Abstract

The beam spreading and evolution behavior of the intensity profile and coherent vortices of partially coherent, four-petal elliptic Gaussian vortex beams propagating in atmospheric turbulence are studied. The analytical expressions for cross-spectral density function, as well as root mean square (rms) beam width, are derived based on the extended Huygens–Fresnel principle. Results showed that, unlike the partially coherent four-petal Gaussian vortex beams, the partially coherent four-petal elliptic Gaussian vortex beam could change its petal number into six. The dependencies of occurrence, appearance, and transition speed from four- to six-petal profile on the topological charge, the beam order, and the ellipticity factor are illustrated. The far field behaviors of partially coherent four-petal elliptic Gaussian vortex beams propagating in atmospheric turbulence and are compared in free space. Beams with larger topological charge, smaller beam order, and larger ellipticity factor were found to be less influenced by atmospheric turbulence. Further, the ellipticity factor can be used as an additional degree of freedom in controlling the conservation distance of coherence vortices’ topological charge.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. Introduction

Propagation characteristics of various laser beams have attracted considerable attention in the past decade. Knowledge of these subjects would contribute to applications in laser communication, remote sensing, and directed energy systems [1]. Numerous types of laser beams have been investigated [1–5]. Among them, there is a growing interest in beams with flower petal structure, which hold great promise in rotational Doppler shift based remote sensing [6,7]. Recently, a laser beam with four petals called the four-petal Gaussian beam has been introduced [8]. Researches on the propagation factor M2, the nonparaxial propagation properties, and the far field vectorial structure of four-petal Gaussian beams have been reported [9–11]. Studies have also been extended to cases with phase singularity or with partial coherence, namely the four-petal Gaussian vortex beam and the partially coherent four-petal Gaussian vortex beam, respectively [12,13]. Special attention has been paid to propagation behavior of these beams in atmospheric and oceanic turbulence [14,15].

It has been shown that vortex beams with fractional orbital angular momentum (OAM) are more resistant to atmospheric thermal blooming [16]. A practical way to generate fractional-OAM optical vortices is to generate an elliptical optical vortex [17]. Elliptical optical vortex beams are also demonstrated to be capable of suppressing turbulence induced scintillations [18]. Researches on the generation, transformation, propagation, and detection of elliptical vortex beams have been carried out [19–22]. However, currently reported works mainly concentrate on Gaussian-like or Bessel-like elliptical vortex beams. To the best of our knowledge, propagating behavior of partially coherent petal-like beams with elliptical optical vortex has not been reported.

In this paper, propagation of partially coherent four-petal elliptic Gaussian vortex beams in atmospheric turbulence is investigated. Analytical formulas for the cross-spectral density function and root mean square (rms) beam width, and method for studying the position of coherence vortices are presented in Sec. 2. Propagation behavior, beam spreading character, and evolution of coherence vortices are analyzed in Sec. 3. Conclusions are drawn in Sec. 4.

2. Propagation theory

According to Gori [23], a sufficient condition for the mathematical form of cross-spectral density functions for optical fields to be genuine is that it can be expressed as:

W(r1,r2,0)=p(v)H*(r1,v)H(r2,v)dv,
where H is an arbitrary kernel and p is a nonnegative function. An instance of this type was given as:
W(r1,r2,0)=τ*(r1)τ(r2)p˜[a(r1r2)],
where τ is a complex profile function, a is a real parameter that plays the role of a scale factor within p˜, and p˜ is the Fourier transformation of arbitrary nonnegative, Fourier transformable function p. In this study, a is set to 1, p is supposed to take the form of a Schell model, and τ takes the form of the four-petal elliptic Gaussian form [12,24], expressed as
τ(x0,y0,0)=(x0y0ω02)2nexp(x02+α2y02ω02)(x02+α2y02ω02)|m|/2exp[imtan1(αy0/x0)],
where n is the order of the four-petal Gaussian beam, ω0 is the waist width, α is the ellipticity factor, m is the topological charge. Then, the cross-spectral density of the corresponding partially coherent four-petal elliptic Gaussian vortex beam at the z=0 plane can be written as:
W(r01,r02,0)=(x01y01ω02)2n(x02y02ω02)2nexp(x012+α2y012+x022+α2y022ω02)(1ω02)|m|[x01isgn(m)αy01]|m|[x02+isgn(m)αy02]|m|exp[(r01r02)22σ2].
Here r01=(x01,y01) and r02=(x02,y02) denote the position vectors at the source plane. σ denotes the transverse coherence width. sgn() is the sign function. || is the symbol of absolute value. Recalling the following equation [25]
(x+iy)M=l=0MM!ill!(Ml)!xMlyl,
Equation (4) can be rewritten as:
W(r01,r02,0)=1ω08n+2|m|m1=0|m|m2=0|m|(|m|m1)(|m|m2)(1)m1[iαsgn(m)]m1+m2x01|m|m1+2ny01m1+2nx02|m|m2+2ny02m2+2nexp(x012+x022ω02)exp(y012+y022ω02/α2)exp[(x01x02)2+(y01y02)22σ2],
where (··) denotes the binomial coefficient defined by (ab)=a!(ab)!b!. Based on the extended Huygens–Fresnel principle [2], the cross-spectral density function of laser beams propagating in atmospheric turbulence can be obtained as:
W(r1,r2,z)=(k2πz)2dr01dr02W(r01,r02,0)exp{ik2z[(r1r01)2(r2r02)2]}exp[Ψ*(r01,r1)+Ψ(r02,r2)],
where r1=(x1,y1) and r2=(x2,y2) denote the two-dimensional position vector at the z plane, k is the wavenumber, Ψ is the random part of the phase of a spherical wave due to turbulence. And Ψ satisfies the following relation [26]:
exp[Ψ*(r01,r1)+Ψ(r02,r2)]=exp{[(r01r02)2+(r01r02)(r1r2)+(r1r2)2]ρ02},
where ρ0=(0.546Cn2k2z)3/5is the coherence length of a spherical wave propagating in the turbulent medium. It should be noted that the quadratic approximation of the two-point spherical wave structure function is employed. By substituting Eqs. (6) and (8) into Eq. (7), and utilizing the relation [25]:
xnexp(px2+2qx)dx=n!exp(q2p)πp(qp)ns=0E[n2]1(n2s)!s!(p4q2)s=πp2ninexp(q2p)(1p)n/2Hn(iqp),
where E[·] denotes the integral part of the enclosed expression, Hn denotes the n-order Hermite polynomial, the derived formula is:
W(r1,r2,z)=(k2πz)21ω08n+2|m|exp[ik2z(x12x22+y12y22)]exp[(x1x2)2+(y1y2)2ρ02]m1=0|m|m2=0|m|(|m|m1)(|m|m2)(1)m1[iαsgn(m)]m1+m2WxWy,
where Wx is given by:
Wx=nx1!πAxπDxexp[1Ax(x2x12ρ02+ikx12z)2+Ex2Dx]s=0E(nx1/2)h=0nx12s1(nx12s)!s!4sAx(nx1s)(nx12sh)(x2x12ρ02+ikx12z)nx12sh(1ρ02+12σ2)h(i2Dx)|m|m2+2n+hH|m|m2+2n+h(iExDx),
With:
nx1=|m|m1+2n,
Ax=1ρ02+12σ2+1ω02+ik2z,
Dx=1Ax(1ρ02+12σ2)2+1ρ02+12σ2+1ω02ik2z,
Ex=[1(1ρ02+12σ2)1Ax]x1x22ρ02+(1ρ02+12σ2)1Axikx12zikx22z,
and Wy is given by:
Wy=ny1!πAyπDyexp[1Ay(y2y12ρ02+iky12z)2+Ey2Dy]s=0E(ny1/2)h=0ny12s1(ny12s)!s!4sAy(ny1s)(ny12sh)(y2y12ρ02+iky12z)ny12sh(1ρ02+12σ2)h(i2Dy)m2+2n+hHm2+2n+h(iEyDy),
Where:
ny1=m1+2n,
and expressions forAy, Dy, Ey are formulated by substituting ω0, x1, x2,Axwith ω0/α, y1, y2,Ay in Eqs. (13)-(15), respectively.

Equations (10)-(17) are the analytical expressions for the cross-spectral density function of partially coherent four-petal elliptic Gaussian vortex beams propagating through atmospheric turbulence. The expression for the average intensity can be obtained by further setting r1=r2=r [2], so that:

I(r,z)=W(r,r,z).

