Statistical properties of a controllable partially coherent radially and azimuthally polarized rotating elliptical Gaussian optical coherence lattice in anisotropic ocean turbulence

Xiang Lv; Chao Sun; Feng Ye; Beibei Ma; Dongmei Deng

doi:10.1364/OE.27.026532

1. Introduction

As is well known, turbulence which is a kind of flow regimes with chaotic changes in flow velocity and pressure is completely different from the laminar flow [1]. So far, a series of propagation characteristics on turbulence has been implemented, including the spectral density [2], the scintillation index [3], the beam drift coherence of the spherical wave [4], the beam profile [5], the power spectrums of oceanic and atmospheric turbulences [6,7] and the spectrum of turbulent water [8]. In addition, turbulence has important applications in free-space optical communication [9,10], imaging systems [11], remote sensing [12], single fluorescent molecule [13]. Wu $et$ $al$ [14] also analyzed theoretically, numerically, and experimentally that the slightly modified image encryption system can be used to achieve arbitrary phase and amplitude of the beam shaping within the limits of stroke range and influence function of the deformable mirrors. In application, the proposed technique can be used to perform mode conversion between optical beams, generate structured light signals for imaging and scanning, and compensate atmospheric turbulence-induced phase and amplitude beam distortions. Because the optical wireless communication underwater media is an innovative and good alternative communication method [14,15], the propagation of optical beams through oceanic turbulence is of great importance. As beams propagate in the oceanic turbulence, the properties are influenced by the turbulence parameters, and the shape of optical fields and the beam quality change during propagation. Therefore, it is essential to study the propagation of light beams in ocean turbulence.

Coherence is served as a fundamental characteristic of the light beam and plays an important role in understanding the internal interactions of light and materials [16]. In the past few years, the light coherence properties have received lots of attention motivated by such various applications as high-resolution imaging [17], coherence holography [18], optical force [19] and solitons research [20]. Compared with completely coherent beams, partially coherent beams have a significant advantage for reducing the inevitable degradation induced by random refractive-index fluctuations of the turbulence [21]. So the propagation of partially coherent laser beams has been a considerable research hotspot in recent years, such as partially coherent flat-topped laser beams [22], partially coherent Airy beams [23] and partially coherent radially polarized doughnut beams [24]. According to the unified theory of coherence and polarization given by Wolf [24], the spectral degree of coherence (DOC), degree of polarization (DOP) of partially coherent electromagnetic beams through the turbulent atmosphere have been studied [25]. On the other hand, an elliptical Gaussian beam can be produced by semiconductor diode laser emitters [26] and its propagation dynamics in the turbulent atmosphere has been explored [27]. The propagation characteristics of the rotating beam through the isotropic optical system [28], the anisotropic medium [29], and the nonlinear medium [30] have caused great concern.

In the past few years, the optical coherent lattice (OCL) generated by a coherent light source with a periodic coherence has become a hot topic [31–35]. For practical applications, the optical lattice can be applied to the time-resolved imaging [31], optical tweezers [32], optical encryption [33], the cold atom interferometer [34] and sorting microfluidic [35]. Rely on the immunity of partially coherent beams for speckle formation in optical imaging [36] and their stability for the turbulence in natural media [37], great efforts have been devoted to investigating the coherent light optical communication, and the transmissions of various coherent laser beams have been studied [38]. In addition, experimental generation of the OCL carrying information has been reported [39]. Liang et al. introduced coherent vector sources with periodic spatial coherence properties as an extension of a recently introduced scalar OCL [40]. However, a limit in the propagations of the OCL in the turbulent atmosphere and the oceanic turbulence with anisotropy [41,42] was that the effect of the turbulence would finally destroy the structure of the OCL, which was regarded as a potential flaw for optical communications. Recently, rotating elliptical Gaussian beam propagating through the atmospheric turbulence and the oceanic turbulence has been studied by Zhang $et$ $al$ [43]. Sun $et$ $al$ [44] also analyzed the propagation properties of partially coherent radially and azimuthally polarized rotating elliptical Gaussian (PCRPREG and PCAPREG) beams through the anisotropic oceanic turbulence, and Huang $et$ $al$ [42] analyzed the average intensity and beam quality of OCL in oceanic turbulence with anisotropy. To date, the combination of the PCRPREG or the PCAPREG function and the OCL has not been researched, which has the potential to modulate the rotation of particles upon propagation and can be fruitful to optical communications in the underwater.

The paper is organized as follows. In Sec. 2, we firstly establish the theoretical model of the partially coherent radially and azimuthally polarized rotating elliptical Gaussian optical coherent lattices (PCRPREGOCL and PCAPREGOCL) in the anisotropic ocean turbulence. Based on the extended Huygens-Fresnel principle and the power spectrum of oceanic turbulence, the analytical expressions for the PCRPREGOCL and the PCAPREGOCL propagating through anisotropy oceanic turbulence are derived. Then in Sec. 3, the second-order statistics propagation properties including the spectrum density, the spectral DOC, the spectral DOP and the intensity Strehl ratio (SR) of the PCRPREGOCL and the PCAPREGOCL in anisotropic ocean turbulence have been discussed in detail. In particular, we deeply study the rotation property and the effects of the lattice constant $d$ on the evolution of the PCRPREGOCL in the anisotropic oceanic turbulence. Finally, we summarize the main results in Sec. 4.

