# Simulation study on the optical processes at deep-sea neutrino telescope
sites

Fan Hu  Zhenyu Wei  Wei Tian  Ziping Ye  Fuyudi Zhang  Zhengyang Sun  Wei Zhi
Qichao Chang  Qiao Xue  Zhuo Li  Donglian Xu<EMAIL_ADDRESS>Department of Astronomy, School of Physics, Peking University, 5 Yiheyuan Rd,
Beijing, China  Tsung-Dao Lee Institute, Shanghai Jiao Tong University, 520
Shengrong Rd, Shanghai, China  The Kavli Institute for Astronomy and
Astrophysics, Peking University, 5 Yiheyuan Rd, Beijing, China  School of
Physics and Astronomy, Shanghai Jiao Tong University, 800 Dongchuan Rd,
Shanghai, China

###### Abstract

The performance of a large-scale water Cherenkov neutrino telescope relies
heavily on the transparency of the surrounding water, quantified by its level
of light absorption and scattering. A pathfinder experiment was carried out to
measure the optical properties of deep seawater in South China Sea with light-
emitting diodes (LEDs) as light sources, photon multiplier tubes (PMTs) and
cameras as photon sensors. Here, we present an optical simulation program
employing the Geant4 toolkit to understand the absorption and scattering
processes in the deep seawater, which helps to extract the underlying optical
properties from the experimental data. The simulation results are compared
with the experimental data and show good agreements. We also verify the
analysis methods that utilize various observables of the PMTs and the cameras
with this simulation program, which can be easily adapted by other neutrino
telescope pathfinder experiments and future large-scale detectors.

###### keywords:

optical process , photon propagation , Monte-Carlo simulation , neutrino
telescope

††journal: NIM - A

##  1 Introduction

Neutrino telescopes observe neutrinos by detecting the faint Cherenkov light
emitted by the secondary charged particles from neutrino-nucleon interactions
[1, 2, 3, 4, 5]. The optical properties of a medium, known primarily as
absorption and scattering, determine the propagation behavior of the Cherenkov
light, which are critical for optimally improving the geometry layout and
angular resolution of a neutrino telescope [6, 7, 8]. Therefore, measuring the
optical properties of the medium at telescope sites in-situ is indispensable
for designing and operating neutrino telescopes [9, 10, 11, 12, 13, 14, 15].

The tRopIcal DEep-sea Neutrino Telescope (TRIDENT) pathfinder experiment
(TRIDENT EXplorer, T-REX for short) [16] measures the optical properties at
the South China Sea, which will be the location of the future TRIDENT neutrino
telescope. The experimental facility is composed of one light emitter module
placed in the middle and two light receiver modules separated aside at
distances of $21.73\pm 0.02\,\mathrm{m}$ and $41.79\pm 0.04\,\mathrm{m}$ from
the emitter, respectively. The emitter module contains LEDs that emit light in
both steady and pulsing modes. Photons emitted from the LEDs are diffused by a
double-layer diffuser and are isotropic in the emitting region [17]. They will
propagate through the vast deep seawater, and a small fraction of them will
register at the receiver modules, observed by the cameras or the PMTs. When
the emitter module works in the steady mode, cameras in the receiver modules
take photographs of the emitter sphere to obtain the spatial distribution of
the received photons. While in the pulsing mode, the PMTs inside the receiver
modules record the arrival time of the photons, which may be delayed due the
scattering processes. A photon arrival time distribution (ATD) is built from
repeating millions of in-situ pulsing experiments. The images from the cameras
and the ATD from the PMTs are then used to extract the optical properties
independently, providing complementary verification of the final experimental
results. Furthermore, the two emitter-receiver pairs with different distances
setup enables a relative measurement to the optical properties of the deep
seawater, which largely cancels out the hidden systematic errors.

Since a photon’s kinematic state (position, direction, and propagation time)
is hard to predict analytically after scattering for a few times, a numerical
simulation is required for an in-depth study to accurately interpret the
experimental results. In this work, we have developed a simulation program to
replicate the T-REX apparatus setup and produce photon propagation experiments
with various optical property parameters in the deep seawater. The simulation
output are then used to reconstruct the experimental observables and verify
the analysis methods.

The paper is organized as follows: in Sec. 2 we briefly overview the optical
physics studied. In Sec. 3, a simulation program based on Geant4 [18, 19] that
simulates the photon propagation in the medium is presented. Then, the
reconstruction of experimental observables using the simulation output is
shown in Sec. 4. The validation of the methods to extract optical properties
from experimental observables with the help of simulation are studied in
detail in Sec. 5. A discussion about the effects of geometry simplification
and the characteristics of the two systems is performed in Sec. 6, and Sec. 7
summarizes the main results of the paper.

