Classification of coherent vortices creation and distance of topological charge conservation in non-Kolmogorov atmospheric turbulence

Jinhong Li; Jun Zeng; Meiling Duan

doi:10.1364/OE.23.011556

1. Introduction

Considerable interest has been exhibited recently in the vortex beams (also known as optical beams with a phase singularity) due to their theoretical importance in the basic science [1, 2] and potential applications in optical tweezers [3, 4], optical communications [5, 6], quantum state manipulation [7], and so on [8, 9]. As pointed out by Gbur and Tyson, the topological charge of the vortex beams propagation through atmospheric turbulence is a robust quantity that could be used as an information carrier in optical communications [10]. Zhu et al. reported the propagation of Bessel-Gaussian beams with optical vortices in atmospheric turbulence and found that the phase singularity rapidly fades away during propagation in atmospheric turbulence [11]. Dipankar et al. have investigated the trajectory of an optical vortex in atmospheric turbulence using numerical simulations in 2009 [12]. A possible method for measuring atmospheric turbulence strength by vortex beam has been suggested by Gu and Gbur [13]. The propagation of Gaussian Schell-model (GSM) vortex beams was dealt with numerically by Li and Lü who found that the beam-width spreading of GSM vortex beams is less influenced by atmospheric turbulence than that of GSM non-vortex beams [14]. The average intensity, the irradiance pattern, the degree of the polarization and scintillation index of vortex beam in atmospheric turbulence have been researched in [15–18]. However, it is worth noting that the classification of coherent vortices creation and the distance of topological charge conservation have been not studied in all the previous works.

In this paper, taking the partially coherent sinh-Gaussian (ShG) vortex beams as an example of partially coherent vortex beams, the classification of coherent vortices creation in non-Kolmogorov atmospheric turbulence will be studied, as well as, the influence of the beams parameters and non-Kolmogorov atmospheric turbulence parameters on the distance of topological charge conservation has been stressed.

2. Theoretical model

The field of an optical vortex beam at the z = 0 plane in the rectangular coordinate system is written as [19]

E (s, z = 0) = u (s) {[s_{x} + i s g n (m) s_{y}]}^{| m |},

where s ≡ (s_x, s_y) is the two-dimensional position vector at the source plane z = 0, sgn(m) denotes the sign function defined as

sgn (m) = {\begin{matrix} 1, \\ 0, \\ - 1, \end{matrix} \begin{matrix}  \end{matrix} \begin{matrix} m > 0, \\ m = 0, \\ m < 0, \end{matrix}

m is the topological charge, u(s) denotes the profile of the background beam envelope, which is assumed to be sinh-Gaussian (ShG) one [20], thus

E (s, 0) = \sin h [Ω_{0} (s_{x} + s_{y})] \exp (- \frac{s_{x}^{2} + s_{y}^{2}}{w_{0}^{2}}) {[s_{x} + i sgn (m) s_{y}]}^{| m |},

where Ω₀ is the parameter associated with the Sh-part, where Ω₀≠0 because U(ρ, z = 0) = 0 if Ω₀ = 0. w₀ denotes the waist width of the Gaussian part, in the following we take m = ± 1.

By introducing a Schell-correlator [21], the cross-spectral density function of the partially coherent ShG vortex beams at the source plane z = 0 is expressed as

\begin{array}{l} W_{0} (s_{1}, s_{2}, 0) = [s_{1 x} s_{2 x} + s_{1 y} s_{2 y} \pm i (s_{1 x} s_{2 y} - s_{2 x} s_{1 y})] \exp (- \frac{s_{1}^{2} + s_{2}^{2}}{w_{0}^{2}}) \\ \times \sin h [Ω_{0} (s_{1 x} + s_{1 y})] \sin h [Ω_{0} (s_{2 x} + s_{2 y})] e x p (\frac{- {(s_{1} - s_{2})}^{2}}{2 σ_{0}^{2}}), \end{array}

where s_i ≡ (s_ix, s_iy) (i = 1, 2), * denotes the complex conjugate, σ₀ denotes the spatial correlation length, and “±” corresponds to m = ± 1.

