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Orbital angular momentum spectra of twisted Laguerre-Gaussian Schell-model beams propagating in weak-to-strong Kolmogorov atmospheric turbulence

Haiyun Wang, Zhaohui Yang, Lin Liu, Yahong Chen, Fei Wang, and Yangjian Cai

Open Access Open Access

Abstract

The presence of atmospheric turbulence in a beam propagation path results in the spread of orbital angular momentum (OAM) modes of laser beams, limiting the performance of free-space optical communications with the utility of vortex beams. The knowledge of the effects of turbulence on the OAM spectrum (also named as spiral spectrum) is thus of utmost importance. However, most of the existing studies considering this effect are limited to the weak turbulence that is modeled as a random complex “screen” in the receiver plane. In this paper, the behavior of the OAM spectra of twisted Laguerre-Gaussian Schell-model (TLGSM) beams propagation through anisotropic Kolmogorov atmospheric turbulence is examined based on the extended Huygens-Fresnel integral which is considered to be applicable in weak-to-strong turbulence. The discrepancies of the OAM spectra between weak and strong turbulence are studied comparatively. The influences of the twist phase and the anisotropy of turbulence on the OAM spectra during propagation are investigated through numerical examples. Our results reveal that the twist phase plays a crucial role in determining the OAM spectra in turbulence, resisting the degeneration of the detection mode weight by appropriately choosing the twist factor, while the effects of the anisotropic factors of turbulence on the OAM spectra seem to be not obvious. Our findings can be applied to the analysis of OAM spectra of laser beams both in weak and strong turbulence.

© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement

1. Introduction

Recently, there is growing interest in laser beams with orbital angular momentum (OAM) owing to their diverse applications in optical manipulation, remote sensing, optical communication and quantum cryptography [1–6]. The study of the OAM in light beams can be traced back to the landmark paper of Allen et al. in 1992 [7]. They revealed that a Laguerre-Gaussian beam with vortex phase exp(ilφ) carries a well-defined OAM lћ per photon, where l is the topological charge and ћ is the reduced Plank constant. Since the vortex phases with different topological charges are orthogonal with each other and in general correspond to OAM eigenstates, the OAM mode forms infinite dimensions in Hilbert space. This provides a protocol for the realization of arbitrary base-N quantum digits to improve the information capacity in optical communications [8,9]. In 2012, Wang et al. proposed an ingenious method, i.e., spatial mode-division multiplexing based on OAM modes, to realize the terabit free-space data transmission [10]. Since then, a great deal of effort has been devoted to the generation, propagation and applications of laser beams with OAM modes (or referred as vortex beams) [2–4,11–13]. However, when a vortex beam propagates through atmosphere, it suffers the atmospheric turbulence-induced degeneration effects such as beam wander, phase distortion and intensity scintillation, resulting in the dispersion of the OAM modes and the crosstalk among the OAM states [14–17]. Hence, the knowledge of the effects of the atmospheric turbulence on the OAM states is vital to understand the OAM mode dispersion. Up to now, a lot of experimental and theoretical work has been carried out to study the propagation characteristics of vortex beams in atmospheric turbulence, including intensity evolution, scintillation index, mode transfer, etc. [18–21]. When calculating the OAM mode dispersion of the vortex beam in turbulence, single phase screen perturbation method [22–27], where the turbulence is modeled as a random phase screen at the receiver and there is no turbulence between the transmitter and the receiver, is one of the widely used approaches. However, this method is limited to the weak fluctuation regime, since it neglects the interaction of the turbulence and OAM modes during propagation. In practical situation, the turbulence is ubiquitous in the propagation channel and the degeneration effects are accumulated as the beam propagates. Multiple phase screen (MPS) method taking this effect into account has been used to examine the OAM mode dispersion of different kinds of laser beams including Laguerre-Gaussian beams and Bessel-Gauss beams [15,28–31], but it is a purely numerical simulation approach, lacking physical insight into the interaction between the OAM mode and the turbulence, and in some cases it is time-consuming, especially for partially spatially coherent sources. Except the MPS method, only a few literature has been devoted to studying OAM spectrum of laser beams beyond weak turbulence [32,33].

On the other hand, it has long been known that partially spatially coherent beams have many advantages [34–36], the most noteworthy of which is that they are less affected by the turbulence-induced disturbance, which makes them ideal candidates for information carriers in free-space optical communications [37–43]. In partially coherent fields, there exists a nontrivial phase named twist phase that offers a degree of freedom to modulate the partially coherent beams [44]. Different from other conventional phases such as quadratic phase and vortex phase, the twist phase is expressed as exp[ikμ0(x1y2-x2y1)] which is non-separated with respect to two positions (x1, y1) and (x2, y2), where μ0 is the twist factor and k is the wavenumber. Perhaps one of the most remarkable properties is that the twist phase induces the beam carrying OAM, resulting in the vortex structure of transverse energy flux across the beam cross-section [45–48] and re-distribution of the OAM spectrum. Another peculiar feature is that the twist phase can further reduce the turbulence-induced intensity scintillations compared to the beams without twist phase under the same conditions [49–52]. It has been lately discovered that twist phase induces classical entanglement [53], which can be maintained on twisted beam propagation through the turbulent atmosphere [54], and gainfully employed in free-space classical and quantum communications protocols, for instance. The twist phase also has found applications in improvement of imaging resolution and boosting entangled photon generation [55,56].

Our aim in this paper is to study the OAM spectra of general models of partially spatially coherent vortex beams carrying twist phase, named twisted Laguerre-Gaussian Schell-model (TLGSM) beams [47], propagation through anisotropic Kolmogorov turbulence. By means of the extended Huygens-Fresnel (eHF) principle and applying the Rytov approximation, analytical expression for the cross spectral density (CSD) of the TLGSM beams at the receiver is derived. The influences of the twist phase and turbulence parameters on the OAM spectra are analyzed through numerical examples. Our analysis is applicable in both weak and strong turbulence regimes.

2. Propagation of twisted Laguerre-Gaussian Schell-model beams through the anisotropic Kolmogorov atmospheric turbulence

Consider a partially coherent beam propagating along the z-axis from source plane (z = 0) to half-space plane (z > 0). The second-order statistics of such beam are characterized by a two-point CSD function $W({{\mathbf{r}_1},{\mathbf{r}_2}} )= \left\langle {E({{\mathbf{r}_1}} ){E^\ast }({{\mathbf{r}_2}} )} \right\rangle$ in space frequency domain, where E(r) denotes a random field realization, ri = (xi, yi), (i = 1, 2) are two position vectors in the source plane (z = 0), perpendicular to the propagation axis. The angle brackets and asterisk denote the ensemble average and complex conjugate, respectively. For TLGSM beams with radial mode 0 and azimuthal mode l, the CSD in the source plane is expressed as [47]

$$\begin{aligned} W({{\mathbf{r}_1},{\mathbf{r}_2}} )&\textrm{ = }{({{r_1}{r_2}} )^{|l |}}\exp [{il({{\varphi_1} - {\varphi_2}} )} ]\exp \left( { - \frac{{\mathbf{r}_1^2 + \mathbf{r}_2^2}}{{4\sigma_0^2}}} \right)\exp \left[ { - \frac{{{{({{\mathbf{r}_1} - {\mathbf{r}_2}} )}^2}}}{{2\delta_0^2}}} \right]\\ &\textrm{ } \times \exp [{ik{\mu_0}({{x_1}{y_2} - {x_2}{y_1}} )} ], \end{aligned}$$
where ${\varphi _i} = \arctan ({y_i}/{x_i})$, (i = 1, 2) are two azimuthal angles. σ0 and δ0 are the beam width and the spatial coherence width, respectively. l is the topological charge. k = 2π/λ is the wavenumber with λ being the wavelength of light. μ0 is the twist factor, a measure of the strength of the twist phase. Its value is bounded by the inequality |μ0|≤(kδ02)-1 to satisfy the non-negative definiteness of the CSD. From the inequality, it shows that the twist phase is exclusive to partially coherent beams, and vanishes in their coherent counterparts (${\delta _0} \to \infty$). When the coherence width tends to infinite value, the TLGSM beams reduce to Laguerre-Gaussian beams, a well-known class of vortex beams. Hence, the TLGSM beams can be regarded as the generalized version of the Laguerre-Gaussian beams.

