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Scintillation of the orbital angular momentum of a Bessel Gaussian beam and its application on multi-parameter multiplexing

Wanjun Wang, Guojun Zhang, Tianchun Ye, Zhensen Wu, and Lu Bai

Open Access Open Access

Abstract

The scintillation of the orbital angular momentum (OAM) of a Bessel Gaussian beam was derived based on the Rytov method to characterize the performance of the OAM communication. Moreover, a multi-parameter demultiplexing method was also proposed which could decode the OAM state, the amplitude and two additional beam width information dimensions. The advantages of the OAM states as the communication carrier over the beam intensity were that the minimum scintillation of the fundamental mode was smaller, and its corresponding radius also diverged slower. The coefficient of variation of the decoding amplitude was approximated to the square root of the radial minimum scintillation, and it provided an estimated decoding precision for the input sample selection. This study not only provided theoretical basis for communication link design, but also had a promising application on the large capacity beam multiplexing in free-space laser communication.

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

1. Introduction

The vortex beams have been widely studied in optical communication for its orbital angular momentum (OAM) could be applied as a new independent information dimension to increase the communication capacity [1–4]. As one crucial parameter to estimating system performance in the laser communication system, the scintillation index was proposed and widely investigated [5–7].

Multiple coaxially vortex beams with different OAM states could provide additional data carriers, and the OAM states can be separated based on the fast Fourier decomposition (FFT) of the wave-front [8–11]. The Bessel Gaussian beam as one type of the vortex beam was also widely studied [12–15], and it could carry more information than that of the OAM beam [3,12]. The information carried by Bessel Gaussian beams was also decoded with the FFT decomposition method [16–19]. However, this method could not make full use of the beam radial information. To increase the communication capacity, mixed multiplexing methods including different wavelengths and OAM in radial direction were studied [20–22]. However, these methods were hard to decode the OAM states in long distance communication, because there was serious overlapping and crosstalk of the OAM caused by the beam divergence after long distance propagation [23–26]. At this circumstance, a multi-parameter demultiplexing method with new information dimension was proposed, and it could decode the OAM state, amplitude and two additional beam widths shown as Fig. 1.

 figure: Fig. 1.

Fig. 1. Multi-parameter demultiplexing in turbulence.

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The beam scintillation theory was widely applied to determine the system performance of the optical communication in turbulence [27,28]. However, the beam OAM crosstalk will cause the intensity diffused into several harmonic modes, and the intensity of the fundamental mode as the signal carrier suffered a decrease [29]. Therefore, the scintillation of the total intensity was difficult to estimate the beam carrier performance. Although the error rate model could characterize the crosstalk strength of the beam harmonic intensity, it was difficult to reveal the influence of the radial signal fluctuation on the decoding precision of the parameters carried by the fundamental mode [29,30]. Moreover, an appropriate sample selection contributed to the improvement of the decoding precision, and the principle of the sample selection for the OAM decoding was less investigated. At this circumstance, the scintillation index of the normalized intensity of the fundamental mode was investigated to characterize the communication performance, and it also provided a theoretical basis for the sampling selection.

In this study, the scintillation index of the OAM state of the Bessel Gaussian beam was derived, and a multi-parameter demultiplexing method was also proposed which could decode the OAM state, the amplitude and two additional beam width information dimensions. Moreover, the relationship between the scintillation and the decoding precision was also analyzed. This study was not only conductive to the communication link design, but also had a promising application on the large capacity beam multiplexing in free-space laser communication.

2. Theory and formulation

2.1 Scintillation index of orbital angular momentum

The field distribution of the Bessel Gaussian beam with radial coordinates r and ${\varphi _r}$ was expressed as Eq. (1) [12,29].

$${U_0}(r,{\varphi _r}) = {J_n}(\beta r)\exp ( - k\alpha {r^2})\exp ( - in{\varphi _r}), $$
where n is the order of the Bessel function, k is the wave number, $\lambda $ is the wavelength, and $k = 2\pi /\lambda $. $\beta $ is the Bessel width. $\alpha = 1/kw_0^2 + i/(2{F_0})$, where ${w_0}$ is the Gaussian source width and ${F_0}$ is the focusing parameter. In this study, the beams are collimated beams and ${F_0} = \infty $.

The scintillation index was defined as Eq. (2) based on Eq. (9) of Chap. 8 in Ref. [6].

$$\sigma _I^2({{\mathbf r},L} )= \frac{{\left\langle {I{{({\mathbf r},L)}^2}} \right\rangle }}{{{{\left\langle {I({\mathbf r},L)} \right\rangle }^2}}} - 1 = \frac{{D[{I({\mathbf r},L)} ]}}{{{M^2}[{I({\mathbf r},L)} ]}}, $$
where $I({\mathbf r},L)$ is the beam intensity. $M[I({\mathbf r},L)]$ and $D[I({\mathbf r},L)]$ are the mean and the variance of the beam intensity.

Based on Eq. (2), the scintillation index of the normalized intensity of the fundamental mode could be expressed as Eq. (3).