In accordance with the extended Huygens-Fresnel principle [22], the cross-spectral density function of partially coherent ShG vortex beams propagating through non-Kolmogorov atmospheric turbulence are given by

\begin{array}{l} W (ρ_{1}, ρ_{2}, z) = {(\frac{k}{2 z π})}^{2} \iint d^{2} s_{1} \iint d^{2} s_{2} W_{0} (s_{1}, s_{2}, 0) \exp {- \frac{i k}{2 z} [{(ρ_{1} - s_{1})}^{2} - {(ρ_{2} - s_{2})}^{2}]} \\ \times 〈 \exp [ψ^{*} (s_{1}, ρ_{1}) + ψ (s_{2}, ρ_{2})] 〉, \end{array}

where z is the propagation distance, k is the wave number related to the wave length λ by k = 2π/λ, ρ_i ≡ (ρ_ix, ρ_iy) is the position vector in the z plane. ψ(s, ρ) is the random part of the complex place of a spherical wave due to the turbulence, and can be expressed as [23]

\begin{array}{l} 〈 \exp [ψ^{*} (s_{1}, ρ_{1}) + ψ (s_{2}, ρ_{2})] 〉 \\ = \exp {- 4 π^{2} k^{2} z \int_{0}^{1} \int_{0}^{\infty} d κ d ξ κ Φ_{n} (κ, α) [1 - J_{0} (k | (1 - ξ) (ρ_{1} - ρ_{2}) + ξ (s_{1} - s_{2}) |)]} \\ = \exp {- T (α, z) [{(ρ_{1} - ρ_{2})}^{2} + (ρ_{1} - ρ_{2}) (s_{1} - s_{2}) + {(s_{1} - s_{2})}^{2}]}, \end{array}

with J₀(●) being the Bessel function of the first kind and zero order, T is the strength of atmospheric turbulence

T (α, z) = \frac{π^{2} k^{2} z}{3} \int_{0}^{\infty} κ^{3} Φ_{n} (κ, α) d κ,

where

Φ_{n} (κ, α)

is the spatial power spectrum of the refractive-index fluctuations of the turbulent medium, having, for the non-Kolmogorov case, the form [24, 25]

Φ_{n} (κ) = A (α) {\tilde{C}}_{n}^{2} \frac{\exp [- (κ^{2} / κ_{m}^{2})]}{{(κ^{2} + κ_{0}^{2})}^{α / 2}}, (0 \leq κ \leq \infty, 3 < α < 4),

A (α) = Γ (α - 1) \cos (α π / 2) / (4 π^{2}),

κ_{0} = 2 π / L_{0},

κ_{m} = {Γ [(5 - α) / 2] A (α) 2 π / 3}^{1 / (α - 5)} / l_{0},

where L₀ and l₀ are the outer and inner scales of atmospheric turbulence, respectively, and

Γ (•)

is the Gamma function, α is the generalized exponent,

{\tilde{C}}_{n}^{2}

is the generalized structure constant with units

m^{3 - α}

[24–26]. On substituting Eqs. (8)-(11) into Eq. (7), we obtain

\begin{array}{l} T (α, z) = \frac{π^{2} k^{2} z}{6 (α - 2)} A (α) {\tilde{C}}_{n}^{2} {\exp (\frac{κ_{0}^{2}}{κ_{m}^{2}}) κ_{m}^{(2 - α)} \\ \times [(α - 2) κ_{m}^{2} + 2 κ_{0}^{2}] Γ (2 - \frac{α}{2}, \frac{κ_{0}^{2}}{κ_{m}^{2}}) - 2 κ_{0}^{4 - α}}, \end{array}

From Eq. (12) we can see that the strength of the turbulence T depends on the generalized structure constant

{\tilde{C}}_{n}^{2}

, the generalized exponent α, the outer and inner scales of turbulence L₀ and l₀.