In the presence of atmospheric turbulence, the propagation of a partially coherent beam from the transmitter to the receiver can be treated by eHF integral, given by [14,41]

$$\begin{aligned} {W_z}({{\boldsymbol{\mathrm{\rho}}_1},{\boldsymbol{\mathrm{\rho}}_2};z} )&= \frac{{{k^2}}}{{4{\pi ^2}{z^2}}}\int_{ - \infty }^\infty {\int_{ - \infty }^\infty {W({{\mathbf{r}_1},{\mathbf{r}_2};0} )} } \exp \left[ {\frac{{ik}}{{2z}}{{({{\mathbf{r}_1} - {\boldsymbol{\mathrm{\rho}}_1}} )}^2} - \frac{{ik}}{{2z}}{{({{\mathbf{r}_2} - {\boldsymbol{\mathrm{\rho}}_2}} )}^2}} \right]\\ &\textrm{ } \times {\left\langle {\exp [{\psi ({{\mathbf{r}_1},{\boldsymbol{\mathrm{\rho}}_1};z} )+ {\psi^ \ast }({{\mathbf{r}_2},{\boldsymbol{\mathrm{\rho}}_2};z} )} ]} \right\rangle _m}{d^2}{\mathbf{r}_1}{d^2}{\mathbf{r}_2}, \end{aligned}$$
where ρ1 = (ρx1, ρy1) and ρ2 = (ρx2, ρy2) are two arbitrary transverse position vectors in the output plane (z > 0). The angle brackets with subscript m stand for the ensemble averaging over the turbulence fluctuations. ψ(r, ρ; z) is turbulence-induced complex perturbations of a spherical wave propagating from (r, 0) to (ρ, z).

To distinguish between the eHF method and the weak turbulence propagation model used in literature [22–27], we plot in Fig. 1 the schematic for the two propagation models. In eHF method, the atmospheric turbulence fills the entire propagation channel as shown in Fig. 1(a), thereby, the turbulence affects the beam’s statistical properties during propagation. While in weak turbulence propagation model, it divides into two steps. The first step is that the beam propagates from the transmitter to the receiver in free space without turbulence. In the second step, the propagated beam in the receiver multiply the turbulence term which is modeled as a single turbulent screen representing the ensemble effects of the turbulence. In the former case, the turbulence effects are gradually accumulated, and they participate in the integration shown in Eq. (2) as beam propagates, but the latter does not. Therefore, the eHF model is considered to be much closer to the real situation of the beam propagation in atmospheric turbulence. Actually, the latter can be regarded as the special case of the former when the turbulence is very weak. The eHF integral is also known to be valid both weak and strong turbulence conditions [14,41]. We emphasize here that besides the above two propagation models, MPS method is another approach to study the propagation characteristics of laser beams in weak-to-strong turbulence. In this study, we only focus on the differences between the eHF method and the weak turbulence propagation model.

 figure: Fig. 1.

Fig. 1. Schematic diagram of (a) the eHF propagation model and (b) weak turbulence propagation model.

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According to [37], the second-order statistics of the complex perturbation induced by the turbulence can be written as the following form

$$\begin{array}{l} {\left\langle {\exp [{\psi ({{\mathbf{r}_1},{\boldsymbol{\mathrm{\rho}}_1};z} )+ {\psi^ \ast }({{\mathbf{r}_2},{\boldsymbol{\mathrm{\rho}}_2};z} )} ]} \right\rangle _m}\\ = \exp \left\{ { - 2\pi {k^2}z\int_0^1 {dt} \int {{d^2}{\boldsymbol{\mathrm{\kappa}}_ \bot }} {\Phi _n}({{\boldsymbol{\mathrm{\kappa}}_ \bot },{\kappa_z} = 0} )\{{1 - \exp [{ - i({t{\boldsymbol{\mathrm{\rho}}_d} \cdot {\boldsymbol{\mathrm{\kappa}}_ \bot } + ({1 - t} ){\mathbf{r}_d} \cdot {\boldsymbol{\mathrm{\kappa}}_ \bot }} )} ]} \}} \right\} \end{array}$$
where ρd = ρ1-ρ2, rd = r1-r2. Φn(κ) is the power spectrum of the refractive index fluctuations of the turbulence, κ =(κx, κy, κz) is spatial frequency vector. The subscript “⊥” denotes the transverse component orthogonal to the z-component. In the derivation of Eq. (3), the Markov approximation that assumes the delta-correlation of the turbulent fluctuations along the beam propagation direction is applied. In anisotropic turbulence, three anisotropy factors μx, μy and μz pertinent to the size of eddies along x, y and z directions were introduced in [26], and their products satisfy µxµyµz = 1 due to the equal volume between isotropic and anisotropic turbulent eddies at the same altitude. By making the variable changes: $\boldsymbol{\mathrm{\kappa}} = ({{{\kappa^{\prime}}_x}/{\mu_x},{{\kappa^{\prime}}_y}/{\mu_y},{{\kappa^{\prime}}_z}/{\mu_z}} )$, ${\mathbf{r}_d} = ({{\mu_x}{{x^{\prime}}_d},{\mu_y}{{y^{\prime}}_d}} )$ and ${\boldsymbol{\mathrm{\rho}}_d} = ({{\mu_x}{{\rho^{\prime}}_{xd}},{\mu_y}{{\rho^{\prime}}_{yd}}} )$. Equation (3) takes the form
$$\begin{array}{l} {\left\langle {\exp [{\psi ({{\mathbf{r}_1},{\boldsymbol{\mathrm{\rho}}_1};z} )+ {\psi^ \ast }({{\mathbf{r}_2},{\boldsymbol{\mathrm{\rho}}_2};z} )} ]} \right\rangle _m}\\ = \exp \left\{ { - \frac{{2\pi {k^2}z}}{{{\mu_x}{\mu_y}}}\int_0^1 {dt} \int {d{{\kappa^{\prime}}_x}d{{\kappa^{\prime}}_y}{\Phi _n}(\boldsymbol{\mathrm{\kappa}} ^{\mathbf\prime}_ \bot)\{{1 - \exp [{ - i({t\boldsymbol{\mathrm{\rho}} ^{\mathbf\prime}_d \cdot \boldsymbol{\mathrm{\kappa}} ^{\mathbf\prime}_ \bot + ({1 - t} ){{\mathbf{r^{\prime}}}_d} \cdot {\boldsymbol{\mathrm{\kappa}}_ \bot }}} )} ]} \}} \right\} \end{array}$$
where $\boldsymbol{\mathrm{\kappa}} ^{\mathbf\prime}_ \bot \equiv ({{{\kappa^{\prime}}_x},{{\kappa^{\prime}}_y}} )$, ${\mathbf{r^{\prime}}_d} \equiv ({{{x^{\prime}}_d},{{y^{\prime}}_d}} )$ and $\boldsymbol{\mathrm{\rho}} ^{\mathbf\prime}_d \equiv ({{{\rho^{\prime}}_{xd}},{{\rho^{\prime}}_{yd}}} )$. In the representation of $\boldsymbol{\mathrm{\kappa}} ^{\mathbf\prime}$, the power spectrum ${\Phi _n}(\boldsymbol{\mathrm{\kappa}} ^{\mathbf\prime}_ \bot )$ is anisotropic. To evaluate Eq. (4), the differential $d{\kappa ^{\prime}_x}d{\kappa ^{\prime}_y}$ changes to $\kappa ^{\prime}d\kappa ^{\prime}d\phi$ in the polar coordinate. After integrating over $\phi$, we obtain the expression
$$\begin{array}{l} {\left\langle {\exp [{\psi ({{\mathbf{r}_1},{\boldsymbol{\mathrm{\rho}}_1};z} )+ {\psi^ \ast }({{\mathbf{r}_2},{\boldsymbol{\mathrm{\rho}}_2};z} )} ]} \right\rangle _m}\\ \textrm{ = }\exp \left\{ { - \frac{{\textrm{4}{\pi^\textrm{2}}{k^2}z}}{{{\mu_x}{\mu_y}}}\int_0^1 {dt} \int_0^\infty {\kappa^{\prime}d\kappa^{\prime}{\Phi _n}({{\boldsymbol{\mathrm{\kappa}}_ \bot }^\prime } )[{1 - {J_\textrm{0}}({|{t{{\mathbf{\rho^{\prime}}}_d} + ({1 - t} ){{\mathbf{r^{\prime}}}_d}} |\boldsymbol{\mathrm{\kappa}} ^{\mathbf\prime}} )} ]} } \right\}, \end{array}$$
where J0 is the zero-order Bessel function, and the parameter t is the integral variable ranging from 0 to 1. We now assume that the region of interest is sufficiently close to the propagation axis or the inner scale of turbulence is much larger than the transverse coherence of the beam propagating a certain distance in turbulence [37]. The Bessel function J0 can approximately expand as the first two terms of Taylor series, i.e., J0(x) = 1 - x2/4. By making use of this condition and integrating over t, we arrive at
$$\begin{array}{r} {\left\langle {\exp [{\psi ({{\mathbf{r}_1},{\boldsymbol{\mathrm{\rho}}_1};z} )+ {\psi^ \ast }({{\mathbf{r}_2},{\boldsymbol{\mathrm{\rho}}_2};z} )} ]} \right\rangle _m} = \exp \left\{ { - \frac{T}{{{\mu_x}{\mu_y}}}({\mathbf{\rho^{\prime}}_d^2 + \mathbf{r^{\prime}}_d^2 + \boldsymbol{\mathrm{\rho}} ^{\mathbf\prime}_d \cdot \mathbf{r^{\prime}}} )} \right\}\\ \textrm{ } = \exp \left[ { - T{\mu_z}\left( {\frac{{\rho_{xd}^2}}{{\mu_x^2}} + \frac{{\rho_{yd}^2}}{{\mu_y^2}} + \frac{{x_d^2}}{{\mu_x^2}} + \frac{{y_d^2}}{{\mu_y^2}} + \frac{{{\rho_{xd}}{x_d}}}{{\mu_x^2}} + \frac{{{\rho_{yd}}{y_d}}}{{\mu_y^2}}} \right)} \right]. \end{array}$$
where $T = \frac{{{\pi ^\textrm{2}}{k^2}z}}{3}\int_0^\infty {{{\kappa ^{\prime}}^3}{\Phi _n}({{\boldsymbol{\mathrm{\kappa}}_ \bot }^\prime } )d\kappa ^{\prime}}$.

