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Set of mathematical models for Bessel-Gauss beams coupling into the parabolic-index fiber under the influence of atmospheric turbulence and random jitter

Shunyuan Shang, Xinyu Li, Wei Deng, Youran Wang, Yiming Han, Hang Su, Huajun Yang, and Ping Jiang

Open Access Open Access

Abstract

The expression of efficiency for Bessel-Gauss (BG) beams coupling into the parabolic fibers (PF) after passing through the Cassegrain antenna system is first derived. The effects of atmospheric turbulence and random jitter of the coupling lens on the efficiency are also taken into account to improve the practical applicability of our model. This article use a BG beam with a wavelength of 1550 mm and fiber with a core radius RF of 50 μm and a relative refractive index difference ζ of 0.01 for simulation testing. The optimal parameters of the antenna system are determined: the radius of the primary mirror and the secondary mirror is 8.33 cm and 1.25 cm, respectively. The coupling efficiency of BG beams of different orders reaches above 94% simultaneously when the lens’s focal length is 7.8 cm. After taking into account the transmission efficiency of the antenna system, the system’s total efficiency for BG beams of different orders averages 76.33%, when the transmission distance is 1 km. The results show that the same degree of turbulence and random jitter have different influences on the coupling efficiency of BG beams of different orders, and lower-order BG beams have better resistance to turbulence and jitter during propagation and coupling. Moreover, the effect of the guided mode field on the coupling efficiency and the resistance to turbulence varies for different values of mode radial index in the fiber p. The guided mode with p = 0 not only enables the BG beams of different orders to achieve the highest transmission efficiency in the coupling system almost simultaneously but also the random jitter and turbulence have less influence on the coupling efficiency of this mode. It means that the BG beams can have higher efficiency when coupled to the mode with p = 0 after long-distance transmission. This property of the fiber mode at p = 0 provides conditions for the simultaneous propagation of multiple BG beams in a parabolic fiber, which provides a theoretical basis for higher transmission capacity. This research work provides a theoretical model for the theoretical study of vortex beams and optical communication, which is beneficial for the design and application of vortex beams and has instructive meaning for practical engineering design.

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

1. Introduction

In the FSO research area, the Cassegrain antenna system has become a widely used optical communication antenna system for its advantages of high broadband and high collimation [1,2]. However, the Cassegrain antenna has a sheltering effect on the beams [1,3], which makes the transmission efficiency of Gaussian beams relatively low. The vortex beam is a kind of beam propagating with doughnut shape, which can not only overcome the beam blocking problem caused by the Cassegrain antennas but also carries the orbital angular momentum (OAM) that supports mode division multiplexing [4,5]. These features and advantages of the vortex beam make it a widely prospective beam for FSO. BG beams of non-zero order are classical vortex beams, their self-recovery capabilities have received a great deal of attention in the field of FSO research.

In contrast, most of the work in the study of beam-to-fiber coupling has been done on single-mode fibers and step-index fibers. Because single-mode fibers can only transmit one mode, after a long period of development, the pipeline resources and single-mode fiber transmission capacity have been almost exhausted. In contrast, compared to single-mode fibers, few-mode fibers, and multimode fibers have larger dimensions, support multiple optical modes, and are more suitable for coupling with vortex beams [6,7]. A step refractive index fiber is one in which the core-to-cladding refractive index change is a step change, while a PF is one in which the core-to-cladding refractive index is a parabolic-index regular change, also known as a square-law fiber or a gradient refractive index fiber. According to the study [8,9], the beam dispersion in PF will be reduced greatly, and the transmission rate is high, suitable for the production and design of few-mode and multimode fibers. Therefore, the study of PF and BG beams coupling characteristics has important research value and theoretical significance.

The current research on BG beams is mostly focused on the calculation of the direct coupling of the ideal BG beams with optical fiber [10], the analysis of the propagation characteristics of the ideal beam in atmospheric turbulence [11] and so on. Although the light field distribution of the BG beams after passing through the Cassegrain antenna system has been studied and analyzed in [12], there is still a lack of the model of the transmission efficiency and the coupling efficiency with the fiber for BG beams passing through the antenna system. In addition, although contemporary research on beam coupling into optical fibers partially considers the effects of atmospheric turbulence and random jitter on coupling efficiency, the research is focused on the study of single-mode fibers and Gaussian beams [10,13,14], and lacks the study of BG beams coupled into PF, which also presents an impediment to the application and design of vortex beams coupling systems. Thus, it is the core task of this work to establish a coupling efficiency calculation model that integrates atmospheric turbulence, lens random jitter, and other factors on the basis of a comprehensive consideration of the whole process of BG beams from emission to final coupling into the PF.

Hence, this work will first analyze the efficiency expression of the BG beams coupled with a PF after passing through the antenna system. Since atmospheric turbulence has a great impact on the transmission quality and efficiency of optical communication, the lens may jitter due to external factors such as the vibration of the satellite platform and the tracking system, random vibrations of the optical platform mount or the boresight error of the beam pointing device [15–17], thus affecting the beam coupling efficiency. In this article, the effects of atmospheric turbulence and random jitter on the coupling efficiency of BG beams are modeled separately. Thus, there are three efficiency calculation models proposed in this paper: the ideal case with no turbulence and no random jitter, the case with turbulence but no random jitter, and the case with random jitter but no atmospheric turbulence. After the simulation, the optimal radius of the antenna’s primary and secondary mirrors and the optimal focal length of the coupling lens will also be determined. Then, this study further analyzes the variation and characteristics of the coupling efficiency when the BG beams of different orders are coupled with different fiber field modes and compare the anti-turbulence and anti-jitter capabilities of different orders of BG beams. In summary, this work will fill an important blank in the vortex beam research area through the establishment of the model, which provides an important theoretical basis for the further development and research of vortex beams in the field of free optical communication. This mathematical calculation model not only integrates several influencing factors but also considers the feasibility of practical assembly and application. On this basis, through the analysis of the coupling characteristics of BG beams with different fiber modes, this work will also provide an important base for the design of vortex beams and PF, which is of great practical significance for engineering applications.

2. Theoretical model

2.1 Parameters table

To enhance the readability and scientificity of the article, some of the key parameters appearing in the text were summarized before the theoretical analysis, and the specific symbols and definitions are shown in the following Table 1 as below:

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Table 1. Parameters Table

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2.2 Overview of BG beams and parabolic-index fiber

In this work, the coupling efficiency of the BG beams after passing through the transmitting and receiving antennas is first investigated. Assuming that the BG beams transmits along the z-axis after it is generated from the light source, the schematic diagram of the whole process is shown as Fig. 1, where $R_A$ and $R_a$ represent the radius of the primary mirror and secondary mirror of the antennas, $\gamma =R_a/R_A$ represents the sheltering rate of the antennas, respectively. $F_1$, and $F_2$ represent the focal length of the primary and secondary mirrors of the Cassegrain antennas, respectively. $L_1$ represents the distance between the source plane and the secondary mirror of the transmitting antenna, $L_2$ represents the distance between the primary mirror of the transmitting antenna and the primary mirror of the receiving antenna, and $L_3$ represents the distance between the secondary mirror of the receiving antenna and the coupling lens, respectively. $f$, $R_L$ represent the focal length and the radius of the coupling lens, respectively.

 figure: Fig. 1.

Fig. 1. The schematic diagram of the whole process of BG Beams coupling into PF after passing through the Cassegrain antenna system and atmospheric turbulence

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The light distribution of BG beams at source plane $E_0$ can be expressed as [18]:

$$\begin{aligned}E_0(r,\theta, z)=\exp\left(-\frac{r^2}{\omega^2}\right)J_{l_E}(k_rr)\exp \left(-\mathrm{i} l_E\theta\right)\exp \left(-\mathrm{i} k z\right)\end{aligned}$$
where $\omega$ denotes the beam waist width of the BG beams, $k$ is the wave number of BG beams in free space, defined by $k=2\pi /\lambda$, $k_r=k\sin (\Gamma )$ is the radial wave number of BG beams, $\Gamma$ is the half-dispersion angle, $J_{l_E}$ represents the $l_E$th-order Bessel function of the first kind. BG beams’ two fundamental quantities, wavelength, and order are represented by $\lambda$ and $l_E$.

