Composite channel modeling for underwater optical wireless communication and analysis of multiple scattering characteristics

Linlin Kou; Jianlei Zhang; Pengwei Zhang; Yi Yang; Fengtao He

doi:10.1364/OE.483234

1. Introduction

Acquiring and transmitting marine data is essential for developing and utilizing marine resources [1]. Owing to the development of technologies such as underwater robots, frogman operations, mobile observation in unmanned underwater vehicles, and underwater sensor networks, technologies with high data–rate transmission and low latency are needed [2]. Underwater optical wireless communication (UOWC) is emerging as an effective solution for high–throughput and large–data underwater communications, owing to its tremendous advantages, such as higher bandwidth, lower time latency, and better security [3]. The UOWC channels are significantly affected by fading due to bubbles, turbulence, absorption, and scattering resulting in a limited effective communication range that is typically < 100 m [4,5]. Therefore, conducting a comprehensive study and developing accurate channel models that capture these degradation effects are crucial.

Extensive research has been conducted to design reliable and comprehensive UOWC systems for modeling the effects of one or more induced factors on the optical properties of underwater channels. For instance, a single–bubble scattering model [6] and a simulation model of optical properties based on the Mie theory were proposed for single– and multi–layer bubbles [7,8], and the scattering properties for the angles 82.7°, 106.2°, and 180° [8–10] were examined. Research has indicated that the movement of ships, the aquatic respiration and decay of marine life, and the breaking of waves can affect the emergence of various–sized bubbles with radii of approximately 10–500 µm in a specific region of the ocean. Accurately calculating the optical scattering characteristics of bubbles with increasing sizes via simulations based on the Mie theory becomes difficult and time–consuming, such bubbles can instead be analyzed via model simulations of geometric–optical approximation (GOA) [11]. Moreover, bubbles always exist as clusters or populations in the natural environment, and the scattering properties of single– and multi–layer bubble populations have been studied according to the Mie theory [7,12]. Unfortunately, the Mie theory and GOA are limited to analyzing the scattering characteristics of single–scattering bubbles or particles of a uniform size. When light is transmitted in a random medium, it undergoes multiple scattering due to particles of various sizes, thus, a channel model that can analyze the optical properties of particles of multiple sizes or different types of scattering is needed.

Accurate channel modeling is critical for effectively evaluating the performance of UOWC systems. Regarding research on UOWC, significant progress has been made both theoretically [13] and experimentally [14,15]. Studies have focused on the effects of particle absorption and scattering channels [16], system–level design and demonstration [17], and even underwater turbulence model simulations [18]. In comparison, research on channel simulation modeling for UOWC with bubble– and particle–induced attenuation is lacking, which was mainly focused of the present study. Moreover, channels generate refraction and reflection–like multiple scattering effects when the channel contains bubbles and particles of different sizes [19]. The effects of bubbles on the optical properties of channels have been investigated, and the fluctuations in the optical signal received in a UOWC channel with bubbles were analyzed [20]. A recent study developed a statistical model of underwater turbulence induced by air bubbles based on real experiments [21,22]. In those studies, the mixed exponential–Gamma distribution and exponential–lognormal distributions were proposed to model the statistics of received power.

In this study, a Monte Carlo algorithm was employed to develop a scattering model for a population containing microbubbles and particles. Specifically, the effects of microbubbles and particles on the optical characteristics of the medium were considered. We also developed a composite channel model for a population with both large and micro–sized bubbles and particles that uses the ray–tracing method to track the path of photons in large bubbles. The optical properties of the composite channel were analyzed under various conditions, including the attenuation coefficient of the ocean and the number and position of the bubbles. We compared the simulation results with experimental data obtained by Zhang [7,12] and Petzold [23] to verify the accuracy of the simulation. Furthermore, we examined the effects of large bubbles at different positions on the channel power at the receiver via simulations and experiments. The effects of the multiple scattering due to bubbles and particles on the optical properties of an optical channel were simulated for the first time in this study. The obtained simulation data and models are of considerable scientific value and practical significance for bubble detection and the development of comprehensive and accurate channel model.

2. Composite channel modeling for UOWC

Existing models for UOWC channels do not consider the multiple scattering effects caused by the combination of bubbles and particles. With both particles and bubbles in the channel, the scattering of the optical signal by the components is superimposed affecting the reliability of the channel. There are underwater bubbles of different sizes; accordingly, the scattering characteristics of microbubbles were investigated according to the Mie theory, and those of large bubbles were simulated via GOA. The scattering model for the microbubbles and particles in this composite channel model is based on the Monte–Carlo algorithm and that for large bubbles is based on ray tracing. The scattering model for an underwater composite channel that considers the effects of large bubbles, microbubbles, and particles on the optical properties of the channel is shown in Fig. 1.

Fig. 1. Scattering model for the underwater composite channel.

Download Full Size | PDF

2.1 Monte–Carlo–based composite channel model for UOWC

Monte–Carlo simulation employs four typical algorithms for modeling photon migration; the most widely used to the albedo–weight method, which is used to avoid computational waste, was used in this simulation [16]. A simplified flowchart outlining the typical Monte–Carlo algorithm and the corresponding composite channel scattering method is presented in Fig. 2.

Fig. 2. Flowchart of the Monte Carlo algorithm and the corresponding composite channel scattering method.

