1 Introduction

In a light wave, the phase singularity can generate an optical vortex. In a given direction, the energy flow rotates around the vortex core; the rotation velocity would be infinite and thus the light intensity must vanish at the center [1]. The optical vortex beam has broad applications in various areas, including particle manipulation [2], multiplex free-space communication [3,4,5] and high-dimensional quantum cryptography [6], and optical tweezers [7]. The linear and nonlinear propagations of the vortices also have a number of of new phenomena, such as vortex collision, four-wave mixing, and modulation instability [8,9,10,11]. The motion of the fluids in the planetary atmospheres and oceans shows abundant global-scale circulation and long-standing vortex phenomena. Some of these coherent vortices are Jupiter’s Red Spot, tropospheric cyclones, hurricanes, and oceanic Gulf Stream rings [12]. The information encoded in the orbital angular momentum (OAM) of light can be extracted after propagation through a 3 km intra-city link with strong turbulence in experiment [13]. The scintillation of the orbital angular momentum multiplexing reduces communication information capacity [14], limits the effective range of the OAM quantum communications [15], and causes cross talk among the OAM modes [4]. The distance for the conservation of the topological charge depends on the turbulence parameters and beam parameters [16]. In addition, the evolution behavior of coherent vortices and spectral degree of polarization vary with different generalized exponent parameter, the generalized refractive index structure parameter, and the beam parameters as well as the propagation distance [16].

In recent years, the interaction of light beams with turbulent atmosphere has been intensively investigated due to their unique important properties in the theoretical aspect and potential applications (in remote sensing, imaging, and optical communication, etc. [17,18,19,20,21,22,23]). The beam profile [24,25,26], the beam wander [27, 28], the beam spreading [29, 30], and the scintillation index [31] in turbulence are also studied in detail. Furthermore, the properties of various optical beams (such as the Gaussian–Schell model (GSM) beams [32], the partially coherent model beam [33], the anti-specular GSM beams [34], the asymmetrical optical beams [35] and so on) propagating in turbulence have been studied. Compared with completely coherent beams, partially coherent beams have significant advantages in reducing unavoidable degradation caused by random refractive index fluctuations of the turbulence [36, 37]. Meanwhile, the spreading of partially coherent vortex beams is less influenced by atmospheric turbulence than that of partially coherent non-vortex beams [38]. The polarization properties of beams have been researched widely in many papers [33, 39,40,41,42,43,44]. Based on the model of the cross-spectral density (CSD) [45], some second-order statistics-correlated beams in anisotropic turbulence were studied in [46,47,48,49]. The degree of polarization (DOP), the degree of coherence (DOC), and the state of polarization (SOP) were introduced in [49, 50]. The intensity pattern of the rotating elliptical chirped Gaussian vortex beam which presents two light spots at the beginning deforms and spins within a certain propagation range [51]. The effects of anisotropic, non-Kolmogorov turbulence on propagating stochastic electromagnetic beam-like fields are discussed for the first time in [52].

However, the investigation on the combination of the high-order vortex, radial polarization, and the rotating elliptical cosine-Gaussian beam has not been reported yet. Here, we deeply study the propagation properties of the radially polarized partially coherent rotating elliptical Gaussian vortex (PCRECGV) beam in non-Kolmogorov turbulence. The analytical formula of \(2\times 2\) CSD matrix of a radially polarized PCRECGV beam propagation through anisotropic turbulence is derived. We studied the influences of the elliptical rotation parameter, vortices, initial waist, and the beam order parameter on the average intensity, the DOP, and SOP of the radially polarized PCRECGV beam in anisotropic turbulence in detail.

2 The theoretical model of a radially polarized PCRECGV beam in anisotropic non-Kolmogorov turbulence

The second-order statistical properties of a partially coherent vector (i.e., electromagnetic) beam can be characterized by the CSD matrix \(\varvec{U}(\varvec{r_{1}}, \varvec{r_{2}}; \omega )\) with elements \(U_{\alpha \beta }(\varvec{r_{1}}, \varvec{r_{2}}; \omega )= \langle E_{\alpha }^{*}(\varvec{r_{1}};\omega )E_{\beta }(\varvec{r_{2}};\omega )\rangle , (\alpha , \beta = x, y)\) [48]. Where the asterisk means the complex conjugate and the angular brackets mean an ensemble average, and \(E_{\alpha }^{*}\) is the electric field component along \(\alpha\)-axis at position vector \(\varvec{r}\). The expression of the CSD matrix of a radially polarized PCRECGV beam in the source plane (z=0) is as follows:

$$\begin{aligned} U_{\alpha \beta }(\varvec{r_{1}},\varvec{r_{2}},\omega )= & {} \frac{\alpha _{1}\beta _{2}}{w^{2}}\exp \left( -\frac{x_{1}^{2}+x_{2}^{2}}{a^{2}w^{2}}-\frac{y_{1}^{2}+y_{2}^{2}}{b^{2}w^{2}}\right. \nonumber \\&\quad \left. -\frac{ix_{2}y_{2}-ix_{1}y_{1}}{c^{2}w^{2}}\right) \nonumber \\&\quad \times \left[ \left( \frac{x_{1}-x_{0}}{w}+i\frac{y_{1}-y_{0}}{w}\right) \left( \frac{x_{2}-x_{0}}{w}\right. \right. \nonumber \\&\quad \left. \left. -i\frac{y_{2}-y_{0}}{w}\right) \right] ^{m}g_{\alpha \beta } \left( \varvec{r_{1}}-\varvec{r_{2}};\omega \right) , \end{aligned}$$
(1)

