# Propagation of Partially Coherent Flat-Topped Vortex Hollow Beams in Anisotropic Turbulent Plasma

Authors: Abdu A. Alkelly; M. A. H. Khaled; Labiba F. Hassan
Journal: International Journal of Optics
 (2022)

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## Abstract

The propagation properties of partially coherent circular flat-topped (FT) vortex hollow/nonvortex beams are studied in anisotropic turbulent plasma. The analytical expression of the optical intensity of these beams is obtained by employing the extended Huygens–Fresnel integral. The effects of the source and turbulent plasma parameters on the intensity distribution of partially coherent circular FT vortex hollow/nonvortex beams are analyzed numerically. The results show that partially coherent circular FT vortex hollow/nonvortex beams will finally converge into a Gaussian intensity profile at increasing propagation distances. The results also showed that the partially coherent FT vortex hollow/nonvortex beams with higher coherence are less affected by anisotropic turbulent plasma than the less coherent beams.


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## Body

## 1. Introduction

The propagation of laser beams in atmospheric turbulence has received great interest in the past decades because of their applications in remote sensing and free-space optical communication systems [1–3]. On the other hand, hypersonic turbulence (due to turbulence in the flow of a high-speed flight vehicle) is an important factor in understanding optical propagation and optical communication [4].In fact, when an aircraft or spacecraft pierces the Earth’s atmosphere with exceedingly high-speed (hypersonic), the gas environment surrounding the hypersonic aircraft will rub against the aircraft body causing a hypersonic plasma sheath to surround the aircraft. Many experiments have confirmed that such a plasma sheath has anisotropic turbulence properties [5, 6]. The presence of anisotropic turbulent plasma sheaths around the vehicles can have a strong influence on the communication characteristics between the vehicles and radars. It can perturb communication and in some circumstances may lead to a disconnection [7, 8].In recent years, the propagation of various laser beams in anisotropic turbulent plasma has been investigated extensively [9–13].Vortices are defined as phase singularities because they consist of a dark center and a spiraling phase front, where the phase is indefinite in the center. In 1992, Allen et al. [14] proved that a vortex beam can carry an orbital angular momentum (OAM). Due to the perpendicularity and completeness of OAM, the OAM has infinite-dimensional Hilbert space and can carry infinite-dimensional information, so it allows for an increase in the channel transmission capacity. Thus, it has been widely used in optical communication systems in free space and turbulent environments [15].In 1994, Gori [16] found the flat-topped beam is derived from the fundamental Gaussian beam by introducing an order of flatness parameter. In medical applications and many practical applications, such as material processing and inertial confinement fusion, the flat-topped spatial beam profile is often required. When the flat-topped beam passes through a spiral phase plate, it becomes a flat-topped vortex hollow beam [17, 18]. The phase of the flat-topped vortex hollow beam can be adjusted by the spiral phase plate. The main advantages of the flat-topped vortex hollow beam are that it carries the orbital angular momentum that has many applications in atomic optics, and the flat-topped vortex hollow beam keeps the dark hollow spot in the far field; these can expand beam applications such as long-distance stealth detection and invisible control. Very recently, some studies have been conducted on the propagation of the partially coherent FT beam and partially coherent flat-topped vortex hollow beam [19–23].To the best of our knowledge, there are no detailed investigations on the propagation of vortex hollow/nonvortex of a partially coherent circular FT beam in anisotropic turbulent plasma. This paper is devoted to studying the beam evolution properties through anisotropic turbulent plasma based on the extended Huygens–Fresnel formula, and the analytical expressions for the intensity are derived. We will focus our studies on the effects of the source and turbulent plasma parameters on the intensity distribution, beam width and beam quality, and the comparison of the intensity distribution versus the propagation distance between partially coherent circular flat-topped vortex hollow and nonvortex beams in anisotropic turbulent plasma.