2. The cross-spectral density matrix of PCRPREGOCL and PCAPREGOCL in anisotropic ocean turbulence

After considering the characteristics of the PCRPREG, the PCAPREG beams and the OCL, we have derived the analytical expression of the PCRPREGOCL and the PCAPREGOCL. Then the cross-spectral density of the PCRPREGOCL in the source plane $z=0$ can be expressed as follows [41,42,44]:

(1)$$\begin{aligned}{\mathbf{W}}_{r}({\mathbf{r}}_{1},{\mathbf{r}}_{2},0)&=A_0^{2}\dfrac{V_{n_{x}}V_{n_{y}}}{\pi}\sum_{n_{x}=0}^{N_{0}}\sum_{n_{y}=0}^{N_{0}}\exp{\bigg[}-\dfrac{2\pi in_{x}(x_1-x_2)}{dw_{0}}- \dfrac{2\pi in_{y}(y_1-y_2)}{dw_{0}}-\dfrac{x_1^{2}+x_2^{2}}{a^{2}w_{0}^{2}}\nonumber\\ &-\dfrac{y_1^{2}+y_2^{2}}{b^{2}w_{0}^{2}}+\dfrac{i(x_2y_2-x_1y_1)}{c^{2}w_{0}^{2}}{\bigg]}\begin{pmatrix} x_{1}x_{2} & x_{1}y_{2} \\ y_{1}x_{2} & y_{1}y_{2} \end{pmatrix}\exp\left[-\frac{(x_1-x_2)^{2}+(y_1-y_2)^{2}}{2\sigma_{0}^{2}}\right], \end{aligned}$$

(2)$$\begin{aligned}{\mathbf{W}}_{\theta}({\mathbf{r}}_{1},{\mathbf{r}}_{2},0)&=A_0^{2}\dfrac{V_{n_{x}}V_{n_{y}}}{\pi}\sum_{n_{x}=0}^{N_{0}}\sum_{n_{y}=0}^{N_{0}}\exp{\bigg[}-\dfrac{2\pi in_{x}(x_1-x_2)}{dw_{0}}- \dfrac{2\pi in_{y}(y_1-y_2)}{dw_{0}}-\dfrac{x_1^{2}+x_2^{2}}{a^{2}w_{0}^{2}}\nonumber\\ &-\dfrac{y_1^{2}+y_2^{2}}{b^{2}w_{0}^{2}}+\dfrac{i(x_2y_2-x_1y_1)}{c^{2}w_{0}^{2}}{\bigg]}\begin{pmatrix}y_{1}y_{2} & -y_{1}x_{2}\\-x_{1}y_{2} & x_{1}x_{2}\end{pmatrix}\exp\left[ -\frac{(x_1-x_2)^{2}+(y_1-y_2)^{2}}{2 \sigma_{0}^{2}}\right], \end{aligned}$$

where ${\mathbf {r}}_{1}=(x_{1},\;y_{1})$ and ${\mathbf {r}}_{2}=(x_{2},\;y_{2})$ are two arbitrary transverse position vectors in the source plane, $\sigma _{0}$ represents spatial coherence width, $w_{0}$ denotes the initial beam waist size of a rotating elliptical gaussian beam, $a$, $b$ and $c$ are the arbitrary real constants, $d$ is a lattice constant; $n_s$ is a non-negative integer ($s=x,\;y$); $V_{n_s}$ represents the power distribution of the pseudo-modes constituting the lattice, and $N_{0}$ indicates the number of lattice lobes, Eqs. (1) and (2) show that the source changes into the fundamental Gaussian beams for $N_{0}=0$. For a vector partially coherent beam in space-frequency domain, the second-order spatial coherence properties can be characterized by the $2 \times 2$ CSD matrix $W_{\alpha \beta }\left ({\mathbf {r}}_{1},{\mathbf {r}}_{2}\right )$ with elements $W_{\alpha \beta }\left ({\mathbf {r}}_{1},{\mathbf {r}}_{2}\right )=< E^{\ast }_{\alpha }\left ({\mathbf {r}}_{1}\right )E_{\beta }\left ({\mathbf {r}}_{2}\right )>$ $\left (\alpha ,\beta =x,\;y\right )$, the asterisk denotes the complex conjugate and the angular brackets represent ensemble average.

Here, we investigate the propagation of the PCRPREGOCL and the PCAPREGOCL in the anisotropic oceanic turbulence. On the basis of the well-known extended Huygens-Fresnel integral, the evolution of the PCRPREGOCL at the receive plane in the anisotropic oceanic turbulence under the paraxial approximation can be expressed as [42]:

(3)$$\begin{aligned}W_{\alpha\beta}({\boldsymbol{\rho}}_{1},{\boldsymbol{\rho}}_{2},\;z)=(\frac{k}{2 \pi z})^{2} \iiiint W_{\alpha \beta}( {\boldsymbol{r}}_{1} , {\boldsymbol{r}}_{2} , 0)\left\langle\psi^{{\ast}}({\boldsymbol{r}}_{1},{\boldsymbol{\rho}}_{1})+\psi({\boldsymbol{r}}_{2},{\boldsymbol{\rho}}_{2})\right\rangle \nonumber\\ \times \exp\left\lbrace -\frac{ik}{2z} \left[ ({\boldsymbol{r}}_{1} -{\boldsymbol{ \rho}}_{1} )^{2} - ( {\boldsymbol{r}}_{2}- {\boldsymbol{ \rho}}_{2} )^{2} \right] \right\rbrace d^{2}{{\boldsymbol{r}}_{1}}d^{2}{{\boldsymbol{r}}_{2}}, \end{aligned}$$

where $k=2\pi /\lambda$ is the wave number with $\lambda$ being the wavelength of the source beam, ${\boldsymbol {\rho }}_{1}=(x_1',\;y_1')$ and ${\boldsymbol {\rho }}_{2}=(x_2',\;y_2')$ are two arbitrary transverse position vectors at receiver plane, $\iiiint d^{2}{{\boldsymbol {r}}_{1}}d^{2}{{\boldsymbol {r}}_{2}}=\iiiint dx_1dy_1dx_2dy_2$. Under quadratic phase approximations, $\left \langle \psi ^{\ast }({\boldsymbol {r}_{1}} , {\boldsymbol {\rho }}_{1})+\psi ({\boldsymbol {r}}_{2} , {\boldsymbol {\rho }}_{2}) \right \rangle$ denotes the ensemble average of the anisotropic turbulent ocean water which can be represented as [15,45]:

(4)$$\left\langle \psi^{{\ast}}({\boldsymbol{r}}_{1} , {\boldsymbol {\rho}}_{1})+\psi({\boldsymbol {r}}_{2} , {\boldsymbol {\rho}}_{2}) \right\rangle= \exp \left[ -\frac{({\boldsymbol{r}}_{1}-{\boldsymbol{r}}_{2})^{2}+({\boldsymbol{r}}_{1}-{\boldsymbol{r}}_{2})\cdot( {\boldsymbol{ \rho}}_{1}- {\boldsymbol{ \rho}}_{2})+( {\boldsymbol{ \rho}}_{1}- {\boldsymbol{ \rho}}_{2})^{2}}{\rho_{o c \xi}^{2}} \right] .$$

In Eq. (4), $\rho _{oc \xi }$ is the lateral coherence length of the spherical wave in the anisotropic turbulent ocean, and the lateral coherence length $\rho _{oc \xi }$ is defined as [46]:

(5)$$\rho_{oc \xi}^{{-}2}=\dfrac{\pi ^{2} k^{2} z \xi^{{-}4}}{3} \int_{0}^{\infty} \kappa '^{3}\tilde{\psi}_{an}(\kappa')d\kappa',$$

with $\int _{0}^{\infty } \kappa '^{3}\tilde {\psi }_{an}(\kappa ')d\kappa '$ and $\xi$ representing the turbulence strength and the anisotropic factor, respectively. We can see that the anisotropic ocean turbulence intensity is inversely related to the coherence length $\rho _{oc \xi }$. This is to say, the bigger $\rho _{oc \xi }$ is, the weaker the turbulence is. Here we only consider the effects of temperature and salinity fluctuations on beams propagation in the anisotropic ocean turbulence with ignoring the scattering and absorption of the ocean water, and the −11/3 power law is picked for the case of study, In the Markov approximation, the two-dimensional refractive index turbulence spatial spectrum model $\tilde {\psi }_{an}(\kappa ')$ for the anisotropic oceanic turbulent is written as [15,24,42,44]:

(6)$$\begin{aligned}\tilde{\psi}_{an}(\kappa')&= 0.388 \times 10^{{-}8} \varepsilon^{{-}1/3}\chi_{T} \xi^{2} \left( \kappa'\right) ^{{-}11/3}\left[ 1+2.35\left( \kappa'\eta\right) ^{2/3}\right]\nonumber\\ &\times\left[ \exp\left({-}A_{T}\delta\right) +\omega^{{-}2}\exp\left({-}A_{S}\delta\right)-2\omega^{{-}1}\exp\left ({-}A_{TS}\delta\right)\right], \end{aligned}$$

where $\varepsilon$ 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 $10^{-10}m^{2}/s^{3}$, $\chi _{T}$ denotes the mean-square temperature dissipation rate taking values in the range from $10^{-10}$ K$^{2}$/s to $10^{-4}$ K$^{2}$/s (K represents the degree Kelvin), $\eta$ is the Kolmogorov micro scale, $\omega$ represents the ratio of the temperature and salinity contribution to the refractive index spectrum, whose value in the ocean water can vary in the interval $[-5, 0]$, with $-5$ and $0$ corresponding to temperature-induced and salinity-induced dominating oceanic turbulence. $A_{T} = 1.863 \times 10^{-2}$ , $A_{S} = 1.9 \times 10^{-4}$ , $A_{TS} = 9.41 \times 10^{-3}$ , $\delta = 8.284 \left ( \kappa '\eta \right )^{4/3} + 12.378 \left ( \kappa '\eta \right )^{2}$ and $\kappa ' = \xi \sqrt { \kappa _{x}^{2}+\kappa _{y}^{2} }$. Substituting Eq. (6) into Eq. (5), the coherence length $\rho _{oc \xi }$ of the spherical wave in anisotropic oceanic turbulence can be obtained [42]:

(7)$$\rho_{oc \xi} = \xi \left| \omega \right| \left[ 1.802 \times 10^{{-}7} k^{2} z \left( \varepsilon\eta\right) ^{{-}1/3} \chi_{T} \left( 0.483 \omega^{2} - 0.835 \omega + 3.380\right) \right] ^{{-}1/2},$$

we can see from Eq. (7) that the decrease of $\varepsilon$ and $\xi$ or the increase of $\chi _{T}$ and $\omega$ leads to the larger coherence length, and then the turbulence strength becomes strong.

Afterwards. on substituting Eq. (7) and Eq. (4) into Eq. (3) with the relation [47]

(8)$$\int_{-\infty}^{\infty} x^{n} \exp \left({-}px^{2} + qx \right) dx = n!\exp\left( \dfrac{q^{2}}{p} \right)\left( \dfrac{q}{p} \right)^{n}\sqrt{\dfrac{\pi}{p}}\sum_{l=0}^{n/2}\dfrac{1}{l!(n-2l)!} \left( \dfrac{q^{2}}{4p} \right)^{l},$$

the analytic expression of the average intensity for the PCRPREGOCL can be expressed as:

(9)$$\begin{aligned} W_{rxx}({\boldsymbol{\rho}_{1}},{\boldsymbol{\rho}_{2}},\;z)=\sum_{n_{x}=0}^{N_{0}}\sum_{n_{y}=0}^{N_{0}}\dfrac{G}{2L_{4}}(q_{3}+\dfrac{q_{3}Q^{2}}{2L_{4}}+q_{4}Q+2q_{5}L_{4})\exp\left(\dfrac{M^{2}}{4L_1}+\dfrac{N^{2}}{4L_2}+\dfrac{P^{2}}{4L_3}+\dfrac{Q^{2}}{4L_4}\right), \end{aligned}$$

