## Abstract

We report the autofocusing behaviors of ring Airy beams (RABs) embedded with two kinds of off-axial vortex singularities. The influences of embedded positions and topological charges of point and *r* vortices on the autofocusing dynamic are numerically and experimentally investigated. The results show that, for the first-order vortex, the embedded position significantly affects the focal field, and once the singularity is located on the main ring of RAB, the symmetric Bessel profile of the focal field will be broken, otherwise the Bessel-like focus can self-heal at the focal plane. However, for the higher-order vortex embedded near the main ring, it will split into several fundamental vortices and then separate with each other along the radial direction under the interaction with the RAB background. Our results hold potential for the practical application of RABs in the atmosphere and other propagation systems with perturbation and even singularities.

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

## 1. Introduction

Recently, researches of ring Airy beams (RABs) with distinctive abruptly autofocusing characteristics are in the ascendant [1–8]. Such a beam manifests itself as maintaining a low-intensity profile before the focal region and abruptly focusing with its energy increased by several orders of magnitude, which was experimentally confirmed in 2011 [1]. This unique property could be implemented in biomedical treatment without damaging the material before the specific target and nonlinear optical processes like optical filamentation [2]. Using the autofocusing effect of RABs to trap and guide microparticles has also been demonstrated [3–6]. However, due to the existent of the Brownian force, transversely trapping the Rayleigh particle within the primary or side rings is difficult to achieve in the region before the focal point [6]. Further researches on RABs about laser ablation, light-bullet, and periodic breather soliton, may provide novel methods for optical micromachining, attosecond physics, and remote sensing [7,8].

Optical vortices (OVs), as a typical phase singularity with null intensity, carrying orbital angular momenta (OAMs) associated with spiral phases [9], have been extensively investigated and applied to various areas, such as optical imaging, optical manipulation, optical communication, quantum information [9–17]. Thereinto, research of propagation dynamic is fundamental and essential for extending the potential applications. Consequently, the evolution dynamics of vortex singularities and the interaction with structured backgrounds have been studied exhaustively [14,15]. Recently, utilizing vortex singularities to manipulate the abruptly autofocusing of RABs has attracted wide attentions, and numerous intriguing phenomena have been proposed [18–20]. For instance, the off-axis *r* vortices nested in RAB will be forced to the region near the center because of the autofocusing property, and yet a pair of vortices will overlap or annihilate at the focal plane if their topological charges are the same or opposite [18]. Likewise, ring Airy Gaussian vortex beams with vortex pairs show similar behaviors during the process of propagation [19]. However, the evolution of off-axial vortex singularity nested near the main ring and the resulted autofocusing of RABs have never been reported, which are of importance for the practical applications of RABs referring to the propagation in atmosphere and other systems with perturbation and even singularity.

In this paper, we investigate the autofocusing of RABs embedded with point and *r* vortex singularities, for the cases that the off-axial vortex singularities are located near the main ring of RABs. The mutual influence of singularity evolution and RABs autofocusing are numerically analyzed and experimentally demonstrated, by discussing the off-axial distance and topological charges of vortex singularities. The vortex singularities near the main ring can significantly affect the autofocusing of RAB, resulting distinctly different focal fields dependent on the singularity kinds and topological charge, which are significantly different from the usual cases of paraxial vortices.

## 2. RABs embedded with vortex singularities

The RAB embedded with vortex singularities can be expressed in the cylindrical coordinates (*r*, *φ*, *z*) as

*E*

_{0}is the amplitude, Ai(

^{.}) depicts the Airy function,

*r*

_{0}determines the radius of the main ring,

*w*is a scaling factor determining the ring width,

*a*is the decaying parameter,

*A*(

*r*,

*φ*,

*z*) denotes the complex amplitude of vortex singularities,

*k*=2π/

*λ*is the wave number and

*λ*is the wavelength. Generally, there are two common types of vortex singularities, of which the complex amplitudes can be expressed as [14]

*R*,

*φ*

_{0}) denotes the location of phase singularity, and

*m*is the topological charge which determines the amount of 2π-phase shifts around the singularity. Equation (2) describes the point vortex whose dark core could be considered as a tiny point. Equation (3) depicts the

*r*vortex whose core size is characterized by its topological charge.

As above description, the evolutions of RAB with single and multiple vortex singularities embedded at the inner position with low energy have been well studied [18–20], and the results show that the autofocusing energy flow constrains and removes these vortex singularities to the axial position, exhibiting the combination or annihilation behaviors. However, it has been demonstrated that the intensity background can significantly affect the dynamic of the vortex [14,15,18,19]. This means that once the embedded vortices are located at the position with high intensity, they will behave distinct evolution dynamics. Therefore, we here focus on the evolution of RAB embedded with vortices near the main lobe, i.e., *R*≈*r*_{0}.

