## Abstract

We report on a novel curvilinear optical vortex beam named twin curvilinear vortex beams (TCVBs) with intensity and phase distribution along a pair of two- or three-dimensional curves, both of which share the same shape and the same topological charge. The TCVBs also possess the character of perfect optical vortex, namely having a size independent of topological charge. We theoretically demonstrate that a TCVB rather than a single-curve vortex beam can be created by the Fourier transform of a cylindrically polarized beam. The behavior of TCVBs generated through our method is investigated by simulation and experiment, including interference experiments for identifying the vortex property of the TCVBs. The TCVBs may find applications in optical tweezers, such as trapping low refractive index particles in the dark region between two curves and driving them moving along the curvilinear trajectory.

© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## Corrections

29 April 2021: A typographical correction was made to the caption of Fig. 9.

## 1. Introduction

Optical vortex beams (VBs) carrying orbital angular momentum (OAM) have been attracting considerable attention owing in recent years for their diverse applications such as optical communication [1–5], optical trapping [6–8], and optical imaging [9–11]. Hence, properties as well as generation of VBs have been studied in a large body of prior literatures [12–14]. Conventional VBs are limited as their ring diameter depends on topological charge value, precluding their use in applications such as coupling multiple OAM beams into an single transmission channel [15] or manipulating microscopic particles via VBs with the same size but different OAM [16]. In 2013, to overcome this limitation of conventional VBs, Ostrovsky et al. proposed the concept of the perfect optical vortex (POV) [17], whose ring diameter is independent of topological charge value. Subsequently, many works on generating POVs were reported, such as phase mask loaded on spatial light modulator (SLM) [18], Pancharatnam-Berry phase elements [19], spiral axicon [20,21] and digital micro-mirror device [22]. At the same time, the researchers extended the concept of POVs to elliptical POVs [23,24], whose shape is changed from ring to ellipse. Thanks to the compelling property of POV beams, many advanced applications have emerged in the optical domain, such as vector vortex beam [25,26], plasmonic metasurfaces [27], microparticle trapping [16] and optical communication [28]. Recently, Pinnell *et al*. pointed out that the degree of ‘perfectness’ of POV depends on the ratio between the ring radius and ring thickness of POV [29]; only when this ratio is much larger than 1 and the topological charge value is not particularly large, the effect of topological charge value on the beam size is very small.

However, the POVs generated by the above methods have only one single bright ring in their cross-section, that is, there is only one main lobe in the intensity profile along the radial direction. In some cases, one single ring will hinder the application of POVs. For instance, a POV with a single bright ring is difficult to trap and drive low-index particles whose refractive index is lower than the surrounding medium. Optical trapping and manipulating of low-index particles are of wide interest in biology and medicine, for example, the drug can be encapsulated in trapped microbubbles and then moved to target tissues to observe the response of target cells to the drug [30]. One strategy to manipulate low-index particles via VBs is to take advantage of a dark region within beams’ cross-section, for example, by utilizing a dark annular region between adjacent bright rings of a higher-order Bessel beams [31] or the dark region surrounded by a POV’s bright ring [32]. Another promising scheme is to design a double-ring POV(DR-POV), which has two closely located rings. A handful of papers have addressed the realization of double-ring POVs through specific techniques, including the Fourier transformation of an azimuthally polarized Bessel beam [33], circular Dammann grating embedded with a spiral phase [34], and superposition of two π-dephased POVs [35]. However, all methods mentioned above can only generate a circular dark region, limiting their further utilization in trapping and rotating low-index particles. In addition, since the dark region created in these methods appears to be closed loop, it allows only a circulation motion of particles. Lately, a method based on the differentiation operation was proposed to generate double-contour plane curves by adding vortex phases to the incident field, and the dark region between the contours can be non-closed [36]. However, the phase gradient along light curves depends on the order of the extra vortex phase and cannot be adjusted on-demand, and split curves are limited to the two-dimensional cases.

