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Intensity and Coherence Characteristics of a Radial Phase-Locked Multi-Gaussian Schell-Model Vortex Beam Array in Atmospheric Turbulence

Intensity and Coherence Characteristics of a Radial Phase-Locked Multi-Gaussian Schell-Model... hv photonics Letter Intensity and Coherence Characteristics of a Radial Phase-Locked Multi-Gaussian Schell-Model Vortex Beam Array in Atmospheric Turbulence Jialu Zhao, Guiqiu Wang, Xiaolu Ma, Haiyang Zhong, Hongming Yin, Yaochuan Wang and Dajun Liu * Department of Physics, College of Science, Dalian Maritime University, Dalian 116026, China; zjl970929@dlmu.edu.cn (J.Z.); gqwang@dlmu.edu.cn (G.W.); maxiaolu@dlmu.edu.cn (X.M.); haae@dlmu.edu.cn (H.Z.); hmyin@dlmu.edu.cn (H.Y.); ycwang@dlmu.edu.cn (Y.W.) * Correspondence: liudajun@dlmu.edu.cn; Tel.: +86-159-4111-5638 Abstract: The theoretical descriptions for a radial phase-locked multi-Gaussian Schell-model vortex (RPLMGSMV) beam array is first given. The normalized intensity and coherence distributions of a RPLMGSMV beam array propagating in free space and atmospheric turbulence are illustrated and analyzed. The results show that a RPLMGSMV beam array with larger total number N or smaller coherence length s can evolve into a beam with better flatness when the beam array translating into the flat-topped profile at longer distance z and the flatness of the flat-topped intensity distribution can be destroyed by the atmospheric turbulence at longer distance z. The coherence distribution of a RPLMGSMV beam array in atmospheric turbulence at the longer distance will have Gaussian distri- bution. The research results will be useful in free space optical communication using a RPLMGSMV beam array. Keywords: average intensity; multi-Gaussian Schell-model source; vortex beam; beam array; atmo- spheric turbulence Citation: Zhao, J.; Wang, G.; Ma, X.; Zhong, H.; Yin, H.; Wang, Y.; Liu, D. Intensity and Coherence Characteristics of a Radial 1. Introduction Phase-Locked Multi-Gaussian Schell-Model Vortex Beam Array in With the development of wireless optical communication and laser radar, the evolu- Atmospheric Turbulence. Photonics tions of laser beams in atmospheric turbulence were widely studied in past years [1]. In past 2021, 8, 5. https://doi.org/10.3390/ years, the properties of fully coherent laser beams in turbulence have been widely analyzed, photonics8010005 such as Gaussian beams [2], Hermite–Gaussian beam [3], Pearcey–Gaussian beam [4], beam array [5–7], Laguerre–Gaussian beam [8], Airy beam [9], hollow beam [10,11], vortex Received: 19 November 2020 beam [12,13], and vortex lattices [14]. From previous studies [1], one can see that partially Accepted: 24 December 2020 coherent beams are resistance to the deleterious effects of turbulence. In past years, the Published: 29 December 2020 properties, including intensity, polarization, and coherence, of Gaussian Schell-model (GSM) beams [15–19] and GSM beam array [20–23] propagating in turbulence were widely Publisher’s Note: MDPI stays neu- studied. On the other hand, the special correlated beams are also be introduced and ana- tral with regard to jurisdictional clai- lyzed, such as non-uniformly correlated beams [24], multi-Gaussian Schell-model (MGSM) ms in published maps and institutio- beams [25–28], cosine-GSM beams [29], multi-cosine-Laguerre–Gaussian correlated Schell- nal affiliations. model beams [30]. Laser arrays can produce the higher power output than single beam and which can have linear, rectangular and radial distributions. The beams correlated with MGSM source can provide flat intensity profiles in the far field [25]. To obtain the Copyright: © 2020 by the authors. Li- flat-topped intensity profiles, the MGSM beam arrays propagating in turbulence are in- censee MDPI, Basel, Switzerland. vestigated, and it is found that the MGSM beam arrays can achieve the better flat-topped This article is an open access article profiles [31,32]. Moreover, the MGSM vortex beam has been introduced and studied. It distributed under the terms and con- shows that the intensity profile of MGSM vortex can be modulated by the topological ditions of the Creative Commons At- charge [33]. Thus, it will be very interesting to consider the laser array composed by tribution (CC BY) license (https:// MGSM vortex beams. In this paper, we extend MGSM vortex beam into the radial phase- creativecommons.org/licenses/by/ locked multi-Gaussian-Schell-model vortex (RPLMGSMV) beam array, and investigate 4.0/). Photonics 2021, 8, 5. https://doi.org/10.3390/photonics8010005 https://www.mdpi.com/journal/photonics Photonics 2021, 8, 5 2 of 12 the intensity and coherence properties of RPLMGSMV beam array propagating in free space and atmospheric turbulence. Moreover, the model of laser arrays with linear and rectangular distributions can also be obtained in the similar analytical approach. 2. Theory Analysis 2.1. Analytical Description of RPLMGSMV Beam Array The electric field distribution of a Gaussian vortex beam at source plane z = 0 is described by 2 2 x + y jMj 0 0 E(r , 0) = [x + isgn(M)y ] exp (1) 0 0 where w is beam waist and M is the topological charge. In this work, laser array with radial distribution will be analyzed as example, the electric field of a radial phase-locked Gaussian vortex beam array with Q beamlets can be given as: " # 2 2 x r + y r 0 qx 0 qy jMj E (r , 0) = x r + isgn(M) y r exp exp ij (2) Q 0 å 0 qx 0 qy q q=1 0 with 2p r = R cos j , r = R sin j , j = q , q = 1, 2, Q (3) qx q qy q q where R is radius; j is the phase of the q-th beamlet; r and r are the center of the q-th n qx qy beamlet element located at z = 0. Considering the unified theory of coherence and polarization [34], the cross spectral density (CSD) of partially coherent beams can be expressed as W(r , r ) = E(r )E (r ) (4) 10 20 10 20 Introducing a MGSM correlation [22], the CSD of a RPLMGSMV beam array with Q beamlets can be written as Q Q W (r , r , 0) = å å exp i j j 20 q2 10 q1 q =1 q =1 1 2 2 2 jMj (x r ) +(y r ) 10 q1x 10 q1y x r + isgn(M) y r exp 10 q1x 10 q1y 2 (5) 2 2 x r + y r jMj ( ) ( ) 20 q2x 20 q2y x r isgn(M) y r exp 20 q2x 20 q2y 2 2 n1 N x r x r y r y r (1) [( ) ( 20 q2x)] [( ) ( 20 q2y)] 1 10 q1x 10 q1y exp 2 2 C n 0 2ns 2ns n=1 where N is total number of terms of MGSM source, s is the coherence length, C is the normalized factor, and can be described by n1 N (1) C = (6) n=1 Figure 1 shows the normalized intensity of a RPLMGSMV beam array at z = 0 for the different Q, one can see that the beamlets of a RPLMGSMV beam array have the hollow center. Photonics 2021, 8, x FOR PEER REVIEW 3 of 13 Photonics 2021, 8, 5 3 of 12 Figure 1 shows the normalized intensity of a RPLMGSMV beam array at z = 0 for the different Q , one can see that the beamlets of a RPLMGSMV beam array have the hollow center. Figure 1. Normalized intensity of a RPLMGSMV beam array at z = 0 for the different Q . (a) Q =3 Figure 1. Normalized intensity of a RPLMGSMV beam array at z = 0 for the different Q. (a) Q = 3, , (b) Q =4 , (c) Q =5 , (d) Q =6 . (b) Q = 4, (c) Q = 5, (d) Q = 6. 2.2. Propagation Analysis 2.2. Propagation Analysis Based on the extended Huygens–Fresnel integral, the CSD of a RPLMGSMV beam Based on the extended Huygens–Fresnel integral, the CSD of a RPLMGSMV beam array propagating in atmospheric turbulence at plane z can be read as [1] array propagating in atmospheric turbulence at plane z can be read as [1] R R R R +¥ +¥ +¥ +¥ W r , r , z = +∞ +∞ +∞ +∞ W r , r , 0 ( ) k ( ) 1 2 10 20 2 2 ¥ ¥ ¥ ¥ 4p z Wz rr,, = W r ,r ,0 () () 12 22 10 20 h  i −∞ −∞ −∞ −∞ 4π z ik 2 ik 2 (7) exp (r r ) + (r r ) 1 10 2 20 2z 2z ik ik  22 (7) ×− exp () r-r + () r -r 110 2 20  22zz hexp [y(r , r ) + y (r , r )]idr dr 10 1 20 2 10 20  ×+ expψψ () rr, (r ,r) ddr r 10 1 20 2 10 20  with " # with 2 2 (r r ) + (r r )(r r ) + (r r ) 10 20 10 20 1 2 1 2 hexp[y(r , r) + y (r , r)]i = exp (8) 10 20  () rr−+() rr− 0 (r−r)+(r−r) ∗ 10 20 10 20 1 2 1 2  expψψ rr,+= r ,r exp − () ( ) (8) 10 20   0 In the above equation, the spatial coher ence length r can be expressed as 3/5 In the above equation, the spatial coherence length can be expressed as 2 2 r = 0.545C k z (9) −35 ρ = 0.545Ck z (9) () 0 n where C is the structure constant of atmospheric turbulence. Substituting C Equation (5) into Equation (7), the CSD of a RPLMGSMV beam array where is the structure constant of atmospheric turbulence. propagating in atmospheric turbulence at plane z can be derived as Substituting Equation (5) into Equation (7), the CSD of a RPLMGSMV beam array h i Q Q 2 2 propagating in atmospheric tu 2 rbulence at plane z can be derived as (x x ) +(y y ) k ik 2 2 1 2 1 2 W (r , r , z) = exp i j j exp r r exp å å 1 2 q1 q2 2 2 2 2z 1 2 4p z q =1 q =1 1 2 (10) jMj jMj l N n1 1 N jMj!i jMj!(i) (1) W(x, z)W(y, z) å å å C n l !(jMjl )! l !(jMjl )! 