Probability property of orbital angular momentum distortion in turbulence

Wanjun Wang; Tianchun Ye; Zhensen Wu

doi:10.1364/OE.445175

1. Introduction

Vortex laser beams have been widely studied in optical communication for their orbital angular momentum (OAM) can be used as an independent information dimension to increase the information capacity [1–3]. The Bessel beam is proposed as one type of vortex beam [4–6]. However, the Bessel beam requires an infinite spatial energy distribution which is unattainable. Therefore, the energy is truncated using the Gaussian window function, and a Bessel Gaussian beam is proposed [7]. Subsequently, the Bessel Gaussian beam propagating through turbulence has been widely studied [8–10].

The field distribution of the laser beam propagating through turbulence has been extensively investigated through experiments and the phase-screen method [11–13]. However, it is difficult for the experiment to sample a large number of the fields in constant strength turbulence, and the precision of the beam field simulated by the phase screen method has seldom been investigated. In addition, the Bessel Gaussian beam has the same problem [14–16]. Therefore, this study investigates the field of the Bessel Gaussian beam in the turbulence based on the phase-screen method with low-frequency compensation [17–19], and the precision of its mean and variance through a comparison with the result derived by the Rytov theory.

Turbulence is detrimental to atmospheric optical communication systems and it causes the beam field distortion. Former scholar investigated the OAM multiplexing of OAM beams, and showed the OAM could be used as an excess dimension to carry the information, increasing the information capacity [20]. However, the turbulence caused the beam distortion and the crosstalk between different order channels which reduces the information capacity [3]. Moreover, even a small crosstalk could cause decoding errors during communication [21,22]. Subsequently, many papers on the beam OAM distortion and its characteristics have been published [23–26]. Although the power ratio distribution is widely accepted to characterize OAM crosstalk [27–29], it cannot present the random-varying property of the beam distortion. In addition, the probability density function (PDF) of the OAM distortion was investigated by an empirical PDF with several undetermined parameters which could be solved by the data fitting method, and there is an appropriate precision of this PDF [30–33]. However, it failed to provide an analytic relationship between parameters of this PDF and the turbulence strength. There have been few studies on the quantitative relationship between turbulence strength and the OAM crosstalk because the analytic expression to characterize the OAM crosstalk is difficult to solve owing to the spatial correlation of the turbulence. Under these circumstances, to characterize the random-varying property of the beam distortion shown in Fig. 1, the PDF of the beam intensity was proposed and established a relationship between the turbulence strength and the harmonic intensity fluctuation. Moreover, the probability of the intensity difference between the fundamental and its adjacent crosstalk modes was also derived to reveal the influence of the turbulence on the OAM decoding error.

Fig. 1. Bessel Gaussian beam propagation in turbulence

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In this study, the probability property of Bessel Gaussian beam distortion in turbulence under weak fluctuation conditions is studied, and a theoretical basis for the application of the Bessel Gaussian beam in laser communication is provided.

2. Theory and formulation

2.1 Phase-screen method

The field distribution of the Bessel Gaussian beam with radial coordinates r and ${\varphi _\textrm{r}}$ can be expressed as Eq. (1) [4,7].

(1)$${U_0}(r,{\varphi _r}) = {J_n}(\beta r)\exp ( - k\alpha {r^2})\exp ( - in{\varphi _r}), $$

where n is the order of the Bessel Gaussian beam, and $\beta $ is the Bessel width parameter. k is the wave number and $k = 2\pi /\lambda $, $\lambda $ is the wavelength. $\alpha $ is a complex parameter related to spot size and phase front radius of the curvature and expressed as $\alpha = 1/kw_0^2 + i/(2{F_0})$, where ${w_0}$ is the Gaussian source width and ${F_0}$ is the focusing parameter. In this paper, the beams are collimated beams and ${F_0} = \infty $.

The field intensity of the Bessel Gaussian beam propagating through the turbulence can be calculated by the phase-screen method and expressed as Eq. (2) [11,18]. The field calculated by this method provides contrast data for the later investigation.

(2)$${U_{m + 1}}({\mathbf r},{L_{m + 1}}) = {\cal F}_2^{ - 1}\{{\exp[{i{\kappa_L}({{L_{m + 1}} - {L_m}} )} ]{\cal F}_2^{}[{U_m}({\mathbf r^{\prime}},{L_m})\exp [{iS({\mathbf r^{\prime}})} ]} \}, $$

where ${L_m}$ is the distance of the mth phase screen, $\boldsymbol{\mathrm{\kappa}}$ is the wave vector of the scalar spatial frequency, and ${\kappa _L}$ is the component of $\boldsymbol{\mathrm{\kappa}}$ along the propagation direction. ${U_m}({\mathbf r^{\prime}},{L_m})$ is the incident field intensity before the mth phase screen. $S({\mathbf r^{\prime}})$ is the FFT-based phase screen. ${{\cal F}_2}$ and ${\cal F}_2^{ - 1}$ are the two-dimensional fast Fourier transform (FFT) and inverse fast Fourier transform, respectively. The phase-screen method is based on the assumption that the imaginary part of $S({\mathbf r})$ is small or can be ignored, and this assumption can be met in most case.

The phase screen is produced based on the theory that the turbulence property could be characterized by the phase fluctuations on the two-dimensional thin slab and it is expressed by Eq. (3).

(3)$$S({\mathbf s})\textrm{ = }\int {\int_{ - \infty }^\infty {{d^2}\kappa } } a(\boldsymbol{\mathrm{\kappa}})\exp (i\boldsymbol{\mathrm{\kappa}} \cdot {\mathbf s})|{G(\boldsymbol{\mathrm{\kappa}})} |, $$

where $G(\boldsymbol{\mathrm{\kappa}})$ is the filter function in Eq. (4) [18]. a is the two-dimensional Gaussian random number, and s is the phase position on the cross section.

(4)$${|{G(\boldsymbol{\mathrm{\kappa}})} |^2} = 2\pi {k^2}\Delta L{\Phi _n}(\kappa ), $$

where ${\Phi _n}(\kappa )$ is the power spectrum and the von Karman spectrum is used as Eq. (5). $\Delta L$ is the interval distance between two phase screens.

