Abstract

In this study, based on the extended Huygens–Fresnel principle, the propagation expressions of circular edge dislocation beams and circular–linear edge dislocation beams were obtained. The propagation properties of the two types of beam were compared in free space and atmospheric turbulence. The results show that, when circular–linear edge dislocation beams propagate in free space or atmospheric turbulence, because the linear edge dislocation is located in different beam locations, circular edge dislocation vanishes or evolves into a pair of optical vortices. However, when circular edge dislocation beams propagate in space, circular edge dislocation exists stably in free space propagation, while it evolves into a pair of optical vortices in atmospheric turbulence propagation. Therefore, the propagation properties of circular edge dislocation can be adjusted by adding linear edge dislocation when circular edge dislocation beams propagate through free space and atmospheric turbulence. This research can be useful for applications in optical communications.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. Introduction

Singular optics is a new branches of optics, in which a series of effects related to phase singularities (wavefront dislocations) in the wave field are studied. Screw dislocation and edge dislocation are two kinds of pure dislocations [1]. Screw dislocation, also known as optical vortex, is related to the orbital angular momentum [2]. Edge dislocation with a π-phase shift is located along a curve in the transverse plane. According to the shape of the curve, there mainly exist circular edge dislocation and linear edge dislocation [3]. Edge dislocation is also related to the orbital angular momentum [4]. Because of the wide application of singular optics in optical communications [512], biological tissue [13], optical tweezers [14,15] and optical imaging [16,17], the attention of many researchers has been drawn to it. Gruneisen et al. proposed optical vortex discrimination with a transmission volume hologram [18], and the emission, which is an incoherent mix of two spatial modes—a Gaussian and a vortex beams—was introduced by Naidoo et al. [19]. Zhang et al. pointed out that perfect vortex beam structures can be adjusted by varying several control parameters [20]. The vortex structure of a non−integer vortex beam was reported by Leach et al. [21]. Ma et al. proposed an optical vortex array called a “circular optical vortex array” [22]. Kaneyasu et al. observed vortex beams using the interference method [23]. The generation of optical vortices has been reported in Refs. [24] and [25].

The propagation of phase singularities has also been widely studied. Monin and Ustinov studied the transformation of Hermite–Gauss beams with an embedded optical vortex [26]. Vortex instability in free space propagation was reported by Lavery [27]. Li et al. studied that the transformation of circular edge dislocations through atmospheric turbulence and free space [28]. Chen et al. studied the propagation properties of two edge dislocations through an astigmatic lens [29]. The propagation of linear edge dislocation and mixed screw–edge dislocations in atmospheric turbulence has been reported [3032]. Vortex beams and edge dislocation beams with wavefront dislocations and phase singularity are a type of beam carrying orbital angular momentum. The propagation properties of different dislocation beams in atmospheric turbulence could lead to wide applications of orbital angular momentum in optical communications.

In this study, more complex dislocation beams were examined. The propagation properties of circular−linear edge dislocation beams were investigated, and a comparative study of the propagation properties of circular edge dislocation beams and circular−linear edge dislocation beams was conducted. In Section 2, the propagation expressions of circular edge dislocation beams and circular–linear edge dislocation beams are given. The propagation properties of the circular edge dislocation and linear edge dislocation are discussed in Sections 3 and 4. Finally, the conclusions of the study are presented.

2. Theoretical model

In the source plane, the field distribution of Laguerre−Gaussian beams can be written as [33,34]

$$E({{\boldsymbol{s}}, \theta , z = 0} )= {\left( {\frac{{\sqrt 2 {\boldsymbol{s}}}}{{{w_0}}}} \right)^m}L_n^m\left( {\frac{{2{{\boldsymbol{s}}^2}}}{{w_0^2}}} \right)\textrm{exp} \left( { - \frac{{{{\boldsymbol{s}}^2}}}{{w_0^2}} + \textrm{i}m\theta } \right),$$
where Lm n(·) denotes the Laguerre polynomial, s and θ denote the radial and azimuthal coordinates, respectively, and w0 is the waist width.

Using the relations between the Laguerre polynomial and Hermite polynomial [35]

$$\begin{aligned} {{\boldsymbol{s}}^m}L_n^m({{{\boldsymbol{s}}^2}} )\textrm{exp} ({\textrm{i}m\theta } ) &= \frac{{{{({ - 1} )}^n}}}{{{2^{2n + m}}n!}}\sum\limits_{t = 0}^n {\sum\limits_{r = 0}^m {{\textrm{i}^r}} } \frac{{n! \; m! }}{{t! \; r! ({n - t} )! \; ({m - r} )!}}\\ &\times {H_{2t + m - r}}({{s_x}} ){H_{2n - 2t + r}}({{s_y}} )\end{aligned},$$
where Hn denotes the Hermite polynomial. Equation (1) can be written as
$$\begin{aligned} E({{\boldsymbol{s}}, z = 0} ) &= \frac{{{{({ - 1} )}^n}}}{{{2^{2n + m}}n!}}\sum\limits_{t = 0}^n {\sum\limits_{r = 0}^m {{\textrm{i}^r}} } \frac{{n! \; m! }}{{t! \; r! ({n - t} )! ({m - r} )!}}{H_{2t + m - r}}\left( {\frac{{\sqrt 2 {s_x}}}{{{w_0}}}} \right)\\ &\times {H_{2n - 2t + r}}\left( {\frac{{\sqrt 2 {s_y}}}{{{w_0}}}} \right)\textrm{exp} \left( { - \frac{{s_x^2 + s_y^2}}{{w_0^2}}} \right) \end{aligned}.$$

When n ≠ 0 and m = 0, Eq. (3) is reduced to the initial field distribution of circular edge dislocation beams which can be expressed as follows

$$E({{\boldsymbol{s}}, z = 0} )= \frac{{{{({ - 1} )}^n}}}{{{2^{2n}}n!}}\sum\limits_{t = 0}^n {\frac{{n!}}{{t!({n - t})!}}} {H_{2t}}\left( {\frac{{\sqrt 2 {s_x}}}{{{w_0}}}} \right){H_{2n - 2t}}\left( {\frac{{\sqrt 2 {s_y}}}{{{w_0}}}} \right)\textrm{exp} \left( { - \frac{{s_x^2 + s_y^2}}{{w_0^2}}} \right).$$
Using Eq. (5) [36]
$${H_n}(x )= \sum\limits_{j = 0}^{[{{\raise0.7ex\hbox{$n$} \mathord{\left/ {\vphantom {n 2}} \right.}\lower0.7ex\hbox{$2$}}} ]} {{{({ - 1} )}^j}} \frac{{n!}}{{j!({n - 2j} )!}}{({2x} )^{n - 2j}},$$
for n = 1, an expression is obtained that carries a circular edge dislocation
$$E({{\boldsymbol{s}}, z = 0} )={-} \frac{2}{{w_0^2}}\left( {s_x^2 + s_y^2 - \frac{{w_0^2}}{2}} \right)\textrm{exp} \left( { - \frac{{s_x^2 + s_y^2}}{{w_0^2}}} \right).$$
In the source plane, the expression of a beam carrying a linear edge dislocation is [37]
$$E({{\boldsymbol{s}}, z = 0} )= \frac{1}{{{w_0}}}({{s_x} - a} )\textrm{exp} \left( { - \frac{{s_x^2 + s_y^2}}{{w_0^2}}} \right),$$
where a is the off−axis distance of the linear edge dislocation.

In this study, the propagation properties of circular edge dislocation and linear edge dislocation were investigated. Therefore, both edge dislocations are nested in Gaussian beams. The new beams are referred to as circular–linear edge dislocation beams, which can be expressed as follows

$$E({{\boldsymbol{s}}, 0} )={-} \frac{2}{{w_0^3}}\left( {s_x^2 + s_y^2 - \frac{{w_0^2}}{2}} \right)({{s_x} - a} )\textrm{exp} \left( { - \frac{{s_x^2 + s_y^2}}{{w_0^2}}} \right).$$

In the source plane z = 0, the cross–spectral density function of circular–linear edge dislocation beams can be written as

$$\begin{aligned} {W_0}({{{\boldsymbol{s}}_1},{{\boldsymbol{s}}_2},z = 0} ) &= \left\langle {E_1^\ast ({{{\boldsymbol{s}}_1},z = 0} ){E_2}({{{\boldsymbol{s}}_2},z = 0} )} \right\rangle \\ &= \frac{{4({{s_{1x}} - a} )({{s_{2x}} - a} )}}{{w_0^6}}\left( {s_{1x}^2 + s_{1y}^2 - \frac{{w_0^2}}{2}} \right)\left( {s_{2x}^2 + s_{2y}^2 - \frac{{w_0^2}}{2}} \right)\\ &\times \textrm{exp} \left( { - \frac{{s_{1x}^2 + s_{1y}^2 + s_{2x}^2 + s_{2y}^2}}{{w_0^2}}} \right) \end{aligned},$$
where * denotes the complex conjugate.

Based on the extended Huygens–Fresnel principle [38], the cross–spectral density function of circular–linear edge dislocation beams propagating in atmospheric turbulence can be written as

$$\begin{aligned} W({{{\boldsymbol{\rho} }_1},{{\boldsymbol{\rho} }_2},z} ) &= {\left( {\frac{k}{{2\mathrm{\pi }z}}} \right)^2}\int {\int {\int {\int {{W_0}} } } } ({{{\boldsymbol{s}}_1},{{\boldsymbol{s}}_2},z = 0} )\textrm{exp} \left\{ { - \frac{{\textrm{i}k}}{{2z}}[{{{({{{\boldsymbol{s}}_1} - {{\boldsymbol{\rho} }_1}} )}^2} - {{({{{\boldsymbol{s}}_2} - {{\boldsymbol{\rho} }_2}} )}^2}} ]} \right\}\\ &\times \left\langle {\textrm{exp} [{{\varphi^\ast }({{{\boldsymbol{\rho} }_1},{{\boldsymbol{s}}_1}} )+ \varphi ({{{\boldsymbol{\rho} }_2},{{\boldsymbol{s}}_2}} )} ]} \right\rangle d{{\boldsymbol{s}}_1}d{{\boldsymbol{s}}_2} \end{aligned},$$
where k denotes the wave number, k = 2π/λ, ρi=(ρix, ρiy) (i = 1, 2) denotes the position vector in the z plane, and <·> is the average over the ensemble of the atmospheric turbulence. Here, $\left\langle {\textrm{exp} [{{\varphi^\ast }({{{\boldsymbol{\rho} }_1},{{\boldsymbol{s}}_1}} )+ \varphi ({{{\boldsymbol{\rho} }_2},{{\boldsymbol{s}}_2}} )} ]} \right\rangle$ can be expressed as follows [39]
$$\left\langle {\textrm{exp} [{{\varphi^\ast }({{{\boldsymbol{\rho} }_1},{{\boldsymbol{s}}_1}} )+ \varphi ({{{\boldsymbol{\rho} }_2},{{\boldsymbol{s}}_2}} )} ]} \right\rangle \approx \textrm{exp} \left[ { - \frac{{{{({{{\boldsymbol{s}}_1} - {{\boldsymbol{s}}_2}} )}^2} + {{({{{\boldsymbol{\rho} }_1} - {{\boldsymbol{\rho} }_2}} )}^2} + ({{{\boldsymbol{s}}_1} - {{\boldsymbol{s}}_2}} )({{{\boldsymbol{\rho} }_1} - {{\boldsymbol{\rho} }_2}} )}}{{\rho_0^2}}} \right],$$
where ${\rho _0} = {({0.545C_n^2{k^2}z} )^{ - {3 / 5}}}$ is the spatial coherence radius of a spherical wave propagating in atmospheric turbulence [40], and $C_n^2$ is the refraction index structure constant [41].

