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Research for propagation properties of LG beam through Cassegrain antenna system in a turbulent atmosphere

Yan Qin, Huajun Yang, Ping Jiang, Weinan Caiyang, Miaofang Zhou, Shengqian Mao, and Biao Cao

Open Access Open Access

Abstract

In this paper, the propagation properties of the Laguerre-Gaussian (LG) beam passing through Cassegrain antenna system in a turbulent atmosphere have been researched. The accurate analytical function for the diffraction field of LG beam passing through Cassegrain antenna, the average intensity, and the cross-talk among different orbital angular momentum (OAM) modes of LG beam passing through Cassegrain antenna in Kolmogorov turbulent have been derived. The simulation results show that LG beam with ring-like distribution is selected to enhance the emission efficiency of Cassegrain antenna. The cross-talk among different OAM modes can remarkably reduce through using Cassegrain antenna. The analysis process can also apply to accurately analyzing the propagation properties of other kinds of beams through different optical systems in turbulent atmosphere.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. Introduction

The Cassegrain antenna system, which has advantages such as large aperture, wideband, small volume, and elimination of aberration, are widely used in free-space optical communication system [1,2]. However, the central part of the incident beam which is reflected by the secondary mirror of the Cassegrain antenna cannot reach the primary mirror, which remarkably reduces the emission efficiency. Some new antenna structures have been designed to reduce the blockage loss and improve the emission efficiency [3–5]. Although these designs are able to reduce even theoretically avoid blockage loss, the complicate structure makes them hard to apply in practice. Compared with designing complicate antenna structures, choosing a new beam like Laguerre-Gaussian (LG) beam to replace Gaussian beam is a more convenient method for avoiding blockage loss, which is demonstrated in [6]. Besides, using LG beam that carries OAM modes for space-optical communication has other significant advantages such as improving data transmission capacity, increasing spectral efficiency, and decreasing inter-beam cross-talk [7,8]. Because the vortex beams are inevitably disturbed when propagation in the turbulent atmosphere, and the initial OAM state may be destroyed and may spread to the adjacent OAM state [9], researchers have been studying the cross-talk among different OAM modes of various vortex beams passing through the atmosphere turbulence, and fruitful results have been proposed [10–13]. Some reduce the cross-talk methods are proposed, such as adopt the adaptive optics [14], using a focusing mirror [15]. However, to the best of our knowledge, current researches only focus on the propagation properties of the vortex beam directly transmitting in turbulence atmosphere without the optical antenna. Some other researches only concentrate on analysis of the diffraction characteristic of the beams passing through the optical antenna in free space without atmosphere turbulence and simplified the antenna system during the calculate. Compare with these papers, this paper studies a more comprehensive and more practical issue: the propagation properties of the LG beam passing through the Cassegrain antenna system in the turbulence atmosphere. In general, this paper has two purposes: a) Improving the transmission efficiency of Cassegrain antenna and reduce the cross-talk among different OAM modes by making Cassegrain antenna and vortex LG beam complement each other; b) Providing a theoretical model that can describe the propagation properties of the LG beam passing through the Cassegrain antenna system in the turbulence atmosphere. And the theoretical model in this paper also can be used to analyze the propagation properties of other beams passing through optical systems in the turbulence atmosphere.

2. Theoretical derivation

The LG beam passing through a Cassegrain antenna system in turbulent atmosphere is illustrated in Fig. 1.

 figure: Fig. 1.

Fig. 1. Schematic of the LG beam passing through a Cassegrain antenna system in the turbulent atmosphere.

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Figure 1 shows that the LG beam is placed before the Cassegrain antenna, then the beam passing through the antenna and transmitting in the turbulent atmosphere. There are many OAM states that can be detected at the receiver plane, although the incident beam carries only one OAM state at the source plane. The central part of the beam reflects by the secondary mirror does not all reach the primary mirror (as shown in the green line), which will reduce the emission efficiency of the antenna. The distance between the beam source and the secondary mirror is L1, and the distance between the primary mirror and the receiver plane is L2.

2.1 Diffraction of the LG beam passing through Cassegrain antenna

In the cylindrical coordinates system, the LG beam at the source plane can be expressed as [16]

$${E_\textrm{0}}({{r_0},{\varphi_0},0} )= {\left( {\sqrt 2 \frac{{{r_0}}}{{{\omega_0}}}} \right)^{{l_\textrm{I}}}}L_p^{{l_\textrm{I}}}\left( {\frac{{2r_0^2}}{{\omega_0^2}}} \right)\exp\left( { - \frac{{r_0^2}}{{\omega_0^2}}} \right)\exp({ - \textrm{i}{l_\textrm{I}}{\varphi_0}} )$$
where r0 and φ0 are the radial and azimuthal coordinates, ω0 is the waist width, $L^{l_1}_p(\cdot)$ is the Laguerre polynomials, lI is the topological charge, and p is the radial index.

Under the framework of paraxial approximation, the complex amplitude of the LG beam passing through the Cassegrain antenna system at the receiver plane can be obtained with the Collins integral [17]

$$\begin{array}{ll} {E_1}({{r_1},{\varphi_1},{z_1}} )&= \frac{\textrm{i}}{{\lambda B}}\exp ({ - \textrm{i}kz} )\int_0^{2\pi } {\int_0^\infty {{E_0}({{r_0},{\varphi_0},0} )t({{r_0};a,b} )} } \\ &\textrm{ } \times \exp \left\{ { - \frac{{\textrm{i}k}}{{2B}}[{Ar_0^2 + Dr_1^2 - 2{r_0}{r_1}\cos ({{\varphi_1} - {\varphi_0}} )} ]} \right\}{r_0}\textrm{d}{r_0}\textrm{d}{\varphi _0} \end{array}$$
where λ is the wavelength in vacuum, k = 2π/λ is the wave number, the A, B, C, and D are the parameters for the ABCD transmission matrix of the Cassegrain antenna. The ABCD transmission matrix is [18]
$$T = \left( {\begin{array}{cc} A&B\\ C&D \end{array}} \right) = \left( {\begin{array}{cc} {\frac{{{f_1}}}{{{f_2}}}}&{\frac{{{f_1}{L_1}}}{{{f_2}}} + \frac{{{f_2}{L_2}}}{{{f_1}}} + {f_1} - {f_2}}\\ 0&{\frac{{{f_2}}}{{{f_1}}}} \end{array}} \right)$$
where f1 and f2 are the focal length of the primary and the secondary mirror, t (r0; a, b) is the aperture function which can be expressed as a finite sum of a Gaussian function [19]:
$$t({{r_0},a,b} )= \sum\limits_{w = 1}^M {{A_w}\left[ {\exp\left( { - \frac{{{B_w}}}{{{b^2}}}r_0^2} \right) - \exp\left( { - \frac{{{B_w}}}{{{a^2}}}r_0^2} \right)} \right]} $$
where a and b are the radius of the secondary and primary mirror, M is the number of Gaussian terms, Aw and Bw are the expansion coefficients and Gaussian coefficients, respectively.

