Vector Partially Coherent Beams With Twisted Sinc-Correlation Structure and Their Statistical Properties

We define a new kind of radially polarized twisted sinc-correlation Schell-model (RPTSCSM) source and establish the source parameter conditions necessary to produce a physical beam. With the help of the extended Huygens-Fresnel integral, Several typical numerical examples are provided to study the influence of the source parameter and turbulent parameter on the statistical properties of such beams on propagation. It is shown that the intensity profiles of such beams always exhibit rotation and self-splitting on propagation, which is caused by the twisted phase of the beams. Moreover, the light intensity splits into an array and maintains its array distribution in free space. While the array gradually evolves to a hollow distribution with a dark core in a turbulence medium, indicating that the light beam may automatically restore the radial polarized light intensity distribution in atmospheric turbulence after a certain distance of propagation. Compared with the intensity, the impact of the twisted factor on the degree of polarization is to produce noticeable distortion around its distribution centre. When propagating in a turbulent atmosphere, the degree of polarization of such beams exhibits well-turbulent resistance. The degree of coherence would also experience self-splitting and rotation caused by the twisted factor, and a turbulent atmosphere has an important influence on the degree of coherence. Our results will benefit multi-particle manipulation and free-space optical communication.

SECTION I. Introduction

An Essential characteristic of laser beams is spatial coherence, which determines the spatial behaviour of light beams on propagation. In terms of lessening the detrimental impacts of atmospheric turbulence, it has been demonstrated that partially coherent beams (PCBs) frequently outperform their fully coherent counterparts [1], [2]. This resistance has a scientific explanation based on a coherent mode representation [3], [4]. The PCBs with a homogeneous correlation structure and their propagation characteristics have been researched extensively. Since Gori established appropriate conditions for devising genuine spatial correlation functions [5], [6], by the selection of different correlation functions, a slew of novel model sources with non-conventional Gaussian correlation functions has been developed due to their unique properties such as self-steering, self-splitting and self-focusing in the far field when propagating in free space [7], [8], [9]. Recently, two kinds of PCBs with Sinc Schell-model function introduced by Z. Mei [10] have attracted considerable interest. These sources carry stable flat and dark hollow profiles in the far field. On this basis, Various beams related to the Sinc Schell-model-correlated function have been widely investigated. Among these beams, cylindrical sinc Gaussian beams, multi-sinc Schell-model beams, electromagnetic sinc Schell-model beams, twisted Sinc-correlation Schell-model beams, electromagnetic sinc Schell-model vortex beams have been of particular interest to researchers [11], [12], [13], [14], [15], [16].

Additionally, the unified theory of coherent polarization closely links coherence and polarization of light field together, which offers a new way to understand the optical statistical properties deeply. According to state of polarization (SOP), Vector beams can be categorized as uniformly polarized and non-uniformly polarized beams based on the state of polarization (SOP). In the past decade, vector beams with spatially nonuniform states of polarization attracted much attention due to their unique and valuable properties [17], [18], [19]. Unlike uniformly polarized beams, such as linearly or elliptically polarized beams, beams with a nonuniform state of polarization are advantageous in many applications [20], [21], [22]. For example, a radially polarized beam, as a typical cylindrical vector beam with a nonuniform state of polarization, has been demonstrated to have a smaller focal spot [21].

In contrast, non-uniformly polarized beams are more resistant to atmospheric turbulence [23]. Therefore, vector partially coherent beams with non-uniformly polarized structures have become the focus of current research [24], [25], [26], [27], [28], [29]. However, these works focus on uniformly correlated (or Schell model) sources. Only a few papers have studied vector partially coherent beams with non-uniformly polarization and non-uniform correlated structures [18], [24]. As non-uniformly polarized and non-uniformly correlated beams have shown advantages separately in optical applications, their combination may lead to the discovery of novel effects.

To address this issue further, We introduce a radially polarized Schell model source class, in which cross-spectral density matrix possesses a sinc-correlation function, including the twisted phase factor. Furthermore, analyze their reliability conditions. After that, utilizing the extended Huygens-Fresnel principle, we investigated these beams' statistical properties, including their intensity, degree of polarization, and degree of coherence on propagation. In Section II, we analyze the reliability of the RPTSCSM beam source, In Section III we derive the theoretical formulation, where the RPTSCSM beams are described according to the cross-spectral density matrix. Section IV provides numerical results and a discussion concerning the propagation characteristics of such beams in turbulent atmosphere. In conclusion, Section V provides a summary of the key findings.

