## Abstract

In this study, an anomalous ring-connected optical vortex array (ARC-OVA) via the superposition of two grafted optical vortices (GOVs) with different topological charges (TCs) has been proposed. Compared with conventional OVAs, the signs and distribution of the OVs can be individually modulated, while the number of OVs remains unchanged. In particular, the positive and negative OVs simultaneously appear in the same intensity ring. Additionally, the size of the dark core occupied by the OV can be modulated, and the specific dark core is shared by a pair of plus–minus OVs. This work deepens our knowledge about connected OVAs and facilitates new potential applications, especially in particle manipulation and optical measurement.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

Because the optical vortex (OV) was confirmed to carry an orbital angular momentum (OAM) [1], it has received considerable attention. The so-called OV beam possesses a spiral phase term of exp(*imφ*) where *m* is the topological charge (TC) and *φ* is the azimuthal coordinate. In the past decade, because of the unique characteristics of the OV, it has been widely studied in many fields such as micro-particle manipulation [2–5], optical communications [6–8], optical measurements [9,10], and optical imaging [11,12]. Because of these diverse applications, an optical vortex array (OVA) has attracted wide interest, which provides additional opportunities for future applications [13–15].

Generally, the OVAs can be classified into two categories; discrete and connected OVAs. The discrete OVAs contain multiple individual OVs which can be designed as different structures [16–21] and the TC can be modulated independently. While, the connected OVAs contain multiple unit OVs (TC = ±1), which superpose into the intensity along a specific distribution. That is, the unit OVs are connected by the intensity. Recently, the connected OVAs have been applied in optical measurements [22,23], and they show prospects in ultra-cold atoms trapping [24,25]. Therefore, the generation, modulation and verification of the connected OVAs is important in related optical fields.

The connected OVAs are usually generated by the superposition of two or more specific optical beams. Researchers have produced several connected OVAs via the superposition of multi-plane beams [26], Gaussian beams [27], Ince-Gaussian beams [28], Hermite-Gaussian beams [29], Laguerre-Gaussian (LG) beams [24,30], and Bessel beams [31,32]. However, since these OVAs are only generated via superposition of two optical beams with specific parameters, mode modulation and potential applications are limited.

To overcome this limitation, we have proposed a ring-connected optical vortex array via the superposition of two concentric perfect OVs with higher modulation [33]. In addition, to enhance mode distribution, an elliptic annular OVA [34] and OVAs arranged along an arbitrary curvilinear path [35,36] have been suggested. These OVAs can be easily generated and modulated using multiple parameters. However, the sign and position of the OVs in the connected OVAs are difficult to modulate. Particularly, the positive and negative OVs do not appear in the same connected OVA at the same time. Hence, to facilitate new applications of the OVAs with controllable sign and distribution of the OVs, developing a novel OVA is necessary.

To address this challenge, we propose a novel OVA called anomalous ring-connected optical vortex array (ARC-OVA). It is based on the coaxial superposition of two grafted optical vortices (GOVs). The ARC-OVA has an anomalous OVs distribution while the number and sign of the OVs can be easily modulated. Further, this method corrects the existing knowledge that the superposition of OV beams with the same TC cannot produce a connection of OVAs. Consequently, we believe that the proposed ARC-OVA will be a motivation for new applications in optical trapping and optical measurement.

## 2. Generation method of ARC-OVA

Initially, the grafted OV(GOV) is generated via combining the existing method [37] with the perfect OV(POV) generation technique [38]. For simplicity, the GOV is generated via grafting two conventional OV beams with different TCs. The initial spiral phase patterns of the GOVs with different grafted TCs are shown in the insets of Figs. 1(a) and 1(b). Moreover, the POV technique is used to obtain the perfect graft. At the observed plane, the complex amplitude of the GOV is expressed as,

*r*,

*θ*) are the polar coordinates of the focal plane, and

*w*

_{g}and

*w*

_{0}are the waists of the Gaussian beams at the initial and observed planes, respectively. The parameter

*R*is the ring radius, rect(.) is the rectangular function, and

*m*are the grafted TCs of the GOV.

