Generation and control of the circle Olver beams

Ziyu Wang; Ziyu Wang; Haobin Yang; Haobin Yang; Danlin Xu; Danlin Xu; Zehong Liang; Zehong Liang; Lingling Shui; Lingling Shui; Lingling Shui; Dongmei Deng; Dongmei Deng

doi:10.1364/OE.483433

1. Introduction

The Olver Beams (OBs), a new class of non-diffracting beams that are derived by solving the scalar Helmholtz-equation, were initially described by Abdelmajid Belafhal and his co-authors in 2015 [1]. More freshly, in 2019, Ioannis M. Besieris pointed out that the Olver functions can be considered as “incomplete” elementary catastrophe integrals [2]. The dynamics of the cusp, the swallowtail and the butterfly optical catastrophes have been well documented recently [3,4]. So we believe that the OBs have potential applications to be explored like the structured light beams based on optical catastrophe, which would be used in particle trapping [5].

Furthermore, the innovative beams come in a variety of orders, such as the finite Airy beam, which is considered to be the zeroth order of the OBs, was originally proposed by G. A. Siviloglou [6,7]. Since then, the Airy beam has gained much attention for some intriguing characteristics such as self-acceleration [8], nondiffraction and self-healing [9]. Inspired by this excellent work, many Airy distortions including the abruptly auto-focusing circle Airy beam (CAB) [10–12], the radially symmetric Airy beam [13] were explored. There was a review article in 2019 that summarized the properties of the Airy beam [14]. Typically, the CAB allows the intensity up right before the focal point where it suddenly enhances by several orders of magnitude [10,11] while keeping an almost constant intensity profile until that very moment. It is demonstrated that this derivative CAB can be utilized in the areas of laser biomedical treatment [9], material processing and so on [14].

On the other hand, optical tweezers which use an optical force to capture and guide particles have recently drawn a lot of interest from the academic community [15–17]. In addition, some reports devote to finding analytical solutions to wave equations to generate spatially accelerating beams for manipulation like the nonparaxial Mathieu and Weber beams propagating along elliptical and parabolic trajectories respectively are proposed by solving the Helmholtz function [18]. Furthermore, some structured light beams can carry orbital angular momentum opening up a new frontier for optical manipulation [16,19,20]. Particularly, demonstrations of particle manipulation can be accomplished using the design of cylindrical vector beams [21], where light-induced force plays an important role in guiding particles on curved trajectories [22] and the concentrated energy distribution on the main regime forms a stable optical trapping force. Due to the auto-focusing nature of such beams, the experimental setup can be further optimized by decreasing bulky setups [23]. Therefore, the OBs as a novel family class of the conical beams and at the same time as a generalization of the ordinary Airy beams, there remains a need to take a deeper look at the OBs and extend the Olver function to cylindrical coordinates. The question we address is whether we can use the modified circle Olver beams (COBs) to achieve auto-focusing, self-healing and even particle trapping.

Therefore, in this paper, by methods of extending the Olver function to cylindrical coordinates, which form the energy redistribution, the COBs endowed with abruptly auto-focusing properties can be generated. The structure of the article is as follows. In Sec. 2, the theoretical method to generate the COBs is shown in detail. Following, we analyze the propagation properties of the COBs with different orders and the effect of the parameters on the COBs, and we confirm that the experimental results are in good agreement with the theoretical ones. What’s more, we demonstrate the self-healing property of diverse orders of the COBs like the CAB. In Sec. 4, we introduce the experimental device for generating the COBs and applying them to capture particles. The experimental results show that the COBs can stably trap particles. Eventually, conclusions are summarized in Sec. 5.

