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Accurate analysis of the efficiency of Bessel Gauss beams passing through two Cassegrain optical antennas in atmospheric turbulence

Shunyuan Shang, Huajun Yang, and Ping Jiang

Open Access Open Access

Abstract

The Bessel Gauss beam has shown good performance in solving occlusion by the secondary mirror of Cassegrain antenna. In this work, the analytical expression for the optical field of the Bessel Gauss beam after passing through the optical communication system comprising two Cassegrain antennas in atmospheric turbulence is derived. The light filed is obtained more precisely by optimising the parameters of the hard-edged optical aperture. And the energy efficiency of the whole system is investigated more accurately taking into account the efficiency of two antennas and the reflection losses. For the 3 order Bessel Gauss beam, the optimal parameters of the system are obtained by calculation. When b = 0.1m, a = 0.0162 m, ηT of Bessel Gauss beams when l = 1 ∼ 5 are 64%, 91%, 96%, 96%, 96%, respectively. At the same time, the light field expressions we have derived allow us to easily analyze the effect of atmospheric turbulence and antenna defocus on the efficiency of the system. So the effect of turbulent atmosphere and antenna defocus on the efficiency of the system and the corresponding reasons are studied as well.

© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement

1. Introduction

Vortex beams have recently become an object of extensive research because of their good performance in free space optical communication systems. A vortex beam is a beam with a special transverse spatial distribution, with a circular distribution of zero intensity on its propagation axis. In addition, its isophase plane tends to spiral forward as it propagates [1]. The vortex light can carry both spin angular momentum (SAM) and orbital angular momentum (OAM) [2]. All the OAM modes of vortex light are orthogonal and there are theoretically an infinite number of modes, making its use in optical communications a good application for improving communication capacity and spectral efficiency compared to conventional techniques [3]. The Bessel Gauss (BG) beam is a vortex beam with self-correcting characteristics [4]. In recent years there has been a lot of research into the generation of BG beams, the generation of BG beams is no longer a problem and there are a variety of methods capable of generating BG beams [5]. The efficiency of the BG beam through the Cassegrain antenna has been studied and can reach an efficiency close to 1 when the range of hard-edge apertures is simply considered and reflection losses are not taken into account [6].

Therefore an accurate analysis of the efficiency of the whole system is necessary to investigate the potential problems or advantages of the application of BG beams in this system. It is also important to calculate the optimal parameters of the system and study the influence of the defocusing of the antenna and the atmospheric turbulence on the system for the application in practice. In this work, the resolved optical field is derived for a BG beam passing through two Cassegrain antennas in atmospheric turbulence. The range of the hard-edge optical aperture function has been optimized by analyzing the secondary mirror occlusion of the Cassegrain antenna, resulting in a more accurate optical field. The efficiency of the whole system is investigated after taking into account the efficiency of the transmitting and receiving antennas as well as the reflection losses of mirrors. The optimum parameters of the system are calculated for a BG beam with $l$ = 3 and on this basis the defocusing of the antenna and the effect of atmospheric turbulence on the system efficiency are investigated.

Our work demonstrates the feasibility of the use of BG beams in this optical system which has important implications for the selection and design of optical systems for vortex beams. Also the optical field expressions we derive are of great importance for research about BG beams in the future.

2. Theoretical model

Figure 1 is an overview diagram of the whole process, Fig. 2 shows the specific meaning of the primary mirror offset, Fig. 3 an overview diagram of the whole calculation process. The parameters of the hard-edged optical aperture are optimised by analysing the range that BG beams can pass through the antenna. The electric field distribution of $l$ order BG beam $E_1$ at the initial plane (z = 0) can be expressed as [7]:

$$E_1\left(r,\theta\right)\ =\ A_0\exp\left(-\frac{{\ r}^2}{\omega_0^2}\right)J_l\left(k_rr\right)\exp\left(-{\mathrm{i}} l\theta\right)$$
where $A_0$ is the constant defining beam energy, $\omega _0$ is the beam’s waist width, $k=2{\pi}/\lambda$ and $k_r=k\sin (\Gamma )$ are the free space wave number and radial wave number, respectively. The wave length and topological charge of the BG beam are represented by $\lambda$ and $l$, respectively.

 figure: Fig. 1.

Fig. 1. Overview of BG beam passing through the optical communication system comprising a Cassegrain transmitting antenna and a Cassegrain receiving antenna in atmospheric turbulence, $b$, $a$ are the radius of the primary and secondary mirrors, $\gamma$ = $b$/$a$ is obscuration rate, $L_1$ denotes the distance from the source plane to the secondary mirror of the transmitting antenna, $L_2$ denotes the distance between the main mirror of the transmitting and receiving antenna, $L_3$ denotes the distance between receiving antenna secondary mirror and receiving plane, $f_1$, $f_2$ denoting the focal length of the primary and secondary mirror, respectively.

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 figure: Fig. 2.

Fig. 2. An overview of the offset of the primary mirror of the antenna, where the dotted line is the position of the primary mirror in the ideal state, and the solid line is the position of the primary mirror after the offset. $\delta _1$ is the offset of the transmitting antenna and $\delta _2$ is the offset of the receiving antenna; (a) transmitting antenna; (b) receiving antenna.

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 figure: Fig. 3.

Fig. 3. Overview diagram of the entire calculation process and secondary mirror obscuration analysis of the Cassegrain antenna; (a) transmitting antenna; (b) receiving antenna.

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The distribution of the electric field after the BG beam passing through the transmitting antenna $E_2$ at the secondary mirror plane of the transmitting antenna ($z_1 = L_1$) can be expressed by the Collins formula [8,9]:

$$\begin{aligned}&E_2\left(\rho_1,\varphi_1\right) =\frac{{\mathrm{i}} k}{2\pi B_1} \exp\left(-{\mathrm{i}} kz_1\right)\int_{0}^{2\pi}\int_{0}^{\infty}H_1(r)E_{1}\left(r,\theta,0\right)\times\\ &\exp\left\{-\frac{{\mathrm{i}} k}{2B}_1\left[A_1r^2-2\rho_1 r\cos\left(\theta-\varphi_1\right)+D_1\rho_1^2\right]\right\} r{\mathrm{d}} r {\mathrm{d}}\theta\end{aligned}$$
The definition of the hard aperture function $H_1(\mathrm r)$ is:
$$H_1(\mathrm{r})=\left\{\begin{array}{ll} 1 & a_2\leq|r| \leq a \\ 0 & \text{others} \end{array}\right.$$
where the $a_2$ = $a f_2/f_1$ and $a$ are the minimum and maximum radius of the transmitting antenna that allows the beam to pass through. According to prior research by J. J. Wn et al., the hard-edged function can be stated in the calculation equation as a superposition of Gaussian functions [10]:
$$H_1({r})=\sum_{\alpha=1}^{M} A_{\alpha}\left[\exp \left(-\frac{B_{\alpha} r^{2}}{a^{2}}\right)-\exp \left(-\frac{B_{\alpha} r^{2}}{a_2^{2}}\right)\right]$$
The value of M is generally valued as 10. $A_{\alpha }$ and $B_{\alpha }$ are the expanded Gaussian coefficients [10]. $A_1$, $B_1$, and $D_1$ stand for the transmission matrix’s parameters for the Cassegrain transmitting antenna. The ABCD transmission matrix of the Cassegrain transmitting antenna can be expressed as [11]:
$$\left[\begin{array}{ll} A_1 & B_1 \\ C_1 & D_1 \end{array}\right]=\left[\begin{array}{ll} 1 & L_{2} \\ 0 & 1 \end{array}\right]\left[\begin{array}{cc} 1 & 0 \\ -\frac{1}{f_{1}} & 1 \end{array}\right]\left[\begin{array}{cc} 1 & f_{1}-f_{2}+\delta_1 \\ 0 & 1 \end{array}\right]\left[\begin{array}{cc} 1 & 0 \\ \frac{1}{f_{2}} & 1 \end{array}\right]\left[\begin{array}{cc} 1 & L_{1} \\ 0 & 1 \end{array}\right]$$
where $\delta _1$ is the offset of the primary mirror of the transmitting antenna. By using the following integral formulas [12],
$$\exp({\mathrm{i}} a\cos\varphi) = \sum_{m={-}\infty}^{+\infty}{{\mathrm{i}}^mJ_m(a)\exp({\mathrm{i}} m\varphi)}$$
$$\int_{0}^{2 \pi} \exp (\operatorname{im} \varphi) {\mathrm{d}} \varphi=\left\{\begin{array}{cc} 2 \pi, & m=0 \\ 0, & m \neq 0 \end{array}\right.$$
$$\int_{0}^{\infty} J_{v}(a t) J_{v}(b t) \exp \left({-}c t^{2}\right) t d t=\frac{1}{2 c} \exp \left(-\frac{a^{2}+b^{2}}{4 c}\right) I_{v}\left(\frac{a b}{2 c}\right)$$
the distribution of BG beam passing through the transmitting antenna can be expressed as [6]:
$$\begin{gathered}E_{2}(\rho_1,\varphi_1) = \frac{{\mathrm{i}}^{l+1} k A_0}{2 B_1} \exp (-{\mathrm{i}} k z_1) \exp \left(-\frac{{\mathrm{i}} k D_1 \rho_1^{2}}{2 B_1}\right) \exp ({\mathrm{i}} l \varphi_1) \sum_{\alpha=1}^{M} A_{\alpha}\times\\ \left\{\frac{1}{S_{1}} \exp \left[-\frac{\left({k r}/{B_1}\right)^{2}+k_{r}^{2}}{4 S_{1}}\right] I_{l}\left(\frac{k_{r} k \rho_1}{2 B_1 S_{1}}\right)\right. \left.-\frac{1}{S_{2}} \exp \left[-\frac{\left({k r}/{B_1}\right)^{2}+k_{r}^{2}}{4 S_{2}}\right] I_{l}\left(\frac{k_{r} k \rho_1}{2 B_1 S_{2}}\right)\right\}\end{gathered}$$
where $S_{1}={1}/{\omega _{0}^{2}}+{B_{\alpha }}/{a^{2}}+{\mathrm {i} k A_1}/({2 B_1})$, $S_{2} = 1/{\omega _{0}^{2}}+{B_{\alpha }}/{a_2^{2}}+{\mathrm {i} k A_1}/({2 B_1})$.