## 2. The Wave Structure Function through the Anisotropic Turbulent Plasma

The wave structure function describing a spherical wave propagating in anisotropic turbulent plasma is shown in Figure1 and can be expressed as [24](1)Dψr1′−r2′=8π2k2z∫01dξ∫0∞dκκΦnκ1−J0κ1−ξr1−r2+ξr1′−r2′,≃2Dspzr1−r22+r1−r2r1′−r2′+r1′−r2′2,where(2)Dspz=π2k2z3∫0∞κ3Φnκdκ,where k=2π/λ is the wave number, in which λ is the incident wavelength and z is the beam propagation distance. Dspz=1/ρ02z, in which ρ0z is the coherence length of a spherical wave propagation in anisotropic turbulent plasma, κ≡κx,κy is the transverse spatial frequency, and Φnκ is the anisotropic refractive index power spectrum of turbulent plasma being transformed from the modified von Karman spectrum and is expressed as follows [25]:(3)Φnκ=a164πn12L02m1−11+100κL02m1exp−κκ0,where n12 is the variance of the refractive index fluctuation, m1 is a constant m1=4−d,d is the fractal dimension of the anisotropic turbulent plasma, and L0 is the outer scale of anisotropic turbulent plasma. Here, a1 is a fitting parameter, which can be expressed as a1=475κ02m1 where κ0=2π/l0m1−0.7 in which l0 represents the inner scale of anisotropic turbulent plasma. The relation between L0 and l0 can be expressed as follows [26]:(4)L0l0=Re3/4,where Re represents the Reynolds number. For the fully developed turbulence in the mixing layer, Re=5×105, d=2.6, and m1=1.4. If the outer scale L0=0.1m, then the inner scale l0=5.3×10−6m. Considering the large-scale asymmetric structure of turbulence eddies on the path, the anisotropic spectrum can be expressed as follows [24]:(5)Φnκ′=a164πn12L02m1−11+100κL02m1exp−κκ0′,where κ′=κ′=ξx2κx2+ξy2κy2+κz2 will be isotropic in the stretched wave number space κ′=κx′,κy′,κz′, κx′=ξxκx, κy′=ξyκy, κz′=κz, and ξx and ξy are two anisotropy parameters representing scale-dependent stretching along the x and y direction, respectively. It is noted that when ξx=ξy=1, Equation (5) reduces itself to isotropic power spectrum Equation (3). Substituting Equation (5) into Equation (2) and applying the Markov approximation, we obtain(6)Dspz=32π3k2za1n12L02m1−13ξx2+ξy2ξx3ξy31100L024U4,5−m1,1100L02κ0×Γ4,where Ua,b,z is the confluent hypergeometric function of the second kind and Γ⋅ is the gamma function.Figure 1
Sketch of hypersonic turbulence (figure from [24]).