2. Theory

To begin, the 1-D finite OBs by considering their exponentially decaying can be expressed by [1]:

(1)$$U_0\left(x_0, z=0\right)=O_m\left(\frac{x_0}{\beta w_0}\right) \exp \left(\alpha \frac{x_0}{w_0}\right),$$

where $w_0$ is the initial width of the fundamental Gaussian beam, $\alpha$ is the truncation parameter to ensure that the finiteness is contained, $\beta$ is the distribution factor, $O_m$ ($\cdot$) means the $n$th-order Olver functions of real $x$ with $m$ = 0, 1, 2… which is provided by Belafhal [1] with the following nondiffracting solutions of the free-space parabolic equation:

(2)$$O_m\left(x_1\right)=\frac{1}{2 \pi} \int_{-\infty}^{+\infty} \exp \left[a(\mathrm{i} u)^ t+\mathrm{i} u x_1\right] \mathrm{d} u,$$

with $t=m+3$, $|a|=\frac {1}{m+3}$. Equation (2) can be calculated numerically using a contour rotation in the complex $u$ plane to guarantee convergence of the integral as $u \rightarrow \pm \infty$, $u \rightarrow u \mathrm {e}^{\mathrm {i} \frac {\pi }{d}}$, where $\mathrm {d}=2 t$.

In the following, we derive the theoretical expression of the COBs in detail. In the paraxial system, the (2 +1) dimensional wave equation is written in the following form:

(3)$$u_z=(i / 2 k )\left(u_{x x}+u_{y y}\right).$$

In order to more conveniently describe a radially symmetrical beam, we express it in cylindrical coordinates. Thus, Eq. (3) can be rewritten as $u_z=(i / 2 k)\left (u_r / r+u_{r r}\right )$ [10], where $u$ denotes the optical wave described in the cylindrical coordinate form, the radius $r$ represents a distance, and $x$, $y$ are the transverse coordinates, and $z$ is the propagation distance. The initial field expression of the COBs in cylindrical coordinates can be written as:

(4)$$u_0(r, \theta, z=0)=A O_m\left(\frac{r_0-r}{\beta w_0}\right) \exp \left(\alpha \frac{r_0-r}{w_0}\right),$$

where the parameter $A$ is a constant, $r_0$ is the main initial radius of the COBs. The propagation along the direction $z$ can be expressed by the Fresnel diffraction integral:

(5)$$u(r, \theta, z)=\int_0^{2 \pi} \int_0^{\infty} \frac{A \rho u(\rho, \varphi, 0) }{i z \lambda} e^ {\frac{i k \left[r^2+\rho^2-2 r \rho \cos (\theta-\varphi)\right]}{2 z}}\,\mathrm{d} \rho \mathrm{d} \varphi.$$

It’s hard to obtain the analytical solutions of the COBs, so we use the split-step Fourier transform method to numerically simulate Eq. (5) [24]. And we set $r_0$ = 0.1mm, $w_0$ = 0.015mm, $A$ = 1 throughout the paper.

3. Theoretical results

To begin, it’s discovered that the properties of the higher order COBs resemble those of the lower order ones after carefully examining significant COBs’ traits. As a result, we focus in this paper on the auto-focusing propagation dynamics of COBs changed with six different orders, for $m$ = 0, 1, 2, 3, 4, 5, respectively. Figures 1(a1)–1(a2) display the transverse intensity distributions of the COBs at the initial plane corresponding to the side-view propagation depicted in Figs. 1(b1)–1(b2). For a clearer and more intuitive effect, we analyze the initial cross-sectional intensity distributions in odd and even modes. Obviously, as the order rises, all rings of the even-order COBs (ECOBs) lose strength, but the percentage of the total energy occupied by the main ring becomes larger. However, the subsequent rings of the odd-order COBs (OCOBs) are concentrated and relatively weak, without interference from the side lobes, the decline after the surge during propagation is monotonic and doesn’t exhibit oscillations, which is beneficial for material processing. With the present parameter settings, all the OCOBs have both inward and outward accelerations unlike the conventional CAB.

Fig. 1. Curves of focusing properties with the 0th-4th orders; (a1)-(a2) the intensity comparisons at the initial plane and the propagation dynamic profiles (b1)-(b2) under splitting parity; (c) focal length of the COBs as a function of orders $m$; (d1)-(d2) intensity contrast $S$ of the COBs. The parameters are: $\alpha$ = 0.01, $\beta$ = 1.