##  2 Optical physics

Electromagnetic waves propagating in a medium deposit energy and excite the
electrons, atoms, molecules, or solid particles in the medium [20]. Upon
excitement, the deposited energy can either be converted to thermal energy, in
a process called absorption, or produce secondary radiation, known as elastic
scattering (named in this work simply as scattering, unless explicitly stated
otherwise) if the radiation remains the same frequency. Though the above
phenomenon is studied and understood in terms of electromagnetic waves, it can
be used to describe the propagation behavior of photons in the quantum world
according to the wave-particle dualism nature of electromagnetic waves.

The scattering processes can be qualitatively classified into two regions with
respect to the relative size between the obstacles and the photon wavelength:
Rayleigh scattering and Mie scattering. Rayleigh scattering refers to the
scattering from small obstacles with radius $r\ll\lambda$, where $\lambda$ is
the photon wavelength. On the other hand, Mie scattering describes the
scattering from obstacles with radius $r\gtrsim\lambda$. The Mie scattering is
mainly caused by the organics in water and usually are the dominant process in
natural water [9, 13].

The angular distribution of scattered photons can be described by a phase
function $\beta(\cos\theta)$, with $\theta$ being the photon deflection angle.
For Rayleigh scattering, $\beta_{\mathrm{Ray}}(\cos\theta)$ can be described
by a dipole radiation model and is symmetric over the polarization direction
of the incident photons. The Mie scattering is much more complex due to the
non-negligible size of the obstacles. Its phase function
$\beta_{\mathrm{Mie}}(\cos\theta)$ has a strong peak in the forward direction
and can be parameterized by the Henyey-Greenstein (HG for short) approximation
[21]. In summary, the pahse functions for Rayleigh and Mie scatterings can be
given, respectively, by:

$\displaystyle\beta_{\mathrm{Ray}}(\cos\theta)$
$\displaystyle=\frac{3}{8}\left(1+\cos{\theta}^{2}\right)$ (1a)
$\displaystyle\beta_{\mathrm{Mie}}(\cos\theta)$
$\displaystyle=\frac{1}{2}\frac{1-\mu^{2}}{(1+\mu^{2}-2\mu\cos{\theta})^{\frac{3}{2}}},$
(1b)

where $\mu$ is the only parameter in the HG formula and has the same value as
the expectation of $\cos\theta$.

The probabilities of photons encounter absorption, Rayleigh scattering, and
Mie scattering are characterized by the mean free paths and are marked as
$\lambda_{\mathrm{abs}}$, $\lambda_{\mathrm{Ray}}$, and
$\lambda_{\mathrm{Mie}}$ respectively. There exist many quantities constructed
to describe the combined effects of absorption and scattering. The most widely
known one is the attenuation length, defined as:

$\frac{1}{\lambda_{\mathrm{att}}}=\frac{1}{\lambda_{\mathrm{abs}}}+\frac{1}{\lambda_{\mathrm{Ray}}}+\frac{1}{\lambda_{\mathrm{Mie}}}.$
(2)

The physical meaning is the exponential decay of the radiance (having a unit
of energy per time per area per solid angle) of a collimated light beam
$I_{\mathrm{beam}}(d)$ over the distance $d$ when propagating in the medium,
since both absorption and scattering can eliminate photons in the light beam.
The decay of $I_{\mathrm{beam}}(d)$ is characterized by the Beer-Lambert law
[22]:

$I_{\mathrm{beam}}(d)=I_{\mathrm{beam}}(0)e^{-d/\lambda_{\mathrm{att}}}.$ (3)

Another typical quantity used in previous experiments [9] is the scattering
angle averaged attenuation length, defined as:

$\frac{1}{\lambda_{\mathrm{att,avg}}}\equiv\frac{1}{\lambda_{\mathrm{abs}}}+\frac{1}{\lambda_{\mathrm{Ray}}}+\frac{1-\mu}{\lambda_{\mathrm{Mie}}},$
(4)

The $\lambda_{\mathrm{att,avg}}$ considers that the strong forward scattering
processes have a tiny influence on the overall behavior of a bulk of photons.
Thus it can describe the overall fade-off of photons in scenarios where the
small changes in the photon direction can be ignored. The effects of the
physical quantities mentioned above will be further discussed by dedicated
simulation studies in Sec. 4 and 5

It should be noted that there exist other optical processes, such as inelastic
scattering [23] in deep seawater. Since their probabilities of occurrence are
orders of magnitude smaller than absorption and elastic scattering, they can
be safely ignored in our study.

