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Twisted splitting and propagation factor of superimposed twisted Hermite-Gaussian Schell-model beams in turbulent atmosphere

Qiangguo Huang, Shaohua Zhang, Zhenglan Zhou, Chen Xie, and Jun Qu

Open Access Open Access

Abstract

We believe this to be a new superposition twisted Hermite-Gaussian Schell-model (STHGSM) beam hat is proposed. Analytic formulas for the intensity distribution and propagation factor of the STHGSM beam in non-Kolmogorov turbulence are derived by utilizing the generalized Huygens-Fresnel principle (HFP) and the Wigner function. The evolution characteristics of STHGSM beams propagating are numerically calculated and analyzed. Our findings indicate that the light intensity of the STHGSM beam gradually undergoes splitting and rotation around the axis during propagation through non-Kolmogorov turbulence, eventually evolving into a diagonal lobe shape at a certain distance of transmission. The anti-turbulence capability of the beam strengthens with higher beam order or twist factor values.

© 2024 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement

1. Introduction

Simon first proposed the notion of the twisted phase of partially coherent beams and introduced partially coherent beams (PCB) with twisted phase in theory [1]. Subsequently, the evolutionary properties of twisted anisotropic GSM beams in dispersive and absorptive media have been thoroughly investigated [2]. In addition, a study was conducted on the spectral characteristics of TGSM beams transmitted through an aperture, revealing a significant influence of the distortion factor on spectral displacement as well as optical switching [3]. During propagation in strongly nonlocal nonlinear media, TGSM beams exhibit novel breather and soliton phenomena [4].

In recent years, there has been widespread study of the novel phenomenon of twisted beams, leading to the exploration of new GSM beams characterized by twisted phases and unconventional structures, which has emerged as a prominent research area. The quality of a beam is impacted by atmospheric turbulence during its propagation. Siegman first proposed to use the propagation factor to describe the beam quality. The M2-factor provides a more scientific and accurate description of beam quality by taking into account the influence of both the beam waist width and far-field divergence angle [5]. A positive definite condition for constructing the cross-spectral density of special spatial correlation structures was first proposed by Gori [6]. After that, he discusses a modeling process that can generate numerous real distortion sources without symmetric constraints [7].

At present, a series of special spatial correlation structure beam models have been proposed one after another. For instance, twisted Gaussian Schell-model array correlated sources have been reported, and their distorted phase can accelerate the diffraction of light beams [8]; the phenomenon of self-reconstruction generated by the twisted Laguerre-Gaussian Schell-model beam passing through a thin lens focusing system [9]; a cylindrical lens was employed by Cai to convert anisotropic GSM beams into TGSM beams [10]. Twisted anisotropic electromagnetic beams can effectively control beam characteristics through adjustments to the distortion factor and topological charge [11]; the Hermite-Gaussian correlation Schell model beam demonstrates a splitting phenomenon during spatial propagation [12]; the partially coherent crescent-shaped multi-vortex correlated Schell-model beam exhibits a crescent-shaped intensity distribution with multiple off-axis vortices on the source plane [13]. The THGCSM beam is characterized by non-circular symmetric coherence, whereas its source amplitude follows a circular Gaussian shape [14]; it was discovered by Peng that the transmission characteristics of RPTPCV beams are significantly impacted by the degree of helicity, characterized by twisted and vortex phases [15]; under ideal isotropic conditions, the orbital angular momentum (OAM) flux density of twisted array beams showcases a central dark hollow circular distribution on the source plane [16]; a general form of THGSM beam was constructed by Qu, and there are more beam parameters to regulate the intensity distribution [17]. In 2023, Cai observed that the twisted phase has a great influence on the OAM in turbulence, and also noted that turbulent atmosphere and twisting phase help to prevent beam splitting during propagation [18,19].

The STHGSM beam is first proposed by our group. The analysis formula of intensity distribution, root-mean-square(rms) angle width, and M2-factor of STHGSM beams are derived theoretically. The calculation shows that the STHGSM beam gradually splits, twists and rotates around the axis during non-Kolmogorov turbulence propagation, and evolves into a diagonal lobe when transmitted to a certain distance. The higher the order of the beam, the stronger its anti-turbulence ability, the larger the distortion factor, and the better its anti-turbulence ability. By appropriately selecting beam or turbulence parameters, the quality of the STHGSM beam can be enhanced. The obtained results hold theoretical significance in areas including directed energy weapons, directional tracking, and long-range optical communication.

2. Theoretical derivation

2.1 Cross-spectral density (CSD) of STHGSM beam

The requirement for the $\textrm{CSD}$ function of beam to satisfy non-negativity and positive definiteness is stated [6]:

$$W({\boldsymbol{\mathrm{\rho}}_1},{\boldsymbol{\mathrm{\rho}}_2},0) = \int {p({\bf v} )} {H^\ast }({\boldsymbol{\mathrm{\rho}}_1},{\bf v})H({\boldsymbol{\mathrm{\rho}}_2},{\bf v}){d^2}{\bf v},$$
where $p({\bf v} )$ is a non-negative weight function, $H(\boldsymbol{\mathrm{\rho}},{\bf v})$ is a kernel function, ${\boldsymbol{\mathrm{\rho}}_1}{\bf = }({x_1},{y_1})$, ${\boldsymbol{\mathrm{\rho}}_2}{\bf = }({x_2},{y_2})$ are two position vectors of the source plane, ${\bf v = }({{v_x},{v_y}} )$ signify the spatial frequency vector. When $p({\bf v} )$ and $H(\boldsymbol{\mathrm{\rho}},{\bf v})$ take different expressions, a series of different special PCB can be generated.

To construct the STHGSM beam, the selected weight function and kernel function are as follows:

$$p({\bf v}) = {C_0}\sum\limits_{t = 0}^{2m} {C_{2m}^t} {({{a_0}{v_x}} )^{nt}}{({{a_0}{v_y}} )^n}^{(2m - t)}\exp ( - g{v_x}^2 - g{v_y}^2),$$
$$H(\boldsymbol{\mathrm{\rho}},{\bf v}) = \exp \left( { - \frac{{{x^2}}}{{\sigma_x^2}} - \frac{{{y^2}}}{{\sigma_y^2}}} \right)\exp \{{ - ({ay + ix} ){v_x} + ({ax - iy} ){v_y}} \},$$
$m$, $n$ are the beam order, while ${\sigma _x}$ and ${\sigma _y}$ are the kernel function's root-mean-square width, g and a are genuine actual integers, and ${C_0}$=$\frac{1}{\pi }{\left( {\frac{i}{2}} \right)^{ - 2mn}}{g^{({mn + 1} )}}$ is the normalized constant.

