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Abstract
Optical vortex arrays (OVAs) have drawn widespread attention owing to their multiple optical vortices and higher dimensions. However, existing OVAs have not yet been utilized to exploit the synergy effect as an entire system, particularly for manipulating multiple particles. Thus, the functionality of OVA should be explored to respond to application requirements. Hence, this study proposes a functional OVA, called cycloid OVA (COVA), based on a combination of cycloid and phase-shift techniques. By modifying the cycloid equation, multiple structural parameters are designed to modulate the structure of the COVAs. Subsequently, versatile and functional COVAs are experimentally generated and modulated. In particular, COVA executes local dynamic modulation, whereas the entire structure remains unchanged. Further, the optical gears are first designed using two COVAs, which exhibit potential for transferring multiple particles. Essentially, OVA is endowed the characteristics and capacity of the cycloid when they meet. This work provides an alternative scheme to generate OVAs, which will open up advanced applications for the complex manipulation, arrangement and transfer of multiple particles.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
In 1992, Allen discovered the orbital angular momentum (OAM) of photon [1]. Since then, the optical vortex (OV) carrying OAM has garnered considerable attention owing to its diverse applications [2,3], such as in optical tweezers [4] and spanners [5], spatial mode multiplexing in optical communications [6], optical sensing [7], metrology [8], and high-dimensional quantum information protocols [9]. Meanwhile, optical vortex array (OVA) with multiple OVs has become a focal point in the structured optical field, thus satisfying the demands for multi-channel optical communications [10–12] and multiple-particle manipulations [13–17].
Typically, several methods are used for OVA generation; these include Dammann grating [18–20], multi-beam interference [21], micro-structure materials [22,23], and spatial light modulator (SLM) [24]. Moreover, checkerboard methods are always employed to improve the image quality [25,26]. The spatial structure of the OVA is typically square, which is successfully used in optical communications based on OAM and tremendously enhances the information capacity [10–12]. Numerous studies have focused on the generation and, structural modulation of OVA, and its applications in particle manipulation. Successively, researchers have developed different OVAs with versatile structures, such as hexagonal [27], triangle [28], close-packed [29], elliptic [30], “Bear” [31], and arbitrary curvilinear OVAs [32,33]. However, these OVAs were generated out of curiosity and creativeness, rather than corresponding to the requirements of the applications.
By contrast, for multiple-particle manipulation [14–17], the OVA function is accumulated by individual OVs in the array. The advantages of the synergistic effect within the individual OVs have not been exploited, and the manipulation capability is limited in the individual OVs in OVA. Consequently, multiple particle transfers along complex paths have not been performed using the aforementioned OVAs. Thus, by taking advantage of the entire function of the array, a novel OVA with functional spatial structures must be constructed to facilitate the advanced applications of OVA.
2. Generation of cycloid optical vortex arrays
To address these challenges, particularly for multiple-particle manipulation, consider the gears in mechanics and the originated cycloid. When OVA meet a cycloid, it empowers the functions and capabilities of the cycloid, thus providing an alternative scheme for constructing the OVA according to different application scenarios. Here, a novel OVA, called cycloid OVA (COVA), was proposed by combining an OVA and cycloid. The modulation properties of the COVAs were experimentally studied. Moreover, the dynamic adjustment of the COVA was demonstrated. Furthermore, to take advantage of the cycloid, an optical gear was designed using multiple COVAs.
