Experimental generation of the polycyclic tornado circular swallowtail beam with self-healing and auto-focusing

Yong Zhang; JiaLong Tu; ShangLing He; YiPing Ding; ZhiLi Lu; You Wu; GuangHui Wang; XiangBo Yang; DongMei Deng

doi:10.1364/OE.446818

1. Introduction

Caustic beams as the concrete manifestation of the catastrophe theory in light phenomenon are the envelopes of families of rays, which tend to a focus spot, curves or surfaces [1]. The most fundamental optical catastrophe [2] is the fold catastrophe characterized by a cubic power in the spectral phase, which can form the phase space by choosing the state and control parameters. Its caustic field was defined by the Airy function, and it was experimentally demonstrated in 2007 [3]. Subsequent to the Airy beams, the Pearcey beam represented by the cusp catastrophe was observed in 2012 [4], which is generated by a quartic power in phase space, and corresponding to the caustic field is described by the Pearcey function. In 2017, $Zannotti$ et al. [5,6] systematically investigated the swallowtail and butterfly catastrophe corresponding to the swallowtail and butterfly beams in the light field. Clearly, each order of the catastrophe forms a specific stable diffraction structure. With the increasing dimensionality of the control parameters, their diffraction structures are given by fold, cusp, swallowtail, hyperbolic umbilic, elliptic umbilic, butterfly, and parabolic umbilic catastrophes [7].

On the other hand, there are some remarkable properties among the great variety of caustic beams. The well-known properties of these caustic beams are autofocusing and self-healing, which have been studied theoretically and demonstrated experimentally [8–11]. In addition, the vortex beams have been drawn considerable attention in recent decades due to their many applications in various fields [12–14]. The optical vortices (OVs) embedded in a light beam are associated with doughnut-like intensity patterns and have phase singularity in the core [15]. They not only have a homogeneous polarized distribution but also carry the orbital angular momentum (OAM) determined by the topological charge [16]. Very recently, people have investigated the propagation dynamics of the vortices imprinted in various background beams. For example, Airy vortex beams [17], Pearcey vortex Gaussian beams [18], Laguerre Gaussian beams [19], partially coherent beams [20], and Bessel vortex beams [21]. The structured beams with OAM have further paved the way toward novel opportunities for scientific research and advanced applications. For example, optical communication [22], high-resolution microscopy [23], and particle manipulation [24]. However, up to now, most of these researches only focus on the properties of the lower-order catastrophe such as Airy and Pearcy beams, which are the representatives of one-dimensional(1D) and two-dimensional (2D), respectively. To the best of our knowledge, the properties of the high-order catastrophe vortex beams have not been explored yet.

The rotation of the structured light has recently attracted lots of attention with the various beams. Zamboni-Rached [25] proposed a new class of nondiffracting beams, namely Frozen beams, which are suitable superpositions of co-propagating Bessel beams, and experimentally demonstrated in 2012 [26]. The evolution of the twisted light fields of Frozen waves has the possibility to enhance the capacity in communication systems [22]. They also used the Frozen waves to trap microparticles through weakly dispersive media over relatively small distances [27]. In contradistinction to Frozen waves, the caustic beams with phase manipulation can increase intensity at focal plane and the distance of the optical trapping of microparticles. The other cases of rotating field, including higher-order Bessel beams [28,29], Laguerre-Gaussian beams [30] and ring Airy beams [31], are performed by superposition of two same co-propagation beams. Besides, Wu $et$ $al$. [32] only numerically calculated the ring-Airy beam with the spiral phase, but they do not experimentally demonstrate them and investigate the self-healing of the beams.

In this paper, we develop a simple and convenient method to generate the chiral optical field of the high-order catastrophe. We discuss the propagation dynamics of the polycyclic tornado circular swallowtail beam (PTCSB) in detail. We numerically and experimentally demonstrate the autofocusing and self-healing of the PTCSB from different perspectives, such as the intensity transverse profile, the peak intensity, the Poynting vector, and the OAM. Last, we obtain a conclusion, simply summarizing the content and listing some potential applications.