Download Full Size | PDF

A Gaussian beam was used to simulate the initialization conditions of the light beam. The probability density function (PDF) p(r₀)_PDF and cumulative distribution function (CDF) p(r₀)_CDF of a Gaussian distribution in terms of the beam radius r₀ and beam waist w₀ are expressed as follows [24,25]:

(1)$$\left\{ \begin{array}{l} p{({{r_0}} )_{PDF}} = \frac{{{e^{({ - {r_0}^2/w_0^2} )}}}}{{{w^2}_0}} \cdot 2{r_0}\\ p{({{r_0}} )_{CDF}} = 1 - \exp \left[ { - \frac{{{r_0}^2}}{{w_0^2}}} \right]. \end{array} \right.\textrm{ }$$

Setting p(r₀) is a random number R and obeys a uniform distribution in the range (0,1), the solution to Eq. (1) can be obtained as:

(2)$${r_0} = {w_0}\sqrt { - \ln ({1 - R} )} \textrm{ }.$$

The initial direction of the photon needs to be obtained from the initial divergence angle θ₀ and azimuthal angle φ₀; it can be calculated as follows [26]:

(3)$$\left\{ \begin{array}{l} f ={-} {w_0}/\phi \\ {\theta_0} ={-} {r_0}/f \end{array} \right.\textrm{ },$$

where $\phi $ represents the divergence half–angle, f represents the focal length of the lens, and w₀ is the beam waist, which is selected according to the type of light source used. The azimuthal angle φ₀ is expressed as:

(4)$${\varphi _0} = 2\pi R\textrm{ },$$

where the radial angle φ₀ is chosen to be normally distributed over [0,2π], and R is a random number in the range of [0,1]. The initial location coordinates (x₀, y₀, z₀) and direction cosines $({{\mu_{{x_0}}},{\mu_{{y_0}}},{\mu_{{z_0}}}} )$ of each photon are calculated as follows:

(5)$$\left\{ \begin{array}{l} ({{x_0},{y_0},{z_0}} )= ({{r_0}\cos {\varphi_0},{r_0}\sin {\varphi_0},0} )\\ ({{\mu_{{x_0}}},{\mu_{{y_0}}},{\mu_{{z_0}}}} )= ({\cos {\varphi_0}\sin {\theta_0},\sin {\varphi_0}\sin {\theta_0},\cos {\theta_0}} ).\end{array} \right.\textrm{ }$$

The distance that of each photon moves from the transmitter to the receiver is determined by the scattering path in the channel, which comprises a few paths. After photon initialization, the step path of each photon s is given as:

(6)$$s ={-} \frac{{\ln (R )}}{{{\mu _t}}},$$

where µ_t is the attenuation coefficient of water, which is the sum of the absorption coefficient µ_a and scattering coefficient µ_s, and R is a random number that follows the uniform distribution and belongs to the range of (0,1). After scattering, the weights W₁ and positions (x₁, y₁, z₁) of the photons are updated as follows:

(7)$$\left\{ \begin{array}{l} {W_1} = {W_0} \cdot {W_{\textrm{albedo}}}\\ ({{x_1},{y_1},{z_1}} )= ({{x_0},{y_0},{z_0}} )+ ({{\mu_{{x_0}}},{\mu_{{y_0}}},{\mu_{{z_0}}}} )\cdot s \end{array} \right.,$$

where W_albedo=µ_s /µ_t is the scattering albedo, which is defined as the ratio of the scattering loss to the total loss, and W₀ is the initial weight, which is set as 1. The propagation direction of each photon is determined by the position (x_m–1, y_m–1, z_m–1) and direction cosines $({{\mu_{{x_m}}},{\mu_{{y_m}}},{\mu_{{z_m}}}} )$ obtained after scattering occurs m times, it is defined as follows [27]:

(8)$$\left[ \begin{array}{@{}l@{}} {\mu_{{z_m}}}\\ {\mu_{{y_m}}}\\ {\mu_{{z_m}}} \end{array} \right] = \left\{ \begin{array}{@{}l@{}} \left[ {\begin{array}{{@{}ccc@{}}} {{\mu_{{x_{m - 1}}}}{\mu_{{z_{m - 1}}}}/\sqrt {1 - {\mu^2}_{{z_{m - 1}}}} }&{ - {\mu_{{y_{m - 1}}}}/\sqrt {1 - {\mu^2}_{{z_{m - 1}}}} }&{{\mu_{{x_{m - 1}}}}}\\ {{\mu_{{y_{m - 1}}}}{\mu_{{z_{m - 1}}}}/\sqrt {1 - {\mu^2}_{{z_{m - 1}}}} }&{{\mu_{{x_{m - 1}}}}/\sqrt {1 - {\mu^2}_{{z_{m - 1}}}} }&{{\mu_{{y_{_{m - 1}}}}}}\\ { - \sqrt {1 - {\mu^2}_{{z_{m - 1}}}} }&0&{{\mu_{{z_{m - 1}}}}} \end{array}} \right] \cdot \left[ {\begin{array}{{@{}c@{}}} {\sin \theta \cos \varphi }\\ {\sin \theta \sin \varphi }\\ {\cos \theta } \end{array}} \right],\textrm{ }{\mu_{{z_{m - 1}}}} < 1\\ sign({{\mu_{{z_{m - 1}}}}} )\left[ {\begin{array}{{@{}c@{}}} {\sin \theta \cos \varphi }\\ {\sin \theta \sin \varphi }\\ {\cos \theta } \end{array}} \right]\qquad \quad \textrm{ },\textrm{ }{\mu_{{z_{m - 1}}}} \approx 1 \end{array} \right.\textrm{ }\textrm{.}$$