In this paper, we introduce a novel method to generate a more general and exotic VB, termed as twin curvilinear VBs (TCVBs). Our scheme enables a cylindrically polarized curvilinear beam to be split into two VBs via a focusing process of lens, making it possible to form a dark region along arbitrarily prescribed trajectory. Consequently, TCVBs can be regarded as a composition of two arbitrary curvilinear VBs with the same shape but slightly different size. Furthermore, we extend the generation of TCVBs from two dimensional (2D) into three dimensional (3D) domain. We believe that benefiting from their characteristic of arbitrarily predefined trajectory with independently tailored distribution of intensity and phase, the proposed TCVBs are promising and competitive in terms of application potential in scenarios of trapping and driving low-index particles.

## 2. Principle of the technique

#### 2.1 Two-dimensional (2D) TCVBs

The proposed technique can be divided into two steps. The first step is to synthesize an input field needed for creating a vector field with customized cylindrical symmetry in state of polarization (SoP). In the second step, the generated cylindrical vector field is converted into a desired TCVB via a focusing process of lens. Basically, the first step follows the design principle for shaping curvilinear scalar/vector beams reported in [37–39], which can be outlined as follows. For generating a 2D curve expressed by $c(t) = [{x_0}(t),{y_0}(t)]$ with $t \in [{t_\textrm{0}},T]$ (*t*_{0} and *T* representing the start and end points of curve, respectively) in the focal plane, the needed incident light field can be derived and calculated according to the following relation [37]

*f*is the focal length of the Fourier lens. At this stage we set $\varphi ({x,y,t} )= 1$ for the generation of 2D TCVBs, and will modify it for the 3D cases to be illustrated in the next section. The complex weight function $g(t)$ in Eq. (1) is responsible for the amplitude and phase to be imposed along the prescribed curve, and can be calculated by

*ℓ*. The parameter

*ℓ*can also be considered the topological charge of a scalar vortex beam, as illustrated later. The complex exponential term in the integrand of Eq. (1) determines the trajectory of the curve. A cylindrical vector field is obtained by converting the scalar field

*E*(

*x*,

*y*) into the vector field ${\bf E}(x,y) = {\bf E}(\rho ,\phi )$ in the following $\rho - \phi $ polar coordinate expression where ${{\bf e}_\phi }$ denotes the unit vector of azimuthally variant and locally linear SoP, expressed by

On substituting Eqs. (3) and (4) into Eq. (5), we obtain the expression for the focal field

where ${{\bf E}_{RCP}}$ and ${{\bf E}_{LCP}}$ are the RCP and LCP components expressed byThe intensity of the focal field is a direct sum of the intensities of two constituent polarized components owing to their orthogonality,

where ${I_{RCP}}(r,\theta ) = {|{{{\bf E}_{RCP}}(r,\theta )} |^2}$ and ${I_{LCP}}(r,\theta ) = {|{{{\bf E}_{LCP}}(r,\theta )} |^2}$.An important feature in the integrands of Eqs. (7) and (8) is the presence of two helical phase terms with opposite helical senses, $\textrm{exp} (i\phi )$ and $\textrm{exp} ( - i\phi )$. Now we will show that two spatially separated curvilinear VBs, named as the TCVB in this paper, can be created in the focal field just thanks to these two helical phase terms. We have performed numerical simulation for validating this argument, because normally the above integrals cannot be evaluated in closed form. As an illustration, we first design a vortex beam with elliptic trajectory by setting:${x_0}(t) = {R_0}\cos (t)$, ${y_0}(t) = 0.8{R_0}\sin (t)$, ${R_0} = 0.26mm$, ${t_0} = 0$ and $T = 2\pi $ (cf. Equations (1) and (2)). Other simulation parameters include that $f = 100mm$ and $\lambda = 532nm$. The simulation result for the resultant focal field is illustrated in Fig. 1. Figures 1(a)–1(b) show the focal plane intensity maps of RCP and LCP components calculated from Eqs. (7) and (8), respectively, and Fig. 1(c) shows their sum, namely the focal plane intensity resulting from the incident light with cylindrical SoP, wherein the topological charge is set as *ℓ* = 5. The appearance of double ellipses shows that the illumination of a vector field with cylindrical SoP represented by Eqs. (3) and (4) indeed produce a TCVB in the focal plane. For comparison, Fig. 1(d) presents the intensity of the same incident field (namely, the distribution of amplitude and phase) but with linear polarization, showing that a single curve is yielded. The vortex nature of this constructed TCVB is confirmed by the phase map shown in Fig. 1(e), wherein the luminance and color of colormap refer to the intensity (Int) and phase of the focal field, respectively. The inset in Figs. 1(a)–1(d) plots the intensity profile along a radial direction marked by the yellow dashed line. The further investigation shows that the dip between two lobes of TCVB significantly depends on the topological charge *ℓ*; the dip becomes shallower when increasing *ℓ*, as shown in Fig. 1(f), but the spacing between two lobes remains roughly unchanged. Our investigation also shows that both the intensity difference between the two peaks and the shallowing of the dip, shown in Fig. 1(f), are mainly due to the truncation on the intensity profile of the incident field by the optical system. Considering that the difference between the two peaks is fairly small, we believe that it will have no noticeable impact on the application of the TCVBs. Besides, the constructed VBs in Fig. 1 also represent a kind of elliptical POVs, since their size is independent of their own topological charge.