2. Theory

The caustic field $C_{n}(a)$ can be defined by catastrophe theory in terms of the standard diffraction integral, which can be defined [33,34]

(1)$$C_{n}=\int_{-\infty}^{+\infty} \exp \left[i p_{n}(a, s)\right] d s,$$

where $p_{n}(a, s)$ can completely determine the properties of the caustic field as the canonical potential function, as the control parameter can change the potential completely, and it can be defined by

(2)$$p_{n}(a, s)=s^{n}+\sum_{j=1}^{n-2} a_{j} s^{j},$$

where the control parameters $a=\left (a_{1}, a_{2}, \ldots, a_{j}\right )$ include all dimensionless control parameters $a_{j}$ with $j=1,2, \ldots, n-2$. When we change the dimensionality of the control parameters, the caustic field possibly becomes a point, curved line, curved surface, or hypersurface.

For the high-order swallowtail catastrophe $C_{5}(a_{1}, a_{2}, a_{3})$, according to the swallowtail catastrophe integral, the swallowtail beam ${Sw}(a_{1}, a_{2}, a_{3})$ can be represented [6]

(3)$${{Sw}}(a_{1}, a_{2}, a_{3})=\int_{-\infty}^{+\infty} \exp \left[i\left(s^{5}+a_{3} s^{3}+a_{2} s^{2}+a_{1} s\right)\right] d s,$$

above the expression, the polynomial function $(s^{5}+a_{3} s^{3}+a_{2} s^{2}+a_{1})$ in Eq. (3) indicates the potential function that determines the diffraction structure of a swallowtail catastrophe.

Under the paraxial optical system, the spatial beam propagating along the $z$-axis can be represented by (2+1) dimensional potential-free Schrodinger equation in free space, which can be described in the form [35]

(4)$$2 i k\frac{\partial u}{\partial z}+\frac{\partial^{2} u}{\partial x^{2}}+\frac{\partial^{2} u}{\partial y^{2}}=0,$$

here, we investigate the propagation characteristics of the PTCSB based on the radially systematic circular swallowtail beam (CSB). It is more convenient to describe it in cylindrical coordinates, and Eq. (4) can be rewritten as

(5)$$2 i k\frac{\partial u}{\partial z}+\frac{\partial^{2} u}{\partial r^{2}}+r^{{-}1} \frac{\partial u}{\partial r}+r^{{-}2} \frac{\partial {^2} u}{\partial \theta {^2}}=0,$$

where $u(r, \theta, z)$ denotes the light field amplitude; $r$ is the radial distance, $\theta$ is the azimuth angle, $z$ is the propagation distance, and $k=2 \pi / \lambda$ is the wave number. The initial electric field of the PTCSB $u(r, \theta, 0)$, superimposed by the annular spiral zone phase (ASZP), can be written in the cylindrical coordinates

(6)$$u(r, \theta, 0)=A_{0} \operatorname{Sw}\left(\frac{r_{0}-r}{ b w_{0}}, 0,0\right) \exp [i \Phi(r, \theta)] q(r, \theta),$$

where $A_{0}$ is a constant amplitude of the initial spatial field, Sw$(\cdot )$ is swallowtail integral, $r_{0}$ is the initial radius of the PTCSB, $w_{0}$ is the initial width of the Gaussian beam, $b$ is a scaling factor that can modulate the intensity distribution of the input beam. In order to ensure that this beam can be realized in reality, the truncation function $q(r, \theta )$ can be indicated

(7)$$q(r, \theta)=\left\{\begin{array}{ll} 1, & 0 \leq r \leq R, 0<\theta \leq 2 \pi \\ 0, & \textrm{other} \end{array}\right.,$$

here, $R$ denotes the radius of the integral aperture. Such ASZP $\Phi (r, \theta )$ can be seen as superposition of multiple phases. The phase distribution can be expressed as [32,36,37]