Download Full Size | PDF

Then we focus on comparing the propagations along the $z$-axis, and we still see evident differences between the OCOBs and the ECOBs which are plotted in Figs. 1(b1)–1(b2). The overall trend is that the COBs focus their energy right before a target while maintaining a low intensity profile until that very moment. Among them, the CAB has a minimal peak length with the most significant surge. And, as the focal length comparison in Fig. 1(c) shows, the focal length of the beam differs from different orders due to the shrinkage of the ring-shaped input caused by the fractional diffraction in the radial direction [25]. The OCOBs possess a smaller focus radius than the ECOBs do which may be caused by the outward acceleration [26]. Generally, the degree of the ECOBs’ focus is more pronounced in even orders which is presented in a clearer manner in Figs. 1(d1)–1(d2) with the peak intensity contrast $S$. The intensity contrast $S$ refers to the increase of the maximum intensity throughout propagation which is expressed by the ratio of $I_m$ to $I_0$. Supposing the beam that focuses more abruptly is a more desirable choice for applications, the above results show that the second order COB performs second only to the CAB in case we look at Fig. 1(d1) in conjunction with Fig. 1(c).

Next, the 0th–2nd order COBs’ numerical simulations and experimental implementations are shown in Fig. 2. For the sake of comparing the 1st and 2nd order COBs which have better performance in the parity order, with the CAB for further analysis. The intensity distributions of these beams at the input plane are clearly displayed in Figs. 2(b1)–2(b3). It is clear that the beams possess a circularly symmetric input profile and develop one high-intensity main-ring and outward or inward of sub-rings which oscillate radially, forming a hollow core surrounded by a series of concentric intensity rings with slightly decreasing width. Then we go a step further to discuss the evolution characteristics of the beams by displaying the intensity profiles in Figs. 2(a1)–2(a3). The pink arrows point to the intensity contrast $S$ versus the propagation distance. Their similar profiles prove the COBs can be considered as an expanded family of abruptly focusing waves. Interestingly, the first few sub-rings are asymmetrically distributed beside the main ring, and obviously the inward acceleration cannot match the outward acceleration because the overall trend of the beam’s propagation appears to coincide with the auto-defocusing beam’s characteristics. In more specific terms, I would say that the beam seems to experience auto-defocusing where the diffraction from the brightest ring is the main element to constitute the apparent focusing in this case. For longer propagation distances, the hollow region disappears at the focal point and grows a disk with concentrated energy which is presented in Figs. 2(d1)–2(d3). We experimentally obtain the cross-sections of the focal planes corresponding to the Figs. 2(c1)–2(c3) by using the experimental device which will be introduced in the Sec. 2. Note that these fit well with our numerical simulations. Above all, the results demonstrate auto-focusing of the COBs.

Fig. 2. (a1)–(a3) Detailed plots of the propagation dynamics of the COBs in the 0th-2nd orders, respectively. The pink arrows denote the intensity contrast $S$ of the COBs versus $z$ axis. Simulation transverse intensity distributions at the input plane (b1)–(b3) and the focus plane (d1)-(d3) correspond to the views in Figs. 2(a1)–2(a3). The solid white line depicts the intensity profile; (c1)-(c3) the corresponding experimental results of Figs. 2(d1)–2(d3). All the parameters are the same as those in Fig. 1.

Download Full Size | PDF

In light of the previously stated parameters, we now evaluate the impact of modifying the scaling factor $\beta$ and the decay parameter $\alpha$ on the characteristics of the COBs. We can see in Figs. 3(a1)–3(a4) that by increasing $\alpha$ from 0.01 to 0.4, all the inward-accelerating rings beside the main ring get weaker. Meanwhile, the outward-accelerating rings of the OCOBs become stronger which won’t show up in the ECOBs. Additionally, the $\alpha$ can ensure containment of the infinite COBs. So when it rises, the whole energy becomes significantly concentrated since the side lobes are almost sacrificed, which are described in Figs. 3(b1)–3(b2). It is also reflected in the fact that the widths of the energy loops are slightly narrowed when the $\beta$ which can adjust the scale between the ring Airy beam and the hollow Gaussian beam gets smaller. In other words, the energy concentration in the starting plane is more tightly packed.

Fig. 3. Effect of different decay factors $\alpha$ and scaling factors $\beta$ on the 1st and 2nd orders COBs, respectively. 2-D intensity diagrams of the 2nd order COB (a1)-(a2) and the 1st order COB (a3)-(a4) at the initial plane with different decay factor $\alpha$, which share the same parameter: $\beta$ = 1; (b1)-(b2) peak intensity along the $x$ axis; (c1)-(c2) peak intensity versus the propagation distance $z$.