Cycloid are divided into: epicycloids and hypocycloids. Based on the coordinate localization technique, individual OVs were arranged evenly on the cycloid trajectories. A perfect optical vortex (POV), which can be obtained by the Fourier transform of the Bessel beam, was selected to ensure the same of the element OVs. In cylindrical coordinates (ρ, φ, z), an ideal Bessel beam is represented as:
where l is the azimuthal index representing the topological charge (TC); Jl is the lth order first kind Bessel function; kr and kz are wave vector components along the radial and axial directions, respectively. If the relationship between the wave-numbers satisfies k = (kr2 + kz2)1/2 = 2π/λ, where λ is the wavelength, then, the Fourier transform is conducted via a convex lens with focal length f. At the Fourier plane, the complex amplitude of the POV is expressed as:According to the shift theorem in the Fourier transform, OVs are precisely positioned on the cycloid trajectory. The complex amplitude of OVA at the SLM plane is expressed as:
3. Results and discussion
The experimental setup herein adopted the same equipment as that in our previous work [30]. The phase mask was generated via multiplication of the COVA phase term obtained from Eq. (3) with the phase term of the blazed grating, which is written into the SLM. According to the properties of the cycloid, COVA can be divided into hypocycloid OVA (HOVA) and epicycloid OVA (EOVA). For a COVA with the same number of OVs, the structures of the COVAs were flexibly changed from HOVA to EOVA by via adjusting the structured parameters, Q, as shown in Fig. 1. The COVA possessed 12 OVs, which were evenly distributed on the cycloid. Figures 1 (a1)–(a3) illustrate the change process of HOVA from tricuspid to circular by decreasing parameter b.
Fig. 1. COVA transformation from HOVA to EOVA. (a1)–(a3) Intensity patterns of HOVA with change of b. (b1)–(b3) Interferograms between the HOVA and a plane wave. (c1)–(c3) Intensity patterns of EOVA. (d1)–(d3) Interferograms between the EOVA and a plane wave. In this case, Q = (1.2, 2.5, 2.5, b, c). Insets are the magnified images of the encircled patterns, 2.5×. For simplicity, the TC of each OV in COVA is set as a unit, l = 1; similar hereinafter.
Evidently, at b = 0.8, the tricuspid HOVA became an equilateral triangle and at b = 0.3 it became a Reuleaux triangle HOVA. The OV on a vertex of the Reuleaux triangle HOVA, the same length as the other OVs on its opposite arc in all directions, as shown by the orange dashed lines in Fig. 1 (a3). When the Reuleaux HOVA rotated, the trajectory was a square, which is suitable for the variable trajectory manipulation of particles. The HOVA became a circle when b = 0 [Fig. 1 (c1)], correspondingly, the parameter c reduced from 2 to 0. By contrast, EOVA was freely transformed when the sign was negative, as in Eq. (5), as illustrated in Figs. 1 (c2) and (c3). During the entire process, the number of OVs in the COVA and the peak number of the COVA were unchanged. Given a smaller interval of b, the COVA can transform more versatile structures of HOVA and EOVA as the researcher’s desires. The interferograms verify the existence of OVs, and the number of forks determines the magnitude of the TC, l = 1, as shown in the second and fourth rows in Fig. 1.
To study the function of the stretch factors a1 and a2, EOVAs were generated along the cardioids, as shown in Fig. 2. During the modulation, the factor a1 was changed from 2.5 to 1.0 at intervals of 0.5, while the other structured parameters remained unchanged. Based on the intensity patterns, the cardioid EOVA was compressed along the horizontal direction when a1 decreased. By contrast, cardioid EOVA was stretched. Similarly, the structure of the COVA could be compressed or stretched along the vertical direction by adjusting the parameter a2.
Fig. 2. Horizontal stretch of the cardioid COVA. (a1)–(a4) Intensity patterns with different stretching factors. (b1)–(b4) Experimental reconstructed phase patterns, white circles represent the positions of OVs. In this case, Q = (1.2, a1, 2.5, 1.2, 2).
Note that the degree of stretch is limited by the size of the cycloid, size and number of OVs in the COVA. If this limitation is exceeded, some OVs superpose and interfere. The corresponding phase patterns were reconstructed by the interferograms between the EOVA and plane wave [34], to reveal the position of the OV and structure of the COVA. The analysis results revealed that, the intensity patterns of the COVA were in good agreement with those of their phase distributions.
Moreover, for a COVA with a specific structure, several modulating factors still exist. In addition to the structure and distribution, the size of each OV is crucial for matching trapped particles. In this case, the size of each OV was modulated by adjusting the cone angle of axicon α [Eq. (3)]. As shown in Figs. 3 (a1)–(a4), a nephroid COVA was generated, and the radii of the OVs were modulated. The change in the radii of the 12 OVs is plotted in Fig. 3 (c), where the radius is taken as the average of ten measurements of each OV along different directions. For the same nephroid COVA, a small fluctuation within the radii of the 12 OVs was observed owing to stray light. Furthermore, the radius of the OV was found to be directly proportional to the cone angle α by fitting the experimental data.