##  3 Simulation program

###  3.1 Set up simulation program using Geant4

Figure 1: Illustration of the geometry for T-REX (left) and in simulation
(right). The receiver module is extended to a receiver shell surrounding the
emitter module to increase the simulation efficiency. The yellow lines
indicate possible photon trajectories. $\vec{x}_{r}$ and $\hat{n}_{r}$ are
photon hit position and direction on the shell, $\theta_{\mathrm{in}}$ is the
incident angle.

The simulation program 111https://github.com/TRIDENT-Neutrino-
Telescope/Pathfinder-Optical-Simulation is built with Geant4 toolkit, a
framework widely used in particle physics community [24, 25, 26]. Geant4
performs particle simulation in a particle-by-particle and step-by-step
manner. Its modularized design helps users to easily apply the physics
processes and extract information from each step.

In this simulation, we assume photons are sufficiently diffused inside the
light emitter module. Thus photons are emitted isotropically from the surface
of the module, which means the radiant intensity follows Lambert’s cosine law.
Once emitted, they propagate through the optical medium and may encounter
scattering and absorption processes. The optical physics described above has
already been implemented in the context of G4OpticalPhysics modules in Geant4
and can be applied directly in simulation. Geant4 can trace every photon till
it is absorbed or arrives at the detector. If a photon successfully arrives at
the detector, its kinematic status will be recorded by the output module.

The main drawback of Geant4 might be its performance in speed when tracing
billions of photons. This can be eased by extending the photon detection area
in simulation. For T-REX, the photons are emitted and diffused by the double-
layer diffusers inside the emitter module. After emitting from the module,
those photons propagate nearly isotropically in the vast seawater environment.
Only a small portion of photons can eventually hit the photon sensors, as
shown in the left panel of Fig. 1. Given that the optical properties of deep
seawater are isotropic, and assuming the photons from the light emitter are
isotropic as well, a photon receiver located at different directions from the
emitter should observe consistent results. In the simulation, we utilize such
symmetry and extend the receiver module to a receiver shell surrounding the
emitter module. The simplified geometry is illustrated in the right panel of
Fig. 1. Such adjustment can efficiently boost the number of received photons
in the simulation and shorten the simulation time. It should be emphasized
that while the light emitting module is not perfectly isotropic in spherical
space, the in-homogeneous effect is calculated by adding a weighting term in
the final result, which will be further described in Sec. 4.2.

In constructing the simulation, we fixed the radius of the central emitter
module to be $20\,\mathrm{cm}$, which corresponds to the radius of the outer
diffuser layer. The refractive index of the deep seawater is set to be 1.34
and works as a time scaling factor in the simulation. Its effect is
perturbatively tunable in the data analysis via rescaling of the photon
arrival times. The other optical parameters as well as the distance between
the light source and receiver are the input parameters of the simulation and
are set in run time.

###  3.2 Simulation input and output

The flexibility of Geant4 framework allows us to explore various effects of
optical properties on the photon propagation process in the medium.

In the simulation input, we set the distance between the light source and
receiver and the optical property parameters of the medium. The program can
take input from a human-readable configure file in YAML format or command line
arguments. The kinematic information of a photon hit is recorded in a per-
photon manner as simulation output, which is recorded in a ROOT file [27].
Table 1 summarizes the input and output of the simulation.

| Variable name | Physical description
---|---|---
Input | $\lambda_{\mathrm{abs}}$ | absorption length
$\lambda_{\mathrm{Ray}}$ | Rayleigh scattering length
$\lambda_{\mathrm{Mie}}$ | Mie scattering length
$\mu$ | the parameter in the HG Mie scattering formula
$d$ | distance between source and receiver
Output | $\vec{x}_{\mathrm{e}}$ | emission position at source shell
$\vec{x}_{\mathrm{r}}$ | hit position at observation shell
$\hat{n}_{\mathrm{r}}$ | photon direction at observation shell
$t$ | hit time at the observation shell
$k$ | number of times that photon get scattered
Table 1: Summary of the simulation input and output.

##  4 Construction of experimental observables

In this section, we show how the experimental observables can be reconstructed
by using the kinematic information of photons from the simulation output. We
launched two simulation runs with emitter-receiver distances of 20 m and 40 m,
respectively. The optical parameter settings used here are summarized in Table
2, which are representative of the deep seawater. [9, 15]. In each run,
$10^{7}$ photons are emitted from the light emitter module.