It should be pointed out that the realization of double VBs has been studied by Liang et al. [33], but their research is constrained to double-ring shaped VBs. By contrast, our scheme is capable of generating arbitrarily curvilinear VBs, regardless of whether their shapes are closed or open. Now let us demonstrate 2D TCVBs with more complicated shapes known as the Superformula curves [40]. A Superformula curve is expressed by the following parametric equation

*I*and

_{RCP}*I*, and total intensity distributions

_{LCP}*I*for the two TCVBs shaped according to Table 1. In contrast, a scalar field having the same complex amplitude but the uniformly linear SoP will yield only a single focal curve, as shown in Figs. 2(a4) and 2(b4), demonstrating again that it is cylindrical SoP that contributes to the formation of twin beams. The phase map of the constructed TCVBs is given in Figs. 2(a5) and 2(b5).

_{TCVB}#### 2.2 Three-dimensional (3D) TCVBs

The generated 2D TCVB in the above section can be generalized to a 3D TCVB along a curve defined by $[{x_0}(t),{y_0}(t),{z_0}(t)]$. To this end, we replace $\varphi ({x,y,t} )$ in Eq. (1) by the following one,

The above phase function actually resembles the function of a vari-focal lens that varies its focal length according to a prescribed manner.

The first 3D example we want to demonstrate is a tilted ring defined by ${x_0}(t) = {R_0}\cos (t)$, ${y_0}(t) = {R_0}\sin (t)$ and ${z_0}(t) = s{R_0}\sin (t)$ with ${R_0} = 0.26mm$, *s* = 22 and $t \in [0,2\pi ]$. It should be noted that the paraxial propagation is assumed in the focusing process, which is a prerequisite of our design method. The paraxial propagation means that the light beam mainly propagates along the *z* direction. In order to obey this paraxial condition, we must specify a much longer longitudinal dimension than its transverse dimension in the focal space. As a result, the *z*-directional scaling parameter *s* needs to be set to a value much larger than 1. We set *s* to be 22 in the experiment to facilitate the display of the results. In fact, as long as the paraxial approximation is satisfied, *s* can take other values. Figure 3 shows the simulation results. Figures 3(a) represent the double tilted-ring VB (i.e. TCVB) produced from the input vector field with cylindrical SoP, while Figs. 3(b) are for a single tilted ring VB resulted from the same light field (amplitude and phase) but with linear polarization. A 3D view of VBs is given in Figs. 3(a1) and 3(b1), and the cross-section intensities at three axial positions ($z ={-} 5.6mm$, 0 *mm* and 5.6 *mm*) are given in Figs. 3(a2-a4) and 3(b2-b4), respectively. An enlarged part marked by the yellow dashed rectangle is also presented in inset, clearly differentiating the TCVB and its single curve counterpart.