By substituting Eq. (2) and (3) into Eq. (1), the CSD function of STHGSM beam on source plane can be obtained:

$$\begin{aligned} &W({\boldsymbol{\mathrm{\rho}}_1},{\boldsymbol{\mathrm{\rho}}_2},0) = \sum\limits_{t = 0}^{2m} {C_{2m}^t} {H_{nt}}\left( {\frac{{ - ({ - ia({{y_1} + {y_2}} )+ {x_2} - {x_1}} )}}{{2\sqrt g }}} \right) \\ &\times {H_{n({2m - t} )}}\left( {\frac{{ - ({ia({{x_2} + {x_1}} )+ {y_2} - {y_1}} )}}{{2\sqrt g }}} \right)\exp \left( { - \frac{{{{({{\boldsymbol{\mathrm{\rho}}_2} - {\boldsymbol{\mathrm{\rho}}_1}} )}^2}}}{{2{\delta_0}^2}}} \right) \\ &\times \exp \left( { - \frac{{({{x_1}^2 + {x_2}^2} )}}{{{w_{0x}}^2}} - \frac{{({{y_1}^2 + {y_2}^2} )}}{{{w_{0y}}^2}}} \right)\exp ({ - iu({{x_1}{y_2} - {x_2}{y_1}} )} ), \end{aligned}$$

In the above formula, we set u = a/g as the twist factor of STHGSM beam. In source plane, the coherence length and beam width can be defined as follows:

$${\delta _0} = {\left( {\frac{{{a^2} + 1}}{{2g}}} \right)^{ - \frac{1}{2}}},\textrm{ }{w_{ox}} = {\left( {\frac{1}{{\sigma_x^2}} - \frac{{{a^2}}}{{2g}}} \right)^{ - \frac{1}{2}}}\textrm{, }{w_{oy}} = {\left( {\frac{1}{{\sigma_y^2}} - \frac{{{a^2}}}{{2g}}} \right)^{ - \frac{1}{2}}},$$

From Eq. (4), we can know that when $m{\bf = }n{\bf \ne }0$ and $u{\bf = }0$, the beam evolves into a SHGCSM beam; when $m{\bf = }n{\bf = }0$ and $u{\bf \ne }0$, the beam degenerates into a TGSM beam, and distortion factor needs to meet inequality requirements [17]:

$$u \le \frac{1}{{\delta _{0x}^2}}\sqrt {\frac{b}{a}} = \frac{1}{{\delta _{0y}^2}}\sqrt {\frac{b}{a}} ,$$

From the expressions (5) and (6), the beam parameters of source plane are constrained to each other, and the CSD needs to satisfy the above conditions; the distortion factor of the STHGSM beam is intimately connected to the parameters a, $b$. when $a = b$, Eq. (6) is consistent with the limiting condition of the TGSM beam.

The analysis formula for $\textrm{CSD}$ function of STHGSM beams undergoing transmission through turbulent media is derived by employing the generalized Huygens-Fresnel principle [20–22], expressed as:

$$\begin{aligned} &W({\boldsymbol{\mathrm{\rho}}_3},{\boldsymbol{\mathrm{\rho}}_4},z) = {\left( {\frac{k}{{2\pi z}}} \right)^2}\int\!\!\!\int {W({\boldsymbol{\mathrm{\rho}}_1},{\boldsymbol{\mathrm{\rho}}_2},0)\exp \left( { - \frac{{ik}}{{2z}}{{({{\boldsymbol{\mathrm{\rho}}_1} - {\boldsymbol{\mathrm{\rho}}_3}} )}^2} + \frac{{ik}}{{2z}}{{({{\boldsymbol{\mathrm{\rho}}_2} - {\boldsymbol{\mathrm{\rho}}_4}} )}^2}} \right)} \\ & \qquad \times \exp \left( { - \frac{{{\pi^2}{k^2}zT}}{3}({{{({{\boldsymbol{\mathrm{\rho}}_3} - {\boldsymbol{\mathrm{\rho}}_4}} )}^2} + ({{\boldsymbol{\mathrm{\rho}}_3} - {\boldsymbol{\mathrm{\rho}}_4}} )({{\boldsymbol{\mathrm{\rho}}_1} - {\boldsymbol{\mathrm{\rho}}_2}} )+ {{({{\boldsymbol{\mathrm{\rho}}_1} - {\boldsymbol{\mathrm{\rho}}_2}} )}^2}} )} \right){d^2}{\boldsymbol{\mathrm{\rho}}_1}{d^2}{\boldsymbol{\mathrm{\rho}}_2} \end{aligned}$$
where $k = 2\pi /\lambda $ is the wavenumber, λ is the wavelength, ${\boldsymbol{\mathrm{\rho}}_3} = ({{\rho_{3x}},{\rho_{3y}}} )$ and ${\boldsymbol{\mathrm{\rho}}_4} = ({{\rho_{4x}},{\rho_{4y}}} )$ represent two distinct points on the receiving plane, and T is represented as [23]:
$$T = \int\limits_0^\infty {{\kappa ^3}} {\Phi _n}(\kappa )d\kappa = 0.1661C_n^2l_0^{ - 1/3},$$

In the equation, $\kappa $ represents spatial wave function, while ${\Phi _n}(\kappa )$ denotes energy spectrum function of refractive index fluctuation [24,25]:

$${\Phi _n}(\kappa )= 0.033C_n^2{\kappa ^{ - 11/3}}\exp ({ - {\kappa^2}/\kappa_m^2} ),$$
where $C_n^2$ represents the refractive index structure constant, ${\kappa _m} = 5.92/{l_0},\,{l_0}$ is the inner scale of turbulence.

Substitute Eqs.  (2), (3), and (8) into (7); after complex calculation, the $\textrm{CSD}$ function of the STHGSM beam is given:

$$\begin{aligned} W({\boldsymbol{\mathrm{\rho}}_3},{\boldsymbol{\mathrm{\rho}}_4},z) &= \sum\limits_{t = 0}^{2m} {C_{2m}^t\frac{{\pi {C_0}}}{{{b_x}{d_x}}}{{\left( {\frac{k}{{2z}}} \right)}^2}{{\left( {\frac{\textrm{i}}{2}} \right)}^{2nm}}} \\ &\times \exp \left[ { - \frac{{ik}}{{2z}}({\boldsymbol{\mathrm{\rho}}_3^2 - \boldsymbol{\mathrm{\rho}}_4^2} )- {T_0}[{{{({{\boldsymbol{\mathrm{\rho}}_3} - {\boldsymbol{\mathrm{\rho}}_4}} )}^2}} ]+ {R_1} + \frac{{{F_1}^2}}{{4{P_1}}} + \frac{{{F_2}^2}}{{4{P_2}}}} \right]\\ &\times {P_1}^{ - ({nt + 1} )/2}{P_2}^{ - ({n({2m - t} )+ 1} )/2}{H_{nt}}\left( {\frac{{i{F_1}}}{{2\sqrt {{P_1}} }}} \right){H_{n({2m - t} )}}\left( {\frac{{i{F_2}}}{{2\sqrt {{P_2}} }}} \right) \end{aligned}$$
where
$$\begin{aligned} &{T_0} = \frac{{{\pi ^2}{k^2}zT}}{3},\textrm{ }{\sigma _x} = {\sigma _y} = \sigma \\ &{b_x} = \left( {\frac{1}{{\sigma_x^2}} + \frac{{ik}}{{2z}} + {T_0}} \right),\textrm{ }{b_y} = \left( {\frac{1}{{\sigma_y^2}} - \frac{{ik}}{{2z}} + {T_0}} \right)\\ &{d_x} = \frac{1}{{\sigma _x^2}} - \frac{{ik}}{{2z}} + {T_0} - \frac{{T_0^2}}{{{b_x}}},\textrm{ }{d_y} = \frac{1}{{\sigma _y^2}} - \frac{{ik}}{{2z}} + {T_0} - \frac{{T_0^2}}{{{b_x}}},\textrm{ }{d_x} = {d_y} \\ & x_3^{\prime} = \frac{{ik{x_3}}}{z} - {T_0}({{x_3} - {x_4}} ),\textrm{ }y_3^{\prime} = \frac{{ik{y_3}}}{z} - {T_0}({{y_3} - {y_4}} )\\ &x_4^{\prime} ={-} \frac{{ik{x_4}}}{z} + {T_0}({{x_3} - {x_4}} ),\textrm{ }y_4^{\prime} ={-} \frac{{ik{y_4}}}{z} + {T_0}({{y_3} - {y_4}} )\\ &{R_1} = \frac{{{{({x_3^{\prime}} )}^2} + {{({y_3^{\prime}} )}^2}}}{{4{b_x}}} + \frac{{{{({{b_x}x_4^{\prime} + x_3^{\prime}{T_0}} )}^2} + {{({y_3^{\prime}{T_0} + {b_x}y_4^{\prime}} )}^2}}}{{4b_x^2{d_x}}}\\ &{P_1} = g + \frac{{1 - {a^2}}}{{4{b_x}}} + \frac{{({1 - {a^2}} )T_0^2}}{{4b_x^2{d_x}}} + \frac{{ - {T_0} - {a^2}}}{{2{b_x}{d_x}}} + \frac{{1 - {a^2}}}{{4{d_x}}} \\ &{P_2} = g + \frac{{1 - {a^2}}}{{4{b_x}}} + \frac{{({1 - {a^2}} )T_0^2}}{{4b_x^2{d_x}}} - \frac{{({{a^2} + 1} ){T_0}}}{{2{b_x}{d_x}}} + \frac{{1 - {a^2}}}{{4{d_x}}}\\ &{F_1} = \frac{{ay_3^{\prime} - ix_3^{\prime}}}{{2{b_x}}} + \frac{{({x_3^{\prime} - x_4^{\prime}} )i{T_0} + ({y_3^{\prime} + y_4^{\prime}} )a{T_0}}}{{2{b_x}{d_x}}} + \frac{{ix_4^{\prime} + ay_4^{\prime}}}{{2{d_x}}} + \frac{{({ay_3^{\prime} - ix_3^{\prime}} )T_0^2}}{{2b_x^2{d_x}}}\\ &{F_2} ={-} \frac{{({ax_3^{\prime} + iy_3^{\prime}} )}}{{2{b_x}}} + \frac{{({y_3^{\prime} - y_4^{\prime}} )i{T_0} + ({ - x_4^{\prime} - x_3^{\prime}} )a{T_0}}}{{2{b_x}{d_x}}} + \frac{{iy_4^{\prime} - ax_4^{\prime}}}{{2{d_x}}} - \frac{{({ax_3^{\prime} + iy_3^{\prime}} )T_0^2}}{{2b_x^2{d_x}}} \end{aligned}$$

