## Abstract

A new class of partially coherent radially and azimuthally polarized rotating elliptical Gaussian (PCRPREG and PCAPREG) beams is introduced. The analytical expressions of the PCRPREG and PCAPREG beams propagating through anisotropy oceanic turbulence are derived based on the extended Huygens-Fresnel principle and the spatial power spectrum of oceanic turbulence. The effects of beam waist size *w*_{0}, coherence width *σ*_{0}, propagation distance *z* and oceanic turbulence parameters on the evolution statistics properties of PCRPREG and PCAPREG beams are studied in detail by numerical simulation. Our results indicate that with the increase of the propagation distance in the far field region, the normalized initial profile with a doughnut-like distribution of PCRPREG and PCAPREG beams gradually converts into a flat-topped one, and finally evolves into a Gaussian-like beam profile. We also find that the salinity-induced turbulence fluctuation makes a greater contribution to the decrease of beam quality compared with the temperature-induced turbulence fluctuation. Furthermore, the full width at half maximum becomes wider for the larger propagation distance *z* and wavelength *λ* or the smaller dissipation rate *ε*. Our work will pave the way for the development of underwater optical communication and underwater laser radar in oceanic environment.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

The evolutions of laser beams in random media have attracted more and more attentions due to their broad applications spanning from laser radar system [1], remote sensing [2], underwater optical communication [3] and imaging [4] to high-resolution microscopy [5] and single fluorescent molecule [6]. So far, a variety of propagation properties have been developed, such as the power spectrums of oceanic and atmospheric turbulences [7, 8] and the spectrum of turbulent water [9]. It was found both theoretically and experimentally that partially coherent beams are less sensitive to the effects of turbulence than fully coherent beams [10, 11], the propagation of partially coherent laser beams has been a considerable hot subject in recent years, such as partially coherent cylindrical vector beam [12], partially coherent flat-topped laser beam [13], partially coherent radially polarized doughnut beam [14] and partially coherent Airy beam [15].

On the other hand, the evolution behavior of various Gaussian beams through the oceanic turbulence also has been studied, such as multi-Gaussian Schell-model beams [16], asymmetrical Gaussian beam [17], Lorentz Gaussian beam [18], Lommel-Gaussian beam [19]. Moreover, the influences of the turbulent characteristic parameters acting on the normalized intensity distribution [12], the conservation of the topological charge [20], the Strehl ratio [21] and orbital angular momentum [22] have been discussed. In 2000, Nikishov proposed an analytical model for such a power spectrum of oceanic turbulence [7]. Subsequently, the temperature and salinity fluctuations of radially polarized beams in oceanic turbulence have been deeply investigated [23, 24]. Interestingly, although both radial and azimuthal polarizations have identical spectral intensity distribution, their polarization characteristics are actually different [25, 26]. Meanwhile, several methods for generating a laser with radial or azimuthal polarization have been reported [27, 28]. Compared with other beams, radially and azimuthally polarized beams have better tight focusing properties and their focusing characteristics remain unchanged during the propagation process [29, 30].

Recently, rotating elliptical Gaussian beam propagating through the atmosphere turbulent and the oceanic turbulencee has been studied by Zhang *et al.* [31, 32]. And Ye *et al.* also analyzed rotating elliptical chirped Gaussian vortex beam through the oceanic turbulence in 2018 [33]. However, to the best of our knowledge, the propagation properties of partially coherent radially and azimuthally polarized rotating elliptical Gaussian (PCRPREG and PCAPREG) beams through the anisotropic oceanic turbulence have not been reported, which have potential application value in underwater wireless optical communication [34] and underwater optical imaging system [35].

Based on the extended Huygens-Fresnel principle and the power spectrum of oceanic turbulence, the analytical expressions for PCRPREG and PCAPREG beams propagating through anisotropy oceanic turbulence are derived. The second-order statistics propagation properties including the normalized spectrum density, the spectral degree of coherence (DOC) and the spectral degree of polarization (DOP) of a PCRPREG and PCAPREG beams in anisotropic oceanic turbulence have been discussed in detail by numerical simulation.

The organization of the paper is as follows. Firstly in Sec. 2, the analytical formula of 2 × 2 cross-spectral density (CSD) matrix of PCRPREG and PCAPREG beams through oceanic turbulence with anisotropy is derived. Then in Sec. 3, we illustrate the evolution behavior of PCRPREG and PCAPREG beams through anisotropic oceanic turbulence numerically. Finally, the main conclusion is drawn in Sec. 4.

