## Abstract

A novel approach to generate a spatially-distributed orbital angular momentum (OAM) beam array based on coherent combining technology is presented. The arrangement of the multiple fundamental Gaussian beams at the initial plane, as well as the intensity weights and the phase distributions of the array beams, is determined by the reversal of Huygens Fresnel diffraction and the greedy algorithm. This method ensures that a vortex beam array is formed at a specified distance, and the distance can be adjusted by phase modulation. The evolution properties of the synthesized beam array near the receiver plane are studied as well to estimate the robustness of the method. The experimental limitations of this technique are discussed, including the maximum number of beams, the relative separation of each beam and the maximum topological charges. The results illustrate that a spatially-distributed OAM beam array can be effectively generated within a finite distance interval, and the distance is adjustable. This new method enables further applications of a structured optical field, such as optical communication and spatial light structuring.

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

## 1. Introduction

Vortex beams, which have helical wave front and ring-shaped intensity structure, can maintain the ring-shaped intensity distribution through propagation and offer great advantages in the realization of long-distance and non-destructive transmission [1]. A phase singularity is formed at the center of the optical vortex beam, where the intensity is null and the phase is undetermined. Vortex beams with different topological charges (TC) carry corresponding different orbital angular momentums (OAM), which make it possible to apply vortex beams as information carriers in optical communication systems based on mode division multiplexing. The fascinating properties of vortex beams bring their great potential in micro-particle manipulation, super-resolution imaging and free-space communication [2–7].

In practical applications, such as structured optical communication and laser processing, multiple vortex beams with controllable OAMs are often required for information encoding, multi-point processing and so on. Therefore, it is necessary to generate multiple vortex beams at the same time, and the importance of generating multiple vortex beams has been widely noticed by researchers and some progresses have been made recently. In 2017, Fu et al. proposed a method to generate multiple vortex beams simultaneously by designing special transmission rate function and polarization sensitive diffraction grating [8]. The generated multiple vortex beams are distributed in the form of rectilinear or square array, and the OAM modes are controllable. In 2018, the concept of digitalized chiral superstructures is proposed by Chen et al. based on cholesteric liquid crystal (CLC), which exhibits unique features in polychromatic and spin-determined phase modulation [9]. An innovative CLC optical vortex processor is developed and the generation of multiple vortex beams with different OAM states is validated.

In addition, high power vortex beams are required for many applications, especially for free-space optical communication systems, which need sufficient signal power to deal with the power loss through propagation [10]. However, the output power of the generated vortex beams by most of the current technical routes is extremely dependent on the key devices such as spatial light modulators, diffractive phase holograms and q-plates, and their power handling is limited. In response to this challenge, coherent beam combining, a widely studied technology, could be taken into consideration [11–17]. To generate a single vortex beam, a new technical route which avoids the dependence of these key devices and makes use of the propagation properties of beams is proposed, that is, generating optical vortices from beam arrays by using coherent combining technology. Wang et al. first analyzed the generation of vortex beams by using coherent beam arrays theoretically in 2009 [18]. Since then, many studies have been carried out. Aksenov et al. presented the generation of Laguerre-Gaussian vortex beams with controllable OAMs, and the evolution of the energy proportion for each mode component of the synthesized beam as well as the influence of atmospheric turbulence are involved [19]. Further study of the statistical characteristics of the synthesized beam propagating through turbulent atmosphere was proposed in 2016 [20]. Moreover, the authors put forward the study in the laws of spatial evolution of the OAM of the synthesized vortex beam and they have analyzed parameters such as total angular momentum and angular momentum density in 2018 [21]. Their works could inspire researchers to study the generation of vortex beams based on coherent combining technology on a deeper level. Besides, different single exotic beams generated from beam arrays have been studied successively. In terms of theoretical research, numerous works concerning the generation of Bessel beams, Airy beams, cylindrical vector beams, anomalous vortex beams and equilateral-polygon-like flat-top focuses have been carried out [22–28]. In experimental aspect, Chu et al. designed the experimental setup for generating Bessel beams and Airy beams based on stochastic parallel gradient descent (SPGD) control loop while Lachinova et al. designed a different feedback control system based on beam-tail interference sensors and fiber-integrated phase shifters [23, 24, 29]. Besides, dark hollow beams and cylindrical vector beams have been generated by coherent combining in experiment [30, 31].