$$\sigma _I^2({n_0}) = D({n_0})/{M^2}({n_0}), $$
where $M({n_0})$ and $D({n_0})$ are the mean and variance of normalized intensity of the beam fundamental mode, and they could be simplified as Eqs. (4) and (5), respectively [29].
$$M({n_0}) = \frac{1}{{{m^\textrm{2}}}}\sum\limits_{k = 1}^m {\sum\limits_{k^{\prime} = 1}^m {\exp[{2{E_1} + {E_2}({{{\mathbf r}_{\mathbf 1}},{{\mathbf r}_{\mathbf 2}}} )} ]} } {\kern 1pt} {\kern 1pt} = \frac{1}{m}{\kern 1pt} {\kern 1pt} \sum\limits_{k = 1}^m {\exp[{2{E_1} + {E_2}({{\theta_k}} )} ]}, $$
where ${\theta _k}\textrm{ = }2\pi k/m$ and ${E_2}(\theta )\textrm{ = }{E_2}(r,r\cos \theta )$. m is the sampling point number.
$$\begin{aligned} D({n_0}) &={-} {M^2} + \frac{1}{{{m^3}}}\sum\limits_{{k_1} = 1}^m {\sum\limits_{{k_2} = 1}^m {\sum\limits_{{k_3} = 1}^m {\exp} {\kern 1pt} [{4{E_1}} {\kern 1pt} } } + {E_2}({{\theta_1}} )+ {E_2}({{\theta_2}} )\\ &+ E_2^\ast ({|{{\theta_1} - {\theta_3}} |} )+ {E_2}({|{{\theta_2} - {\theta_3}} |} )+ {E_3}({{\theta_3}} )+ {E_3^\ast ({|{{\theta_1} - {\theta_2}} |} )} ]\end{aligned}, $$
where ${\theta _1}\textrm{ = }2\pi {k_1}/m$, ${\kern 1pt} {\theta _2}\textrm{ = }2\pi {k_2}/m$, ${\kern 1pt} {\theta _3}\textrm{ = }2\pi {k_3}/m$ and ${E_3}(\theta )\textrm{ = }{E_3}(r,r\cos \theta )$. ${E_1}$, ${E_2}$ and ${E_3}$ are the turbulence statistical moments of Bessel Gaussian beams which can be derived as Eqs. (6), (7) and (8) with the same method as Ref. [29,31,33]. And the phase can be expressed as Eq. (9).
$${E_1}({\mathbf r},{\mathbf r}) ={-} \pi {k^2}\int_0^L {d\eta } \int {\int_{ - \infty }^\infty {{d^2}\kappa } } {\Phi _n}(\kappa ), $$
$$\begin{aligned} {E_2}({{\mathbf r}_1},{{\mathbf r}_2}) &= 2\pi {k^2}{\left[ {{J_n}\left( {\frac{{\beta {r_1}}}{{1 + 2i\alpha L}}} \right)} \right]^{ - 1}}{\left[ {J_n^\ast \left( {\frac{{\beta {r_2}}}{{1 + 2i\alpha L}}} \right)} \right]^{ - 1}}\int_0^L {d\eta } \int {\int_{ - \infty }^\infty {{d^2}\kappa } } {\Phi _n}(\kappa )\\ &\times \exp \left[ {i{\mathbf K} \cdot (\gamma {{\mathbf r}_1} - \gamma \ast {{\mathbf r}_2}) - \frac{{i{\kappa^2}}}{{2k}}(\gamma - \gamma \ast )(L - \eta )} \right]\exp [{ - in({{\varphi_{K{r_1}}} - {\varphi_{K{r_2}}} - {\varphi_{{r_\textrm{1}}}} + {\varphi_{{r_2}}}} )} ]\\ &\times {J_n}\left[ { - \frac{{\beta (L - \eta )}}{{k({1 + 2i\alpha L} )}}\left|{{\mathbf K} - \frac{{k{{\mathbf r}_1}}}{{L - \eta }}} \right|} \right]J_n^\ast \left[ { - \frac{{\beta (L - \eta )}}{{k({1 + 2i\alpha L} )}}\left|{{\mathbf K} - \frac{{k{{\mathbf r}_2}}}{{L - \eta }}} \right|} \right] \end{aligned}$$
$$\begin{aligned} {E_3}({{\mathbf r}_1},{{\mathbf r}_2}) &={-} 2\pi {k^2}{\left[ {{J_n}\left( {\frac{{\beta {r_1}}}{{1 + 2i\alpha L}}} \right)} \right]^{ - 1}}{\left[ {{J_n}\left( {\frac{{\beta {r_2}}}{{1 + 2i\alpha L}}} \right)} \right]^{ - 1}}\int_0^L {d\eta } \int {\int_{ - \infty }^\infty {{d^2}\kappa } {\Phi _n}(\kappa )} \\ &\times \exp \left[ {i\gamma {\mathbf K} \cdot ({{\mathbf r}_1} - {{\mathbf r}_2}) - \frac{{i{\kappa^2}\gamma }}{k}(L - \eta )} \right]\exp [{ - in({{\varphi_{K{r_\textrm{1}}}} + {\varphi_{ - K{r_\textrm{2}}}} - {\varphi_{{r_\textrm{1}}}} - {\varphi_{{r_2}}}} )} ]\\ &\times {J_n}\left[ { - \frac{{\beta (L - \eta )}}{{k({1 + 2i\alpha L} )}}\left|{{\mathbf K} - \frac{{k{{\mathbf r}_1}}}{{L - \eta }}} \right|} \right]{J_n}\left[ { - \frac{{\beta (L - \eta )}}{{k({1 + 2i\alpha L} )}}\left|{{\mathbf K}\textrm{ + }\frac{{k{{\mathbf r}_2}}}{{L - \eta }}} \right|} \right] \end{aligned}$$
$$\exp ({i{\varphi_{\kappa r}}} )= \frac{{\kappa (L - z)\exp (i{\varphi _\kappa }) - rkexp (i{\varphi _r})}}{{|{\kappa (L - z)\exp (i{\varphi_\kappa }) - rk\exp (i{\varphi_r})} |}}, $$
where $\gamma = (1 + 2i\alpha \eta )/(1 + 2i\alpha L)$, and the symbol | | is the modulus of vectors. ${\varphi _{\kappa r}}$ is the angle of the vector ${\mathbf K} - k{(L - \eta )^{ - 1}}{\mathbf r}$. ${\Phi _n}(\kappa )$ is the power spectrum and the von Karman spectrum is used as Eq. (10) based on Eq. (19) of Chap. 3 in Ref. [6].
$${\Phi _n}(\kappa ) = 0.033C_n^2{({{\kappa^2} + \kappa_0^2} )^{ - 11/6}}\exp ({ - {\kappa^2}/\kappa_m^2} ), $$
where ${\kappa _m} = 5.92/{l_0}$ and ${\kappa _0} = 2\pi /{L_0}$. The ${l_0}$ and ${L_0}$ are the inner scale and the outer scale of the turbulence. $C_n^2$ is the refractive index structure constant.

In weak fluctuation condition, the mean and variance of the normalized intensity of the beam basic mode can be approximated as Eqs. (4) and (5), respectively (see appendix A).

$$M({n_0}) = 1 + 2{E_1} + \sum\limits_{k = 1}^m {{E_2}({{\theta_k}} )/m}, $$
$$D({n_0}) = \frac{2}{m}\sum\limits_{k = 1}^m {Re [{{E_2}({{\theta_k}} )+ {E_3}({{\theta_k}} )} ]}, $$
where ${\theta _k}\textrm{ = }2\pi k/m$.