(10)$$\begin{aligned} W_{rxy}({\boldsymbol{\rho}_{1}},{\boldsymbol{\rho}_{2}},\;z)=\sum_{n_{x}=0}^{N_{0}}\sum_{n_{y}=0}^{N_{0}}\dfrac{G}{2L_{4}}(q_{6}+\dfrac{q_{6}Q^{2}}{2L_{4}}+q_{7}Q)\exp\left(\dfrac{M^{2}}{4L_1}+\dfrac{N^{2}}{4L_2}+\dfrac{P^{2}}{4L_3}+\dfrac{Q^{2}}{4L_4}\right), \end{aligned}$$

(11)$$\begin{aligned} W_{ryx}({\boldsymbol{\rho}_{1}},{\boldsymbol{\rho}_{2}},\;z) =W_{rxy}^{{\ast}}( {\boldsymbol{\rho}_{1}},{\boldsymbol{\rho}_{2}},\;z) \end{aligned},$$

(12)$$\begin{aligned} W_{ryy}({\boldsymbol{\rho}_{1}},{\boldsymbol{\rho}_{2}},\;z)=\sum_{n_{x}=0}^{N_{0}}\sum_{n_{y}=0}^{N_{0}}\dfrac{G}{2L_{4}}(q_{8}+\dfrac{q_{8}Q^{2}}{2L_{4}}+q_{9}Q)\exp\left(\dfrac{M^{2}}{4L_1}+\dfrac{N^{2}}{4L_2}+\dfrac{P^{2}}{4L_3}+\dfrac{Q^{2}}{4L_4}\right), \end{aligned}$$

with

T=\dfrac {1}{\rho _{o c \xi }^{2}}, G=\dfrac {A_0^{2}k^{2}V_{n_{x}}V_{n_{y}}}{4{\pi }z^{2}w_{0}^{2}\sqrt {L_{1}L_{2}L_{3}L_{4}}}\exp \Big [-\dfrac {ik}{2z}(\boldsymbol {\rho _{1}}^{2}-\boldsymbol {\rho _{2}}^{2})-T\left ( \boldsymbol {\rho _{1}} - \boldsymbol {\rho _{2}} \right )^{2}\Big ],

u=T+\dfrac {1}{2 \sigma _{0}^{2}}, v=\dfrac {i}{c^{2}w_{0}^{2}}, L_{1}=\dfrac {ik}{2z}+\dfrac {1}{{a^{2}}{w^{2}}}+u, M=\dfrac {-2i{\pi }n_{\textrm {x}}}{dw}+\dfrac {ikx_{1}'}{z}+T(x_{2}'-x_{1}'),

L_{2}=\dfrac {1}{{b^{2}}{w^{2}}}+\dfrac {ik}{2z}+u-\dfrac {v^{2}}{4{L_{1}}}, N=\dfrac {-2i{\pi }n_{\textrm {y}}}{dw}+\dfrac {iky_{1}'}{z}+T(y_{2}'-y_{1}')-\dfrac {vM}{2L_{1}}, R=1-\dfrac {u^{2}}{L_{1}L_{2}},

L_{3}=\dfrac {1}{a^{2}w^{2}}-\dfrac {ik}{2z}+u-\dfrac {u^{2}}{{L_{1}}}-\dfrac {v^{2}u^{2}}{4L_{1}^{2}L_{2}}, P=\dfrac {2i{\pi }n_{\textrm {x}}}{dw}-\dfrac {ikx_{2}'}{z}+T(x_{1}'-x_{2}')+\dfrac {uM}{L_{1}}-\dfrac {uvN}{2L_{1}L_{2}},

L_{4}=\dfrac {1}{b^{2}w^{2}}-\dfrac {ik}{2z}+u-\dfrac {u^{2}}{{L_{2}}}-\dfrac {v^{2}R^{2}}{4L_{3}}, Q=\dfrac {2i{\pi }n_{\textrm {y}}}{dw}-\dfrac {iky_{2}'}{z}+T(y_{1}'-y_{2}')+\dfrac {uN}{L_{2}}+\dfrac {vPR}{2L_{3}},

q_{1}=\dfrac {M}{2L_{1}}, q_{4}=\dfrac {q_{2}vPR}{2L_{3}^{2}}+\dfrac {q_{1}vR}{2L_{3}}-\dfrac {vuP}{4L_{1}L_{2}L_{3}}, q_{7}=\dfrac {q_{2}P}{2L_{3}}+q_{1},

q_{2}=\dfrac {u}{L_{1}}+\dfrac {v^{2}u}{4L_{1}^{2}L_{2}}, q_{5}=\dfrac {q_{2}}{2L_{3}}+\dfrac {q_{2}P^{2}}{4L_{3}^{2}}+\dfrac {q_{1}P}{2L_{3}}, q_{8}=\dfrac {u}{L_{2}}-\dfrac {v^{2}uR}{4L_{1}L_{2}L_{3}},

q_{3}=\dfrac {q_{2}v^{2}R^{2}}{4L_{3}^{2}}-\dfrac {v^{2}uR}{4L_{1}L_{2}L_{3}}, q_{6}=\dfrac {q_{2}vR}{2L_{3}}-\dfrac {vu}{2L_{1}L_{2}}, q_{9}=\dfrac {N}{2L_{2}}-\dfrac {vuP}{4L_{1}L_{2}L_{3}}.