Download Full Size | PDF

Further, as illustrated in Figs. 3(c1)–3(c2), with the enhancement of $\beta$, the focusing position is slightly offset from the original plane and the depth of focus is shorted sharply. The reason is that the scaling factor determines the beams’ spreading length of acceleration. Along with the increase of the focusing distance, the peak intensity of the focal point will be restrained with a larger focal spot. In the opposite sense, the focusing distance gets smaller during propagation with a more slender depth of focus in the auto-focusing plane when the value of $\beta$ is reduced.

A well-known feature of the CAB is its self-healing property. We will wonder if other order beams still have self-healing property, and if so, how do they perform? So in the following we investigate the capability of the COBs to self-heal. It has been demonstrated that the angularly blocked CAB does not self-heal until its own focus [27]. Only after the CAB self-heals clearly and the intensity distribution clears its focus does it recover symmetry. Hence, to explore the self-healing property, we define a sector-shaped block function at the initial plane that serves as an opaque obstacle to partially block the optical beam [28]:

(6)$$P H(\theta, z=0)=\left\{\begin{array}{c} 0.7, \phi \in[-\frac{\pi }{6}, \frac{\pi}{6}] \\ 1, \text{ otherwise }\end{array}\right.,$$

where $\phi$ is the angle spread of the semi-transparent block. According to the aforementioned method, we can map the phase of the blocked CAB. Figure 6 (a) introduces the procedure for generating the phase of the CAB, which serves as a presentative for other COBs.

Additionally, we compute the transverse Poynting vector $\vec {S}$ to verify that it may account for the self-healing characteristics of the beam. It can be expressed as [9]:

(7)$$\langle\vec{S}\rangle=\frac{c}{8 \pi}\left[i \omega\left(u_b \nabla_{{\perp}} u_b^*-u_b^* \nabla_{{\perp}} u_b \right)+2 \omega k|u_b|^2 \vec{e}_z\right],$$

where the transverse hamiltonian operator $\nabla _{\perp }=\vec {e}_x \frac {\partial }{\partial x}+ \vec {e}_y \frac {\partial }{\partial y}, \vec {e}_x, \vec {e}_y$ and $\vec {e}_z$ are the unit vectors along the $x$, $y$ and $z$ direction; $u_{\mathrm {b}}= u PH$ , $*$ is the complex conjugate. Now we calculate the transverse Poynting vector $\vec {S}$ at the planes of $z$ = 0dm, 0.05dm, 0.1dm with the sector-shaped obstacle positioned at $z = 0$dm. As seen in Figs. 4(c1)–4(e3), the COBs display very low light intensity in the blocked area, and the light intensity around the blocked part is also fainter, assuming the arrows point in the direction of the transverse energy flow. It clearly delineates the energy circulates from the side unblocked portion to the annular distribution of light in the blocked region as the propagating distance increases as a way to compensate for the low intensity of that area. Overall, the energy flow in all blocked parts will be weaker than that in the symmetrical regions. And the convergence diminishes with the increase of the propagation distance, indicating that the self-healing process is progressively complete. So although the surged energy is weak, it is the energy flowing from the unblocked region that reconstructs the blocked region which indicates that different orders of the COBs have the same self-healing property. This broadens the theoretical basis for the practical application of the light beam in self-imaging.

Fig. 4. (a) The process for generating the phase of the COBs (take the CAB as a representative); (b) the similarity $F$ between the COBs and the blocked COBs along $z$ coordinate with different orders; (c)-(e) the transverse Poynting vector of the blocked COBs for different orders: m = 0 (c), m = 1 (d), m = 2 (e), under the same obstruction’s position: z = 0dm; the numbers 1-3 in (c)-(e) denote at the z = 0dm, 0.05dm, 0.1dm planes, respectively.