Parameters | $\lambda_{\mathrm{a}}$ | $\lambda_{\mathrm{Ray}}$ | $\lambda_{\mathrm{Mie}}$ | $\mu$
---|---|---|---|---
Values | 27 m | 200 m | 70 m | 0.97
Table 2: Summary of the default optical parameter settings in simulation.
Those parameters are for clean seawater. [9, 15, 16]

###  4.1 Camera images

Figure 2: Upper: illustration for the possible trajectories of photons emitted
from the light source. The dashed line represents the viewing angle of the
camera’s field of view. Trajectories illustrated are: (a) directly arriving
photon, (b) photon that is scattered into the camera, (c) photon that gets
scattered out, (d) photon that is scattered into the camera but is outside the
field of view. Among them, only (a) and (b) can be recorded by the camera.
Lower: geometry of pinhole camera model. All photons are focused on the CMOS
film, and no bokeh effect is simulated. The hit radius at film
$r_{\mathrm{film}}$ corresponds to the incident angle of photon
$\theta_{\mathrm{in}}$, as previously shown in Fig. 1.

In T-REX, a camera is housed in the light receiver module and takes
photographs for the light emitter module when it is glowing in steady mode
[28]. Once emitted from the source, photons can go through different
trajectories, as shown in the upper panel of Fig. 2, and some of them arrive
at the camera and may be recorded by the CMOS film of the camera to form an
image.

In our simulation, we can also construct the image by using the photon
direction $\hat{n}_{\mathrm{r}}$ and position $\vec{x}_{\mathrm{r}}$ at the
receiver sphere. This is done according to the pinhole camera model, as
abstracted in the lower panel of Fig. 2. The incident angle of the photon at
the lens $\theta_{\mathrm{in}}$ can be evaluated by the geometry relation:

$\theta_{\mathrm{in}}=\arccos\left(\hat{n}_{\mathrm{r}}\cdot\frac{\vec{x}_{\mathrm{r}}}{|\vec{x}_{\mathrm{r}}|}\right).$
(5)

And the radial coordinate of the photon hit point at the camera film
$r_{\mathrm{film}}$ can be derived as:

$r_{\mathrm{film}}=f_{\mathrm{lens}}\sin{\theta_{\mathrm{in}}},$ (6)

where we have used $f_{\mathrm{lens}}$ as the focus of the lens. For the case
in T-REX, $f_{\mathrm{lens}}=25~{}\mathrm{mm}$ and the pixel size in CMOS is
$3.45~{}\mathrm{\mu m}$. Thus one pixel corresponds to an incident angle of
$1.38\times 10^{-4}~{}\mathrm{rad}$ [28].

Figure 3: Gray value distribution. The central plateau in the gray value curve
corresponds to the image of the light emitter module, while the tail
corresponds to the halo caused by the scattering processes.

The distribution of radius of hit points at film $r_{\mathrm{film}}$ is shown
in Fig. 3, with two emitter-receiver distances. The linearity of the CMOS
response allows us to convert the number of hits into the gray value of
images. The central plateau in the gray value curve corresponds to the image
of the light emitter module.

Assuming the light source is symmetric in the axial direction, we generate the
camera image according to the distribution of $r_{\mathrm{film}}$. Figure 4
shows an example of the resulting image, which consists of a bright central
sphere with a surrounding halo of scattered light. The halo is caused by
scattering light and has a gray value of $\sim 2\%$ compared to the bright
central sphere.

Figure 4: Camera image from simulation. A pinhole camera is taking photons for
the light source at a distance of 20 m. The gray value is shown in log scale
to illustrate the gradient of the halo.

###  4.2 Photon arrival time distribution in PMT

The PMTs in the receiver module measure the ATD of received photons. The
photons that have undergone scattering processes arrive late and form a long
tail in the distribution. The scattering effect of the medium can be resolved
by fitting the shape and amplitude of the tail [9, 10]. The ATD can be
calculated using the photon hit time $t$ recorded in the simulation output, as
shown in Fig. 5. Since the PMTs in T-REX have a transient time spread (TTS) of
about $4~{}\mathrm{ns}$ [29], we have convolved the ATD with a Gaussian kernel
with full width at half maximum (FWHM) equal to $4~{}\mathrm{ns}$ to simulate
the effect of TTS.

The 3D printed supporter inside the emitter module brings asymmetry to the
light intensity in latitude direction [17]. Such an effect can be studied in
simulation by deriving the relative latitude of the photon’s emitting point
compared to the photon’s receiving point $\theta_{\mathrm{e}}$, defined as:

$\theta_{\mathrm{e}}=\arccos\left(\frac{\vec{x}_{\mathrm{r}}}{|\vec{x}_{\mathrm{r}}|}\cdot\frac{\vec{x}_{\mathrm{e}}}{|\vec{x}_{\mathrm{e}}|}\right).$
(7)

As shown in the right panel of Fig. 5, photons from different
$\theta_{\mathrm{e}}$ have distinct ATDs. The photons from the negative
$\theta_{\mathrm{e}}$ area, which face opposite to the PMT, can only arrive at
PMT after being scattered. Thus the ATD of those photons has no peak at the
geometric expected time but a long tail. Once we have evaluated
$\theta_{\mathrm{e}}$, we can add weighting terms to the photons from
different $\theta_{\mathrm{e}}$ to simulate the asymmetry effect of the
emitter module in the analysis.