When the turbulence strength is small, the mean of the normalized intensity of the fundamental mode is close to 1, and the scintillation index of the normalized intensity of the beam fundamental mode could be approximated as Eq. (13) by submitting the Eqs. (11) and (12) into Eq. (2).

$$\sigma _I^2({n_0}) \approx \frac{2}{m}\sum\limits_{k = 1}^m {{\mathrm{Re}} [{{E_2}({{\theta_k}} )+ {E_3}({{\theta_k}} )} ]}. $$

The scintillation index of the beam intensity could be expressed as Eq.(14) based on Eq. (13) in Chap. 8 in Ref. [6], and it is larger than that of the fundamental mode when the turbulence is weak (see appendix B).

$$\sigma _I^2 = 2{\textrm{Re}} [{{E_2}({{\mathbf r},{\mathbf r}} )+ {E_3}({{\mathbf r},{\mathbf r}} )} ]. $$

2.2 Multiple parameter multiplexing method

The field distribution of Bessel Gaussian beams on a receiver plane in the turbulence can be expressed as Eq. (15) based on the Rytov approximation as the Eq. (27) of Chap. 6 in Ref. [6].

$$U({\mathbf r},L) = {U_{free}}({\mathbf r},L)\exp [{\Psi ({\mathbf r},L)} ], $$
where ${U_{free}}({\mathbf r},L)$ is the field intensity in the free-space and expressed as Eq. (16) [32].
$${U_{free}}({\mathbf r},L) = A\frac{{\exp (ikL)}}{{1 + 2i\alpha L}}\exp \left[ { - \frac{{i\beta_{}^2L + 2\alpha {k^2}{r^2}}}{{2k({1 + 2i\alpha L} )}}} \right]{J_n}\left( {\frac{{\beta r}}{{1 + 2i\alpha L}}} \right)\exp ( - in{\varphi _r}).$$

The turbulence has less influence on the intensity angular distribution of Bessel Gaussian beam [6,33]. Moreover, the average turbulence influence is the function of the propagation distance as Eq. (17) based on the Eq. (27) in Chap. 6 in Ref. [6]. Therefore, assume that the angular symmetry of the beam field on the cross section is less influenced by the turbulence.

$$\left\langle {\exp [{\Psi ({\mathbf r},L)} ]} \right\rangle = \exp ({{E_1}} )\approx \exp ({ - 0.3908C_n^2{k^2}\kappa_0^{ - 5/3}L} ). $$

By submitting turbulence influence Eq. (17) into Eq. (15), the field distribution of the Bessel Gaussian in the turbulence could be expressed as Eq. (18).

$$U({\mathbf r},L) = {a_n}(r)\exp ( - in{\varphi _r}), $$
where ${a_n}(r)$ is the complex amplitude and expressed as Eq. (19).
$${a_n}(r) = A\frac{{\exp (ikL)}}{{1 + 2i\alpha L}}\exp \left( { - \frac{{i\beta_{}^2L}}{{2k + 4ik\alpha L}}} \right)\exp \left[ { - \frac{{2\alpha {k^2}{r^2}}}{{2k({1 + 2i\alpha L} )}}} \right]{J_n}\left( {\frac{{\beta r}}{{1 + 2i\alpha L}}} \right)\exp [{\Psi (L)} ].$$

The complex amplitude could also be calculated by the inverse fast Fourier transform of the field intensity shown as Eq. (18) and expressed as Eq. (20).

$${a_n}(r) = {{\cal F}^{ - 1}}[{{U_t}({r_m},{r_n},L)} ], $$
where ${U_t}({r_m},{r_n},L)$ is field intensity of the samplings, and it is calculated by the phase screen method in this study [29].

To make full use of the multidimensional information of the Bessel Gaussian beam, the beam amplitude and the two beam width parameters in Eq. (19) were selected to carry the information and represented as Eq. (21).

$${a_n}(r) = {C_1}\exp \left( { - \frac{{{C_3}k{r^2}}}{{1 + 2i{C_3}L}}} \right){J_n}\left( {\frac{{{C_2}r}}{{1 + 2i{C_3}L}}} \right), $$
where ${C_1}$, ${C_2}$ and ${C_3}$ are the parameters to be decoded. The demultiplexing method is designed based on that the complex amplitude of samplings as Eq. (20) is equal to the theoretical value as Eq. (19). Therefore, given ${a_n}(r)$ along the radius, complex parameters ${C_1}$, ${C_2}$ and ${C_3}$ can be obtained by searching the optimal solution with the nonlinear curve-fitting method in least-squares sense. The decoded parameters were complex numbers, and its magnitude was used approximate to the information carried by the beam as Eqs. (22) and (23).
$$A = \frac{1}{{\left\langle A \right\rangle }}\left|{{C_1}(1 + 2i{C_3}L)\exp \left( {\frac{{iC_2^2L}}{{2k + 4ik{C_3}L}}} \right)} \right|, $$
$$\beta = {\kern 1pt} |{{C_2}} |, \alpha = {\kern 1pt} {\kern 1pt} |{{C_3}} |,$$
where | | represents the magnitude of the complex number in Eqs. (22) and (23). $\left\langle A \right\rangle $ is the mean of the beam amplitude. The turbulence influence $\Psi (L)$ is unknown. Therefore, only the normalized amplitude could be decoded.

The bounds of decoding parameters are shown in Table 1. The decoding precision could be improved through fine-tuning the bound of the amplitude. The instantaneous influence of the turbulence mainly caused parts of the signal fluctuation rather than the whole intensity attenuation. Therefore, the parameters in free-space were selected as initial value of the demutiplexing method.

Tables Icon

Table 1. Bounds of parameters for different methods

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Based on Eqs. (19) and (21), the parameter T in Table 1 is expressed as Eq. (24).

$$T = \frac{{\exp (ikL)}}{{1 + 2i{k^{ - 1}}w_0^{ - 2}L}}\exp \left( { - \frac{{i{\beta^2}L}}{{2k + 4iw_0^{ - 2}L}}} \right). $$

In this study, the coefficient of variation was selected to characterize the fluctuation of decoding parameters, and it was defined as the standard deviation divided by the mean. The coefficient of variation was the positive square root of the scintillation index as Eq. (2).

3. Results and discussion

In this study, without special annotation, sampling point number around the circumference is $m = 72$, the wavelength is $\lambda = 1.55\mu m$. The turbulence scale is ${l_0} = 0.02m$ and ${L_0} = 2m$.

Figure 2 showed the scintillation of the beam normalized intensity and that of the normalized intensity of the fundamental mode. BG in figures represented Bessel Gaussian beam. The scintillation index was derived by the Rytov method, and it was also calculated by the phase screen method with an acceptable precision [29]. The scintillation calculated by the phase screen method was an average value of 1000 samplings. The scintillation derived by the Rytov method coincided well with that calculated by the phase screen method neglecting the paraxial data which did not meet the applicable condition of the Rytov method [31]. Moreover, the approximated scintillation index also had a good agreement with that derived by the Rytov method when the refractive index structure constant was small. There was a minimum scintillation index in the radial direction [29,7].

 figure: Fig. 2.

Fig. 2. Scintillation of beam intensity and that of normalized intensity of fundamental mode (a) $C_n^2 = 1 \times {10^{ - 15}}$, (b) $C_n^2 = 1 \times {10^{ - 16}}$.