With the widespread application of coherent beam combining technology in generating structured optical field, an interesting question arises: Is it possible to generate vortex beam arrays by the coherent combining of fundamental Gaussian array beams? In this paper, we propose a novel method to generate a vortex beam array with controllable OAMs based on the coherent combining of fundamental Gaussian array beams for the first time (to our knowledge). Specifically, we determine the configuration and parameters of the fundamental Gaussian array beams at the transverse plane for generating the target structured optical field by the reversal of Huygens Fresnel diffraction and the greedy algorithm [32, 33]. We investigate the propagation properties by numerical simulation and demonstrate that the synthesized beam array is in good agreement with the ideal vortex beam array within a specified distance interval. This new technique of generating vortex beam arrays has useful application in optical communication and spatial light structuring.

## 2. Principle and method

Under the paraxial approximation, the near-field optical wave propagation can be described by the Huygens Fresnel diffraction integral as [34]

*ξ*,

*η*) and (

*x*,

*y*) are the coordinates of the source plane and the receiver plane in the Cartesian coordinate system, respectively.

*U*represents the complex amplitude of the optical field,

*λ*is the optical wavelength and

*L*accounts for the propagation distance.

According to the idea provided by Chen et al. in 2016, the optical field at the source plane could be calculated when the optical field at the receiver plane is given [32]. We transform the expression of Eq. (1) into a Fourier transform (FT) form, and the optical field at the source plane could be derived by transposition and inverse Fourier transform (IFT) as

The method to generate vortex beam arrays makes use of the reversal of Huygens Fresnel diffraction integral, thus the intensity and phase distributions at the source plane are calculated when a well-designed vortex beam array is set to the optical field at the receiver plane. To simplify the optical field at the source plane and make it possible for application, a greedy algorithm is involved to extract main information of the intensity distribution. We assume that the fundamental Gaussian beams locate at different positions constitute a set of independent basis functions to compose the initial optical field, and the parameters of each Gaussian beam are acquired for fitting the intensity distribution at the source plane. The process of determining the parameters of the Gaussian beam array can be abstracted as a constrained optimization problem, i.e.

where *A _{j}* ,

*w*,

_{0j}*d*and (

_{j}*a*,

_{j}*b*) are the amplitude, waist width, aperture diameter and central position of the

_{j}*j*th-order Gaussian beam, respectively, and

*N*is the total number of the beamlets at the source plane.

*I*and

_{ideal}*I*are the intensities of the ideal and the synthesized optical field at the source plane.

_{array}The constrained optimization problem described in Eq. (3) means that one of the goals of our technique is to construct the optical field as close to the ideal optical field at the source plane as possible with a Gaussian beam array. In other words, to solve the constrained optimization problem is to determine the parameters of the Gaussian beam array. In this paper, the greedy algorithm is used to solve the constrained optimization problem: we first divide the intensity pattern of the ideal optical field at the source plane into sub-regions, and then determine the amplitude and the center position of each Gaussian beam according to the value and position of the intensity peak within each sub-region. Subsequently, we calculate other parameters of each Gaussian beam such as waist width and aperture diameter one by one, and the setup of the Gaussian beam array can be finally obtained. The optical field of multiple Gaussian beams located at the source plane is expressed as

*U*and

_{ideal}*U*are the complex amplitudes of the ideal and the synthesized optical field at the source plane.