(5)$${\Phi _n}(\kappa ) = 0.033C_n^2\frac{{\exp ({ - {\kappa^2}/\kappa_m^2} )}}{{{{({{\kappa^2} + \kappa_0^2} )}^{11/6}}}}, $$

where ${\kappa _m} = 2\pi /{l_0}$. and ${\kappa _0} = 2\pi /{L_0}$. l ₀ and L ₀ are the inner and outer scales of the turbulence. L ₀=2m and l ₀=0.02m. In this study, the beam propagates in a horizontal path and $C_n^2$ is a constant along the propagation path.

The turbulent phase screen in Eq. (3) could be discretized and expressed by Eq. (6).

(6)$$\begin{aligned}S(m\Delta x,n\Delta y) &= \sum\limits_{m^{\prime} ={-} N/2}^{N/2 - 1} {\sum\limits_{n^{\prime} ={-} N/2}^{N/2 - 1} {\frac{{a(m^{\prime},n^{\prime})}}{{\sqrt {\Delta {\kappa _x}\Delta {\kappa _y}} }}\exp \left( {\frac{{i2\pi mm^{\prime}}}{{{N_m}}} + \frac{{i2\pi nn^{\prime}}}{{{N_n}}}} \right)} } \int_{m^{\prime}\Delta {\kappa _x}}^{({m^{\prime} + 1} )\Delta {\kappa _x}}\nonumber\\&\quad\times {\int_{n^{\prime}\Delta {\kappa _y}}^{({n^{\prime}\textrm{ + 1}} )\Delta {\kappa _y}} {d{\kappa _x}d{\kappa _y}} } G({\kappa _x},{\kappa _y})\end{aligned}$$

where $\Delta x$ and $\Delta y$ are the sample intervals of the phase screen with the relationship that $\Delta x = {D_x}/{N_m}$ and $\Delta y = {D_y}/{N_n}$. ${D_x}$ and ${D_y}$ are the weight and the height of the phase screen. ${N_m}$ and ${N_n}$ are the sample numbers. $\Delta {\kappa _x}$ and $\Delta {\kappa _y}$ are the sample intervals in the wavenumber domain, and $\Delta {\kappa _x} = {\kappa _x}/{N_m}$, $\Delta {\kappa _y} = {\kappa _y}/{N_n}$. ${\kappa _x}$ and ${\kappa _y}$ are the weight and the height of the phase screen in the wavenumber domain, and ${\kappa _x} = 2\pi /\Delta x$, ${\kappa _y} = 2\pi /\Delta y$. $a(m^{\prime},n^{\prime})$ is the Gaussian noise with the real ${a_{real}}$ and imaginary parts ${a_{imag}}$, and both parts are the standard normal distribution random numbers as shown in Eq. (7) [34,35].

(7)$$a(m^{\prime},n^{\prime}) = {a_{real}}(m^{\prime},n^{\prime}) + i{a_{imag}}(m^{\prime},n^{\prime}). $$

Based on assumption that the filter function is invariant in the integral interval, the phase screen in Eq. (6) could be approximated as Eq. (8).

(8)$$S(m\Delta x,n\Delta y)\textrm{ = }\sqrt {\Delta {\kappa _x}\Delta {\kappa _y}} \sum\limits_{m^{\prime} ={-} N/2}^{N/2} {\sum\limits_{n^{\prime} ={-} N/2}^{N/2} {a(m^{\prime},n^{\prime})} } G({m^{\prime}\Delta {\kappa_x},n^{\prime}\Delta {\kappa_y}} )\exp \left( {\frac{{i2\pi mm^{\prime}}}{{{N_m}}} + \frac{{i2\pi nn^{\prime}}}{{{N_n}}}} \right). $$

The simulation data calculated by this phase-screen method must be adjusted for the undersampling of the low-frequency component. In this study, the compensation matrix which is the ratio between the integral value of the filter function and the FFT sample value is provided, as shown in Eq. (9) [36]. Therefore, the phase-screen method with this compensation matrix calculates fast and meets the requirement of this study which needs large amounts of samplings.

(9)$${M_c}(m^{\prime},n^{\prime}) = \frac{{\int_{m^{\prime}\Delta {\kappa _x}}^{({m^{\prime} + 1} )\Delta {\kappa _x}} {\int_{n^{\prime}\Delta {\kappa _y}}^{({n^{\prime}\textrm{ + 1}} )\Delta {\kappa _y}} {d{\kappa _x}d{\kappa _y}} } \exp [{ - {{({\kappa_x^2 + \kappa_y^2} )} / {({2\kappa_m^2} )}}} ]{{({\kappa_x^2 + \kappa_y^2 + \kappa_0^2} )}^{ - 11/12}}}}{{\Delta {\kappa _x}\Delta {\kappa _y}\exp \{{ - {{[{{{({m^{\prime}\Delta {\kappa_x}} )}^2} + {{({n^{\prime}\Delta {\kappa_y}} )}^2}} ]} / {({2\kappa_m^2} )}}} \}{{[{{{({m^{\prime}\Delta {\kappa_x}} )}^2} + {{({n^{\prime}\Delta {\kappa_y}} )}^2} + \kappa_0^2} ]}^{ - 11/12}}}}. $$

The compensation matrix is related to the turbulence scale and the sample interval. In general, the phase screen size is constant, and the turbulence structure remains the same. Therefore, only one compensation matrix is sufficient to produce multiple phase screens. The phase screen could be expressed as Eq. (10).

(10)$$\begin{aligned} S(m\Delta x,n\Delta y)&\textrm{ = }\sqrt {\Delta {\kappa _x}\Delta {\kappa _y}} \sum\limits_{m^{\prime} ={-} N/2}^{N/2 - 1} {\sum\limits_{n^{\prime} ={-} N/2}^{N/2 - 1} {a({m^{\prime},n^{\prime}} )} } {M_c}({m^{\prime},n^{\prime}} )\\ &\times G({m^{\prime}\Delta {\kappa_x},n^{\prime}\Delta {\kappa_y}} )\exp ({i2\pi mm^{\prime}/{N_m} + i2\pi nn^{\prime}/{N_n}} )\end{aligned}. $$

2.2 Turbulence moment of Bessel Gaussian beam

The Rytov method could be applied under weak fluctuation conditions as Eq. (11) and this study is also on this basis.

(11)$$\sigma _R^2 < 1 \quad {\textrm{and}} \quad \sigma _R^2{\Lambda ^{5/6}} \lt 1,$$

where $\sigma _R^2$ is the Rytov variance and $\sigma _R^2 = \textrm{1}\textrm{.23}C_n^2{k^{7/6}}{L^{11/6}}$, and $\Lambda = 2L/(kw_0^2)$ is the Gaussian beam parameter characterizing the spot size at the receiver plane.