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Figure 3 illustrated the scintillation index variation with the refractive index structure constant. The scintillation index enlarged with the turbulence strength increasing. The scintillation index derived by the Rytov method coincided well with that calculated by the phase screen method. Moreover, when the refractive index structure constant was small, the approximated scintillation index also had a good agreement with that derived by the Rytov method, and the scintillation index of the fundamental mode was smaller than that of the beam intensity which was proved in appendix B. With the turbulence strength increasing, the beam crosstalk became stronger which caused the beam power diffused to other crosstalk mode, and the mean of the normalized intensity of the fundamental mode decreased faster along the radius shown in Fig. 5(b) [29]. Therefore, the scintillation of the fundamental mode as Eq. (3) will become larger than that of the beam intensity with the refractive index structure constant increasing.

 figure: Fig. 3.

Fig. 3. Scintillation variation with the refractive index structure constant (a) L = 1000 m, (b) L = 2000 m.

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Figure 4 showed the scintillation index variation with the Rytov variance which was defined as $\sigma _R^2 = \textrm{1}\textrm{.23}C_n^2{k^{7/6}}{L^{11/6}}$ [6,29]. The radius was selected at the abscissa of the minimum scintillation index as Fig. 4(b). The radius of the minimum scintillation of the intensity was approximately unchanged with the turbulence strength. The beam diverged with the propagation distance increasing, and the radius of the minimum scintillation also enlarged. Although the scintillation of the beam at different propagation distance and in different strength turbulence could be similar when their Rytov variance was approximately the same, there was a bias between their radiuses. With the beam propagation distance increasing, the mean of the normalized intensity of the fundamental mode also decreased faster along the radius as Eq. (44) [29], which was similar to the mean variation with the turbulence strength shown in Fig. 5(b), and the radius of the minimum scintillation became smaller. Therefore, the radius of the minimum scintillation of the fundamental mode diverged slower than that of the beam intensity with the propagation distance increasing. Moreover, the minimum scintillation of the fundamental mode was also smaller than that of the intensity when the Rytov variance was small. Therefore, the OAM states as the communication carrier had a better performance on the decoding precision and the communication link design.

 figure: Fig. 4.

Fig. 4. (a) Scintillation index variation with Rytov variance; (b) corresponding radius.

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Figure 5(a) illustrated the radial minimum scintillation of the different order beam. The scintillation of the fundamental mode of the low order beam was smaller for the low order beam diverged slowly and its energy was more concentrated [31]. Moreover, the radius of the minimum scintillation of the fundamental mode of the different order beam also became smaller with the turbulence strength increasing. Figure 5(b) showed the statistics comparison of the normalized intensity. The turbulence only caused the beam divergence instead of the beam total power attenuation [31]. Therefore, the mean of the normalized intensity of the fundamental mode decreased to a trough, and then increased along the radial direction. However, the mean of the normalized intensity of the fundamental mode decreased along the radial direction, because the beam power diffused to other crosstalk mode and the beam coherence intensity reduced with the spatial correlation length increasing as Eq. (44) [29]. Moreover, with the turbulence strength increasing, the strengthen beam crosstalk caused more power diffuse to other harmonic mode. Therefore, the radial decrease of the mean normalized intensity of the fundamental mode became faster as Eq. (44), and the radius of the minimum scintillation of the fundamental mode also became smaller.

 figure: Fig. 5.

Fig. 5. (a) Different order beam scintillation; (b) normalized statistics comparison.

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Figure 6 showed the coefficient of variation of the beam decoding parameters varying with the beam width, and the influence of the two selection methods of input samplings on the coefficient of variation was also analyzed. Moreover, the coefficient of variation of the different width beam could also provide a theoretical support for the carrier beam selection. The solid legends represented the coefficient of variation varying with the Bessel width, and its abscissa was at the bottom of the figure. The hollow legends represented the coefficient of variation varying with the Gaussian width, and its abscissa was at the top of the figure. The coefficient of variation was defined as the standard deviation divided by the mean, and its square was the scintillation index. Tcv in the figures represented the theoretical coefficient of variation, which was equal to the square root of the radial minimum scintillation index of the fundamental mode derived by the Rytov method as Eq. (3). The derived minimum scintillation of the fundamental mode changed slowly with the beam width, and its corresponding radius was illustrated in Fig. 8(a). In this study, the coefficient of variation of the decoding parameters was an average value with 1000 samplings.

 figure: Fig. 6.

Fig. 6. Coefficient of variation decoded with different input samplings area. (a) Area with intensity larger than 50% of maximum intensity; (b) area with scintillation smaller than 110% of minimum scintillation.

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The intensity fluctuation and the beam wander were two major influence on the performance of the beam communication [6,34], and these two disadvantages could be improved through selecting less influenced samples as the input of the demultiplexing method. When the beam width was large, the precision of the decoding parameter was acceptable with input samplings in the area where the beam intensity was larger than half of maximum intensity as Fig. 6(a). Samplings in the area where the scintillation was smaller than 110% of the minimum scintillation could also be selected as the input of the demultiplexing method as Fig. 6(b), and the decoding precision was higher than that in Fig. 6(a) when the beam width was small. Therefore, an appropriate sample selection was conductive to the improvement of the decoding precision. However, the scintillation was a statistic moment and it was difficult to obtain through a transient sampling. Therefore, samplings in the area where the intensity was larger than a stated percentage of the maximum intensity, was recommended to be selected as the input of the demutiplexing method, and it was conductive to improve the decoding precision of the time-varying signal. The fluctuation of the beam width was much smaller than that of the beam amplitude in general, and the coefficient of variation of the decoding amplitude was approximated to the square root of the radial minimum scintillation of the fundamental mode derived by the Rytov method. Moreover, the decoding parameters were an average value of different radius, and the decoding precision could be slightly smaller than square root of the derived radial minimum scintillation. Therefore, the scintillation of the fundamental mode derived by the Rytov method could be used to estimate the performance of the communication system, and it also provided a theoretical instruction for the input sampling selection.

Figure 7 illustrated the coefficient of variation varying with the turbulence scale. The turbulence scale had little influence on the radius of the minimum intensity scintillation. The theoretical minimum scintillation of the fundamental mode which was the square of the coefficient of variation, increased to a constant with the inner scale decreasing or the outer scale increasing, and so was the coefficient of variation of the decoding parameters. Moreover, the radius of the minimum scintillation of the normalized intensity of the fundamental mode decreased with the inner or outer scale increasing.

 figure: Fig. 7.

Fig. 7. Coefficient of variation (a) with inner scale, (b) with outer scale.