_{array}Substitute Eq. (4) to Eq. (1), the optical field generated by the beam array at the receiver plane can be finally derived as

Generally, the generation of a four-vortex-beam array is taken as an example. The setup for the vortex beam array generation is shown in Figs. 1(a)-1(c). Figure 1(a) presents the optical field calculated by the reversal of Huygens Fresnel diffraction at the source plane, and Fig. 1(b) shows the configuration and phase distribution of the Gaussian beam array at the source plane. The designed Gaussian beam array propagates and superposes coherently in free space, and finally forms the rectangular array of four vortex beams, as schemed in Fig. 1(c).

## 3. Numerical simulation results and discussion

From Eq. (2), the optical field at the source plane to generate a vortex beam array can be obtained. The coherent combining of the vortex beam array is involved to illustrate the differences of the intensity and phase distributions between the forward and the reverse propagation of the vortex beam array at the same distance symmetrically. Then, a Gaussian beam array is arranged to fit the optical fields obtained from Eq. (2) and further generates a vortex beam array. Subsequently, the generation and evolution of the vortex beam array are studied by numerical simulation, and the validity and the robustness of the method are demonstrated. Finally, the experimental limitations of this technique are discussed in detail.

#### 3.1 The reversal of Huygens Fresnel diffraction

The intensity and phase distributions in the *x*-*y* plane of the forward propagation, the reverse propagation and the differences between the two types of propagations are presented in Fig. 2(a), 2(b) and 2(c), respectively. A rectangular array of four vortex beams is set at the reference plane. The wavelength, waist width, TC and transverse distance of vortex beams are 1.06*μm*, 5*mm*, 1 and 20*mm*, respectively. Substituting the vortex array structured optical field at the reference plane into the Eq. (1) and Eq. (2), the optical field at the same longitudinal distance (2000*m*) is calculated based on the Huygens Fresnel diffraction integral and the reversal of Huygens Fresnel diffraction integral.

In Fig. 2(a) and 2(b), similar intensity and phase distributions of the forward propagation and the reverse propagation can be observed. In contrast to the well-studied coherent combining of multiple parallel linearly polarized and phase-locked Gaussian beams which has the intensity peak on axis and concentrated energy, the intensity distribution of the vortex beam array after propagating and coherent combining has a dark hollow pattern on axis and the energy is separated into outside sidelobes. A phase singularity is located at the center position, where the phase changes from -π to π continuously in a screw type (for TC = 1). The helical structured phase pattern depends on the TC of the vortex beams at the reference plane. Besides, the branch cuts, which correspond to the zero-intensity “lattice”, divide the phase pattern into sub-regions. In comparison, the optical field calculated by the reversal of diffraction has resemble intensity distribution and same clockwise phase rotation at the central position. More detailed differences between the forward and the reverse propagation are shown in Fig. 2(c). Specifically, a subtle angular misalignment in the intensity patterns can be observed while a flat-top profile of phase difference exists within the central sub-region divided by branch cuts. In a nutshell, the intensity distributions are similar and the inner regional phase distributions have same direction of rotation in the case of the forward and the reverse propagation. However, self-reproduction of the optical field cannot be observed through propagation along *z*-axis because of the non-reversibility in the phase evolution.

#### 3.2 Setup of the beam array at the source plane

From the analysis above, the intensity pattern generated by the reversal of Huygens Fresnel diffraction contains multiple petals, which carry substantial information and have potential to reproduce the vortex beam array. It means that the intensity pattern can be divided into sub-regions as is shown in Fig. 3(a), and it is conducive to solve the constrained optimization problem described in Eq. (3) by using the greedy algorithm. Specifically, the value and position of the intensity peak within each sub-region (local optimums), which determine the amplitude and the center position of each Gaussian beam, can be obtained accurately. Other parameters such as waist width and aperture diameter can be further calculated by obtaining the positions of 1/*e*^{2} and 0.01 peak intensity. Figure (b) depicts the arrangement of the Gaussian beam array, and the center position of each beamlet is refined symmetrically at the end of the algorithm. Note that the greedy algorithm refers to obtaining a series of local optimal choices, and then synthesizing the solutions of sub-problems to obtain the global solution of the original problem, thus a perfectly accurate decomposition of the intensity pattern in Fig. 3(a) may not be obtained. However, the configuration of the Gaussian beam array in Fig. 3(b) is still reasonable balancing the number of beamlets (corresponds to the experimental feasibility) and accuracy of the approximate, as is shown in Fig. 3(c).