Download Full Size | PDF

Next, we consider the effect of different orders $m$ on the self-healing ability of the COBs quantitatively. The self-healing ability of the COBs can be quantified by comparing the similarity between the COBs and the blocked COBs, which is defined as [29]:

(8)$$F(z_b, z)=\frac{\iint_{-\infty}^{\infty} I(z) I_b(z_b, z) d x d y}{\sqrt{\iint_{-\infty}^{\infty} I^2(z) d x d y \iint_{-\infty}^{\infty} I_b^2(z_b, z) d x d y}},$$

where $I(z)$ denotes the transverse intensity of the $\mathrm {CAB}$ without the block and $I_b(z_b, z)$ is the CAB’s transverse intensity when it is blocked. Figure 4(b) exhibits the capability of diverse orders of the $\mathrm {COBs}$ like $m$ = 0, 1, 2, 3, 4, in terms of propagation distance as the independent variable where the block is placed at $\mathrm {z}$ = 0dm. According to all of the curves in Fig. 4(b), these blocked COBs reform faster at first until they reach the maximum, then slightly decline, and finally be similar to stabilize. Such behavior means that there exists different optimal positions to obtain the self-healing effect. What’s more, the maximum recovery values reached by the OCOBs’ self-healing are smaller than those of the ECOBs, implying that when the OCOBs encounter obstacles during propagation, the integrity of the light beam will be compromised more. It is at a disadvantage in the processing of some precision instruments. However, the capabilities of the OCOBs are stable because the curves are relatively smooth.

4. Experimental setup and demonstrations for particle trapping

The experimental setup is sketched in Fig. 5, and we use the theoretical analysis performed beforehand to experimentally generate the COBs. Firstly, the laser emits a linearly polarized Gaussian beam with a wavelength $\lambda$ = 632nm. Then the beam passes through a half-wave plate and a Glan-Taylor prism, which controls the beam’s power as well as adjusts the polarization direction to meet the needs of the spatial light modulator (SLM) [30]. Following that, L1 and L2 are combined as an expander mainly used to expand the beam. Immediately after being collimated the beam will be reflected by the SLM (Santec SLM-200 with 1900 $\times$ 1200 pixel resolution), which is utilized to load the phase hologram to reconstruct the target beam. After going through the SLM, the beam is sent through the 4f filter system which is consisted of two lenses (L3 and L4) and a circular diaphragm, and we choose the positive first-order interference fringes on the aperture (AP).

Fig. 5. Experimental setup: reflective spatial light modulator (SLM), half-wave plate (HWP), glan–Taylor prism (GP), expander (L1 $\sim$ L2, lenses), 4f system (L3 $\sim$ L4, lenses), aperture (AP), charge-coupled device (CCD), delay system (M1 $\sim$ M4, mirrors), beam splitter (BS), white light source (Illuminator), sample, dichroic mirror (DM), filter, complementary metal oxide semiconductor camera (CMOS).

Download Full Size | PDF

Then, the beam is divided into two parts by passing through a beam splitter: one is sent into the charge-coupled device (CCD) which aids in obtaining information about the cross-section of the beam, and the other is transmitted to the oil-immersion objective (100$\times$, 60$\times$, NA = 1.25) by a dichroic mirror (DM) to illuminate the sample that consists of 4 $\mu$m polystyrene beads suspended in distilled water and sandwiched between two thin glass plates [31]. The DM and the optical filter can block the beam entering the complementary metal-oxide semiconductor (CMOS) camera while allowing white light to pass through. The white light illuminates the sample for obtaining the imaging of the particles from the opposite direction achieved by the CMOS camera. Additionally, with the help of the delay system [32] composed by four mirrors set on a translation stage, the optical path can be controllable. So that by vertically moving the delay system, the desired light patterns at different $z$-axial locations can be obtained by the CCD [33]. Through the realization of the above optical path, the light beam can be generated and even applied in imaging of the particle trapping situation, which assists us to check our theoretical simulations.

As we all know, one of the proposed applications of the CAB with angular acceleration is for optical trapping [23,34]. Below, we explore such a possibility using our other orders of the COBs for manipulating particles. As indicated in Ref. [15], such an auto-focusing beam with angular acceleration can exert gradient forces. The gradient pressures can drive particles in regions of the highest intensity because the refractive index of silica particles in a fluid environment is higher than that of surroundings. In this article, we increase the laser output power to 4.8W, so that the gradient forces are capable of defeating the Brownian motion. In this way, the beam is like dragging the particle in motion to the high-intensity region and finally trapping it, completing the particle capture.

