A. *Power Spectrum of Oceanic Turbulence Fluctuations*

Nikishov’s power spectrum [11] is commonly preferred in analyzing optical wave propagation in oceanic turbulence and have important consequences on various aspects of optical wave propagation through the ocean such as change in the mean field and irradiance, spot size (i.e., footprint of the received optical beam), beam wander and particularly scintillation. The power spectrum model for refractive index fluctuations of isotropic and homogeneous oceanic water that is based on salinity, temperature and coupling fluctuations is represented by [11]

$$\varPhi \left(\kappa \right)=0.388\times {10}^{-8}{\kappa }^{-11/3}{\epsilon }^{-1/3}\left[1+2.35{\left(\kappa \eta \right)}^{2/3}\right]\frac{{X}_{T}}{{\omega }^{2}}\left({\omega }^{2}{e}^{-{A}_{T}\delta }+{e}^{-{A}_{S}\delta }-2\omega {e}^{-{A}_{TS}\delta }\right),$$

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where \(\kappa\) is the spatial frequency, \(\epsilon\) is the rate of dissipation of kinetic energy per unit mass of fluid, \(\eta\) is the Kolmogorov microscale (inner scale), \({X}_{T}\) is the rate of dissipation of mean-squared temperature, \(\omega\) is a unitless parameter providing the ratio of temperature to salinity contributions to the refractive index spectrum, \({A}_{T}=1.863\times {10}^{-2}\), \({A}_{S}=1.9\times {10}^{-4}\), \({A}_{TS}=9.41\times {10}^{-3}\) and \(\delta \left(\kappa ,\eta \right)=8.284{\left(\kappa \eta \right)}^{4/3}+12.978{\left(\kappa \eta \right)}^{2}\). It should be noted that \(\eta\) varies in the range of \(0.01-6\times {10}^{-5}\text{ m}\) and \(\epsilon\) varies in the range of \({10}^{-10}-{10}^{-1}{\text{m}}^{2}/{\text{s}}^{\text{3}}\), depending on the layers of ocean (abyssal ocean to most actively turbulent region i.e., near ocean surface). In a similar fashion, \({X}_{T}\) ranges from \({10}^{-10}{\text{K}}^{2}/\text{s}\) to \({10}^{-4}{\text{K}}^{2}/\text{s}\) and the unitless parameter \(\omega\) is a crucial parameter since it emphasize the relative strength of temperature and salinity fluctuations, \(\omega\) ranges from − 5 dominating temperature-induced turbulence to 0 dominating salinity-induced turbulence [6,7].

B. *Gaussian Beam Propagation through Oceanic Turbulence*

1. Received Optical Intensity

The beam field of the Gaussian source laser at the transmitter spatial plane is given by [8]

$$u\left(s\right)={A}_{s}\text{e}\text{x}\text{p}\left[-\frac{1}{2}k\left(\frac{1}{k{\alpha }_{s}^{2}}+\frac{i}{{F}_{0}}\right){\mathbf{s}}^{2}\right]$$

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where \({A_s}\) is the field amplitude, \(k=2\pi /\lambda\) is the wave number, \({\alpha }_{s}\) is the laser source size, \(i=\sqrt{-1}\), \(\lambda\) is the wavelength, \(\mathbf{s}=\left({s}_{x},{s}_{y}\right)\) denotes the source transverse coordinates, \({F}_{0}\) is the phase front radius of curvature. In this study, \({F}_{0}\) is chosen as infinity to represent the collimated beam. To find the received average optical intensity on the axis, we use the extended Huygens-Fresnel principle in Eq. (3) which involves the convolution of source field Eq. (2) with the spherical wave response of the turbulent medium [25], which is

$$\begin{array}{c}⟨I\left(z=L\right)⟩=\frac{1}{{\left(\lambda L\right)}^{2}}\underset{-\infty }{\overset{\infty }{\int }}\underset{-\infty }{\overset{\infty }{\int }}\underset{-\infty }{\overset{\infty }{\int }}\underset{-\infty }{\overset{\infty }{\int }}\mathbf{d}{\mathbf{s}}_{1}^{2}\text{\hspace{0.17em}}\mathbf{d}{\mathbf{s}}_{2}^{2}\text{\hspace{0.17em}}u\left({s}_{1}\right){u}^{*}\left({s}_{2}\right)\\ \times \text{e}\text{x}\text{p}\left\{\frac{ik}{2L}\left[{\left({\mathbf{s}}_{1}\right)}^{2}-{\left({\mathbf{s}}_{2}\right)}^{2}\right]-{\rho }_{0}^{-2}{\left({\mathbf{s}}_{1}-{\mathbf{s}}_{2}\right)}^{2}\right\}\end{array}$$