Figure 5: Photon arrival time distribution (ATD) measured in PMT at a distance
of 20 m. A time smearing with FWHM of $4~{}\mathrm{ns}$ is added. In the left
panel, the photons are divided into scattered and direct groups. A long
scattering tail can be observed in the scattered group. In the right panel,
the photons are divided into groups according to $\theta_{\mathrm{e}}$. The
values in the legend represents the binning of $\theta_{\mathrm{e}}$.

###  4.3 Comparison with experimental results

Figure 6: Comparison between the simulation data and experimental data for the
photon arrival time distribution (ATD) measured by the PMT system. The black
dots correspond to the data taken from the T-REX. The curves correspond to the
simulation results after calibration. The simulation input parameters are the
combination of: $\lambda_{a}=27\,\mathrm{m}$,
$\lambda_{\mathrm{Ray}}=[120\,\mathrm{m},180\,\mathrm{m},240\,\mathrm{m}]$,
$\lambda_{\mathrm{Mie}}=[30\,\mathrm{m},60\,\mathrm{m},90\,\mathrm{m}]$ and
$\langle\cos\theta\rangle=[0.90,0.94,0.98]$, summing up to 27 different
configurations. The blue and red curves represent the minimal and maximal
scattering cases. The two green dashed lines indicate the dominantly
scattering tail region.

The simulation results can be compared with the measurements in T-REX after
meticulous calibration. For the PMT system, the time profile of the pulsing
LEDs, the response of PMTs, and readout electronics are calibrated in the
laboratory during the T-REX [29]. After considering those effects, the
simulation result matches well with the experimental data, as shown in Fig. 6.
The curves from simulation with different optical parameters can cover the
dominantly scattering tail in between $250\,\mathrm{ns}$ and
$350\,\mathrm{ns}$ observed in the experiment data, where the scattering
effect is mostly revealed.

A detailed discussion about the $\chi^{2}$ method that compares the
experimental data with calibrated simulation results to obtain the best fit
optical properties are presented in [16].

##  5 Validation of analysis methods

Apart from the brute force $\chi^{2}$ fitting method mentioned above, the
optical properties of the medium can also be evaluated by constructing concise
expressions. However, theoretical statistic analysis of photons’ behavior
without low-order approximation appears intricate when both absorption and
scattering processes exist and tangle with each other. Any experimental
macroscopic observable is engendered under the mixture of two distinct optical
processes. This makes numerical simulation a better way to conduct viable
analysis. With the help of the simulation program developed above, we can
exploit how to measure the optical properties of the medium.

Note that we are going to construct a variety of effective attenuation lengths
directly from the simulated observables in the following context. Those
physical quantities are marked by a hat, e.g., $\hat{\lambda}_{\mathrm{att}}$,
to be distinguished from the canonical quantities defined in Sec. 2.

###  5.1 Analysis methods for the camera system

The camera system takes photographs of the steady emitting light source. It
measures the directions of arrived photons by their positions at the film, as
shown in Fig. 2. The gray value of the image observed by the camera system
corresponds to the radiance of the light field in physics.

We performed numerical simulation using the input optical parameters shown in
Table 2 with various distances between the source and receiver, ranging from
$15\,\mathrm{m}$ to $55\,\mathrm{m}$, covering the distance range in the
experiment. It is verified in simulation that the photons composing the
central radiance, $I_{\mathrm{center}}$, represented by the gray value of the
pixels at the center part of the image (Fig. 3), are mainly directly arriving
photons that have not undergone any scatterings during propagation. As shown
in Fig. 7, the ratio of the directly arriving photons $\alpha$ in
$I_{\mathrm{center}}$ has a stable value of $\alpha=92\pm 1\%$ in the distance
range.

Figure 7: The ratio of scattered photons in the central region of a camera
image observed at different distances. It shows that the ratio remains
relatively constant over different distances. Figure 8: The gray scale of
central pixels ($I_{\mathrm{center}}$) as a function of source-receiver
distance. The input optical properties in simulation is summarized in Table 2,
i.e., the attenuation length is $17.75\,\mathrm{m}$. The fitted attenuation
length according to Equation 8 is $\hat{\lambda}_{\mathrm{att}}=18.13\pm
0.18\,\mathrm{m}$, where the error comes from the fitting.