To demonstrate the capability of our proposed method for generating arbitrary 3D curve, Fig. 4 presents the simulation results of a 3D Archimedean spiral defined by the coordinate parameters ${x_0}(t) ={-} {R_0}t\cos (10t)$, ${y_0}(t) ={-} {R_0}t\sin (10t)$ and ${z_0}(t) = s{R_0}[0.5 - {(1 - {t^2})^{1/2}}]$, with $t \in [0,1]$, ${R_0} = 0.26mm$ and $s = 22$. Figure 4(a) show the Archimedean spiral TCVB produced from the input vector field with cylindrical SoP, while Fig. 4(b) show a single VB produced from the same input field but with linear polarization. Figures 4(a1)-4(a4) and 4(b1)-4(b4) show a 3D view of VBs and their cross-section intensities at three respective axial positions ($z ={-} 2.5mm$, $- 1.0mm$ and 1.0 *mm*), demonstrating again that imposing cylindrical SoP on an input light field enable its original single VB to split into a double VB in 3D domain.

## 3. Experimental results and discussion

The experimental arrangement for creating the TCVBs is depicted in Fig. 5. A collimated laser beam with the wavelength of 532 nm is incident on a spatial light modulator (SLM). The SLM (HOLOEYE Leto, 6.4µm pixel pitch, 1920 × 1080 resolution) is utilized to address a phase-only computer-generated hologram (CGH) which is produced from the complex amplitude *E*(*x*,*y*) in Eq. (1) through a cosine-grating encoding method [41]. The vertically polarized incident beam is modulated by the SLM, and the desired light field is realized after a 4-*f* optical filtering system consisting of two lenses (L1 and L2, with focal length of 400 mm and 300 mm, respectively) and a filter. The filter includes two holes which are positioned at the +1st diffraction order in the *x*- and *y*-axis, respectively, one responsible for the passage of Fourier spectrum of the encoded field and the other enabling the passage of a reference beam. We employ a polarization converter called the S-waveplate (super-structured space-variant waveplate) to convert the output beam of the 4-*f* system, which is itself linearly polarized, into a cylindrically polarized one. The cylindrically polarized field is finally focused by the Fourier lens L3 with a focal length of 100 mm into its focal region, yielding the designed TCVBs. A charge coupled device (CCD) camera placed in the focal space is used to capture the intensity patterns, and moves forth and back along the *z* direction via a translational stage to record cross-sectional intensities at different *z* positions. To manifest the helical phase embedded in resultant TCVBs, we also prepare a reference beam that is intended to interfere with the constructed TCVBs, as will be demonstrated below in the experimental examples. The prepared reference beam, marked by the dashed red-color path in Fig. 5, is optionally passed through the optical system, depending on the opening/closing of the filter aperture. The constructed TCVB and the reference beam are combined collinearly by using a Ronchi grating (G) to form interference pattern.

Figure 6 gives the experimental results of generated 2D curvilinear VBs, including a ring, an ellipse, an Archimedean spiral, a Pentagon of ${\bf q} = (1,1,10,6,6,5)$ and a Spiral of ${\bf q} = (1,1,250,100,100,6)$ with $\alpha (t) = \textrm{exp} ({0.2t} )$, in Figs. 6(a)–6(d). Figures 6(a1)-6(a5) show the single-curve VBs resulting from light fields with linear polarization, whereas Figs. 6(b)–6(d) correspond to the TCVBs generated after the linear polarization is converted into the cylindrical polarization, convincing the performance of cylindrical polarization on resulting in the twin beams. In particular, Fig. 6 also shows that the sizes of these TCVBs remain the same for different topological charges of 1, 5, and 10, indicating the TCVBs hold the nature of the so-called “perfect vortex”.

