Abstract

Analytical formulas for the cross-spectral density matrix of a partially coherent radially polarized Laguerre–Gaussian vortex (PCRPLGV) beam in anisotropic atmospheric turbulence are derived based on the extended Huygens-Fresnel principle. The evolution laws of statistical properties of a PCRPLGV beam in turbulence, such as the average intensity, degree of coherence (DOC) and degree of polarization (DOP), are investigated in detail. It is found that the atmospheric turbulence induces degeneration of the intensity distribution of a PCRPLGV beam on propagation, and some new properties, such as self-shaping and self-rotating, will appear on propagation due to vortex phase. In addition, in order to verify our theoretical results, we combined the complex screen method and multi-phase screens method to simulate the propagation of PCRPLGV beam in atmospheric turbulence. And the studies indicate that the simulation results are consistent with the theoretical predictions. Our results will be useful in some potential applications, such lidar detection, remote sensing and free-space optical communications.

© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. Introduction

Free space optical communication (FSOC) is a branch developed in microwave communication, which generally refers to the use of laser as the carrier of communication technology. In 1992, Allen et. al. found that each photon in a beam with a phase factor exp(imθ) has an orbital angular momentum (OAM) of ħm, where m is the topological charge carried by the beam, ħ is the reduced Planck constant, and this kind of beam with OAM is called vortex beam [1]. In optical information transmission, the advantage of vortex beam is that OAM coding or OAM multiplexing can be used to greatly improve the channel capacity of communication system [2,3]. Laguerre–Gaussian (LG) beam is a classical vortex beam and carry the OAM. LG beams have found applications ranging from optical manipulation of Bose-Einstein condensates to the optical imaging, biomedicine, quantum information coding and other fields [410].

It is well known that the atmospheric turbulence will cause a series of harmful turbulence effects such as light intensity scintillation, phase distortion and so on, which will lead to the increase of the communication error rate and reduce the channel capacity of the system. On the other hand, atmospheric turbulence is usually assumed to be statistical homogeneous and isotropic in most cases. However, in actual atmospheric measurements, a large number of experimental data show that the atmospheric turbulence spectrum deviates from the Kolmogorov turbulence spectrum. Therefore, Toselli et al. firstly proposed a non-Kolmogorov turbulence spectrum model, taking into account the effects of inner and outer scales of turbulence [11]. In 2014, Toselli systematically introduced the concept of anisotropy at different atmospheric turbulence scales using anisotropic factors [12].

In the past few decades, a great deal of researches about the effects of atmospheric turbulence on beams and the anti-turbulence ability of beams have been reported [1322]. It can be found that partially coherent (PC) beam has more advantages than the fully coherent beam. For example, the PC beam is less affected by the turbulent atmosphere, and is more stable than the fully coherent beam. The PC beam can effectively eliminate some harmful phenomena such as speckle and non-uniform intensity distribution [1416,23]. Therefore, PC beams are widely used in optical communication, laser scanning, holographic technology, optical capture [1318]. Besides, the non-uniformly polarized beams [radially polarized (RP) beam, azimuthally polarized beam, etc.] can also effectively reduce the turbulence effect caused by atmospheric turbulence [1922]. Gu et.al found that the non-uniformly polarized light has an advantage over linearly polarized light in reducing the scintillation caused by turbulence [19,20]. In addition, the literatures also indicate that PC vortex beam can not only improve the communication capacity, but also has a strong ability to resist atmospheric turbulence [2429]. Therefore, we investigate a partially coherent radially polarized Laguerre–Gaussian vortex (PCRPLGV) beam, which contains three anti-turbulence structures: coherence, polarization and phase structures. To the best of our knowledge, the propagation properties of a PCRPLGV beam in anisotropic atmospheric turbulence have not been studied so far.

Therefore, in this paper, we first derive the analytical formulas for the cross-spectral density function of PCRPLGV beam through the anisotropic atmospheric turbulence based on the extended Huygens-Fresnel principle, then study the statistical properties including average intensity, degree of coherence (DOC) and degree of polarization (DOP) of a PCRPLGV beam and analyze the evolution laws. Finally, we use the multi-phase screen method to verify the theoretical results. And the simulation results agree well with the theoretical results.

2. Theoretical model

The expression of the electric field of a standard LG beam in cylindrical coordinate system is expressed as [26,30]:

$$E({r,\theta ;0} )= {\left( {\frac{{\sqrt 2 r}}{{{w_0}}}} \right)^m}\textrm{exp} \left( { - \frac{{{r^2}}}{{w_0^2}}} \right)L_n^m\left( {\frac{{2{r^2}}}{{w_0^2}}} \right)\textrm{exp} ({im\theta } ),$$
where, r and θ are the radial coordinate and azimuthal coordinate, respectively; w0 is the beam waist for the fundamental Gaussian beam. $L_n^m(\cdot)$ denotes the Laguerre polynomial with m being the topological charge number and n being the radial index.

The LG mode can be expressed as the superposition of the Hermite–Gaussian modes [26,30,31]:

$$\textrm{exp} ({im\theta } ){r^m}L_n^m({{r^2}} )= \frac{{{{({ - 1} )}^n}}}{{{2^{2n + m}}n!}}\sum\limits_{t = 0}^n {\sum\limits_{s = 0}^m {{i^s}} } \left( {\begin{array}{{c}} n\\ t \end{array}} \right)\left( {\begin{array}{{c}} m\\ s \end{array}} \right){H_{2t + m - s}}(x ){H_{2n - 2t + s}}(y ),$$
where H2t+m−s(x) and H2n−2t+s(y) represent the Hermite polynomials, $\left( {\begin{array}{{c}} n\\ t \end{array}} \right) $ and $\left( {\begin{array}{{c}} m\\ s \end{array}} \right)$ are the binomial coefficients. So, the electric field of radially polarized Laguerre–Gaussian vortex (RPLGV) beam can be written in the form with Hermite–Gaussian mode in Cartesian coordinates as follows:
$$\begin{aligned} {\boldsymbol E}({x,y;0} )&= \frac{x}{{{w_0}}}\textrm{exp} \left( { - \frac{{{r^2}}}{{w_0^2}}} \right)\frac{{{{({ - 1} )}^n}}}{{{2^{2n + m}}n!}}\sum\limits_{t = 0}^n {\sum\limits_{s = 0}^m {{i^s}} } \left( {\begin{array}{{c}} n\\ t \end{array}} \right)\left( {\begin{array}{{c}} m\\ s \end{array}} \right){H_{2t + m - s}}\left( {\frac{x}{{{a_0}}}} \right){H_{2n - 2t + s}}\left( {\frac{y}{{{a_0}}}} \right){{\boldsymbol e}_x}\\ &+ \frac{y}{{{w_0}}}\textrm{exp} \left( { - \frac{{{r^2}}}{{w_0^2}}} \right)\frac{{{{({ - 1} )}^n}}}{{{2^{2n + m}}n!}}\sum\limits_{t = 0}^n {\sum\limits_{s = 0}^m {{i^s}} } \left( {\begin{array}{{c}} n\\ t \end{array}} \right)\left( {\begin{array}{{c}} m\\ s \end{array}} \right){H_{2t + m - s}}\left( {\frac{x}{{{a_0}}}} \right){H_{2n - 2t + s}}\left( {\frac{y}{{{a_0}}}} \right){{\boldsymbol e}_y}, \end{aligned}$$
with a0 = w0/${\sqrt 2}$. ex and ey represent the unit vectors in the x and y directions, respectively. r = (x, y) is arbitrary transverse position vector.

Then, let’s consider a PCRPLGV beam. Its second-order statistical properties in the plane z = 0 are generally characterized by a 2×2 cross spectral density matrix (CSDM) [21,27,32]:

$$\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\leftrightarrow$}} \over W} ({{{\boldsymbol r}_1},{{\boldsymbol r}_2};0} )= \left[ {\begin{array}{{cc}} {{W_{xx}}({{{\boldsymbol r}_1},{{\boldsymbol r}_2};0} )}&{{W_{xy}}({{{\boldsymbol r}_1},{{\boldsymbol r}_2};0} )}\\ {{W_{yx}}({{{\boldsymbol r}_1},{{\boldsymbol r}_2};0} )}&{{W_{yy}}({{{\boldsymbol r}_1},{{\boldsymbol r}_2};0} )} \end{array}} \right],$$
where r1= (x1, y1) and r2= (x2, y2) are positions of two points in the source plane. The elements of CSDM are: Wαβ (r1, r2;0) = 〈Eα (r1;0) Eβ*(r2;0)〉m, (α, β = x, y), with Ex and Ey being the two components of the stochastic electric vector with respect to two mutually orthogonal x and y directions, 〈·〉m denoting the ensemble average and * representing the complex conjugate. For a PC beam with Gaussian Schell-model correlations, the elements of CSDM can be expressed in the following form [26,30]:
$${W_{\alpha \beta }}({{{\boldsymbol r}_1},{{\boldsymbol r}_2};0} )= {E_\alpha }({{{\boldsymbol r}_1};0} )E_\beta ^\ast ({{{\boldsymbol r}_2};0} ){g_{\alpha \beta }}({{{\boldsymbol r}_1} - {{\boldsymbol r}_2}} ),$$
where gαβ (r1r2) denote the spectral degree of coherence with the following expression [30]:
$${g_{\alpha \beta }}({{{\boldsymbol r}_1} - {{\boldsymbol r}_2}} )= {B_{\alpha \beta }}\textrm{exp} \left[ { - \frac{{{{({{x_1} - {x_2}} )}^2} + {{({{y_1} - {y_2}} )}^2}}}{{2\delta_{\alpha \beta }^2}}} \right].$$
Bαβ are the complex correlation coefficient which are used to describe the degree of correlation between two components of the stochastic electric vector. Bxx = Byy= 1, Bxy= Byx* = |Bxy|exp() (|Bxy| ≤ 1), φ is the initial phase difference between Ex and Ey component. δαβ is initial auto-correlation lengths (for α = β) or initial mutual correlation lengths (for αβ). For a Schell-model source, it should be noted that δαβ and Bαβ must satisfy: max {δxx, δyy} ≤ δxy ≤ min {δxx/[|Bxy|]1/2, δyy/[|Bxy|]1/2} [21].

So, substituting Eqs. (3) and (6) into Eq. (5), we can obtain the elements of CSDM for PCRPLGV beams as follows:

$$\begin{aligned} {W_{\alpha \beta }}({{{\boldsymbol r}_1},{{\boldsymbol r}_2},0} )&= \frac{{{\alpha _1}{\beta _2}}}{{w_0^2}}\textrm{exp} \left( { - \frac{{r_1^2 + r_2^2}}{{w_0^2}}} \right)\textrm{exp} \left[ { - \frac{{{{({{{\boldsymbol r}_1} - {{\boldsymbol r}_2}} )}^2}}}{{2\delta_{\alpha \beta }^2}}} \right]\frac{1}{{{2^{4n + 2m}}{{({n!} )}^2}}}\sum\limits_{t = 0}^n {\sum\limits_{s = 0}^m {\sum\limits_{\mu = 0}^n {\sum\limits_{\nu = 0}^m {{{({{i^s}} )}^\ast }{i^v}} } } } \\ &\left( {\begin{array}{{c}} n\\ t \end{array}} \right)\left( {\begin{array}{{c}} m\\ s \end{array}} \right)\left( {\begin{array}{{c}} n\\ \mu \end{array}} \right)\left( {\begin{array}{{c}} m\\ \nu \end{array}} \right){H_{2t + m - s}}\left( {\frac{{{x_1}}}{{{a_0}}}} \right){H_{2\mu + m - \nu }}\left( {\frac{{{x_2}}}{{{a_0}}}} \right){H_{2n - 2t + s}}\left( {\frac{{{y_1}}}{{{a_0}}}} \right){H_{2n - 2\mu + \nu }}\left( {\frac{{{y_2}}}{{{a_0}}}} \right). \end{aligned}$$

In order to calculate conveniently, in this paper, we take δxx = δyy = δxy = δyx = δ0 and Bxy = 1, and the off-diagonal elements of the CSDM for PCRPLGV beam are set to be zero, i.e., Wxy (r1, r2;0) = Wyx (r1, r2;0) = 0.