(8)$$\Phi(r, \theta)=\sum_{i=1}^{m} \varsigma_{i}(r, \theta),$$

here, $m$ is the number of subphases. Each layer subphase can be written as

(9)$$\varsigma_{i}(r, \theta)=\left\{\begin{array}{ll} \beta r, & r_{i}^{ir}<r<r_{i}^{or} \\ \gamma, & r_{i}^{ie}<r<r_{i}^{oe} \\ \ell_{i} \theta+\alpha r, & r_{i}^{is}<r<r_{i}^{os} \end{array},\right.$$

where $\varsigma _{i}(r, \theta )$ represents the $i$th subphase; $\beta r$ denotes the radial phase, $\gamma$ is the equiphase, $\ell _{i}\theta +\alpha r$ is the spiral phase; $\ell _{i}$ is the topological charge for the ith subphase, respectively; $\alpha$ and $\beta$ are the tunable radial shift factors; $r_{i}^{ir}, r_{i}^{ie}, r_{i}^{is}$ are the inner radii of the ith radial phase, equiphase, and spiral phase, respectively; $r_{i}^{o r}, r_{i}^{oe}, r_{i}^{os}$ are the outer radii of the ith radial phase, equiphase, and spiral phase, respectively. In order to specifically explain the construction process of the subphase, we define the ring width of the each subphase, which are described as $r_{rpi}=r_{i}^{o r}-r_{i}^{ir}$, $r_{epi}=r_{i}^{oe}-r_{i}^{ie}$, and $r_{spi}=r_{i}^{os}-r_{i}^{is}$. Due to the continuity of the these annular phases, $r_{i}^{ie}=r_{i}^{or}$ and $r_{i}^{is}=r_{i}^{oe}$.

Figure 1 illustrates the generation process of the PTCSB phase. The ASZP is formed by superposing two-layer chiral subphases (SP1 and SP2). It is clear that the subphases are constituted by three parts consisting of the radial phase, the equiphase and the spiral phase, respectively. The case of superposing the annular spiral phase is similar to produce the interference between vortex beam and plane wave [38]. In addition, by changing the radial shift factors $\alpha$ and $\beta$ to offset radially the optical vortex, the interference with the plane wave can be extended [39]. Finally, the phase patterns of the PTCSB are generated by superposing ASZP and the phase of CSB, as illustrated in Figs. 1(b1)–1(b3).

Fig. 1. (a1)-(a3) Process for generating the ASZP with two SPs, $\alpha =0.001 \pi, \beta =0.6 \pi, \gamma =0$, $\ell _{1}=\ell _{2}=8$, $r_{1}^{ir}$=2 mm, $r_{rp1}=r_{rp2}=2$ mm, $r_{ep1}=r_{ep2}=5$ mm, $r_{sp1}=r_{sp2}=2$ mm. (b1)-(b3) Process for generating the phase of the PTCSB with $b$=0.3, $w_{0}=1$ mm.

Download Full Size | PDF

Now we discuss the propagation properties of the PTCSB by substituting Eq.(4) into the diffraction integral formula assuming paraxial propagation [40]

(10)$$u(r, \theta, z)=\int_{0}^{2 \pi} \int_{0}^{R} \frac{A_{0} ku(r, \varphi, 0) \rho}{2 i \pi z} exp\left \{{\frac{ik\left[r^{2}+\rho^{2}-2 r \rho \operatorname{cos}(\theta-\varphi)\right]}{2 z}} \right \} d \rho d \varphi.$$

It is difficult to obtain the solution for the three-dimensional optical field $u(r, \theta, z)$. Fortunately, the split-step Fourier transfer approach makes us simulate the propagation of the PTCSB numerically [41]. In the following analysis, we assume the $\lambda =532$ nm, $\mathrm {R}=20 mm, \alpha =0.001 \pi, \beta =0.6 \pi, r_{0}=5$mm, $r_{1}^{ir}=2$ mm, and $z_{R}=kw_{0}^2$.

