This is associated with the fact that less received photons correspond to fewer values in the received power histogram that estimate the magnitude response.  
From Figures 3 and 4 we see that increasing the aperture size and FoV increases the received power gain. Since ocean and coastal waters are less turbid, resulting in fewer scattering events, photons are scattered at relatively larger angles. Therefore, a receiver with a wider aperture and FoV captures a higher number of photons in these waters. The overall power gains are relatively higher than the harbor water because the collected photons are not critically absorbed. It is intuitive based on the nature of the underwater channel and lesser scattering events. It also implies that the multipath attenuation is comparatively lower in the ocean and coastal water than harbor water; thus, the photons have sufficient power to reach the receiver. Hence, the effects of temporal dispersion in the ocean and coastal water can be controlled with wide receivers.  
In Figure 5, we see that the change in aperture size and FoV does not affect the channel's response. It implies that capturing a higher number of photons alone does not help. In harbor water, photons scatter at a shorter distance due to the greater number of scattering events; therefore, received photons are severely scattered and are eventually absorbed. Moreover, smaller scattering  
\begin{table}
\begin{tabular}{l l l} \hline \hline
**Coefficients** & **Symbol** & **Value** \\ \hline Half-angle beam divergence & \(b_{div}\) & 1.5 mrad \\ Beam waist & \(b_{unist}\) & 1 mm \\ Light source wavelength & \(\lambda\) & 532 nm \\ Link distance & \(d\) & 3 m \\ Water refractive index & \(n_{w}\) & 1.33 \\ Field-of-views & FoV & \(1^{\circ}\), \(18^{\circ}\), \(45^{\circ}\), \(90^{\circ}\), \(180^{\circ}\) \\ Apertures & \(D_{A}\) & \(1^{\prime\prime}\), \(5^{\prime\prime}\), \(9^{\prime\prime}\) \\ No. of evaluation points & \(n\) & 512 \\ \hline \hline \end{tabular}
\end{table}
Table 3: Important parameters used for temporal dispersion characterization and MC-based channel simulation.  
angles cause photons to stay close to the optical axis; therefore, wide receivers are not the optimal solution in minimizing multipath attenuation in harbor water. Hence, it is multiple scattering that contributes to the temporal dispersion of varying degrees for all water types.