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

e propagation properties of partially coherent circular at-topped (FT) vortex hollow/nonvortex beams are studied in anisotropic turbulent plasma.e analytical expression of the optical intensity of these beams is obtained by employing the extended Huygens–Fresnel integral. e eects of the source and turbulent plasma parameters on the intensity distribution of partially coherent circular FT vortex hollow/nonvortex beams are analyzed numerically. e results show that partially coherent circular FT vortex hollow/nonvortex beams will nally converge into a Gaussian intensity prole at increasing propagation distances. e results also showed that the partially coherent FT vortex hollow/nonvortex beams with higher coherence are less aected by anisotropic turbulent plasma than the less coherent beams.


Introduction
e 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][2][3]. On the other hand, hypersonic turbulence (due to turbulence in the ow of a highspeed ight 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 con rmed that such a plasma sheath has anisotropic turbulence properties [5,6]. e presence of anisotropic turbulent plasma sheaths around the vehicles can have a strong inuence 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][10][11][12][13].
Vortices are de ned as phase singularities because they consist of a dark center and a spiraling phase front, where the phase is inde nite 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 in nite-dimensional Hilbert space and can carry in nite-dimensional information, so it allows for an increase in the channel transmission capacity. us, it has been widely used in optical communication systems in free space and turbulent environments [15].
In 1994, Gori [16] found the at-topped beam is derived from the fundamental Gaussian beam by introducing an order of atness parameter. In medical applications and many practical applications, such as material processing and inertial con nement fusion, the at-topped spatial beam pro le is often required. When the at-topped beam passes through a spiral phase plate, it becomes a at-topped vortex hollow beam [17,18]. e phase of the at-topped vortex hollow beam can be adjusted by the spiral phase plate. e main advantages of the at-topped vortex hollow beam are that it carries the orbital angular momentum that has many applications in atomic optics, and the at-topped vortex hollow beam keeps the dark hollow spot in the far eld; 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 at-topped vortex hollow beam [19][20][21][22][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. is 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 e ects 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 at-topped vortex hollow and nonvortex beams in anisotropic turbulent plasma.

The Wave Structure Function through the
Anisotropic Turbulent Plasma e wave structure function describing a spherical wave propagating in anisotropic turbulent plasma is shown in Figure 1 and can be expressed as [24] where where k 2π/λ is the wave number, in which λ is the incident wavelength and z is the beam propagation distance. D sp (z) 1/ρ 2 0 (z), in which ρ 0 (z) is the coherence length of a spherical wave propagation in anisotropic turbulent plasma, κ ≡ (κ x , κ y ) is the transverse spatial frequency, and Φ n (κ) is the anisotropic refractive index power spectrum of turbulent plasma being transformed from the modi ed von Karman spectrum and is expressed as follows [25]: where n 2 1 is the variance of the refractive index uctuation, m 1 is a constant m 1 4 − d,d is the fractal dimension of the anisotropic turbulent plasma, and L 0 is the outer scale of anisotropic turbulent plasma. Here, a 1 is a tting parameter, which can be expressed as a 1 475(κ 0 ) 2m 1 where κ 0 (2π/l 0 ) m 1 − 0.7 in which l 0 represents the inner scale of anisotropic turbulent plasma. e relation between L 0 and l 0 can be expressed as follows [26]: where R e represents the Reynolds number. For the fully developed turbulence in the mixing layer, R e 5 × 10 5 , d 2.6, and m 1 1.4. If the outer scale L 0 0.1m, then the inner scale l 0 5.3 × 10 − 6 m. Considering the large-scale asymmetric structure of turbulence eddies on the path, the anisotropic spectrum can be expressed as follows [24]: where κ ′ |κ ′ | ξ 2 x κ 2 x + ξ 2 y κ 2 y + κ 2 z will be isotropic in the stretched wave number space κ ′ (κ x ′ , κ y ′ , κ z ′ ), κ x ′ ξ x κ x , κ y ′ ξ y κ y , κ z ′ κ z , and ξ x and ξ y are two anisotropy parameters representing scale-dependent stretching along the x and y direction, respectively. It is noted that when ξ x ξ y 1, Equation (5) reduces itself to isotropic power spectrum Equation (3). Substituting Equation (5) into Equation (2) and applying the Markov approximation, we obtain where U(a, b, z) is the con uent hypergeometric function of the second kind and Γ(·) is the gamma function.

The Partially Coherent Flat-Topped Vortex Hollow Beam of Circular Symmetry in Anisotropic Turbulent Plasma
In the Cartesian coordinate system, the electric eld of a circular at-topped vortex hollow beam at the source plane (z 0) is expressed as follows [17]: z x Figure 1: Sketch of hypersonic turbulence ( gure from [24]).