Substituting Eq. (1) and Eq. (4) into Eq. (2), the optical field of the LG beam after passing through the Cassegrain antenna system can be expressed as follow:

$$\begin{array}{ll} {E_1}({{r_1},{\varphi_1},{z_1}} )&= \frac{{\textrm{i}k}}{{2B}}{\left( {\frac{{\textrm{i}k{r_1}}}{{\sqrt 2 {\omega_0}B}}} \right)^{{l_\textrm{I}}}}\exp ({ - \textrm{i}k{z_1}} )\exp({ - \textrm{i}{l_\textrm{I}}{\varphi_1}} )\exp \left( { - \frac{{\textrm{i}kD}}{{2B}}r_1^2} \right)\\ &\textrm{ } \times \left\{ \begin{array}{l} \sum\limits_{w = 1}^M {{A_w}M_w^{ - {l_\textrm{I}} - p - 1}{{({{M_w} - 2W} )}^p}\exp \left( { - \frac{{{k^2}r_1^2}}{{4{B^2}{M_w}}}} \right)L_p^{{l_\textrm{I}}}\left[ {\frac{{{k^2}r_1^2}}{{2\omega_0^2{B^2}{M_w}({2W - {M_w}} )}}} \right]} \\ - \sum\limits_{w = 1}^M {{A_w}m_w^{ - {l_\textrm{I}} - p - 1}{{({{m_w} - 2W} )}^p}\exp \left( { - \frac{{{k^2}r_1^2}}{{4{B^2}{m_w}}}} \right)L_p^{{l_\textrm{I}}}\left[ {\frac{{{k^2}r_1^2}}{{2\omega_0^2{B^2}{m_w}({2W - {m_w}} )}}} \right]} \end{array} \right\} \end{array}$$
where $W=1/{\omega}^2_0$, Mw=W + Bw/b2+iAk/(2B), mw=W + Bw/a2+iAk/(2B). The following relations [20] are used to obtain Eq. (5)
$$\exp ({\textrm{i}\alpha \cos \theta } )= \sum\limits_{n ={-} \infty }^\infty {{\textrm{i}^n}{J_n}(\alpha )\exp ({ - \textrm{i}n\theta } )} $$
$$\frac{1}{{2\pi }}\int_0^{2\pi } {\exp ({ - \textrm{i}n\theta } )} \textrm{d}\theta = \left\{ {\begin{array}{c} 1\\ 0 \end{array}} \right.\textrm{ }\begin{array}{c} {n = 0}\\ {n \ne 0} \end{array}$$
$$\int_0^\infty {{x^{v + 1}}\exp ({ - \beta {x^2}} )L_n^v({\alpha {x^2}} ){J_v}({xy} )\textrm{d}x = {2^{ - v - 1}}{\beta ^{ - v - n - 1}}{{({\beta - \alpha } )}^n}{y^v}\exp \left( { - \frac{{{y^2}}}{{4\beta }}} \right)L_n^v\left[ {\frac{{\alpha {y^2}}}{{4\beta ({\alpha - \beta } )}}} \right]} $$
The emission efficiency of the Cassegrain antenna can be expressed as:
$$\eta = \frac{{\int_0^{2\pi } {\int_0^{ + \infty } {{I_1}({{r_1},{\varphi_1},{z_1}} ){r_1}\textrm{d}{r_1}\textrm{d}{\varphi _1}} } }}{{\int_0^{2\pi } {\int_0^{ + \infty } {{I_0}({{r_0},{\varphi_0},0} ){r_0}\textrm{d}{r_0}\textrm{d}{\varphi _0}} } }}$$
where $I_{0}\left(r_{0}, \varphi_{0}, 0\right)=E_{0}\left(r_{0}, \varphi_{0}, 0\right) E_{0}^{*}\left(r_{0}, \varphi_{0}, 0\right)$ is the intensity of the LG beam at the source plane, $I_{1}\left(r_{1}, \varphi_{1}, z_{1}\right)=E_{1}\left(r_{1}, \varphi_{1}, z_{1}\right) E_{1}^{*}\left(r_{1}, \varphi_{1}, z_{1}\right)$ is the intensity of diffraction field of the LG beam propagating through the Cassegrain antenna system, and the asterisk denotes the complex conjugation.

2.2 LG beam passing through the Cassegrain antenna system in a turbulent atmosphere

The LG modes could expand into a finite sum of Hermite-Gaussian modes, and the LG beam at the source plane can be expressed as [21]

$${E_0}({{x_0},{y_0},0} )= \frac{{{{({ - 1} )}^p}}}{{{2^{2p + {l_\textrm{I}}}}p!}}\sum\limits_{t = 0}^p {\sum\limits_{s = 0}^{{l_\textrm{I}}} {{i^s}\left( {\begin{array}{c} p\\ t \end{array}} \right)\left( {\begin{array}{c} {{l_\textrm{I}}}\\ s \end{array}} \right)} } {H_{2t + {l_I} - s}}\left( {\frac{{\sqrt 2 {x_0}}}{{{\omega_0}}}} \right){H_{2p - 2t + s}}\left( {\frac{{\sqrt 2 {y_0}}}{{{\omega_0}}}} \right)\exp \left( { - \frac{{{x_0}^2 + {y_0}^2}}{{\omega_0^2}}} \right)$$
where H(x) is the Hermite polynomial.

The average intensity distribution at the receiver plane can be expressed as [22]

$$\begin{array}{l} I\left\langle {({{x_1},{y_1},{z_1}} )} \right\rangle = \frac{{{k^2}}}{{4{\pi ^2}{B^2}}}\int_0^\infty {\int_0^\infty {\int_0^\infty {\int_0^\infty {{E_0}({{x_0},{y_0},0} )t({{x_0},{y_0};a,b} )E_0^ \ast ({{{x^{\prime}}_0},{{y^{\prime}}_0},0} ){t^ \ast }({{{x^{\prime}}_0},{{y^{\prime}}_0};a,b} )} } } } \;\\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; \times \exp \left\{ { - \frac{{\textrm{i}k}}{{2B}}[{A({x_0^2 + y_0^2} )- A({x^{\prime 2}_0 + y^{\prime 2}_0} )- 2({{x_0}{x_1} + {y_0}{y_1}} )+ 2({{{x^{\prime}}_0}{x_1} + {{y^{\prime}}_0}{y_1}} )} ]} \right\}\\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \times \left\langle {\exp [{\psi ({{x_0},{y_0},{z_1}} )+ {\psi^ \ast }({{{x^{\prime}}_0},{{y^{\prime}}_0},{z_1}} )} ]} \right\rangle \textrm{d}{x_0}\textrm{d}{y_0}\textrm{d}{{x^{\prime}}_0}\textrm{d}{{y^{\prime}}_0} \end{array}$$
where the 〈·〉 indicates the ensemble average, 〈exp[ψ(x0, y0, z1)+ ψ*(x0′, y0′, z1)]〉 denotes the ensemble average over the medium statistics covering the log-amplitude and phase fluctuations due to the turbulent atmosphere, which can be expressed as [22]
$$\left\langle {\exp [{\psi ({{x_0},{y_0},{z_1}} )+ {\psi^ \ast }({{{x^{\prime}}_0},{{y^{\prime}}_0},{z_1}} )} ]} \right\rangle = \exp \left[ { - \frac{{{{({{x_0} - {{x^{\prime}}_0}} )}^2}}}{{\rho_0^2}} - \frac{{{{({{y_0} - {{y^{\prime}}_0}} )}^2}}}{{\rho_0^2}}} \right]$$
where ρ0 is the coherence length in turbulent atmosphere. The spatial coherence length ρ0 of Kolmogorov turbulence is as following [8]:
$${\rho _0} = {({0.545C_n^2{k^2}{z_1}} )^{ - 3/5}}$$
where $C^2_n$ is the structure constant with units m−2/3.

The t (x0, y0; a, b) is the aperture function under the Cartesian coordinate system which can be expressed as:

$$t({{x_0},{y_0};a,b} )= \sum\limits_{w = 1}^M {{A_w}\left\{ {\exp \left[ { - \frac{{{B_w}}}{{{b^2}}}({x_0^2 + y_0^2} )} \right] - \exp \left[ { - \frac{{{B_w}}}{{{a^2}}}({x_0^2 + y_0^2} )} \right]} \right\}} $$
Substituting Eq. (10), Eq. (12) and Eq. (14) into Eq. (11), the average intensity at the receiver plane of the LG beam passing through a Cassegrain antenna system in the turbulent atmosphere can be written as follow:
$$\begin{array}{l} I\left\langle {({{x_1},{y_1},{z_1}} )} \right\rangle = \frac{{{k^2}}}{{4{B^2}}}\frac{{{{({ - 1} )}^{2p}}}}{{{2^{4p + 2{l_\textrm{I}}}}{{({p!} )}^2}}}\sum\limits_{{t_1} = 0}^p {\sum\limits_{{t_2} = 0}^p {\sum\limits_{{s_1} = 0}^{{l_\textrm{I}}} {\sum\limits_{{s_2} = 0}^{{l_\textrm{I}}} {\sum\limits_{{m_1} = 0}^{\left[ {\frac{{2{t_1} + {l_\textrm{I}} - {s_1}}}{2}} \right]} {\sum\limits_{{m_2} = 0}^{\left[ {\frac{{2p - 2{t_1} + {s_1}}}{2}} \right]} {\sum\limits_{{m_3} = 0}^{\left[ {\frac{{2{t_2} + {l_\textrm{I}} - {s_2}}}{2}} \right]} {\sum\limits_{{m_4} = 0}^{\left[ {\frac{{2p - 2{t_2} + {s_2}}}{2}} \right]} {\sum\limits_{{k_1} = 0}^{2{t_1} + {l_\textrm{I}} - {s_1} - 2{m_1}} {\sum\limits_{{k_2} = 0}^{2p - 2{t_1} + {s_1} - 2{m_2}} {\sum\limits_{{n_1} = 0}^{\left[ {\frac{{{k_1}}}{2}} \right]} {} } } } } } } } } } } \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \times \sum\limits_{{n_2} = 0}^{\left[ {\frac{{{k_2}}}{2}} \right]} {\sum\limits_{w = 1}^M {\sum\limits_{v = 1}^M {{A_w}A_v^ \ast \left( {\begin{array}{c} p\\ {{t_1}} \end{array}} \right)\left( {\begin{array}{c} p\\ {{t_2}} \end{array}} \right)\left( {\begin{array}{c} {{l_\textrm{I}}}\\ {{s_1}} \end{array}} \right)\left( {\begin{array}{c} {{l_\textrm{I}}}\\ {{s_2}} \end{array}} \right)\left( {\begin{array}{c} {2{t_1} + {l_\textrm{I}} - {s_1} - 2{m_1}}\\ {{k_1}} \end{array}} \right)\left( {\begin{array}{c} {2p - 2{t_1} + {s_1} - 2{m_2}}\\ {{k_2}} \end{array}} \right)\frac{{{k_1}!}}{{{n_1}!({{k_1} - 2{n_1}} )!}}} } \;\;} \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \times {\kern 1pt} \frac{{{k_2}!}}{{{n_2}!({{k_2} - 2{n_2}} )!}}\frac{{({2{t_1} + {l_\textrm{I}} - {s_1}} )!}}{{{m_1}!({2{t_1} + {l_\textrm{I}} - {s_1} - 2{m_1}} )!}}\frac{{({2p - 2{t_1} + {s_1}} )!}}{{{m_2}!({2p - 2{t_1} + {s_1} - 2{m_2}} )!}}\frac{{({2{t_2} + {l_\textrm{I}} - {s_2}} )!}}{{{m_3}!({2{t_2} + {l_\textrm{I}} - {s_2} - 2{m_3}} )!}}\\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \times \frac{{({2p - 2{t_2} + {s_2}} )!}}{{{m_4}!({2p - 2{t_2} + {s_2} - 2{m_4}} )!}}{({ - 1} )^{{m_1} + {m_2} + {m_3} + {m_4} + {n_1} + {n_2} + {s_2}}}{\textrm{i}^{2{m_1} + 2{m_2} + 2{m_3} + 2{m_4} + {s_1} + {s_2} - 2{l_\textrm{I}} - 4p}}{\left( {\frac{1}{{\rho_0^2}}} \right)^{{k_1} + {k_2} - 2{n_1} - 2{n_2}}}{\kern 1pt} \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \times {\sqrt 2 ^{{k_1} + {k_2} - 2{n_1} - 2{n_2} - 2{m_3} - 2{m_4} + {l_\textrm{I}} + 2p}}{\left( {\frac{1}{{{\omega_0}}}} \right)^{2{l_\textrm{I}} + 4p - 2{m_1} - 2{m_2} - 2{m_3} - 2{m_4}}}\{{{G_1} - {G_2} - {G_3} + {G_4}} \}\\ \textrm{ } \end{array}$$
where
$$\begin{array}{l} {G_j} = {\left( {\frac{1}{{{M_j}}}} \right)^{2p + {l_\textrm{I}} + 2 - 2{m_1} - 2{m_2} + {k_1} + {k_2} - 2{n_1} - 2{n_2}}}{\left( {\frac{1}{{{N_j}}}} \right)^{2p + {l_\textrm{I}} + 2 - 2{m_3} - 2{m_4} + {k_1} + {k_2} - 2{n_1} - 2{n_2}}}\exp \left[ {{{\left( {\frac{{P{x_1}}}{{{N_j}M_j^2\rho_0^2}} - \frac{{P{x_1}}}{{{N_j}}}} \right)}^2}} \right]\\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \times \textrm{exp}\left[ {{{\left( {\frac{{P{y_1}}}{{{N_j}M_j^2\rho_0^2}} - \frac{{P{y_1}}}{{{N_j}}}} \right)}^2}} \right]\exp \left[ {{{\left( {\frac{P}{{{M_j}}}} \right)}^2}({x_1^2 + y_1^2} )} \right]{H_{2{t_2} + {l_\textrm{I}} - {s_2} - 2{m_3} + {k_1} - 2{n_1}}}\left( {\frac{{Q{x_1}}}{{\sqrt 2 {N_j}}} - \frac{{Q{x_1}}}{{\sqrt 2 {N_j}M_j^2\rho_0^2}}} \right)\\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \times {H_{2p - 2{t_2} + {s_2} - 2{m_4} + {k_2} - 2{n_2}}}\left( {\frac{{Q{y_1}}}{{\sqrt 2 {N_j}}} - \frac{{Q{y_1}}}{{\sqrt 2 {N_j}M_j^2\rho_0^2}}} \right){H_{2{t_1} + {l_\textrm{I}} - {s_1} - 2{m_1} - {k_1}}}\left( { - \frac{{Q{x_1}}}{{{M_j}}}} \right){H_{2p - 2{t_1} + {s_1} - 2{m_2} - {k_2}}}\left( { - \frac{{Q{y_1}}}{{{M_j}}}} \right){\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \end{array}\textrm{ ,}\quad\quad ({j = 1,2,3,4} ),$$
and $P = \frac{{\textrm{i}k}}{{\textrm{2}B}}$, $Q = \frac{k}{{\sqrt 2} B}$, ${{M}_{1}}{ = }{{M}_{2}}{ = }\sqrt {\frac{{1}}{{{\omega }_{0}^{2}}}{ + PA + }\frac{{1}}{{{\rho }_{0}^{2}}}{ + }\frac{{{{B}_{w}}}}{{{{b}^{2}}}}} $, ${{M}_{3}}{ = }{{M}_{4}}{ = }\sqrt {\frac{{1}}{{{\omega }_{0}^{2}}}{ + PA + }\frac{{1}}{{{\rho }_{0}^{2}}}{ + }\frac{{{{B}_{w}}}}{{{{a}^{2}}}}} $, ${{N}_{1}}{ = }\sqrt {\frac{{1}}{{{\omega }_{0}^{2}}} - \frac{{1}}{{{M}_{1}^{2}{\rho }_{0}^{4}}} - {PA + }\frac{{1}}{{{\rho }_{0}^{2}}}{ + }\frac{{{B}_{v}^\ast }}{{{{b}^{2}}}}} $, ${{N}_{2}}{ = }\sqrt {\frac{{1}}{{{\omega }_{0}^{2}}} - \frac{{1}}{{{M}_{2}^{2}{\rho }_{0}^{4}}} - {PA + }\frac{{1}}{{{\rho }_{0}^{2}}}{ + }\frac{{{B}_{v}^\ast }}{{{{a}^{2}}}}} $, $\; {{N}_{3}}{ = }\sqrt {\frac{{1}}{{{\omega }_{0}^{2}}} - \frac{{1}}{{{M}_{3}^{2}{\rho }_{0}^{4}}} - {PA + }\frac{{1}}{{{\rho }_{0}^{2}}}{ + }\frac{{{B}_{v}^\ast }}{{{{b}^{2}}}}} $, $\; {{N}_{4}}{ = }\sqrt {\frac{{1}}{{{\omega }_{0}^{2}}} - \frac{{1}}{{{M}_{4}^{2}{\rho }_{0}^{4}}} - {PA + }\frac{{1}}{{{\rho }_{0}^{2}}}{ + }\frac{{{B}_{v}^\ast }}{{{{a}^{2}}}}} $.