The expression for the root mean square (rms) beam width [27], defined as

ω(z)=r2I(r,z)d2rI(r,z)d2r,
is then derived as follows:
ω(z)=ω1x+ω1yω2,
Where:
ω2=(k2πz)21ω08n+2|m|m1=0|m|m2=0|m|(|m|m1)(|m|m2)(1)m1[iαsgn(m)]m1+m2RxRy,
ω1x=(k2πz)21ω08n+2|m|m1=0|m|m2=0|m|(|m|m1)(|m|m2)(1)m1[iαsgn(m)]m1+m2R2xRy,
ω1y=(k2πz)21ω08n+2|m|m1=0|m|m2=0|m|(|m|m1)(|m|m2)(1)m1[iαsgn(m)]m1+m2RxR2y,
Where:
Rx=nx1!πAxπDxs=0E(nx1/2)h=0nx12st=0E[(|m|m2+2n+h)/2]Gx[1nx2+2E(nx2/2)]Γ(nx2+12)/(Fx)nx2+12,
Ry=ny1!πAyπDys=0E(ny1/2)h=0ny12st=0E[(m2+2n+h)/2]Gy[1ny2+2E(ny2/2)]Γ(ny2+12)/(Fy)ny2+12,
R2x=nx1!πAxπDxs=0E(nx1/2)h=0nx12st=0E[(|m|m2+2n+h)/2]Gx[1nx2+2E(nx2/2)]Γ(nx2+32)/(Fx)nx2+32,
R2y=ny1!πAyπDys=0E(ny1/2)h=0ny12st=0E[(m2+2n+h)/2]Gy[1ny2+2E(ny2/2)]Γ(ny2+32)/(Fy)ny2+32,
With Γ() represents the gamma function, and:

Fx=k24Dxz2(1Axρ02+12Axσ21)2k24Axz2,
Gx=1(nx12s)!s!4sAxnx1s(nx12sh)(1ρ02+12σ2)h(i2Dx)|m|m2+2n+h(|m|m2+2n+h)!(1)t(|m|m2+2n+h2t)!t![1Dx(1Axρ02+12Axσ21)kz]|m|m2+2n+h2t(ik2z)|m|m1+2nh2s,
nx2=m2+4n2t+2|m|m12s,
Fy=k24Dyz2(1Ayρ02+12Ayσ21)2k24Ayz2,
Gy=1(ny12s)!s!4sAyny1s(ny12sh)(1ρ02+12σ2)h(i2Dy)m2+2n+h(m2+2n+h)!(1)t(m2+2n+h2t)!t![1Dy(1Ayρ02+12Ayσ21)kz]m2+2n+h2t(ik2z)m1+2nh2s,
ny2=m1+m2+4n2t2s.

The position of coherence vortices is determined by [28]:

Re[μ(r1,r2,z)]=0,
And:
Im[μ(r1,r2,z)]=0,
where Re() and Im() are the real and imaginary part of the enclosed expression, respectively, and μ(r1,r2,z) is the spectral degree of coherence [2], defined as:

μ(r1,r2,z)=W(r1,r2,z)I(r1,z)I(r2,z).

The conservation distance of coherence vortices topological charge is defined to be the maximum propagation distance for the detected total coherence vortices topological charge keeps unchanged [29]. The total coherence vortices topological charge at the detector is calculated by evaluating [ϕ(r1)dl]/(2π) around the perimeter of the detector aperture, where ϕ(r1) denotes the phase of μ(r1,r2,z) with r2 and z fixed. The conservation distance of coherence vortices topological charge provides a measure about how distant the detected beam could maintain its spatial two-point correlation singularity.

To compare the analytical result with those given in Ref [14], it should be noted that the choice of expressions for initial field are not exactly the same. The form of Eqs. (3) and (4) follows the customs in Refs [30]. and [31]. However, by substituting m into M, 2σ2 into σ2, setting α=1, and multiplied by ω02|m|, Eq. (4) in the current work are found to be the same as Eq. (4) in Ref [14]. And correspondingly, by substituting m into M, 2σ2 into σ2, setting r1=r2=r, multiplied by ω02|m|, and changing the name of some local index variables, the analytical result for average intensity for the special case of α=1 in this work is found to be the same as the result in Ref [14]. According to this correspondence, we set ω0=20mm, λ=1.064μm, Cn2=1013m-2/3, α=1, σ=7.072mm, m=1 and n=1, the calculated normalized intensity profiles at z=100m, 500m, 1000m, 5000m are found to be the same as Fig. 1 in Ref [14].

 figure: Fig. 1

Fig. 1 Evolution of normalized intensity distribution of a partially coherent four-petal elliptic Gaussian vortex beam propagating through turbulent atmosphere for (a) z = 200m, (b) z = 300m, (c) z = 500m, (d) z = 1000m, (e) z = 2000m, (f) z = 50000m, respectively. The calculation parameters are seen in the text. All subfigures share the same coordinate axis direction. All intensity data are normalized by the peak intensity value of the initial wave at z = 0. The peak values Im of each subfigure are 0.754, 0.564, 0.385, 0.188, 0.060, and 1.61 × 10−5 for (a)-(f), respectively.

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3. Numerical examples and analysis

The average intensity, beam spreading, and evolution of coherence vortices of partially coherent four-petal elliptic Gaussian vortex beams propagating through turbulent atmosphere are numerically studied based on formulas in Sec. 2. The calculation parameters were set as ω0=20mm, λ=1.315μm, Cn2=1014m2/3, α=1.5, σ=40mmunless otherwise specified.

The normalized average intensity of partially coherent four-petal elliptic Gaussian vortex beam propagating through turbulent atmosphere with m=1, n=1 is shown in Fig. 1. The average intensity experiences several stages of evolution. At the beginning, the initial four-petal profile gradually deformed into a six-petal profile in the short propagation distance. This is different from the partially coherent four-petal Gaussian vortex beam, whose number of petals and sidelobes would always be an integer multiple of four in this stage [14]. With the propagation distance increasing, the dark hollow center of the partially coherent four-petal elliptic Gaussian vortex beam will shrink due to the expansion of each petal. And at last, the beam will evolve into a Gaussian-like beam in the far field. The minimal distance at which the beam center intensity equals to its maximum intensity, denoted by zflat, is 2.51km. This parameter can be considered as an indicator of how fast the beam evolves.

Figure 2 compares the normalized average intensity of partially coherent four-petal elliptic Gaussian vortex beams propagating through turbulent atmosphere with n=1 and different m. It can be found that the partially coherent four-petal elliptic Gaussian vortex beams with different m have similar evolution behaviors, but as the topological charge m is getting larger, the intensity profile rotation around the z axis is getting faster. For m=2, the transition from four-petal to six-petal profile is slower than m=1. For m>2, there is no obvious stage with six-petal pattern within the whole propagation range. All beams with different m will evolve into Gaussian-like beams in the far field. zflat are 5.28km, 5.49km, 3.81km, 7.10km for m=2, 3, 4, and 5, respectively. Noticing that in Eq. (3) the amplitude profile is of reflective symmetry with respect to the x axis and the y axis, while the phase profile is of rotational symmetry of order m, zflat exhibits more complex trends with the increase of m rather than to be monotonic.

 figure: Fig. 2

Fig. 2 Normalized intensity distribution of partially coherent four-petal elliptic Gaussian vortex beams propagating through turbulent atmosphere with n = 1 and different m. (a-c) m = 2, (d-f) m = 3, (g-i) m = 4, (j-l) m = 5, (a, d, g, j) z = 500m, (b, e, h, k) z = 1000m, (c, f, i, l) z = 50000m. All intensity data are normalized by the peak intensity value of the initial wave at z = 0. The peak values Im of each subfigure are 0.350, 0.201, 1.73 × 10−5, 0.373, 0.183, 1.82 × 10−5, 0.389, 0.165, 1.88 × 10−5, 0.401, 0.153, and 1.92 × 10−5 for (a)-(l), respectively.

