Open Access
Add to BibSonomy
Abstract
High-power vortex beams have extensive applications in optical communication, nonlinear frequency conversion, and laser processing. To overcome a single beam’s power limitation, generating vortex beams, based on a phased beam array, is an intuitive idea that requires locking each beamlet’s phase to a specific different value. Conventionally, the intensity profiles of the focal plane (far field) are used for extracting the cost functions in active phase control systems. However, as for generating vortex beams, the cost function extraction method at the focal plane suffers because the same intensity profile of the beam array could correspond to different phase distributions in near field. Thus, the accurate phase control signals are difficult to obtain. In this paper, a new concept of extracting cost functions at the non-focal-plane is firstly presented and analyzed in detail by numerical simulation. This cost function extraction method is an efficient way of generating vortex beams with different topological charges, including second-order Bessel-Gaussian beams. The new concept could provide a valuable reference and contribute to the practical implementation of generating vortex beams by coherent beam combining technology.
© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Owing to the helical wavefront and ring-shaped intensity structure, vortex beams carrying orbital angular momentum [1,2] offer advantages to the applications of free-space optical communication [3–7], super-resolution imaging [8], optical manipulation [9–13], tight focusing [14,15] and laser matter interaction [16–18]. The generation and propagation properties of vortex beams have been investigated intensively in recent years [19–24]. Especially in some special applications, such as laser ablation, material processing, nonlinear frequency conversion and satellite-to-ground communications, high power vortex beams are strongly required [25–27]. Coherent beam combining (CBC), as a widely studied technology, could break the power limitation of a single laser beam while maintaining good beam quality [28–33]. Besides, the propagation properties of beam arrays through free space, atmospheric turbulence and oceanic turbulence have been investigated extensively [34–38]. Recently, to enhance the output power, the CBC technology has been incorporated into vortex beams generation, which has been widely investigated [39–44]. The essence of the method is tailoring the intensity distribution and relative phase of each element in a phased array to construct different wavefronts and generate various structured light fields.
Based on the CBC technology, theoretical works for generating various kinds of structured light fields have been carried out [39,41,42,45–51]. These theoretical studies were all based on a premise, i.e. no dynamic phase noise in the CBC systems—in other words, the phase difference between each beamlet is constant. However, this premise is unpractical in experiments, especially operation in high power systems with serious thermal and environmental fluctuations. In experiment, Yu et al. realized the generation of second-order Bessel-Gaussian vortex beams based on a low-noise, low-power He-Ne laser array, and phase control is not considered in their work [43]. To eliminate the influence of dynamic phase noise, Zheng et al. proposed a method for generating dark hollow beams by a coherent laser array based on an adaptive optics servo fed at the focal plane [52]. In the following sections, we would illustrate that the servo fed at the focal plane for extracting cost functions suffers from the fact that the same intensity profile of the beam array could correspond to different phase distributions in near field, thus the method is difficult to generate vortex beams. To overcome this difficulty, Lachinova et al. proposed an efficient phase retrieving method in the year of 2013 [40], and it is confirmed by Aksenov et al. in 2018 based on an array of six beamlets in experiment [44]. As for this method, a feedback phase control system based on an elaborate network of beam-tail interference sensors and fiber-integrated phase shifters is constructed. The interference sensors outputs determined by the overlapping of beam-tails are sent to the feedback controller for real-time phase modulation.
In this paper, we propose a simple and robust method for eliminating the influence of phase noise and avoiding the misleading phase control in the CBC systems for generating high power vortex beams. Our method is based on the concept of extracting cost functions at the non-focal-plane, and no complex phase demodulation circuit as well as only one intensity profile detector is required in the system. By formula derivation and Monte-Carlo Simulation, we illustrate that vortex beams could not be efficiently generated from the conventional phased arrays fed at the focal plane. The phase locking for the generation of vortex beams with different topological charges (TCs) including second-order Bessel-Gaussian beams is demonstrated through numerical simulation. The concept proposed here is highly available and offers a significant practical reference on generating high power vortex beams with various TCs based on CBC technology.