Because the scattering direction of the photons is determined by the scattering angle θ and azimuthal angle φ, the azimuthal angles φ of all the photons were assumed to be uniformly distributed over the interval [0,2π]. The scattering angle θ can be estimated using the total volume scattering function (VSF) β_diff(θ) of the composite channel, which is expressed as [28]:

(9)$${\beta _{\textrm{diff}}}(\theta )= {\beta _{\textrm{s\_par}}}(\theta )\textrm{ + }{\beta _{\textrm{s\_bub}}}(\theta )= {\mu _\textrm{s}} \cdot {\overline \beta _{\textrm{s\_par}}}(\theta )+ {\mu _{\textrm{s\_bub}}} \cdot {\overline \beta _{\textrm{s\_bub}}}(\theta ),$$

where µ_s and µ_{s_bub} are the scattering coefficients of the particles and bubbles, respectively, and ${\overline \beta _{\textrm{s\_par}}}(\theta )$ and ${\overline \beta _{\textrm{s\_bub}}}(\theta )$ are the scattering phase functions of the particles and bubbles, respectively. The scattering measurements performed by Petzold [23] focused on the absorption and scattering of particles in water, as shown in Table 1. In our simulations, parameters such as the absorption coefficient µ_a, scattering coefficient µ_s, and attenuation coefficient µ_t of the particles were approximated according to the measurements of Petzold.

Table 1. Measured parameters for different types of water

View Table | View all tables in this article

The Henyey–Greenstein (HG) phase function ${\overline \beta _{\textrm{s\_par}}}(\theta )$, which is used to describe the scattering angle distribution of photons following a collision with particles, which is expressed as [1]:

(10)$${\overline \beta _{\textrm{s\_par}}}(\theta )= \frac{1}{{4\pi }}\frac{{1 - {g^2}}}{{{{({1 - {g^2} + 2g\cos \theta } )}^{\frac{3}{2}}}}},$$

where $g = \overline {\cos \theta } $ is the anisotropy factor and the mean cosine of the scattering angle, which ranges from –1 to 1. When g = 1, there is complete forward scattering, and when g=–1, there is complete backscattering.

The optical properties of the bubble population were calculated as follows [6]:

(11)$${\beta _{\textrm{s\_bub}}}(\theta )\textrm{ = }\int_{{r_{\min }}}^{{r_{\max }}} {{Q_\beta }({\theta ,r} )} \pi {r^2}n(r )dr\textrm{ ,}$$

where β_diff(θ) is the VSF of the bubble population (m^–1sr^–1); ${Q_\beta }({\theta ,r} )$ represents the scattering efficiency per unit solid angle in the direction θ for a particle of size r, which was calculated using the Mie theory; n(r) represent the bubble size distribution, the number of bubbles per unit volume per unit radius interval with radius r(m^–3µm^–1), and r_max and r_min represent the maximum and minimum radii in the bubble size distribution, respectively. By integrating ${\beta _{\textrm{s\_bub}}}(\theta )$ over all the angles, we obtained the following expression for the scattering coefficient µ_{s_bub}(m^–1):

(12)$${\mu _{\textrm{s\_bub}}}\textrm{ = }\int_{4\pi } {{\beta _{\textrm{s\_bub}}}(\theta )d\theta } = 2\pi \int_{\theta = 0}^{2\pi } {{\beta _{\textrm{s\_bub}}}(\theta )\sin (\theta )d\theta } ,$$

where the scattering phase function ${\overline \beta _{\textrm{s\_bub}}}(\theta )$ is a normalized VSF that describes the angular probability of scattering as a PDF [26]:

(13)$${\overline \beta _{\textrm{s\_bub}}}(\theta )\textrm{ = }\frac{{{\beta _{\textrm{s\_bub}}}(\theta )}}{{{\mu _{\textrm{s\_bub}}}}}.$$

Since it was difficult to analytically obtain the inverse function of the total scattering phase function β_diff(θ), we applied the rejection sampling technique to appropriately generate the scattering angle θ in our simulations. The scattering angle θ for the composite channel was obtained via rejection sampling, as shown in Fig. 3.

Fig. 3. Rejection sampling.

Download Full Size | PDF

The parameter M was introduced, the helper sampling function q(x) was chosen, a random sample x_i was produced satisfying $\forall Mq({{x_i}} )\ge p({{x_i}} )$, and a random number u that following the uniform distribution range of [0,1] was chosen, if u_i < p(x_i)/Mq(x_i) was satisfied the sample was accepted, otherwise, it was rejected. These steps were repeated several times until a random sample that fit the p(x) distribution was obtained. In this simulation, the HG function ${\overline \beta _{\textrm{s\_par}}}(\theta )$ was selected as the known distribution q(x), the total VSF β_diff(θ) was p(x), and M was 100.

The bubble size distribution $n(r )$ in Eq. (11) is expressed as:

(14)$$n(r )= {N_0} \cdot p(r ),$$

where N₀ represents the bubble number density in a unit volume of water (m^–3), and p(r) represents the PDF at the radius. The bubble size distributions were initially measured using photographic [29], holographic [30], optical inversion [31], and acoustic backscattering [32] methods, and the photographic method revealed that the distribution of the bubble populations followed a normal distribution with peaks between approximately 40 and 100 µm. In some studies, this variation was attributed to the presence of a flat peak region in the region with bubbles of medium radii, with rapid shrinkage of the large and small radius regions at both terminals. The distribution of a bubble population can be expressed as follows [29,33]:

(15)$$p(r )= \left\{ \begin{array}{l} {c_1} \cdot {r^4}\textrm{ }\qquad 0 \le {r_a}\\ {c_2}\textrm{ }\qquad{r_a} < r < {r_b}\\ {c_3} \cdot {r^{ - 4}}\textrm{ }\qquad{r_b} \le r \end{array} \right.,$$

where r_a and r_b represent the radii of the bubbles that define the limits of the plateau, and c₁, c₂, and c₃ are uniquely determined by r_a and r_b. In this simulation, c₁, c₂, and c₃ were 6.25 × 10^–10, 0.015, and 1.5 × 10^–6, respectively, and r_a and r_b were 70 and 100 µm, respectively.

