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Abstract
We demonstrate a universal approach for generating high-order diffraction catastrophe beams, specifically for Swallowtail-type beams (abbreviated as Swallowtail beams), using diffraction catastrophe theory that was defined by potential functions depending on the control and state parameters. The three-dimensional curved caustic surfaces of these Swallowtail catastrophe beams are derived by the potential functions. Such beams are generated by mapping the cross sections of the high-order control parameter space to the corresponding transverse plane. Owing to the flexibility of the high-order diffraction catastrophe, these Swallowtail beams can be tuned to a diverse range of optical light structures. Owing to the similarity in their frequency spectra, we found that the Swallowtail beams change into low-order Pearcey beams under given conditions during propagation. Our experimental results are in close agreement with our simulated results. Such fantastic catastrophe beams that can propagate along curved trajectories are likely to give rise to new applications in micromachining and optical manipulation, furthermore, these diverse caustic beams will pave the way for the tailoring of arbitrarily accelerating caustic beams.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
The various accelerating beams propagating along desired trajectories have received considerable attention [1–14], and give rise to potential applications such as laser micromachining [15], optical manipulation [16], microscopy imaging [17], light bullets [18], and plasma generation [19]. The best-known examples of these accelerating beams are Airy beams [1], Mathieu and Weber beams [20]. These accelerating beams are typically associated with optical caustics [21,22], where the intensity maxima of these beams are localized and follow curved trajectories. With respect to geometrical optics, caustics are points or curves in two- dimensional (2D) space, (or surfaces in 3D space) where light rays are focused. That is to say, caustics are concerned with the envelopes of families of light rays. With respect to wave optics, caustics are regions where a light field transitions from an oscillatory behavior in the lit region (due to intense interference) to exponentially decaying behavior in the shadow region as an evanescent field. Optical caustics can be employed to tailor diverse artificial beams following various curved paths including convex [23], spiral [4], and arbitrary trajectories [24]. The caustic approach has also proven successful in generating other types of structured light, including propagation-invariant beams [25], autofocusing beams [26–30], intensity-symmetric beams [31,32], bottle shaped beams [33].
Optical caustics are specific manifestations of diffraction catastrophes [21]. In order of the increasing dimensionality (d) of their control parameter spaces, there are fold, cusp, swallowtail, butterfly, elliptical umbilic, hyperbolic umbilic, and parabolic umbilic diffraction catastrophes. Each catastrophe forms a stable diffraction structure that is characterized by an abrupt change in the critical points of the diffraction structure [34]. In general, catastrophes cause caustics to appear as points, lines, surfaces, or hypersurfaces. In 1D space, an Airy beam, which propagates along a smooth parabolic trajectory, is the simplest fold catastrophe. The Airy beam is defined by the diffraction catastrophe integral known as the Airy function [1]: $Ai(X) = \frac{1}{{2\pi }}\int_{ - \infty }^{ + \infty } {\exp [i({s^3}/3 + Xs)]{\kern 1pt} } ds{\kern 1pt}$. A fold caustic is formed when two rays exist at any point on one side of the caustic, and do not exist on the other side. In a 2D space, Pearcey beams, which are defined by the cusp catastrophe, are represented via the Pearcey function [35]. Pearcey beams exhibit the intriguing properties of auto-focusing and form-invariance. Despite the recent progress in exploring the possibilities presented by caustic beams, only the low-order catastrophe Airy beams (1D, fold catastrophe) and Pearcey beams (2D, cusp catastrophe) have been experimentally generated. To date, there has been little success in generating high-order catastrophes (i.e., when caustics appear as surfaces or hypersurfaces; $d \ge 3$).
In this work, we develop a universal approach to realizing high-order catastrophe beams, specifically Swallowtail-type beams (hereafter this text will be abbreviated as Swallowtail beams), using diffraction catastrophe integral [21,36]. Here, the Swallowtail beam caustic manifests as a high-intensity curved surface in 3D space. Owing to the flexibility of high-order diffraction catastrophes, Swallowtail beams can be tuned to produce a wide variety of optical light structures. Experimentally, Swallowtail beams are generated by projecting the cross sections of a 3D space onto the corresponding 2D plane; and our experimental results are concordant with the model simulations for these beams. In our investigation of Swallowtail beams, we found that due to similarities in their frequency spectra, these beams may evolve into low-order Pearcey beams during propagation. We also demonstrate, both analytically and experimentally, that the amplitude of the Fourier spectrum of a Swallowtail beam can be expressed using polynomials. Using our findings, it is fairly straightforward to generate caustic beams using spectral phase engineering. Such exotic high-order catastrophe beams will have many applications in the fields of micromachining and optical manipulation. Exploring these diverse caustic beams represents a significant step forward in the ultimate goal of optically moving particles along desired trajectories.