In the experiment, we use the aforementioned classical particle capture device and select specific planes during propagation as the capture surface to successively verify the ability of the second and fourth order COBs to capture particles. More narrowly, by interfering the initial field of the COB with a plane wave, the appropriate phase mask can be created which is given by

(9)$$\psi_{\mathrm{SLM}}=\left|1+u_0 \exp \left(i 2 \pi f_x x\right)\right|^2,$$

where $f_x$ represents the grating frequency and characters the angle of diffraction for the light reflected by the SLM in the $x$ direction. We can change the value of $f_x$ to prevent the diffraction fringes from overlapping on the spectrum plane in order for the 4f filter system to choose the positive first-order fringe. After imposing the phase masks onto the SLM, the COB is generated by the experimental setup in Fig. 3 and the corresponding transverse intensity evolutions are obtained by steadily moving the mirrors (M2, M3) in the delay system. Therefor, we can choose idea planes to achieve optical trapping. As shown in Figs. 6(a1)–6(a2), the panel illustrates how the particles move and are trapped. We load the phase mask of the 2nd COB which is shown in Fig. 6(a3) on the SLM and move the delay system to pick the special plane. The experimentally picked plane shown in Fig. 6(a4) is at a certain distance from the focal point where a bright spot is distributed in the center as a result of the initial energy auto-focuses, surrounded by a ring-like distribution of lower energy than the initial one. As displayed in Figs. 6(b1)–6(b4), the particles are originally arranged in a disorderly manner, and in just a few tens of seconds the two particles are gradually subjected to the position of a circle. It appears that the particles tend to form ring patterns to match the main lobes of the CAB. Furthermore, when we select a different plane with fewer particles and slowly move the sample, we notice that only the two particles circled in yellow remain steadily and seemingly fixed in the same region without moving. We use the particles around them as a reference, and it is clear that the other particles move independently, indicating that these two particles are steadily captured at the center spot. The two phenomena agree with the intensity distribution of our selected plane.

Fig. 6. Experimental demonstrations of particle trapping using the COB with m = 2. (a1)-(a2) Illustration of particle motion guided by the light; (a3) the phase mask of the 2nd COB; (a4) numerical simulation and experimental snapshot of the imaging plane of the objective; (b1)–(b4) snapshots of particles trapped at the trapping plane like the situation (a1); and (c1)–(c3) snapshots of particles trapped at the trapping plane like the situation (a2). The arrows depict the direction of motion of the particles circled by solid yellow or blue circles.

Download Full Size | PDF

In Fig. 7, we utilize the 100$\times$ oiled objective in this case to make the trapping process more clear and we shine the selected beam to the upper left area of the photographed plane of the slide. From the experimental results displayed in Figs. 7(b1)–7(b4) and Figs. 7(c1)–7(c3), we know that the particles are rapidly dragged to the upper left corner since we slightly increase the laser power to 5.2W. The dragging force is very strong, and the particles are even dragged out of the photographed plane, showing the sinking phenomenon. The fuzzy particle circled in red is sinking. Above all, the results are in agreement with the theoretical predictions, demonstrating that the other orders COBs also have a good capability of trapping particles.

Fig. 7. Particles trapping process of the COB with m = 4. The red circle marks the location of the sinking particle.

Download Full Size | PDF

5. Conclusion

In summary, we explore the propagating dynamics of the COBs with low orders. Firstly, the new classes of cylindrical vector beams are proposed by extending the Olver function to cylindrical coordinates. Furthermore, we successively exhibit the auto-focusing and self-healing properties of the COBs during propagation in the free space. The COBs can achieve both inward and outward accelerations which will be helpful to trap particles. What’s more, the focusing process and the motion trajectory can be adjusted by choosing the scaling factor $\beta$ and the decay factor $\alpha$. We also present the experimental results of the focus plane which fit well with the numerical simulations. Following, we give a brief introduction of our experimental setup. Finally, we experimentally demonstrate the stable particle trappings with the ECOBs at the auto-focusing plane which fit well with the theoretical predictions. These intriguing features of the COBs offer possibilities for the development of optical tweezers.

Funding

National Natural Science Foundation of China (11775083, 12174122); Natural Science Foundation of Guangdong Province (2022A1515011482); Program of Innovation and Entrepreneurship for Undergraduates; Special Funds for the Cultivation of Guangdong College Students Scientific and Technological Innovation (“Climbing Program”Special Funds) (pdjh2022a0129); Extracurricular Scientific Program of School of Information and Opto-electronic Science and Engineering, South China Normal University (22GDGB01).