When a PCRPLGV beam propagates through the atmospheric turbulence, the elements of CSDM in the received plane can be expressed by using the extended Huygens-Fresnel principle as following [21,2527]:

$$\begin{aligned} {W_{\alpha \beta }}({{{\boldsymbol \rho }_1},{{\boldsymbol \rho }_2};z} )&= {\left( {\frac{k}{{2\pi z}}} \right)^2}\int\!\!\!\int {\int\!\!\!\int {{W_{\alpha \beta }}({{{\boldsymbol r}_1},{{\boldsymbol r}_2},0} )} } \\ &\times \textrm{exp} \left[ { - \frac{{ik{{({{{\boldsymbol \rho }_1} - {{\boldsymbol r}_1}} )}^2}}}{{2z}} + \frac{{ik{{({{{\boldsymbol \rho }_2} - {{\boldsymbol r}_2}} )}^2}}}{{2z}} - \frac{{{D_w}({{{\boldsymbol r}_1},{{\boldsymbol r}_2},{{\boldsymbol \rho }_1},{{\boldsymbol \rho }_2};z} )}}{2}} \right]{d^2}{{\boldsymbol r}_1}{d^2}{{\boldsymbol r}_2}, \end{aligned}$$
where ρ1 = (u1, v1) and ρ2 = (u2, v2) are positions of two points in the received plane, k=2π/λ represents the wave number with λ being the wavelength and z denotes the propagation distance. Dw (r1, r2, ρ1, ρ1; z) denoting a two-point spherical wave structure function which contain the influence of atmospheric turbulence is given by [2527]:
$${D_w}({{{\boldsymbol r}_1},{{\boldsymbol r}_2},{{\boldsymbol \rho }_1},{{\boldsymbol \rho }_2};z} )\approx \frac{{{\pi ^2}{k^2}z}}{3}[{{{({{{\boldsymbol \rho }_1} - {{\boldsymbol \rho }_2}} )}^2} + ({{{\boldsymbol \rho }_1} - {{\boldsymbol \rho }_2}} )({{{\boldsymbol r}_1} - {{\boldsymbol r}_2}} )+ {{({{{\boldsymbol r}_1} - {{\boldsymbol r}_2}} )}^2}} ]\int_0^\infty {{{\boldsymbol \kappa }^3}{\Phi _n}({\boldsymbol \kappa } )d{\boldsymbol \kappa }} ,$$
where Фn(κ) represents the spatial power spectrum of the refractive-index fluctuations of atmospheric turbulence with κ = (κx, κy, κz) denoting the spatial wavenumber. The coherence length ρ0 of a spherical wave propagating in turbulence is defined as:
$${\rho _0} = {\left( {\frac{1}{3}{\pi^2}{k^2}zT} \right)^{{{ - 1} / 2}}}, \;\;\;T = \int_0^\infty {{{\boldsymbol \kappa }^3}{\Phi _n}({\boldsymbol \kappa } )d{\boldsymbol \kappa }} .$$

Substituting from Eqs. (7) and (9), (10) into Eq. (8), we can obtain the formula:

$$\begin{aligned} &{W_{\alpha \beta }}({{{\boldsymbol \rho }_1},{{\boldsymbol \rho }_2};z} )= {\left( {\frac{k}{{2\pi z{w_0}}}} \right)^2}\textrm{exp} \left[ { - \frac{{ik}}{{2z}}({{\boldsymbol \rho }_1^2 - {\boldsymbol \rho }_2^2} )} \right]\textrm{exp} \left[ { - \frac{1}{{\rho_0^2}}{{({{{\boldsymbol \rho }_1} - {{\boldsymbol \rho }_2}} )}^2}} \right]\\ &\quad \times \frac{1}{{{2^{4n + 2m}}{{({n!} )}^2}}}\sum\limits_{t = 0}^n {\sum\limits_{s = 0}^m {\sum\limits_{\mu = 0}^n {\sum\limits_{\nu = 0}^m {{{({{i^s}} )}^\ast }{i^v}} } } } \left( {\begin{array}{{c}} n\\ t \end{array}} \right)\left( {\begin{array}{{c}} m\\ s \end{array}} \right)\left( {\begin{array}{{c}} n\\ \mu \end{array}} \right)\left( {\begin{array}{{c}} m\\ \nu \end{array}} \right)\\ &\quad \int\!\!\!\int {\int\!\!\!\int {{\alpha _1}{\beta _2}} } {H_{2t + m - s}}\left( {\frac{{{x_1}}}{{{a_0}}}} \right)\textrm{exp} \left[ { - \left( {\frac{1}{{w_0^2}} + \frac{1}{{2\delta_{\alpha \beta }^2}} + \frac{{ik}}{{2z}} + \frac{1}{{\rho_0^2}}} \right)x_1^2 + \left( {\frac{{{x_2}}}{{\delta_{\alpha \beta }^2}} + \frac{{2{x_2}}}{{\rho_0^2}} + \frac{{ik{u_1}}}{z} - \frac{{{u_1} - {u_2}}}{{\rho_0^2}}} \right){x_1}} \right]\\ &\quad \times {H_{2\mu + m - \nu }}\left( {\frac{{{x_2}}}{{{a_0}}}} \right)\textrm{exp} \left[ { - \left( {\frac{1}{{w_0^2}} + \frac{1}{{2\delta_{\alpha \beta }^2}} - \frac{{ik}}{{2z}} + \frac{1}{{\rho_0^2}}} \right)x_2^2 + \left( { - \frac{{ik{u_2}}}{z} + \frac{{{u_1} - {u_2}}}{{\rho_0^2}}} \right){x_2}} \right]\\ &\quad \times {H_{2n - 2t + s}}\left( {\frac{{{y_1}}}{{{a_0}}}} \right)\textrm{exp} \left[ { - \left( {\frac{1}{{w_0^2}} + \frac{1}{{2\delta_{\alpha \beta }^2}} + \frac{{ik}}{{2z}} + \frac{1}{{\rho_0^2}}} \right)y_1^2 + \left( {\frac{{{y_2}}}{{\delta_{\alpha \beta }^2}} + \frac{{2{y_2}}}{{\rho_0^2}} + \frac{{ik{v_1}}}{z} - \frac{{{v_1} - {v_2}}}{{\rho_0^2}}} \right){y_1}} \right]\\ &\quad \times {H_{2n - 2\mu + \nu }}\left( {\frac{{{y_2}}}{{{a_0}}}} \right)\textrm{exp} \left[ { - \left( {\frac{1}{{w_0^2}} + \frac{1}{{2\delta_{\alpha \beta }^2}} - \frac{{ik}}{{2z}} + \frac{1}{{\rho_0^2}}} \right)y_2^2 + \left( { - \frac{{ik{v_2}}}{z} + \frac{{{u_1} - {u_2}}}{{\rho_0^2}}} \right){y_2}} \right]d{x_1}d{y_1}d{x_2}d{y_2}. \end{aligned}$$

Using the following integral and expansion formulas [26,30,33]:

$${H_n}(x )= \sum\limits_{k = 0}^{[{{n / 2}} ]} {\frac{{{{({ - 1} )}^k}n!}}{{k!({n - 2k} )!}}{{({2x} )}^{n - 2k}}} ,$$
$$\int_{ - \infty }^{ + \infty } {{x^n}} \textrm{exp} [{ - {{({x - \beta } )}^2}} ]dx = \sqrt \pi {({2i} )^{ - n}}{H_n}({i\beta } ),$$
$${H_n}({x + y} )= {\left( {\frac{1}{2}} \right)^{{n / 2}}}\sum\limits_{k = 0}^n {\left( {\begin{array}{{c}} n\\ k \end{array}} \right){H_k}} \left( {\sqrt 2 x} \right){H_{n - k}}\left( {\sqrt 2 y} \right),$$
$$\int_{ - \infty }^\infty {{H_n}} ({ax} )\textrm{exp} [{ - {{({x - y} )}^2}} ]dx = \sqrt \pi {({1 - {a^2}} )^{{n / 2}}}{H_n}\left[ {\frac{{ay}}{{{{({1 - {a^2}} )}^{{1 / 2}}}}}} \right],$$
after the tedious integral, the analytical formulas for the CSDM [Wxx (r1, r2;0) and Wyy (r1, r2;0)] of the PCRPLGV beam through the turbulent atmosphere are given by:
$$\begin{array}{l} {W_{xx}}({{{\boldsymbol \rho }_1},{{\boldsymbol \rho }_2};z} )= {\left( {\frac{k}{{2z{w_0}}}} \right)^2}\textrm{exp} \left[ { - \frac{{ik}}{{2z}}({{\boldsymbol \rho }_1^2 - {\boldsymbol \rho }_2^2} )} \right]\textrm{exp} \left[ { - \frac{1}{{\rho_0^2}}{{({{{\boldsymbol \rho }_1} - {{\boldsymbol \rho }_2}} )}^2}} \right]\frac{1}{{{2^{4n + 2m}}{{({n!} )}^2}}}\\ \sum\limits_{t = 0}^n {\sum\limits_{s = 0}^m {\sum\limits_{\mu = 0}^n {\sum\limits_{\nu = 0}^m {{{({{i^s}} )}^ \ast }{i^\nu }} } } } \left( {\begin{array}{{c}} n\\ t \end{array}} \right)\left( {\begin{array}{{c}} m\\ s \end{array}} \right)\left( {\begin{array}{{c}} n\\ \mu \end{array}} \right)\left( {\begin{array}{{c}} m\\ \nu \end{array}} \right)\\ \sum\limits_{a = 0}^{[{{{({2t + m - s} )} / 2}} ]} {\sum\limits_{b = 0}^{2t + m - s - 2a + 1} {\sum\limits_{c = 0}^{[{{b / 2}} ]} {\sum\limits_{d = 0}^{[{{{({2\mu + m - \nu } )} / 2}} ]} {\sum\limits_{e = 0}^{2n - 2t + s} {\sum\limits_{f = 0}^{[{{e / 2}} ]} {\sum\limits_{g = 0}^{[{{{({2n - 2\mu + \nu } )} / 2}} ]} {\left( {\begin{array}{{c}} {2t + m - s - 2a + 1}\\ b \end{array}} \right)\left( {\begin{array}{{c}} {2n - 2t + s}\\ e \end{array}} \right)} } } } } } } \\ \frac{{{{(i )}^{ - 2m - 2n + s - 2t + 4a + 2c + 4d - e + 4f + 4g - 2}}b!e!({2t + m - s} )!({2\mu + m - \nu } )!({2n - 2\mu + \nu } )!}}{{a!c!d!f!g!({b - 2c} )!({e - 2f} )!({2t + m - s - 2a} )!({2\mu + m - \nu - 2d} )!({2n - 2\mu + \nu - 2g} )!}}\\ {\sqrt 2 ^{ - m - 2n + 2a - b + 2c - e + 2f - 5}}{\left( {\frac{1}{{{a_0}}}} \right)^{2m + 4n - 2a - 2d - 2g}}{\left( {\frac{1}{{\sqrt A }}} \right)^{m + 4n + s - 2t - 2a + b - 2c + 3}}\\ {\left( {\frac{1}{{\sqrt {{A^2}a_0^2 - A} }}} \right)^{ - 2n - s + 2t + e - 2f}}{(B )^{b - 2c + e - 2f}}{\left( {\frac{1}{{\sqrt D }}} \right)^{m + 2n + b - 2c - 2d + e - 2f - 2g + 3}}\\ \times \textrm{exp} \left( {\frac{{C_{1x}^2}}{{4A}}} \right)\textrm{exp} \left( {\frac{{{E^2}}}{{4D}}} \right)\textrm{exp} \left( {\frac{{C_{1y}^2}}{{4A}}} \right)\textrm{exp} \left( {\frac{{{F^2}}}{{4D}}} \right)\\ {H_{2t + m - s - 2a + 1 - b}}\left( {\frac{{\sqrt 2 i{C_{1x}}}}{{2\sqrt A }}} \right){H_{2\mu + m - \nu + b - 2c - 2d + 1}}\left( {\frac{{iE}}{{2\sqrt D }}} \right)\\ {H_{2n - 2t + s - e}}\left( {\frac{{\sqrt 2 {C_{1y}}}}{{2\sqrt {{A^2}a_0^2 - A} }}} \right){H_{2n - 2\mu + \nu + e - 2f - 2g}}\left( {\frac{{iF}}{{2\sqrt D }}} \right), \end{array}$$
and
$$\begin{array}{l} {W_{yy}}({{{\boldsymbol \rho }_1},{{\boldsymbol \rho }_2};z} )= {\left( {\frac{k}{{2z{w_0}}}} \right)^2}\textrm{exp} \left[ { - \frac{{ik}}{{2z}}({{\boldsymbol \rho }_1^2 - {\boldsymbol \rho }_2^2} )} \right]\textrm{exp} \left[ { - \frac{1}{{\rho_0^2}}{{({{{\boldsymbol \rho }_1} - {{\boldsymbol \rho }_2}} )}^2}} \right]\frac{1}{{{2^{4n + 2m}}{{({n!} )}^2}}}\\ \sum\limits_{t = 0}^n {\sum\limits_{s = 0}^m {\sum\limits_{\mu = 0}^n {\sum\limits_{\nu = 0}^m {{{({{i^s}} )}^ \ast }{i^\nu }} } } } \left( {\begin{array}{{c}} n\\ t \end{array}} \right)\left( {\begin{array}{{c}} m\\ s \end{array}} \right)\left( {\begin{array}{{c}} n\\ \mu \end{array}} \right)\left( {\begin{array}{{c}} m\\ \nu \end{array}} \right)\\ \sum\limits_{a = 0}^{2t + m - s} {\sum\limits_{b = 0}^{[{{a / 2}} ]} {\sum\limits_{c = 0}^{[{{{({2\mu + m - \nu } )} / 2}} ]} {\sum\limits_{d = 0}^{[{{{({2n - 2t + s} )} / 2}} ]} {\sum\limits_{e = o}^{2n - 2t + s - 2d + 1} {\sum\limits_{f = 0}^{[{{e / 2}} ]} {\sum\limits_{g = 0}^{[{{{({2n - 2\mu + \nu } )} / 2}} ]} {\left( {\begin{array}{{c}} {2t + m - s}\\ a \end{array}} \right)} } \left( {\begin{array}{{c}} {2n - 2t + s - 2d + 1}\\ e \end{array}} \right)} } } } } \\ \frac{{{{(i )}^{ - m - 4n - s + 2t - a + 4b + 4c + 4d + 2f + 4g - 2}}a!e!({2\mu + m - \nu } )!({2n - 2t + s} )!({2n - 2\mu + \nu } )!}}{{b!c!d!f!g!({a - 2b} )!({e - 2f} )!({2\mu + m - \nu - 2c} )!({2n - 2t + s - 2d} )!({2n - 2\mu + \nu - 2g} )!}}\\ {\sqrt 2 ^{ - m - 2n - a + 2b + 2d - e + 2f - 5}}{\left( {\frac{1}{{{a_0}}}} \right)^{2m + 4n - 2c - 2d - 2g}}{\left( {\frac{1}{{\sqrt A }}} \right)^{2m + 2n - s + 2t - 2d + e - 2f + 3}}\\ {\left( {\frac{1}{{\sqrt {{A^2}a_0^2 - A} }}} \right)^{ - m + s - 2t + a - 2b}}{(B )^{a - 2b + e - 2f}}{\left( {\frac{1}{{\sqrt D }}} \right)^{m + 2n + a - 2b - 2c + e - 2f - 2g + 3}}\\ \textrm{exp} \left( {\frac{{C_{1x}^2}}{{4A}}} \right)\textrm{exp} \left( {\frac{{{E^2}}}{{4D}}} \right)\textrm{exp} \left( {\frac{{C_{1y}^2}}{{4A}}} \right)\textrm{exp} \left( {\frac{{{F^2}}}{{4D}}} \right)\\ {H_{2t + m - s - a}}\left( {\frac{{\sqrt 2 {C_{1x}}}}{{2\sqrt {{A^2}a_0^2 - A} }}} \right){H_{2\mu + m - \nu + a - 2b - 2c}}\left( {\frac{{iE}}{{2\sqrt D }}} \right)\\ {H_{2n - 2t + s - 2d + 1 - e}}\left( {\frac{{\sqrt 2 i{C_{1y}}}}{{2\sqrt A }}} \right){H_{2n - 2\mu + \nu + e - 2f - 2g + 1}}\left( {\frac{{iF}}{{2\sqrt D }}} \right), \end{array}$$
where
$$\begin{array}{l} A = \frac{1}{{w_0^2}} + \frac{1}{{2\delta _0^2}} + \frac{{ik}}{{2z}} + \frac{1}{{\rho _0^2}},\;\;\; B = \frac{1}{{\delta _0^2}} + \frac{2}{{\rho _0^2}}, \\ {C_{1x}} = \frac{{ik{u_1}}}{z} - \frac{{{u_1} - {u_2}}}{{\rho _0^2}},\;\;\;\;\;\; {C_{1y}} = \frac{{ik{v_1}}}{z} - \frac{{{v_1} - {v_2}}}{{\rho _0^2}}, \\ {C_{2x}} ={-} \frac{{ik{u_2}}}{z} + \frac{{{u_1} - {u_2}}}{{\rho _0^2}},\;\;\;\;\;\; {C_{2y}} ={-} \frac{{ik{v_2}}}{z} + \frac{{{v_1} - {v_2}}}{{\rho _0^2}},\\ D ={-} \frac{{{B^2}}}{{4A}} + {A^ \ast },\;\;\; E = \frac{{B{C_{1x}}}}{{2A}} + {C_{2x}},\;\;\; F = \frac{{B{C_{1y}}}}{{2A}} + {C_{2y}}. \end{array}$$