By using the same method, the parameters of the Gaussian beam array to generate the vortex beam array carrying a specified OAM can be determined. The generation of the vortex beam arrays with TC = ± 1 and TC = ± 2 are studied and the setups of the Gaussian beam arrays at the source plane are presented in Fig. 4.

#### 3.3 Generation and evolution of the vortex beam array

The Gaussian beam array at the source plane, as discussed above, can form optical vortices through propagation. Based on the optical fields at the source plane in Fig. 4, the intensity and phase distributions at the receiver plane are shown as follow.

Figure 5 shows the vortex beam arrays with different OAMs generated from the Gaussian beam array by using coherent combining technology. Specifically, annular dark hollow shaped beams are observed in the intensity pattern, while the corresponding phase singularities exist. Positive or negative value of TC determines the phase rotating direction around the phase singularity. Besides, the dark hollow region of each vortex beam becomes larger for a larger absolute value of TC. Generation of the vortex beam array at the receiver plane is demonstrated, though subtle angular misalignments exist in both intensity and phase patterns because of the information distortion in the fitting process using multiple Gaussian beams.

In a real implementation, the setup and parameters of the Gaussian beam array should be adjusted when a different distance is required. There are two ways to adjust the distance of the generated vortex beam array. One is to rearrange the position of each Gaussian beam at the source plane and adjust its waist width simultaneously, which might be challenging, while another is to adjust the phase distribution of each Gaussian beam, which could be easily done. Hence, the latter way is preferable out of consideration for the experimental feasibility.

Based on the setup of the Gaussian beam array at the source plane shown in Fig. 4 (a), the vortex beam array could be generated at different distances by modulating the phase distribution of the Gaussian beam array as is shown in Fig. 6. The waist width of each generated vortex beam is different when a different distance is required, while the relative separation (the ratio of the separation to the waist width) of each beam remains the same. Therefore, the technique could generate vortex beam arrays at different distances by phase modulation.

In Fig. 7, the evolution properties of the synthesized beam array are presented. The vortex beam array with TC = 1 is generated at the receiver plane (*z* = *L* = 2km) and Δ*L* represents the distance between the position of the observation plane and the receiver plane. It is clear that the distortion in the intensity and phase distributions occurs as the observation plane moving away from the receiver plane. Moreover, the multiple ring-shaped intensity patterns finally vanish while the multiple phase singularities concentrating and forming single phase singularity at the central position gradually. We should note that the helical phase structure can maintain for a longer distance compared with the intensity distribution. Similar phenomenon during the generation of single vortex beams based on coherent combining has been discussed by A. P. Aksenov et al. in 2015 [19]. To conclude, without phase modulation mentioned above, the available distance interval to generate a vortex beam array is larger than 0.05*L* but is finite as well, and therefore, the robustness of the novel technique is considerable.

#### 3.4 Experimental limitations

The position and parameters of each Gaussian beam at the source plane could be calculated through the reversal of Huygens Fresnel diffraction and the greedy algorithm when the expected vortex beam array is given. The number, separations and TCs of the expected vortex beams have effect on the number and positions of the required Gaussian beams, which could bring about the experimental limitations of this technique. Considering the current level of coherent combining technology, 101 optical elements could be coherent combined as is reported in [35]. Hence, the limitations of the number, separations and TCs of the generated vortex beams could be studied based on the number limitation of the required Gaussian beams at the source plane.