The expression for the light field distribution of BG beams after passing through the Cassegrain antenna system $E(r,\theta )$ is as follows [12]:

$$\small \begin{aligned} E & (r, \theta,z)= \frac{\mathrm{i}^{{2l_E+2}} k^2}{4 B_2 B_1} \exp \left(-\mathrm{i} k z\right) \exp \left(-\mathrm{i} l_E \theta\right) \exp \left(\frac{\mathrm{i} k D_2}{2 B_2} r^2\right) \sum_{\alpha_2=1}^M \sum_{\alpha_1=1}^M {U_{\alpha_2} U_{\alpha_1} }T_{{r,\alpha_1,\alpha _2}} \end{aligned}$$
$$\small \begin{aligned} & T_{{r,\alpha_1,\alpha _2}}= \\ & \left\{ \begin{aligned} & \frac{1}{S_{1\alpha _1}M_{1\alpha _1,\alpha _2}}\exp \left( -\frac{k_{r}^{2}}{4S_{1\alpha _1}} \right) \exp \left( \frac{N_{1\alpha _1}^{2}}{4M_{1\alpha _1,\alpha _2}} \right) \exp \left[ -\frac{\left( k{r} \right) ^2}{4M_{1\alpha _1,\alpha _2}B_{2}^{2}} \right] J_{l_E}\left( \frac{N_{1\alpha _1}k{r}}{2M_{1\alpha _1,\alpha _2}B_2} \right) \\ & -\frac{1}{S_{2\alpha _1}M_{2\alpha _1,\alpha _2}}\exp \left( -\frac{k_{r}^{2}}{4S_{2\alpha _1}} \right) \exp \left( \frac{N_{2\alpha _1}^{2}}{4M_{2\alpha _1,\alpha _2}} \right) \exp \left[ -\frac{\left( k{r} \right) ^2}{4M_{2\alpha _1,\alpha _2}B_{2}^{2}} \right] J_{l_E}\left( \frac{N_{2\alpha _1}k{r}}{2M_{2\alpha _1,\alpha _2}B_2} \right)\\ & -\frac{1}{S_{1\alpha _1}M_{3\alpha _1,\alpha _2}}\exp \left( -\frac{k_{r}^{2}}{4S_{1\alpha _1}} \right) \exp \left( \frac{N_{1\alpha _1}^{2}}{4M_{3\alpha _1,\alpha _2}} \right) \exp \left[ -\frac{\left( k{r} \right) ^2}{4M_{3\alpha _1,\alpha _2}B_{2}^{2}} \right] J_{l_E}\left( \frac{N_{1\alpha _1}k{r}}{2M_{3\alpha _1,\alpha _2}B_2} \right)\\ & +\frac{1}{S_{2\alpha _1}M_{4\alpha _1,\alpha _2}}\exp \left( -\frac{k_{r}^{2}}{4S_{2\alpha _1}} \right) \exp \left( \frac{N_{2\alpha _1}^{2}}{4M_{4\alpha _1,\alpha _2}} \right) \exp \left[ -\frac{\left( k{r} \right) ^2}{4M_{4\alpha _1,\alpha _2}B_{2}^{2}} \right] J_{l_E}\left( \frac{N_{2\alpha _1}k{r}}{2M_{4\alpha _1,\alpha _2}B_2} \right) \\ \end{aligned} \right\} \end{aligned}$$
where $S_{1_{\alpha _1}}={1}/{\omega ^{2}}+{{V_{\alpha _1}}}/{{R_a}^{2}}+{\mathrm {i} k A_1}/({2 B_1})$, $S_{2_{\alpha _1}} = 1/{\omega ^{2}}+{{V_{\alpha _1}}}/{{R_p}^{2}}+{\mathrm {i} k A_1}/({2 B_1})$, $N_{1_{\alpha _1}}={k_{r} k}/({2 B_1 S_{1_{\alpha _1}}})$, $N_{2_{\alpha _1}}={k_{r} k}/({2 B_1 S_{2_{\alpha _1}}})$, $M_{1_{\alpha _1,\alpha _2}}={{V_{\alpha _2}}}/{{R_A}^{2}}+{k^{2}}/({4 S_{1_{\alpha _1}} {B_1}^{2}})+{\mathrm {i} k D_1}/({2 B_1})+{\mathrm {i} k A_{2}}/({2 B_{2}})$, $M_{2_{\alpha _1,\alpha _2}}={{V_{\alpha _2}}}/{{R_A}^{2}}+{k^{2}}/({4 S_{2_{\alpha _1}} {B_1}^{2}})+{\mathrm {i} k D_1}/({2 B_1})+{\mathrm {i} k A_{2}}/({2 B_{2}})$, $M_{3_{\alpha _1,\alpha _2}}={{V_{\alpha _2}}}/{{R_a}^{2}}+{k^{2}}/({4 S_{1_{\alpha _1}} {B_1}^{2}})+{\mathrm {i} k D_1}/({2 B_1})+{\mathrm {i} k A_{2}}/({2 B_{2}})$, $M_{4_{\alpha _1,\alpha _2}}={{V_{\alpha _2}}}/{{R_a}^{2}}+{k^{2}}/({4 S_{2_{\alpha _1}} {B_1}^{2}})+{\mathrm {i} k D_1}/({2 B_1})+{\mathrm {i} k A_{2}}/({2 B_{2}})$.

Here the apparatus function is expanded into the form of summation of multiple exponential terms, and for the convenience of solution and derivation, only the $M$ terms are chosen for derivation. The value of $M$ is generally valued as 10 [2,19]. And due to the existence of two apparatus, there is a double summation. ${{V_{\alpha }}}$ and ${{U_{\alpha }}}$ are the expanded Gaussian coefficients [2,20]. $\alpha$ is an ordinal number in the summation process, which is represented differently in different summation processes, $\alpha _1$, $\alpha _2$, $\alpha _3$ and $\alpha _4$ appear in this work. $r$ is the radial coordinate of the polar coordinate system, which has different representations in different optical fields, $r_1$ and $r_2$ appearing in the following. $A_1$, $B_1$, and $D_1$ stand for the transmission matrix’s parameters for the Cassegrain transmitting antenna, $A_2$, $B_2$, and $D_2$ stand for the transmission matrix’s parameters for the Cassegrain receiving antenna, respectively.

$$\left[\begin{array}{ll} A_1 & B_1 \\ C_1 & D_1 \end{array}\right]=\left[\begin{array}{ll} 1 & L_{2} \\ 0 & 1 \end{array}\right]\left[\begin{array}{cc} 1 & 0 \\ -\frac{1}{F_{1}} & 1 \end{array}\right]\left[\begin{array}{cc} 1 & F_{1}-F_{2} \\ 0 & 1 \end{array}\right]\left[\begin{array}{cc} 1 & 0 \\ \frac{1}{F_{2}} & 1 \end{array}\right]\left[\begin{array}{cc} 1 & L_{1} \\ 0 & 1 \end{array}\right]$$
$$\left[\begin{array}{ll} A_2 & B_2 \\ C_2 & D_2 \end{array}\right]=\left[\begin{array}{ll} 1 & L_{3} \\ 0 & 1 \end{array}\right]\left[\begin{array}{cc} 1 & 0 \\ \frac{1}{F_{2}} & 1 \end{array}\right]\left[\begin{array}{cc} 1 & F_{1}-F_{2} \\ 0 & 1 \end{array}\right]\left[\begin{array}{cc} 1 & 0 \\ -\frac{1}{F_{1}} & 1 \end{array}\right]$$

According to [8,21], $F\left ( r,\theta \right )$ is the mode of PF and the expression for the mode in PF is obtained by solving the Helmholtz equation in cylindrical coordinates. For PF, the classical form of the guided mode field expression is the Laguerre-Gauss polynomial. So, the back-propagated field distribution of PF to the lens can be expressed as:

$${F_{p,l_F}}(r, \theta)=\frac{B_{p l_F}}{\omega_a}\left(\sqrt{2} \frac{r}{\omega_a}\right)^{l_F} L_p^{l_F}\left(\frac{2 r^2}{\omega_a^2}\right) \exp \left(-\frac{r^2}{\omega_a^2}\right) \exp(-\mathrm{i} l_F \theta)$$
where ${B_{p l_F}}= \sqrt { 2 p ! / ( \pi {{ ( l_E + p ) !}} ) }$ denotes the normalization constant, $L_p^{{{l_E}}}$ denotes the associated Laguerre polynomial expression, $p$ is the mode radial index in the fiber, $\omega _a= 2 f / ( \pi \omega _0 )$ represents the radius of the back-propagated mode-field of the fiber, $\omega _0=\sqrt {2 {R_F} /\left (k \tau _0 \sqrt {2 \zeta }\right )}$ is the radius of the field in the fiber and $l_F$ denotes the order of mode in the light field. For PF, the variation of core refractive index with radius is a parabola function. The refractive index of the fiber is maximum at the core and decreases parabolically with increasing radius. The expression for the variation of refractive index with radius in the fiber core is [22]:
$$\tau_r ^ { 2 } ( r ) = \tau _ { 0 } ^ { 2 } [ 1 - 2 \zeta ( \frac { r } { {R_F} } ) ^ { 2 } ]$$
where $\tau _ { 0 }$ is the core refractive index, $\zeta$ is the relative refractive index difference, ${R_F}$ is the core radius of the fiber field.

