## Abstract

We propose and experimentally demonstrate annular arrayed-Airy beams (AAABs) carrying vortex arrays by combining multiple beams. The propagation dynamics and abrupt autofocusing property are studied. The focal intensity can be greatly increased by two orders of magnitude by increasing vortex array number. Furthermore, the autofocusing property is also enhanced significantly. This tightly autofocusing property would be advantageous for the generation of high intensity laser, optical manipulation, medical treatments, and nonlinear effects.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

Airy beams have attracted significant attention since they were first predicted [1] and observed in 2007 [2]. They have exhibited a variety of intriguing properties, including diffraction-free, self-healing and self-acceleration. Perhaps the most prominent property is their ability to accelerate along parabolic trajectories. The lateral acceleration causes a type of abruptly autofocusing beams (AABs) [3]. Classic AABs are referred to as circular Airy beams (CABs) [3–6] described by a circular Airy function, or circular caustic beams [7–10]. Recently, Yuan *et al* generated abruptly autofocusing CABs with an annular arrayed-waveguide [11]. The focal intensity pattern of classic CABs is a solid focal spot, which is not applicable for trapping micro particles whose refractive index is less than that of the surrounding media. And also, such beams are incapable of rotating particles. Consequently, CABs with optical vortices (OVs) were proposed [12,13]. OVs are characterized by hollow-core intensity distributions, phase singularities and orbital angular momentum (OAM) [14]. The OAM carried by individual photons can deliver energy to particles and cause them to rotate [15]. Besides, other autofocusing beams with special intensity [16,17], phase [18,19], or polarization [20], have been studied. The developments of CABs have attracted significant interest owing to their unusual autofocusing property, which is especially beneficial for applications in optical trapping [21], light bullet [22], optical communication [23], medical treatments [24], multi-photon polymerization [25], and nonlinear effects [26].

The autofocusing property of light beams is an issue of great practical importance [27], especially for the generation of high intensity laser beams. Lu *et al*. demonstrated that the abruptly autofocusing property can be increased up to a factor of three by blocking the first ring of the CAB [28]. Subsequently, modified circular Airy beam was proposed to enhance the autofocusing property by introducing two apodization factors [29]. Recently, the autofocusing property for symmetric Airy beam (SAB) [30] were enhanced compared to that of autofocusing dual Airy beam (DAB) [31]. However, these beams including CAB, SAB and DAB are single beams, and they do not improve total laser intensity. Recently, the superposition of one-dimensional (1D) Airy beams was discussed [32].

In this letter, different from classic CABs, we propose a 2D multi-Airy beam synthesis approach for constructing annular arrayed-Airy beams (AAABs) carrying vortex arrays (VAs) [33,34], to improve greatly total laser intensity. Furthermore, the autofocusing property will significantly be enhanced. These excellent properties could be especially advantageous for the generation of high intensity laser, optical trapping, biomedical treatment.

## 2. Theory

The generation model for an AAAB carrying a spiral phase is depicted in Fig. 1(a). Multiple Airy beams are distributed uniformly along a circle, to generate an AAAB, where, the main lobes of Airy beams form a main annulus with a radius of *r*_{0}. Then, the AAAB carrying an OV can be generated by imposing an OV [denoted by the white dot in Fig. 1(a)] on the AAAB, and expressed in Cartesian coordinates as [35]:

Here, *j* = 1, 2, 3…, is a summation index, *n* is the Airy array number, *d* denotes the transverse displacement, and *r*_{0} = $\sqrt{2}d$. Here, *Ai*($\cdot $) denotes an Airy function and 0<*a*<1 is a decay parameter which determines the propagation distance. *cAi*(*X _{j}/w_{0}*)exp(

*aX*) denotes the

_{j}/w_{0}*j*one dimensional Airy beam which is obtained by rotating

^{th}*cAi*(

*X*

_{1}/

*w*

_{0})exp(

*aX*

_{1}/

*w*

_{0}) by 2π(

*j-*1)/

*n*radians. The term

*cAi*(

*Y*/

_{j}*w*

_{0})exp(

*aY*/

_{j}*w*

_{0}) represents another 1D Airy beam, which is perpendicular to

*cAi*(

*X*/

_{j}*w*

_{0})exp(

*aX*/

_{j}*w*

_{0}).