The average intensity, DOC and DOP of PCRPLGV beam through atmospheric turbulence are expressed as [21]:

$$I({{\boldsymbol \rho },{\boldsymbol \rho };z} )= {W_{xx}}({{\boldsymbol \rho },{\boldsymbol \rho };z} )+ {W_{yy}}({{\boldsymbol \rho },{\boldsymbol \rho };z} ),$$
$$\mu ({{{\boldsymbol \rho }_\textrm{1}},{{\boldsymbol \rho }_2};z} )= \frac{{Tr\left[ {\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\leftrightarrow$}} \over W} ({{{\boldsymbol \rho }_\textrm{1}},{{\boldsymbol \rho }_2};z} )} \right]}}{{\sqrt {Tr\left[ {\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\leftrightarrow$}} \over W} ({{{\boldsymbol \rho }_\textrm{1}},{{\boldsymbol \rho }_1};z} )} \right]Tr\left[ {\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\leftrightarrow$}} \over W} ({{{\boldsymbol \rho }_2},{{\boldsymbol \rho }_2};z} )} \right]} }},$$
and
$$P({{\boldsymbol \rho },{\boldsymbol \rho };z} )= \sqrt {1 - \frac{{4Det\left[ {\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\leftrightarrow$}} \over W} ({{\boldsymbol \rho },{\boldsymbol \rho }\textrm{;}z} )} \right]}}{{{{\left\{ {Tr\left[ {\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\leftrightarrow$}} \over W} ({{\boldsymbol \rho },{\boldsymbol \rho }\textrm{;}z} )} \right]} \right\}}^\textrm{2}}}}} .$$
where Tr and Det represent the trace and the determinant of matrix, respectively.

3. Simulation model

Multi-phase screen is the most widely used to simulate the propagation of light waves through atmospheric turbulence. Multi-phase screen simulates light propagation in atmospheric turbulence by replacing a continuous random medium with a series of phase screens (PS) spaced Δz, as shown in Fig. 1. The phase disturbance caused by one PS is equivalent to the cumulative phase disturbance effect of turbulence after the propagation of Δz distance, and the light wave reaches the received plane after the phase modulation of the multilayer PSs.

 figure: Fig. 1.

Fig. 1. Schematic diagram of multi-phase screen method. PS: phase screen.

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The whole simulation process can be expressed by the following formula [34]:

$${E_i}({x,y;{z_i}} )= {F^{ - 1}}\left\{ {F\{{{E_{i - 1}}({x,y;{z_{i - 1}}} )\textrm{exp} [{i\phi ({x,y;{z_i}} )} ]} \}\textrm{exp} \left[ { - \frac{{i\mathrm{\Delta }z}}{{2k}}({\kappa_x^2 + \kappa_y^2} )} \right]} \right\},$$
where zi−1 and zi represent the position of the (i−1)-th and i-th PS, Ei−1 (x, y; zi−1) and Ei (x, y; zi) is the fully coherent light field on the (i–1)-th and i-th PS, ϕ (x, y; zi) denotes phase distribution on the i-th PS, F and F−1 denote the Fourier transform and inverse Fourier transform, respectively. Spectrum inversion is the most commonly used method to construct the PS. The basic idea of spectral inversion method is to filter a complex Gaussian random number matrix with the power spectrum of atmospheric turbulence, and then use inverse Fourier transform to get the phase disturbance induced by atmospheric turbulence. The specific method for generating PS is shown in [29].

However, the multi-phase screen method cannot be directly applied to PC beam, because PC beam is not described by electric field but by CSD. In the actual simulation, we first set a complex screen (CS) on the light source surface of the fully coherent beam to synthesize the PC beam. The fabrication of the CS is the similar to the turbulent phase screen except that the spectral density function is different. The amplitude and phase of the light source can be modulated simultaneously by the CS, and the correlation function of the CS is strictly equal to the correlation function of the PC beam, so it can be used to simulate the PC beam with any coherent structure [29,35].

Using CS method to synthesize PC source, a fully coherent beam modulated by a CS can be expressed as:

$$E({x,y} )= {E_0}({x,y} )T({x,y} ),$$
where E0(x, y) is the incident electric field, T(x, y) denotes the CS with certain spatial correlation characteristics.

Firstly, we take Fourier transforms of the spectral degree of coherence to get the spatial frequency spectrum of it:

$$S({{f_x},{f_y}} )= \int\!\!\!\int {g({\Delta {r_x},\Delta {r_y}} )\textrm{exp} [{2\pi i({{f_x}\Delta {r_x} + {f_y}\Delta {r_y}} )} ]d\Delta {r_x}d\Delta {r_y}} ,$$
where Δrx = x1x2, Δry = y1y2, fx and fy are the spatial frequency. Then, we use a zero-mean, unit-variance, complex Gaussian random function to filter S (fx, fy):
$$T^{\prime}({{f_x},{f_y}} )= {a_r}({{f_x},{f_y}} )\sqrt {S({{f_x},{f_y}} )} .$$

So far, the T′(fx, fy) is now the Fourier transform of T (x, y). So, we can get the CS T (x, y) by taking the inverse Fourier transform of the T′(fx, fy). And this method is also suitable for generating vector PC beams. We only need to modulate each component of electric field of the vector beam with CS, respectively.

In summary, the logic process of numerical simulation of PC beams in atmospheric turbulence is described as follows. (1) Synthesize the corresponding turbulence PS and fix it on the propagation path. (2) J1 CSs are used to synthesize a PC beam, and the electric field of each frame passes through the above PS fixed on the propagation path separately. (3) The average value of the light intensity of J1 frames on the received plane is a frame PC beam after propagating the atmospheric turbulence. (4) Cycle steps (1)–(3) J2 times to get the result of J2 frames PC beam through the atmospheric turbulence, and the J2 frames results can be used to analyze and calculate the statistical characteristics of PC beams through the atmospheric turbulence. J1 and J2 denote the number of CS and PS changes, respectively.

 figure: Fig. 2.

Fig. 2. Normalized average intensities Ix/Ixmax, Iy/Iymax and I/Imax of a PCRPLGV beam at different propagation distances in atmospheric turbulence. n=1, m=0 and $\tilde{C}_n^{2^\ast} $ = 1×10−14 m3–α.

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4. Numerical examples and discussions

In this paper, we chose the anisotropic non-Kolmogorov power spectrum as the atmospheric turbulence model, and its spatial power spectrum Фn(κ) is expressed [21,36]:

$${\Phi _n}({{\kappa_x},{\kappa_y},{\kappa_z}} )= \frac{{A(\alpha )\tilde{C}_n^2{\xi _x}{\xi _y}}}{{{{({\xi_x^2\kappa_x^2 + \xi_x^2\kappa_y^2 + \kappa_z^2 + \kappa_0^2} )}^{{\alpha / 2}}}}}\textrm{exp} \left( { - \frac{{\xi_x^2\kappa_x^2 + \xi_y^2\kappa_y^2 + \kappa_z^2}}{{\kappa_m^2}}} \right),$$
where ξx and ξy are the anisotropic factors in x and y direction, respectively, α is the power law exponent, $\tilde{C}_n^2 $ is the generalized refractive-index structure parameter with units m3-α, κx, κy and κz are three components of the spatial wavenumber κ along the x, y and z directions, κ0=2π/L0 with L0 being the outer scale of atmospheric turbulence, and κm=c(α)/l0 with l0 being the inner scale of atmospheric turbulence. According to Markov approximation, κz in Eq. (27) can be ignored. A(α)=[Г(α–1)cos(απ/2)]/(4π2), c(α)={[2πA(α)Г(5/2–α/2)]/3}1/(α−5), where α is the generalized exponent parameter and Г(·) is the gamma function.

Substituting Eq. (26) into Eq. (10), the turbulence quantity T can be easily obtained by [21,36,37]:

$$T = {\kern 1pt} \frac{{\xi _x^2 + \xi _y^2}}{{\xi _x^2\xi _y^2}} \times \frac{{A(\alpha )\tilde{C}_n^2}}{{4({\alpha - 2} )}}\left\{ {[{2\kappa_0^2\kappa_m^{2 - \alpha } + ({\alpha - 2} )\kappa_m^{4 - \alpha }} ]\textrm{exp} \left( {\frac{{\kappa_0^2}}{{\kappa_m^2}}} \right)\varGamma \left( {2 - \frac{\alpha }{2},\frac{{\kappa_0^2}}{{\kappa_m^2}}} \right) - 2\kappa_0^{4 - \alpha }} \right\},$$
where Г (·, ·) is an incomplete gamma function.

In the calculation, we have fixed some parameters: w0=10mm, λ=632.8nm, δ0=20mm, l0=10 mm, L0=100 m, α=3.6. These values will be used as calculation parameters unless otherwise stated. For convenience, we set $\tilde{C}_{n}^{2^{*}}=\tilde{C}_{n}^{2}\left(\xi_{x}^{2}+\xi_{y}^{2}\right) /\left(2 \xi_{x}^{2} \xi_{y}^{2}\right)$. In the simulation of beam propagation using multi-phase screen, the distance between each two PS is Δz=50m, and the number of data acquisition samples is set to 512×512. We take J1=200, that is, 1 frame instantaneous intensity of PCRPLGV beam through the atmospheric turbulence is simulated by the average intensity over 200 cycles. And the conversion times of PS are J2=200, i.e., 200 frames instantaneous light intensity of PCRPLGV beam through the atmospheric turbulence are collected at the received plane, and the average of these 200 frames instantaneous intensity is the final average intensity of PCRPLGV beam through the atmospheric turbulence.

 figure: Fig. 3.

Fig. 3. Normalized average intensities Ix/Ixmax, Iy/Iymax and I/Imax of a PCRPLGV beam at different propagation distances in atmospheric turbulence. n=1, m=2 and $\tilde{C}_n^{2^\ast} $ = 1×10−14 m3–α.

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In Fig. 2 we have calculate the normalized average intensities Ix/Ixmax (Ix=Wxx(ρ, ρ; z)), Iy/Iymax (Iy=Wyy(ρ, ρ; z)) and I/Imax of a PCRPLGV beam with n=1 and m=0 at several different propagation distances in atmospheric turbulence. One finds that the initial intensity distributions of Ix/Ixmax and Iy/Iymax of a PCRPLGV beam obey the classic two-beamlets array distribution along the x or y direction of RP beam. The initial beam profiles of I/Imax show a multi-ring dark hollow beam profile and the intensity of the outer ring is greater than that of the inner ring; that is quite different from those of a partially coherent Laguerre–Gaussian (PCLG) beam, whose beam profiles display a multi-ring dark hollow beam profile but the intensity of the outer ring is weaker than that of the inner ring. The intensity distributions of Ix/Ixmax and Iy/Iymax of a PCRPLGV beam will degenerate into elliptical Gaussian distributions with the increase of propagation distance and eventually evolve into Gaussian distributions in the far field (z=15km). The intensity distributions of I/Imax exhibit the self-shaping property, and a PCRPLGV beam will gradually involve from a multi-ring dark hollow distribution into a circular Gaussian distribution on propagation.