Substituting Eq. (9) into Eq. (10), one obtains the analytical expressions for the cross–spectral density function of circular–linear edge dislocation beams propagating in atmospheric turbulence, which can be written as

$$W({{{\boldsymbol{\rho} }_1},{{\boldsymbol{\rho} }_2},z} )= \frac{{4B}}{{w_0^6}}\textrm{exp} \left[ { - \frac{{{{({{\rho_{1x}} - {\rho_{2x}}} )}^2} + {{({{\rho_{1y}} - {\rho_{2y}}} )}^2}}}{{\rho_0^2}} - \frac{{\textrm{i}k}}{{2z}}({\rho_{1x}^2 + \rho_{1y}^2 - \rho_{2x}^2 - \rho_{2y}^2} )} \right],$$
where
$$B = \frac{{{k^2}}}{{4C\pi {z^2}}}\textrm{exp} \left( {\frac{{{D^2} + {E^2}}}{C}} \right)\sum\limits_{t = 0}^6 {{F_t}} {G_t},$$
and
$$C = \frac{1}{{w_0^2}} + \frac{{ik}}{{2z}} + \frac{1}{{\rho _0^2}},$$
$$D = \frac{{\textrm{i}k{\rho _{1y}}}}{{2z}} - \frac{{{\rho _{1y}} - {\rho _{2y}}}}{{2\rho _0^2}},$$
$$E = \frac{{\textrm{i}k{\rho _{1x}}}}{{2z}} - \frac{{{\rho _{1x}} - {\rho _{2x}}}}{{2\rho _0^2}},$$
$$\begin{aligned} {F_0} &= \frac{{aE{R_1}}}{C} + {R_2}\left( {\frac{{a{E^3}}}{{{C^3}}} + \frac{{3aE}}{{2{C^2}}} - \frac{{{a^2}}}{{2C}} - \frac{{{a^2}{E^2}}}{{{C^2}}} + aJ} \right)\\ &+ \frac{{{a^2}({2{H_4} - w_0^2{H_2}} )}}{{2{C^2}\rho _0^4}} - \frac{{{a^2}D({2{H_3} + w_0^2{H_1}} )}}{{{C^2}\rho _0^2}} \end{aligned},$$
$$\begin{aligned} {F_1} &= \frac{E}{C}\left( { - {R_1} - \frac{{w_0^2D{H_1}}}{{{C^2}\rho_0^2}}} \right) + \frac{{a{R_1}}}{{C\rho _0^2}} - \frac{{a({2{H_4} - w_0^2{H_2}} )}}{{2{C^2}\rho _0^4}} - \frac{{2aD({{H_3} - w_0^2{H_1}} )}}{{{C^2}\rho _0^2}}\\ &+ {R_2}\left( {\frac{{3a{E^2}}}{{{C^3}\rho_0^2}} + \frac{{3a}}{{2{C^2}\rho_0^2}} + \frac{a}{{2C}} + \frac{{a{E^2}}}{{{C^2}}} - \frac{{{E^3}}}{{{C^3}}} - \frac{{3E}}{{2{C^2}}} - \frac{{2{a^2}E}}{{{C^2}\rho_0^2}} - J} \right) \end{aligned},$$
$$\begin{aligned} {F_2} &= {R_2}\left( {\frac{{3aE}}{{{C^3}\rho_0^4}} + \frac{{2aE}}{{{C^2}\rho_0^2}} - \frac{{3{E^2}}}{{{C^3}\rho_0^2}} - \frac{3}{{2{C^2}\rho_0^2}} - \frac{{{a^2}}}{{{C^2}\rho_0^4}}} \right) - \frac{1}{{C\rho _0^2}}\left( {{R_1} + \frac{{w_0^2D{H_1}}}{{{C^2}\rho_0^2}}} \right)\\ &+ a{H_0}\left( {\frac{a}{{2C}} + \frac{{a{E^2}}}{{{C^2}}} - \frac{{{E^3}}}{{{C^3}}} - \frac{{3E}}{{2{C^2}}}} \right) - \frac{{aE{R_3}}}{C} - a{H_0}J + \frac{{{a^2}{H_2}}}{{{C^2}\rho _0^4}} + \frac{{2{a^2}D{H_1}}}{{{C^2}\rho _0^2}} \end{aligned},$$
$$\begin{aligned} {F_3} &= {R_2}\left( {\frac{a}{{{C^3}\rho_0^6}} + \frac{a}{{{C^2}\rho_0^4}} - \frac{{3E}}{{{C^3}\rho_0^4}}} \right) - \frac{{2aD{H_1}}}{{{C^2}\rho _0^2}} + \frac{{2{a^2}E{H_0}}}{{{C^2}\rho _0^2}} + {R_3}\left( {\frac{E}{C} - \frac{a}{{C\rho_0^2}}} \right)\\ &- a{H_0}\left( {\frac{{3{E^3}}}{{{C^3}\rho_0^2}} + \frac{3}{{2{C^2}\rho_0^2}} + \frac{1}{{2C}} + \frac{{{E^2}}}{{{C^2}}}} \right) + {H_0}\left( {\frac{{{E^3}}}{{{C^3}}} + \frac{{3E}}{{2{C^2}}} + J} \right) - \frac{{a{H_2}}}{{{C^2}\rho _0^4}} \end{aligned},$$
$${F_4} = - \frac{{{R_2}}}{{{C^3}\rho _0^6}} - \frac{{3aE{H_0}}}{{{C^3}\rho _0^4}} + {H_0}\left( {\frac{{3{E^2}}}{{{C^3}\rho_0^2}} + \frac{3}{{2{C^2}\rho_0^2}}} \right) + \frac{{{a^2}{H_0}}}{{{C^2}\rho _0^4}} - \frac{{2aE{H_0}}}{{{C^2}\rho _0^2}} + \frac{{{R_3}}}{{C\rho _0^2}},$$
$${F_5} = - \frac{{a{H_0}}}{{{C^3}\rho _0^6}} + \frac{{3E{H_0}}}{{{C^3}\rho _0^4}} - \frac{{a{H_0}}}{{{C^2}\rho _0^4}},$$
$${F_6} = \frac{{{H_0}}}{{{C^3}\rho _0^6}},$$
$${G_t} = t!\textrm{exp} \left( {\frac{{Q_1^2}}{P}} \right)\sqrt {\frac{\mathrm{\pi }}{P}} {\left( {\frac{{{Q_1}}}{P}} \right)^t} \times \sum\limits_{K = 0}^{E\left[ {\frac{t}{2}} \right]} {\frac{1}{{({t - 2K} )!K!}}{{\left( {\frac{P}{{4Q_1^2}}} \right)}^K}}, \qquad (t = 0\textrm{, }1\cdot{\cdot}\cdot6)$$
$${H_l} = l!\textrm{exp} \left( {\frac{{Q_2^2}}{P}} \right)\sqrt {\frac{\mathrm{\pi }}{P}} {\left( {\frac{{{Q^2}}}{P}} \right)^l} \times \sum\limits_{K = 0}^{E\left[ {\frac{l}{2}} \right]} {\frac{1}{{({l - 2K} )!K!}}{{\left( {\frac{P}{{4Q_2^2}}} \right)}^K}} , \qquad (l = 0\textrm{, }1\cdot{\cdot}\cdot4)$$
$$I ={-} \frac{{w_0^2}}{2} + \frac{1}{{2C}} + \frac{{{D^2}}}{{{C^2}}},$$
$$J = \frac{{aw_0^2}}{2} - \frac{a}{{2C}} + \frac{{a{D^2}}}{{{C^2}}},$$
$$P = \frac{1}{{w_0^2}} - \frac{{ik}}{{2z}} + \frac{1}{{\rho _0^2}} - \frac{1}{{C\rho _0^4}},$$
$${Q_1} ={-} \frac{{\textrm{i}k{\rho _{\textrm{2}y}}}}{{2z}} + \frac{{{\rho _{1y}} - {\rho _{2y}}}}{{2\rho _0^2}} + \frac{D}{{C\rho _0^2}},$$
$${Q_2} ={-} \frac{{\textrm{i}k{\rho _{\textrm{2}x}}}}{{2z}} + \frac{{{\rho _{1x}} - {\rho _{2x}}}}{{2\rho _0^2}} + \frac{E}{{C\rho _0^2}},$$
$${R_1} = \frac{{w_0^2{H_0}I}}{2} - \frac{{{H_4}}}{{{C^2}\rho _0^4}} - \frac{{2D{H_3}}}{{{C^2}\rho _0^2}} + \frac{{w_0^2{H_2}}}{{2{C^2}\rho _0^4}} - {H_2}I,$$
$${R_2} = \frac{{w_0^2{H_0}}}{2} - {H_2},$$
$${R_3} = \frac{{{H_2}}}{{{C^2}\rho _0^4}} + \frac{{2D{H_1}}}{{{C^2}\rho _0^2}} + {H_0}I.$$

When $C_n^2$ = 0, Eq. (12) is simplified as the analytical expressions for the cross–spectral density function of circular–linear edge dislocation beams propagating in free space. The selection of the parameters w0, k, a and ρ1 of Eq. (12) for simulation refers to previous research [42,43].

When ρ1= ρ2= ρ, one obtains the expression of the light intensity of circular−linear edge dislocation beams as follows

$$I({{\boldsymbol{\rho} },z} )= W({{{\boldsymbol{\rho} }_1},{{\boldsymbol{\rho} }_2},z} ).$$

The analytical expressions for the cross–spectral density function of circular edge dislocation beams can be found in a previous report [28].

The spectral degree of coherence is defined as [44]

$$\mu ({{{\boldsymbol{\rho} }_1},{{\boldsymbol{\rho} }_2},z} )= \frac{{W({{{\boldsymbol{\rho} }_1},{{\boldsymbol{\rho} }_2},z} )}}{{{{[{I({{{\boldsymbol{\rho} }_1},z} )I({{{\boldsymbol{\rho} }_2},z} )} ]}^{{1 / 2}}}}}.$$

The position of phase singularities is determined using Eqs. (27) and (28) [45]

$$\textrm{Im}[{\mu ({{{\boldsymbol{\rho} }_1},{{\boldsymbol{\rho} }_2},z} )} ]= 0, $$
$$\textrm{R}\textrm{e}[{\mu ({{{\boldsymbol{\rho} }_1},{{\boldsymbol{\rho} }_2},z} )} ]= 0, $$
where Im and Re denote the imaginary and real parts of $\mu ({{{\boldsymbol{\rho} }_1},{{\boldsymbol{\rho} }_2},z} )$, respectively. The sign of an optical vortex can be obtained by the vorticity of the phase contours around singularities [46].

3. Circular edge dislocation beams and circular−linear edge dislocation beams propagating through free space

Figure 1 shows the phase distribution (a−d) and normalized intensity distribution (e−h) of circular−linear edge dislocation beams in the source plane and free space propagation. The calculation parameters are λ = 1.06 µm, w0 = 3 cm, C2 n = 0, ρ1 = (2 cm, 2 cm), and a = 0. Figures 1(a, e) show that circular–linear edge dislocation beams have a linear edge dislocation (denoted as A) and a circular edge dislocation (denoted as B) in the source plane, and the light intensity at the circular edge dislocation and linear edge dislocation is zero. Figure 1(a) also shows that the linear edge dislocation passes the center of the circular edge dislocation. Figures 1(b−d) and Figs. 1(f−h) show that, when circular–linear edge dislocation beams propagate in free space, the linear edge dislocation A remains stable, while the circular edge dislocation B vanishes.

 

Fig. 1. Phase distribution (a−d) and normalized intensity distribution (e−h) of circular−linear edge dislocation beams with a = 0 in the source plane and propagating through free space at the propagation distance z = 1, 7, and 10 km.

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Figure 2 shows a comparison of the phase distributions of circular–linear edge dislocation beams (a) and circular edge dislocation beams (b) in the source plane and free space propagation, where a = 1 cm (a), and the other calculation parameters are the same as those in Fig. 1. In Fig. 2(a), the linear edge dislocation is denoted as C and the circular edge dislocation is denoted as D in the source plane. In contrast to Fig. 1(a), the linear edge dislocation does not pass the center of the circular edge dislocation. When circular–linear edge dislocation beams propagate in free space, at z = 1 km, the linear edge dislocation C evolves into a pair of optical vortices (C+ and C), whose topological charges are +1 and −1, respectively. The circular edge dislocation D also evolves into a pair of optical vortices (D+ and D) with topological charges +1 and −1, respectively. The optical vortices C+, C, D+, and D still exist at z = 7 km. When circular–linear edge dislocation beams continue to propagate, at z = 10 km, the optical vortices C+ and C have annihilated, while the optical vortices D+ and D remain stable. Figure 2(b) shows that the circular edge dislocation beams have a circular edge dislocation (denoted as E) in the source plane. The circular edge dislocation E is stable when circular edge dislocation beams propagate through free space.