where a and b are elliptical parameters that describe the shape of the radially polarized PCRECGV beam, c is an elliptical rotation parameter that specifies the beam rotates 90\(^\circ\) in a distance far enough, w is the initial beam waist of the Gaussian beam, m is the topological charge of vortices, \((x_{0},y_{0})\) means the position of the off-axis vortex, \(\alpha _1=(x_1, y_1),\) \(\beta _2=(x_2, y_2),\) and the spectral degree of coherence is given by [49]

$$\begin{aligned}&g_{\alpha \beta }(\varvec{r_{1}}-\varvec{r_{2}};\omega )=\nonumber \\&\quad \exp \left[ -\frac{(\varvec{r_{1}}-\varvec{r_{2}})^{2}}{2\sigma _{0}^{2}}\right] \cos \left[ \frac{n\sqrt{2\pi }(x_{1}-x_{2})}{\sigma _{0}}\right] \cos \nonumber \\&\quad \left[ \frac{n\sqrt{2\pi }(y_{1}-y_{2})}{\sigma _{0}}\right] , \end{aligned}$$
(2)

where \(\sigma _{0}\) represents spatial coherence width, \(\varvec{r_{1}}\equiv (x_{1},y_{1})\), \(\varvec{r_{2}}\equiv (x_{2},y_{2})\) are two arbitrary transverse position vectors in the source plane, and n is the beam order factor. When \(n=0\), a radially polarized PCRECGV beam reduces to a conventional radial polarization rotating elliptical GSM beam with vortex. When \(n\ne 0\), the beamlets of the radially polarized PCRECGV beam vary with n, and the radially polarized PCRECGV beam can spilt into four beamlets after a distance.

Within the validity of the paraxial approximation and the extended Huygens–Fresnel integral, the elements of the CSD at the receiver plane can be expressed as follows [16, 17, 29, 46, 48]:

$$\begin{aligned} U_{\alpha \beta }(\varvec{\rho _{1}},\varvec{\rho _{2}},\omega )= & {} \left( \frac{k}{2\pi z}\right) ^{2}\nonumber \\&\quad \int \int \int \int \ U_{\alpha \beta }(\varvec{r_{1}},\varvec{r_{2}},\omega )\nonumber \\&\quad \bigg \langle \exp \bigg [\psi (\varvec{r_{1}},\varvec{\rho _{1}},\omega )+\psi ^{*}\bigg (\varvec{r_{2}},\varvec{\rho _{2}},\omega \bigg )\bigg ]\bigg \rangle \nonumber \\&\quad \times \exp \left\{ -\frac{ik}{2z}\left[ \bigg (\varvec{r_{1}}^{2}-\varvec{r_{2}}^{2}\bigg )-2\bigg (\varvec{r_{1}}\cdot \varvec{\rho _{1}}-\varvec{r_{2}}\cdot \varvec{\rho _{2}}\bigg )\right. \right. \nonumber \\&\quad \left. \left. +\bigg (\varvec{\rho _{1}}^{2}-\varvec{\rho _{2}}^{2}\bigg )\right] \right\} d^{2}\varvec{r_{1}}d^{2}\varvec{r_{2}}, \end{aligned}$$
(3)

where \(k = 2\pi /\lambda\) is the wave number, \(\lambda\) is the wavelength, \(\psi\) represents the phase distortion of a monochromatic spherical wave in the turbulent atmosphere, and \(\varvec{\rho _{1}}=(u_{1},v_{1})\) and \(\varvec{\rho _{2}}=(u_{2},v_{2})\) are two arbitrary transverse position vectors at the receiver plane. The average turbulence phase perturbation can be expressed with quadratic phase approximations [46, 48, 50]:

$$\begin{aligned}&\bigg \langle \exp \bigg [\psi \bigg (\varvec{r_{1}},\varvec{\rho _{1}},\omega \bigg )+\psi ^{*}\bigg (\varvec{r_{2}},\varvec{\rho _{2}},\omega \bigg )\bigg ]\bigg \rangle =\nonumber \\&\quad \exp \bigg [-\frac{\pi ^{2}k^{2}z}{3}\bigg (\mathbf{r }_{\varDelta }^{2}+\varvec{r_{\varDelta }}\cdot \varvec{\rho }_{\varDelta }+\varvec{\rho }_{\varDelta }^{2}\bigg )\nonumber \\&\quad \int _{0}^{\infty }\kappa ^{3}\varPhi _{n}\bigg (\kappa ,\alpha \bigg )d\kappa \bigg ], \end{aligned}$$
(4)

where \(\varvec{r}_{\varDelta }\equiv \varvec{r_{1}}-\varvec{r_{2}}\), \(\varvec{\rho }_{\varDelta }\equiv \varvec{\rho _{1}}-\varvec{\rho _{2}}\), \(\varPhi _{n}(\kappa ,\alpha )\) means the three-dimensional power spectrum of the refractive index fluctuations, and \(\kappa\) is the magnitude of the spatial wave number. Here, the expression of the anisotropic non-Kolmogorov power spectrum and an effective anisotropic factor \(\zeta _\mathrm{eff}\) introduced in [17, 46]:

$$\begin{aligned} \varPhi _{n}(\kappa ,\alpha )=\frac{A(\alpha ){\widetilde{C}}_{n}^{2}\zeta _\mathrm{eff}^{2}}{\bigg (\zeta _\mathrm{eff}^{2}\kappa _{xy}^{2}+\kappa _{z}^{2}+\kappa _{0}^{2}\bigg )^{\alpha /2}}\exp \left( -\frac{\zeta _\mathrm{eff}^{2}\kappa _{xy}^{2}+\kappa _{z}^{2}}{\kappa _{H}^{2}}\right) , \end{aligned}$$
(5)

where \(\kappa =\sqrt{\zeta _\mathrm{eff}^{2}(\kappa _{x}^{2}+\kappa _{y}^{2})+\kappa _{z}^{2}}=\sqrt{\zeta _\mathrm{eff}^{2}\kappa _{xy}^{2}+\kappa _{z}^{2}}\), \(A(\alpha )=\frac{\varGamma (\alpha -1)}{4\pi ^{2}}\cos (\frac{\pi \alpha }{2}),\) \(\alpha (3<\alpha <4)\) is the power law, and under the condition of \(\alpha\) = 3.1 the disturbance of turbulence is the largest [52]. \({\widetilde{C}}_{n}^{2}= \beta {C}_{n}^{2}\) is a generalized structure parameter with unit \(\mathrm{m}^{3-\alpha }\) and \(\beta\) is a dimensional constant with unit \(\mathrm{m}^{11/3-\alpha }\), \(C_{n}^{2}\) represents the structure constant of the refractive index fluctuations with unit \(\mathrm{m}^{-2/3}\), \(\kappa _{0}= 2\pi /L_{0}\), \(\kappa _{H}=C(\alpha )/l_{0}\), \(L_{0}\) means the outer scale of turbulence, \(l_{0}\) means the inner scale of turbulence, and \(C(\alpha )\) is defined as

$$\begin{aligned} C(\alpha )=\left[ \pi A(\alpha )\varGamma \left( \frac{3}{2}-\frac{\alpha }{2}\right) \left( \frac{3-\alpha }{3}\right) \right] ^{1/(\alpha -5)}, \end{aligned}$$
(6)

here \(\varGamma (\alpha )\) is the Gamma function. We can get \({\widetilde{C}}_{n}^{2}(h)\) by means of the typical Hufnagel–Valley (H–V) model [17, 46]:

$$\begin{aligned} {\widetilde{C}}_{n}^{2}(h)= & {} 0.00594\left( \frac{V_{s}}{27}\right) ^{2}(10^{-5}h)^{10}e^{\left( \frac{-h}{1000}\right) }\nonumber \\&\quad +2.7\times 10^{-16}e^{\left( \frac{-h}{1500}\right) }+{{\widetilde{C}}}_{n}^{2}e^{\left( \frac{-h}{100}\right) }, \end{aligned}$$
(7)

where \(h_{0}\) and H are the heights above the ground in which transmitter and receiver are located. In this paper, we replace \({{\widetilde{C}}}_{n}^{2}\) with the average value \(\overline{{{\widetilde{C}}}_{n}^{2}}\) in considering the vertical profile. The integration in Eq. (4) is derived as follows  [16, 48]:

$$\begin{aligned} \begin{aligned} T_{ami}=&\frac{\pi ^{2}k^{2}z}{3}\int _{0}^{\infty }\kappa ^{3}\varPhi _{n}(\kappa ,\alpha )d\kappa \\ =&\frac{\pi ^{2}k^{2}z\zeta _\mathrm{eff}^{2-\alpha }}{6(\alpha -2)}A(\alpha )\overline{{{\widetilde{C}}}_{n}^{2}}\\&\quad \big [{\tilde{\kappa }}_{H}^{2-\alpha }\beta \exp \left( \frac{\kappa _{0}^{2}}{\kappa _{H}^{2}}\right) \varGamma \left( 2-\frac{\alpha }{2},\frac{\kappa _{0}^{2}}{\kappa _{H}^{2}}\right) -2{\widetilde{\kappa }}_{0}^{A-\alpha }\big ], \end{aligned} \end{aligned}$$
(8)

where \({\widetilde{\kappa }}_{0}= \kappa _{0}/\kappa _\mathrm{eff}\), \({\widetilde{\kappa }}_{H}= \kappa _{H}/\kappa _\mathrm{eff}\), \(\beta = 2{\widetilde{\kappa }}_{0}^{2}+(\alpha -2){\widetilde{\kappa }}_{H}^{2}\) and \(\varGamma (\ldots )\) is the incomplete Gamma function. The anisotropy turbulence is rescaled by the factor \(\zeta _\mathrm{eff}^{2-\alpha }\). By substituting Eqs. (1), (2), (4) into Eq. (3), the elements of the CSD matrix of a radially polarized PCRECGV beam in turbulent atmosphere is as follows:

$$\begin{aligned} U_{\alpha \beta }(\varvec{\rho _{1}},\varvec{\rho _{2}},\omega )=\sum CF(e,f,s,t,C,m), \end{aligned}$$
(9)