*s*=

_{x}*x*/

*x*

_{0}and

*s*=

_{y}*y*/

*y*

_{0}are dimensionless transverse and longitudinal coordinates, respectively. The parameter

*c*is a constant related to the beam power [5],

*w*

_{0}is a scaling factor,

*i*= $\sqrt{-1}$,

*l*denotes the topological charge of the OVs,

*sign*($\cdot $) is the sign function which determines clockwise or anti-clockwise vorticity, (

*S*,

_{x}*S*) denote the position of the OVs.

_{y}The evolution of such beam can be expressed as a function of the normalized propagation distance *ξ* = *z*/*kx*_{0}^{2} using a Fresnel diffraction integral, where *k* = 2*π*/*λ* is the wavenumber. Figure 2(a) exhibits the side-view propagation of an AAAB carrying an OV for the case of *l* = 1 and *n* = 100, where the OV is imposed at the origin (0,0). Here, we observe that these Airy beams propagate along parabolic trajectories and autofocus at the same focal point, where the intensity abruptly increases due to lateral acceleration of Airy beams. Figure 2(b) displays a computer-generated hologram imposed on the SLM, which is generated using the interference between a plane wave and the AAAB carrying an OV. Figures 2(c1)-2(c3) display numerical intensity distributions in *ξ* = 0, 150, and 208 [focal length *ξ _{f}* ] planes, respectively. The main annulus radius decreases with increasing propagation distance before reaching the focal plane. The AAAB carrying an OV evolves into a focused annulus with zero intensity in its center at the focal plane [the enlarged inset image in Fig. 2(c3)]. Such structured light field with a zero-intensity area surrounded by a high-intensity annulus is particularly suitable for rotating and trapping microparticles. Figures 2(d1)-2(d3) depict phase structures corresponding to Figs. 2(c1)-2(c3). These patterns clearly display phase singularities which rotate clockwise during propagation before reaching the focal plane.

AAABs carrying an OV can be generated using a computer-controlled reflective spatial light modulator (SLM), as shown in Fig. 1(b). A He-Ne laser (632 nm) is used to produce a collimated and expanded Gaussian beam with a FWHM of 8.6 mm. The Gaussian beam is then reflected by the SLM (pixel size of 8μm; 1920 × 1080 pixels), on which a computer-generated hologram (CGH) is imposed. The modulated and reflected beams perform a Fourier transform using a Fourier lens (FLs) with a focal length of *f* = 300 mm. The AAAB with an OV is then recorded using a CCD (pixel size of 1.4 μm; 4608 × 3288 pixels) located on the back focal plane of FLs. The CGH image is formed by the interference between the plane wave and ${\varphi}_{n}({s}_{x},{s}_{y},0)$.

Experimental intensity distributions in *z* = 0,149 and 206 mm planes are shown in Figs. 2(e1)-2(e3), respectively. The size this focused annulus is approximately 9 μm in the focal plane [Fig. 2(e3)]. Such optical field structures also allow such beams to trap living cells while reducing thermal damage on these living cells. These experimental results agree with numerical simulations. The following parameters are assumed in all simulations unless otherwise specified: *a* = 0.1, *w*_{0} = 0.08, *r*_{0} = 0.85, *λ* = 632.8 × 10^{−6}mm, *x*_{0} = *y*_{0} = 10μm, and *c* = 100. Figure 2(a) clearly depicts the autofocusing property of an AAAB with an OV. To describe the autofocusing property, we define the maximum intensity of the initial plane as *I*_{0} [Fig. 2(c1)], and the maximum intensity at an arbitrary propagation plane is *I*_{m} [Fig. 2(c2)]. The autofocusing property during propagation can be described by the ratio of *K* = *I*_{m}*/I*_{0}. The maximum intensity at focal plane is considered as *I*_{max} [Fig. 2(c3)]. This value *I*_{m} decreases with an oscillatory declination after the focal point, caused by the interaction between subsequent rings [Fig. 2(a)].

