Radial phased-locked partially coherent flat-topped vortex beam array in non-Kolmogorov medium

Huilong Liu; Yanfei Lü; Jing Xia; Dong Chen; Wei He; Xiaoyun Pu

doi:10.1364/OE.24.019695

1. Introduction

Recently, the propagation of partially coherent beams propagating through atmospheric turbulence and non-Kolmogorov turbulence has been extensively investigated because of their wide applications in free-space optical communications, remote sensing and tracking [1–12]. It can be found that the propagation properties of partially coherent vortex beam is less affected by atmospheric turbulence than partially coherent vortex-free, and the topological charge of the vortex beam carrying the orbital angular momentum can be used as the information carrier in optical communication [13, 14]. Accordingly, all kinds of vortex beams have received much attention, such as Laguerre-Gaussian correlated Schell-model vortex beam [15], multi-Gaussian Schell-model vortex beam [16], terahertz vortex beam [17], anomalous vortex beam [18], Lorentz-Gauss vortex beam [19], Airy–Gaussian vortex beam [20], circularly polarized vortex beam [21], Hermite–Gaussian vortex beam [22], flat-topped vortex beam [23], four-petal Gaussian vortex beam [24] and so on.

On the other hand, the output of low beam power of a single laser beam limits its applications. Accordingly, coherent beam arrays have been proposed to provide the efficient high-power output [25–27]. Naturally, various beam arrays have been studied, including rectangular Lorentz beam array [28], radial phased-locked Lorentz beam array [29], radial Gaussian beam array [30], radial Gaussian Schell-model array beams [31], hollow beam array [32], flat-topped beam array [33]. Until now, less work has been reported that a phased-locked partially coherent flat-topped vortex beam array propagates in non-Kolmogorov medium. In this paper, taking the atmospheric turbulence as a typical example of non-Kolmogorov medium, we study the properties of spreading of a radial phased-locked partially coherent flat-topped vortex beam array propagating through non-Kolmogorov medium. It is believed that the results will benefit to study propagation properties of partially coherent flat-topped vortex beam array.

2. Propagation theory

The optical field of the m-th flat-topped vortex beam element in the source plane z = 0 takes the form as [29,32,34]

\begin{matrix} E_{m} ({ρ^{'}}_{x}, {ρ^{'}}_{y}, 0) = {[{ρ^{'}}_{x} - a_{m x} + i sgn (l) ({ρ^{'}}_{y} - a_{m y})]}^{| l |} \\ \sum_{n = 1}^{N} \frac{{(- 1)}^{n - 1}}{N} (\begin{matrix} N \\ n \end{matrix}) \exp (- \frac{n {({ρ^{'}}_{x} - a_{m x})}^{2}}{w^{2}} - \frac{n {({ρ^{'}}_{y} - a_{m y})}^{2}}{w^{}}) \exp (i φ_{m}) . \end{matrix}

where a_mx = rcosφ_m and a_my = rsinφ_m are the center of the m-th beamlet located at the source plane, r is the radius and φ_m is the initial phase of the m-th beamlet. φ_m = mφ₀ = 2mπ/M, m = 1,2,3,4…M. M is beamlet number. N denotes the beam order and w is the waist width in Gaussian part, and l is the topological charge, sgn(l) specifies the sign function(l>0, sgn(l) = 1; l = 0, sgn(l) = 0; l<0,sgn(l) = −1). In the following, we restricted ourselves to the case of l = 1.Intensity distribution of a radial phased-locked flat-topped vortex beam array in the source plane is shown in Fig. 1.

Fig. 1 Intensity distribution of a radial phased-locked flat-topped beam vortex array in the source plane.

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Introducing a Schell model correlator, the cross-spectral density of the corresponding m-th partially coherent flat-topped vortex beam element at the z = 0 plane can be written as [34–37]

W_{m}^{(0)} ({ρ^{'}}_{1}, {ρ^{'}}_{2}, 0) = \sqrt{I_{m} ({ρ^{'}}_{1 x}, {ρ^{'}}_{1 y}, 0) I_{m} ({ρ^{'}}_{2 x}, {ρ^{'}}_{2 y}, 0)} \exp [- \frac{(n_{1} + n_{2}) {({ρ^{'}}_{1} - {ρ^{'}}_{2})}^{2}}{4 σ_{0}^{2}}] .

where I_m(ρ'_ix, ρ'_iy,0) (i = 1,2) is intensity of m-th flat-topped vortex beam element, andρ'₁ = (ρ'₁_x, ρ'₁_y),ρ'₂ = (ρ'₂_x, ρ'₂_y) are positions of two points at the z = 0 plane. σ₀ denotes the correlation length.The cross-spectral density of a radial phased-locked partially coherent flat-topped vortex beam array composed of M beamlets in the initial plane z = 0 is given by

W^{(0)} ({ρ^{'}}_{1}, {ρ^{'}}_{2}, 0) = \sum_{m = 1}^{M} W_{m}^{(0)} ({ρ^{'}}_{1}, {ρ^{'}}_{2}, 0) .