3. Results and Discussions

The experimental realization of the PTCSB is implemented through a computer-generated hologram reproduced by the reflective spatial light modulator (rSLM, 1900 1200 pixels ), as schematically presented in Fig. 2. A linearly polarized Gaussian beam at 532 nm is emitted from the He-Ne laser. When the beam passes through the beam expander, the beam is collimated and expanded, and then reflected by a mirror. Then, by imprinting a phase mask containing the formation of the proposed PTCSB on a computer-controlled rSLM, the expanded beam can be constructed that utilizes interference between the plane wave and the swallowtail beam. For the purpose of selecting the desired beams, a 4f system consisting of two thin lenses ($L_{1}$ with its focal length $f_{1}$=300 mm and $L_{2}$ with its focal length $f_{2}$=500 mm) with a circular aperture (CA) is established. Besides, the function that is used to produce the pattern of interference fringes on the rSLM can be adjusted by the number and the width of the interference fringes to make the modulated beam clearly. The positive first-order fringes from the rSLM can be regarded as our desired PTCSBs. This method can efficiently and feasibly reproduce various optical beams compared with conversational methods [42]. Last, we employ the beam quality analyzer (BQA) to measure the intensity distributions at different propagation distances.

Fig. 2. Experimental setup for generating the PTCSB. BE, beam expander; M, mirror; rSLM, reflective spatial light modulator (Santec SLM-200); $L_{1}$, $L_{2}$, thin lens; CA, circular aperture; BQA, beam quality analyzer. The $4f$ system consisted of two lens implements to choose the positive first-order spatial-spectral fringes.

Download Full Size | PDF

Fristly, we display a typical simulation results in Fig. 5. In our simulation, the ASZP with topological charge $\ell _{1}=0$ is superposed on CSB, and multioptical bottles marked in the white oval in Fig. 5(a) are formed. The PTCSB inherits the autofocusing property from the CSB, but it does not exhibit the rotating characteristic during the propagation. We define the intensity contrast values to clearly claim the intensity change, which are described by $I_{m}/I_{0}$, where $I_{m}$ is the maximum intensity at an arbitrary plane during the propagation and $I_{0}$ is the maximum intensity of the PTCSB at the initial plane. We can find that the intensity contrast values remain almost constant before reaching the autofocusing plane, then abruptly increase when the PTCSB reaches the autofocusing plane. Finally, the intensity contrast values decrease with a slightly oscillation because of the interference between the main ring and the sub-ring.

Figure 4 depicts the propagation characteristics of the PTCSB in free space. Obviously, the PTCSB has a circularly symmetric input profile that consists of a dark disk and a series of concentric intensity rings with decreased oscillating intensity, which is similar to the input profile of the CSB. From Figs. 4(c1)–4(c2), it can be clearly found that the optical field can rotate continuously and becomes a tornado pattern automatically. As the propagation distance increases, the tornado pattern becomes firstly smaller, and then becomes larger. In addition, we find that the rotation direction of the optical field occurs to change after the autofocusing plane, and the rotation direction of the optical field is clockwise in Fig. 4(c2) marked by the white curved arrows, which is clearly displayed the direction-rotating. However, the optical field begins to rotate counterclockwise after this position, shown in Figs. 4(c3)–4(c4). This phenomenon is similar to that reported in Ref. [4] because of the inversion of the swallowtail function at the autofocusing plane.

Fig. 3. The propagation dynamics of the PTCSB with the topological charge $\ell _{1}=0$, $\alpha =0.001 \pi, \beta =0.6 \pi, \gamma =0$, $\ell _{1}$=8, $r_{1}^{ir}$=2 mm, $r_{rp1}=2$ mm, $r_{ep1}=5$ mm, $r_{sp1}=2$ mm, $b$=0.3, and $w_{0}=1$ mm. (a) side view of the PTCSB numerical propagation; (b) corresponding to the intensity contrast values. (c1)-(c2) Snapshots view of transverse intensity patterns of the PTCSB at planes $1-4$ marked in Fig. 3(a), respectively.

Download Full Size | PDF

Fig. 4. Propagation dynamics of the PTCSB with $\alpha =0.001 \pi, \beta =0.6 \pi, \gamma =0$, $\ell _{1}$=8, $r_{1}^{ir}$=2 mm, $r_{rp1}=2$ mm, $r_{ep1}=5$ mm, $r_{sp1}=2$ mm, $b$=0.3, and $w_{0}=1$ mm. (a) Numerical side view of the PTCSB propagation; (b) the intensity contrast values of the PTCSB along the z direction. (c1)-(c4) Snapshots of the transverse intensity pattern of the PTCSB numerically at planes $1-4$ marked in Fig. 5(a), respectively; (d1)-(d4) the corresponding experimentally results. The white curved arrows denote the rotation direction of the optical fields.

