## Appendix A Major Processes in Monte Carlo Simulation  
In this Appendix, we introduce the three major processes of Monte Carlo simulation for photon tracking, including initialization, photon propagation and photon reception, which are based on the description in [27; 45].  
(I) _Initialization_. As depict in Figure 3, the simulation geometry is defined by a cartesian coordinate system where a set of photons are assumed to be launched on x/y plane and propagate along the z-axis. Each photon is determined by its location \((x,y,z)\) and direction \(\mu_{x},\mu_{y},\mu_{z}\), where \(\mu_{x}=\cos\theta_{x},\mu_{y}=\cos\theta_{y},\mu_{z}=\cos\theta_{z}\). \(\theta_{x},\theta_{y},\theta_{z}\) are the angles between the direction vector of the photon and the x, y, and z-axes, respectively. The initial weight of each photon is defined as 1 to record the power loss of them. The initial position of each photon is generated according to \(\mathcal{U}[0,\omega_{0}]\), where \(\omega_{0}\) represents the beam width and \(\mathcal{U}[a,b]\) represents the uniform distribution between \(a\) and \(b\). The initial direction of each photon is generated according to \(\mathcal{U}[-\theta_{0,max},\theta_{0,max}]\) for \(\theta_{0}\) and \(\mathcal{U}[0,2\pi]\) for \(\phi_{0}\), where \(\theta_{0,max}\) represents the maximum initial divergence angle, \(\theta_{0}\) denotes the initial zenith angle and \(\phi_{0}\) denotes the initial azimuthal angle. Therefore the starting direction of a photon can be calculated by  
\[\mu_{x} =\sin\theta_{0}\cos\phi_{0},\] \[\mu_{y} =\sin\theta_{0}\sin\phi_{0}, \tag{1}\] \[\mu_{z} =\cos\theta_{0}.\]  
(II) _Photon Propagation_. The transmitted photon travels a random distance of \(\delta\) before interacting with a particle in water. The value of \(\delta\) for each photon is selected according to \(\delta=-\log(\chi_{\delta})/c\), where \(\chi_{\delta}\) is a random variable of distribution \(\mathcal{U}[0,1]\). Besides, a part of photon weight is lost when interacting with the particle, and the new weight \(W_{post}\) is calculated by \(W_{post}=W_{pre}(1-a/c)\). After interacting with the particle, the propagation direction of photon is changed and the new direction is determined by regenerating the zenith angle \(\theta\) and azimuthal angle \(\phi\). The regenerated angle \(\phi\) is considered to be a random variable of distribution \(\mathcal{U}[0,2\pi]\), and the regenerated angle \(\theta\) is selected based on Henyey-Greenstein model, which is a commonly used scattering phase function to model the trajectory of the scattered photon in seawater [27; 35].  
(III) _Photon Reception._ All the steps mentioned in II are cyclically repeated until the photon reaches the receiver plane or the photon weight is too small and negligible. The receiver is defined on the x/y plane and perpendicular to the beam axis. Please note that only the arrived photons within the receiver aperture and with incident angles less than the half angle of receiver FOV are considered to be received, otherwise they are considered to be lost.  
After the above processes, we can obtain the weights of the photons that reach the receiving plane and the photons that are successfully received, as well as the Cartesian coordinates and incident angles of these photons when intersecting with the receiving plane.