2. Principle and method
Assume a coherent laser array consisted of N beamlets, which are linearly polarized fundamental Gaussian modes of the same amplitude and waist width, and truncated by circular apertures with radius R. The complex amplitude of the beam array at the source plane is given by
where A0, w0, ϕ j and (xj , yj) are the amplitude, waist width, initial phase and central position of the jth shifted Gaussian beam, respectively. Tj (x, y, 0) represents the window function of the jth circular aperture and could be expanded by complex Gaussian function as [53]where Bt and Ct account for the expansion coefficients, and t is the expansion order. Substitute Eq. (2) to Eq. (1), the optical field at the source plane is expressed asThe complex amplitude of the jth beamlet propagating through an optical system parameterized by an ABCD transfer matrix could be described by the Huygens–Fresnel diffraction integral as [54]
where (x, y) and (ξ, η) are the coordinates of the source plane and the receiver plane in the Cartesian coordinate system, respectively. λ is the optical wavelength, k = 2π /λ is the wave number, and L0 accounts for the axial optical distance between the source plane and the receiver plane. Therefore, the optical field at the receiver plane could be derived asand the intensity distribution isThe tiled aperture CBC system for generating vortex beams based on our phase locking method is schemed in Fig. 1. A linearly polarized seed laser (SL) is amplified by a pre-amplifier (PA) and is then split into N channels by a fiber splitter (FS). The laser beam of each channel is coupled to a fiber phase modulator (FPM), and is sent to cascaded fiber amplifiers (FAs) subsequently to boost the output power. After power scaling, the laser beams pass through a collimator array, which determines the position of each beamlet at the source plane, and then propagate in free space. The collimated beam array is split into two parts by a high reflective mirror (HRM): the high power part would generate a vortex beam in the far-field, while the low power part is sampled for observing the beam profile and extracting the cost function. Specifically, the low power part is sent to a focus lens (FL) via another HRM, and the focused beam array is split by a beam splitter (BS). Part of the beam array is coupled into a CCD locates at the focal plane of the FL, and the far-field beam profile could be displayed by the labtop computer connected to the CCD. The other part of the beam array is received by a camera locates at the non-focal-plane for extracting the cost function. The signal collected by the camera is observed by an oscilloscope and is sent to a FPGA controller. The FPGA controller produces output control voltages, which are applied to the FPMs for phase locking. In conventional tiled aperture CBC systems, the stochastic parallel gradient descent (SPGD) algorithm is widely implemented, which has been investigated in detail in previous works [30,31,55]. Throughout the phase locking process, the SPGD algorithm is performed in the FPGA controller.
Fig. 1 The sketch for the generation of a vortex beam based on coherent combining technology. (SL: seed laser; PA: pre-amplifier; FS: fiber splitter; FPM: fiber phase modulator; FAs: fiber amplifiers; HRM: high reflective mirror; FL: focus lens; BS: beam splitter.)
The transfer matrix of the optical lens system for coherent beam combining could be simplified as
where f represents the focal length of the FL and l is the distance between the collimator array and the FL.The essence of our method is determining the cost function of the SPGD algorithm based on the intensity profile at the non-focal-plane (where L≠f), and the necessity for extracting the cost function at the non-focal-plane rather than focal plane would be discussed in detail in the next section. Substitute Eq. (7) to Eq. (5) and Eq. (6), the intensity distribution of the non-focal-plane (camera) could be obtained as INFP (ξ,η, z = ZL), where ZL = l + L, and the cost function would be defined subsequently.
In CBC systems, the power-in-the-bucket (PIB), which describes the energy encircled in an on-axis circular area at the receiver plane, has been widely used as the cost function [30,56–58]. Draw from the concept, in CBC systems for generating vortex beams, the “bucket” should be redefined. Specifically, the generalized “bucket” for calculating the integral of intensity should be defined as an area contains points of relative high power density from the intensity profile, rather than an on-axis circle. Here, the generalized “bucket” could be defined by a proportion parameter κ (0<κ <1) as
The “bucket” with κ represents an area where the proportion of the power density and the peak power density is over κ, and accordingly, the cost function JPIGB could be defined as the normalized power-in-the generalized “bucket” (PIGB): the proportion of the PIGB at the condition of the instantaneous intensity profile and the expected intensity profile (without dynamic phase noise), i.e.The control voltages applied to the FPMs update simultaneously to adjust the phase of each beamlet, and during the iteration, the cost function JPIGB evolves to its extremum step by step, thus the phase locking for generating vortex beams could be achieved.
