Underwater wireless optical communication (UWOC) has come to the forefront as a promising alternative technology to communicate under water with high data rates over relatively medium transmission spans [1–3]. This rapid emergence of UWOC is enabled by high-performing visible light-emitting diodes) and laser diodes (LDs) that have the lowest attenuation in seawater [4,5]. However, the aquatic environment is optically very challenging. One of the key factors that significantly affects the propagation of optical signals in seawater is the underwater optical turbulence (UOT). UOT arises from the refractive index fluctuations due to temperature, salinity, and density variations in the underwater environment [6]. These variations in the refractive index along the propagation path translate into large fluctuations (scintillations) in the intensity of the optical signal at the receiver which ultimately degrade the performance of UWOC links and image quality of underwater imaging systems [7,8].

Ocean temperature frontal zones (TFZs) such as the Antarctic Circumpolar Current, Gulf Stream, Kuroshio, and Labrador Currents where the mean gradient of the sea surface temperature (SST) varies between 0.1 and $1.5°\mathrm{C}·{\mathrm{km}}^{-1}$ are a few examples of thermally induced turbulent UWOC channels. Other sources of ocean temperature gradient include the rapid influxes of glacial fresh water and extratropical cyclones [9]. Note that SST is not only defined as the water temperature close to the ocean’s surface, but also includes temperatures from depth down to 20 m below the sea surface. Recent research efforts have investigated and characterized the effects of optical turbulence on UWOC channels, mainly from the theoretical point of view. Using the Rytov theory, Korotkova and co-workers have evaluated the scintillation index of an optical plane and spherical waves in a weak turbulent water channel [10]. The bit error rate (BER) performance of UWOC systems with a log-normal distribution as a channel fading model and the BER of focused Gaussian beams in weak turbulence were investigated in Refs. [11,12], respectively.

Detailed experimental studies on the statistical distribution of turbulence-induced fading for UWOC channels are very sparse. A probability density function (PDF) that describes all UWOC channel conditions is not yet well established. Early experimental work by Bissonnette in a 1.5 m deep water tank with an unstable vertical temperature gradient showed that an index of refraction structure function only obeyed the 2/3 power law of the Kolmogorov’s theoretical model in a limited range of 2–10 mm [13]. Very recently, measured data were used to model and obtain the PDFs of salty and bubbly underwater channels under various channel conditions [14].

Most of these emerging UWOC papers have not considered the effect of temperature-induced turbulence. The goal of this Letter is to experimentally study weak temperature-induced turbulence in underwater optical channels and develop new accurate models to describe their statistical properties. We propose the generalized gamma distribution (GGD) to describe both non-turbulent thermally uniform underwater optical channels and underwater optical channels with temperature gradients.

Figure 1 shows the actual photograph of the setup used to study the thermally induced turbulent UWOC channels. The water fills about three-fourths of the tank of the dimensions $1\mathrm{m}\times 0.6\mathrm{m}\times 0.6\mathrm{m}$ along $x$, $y$, and $z$, respectively. The fresh tap water has an estimated attenuation coefficient of $0.071{\mathrm{m}}^{-1}$ [15], similar to clear ocean water. Two identical refrigerated and heating circulators (Julabo-F12 chiller) are used to change the water channel temperature conditions, such as mixing water of different temperatures or maintaining a uniform temperature throughout the tank. The transmitter light source is a 15 mW, transistor outline (TO)-can packaged and fiber-pigtailed green LD (LP520-SF15) emitting at a wavelength of about 515 nm. The laser beam was collimated by a 25 mm plano–convex lens and passed through the water medium. The output irradiance was collected using a 75 mm lens and focused into a silicon photodiode (Thorlabs DET36A) which, in turn, fed into a Tektronix mixed domain oscilloscope for intensity fluctuation measurements. We acquired 100 K samples of intensity fluctuation data with a sampling rate of $100\mathrm{S}/\mathrm{s}$.

 

Fig. 1. Photograph of the experimental setup used to study thestatistics of the temperature-induced turbulence in an underwater wireless optical channel: LD, laser diode; PD, photodetector.

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We propose the GGD to characterize the temperature-induced fading of weakly turbulent underwater wireless optical channels. The GGD is an important probability distribution, especially in reliability modeling and analysis, due to its flexibility. The PDF of GGD is expressed as [16]

Many commonly used probability distributions are special cases of the GGD. For example, the gamma distribution can be expressed as $f(I;a,b,1)$, and the exponential distribution can be expressed as $f(I;1,b,1)$. If $a=1$, the GGD becomes the Weibull distribution. The scintillation index ${\sigma }_I^2$ of the GGD, which is defined as the normalized variance of intensity fluctuations, is given by

We used the maximum likelihood method to obtain estimates of the GGD parameters that can be numerically computed using the wafo toolbox [17].