where \(U_{xx}\) and \(U_{yy}\) are the intensities along the x- and y-axis. \(C_{n}^{m}\) is the formula of combinatorial numbers, \(C=(-1)^{g_{1}}i^{g_2}N_1^{m_6}N_2^{g_3}C_{m}^{m_1}C_{m_1}^{m_2}C_{m_2}^{m_3}C_{m_3}^{m_4}C_{m_4}^{m_5}C_{m_5}^{m_6}C_{m_6}^{m_7}C_{m_7}^{m_8}\), \(N_1=x_0+iy_0\), \(N_2=x_0-iy_0\), \(g_{1}=m_1-m_3+m_5-m_7\), \(g_2=m_1-m_2+m_3-m_{4}+m_5-m_6+m_7-m_8\), \(g_{3}=m_2-m_3+m_5-m_6+m_{8}\):

$$\begin{aligned}&F(e,f,s,t,C,m)=V(\varvec{\rho _{1}},\varvec{\rho _{2}},\omega )\nonumber \\&\quad \frac{e_1!f_1!s!t_1!\pi ^2}{2^{e_{1}+f_1+s+t_1}\sqrt{\eta _{11}\eta _{12}\alpha _{21}\eta _{22}}}\sum ^{1}_{A=-1}\sum ^{1}_{B=-1}\varDelta _{11}\varDelta _{12}\varDelta _{21}\varDelta _{22} \nonumber \\&\quad \times \exp \left[ \frac{1}{4}\left( \frac{\lambda _{11}^{2}}{\eta _{11}} +\frac{\lambda _{12}^{2}}{\eta _{12}}\right. \right. \nonumber \\&\quad \left. \left. +\frac{\beta _{21}^2}{\alpha _{21}}+\frac{\beta _{22}^2}{\eta _{22}}+\frac{\beta _{21}^2\xi ^2}{4\eta _{22}\alpha _{21}^2}-\frac{\beta _{21}\beta _{22}\xi }{\eta _{22}\alpha _{21}}\right) \right] , \end{aligned}$$
(10)

where

$$\begin{aligned}&V({\varvec{{\rho _{1}}}},{\varvec{{\rho _{2}}}},\omega )=\frac{k^{2}A_{0}^{2}}{4z^{2}\pi ^{2}w^{2+2m}}\exp \left[ -T_{ami}\varvec{\rho }_{\varDelta }^{2}-\frac{ik}{2z}(\varvec{\rho _{1}^{2}}-\varvec{\rho _{2}^{2}})\right] ,\\&\quad \gamma =\frac{i}{c^{2}w^{2}}, \xi =2\mathrm{Tami}+\frac{1}{\sigma _{0}^{2}},&\\&\varDelta _{11}=\sum _{l_{11}=0}^{e_{1}-2k_{11}}\frac{\eta _{11}^{k_{11}-e_{1}}}{(e_{1}-2k_{11}-l_{11})!l_{11}!k_{11}!}\lambda _{11}^{e_{1}-2k_{11}-l_{11}}\theta _{11}^{l_{11}},\\&\quad \varDelta _{12}=\sum ^{f_{11}}_{k_{12}=0}\frac{\eta _{12}^{k_{12}-f_1}}{(f_1-2k_{12})!k_{12}!}\lambda _{12}^{f_1-2k_{12}},&\\&\varDelta _{21}=\sum _{k_{21}=0}^{s_{11}}\sum _{r_{21}=0}^{s-2k_{21}}\sum _{l_{21}=0}^{r_{21}}\\&\quad \frac{\alpha _{21}^{k_{21}-s}}{(s-2k_{21}-r_{21})!(r_{21}-l_{21})!k_{21}!}\beta _{21}^{s-2k_{21}-r_{21}}\xi ^{r_{21}-l_{21}}(-\gamma )^{l_{21}},\\&\varDelta _{22}=\sum _{k_{22}=0}^{t_{11}}\sum _{r_{22}=0}^{t_1-2k_{22}}\sum _{l_{22}=0}^{r_{22}}\\&\quad \frac{\eta _{22}^{k_{22}-t_1}}{(t_1-2k_{22}-r_{22})!(r_{22}-l_{22})!(k_{22})!}\\&\quad \left( \beta _{22}-\frac{\beta _{21}\xi }{2\alpha _{21}}\right) ^{t_1-2k_{22}-r_{22}}\left( \frac{-\gamma \xi }{2\alpha _{21}}\right) ^{r_{22}-l_{22}}(-\gamma )^{l_{22}},&\end{aligned}$$

where e, f, s, t are the orders of parameters, and m means the topological charges of vortices (\(m=0,1,2 \ldots\)), \(e=m-m_3\); \(f=m-m_1+m_3-m_4+m_6-m_7\); \(s=m_3-m_6\); \(t=m_1-m_2+m_4-m_5+m_7-m_8\); in the calculation, \(t_1=t+r_{21}-l_{21}\), \(t_{11}=\lfloor \frac{t_1}{2}\rfloor\), \(e_1=e+l_{21}+r_{22}-l_{22}\), \(e_{11}=\lfloor \frac{e_1}{2}\rfloor\), \(f_1=f+l_{22}+l_{12}\), \(f_{11}=\lfloor \frac{f_1}{2}\rfloor\), and \(y_1=\lfloor \frac{x_1}{2}\rfloor\) means that \(y_1\) is the lower bound of \(\frac{x_1}{2}\). And the other parameters are as follows:

$$\begin{aligned} \alpha _{11}&=\frac{1}{a^{2}w^{2}}+\frac{\xi }{2}+\frac{ik}{2z},\\ \quad \alpha _{12}&=\frac{1}{a^{2}w^{2}}+\frac{\xi }{2}-\frac{ik}{2z},\\ \quad \alpha _{21}&=\frac{1}{b^{2}w^{2}}+\frac{\xi }{2}+\frac{ik}{2z}, \alpha _{22}=\frac{1}{b^{2}w^{2}}+\frac{\xi }{2}-\frac{ik}{2z},\\ \beta _{11}&=\frac{in\sqrt{2\pi }A}{\sigma _{0}}-{u_{\varDelta }}\mathrm{Tami} + \frac{iku_1}{z},\\ \quad \beta _{12}&=-\frac{in\sqrt{2\pi }A}{\sigma _{0}}+{u_{\varDelta }}\mathrm{Tami}-\frac{iku_2}{z},\\ \quad \theta _{11}&=\xi -\frac{\xi \gamma ^2}{4\eta _{22}\alpha _{21}},\\ \quad \beta _{21}&=\frac{in\sqrt{2\pi }B}{\sigma _{0}}-{u_{\varDelta }}\mathrm{Tami}+\frac{ikv_1}{z},\\ \quad \beta _{22}&=-\frac{in\sqrt{2\pi }B}{\sigma _{0}}+{u_{\varDelta }}\mathrm{Tami}-\frac{ikv_1}{z},\\ \lambda _{12}&=\beta _{12}+\frac{\beta _{22}\gamma }{2\alpha _{22}}+\frac{\beta _{21}\xi \gamma }{4\eta _{22}\alpha _{21}}+\frac{\lambda _{11}\theta _{11}}{2\eta _{11}},\\ \quad \lambda _{11}&=\beta _{11}-\frac{\beta _{21}\gamma }{2\alpha _{21}}-\frac{\beta _{22}\xi \gamma }{4\eta _{22}\alpha _{21}}-\frac{\beta _{22}\xi ^{2}\gamma }{8\eta _{22}\alpha _{21}^{2}},\\ \eta _{22}&=\alpha _{22}-\frac{\xi ^2}{4\alpha _{21}},\eta _{11}=\alpha _{11}-\frac{\gamma ^{2}}{4\alpha _{21}}-\frac{\xi ^2\gamma ^2}{16\eta _{22}\alpha _{21}^2},\\ \quad \eta _{12}&=\alpha _{12}-\frac{\gamma ^{2}}{4\eta _{22}}-\frac{\theta _{11}^2}{4\eta _{11}}. \end{aligned}$$

The average intensity and the spectral DOP are given by the formulas [46,47,48]:

$$\begin{aligned} I(\varvec{\rho })= & {} U_{xx}(\varvec{\rho },\omega )+U_{yy}(\varvec{\rho },\omega ), \end{aligned}$$
(11)
$$\begin{aligned} P(\varvec{\rho },\omega )= & {} \sqrt{1-\frac{4{\mathrm{Det}}\bigg [\overleftrightarrow {U}\bigg (\varvec{\rho },\omega \bigg )\bigg ]}{\left\{ Tr[\overleftrightarrow {U} (\varvec{\rho },\omega )] \right\} ^{2}}}, \end{aligned}$$
(12)

where \(\varvec{\rho _{1}}=\varvec{\rho _{2}}=\varvec{\rho }\), \(\overleftrightarrow {U}\) means \(2\times 2\) CSD matrix, Det [\(\ldots\)] denotes the determinant of the matrix, and Tr [\(\ldots\)] stands for the trace of the matrix. The spectral DOC [32] of a vector beam in any transverse plane is defined by

$$\begin{aligned} \mu (\varvec{\rho _{1}},\varvec{\rho _{2}},\omega )=\frac{Tr[\overleftrightarrow {U}(\varvec{\rho _{1}},\varvec{\rho _{2}},\omega )]}{\sqrt{Tr[\overleftrightarrow {U} (\varvec{\rho _{1}},\varvec{\rho _{1}},\omega )]Tr[\overleftrightarrow {U} (\varvec{\rho _{2}},\varvec{\rho _{2}},\omega )]}}. \end{aligned}$$
(13)

3 Numerical analysis

In this paper, we choose (unless other values of parameters are specified): \(w=4\) cm, \(A_{0}=1\), \(\lambda =1.55 \mu\) m, the position of the vortex \((x_0=0.01\) m, \(y_0=0.01\) m), \({\widetilde{C}}_{n}^{2}=1.7\times 10^{-14}\)m\(^{3-\alpha }\), \(v_{s}=21\) m/s, \(l_{0}=1\) mm, \(L_{0}=10\) m, \(h_{0}=0\), \(H=30\) km, \(\alpha =3.5\), \(\zeta _\mathrm{eff}=3\). The average refractive index structure constant is \(\overline{{\widetilde{C}}_{n}^{2}}=7.5\times 10^{-17}\)m\(^{3-\alpha }\).

