Propagation of Rectangular Multi-Gaussian Schell-Model Array Beams through Free Space and Non-Kolmogorov Turbulence

: In this paper, rectangular multi-Gaussian Schell-model (MGSM) array beams, which consists N × D beams in rectangular symmetry, are ﬁrst introduced. The analytical expressions of MGSM array beams propagating through free space and non-Kolmogorov turbulence are derived. The propagation properties, such as normalized average intensity and e ﬀ ective beam sizes of MGSM array beams are investigated and analyzed. It is found that the propagation properties of MGSM array beams depend on the parameters of the MGSM source and turbulence. It can also be seen that the beam size of Gaussian beams translated by MGSM array beams will become larger as the total number of terms, M, increases or coherence length, σ , decreases, and the beam in stronger non-Kolmogorov turbulence (larger α and l 0 , or smaller L 0 ) will also have a larger beam size.

However, the array beams were widely studied because they can provide higher power output than single beam. The Talbot effect, which describes the periodic beam propagating in studies of self-imaging, was investigated [23,24], and array beams with orbital angular momentum can also be generated based on Talbot effect [25,26]. In free space communication, the properties of coherent array beams propagating in free space and turbulent atmosphere have been widely investigated [27][28][29][30]. According to previous work, the partially coherent beams can reduce the effects of turbulent atmosphere [31], thus the evolutions of partially coherent array beams in turbulence, such as Gaussian Schell-model array beams [3,[32][33][34], radial phase-locked array beams [35][36][37][38][39][40], and optical coherence vortex lattices [41], were also investigated. The multi-Gaussian Schell-model (MGSM) beams will evolve into flat-topped beams, thus the beams generated by a MGSM source have attracted much attention [42][43][44][45][46][47][48]. Considering the special properties of MGSM beams, it will thus be very interesting to investigate the array beams generated by a MGSM source. In this paper, we first introduce the model of a rectangular MGSM array beam, and then investigate the evolutions of MGSM array beams propagating through free space and non-Kolmogorov turbulence. 2 where N and D are odd numbers and represent the orders of individual off-axis MGSM beams along the xand y-axes, X 0 and Y 0 denote separations distances along the xand y-axes, and w 0 represents the beam waist. The contour graphs of N × D MGSM array beams in source plane z = 0 are plotted in Figure 1, which shows that the distance between the different beamlets will increase as the off-axis parameters X 0 and Y 0 increase.
introduce the model of a rectangular MGSM array beam, and then investigate the evolutions of MGSM array beams propagating through free space and non-Kolmogorov turbulence.

Analytical Expressions of MGSM Array Beams
Recalling the descriptions of MGSM beams, the spectral degree of coherence of the MGSM source takes the form as follows [49]: 2  2  10  20  10  20  10 20  2  2  1  0   1 where 0 C represents normalization coefficient, σ is the initial correlation coherence length, M is total number of terms of MGSM source, and Considering the diagram of a N D × array beam in rectangular symmetry, the cross-spectral density (CSD) of a rectangular N D × MGSM array beam generated by a MGSM source is described by the following: where N and D are odd numbers and represent the orders of individual off-axis MGSM beams along the x-and y-axes, 0 X and 0 Y denote separations distances along the x-and y-axes, and 0 w represents the beam waist. The contour graphs of N D × MGSM array beams in source plane z = 0 are plotted in Figure 1, which shows that the distance between the different beamlets will increase as the off-axis parameters 0 X and 0 Y increase.