 

Fig. 2. Comparison of phase distribution through free space of circular–linear edge dislocation beams with a = 1 (a) and circular edge dislocation beams (b); ⭘ and ⬤ denote that the topological charges are –1 and +1, respectively.

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As shown in Figs. 1 and 2, when circular–linear edge dislocation beams propagate through free space, linear edge dislocation evolves into a pair of optical vortices with opposite topological charges. When the distance of propagation is far enough, the pair of optical vortices is annihilated. Circular edge dislocation also evolves into a pair of optical vortices. As the transmission distance increases, the pair of optical vortices remains stable. In particular, when a linear edge dislocation passes through the center of the circular edge dislocation in the source plane, the linear edge dislocation is stable, while the circular edge dislocation vanishes in free space transmission. It was also found that, when linear edge dislocation is added to circular edge dislocation beams, circular edge dislocation cannot be transmitted stably in free space.

4. Circular edge dislocation beams and circular−linear edge dislocation beams propagating through atmospheric turbulence

Figure 3 shows a comparison of the phase distributions of circular–linear edge dislocation beams (a, b) and circular edge dislocation beams (c) in the source plane and atmospheric turbulence propagation, where the calculation parameters are a = 0 (a), a = 1 cm (b) and

 

Fig. 3. Comparison of phase distribution through atmospheric turbulence of circular–linear edge dislocation beams with a = 0 (a), circular–linear edge dislocation beams with a = 1 (b), and circular edge dislocation beams (c); ⭘ and ⬤ denote that the topological charges are –1 and +1, respectively.

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C2 n = 1×10−16 m−2/3. The other calculation parameters are the same as those in Fig. 1. Figure 3(a) indicates that, when a = 0—that is, when the linear edge dislocation passes through the center of the circular edge dislocation—the linear edge dislocation A and the circular edge dislocation B vanish in atmospheric turbulence transmission. Figure 3(b) shows that, when the linear edge dislocation does not pass through the center of the circular edge dislocation in the source plane, the propagation properties of circular–linear edge dislocation beams through atmospheric turbulence are similar to those in Fig. 2(a). The difference is that, compared with Fig. 2(a), at z = 7 km, the optical vortices C+ and C still exist in free space transmission, while the optical vortices C+ and C have annihilated in atmospheric turbulence transmission. Figure 3(c) shows that, when circular edge dislocation beams propagate in atmospheric turbulence, the circular edge dislocation E evolves into a pair of optical vortices (E+ and E), which are annihilated with the increase in transmission distance.

Figure 3 shows that when circular–linear edge dislocation beams propagate in atmospheric turbulence and a ≠ 0, the propagation properties of circular–linear edge dislocation beams are similar to those in free space. The difference is that the existence of atmospheric turbulence accelerates the evolution process. In particular, when linear edge dislocation passes through the center of circular edge dislocation in the source plane, circular edge dislocation and linear edge dislocation vanish in atmospheric turbulence transmission. Moreover, when linear edge dislocation is added to circular edge dislocation beams, the propagation properties of circular edge dislocation through atmospheric turbulence can be adjusted.

5. Conclusions

Based on the extended Huygens−Fresnel principle, the propagation properties of circular edge dislocation beams and circular−linear edge dislocation beams were studied in detail. The results show that, when circular–linear edge dislocation beams propagate in free space and atmospheric turbulence, circular edge dislocation and linear edge dislocation each evolve into a pair of optical vortices. As the beam propagates, the pair of optical vortices that evolve from linear edge dislocation is annihilated, while the pair of optical vortices that evolve from circular edge dislocation transmits stably. In particular, when the linear edge dislocation passes through the center of the circular edge dislocation in the source plane, the linear edge dislocation remains stable, and the circular edge dislocation vanishes in free space transmission. However, circular edge dislocation and linear edge dislocation vanish in atmospheric turbulence transmission. In addition, when circular edge dislocation beams propagate through free space and atmospheric turbulence, their propagation properties can be adjusted by adding linear edge dislocation. In addition, using different wavelengths, the propagation of circular−linear edge dislocation beams and circular edge dislocation beams was numerically simulated. It was found that the conclusion is still valid.

The conclusion of this study is helpful to deepen the understanding of the propagation of edge dislocation and to control the transmission of edge dislocation. It also has certain theoretical significance for the application of optical vortices in optical communications.

Funding

National Natural Science Foundation of China (61875156, 61475123); Higher Education Discipline Innovation Project (B17035).

Disclosures

The authors declare no conflicts of interest.

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7. Q. Wang, J. Z. Li, and W. X. Xie, “Spiraling elliptic Laguerre–Gaussian soliton in isotropic nonlocal competing cubic-quintic nonlinear media,” Appl. Phys. B 124(6), 104 (2018). [CrossRef]  

8. M. Cheng, L. Guo, J. Li, and Y. Zhang, “Channel capacity of the OAM-based free-space optical communication links with Bessel-Gauss beams in turbulent ocean,” IEEE Photonics J. 8(1), 1–11 (2016). [CrossRef]  

9. J. Wang, J. Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, and M. Tur, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6(7), 488–496 (2012). [CrossRef]  

10. Z. Shao, J. Zhu, Y. Chen, Y. Zhang, and S. Yu, “Spin-orbit interaction of light induced by transverse spin angular momentum engineering,” Nat. Commun. 9(1), 926 (2018). [CrossRef]  

11. A. E. Willner, A. F. Molisch, C. Bao, G. Xie, H. Huang, J. Wang, L. Li, M. Tur, M. P. J. Lavery, and N. Ahmed, “Optical communications using orbital angular momentum beams,” Adv. Opt. Photonics 7(1), 66–106 (2015). [CrossRef]  

12. T. Su, R. P. Scott, S. S. Djordjevic, N. K. Fontaine, D. J. Geisler, X. Cai, and S. J. Yoo, “Demonstration of free space coherent optical communication using integrated silicon photonic orbital angular momentum devices,” Opt. Express 20(9), 9396–9402 (2012). [CrossRef]  

13. M. Luo, Q. Chen, L. Hua, and D. Zhao, “Propagation of stochastic electromagnetic vortex beams through the turbulent biological tissues,” Phys. Lett. A 378(3), 308–314 (2014). [CrossRef]  

14. M. Padgett and R. Bowman, “Tweezers with a twist,” Nat. Photonics 5(6), 343–348 (2011). [CrossRef]  

15. D. Bradshaw and D. Andrews, “Manipulating particles with light: radiation and gradient forces,” Eur. J. Phys. 38(3), 034008 (2017). [CrossRef]  

16. A. Chmyrov, J. Keller, T. Grotjohann, M. Ratz, E. D’Este, S. Jakobs, C. Eggeling, and S. W. Hell, “Nanoscopy with more than 100,000 ‘doughnuts’,” Nat. Methods 10(8), 737–740 (2013). [CrossRef]  

17. A. Aleksanyan, N. Kravets, and E. Brasselet, “Multiple-star system adaptive vortex coronagraphy using a liquid crystal light valve,” Phys. Rev. Lett. 118(20), 203902 (2017). [CrossRef]  

18. M. T. Gruneisen, R. C. Dymale, K. E. Stoltenberg, and N. Steinhoff, “Optical vortex discrimination with a transmission volume hologram,” New J. Phys. 13(8), 083030 (2011). [CrossRef]  

19. D. Naidoo, A. Harfouche, M. Fromager, K. Ait-Ameur, and A. Forbes, “Emission of a propagation invariant flat-top beam from a microchip laser,” J. Lumin. 170(3), 750–754 (2016). [CrossRef]  

20. Y. Zhang, W. Liu, J. Gao, and X. Yang, “Generating focused 3D perfect vortex beams by plasmonic metasurfaces,” Adv. Opt. Mater. 6(4), 1701228 (2018). [CrossRef]  

21. J. Leach, E. Yao, and M. J. Padgett, “Observation of the vortex structure of a non-integer vortex beam,” New J. Phys. 6, 71 (2004). [CrossRef]  

22. H. Ma, X. Li, Y. Tai, H. Li, J. Wang, M. Tang, J. Tang, Y. Wang, and Z. Nie, “Generation of circular optical vortex array,” Ann. Phys. 529(12), 1700285 (2017). [CrossRef]  

23. T. Kaneyasu, Y. Hikosaka, M. Fujimoto, H. Iwayama, M. Hosaka, E. Shigemasa, and M. Katoh, “Observation of an optical vortex beam from a helical undulator in the XUV region,” J. Synchrotron Radiat. 24(5), 934–938 (2017). [CrossRef]  

24. Z. Qiao, G. Xie, Y. Wu, P. Yuan, J. Ma, L. Qian, and D. Fan, “Generating high-charge optical vortices directly from laser up to 288th order,” Laser Photonics Rev. 12(8), 1800019 (2018). [CrossRef]  

25. G. M. Thomas, A. Minassian, and M. J. Damzen, “Optical vortex generation from a diode-pumped alexandrite laser,” Laser Phys. Lett. 15(4), 045804 (2018). [CrossRef]  

26. E. O. Monin and A. V. Ustinov, “Optical vortex generation from a diode-pumped alexandrite laser,” J. Phys.: Conf. Ser. 1038(1), 012039 (2018). [CrossRef]  

27. M. P. J. Lavery, “Vortex instability in turbulent free-space propagation,” New J. Phys. 20(4), 043023 (2018). [CrossRef]  

28. J. Li, P. Gao, K. Cheng, and M. Duan, “Dynamic evolution of circular edge dislocations in free space and atmospheric turbulence,” Opt. Express 25(3), 2895–2908 (2017). [CrossRef]  

29. H. Chen, Z. Gao, H. Yang, S. Xiao, F. Wang, X. Huang, and X. Liu, “Evolution behavior of two edge dislocations passing through an astigmatic lens,” J. Mod. Opt. 59(21), 1863–1872 (2012). [CrossRef]  

30. P. Gao, L. Bai, Z. Wang, J. Li, and L. Guo, “Evolution behavior of mixed screw-edge dislocations propagating through atmospheric turbulence,” Appl. Phys. B 124(9), 183 (2018). [CrossRef]  

31. P. Gao, L. Bai, Z. Wang, Z. Wu, and L. Guo, “Evolution behavior of mixed higher-order optical vortex-edge dislocations propagating through atmospheric turbulence,” IEEE Photonics J. 10(3), 1–10 (2018). [CrossRef]  

32. P. Gao, L. Bai, Z. Wu, and L. Guo, “Evolution of linear edge dislocation in atmospheric turbulence and free space,” J. Mod. Opt. 66(1), 17–25 (2019). [CrossRef]  

33. E. Zauderer, “Complex argument Hermite–Gaussian and Laguerre–Gaussian beams,” J. Opt. Soc. Am. A 3(4), 465–469 (1986). [CrossRef]  

34. J. Qu, Y. Zhong, Z. Cui, and Y. Cai, “Elegant Laguerre–Gaussian beam in a turbulent atmosphere,” Opt. Commun. 283(14), 2772–2781 (2010). [CrossRef]  

35. I. Kimel and L. R. Elias, “Relations between Hermite and Laguerre Gaussian modes,” IEEE J. Quantum Electron. 29(9), 2562–2567 (1993). [CrossRef]  

36. I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products (Academic Press, 2007).

37. D. V. Petrov, “Splitting of an edge dislocation by an optical vortex,” Opt. Quantum Electron. 34(8), 759–773 (2002). [CrossRef]  

38. L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media (SPIE Press, 2005).

39. S. C. H. Wang and M. A. Plonus, “Optical beam propagation for a partially coherent source in the turbulent atmosphere,” J. Opt. Soc. Am. 69(9), 1297–1304 (1979). [CrossRef]  

40. Y. Zhang, Y. Dong, W. Wen, F. Wang, and Y. Cai, “Spectral shift of a partially coherent standard or elegant Laguerre–Gaussian beam in turbulent atmosphere,” J. Mod. Opt. 60(5), 422–430 (2013). [CrossRef]  

41. J. I. Davis, “Consideration of atmospheric turbulence in laser systems design,” Appl. Opt. 5(1), 139–147 (1966). [CrossRef]  

42. Y. Huang, F. Wang, Z. Gao, and B. Zhang, “Propagation properties of partially coherent electromagnetic hyperbolic-sine-Gaussian vortex beams propagation through non-Kolmogorov turbulence,” Opt. Express 23(2), 1088–1102 (2015). [CrossRef]  

43. J. Li and B. Lü, “The transformation of an edge dislocation in atmospheric turbulence,” Opt. Commun. 284(1), 1–7 (2011). [CrossRef]  

44. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University Press, 1995).