Figure 8(a) describes the number of Gaussian beams required to generate the four-vortex-beam arrays (TC = 1) with different separations. The separation of each vortex beam should be less than 10*σ*. Specifically, *N _{G}* increases as

*d*increases, and the tendency could be fitted with a quadratic curve:

_{V}*y*=

*x*

^{2}-1.3

*x*+ 5.9. This phenomenon is very similar with the law in coherent combination of multiple Gaussian beams, i.e. the number and energy proportion of the sidelobes at the receiver plane increase as the separation of each Gaussian beam at the source plane increases (the fill factor of the beam array decreases) [36]. On the one hand, vortex beam arrays with larger separations are more feasible and require more Gaussian beams, which could bring higher power. On the other hand, with the increase of the number of Gaussian beams, the application of coherent combining technology becomes more difficult. To conclude, the separations and power of vortex beam arrays and the number of the required Gaussian beams should be weighed in practical application.

Based on the above discussion, the expected vortex beams should be set as near as possible to study the effect of their TCs and number on the minimum number of the required Gaussian beams. Figure 8(b) describes the minimum number of Gaussian beams required to generate the four-vortex-beam arrays with different TCs. The possible range of TC is from −9 to 9. Specifically, *N _{G}* increases as the absolute value of TC increases, and the tendency could be fitted with a linear curve:

*y*= 11

*x*-1.9. Since vortex beams with larger TCs have larger dark hollow area, the optical fields calculated by the reversal of Huygens Fresnel diffraction contain more sidelobes, and as a result, the technique requires more Gaussian beams at the source plane.

As is shown in Fig. 9, the minimum number of the required Gaussian beams increases as the number of generated vortex beams increases, and the increase trend becomes slower when the number of generated vortex beams exceeds 9. Moreover, the advantages of the technique in high power applications would gradually disappear with the increase of the number of generated vortex beams (the power provided by Gaussian beams becomes more limited), unless the relative separation of each vortex beam increases.

In a nutshell, the setup of the Gaussian beam array required to generate multiple vortex beams is determined by the setup and parameters of the expected vortex beam array. The number of the vortex beams, the separation and TC of each vortex beam have effect on the number of the Gaussian beams at the source plane together, and the number of the Gaussian beams brings about the experimental limitations. Hence, the number of the vortex beams, the separation and TC of each vortex beam should be fully considered when the experimental limitations are discussed in different cases.

## 4. Discussion of application

Based on the approach introduced above to generate vortex beam arrays, the potential application of the approach can be further discussed. One of the potential applications is free-space optical communication. As is known, multiple OAM states of vortex beams can be used as carriers for multiple data streams, thus the capacity of the optical communication system can be increased. Furthermore, vortex beam arrays, which consist of multiple optical vortices, could be used for optical communication systems, and the carriers for data streams exist in different positions simultaneously.

The schematic diagram of free-space optical communication based on the vortex beam array generated by coherent combining technology is shown in Fig. 10. Different Gaussian beam arrays with phases controlled by the phase modulators are set at the source plane 1 and the source plane 2, which can form the optical vortices with TC = 1 and TC = 2, respectively. The output beams from the source plane 1 and the source plane 2 are combined by a non-polarizing beam splitter (NPBS), and then generate two vortex beam arrays through free-space propagation at the receiver plane. The capacity of the optical communication system could be increased by mode division multiplexing at different positions, while space division multiplexing might not be achieved because that all the vortex beams with the same TC generated from their corresponding source plane carry the same information. The capacity of the system can be further improved by adding NPBSs in the optical path. Although the power loss in the process of beam combination by using NPBSs might be difficult to avoid, the technique could still be useful in optical communication. Firstly, the technique could generate high power vortex beams, thus the power loss could be acceptable to some extent. Besides, the fidelity of information in optical communication is extremely important, and our technique could be beneficial for checking the accuracy of the information. Specifically, each beam of the vortex beam array generated from the corresponding encoded source plane has the same TC and carries the same information, thus the accuracy of the received information could be cross-checked by the received information at different positions. Moreover, the loss beams could be further used to check the accuracy of the received information as well.