SECTION II. RPTSCSM Source

The second-order correlation properties of a statistically RPTSCSM source at a pair of points r₁and r₂ in the source plane z=0 can be described in terms of the cross-spectral density (CSD) function as [30]

$$\begin{equation*} {W}_{\alpha \beta }({r}_1,{r}_2) = \left\langle {E_\alpha ^ * ({r}_1){E}_\beta ({r}_2)} \right\rangle,\,\left({\alpha,\beta = x,y} \right) \tag{1} \end{equation*}$$

The terms E_x and E_y in (1) denote two components of the random electric vector along the x and y axes that are mutually orthogonal, respectively. The complex conjugate is shown by the asterisk, while the angular brackets indicate the average of a monochromatic ensemble.

According to the findings of Gori et al. [5], [6], the integral representation of the form satisfies the non-negative definiteness requirements for physically realizable CSD matrices

$$\begin{equation*} {W}_{\alpha \beta }({r}_1,{r}_2) = \int{{{p}_{\alpha \beta }\left(\nu \right)}}H_\alpha ^ * ({r}_1,\nu){H}_\beta ({r}_2,\nu)\mathrm{d}\nu, \tag{2} \end{equation*}$$ where p_αβ(v) is a non-negative weight function, v is a 2D vector in the Fourier plane H_α(r_1, v) and H_β(r_2, v) are two arbitrary kernels which can have plenty of choices, and each choice is likely to lead to distinct classes of CSD matrices. In order to generate the vector sources with rotating intensity, we consider the kernel as [22] $$\begin{align*} {H}_{\alpha (\beta)}(r,v) =& {A}_{\alpha (\beta)}\frac{{\alpha (\beta)}}{{{\sigma }^2}}\exp \left[ {{{ - {r}^2} \mathord{\left/ {\vphantom {{ - {r}^2} {{\sigma }^2}}} \right. } {{\sigma }^2}}} \right]\\ &\times \exp \left\{ { - \left[ {\left({uy + ix} \right){v}_x - \left({ux - iy} \right){v}_y} \right]} \right\}, \tag{3} \end{align*}$$ where σ is the beam width, and u denotes the twisted factor. A_α is the amplitude of the electric field's component along α direction. Furthermore, the kernel function indicates that the electromagnetic field is a radial polarization component with a twisted phase. the weight function p_αβ(v) is given as [15] $$\begin{align*} {p}_{\alpha \beta }(v) = {B}_{\alpha \beta }\frac{{{\delta }_\alpha {\delta }_\beta }} {{{{\rm{\pi }}}^2}}\text{rect}\left(\frac{{{\delta }_x{v}_x}}{{\rm{\pi }}}\right)\text{rect}\left(\frac{{{\delta }_y{v}_y}}{{\rm{\pi }}}\right), \tag{4} \end{align*}$$ where δ_x and δ_y denote coherence width, B_αβ is the complex correlation coefficient. Substituting (3) and (4) into (2), and applying the following integral formula $$\begin{equation*} \int{{f(x)}} \cdot {\rm{rect(2\pi i}}v{\rm{)d}}x = \sin c(x). \tag{5} \end{equation*}$$

We can quickly obtain the CSD of RPTSCSM beams as the following expression.

$$\begin{align*} {W}_{\alpha \beta }({r}_1,{r}_2) & = {A}_\alpha {A}_\beta {B}_{\alpha \beta }\exp \left({ - \frac{{r_1^2 + r_2^2}}{{2{\sigma }^2}}} \right)\\ & \quad \cdot \sin c\left[ {\frac{{({x}_1 - {x}_2) + iu({y}_1 + {y}_2)}}{{2{\delta }_x}}} \right]\sin c\\ &\quad \times \left[ {\frac{{({y}_1 - {y}_2) + iu({x}_1 + {x}_2)}}{{2{\delta }_y}}} \right] \cdot \tag{6} \end{align*}$$

Equation (6) denotes the radially polarized twisted sinc Schell-model beams.

To guarantee the physically realizable field described in (6), It is necessary to establish the restrictions of the source parameters. The CSD matrices must be quasi-Hermitian [6]. Specifically, W_αβ (r₁, r₂) = W_βα^*(r₁, r₂). Secondly, the non-negative definiteness constraint for the genuine CSD matrices means the following inequalities.