According to the Beer-Lambert law [22], the attenuation length can be measured
by comparing the radiance of directly arriving photons at two distances. Here,
we use the observed central gray value to fit the attenuation length:

$I_{\mathrm{center}}(d)\propto\frac{1}{\alpha}\exp(-\frac{d}{\hat{\lambda}_{\mathrm{att}}}),$
(8)

The verification of this method is shown in Fig. 8, where we have assumed the
ratio of directly arriving photons $\alpha$ is constant over the distance. It
can be found that fitting result of $\hat{\lambda}_{\mathrm{att}}=18.13\pm
0.18\,\mathrm{m}$ is close to the canonical attenuation length
$\lambda_{\mathrm{att}}=17.75\,\mathrm{m}$. The attenuation length obtained
using this method is about $2\%$ larger than the canonical attenuation length
due to the slightly increasing ratio of scattered photons over the distance.
The bias of this method is only a minor effect compared to other systematic
errors in the measurement of attenuation length in the actual experiment. If
we take this effect into account and select only direct photons in simulation,
the fitting result will be $\hat{\lambda}_{\mathrm{att}}=17.85\pm
0.10\,\mathrm{m}$, which is consistent with the canonical attenuation length.

###  5.2 Analysis methods for the PMT system

Figure 9: Attenuation lengths with different definitions as a function of
Rayleigh scattering length. The optical properties except Rayleigh scattering
length are the same as Table 2. The curves for $\lambda_{\mathrm{att}}$,
$\lambda_{\mathrm{att,avg}}$ and $\lambda_{\mathrm{abs}}$ are directly
evaluated from simulation truth by their definitions. The red dots represent
the measured effective attenuation length using the method described by
Equation 9. The blue dots are similar to red dots but use the photon numbers
from a $3\times$ TTS time window around the peak instead of the whole time
window.  Figure 10: The ratio of scattering photons in each time bin at 20 m
and 40 m. The dashed line shows the range of integrated bins for the peak-only
method: from $-3\times\mathrm{TTS}$ to $+3\times\mathrm{TTS}$. At times after
$3\times\mathrm{TTS}$, almost all photons have encountered scattering during
propagation.

The PMT system measures the number of arrived photons as a function of time.
It can also derive an effective attenuation length
$\hat{\lambda}_{\mathrm{att,eff}}^{\mathrm{tot}}$ by comparing the total
number of received photons at two distances:

$\hat{\lambda}_{\mathrm{att,eff}}^{\mathrm{tot}}=(d_{1}-d_{2})/\ln\left(\frac{N_{\mathrm{2}}^{\mathrm{tot}}d_{2}^{2}}{N_{\mathrm{1}}^{\mathrm{tot}}d_{1}^{2}}\right),$
(9)

where $N_{\mathrm{1}}^{\mathrm{tot}}$ and $N_{\mathrm{2}}^{\mathrm{tot}}$ are
the total number of photons received by PMTs at distances of $d_{1}=20\,m$ and
$d_{2}=40\,m$, respectively.

Some previous studies [9, 15] use
$\hat{\lambda}_{\mathrm{att,eff}}^{\mathrm{tot}}$ (Equation 9) to estimate the
canonical attenuation length $\lambda_{\mathrm{att}}$ (Equation 2). To verify
this method, we performed multiple simulation runs with various Rayleigh
scattering lengths $\lambda_{\mathrm{Ray}}$ and recorded outputs at distances
of 20 m and 40 m. The other optical parameters in simulation remain the same
as Table 2. Our simulation shows that their deviation can be as significant as
$\sim 40\%$, shown in Fig. 9. The deviation between
$\hat{\lambda}_{\mathrm{att,eff}}^{\mathrm{tot}}$ and $\lambda_{\mathrm{att}}$
is caused by the large ratio of scattered photons observed by PMTs, which
increases with the emitter-receiver distance, as shown in the left panel of
Fig. 5 and Fig. 10. We find that the ratios of scattered photons among all
photons observed by the PMTs are 43% and 67% in distances of 20 m and 40 m,
respectively. Since our light source is not beamed to the receiver but is
isotropically emitting photons in all directions, some photons that initially
emitted towards a different direction to sensor might be received by the PMT,
which has a large viewing angle, after encountering multiple times scattering.
Our study shows that $\hat{\lambda}_{\mathrm{att,eff}}^{\mathrm{tot}}$
measured by the above method can not be used to estimate the canonical
attenuation length of the medium.