Another potential application of our approach is spatial light structuring. The combination of multiple vortex beams with orthogonal OAMs has been studied by G. Xie et al. in 2017, and a novel intensity distribution (high localized power) is created in the near field [37]. Based on this study, a high localized power beam array can be generated at a desired distance, rather than being limited in the near field.

High localized power intensity pattern can be achieved by coherently combining multiple vortex beams with orthogonal OAMs, thus a high localized power beam array can be structured by coherently combining vortex beam arrays with orthogonal OAMs. The approach to generate vortex beam arrays with desired TCs has been introduced in Sec. 2. Following this thought, the desired optical field can be structured from the Gaussian beam array at the source plane by controlling the phase distribution. In Fig. 11, the setup of the Gaussian beam array, intensity distribution and phase distribution are determined by the reversal of Huygens Fresnel diffraction and the greedy algorithm. The phase distribution in Fig. 11(b) is the sum of the phase distributions to generate the vortex beam arrays with OAM −2 to 2. The intensity profile of the structured beam is shown in Fig. 11(c), in connect with the result in [37].

## 5. Conclusion

In this paper, a novel method to generate vortex beam arrays is proposed and its validity is demonstrated. A formula to calculate the reversal of Huygens Fresnel diffraction has been derived. The greedy algorithm is used for fitting the optical field to further generate a vortex beam array by using multiple fundamental Gaussian beams, and the parameters of the beam array at the source plane are determined. The synthesized vortex beam array with different TCs is studied in detail by simulation, and the feasibility of this new method is illustrated. In addition, the propagating properties of the synthesized beams near the receiver plane are analyzed to estimate the available range and robustness of the method. Within a considerable distance interval, the vortex beam array with controllable OAMs can be generated from Gaussian beam array by coherent beam combining technology, and the distance can be adjusted by phase modulation. Our results may have potential applications in optical communication and spatial light structuring.

## Funding

National Natural Science Foundation of China (NSFC) (No. 61405255)

## Acknowledgment

The authors are very thankful to the reviewers for their valuable comments.

## References and links

**1. **J. F. Nye and M. V. Berry, “Dislocations in wave trains,” Proc. R. Soc. Lond. A Math. Phys. Sci. **336**(1605), 165–190 (1974). [CrossRef]

**2. **L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A **45**(11), 8185–8189 (1992). [CrossRef] [PubMed]

**3. **G. Gibson, J. Courtial, M. Padgett, M. Vasnetsov, V. Pas’ko, S. Barnett, and S. Franke-Arnold, “Free-space information transfer using light beams carrying orbital angular momentum,” Opt. Express **12**(22), 5448–5456 (2004). [CrossRef] [PubMed]

**4. **J. Wang, J. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics **6**(7), 488–496 (2012). [CrossRef]

**5. **L. Paterson, M. P. MacDonald, J. Arlt, W. Sibbett, P. E. Bryant, and K. Dholakia, “Controlled rotation of optically trapped microscopic particles,” Science **292**(5518), 912–914 (2001). [CrossRef] [PubMed]

**6. **S. Bernet, A. Jesacher, S. Fürhapter, C. Maurer, and M. Ritsch-Marte, “Quantitative imaging of complex samples by spiral phase contrast microscopy,” Opt. Express **14**(9), 3792–3805 (2006). [CrossRef] [PubMed]

**7. **M. Šiler, P. Jákl, O. Brzobohatý, and P. Zemánek, “Optical forces induced behavior of a particle in a non-diffracting vortex beam,” Opt. Express **20**(22), 24304–24319 (2012). [CrossRef] [PubMed]

**8. **S. Fu, T. Wang, Z. Zhang, Y. Zhai, and C. Gao, “Selective acquisition of multiple states on hybrid Poincare sphere,” Appl. Phys. Lett. **110**(19), 191102 (2017). [CrossRef]