In the similar way, we obtain the following formulas for CSD matrix of a PCAPREGOCL propagating in the anisotropy oceanic turbulence [44]:

(13)$$\begin{aligned}W_{\theta xx}( {\boldsymbol{\rho}_{1}},{\boldsymbol{\rho}_{2}},\;z) &=W_{r yy}( {\boldsymbol{\rho}_{1}},{\boldsymbol{\rho}_{2}},\;z),W_{\theta yy}( {\boldsymbol{\rho}_{1}},{\boldsymbol{\rho}_{2}},\;z) =W_{r xx}( {\boldsymbol{\rho}_{1}},{\boldsymbol{\rho}_{2}},\;z),\nonumber\\ &W_{\theta xy}( {\boldsymbol{\rho}_{1}},{\boldsymbol{\rho}_{2}},\;z) =W_{\theta yx}( {\boldsymbol{\rho}_{1}},{\boldsymbol{\rho}_{2}},\;z)={-}W_{r xy}( {\boldsymbol{\rho}_{1}},{\boldsymbol{\rho}_{2}},\;z). \end{aligned}$$

Substituting ${\boldsymbol {\rho }_{1}}={\boldsymbol {\rho }_{2}}={\boldsymbol {\rho }}$ into Eqs. (9)–(12), the spectral density $I \left ( \boldsymbol {\rho } , z \right )$ , the spectral DOC $\mu (\boldsymbol {\rho }_{1},\boldsymbol {\rho }_{2},\;z)$ and the spectral DOP $P_{0}\left ( \boldsymbol {\rho } , z \right )$ at the receiver plane can be expressed by the formulas as follows [25]:

(14)$$I \left( \boldsymbol{\rho} , z \right) = \textrm{Tr} \mathbf{W} \left( \boldsymbol{\rho} , \boldsymbol{\rho} ; z \right) = W_{rxx}\left( \boldsymbol{\rho} , \boldsymbol{\rho} , z \right) + W_{ryy}\left( \boldsymbol{\rho} , \boldsymbol{\rho} , z \right),$$

(15)$$\mu (\boldsymbol{\rho}_{1},\boldsymbol{\rho}_{2},\;z)=\dfrac{\textrm{Tr} \mathbf{W} \left( \boldsymbol{\rho}_{1} , \boldsymbol{\rho}_{2} ; z \right)}{ \sqrt{ I \left( \boldsymbol{\rho}_{1} , z \right)} \sqrt{ I \left( \boldsymbol{\rho}_{2} , z \right)}},$$

(16)$$P_{0}\left( \boldsymbol{\rho} , z \right) = \sqrt{ 1 - \dfrac{4 \textrm{Det} \mathbf{W} \left( \boldsymbol{\rho} , \boldsymbol{\rho} ; z \right)}{ \left[ \textrm{Tr} \mathbf{W} \left( \boldsymbol{\rho} , \boldsymbol{\rho} ; z \right) \right]^{2} } } = \sqrt{ 1 - \dfrac{4\left( W_{rxx}W_{ryy}-W_{rxy}W_{ryx} \right) }{ \left( W_{rxx}+W_{ryy}\right)^{2} }},$$

where ${\rm Tr}$ and ${\rm Det}$ denote the trace and determinant of the matrix. Eq. (14) shows that the average intensity of the PCRPREGOCL depends on the initial pattern waist $w_{0}$, the lattice constant $d$, the propagation distance $z$, the number of lattice lobes $N_{0}$ and the anisotropic ocean turbulence parameters.

Equations (13)–(16) indicate that the PCRPREGOCL and the PCAPREGOCL have the same evolution properties.

3. Results and discussions

According to the analytical expression derived in the previous section, the behaviors of the second-order statistics properties of the PCRPREGOCL through the anisotropic oceanic turbulence by a set of numerical examples are analyzed in this section. In our simulation, some parameters are chosen as: $a=2$, $b=0.5$, $c=0.1$, $d=0.01m$, $w_{0}=0.03m$, $A_0=1$ and the Kolmogorov micro scale (inner scale) $\eta =0.001m$, $\sigma _{0}=3mm$, $\lambda =532nm$, $\xi =1.0$, $\varepsilon =10^{-5} m^{2}/s^{3}$, $\chi _{T}=10^{-9}K^{2}/s$, $\omega =-2.5$. Hereafter, the parameters are the same as those aforesaid except other stated. In particular, the wavelength is fixed at $532nm$ since the green light is more suitable in the underwater channel. In order to conveniently calculate the results, we use the Rayleigh distance $Z_{R}=kw_{0}^{2}/2=5.3km$ here.

Figure 1 represents the evolution of the spectral density of the PCRPRFGOCL with propagation distance in the anisotropic ocean turbulence. In the case of lattice constant $d=0.01m$, $N_{0}=5$ and elliptic coefficients $a=2$, $b=0.5$ and $c=0.1$, it can be seen that the initial elliptical Gaussian pattern gradually splits into the Gaussian array and then evolves to a rectangular shape and the pattern rotates during propagation and this phenomenon is known as the periodic reciprocity, however, Liu $et$ $al$ [43] have found that periodic reciprocity will be broken down at a certain distance after propagation in the turbulence. Interestingly, the beam center is not at the center of the coordinate, but is moving to the right, and each sub-pattern of the array rotates counterclockwise around its pattern center in a synchronous motion within a certain range of propagation (about $0.3Z_R$). From Fig. 1(a), initially, the major axis is parallel to the x-axis in the initial plane while parallel to the y-axis at the distance of $0.3Z_R$, as shown in Fig. 1(d). The physics behind the phenomenon of rotation is the nonuniformity of lateral energy flow [48]. As the propagation distance increases, the angular momentum and the angular velocity of the PCRPREGOCL gradually decrease to zero, so the intensity mode becomes non-rotating and the maximum rotation angle is $90^{\circ }$. More specifically described in Fig. 1(c), the initial elliptical Gaussian pattern splits into an array of $(N+1)\times (N+1)$ Gaussian patterns, and the angle between the patterns’ major axis and the $x$ -axis of each sub-pattern is about $45^{\circ }$ at $z=0.02$ $Z_R$ (about $108m$). Subsequently, it can be seen from Figs. 1(a)–1(d) that as the propagation distance increases, the angle of rotation changes slower and slower. Moreover, the elliptical Gaussian array becomes an array of strip patterns and gradually becomes a rectangular shape pattern and finally back to an ellipse Gaussian pattern with the propagation distance increasing.