$$\begin{equation*} {p}_{\alpha \alpha }{\rm{(}}v{\rm{)}} \geq {\rm{0,}}\,{p}_{xx}{\rm{(}}v{\rm{)}} \cdot {p}_{yy}{\rm{(}}v{\rm{)}} - {p}_{xy}{\rm{(}}v{\rm{)}} \cdot {p}_{yx}{\rm{(}}v{\rm{)}} \geq {0}, \tag{7} \end{equation*}$$ have to be satisfied for any v. We readily find from (4) that inequality (8) is always satisfied, and on substituting from (4) into inequalities (8), it has been derived the following inequality relations. $$\begin{equation*} {\delta }_{xx}{\delta }_{yy}{\rm{rect (}}{\delta }_{xx}{v}_x{\rm{)rect (}}{\delta }_{yy}{v}_y{\rm{)}} \geq {\left| {{B}_{xy}} \right|}^2\delta _{xy}^2{\left[ {{\rm{rect (}}{\delta }_{xy}v{\rm{)}}} \right]}^2, \tag{8} \end{equation*}$$ where rect(x) is the Sinc function, when │x│≤1/2, it equals 1; otherwise, it equals 0. It is simply determined that the realizability condition is represented by the following fork inequality [31] $$\begin{equation*} \max \left\{ {{\delta }_{xx},{\delta }_{yy}} \right\} \leq {\delta }_{xy} \leq \frac{{\sqrt {{\delta }_{xx}{\delta }_{yy}} }}{{\left| {{B}_{xy}} \right|}}, \tag{9} \end{equation*}$$ finally, as a radially polarized, such beam is linearly polarized at any point in the original plane, which indicates that the minor semi-axes of the polarization ellipse equals to zero, and the polarization orientation angle at any location should meet θ(x,y)=arctan(y/x) [24]. A polarization ellipse can describe the state of polarization (SOP). For this reason, the RPTSCSM beams must additionally meet [28] $$\begin{align*} {B}_{xy} &= {B}_{yx} = 1, \tag{10}\\ {\delta }_{xx} &= {\delta }_{yy} = {\delta }_{yx} = \delta, \tag{11} \end{align*}$$

SECTION III. Modelling of CSD Matrix of the RPTSCSM Sources in Linear Random Medium

Assume the source in (6) produces a beam-like field propagating into a turbulent atmosphere-filled half-space with z>0. Paraxial propagation of the CSD of RPTSCSM beams can be described by the extended Huygens-Fresnel integral