The distribution of the electric field after the BG beam passing through the transmitting and receiving antenna $E_3$ at receiver plane ($z_3 = L_1+L_2+L_3+2f_2-2f_1$) can be expressed by the Collins formula [8,9]:

$$\begin{gathered}E_3\left(\rho_2,\varphi_2\right) =\frac{{\mathrm{i}} k}{2\pi B_2} \exp\left(-{\mathrm{i}} kz_2\right)\int_{0}^{2\pi}\int_{0}^{\infty}H_2(\rho_1)E_{2}\left(\rho_1,\varphi_1,0\right)\times\\ \exp\left\{-\frac{{\mathrm{i}} k}{2B_2}\left[A_2\rho_1^2-2\rho_2 \rho_1\cos\left(\varphi_1-\varphi_2\right)+D_1\rho_2^2\right]\right\} \rho_1{\mathrm{d}}\rho_1{\mathrm{d}}\varphi_1\end{gathered}$$
The definition of the hard aperture function $H_2(\rho _1)$ is:
$$H_2(\rho_1)=\left\{\begin{array}{ll} 1 & b_2\leq|\rho_1| \leq b \\ 0 & \text{others} \end{array}\right.$$
where the $a$ and $b_2$ = $bf_2/f_1$ are the minimum and maximum radius of the receiving antenna that allows the beam to pass through. The hard-edged function can be stated in the calculation equation as a superposition of Gaussian functions [10]:
$$H_2({\rho_1})=\sum_{\alpha=1}^{M} A_{\alpha}\left[\exp \left(-\frac{B_{\alpha} \rho_1^{2}}{a^{2}}\right)-\exp \left(-\frac{B_{\alpha} \rho_1^{2}}{b_2^{2}}\right)\right]$$
where $A_2$, $B_2$, and $D_2$ stand for the transmission matrix’s parameters for the Cassegrain receiving antenna. The ABCD transmission matrix of the Cassegrain receiving antenna can be expressed as [11]:
$$\left[\begin{array}{ll} A_2 & B_2 \\ C_2 & D_2 \end{array}\right]=\left[\begin{array}{ll} 1 & L_{3} \\ 0 & 1 \end{array}\right]\left[\begin{array}{cc} 1 & 0 \\ \frac{1}{f_{2}} & 1 \end{array}\right]\left[\begin{array}{cc} 1 & f_{1}-f_{2}+\delta_2 \\ 0 & 1 \end{array}\right]\left[\begin{array}{cc} 1 & 0 \\ -\frac{1}{f_{1}} & 1 \end{array}\right]$$
where $\delta _2$ is the offset of the primary mirror of the receiving antenna. By using the integral formulas mentioned above and below [12],
$$\int_{0}^{\infty} I_{v}(a t) J_{v}(b t) \exp \left({-}c t^{2}\right) t {\mathrm{d}} t=\frac{1}{2 c} \exp \left(\frac{a^{2}-b^{2}}{4 c}\right) J_{v}\left(\frac{a b}{2 c}\right)$$
the distribution of the electric field after the BG beam passing through the transmitting and receiving antenna $E_3$ can be expressed as:
$$\begin{aligned} E_{3}\left(\rho_{2}, \varphi_{2}\right)= & \frac{{\mathrm{i}}^{l+2} k^{2} A_{0}}{4 B_{2} B_1} \exp \left(-{\mathrm{i}} k z_{2}\right) \exp \left(-{\mathrm{i}} l \varphi_{2}\right) \exp \left(\frac{{\mathrm{i}} k D_{2}}{2 B_{2}} \rho_{2}^{2}\right)\times \\ \sum_{\alpha_{2}=1}^{M} \sum_{\alpha_1=1}^{M} A_{\alpha_{2}} A_{\alpha_1} & \left\{\frac{1}{S_{1} M_{1}} \exp \left(-\frac{k_{r}^{2}}{4 S_{1}}\right) \exp \left(\frac{N_{1}^{2}}{4 M_{1}}\right) \exp \left[-\frac{\left(k \rho_{2}\right)^{2}}{4 M_{1} B_{2}^{2}}\right] J_{l}\left(\frac{N_{1} k \rho_{2}}{2 M_{1} B_{2}}\right)\right.\\ & -\frac{1}{S_{2} M_{2}} \exp \left(-\frac{k_{r}^{2}}{4 S_{2}}\right) \exp \left(\frac{N_{2}^{2}}{4 M_{2}}\right) \exp \left[-\frac{\left(k \rho_{2}\right)^{2}}{4 M_{2} B_{2}^{2}}\right] J_{l}\left(\frac{N_{2} k \rho_{2}}{2 M_{2} B_{2}}\right) \\ & -\frac{1}{S_{1} M_{3}} \exp \left(-\frac{k_{r}^{2}}{4 S_{1}}\right) \exp \left(\frac{N_{1}^{2}}{4 M_{3}}\right) \exp \left[-\frac{\left(k \rho_{2}\right)^{2}}{4 M_{3} B_{2}^{2}}\right] J_{l}\left(\frac{N_{1} k \rho_{2}}{2 M_{3} B_{2}}\right) \\ & \left.+\frac{1}{S_{2} M_{4}} \exp \left(-\frac{k_{r}^{2}}{4 S_{2}}\right) \exp \left(\frac{N_{2}^{2}}{4 M_{4}}\right) \exp \left[-\frac{\left(k \rho_{2}\right)^{2}}{4 M_{4} B_{2}^{2}}\right] J_{l}\left(\frac{N_{2} k \rho_{2}}{2 M_{4} B_{2}}\right)\right\} \end{aligned}$$
where $N_{1}={k_{r} k}/({2 B_1 S_{1}})$, $N_{2}={k_{r} k}/({2 B_1 S_{2}})$, $M_{1}={B_{\alpha _{2}}}/{b_2^{2}}+{k^{2}}/({4 S_{1} B_1^{2}})+{{\mathrm {i}} k D_1}/({2 B_1})+{{\mathrm {i}} k A_{2}}/({2 B_{2}})$, $M_{2}={B_{\alpha _{2}}}/{b_2^{2}}+{k^{2}}/({4 S_{2} B_1^{2}})+{{\mathrm {i}} k D_1}/({2 B_1})+{{\mathrm {i}} k A_{2}}/({2 B_{2}})$, $M_{3}={B_{\alpha _{2}}}/{a^{2}}+{k^{2}}/({4 S_{1} B_1^{2}})+{{\mathrm {i}} k D_1}/({2 B_1})+{{\mathrm {i}} k A_{2}}/({2 B_{2}})$, $M_{4}={B_{\alpha _{2}}}/{a^{2}}+{k^{2}}/({4 S_{2} B_1^{2}})+{{\mathrm {i}} k D_1}/({2 B_1})+{{\mathrm {i}} k A_{2}}/({2 B_{2}})$