Let us also make a remark concerning the phenomenon that higher topological charge leads to lower intensity contrast of the dip between the twin bright curves with the bright curves, which can be seen more clearly in the numerical simulation shown by Fig. 1(f) and can be visualized in experimental results for higher topological charges (say, *ℓ* > 20, but not shown here for conciseness). The shallowing of the dip is detrimental to the stable trapping of low-index particles. We have found by simulation that this problem can be overcome by increasing the polarization topological charge of the S-waveplate, and further experimental investigations are still underway.

To further verify vortex properties of the generated TCVBs, we experimentally identify the topological charge value through interferometry. To visualize the helically varying phase along the trajectory of TCVBs, we prepare a reference beam following the design procedure of TCVBs so that the reference beam holds the same distribution as the TCVB to be tested except that the reference beam has a plane phase. Two complex amplitudes of the TCVB and reference beam are encoded into a 2D cosine-grating-like CGH. The two fields reconstructed at the +1st diffraction order in the *x*- and *y*-axis, respectively, are picked off with the filter of the 4-*f* system shown in Fig. 5, and interfere with each other on the sensor area of the CCD. The corresponding interference patterns of the TCVB examples in Fig. 6 are shown in Fig. 7. As can be seen in Fig. 7 for the topological charges *ℓ* = 1, 5, and 10, the corresponding results show that the paired light curves in the TCVBs are simultaneously divided into 1, 5, and 10 segments, respectively, implying that the two light curves possess the same topological charge. It should also be pointed out that our proposed method cannot generate TCVBs with each curve having a different topological charge.

As the incident light is cylindrically polarized, its SoP will undergo a change after being focused into split light curves. To visualize the SoP distribution of the focal field, we measure the Stokes parameters (S0, S1, S2, S3) (the definition of the Stokes parameters is omitted here for brevity and can be found, for example, in [42]) of the created TCVBs. Figure 8 gives the measurement results about the Stokes parameters of the TCVBs corresponding to *ℓ* = 5 in Fig. 6. As analyzed in the preceding section, a cylindrically polarized incident field expressed by Eq. (3) can be decomposed into left- and right-handed circularly polarized components and is split into two curves in the focal space. Also, the intensity distributions of focused fields shown in Figs. 1 and 2 indicate that in the outer curve the right-hand polarized component exhibits much higher intensity than the left-hand polarized component, and vice versa in the inner curve. According to the vector superposition theory [42,43], it is known that the two curves in TCVBs are no longer of locally linear SoP, but locally elliptical SoP, and the SoP of twin curves are located on either side of the equator of the Poincaré sphere, respectively, as depicted by S3 shown in Figs. 8(d).

Next, we present experimental results for 3D TCVBs. Two 3D TCVBs, each carrying the vortex phase of *ℓ* = 5 but one having the shapes of twin tilted-rings and the other having twin Archimedean spirals, are experimentally generated and shown in Figs. 9(a) and 10(a), respectively. By moving the CCD back and forth along the optical axis, we can record the longitudinal evolution of the 3D VBs upon propagation. The geometrical structure of beams is depicted in the leftmost column of the figures, and cross-section intensities at three axial positions are shown in the remaining columns. For comparison, Figs. 9(b) and 10(b) show the single curved VBs. The transformation of the single into the twin curved VB is fulfilled by inserting the polarization converter (PC) into the optical path shown in Fig. 5, which converts a linear polarization into an expected cylindrical polarization. The detailed dynamical evolution of the beams can also be found in Visualization 1, Visualization 2, Visualization 3 and Visualization 4 which correspond to the respective VBs in Figs. 9 and 10.

Finally, we show the vortex nature of the generated 3D TCVBs through interferometry. The experimental method is the same as that for the 2D TCVBs illustrated above. The required reference beam is prepared following the design procedure of TCVBs so that the reference beam holds the same distribution as the 3D TCVB to be tested except that the reference beam has a plane phase. Figure 11 gives the interference patterns resulted from the superposition of the 3D TCVBs and their reference counterparts; Figs. 11(a) and 11(b) present the results corresponding to the TCVBs shown in Figs. 9 and 10, respectively. The dynamical evolution of the beam’s interference pattern along the *z* direction can be found in Visualization 5 and Visualization 6. Again, it is seen that the paired light curves in the 3D TCVBs appear to be simultaneously segmented, implying that the two light curves possess the same phase gradient along their respective trajectories.

