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

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Abstract

The analytical expressions for the cross-spectral density, the average intensity and the complex degree of spatial coherence of a radial phased-locked partially coherent flat-topped vortex beam array propagating through non-Kolmogorov medium are obtained by using the extended Huygens–Fresnel principle. The evolution behaviors of a radial phased-locked partially coherent flat-topped vortex beam array propagating through non-Kolmogorov medium are studied in detail. It is shown that the evolution behaviors of average intensity depend on beam parameters including the spatial correlation length, the radius of the beam array, as well as the propagation distance. A radial phased-locked partially coherent flat-topped vortex beam array with high coherence evolves more rapidly than that with low coherence.

© 2016 Optical Society of America

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]

Em(ρx,ρy,0)=[ρxamx+isgn(l)(ρyamy)]|l|n=1N(1)n1N(Nn)exp(n(ρxamx)2w2n(ρyamy)2w)exp(iφm).
where amx = rcosφm and amy = rsinφm are the center of the m-th beamlet located at the source plane, r is the radius and φm is the initial phase of the m-th beamlet. φm = mφ0 = 2/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.

 figure: 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]

Wm(0)(ρ1,ρ2,0)=Im(ρ1x,ρ1y,0)Im(ρ2x,ρ2y,0)exp[(n1+n2)(ρ1ρ2)24σ02].
where Im'ix, ρ'iy,0) (i = 1,2) is intensity of m-th flat-topped vortex beam element, andρ'1 = (ρ'1x, ρ'1y),ρ'2 = (ρ'2x, ρ'2y) 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)(ρ1,ρ2,0)=m=1MWm(0)(ρ1,ρ2,0).
Substituting Eqs. (1) and (2) into Eq. (3), we can obtain

W(0)(ρ1,ρ2,0)={ρ1xρ2x+ρ1yρ2yamx(ρ1x+ρ2x)amy(ρ1y+ρ2y)+i[ρ1xρ2yρ2xρ1yamy(ρ1xρ2x)+amx(ρ1yρ2y)]+amx2+amy2}×m=1Mn1=1Nn2=1N(1)n1+n22N2(Nn1)(Nn2)exp[n1w2(ρ1x2+ρ1y2)n2w2(ρ2x2+ρ2y2)]×exp[2amxw2(n1ρ1x+n2ρ2x)+2amyw2(n1ρ1y+n2ρ2y)]exp(iφm)×exp[(amx2+amy2)(n1+n2)w2]exp[(n1+n2)4σ02(ρ1ρ2)2].

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

W(ρ1,ρ2,z)=k24π2z2++++W(0)(ρ1,ρ2,0)×exp{ik2z[(ρ1ρ1)2(ρ2ρ2)2]}×exp[ψ*(ρ1,ρ1)+ψ(ρ2,ρ2)]d2ρ1d2ρ2.
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]
exp[ψ*(ρ1,ρ1)+ψ(ρ2,ρ2)]=exp{4π2k2z010+dκdξΦn(κ,α)[1J0(κ|(1ξ)(ρ1ρ2)+ξ(ρ1ρ2)|)]}=exp{π2k2z30+κ3Φn(κ,α)dκ[(ρ1ρ2)2+(ρ1ρ2)(ρ1ρ2)+(ρ1ρ2)2]}.
where Φn(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]
Φn(κ,α)=A(α)C˜n2(κ2+κ02)α/2exp(κ2κm2).
where α denotes the generalized exponent factor of non-Kolmogorov turbulence, 3<α<4. 0≤κ<∞. C˜n2 is the generalized structure constant with units m3-α.

The terms can be given by κ0 = 2π/L0 (L0-outer scale of the atmospheric turbulence), κm = 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=π2k2z30+κ3Φn(κ,α)dκ=π2k2zA(α)C˜n26(α2)[βexp(κ02κm2)Γ(4α2,κ02κm2)κm2α2κ04α].
where β = 2(κ02κm2) + ακm2, 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

W(ρ1,ρ2,z)=k24π2z2exp[ik2z(ρ12ρ22)]exp[K(x1x2)2K(y1y2)2]×m=1Mn1=1Nn2=1N(1)n1+n22N2(Nn1)(Nn2)++++d2ud2vexp[(amx2+amy2)(n1+n2)w2]×[uv42amxux2amyuy+i(vxuyuxvyamyvx+amxvy)+amx2+amy2]×exp[2amx(n1+n2)w2ux+amx(n1n2)w2vx+2amy(n1+n2)w2uy+amy(n1n2)w2vy]×exp[ik(ρ12ρ22)zu]exp{[ik(ρ12+ρ22)2zK(ρ1ρ2)]v}×exp(n1+n2w2u2)exp(Qv2Puv)exp(iφm),
where

P=n1n2w2+ikz,
Q=n1+n24w2+n1+n24σ02+K.

Recalling the integral formula [40]

xnexp(Ax2+Bx)dx=n!exp(A2B)πA(AB)nm=0[n/2]1m!(n2m)!(A4B2)m.
with [n/2] being the integral part of the enclosed expression. After tedious but straightforward integral calculations, we can obtain
W(ρ1,ρ2,z)=(k2πz)2exp[ik2z(ρ12ρ22)]exp[K(x1x2)2K(y1y2)2]×m=1Mn1=1Nn2=1N(1)n1+n22N2(Nn1)(Nn2)exp(iφm)exp[(amx2+amy2)(n1+n2)w2]×{H1(1+2Cx2A)+H1(1+2Cy2A)Hxy4(1E+2Dx2E2)Hxy4(1E+2Dy2E2)4amxH1Cx4amyH1Cy+i[Gyx(DxCyEA)Gxy(DyCxEA)amyHxy(DxE)+amxHxy(DyE)]+amx2Hxy+amy2Hxy},
where
H1=π22A2QBxByexp(Cx2+Cy2A),
Hxy=π2w22E(n1+n2)exp{[ik(x1x2)2z+2amx(n1+n2)2w2]2w2n1+n2}×exp{[ik(y1y2)2z+2amy(n1+n2)2w2]2w2n1+n2}exp(Dx2+Dy2E),
Gyx=π2wAEQ(n1+n2)Byexp{[ik(x1x2)2z+2amx(n1+n2)2w2]2w2n1+n2}exp(Dx2E+Cy2A),
Gxyπ2wAEQ(n1+n2)Bxexp{[ik(y1y2)2z+2amy(n1+n2)2w2]2w2n1+n2}exp(Dy2E+Cx2A),
Bj=exp[a2mj(n1n2)24Qw4]exp[ikamj(n1n2)4Qw2z(j1+j2)]×exp[k216Qz2(j1+j2)2]exp[ikz4QzK(j12j22)]×exp[amj(n1n2)2Qw2K(j1j2)]exp[14QK2(j1j2)2],
Cj=amjP(n1n2)4Qw2ikP(j1+j2)8Qz+P4QK(j1j2)+ik(j1j2)2z+2amj(n1+n2)2w2,
Dj=amj(n1n2)2w2+ik(j1+j2)4z12K(j1j2)ikPw2(j1j2)4z(n1+n2)amjP2,
A=n1+n2w2P24Q,
E=QP2w24(n1+n2).
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
I(ρ,z)=W(ρ,ρ,z)=W(x,y,x,y,z)=(k2πz)2m=1Mn1=1Nn2=1N(1)n1+n22N2(Nn1)(Nn2)exp(iφm)exp[(amx2+amy2)(n1+n2)w2]×{2H1(1+Cx2+Cy2A)Hxy2(1E+Dx2+Dy2E2)4H1(amxCx+amyCy)+i[(GyxDxCyGxyDyCxEA)+HxyE(amxDyamyDx)]+Hxy(amx2+amy2)},
with

Cj=amjP(n1n2)4Qw2ikP4Qzj+amj(n1+n2)w2,
Dj=amj(n1n2)2w2+ik2zjamjP2,
Hxy=π2w22E(n1+n2)exp[(n1+n2)w2(amx2+amy2)]exp(Dx2+Dy2E),
Gyx=π2wAEQ(n1+n2)Byexp[amx2(n1+n2)w2]exp(Dx2E+Cy2A),
Gxy=π2wAEQ(n1+n2)Bxexp[amy2(n1+n2)w2]exp(Dy2E+Cx2A),
H1=π22A2QBxByexp(Cx2+Cy2A),
Bj=exp[a2mj(n1n2)24Qw4]exp[ikamj(n1n2)2Qw2zj]exp[k24Qz2j2].