Fig. 1
figure 1

a1–c1 Are the average intensity distribution of initial radially polarized PCRECGV beam; others are phase distribution with several different c and m. The parameters are chosen as \(a=1, b=1, c=1, n=2\), a1– a4 and c1–c4 \(x_0=0.1\) m, \(y_0=0.1\) m; and b1–b4 \(x_0=0\), \(y_0=0\)

Full size image

We show the average intensity and the phase of a radially polarized PCRECGV beam (components \(U_x\) and \(U_y\)) in Fig. 1. Figure 1a1–b1 shows the average intensity of \(U_x\) with vortices in different positions, and Fig. 1c1 shows the average intensity of \(U_y\). We can see from Fig. 1 that the phase distribution changes with the elliptical rotation parameter c. As c increases, the phase gradually attains stability. When the elliptical rotation parameter is big enough, and the phase distribution of the radially polarized PCRECGV beam will not change. We can get the result that the phase of the radially polarized PCRECGV beam is effected by the elliptical rotation parameter, and the smaller the c, the bigger is the influence.

To further reveal the evolution properties of a radially polarized PCRECGV beam in the anisotropic turbulence, the peak intensity of the radially polarized PCRECGV beam with various structure constants of the turbulence is elucidated in Fig. 2. It is not difficult to find that (Fig. 2a) the peak intensity decreases gradually with the beam propagating through the anisotropic turbulence. Nevertheless, the peak intensity of the radially polarized PCRECGV beam with a smaller \({{\widetilde{C}}}_{n}^{2}\) will decrease slowly. In addition, the intensity distribution of the radially polarized PCRECGV beam varies with the structure constant at a long propagation distance in Fig. 2a1–a3, which also manifests that peak average intensity of the radially polarized PCRECGV beam in the weak turbulent atmosphere is larger than that in the strong turbulent atmosphere during the long-distance propagation.

Fig. 2
figure 2

The peak intensity of a radially polarized PCRECGV beam with several different \({{\widetilde{C}}}_{n}^{2}\). a1–a3 are the average intensity distribution of the radially polarized PCRECGV beam at \(z=1\) km, and corresponding to \({{\widetilde{C}}}_{n}^{2}\)=\(1.7\times 10^{-13}\hbox {m}^{\frac{1}{2}}\); \(1.7\times 10^{-10}\hbox {m}^{\frac{1}{2}}\); \(1.7\times 10^{-9}\hbox {m}^{\frac{1}{2}}\). Other parameters are the same as those in Fig. 1a1

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3.1 The vortices at one location

Fig. 3
figure 3

The average intensity distribution of a radially polarized PCRECGV beam with several different z. a1–a4 \(m=1,\) b1–b4 \(m=3,\) c1–c4 \(m=5\). Other parameters are the same as those in Fig. 1a1

Full size image

Figure 3 describes that the average intensity distribution with different distances and the topological charges. One can find that the distribution of a radially polarized PCRECGV beam gradually becomes a four-beamlet array state as z increases, which means that the beam order parameter can make the beam split. Meanwhile, when the transmission distance of the radially polarized PCRECGV beam increases, the beam rotates during the propagation due to the rotation factor. At the same time, each lobe has a vortex, which is significantly different from those conventional radially polarized GSM beams. The generation, annihilation, and evolution of the optical vortex in atmosphere and the transmission problem of topological charge in the atmosphere were discussed in [53]. We can see that a radially polarized PCRECGV beam gradually decomposes into a four-beamlet array distribution with four vortices and then the vortices of four-beamlets annihilate as z increases. There are several physical interpretations, one is that the vortex deflects from its original direction, the other is that its speed does not coincide with the radially polarized rotating elliptical cosine-Gaussian beam, and the last one is that the radially polarized PCRECGV beam’s vortex and some new vortices generated during transmission counteract each other. For a fixed z, the dark average intensity areas of the radially polarized PCRECGV beam get lager as m increases and the average intensity concentrates on the area where there is no vortex. As z increases, the radially polarized PCRECGV beam still splits into four beamlets carrying high-order vortices. We take the dashed line (Fig. 3a4–c4) as the baseline; when the beam travels a long distance, the radially polarized PCRECGV beam can shift to the direction of the absence of the vortices’ location. Meanwhile, the offset distances are longer as m increases. These phenomena illustrate that the manipulation can be achieved in the radially polarized PCRECGV beam, and when the topological charge of the vortices is higher, the controllability effect is better. This indicates that the average intensity distributions are greatly influenced by high-order vortices.

Fig. 4
figure 4

The average intensity of a radially polarized PCRECGV beam at \(z=0.5\) km, with several different initial beam waist w. Other parameters are the same as those in Fig. 1a1

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Figure 4 plots the transverse average intensity of the radially polarized PCRECGV beam through non-Kolmogorov turbulence with different w in \(z=0.5\) km. The spectrum density distribution is the most concentrated and the minimum is the biggest when \(w=0.02\) m by changing the initial beam width, as shown in [47]. One can see from Fig. 4 that the average intensity distribution of the radially polarized PCRECGV beam changes greatly with different w. When \(w=0.01\) m in Fig. 4a1, we see a big centrosymmetric beam clearly with four vortex points caused by the splitting and one coherent point at the center. We know that when the beam is in the far field, the larger the initial beam width, the smaller is the effective beam width [50]. In Fig. 4a2–a4, the concentration of the radially polarized PCRECGV beam gradually decreases as w increases. We can see from Fig. 4b1–b4 that when w continues to increase, the distance between each beamlet decreases until the beamlets reconnect again. As the initial waist continues to grow, the effect of vortex gradually decreases, and the shape of the radially polarized PCRECGV beam gradually approximates to that of the shape with the transmission distance less than \(z=0.5\) km. In other words, the beam can transmit farther and does not separate from vortices when the value of w is bigger. We can conclude that the shape of the beam at the same transmission distance can be adjusted by changing the original beam waist.