To investigated the beam distortion in turbulence, E ₁, E ₂ and E ₃ representing the turbulence statistical moments of the Bessel Gaussian beam are introduced and they can be derived as Eqs. (12), (13) and (14) with the same method as Ref. [37,38]. Moreover, to characterize the spatial correlation of the turbulence, the turbulence moments are expressed as ${E_2}({{\mathbf r}_1},{{\mathbf r}_2})$ and ${E_3}({{\mathbf r}_1},{{\mathbf r}_2})$ with different spatial locations r_i.

(12)$${E_1}({\mathbf r},{\mathbf r}) ={-} \pi {k^2}\int_0^L {d\eta } \int {\int_{ - \infty }^\infty {{d^2}\kappa } } {\Phi _n}(\kappa ), $$

(13)$$\begin{aligned} {E_2}({{\mathbf r}_1},{{\mathbf r}_2}) &= 2\pi {k^2}{\left[ {{J_n}\left( {\frac{{\beta {r_1}}}{{1 + 2i\alpha L}}} \right)} \right]^{ - 1}}{\left[ {J_n^\ast \left( {\frac{{\beta {r_2}}}{{1 + 2i\alpha L}}} \right)} \right]^{ - 1}}\int_0^L {d\eta } \int {\int_{ - \infty }^\infty {{d^2}\kappa } } {\Phi _n}(\kappa )\\ &\times \exp \left[ {i{\mathbf K} \cdot (\gamma {{\mathbf r}_1} - \gamma \ast {{\mathbf r}_2}) - \frac{{i{\kappa^2}}}{{2k}}(\gamma - \gamma \ast )(L - \eta )} \right]\exp [{ - in({{\varphi_{K{r_1}}} - {\varphi_{K{r_2}}}} )} ]\\ &\times {J_n}\left[ { - \frac{{\beta (L - \eta )}}{{k({1 + 2i\alpha L} )}}\left|{{\mathbf K} - \frac{{k{{\mathbf r}_1}}}{{L - \eta }}} \right|} \right]J_n^\ast \left[ { - \frac{{\beta (L - \eta )}}{{k({1 + 2i\alpha L} )}}\left|{{\mathbf K} - \frac{{k{{\mathbf r}_2}}}{{L - \eta }}} \right|} \right] \end{aligned}, $$

(14)$$\begin{aligned} {E_3}({{\mathbf r}_1},{{\mathbf r}_2}) &={-} 2\pi {k^2}{\left[ {{J_n}\left( {\frac{{\beta {r_1}}}{{1 + 2i\alpha L}}} \right)} \right]^{ - 1}}{\left[ {{J_n}\left( {\frac{{\beta {r_2}}}{{1 + 2i\alpha L}}} \right)} \right]^{ - 1}}\int_0^L {d\eta } \int {\int_{ - \infty }^\infty {{d^2}\kappa } {\Phi _n}(\kappa )} \\ &\times \exp \left[ {i\gamma {\mathbf K} \cdot ({{\mathbf r}_1} - {{\mathbf r}_2}) - \frac{{i{\kappa^2}\gamma }}{k}(L - \eta )} \right]\exp [{ - in({{\varphi_{K{r_\textrm{1}}}} + {\varphi_{ - K{r_\textrm{2}}}} - {\varphi_{{r_\textrm{1}}}} - {\varphi_{{r_2}}}} )} ]\\ &\times {J_n}\left[ { - \frac{{\beta (L - \eta )}}{{k({1 + 2i\alpha L} )}}\left|{{\mathbf K} - \frac{{k{{\mathbf r}_1}}}{{L - \eta }}} \right|} \right]{J_n}\left[ { - \frac{{\beta (L - \eta )}}{{k({1 + 2i\alpha L} )}}\left|{{\mathbf K}\textrm{ + }\frac{{k{{\mathbf r}_2}}}{{L - \eta }}} \right|} \right] \end{aligned}, $$

The phase can be expressed as Eq. (15).

(15)$$\exp ({i{\varphi_{\kappa r}}} )= \frac{{\kappa (L - z)\exp (i{\varphi _\kappa }) - rk \exp (i{\varphi _r})}}{{|{\kappa (L - z)\exp (i{\varphi_\kappa }) - rk \exp (i{\varphi_r})} |}}. $$

The mean field intensity and the average intensity derived from the Rytov theory can be expressed as Eqs. (16) and (17) [39].

(16)$$\left\langle {U({\mathbf r})} \right\rangle = {U_{free}}({\mathbf r},L)\exp [{ - {E_1}({\mathbf r},{\mathbf r})} ], $$

(17)$$\left\langle {I({\mathbf r},L)} \right\rangle = {|{{U_{free}}({\mathbf r},L)} |^2}\exp [{2{E_1}({\mathbf r},{\mathbf r}) + {E_2}({\mathbf r},{\mathbf r})} ], $$

where U_free is the field distribution of the Bessel Gaussian beam on a receiver plane in free space, and expressed as Eq. (18) [40].

(18)$${U_{free}}({\mathbf r},L) = {A_n}\frac{{\exp (ikL)}}{{1 + 2i\alpha L}}\exp \left[ { - \frac{{i\beta_{}^2L + 2\alpha {k^2}{r^2}}}{{2k({1 + 2i\alpha L} )}}} \right]{J_n}\left( {\frac{{\beta r}}{{1 + 2i\alpha L}}} \right)\exp ( - in{\varphi _r}). $$

2.3 Beam distortion characterization

The random-varying distortion of the beam in turbulence can be characterized by the probability of its harmonic intensity fluctuation. The harmonic mode of the Bessel Gaussian beam in this study consists of the fundamental mode and the crosstalk mode.

2.3.1 Distortion measured by power ratio

The field of the Bessel Gaussian beam at the receiver plane can be decomposed to a Fourier series expression as Eq. (19) [27,28].

(19)$$U({\mathbf r},L) = \sum\limits_{n ={-} \infty }^\infty {{c_n}(r)} \exp ( - in{\varphi _r}), $$

where ${c_n}(r) = {(2\pi )^{ - 1}}\int {d{\varphi _r}} U({\mathbf r},L)\exp (in{\varphi _r})$ and ${c_n}(r)$ is the amplitude component of the nth-order harmonic wave. The power ratio or the power weight of the nth-order OAM beam is used to measure the OAM distortion, as shown in Eq. (20) [27,28,41].