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, amx = 0 and amy = 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, Cn2 = 10−14m3-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 Cn2. Comparing Fig. 6(a) with Fig. 4(c), even though the structure constant of the atmospheric turbulence (Cn2 = 10−14m3-α in Fig. 4 and Cn2 = 10−12m3-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 × 10−12m3-α. Until the structure constant of the atmospheric turbulence increases to 10−10m3-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.....

 figure: Fig. 2

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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 figure: Fig. 3

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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 figure: Fig. 4

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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 figure: Fig. 5

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

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 figure: Fig. 6

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) Cn2 = 10−12m3-α (b) Cn2 = 5 × 10−12m3-α(c) Cn2 = 10−11m3-α (d) Cn2 = 10−10m3-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).

 figure: Fig. 7

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.....

 figure: Fig. 8

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 Kmax is obtained for α = 3.073 by letting K/α=0. As a result, the turbulence is the strongest and on-axis normalized intensity is smallest for α = 3.073.

 figure: Fig. 9

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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 figure: Fig. 10

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.

 figure: Fig. 11

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)).

 figure: Fig. 12

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.

 figure: Fig. 13

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(x1,x2,z)=W(x1,x2,z)I(x1,z)I(x2,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)=Ω1Ω2Ω3,
where
Ωp=m=1Mn1=1Nn2=1N(1)n1+n22N2(Nn1)(Nn2)exp(iφm)exp[(amx2+amy2)(n1+n2)w2]×{2Hp1(1+Cpx2+C2A)Hpxy2E(1+Dpx2+D2E)4Hp1(amxCpx+amyC)+i[(GpyxDxCGpxyDCpxEA)+HpxyE(amxDamyDpx)]+Hpxy(amx2+amy2)},
Hp1=π22A2QBpxexp[a2my(n1n2)24Qw4]exp(Cpx2+C2A),
H1xy=π2w22E(n1+n2)exp{[ikzx+2amx(n1+n2)2w2]2w2n1+n2}exp[amy2(n1+n2)w2]exp(D1x2+D2E),
G1yx=π2wAEQ(n1+n2)exp[a2my(n1n2)24Qw4]exp{[ikxz+2amx(n1+n2)2w2]2w2n1+n2}exp(D1x2E+C2A),
G1xyπ2wAEQ(n1+n2)B1xexp[amy2(n1+n2)w2]exp(D2E+C1x2A),
B1x=exp[a2mx(n1n2)24Qw4]exp[amxK(n1n2)Qw2x]exp(K2Qx2),
C1x=amxP(n1n2)4Qw2+2amx(n1+n2)2w2(PK2Q+ikz)x,
D1x=amx(n1n2)2w2amxP2+[K+ikPw22z(n1+n2)]x,
Hqxy=π2w22E(n1+n2)exp[(n1+n2)w2(amx2+amy2)]exp(Dqx2+D2E),
Gqyx=π2wAEQ(n1+n2)exp[a2my(n1n2)24Qw4]exp[amx2(n1+n2)w2]exp(Dqx2E+C2A),
Gqxy=π2wAEQ(n1+n2)Bqxexp[amy2(n1+n2)w2]exp(D2E+Cqx2A),
Bqx=exp[a2mx(n1n2)24Qw4]exp[ikamx(n1n2)2Qw2zx]exp[(1)qk24Qz2x2],
Cqx=amxP(n1n2)4Qw2+2amx(n1+n2)2w2+(1)qikP4Qzx,
Dqx=amx(n1n2)2w2amxP2+(1)q+1ik2zx,
C=amyP(n1n2)4Qw2+2amy(n1+n2)2w2,
D=amy(n1n2)2w2amyP2.
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.

 figure: Fig. 14

Fig. 14 The complex degree of spatial coherence, (a) at several z values with δ = 5 mm, r = 3cm, Cn2 = 10−14m3-α, (b) at several δ values with r = 3cm, z = 1 km, Cn2 = 10−14m3-α, (c) at several r values with δ = 5 mm, z = 1 km, Cn2 = 10−14m3-α, (d) at several Cn2 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).

References and links

1. T. Shirai, A. Dogariu, and E. Wolf, “Mode analysis of spreading of partially coherent beams propagating through atmospheric turbulence,” J. Opt. Soc. Am. A 20(6), 1094–1102 (2003). [CrossRef]   [PubMed]  

2. T. Shirai, A. Dogariu, and E. Wolf, “Directionality of Gaussian Schell-model beams propagating in atmospheric turbulence,” Opt. Lett. 28(8), 610–612 (2003). [CrossRef]   [PubMed]  

3. X. Wang, M. Yao, Z. Qiu, X. Yi, and Z. Liu, “Evolution properties of Bessel-Gaussian Schell-model beams in non-Kolmogorov turbulence,” Opt. Express 23(10), 12508–12523 (2015). [CrossRef]   [PubMed]  

4. Y. Cai and S. He, “Propagation of a partially coherent twisted anisotropic Gaussian Schell-model beam in a turbulent atmosphere,” Appl. Phys. Lett. 89(4), 041117 (2006). [CrossRef]  

5. X. Du, D. Zhao, and O. Korotkova, “Changes in the statistical properties of stochastic anisotropic electromagnetic beams on propagation in the turbulent atmosphere,” Opt. Express 15(25), 16909–16915 (2007). [CrossRef]   [PubMed]  

6. Y. Dan and B. Zhang, “Beam propagation factor of partially coherent flat-topped beams in a turbulent atmosphere,” Opt. Express 16(20), 15563–15575 (2008). [CrossRef]   [PubMed]  

7. G. Wu, H. Guo, S. Yu, and B. Luo, “Spreading and direction of Gaussian-Schell model beam through a non-Kolmogorov turbulence,” Opt. Lett. 35(5), 715–717 (2010). [CrossRef]   [PubMed]  

8. M. Tang and D. Zhao, “Spectral changes in stochastic electromagnetic beams propagating through turbulent ocean,” Opt. Commun. 312(1), 89–93 (2014). [CrossRef]  

9. O. Korotkova and E. Shchepakina, “Rectangular Multi-Gaussian Schell-Model beams in atmospheric turbulence,” J. Opt. 16(4), 045704 (2014). [CrossRef]  

10. O. Korotkova, L. C. Andrews, and R. L. Phillips, “Speckle propagation through atmospheric turbulence: effects of a random phase screen at the source,” Proc. SPIE 4821, 98–109 (2002). [CrossRef]  

11. O. Korotkova and E. Shchepakina, “Color changes in stochastic light fields propagating in non-Kolmogorov turbulence,” Opt. Lett. 35(22), 3772–3774 (2010). [CrossRef]   [PubMed]  

12. E. Shchepakina and O. Korotkova, “Second-order statistics of stochastic electromagnetic beams propagating through non-Kolmogorov turbulence,” Opt. Express 18(10), 10650–10658 (2010). [CrossRef]   [PubMed]  

13. 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]  

14. T. Wang, J. Pu, and Z. Chen, “Propagation of partially coherent vortex beams in a turbulent atmosphere,” Opt. Eng. 47(3), 036002 (2008). [CrossRef]  

15. Y. Chen, F. Wang, C. Zhao, and Y. Cai, “Experimental demonstration of a Laguerre-Gaussian correlated Schell-model vortex beam,” Opt. Express 22(5), 5826–5838 (2014). [CrossRef]   [PubMed]  

16. Y. Zhang, L. Liu, C. Zhao, and Y. Cai, “Multi-Gaussian Schell-model vortex beam,” Phys. Lett. A 378(9), 750–754 (2014). [CrossRef]  

17. J. He, X. Wang, D. Hu, J. Ye, S. Feng, Q. Kan, and Y. Zhang, “Generation and evolution of the terahertz vortex beam,” Opt. Express 21(17), 20230–20239 (2013). [CrossRef]   [PubMed]  

18. 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]  

19. Y. Ni and G. Zhou, “Propagation of a Lorentz–Gauss vortex beam through a paraxial ABCD optical system,” Opt. Commun. 291(15), 19–25 (2013). [CrossRef]  

20. B. Chen, C. Chen, X. Peng, and D. Deng, “Propagation of Airy Gaussian vortex beams through slabs of right-handed materials and left-handed materials,” J. Opt. Soc. Am. B 32(1), 173–178 (2015). [CrossRef]  

21. Q. Zhan, “Properties of circularly polarized vortex beams,” Opt. Lett. 31(7), 867–869 (2006). [CrossRef]   [PubMed]  

22. V. V. Kotlyar, A. A. Kovalev, and A. P. Porfirev, “Vortex Hermite-Gaussian laser beams,” Opt. Lett. 40(5), 701–704 (2015). [CrossRef]   [PubMed]  