The following relations [20] are used to obtain Eq. (15)

$${H_n}(x )= \sum\limits_{m = 0}^{\left[ {\frac{n}{2}} \right]} {{{({ - 1} )}^m}\frac{{n!}}{{m!({n - 2m} )!}}{{({2x} )}^{n - 2m}}} $$
$$\int_0^\infty {{x^n}\exp [{ - {{({x - \beta } )}^2}} ]\textrm{d}x = {{({2\textrm{i}} )}^{ - n}}\sqrt \pi {H_n}({\textrm{i}\beta } )} $$
$${H_n}({x + y} )= \frac{1}{{{2^{{\raise0.7ex\hbox{$n$} \!\mathord{\left/ {\vphantom {n 2}} \right.}\!\lower0.7ex\hbox{$2$}}}}}}\sum\limits_{k = 0}^n {\left( {\begin{array}{c} n\\ k \end{array}} \right){H_k}\left( {\sqrt 2 x} \right){H_{n - k}}\left( {\sqrt 2 y} \right)} $$

2.3 Cross-talk among different OAM modes

The cross-talk among different OAM modes at the receiver plane can be described by the transmission probability spectrum of OAM mode which can be expressed as [15]:

$${P_{{l_\textrm{I}},{l_\textrm{R}}}} = \frac{{{\omega _{{l_\textrm{I}},{l_\textrm{R}}}}}}{{\sum\nolimits_{l^{\prime} ={-} \infty }^{l^{\prime} = \infty } {{\omega _{{l_\textrm{I}},{{l^{\prime}}_\textrm{R}}}}} }}$$
where the lI is the initial topological charge at the source plane, lR and l′R is the received topological charge at receiver plane, and ωlI, lRis the mode weight for each OAM mode at the receiver plane, is defined as [15]:
$${\omega _{{l_\textrm{I}},{l_\textrm{R}}}} = \int_0^{{R_0}} {\left\langle {{{|{{a_{{l_I},{l_R}}}({{r_1},{z_1}} )} |}^2}} \right\rangle } {r_1}\textrm{d}{r_1}$$
where R0 is the radius of the receiving aperture, 〈alI,lR(r1, z1)〉 is the probability density of the OAM mode of the LG beam which is expressed as [15]:
$$\begin{array}{ll} \left\langle {{{|{{a_{{l_\textrm{I}},{l_\textrm{R}}}}({{r_1},{z_1}} )} |}^2}} \right\rangle &= \frac{1}{{2\pi }}\int_0^{2\pi } {\int_0^{2\pi } {\left\langle {{E_1}({{r_1},{\varphi_1},{z_1}} )E_1^\ast ({{r_1},{{\varphi^{\prime}}_1},{z_1}} )} \right\rangle } } \exp [{ - \textrm{i}l({{{\varphi^{\prime}}_1} - {\varphi_1}} )} ]\\ &\textrm{ } \times \left\langle {\exp [{\psi ({{r_1},{\varphi_1},{z_1}} )+ {\psi^\ast }({{r_1},{{\varphi^{\prime}}_1},z{}_1} )} ]} \right\rangle \textrm{d}{\varphi _1}\textrm{d}{{\varphi ^{\prime}}_1} \end{array}$$
where
$${\kern 1pt} \left\langle {\exp [{\psi ({{r_1},{\varphi_1},{z_1}} )+ {\psi^\ast }({{r_1},{{\varphi^{\prime}}_1},z{}_1} )} ]} \right\rangle = \exp \left[ { - \frac{{2r_1^2 - 2r_1^2\cos ({{\varphi_1} - {{\varphi^{\prime}}_1}} )}}{{\rho_0^2}}} \right]$$
We substitute Eqs. (5) and (22) into Eq. (21), the probability density of the OAM mode of the LG beam been expressed as
$$\begin{array}{l} \left\langle {{{|{{a_{{l_\textrm{I}},{l_\textrm{R}}}}({{r_1},{z_1}} )} |}^2}} \right\rangle = 2\pi {\left( {\frac{k}{{2B}}} \right)^2}{\left( {\frac{{k{r_1}}}{{\sqrt 2 {\omega_0}B}}} \right)^{2{l_\textrm{I}}}}\exp \left( { - \frac{{2r_1^2}}{{\rho_0^2}}} \right){I_{{l_\textrm{I}} - {l_\textrm{R}}}}\left( {\frac{{2r_1^2}}{{\rho_0^2}}} \right)\\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \times \sum\limits_{w = 1}^M {\sum\limits_{v = 1}^M {\left\{ \begin{array}{l} {E_w}E_v^ \ast \exp [{ - ({{D_w} + D_v^ \ast } )r_1^2} ]L_p^{{l_\textrm{I}}}({{F_w}r_1^2} )L_p^{{l_\textrm{I}}}({F_v^ \ast r_1^2} )\\ - {E_w}e_v^ \ast \exp [{ - ({{D_w} + d_v^ \ast } )r_1^2} ]L_p^{{l_\textrm{I}}}({{F_w}r_1^2} )L_p^{{l_\textrm{I}}}({f_v^ \ast r_1^2} )\\ - {e_w}E_v^ \ast \exp [{ - ({{d_w} + D_v^ \ast } )r_1^2} ]L_p^{{l_\textrm{I}}}({{f_w}r_1^2} )L_p^{{l_\textrm{I}}}({F_v^ \ast r_1^2} )\\ + {e_w}e_v^ \ast \exp [{ - ({{d_w} + d_v^ \ast } )r_1^2} ]L_p^{{l_\textrm{I}}}({{f_w}r_1^2} )L_p^{{l_\textrm{I}}}({f_v^ \ast r_1^2} )\end{array} \right\}} } \end{array}$$
where ${ D}_{ w}{ = }\displaystyle{{{ k}^{ 2}} \over {{ 4}{ B}^{ 2}{ M}_{ w}}}$, ${{d}_{w}}{ = }\frac{{{{k}^{2}}}}{{{4}{{B}^{2}}{{m}_{w}}}}$, ${D}_{v}^{\ast }{ = }\frac{{{{k}^{2}}}}{{{4}{{B}^{2}}{M}_{v}^{\ast }}}$, ${d}_{v}^{\ast }{ = }\frac{{{{k}^{2}}}}{{{4}{{B}^{2}}{m}_{v}^{\ast }}}$, ${ M}_{ w}{ = }\displaystyle{{ 1} \over {{ \omega }_{ 0}^{ 2} }}{ + }\displaystyle{{{ B}_{ w}} \over {{ b}^{ 2}}}{ + }\displaystyle{{{ iAk}} \over {{ 2B}}}$, ${ m}_{ w}{ = }\displaystyle{{ 1} \over {{ \omega }_{ 0}^{ 2} }}{ + }\displaystyle{{{ B}_{ w}} \over {{ a}^{ 2}}}{ + }\displaystyle{{{ iAk}} \over {{ 2B}}}$, $\; { M}_{ v}^{ *} { = }\displaystyle{{ 1} \over {{ \omega }_{ 0}^{ 2} }}{ + }\displaystyle{{{ B}_{ v}^{ *} } \over {{ b}^{ 2}}}-\displaystyle{{{ iAk}} \over {{ 2B}}}$, ${m}_{v}^{\ast }{ = }\frac{{1}}{{{\omega }_{0}^{2}}}{ + }\frac{{{B}_{v}^{\ast }}}{{{{a}^{2}}}} - \frac{{{iAk}}}{{{2B}}}$, ${{E}_{w}}{ = }{{A}_{w}}{M}_{w}^{{ - p - }{{l}_{I}}{ - 1}}{\left( {{{M}_{w}} - \frac{{2}}{{{\omega }_{0}^{2}}}} \right)^{p}}$, ${{e}_{w}}{ = }{{A}_{w}}{m}_{w}^{{ - p - }{{l}_{I}}{ - 1}}{\left( {{{m}_{w}} - \frac{{2}}{{{\omega }_{0}^{2}}}} \right)^{p}}$, ${E}_{v}^{\ast }{ = A}_{v}^{\ast }{M}_{v}^{{\ast{-} p - }{{l}_{I}}{ - 1}}{\left( {{M}_{v}^{\ast } - \frac{{2}}{{{\omega }_{0}^{2}}}} \right)^{p}}$, ${e}_{v}^{\ast }{ = A}_{v}^{\ast }{m}_{v}^{{\ast{-} p - }{{l}_{I}}{ - 1}}{\left( {{m}_{v}^{\ast } - \frac{{2}}{{{\omega }_{0}^{2}}}} \right)^{p}}$, ${{F}_{w}}{ = }\frac{{{{k}^{2}}}}{{{2\omega }_{0}^{2}{{B}^{2}}{{M}_{w}}\left( {\frac{{2}}{{{\omega }_{0}^{2}}} - {{M}_{w}}} \right)}}$, ${{f}_{w}}{ = }\frac{{{{k}^{2}}}}{{{2\omega }_{0}^{2}{{B}^{2}}{{m}_{w}}\left( {\frac{{2}}{{{\omega }_{0}^{2}}} - {{m}_{w}}} \right)}}$, ${F}_{v}^{\ast }{ = }\frac{{{{k}^{2}}}}{{{2\omega }_{0}^{2}{{B}^{2}}{M}_{v}^{\ast }\left( {\frac{{2}}{{{\omega }_{0}^{2}}} - {M}_{v}^{\ast }} \right)}}$, ${f}_{v}^{\ast }{ = }\frac{{{{k}^{2}}}}{{{2\omega }_{0}^{2}{{B}^{2}}{m}_{v}^{\ast }\left( {\frac{{2}}{{{\omega }_{0}^{2}}} - {m}_{v}^{\ast }} \right)}}$.