_{n’}Furthermore, the ARC-OVA is generated via superposition of two concentric GOVs. The complex amplitude of an ARC-OVA can be expressed as:

where*E*(

_{a}*r*,

*θ*) and

*E*(

_{b}*r*,

*θ*) are the inner and outer GOVs, respectively. As shown in Figs. 1(a) and 1(b), the two concentric GOV beams (

*E*and

_{a}*E*) have the same ring width, 2

_{b}*w*

_{0}, but different radii

*R*

_{a}and

*R*

_{b}. The grafted TCs of the two GOVs are given as

*m*

_{a1}= −3,

*m*

_{a2}= 1 and

*m*

_{b1}= 3,

*m*

_{b2}= −1, respectively. To ensure the superposition of these two GOVs,

*R*

_{b}−

*R*

_{a}must be less than 2

*w*

_{0}[33], while the radius of the ARC-OVA is

*R*

_{0}= (

*R*

_{a}+

*R*

_{b}) / 2. Note that the OVs will vanish if

*R*

_{a}=

*R*

_{b}[33]. In addition, the two grafted TCs of the GOV must satisfy the relation |

*m*+

_{n’}*m*

_{n’}_{+1}| = 2

*c*(

*c*= 1,2,3…) to ensure that the GOV has a constant intensity ring without a gap.

Finally, the ARC-OVA is generated as shown in Fig. 1(c). Meanwhile, the corresponding phase patterns are shown in the second row of Fig. 1. The total number of the OVs is 4, which agrees with the expression *N* = *N*_{1} + *N*_{2} = |*m*_{b1} − *m*_{a1}| / 2 + |*m*_{b2} − *m*_{a2}| / 2, where *N*_{1} and *N*_{2} represent the number of the OVs in the lower and upper halves in the light ring, respectively. Figure 1(f) demonstrates the phase pattern of the ARC-OVA where the positive and negative OVs are marked by white and black circles, respectively. The sign of the OVs is determined by the direction of the phase increase surrounding the singular point [33,35]. As shown in the insets in Fig. 1(f), if the phase anticlockwise increases from 0 to 2π, the sign is positive. Conversely, the sign is negative. Furthermore, the sign of each OV in the lower and upper halves is determined by the sign of *m*_{b1}-*m*_{a1} and *m*_{b2}-*m*_{a2}, respectively.

## 3. Experimental setup

To experimentally produce the ARC-OVA, we employ the Fourier transform of the Bessel Gaussian beam [38] using an axicon. For a flexible approach, the transmittance function of the complex amplitude of the axicon is applied to the spatial light modulator (SLM). The phase mask in the SLM plus the spiral phase term of the GOV is given by the following formula,

_{a}and GOV

_{b}, respectively. (

*ρ*,

*φ*) are the polar coordinates in the SLM plane,

*k*is the wave number,

*n*and

*α*are the refractive index and the cone angle of the axicon, respectively. Here, the radius and width of the light ring are controlled via adjusting the cone angle of the axicon,

*α,*and the incident Gaussian beam waist,

*w*

_{g}.

The schematic of the experimental setup is shown in Fig. 2. The experimental setup includes the generated optical path of the ARC-OVA and an interference optical path to determine the presence of the OVs in the ARC-OVA. A solid-state laser with a wavelength of 532 nm (Laserwave Co. Ltd) is transformed into an expanded beam by passing through a micro-objective and a convex lens L1 (*f*_{1} = 100 mm). The aperture A1 is used to obtain the central part of the expanded beam. The expanded beam is split into two beams using a beam splitter BS1. An output beam is displayed on the SLM (HOLOEYE, PLUTO-VIS-016, pixel size: 8 μm × 8 μm) giving the phase mask in Eq. (3). Two polarizers P1 and P2 are used to modulate a linearly polarized beam and to eliminate the parasitic light [20], respectively. The convex lens L2 generates the Fourier transform and the aperture A2 keeps the +1st diffraction order. Thus, the ARC-OVA is generated on the Fourier plane of L2 (point O).