Introducing two variables of integration μ = (s₁ + s₂)/2, ν = s₁- s₂, and recalling integral formulas [27]

\int \exp (- p x^{2} + 2 q x) d x = \sqrt{\frac{π}{p}} \exp (\frac{q^{2}}{p}),

\int x \cdot \exp (- p x^{2} + 2 q x) d x = \sqrt{\frac{π}{p}} (\frac{q}{p}) \exp (\frac{q^{2}}{p}),

\int x^{2} \cdot \exp (- p x^{2} + 2 q x) d x = \frac{1}{2 p} \sqrt{\frac{π}{p}} (1 + \frac{2 q^{2}}{p}) \exp (\frac{q^{2}}{p}),

on substituting Eqs. (4) and (6) into Eq. (5), (10) the cross-spectral density function of partially coherent ShG vortex beams propagating through non-Kolmogorov atmospheric turbulence is written as

\begin{array}{l} W (ρ_{1}, ρ_{2}, z) = \frac{k^{2}}{16 A C z^{2}} \exp [- \frac{i k}{2 z} (ρ_{1}^{2} - ρ_{2}^{2})] \exp [- T (α, z) {(ρ_{1} - ρ_{2})}^{2}] \\ \times (M_{1} + M_{2} - M_{3} - M_{4}), \end{array}

where

M_{1} = (\frac{E_{x}^{2} + E_{y}^{2}}{C^{2}} + \frac{1}{C} - \frac{I_{x}^{2} + I_{y}^{2}}{4 H^{2}} - \frac{1}{4 H} \pm i \frac{I_{x} E_{y} - E_{x} I_{y}}{C H}) e x p (\frac{B_{x}^{2} + B_{y}^{2}}{4 A} + \frac{E_{x}^{2} + E_{y}^{2}}{C}),

M_{3} = (\frac{G_{x}^{2} + G_{y}^{2}}{C^{2}} + \frac{1}{C} - \frac{J_{x}^{2} + J_{y}^{2}}{4 H^{2}} - \frac{1}{4 H} \pm i \frac{J_{x} G_{y} - G_{x} J_{y}}{C H}) e x p (\frac{F_{x}^{2} + F_{y}^{2}}{4 A} + \frac{G_{x}^{2} + G_{y}^{2}}{C}),

A = \frac{1}{2 w_{0}^{2}} + \frac{1}{2 σ_{0}^{2}} + T (α, z),

B_{x} = \frac{i k}{2 z} (ρ_{1 x} + ρ_{2 x}) - T (α, z) (ρ_{1 x} - ρ_{2 x}),

C = \frac{2}{w_{0}^{2}} + \frac{k^{2}}{4 A z^{2}},

D_{x} = \frac{i k}{z} (ρ_{1 x} - ρ_{2 x}) + 2 Ω_{0},

E_{x} = \frac{1}{2} (D_{x} - \frac{i k}{2 A z} B_{x}),

F_{x} = B_{x} + Ω_{0},

G_{x} = \frac{1}{2} [\frac{i k}{z} (ρ_{1 x} - ρ_{2 x}) - \frac{i k F_{x}}{2 A z}],

H = A + \frac{k^{2} w_{0}^{2}}{8 z^{2}},

I_{x} = \frac{1}{2} (B_{x} - \frac{i k w_{0}^{2} D_{x}}{4 z}),

J_{x} = \frac{1}{2} [F_{x} + \frac{k^{2} w_{0}^{2} (ρ_{1 x} - ρ_{2 x})}{4 z^{2}}] .

Due to the symmetry, B_y, D_y, E_y, I_y, F_y, G_y and J_y can be obtained by replacement of ρ₁_x and ρ₂_x in B_x, D_x, E_x, I_x, F_x, G_x and J_x with ρ₁_y and ρ₂_y. M₂ and M₄ can be obtained by replacement of Ω₀ M₁ and M₃ with -Ω₀. “±” in M₁ and M₃ corresponds to m = ± 1, respectively.