Propagation Theory
Considering the extended Huygens-Fresnel integral, the CSD of N × D MGSM array beams in non-Kolmogorov turbulent atmosphere at the plane z is written as follows [15][16][17][18][19][20]: where k = 2π/λ is the wave number with λ as the wavelength. The definitions of ψ(r 0 , r) can be found in reference [19], and as follows: , κ is the magnitude of spatial frequency, and α is the power law exponent. The non-Kolmogorov power spectrum Φ(κ, α) can be described by the following [15][16][17]: where C 2 n is the turbulent structure constant with units m 3−α ; κ 0 = 2π/L 0 , where L 0 is the outer scale; and κ m = c(α)/l 0 , where l 0 is the inner scale; and where Γ(x) denotes the Gamma function. Substituting Equations (6)-(8) into the definition of 1/ρ 2 0 , we can obtain the following: where Γ(x, y) represents the incomplete Gamma function. Thus, the propagation CSD of N × D MGSM array beams in non-Kolmogorov turbulent atmosphere are derived as follows: where W ND (x, z) and W ND (y, z) are given as follows: with a = 1 Appl. Sci. 2020, 10, 450 4 of 11 where j = x or y in all locations of equations. When j = x, the symbols h and Λ represent n and X in all equations, respectively. When j = y, the symbols h and Λ represent d and Y in all equations, respectively. The intensity [50] and effective beam widths [51] along the xand y-axes of N × D MGSM array beams propagating through non-Kolmogorov turbulent atmosphere can be written as follows: Substituting Equations (10)- (14) into Equation (16), the effective beam widths of a N × D MGSM array beam propagating through non-Kolmogorov turbulent atmosphere can be obtained as follows: where with Due to the symmetrical characteristics, the effective beam size W x and W y will have similar properties. Thus, only the effective beam size W x is investigated in the following numerical analysis.
The normalized intensity of N × D MGSM array beams propagating through free space (C 2 n = 0) and non-Kolmogorov turbulent atmosphere are illustrated in Figures 2 and 3, respectively. According to Figure 2, we can see that the N × D MGSM array beams in free space will maintain the rectangular distribution of beamlets in the short propagation distance. As z increases, the individual beamlets will gradually translate into a flat-topped beam caused by the influence of the MGSM sources. As z further increases, the individual flat-topped beams will overlap with each other. Eventually, the MGSM array beam will translate a solid beam; at last, the MGSM array beams will involve into a beam with flat-topped profile. While N × D MGSM array beams propagating through non-Kolmogorov turbulent atmosphere will have similar evolution properties with array beams in free space, the individual beamlets in non-Kolmogorov turbulent atmosphere will overlap with each other more rapidly than the array beams in free space at the same distance, z = 200 (Figures 2b and 3a). As z further increases, the array beams in non-Kolmogorov turbulent atmosphere will evolve into a Gaussian beam. The phenomenon in Figures 2 and 3 can be explained as the general beams propagation in atmosphere can translate into Gaussian beams [52]. To view the effects of non-Kolmogorov turbulent atmosphere, the cross sections of N × D MGSM array beams with N = D = 3 through non-Kolmogorov turbulent atmosphere for the different C 2 n are shown in Figure 4. It can be seen that the MGSM array beams propagating through non-Kolmogorov turbulent atmosphere with a larger C 2 n will translate into a larger Gaussian-like beam spot more rapidly than in free space (C 2 n = 0) at the same distance z.
Appl. Sci. 2020, 10, x FOR PEER REVIEW 6 of 11          Figure 5. As can be found, the MGSM array beams with smaller σ will evolve into a Gaussian beam more rapidly with increasing propagation distance z, and will have a larger spot. To investigate the effective beam size of a Gaussian beam translated by the MGSM array beam, Figure 6 gives the effective beam size x W of the Gaussian beam for different 2 n C and σ . One can see that x W of a Gaussian beam obtained by a MGSM array beam will increase with the increase of 2 n C or the decrease of σ . The effects of coherence length, σ, on N × D MGSM array beams with N = D = 3 propagating through non-Kolmogorov turbulent atmosphere are illustrated in Figure 5. As can be found, the MGSM array beams with smaller σ will evolve into a Gaussian beam more rapidly with increasing propagation distance z, and will have a larger spot. To investigate the effective beam size of a Gaussian beam translated by the MGSM array beam, Figure 6 gives the effective beam size W x of the Gaussian beam for different C 2 n and σ. One can see that W x of a Gaussian beam obtained by a MGSM array beam will increase with the increase of C 2 n or the decrease of σ.