Figures 3(a) and 3(b) provide a comparison of the initial and focal intensity profiles, respectively, in the cases of *l* = 0,1,2. Their autofocusing properties (*K*) are plotted in Fig. 3(c). Where, all other parameters are the same as those in Fig. 2. Here, we observe that *K* at the focal plane is 50 for *l* = 1, however, it is approximately 110 for *l* = 2. *I*_{max} becomes nearly twice for *l* = 2 compared to *l* = 1 in the case. Consequently, the autofocusing property can be significantly enhanced by increasing the topological charge in this case. The abruptly increased *K* in focal plane is mainly because the beam with a greater *l* will cause a smaller *I*_{0} and a greater *I*_{max}.

We also see that *K*-curves remain almost to be uniform in the initial propagation stage. This is because these Airy beams accelerate laterally without diffraction, and the intensity *I*_{m} and *I*_{0} remains unchanged, such that, *K*-curve appear linear at initial stage. Subsequently, it abruptly increases at the focal plane, and here *I*_{max} for *l* = 2 can be enhanced by a factor of 110, which is much higher than that of AAAB. Finally, *K* decreases oscillatorily. This can be explained that those sidelobes of Airy beams encounter and interfere with each other after the focal length, the intensity *I*_{m} and *I*_{0} fluctuate greatly, consequently, *K*-curve appears oscillating [Fig. 3(c)]. The zero-intensity area in focal plane becomes larger with increasing *l*. As expected, the ring-shaped intensity pattern in focal plane evolves into a focal point when *l* = 0 [Fig. 3(b)]. The normalized focal length *ξ _{f}* is almost consistent despite different

*l*[Fig. 3(c)].

Figures 4(a) and 4(b) exhibit the initial and focal intensity profiles, respectively, in the cases of *n* = 50 and 100. Here, *I*_{max} is nearly 55 times *I*_{0} for *n* = 100. Moreover, both *I*_{0} and *I*_{max} increase almost by a factor of 5 for *n* = 100, compared to *n* = 50 in the case. Such that, multi-Airy beams synthesis is an effective approach for enhancing laser intensity. Figure 4(c) depicts the comparison of autofocusing property during propagation for *n* = 50 and 100. We see that the two *K*-curves coincide completely. This is because both *I*_{0} and *I*_{m} increase in the same proportion with increasing Airy array number *n*, consequently, the ratio *I*_{m}*/I*_{0} remains unchanged.

## 3. Results and discussions

To improve the autofocusing property of such beams, we further develop AAABs carrying VAs. Where, multiple optical vortices are arranged uniformly and symmetrically along a circle, to form a circular VA, and are imposed on an AAAB. Then, such beam carrying a VA can be expressed as [35]:

Here, *m* = 8 is VA number (an even integer). $({S}_{{x}_{t}},{S}_{{y}_{t}})$ denote the positions of the *t*^{th} off-axis vortex. The eight off-axis vortices are distributed uniformly and symmetrically along a circle with a radius of *r*_{1} = 0.5 [Fig. 5(d1)]. Here, all *l* are identical including the vorticity in this case (*l* = 1 and *n* = 100). Figures 5(a) and 5(b) exhibit the side-view propagation and the computer-generated hologram for generating such beam. Clearly, such beam also possesses abruptly autofocusing property. Figures 5(c1)-5(c3) display the intensity distributions for *ξ* = 0, 150 and 208 planes, respectively. The initial intensity pattern [Fig. 5(c1)] is obviously different from that of Fig. 2(c1), and we find that the position of *I*_{0} is no longer located on the internal main annulus, but the peripheral annulus. This is caused by the off-axis VAs, which compel the energy of internal main annulus to move toward outside. Because the center intensity of vortex must be zero, such that this property inevitably compels the internal intensity move outside when these vortices are located near the internal ring. Meanwhile, the hollow area [Fig. 5(c3)] in focal plane become larger compared to that of Fig. 2(c3) as a result of the off-axis VAs. The corresponding phase patterns are plotted in Figs. 5(d1)-5(d3), where the white circles in Fig. 5(d1) depict the positions of the phase singularities. We see that these vortices rotate around and move toward the optical axis during propagation, i.e., they also follow a curved trajectory. However, these vortices do not meet at a same focal point but are offset by a small spacing, which is reflected by the size of the zero-intensity area [Fig. 5(c3)]. Figures 5(e1)-5(e3) exhibit the experimental intensity distributions in z = 0,149,206 mm planes.