Fig. 1. The spectral intensity distribution of the PCRPREGOCL for different propagation distances: (a) $z=0Z_R$, (b) $z=0.003Z_R$, (c) $z=0.02Z_R$, (d) $z=0.3Z_R$, (e) $z=1.5Z_R$, (f) $z=6Z_R$.

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Such a situation is indicated in Fig. 2. Numerical analysis is performed by increasing the lattice constant to $d=0.1$ with the other parameters being the same as before. It can be seen from Fig. 2 that the lattice constant $d$ can control the evolution mode of the spectral intensity pattern upon propagation for the PCRPREGOCL in addition to reappearing the periodicity reciprocity [44]. More specifically, the initial elliptical Gaussian pattern gradually splits into (2N+1) elliptical Gaussian sub-patterns under the action of rotation and periodic reciprocity whose angle between the major axis and the $x$ -axis is approximately $45^{\circ }$, as shown in Fig. 2(b). Then, when the periodic reciprocity completely breaks down under the influence of the anisotropic ocean turbulence and the propagation distance, the angular momentum and the angular velocity of the PCRPREGOCL gradually decrease to zero, and the PCRPREGOCL is no longer an array of patterns, but also becomes a non-rotating rectangular beam, as illustrated in Fig. 2(c). Finally, the PCRPREGOCL can maintain the rectangular lattice shape over a certain propagation distance, while this shape will eventually change into the elliptical Gaussian pattern, as shown in Fig. 2(d).

Fig. 2. The evolution of the spectral intensity for the PCRPREGOCL with $d=0.1$ at different propagation distances in the anisotropic ocean turbulence: (a) $z=0Z_R$, (b) $z=0.02Z_R$, (c) $z=1Z_R$, (d) $z=5Z_R$.

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Further research on the propagation behavior of the PCRPREGOCL shows that the elliptic parameter $c$ plays an active role in changing the velocity of rotation, while the lattice constant $d$ plays a positive role in altering the periodicity reciprocity. The aforementioned PCRPREGOCL becomes a non-rotating pattern when it propagates to about $0.3Z_R$ and the maximum rotation angle is $90^{\circ }$. As can be seen from Fig. 3 sections (a1 to a2), when $c$ is small, the initial PCRPREGOCL spectral intensity pattern spins so fast that the rotation angle reaches to $90^{\circ }$ before $z=1.2Z_R$. Therefore, there is no periodic reciprocity in the PCRPREGOCL, but with the increase of $c$, its velocity of rotation becomes slower, and at the same time, the periodic reciprocity can reappear. Moreover, the lattice constant $d$ does not change the rotating velocity of the PCRPREGOCL, but as the lattice constant $d$ increases, the destruction of the periodic reciprocity will accelerate, as indicated in Fig. 3 sections (a3 to d3) and (a4 to d4). In other words, the periodicity reciprocity of the PCRPREGOCL in the anisotropic ocean turbulence can be reproduced with the decrease of the lattice constant $d$ and disappears after propagating through a certain distance.

Fig. 3. The evolution of the spectral intensity of the PCRPREGOCL with different elliptical coefficients $c$ and lattice constants $d$ in the anisotropic ocean turbulence at different propagation distances: (a1)-(d1) $z=0.01Z_R$, (a2)-(d2) $z=1.2Z_R$ and with different elliptical coefficients: (a1) and (a2) $c=0.01$, (b1) and (b2) $c=0.06$, (c1) and (c2) $c=0.1$, (d1) and (d2) $c=0.5$. (a3)-(d3) $z=0.01Z_R$, (a4)-(d4) $z=0.5Z_R$ and with different lattice constants: (a3) and (a4) $c=0.005$, (b3) and (b4) $d=0.01$, (c3) and (c4) $d=0.05$, (d3) and (d4) $d=0.2$.

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Figure 4 shows the influence of ocean parameters on the evolution of the PCRPREGOCL spectral density at $z=4Z_R$. Figure 4(a) gives the spectral intensity of the PCRPREGOCL under different $\varepsilon$, and the larger $\varepsilon$ leads to the emergence of the periodicity reciprocity, thus the distance for the PCRPREGOCL holding the periodicity reciprocity increases, also the similar conclusion can be obtained from Figs. 4(c) and 4(e) where the decrease of $\xi$ and $\eta$ makes the PCRPREGOCL begin to show the periodicity reciprocity, however, the opposite results are presented for a larger anisotropic parameter in Figs. 4(b) and 4(d) comparing with Fig. 4(e), and with the increase of $\omega$ and $\chi _{T}$, the velocity of evolving into Gaussian patterns will accelerate. From Eq. (7), the increase of $\varepsilon$ and $\xi$ or the decreases of $\chi _{T}$ and $\omega$ leads to the larger coherence length, and then the turbulence strength becomes weak. Therefore one result can be concluded that the periodicity reciprocity preservation distance for the PCRPREGOCL decreases for the stronger oceanic turbulence.

Fig. 4. The spectral intensity of the PCRPREGOCL in the anisotropic ocean turbulence with different $\varepsilon$ in (a), $\chi _{T}$ in (b), $\eta$ in (c), $\omega$ in (d) and $\xi$ in (e) at $z=4Z_R$ .