The following integral formula [26] is used in the calculation:

$$\int\limits_{ - \infty }^{ + \infty } {\exp ({ - p{x^2} \pm qx} )} dx = \sqrt {\frac{\pi }{p}} \exp \left( {\frac{{{q^2}}}{{4p}}} \right)$$
$$\int\limits_{ - \infty }^{ + \infty } {{x^m}\exp ({ - p{x^2} - qx} )} dx = \sqrt \pi {\left( {\frac{i}{2}} \right)^m}{p^{ - ({m + 1} )/2}}\exp \left( {\frac{{{q^2}}}{{4p}}} \right){H_m}\left( {\frac{{iq}}{{2\sqrt p }}} \right)$$

2.2 Rms angle width and propagation factor of STHGSM beams

Propagation factor represents a significant covariate to measure the quality of beam in propagation, and following the generalized HFP, $\textrm{CSD}$ function of the beam within turbulence is given by [27]:

$$\begin{aligned} W({{\boldsymbol{\mathrm{\rho}}_s},{\boldsymbol{\mathrm{\rho}}_d},z} ) &= {\left( {\frac{k}{{2\pi z}}} \right)^2}\int\!\!\!\int {} W({{{\bf r}_s},{{\bf r}_d},0} )\\ &\times \exp \left( {\frac{{ik}}{z}({{\boldsymbol{\mathrm{\rho}}_s} - {{\bf r}_s}} )({{\boldsymbol{\mathrm{\rho}}_d} - {{\bf r}_d}} )- H({{\boldsymbol{\mathrm{\rho}}_d},{{\bf r}_d},z} )} \right){d^2}{{\bf r}_s}{d^2}{{\bf r}_d} \end{aligned}$$
where $H({{\boldsymbol{\mathrm{\rho}}_d},{{\bf r}_d},z} )$ denotes the impact of non-Kolmogorov turbulence. In the process of integral calculation, Eq. (15) is used:
$$\begin{aligned} &{x_s} = ({x_1} + {x_2})/2,{y_s} = ({y_1} + {y_2})/2\textrm{ ,}{x_d} = {x_1} - {x_2},{y_d} = {y_1} - {y_2} \\ &{\rho _{sx}} = ({\rho _{3x}} + {\rho _{4x}})/2,{\rho _{sy}} = ({\rho _{3y}} + {\rho _{4y}})/2,{\rho _{dx}} = {\rho _{3x}} - {\rho _{4x}}\textrm{ },{\rho _{dy}} = {\rho _{3y}} - {\rho _{4y}} \end{aligned}$$

According to a series of transformations shown in Ref. [28], Eq. (14) can be written as:

$$\begin{aligned} W({{\boldsymbol{\mathrm{\rho}}_s},{\boldsymbol{\mathrm{\rho}}_d},z} ) &= \frac{1}{{{{(2\pi )}^2}}}\int\!\!\!\int {} W\left( {{\bf r}_s^{\prime},{\boldsymbol{\mathrm{\rho}}_d} + \frac{z}{k}{{\bf \kappa }_d},0} \right){d^2}{\bf r}_s^{\prime}{d^2}{\boldsymbol{\mathrm{\kappa}}_d}\\ &\times \exp \left( { - i{\boldsymbol{\mathrm{\rho}}_s} \cdot {\boldsymbol{\mathrm{\kappa}}_d} + i{\bf r}_s^{\prime} \cdot {\boldsymbol{\mathrm{\kappa}}_d} - \frac{{{\pi^2}{k^2}z}}{3}\left( {3\boldsymbol{\mathrm{\rho}}_d^2 + 3\frac{z}{k}{\boldsymbol{\mathrm{\rho}}_d}{\boldsymbol{\mathrm{\kappa}}_d} + \frac{{{z^2}}}{{{k^2}}}\boldsymbol{\mathrm{\kappa}}_d^2} \right)} \right) \end{aligned}$$
${\boldsymbol{\mathrm{\kappa}}_d}$ represents the position vector.

The Wigner distribution function ($\textrm{WDF}$) of beams can be represented using CSD [29]:

$$h({{\boldsymbol{\mathrm{\rho}}_s},{\bf \theta },z} )= {\left( {\frac{k}{{2\pi }}} \right)^2}\int_{ - \infty }^\infty {} \int_{ - \infty }^\infty {W({{\boldsymbol{\mathrm{\rho}}_s},{\boldsymbol{\mathrm{\rho}}_d},z} )} \exp ({ - ik{\bf \theta }{\boldsymbol{\mathrm{\rho}}_d}} ){d^2}{\boldsymbol{\mathrm{\rho}}_d}$$
where $k{\theta _x}$ and $k{\theta _y}$ represent the wave vector components along the $x$ and $y$-axis respectively, and ${\bf \theta } = ({{\theta_x},{\theta_y}} )$ is the angle between the relevant coordinate and the z-direction.