After obtaining the expression of the BG beam on the receiving plane, the system efficiency under ideal conditions can be calculated. But to analyze the influence of atmospheric turbulence on the system, it is necessary to recalculate the energy distribution of the light field. Considering the short propagation distance of the beam from the source plane to the process through the transmitting antenna, to simplify the calculation, the influence of atmospheric turbulence on the system efficiency in this process is ignored. In this case, only the influence of atmospheric turbulence on the system efficiency after being transmitted from the transmitting antenna and received by the receiving antenna needs to be considered. By using the extended Huygens-Fresnel principle, the average intensity distribution at the receiver plane when considering the atmospheric turbulence can be expressed as [13,14]:

$$\begin{array}{c} \langle I(R, \Theta, L)\rangle=\int_{0}^{2 \pi} \int_{0}^{\infty} \int_{0}^{2 \pi} \int_{0}^{\infty} H\left(r_{1}\right) E_{2}\left(r_{1}, \theta_{1}, z\right)\times \\ \exp \left\{-\frac{{\mathrm{i}} k}{2 B_{2}}\left[A_{2} r_{1}^{2}-2 R r_{1} \cos \left(\theta_{1}-\Theta\right)+D_{2} R^{2}\right]\right\} \times\\ \left\{H\left(r_{2}\right) E_{2}\left(r_{2}, \theta_{2}, z\right) \exp \left\{-\frac{i k}{2 B_{2}}\left[A_{2} r_{2}^{2}-2 R r_{2} \cos \left(\theta_{2}-\Theta\right)+D_{2} R^{2}\right]\right\}\right)^{*} \times\\ \left\langle\psi\left(R, \Theta, r_{1}, \theta_{1}\right)+\psi^{*}\left(R, \Theta, r_{2}, \theta_{2}\right)\right\rangle r_{1} r_{2} {\mathrm{d}} r_{1} {\mathrm{d}} r_{2} {\mathrm{d}} \theta_{1} {\mathrm{d}} \theta_{2} \end{array}$$
where $L = L_2$ represents the propagation distance, $R$ and $\Theta$ represent the polar coordinates at the receiver plane, and the asterisk represents the complex conjugation. In atmospheric turbulence, the symbol $\left \langle \cdot \right \rangle$ signifies the ensemble average. $\left \langle \psi \left (R, \Theta, r_{1}, \theta _{1}\right )+\psi ^{*}\left (R, \Theta, r_{2}, \theta _{2}\right )\right \rangle$, which can be represented as [15]:
$$\left\langle\psi\left(R, \Theta, r_{1}, \theta_{1}\right)+\psi^{*}\left(R, \Theta, r_{2}, \theta_{2}\right)\right\rangle=\exp \left\{-\frac{1}{\rho_{0}^{2}}\left[r_{1}^{2}+r_{2}^{2}-2 r_{1} r_{2} \cos \left(\theta_{1}-\theta_{2}\right)\right]\right\}$$
where $\rho _{0}=\left (0.545 {C_{n}^{2}} k^{2} L_2\right )^{-3 / 5}$ denotes the spatial coherence length of the Kolmogorov turbulence, $C_{n}^{2}$ is the average value of the atmospheric refractive index structure constant along the propagation path with units $m^{-2/3}$.