0 1 1 2 2 l =0 l =0 n=1 1 2 Photonics 2021, 8, 5 4 of 12 with h  i h i 1 ik 2 2 ik W (x, z) = exp r r exp r r exp 2 x r x r q x q x 1 q x 2 q x 2 2 q x q x 2 1 2z 2 2z 1 r 1 jMjl (x x ) r r 2 ( q x q x) 1 p 1 1 2 exp ( M l )! j j a a " # jMjl [ ] x x +2 r r ( ) k 1 2 q x q x 1 ik ik 1 a 1 1 2 exp x r (11) q x å 1 2 a 2z 2z 1 4 k !(jMjl 2k )! 2r 1 1 1 k =0 jMjl 2k s s jMjl 2k 1 1 1 1 1 1 x x +2 r r M l 2k ! ( q x q x) (j j ) 1 2 1 1 ik ik 1 2 1 1 å x r + 1 q x 2 2 2 2z 2z 1 s !(jMjl 2k s )! 2r 2ns r 1 1 1 1 0 0 s=0 jMjl +s q   2 c ic p i x x p p exp H jMjl +s b b 2 1 2 b b h  i h i 1 ik 2 2 ik W (y, z) = exp r r exp r r exp 2 y r y r q y q y q y q y 2 q y q y 1 2 1 2 2z 2z 1 2 1 2 (y y ) r r 1 ( q y q y) 1 2 2 p 1 exp (l )! 2 1 a a " # [ ] y y +2 r r ( ) k 1 2 q y q y 1 ik ik 1 a 2 1 2 exp y r q y å (12) 1 2 a 2z 2z 1 4 y k !(l 2k )! 2r 2 1 2 k =0 l 2k s s 1 2 2 2 l 2k 1 2 y y +2 r r (l 2k )! 1 2 ( q y q y) ik ik 1 1 1 2 1 2 y r + 1 q y 2 2 2 s !(l 2k s )! 2z 2z 2r 2ns r 3 1 2 2 0 0 s =0 q  l +s 2 2 c ic p i y y p p exp H l +s 2 2 b b 2 b b where 1 1 1 ik a = + + + (13) 2 2 2 2ns 2z w r 0 0 1 1 1 ik 1 1 1 b = + + + (14) 2 2 2 2 2 2ns 2z a 2ns w r r 0 0 0 x x +2 r r ( q x q x) 1 2 ik ik 2 c = r x + x q x 2 2z 2 2z 2r (15) x x +2 r r ( q x q x) 1 2 1 1 1 ik ik 2 + + x r q x 2 2 1 2 a 2z 2z 1 2ns r 2r 0 0 y y +2(r r ) 1 2 q y q y ik ik 1 2 c = r y + y q y 2 2 2z 2 2z 2r (16) y y +2(r r ) 1 2 q y q y 1 1 1 ik ik 1 2 + + y r q y 2 2 1 2 a 2z 2z 1 2ns r 2r 0 0 In the derivations of Equation (10), the following equations has been applied [35] M!i Ml l (x + iy) = x y (17) l!(M l)! l=0 +¥ p i b ib n 2 x exp ax + 2bx dx = p exp H p (18) a a ¥ 2 a a [ ] (1) n! n2l H (x) = (2x) (19) l!(n 2l)! l=0 Photonics 2021, 8, 5 5 of 12 When r = r = r in Equation (10), the intensity of a RPLMGSMV beam array 1 2 propagating in atmospheric turbulence is written as Q Q jMj jMj l l 2 1 2 jMj![isgn(M)] jMj![isgn(M)] k 1 I(r, z) = å å å å 2 2 4p z l !(jMjl )! l !(jMjl )! 0 1 1 2 2 q =1 q =1 l =0 l =0 1 2 2 (20) N n1 (1) exp i j j I(x, z)I(y, z) q1 q2 n=1 where h  i h i 1 ik 2 2 ik I(x, z) = exp r r exp r r exp 2 r r x q x q x q x q x 2 q x q x 1 2 2z 2z 1 2 1 2 " # jMjl 1 r r q x q x p 1 1 ik ik 1 2 (jMj l )! exp x r q x 1 2 a a a 2z 2z 1 jMjl [ ] jMjl 2k 2  1 1 k (jMjl 2k )! 1 a 1 (21) 1 1 å å k !(jMjl 2k )! 4 s !(jMjl 2k s )! 1 1 1 1 1 1 1 k =0 s =0 jMjl 2k s s 1 1 1 1 r r q x q x ik ik 1 1 1 2 x r + q x 2 2 2 2z 2z r 2ns r 0 0 jMjl +s 2 1 p i ic xx xx p p exp H jMjl +s b b 1 2 b b h  i h i 1 ik ik 2 2 I(y, z) = exp r r exp r r exp 2 r r y 2 q y q y q y q y 1 2 q y q y 1 2 2z 1 2 2z " # q   2 r r q y q y p 1 1 ik ik 1 2 l ! exp y r 1 q y 2 a a a 2z 2z [ ] l 2k 2  1 2 k (l 2k )! 1 a 2 1 2 (22) å å k !(l 2k )! 4 s !(l 2k s )! 2 1 2 2 1 2 2 k =0 s =0 l 2k s s 1 2 2 2 r r q y q y ik ik 1 1 1 2 y r + q y 2 2 2 2z 2z 1 2ns r r 0 0 l +s 2 2 ic yy yy p i p p exp H l +s b b 2 2 b b with ! ! r r r r ik ik 1 1 1 ik ik q x q x q x q x 1 2 1 2 c = r x + + + x r (23) xx q x q x 2 1 2 2 2 2 2z 2z r a 2ns r 2z 2z r 0 0 0 ! ! ik ik r r 1 1 1 ik ik r r q y q y q y q y 1 2 1 2 c = r y + + + y r (24) yy q y q y 2 1 2 2 2 2 2z 2z a 2ns 2z 2z r r r 0 0 0 The degree of coherence for a RPLMGSMV beam array propagating in atmospheric turbulence at plane z is given as [34] W(r , r , z) 1 2 m(r , r , z) = (25) 1 2 1/2 [W(r , r , z)W(r , r , z)] 1 1 2 2 3. Numerical Results and Discussions In this section, the intensity and coherence distributions of a RPLMGSMV beam array in free space will firstly be investigated, and then the influences of atmospheric turbulence Photonics 2021, 8, 5 6 of 12 on intensity and coherence distributions of a RPLMGSMV beam array will be discussed. The relevant parameters in numerical simulations are selected as l = 532 nm, w = 1 cm, s = 1 mm, N = 10, M = 1 and R = 5 cm without other explanations. The normalized intensity of a RPLMGSMV beam array with Q = 5 in free space at the different distances is illustrated in Figure 2. As can be seen from Figures 2a and 1c, the dark hollow center of beamlets of a RPLMGSMV beam array will evolve into a Gaussian-like beam at z = 50 m (Figure 2a), while the beamlets have the dark hollow center at z = 0 (Figure 1c); The reason that the dark hollow profile translating into the Gaussian beam can be explained as the effect of initial coherence length [33]. As the z increases further, the Gaussian-like beamlets can evolve into a beam with flat-topped profile (Figure 2b), and the beamlets will also begin to overlap with each other (Figure 2b); thus, a RPLMGSMV beam array can translate into a beam with Gaussian-like intensity distribution; at last, the RPLMGSMV beam array can evolve from beam array into the beam with flat-topped inten- sity distribution at longer distance z (Figure 2d). The phenomenon whereby a RPLMGSMV Photonics 2021, 8, x FOR PEER REVIEW 7 of 13 evolves into a beam with flat-topped profile is dominated by MGSM correlated function, and similar evolutions can also be found in the previous reports [25–28,33]. Figure 2. Normalized intensity of a RPLMGSMV beam array with Q = 5 in free space. (a) z = 50 m, Figure 2. Normalized intensity of a RPLMGSMV beam array with Q = 5 in free space. (a) (b) z = 200 m, (c) z = 400 m, (d) z = 900 m. z =50m , (b) z = 200m , (c) z = 400m , (d) z = 900 m . To view the action of Q on intensity distribution, normalized intensity of a RPLMGSMV beam array with Q = 4 in free space are illustrated in Figure 3. As z increases, it is found To view the action of Q on intensity distribution, normalized intensity of a that the evolution of intensity distributions of a RPLMGSMV beam array with Q = 4 RPLMGSMV beam array with Q = 4 in free space are illustrated in Figure 3. As z in- are almost the same with a RPLMGSMV beam array with Q = 5 (Figure 2), the beamlets of beam array will lose the dark hollow profile and become a beam with Gaussian-like creases, it is found that the evolution of intensity distributions of a RPLMGSMV beam beam distribution, the beam array with Q = 4 will translate into the flat-topped profile array with Q = 4 are almost the same with a RPLMGSMV beam array with Q = 5 (Fig- (Figure 3b). Moreover, the flat-topped profile of a RPLMGSMV beam array is dominated by ure 2), the beamlets of beam array will lose the dark hollow profile and become a beam the MGSM sources at longer distance [33]. By comparing Figures 2 and 3, we can conclude with Gaussian-like beam distribution, the beam array with Q = 4 will translate into the flat-topped profile (Figure 3b). Moreover, the flat-topped profile of a RPLMGSMV beam array is dominated by the MGSM sources at longer distance [33]. By comparing Figures 2 and 3, we can conclude that a RPLMGSMV beam array with the different Q will evolve form beam array into flat-topped profile due to the action of MGSM source. Figure 3. Normalized intensity of a RPLMGSMV beam array with Q =4 in free space. (a) , (b) . z = 50 m z =1000m To view the effects of source parameters on the evolutions of intensity of a RPLMGSMV beam array propagating in free space, the cross sections (y = 0) of the nor- malized intensity of a RPLMGSMV beam array with in free space for source pa- Q = 4 rameters σ , N and M at the different distances are shown in Figures 4–6, respectively. Photonics 2021, 8, x FOR PEER REVIEW 7 of 13 Figure 2. Normalized intensity of a RPLMGSMV beam array with in free space. (a) Q = 5 z =50m , (b) z = 200m , (c) z = 400m , (d) z = 900 m . To view the action of on intensity distribution, normalized intensity of a RPLMGSMV beam array with Q = 4 in free space are illustrated in Figure 3. As z in- creases, it is found that the evolution of intensity distributions of a RPLMGSMV beam array with Q = 4 are almost the same with a RPLMGSMV beam array with Q = 5 (Fig- ure 2), the beamlets of beam array will lose the dark hollow profile and become a beam with Gaussian-like beam distribution, the beam array with Q = 4 will translate into the flat-topped profile (Figure 3b). Moreover, the flat-topped profile of a RPLMGSMV beam Photonics 2021, 8, 5 7 of 12 array is dominated by the MGSM sources at longer distance [33]. By comparing Figures 2 and 3, we can conclude that a RPLMGSMV beam array with the different Q will evolve that a RPLMGSMV beam array with the different Q will evolve form beam array into form beam array into flat-topped profile due to the action of MGSM source. flat-topped profile due to the action of MGSM source. Figure 3. Normalized intensity of a RPLMGSMV beam array with Q = 4 in free space. (a) z = 50 m, Figure 3. Normalized intensity of a RPLMGSMV beam array with Q =4 in free space. (a) (b) z = 1000 m. z = 50 m , (b) z =1000m . To view the effects of source parameters on the evolutions of intensity of a RPLMGSMV beam array propagating in free space, the cross sections (y = 0) of the normalized intensity of a RPLMGSMV beam array with Q = 4 in free space for source parameters s, N and M at To view the effects of source parameters on the evolutions of intensity of a the different distances are shown in Figures 4–6, respectively. From Figure 4, it is found