propagation distance z, and will have a larger spot. To investigate the effective beam size of a Gaussian beam translated by the MGSM array beam, Figure 6 gives the effective beam size x W of the Gaussian beam for different 2 n C and σ . One can see that x W of a Gaussian beam obtained by a MGSM array beam will increase with the increase of 2 n C or the decrease of σ . By changing the parameters M and N D = , the effective beam size for MGSM array beams propagating through non-Kolmogorov turbulent atmosphere is shown in Figure 7. As can be found, when the propagation distance z remains the same, the Gaussian beam translated by the MGSM array beams with larger M can have a larger effective beam size. When =1 M , the MGSM array beam will reduce into a GSM array beam, which will have the smallest effective beam size. While, in the influence of N D = , we can find that as z increases, the effects of N D = on the effective beam size of the Gaussian beam will disappear. These results mean that the MGSM array beams with different N D = will have almost the same effective beam size. Thus, the larger the N D = , the higher the power the Gaussian beam will have. This phenomenon is beneficial for free space optical communications using MGSM array beams. By changing the parameters M and N = D, the effective beam size for MGSM array beams propagating through non-Kolmogorov turbulent atmosphere is shown in Figure 7. As can be found, when the propagation distance z remains the same, the Gaussian beam translated by the MGSM array beams with larger M can have a larger effective beam size. When M = 1, the MGSM array beam will reduce into a GSM array beam, which will have the smallest effective beam size. While, in the influence of N = D, we can find that as z increases, the effects of N = D on the effective beam size of the Gaussian beam will disappear. These results mean that the MGSM array beams with different N = D will have almost the same effective beam size. Thus, the larger the N = D, the higher the power the Gaussian beam will have. This phenomenon is beneficial for free space optical communications using MGSM array beams.  The influences of non-Kolmogorov turbulent atmosphere parameters α , 0 L , and 0 l on x W are shown in Figure 8. When z increases to 1000m z = , the MGSM array beams will become the Gaussian   The influences of non-Kolmogorov turbulent atmosphere parameters α , 0 L , and 0 l on x W are shown in Figure 8. When z increases to 1000m z = , the MGSM array beams will become the Gaussian beam, the effective beam size, x W , of the Gaussian-like beam translated by the MGSM array beams will increase with α , and a decrease of 0 l or an increase of 0 L . The phenomenon can be explained as follows: The strength of non-Kolmogorov turbulent atmosphere depends on α , 0 L , and 0 l . The influences of non-Kolmogorov turbulent atmosphere parameters α, L 0 , and l 0 on W x are shown in Figure 8. When z increases to z = 1000 m, the MGSM array beams will become the Gaussian beam, the effective beam size, W x , of the Gaussian-like beam translated by the MGSM array beams will increase with α, and a decrease of l 0 or an increase of L 0 . The phenomenon can be explained as follows: The strength of non-Kolmogorov turbulent atmosphere depends on α, L 0 , and l 0 .  The influences of non-Kolmogorov turbulent atmosphere parameters α , 0 L , and 0 l on x W are shown in Figure 8. When z increases to 1000m z = , the MGSM array beams will become the Gaussian beam, the effective beam size, x W , of the Gaussian-like beam translated by the MGSM array beams will increase with α , and a decrease of 0 l or an increase of 0 L . The phenomenon can be explained as follows: The strength of non-Kolmogorov turbulent atmosphere depends on α , 0 L , and 0 l .

Conclusions
In conclusion, the rectangular MGSM array beams composed by N × D MGSM beams in rectangular symmetry are given, and the intensity and effective beam size of MGSM array beams in free space and in non-Kolmogorov turbulent atmosphere were investigated. Based on the derived equations, the results show that the MGSM array beams propagating through non-Kolmogorov turbulent atmosphere will become a Gaussian beam caused by the non-Kolmogorov turbulent atmosphere, while array beams propagating through free space will translate into flat-topped beams. With an increase in C 2 n and L 0 , or and decrease of α and l 0 , the Gaussian-like beam will have a larger beam size. With M increases or σ decreases, the array beams will also evolve into beams with a larger spot. The obtained results may be beneficial for free space optical communication.