By using the integral formulas mentioned above and below [12],

$$\int_{0}^{2 \pi} \exp [-{\mathrm{i}} l \theta+2 a R r \cos (\varphi-\theta)] {\mathrm{d}} \theta=2 \pi \exp (-{\mathrm{i}} l \theta) I_{l}(2 a R r)$$
the average intensity distribution at the receiver plane when considering the atmospheric turbulence can be expressed as:
$$\begin{gathered}\langle I(R, \Theta, L)\rangle = \frac{k^{4} A_{0}^{2}}{16 B_{2}^{2} B^{2}} \sum_{\alpha_{4} = 1}^{M} \sum_{\alpha_{3} = 1}^{M} \sum_{\alpha_{2} = 1}^{M} \sum_{\alpha_{1} = 1}^{M} A_{\alpha_{4}}^{*} A_{\alpha_{3}}^{*} A_{\alpha_{2}} A_{\alpha_{1}}\times\\ \left(T_{1}-T_{2}-T_{3}+T_{4}-T_{5}+T_{6}+T_{7}-T_{8}-T_{9}+T_{10}+T_{10}-T_{12}+T_{13}-T_{14}-T_{15}+T_{16}\right)\end{gathered}$$
where $P_{1}={B_{\alpha _{2}}}/{b_2^{2}}+{k^{2}}/({4 S_{1} B^{2}})+{{\mathrm {i}} k D_1}/({2 B_1})+{{\mathrm {i}} k A_{2}}/({2 B_{2}})+{1}/{\rho _{0}^{2}}$, $P_{2}={B_{\alpha _{2}}}/{b_2^{2}}+{k^{2}}/({4 S_{2} B^{2}})+{{\mathrm {i}} k D_1}/({2 B_1})+{{\mathrm {i}} k A_{2}}/({2 B_{2}})+{1}/{\rho _{0}^{2}}$, $P_{3}={B_{\alpha _{2}}}/{a^{2}}+{k^{2}}/({4 S_{1} B^{2}})+{{\mathrm {i}} k D_1}/({2 B_1})+{{\mathrm {i}} k A_{2}}/({2 B_{2}})+{1}/{\rho _{0}^{2}}$, $P_{4}={B_{\alpha _{2}}}/{a^{2}}+{k^{2}}/({4 S_{2} B^{2}})+{{\mathrm {i}} k D_1}/({2 B_1})+{{\mathrm {i}} k A_{2}}/({2 B_{2}})+{1}/{\rho _{0}^{2}}$,
$$\small \begin{aligned} T_{1}=\frac{1}{S_{1} M_{1} S_{1}^{*} M_{1}^{*}} \exp \left[\frac{N_{1}^{2}}{4 M_{1}}+\frac{N_{1}^{* 2}}{4 M_{1}^{*}}-\frac{k_{r}^{2}}{4 S_{1}}-\frac{k_{r}^{2}}{4 S_{1}^{*}}-\frac{(k R)^{2}}{4 M_{1} B_{2}^{2}}-\frac{(k R)^{2}}{4 M_{1}^{*} B_{2}^{2}}\right] J_{l}\left(\frac{N_{1} k R}{2 M_{1} B_{2}}\right) J_{l}\left(\frac{N_{1}^{*} k R}{2 M_{1}^{*} B_{2}}\right), \end{aligned}$$
$$\small \begin{aligned} T_{2}=\frac{1}{S_{1} M_{1} S_{2}^{*} M_{2}^{*}} \exp \left[\frac{N_{1}^{2}}{4 M_{1}}+\frac{N_{2}^{* 2}}{4 M_{2}^{*}}-\frac{k_{r}^{2}}{4 S_{1}}-\frac{k_{r}^{2}}{4 S_{2}^{*}}-\frac{(k R)^{2}}{4 M_{1} B_{2}^{2}}-\frac{(k R)^{2}}{4 M_{2}^{*} B_{2}^{2}}\right] J_{l}\left(\frac{N_{1} k R}{2 M_{1} B_{2}}\right) J_{l}\left(\frac{N_{2}^{*} k R}{2 M_{2}^{*} B_{2}}\right), \end{aligned}$$
$$\small \begin{aligned} T_{3}=\frac{1}{S_{1} M_{1} S_{1}^{*} M_{3}^{*}}\exp \left[\frac{N_{1}^{2}}{4 M_{1}}+\frac{N_{1}^{* 2}}{4 M_{3}^{*}}-\frac{k_{r}^{2}}{4 S_{1}}-\frac{k_{r}^{2}}{4 S_{1}^{*}}-\frac{(k R)^{2}}{4 M_{1} B_{2}^{2}}-\frac{(k R)^{2}}{4 M_{3}^{*} B_{2}^{2}}\right] J_{l}\left(\frac{N_{1} k R}{2 M_{1} B_{2}}\right) J_{l}\left(\frac{N_{1}^{*} k R}{2 M_{3}^{*} B_{2}}\right), \end{aligned}$$
$$\small \begin{aligned} T_{4}=\frac{1}{S_{1} M_{1} S_{2}^{*} M_{4}^{*}} \exp \left[\frac{N_{1}^{2}}{4 M_{1}}+\frac{N_{2}^{* 2}}{4 M_{4}^{*}}-\frac{k_{r}^{2}}{4 S_{1}}-\frac{k_{r}^{2}}{4 S_{2}^{*}}-\frac{(k R)^{2}}{4 M_{1} B_{2}^{2}}-\frac{(k R)^{2}}{4 M_{4}^{*} B_{2}^{2}}\right] J_{l}\left(\frac{N_{1} k R}{2 M_{1} B_{2}}\right) J_{l}\left(\frac{N_{2}^{*} k R}{2 M_{4}^{*} B_{2}}\right), \end{aligned}$$
$$\small \begin{aligned} T_{5}=\frac{1}{S_{2} M_{2} S_{1}^{*} M_{1}^{*}} \exp \left[\frac{N_{2}^{2}}{4 M_{2}}+\frac{N_{1}^{* 2}}{4 M_{1}^{*}}-\frac{k_{r}^{2}}{4 S_{2}}-\frac{k_{r}^{2}}{4 S_{1}^{*}}-\frac{(k R)^{2}}{4 M_{2} B_{2}^{2}}-\frac{(k R)^{2}}{4 M_{1}^{*} B_{2}^{2}}\right] J_{l}\left(\frac{N_{2} k R}{2 M_{2} B_{2}}\right) J_{l}\left(\frac{N_{1}^{*} k R}{2 M_{1}^{*} B_{2}}\right), \end{aligned}$$
$$\small \begin{aligned} T_{6}=\frac{1}{S_{2} M_{2} S_{2}^{*} M_{2}^{*}} \exp \left[\frac{N_{2}^{2}}{4 M_{2}}+\frac{N_{2}^{* 2}}{4 M_2^{*}}-\frac{k_{r}^{2}}{4 S_{2}}-\frac{k_{r}^{2}}{4 S_{2}^{*}}-\frac{(k R)^{2}}{4 M_{2} B_{2}^{2}}-\frac{(k R)^{2}}{4 M_{2}^{*} B_{2}^{2}}\right] J_{l}\left(\frac{N_{2} k R}{2 M_{2} B_{2}}\right) J_{l}\left(\frac{N_{2}^{*} k R}{2 M_{2}^{*} B_{2}}\right), \end{aligned}$$
$$\small \begin{aligned} T_{7}=\frac{1}{S_{2} M_{2} S_{1}^{*} M_{3}^{*}} \exp \left[\frac{N_{2}^{2}}{4 M_{2}}+\frac{N_{1}^{* 2}}{4 M_{3}^{*}}-\frac{k_{r}^{2}}{4 S_{2}}-\frac{k_{r}^{2}}{4 S_{1}^{*}}-\frac{(k R)^{2}}{4 M_{2} B_{2}^{2}}-\frac{(k R)^{2}}{4 M_{3}^{*} B_{2}^{2}}\right] J_{l}\left(\frac{N_{2} k R}{2 M_{2} B_{2}}\right) J_{l}\left(\frac{N_{1}^{*} k R}{2 M_{3}^{*} B_{2}}\right), \end{aligned}$$
$$\small \begin{aligned} T_{8}=\frac{1}{S_{2} M_{2} S_{2}^{*} M_{4}^{*}} \exp \left[\frac{N_{2}^{2}}{4 M_{2}}+\frac{N_{2}^{* 2}}{4 M_4^{*}}-\frac{k_{r}^{2}}{4 S_{2}}-\frac{k_{r}^{2}}{4 S_{2}^{*}}-\frac{(k R)^{2}}{4 M_{2} B_{2}^{2}}-\frac{(k R)^{2}}{4 M_{4}^{*} B_{2}^{2}}\right] J_{l}\left(\frac{N_{2} k R}{2 M_{2} B_{2}}\right) J_{l}\left(\frac{N_{2}^{*} k R}{2 M_{4}^{*} B_{2}}\right), \end{aligned}$$
$$\small \begin{aligned} T_{9}=\frac{1}{S_{1} M_{3} S_{1}^{*} M_{1}^{*}} \exp \left[\frac{N_{1}^{2}}{4 M_{3}}+\frac{N_{1}^{* 2}}{4 M_{1}^{*}}-\frac{k_{r}^{2}}{4 S_{1}}-\frac{k_{r}^{2}}{4 S_{1}^{*}}-\frac{(k R)^{2}}{4 M_{3} B_{2}^{2}}-\frac{(k R)^{2}}{4 M_{1}^{*} B_{2}^{2}}\right] J_{l}\left(\frac{N_{1} k R}{2 M_{3} B_{2}}\right) J_{l}\left(\frac{N_{1}^{*} k R}{2 M_{1}^{*} B_{2}}\right), \end{aligned}$$
$$\small \begin{aligned} T_{10}=\frac{1}{S_{1} M_{3} S_{2}^{*} M_{2}^{*}} \exp \left[\frac{N_{1}^{2}}{4 M_{3}}+\frac{N_{2}^{* 2}}{4 M_{2}^{*}}-\frac{k_{r}^{2}}{4 S_{1}}-\frac{k_{r}^{2}}{4 S_{1}^{*}}-\frac{(k R)^{2}}{4 M_{3} B_{2}^{2}}-\frac{(k R)^{2}}{4 M_{2}^{*} B_{2}^{2}}\right] J_{l}\left(\frac{N_{1} k R}{2 M_{3} B_{2}}\right) J_{l}\left(\frac{N_{2}^{*} k R}{2 M_{2}^{*} B_{2}}\right), \end{aligned}$$
$$\small \begin{aligned} T_{11}=\frac{1}{S_{1} M_{3} S_{1}^{*} M_{3}^{*}}\exp \left[\frac{N_{1}^{2}}{4 M_{3}}+\frac{N_{1}^{* 2}}{4 M_{3}^{*}}-\frac{k_{r}^{2}}{4 S_{1}}-\frac{k_{r}^{2}}{4 S_{1}^{*}}-\frac{(k R)^{2}}{4 M_{3} B_{2}^{2}}-\frac{(k R)^{2}}{4 M_{3}^{*} B_{2}^{2}}\right] J_{l}\left(\frac{N_{1} k R}{2 M_{3} B_{2}}\right) J_{l}\left(\frac{N_{1}^{*} k R}{2 M_{3}^{*} B_{2}}\right), \end{aligned}$$
$$\small \begin{aligned} T_{12}=\frac{1}{S_{1} M_{3} S_{2}^{*} M_{4}^{*}}\exp \left[\frac{N_{1}^{2}}{4 M_{3}}+\frac{N_{2}^{* 2}}{4 M_{4}^{*}}-\frac{k_{r}^{2}}{4 S_{1}}-\frac{k_{r}^{2}}{4 S_{1}^{*}}-\frac{(k R)^{2}}{4 M_{3} B_{2}^{2}}-\frac{(k R)^{2}}{4 M_{4}^{*} B_{2}^{2}}\right] J_{l}\left(\frac{N_{1} k R}{2 M_{3} B_{2}}\right) J_{l}\left(\frac{N_{2}^{*} k R}{2 M_{4}^{*} B_{2}}\right), \end{aligned}$$
$$\small \begin{aligned} T_{13}=\frac{1}{S_{2} M_{4} S_{1}^{*} M_{1}^{*}}\exp \left[\frac{N_{2}^{2}}{4 M_{4}}+\frac{N_{1}^{* 2}}{4 M_{1}^{*}}-\frac{k_{r}^{2}}{4 S_{1}}-\frac{k_{r}^{2}}{4 S_{1}^{*}}-\frac{(k R)^{2}}{4 M_{4} B_{2}^{2}}-\frac{(k R)^{2}}{4 M_{1}^{*} B_{2}^{2}}\right] J_{l}\left(\frac{N_{2} k R}{2 M_{4} B_{2}}\right) J_{l}\left(\frac{N_{1}^{*} k R}{2 M_{1}^{*} B_{2}}\right), \end{aligned}$$
$$\small \begin{aligned} T_{14}=\frac{1}{S_{2} M_{4} S_{2}^{*} M_{2}^{*}} \exp \left[\frac{N_{2}^{2}}{4 M_{4}}+\frac{N_{2}^{* 2}}{4 M_{2}^{*}}-\frac{k_{r}^{2}}{4 S_{1}}-\frac{k_{r}^{2}}{4 S_{2}^{*}}-\frac{(k R)^{2}}{4 M_{4} B_{2}^{2}}-\frac{(k R)^{2}}{4 M_{2}^{*} B_{2}^{2}}\right] J_{l}\left(\frac{N_{2} k R}{2 M_{4} B_{2}}\right) J_{l}\left(\frac{N_{2}^{*} k R}{2 M_{2}^{*} B_{2}}\right), \end{aligned}$$
$$\small \begin{aligned} T_{15}=\frac{1}{S_{2} M_{4} S_{1}^{*} M_{3}^{*}}\exp \left[\frac{N_{2}^{2}}{4 M_{4}}+\frac{N_{1}^{* 2}}{4 M_{3}^{*}}-\frac{k_{r}^{2}}{4 S_{1}}-\frac{k_{r}^{2}}{4 S_{1}^{*}}-\frac{(k R)^{2}}{4 M_{4} B_{2}^{2}}-\frac{(k R)^{2}}{4 M_{3}^{*} B_{2}^{2}}\right] J_{l}\left(\frac{N_{2} k R}{2 M_{4} B_{2}}\right) J_{l}\left(\frac{N_{1}^{*} k R}{2 M_{3}^{*} B_{2}}\right), \end{aligned}$$
$$\small \begin{aligned} T_{16}=\frac{1}{S_{2} M_{4} S_{2}^{*} M_{4}^{*}}\exp \left[\frac{N_{2}^{2}}{4 M_{4}}+\frac{N_{2}^{* 2}}{4 M_{4}^{*}}-\frac{k_{r}^{2}}{4 S_{1}}-\frac{k_{r}^{2}}{4 S_{2}^{*}}-\frac{(k R)^{2}}{4 M_{4} B_{2}^{2}}-\frac{(k R)^{2}}{4 M_{4}^{*} B_{2}^{2}}\right] J_{l}\left(\frac{N_{2} k R}{2 M_{4} B_{2}}\right) J_{l}\left(\frac{N_{2}^{*} k R}{2 M_{4}^{*} B_{2}}\right), \end{aligned}$$