(20)$${\omega _n} = \frac{{\int_0^\infty {{{|{{c_n}(r)} |}^2}rdr} }}{{\sum\limits_{n = 0}^\infty {\int_0^\infty {{{|{{c_n}(r)} |}^2}rdr} } }}{\kern 1pt}. $$

2.3.2 Statistics of harmonic intensity

The field of the Bessel Gaussian beam at the receiver plane expressed as Eq. (31) in Chapter 5 of Ref. [39] is rewritten here as Eq. (21) for simplicity.

(21)$$U({r,{\varphi_r}} )= {U_{free}}({r,{\varphi_r}} )\exp(\psi ), $$

where $\psi$ is the phase fluctuation caused by the turbulence.

To characterize the random-varying property of the turbulence influence, the normalized complex amplitude of nth-order harmonic component $T(n )$ is proposed as Eq. (22). The square of its magnitude $|T(n){|^2}$ is the harmonic normalized intensity and is expressed as Eq. (23). Based on Eq. (20), the fluctuation of the harmonic normalized intensity could characterize the beam distortion when the beam radius is constant.

(22)$$T(n) = {c_n}(r)/Nc, $$

(23)$$\begin{aligned} {|{T(n)} |^2} &= {|{{{\cal F}^{ - 1}}[{\exp({ - i{n_0}{\varphi_r} + \psi } )} ]{\kern 1pt} } |^2}\\ & = {\left|{\frac{1}{m}\sum\limits_{k = 1}^m {\exp\left( { - i\frac{{2\pi {n_0}}}{m}k + {\psi_k}} \right)\exp \left( {i\frac{{2\pi }}{m}nk} \right)} } \right|^2} \end{aligned}, $$

where m is the number of sampling points along the circumference and m=180. In addition, the polar angle ${\varphi _r}$ is decomposed into m equal parts as ${\varphi _{rk}} = 2\pi k/m$ and the initiation angle is ${\varphi _{r1}} = 2\pi /m$. ${\psi _k}$ is the perturbed phase at different locations. Nc is the normalized factor and Nc=|U_free| in this study.

The mean and variance of the normalized intensity fluctuation $|T(n){|^2}$ are derived and expressed as Eqs. (24) and (25), respectively:

(24)$$M(n) = \left\langle {{{|{T(n )} |}^2}} \right\rangle = \left\langle {\frac{1}{{{m^2}}}\sum\limits_{k = 1}^m {\sum\limits_{k^{\prime} = 1}^m {\exp({{\psi_k} + \psi_{k^{\prime}}^\ast } )} \exp\left[ { - i\frac{{2\pi }}{m}({{n_0} - n} )({k - k^{\prime}} )} \right]} } \right\rangle, $$

(25)$$\begin{aligned} D(n) &= \left\langle {{{|{T(n )} |}^4}} \right\rangle - M{(n)^2}\\ & \textrm{ = }\frac{1}{{{m^4}}}\left\langle {\sum\limits_{{k_1} = 1}^m {\sum\limits_{{k_2} = 1}^m {\sum\limits_{{k_3} = 1}^m {\sum\limits_{{k_4} = 1}^m {\exp({{\psi_{{k_1}}} + \psi_{{k_2}}^\ast{+} {\psi_{{k_3}}} + \psi_{{k_4}}^\ast } )} } {\kern 1pt} {\kern 1pt} } } } \right.{\kern 1pt} {\kern 1pt} \\ & \times \left. {\exp\left[ { - i\frac{{2\pi }}{m}({{n_0} - n} )({{k_1} - {k_2} + {k_3} - {k_4}} )} \right]} \right\rangle - M{(n)^2} \end{aligned}. $$

Based on the Rytov theory expressed by Eqs. (16) and (17) in Chapter 6 [39], the second- and fourth-order moments of the turbulence follow the relations expressed as Eqs. (26) and (27).

(26)$${\kern 1pt} {\kern 1pt} \left\langle {\exp[{\psi ({{{\mathbf r}_{\mathbf 1}}} )+ \psi_{}^\ast ({{{\mathbf r}_{\mathbf 2}}} )} ]} \right\rangle \textrm{ = }\exp[{2{E_1} + {E_2}({{{\mathbf r}_{\mathbf 1}},{{\mathbf r}_{\mathbf 2}}} )} ], $$

(27)$$\begin{array}{l} {\kern 1pt} {\kern 1pt} \left\langle {\exp[{\psi ({{{\mathbf r}_{\mathbf 1}}} )+ {\psi^\ast }({{{\mathbf r}_{\mathbf 2}}} )+ \psi ({{{\mathbf r}_3}} )+ {\psi^\ast }({{{\mathbf r}_4}} )} ]} \right\rangle \\ \textrm{ = }\exp[{4{E_1} + {E_2}({{{\mathbf r}_{\mathbf 1}},{{\mathbf r}_{\mathbf 2}}} )+ {E_2}({{{\mathbf r}_{\mathbf 1}},{{\mathbf r}_4}} )+ {E_2}({{{\mathbf r}_3},{{\mathbf r}_{\mathbf 2}}} )+ {E_2}({{{\mathbf r}_3},{{\mathbf r}_4}} )+ {E_3}({{{\mathbf r}_{\mathbf 1}},{{\mathbf r}_3}} )+ E_3^\ast ({{{\mathbf r}_2},{{\mathbf r}_4}} )} ]\end{array}. $$

By substituting Eqs. (26) and (27) into Eqs. (24) and (25), the mean and variance of the harmonic normalized intensity fluctuation can be expressed as Eqs. (28) and (29), respectively:

(28)$$M(n) = \frac{1}{{{m^2}}}\sum\limits_{k = 1}^m {\sum\limits_{k^{\prime} = 1}^m {\exp\left[ { - i\frac{{2\pi }}{m}({{n_0} - n} )({k - k^{\prime}} )} \right]\exp[{2{E_1} + {E_2}({k,k^{\prime}} )} ]} }, $$

(29)$$\begin{array}{l} D(n)\textrm{ = } - M{(n)^2} + \frac{1}{{{m^4}}}\sum\limits_{{k_1} = 1}^m {\sum\limits_{{k_2} = 1}^m {\sum\limits_{{k_3} = 1}^m {\sum\limits_{{k_4} = 1}^m {\exp\left[ { - i\frac{{2\pi }}{m}({{n_0} - n} )({{k_1} - {k_2} + {k_3} - {k_4}} )} \right]} } {\kern 1pt} {\kern 1pt} } } \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \times \exp[{4{E_1} + {E_2}({{k_1},{k_2}} )+ {E_2}({{k_1},{k_4}} )+ {E_2}({{k_3},{k_2}} )+ {E_2}({{k_3},{k_4}} )+ {E_3}({{k_1},{k_3}} )+ E_3^\ast ({{k_2},{k_4}} )} ]\end{array}. $$