2.2 Scattering modeling of large bubbles, microbubbles, and particles based on ray tracing

The large–bubble scattering model is based on geometrical optics; thus, it is suitable for Monte–Carlo ray tracing. The algorithmic process of the model is briefly described as follows. The scattering path of each photon from the transmitter (Tx) to the receiver (Rx) can be considered a virtual light ray. When the light intersects a large bubble, each photon is divided into several new photons with different directions and weights. The scattering direction and weight of each photon are determined by Fresnel's theory and Snell's law. The photons are tracked, and their parameters are calculated until they reach the receiver (Rx). The previous step is repeated until all the preset photons are scattered. Flowchart of the composite channel for particles, microbubbles, and large bubbles is shown in Fig. 4.

Fig. 4. Flowchart of the composite channel with particles, microbubbles, and large bubbles.

Download Full Size | PDF

The scattering properties of large bubbles are the basis of GOA, and when photons are scattered by large bubbles, they are reflected and refracted on the bubble surfaces, as shown in Fig. 5, where p represent the interaction time between the photon and bubble, Snell’s law determines how these reflected and refracted photons travel. A similar phenomenon occurred inside each bubble, part of the photon energy was refracted out of the bubble and another part was reflected many times on the inner surface of the bubble. As p increased, the weight of photon inside the bubble approached an infinitesimal value. The scattering path of the photons is shown in Fig. 5.

Fig. 5. Optical paths of light scattering due to bubbles.

Download Full Size | PDF

The directional vector of the reflected light $({{\mu_{fx}},{\mu_{fy}},{\mu_{fz}}} )$ can be expressed as follows:

(16)$$\left\{ \begin{array}{l} {\mu_{fx}} = {\mu_{{x_i}}} - 2 \cdot \left( {{\mu_{{x_i}}} \cdot \frac{{X - {a_{sphere}}}}{{{S_r}}}} \right) \cdot \frac{{X - {a_{sphere}}}}{{{S_r}}}\\ {\mu_{fy}} = {\mu_{{y_i}}} - 2 \cdot \left( {{\mu_{{y_i}}} \cdot \frac{{Y - {b_{sphere}}}}{{{S_r}}}} \right) \cdot \frac{{Y - {b_{sphere}}}}{{{S_r}}}\\ {\mu_{fz}} = {\mu_{{z_i}}} - 2 \cdot \left( {{\mu_{{z_i}}} \cdot \frac{{Z - {c_{sphere}}}}{{{S_r}}}} \right) \cdot \frac{{Z - {c_{sphere}}}}{{{S_r}}} \end{array} \right.,$$

where (a_sphere, b_sphere, c_sphere) and S_r are the center coordinates and radius of each bubble, respectively, with the spatial distribution of the large bubbles being determined by their number, position, and size; (X, Y, Z) represents the intersection of the bubbles and photons step length, and $({{\mu_{{x_i}}},{\mu_{{y_i}}},{\mu_{{z_i}}}} )$ represents the incident direction. The incident light p = 0 was decomposed into rays characterized by a scattering order p = 1, 2, 3, …, and the corresponding complementary incident θ_i and refraction angles θ_r are related by Snell's law as follows:

(17)$${\theta _r} = \arcsin ({m \cdot \sin {\theta_i}} ),$$

where m is the relative refractive index, and the refractive indices of water and a bubble are 1.33 and 1, respectively. The refracted light (µ_rx, µ_ry, µ_rz) is calculated as follows:

(18)$$\left( {\begin{array}{{ccc}} {{\mu_{{x_i}}}}&{{\mu_{{y_i}}}}&{{\mu_{{z_i}}}}\\ {{n_x}}&{{n_y}}&{{n_z}}\\ {{\mu_{rx}}}&{{\mu_{ry}}}&{{\mu_{rz}}} \end{array}} \right) = 0,$$

(19)$${n_x} \cdot {\mu _{rx}} + {n_y} \cdot {\mu _{ry}} + {n_z} \cdot {\mu _{rz}} = \cos {\theta _r},$$

(20)$$\mu_x^{{\prime}2} + \mu_y^{{\prime}2} + \mu_z^{{\prime}2} = 1\textrm{ }.$$

When the light was incident on a large bubble, GOA was used to determine whether a photon was scattered by a bubble to simulate the interactions with the photons and bubbles. Fresnel’s theory was used to adjust the weight of each photon as it was scattered by each bubble, which is given as follows:

(21)$$\left\{ \begin{array}{l} {R_s} = {\left[ {\frac{{\sin ({{\theta_i} - {\theta_r}} )}}{{\sin ({{\theta_i} + {\theta_r}} )}}} \right]^2}\\ {R_p} = {\left[ {\frac{{\tan ({{\theta_i} - {\theta_r}} )}}{{\tan ({{\theta_i} + {\theta_r}} )}}} \right]^2}. \end{array} \right.\textrm{ }$$

The refracted weight W_T and reflected weight W_R are expressed as follows:

(22)$$\left\{ \begin{array}{l} {W_R} = W \cdot R = W \cdot \left( {\frac{1}{2} \cdot ({{R_s} + {R_p}} )} \right)\\ {W_T} = W \cdot T = W \cdot ({1 - R} )\end{array} \right.\textrm{ }.$$

After a few moves, the reflected weight W_R decreased to low levels that were no longer beneficial for the simulations even if further calculations were performed. Considering the calculation efficiency and accuracy, we set the weight threshold to 10^–7 for this simulation.