The efficiency of the transmitting antenna $\eta _1$ can be expressed as:

$$\eta_1=\frac{ \int_{0}^{2 \pi}\int_{a}^{b} E_{2}(\rho_1, \varphi_1, 0) E_{2}^{*}(\rho_1, \varphi_1, 0) \rho_1 d \rho_1 d \varphi_1}{ \int_{0}^{2 \pi}\int_{0}^{\infty} E_{1}(r, \theta, 0) E_{1}^{*}(r, \theta, 0) r {\mathrm{d}} r {\mathrm{d}} \theta}\times\beta^2$$

The efficiency of the receiving antenna $\eta _2$ can be expressed as:

$$\eta_{2}=\begin{cases} \frac{\int_{0}^{2 \pi} \int_{0}^{b}E_3 E_3^*R d R d \Theta}{\int_{0}^{2 \pi} \int_{a}^{b} E_{2}\left(\rho_{1}, \varphi_{1}, 0\right) E_{2}^{*}\left(\rho_{1}, \varphi_{1}, 0\right) \rho_{1} d \rho_{1} d \varphi_{1}} \times \beta^{2}, \text{not considering atmospheric turbulence} \\ \frac{\int_{0}^{2 \pi} \int_{0}^{b}<I(R, \Theta, L)>R d R d \Theta}{\int_{0}^{2 \pi} \int_{a}^{b} E_{2}\left(\rho_{1}, \varphi_{1}, 0\right) E_{2}^{*}\left(\rho_{1}, \varphi_{1}, 0\right) \rho_{1} d \rho_{1} d \varphi_{1}} \times \beta^{2}, \text{considering atmospheric turbulence} \end{cases}$$
where $\beta$ denotes the antenna’s reflectivity. Assuming the antireflection coating is used, which is a porous film created by spin-coating a cellulose acetate butyrate (CAB) solution in tetrahydrofuran under humid conditions, $\beta$ can be 99.12$\%$ [16]. The total transmission efficiency of the system $\eta _T$ can be expressed as:
$$\eta_T=\eta_1\eta_2$$

3. Numerical calculations and results analysis

First the parameters are set as: $\lambda = 1550$ nm, $\omega _0 = 5$ mm, $\Gamma = 10^{-5}$, $f_1 = 0.5$ m, $f_2 = 0.1$ m, $L_1 = 1$ m, $L_2 = 1000$ m, $L_3 = 1$ m. To obtain the optimal parameters of the system, i.e., the parameters that make the total transmission efficiency of the system maximum. Total transmission efficiency of the system $\eta _T$ is plotted as a function of apertures of primary and secondary mirrors of the antennas.

Figure 4(a) shows that when the value of $a$ remains constant, the total transmission efficiency of the system $\eta _T$ increases continuously as $b$ increases. When the value of $b$ remains constant, the total transmission efficiency of the system $\eta _T$ increases and then decreases as $a$ increases. $\eta _T$ is maximised when the value of $a$ is within a certain range. This range keeps getting larger as $b$ increases. And in a relatively small range, the total transmission efficiency of the system $\eta _T$ can reach above 96$\%$. However, considering that BG beams of different orders have different light field distributions, a more suitable antenna parameter needs to be chosen in order to make BG beams of other orders also perform well. Clearly there is a wider range of values of $a$ when $b$ = 0.1 m which makes the system most efficient for third order BG beams. So $b$ is set to 0.1 m and then the total transmission efficiency of the system of the BG beam of different orders is plotted as a function of shading rate $\gamma$.

 figure: Fig. 4.

Fig. 4. (a) The total transmission efficiency of the system $\eta _T$ as a function of the primary and secondary apertures of the antenna $a$ and $b$, when $l$ = 3; (b) total transmission efficiency of the system $\eta _T$ of the BG beam with different orders as a function of shading rate $\gamma$.

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Figure 4(b) shows that as the $\gamma$ increases, $\eta _T$ first increases and then decreases. And as $l$ increases, the maximum efficiency of the BG beam increases and can be maintained in a larger range of $\gamma$. At the same time, $\eta _T$ shows a step-down as the $\gamma$ decreases, which is a result of the particular energy distribution of the BG beam, i.e. the ring-shaped distribution. To have a good efficiency for different orders of BG beams, $\gamma$ of the system is set to 0.162. So the optimal parameters of the system are $b$ = 0.1 m, $a$ = 0.0162 m. In this case, $\eta _T$ of BG beams when $l$ = 0-5 are 24$\%$, 64$\%$, 91$\%$, 96$\%$, 96$\%$, 96$\%$, respectively

After obtaining the optimal parameters for the system in the ideal case, it is necessary to study the effect of antenna defocus on the system. Considering that the BG beam propagates a relatively short distance before reaching the receiving plane after passing through the receiving antenna, i.e., $L_3$ is very small. So the effect of $\delta _2$ on the efficiency is not great. In contrast, the BG propagates a long distance after passing through the transmitting antenna, i.e., $L_2$ is relatively big. So the offset of the primary mirror of the transmitting antenna $\delta _1$ can have a significant effect on the system.

Figure 5(a) shows that total transmission efficiency of the system decreases as the absolute value of the $\delta _1$ increases. When the $\delta _1$ is within a certain range, the total transmission efficiency of the system does not change when $l$ = 5. At the same time, the larger the $l$, the faster the total transmission efficiency of the system decreases. However, for the case $l$ = 0 - 4, the total transmission efficiency of the system appears to increase and then decrease as the absolute value of $\delta _1$ increases. To investigate the causes of this phenomenon, the first-order BG beam was chosen for study cause this phenomenon occurs most severely when $l$ = 1. The transmitting antenna efficiency $\eta _1$, the receiving antenna efficiency $\eta _2$ and the total transmission efficiency of the system $\eta _T$ of the first-order BG beam are plotted as a function of delta. Figure 5(b) shows that the efficiency of the transmitting antenna $\eta _1$ has the same trend as the total transmission efficiency of the system $\eta _T$ and therefore the efficiency of the transmission is the main cause of this phenomenon. And when the sign of $\delta _1$ is different, $\eta _T$ shows different phenomena with $\delta _1$ increasing. When $\delta _1$ is less than zero, $\eta _T$ first increases and then decreases; when $\delta _1$ is greater than zero, $\eta _T$ first decreases, then increases and then decreases. To investigate the reasons for this phenomenon in the efficiency of the transmitting antenna $\eta _1$, the values of $\delta _1$ at several particular turning points were recorded. At the same time, the distribution of the optical field of $E_2$ in these cases are plotted.

 figure: Fig. 5.

Fig. 5. (a) The total transmission efficiency of the system $\eta _T$ as a function of the offset $\delta _1$; (b) the transmitting antenna efficiency $\eta _1$, the receiving antenna efficiency $\eta _2$ and the total transmission efficiency of the system $\eta _T$ of the first-order BG beam as a function of $\delta _1$.

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Figure 6 shows that the offset of the main mirror of the transmitting antenna results in a change in the energy distribution of the BG beam, but the range of the receiving antenna is constant and therefore the efficiency of the receiving antenna changes. When $\delta _1$ = - 0.18 mm, compared to $\delta _1$ = 0, more of the energy of the receiving antenna is distributed within the range that can be received by the receiving antenna, and the efficiency of the transmitting antenna $\eta _1$ increased. Conversely, when $\delta _1$ = 0.28 mm, less energy can be received by the receiving antenna and therefore $\eta _1$ reduced. To investigate the reason for the change in energy distribution of the BG beam, the propagation simulation of the BG beam at different offsets is plotted.

 figure: Fig. 6.