45. G. Gbur and T. D. Visser, “Coherence vortices in partially coherent beams,” Opt. Commun. 222(1-6), 117–125 (2003). [CrossRef]  

46. I. Freund and N. Shvartsman, “Wave-field phase singularities: The sign principle,” Phys. Rev. A 50(6), 5164–5172 (1994). [CrossRef]  

References

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  1. J. F. Nye and M. V. Berry, “Dislocations in waves trains,” Proc. Roy. Soc. A Math. Phys. Eng. Sci. 336(1605), 165–190 (1974).
  2. L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992).
    [Crossref]
  3. M. S. Soskin and M. V. Vasnetsov, “Singular optics,” Prog. Opt. 42(9), 219–276 (2001).
    [Crossref]
  4. M. Krenn, R. Fickler, M. Fink, J. Handsteiner, M. Malik, T. Scheidl, R. Ursin, and A. Zeilinger, “Communication with spatially modulated light through turbulent air across Vienna,” New J. Phys. 16(11), 113028 (2014).
    [Crossref]
  5. W. Zhang, L. Wang, W. Wang, and S. Zhao, “Propagation property of Laguerre-Gaussian beams carrying fractional orbital angular momentum in an underwater channel,” OSA Continuum 2(11), 3281–3287 (2019).
    [Crossref]
  6. Q. Liang, D. Yang, Y. Zhang, Y. Zheng, and L. Hu, “Probability of orbital angular momentum modes carried by a finite energy frozen wave in turbulent seawater,” OSA Continuum 3(9), 2429–2440 (2020).
    [Crossref]
  7. Q. Wang, J. Z. Li, and W. X. Xie, “Spiraling elliptic Laguerre–Gaussian soliton in isotropic nonlocal competing cubic-quintic nonlinear media,” Appl. Phys. B 124(6), 104 (2018).
    [Crossref]
  8. M. Cheng, L. Guo, J. Li, and Y. Zhang, “Channel capacity of the OAM-based free-space optical communication links with Bessel-Gauss beams in turbulent ocean,” IEEE Photonics J. 8(1), 1–11 (2016).
    [Crossref]
  9. J. Wang, J. Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, and M. Tur, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6(7), 488–496 (2012).
    [Crossref]
  10. Z. Shao, J. Zhu, Y. Chen, Y. Zhang, and S. Yu, “Spin-orbit interaction of light induced by transverse spin angular momentum engineering,” Nat. Commun. 9(1), 926 (2018).
    [Crossref]
  11. A. E. Willner, A. F. Molisch, C. Bao, G. Xie, H. Huang, J. Wang, L. Li, M. Tur, M. P. J. Lavery, and N. Ahmed, “Optical communications using orbital angular momentum beams,” Adv. Opt. Photonics 7(1), 66–106 (2015).
    [Crossref]
  12. T. Su, R. P. Scott, S. S. Djordjevic, N. K. Fontaine, D. J. Geisler, X. Cai, and S. J. Yoo, “Demonstration of free space coherent optical communication using integrated silicon photonic orbital angular momentum devices,” Opt. Express 20(9), 9396–9402 (2012).
    [Crossref]
  13. M. Luo, Q. Chen, L. Hua, and D. Zhao, “Propagation of stochastic electromagnetic vortex beams through the turbulent biological tissues,” Phys. Lett. A 378(3), 308–314 (2014).
    [Crossref]
  14. M. Padgett and R. Bowman, “Tweezers with a twist,” Nat. Photonics 5(6), 343–348 (2011).
    [Crossref]
  15. D. Bradshaw and D. Andrews, “Manipulating particles with light: radiation and gradient forces,” Eur. J. Phys. 38(3), 034008 (2017).
    [Crossref]
  16. A. Chmyrov, J. Keller, T. Grotjohann, M. Ratz, E. D’Este, S. Jakobs, C. Eggeling, and S. W. Hell, “Nanoscopy with more than 100,000 ‘doughnuts’,” Nat. Methods 10(8), 737–740 (2013).
    [Crossref]
  17. A. Aleksanyan, N. Kravets, and E. Brasselet, “Multiple-star system adaptive vortex coronagraphy using a liquid crystal light valve,” Phys. Rev. Lett. 118(20), 203902 (2017).
    [Crossref]
  18. M. T. Gruneisen, R. C. Dymale, K. E. Stoltenberg, and N. Steinhoff, “Optical vortex discrimination with a transmission volume hologram,” New J. Phys. 13(8), 083030 (2011).
    [Crossref]
  19. D. Naidoo, A. Harfouche, M. Fromager, K. Ait-Ameur, and A. Forbes, “Emission of a propagation invariant flat-top beam from a microchip laser,” J. Lumin. 170(3), 750–754 (2016).
    [Crossref]
  20. Y. Zhang, W. Liu, J. Gao, and X. Yang, “Generating focused 3D perfect vortex beams by plasmonic metasurfaces,” Adv. Opt. Mater. 6(4), 1701228 (2018).
    [Crossref]
  21. J. Leach, E. Yao, and M. J. Padgett, “Observation of the vortex structure of a non-integer vortex beam,” New J. Phys. 6, 71 (2004).
    [Crossref]
  22. H. Ma, X. Li, Y. Tai, H. Li, J. Wang, M. Tang, J. Tang, Y. Wang, and Z. Nie, “Generation of circular optical vortex array,” Ann. Phys. 529(12), 1700285 (2017).
    [Crossref]
  23. T. Kaneyasu, Y. Hikosaka, M. Fujimoto, H. Iwayama, M. Hosaka, E. Shigemasa, and M. Katoh, “Observation of an optical vortex beam from a helical undulator in the XUV region,” J. Synchrotron Radiat. 24(5), 934–938 (2017).
    [Crossref]
  24. Z. Qiao, G. Xie, Y. Wu, P. Yuan, J. Ma, L. Qian, and D. Fan, “Generating high-charge optical vortices directly from laser up to 288th order,” Laser Photonics Rev. 12(8), 1800019 (2018).
    [Crossref]
  25. G. M. Thomas, A. Minassian, and M. J. Damzen, “Optical vortex generation from a diode-pumped alexandrite laser,” Laser Phys. Lett. 15(4), 045804 (2018).
    [Crossref]
  26. E. O. Monin and A. V. Ustinov, “Optical vortex generation from a diode-pumped alexandrite laser,” J. Phys.: Conf. Ser. 1038(1), 012039 (2018).
    [Crossref]
  27. M. P. J. Lavery, “Vortex instability in turbulent free-space propagation,” New J. Phys. 20(4), 043023 (2018).
    [Crossref]
  28. J. Li, P. Gao, K. Cheng, and M. Duan, “Dynamic evolution of circular edge dislocations in free space and atmospheric turbulence,” Opt. Express 25(3), 2895–2908 (2017).
    [Crossref]
  29. H. Chen, Z. Gao, H. Yang, S. Xiao, F. Wang, X. Huang, and X. Liu, “Evolution behavior of two edge dislocations passing through an astigmatic lens,” J. Mod. Opt. 59(21), 1863–1872 (2012).
    [Crossref]
  30. P. Gao, L. Bai, Z. Wang, J. Li, and L. Guo, “Evolution behavior of mixed screw-edge dislocations propagating through atmospheric turbulence,” Appl. Phys. B 124(9), 183 (2018).
    [Crossref]
  31. P. Gao, L. Bai, Z. Wang, Z. Wu, and L. Guo, “Evolution behavior of mixed higher-order optical vortex-edge dislocations propagating through atmospheric turbulence,” IEEE Photonics J. 10(3), 1–10 (2018).
    [Crossref]
  32. P. Gao, L. Bai, Z. Wu, and L. Guo, “Evolution of linear edge dislocation in atmospheric turbulence and free space,” J. Mod. Opt. 66(1), 17–25 (2019).
    [Crossref]
  33. E. Zauderer, “Complex argument Hermite–Gaussian and Laguerre–Gaussian beams,” J. Opt. Soc. Am. A 3(4), 465–469 (1986).
    [Crossref]
  34. J. Qu, Y. Zhong, Z. Cui, and Y. Cai, “Elegant Laguerre–Gaussian beam in a turbulent atmosphere,” Opt. Commun. 283(14), 2772–2781 (2010).
    [Crossref]
  35. I. Kimel and L. R. Elias, “Relations between Hermite and Laguerre Gaussian modes,” IEEE J. Quantum Electron. 29(9), 2562–2567 (1993).
    [Crossref]
  36. I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products (Academic Press, 2007).
  37. D. V. Petrov, “Splitting of an edge dislocation by an optical vortex,” Opt. Quantum Electron. 34(8), 759–773 (2002).
    [Crossref]
  38. L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media (SPIE Press, 2005).
  39. S. C. H. Wang and M. A. Plonus, “Optical beam propagation for a partially coherent source in the turbulent atmosphere,” J. Opt. Soc. Am. 69(9), 1297–1304 (1979).
    [Crossref]
  40. Y. Zhang, Y. Dong, W. Wen, F. Wang, and Y. Cai, “Spectral shift of a partially coherent standard or elegant Laguerre–Gaussian beam in turbulent atmosphere,” J. Mod. Opt. 60(5), 422–430 (2013).
    [Crossref]
  41. J. I. Davis, “Consideration of atmospheric turbulence in laser systems design,” Appl. Opt. 5(1), 139–147 (1966).
    [Crossref]
  42. Y. Huang, F. Wang, Z. Gao, and B. Zhang, “Propagation properties of partially coherent electromagnetic hyperbolic-sine-Gaussian vortex beams propagation through non-Kolmogorov turbulence,” Opt. Express 23(2), 1088–1102 (2015).
    [Crossref]
  43. J. Li and B. Lü, “The transformation of an edge dislocation in atmospheric turbulence,” Opt. Commun. 284(1), 1–7 (2011).
    [Crossref]
  44. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University Press, 1995).
  45. G. Gbur and T. D. Visser, “Coherence vortices in partially coherent beams,” Opt. Commun. 222(1-6), 117–125 (2003).
    [Crossref]
  46. I. Freund and N. Shvartsman, “Wave-field phase singularities: The sign principle,” Phys. Rev. A 50(6), 5164–5172 (1994).
    [Crossref]

2020 (1)

2019 (2)

W. Zhang, L. Wang, W. Wang, and S. Zhao, “Propagation property of Laguerre-Gaussian beams carrying fractional orbital angular momentum in an underwater channel,” OSA Continuum 2(11), 3281–3287 (2019).
[Crossref]

P. Gao, L. Bai, Z. Wu, and L. Guo, “Evolution of linear edge dislocation in atmospheric turbulence and free space,” J. Mod. Opt. 66(1), 17–25 (2019).
[Crossref]

2018 (9)

P. Gao, L. Bai, Z. Wang, J. Li, and L. Guo, “Evolution behavior of mixed screw-edge dislocations propagating through atmospheric turbulence,” Appl. Phys. B 124(9), 183 (2018).
[Crossref]

P. Gao, L. Bai, Z. Wang, Z. Wu, and L. Guo, “Evolution behavior of mixed higher-order optical vortex-edge dislocations propagating through atmospheric turbulence,” IEEE Photonics J. 10(3), 1–10 (2018).
[Crossref]

Y. Zhang, W. Liu, J. Gao, and X. Yang, “Generating focused 3D perfect vortex beams by plasmonic metasurfaces,” Adv. Opt. Mater. 6(4), 1701228 (2018).
[Crossref]

Z. Qiao, G. Xie, Y. Wu, P. Yuan, J. Ma, L. Qian, and D. Fan, “Generating high-charge optical vortices directly from laser up to 288th order,” Laser Photonics Rev. 12(8), 1800019 (2018).
[Crossref]

G. M. Thomas, A. Minassian, and M. J. Damzen, “Optical vortex generation from a diode-pumped alexandrite laser,” Laser Phys. Lett. 15(4), 045804 (2018).
[Crossref]

E. O. Monin and A. V. Ustinov, “Optical vortex generation from a diode-pumped alexandrite laser,” J. Phys.: Conf. Ser. 1038(1), 012039 (2018).
[Crossref]