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Figure 3 compares the normalized average intensity of partially coherent four-petal elliptic Gaussian vortex beams propagating through turbulent atmosphere with m=1 and different n. It can be seen that as the four-petal Gaussian beam order n is getting larger, the transition from four-petal to six-petal profile is getting slower, and the intensity profile rotation speed around the z axis keeps unchanged. All beams with different n will evolve into Gaussian-like beams in the far field. zflat are 2.24km, 2.28km, 2.41km, 2.58km for n=2, 3, 4, and 5, respectively.

 figure: Fig. 3

Fig. 3 Normalized intensity distribution of partially coherent four-petal elliptic Gaussian vortex beams propagating through turbulent atmosphere with m = 1 and different n. (a-c) n = 2, (d-f) n = 3, (g-i) n = 4, (j-l) n = 5, (a, d, g, j) z = 500m, (b, e, h, k) z = 1000m, (c, f, i, l) z = 50000m. All intensity data are normalized by the peak intensity value of the initial wave at z = 0. The peak values Im of each subfigure are 0.291, 0.147, 1.57 × 10−5, 0.255, 0.113, 1.55 × 10−5, 0.260, 0.0936, 1.55 × 10−5, 0.255, 0.0865, and 1.54 × 10−5 for (a)-(l), respectively.

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Figure 4 compares the normalized average intensity of partially coherent four-petal elliptic Gaussian vortex beams propagating through turbulent atmosphere with m=1,n=1, and different α. It can be seen that while α<1, the four-petal profile would morph into six-petal profile with three along x direction and two along y direction, and while α>1, would morph into six-petal profile with two along x direction and three along y direction. For α>1, the transition from four-petal to six-petal profile is getting faster with the increasing of α. All beams with different α will evolve into Gaussian-like beams in the far field. Further computation shows that zflat are 3.41km, 2.88km, 2.27km, 1.86km, 1.68km, 1.83km, 2.23km, 2.91km, 3.30km for α=0.5, 0.6, 0.75, 0.9, 1, 1.1, 1.33, 1.75, and 2, respectively. It is found that for α<1, zflat will decrease as α increases, and for α>1, zflat will increase as α increases.

 figure: Fig. 4

Fig. 4 Normalized intensity distribution of partially coherent four-petal elliptic Gaussian vortex beams propagating through turbulent atmosphere with m = 1, n = 1 and different α. (a-c) α = 0.5, (d-f) α = 0.75, (g-i) α = 1.33, (j-l) α = 2, (a, d, g, j) z = 500m, (b, e, h, k) z = 1000m, (c, f, i, l) z = 50000m. All intensity data are normalized by the peak intensity value of the initial wave at z = 0. The peak values Im of each subfigure are 0.708, 0.373, 5.17 × 10−5, 0.630, 0.270, 3.41 × 10−5, 0.377, 0.200, 1.84 × 10−5, 0.407, 0.138, and 1.15 × 10−5 for (a)-(l), respectively.

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Figure 5 shows the normalized average intensity of partially coherent four-petal elliptic Gaussian vortex beams propagating in free space. By comparing Figs. 1-4 with Fig. 5, it can be found that the beam profiles for short propagation distances (e.g. less than 1000m) in free space are similar with those propagating in turbulence. This is because in the considered case, the turbulence is relatively weak, and for short propagation distances, the phase distortion caused by the turbulence is small, and thus have little effect on the propagated beam. For m=1 and n=1, there is transition from four petal profile to six petal profile. For m=5 and n=1, there is no obvious stage with six-petal pattern within the whole propagation range. Often, the transition from four-petal profile to six-petal profile takes place at short propagation distances, thus this phenomenon will be similar no matter the beam is propagated in free space or weak turbulence. Nevertheless, for far field propagation the beam profiles in free space are different from those in turbulence. The far field of partially coherent four-petal elliptic Gaussian vortex beam propagating in free space exhibits petal or bud-like pattern with a dip in the center, while those propagating in turbulence will always evolve into Gaussian-like patterns. Both beams in Fig. 5 do not have valid zflat value within the whole considered range (0<z<50000m).

 figure: Fig. 5

Fig. 5 Normalized intensity distribution of partially coherent four-petal elliptic Gaussian vortex beams propagating in free space. (a-c) m = 1, n = 1, (d-f) m = 5, n = 1, (a, d) z = 500m, (b, e) z = 1000m, (c, f) z = 50000m. All intensity data are normalized by the peak intensity value of the initial wave at z = 0. The peak values Im of each subfigure are 0.406, 0.218, 1.62 × 10−4, 0.414, 0.176, and 1.04 × 10−4 for (a)-(f), respectively.

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Figure 6 compares the normalized average intensity of partially coherent four-petal elliptic Gaussian vortex beams propagating through turbulent atmosphere with m=1, n=1, and different σ. zflat are 1.60km and 0.90km for σ=20mm and 10mm, respectively. Associating with Fig. 1, it can be seen that as σ is getting smaller, zflat is getting smaller, and each of the six-petals would getting expanded and connected to its neighbors earlier. For σ much smaller than ω0, the petals would merge into two, and there is no obvious stage with six-petal pattern within the whole propagation range. The petal numbers of these beams around z=500m are different from those of the partially coherent four-petal Gaussian vortex beams [14].

 figure: Fig. 6

Fig. 6 Normalized intensity distribution of partially coherent four-petal elliptic Gaussian vortex beams propagating through turbulent atmosphere with m = 1, n = 1 and different σ. (a-c) σ = 20mm, (d-f) σ = 10mm, (a, d) z = 500m, (b, e) z = 1000m, (c, f) z = 50000m. All intensity data are normalized by the peak intensity value of the initial wave at z = 0. The peak values Im of each subfigure are 0.296, 0.139, 1.57 × 10−5, 0.191, 0.0977, 1.44 × 10−5 for (a)-(f), respectively.

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Figures 7(a)-7(c) plot the relative normalized rms beam width ω(z)/ωfree(z) of partially coherent four-petal elliptic Gaussian vortex beams with different m, different n, and different α, respectively. Here ωfree(z) denotes the rms beam width while propagating in free space. We see that all curves increase monotonically with the propagation distance z, meaning the beam spreading in turbulence is always faster than in free space. In the far field, the relative normalized rms beam width decreases with increasing m, decreasing n, and increasing α. This means beams with larger m, smaller n, and larger α will be less influenced by atmospheric turbulence. Note that in Figs. 6(b) and 6(c), there are intersections between curves. We attempted to explain this by analysing a similar but simpler example. Consider the case of m=0, and n=0, the expression for ω(z) can be simplified as:

ω(z)=(14+14α2)ω02+(α2+1ω02+2σ2)(zk)2+4ρ02(zk)2=I+II+III.
The first term I under the square root on the right-hand side represents the original beam width corresponding to z=0,. The second term II describes the beam width spreading due to the free-space diffraction, and the third term III characterizes the spreading owing to the atmospheric turbulence. Noticing that α is contained in the denominator of term I and the numerator of term II, respectively. This leads to the change in the relative size order of ω(z)/ωfree(z) in different propagation distances.

 figure: Fig. 7

Fig. 7 relative normalized rms beam width of partially coherent four-petal elliptic Gaussian vortex beams with (a) n = 1, α = 1.5, and different m, (b) m = 1, α = 1.5, and different n, and (c) m = 1, n = 1, and different α. Insets show enlarged view of curves for z<6000m.

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Figure 8 shows typical curves corresponding to solutions of Eqs. (34) and (35). In the current case m=1, n=1, andr1=(x1,y1) is set to be (3cm, 6cm). One can find that as the propagation distance increases, the number of coherence vortices will change. The conservation distance d of partially coherent four-petal elliptic Gaussian vortex beams propagating through atmospheric turbulence for different values of α is summarized in Table 1, where the other calculation parameters are the same as those in Fig. 7. From Table 1, it follows that two local maxima occur at about α=0.5 and α=1.7. Thus, the ellipticity factor can be utilized as an additional degree of freedom in controlling the conservation distance of coherence vortices topological charge.

 figure: Fig. 8

Fig. 8 Curves of Re μ = 0 and Im μ = 0 of a partially coherent four-petal elliptic Gaussian vortex beam with m = 1, n = 1 and (a) z = 200m, (b) z = 500m and (c) z = 1000m.

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Tables Icon

Table 1. Conservation distance d for different values of α

The change of detected total topological charge during propagation is caused by the evolution and movement of the vortices. According to Gbur [29], as the beam propagates, the vortex “wanders”. If it wanders outside the perimeter of the detector aperture, the topological charge will not be measured by the detector. For 0.1α0.5 and 1.7α2, the vortex getting out of the aperture comes from the pair created within (0.5cm,0.5cm)×(2cm,0), while for 0.6α1.6, comes from the pair created within (0,2cm)×(0.5cm,0). That is to say, different vortex movements are responsible for the conservation distance for different range of α. This corresponds to the fact that α=0.5 and 1.7 are extreme points. Note that a square aperture with side length of 4cm is used in Table 1. If we change the aperture to be circular with diameter of 4cm, the trend of the conservation distance with respect to α keeps unchanged, while the two local maxima still at about α=0.5 and 1.7.