Fig. 6. Light field distribution of the BG beam before it passes through the receiving antenna when $l$ = 1, $\delta _1$ = -0.18 mm, 0, 0.28 mm.

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Figure 7 shows that when the antenna is assembled properly, the BG beam will diverge, but most of the beam will still be picked up by the receiving antenna. When delta is less than zero, the divergence of the beam becomes greater, so the beam energy distribution shifts towards where r increases. At first, this shift allows more of the beam to enter the receiving antenna’s range, as in the case of $\delta _1$ = - 0.18 mm. However, as $\delta _1$ continues to increase, the beam moves out of the receiving range of the antenna, leading to a reduction in system efficiency, as in the case of $\delta _1$ = - 0.43 mm. When $\delta _1$ is greater than zero, the beam converges, which causes the energy distribution of the beam to move in the direction of decreasing $r$. Initially, this causes the beam to move out of the receiving range of the receiving antenna, resulting in a reduction in system efficiency as in the case of $\delta _1$ = 0.28 mm. However, as $\delta _1$ increases, the beam converges and then diverges, resulting in the same phenomenon as when the $\delta _1$ is less than zero. So Fig. 5(a) has the properties of symmetry.

 figure: Fig. 7.

Fig. 7. Simulation of the propagation of the BG beam when $l$ = 1, $\delta$ = - 0.43 mm, - 0.18 mm, 0 mm, 0.28 mm, 0.94 mm

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Figure 8(a) shows that the total transmission efficiency of the system is decreasing as the propagation distance increases. However, for BG beams with $l$ = 0 to 3, the total transmission efficiency of the system increases and then decreases as the propagation distance increases. For example, for a BG beam with $l$ = 3, the total transmission efficiency of the system first decreases and then increases between 100 m and 300 m. In orders to analyse the causes of this phenomenon, the first-order BG beam, where this phenomenon is most evident, was selected for study. The transmitting antenna efficiency and receiving antenna efficiency as well as the total transmission efficiency of the system as a function of transmission distance are plotted for the first order BG beam. Figure 8(b) shows that as the propagation distance increases, the efficiency of the transmitting antenna and the total transmission efficiency of the system follow a similar trend, so the efficiency of the transmitting antenna is the main cause of the above phenomenon. To investigate the causes of the change in efficiency of the transmitting antenna, the light field distribution of the BG beam before it enters the receiving antenna is plotted. The peak after the rise of the efficiency of the transmitting antenna and the transmission distance when the total transmission efficiency of the system is reduced to 43$\%$ were chosen for the study.

 figure: Fig. 8.

Fig. 8. (a) The total transmission efficiency of the system $\eta _T$ as a function of the offset $L_2$; (b) the transmitting antenna efficiency $\eta _1$, the receiving antenna efficiency $\eta _2$ and the total transmission efficiency of the system $\eta _T$ of the first-order BG beam as a function of $L_2$.

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Figure 9 shows that as the propagation distance increases, the energy of the beam moves in the direction of increasing $r$. At certain locations this may make the system more efficient as more of the beam can be picked up by the receiving antenna.

 figure: Fig. 9.

Fig. 9. Light field distribution of the BG beam before it passes through the receiving antenna when $l$ = 1, $L_2$ = 0, 1.5km, 3 km.

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4. Conclusion

In the work, an expression for the optical field of the BG beam after it has passed through the optical communication system comprising two antennas in atmospheric turbulence is derived. The difference from the traditional approach is that in the process of this derivation, the range of the Hard-edge aperture has been carefully calculated so that the light filed is obtained more precisely. And the total transmission efficiency of the systemm is calculated, taking into account the reflection losses and the efficiency of the two antennas, which is important for designing high-performance systems. Also the effect of turbulence and the defocusing of the main mirror of the antenna on the total transmission efficiency of the system is investigated and analysed by the distribution of the light field. Through calculation and analysis, the optimal parameters of the system are obtained for a third order BG beam. When $b$ = 0.1m, $a$ = 0.0162 m, $\eta _T$ of BG beams when $l$ = 1 $\sim$ 5 are 64$\%$, 91$\%$, 96$\%$, 96$\%$, 96$\%$, respectively. Turbulent atmospheres and defocusing of the antenna can lead to a reduction in the total transmission efficiency of the system. In addition, turbulent atmospheres and defocusing of the antenna both may lead to an increase and then a decrease in the total transmission efficiency of the system, and the corresponding causes are found by propagation diagram and the light field distribution of BG beams. Our work has proven the practicability of the use of BG beams in this optical system. And we found that the collimation of the BG beam is an important factor in the total transmission efficiency of the system.

Funding

National Natural Science Foundation of China (11574042); Natural Science Foundation of Sichuan Province (2022NSFSC0561).

Disclosures

The authors declare no conflicts of interest.

Data availability

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

References

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

7. Z. Cui, Y. Hui, W. Ma, W. Zhao, and Y. Han, “Dynamical characteristics of laguerre–gaussian vortex beams upon reflection and refraction,” J. Opt. Soc. Am. B 37(12), 3730 (2020). [CrossRef]  

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

9. Y. Cai and S. He, “Propagation of hollow gaussian beams through apertured paraxial optical systems,” J. Opt. Soc. Am. A 23(6), 1410–1418 (2006). [CrossRef]  

10. 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]  

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

12. I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products (Academic Press, 2014).

13. S. Jian-Hua, C. Zi-Yang, and P. Ji-Xiong, “Polarization Changes of Partially Coherent Electromagnetic Vortex Beams Propagating in Turbulent Atmosphere,” Chin. Phys. Lett. 26(2), 024207 (2009). [CrossRef]  

14. H. Wu, S. Sheng, Z. Huang, S. Zhao, H. Wang, Z. Sun, and X. Xu, “Study on power coupling of annular vortex beam propagating through a two-Cassegrain-telescope optical system in turbulent atmosphere,” Opt. Express 21(4), 4005 (2013). [CrossRef]  

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16. X. Zhang, H. Yang, P. Jiang, M. Zhou, W. Caiyang, Y. Qin, and B. Cao, “Cassegrain antenna for a laser diode source using an e-a system to improve the transmission efficiency,” Appl. Opt. 60(23), 6829–6836 (2021). [CrossRef]  

Data availability

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

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Figures (9)
Fig. 1.
Fig. 1. Overview of BG beam passing through the optical communication system comprising a Cassegrain transmitting antenna and a Cassegrain receiving antenna in atmospheric turbulence, $b$, $a$ are the radius of the primary and secondary mirrors, $\gamma$ = $b$/$a$ is obscuration rate, $L_1$ denotes the distance from the source plane to the secondary mirror of the transmitting antenna, $L_2$ denotes the distance between the main mirror of the transmitting and receiving antenna, $L_3$ denotes the distance between receiving antenna secondary mirror and receiving plane, $f_1$, $f_2$ denoting the focal length of the primary and secondary mirror, respectively.

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Fig. 2.
Fig. 2. An overview of the offset of the primary mirror of the antenna, where the dotted line is the position of the primary mirror in the ideal state, and the solid line is the position of the primary mirror after the offset. $\delta _1$ is the offset of the transmitting antenna and $\delta _2$ is the offset of the receiving antenna; (a) transmitting antenna; (b) receiving antenna.

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Fig. 3.
Fig. 3. Overview diagram of the entire calculation process and secondary mirror obscuration analysis of the Cassegrain antenna; (a) transmitting antenna; (b) receiving antenna.

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Fig. 4.
Fig. 4. (a) The total transmission efficiency of the system $\eta _T$ as a function of the primary and secondary apertures of the antenna $a$ and $b$, when $l$ = 3; (b) total transmission efficiency of the system $\eta _T$ of the BG beam with different orders as a function of shading rate $\gamma$.

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Fig. 5.
Fig. 5. (a) The total transmission efficiency of the system $\eta _T$ as a function of the offset $\delta _1$; (b) the transmitting antenna efficiency $\eta _1$, the receiving antenna efficiency $\eta _2$ and the total transmission efficiency of the system $\eta _T$ of the first-order BG beam as a function of $\delta _1$.

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Fig. 6.
Fig. 6. Light field distribution of the BG beam before it passes through the receiving antenna when $l$ = 1, $\delta _1$ = -0.18 mm, 0, 0.28 mm.