## 3. Experiment and results

#### 3.1 Autofocusing of RAB embedded with a point vortex

For the sake of precisely controlling the off-axial position of a vortex singularity, we directly generate a RAB embedded with vortex singularities in experiment. The experimental setup is illustrated in Fig. 1(a). A horizontally polarized input beam (from Ar^{3+} laser with wavelength *λ *= 514.5 nm) is firstly divided into two parts by a beam splitter (BS), the transmitted part is used as a reference beam, the reflected one is projected on a reflect-type spatial light modulator (SLM) which carries a two-dimensional computer-generated hologram (CGH) [21–23], then orderly passes through the BS again and a 4*f* filter system, which allows the first diffracting order passing through. In practice, the SLM is placed at the front focal plane of the 4*f* system, and at the back focal plane, a RAB embedded with vortex singularities are obtained. To measure the phase structure, the reference beam is reflected by two mirrors (M_{1} and M_{2}), and then interfere with the object beam. The interferogram is observed by a charge-coupled device (CCD), from which the phase structure is obtained according to the digital holography [23]. In experiment, the parameters are selected as *r*_{0} *=* 876.65µm, *w* *=* 51.1381µm, *a* *=* 0.0486. For this special case, the radius of the main ring’s peak is *q* = 929.25µm, and we regard the value of *q* as the benchmark of the off-axis distance. Because of the rotation symmetry of RAB, here we set the displacement of the off-axial vortex on the *x*-axis, i.e., *φ*_{0}=0.

Figure 1(b) shows the amplitude profile of RAB along the *x*-axis, wherein five points depict the locations of vortex singularities. For comparison, we first observe the normal focusing of RAB without vortex singularity. Figures 1(c) and 1(d) show the measured distributions of intensity and phase at the initial plane, respectively. In free space, all lobes propagate in a parabolic trajectory before the focus (*z*=41.63 cm) [1], and then focus into a zero-order Bessel field at the focal plane, as shown in Fig. 1(e) [3].

As we all know, the RAB has a radial cubic phase and its energy flow points to the beam center at the initial plane. During the propagation, the convergence of energy flow leads to autofocusing behavior. Compared with the evenly spaced annular phase structure of a Bessel beam [24], the phase distribution of RAB is parabolic-like, which means that the RAB has a very wide (low slope) annular region (flat phase region) near the main ring, as shown in Fig. 1(d), which has a phase gradient from inside to outside when *R *< *q*, and yet opposite gradient in the outside region.

Next, we observe the influences of off-axial point vortex on the focal field. The intensity distribution and phase contour of the RAB embedded with a first-order point vortex are theoretically calculated and experimentally confirmed in Fig. 2. The top row corresponds to the measured intensity and phase distributions at the initial plane, where the white dashed circles depict the locations of singularities, and the relative positions on the profile of the RAB amplitude are represented by the black spots in Fig. 1(b). The middle row displays the measured intensity and phase distributions at the focal plane, and it is shown that as *R* increases, the focus could be observed at *z*=41.63 cm, this means that the focal distance does not change. Figures 2(a_{3})–2(e_{3}) and 2(a_{4})–2(e_{4}) show the numerically simulated intensity and phase profiles at the focal plane, respectively. When *R*=0, the focal field presents a hollow ring structure instead of a solid spot, meaning that the point vortex always stays on the *z*-axis for the optical forces [25] canceling out. This phenomenon was reported by Jiang [18], but here we pay more attention to the situation that the vortex nested near the main lobe, where the field intensity is significantly enhanced. When *R* radially increases but below *q*, i.e., *R *< *q*, the symmetric Bessel profile is broken, and yet the dark core is still located near the center of the focal plane. However, when *R *≥ *q*, as it increases, the vortex singularity whose shape looks like a fork moves away from the center. Remarkably, the focal field appears a Bessel-like pattern when *R* equals to 1.2*q*, which indicates the intensity pattern will self-heal under the effects of off-axial fundamental phase singularity. From the numerical simulation, we can see that the results are consistent with the outcome of the experiment.

The phenomenon elucidates that the embedded position of phase singularity could significantly influence the autofocusing of RAB. From the circumstances of the focal plane, we find that the fundamental point vortex inside the main ring will be forced to the region near the center, because of the inward energy flow caused by the radial cubic phase, but the point vortex outside the main ring couldn’t approach to the center, it means that the flat phase region could be regarded as a barrier-like effect.

The situations corresponding to the RABs embedded with higher-order point vortices are shown in Fig. 3. The top and below rows depict the second-order and fifth-order vortices, respectively, and the corresponding numerical simulations of the second-order vortices are shown in the middle row. Obviously, the dark core at the center of the focal plane is bigger than the status of the first-order one, owing to the increase of the topological charge. When *R*=0, similarly, the autofocusing energy flow constrains the vortex at the axial position. Nevertheless, different from the traditional phenomena, when the displacement of vortex singularity near the main lobe, i.e., *R*≈*q*, the innermost ring at the focal plane becomes smaller, indicating that the high-order vortex split into multiple low-order vortices, which breaks the integrality of the Airy ring, as shown in the top row of Fig. 3.