$$\begin{align*} {W}_{\alpha \beta }({\rho }_1,{\rho }_2,z) =& {{\left({\frac{k}{{2{\rm{\pi }}z}}} \right)}}^2\iint {W}_{\alpha \beta }({r}_1,{r}_2)\\ & \cdot \exp \left[ { - ik\frac{{{{({\rho }_1 - {r}_1)}}^2 - {{({\rho }_2 - {r}_2)}}^2}}{{2z}}} \right]\\ & { \cdot {{\left\langle {\exp \left[ {\psi \left({{\rho }_1,{r}_1} \right) + {\psi }^{\rm{*}}\left({{\rho }_2,{r}_2} \right)} \right]} \right\rangle }}_R{\rm{\ d}}{r}_1{\rm{\ d}}{r}_2}, \tag{12} \end{align*}$$ where ψ stands for the complex phase perturbation caused by the medium, ${\langle \cdot \rangle }_R$ implies averaging over the ensemble of statistical realizations of the turbulence. Under quadratic phase approximations, ${\langle \cdot \rangle }_R$ can be rewritten as $$\begin{align*} & {\left\langle {\exp \left[ {\psi ({\rho }_1,{r}_1) + {\psi }^ * ({\rho }_2,{r}_2)} \right]} \right\rangle }_R \\ & = \exp \left\{ { - \frac{{{{\rm{\pi }}}^2{k}^2z}}{3}\int_{0}^{\infty }{{{\kappa }^3{\phi }_n(\kappa)}}} \right.\\ & \vphantom{{ - \frac{{{{\rm{\pi }}}^2{k}^2z}}{3}\int_{0}^{\infty }{{{\kappa }^3{\phi }_n(\kappa)}}}}\mathop {}\nolimits^{} \mathop {}\nolimits^{} \mathop {}\nolimits^{} \left. {\mathop {}\nolimits^{} \cdot \left[ {{{({\rho }_1 - {\rho }_2)}}^2 + ({\rho }_1 - {\rho }_2)({r}_1 - {r}_2) + {{({r}_1 - {r}_2)}}^2} \right]} \right\}\mathrm{d}\kappa, \tag{13} \end{align*}$$ with к being the magnitude of the spatial wave number, and ${\phi }_n(\kappa)$ is the spatial power spectrum of refractive-index fluctuation representing anisotropic atmosphere characteristics. Here, we choose the van Karman power spectrum [8], [18] $$\begin{equation*} {\Phi }_{\mathrm{n}}(\kappa) = = 0.033C_n^2{({\kappa }^2 + \kappa _0^2)}^{{{ - 11} \mathord{\left/ {\vphantom {{ - 11} 6}} \right. } 6}}\exp \left({{{\kappa }^2} \mathord{\left/ {\vphantom {{{\kappa }^2} {{\kappa }_m}}} \right. } {{\kappa }_m}}\right), \tag{14} \end{equation*}$$ where C²_n represents a generalized refractive-index structure parameter, $\underline{K}$₀ =2π/L₀ and $\underline{K}$_m=5.92/l₀ with L₀ and l₀ being the outer and inner scale of turbulence, respectively. By substituting (2) into (12) and performing mathematical calculation, we can obtain $$\begin{equation*} {W}_{\alpha \beta }\left({{\rho }_1,{\rho }_2,z} \right) = \frac{{{\pi }^4{\sigma }^2}}{{4{a}^{*2}{b}^2{\lambda }^2{z}^2}}\exp \left[ {\frac{{ik}}{{2z}}\left({\rho _2^2 - \rho _1^2} \right){\rm{ + }}T{{({\rho }_1 - {\rho }_2)}}^2} \right]\int{{\int{{p(v)}}}}{H}_{\alpha \beta }\mathrm{d}{v}_x\mathrm{d}{v}_y, \tag{15} \end{equation*}$$ with eq. (16–​18) shown at the bottom of this page $$\begin{align*} {H}_{xx}({v}_x,{v}_y) = &\left\{ {2T + \left[ {{f}_1({\rho }_{1x},{\rho }_{2x}) - u{v}_y + i{v}_x} \right]\left[ {{f}_2({\rho }_{1x},{\rho }_{2x}) + ud{v}_y + ie{v}_x} \right] + \frac{T}{b}{{\left[ {{f}_2({\rho }_{1x},{\rho }_{2x}) + ud{v}_y + ie{v}_x} \right]}}^2} \right\}\\ & \exp \left\{ {\frac{1}{{4{a}^ * }}{{\left[ {{f}_1({\rho }_{1x},{\rho }_{2x}) - u{v}_y + i{v}_x} \right]}}^2 + \frac{1}{{4b}}{{\left[ {{f}_2({\rho }_{1x},{\rho }_{2x}) + ud{v}_y + ie{v}_x} \right]}}^2} \right\}\\ & \cdot \exp \left\{ {\frac{1}{{4{a}^ * }}{{\left[ {{f}_1({\rho }_{1y},{\rho }_{2y}) + u{v}_x + i{v}_y} \right]}}^2 + \frac{1}{{4b}}\left[ {{f}_2({\rho }_{1y},{\rho }_{2y}) - ud{v}_x + ie{v}_y} \right]{)}^2} \right\}, \tag{16}\\ {H}_{yy} =& \left\{ {2T + \left[ {{f}_1({\rho }_{1y},{\rho }_{2y}) + u{v}_x + i{v}_y} \right]} \right. \cdot \left[ {{f}_2({\rho }_{1y},{\rho }_{2y}) - ud{v}_x + ie{v}_y} \right]\left. { + \frac{T}{b}{{\left[ {{f}_2({\rho }_{1y},{\rho }_{2y}) - ud{v}_x + ie{v}_y} \right]}}^2} \right\}\\ & \exp \left\{ {\frac{1}{{4{a}^ * }}{{\left[ {{f}_1({\rho }_{1y},{\rho }_{2y}) + u{v}_x + i{v}_y} \right]}}^2} \right\}\exp \left\{ {\frac{1}{{4b}}{{\left[ {{f}_2({\rho }_{1y},{\rho }_{2y}) - ud{v}_x + ie{v}_y} \right]}}^2} \right\}\\ & \exp \left\{ {\frac{1}{{4{a}^ * }}{{\left[ {{f}_1({\rho }_{1x},{\rho }_{2x}) - u{v}_y + i{v}_x} \right]}}^2} \right\}\exp \left\{ {\frac{1}{{4b}}{{\left[ {{f}_2({\rho }_{1x},{\rho }_{2x}) + ud{v}_y + ie{v}_x} \right]}}^2} \right\}, \tag{17}\\ {H}_{xy} =& \frac{{{\pi }^2}}{{4a{}^{*2}{b}^2}}\left[ {{f}_2({\rho }_{1x},{\rho }_{2x}) + ud{v}_y + i{d}_3{v}_x} \right]\left\{ {\left[ {{f}_1({\rho }_{1y},{\rho }_{2y}) + u{v}_x + i{v}_y} \right] + \frac{T}{b}\left[ {{f}_2({\rho }_{1y},{\rho }_{2y}) - ud{v}_x + ie{v}_y} \right]} \right\}\\ & \exp \left\{ {\frac{1}{{4{a}^ * }}{{\left[ {{f}_1({\rho }_{1x},{\rho }_{2x}) - u{v}_y + i{v}_x} \right]}}^2} \right\}\exp \left\{ {\frac{1}{{4b}}{{\left[ {{f}_2({\rho }_{1x},{\rho }_{2x}) + ud{v}_y + ie{v}_x} \right]}}^2} \right\}\\ & \exp \left\{ {\frac{1}{{4{a}^ * }}{{\left[ {{f}_1({\rho }_{1y},{\rho }_{2y}) + u{v}_x + i{v}_y} \right]}}^2} \right\}\exp \left\{ {\frac{1}{{4b}}{{\left[ {{f}_2({\rho }_{1y},{\rho }_{2y}) - ud{v}_x + ie{v}_y} \right]}}^2} \right\}, \tag{18} \end{align*}$$ where $$\begin{align*} {f}_1({\rho }_{1x},{\rho }_{2x}) &= \frac{{ik}}{z}{\rho }_{2x} - T\left({{\rho }_{2x} - {\rho }_{1x}} \right),\,{f}_1({\rho }_{1y},{\rho }_{2y}) = \frac{{ik}}{z}{\rho }_{2y} - T\left({{\rho }_{2y} - {\rho }_{1y}} \right),\\ {f}_2({\rho }_{1x},{\rho }_{2x})& = \frac{{ik}}{z}\left({{\rho }_{1x} + \frac{T}{{{a}^ * }}{\rho }_{2x}} \right) - c\left({{\rho }_{2x} - {\rho }_{1x}} \right),\,{f}_2({\rho }_{1y},{\rho }_{2y}) = \frac{{ik}}{z}\left({{\rho }_{1y} + \frac{T}{{{a}^ * }}{\rho }_{2y}} \right) - c\left({{\rho }_{2y} - {\rho }_{1y}} \right), \tag{19}\\ a& = \left({\frac{1}{{{\sigma }^2}} + \frac{{ik}}{{2z}} - T} \right),\,b = a + \frac{{{T}^2}}{{{a}^ * }},c = T\left({1 + \frac{T}{{{a}^ * }}} \right),\,d = 1 - \frac{T}{{{a}^ * }},\,e = 1 + \frac{T}{{{a}^ * }}, \tag{20}\\ T &= \frac{{4{{\rm{\pi }}}^2{k}^2{z}^2}}{3}\int{{{\kappa }^3}}{\Phi }_n(\kappa){\mathrm{d}}^2\kappa . \tag{21} \end{align*}$$