As theoretically analyzed and experimentally demonstrated above, it is the cylindrical polarization that enables a 2D or 3D curvilinear VB to split into a pair of VBs while retaining their original shape. Since the conversion from a light field with linear polarization to its counterpart field with cylindrical polarization can be easily achieved by employing a commercially-available S-waveplate, our method provides an easy-to-implement solution to the generation of TCVBs. We believe that such a kind of TCVBs holding a highly confined dark region between the paired beams, together with the phase gradient owing to their vortex phase, can benefit optical trapping and guiding of low-refractive-index particles. The study on optical micromanipulation of particles using designed TCVBs is on-going, and relevant results will be reported in the future. Our study also shows that the polarization, in addition to the amplitude and phase, can play a key role in governing the evolution behavior of the light beam. As a consequence, we can make full use of the polarization of light field for beam shaping and tailoring [44].

## 4. Conclusion

In summary, we have proposed a method to generate a novel curvilinear VB, called the TCVBs, whose intensity and phase are prescribed along arbitrary 2D or 3D paired curves. We theoretically elaborate on how a TCVB rather than a single-curve VB is created by the Fourier transform of a cylindrically polarized beam. Simulation and experimental results show that such a vortex beam has two closely located light curves with arbitrary shape independent of topological charge. Moreover, we carried out optical experiment to corroborate topological charge value of the TCVBs, confirming that two light curves of the TCVBs possess the identical vortex nature. The proposed TCVBs can be appreciated in trapping and driving low-refractive-index particle along a 2D or 3D trajectory.

## Funding

National Natural Science Foundation of China ( 91750202, 11922406); National Key Research and Development Program of China (2018YFA0306200, 2017YFA0303700).

## 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. **A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, “Entanglement of the orbital angular momentum states of photons,” Nature **412**(6844), 313–316 (2001). [CrossRef]

**2. **C. Perumangatt, N. Lal, A. Anwar, S. G. Reddy, and R. P. Singh, “Quantum information with even and odd states of orbital angular momentum of light and odd states of orbital angular,” Phys. Lett. A **381**(22), 1858–1865 (2017). [CrossRef]

**3. **J. Wang, “Advances in communications using optical vortices,” Photonics Res. **4**(5), B14–B28 (2016). [CrossRef]

**4. **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]

**5. **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]

**6. **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]

**7. **M. Padgett and R. Bowman, “Tweezers with a twist,” Nat. Photonics **5**(6), 343–348 (2011). [CrossRef]

**8. **J. Ng, Z. F. Lin, and C. T. Chan, “Theory of Optical Trapping by an Optical Vortex Beam,” Phys. Rev. Lett. **104**(10), 103601 (2010). [CrossRef]

**9. **F. Tamburini, G. Anzolin, G. Umbriaco, A. Bianchini, and C. Barbieri, “Overcoming the Rayleigh criterion limit with optical vortices,” Phys. Rev. Lett. **97**(16), 163903 (2006). [CrossRef]

**10. **P. S. Tan, X. C. Yuan, G. H. Yuan, and Q. Wang, “High-resolution wide-field standing-wave surface plasmon resonance fluorescence microscopy with optical vortices,” Appl. Phys. Lett. **97**(24), 241109 (2010). [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. **N. Radwell, R. D. Hawley, J. B. Gotte, and S. Franke-Arnold, “Achromatic vector vortex beams from a glass cone,” Nat. Commun. **7**(1), 10564 (2016). [CrossRef]

**13. **X. W. Wang, Z. Q. Nie, Y. Liang, J. Wang, T. Li, and B. H. Jia, “Recent advances on optical vortex generation,” Nanophotonics **7**(9), 1533–1556 (2018). [CrossRef]

**14. **X. R. Liu, Y. Li, Y. H. Han, D. Deng, and D. Z. Zhu, “High order perfect optical vortex shaping,” Opt. Commun. **435**, 93–96 (2019). [CrossRef]