4. Conclusions

In this paper, taking the partially coherent four-petal elliptic Gaussian vortex beam as an example of partially coherent petal-like beams with elliptical optical vortex, the evolution of average intensity and coherence vortices, and the beam spreading behavior have been studied.

It is found that, the initial four-petal profile of partially coherent four-petal elliptic Gaussian vortex beams could deform into six-petal profiles in short propagation distance, and this is different from the partially coherent four-petal Gaussian vortex beam. The occurence, the appearance, and the transition speed from four-petal to six-petal profile depend on m, n, and α. The rotation speed of the near field intensity profile around the z axis depends mainly on m.

The far field distributions of partially coherent four-petal elliptic Gaussian vortex beams through atmospheric turbulence are always Gaussian-like patterns. Quite differently, those propagating in free space will exhibit petal or bud-like patterns with dip in the center.

In the far field, the relative normalized rms beam width ω(z)/ωfree(z) decreases with increasing m, decreasing n, and increasing α. This means beams with larger m, smaller n, and larger α will be less influenced by atmospheric turbulence. In short propagation distances, however, the relative size order of ω(z)/ωfree(z) with different n and α might alter.

As the propagation distance increases, the number of coherence vortices will change. It is found that the conservation distance of coherence vortices topological charge depends on α, so that the ellipticity factor can be utilized as an additional degree of freedom in controlling the conservation distance.

Funding

National Natural Science Foundation of China (61875197, 61205139, 11705207, 21573218); Youth Innovation Promotion Association CAS (2016168); Scientific Innovative Foundation of Chinese Academy of Sciences (CXJJ-17S056); Strategic Priority Research Program of the Chinese Academy of Sciences (XDB17010300); Major Program of National Natural Science Foundation of China (21590803).

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3. 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]  

4. W. Nelson, P. Sprangle, and C. C. Davis, “Atmospheric propagation and combining of high-power lasers,” Appl. Opt. 55(7), 1757–1764 (2016). [CrossRef]   [PubMed]  

5. M. Yousefi, S. Golmohammady, A. Mashal, and F. D. Kashani, “Analyzing the propagation behavior of scintillation index and bit error rate of a partially coherent flat-topped laser beam in oceanic turbulence,” J. Opt. Soc. Am. A 32(11), 1982–1992 (2015). [CrossRef]   [PubMed]  

6. M. P. Lavery, F. C. Speirits, S. M. Barnett, and M. J. Padgett, “Detection of a spinning object using light’s orbital angular momentum,” Science 341(6145), 537–540 (2013). [CrossRef]   [PubMed]  

7. M. Lavery, S. Barnett, F. Speirits, and M. Padgett, “Observation of the rotational Doppler shift of a white-light, orbital-angular-momentum-carrying beam backscattered from a rotating body,” Optica 1(1), 1–4 (2014). [CrossRef]  

8. K. Duan and B. Lü, “Four-petal Gaussian beams and their propagation,” Opt. Commun. 261(2), 327–331 (2006). [CrossRef]  

9. G. Zeng-Hui and L. Bai-Da, “Vectorial nonparaxial four-petal Gaussian beams and their propagation in free space,” Chin. Phys. Lett. 23(8), 2070–2073 (2006). [CrossRef]  

10. Z. Guo-Quan and F. Yan, “M2 factor of four-petal Gaussian beam,” Chin. Phys. B 17(10), 3708–3712 (2008). [CrossRef]  

11. X. Long, K. Lu, Y. Zhang, J. Guo, and K. Li, “Vectorial structure of a hard-edged-diffracted four-petal Gaussian beam in the far field,” Opt. Commun. 283(23), 4586–4593 (2010). [CrossRef]  

12. L. Guo, Z. Tang, and W. Wan, “Propagation of four-petal Gaussian vortex beam through a paraxial ABCD optical system,” Optik (Stuttg.) 125(19), 5542–5545 (2014). [CrossRef]  

13. D. Liu, H. Zhong, G. Wang, H. Yin, and Y. Wang, “Nonparaxial propagation of a partially coherent four-petal Gaussian vortex beam,” Optik (Stuttg.) 158, 451–459 (2018). [CrossRef]  

14. D. Liu, Y. Wang, and H. Yin, “Propagation properties of partially coherent four-petal Gaussian vortex beams in turbulent atmosphere,” Opt. Laser Technol. 78, 95–100 (2016). [CrossRef]  

15. D. Liu, Y. Wang, G. Wang, X. Luo, and H. Yin, “Propagation properties of partially coherent four-petal Gaussian vortex beams in oceanic turbulence,” Laser Phys. 27(1), 016001 (2017). [CrossRef]  

16. M. A. Molchan, E. V. Doktorov, and R. A. Vlasov, “Propagation of vector fractional charge Laguerre-Gaussian light beams in the thermally nonlinear moving atmosphere,” Opt. Lett. 35(5), 670–672 (2010). [CrossRef]   [PubMed]  

17. V. Kotlyar and A. Kovalev, “Controlling orbital angular momentum of an optical vortex by varying its ellipticity,” Opt. Commun. 410(1), 202–205 (2018). [CrossRef]  

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

19. A. Kovalev, V. Kotlyar, and A. Porfirev, “A highly efficient element for generating elliptic perfect optical vortices,” Appl. Phys. Lett. 110(26), 261102 (2017). [CrossRef]  

20. H. Yan and B. Lü, “The transformation of canonical vortices embedded in a general elliptical Gaussian beam to noncanonical vortices,” J. Opt. A, Pure Appl. Opt. 11(1), 015702 (2009). [CrossRef]  

21. Y. Luo and B. Lü, “Far-field properties of electromagnetic elliptical Gaussian vortex beams,” Opt. Commun. 283(19), 3578–3584 (2010). [CrossRef]  

22. Y. Liu and J. Pu, “Measuring the orbital angular momentum of elliptical vortex beams by using a slit hexagon aperture,” Opt. Commun. 284(10-11), 2424–2429 (2011). [CrossRef]  

23. F. Gori and M. Santarsiero, “Devising genuine spatial correlation functions,” Opt. Lett. 32(24), 3531–3533 (2007). [CrossRef]   [PubMed]  

24. Z. Chen and D. Zhao, “Focusing of an elliptic vortex beam by a square Fresnel zone plate,” Appl. Opt. 50(15), 2204–2210 (2011). [CrossRef]   [PubMed]  

25. H. D. A. Jeffrey, Handbook of Mathematical Formulas and Integrals, 4th ed. (Academic, 2008).

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

27. 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]  

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

29. 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]  

30. 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]  

31. X. Li, F. Wang, and Y. Cai, “An alternative model for a partially coherent elliptical dark hollow beam,” Opt. Laser Technol. 43(3), 577–585 (2011). [CrossRef]  