The mean and variance of the intensity fluctuation of the beam fundamental mode can be simplified as Eqs. (30) and (31), respectively:

(30)$$M({n_0}) = \frac{1}{{{m^\textrm{2}}}}\sum\limits_{k = 1}^m {\sum\limits_{k^{\prime} = 1}^m {\exp[{2{E_1} + {E_2}({{{\mathbf r}_{\mathbf 1}},{{\mathbf r}_{\mathbf 2}}} )} ]} } {\kern 1pt} {\kern 1pt} = \frac{1}{m}{\kern 1pt} {\kern 1pt} \sum\limits_{k = 1}^m {\exp[{2{E_1} + {E_2}({{\theta_k}} )} ]}, $$

where ${\theta _k}\textrm{ = }2\pi k/m$ and ${E_2}(\theta )\textrm{ = }{E_2}[{r,r\cos (\theta )} ]$.

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

where ${\theta _i}\textrm{ = }2\pi {k_i}/m$, ${E_2}(\theta )\textrm{ = }{E_2}[{r,r\cos (\theta )} ]$ and ${E_3}(\theta )\textrm{ = }{E_3}[{r,r\cos (\theta )} ]$.

2.3.3 Probability density function of harmonic intensity

The PDF of the harmonic intensity can not only characterize the beam OAM distortion, but can also be used as the applicable condition of beam multiplexing. Due to the spatial correlation of the turbulence, the analytical expression of the PDF is difficult to derive through the mathematical deduction. Therefore, it is approximated by the Gamma distribution which has a better performance in the commonly used simple distribution to approximate the PDF of the intensity of the crosstalk mode. On this assumption, the precision of the PDF of the crosstalk mode was investigated, and the PDF of the fundamental mode was derived.

The lognormal Gaussian distribution has been widely used to predict the nature of the intensity fluctuations [39,42], and it can be approximated by the Gaussian distribution by neglecting the value less than 0, and this approximation could be met under weak fluctuation conditions. Therefore, the PDF of the total normalized intensity of all harmonic modes is approximated by a Gaussian distribution expressed as Eq. (32).

(32)$${\kern 1pt} {f_I}(x) = \frac{1}{{\sqrt {2\pi \sigma _0^2} }}\exp \left[ { - \frac{{{{(x - {m_0})}^2}}}{{2\sigma_0^2}}} \right], $$

where ${\kern 1pt} {m_0}$ and $\sigma _0^2{\kern 1pt}$ is the mean and variance as Eqs. (33) and (34).

(33)$$\begin{aligned} {m_0} &= \left\langle {\sum\limits_{n = 0}^{m/2} {{{|{T(n )} |}^2}} } \right\rangle \\ &= \frac{1}{{{m^2}}}\sum\limits_{k = 1}^m {\sum\limits_{k^{\prime} = 1}^m {\exp [{2{E_1} + {E_2}({k,k^{\prime}} )} ]} } \exp\left[ { - i\frac{{2\pi {n_0}}}{m}({k - k^{\prime}} )} \right]\frac{{1 - \exp[{\pi i({k - k^{\prime}} )} ]}}{{1 - \exp[{2\pi i({k - k^{\prime}} )/m} ]}} \end{aligned}, $$

(34)$$\begin{aligned} \sigma _0^2 &= \left\langle {\sum\limits_{n = 0}^{m/2} {{{|{T(n )} |}^2}\sum\limits_{n^{\prime} = 0}^{m/2} {{{|{T({n^{\prime}} )} |}^2}} } } \right\rangle - {\left\langle {\sum\limits_{n = 0}^{m/2} {{{|{T(n )} |}^2}} } \right\rangle ^2}\\ &= \frac{1}{{{m^4}}}\sum\limits_{{k_1} = 1}^m {\sum\limits_{{k_2} = 1}^m {\sum\limits_{{k_3} = 1}^m {\sum\limits_{{k_4} = 1}^m {\left\langle {\exp({{\psi_{{k_1}}} + \psi_{{k_2}}^\ast{+} {\psi_{{k_3}}} + \psi_{{k_4}}^\ast } )} \right\rangle } } {\kern 1pt} {\kern 1pt} } } \exp\left[ { - i\frac{{2\pi {n_0}}}{m}({{k_1} - {k_2} + {k_3} - {k_4}} )} \right]\\ &\times \frac{{1 - \exp[{\pi i({{k_1} - {k_2}} )} ]}}{{1 - \exp[{2\pi i({{k_1} - {k_2}} )/m} ]}}\frac{{1 - \exp[{\pi i({{k_3} - {k_4}} )} ]}}{{1 - \exp[{2\pi i({{k_3} - {k_4}} )/m} ]}} - m_0^2 \end{aligned}, $$

where the fourth-order statistical moment can be expressed as Eq. (35) derived by Eq. (27).

(35)$$\begin{array}{l} \left\langle {\exp({{\psi_{{k_1}}} + \psi_{{k_2}}^\ast{+} {\psi_{{k_3}}} + \psi_{{k_4}}^\ast } )} \right\rangle \\ \textrm{ = }\exp[{4{E_1} + {E_2}({{k_1},{k_2}} )+ {E_2}({{k_1},{k_4}} )+ {E_2}({{k_3},{k_2}} )+ {E_2}({{k_3},{k_4}} )+ {E_3}({{k_1},{k_3}} )+ E_3^\ast ({{k_2},{k_4}} )} ]\end{array}. $$

The Gamma distribution has better performance in the commonly used simple distribution and it is selected to approximate the PDF of the crosstalk mode intensity, as shown in Eq. (36) [43,44].

(36)$${f_C}(x )\textrm{ = }\frac{{M(n )/D(n )}}{{\Gamma [{M_{}^2(n )/D(n )} ]}}{\left[ {\frac{{M(n )x}}{{D(n )}}} \right]^{M_{}^2/D - 1}}\exp \left[ { - \frac{{M(n )x}}{{D(n )}}} \right], $$

where ${f_C}$ is the PDF of the normalized intensity of the nth-order crosstalk mode.