3.2 Vortices at different locations

Fig. 5
figure 5

Transmission state of the radially polarized PCRECGV beam with the peak average intensity for different values of n, z with two vortices in (0, 0.06 m), (\(0,-0.06\) m). Other parameters are the same as those in Fig. 1a1 except \(b=2, m=2\)

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Fig. 6
figure 6

The average intensity distribution of a radially polarized PCRECGV beam with several different c at \(z=1\) km. Other parameters are the same as those in Fig. 5

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By changing the position and the number of vortices, the radially polarized PCRECGV beam can evolve into some interesting patterns during transmission, which has some potential practical significances for optical manipulation and optical information transfer. Figure 5 illustrates the average intensity evolution of the radially polarized PCRECGV beam for different n and z with two vortices in (0,0.06 m) and (\(0,-0.06\) m). Owing to diffraction of the background beam, the vortices not only rotate but also move radially outward as the beam propagates. When the transmission distance is very close (Fig. 5a1–c1), the change of the average intensity is little affected by the beam order parameter. The average intensity of the radially polarized PCRECGV beam in the middle is weak and on both sides is strong during the transmission in Fig. 5. It is worth mentioning that the anticlockwise rotation phenomena are caused by the inhomogeneity of the transverse energy flow [54]. For the case of \(n=1\), the phenomena of splitting and rotation of the radially polarized PCRECGV beam appear with increasing propagation distance. Due to the action of vortex and energy flow, the position of the maximum average intensity changes. When \(z\le 5\) km, the radially polarized PCRECGV beam does not fully split out and the light average intensity still mainly concentrates in the center. For the case of \(n=2\), with increasing propagation distance, the two vortices divide into eight vortices, and the vortices operate separately on each small beamlet. When \(z=5\) km, the radially polarized PCRECGV beam divides into four beamlets with weak average intensity in intermediate and large average intensity on both sides. Next, we focus on the spreading of a partially coherent vortex beam propagating in anisotropic turbulence with different rotation factors in Fig. 6. When c is small, the effect of the rotation parameter exerts a tremendous influence on the radially polarized PCRECGV beam. At this time, the average intensity distribution of the radially polarized PCRECGV beam is not stable and has a large angle of inclination in the fixed transmission distance. With the increase in c, the state of light average intensity tends to be stable which means that the bigger the rotation parameter, the slower is the rotation speed of the radially polarized PCRECGV beam.

Fig. 7
figure 7

Transmission state of the radially polarized PCRECGV beam with the peak average intensity for different values of n, z. a1–a4 with no vortex; b1–b4 with four vortices in (0.03 m, 0.03 m), (\(-0.03\) m, \(-0.03\) m), (\(-0.03\) m, 0.03 m), (0.03m, \(-0.03\) m); c1–c4 with four vortices in (0.05 m, 0.05 m), (\(-0.05\) m, \(-0.05\) m), (\(-0.05\) m, 0.05 m), (0.05 m, \(-0.05\) m). Other parameters are the same as those in Fig. 1a1, except \(n=1, m=4\)

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To further explore the novel transmission mode of the radially polarized PCRECGV beam in anisotropic non-Kolmogorov turbulence, we draw the average intensity of the radially polarized PCRECGV beam with four vortices in Fig. 7. For comparison, we show a radially polarized partially coherent rotating elliptical cosine-Gaussian (RPPCRECG) beam propagates in anisotropic turbulence, and the influence of turbulence on the beam spreading is obvious in Fig. 7a1–a4. Figure 7b1–b4 are the average intensity distribution of the radially polarized PCRECGV beam with four vortices near the center of the beam. When the initial beam transmits at a closer distance (Fig. 7b1), the average intensity shows a symmetrical button shape which the zero average intensity points of four vortices and a coherent point. Taking the origin as the center, the light average intensity on the positive and negative x, y axis is larger. For the case of \(z=0.5\) km, the average intensity on the axis begins to rotate and diffract. As the average intensity changing shows in Fig. 7b3–b4, the rotation angles continue to increase over longer propagation distances. To attain more effective and diversified transmission mode, we change the position of vortices in Fig. 7c1–c4. When the distance is closer, the shape of the average intensity is a diamond with a central point of coherence in Fig. 7c1. During the transmission, the pattern of the beam is very similar to that without vortex, due to the distance between the vortices and the center of the beam is little farther. We can draw a conclusion that when the positions of vortices are closer to the center, the beamlets do not totally separate out during the transmission (\(z\le 5\) km), and the central light average intensity exists; when the vortex is located at a position little far from the center, the beamlets separate out during the transmission and the central optical average intensity disappears. Then, the existence of the vortices can control the shape of the beam and make the light average intensity more concentrated during transmission. This conclusion also applies to the cases of more vortices.