23. H. Liu, Y. Lü, J. Xia, X. Pu, and L. Zhang, “Flat-topped vortex hollow beam and its propagation properties,” J. Opt. 17(7), 075606 (2015). [CrossRef]  

24. D. Liu, Y. Wang, and H. Yin, “Evolution properties of partially coherent four-petal Gaussian vortex beams propagating in uniaxial crystals orthogonal to the optical axis,” J. Opt. Soc. Am. A 32(9), 1683–1690 (2015). [CrossRef]   [PubMed]  

25. L. Wang, L. Wang, and S. Zhu, “Formation of optical vortices using coherent laser beam array,” Opt. Commun. 282(6), 1088–1094 (2009). [CrossRef]  

26. X. Du and D. Zhao, “Statistical properties of correlated radial stochastic electromagnetic array beams on propagation,” Opt. Commun. 282(10), 1993–1997 (2009). [CrossRef]  

27. X. Du and D. Zhao, “Propagation characteristics of stochastic electromagnetic array beams,” Appl. Phys. B 93(4), 901–905 (2008). [CrossRef]  

28. P. Zhou, X. Wang, Y. Ma, and Z. Liu, “Propagation properties of a Lorentz beam array,” Appl. Opt. 49(13), 2497–2503 (2010). [CrossRef]  

29. G. Zhou, “Propagation of a radial phased-locked Lorentz beam array in turbulent atmosphere,” Opt. Express 19(24), 24699–24711 (2011). [CrossRef]   [PubMed]  

30. H. Tang, B. Ou, B. Luo, H. Guo, and A. Dang, “Average spreading of a radial gaussian beam array in non-Kolmogorov turbulence,” J. Opt. Soc. Am. A 28(6), 1016–1021 (2011). [CrossRef]   [PubMed]  

31. Y. Zhu, D. Zhao, and X. Du, “Propagation of stochastic Gaussian-Schell model array beams in turbulent atmosphere,” Opt. Express 16(22), 18437–18442 (2008). [CrossRef]   [PubMed]  

32. K. Wang and C. Zhao, “Propagation properties of a radial phased-locked partially coherent anomalous hollow beam array in turbulent atmosphere,” Opt. Laser Technol. 57, 44–51 (2014). [CrossRef]  

33. Y. Yuan, Y. Cai, H. Eyyuboğlu, Y. Baykal, and J. Chen, “Propagation factor of partially coherent flat-topped beam array in free space and turbulent atmosphere,” Opt. Lasers Eng. 50(5), 752–759 (2012). [CrossRef]  

34. X. He and B. Lü, “Propagation of partially coherent flat-topped vortex beams through non-Kolmogorov, atmospheric turbulence,” J. Opt. Soc. Am. A 28(9), 1941–1948 (2011). [CrossRef]  

35. Y. Huang, B. Zhang, Z. Gao, G. Zhao, and Z. Duan, “Evolution behavior of Gaussian Schell-model vortex beams propagating through oceanic turbulence,” Opt. Express 22(15), 17723–17734 (2014). [CrossRef]   [PubMed]  

36. X. He and B. Lü, “Propagation properties of partially coherent Hermite-Gaussian beams through non-Kolmogorov turbulence,” Chin. Phys. B 20(9), 244–252 (2011). [CrossRef]  

37. M. Tang and D. Zhao, “Propagation of multi-Gaussian Schell-model vortex beams in isotropic random media,” Opt. Express 23(25), 32766–32776 (2015). [CrossRef]   [PubMed]  

38. G. Gbur and E. Wolf, “Spreading of partially coherent beams in random media,” J. Opt. Soc. Am. A 19(8), 1592–1598 (2002). [CrossRef]   [PubMed]  

39. 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]  

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

41. X. Chu, C. Qiao, and X. Feng, “The effect of non-Kolmogorov turbulence on the propagation of cosh-Gaussian beam,” Opt. Commun. 283(18), 3398–3403 (2010). [CrossRef]  

42. 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]  

43. A. Friberg and J. Turunen, “Imaging of Gaussian Schell-model sources,” J. Opt. Soc. Am. A 5(5), 713–720 (1988). [CrossRef]  

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

45. H. F. Schouten, G. Gbur, T. D. Visser, and E. Wolf, “Phase singularities of the coherence functions in Young’s interference pattern,” Opt. Lett. 28(12), 968–970 (2003). [CrossRef]   [PubMed]  

46. A. Dogariu and S. Amarande, “Propagation of partially coherent beams: turbulence-induced degradation,” Opt. Lett. 28(1), 10–12 (2003). [CrossRef]   [PubMed]  

47. 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]  

References

  • View by:

  1. T. Shirai, A. Dogariu, and E. Wolf, “Mode analysis of spreading of partially coherent beams propagating through atmospheric turbulence,” J. Opt. Soc. Am. A 20(6), 1094–1102 (2003).
    [Crossref] [PubMed]
  2. T. Shirai, A. Dogariu, and E. Wolf, “Directionality of Gaussian Schell-model beams propagating in atmospheric turbulence,” Opt. Lett. 28(8), 610–612 (2003).
    [Crossref] [PubMed]
  3. X. Wang, M. Yao, Z. Qiu, X. Yi, and Z. Liu, “Evolution properties of Bessel-Gaussian Schell-model beams in non-Kolmogorov turbulence,” Opt. Express 23(10), 12508–12523 (2015).
    [Crossref] [PubMed]
  4. Y. Cai and S. He, “Propagation of a partially coherent twisted anisotropic Gaussian Schell-model beam in a turbulent atmosphere,” Appl. Phys. Lett. 89(4), 041117 (2006).
    [Crossref]
  5. X. Du, D. Zhao, and O. Korotkova, “Changes in the statistical properties of stochastic anisotropic electromagnetic beams on propagation in the turbulent atmosphere,” Opt. Express 15(25), 16909–16915 (2007).
    [Crossref] [PubMed]
  6. Y. Dan and B. Zhang, “Beam propagation factor of partially coherent flat-topped beams in a turbulent atmosphere,” Opt. Express 16(20), 15563–15575 (2008).
    [Crossref] [PubMed]
  7. G. Wu, H. Guo, S. Yu, and B. Luo, “Spreading and direction of Gaussian-Schell model beam through a non-Kolmogorov turbulence,” Opt. Lett. 35(5), 715–717 (2010).
    [Crossref] [PubMed]
  8. M. Tang and D. Zhao, “Spectral changes in stochastic electromagnetic beams propagating through turbulent ocean,” Opt. Commun. 312(1), 89–93 (2014).
    [Crossref]
  9. O. Korotkova and E. Shchepakina, “Rectangular Multi-Gaussian Schell-Model beams in atmospheric turbulence,” J. Opt. 16(4), 045704 (2014).
    [Crossref]
  10. O. Korotkova, L. C. Andrews, and R. L. Phillips, “Speckle propagation through atmospheric turbulence: effects of a random phase screen at the source,” Proc. SPIE 4821, 98–109 (2002).
    [Crossref]
  11. O. Korotkova and E. Shchepakina, “Color changes in stochastic light fields propagating in non-Kolmogorov turbulence,” Opt. Lett. 35(22), 3772–3774 (2010).
    [Crossref] [PubMed]
  12. E. Shchepakina and O. Korotkova, “Second-order statistics of stochastic electromagnetic beams propagating through non-Kolmogorov turbulence,” Opt. Express 18(10), 10650–10658 (2010).
    [Crossref] [PubMed]
  13. 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]
  14. T. Wang, J. Pu, and Z. Chen, “Propagation of partially coherent vortex beams in a turbulent atmosphere,” Opt. Eng. 47(3), 036002 (2008).
    [Crossref]
  15. Y. Chen, F. Wang, C. Zhao, and Y. Cai, “Experimental demonstration of a Laguerre-Gaussian correlated Schell-model vortex beam,” Opt. Express 22(5), 5826–5838 (2014).
    [Crossref] [PubMed]
  16. Y. Zhang, L. Liu, C. Zhao, and Y. Cai, “Multi-Gaussian Schell-model vortex beam,” Phys. Lett. A 378(9), 750–754 (2014).
    [Crossref]
  17. J. He, X. Wang, D. Hu, J. Ye, S. Feng, Q. Kan, and Y. Zhang, “Generation and evolution of the terahertz vortex beam,” Opt. Express 21(17), 20230–20239 (2013).
    [Crossref] [PubMed]
  18. 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]
  19. Y. Ni and G. Zhou, “Propagation of a Lorentz–Gauss vortex beam through a paraxial ABCD optical system,” Opt. Commun. 291(15), 19–25 (2013).
    [Crossref]
  20. B. Chen, C. Chen, X. Peng, and D. Deng, “Propagation of Airy Gaussian vortex beams through slabs of right-handed materials and left-handed materials,” J. Opt. Soc. Am. B 32(1), 173–178 (2015).
    [Crossref]
  21. Q. Zhan, “Properties of circularly polarized vortex beams,” Opt. Lett. 31(7), 867–869 (2006).
    [Crossref] [PubMed]
  22. V. V. Kotlyar, A. A. Kovalev, and A. P. Porfirev, “Vortex Hermite-Gaussian laser beams,” Opt. Lett. 40(5), 701–704 (2015).
    [Crossref] [PubMed]
  23. H. Liu, Y. Lü, J. Xia, X. Pu, and L. Zhang, “Flat-topped vortex hollow beam and its propagation properties,” J. Opt. 17(7), 075606 (2015).
    [Crossref]
  24. D. Liu, Y. Wang, and H. Yin, “Evolution properties of partially coherent four-petal Gaussian vortex beams propagating in uniaxial crystals orthogonal to the optical axis,” J. Opt. Soc. Am. A 32(9), 1683–1690 (2015).
    [Crossref] [PubMed]
  25. L. Wang, L. Wang, and S. Zhu, “Formation of optical vortices using coherent laser beam array,” Opt. Commun. 282(6), 1088–1094 (2009).
    [Crossref]
  26. X. Du and D. Zhao, “Statistical properties of correlated radial stochastic electromagnetic array beams on propagation,” Opt. Commun. 282(10), 1993–1997 (2009).
    [Crossref]
  27. X. Du and D. Zhao, “Propagation characteristics of stochastic electromagnetic array beams,” Appl. Phys. B 93(4), 901–905 (2008).
    [Crossref]
  28. P. Zhou, X. Wang, Y. Ma, and Z. Liu, “Propagation properties of a Lorentz beam array,” Appl. Opt. 49(13), 2497–2503 (2010).
    [Crossref]
  29. G. Zhou, “Propagation of a radial phased-locked Lorentz beam array in turbulent atmosphere,” Opt. Express 19(24), 24699–24711 (2011).
    [Crossref] [PubMed]
  30. H. Tang, B. Ou, B. Luo, H. Guo, and A. Dang, “Average spreading of a radial gaussian beam array in non-Kolmogorov turbulence,” J. Opt. Soc. Am. A 28(6), 1016–1021 (2011).
    [Crossref] [PubMed]
  31. Y. Zhu, D. Zhao, and X. Du, “Propagation of stochastic Gaussian-Schell model array beams in turbulent atmosphere,” Opt. Express 16(22), 18437–18442 (2008).
    [Crossref] [PubMed]
  32. K. Wang and C. Zhao, “Propagation properties of a radial phased-locked partially coherent anomalous hollow beam array in turbulent atmosphere,” Opt. Laser Technol. 57, 44–51 (2014).
    [Crossref]
  33. Y. Yuan, Y. Cai, H. Eyyuboğlu, Y. Baykal, and J. Chen, “Propagation factor of partially coherent flat-topped beam array in free space and turbulent atmosphere,” Opt. Lasers Eng. 50(5), 752–759 (2012).
    [Crossref]
  34. X. He and B. Lü, “Propagation of partially coherent flat-topped vortex beams through non-Kolmogorov, atmospheric turbulence,” J. Opt. Soc. Am. A 28(9), 1941–1948 (2011).
    [Crossref]
  35. Y. Huang, B. Zhang, Z. Gao, G. Zhao, and Z. Duan, “Evolution behavior of Gaussian Schell-model vortex beams propagating through oceanic turbulence,” Opt. Express 22(15), 17723–17734 (2014).
    [Crossref] [PubMed]
  36. X. He and B. Lü, “Propagation properties of partially coherent Hermite-Gaussian beams through non-Kolmogorov turbulence,” Chin. Phys. B 20(9), 244–252 (2011).
    [Crossref]
  37. M. Tang and D. Zhao, “Propagation of multi-Gaussian Schell-model vortex beams in isotropic random media,” Opt. Express 23(25), 32766–32776 (2015).
    [Crossref] [PubMed]
  38. G. Gbur and E. Wolf, “Spreading of partially coherent beams in random media,” J. Opt. Soc. Am. A 19(8), 1592–1598 (2002).
    [Crossref] [PubMed]
  39. 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]
  40. I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products (Academic, 2007).
  41. X. Chu, C. Qiao, and X. Feng, “The effect of non-Kolmogorov turbulence on the propagation of cosh-Gaussian beam,” Opt. Commun. 283(18), 3398–3403 (2010).
    [Crossref]
  42. 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]
  43. A. Friberg and J. Turunen, “Imaging of Gaussian Schell-model sources,” J. Opt. Soc. Am. A 5(5), 713–720 (1988).
    [Crossref]
  44. G. Gbur and T. Visser, “Coherence vortices in partially coherent beams,” Opt. Commun. 222(1–6), 117–125 (2003).
    [Crossref]
  45. H. F. Schouten, G. Gbur, T. D. Visser, and E. Wolf, “Phase singularities of the coherence functions in Young’s interference pattern,” Opt. Lett. 28(12), 968–970 (2003).
    [Crossref] [PubMed]
  46. A. Dogariu and S. Amarande, “Propagation of partially coherent beams: turbulence-induced degradation,” Opt. Lett. 28(1), 10–12 (2003).
    [Crossref] [PubMed]
  47. 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]

2015 (6)

2014 (6)

Y. Huang, B. Zhang, Z. Gao, G. Zhao, and Z. Duan, “Evolution behavior of Gaussian Schell-model vortex beams propagating through oceanic turbulence,” Opt. Express 22(15), 17723–17734 (2014).
[Crossref] [PubMed]

K. Wang and C. Zhao, “Propagation properties of a radial phased-locked partially coherent anomalous hollow beam array in turbulent atmosphere,” Opt. Laser Technol. 57, 44–51 (2014).
[Crossref]

M. Tang and D. Zhao, “Spectral changes in stochastic electromagnetic beams propagating through turbulent ocean,” Opt. Commun. 312(1), 89–93 (2014).
[Crossref]

O. Korotkova and E. Shchepakina, “Rectangular Multi-Gaussian Schell-Model beams in atmospheric turbulence,” J. Opt. 16(4), 045704 (2014).
[Crossref]

Y. Chen, F. Wang, C. Zhao, and Y. Cai, “Experimental demonstration of a Laguerre-Gaussian correlated Schell-model vortex beam,” Opt. Express 22(5), 5826–5838 (2014).
[Crossref] [PubMed]

Y. Zhang, L. Liu, C. Zhao, and Y. Cai, “Multi-Gaussian Schell-model vortex beam,” Phys. Lett. A 378(9), 750–754 (2014).
[Crossref]

2013 (3)

2012 (1)

Y. Yuan, Y. Cai, H. Eyyuboğlu, Y. Baykal, and J. Chen, “Propagation factor of partially coherent flat-topped beam array in free space and turbulent atmosphere,” Opt. Lasers Eng. 50(5), 752–759 (2012).
[Crossref]

2011 (4)

2010 (5)

2009 (2)

L. Wang, L. Wang, and S. Zhu, “Formation of optical vortices using coherent laser beam array,” Opt. Commun. 282(6), 1088–1094 (2009).
[Crossref]

X. Du and D. Zhao, “Statistical properties of correlated radial stochastic electromagnetic array beams on propagation,” Opt. Commun. 282(10), 1993–1997 (2009).
[Crossref]

2008 (6)

2007 (3)

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]

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]

X. Du, D. Zhao, and O. Korotkova, “Changes in the statistical properties of stochastic anisotropic electromagnetic beams on propagation in the turbulent atmosphere,” Opt. Express 15(25), 16909–16915 (2007).
[Crossref] [PubMed]

2006 (2)

Y. Cai and S. He, “Propagation of a partially coherent twisted anisotropic Gaussian Schell-model beam in a turbulent atmosphere,” Appl. Phys. Lett. 89(4), 041117 (2006).
[Crossref]

Q. Zhan, “Properties of circularly polarized vortex beams,” Opt. Lett. 31(7), 867–869 (2006).
[Crossref] [PubMed]

2003 (5)

2002 (2)

O. Korotkova, L. C. Andrews, and R. L. Phillips, “Speckle propagation through atmospheric turbulence: effects of a random phase screen at the source,” Proc. SPIE 4821, 98–109 (2002).
[Crossref]

G. Gbur and E. Wolf, “Spreading of partially coherent beams in random media,” J. Opt. Soc. Am. A 19(8), 1592–1598 (2002).
[Crossref] [PubMed]

1988 (1)

Amarande, S.