3. Results and discussion

We investigated the scattering characteristics for a population of microbubbles, particles, and for a population of large bubbles and particles, to quantify the effects of both types of bubbles on the channel. In this section, we discuss the inherent optical properties of the composite channel for different densities, locations, and water types (the parameters for different types are presented in Table 1) and analyze the simulation results.

3.1 Performance of the model for microbubbles and particles

The VSFs of the measured and simulated data were compared, and the results are shown in Fig. 6.

Fig. 6. VSF for three types of water measured by Petzold, experimental values measured by Zhang, the microbubble population, and simulation data.

Download Full Size | PDF

The simulated data obtained from the experiment of [12], wherein the VSF in the North Atlantic was tested at high wind speeds on 26 July, were used as a control group. The bubble size distribution model used in the simulation was that described in Eq. (2), the bubble radius was 30–300 µm, the bubble number density was N₀= 9.1 × 10⁴, and the attenuation coefficient for the composite channel measured by Zhang was 5.16 × 10^–4, in simulation is 4.143 × 10^–4, as shown in [7]. The VSFs for three common types of water deep clear ocean (Tongue of the Ocean), coastal ocean (off Southern California), and harbor water (San Diego Harbor)— were tested using data from Petzold [23]. Because the North Atlantic is a deep ocean, and the scattering coefficients of the water taken from the clear ocean (I)–type water (Table 1), and the VSF of the particles in the simulation was the HG scattering phase function (g = 0.924). As shown in Fig. 6, the simulated data of the total VSF of the composite channel (yellow line) were close to the measured data of the total VSF (black point). However, the deviations were larger around small angles in the range of 0°–20° and large angles in the range of 150°–180°, owing to the VSF of the particles in the simulation by the HG function, which deviated from the measured values The two–term Henyey–Greenstein (TTHG) scattering phase function [34] and Fournier–Forand (FF) scattering phase function [35] have been proposed for addressing this issue. Additionally, the attenuation coefficient of the measured water was unknown, therefore, there were deviations between the simulated and measured data.

Figure 7 shows the scattering phase function for different number densities, N₀ = 4 × 10⁴, 7.5 × 10⁵, 3 × 10⁶, and 1 × 10⁷, along with the HG function (g = 0.924) and the phase function of the bubble population wherein the bubble sizes ranged between 30 and 300 µm. The ordinate is presented on a logarithmic scale for better visualization of the differences between the phase functions for small values of θ. With an increase in the number of bubbles, the scattering phase function of the composite channel approached that of the bubble population. As the number of bubbles increased, the VSF of composite channel approaches the VSF of the bubbles. This is because as the number of bubbles increases, their contribution to the optical properties of the channel increases. The prominent peak of the VSF observed at the critical scattering angle of 82.7° becomes evident after the emergence of microbubbles, which agrees with the geometrical optics conclusion that this peak is caused by total reflection. Additionally, the peak of the bubble critical angle reflects the property that the relative refractive index of bubbles in water is < 1, which can be used to distinguish the existence of bubbles.

Fig. 7. Scattering phase function of the composite channel for different number densities and water types.

Download Full Size | PDF

The optical properties of the channel were further investigated. Table 2 represent the attenuation coefficient µ_{t_com} (m^–1) and albedo W_albedo of the composite channel for four number densities N₀ and three water types of. According to the bubble size distribution measured by O’Hern [30], the number density of bubbles is between 4 × 10⁴ and 2.17 × 10⁷ at high and low wind speeds.

Table 2. Attenuation coefficient of composite channel µ_{t_com} and albedo of the composite channel W_albedo

View Table | View all tables in this article

Figure 8 shows the simulated channel impulse response (CIR) for the clear ocean (I)–type, coastal ocean (II)–type, and harbor waters (III)–type waters, and the distance is 20 m. The different types of waters exhibited similar behaviors of the delay profile, whereas the CIR for the harbor water (III) exhibited the highest level of delayed photons. Light was subjected to the effects of multiple scattering, which causes the photons to arrive at the receiver at different times, resulting in temporal dispersion and a penalty in the channel bandwidth. With an increase in the number of bubbles, more photon scattering occurs, resulting in a greater delay in the CIR. As the attenuation coefficient increased, the scattering step length of each photon decreased, the number of scattering events increased, and the total length of photons traveling in the channel increased. The attenuation coefficients of clear ocean(I)–type water and harbor water (III)–type were 0.1636 and 2.2026, respectively, for a bubble number density of 1 × 10⁷, indicating that the distance traveled by photons decreased by 0.5 and 0.01 m, respectively, for each scattering on average. As the number of scatterings increased, the scattering pathlength also increased, thus, a larger number of bubbles corresponded to a higher CIR delay.

Fig. 8. CIRs for (a) clear ocean (I)–type water; (b) coastal ocean (II)–type water; and harbor (III)–type water.