The coupling efficiency $\eta _c$ of the BG beams to the PF is defined as the ratio of $P_c$ to $P_r$, $P_c$ is energy function of BG beams coupling with the fiber but without random jitter and atmospheric turbulence. The coupling efficiency expression can be expanded as [23]:

$$\eta_c=\frac{P_c}{P_r}=\frac{{\left |\int _ { A } E ( r, \theta ) F ^ { * } ( r,\theta ) rd rd\theta \right |}^2}{\int_0^{R_L}{}\int_0^{2\pi}\left| E\left( r,\theta \right) \right|^2d\theta_{}rdr\int_0^{R_L}\int_0^{2\pi}\left| F\left( r,\theta \right) \right|^2d\theta rdr}$$

2.3 Effect of atmospheric turbulence on the coupling efficiency

After the derivation of the coupling efficiency in the ideal case, the next step in this work is to derive a model for calculating the coupling efficiency under the influence of atmospheric turbulence. According to the research [24], since free optical communication is exposed to the atmosphere, its communication quality and transmission efficiency are largely affected by environmental factors such as atmospheric turbulence. Therefore, it is necessary to consider the effect of turbulence on the coupling efficiency, and this section will therefore establish a further computational model of the coupling efficiency of a BG beams passing through an antenna system under the influence of turbulence.

Figure 2 shows a schematic diagram of the effect of atmospheric turbulence on the transmission of BG beams in the atmosphere.

 figure: Fig. 2.

Fig. 2. Schematic diagram of BG beams with phase distortion under the influence of atmospheric turbulence coupled to the fiber after passing through the coupling lens

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In order to improve the channel capacity, it is necessary to ensure the simultaneous transmission of multiple modes of vortex beam with high efficiency, and considering the simplicity of the model, this work uses the average value for calculation. The coupling efficiency under the influence of atmospheric turbulence $\eta _{cT}$ of the BG beams to the PF is defined as the ratio of $P_{cT}$ to $P_r$, which can be expressed as [25]:

$${ \eta_{cT}} =\frac{P_{cT}}{P_r}=\frac{\int_0^{R_L}\int_0^{R_L}\int_0^{2\pi}\int_0^{2\pi}{W\left(r_1,\theta_1,r_2,\theta_2\right)} F^*\left(r_1,\theta_1 \right) F\left(r_2,\theta_{2} \right) d\theta_1 d\theta_2r_1dr_1r_2dr_2}{\int_0^{R_L}{}\int_0^{2\pi}\left| E\left( r,\theta \right) \right|^2d\theta_{}rdr\int_0^{R_L}\int_0^{2\pi}\left| F\left( r,\theta \right) \right|^2d\theta rdr}$$

According to the extended Huygens-Fresnel principle, the light originating from the transmit aperture can be described by a superposition of spherical waves. Hence, the coherence radius for spherical waves is used in the following, and the mutual coherence function of BG beams is a mutual coherence function of ${E(r,\theta )}$, which can be expressed as [25]:

$$W\left(r_1,\theta_1,r_2,\theta_2\right)=E\left(r_1,\theta_1\right) E^*\left(r_2,\theta_{2}\right)\exp\left\langle \psi \left( r_1,\theta_1 \right) +\psi^*\left( r_2,\theta_{2} \right)\right\rangle$$
where $\left\langle \cdot \right\rangle$ denotes the average value of the operator, $F\left ( r, \theta \right )$ denotes the field distribution of the guided-mode field of the PF back propagates to the coupling plane, and $*$ denotes the conjugate of the original physical operator. Since the calculation of the denominator part is a simple double integral multiplication, the main derivation and analysis in this work focus on the expression of the energy function after coupling the beam to the fiber, $P_{cT}$.

According to the atmospheric turbulence structure determined above and the previous research [25,26]. For beams transmitted in isotropic atmospheric turbulence, the exponential term part of the partial coherence function can be approximated using the following equation:

$$\exp \left\langle\psi\left(r_1, \theta_1\right)+\psi^*\left(r_2, \theta_2\right)\right\rangle \cong \exp \left\{-\frac{1}{\rho_0^2}\left[r_1^2+r_2^2-2 r_1 r_2 \cos \left(\theta_1-\theta_2\right)\right]\right\}$$
where $k$ is the wave number of the beam, and $\rho _0$ denotes the coherence length of the Kolmogorov turbulence in turbulent atmosphere, which can be expressed as [24,27]:
$$\rho _{0}=(0.545k^{2}\int _{0}^{L_2}C_{n}^{2}(z)dz)^{{-}3/5}$$

Since $L_1$ and $L_3$ are very short, the effect of turbulence can be neglected, thus the total propagation distance of the beam is approximated as $L_2$.

Based on [27], $C_n^2$ is the average value of the atmospheric refractive index structure constant with units of $m^{-2/3}$, which is determined by the following equation, $h$ denotes height above ground, $v$ denotes wind speed, and $G_\omega$ is the variable related to wind speed.

$$\small C_{ n } ^ { 2 } ( v , h ) = 0.00594 ( v / 27 ) ^ { 2 } ( 10 ^ { - 5 } h ) ^ { 10 } \exp ( - h / 1000 ) + 2.7 \times 10 ^ { - 16 } \exp ( - h / 1500 ) + {G_\omega} \exp ( - h / 100 )$$

Based on [24,27], in isotropic turbulent, $\rho _{0}$ is generally approximated as a constant:

$$\rho_ { 0 } = ( 0.545 C_ {n} ^ {2 } k ^ { 2 } L_2 ) ^ { - 3 / 5 }$$

Therefore, the mutual coherence function of the beam under the influence of atmospheric turbulence can be expressed as:

$${W}\left(r_1,\theta_1,r_2,\theta_2\right)=E\left(r_1,\theta_1\right) E^*\left(r_2,\theta_{2}\right) \exp \left\{-\frac{1}{\rho_0^2}\left[r_1^2+r_2^2-2 r_1 r_2 \cos \left(\theta_1-\theta_2\right)\right]\right\}$$

The coupling energy under the influence of turbulence $P_{cT}$ can be expressed as:

$$P_{cT}=\int_0^{R_L}\int_0^{R_L}\int_0^{2\pi}\int_0^{2\pi}{{W}\left(r_1,\theta_1,r_2,\theta_2\right)} F^*\left(r_1,\theta_1 \right) F\left(r_2,\theta_{2} \right) d\theta_1 d\theta_2r_1dr_1r_2dr_2$$

Substitution of the $W_t$ and $F_{pl_F}$ into (16), and using the following equations [28],

$$\exp\left( \mathrm{i} a\cos\theta \right) = \sum_{n={-}\infty}^{\infty}{\mathrm{i}^nJ_n\left( a \right) \exp\left({-}in\theta \right)}$$
$$\int_0^{2 \pi} \exp ({-}i n \theta)=\left\{\begin{array}{c} 2 \pi, n=0 \\ 0, n \neq 0 \end{array}\right.$$