Andrews, L. C.

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]

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]

O. Korotkova, L. C. Andrews, and R. L. Phillips, “Speckle propagation through atmospheric turbulence: effects of a random phase screen at the source,” Proc. SPIE 4821, 98–109 (2002).
[Crossref]

Baykal, Y.

Y. Yuan, Y. Cai, H. Eyyuboğlu, Y. Baykal, and J. Chen, “Propagation factor of partially coherent flat-topped beam array in free space and turbulent atmosphere,” Opt. Lasers Eng. 50(5), 752–759 (2012).
[Crossref]

Cai, Y.

Y. Chen, F. Wang, C. Zhao, and Y. Cai, “Experimental demonstration of a Laguerre-Gaussian correlated Schell-model vortex beam,” Opt. Express 22(5), 5826–5838 (2014).
[Crossref] [PubMed]

Y. Zhang, L. Liu, C. Zhao, and Y. Cai, “Multi-Gaussian Schell-model vortex beam,” Phys. Lett. A 378(9), 750–754 (2014).
[Crossref]

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]

Y. Yuan, Y. Cai, H. Eyyuboğlu, Y. Baykal, and J. Chen, “Propagation factor of partially coherent flat-topped beam array in free space and turbulent atmosphere,” Opt. Lasers Eng. 50(5), 752–759 (2012).
[Crossref]

Y. Cai and S. He, “Propagation of a partially coherent twisted anisotropic Gaussian Schell-model beam in a turbulent atmosphere,” Appl. Phys. Lett. 89(4), 041117 (2006).
[Crossref]

Chen, B.

Chen, C.

Chen, J.

Y. Yuan, Y. Cai, H. Eyyuboğlu, Y. Baykal, and J. Chen, “Propagation factor of partially coherent flat-topped beam array in free space and turbulent atmosphere,” Opt. Lasers Eng. 50(5), 752–759 (2012).
[Crossref]

Chen, Y.

Chen, Z.

T. Wang, J. Pu, and Z. Chen, “Propagation of partially coherent vortex beams in a turbulent atmosphere,” Opt. Eng. 47(3), 036002 (2008).
[Crossref]

Chu, X.

X. Chu, C. Qiao, and X. Feng, “The effect of non-Kolmogorov turbulence on the propagation of cosh-Gaussian beam,” Opt. Commun. 283(18), 3398–3403 (2010).
[Crossref]

Dan, Y.

Dang, A.

Deng, D.

Dogariu, A.

Dong, Y.

Du, X.

X. Du and D. Zhao, “Statistical properties of correlated radial stochastic electromagnetic array beams on propagation,” Opt. Commun. 282(10), 1993–1997 (2009).
[Crossref]

X. Du and D. Zhao, “Propagation characteristics of stochastic electromagnetic array beams,” Appl. Phys. B 93(4), 901–905 (2008).
[Crossref]

Y. Zhu, D. Zhao, and X. Du, “Propagation of stochastic Gaussian-Schell model array beams in turbulent atmosphere,” Opt. Express 16(22), 18437–18442 (2008).
[Crossref] [PubMed]

X. Du, D. Zhao, and O. Korotkova, “Changes in the statistical properties of stochastic anisotropic electromagnetic beams on propagation in the turbulent atmosphere,” Opt. Express 15(25), 16909–16915 (2007).
[Crossref] [PubMed]

Duan, Z.

Eyyuboglu, H.

Y. Yuan, Y. Cai, H. Eyyuboğlu, Y. Baykal, and J. Chen, “Propagation factor of partially coherent flat-topped beam array in free space and turbulent atmosphere,” Opt. Lasers Eng. 50(5), 752–759 (2012).
[Crossref]

Feng, S.

Feng, X.

X. Chu, C. Qiao, and X. Feng, “The effect of non-Kolmogorov turbulence on the propagation of cosh-Gaussian beam,” Opt. Commun. 283(18), 3398–3403 (2010).
[Crossref]

Ferrero, V.

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]

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]

Friberg, A.

Gao, Z.

Gbur, G.

Guo, H.

He, J.

He, S.

Y. Cai and S. He, “Propagation of a partially coherent twisted anisotropic Gaussian Schell-model beam in a turbulent atmosphere,” Appl. Phys. Lett. 89(4), 041117 (2006).
[Crossref]

He, X.

X. He and B. Lü, “Propagation properties of partially coherent Hermite-Gaussian beams through non-Kolmogorov turbulence,” Chin. Phys. B 20(9), 244–252 (2011).
[Crossref]

X. He and B. Lü, “Propagation of partially coherent flat-topped vortex beams through non-Kolmogorov, atmospheric turbulence,” J. Opt. Soc. Am. A 28(9), 1941–1948 (2011).
[Crossref]

Hu, D.

Huang, Y.

Kan, Q.

Korotkova, O.

Kotlyar, V. V.

Kovalev, A. A.

Liu, D.

Liu, H.

H. Liu, Y. Lü, J. Xia, X. Pu, and L. Zhang, “Flat-topped vortex hollow beam and its propagation properties,” J. Opt. 17(7), 075606 (2015).
[Crossref]

Liu, L.

Y. Zhang, L. Liu, C. Zhao, and Y. Cai, “Multi-Gaussian Schell-model vortex beam,” Phys. Lett. A 378(9), 750–754 (2014).
[Crossref]

Liu, Z.

Lü, B.

X. He and B. Lü, “Propagation properties of partially coherent Hermite-Gaussian beams through non-Kolmogorov turbulence,” Chin. Phys. B 20(9), 244–252 (2011).
[Crossref]

X. He and B. Lü, “Propagation of partially coherent flat-topped vortex beams through non-Kolmogorov, atmospheric turbulence,” J. Opt. Soc. Am. A 28(9), 1941–1948 (2011).
[Crossref]

Lü, Y.

H. Liu, Y. Lü, J. Xia, X. Pu, and L. Zhang, “Flat-topped vortex hollow beam and its propagation properties,” J. Opt. 17(7), 075606 (2015).
[Crossref]

Luo, B.

Ma, Y.

Ni, Y.

Y. Ni and G. Zhou, “Propagation of a Lorentz–Gauss vortex beam through a paraxial ABCD optical system,” Opt. Commun. 291(15), 19–25 (2013).
[Crossref]

Ou, B.

Peng, X.

Phillips, R. L.

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]

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]

O. Korotkova, L. C. Andrews, and R. L. Phillips, “Speckle propagation through atmospheric turbulence: effects of a random phase screen at the source,” Proc. SPIE 4821, 98–109 (2002).
[Crossref]

Porfirev, A. P.

Pu, J.

T. Wang, J. Pu, and Z. Chen, “Propagation of partially coherent vortex beams in a turbulent atmosphere,” Opt. Eng. 47(3), 036002 (2008).
[Crossref]

Pu, X.

H. Liu, Y. Lü, J. Xia, X. Pu, and L. Zhang, “Flat-topped vortex hollow beam and its propagation properties,” J. Opt. 17(7), 075606 (2015).
[Crossref]

Qiao, C.

X. Chu, C. Qiao, and X. Feng, “The effect of non-Kolmogorov turbulence on the propagation of cosh-Gaussian beam,” Opt. Commun. 283(18), 3398–3403 (2010).
[Crossref]

Qiu, Z.

Schouten, H. F.

Shchepakina, E.

Shirai, T.

Tang, H.

Tang, M.

M. Tang and D. Zhao, “Propagation of multi-Gaussian Schell-model vortex beams in isotropic random media,” Opt. Express 23(25), 32766–32776 (2015).
[Crossref] [PubMed]

M. Tang and D. Zhao, “Spectral changes in stochastic electromagnetic beams propagating through turbulent ocean,” Opt. Commun. 312(1), 89–93 (2014).
[Crossref]

Toselli, I.

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]

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]

Turunen, J.

Tyson, R. K.

Visser, T.

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

Visser, T. D.

Wang, F.

Wang, K.

K. Wang and C. Zhao, “Propagation properties of a radial phased-locked partially coherent anomalous hollow beam array in turbulent atmosphere,” Opt. Laser Technol. 57, 44–51 (2014).
[Crossref]

Wang, L.