Figure 3 displays the normalized average intensities Ix/Ixmax, Iy/Iymax and I/Imax of a PCRPLGV beam with n=1 and m=2 at several different propagation distances in atmospheric turbulence. Compared with Fig. 2, it can be easily found that the PCRPLGV beam with large m has a larger light spot. From Fig. 3, we can find that the topological charge number m will strongly affect the evolution of the intensity distributions. The topological charge number will cause the self-rotating (or self-distorting) of the distributions of Ix/Ixmax and Iy/Iymax in the propagation, so that they are no longer distributed along the x or y direction. When m≠0, the vortex phase part [exp(imθ)] of PCRPLGV beam will impose an orbital angular momentum (OAM) of on the beam, which causes the beam to rotate on propagation in turbulence. In Ref. [38], the radially polarized twisted Gaussian Schell-model (RPTGSM) beam studied by Peng et.al. also showed a similar phenomenon. Compared with Fig. 2 with Fig. 3, we can find that the distributions of Ix/Ixmax and Iy/Iymax gradually evolve into elliptic Gaussian distribution on propagation, the distributions of Ix/Ixmax and Iy/Iymax of PCRPLGV beam with m=0 have higher ellipticity than those of PCRPLGV beam with m=2. This indicates that PCRPLGV beam with high m will evolve into Gaussian distribution more quickly in atmospheric turbulence, which means vortex phase plays an important role in self-shaping for PCRPLGV beam.

 figure: Fig. 4.

Fig. 4. Normalized average intensity distribution of a PCRPLGV beam in free space and atmospheric turbulence. n=1, m=1; (a1)–(a8): $\tilde{C}_n^{2^\ast} $ = 0; (b1)–(b8): $\tilde{C}_n^{2^\ast} $ = 2×10−15 m3−α.

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 figure: Fig. 5.

Fig. 5. Simulation of normalized average intensity distribution of a PCRPLGV beam in free space and atmospheric turbulence by using multi-phase screen method. n=1, m=1; (a1)–(a8): $\tilde{C}_n^{2^\ast} $ = 0; (b1)–(b8): $\tilde{C}_n^{2^\ast} $ = 2×10−15 m3–α.

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Figure 4 shows the normalized average intensity distribution of a PCRPLGV beam with n=1and m=1 in free space and atmospheric turbulence ($\tilde{C}_n^{2^\ast} $ = 2×10−15 m3–α). In free space (see Fig. 4(a1)–(a8)), the evolution process of the PCRPLGV beam can be summarized as follows: with the increase of the propagation distance, the intensity of the dark center region gradually increases, the dark ring gradually disappears, the outer ring with the largest intensity gradually weakens, and finally the beam spot evolves into a Gaussian like distribution. Compared Fig. 4(a1)–(a4) and Fig. 4(b1)–(b4), it can be found that atmospheric turbulence has less influence on the evolution of the beam over a short propagation distance (z<0.4km). We can also find in Fig. 4(b5)–(b8), with the further increase of the propagation distance (z>0.8km), the evolution speed of the beam in atmospheric turbulence increases obviously, and the beam profile tends to the Gaussian distribution. This is consistent with the conclusion in [32]: the far-field kurtosis parameter (K parameter) of any beam in atmospheric turbulence is equal to 2, which means that any beam will evolve into Gaussian beam in atmospheric turbulence. Then, we use the multi-phase screen method to simulate the propagation of PCRPLGV beam in free space and atmospheric turbulence as shown in Fig. 5. The parameters in the simulation are same as those in Fig. 4. The simulation results show that the simulation agrees well with the theory in both free space and atmospheric turbulence, which implies that our simulation model is valid and the theoretical results derived are correct.

The influence of atmospheric turbulence parameters on PCRPLGV beam is shown in Fig. 6. It can be seen from Fig. 6(a1)–(a4) that the smaller the anisotropy factors is, the closer PCRPLGV beam tends to the Gaussian distribution. This shows that the turbulence intensity decreases with the increase of the anisotropy factors ξx and ξy, and the smaller the influence on the beam. Figure 6(b1)–(b4) show that with the increase of power law exponent α, the influence of turbulence on the beam first increases and then decreases.

In Fig. 7 we have illustrated the DOC distribution of PCRPLGV beam with different m and n at several propagation distances in atmospheric turbulence. It is well known that the DOC of PCRPLGV beam on the initial plane satisfies the Gaussian distribution. As can be seen from Fig. 6, DOC will appear multi-level rings in the propagation, and the number of rings will increase with the increase of propagation distance. With the further increase of the propagation distance, the number of rings will gradually decrease, and finally the DOC of PCRPLGV beam will degenerate into a Gaussian distribution again. In Fig. 7, we also find that in the propagation process, the relationship between the maximum number of rings of DOC distribution of PCRPLGV beam and radial index and topological charge of the beam satisfies 2n + m + 1. In addition, in [21], we have studied the evolution law of DOC for partially coherent radially polarized (PCRP) beam, it fits well with what the Fig. 7(a1)–(a4) shows (the PCRPLGV beam with m = n=0 reduces to a PCRP beam). In order to verify our theoretical results, the corresponding simulations are shown in Fig. 8. It is found that the simulation results agree well with the theory given by in Fig. 7.

In Fig. 9, we have studied the influence of atmospheric turbulence parameters on the DOC distribution of PCRPLGV beam. We can find that it has the same rules with Fig. 6. Combined with Fig. 6, we can also find that the average intensity and the DOC of PCRPLGV beam are more affected by various anisotropy factors.

Figure 10 displays the change of DOP of PCRPLGV beam at (2cm,0) with propagation distance under different conditions. We can see that from Fig. 10, when m≠0, the DOP of PCRPLGV beam at (2cm,0) drops rapidly from 1 to 0, then rises, and finally decreases to 0 with the increase of propagation distance. The bigger m is, the faster the DOP drops to 0, but the greater the subsequent rise [see Fig. 10(a)]. In Fig. 10(b), it could be found when m=0, DOP will decrease with the increase of propagation distance, and the bigger n is, the faster DOP decreases. And Figs.10 (c)-(f) indicate that in the DOP rising stage, the rising amplitude will increase with the increase of the initial coherent length of PCRPLGV beam, the anisotropy factor ξx and ξy and power law exponent α (α>3.1), the decrease of $\tilde{C}_n^{2^\ast} $.

 figure: Fig. 6.

Fig. 6. Normalized average intensities of PCRPLGV beam at z = 2 km in atmospheric turbulence with different anisotropic factors and power law exponent. $\tilde{C}_n^{2} $ = 1×10−14 m3−α; (b1)–(b4): ξx=2, ξy=3.

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 figure: Fig. 7.

Fig. 7. The DOC distribution of a PCRPLGV beam at different propagation distances in atmospheric turbulence. $\tilde{C}_n^{2^\ast} $ = 1×10−15 m3–α.

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 figure: Fig. 8.

Fig. 8. Simulation of DOC distribution of a PCRPLGV beam at different propagation distances in atmospheric turbulence. $\tilde{C}_n^{2^\ast} $ = 1×10−15 m3–α.

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 figure: Fig. 9.

Fig. 9. The DOC distribution of PCRPLGV beam at z = 2 km in atmospheric turbulence with different anisotropic factors and power law exponent. $\tilde{C}_n^{2} $ = 1×10−14 m3−α; (b1)–(b4): ξx=2, ξy=3.

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 figure: Fig. 10.

Fig. 10. The DOP of PCRPLGV beam at (2 cm,0) vs propagation distance z in different conditions. (a) n = 0, $\tilde{C}_n^{2^\ast} $ = 2×10−15 m3–α, δ0 = 20 mm; (b) m = 0, $\tilde{C}_n^{2^\ast} $ = 2×10−15 m3–α, δ0 = 20 mm; (c) n = 0, m = 1, $\tilde{C}_n^{2^\ast} $ = 2×10−15 m3–α; (d) n = 0, m = 1, δ0 = 20 mm; (e) n = 0, m = 1, $\tilde{C}_n^{2} $= 1×10−14 m3–α, δ0 = 20 mm; (f) n = 0, m = 1, $\tilde{C}_n^{2^\ast} $ = 2×10−15 m3–α, δ0 = 20 mm.

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5. Conclusions

In this paper, we have derived the analytical expressions for the diagonal elements of CSDM of a PCRPLGV beam in anisotropic atmospheric turbulence, and studied the evolution characteristics of light intensity, DOC and DOP of PCRPLGV beam on propagation. It’s shown that the initial beam profiles of PCRPLGV beam show a multi-ring dark hollow beam profile and the intensity of the outer ring is greater than that of the inner ring, which is quite different from those of the PCLG beam. The intensity distributions of Ix/Ixmax and Iy/Iymax of PCRPLGV beam in initial plane obey the classic two-beamlets array distribution along the x or y direction of RP beam, and they will degenerate into elliptical Gaussian distributions with the increase of propagation distance and eventually evolve into Gaussian distributions in the far field. Another finds that the topological charge number m will cause the rotation of the distributions of Ix/Ixmax and Iy/Iymax on propagation. The beam profile of PCRPLGV beam with high m will evolve into Gaussian distribution more quickly in atmospheric turbulence. The DOC of PCRPLGV beam will appear multi-level rings in the propagation, and the number of rings will increase with the increase of propagation distance. With the further increase of the propagation distance, the DOC will degenerate into a Gaussian distribution again. It can be found that the relationship between the maximum number of rings of DOC distribution of PCRPLGV beam and radial index and topological charge of the beam satisfies 2n + m + 1. And we have also found that the DOP of PCRPLGV beam is greatly affected by the topological charge m. In addition, in order to verify our theoretical results, we combined the complex screen method and multi-phase screen method to simulate the propagation of PCRPLGV beam in atmospheric turbulence. And the simulation results are in good agreement with the theoretical results. Compared with the general PCLG beam, PCRPLGV beam shows some advantages and unique propagation characteristics. Thus, PCRPLGV beam may find potential applications in laser lidar and laser communication.

Funding

Civil Aviation University of China (J2019-063, JG2019-19-03); Department of Science and Technology of Sichuan Province (2019YJ0470).

Disclosures

The authors declare no conflicts of interest.

Data availability

No data were generated or analyzed in the presented research.