Download Full Size | PDF

Figure 9 presents a comparison of the beam spread in the receiver for the clear ocean (I)–type and coastal ocean (II)–type with an emitted beam divergence of 0.0049 rad, a beam width of 0.0078 m, and 2 × 10⁵ photons, the channel distance is 10 m. With an increase in the number of bubbles, the beam spread extended and the light spot boundary became fuzzier and the received power is decrease regarding the clear ocean (I) and coastal ocean (II)–types water.

Fig. 9. Beam spread in clear ocean (I)–type and coastal ocean (II)–type waters for different number densities (${N_0}$) and a channel distance of 10 m.

Download Full Size | PDF

3.2 Scattering characteristics of large bubbles and particles

We evaluated the composite channel model for large bubbles at different locations and numbers via simulations and experiments. Figure 10 shows the experimental setup for the received optical power test with the large bubble at different position. The length of the water tank was L = 0.5 m, and the bubble radius was approximately d = 2.5, 3.5 and 5 mm. The test bubbles were generated by an air pump with a tube with air holes. The sizes of these holes were approximately 2.5, 3.5, and 5 mm, and the bubbles were positioned in the water at distance of 0.1, 0.2, 0.3 and 0.4 m from the water tank, Table 3 represents the parameters of the experimental equipment.

Fig. 10. Experimental setup for the received optical power test with the bubbles at different positions.

Download Full Size | PDF

Table 3. Parameters of the experimental equipment

View Table | View all tables in this article

The bubble changed in shape from spherical to spheroid after rising because of the friction. As bubble movement was complex, we assumed that the large bubbles were spherical in our calculations. The rising velocity of the bubble v (m/s), is given as follows [22]:

(23)$$v = \left\{ \begin{array}{l} \frac{{g \cdot \rho \cdot R_i^2}}{{3\mu }},\qquad \quad \quad \quad \quad \quad \quad \mathrm{\ 0\ < }{R_i} < 0.08015\textrm{ mm}\\ 0.408 \cdot {g^{\frac{5}{6}}} \cdot {\left( {\frac{\rho }{\mu }} \right)^{\frac{2}{3}}}{R_i^{\frac{2}{3}}}, \quad \textrm{ }0.08015 \le {R_i} < 0.575\textrm{ mm}\\ \sqrt {\frac{{1.07 \cdot {\sigma_s}}}{{\rho \cdot {R_i}}} + 1.01 \cdot g \cdot {R_i}} , \quad \textrm{ }{R_i} \ge 0.575\textrm{ mm} \end{array} \right.,$$

where R_i represents the bubble radius (m), ρ represents the density of the water, σ_s represents the surface tension of the water, µ represents the viscosity of the water, and g is the gravitational acceleration. In the experiment, the light beam was 0.15 m from the bottom of the tank, and the bubble velocities were 0.236, 0.238, and 0.2551 m/s for bubble radii of 2.5, 3.5, and 5 mm, respectively.

Figure 11 shows the effects of bubbles with radii of 2.5, 3.5, and 5 mm on the received power at distances of 0.1, 0.2, 0.3, and 0.4 m from the water tank, where “sim” (solid line) represents simulated data and “exp” (dashed line) denotes experimental data. We selected clear ocean (I)–type from Table 1 for the simulation, which has a weaker attenuation coefficient and a smallest particle attenuation effect for the entire channel. In fact, since the bubbles get larger as they rose, implying that the bubbles reaching the test point were larger than the original size, resulting in the simulation results for thereceived power exceeding the test results. In addition, both simulated and experimental data show the trend that the closer the bubble is to the receiving end, the more power it can receive, which is accord with the GOA. Therefore, bubbles could not simply simulate as aperture–obstructed models.

Fig. 11. Received power for different positions of the bubble in the channel.

Download Full Size | PDF

Figure 12 shows the effects of the distance L from the bubble center to the beam center and radii of the bubbles on the received power. When the bubbles were located at the same position, the effect of the radius on the received power was more pronounced. The received power at the receiver was approximately 73% of the total power when the bubble size was 7 mm and the bubble was 1 mm from the beam center, and it was approximately 79% of the total power when the bubble was 7 mm from the beam center. The received power decreased as the bubble radius increased and the distance from the bubble to the beam center position decreased, when the channel distance was being determined. The scattering model for large bubbles can accurately simulate the effects of different locations, numbers, and sizes of bubbles on the received power fluctuations.

Fig. 12. Received power for different positions and radii of the bubble in the channel.

Download Full Size | PDF

The relative power for different bubble numbers and distances L is shown in Fig. 13. The bubbles significantly degraded the performance of the UOWC links. The optical intensity decreased rapidly to 0 as the number of bubbles increased. The received power decreased as the density of bubbles increased, increasing the probability of photon scattering due to bubbles. In addition, and the photon scattering time and optical intensity decreased with an increase in the number of bubbles, and the optical intensity decreases slowly, thus, a larger number of bubbles that can be penetrated by light corresponds to a longer transmission distance.

Fig. 13. Received power for varying positions of the bubble in the channel.

Download Full Size | PDF

4. Conclusion

We modeled the optical properties of an UOWC channel containing both bubbles and particles. Simulations indicated that the number of bubbles significantly affects, particularly in waters with small attenuation. Our experimental and simulation results indicated that the optical properties of the composite channel model closely matched empirical data. These findings provide valuable references for link design in subsequent UOWC applications. Moreover, our study highlights the distinct effects of bubble– and particle–induced multiple scattering on the optical properties of the channel. It is crucial to consider the effects of both bubbles and particles on the optical performance of UOWC channels in practical applications.