The following relation [20] are used to obtain Eq. (23)

$$\int_0^{2\pi } {\exp [{ - {i}l\theta + 2\beta \cos ({\theta - \varphi } )} ]} {d}x = 2\pi \exp ({ - {i}l\varphi } ){I_l}({2\beta } )$$

3. Numerical results and discussion

In this section, the propagation properties of the LG beam passing through the Cassegrain antenna system in the turbulent atmosphere have been analyzed. The parameters of the Cassegrain antenna and beam source are set as following: f1=30 cm, f2=6 cm, the distance between the primary mirror and secondary mirror is 24 cm [18], ω0=10 mm, λ=1550 nm. The structure constant $\; {C}_{n}^{2}$=5×10−15 m−2/3 [8].

The emission efficiency of the Cassegrain antenna is calculated with the help of Eq. (9). The previous study shows that the size of the primary and secondary mirrors has a significant influence on antenna emission efficiency [6,18]. We study the variation of antenna emission efficiency with varying the size of the primary and secondary mirrors to adopt appropriate parameters for the following study. The Cassegrain antenna emission efficiency as a function of the ratio a/b for different lI, which is shown in Fig. 2.

 figure: Fig. 2.

Fig. 2. Emission efficiency of the Cassegrain antenna as a function of ratios a/b for different lI with b = 25 mm and z = 100 m.

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Figure 2 shows that, for the LG beam, with the ratio a/b increasing, the Cassegrain antenna emission efficiency almost unchanged and then decreases. With the value of topological charge increases, the Cassegrain antenna emission efficiency rises first and then falls under the same ratio a/b. This changing trend of the Cassegrain antenna emission efficiency is consistent with the previous research results [6]. This phenomenon occurs because different parameters of the primary and secondary mirror ratio will cause the different secondary mirror occlusion ratio, and more beams would be blocked with the radius of the secondary mirror increasing under the same central dark spot radius of the LG beam. The previous studying shows that with the topological charge increasing, the central dark spot radius of the LG beam increasing [23]. Under the larger topological charge, more beams would be bypass with the radius of the secondary mirror increasing. With the topological charge increase further, part of beam spot exceeds the outer dimensions of the Cassegrain antenna system, and it causes the emission efficiency of the Cassegrain antenna decreases. Therefore, determining the parameters of the primary and secondary mirror based on the radius of central dark spot of LG beam can greatly improve the transmission efficiency of the Cassegrain antenna.

The antenna parameters are selected as a = 5 mm and b = 25 mm for the following research, and the emission efficiency of the antenna is 91.07%.

Figure 3 shows the intensity and phase distribution of the LG beam passing through the Cassegrain antenna with different transmission distances which is calculated with the help of Eq. (5). Figure 3(a) shows the intensity distribution of the LG beam passing through the Cassegrain antenna under different transmission distances in the x-o-z plane. The chromaticity bar represents the intensity and phase distribution change of each of the graphs from Fig. 3(a) to Fig. 3(c5).

 figure: Fig. 3.

Fig. 3. Intensity and phase distribution of the LG beam passing through the Cassegrain antenna when a = 5 mm, b = 25 mm, p = 0, lI=1 with different transmission distance: (a) in x-o-z plane, (b1) - (c5) in x-o-y plane and (b1), (c1), z = 0 m, (b2), (c2) z = 10 m, (b3), (c3) z = 100 m, (b4), (c4) z = 1000 m, (b5), (c5) z = 10000 m.

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Figure 3(a) shows that due to the diffraction effect, the beam radius increases significantly and the intensity of the beam decrease with the transmission distance increase. Figure 3(b1) – 3(b5) and Figs. 3(c1)–3(c5) show that with the transmission distance increase, the diffraction fringe and the phase distribution also change. Figs. 3(b1) and 3(b2) show that the radius of the beam increasing significantly when the LG beam passing through the Cassegrain antenna. When the transmission distance is 100 m, the center of the LG beam has diffraction rings as shown in Fig. 3(b3). With the transmission distance increases, the outside of the LG beam has some diffraction rings as shown in Fig. 3(b4). And with the transmission distance increases further, the diffraction rings become blurred as shown in Fig. 3(b5).

The average intensity distribution of the LG beam passing through the Cassegrain antenna in the turbulent atmosphere is calculated with the help of Eq. (15). Figures 4(a1)–4(c) show the average intensity distribution of the LG beam passing through the Cassegrain antenna in the turbulent atmosphere and free space with different transmission distances. In Fig. 4, the average intensity distribution of the LG beam passing through the Cassegrain antenna in the turbulent atmosphere is shown in Figs. 4(b1), 4(b2) and 4(b3), in free space as shown in Figs. 4(a1), 4(a2) and 4(a3). The chromaticity bar represents the intensity distribution change based on Fig. 4(a1). In Fig. 4(c), the solid curves show the LG beam passing through the Cassegrain antenna in the turbulent atmosphere, and the dash curves show the LG beam passing through the Cassegrain antenna in free space.

 figure: Fig. 4.

Fig. 4. Average intensity distribution of the LG beam passing through the Cassegrain antenna in the free space (a1) – (a3), in the turbulent atmosphere (b1) – (b2), and the average intensity distribution curve (c) with different transmission distance when lI=1.

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Figures 4(a1)–4(b3) show that with the transmission distance increases, the radius of the LG beam passing through the Cassegrain antenna expanding both in the turbulence atmosphere and in free space. Compare with transmission in free space, the turbulence atmosphere does not change the LG beam average intensity distribution trend, but it makes the beam shape becomes blurred. Figure 4(c) shows that compare with the LG beam passing through the Cassegrain antenna in free space, the turbulent atmosphere makes the beam become flatter, the average intensity of beam decreases, and the central dark spot appears beam distribution.

The cross-talk among different OAM modes at the receiver plane is calculated with the help of Eq. (19). Figure 5 shows the received OAM mode transmission probability spectrum of the LG beam transmission in turbulence atmosphere with different transmission distances. Figure 5(a) shows the received OAM mode transmission probability spectrum of the LG beam transmitting with the Cassegrain antenna in turbulence atmosphere, and Fig. 5(b) shows the received OAM mode transmission probability spectrum of the LG beam transmitting without the Cassegrain antenna in turbulence atmosphere. The initial topological charge lI=1, the received topological charge lR=1, 2, 3, 4, 5 with the radius of the receiving aperture R0=0.1 m.

 figure: Fig. 5.

Fig. 5. The received OAM modes transmission probability spectrum of the LG beam transmitting with (a) and without (b) Cassegrain antenna in turbulent atmosphere.