References

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  1. L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media (SPIE, 1998).
  2. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).
  3. 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]
  4. W. Nelson, P. Sprangle, and C. C. Davis, “Atmospheric propagation and combining of high-power lasers,” Appl. Opt. 55(7), 1757–1764 (2016).
    [Crossref] [PubMed]
  5. M. Yousefi, S. Golmohammady, A. Mashal, and F. D. Kashani, “Analyzing the propagation behavior of scintillation index and bit error rate of a partially coherent flat-topped laser beam in oceanic turbulence,” J. Opt. Soc. Am. A 32(11), 1982–1992 (2015).
    [Crossref] [PubMed]
  6. M. P. Lavery, F. C. Speirits, S. M. Barnett, and M. J. Padgett, “Detection of a spinning object using light’s orbital angular momentum,” Science 341(6145), 537–540 (2013).
    [Crossref] [PubMed]
  7. M. Lavery, S. Barnett, F. Speirits, and M. Padgett, “Observation of the rotational Doppler shift of a white-light, orbital-angular-momentum-carrying beam backscattered from a rotating body,” Optica 1(1), 1–4 (2014).
    [Crossref]
  8. K. Duan and B. Lü, “Four-petal Gaussian beams and their propagation,” Opt. Commun. 261(2), 327–331 (2006).
    [Crossref]
  9. G. Zeng-Hui and L. Bai-Da, “Vectorial nonparaxial four-petal Gaussian beams and their propagation in free space,” Chin. Phys. Lett. 23(8), 2070–2073 (2006).
    [Crossref]
  10. Z. Guo-Quan and F. Yan, “M2 factor of four-petal Gaussian beam,” Chin. Phys. B 17(10), 3708–3712 (2008).
    [Crossref]
  11. X. Long, K. Lu, Y. Zhang, J. Guo, and K. Li, “Vectorial structure of a hard-edged-diffracted four-petal Gaussian beam in the far field,” Opt. Commun. 283(23), 4586–4593 (2010).
    [Crossref]
  12. L. Guo, Z. Tang, and W. Wan, “Propagation of four-petal Gaussian vortex beam through a paraxial ABCD optical system,” Optik (Stuttg.) 125(19), 5542–5545 (2014).
    [Crossref]
  13. D. Liu, H. Zhong, G. Wang, H. Yin, and Y. Wang, “Nonparaxial propagation of a partially coherent four-petal Gaussian vortex beam,” Optik (Stuttg.) 158, 451–459 (2018).
    [Crossref]
  14. D. Liu, Y. Wang, and H. Yin, “Propagation properties of partially coherent four-petal Gaussian vortex beams in turbulent atmosphere,” Opt. Laser Technol. 78, 95–100 (2016).
    [Crossref]
  15. D. Liu, Y. Wang, G. Wang, X. Luo, and H. Yin, “Propagation properties of partially coherent four-petal Gaussian vortex beams in oceanic turbulence,” Laser Phys. 27(1), 016001 (2017).
    [Crossref]
  16. M. A. Molchan, E. V. Doktorov, and R. A. Vlasov, “Propagation of vector fractional charge Laguerre-Gaussian light beams in the thermally nonlinear moving atmosphere,” Opt. Lett. 35(5), 670–672 (2010).
    [Crossref] [PubMed]
  17. V. Kotlyar and A. Kovalev, “Controlling orbital angular momentum of an optical vortex by varying its ellipticity,” Opt. Commun. 410(1), 202–205 (2018).
    [Crossref]
  18. X. Liu and J. Pu, “Investigation on the scintillation reduction of elliptical vortex beams propagating in atmospheric turbulence,” Opt. Express 19(27), 26444–26450 (2011).
    [Crossref] [PubMed]
  19. A. Kovalev, V. Kotlyar, and A. Porfirev, “A highly efficient element for generating elliptic perfect optical vortices,” Appl. Phys. Lett. 110(26), 261102 (2017).
    [Crossref]
  20. H. Yan and B. Lü, “The transformation of canonical vortices embedded in a general elliptical Gaussian beam to noncanonical vortices,” J. Opt. A, Pure Appl. Opt. 11(1), 015702 (2009).
    [Crossref]
  21. Y. Luo and B. Lü, “Far-field properties of electromagnetic elliptical Gaussian vortex beams,” Opt. Commun. 283(19), 3578–3584 (2010).
    [Crossref]
  22. Y. Liu and J. Pu, “Measuring the orbital angular momentum of elliptical vortex beams by using a slit hexagon aperture,” Opt. Commun. 284(10-11), 2424–2429 (2011).
    [Crossref]
  23. F. Gori and M. Santarsiero, “Devising genuine spatial correlation functions,” Opt. Lett. 32(24), 3531–3533 (2007).
    [Crossref] [PubMed]
  24. Z. Chen and D. Zhao, “Focusing of an elliptic vortex beam by a square Fresnel zone plate,” Appl. Opt. 50(15), 2204–2210 (2011).
    [Crossref] [PubMed]
  25. H. D. A. Jeffrey, Handbook of Mathematical Formulas and Integrals, 4th ed. (Academic, 2008).
  26. S. C. H. Wang and M. A. Plonus, “Optical beam propagation for a partially coherent source in the turbulent atmosphere,” J. Opt. Soc. Am. 69(9), 1297–1304 (1979).
    [Crossref]
  27. 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]
  28. G. Gbur and T. D. Visser, “Coherence vortices in partially coherent beams,” Opt. Commun. 222(1–6), 117–125 (2003).
    [Crossref]
  29. 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]
  30. 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]
  31. X. Li, F. Wang, and Y. Cai, “An alternative model for a partially coherent elliptical dark hollow beam,” Opt. Laser Technol. 43(3), 577–585 (2011).
    [Crossref]

2018 (2)

D. Liu, H. Zhong, G. Wang, H. Yin, and Y. Wang, “Nonparaxial propagation of a partially coherent four-petal Gaussian vortex beam,” Optik (Stuttg.) 158, 451–459 (2018).
[Crossref]

V. Kotlyar and A. Kovalev, “Controlling orbital angular momentum of an optical vortex by varying its ellipticity,” Opt. Commun. 410(1), 202–205 (2018).
[Crossref]

2017 (2)

A. Kovalev, V. Kotlyar, and A. Porfirev, “A highly efficient element for generating elliptic perfect optical vortices,” Appl. Phys. Lett. 110(26), 261102 (2017).
[Crossref]

D. Liu, Y. Wang, G. Wang, X. Luo, and H. Yin, “Propagation properties of partially coherent four-petal Gaussian vortex beams in oceanic turbulence,” Laser Phys. 27(1), 016001 (2017).
[Crossref]

2016 (2)

D. Liu, Y. Wang, and H. Yin, “Propagation properties of partially coherent four-petal Gaussian vortex beams in turbulent atmosphere,” Opt. Laser Technol. 78, 95–100 (2016).
[Crossref]

W. Nelson, P. Sprangle, and C. C. Davis, “Atmospheric propagation and combining of high-power lasers,” Appl. Opt. 55(7), 1757–1764 (2016).
[Crossref] [PubMed]

2015 (1)

2014 (2)

M. Lavery, S. Barnett, F. Speirits, and M. Padgett, “Observation of the rotational Doppler shift of a white-light, orbital-angular-momentum-carrying beam backscattered from a rotating body,” Optica 1(1), 1–4 (2014).
[Crossref]

L. Guo, Z. Tang, and W. Wan, “Propagation of four-petal Gaussian vortex beam through a paraxial ABCD optical system,” Optik (Stuttg.) 125(19), 5542–5545 (2014).
[Crossref]

2013 (1)

M. P. Lavery, F. C. Speirits, S. M. Barnett, and M. J. Padgett, “Detection of a spinning object using light’s orbital angular momentum,” Science 341(6145), 537–540 (2013).
[Crossref] [PubMed]

2011 (5)

2010 (3)

Y. Luo and B. Lü, “Far-field properties of electromagnetic elliptical Gaussian vortex beams,” Opt. Commun. 283(19), 3578–3584 (2010).
[Crossref]

M. A. Molchan, E. V. Doktorov, and R. A. Vlasov, “Propagation of vector fractional charge Laguerre-Gaussian light beams in the thermally nonlinear moving atmosphere,” Opt. Lett. 35(5), 670–672 (2010).
[Crossref] [PubMed]

X. Long, K. Lu, Y. Zhang, J. Guo, and K. Li, “Vectorial structure of a hard-edged-diffracted four-petal Gaussian beam in the far field,” Opt. Commun. 283(23), 4586–4593 (2010).
[Crossref]

2009 (1)

H. Yan and B. Lü, “The transformation of canonical vortices embedded in a general elliptical Gaussian beam to noncanonical vortices,” J. Opt. A, Pure Appl. Opt. 11(1), 015702 (2009).
[Crossref]

2008 (2)

2007 (2)

2006 (2)

K. Duan and B. Lü, “Four-petal Gaussian beams and their propagation,” Opt. Commun. 261(2), 327–331 (2006).
[Crossref]

G. Zeng-Hui and L. Bai-Da, “Vectorial nonparaxial four-petal Gaussian beams and their propagation in free space,” Chin. Phys. Lett. 23(8), 2070–2073 (2006).
[Crossref]

2003 (2)

1979 (1)

Bai-Da, L.

G. Zeng-Hui and L. Bai-Da, “Vectorial nonparaxial four-petal Gaussian beams and their propagation in free space,” Chin. Phys. Lett. 23(8), 2070–2073 (2006).
[Crossref]

Barnett, S.

Barnett, S. M.

M. P. Lavery, F. C. Speirits, S. M. Barnett, and M. J. Padgett, “Detection of a spinning object using light’s orbital angular momentum,” Science 341(6145), 537–540 (2013).
[Crossref] [PubMed]

Cai, Y.

X. Li, F. Wang, and Y. Cai, “An alternative model for a partially coherent elliptical dark hollow beam,” Opt. Laser Technol. 43(3), 577–585 (2011).
[Crossref]

Chen, Z.