that RPLMGSMV beam array propagating in free space, the cross sections (y = 0) of the nor- the RPLMGSMV beam array with larger s will lose the dark hollow distribution slower than the beam array with smaller s. So, the smaller coherence length s will accelerate malized intensity of a RPLMGSMV beam array with Q = 4 in free space for source pa- the evolutions of beam array translating into flat-topped profile, and the beam array with smaller s will have the better flatness when the beam array translating into flat-topped rameters σ , N and M at the different distances are shown in Figures 4–6, respectively. intensity profile at the longer distance z (Figure 4d). Thus, it can conclude that the speed of a beam array translating into the flat-topped profile can be dominated by setting different s of MGSM source. Figure 5 shows that the RPLMGSMV beam array with larger N will evolve from beam array into flat-topped beam faster, and which will have the better flatness when the beam array translating into the flat-topped beam at last (Figure 5d). The flatness of flat-topped profile is dominated by the total number N of MGSM source, and the similar results can also be seen in the previous work [25]. Thus, from previous discussions, one can conclude that the flatness of flat-topped profile generating by RPLMGSMV beam array can be modulated by the parameters s and N in the far field. The better the flatness of flat-topped profile is, the more power can be received by the same receiver, this is helpful for received power of free space optical communication. One can see from Figure 6 that the RPLMGSMV beam array with larger M will have the larger dark hollow center at z = 0 (Figure 6a), while the influences of different M on the intensity distribution will disappear as the beam array evolve into the flat-topped beam at the longer distance z (Figure 6d). Thus, we can conclude that the flat-topped profile is not correlated with topological charge in the far field. Photonics 2021, 8, x FOR PEER REVIEW 8 of 13 From Figure 4, it is found that the RPLMGSMV beam array with larger σ will lose the dark hollow distribution slower than the beam array with smaller . So, the smaller co- herence length σ will accelerate the evolutions of beam array translating into flat- topped profile, and the beam array with smaller σ will have the better flatness when the beam array translating into flat-topped intensity profile at the longer distance z (Figure 4d). Thus, it can conclude that the speed of a beam array translating into the flat-topped profile can be dominated by setting different σ of MGSM source. Figure 5 shows that the RPLMGSMV beam array with larger N will evolve from beam array into flat-topped beam faster, and which will have the better flatness when the beam array translating into the flat-topped beam at last (Figure 5d). The flatness of flat-topped profile is dominated by the total number N of MGSM source, and the similar results can also be seen in the previous work [25]. Thus, from previous discussions, one can conclude that the flatness of flat-topped profile generating by RPLMGSMV beam array can be modulated by the parameters σ and N in the far field. The better the flatness of flat-topped profile is, the more power can be received by the same receiver, this is helpful for received power of free space optical communication. One can see from Figure 6 that the RPLMGSMV beam array with larger M will have the larger dark hollow center at z = 0 (Figure 6a), while the influences of different M on the intensity distribution will disappear as the beam array Photonics 2021, 8, 5 8 of 12 evolve into the flat-topped beam at the longer distance z (Figure 6d). Thus, we can con- clude that the flat-topped profile is not correlated with topological charge in the far field. Photonics 2021, 8, x FOR PEER REVIEW 9 of 13 Figure 4. Figure 4. CCr ross oss se sections ctions (y( = y 0) = 0) of normalize of normalized d intensity of intensity of a R a P RPLMGSMV LMGSMV be beam am array with array withQQ==4 4 in free space for the different s. (a) z = 50 m, (b) z = 200 m, (c) z = 400 m, (d) z = 1000 m. in free space for the different σ . (a) z = 50 m , (b) z = 200m , (c) z = 400m , (d) z =1000m . Figure 5. Cross sections (y = 0) of the normalized intensity of a RPLMGSMV beam array with Figure 5. Cross sections (y = 0) of the normalized intensity of a RPLMGSMV beam array with Q = 4 Q =4 in free space for the different N . (a) , (b) , (c) , (d) z = 50 m z = 200m z = 400m in free space for the different N. (a) z = 50 m, (b) z = 200 m, (c) z = 400 m, (d) z = 1000 m. z =1000m . Figure 7 illustrates the average intensity of a RPLMGSMV beam array with Q =4 propagating in free space and atmospheric turbulence for different . As can be seen that in Figure 7b, the flatness of the flat-topped profile obtained at the longer distance is poorer than a RPLMGSMV beam array in free space, and the larger the is, the poorer the flatness of flat-topped profile is. The phenomenon where the flat-topped profile is be- coming poor in atmospheric turbulence can be explained by the influences of atmospheric turbulence. Photonics 2021, 8, x FOR PEER REVIEW 9 of 13 Figure 5. Cross sections (y = 0) of the normalized intensity of a RPLMGSMV beam array with Q =4 in free space for the different N . (a) , (b) , (c) , (d) z = 50 m z = 200m z = 400m z =1000m Figure 7 illustrates the average intensity of a RPLMGSMV beam array with Q =4 propagating in free space and atmospheric turbulence for different . As can be seen that in Figure 7b, the flatness of the flat-topped profile obtained at the longer distance is poorer than a RPLMGSMV beam array in free space, and the larger the is, the poorer Photonics 2021, 8, 5 9 of 12 the flatness of flat-topped profile is. The phenomenon where the flat-topped profile is be- coming poor in atmospheric turbulence can be explained by the influences of atmospheric turbulence. Figure 6. Cross sections (y = 0) of the normalized intensity of a RPLMGSMV beam array with Q = 4 in free space for the different M. (a) z = 0 m, (b) z = 100 m, (c) z = 200 m, (d) z = 1000 m. Photonics 2021, 8, x FOR PEER REVIEW 10 of 13 Figure 7 illustrates the average intensity of a RPLMGSMV beam array with Q = 4 propagating in free space and atmospheric turbulence for different C . As can be seen that in Figure 7b, the flatness of the flat-topped profile obtained at the longer distance is poorer Figure 6. Cross sections (y = 0) of the normalized intensity of a RPLMGSMV beam array with than a RPLMGSMV beam array in free space, and the larger the C is, the poorer the flatness Q =4 in free space for the different M . (a) , (b) , (c) , (d) z =0m z =100 m z = 200m of flat-topped profile is. The phenomenon where the flat-topped profile is becoming poor z in =100 atmospheric 0m . turbulence can be explained by the influences of atmospheric turbulence. Figure 7. Cross sections of a RPLMGSMV beam array with Q = 4Qin =atmospheric 4 turbulence for the Figure 7. Cross sections of a RPLMGSMV beam array with in atmospheric turbulence for different C . (a) z = 100 m, (b) z = 1000 m. the different C . (a) z =100m , (b) z =1000m . Figure 8 gives the cross-sections of degree of coherence of a RPLMGSMV beam array Figure 8 gives the cross-sections of degree of coherence of a RPLMGSMV beam array with Q = 5 in free space (Figure 8a–c) and atmospheric turbulence at z = 1000 m for the dif Qfer =5 ent M, s, N and C . From Figure 8a,b, it is seen that the coherence properties with in free space (Figure 8a–c) and atmospheric turbulence at z =1000m for the of a RPLMGSMV beam array can be affected by the M and s. Meanwhile, the effects σ N of total number N on the coher nence distribution is not found (Figure 8c). Further, when different , , and . From Figure 8a,b, it is seen that the coherence properties x x is smaller, the influences of M and s are less. In the analysis of the influences 1 2 σ of a RPLMGSMV beam array can be affected by the and . Meanwhile, the effects of atmospheric turbulence on coherence, it is seen that the coherence distribution of a of total number on the coherence distribution is not found (Figure 8c). Further, when RPLMGSMV beam array in atmospheric turbulence with larger C will have the Gaussian x − x distribution (Figure 8d). Meanwhile, the same beam in free pace will have the irregular is smaller, the influences of and are less. In the analysis of the influences of atmospheric turbulence on coherence, it is seen that the coherence distribution of a RPLMGSMV beam array in atmospheric turbulence with larger will have the Gauss- ian distribution (Figure 8d). Meanwhile, the same beam in free pace will have the irregular coherence distribution. To view the influences of z on coherence distribution, the coher- Q =5 ence of a RPLMGSMV beam array with at the different distance z is illustrated in Figure 9. One can see from Figure 9 that a RPLMGSMV beam array at the longer propa- gation distance will have