**15. **A. E. Willner, J. Wang, and H. Huang, “A different angle on light communications,” Science **337**(6095), 655–656 (2012). [CrossRef]

**16. **M. Z. Chen, M. Mazilu, Y. Arita, E. M. Wright, and K. Dholakia, “Dynamics of microparticles trapped in a perfect vortex beam,” Opt. Lett. **38**(22), 4919–4922 (2013). [CrossRef]

**17. **A. S. Ostrovsky, C. Rickenstorff-Parrao, and V. Arrizon, “Generation of the “perfect” optical vortex using a liquid-crystal spatial light modulator,” Opt. Lett. **38**(4), 534–536 (2013). [CrossRef]

**18. **J. Garcia-Garcia, C. Rickenstorff-Parrao, R. Ramos-Garcia, V. Arrizon, and A. S. Ostrovsky, “Simple technique for generating the perfect optical vortex,” Opt. Lett. **39**(18), 5305–5308 (2014). [CrossRef]

**19. **Y. C. Liu, Y. G. Ke, J. X. Zhou, Y. Y. Liu, H. L. Luo, S. C. Wen, and D. Y. Fan, “Generation of perfect vortex and vector beams based on Pancharatnam-Berry phase elements,” Sci. Rep. **7**(1), 44096 (2017). [CrossRef]

**20. **M. V. Jabir, N. A. Chaitanya, A. Aadhi, and G. K. Samanta, “Generation of “perfect” vortex of variable size and its effect in angular spectrum of the down-converted photons,” Sci. Rep. **6**(1), 21877 (2016). [CrossRef]

**21. **V. V. Kotlyar, A. A. Kovalev, and A. P. Porfirev, “Optimal phase element for generating a perfect optical vortex,” J. Opt. Soc. Am. A **33**(12), 2376–2384 (2016). [CrossRef]

**22. **Y. Chen, Z. X. Fang, Y. X. Ren, L. Gong, and R. D. Lu, “Generation and characterization of a perfect vortex beam with a large topological charge through a digital micromirror device,” Appl. Opt. **54**(27), 8030–8035 (2015). [CrossRef]

**23. **A. A. Kovalev, V. V. Kotlyar, and A. P. Porfirev, “A highly efficient element for generating elliptic perfect optical vortices,” Appl. Phys. Lett. **110**(26), 261102 (2017). [CrossRef]

**24. **X. Z. Li, H. X. Ma, C. L. Yin, J. Tang, H. H. Li, M. M. Tang, J. G. Wang, Y. P. Tai, X. F. Li, and Y. S. Wang, “Controllable mode transformation in perfect optical vortices,” Opt. Express **26**(2), 651–662 (2018). [CrossRef]

**25. **L. Li, C. L. Chang, C. J. Yuan, S. T. Feng, S. P. Nie, Z. C. Ren, H. T. Wang, and J. P. Ding, “High efficiency generation of tunable ellipse perfect vector beams,” Photonics Res. **6**(12), 1116–1123 (2018). [CrossRef]

**26. **R. Xu, P. Chen, J. Tang, W. Duan, S. J. Ge, L. L. Ma, R. X. Wu, W. Hu, and Y. Q. Lu, “Perfect Higher-Order Poincare Sphere Beams from Digitalized Geometric Phases,” Phys. Rev. A **10**(3), 034061 (2018). [CrossRef]

**27. **Y. C. Zhang, W. W. Liu, J. Gao, and X. D. Yang, “Generating Focused 3D Perfect Vortex Beams By Plasmonic Metasurfaces,” Adv. Opt. Mater. **6**(4), 1701228 (2018). [CrossRef]

**28. **M. K. Karahroudi, S. A. Moosavi, A. Mobashery, B. Parmoon, and H. Saghafifar, “Performance evaluation of perfect transmission of optical vortices in an underwater optical communication system,” Appl. Opt. **57**(30), 9148–9154 (2018). [CrossRef]