M. P. J. Lavery, “Vortex instability in turbulent free-space propagation,” New J. Phys. 20(4), 043023 (2018).
[Crossref]

Q. Wang, J. Z. Li, and W. X. Xie, “Spiraling elliptic Laguerre–Gaussian soliton in isotropic nonlocal competing cubic-quintic nonlinear media,” Appl. Phys. B 124(6), 104 (2018).
[Crossref]

Z. Shao, J. Zhu, Y. Chen, Y. Zhang, and S. Yu, “Spin-orbit interaction of light induced by transverse spin angular momentum engineering,” Nat. Commun. 9(1), 926 (2018).
[Crossref]

2017 (5)

D. Bradshaw and D. Andrews, “Manipulating particles with light: radiation and gradient forces,” Eur. J. Phys. 38(3), 034008 (2017).
[Crossref]

A. Aleksanyan, N. Kravets, and E. Brasselet, “Multiple-star system adaptive vortex coronagraphy using a liquid crystal light valve,” Phys. Rev. Lett. 118(20), 203902 (2017).
[Crossref]

J. Li, P. Gao, K. Cheng, and M. Duan, “Dynamic evolution of circular edge dislocations in free space and atmospheric turbulence,” Opt. Express 25(3), 2895–2908 (2017).
[Crossref]

H. Ma, X. Li, Y. Tai, H. Li, J. Wang, M. Tang, J. Tang, Y. Wang, and Z. Nie, “Generation of circular optical vortex array,” Ann. Phys. 529(12), 1700285 (2017).
[Crossref]

T. Kaneyasu, Y. Hikosaka, M. Fujimoto, H. Iwayama, M. Hosaka, E. Shigemasa, and M. Katoh, “Observation of an optical vortex beam from a helical undulator in the XUV region,” J. Synchrotron Radiat. 24(5), 934–938 (2017).
[Crossref]

2016 (2)

D. Naidoo, A. Harfouche, M. Fromager, K. Ait-Ameur, and A. Forbes, “Emission of a propagation invariant flat-top beam from a microchip laser,” J. Lumin. 170(3), 750–754 (2016).
[Crossref]

M. Cheng, L. Guo, J. Li, and Y. Zhang, “Channel capacity of the OAM-based free-space optical communication links with Bessel-Gauss beams in turbulent ocean,” IEEE Photonics J. 8(1), 1–11 (2016).
[Crossref]

2015 (2)

A. E. Willner, A. F. Molisch, C. Bao, G. Xie, H. Huang, J. Wang, L. Li, M. Tur, M. P. J. Lavery, and N. Ahmed, “Optical communications using orbital angular momentum beams,” Adv. Opt. Photonics 7(1), 66–106 (2015).
[Crossref]

Y. Huang, F. Wang, Z. Gao, and B. Zhang, “Propagation properties of partially coherent electromagnetic hyperbolic-sine-Gaussian vortex beams propagation through non-Kolmogorov turbulence,” Opt. Express 23(2), 1088–1102 (2015).
[Crossref]

2014 (2)

M. Luo, Q. Chen, L. Hua, and D. Zhao, “Propagation of stochastic electromagnetic vortex beams through the turbulent biological tissues,” Phys. Lett. A 378(3), 308–314 (2014).
[Crossref]

M. Krenn, R. Fickler, M. Fink, J. Handsteiner, M. Malik, T. Scheidl, R. Ursin, and A. Zeilinger, “Communication with spatially modulated light through turbulent air across Vienna,” New J. Phys. 16(11), 113028 (2014).
[Crossref]

2013 (2)

A. Chmyrov, J. Keller, T. Grotjohann, M. Ratz, E. D’Este, S. Jakobs, C. Eggeling, and S. W. Hell, “Nanoscopy with more than 100,000 ‘doughnuts’,” Nat. Methods 10(8), 737–740 (2013).
[Crossref]

Y. Zhang, Y. Dong, W. Wen, F. Wang, and Y. Cai, “Spectral shift of a partially coherent standard or elegant Laguerre–Gaussian beam in turbulent atmosphere,” J. Mod. Opt. 60(5), 422–430 (2013).
[Crossref]

2012 (3)

T. Su, R. P. Scott, S. S. Djordjevic, N. K. Fontaine, D. J. Geisler, X. Cai, and S. J. Yoo, “Demonstration of free space coherent optical communication using integrated silicon photonic orbital angular momentum devices,” Opt. Express 20(9), 9396–9402 (2012).
[Crossref]

J. Wang, J. Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, and M. Tur, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6(7), 488–496 (2012).
[Crossref]

H. Chen, Z. Gao, H. Yang, S. Xiao, F. Wang, X. Huang, and X. Liu, “Evolution behavior of two edge dislocations passing through an astigmatic lens,” J. Mod. Opt. 59(21), 1863–1872 (2012).
[Crossref]

2011 (3)

M. Padgett and R. Bowman, “Tweezers with a twist,” Nat. Photonics 5(6), 343–348 (2011).
[Crossref]

M. T. Gruneisen, R. C. Dymale, K. E. Stoltenberg, and N. Steinhoff, “Optical vortex discrimination with a transmission volume hologram,” New J. Phys. 13(8), 083030 (2011).
[Crossref]

J. Li and B. Lü, “The transformation of an edge dislocation in atmospheric turbulence,” Opt. Commun. 284(1), 1–7 (2011).
[Crossref]

2010 (1)

J. Qu, Y. Zhong, Z. Cui, and Y. Cai, “Elegant Laguerre–Gaussian beam in a turbulent atmosphere,” Opt. Commun. 283(14), 2772–2781 (2010).
[Crossref]

2004 (1)

J. Leach, E. Yao, and M. J. Padgett, “Observation of the vortex structure of a non-integer vortex beam,” New J. Phys. 6, 71 (2004).
[Crossref]

2003 (1)

G. Gbur and T. D. Visser, “Coherence vortices in partially coherent beams,” Opt. Commun. 222(1-6), 117–125 (2003).
[Crossref]

2002 (1)

D. V. Petrov, “Splitting of an edge dislocation by an optical vortex,” Opt. Quantum Electron. 34(8), 759–773 (2002).
[Crossref]

2001 (1)

M. S. Soskin and M. V. Vasnetsov, “Singular optics,” Prog. Opt. 42(9), 219–276 (2001).
[Crossref]

1994 (1)

I. Freund and N. Shvartsman, “Wave-field phase singularities: The sign principle,” Phys. Rev. A 50(6), 5164–5172 (1994).
[Crossref]

1993 (1)

I. Kimel and L. R. Elias, “Relations between Hermite and Laguerre Gaussian modes,” IEEE J. Quantum Electron. 29(9), 2562–2567 (1993).
[Crossref]

1992 (1)

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992).
[Crossref]

1986 (1)

1979 (1)

1974 (1)

J. F. Nye and M. V. Berry, “Dislocations in waves trains,” Proc. Roy. Soc. A Math. Phys. Eng. Sci. 336(1605), 165–190 (1974).

1966 (1)

Ahmed, N.

A. E. Willner, A. F. Molisch, C. Bao, G. Xie, H. Huang, J. Wang, L. Li, M. Tur, M. P. J. Lavery, and N. Ahmed, “Optical communications using orbital angular momentum beams,” Adv. Opt. Photonics 7(1), 66–106 (2015).
[Crossref]

J. Wang, J. Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, and M. Tur, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6(7), 488–496 (2012).
[Crossref]

Ait-Ameur, K.

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Y. Zhang, W. Liu, J. Gao, and X. Yang, “Generating focused 3D perfect vortex beams by plasmonic metasurfaces,” Adv. Opt. Mater. 6(4), 1701228 (2018).
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P. Gao, L. Bai, Z. Wu, and L. Guo, “Evolution of linear edge dislocation in atmospheric turbulence and free space,” J. Mod. Opt. 66(1), 17–25 (2019).
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P. Gao, L. Bai, Z. Wang, J. Li, and L. Guo, “Evolution behavior of mixed screw-edge dislocations propagating through atmospheric turbulence,” Appl. Phys. B 124(9), 183 (2018).
[Crossref]

P. Gao, L. Bai, Z. Wang, Z. Wu, and L. Guo, “Evolution behavior of mixed higher-order optical vortex-edge dislocations propagating through atmospheric turbulence,” IEEE Photonics J. 10(3), 1–10 (2018).
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J. Li, P. Gao, K. Cheng, and M. Duan, “Dynamic evolution of circular edge dislocations in free space and atmospheric turbulence,” Opt. Express 25(3), 2895–2908 (2017).
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A. Chmyrov, J. Keller, T. Grotjohann, M. Ratz, E. D’Este, S. Jakobs, C. Eggeling, and S. W. Hell, “Nanoscopy with more than 100,000 ‘doughnuts’,” Nat. Methods 10(8), 737–740 (2013).
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M. T. Gruneisen, R. C. Dymale, K. E. Stoltenberg, and N. Steinhoff, “Optical vortex discrimination with a transmission volume hologram,” New J. Phys. 13(8), 083030 (2011).
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P. Gao, L. Bai, Z. Wu, and L. Guo, “Evolution of linear edge dislocation in atmospheric turbulence and free space,” J. Mod. Opt. 66(1), 17–25 (2019).
[Crossref]

P. Gao, L. Bai, Z. Wang, Z. Wu, and L. Guo, “Evolution behavior of mixed higher-order optical vortex-edge dislocations propagating through atmospheric turbulence,” IEEE Photonics J. 10(3), 1–10 (2018).
[Crossref]

P. Gao, L. Bai, Z. Wang, J. Li, and L. Guo, “Evolution behavior of mixed screw-edge dislocations propagating through atmospheric turbulence,” Appl. Phys. B 124(9), 183 (2018).
[Crossref]

M. Cheng, L. Guo, J. Li, and Y. Zhang, “Channel capacity of the OAM-based free-space optical communication links with Bessel-Gauss beams in turbulent ocean,” IEEE Photonics J. 8(1), 1–11 (2016).
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M. Krenn, R. Fickler, M. Fink, J. Handsteiner, M. Malik, T. Scheidl, R. Ursin, and A. Zeilinger, “Communication with spatially modulated light through turbulent air across Vienna,” New J. Phys. 16(11), 113028 (2014).
[Crossref]

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D. Naidoo, A. Harfouche, M. Fromager, K. Ait-Ameur, and A. Forbes, “Emission of a propagation invariant flat-top beam from a microchip laser,” J. Lumin. 170(3), 750–754 (2016).
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A. Chmyrov, J. Keller, T. Grotjohann, M. Ratz, E. D’Este, S. Jakobs, C. Eggeling, and S. W. Hell, “Nanoscopy with more than 100,000 ‘doughnuts’,” Nat. Methods 10(8), 737–740 (2013).
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T. Kaneyasu, Y. Hikosaka, M. Fujimoto, H. Iwayama, M. Hosaka, E. Shigemasa, and M. Katoh, “Observation of an optical vortex beam from a helical undulator in the XUV region,” J. Synchrotron Radiat. 24(5), 934–938 (2017).
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T. Kaneyasu, Y. Hikosaka, M. Fujimoto, H. Iwayama, M. Hosaka, E. Shigemasa, and M. Katoh, “Observation of an optical vortex beam from a helical undulator in the XUV region,” J. Synchrotron Radiat. 24(5), 934–938 (2017).
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Hua, L.

M. Luo, Q. Chen, L. Hua, and D. Zhao, “Propagation of stochastic electromagnetic vortex beams through the turbulent biological tissues,” Phys. Lett. A 378(3), 308–314 (2014).
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A. E. Willner, A. F. Molisch, C. Bao, G. Xie, H. Huang, J. Wang, L. Li, M. Tur, M. P. J. Lavery, and N. Ahmed, “Optical communications using orbital angular momentum beams,” Adv. Opt. Photonics 7(1), 66–106 (2015).
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J. Wang, J. Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, and M. Tur, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6(7), 488–496 (2012).
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H. Chen, Z. Gao, H. Yang, S. Xiao, F. Wang, X. Huang, and X. Liu, “Evolution behavior of two edge dislocations passing through an astigmatic lens,” J. Mod. Opt. 59(21), 1863–1872 (2012).
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Iwayama, H.