more regular coherence distribution. One can conclude that the spectral degree of coherence of a RPLMGSMV beam array in atmospheric turbulence will have a Gaussian distribution at longer distance. Photonics 2021, 8, x FOR PEER REVIEW 10 of 13 Figure 6. Cross sections (y = 0) of the normalized intensity of a RPLMGSMV beam array with Q =4 in free space for the different M . (a) z =0m , (b) z =100 m , (c) z = 200m , (d) z =1000m . Q =4 Figure 7. Cross sections of a RPLMGSMV beam array with in atmospheric turbulence for the different C . (a) z =100m , (b) z =1000m . Figure 8 gives the cross-sections of degree of coherence of a RPLMGSMV beam array Q =5 with in free space (Figure 8a–c) and atmospheric turbulence at z =1000m for the σ N M n different , , and . From Figure 8a,b, it is seen that the coherence properties of a RPLMGSMV beam array can be affected by the and . Meanwhile, the effects of total number on the coherence distribution is not found (Figure 8c). Further, when x − x 12 M is smaller, the influences of and are less. In the analysis of the influences of atmospheric turbulence on coherence, it is seen that the coherence distribution of a Photonics 2021, 8, 5 10 of 12 RPLMGSMV beam array in atmospheric turbulence with larger will have the Gauss- ian distribution (Figure 8d). Meanwhile, the same beam in free pace will have the irregular coherence distribution. To view the influences of z on coherence distribution, the coher- coherence distribution. To view the influences of z on coherence distribution, the coherence Q =5 ence of of a RPLMGSMV a RPLMGSMV beam beam a array rray wi with Q th = 5 at the at the d different iffer distance ent distance z is illustrated z is illustrat in e Figur d in e 9. One can see from Figure 9 that a RPLMGSMV beam array at the longer propagation Figure 9. One can see from Figure 9 that a RPLMGSMV beam array at the longer propa- distance will have more regular coherence distribution. One can conclude that the spectral gation distance will have more regular coherence distribution. One can conclude that the degree of coherence of a RPLMGSMV beam array in atmospheric turbulence will have a spectral degree of coherence of a RPLMGSMV beam array in atmospheric turbulence will Gaussian distribution at longer distance. have a Gaussian distribution at longer distance. Photonics 2021, 8, x FOR PEER REVIEW 11 of 13 Photonics 2021, 8, x FOR PEER REVIEW 11 of 13 Figure 8. Modulus of the coherence of a RPLMGSMV beam array with Q =5 in the free space Figure 8. Figure Mod 8. Modulus ulus of the of coh the e coher rence of ence a RPLMGSMV bea of a RPLMGSMVm beam array with array with Q =Q5 = in the free 5 in the fr space ee space and and atmospheric turbulence at the distance z =1000m . (a)different M , (b) different σ , (c) dif- atmospheric turbulence at the distance z = 1000 m. (a) different M, (b) different s, (c) different N, and atmospheric turbulence at the distance z =1000m . (a)different M , (b) different σ , (c) dif- ferent N, (d) different 2C . (d) different C . ferent N, (d) different C . Q =5 Figure 9. Figure 9. Mod Modulus ulus of the of the cohcoher erence of ence a RPLMGSMV bea of a RPLMGSMVm beam array with array with Q for the different = 5 for the different Figure 9. Modulus of the coherence of a RPLMGSMV beam array with Q =5 for the different distance z. (a) free space, (b) atmospheric turbulence. distance z. (a) free space, (b) atmospheric turbulence. distance z. (a) free space, (b) atmospheric turbulence. 4. Conclusions 4. Conclusions 4. Conclusions In this paper, the analytical description of a RPLMGSMV beam array generated by In this paper, the analytical description of a RPLMGSMV beam array generated by In this paper, the analytical description of a RPLMGSMV beam array generated by MGSM sources is introduced and analyzed. Based on the extended Huygens–Fresnel MGSM sources is introduced and analyzed. Based on the extended Huygens–Fresnel in- MGSM sources is introduced and analyzed. Based on the extended Huygens–Fresnel in- integral, the CSD of a RPLMGSMV beam array propagating in atmospheric turbulence tegral, the CSD of a RPLMGSMV beam array propagating in atmospheric turbulence is tegral, the CSD of a RPLMGSMV beam array propagating in atmospheric turbulence is is derived. The evolutions of intensity and coherence properties of a RPLMGSMV beam derived. The evolutions of intensity and coherence properties of a RPLMGSMV beam ar- derived. The evolutions of intensity and coherence properties of a RPLMGSMV beam ar- array propagating in free space and atmospheric turbulence are analyzed in detail. It ray propagating in free space and atmospheric turbulence are analyzed in detail. It is seen ray propagating in free space and atmospheric turbulence are analyzed in detail. It is seen is seen that a RPLMGSMV beam array propagating in free space can gradually lose the that a RPLMGSMV beam array propagating in free space can gradually lose the initial that a RPLMGSMV beam array propagating in free space can gradually lose the initial initial intensity distribution of beamlets, and evolve from the beam array into a beam with intensity distribution of beamlets, and evolve from the beam array into a beam with a flat- intensity distribution of beamlets, and evolve from the beam array into a beam with a flat- topped profile due to the action of MGSM source as z increases. In the far field, when the topped profile due to the action of MGSM source as z increases. In the far field, when the total number N is larger or the coherence length is smaller, the flatness of flat-topped total number N is larger or the coherence length σ is smaller, the flatness of flat-topped profile of a RPLMGSMV beam array will be better, this is useful for the free space optical profile of a RPLMGSMV beam array will be better, this is useful for the free space optical communication. When a RPLMGSMV beam array propagates in atmospheric turbulence, communication. When a RPLMGSMV beam array propagates in atmospheric turbulence, the flatness of the flat-topped profile can be dominated by the atmospheric turbulence, the flatness of the flat-topped profile can be dominated by the atmospheric turbulence, the flatness will become poor. It is also found that the coherence distribution of a the flatness will become poor. It is also found that the coherence distribution of a RPLMGSMV beam array in atmospheric turbulence will have Gaussian distribution at the RPLMGSMV beam array in atmospheric turbulence will have Gaussian distribution at the longer distance. longer distance. Author Contributions: Data curation, J.Z. and X.M; writing—original draft preparation, J.Z., G.W., Author Contributions: Data curation, J.Z. and X.M; writing—original draft preparation, J.Z., G.W., Y.W., and D.L.; writing—review and editing, H.Z. and H.Y.; supervision, G.W., Y.W., and D.L.; pro- Y.W., and D.L.; writing—review and editing, H.Z. and H.Y.; supervision, G.W., Y.W., and D.L.; pro- ject administration, G.W. and D.L. All authors have read and agreed to the published version of the ject administration, G.W. and D.L. All authors have read and agreed to the published version of the manuscript. manuscript. Funding: This research was supported by National Natural Science Foundation of China [11604038, Funding: This research was supported by National Natural Science Foundation of China [11604038, 11875096, 11404048], and the Fundamental Research Funds for the Central Universities 11875096, 11404048], and the Fundamental Research Funds for the Central Universities [3132020175]. [3132020175]. Photonics 2021, 8, 5 11 of 12 a flat-topped profile due to the action of MGSM source as z increases. In the far field, when the total number N is larger or the coherence length s is smaller, the flatness of flat-topped profile of a RPLMGSMV beam array will be better, this is useful for the free space optical communication. When a RPLMGSMV beam array propagates in atmospheric turbulence, the flatness of the flat-topped profile can be dominated by the atmospheric turbulence, the flatness will become poor. It is also found that the coherence distribution of a RPLMGSMV beam array in atmospheric turbulence will have Gaussian distribution at the longer distance. Author Contributions: Data curation, J.Z. and X.M.; writing—original draft preparation, J.Z., G.W., Y.W., and D.L.; writing—review and editing, H.Z. and H.Y.; supervision, G.W., Y.W., and D.L.; project administration, G.W. and D.L. All authors have read and agreed to the published version of the manuscript. Funding: This research was supported by National Natural Science Foundation of China [11604038, 11875096, 11404048], and the Fundamental Research Funds for the Central Universities [3132020175]. Acknowledgments: The authors express their appreciation to the anonymous reviewers for their valuable suggestions. Conflicts of Interest: The authors declare no conflict of interest. References 1. Wang, F.; Liu, X.L.; Cai, Y.J. Propagation of partially coherent beam in turbulent atmosphere: A review. Prog. Electromagn. 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Propagation of Rectangular Multi-Gaussian Schell-Model Array Beams through Free Space and Non-Kolmogorov Turbulence. Appl. Sci. 2020, 10, 450. [CrossRef] 33. Zhang, Y.T.; Liu, L.; Zhao, C.L.; Cai, Y.J. Multi-Gaussian Schell-model vortex beam. Phys. Lett. A 2014, 378, 750–754. [CrossRef] 34. Wolf, E. Unified theory of coherence and polarization of random electromagnetic beams. Phys. Lett. A 2003, 312, 263–267. [CrossRef] 35. Jeffrey, A.; Dai, H.H. Handbook of Mathematical Formulas and Integrals, 4th ed.; Academic Press Inc.: Cambridge, MA, USA, 2008. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Photonics Multidisciplinary Digital Publishing Institute