After complicated calculations and derivations, $P_c$ can be expressed as:

$$\scriptsize \begin{aligned} & P_{cT}=\int_0^{R_L}\int_0^{R_L}\sum_{\alpha _4=1}^M\sum_{\alpha _3=1}^M\sum_{\alpha _2=1}^M\sum_{\alpha _1=1}^M{{{U_{\alpha _4}}^*{U_{\alpha _3}}^*{U_{\alpha _2}}{U_{\alpha _1}}}}\left( \frac{k^2\pi B_{pl_F}}{2B_2B_1\omega _a} \right) ^2\exp\left\{ -\frac{r_{1}^{2}+r_{2}^{2}}{\rho _{0}^{2}} \right\} \exp \left[ \frac{\mathrm{i}kD_2}{2B_2}\left( r_{1}^{2}-r_{2}^{2} \right) \right]\\ & I_{l_E-l_F}\left( \frac{2r_1r_2}{\rho _{0}^{2}} \right) \left[ (\frac{\sqrt{2}r_2}{\omega _a})^{l_F}L_{p}^{l_F}\left( \frac{2{r_2}^2}{\omega _{a}^{2}} \right) \exp\left( -\frac{{r_2}^2}{\omega _{a}^{2}} \right) \right] \left[ (\frac{\sqrt{2}r_1}{\omega _a})^{l_F}L_{p}^{l_F}\left( \frac{2{r_1}^2}{\omega _{a}^{2}} \right) \exp\left( -\frac{{r_1}^2}{\omega _{a}^{2}} \right) \right]^*{T_{r_1,\alpha_1,\alpha _2}T^*_{r_2,\alpha _3,\alpha _4}}r_1r_2dr_1dr_2 \end{aligned}$$

2.4 Effect of coupling lens jitter on coupling efficiency

Then, this article derives a model for calculating the coupling efficiency in the presence of random jitter in the lens but without the influence of atmospheric turbulence. Since random jitter of the coupling lens can occur due to various reasons during the actual utilization, including but not limited to the vibration of the satellite platform, random vibrations of the optical platform mount [15,16], and so on, it is necessary to analyze the effect of random jitter on the coupling efficiency. According to [14,16,29], this section assumes that the error angle at the assembly is zero and the misalignment between the lens focus and the fiber center is only generated by the random jitter of the lens, then the probability density distribution of the random jitter angle of the lens is:

$$\left.{f_{\Delta \theta}}(\Delta \theta)\right|_{\phi=0}=\dfrac{\Delta \theta}{\sigma^2}\text{exp}\left(-\dfrac{\Delta \theta^2}{2\sigma^2}\right).$$
where $\Delta \theta$ denotes the random jitter angle of the coupling lens, and $\sigma ^2$ denotes the variance of the random jitter angle.

As shown in Fig. 3, the random jitter is a lateral offset, and the focus of the incident beam is also on the fiber coupling plane.

 figure: Fig. 3.

Fig. 3. The effect of random jitter of the lens on the coupling process. (A) is a profile view of the jitter process (B) is a schematic diagram showing the offset of the fiber plane from the optical field as seen from the z-direction

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According to [11], we use $\Delta r= f\Delta \theta$ to characterize the effect of random jitter for computational simplicity $\Delta r$ is the offset distance between the center of the fiber termination and the center of the beam after equivalence, $f$ is the focal length of the coupling lens. Then the probability density distribution of $\Delta r$ is:

$${f_{\Delta r}}(\Delta r)=\dfrac{\Delta r}{\sigma_r^2}\text{exp}\left(-\dfrac{\Delta r^2}{2\sigma_r^2}\right).$$

$\sigma _r^2$ is the variance of the $\Delta r$. Under the influence of random jitter, an exponential term factor is added to the expression of the mode of the back-propagated fiber to the lens according to [14]:

$$\small F_J(r,\theta, \Delta r )=\left[ \frac{B_{pl_F}}{\omega _a}(\frac{\sqrt{2}r}{\omega _a})^{l_F}L_{p}^{l_F}\left( \frac{2r^2}{\omega _{a}^{2}} \right) \exp\left( -\frac{r^2}{\omega _{a}^{2}} \right) \exp(-\mathrm{i} l_F\theta ) \right] \exp \left[ \mathrm{i}\frac{2\pi}{\lambda f}{\text{cos}}(\theta -\Omega )r\Delta r \right]$$

Hence, the expression for the energy function of the BG beams coupled to the fiber under the influence of random jitter $P_{cJ}$ is:

$$\begin{aligned} P_{cJ}=\int_0^{R_L} & {\int_0^{R_L}}\int_0^{2\pi}\int_0^{2\pi}{\int_0^{\infty}{\int_0^{\infty}}}E\left( r_1,\theta _1 \right) F_J^*(r_1,\theta _1,\Delta r_1)f(\Delta r_1)\\ & {E^*}\left( r_2,\theta _2 \right) F_J^{}(r_2,\theta _2,\Delta r_2)f(\Delta r_2)r_1r_2dr_1dr_2d\theta _1d\theta _2d\varDelta r_1d\varDelta r_2 \end{aligned}$$

Substituting both the probability density distribution $f(\Delta r)$ and the modified fiber field expression $F_J(r,\theta )$ into the coupling energy calculation equation, we can obtain the detailed expression of the $P_c$. While studying the effect of random jitter on the coupling efficiency of the vortex beams, due to the complexity of the computational process and the difficulty of the solution, this article only consider the case where $l_F$=$l_E$.

The Bessel function properties and the integral formulas as above and below are applied to obtain $P_c$ as a dual integral expression with respect to $r_1$ and $r_2$ [28,30].

$$\begin{aligned} & \exp (a \cos \theta) =\sum_{n={-}\infty}^{\infty} I_n(a) \exp ({-}i n \theta) \\ \int_0^{\infty} x\exp & \left(-\alpha x^2\right) J_0(\beta x) d x={\frac{1}{2 \alpha} \exp \left(\frac{-\beta^2}{4 \alpha}\right)} \end{aligned}$$

The final expression for the $P_{cJ}$ is obtained as:

$$\scalebox{0.9}{$\begin{aligned} P_{cJ}= & \int_0^{R_L}{}\int_0^{R_L}\sum_{\alpha _4=1}^M\sum_{\alpha _3=1}^M\sum_{\alpha _2=1}^M\sum_{\alpha _1=1}^M{{{U_{\alpha _4}}^*{U_{\alpha _3}}^*{U_{\alpha _2}}{U_{\alpha _1}}}}\left( \frac{k^2\pi B_{pl_F}}{2B_2B_1\omega _a} \right) ^2(\frac{2r_1r_2}{{\omega _a}^2})^{l_F}\exp \left[ \frac{\mathrm{i}kD_2}{2B_2}\left( r_{1}^{2}-r_{2}^{2} \right) \right]\\\times & \exp\left\{ -\left[ \frac{1}{\omega _{a}^{2}}+\left( \frac{\sqrt{2}\pi \sigma}{\lambda f} \right) ^2 \right] \left( {r_1}^2+{r_2}^2 \right) \right\} \left[ L_{p}^{l_F}\left( \frac{2{r_1}^2}{\omega _{a}^{2}} \right) \right] ^*\left[ L_{p}^{l_F}\left( \frac{2{r_2}^2}{\omega _{a}^{2}} \right) \right] {T_{r_1,\alpha_1,\alpha _2}T^*_{r_2,\alpha _3,\alpha _4}}r_1r_2dr_1dr_2 \end{aligned}$}$$

The expression for the fiber coupling efficiency under the influence of random jitter $\eta _{c J}$ is

$$\eta_{cJ}=\frac{P_{c J}}{P_r}$$

2.5 Total transmission efficiency of the beam

The efficiency of the beam and fiber coupling of the BG beams after passing through the transmitting and receiving antennas and the different influencing factors were studied above. In this section, the calculation model of the entire transmission efficiency of the BG beams from generation to coupling fiber is established. According to the study [31], the efficiency of the BG beams passing through the Cassegrain antenna system is:

$$\eta_A=\epsilon^4\frac{P_r}{P_0}=\epsilon^4\frac{\int_0^{2 \pi} \int_0^{R_A} E\left(r^{\prime}, \theta^{\prime}\right) E^*\left(r^{\prime}, \theta^{\prime}\right) r^{\prime} d r^{\prime} d \theta^{\prime}}{\int_0^{2 \pi} \int_0^{\infty} E_0(r, \theta) E_0^*(r, \theta) r d r d \theta}$$
where $E\left (r^{\prime },\theta ^{\prime }\right )$ is the light field function of the BG beams after passing through the transmitting and receiving antennas, ${R_A}$ is the radius of the primary mirror of the antenna, $\epsilon$ denotes the reflectivity of the antenna mirror (both transmitting and receiving antennas have two mirrors), and according to to [2], $\epsilon = 99.12{\%}$. ${P_0}$ denotes the energy function of the ideal BG beams at the light source. On this basis, we define the expression for the total transmission efficiency of the BG beams over the entire free-space communication system process without turbulence and random jitter $\eta _S$, the total transmission efficiency with random jitter but without turbulence $\eta _{SJ}$ and the total transmission efficiency with turbulence but without random jitter $\eta _{ST}$ are:
$$\left\{\begin{array}{l} \eta_{SJ}=\eta_A \cdot \eta_{C J}, \text{With random jitter but without turbulence } \\ \eta_{ST}=\eta_A \cdot \eta_{C T}, \text{ With turbulence but without random jitter }\\ \eta_{S}=\eta_A \cdot \eta_{C}, \text{ Without turbulence and random jitter } \end{array}\right.$$

3. Results and analysis

The optimal parameters of the system should be determined first to accurately analyze the effects of different factors on the coupling efficiency. After determining the optimal parameters of the antenna system as well as the coupling system, we will also analyze the effects of propagation distance and atmospheric turbulence on the coupling efficiency, the relationship between the coupling efficiency and the intensity of random jitter, and the degree of crosstalk between different modes of the BG beams.