Equation (16) is substituted into Eq. (17) to derive the $\textrm{WDF}$ of STHGSM beam as:

$$\begin{aligned} &h({{\boldsymbol{\mathrm{\rho}}_s},{\bf \theta },z} )= \frac{{{k^2}\pi }}{{16{\pi ^4}}}\sum\limits_{t = 0}^{2m} {C_{2m}^t\sum\limits_{{k_1} = 0}^{n({2m - t} )} {} \sum\limits_{{j_1} = 0}^{[{{k_1}/2} ]} {} \sum\limits_{{k_2} = 0}^{nt} {} \sum\limits_{{j_2} = 0}^{[{{k_2}/2} ]} {} } \frac{1}{{{2^{nm}}}}{({ - 1} )^{{j_1} + {j_2} + {k_1} - 2{j_1}}}\\ &\times \frac{{{k_1}!}}{{{j_1}!({{k_1} - 2{j_1}} )!}}\frac{{{k_2}!}}{{{j_2}!({{k_2} - 2{j_2}} )!}}{\left( {\frac{{2ia\sqrt 2 }}{{\sqrt g }}} \right)^{{k_1} - 2{j_1} + {k_2} - 2{j_2}}}{\left( {\frac{2}{{{w_{0x}}^2}}} \right)^{ - ({{k_1} - 2{j_1} + 1} )/2 - ({{k_2} - 2{j_2} + 1} )/2}}\\ &\times {\left( {\frac{i}{2}} \right)^{{k_1} - 2{j_1} + {k_2} - 2{j_2}}}\int\!\!\!\int {\int\!\!\!\int {} } {H_{n({2m - t} )- {k_1}}}\left( {\sqrt 2 \frac{{\left( {{\rho_{dy}} + \frac{z}{k}{\kappa_{dy}}} \right)}}{{2\sqrt g }}} \right){H_{nt - {k_2}}}\left( {\sqrt 2 \frac{{\left( {{\rho_{dx}} + \frac{z}{k}{\kappa_{dx}}} \right)}}{{2\sqrt g }}} \right)\\ &\times {H_{{k_1} - 2{j_1}}}\left( {\frac{{{u_0}\left( {{\rho_{dy}} + \frac{z}{k}{\kappa_{dy}}} \right) + {\kappa_{dx}}}}{{2\sqrt {\frac{2}{{{w_{0x}}^2}}} }}} \right){H_{{k_2} - 2{j_2}}}\left( {\frac{{ - {u_0}\left( {{\rho_{dx}} + \frac{z}{k}{\kappa_{dx}}} \right) + {\kappa_{dy}}}}{{2\sqrt {\frac{2}{{{w_{0y}}^2}}} }}} \right)\\ &\times \exp \left( {\frac{{{u_0}{w_{0x}}^2}}{4}({{\rho_{dx}}{\kappa_{dy}} - {\rho_{dy}}{\kappa_{dx}}} )} \right)\exp ({ - {E_1}\boldsymbol{\mathrm{\rho}}_d^2 - {G_1}\boldsymbol{\mathrm{\kappa}}_d^2 - {N_1}{\boldsymbol{\mathrm{\rho}}_d}{\boldsymbol{\mathrm{\kappa}}_d} - ik{\bf \theta }{\boldsymbol{\mathrm{\rho}}_d} - i{\boldsymbol{\mathrm{\rho}}_s}{\boldsymbol{\mathrm{\kappa}}_d}} ){d^2}{\boldsymbol{\mathrm{\kappa}}_d}{d^2}{\boldsymbol{\mathrm{\rho}}_d} \end{aligned}$$
where
$${E_1} = \left( {\frac{1}{{2{\delta_0}^2}} + \frac{1}{{2{w_{0x}}^2}} + \frac{{u_0^2{w_{0x}}^2}}{8} + {\pi^2}{k^2}zT} \right)\,\,{N_1} = \left( {\frac{z}{{{\delta_0}^2k}} + \frac{z}{{{w_{0x}}^2k}} + \frac{{u_0^2{w_{0x}}^2z}}{{4k}} + {\pi^2}k{z^2}T} \right)$$
$${G_1} = \left( {\frac{{{z^2}}}{{2{\delta_0}^2{k^2}}} + \frac{{{z^2}}}{{2{w_{0x}}^2{k^2}}} + \frac{{u_0^2{w_{0x}}^2{z^2}}}{{8{k^2}}} + \frac{{{w_{0x}}^2}}{8} + \frac{{{\pi^2}{z^3}T}}{3}} \right)$$

Formula (13) and the following formula are used in the calculation:

$${H_n}({x + y} )= \frac{1}{{{2^{n/2}}}}\sum\limits_{k = 0}^n {} {H_k}\left( {\sqrt 2 x} \right){H_{n - k}}\left( {\sqrt 2 y} \right)$$

The expression for the ${n_1} + {n_2} + {m_1} + {m_2}$ -order $\textrm{WDF}$ of the beam is given by [30]:

$$\left\langle {{x^{{n_1}}}{y^{{n_2}}}\theta_x^{{m_1}}\theta_y^{{m_2}}} \right\rangle = \frac{1}{p}\int\!\!\!\int {} {x^{{n_1}}}{y^{{n_2}}}\theta _x^{{m_1}}\theta _y^{{m_2}}h({{\boldsymbol{\mathrm{\rho}}_s},{\bf \theta },z} ){d^2}{\boldsymbol{\mathrm{\rho}}_s}{d^2}{\bf \theta }$$
$$P = \int\!\!\!\int {} h({{\boldsymbol{\mathrm{\rho}}_s},{\bf \theta },z} ){d^2}{\boldsymbol{\mathrm{\rho}}_s}{d^2}{\bf \theta }$$
$${\theta _N}(z )= {\left( {\left\langle {{{\bf \theta }^2}} \right\rangle } \right)^{1/2}} = {\left( {\left\langle {\theta_x^2} \right\rangle + \left\langle {\theta_y^2} \right\rangle } \right)^{1/2}}$$
$${M^2}(z )= k{\left( {\left\langle {{\boldsymbol{\mathrm{\rho}}^2}} \right\rangle \left\langle {{{\bf \theta }^2}} \right\rangle - {{\left\langle {\boldsymbol{\mathrm{\rho}} \cdot {\bf \theta }} \right\rangle }^2}} \right)^{1/2}}$$

In our calculations, we utilize the following integral equation [27]:

$$\delta (s )= \frac{1}{{2\pi }}\int\limits_{ - \infty }^\infty {} \exp ({ - isx} )dx$$
$${\delta ^n}(s )= \frac{1}{{2\pi }}\int\limits_{ - \infty }^\infty {} {({ - ix} )^n}\exp ({ - isx} )dx\textrm{ }({n = 1,2} )$$
$$\int\limits_{ - \infty }^\infty {} f(x ){\delta ^n}(x )dx = {({ - 1} )^n}\textrm{ }{f^n}(0 )\textrm{ }({n = 1,2} )$$

From Eqs. (18)–(27), we derive analysis formulas for the $\textrm{rms}$ angle width and the M2-factor of STHGSM beams.

3. Numerical results

As illustrated in Fig. 1, the intensity distribution of STHGSM beam in source plane is simulated by varying beam orders and twist factors. From Fig. 1 it can be seen that distortion factor and order impact the splitting of STHGSM beams. The STHGSM beam with a large twist factor has a more obvious degree of beam splitting. As the beam order increases, the beam splitting becomes more pronounced. The beam can be split into two or four flaps that are distorted and rotated by adjustment of the distortion factor and beam order values [17]. When m = n = 0, the beam degenerates into a TGSM beam, exhibiting a Gaussian intensity distribution that does not split with increasing distortion factor. When u = 0, the STHGSM beam degenerates into a SHGCSM beam, maintaining a Gaussian-shaped intensity distribution. Conversely, when u≠0, the degree of beam splitting increases with higher beam orders.

 figure: Fig. 1.