Intensity and Coherence Characteristics of a Radial Phase-Locked Multi-Gaussian Schell-Model Vortex Beam Array in Atmospheric Turbulence

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hv photonics Letter Intensity and Coherence Characteristics of a Radial Phase-Locked Multi-Gaussian Schell-Model Vortex Beam Array in Atmospheric Turbulence Jialu Zhao, Guiqiu Wang, Xiaolu Ma, Haiyang Zhong, Hongming Yin, Yaochuan Wang and Dajun Liu * Department of Physics, College of Science, Dalian Maritime University, Dalian 116026, China; zjl970929@dlmu.edu.cn (J.Z.); gqwang@dlmu.edu.cn (G.W.); maxiaolu@dlmu.edu.cn (X.M.); haae@dlmu.edu.cn (H.Z.); hmyin@dlmu.edu.cn (H.Y.); ycwang@dlmu.edu.cn (Y.W.) * Correspondence: liudajun@dlmu.edu.cn; Tel.: +86-159-4111-5638 Abstract: The theoretical descriptions for a radial phase-locked multi-Gaussian Schell-model vortex (RPLMGSMV) beam array is first given. The normalized intensity and coherence distributions of a RPLMGSMV beam array propagating in free space and atmospheric turbulence are illustrated and analyzed. The results show that a RPLMGSMV beam array with larger total number N or smaller coherence length s can evolve into a beam with better flatness when the beam array translating into the flat-topped profile at longer distance z and the flatness of the flat-topped intensity distribution can be destroyed by the atmospheric turbulence at longer distance z. The coherence distribution of a RPLMGSMV beam array in atmospheric turbulence at the longer distance will have Gaussian distri- bution. The research results will be useful in free space optical communication using a RPLMGSMV beam array. Keywords: average intensity; multi-Gaussian Schell-model source; vortex beam; beam array; atmo- spheric turbulence Citation: Zhao, J.; Wang, G.; Ma, X.; Zhong, H.; Yin, H.; Wang, Y.; Liu, D. Intensity and Coherence Characteristics of a Radial 1. Introduction Phase-Locked Multi-Gaussian Schell-Model Vortex Beam Array in With the development of wireless optical communication and laser radar, the evolu- Atmospheric Turbulence. Photonics tions of laser beams in atmospheric turbulence were widely studied in past years [1]. In past 2021, 8, 5. https://doi.org/10.3390/ years, the properties of fully coherent laser beams in turbulence have been widely analyzed, photonics8010005 such as Gaussian beams [2], Hermite–Gaussian beam [3], Pearcey–Gaussian beam [4], beam array [5–7], Laguerre–Gaussian beam [8], Airy beam [9], hollow beam [10,11], vortex Received: 19 November 2020 beam [12,13], and vortex lattices [14]. From previous studies [1], one can see that partially Accepted: 24 December 2020 coherent beams are resistance to the deleterious effects of turbulence. In past years, the Published: 29 December 2020 properties, including intensity, polarization, and coherence, of Gaussian Schell-model (GSM) beams [15–19] and GSM beam array [20–23] propagating in turbulence were widely Publisher’s Note: MDPI stays neu- studied. On the other hand, the special correlated beams are also be introduced and ana- tral with regard to jurisdictional clai- lyzed, such as non-uniformly correlated beams [24], multi-Gaussian Schell-model (MGSM) ms in published maps and institutio- beams [25–28], cosine-GSM beams [29], multi-cosine-Laguerre–Gaussian correlated Schell- nal affiliations. model beams [30]. Laser arrays can produce the higher power output than single beam and which can have linear, rectangular and radial distributions. The beams correlated with MGSM source can provide flat intensity profiles in the far field [25]. To obtain the Copyright: © 2020 by the authors. Li- flat-topped intensity profiles, the MGSM beam arrays propagating in turbulence are in- censee MDPI, Basel, Switzerland. vestigated, and it is found that the MGSM beam arrays can achieve the better flat-topped This article is an open access article profiles [31,32]. Moreover, the MGSM vortex beam has been introduced and studied. It distributed under the terms and con- shows that the intensity profile of MGSM vortex can be modulated by the topological ditions of the Creative Commons At- charge [33]. Thus, it will be very interesting to consider the laser array composed by tribution (CC BY) license (https:// MGSM vortex beams. In this paper, we extend MGSM vortex beam into the radial phase- creativecommons.org/licenses/by/ locked multi-Gaussian-Schell-model vortex (RPLMGSMV) beam array, and investigate 4.0/). Photonics 2021, 8, 5. https://doi.org/10.3390/photonics8010005 https://www.mdpi.com/journal/photonics Photonics 2021, 8, 5 2 of 12 the intensity and coherence properties of RPLMGSMV beam array propagating in free space and atmospheric turbulence. Moreover, the model of laser arrays with linear and rectangular distributions can also be obtained in the similar analytical approach. 2. Theory Analysis 2.1. Analytical Description of RPLMGSMV Beam Array The electric field distribution of a Gaussian vortex beam at source plane z = 0 is described by 2 2 x + y jMj 0 0 E(r , 0) = [x + isgn(M)y ] exp (1) 0 0 where w is beam waist and M is the topological charge. In this work, laser array with radial distribution will be analyzed as example, the electric field of a radial phase-locked Gaussian vortex beam array with Q beamlets can be given as: " # 2 2 x r + y r 0 qx 0 qy jMj E (r , 0) = x r + isgn(M) y r exp exp ij (2) Q 0 å 0 qx 0 qy q q=1 0 with 2p r = R cos j , r = R sin j , j = q , q = 1, 2, Q (3) qx q qy q q where R is radius; j is the phase of the q-th beamlet; r and r are the center of the q-th n qx qy beamlet element located at z = 0. Considering the unified theory of coherence and polarization [34], the cross spectral density (CSD) of partially coherent beams can be expressed as W(r , r ) = E(r )E (r ) (4) 10 20 10 20 Introducing a MGSM correlation [22], the CSD of a RPLMGSMV beam array with Q beamlets can be written as Q Q W (r , r , 0) = å å exp i j j 20 q2 10 q1 q =1 q =1 1 2 2 2 jMj (x r ) +(y r ) 10 q1x 10 q1y x r + isgn(M) y r exp 10 q1x 10 q1y 2 (5) 2 2 x r + y r jMj ( ) ( ) 20 q2x 20 q2y x r isgn(M) y r exp 20 q2x 20 q2y 2 2 n1 N x r x r y r y r (1) [( ) ( 20 q2x)] [( ) ( 20 q2y)] 1 10 q1x 10 q1y exp 2 2 C n 0 2ns 2ns n=1 where N is total number of terms of MGSM source, s is the coherence length, C is the normalized factor, and can be described by n1 N (1) C = (6) n=1 Figure 1 shows the normalized intensity of a RPLMGSMV beam array at z = 0 for the different Q, one can see that the beamlets of a RPLMGSMV beam array have the hollow center. Photonics 2021, 8, x FOR PEER REVIEW 3 of 13 Photonics 2021, 8, 5 3 of 12 Figure 1 shows the normalized intensity of a RPLMGSMV beam array at z = 0 for the different Q , one can see that the beamlets of a RPLMGSMV beam array have the hollow center. Figure 1. Normalized intensity of a RPLMGSMV beam array at z = 0 for the different Q . (a) Q =3 Figure 1. Normalized intensity of a RPLMGSMV beam array at z = 0 for the different Q. (a) Q = 3, , (b) Q =4 , (c) Q =5 , (d) Q =6 . (b) Q = 4, (c) Q = 5, (d) Q = 6. 2.2. Propagation Analysis 2.2. Propagation Analysis Based on the extended Huygens–Fresnel integral, the CSD of a RPLMGSMV beam Based on the extended Huygens–Fresnel integral, the CSD of a RPLMGSMV beam array propagating in atmospheric turbulence at plane z can be read as [1] array propagating in atmospheric turbulence at plane z can be read as [1] R R R R +¥ +¥ +¥ +¥ W r , r , z = +∞ +∞ +∞ +∞ W r , r , 0 ( ) k ( ) 1 2 10 20 2 2 ¥ ¥ ¥ ¥ 4p z Wz rr,, = W r ,r ,0 () () 12 22 10 20 h  i −∞ −∞ −∞ −∞ 4π z ik 2 ik 2 (7) exp (r r ) + (r r ) 1 10 2 20 2z 2z ik ik  22 (7) ×− exp () r-r + () r -r 110 2 20  22zz hexp [y(r , r ) + y (r , r )]idr dr 10 1 20 2 10 20  ×+ expψψ () rr, (r ,r) ddr r 10 1 20 2 10 20  with " # with 2 2 (r r ) + (r r )(r r ) + (r r ) 10 20 10 20 1 2 1 2 hexp[y(r , r) + y (r , r)]i = exp (8) 10 20  () rr−+() rr− 0 (r−r)+(r−r) ∗ 10 20 10 20 1 2 1 2  expψψ rr,+= r ,r exp − () ( ) (8) 10 20   0 In the above equation, the spatial coher ence length r can be expressed as 3/5 In the above equation, the spatial coherence length can be expressed as 2 2 r = 0.545C k z (9) −35 ρ = 0.545Ck z (9) () 0 n where C is the structure constant of atmospheric turbulence. Substituting C Equation (5) into Equation (7), the CSD of a RPLMGSMV beam array where is the structure constant of atmospheric turbulence. propagating in atmospheric turbulence at plane z can be derived as Substituting Equation (5) into Equation (7), the CSD of a RPLMGSMV beam array h i Q Q 2 2 propagating in atmospheric tu 2 rbulence at plane z can be derived as (x x ) +(y y ) k ik 2 2 1 2 1 2 W (r , r , z) = exp i j j exp r r exp å å 1 2 q1 q2 2 2 2 2z 1 2 4p z q =1 q =1 1 2 (10) jMj jMj l N n1 1 N jMj!i jMj!