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. Belafhal, L. Ez-Zariy, S. Hennani, and H. Nebdi, “Theoretical introduction and generation method of a novel nondiffracting waves: Olver beams,” Opt. Photonics J. 05(07), 234–246 (2015). [CrossRef]

2. I. M. Besieris, A. M. Shaarawi, and B. Davis, “A note on self-accelerating olver and olver-gauss beams,” Opt. Photonics J. 09(01), 1–7 (2019). [CrossRef]

3. A. Zannotti, F. Diebel, M. Boguslawski, and C. Denz, “Optical catastrophes of the swallowtail and butterfly beams,” New J. Phys. 19(5), 053004 (2017). [CrossRef]

4. A. Zannotti, F. Diebel, and C. Denz, “Dynamics of the optical swallowtail catastrophe,” Optica 4(10), 1157–1162 (2017). [CrossRef]

5. J. Jiang, D. Xu, Z. Mo, X. Cai, H. Huang, Y. Zhang, H. Yang, H. Huang, Y. Wu, L. Shui, and D. Deng, “Generation and control of tornado waves by means of ring swallowtail vortex beams,” Opt. Express 30(7), 11331–11344 (2022). [CrossRef]

6. G. A. Siviloglou and D. N. Christodoulides, “Accelerating finite energy airy beams,” Opt. Lett. 32(8), 979–981 (2007). [CrossRef]

7. G. Siviloglou, J. Broky, A. Dogariu, and D. Christodoulides, “Observation of accelerating airy beams,” Phys. Rev. Lett. 99(21), 213901 (2007). [CrossRef]

8. G. Siviloglou, J. Broky, A. Dogariu, and D. Christodoulides, “Ballistic dynamics of airy beams,” Opt. Lett. 33(3), 207–209 (2008). [CrossRef]

9. J. Broky, G. A. Siviloglou, A. Dogariu, and D. N. Christodoulides, “Self-healing properties of optical airy beams,” Opt. Express 16(17), 12880–12891 (2008). [CrossRef]

10. N. K. Efremidis and D. N. Christodoulides, “Abruptly autofocusing waves,” Opt. Lett. 35(23), 4045–4047 (2010). [CrossRef]

11. D. G. Papazoglou, N. K. Efremidis, D. N. Christodoulides, and S. Tzortzakis, “Observation of abruptly autofocusing waves,” Opt. Lett. 36(10), 1842–1844 (2011). [CrossRef]

12. S. Hong, X. Yang, K. Zhou, Y. Wu, X. Fu, and D. Deng, “Ring airy-like beams along predesigned parabolic trajectories,” Ann. Phys. 532(7), 2000130 (2020). [CrossRef]

13. C. Xu, H. Hu, Y. Liu, and D. Deng, “Radially polarized symmetric airy beam,” Opt. Lett. 45(6), 1451–1454 (2020). [CrossRef]

14. N. K. Efremidis, Z. Chen, M. Segev, and D. N. Christodoulides, “Airy beams and accelerating waves: an overview of recent advances,” Optica 6(5), 686–701 (2019). [CrossRef]

15. P. Zhang, J. Prakash, Z. Zhang, M. S. Mills, N. K. Efremidis, D. N. Christodoulides, and Z. Chen, “Trapping and guiding microparticles with morphing autofocusing airy beams,” Opt. Lett. 36(15), 2883–2885 (2011). [CrossRef]

16. E. Otte and C. Denz, “Optical trapping gets structure: Structured light for advanced optical manipulation,” Appl. Phys. Rev. 7(4), 041308 (2020). [CrossRef]

17. R. A. Suarez, L. A. Ambrosio, A. A. Neves, M. Zamboni-Rached, and M. R. Gesualdi, “Experimental optical trapping with frozen waves,” Opt. Lett. 45(9), 2514–2517 (2020). [CrossRef]

18. Y. Zhang, Z. Mo, D. Xu, S. He, Y. Ding, Q. Huang, Z. Lu, and D. Deng, “Circular mathieu and weber autofocusing beams,” Opt. Lett. 47(12), 3059–3062 (2022). [CrossRef]