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Figure 5 presents the curves of the normalized spectral density distribution $I/I_{max}$ ($I_{max}$ —maximum spectral density) versus $x$ for the PCRPREGOCL through the anisotropic oceanic turbulence by controlling different variables. It can be seen from Fig. 5(a) that in the initial plane, the normalized spectral density distribution of the central portion is zero, and the initial symmetry is about $y$ axis and two peaks at both sides, as the propagation distance increases further, it evolves into a Gaussian-like unimodal distribution and the center of the peak is not at $x=0$ due to the properties of the optical coherent lattice. During the evolution in Fig. 5(a), the distance between the initial two peaks is almost invariable when it is still bimodal, but with the propagation distance increasing, its minimum rises to 1 until it becomes unimodal distribution. As shown in Figs. 5(b) and 5(c), the smaller $\chi _{T}$ and $w_{0}$ are, the faster the change of the beam spectral density distribution is, and the PCRPREGOCL is more likely to evolving into a Gaussian-like unimodal distribution, and that the smaller $\lambda$ and $\sigma _{0}$ are, the slower the change of the beam spectral density distribution is, as shown in Figs. 5(d) and 5(e). We find out that the normalized spectral density distribution is the most concentrated when $w_{0}=0.06m$ by changing the initial pattern width, as shown in Fig. 5(c). The physical characteristic behind this phenomenon is that the rotation effect causes the effective beam width of the PCRPREGOCL in the $x$ direction to have a larger initial beam width, which increases more slowly over a certain propagation distance. Moreover, with a bigger $\sigma _{0}$, the normalized spectrum density distributes more concentrated, as depicted in Fig. 5(e). From Figs. 5(b)–5(e), it can be seen that the evolution behavior of the normalized spectral density distribution of the PCRPREGOCL through the anisotropic ocean turbulence is deeply affected by anisotropic ocean turbulence parameters including $\chi _{T}$, $w_{0}$, $\lambda$ and $\sigma _{0}$. Whereas, the anisotropic factor plays a less definitive role in normalized spectral density distribution, as depicted in Fig. 5(f). In addition, we discover that the effect of the anisotropic ocean turbulent factor is much smaller than that of the PCRPREGOCL’s intrinsical parameters within a short propagation range. From Fig. 5, it can be seen that the normalized spectral density distribution of the PCRPREGOCL changes from the initial bimodal distribution to the unimodal distribution. The evolution of the spectrum density is dominated due to the rotation of the PCRPREGOCL, further illustrating that the PCRPREGOCL has a better capacity of anti-interference for turbulent disturbance.

Fig. 5. The normalized spectral density distribution of the PCRPREGOCL as a function of $x$ through the anisotropic oceanic turbulence with different $z$ in (a), $\chi _{T}$ in (b), $w_{0}$ in (c), $\lambda$ in (d), $\sigma _{0}$ in (e) and $\varepsilon$ in (f) at $z=100m$ except (a).

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To learn more about the partial coherence properties of the PCRPREGOCL through anisotropic ocean turbulence, the variation of the spectral DOC is displayed in Fig. 6. One can see from Fig. 6(a) that the $\mu$ decreases quickly from 1 to a negative value and increases slowly from a negative value to 0 with the increase $x$, and the larger the wavelength $\lambda$ is, the smaller the minimum value of $\mu$ is. With the increase of the propagation distance $z$, the $\mu$ decreases from 1 to −1, and then increases to 1, and as the wavelength $\lambda$ decreases, the minimum value of $\mu$ has a very small increase, as shown in Fig. 6(b). It is not difficult to find from Figs. 6(a) and 6(b) that the choice of wavelength has a great influence on the DOC of the PCRPREGOCL. The effects of the anisotropic oceanic turbulence factors on the $\mu$ are shown in Figs. 6(c)–6(f). As can be seen from Figs. 6(c)–6(e), when the $\omega$ is less than $-2$, the $\mu$ is almost a fixed value. The $\mu$ gradually increases with the increase of $\omega$, and the increasing trend becomes sharp for $\omega$ close to 0. Namely, the salinity-induced turbulence fluctuation makes a greater contribution to the decrease of the beam quality compared with the temperature-induced turbulence fluctuation. As can be seen from Fig. 6(f), the $\mu$ oscillates back and forth between −1 and 1 as the propagation distance $z$ increases. Whereas, the anisotropic factor $\xi$ plays a less definitive role in the spectral DOC.

Fig. 6. The spectral DOC of the PCRPREGOCL through the anisotropic oceanic turbulence, (a) versus $x$ with different $\lambda$ at $z=50m$, (b) versus $z$ with different $\lambda$, (c) versus $\omega$ with different $\xi$ at $z=50m$, (d) versus $\omega$ with different $\varepsilon$ at $z=50m$, (e) versus $\omega$ with different $z$ and (f) versus $z$ with different $\xi$.

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The spectral DOP of the PCRPREGOCL and the corresponding cross line $(y'=0)$ with different parameters through the anisotropic ocean turbulence are investigated in Fig. 7 and Fig. 8. The DOP of the PCRPREGOCL equals 1 for all the points across the entire transverse plane when $z=0$, which is manifested in Fig. 7 sections (a1 to a3). One can see from Fig. 7 sections (b1 to f1) that the distribution of the DOP appears a unimodal distribution of handstands and the DOP distribution curve reaches the lowest point when $x=0$, the peaks are increasing with the propagation distance $z$ increasing, but the distribution is nearly unchanged, which implies that a radial polarization structure of a PCRPREGOCL is destroyed during the propagation in an anisotropic oceanic turbulence. We also find out that the width of the dip increases as the increases of the propagation distance. However, the DOP distribution evolves from a unimodal distribution to a bimodal distribution, and further side lobes appear in the middle of the bimodal with the propagation distance increasing, as is depicted in Fig. 7 sections (b3 to f3). Furthermore, one also finds from Figs. 8(a)–8(c) that the DOP curve is symmetric about the $x=0$ when $N_{0}=0$. Moreover, the DOP curve is symmetric around a fixed value instead of $x=0$ when $N_{0}=5$, and the larger the wavelength $\lambda$ is and the smaller the $\varepsilon$ is, the bigger fixed value is in Figs. 8(d)–8(f). As Fig. 8 displays, the full width at half maximum is greater for the larger $\omega$ and $\lambda$ and the smaller $\varepsilon$.