(i) (1) W(x, z)W(y, z) å å å C n l !(jMjl )! l !(jMjl )! 0 1 1 2 2 l =0 l =0 n=1 1 2 Photonics 2021, 8, 5 4 of 12 with h  i h i 1 ik 2 2 ik W (x, z) = exp r r exp r r exp 2 x r x r q x q x 1 q x 2 q x 2 2 q x q x 2 1 2z 2 2z 1 r 1 jMjl (x x ) r r 2 ( q x q x) 1 p 1 1 2 exp ( M l )! j j a a " # jMjl [ ] x x +2 r r ( ) k 1 2 q x q x 1 ik ik 1 a 1 1 2 exp x r (11) q x å 1 2 a 2z 2z 1 4 k !(jMjl 2k )! 2r 1 1 1 k =0 jMjl 2k s s jMjl 2k 1 1 1 1 1 1 x x +2 r r M l 2k ! ( q x q x) (j j ) 1 2 1 1 ik ik 1 2 1 1 å x r + 1 q x 2 2 2 2z 2z 1 s !(jMjl 2k s )! 2r 2ns r 1 1 1 1 0 0 s=0 jMjl +s q   2 c ic p i x x p p exp H jMjl +s b b 2 1 2 b b h  i h i 1 ik 2 2 ik W (y, z) = exp r r exp r r exp 2 y r y r q y q y q y q y 2 q y q y 1 2 1 2 2z 2z 1 2 1 2 (y y ) r r 1 ( q y q y) 1 2 2 p 1 exp (l )! 2 1 a a " # [ ] y y +2 r r ( ) k 1 2 q y q y 1 ik ik 1 a 2 1 2 exp y r q y å (12) 1 2 a 2z 2z 1 4 y k !(l 2k )! 2r 2 1 2 k =0 l 2k s s 1 2 2 2 l 2k 1 2 y y +2 r r (l 2k )! 1 2 ( q y q y) ik ik 1 1 1 2 1 2 y r + 1 q y 2 2 2 s !(l 2k s )! 2z 2z 2r 2ns r 3 1 2 2 0 0 s =0 q  l +s 2 2 c ic p i y y p p exp H l +s 2 2 b b 2 b b where 1 1 1 ik a = + + + (13) 2 2 2 2ns 2z w r 0 0 1 1 1 ik 1 1 1 b = + + + (14) 2 2 2 2 2 2ns 2z a 2ns w r r 0 0 0 x x +2 r r ( q x q x) 1 2 ik ik 2 c = r x + x q x 2 2z 2 2z 2r (15) x x +2 r r ( q x q x) 1 2 1 1 1 ik ik 2 + + x r q x 2 2 1 2 a 2z 2z 1 2ns r 2r 0 0 y y +2(r r ) 1 2 q y q y ik ik 1 2 c = r y + y q y 2 2 2z 2 2z 2r (16) y y +2(r r ) 1 2 q y q y 1 1 1 ik ik 1 2 + + y r q y 2 2 1 2 a 2z 2z 1 2ns r 2r 0 0 In the derivations of Equation (10), the following equations has been applied [35] M!i Ml l (x + iy) = x y (17) l!(M l)! l=0 +¥ p i b ib n 2 x exp ax + 2bx dx = p exp H p (18) a a ¥ 2 a a [ ] (1) n! n2l H (x) = (2x) (19) l!(n 2l)! l=0 Photonics 2021, 8, 5 5 of 12 When r = r = r in Equation (10), the intensity of a RPLMGSMV beam array 1 2 propagating in atmospheric turbulence is written as Q Q jMj jMj l l 2 1 2 jMj![isgn(M)] jMj![isgn(M)] k 1 I(r, z) = å å å å 2 2 4p z l !(jMjl )! l !(jMjl )! 0 1 1 2 2 q =1 q =1 l =0 l =0 1 2 2 (20) N n1 (1) exp i j j I(x, z)I(y, z) q1 q2 n=1 where h  i h i 1 ik 2 2 ik I(x, z) = exp r r exp r r exp 2 r r x q x q x q x q x 2 q x q x 1 2 2z 2z 1 2 1 2 " # jMjl 1 r r q x q x p 1 1 ik ik 1 2 (jMj l )! exp x r q x 1 2 a a a 2z 2z 1 jMjl [ ] jMjl 2k 2  1 1 k (jMjl 2k )! 1 a 1 (21) 1 1 å å k !(jMjl 2k )! 4 s !(jMjl 2k s )! 1 1 1 1 1 1 1 k =0 s =0 jMjl 2k s s 1 1 1 1 r r q x q x ik ik 1 1 1 2 x r + q x 2 2 2 2z 2z r 2ns r 0 0 jMjl +s 2 1 p i ic xx xx p p exp H jMjl +s b b 1 2 b b h  i h i 1 ik ik 2 2 I(y, z) = exp r r exp r r exp 2 r r y 2 q y q y q y q y 1 2 q y q y 1 2 2z 1 2 2z " # q   2 r r q y q y p 1 1 ik ik 1 2 l ! exp y r 1 q y 2 a a a 2z 2z [ ] l 2k 2  1 2 k (l 2k )! 1 a 2 1 2 (22) å å k !(l 2k )! 4 s !(l 2k s )! 2 1 2 2 1 2 2 k =0 s =0 l 2k s s 1 2 2 2 r r q y q y ik ik 1 1 1 2 y r + q y 2 2 2 2z 2z 1 2ns r r 0 0 l +s 2 2 ic yy yy p i p p exp H l +s b b 2 2 b b with ! ! r r r r ik ik 1 1 1 ik ik q x q x q x q x 1 2 1 2 c = r x + + + x r (23) xx q x q x 2 1 2 2 2 2 2z 2z r a 2ns r 2z 2z r 0 0 0 ! ! ik ik r r 1 1 1 ik ik r r q y q y q y q y 1 2 1 2 c = r y + + + y r (24) yy q y q y 2 1 2 2 2 2 2z 2z a 2ns 2z 2z r r r 0 0 0 The degree of coherence for a RPLMGSMV beam array propagating in atmospheric turbulence at plane z is given as [34] W(r , r , z) 1 2 m(r , r , z) = (25) 1 2 1/2 [W(r , r , z)W(r , r , z)] 1 1 2 2 3. Numerical Results and Discussions In this section, the intensity and coherence distributions of a RPLMGSMV beam array in free space will firstly be investigated, and then the influences of atmospheric turbulence Photonics 2021, 8, 5 6 of 12 on intensity and coherence distributions of a RPLMGSMV beam array will be discussed. The relevant parameters in numerical simulations are selected as l = 532 nm, w = 1 cm, s = 1 mm, N = 10, M = 1 and R = 5 cm without other explanations. The normalized intensity of a RPLMGSMV beam array with Q = 5 in free space at the different distances is illustrated in Figure 2. As can be seen from Figures 2a and 1c, the dark hollow center of beamlets of a RPLMGSMV beam array will evolve into a Gaussian-like beam at z = 50 m (Figure 2a), while the beamlets have the dark hollow center at z = 0 (Figure 1c); The reason that the dark hollow profile translating into the Gaussian beam can be explained as the effect of initial coherence length [33]. As the z increases further, the Gaussian-like beamlets can evolve into a beam with flat-topped profile (Figure 2b), and the beamlets will also begin to overlap with each other (Figure 2b); thus, a RPLMGSMV beam array can translate into a beam with Gaussian-like intensity distribution; at last, the RPLMGSMV beam array can evolve from beam array into the beam with flat-topped inten- sity distribution at longer distance z (Figure 2d). The phenomenon whereby a RPLMGSMV Photonics 2021, 8, x FOR PEER REVIEW 7 of 13 evolves into a beam with flat-topped profile is dominated by MGSM correlated function, and similar evolutions can also be found in the previous reports [25–28,33]. Figure 2. Normalized intensity of a RPLMGSMV beam array with Q = 5 in free space. (a) z = 50 m, Figure 2. Normalized intensity of a RPLMGSMV beam array with Q = 5 in free space. (a) (b) z = 200 m, (c) z = 400 m, (d) z = 900 m. z =50m , (b) z = 200m , (c) z = 400m , (d) z = 900 m . To view the action of Q on intensity distribution, normalized intensity of a RPLMGSMV beam array with Q = 4 in free space are illustrated in Figure 3. As z increases, it is found To view the action of Q on intensity distribution, normalized intensity of a that the evolution of intensity distributions of a RPLMGSMV beam array with Q = 4 RPLMGSMV beam array with Q = 4 in free space are illustrated in Figure 3. As z in- are almost the same with a RPLMGSMV beam array with Q = 5 (Figure 2), the beamlets of beam array will lose the dark hollow profile and become a beam with Gaussian-like creases, it is found that the evolution of intensity distributions of a RPLMGSMV beam beam distribution, the beam array with Q = 4 will translate into the flat-topped profile array with Q = 4 are almost the same with a RPLMGSMV beam array with Q = 5 (Fig- (Figure 3b). Moreover, the flat-topped profile of a RPLMGSMV beam array is dominated by ure 2), the beamlets of beam array will lose the dark hollow profile and become a beam the MGSM sources at longer distance [33]. By comparing Figures 2 and 3, we can conclude with Gaussian-like beam distribution, the beam array with Q = 4 will translate into the flat-topped profile (Figure 3b). Moreover, the flat-topped profile of a RPLMGSMV beam array is dominated by the MGSM sources at longer distance [33]. By comparing Figures 2 and 3, we can conclude that a RPLMGSMV beam array with the different Q will evolve form beam array into flat-topped profile due to the action of MGSM source. Figure 3. Normalized intensity of a RPLMGSMV beam array with Q =4 in free space. (a) , (b) . z = 50 m z =1000m To view the effects of source parameters on the evolutions of intensity of a RPLMGSMV beam array propagating in free space, the cross sections (y = 0) of the nor- malized intensity of a RPLMGSMV beam array with in free space for source pa- Q = 4 rameters σ , N and M at the different distances are shown in Figures 4–6, respectively. Photonics 2021, 8, x FOR PEER REVIEW 7 of 13 Figure 2. Normalized intensity of a RPLMGSMV beam array with in free space. (a) Q = 5 z =50m , (b) z = 200m , (c) z = 400m , (d) z = 900 m . To view the action of on intensity distribution, normalized intensity of a RPLMGSMV beam array with Q = 4 in free space are illustrated in Figure 3. As z in- creases, it is found that the evolution of intensity distributions of a RPLMGSMV beam array with Q = 4 are almost the same with a RPLMGSMV beam array with Q = 5 (Fig- ure 2), the beamlets of beam array will lose the dark hollow profile and become a beam with Gaussian-like beam distribution, the beam array with Q = 4 will translate into the flat-topped profile (Figure 3b). Moreover, the flat-topped profile of a RPLMGSMV beam Photonics 2021, 8, 5 7 of 12 array is dominated by the MGSM sources at longer distance [33]. By comparing Figures 2 and 3, we can conclude that a RPLMGSMV beam array with the different Q will evolve that a RPLMGSMV beam array with the different Q will evolve form beam array into form beam array into flat-topped profile due to the action of MGSM source. flat-topped