On substituting Eq. (1) and Eq. (6) into Eq. (2), and after some tedious but straightforward integrating, we obtain the analytical expression for the CSD of the TLGSM beams at the receiver

$$\scalebox{0.9}{$\begin{array}{l} {W_z}({{\boldsymbol{\mathrm{\rho}}_1},{\boldsymbol{\mathrm{\rho}}_2};z} )= \frac{{A{\pi ^\textrm{2}}}}{{{\lambda ^\textrm{2}}{z^2}}}\exp \left[ {\frac{{ik}}{{2z}}({\boldsymbol{\mathrm{\rho}}_1^2 - \boldsymbol{\mathrm{\rho}}_2^2} )} \right]\exp [{ - {T_x}{{({{\rho_{1x}} - {\rho_{2x}}} )}^2} - {T_y}{{({{\rho_{1y}} - {\rho_{2y}}} )}^2}} ]\\ \;\;\;\;\times \exp \left( {\frac{{\Delta _{x1}^2}}{{4({{N_\textrm{1}}\textrm{ + }{T_x}} )}} + \frac{{\Delta _{y1}^2}}{{4({{N_\textrm{1}} + {T_y}} )}}} \right)\exp \left( {\frac{{\Delta _{x3}^2}}{{4{N_\textrm{2}}}} + \frac{{\Delta _{y3}^2}}{{4{N_3}}}} \right)\\ \;\;\;\;\times \sum\limits_{c1 = 0}^l {\sum\limits_{c2 = 0}^l {\sum\limits_{p1 = 0}^{l - c1} {\sum\limits_{p\textrm{2} = 0}^{l - c1 - p\textrm{1}} {\sum\limits_{p3 = 0}^{c1} {\sum\limits_{p4 = 0}^{c1 - p3} {\sum\limits_{m1 = 0}^{[{p1/2} ]} {\sum\limits_{m\textrm{2} = 0}^{[{p\textrm{2}/2} ]} {\sum\limits_{m3 = 0}^{[{p3/2} ]} {\sum\limits_{m4 = 0}^{[{p4/2} ]} {\sum\limits_{p5 = 0}^{\alpha 1} {\sum\limits_{m5 = 0}^{[{({\alpha 1 - p5} )/2} ]} {{{({2i} )}^{ - ({\alpha 1 + \alpha 2\textrm{ + }l} )}}{{({ - 1} )}^{m1 + m2 + m3 + m4 + m5}}} } } } } } } } } } } } \\ \;\;\;\;\times \frac{1}{{{2^{l - ({p1 + p3} )/2 + \alpha 1/2}}}}{({{N_\textrm{1}} + {T_x}} )^{({c1 - l - 1} )/2}}{({{N_\textrm{1}} + {T_y}} )^{ - ({c1 + 1} )/2}}N_2^{ - ({\alpha 1 + 1} )/2}N_3^{ - ({1 + \alpha 2} )/2}\\ \;\;\;\;\times \left( {\begin{array}{c} {l - c1}\\ {p1} \end{array}} \right)\left( {\begin{array}{c} {l - c1 - p\textrm{1}}\\ {p\textrm{2}} \end{array}} \right)\left( {\begin{array}{c} {c1}\\ {p3} \end{array}} \right)\left( {\begin{array}{c} {c1 - p3}\\ {p4} \end{array}} \right)\left( {\begin{array}{c} {\alpha 1}\\ {p5} \end{array}} \right)\\ \;\;\;\;\times \frac{{l!{i^{c1}}}}{{c1!({l - c1} )!}}\frac{{l!{{({ - i} )}^{c2}}}}{{c2!({l - c2} )!}}\frac{{p1!}}{{m1!({p1 - 2m1} )!}}\frac{{p2!}}{{m2!({p2 - 2m2} )!}}\\ \;\;\;\;\times \frac{{p3!}}{{m3!({p3 - 2m3} )!}}\frac{{p4!}}{{m4!({p4 - 2m4} )!}}\frac{{({\alpha 1 - p5} )!}}{{m5!({\alpha 1 - p5 - 2m5} )!}}\\ \;\;\;\;\times {\left( {\frac{{\sqrt 2 i{\Delta _{x2}}}}{{\sqrt {{N_\textrm{1}}\textrm{ + }{T_x}} }}} \right)^{p1 - 2m1}}{\left( {\frac{{2k{\mu_0}}}{{\sqrt {{N_\textrm{1}} + {T_y}} }}} \right)^{p4 - 2m4}}{\left( { - \frac{{2k{\mu_0}}}{{\sqrt {{N_\textrm{1}}\textrm{ + }{T_x}} }}} \right)^{p2 - 2m2}}{\left( {\frac{{\sqrt 2 i{\Delta _{y2}}}}{{\sqrt {{N_\textrm{1}} + {T_y}} }}} \right)^{p3 - 2m3}}{\left( {\frac{{\sqrt 2 i{\Delta _{x\textrm{4}}}}}{{\sqrt {{N_\textrm{2}}} }}} \right)^{\alpha 1 - p5 - 2m5}}\\ \;\;\;\;\times {H_{l - c1 - p\textrm{1} - p\textrm{2}}}\left( {\frac{{i{\Delta _{x\textrm{1}}}}}{{\sqrt {{N_\textrm{1}}\textrm{ + }{T_x}} }}} \right){H_{c1 - p3 - p4}}\left( {\frac{{i{\Delta _{y1}}}}{{\sqrt {{N_\textrm{1}} + {T_y}} }}} \right){H_{p5}}\left( {\frac{{i{\Delta _{x\textrm{3}}}}}{{\sqrt {2{N_\textrm{2}}} }}} \right){H_{\alpha 2}}\left( {\frac{{i{\Delta _{y\textrm{3}}}}}{{\textrm{2}\sqrt {{N_3}} }}} \right), \end{array}$}$$
with
$$\begin{array}{l} {T_x}\textrm{ = }\frac{{T{\mu _z}}}{{\mu _x^2}}\textrm{, }{T_y} = \frac{{T{\mu _z}}}{{\mu _y^2}},\textrm{ }{\Delta _{x\textrm{1}}}\textrm{ = } - \frac{{ik{\rho _{1x}}}}{z} - {T_x}{\rho _{xd}}\textrm{, }{\Delta _{x2}} = \frac{1}{{\delta _0^2}} + 2{T_x}\textrm{,}\\ {\Delta _{x\textrm{3}}}\textrm{ = }\frac{{{\Delta _{x1}}{\Delta _{x2}}}}{{2({{N_\textrm{1}}\textrm{ + }{T_x}} )}} - \frac{{ik{\mu _0}{\Delta _{y1}}}}{{2({{N_\textrm{1}} + {T_y}} )}} + \frac{{ik{\rho _{2x}}}}{z} + {T_x}{\rho _{xd}}\textrm{, }{\Delta _{x\textrm{4}}} = \frac{{ik{\mu _0}{\Delta _{x2}}}}{{2({{N_\textrm{1}}\textrm{ + }{T_x}} )}} - \frac{{ik{\mu _0}{\Delta _{y2}}}}{{2({{N_\textrm{1}} + {T_y}} )}},\\ {\Delta _{y\textrm{1}}}\textrm{ = } - \frac{{ik{\rho _{1y}}}}{z} - {T_y}{\rho _{yd}}\textrm{, }{\Delta _{y2}} = \frac{1}{{\delta _0^2}} + 2{T_y}\textrm{,}\\ {\Delta _{y\textrm{3}}}\textrm{ = }\frac{{{\Delta _{y1}}{\Delta _{y2}}}}{{2({{N_\textrm{1}} + {T_y}} )}} + \frac{{ik{\mu _0}{\Delta _{x\textrm{1}}}}}{{2({{N_\textrm{1}}\textrm{ + }{T_x}} )}} + \frac{{{\Delta _{x\textrm{3}}}{\Delta _{x\textrm{4}}}}}{{2{N_\textrm{2}}}} + \frac{{ik{\rho _{2y}}}}{z} + {T_y}{\rho _{yd}},\\ {N_\textrm{1}}\textrm{ = }\left( {\frac{1}{{4\sigma_0^2}} + \frac{1}{{2\delta_0^2}} - \frac{{ik}}{{2z}}} \right)\textrm{, }{N_\textrm{2}}\textrm{ = }N_\textrm{1}^\mathrm{\ast} + {T_x} - \frac{{\Delta _{x2}^2}}{{4({{N_\textrm{1}}\textrm{ + }{T_x}} )}} + \frac{{{k^2}\mu _0^2}}{{4({{N_\textrm{1}} + {T_y}} )}}\textrm{,}\\ {N_3}\textrm{ = }N_\textrm{1}^\mathrm{\ast } + {T_y} - \frac{{\Delta _{x4}^2}}{{4{N_\textrm{2}}}} - \frac{{\Delta _{y2}^2}}{{4({{N_\textrm{1}} + {T_y}} )}} + \frac{{{k^2}\mu _0^2}}{{4({{N_\textrm{1}}\textrm{ + }{T_x}} )}},\\ {\alpha _\textrm{1}} = l + p1 + p4 - c2 - 2m1 - 2m4,\\ {\alpha _2} = c2 + p2 + p3 + \alpha 1 - 2m2 - 2m3 - p5 - 2m5. \end{array}$$