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References

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  1. L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992).
    [Crossref]
  2. G. Gibson, J. Courtial, M. J. Padgett, M. Vasnetsov, V. Pas’ko, S. M. Barnett, and S. Franke-Arnold, “Free-space information transfer using light beams carrying orbital angular momentum,” Opt. Express 12(22), 5448–5456 (2004).
    [Crossref]
  3. J. Wang, J. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6(7), 488–496 (2012).
    [Crossref]
  4. G. Grier D, “A revolution in optical manipulation,” Nature 424(6950), 810–816 (2003).
    [Crossref]
  5. M. P. MacDonald, L. Paterson, K. Volke-Sepulveda, J. Arlt, W. Sibbett, and K. Dholakia, “Creation and manipulation of three-dimensional optically trapped structures,” Science 296(5570), 1101–1103 (2002).
    [Crossref]
  6. G. A. Swartzlander, “Peering into darkness with a vortex spatial filter,” Opt. Lett. 26(8), 497–499 (2001).
    [Crossref]
  7. B. Melo, I. Brandao, B. P. da Silva, R. B. Rodrigues, A. Z. Khoury, and T. Guerreiro, “Optical trapping in a dark focus,” Phys. Rev. Appl. 14(3), 034069 (2020).
    [Crossref]
  8. M. Luo, D. Sun, Y. Yang, S. Liu, J. Wu, Z. Ma, S. Sun, and X. Sun, “Three-dimensional isotropic STED microscopy generated by 4 pi focusing of a radially polarized vortex Laguerre-Gaussian beam,” Opt. Commun. 463, 125434 (2020).
    [Crossref]
  9. S. Liu, S. Liu, C. Yang, Z. Xu, Y. Li, Y. Li, Z. Zhou, G. Guo, and B. Shi, “Classical simulation of high-dimensional entanglement by non-separable angular-radial modes,” Opt. Express 27(13), 18363–18375 (2019).
    [Crossref]
  10. X. Liu, J. Li, Q. Zhang, G. Pang, and D. J. Gelmecha, “Revolution and spin of a particle induced by an orbital-angular-momentum-carrying Laguerre-Gaussian beam in a dielectric chiral medium,” Phys. Rev. A 98(5), 053847 (2018).
    [Crossref]
  11. I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Angle of Arrival Fluctuations for Free Space Laser Beam Propagation through non Kolmogorov turbulence,” Proc. of SPIE 6551, 65510E (2007).
    [Crossref]
  12. I. Toselli, “Introducing the concept of anisotropy at different scales for modeling optical turbulence,” J. Opt. Soc. Am. A 31(8), 1868–1875 (2014).
    [Crossref]
  13. Y. Wu, H. Mei, C. Dai, F. Zhao, and H. Wei, “Design and analysis of performance of FSO communication system based on partially coherent beams,” Opt. Commun. 472, 126041 (2020).
    [Crossref]
  14. B. Zhang, H. Huang, C. Xie, S. Zhu, and Z. Li, “Twisted rectangular Laguerre-Gaussian correlated sources in anisotropic turbulent atmosphere,” Opt. Commun. 459, 125004 (2020).
    [Crossref]
  15. M. Zhou, W. Fan, and G. Wu, “Evolution properties of the orbital angular momentum spectrum of twisted Gaussian Schell-model beams in turbulent atmosphere,” J. Opt. Soc. Am. A 37(1), 142–148 (2020).
    [Crossref]
  16. X. Zhou, Z. Zhou, P. Tian, and X. Yuan, “Numerical research on partially coherent flat-topped beam propagation through atmospheric turbulence along a slant path,” Appl. Optics 58(34), 9443–9454 (2019).
    [Crossref]
  17. K. Wu, Y. Huai, T. Zhao, and Y. Jin, “Propagation of partially coherent four-petal elliptic Gaussian vortex beams in atmospheric turbulence,” Opt. Express 26(23), 30061–30075 (2018).
    [Crossref]
  18. J. Yu, Y. Cai, and G. Gbur, “Rectangular Hermite non-uniformly correlated beams and its propagation properties,” Opt. Express 26(21), 27894–27906 (2018).
    [Crossref]
  19. Y. Gu, O. Korotkova, and G. Gbur, “Scintillation of nonuniformly polarized beams in atmospheric turbulence,” Opt. Lett. 34(15), 2261–2263 (2009).
    [Crossref]
  20. Y. Gu and G. Gbur, “Reduction of Turbulence-induced Scintillation by Nonuniformly Polarized Beam Arrays,” Opt. Lett. 37(9), 1553–1555 (2012).
    [Crossref]
  21. L. Zhao, Y. Xu, and S. Yang, “Statistical properties of partially coherent vector beams propagating through anisotropic atmospheric turbulence,” Optik 227, 166115 (2021).
    [Crossref]
  22. M. Cheng, L. Guo, J. Li, X. Yan, K. Dong, and Y. You, “Average intensity and spreading of a radially polarized multi-Gaussian Schell-model beam in anisotropic turbulence,” J. Quant. Spectrosc. Radiat. Transf. 218, 12–20 (2018).
    [Crossref]
  23. G. P. Berman, V. N. Gorshkov, and S. V. Torous, “Scintillation reduction for laser beams propagating through turbulent atmosphere,” J. Phys. B-At. Mol. Opt. 44(5), 055402 (2011).
    [Crossref]
  24. J. Li, W. Wang, M. Duan, and J. Wei, “Influence of non-Kolmogorov atmospheric turbulence on the beam quality of vortex beams,” Opt. Express 24(18), 20413–20423 (2016).
    [Crossref]
  25. T. Yang, Y. Xu, H. Tian, D. Die, Q. Du, B. Zhang, and Y. Dan, “Propagation of partially coherent Laguerre Gaussian beams through inhomogeneous turbulent atmosphere,” J. Opt. Soc. Am. A 34(5), 713–720 (2017).
    [Crossref]
  26. Y. Xu, Y. Li, and X. Zhao, “Intensity and effective beam width of partially coherent Laguerre–Gaussian beams through a turbulent atmosphere,” J. Opt. Soc. Am. A 32(9), 1623–1630 (2015).
    [Crossref]
  27. Y. Xu and Y. Dan, “Statistical properties of electromagnetic anomalous vortex beam with orbital angular momentum in atmospheric turbulence,” Optik 179, 654–664 (2019).
    [Crossref]
  28. H. T. Eyyuboglu, “Scintillation behavior of vortex beams in strong turbulence region,” J. Mod. Opt. 63(21), 2374–2381 (2016).
    [Crossref]
  29. J. Yu, Y. Huang, F. Wang, X. Liu, G. Gbur, and Y. Cai, “Scintillation properties of a partially coherent vector beam with vortex phase in turbulent atmosphere,” Opt. Express 27(19), 26676–26688 (2019).
    [Crossref]
  30. F. Wang, Y. Cai, and O. Korotkova, “Partially coherent standard and elegant Laguerre-Gaussian beams of all orders,” Opt. Express 17(25), 22366–22379 (2009).
    [Crossref]
  31. K. Sidoro and R. E. Luis, “Relations between Hermite and Laguerre Gaussian modes,” IEEE J. Quantum Electron. 29(9), 2562–2567 (1993).
    [Crossref]
  32. Y. Xu, Y. Dan, J. Yu, and Y. Cai, “Kurtosis parameter K of arbitrary electromagnetic beams propagating through non-Kolmogorov turbulence,” J. Mod. Opt. 64(19), 1976–1987 (2017).
    [Crossref]
  33. M. Abramowitz and I. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (U. S. Department of Commerce, 1970).
  34. K. Yong, S. Tang, X. Yang, and R. Zhang, “Propagation characteristics of a ring Airy vortex beam in slant atmospheric turbulence,” J. Opt. Soc. Am. B 38(5), 1510–1517 (2021).
    [Crossref]
  35. S. Basu, M. W. Hyde, X. Xiao, D. G. Voelz, and O. Korotkova, “Computational approaches for generating electromagnetic Gaussian Schell-model sources,” Opt. Express 22(26), 31691–31707 (2014).
    [Crossref]
  36. L. C. Andrews, R. L. Phillips, and R. Crabbs, “Propagation of a Gaussian-beam wave in general anisotropic turbulence,” Proc. SPIE 9224, 922402 (2014).
    [Crossref]
  37. M. Cheng, L. Guo, J. Li, and Q. Huang, “Propagation properties of an optical vortex carried by a Bessel–Gaussian beam in anisotropic turbulence,” J. Opt. Soc. Am. A 33(8), 1442–1450 (2016).
    [Crossref]
  38. X. Peng, L. Liu, J. Yu, X. Liu, Y. Cai, Y. Baykal, and W. Li, “Propagation of a radially polarized twisted Gaussian Schell-model beam in turbulent atmosphere,” J. Opt. 18(12), 125601 (2016).
    [Crossref]

2021 (2)

L. Zhao, Y. Xu, and S. Yang, “Statistical properties of partially coherent vector beams propagating through anisotropic atmospheric turbulence,” Optik 227, 166115 (2021).
[Crossref]

K. Yong, S. Tang, X. Yang, and R. Zhang, “Propagation characteristics of a ring Airy vortex beam in slant atmospheric turbulence,” J. Opt. Soc. Am. B 38(5), 1510–1517 (2021).
[Crossref]

2020 (5)

B. Melo, I. Brandao, B. P. da Silva, R. B. Rodrigues, A. Z. Khoury, and T. Guerreiro, “Optical trapping in a dark focus,” Phys. Rev. Appl. 14(3), 034069 (2020).
[Crossref]

M. Luo, D. Sun, Y. Yang, S. Liu, J. Wu, Z. Ma, S. Sun, and X. Sun, “Three-dimensional isotropic STED microscopy generated by 4 pi focusing of a radially polarized vortex Laguerre-Gaussian beam,” Opt. Commun. 463, 125434 (2020).
[Crossref]

Y. Wu, H. Mei, C. Dai, F. Zhao, and H. Wei, “Design and analysis of performance of FSO communication system based on partially coherent beams,” Opt. Commun. 472, 126041 (2020).
[Crossref]

B. Zhang, H. Huang, C. Xie, S. Zhu, and Z. Li, “Twisted rectangular Laguerre-Gaussian correlated sources in anisotropic turbulent atmosphere,” Opt. Commun. 459, 125004 (2020).
[Crossref]

M. Zhou, W. Fan, and G. Wu, “Evolution properties of the orbital angular momentum spectrum of twisted Gaussian Schell-model beams in turbulent atmosphere,” J. Opt. Soc. Am. A 37(1), 142–148 (2020).
[Crossref]

2019 (4)

X. Zhou, Z. Zhou, P. Tian, and X. Yuan, “Numerical research on partially coherent flat-topped beam propagation through atmospheric turbulence along a slant path,” Appl. Optics 58(34), 9443–9454 (2019).
[Crossref]

S. Liu, S. Liu, C. Yang, Z. Xu, Y. Li, Y. Li, Z. Zhou, G. Guo, and B. Shi, “Classical simulation of high-dimensional entanglement by non-separable angular-radial modes,” Opt. Express 27(13), 18363–18375 (2019).
[Crossref]

Y. Xu and Y. Dan, “Statistical properties of electromagnetic anomalous vortex beam with orbital angular momentum in atmospheric turbulence,” Optik 179, 654–664 (2019).
[Crossref]

J. Yu, Y. Huang, F. Wang, X. Liu, G. Gbur, and Y. Cai, “Scintillation properties of a partially coherent vector beam with vortex phase in turbulent atmosphere,” Opt. Express 27(19), 26676–26688 (2019).
[Crossref]

2018 (4)

M. Cheng, L. Guo, J. Li, X. Yan, K. Dong, and Y. You, “Average intensity and spreading of a radially polarized multi-Gaussian Schell-model beam in anisotropic turbulence,” J. Quant. Spectrosc. Radiat. Transf. 218, 12–20 (2018).
[Crossref]

X. Liu, J. Li, Q. Zhang, G. Pang, and D. J. Gelmecha, “Revolution and spin of a particle induced by an orbital-angular-momentum-carrying Laguerre-Gaussian beam in a dielectric chiral medium,” Phys. Rev. A 98(5), 053847 (2018).
[Crossref]

K. Wu, Y. Huai, T. Zhao, and Y. Jin, “Propagation of partially coherent four-petal elliptic Gaussian vortex beams in atmospheric turbulence,” Opt. Express 26(23), 30061–30075 (2018).
[Crossref]

J. Yu, Y. Cai, and G. Gbur, “Rectangular Hermite non-uniformly correlated beams and its propagation properties,” Opt. Express 26(21), 27894–27906 (2018).
[Crossref]

2017 (2)

T. Yang, Y. Xu, H. Tian, D. Die, Q. Du, B. Zhang, and Y. Dan, “Propagation of partially coherent Laguerre Gaussian beams through inhomogeneous turbulent atmosphere,” J. Opt. Soc. Am. A 34(5), 713–720 (2017).
[Crossref]

Y. Xu, Y. Dan, J. Yu, and Y. Cai, “Kurtosis parameter K of arbitrary electromagnetic beams propagating through non-Kolmogorov turbulence,” J. Mod. Opt. 64(19), 1976–1987 (2017).
[Crossref]

2016 (4)

J. Li, W. Wang, M. Duan, and J. Wei, “Influence of non-Kolmogorov atmospheric turbulence on the beam quality of vortex beams,” Opt. Express 24(18), 20413–20423 (2016).
[Crossref]

M. Cheng, L. Guo, J. Li, and Q. Huang, “Propagation properties of an optical vortex carried by a Bessel–Gaussian beam in anisotropic turbulence,” J. Opt. Soc. Am. A 33(8), 1442–1450 (2016).
[Crossref]

X. Peng, L. Liu, J. Yu, X. Liu, Y. Cai, Y. Baykal, and W. Li, “Propagation of a radially polarized twisted Gaussian Schell-model beam in turbulent atmosphere,” J. Opt. 18(12), 125601 (2016).
[Crossref]

H. T. Eyyuboglu, “Scintillation behavior of vortex beams in strong turbulence region,” J. Mod. Opt. 63(21), 2374–2381 (2016).
[Crossref]

2015 (1)

2014 (3)

2012 (2)

Y. Gu and G. Gbur, “Reduction of Turbulence-induced Scintillation by Nonuniformly Polarized Beam Arrays,” Opt. Lett. 37(9), 1553–1555 (2012).
[Crossref]

J. Wang, J. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6(7), 488–496 (2012).
[Crossref]

2011 (1)

G. P. Berman, V. N. Gorshkov, and S. V. Torous, “Scintillation reduction for laser beams propagating through turbulent atmosphere,” J. Phys. B-At. Mol. Opt. 44(5), 055402 (2011).
[Crossref]

2009 (2)

2007 (1)

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Angle of Arrival Fluctuations for Free Space Laser Beam Propagation through non Kolmogorov turbulence,” Proc. of SPIE 6551, 65510E (2007).
[Crossref]

2004 (1)

2003 (1)

G. Grier D, “A revolution in optical manipulation,” Nature 424(6950), 810–816 (2003).
[Crossref]

2002 (1)

M. P. MacDonald, L. Paterson, K. Volke-Sepulveda, J. Arlt, W. Sibbett, and K. Dholakia, “Creation and manipulation of three-dimensional optically trapped structures,” Science 296(5570), 1101–1103 (2002).
[Crossref]

2001 (1)

1993 (1)

K. Sidoro and R. E. Luis, “Relations between Hermite and Laguerre Gaussian modes,” IEEE J. Quantum Electron. 29(9), 2562–2567 (1993).
[Crossref]

1992 (1)

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992).
[Crossref]

Abramowitz, M.

M. Abramowitz and I. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (U. S. Department of Commerce, 1970).

Ahmed, N.

J. Wang, J. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6(7), 488–496 (2012).
[Crossref]

Allen, L.

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992).
[Crossref]

Andrews, L. C.

L. C. Andrews, R. L. Phillips, and R. Crabbs, “Propagation of a Gaussian-beam wave in general anisotropic turbulence,” Proc. SPIE 9224, 922402 (2014).
[Crossref]

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Angle of Arrival Fluctuations for Free Space Laser Beam Propagation through non Kolmogorov turbulence,” Proc. of SPIE 6551, 65510E (2007).
[Crossref]

Arlt, J.

M. P. MacDonald, L. Paterson, K. Volke-Sepulveda, J. Arlt, W. Sibbett, and K. Dholakia, “Creation and manipulation of three-dimensional optically trapped structures,” Science 296(5570), 1101–1103 (2002).
[Crossref]

Barnett, S. M.

Basu, S.

Baykal, Y.

X. Peng, L. Liu, J. Yu, X. Liu, Y. Cai, Y. Baykal, and W. Li, “Propagation of a radially polarized twisted Gaussian Schell-model beam in turbulent atmosphere,” J. Opt. 18(12), 125601 (2016).
[Crossref]

Beijersbergen, M. W.

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992).
[Crossref]

Berman, G. P.

G. P. Berman, V. N. Gorshkov, and S. V. Torous, “Scintillation reduction for laser beams propagating through turbulent atmosphere,” J. Phys. B-At. Mol. Opt. 44(5), 055402 (2011).
[Crossref]

Brandao, I.