Davis, C. C.

Dogariu, A.

Doktorov, E. V.

Du, X.

Duan, K.

K. Duan and B. Lü, “Four-petal Gaussian beams and their propagation,” Opt. Commun. 261(2), 327–331 (2006).
[Crossref]

Gbur, G.

Golmohammady, S.

Gori, F.

Guo, J.

X. Long, K. Lu, Y. Zhang, J. Guo, and K. Li, “Vectorial structure of a hard-edged-diffracted four-petal Gaussian beam in the far field,” Opt. Commun. 283(23), 4586–4593 (2010).
[Crossref]

Guo, L.

L. Guo, Z. Tang, and W. Wan, “Propagation of four-petal Gaussian vortex beam through a paraxial ABCD optical system,” Optik (Stuttg.) 125(19), 5542–5545 (2014).
[Crossref]

Guo-Quan, Z.

Z. Guo-Quan and F. Yan, “M2 factor of four-petal Gaussian beam,” Chin. Phys. B 17(10), 3708–3712 (2008).
[Crossref]

He, X.

Kashani, F. D.

Korotkova, O.

Kotlyar, V.

V. Kotlyar and A. Kovalev, “Controlling orbital angular momentum of an optical vortex by varying its ellipticity,” Opt. Commun. 410(1), 202–205 (2018).
[Crossref]

A. Kovalev, V. Kotlyar, and A. Porfirev, “A highly efficient element for generating elliptic perfect optical vortices,” Appl. Phys. Lett. 110(26), 261102 (2017).
[Crossref]

Kovalev, A.

V. Kotlyar and A. Kovalev, “Controlling orbital angular momentum of an optical vortex by varying its ellipticity,” Opt. Commun. 410(1), 202–205 (2018).
[Crossref]

A. Kovalev, V. Kotlyar, and A. Porfirev, “A highly efficient element for generating elliptic perfect optical vortices,” Appl. Phys. Lett. 110(26), 261102 (2017).
[Crossref]

Lavery, M.

Lavery, M. P.

M. P. Lavery, F. C. Speirits, S. M. Barnett, and M. J. Padgett, “Detection of a spinning object using light’s orbital angular momentum,” Science 341(6145), 537–540 (2013).
[Crossref] [PubMed]

Li, K.

X. Long, K. Lu, Y. Zhang, J. Guo, and K. Li, “Vectorial structure of a hard-edged-diffracted four-petal Gaussian beam in the far field,” Opt. Commun. 283(23), 4586–4593 (2010).
[Crossref]

Li, X.

X. Li, F. Wang, and Y. Cai, “An alternative model for a partially coherent elliptical dark hollow beam,” Opt. Laser Technol. 43(3), 577–585 (2011).
[Crossref]

Liu, D.

D. Liu, H. Zhong, G. Wang, H. Yin, and Y. Wang, “Nonparaxial propagation of a partially coherent four-petal Gaussian vortex beam,” Optik (Stuttg.) 158, 451–459 (2018).
[Crossref]

D. Liu, Y. Wang, G. Wang, X. Luo, and H. Yin, “Propagation properties of partially coherent four-petal Gaussian vortex beams in oceanic turbulence,” Laser Phys. 27(1), 016001 (2017).
[Crossref]

D. Liu, Y. Wang, and H. Yin, “Propagation properties of partially coherent four-petal Gaussian vortex beams in turbulent atmosphere,” Opt. Laser Technol. 78, 95–100 (2016).
[Crossref]

Liu, X.

Liu, Y.

Y. Liu and J. Pu, “Measuring the orbital angular momentum of elliptical vortex beams by using a slit hexagon aperture,” Opt. Commun. 284(10-11), 2424–2429 (2011).
[Crossref]

Long, X.

X. Long, K. Lu, Y. Zhang, J. Guo, and K. Li, “Vectorial structure of a hard-edged-diffracted four-petal Gaussian beam in the far field,” Opt. Commun. 283(23), 4586–4593 (2010).
[Crossref]

Lu, K.

X. Long, K. Lu, Y. Zhang, J. Guo, and K. Li, “Vectorial structure of a hard-edged-diffracted four-petal Gaussian beam in the far field,” Opt. Commun. 283(23), 4586–4593 (2010).
[Crossref]

Lü, B.

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]

Y. Luo and B. Lü, “Far-field properties of electromagnetic elliptical Gaussian vortex beams,” Opt. Commun. 283(19), 3578–3584 (2010).
[Crossref]

H. Yan and B. Lü, “The transformation of canonical vortices embedded in a general elliptical Gaussian beam to noncanonical vortices,” J. Opt. A, Pure Appl. Opt. 11(1), 015702 (2009).
[Crossref]

K. Duan and B. Lü, “Four-petal Gaussian beams and their propagation,” Opt. Commun. 261(2), 327–331 (2006).
[Crossref]

Luo, X.

D. Liu, Y. Wang, G. Wang, X. Luo, and H. Yin, “Propagation properties of partially coherent four-petal Gaussian vortex beams in oceanic turbulence,” Laser Phys. 27(1), 016001 (2017).
[Crossref]

Luo, Y.

Y. Luo and B. Lü, “Far-field properties of electromagnetic elliptical Gaussian vortex beams,” Opt. Commun. 283(19), 3578–3584 (2010).
[Crossref]

Mashal, A.

Molchan, M. A.

Nelson, W.

Padgett, M.

Padgett, M. J.

M. P. Lavery, F. C. Speirits, S. M. Barnett, and M. J. Padgett, “Detection of a spinning object using light’s orbital angular momentum,” Science 341(6145), 537–540 (2013).
[Crossref] [PubMed]

Plonus, M. A.

Porfirev, A.

A. Kovalev, V. Kotlyar, and A. Porfirev, “A highly efficient element for generating elliptic perfect optical vortices,” Appl. Phys. Lett. 110(26), 261102 (2017).
[Crossref]

Pu, J.

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

Y. Liu and J. Pu, “Measuring the orbital angular momentum of elliptical vortex beams by using a slit hexagon aperture,” Opt. Commun. 284(10-11), 2424–2429 (2011).
[Crossref]

Santarsiero, M.

Shirai, T.

Speirits, F.

Speirits, F. C.

M. P. Lavery, F. C. Speirits, S. M. Barnett, and M. J. Padgett, “Detection of a spinning object using light’s orbital angular momentum,” Science 341(6145), 537–540 (2013).
[Crossref] [PubMed]

Sprangle, P.

Tang, Z.

L. Guo, Z. Tang, and W. Wan, “Propagation of four-petal Gaussian vortex beam through a paraxial ABCD optical system,” Optik (Stuttg.) 125(19), 5542–5545 (2014).
[Crossref]

Tyson, R. K.

Visser, T. D.

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

Vlasov, R. A.

Wan, W.

L. Guo, Z. Tang, and W. Wan, “Propagation of four-petal Gaussian vortex beam through a paraxial ABCD optical system,” Optik (Stuttg.) 125(19), 5542–5545 (2014).
[Crossref]

Wang, F.

X. Li, F. Wang, and Y. Cai, “An alternative model for a partially coherent elliptical dark hollow beam,” Opt. Laser Technol. 43(3), 577–585 (2011).
[Crossref]

Wang, G.

D. Liu, H. Zhong, G. Wang, H. Yin, and Y. Wang, “Nonparaxial propagation of a partially coherent four-petal Gaussian vortex beam,” Optik (Stuttg.) 158, 451–459 (2018).
[Crossref]

D. Liu, Y. Wang, G. Wang, X. Luo, and H. Yin, “Propagation properties of partially coherent four-petal Gaussian vortex beams in oceanic turbulence,” Laser Phys. 27(1), 016001 (2017).
[Crossref]

Wang, S. C. H.

Wang, Y.

D. Liu, H. Zhong, G. Wang, H. Yin, and Y. Wang, “Nonparaxial propagation of a partially coherent four-petal Gaussian vortex beam,” Optik (Stuttg.) 158, 451–459 (2018).
[Crossref]

D. Liu, Y. Wang, G. Wang, X. Luo, and H. Yin, “Propagation properties of partially coherent four-petal Gaussian vortex beams in oceanic turbulence,” Laser Phys. 27(1), 016001 (2017).
[Crossref]

D. Liu, Y. Wang, and H. Yin, “Propagation properties of partially coherent four-petal Gaussian vortex beams in turbulent atmosphere,” Opt. Laser Technol. 78, 95–100 (2016).
[Crossref]

Wolf, E.