We obtain the CSD matrix of RPTSCSM beams at any propagation distance by evaluating (15) shown at the bottom of this page. Setting ρ₁ = ρ₂ =ρ, the spectral intensity for RPTSCSM beams is given as

$$\begin{equation*} S(\rho,z) = {W}_{xx}(\rho,\rho,z) + {W}_{yy}(\rho,\rho,z), \tag{22} \end{equation*}$$ the spectral degree of polarization (DOP) and the spectral degree of coherence (DOC) of RPHNUCA beams can be calculated by the expression $$\begin{align*} P(\rho,z) &= \sqrt {1 - \frac{ 4\text{Det}\left[ {W}(\rho,\rho,z) \right]} { \left\{ {\text{Tr}\left[ {W} (\rho,\rho,z) \right]} \right\}^2 }}, \tag{23}\\ \mu \left({{\rho }_1,{\rho }_2,z} \right) &= \frac{{\text{Tr}W\left({{\rho }_1,{\rho }_2,z} \right)}} {{\sqrt {\text{Tr}W\left({{\rho }_1,{\rho }_1,z} \right)} \sqrt {\text{Tr}W\left({{\rho }_2,{\rho }_2,z} \right)} }}, \tag{24} \end{align*}$$ where Det and Tr Stand for the determinant and the matrix trace. Applying (22)–(24), we can study the statistical properties in free space and the turbulent atmosphere.

SECTION IV. The Statistical Properties of RPTSCSM Beams in Free Space and Turbulent Atmosphere

Subsequently, The behavior of RPTSCSM beams propagating in free space and a turbulent atmosphere will be demonstrated. Unless stated in the figure captions otherwise, we assume that following the relevant initial parameters of RPTSCSM beams and turbulence are set as λ=632.8nm, L₀=1m, l₀=1 mm, and C_n²=10⁻¹⁵m^−2/3.