Funding

Joint Fund of the Ministry of Education for Equipment Pre-research (8091B032130); National Natural Science Foundation of China (61805199); Shaanxi Provincial Technology Innovation Guidance Special Fund (2020TG-001); Foundation from the State Key Laboratory of Underwater Information and Control.

Disclosures

The authors declare no conflicts of interest.

Data availability

Data in the study are available upon request.

References

1. R. M. G. Kraemer, L. M. Pessoa, and H. M. Salgado, Monte–Carlo Radiative Transfer Modeling of Underwater Channel (IntechOpen, 2020).

2. S. McPhail, “Autosub6000: A deep diving long range AUV,” J. Bionic Eng. 6(1), 55–62 (2009). [CrossRef]

3. W. H. W. Hassan, M. S. Sabril, F. Jasman, and S. M. Idrus, “Experimental study of light wave propagation for Underwater Optical Wireless Communication (UOWC),” J. Commun 17, 23–29 (2022). [CrossRef]

4. H. Kaushal and G. Kaddoum, “Underwater optical wireless communication,” IEEE Access 4, 1518–1547 (2016). [CrossRef]

5. F. Akhoundi, M. V. Jamali, N. B. Hassan, H. Beyranvand, A. Minoofar, and J. A. Salehi, “Cellular underwater wireless optical CDMA network: Potentials and challenges,” IEEE Access 4, 4254–4268 (2016). [CrossRef]

6. H. C. Hulst and H. C. van de Hulst, Light Scattering by Small Particles (Courier Corporation, 1981).

7. X. Zhang, M. Lewis, and B. Johnson, “Influence of bubbles on scattering of light in the ocean,” Appl. Opt. 37(27), 6525–6536 (1998). [CrossRef]

8. L. M. Philip, C. B. Stuart, and E. D. Cleon, “Scattering of Light by a coated bubble in water near the critical and brewster scattering angles,” in Proc.SPIE (1988), pp. 308–316.

9. P. L. Marston, “Critical angle scattering by a bubble: physical–optics approximation and observations,” J. Opt. Soc. Am. 69(9), 1205–1211 (1979). [CrossRef]

10. D. S. Langley and P. L. Marston, “Glory in optical backscattering from air bubbles,” Phys. Rev. Lett. 47(13), 913–916 (1981). [CrossRef]

11. G. E. Davis, “Scattering of light by an air bubble in water,” J. Opt. Soc. Am. 45(7), 572–581 (1955). [CrossRef]

12. X. Zhang, M. Lewis, M. Lee, B. Johnson, and G. Korotaev, “The volume scattering function of natural bubble populations,” Limnol. Oceanogr. 47(5), 1273–1282 (2002). [CrossRef]

13. C. D. Mobley, Light and Water: Radiative Transfer in Natural Waters (Academic Press, 1994).

14. L. J. Johnson, F. Jasman, R. J. Green, and M. S. Leeson, “Recent advances in underwater optical wireless communications,” Underw. Technol. 32(3), 167–175 (2014). [CrossRef]

15. H. M. Oubei, C. Shen, A. Kammoun, E. Zedini, K–H Park, X. Sun, G. Liu, C. H. Kang, T. K. Ng, M–S Alouini, and B. S. Ooi, “Light based underwater wireless communications,” Jpn. J. Appl. Phys. 57(8S2), 08PA06 (2018). [CrossRef]

16. A. Sassaroli and F. Martelli, “Equivalence of four Monte–Carlo methods for photon migration in turbid media,” J. Opt. Soc. Am. A 29(10), 2110–2117 (2012). [CrossRef]

17. S. Arnon, “Underwater optical wireless communication network,” Opt. Eng 49(1), 015001 (2010). [CrossRef]

18. C. T. Geldard, J. Thompson, and W. O. Popoola, “Effects of turbulence induced scattering on underwater optical wireless communications,” arXiv, arXiv:2008.01152 (2020). [CrossRef]

19. D. Exerowa and P. M. Kruglyakov, Foam and Foam Films: Theory, Experiment, Application (Elsevier, 1997).

20. D. Chen, J. Wang, S. Li, and Z. Xu, “Effects of air bubbles on underwater optical wireless communication,” Chin. Opt. Lett. 17(10), 100008 (2019). [CrossRef]

21. M. V. Jamali, A. Mirani, A. Parsay, B. Abolhassani, P. Nabavi, A. Chizari, P. Khorramshahi, S. Abdollahramezani, and J. A. Salehi, “Statistical studies of fading in underwater wireless optical channels in the presence of air bubble, temperature, and salinity random variations,” IEEE Trans. Commun. 66, 1 (2018). [CrossRef]

22. M. Shin, K–H Park, and M–S Alouini, “Statistical modeling of the impact of underwater bubbles on an optical wireless channel,” IEEE Open J. Commun. Soc. 1, 808–818 (2020). [CrossRef]

23. T. J. Petzold, “Volume scattering functions for selected ocean waters,” La Jolla Ca Visibility Lab 72–78 (1972).

24. R. Qadar, M. K. Kasi, and F. A. Kakar, “Monte–Carlo based estimation and performance evaluation of temporal channel behavior of UOWC under multiple scattering,” in OCEANS 2017 – Anchorage (IEEE, 2017), pp. 1–7.