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Figure 5 shows that when transmitting a single topological charge, many OAM modes can be detected on the receiver plane. The variation trend of received OAM modes transmission probability spectrum of the LG beam transmitting with Cassegrain antenna and without Cassegrain antenna in turbulent atmosphere is different. The received OAM modes transmission probability spectrum decreases with the transmission distance increase both with and without Cassegrain antenna when lI=lR=1. The detection probability at the receiver plane of using the Cassegrain antenna and not using the Cassegrain antenna is 64.99% and 51.83% with the transmission distance is 1 km when lI=lR=1. The difference of detection probability of received OAM mode for lR=1 to 5 is much the same when the LG beam transmitting without Cassegrain antenna in turbulent atmosphere with the transmission distance is 10 km.

The received OAM mode transmission probability spectrum difference between with and without Cassegrain antenna (DP=Pwith-Pwithout) as shown in Fig. 6.

 figure: Fig. 6.

Fig. 6. The difference of the received OAM mode transmission probability spectrum between with and without the Cassegrain antenna (DP=Pwith-Pwithout) when R0 = 0.1 m.

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Figure 6 shows that the DP is greater than zero when the transmission distance greater than 0.7 km, it means that the OAM mode crosstalk is reduced through using the Cassegrain antenna when the transmission distance is long. The reason for this phenomenon may be the beam radius will increase significantly when the LG beam passing through the Cassegrain antenna. When the transmission distance is increasing further, using the Cassegrain antenna can reduce the divergence angle of the beam which compared with not using Cassegrain antenna. The DP is greater than 10% when the transmission distance between 1 km to 4.5 km with lI=lR=1. This simulation result shows that the crosstalk reduction significant through using the Cassegrain antenna in long-distance communication transmission.

Figure 7 shows the received OAM mode transmission probability spectrum of the LG beam transmission in turbulence atmosphere with different receiver aperture radius R0 when the transmission distance is 10 km. Figures 7(a) and 7(c) are the LG beam transmitting in turbulent atmosphere with Cassegrain antenna, Figs. 7(b) and 7(d) are the LG beam transmitting in turbulent atmosphere without Cassegrain antenna. In Figs. 7(a) and 7(b), receiver aperture radius R0=0.1 m, in Figs. 7(c) and 7(d), receiver aperture radius R0=1 m. The initial topological charge set as lI=0 to 4, and the received topological charge lR=−4 to 4.

 figure: Fig. 7.

Fig. 7. The received OAM mode transmission probability spectrum with different value of the receiver aperture radius R0=(a), (b) 0.1 m, (c), (d) 1 m, (a) (c) with Cassegrain antenna, (b) (d) without Cassegrain antenna when z = 10 km.

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Figure 7 shows that through using the Cassegrain antenna and under the receiver aperture radius R0 = 0.1 m [as shown in Fig. 7(a)], the difference of the detection probability for receiver OAM mode is larger with the same initial OAM mode, it suggests that the cross-talk is weaker. When the LG beam transmits without Cassegrain antenna and R0=1 m [as shown in Fig. 7(d)], the detection probabilities of received different OAM modes are almost the same under the same initial OAM mode, which suggests that it is hard to separate different topological charges at receiver plane. The difference of the detection probability for receiver OAM mode goes down when the value of the initial topological charge increasing, it means the cross-talk of OAM mode at the receiver plane increasing. The simulation result shows that the value of the receiver aperture radius also influences the cross-talk of OAM modes at receiver plane.

Under the difference receiver aperture radius, the DP are shown in Fig. 8.

 figure: Fig. 8.

Fig. 8. The DP when the transmission distance is 1 km with different aperture radius R0.

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Figure 8 shows that the detection probability of using the Cassegrain antenna still large than not using the Cassegrain antenna under different receiver aperture radius and topological charges. When the receiver aperture radius large than 0.036 m, the DP will decrease with topological charges increase. When the value of the receiver aperture radius is in the range of 0.05 to 0.12 m, the DP is increasing with the receiver aperture radius increase. If the receiver aperture radius large than 0.12 m, the DP remains almost unchanged. The simulation result shows that using the Cassegrain antenna not only decrease the cross-talk among different OAM modes but also facilitates a more flexible selection of the receiving screen.

4. Conclusions

In this paper, the propagation properties of the LG beam passing through a Cassegrain antenna system in turbulence atmosphere have been researched. With the transmission distance increase, the optical field of the LG beam after passing through the Cassegrain antenna will first appear diffraction fringe, and the diffraction fringe becomes fuzzy with transmission distance increase further. Through using the LG beam and adopting appropriate parameters of the Cassegrain antenna, the emission efficiency of the Cassegrain antenna enhances. By the use of Cassegrain antenna, the cross-talk among different OAM modes at the receiver plane decreases significantly when transmitting over a long distance with the emission efficiency of the Cassegrain antenna is 91.07%. The receiver aperture radius also influences the received OAM mode transmission probability spectrum. By comparison, using the Cassegrain antenna can make the selection of the receiver screen more flexible. The analysis process proposed in this paper can also apply to analyze the propagation properties of other kinds of beams through different optical systems in turbulence atmosphere.

Funding

National Natural Science Foundation of China (11574042, 61271167).

Disclosures

The authors declare no conflicts of interest.

References

1. J. Y. Guo, P. Jiang, H. J. Yang, Y. Niu, and J. W. Xianyu, “Transmitting characteristic analysis of antenna with off-axial parabolic rotating surfaces configuration,” Appl. Opt. 56(9), 2455–2461 (2017). [CrossRef]  

2. M. Y. Yu, P. Jiang, H. J. Yang, Y. C. Huang, X. J. Ma, and B. Wang, “Improving transmission efficiency of Cassegrain antenna,” Optik 126(7-8), 795–798 (2015). [CrossRef]  

3. W. N. Caiyang, H. J. Yang, P. Jiang, W. S. He, Y. Tian, and X. Chen, “Cassegrain antenna with a semitransparent secondary reflector,” Appl. Opt. 56(17), 5080–5085 (2017). [CrossRef]  

4. L. Zhang, L. Chen, H. J. Yang, P. Jiang, S. Q. Mao, and W. Caiyang, “Hyperbola–parabola primary mirror in Cassegrain optical antenna to improve transmission efficiency,” Appl. Opt. 54(24), 7148–7153 (2015). [CrossRef]  

5. M. F. Zhou, H. J. Yang, P. Jiang, Y. Qin, W. N. Caiyang, S. Q. Mao, and B. Cao, “Structure design of a conic lens pair for improving the transmission efficiency of a Cassegrain antenna,” Appl. Opt. 58(13), 3410–3417 (2019). [CrossRef]  

6. X. Ke, J. Wang, M. Wang, and Z. Tan, “Diffraction characteristics of a Laguerre-Gaussian beam through a Maksutov-Cassegrain optical system,” Appl. Opt. 57(10), 2570–2576 (2018). [CrossRef]  

7. A. E. Willner, H. Huang, Y. Yan, Y. Ren, N. Ahmed, G. Xie, C. Bao, L. Li, Y. Cao, Z. Zhao, J. Wang, M. P. J. Lavery, M. Tur, S. Ramachandran, A. F. Molisch, N. Ashrafi, and S. Ashrafi, “Optical communications using orbital angular momentum beams,” Adv. Opt. Photonics 7(1), 66–106 (2015). [CrossRef]  

8. Y. Yuan, D. Liu, Z. Zhou, H. Xu, J. Qu, and Y. Cai, “Optimization of the probability of orbital angular momentum for Laguerre-Gaussian beam in Kolmogorov and non-Kolmogorov turbulence,” Opt. Express 26(17), 21861–21871 (2018). [CrossRef]  

9. C. Paterson, “Atmospheric turbulence and orbital angular momentum of single photons for optical communication,” Phys. Rev. Lett. 94(15), 153901 (2005). [CrossRef]  

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

11. Y. Zhou, Y. Yuan, J. Qu, and W. Huang, “Propagation properties of Laguerre-Gaussian correlated Schell-model beam in non-Kolmogorov turbulence,” Opt. Express 24(10), 10682–10693 (2016). [CrossRef]  