For ${\tilde{C}}_{n}^{2} = 0$ , Eq. (16) reduces to the cross-specral density function of coherent ShG vortex beams in free space, and it is expressed as

W_{f r e e} (ρ_{1}, ρ_{2}, z) = \frac{k^{2}}{16 A_{0} C_{0} z^{2}} e x p [- \frac{i k}{2 z} (ρ_{1}^{2} - ρ_{2}^{2})] (M_{10} + M_{20} - M_{30} - M_{40}),

by letting

{\tilde{C}}_{n}^{2} = 0

in A, C, M₁, M₂, M₃, M₄, we can obtain A₀, C₀, M₁₀, M₂₀, M₃₀, M₄₀, respectively.

The spectral degree of coherence is defined as [28]

μ (ρ_{1}, ρ_{2}, z) = \frac{W (ρ_{1}, ρ_{2}, z)}{{[I (ρ_{1}, z) I (ρ_{2}, z)]}^{1 / 2}},

where I(ρ_i, z) = W(ρ_i, ρ_i, z) (i = 1, 2) stands for the spectral intensity. The position of coherence vortices is determined by

R e [μ (ρ_{1}, ρ_{2}, z)] = 0,

I m [μ (ρ_{1}, ρ_{2}, z)] = 0,

where Re and Im denote the real and imaginary parts of

μ (ρ_{1}, ρ_{2}, z)

, respectively. The topological charge and its sign of coherent vortices are determined by the vorticity of phase contours around singularities [29], namely, when varying phase in counterclockwise direction and clockwise direction, the sign of the topological charge corresponds to “+” and “-”, respectively, and phase changes 2mπ, the corresponding topological charge is m.

3. Classification of coherent vortices creation

Figure 1 represents the position of coherent vortices of partially coherent ShG vortex beams with m = + 1 propagating through (a)-(b) free space and (c)-(d) non-Kolmogorov atmospheric turbulence, and illustrations give the contour lines of phase which correspond to the coherent vortices. The calculation parameters are λ = 1.06μm, w₀ = 3cm, σ₀ = 4cm, Ω₀ = 30, ${\tilde{C}}_{n}^{2}$ = 10⁻¹⁴m^3-α, α = 3.2, l₀ = 1cm, L₀ = 1m. From Figs. 1(a) and 1(b) we note that at the beginning there is only one coherent vortex (being marked as no.1 in the Fig. 1) with topological charge is + 1. With the increment of the propagation distance, a new coherent vortex (no.2) with m = −1 is present at z = 1.25km, so the total topological charge is “0”. Namely, the topological charge has no longer conserved and the distance z_c of topological charge conservation is 1.25km. Later a pair of coherent vortices appear at z = 3km whose topological charge is + 1 and −1 (no.3 and no.4). Among the four coherent vortices, no.1 is inherent coherent vortex of the vortex beam, but no.2, no.3 and no.4 are created by the partially coherent ShG vortex beams when propagating through free space. From Figs. 1(c) and 1(d) we see that there also exsit no.1, no.2, no.3 and no.4 coherent vortices for partially coherent ShG vortex beams in atmospheric turbulence, which is similar to Figs. 1(a) and 1(b). However, the distance z_c of topological charge conservation is 0.5km in Figs. 1(c) and 1(d). In addition, two pairs of coherent vortices marked as no.5, no.6, no.7 and no.8 appear at z = 1km and z = 2.25km, respectively, and they will annihilate subsequently. Nos.5-8 coherent vortices are created in pairs for partially coherent ShG vortex beams in atmospheric turbulence, and their creation are different from that of nos. 1-4 obviously. It is indicated that nos.5-8 coherent vortices are created by the non-Kolmogorov atmospheric turbulence inducing the vortex beams.