A 2*f*-2*f* system (L3 and L4) is used to record the images at point O and easily arrange the interference elements. Finally, the intensity patterns of the ARC-OVA are projected by a charge coupled device (CCD) camera (Basler acA1600-60gc, pixel size of 4.5 μm × 4.5 μm). The other beam after BS1 is modified to a spherical wave as a reference beam through the lens L5(*f*_{5} = 75 mm). The reference optical path is obtained with the beam splitter of BS2. The reference beam determines the presence of the OVs in the ARC-OVA and the interference patterns are also projected by the CCD camera.

## 4. Results and discussions

The expected ARC-OVAs are theoretically and experimentally demonstrated as shown in Fig. 3. Compared with conventional OVAs [33], the grafted TCs, i.e., *m*_{a1} of ARC-OVAs vary from 4 to 1, and *m*_{a2} of ARC-OVAs vary from 4 to 7, and the grafted TCs i.e., *m*_{b1} = *m*_{b2} = −4, are set to be constant. Similarly, the equivalent TC (ETC) of the GOV_{a} is kept constant at (*m*_{a1} + *m*_{a2}) / 2 = 4 [39,40]. In addition, the GOV_{b} can be considered as a conventional OV with TC = −4. Furthermore, to achieve the overlapping ratio of the two GOV beams up to 36% [33], the ring radii of the two GOVs are defined by the cone angle of the axicon as *α*_{a} = 0.08 rad and *α*_{b} = 0.09 rad, respectively.

The result shows that the OVAs in Figs. 3(a1)–3(a4) correspond to the conventional OVA. The theoretical and experimental intensity patterns of the corresponding OVAs are shown in the first and second rows of Fig. 3. The experimental results, which are in agreement with the theoretical results, confirm that the ARC-OVAs are successfully generated. The number of the OVs for the conventional OVA and ARC-OVAs satisfies the relation *N* = *N*_{1} + *N*_{2} = |*m*_{b1} − *m*_{a1}| / 2 + |*m*_{b2} − *m*_{a2}| / 2 = 8. The distributions of the OVs on the light ring are uniform in the conventional OVA. On the contrary, the OVs distributions of the ARC-OVAs vary. In particular, the number of the OVs in the upper half of the light ring increases as the grafted TCs *m*_{a2} increases, while the number of the OVs in the lower half of the light ring decreases as the grafted TCs *m*_{a1} decreases. The corresponding phase patterns are shown in Figs. 3(a4)–3(d4), which show the different distributions of OVs on the intensity rings. Obviously, when *N*_{1} and *N*_{2} are both half-integers, a whole OV still forms on the ring, which is indicated by the white circles in Figs. 3(b1) and 3(d1).

To quantitatively determine the characteristics of the OV distribution of the proposed ARC-OVA, we investigated the center profiles of their theoretical and experimental intensity rings, and are shown in Figs. 3(a3)–3(d3). On the horizontal axis, intervals of [-π, 0] and [0, π] indicate the lower and upper halves of the ARC-OVAs, respectively. The valleys on the curves indicate the positions of the OVs and the distance between two adjacent valleys represent the formed angle between two adjacent OVs. For conventional OVA, the distance between two adjacent valleys is equal to π/4. While, the distance between two adjacent OVs for ARC-OVA, is uneven and determined by the relations 2π / |*m*_{b1} − *m*_{a1}| and 2π / |*m*_{b2} − *m*_{a2}| within [-π, 0] and [0, π], respectively. The results show that the distribution of the OVs of the ARC-OVA can be easily modulated, as the number of the OVs is kept constant.

The fact that the OVA cannot be formed via superposition of two OVs with the same TC is well known [33]. However, the ARC-OVAs were formed via superposition of two GOVs with the same ETC = 5 as shown in Fig. 4. The OVs appear on the light ring due to the different grafted TCs on the superposed semicircles. The analysis shows that the number of the OVs satisfies the relation *N* = *N*_{1} + *N*_{2} = |*m*_{b1} − *m*_{a1}| / 2 + |*m*_{b2} − *m*_{a2}| / 2.