Substituting Eqs. (1) and (2) into Eq. (3), we can obtain

\begin{matrix} W^{(0)} ({ρ^{'}}_{1}, {ρ^{'}}_{2}, 0) = {{ρ^{'}}_{1 x} {ρ^{'}}_{2 x} + {ρ^{'}}_{1 y} {ρ^{'}}_{2 y} - a_{m x} ({ρ^{'}}_{1 x} + {ρ^{'}}_{2 x}) - a_{m y} ({ρ^{'}}_{1 y} + {ρ^{'}}_{2 y}) \\ + i [{ρ^{'}}_{1 x} {ρ^{'}}_{2 y} - {ρ^{'}}_{2 x} {ρ^{'}}_{1 y} - a_{m y} ({ρ^{'}}_{1 x} - {ρ^{'}}_{2 x}) + a_{m x} ({ρ^{'}}_{1 y} - {ρ^{'}}_{2 y})] + a_{m x}^{2} + a_{m y}^{2}} \\ \times \sum_{m = 1}^{M} \sum_{n_{1} = 1}^{N} \sum_{n_{2} = 1}^{N} \frac{{(- 1)}^{n_{1} + n_{2} - 2}}{N^{2}} (\begin{matrix} N \\ n_{1} \end{matrix}) (\begin{matrix} N \\ n_{2} \end{matrix}) \exp [- \frac{n_{1}}{w^{2}} ({ρ^{'}}_{1 x}^{2} + {ρ^{'}}_{1 y}^{2}) - \frac{n_{2}}{w^{2}} ({ρ^{'}}_{2 x}^{2} + {ρ^{'}}_{2 y}^{2})] \\ \times \exp [\frac{2 a_{m x}}{w^{2}} (n_{1} {ρ^{'}}_{1 x} + n_{2} {ρ^{'}}_{2 x}) + \frac{2 a_{m y}}{w^{2}} (n_{1} {ρ^{'}}_{1 y} + n_{2} {ρ^{'}}_{2 y})] \exp (i φ_{m}) \\ \times \exp [- \frac{(a_{m x}^{2} + a_{m y}^{2}) (n_{1} + n_{2})}{w^{2}}] \exp [- \frac{(n_{1} + n_{2})}{4 σ_{0}^{2}} {({ρ^{'}}_{1} - {ρ^{'}}_{2})}^{2}] . \end{matrix}

According to the extended Huygens–Fresnel principle, which describes the interaction of waves with linear random media, the cross-spectral density function at the z plane (z>0) of a radial phased-locked partially coherent flat-topped vortex beam array through non-Kolmogorov medium is written as

\begin{matrix} W (ρ_{1}, ρ_{2}, z) = \frac{k^{2}}{4 π^{2} z^{2}} \int_{- \infty}^{+ \infty} \int_{- \infty}^{+ \infty} \int_{- \infty}^{+ \infty} \int_{- \infty}^{+ \infty} W^{(0)} ({ρ^{'}}_{1}, {ρ^{'}}_{2}, 0) \\ \times \exp {- \frac{i k}{2 z} [{(ρ_{1} - {ρ^{'}}_{1})}^{2} - {(ρ_{2} - {ρ^{'}}_{2})}^{2}]} \\ \times 〈 \exp [ψ^{*} (ρ_{1}, {ρ^{'}}_{1}) + ψ (ρ_{2}, {ρ^{'}}_{2})] 〉 d^{2} {ρ^{'}}_{1} d^{2} {ρ^{'}}_{2} . \end{matrix}

where k = 2π/λ is the wave number with λ being the wavelength of the light, ψ denotes the complex phase perturbation due to the random medium. 〈…〉 stands for averaging over the ensemble of statistical realizations of the turbulent medium. The asterisk specifies the complex conjugate. ρ₁ = (x₁_, y₁),ρ₂ = (x₂_, y₂) denotes the two-dimensional position vector at the z>0 plane. The term in the sharp brackets in Eq. (5) can be read as [38]

\begin{matrix} 〈 \exp [ψ^{*} (ρ_{1}, {ρ^{'}}_{1}) + ψ (ρ_{2}, {ρ^{'}}_{2})] 〉 \\ = \exp {- 4 π^{2} k^{2} z \int_{0}^{1} \int_{0}^{+ \infty} d κ d ξ Φ_{n} (κ, α) [1 - J_{0} (κ | (1 - ξ) (ρ_{1} - ρ_{2}) + ξ ({ρ^{'}}_{1} - {ρ^{'}}_{2}) |)]} \\ = \exp {- \frac{π^{2} k^{2} z}{3} \int_{0}^{+ \infty} κ^{3} Φ_{n} (κ, α) d κ [{({ρ^{'}}_{1} - {ρ^{'}}_{2})}^{2} + ({ρ^{'}}_{1} - {ρ^{'}}_{2}) (ρ_{1} - ρ_{2}) + {(ρ_{1} - ρ_{2})}^{2}]} . \end{matrix}

where Φ_n(κ,α) is the one-dimensional spatial power spectrum of the refractive-index fluctuations of random medium, κ being spatial frequency. J₀(·) denotes the Bessel function of the first kind and zero order. We employ the power spectrum introduced by Toselli et al. [39]

Φ_{n} (κ, α) = \frac{A (α) {\tilde{C}}_{n}^{2}}{{(κ^{2} + κ_{0}^{2})}^{α / 2}} \exp (- \frac{κ^{2}}{κ_{m}^{2}}) .

where α denotes the generalized exponent factor of non-Kolmogorov turbulence, 3<α<4. 0≤κ<∞.

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

is the generalized structure constant with units m^3-^α.

The terms can be given by κ₀ = 2π/L₀ (L₀-outer scale of the atmospheric turbulence), κ_m = C(α)/l₀ (l₀-inner scale of the atmospheric turbulence) with C(α) = (2πA(α)Γ(2.5-α/2)/3)^1/(^α-⁵⁾ and A(α) = Γ(α-1)cos(απ/2)/(4π²). Γ(·) is the Gamma function. According to the assumption, the integral in Eq. (6) can be rewritten as

K = \frac{π^{2} k^{2} z}{3} \int_{0}^{+ \infty} κ^{3} Φ_{n} (κ, α) d κ = \frac{π^{2} k^{2} z A (α) {\tilde{C}}_{n}^{2}}{6 (α - 2)} [β \exp (\frac{κ_{0}^{2}}{κ_{m}^{2}}) Γ (\frac{4 - α}{2}, \frac{κ_{0}^{2}}{κ_{m}^{2}}) κ_{m}^{2 - α} - 2 κ_{0}^{4 - α}] .

where β = 2(κ₀²–κ_m²) + ακ_m², and Γ(·, ·) is the incomplete Gamma function.