In Eq. (7), Hn denotes n-order Hermite polynomial. In the derivation of Eq. (7), the following integral and expansion formulas are applied

$$\int_{ - \infty }^\infty {{x^\alpha }} \exp [{ - {{({x - \beta } )}^2}} ]dx = {({2i} )^{ - \alpha }}\sqrt \pi {H_\alpha }({i\beta } ),$$
$${H_\alpha }({x + \beta } )= \frac{1}{{{2^{\alpha /2}}}}\sum\limits_{p = 0}^\alpha {\left( {\begin{array}{c} \alpha \\ p \end{array}} \right)} {H_p}\left( {\sqrt 2 x} \right){H_{\alpha - p}}\left( {\sqrt 2 \beta } \right),$$
$${H_n}({{x_1}} )= \sum\limits_{m = 0}^{[{n/2} ]} {{{({ - 1} )}^m}\frac{{n!}}{{m!({n - 2m} )!}}{{({2{x_1}} )}^{n - 2m}}} .$$

Let us pay attention to the OAM spectra of the TLGSM beams at the receiver plane after propagation through the turbulence. The OAM spectrum refers the energy distribution of the OAM modes contained in a light beam as a function of the topological charge. For a partially coherent beam, a realization of electric field E(ρ,φ,z) can be deconstructed into OAM eigenfunctions as [57]

$$E(\rho ,\varphi ,z) = \frac{1}{{\sqrt {2\pi } }}\sum\limits_{p = 0}^\infty {\sum\limits_{m ={-} \infty }^\infty {{a_{pm}}{R_p}(\rho ,z){e^{im\varphi }}} } ,$$
where apm are complex coefficients. Rp(ρ, z) comprise a radial basis set. On the basis of Eq. (12), the average energy of the OAM mode m contained in the partially coherent beam is evaluated as the following formula
$$\begin{aligned} {C_m} = \sum\limits_{p = 0}^\infty {\left\langle {{{|{{a_{pm}}} |}^2}} \right\rangle } &= \frac{1}{{2\pi }}\int\limits_0^\infty {\int\limits_0^\infty {\int\limits_0^{2\pi } {\int\limits_0^{2\pi } {\left\langle {E({\rho_1},{\varphi_1},z){E^\ast }({\rho_2},{\varphi_2},z)} \right\rangle {e^{ - im{\varphi _1} + im{\varphi _2}}}} } } } \\ &\textrm{ } \times \sum\limits_{p = 0}^\infty {{R_p}({\rho _1},z)} R_p^\ast ({\rho _2},z){\rho _1}{\rho _2}d{\rho _1}d{\rho _2}d{\varphi _1}d{\varphi _2}, \end{aligned}$$
where the angle brackets represent the ensemble averaging over the field fluctuations. By applying the completeness of the radial basis $\sum\limits_{p = 0}^\infty {{R_p}({\rho _1},z)} R_p^\ast ({\rho _2},z) = \delta ({\rho _1} - {\rho _2})/{\rho _1}$ and integrating over ρ2, Eq. (13) reduces to
$${C_m} = \frac{1}{{2\pi }}\int\limits_0^\infty {\int\limits_0^{2\pi } {\int\limits_0^{2\pi } {W(\rho ,{\varphi _1},\rho ,{\varphi _2};z){e^{ - im{\varphi _1} + im{\varphi _2}}}} } } \rho d\rho d{\varphi _1}d{\varphi _2}.$$

In Eq. (14), the definition of the CSD function $W({\rho _1},{\varphi _1},{\rho _2},{\varphi _2};z) = \left\langle {E({\rho_1},{\varphi_1},z){E^\ast }({\rho_2},{\varphi_2},z)} \right\rangle$ is applied. Equation (14) establishes the relation between the energy of OAM mode m and the CSD in the receiver plane, which provides a way to evaluate the OAM spectrum of a partially coherent beam when the CSD is known.

3. Numerical results and analysis

In this section, as numerical examples, we will examine the behavior of the OAM spectra of the TLGSM beams after propagation through anisotropic Kolmogorov turbulence from weak to strong regime based on the derived formula in section 2.