Fig. 1 Position of coherent vortices of partially coherent ShG vortex beams propagating through (a)-(b) free space and (c)-(d) non-Kolmogorov atmospheric turbulence, ‘●’ m = + 1 and ‘○’m = −1. The illustrations give the contour lines of phase which correspond to the coherent vortices.

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As shown in Fig. 1, the coherent vortices are grouped into three classes according to the creation: the first is their inherent coherent vortices of the vortex beam, as suggested as no.1 coherent vortex; the second is created by vortex beams when propagating in free space, as shown as nos.2-4 coherent vortices; the third is created by the non-Kolmogorov atmospheric turbulence inducing the vortex beam, namely, the coherent vortex is the outcome of the combination of vortex beam with the non-Kolmogorov atmospheric turbulence, as shown as nos.5-8 coherent vortices in Figs. 1 (c) and 1(d).

4. Distance of topological charge conservation

The position of coherent vortices of partially coherent ShG vortex beams with m = + 1 propagating through non-Kolmogorov atmospheric turbulence are shown in Fig. 2 for the general structure constant ${\tilde{C}}_{n}^{2}$ = 10⁻¹³ m^3-α, and the other calculation parameters are the same as in Fig. 1. It is seen that the coherent vortex of no.2 appears at the propagation distance z = 0.25km, i.e., the distance z_c of topological charge conservation is 0.25km. A comparison between Figs. 1 and 2 shows that the larger the general structure constant ${\tilde{C}}_{n}^{2}$ is, the strongerthe atmospheric turbulence is, the smaller the distance z_c of the topological charge conservation is, e.g., for ${\tilde{C}}_{n}^{2}$ = 0, 10⁻¹⁴m^-2/3, and 10⁻¹³m^-2/3, z_c = 1.25, 0.5 and 0.25 km, respectively.

Fig. 2 The position of coherent vortices of partially coherent ShG vortex beams propagating through non-Kolmogorov atmospheric turbulence for ${\tilde{C}}_{n}^{2}$ = 10⁻¹³m^-2/3.

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Figure 3 represents the distance z_c of topological charge conservation versus (a) the general structure constant ${\tilde{C}}_{n}^{2}$ , (b) the general exponent α, (c) the inner scale of turbulence l₀ and (d) the outer scale of turbulence L₀ for partially coherent ShG vortex beams with m = + 1 in non-Kolmogorov atmospheric turbulence, and the other calculation parameters are the same as in Fig. 1. As can be seen, the distance of topological charge conservation depends on the general structure constant ${\tilde{C}}_{n}^{2}$ , the general exponent α and the inner scale of turbulence l₀. With the decrement of the generalized structure constant ${\tilde{C}}_{n}^{2}$ , as well as the increment of the generalized exponent α and the inner scale of turbulence l₀, the distance z_c of topological charge conservation will increase for partially coherent ShG vortex beams in non-Kolmogorov atmospheric turbulence. Figure 3 (d) indicates that the outer scale of turbulence L₀ has no effect on the distance z_c of topological charge conservation.

Fig. 3 The distance z_c of topological charge conservation versus (a) the general structure constant ${\tilde{C}}_{n}^{2}$ , (b) the general exponenta α, (c) the inner scale of turbulence l₀ and (d) the outer scale of turbulence L₀.

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The distance z_c of topological charge conservation versus (a) the spatial correlation length σ₀, (b) the waist width w₀ and (c) the Sh-part parameters Ω₀ are depicted in Fig. 4 for partially coherent ShG vortex beams with m = + 1 in non-Kolmogorov atmospheric turbulence, and the other calculation parameters are the same as in Fig. 1. From Fig. 4 it is seen that the distance z_c of topological charge conservation increases with increasing the spatial correlation length σ₀ or decreasing the waist width w₀, i.e., the bigger the spatial correlation length σ₀ is, the smaller the waist width w₀ is, the bigger the distance z_c of topological charge conservation is. Besides, the Sh-part parameters Ω₀ has no effect on the distance z_c of topological charge conservation.