We introduce two new variables of integration u = (ρ'₁ + ρ'₂)/2, v = ρ'₁-ρ'₂ and substitute Eqs. (4) and (6) into Eq. (5), Eq. (5) can be given by

\begin{matrix} W (ρ_{1}, ρ_{2}, z) = \frac{k^{2}}{4 π^{2} z^{2}} \exp [- \frac{i k}{2 z} (ρ_{1}^{2} - ρ_{2}^{2})] \exp [- K {(x_{1} - x_{2})}^{2} - K {(y_{1} - y_{2})}^{2}] \\ \times \sum_{m = 1}^{M} \sum_{n_{1} = 1}^{N} \sum_{n_{2} = 1}^{N} \frac{{(- 1)}^{n_{1} + n_{2} - 2}}{N^{2}} (\begin{matrix} N \\ n_{1} \end{matrix}) (\begin{matrix} N \\ n_{2} \end{matrix}) \int_{- \infty}^{+ \infty} \int_{- \infty}^{+ \infty} \int_{- \infty}^{+ \infty} \int_{- \infty}^{+ \infty} d^{2} u d^{2} v \exp [- \frac{(a_{m x}^{2} + a_{m y}^{2}) (n_{1} + n_{2})}{w^{2}}] \\ \times [u - \frac{v}{4} - 2 a_{m x} u_{x} - 2 a_{m y} u_{y} + i (v_{x} u_{y} - u_{x} v_{y} - a_{m y} v_{x} + a_{m x} v_{y}) + a_{m x}^{2} + a_{m y}^{2}] \\ \times \exp [\frac{2 a_{m x} (n_{1} + n_{2})}{w^{2}} u_{x} + \frac{a_{m x} (n_{1} - n_{2})}{w^{2}} v_{x} + \frac{2 a_{m y} (n_{1} + n_{2})}{w^{2}} u_{y} + \frac{a_{m y} (n_{1} - n_{2})}{w^{2}} v_{y}] \\ \times \exp [\frac{i k (ρ_{1}^{2} - ρ_{2}^{2})}{z} u] \exp {[\frac{i k (ρ_{1}^{2} + ρ_{2}^{2})}{2 z} - K (ρ_{1} - ρ_{2})] v} \\ \times \exp (- \frac{n_{1} + n_{2}}{w^{2}} u^{2}) \exp (- Q v^{2} - P u v) \exp (i φ_{m}), \end{matrix}

where

P = \frac{n_{1} - n_{2}}{w^{2}} + \frac{i k}{z},

Q = \frac{n_{1} + n_{2}}{4 w^{2}} + \frac{n_{1} + n_{2}}{4 σ_{0}^{2}} + K .

Recalling the integral formula [40]

\int_{- \infty}^{\infty} x^{n} \exp (- A x^{2} + B x) d x = n! \exp (\frac{A^{2}}{B}) \sqrt{\frac{π}{A}} {(\frac{A}{B})}^{n} \sum_{m = 0}^{[n / 2]} \frac{1}{m! (n - 2 m)!} {(\frac{A}{4 B^{2}})}^{m} .

with [n/2] being the integral part of the enclosed expression. After tedious but straightforward integral calculations, we can obtain

\begin{matrix} W (ρ_{1}, ρ_{2}, z) = {(\frac{k}{2 π z})}^{2} \exp [- \frac{i k}{2 z} (ρ_{1}^{2} - ρ_{2}^{2})] \exp [- K {(x_{1} - x_{2})}^{2} - K {(y_{1} - y_{2})}^{2}] \\ \times \sum_{m = 1}^{M} \sum_{n_{1} = 1}^{N} \sum_{n_{2} = 1}^{N} \frac{{(- 1)}^{n_{1} + n_{2} - 2}}{N^{2}} (\begin{matrix} N \\ n_{1} \end{matrix}) (\begin{matrix} N \\ n_{2} \end{matrix}) \exp (i φ_{m}) \exp [- \frac{(a_{m x}^{2} + a_{m y}^{2}) (n_{1} + n_{2})}{w^{2}}] \\ \times {H_{1} (1 + \frac{2 C_{x}^{2}}{A}) + H_{1} (1 + \frac{2 C_{y}^{2}}{A}) - \frac{H_{x y}}{4} (\frac{1}{E} + \frac{2 D_{x}^{2}}{E^{2}}) - \frac{H_{x y}}{4} (\frac{1}{E} + \frac{2 D_{y}^{2}}{E^{2}}) - 4 a_{m x} H_{1} C_{x} \\ - 4 a_{m y} H_{1} C_{y} + i [G_{y x} (\frac{D_{x} C_{y}}{E A}) - G_{x y} (\frac{D_{y} C_{x}}{E A}) - a_{m y} H_{x y} (\frac{D_{x}}{E}) + a_{m x} H_{x y} (\frac{D_{y}}{E})] + a_{m x}^{2} H_{x y} + a_{m y}^{2} H_{x y}}, \end{matrix}

where

H_{1} = \frac{π^{2}}{2 A^{2} Q} B_{x} B_{y} \exp (\frac{C_{x}^{2} + C_{y}^{2}}{A}),

\begin{matrix} H_{x y} = \frac{π^{2} w^{2}}{2 E (n_{1} + n_{2})} \exp {{[\frac{i k (x_{1} - x_{2})}{2 z} + \frac{2 a_{m x} (n_{1} + n_{2})}{2 w^{2}}]}^{2} \frac{w^{2}}{n_{1} + n_{2}}} \\ \times \exp {{[\frac{i k (y_{1} - y_{2})}{2 z} + \frac{2 a_{m y} (n_{1} + n_{2})}{2 w^{2}}]}^{2} \frac{w^{2}}{n_{1} + n_{2}}} \exp (\frac{D_{x}^{2} + D_{y}^{2}}{E}), \end{matrix}