L. Wang, L. Wang, and S. Zhu, “Formation of optical vortices using coherent laser beam array,” Opt. Commun. 282(6), 1088–1094 (2009).
[Crossref]

L. Wang, L. Wang, and S. Zhu, “Formation of optical vortices using coherent laser beam array,” Opt. Commun. 282(6), 1088–1094 (2009).
[Crossref]

Wang, T.

T. Wang, J. Pu, and Z. Chen, “Propagation of partially coherent vortex beams in a turbulent atmosphere,” Opt. Eng. 47(3), 036002 (2008).
[Crossref]

Wang, X.

Wang, Y.

Wolf, E.

Wu, G.

Xia, J.

H. Liu, Y. Lü, J. Xia, X. Pu, and L. Zhang, “Flat-topped vortex hollow beam and its propagation properties,” J. Opt. 17(7), 075606 (2015).
[Crossref]

Yang, Y.

Yao, M.

Ye, J.

Yi, X.

Yin, H.

Yu, S.

Yuan, Y.

Y. Yuan, Y. Cai, H. Eyyuboğlu, Y. Baykal, and J. Chen, “Propagation factor of partially coherent flat-topped beam array in free space and turbulent atmosphere,” Opt. Lasers Eng. 50(5), 752–759 (2012).
[Crossref]

Zhan, Q.

Zhang, B.

Zhang, L.

H. Liu, Y. Lü, J. Xia, X. Pu, and L. Zhang, “Flat-topped vortex hollow beam and its propagation properties,” J. Opt. 17(7), 075606 (2015).
[Crossref]

Zhang, Y.

Zhao, C.

Y. Zhang, L. Liu, C. Zhao, and Y. Cai, “Multi-Gaussian Schell-model vortex beam,” Phys. Lett. A 378(9), 750–754 (2014).
[Crossref]

Y. Chen, F. Wang, C. Zhao, and Y. Cai, “Experimental demonstration of a Laguerre-Gaussian correlated Schell-model vortex beam,” Opt. Express 22(5), 5826–5838 (2014).
[Crossref] [PubMed]

K. Wang and C. Zhao, “Propagation properties of a radial phased-locked partially coherent anomalous hollow beam array in turbulent atmosphere,” Opt. Laser Technol. 57, 44–51 (2014).
[Crossref]

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]

Zhao, D.

M. Tang and D. Zhao, “Propagation of multi-Gaussian Schell-model vortex beams in isotropic random media,” Opt. Express 23(25), 32766–32776 (2015).
[Crossref] [PubMed]

M. Tang and D. Zhao, “Spectral changes in stochastic electromagnetic beams propagating through turbulent ocean,” Opt. Commun. 312(1), 89–93 (2014).
[Crossref]

X. Du and D. Zhao, “Statistical properties of correlated radial stochastic electromagnetic array beams on propagation,” Opt. Commun. 282(10), 1993–1997 (2009).
[Crossref]

X. Du and D. Zhao, “Propagation characteristics of stochastic electromagnetic array beams,” Appl. Phys. B 93(4), 901–905 (2008).
[Crossref]

Y. Zhu, D. Zhao, and X. Du, “Propagation of stochastic Gaussian-Schell model array beams in turbulent atmosphere,” Opt. Express 16(22), 18437–18442 (2008).
[Crossref] [PubMed]

X. Du, D. Zhao, and O. Korotkova, “Changes in the statistical properties of stochastic anisotropic electromagnetic beams on propagation in the turbulent atmosphere,” Opt. Express 15(25), 16909–16915 (2007).
[Crossref] [PubMed]

Zhao, G.

Zhou, G.

Y. Ni and G. Zhou, “Propagation of a Lorentz–Gauss vortex beam through a paraxial ABCD optical system,” Opt. Commun. 291(15), 19–25 (2013).
[Crossref]

G. Zhou, “Propagation of a radial phased-locked Lorentz beam array in turbulent atmosphere,” Opt. Express 19(24), 24699–24711 (2011).
[Crossref] [PubMed]

Zhou, P.

Zhu, S.

L. Wang, L. Wang, and S. Zhu, “Formation of optical vortices using coherent laser beam array,” Opt. Commun. 282(6), 1088–1094 (2009).
[Crossref]

Zhu, Y.

Appl. Opt. (1)

Appl. Phys. B (1)

X. Du and D. Zhao, “Propagation characteristics of stochastic electromagnetic array beams,” Appl. Phys. B 93(4), 901–905 (2008).
[Crossref]

Appl. Phys. Lett. (1)

Y. Cai and S. He, “Propagation of a partially coherent twisted anisotropic Gaussian Schell-model beam in a turbulent atmosphere,” Appl. Phys. Lett. 89(4), 041117 (2006).
[Crossref]

Chin. Phys. B (1)

X. He and B. Lü, “Propagation properties of partially coherent Hermite-Gaussian beams through non-Kolmogorov turbulence,” Chin. Phys. B 20(9), 244–252 (2011).
[Crossref]

J. Opt. (2)

O. Korotkova and E. Shchepakina, “Rectangular Multi-Gaussian Schell-Model beams in atmospheric turbulence,” J. Opt. 16(4), 045704 (2014).
[Crossref]

H. Liu, Y. Lü, J. Xia, X. Pu, and L. Zhang, “Flat-topped vortex hollow beam and its propagation properties,” J. Opt. 17(7), 075606 (2015).
[Crossref]

J. Opt. Soc. Am. A (8)

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

Opt. Commun. (6)

L. Wang, L. Wang, and S. Zhu, “Formation of optical vortices using coherent laser beam array,” Opt. Commun. 282(6), 1088–1094 (2009).
[Crossref]

X. Du and D. Zhao, “Statistical properties of correlated radial stochastic electromagnetic array beams on propagation,” Opt. Commun. 282(10), 1993–1997 (2009).
[Crossref]

Y. Ni and G. Zhou, “Propagation of a Lorentz–Gauss vortex beam through a paraxial ABCD optical system,” Opt. Commun. 291(15), 19–25 (2013).
[Crossref]

M. Tang and D. Zhao, “Spectral changes in stochastic electromagnetic beams propagating through turbulent ocean,” Opt. Commun. 312(1), 89–93 (2014).
[Crossref]

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

X. Chu, C. Qiao, and X. Feng, “The effect of non-Kolmogorov turbulence on the propagation of cosh-Gaussian beam,” Opt. Commun. 283(18), 3398–3403 (2010).
[Crossref]

Opt. Eng. (1)

T. Wang, J. Pu, and Z. Chen, “Propagation of partially coherent vortex beams in a turbulent atmosphere,” Opt. Eng. 47(3), 036002 (2008).
[Crossref]

Opt. Express (10)

Y. Chen, F. Wang, C. Zhao, and Y. Cai, “Experimental demonstration of a Laguerre-Gaussian correlated Schell-model vortex beam,” Opt. Express 22(5), 5826–5838 (2014).
[Crossref] [PubMed]

E. Shchepakina and O. Korotkova, “Second-order statistics of stochastic electromagnetic beams propagating through non-Kolmogorov turbulence,” Opt. Express 18(10), 10650–10658 (2010).
[Crossref] [PubMed]

X. Wang, M. Yao, Z. Qiu, X. Yi, and Z. Liu, “Evolution properties of Bessel-Gaussian Schell-model beams in non-Kolmogorov turbulence,” Opt. Express 23(10), 12508–12523 (2015).
[Crossref] [PubMed]

X. Du, D. Zhao, and O. Korotkova, “Changes in the statistical properties of stochastic anisotropic electromagnetic beams on propagation in the turbulent atmosphere,” Opt. Express 15(25), 16909–16915 (2007).
[Crossref] [PubMed]

Y. Dan and B. Zhang, “Beam propagation factor of partially coherent flat-topped beams in a turbulent atmosphere,” Opt. Express 16(20), 15563–15575 (2008).
[Crossref] [PubMed]

G. Zhou, “Propagation of a radial phased-locked Lorentz beam array in turbulent atmosphere,” Opt. Express 19(24), 24699–24711 (2011).
[Crossref] [PubMed]

Y. Zhu, D. Zhao, and X. Du, “Propagation of stochastic Gaussian-Schell model array beams in turbulent atmosphere,” Opt. Express 16(22), 18437–18442 (2008).
[Crossref] [PubMed]

J. He, X. Wang, D. Hu, J. Ye, S. Feng, Q. Kan, and Y. Zhang, “Generation and evolution of the terahertz vortex beam,” Opt. Express 21(17), 20230–20239 (2013).
[Crossref] [PubMed]