T. Kaneyasu, Y. Hikosaka, M. Fujimoto, H. Iwayama, M. Hosaka, E. Shigemasa, and M. Katoh, “Observation of an optical vortex beam from a helical undulator in the XUV region,” J. Synchrotron Radiat. 24(5), 934–938 (2017).
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A. Chmyrov, J. Keller, T. Grotjohann, M. Ratz, E. D’Este, S. Jakobs, C. Eggeling, and S. W. Hell, “Nanoscopy with more than 100,000 ‘doughnuts’,” Nat. Methods 10(8), 737–740 (2013).
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T. Kaneyasu, Y. Hikosaka, M. Fujimoto, H. Iwayama, M. Hosaka, E. Shigemasa, and M. Katoh, “Observation of an optical vortex beam from a helical undulator in the XUV region,” J. Synchrotron Radiat. 24(5), 934–938 (2017).
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T. Kaneyasu, Y. Hikosaka, M. Fujimoto, H. Iwayama, M. Hosaka, E. Shigemasa, and M. Katoh, “Observation of an optical vortex beam from a helical undulator in the XUV region,” J. Synchrotron Radiat. 24(5), 934–938 (2017).
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A. Chmyrov, J. Keller, T. Grotjohann, M. Ratz, E. D’Este, S. Jakobs, C. Eggeling, and S. W. Hell, “Nanoscopy with more than 100,000 ‘doughnuts’,” Nat. Methods 10(8), 737–740 (2013).
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I. Kimel and L. R. Elias, “Relations between Hermite and Laguerre Gaussian modes,” IEEE J. Quantum Electron. 29(9), 2562–2567 (1993).
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A. Aleksanyan, N. Kravets, and E. Brasselet, “Multiple-star system adaptive vortex coronagraphy using a liquid crystal light valve,” Phys. Rev. Lett. 118(20), 203902 (2017).
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M. Krenn, R. Fickler, M. Fink, J. Handsteiner, M. Malik, T. Scheidl, R. Ursin, and A. Zeilinger, “Communication with spatially modulated light through turbulent air across Vienna,” New J. Phys. 16(11), 113028 (2014).
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H. Ma, X. Li, Y. Tai, H. Li, J. Wang, M. Tang, J. Tang, Y. Wang, and Z. Nie, “Generation of circular optical vortex array,” Ann. Phys. 529(12), 1700285 (2017).
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P. Gao, L. Bai, Z. Wang, J. Li, and L. Guo, “Evolution behavior of mixed screw-edge dislocations propagating through atmospheric turbulence,” Appl. Phys. B 124(9), 183 (2018).
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J. Li, P. Gao, K. Cheng, and M. Duan, “Dynamic evolution of circular edge dislocations in free space and atmospheric turbulence,” Opt. Express 25(3), 2895–2908 (2017).
[Crossref]

M. Cheng, L. Guo, J. Li, and Y. Zhang, “Channel capacity of the OAM-based free-space optical communication links with Bessel-Gauss beams in turbulent ocean,” IEEE Photonics J. 8(1), 1–11 (2016).
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A. E. Willner, A. F. Molisch, C. Bao, G. Xie, H. Huang, J. Wang, L. Li, M. Tur, M. P. J. Lavery, and N. Ahmed, “Optical communications using orbital angular momentum beams,” Adv. Opt. Photonics 7(1), 66–106 (2015).
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H. Ma, X. Li, Y. Tai, H. Li, J. Wang, M. Tang, J. Tang, Y. Wang, and Z. Nie, “Generation of circular optical vortex array,” Ann. Phys. 529(12), 1700285 (2017).
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Liu, W.

Y. Zhang, W. Liu, J. Gao, and X. Yang, “Generating focused 3D perfect vortex beams by plasmonic metasurfaces,” Adv. Opt. Mater. 6(4), 1701228 (2018).
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H. Chen, Z. Gao, H. Yang, S. Xiao, F. Wang, X. Huang, and X. Liu, “Evolution behavior of two edge dislocations passing through an astigmatic lens,” J. Mod. Opt. 59(21), 1863–1872 (2012).
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J. Li and B. Lü, “The transformation of an edge dislocation in atmospheric turbulence,” Opt. Commun. 284(1), 1–7 (2011).
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M. Luo, Q. Chen, L. Hua, and D. Zhao, “Propagation of stochastic electromagnetic vortex beams through the turbulent biological tissues,” Phys. Lett. A 378(3), 308–314 (2014).
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H. Ma, X. Li, Y. Tai, H. Li, J. Wang, M. Tang, J. Tang, Y. Wang, and Z. Nie, “Generation of circular optical vortex array,” Ann. Phys. 529(12), 1700285 (2017).
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Z. Qiao, G. Xie, Y. Wu, P. Yuan, J. Ma, L. Qian, and D. Fan, “Generating high-charge optical vortices directly from laser up to 288th order,” Laser Photonics Rev. 12(8), 1800019 (2018).
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M. Krenn, R. Fickler, M. Fink, J. Handsteiner, M. Malik, T. Scheidl, R. Ursin, and A. Zeilinger, “Communication with spatially modulated light through turbulent air across Vienna,” New J. Phys. 16(11), 113028 (2014).
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A. E. Willner, A. F. Molisch, C. Bao, G. Xie, H. Huang, J. Wang, L. Li, M. Tur, M. P. J. Lavery, and N. Ahmed, “Optical communications using orbital angular momentum beams,” Adv. Opt. Photonics 7(1), 66–106 (2015).
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H. Ma, X. Li, Y. Tai, H. Li, J. Wang, M. Tang, J. Tang, Y. Wang, and Z. Nie, “Generation of circular optical vortex array,” Ann. Phys. 529(12), 1700285 (2017).
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J. F. Nye and M. V. Berry, “Dislocations in waves trains,” Proc. Roy. Soc. A Math. Phys. Eng. Sci. 336(1605), 165–190 (1974).

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M. Padgett and R. Bowman, “Tweezers with a twist,” Nat. Photonics 5(6), 343–348 (2011).
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J. Leach, E. Yao, and M. J. Padgett, “Observation of the vortex structure of a non-integer vortex beam,” New J. Phys. 6, 71 (2004).
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Z. Qiao, G. Xie, Y. Wu, P. Yuan, J. Ma, L. Qian, and D. Fan, “Generating high-charge optical vortices directly from laser up to 288th order,” Laser Photonics Rev. 12(8), 1800019 (2018).
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J. Qu, Y. Zhong, Z. Cui, and Y. Cai, “Elegant Laguerre–Gaussian beam in a turbulent atmosphere,” Opt. Commun. 283(14), 2772–2781 (2010).
[Crossref]

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A. Chmyrov, J. Keller, T. Grotjohann, M. Ratz, E. D’Este, S. Jakobs, C. Eggeling, and S. W. Hell, “Nanoscopy with more than 100,000 ‘doughnuts’,” Nat. Methods 10(8), 737–740 (2013).
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J. Wang, J. Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, and M. Tur, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6(7), 488–496 (2012).
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I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products (Academic Press, 2007).

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M. Krenn, R. Fickler, M. Fink, J. Handsteiner, M. Malik, T. Scheidl, R. Ursin, and A. Zeilinger, “Communication with spatially modulated light through turbulent air across Vienna,” New J. Phys. 16(11), 113028 (2014).
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Shao, Z.

Z. Shao, J. Zhu, Y. Chen, Y. Zhang, and S. Yu, “Spin-orbit interaction of light induced by transverse spin angular momentum engineering,” Nat. Commun. 9(1), 926 (2018).
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T. Kaneyasu, Y. Hikosaka, M. Fujimoto, H. Iwayama, M. Hosaka, E. Shigemasa, and M. Katoh, “Observation of an optical vortex beam from a helical undulator in the XUV region,” J. Synchrotron Radiat. 24(5), 934–938 (2017).
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[Crossref]

Steinhoff, N.

M. T. Gruneisen, R. C. Dymale, K. E. Stoltenberg, and N. Steinhoff, “Optical vortex discrimination with a transmission volume hologram,” New J. Phys. 13(8), 083030 (2011).
[Crossref]

Stoltenberg, K. E.

M. T. Gruneisen, R. C. Dymale, K. E. Stoltenberg, and N. Steinhoff, “Optical vortex discrimination with a transmission volume hologram,” New J. Phys. 13(8), 083030 (2011).
[Crossref]

Su, T.

Tai, Y.

H. Ma, X. Li, Y. Tai, H. Li, J. Wang, M. Tang, J. Tang, Y. Wang, and Z. Nie, “Generation of circular optical vortex array,” Ann. Phys. 529(12), 1700285 (2017).
[Crossref]

Tang, J.

H. Ma, X. Li, Y. Tai, H. Li, J. Wang, M. Tang, J. Tang, Y. Wang, and Z. Nie, “Generation of circular optical vortex array,” Ann. Phys. 529(12), 1700285 (2017).
[Crossref]

Tang, M.

H. Ma, X. Li, Y. Tai, H. Li, J. Wang, M. Tang, J. Tang, Y. Wang, and Z. Nie, “Generation of circular optical vortex array,” Ann. Phys. 529(12), 1700285 (2017).
[Crossref]

Thomas, G. M.

G. M. Thomas, A. Minassian, and M. J. Damzen, “Optical vortex generation from a diode-pumped alexandrite laser,” Laser Phys. Lett. 15(4), 045804 (2018).
[Crossref]

Tur, M.

A. E. Willner, A. F. Molisch, C. Bao, G. Xie, H. Huang, J. Wang, L. Li, M. Tur, M. P. J. Lavery, and N. Ahmed, “Optical communications using orbital angular momentum beams,” Adv. Opt. Photonics 7(1), 66–106 (2015).
[Crossref]

J. Wang, J. Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, and M. Tur, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6(7), 488–496 (2012).
[Crossref]

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M. Krenn, R. Fickler, M. Fink, J. Handsteiner, M. Malik, T. Scheidl, R. Ursin, and A. Zeilinger, “Communication with spatially modulated light through turbulent air across Vienna,” New J. Phys. 16(11), 113028 (2014).
[Crossref]

Ustinov, A. V.

E. O. Monin and A. V. Ustinov, “Optical vortex generation from a diode-pumped alexandrite laser,” J. Phys.: Conf. Ser. 1038(1), 012039 (2018).
[Crossref]

Vasnetsov, M. V.

M. S. Soskin and M. V. Vasnetsov, “Singular optics,” Prog. Opt. 42(9), 219–276 (2001).
[Crossref]

Visser, T. D.

G. Gbur and T. D. Visser, “Coherence vortices in partially coherent beams,” Opt. Commun. 222(1-6), 117–125 (2003).
[Crossref]

Wang, F.

Y. Huang, F. Wang, Z. Gao, and B. Zhang, “Propagation properties of partially coherent electromagnetic hyperbolic-sine-Gaussian vortex beams propagation through non-Kolmogorov turbulence,” Opt. Express 23(2), 1088–1102 (2015).
[Crossref]

Y. Zhang, Y. Dong, W. Wen, F. Wang, and Y. Cai, “Spectral shift of a partially coherent standard or elegant Laguerre–Gaussian beam in turbulent atmosphere,” J. Mod. Opt. 60(5), 422–430 (2013).
[Crossref]

H. Chen, Z. Gao, H. Yang, S. Xiao, F. Wang, X. Huang, and X. Liu, “Evolution behavior of two edge dislocations passing through an astigmatic lens,” J. Mod. Opt. 59(21), 1863–1872 (2012).
[Crossref]

Wang, J.

H. Ma, X. Li, Y. Tai, H. Li, J. Wang, M. Tang, J. Tang, Y. Wang, and Z. Nie, “Generation of circular optical vortex array,” Ann. Phys. 529(12), 1700285 (2017).
[Crossref]

A. E. Willner, A. F. Molisch, C. Bao, G. Xie, H. Huang, J. Wang, L. Li, M. Tur, M. P. J. Lavery, and N. Ahmed, “Optical communications using orbital angular momentum beams,” Adv. Opt. Photonics 7(1), 66–106 (2015).
[Crossref]

J. Wang, J. Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, and M. Tur, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6(7), 488–496 (2012).
[Crossref]

Wang, L.

Wang, Q.

Q. Wang, J. Z. Li, and W. X. Xie, “Spiraling elliptic Laguerre–Gaussian soliton in isotropic nonlocal competing cubic-quintic nonlinear media,” Appl. Phys. B 124(6), 104 (2018).
[Crossref]

Wang, S. C. H.

Wang, W.

Wang, Y.