To determine the sign of the OVs, the phase patterns of the ARC-OVAs are demonstrated and shown in Figs. 4(a3)–4(d3). Because (*m*_{b1} − *m*_{a1}) > 0 and (*m*_{b2} − *m*_{a2}) < 0, the OVs in the lower half of the ARC-OVA have positive signs while the OVs in the upper half have negative signs. Note that a specific dark core is formed at the center-right side of the ARC-OVAs as shown in Figs. 4(a1) and 4(c1). The dark core contains a pair of negative and positive OVs because the values of (*m*_{b1} − *m*_{a1}) and (*m*_{b2} − *m*_{a2}) are half-integers with inverse signs as shown in Figs. 4(a3) and 4(d3). This result disproves existing knowledge that an OVA cannot be generated via the superposition of two similar OV beams, and that a dark core only corresponds to an OV on the OVA.

To experimentally determine the presence of the OVs, the interference patterns between the ARC-OVAs and a spherical wave are illustrated and shown in Fig. 5. Clearly, the forks appear on the position of each dark core because of the nature of the OV. The details are shown as the 2× magnification of specific forks in Fig. 5. Moreover, the number of the forks is equal to that of the OVs. The clockwise and anticlockwise forks marked by the red and blue dashed box show the existence of the positive and negative OVs, respectively.

As observed in Fig. 4, the positive and negative OVs simultaneously appear on the light ring of the ARC-OVA. This is different from the conventional OVA pattern, thus making it a novel discovery. Figure 6 shows the independent modulation of distribution and sign of the OVs in the upper and lower halves of the ARC-OVAs. In comparison, the modulation of the OVs distribution is observed on the ARC-OVA with the same odevity of the superimposed grafted TCs, which is shown in the first and second rows of Fig. 6. As expected, the numbers and distributions of the OVs in the lower and upper halves are easily modulated. The size of the dark core is inversely proportional to the OVs number. The corresponding phase patterns shown in Figs. 6(a2)–6(d2) confirmed that all the OVs are negative.

To modulate the sign of the OVs, the odevity of the grafted TCs is reset to ensure the opposite odevity between (*m*_{b1} − *m*_{a1}) and (*m*_{b2} − *m*_{a2}), which are shown in the third and fourth rows of Fig. 6. The intensity patterns show that the position and the number of the dark core remain unchanged between the first and third rows of Fig. 6. However, the sign of OVs is opposite in the upper and lower halves of the ARC-OVAs as shown in Figs. 6(a4)–6(d4). Consequently, the ARC-OVA shows significant properties. For example, the number, distribution and signs of OVs have independent modulation.

For the aforementioned ARC-OVAs, the initial phase of the two superposed GOVs is zero. Considering the modulation of the initial phase difference between the two GOVs, the complex amplitude of the ARC-OVA is rewritten as,

where*Ψ*

_{0}is the initial phase difference. Figure 7 shows the experimental results with

*Ψ*

_{0}increasing from 0 to 2π by an interval of π/2. The bold white lines indicate the original position of two specific OVs in the upper and lower halves of the ARC-OVAs, respectively. The blue and orange dotted lines indicate the transition of the specific OVs. With the phase difference

*Ψ*

_{0}increasing from 0 to 2π, the OVs rotate in a clockwise direction along the light ring (for details, see Visualization 1). By calculation, we find that the rotational speed of the upper OVs is greater than that of the lower OVs. The rotation angles are determined by the relations

*θ*

_{1}=

*Ψ*

_{0}/ |

*m*

_{b1}–

*m*

_{a1}| and

*θ*

_{2}=

*Ψ*

_{0}/ |

*m*

_{b2}–

*m*

_{a2}|, respectively. In addition, the direction of the rotation is determined by the sign of the phase difference

*Ψ*

_{0}. If the sign is negative, then the direction of the rotation will be anticlockwise.