Y. Huang, B. Zhang, Z. Gao, G. Zhao, and Z. Duan, “Evolution behavior of Gaussian Schell-model vortex beams propagating through oceanic turbulence,” Opt. Express 22(15), 17723–17734 (2014).
[Crossref] [PubMed]

M. Tang and D. Zhao, “Propagation of multi-Gaussian Schell-model vortex beams in isotropic random media,” Opt. Express 23(25), 32766–32776 (2015).
[Crossref] [PubMed]

Opt. Laser Technol. (1)

K. Wang and C. Zhao, “Propagation properties of a radial phased-locked partially coherent anomalous hollow beam array in turbulent atmosphere,” Opt. Laser Technol. 57, 44–51 (2014).
[Crossref]

Opt. Lasers Eng. (1)

Y. Yuan, Y. Cai, H. Eyyuboğlu, Y. Baykal, and J. Chen, “Propagation factor of partially coherent flat-topped beam array in free space and turbulent atmosphere,” Opt. Lasers Eng. 50(5), 752–759 (2012).
[Crossref]

Opt. Lett. (8)

Phys. Lett. A (1)

Y. Zhang, L. Liu, C. Zhao, and Y. Cai, “Multi-Gaussian Schell-model vortex beam,” Phys. Lett. A 378(9), 750–754 (2014).
[Crossref]

Proc. SPIE (3)

O. Korotkova, L. C. Andrews, and R. L. Phillips, “Speckle propagation through atmospheric turbulence: effects of a random phase screen at the source,” Proc. SPIE 4821, 98–109 (2002).
[Crossref]

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]

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]

Other (1)

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

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Figures (14)

Fig. 1
Fig. 1 Intensity distribution of a radial phased-locked flat-topped beam vortex array in the source plane.
Fig. 2
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.
Fig. 3
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.
Fig. 4
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.
Fig. 5
Fig. 5 Normalized intensity in y-direction at different propagation distance z.
Fig. 6
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) Cn2 = 10−12m3-α (b) Cn2 = 5 × 10−12m3-α(c) Cn2 = 10−11m3-α (d) Cn2 = 10−10m3-α.
Fig. 7
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.
Fig. 8
Fig. 8 Normalized intensity in y-direction for different values of σ0 at the different propagation distance z.(a)z = 500m (b)z = 1000m.
Fig. 9
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.
Fig. 10
Fig. 10 K versus α for different propagation distance z.
Fig. 11
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.
Fig. 12
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.
Fig. 13
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.
Fig. 14
Fig. 14 The complex degree of spatial coherence, (a) at several z values with δ = 5 mm, r = 3cm, Cn2 = 10−14m3-α, (b) at several δ values with r = 3cm, z = 1 km, Cn2 = 10−14m3-α, (c) at several r values with δ = 5 mm, z = 1 km, Cn2 = 10−14m3-α, (d) at several Cn2 values with δ = 5 mm, z = 1 km, r = 3 cm.

Equations (48)