## 3. The Partially Coherent Flat-Topped Vortex Hollow Beam of Circular Symmetry in Anisotropic Turbulent Plasma

In the Cartesian coordinate system, the electric field of a circular flat-topped vortex hollow beam at the source planez=0 is expressed as follows [17]:(7)Er′,0=∑n=1N−1n−1NN!n!N−n!exp−nx′2+y′2w02x′+iy′w0l,where r′=x′,y′ is the position vector at the source plane, N is the order of the circular flat-topped vortex hollow beam, w0 denotes the beam width, and l is the topological charge. Figure 2(b) shows the normalized intensity distribution of the circular flat-topped vortex hollow beam for different values of N; when l=0, Equation (7) will reduce to the electric field of a flat-topped nonvortex beam (Figure 2(a)).Figure 2
Normalized intensity distribution at the source plane before beam passing through anisotropic turbulent plasma forw0=1cm.(a) Circular flat-topped nonvortex beam and (b) circular flat-topped vortex hollow beam.
(a)(b)The cross-spectral density characterizes the spatial correlations of the field at pairs of pointsr1′ and r2′ at the source plane and can be defined as follows [27]:(8)Wr1′,r2′,0=Er1′,0E∗r2′,0,=Ar1′Ar2′gr1′−r2′,where Eri′,0 is the electric field of a fully coherent optical vortex beam and ri′ is the position vector at the source plane, i=1,2. The angular brackets  denote an ensemble average, while the asterisk ∗ denotes the complex conjugate, Ari′ represents the amplitude, and gr1′−r2′ denotes the correlation function between two points r1′ and r2′, which can be expressed as follows [27]:(9)gr1′−r2′=exp−r1′−r2′22σ2,where σ denotes the spatial coherence length. Then the cross-spectral density of a partially coherent flat-topped vortex hollow beam of circular symmetry at the source plane can be calculated as follows:(10)Wr1′,r2′,0=∑m=1N∑n=1N−1m+nN2N!m!N−m!N!n!N−n!exp−mx1′2+y1′2w02−nx2′2+y2′2w02×x1′+iy1′w0lx2′−iy2′w0lexp−r1′−r2′22σ2.By using the extended Huygens–Fresnel integral formula, the cross-spectral density of the partially coherent circular flat-topped vortex hollow beam in anisotropic turbulent plasma can be expressed as follows [25, 28]:(11)Woutr1,r2,z=k2πz2∬Winr1′,r2′,0exp−ik2zr1−r1′2−r2−r2′2×expψ∗r1′,r1,z+ψr2′,r2,zd2r1′d2r2′,where ψ is the random part of the complex phase of a spherical wave propagating through anisotropic turbulent plasma. Over the ensemble of the statistical realization of the random medium, we can write the equation as follows [29]:(12)expψ∗r1′,r1,z+ψr2′,r2,z=exp−Dψr1′−r2′2≃−Dspzr1−r22+r1−r2r1′−r2′+r1′−r2′2,where J0 is the zero-order Bessel function and Dψr1′−r2′ is the wave structure function through anisotropic turbulent plasma, which is given in Equation (1). Now, substituting Equations (10) and (12) into Equation (11), we get(13)Woutr1,r2,z=k2πz2∑m=1N∑n=1N−1m+nN2N!m!N−m!N!n!N−n!∬exp−mx1′2+y1′2w02−nx2′2+y2′2w02×x1′+iy1′w0lx2′−iy2′w0lexp−r1′−r2′22σ2exp−ik2zr1−r1′2−r2−r2′2×exp−Dspzr1−r22+r1−r2r1′−r2′+r1′−r2′2d2r1′d2r2′,Since the optical intensity of the partially coherent light is given byIr,z=Wr,r,z [27], then the optical intensity of a partially coherent circular flat-topped vortex hollow beam can be written as(14)Ioutr,z=k2πz2∑m=1N∑n=1N−1m+nN2N!m!N−m!N!n!N−n!∬exp−mx1′2+y1′2w02−nx2′2+y2′2w02×x1′x2′+y1′y2′+ix2′y1′−x1′y2′w02lexp−r1′−r2′22σ2exp−ik2zr−r1′2−r−r2′2×exp−Dspzr1′−r2′2d2r1′d2r2′,Using the following expression [30], we get(15)x+iyn=∑t=0nn!t!n−t!xn−tiyt,and by applying the integral formula [31], we derive the following equation:(16)∫−∞∞xnexp−px2+2qxdx=n!πpqpnexpq2p∑k=0n/21k!n−2k!p4q2k.Using the following expression for the Hermite functionHqx [30], we get(17)Hnx=∑k=0n/2−1kn!k!n−2k!2xn−2k.The integration in Equation (14) can be evaluated, and finally, we obtain(18)Ioutr,z=k2πz2∑m=1N∑n=1N−1m+nN2N!m!N−m!N!n!N−n!∑t=0litl!t!l−t!1w0l∑s=0l−isl!s!l−s!1w0l×l−t!πa11a1l−texp−1a1k2zx2∑d1=0l−t/21d1!l−t−2d1!a14d1∑t1=0l−t−2d1l−t−2d1!t1!l−t−2d1−t1!×ik2zxl−t−2d1−t112σ2+Dspzt12−l+s−t1il−s+t1expc12b1πb11b1l−s+t1Hl−s+t1−ic1b1×t!πa21a2texp−1a2k2zy2∑d2=0t/21d2!t−2d2!a24d2∑t2=0t−2d2t−2d2!t2!t−2d2−t2!ik2zyt−2d2−t2×12σ2+Dspzt22−s−t2is+t2expc22b2πb21b2s+t2Hs+t2−ic2b2.Equation (18) is the analytical expression of the optical intensity of the partially coherent circular FT vortex hollow beam in anisotropic plasma turbulence, where(19)a1=a2=nw02+12σ2+ik2z+Dspz,b1=b2=mw02+12σ2−ik2z+Dspz−1a112σ2+Dspz2,c1=1a112σ2+Dspzik2zx−ik2zx,c2=1a112σ2+Dspzik2zy−ik2zy.