G_{y}_{x} = \frac{π^{2} w}{\sqrt{A E Q (n_{1} + n_{2})}} B_{y} \exp {{[\frac{i k (x_{1} - x_{2})}{2 z} + \frac{2 a_{m x} (n_{1} + n_{2})}{2 w^{2}}]}^{2} \frac{w^{2}}{n_{1} + n_{2}}} \exp (\frac{D_{x}^{2}}{E} + \frac{C_{y}^{2}}{A}),

G_{x y} \frac{π^{2} w}{\sqrt{A E Q (n_{1} + n_{2})}} B_{x} \exp {{[\frac{i k (y_{1} - y_{2})}{2 z} + \frac{2 a_{m y} (n_{1} + n_{2})}{2 w^{2}}]}^{2} \frac{w^{2}}{n_{1} + n_{2}}} \exp (\frac{D_{y}^{2}}{E} + \frac{C_{x}^{2}}{A}),

\begin{matrix} B_{j} = \exp [\frac{a^{2}_{m j} {(n_{1} - n_{2})}^{2}}{4 Q w^{4}}] \exp [\frac{i k a_{m j} (n_{1} - n_{2})}{4 Q w^{2} z} (j_{1} + j_{2})] \\ \times \exp [- \frac{k^{2}}{16 Q z^{2}} {(j_{1} + j_{2})}^{2}] \exp [- \frac{i k z}{4 Q z} K (j_{1}^{2} - j_{2}^{2})] \\ \times \exp [- \frac{a_{m j} (n_{1} - n_{2})}{2 Q w^{2}} K (j_{1} - j_{2})] \exp [\frac{1}{4 Q} K^{2} {(j_{1} - j_{2})}^{2}], \end{matrix}

C_{j} = - \frac{a_{m j} P (n_{1} - n_{2})}{4 Q w^{2}} - \frac{i k P (j_{1} + j_{2})}{8 Q z} + \frac{P}{4 Q} K (j_{1} - j_{2}) + \frac{i k (j_{1} - j_{2})}{2 z} + \frac{2 a_{m j} (n_{1} + n_{2})}{2 w^{2}},

D_{j} = \frac{a_{m j} (n_{1} - n_{2})}{2 w^{2}} + \frac{i k (j_{1} + j_{2})}{4 z} - \frac{1}{2} K (j_{1} - j_{2}) - \frac{i k P w^{2} (j_{1} - j_{2})}{4 z (n_{1} + n_{2})} - \frac{a_{m j} P}{2},

A = \frac{n_{1} + n_{2}}{w^{2}} - \frac{P^{2}}{4 Q},

E = Q - \frac{P^{2} w^{2}}{4 (n_{1} + n_{2})} .

and j = x or y (hereafter) refers to receiver plane coordinates. Substituting ρ₁ = ρ₂ = ρ into Eq. (13), the average intensity at any point (ρ, z) can be obtained

\begin{matrix} I (ρ, z) = W (ρ, ρ, z) = W (x, y, x, y, z) \\ = {(\frac{k}{2 π z})}^{2} \sum_{m = 1}^{M} \sum_{n_{1} = 1}^{N} \sum_{n_{2} = 1}^{N} \frac{{(- 1)}^{n_{1} + n_{2} - 2}}{N^{2}} (\begin{matrix} N \\ n_{1} \end{matrix}) (\begin{matrix} N \\ n_{2} \end{matrix}) \exp (i φ_{m}) \exp [- \frac{(a_{m x}^{2} + a_{m y}^{2}) (n_{1} + n_{2})}{w^{2}}] \\ \times {2 {H^{'}}_{1} (1 + \frac{{C^{'}}_{x}^{2} + {C^{'}}_{y}^{2}}{A}) - \frac{{H^{'}}_{x y}}{2} (\frac{1}{E} + \frac{{D^{'}}_{x}^{2} + {D^{'}}_{y}^{2}}{E^{2}}) - 4 {H^{'}}_{1} (a_{m x} {C^{'}}_{x} + a_{m y} {C^{'}}_{y}) \\ + i [(\frac{{G^{'}}_{y x} {D^{'}}_{x} {C^{'}}_{y} - {G^{'}}_{x y} {D^{'}}_{y} {C^{'}}_{x}}{E A}) + \frac{{H^{'}}_{x y}}{E} (a_{m x} {D^{'}}_{y} - a_{m y} {D^{'}}_{x})] + {H^{'}}_{x y} (a_{m x}^{2} + a_{m y}^{2})}, \end{matrix}

with

{C^{'}}_{j} = - \frac{a_{m j} P (n_{1} - n_{2})}{4 Q w^{2}} - \frac{i k P}{4 Q z} j + \frac{a_{m j} (n_{1} + n_{2})}{w^{2}},

{D^{'}}_{j} = \frac{a_{m j} (n_{1} - n_{2})}{2 w^{2}} + \frac{i k}{2 z} j - \frac{a_{m j} P}{2},

{H^{'}}_{x y} = \frac{π^{2} w^{2}}{2 E (n_{1} + n_{2})} \exp [\frac{(n_{1} + n_{2})}{w^{2}} (a_{m x}^{2} + a_{m y}^{2})] \exp (\frac{{D^{'}}_{x}^{2} + {D^{'}}_{y}^{2}}{E}),