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Figure 8 showed the decoding precision of the signal carried by the Bessel Gaussian beam with different number sub beams. Through the comparison between the precision in Fig. 8(a) and that in Fig. 6(a), the decoding precision could be improved by reducing the number of the decoding parameters, but there was a disadvantage that it will also reduce the communication capacity. When the beam width was larger, the decoding precision of the information carried by the beam with a constant Gaussian width was higher than that carried by a constant Bessel width beam. Therefore, the Bessel width was recommended as the information carrier. The multiple Bessel Gaussian beam with 3rd and 5th order sub beams was investigated in Fig. 8(b). The normalized constants as ${A_0}$ and ${\beta _0}$ were captioned in Fig. 8(b). Cv represented the coefficient of variation of the normalized decoding parameters which was equal to decoded parameters divided by input parameters. The demultiplexing method could also decode the information carried by the multiple Bessel Gaussian beams, and the communication capacity could be multiplied through increasing the number of the sub beams.

 figure: Fig. 8.

Fig. 8. Decoding precision of the demultiplexing parameters (a) single beam, (b) two sub beam.

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4. Conclusion

In this study, the scintillation of the OAM of Bessel Gaussian beam was derived under weak fluctuation conditions to characterize the performance of the OAM multiplexing communication. Moreover, a multi-parameter multiplexing method including new information dimensions was proposed to increase the communication capacity.

The scintillation of the normalized intensity of fundamental mode was derived based on the Rytov method, and an approximated method of the scintillation index was also proposed. The scintillation derived by the Rytov method coincided well with that calculated by the phase screen method, and it increased with the turbulence strength increasing. The scintillation increased to a constant with the inner scale decreasing or the outer scale increasing. The scintillation of the fundamental mode was smaller than that of the beam intensity when the turbulence strength was small. Moreover, the radius of the minimum scintillation of the fundamental mode decreased with the turbulence strength increasing, and it also diverged slower than that of the intensity with the propagation distance increasing. Therefore, the OAM states as the communication carrier had a better performance on the decoding precision and the communication link design.

A multiple-parameter demultiplexing method was proposed which could decode the OAM state, amplitude and two additional beam width parameters. Samplings in the area where the intensity was larger than a stated percentage of the maximum intensity was recommended to be selected as the input of the demutiplexing method to overcome the influence of the intensity scintillation and beam wander. The perturbation of the decoding amplitude was larger than that of the beam width. Moreover, with appropriate samplings as the input of the demultiplexing method, the coefficient of variation of the decoding amplitude could be close to the square root of the radial minimum scintillation of the fundamental mode derived by the Rytov method. Therefore, the scintillation could also be used to estimate the decoding precision as a theoretical basis for the sample selection.

This study on the OAM scintillation and multi-parameter multiplexing method not only provided theoretical basis for to the communication link design, but also had a promising application on the large capacity of the beam multiplexing in free-space laser communication.

Appendix A

The turbulence moment was close to zero in weak fluctuation condition. Therefore, the mean and variance of the beam basic mode can be approximated as Eqs. (25) and (26) by submitting Eq. (27) into Eqs. (4) and (5).

$$M({n_0}) \approx \frac{1}{m}{\kern 1pt} {\kern 1pt} \sum\limits_{k = 1}^m {[{1 + 2{E_1} + {E_2}({{\theta_k}} )} ]} = 1 + 2{E_1} + \sum\limits_{k = 1}^m {{E_2}({{\theta_k}} )/m}, $$
$$\begin{aligned} D({n_0}) &\approx{-} {M^2} + \frac{1}{{{m^3}}}\sum\limits_{{k_1} = 1}^m {\sum\limits_{{k_2} = 1}^m {\sum\limits_{{k_3} = 1}^m {[{1 + 4{E_1} + {E_2}({{\theta_1}} )} } {\kern 1pt} } } + {E_2}({{\theta_2}} )\\ &+ E_2^\ast ({|{{\theta_1} - {\theta_3}} |} )+ {E_2}({|{{\theta_2} - {\theta_3}} |} )+ {E_3}({{\theta_3}} )+ {E_3^\ast ({|{{\theta_1} - {\theta_2}} |} )} ]\end{aligned}, $$
$$\exp (z) \approx 1 + z, {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} z \ll 1. $$

The quadratic product term of the square mean was far less than the lower order terms, and the square of mean could be approximated as Eq. (28).

$${M^2} \approx 1 + 4{E_1} + 2\sum\limits_{k = 1}^m {{E_2}({{\theta_k}} )/m}. $$

Decomposed summation term in Eq. (26) and summed each item separately as Eqs. (29), (30) and (31).

$$\frac{1}{{{m^3}}}\sum\limits_{{k_1} = 1}^m {\sum\limits_{{k_2} = 1}^m {\sum\limits_{{k_3} = 1}^m {({1 + 4{E_1}} )} {\kern 1pt} } } = 1 + 4{E_1}, $$
$$\frac{1}{{{m^3}}}\sum\limits_{{k_1} = 1}^m {\sum\limits_{{k_2} = 1}^m {\sum\limits_{{k_3} = 1}^m {{E_j}({{\theta_i}} )} {\kern 1pt} } } = \frac{1}{m}\sum\limits_{k = 1}^m {{E_j}({{\theta_k}} )} ,{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} j = 2,3;{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \textrm{ }i = 1,2,3, $$
$$\frac{1}{{{m^3}}}\sum\limits_{{k_1} = 1}^m {\sum\limits_{{k_2} = 1}^m {\sum\limits_{{k_3} = 1}^m {{E_2}({|{{\theta_2} - {\theta_3}} |} )} {\kern 1pt} } } = \frac{1}{{{m^2}}}\sum\limits_{{k_2} = 1}^m {\sum\limits_{{k_3} = 1}^m {{E_2}({|{{\theta_2} - {\theta_3}} |} )} {\kern 1pt} }. $$

There was a $2\pi$ periodicity of angular angle of the turbulence statistic moment as Eq. (32).

$${E_2}({|{2\pi {k_2}/m - 2\pi {k_3}/m} |} )= \left\{ {\begin{array}{c} {{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {E_2}({|{2\pi {k_2}/m - 2\pi {k_3}/m} |} ), {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {k_2} \ge {k_3}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} }\\ {{E_2}({|{2\pi {k_2}/m - 2\pi {k_3}/m + 2\pi } |} ){\kern 1pt} {\kern 1pt} , {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {k_2} < {k_3}{\kern 1pt} } \end{array}}. \right.$$

By submitting Eq. (32) in to Eq. (31), the sum item could be simplified as Eq. (33).

$$\frac{1}{{{m^3}}}\sum\limits_{{k_1} = 1}^m {\sum\limits_{{k_2} = 1}^m {\sum\limits_{{k_3} = 1}^m {{E_2}({|{2\pi {k_2}/m - 2\pi {k_3}/m} |} )} {\kern 1pt} } } = \frac{1}{m}\sum\limits_{k = 1}^m {{E_2}({2\pi k/m} )}. $$

Other items of the turbulence statistic moment in Eq. (26) could be simplified with the same method as Eqs. (32) and (33).