First, we investigated the effect of temperature change in the water tank. We varied the water temperature from 15°C to 40°C in increments of 5°C. Note that in this study the temperature is uniform throughout the tank. In other words, there is no temperature gradient within the tank for a given test temperature. The temperature uniformity is maintained within 0.1°C by using one Julabo-F12 chiller that circulates water at constant temperature, pressure, and low speed. When the water temperature is uniform throughout the tank, the received intensity of the laser beam is best described by the simple gamma distribution which is a special case of the GGD, as mentioned above. Figure 2(a) shows the fitness of gamma distribution with the acquired data histogram for a 25°C water channel temperature test condition. We observe a very good agreement with measured data. Although a histogram of only one temperature is shown in Fig. 2(a), a good fitness was obtained for each temperature up to 40°C. It is important to mention that the shape of the histogram depends on the choice of the number of bins. In our work, we used the Sturges rule to calculate the number of bins [18]. Figure 2(b) illustrates the scintillation index of the optical signal as a function of water channel temperature. The scintillation index increases as the water temperature increases. This behavior can be explained by the dipole polarization of water molecules using the Lorentz–Lorenz law that relates the refractive index of a substance to its polarizability [19]. The increase in temperature leads to more thermal collision of water molecules, weakening their bonding forces and making it harder for their dipole moments to align. As a result, the refractive index which is directly proportional to the square root of the medium’s electric susceptibility decreases [20].

 

Fig. 2. (a) Acquired data histogram along with the gamma PDF for 25°C water temperature—${\sigma }_{I\_\text{measured}}^2=7.0622\times {10}^{-4}$, $a=1.412\times {10}^3$, $b=7.0822\times {10}^{-4}$, and ${\sigma }_I^2=7.1649\times {10}^{-4}$. (b) Scintillation index as a function of water temperature (solid squares for measurements and closed circles for calculations). The dashed lines are guides to the eye.

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In the second phase of the temperature-dependent UWOC channel study, we investigated the effect of the water temperature gradient. A horizontal temperature gradient was created by mixing two water samples of different temperatures from both ends (Inlet 1 & 2 in Fig. 1) of the tank along the optical axis. The mixture creates a refractive index difference in the propagation path that severely affects the beam as it propagates. The temperature difference between the two ends of the tank was varied from 5°C to 20°C. Note that the mean temperature in the tank was kept constant at 25°C. Table 1 shows different temperature values used to create the thermal gradient along the 1 m water tank.

Table 1. Different Temperature Values Used to Create Temperature Gradients in the Water Channel

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We use the statistical method $R^2$ test [21] to quantitatively describe the fitness of GGD to experimental data. Note that $R^2$ has a value that ranges from 0 to 1. An $R^2$ of 1 indicates that the model perfectly fits the data. Here, we should also point out that $R^2$ depends on the number of bins. Table 2 summarizes the results of $R^2$ test for three different statistical distributions under the four temperature gradient levels. Note that the scintillation index strongly increases with the temperature gradient, but is still within the weak turbulence regime. For low gradient levels (${\sigma }_I^2\le 0.001$), the simple gamma distribution models well the intensity fluctuations, but it loses accuracy very rapidly as the temperature gradient increases and the channel becomes more turbulent. Meanwhile, the Weibull distribution acceptably predicts the fluctuations for all gradient levels. In contrast to gamma and Weibull distributions, the GGD has an excellent goodness of fit for all values of scintillation index. Figure 3 better illustrates the fit of all three distributions with the measured data histograms for all four gradient levels. It is worth mentioning that the maximum gradient level used in our experiment is at least 10 times higher than the SST of Gulf Stream and Kuroshio currents. In addition, it is large enough to represent almost any realistic weak turbulence scenario encountered in the ocean where even the largest temperature difference between the reservoir of warm water at the surface and the reservoir of cold water deeper in the ocean varies between 22°C and 25°C [22].

 

Fig. 3. Histograms of the measured data along with the PDF of various distributions under different temperature gradient levels: (a) $0.05°\mathrm{C}·{\mathrm{cm}}^{-1}$, (b) $0.10°\mathrm{C}·{\mathrm{cm}}^{-1}$, (c) $0.15°\mathrm{C}·{\mathrm{cm}}^{-1}$, and (d) $0.20°\mathrm{C}·{\mathrm{cm}}^{-1}$.

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Table 2. $R^2$ Test Measure Results for Different Distributions under Four Different Underwater Channel Temperature Gradient Levels

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In this Letter, we experimentally investigated the statistical distribution of optical intensity fluctuations caused by thermally induced underwater wireless optical channels. The GGD is found to precisely fit the measured data under all weak turbulence channel conditions. However, for UWOC channels with low temperature gradients (${\sigma }_I^2\le 0.001$), the simple gamma distribution can be used to characterize the PDF of the received signal. This channel model paves the way for developing a comprehensive and unified model to predict the performance of UWOC links under all turbulence conditions.