Fig. 4 The distance z_c of topological charge conservation versus (a) the spatial correlation length σ₀, (b) the waist width w₀, and (c) the Sh-part parameters Ω₀.

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5. Conclusion

In this paper, based on the non-Kolmogorov spectrum, using the extended Huygens-Fresnel principle, the analytical expressions for the cross-spectral density function of partially coherent ShG vortex beams propagating through free space and non-Kolmogorov atmospheric turbulence have been derived, and used to study the classification of coherent vortices creation and the influence of non-Kolmogorov atmospheric turbulence parameters ( ${\tilde{C}}_{n}^{2}$ , α, l₀ and L₀) and beams parameters (σ₀, w₀ and Ω₀) on distance of topological charge conservation. According to the creation, the coherent vortices are grouped into three classes: the first is the inherent coherent vortices of the vortex beams, the second is created by the vortex beams when propagating through free space, and the third is created by the non-Kolmogorov atmospheric turbulence inducing the vortex beams. With the decrement of the generalized structure constant ${\tilde{C}}_{n}^{2}$ and the waist width w₀, as well as the increment of the generalized exponent α, the inner scale of turbulence l₀ and the spatial correlation length σ₀, the distance z_c of topological charge conservation will increase for partially coherent ShG vortex beams in non-Kolmogorov atmospheric turbulence. Besides, the outer scale of turbulence L₀ and the Sh-part parameters Ω₀ have no effect on the distance z_c of topological charge conservation. These results have potential applications in optical vortex communication systems.

Acknowledgment

This work was supported by the National Natural Science Foundation of China (Grant nos. 61405136 and 61178067), and the Natural Science Foundation for Young Scientists of Shanxi Province (Grant nos. 2012021016 and 2013021010-4).

References and links

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

2. B. J. McMorran, A. Agrawal, I. M. Anderson, A. A. Herzing, H. J. Lezec, J. J. McClelland, and J. Unguris, “Electron vortex beams with high quanta of orbital angular momentum,” Science 331(6014), 192–195 (2011). [CrossRef] [PubMed]

3. A. T. O’Neil, I. MacVicar, L. Allen, and M. J. Padgett, “Intrinsic and extrinsic nature of the orbital angular momentum of a light beam,” Phys. Rev. Lett. 88(5), 053601 (2002). [CrossRef] [PubMed]

4. N. B. Simpson, K. Dholakia, L. Allen, and M. J. Padgett, “Mechanical equivalence of spin and orbital angular momentum of light: an optical spanner,” Opt. Lett. 22(1), 52–54 (1997). [CrossRef] [PubMed]

5. B. Thidé, H. Then, J. Sjöholm, K. Palmer, J. Bergman, T. D. Carozzi, Y. N. Istomin, N. H. Ibragimov, and R. Khamitova, “Utilization of photon orbital angular momentum in the low-frequency radio domain,” Phys. Rev. Lett. 99(8), 087701 (2007). [CrossRef] [PubMed]

6. C. Paterson, “Atmospheric turbulence and orbital angular momentum of single photons for optical communication,” Phys. Rev. Lett. 94(15), 153901 (2005). [CrossRef] [PubMed]

7. M. F. Andersen, C. Ryu, P. Cladé, V. Natarajan, A. Vaziri, K. Helmerson, and W. D. Phillips, “Quantized rotation of atoms from photons with orbital angular momentum,” Phys. Rev. Lett. 97(17), 170406 (2006). [CrossRef] [PubMed]

8. J. Verbeeck, H. Tian, and P. Schattschneider, “Production and application of electron vortex beams,” Nature 467(7313), 301–304 (2010). [CrossRef] [PubMed]