Fig. 3. Nephroid COVAs with different radii and spacing distances of OVs. (a1)–(a4) Intensity patterns of COVAs with different radii of OVs arranged on the same nephroid. (b1)–(b4) Spacing modulation of (a1)–(a4) via adjusting the expansion factor, D = 1.2, 1.3, 1.4 and 1.5. (c) Radii of the OVs in nephroid COVA VS the cone angle and the number of OVs. (d) Change of the spacing distance (marked as I, E, F, G, and K) between OVs in the COVA.
However, interference will be occurred between two adjacent OVs with the greater radii on the unchanged trajectory of the cycloid. To obtain an OV with a greater radius, the entire structures of the COVAs in Figs. 3 (a1)–(a4) were expanded via adjusting the expansion factor, D, as shown in Figs. 3 (b1)–(b4). To quantitatively characterize the expansion, five principal segments of the nephroid COVA, marked as I, E, F, G, and K, were determined and the results are illustrated in Fig. 3 (d). Owing to the symmetry of the nephroid COVA, same values were observed between E and F, and between G and K; that E = F and G = K in each pattern of Figs. 3 (b1)–(b5). Furthermore, the distances of the principal segments increased linearly with D, as shown in Fig. 3 (d). Simultaneously, the radii of the OVs were modulated in the same manner as in the first row of Fig. 3. In this case, the combination modulations of the element (OV radius) and the entire structure of the COVA were executed. These properties of COVA provide additional flexibility for some advanced applications, such as multiple particle trapping and even expansion of their distances.
When multiple particles are manipulated, several of them always need to move to form a specific arrangement [16]. Figure 4 (Multimedia view) illustrates the capacity of local dynamic modulation of the astroid COVA while the entire structure remains unchanged. In the astroid COVA, four OVs were fixed during the dynamic modulation process, which are marked as 1, 2, 3, and 4. The while other four OVs (d1–d4) rotated on the edge curves of the astroid cycloid with the same angular velocity. Dynamic modulation was realized via adjusting the orientation angular matrix of the OVs in the array, as follows.
where d1 = π/4 – β, d2 = 3π/4 – β, d3 = 5π/4 – β and d4 = 7π/4 – $\beta,\beta \in[0, \pi]$. The four OVs anticlockwise slid on the astroid cycloid for a quarter of the period, as shown in Fig. 4. Evidently from Visualization 1, explicit interference occurred when the two OVs were partially superposed. However, the OVs were still doughnut structures when two OVs were completely superposed [Fig. 4 (a3)]. This interference did not seem to have happened. Is it true? Absolutely not. In inherent concepts, there are several bright fringes when the interference occurs between OV beam and the other beam, as shown in Fig. 1 and the results in Ref. [29]. Differently, according to the theoretical and intensity analyses, each doughnut in Fig. 4 (a3) corresponds to an entire bright fringe of the enhanced interference, i. e., there is only one bright fringe. However, the intensity was greater than that of the non-superposed single doughnut. Furthermore, TC was still a unit, as shown in Fig. 3 (b3), which is consistent with the existing rule: TC = (l1 + l2)/2 for interference between two OV beams [35].Fig. 4. Local dynamic modulation of the astroid COVA. (a1)–(a4) Intensity patterns in different dynamic states. (b1)–(b4) Experimental reconstructed phase patterns. For the detailed dynamic modulation process (see Visualization 1).
In mechanics, gear is the most important transfer structure via a cycloid; it can execute an arbitrary complex transmission. Optical gears can be designed using two or more COVAs. By carefully designing the parameters, as in the captions of Fig. 5 (multimedia view), the basic structure of the optical gear was constructed using a HOVA and EOVA. Figure 5 illustrates two COVA gears with different rotation directions, structures, and OV numbers. Different rotation directions were realized, as shown in the first and second rows in Fig. 5. To match the teeth of the HOVA and EOVA, the convex curve of the EOVA was equal to the concave curve of the HOVA, marked by red curves in each pattern. The ratio between the numbers of teeth in the HOVA and EOVA was used to determine the ratio of their rotation velocities. The ratio values for these two optical gears were 5/3 = 1.67 and 6/4 = 1.5, respectively.