profile due to the action of MGSM source. Figure 3. Normalized intensity of a RPLMGSMV beam array with Q = 4 in free space. (a) z = 50 m, Figure 3. Normalized intensity of a RPLMGSMV beam array with Q =4 in free space. (a) (b) z = 1000 m. z = 50 m , (b) z =1000m . To view the effects of source parameters on the evolutions of intensity of a RPLMGSMV beam array propagating in free space, the cross sections (y = 0) of the normalized intensity of a RPLMGSMV beam array with Q = 4 in free space for source parameters s, N and M at To view the effects of source parameters on the evolutions of intensity of a the different distances are shown in Figures 4–6, respectively. From Figure 4, it is found that RPLMGSMV beam array propagating in free space, the cross sections (y = 0) of the nor- the RPLMGSMV beam array with larger s will lose the dark hollow distribution slower than the beam array with smaller s. So, the smaller coherence length s will accelerate malized intensity of a RPLMGSMV beam array with Q = 4 in free space for source pa- the evolutions of beam array translating into flat-topped profile, and the beam array with smaller s will have the better flatness when the beam array translating into flat-topped rameters σ , N and M at the different distances are shown in Figures 4–6, respectively. intensity profile at the longer distance z (Figure 4d). Thus, it can conclude that the speed of a beam array translating into the flat-topped profile can be dominated by setting different s of MGSM source. Figure 5 shows that the RPLMGSMV beam array with larger N will evolve from beam array into flat-topped beam faster, and which will have the better flatness when the beam array translating into the flat-topped beam at last (Figure 5d). The flatness of flat-topped profile is dominated by the total number N of MGSM source, and the similar results can also be seen in the previous work [25]. Thus, from previous discussions, one can conclude that the flatness of flat-topped profile generating by RPLMGSMV beam array can be modulated by the parameters s and N in the far field. The better the flatness of flat-topped profile is, the more power can be received by the same receiver, this is helpful for received power of free space optical communication. One can see from Figure 6 that the RPLMGSMV beam array with larger M will have the larger dark hollow center at z = 0 (Figure 6a), while the influences of different M on the intensity distribution will disappear as the beam array evolve into the flat-topped beam at the longer distance z (Figure 6d). Thus, we can conclude that the flat-topped profile is not correlated with topological charge in the far field. Photonics 2021, 8, x FOR PEER REVIEW 8 of 13 From Figure 4, it is found that the RPLMGSMV beam array with larger σ will lose the dark hollow distribution slower than the beam array with smaller . So, the smaller co- herence length σ will accelerate the evolutions of beam array translating into flat- topped profile, and the beam array with smaller σ will have the better flatness when the beam array translating into flat-topped intensity profile at the longer distance z (Figure 4d). Thus, it can conclude that the speed of a beam array translating into the flat-topped profile can be dominated by setting different σ of MGSM source. Figure 5 shows that the RPLMGSMV beam array with larger N will evolve from beam array into flat-topped beam faster, and which will have the better flatness when the beam array translating into the flat-topped beam at last (Figure 5d). The flatness of flat-topped profile is dominated by the total number N of MGSM source, and the similar results can also be seen in the previous work [25]. Thus, from previous discussions, one can conclude that the flatness of flat-topped profile generating by RPLMGSMV beam array can be modulated by the parameters σ and N in the far field. The better the flatness of flat-topped profile is, the more power can be received by the same receiver, this is helpful for received power of free space optical communication. One can see from Figure 6 that the RPLMGSMV beam array with larger M will have the larger dark hollow center at z = 0 (Figure 6a), while the influences of different M on the intensity distribution will disappear as the beam array Photonics 2021, 8, 5 8 of 12 evolve into the flat-topped beam at the longer distance z (Figure 6d). Thus, we can con- clude that the flat-topped profile is not correlated with topological charge in the far field. Photonics 2021, 8, x FOR PEER REVIEW 9 of 13 Figure 4. Figure 4. CCr ross oss se sections ctions (y( = y 0) = 0) of normalize of normalized d intensity of intensity of a R a P RPLMGSMV LMGSMV be beam am array with array withQQ==4 4 in free space for the different s. (a) z = 50 m, (b) z = 200 m, (c) z = 400 m, (d) z = 1000 m. in free space for the different σ . (a) z = 50 m , (b) z = 200m , (c) z = 400m , (d) z =1000m . Figure 5. Cross sections (y = 0) of the normalized intensity of a RPLMGSMV beam array with Figure 5. Cross sections (y = 0) of the normalized intensity of a RPLMGSMV beam array with Q = 4 Q =4 in free space for the different N . (a) , (b) , (c) , (d) z = 50 m z = 200m z = 400m in free space for the different N. (a) z = 50 m, (b) z = 200 m, (c) z = 400 m, (d) z = 1000 m. z =1000m . Figure 7 illustrates the average intensity of a RPLMGSMV beam array with Q =4 propagating in free space and atmospheric turbulence for different . As can be seen that in Figure 7b, the flatness of the flat-topped profile obtained at the longer distance is poorer than a RPLMGSMV beam array in free space, and the larger the is, the poorer the flatness of flat-topped profile is. The phenomenon where the flat-topped profile is be- coming poor in atmospheric turbulence can be explained by the influences of atmospheric turbulence. Photonics 2021, 8, x FOR PEER REVIEW 9 of 13 Figure 5. Cross sections (y = 0) of the normalized intensity of a RPLMGSMV beam array with Q =4 in free space for the different N . (a) , (b) , (c) , (d) z = 50 m z = 200m z = 400m z =1000m Figure 7 illustrates the average intensity of a RPLMGSMV beam array with Q =4 propagating in free space and atmospheric turbulence for different . As can be seen that in Figure 7b, the flatness of the flat-topped profile obtained at the longer distance is poorer than a RPLMGSMV beam array in free space, and the larger the is, the poorer Photonics 2021, 8, 5 9 of 12 the flatness of flat-topped profile is. The phenomenon where the flat-topped profile is be- coming poor in atmospheric turbulence can be explained by the influences of atmospheric turbulence. Figure 6. Cross sections (y = 0) of the normalized intensity of a RPLMGSMV beam array with Q = 4 in free space for the different M. (a) z = 0 m, (b) z = 100 m, (c) z = 200 m, (d) z = 1000 m. Photonics 2021, 8, x FOR PEER REVIEW 10 of 13 Figure 7 illustrates the average intensity of a RPLMGSMV beam array with Q = 4 propagating in free space and atmospheric turbulence for different C . As can be seen that in Figure 7b, the flatness of the flat-topped profile obtained at the longer distance is poorer Figure 6. Cross sections (y = 0) of the normalized intensity of a RPLMGSMV beam array with than a RPLMGSMV beam array in free space, and the larger the C is, the poorer the flatness Q =4 in free space for the different M . (a) , (b) , (c) , (d) z =0m z =100 m z = 200m of flat-topped profile is. The phenomenon where the flat-topped profile is becoming poor z in =100 atmospheric 0m . turbulence can be explained by the influences of atmospheric turbulence. Figure 7. Cross sections of a RPLMGSMV beam array with Q = 4Qin =atmospheric 4 turbulence for the Figure 7. Cross sections of a RPLMGSMV beam array with in atmospheric turbulence for different C . (a) z = 100 m, (b) z = 1000 m. the different C . (a) z =100m , (b) z =1000m . Figure 8 gives the cross-sections of degree of coherence of a RPLMGSMV beam array Figure 8 gives the cross-sections of degree of coherence of a RPLMGSMV beam array with Q = 5 in free space (Figure 8a–c) and atmospheric turbulence at z = 1000 m for the dif Qfer =5 ent M, s, N and C . From Figure 8a,b, it is seen that the coherence properties with in free space (Figure 8a–c) and atmospheric turbulence at z =1000m for the of a RPLMGSMV beam array can be affected by the M and s. Meanwhile, the effects σ N of total number N on the coher nence distribution is not found (Figure 8c). Further, when different , , and . From Figure 8a,b, it is seen that the coherence properties x x is smaller, the influences of M and s are less. In the analysis of the influences 1 2 σ of a RPLMGSMV beam array can be affected by the and . Meanwhile, the effects of atmospheric turbulence on coherence, it is seen that the coherence distribution of a of total number on the coherence distribution is not found (Figure 8c). Further, when RPLMGSMV beam array in atmospheric turbulence with larger C will have the Gaussian x − x distribution (Figure 8d). Meanwhile, the same beam in free pace will have the irregular is smaller, the influences of and are less. In the analysis of the influences of atmospheric turbulence on coherence, it is seen that the coherence distribution of a RPLMGSMV beam array in atmospheric turbulence with larger will have the Gauss- ian distribution (Figure 8d). Meanwhile, the same beam in free pace will have the irregular coherence distribution. To view the influences of z on coherence distribution, the coher- Q =5 ence of a RPLMGSMV beam array with at the different distance z is illustrated in Figure 9. One can see from Figure 9 that a RPLMGSMV beam array at the longer propa- gation distance will have more regular coherence distribution. One can conclude that the spectral degree of coherence of a RPLMGSMV beam array in atmospheric turbulence will have a Gaussian distribution at longer distance. Photonics 2021, 8, x FOR PEER REVIEW 10 of 13 Figure 6. Cross sections (y = 0) of the normalized intensity of a RPLMGSMV beam array with Q =4 in free space for the different M . (a) z =0m , (b) z =100 m , (c) z = 200m , (d) z =1000m . Q =4 Figure 7. Cross sections of a RPLMGSMV beam array with in atmospheric turbulence for the different C . (a) z =100m , (b) z =1000m . Figure 8 gives the cross-sections of degree of coherence of a RPLMGSMV beam array Q =5 with in free space (Figure 8a–c) and atmospheric turbulence at z =1000m for the σ N M n different , , and . From Figure 8a,b, it is seen that the coherence properties of a RPLMGSMV beam array can be affected by the and . Meanwhile, the effects of total number on the coherence distribution is not found (Figure 8c). Further, when x − x 12 M is smaller, the influences of and are less. In the analysis of the influences of atmospheric turbulence on coherence, it is seen that the coherence distribution of a Photonics 2021, 8, 5 10 of 12 RPLMGSMV beam array in atmospheric turbulence with larger will have the Gauss- ian distribution (Figure 8d). Meanwhile, the same beam in free pace will have the irregular coherence distribution. To view the influences of z on coherence distribution, the coher- coherence distribution. To view the influences of z on coherence distribution, the coherence Q =5 ence of of a RPLMGSMV a RPLMGSMV beam beam a array rray wi with Q th = 5 at the at the d different iffer distance ent distance z is illustrated z is illustrat in e Figur d in e 9. One can see from Figure 9 that a RPLMGSMV beam array at the longer propagation Figure 9. One can see from Figure 9 that a RPLMGSMV beam array at the longer propa- distance will have more regular coherence distribution. One can conclude that the spectral gation distance will have more regular coherence distribution. One can conclude that the degree of coherence of a RPLMGSMV beam array in atmospheric turbulence will have a spectral degree of coherence of a RPLMGSMV beam array in atmospheric turbulence will Gaussian distribution at longer distance. have a Gaussian distribution at longer distance. Photonics 2021, 8, x FOR PEER REVIEW 11 of 13 Photonics 2021, 8, x FOR PEER REVIEW 11 of 13 Figure 8. Modulus of the coherence of a RPLMGSMV beam array with Q =5 in the free space Figure 8. Figure Mod 8. Modulus ulus of the of coh the e coher rence of ence a RPLMGSMV bea of a RPLMGSMVm beam array with array with Q =Q5 = in the free 5 in the fr space ee space and and atmospheric turbulence at the distance z =1000m . (a)different M , (b) different σ , (c) dif- atmospheric turbulence at the distance z = 1000 m. (a) different M, (b) different s, (c) different N, and atmospheric turbulence at the distance z =1000m . (a)different M , (b) different σ , (c) dif- ferent N, (d) different 2C . (d) different C . ferent N, (d) different C . Q =5 Figure 9. Figure 9. Mod Modulus ulus of the of the cohcoher erence of ence a RPLMGSMV bea of a RPLMGSMVm beam array with array with Q for the different = 5 for the different Figure 9. Modulus of the coherence of a RPLMGSMV beam array with Q =5 for the different distance z. (a) free space, (b) atmospheric turbulence. distance z. (a) free space, (b) atmospheric turbulence. distance z. (a) free space, (b) atmospheric turbulence. 4. Conclusions 4. Conclusions 4. Conclusions In this paper, the analytical description of a RPLMGSMV beam array generated by In this paper, the analytical description of a RPLMGSMV beam array generated by In this paper, the analytical description of a RPLMGSMV beam array generated by MGSM sources is introduced and analyzed. Based on the extended Huygens–Fresnel MGSM sources is introduced and analyzed. Based on the extended Huygens–Fresnel in- MGSM sources is introduced and analyzed. Based on the extended Huygens–Fresnel in- integral, the CSD of a RPLMGSMV beam array propagating in atmospheric turbulence tegral, the CSD of a RPLMGSMV beam array propagating in atmospheric turbulence is tegral, the CSD of a RPLMGSMV beam array propagating in atmospheric turbulence is is derived. The evolutions of intensity and coherence properties of a RPLMGSMV beam derived. The evolutions of intensity and coherence properties of a RPLMGSMV beam ar- derived. The evolutions of intensity and coherence properties of a RPLMGSMV beam ar- array propagating in free space and atmospheric turbulence are analyzed in detail. It ray propagating in free space and atmospheric turbulence are analyzed in detail. It is seen ray propagating in free space and atmospheric turbulence are analyzed in detail. It is seen is seen that a RPLMGSMV beam array propagating in free space can gradually lose the that a RPLMGSMV beam array propagating in free space can gradually lose the initial that a RPLMGSMV beam array propagating in free space can gradually lose the initial initial intensity distribution of beamlets, and evolve from the beam array into a beam with intensity distribution of beamlets, and evolve from the beam array into a beam with a flat- intensity distribution of beamlets, and evolve from the beam array into a beam with a flat- topped profile due to the action of MGSM source as z increases. In the far field, when the topped profile due to the action of MGSM source as z increases. In the far field, when the total number N is larger or the coherence length is smaller, the flatness of flat-topped total number N is larger or the coherence length σ is smaller, the flatness of flat-topped profile of a RPLMGSMV beam array will be better, this is useful for the free space optical profile of a RPLMGSMV beam array will be better, this is useful for the free space optical communication. When a RPLMGSMV beam array propagates in atmospheric turbulence, communication. When a RPLMGSMV beam array propagates in atmospheric turbulence, the flatness of the flat-topped profile can be dominated by the atmospheric turbulence, the flatness of the flat-topped profile can be dominated by the atmospheric turbulence, the flatness will become poor. It is also found that the coherence distribution of a the flatness will become poor. It is also found that the coherence distribution of a RPLMGSMV beam array in atmospheric turbulence will have Gaussian distribution at the RPLMGSMV beam array in atmospheric turbulence will have Gaussian distribution at the longer distance. longer distance. Author Contributions: Data curation, J.Z. and X.M; writing—original draft preparation, J.Z., G.W., Author Contributions: Data curation, J.Z. and X.M; writing—original draft preparation, J.Z., G.W., Y.W., and D.L.; writing—review and editing, H.Z. and H.Y.; supervision, G.W., Y.W., and D.L.; pro- Y.W., and D.L.; writing—review and editing, H.Z. and H.Y.; supervision, G.W., Y.W., and D.L.; pro- ject administration, G.W. and D.L. All authors have read and agreed to the published version of the ject administration, G.W. and D.L. All authors have read and agreed to the published version of the manuscript. manuscript. Funding: This research was supported by National Natural Science Foundation of China [11604038, Funding: This research was supported by National Natural Science Foundation of China [11604038, 11875096, 11404048], and the Fundamental Research Funds for the Central Universities 11875096, 11404048], and the Fundamental Research Funds for the Central Universities [3132020175]. [3132020175]. Photonics 2021, 8, 5 11 of 12 a flat-topped profile due to the action of MGSM source as z increases. In the far field, when the total number N is larger or the coherence length s is smaller, the flatness of flat-topped profile of a RPLMGSMV beam array will be better, this is useful for the free space optical communication. When a RPLMGSMV beam array propagates in atmospheric turbulence, the flatness of the flat-topped profile can be dominated by the atmospheric turbulence, the flatness will become poor. 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Photonics – Multidisciplinary Digital Publishing Institute

Published: Dec 29, 2020

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