## 2. The CSD matrix of PCRPREG and PCAPREG beams in oceanic turbulence with anisotropy

In the Cartesian coordinate system, the vectorial electric field of radially polarized rotating elliptical Gaussian (RPREG) and azimuthally polarized rotating elliptical Gaussian (APREG) beams can be expressed as the coherent superposition of a TEM_{01}with a polarization direction parallel to the x-axis and a TEM_{10} with a polarization direction parallel to the y-axis [25, 26, 36, 37]:

*w*

_{0}denotes the beam waist size of a rotating elliptical gaussian beam, a, b and c are the arbitrary real constants, ${\overrightarrow{\mathbf{e}}}_{x}$ and ${\overrightarrow{\mathbf{e}}}_{y}$ represent the unit vectors in the

*x*and

*y*directions, respectively. Figure 1 shows the RPREG (a) and APREG (b) beams at the initial plane.

Here, the vertical electric field part can be ignored under the paraxial condition. For a vector partially coherent beam in space-frequency domain, the second-order spatial coherence properties can be characterized by the 2 × 2 CSD matrix of the electric field, defined by the formula [38]

The elements of the CSD matrix for PCRPREG and PCAPREG beams at source plane $\left(z=0\right)$ can be expressed as follows [39]:

*σ*

_{0}represents spatial coherence width.

Here, we investigate the propagation of the PCRPREG and PCAPREG beams in the oceanic turbulence. By applying the well-known extended Huygens-Fresnel integral formula, the element of the CSD matrix ${W}_{\alpha \beta}\left({\overrightarrow{\mathbf{r}}}_{1},{\overrightarrow{\mathbf{r}}}_{2}\right)$ at the receiver plane can be expressed as [21, 23, 40]:

*λ*being the wavelength of the source beam. Under quadratic phase approximations, $\psi \left(\overrightarrow{{\mathit{r}}_{1}},{\overrightarrow{\mathit{\rho}}}_{1}\right)+{\psi}^{*}\left({\overrightarrow{\mathit{r}}}_{2},{\overrightarrow{\mathit{\rho}}}_{2}\right)$ denotes the ensemble average of the turbulent ocean water which can be represented as [41]:

*ξ*representing the turbulence strength and the anisotropic factor, respectively. We assume that the anisotropy exists only along the propagating direction

*z*of the beam, using an effective anisotropic factor

*ξ*to present the anisotropic power spectrum of oceanic turbulence. This is to say, the smaller ${\rho}_{oc\xi}$ means the stronger turbulent strength. For the sake of simplicity, we only consider the influence of temperature and salinity fluctuations on the propagation of beams in the ocean turbulence, and ignore the scattering and absorption of the ocean water. In the Markov approximation, the two-dimensional refractive index turbulence spatial spectrum model of the anisotropic oceanic turbulent is written as [43]:

*η*is the Kolmogorov micro scale,

*χ*is the rate of dissipation of mean square temperature and ranges from ${10}^{-4}{K}^{2}/s$ to ${10}^{-10}{K}^{2}/s$,

_{T}*ε*is the rate of dissipation of turbulent kinetic energy per unit mass of fluid which may vary from ${10}^{-4}{m}^{2}/{s}^{3}$ to ${10}^{-10}{m}^{2}/{s}^{3}$,

*ω*represents the ratio of the temperature and salinity contribution to the refractive index spectrum, which in the ocean water the value can range from $-5$ to 0, $-5$ corresponding to the situation when salinity-driven turbulence prevails, 0 corresponding to the case when temperature-driven turbulence dominates.

By using the relation [44]

Eq. (6) can be analytically integrated as:

In the similar way, we obtain the following formulas for CSD matrix of a PCAPREG beam propagating in the anisotropy oceanic turbulence:

The spectral density and the spectral DOC at the output plane are given by the formulas [45]:

*Tr*stands for the trace of the matrix.

The spectral DOP of the PCAPREG beam is defined as [45, 46]:

Equations (16)-(20) indicate that the PCRPREG and PCAPREG beams has the same evolution properties.

## 3. Results and discussions

In this section, we will now analyze the behavior of the second-order statistics properties of a PCRPREG beam through anisotropic oceanic turbulence by a set of numerical examples based on the analytical formulas derived in the previous section. Unless other values are specified in caption, the parameters of the source and the oceanic are set as: $a=2.0$, $b=1.8$, $c=1.7$, ${w}_{0}=1.5mm$, ${\sigma}_{0}=2.5mm$, $\lambda =416nm$, $\xi =2.0$, $\omega =-2.5$, $\epsilon ={10}^{-5}{m}^{2}/{s}^{3}$, $eta=1mm$, ${\chi}_{T}={10}^{-9}{K}^{2}/s$.