**29. **J. Pinnell, V. Rodríguez-Fajardo, and A. Forbes, “How perfect are perfect vortex beams?” Opt. Lett. **44**(22), 5614–5617 (2019). [CrossRef]

**30. **E. Spyratou, M. Makropoulou, E. Mourelatou, and C. Demetzos, “Biophotonic techniques for manipulation and characterization of drug delivery nanosystems in cancer therapy,” Cancer Lett. **327**(1-2), 111–122 (2012). [CrossRef]

**31. **V. Garces-Chavez, K. Volke-Sepulveda, S. Chavez-Cerda, W. Sibbett, and K. Dholakia, “Transfer of orbital angular momentum to an optically trapped low-index particle,” Phys. Rev. A **66**(6), 063402 (2002). [CrossRef]

**32. **Y. S. Liang, M. Lei, S. H. Yan, M. M. Li, Y. A. Cai, Z. J. Wang, X. H. Yu, and B. L. Yao, “Rotating of low-refractive-index microparticles with a quasi-perfect optical vortex,” Appl. Opt. **57**(1), 79–84 (2018). [CrossRef]

**33. **Y. S. Liang, S. H. Yan, M. R. He, M. M. Li, Y. N. Cai, Z. J. Wang, M. Lei, and B. L. Yao, “Generation of a double-ring perfect optical vortex by the Fourier transform of azimuthally polarized Bessel beams,” Opt. Lett. **44**(6), 1504–1507 (2019). [CrossRef]

**34. **J. J. Yu, C. F. Miao, J. Wu, and C. H. Zhou, “Circular Dammann gratings for enhanced control of the ring profile of perfect optical vortices,” Photonics Res. **8**(5), 648–658 (2020). [CrossRef]

**35. **C. Rickenstorff, “L. del Carmen Gomez-Pavon, C. Teresa Sosa-Sanchez, and G. Silva-Ortigoza, “Paraxial and tightly focused behaviour of the double ring perfect optical vortex,” Opt. Express **28**(19), 28713–28726 (2020). [CrossRef]

**36. **S. Khonina and A. Porfirev, “Generation of multi-contour plane curves using vortex beams,” Optik **229**, 166299 (2021). [CrossRef]

**37. **J. A. Rodrigo, T. Alieva, E. Abramochkin, and I. Castro, “Shaping of light beams along curves in three dimensions,” Opt. Express **21**(18), 20544–20555 (2013). [CrossRef]

**38. **C. L. Chang, Y. Gao, J. P. Xia, S. P. Nie, and J. P. Ding, “Shaping of optical vector beams in three dimensions,” Opt. Lett. **42**(19), 3884–3887 (2017). [CrossRef]

**39. **J. A. Rodrigo and T. Alieva, “Vector polymorphic beam,” Sci. Rep. **8**(1), 7698 (2018). [CrossRef]

**40. **J. Gielis, “A generic geometric transformation that unifies a wide range of natural and abstract shapes,” Am. J. Bot. **90**(3), 333–338 (2003). [CrossRef]

**41. **V. Arrizon, U. Ruiz, R. Carrada, and L. A. Gonzalez, “Pixelated phase computer holograms for the accurate encoding of scalar complex fields,” J. Opt. Soc. Am. A **24**(11), 3500–3507 (2007). [CrossRef]

**42. **Z. Chen, T. Zeng, B. Qian, and J. Ding, “Complete shaping of optical vector beams,” Opt. Express **23**(14), 17701–17710 (2015). [CrossRef]

**43. **X.-L. Wang, J. Ding, W.-J. Ni, C.-S. Guo, and H.-T. Wang, “Generation of arbitrary vector beams with a spatial light modulator and a common path interferometric arrangement,” Opt. Lett. **32**(24), 3549–3551 (2007). [CrossRef]

**44. **H. Chen, Z. Zheng, B.-F. Zhang, J. Ding, and H.-T. Wang, “Polarization structuring of focused field through polarization-only modulation of incident beam,” Opt. Lett. **35**(16), 2825–2827 (2010). [CrossRef]