Submitting Eqs. (28), (29), (30) and (33) into (26), the variance of the beam basic mode can be simplified as Eq. (34).

$$D({n_0}) = \frac{2}{m}{\textrm{Re}} \left[ {\sum\limits_{k = 1}^m {{E_2}({{\theta_k}} )+ {E_3}({{\theta_k}} )} } \right]. $$

Appendix B

Considering the applicable condition of the Rytov method, the inequality $|{k{\mathbf r}} |\gg |{{\mathbf K}({L - \eta } )} |$ could be met at most of the integration interval [31], and there are two approximations as Eq. (35).

$${J_n}\left[ { - \frac{{\beta (L - \eta )}}{{k({1 + 2i\alpha L} )}}\left|{{\mathbf K} - \frac{{k{\mathbf r}}}{{L - \eta }}} \right|} \right] \approx {J_n}\left( { - \frac{{\beta r}}{{1 + 2i\alpha L}}} \right), {\varphi _{Kr}} \approx {\varphi _r}.$$

There was an integral as Eq. (36) coming from the combination of 3.937(1) and (2) [35].

$$\int_0^{2\pi } {dx} \exp ({\beta cosx} )\exp ({ - inx} )= 2\pi {I_n}(\beta ). $$

The turbulence statistic moment could be approximated as Eqs. (37) and (38) by submitting Eqs. (35) and (36) into Eqs. (7) and (8).

$${E_2}({{\mathbf r}_1},{{\mathbf r}_2}) = 4{\pi ^2}{k^2}\int_0^L {d\eta } \int_0^\infty {\kappa d\kappa } {\Phi _n}(\kappa )\exp \left[ { - \frac{{i{\kappa^2}(\gamma - \gamma \ast )(L - \eta )}}{{2k}}} \right]{J_0}({\kappa |{\gamma {{\mathbf r}_1} - \gamma \ast {{\mathbf r}_2}} |} ),$$
$${E_3}({{\mathbf r}_1},{{\mathbf r}_2}) ={-} 4{\pi ^2}{k^2}\int_0^L {d\eta } \int {\kappa d\kappa {\Phi _n}(\kappa )} \exp \left[ { - \frac{{i{\kappa^2}\gamma (L - \eta )}}{k}} \right]{J_0}({\gamma \kappa |{{{\mathbf r}_1} - {{\mathbf r}_2}} |} ).$$

The wave number k was large, and $\kappa$ was small for the limitation of ${\Phi _n}(\kappa )$ in general. Therefore, there were two approximations as Eq.(39) in the beam bright area.

$${J_0}(z) \approx 1 - {z^2}/4, \exp (z) \approx 1 + z, z \ll 1.$$

Neglecting of the quadratic product term, the turbulence statistic moment could be approximated to Eqs. (40) and (41) through submitting Eq. (39) into Eqs. (37) and (38).

$${E_2}({{\mathbf r}_1},{{\mathbf r}_2}) = 4{\pi ^2}{k^2}\int_0^L {d\eta } \int_0^\infty {\kappa d\kappa } {\Phi _n}(\kappa )\left[ {1 - \frac{{2\alpha {\kappa^2}{{(L - \eta )}^2}}}{{k({1 + 4{\alpha^2}{L^2}} )}} - \frac{{{\kappa^2}{{|{\gamma {{\mathbf r}_1} - \gamma \ast {{\mathbf r}_2}} |}^2}}}{4}} \right],$$
$$\begin{aligned} {E_3}({{\mathbf r}_1},{{\mathbf r}_2}) &={-} 4{\pi ^2}{k^2}\int_0^L {d\eta } \int {\kappa d\kappa {\Phi _n}(\kappa )} \\ &\times \left[ {1 - \frac{{2\alpha {\kappa^2}{{(L - \eta )}^2}}}{{k({1 + 4{\alpha^2}{L^2}} )}} - \frac{{{\gamma^2}{\kappa^2}{{|{{{\mathbf r}_1} - {{\mathbf r}_2}} |}^2}}}{4} - \frac{{i{\kappa^2}(L - \eta )(1 + 4{\alpha^2}\eta L)}}{{k({1 + 4{\alpha^2}{L^2}} )}}} \right]. \end{aligned}$$

The scintillation index could be approximated as Eq. (42) through submitting Eqs. (40) and (41) into Eq. (13).

$$\sigma _I^2({n_0}) = \frac{{2{\pi ^2}{k^2}}}{m}\int_0^L {d\eta } \int_0^\infty {\kappa d\kappa } {\Phi _n}(\kappa )\sum\limits_{k = 1}^m {\textrm{Re}} [{ - {\kappa^2}{{|{\gamma {{\mathbf r}_1} - \gamma \ast {{\mathbf r}_2}} |}^2} + {\gamma^2}{\kappa^2}{{|{{{\mathbf r}_1} - {{\mathbf r}_2}} |}^2}} ].$$

The sampling points were selected on one circumference and there was a relationship as Eq. (43).

$$|{{{\mathbf r}_1}} |= |{{{\mathbf r}_2}} |= r, \quad {{\mathbf r}_1} \cdot {{\mathbf r}_2} = {r^2}\cos (\theta ).$$
where r is the radius and $\theta$ is the included angle between two vectors.

The mean of the normalized intensity of the fundamental mode could be approximated as Eq. (44) by submitting the Eqs. (40) and (43) into Eq. (11).

$$\begin{aligned} M({n_0}) &= 1 + 4{\pi ^2}{k^2}\int_0^\infty {d\kappa } \kappa {\Phi _n}(\kappa )\left[ { - \frac{{2\alpha {\kappa^2}{L^3}}}{{3k + 12k{\alpha^2}{L^2}}}} \right.\\ &- \frac{{{\kappa ^2}{r^2}}}{2}\left. {\left( {\frac{{L + 8{\alpha^2}{L^3}/3 + 16{\alpha^4}{L^5}/3}}{{1 + 8{\alpha^2}{L^2} + 16{\alpha^4}{L^4}}} - \frac{{1 + 4{\alpha^2}{L^3}/3}}{{1 + 4{\alpha^2}{L^2}}}\sum\limits_{k = 0}^{m - 1} {\frac{{cos{\theta_k}}}{m}} } \right)} \right]. \end{aligned}$$

The scintillation index of the fundamental mode could be derived as Eq. (45) by submitting Eq. (43) into Eq. (42).

$$\sigma _I^2({n_0}) = \frac{{32{\pi ^2}{k^2}{\alpha ^2}{r^2}}}{{m{{({1 + 4{\alpha^2}{L^2}} )}^2}}}\int_0^L {d\eta } \int_0^\infty {d\kappa } {\kappa ^3}{({L - \eta } )^2}{\Phi _n}(\kappa )\sum\limits_{k = 1}^m {\cos ({{\theta_k}} )}. $$

The scintillation index of the intensity as Eq. (14) could be approximated as Eq. (45) with ${\theta _k} = 0$, and the cosine function was largest at this moment. Therefore, the scintillation index of the fundamental mode was smaller than that of the intensity.