B. Melo, I. Brandao, B. P. da Silva, R. B. Rodrigues, A. Z. Khoury, and T. Guerreiro, “Optical trapping in a dark focus,” Phys. Rev. Appl. 14(3), 034069 (2020).
[Crossref]

Cai, Y.

Cheng, M.

M. Cheng, L. Guo, J. Li, X. Yan, K. Dong, and Y. You, “Average intensity and spreading of a radially polarized multi-Gaussian Schell-model beam in anisotropic turbulence,” J. Quant. Spectrosc. Radiat. Transf. 218, 12–20 (2018).
[Crossref]

M. Cheng, L. Guo, J. Li, and Q. Huang, “Propagation properties of an optical vortex carried by a Bessel–Gaussian beam in anisotropic turbulence,” J. Opt. Soc. Am. A 33(8), 1442–1450 (2016).
[Crossref]

Courtial, J.

Crabbs, R.

L. C. Andrews, R. L. Phillips, and R. Crabbs, “Propagation of a Gaussian-beam wave in general anisotropic turbulence,” Proc. SPIE 9224, 922402 (2014).
[Crossref]

da Silva, B. P.

B. Melo, I. Brandao, B. P. da Silva, R. B. Rodrigues, A. Z. Khoury, and T. Guerreiro, “Optical trapping in a dark focus,” Phys. Rev. Appl. 14(3), 034069 (2020).
[Crossref]

Dai, C.

Y. Wu, H. Mei, C. Dai, F. Zhao, and H. Wei, “Design and analysis of performance of FSO communication system based on partially coherent beams,” Opt. Commun. 472, 126041 (2020).
[Crossref]

Dan, Y.

Y. Xu and Y. Dan, “Statistical properties of electromagnetic anomalous vortex beam with orbital angular momentum in atmospheric turbulence,” Optik 179, 654–664 (2019).
[Crossref]

T. Yang, Y. Xu, H. Tian, D. Die, Q. Du, B. Zhang, and Y. Dan, “Propagation of partially coherent Laguerre Gaussian beams through inhomogeneous turbulent atmosphere,” J. Opt. Soc. Am. A 34(5), 713–720 (2017).
[Crossref]

Y. Xu, Y. Dan, J. Yu, and Y. Cai, “Kurtosis parameter K of arbitrary electromagnetic beams propagating through non-Kolmogorov turbulence,” J. Mod. Opt. 64(19), 1976–1987 (2017).
[Crossref]

Dholakia, K.

M. P. MacDonald, L. Paterson, K. Volke-Sepulveda, J. Arlt, W. Sibbett, and K. Dholakia, “Creation and manipulation of three-dimensional optically trapped structures,” Science 296(5570), 1101–1103 (2002).
[Crossref]

Die, D.

Dolinar, S.

J. Wang, J. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6(7), 488–496 (2012).
[Crossref]

Dong, K.

M. Cheng, L. Guo, J. Li, X. Yan, K. Dong, and Y. You, “Average intensity and spreading of a radially polarized multi-Gaussian Schell-model beam in anisotropic turbulence,” J. Quant. Spectrosc. Radiat. Transf. 218, 12–20 (2018).
[Crossref]

Du, Q.

Duan, M.

Eyyuboglu, H. T.

H. T. Eyyuboglu, “Scintillation behavior of vortex beams in strong turbulence region,” J. Mod. Opt. 63(21), 2374–2381 (2016).
[Crossref]

Fan, W.

Fazal, I. M.

J. Wang, J. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6(7), 488–496 (2012).
[Crossref]

Ferrero, V.

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Angle of Arrival Fluctuations for Free Space Laser Beam Propagation through non Kolmogorov turbulence,” Proc. of SPIE 6551, 65510E (2007).
[Crossref]

Franke-Arnold, S.

Gbur, G.

Gelmecha, D. J.

X. Liu, J. Li, Q. Zhang, G. Pang, and D. J. Gelmecha, “Revolution and spin of a particle induced by an orbital-angular-momentum-carrying Laguerre-Gaussian beam in a dielectric chiral medium,” Phys. Rev. A 98(5), 053847 (2018).
[Crossref]

Gibson, G.

Gorshkov, V. N.

G. P. Berman, V. N. Gorshkov, and S. V. Torous, “Scintillation reduction for laser beams propagating through turbulent atmosphere,” J. Phys. B-At. Mol. Opt. 44(5), 055402 (2011).
[Crossref]

Grier D, G.

G. Grier D, “A revolution in optical manipulation,” Nature 424(6950), 810–816 (2003).
[Crossref]

Gu, Y.

Guerreiro, T.

B. Melo, I. Brandao, B. P. da Silva, R. B. Rodrigues, A. Z. Khoury, and T. Guerreiro, “Optical trapping in a dark focus,” Phys. Rev. Appl. 14(3), 034069 (2020).
[Crossref]

Guo, G.

Guo, L.

M. Cheng, L. Guo, J. Li, X. Yan, K. Dong, and Y. You, “Average intensity and spreading of a radially polarized multi-Gaussian Schell-model beam in anisotropic turbulence,” J. Quant. Spectrosc. Radiat. Transf. 218, 12–20 (2018).
[Crossref]

M. Cheng, L. Guo, J. Li, and Q. Huang, “Propagation properties of an optical vortex carried by a Bessel–Gaussian beam in anisotropic turbulence,” J. Opt. Soc. Am. A 33(8), 1442–1450 (2016).
[Crossref]

Huai, Y.

Huang, H.

B. Zhang, H. Huang, C. Xie, S. Zhu, and Z. Li, “Twisted rectangular Laguerre-Gaussian correlated sources in anisotropic turbulent atmosphere,” Opt. Commun. 459, 125004 (2020).
[Crossref]

J. Wang, J. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6(7), 488–496 (2012).
[Crossref]

Huang, Q.

Huang, Y.

Hyde, M. W.

Jin, Y.

Khoury, A. Z.

B. Melo, I. Brandao, B. P. da Silva, R. B. Rodrigues, A. Z. Khoury, and T. Guerreiro, “Optical trapping in a dark focus,” Phys. Rev. Appl. 14(3), 034069 (2020).
[Crossref]

Korotkova, O.

Li, J.

M. Cheng, L. Guo, J. Li, X. Yan, K. Dong, and Y. You, “Average intensity and spreading of a radially polarized multi-Gaussian Schell-model beam in anisotropic turbulence,” J. Quant. Spectrosc. Radiat. Transf. 218, 12–20 (2018).
[Crossref]

X. Liu, J. Li, Q. Zhang, G. Pang, and D. J. Gelmecha, “Revolution and spin of a particle induced by an orbital-angular-momentum-carrying Laguerre-Gaussian beam in a dielectric chiral medium,” Phys. Rev. A 98(5), 053847 (2018).
[Crossref]

J. Li, W. Wang, M. Duan, and J. Wei, “Influence of non-Kolmogorov atmospheric turbulence on the beam quality of vortex beams,” Opt. Express 24(18), 20413–20423 (2016).
[Crossref]

M. Cheng, L. Guo, J. Li, and Q. Huang, “Propagation properties of an optical vortex carried by a Bessel–Gaussian beam in anisotropic turbulence,” J. Opt. Soc. Am. A 33(8), 1442–1450 (2016).
[Crossref]

Li, W.

X. Peng, L. Liu, J. Yu, X. Liu, Y. Cai, Y. Baykal, and W. Li, “Propagation of a radially polarized twisted Gaussian Schell-model beam in turbulent atmosphere,” J. Opt. 18(12), 125601 (2016).
[Crossref]

Li, Y.

Li, Z.

B. Zhang, H. Huang, C. Xie, S. Zhu, and Z. Li, “Twisted rectangular Laguerre-Gaussian correlated sources in anisotropic turbulent atmosphere,” Opt. Commun. 459, 125004 (2020).
[Crossref]

Liu, L.

X. Peng, L. Liu, J. Yu, X. Liu, Y. Cai, Y. Baykal, and W. Li, “Propagation of a radially polarized twisted Gaussian Schell-model beam in turbulent atmosphere,” J. Opt. 18(12), 125601 (2016).
[Crossref]

Liu, S.

Liu, X.

J. Yu, Y. Huang, F. Wang, X. Liu, G. Gbur, and Y. Cai, “Scintillation properties of a partially coherent vector beam with vortex phase in turbulent atmosphere,” Opt. Express 27(19), 26676–26688 (2019).
[Crossref]

X. Liu, J. Li, Q. Zhang, G. Pang, and D. J. Gelmecha, “Revolution and spin of a particle induced by an orbital-angular-momentum-carrying Laguerre-Gaussian beam in a dielectric chiral medium,” Phys. Rev. A 98(5), 053847 (2018).
[Crossref]

X. Peng, L. Liu, J. Yu, X. Liu, Y. Cai, Y. Baykal, and W. Li, “Propagation of a radially polarized twisted Gaussian Schell-model beam in turbulent atmosphere,” J. Opt. 18(12), 125601 (2016).
[Crossref]

Luis, R. E.

K. Sidoro and R. E. Luis, “Relations between Hermite and Laguerre Gaussian modes,” IEEE J. Quantum Electron. 29(9), 2562–2567 (1993).
[Crossref]

Luo, M.

M. Luo, D. Sun, Y. Yang, S. Liu, J. Wu, Z. Ma, S. Sun, and X. Sun, “Three-dimensional isotropic STED microscopy generated by 4 pi focusing of a radially polarized vortex Laguerre-Gaussian beam,” Opt. Commun. 463, 125434 (2020).
[Crossref]

Ma, Z.

M. Luo, D. Sun, Y. Yang, S. Liu, J. Wu, Z. Ma, S. Sun, and X. Sun, “Three-dimensional isotropic STED microscopy generated by 4 pi focusing of a radially polarized vortex Laguerre-Gaussian beam,” Opt. Commun. 463, 125434 (2020).
[Crossref]

MacDonald, M. P.

M. P. MacDonald, L. Paterson, K. Volke-Sepulveda, J. Arlt, W. Sibbett, and K. Dholakia, “Creation and manipulation of three-dimensional optically trapped structures,” Science 296(5570), 1101–1103 (2002).
[Crossref]

Mei, H.

Y. Wu, H. Mei, C. Dai, F. Zhao, and H. Wei, “Design and analysis of performance of FSO communication system based on partially coherent beams,” Opt. Commun. 472, 126041 (2020).
[Crossref]

Melo, B.

B. Melo, I. Brandao, B. P. da Silva, R. B. Rodrigues, A. Z. Khoury, and T. Guerreiro, “Optical trapping in a dark focus,” Phys. Rev. Appl. 14(3), 034069 (2020).
[Crossref]

Padgett, M. J.

Pang, G.

X. Liu, J. Li, Q. Zhang, G. Pang, and D. J. Gelmecha, “Revolution and spin of a particle induced by an orbital-angular-momentum-carrying Laguerre-Gaussian beam in a dielectric chiral medium,” Phys. Rev. A 98(5), 053847 (2018).
[Crossref]

Pas’ko, V.

Paterson, L.

M. P. MacDonald, L. Paterson, K. Volke-Sepulveda, J. Arlt, W. Sibbett, and K. Dholakia, “Creation and manipulation of three-dimensional optically trapped structures,” Science 296(5570), 1101–1103 (2002).
[Crossref]

Peng, X.

X. Peng, L. Liu, J. Yu, X. Liu, Y. Cai, Y. Baykal, and W. Li, “Propagation of a radially polarized twisted Gaussian Schell-model beam in turbulent atmosphere,” J. Opt. 18(12), 125601 (2016).
[Crossref]

Phillips, R. L.

L. C. Andrews, R. L. Phillips, and R. Crabbs, “Propagation of a Gaussian-beam wave in general anisotropic turbulence,” Proc. SPIE 9224, 922402 (2014).
[Crossref]

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Angle of Arrival Fluctuations for Free Space Laser Beam Propagation through non Kolmogorov turbulence,” Proc. of SPIE 6551, 65510E (2007).
[Crossref]

Ren, Y.

J. Wang, J. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6(7), 488–496 (2012).
[Crossref]

Rodrigues, R. B.

B. Melo, I. Brandao, B. P. da Silva, R. B. Rodrigues, A. Z. Khoury, and T. Guerreiro, “Optical trapping in a dark focus,” Phys. Rev. Appl. 14(3), 034069 (2020).
[Crossref]

Shi, B.

Sibbett, W.

M. P. MacDonald, L. Paterson, K. Volke-Sepulveda, J. Arlt, W. Sibbett, and K. Dholakia, “Creation and manipulation of three-dimensional optically trapped structures,” Science 296(5570), 1101–1103 (2002).
[Crossref]

Sidoro, K.

K. Sidoro and R. E. Luis, “Relations between Hermite and Laguerre Gaussian modes,” IEEE J. Quantum Electron. 29(9), 2562–2567 (1993).
[Crossref]

Spreeuw, R. J. C.

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992).
[Crossref]

Stegun, I.

M. Abramowitz and I. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (U. S. Department of Commerce, 1970).

Sun, D.

M. Luo, D. Sun, Y. Yang, S. Liu, J. Wu, Z. Ma, S. Sun, and X. Sun, “Three-dimensional isotropic STED microscopy generated by 4 pi focusing of a radially polarized vortex Laguerre-Gaussian beam,” Opt. Commun. 463, 125434 (2020).
[Crossref]

Sun, S.

M. Luo, D. Sun, Y. Yang, S. Liu, J. Wu, Z. Ma, S. Sun, and X. Sun, “Three-dimensional isotropic STED microscopy generated by 4 pi focusing of a radially polarized vortex Laguerre-Gaussian beam,” Opt. Commun. 463, 125434 (2020).
[Crossref]

Sun, X.