The PDF of the intensity fluctuation of the fundamental mode can be derived as the PDF of the intensity difference between the sum of the harmonic normalized intensity and the sum of the crosstalk mode intensity, as expressed in Eq. (37).

(37)$${\kern 1pt} {f_B}(z )\textrm{ = }\int_0^\infty {dx} {f_{{S_c}}}(x ){f_I}({x + z} ), $$

where ${f_B}$ is the PDF of the normalized intensity of the fundamental mode, ${f_{{S_c}}}$ is the PDF of the sum of the crosstalk mode intensity, and ${f_I}$ is the PDF of the sum of the harmonic normalized intensity.

The distribution of the sum of the Gamma varieties can be approximated by a single Gamma function [45], and the mean can be expressed as Eq. (38).

(38)$${m_c} = \left\langle {\sum\limits_{n = 0}^{m/2} {{{|{T(n )} |}^2} - {{|{T({{n_0}} )} |}^2}} } \right\rangle = {m_0} - M({{n_0}} ). $$

The variance of the sum of the crosstalk mode intensity $\sigma _c^2$ can be calculated by solving the equation that the variance obtained from the PDF in Eq. (40) is equal to the variance derived by the Rytov method in Eq. (31). For convenience, this variance $\sigma _c^2$ can be approximated by the variance of the fundamental mode, as shown in Eq. (39) for the sum of the harmonic intensity is close to a constant with a very small variance.

(39)$$\sigma _c^2 = D({n_0}). $$

Substituting Eqs. (32), (36) in this study and 3.462, 9.240 in Ref. [46] into Eq. (37), the PDF of the beam fundamental mode intensity fluctuation can be derived as Eq. (40).

(40)$$\begin{aligned} {\kern 1pt} {f_B}(z )&= \frac{1}{{2\sqrt \pi }}\frac{1}{{\Gamma ({m_c^2/\sigma_c^2} )}}{\left( {\frac{{\sqrt {2\sigma_0^2} {m_c}}}{{\sigma_c^2}}} \right)^{m_c^2/\sigma _c^2}}\exp \left[ { - \frac{{{{({z - {m_0}} )}^2}}}{{2\sigma_0^2}}} \right]\\ &\times \left\{ {\sqrt {\frac{1}{{2\sigma_0^2}}} } \right.\Gamma \left( {\frac{{m_c^2}}{{2\sigma_c^2}}} \right){{\kern 1pt} _1}{F_1}\left[ {\frac{{m_c^2}}{{2\sigma_c^2}},\frac{1}{2},\frac{{\sigma_0^2}}{2}{{\left( {\frac{{z - {m_0}}}{{\sigma_0^2}} + \frac{{{m_c}}}{{\sigma_c^2}}} \right)}^2}} \right]\\ &- \left( {\frac{{z - {m_0}}}{{\sigma_0^2}} + \frac{{{m_c}}}{{\sigma_c^2}}} \right)\Gamma \left( {\frac{1}{2} + \frac{{{m_c}^2}}{{2\sigma_c^2}}} \right){{\kern 1pt} _1}{F_1}\left. {\left[ {\frac{1}{2} + \frac{{m_c^2}}{{2\sigma_c^2}},\frac{3}{2},\frac{{\sigma_0^2}}{2}{{\left( {\frac{{z - {m_0}}}{{\sigma_0^2}} + \frac{{{m_c}}}{{\sigma_c^2}}} \right)}^2}} \right]} \right\} \end{aligned}. $$

The PDF of the intensity difference between the fundamental and the adjacent-crosstalk mode which is adjacent to the fundamental mode is derived using the same method as Eq. (40), and it is expressed as Eq. (41).

(41)$$\begin{aligned} {\kern 1pt} {f_E}(z )&\textrm{ = }\frac{1}{{2\sqrt \pi }}\frac{1}{{\Gamma ({m_e^2/\sigma_e^2} )}}{\left( {\frac{{\sqrt {2\sigma_0^2} {m_e}}}{{\sigma_e^2}}} \right)^{m_e^2/\sigma _e^2}}\exp \left[ { - \frac{{{{({z - {m_0}} )}^2}}}{{2\sigma_0^2}}} \right]\\ &\times \left\{ {\sqrt {\frac{1}{{2\sigma_0^2}}} } \right.\Gamma \left( {\frac{{m_e^2}}{{2\sigma_e^2}}} \right){{\kern 1pt} _1}{F_1}\left[ {\frac{{m_e^2}}{{2\sigma_e^2}},\frac{1}{2},\frac{{\sigma_0^2}}{2}{{\left( {\frac{{z - {m_0}}}{{\sigma_0^2}} + \frac{{{m_e}}}{{\sigma_e^2}}} \right)}^2}} \right]\\ &- \left( {\frac{{z - {m_0}}}{{\sigma_0^2}} + \frac{{{m_e}}}{{\sigma_e^2}}} \right)\Gamma \left( {\frac{1}{2} + \frac{{m_e^2}}{{2\sigma_e^2}}} \right){{\kern 1pt} _1}{F_1}\left. {\left[ {\frac{1}{2} + \frac{{m_e^2}}{{2\sigma_e^2}},\frac{3}{2},\frac{{\sigma_0^2}}{2}{{\left( {\frac{{z - {m_0}}}{{\sigma_0^2}} + \frac{{{m_e}}}{{\sigma_e^2}}} \right)}^2}} \right]} \right\} \end{aligned}, $$

where ${f_E}$ is the PDF of the intensity difference, ${m_e}$ is the mean expressed as Eq. (42), and $\sigma _e^2$ is the variance approximated as Eq. (43).

(42)$${m_e} = {m_0} - M({{n_0}} )+ M({{n_0} + 1} ), $$

(43)$$\begin{aligned} \sigma _e^2 &= \left\langle {({{{|{T({{n_0}} )} |}^2} - {{|{T(n )} |}^2}} )({{{|{T({{n_0}} )} |}^2} - {{|{T(n )} |}^2}} )} \right\rangle - {[{M({{n_0}} )- M({{n_0} + 1} )} ]^2}\\ &= D({{n_0}} )+ D({{n_0} + 1} )+ 2M({{n_0}} )M({{n_0} + 1} )\\ &- \frac{2}{{{m^4}}}{\left|{\sum\limits_{{k_1} = 1}^m {\sum\limits_{{k_2} = 1}^m {\sum\limits_{{k_3} = 1}^m {\sum\limits_{{k_4} = 1}^m {\left\langle {\exp({{\psi_{{k_1}}} + \psi_{{k_2}}^\ast{+} {\psi_{{k_3}}} + \psi_{{k_4}}^\ast } )} \right\rangle \exp \left[ {i\frac{{2\pi }}{m}({{k_1} - {k_2}} )} \right]} } {\kern 1pt} {\kern 1pt} } } } \right|^2} \end{aligned}. $$

The decoding error probability ${P_E}$ is derived from the integral of Eq. (41) and expressed as Eq. (44).