From the phase distributions, we find that the second-order vortex splits into two fundamental vortices, thereinto, one remains near the center and another one moves along the *x*-axis as the off-axis displacement increases. It is noteworthy that, these topological charges are conserved, as mentioned in [26]. In addition, from the insets of Figs. 3(d_{1})–3(e_{1}) and 3(d_{2})–3(e_{2}), we can find that the energy is concentrated on the top of the innermost ring, meaning that the fundamental vortex maintains the dark core actually at the bottom right of the center. Whereas for the fifth-order point vortex, the hollow core size decreases with the increase of vortex displacement, indicating the reduction of vortex number. On the other hand, the phase shifts along the white dashed circles gradually reduce from 10π [Fig. 3(f)] to 4π [Fig. 3(j)], which similarly confirms the reduction of vortex number. Notably, the embedded location of the vortex determines the degree of separation of the singularity.

We consider that the propagation of high-order point vortex is instable, under the effect of autofocusing and the sloping field of RAB, the higher-order vortex splits into multiple fundamental vortices which repel each other. Differently, the interaction between two positive OVs in a Gaussian background has been reported that, these vortices move along the direction perpendicular to the sloping background [14]. However, for the RAB background, the gradient of the amplitude near the main ring is much greater than that of the interior region, as a result, the vortex singularity couldn’t get across this barrier. In addition, the direction of optical forces outside of the flat phase region is opposite to the previous case, which breaks the spiral phase structure, and the focal field energy concentrates on the upper part of the center, which means that the singularities near the center clockwise rotate π/2 in the autofocusing process [26]. The increasing of the off-axis distance and the topological charge aggravates the intensity symmetry breaking [27]. Compared the focal intensity patterns of RAB with the fractional vortex Bessel beam [28,29], although the asymmetric shapes of them are similar, the fundamental singularities distributions of the latter are dispersed. And there is a transverse gap caused by the abrupt phase change on the *x*-axis, the intensity near the gap is bigger than that in the other regions, which is opposite to the case in our paper.

#### 3.2 Autofocusing of RAB embedded with an r vortex

Unlike the phase perturbation of a point vortex, *r* vortex modulates both of the phase and the amplitude of the beams. Under the same consideration, we experimentally observe the autofocusing field of RAB embedded with first-order *r* vortex, and the corresponding results are shown in Fig. 4. Where the top and below rows are the distributions at the initial and focal planes, respectively. From the insets of Figs. 4(b_{1})–4(e_{1}), we can find that the amplitude modulation of *r* vortex breaks the rotational symmetry of the RAB when *R*≠0. As *R* increases, the autofocusing intensity distributions barely vary, and the focal fields present a dark core near the center, which is similar to the results in [18], where the off-axis *r* vortex in the main ring will move to the center at the focal plane. Here, we consider that the barrier-like effect of the flat phase region may be compromised by the symmetric broken induced by the *r* vortex. Because the gradient of the intensity is reduced, the autofocusing effect is strong enough to push the vortex to the center in all cases.

The experiment results for the case of second-order *r* vortex are shown in Fig. 5. Where the top and below rows correspond to the distributions at the initial and focal planes, respectively. Obviously, the second-order vortex splits into two fundamental vortices at the focal plane, as shown in Fig. 5(b_{2}). Likewise, the focal field keeps an identical profile with the variation of off-axial distance. In contrast to the first-order *r* vortex, as the increase of topological charge, the symmetry broken is even more serious, as a result, one fundamental vortex moves away from the center. In addition, the rotations of the singularities are in accordance with the case of the point vortices, which makes the energy concentrates on the top of the center.

Compared with the autofocusing characteristics of the RAB embedded with second-order point vortex, it can be seen that the effect of background field causes the splitting phenomenon, and the singularities near the coordinate origin will rotate with the increase of propagating distance. The remarkable difference appears in the autofocusing of RABs embedded with the first-order vortex, which are the most possible cases in the real environment, that the *r* vortex couldn’t reappear Bessel-like pattern even *R*>1.2*q*. As the location of singularity is away from the main ring, the autofocusing effect is more weaken and the distance between two separated vortices becomes farther. Therefore, research of RAB embedded with kinds of OVs may be instrumental in the development of optical perturbation and free-space communication. For example, we could load information to the beams by the OVs, and then measure the phase contour and get the message carried by the OVs arbitrarily.

## 4. Conclusion

In conclusion, the topological charge, vortex type, and the off-axial position of the vortex singularities significantly influence the autofocusing behaviors of RAB. The interaction with the RAB background constrains the first-order point vortex and removes it toward the center, but the barrier-like effect of the flat phase region blocks the movement of the vortices that split from the high-order vortex, and the singularities near the center always rotate π/2 when the beam comes to the focal field. Nevertheless, the off-axial distance of the *r* vortex hardly affects the autofocusing intensity distributions regardless of the topological charge. We believe that this study about the propagation dynamics of RAB embedded with the OVs could contribute to perfect the basic theory of singular optics.

## Funding

National Natural Science Foundation of China (11634010, 11774289, 61675168); National Key Research and Development Program of China (2017YFA0303800); Joint Fund of the National Natural Science Foundation Committee of China Academy of Engineering Physics (U1630125); Seed Foundation of Innovation and Creation for Graduate Students in Northwestern Polytechnical University, China(ZZ2019217).

## Disclosures

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

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