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Fig. 7.
Fig. 7. Simulation of the propagation of the BG beam when $l$ = 1, $\delta$ = - 0.43 mm, - 0.18 mm, 0 mm, 0.28 mm, 0.94 mm

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Fig. 8.
Fig. 8. (a) The total transmission efficiency of the system $\eta _T$ as a function of the offset $L_2$; (b) the transmitting antenna efficiency $\eta _1$, the receiving antenna efficiency $\eta _2$ and the total transmission efficiency of the system $\eta _T$ of the first-order BG beam as a function of $L_2$.

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Fig. 9.
Fig. 9. Light field distribution of the BG beam before it passes through the receiving antenna when $l$ = 1, $L_2$ = 0, 1.5km, 3 km.

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Equations (38)

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E 1 ( r , θ )   =   A 0 exp ⁡ ( −   r 2 ω 0 2 ) J l ( k r r ) exp ⁡ ( − i l θ )
E 2 ( ρ 1 , φ 1 ) = i k 2 π B 1 exp ⁡ ( − i k z 1 ) ∫ 0 2 π ∫ 0 ∞ H 1 ( r ) E 1 ( r , θ , 0 ) × exp ⁡ { − i k 2 B 1 [ A 1 r 2 − 2 ρ 1 r cos ⁡ ( θ − φ 1 ) + D 1 ρ 1 2 ] } r d r d θ
H 1 ( r ) = { 1 a 2 ≤ | r | ≤ a 0 others
H 1 ( r ) = ∑ α = 1 M A α [ exp ⁡ ( − B α r 2 a 2 ) − exp ⁡ ( − B α r 2 a 2 2 ) ]
[ A 1 B 1 C 1 D 1 ] = [ 1 L 2 0 1 ] [ 1 0 − 1 f 1 1 ] [ 1 f 1 − f 2 + δ 1 0 1 ] [ 1 0 1 f 2 1 ] [ 1 L 1 0 1 ]
exp ⁡ ( i a cos ⁡ φ ) = ∑ m = − ∞ + ∞ i m J m ( a ) exp ⁡ ( i m φ )
∫ 0 2 π exp ⁡ ( im ⁡ φ ) d φ = { 2 π , m = 0 0 , m ≠ 0
∫ 0 ∞ J v ( a t ) J v ( b t ) exp ⁡ ( − c t 2 ) t d t = 1 2 c exp ⁡ ( − a 2 + b 2 4 c ) I v ( a b 2 c )
E 2 ( ρ 1 , φ 1 ) = i l + 1 k A 0 2 B 1 exp ⁡ ( − i k z 1 ) exp ⁡ ( − i k D 1 ρ 1 2 2 B 1 ) exp ⁡ ( i l φ 1 ) ∑ α = 1 M A α × { 1 S 1 exp ⁡ [ − ( k r / B 1 ) 2 + k r 2 4 S 1 ] I l ( k r k ρ 1 2 B 1 S 1 ) − 1 S 2 exp ⁡ [ − ( k r / B 1 ) 2 + k r 2 4 S 2 ] I l ( k r k ρ 1 2 B 1 S 2 ) }
E 3 ( ρ 2 , φ 2 ) = i k 2 π B 2 exp ⁡ ( − i k z 2 ) ∫ 0 2 π ∫ 0 ∞ H 2 ( ρ 1 ) E 2 ( ρ 1 , φ 1 , 0 ) × exp ⁡ { − i k 2 B 2 [ A 2 ρ 1 2 − 2 ρ 2 ρ 1 cos ⁡ ( φ 1 − φ 2 ) + D 1 ρ 2 2 ] } ρ 1 d ρ 1 d φ 1
H 2 ( ρ 1 ) = { 1 b 2 ≤ | ρ 1 | ≤ b 0 others
H 2 ( ρ 1 ) = ∑ α = 1 M A α [ exp ⁡ ( − B α ρ 1 2 a 2 ) − exp ⁡ ( − B α ρ 1 2 b 2 2 ) ]
[ A 2 B 2 C 2 D 2 ] = [ 1 L 3 0 1 ] [ 1 0 1 f 2 1 ] [ 1 f 1 − f 2 + δ 2 0 1 ] [ 1 0 − 1 f 1 1 ]
∫ 0 ∞ I v ( a t ) J v ( b t ) exp ⁡ ( − c t 2 ) t d t = 1 2 c exp ⁡ ( a 2 − b 2 4 c ) J v ( a b 2 c )
E 3 ( ρ 2 , φ 2 ) = i l + 2 k 2 A 0 4 B 2 B 1 exp ⁡ ( − i k z 2 ) exp ⁡ ( − i l φ 2 ) exp ⁡ ( i k D 2 2 B 2 ρ 2 2 ) × ∑ α 2 = 1 M ∑ α 1 = 1 M A α 2 A α 1 { 1 S 1 M 1 exp ⁡ ( − k r 2 4 S 1 ) exp ⁡ ( N 1 2 4 M 1 ) exp ⁡ [ − ( k ρ 2 ) 2 4 M 1 B 2 2 ] J l ( N 1 k ρ 2 2 M 1 B 2 ) − 1 S 2 M 2 exp ⁡ ( − k r 2 4 S 2 ) exp ⁡ ( N 2 2 4 M 2 ) exp ⁡ [ − ( k ρ 2 ) 2 4 M 2 B 2 2 ] J l ( N 2 k ρ 2 2 M 2 B 2 ) − 1 S 1 M 3 exp ⁡ ( − k r 2 4 S 1 ) exp ⁡ ( N 1 2 4 M 3 ) exp ⁡ [ − ( k ρ 2 ) 2 4 M 3 B 2 2 ] J l ( N 1 k ρ 2 2 M 3 B 2 ) + 1 S 2 M 4 exp ⁡ ( − k r 2 4 S 2 ) exp ⁡ ( N 2 2 4 M 4 ) exp ⁡ [ − ( k ρ 2 ) 2 4 M 4 B 2 2 ] J l ( N 2 k ρ 2 2 M 4 B 2 ) }
⟨ I ( R , Θ , L ) ⟩ = ∫ 0 2 π ∫ 0 ∞ ∫ 0 2 π ∫ 0 ∞ H ( r 1 ) E 2 ( r 1 , θ 1 , z ) × exp ⁡ { − i k 2 B 2 [ A 2 r 1 2 − 2 R r 1 cos ⁡ ( θ 1 − Θ ) + D 2 R 2 ] } × { H ( r 2 ) E 2 ( r 2 , θ 2 , z ) exp ⁡ { − i k 2 B 2 [ A 2 r 2 2 − 2 R r 2 cos ⁡ ( θ 2 − Θ ) + D 2 R 2 ] } ) ∗ × ⟨ ψ ( R , Θ , r 1 , θ 1 ) + ψ ∗ ( R , Θ , r 2 , θ 2 ) ⟩ r 1 r 2 d r 1 d r 2 d θ 1 d θ 2
⟨ ψ ( R , Θ , r 1 , θ 1 ) + ψ ∗ ( R , Θ , r 2 , θ 2 ) ⟩ = exp ⁡ { − 1 ρ 0 2 [ r 1 2 + r 2 2 − 2 r 1 r 2 cos ⁡ ( θ 1 − θ 2 ) ] }
∫ 0 2 π exp ⁡ [ − i l θ + 2 a R r cos ⁡ ( φ − θ ) ] d θ = 2 π exp ⁡ ( − i l θ ) I l ( 2 a R r )
⟨ I ( R , Θ , L ) ⟩ = k 4 A 0 2 16 B 2 2 B 2 ∑ α 4 = 1 M ∑ α 3 = 1 M ∑ α 2 = 1 M ∑ α 1 = 1 M A α 4 ∗ A α 3 ∗ A α 2 A α 1 × ( T 1 − T 2 − T 3 + T 4 − T 5 + T 6 + T 7 − T 8 − T 9 + T 10 + T 10 − T 12 + T 13 − T 14 − T 15 + T 16 )