{G^{'}}_{y x} = \frac{π^{2} w}{\sqrt{A E Q (n_{1} + n_{2})}} {B^{'}}_{y} \exp [\frac{a_{m x}^{2} (n_{1} + n_{2})}{w^{2}}] \exp (\frac{{D^{'}}_{x}^{2}}{E} + \frac{{C^{'}}_{y}^{2}}{A}),

{G^{'}}_{x y} = \frac{π^{2} w}{\sqrt{A E Q (n_{1} + n_{2})}} {B^{'}}_{x} \exp [\frac{a_{m y}^{2} (n_{1} + n_{2})}{w^{2}}] \exp (\frac{{D^{'}}_{y}^{2}}{E} + \frac{{C^{'}}_{x}^{2}}{A}),

{H^{'}}_{1} = \frac{π^{2}}{2 A^{2} Q} {B^{'}}_{x} {B^{'}}_{y} \exp (\frac{{C^{'}}_{x}^{2} + {C^{'}}_{y}^{2}}{A}),

{B^{'}}_{j} = \exp [\frac{a^{2}_{m j} {(n_{1} - n_{2})}^{2}}{4 Q w^{4}}] \exp [\frac{i k a_{m j} (n_{1} - n_{2})}{2 Q w^{2} z} j] \exp [- \frac{k^{2}}{4 Q z^{2}} j^{2}] .

Equations (13)-(22) provide the analytical expressions for the cross spectral density of a radial phased-locked partially coherent flat-topped vortex beam array composed of M beamlets with l = 1 propagating through non-Kolmogorov atmospheric turbulence. We can see from Eq. (13) that W(ρ₁,ρ₂,z) at the z (z>0) plane is closely related to the beam order N, correlation length σ₀, waist width w, and to K with the outer scale L₀, inner scale l₀, and the generalized exponent factor α. For M = 1, N = 1, Eq. (13) reduces to the cross-spectral density of a Gaussian vortex beam with l = 1 propagating through non-Kolmogorov atmospheric turbulence. By setting the parameters M = 1, a_mx = 0 and a_my = 0, Eq. (13) can be reduced to Eq. (17) with m = 1 in [34], He et al...., namely, the cross-spectral density of partially coherent flat-topped vortex beams through non-Kolmogorov atmospheric turbulence. Equations. (23)-(30) provide the analytical expressions for the average intensity of a radial phased-locked partially coherent flat-topped vortex beam array composed of M beamlets with l = 1 propagating through non-Kolmogorov atmospheric turbulence.

3. Numerical examples and analysis

In this section, the average intensity of a radial phased-locked partially coherent flat-topped vortex beam array composed of M beamlets through non-Kolmogorov atmospheric turbulence are calculated by using the formula derived above. Unless specified in captions, the parameters of the source and the medium used in calculations are chosen as follow: M = 6, N = 2, λ = 632 nm, σ₀ = 1mm, w = 1cm, L₀ = 1m, l₀ = 0.01m, α = 3.8, C_n² = 10⁻¹⁴m^3-α.

Figure 2, Fig. 3, and Fig. 4 show the contour graph of normalized intensity distribution of a radial phased-locked partially coherent flat-topped vortex beam array with different r at several different propagation distances in non-Kolmogorov atmospheric turbulence. The 2-D normalized intensity distribution of y-direction is also plotted in Fig. 5. In Fig. 2, the parameter r = 2 cm is chosen, it can be seen in the near field that there is a hollow region in the contour graph of normalized intensity distribution, and the inner distribution in the contour graph of normalized intensity takes on a hexagonal screw, each side of which attaches a rectangular beam evolved by initial flat-topped vortex beam element. When the propagation distance z increases, the hollow region disappears, and the outer distribution in the contour graph of normalized intensity takes on a plum blossom with six petals, and the distribution becomes a Gaussian-shaped profile in the far field. The on-axis intensity increases from zero to the maximum value. In Fig. 3, the parameter r = 3 cm is chosen, from Fig. 3(a), one can find that the distribution with dark region of hexagon attached by six hollow beams is shaped. With the increase of the propagation distance z, the whole distribution in the contour graph of normalized intensity takes on a hexagonal screw and the beam array changes from central dark beam to flat-topped beam and to the Gaussian-like beam. In Fig. 4, the parameter r = 5 cm is chosen, we can see that, in the near field, the each beamlet of beam array evolves independently from vortex hollow bean into flat-topped beam. When the propagation distance z increases, the flat-topped beam petals become interlinked Gaussian-like beams, and a hollow beam with the outer distribution being a hexagon is formed. From Fig. 4(d), it is obvious that a good flat-topped beam can be obtained. In Fig. 5, the parameters are same as those in Fig. 4, the distribution of normalized intensity in y-direction at different propagation distance z is shown. We can clearly see that the distribution of normalized intensity of the radial phased-locked flat-topped vortex beam array changes from a hollow beam to quasi-hollow beam, and to flat-topped beam, and to Gaussian-like beam, eventually, to Gaussian beam [41]. Comparing Fig. 2 with Fig. 3 and Fig. 4, the larger the parameter r is, the longer the axial propagation distance z within which the beam array is a hollow beam and the flat-topped beam is evolved is. In Fig. 6, we explore the influences of the structure constant of the atmospheric turbulence on propagation properties of a radial phase-locked partially coherent flat-topped vortex beam array. The parameters are same as those in Fig. 4 except z = 0.8 km and C_n². Comparing Fig. 6(a) with Fig. 4(c), even though the structure constant of the atmospheric turbulence (C_n² = 10⁻¹⁴m^3-α in Fig. 4 and C_n² = 10⁻¹²m^3-α in Fig. 6(a)) is quite different, the distribution of normalized intensity is the same completely. From the Fig. 6(b), one can find that the on-axis intensity is still zero when the structure constant of the atmospheric turbulence increases to 5 × 10⁻¹²m^3-α. Until the structure constant of the atmospheric turbulence increases to 10⁻¹⁰m^3-α, the Gaussian distribution of normalized intensity doesn’t appear. There is little influence of the structure constant on a radial phased-locked partially coherent flat-topped vortex beam array propagating through non-Kolmogorov atmospheric turbulence, which can be obtained by means of comparison with anomalous hollow beam array in [32], Wang et al.....