Fig. 5. Optical gear constructed by two COVAs: a HOVA and EOVA. (a) Rotation toward each other in the same direction. Q1 = (1.2, 2.5, 2.5, 0.7, 4), N1 = 15; Q2 = (1.0, 2.5, 2.5, 0.7, 4), N2 = 12, h = 1.88 mm, g = 0.438 mm and r = 0.125 mm. For the detailed dynamic modulation process (see Visualization 2). (b) Rotation in the reverse directions of the optical gear. Q1 = (1.25, 2.5, 2.5, 0.7, 5), N1 = 12; Q2 = (1.0, 2.5, 2.5, 0.7, 5), N2 = 12, h = 2.50 mm, g = 0.437 mm and r = 0.115 mm. For the detailed dynamic modulation process (see Visualization 3). Arrows represent the rotation directions of the gears. Red curves indicate the orientation of the optical gear in rotation. Subscripts 1 and 2 represent the HOVA and EOVA, respectively.
For particle manipulation, the COVA gear can be used for particle transport and transfer from HOVA to EOVA, and vice versa. However, several factors must be considered before executing a transfer action. For example, to transfer particles, the size of the trapped particle (radius, rp) should be greater than the nearest distance between two adjacent OVs on HOVA and EOVA, as indicated by the orange dashed line in Figs. 5 (a4) and (b4). In this case, the condition should be satisfied as rp > (g - 2r)/2. Further, the transfer efficiency is determined by the number of matched pairs of OVs when the tooth is well matched, as the state in Fig. 5 (a2) and (b2). Moreover, the transfer is affected by the light force distribution, magnitude of the OAM, and intensity of the OVs, which should be further studied in the near future. Nonetheless, the COVA gear exhibits the capacity for particle transfer, thus providing a platform for the complex manipulation and simultaneous transfer of multiple particles.
As previously mentioned, when the optical vortex array meets the cycloid, COVAs are generated and modulated, and their structures are complex, versatile, and flexible. Remarkably, OVA is endowed the characteristics and capacity of the cycloid, including multiple adjusting parameters and versatile functions of mechanical transfer. Based on this aspiration, to the best of our knowledge, OVA first possesses a functional structure, which combines the advantages of the OAM of individual OV and the properties of the entire array. In extended studies and applications, complex manipulation and functional arrangements of multiple particles are freely achieved based on this proposed novel scheme. Moreover, particle transfer can be executed via exploiting the COVA gears under different circumstances and complex paths of bioscience and medical biology.
4. Conclusions
In summary, to exploit the capacity of the entire OVA structure, a novel cycloid OVA, referred to as COVA, was proposed; in COVA the OVs are evenly distributed on the cycloid trajectory. Five structured parameters of COVA were proposed, and their functions, including expansion, stretching, and cycloid factors, were studied in detail. Partial dynamic modulation of the OVs was conducted while the cycloid structure remained unchanged. Furthermore, optical gears were designed via multiple COVAs and exhibited the capacity to transfer multiple particles. Thus, OVA is energized by the cycloids when they meet. In future, the OVA should be connected with other functional structures for broad and more advanced applications. This work provides a distinct scheme to generate OVAs, thus faciliting potential applications for the complex manipulation and transfer of multi-particle systems.