T 1 = 1 S 1 M 1 S 1 ∗ M 1 ∗ exp ⁡ [ N 1 2 4 M 1 + N 1 ∗ 2 4 M 1 ∗ − k r 2 4 S 1 − k r 2 4 S 1 ∗ − ( k R ) 2 4 M 1 B 2 2 − ( k R ) 2 4 M 1 ∗ B 2 2 ] J l ( N 1 k R 2 M 1 B 2 ) J l ( N 1 ∗ k R 2 M 1 ∗ B 2 ) ,
T 2 = 1 S 1 M 1 S 2 ∗ M 2 ∗ exp ⁡ [ N 1 2 4 M 1 + N 2 ∗ 2 4 M 2 ∗ − k r 2 4 S 1 − k r 2 4 S 2 ∗ − ( k R ) 2 4 M 1 B 2 2 − ( k R ) 2 4 M 2 ∗ B 2 2 ] J l ( N 1 k R 2 M 1 B 2 ) J l ( N 2 ∗ k R 2 M 2 ∗ B 2 ) ,
T 3 = 1 S 1 M 1 S 1 ∗ M 3 ∗ exp ⁡ [ N 1 2 4 M 1 + N 1 ∗ 2 4 M 3 ∗ − k r 2 4 S 1 − k r 2 4 S 1 ∗ − ( k R ) 2 4 M 1 B 2 2 − ( k R ) 2 4 M 3 ∗ B 2 2 ] J l ( N 1 k R 2 M 1 B 2 ) J l ( N 1 ∗ k R 2 M 3 ∗ B 2 ) ,
T 4 = 1 S 1 M 1 S 2 ∗ M 4 ∗ exp ⁡ [ N 1 2 4 M 1 + N 2 ∗ 2 4 M 4 ∗ − k r 2 4 S 1 − k r 2 4 S 2 ∗ − ( k R ) 2 4 M 1 B 2 2 − ( k R ) 2 4 M 4 ∗ B 2 2 ] J l ( N 1 k R 2 M 1 B 2 ) J l ( N 2 ∗ k R 2 M 4 ∗ B 2 ) ,
T 5 = 1 S 2 M 2 S 1 ∗ M 1 ∗ exp ⁡ [ N 2 2 4 M 2 + N 1 ∗ 2 4 M 1 ∗ − k r 2 4 S 2 − k r 2 4 S 1 ∗ − ( k R ) 2 4 M 2 B 2 2 − ( k R ) 2 4 M 1 ∗ B 2 2 ] J l ( N 2 k R 2 M 2 B 2 ) J l ( N 1 ∗ k R 2 M 1 ∗ B 2 ) ,
T 6 = 1 S 2 M 2 S 2 ∗ M 2 ∗ exp ⁡ [ N 2 2 4 M 2 + N 2 ∗ 2 4 M 2 ∗ − k r 2 4 S 2 − k r 2 4 S 2 ∗ − ( k R ) 2 4 M 2 B 2 2 − ( k R ) 2 4 M 2 ∗ B 2 2 ] J l ( N 2 k R 2 M 2 B 2 ) J l ( N 2 ∗ k R 2 M 2 ∗ B 2 ) ,
T 7 = 1 S 2 M 2 S 1 ∗ M 3 ∗ exp ⁡ [ N 2 2 4 M 2 + N 1 ∗ 2 4 M 3 ∗ − k r 2 4 S 2 − k r 2 4 S 1 ∗ − ( k R ) 2 4 M 2 B 2 2 − ( k R ) 2 4 M 3 ∗ B 2 2 ] J l ( N 2 k R 2 M 2 B 2 ) J l ( N 1 ∗ k R 2 M 3 ∗ B 2 ) ,
T 8 = 1 S 2 M 2 S 2 ∗ M 4 ∗ exp ⁡ [ N 2 2 4 M 2 + N 2 ∗ 2 4 M 4 ∗ − k r 2 4 S 2 − k r 2 4 S 2 ∗ − ( k R ) 2 4 M 2 B 2 2 − ( k R ) 2 4 M 4 ∗ B 2 2 ] J l ( N 2 k R 2 M 2 B 2 ) J l ( N 2 ∗ k R 2 M 4 ∗ B 2 ) ,
T 9 = 1 S 1 M 3 S 1 ∗ M 1 ∗ exp ⁡ [ N 1 2 4 M 3 + N 1 ∗ 2 4 M 1 ∗ − k r 2 4 S 1 − k r 2 4 S 1 ∗ − ( k R ) 2 4 M 3 B 2 2 − ( k R ) 2 4 M 1 ∗ B 2 2 ] J l ( N 1 k R 2 M 3 B 2 ) J l ( N 1 ∗ k R 2 M 1 ∗ B 2 ) ,
T 10 = 1 S 1 M 3 S 2 ∗ M 2 ∗ exp ⁡ [ N 1 2 4 M 3 + N 2 ∗ 2 4 M 2 ∗ − k r 2 4 S 1 − k r 2 4 S 1 ∗ − ( k R ) 2 4 M 3 B 2 2 − ( k R ) 2 4 M 2 ∗ B 2 2 ] J l ( N 1 k R 2 M 3 B 2 ) J l ( N 2 ∗ k R 2 M 2 ∗ B 2 ) ,
T 11 = 1 S 1 M 3 S 1 ∗ M 3 ∗ exp ⁡ [ N 1 2 4 M 3 + N 1 ∗ 2 4 M 3 ∗ − k r 2 4 S 1 − k r 2 4 S 1 ∗ − ( k R ) 2 4 M 3 B 2 2 − ( k R ) 2 4 M 3 ∗ B 2 2 ] J l ( N 1 k R 2 M 3 B 2 ) J l ( N 1 ∗ k R 2 M 3 ∗ B 2 ) ,
T 12 = 1 S 1 M 3 S 2 ∗ M 4 ∗ exp ⁡ [ N 1 2 4 M 3 + N 2 ∗ 2 4 M 4 ∗ − k r 2 4 S 1 − k r 2 4 S 1 ∗ − ( k R ) 2 4 M 3 B 2 2 − ( k R ) 2 4 M 4 ∗ B 2 2 ] J l ( N 1 k R 2 M 3 B 2 ) J l ( N 2 ∗ k R 2 M 4 ∗ B 2 ) ,
T 13 = 1 S 2 M 4 S 1 ∗ M 1 ∗ exp ⁡ [ N 2 2 4 M 4 + N 1 ∗ 2 4 M 1 ∗ − k r 2 4 S 1 − k r 2 4 S 1 ∗ − ( k R ) 2 4 M 4 B 2 2 − ( k R ) 2 4 M 1 ∗ B 2 2 ] J l ( N 2 k R 2 M 4 B 2 ) J l ( N 1 ∗ k R 2 M 1 ∗ B 2 ) ,
T 14 = 1 S 2 M 4 S 2 ∗ M 2 ∗ exp ⁡ [ N 2 2 4 M 4 + N 2 ∗ 2 4 M 2 ∗ − k r 2 4 S 1 − k r 2 4 S 2 ∗ − ( k R ) 2 4 M 4 B 2 2 − ( k R ) 2 4 M 2 ∗ B 2 2 ] J l ( N 2 k R 2 M 4 B 2 ) J l ( N 2 ∗ k R 2 M 2 ∗ B 2 ) ,
T 15 = 1 S 2 M 4 S 1 ∗ M 3 ∗ exp ⁡ [ N 2 2 4 M 4 + N 1 ∗ 2 4 M 3 ∗ − k r 2 4 S 1 − k r 2 4 S 1 ∗ − ( k R ) 2 4 M 4 B 2 2 − ( k R ) 2 4 M 3 ∗ B 2 2 ] J l ( N 2 k R 2 M 4 B 2 ) J l ( N 1 ∗ k R 2 M 3 ∗ B 2 ) ,
T 16 = 1 S 2 M 4 S 2 ∗ M 4 ∗ exp ⁡ [ N 2 2 4 M 4 + N 2 ∗ 2 4 M 4 ∗ − k r 2 4 S 1 − k r 2 4 S 2 ∗ − ( k R ) 2 4 M 4 B 2 2 − ( k R ) 2 4 M 4 ∗ B 2 2 ] J l ( N 2 k R 2 M 4 B 2 ) J l ( N 2 ∗ k R 2 M 4 ∗ B 2 ) ,
η 1 = ∫ 0 2 π ∫ a b E 2 ( ρ 1 , φ 1 , 0 ) E 2 ∗ ( ρ 1 , φ 1 , 0 ) ρ 1 d ρ 1 d φ 1 ∫ 0 2 π ∫ 0 ∞ E 1 ( r , θ , 0 ) E 1 ∗ ( r , θ , 0 ) r d r d θ × β 2
η 2 = { ∫ 0 2 π ∫ 0 b E 3 E 3 ∗ R d R d Θ ∫ 0 2 π ∫ a b E 2 ( ρ 1 , φ 1 , 0 ) E 2 ∗ ( ρ 1 , φ 1 , 0 ) ρ 1 d ρ 1 d φ 1 × β 2 , not considering atmospheric turbulence ∫ 0 2 π ∫ 0 b < I ( R , Θ , L ) > R d R d Θ ∫ 0 2 π ∫ a b E 2 ( ρ 1 , φ 1 , 0 ) E 2 ∗ ( ρ 1 , φ 1 , 0 ) ρ 1 d ρ 1 d φ 1 × β 2 , considering atmospheric turbulence
η T = η 1 η 2
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