A. Spectral Density

Fig. 1 presents the evolution of the spectral intensity of such beams with different values of the twisted factor. We can observe from fig. (a0) that the intensity profile of such beams is four-lobed bright spots with a circularly symmetric hollow shape on the source plane. As propagation distance increases, Fig. 1 shows the evolution of these beams' intensities for various twist factor values. We can observe from fig. (a0) that the intensity profile of such beams is four-lobed bright spots with a circularly symmetric hollow shape on the source plane. As propagation distance increases, two significant impacts on the intensity induced by the twisted factor gradually emerge. First, the twisted factor causes the light intensity to rotate 90 degrees. When twist factor u>0, light spots would rotate clockwise. Otherwise, they would rotate anticlockwise. Secondly, the self-splitting phenomenon of the intensity distribution can also be caused by the twisted factor. This is because the intensity distribution of a radially polarized beam is a hollow dark core at the source plane. Due to the rotation of the twisted phase factor and sinc correlation function produced by the Fourier transform of the rectangular function, the beam self-splits on propagation. As shown in Fig. 1, the light spots gradually expand their distribution scopes and slit into a 2 × 2 isotropic array on propagation. which entirely different from those of sinc Schell-model beams reported in Ref. [11], where sinc Schell-model beams would gradually change from a Gaussian profile to a flat profile with increasing propagation distance.

Fig. 1.

The normalized intensity of RPTSCSM beams with δ = 1 mm, σ = 1mm for different propagating distance in free space. In the figure, (a0) z = 0, u = −2; (a1)–(d1) u = −2; (a2)–(d2) u = 2.

Fig. 2 illustrates how the twisted factor affects the spectral intensity distribution of RPTSCSM beams at z=10m. with increasing values of the twisted factor, the light intensity distribution gradually splits from the Gaussian distribution into an isotropic array, and the array spacing becomes wider while maintaining rotation.

Fig. 2.

The normalized intensity of RPTSCSM beams with δ = 1 mm, σ = 1mm at propagating distance z = 10 m for different twisted factors in free space. (a) u = 0; (b) u = −0.5; (c) u = −1; (d) u = −1.5; (e) u = −2; (f) u = −2.5.

Fig. 3 exhibits the rotation angle and normalized intensity of RPTSCSM beams for different beam parameters in free space. For simplicity, we choose the y = 0 plane at propagating distance z = 5 km in Fig. 3(b) and (c). As described in Fig 3(a), the twisted factor causes the intensity rotation of the RPTSCSM beams on propagation. The speed at which the light intensity spins increases with greater values of the twisted factor. We can find from Fig. 3(b) and (c) that the light spots slit into two peaks along the x-direction. With an increase of the coherence width δ and beam width σ, the two peaks extend along the x-axis and the spacing of the peaks gradually broadens, which means that the beams form a stable array due to the influence of the twisted factor. The beam parameters can effectively adjust the array distribution.

Fig. 3.

Illustrates the rotation angle and normalized intensity of RPTSCSM beams for different beam parameters in free space. (a) σ = 1 mm, δ = 1 mm; (b) σ = 1 mm, u = −2, z = 5 km; (c) δ = 1 mm, u = −2, z = 5 km.

To more clearly show the intensity evolution of RPTSCSM beams propagating in a turbulent atmosphere. Fig. 4 presents the intensity distribution of RPTSCSM beams for different propagation distances. As observed in Fig. 4, the initial radial polarized light intensity profile transforms into a four-lobe circular distribution with a hollow dark core on the original plane due to the manipulation of the twisted factor and sinc correlation function. With the increase of propagation distance, the light intensity distribution splits and rotates. In the near field, the light intensity splits into a four-lobe array on propagation due to the induction of the twisted factor. When the propagation distance continues to increase, the cumulative effect of turbulence is gradually significant. In the far field, the light spots gradually converge toward the centre, and the array develops into a Gaussian intensity profile. Finally, intensity distribution gradually becomes hollow with a dark core, indicating that the light beam may automatically restore the radial polarized light intensity distribution in atmospheric turbulence after a certain distance of propagation. We may effectively control the beam propagation by reasonably choosing the beam's initial parameters.

Fig. 4.

The normalized intensity of RPTCSM beams with δ = 1 mm, σ = 1 mm, u = -2, C²_n=10⁻¹⁴m^−2/3 for different propagating distance in the turbulent atmosphere.

B. DOP of RPTSCSM Beams

Next, we focus on the variations in the degree of polarization of RPTSCSM beams. As shown in Fig. 5, the DOP distribution of RPTSCSM beams in free space is given. For radially polarized sinc Schell-model beams (seen in Fig. 5(a1)–​(d1)), we can see that the spectral degree of the polarization presents a radially polarized distribution in the source plane, which is similar to that of a general radially polarized partially coherent beams [9].