Fig. 2 Contour graphs of normalized intensity of a radial phased-locked partially coherent flat-topped vortex beam array with several different propagation distances through non-Kolmogorov atmospheric turbulence. r = 2 cm. (a) z = 0.2 km (b) z = 0.3 km (c) z = 0.4 km (d) z = 0.5 km.

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Fig. 3 Contour graphs of normalized intensity of a radial phased-locked partially coherent flat-topped vortex beam array with several different propagation distances through non-Kolmogorov atmospheric turbulence. r = 3 cm. (a) z = 0.2 km (b) z = 0.3 km (c) z = 0.8 km(d) z = 1 km.

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Fig. 4 Contour graphs of normalized intensity of a radial phased-locked partially coherent flat-topped vortex beam array with several different propagation distances through non-Kolmogorov atmospheric turbulence. r = 5 cm. (a) z = 0.3 km (b) z = 0.5 km (c) z = 0.8 km (d) z = 1.35 km.

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Fig. 5 Normalized intensity in y-direction at different propagation distance z.

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Fig. 6 Contour graphs of normalized intensity of a radial phased-locked partially coherent flat-topped vortex beam array with several different structure constants through non-Kolmogorov atmospheric turbulence. r = 5cm, z = 0.8km (a) C_n² = 10⁻¹²m^3-α (b) C_n² = 5 × 10⁻¹²m^3-α(c) C_n² = 10⁻¹¹m^3-α (d) C_n² = 10⁻¹⁰m^3-α.

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Figure 7 shows the contour graphs of normalized intensity distribution of a radial phased-locked partially coherent flat-topped vortex beam array with different beamlet number through non-Kolmogorov atmospheric turbulence. One can see that, when the number of the beamlet is large enough, centrally dark region of partially coherent flat-topped vortex beam array always takes on round distribution instead of polygonal distribution. With the increasing beamlet number M, the central intensity evolution of each beam element of the radial phased-locked partially coherent flat-topped vortex beam array is of interest, presenting hollow beam in Fig. 7(a), round flat-topped beam in Fig. 7(b), elliptical flat-topped beam in Fig. 7(c) and triangular flat-topped beam in Fig. 7(d).

Fig. 7 Contour graphs of normalized intensity of a radial phased-locked partially coherent flat-topped vortex beam array with different beamlet numbers through non-Kolmogorov atmospheric turbulence. r = 3cm, z = 0.2km (a) M = 6 (b) M = 8 (c) M = 10 (d) M = 12.

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Figure 8 shows the normalized intensity in y-direction for different values of σ₀ at the different propagation distance z. In Fig. 8, the effects of the correlation length on the propagation properties of a radial phased-locked partially coherent flat-topped vortex beam array are explored. We can see that a radial phased-locked partially coherent flat-topped vortex beam array with high coherence evolve more rapidly than a radial phased-locked partially coherent flat-topped vortex beam array with low coherence. That is to say, the partially coherent beams with high initial coherence spread more widely than a partially coherent beam with low initial coherence, which means that the beam spot spreads more rapidly for high coherence, this point is consistent with the results in [4], Cai et al.....

Fig. 8 Normalized intensity in y-direction for different values of σ₀ at the different propagation distance z.(a)z = 500m (b)z = 1000m.

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Figure 9 shows on-axis normalized intensity of a radial phased-locked partially coherent flat-topped vortex beam array through non-Kolmogorov turbulence versus generalized exponent factor α for different values of N at propagation distance z = 1km. Figure 10 shows K versus α for different propagation distance z.At a fixed exponent factor α in Fig. 9, the larger N is, the smaller on-axis normalized intensity is. For α>3.8, the few effects of N on on-axis normalized intensity can be found. We can see that from Fig. 9,for the same N and 3<α<3 073, the larger exponent factor α leads to the smaller on-axis normalized intensity. For α = 3.073, the smallest on-axis normalized intensity is achieved. For 3.073<α<4, however, the larger exponent factor α is, the larger on-axis normalized intensity is. This result can be explained in Fig. 10. K describes the strength of the turbulence, which means the larger K corresponds to the stronger turbulence [36]. The maximum K_max is obtained for α = 3.073 by letting $\partial K / \partial α = 0$ . As a result, the turbulence is the strongest and on-axis normalized intensity is smallest for α = 3.073.

Fig. 9 On-axis normalized intensity of a radial phased-locked partially coherent flat-topped vortex beam array through non-Kolmogorov turbulence versus generalized exponent factor α for different values of N.

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Fig. 10 K versus α for different propagation distance z.

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Figure 11 shows the y-direction intensity of a radial phased-locked partially coherent flat-topped vortex beam array through non-Kolmogorov turbulence for different exponent α, beamlet number M, and propagation distance z. One can see that, as beamlet number M increases, the intensity of a radial phased-locked partially coherent flat-topped vortex beam array increases. For α = 11/3 = 3.67, the non-Kolmogorov turbulence reduces to Kolmogorov turbulence [42]. It is obvious that evolution behaviors of a radial phased-locked partially coherent flat-topped vortex beam array propagating through non-Kolmogorov turbulence are different than that propagating through Kolmogorov turbulence. Furthermore, The intensity decreases with the increasing propagation distance z.