Fig. 1. Intensity distribution of STHGSM beams with varying beam orders and twist factors. where parameters include: λ=632.8 nm, g = 0.05 $c{m^2}$, ${\sigma _x}$=${\sigma _y}$=0.09 cm, $C_n^2$=${10^{ - 15}}\,{m^{3 - \alpha }}$, and ${l_0}$=1 mm.

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To investigate the impact of the root-mean-square beam width and parameter g on the normalized light intensity distribution, we conducted simulations for multiple sets of light intensity results under various ${\sigma _x}$, ${\sigma _y}$ and parameters g, as illustrated in Fig. 2. With a distortion factor (u = 20 $c{m^{ - 2}}$) and beam orders (m = n = 4), the intensity distribution of STHGSM beam predominantly manifests as four concentrated spots, and the intensity gradually evolves into a symmetrical and uniform distribution on the source plane. The row of Fig. 2(a-d) reflects the evolution of light intensity distribution with the parameter g. The larger the parameter g of the STHGSM beam, the more obvious the degree of beam splitting; from columns (1-4), it is evident that as the root mean-squared widths (${\sigma _x}$=${\sigma _y}$)increase, the degree of beam splitting becomes more pronounced.

 figure: Fig. 2.

Fig. 2. Intensity distribution of STHGSM beams with varying parameters g, ${\sigma _x}$ and ${\sigma _y}$. where parameters include: λ=632.8 nm, $C_n^2$=${10^{ - 15}}\,{m^{3 - \alpha }}$, u = 20 $c{m^{ - 2}}$, m = n = 4, and ${l_0}$=1 mm.

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Figure 3 simulates the spatial light intensity at a propagation distance of 25 meters, aiming to investigate the evolutionary patterns of spatial intensity distribution of STHGSM beam under non-Kolmogorov turbulent propagation concerning various beam parameters. In Fig. 3, we can observe that when the beam propagates at z = 25 m, the light intensity distribution of STHGSM beam exhibits spins along its transmission axis. Analysis of the rows in Fig. 3(a-d) reveals that a higher twist factor results in a more pronounced degree of beam splitting; from the rows presented in Fig. 3(b) and (c), it is evident that the distortion factor significantly impacts the direction of rotation of the light field distribution, leading to the rotation of the beam around the transmission axis during propagation. When m = n = 0, the intensity distribution of STHGSM beam maintains a Gaussian shape, and the beam does not show the characteristic of self-splitting; from columns (1-4), it is apparent that beam order influences the degree of beam splitting, and the higher beam order is, the more dramatic the degree of beam splitting is.

 figure: Fig. 3.

Fig. 3. Intensity distribution of STHGSM beams under non-Kolmogorov turbulence with varying distortion factors u and order. where parameters include: $\lambda $=632.8 nm, g = 0.05 $c{m^2}$, ${\sigma _x}$=${\sigma _y}$=0.09 cm, $C_n^2$=${10^{ - 15}}\,{m^{3 - \alpha }}$, ${l_0}$=1 mm.

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Figure 4 illustrates the light intensity distribution of STHGSM beams at various distances within non-Kolmogorov turbulence. It is evident that along the direction of light propagation, the beam gradually splits and rotates around the axis during non-Kolmogorov turbulent propagation. From Fig. 4(a), the light intensity distribution in the source plane has a Gaussian shape when u = 0. As the transmission distance z increases, the STHGSM beam shows a self-splitting phenomenon; Fig. 4(a-c) reflects the impact of different twist factors on the light intensity distribution of the STHGSM beam. The greater the distortion factor, the more pronounced STHGSM beam splitting, enhancing the beam's resistance to turbulence. From Fig. 4(c) and (d), it is evident that when the beam possesses the same distortion factor, a higher beam order results in a more pronounced degree of splitting, thereby enhancing the anti-turbulence effectiveness. Compared with Fig. 4(c) and (e), when u = 40 $c{m^{ - 2}}$, the beam splitting with a larger root-mean-squared width is more obvious.

 figure: Fig. 4.

Fig. 4. The intensity distribution of STHGSM beams with varying beam parameters. The (a-c) row parameters are: $\lambda $=632.8 nm, g = 0.05 $c{m^2}$, ${\sigma _x}$=${\sigma _y}$=0.09 cm, m = n = 4, $C_n^2$=${10^{ - 15}}\,{m^{3 - \alpha }}$, ${l_0}$=1 mm.

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Figure 5 explores the light-intensity evolution of STHGSM beams in the long-distance propagation of non-Kolmogorov turbulence. After examining Fig. 5(a), it is evident that light intensity in source plane assumes a Gaussian shape when u = 0, and as transmission distance increases, the STHGSM beam exhibits a phenomenon of distortion. Figure 5(a-c) illustrates how the light intensity distribution varies with twist factor. A larger twist factor correlates with a more pronounced beam splitting, thereby enhancing its resistance to turbulence effects. Figure 5(c) and (d) reflect the impact of different beam orders on the light intensity during non-Kolmogorov turbulence transmission. A higher beam order correlates with a more pronounced splitting degree and enhanced resistance to turbulence effects. Figure 5(c), (e) shows that when the beam has the same twist factor, the beam with the root-mean-squared width ${\sigma _x} = \,{\sigma _y}$ is larger, and the splitting is more obvious.

 figure: Fig. 5.

Fig. 5. The intensity distribution of STHGSM beams with varying beam parameters. The (a-c) row parameters are: $\lambda $=632.8 nm, g = 0.05 $c{m^2}$, ${\sigma _x}$=${\sigma _y}$=0.09 cm, m = n = 4, $C_n^2$=${10^{ - 15}}\,{m^{3 - \alpha }}$, ${l_0}$=1 mm.

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When the STHGSM beam propagates to a certain distance, the light intensity is gradually allocated to the upper and lower diagonal lobes under the action of turbulence. The refractive index structure constant serves as a measure of turbulence intensity. Therefore, we conduct a numerical analysis to investigate the evolution of STHGSM beams within non-Kolmogorov turbulence transmission under varying refractive index structure constants. From Fig. 6, it is evident that the STHGSM beam progressively diverges and undergoes slight rotation as the transmission distance increases. Upon transmission to a specific distance, the light intensity predominantly concentrates on the two lobes situated along the upper and lower diagonals, attributed to the combined effects of turbulence intensity and twist factor.

 figure: Fig. 6.

Fig. 6. The intensity distribution of STHGSM beams under non-Kolmogorov turbulence conditions with varying $C_n^2$. The (a-c) row calculation parameters are: $\lambda $=632.8 nm, g = 0.05 $c{m^2}$, ${\sigma _x}$=${\sigma _y}$=0.09 cm, m = n = 4, $u$= 40 $c{m^{ - 2}}$, ${l_0}$=1 mm.

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Figure 7 reflects the variation of normalized root-mean-square angle width of STHGSM beams with different optical orders and wavelengths with propagation distance under non-Kolmogorov turbulence. From Fig. 7(a), the STHGSM beam is degraded to a TGSM beam when m = n = 0. We observed that higher beam orders are associated with smaller rms angle widths, resulting in diminished influence from non-Kolmogorov turbulence and an improved beam’s anti-turbulence capability. In Fig. 7(b), the trend of the rms angle width variation with respect to wavelength is presented. It is observed that as the wavelength decreases, the rms angle width increases, indicating a stronger impact of no-Kolmogorov turbulence.

 figure: Fig. 7.