Fig. 5.

The distribution of spectral degree of the polarization of RPTSCSM beams with δ = 1 mm, σ = 1 mm for different propagating distances in free space. In the figure, (a1)–(d1) u = 0; (a2)–(d2) u = −2; (a3)–(d3) u = 2.

For radially polarized twisted sinc Schell-model beams, the distribution of spectral degree of the polarization is twisted, and the twisted direction is closely related to the twisted factor. When the twisted factor is a positive value, the distribution of spectral degree of Polarization would twist clockwise. Otherwise, it would twist anticlockwise. In addition, from Fig. 5(a2)–​(d2) and (a3)–​(d3), we also can see that as the propagation distance increases, it tends to be the same for all types of beam (seen in Fig.(d1)-(d3)). These show that we can construct this new kind of twisted beam by choosing an appropriate twist factor.

Fig. 6 depicts the effect of the twisted factor on the DOP of RPTSCSM beams at z=10m. We see that the spectral degree of the polarization is distorted around its distribution centre, and as values of the twisted factor increase, the distortion of the degree of spectral polarization becomes stronger.

Fig. 6.

The distribution of spectral degree of the polarization of RPTSCSM beams with δ = 1 mm, σ = 1mm at propagating distance z = 10m for different twisted factors in free space. (a) u = 0; (b) u = −0.5; (c) u = −1; (d) u = −1.5; (e) u = −2; (f) u = −2.5.

Fig. 7 presents the distribution of spectral degree of the polarization of RPTSCSM beams with u = −2 at propagating distance z = 5 km for different beam widths and coherent widths in free space. We can see that coherence width and beam width affect the distribution of spectral degrees of polarization. Compared with the coherence width in Fig. 7(a), the influence of beam width is more significant in Fig. 7(b).

Fig. 7.

The distribution of spectral degree of RPTSCSM beams with u = −2 at propagating distance z = 5 km for different beam widths and coherent widths in free space. (a) σ = 1 mm, u = −2; (b) δ = 1 mm, u = −2.

As shown in Fig. 8, the effect of a turbulent atmosphere on the DOP of RPTCSM beams at propagation distance z = 0.5 km is given. We can find Fig. 8 that, for the case of Fig. 8(a), in the weakest turbulence, the distribution of the degree of polarization is similar to the case of free space. With the strengthening of turbulence, the prominent central region gradually disappears, as shown in Fig. 8(b) and (c). For the case of Fig. 8(d), (e) and (f) of the stronger turbulence, the distribution of spectral degree of polarization tends to be the same. It also shows that the degree of polarization of RPTSCSM beams exhibits well-turbulent resistance.

Fig. 8.

The distribution of spectral degree of polarization of RPTSCSM beams with δ = 1 mm, σ = 1 mm, u = −2 at propagating distance z = 0.5 km for different values of atmosphere parameters length in the turbulent atmosphere. (a) C²_n=10⁻¹⁷m^−2/3; (b) C²_n=10⁻¹⁶m^−2/3; (c) C²_n=5×10⁻¹⁶m^−2/3; (d) C²_n=10⁻¹⁵m^−2/3; (e) C²_n=10⁻¹⁴m^−2/3; (e) C²_n=10⁻¹³m^−2/3.

C. DOC of RPTSCSM Beams

In this section, we will analyze the change of the DOC of RPTSCSM beams. Fig. 9 illustrates the behavior of the spectral degree of coherence of RPTSCSM beams as a function of x = x₂-x₁ and y = y₂-y₁ between two points concerning the optical axis, i.e., x₁=x₂=-x/2, y₁=-y₂=-y/2. We can see from Fig. 9 that, two effects are occurring with the spectral degree of coherence. First, as the propagation distance increases, spectral degree of coherence gradually changes from a Gaussian profile into a rectangular distribution, and the array distribution around the center part would gradually appears. Secondly, due to the effect of the twisted factor during propagation, the DOC rotates, and the rotation direction is related to the positive and negative values of the twisted factor.

Fig. 9.

The distribution of degree of spectral coherence of RPTCSM beams with δ = 1 mm, σ = 1 mm, u = −2 for different propagating distances in free space. In the figure, (a) z = 0 m; (b) z = 5 m; (c) z = 7 m; (d) z = 10 m; (e) z = 25 m; (f) z = 50m (g) z = 100 m; (h) z = 500 m.