Figure 2 presents normalized spectral density $I\left(x,0,z\right)/{I}_{max}\left(x,0,0\right)$ of a PCRPREG beam at several propagation distances in anisotropic oceanic turbulence for two different values of *χ _{T}*. With the increase of propagation distance

*z*, the doughnut beam profile converts into a flat-topped profile, as appreciably depicted in Figs. 2(a)-2(d). It is observed from Figs. 2(d)-2(f) that the flat-topped profile evolves into a Gaussian-like beam profile as the further increase of

*z*. Besides, the larger the rate of dissipation of mean square temperature

*χ*is, the faster the beam profile is close to the Gaussian-like distribution. The reason for this phenomenon is that as the transmission distance of the doughnut beam increases, the doughnut beam expands due to turbulence and vacuum diffraction, among which the expansion towards the center of the ring causes the spot center to collapse [48].

_{T}Figure 3 shows the evolutions of the spectrum density distribution and the corresponding cross line of a PCRPREG beam at several different propagation distances through anisotropic oceanic turbulence. Figures 3(a1)-3(f1) are three-dimensional diagrams corresponding to Figs. 3(a)-3(f) under the same conditions. It is significant that the spectrum density pattern has an elliptical annular shape in the initial plane whose major axis is parallel to the x-axis, which gradually rotates clockwise and converts into a quasi-circle annular shape with the increase of propagation distance z. Ultimately, the major axis of the spectrum density pattern is parallel to the y-axis in the far field region. In the oceanic turbulence, the spectrum density of a PCRPREG beam changes from an elliptical dark hollow beam profile to an elliptical flat-topped profile at certain propagation distances. Meanwhile, the elliptical flat-topped profile becomes a Gaussian-like beam profile in the far field region. The above phenomenon can be interpreted as the fact that the stronger oceanic turbulence is, the larger the degradation of the coherence is. One also finds from Fig. 3 that the azimuthal symmetry of the transverse intensity pattern is not affected by the oceanic turbulence although the elliptical dark hollow beam profile disappears due to the anisotropic influence of the oceanic turbulence.

Figure 4 plots normalized spectral density distribution $I/{I}_{max}$ (${I}_{max}$‒maximum spectral density) of a PCRPREG beam for different values of (a) propagation distance *z*, (b) wavelength *λ*, (c) spatial coherence width *σ*_{0}, (d) transverse beam size *w*_{0}, (e) rate of dissipation of mean-square temperature *χ _{T}*, (f) anisotropic factor

*ξ*. It is observed from Fig. 4(a) that the normalized average spectral density distribution of a PCRPREG beam through anisotropic oceanic turbulence undergoes several stages of evolution analogous to the propagation of a Guassian Schell-model vortex beam through oceanic turbulence [21]. At the

*z*= 0 plane, the normalized average spectral density distribution in the center part is zero, where the phase becomes singular [20]. With the increase of the propagation distance, a flat-topped spectral density profile occurs, and then evolves into a Gaussian-like beam profile. In Figs. 4(b)-4(e), the evolution behavior of normalized average spectral density distribution of a PCRPREG beam through anisotropic oceanic turbulence is profoundly effected by the anisotropic oceanic turbulence parameters including

*λ*,

*σ*

_{0},

*w*

_{0}, and

*χ*. The smaller wavelength

_{T}*λ*and

*w*

_{0}are and the larger

*σ*

_{0}and

*χ*are, the faster the change of the beam spectral density distribution is. One can find from Eq. (10) and Fig. 4(e) that the larger

_{T}*χ*is, the stronger the oceanic turbulence is. Whereas, the anisotropic factor

_{T}*ξ*plays a less definitive role in normalized average spectral density distribution, as depicted in Fig. 4(f). In the meantime, it is shown that the longer propagation distance

*z*is, the larger wavelength

*λ*and

*χ*are and the smaller

_{T}*σ*

_{0}and

*w*

_{0}are, the larger the spreading of the beam width is, but the effect caused by anisotropic factor is not significant.