The power spectrum Φn(κ) of the turbulence is adopted as [14,26]

$${\Phi _n}(\boldsymbol{\mathrm{\kappa}} )= 0.033C_n^2\frac{{{\mu _x}{\mu _y}{\mu _z}\exp [{ - ({\mu_x^2\kappa_x^2 + \mu_y^2\kappa_y^2 + \mu_z^2\kappa_z^2} )/\kappa_m^2} ]}}{{{{({\mu_x^2\kappa_x^2 + \mu_y^2\kappa_y^2 + \mu_z^2\kappa_z^2 + \kappa_0^2} )}^{11/6}}}},$$
where $C_n^2$ is the structure constant of turbulence with units m−2/3, ${\kappa _0} = 2\pi /{L_0}$, ${\kappa _m} = 5.92/{l_0}$ with L0 and l0 being the outer and inner scale of the turbulence, respectively. Note that when µx=µy=µz = 1, Eq. (15) reduces to the von Kármán spectrum.

By applying Eq. (15), the parameter T has found the analytical solution, i.e.,

$$T = 0.0033{\pi ^2}{k^2}zC_n^2[{\eta \kappa_m^{ - 5/3}\exp ({\kappa_0^2/\kappa_m^2} ){\Gamma _1}({1/6,\kappa_0^2/\kappa_m^2} )- 2\kappa_0^{1/3}} ],$$
where $\eta = 2\kappa _0^2 + 5/3\kappa _m^2$, ${\Gamma _1}$ denotes incomplete gamma function.

In the following numerical analysis, the parameters are chosen to be $\lambda = 1550\textrm{nm,} l = 1,$ ${\sigma _0} = 1\textrm{cm,} {L_0} = 1\textrm{m,}$ ${l_0} = 1\textrm{cm}$ and $C_n^2 = 2 \times {10^{ - 14}}{\textrm{m}^{ - 2/3}}$ unless specified otherwise, but the propagation distance will be varied.

3.1 OAM spectra of TLGSM beams in the source plane

We first examine the OAM spectra of the TLGSM beams in the source plane (z = 0). The numerical results of the OAM spectra with different coherence widths δ0 and the twist factor µ0 as a function of Δl = m-l are illustrated in Figs. 2 and 3, respectively. The mode weight Pm is normalized by its total energy carried by the beam, i.e., ${P_m} = {C_m}/\sum\nolimits_{m ={-} \infty }^\infty {{C_m}}$. In the absence of the twist phase, the spectra are of symmetry with respect to Δl = 0, regardless of the coherence width. Nevertheless, as the coherence width decreases, the spectra become more dispersed, which implies the energy of the central mode (Δl = 0) is significantly reduced, and other OAM modes (Δl≠0) gradually increase. The total OAM calculated from $\sum\nolimits_{m ={-} \infty }^\infty {{P_m}m\hbar } = l\hbar$ is conserved. From Fig. 3, one finds that the symmetric distribution of the OAM spectra is broken owing to the existence of the twist phase. Both of the magnitude and the sign of the twist factor have great influences on the OAM spectra. The first row of Fig. 3 indicates that µ0 ≤ 0, and the second row represents µ0 ≥ 0. When the twist factor is negative, the OAM modes with Δl > 0 dominate the major role. The larger the values of the twist factor is, the larger the OAM mode weights with Δl > 0 are. The OAM mode Δl = 0 is no longer the dominant mode if the twist factor is larger enough [see Fig. 3(d) and 3(d1)]. When the twist factor is positive, the situation is reversed. The asymmetric distribution of the OAM spectra is attribute to the interaction of the vortex phase and the twist phase. Different from the vortex phase that induces the well-defined OAM only proportional to topological l, the OAM induced by the twist phase depends on the twist factor and the beam width, and varies continuously with the two parameters. According to [47], the time-average OAM of the TLGSM beams per photon is ${J_z} = [l - 2{\mu _0}k\sigma _0^2 - 2{\mu _0}k\sigma _0^2|l|]\hbar$. The existence of the twist phase greatly enhances the total OAM, which results the re-distribution of the OAM spectra.

 figure: Fig. 2.

Fig. 2. OAM spectra of the TLGSM beam for different values of δ0 with µ0 = 0 in the source plane.

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 figure: Fig. 3.

Fig. 3. OAM spectra of the TLGSM beam for different values of twist factor µ0 with δ0 = 0.01 m in the source plane. (a)-(d): µ0 ≤ 0; (a1)-(d1): µ0 ≥ 0. Note that (a) and (a1) all represent the OAM spectra in the case µ0 = 0.

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3.2 OAM spectra of TLGSM beams in weak-to-strong anisotropic atmospheric turbulence

We now consider the evolution of the OAM spectra of the TLGSM beams in isotropic Kolmogorov turbulence. In this case, the anisotropic factors of the turbulence are µx=µy=µz = 1. Also, we fix the coherence width at δ0 = 0.015 m. Figure 4 presents the numerical results of the OAM spectra of the TLGSM beams with different twist factors at four different propagation distances. The blue histograms correspond to the results calculated from eHF integration (Method A). For comparison, the pink histograms obtained from the weak turbulence propagation model (Method B) are also illustrated. To judge the strength of the turbulence, the Rytov variance $\sigma _R^2 = 1.23C_n^2{k^{7/6}}{z^{11/6}}$ which corresponds to the scintillation index of a plane wave is used. Weak fluctuations are associated with $\sigma _R^2 < 1$, moderate fluctuation conditions are characterized by $\sigma _R^2 \sim 1$, and strong fluctuations are associated with $\sigma _R^2 > 1$. The Rytov variance in our calculation for z = 50 m, 1 km, 1.5 km and 2 km are 0.002, 0.398, 0.837 and 1.419, respectively, ranging from weak to strong fluctuations. When the turbulence is very weak $\sigma _R^2 \sim 0.002$, the method A and the method B lead to the same results [see Fig. 4(a)–4(a2)]. As the propagation distance increases, i.e., the effect of the turbulence is accumulated, the discrepancies of the OAM spectra between two methods become obvious. The differences of the central mode (Δl = 0) in the case of µ0 = 0 between two methods are 0.029, 0.019 and 0.014, at propagation distance z = 1 km, 1.5 km and 2.0 km, respectively. It is found in Fig. 4(b)–4(d) that the spectra become asymmetric with respect to Δl = 0 when the turbulence is relative strong ($\sigma _R^2 > 0.4$). We attribute this phenomenon to the interaction between the partial spatial coherence of the beam source and the atmospheric turbulence during beam’s propagation. In addition, the results show that the sign of the twist factor has a tremendous influence on the OAM spectra, especially in the strong fluctuation regime. As shown in Fig. 4(d1) and (d2), the OAM spectra with the negative twist factor are much more dispersive than those of the positive twist factor. Owing to the dispersion of the OAM spectra in negative case, the differences of the OAM spectra between two methods become small.

 figure: Fig. 4.

Fig. 4. Distribution of the OAM spectra of the TLGSM beam with different values of µ0 at several propagation distances z in an isotropic Kolmogorov turbulence atmosphere (µx =µy=µz = 1). Method A: eHF propagation model; Method B: weak turbulence propagation model. (a)-(d): µ0 = 0; (a1)-(d1): µ0 < 0; (a2)-(d2): µ0 > 0.

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Figure 5 presents the variation of central mode Pm = l with the propagation distance z for different values of the twist factor µ0. The dashed curves and the dotted curves are the results from the method A and the method B, respectively. The values of the mode weight obtained from the eHF method are always smaller than those obtained from the weak turbulence propagation method. The reason is that the latter method ignores the interaction between the turbulence and the laser beam during propagation. It is also clearly shown that when the propagation distance is about in the range from z = 0.25 km to z = 1.25 km, the differences between the two methods are the most significant.

 figure: Fig. 5.

Fig. 5. Variation of Pm = l with the propagation distances z in isotropic Kolmogorov turbulence atmosphere for different values of the twist factor µ0. A: eHF propagation model; B: weak turbulence propagation model.