Download Full Size | PDF

On the other hand, the PTCSB forms a hollow core surrounded by the tornado shape with a helical wave-front. In this case, this light field with a helical phase structure and the phase gradient can generate an azimuthal component. The property of the PTCSB is very promising application in biomedical treatment field [43] and optical tweezers micromanipulations [27], which can avoid damaging the surrounding cells before reaching the particular target. Additionally, the intensity that is sharply increased also enhances the optical trapping force to obtain greater stability. The corresponding experimental results at planes $1-4$ marked in Fig. 4(a) display in Figs. 4(d1)–4(d4), respectively. The numerical intensity distributions agree with those of the experimental results. Therefore, the PTCSB exhibits visibly autofocusing property numerically and experimentally.

To investigate the physical insight of the rotating characteristic of the PTCSB, we show the transverse component of the Poynting vector and the OAM of the PTCSB at the various planes along the propagation direction. The Poynting vector for a linear polarized field can be defined by [44]

(11)$$\langle\vec{S}\rangle=\frac{c}{4 \pi}\langle\vec{E} \times \vec{B}\rangle=\frac{c}{8 \pi}\left[i \omega\left(u \nabla_{{\perp}} u^{*}-u^{*} \nabla_{{\perp}} u\right)+2 \omega k|u|^{2} \vec{e}_{z}\right],$$

where $\nabla _{\perp }=\vec {e}_{x} \partial / \partial x+\vec {e}_{y} \partial / \partial y$, $\vec {e}_{x}, \vec {e}_{y}$ and $\vec {e}_{z}$ are the unit vectors along the $x, y$, and $z$ directions, respectively; $u$ refers to the complex amplitude of the optical field; $\omega$ represents the angular frequency; $c$ indicates the light speed in vacuum; $\vec E$ and $\vec B$ are the electric and the magnetic fields, respectively; and the symbol $*$ denotes the complex conjugate. It is found that the arrows of side lobes are denser than those between the side lobes, and the rotating direction of pattern of the Poynting vector is also clockwise in the Fig. 5(a1). Obviously, the denser the arrows are, the larger the densities are. As the beam propagates further, it becomes more denser in the singularity of the beam before reaching the focusing spot. When the PTCSB reaches the focusing plane, the main direction of the Poynting vector begins to become from the outside to inside along the counterclockwise direction. The densities of the arrows also become sparser, indicating that the energy is beginning to diverge outward. The inversion flow of the energy confirms the rotation of the PTCSB.

Fig. 5. Poynting vector and angular momentum density of the PTCSB with topological charge $\ell _{1}=8$ at $z=50z_{R}, 105z_{R}, 117z_{R},$ and $125z_{R}$ planes, respectively. (a1)-(a4) energy density flow (background) and transverse energy density flow (arrows) of the PTCSB. (b1)-(b4) all are the same as those in (a1)-(a4) except for the angular momentum density (background) and transverse angular momentum density flow (arrows). All other parameters are the same as those in Fig. 4.

Download Full Size | PDF

Different from the Poynting vector, the handedness of the OAM is more uniform and orderly. The OAM is written by [45]

(12)$$\langle\vec{J}\rangle=\vec{r} \times\langle\vec{E} \times \vec{B}\rangle=\frac{\omega}{2}\left(2 y k|u|^{2}-i z S_{y}\right) \overrightarrow{e_{x}}+\left(2 z S_{x}-2 y k|u|^{2}\right) \overrightarrow{e_{y}}+i\left(x S_{y}-y S_{x}\right) \overrightarrow{e_{z}}$$

where $S_{x}$ and $S_{y}$ are the energy flux densities in $x$ and $y$ directions, respectively. As the propagation distance increases, the singularity of the OAM in the center becomes smaller, but the rotation direction of that is unchanging. Note that the distribution of the OAM density flow represents the change of the vortex at the on-axis singularity during propagation of the PTCSB. These phenomenons about the energy and the OAM density flow claim further the beam-rotating characteristic derived above.