M. Luo, D. Sun, Y. Yang, S. Liu, J. Wu, Z. Ma, S. Sun, and X. Sun, “Three-dimensional isotropic STED microscopy generated by 4 pi focusing of a radially polarized vortex Laguerre-Gaussian beam,” Opt. Commun. 463, 125434 (2020).
[Crossref]

Swartzlander, G. A.

Tang, S.

Tian, H.

Tian, P.

X. Zhou, Z. Zhou, P. Tian, and X. Yuan, “Numerical research on partially coherent flat-topped beam propagation through atmospheric turbulence along a slant path,” Appl. Optics 58(34), 9443–9454 (2019).
[Crossref]

Torous, S. V.

G. P. Berman, V. N. Gorshkov, and S. V. Torous, “Scintillation reduction for laser beams propagating through turbulent atmosphere,” J. Phys. B-At. Mol. Opt. 44(5), 055402 (2011).
[Crossref]

Toselli, I.

I. Toselli, “Introducing the concept of anisotropy at different scales for modeling optical turbulence,” J. Opt. Soc. Am. A 31(8), 1868–1875 (2014).
[Crossref]

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Angle of Arrival Fluctuations for Free Space Laser Beam Propagation through non Kolmogorov turbulence,” Proc. of SPIE 6551, 65510E (2007).
[Crossref]

Tur, M.

J. Wang, J. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6(7), 488–496 (2012).
[Crossref]

Vasnetsov, M.

Voelz, D. G.

Volke-Sepulveda, K.

M. P. MacDonald, L. Paterson, K. Volke-Sepulveda, J. Arlt, W. Sibbett, and K. Dholakia, “Creation and manipulation of three-dimensional optically trapped structures,” Science 296(5570), 1101–1103 (2002).
[Crossref]

Wang, F.

Wang, J.

J. Wang, J. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6(7), 488–496 (2012).
[Crossref]

Wang, W.

Wei, H.

Y. Wu, H. Mei, C. Dai, F. Zhao, and H. Wei, “Design and analysis of performance of FSO communication system based on partially coherent beams,” Opt. Commun. 472, 126041 (2020).
[Crossref]

Wei, J.

Willner, A. E.

J. Wang, J. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6(7), 488–496 (2012).
[Crossref]

Woerdman, J. P.

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992).
[Crossref]

Wu, G.

Wu, J.

M. Luo, D. Sun, Y. Yang, S. Liu, J. Wu, Z. Ma, S. Sun, and X. Sun, “Three-dimensional isotropic STED microscopy generated by 4 pi focusing of a radially polarized vortex Laguerre-Gaussian beam,” Opt. Commun. 463, 125434 (2020).
[Crossref]

Wu, K.

Wu, Y.

Y. Wu, H. Mei, C. Dai, F. Zhao, and H. Wei, “Design and analysis of performance of FSO communication system based on partially coherent beams,” Opt. Commun. 472, 126041 (2020).
[Crossref]

Xiao, X.

Xie, C.

B. Zhang, H. Huang, C. Xie, S. Zhu, and Z. Li, “Twisted rectangular Laguerre-Gaussian correlated sources in anisotropic turbulent atmosphere,” Opt. Commun. 459, 125004 (2020).
[Crossref]

Xu, Y.

L. Zhao, Y. Xu, and S. Yang, “Statistical properties of partially coherent vector beams propagating through anisotropic atmospheric turbulence,” Optik 227, 166115 (2021).
[Crossref]

Y. Xu and Y. Dan, “Statistical properties of electromagnetic anomalous vortex beam with orbital angular momentum in atmospheric turbulence,” Optik 179, 654–664 (2019).
[Crossref]

T. Yang, Y. Xu, H. Tian, D. Die, Q. Du, B. Zhang, and Y. Dan, “Propagation of partially coherent Laguerre Gaussian beams through inhomogeneous turbulent atmosphere,” J. Opt. Soc. Am. A 34(5), 713–720 (2017).
[Crossref]

Y. Xu, Y. Dan, J. Yu, and Y. Cai, “Kurtosis parameter K of arbitrary electromagnetic beams propagating through non-Kolmogorov turbulence,” J. Mod. Opt. 64(19), 1976–1987 (2017).
[Crossref]

Y. Xu, Y. Li, and X. Zhao, “Intensity and effective beam width of partially coherent Laguerre–Gaussian beams through a turbulent atmosphere,” J. Opt. Soc. Am. A 32(9), 1623–1630 (2015).
[Crossref]

Xu, Z.

Yan, X.

M. Cheng, L. Guo, J. Li, X. Yan, K. Dong, and Y. You, “Average intensity and spreading of a radially polarized multi-Gaussian Schell-model beam in anisotropic turbulence,” J. Quant. Spectrosc. Radiat. Transf. 218, 12–20 (2018).
[Crossref]

Yan, Y.

J. Wang, J. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6(7), 488–496 (2012).
[Crossref]

Yang, C.

Yang, J.

J. Wang, J. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6(7), 488–496 (2012).
[Crossref]

Yang, S.

L. Zhao, Y. Xu, and S. Yang, “Statistical properties of partially coherent vector beams propagating through anisotropic atmospheric turbulence,” Optik 227, 166115 (2021).
[Crossref]

Yang, T.

Yang, X.

Yang, Y.

M. Luo, D. Sun, Y. Yang, S. Liu, J. Wu, Z. Ma, S. Sun, and X. Sun, “Three-dimensional isotropic STED microscopy generated by 4 pi focusing of a radially polarized vortex Laguerre-Gaussian beam,” Opt. Commun. 463, 125434 (2020).
[Crossref]

Yong, K.

You, Y.

M. Cheng, L. Guo, J. Li, X. Yan, K. Dong, and Y. You, “Average intensity and spreading of a radially polarized multi-Gaussian Schell-model beam in anisotropic turbulence,” J. Quant. Spectrosc. Radiat. Transf. 218, 12–20 (2018).
[Crossref]

Yu, J.

J. Yu, Y. Huang, F. Wang, X. Liu, G. Gbur, and Y. Cai, “Scintillation properties of a partially coherent vector beam with vortex phase in turbulent atmosphere,” Opt. Express 27(19), 26676–26688 (2019).
[Crossref]

J. Yu, Y. Cai, and G. Gbur, “Rectangular Hermite non-uniformly correlated beams and its propagation properties,” Opt. Express 26(21), 27894–27906 (2018).
[Crossref]

Y. Xu, Y. Dan, J. Yu, and Y. Cai, “Kurtosis parameter K of arbitrary electromagnetic beams propagating through non-Kolmogorov turbulence,” J. Mod. Opt. 64(19), 1976–1987 (2017).
[Crossref]

X. Peng, L. Liu, J. Yu, X. Liu, Y. Cai, Y. Baykal, and W. Li, “Propagation of a radially polarized twisted Gaussian Schell-model beam in turbulent atmosphere,” J. Opt. 18(12), 125601 (2016).
[Crossref]

Yuan, X.

X. Zhou, Z. Zhou, P. Tian, and X. Yuan, “Numerical research on partially coherent flat-topped beam propagation through atmospheric turbulence along a slant path,” Appl. Optics 58(34), 9443–9454 (2019).
[Crossref]

Zhang, B.

B. Zhang, H. Huang, C. Xie, S. Zhu, and Z. Li, “Twisted rectangular Laguerre-Gaussian correlated sources in anisotropic turbulent atmosphere,” Opt. Commun. 459, 125004 (2020).
[Crossref]

T. Yang, Y. Xu, H. Tian, D. Die, Q. Du, B. Zhang, and Y. Dan, “Propagation of partially coherent Laguerre Gaussian beams through inhomogeneous turbulent atmosphere,” J. Opt. Soc. Am. A 34(5), 713–720 (2017).
[Crossref]

Zhang, Q.

X. Liu, J. Li, Q. Zhang, G. Pang, and D. J. Gelmecha, “Revolution and spin of a particle induced by an orbital-angular-momentum-carrying Laguerre-Gaussian beam in a dielectric chiral medium,” Phys. Rev. A 98(5), 053847 (2018).
[Crossref]

Zhang, R.

Zhao, F.

Y. Wu, H. Mei, C. Dai, F. Zhao, and H. Wei, “Design and analysis of performance of FSO communication system based on partially coherent beams,” Opt. Commun. 472, 126041 (2020).
[Crossref]

Zhao, L.

L. Zhao, Y. Xu, and S. Yang, “Statistical properties of partially coherent vector beams propagating through anisotropic atmospheric turbulence,” Optik 227, 166115 (2021).
[Crossref]

Zhao, T.

Zhao, X.

Zhou, M.

Zhou, X.

X. Zhou, Z. Zhou, P. Tian, and X. Yuan, “Numerical research on partially coherent flat-topped beam propagation through atmospheric turbulence along a slant path,” Appl. Optics 58(34), 9443–9454 (2019).
[Crossref]

Zhou, Z.

X. Zhou, Z. Zhou, P. Tian, and X. Yuan, “Numerical research on partially coherent flat-topped beam propagation through atmospheric turbulence along a slant path,” Appl. Optics 58(34), 9443–9454 (2019).
[Crossref]

S. Liu, S. Liu, C. Yang, Z. Xu, Y. Li, Y. Li, Z. Zhou, G. Guo, and B. Shi, “Classical simulation of high-dimensional entanglement by non-separable angular-radial modes,” Opt. Express 27(13), 18363–18375 (2019).
[Crossref]

Zhu, S.

B. Zhang, H. Huang, C. Xie, S. Zhu, and Z. Li, “Twisted rectangular Laguerre-Gaussian correlated sources in anisotropic turbulent atmosphere,” Opt. Commun. 459, 125004 (2020).
[Crossref]

Appl. Optics (1)

X. Zhou, Z. Zhou, P. Tian, and X. Yuan, “Numerical research on partially coherent flat-topped beam propagation through atmospheric turbulence along a slant path,” Appl. Optics 58(34), 9443–9454 (2019).
[Crossref]

IEEE J. Quantum Electron. (1)

K. Sidoro and R. E. Luis, “Relations between Hermite and Laguerre Gaussian modes,” IEEE J. Quantum Electron. 29(9), 2562–2567 (1993).
[Crossref]

J. Mod. Opt. (2)

Y. Xu, Y. Dan, J. Yu, and Y. Cai, “Kurtosis parameter K of arbitrary electromagnetic beams propagating through non-Kolmogorov turbulence,” J. Mod. Opt. 64(19), 1976–1987 (2017).
[Crossref]

H. T. Eyyuboglu, “Scintillation behavior of vortex beams in strong turbulence region,” J. Mod. Opt. 63(21), 2374–2381 (2016).
[Crossref]

J. Opt. (1)

X. Peng, L. Liu, J. Yu, X. Liu, Y. Cai, Y. Baykal, and W. Li, “Propagation of a radially polarized twisted Gaussian Schell-model beam in turbulent atmosphere,” J. Opt. 18(12), 125601 (2016).
[Crossref]

J. Opt. Soc. Am. A (5)

J. Opt. Soc. Am. B (1)

J. Phys. B-At. Mol. Opt. (1)

G. P. Berman, V. N. Gorshkov, and S. V. Torous, “Scintillation reduction for laser beams propagating through turbulent atmosphere,” J. Phys. B-At. Mol. Opt. 44(5), 055402 (2011).
[Crossref]

J. Quant. Spectrosc. Radiat. Transf. (1)

M. Cheng, L. Guo, J. Li, X. Yan, K. Dong, and Y. You, “Average intensity and spreading of a radially polarized multi-Gaussian Schell-model beam in anisotropic turbulence,” J. Quant. Spectrosc. Radiat. Transf. 218, 12–20 (2018).
[Crossref]

Nat. Photonics (1)

J. Wang, J. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6(7), 488–496 (2012).
[Crossref]

Nature (1)

G. Grier D, “A revolution in optical manipulation,” Nature 424(6950), 810–816 (2003).
[Crossref]

Opt. Commun. (3)

M. Luo, D. Sun, Y. Yang, S. Liu, J. Wu, Z. Ma, S. Sun, and X. Sun, “Three-dimensional isotropic STED microscopy generated by 4 pi focusing of a radially polarized vortex Laguerre-Gaussian beam,” Opt. Commun. 463, 125434 (2020).
[Crossref]

Y. Wu, H. Mei, C. Dai, F. Zhao, and H. Wei, “Design and analysis of performance of FSO communication system based on partially coherent beams,” Opt. Commun. 472, 126041 (2020).
[Crossref]

B. Zhang, H. Huang, C. Xie, S. Zhu, and Z. Li, “Twisted rectangular Laguerre-Gaussian correlated sources in anisotropic turbulent atmosphere,” Opt. Commun. 459, 125004 (2020).
[Crossref]

Opt. Express (8)

K. Wu, Y. Huai, T. Zhao, and Y. Jin, “Propagation of partially coherent four-petal elliptic Gaussian vortex beams in atmospheric turbulence,” Opt. Express 26(23), 30061–30075 (2018).
[Crossref]

J. Yu, Y. Cai, and G. Gbur, “Rectangular Hermite non-uniformly correlated beams and its propagation properties,” Opt. Express 26(21), 27894–27906 (2018).
[Crossref]

S. Liu, S. Liu, C. Yang, Z. Xu, Y. Li, Y. Li, Z. Zhou, G. Guo, and B. Shi, “Classical simulation of high-dimensional entanglement by non-separable angular-radial modes,” Opt. Express 27(13), 18363–18375 (2019).
[Crossref]

G. Gibson, J. Courtial, M. J. Padgett, M. Vasnetsov, V. Pas’ko, S. M. Barnett, and S. Franke-Arnold, “Free-space information transfer using light beams carrying orbital angular momentum,” Opt. Express 12(22), 5448–5456 (2004).
[Crossref]