3. Numerical simulation results and discussion
3.1 Necessity of phase locking at non-focal-plane
Generally, the signal processed by the FPGA controller is collected at the focal plane, and the cost function is extracted from the far-field intensity profile of the beam array. However, for different phase coherent combining to generate vortex beams, it is necessary to illustrate that whether there is one-to-one correspondence between the intensity profile at the focal plane and the relative phase of each beamlet. If different relative phase distributions at the source plane have the same intensity profile at the focal plane, the relative phase of each beamlet would not always convergent to a certain value after iteration, and the phase locking system would be instable in experiments. Substitute Eq. (7) to Eq. (5) and Eq. (6), the intensity distributions of the focal plane (where z = Zf = l + f) when l << f could be obtained as
withWithout loss of generality, a tiled aperture CBC system for generating vortex beams with TC = ± 1 is taken as an example. The N beamlets at the source plane are located symmetrically to construct a radial laser array as shown in Fig. 2(a), and the piston phase of each beamlet should be set to approximate the vortex phase distribution. The parameters of the beam arrays for generating the vortex beams with TC = ± 1 are N = 6, ϕm = ± π (m-1)/3, xm = ρ cos(π m/3), ym = ρ sin(π m/3), respectively, where ρ is the radius of the ring and m denotes the order of the beamlet. Substitute the phase and position of each beamlet to Eq. (10) and Eq. (11), we could manifest that the expressions of the intensity profile at the focal plane are the same for generating the vortex beams with TC = 1 and TC = −1. Hence, the intensity profile at the focal plane could not be used for extracting cost functions.
In order to find a reasonable plane for extracting cost functions, we suppose that a plane behind the focal plane could meet our requirements. Here, we use the Monte-Carlo Simulation to demonstrate that this non-focal-plane is proper for extracting cost functions. The cost function is defined as the correlation coefficient of the expected (calculated by the parameters mentioned above, as shown in Figs. 2(b) and 2(c)) and the instantaneous intensity profiles at the receiver plane. In the Monte-Carlo Simulation, the SPGD algorithms provide with a method to find the relative phase solutions for a certain intensity profile at the receiver plane by iterations.
The results of the Monte-Carlo Simulation (500 times) to find the relative phase solutions for the intensity profiles of the generated vortex beam at the focal plane and the same beam at the non-focal-plane are shown in Figs. 3(a)-3(c) and Figs. 3(d)-3(f), respectively. The wavelength, waist width, and aperture diameter of the input beamlets are 1.06μm, 10.24mm and 23mm, respectively. The focal length is 20m, and the radius of the ring-shaped array is 25mm. The non-focal-plane is 0.6m behind the focal plane. The iteration curves for the cost function are depicted in Fig. 3 (a) and Fig. 3(d). It could be noticed that the correlation coefficient at the focal plane could converge to 1 for every iteration, while local optimums occur in 10 times at the non-focal-plane. Figures 3(b) and 3(e) present the convergence curves for the relative piston phase of each beamlet (the red solid, yellow dotted, cyan dotted, green dashed, pink dashed and blue dashed curves denote the relative phase of beamlet from No. 1 to No. 6), and the cases for trapping in local optimums in Fig. 3(d) are excluded to avoid bringing about extraneous solutions. The results show that the relative phase of each beamlet could converge a certain value for the non-focal-plane case, while the relative phase solutions for the focal plane case are not unique. To be more specific, different solutions could be clustered and visualized by dimensionality reduction, as shown in Fig. 3(c) and Fig. 3(f). The extraneous solutions, which account for 2% of all relative phase solutions, are separated from the solutions of the non-focal-plane intensity profile in Fig. 3(f), while no extraneous solution exists for the focal plane case. However, nearly 50% of relative phase solutions could not meet the requirements for generating vortex beams with TC = 1. On the contrary, beamlets of these phase solutions could generate vortex beams with TC = −1. The results also demonstrate that the intensity profile at the focal plane are the same for generating the vortex beams with TC = 1 and TC = −1, which is consistent with the conclusion of formula derivation.