## 4. Numerical Results and Discussion

In this section, we numerically investigate the effects of the source and turbulent plasma parameters on the evolution properties of partially coherent circular FT vortex hollow and nonvortex beams and the comparison of the intensity distribution versus the propagation distance between partially coherent circular FT vortex hollow and non-vortex beams in anisotropic turbulent plasma. The beam parameters are taken as follows (unless the other values of parameters are specified in the figure): the order of the circular FT beamN=3, the topological charge for the vortex hollow beam l=1and nonvortex beam l=0, the beam width w0=1cm, the wavelength λ=1550nm, and the spatial coherence length σ=1mm, whereas the anisotropic turbulent plasma parameters are taken as follows: the outer and inner scales of the turbulence L0=0.1m and l0=5×10−6m, respectively, the refractive index fluctuation variance n12=0.73×10−20, and anisotropy parameters ξx=2, ξy=1 [24].


### 4.1. The Effects of the Turbulent Plasma Parameters on Beam Evolution

Figures3–5 have presented the intensity distribution of partially coherent FT vortex hollow/nonvortex beams under the influence of anisotropic turbulent plasma parameters. From Figure 3, we can see that by increasing the anisotropic parameter ξx, the axial intensity distribution of both nonvortex and vortex hollow beams increases, while the beam size becomes narrower as shown in Figures 3(a) and 3(b). The intensity distribution in Figure 3(a) shows a circular FT beam profile evolving into a Gauss-like beam as a result of the decreasing anisotropy parameter ξx, and from Figure 3(b), it can be seen that decreasing the anisotropy parameter ξx leads to a decrease in the dark hollow center of the vortex hollow beam; the profile of the partially coherent FT vortex hollow beam evolves into the Gauss-like beam for the lower value of ξx. These results indicate that the anisotropy in turbulent plasma reduces the turbulence effect on the evolution behavior of intensity distribution.Figure 3
Cross-sections of the intensity distribution of the partially coherent circular: (a) flat-topped nonvortex beam and (b) flat-topped vortex hollow beam in anisotropic turbulent plasma for different values of the anisotropy parameterξx (z=1m).
(a)(b)Figure 4
Cross-sections of the intensity distribution of the partially coherent circular: (a) flat-topped nonvortex beam and (b) flat-topped vortex hollow beam in anisotropic turbulent plasma for different values of the outer scaleL0 (z=1m).
(a)(b)Figure 5
Cross-sections of the intensity distribution of the partially coherent circular: (a) flat-topped nonvortex beam and (b) flat-topped vortex hollow beam in anisotropic turbulent plasma for different values of the refractive index fluctuation variancen12 (z=1m).
(a)(b)Figure4 shows the variations of the partially coherent FT nonvortex and vortex hollow beams in anisotropic turbulent plasma for different values of the outer scale of the plasma turbulence L0. It is observed that the profiles of partially coherent FT nonvortex and vortex hollow beams become smaller with increasing L0, while the axial intensity drops as L0 decreases as shown in Figures 3(a) and 3(b). We can see from Figure 4(a) that the intensity distribution profile for a circular flat-topped beam evolves into a Gauss-like beam with the decrease of L0. For a partially coherent FT vortex hollow beam, Figure 4(b) indicates that when the outer scale of the turbulence enlarges, the dark hollow center of the beam increases.Figures5(a) and 5(b) show the partially coherent FT vortex hollow and nonvortex beam in anisotropic turbulent plasma for different values of the variance of the refractive index n12. From these figures, we can see that the axial intensity drops as n12 increases, and the beam size (FT nonvortex and FT vortex hollow beams) becomes wide. The intensity distribution in Figure 5(a) presents a circular flat-topped beam profile that evolves into a Gauss-like beam as a result of increasing the variance of the refractive index n12. One can see from Figure 5(b) that for a small variance of the refractive index n12, the intensity distribution retains a dark hollow pattern. In this case, the effect of turbulent plasma on the beam is not noticeable, while for a large variance of the refractive index n12, the influence of the variance of the refractive index n12 on the propagation of the beam becomes more noticeable, and the profile of the hollow intensity is destroyed and becomes Gaussian distribution.