Fig. 8
figure 8

The modulus of the spectral DOP \(\mu (u,v,0,0)\) distribution of a radially polarized PCRECGV beam at different propagation distances. Other parameters are the same as those in Fig. 1a1

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Fig. 9
figure 9

The modulus of the spectral SOP distribution of a radially polarized PCRECGV beam at different propagation distances and carrying different vortices. a1–a4 no vortex; b1–b4 one vortex at (0.01 m, 0.01 m); b1–b4 two vortices at (0.03 m, 0.03 m), (\(-0.03\) m, \(-0.03\) m); other parameters are the same as those in Fig. 1a1

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The spectral DOP of the radially polarized PCRECGV beam with different distances through non-Kolmogorov turbulence is investigated in Fig. 8. When z is small, the radially polarized PCRECGV beam does not separate and the vortex only works at the point (0.01 m,0.01 m). As z increases, the radially polarized PCRECGV beam is dividing, meantime the DOP of the radially polarized PCRECGV beam changes at the position of the vortex. The radially polarized PCRECGV beam rotates and splits, and the vortex divides into four parts acting on the beam as z continues to increase. Because of the effect of vortices, the spectral DOP of the radially polarized PCRECGV beam is not symmetrical any more. To observe the change of the spectral SOP of the radially polarized PCRECGV beam intuitively, we choose three situations at different positions of vortices [(no vortex, a vortex at (0.01 m,0.01 m), two vortices at (0.03 m,0.03 m) and (\(-0.03\) m,\(-0.03\) m)], which are shown as Fig. 9. First, we focus on the situation without vortex in Fig. 9a1–a4. When z is small, the elliptical radial polarization structure of a radially polarized RPPCRECG beam remains well, and the state is broken as z increases. In the far field, the radially polarized RPPCRECG beam divides into four beamlets and each beam has the radial polarization structure. For the case of vortex existence, we can see that the radially polarized PCRECGV beam with one vortex has the same spectral SOP as that of the RPPCRECG beam at \(z=0\) (Fig. 9a1–c1). When increasing propagation distance, the spectral SOP is destroyed due to the effect of vortices. The directions and rules of the spectral SOP change, and the spectral SOP is not a strict radial polarization structure on each beamlet. The more vortices the beam has, the more serious the spectral SOP damaged is in the far field. We obtain an important conclusion that when the transmission distance is relatively close, spectral SOP is not affected by vortices. The degree of radial polarization structure is damaged in far field closely relates to the vortices numbers. But if we take more topological charges of the vortices in a position, we can get a similar situation in that the spectral SOP is damaged more seriously as the topological charges of vortices increase. The results also can be extrapolated to the cases of more and higher-order vortices.

3.3 Numerical experiment analysis

Fig. 10
figure 10

The numerical experiment of a radially polarized PCRECGV beam at different propagation distances and carrying four vortices. The parameters are the same as those in Fig. 7, a1–a4 \(z=0\), b1–b4: \(z=0.5\) km, c1–c4 \(z=3\) km

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We show the numerical experiment results of the radially polarized PCRECGV beam for studying its evolution properties. Figure 10a1–c1 shows the average intensity patterns of the initial input of the radially polarized PCRECGV beam. Figure 10a2–c2 shows the interference average intensity diagram of the initial plane beam. In the numerical experimental generation [48, 55], an initial beam has been launched to reconstruct the off-axis computer-generated hologram of the desired beam profiles as in Fig. 10a3–c3. We can obtain the transverse average intensity patterns at the input plane as in Fig. 10a4–c4. It is not difficult to see that the shapes of interference average intensity diagrams are consistent with the initial input intensity. We can get the result that our numerical experiments are consistent with the analytic solutions.

4 Summary

In summary, we investigate the statistics of a radially polarized PCRECGV beam in anisotropic turbulence with elliptical parameters and vortices. By performing numeric simulations for a radially polarized PCRECGV beam, we also demonstrate the specifics of the propagation and the polarization processes. The results show that the spectral SOP of the radially polarized PCRECGV beam and the evolution behavior of average intensity vary for different values of the parameters m, w, c and n in non-Kolmogorov turbulence. When the radially polarized PCRECGV beam travels in anisotropic turbulence, the average intensity distributions are greatly influenced by high-order vortices. The peak average intensity of the radially polarized PCRECGV beam in the weak turbulent atmosphere is larger than that in the strong turbulent atmosphere during a long-distance propagation. We observe that a radially polarized PCRECGV beam decomposes into a four-beamlet array distribution with four vortices and rotates gradually, and then the vortices of beamlets annihilate as z increases. The modes of the radially polarized PCRECGV beam at the same transmission distance can be adjusted by changing the original beam waist. By changing the position and number of vortices, the radially polarized PCRECGV beam can have some interesting shapes during transmission, which has some potential practical significances for optical manipulation and optical information transfer. The examples of two vortices and four vortices provide good illustrations that the pattern of light average intensity distribution has a great relationship with the position of vortex. The spectral SOP of a radially polarized PCRECGV beam remains well when z is small, and the more vortices the beam has, the more serious the spectral SOP damaged is in the far field. Our research results can be extrapolated to the cases of more and higher-order vortices beams propagating in anisotropic turbulence. Our work is helpful in optical trapping, polarization communications, remote polarization sensing, etc.