Funding
State Key Laboratory of Transient Optics and Photonics (SKLST202216); Key Scientific Research Projects of Institutions of Higher Learning of the Henan Province Education Department (21zx002); National Natural Science Foundation of China (11974102, 12274116); Natural Science Foundation of Henan Province (232300421019).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
References
1. 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]
2. J. Chen, C. H. Wan, and Q. W. Zhan, “Engineering photonic angular momentum with structured light: a review,” Adv. Photonics 3(06), 064001 (2021). [CrossRef]
3. J. Wang, “High-dimensional orbital angular momentum comb,” Adv. Photonics 4(05), 050501 (2022). [CrossRef]
4. Y. J. Yang, Y. X. Ren, M. Z. Chen, Y. Arita, and C. Rosales-Guzman, “Optical trapping with structured light,” Adv. Photonics 3(03), 034001 (2021). [CrossRef]
5. M. Padgett and R. Bowman, “Tweezers with a twist,” Nat. Photonics 5(6), 343–348 (2011). [CrossRef]
6. J. Wang, “Advances in communications using optical vortices,” Photonics Res. 4(5), B14–B28 (2016). [CrossRef]
7. G. Milione, T. Wang, J. Han, and L. Bai, “Remotely sensing an object’s rotational orientation using the orbital angular momentum of light,” Chin. Opt. Lett. 15(3), 030012 (2017). [CrossRef]
8. W. Wang, T. Yokozeki, R. Ishijima, M. Takeda, and S. G. Hanson, “Optical vortex metrology based on the core structures of phase singularities in Laguerre-Gauss transform of a speckle pattern,” Opt. Express 14(22), 10195–10206 (2006). [CrossRef]
9. X. L. Wang, X. D. Cai, Z. E. Su, M. C. Chen, D. Wu, L. Li, N. L. Liu, C. Y. Lu, and J. W. Pan, “Quantum teleportation of multiple degrees of freedom of a single photon,” Nature 518(7540), 516–519 (2015). [CrossRef]
10. L. Zhu, A. D. Wang, M. L. Deng, B. Lu, and X. J. Guo, “Experimental demonstration of multiple dimensional coding decoding for image transfer with controllable vortex arrays,” Sci. Rep. 11(1), 12012 (2021). [CrossRef]
11. X. K. Li, Y. Li, X. N. Zeng, and Y. H. Han, “Perfect optical vortex array for optical communication based on orbital angular momentum shift keying,” J. Opt. 20(12), 125604 (2018). [CrossRef]
12. S. H. Li and J. Wang, “Experimental demonstration of optical interconnects exploiting orbital angular momentum array,” Opt. Express 25(18), 21537–21547 (2017). [CrossRef]
13. X. Z. Li, H. X. Ma, H. Zhang, M. M. Tang, H. H. Li, J. Tang, and Y. S. Wang, “Is it possible to enlarge the trapping range of optical tweezers via a single beam,” Appl. Phys. Lett. 114(8), 081903 (2019). [CrossRef]
14. K. Ladavac and D. Grier, “Microoptomechanical pumps assembled and driven by holographic optical vortex arrays,” Opt. Express 12(6), 1144–1149 (2004). [CrossRef]
15. C. S. Guo, Y. N. Yu, and Z. P. Hong, “Optical sorting using an array of optical vortices with fractional topological charge,” Opt. Commun. 283(9), 1889–1893 (2010). [CrossRef]
16. X. Li, Y. Zhou, Y. N. Cai, Y. N. Zhang, S. H. Yan, M. M. Li, R. Z. Li, and B. L. Yao, “Generation of Hybrid Optical Trap Array by Holographic Optical Tweezers,” Front. Phys. 9, 591747 (2021). [CrossRef]
17. T. Wu, T. A. Nieminen, S. Mohanty, J. Miotke, R. L. Meyer, H. Rubinsztein-Dunlop, and M. W. Berns, “A photon-driven micromotor can direct nerve fibre growth,” Nat. Photonics 6(1), 62–67 (2012). [CrossRef]
18. 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]
19. T. Lei, M. Zhang, Y. R. Li, P. Jia, G. N. Liu, X. G. Xu, Z. H. Li, C. J. Min, J. Lin, C. Y. Yu, H. B. Niu, and X. C. Yuan, “Massive individual orbital angular momentum channels for multiplexing enabled by Dammann gratings,” Light Sci. Appl.4(3), e257 (2015). [CrossRef]