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To quantitatively assess the influence of the twist factor on the detection probabilities of the center mode in turbulence, the parameter of drop proportion D defined as 1 – Pm (z) / Pm (z = 0) is used. The value of the D is range from 0 to 1. D = 0 denotes “no drop” of the detection probabilities of the mode m, while the probabilities are completely lost for D = 1. Table 1 shows the drop proportion of Pm = l at several propagation distances z with different values of the twist factor µ0. It is found that in weak-to-moderate turbulence, the TLGSM beams with negative twist factors are superior to those of zero and positive twist factors since the value of D is smaller among the three cases under the same conditions. However, the situation is reversed in strong turbulence, where the TLGSM beams with positive twist factors become the best choice. Therefore, properly modulating twist factors will improve the efficiency of the detection probabilities of the certain OAM modes under different strengths of the turbulence.

Tables Icon

Table 1. Drop proportion of the central OAM mode at several propagation distances z with different values of the twist factor

View Table

Let us pay attention to the behavior of the OAM spectra of the TLGSM beams propagating in anisotropic atmospheric turbulence. In the calculations, we make the following restrictions on the anisotropy factors: (1) µxµyµz = 1; (2) When µx>µy, we have µz=µy, otherwise we set µz=µx. The other parameters are the same as those in Fig. 4 if the values are not specified.

Figure 6 demonstrates the influence of the µx /µy on the OAM spectra of the TLGSM beams with different twist factors at propagation distance z = 1.5 km. The first row indicates µ0 = 0, the second row and the third row represent µ0 < 0 and µ0 > 0, respectively. The results show that the ratio of µx / µy has only a little effect on the distribution of the OAM spectra. The value of the central mode m = l is almost unchanged under different µx / µy. Figure 7 presents the changes of the weight of the central mode Pm = l with the propagation distances z under different values of µx / µy. The mode weight Pm = l with µx / µy = 1 is always the minimum if the beam parameters and the propagation distance are the same, which implies that the anisotropy of the turbulence has less effect on the OAM mode dispersions. Especially in moderate-strong turbulence, the values of the Pm = l in isotropic turbulence are significantly smaller than those in anisotropic turbulence.

 figure: Fig. 6.

Fig. 6. OAM spectra of TLGSM beams with different values of µ0 and µx /µy at z = 1.5 km in anisotropic turbulence. (a)-(d): µ0 = 0; (a1)-(d1): µ0 < 0; (a2)-(d2): µ0 > 0.

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 figure: Fig. 7.

Fig. 7. Variation of the central mode Pm = l with the propagation distances z in isotropic and anisotropic turbulence for different values of the twist factor µ0 and the anisotropic factor µx /µy.

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4. Conclusion

In summary, we have studied the OAM spectra of the TLGSM beams propagating in isotropic and anisotropic Kolmogorov turbulence. The method is based on the eHF integral that is applicable both in weak and strong turbulence. In the source plane, the OAM spectra of the TLGSM beams are closely related to the sign of the twist factor. When the twist factor is negative, the mode weights larger than the topological l occupies the main positions, while the situation is reversed for the positive values of the twist factor. As the coherence width decreases, the distribution of the OAM spectra is broadened. Further, the dependence of the sign of the twist factor on the OAM spectra of the TLGSM beams in isotropic turbulence is examined through numerical results. The results are compared with those obtained from the weak turbulence model. It reveals that in very weak turbulence, the two methods lead to the same results of the OAM spectra. In moderate turbulence, there exists significant differences between two methods. However, the results of the two methods seem to be the same when the turbulence is strong enough. Further, we found that by appropriately choosing the twist factor, the central mode weight is less affected by the turbulence, compared to the TLGSM beam without twist phase under the same conditions. The influence of the anisotropic factors of the turbulence on the OAM spectra of the TLGSM beam is also analyzed, but the results show that there is only a little effect on the OAM spectra. Our results could have prospective applications in OAM communication and optical detection in both weakly and strongly turbulent atmosphere.

Funding

National Key Research and Development Program of China (2019YFA0705000, 2022YFA1404800); National Natural Science Foundation of China (11874046, 11904247, 11974218, 12104263, 12174279, 12192254, 92250304); Local Science and Technology Development Project of the Central Government (YDZX20203700001766).

Disclosures

The authors declare no conflicts of interest.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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Figures (7)
Fig. 1.
Fig. 1. Schematic diagram of (a) the eHF propagation model and (b) weak turbulence propagation model.

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Fig. 2.
Fig. 2. OAM spectra of the TLGSM beam for different values of δ0 with µ0 = 0 in the source plane.

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Fig. 3.
Fig. 3. OAM spectra of the TLGSM beam for different values of twist factor µ0 with δ0 = 0.01 m in the source plane. (a)-(d): µ0 ≤ 0; (a1)-(d1): µ0 ≥ 0. Note that (a) and (a1) all represent the OAM spectra in the case µ0 = 0.

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Fig. 4.
Fig. 4. Distribution of the OAM spectra of the TLGSM beam with different values of µ0 at several propagation distances z in an isotropic Kolmogorov turbulence atmosphere (µx =µy=µz = 1). Method A: eHF propagation model; Method B: weak turbulence propagation model. (a)-(d): µ0 = 0; (a1)-(d1): µ0 < 0; (a2)-(d2): µ0 > 0.

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Fig. 5.
Fig. 5. Variation of Pm = l with the propagation distances z in isotropic Kolmogorov turbulence atmosphere for different values of the twist factor µ0. A: eHF propagation model; B: weak turbulence propagation model.

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Fig. 6.
Fig. 6. OAM spectra of TLGSM beams with different values of µ0 and µx /µy at z = 1.5 km in anisotropic turbulence. (a)-(d): µ0 = 0; (a1)-(d1): µ0 < 0; (a2)-(d2): µ0 > 0.

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Fig. 7.
Fig. 7. Variation of the central mode Pm = l with the propagation distances z in isotropic and anisotropic turbulence for different values of the twist factor µ0 and the anisotropic factor µx /µy.

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Tables (1)

Tables Icon

Table 1. Drop proportion of the central OAM mode at several propagation distances z with different values of the twist factor

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Equations (16)