2. Theory
In optics, according to catastrophe theory, a caustic field ${C_n}(a){\kern 1pt}$ is represented using a standard diffraction catastrophe integral [21]:
Where the properties of the caustic field are determined by ${p_n}(a,s)$, the canonical potential function. The potential function is defined as
Where, s is the state variable and the vector $a = ({a_1},{a_2},\ldots ,{a_j})$ represents all of the dimensionless control parameters ${a_j}$ with $j = 1,2,\ldots ,n - 2$. The control parameter space a is equivalent to the real space (X,Y,Z), which corresponds to a dimension of $n \ge 3$. Because each control space has a variety of possible dimensionalities, the caustic field may emerge as a point, line, surface, or hypersurface.
For the high-order swallowtail catastrophe ${C_5}(X,Y,Z){\kern 1pt}$, a Swallowtail beam (SB) $Sw(X,Y,Z)$ is defined by the swallowtail catastrophe integral:
Where $X = x/{x_0}$, $Y = y/{y_0}$, and $Z = z/{z_0}$ represent dimensionless coordinates in space and ${x_0},\,{y_0}$ and ${z_0}$ are arbitrary scaling factors. The polynomial ${s^5} + Z{s^3} + Y{s^2} + Xs$ in Eq. (3) denotes the potential function ${p_n}(a,s)$ that determines the diffraction structure of the swallowtail catastrophe. When d=3, integrating Eq. (3) produces the swallowtail field, caustic of which manifests as a 3D curved surface. Therefore, the expression of a given caustic depends on the d value of the control parameter space. When $d > 3$, the caustic appears as hypersurface. For example, the butterfly catastrophe caustic, where d=4, manifests as hypersurface.
To investigate these high-order catastrophe fields, we project the cross-sections of the 3D Swallowtail beam onto the 2D transverse plane. In this framework, we define $a = ({a_1},{a_2},{a_3})$, where each element of a corresponds to a dimension in real space $(X,Y,Z)$. In the swallowtail catastrophe, we set one of the three control parameters to be constant, leaving the two remaining control parameters as the corresponding space coordinates. We can derive three categories of 2D Swallowtail beams by projecting the high-order catastrophe onto the low-order control parameter space. Specifically, $Sw(X,Y,{a_3})$ can be expressed as
Figures 1(a1)–1(a3) show the numerical intensity distributions for $Sw(X,Y,{a_3})$ when ${a_3} ={-} 4$, 0, and 8, respectively. Because the control parameter ${a_3}$ generates different diffraction catastrophe structures, changing ${a_3}$ results in diverse light structures in the $Sw(X,Y,{a_3})$ beam. The corresponding phase structures are displayed in Figs. 1(b1)–1(b3).
Fig. 1. (a1)–(a3) Numerical intensity distributions and (b1)–(b3) Corresponding phase structures for $Sw(X,Y,{a_3})$ when ${a_3} ={-} 4{\kern 1pt}$, 0, and 8, respectively.
To visualize these 3D caustics, we introduce the potential function of catastrophe theory into optics field. Catastrophe theory describes the key potential functions which plays an important role, and caustics are defined as abrupt transitions of the optical system. At these transition [21], the potential function ${p_n}(a,s)$ is stationary in s when ${a_j}$ is constant, and the caustics are stable diffraction structures defined by the singular mapping of the potential function ${p_n}(a,s)$. Consequently, the first and second derivatives of the potential function must be equal to zero [37,38]:
Applying Eqs. (5.1) and (5.2) to the swallowtail catastrophe potential function of ${s^5} + Z{s^3} + Y{s^2} + Xs$, we derive the following parameter equations that describe the caustics for the swallowtail catastrophe:
Figures 2(a) and 2(b) visualize the swallowtail catastrophe caustics from two different perspectives. The red, yellow and blue lines represent the mapping of the 3D caustics into 2D cross section in the X-Y plane when ${a_3} < 0,\,{a_3} = 0$ and ${a_3} > 0$, respectively. These caustic cross sections geometries are very similar to the maximum intensity distributions shown in Figs. 1(a1)-1(a3). Furthermore, the projected caustic structures in X-Y plane are symmetric about the X axis when coordinate space $(X,Y,Z)$ corresponds to the control parameter space $({a_1},{a_2},{a_3})$.