Fig. 11 The y-direction intensity of a radial phased-locked partially coherent flat-topped vortex beam array through non-Kolmogorov turbulence for different exponent α, beamlet number M, and propagation distance z.

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Figure 12 shows the y-direction intensity of a radial phased-locked partially coherent flat-topped vortex beam array through non-Kolmogorov turbulence for different N and exponent α at a given propagation distance z = 800m. We can see that the flattened degree of a radial phased-locked partially coherent flat-topped vortex beam array is higher with the increase of N (see Fig. 12(a)). The flattened degree of a radial phased-locked partially coherent flat-topped vortex beam array propagating through Kolmogorov turbulence is higher than propagating through non-Kolmogorov turbulence (see Fig. 12(b)).

Fig. 12 The y-direction intensity of a radial phased-locked partially coherent flat-topped vortex beam array through non-Kolmogorov turbulence for different N and exponent α.

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Figure 13 shows the y-direction intensity of a radial phased-locked partially coherent flat-topped vortex beam array through non-Kolmogorov turbulence at propagation distance z = 3km for different exponent α, outer scale L₀, and inner scale l₀. We can see that the influences of outer scale L₀ on evolution behaviors of a radial phased-locked partially coherent flat-topped vortex beam array are less obvious, however, the influences of inner scale l₀ on that are apparent. According to Fig. 10, the turbulence with α = 3.1 is stronger than the case α = 3.67, therefore, the intensity for α = 3.1 is smaller than the case α = 3.67. The larger intensity can be finished with smaller outer scale L₀ or larger inner scale l₀.

Fig. 13 The y-direction intensity of a radial phased-locked partially coherent flat-topped vortex beam array through non-Kolmogorov turbulence for different exponent α, outer scale L₀, and inner scale l₀.

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4. The complex degree of spatial coherence

The complex degree of spatial coherence of two points Q(x₁,z) and Q(x₂,z) at receiver plane is defined as [43]

μ (x_{1}, x_{2}, z) = \frac{W (x_{1}, x_{2}, z)}{\sqrt{I (x_{1}, z) I (x_{2}, z)}} .

Substituting Eq. (13) and Eq. (23) into Eq. (31), we just consider the complex degree of spatial coherence of two symmetric points Q(-x,z) and Q(x,z) at z plane, the analytic expression of complex degree of spatial coherence of a radial phased-locked partially coherent flat-topped vortex beam array through non-Kolmogorov atmospheric turbulence can be obtained as follow

μ (- x, x, z) = \frac{Ω_{1}}{\sqrt{Ω_{2} Ω_{3}}},

where

\begin{matrix} Ω_{p} = \sum_{m = 1}^{M} \sum_{n_{1} = 1}^{N} \sum_{n_{2} = 1}^{N} \frac{{(- 1)}^{n_{1} + n_{2} - 2}}{N^{2}} (\begin{matrix} N \\ n_{1} \end{matrix}) (\begin{matrix} N \\ n_{2} \end{matrix}) \exp (i φ_{m}) \exp [- \frac{(a_{m x}^{2} + a_{m y}^{2}) (n_{1} + n_{2})}{w^{2}}] \\ \times {2 H_{p 1} (1 + \frac{C_{p x}^{2} + C^{2}}{A}) - \frac{H_{p x y}}{2 E} (1 + \frac{D_{p x}^{2} + D^{2}}{E}) - 4 H_{p 1} (a_{m x} C_{p x} + a_{m y} C) \\ + i [(\frac{G_{p y x} D_{x} C - G_{p x y} D C_{p x}}{E A}) + \frac{H_{p x y}}{E} (a_{m x} D - a_{m y} D_{p x})] + H_{p x y} (a_{m x}^{2} + a_{m y}^{2})}, \end{matrix}

H_{p 1} = \frac{π^{2}}{2 A^{2} Q} B_{p x} \exp [\frac{a^{2}_{m y} {(n_{1} - n_{2})}^{2}}{4 Q w^{4}}] \exp (\frac{C_{p x}^{2} + C^{2}}{A}),

H_{1 x y} = \frac{π^{2} w^{2}}{2 E (n_{1} + n_{2})} \exp {{[- \frac{i k}{z} x + \frac{2 a_{m x} (n_{1} + n_{2})}{2 w^{2}}]}^{2} \frac{w^{2}}{n_{1} + n_{2}}} \exp [a_{m y}^{2} \frac{(n_{1} + n_{2})}{w^{2}}] \exp (\frac{D_{1 x}^{2} + D^{2}}{E}),

G_{1 y}_{x} = \frac{π^{2} w}{\sqrt{A E Q (n_{1} + n_{2})}} \exp [\frac{a^{2}_{m y} {(n_{1} - n_{2})}^{2}}{4 Q w^{4}}] \exp {{[- \frac{i k x}{z} + \frac{2 a_{m x} (n_{1} + n_{2})}{2 w^{2}}]}^{2} \frac{w^{2}}{n_{1} + n_{2}}} \exp (\frac{D_{1 x}^{2}}{E} + \frac{C^{2}}{A}),

G_{1 x y} \frac{π^{2} w}{\sqrt{A E Q (n_{1} + n_{2})}} B_{1 x} \exp [a_{m y}^{2} \frac{(n_{1} + n_{2})}{w^{2}}] \exp (\frac{D^{2}}{E} + \frac{C_{1 x}^{2}}{A}),

B_{1 x} = \exp [\frac{a^{2}_{m x} {(n_{1} - n_{2})}^{2}}{4 Q w^{4}}] \exp [\frac{a_{m x} K (n_{1} - n_{2})}{Q w^{2}} x] \exp (\frac{K^{2}}{Q} x^{2}),