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$$\begin{aligned} W({{\mathbf{r}_1},{\mathbf{r}_2}} )&\textrm{ = }{({{r_1}{r_2}} )^{|l |}}\exp [{il({{\varphi_1} - {\varphi_2}} )} ]\exp \left( { - \frac{{\mathbf{r}_1^2 + \mathbf{r}_2^2}}{{4\sigma_0^2}}} \right)\exp \left[ { - \frac{{{{({{\mathbf{r}_1} - {\mathbf{r}_2}} )}^2}}}{{2\delta_0^2}}} \right]\\ &\textrm{ } \times \exp [{ik{\mu_0}({{x_1}{y_2} - {x_2}{y_1}} )} ], \end{aligned}$$
$$\begin{aligned} {W_z}({{\boldsymbol{\mathrm{\rho}}_1},{\boldsymbol{\mathrm{\rho}}_2};z} )&= \frac{{{k^2}}}{{4{\pi ^2}{z^2}}}\int_{ - \infty }^\infty {\int_{ - \infty }^\infty {W({{\mathbf{r}_1},{\mathbf{r}_2};0} )} } \exp \left[ {\frac{{ik}}{{2z}}{{({{\mathbf{r}_1} - {\boldsymbol{\mathrm{\rho}}_1}} )}^2} - \frac{{ik}}{{2z}}{{({{\mathbf{r}_2} - {\boldsymbol{\mathrm{\rho}}_2}} )}^2}} \right]\\ &\textrm{ } \times {\left\langle {\exp [{\psi ({{\mathbf{r}_1},{\boldsymbol{\mathrm{\rho}}_1};z} )+ {\psi^ \ast }({{\mathbf{r}_2},{\boldsymbol{\mathrm{\rho}}_2};z} )} ]} \right\rangle _m}{d^2}{\mathbf{r}_1}{d^2}{\mathbf{r}_2}, \end{aligned}$$
$$\begin{array}{l} {\left\langle {\exp [{\psi ({{\mathbf{r}_1},{\boldsymbol{\mathrm{\rho}}_1};z} )+ {\psi^ \ast }({{\mathbf{r}_2},{\boldsymbol{\mathrm{\rho}}_2};z} )} ]} \right\rangle _m}\\ = \exp \left\{ { - 2\pi {k^2}z\int_0^1 {dt} \int {{d^2}{\boldsymbol{\mathrm{\kappa}}_ \bot }} {\Phi _n}({{\boldsymbol{\mathrm{\kappa}}_ \bot },{\kappa_z} = 0} )\{{1 - \exp [{ - i({t{\boldsymbol{\mathrm{\rho}}_d} \cdot {\boldsymbol{\mathrm{\kappa}}_ \bot } + ({1 - t} ){\mathbf{r}_d} \cdot {\boldsymbol{\mathrm{\kappa}}_ \bot }} )} ]} \}} \right\} \end{array}$$
$$\begin{array}{l} {\left\langle {\exp [{\psi ({{\mathbf{r}_1},{\boldsymbol{\mathrm{\rho}}_1};z} )+ {\psi^ \ast }({{\mathbf{r}_2},{\boldsymbol{\mathrm{\rho}}_2};z} )} ]} \right\rangle _m}\\ = \exp \left\{ { - \frac{{2\pi {k^2}z}}{{{\mu_x}{\mu_y}}}\int_0^1 {dt} \int {d{{\kappa^{\prime}}_x}d{{\kappa^{\prime}}_y}{\Phi _n}(\boldsymbol{\mathrm{\kappa}} ^{\mathbf\prime}_ \bot)\{{1 - \exp [{ - i({t\boldsymbol{\mathrm{\rho}} ^{\mathbf\prime}_d \cdot \boldsymbol{\mathrm{\kappa}} ^{\mathbf\prime}_ \bot + ({1 - t} ){{\mathbf{r^{\prime}}}_d} \cdot {\boldsymbol{\mathrm{\kappa}}_ \bot }}} )} ]} \}} \right\} \end{array}$$
$$\begin{array}{l} {\left\langle {\exp [{\psi ({{\mathbf{r}_1},{\boldsymbol{\mathrm{\rho}}_1};z} )+ {\psi^ \ast }({{\mathbf{r}_2},{\boldsymbol{\mathrm{\rho}}_2};z} )} ]} \right\rangle _m}\\ \textrm{ = }\exp \left\{ { - \frac{{\textrm{4}{\pi^\textrm{2}}{k^2}z}}{{{\mu_x}{\mu_y}}}\int_0^1 {dt} \int_0^\infty {\kappa^{\prime}d\kappa^{\prime}{\Phi _n}({{\boldsymbol{\mathrm{\kappa}}_ \bot }^\prime } )[{1 - {J_\textrm{0}}({|{t{{\mathbf{\rho^{\prime}}}_d} + ({1 - t} ){{\mathbf{r^{\prime}}}_d}} |\boldsymbol{\mathrm{\kappa}} ^{\mathbf\prime}} )} ]} } \right\}, \end{array}$$
$$\begin{array}{r} {\left\langle {\exp [{\psi ({{\mathbf{r}_1},{\boldsymbol{\mathrm{\rho}}_1};z} )+ {\psi^ \ast }({{\mathbf{r}_2},{\boldsymbol{\mathrm{\rho}}_2};z} )} ]} \right\rangle _m} = \exp \left\{ { - \frac{T}{{{\mu_x}{\mu_y}}}({\mathbf{\rho^{\prime}}_d^2 + \mathbf{r^{\prime}}_d^2 + \boldsymbol{\mathrm{\rho}} ^{\mathbf\prime}_d \cdot \mathbf{r^{\prime}}} )} \right\}\\ \textrm{ } = \exp \left[ { - T{\mu_z}\left( {\frac{{\rho_{xd}^2}}{{\mu_x^2}} + \frac{{\rho_{yd}^2}}{{\mu_y^2}} + \frac{{x_d^2}}{{\mu_x^2}} + \frac{{y_d^2}}{{\mu_y^2}} + \frac{{{\rho_{xd}}{x_d}}}{{\mu_x^2}} + \frac{{{\rho_{yd}}{y_d}}}{{\mu_y^2}}} \right)} \right]. \end{array}$$
$$\scalebox{0.9}{$\begin{array}{l} {W_z}({{\boldsymbol{\mathrm{\rho}}_1},{\boldsymbol{\mathrm{\rho}}_2};z} )= \frac{{A{\pi ^\textrm{2}}}}{{{\lambda ^\textrm{2}}{z^2}}}\exp \left[ {\frac{{ik}}{{2z}}({\boldsymbol{\mathrm{\rho}}_1^2 - \boldsymbol{\mathrm{\rho}}_2^2} )} \right]\exp [{ - {T_x}{{({{\rho_{1x}} - {\rho_{2x}}} )}^2} - {T_y}{{({{\rho_{1y}} - {\rho_{2y}}} )}^2}} ]\\ \;\;\;\;\times \exp \left( {\frac{{\Delta _{x1}^2}}{{4({{N_\textrm{1}}\textrm{ + }{T_x}} )}} + \frac{{\Delta _{y1}^2}}{{4({{N_\textrm{1}} + {T_y}} )}}} \right)\exp \left( {\frac{{\Delta _{x3}^2}}{{4{N_\textrm{2}}}} + \frac{{\Delta _{y3}^2}}{{4{N_3}}}} \right)\\ \;\;\;\;\times \sum\limits_{c1 = 0}^l {\sum\limits_{c2 = 0}^l {\sum\limits_{p1 = 0}^{l - c1} {\sum\limits_{p\textrm{2} = 0}^{l - c1 - p\textrm{1}} {\sum\limits_{p3 = 0}^{c1} {\sum\limits_{p4 = 0}^{c1 - p3} {\sum\limits_{m1 = 0}^{[{p1/2} ]} {\sum\limits_{m\textrm{2} = 0}^{[{p\textrm{2}/2} ]} {\sum\limits_{m3 = 0}^{[{p3/2} ]} {\sum\limits_{m4 = 0}^{[{p4/2} ]} {\sum\limits_{p5 = 0}^{\alpha 1} {\sum\limits_{m5 = 0}^{[{({\alpha 1 - p5} )/2} ]} {{{({2i} )}^{ - ({\alpha 1 + \alpha 2\textrm{ + }l} )}}{{({ - 1} )}^{m1 + m2 + m3 + m4 + m5}}} } } } } } } } } } } } \\ \;\;\;\;\times \frac{1}{{{2^{l - ({p1 + p3} )/2 + \alpha 1/2}}}}{({{N_\textrm{1}} + {T_x}} )^{({c1 - l - 1} )/2}}{({{N_\textrm{1}} + {T_y}} )^{ - ({c1 + 1} )/2}}N_2^{ - ({\alpha 1 + 1} )/2}N_3^{ - ({1 + \alpha 2} )/2}\\ \;\;\;\;\times \left( {\begin{array}{c} {l - c1}\\ {p1} \end{array}} \right)\left( {\begin{array}{c} {l - c1 - p\textrm{1}}\\ {p\textrm{2}} \end{array}} \right)\left( {\begin{array}{c} {c1}\\ {p3} \end{array}} \right)\left( {\begin{array}{c} {c1 - p3}\\ {p4} \end{array}} \right)\left( {\begin{array}{c} {\alpha 1}\\ {p5} \end{array}} \right)\\ \;\;\;\;\times \frac{{l!