By restricting the integration window to $3\times\mathrm{TTS}$ around the peak
of ATD instead of the whole windows, we can derive another experimental
observable $\hat{\lambda}_{\mathrm{att,eff}}^{\mathrm{peak}}$. As shown in
Fig. 9, we find that $\hat{\lambda}_{\mathrm{att,eff}}^{\mathrm{peak}}$ has a
similar value to the scattering angle averaged attenuation length
$\lambda_{\mathrm{att,avg}}$ (Equation 4), and their relative error is within
$3\%$ in the cases of our input optical parameters (Table 2). As discussed in
Sec. 2, $\lambda_{\mathrm{att,avg}}$ is larger than $\lambda_{\mathrm{att}}$
because it considers the strong forwardness of the deflection angle in Mie
scattering. In the case of PMT measurement, the photons arriving around the
peak time are either non-scattered or slighted scattered, thus their number is
approximately corresponding to $\lambda_{\mathrm{att,avg}}$.

Despite the difficulties in measuring the canonical attenuation length using
the PMT system, the absorption length of the medium can be measured well using
the timing ability of PMT:

$\int_{-\infty}^{\infty}\mathcal{N}_{1}(t)e^{\frac{ct}{n\lambda_{\mathrm{abs}}}}~{}\mathrm{d}t\times
d_{1}^{2}=\int_{-\infty}^{\infty}\mathcal{N}_{2}(t)e^{\frac{ct}{n\lambda_{\mathrm{abs}}}}~{}\mathrm{d}t\times
d_{2}^{2},$ (10)

where $t$ is arrival time of photon hits,
$\mathcal{N}(t)=\frac{\mathrm{d}N}{\mathrm{d}t}$ is the ATD (number density
per unit time) of photons observed by PMT, $n$ is the refractive index of the
medium, $c$ is the speed of the light, and $d$ is the emitter-receiver
distance. $i$ indicates the two measurements at a distance of $20\,\mathrm{m}$
and $40\,\mathrm{m}$. This measurement utilizes the timing ability of PMTs,
which can be used to convert the photon arrival time to the trajectory length
$l$ of the photons using relation $l=ct/n$. The probability of the photon not
being absorbed through a trajectory length corresponds to $t$ is
$e^{\frac{ct}{n\lambda_{\mathrm{abs}}}}$ and is used as the weighting term in
Equation 10.

An advantage of using Equation 10 is its invariance under convolution. Due to
the transient time spread of PMT and the LED emitting pulse width, the
observed ATD $N_{\mathrm{obs}}(t)$ is smeared. Assuming the smearing kernel
function $G(\tau)$ is the same for the two PMTs, it can be proved that such a
smearing effect can bring only a form factor
$\mathcal{G}=\int_{-\infty}^{\infty}G(\tau)e^{\frac{c\tau}{n\lambda_{\mathrm{abs}}}}~{}\mathrm{d}\tau$:

$\displaystyle~{}~{}~{}\int_{-\infty}^{\infty}\mathcal{N}_{\mathrm{obs}}(t)e^{\frac{ct}{n\lambda_{\mathrm{abs}}}}~{}\mathrm{d}t$
(11)
$\displaystyle=\int_{-\infty}^{\infty}G(\tau)\mathcal{N}(t-\tau)e^{\frac{ct}{n\lambda_{\mathrm{abs}}}}~{}\mathrm{d}t\mathrm{d}\tau$
$\displaystyle=\int_{-\infty}^{\infty}G(\tau)e^{\frac{c\tau}{n\lambda_{\mathrm{abs}}}}~{}\mathrm{d}\tau\times\int_{-\infty}^{\infty}\mathcal{N}(t-\tau)e^{\frac{c(t-\tau)}{n\lambda_{\mathrm{abs}}}}~{}\mathrm{d}t$
$\displaystyle=\mathcal{G}\int_{-\infty}^{\infty}\mathcal{N}(t)e^{\frac{ct}{n\lambda_{\mathrm{abs}}}}~{}\mathrm{d}t$

Thus Equation 10 remains valid after replacing the original ATD
$\mathcal{N}(t)$ by the smeared distribution $\mathcal{N}_{\mathrm{obs}}(t)$.

Figure 11: Illustration for measuring absorption length in the PMT system.
Left panel: the ATD observed by PMT with and without absorption effect. The
PMT is located 20 m away from the light source, and no time smearing effect is
simulated. The distribution with absorption can be perfectly corrected to the
distribution without absorption. Right panel: The ATD observed by PMT in 20 m
and 40 m considering time smearing and absorption effect. The total number of
photons after absorption correction (integration of green and red curves) at
two distances matches well.

The above method can be verified in the simulation. We simulate the ATD with
two settings, one with the absorption process and the other without. The
distribution with absorption can be corrected to the distribution without
absorption using the $e^{\frac{ct}{n\lambda_{\mathrm{abs}}}}$ weighting term,
as shown in the left panel of Fig. 11. The comparison between the left and
right side of Equation 10 is shown in the right panel of Fig. 11, where we
have shown the measured ATD at a distance of 20 m and 40 m and their
absorption corrected distribution. The number of photon hits obtained by
integrating the ATD at 40 m and 20 m distances matches well after the
absorption correction, with the former only $\sim 3\%$ smaller than the
latter. Because we have ignored some late-arrived photons in the analysis
since the background noise might cover those photons in the actual experiment.