3.1 Optimal parameters of the Cassegrain antenna system and coupling lens

According to previous work [2] [8] [12], partial parameters of system can be set as: $\lambda = 1550$ nm, $\Gamma = 10^{-5}$, $F_1$ = 0.5 m, $L_1 = 1$ m, $L_3 =$1 m, $\tau _0=1.46$, $\zeta =0.01$, ${R_F} = 50\mu$m, and $C_n^2=10^{-15}\text {m}^{-2/3}$. Lenses are assumed to have the same radius as the secondary mirror of the antenna $R_L=R_a$, and the ratio of the focal length of the primary mirror of the antennas to the secondary mirror of the antennas $F_1/F_2$ is set as the same as $\gamma$ make full use of the antennas.

Although atmospheric turbulence and random jitter may affect the antenna parameters, the focus of the article is not solving for the optimal antenna parameters, so the effects of atmospheric turbulence and lens jitter on the antenna parameters are ignored to simplify the model calculations. It is assumed that the propagation distance of BG beams $L_2$ = 10 m. In this section, the radius of the primary and secondary mirrors of the antenna are first determined. The average transmission efficiency of all modes is chosen as the evaluation metric, which can ensure a lower depletion of the antenna system. A three-dimensional plot of antenna transmission efficiency versus different radii of the secondary mirror and sheltering ratio conditions is obtained Fig. 4. Under ideal conditions, the beam transmission efficiency would be maximum when the radius of the primary mirror is infinite and the sheltering rate is close to zero. However, the smaller the sheltering ratio means that it is more difficult to guarantee accuracy in the actual assembly process, thus leading to large assembly errors, which also has a great impact on the beam transmission efficiency. Thus we choose a compromise ratio based on the results in the figure, i.e., $\gamma =0.15$, $R_a$ is $1.25$ cm, and the $R_A$ is $8.33$ cm in this case. The focal length of the primary mirror $F_1$ and the focal length of the secondary mirror $F_2$ are $0.5$ m and $0.0625$ m. It can be seen that the antenna transmission efficiency shows a trend of increasing first and then decreasing with the radius of the secondary mirror, which is related to the radius of the BG beams. And it can be seen that when $R_a$ is small, the effect of sheltering rate on antenna efficiency is very weak, while when the radius of the secondary mirror increases, the effect of $\gamma$ on antenna efficiency is more significant.

 figure: Fig. 4.

Fig. 4. Average transmission efficiency of the BG beams of $l_E=1-5$ passing through the antenna without the effect of turbulence as a function of the radius of the secondary mirror $R_a$ and the sheltering rate $\gamma$

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After obtaining the optimal antenna radius, the optimal focal length of the coupling lens is investigated. From the expression of the LG mode and the properties of the fiber field we know that BG beams of order $l_E$ will couple with fiber field modes that have the same $l_F$ but different $p$, and thus the efficiency of the BG beams coupling to the fibers for different $l_E$ values as a function of the lens focal length for the cases of $p$ = 0, 1, and 2 is ploted, which can be seen in Fig. 5.

 figure: Fig. 5.

Fig. 5. The relationship between the coupling efficiency of the BG beams $\eta _c$ after passing through the antenna system with the PF and the focal length of the lens $f$ in different cases $l_F=l_E$. (A1) $p=0$, (A2) $p=1$, (A3) $p=2$, (B) The efficiency of coupling the BG beams with fiber modes of different $p$ when $l_E=l_F=3$

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As shown in Fig. 5(A1), when $p$ = 0, BG beams of different orders can have high coupling efficiency with the fiber, especially at $f=7.80$ cm, the values of the coupling efficiency of BG beams with $l_E=2,3,4,5$ reach the peak at the same time, and the value of the coupling efficiency of BG beams with $l_E=1$ also reaches nearly $94{\%}$. As can be seen from Fig. 5(A1-A3), the BG beams coupling efficiency of all orders shows a decreasing trend as $p$ increases. And the optimal focal length for $p$=1,2 are $4.24$cm and $4.05$cm.

After calculating the average coupling efficiency for different orders the best focal length of the lens for $p=1$ is chosen to be $4.24$ cm and the best focal length for $p=2$ is $4.05$ cm. Figure 5(B) shows the variation of coupling efficiency with lens focal length when coupling the BG beams into the fiber at $l_E=3=l_F$. We can assume that the total efficiency of coupling $l$-th order BG beams to the PF is the sum of the coupling efficiency of the modes with the same $l$ but different $p$. In addition, through Fig. 5(B), we also figure out that the appropriate selection of the focal length of the coupling lens not only makes the coupling efficiency of the BG beams for $p=0$ close to 1, but also the coupling efficiency for other kinds of $p$ is close to zero, which indicates that by calculating the appropriate focal length of the coupling lens in the BG beams coupling process can better avoid the beam coupling with low-efficiency fiber mode, and effectively improve the coupling efficiency of the BG beams in PF.

Fig. 6 shows the visualization of a BG beam propagating in a fiber at different orders without considering phase change. Fig. 6(A1, B1) shows the optical field distribution of BG beams for $l_E=1,3$. Figure 6(A2-A4) shows the distribution of fiber guide field for different $p$ when $l_F=1$. Figure 6(B2-B4) shows the distribution of fiber guide field for different $p$ when $l_F=3$. It can be seen that when $l_F=l_E$, the smaller the $p$, the higher the overlap between the BG beams and the parabolic-index fiber guide field, and when $p=3$ the BG beams is mostly in the hollow range of the guide field, which explains why the coupling efficiency decreases with the increase of $p$. When studying different $l_E$ and $L_F$ cases, it can be found that the aperture range of higher-order BG beams is larger, and the radius of the guided-mode field in the fiber is significantly larger for $l_F=3$ than that for $l_F=1$, which makes the overlap between the fiber field and the beam to be lower, which explains the low coupling efficiency of higher-order BG beams to the fiber to some extent.

 figure: Fig. 6.

Fig. 6. (A1,B1) Diagrams of the optical field distribution of the BG beams when $l_E=1,3$ and the distribution of the fiber guide mode field patterns with different $p$,(A2-A4) $l_F=1$, (B2-B4) $l_F=3$

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In order to show the results of the model calculation more detailed, Table 2 shows the transmission efficiency, coupling efficiency, and total efficiency of different orders of BG beams for the ideal case when $p=0$, $f=7.8$ cm, and the propagation distance $L_2$ is $1$ km without the effect of turbulence.

Tables Icon

Table 2. $\eta _A$, $\eta _c$, and $\eta _S$ for different orders of BG beams propagating 1 km without the effect of turbulence.

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3.2 Effect of the distance transmitted in atmospheric turbulence on the coupling efficiency of the BG beams

The optimal antenna radius, antenna mirror focal length and coupling lens focal length at different $p$ have been determined above, and this work proceeds to analyze the effect of the distance the beam propagates in atmospheric turbulence on the coupling efficiency $\eta _{cT}$. Since the free-space optics communication system is greatly influenced by the environment, and the farther the beam is transmitted in the atmosphere, the greater the effect of turbulence. Thus, it is necessary to study the coupling efficiency of the beam at different transmission distances and different p-values.