(44)$${\kern 1pt} {P_E}(z < 0) = \int_{ - \infty }^0 {dz} {f_E}(z ). $$

3. Result and analysis

In this paper, without special annotation in figures, parameters are selected as Table 1.

Table 1. Parameter selection.

View Table

Figure 2 shows a sampling result of third order Bessel Gaussian beam calculated by the phase-screen method. There is the spiral symmetry distribution of the beam phase and its angular frequency is equal to the incident beam order in Fig. 2(b). In this strength turbulence, the power component of the crosstalk mode is very small as shown in Fig. 2(c), thus the beam distortion is small.

Fig. 2. Third order Bessel Gaussian beam (a) Intensity, (b) Phase, (c) Power ratio.

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Figure 3(a), 3(b) and 3(c) show the relative error of the mean and standard deviation of the imaginary part of the normalized field intensity, which is equal to the field divided by the theoretical mean field expressed as Eq. (16). The field counted are in the area where the imaginary part of the beam field is larger than half of the maximum imaginary part of the field.

Fig. 3. (a) Relative error varying with sample number, (b) Relative error varying with refractive index structure constant, (c) Relative error varying phase screen size, (d) Statistics varying with sampling point number.

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The relative error of the mean is small. The standard deviation tends to become more stable as the sample number increases. Given a finite size screen, using the low-frequency compensation, the precision of the field calculated by the phase-screen method could be improved, and the relative error of the standard deviation is less than 10%, which is much better than the result without the low-frequency compensation, as shown in Fig. 3(a).

Figure 3(b) illustrates the relative error varying with refractive index structure constant. The relative error of the standard deviation enlarges with the refractive index constant increasing. The field as Eq. (16) is more influenced by the turbulence than the intensity in Eq. (17), and the mean field is extremely small after the beam propagating through larger fluctuation turbulence. Therefore, the chaotic behavior of stronger fluctuation turbulence may cause the average sampling field calculated by the phase-screen method not converge to the theoretical mean field.

Figure 3(c) shows the screen size influence on the result precision. The relative error decreases with the screen size increasing. In theory, the phase screen method requires an infinite-size screen to simulate the turbulence, which is not feasible. Therefore, there is still a little bias of the standard deviation from the theoretical value. Considering the result precision and the calculation time, 1.2m was selected as the screen size and the spatial resolution of the phase screen was 1024 × 1024. The precision of the field calculated by the phase-screen method with this size screen is better than 90%.

Figure 3(d) illustrates the mean and standard deviation of the normalized intensity of the beam fundamental mode expressed as Eqs. (30) and (31) varying with sampling point number. The normalized intensity is equal to the harmonic intensity divided by the theoretical intensity in free space. The mean and standard deviation of the normalized intensity decoded with 180 sampling points are selected as the normalized parameters. With the sampling point increasing, the mean and standard are both tends to a constant and it provides a theoretical basis for the sampling point selection. In order to make full use of the large number of the pixel points in the phase screen, 180 sampling points are selected along the circumference in this study. The fundamental mode equals to the incident beam order annotated in Fig. 3(d) and the same annotations are also shown in Fig. 4 and Fig. 6.

Fig. 4. Mean and standard deviation of the normalized intensity fluctuation of the fundamental mode (a) comparison of different methods, (b) comparison of different beam orders.

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Figure 4(a) illustrates the mean and standard deviation of the normalized intensity of the fundamental beam mode. Nsd is the normalized standard deviation, which is equal to 20 times the standard deviation. There is a good agreement between the results of the two methods, and the maximum difference of the standard deviation is approximately 5%. Therefore, the Rytov theory has an appropriate performance for estimating the statistics of the beam harmonic intensity under weak fluctuation conditions, and it is faster than the phase-screen method which requires thousands of samples. As the beam radius increases, the mean decreases slowly, and the field intensity at the beam edge or the beam center is more influenced by the turbulence because the standard deviation is larger. Therefore, the field intensity at the area around the minimum standard deviation should be selected to decode the information carried by the beam OAM instead of the data on the beam edge. Figure 4(b) shows the mean and standard deviation of the different order beam intensity calculated by Rytov theory. The minimum standard deviation of the low-order beam is smaller than that of the high-order beam because the high-order beam diverges faster [38,40].

Figure 5 shows the mean and standard deviation of the harmonic normalized intensity of the third order Bessel Gaussian. The theoretical statistics derived by the Rytov theory coincide well with the results of the phase-screen method. The statistics of the harmonic intensity is the Fourier decomposition of the beam intensity, and the values of the adjacent-crosstalk mode which is adjacent to the fundamental mode are much larger than those of other crosstalk modes. Therefore, the main influence on the OAM decoding is the intensity of the adjacent-crosstalk mode.

Fig. 5. Statistics distribution of harmonic normalized intensity (a) mean (b) standard deviation

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Figure 6 illustrates the mean and standard deviation of the normalized intensity of the fundamental mode varying with the refractive index structure constant. The statistics derived by the Rytov theory agree well with the results obtained by the phase-screen method. With the turbulence strength increasing, the beam diverges significantly, accompanied by a decrease in the normalized intensity. The standard deviation varies more complex because it is not only related to the turbulence strength but also the sampling point location on the cross section, as shown in Fig. 4(b). Moreover, based on Fig. 4, the approximately beam radius of the minimum intensity fluctuation of different order beam at the radial direction were selected as the first two groups of curves in Fig. 6, and the variances at these place both enlarged with the turbulence strength increasing. In general, the beam distortion was larger in stronger turbulence because its mean intensity was smaller and the variance was larger.

Fig. 6. Statistics of normalized intensity of fundamental mode varying with refractive index structure constant, (a) mean (b) standard deviation.

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Figure 7 shows the mean and standard deviation of the intensity difference between the fundamental and adjacent-crosstalk mode. The statistics calculated by the Rytov method agree well with those of the phase-screen method. The variation of statistics is similar to that shown in Fig. 6. The mean and variance of the intensity difference are derived as a basis to investigate the probability of the OAM decoding error.