Yan, F.

Z. Guo-Quan and F. Yan, “M2 factor of four-petal Gaussian beam,” Chin. Phys. B 17(10), 3708–3712 (2008).
[Crossref]

Yan, H.

H. Yan and B. Lü, “The transformation of canonical vortices embedded in a general elliptical Gaussian beam to noncanonical vortices,” J. Opt. A, Pure Appl. Opt. 11(1), 015702 (2009).
[Crossref]

Yin, H.

D. Liu, H. Zhong, G. Wang, H. Yin, and Y. Wang, “Nonparaxial propagation of a partially coherent four-petal Gaussian vortex beam,” Optik (Stuttg.) 158, 451–459 (2018).
[Crossref]

D. Liu, Y. Wang, G. Wang, X. Luo, and H. Yin, “Propagation properties of partially coherent four-petal Gaussian vortex beams in oceanic turbulence,” Laser Phys. 27(1), 016001 (2017).
[Crossref]

D. Liu, Y. Wang, and H. Yin, “Propagation properties of partially coherent four-petal Gaussian vortex beams in turbulent atmosphere,” Opt. Laser Technol. 78, 95–100 (2016).
[Crossref]

Yousefi, M.

Zeng-Hui, G.

G. Zeng-Hui and L. Bai-Da, “Vectorial nonparaxial four-petal Gaussian beams and their propagation in free space,” Chin. Phys. Lett. 23(8), 2070–2073 (2006).
[Crossref]

Zhang, Y.

X. Long, K. Lu, Y. Zhang, J. Guo, and K. Li, “Vectorial structure of a hard-edged-diffracted four-petal Gaussian beam in the far field,” Opt. Commun. 283(23), 4586–4593 (2010).
[Crossref]

Zhao, D.

Zhong, H.

D. Liu, H. Zhong, G. Wang, H. Yin, and Y. Wang, “Nonparaxial propagation of a partially coherent four-petal Gaussian vortex beam,” Optik (Stuttg.) 158, 451–459 (2018).
[Crossref]

Appl. Opt. (2)

Appl. Phys. Lett. (1)

A. Kovalev, V. Kotlyar, and A. Porfirev, “A highly efficient element for generating elliptic perfect optical vortices,” Appl. Phys. Lett. 110(26), 261102 (2017).
[Crossref]

Chin. Phys. B (1)

Z. Guo-Quan and F. Yan, “M2 factor of four-petal Gaussian beam,” Chin. Phys. B 17(10), 3708–3712 (2008).
[Crossref]

Chin. Phys. Lett. (1)

G. Zeng-Hui and L. Bai-Da, “Vectorial nonparaxial four-petal Gaussian beams and their propagation in free space,” Chin. Phys. Lett. 23(8), 2070–2073 (2006).
[Crossref]

J. Opt. A, Pure Appl. Opt. (1)

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Optica (1)

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Figures (8)

Fig. 1
Fig. 1 Evolution of normalized intensity distribution of a partially coherent four-petal elliptic Gaussian vortex beam propagating through turbulent atmosphere for (a) z = 200m, (b) z = 300m, (c) z = 500m, (d) z = 1000m, (e) z = 2000m, (f) z = 50000m, respectively. The calculation parameters are seen in the text. All subfigures share the same coordinate axis direction. All intensity data are normalized by the peak intensity value of the initial wave at z = 0. The peak values I m of each subfigure are 0.754, 0.564, 0.385, 0.188, 0.060, and 1.61 × 10−5 for (a)-(f), respectively.
Fig. 2
Fig. 2 Normalized intensity distribution of partially coherent four-petal elliptic Gaussian vortex beams propagating through turbulent atmosphere with n = 1 and different m. (a-c) m = 2, (d-f) m = 3, (g-i) m = 4, (j-l) m = 5, (a, d, g, j) z = 500m, (b, e, h, k) z = 1000m, (c, f, i, l) z = 50000m. All intensity data are normalized by the peak intensity value of the initial wave at z = 0. The peak values I m of each subfigure are 0.350, 0.201, 1.73 × 10−5, 0.373, 0.183, 1.82 × 10−5, 0.389, 0.165, 1.88 × 10−5, 0.401, 0.153, and 1.92 × 10−5 for (a)-(l), respectively.
Fig. 3
Fig. 3 Normalized intensity distribution of partially coherent four-petal elliptic Gaussian vortex beams propagating through turbulent atmosphere with m = 1 and different n. (a-c) n = 2, (d-f) n = 3, (g-i) n = 4, (j-l) n = 5, (a, d, g, j) z = 500m, (b, e, h, k) z = 1000m, (c, f, i, l) z = 50000m. All intensity data are normalized by the peak intensity value of the initial wave at z = 0. The peak values I m of each subfigure are 0.291, 0.147, 1.57 × 10−5, 0.255, 0.113, 1.55 × 10−5, 0.260, 0.0936, 1.55 × 10−5, 0.255, 0.0865, and 1.54 × 10−5 for (a)-(l), respectively.
Fig. 4
Fig. 4 Normalized intensity distribution of partially coherent four-petal elliptic Gaussian vortex beams propagating through turbulent atmosphere with m = 1, n = 1 and different α. (a-c) α = 0.5, (d-f) α = 0.75, (g-i) α = 1.33, (j-l) α = 2, (a, d, g, j) z = 500m, (b, e, h, k) z = 1000m, (c, f, i, l) z = 50000m. All intensity data are normalized by the peak intensity value of the initial wave at z = 0. The peak values I m of each subfigure are 0.708, 0.373, 5.17 × 10−5, 0.630, 0.270, 3.41 × 10−5, 0.377, 0.200, 1.84 × 10−5, 0.407, 0.138, and 1.15 × 10−5 for (a)-(l), respectively.
Fig. 5
Fig. 5 Normalized intensity distribution of partially coherent four-petal elliptic Gaussian vortex beams propagating in free space. (a-c) m = 1, n = 1, (d-f) m = 5, n = 1, (a, d) z = 500m, (b, e) z = 1000m, (c, f) z = 50000m. All intensity data are normalized by the peak intensity value of the initial wave at z = 0. The peak values I m of each subfigure are 0.406, 0.218, 1.62 × 10−4, 0.414, 0.176, and 1.04 × 10−4 for (a)-(f), respectively.
Fig. 6
Fig. 6 Normalized intensity distribution of partially coherent four-petal elliptic Gaussian vortex beams propagating through turbulent atmosphere with m = 1, n = 1 and different σ. (a-c) σ = 20mm, (d-f) σ = 10mm, (a, d) z = 500m, (b, e) z = 1000m, (c, f) z = 50000m. All intensity data are normalized by the peak intensity value of the initial wave at z = 0. The peak values I m of each subfigure are 0.296, 0.139, 1.57 × 10−5, 0.191, 0.0977, 1.44 × 10−5 for (a)-(f), respectively.
Fig. 7
Fig. 7 relative normalized rms beam width of partially coherent four-petal elliptic Gaussian vortex beams with (a) n = 1, α = 1.5, and different m, (b) m = 1, α = 1.5, and different n, and (c) m = 1, n = 1, and different α. Insets show enlarged view of curves for z<6000m.
Fig. 8
Fig. 8 Curves of Re μ = 0 and Im μ = 0 of a partially coherent four-petal elliptic Gaussian vortex beam with m = 1, n = 1 and (a) z = 200m, (b) z = 500m and (c) z = 1000m.