9. Y. Yang, Y. Dong, C. Zhao, and Y. Cai, “Generation and propagation of an anomalous vortex beam,” Opt. Lett. 38(24), 5418–5421 (2013). [CrossRef] [PubMed]

10. G. Gbur and R. K. Tyson, “Vortex beam propagation through atmospheric turbulence and topological charge conservation,” J. Opt. Soc. Am. A 25(1), 225–230 (2008). [CrossRef] [PubMed]

11. K. Zhu, G. Zhou, X. Li, X. Zheng, and H. Tang, “Propagation of Bessel-Gaussian beams with optical vortices in turbulent atmosphere,” Opt. Express 16(26), 21315–21320 (2008). [CrossRef] [PubMed]

12. A. Dipankar, R. Marchiano, and P. Sagaut, “Trajectory of an optical vortex in atmospheric turbulence,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 80(4), 046609 (2009). [CrossRef] [PubMed]

13. Y. Gu and G. Gbur, “Measurement of atmospheric turbulence strength by vortex beam,” Opt. Commun. 283(7), 1209–1212 (2010). [CrossRef]

14. J. Li and B. Lü, “Propagation of Gaussian Schell-model vortex beams through atmospheric turbulence and evolution of coherent vortices,” J. Opt. A 11(4), 045710 (2009). [CrossRef]

15. Z. Chen, C. Li, P. Ding, J. Pu, and D. Zhao, “Experimental investigation on the scintillation index of vortex beams propagating in simulated atmospheric turbulence,” Appl. Phys. B 107(2), 469–472 (2012). [CrossRef]

16. G. Zhou, Y. Cai, and X. Chu, “Propagation of a partially coherent hollow vortex Gaussian beam through a paraxial ABCD optical system in turbulent atmosphere,” Opt. Express 20(9), 9897–9910 (2012). [PubMed]

17. X. Liu and J. Pu, “Investigation on the scintillation reduction of elliptical vortex beams propagating in atmospheric turbulence,” Opt. Express 19(27), 26444–26450 (2011). [CrossRef] [PubMed]

18. M. Bastiaans, “Wigner distribution function and its application to first-order optics,” J. Opt. Soc. Am. 69(12), 1710–1716 (1979). [CrossRef]

19. F. Flossmann, U. T. Schwarz, and M. Maier, “Propagation dynamics of optical vortices in Laguerre-Gaussian beams,” Opt. Commun. 250(4-6), 218–230 (2005). [CrossRef]

20. J. Li, A. Yang, and B. Lü, “Comparative study of the beam-width spreading of partially coherent Hermite-sinh-Gaussian beams in atmospheric turbulence,” J. Opt. Soc. Am. A 25(11), 2670–2679 (2008). [PubMed]

21. M. Zahid and M. S. Zubairy, “Directionality of partially coherent Bessel-Gauss beams,” Opt. Commun. 70(5), 361–364 (1989). [CrossRef]

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

23. 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]

24. I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Scintillation index of optical plane wave propagating through non-Kolmogorov moderate-strong turbulence,” Proc. SPIE 6747, 67470B (2007). [CrossRef]

25. I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Angle of arrival fluctuations for free space laser beam propagation through non Kolmogorov turbulence,” Proc. SPIE 6551, 65510E (2007). [CrossRef]

26. O. Korotkova and E. Shchepakina, “Tuning the spectral composition of random beams propagating in free space and in a turbulent atmosphere,” J. Opt. 15(7), 075714 (2013). [CrossRef]

27. I. S. Gradshteyn, I. M. Ryzhik, A. Jeffrey, and D. Zwillinger, Table of Integrals, Series, and Products (Elsevier, 2007).

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

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

Classification of coherent vortices creation and distance of topological charge conservation in non-Kolmogorov atmospheric turbulence

Abstract

1. Introduction

2. Theoretical model

3. Classification of coherent vortices creation

4. Distance of topological charge conservation

5. Conclusion

Acknowledgment

References and links

Cited By

Figures (4)

Equations (32)

Optics Express