Fig. 7. Statistics of intensity difference.

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Figure 8 illustrates the PDF of the sum of the harmonic normalized intensity. The PDF obtained from the phase-screen method agrees well with the logarithmic normal distribution, which is widely used to characterize the intensity distribution [39,42]. The Gaussian distribution also agrees well with other distributions by neglecting the value less than 0 and this approximation is generally satisfied under weak fluctuation conditions. Therefore, the Gaussian distribution is applied in this study to approximate the PDF of the sum of the harmonic intensity.

Fig. 8. PDF of sum of harmonic intensity.

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Figure 9 shows the PDF of the normalized intensity of the crosstalk mode. The PDF of the harmonic intensity fluctuation is the joint PDF of a series of random variables with a complex PDF, and there is a spatial correlation of variables at different locations caused by the turbulence. Therefore, the analytical expression of this PDF is difficult to derive through the mathematical deduction. A widely applied method to solve this problem is to approximate this PDF by some commonly used distribution [39,42]. Fortunately, the PDF obtained from the phase-screen method is close to the Gamma distribution. Under this assumption, the Gamma distribution is selected to approximate the PDF of the crosstalk mode intensity with the mean and variance derived by the Rytov theory, and it is also appropriate to approximate the PDF of the normalized intensity of other high order crosstalk mode as Fig. 9(b).

Fig. 9. PDF of normalized intensity of crosstalk mode (a) adjacent-crosstalk mode intensity, (b) high order crosstalk mode.

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Figure 10 shows the PDF of the beam normalized intensity of the fundamental mode. The PDF based on the mean and variance derived by Rytov theory agrees well with that of the phase-screen method. Their bias enlarges with turbulence strength increasing because the precision of the intensity becomes reduced with the turbulence strength increasing as the description below Fig. 3(b) and Fig. 6. The variance of the sum of the crosstalk mode intensity can be approximated by the variance of the fundamental mode intensity because the sum of their intensities is close to a constant with a sufficiently small variance. In terms of fitting precision, this approximated method performs well, as shown in Fig. 10(b). The PDF establishes a relationship between the beam intensity perturbation and turbulence strength to characterize the random-varying property of beam distortion.

Fig. 10. PDF of normalized intensity of fundamental mode (a) fundamental mode n=3, (b) fundamental mode n=0.

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Figure 11(a) shows the PDF of the intensity difference between the fundamental and adjacent-crosstalk mode. This PDF is inapplicable of being solved directly by the joint PDF of the intensity difference between the fundamental and adjacent-crosstalk mode, because there is a restriction that their sum intensity must be less than the beam total intensity. Therefore, the PDF of the intensity difference is derived with the same method as that of the fundamental mode intensity, and this PDF with the statistics derived by the Rytov method agrees well with that of the phase-screen method. The probability that the intensity of the fundamental mode is both larger than that of the adjacent-crosstalk mode and smaller than that of other crosstalk mode is low enough to be neglected under weak fluctuation conditions. Moreover, there is the strong correlation of the harmonic intensity of the two adjacent-crosstalk modes. Therefore, the probability of the OAM decoding error could be approximated by the probability of the intensity difference between the fundamental and either of the two adjacent-crosstalk modes. The probability can be calculated through the integral of the PDF with its upper limit equaling to 0, and it enlarges with the turbulence refractive index structure constant increasing, as shown in Fig. 11(b). The probability relationship can be used as an applicable condition to calculate the maximum turbulence strength or the longest propagation distance with user-defined precision in laser communication.

Fig. 11. (a) PDF of intensity difference, (b) probability of decoding error

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4. Conclusion

In this study, the probability density function of the harmonic intensity fluctuation of the Bessel Gaussian beam in turbulence under weak fluctuation conditions has been investigated to characterize the random-varying performance of the OAM distortion.

The field intensity of the Bessel Gaussian beam as the contrasting data was sampled massively using the phase-screen method with the low-frequency compensation. The precision of the mean field intensity is better than 90% through comparison with the average field derived by the Rytov theory. The precision study of the phase-screen method provides theoretical support for the PDF investigation and the phase screen size selection.

The PDF of the harmonic intensity fluctuation is proposed based on the mean and variance derived by the Rytov theory, and the precision of these statistics is better than 94% through a comparison with those calculated by the phase-screen method. By analyzing millions of the field intensity samples in the phase-screen method, the Gamma distribution performs better in the commonly used simple distribution to approximate the PDF of the crosstalk mode intensity. Based on this, the PDF of the fundamental mode intensity has been derived with the mean and variance derived by the Rytov theory. These PDF establish a relationship between the intensity fluctuation and turbulence strength to characterize the random-varying property of the beam OAM distortion. Moreover, the PDF of the difference intensity between the fundamental and its adjacent mode was derived, and the probability of the beam OAM decoding error was also provided. These probability models could quantitatively characterize the turbulence influence on the OAM decoding error and fast estimate the maximum turbulence strength or the longest propagation distance with the user-defined precision in OAM communication.

In this paper, the probability property of the Bessel Gaussian beam in turbulence under weak fluctuation conditions has been investigated to provide the theoretical basis for the beam multiplexing. Moreover, the probability properties are also beneficial for the selection of signal parameters and the communication link design in laser communication.

Disclosures

The authors declare no conflicts of interest.

Data availability

The data that support the findings of this study are available from the corresponding author upon reasonable request.

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Beam parameters	Symbols	value
wavelength	$λ$	1.55 × 10⁻⁶ m
propagation distance	L	1000 m
interval distance	$Δ L = L / 10$	100 m
Bessel width	$β$	200 m^-1
Gaussian width	$ω_{0}$	0.02 m
sampling point number	m	180
phase screen size	D	1.2 m
phase screen resolution	N_m × N_n	1024 × 1024
refractive index structure constant	$C_{n}^{2}$	1.0 × 10⁻¹⁵ m^-2/3
sample number	N	10000

Probability property of orbital angular momentum distortion in turbulence

Abstract

1. Introduction

2. Theory and formulation

2.1 Phase-screen method

2.2 Turbulence moment of Bessel Gaussian beam

2.3 Beam distortion characterization

2.3.1 Distortion measured by power ratio

2.3.2 Statistics of harmonic intensity

2.3.3 Probability density function of harmonic intensity

3. Result and analysis

4. Conclusion

Disclosures

Data availability

References

Data availability

Cited By

Figures (11)

Tables (1)

Equations (44)

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