H. Ma, X. Li, Y. Tai, H. Li, J. Wang, M. Tang, J. Tang, Y. Wang, and Z. Nie, “Generation of circular optical vortex array,” Ann. Phys. 529(12), 1700285 (2017).
[Crossref]

Wang, Z.

P. Gao, L. Bai, Z. Wang, J. Li, and L. Guo, “Evolution behavior of mixed screw-edge dislocations propagating through atmospheric turbulence,” Appl. Phys. B 124(9), 183 (2018).
[Crossref]

P. Gao, L. Bai, Z. Wang, Z. Wu, and L. Guo, “Evolution behavior of mixed higher-order optical vortex-edge dislocations propagating through atmospheric turbulence,” IEEE Photonics J. 10(3), 1–10 (2018).
[Crossref]

Wen, W.

Y. Zhang, Y. Dong, W. Wen, F. Wang, and Y. Cai, “Spectral shift of a partially coherent standard or elegant Laguerre–Gaussian beam in turbulent atmosphere,” J. Mod. Opt. 60(5), 422–430 (2013).
[Crossref]

Willner, A. E.

A. E. Willner, A. F. Molisch, C. Bao, G. Xie, H. Huang, J. Wang, L. Li, M. Tur, M. P. J. Lavery, and N. Ahmed, “Optical communications using orbital angular momentum beams,” Adv. Opt. Photonics 7(1), 66–106 (2015).
[Crossref]

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L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992).
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Wolf, E.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University Press, 1995).

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Z. Qiao, G. Xie, Y. Wu, P. Yuan, J. Ma, L. Qian, and D. Fan, “Generating high-charge optical vortices directly from laser up to 288th order,” Laser Photonics Rev. 12(8), 1800019 (2018).
[Crossref]

Wu, Z.

P. Gao, L. Bai, Z. Wu, and L. Guo, “Evolution of linear edge dislocation in atmospheric turbulence and free space,” J. Mod. Opt. 66(1), 17–25 (2019).
[Crossref]

P. Gao, L. Bai, Z. Wang, Z. Wu, and L. Guo, “Evolution behavior of mixed higher-order optical vortex-edge dislocations propagating through atmospheric turbulence,” IEEE Photonics J. 10(3), 1–10 (2018).
[Crossref]

Xiao, S.

H. Chen, Z. Gao, H. Yang, S. Xiao, F. Wang, X. Huang, and X. Liu, “Evolution behavior of two edge dislocations passing through an astigmatic lens,” J. Mod. Opt. 59(21), 1863–1872 (2012).
[Crossref]

Xie, G.

Z. Qiao, G. Xie, Y. Wu, P. Yuan, J. Ma, L. Qian, and D. Fan, “Generating high-charge optical vortices directly from laser up to 288th order,” Laser Photonics Rev. 12(8), 1800019 (2018).
[Crossref]

A. E. Willner, A. F. Molisch, C. Bao, G. Xie, H. Huang, J. Wang, L. Li, M. Tur, M. P. J. Lavery, and N. Ahmed, “Optical communications using orbital angular momentum beams,” Adv. Opt. Photonics 7(1), 66–106 (2015).
[Crossref]

Xie, W. X.

Q. Wang, J. Z. Li, and W. X. Xie, “Spiraling elliptic Laguerre–Gaussian soliton in isotropic nonlocal competing cubic-quintic nonlinear media,” Appl. Phys. B 124(6), 104 (2018).
[Crossref]

Yan, Y.

J. Wang, J. Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, and M. Tur, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6(7), 488–496 (2012).
[Crossref]

Yang, D.

Yang, H.

H. Chen, Z. Gao, H. Yang, S. Xiao, F. Wang, X. Huang, and X. Liu, “Evolution behavior of two edge dislocations passing through an astigmatic lens,” J. Mod. Opt. 59(21), 1863–1872 (2012).
[Crossref]

Yang, J. Y.

J. Wang, J. Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, and M. Tur, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6(7), 488–496 (2012).
[Crossref]

Yang, X.

Y. Zhang, W. Liu, J. Gao, and X. Yang, “Generating focused 3D perfect vortex beams by plasmonic metasurfaces,” Adv. Opt. Mater. 6(4), 1701228 (2018).
[Crossref]

Yao, E.

J. Leach, E. Yao, and M. J. Padgett, “Observation of the vortex structure of a non-integer vortex beam,” New J. Phys. 6, 71 (2004).
[Crossref]

Yoo, S. J.

Yu, S.

Z. Shao, J. Zhu, Y. Chen, Y. Zhang, and S. Yu, “Spin-orbit interaction of light induced by transverse spin angular momentum engineering,” Nat. Commun. 9(1), 926 (2018).
[Crossref]

Yuan, P.

Z. Qiao, G. Xie, Y. Wu, P. Yuan, J. Ma, L. Qian, and D. Fan, “Generating high-charge optical vortices directly from laser up to 288th order,” Laser Photonics Rev. 12(8), 1800019 (2018).
[Crossref]

Yue, Y.

J. Wang, J. Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, and M. Tur, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6(7), 488–496 (2012).
[Crossref]

Zauderer, E.

Zeilinger, A.

M. Krenn, R. Fickler, M. Fink, J. Handsteiner, M. Malik, T. Scheidl, R. Ursin, and A. Zeilinger, “Communication with spatially modulated light through turbulent air across Vienna,” New J. Phys. 16(11), 113028 (2014).
[Crossref]

Zhang, B.

Zhang, W.

Zhang, Y.

Q. Liang, D. Yang, Y. Zhang, Y. Zheng, and L. Hu, “Probability of orbital angular momentum modes carried by a finite energy frozen wave in turbulent seawater,” OSA Continuum 3(9), 2429–2440 (2020).
[Crossref]

Z. Shao, J. Zhu, Y. Chen, Y. Zhang, and S. Yu, “Spin-orbit interaction of light induced by transverse spin angular momentum engineering,” Nat. Commun. 9(1), 926 (2018).
[Crossref]

Y. Zhang, W. Liu, J. Gao, and X. Yang, “Generating focused 3D perfect vortex beams by plasmonic metasurfaces,” Adv. Opt. Mater. 6(4), 1701228 (2018).
[Crossref]

M. Cheng, L. Guo, J. Li, and Y. Zhang, “Channel capacity of the OAM-based free-space optical communication links with Bessel-Gauss beams in turbulent ocean,” IEEE Photonics J. 8(1), 1–11 (2016).
[Crossref]

Y. Zhang, Y. Dong, W. Wen, F. Wang, and Y. Cai, “Spectral shift of a partially coherent standard or elegant Laguerre–Gaussian beam in turbulent atmosphere,” J. Mod. Opt. 60(5), 422–430 (2013).
[Crossref]

Zhao, D.

M. Luo, Q. Chen, L. Hua, and D. Zhao, “Propagation of stochastic electromagnetic vortex beams through the turbulent biological tissues,” Phys. Lett. A 378(3), 308–314 (2014).
[Crossref]

Zhao, S.

Zheng, Y.

Zhong, Y.

J. Qu, Y. Zhong, Z. Cui, and Y. Cai, “Elegant Laguerre–Gaussian beam in a turbulent atmosphere,” Opt. Commun. 283(14), 2772–2781 (2010).
[Crossref]

Zhu, J.

Z. Shao, J. Zhu, Y. Chen, Y. Zhang, and S. Yu, “Spin-orbit interaction of light induced by transverse spin angular momentum engineering,” Nat. Commun. 9(1), 926 (2018).
[Crossref]

Adv. Opt. Mater. (1)

Y. Zhang, W. Liu, J. Gao, and X. Yang, “Generating focused 3D perfect vortex beams by plasmonic metasurfaces,” Adv. Opt. Mater. 6(4), 1701228 (2018).
[Crossref]

Adv. Opt. Photonics (1)

A. E. Willner, A. F. Molisch, C. Bao, G. Xie, H. Huang, J. Wang, L. Li, M. Tur, M. P. J. Lavery, and N. Ahmed, “Optical communications using orbital angular momentum beams,” Adv. Opt. Photonics 7(1), 66–106 (2015).
[Crossref]

Ann. Phys. (1)

H. Ma, X. Li, Y. Tai, H. Li, J. Wang, M. Tang, J. Tang, Y. Wang, and Z. Nie, “Generation of circular optical vortex array,” Ann. Phys. 529(12), 1700285 (2017).
[Crossref]

Appl. Opt. (1)

Appl. Phys. B (2)

P. Gao, L. Bai, Z. Wang, J. Li, and L. Guo, “Evolution behavior of mixed screw-edge dislocations propagating through atmospheric turbulence,” Appl. Phys. B 124(9), 183 (2018).
[Crossref]

Q. Wang, J. Z. Li, and W. X. Xie, “Spiraling elliptic Laguerre–Gaussian soliton in isotropic nonlocal competing cubic-quintic nonlinear media,” Appl. Phys. B 124(6), 104 (2018).
[Crossref]

Eur. J. Phys. (1)

D. Bradshaw and D. Andrews, “Manipulating particles with light: radiation and gradient forces,” Eur. J. Phys. 38(3), 034008 (2017).
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I. Kimel and L. R. Elias, “Relations between Hermite and Laguerre Gaussian modes,” IEEE J. Quantum Electron. 29(9), 2562–2567 (1993).
[Crossref]

IEEE Photonics J. (2)

P. Gao, L. Bai, Z. Wang, Z. Wu, and L. Guo, “Evolution behavior of mixed higher-order optical vortex-edge dislocations propagating through atmospheric turbulence,” IEEE Photonics J. 10(3), 1–10 (2018).
[Crossref]

M. Cheng, L. Guo, J. Li, and Y. Zhang, “Channel capacity of the OAM-based free-space optical communication links with Bessel-Gauss beams in turbulent ocean,” IEEE Photonics J. 8(1), 1–11 (2016).
[Crossref]

J. Lumin. (1)

D. Naidoo, A. Harfouche, M. Fromager, K. Ait-Ameur, and A. Forbes, “Emission of a propagation invariant flat-top beam from a microchip laser,” J. Lumin. 170(3), 750–754 (2016).
[Crossref]

J. Mod. Opt. (3)

H. Chen, Z. Gao, H. Yang, S. Xiao, F. Wang, X. Huang, and X. Liu, “Evolution behavior of two edge dislocations passing through an astigmatic lens,” J. Mod. Opt. 59(21), 1863–1872 (2012).
[Crossref]

P. Gao, L. Bai, Z. Wu, and L. Guo, “Evolution of linear edge dislocation in atmospheric turbulence and free space,” J. Mod. Opt. 66(1), 17–25 (2019).
[Crossref]

Y. Zhang, Y. Dong, W. Wen, F. Wang, and Y. Cai, “Spectral shift of a partially coherent standard or elegant Laguerre–Gaussian beam in turbulent atmosphere,” J. Mod. Opt. 60(5), 422–430 (2013).
[Crossref]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (1)

J. Phys.: Conf. Ser. (1)

E. O. Monin and A. V. Ustinov, “Optical vortex generation from a diode-pumped alexandrite laser,” J. Phys.: Conf. Ser. 1038(1), 012039 (2018).
[Crossref]

J. Synchrotron Radiat. (1)

T. Kaneyasu, Y. Hikosaka, M. Fujimoto, H. Iwayama, M. Hosaka, E. Shigemasa, and M. Katoh, “Observation of an optical vortex beam from a helical undulator in the XUV region,” J. Synchrotron Radiat. 24(5), 934–938 (2017).
[Crossref]

Laser Photonics Rev. (1)

Z. Qiao, G. Xie, Y. Wu, P. Yuan, J. Ma, L. Qian, and D. Fan, “Generating high-charge optical vortices directly from laser up to 288th order,” Laser Photonics Rev. 12(8), 1800019 (2018).
[Crossref]

Laser Phys. Lett. (1)

G. M. Thomas, A. Minassian, and M. J. Damzen, “Optical vortex generation from a diode-pumped alexandrite laser,” Laser Phys. Lett. 15(4), 045804 (2018).
[Crossref]

Nat. Commun. (1)

Z. Shao, J. Zhu, Y. Chen, Y. Zhang, and S. Yu, “Spin-orbit interaction of light induced by transverse spin angular momentum engineering,” Nat. Commun. 9(1), 926 (2018).
[Crossref]

Nat. Methods (1)