Fig. 7. Variation in the rms angle width of STHGSM beams with differing orders and wavelengths. The common parameters include: $u$= 40 $c{m^{ - 2}}$, g = 0.05 $c{m^2}$, ${\sigma _x}$=${\sigma _y}$=0.09 cm, ${l_0}$=1 mm,, $C_n^2$=4*${10^{ - 15}}\,{m^{3 - \alpha }}$. (a) $\lambda $=632.8 nm; (b) m = n = 2

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Figure 8 illustrates the variation in the rms angle width of STHGSM beams across various refractive index structure constants and turbulent inner scales within non-Kolmogorov turbulence. As depicted in Fig. 8(a), beams characterized by smaller refractive index structure constants exhibit reduced rms angle widths, indicating lesser susceptibility to non-Kolmogorov turbulence effects. From Fig. 8(b), it is evident that a smaller turbulence scale of the beam corresponds to a larger rms angle width, highlighting a heightened impact of turbulence.

 figure: Fig. 8.

Fig. 8. The variation of the rms angle width of STHGSM beams with various refractive index structure constants and turbulent inner scales. where parameters include: $\lambda $=632.8 nm, g = 0.05 $c{m^2}$, ${\sigma _x}$=${\sigma _y}$=0.09 cm, $u$= 40 $c{m^{ - 2}}$, m = n = 2. (a) ${l_0}$=1 mm; (b) $C_n^2$=4*${10^{ - 15}}\,{m^{3 - \alpha }}$

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To investigate the impact of beam quality on the transmission of STHGSM beams under non-Kolmogorov turbulence, we conducted numerical calculations of the normalized M2-factor. Figures 9–11 present the results of these calculations. It is observed that the propagation factor increases with the transmission distance, indicating a decrease in beam quality due to turbulence effects. Figure 9 reflects the variation of M2-factor and propagation distance z of STHGSM beams with different beam orders and distortion factors. From Fig. 9(a), it is evident that as the beam order increases, M2-factor of the STHGSM beam decreases, indicating higher resistance to turbulence. Figure 9(b) demonstrates that an increased twist factor of the STHGSM beam correlates with a reduced normalized M2-factor during transmission, suggesting enhanced turbulence resilience of the beam.

 figure: Fig. 9.

Fig. 9. The impact of varying distortion factor and order on the propagation factor of STHGSM beam. Where parameters include: $\lambda $=632.8 nm, g = 0.05 $c{m^2}$, ${\sigma _x}$=${\sigma _y}$=0.09 cm, $C_n^2$=$4\ast {10^{ - 15}}\,{m^{3 - \alpha }}$, ${l_0}$=1 mm. (a) $u$= 40 $c{m^{ - 2}}$;(b) m = n = 4

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 figure: Fig. 10.

Fig. 10. The influence of different parameters g, ${\sigma _x}$, ${\sigma _y}$, as well as wavelength on the ${M^\textrm{2}}$-factor of STHGSM beam. The common parameters include: m = n = 4, u = 40 $c{m^{ - 2}}$, $C_n^2$=$4\ast {10^{ - 15}}\,{m^{3 - \alpha }}$, ${l_0}$=1 mm. (a)$\lambda $=632.8 nm, ${\sigma _x}$=${\sigma _y}$=0.09 cm; (b)$\lambda $=632.8 nm, g = 0.05 $c{m^2}$

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 figure: Fig. 11.

Fig. 11. The influence of different refractive index structure constants and turbulence inner scale on the M2-factor of STHGSM beams. The common parameters include: m = n = 4, u = 40 $c{m^{ - 2}}$, g = 0.05 $c{m^2}$, ${\sigma _x}$=${\sigma _y}$=0.09 cm, $\lambda $=632.8 nm.(a) ${l_0}$=1 mm; (b) $C_n^2$=$4\ast {10^{ - 15}}\,{m^{3 - \alpha }}$

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Figure 10 reflects the variation of normalized M2-factor and propagation distance z of STHGSM beams with different parameters g and root-mean-squared width. From Fig. 10(a), it is observed that increasing the parameter g of the STHGSM beam leads to a decrease in the propagation factor during transmission, indicating enhanced turbulence resistance of the beam. Similarly, in Fig. 10(b), M2-factor of the STHGSM beam diminishes as the root mean-squared width increases during turbulent propagation, highlighting the beam's improved ability to withstand turbulence effects.

Given the substantial impact of turbulence parameters on the transmission characteristics of STHGSM beams under turbulent environments, Fig. 11 investigates how the normalized M2-factor of STHGSM beams changes as they propagate through non-Kolmogorov turbulence, considering various refractive index constants $C_n^2$ and turbulence inner scales. From Fig. 11(a), we can see that as the $C_n^2$ increases, M2-factor of the STHGSM beam becomes larger and beam quality becomes worse. As depicted in Fig. 11(b), an increase in the inner scale of turbulence correlates with a decrease in the M2-factor of the STHGSM beam, thereby enhancing the beam's resilience to the effects of turbulence.

4. Summary

In this paper, a class of STHGSM beams is theoretically proposed. The analysis formula of intensity distribution, rms angle width, and M2-factor of STHGSM beams within non-Kolmogorov turbulence are derived. The STHGSM beam provides more beam parameters and turbulence parameters, which can better control the light intensity distribution within source plane and turbulence media. The transmission characteristics of STHGSM beams under non-Kolmogorov turbulence are numerically calculated and conducted. The findings indicate that by adjusting the beam parameters, the STHGSM beam can be divided into either two or four lobes. The light intensity distribution of STHGSM beam rotates around its axis, and the positive or negative distortion factor affects rotation direction of a light field. STHGSM beams also demonstrate inherent twisting and splitting attributes. Under turbulence, the light intensity gradually evolves from four lobes to upper and lower diagonal lobes. Furthermore, we conduct quantitative calculations to determine the normalized M2-factor of STHGSM beam and investigate how the quality of the STHGSM beam varies with both beam parameters and turbulence parameters. The conclusion we have obtained indicates: the STHGSM beams with larger distortion factor, root-mean-square width, parameter g, turbulent inner scale, and higher beam order have strong anti-turbulence ability. while the larger $C_n^2$ constant has a negative effect on the beam quality. Our research group believes that the self-splitting beam holds promising potential for applications in optical communication.

Funding

National Natural Science Foundation of China (12074005).

Disclosures

The authors declare no conflicts of interest.

Data availability

The 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.

References

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Data availability

The 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 (11)
Fig. 1.
Fig. 1. Intensity distribution of STHGSM beams with varying beam orders and twist factors. where parameters include: λ=632.8 nm, g = 0.05 $c{m^2}$, ${\sigma _x}$=${\sigma _y}$=0.09 cm, $C_n^2$=${10^{ - 15}}\,{m^{3 - \alpha }}$, and ${l_0}$=1 mm.

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Fig. 2.
Fig. 2. Intensity distribution of STHGSM beams with varying parameters g, ${\sigma _x}$ and ${\sigma _y}$. where parameters include: λ=632.8 nm, $C_n^2$=${10^{ - 15}}\,{m^{3 - \alpha }}$, u = 20 $c{m^{ - 2}}$, m = n = 4, and ${l_0}$=1 mm.