As *Ψ*_{0} increases, the size of the dark core changes when it passes through the horizontal dashed line. As illustrated in Fig. 8(a), two dark cores of the OVs are selected to study the size change, which are indicated by A and B as the initial phase difference *Ψ*_{0} = 0. The sizes of these two dark cores are defined by the arc angles of *β*_{A} and *β*_{B}, respectively. Figure 8(b) shows the change of the dark cores with an increase in *Ψ*_{0}.

During rotation in upper or lower half-circles, the arc angle of the dark core remains unchanged, which is indicated by the shadow areas in Fig. 8(b). While the two dark cores start to rotate in the opposite half, the *β*_{A} increases with increasing *Ψ*_{0}, as *β*_{B} decreases. By comparison, there is a linear relationship between the arc angle and the initial phase difference, such that *β*_{A} = 0.0583 + 0.0284*Ψ*_{0}, and *β*_{B} = 0.2482 − 0.0264*Ψ*_{0}, respectively. And the correlation coefficients are given as 0.9875 and −0.9864, respectively. This result shows that the size of the dark core occupied by the specific OV can be linearly modulated via changing the initial phase difference.

Without loss of generality, the ARC-OVAs are also generated when the grafted TCs are half-integers. Figure 9 show the intensity and phase distribution of the ARC-OVAs. Results show that each OV forms a complete dark core and the number of the OVs satisfies the relation *N* = *N*_{1} + *N*_{2} = |*m*_{b1} − *m*_{a1}| / 2 + |*m*_{b2} − *m*_{a2}| / 2. The insets (profiles) in the top and middle rows of Fig. 9 show that a dark core always appeared in the center on the right side due to the value of *N*_{1} and *N*_{2} as half-integers. The experimental intensity patterns are in agreement with the theoretical results. The sign of all OVs is negative, and is determined by the sign of (*m*_{b1} − *m*_{a1}) and (*m*_{b2} − *m*_{a2}).

The proposed method has successfully produced the expected ARC-OVA, which can modulate the number, distribution and sign of the OVs. However, the sign of the OVs determines the control of subsections on the ring as a first step. In future research, we will attempt to modulate the sign of each OV separately. One feasible approach is to apply the GOV with multiple OV beams graft. In combination with arbitrary curve and mode transformation techniques, the ARC-OVA can be transformed into other structures, such as, ellipse, multi-rings, and star structures. Further, the ARC-OVA obtained in the potential applications should be a strong motivation.

## 5. Conclusions

In conclusion, we have proposed an anomalous ring-connected OVA (ARC-OVA). Using the superposition of two GOVs with different grafted TCs, the sign and distribution of the OVs have been successfully modulated, which overcome the limitation of the conventional OVAs. The number of the OVs in the ARC-OVA is determined by the formula, *N* = *N*_{1} + *N*_{2} = |*m*_{b1} – *m*_{a1}| / 2 + |*m*_{b2} – *m*_{a2}| / 2. The signs of the OVs in the lower and upper halves are determined by the signs of (*m*_{b1} – *m*_{a1}) and (*m*_{b2} – *m*_{a2}), respectively. Given an initial phase difference, the OVs have different rotation speeds in the upper and lower halves of the ARC-OVA ring. Using this anomalous ring-connected method, more versatile connected OVAs can be obtained, with subsequent potential applications in optical trapping and optical manipulation.

## Funding

National Natural Science Foundation of China (11974102); State Key Laboratory of Transient Optics and Photonics (SKLST201901).

## Disclosures

The authors declare that there are no conflicts of interest related to this article.