Figure 7(A1-A3) show the coupling efficiency versus transmission distance for BG beams with different $l_E$ for $p=0, 1, 2$, respectively, while Fig. 7(B) shows the normalized coupling efficiency versus transmission distance for BG beams with $l_E=3=l_F$ for different $p$ and the corresponding optimal lens focal lengths.

 figure: Fig. 7.

Fig. 7. The relationship between the coupling efficiency of the BG beams $\eta _c$ with the PF and the propagation distance $L_2$ in different cases, $l_F=l_E$. (A1) $p=0$, (A2) $p=1$, (A3) $p=2$, (B) The variation of normalized coupling efficiency of the BG beams $\eta _{cTN}$ with fiber modes of different $p$ when $l_E=l_F=3$ with transmission distance.

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The results in Fig. 7(A1) show that when $p=0$, the coupling efficiency of BG beams of different orders is reduced more obviously. When the transmission distance $L_2$ is $2\sim 4$ km, the coupling efficiency of BG beams decreases faster, and the coupling efficiency of BG beams of different orders decreases to about 22% when the transmission distance reaches 6 km. However, the coupling efficiency of the 1st order BG beams in the close transmission is not as high as that of the other orders, but the coupling efficiency of the 1st order BG beams in the long-range transmission is significantly improved compared with that of the other orders. The results also show that the coupling efficiency of the 1st-order beam decreases more slowly with distance. Which means the lower the order of the BG beam has a stronger resistance to turbulence effects during transmission.

When $p=1$, Fig. 7(A2) illustrates that the coupling efficiency of different orders also decreases to a larger extent, but the coupling efficiency of the 1st-order BG beams are still higher than that of the beams of other orders. Figure 7(A3) shows the situation when $p=2$, the coupling efficiency of the 1st-order beam decreases significantly slower, and similar to the situation when $p=0$, the coupling efficiency of the 1st-order beam in long-range transmission has a significant improvement over the coupling efficiency of the other orders. Overall, when $p=0, 1, 2$, the beams of different orders fluctuate to a large extent at a low transmission distance of $0\sim 1$ km, among which the vibration of the 1st-order beam is the strongest. In order to compare more clearly the rate of change of beam coupling efficiency for different $p$, the normalized coupling efficiency variation is plotted for BG beams with $l_E=3$ coupled with different $p$ as an example. For $l_E=l_F=3$, the normalized coupling efficiency decays considerably in the course of $1\sim 3$ km, but the normalized coupling efficiency at $p=0$ decreases more slowly and maintains a higher efficiency for long-range transmission. Thus, the analysis shows that the BG beams with $l_E=1$ not only have higher long-range coupling efficiency but also is more resistant to turbulence. And for the BG beam with $l_E=1$, higher efficiency can be achieved when coupling to the fiber mode of $p=0$.

3.3 Effect of different variances of random jitter on coupling efficiency

In this work, the effect of the random jitter of the lens on the coupling efficiency has been modeled and derived. In this section, the effect of random jitter of different magnitudes on the coupling efficiency is analyzed and compared. The ratio of the standard deviation $\sigma _r$ to the radius of the mode field $\omega _0$ is used as the object of our study. To observe the variation of efficiency at different $p$, the analysis is performed separately using the focal length of the lens determined in Section 3.1. Fig. 8(A-C) show the coupling efficiency of BG beams versus random jitter for different $l_E$ when $p$ is 0, 1, and 2, respectively.

 figure: Fig. 8.

Fig. 8. Effect of the intensity of random jitter on the coupling efficiency of BG beams $\eta _c$ of different orders $l_E$ with fiber fields of different $p$ when $l_F=l_E$. (A):$p=0$, (B):$p=1$, (C):$p=2$.

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Figure 8(A) shows that when $p=0$, the coupling efficiency of BG beams of different orders decreases as the random jitter variance increases. When the variance also varies from $0\sim 0.3\omega _0$, the coupling efficiency decreases significantly faster for higher order BG beams, while for BG beams with $l_E=1$, the coupling efficiency does not change significantly and still retains a coupling efficiency of $50{\%}$ or more. And as the BG beam order increases, the random jitter has an increasing effect on the beam coupling efficiency. Figure 8(B) shows that when $p=1$, the coupling efficiency decreases more rapidly for BG beams of different orders than $p=0$. And the coupling efficiency at $l_E=1$ has a higher coupling efficiency compared with other orders. The analysis of the case at $p=2$ in Fig. 8(C) shows that the coupling efficiency decays faster with the increase of the random jitter, and the beam coupling efficiency of all orders is close to 0 when $\sigma _r=0.3\omega _0$.

And Fig. 9 shows the trend of normalized coupling efficiency for different $p$ with $l_E=l_F=3$ as an example.

 figure: Fig. 9.

Fig. 9. The variation of normalized coupling efficiency of the BG beams under the influence of random jitter $\eta _{cJN}$ with fiber modes of different $p$ when $l_E=l_F=3$ with different $\sigma _r$.

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It is obvious from the figure that as the random jitter increases, the 3rd order BG beams coupling efficiency with $p=0$ decreases significantly slower than the other two. This is not only consistent with the results analyzed in Fig. 8, but also fully illustrates that the mode with $p=0$ has good jitter resistance, so it is important to ensure that the BG beams are coupled with the fiber mode with $p=0$ as much as possible in the practical application.

4. Conclusion

In this work, a mathematical computational model of the BG beams passing through the Cassegrain antenna system and then coupled with a PF is derived, thus achieving the study and modeling of the whole process from generation to reception of the vortex beams. The influence of antenna sizes, atmospheric turbulence, and random jitter of the coupling system on the coupling efficiency and antenna transmission efficiency are also fully considered. The optimal size of antenna primary and secondary mirrors, the optimal focal length of the coupling lens, as well as analyzes are also determined. The characteristics of different guide mode fields in PF are compared. The results show that the optimal radius of the primary and secondary mirrors are $1.25$ cm and $8.33$ cm for BG beams with a wavelength of 1550 nm. The optimal focal length of the coupling lens varies with the BG beams coupled to PF field patterns of different $p$. For $p=0, 1, 2$, the corresponding optimal lens focal lengths are $7.8$ cm, 4.24 cm, and $4.05$ cm, respectively. After taking into account the transmission efficiency of the antenna system, the total efficiency of the system for BG beams of different orders averages 76.33${\%}$, when the transmission distance is 1 km.

The results show that the resistance to turbulence and random jitter of the BG beam decreases continuously as the order of the beam increases. When a BG beam of a certain order is coupled to the fiber, when lE = lF, the fiber will have multiple modes due to the different p parameters. It is found that the coupling efficiency is greatest when the beam is coupled to a mode with p = 0, and the coupling efficiency is influenced least by turbulence and random jitter. More importantly, when lE = lF, the maximum values of BG beams coupled to the fiber p = 0 mode for different orders correspond to almost the same lens focal length, which provides a guide for the simultaneous propagation of BG beams of different orders. Since the maximum efficiency of the BG beam coupled to the fiber p = 0, 1, and 2 modes correspond to different lens focal lengths, and the p = 0 mode has so many advantages, we recommend choosing the optimal parameters for the p = 0 mode in practical applications.

Funding

National Natural Science Foundation of China (11574042); Natural Science Foundation of Sichuan Province (501100018542) (2022NSFSC0561).

Disclosures

The authors declare no conflicts of interest.

Data availability

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

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Data availability

Data underlying the results presented in this work 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. The schematic diagram of the whole process of BG Beams coupling into PF after passing through the Cassegrain antenna system and atmospheric turbulence

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Fig. 2.
Fig. 2. Schematic diagram of BG beams with phase distortion under the influence of atmospheric turbulence coupled to the fiber after passing through the coupling lens

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Fig. 3.
Fig. 3. The effect of random jitter of the lens on the coupling process. (A) is a profile view of the jitter process (B) is a schematic diagram showing the offset of the fiber plane from the optical field as seen from the z-direction

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Fig. 4.
Fig. 4. Average transmission efficiency of the BG beams of $l_E=1-5$ passing through the antenna without the effect of turbulence as a function of the radius of the secondary mirror $R_a$ and the sheltering rate $\gamma$

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Fig. 5.
Fig. 5. The relationship between the coupling efficiency of the BG beams $\eta _c$ after passing through the antenna system with the PF and the focal length of the lens $f$ in different cases $l_F=l_E$ . (A1) $p=0$ , (A2) $p=1$ , (A3) $p=2$ , (B) The efficiency of coupling the BG beams with fiber modes of different $p$ when $l_E=l_F=3$

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Fig. 6.
Fig. 6. (A1,B1) Diagrams of the optical field distribution of the BG beams when $l_E=1,3$ and the distribution of the fiber guide mode field patterns with different $p$ ,(A2-A4) $l_F=1$ , (B2-B4) $l_F=3$

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Fig. 7.
Fig. 7. The relationship between the coupling efficiency of the BG beams $\eta _c$ with the PF and the propagation distance $L_2$ in different cases, $l_F=l_E$ . (A1) $p=0$ , (A2) $p=1$ , (A3) $p=2$ , (B) The variation of normalized coupling efficiency of the BG beams $\eta _{cTN}$ with fiber modes of different $p$ when $l_E=l_F=3$ with transmission distance.