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where *L* is link distance, * is the complex conjugate, \({\rho }_{0}\) is the coherence length of the spherical wave propagating through the ocean, which is given by [26,27]

$$\begin{array}{c}{\rho }_{0}={\left[\frac{{\pi }^{2}{k}^{2}L}{3}\underset{0}{\overset{\infty }{\int }}{\kappa }^{3}\varPhi \left(\kappa \right)d\kappa \right]}^{-1/2}\\ ={\left[1.28\times {10}^{-8}{k}^{2}L{\eta }^{-1/3}{\epsilon }^{-1/3}{X}_{T}\left(6.78+47.57{w}^{-2}-17.67{w}^{-1}\right)\right]}^{-1/2}\end{array}$$

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Inserting the source field Eq. (2) into the convolutional integral, Eq. (3), and solving Eq. (3) by the repeated use of Eq. (3.323.2) of [28], we find the average received optical intensity underwater [29] as

$$⟨I\left(z=L\right)⟩=\frac{{A}_{s}^{2}{\left(\frac{k{\alpha }_{s}}{2L}\right)}^{2}}{\left(\frac{1}{4{\alpha }_{s}^{2}}+\frac{1}{{\rho }_{0}^{2}}+\frac{{k}^{2}{\alpha }_{s}^{2}}{4{L}^{2}}\right)}\text{.}$$

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In the absence of turbulence, the wavefront of the laser beam is not distorted and thus the spatial coherence is not affected. To find the received optical intensity in free space (i.e., without turbulence), coherence length in Eq. (4) is chosen as infinite \({\rho }_{0}\to \infty\), and therefore Eq. (5) reduces to Eq. (52) of Chap. 4 of [30] which is

$${I}_{0}=\frac{{A}_{s}^{2}{\left(\frac{k{\alpha }_{s}}{2L}\right)}^{2}}{\left(\frac{1}{4{\alpha }_{s}^{2}}+\frac{{k}^{2}{\alpha }_{s}^{2}}{4{L}^{2}}\right)}\text{=}{A}_{s}^{2}\frac{1}{{\varTheta }_{0}^{2}+{\varLambda }_{0}^{2}}\text{.}$$

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where \({\varTheta }_{0}=1-L/{F}_{0}\), \({\varLambda }_{0}=L/\left(k{\alpha }_{s}^{2}\right)\) are the transmitter plane parameters of the Gaussian beam, \({\varTheta }_{0}=1\) corresponds to the collimated Gaussian beam.

2. Scintillation Index

Scintillation is described as the fluctuations in the received intensity and the normalized variance of intensity fluctuations is called scintillation index. Under weak intensity fluctuations, the scintillation index is approximately equal to the log-irradiance variance, \({\sigma }_{\text{l}\text{n}I}^{2}\). Using the Rytov theory and assuming Gaussian beam propagation, \({\sigma }_{\text{l}\text{n}I}^{2}\) is expressed in terms of the log amplitude variance \({\sigma }_{\chi }^{2}\) [30] as

$$\begin{array}{c}{\sigma }_{\text{l}\text{n}I}^{2}=4{\sigma }_{\chi }^{2}\\ \text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{=}\text{ }8{\pi }^{2}{k}^{2}L\underset{0}{\overset{1}{\int }}\underset{0}{\overset{\infty }{\int }}d\xi \kappa d\kappa \varPhi \left(\kappa \right)exp\left(-\frac{{\varLambda }_{1}L{\kappa }^{2}{\xi }^{2}}{k}\right)\left\{1-\text{c}\text{o}\text{s}\left[\frac{L{\kappa }^{2}}{k}\xi \left(1-{\stackrel{-}{\varTheta }}_{1}\xi \right)\right]\right\}\text{,}\end{array}$$

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where is distance parameter, \(\xi =1-z/L\), \({\varLambda }_{1}={\varLambda }_{0}/\left({\varTheta }_{0}^{2}+{\varLambda }_{0}^{2}\right)\),\({\stackrel{-}{\varTheta }}_{1}=1-{\varTheta }_{1}\), \({\varTheta }_{1}={\varTheta }_{0}/\left({\varTheta }_{0}^{2}+{\varLambda }_{0}^{2}\right)\).

C. *Signal Noise in Underwater*

There are many sources of noise, namely background noise resulting from ambient light in underwater, thermal noise, shot noise in the presence of received signal and photodiode dark current noise. These noises degrade the performance of the OWC system in underwater. The total noise variance is defined as the combination of the aforementioned noises and is given by [31]

$${\sigma }^{2}={\sigma }_{Bg}^{2}+{\sigma }_{th}^{2}+{\sigma }_{sn}^{2}\text{+}{\sigma }_{dc}^{2}$$