##  6 Discussion

###  6.1 Effects of geometry simplification in simulation

The geometry of the simulation program developed in this work assumes an
isotropic symmetry of the light source and extends the receiver shell to
increase the simulation efficiency, as shown in Fig. 1. While the actual light
source is not isotropic due to the penetrator, vacuum port, and other
mechanical structures, the data analysis performed always integrates over
$\phi$ direction, and the anisotropy in $\theta$ direction can be calibrated
in the laboratory and implemented on simulation data. The occlusions caused by
apparatus such as submarine cable have effects of less than 0.1% for T-REX and
are neglected in the simulation.

The boundary process of photons when going through air-glass and glass-water
interfaces are not simulated either. While the reflection on the interfaces
can reduce the number of photons detector in the photon sensors, our relative
measurement strategy can eliminate this effect. The effect caused refraction
of photons is more complicated than the reflection. Since the thickness of the
glass shell is uniform, the air-glass and glass-water interfaces can be merged
into a single air-water interface when studying the refraction effect.
Detailed optical analysis shows that while the interface between the emitter
and water can shrink the image size of the emitter for an observer in water,
the interface between the receiver and water can expand the image size
observed by the camera [28]. So those interfaces do not influence the final
image size observed by the camera.

###  6.2 Difference between the two systems

T-REX experiment utilized both the camera system and the PMT system to measure
the optical properties of deep seawater. In our numerical simulation, we
constructed various physical quantities from the observables measured by the
two systems, as presented in Sec. 5.

The two photon sensors observe the light field at two different distances.
Their instrumental properties cause the difference in the optical parameters
to which they are sensitive. The camera system features high angular
resolution to the arrived photons. The gray values in the image taken by the
camera corresponds to the radiance of the steady light field:

$\mathrm{G}(x,y)\propto I(\theta_{\mathrm{in}}),$ (12)

where $\mathrm{G}$ is the gray value of the image, $(x,y)$ is the coordinate
at the image and is related to the incident angle of photons
$\theta_{\mathrm{in}}$ using the geometric relation shown in the lower panel
of Fig. 2. The camera system can effectively select directly arriving photons
using direction information, i.e., using only the gray values at the center of
the image. Thus, the camera system is capable of measuring the attenuation
length.

The PMTs can not distinguish photons from different directions and thus can
only measure the irradiance, which is the integral of radiance over all
directions. On the other hand, the timing ability enables PMTs to measure the
arrival time of the photons from a pulsing light source. The PMT system can
approximately measure the scattering angle averaged attenuation length by
comparing the number of photons received in a time window of
$3\times\mathrm{TTS}$ around the peak at two distances, since the arrival time
of those photons indicates that they can not by strongly scattered. The extra
trajectory length induced by the scattering effect can be well calibrated
using the arrival time measured by the PMTs. Those properties make the PMT
system suitable for measuring the absorption length of the deep seawater.

##  7 Summary

In this paper, we introduce a program based on Geant4 toolkit that simulates
photon propagation in the deep seawater. The simulation results are compared
to the experimental data taken during the T-REX expedition in South China Sea
and show good agreement for both the PMTs and the cameras. The program is used
to extract the key optical parameters at the selected site, which is crucial
for obtaining accurate detector simulation and optimizing detector design of
the TRIDENT telescope.

Our simulation provides a thorough examination of different optical
parameters. We evaluate the analysis methods using different observables from
the PMTs and the cameras. The differences between the effective attenuation
length - measured by the PMT using photon counts, and the canonical
attenuation length - measured by the camera using central gray value of the
photographs, were also resolved. It is shown that the effective attenuation
length can get up to $\sim 40\%$ larger than the canonical attenuation length
if all photons observed by a PMT are counted.

The simulation program developed can be employed by other experiments with
PMTs or cameras to measure optical properties for neutrino telescopes in deep
lake water or glacial ice. It can also be easily adapted for application in
the optical calibration system for the next-generation neutrino telescopes.

## Acknowledgements

We are grateful to Jun Guo and Iwan Morton-Blake for their help to improve
this paper.

This work is supported by the Ministry of Science and Technology of China [No.
2022YFA1605500]; Office of Science and Technology, Shanghai Municipal
Government [No. 22JC1410100]; the Natural Science Foundation of China [No.
11773003, U1931201]; and the China Manned Space Project [No. CMS-
CSST-2021-B11].

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