Fig. 3 Monte-Carlo Simulation for determining the relative phase solutions. Convergence curves for (a) the cost function and (b) the relative phase of each beamlet, and (c) clustering of the relative phase solutions for the focal plane case. Convergence curves for (d) the cost function and (e) the relative phase of each beamlet, and (f) clustering of the relative phase solutions for the non-focal-plane case.
In a nutshell, the intensity profile at the focal plane could be the same for different clusters of relative phase solutions, and there is a 50% probability for generating a vortex beam with an unexpected TC, thus the cost function should by no means be obtained at the focal plane. Furthermore, difficulties in distinguishing conjugated relative phase solutions can be overcome at the non-focal-plane, from where the cost function could be extracted.
3.2 Cost function of non-focal-plane phase locking method
From the analysis above, the intensity profile at the non-focal-plane could more accurately reflect the phase distribution of the beam array at the source plane, and therefore, the cost function is supposed to be obtained at the non-focal-plane. Figure 4 shows the intensity profiles and phase distributions of the generated vortex beams with TC = 1 and TC = −1 at a non-focal-plane 0.6m behind the focal plane. Phase singularities, screw type wavefronts and hollow intensity profiles could be observed at the non-focal-plane. Different from the intensity profile of the focal plane, which has been discussed in previous studies [39,41], the energy would diffuse to the sidelobes from the inner hollow beam on axis, and the inner annular hollow beam would evolve into a hexagonal hollow beam when the receiver plane is moved from the focal plane. The positions of sidelobes as well as the inner hexagonal hollow beams are different with different rotation directions of the wavefronts, thus the vortex beams with TC = 1 and TC = −1 could be distinguished. To make the most of such differences in sidelobes, the receiver plane to obtain the intensity profiles for extracting cost functions is set at the non-focal-plane, where the peak intensity of each sidelobe is comparable with the peak intensity of the inner hollow beam.
Fig. 4 Intensity profiles of the generated vortex beams with (a) TC = 1 and (c) TC = −1 at the non-focal-plane. Phase distributions of the generated vortex beams with (b) TC = 1 and (d) TC = −1 at the non-focal-plane.
The cost function has been defined as the proportion of the normalized PIGB at the non-focal-plane, and the size of the generalized “bucket” could influence the efficiency of phase locking. Hence, we should determine an appropriate κ by fully considering the convergence speed and convergence accuracy. For generating vortex beams with TC = 1, the generalized “buckets” with different κ are depicted in Figs. 5(a)-5(c).
Fig. 5 Generalized “buckets” for generating vortex beams (TC = 1) with (a) κ = 0.5, (b) κ = 0.7 and (c) κ = 0.9.
Figure 6 describes the dependence of the convergence speed on the shape of the generalized “bucket”. The red solid curves in Figs. 6(a)-6(c) account for the average convergence curve over 1000 times simulation for κ = 0.5, κ = 0.7 and κ = 0.9, respectively. Local optimum occurs when κ = 0.5, as shown in Figs. 6(a) and 6(d). With the increase of κ, the area of the “bucket” becomes smaller while the acceleration of convergence could be observed. The extraneous relative phase solutions occur in section 3.1 could be avoided, as shown in Figs. 6(e) and 6(f). In Figs. 6(b) and 6(c), the cost function could always be convergent within 150 steps, while the advantages in convergence speed of κ = 0.9 over κ = 0.7 is not obvious.
Fig. 6 Simulation (1000 times) for determining the dependences of the convergence speed on the shape of the generalized “bucket”. Convergence curves for cost function with (a) κ = 0.5 (b) κ = 0.7, and (c) κ = 0.9. Convergence curves for the relative phase of each beamlet when (d) κ = 0.5 (e) κ = 0.7, and (f) κ = 0.9.