12. X. Yan, L. Guo, M. Cheng, J. Li, Q. Huang, and R. Sun, “Probability density of orbital angular momentum mode of autofocusing Airy beam carrying power-exponent-phase vortex through weak anisotropic atmosphere turbulence,” Opt. Express 25(13), 15286–15298 (2017). [CrossRef]  

13. Y. Yuan, T. Lei, S. Gao, X. Weng, L. Du, and X-C. Yuan, “The orbital angular momentum spreading for cylindrical vector beams in turbulent atmosphere,” IEEE Photonics J. 9(2), 1–10 (2017). [CrossRef]  

14. N. Li, X. Chu, P. Zhang, X. Feng, C. Fan, and C. Qiao, “Compensation for the orbital angular momentum of a vortex beam in turbulent atmosphere by adaptive optics,” Opt. Laser Technol. 98, 7–11 (2018). [CrossRef]  

15. M. Y. Zhou, Y. Q. Zhou, and G. F. Wu, “Reducing the cross-talk among different orbital angular momentum modes in turbulent atmosphere by using a focusing mirror,” Opt. Express 27(7), 10280–10287 (2019). [CrossRef]  

16. F. J. Salgado-Remacha, “Laguerre-Gaussian beam shaping by binary phase plates as illumination sources in micro-optics,” Appl. Opt. 53(29), 6782–6788 (2014). [CrossRef]  

17. S. A. Collins, “Lens-system diffraction integral written in terms of matrix optics,” J. Opt. Soc. Am. 60(9), 1168–1177 (1970). [CrossRef]  

18. M. Y. Yu, H. J. Yang, P. Jiang, Y. Zhang, L. Chen, and S. Q. Mao, “On-axial defocused characteristic analysis for Cassegrain antenna in optical communication,” Optik 127(4), 1734–1737 (2016). [CrossRef]  

19. J. J. Wen and M. A. Breazeale, “A diffraction beam field expressed as the superposition of Gaussian beams,” J. Acoust. Soc. Am. 83(5), 1752–1756 (1988). [CrossRef]  

20. I. S. Gradshteyn and I. M. Ryzhik, Table of integrals, series, and products, 6th ed. (Academic, 2000).

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

22. X. Chu and G. Zhou, “Power coupling of a two-Cassegrain-telescopes system in turbulent atmosphere in a slant path,” Opt. Express 15(12), 7697–7707 (2007). [CrossRef]  

23. Y. Ohtake, T. Ando, N. Fukuchi, N. Matsumoto, H. Ito, and T. Hara, “Universal generation of higher-order multiringed Laguerre-Gaussian beams by using a spatial light modulator,” Opt. Lett. 32(11), 1411–1413 (2007). [CrossRef]  

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Figures (8)
Fig. 1.
Fig. 1. Schematic of the LG beam passing through a Cassegrain antenna system in the turbulent atmosphere.

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Fig. 2.
Fig. 2. Emission efficiency of the Cassegrain antenna as a function of ratios a/b for different lI with b = 25 mm and z = 100 m.

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Fig. 3.
Fig. 3. Intensity and phase distribution of the LG beam passing through the Cassegrain antenna when a = 5 mm, b = 25 mm, p = 0, lI=1 with different transmission distance: (a) in x-o-z plane, (b1) - (c5) in x-o-y plane and (b1), (c1), z = 0 m, (b2), (c2) z = 10 m, (b3), (c3) z = 100 m, (b4), (c4) z = 1000 m, (b5), (c5) z = 10000 m.

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Fig. 4.
Fig. 4. Average intensity distribution of the LG beam passing through the Cassegrain antenna in the free space (a1) – (a3), in the turbulent atmosphere (b1) – (b2), and the average intensity distribution curve (c) with different transmission distance when lI=1.

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Fig. 5.
Fig. 5. The received OAM modes transmission probability spectrum of the LG beam transmitting with (a) and without (b) Cassegrain antenna in turbulent atmosphere.

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Fig. 6.
Fig. 6. The difference of the received OAM mode transmission probability spectrum between with and without the Cassegrain antenna (DP=Pwith-Pwithout) when R0 = 0.1 m.

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Fig. 7.
Fig. 7. The received OAM mode transmission probability spectrum with different value of the receiver aperture radius R0=(a), (b) 0.1 m, (c), (d) 1 m, (a) (c) with Cassegrain antenna, (b) (d) without Cassegrain antenna when z = 10 km.

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Fig. 8.
Fig. 8. The DP when the transmission distance is 1 km with different aperture radius R0.

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Equations (25)