International Journal of Optics
where r N is the order of the circular at-topped vortex hollow beam, w 0 denotes the beam width, and l is the topological charge. Figure 2(b) shows the normalized intensity distribution of the circular at-topped vortex hollow beam for di erent values of N; when l 0, Equation (7) will reduce to the electric eld of a at-topped nonvortex beam (Figure 2(a)). e cross-spectral density characterizes the spatial correlations of the eld at pairs of points r 1 ′ and r 2 ′ at the source plane and can be de ned as follows [27]: where E(r i ′ , 0) is the electric eld of a fully coherent optical vortex beam and r i ′ is the position vector at the source plane, i 1, 2. e angular brackets denote an ensemble average, while the asterisk * denotes the complex conjugate, A(r i ′ ) represents the amplitude, and g(r 1 ′ − r 2 ′ ) denotes the correlation function between two points r 1 ′ and r 2 ′ , which can be expressed as follows [27]: where σ denotes the spatial coherence length. en the cross-spectral density of a partially coherent at-topped vortex hollow beam of circular symmetry at the source plane can be calculated as follows: By using the extended Huygens-Fresnel integral formula, the cross-spectral density of the partially coherent circular at-topped vortex hollow beam in anisotropic turbulent plasma can be expressed as follows [25,28]: 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]: where J 0 is the zero-order Bessel function and D ψ (r 1 ′ − r 2 ′ ) 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 Since the optical intensity of the partially coherent light is given by I(r, z) � W(r, r, z) [27], then the optical intensity of a partially coherent circular flat-topped vortex hollow beam can be written as Using the following expression [30], we get and by applying the integral formula [31], we derive the following equation: Using the following expression for the Hermite function e integration in Equation (14) can be evaluated, and finally, we obtain Equation (18) is the analytical expression of the optical intensity of the partially coherent circular FT vortex hollow beam in anisotropic plasma turbulence, where x ,

Numerical Results and Discussion
In this section, we numerically investigate the e ects 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. e beam parameters are taken as follows (unless the other values of parameters are speci ed in the gure): the order of the circular FT beam N 3, the topological charge for the vortex hollow beam l 1and nonvortex beam l 0, the beam width w 0 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 L 0 0.1m and l 0 5 × 10 − 6 m, respectively, the refractive index uctuation variance n 2 1 0.73 × 10 − 20 , and anisotropy parameters ξ x 2, ξ y 1 [24].

e E ects of the Turbulent Plasma Parameters on Beam
Evolution. Figures 3-5 have presented the intensity distribution of partially coherent FT vortex hollow/nonvortex beams under the in uence of anisotropic turbulent plasma parameters. From Figure 3, we can see that by increasing the anisotropic parameter ξ x , 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). e intensity distribution in Figure 3(a) shows a circular FT beam pro le evolving into a Gauss-like beam as a result of the decreasing anisotropy parameter ξ x , and from Figure 3(b), it can be seen that decreasing the anisotropy   parameter ξ x leads to a decrease in the dark hollow center of the vortex hollow beam; the pro le of the partially coherent FT vortex hollow beam evolves into the Gauss-like beam for the lower value of ξ x . ese results indicate that the anisotropy in turbulent plasma reduces the turbulence e ect on the evolution behavior of intensity distribution. Figure 4 shows the variations of the partially coherent FT nonvortex and vortex hollow beams in anisotropic turbulent plasma for di erent values of the outer scale of the plasma turbulence L 0 . It is observed that the pro les of partially coherent FT nonvortex and vortex hollow beams become smaller with increasing L 0 , while the axial intensity drops as L 0 decreases as shown in Figures 3(a) and 3(b). We can see  International Journal of Optics 7 from Figure 4(a) that the intensity distribution pro le for a circular at-topped beam evolves into a Gauss-like beam with the decrease of L 0 . 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. Figures 5(a) and 5(b) show the partially coherent FT vortex hollow and nonvortex beam in anisotropic turbulent plasma for di erent values of the variance of the refractive index n 2 1 . From these gures, we can see that the axial intensity drops as n 2 1 increases, and the beam size (FT nonvortex and FT vortex hollow beams) becomes wide. e intensity distribution in Figure 5(a) presents a circular attopped beam pro le that evolves into a Gauss-like beam as a result of increasing the variance of the refractive index n 2 1 . One can see from Figure 5(b) that for a small variance of the refractive index n 2 1 , the intensity distribution retains a dark hollow pattern. In this case, the e ect of turbulent plasma on the beam is not noticeable, while for a large variance of the refractive index n 2 1 , the in uence of the variance of the refractive index n 2 1 on the propagation of the beam becomes more noticeable, and the pro le of the hollow intensity is destroyed and becomes Gaussian distribution. Figures 6 and 7 present the modi cations on the intensity distribution of partially coherent FT vortex hollow and nonvortex beams under the in uence of the source parameters. To investigate the in uences 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 di erent 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. erefore, when the partially coherent FT vortex hollow beam propagates in anisotropic turbulent plasma, the change of the intensity distribution is a ected by the spatial coherence of the source.   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. Figure 8 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 9, 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. e 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].

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 parameter ξ x and outer scale L 0 or increase in the refractive index fluctuation variance 〈n 2 1 〉. 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.
Data Availability e extended Huygens-Fresnel integral that we applied to obtain the analytical expression of the optical intensity of beams in turbulent plasma is taken from reference [25]. e wave structure function of a spherical wave propagating in the anisotropic turbulent plasma is taken from previously reported studies, which have been cited in [24]. e analytical expression of the optical intensity of the partially coherent circular flat-topped vortex hollow beam in anisotropic plasma turbulence in Equation (18) is obtained by using the famous mathematical expressions and integral formula found in references [30,31].