19. R. A. Suarez, A. A. Neves, and M. R. Gesualdi, “Generation and characterization of an array of airy-vortex beams,” Opt. Commun. 458, 124846 (2020). [CrossRef]

20. I. S. Yepes, T. A. Vieira, R. A. Suarez, S. R. Fernandez, and M. R. Gesualdi, “Phase and intensity analysis of non-diffracting beams via digital holography,” Opt. Commun. 437, 121–127 (2019). [CrossRef]

21. V. Shvedov, A. R. Davoyan, C. Hnatovsky, N. Engheta, and W. Krolikowski, “A long-range polarization-controlled optical tractor beam,” Nat. Photonics 8(11), 846–850 (2014). [CrossRef]

22. Y. Yang, Y. Ren, M. Chen, Y. Arita, and C. Rosales-Guzmán, “Optical trapping with structured light: a review,” Adv. Photonics 3(03), 034001 (2021). [CrossRef]

23. M. Chen, S. Huang, X. Liu, Y. Chen, and W. Shao, “Optical trapping and rotating of micro-particles using the circular airy vortex beams,” Appl. Phys. B 125(10), 184 (2019). [CrossRef]

24. T. Chung and T. Kim, “Engineering optics with mat lab,” (2006).

25. S. He, B. A. Malomed, D. Mihalache, X. Peng, X. Yu, Y. He, and D. Deng, “Propagation dynamics of abruptly autofocusing circular airy gaussian vortex beams in the fractional schrödinger equation,” Chaos, Solitons Fractals 142, 110470 (2021). [CrossRef]

26. P. Zhang, J. Prakash, Z. Zhang, Y. Hu, N. K. Efremidis, V. Kajorndejnukul, D. N. Christodoulides, and Z. Chen, “Observation of auto-focusing radially symmetric airy beams,” in Quantum Electronics and Laser Science Conference, (Optica Publishing Group, 2011), p. QThS7.

27. N. Li, Y. Jiang, K. Huang, and X. Lu, “Abruptly autofocusing property of blocked circular airy beams,” Opt. Express 22(19), 22847–22853 (2014). [CrossRef]

28. L. Chen, J. Wen, D. Sun, and L.-G. Wang, “Self-healing property of focused circular airy beams,” Opt. Express 28(24), 36516–36526 (2020). [CrossRef]

29. X. Chu and W. Wen, “Quantitative description of the self-healing ability of a beam,” Opt. Express 22(6), 6899–6904 (2014). [CrossRef]

30. X. Zhou, Z. Pang, and D. Zhao, “Generalized ring pearcey beams with tunable autofocusing properties,” Ann. Phys. 533(7), 2100110 (2021). [CrossRef]

31. P. Zhang, D. Hernandez, D. Cannan, Y. Hu, S. Fardad, S. Huang, J. C. Chen, D. N. Christodoulides, and Z. Chen, “Trapping and rotating microparticles and bacteria with moiré-based optical propelling beams,” Biomed. Opt. Express 3(8), 1891–1897 (2012). [CrossRef]

32. W. Yan, Y. Gao, Z. Yuan, Z. Wang, Z.-C. Ren, X.-L. Wang, J. Ding, and H.-T. Wang, “Non-diffracting and self-accelerating bessel beams with on-demand tailored intensity profiles along arbitrary trajectories,” Opt. Lett. 46(7), 1494–1497 (2021). [CrossRef]

33. X. Liu, Y. E. Monfared, R. Pan, P. Ma, Y. Cai, and C. Liang, “Experimental realization of scalar and vector perfect laguerre–gaussian beams,” Appl. Phys. Lett. 119(2), 021105 (2021). [CrossRef]

34. A. P. Porfirev, “Laser manipulation of airborne microparticles behind non-transparent obstacles with the help of circular airy beams,” Appl. Opt. 60(3), 670–675 (2021). [CrossRef]

Generation and control of the circle Olver beams

Abstract

Corrections

1. Introduction

2. Theory

3. Theoretical results

4. Experimental setup and demonstrations for particle trapping

5. Conclusion

Funding

Disclosures

Data availability

References

Data availability

Cited By

Figures (7)

Equations (9)

Optics Express