For control parameters $a = ({a_1},{a_2}{\kern 1pt} )$ corresponding to 2D transverse coordinates $(X,Y)$, Pearcey beams are described as ${C_4}(X,Y) = Pe(X,Y) = \int_{ - \infty }^{ + \infty } {\exp [i({s^4} + Y{s^2} + Xs)]} {\kern 1pt} ds$ for $n = 4$ [35]. However, for control parameter $a = {a_1}$ corresponding to the 1D transverse coordinate X, Airy beams, which are defined by fold catastrophe, are expressed as ${C_3}(X) = Ai(X) = \frac{1}{{2\pi }}\int_{ - \infty }^{ + \infty } {\exp [i({s^3}/3 + Xs)]} {\kern 1pt} {\kern 1pt} ds$ when $n = 3$. Consequently, 2D Airy beams are practically realized by multiplying two orthogonal and separable 1D Airy beams: $Ai(X,Y) = Ai(X) \cdot Ai(Y)$. In this case, only the 1D Airy beam exhibits a general structure dictated by a fold catastrophe.
We can explore other properties of a 2D Swallowtail beam by holding different control parameters constant. $Sw({a_1},Y,Z)$ and $Sw(X,{a_2},Z)$ beams also demonstrate diverse exotic light field structures when they are projected into their corresponding 2D planes. The numerical intensity distributions of $Sw({a_1},Y,Z)$ (Figs. 3(a1)–3(a3)) and $Sw(X,{a_2},Z)$ (Fig. 3(b1)–3(b3)) beams are consistent with the 3D caustics shown in Fig. 2. Note that Z no longer represents the conventionally defined propagation distance because the high-order Swallowtail beams are expressed as 3D structured light beams. The flexibility of these high-order swallowtail catastrophes will be an important aspect in future efforts to optically manipulate or move particles along desired trajectories.
Fig. 3. Numerical intensity distributions for (a1–a3) $Sw({a_1},Y,Z){\kern 1pt}$ beams when ${a_1} ={-} 4$, 0, and 8, respectively, and (b1–b3) $Sw(X,{a_2},Z){\kern 1pt}$ beams when ${a_2} ={-} 4$, 0, and 4, respectively.
3. Experimental results and discussions
As shown in Fig. 4, Swallowtail beams can be experimentally created using computer-controlled reflective phase-only spatial light modulator (SLM, Holoeye, Pluto, 1920×1080 pixels). We used a He-Ne laser with $\lambda$=632 nm generates an expanded and collimated Gaussian beam with a FWHM of 8.6 mm. That beam is then reflected by the SLM, where we maintained a pre-designed computer-generated hologram (CGH). Finally, using a movable CCD (4608×3288 pixels), we recorded the intensity distribution for a Swallowtail beam with a 4f-system that has an appropriate filter. Here, the computer-generated hologram was generated by the interference between a Swallowtail beam and a plane wave.
Fig. 4. Experimental setup. BE: beam expander, BS: beam splitter, L: lens, FF: Fourier filter, SLM: spatial light modulator.
A $Sw(X,Y,{a_3})$ beam evolves in two spatial dimensions according to the paraxial approximation of the angular spectrum integral [31]:
Where $\widetilde {Sw}({K_X},{K_Y}) = {\kern 1pt} \delta ({K_X}^2 - {K_Y}) \cdot {e^{i({K_X}^5 + {a_3}{K_X}^3)}}{\kern 1pt} {\kern 1pt}$ denotes the angular spectrum of the optical field $Sw(X,Y,{a_3},\xi = 0)$, and $2\pi$ scaling factor is neglected in the spectrum for reasons of clarity. Here, $\delta ({\cdot} )$ represents the Dirac function, $\xi = {{{\xi _p}} / {(2Kz_0^2)}}$ represents a dimensionless propagation distance, where ${\xi _p}$ is the propagation distance and ${\xi _0}$ is arbitrary scaling factor. ${K_X}$ and ${K_Y}$ are the dimensionless spatial frequencies in X- and Y- directions, respectively, and $K = \sqrt {{K_X}^2 + {K_Y}^2 + {K_Z}^2} $ represents the dimensionless wavenumber, where ${a_3} = 0$.