To further determine a proper κ for phase locking, the convergence accuracy should be taken into consideration. Figure 7 presents the average intensity profiles and phase distributions of the vortex beams generated from ideal coherent beam array, phase locked array with κ = 0.7 and phase locked array with κ = 0.9 at the focal plane. Similar helical phase structures with TC = 1 could be formed as shown in Figs. 7(d)-7(f). However, in Figs. 7(b) and 7(c), the differences of the intensity profiles could be observed. The generalized “bucket” with κ = 0.9 is thinner in shape, and consequently, more concentrated in intensity in local area (close to the y-axis). The correlation coefficient of the vortex beams in Figs. 7(a) and 7(c) is 0.986, which is lower than 0.999, as the correlation coefficient of the vortex beams in Figs. 7(a) and 7(b). Therefore, the generalized “bucket” with κ = 0.7 could ensure that the generation of vortex beams with TC = 1 is more accurate.
Fig. 7 Simulation (500 times) for determining the dependences of the convergence accuracy on the shape of the generalized “bucket”. (a), (b) and (c) represent the average intensity profiles of the vortex beams generated from ideal coherent beam array, phase locked array with κ = 0.7 and phase locked array with κ = 0.9, respectively. (d), (e) and (f) are the average phase distributions of the vortex beams generated from ideal coherent beam array, phase locked array with κ = 0.7 and phase locked array with κ = 0.9, respectively.
In conclusion, the feasibility of the redefined cost function is demonstrated, and the proportion parameter κ to determine the “bucket” would be set to 0.7 when generating vortex beams with TC = ± 1 in order to achieve relatively high convergence speed and accuracy. Furthermore, for generating vortex beams with other TCs, a tradeoff should be performed to balance the convergence speed and accuracy when selecting a proper κ.
3.3 Generation of various vortex beams by non-focal-plane phase locking method
From the above section, it is clear that the non-focal-plane phase locking with the defined cost function based on the generalized “bucket” has potential in generating vortex beams with helical wavefronts. To confirm that the method could be used for generating vortex beams with different TCs, numerical simulation is performed. Figures 8(a)-8(c) show the intensity and phase distributions of the beam array at the source plane to generate the vortex beams with TC = −1, TC = 2 and TC = 3, and the radius of the arrays is 25mm, 31.25mm and 50mm, respectively.
Fig. 8 Intensity (upper) and phase (below) distributions of the beam arrays at the source plane to generate the vortex beams with (a) TC = −1, (b) TC = 2 and (c) TC = 3.
By using the method at the condition of κ = 0.7, κ = 0.7 and κ = 0.8, vortex beams with TC = −1, TC = 2 and TC = 3 could be obtained as shown in Fig. 9. The distances between the non-focal-plane (behind the focal plane) and the focal plane are 0.6m, 0.48m and 0.3m, respectively. In Fig. 9, from top to bottom, the TCs of the generated vortex beams are in turn −1, 2 and 3. The first column depicts the generalized “buckets”, and the second to the fourth correspond to the convergence curves of the cost function, the average intensity profiles and phase distributions of the generated vortex beams at the focal plane.
Fig. 9 Generation of vortex beams based on non-focal-plane CBC and SPGD. From top to bottom, the TCs are in turn −1, 2 and 3, respectively. From left to right, they are, in turn, the generalized “buckets”, the convergence curves of the cost function, the average intensity profiles and phase distributions.
Noted that a second-order Bessel-Gaussian (BG) beam, as a special case of vortex beam, could also be generated based on a radial Gaussian beam array [47]. The optical field of the ideal second-order BG beam at the source plane is given by
where J2 represents the second-order Bessel function of the first kind, r0 and θ0 are the radial and azimuthal coordinates of the source plane in the polar coordinate system, respectively. At the condition of b = 62.5mm and a = 48m−1, the parameters of the corresponding radial beam array are given as N = 16, ρ = 62.5mm. Figures 10(a) and 10(b) present the setup of the radial beam array at the source plane.Fig. 10 Generation of second-order BG beams based on non-focal-plane CBC and SPGD. (a) Intensity profile and (b) phase distribution of the beam array at the source plane. (c) Generalized “bucket” with κ = 0.7. (d) Convergence curves of the cost function. (e) Intensity profile and (f) phase distribution of the generated BG beams. (g) Intensity profile and (h) phase distribution of ideal second-order BG beams.