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where \({\sigma }_{Bg}^{2}=2q{R}_{p}{P}_{Bg}{B}_{f}\) is the background noise variance, \(q=1.\text{60217662}\times {10}^{-19}\) is the electronic charge in Coulomb, *R**p* is the responsivity of the PIN photodetector, \({P}_{Bg}\) is the background noise power in watts, \({B}_{f}\) is the post detection electrical filter bandwidth in Hz, \({\sigma }_{th}^{2}=4{k}_{B}{T}_{e}{F}_{N}{B}_{f}/{R}_{L}\) is the thermal noise variance, \({k}_{B}=1.3807\times {10}^{-23}\) is the Boltzmann’s constant in Joules per degree Kelvin, \({T}_{e}\) is the equivalent temperature in degrees Kelvin (oK), \({F}_{N}\) is the noise factor of the system, \({R}_{L}\) is the equivalent load resistor in ohm, \({\sigma }_{sn}^{2}=2q{R}_{p}⟨I⟩{B}_{f}\) is the shot-noise variance, \(⟨I⟩\) is the received signal power given in Eq. (5), \({\sigma }_{dc}^{2}=2q{I}_{dc}{B}_{f}\) is the dark current noise, \({I}_{dc}\) in Amp is the dark current of the photodiode.

D. *BER Performance of Receiver Diversity System in Log-Normal Channels*

1. Equal Gain Combining

EGC system samples each receiver photodetector coherently with equal weights. The sum of irradiances, having statistically independent signal fades causes a decrease in the system noise. Note that the sum of irradiances with equal weights confirms the log-normal distribution. BER of an optical wireless communication system employing EGC, and BPSK subcarrier intensity modulation is defined by [32]

$${P}_{e\left(EGC\right)}=\frac{1}{\sqrt{\pi }}\sum _{j=1}^{n}{w}_{j}Q\left[{K}_{1}\text{e}\text{x}\text{p}\left({x}_{j}\sqrt{2{\sigma }_{U}^{2}}+{\mu }_{U}\right)\right]$$

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where \({\left[{w}_{j}\right]}_{j=1}^{n}\) and \({\left[{x}_{j}\right]}_{j=1}^{n}\), whose values are found in [33], are the weight factors and the zeros of an n-th-order Hermite polynomial, \(Q\left(x\right)=0.5erfc\left(x/\sqrt{2}\right)\), \(erfc\left(.\right)\) is the complementary error function, \({K}_{1}={R}_{p}{I}_{0}{A}_{s}/\sqrt{2{N}^{2}{\sigma }^{2}}\), \({I}_{0}\) is the received intensity in the absence of turbulence given in Eq. (6), \({\sigma }_{U}^{2}=\text{l}\text{n}\left\{1+\left[\text{e}\text{x}\text{p}\left({\sigma }_{\text{l}\text{n}I}^{2}\right)-1\right]/N\right\}\), \(\sigma _{{\ln I}}^{2}\) is the variance of log irradiance, is the number of separated photodetectors, and \({\mu }_{U}=\text{l}\text{n}\left(N\right)-\frac{1}{2}{\sigma }_{U}^{2}\).

2. Maximum Ratio Combining

MRC is an effective combining scheme for receiver arrays to combat underwater turbulence and optical noise. MRC requires estimation of the received irradiance and phase at each branch on the array. Weighted branches with signal, having statistically independent signal fades, are co-phased and summed coherently, and thus the overall noise in the system is attenuated. BER of an OWC system employing MRC and BPSK subcarrier intensity modulation is given by [32]

$${P}_{e\left(MRC\right)}=\frac{1}{\pi }\underset{0}{\overset{\pi /2}{\int }}{\left[S\left(\theta \right)\right]}^{N}d\theta$$

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where

$$S\left(\theta \right)\approx \frac{1}{\sqrt{\pi }}\sum _{j=1}^{n}{w}_{j}\text{e}\text{x}\text{p}\left\{-\frac{{K}_{0}^{2}}{2{\text{s}\text{i}\text{n}}^{2}\theta }\text{e}\text{x}\text{p}\left[2\left({x}_{j}\sqrt{2{\sigma }_{\text{l}\text{n}I}^{2}}-\frac{{\sigma }_{\text{l}\text{n}I}^{2}}{2}\right)\right]\right\}$$

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and \({K}_{0}={R}_{p}{I}_{0}{A}_{s}/\sqrt{2N{\sigma }^{2}}\).

3. Selection Combining

In SelC technique, combiner selects the branch that has the strongest SNR level i.e., probably exposed to less signal fade, and therefore SNR level at each branch is estimated and compared relatively. In addition, SelC requires only one of the branches, which makes the complexity of the system remarkably reduced compared to EGC and MRC. The unconditional BER for an OWC system employing SelC and BPSK subcarrier intensity modulation is defined by [32]

$${P}_{e\left(SelC\right)}=\frac{N}{{2}^{N-1}\sqrt{\pi }}{\sum _{j=1}^{n}{w}_{j}\left[1+erf\left({x}_{j}\right)\right]}^{N-1}Q\left[{K}_{0}\text{e}\text{x}\text{p}\left({x}_{j}\sqrt{2{\sigma }_{\text{l}\text{n}I}^{2}}-{\sigma }_{\text{l}\text{n}I}^{2}/2\right)\right]$$

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.