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E 0 ( r 0 , φ 0 , 0 ) = ( 2 r 0 ω 0 ) l I L p l I ( 2 r 0 2 ω 0 2 ) exp ⁡ ( − r 0 2 ω 0 2 ) exp ⁡ ( − i l I φ 0 )
E 1 ( r 1 , φ 1 , z 1 ) = i λ B exp ⁡ ( − i k z ) ∫ 0 2 π ∫ 0 ∞ E 0 ( r 0 , φ 0 , 0 ) t ( r 0 ; a , b )   × exp ⁡ { − i k 2 B [ A r 0 2 + D r 1 2 − 2 r 0 r 1 cos ⁡ ( φ 1 − φ 0 ) ] } r 0 d r 0 d φ 0
T = ( A B C D ) = ( f 1 f 2 f 1 L 1 f 2 + f 2 L 2 f 1 + f 1 − f 2 0 f 2 f 1 )
t ( r 0 , a , b ) = ∑ w = 1 M A w [ exp ⁡ ( − B w b 2 r 0 2 ) − exp ⁡ ( − B w a 2 r 0 2 ) ]
E 1 ( r 1 , φ 1 , z 1 ) = i k 2 B ( i k r 1 2 ω 0 B ) l I exp ⁡ ( − i k z 1 ) exp ⁡ ( − i l I φ 1 ) exp ⁡ ( − i k D 2 B r 1 2 )   × { ∑ w = 1 M A w M w − l I − p − 1 ( M w − 2 W ) p exp ⁡ ( − k 2 r 1 2 4 B 2 M w ) L p l I [ k 2 r 1 2 2 ω 0 2 B 2 M w ( 2 W − M w ) ] − ∑ w = 1 M A w m w − l I − p − 1 ( m w − 2 W ) p exp ⁡ ( − k 2 r 1 2 4 B 2 m w ) L p l I [ k 2 r 1 2 2 ω 0 2 B 2 m w ( 2 W − m w ) ] }
exp ⁡ ( i α cos ⁡ θ ) = ∑ n = − ∞ ∞ i n J n ( α ) exp ⁡ ( − i n θ )
1 2 π ∫ 0 2 π exp ⁡ ( − i n θ ) d θ = { 1 0   n = 0 n ≠ 0
∫ 0 ∞ x v + 1 exp ⁡ ( − β x 2 ) L n v ( α x 2 ) J v ( x y ) d x = 2 − v − 1 β − v − n − 1 ( β − α ) n y v exp ⁡ ( − y 2 4 β ) L n v [ α y 2 4 β ( α − β ) ]
η = ∫ 0 2 π ∫ 0 + ∞ I 1 ( r 1 , φ 1 , z 1 ) r 1 d r 1 d φ 1 ∫ 0 2 π ∫ 0 + ∞ I 0 ( r 0 , φ 0 , 0 ) r 0 d r 0 d φ 0
E 0 ( x 0 , y 0 , 0 ) = ( − 1 ) p 2 2 p + l I p ! ∑ t = 0 p ∑ s = 0 l I i s ( p t ) ( l I s ) H 2 t + l I − s ( 2 x 0 ω 0 ) H 2 p − 2 t + s ( 2 y 0 ω 0 ) exp ⁡ ( − x 0 2 + y 0 2 ω 0 2 )
I ⟨ ( x 1 , y 1 , z 1 ) ⟩ = k 2 4 π 2 B 2 ∫ 0 ∞ ∫ 0 ∞ ∫ 0 ∞ ∫ 0 ∞ E 0 ( x 0 , y 0 , 0 ) t ( x 0 , y 0 ; a , b ) E 0 ∗ ( x ′ 0 , y ′ 0 , 0 ) t ∗ ( x ′ 0 , y ′ 0 ; a , b ) × exp ⁡ { − i k 2 B [ A ( x 0 2 + y 0 2 ) − A ( x 0 ′ 2 + y 0 ′ 2 ) − 2 ( x 0 x 1 + y 0 y 1 ) + 2 ( x ′ 0 x 1 + y ′ 0 y 1 ) ] } × ⟨ exp ⁡ [ ψ ( x 0 , y 0 , z 1 ) + ψ ∗ ( x ′ 0 , y ′ 0 , z 1 ) ] ⟩ d x 0 d y 0 d x ′ 0 d y ′ 0
⟨ exp ⁡ [ ψ ( x 0 , y 0 , z 1 ) + ψ ∗ ( x ′ 0 , y ′ 0 , z 1 ) ] ⟩ = exp ⁡ [ − ( x 0 − x ′ 0 ) 2 ρ 0 2 − ( y 0 − y ′ 0 ) 2 ρ 0 2 ]
ρ 0 = ( 0.545 C n 2 k 2 z 1 ) − 3 / 5
t ( x 0 , y 0 ; a , b ) = ∑ w = 1 M A w { exp ⁡ [ − B w b 2 ( x 0 2 + y 0 2 ) ] − exp ⁡ [ − B w a 2 ( x 0 2 + y 0 2 ) ] }
I ⟨ ( x 1 , y 1 , z 1 ) ⟩ = k 2 4 B 2 ( − 1 ) 2 p 2 4 p + 2 l I ( p ! ) 2 ∑ t 1 = 0 p ∑ t 2 = 0 p ∑ s 1 = 0 l I ∑ s 2 = 0 l I ∑ m 1 = 0 [ 2 t 1 + l I − s 1 2 ] ∑ m 2 = 0 [ 2 p − 2 t 1 + s 1 2 ] ∑ m 3 = 0 [ 2 t 2 + l I − s 2 2 ] ∑ m 4 = 0 [ 2 p − 2 t 2 + s 2 2 ] ∑ k 1 = 0 2 t 1 + l I − s 1 − 2 m 1 ∑ k 2 = 0 2 p − 2 t 1 + s 1 − 2 m 2 ∑ n 1 = 0 [ k 1 2 ] × ∑ n 2 = 0 [ k 2 2 ] ∑ w = 1 M ∑ v = 1 M A w A v ∗ ( p t 1 ) ( p t 2 ) ( l I s 1 ) ( l I s 2 ) ( 2 t 1 + l I − s 1 − 2 m 1 k 1 ) ( 2 p − 2 t 1 + s 1 − 2 m 2 k 2 ) k 1 ! n 1 ! ( k 1 − 2 n 1 ) ! × k 2 ! n 2 ! ( k 2 − 2 n 2 ) ! ( 2 t 1 + l I − s 1 ) ! m 1 ! ( 2 t 1 + l I − s 1 − 2 m 1 ) ! ( 2 p − 2 t 1 + s 1 ) ! m 2 ! ( 2 p − 2 t 1 + s 1 − 2 m 2 ) ! ( 2 t 2 + l I − s 2 ) ! m 3 ! ( 2 t 2 + l I − s 2 − 2 m 3 ) ! × ( 2 p − 2 t 2 + s 2 ) ! m 4 ! ( 2 p − 2 t 2 + s 2 − 2 m 4 ) ! ( − 1 ) m 1 + m 2 + m 3 + m 4 + n 1 + n 2 + s 2 i 2 m 1 + 2 m 2 + 2 m 3 + 2 m 4 + s 1 + s 2 − 2 l I − 4 p ( 1 ρ 0 2 ) k 1 + k 2 − 2 n 1 − 2 n 2 × 2 k 1 + k 2 − 2 n 1 − 2 n 2 − 2 m 3 − 2 m 4 + l I + 2 p ( 1 ω 0 ) 2 l I + 4 p − 2 m 1 − 2 m 2 − 2 m 3 − 2 m 4 { G 1 − G 2 − G 3 + G 4 }  
G j = ( 1 M j ) 2 p + l I + 2 − 2 m 1 − 2 m 2 + k 1 + k 2 − 2 n 1 − 2 n 2 ( 1 N j ) 2 p + l I + 2 − 2 m 3 − 2 m 4 + k 1 + k 2 − 2 n 1 − 2 n 2 exp ⁡ [ ( P x 1 N j M j 2 ρ 0 2 − P x 1 N j ) 2 ] × exp [ ( P y 1 N j M j 2 ρ 0 2 − P y 1 N j ) 2 ] exp ⁡ [ ( P M j ) 2 ( x 1 2 + y 1 2 ) ] H 2 t 2 + l I − s 2 − 2 m 3 + k 1 − 2 n 1 ( Q x 1 2 N j − Q x 1 2 N j M j 2 ρ 0 2 ) × H 2 p − 2 t 2 + s 2 − 2 m 4 + k 2 − 2 n 2 ( Q y 1 2 N j − Q y 1 2 N j M j 2 ρ 0 2 ) H 2 t 1 + l I − s 1 − 2 m 1 − k 1 ( − Q x 1 M j ) H 2 p − 2 t 1 + s 1 − 2 m 2 − k 2 ( − Q y 1 M j )  , ( j = 1 , 2 , 3 , 4 ) ,
H n ( x ) = ∑ m = 0 [ n 2 ] ( − 1 ) m n ! m ! ( n − 2 m ) ! ( 2 x ) n − 2 m
∫ 0 ∞ x n exp ⁡ [ − ( x − β ) 2 ] d x = ( 2 i ) − n π H n ( i β )
H n ( x + y ) = 1 2 n / n 2 2 ∑ k = 0 n ( n k ) H k ( 2 x ) H n − k ( 2 y )
P l I , l R = ω l I , l R ∑ l ′ = − ∞ l ′ = ∞ ω l I , l ′ R
ω l I , l R = ∫ 0 R 0 ⟨ | a l I , l R ( r 1 , z 1 ) | 2 ⟩ r 1 d r 1
⟨ | a l I , l R ( r 1 , z 1 ) | 2 ⟩ = 1 2 π ∫ 0 2 π ∫ 0 2 π ⟨ E 1 ( r 1 , φ 1 , z 1 ) E 1 ∗ ( r 1 , φ ′ 1 , z 1 ) ⟩ exp ⁡ [ − i l ( φ ′ 1 − φ 1 ) ]   × ⟨ exp ⁡ [ ψ ( r 1 , φ 1 , z 1 ) + ψ ∗ ( r 1 , φ ′ 1 , z 1 ) ] ⟩ d φ 1 d φ ′ 1
⟨ exp ⁡ [ ψ ( r 1 , φ 1 , z 1 ) + ψ ∗ ( r 1 , φ ′ 1 , z 1 ) ] ⟩ = exp ⁡ [ − 2 r 1 2 − 2 r 1 2 cos ⁡ ( φ 1 − φ ′ 1 ) ρ 0 2 ]
⟨ | a l I , l R ( r 1 , z 1 ) | 2 ⟩ = 2 π ( k 2 B ) 2 ( k r 1 2 ω 0 B ) 2 l I exp ⁡ ( − 2 r 1 2 ρ 0 2 ) I l I − l R ( 2 r 1 2 ρ 0 2 ) × ∑ w = 1 M ∑ v = 1 M { E w E v ∗ exp ⁡ [ − ( D w + D v ∗ ) r 1 2 ] L p l I ( F w r 1 2 ) L p l I ( F v ∗ r 1 2 ) − E w e v ∗ exp ⁡ [ − ( D w + d v ∗ ) r 1 2 ] L p l I ( F w r 1 2 ) L p l I ( f v ∗ r 1 2 ) − e w E v ∗ exp ⁡ [ − ( d w + D v ∗ ) r 1 2 ] L p l I ( f w r 1 2 ) L p l I ( F v ∗ r 1 2 ) + e w e v ∗ exp ⁡ [ − ( d w + d v ∗ ) r 1 2 ] L p l I ( f w r 1 2 ) L p l I ( f v ∗ r 1 2 ) }
∫ 0 2 π exp ⁡ [ − i l θ + 2 β cos ⁡ ( θ − φ ) ] d x = 2 π exp ⁡ ( − i l φ ) I l ( 2 β )
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