Figure 6 presents numerical results associated with four different cases of the PTCSB, where the tolopogical charge takes the values $\ell _{1}=4, \ell _{1}=8, \ell _{1}=\ell _{2}=8,$ and $\ell _{1}=-8, \ell _{2}=8$, respectively. Figures 6(a1)–6(a4) show the normalized transverse intensity distribution at $z=105\,z_{R}$ plane with the different cases. Figures 6(b1)–6(b4) are the corresponding intensity contrast values. When $\ell _{1}=4$, we can observe that there are 4 spiral stripes, while there are 8 spiral stripes when $\ell _{1}=8$. With the increase of the topological charge, the number of the spiral stripes also increases. Since the number of the spiral stripes of the transverse intensities is the same as the value of the topological charge, we can utilize the method to measure the topological charge of optical vortices. Additionally, we learn that the rotation direction of the inside lobe is opposite to that of the outside lobe compared with Fig. 6(a3) and Fig. 6(a4), which shows that the signs of the topological charge are determined the rotation direction of the PTCSB. Therefore, the chirality properties of the PTCSB for each layer can be controlled by adjusting the signs of the topological charge.

Fig. 6. (al)-(a4) The normalized transverse intensity distribution at $z=105z_{R}$ plane with the topological charge $\ell _{1}=4, \ell _{1}=8, \ell _{1}=\ell _{2}=8,$ and $\ell _{1}=-8, \ell _{2}=8$, respectively; (b1)-(b4) corresponding intensity profile. All other parameters are same as those in Fig. 4.

Download Full Size | PDF

We also demonstrate the self-healing ability of the PTCSB through numerical simulation and experiment. Figures 7(c1)–7(c4) show numerical results of the intensity distribution of the PTCSB by passing through a sector-shaped opaque obstacle associated with the center angle $\theta =\pi / 6$ at planes $1-4$ marked in Fig. 7(a), respectively. It can be described that the sector-shaped block is placed at the initial plane $z =0$, which causes its first few rings to be partially blocked. We can find that the region of the unblocked PTCSB rotates into the defect region during propagation, which is gradually reconstructed the blocked region, as shown in Figs. 7(c1)–7(c2). However, the focus position of the unblocked PTCSB is larger than that of the blocked PTCSB compared with Fig. 4(c4), and then the spiral shape of the intensity distribution is recovered before the focus position. Our experimental results agree reasonably well with the numerical results, as shown in Figs. 7(d1)–7(d4).

Fig. 8 displays the normalized intensity of the PTCSB passing through a sector-shaped opaque obstacle with different center angles and $b$. We learn that when the PTCSB propagates through the sector-shaped opaque obstacle with the center angle $\theta =\pi / 6$, the blocked PTCSB with $b=0.3$ mm begins to finish the autofocus at $z=118z_{R}$ plane, while that with $b=0.2$ mm disappears at $z=71.25z_{R}$ plane, which means that the waist width has obvious effect on the focal distance of the PTCSB and the focal position is larger with the increase of $b$. Additionally, as the center angle increases to $\pi /3$, the depth of the autofocus, which is defined as the distance corresponding to the light intensity being 0.705 times the maximum light intensity, is almost a constant.

Fig. 7. Propagation dynamics of the blocked PTCSB passing through a sector-shaped opaque obstacle with the center angle $\theta =\pi / 6$. (a) Numerical side view of the blocked PTCSB propagation; (b) the intensity contrast values of the PTCSB along the $z$ direction. (c1)-(c4) Snapshots of the transverse intensity pattern of the blocked numerically planes $1-4$ marked in Fig. 7(a), respectively; (d1)-(d4) the corresponding experimentally results. All other parameters are the same as those in Fig. 4.

Download Full Size | PDF

Fig. 8. The intensity contrast values of the blocked PTCSB with different center angles and waist widths. All other parameters are the same as those in Fig. 4.