## 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. **D. G. Grier, “A revolution in optical manipulation,” Nature **424**(6950), 810–816 (2003). [CrossRef]

**3. **Y. Q. Zhang, X. J. Dou, Y. M. Dai, X. Y. Wang, C. J. Min, and X. C. Yuan, “All-optical manipulation of micrometer-sized metallic particles,” Photonics Res. **6**(2), 66–71 (2018). [CrossRef]

**4. **H. P. Wang, L. Q. Tang, J. N. Ma, H. W. Hao, X. Y. Zheng, D. H. Song, Y. Hu, Y. G. Li, and Z. G. Chen, “Optical clearing and shielding with fan-shaped vortex beams,” APL Photonics **5**(1), 016102 (2020). [CrossRef]

**5. **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]

**6. **J. Wang, J. Y. 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]

**7. **N. Bozinovic, Y. Yue, Y. X. Ren, M. Tur, P. Kristensen, H. Huang, A. E. Willner, and S. Ramachandran, “Terabit-scale orbital angular momentum mode division multiplexing in fibers,” Science **340**(6140), 1545–1548 (2013). [CrossRef]

**8. **K. Rottwitt, J. G. Koefoed, K. Ingerslev, and P. Kristensen, “Inter-modal Raman amplification of OAM fiber modes,” APL Photonics **4**(3), 030802 (2019). [CrossRef]

**9. **H. X. Ma, X. Z. Li, Y. P. Tai, H. H. Li, J. G. Wang, M. M. Tang, Y. S. Wang, J. Tang, and Z. G. Nie, “In situ measurement of the topological charge of a perfect vortex using the phase shift method,” Opt. Lett. **42**(1), 135–138 (2017). [CrossRef]

**10. **Y. J. Yang, Q. Zhao, L. L. Liu, Y. D. Liu, C. Rosales-Guzmán, and C. W. Qiu, “Manipulation of orbital-angular-momentum spectrum using pinhole plates,” Phys. Rev. Appl. **12**(6), 064007 (2019). [CrossRef]

**11. **A. Aleksanyan, N. Kravets, and E. Brasselet, “Multiple-star system adaptive vortex coronagraphy using a liquid crystal light valve,” Phys. Rev. Lett. **118**(20), 203902 (2017). [CrossRef]

**12. **X. D. Qiu, F. S. Li, W. H. Zhang, Z. H. Zhu, and L. X. Chen, “Spiral phase contrast imaging in nonlinear optics: seeing phase objects using invisible illumination,” Optica **5**(2), 208–212 (2018). [CrossRef]

**13. **Y. Q. Zhang, Z. K. Wu, C. Z. Yuan, X. Yao, K. Q. Lu, M. Belic, and Y. P. Zhang, “Optical vortices induced in nonlinear multilevel atomic vapors,” Opt. Lett. **37**(21), 4507–4509 (2012). [CrossRef]

**14. **D. S. Ding, W. Zhang, S. Shi, Z. Y. Zhou, Y. Li, B. S. Shi, and G. C. Guo, “High-dimensional entanglement between distant atomic-ensemble memories,” Light: Sci. Appl. **5**(10), e16157 (2016). [CrossRef]

**15. **Y. X. Qian, Y. L. Shi, W. M. Jin, F. R. Hu, and Z. J. Ren, “Annular arrayed-Airy beams carrying vortex arrays,” Opt. Express **27**(13), 18085–18093 (2019). [CrossRef]

**16. **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]

**17. **S. Y. Fu, T. L. Wang, and C. Q. Gao, “Perfect optical vortex array with controllable diffraction order and topological charge,” J. Opt. Soc. Am. A **33**(9), 1836–1842 (2016). [CrossRef]

**18. **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 (2016). [CrossRef]

**19. **X. D. Qiu, F. S. Li, H. G. Liu, X. F. Chen, and L. X. Chen, “Optical vortex copier and regenerator in the Fourier domain,” Photonics Res. **6**(6), 641–646 (2018). [CrossRef]

**20. **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]

**21. **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]

**22. **W. Wang, S. G. Hanson, Y. Miyamoto, and M. Takeda, “Experimental investigation of local properties and statistics of optical vortices in random wave fields,” Phys. Rev. Lett. **94**(10), 103902 (2005). [CrossRef]