J. Li, W. Wang, M. Duan, and J. Wei, “Influence of non-Kolmogorov atmospheric turbulence on the beam quality of vortex beams,” Opt. Express 24(18), 20413–20423 (2016).
[Crossref]

J. Yu, Y. Huang, F. Wang, X. Liu, G. Gbur, and Y. Cai, “Scintillation properties of a partially coherent vector beam with vortex phase in turbulent atmosphere,” Opt. Express 27(19), 26676–26688 (2019).
[Crossref]

F. Wang, Y. Cai, and O. Korotkova, “Partially coherent standard and elegant Laguerre-Gaussian beams of all orders,” Opt. Express 17(25), 22366–22379 (2009).
[Crossref]

S. Basu, M. W. Hyde, X. Xiao, D. G. Voelz, and O. Korotkova, “Computational approaches for generating electromagnetic Gaussian Schell-model sources,” Opt. Express 22(26), 31691–31707 (2014).
[Crossref]

Opt. Lett. (3)

Optik (2)

L. Zhao, Y. Xu, and S. Yang, “Statistical properties of partially coherent vector beams propagating through anisotropic atmospheric turbulence,” Optik 227, 166115 (2021).
[Crossref]

Y. Xu and Y. Dan, “Statistical properties of electromagnetic anomalous vortex beam with orbital angular momentum in atmospheric turbulence,” Optik 179, 654–664 (2019).
[Crossref]

Phys. Rev. A (2)

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992).
[Crossref]

X. Liu, J. Li, Q. Zhang, G. Pang, and D. J. Gelmecha, “Revolution and spin of a particle induced by an orbital-angular-momentum-carrying Laguerre-Gaussian beam in a dielectric chiral medium,” Phys. Rev. A 98(5), 053847 (2018).
[Crossref]

Phys. Rev. Appl. (1)

B. Melo, I. Brandao, B. P. da Silva, R. B. Rodrigues, A. Z. Khoury, and T. Guerreiro, “Optical trapping in a dark focus,” Phys. Rev. Appl. 14(3), 034069 (2020).
[Crossref]

Proc. of SPIE (1)

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Angle of Arrival Fluctuations for Free Space Laser Beam Propagation through non Kolmogorov turbulence,” Proc. of SPIE 6551, 65510E (2007).
[Crossref]

Proc. SPIE (1)

L. C. Andrews, R. L. Phillips, and R. Crabbs, “Propagation of a Gaussian-beam wave in general anisotropic turbulence,” Proc. SPIE 9224, 922402 (2014).
[Crossref]

Science (1)

M. P. MacDonald, L. Paterson, K. Volke-Sepulveda, J. Arlt, W. Sibbett, and K. Dholakia, “Creation and manipulation of three-dimensional optically trapped structures,” Science 296(5570), 1101–1103 (2002).
[Crossref]

Other (1)

M. Abramowitz and I. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (U. S. Department of Commerce, 1970).

Data availability

No data were generated or analyzed in the presented research.

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Figures (10)

Fig. 1.
Fig. 1. Schematic diagram of multi-phase screen method. PS: phase screen.
Fig. 2.
Fig. 2. Normalized average intensities Ix/Ixmax, Iy/Iymax and I/Imax of a PCRPLGV beam at different propagation distances in atmospheric turbulence. n=1, m=0 and $\tilde{C}_n^{2^\ast} $ = 1×10−14 m3–α.
Fig. 3.
Fig. 3. Normalized average intensities Ix/Ixmax, Iy/Iymax and I/Imax of a PCRPLGV beam at different propagation distances in atmospheric turbulence. n=1, m=2 and $\tilde{C}_n^{2^\ast} $ = 1×10−14 m3–α.
Fig. 4.
Fig. 4. Normalized average intensity distribution of a PCRPLGV beam in free space and atmospheric turbulence. n=1, m=1; (a1)–(a8): $\tilde{C}_n^{2^\ast} $ = 0; (b1)–(b8): $\tilde{C}_n^{2^\ast} $ = 2×10−15 m3−α.
Fig. 5.
Fig. 5. Simulation of normalized average intensity distribution of a PCRPLGV beam in free space and atmospheric turbulence by using multi-phase screen method. n=1, m=1; (a1)–(a8): $\tilde{C}_n^{2^\ast} $ = 0; (b1)–(b8): $\tilde{C}_n^{2^\ast} $ = 2×10−15 m3–α.
Fig. 6.
Fig. 6. Normalized average intensities of PCRPLGV beam at z = 2 km in atmospheric turbulence with different anisotropic factors and power law exponent. $\tilde{C}_n^{2} $ = 1×10−14 m3−α; (b1)–(b4): ξx=2, ξy=3.
Fig. 7.
Fig. 7. The DOC distribution of a PCRPLGV beam at different propagation distances in atmospheric turbulence. $\tilde{C}_n^{2^\ast} $ = 1×10−15 m3–α.
Fig. 8.
Fig. 8. Simulation of DOC distribution of a PCRPLGV beam at different propagation distances in atmospheric turbulence. $\tilde{C}_n^{2^\ast} $ = 1×10−15 m3–α.
Fig. 9.
Fig. 9. The DOC distribution of PCRPLGV beam at z = 2 km in atmospheric turbulence with different anisotropic factors and power law exponent. $\tilde{C}_n^{2} $ = 1×10−14 m3−α; (b1)–(b4): ξx=2, ξy=3.
Fig. 10.
Fig. 10. The DOP of PCRPLGV beam at (2 cm,0) vs propagation distance z in different conditions. (a) n = 0, $\tilde{C}_n^{2^\ast} $ = 2×10−15 m3–α, δ0 = 20 mm; (b) m = 0, $\tilde{C}_n^{2^\ast} $ = 2×10−15 m3–α, δ0 = 20 mm; (c) n = 0, m = 1, $\tilde{C}_n^{2^\ast} $ = 2×10−15 m3–α; (d) n = 0, m = 1, δ0 = 20 mm; (e) n = 0, m = 1, $\tilde{C}_n^{2} $= 1×10−14 m3–α, δ0 = 20 mm; (f) n = 0, m = 1, $\tilde{C}_n^{2^\ast} $ = 2×10−15 m3–α, δ0 = 20 mm.

Equations (27)

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E(r,θ;0)=(2rw0)mexp(r2w02)Lnm(2r2w02)exp(imθ),
exp(imθ)rmLnm(r2)=(1)n22n+mn!t=0ns=0mis(nt)(ms)H2t+ms(x)H2n2t+s(y),
E(x,y;0)=xw0exp(r2w02)(1)n22n+mn!t=0ns=0mis(nt)(ms)H2t+ms(xa0)H2n2t+s(ya0)ex+yw0exp(r2w02)(1)n22n+mn!t=0ns=0mis(nt)(ms)H2t+ms(xa0)H2n2t+s(ya0)ey,
W(r1,r2;0)=[Wxx(r1,r2;0)Wxy(r1,r2;0)Wyx(r1,r2;0)Wyy(r1,r2;0)],
Wαβ(r1,r2;0)=Eα(r1;0)Eβ(r2;0)gαβ(r1r2),
gαβ(r1r2)=Bαβexp[(x1x2)2+(y1y2)22δαβ2].
Wαβ(r1,r2,0)=α1β2w02exp(r12+r22w02)exp[(r1r2)22δαβ2]124n+2m(n!)2t=0ns=0mμ=0nν=0m(is)iv(nt)(ms)(nμ)(mν)H2t+ms(x1a0)H2μ+mν(x2a0)H2n2t+s(y1a0)H2n2μ+ν(y2a0).
Wαβ(ρ1,ρ2;z)=(k2πz)2Wαβ(r1,r2,0)×exp[ik(ρ1r1)22z+ik(ρ2r2)22zDw(r1,r2,ρ1,ρ2;z)2]d2r1d2r2,
Dw(r1,r2,ρ1,ρ2;z)π2k2z3[(ρ1ρ2)2+(ρ1ρ2)(r1r2)+(r1r2)2]0κ3Φn(κ)dκ,
ρ0=(13π2k2zT)1/2,T=0κ3Φn(κ)dκ.
Wαβ(ρ1,ρ2;z)=(k2πzw0)2exp[ik2z(ρ12ρ22)]exp[1ρ02(ρ1ρ2)2]×124n+2m(n!)2t=0ns=0mμ=0nν=0m(is)iv(nt)(ms)(nμ)(mν)α1β2H2t+ms(x1a0)exp[(1w02+12δαβ2+ik2z+1ρ02)x12+(x2δαβ2+2x2ρ02+iku1zu1u2ρ02)x1]×H2μ+mν(x2a0)exp[(1w02+12δαβ2ik2z+1ρ02)x22+(iku2z+u1u2ρ02)x2]×H2n2t+s(y1a0)exp[(1w02+12δαβ2+ik2z+1ρ02)y12+(y2δαβ2+2y2ρ02+ikv1zv1v2ρ02)y1]×H2n2μ+ν(y2a0)exp[(1w02+12δαβ2ik2z+1ρ02)y22+(ikv2z+u1u2ρ02)y2]dx1dy1dx2dy2.
Hn(x)=k=0[n/2](1)kn!k!(n2k)!(2x)n2k,
+xnexp[(xβ)2]dx=π(2i)nHn(iβ),
Hn(x+y)=(12)n/2k=0n(nk)Hk(2x)Hnk(2y),
Hn(ax)exp[(xy)2]dx=π(1a2)n/2Hn[ay(1a2)1/2],
Wxx(ρ1,ρ2;z)=(k2zw0)2exp[ik2z(ρ12ρ22)]exp[1ρ02(ρ1ρ2)2]124n+2m(n!)2t=0ns=0mμ=0nν=0m(is)iν(nt)(ms)(nμ)(mν)a=0[(2t+ms)/2]b=02t+ms2a+1c=0[b/2]d=0[(2μ+mν)/2]e=02n2t+sf=0[e/2]g=0[(2n2μ+ν)/2](2t+ms2a+1b)(2n2t+se)(i)2m2n+s2t+4a+2c+4de+4f+4g2b!e!(2t+ms)!(2μ+mν)!(2n2μ+ν)!a!c!d!f!g!(b2c)!(e2f)!(2t+ms2a)!(2μ+mν2d)!(2n2μ+ν2g)!2m2n+2ab+2ce+2f5(1a0)2m+4n2a2d2g(1A)m+4n+s2t2a+b2c+3(1A2a02A)2ns+2t+e2f(B)b2c+e2f(1D)m+2n+b2c2d+e2f2g+3×exp(C1x24A)exp(E24D)exp(C1y24A)exp(F24D)H2t+ms2a+1b(2iC1x2A)H2μ+mν+b2c2d+1(iE2D)H2n2t+se(2C1y2A2a02A)H2n2μ+ν+e2f2g(iF2D),
Wyy(ρ1,ρ2;z)=(k2zw0)2exp[ik2z(ρ12ρ22)]exp[1ρ02(ρ1ρ2)2]124n+2m(n!)2t=0ns=0mμ=0nν=0m(is)iν(nt)(ms)(nμ)(mν)a=02t+msb=0[a/2]c=0[(2μ+mν)/2]d=0[(2n2t+s)/2]e=o2n2t+s2d+1f=0[e/2]g=0[(2n2μ+ν)/2](2t+msa)(2n2t+s2d+1e)(i)m4ns+2ta+4b+4c+4d+2f+4g2a!e!(2μ+mν)!(2n2t+s)!(2n2μ+ν)!b!c!d!f!g!(a2b)!(e2f)!(2μ+mν2c)!(2n2t+s2d)!(2n2μ+ν2g)!2m2na+2b+2de+2f5(1a0)2m+4n2c2d2g(1A)2m+2ns+2t2d+e2f+3(1A2a02A)m+s2t+a2b(B)a2b+e2f(1D)m+2n+a2b2c+e2f2g+3exp(C1x24A)exp(E24D)exp(C1y24A)exp(F24D)H2t+msa(2C1x2A2a02A)H2μ+mν+a2b2c(iE2D)H2n2t+s2d+1e(2iC1y2A)H2n2μ+ν+e2f2g+1(iF2D),
A=1w02+12δ02+ik2z+1ρ02,B=1δ02+2ρ02,C1x=iku1zu1u2ρ02,C1y=ikv1zv1v2ρ02,C2x=iku2z+u1u2ρ02,C2y=ikv2z+v1v2ρ02,D=B24A+A,E=BC1x2A+C2x,F=BC1y2A+C2y.
I(ρ,ρ;z)=Wxx(ρ,ρ;z)+Wyy(ρ,ρ;z),
μ(ρ1,ρ2;z)=Tr[W(ρ1,ρ2;z)]Tr[W(ρ1,ρ1;z)]Tr[W(ρ2,ρ2;z)],
P(ρ,ρ;z)=14Det[W(ρ,ρ;z)]{Tr[W(ρ,ρ;z)]}2.
Ei(x,y;zi)=F1{F{Ei1(x,y;zi1)exp[iϕ(x,y;zi)]}exp[iΔz2k(κx2+κy2)]},
E(x,y)=E0(x,y)T(x,y),
S(fx,fy)=g(Δrx,Δry)exp[2πi(fxΔrx+fyΔry)]dΔrxdΔry,
T(fx,fy)=ar(fx,fy)S(fx,fy).
Φn(κx,κy,κz)=A(α)C~n2ξxξy(ξx2κx2+ξx2κy2+κz2+κ02)α/2exp(ξx2κx2+ξy2κy2+κz2κm2),
T=ξx2+ξy2ξx2ξy2×A(α)C~n24(α2){[2κ02κm2α+(α2)κm4α]exp(κ02κm2)Γ(2α2,κ02κm2)2κ04α},