Download Full Size | PDF

4. Conclusion

In summary, we numerically and experimentally investigate the autofocusing and self-healing properties of the PTCSB. Unlike the properties of the CSB, the PTCSB generated through the appropriate phase modulation presents a tornado wave during the propagation. We further study the evolution process of the self-healing after passing through a sector-shaped opaque obstacle. Our numerical anticipations have been supported by the experimental results well. We discuss the Poynting vectors and OAM to illustrate the rotating mechanism of the beam clearly. Moreover, when the scale factor $b$ is larger, the focal point is larger, and the intensity of the focal point will be affected by the center angle of the sector-shaped opaque, but the focusing depth will be almost unchanged. This new type of the beam may be useful in some applications of optical spanners and optical communication.

Funding

National Natural Science Foundation of China (11374108, 11775083, 12174122); Guangzhou Municipal Science and Technology Project (2019050001); Natural Science Foundation of Guangdong Province (2018A030313480).

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. P. Vaveliuk, A. Lencina, and O. Martinez-Matos, “Caustic beams from unusual powers of the spectral phase,” Opt. Lett. 42(19), 4008–4011 (2017). [CrossRef]

2. P. Vaveliuk, A. Lencina, J. A. Rodrigo, and O. M. Matos, “Caustics, catastrophes, and symmetries in curved beams,” Phys. Rev. A 92(3), 033850 (2015). [CrossRef]

3. G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Observation of Accelerating Airy Beams,” Phys. Rev. Lett. 99(21), 213901 (2007). [CrossRef]

4. J. D. Ring, J. Lindberg, A. Mourka, M. Mazilu, K. Dholakia, and M. R. Dennis, “Auto-focusing and self-healing of Pearcey beams,” Opt. Express 20(17), 18955–18966 (2012). [CrossRef]

5. 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]

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

7. H. Teng, Y. Qian, Y. Lan, and W. Cui, “Swallowtail-type diffraction catastrophe beams,” Opt. Express 29(3), 3786–3794 (2021). [CrossRef]

8. 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]

9. X. Chen, D. Deng, J. Zhuang, X. Peng, D. Li, L. Zhang, F. Zhao, X. Yang, H. Liu, and G. Wang, “Focusing properties of circle Pearcey beams,” Opt. Lett. 43(15), 3626–3629 (2018). [CrossRef]

10. P. Vaveliuk, Ó. Martínez-Matos, Y. Ren, and R. Lu, “Dual behavior of caustic optical beams facing obstacles,” Phys. Rev. A 95(6), 063838 (2017). [CrossRef]

11. H. Teng, Y. Qian, Y. Lan, and Y. Cai, “Abruptly autofocusing circular swallowtail beams,” Opt. Lett. 46(2), 270–273 (2021). [CrossRef]

12. L. Allen, V. E. Lembessis, and M. Babiker, “Spin-orbit coupling in free-space Laguerre-Gaussian light beams,” Phys. Rev. A 53(5), R2937–R2939 (1996). [CrossRef]

13. Z. Wang, N. Zhang, and X. Yuan, “High-volume optical vortex multiplexing and de-multiplexing for free-space optical communication,” Opt. Express 19(2), 482–492 (2011). [CrossRef]

14. S. Tao, X. Yuan, J. Lin, X. Peng, and H. Niu, “Fractional optical vortex beam induced rotation of particles,” Opt. Express 13(20), 7726–7731 (2005). [CrossRef]

15. Z. Liu, Y. Liu, Y. Ke, Y. Liu, W. Shu, H. Luo, and S. Wen, “Generation of arbitrary vector vortex beams on hybrid-order Poincaré sphere,” Photonics Res. 5(1), 15–21 (2017). [CrossRef]

16. Y. Wen, I. Chremmos, Y. Chen, J. Zhu, Y. Zhang, and S. Yu, “Spiral Transformation for High-Resolution and Efficient Sorting of Optical Vortex Modes,” Phys. Rev. Lett. 120(19), 193904 (2018). [CrossRef]

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

18. X. Chen, D. Deng, G. Wang, X. Yang, and H. Liu, “Abruptly autofocused and rotated circular chirp Pearcey Gaussian vortex beams,” Opt. Lett. 44(4), 955–958 (2019). [CrossRef]