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E m ( ρ x , ρ y ,0)= [ ρ x a mx +isgn( l )( ρ y a my ) ] | l | n=1 N ( 1 ) n1 N ( N n )exp( n ( ρ x a mx ) 2 w 2 n ( ρ y a my ) 2 w )exp( i φ m ).
W m ( 0 ) ( ρ 1 , ρ 2 ,0 )= I m ( ρ 1x , ρ 1y ,0) I m ( ρ 2x , ρ 2y ,0) exp[ ( n 1 + n 2 ) ( ρ 1 ρ 2 ) 2 4 σ 0 2 ].
W ( 0 ) ( ρ 1 , ρ 2 ,0 )= m=1 M W m ( 0 ) ( ρ 1 , ρ 2 ,0 ) .
W ( 0 ) ( ρ 1 , ρ 2 ,0 )={ ρ 1x ρ 2x + ρ 1y ρ 2y a mx ( ρ 1x + ρ 2x ) a my ( ρ 1y + ρ 2y ) + i[ ρ 1x ρ 2y ρ 2x ρ 1y a my ( ρ 1x ρ 2x )+ a mx ( ρ 1y ρ 2y ) ]+ a mx 2 + a my 2 } × m=1 M n 1 =1 N n 2 =1 N ( 1 ) n 1 + n 2 2 N 2 ( N n 1 )( N n 2 ) exp[ n 1 w 2 ( ρ 1x 2 + ρ 1y 2 ) n 2 w 2 ( ρ 2x 2 + ρ 2y 2 ) ] ×exp[ 2 a mx w 2 ( n 1 ρ 1x + n 2 ρ 2x )+ 2 a my w 2 ( n 1 ρ 1y + n 2 ρ 2y ) ]exp( i φ m ) ×exp[ ( a mx 2 + a my 2 )( n 1 + n 2 ) w 2 ]exp[ ( n 1 + n 2 ) 4 σ 0 2 ( ρ 1 ρ 2 ) 2 ].
W( ρ 1 , ρ 2 ,z )= k 2 4 π 2 z 2 + + + + W ( 0 ) ( ρ 1 , ρ 2 ,0 ) ×exp{ ik 2z [ ( ρ 1 ρ 1 ) 2 ( ρ 2 ρ 2 ) 2 ] } × exp[ ψ * ( ρ 1 , ρ 1 )+ψ( ρ 2 , ρ 2 ) ] d 2 ρ 1 d 2 ρ 2 .
exp[ ψ * ( ρ 1 , ρ 1 )+ψ( ρ 2 , ρ 2 ) ] =exp{ 4 π 2 k 2 z 0 1 0 + dκdξ Φ n ( κ,α )[ 1 J 0 ( κ| ( 1ξ )( ρ 1 ρ 2 )+ξ( ρ 1 ρ 2 ) | ) ] } =exp{ π 2 k 2 z 3 0 + κ 3 Φ n ( κ,α )dκ[ ( ρ 1 ρ 2 ) 2 +( ρ 1 ρ 2 )( ρ 1 ρ 2 )+ ( ρ 1 ρ 2 ) 2 ] }.
Φ n ( κ,α )= A( α ) C ˜ n 2 ( κ 2 + κ 0 2 ) α/2 exp( κ 2 κ m 2 ).
K= π 2 k 2 z 3 0 + κ 3 Φ n ( κ,α )dκ= π 2 k 2 zA( α ) C ˜ n 2 6( α2 ) [ βexp( κ 0 2 κ m 2 )Γ( 4α 2 , κ 0 2 κ m 2 ) κ m 2α 2 κ 0 4α ].
W( ρ 1 , ρ 2 ,z )= k 2 4 π 2 z 2 exp[ ik 2z ( ρ 1 2 ρ 2 2 ) ]exp[ K ( x 1 x 2 ) 2 K ( y 1 y 2 ) 2 ] × m=1 M n 1 =1 N n 2 =1 N ( 1 ) n 1 + n 2 2 N 2 ( N n 1 )( N n 2 ) + + + + d 2 u d 2 v exp[ ( a mx 2 + a my 2 )( n 1 + n 2 ) w 2 ] ×[ u v 4 2 a mx u x 2 a my u y +i( v x u y u x v y a my v x + a mx v y )+ a mx 2 + a my 2 ] ×exp[ 2 a mx ( n 1 + n 2 ) w 2 u x + a mx ( n 1 n 2 ) w 2 v x + 2 a my ( n 1 + n 2 ) w 2 u y + a my ( n 1 n 2 ) w 2 v y ] ×exp[ ik( ρ 1 2 ρ 2 2 ) z u ]exp{ [ ik( ρ 1 2 + ρ 2 2 ) 2z K( ρ 1 ρ 2 ) ]v } ×exp( n 1 + n 2 w 2 u 2 )exp( Q v 2 Puv )exp( i φ m ),
P= n 1 n 2 w 2 + ik z ,
Q= n 1 + n 2 4 w 2 + n 1 + n 2 4 σ 0 2 +K.
x n exp( A x 2 +Bx ) dx=n!exp( A 2 B ) π A ( A B ) n m=0 [ n/2 ] 1 m!( n2m )! ( A 4 B 2 ) m .
W( ρ 1 , ρ 2 ,z )= ( k 2πz ) 2 exp[ ik 2z ( ρ 1 2 ρ 2 2 ) ]exp[ K ( x 1 x 2 ) 2 K ( y 1 y 2 ) 2 ] × m=1 M n 1 =1 N n 2 =1 N ( 1 ) n 1 + n 2 2 N 2 ( N n 1 )( N n 2 ) exp( i φ m )exp[ ( a mx 2 + a my 2 )( n 1 + n 2 ) w 2 ] ×{ H 1 ( 1+ 2 C x 2 A )+ H 1 ( 1+ 2 C y 2 A ) H xy 4 ( 1 E + 2 D x 2 E 2 ) H xy 4 ( 1 E + 2 D y 2 E 2 )4 a mx H 1 C x 4 a my H 1 C y +i[ G yx ( D x C y EA ) G xy ( D y C x EA ) a my H xy ( D x E )+ a mx H xy ( D y E ) ]+ a mx 2 H xy + a my 2 H xy },
H 1 = π 2 2 A 2 Q B x B y exp( C x 2 + C y 2 A ),
H xy = π 2 w 2 2E( n 1 + n 2 ) exp{ [ ik( x 1 x 2 ) 2z + 2 a mx ( n 1 + n 2 ) 2 w 2 ] 2 w 2 n 1 + n 2 } ×exp{ [ ik( y 1 y 2 ) 2z + 2 a my ( n 1 + n 2 ) 2 w 2 ] 2 w 2 n 1 + n 2 }exp( D x 2 + D y 2 E ),
G y x = π 2 w AEQ( n 1 + n 2 ) B y exp{ [ ik( x 1 x 2 ) 2z + 2 a mx ( n 1 + n 2 ) 2 w 2 ] 2 w 2 n 1 + n 2 }exp( D x 2 E + C y 2 A ),
G xy π 2 w AEQ( n 1 + n 2 ) B x exp{ [ ik( y 1 y 2 ) 2z + 2 a my ( n 1 + n 2 ) 2 w 2 ] 2 w 2 n 1 + n 2 }exp( D y 2 E + C x 2 A ),
B j =exp[ a 2 mj ( n 1 n 2 ) 2 4Q w 4 ]exp[ ik a mj ( n 1 n 2 ) 4Q w 2 z ( j 1 + j 2 ) ] ×exp[ k 2 16Q z 2 ( j 1 + j 2 ) 2 ]exp[ ikz 4Qz K( j 1 2 j 2 2 ) ] ×exp[ a mj ( n 1 n 2 ) 2Q w 2 K( j 1 j 2 ) ]exp[ 1 4Q K 2 ( j 1 j 2 ) 2 ],
C j = a mj P( n 1 n 2 ) 4Q w 2 ikP( j 1 + j 2 ) 8Qz + P 4Q K( j 1 j 2 )+ ik( j 1 j 2 ) 2z + 2 a mj ( n 1 + n 2 ) 2 w 2 ,
D j = a mj ( n 1 n 2 ) 2 w 2 + ik( j 1 + j 2 ) 4z 1 2 K( j 1 j 2 ) ikP w 2 ( j 1 j 2 ) 4z( n 1 + n 2 ) a mj P 2 ,
A= n 1 + n 2 w 2 P 2 4Q ,
E=Q P 2 w 2 4( n 1 + n 2 ) .
I( ρ,z )=W( ρ,ρ,z )=W( x,y,x,y,z ) = ( k 2πz ) 2 m=1 M n 1 =1 N n 2 =1 N ( 1 ) n 1 + n 2 2 N 2 ( N n 1 )( N n 2 ) exp( i φ m )exp[ ( a mx 2 + a my 2 )( n 1 + n 2 ) w 2 ] ×{ 2 H 1 ( 1+ C x 2 + C y 2 A ) H xy 2 ( 1 E + D x 2 + D y 2 E 2 )4 H 1 ( a mx C x + a my C y ) + i[ ( G yx D x C y G xy D y C x EA )+ H xy E ( a mx D y a my D x ) ]+ H xy ( a mx 2 + a my 2 ) },
C j = a mj P( n 1 n 2 ) 4Q w 2 ikP 4Qz j+ a mj ( n 1 + n 2 ) w 2 ,
D j = a mj ( n 1 n 2 ) 2 w 2 + ik 2z j a mj P 2 ,
H xy = π 2 w 2 2E( n 1 + n 2 ) exp[ ( n 1 + n 2 ) w 2 ( a mx 2 + a my 2 ) ]exp( D x 2 + D y 2 E ),
G yx = π 2 w AEQ( n 1 + n 2 ) B y exp[ a mx 2 ( n 1 + n 2 ) w 2 ]exp( D x 2 E + C y 2 A ),
G xy = π 2 w AEQ( n 1 + n 2 ) B x exp[ a my 2 ( n 1 + n 2 ) w 2 ]exp( D y 2 E + C x 2 A ),
H 1 = π 2 2 A 2 Q B x B y exp( C x 2 + C y 2 A ),
B j =exp[ a 2 mj ( n 1 n 2 ) 2 4Q w 4 ]exp[ ik a mj ( n 1 n 2 ) 2Q w 2 z j ]exp[ k 2 4Q z 2 j 2 ].
μ( x 1 , x 2 ,z )= W( x 1 , x 2 ,z ) I( x 1 ,z )I( x 2 ,z ) .
μ( x,x,z )= Ω 1 Ω 2 Ω 3 ,
Ω p = m=1 M n 1 =1 N n 2 =1 N ( 1 ) n 1 + n 2 2 N 2 ( N n 1 )( N n 2 ) exp( i φ m )exp[ ( a mx 2 + a my 2 )( n 1 + n 2 ) w 2 ] ×{ 2 H p1 ( 1+ C px 2 + C 2 A ) H pxy 2E ( 1+ D px 2 + D 2 E )4 H p1 ( a mx C px + a my C ) + i[ ( G pyx D x C G pxy D C px EA )+ H pxy E ( a mx D a my D px ) ]+ H pxy ( a mx 2 + a my 2 ) },
H p1 = π 2 2 A 2 Q B px exp[ a 2 my ( n 1 n 2 ) 2 4Q w 4 ]exp( C px 2 + C 2 A ),
H 1xy = π 2 w 2 2E( n 1 + n 2 ) exp{ [ ik z x+ 2 a mx ( n 1 + n 2 ) 2 w 2 ] 2 w 2 n 1 + n 2 }exp[ a my 2 ( n 1 + n 2 ) w 2 ]exp( D 1x 2 + D 2 E ),
G 1y x = π 2 w AEQ( n 1 + n 2 ) exp[ a 2 my ( n 1 n 2 ) 2 4Q w 4 ]exp{ [ ikx z + 2 a mx ( n 1 + n 2 ) 2 w 2 ] 2 w 2 n 1 + n 2 }exp( D 1x 2 E + C 2 A ),
G 1xy π 2 w AEQ( n 1 + n 2 ) B 1x exp[ a my 2 ( n 1 + n 2 ) w 2 ]exp( D 2 E + C 1x 2 A ),
B 1x =exp[ a 2 mx ( n 1 n 2 ) 2 4Q w 4 ]exp[ a mx K( n 1 n 2 ) Q w 2 x ]exp( K 2 Q x 2 ),
C 1x = a mx P( n 1 n 2 ) 4Q w 2 + 2 a mx ( n 1 + n 2 ) 2 w 2 ( PK 2Q + ik z )x,
D 1x = a mx ( n 1 n 2 ) 2 w 2 a mx P 2 +[ K+ ikP w 2 2z( n 1 + n 2 ) ]x,
H qxy = π 2 w 2 2E( n 1 + n 2 ) exp[ ( n 1 + n 2 ) w 2 ( a mx 2 + a my 2 ) ]exp( D qx 2 + D 2 E ),
G qy x = π 2 w AEQ( n 1 + n 2 ) exp[ a 2 my ( n 1 n 2 ) 2 4Q w 4 ]exp[ a mx 2 ( n 1 + n 2 ) w 2 ]exp( D qx 2 E + C 2 A ),
G qxy = π 2 w AEQ( n 1 + n 2 ) B qx exp[ a my 2 ( n 1 + n 2 ) w 2 ]exp( D 2 E + C qx 2 A ),
B qx =exp[ a 2 mx ( n 1 n 2 ) 2 4Q w 4 ]exp[ ik a mx ( n 1 n 2 ) 2Q w 2 z x ]exp[ ( 1 ) q k 2 4Q z 2 x 2 ],
C qx = a mx P( n 1 n 2 ) 4Q w 2 + 2 a mx ( n 1 + n 2 ) 2 w 2 + ( 1 ) q ikP 4Qz x,
D qx = a mx ( n 1 n 2 ) 2 w 2 a mx P 2 + ( 1 ) q+1 ik 2z x,
C= a my P( n 1 n 2 ) 4Q w 2 + 2 a my ( n 1 + n 2 ) 2 w 2 ,
D= a my ( n 1 n 2 ) 2 w 2 a my P 2 .

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