### 4.2. The Effects of the Source Parameters on Beam Evolution

Figures6 and 7 present the modifications on the intensity distribution of partially coherent FT vortex hollow and nonvortex beams under the influence of the source parameters. To investigate the influences of spatial coherence length σ on the spreading properties of partially coherent FT nonvortex and vortex hollow beams, the cross-sections of the intensity distribution of partially coherent FT nonvortex and vortex hollow beams propagating in anisotropic turbulent plasma for different values of the spatial coherence length σ are given in Figure 6. From Figures 6(a) and 6(b), we can see that the axial intensity drops as σ decreases, while the beam size does not change. As shown in Figure 6(b), increasing the spatial coherence length σ increases the dark hollow center of the partially coherent FT vortex hollow beam. Therefore, when the partially coherent FT vortex hollow beam propagates in anisotropic turbulent plasma, the change of the intensity distribution is affected by the spatial coherence of the source.Figure 6
Cross-sections of the intensity distribution of the partially coherent circular: (a) flat-topped nonvortex beam and (b) flat-topped vortex hollow beam in anisotropic turbulent plasma for different values of the spatial coherence lengthσ (z=1m).
(a)(b)Figure 7
Cross-sections of the intensity distribution of the partially coherent circular: (a) flat-topped nonvortex beam and (b) flat-topped vortex hollow beam in anisotropic turbulent plasma for different values of the order of the circular flat-topped beamN (z=1m).
(a)(b)Figure7 shows the cross-sections of the intensity distribution of partially coherent circular FT nonvortex and vortex hollow beams for different values of the order of the circular flat-topped beam N. From Figures 7(a) and 7(b), we can see that the beam size becomes wider with the increase in N, and it is found that the partially coherent circular nonvortex and vortex hollow beams with larger orders of the circular flat-topped beam can keep the beam profile better when the beam propagates in turbulent plasma. From Figure 7(b), it is seen that increasing N can increase the dark hollow center of the FT vortex hollow beam.Figure8 show cross-sections of normalized intensities for partially coherent circular flat-topped vortex hollow beams propagating through turbulent plasma for the different topological charge l. As can be seen, the beam with larger l has the larger dark hollow center in the near field (Figure 8(a)), and the beam with the different l will evolve into a Gauss-like beam with an increase in the propagation distance (Figure 8(b)) due to the influence of turbulent plasma. It can be noted that the beam with larger l will lose its initial dark hollow center slowly.Figure 8
Cross-sections of the intensity distribution of the partially coherent circular flat-topped vortex hollow beam in anisotropic turbulent plasma for different values of the topological chargel at (a)z=1m and (b)z=2m.
(a)(b)


### 4.3. The Comparison of the Intensity Distribution versus the Propagation Distance

From Figure9, we can see that the intensity of both partially coherent FT nonvortex and vortex hollow beams gradually decreases and becomes weaker with increasing propagation distance z, and also, one can find that the beam significantly expands during the propagation. For comparison, the corresponding results of partially coherent flat-topped nonvortex and vortex hollow beams are shown together propagating in anisotropic turbulent plasma for different values of propagation distance z as shown in Figure 10. From Figures 10(a)–10(d), as can be seen, the partially coherent flat-topped vortex hollow beam propagating in turbulent plasma can keep its initial dark center at a short propagation distance (Figure 10(a)). As the propagation distance z increases, the beam will lose its initial dark center and evolve into a Gaussian-like beam. The reason for this phenomenon is that as the transmission distance of the dark hollow center beam increases, the dark hollow center beam expands due to turbulence and diffraction, among which the expansion towards the center of the dark hollow causes the spot center to breakdown. It is worth noting that a general optical beam will finally converge into a Gaussian intensity profile at increasing propagation distances, and this result is in a good agreement as shown by the authors in [32].Figure 9
The intensity distribution of the partially coherent circular flat-topped nonvortex beam (top row) and vortex hollow beam (bottom row) in anisotropic turbulent plasma for different values of propagation distancesz.Figure 10
The normalized intensity distribution of the partially coherent circular FT vortex hollow beam and FT nonvortex beam in anisotropic turbulent plasma for different values of propagation distances: (a)z=0m, (b)z=1m, (c)z=1.3m, and (d)z=3m.
(a)(b)(c)(d)