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Fig. 3.
Fig. 3. Intensity distribution of STHGSM beams under non-Kolmogorov turbulence with varying distortion factors u and order. where parameters include: $\lambda $=632.8 nm, g = 0.05 $c{m^2}$, ${\sigma _x}$=${\sigma _y}$=0.09 cm, $C_n^2$=${10^{ - 15}}\,{m^{3 - \alpha }}$, ${l_0}$=1 mm.

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Fig. 4.
Fig. 4. The intensity distribution of STHGSM beams with varying beam parameters. The (a-c) row parameters are: $\lambda $=632.8 nm, g = 0.05 $c{m^2}$, ${\sigma _x}$=${\sigma _y}$=0.09 cm, m = n = 4, $C_n^2$=${10^{ - 15}}\,{m^{3 - \alpha }}$, ${l_0}$=1 mm.

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Fig. 5.
Fig. 5. The intensity distribution of STHGSM beams with varying beam parameters. The (a-c) row parameters are: $\lambda $=632.8 nm, g = 0.05 $c{m^2}$, ${\sigma _x}$=${\sigma _y}$=0.09 cm, m = n = 4, $C_n^2$=${10^{ - 15}}\,{m^{3 - \alpha }}$, ${l_0}$=1 mm.

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Fig. 6.
Fig. 6. The intensity distribution of STHGSM beams under non-Kolmogorov turbulence conditions with varying $C_n^2$. The (a-c) row calculation parameters are: $\lambda $=632.8 nm, g = 0.05 $c{m^2}$, ${\sigma _x}$=${\sigma _y}$=0.09 cm, m = n = 4, $u$= 40 $c{m^{ - 2}}$, ${l_0}$=1 mm.

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Fig. 7.
Fig. 7. Variation in the rms angle width of STHGSM beams with differing orders and wavelengths. The common parameters include: $u$= 40 $c{m^{ - 2}}$, g = 0.05 $c{m^2}$, ${\sigma _x}$=${\sigma _y}$=0.09 cm, ${l_0}$=1 mm,, $C_n^2$=4*${10^{ - 15}}\,{m^{3 - \alpha }}$. (a) $\lambda $=632.8 nm; (b) m = n = 2

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Fig. 8.
Fig. 8. The variation of the rms angle width of STHGSM beams with various refractive index structure constants and turbulent inner scales. where parameters include: $\lambda $=632.8 nm, g = 0.05 $c{m^2}$, ${\sigma _x}$=${\sigma _y}$=0.09 cm, $u$= 40 $c{m^{ - 2}}$, m = n = 2. (a) ${l_0}$=1 mm; (b) $C_n^2$=4*${10^{ - 15}}\,{m^{3 - \alpha }}$

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Fig. 9.
Fig. 9. The impact of varying distortion factor and order on the propagation factor of STHGSM beam. Where parameters include: $\lambda $=632.8 nm, g = 0.05 $c{m^2}$, ${\sigma _x}$=${\sigma _y}$=0.09 cm, $C_n^2$=$4\ast {10^{ - 15}}\,{m^{3 - \alpha }}$, ${l_0}$=1 mm. (a) $u$= 40 $c{m^{ - 2}}$;(b) m = n = 4

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Fig. 10.
Fig. 10. The influence of different parameters g, ${\sigma _x}$, ${\sigma _y}$, as well as wavelength on the ${M^\textrm{2}}$-factor of STHGSM beam. The common parameters include: m = n = 4, u = 40 $c{m^{ - 2}}$, $C_n^2$=$4\ast {10^{ - 15}}\,{m^{3 - \alpha }}$, ${l_0}$=1 mm. (a)$\lambda $=632.8 nm, ${\sigma _x}$=${\sigma _y}$=0.09 cm; (b)$\lambda $=632.8 nm, g = 0.05 $c{m^2}$

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Fig. 11.
Fig. 11. The influence of different refractive index structure constants and turbulence inner scale on the M2-factor of STHGSM beams. The common parameters include: m = n = 4, u = 40 $c{m^{ - 2}}$, g = 0.05 $c{m^2}$, ${\sigma _x}$=${\sigma _y}$=0.09 cm, $\lambda $=632.8 nm.(a) ${l_0}$=1 mm; (b) $C_n^2$=$4\ast {10^{ - 15}}\,{m^{3 - \alpha }}$

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Equations (28)