20. J. J. Yu, C. H. Zhou, Y. C. Lu, J. Wu, L. W. Zhu, and W. Jia, “Square lattices of quasi-perfect optical vortices generated by two-dimensional encoding continuous-phase gratings,” Opt. Lett. 40(11), 2513–2516 (2015). [CrossRef]
21. J. P. Yuan, H. F. Zhang, C. H. Wu, L. R. Wang, L. T. Xiao, and S. T. Jia, “Tunable optical vortex array in a two-dimensional electromagnetically induced atomic lattice,” Opt. Lett. 46(17), 4184–4187 (2021). [CrossRef]
22. X. D. Qiao, B. Midya, Z. H. Gao, Z. F. Zhang, H. Q. Zhao, T. W. Wu, J. Yim, R. Agarwal, N. M. Litchinitser, and L. Feng, “Higher-dimensional supersymmetric microlaser arrays,” Science 372(6540), 403–408 (2021). [CrossRef]
23. J. Kim, J. Seong, Y. Yang, S. W. Moon, T. Badloe, and J. Rho, “Tunable metasurfaces towards versatile metalenses and metaholograms,” Adv. Photonics 4(02), 024001 (2022). [CrossRef]
24. G. X. Wei, L. L. Lu, and C. S. Guo, “Generation of optical vortex array based on the fractional Talbot effect,” Opt. Commun. 282(14), 2665–2669 (2009). [CrossRef]
25. D. Deng, Y. Li, Y. H. Han, X. Y. Su, J. F. Ye, J. M. Gao, Q. Q. Sun, and S. L. Qu, “Perfect vortex in three-dimensional multifocal array,” Opt. Express 24(25), 28270–28278 (2016). [CrossRef]
26. A. Sabatyan and J. Rafighdoost, “Grating- and checkerboard-based zone plates as an optical array generator with a favorable beam shape,” Appl. Opt. 56(19), 5355–5359 (2017). [CrossRef]
27. P. Chen, S. J. Ge, W. Duan, B. Y. Wei, G. X. Cui, W. Hu, and Y. C. Lu, “Digitalized Geometric Phases for Parallel Optical Spin and Orbital Angular Momentum Encoding,” ACS Photonics 4(6), 1333–1338 (2017). [CrossRef]
28. X. D. Qiu, F. S. Li, H. G. Liu, X. G. Chen, and L. X. Chen, “Optical vortex copier and regenerator in the Fourier domain,” Photonics Res. 6(6), 641–646 (2018). [CrossRef]
29. X. Z. Li, H. X. Ma, H. Zhang, Y. P. Tai, H. H. Li, M. M. Tang, J. G. Wang, J. Tang, and Y. J. Cai, “Close-packed optical vortex lattices with controllable structures,” Opt. Express 26(18), 22965–22975 (2018). [CrossRef]
30. Y. K. Wang, H. X. Ma, L. H. Zhu, Y. P. Tai, and X. Z. Li, “Orientation-selective elliptic optical vortex array,” Appl. Phys. Lett. 116(1), 011101 (2020). [CrossRef]
31. H. Wang, S. Y. Fu, and C. Q. Gao, “Tailoring a complex perfect optical vortex array with multiple selective degrees of freedom,” Opt. Express 29(7), 10811–10824 (2021). [CrossRef]
32. L. Li, C. L. Chang, X. Z. Yuan, C. J. Yuan, S. T. Feng, S. P. Nie, and J. P. Ding, “Generation of optical vortex array along arbitrary curvilinear arrangement,” Opt. Express 26(8), 9798–9812 (2018). [CrossRef]
33. J. T. Hu, Y. P. Lan, H. H. Fan, W. N. Ye, P. Q. Zeng, Y. X. Qian, and X. Z. Li, “Generation and manipulation of multi-twisted beams via azimuthal shift factors,” Appl. Phys. Lett. 121(22), 221103 (2022). [CrossRef]
34. R. Z. Zhang, N. Z. Hu, H. B. Zhou, K. H. Zou, X. Z. Su, Y. Y. Zhou, H. Q. Song, K. Pang, H. Song, A. Minoofar, Z. Zhao, C. Liu, K. Manukyan, A. Almaiman, B. Lynn, R. W. Boyd, M. Tur, and A. E. Willner, “Turbulence-resilient pilot-assisted self-coherent free-space optical communications using automatic optoelectronic mixing of many modes,” Nat. Photonics 15(10), 743–750 (2021). [CrossRef]
35. V. V. Kotlyar, A. A. Kovalev, P. Amiri, P. Soltani, and S. Rasouli, “Topological charge of two parallel Laguerre-Gaussian beams,” Opt. Express 29(26), 42962–42977 (2021). [CrossRef]