**23. **X. Z. Li, Y. P. Tai, L. P. Zhang, H. J. Li, and L. B. Li, “Characterization of dynamic random process using optical vortex metrology,” Appl. Phys. B **116**(4), 901–909 (2014). [CrossRef]

**24. **S. Franke-Arnold, J. Leach, M. J. Padgett, V. E. Lembessis, D. Ellinas, A. J. Wright, J. M. Girkin, P. Öhberg, and A. S. Arnold, “Optical ferris wheel for ultracold atoms,” Opt. Express **15**(14), 8619–8625 (2007). [CrossRef]

**25. **A. S. Arnold, “Extending dark optical trapping geometries,” Opt. Lett. **37**(13), 2505–2507 (2012). [CrossRef]

**26. **K. O’Holleran, M. J. Padgett, and M. R. Dennis, “Topology of optical vortex lines formed by the interference of three, four, and five plane waves,” Opt. Express **14**(7), 3039–3044 (2006). [CrossRef]

**27. **P. Vaity, A. Aadhi, and R. Singh, “Formation of optical vortices through superposition of two Gaussian beams,” Appl. Opt. **52**(27), 6652–6656 (2013). [CrossRef]

**28. **S. C. Chu, C. S. Yang, and K. Otsuka, “Vortex array laser beam generation from a Dove prism-embedded unbalanced Mach-Zehnder interferometer,” Opt. Express **16**(24), 19934–19949 (2008). [CrossRef]

**29. **Y. C. Lin, T. H. Lu, K. F. Huang, and Y. F. Chen, “Generation of optical vortex array with transformation of standing-wave Laguerre-Gaussian mode,” Opt. Express **19**(11), 10293–10303 (2011). [CrossRef]

**30. **S. J. Huang, Z. Miao, C. He, F. F. Pang, Y. C. Li, and T. Y. Wang, “Composite vortex beams by coaxial superposition of Laguerre–Gaussian beams,” Opt. Laser. Eng. **78**, 132–139 (2016). [CrossRef]

**31. **R. Vasilyeu, A. Dudley, N. Khilo, and A. Forbes, “Generating superpositions of higher–order Bessel beams,” Opt. Express **17**(26), 23389–23395 (2009). [CrossRef]

**32. **A. Dudley and A. Forbes, “From stationary annular rings to rotating Bessel beams,” J. Opt. Soc. Am. A **29**(4), 567–573 (2012). [CrossRef]

**33. **H. X. Ma, X. Z. Li, Y. P. Tai, H. H. Li, J. G. Wang, M. M. Tang, J. Tang, Y. S. Wang, and Z. G. Nie, “Generation of circular optical vortex array,” Ann. Phys. **529**(12), 1700285 (2017). [CrossRef]

**34. **H. X. Ma, X. Z. Li, H. Zhang, J. Tang, Z. G. Nie, H. H. Li, M. M. Tang, J. G. Wang, Y. P. Tai, and Y. S. Wang, “Adjustable elliptic annular optical vortex array,” IEEE Photonics Technol. Lett. **30**(9), 813–816 (2018). [CrossRef]

**35. **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]

**36. **C. L. Chang, L. Li, Y. Gao, S. P. Nie, Z. C. Ren, J. P. Ding, and H. T. Wang, “Tunable polarization singularity array enabled using superposition of vector curvilinear beams,” Appl. Phys. Lett. **114**(4), 041101 (2019). [CrossRef]

**37. **H. Zhang, X. Z. Li, H. X. Ma, M. M. Tang, H. H. Li, J. Tang, and Y. J. Cai, “Grafted optical vortex with controllable orbital angular momentum distribution,” Opt. Express **27**(16), 22930–22938 (2019). [CrossRef]

**38. **P. Vaity and L. Rusch, “Perfect vortex beam: Fourier transformation of a Bessel beam,” Opt. Lett. **40**(4), 597–600 (2015). [CrossRef]

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

**40. **G. Gbur, “Fractional vortex Hilbert's hotel,” Optica **3**(3), 222–225 (2016). [CrossRef]