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W ( ρ 1 , ρ 2 , 0 ) = ∫ p ( v ) H ∗ ( ρ 1 , v ) H ( ρ 2 , v ) d 2 v ,
p ( v ) = C 0 ∑ t = 0 2 m C 2 m t ( a 0 v x ) n t ( a 0 v y ) n ( 2 m − t ) exp ⁡ ( − g v x 2 − g v y 2 ) ,
H ( ρ , v ) = exp ⁡ ( − x 2 σ x 2 − y 2 σ y 2 ) exp ⁡ { − ( a y + i x ) v x + ( a x − i y ) v y } ,
W ( ρ 1 , ρ 2 , 0 ) = ∑ t = 0 2 m C 2 m t H n t ( − ( − i a ( y 1 + y 2 ) + x 2 − x 1 ) 2 g ) × H n ( 2 m − t ) ( − ( i a ( x 2 + x 1 ) + y 2 − y 1 ) 2 g ) exp ⁡ ( − ( ρ 2 − ρ 1 ) 2 2 δ 0 2 ) × exp ⁡ ( − ( x 1 2 + x 2 2 ) w 0 x 2 − ( y 1 2 + y 2 2 ) w 0 y 2 ) exp ⁡ ( − i u ( x 1 y 2 − x 2 y 1 ) ) ,
δ 0 = ( a 2 + 1 2 g ) − 1 2 ,   w o x = ( 1 σ x 2 − a 2 2 g ) − 1 2 ,  w o y = ( 1 σ y 2 − a 2 2 g ) − 1 2 ,
u ≤ 1 δ 0 x 2 b a = 1 δ 0 y 2 b a ,
W ( ρ 3 , ρ 4 , z ) = ( k 2 π z ) 2 ∫ ∫ W ( ρ 1 , ρ 2 , 0 ) exp ⁡ ( − i k 2 z ( ρ 1 − ρ 3 ) 2 + i k 2 z ( ρ 2 − ρ 4 ) 2 ) × exp ⁡ ( − π 2 k 2 z T 3 ( ( ρ 3 − ρ 4 ) 2 + ( ρ 3 − ρ 4 ) ( ρ 1 − ρ 2 ) + ( ρ 1 − ρ 2 ) 2 ) ) d 2 ρ 1 d 2 ρ 2
T = ∫ 0 ∞ κ 3 Φ n ( κ ) d κ = 0.1661 C n 2 l 0 − 1 / 3 ,
Φ n ( κ ) = 0.033 C n 2 κ − 11 / 3 exp ⁡ ( − κ 2 / κ m 2 ) ,
W ( ρ 3 , ρ 4 , z ) = ∑ t = 0 2 m C 2 m t π C 0 b x d x ( k 2 z ) 2 ( i 2 ) 2 n m × exp ⁡ [ − i k 2 z ( ρ 3 2 − ρ 4 2 ) − T 0 [ ( ρ 3 − ρ 4 ) 2 ] + R 1 + F 1 2 4 P 1 + F 2 2 4 P 2 ] × P 1 − ( n t + 1 ) / 2 P 2 − ( n ( 2 m − t ) + 1 ) / 2 H n t ( i F 1 2 P 1 ) H n ( 2 m − t ) ( i F 2 2 P 2 )
T 0 = π 2 k 2 z T 3 ,   σ x = σ y = σ b x = ( 1 σ x 2 + i k 2 z + T 0 ) ,   b y = ( 1 σ y 2 − i k 2 z + T 0 ) d x = 1 σ x 2 − i k 2 z + T 0 − T 0 2 b x ,   d y = 1 σ y 2 − i k 2 z + T 0 − T 0 2 b x ,   d x = d y x 3 ′ = i k x 3 z − T 0 ( x 3 − x 4 ) ,   y 3 ′ = i k y 3 z − T 0 ( y 3 − y 4 ) x 4 ′ = − i k x 4 z + T 0 ( x 3 − x 4 ) ,   y 4 ′ = − i k y 4 z + T 0 ( y 3 − y 4 ) R 1 = ( x 3 ′ ) 2 + ( y 3 ′ ) 2 4 b x + ( b x x 4 ′ + x 3 ′ T 0 ) 2 + ( y 3 ′ T 0 + b x y 4 ′ ) 2 4 b x 2 d x P 1 = g + 1 − a 2 4 b x + ( 1 − a 2 ) T 0 2 4 b x 2 d x + − T 0 − a 2 2 b x d x + 1 − a 2 4 d x P 2 = g + 1 − a 2 4 b x + ( 1 − a 2 ) T 0 2 4 b x 2 d x − ( a 2 + 1 ) T 0 2 b x d x + 1 − a 2 4 d x F 1 = a y 3 ′ − i x 3 ′ 2 b x + ( x 3 ′ − x 4 ′ ) i T 0 + ( y 3 ′ + y 4 ′ ) a T 0 2 b x d x + i x 4 ′ + a y 4 ′ 2 d x + ( a y 3 ′ − i x 3 ′ ) T 0 2 2 b x 2 d x F 2 = − ( a x 3 ′ + i y 3 ′ ) 2 b x + ( y 3 ′ − y 4 ′ ) i T 0 + ( − x 4 ′ − x 3 ′ ) a T 0 2 b x d x + i y 4 ′ − a x 4 ′ 2 d x − ( a x 3 ′ + i y 3 ′ ) T 0 2 2 b x 2 d x
∫ − ∞ + ∞ exp ⁡ ( − p x 2 ± q x ) d x = π p exp ⁡ ( q 2 4 p )
∫ − ∞ + ∞ x m exp ⁡ ( − p x 2 − q x ) d x = π ( i 2 ) m p − ( m + 1 ) / 2 exp ⁡ ( q 2 4 p ) H m ( i q 2 p )
W ( ρ s , ρ d , z ) = ( k 2 π z ) 2 ∫ ∫ W ( r s , r d , 0 ) × exp ⁡ ( i k z ( ρ s − r s ) ( ρ d − r d ) − H ( ρ d , r d , z ) ) d 2 r s d 2 r d
x s = ( x 1 + x 2 ) / 2 , y s = ( y 1 + y 2 ) / 2  , x d = x 1 − x 2 , y d = y 1 − y 2 ρ s x = ( ρ 3 x + ρ 4 x ) / 2 , ρ s y = ( ρ 3 y + ρ 4 y ) / 2 , ρ d x = ρ 3 x − ρ 4 x   , ρ d y = ρ 3 y − ρ 4 y
W ( ρ s , ρ d , z ) = 1 ( 2 π ) 2 ∫ ∫ W ( r s ′ , ρ d + z k κ d , 0 ) d 2 r s ′ d 2 κ d × exp ⁡ ( − i ρ s ⋅ κ d + i r s ′ ⋅ κ d − π 2 k 2 z 3 ( 3 ρ d 2 + 3 z k ρ d κ d + z 2 k 2 κ d 2 ) )
h ( ρ s , θ , z ) = ( k 2 π ) 2 ∫ − ∞ ∞ ∫ − ∞ ∞ W ( ρ s , ρ d , z ) exp ⁡ ( − i k θ ρ d ) d 2 ρ d
h ( ρ s , θ , z ) = k 2 π 16 π 4 ∑ t = 0 2 m C 2 m t ∑ k 1 = 0 n ( 2 m − t ) ∑ j 1 = 0 [ k 1 / 2 ] ∑ k 2 = 0 n t ∑ j 2 = 0 [ k 2 / 2 ] 1 2 n m ( − 1 ) j 1 + j 2 + k 1 − 2 j 1 × k 1 ! j 1 ! ( k 1 − 2 j 1 ) ! k 2 ! j 2 ! ( k 2 − 2 j 2 ) ! ( 2 i a 2 g ) k 1 − 2 j 1 + k 2 − 2 j 2 ( 2 w 0 x 2 ) − ( k 1 − 2 j 1 + 1 ) / 2 − ( k 2 − 2 j 2 + 1 ) / 2 × ( i 2 ) k 1 − 2 j 1 + k 2 − 2 j 2 ∫ ∫ ∫ ∫ H n ( 2 m − t ) − k 1 ( 2 ( ρ d y + z k κ d y ) 2 g ) H n t − k 2 ( 2 ( ρ d x + z k κ d x ) 2 g ) × H k 1 − 2 j 1 ( u 0 ( ρ d y + z k κ d y ) + κ d x 2 2 w 0 x 2 ) H k 2 − 2 j 2 ( − u 0 ( ρ d x + z k κ d x ) + κ d y 2 2 w 0 y 2 ) × exp ⁡ ( u 0 w 0 x 2 4 ( ρ d x κ d y − ρ d y κ d x ) ) exp ⁡ ( − E 1 ρ d 2 − G 1 κ d 2 − N 1 ρ d κ d − i k θ ρ d − i ρ s κ d ) d 2 κ d d 2 ρ d
E 1 = ( 1 2 δ 0 2 + 1 2 w 0 x 2 + u 0 2 w 0 x 2 8 + π 2 k 2 z T ) N 1 = ( z δ 0 2 k + z w 0 x 2 k + u 0 2 w 0 x 2 z 4 k + π 2 k z 2 T )
G 1 = ( z 2 2 δ 0 2 k 2 + z 2 2 w 0 x 2 k 2 + u 0 2 w 0 x 2 z 2 8 k 2 + w 0 x 2 8 + π 2 z 3 T 3 )
H n ( x + y ) = 1 2 n / 2 ∑ k = 0 n H k ( 2 x ) H n − k ( 2 y )
⟨ x n 1 y n 2 θ x m 1 θ y m 2 ⟩ = 1 p ∫ ∫ x n 1 y n 2 θ x m 1 θ y m 2 h ( ρ s , θ , z ) d 2 ρ s d 2 θ
P = ∫ ∫ h ( ρ s , θ , z ) d 2 ρ s d 2 θ
θ N ( z ) = ( ⟨ θ 2 ⟩ ) 1 / 2 = ( ⟨ θ x 2 ⟩ + ⟨ θ y 2 ⟩ ) 1 / 2
M 2 ( z ) = k ( ⟨ ρ 2 ⟩ ⟨ θ 2 ⟩ − ⟨ ρ ⋅ θ ⟩ 2 ) 1 / 2
δ ( s ) = 1 2 π ∫ − ∞ ∞ exp ⁡ ( − i s x ) d x
δ n ( s ) = 1 2 π ∫ − ∞ ∞ ( − i x ) n exp ⁡ ( − i s x ) d x   ( n = 1 , 2 )
∫ − ∞ ∞ f ( x ) δ n ( x ) d x = ( − 1 ) n   f n ( 0 )   ( n = 1 , 2 )
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