## 4.1. The Effects of the Turbulent Plasma Parameters on Beam Evolution

Figures3–5 have presented the intensity distribution of partially coherent FT vortex hollow/nonvortex beams under the influence of anisotropic turbulent plasma parameters. From Figure 3, we can see that by increasing the anisotropic parameter ξx, the axial intensity distribution of both nonvortex and vortex hollow beams increases, while the beam size becomes narrower as shown in Figures 3(a) and 3(b). The intensity distribution in Figure 3(a) shows a circular FT beam profile evolving into a Gauss-like beam as a result of the decreasing anisotropy parameter ξx, and from Figure 3(b), it can be seen that decreasing the anisotropy parameter ξx leads to a decrease in the dark hollow center of the vortex hollow beam; the profile of the partially coherent FT vortex hollow beam evolves into the Gauss-like beam for the lower value of ξx. These results indicate that the anisotropy in turbulent plasma reduces the turbulence effect on the evolution behavior of intensity distribution.Figure 3
Cross-sections of the intensity distribution of the partially coherent circular: (a) flat-topped nonvortex beam and (b) flat-topped vortex hollow beam in anisotropic turbulent plasma for different values of the anisotropy parameterξx (z=1m).
(a)(b)Figure 4
Cross-sections of the intensity distribution of the partially coherent circular: (a) flat-topped nonvortex beam and (b) flat-topped vortex hollow beam in anisotropic turbulent plasma for different values of the outer scaleL0 (z=1m).
(a)(b)Figure 5
Cross-sections of the intensity distribution of the partially coherent circular: (a) flat-topped nonvortex beam and (b) flat-topped vortex hollow beam in anisotropic turbulent plasma for different values of the refractive index fluctuation variancen12 (z=1m).
(a)(b)Figure4 shows the variations of the partially coherent FT nonvortex and vortex hollow beams in anisotropic turbulent plasma for different values of the outer scale of the plasma turbulence L0. It is observed that the profiles of partially coherent FT nonvortex and vortex hollow beams become smaller with increasing L0, while the axial intensity drops as L0 decreases as shown in Figures 3(a) and 3(b). We can see from Figure 4(a) that the intensity distribution profile for a circular flat-topped beam evolves into a Gauss-like beam with the decrease of L0. For a partially coherent FT vortex hollow beam, Figure 4(b) indicates that when the outer scale of the turbulence enlarges, the dark hollow center of the beam increases.Figures5(a) and 5(b) show the partially coherent FT vortex hollow and nonvortex beam in anisotropic turbulent plasma for different values of the variance of the refractive index n12. From these figures, we can see that the axial intensity drops as n12 increases, and the beam size (FT nonvortex and FT vortex hollow beams) becomes wide. The intensity distribution in Figure 5(a) presents a circular flat-topped beam profile that evolves into a Gauss-like beam as a result of increasing the variance of the refractive index n12. One can see from Figure 5(b) that for a small variance of the refractive index n12, the intensity distribution retains a dark hollow pattern. In this case, the effect of turbulent plasma on the beam is not noticeable, while for a large variance of the refractive index n12, the influence of the variance of the refractive index n12 on the propagation of the beam becomes more noticeable, and the profile of the hollow intensity is destroyed and becomes Gaussian distribution.