In Fig. 10, we illustrate the evolution of the spectral degree of coherence of RPTCSM beams at propagating distance z = 10 m for different twisted factors in free space. It can find that, as values of the twist factor increase, the spectral degree of coherence gradually changes from the Gaussian-like distribution to an array distribution, and the array rotates around its center part. Fig. 11 presents the DOC rotation angle change and DOC distribution of RPTCSM beams with u = −2 at propagating distance z = 5 km for different beam widths and coherence widths in free space.

Fig. 10.

The DOC evolution of RPTCSM beams with δ = 1 mm, σ = 1mm at propagating distance z=10m for different twisted factors in free space. (a) u = 0; (b) u = −0.5; (c) u = −1; (d) u = −1.5; (e) u = −2; (f) u = −2.5.

Fig. 11.

Illustrates the DOC rotation angle change and DOC distribution of RPTCSM beams under different beam parameters in free space (a) σ=1 mm, δ=1mm; (b) σ=1 mm, u=−2, z=5 km; (c) δ=1 mm, u=−2, z=5 km.

We can see from Fig. 11(a) that when the twisted factor increases, the DOC distribution rotates more slowly in the opposite direction to the spectral intensity rotation during propagation. Furthermore, from Fig. 11(b) and (c), we can also find that, as either the coherence width decreases or the beam width increases,the profile of the DOC fluctuation gradually weaken and finally take on a Gaussian shape. Compared with the beam width in Fig. 11(b), the influence of coherence width is more evident in Fig. 11(a).

Fig. 12 shows the impact of a turbulent atmosphere on the DOC of RPTCSM beams at propagation distance z = 0.5 km. We can find from Fig. 8 that the parameter of a turbulent atmosphere plays a vital role in determining the DOC distribution of RPTCSM beams. When turbulence is relatively weak (seen in Fig. 12(a)–​(d)), the distribution of the DOC is similar to that in free space. As turbulence effects intensify, the array distribution of the DOC gradually disappears, and such distribution finally changes from a Gaussian profile to a cross-like distribution with four side-lobes (seen in Fig. 12(e)–​(h)).

Fig. 12.

The distribution of spectral degree of coherence of RPTCSM beams with δ = 1 mm, σ = 1 mm, u = −2 at propagating distance z = 0.5 km for different values of atmosphere parameters in the turbulent atmosphere. (a) C²_n=10⁻¹⁷m^−2/3; (b) C²_n=10⁻¹⁶m^−2/3; (c) C²_n=5×10⁻¹⁶m^−2/3; (d) C²_n=10⁻¹⁵m^−2/3; (e) C²_n=10⁻¹⁴m^−2/3; (f) C²_n=10⁻¹³m^−2/3.

D. The Scheme for the Experimental Research

At first, a linearly polarized beam focused by a thin lens illuminates a rotating ground-glass plate (RGGP), producing a partially coherent beam with Gaussian statistics. And then, after collimating by another thin lens and passing through the Gaussian amplitude filter (GAF), the transmitted beams turn into a linearly polarized Gaussian Schell-model (GSM) beam. Second, after going via a radial polarization converter (RPC), the GSM beam transforms into a radially partly coherent (RPGSM) beam [32]. Thirdly, two sets of three cylindrical lenses that perform imaging in one transverse coordinate and a Fourier transform in the other can be used to create the twist phase of an RPGSM beam. The beam parameters satisfy specific requirements when the latter group is rotated by 45 degrees around the axis. The system converts the transmitted beam into a PPGSM beam with the twist phase [33]. Finally, the RPTSCSM beams transform into a radially polarized twisted sinc-correlation Schell-model (RPTSCSM) by Fourier transform of the rectangular aperture diffraction screen [34].

SECTION V. Conclusion

In this work, we introduce a class of radially polarized partially coherent sources with twisted sinc-correlation and derive the source parameter conditions required to generate a physical beam. To particularly study the influence of the source characteristics and turbulent parameters on the statistical properties of such beams on propagation, several typical numerical examples are given. Due to the twisted effect, the intensity profile of such beams always splits into a 2×2 isotropic array and rotates to 90 degrees in free space and weaker turbulence medium. Moreover, the array gradually evolves to a hollow distribution with a dark core propagating in the turbulence medium. It indicates that the light beam may automatically restore the radial polarized light intensity distribution in atmospheric turbulence after a certain propagation distance. The distribution of the DOP is closely related to the twisted factor. With appropriate values of the twisted factor, the DOP would be twisted to a certain degree. Moreover, when propagating in a turbulent atmosphere, the DOP of such beams exhibits well-turbulent resistance. Additionally, the self-splitting and rotation induced by the twisted factor would occur in the DOC, and turbulent atmosphere has an important influence on the DOC. These results will benefit multi-particle manipulation and free-space optical communication.

Conflict of Interest: The authors declare that they have no conflict of interest.