C_{1 x} = - \frac{a_{m x} P (n_{1} - n_{2})}{4 Q w^{2}} + \frac{2 a_{m x} (n_{1} + n_{2})}{2 w^{2}} - (\frac{P K}{2 Q} + \frac{i k}{z}) x,

D_{1 x} = \frac{a_{m x} (n_{1} - n_{2})}{2 w^{2}} - \frac{a_{m x} P}{2} + [K + \frac{i k P w^{2}}{2 z (n_{1} + n_{2})}] x,

H_{q x y} = \frac{π^{2} w^{2}}{2 E (n_{1} + n_{2})} \exp [\frac{(n_{1} + n_{2})}{w^{2}} (a_{m x}^{2} + a_{m y}^{2})] \exp (\frac{D_{q x}^{2} + D^{2}}{E}),

G_{q y}_{x} = \frac{π^{2} w}{\sqrt{A E Q (n_{1} + n_{2})}} \exp [\frac{a^{2}_{m y} {(n_{1} - n_{2})}^{2}}{4 Q w^{4}}] \exp [\frac{a_{m x}^{2} (n_{1} + n_{2})}{w^{2}}] \exp (\frac{D_{q x}^{2}}{E} + \frac{C^{2}}{A}),

G_{q x y} = \frac{π^{2} w}{\sqrt{A E Q (n_{1} + n_{2})}} B_{q x} \exp [\frac{a_{m y}^{2} (n_{1} + n_{2})}{w^{2}}] \exp (\frac{D^{2}}{E} + \frac{C_{q x}^{2}}{A}),

B_{q x} = \exp [\frac{a^{2}_{m x} {(n_{1} - n_{2})}^{2}}{4 Q w^{4}}] \exp [- \frac{i k a_{m x} (n_{1} - n_{2})}{2 Q w^{2} z} x] \exp [{(- 1)}^{q} \frac{k^{2}}{4 Q z^{2}} x^{2}],

C_{q x} = - \frac{a_{m x} P (n_{1} - n_{2})}{4 Q w^{2}} + \frac{2 a_{m x} (n_{1} + n_{2})}{2 w^{2}} + {(- 1)}^{q} \frac{i k P}{4 Q z} x,

D_{q x} = \frac{a_{m x} (n_{1} - n_{2})}{2 w^{2}} - \frac{a_{m x} P}{2} + {(- 1)}^{q + 1} \frac{i k}{2 z} x,

C = - \frac{a_{m y} P (n_{1} - n_{2})}{4 Q w^{2}} + \frac{2 a_{m y} (n_{1} + n_{2})}{2 w^{2}},

D = \frac{a_{m y} (n_{1} - n_{2})}{2 w^{2}} - \frac{a_{m y} P}{2} .

with p = 1,2,3 and q = 2,3.

According to the expressions derived above, the complex degree of spatial coherence of a radial phased-locked partially coherent flat-topped vortex beam array through non-Kolmogorov atmospheric turbulence is shown in Fig. 14. From Fig. 14, one can find that the complex degree of spatial coherence is not a function decreasing with the increase of x, but a function with oscillatory phenomenon. Furthermore, the points of μ(-x,x,z) = 0 can be found, i.e., coherence vortices, which show that the two axisymmetric points along the z-axis are incoherent completely. This is called as phase singularity phenomenon [44,45]. Figure 14(a) shows the relationship between the complex degree of spatial coherence and x/w, one sees that the oscillation intensity of the complex degree of spatial coherence increases with the increasing the propagation distance z. From Fig. 14(b) and Fig. 14(c), one sees that although the complex degree of spatial coherence varies slightly with the source correlation width σ₀, it alters significantly with the changes of r, and the larger the value of r is, the faster the oscillation intensity decreases. From Fig. 14(d), one sees that there is little influence of structure constant on the complex degree of spatial coherence, which agrees with the result discussed above.

Fig. 14 The complex degree of spatial coherence, (a) at several z values with δ = 5 mm, r = 3cm, C_n² = 10⁻¹⁴m^3-α, (b) at several δ values with r = 3cm, z = 1 km, C_n² = 10⁻¹⁴m^3-α, (c) at several r values with δ = 5 mm, z = 1 km, C_n² = 10⁻¹⁴m^3-α, (d) at several C_n² values with δ = 5 mm, z = 1 km, r = 3 cm.

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5. Concluding remarks

In this paper, we have studied the propagation properties of average intensity of a radial phased-locked partially coherent flat-topped vortex beam array composed of M beamlets with l = 1 propagating through non-Kolmogorov atmospheric turbulence in detail. It has been shown analytically and numerically that the normalized intensity of a radial phased-locked partially coherent flat-topped vortex beam array changes from a hollow beam to quasi-hollow beam, and to flat-topped beam, and to Gaussian-like beam, and to Gaussian beam. One can see that the evolution behavior of average intensity depends on beam parameters including the spatial correlation length σ₀, the radius r, as well as the propagation distance z. A radial phased-locked partially coherent flat-topped vortex beam array with high coherence evolves more rapidly than that with low coherence. If one wants to get a dark hollow beam or a flat-topped beam in the far-field, the parameter should be appropriate, e.g., r is large enough.

The flattened degree of a radial phased-locked partially coherent flat-topped vortex beam array is higher with the increase of N during propagation process, which would be useful for optical processing and inertial confinement fusion [1,46,47].

Funding

National Natural Science Foundation of China (NSFC) (No. 61475026, No. 61275135, No. 61108029).

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Radial phased-locked partially coherent flat-topped vortex beam array in non-Kolmogorov medium

Abstract

1. Introduction

2. Propagation theory

3. Numerical examples and analysis

4. The complex degree of spatial coherence

5. Concluding remarks

Funding

References and links

Cited By

Figures (14)

Equations (48)

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