{i^{c1}}}}{{c1!({l - c1} )!}}\frac{{l!{{({ - i} )}^{c2}}}}{{c2!({l - c2} )!}}\frac{{p1!}}{{m1!({p1 - 2m1} )!}}\frac{{p2!}}{{m2!({p2 - 2m2} )!}}\\ \;\;\;\;\times \frac{{p3!}}{{m3!({p3 - 2m3} )!}}\frac{{p4!}}{{m4!({p4 - 2m4} )!}}\frac{{({\alpha 1 - p5} )!}}{{m5!({\alpha 1 - p5 - 2m5} )!}}\\ \;\;\;\;\times {\left( {\frac{{\sqrt 2 i{\Delta _{x2}}}}{{\sqrt {{N_\textrm{1}}\textrm{ + }{T_x}} }}} \right)^{p1 - 2m1}}{\left( {\frac{{2k{\mu_0}}}{{\sqrt {{N_\textrm{1}} + {T_y}} }}} \right)^{p4 - 2m4}}{\left( { - \frac{{2k{\mu_0}}}{{\sqrt {{N_\textrm{1}}\textrm{ + }{T_x}} }}} \right)^{p2 - 2m2}}{\left( {\frac{{\sqrt 2 i{\Delta _{y2}}}}{{\sqrt {{N_\textrm{1}} + {T_y}} }}} \right)^{p3 - 2m3}}{\left( {\frac{{\sqrt 2 i{\Delta _{x\textrm{4}}}}}{{\sqrt {{N_\textrm{2}}} }}} \right)^{\alpha 1 - p5 - 2m5}}\\ \;\;\;\;\times {H_{l - c1 - p\textrm{1} - p\textrm{2}}}\left( {\frac{{i{\Delta _{x\textrm{1}}}}}{{\sqrt {{N_\textrm{1}}\textrm{ + }{T_x}} }}} \right){H_{c1 - p3 - p4}}\left( {\frac{{i{\Delta _{y1}}}}{{\sqrt {{N_\textrm{1}} + {T_y}} }}} \right){H_{p5}}\left( {\frac{{i{\Delta _{x\textrm{3}}}}}{{\sqrt {2{N_\textrm{2}}} }}} \right){H_{\alpha 2}}\left( {\frac{{i{\Delta _{y\textrm{3}}}}}{{\textrm{2}\sqrt {{N_3}} }}} \right), \end{array}$}$$
$$\begin{array}{l} {T_x}\textrm{ = }\frac{{T{\mu _z}}}{{\mu _x^2}}\textrm{, }{T_y} = \frac{{T{\mu _z}}}{{\mu _y^2}},\textrm{ }{\Delta _{x\textrm{1}}}\textrm{ = } - \frac{{ik{\rho _{1x}}}}{z} - {T_x}{\rho _{xd}}\textrm{, }{\Delta _{x2}} = \frac{1}{{\delta _0^2}} + 2{T_x}\textrm{,}\\ {\Delta _{x\textrm{3}}}\textrm{ = }\frac{{{\Delta _{x1}}{\Delta _{x2}}}}{{2({{N_\textrm{1}}\textrm{ + }{T_x}} )}} - \frac{{ik{\mu _0}{\Delta _{y1}}}}{{2({{N_\textrm{1}} + {T_y}} )}} + \frac{{ik{\rho _{2x}}}}{z} + {T_x}{\rho _{xd}}\textrm{, }{\Delta _{x\textrm{4}}} = \frac{{ik{\mu _0}{\Delta _{x2}}}}{{2({{N_\textrm{1}}\textrm{ + }{T_x}} )}} - \frac{{ik{\mu _0}{\Delta _{y2}}}}{{2({{N_\textrm{1}} + {T_y}} )}},\\ {\Delta _{y\textrm{1}}}\textrm{ = } - \frac{{ik{\rho _{1y}}}}{z} - {T_y}{\rho _{yd}}\textrm{, }{\Delta _{y2}} = \frac{1}{{\delta _0^2}} + 2{T_y}\textrm{,}\\ {\Delta _{y\textrm{3}}}\textrm{ = }\frac{{{\Delta _{y1}}{\Delta _{y2}}}}{{2({{N_\textrm{1}} + {T_y}} )}} + \frac{{ik{\mu _0}{\Delta _{x\textrm{1}}}}}{{2({{N_\textrm{1}}\textrm{ + }{T_x}} )}} + \frac{{{\Delta _{x\textrm{3}}}{\Delta _{x\textrm{4}}}}}{{2{N_\textrm{2}}}} + \frac{{ik{\rho _{2y}}}}{z} + {T_y}{\rho _{yd}},\\ {N_\textrm{1}}\textrm{ = }\left( {\frac{1}{{4\sigma_0^2}} + \frac{1}{{2\delta_0^2}} - \frac{{ik}}{{2z}}} \right)\textrm{, }{N_\textrm{2}}\textrm{ = }N_\textrm{1}^\mathrm{\ast} + {T_x} - \frac{{\Delta _{x2}^2}}{{4({{N_\textrm{1}}\textrm{ + }{T_x}} )}} + \frac{{{k^2}\mu _0^2}}{{4({{N_\textrm{1}} + {T_y}} )}}\textrm{,}\\ {N_3}\textrm{ = }N_\textrm{1}^\mathrm{\ast } + {T_y} - \frac{{\Delta _{x4}^2}}{{4{N_\textrm{2}}}} - \frac{{\Delta _{y2}^2}}{{4({{N_\textrm{1}} + {T_y}} )}} + \frac{{{k^2}\mu _0^2}}{{4({{N_\textrm{1}}\textrm{ + }{T_x}} )}},\\ {\alpha _\textrm{1}} = l + p1 + p4 - c2 - 2m1 - 2m4,\\ {\alpha _2} = c2 + p2 + p3 + \alpha 1 - 2m2 - 2m3 - p5 - 2m5. \end{array}$$
$$\int_{ - \infty }^\infty {{x^\alpha }} \exp [{ - {{({x - \beta } )}^2}} ]dx = {({2i} )^{ - \alpha }}\sqrt \pi {H_\alpha }({i\beta } ),$$
$${H_\alpha }({x + \beta } )= \frac{1}{{{2^{\alpha /2}}}}\sum\limits_{p = 0}^\alpha {\left( {\begin{array}{c} \alpha \\ p \end{array}} \right)} {H_p}\left( {\sqrt 2 x} \right){H_{\alpha - p}}\left( {\sqrt 2 \beta } \right),$$
$${H_n}({{x_1}} )= \sum\limits_{m = 0}^{[{n/2} ]} {{{({ - 1} )}^m}\frac{{n!}}{{m!({n - 2m} )!}}{{({2{x_1}} )}^{n - 2m}}} .$$
$$E(\rho ,\varphi ,z) = \frac{1}{{\sqrt {2\pi } }}\sum\limits_{p = 0}^\infty {\sum\limits_{m ={-} \infty }^\infty {{a_{pm}}{R_p}(\rho ,z){e^{im\varphi }}} } ,$$
$$\begin{aligned} {C_m} = \sum\limits_{p = 0}^\infty {\left\langle {{{|{{a_{pm}}} |}^2}} \right\rangle } &= \frac{1}{{2\pi }}\int\limits_0^\infty {\int\limits_0^\infty {\int\limits_0^{2\pi } {\int\limits_0^{2\pi } {\left\langle {E({\rho_1},{\varphi_1},z){E^\ast }({\rho_2},{\varphi_2},z)} \right\rangle {e^{ - im{\varphi _1} + im{\varphi _2}}}} } } } \\ &\textrm{ } \times \sum\limits_{p = 0}^\infty {{R_p}({\rho _1},z)} R_p^\ast ({\rho _2},z){\rho _1}{\rho _2}d{\rho _1}d{\rho _2}d{\varphi _1}d{\varphi _2}, \end{aligned}$$
$${C_m} = \frac{1}{{2\pi }}\int\limits_0^\infty {\int\limits_0^{2\pi } {\int\limits_0^{2\pi } {W(\rho ,{\varphi _1},\rho ,{\varphi _2};z){e^{ - im{\varphi _1} + im{\varphi _2}}}} } } \rho d\rho d{\varphi _1}d{\varphi _2}.$$
$${\Phi _n}(\boldsymbol{\mathrm{\kappa}} )= 0.033C_n^2\frac{{{\mu _x}{\mu _y}{\mu _z}\exp [{ - ({\mu_x^2\kappa_x^2 + \mu_y^2\kappa_y^2 + \mu_z^2\kappa_z^2} )/\kappa_m^2} ]}}{{{{({\mu_x^2\kappa_x^2 + \mu_y^2\kappa_y^2 + \mu_z^2\kappa_z^2 + \kappa_0^2} )}^{11/6}}}},$$
$$T = 0.0033{\pi ^2}{k^2}zC_n^2[{\eta \kappa_m^{ - 5/3}\exp ({\kappa_0^2/\kappa_m^2} ){\Gamma _1}({1/6,\kappa_0^2/\kappa_m^2} )- 2\kappa_0^{1/3}} ],$$
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