As is shown in Fig. 10, the phasing locking for the generation of second-order BG beams is achieved by our method. The non-focal-plane is 0.3m behind the focal plane and the generalized “bucket” is depicted in Fig. 10(c) with κ = 0.7. As a result, the cost function could always be convergent as shown in Fig. 10(d). Compare Figs. 10(e) and 10(f) with 10(g) and 10(h), the phase structures of the generated BG beams and the ideal second-order BG beams are similar, and the correlation coefficient of the intensity profiles is up to 0.964.
4. Conclusion
In this paper, a new concept of extracting cost functions at the non-focal-plane is proposed and implemented to the tiled aperture CBC systems for generating vortex beams. We clarify the difficulties of feeding the phased beam array at the focal plane and illustrate the feasibility of extracting cost functions at the non-focal-plane by formula derivation and Monte-Carlo Simulation. We determine the cost function based on the definition of PIGB, and then demonstrate the phase locking for generating vortex beams with different TCs as well as second-order BG beams. This concept offers an opportunity for generating high power vortex beams based on CBC technology from theory to experiment and has potential in guiding the generation of various structured light fields, which deserves further research.
Funding
National Natural Science Foundations of China (NSFC) (61705264 and No. 61705265).
Acknowledgment
The authors are very thankful to the reviewers for their valuable comments.
References
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. Y. Yan, G. Xie, M. P. Lavery, H. Huang, N. Ahmed, C. Bao, Y. Ren, Y. Cao, L. Li, Z. Zhao, A. F. Molisch, M. Tur, M. J. Padgett, and A. E. Willner, “High-capacity millimetre-wave communications with orbital angular momentum multiplexing,” Nat. Commun. 5(1), 4876 (2014). [CrossRef] [PubMed]
6. G. Gbur and R. K. Tyson, “Vortex beam propagation through atmospheric turbulence and topological charge conservation,” J. Opt. Soc. Am. A 25(1), 225–230 (2008). [CrossRef] [PubMed]
7. V. A. Soifer, O. Korotkova, S. N. Khonina, and E. A. Shchepakina, “Vortex beams in turbulent media: review,” Comput. Opt. 40(5), 605–624 (2016). [CrossRef]
8. 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]
9. 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]
10. M. Dienerowitz, M. Mazilu, P. J. Reece, T. F. Krauss, and K. Dholakia, “Optical vortex trap for resonant confinement of metal nanoparticles,” Opt. Express 16(7), 4991–4999 (2008). [CrossRef] [PubMed]
11. 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]
12. S. N. Khonina, R. V. Skidanov, V. V. Kotlyar, V. A. Soifer, and J. Turunen, “DOE-generated laser beams with given orbital angular moment: application for micromanipulation,” Proc. SPIE Int. Soc. Opt. Eng. 5962, 59622W (2005). [CrossRef]
13. N. Chattrapiban, E. A. Rogers, I. V. Arakelyan, R. Roy, and W. T. Hill III, “Laser beams with embedded vortices: tools for atom optics,” J. Opt. Soc. Am. B 23(1), 94–103 (2006). [CrossRef]
14. P. Török and P. Munro, “The use of Gauss-Laguerre vector beams in STED microscopy,” Opt. Express 12(15), 3605–3617 (2004). [CrossRef] [PubMed]
15. S. N. Khonina, “Simple phase optical elements for narrowing of a focal spot in high-numerical-aperture conditions,” Opt. Eng. 52(9), 091711 (2013). [CrossRef]