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Fig. 8.
Fig. 8. Effect of the intensity of random jitter on the coupling efficiency of BG beams $\eta _c$ of different orders $l_E$ with fiber fields of different $p$ when $l_F=l_E$ . (A): $p=0$ , (B): $p=1$ , (C): $p=2$ .

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Fig. 9.
Fig. 9. The variation of normalized coupling efficiency of the BG beams under the influence of random jitter $\eta _{cJN}$ with fiber modes of different $p$ when $l_E=l_F=3$ with different $\sigma _r$ .

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Tables (2)

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Table 1. Parameters Table

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Table 2. $\eta _A$ , $\eta _c$ , and $\eta _S$ for different orders of BG beams propagating 1 km without the effect of turbulence.

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Equations (28)

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$$\begin{aligned}E_0(r,\theta, z)=\exp\left(-\frac{r^2}{\omega^2}\right)J_{l_E}(k_rr)\exp \left(-\mathrm{i} l_E\theta\right)\exp \left(-\mathrm{i} k z\right)\end{aligned}$$
$$\small \begin{aligned} E & (r, \theta,z)= \frac{\mathrm{i}^{{2l_E+2}} k^2}{4 B_2 B_1} \exp \left(-\mathrm{i} k z\right) \exp \left(-\mathrm{i} l_E \theta\right) \exp \left(\frac{\mathrm{i} k D_2}{2 B_2} r^2\right) \sum_{\alpha_2=1}^M \sum_{\alpha_1=1}^M {U_{\alpha_2} U_{\alpha_1} }T_{{r,\alpha_1,\alpha _2}} \end{aligned}$$
$$\small \begin{aligned} & T_{{r,\alpha_1,\alpha _2}}= \\ & \left\{ \begin{aligned} & \frac{1}{S_{1\alpha _1}M_{1\alpha _1,\alpha _2}}\exp \left( -\frac{k_{r}^{2}}{4S_{1\alpha _1}} \right) \exp \left( \frac{N_{1\alpha _1}^{2}}{4M_{1\alpha _1,\alpha _2}} \right) \exp \left[ -\frac{\left( k{r} \right) ^2}{4M_{1\alpha _1,\alpha _2}B_{2}^{2}} \right] J_{l_E}\left( \frac{N_{1\alpha _1}k{r}}{2M_{1\alpha _1,\alpha _2}B_2} \right) \\ & -\frac{1}{S_{2\alpha _1}M_{2\alpha _1,\alpha _2}}\exp \left( -\frac{k_{r}^{2}}{4S_{2\alpha _1}} \right) \exp \left( \frac{N_{2\alpha _1}^{2}}{4M_{2\alpha _1,\alpha _2}} \right) \exp \left[ -\frac{\left( k{r} \right) ^2}{4M_{2\alpha _1,\alpha _2}B_{2}^{2}} \right] J_{l_E}\left( \frac{N_{2\alpha _1}k{r}}{2M_{2\alpha _1,\alpha _2}B_2} \right)\\ & -\frac{1}{S_{1\alpha _1}M_{3\alpha _1,\alpha _2}}\exp \left( -\frac{k_{r}^{2}}{4S_{1\alpha _1}} \right) \exp \left( \frac{N_{1\alpha _1}^{2}}{4M_{3\alpha _1,\alpha _2}} \right) \exp \left[ -\frac{\left( k{r} \right) ^2}{4M_{3\alpha _1,\alpha _2}B_{2}^{2}} \right] J_{l_E}\left( \frac{N_{1\alpha _1}k{r}}{2M_{3\alpha _1,\alpha _2}B_2} \right)\\ & +\frac{1}{S_{2\alpha _1}M_{4\alpha _1,\alpha _2}}\exp \left( -\frac{k_{r}^{2}}{4S_{2\alpha _1}} \right) \exp \left( \frac{N_{2\alpha _1}^{2}}{4M_{4\alpha _1,\alpha _2}} \right) \exp \left[ -\frac{\left( k{r} \right) ^2}{4M_{4\alpha _1,\alpha _2}B_{2}^{2}} \right] J_{l_E}\left( \frac{N_{2\alpha _1}k{r}}{2M_{4\alpha _1,\alpha _2}B_2} \right) \\ \end{aligned} \right\} \end{aligned}$$
$$\left[\begin{array}{ll} A_1 & B_1 \\ C_1 & D_1 \end{array}\right]=\left[\begin{array}{ll} 1 & L_{2} \\ 0 & 1 \end{array}\right]\left[\begin{array}{cc} 1 & 0 \\ -\frac{1}{F_{1}} & 1 \end{array}\right]\left[\begin{array}{cc} 1 & F_{1}-F_{2} \\ 0 & 1 \end{array}\right]\left[\begin{array}{cc} 1 & 0 \\ \frac{1}{F_{2}} & 1 \end{array}\right]\left[\begin{array}{cc} 1 & L_{1} \\ 0 & 1 \end{array}\right]$$
$$\left[\begin{array}{ll} A_2 & B_2 \\ C_2 & D_2 \end{array}\right]=\left[\begin{array}{ll} 1 & L_{3} \\ 0 & 1 \end{array}\right]\left[\begin{array}{cc} 1 & 0 \\ \frac{1}{F_{2}} & 1 \end{array}\right]\left[\begin{array}{cc} 1 & F_{1}-F_{2} \\ 0 & 1 \end{array}\right]\left[\begin{array}{cc} 1 & 0 \\ -\frac{1}{F_{1}} & 1 \end{array}\right]$$
$${F_{p,l_F}}(r, \theta)=\frac{B_{p l_F}}{\omega_a}\left(\sqrt{2} \frac{r}{\omega_a}\right)^{l_F} L_p^{l_F}\left(\frac{2 r^2}{\omega_a^2}\right) \exp \left(-\frac{r^2}{\omega_a^2}\right) \exp(-\mathrm{i} l_F \theta)$$
$$\tau_r ^ { 2 } ( r ) = \tau _ { 0 } ^ { 2 } [ 1 - 2 \zeta ( \frac { r } { {R_F} } ) ^ { 2 } ]$$
$$\eta_c=\frac{P_c}{P_r}=\frac{{\left |\int _ { A } E ( r, \theta ) F ^ { * } ( r,\theta ) rd rd\theta \right |}^2}{\int_0^{R_L}{}\int_0^{2\pi}\left| E\left( r,\theta \right) \right|^2d\theta_{}rdr\int_0^{R_L}\int_0^{2\pi}\left| F\left( r,\theta \right) \right|^2d\theta rdr}$$
$${ \eta_{cT}} =\frac{P_{cT}}{P_r}=\frac{\int_0^{R_L}\int_0^{R_L}\int_0^{2\pi}\int_0^{2\pi}{W\left(r_1,\theta_1,r_2,\theta_2\right)} F^*\left(r_1,\theta_1 \right) F\left(r_2,\theta_{2} \right) d\theta_1 d\theta_2r_1dr_1r_2dr_2}{\int_0^{R_L}{}\int_0^{2\pi}\left| E\left( r,\theta \right) \right|^2d\theta_{}rdr\int_0^{R_L}\int_0^{2\pi}\left| F\left( r,\theta \right) \right|^2d\theta rdr}$$
$$W\left(r_1,\theta_1,r_2,\theta_2\right)=E\left(r_1,\theta_1\right) E^*\left(r_2,\theta_{2}\right)\exp\left\langle \psi \left( r_1,\theta_1 \right) +\psi^*\left( r_2,\theta_{2} \right)\right\rangle$$