Tables (1)

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Table 1 Conservation distance d for different values of α

Equations (37)

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W( r 1 , r 2 ,0 )= p( v ) H * ( r 1 ,v )H( r 2 ,v )dv,
W( r 1 , r 2 ,0 )= τ * ( r 1 )τ( r 2 ) p ˜ [ a( r 1 r 2 ) ],
τ( x 0 , y 0 ,0 )= ( x 0 y 0 ω 0 2 ) 2n exp( x 0 2 + α 2 y 0 2 ω 0 2 ) ( x 0 2 + α 2 y 0 2 ω 0 2 ) | m |/2 exp[ im tan 1 ( α y 0 / x 0 ) ],
W( r 01 , r 02 ,0 )= ( x 01 y 01 ω 0 2 ) 2n ( x 02 y 02 ω 0 2 ) 2n exp( x 01 2 + α 2 y 01 2 + x 02 2 + α 2 y 02 2 ω 0 2 ) ( 1 ω 0 2 ) | m | [ x 01 isgn( m )α y 01 ] | m | [ x 02 +isgn( m )α y 02 ] | m | exp[ ( r 01 r 02 ) 2 2 σ 2 ].
( x+iy ) M = l=0 M M! i l l!( Ml )! x Ml y l ,
W( r 01 , r 02 ,0 )= 1 ω 0 8n+2| m | m 1 =0 | m | m 2 =0 | m | ( | m | m 1 ) ( | m | m 2 ) ( 1 ) m 1 [ iαsgn( m ) ] m 1 + m 2 x 01 | m | m 1 +2n y 01 m 1 +2n x 02 | m | m 2 +2n y 02 m 2 +2n exp( x 01 2 + x 02 2 ω 0 2 )exp( y 01 2 + y 02 2 ω 0 2 / α 2 )exp[ ( x 01 x 02 ) 2 + ( y 01 y 02 ) 2 2 σ 2 ],
W( r 1 , r 2 ,z )= ( k 2πz ) 2 d r 01 d r 02 W( r 01 , r 02 ,0 ) exp{ ik 2z [ ( r 1 r 01 ) 2 ( r 2 r 02 ) 2 ] } exp[ Ψ * ( r 01 , r 1 )+Ψ( r 02 , r 2 ) ] ,
exp[ Ψ * ( r 01 , r 1 )+Ψ( r 02 , r 2 ) ] =exp{ [ ( r 01 r 02 ) 2 +( r 01 r 02 )( r 1 r 2 )+ ( r 1 r 2 ) 2 ] ρ 0 2 },
x n exp( p x 2 +2qx )dx=n!exp( q 2 p ) π p ( q p ) n s=0 E[ n 2 ] 1 (n2s)!s! ( p 4 q 2 ) s = π p 2 n i n exp( q 2 p ) ( 1 p ) n/2 H n ( iq p ),
W( r 1 , r 2 ,z )= ( k 2πz ) 2 1 ω 0 8n+2| m | exp[ ik 2z ( x 1 2 x 2 2 + y 1 2 y 2 2 ) ] exp[ ( x 1 x 2 ) 2 + ( y 1 y 2 ) 2 ρ 0 2 ] m 1 =0 | m | m 2 =0 | m | ( | m | m 1 ) ( | m | m 2 ) ( 1 ) m 1 [ iαsgn( m ) ] m 1 + m 2 W x W y ,
W x = n x1 ! π A x π D x exp[ 1 A x ( x 2 x 1 2 ρ 0 2 + ik x 1 2z ) 2 + E x 2 D x ] s=0 E( n x1 /2 ) h=0 n x1 2s 1 ( n x1 2s )!s! 4 s A x ( n x1 s ) ( n x1 2s h ) ( x 2 x 1 2 ρ 0 2 + ik x 1 2z ) n x1 2sh ( 1 ρ 0 2 + 1 2 σ 2 ) h ( i 2 D x ) | m | m 2 +2n+h H | m | m 2 +2n+h ( i E x D x ),
n x1 =| m | m 1 +2n,
A x = 1 ρ 0 2 + 1 2 σ 2 + 1 ω 0 2 + ik 2z ,
D x = 1 A x ( 1 ρ 0 2 + 1 2 σ 2 ) 2 + 1 ρ 0 2 + 1 2 σ 2 + 1 ω 0 2 ik 2z ,
E x =[ 1( 1 ρ 0 2 + 1 2 σ 2 ) 1 A x ] x 1 x 2 2 ρ 0 2 +( 1 ρ 0 2 + 1 2 σ 2 ) 1 A x ik x 1 2z ik x 2 2z ,
W y = n y1 ! π A y π D y exp[ 1 A y ( y 2 y 1 2 ρ 0 2 + ik y 1 2z ) 2 + E y 2 D y ] s=0 E( n y1 /2 ) h=0 n y1 2s 1 ( n y1 2s )!s! 4 s A y ( n y1 s ) ( n y1 2s h ) ( y 2 y 1 2 ρ 0 2 + ik y 1 2z ) n y1 2sh ( 1 ρ 0 2 + 1 2 σ 2 ) h ( i 2 D y ) m 2 +2n+h H m 2 +2n+h ( i E y D y ),
n y1 = m 1 +2n,
I( r,z )=W( r,r,z ).
ω(z)= r 2 I( r,z ) d 2 r I( r,z ) d 2 r ,
ω(z)= ω 1x + ω 1y ω 2 ,
ω 2 = ( k 2πz ) 2 1 ω 0 8n+2| m | m 1 =0 | m | m 2 =0 | m | ( | m | m 1 ) ( | m | m 2 ) ( 1 ) m 1 [ iαsgn( m ) ] m 1 + m 2 R x R y ,
ω 1x = ( k 2πz ) 2 1 ω 0 8n+2| m | m 1 =0 | m | m 2 =0 | m | ( | m | m 1 ) ( | m | m 2 ) ( 1 ) m 1 [ iαsgn( m ) ] m 1 + m 2 R 2x R y ,
ω 1y = ( k 2πz ) 2 1 ω 0 8n+2| m | m 1 =0 | m | m 2 =0 | m | ( | m | m 1 ) ( | m | m 2 ) ( 1 ) m 1 [ iαsgn( m ) ] m 1 + m 2 R x R 2y ,
R x = n x1 ! π A x π D x s=0 E( n x1 /2 ) h=0 n x1 2s t=0 E[ ( | m | m 2 +2n+h )/2 ] G x [ 1 n x2 +2E( n x2 /2 ) ]Γ( n x2 +1 2 )/ ( F x ) n x2 +1 2 ,
R y = n y1 ! π A y π D y s=0 E( n y1 /2 ) h=0 n y1 2s t=0 E[ ( m 2 +2n+h )/2 ] G y [ 1 n y2 +2E( n y2 /2 ) ]Γ( n y2 +1 2 )/ ( F y ) n y2 +1 2 ,
R 2x = n x1 ! π A x π D x s=0 E( n x1 /2 ) h=0 n x1 2s t=0 E[ ( | m | m 2 +2n+h )/2 ] G x [ 1 n x2 +2E( n x2 /2 ) ]Γ( n x2 +3 2 )/ ( F x ) n x2 +3 2 ,
R 2y = n y1 ! π A y π D y s=0 E( n y1 /2 ) h=0 n y1 2s t=0 E[ ( m 2 +2n+h )/2 ] G y [ 1 n y2 +2E( n y2 /2 ) ]Γ( n y2 +3 2 )/ ( F y ) n y2 +3 2 ,
F x = k 2 4 D x z 2 ( 1 A x ρ 0 2 + 1 2 A x σ 2 1 ) 2 k 2 4 A x z 2 ,
G x = 1 ( n x1 2s )!s! 4 s A x n x1 s ( n x1 2s h ) ( 1 ρ 0 2 + 1 2 σ 2 ) h ( i 2 D x ) | m | m 2 +2n+h ( | m | m 2 +2n+h )! ( 1 ) t ( | m | m 2 +2n+h2t )!t! [ 1 D x ( 1 A x ρ 0 2 + 1 2 A x σ 2 1 ) k z ] | m | m 2 +2n+h2t ( ik 2z ) | m | m 1 +2nh2s ,
n x2 = m 2 +4n2t+2| m | m 1 2s,
F y = k 2 4 D y z 2 ( 1 A y ρ 0 2 + 1 2 A y σ 2 1 ) 2 k 2 4 A y z 2 ,
G y = 1 ( n y1 2s )!s! 4 s A y n y1 s ( n y1 2s h ) ( 1 ρ 0 2 + 1 2 σ 2 ) h ( i 2 D y ) m 2 +2n+h ( m 2 +2n+h )! ( 1 ) t ( m 2 +2n+h2t )!t! [ 1 D y ( 1 A y ρ 0 2 + 1 2 A y σ 2 1 ) k z ] m 2 +2n+h2t ( ik 2z ) m 1 +2nh2s ,
n y2 = m 1 + m 2 +4n2t2s.
Re[ μ( r 1 , r 2 ,z ) ]=0,
Im[ μ( r 1 , r 2 ,z ) ]=0,
μ( r 1 , r 2 ,z )= W( r 1 , r 2 ,z ) I( r 1 ,z )I( r 2 ,z ) .
ω( z )= ( 1 4 + 1 4 α 2 ) ω 0 2 +( α 2 +1 ω 0 2 + 2 σ 2 ) ( z k ) 2 + 4 ρ 0 2 ( z k ) 2 = I+II+III .

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