## 4.2. The Effects of the Source Parameters on Beam Evolution

Figures6 and 7 present the modifications on the intensity distribution of partially coherent FT vortex hollow and nonvortex beams under the influence of the source parameters. To investigate the influences of spatial coherence length σ on the spreading properties of partially coherent FT nonvortex and vortex hollow beams, the cross-sections of the intensity distribution of partially coherent FT nonvortex and vortex hollow beams propagating in anisotropic turbulent plasma for different values of the spatial coherence length σ are given in Figure 6. From Figures 6(a) and 6(b), we can see that the axial intensity drops as σ decreases, while the beam size does not change. As shown in Figure 6(b), increasing the spatial coherence length σ increases the dark hollow center of the partially coherent FT vortex hollow beam. Therefore, when the partially coherent FT vortex hollow beam propagates in anisotropic turbulent plasma, the change of the intensity distribution is affected by the spatial coherence of the source.Figure 6
Cross-sections of the intensity distribution of the partially coherent circular: (a) flat-topped nonvortex beam and (b) flat-topped vortex hollow beam in anisotropic turbulent plasma for different values of the spatial coherence lengthσ (z=1m).
(a)(b)Figure 7
Cross-sections of the intensity distribution of the partially coherent circular: (a) flat-topped nonvortex beam and (b) flat-topped vortex hollow beam in anisotropic turbulent plasma for different values of the order of the circular flat-topped beamN (z=1m).
(a)(b)Figure7 shows the cross-sections of the intensity distribution of partially coherent circular FT nonvortex and vortex hollow beams for different values of the order of the circular flat-topped beam N. From Figures 7(a) and 7(b), we can see that the beam size becomes wider with the increase in N, and it is found that the partially coherent circular nonvortex and vortex hollow beams with larger orders of the circular flat-topped beam can keep the beam profile better when the beam propagates in turbulent plasma. From Figure 7(b), it is seen that increasing N can increase the dark hollow center of the FT vortex hollow beam.Figure8 show cross-sections of normalized intensities for partially coherent circular flat-topped vortex hollow beams propagating through turbulent plasma for the different topological charge l. As can be seen, the beam with larger l has the larger dark hollow center in the near field (Figure 8(a)), and the beam with the different l will evolve into a Gauss-like beam with an increase in the propagation distance (Figure 8(b)) due to the influence of turbulent plasma. It can be noted that the beam with larger l will lose its initial dark hollow center slowly.Figure 8
Cross-sections of the intensity distribution of the partially coherent circular flat-topped vortex hollow beam in anisotropic turbulent plasma for different values of the topological chargel at (a)z=1m and (b)z=2m.
(a)(b)


## 4.3. The Comparison of the Intensity Distribution versus the Propagation Distance

From Figure9, we can see that the intensity of both partially coherent FT nonvortex and vortex hollow beams gradually decreases and becomes weaker with increasing propagation distance z, and also, one can find that the beam significantly expands during the propagation. For comparison, the corresponding results of partially coherent flat-topped nonvortex and vortex hollow beams are shown together propagating in anisotropic turbulent plasma for different values of propagation distance z as shown in Figure 10. From Figures 10(a)–10(d), as can be seen, the partially coherent flat-topped vortex hollow beam propagating in turbulent plasma can keep its initial dark center at a short propagation distance (Figure 10(a)). As the propagation distance z increases, the beam will lose its initial dark center and evolve into a Gaussian-like beam. The reason for this phenomenon is that as the transmission distance of the dark hollow center beam increases, the dark hollow center beam expands due to turbulence and diffraction, among which the expansion towards the center of the dark hollow causes the spot center to breakdown. It is worth noting that a general optical beam will finally converge into a Gaussian intensity profile at increasing propagation distances, and this result is in a good agreement as shown by the authors in [32].Figure 9
The intensity distribution of the partially coherent circular flat-topped nonvortex beam (top row) and vortex hollow beam (bottom row) in anisotropic turbulent plasma for different values of propagation distancesz.Figure 10
The normalized intensity distribution of the partially coherent circular FT vortex hollow beam and FT nonvortex beam in anisotropic turbulent plasma for different values of propagation distances: (a)z=0m, (b)z=1m, (c)z=1.3m, and (d)z=3m.
(a)(b)(c)(d)


## 5. Conclusions

In this paper, the partially coherent circular FT vortex hollow/nonvortex beams propagating in anisotropic turbulent plasma have been investigated by using the extended Huygens–Fresnel diffraction integral. We have analyzed and discussed the dependence of the beam evolution on the anisotropic turbulent plasma parameters, source parameters, and the propagation distance by using numerical examples. It can be seen from our results that the partially coherent circular FT vortex hollow/nonvortex beams evolve into Gaussian-like beams because of a decrease in the anisotropy parameterξx and outer scale L0 or increase in the refractive index fluctuation variance n12. It is found that when the beam propagates in anisotropic turbulent plasma, the partially coherent FT vortex hollow/nonvortex beams with high order N and topological charge l of the circular FT beam can keep the beam profile better. It is to be noted that these results may be useful for optical communications.



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