16. K. Toyoda, K. Miyamoto, N. Aoki, R. Morita, and T. Omatsu, “Using optical vortex to control the chirality of twisted metal nanostructures,” Nano Lett. 12(7), 3645–3649 (2012). [CrossRef] [PubMed]
17. J. J. Nivas, H. Shutong, K. K. Anoop, A. Rubano, R. Fittipaldi, A. Vecchione, D. Paparo, L. Marrucci, R. Bruzzese, and S. Amoruso, “Laser ablation of silicon induced by a femtosecond optical vortex beam,” Opt. Lett. 40(20), 4611–4614 (2015). [CrossRef] [PubMed]
18. S. Syubaev, A. Zhizhchenko, A. Kuchmizhak, A. Porfirev, E. Pustovalov, O. Vitrik, Y. Kulchin, S. Khonina, and S. Kudryashov, “Direct laser printing of chiral plasmonic nanojets by vortex beams,” Opt. Express 25(9), 10214–10223 (2017). [CrossRef] [PubMed]
19. A. E. Willner, H. Huang, Y. Yan, Y. Ren, N. Ahmed, G. Xie, C. Bao, L. Li, Y. Cao, Z. Zhao, J. Wang, M. P. J. Lavery, M. Tur, S. Ramachandran, A. F. Molisch, N. Ashrafi, and S. Ashrafi, “Optical communications using orbital angular momentum beams,” Adv. Opt. Photonics 7(1), 66–106 (2015). [CrossRef]
20. H. Rubinsztein-Dunlop, A. Forbes, M. V. Berry, M. R. Dennis, D. L. Andrews, M. Mansuripur, C. Denz, C. Alpmann, P. Banzer, T. Bauer, E. Karimi, L. Marrucci, M. Padgett, M. Ritsch-Marte, N. M. Litchinitser, N. P. Bigelow, C. Rosales-Guzmán, A. Belmonte, J. P. Torres, T. W. Neely, M. Baker, R. Gordon, A. B. Stilgoe, J. Romero, A. G. White, R. Fickler, A. E. Willner, G. Xie, B. McMorran, and A. M. Weiner, “Roadmap on structured light,” J. Opt. 19(1), 013001 (2017). [CrossRef]
21. 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]
22. X. Li, H. Ma, H. Zhang, Y. Tai, H. Li, M. Tang, J. Wang, J. Tang, and Y. Cai, “Close-packed optical vortex lattices with controllable structures,” Opt. Express 26(18), 22965–22975 (2018). [CrossRef] [PubMed]
23. Y. Shen, Z. Wan, X. Fu, Q. Liu, and M. Gong, “Vortex lattices with transverse-mode-locking states switching in a large-aperture off-axis-pumped solid-state laser,” J. Opt. Soc. Am. B 35(12), 2940–2944 (2018). [CrossRef]
24. Y. Shen, X. Fu, and M. Gong, “Truncated triangular diffraction lattices and orbital-angular-momentum detection of vortex SU(2) geometric modes,” Opt. Express 26(20), 25545–25557 (2018). [CrossRef] [PubMed]
25. J. Hamazaki, R. Morita, K. Chujo, Y. Kobayashi, S. Tanda, and T. Omatsu, “Optical-vortex laser ablation,” Opt. Express 18(3), 2144–2151 (2010). [CrossRef] [PubMed]
26. A. Chaitanya, A. Aadhi, M. V. Jabir, and G. K. Samanta, “Frequency-doubling characteristics of high-power, ultrafast vortex beams,” Opt. Lett. 40(11), 2614–2617 (2015). [CrossRef] [PubMed]
27. 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]
28. T. Y. Fan, “Laser beam combining for high-power, high-radiance sources,” IEEE J. Sel. Top. Quantum Electron. 11(3), 567–577 (2005). [CrossRef]
29. G. D. Goodno, H. Komine, S. J. McNaught, S. B. Weiss, S. Redmond, W. Long, R. Simpson, E. C. Cheung, D. Howland, P. Epp, M. Weber, M. McClellan, J. Sollee, and H. Injeyan, “Coherent combination of high-power, zigzag slab lasers,” Opt. Lett. 31(9), 1247–1249 (2006). [CrossRef] [PubMed]
30. 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]
31. 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]
32. 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]
33. 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]
34. B. Lü and H. Ma, “Beam propagation properties of radial laser arrays,” J. Opt. Soc. Am. A 17(11), 2005–2009 (2000). [CrossRef] [PubMed]
35. Y. Cai, Y. Chen, H. T. Eyyuboğlu, and Y. Baykal, “Propagation of laser array beams in a turbulent atmosphere,” Appl. Phys. B 88(3), 467–475 (2007). [CrossRef]