Figure 6 plots autofocusing properties for VAs with identical vorticity (*n* = 100). The red *K*-curve denotes the VAs with *m* = 8 for the case of *a* = 0.35, and the blue and green *K*-curves are the VAs with *m* = 2 and an on-axis vortex, respectively, for the case of *a* = 0.1. We see that VA for *m* = 8 increase *I*_{max} by a factor of 212 compared to *I*_{0} in this case, and the value *K* in focal plane is much higher than that for *m* = 2, and nearly 4.2 times that of an on-axis vortex. The sharply increased *K* in focal plane is because *I*_{0} becomes smaller owing the drive by the VAs, meanwhile, *I*_{max} increases due to the dual action of Airy self-acceleration and the drive of VAs. Such that, autofocusing property can be significantly enhanced by introducing VAs and controlling appropriately the decay factor *a*. The abrupt increase in intensity carrying OAM contribute to trap living particles.

Further on, we assume the vorticity for the two symmetric distributed vortices in VAs is opposite. Specifically, the two white circles [Fig. 7(a1)] denote the two symmetrically distributed vortex pairs, which is characterized by white arrow lines. Other parameters are same as Fig. 5. Figures 7(a1)-7(a3) show the phase patterns in *ξ* = 0, 150, and 208 planes, respectively, and corresponding intensity distributions are displayed in Figs. 7(b1)-7(b3). We note that the VA interact during propagation, and then they completely converge in focal plane, finally, the phase singularities disappear [Fig. 7(a3)]. Such that, the focal intensity pattern evolves into a focal spot [Fig. 7(b3)], instead of an annulus. This is mainly because the vortex pairs blend fully in the focal plane, and finally annihilate owing to their opposite vorticity, and they no longer appear after the focal plane. Figures 7(c1)-7(c3) show experimental results at *z* = 0,149,206mm, respectively. The diameter of the generated focal spot can be is approximately 6 μm, and its size can be of the order of wavelength by using a suitable numerical aperture. This tightly autofocusing property would be especially beneficial for biomedical treatment, optical trapping and nonlinear effects.

Figure 8 depicts the comparison of *K*-curves in the case of opposite vorticity. Where, other parameters are identical to Fig. 6. We see that value *K* for *m* = 8 in focal plane is much larger than that *m* = 0 (correspond to an AAAB), and approximately 8 times that of AAAB. This enhanced abrupt autofocusing property is especially advantageous for medical treatments and the ignition of nonlinear effects. Additionally, those cases that a vortex or VAs is imposed at outside of the main annulus are more complicated, and would be discussed in future. The aforementioned results can be extended to non-paraxial regime [36,37], and further we would exhibit these results in our future work.

## 4. Conclusions

In conclusion, we have developed a multi-Airy beam synthesis approach for generating autofocusing AAABs carrying VAs. The focal intensity can be greatly improved by two orders of magnitude. We demonstrate that the autofocusing property can be significantly enhanced by introducing VAs. These excellent properties would be helpful for the generation of high intensity laser, optical trapping, biomedical treatment.

## Funding

Zhejiang Provincial Natural Science Foundation (Grant No. LY17A040002), the National Natural Science Foundation of China (Grant No. 61377014 and 11674288), the Key Laboratory of Optical Field Manipulation of Zhejiang Province.

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