**9. **P. Chen, L. L. Ma, W. Duan, J. Chen, S. J. Ge, Z. H. Zhu, M. J. Tang, R. Xu, W. Gao, T. Li, W. Hu, and Y. Q. Lu, “Digitalizing self-assembled chiral superstructures for optical vortex processing,” Adv. Mater. **30**(10), 1705865 (2018). [CrossRef] [PubMed]

**10. **G. Xie, L. Li, Y. Ren, H. Huang, Y. Yan, N. Ahmed, Z. Zhao, M. P. J. Lavery, N. Ashrafi, S. Ashrafi, R. Bock, M. Tur, A. F. Molisch, and A. E. Willner, “Performance metrics and design considerations for a free-space optical orbital-angular-momentum-multiplexed communication link,” Optica **2**(4), 357–364 (2015). [CrossRef]

**11. **T. Y. Fan, “Laser beam combining for high-power, high-radiance sources,” IEEE J. Sel. Top. Quantum Electron. **11**(3), 567–577 (2005). [CrossRef]

**12. **P. Zhou, Z. Liu, X. Wang, Y. Ma, H. Ma, X. Xu, and S. Guo, “Coherent beam combining of fiber amplifiers using stochastic parallel gradient descent algorithm and its application,” IEEE J. Sel. Top. Quantum Electron. **15**(2), 248–256 (2009). [CrossRef]

**13. **Y. Ma, P. Zhou, X. Wang, H. Ma, X. Xu, L. Si, Z. Liu, and Y. Zhao, “Coherent beam combination with single frequency dithering technique,” Opt. Lett. **35**(9), 1308–1310 (2010). [CrossRef] [PubMed]

**14. **C. X. Yu, S. J. Augst, S. M. Redmond, K. C. Goldizen, D. V. Murphy, A. Sanchez, and T. Y. Fan, “Coherent combining of a 4 kW, eight-element fiber amplifier array,” Opt. Lett. **36**(14), 2686–2688 (2011). [CrossRef] [PubMed]

**15. **Z. Liu, P. Zhou, X. Xu, X. Wang, and Y. Ma, “Coherent beam combining of high power fiber lasers: progress and prospect,” Sci. Chin. Technol. **56**(7), 1597–1606 (2013). [CrossRef]

**16. **A. Flores, I. Dajani, R. Holten, T. Ehrenreich, and B. Anderson, “Multi-kilowatt diffractive coherent combining of pseudorandom-modulated fiber amplifiers,” Opt. Eng. **55**(9), 096101 (2016). [CrossRef]

**17. **Z. Liu, P. Ma, R. Su, R. Tao, Y. Ma, X. Wang, and P. Zhou, “High-power coherent beam polarization combination of fiber lasers: progress and prospect,” J. Opt. Soc. Am. B **34**(3), A7–A14 (2017). [CrossRef]

**18. **L. Wang, L. Wang, and S. Zhu, “Formation of optical vortices using coherent laser beam arrays,” Opt. Commun. **282**(6), 1088–1094 (2009). [CrossRef]

**19. **V. P. Aksenov, V. V. Dudorov, and V. V. Kolosov, “Characterization of vortex beams synthesized on the basis of a fiber laser array,” Proc. SPIE **9680**, 96802D (2015).

**20. **V. P. Aksenov, V. V. Dudorov, and V. V. Kolosov, “Statistical characteristics of common and synthesized vortex beams in a turbulent atmosphere,” Proc. SPIE **10035**, 100352P (2016). [CrossRef]

**21. **V. P. Aksenov, V. V. Dudorov, G. A. Filimonov, V. V. Kolosov, and V. Y. Venediktov, “Vortex beams with zero orbital angular momentum and non-zero topological charge,” Opt. Laser Technol. **104**, 159–163 (2018). [CrossRef]