25. Z. Du, W. Ge, and G. Song, “Partially pruned DNN coupled with parallel Monte–Carlo algorithm for path loss prediction in underwater wireless optical channels,” Opt. Express 30(8), 12835–12847 (2022). [CrossRef]

26. W. C. Cox Jr, “Simulation, Modeling, and Design of Underwater Optical Communication Systems,” (North Carolina State University, 2012).

27. L. Wang and S. L. Jacques, “Monte–Carlo modeling of light transport in multi–layered tissues in standard C,” The University of Texas, MD Anderson Cancer Center, Houston 4, 2304–2314 (1992).

28. D. Stramski, A. Bricaud, and A. Morel, “Modeling the inherent optical properties of the ocean based on the detailed composition of the planktonic community,” Appl. Opt. 40(18), 2929–2945 (2001). [CrossRef]

29. B. Johnson and R. Cooke, “Bubble populations and spectra in coastal waters: A photographic approach,” J. Geophys. Res. 84(C7), 3761–3766 (1979). [CrossRef]

30. T. O’Hern, L. d’AGOAstino, and A. Acosta, “Comparison of holographic and Coulter counter measurements of cavitation nuclei in the ocean,” J. Fluids Eng. 110(2), 200–207 (1988). [CrossRef]

31. M. V. Su, D. Todoroff, and J. Cartmill, “Laboratory comparisons of acoustic and optical sensors for microbubble measurement,” J. Atmos. Oceanic Technol. 11(1), 170–181 (1994). [CrossRef]

32. A. Pandit, J. Varley, R. Thorpe, and J. Davidson, “Measurement of bubble size distribution: an acoustic technique,” Chem. Eng. Sci. 47(5), 1079–1089 (1992). [CrossRef]

33. G. Crawford and D. Farmer, “On the spatial distribution of ocean bubbles,” J. Geophys. Res. 92(C8), 8231–8243 (1987). [CrossRef]

34. V. I. Haltrin, “One–parameter two–term Henyey–Greenstein phase function for light scattering in seawater,” Appl. Opt. 41(6), 1022–1028 (2002). [CrossRef]

35. G. R. Fournier and J. L. Forand, “Analytic phase function for ocean water,” Proc. SPIE 2258, 1–8 (1994). [CrossRef]

Type of water	Absorption coefficient $μ_{a} (m^{- 1})$	Scattering coefficient $μ_{s} (m^{- 1})$	Attenuation coefficient $μ_{t} (m^{- 1})$	Albedo $W_{albedo}$
Clear ocean (I)	0.114	0.037	0.151	0.245
Coastal ocean (II)	0.179	0.219	0.398	0.731
Harbor water (III)	0.366	1.824	2.190	0.833

	Attenuation coefficient of the composite channel $μ_{t\_com} (m^{- 1})$				Albedo $W_{albedo}$
	$4 \times 10^{4}$	$7.5 \times 10^{5}$	$3 \times 10^{6}$	$1 \times 10^{7}$	$4 \times 10^{4}$	$7.5 \times 10^{5}$	$3 \times 10^{6}$	$1 \times 10^{7}$
Clear ocean (I)	0.1513	0.1521	0.1548	0.1636	0.2453	0.2497	0.2634	0.3030
Coastal ocean (II)	0.3984	0.3989	0.4018	0.4106	0.5503	0.5513	0.5545	0.5640
Harbor water (III)	2.1901	2.1909	2.1938	2.2026	0.8329	0.8329	0.8332	0.8338

Parameters	value
Water tank	0.2 × 0.3 × 0.5 m³
Wavelength	520 nm
Transmitted power	2 mW
Bubble radius ( $R_{i}$ )	2.5, 3.5 and 5 mm
Distance between laser diode and bottom of water tank	0.5 m
Air pump flow rate	1.8 L/ min
Optical power meter	Thorlabs – S142C Integrating Sphere Photodiode Power Sensor

Type of water	Absorption coefficient $μ_{a} (m^{- 1})$	Scattering coefficient $μ_{s} (m^{- 1})$	Attenuation coefficient $μ_{t} (m^{- 1})$	Albedo $W_{albedo}$
Clear ocean (I)	0.114	0.037	0.151	0.245
Coastal ocean (II)	0.179	0.219	0.398	0.731
Harbor water (III)	0.366	1.824	2.190	0.833

	Attenuation coefficient of the composite channel $μ_{t\_com} (m^{- 1})$				Albedo $W_{albedo}$
	$4 \times 10^{4}$	$7.5 \times 10^{5}$	$3 \times 10^{6}$	$1 \times 10^{7}$	$4 \times 10^{4}$	$7.5 \times 10^{5}$	$3 \times 10^{6}$	$1 \times 10^{7}$
Clear ocean (I)	0.1513	0.1521	0.1548	0.1636	0.2453	0.2497	0.2634	0.3030
Coastal ocean (II)	0.3984	0.3989	0.4018	0.4106	0.5503	0.5513	0.5545	0.5640
Harbor water (III)	2.1901	2.1909	2.1938	2.2026	0.8329	0.8329	0.8332	0.8338

Composite channel modeling for underwater optical wireless communication and analysis of multiple scattering characteristics

Abstract

1. Introduction

2. Composite channel modeling for UOWC

2.1 Monte–Carlo–based composite channel model for UOWC

2.2 Scattering modeling of large bubbles, microbubbles, and particles based on ray tracing

3. Results and discussion

3.1 Performance of the model for microbubbles and particles

3.2 Scattering characteristics of large bubbles and particles

4. Conclusion

Funding

Disclosures

Data availability

References

Data availability

Cited By

Figures (13)

Tables (3)

Equations (23)

Optics Express