Figure 5(a) illustrates the evolution of the transverse DOCs of the PCRPREG beam between two points ${\rho}_{1}=\left(u,0\right)$ and ${\rho}_{2}=\left(-u,0\right)$ at $z=100m$. It can be clearly seen that the *μ* decreases quickly from 1 to -0.4 and then increases slowly from -0.4 to 0 with the increase of the transverse distance *u*. As the wavelength decreases, the minimum value appears in a shorter transverse distance. In Figs. 5(b)-5(f), two points are fixed at ${\rho}_{1}=\left(0.3cm,0.3cm\right)$ and ${\rho}_{2}=\left(-0.3cm,-0.3cm\right)$, the DOCs of the PCRPREG beam are interpreted. For convenience, we choose $z=100m$ for analysis in Fig. 5(c). With the increase of the propagation distance *z*, *μ* decreases rapidly from 0 to -0.3, and then increases progressively. The *μ* changes more significantly with the increase of the wavelength, as shown in Fig. 5(b). It is easy to find from Figs. 5(a) and 5(b) that the choice of wavelength has a great influence on the DOCs of the PCRPREG beam. The effects of the anisotropic oceanic turbulence on the *μ* are shown in Figs. 5(c)-5(d). Obviously, the *μ* gradually increases with the increase of *ω*, and the increasing trend becomes sharp for *ω* close to 0 value. Namely, the salinity-induced turbulence fluctuation makes a greater contribution to the decrease of beam quality compared with the temperature-induced turbulence fluctuation [21]. On the other hand, the decreasing *ε* and the increasing *z* lead to the increasing *μ*. One can see from Figs. 5(e) and 5(f) that the spectral DOC *μ* reaches the minimal value and then increases quickly. The larger *ξ* is and the smaller *ω* is, the greater the change of the spectral DOC *μ* is. As the propagation distance increases, the*μ* decreases from the value of 0 of the initial plane to the minimum value, and then returns to the initial value of 0, which is due to the influence of ocean turbulence as displayed in Figs. 5(b), 5(e) and 5(f). The above phenomenon can be explained by Eq. (10), it is not difficult to find that the smaller *ω* and the larger *ξ* are, the stronger the oceanic turbulence is. The spectral DOCs of the PCRPREG beam depend on the oceanic turbulence strength.

To study about the influence of the oceanic turbulence on the polarization properties of a PCRPREG beam in an anisotropic oceanic turbulence. We calculate the DOP of a PCRPREG beam and the corresponding cross line $\left(v=0\right)$ at several propagation distances as shown in Fig. 6. The DOP of a PCRPREG beam equals 1 for all the points across the entire transverse plane when *z* = 0, which is manifested in Fig. 6. In the anisotropic oceanic turbulence, one can see from Fig. 6 that a dip appears in the distribution of the DOP, where the DOP distribution curve reaches the lowest point when *u* = 0, but as the propagation distance z increases, the minimum of the spectrum DOP decreases to $0.2$ and the distribution is nearly unchanged. We also find that the width of the dip increases with the increasing of propagation distance, which implies that a radial polarization structure of a PCRPREG beam is destroyed during the propagation in an anisotropic oceanic turbulence. Furthermore, one also finds from Fig. 7 that the DOP curve is symmetric about the v axis, and the full width at half maximum is greater for the larger the ratio of the temperature and salinity contributions to the refractive index sprctrum *ω* (a) and wavelength *λ* (b) and the smaller the rate of dissipation of turbulent kinetic energy per unit mass of fluid *ε* (c) .

Finally, to learn more about the polarization properties, we calculate the orientation angle of a PCRPREG beam in anisotropic oceanic turbulence at several propagation distances. Figure 8 presents the radial polarization structure is destroyed due to the effects of ocean turbulence. It is interesting to note that the center of polarization structure changes most significantly with the increase of propagation distance.

## 4. Conclusion

In this paper, the analytical formulae for PCRPREG and PCAPREG beams propagating through anisotropy oceanic turbulence are derived based on the extended Huygens-Fresnel principle and the power spectrum of oceanic turbulence. The propagation evolution properties of PCRPREG and PCAPREG beams in oceanic turbulence have been discussed in detail by numerical simulation. Our results show that the propagation properties of the beam are closely related to beam waist size *w*_{0}, coherence width *σ*_{0},propagation distance *z* and the oceanic turbulence parameters. The initial profile of PCRPREG and PCAPREG beams gradually disappears with a flat-topped one occurring, and finally the evolves into a Gaussian-like beam profile with the increase of the propagation distance in the far field region. Obviously, the longer propagation distance *z* is, the larger wavelength *λ* and the rate of dissipation of mean square temperature *χ _{T}* are and the smaller spatial coherence width

*σ*

_{0}and the beam waist size

*w*

_{0}are, the larger the spreading of the beam width is. In addition, the salinity-induced turbulence fluctuation makes a greater contribution to the decrease of beam quality comparing with the temperature-induced turbulence fluctuation. The larger the ratio of the temperature and salinity contributions to the refractive index sprectrum

*ω*and wavelength

*λ*are or the smaller the rate of dissipation of turbulent kinetic energy per unit mass of fluid

*ε*is, the greater the full width at half maximum is. It is rather remarkable that the radial polarization structure of a PCRPREG beam is destroyed due to the anisotropic influence of the oceanic turbulence. It is generally believed that our work may have potential applications in underwater optical comminication system.

## Funding

National Natural Science Foundation of China (NSFC) (11775083, 11374108); Innovation Project of Graduate School of South China Normal University (2018LKXM043).

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