Funding

National Natural Science Foundation of China (62201566).

Disclosures

The authors declare no conflicts of interest.

Data availability

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

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Data availability

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

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Figures (8)
Fig. 1.
Fig. 1. Multi-parameter demultiplexing in turbulence.

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Fig. 2.
Fig. 2. Scintillation of beam intensity and that of normalized intensity of fundamental mode (a) $C_n^2 = 1 \times {10^{ - 15}}$, (b) $C_n^2 = 1 \times {10^{ - 16}}$.

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Fig. 3.
Fig. 3. Scintillation variation with the refractive index structure constant (a) L = 1000 m, (b) L = 2000 m.

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Fig. 4.
Fig. 4. (a) Scintillation index variation with Rytov variance; (b) corresponding radius.

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Fig. 5.
Fig. 5. (a) Different order beam scintillation; (b) normalized statistics comparison.

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Fig. 6.
Fig. 6. Coefficient of variation decoded with different input samplings area. (a) Area with intensity larger than 50% of maximum intensity; (b) area with scintillation smaller than 110% of minimum scintillation.

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Fig. 7.
Fig. 7. Coefficient of variation (a) with inner scale, (b) with outer scale.

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Fig. 8.
Fig. 8. Decoding precision of the demultiplexing parameters (a) single beam, (b) two sub beam.

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Tables (1)

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Table 1. Bounds of parameters for different methods

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Equations (45)