A. Chmyrov, J. Keller, T. Grotjohann, M. Ratz, E. D’Este, S. Jakobs, C. Eggeling, and S. W. Hell, “Nanoscopy with more than 100,000 ‘doughnuts’,” Nat. Methods 10(8), 737–740 (2013).
[Crossref]

Nat. Photonics (2)

M. Padgett and R. Bowman, “Tweezers with a twist,” Nat. Photonics 5(6), 343–348 (2011).
[Crossref]

J. Wang, J. Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, and M. Tur, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6(7), 488–496 (2012).
[Crossref]

New J. Phys. (4)

M. Krenn, R. Fickler, M. Fink, J. Handsteiner, M. Malik, T. Scheidl, R. Ursin, and A. Zeilinger, “Communication with spatially modulated light through turbulent air across Vienna,” New J. Phys. 16(11), 113028 (2014).
[Crossref]

M. T. Gruneisen, R. C. Dymale, K. E. Stoltenberg, and N. Steinhoff, “Optical vortex discrimination with a transmission volume hologram,” New J. Phys. 13(8), 083030 (2011).
[Crossref]

J. Leach, E. Yao, and M. J. Padgett, “Observation of the vortex structure of a non-integer vortex beam,” New J. Phys. 6, 71 (2004).
[Crossref]

M. P. J. Lavery, “Vortex instability in turbulent free-space propagation,” New J. Phys. 20(4), 043023 (2018).
[Crossref]

Opt. Commun. (3)

J. Qu, Y. Zhong, Z. Cui, and Y. Cai, “Elegant Laguerre–Gaussian beam in a turbulent atmosphere,” Opt. Commun. 283(14), 2772–2781 (2010).
[Crossref]

J. Li and B. Lü, “The transformation of an edge dislocation in atmospheric turbulence,” Opt. Commun. 284(1), 1–7 (2011).
[Crossref]

G. Gbur and T. D. Visser, “Coherence vortices in partially coherent beams,” Opt. Commun. 222(1-6), 117–125 (2003).
[Crossref]

Opt. Express (3)

Opt. Quantum Electron. (1)

D. V. Petrov, “Splitting of an edge dislocation by an optical vortex,” Opt. Quantum Electron. 34(8), 759–773 (2002).
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OSA Continuum (2)

Phys. Lett. A (1)

M. Luo, Q. Chen, L. Hua, and D. Zhao, “Propagation of stochastic electromagnetic vortex beams through the turbulent biological tissues,” Phys. Lett. A 378(3), 308–314 (2014).
[Crossref]

Phys. Rev. A (2)

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992).
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A. Aleksanyan, N. Kravets, and E. Brasselet, “Multiple-star system adaptive vortex coronagraphy using a liquid crystal light valve,” Phys. Rev. Lett. 118(20), 203902 (2017).
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Proc. Roy. Soc. A Math. Phys. Eng. Sci. (1)

J. F. Nye and M. V. Berry, “Dislocations in waves trains,” Proc. Roy. Soc. A Math. Phys. Eng. Sci. 336(1605), 165–190 (1974).

Prog. Opt. (1)

M. S. Soskin and M. V. Vasnetsov, “Singular optics,” Prog. Opt. 42(9), 219–276 (2001).
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Other (3)

L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media (SPIE Press, 2005).

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products (Academic Press, 2007).

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University Press, 1995).

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Figures (3)

Fig. 1.
Fig. 1. Phase distribution (a−d) and normalized intensity distribution (e−h) of circular−linear edge dislocation beams with a = 0 in the source plane and propagating through free space at the propagation distance z = 1, 7, and 10 km.
Fig. 2.
Fig. 2. Comparison of phase distribution through free space of circular–linear edge dislocation beams with a = 1 (a) and circular edge dislocation beams (b); ⭘ and ⬤ denote that the topological charges are –1 and +1, respectively.
Fig. 3.
Fig. 3. Comparison of phase distribution through atmospheric turbulence of circular–linear edge dislocation beams with a = 0 (a), circular–linear edge dislocation beams with a = 1 (b), and circular edge dislocation beams (c); ⭘ and ⬤ denote that the topological charges are –1 and +1, respectively.

Equations (37)

Equations on this page are rendered with MathJax. Learn more.

E ( s , θ , z = 0 ) = ( 2 s w 0 ) m L n m ( 2 s 2 w 0 2 ) exp ( s 2 w 0 2 + i m θ ) ,
s m L n m ( s 2 ) exp ( i m θ ) = ( 1 ) n 2 2 n + m n ! t = 0 n r = 0 m i r n ! m ! t ! r ! ( n t ) ! ( m r ) ! × H 2 t + m r ( s x ) H 2 n 2 t + r ( s y ) ,
E ( s , z = 0 ) = ( 1 ) n 2 2 n + m n ! t = 0 n r = 0 m i r n ! m ! t ! r ! ( n t ) ! ( m r ) ! H 2 t + m r ( 2 s x w 0 ) × H 2 n 2 t + r ( 2 s y w 0 ) exp ( s x 2 + s y 2 w 0 2 ) .
E ( s , z = 0 ) = ( 1 ) n 2 2 n n ! t = 0 n n ! t ! ( n t ) ! H 2 t ( 2 s x w 0 ) H 2 n 2 t ( 2 s y w 0 ) exp ( s x 2 + s y 2 w 0 2 ) .
H n ( x ) = j = 0 [ n / n 2 2 ] ( 1 ) j n ! j ! ( n 2 j ) ! ( 2 x ) n 2 j ,
E ( s , z = 0 ) = 2 w 0 2 ( s x 2 + s y 2 w 0 2 2 ) exp ( s x 2 + s y 2 w 0 2 ) .
E ( s , z = 0 ) = 1 w 0 ( s x a ) exp ( s x 2 + s y 2 w 0 2 ) ,
E ( s , 0 ) = 2 w 0 3 ( s x 2 + s y 2 w 0 2 2 ) ( s x a ) exp ( s x 2 + s y 2 w 0 2 ) .
W 0 ( s 1 , s 2 , z = 0 ) = E 1 ( s 1 , z = 0 ) E 2 ( s 2 , z = 0 ) = 4 ( s 1 x a ) ( s 2 x a ) w 0 6 ( s 1 x 2 + s 1 y 2 w 0 2 2 ) ( s 2 x 2 + s 2 y 2 w 0 2 2 ) × exp ( s 1 x 2 + s 1 y 2 + s 2 x 2 + s 2 y 2 w 0 2 ) ,
W ( ρ 1 , ρ 2 , z ) = ( k 2 π z ) 2 W 0 ( s 1 , s 2 , z = 0 ) exp { i k 2 z [ ( s 1 ρ 1 ) 2 ( s 2 ρ 2 ) 2 ] } × exp [ φ ( ρ 1 , s 1 ) + φ ( ρ 2 , s 2 ) ] d s 1 d s 2 ,
exp [ φ ( ρ 1 , s 1 ) + φ ( ρ 2 , s 2 ) ] exp [ ( s 1 s 2 ) 2 + ( ρ 1 ρ 2 ) 2 + ( s 1 s 2 ) ( ρ 1 ρ 2 ) ρ 0 2 ] ,
W ( ρ 1 , ρ 2 , z ) = 4 B w 0 6 exp [ ( ρ 1 x ρ 2 x ) 2 + ( ρ 1 y ρ 2 y ) 2 ρ 0 2 i k 2 z ( ρ 1 x 2 + ρ 1 y 2 ρ 2 x 2 ρ 2 y 2 ) ] ,
B = k 2 4 C π z 2 exp ( D 2 + E 2 C ) t = 0 6 F t G t ,
C = 1 w 0 2 + i k 2 z + 1 ρ 0 2 ,
D = i k ρ 1 y 2 z ρ 1 y ρ 2 y 2 ρ 0 2 ,
E = i k ρ 1 x 2 z ρ 1 x ρ 2 x 2 ρ 0 2 ,
F 0 = a E R 1 C + R 2 ( a E 3 C 3 + 3 a E 2 C 2 a 2 2 C a 2 E 2 C 2 + a J ) + a 2 ( 2 H 4 w 0 2 H 2 ) 2 C 2 ρ 0 4 a 2 D ( 2 H 3 + w 0 2 H 1 ) C 2 ρ 0 2 ,
F 1 = E C ( R 1 w 0 2 D H 1 C 2 ρ 0 2 ) + a R 1 C ρ 0 2 a ( 2 H 4 w 0 2 H 2 ) 2 C 2 ρ 0 4 2 a D ( H 3 w 0 2 H 1 ) C 2 ρ 0 2 + R 2 ( 3 a E 2 C 3 ρ 0 2 + 3 a 2 C 2 ρ 0 2 + a 2 C + a E 2 C 2 E 3 C 3 3 E 2 C 2 2 a 2 E C 2 ρ 0 2 J ) ,
F 2 = R 2 ( 3 a E C 3 ρ 0 4 + 2 a E C 2 ρ 0 2 3 E 2 C 3 ρ 0 2 3 2 C 2 ρ 0 2 a 2 C 2 ρ 0 4 ) 1 C ρ 0 2 ( R 1 + w 0 2 D H 1 C 2 ρ 0 2 ) + a H 0 ( a 2 C + a E 2 C 2 E 3 C 3 3 E 2 C 2 ) a E R 3 C a H 0 J + a 2 H 2 C 2 ρ 0 4 + 2 a 2 D H 1 C 2 ρ 0 2 ,
F 3 = R 2 ( a C 3 ρ 0 6 + a C 2 ρ 0 4 3 E C 3 ρ 0 4 ) 2 a D H 1 C 2 ρ 0 2 + 2 a 2 E H 0 C 2 ρ 0 2 + R 3 ( E C a C ρ 0 2 ) a H 0 ( 3 E 3 C 3 ρ 0 2 + 3 2 C 2 ρ 0 2 + 1 2 C + E 2 C 2 ) + H 0 ( E 3 C 3 + 3 E 2 C 2 + J ) a H 2 C 2 ρ 0 4 ,
F 4 = R 2 C 3 ρ 0 6 3 a E H 0 C 3 ρ 0 4 + H 0 ( 3 E 2 C 3 ρ 0 2 + 3 2 C 2 ρ 0 2 ) + a 2 H 0 C 2 ρ 0 4 2 a E H 0 C 2 ρ 0 2 + R 3 C ρ 0 2 ,
F 5 = a H 0 C 3 ρ 0 6 + 3 E H 0 C 3 ρ 0 4 a H 0 C 2 ρ 0 4 ,
F 6 = H 0 C 3 ρ 0 6 ,
G t = t ! exp ( Q 1 2 P ) π P ( Q 1 P ) t × K = 0 E [ t 2 ] 1 ( t 2 K ) ! K ! ( P 4 Q 1 2 ) K , ( t = 0 1 6 )
H l = l ! exp ( Q 2 2 P ) π P ( Q 2 P ) l × K = 0 E [ l 2 ] 1 ( l 2 K ) ! K ! ( P 4 Q 2 2 ) K , ( l = 0 1 4 )
I = w 0 2 2 + 1 2 C + D 2 C 2 ,
J = a w 0 2 2 a 2 C + a D 2 C 2 ,
P = 1 w 0 2 i k 2 z + 1 ρ 0 2 1 C ρ 0 4 ,
Q 1 = i k ρ 2 y 2 z + ρ 1 y ρ 2 y 2 ρ 0 2 + D C ρ 0 2 ,
Q 2 = i k ρ 2 x 2 z + ρ 1 x ρ 2 x 2 ρ 0 2 + E C ρ 0 2 ,
R 1 = w 0 2 H 0 I 2 H 4 C 2 ρ 0 4 2 D H 3 C 2 ρ 0 2 + w 0 2 H 2 2 C 2 ρ 0 4 H 2 I ,
R 2 = w 0 2 H 0 2 H 2 ,
R 3 = H 2 C 2 ρ 0 4 + 2 D H 1 C 2 ρ 0 2 + H 0 I .
I ( ρ , z ) = W ( ρ 1 , ρ 2 , z ) .
μ ( ρ 1 , ρ 2 , z ) = W ( ρ 1 , ρ 2 , z ) [ I ( ρ 1 , z ) I ( ρ 2 , z ) ] 1 / 2 .
Im [ μ ( ρ 1 , ρ 2 , z ) ] = 0 ,
R e [ μ ( ρ 1 , ρ 2 , z ) ] = 0 ,

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