19. W. B. Wang, R. Gozali, L. Shi, L. Lindwasser, and R. R. Alfano, “Deep transmission of Laguerre-Gaussian vortex beams through turbid scattering media,” Opt. Lett. 41(9), 2069–2072 (2016). [CrossRef]

20. Y. Zhang, Y. Cai, and G. Gbur, “Control of orbital angular momentum with partially coherent vortex beams,” Opt. Lett. 44(15), 3617–3620 (2019). [CrossRef]

21. Y. Yang, X. Zhu, J. Zeng, X. Lu, C. Zhao, and Y. Cai, “Anomalous Bessel vortex beam: modulating orbital angular momentum with propagation,” Nanophotonics 7(3), 677–682 (2018). [CrossRef]

22. A. H. Dorrah, M. Zamboni-Rached, and M. Mojahedi, “Controlling the topological charge of twisted light beams with propagation,” Phys. Rev. A 93(6), 063864 (2016). [CrossRef]

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

24. R. A. Suarez, A. A. Neves, and M. R. Gesualdi, “Optical trapping with non-diffracting Airy beams array using a holographic optical tweezers,” Opt. Laser Technol. 135, 106678 (2021). [CrossRef]

25. M. Zamboni-Rached, “Stationary optical wave fields with arbitrary longitudinal shape by superposing equal frequency Bessel beams: Frozen Waves,” Opt. Express 12(17), 4001–4006 (2004). [CrossRef]

26. T. A. Vieira, M. R. R. Gesualdi, and M. Zamboni-Rached, “Frozen waves: experimental generation,” Opt. Lett. 37(11), 2034–2036 (2012). [CrossRef]

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

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

29. C. Schulze, F. S. Roux, A. Dudley, R. Rop, M. Duparré, and A. Forbess, “Accelerated rotation with orbital angular momentum modes,” Phys. Rev. A 91(4), 043821 (2015). [CrossRef]

30. J. Webster, C. Rosales-Guzmán, and A. Forbes, “Radially dependent angular acceleration of twisted light,” Opt. Lett. 42(4), 675–678 (2017). [CrossRef]

31. A. Brimis, K. G. Makris, and D. G. Papazoglou, “Tornado waves,” Opt. Lett. 45(2), 280–283 (2020). [CrossRef]

32. Y. Wu, C. Xu, Z. Lin, H. Qiu, X. Fu, K. Chen, and D. Deng, “Abruptly autofocusing polycyclic tornado ring Airy beam,” New J. Phys. 22(9), 093045 (2020). [CrossRef]

33. I. Stewart, “Catastrophe theory in physics,” Rep. Prog. Phys. 45(2), 185–221 (1982). [CrossRef]

34. Y. A. Kravtsov and Y. I. Orlov, Caustics, catastrophes and wave fields (Springer Science & Business Media, 2012).

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

36. Y. Lan, Y. Qian, and Z. Ren, “Tunable Polycyclic Chiral Beams,” Ann. Phys. 532(4), 1900530 (2020). [CrossRef]

37. K. Chen, H. Qiu, Y. Wu, Z. Lin, H. Huang, S. Ling, D. Deng, H. Liu, and Z. Chen, “Generation and control of dynamically tunable circular Pearcey beams with annular spiral-zone phase,” Sci. China: Phys., Mech. Astron. 64(10), 104211 (2021). [CrossRef]

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

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

40. M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, 1980).

41. T. Poon and T. Kim, Engineering optics with MATLAB (world scientific).

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

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

44. Y. Liu, C. Xu, Z. Lin, Y. Wu, Y. Wu, L. Wu, and D. Deng, “Auto-focusing and self-healing of symmetric odd-Pearcey Gauss beams,” Opt. Lett. 45(11), 2957–2960 (2020). [CrossRef]

45. H. I. Sztul and R. R. Alfano, “The Poynting vector and angular momentum of Airy beams,” Opt. Express 16(13), 9411–9416 (2008). [CrossRef]

Experimental generation of the polycyclic tornado circular swallowtail beam with self-healing and auto-focusing

Abstract

1. Introduction

2. Theory

3. Results and Discussions

4. Conclusion

Funding

Disclosures

Data availability

References

Data availability

Cited By

Figures (8)

Equations (12)

Optics Express