36. X. Du and D. Zhao, “Statistical properties of correlated radial stochastic electromagnetic array beams on propagation,” Opt. Commun. 282(10), 1993–1997 (2009). [CrossRef]
37. X. Ji, H. T. Eyyuboğlu, and Y. Baykal, “Influence of turbulence on the effective radius of curvature of radial Gaussian array beams,” Opt. Express 18(7), 6922–6928 (2010). [CrossRef] [PubMed]
38. L. Lu, Z. Wang, J. Zhang, P. Zhang, C. Qiao, C. Fan, and X. Ji, “Average intensity of M × N Gaussian array beams in oceanic turbulence,” Appl. Opt. 54(25), 7500–7507 (2015). [CrossRef] [PubMed]
39. L. Wang, L. Wang, and S. Zhu, “Formation of optical vortices using coherent laser beam arrays,” Opt. Commun. 282(6), 1088–1094 (2009). [CrossRef]
40. S. L. Lachinova and M. A. Vorontsov, “Exotic laser beam engineering with coherent fiber-array systems,” J. Opt. 15(10), 105501 (2013). [CrossRef]
41. V. P. Aksenov, V. V. Dudorov, and V. V. Kolosov, “Properties of vortex beams formed by an array of fibre lasers and their propagation in a turbulent atmosphere,” Quant. Elect. 46(8), 726–732 (2016). [CrossRef]
42. 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]
43. T. Yu, H. Xia, Z. Fan, W. Xie, P. Zhang, J. Liu, and X. Chen, “Generation of Bessel-Gaussian vortex beam by combining technology,” Wuli Xuebao 67(13), 134203 (2018).
44. V. P. Aksenov, V. V. Dudorov, V. V. Kolosov, M. E. Levitsky, T. D. Petukhov, and A. P. Rostov, “Generation of vortex and partially coherent laser beams based on fiber array coherent combining,” Proc. SPIE 10787, 107870M (2018).
45. 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]
46. 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]
47. 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]
48. 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]
49. 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]
50. 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]
51. T. Hou, D. Zhi, R. Tao, Y. Ma, P. Zhou, and Z. Liu, “Spatially-distributed orbital angular momentum beam array generation based on greedy algorithms and coherent combining technology,” Opt. Express 26(12), 14945–14958 (2018). [CrossRef] [PubMed]
52. 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]
53. Y. Zhang, “Nonparaxial propagation analysis of elliptical Gaussian beams diffracted by a circular aperture,” Opt. Commun. 248(4–6), 317–326 (2005). [CrossRef]
54. S. A. Collins Jr., “Lens-system diffraction integral written in terms of matrix optics,” J. Opt. Soc. Am. 60(9), 1168–1177 (1970). [CrossRef]
55. R. Su, P. Zhou, X. Wang, H. Zhang, and X. Xu, “Active coherent beam combining of a five-element, 800 W nanosecond fiber amplifier array,” Opt. Lett. 37(19), 3978–3980 (2012). [CrossRef] [PubMed]
56. M. A. Vorontsov and S. L. Lachinova, “Laser beam projection with adaptive array of fiber collimators. I. Basic considerations for analysis,” J. Opt. Soc. Am. A 25(8), 1949–1959 (2008). [CrossRef] [PubMed]
57. S. L. Lachinova and M. A. Vorontsov, “Laser beam projection with adaptive array of fiber collimators. II. Analysis of atmospheric compensation efficiency,” J. Opt. Soc. Am. A 25(8), 1960–1973 (2008). [CrossRef] [PubMed]
58. T. Weyrauch, M. A. Vorontsov, G. W. Carhart, L. A. Beresnev, A. P. Rostov, E. E. Polnau, and J. J. Liu, “Experimental demonstration of coherent beam combining over a 7 km propagation path,” Opt. Lett. 36(22), 4455–4457 (2011). [CrossRef] [PubMed]