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U 0 ( r , φ r ) = J n ( β r ) exp ⁡ ( − k α r 2 ) exp ⁡ ( − i n φ r ) ,
σ I 2 ( r , L ) = ⟨ I ( r , L ) 2 ⟩ ⟨ I ( r , L ) ⟩ 2 − 1 = D [ I ( r , L ) ] M 2 [ I ( r , L ) ] ,
σ I 2 ( n 0 ) = D ( n 0 ) / M 2 ( n 0 ) ,
M ( n 0 ) = 1 m 2 ∑ k = 1 m ∑ k ′ = 1 m exp ⁡ [ 2 E 1 + E 2 ( r 1 , r 2 ) ] = 1 m ∑ k = 1 m exp ⁡ [ 2 E 1 + E 2 ( θ k ) ] ,
D ( n 0 ) = − M 2 + 1 m 3 ∑ k 1 = 1 m ∑ k 2 = 1 m ∑ k 3 = 1 m exp [ 4 E 1 + E 2 ( θ 1 ) + E 2 ( θ 2 ) + E 2 ∗ ( | θ 1 − θ 3 | ) + E 2 ( | θ 2 − θ 3 | ) + E 3 ( θ 3 ) + E 3 ∗ ( | θ 1 − θ 2 | ) ] ,
E 1 ( r , r ) = − π k 2 ∫ 0 L d η ∫ ∫ − ∞ ∞ d 2 κ Φ n ( κ ) ,
E 2 ( r 1 , r 2 ) = 2 π k 2 [ J n ( β r 1 1 + 2 i α L ) ] − 1 [ J n ∗ ( β r 2 1 + 2 i α L ) ] − 1 ∫ 0 L d η ∫ ∫ − ∞ ∞ d 2 κ Φ n ( κ ) × exp ⁡ [ i K ⋅ ( γ r 1 − γ ∗ r 2 ) − i κ 2 2 k ( γ − γ ∗ ) ( L − η ) ] exp ⁡ [ − i n ( φ K r 1 − φ K r 2 − φ r 1 + φ r 2 ) ] × J n [ − β ( L − η ) k ( 1 + 2 i α L ) | K − k r 1 L − η | ] J n ∗ [ − β ( L − η ) k ( 1 + 2 i α L ) | K − k r 2 L − η | ]
E 3 ( r 1 , r 2 ) = − 2 π k 2 [ J n ( β r 1 1 + 2 i α L ) ] − 1 [ J n ( β r 2 1 + 2 i α L ) ] − 1 ∫ 0 L d η ∫ ∫ − ∞ ∞ d 2 κ Φ n ( κ ) × exp ⁡ [ i γ K ⋅ ( r 1 − r 2 ) − i κ 2 γ k ( L − η ) ] exp ⁡ [ − i n ( φ K r 1 + φ − K r 2 − φ r 1 − φ r 2 ) ] × J n [ − β ( L − η ) k ( 1 + 2 i α L ) | K − k r 1 L − η | ] J n [ − β ( L − η ) k ( 1 + 2 i α L ) | K  +  k r 2 L − η | ]
exp ⁡ ( i φ κ r ) = κ ( L − z ) exp ⁡ ( i φ κ ) − r k e x p ( i φ r ) | κ ( L − z ) exp ⁡ ( i φ κ ) − r k exp ⁡ ( i φ r ) | ,
Φ n ( κ ) = 0.033 C n 2 ( κ 2 + κ 0 2 ) − 11 / 6 exp ⁡ ( − κ 2 / κ m 2 ) ,
M ( n 0 ) = 1 + 2 E 1 + ∑ k = 1 m E 2 ( θ k ) / m ,
D ( n 0 ) = 2 m ∑ k = 1 m R e [ E 2 ( θ k ) + E 3 ( θ k ) ] ,
σ I 2 ( n 0 ) ≈ 2 m ∑ k = 1 m R e [ E 2 ( θ k ) + E 3 ( θ k ) ] .
σ I 2 = 2 Re [ E 2 ( r , r ) + E 3 ( r , r ) ] .
U ( r , L ) = U f r e e ( r , L ) exp ⁡ [ Ψ ( r , L ) ] ,
U f r e e ( r , L ) = A exp ⁡ ( i k L ) 1 + 2 i α L exp ⁡ [ − i β 2 L + 2 α k 2 r 2 2 k ( 1 + 2 i α L ) ] J n ( β r 1 + 2 i α L ) exp ⁡ ( − i n φ r ) .
⟨ exp ⁡ [ Ψ ( r , L ) ] ⟩ = exp ⁡ ( E 1 ) ≈ exp ⁡ ( − 0.3908 C n 2 k 2 κ 0 − 5 / 3 L ) .
U ( r , L ) = a n ( r ) exp ⁡ ( − i n φ r ) ,
a n ( r ) = A exp ⁡ ( i k L ) 1 + 2 i α L exp ⁡ ( − i β 2 L 2 k + 4 i k α L ) exp ⁡ [ − 2 α k 2 r 2 2 k ( 1 + 2 i α L ) ] J n ( β r 1 + 2 i α L ) exp ⁡ [ Ψ ( L ) ] .
a n ( r ) = F − 1 [ U t ( r m , r n , L ) ] ,
a n ( r ) = C 1 exp ⁡ ( − C 3 k r 2 1 + 2 i C 3 L ) J n ( C 2 r 1 + 2 i C 3 L ) ,
A = 1 ⟨ A ⟩ | C 1 ( 1 + 2 i C 3 L ) exp ⁡ ( i C 2 2 L 2 k + 4 i k C 3 L ) | ,
β = | C 2 | , α = | C 3 | ,
T = exp ⁡ ( i k L ) 1 + 2 i k − 1 w 0 − 2 L exp ⁡ ( − i β 2 L 2 k + 4 i w 0 − 2 L ) .
M ( n 0 ) ≈ 1 m ∑ k = 1 m [ 1 + 2 E 1 + E 2 ( θ k ) ] = 1 + 2 E 1 + ∑ k = 1 m E 2 ( θ k ) / m ,
D ( n 0 ) ≈ − M 2 + 1 m 3 ∑ k 1 = 1 m ∑ k 2 = 1 m ∑ k 3 = 1 m [ 1 + 4 E 1 + E 2 ( θ 1 ) + E 2 ( θ 2 ) + E 2 ∗ ( | θ 1 − θ 3 | ) + E 2 ( | θ 2 − θ 3 | ) + E 3 ( θ 3 ) + E 3 ∗ ( | θ 1 − θ 2 | ) ] ,
exp ⁡ ( z ) ≈ 1 + z , z ≪ 1.
M 2 ≈ 1 + 4 E 1 + 2 ∑ k = 1 m E 2 ( θ k ) / m .
1 m 3 ∑ k 1 = 1 m ∑ k 2 = 1 m ∑ k 3 = 1 m ( 1 + 4 E 1 ) = 1 + 4 E 1 ,
1 m 3 ∑ k 1 = 1 m ∑ k 2 = 1 m ∑ k 3 = 1 m E j ( θ i ) = 1 m ∑ k = 1 m E j ( θ k ) , j = 2 , 3 ;   i = 1 , 2 , 3 ,
1 m 3 ∑ k 1 = 1 m ∑ k 2 = 1 m ∑ k 3 = 1 m E 2 ( | θ 2 − θ 3 | ) = 1 m 2 ∑ k 2 = 1 m ∑ k 3 = 1 m E 2 ( | θ 2 − θ 3 | ) .
E 2 ( | 2 π k 2 / m − 2 π k 3 / m | ) = { E 2 ( | 2 π k 2 / m − 2 π k 3 / m | ) , k 2 ≥ k 3 E 2 ( | 2 π k 2 / m − 2 π k 3 / m + 2 π | ) , k 2 < k 3 .
1 m 3 ∑ k 1 = 1 m ∑ k 2 = 1 m ∑ k 3 = 1 m E 2 ( | 2 π k 2 / m − 2 π k 3 / m | ) = 1 m ∑ k = 1 m E 2 ( 2 π k / m ) .
D ( n 0 ) = 2 m Re [ ∑ k = 1 m E 2 ( θ k ) + E 3 ( θ k ) ] .
J n [ − β ( L − η ) k ( 1 + 2 i α L ) | K − k r L − η | ] ≈ J n ( − β r 1 + 2 i α L ) , φ K r ≈ φ r .
∫ 0 2 π d x exp ⁡ ( β c o s x ) exp ⁡ ( − i n x ) = 2 π I n ( β ) .
E 2 ( r 1 , r 2 ) = 4 π 2 k 2 ∫ 0 L d η ∫ 0 ∞ κ d κ Φ n ( κ ) exp ⁡ [ − i κ 2 ( γ − γ ∗ ) ( L − η ) 2 k ] J 0 ( κ | γ r 1 − γ ∗ r 2 | ) ,
E 3 ( r 1 , r 2 ) = − 4 π 2 k 2 ∫ 0 L d η ∫ κ d κ Φ n ( κ ) exp ⁡ [ − i κ 2 γ ( L − η ) k ] J 0 ( γ κ | r 1 − r 2 | ) .
J 0 ( z ) ≈ 1 − z 2 / 4 , exp ⁡ ( z ) ≈ 1 + z , z ≪ 1.
E 2 ( r 1 , r 2 ) = 4 π 2 k 2 ∫ 0 L d η ∫ 0 ∞ κ d κ Φ n ( κ ) [ 1 − 2 α κ 2 ( L − η ) 2 k ( 1 + 4 α 2 L 2 ) − κ 2 | γ r 1 − γ ∗ r 2 | 2 4 ] ,
E 3 ( r 1 , r 2 ) = − 4 π 2 k 2 ∫ 0 L d η ∫ κ d κ Φ n ( κ ) × [ 1 − 2 α κ 2 ( L − η ) 2 k ( 1 + 4 α 2 L 2 ) − γ 2 κ 2 | r 1 − r 2 | 2 4 − i κ 2 ( L − η ) ( 1 + 4 α 2 η L ) k ( 1 + 4 α 2 L 2 ) ] .
σ I 2 ( n 0 ) = 2 π 2 k 2 m ∫ 0 L d η ∫ 0 ∞ κ d κ Φ n ( κ ) ∑ k = 1 m Re [ − κ 2 | γ r 1 − γ ∗ r 2 | 2 + γ 2 κ 2 | r 1 − r 2 | 2 ] .
| r 1 | = | r 2 | = r , r 1 ⋅ r 2 = r 2 cos ⁡ ( θ ) .
M ( n 0 ) = 1 + 4 π 2 k 2 ∫ 0 ∞ d κ κ Φ n ( κ ) [ − 2 α κ 2 L 3 3 k + 12 k α 2 L 2 − κ 2 r 2 2 ( L + 8 α 2 L 3 / 3 + 16 α 4 L 5 / 3 1 + 8 α 2 L 2 + 16 α 4 L 4 − 1 + 4 α 2 L 3 / 3 1 + 4 α 2 L 2 ∑ k = 0 m − 1 c o s θ k m ) ] .
σ I 2 ( n 0 ) = 32 π 2 k 2 α 2 r 2 m ( 1 + 4 α 2 L 2 ) 2 ∫ 0 L d η ∫ 0 ∞ d κ κ 3 ( L − η ) 2 Φ n ( κ ) ∑ k = 1 m cos ⁡ ( θ k ) .
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