**22. **C. Y. Hwang, K. Y. Kim, and B. Lee, “Bessel-like beam generation by superposing multiple Airy beams,” Opt. Express **19**(8), 7356–7364 (2011). [CrossRef] [PubMed]

**23. **X. Chu, Q. Sun, J. Wang, P. Lü, W. Xie, and X. Xu, “Generating a Bessel-Gaussian beam for the application in optical engineering,” Sci. Rep. **5**(1), 18665 (2016). [CrossRef] [PubMed]

**24. **X. Chu, Z. Liu, and P. Zhou, “Generation of a high-power Airy beam by coherent combining technology,” Laser Phys. Lett. **10**(12), 125102 (2013). [CrossRef]

**25. **D. Zhi, R. Tao, P. Zhou, Y. Ma, W. Wu, X. Wang, and L. Si, “Propagation of ring Airy Gaussian beams with optical vortices through anisotropic non-Kolmogorov turbulence,” Opt. Commun. **387**, 157–165 (2017). [CrossRef]

**26. **R. S. Kurti, K. Halterman, R. K. Shori, and M. J. Wardlaw, “Discrete Cylindrical Vector Beam Generation from an Array of Optical Fibers,” Opt. Express **17**(16), 13982–13988 (2009). [CrossRef] [PubMed]

**27. **Y. Yang, Y. Dong, C. Zhao, and Y. Cai, “Generation and propagation of an anomalous vortex beam,” Opt. Lett. **38**(24), 5418–5421 (2013). [CrossRef] [PubMed]

**28. **X. Wang, B. Zhu, Y. Dong, S. Wang, Z. Zhu, F. Bo, and X. Li, “Generation of equilateral-polygon-like flat-top focus by tightly focusing radially polarized beams superposed with off-axis vortex arrays,” Opt. Express **25**(22), 26844–26852 (2017). [CrossRef] [PubMed]

**29. **S. L. Lachinova and M. A. Vorontsov, “Exotic laser beam engineering with coherent fiber-array systems,” J. Opt. **15**(10), 105501 (2013). [CrossRef]

**30. **Y. Zheng, X. Wang, F. Shen, and X. Li, “Generation of dark hollow beam via coherent combination based on adaptive optics,” Opt. Express **18**(26), 26946–26958 (2010). [CrossRef] [PubMed]

**31. **P. Ma, P. Zhou, Y. Ma, X. Wang, R. Su, and Z. Liu, “Generation of azimuthally and radially polarized beams by coherent polarization beam combination,” Opt. Lett. **37**(13), 2658–2660 (2012). [CrossRef] [PubMed]

**32. **Z. Chen, T. Zeng, and J. Ding, “Reverse engineering approach to focus shaping,” Opt. Lett. **41**(9), 1929–1932 (2016). [CrossRef] [PubMed]

**33. **T. H. Cormen, C. E. Leiserson, R. L. Rivest, and C. Stein, *Introduction to algorithms* (Massachusetts Institute of Technology, 2009), Chap. 16.

**34. **J. W. Goodman, *Introduction to Fourier Optics* (McGraw-Hill, 1968), Chap. 4.

**35. ** Communications Office MIT Lincoln Laboratory, “MIT Lincoln Laboratory demonstrates novel laser at technology expo,” http://www.ll.mit.edu/news/wait-what.html (2015).

**36. **M. R. Andrew and R. W. Berdine, *Introduction to High Power Fiber Lasers* (NM: Directed Energy Professional Society, 2009), Chap. 9.

**37. **G. Xie, C. Liu, L. Li, Y. Ren, Z. Zhao, Y. Yan, N. Ahmed, Z. Wang, A. J. Willner, C. Bao, Y. Cao, P. Liao, M. Ziyadi, A. Almaiman, S. Ashrafi, M. Tur, and A. E. Willner, “Spatial light structuring using a combination of multiple orthogonal orbital angular momentum beams with complex coefficients,” Opt. Lett. **42**(5), 991–994 (2017). [CrossRef] [PubMed]