Fig. 7. The spectral DOP of the PCRPREGOCL and the corresponding cross line $(y'=0)$ with different the number of lattice lobes $N_{0}$ in the anisotropic ocean turbulence at different propagation distances: (a1)-(f1) the corresponding cross line $(y'=0)$ with the number of lattice lobes $N_{0}=0$ for different values of $z$, (a2)-(f2) the spectral DOP with $N_{0}=0$ for different values of $z$, (a3)-(f3) the spectral DOP with $N_{0}=5$ for different values of $z$.

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Fig. 8. Cross lines $(y'=0)$ of the DOP of the PCRPREGOCL for difference values of $\omega$, $\lambda$ and $\varepsilon$: (a)-(c) with $N_{0}=0$, (d)-(f) with $N_{0}=5$.

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To evaluate the beam quality of the PCRPREGOCL in the anisotropic ocean turbulence, the SR is investigated next. Generally, the SR is identified as a vital parameter of the beam quality which implies that how much the peak intensity decreases, which is given by [49,50]

(17)$$SR = \dfrac{I_{\textrm{max}}}{I_{0\textrm{max}}},$$

where $I_{\textrm {max}}$ and $I_{0\textrm {max}}$ are the peak intensities in the anisotropic ocean turbulence and the free space, respectively.

Figure 9 shows the evolution of the SR versus $z$ with several anisotropic ocean turbulence parameters, elliptical coefficients and the number of lattice lobes for the PCRPREGOCL, in which the change trend of SR first decreases rapidly, then increases rapidly and finally decreases slowly. It indicates that the beam quality becomes worse during the propagation in the anisotropic ocean turbulence. However, the declining pace of the SR becomes slower with $c=0.06$ and $c=0.08$ than that with $c=0.1$. Meanwhile, there are three dips: $z=40m$ when $c=0.06$, $z=80m$ when $c=0.08$ and $z=120m$ when $c=0.1$. Surprisingly, the angle between the long-axis and $x$-axis of the PCRPREGOCL sub-beams just comes out to be $45^{\circ }$ at the corresponding location of the dips mentioned above. As can be seen from Fig. 1 and Fig. 2, when the PCRPREGOCL rotates to $45^{\circ }$, the light intensity will average to each sub-beam, $I_{\textrm {max}}$ will become smaller, so SR of the PCRPREGOCL through the anisotropic ocean turbulence will appear a dip. They all have a dip, and the smaller $c$, the earlier the dip appears along with a smaller value of the SR, as is depicted in Fig. 9(a). From Fig. 9(b), they all have a dip with the same propagation distance, and that the smaller $\chi _{T}$ is, the larger the value of the dip is. In general, the greater the value of $\chi _{T}$ is, the stronger the turbulence is. Whereas, the $\omega$ and $N_{0}$ play a less definitive role in the evolution of the SR, as depicted in Fig. 5(f). From the analysis in Fig. 9, SR decreases the slowest and the beam quality is the best in the weak ocean turbulence.

Fig. 9. The SR of the PCRPREGOCL through the anisotropic ocean turbulence from $0$ to $100m$ with different $c$ in (a), $\chi _{T}$ in (b), $\omega$ in (c) and $N_{0}$ in (d).

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4. Conclusion

In this paper, we have introduced a new kind of the OCL with a controllable rotation during the propagation in the anisotropic ocean turbulence, and that the analytical formulae for the PCRPREGOCL and the PCAPREGOCL propagating through the anisotropy oceanic turbulence are derived based on the extended Huygens-Fresnel principle and the power spectrum of the oceanic turbulence. The second-order statistics propagation properties including the spectrum density, the DOC, the DOP and the SR of the PCRPREGOCL and the PCAPREGOCL in the anisotropic oceanic turbulence have been discussed in detail. Our results show that the propagation properties of the PCRPREGOCL are closely related to the waist size $w_{0}$, the coherence width $\sigma _{0}$, the propagation distance $z$, the lattice constant $d$ and the elliptical coefficient $c$ and the oceanic turbulence parameters. It deepens our insight that the lattice constant $d$ and the elliptical coefficient $c$ are capable to alter the evolution of the average intensity profile under the effect of the outer oceanic turbulence, the beam rotation and the inner periodicity reciprocity. With the propagating distance increasing, the intensity pattern of the PCRPREGOCL goes through the following steps: a double ellipse Gaussian pattern, ellipse Gaussian array and a single flat-topped Gaussian pattern then back to a single ellipse Gaussian pattern by altering a proper lattice constant $d$. Interestingly, the pattern center is not at the center of the coordinate, but is moving to the right as the propagating distance increases. The behavior of the PCRPREGOCL within the rotation-preservation distance is deeply investigated and the rotation velocity of the PCRPREGOCL can be controlled by adjusting the elliptical coefficient $c$. The larger the wavelength $\lambda$ and the spatial coherence width $\sigma _{0}$ are and the smaller the rate of dissipation of the mean square temperature $\chi _{T}$ and the pattern waist size $w_{0}$ are, the faster the evolving of the pattern is. In addition, the full width at half maximum is greater for the larger $\omega$ and $\lambda$ and the smaller $\varepsilon$. Moreover, we also find that the elliptical coefficient $c$ is smaller and the oceanic turbulence is weaker, the value of SR decreases more slowly and the beam quality is better. It is rather remarkable that the radial polarization structure and the periodic reciprocity of a PCRPREGOCL are destroyed due to the anisotropic influence of the oceanic turbulence. So our results will spark a renewed interest in the OCL and may have potential applications in the underwater optical communication and trapping.

Funding

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

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Statistical properties of a controllable partially coherent radially and azimuthally polarized rotating elliptical Gaussian optical coherence lattice in anisotropic ocean turbulence

Abstract

1. Introduction

2. The cross-spectral density matrix of PCRPREGOCL and PCAPREGOCL in anisotropic ocean turbulence

3. Results and discussions

4. Conclusion

Funding

References

Cited By

Figures (9)

Equations (25)

Optics Express