$$\exp \left\langle\psi\left(r_1, \theta_1\right)+\psi^*\left(r_2, \theta_2\right)\right\rangle \cong \exp \left\{-\frac{1}{\rho_0^2}\left[r_1^2+r_2^2-2 r_1 r_2 \cos \left(\theta_1-\theta_2\right)\right]\right\}$$
$$\rho _{0}=(0.545k^{2}\int _{0}^{L_2}C_{n}^{2}(z)dz)^{{-}3/5}$$
$$\small C_{ n } ^ { 2 } ( v , h ) = 0.00594 ( v / 27 ) ^ { 2 } ( 10 ^ { - 5 } h ) ^ { 10 } \exp ( - h / 1000 ) + 2.7 \times 10 ^ { - 16 } \exp ( - h / 1500 ) + {G_\omega} \exp ( - h / 100 )$$
$$\rho_ { 0 } = ( 0.545 C_ {n} ^ {2 } k ^ { 2 } L_2 ) ^ { - 3 / 5 }$$
$${W}\left(r_1,\theta_1,r_2,\theta_2\right)=E\left(r_1,\theta_1\right) E^*\left(r_2,\theta_{2}\right) \exp \left\{-\frac{1}{\rho_0^2}\left[r_1^2+r_2^2-2 r_1 r_2 \cos \left(\theta_1-\theta_2\right)\right]\right\}$$
$$P_{cT}=\int_0^{R_L}\int_0^{R_L}\int_0^{2\pi}\int_0^{2\pi}{{W}\left(r_1,\theta_1,r_2,\theta_2\right)} F^*\left(r_1,\theta_1 \right) F\left(r_2,\theta_{2} \right) d\theta_1 d\theta_2r_1dr_1r_2dr_2$$
$$\exp\left( \mathrm{i} a\cos\theta \right) = \sum_{n={-}\infty}^{\infty}{\mathrm{i}^nJ_n\left( a \right) \exp\left({-}in\theta \right)}$$
$$\int_0^{2 \pi} \exp ({-}i n \theta)=\left\{\begin{array}{c} 2 \pi, n=0 \\ 0, n \neq 0 \end{array}\right.$$
$$\scriptsize \begin{aligned} & P_{cT}=\int_0^{R_L}\int_0^{R_L}\sum_{\alpha _4=1}^M\sum_{\alpha _3=1}^M\sum_{\alpha _2=1}^M\sum_{\alpha _1=1}^M{{{U_{\alpha _4}}^*{U_{\alpha _3}}^*{U_{\alpha _2}}{U_{\alpha _1}}}}\left( \frac{k^2\pi B_{pl_F}}{2B_2B_1\omega _a} \right) ^2\exp\left\{ -\frac{r_{1}^{2}+r_{2}^{2}}{\rho _{0}^{2}} \right\} \exp \left[ \frac{\mathrm{i}kD_2}{2B_2}\left( r_{1}^{2}-r_{2}^{2} \right) \right]\\ & I_{l_E-l_F}\left( \frac{2r_1r_2}{\rho _{0}^{2}} \right) \left[ (\frac{\sqrt{2}r_2}{\omega _a})^{l_F}L_{p}^{l_F}\left( \frac{2{r_2}^2}{\omega _{a}^{2}} \right) \exp\left( -\frac{{r_2}^2}{\omega _{a}^{2}} \right) \right] \left[ (\frac{\sqrt{2}r_1}{\omega _a})^{l_F}L_{p}^{l_F}\left( \frac{2{r_1}^2}{\omega _{a}^{2}} \right) \exp\left( -\frac{{r_1}^2}{\omega _{a}^{2}} \right) \right]^*{T_{r_1,\alpha_1,\alpha _2}T^*_{r_2,\alpha _3,\alpha _4}}r_1r_2dr_1dr_2 \end{aligned}$$
$$\left.{f_{\Delta \theta}}(\Delta \theta)\right|_{\phi=0}=\dfrac{\Delta \theta}{\sigma^2}\text{exp}\left(-\dfrac{\Delta \theta^2}{2\sigma^2}\right).$$
$${f_{\Delta r}}(\Delta r)=\dfrac{\Delta r}{\sigma_r^2}\text{exp}\left(-\dfrac{\Delta r^2}{2\sigma_r^2}\right).$$
$$\small F_J(r,\theta, \Delta r )=\left[ \frac{B_{pl_F}}{\omega _a}(\frac{\sqrt{2}r}{\omega _a})^{l_F}L_{p}^{l_F}\left( \frac{2r^2}{\omega _{a}^{2}} \right) \exp\left( -\frac{r^2}{\omega _{a}^{2}} \right) \exp(-\mathrm{i} l_F\theta ) \right] \exp \left[ \mathrm{i}\frac{2\pi}{\lambda f}{\text{cos}}(\theta -\Omega )r\Delta r \right]$$
$$\begin{aligned} P_{cJ}=\int_0^{R_L} & {\int_0^{R_L}}\int_0^{2\pi}\int_0^{2\pi}{\int_0^{\infty}{\int_0^{\infty}}}E\left( r_1,\theta _1 \right) F_J^*(r_1,\theta _1,\Delta r_1)f(\Delta r_1)\\ & {E^*}\left( r_2,\theta _2 \right) F_J^{}(r_2,\theta _2,\Delta r_2)f(\Delta r_2)r_1r_2dr_1dr_2d\theta _1d\theta _2d\varDelta r_1d\varDelta r_2 \end{aligned}$$
$$\begin{aligned} & \exp (a \cos \theta) =\sum_{n={-}\infty}^{\infty} I_n(a) \exp ({-}i n \theta) \\ \int_0^{\infty} x\exp & \left(-\alpha x^2\right) J_0(\beta x) d x={\frac{1}{2 \alpha} \exp \left(\frac{-\beta^2}{4 \alpha}\right)} \end{aligned}$$
$$\scalebox{0.9}{$\begin{aligned} P_{cJ}= & \int_0^{R_L}{}\int_0^{R_L}\sum_{\alpha _4=1}^M\sum_{\alpha _3=1}^M\sum_{\alpha _2=1}^M\sum_{\alpha _1=1}^M{{{U_{\alpha _4}}^*{U_{\alpha _3}}^*{U_{\alpha _2}}{U_{\alpha _1}}}}\left( \frac{k^2\pi B_{pl_F}}{2B_2B_1\omega _a} \right) ^2(\frac{2r_1r_2}{{\omega _a}^2})^{l_F}\exp \left[ \frac{\mathrm{i}kD_2}{2B_2}\left( r_{1}^{2}-r_{2}^{2} \right) \right]\\\times & \exp\left\{ -\left[ \frac{1}{\omega _{a}^{2}}+\left( \frac{\sqrt{2}\pi \sigma}{\lambda f} \right) ^2 \right] \left( {r_1}^2+{r_2}^2 \right) \right\} \left[ L_{p}^{l_F}\left( \frac{2{r_1}^2}{\omega _{a}^{2}} \right) \right] ^*\left[ L_{p}^{l_F}\left( \frac{2{r_2}^2}{\omega _{a}^{2}} \right) \right] {T_{r_1,\alpha_1,\alpha _2}T^*_{r_2,\alpha _3,\alpha _4}}r_1r_2dr_1dr_2 \end{aligned}$}$$
$$\eta_{cJ}=\frac{P_{c J}}{P_r}$$
$$\eta_A=\epsilon^4\frac{P_r}{P_0}=\epsilon^4\frac{\int_0^{2 \pi} \int_0^{R_A} E\left(r^{\prime}, \theta^{\prime}\right) E^*\left(r^{\prime}, \theta^{\prime}\right) r^{\prime} d r^{\prime} d \theta^{\prime}}{\int_0^{2 \pi} \int_0^{\infty} E_0(r, \theta) E_0^*(r, \theta) r d r d \theta}$$
$$\left\{\begin{array}{l} \eta_{SJ}=\eta_A \cdot \eta_{C J}, \text{With random jitter but without turbulence } \\ \eta_{ST}=\eta_A \cdot \eta_{C T}, \text{ With turbulence but without random jitter }\\ \eta_{S}=\eta_A \cdot \eta_{C}, \text{ Without turbulence and random jitter } \end{array}\right.$$
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