Figures 5(a1)–5(c1) demonstrate the numerical intensity distributions for ξ = 0, 0.7, and 1.5 planes, respectively. $Sw(X,Y,{a_3},\xi )$ beams degenerate into low-order Pearcey beams during propagation because the Fourier spectrum of $Sw(X,Y,{a_3},\xi = 0)$ is similar to that of a Pearcey beam. The Fourier spectrum of a Pearcey beam is $\widetilde {Pe}({K_X},{K_Y}) = {\kern 1pt} \delta ({K_X}^2 - {K_Y}) \cdot {e^{i{K_X}^4}}{\kern 1pt} {\kern 1pt}$, while the Fourier spectrum of a Swallowtail beam can be expressed by $\widetilde {Sw}({K_X},{K_Y}) = {\kern 1pt} \widetilde {Pe}({K_X},{K_Y}) \cdot {e^{i({K_X}^5 + {a_3}{K_X}^3 - {K_X}^4)}}$. Consequently, the propagation field of a Swallowtail beam can be written as:
Where $\phi = {K_X}^5 + {a_3}{K_X}^3 - {K_X}^4$ is regarded as the extra-generated phase resulted from propagation. According to angular spectrum theory, the propagation field only changes the spectral phase by a certain amount compared with initial field. This actually means a beam has moved a certain distance in real space. Equation (8) indicates that the propagation field of a Swallowtail beam manifests as the propagation field of a Pearcey beam. And that, Pearcey beams show shape-invariant structure distributions during propagation, consequently, the Swallowtail beam inevitably involves into Pearcey beam during propagation. Figures 5(a2)-5(c2) display our experimental results for ${\xi _p}$=0, 139 and 299 mm planes, where $\lambda = \textrm{632}.\textrm{8} \times \textrm{1}{0^{ - \textrm{6}}}mm,\,{a_3} = 0$ and ${x_0} = {y_0} = {z_0} = {\xi _\textrm{0}}\textrm{ = }0.1{\kern 1pt} {\kern 1pt} {\kern 1pt} mm$. Our experimental results are in good agreement with our numerical results. With this control parameter, note that the Swallowtail beam $Sw(X,{a_2},Z)$ will evolve into a higher-order butterfly beam, which produces a similar Fourier spectrum to the ones shown previously.
Fig. 5. $Sw(X,Y,{a_3},\xi )$ beam. (a1)–(c1) Numerical intensity distributions for a $Sw(X,Y,{a_3},\xi )$ beam for planes of ξ=0, 0.7, and 1.5. (a2)–(c2) Experimental $Sw(X,Y,{a_3},\xi )$ results for planes at ${\xi _p}$=0, 139, and 299 mm.
To better discuss the propagation of Swallowtail beams, the dynamics of $Sw(X,Y,{a_3})$ beam can also be described by their propagating caustics. Here $Sw(X,Y,{a_3},\xi )$ can be rewritten as:
Visibly, the spectral amplitude $\delta ({K_X}^2 - {K_Y})$ obeys a parabolic distribution that satisfy ${K_Y} = {K_X}^2$. Equation (9) is a noncanonical fifth-order exponential function. To simplify the swallowtail integral $Sw(X,Y,{a_3},\xi )$ in Eq. (9), we substitute $s + \xi m$ for ${K_X}$ in order to suppress the next-to-leading order term after applying the Tschirnhaus transform [39]:
Where, ${C_1} = X + 2mY\xi + m({3{a_3}m - 2} ){\xi ^2} - 15{m^4}{\xi ^4}{\kern 1pt} ,\,{C_2} = Y + ({3{a_3}m - 1} )\xi - 20{m^3}{\xi ^3}{\kern 1pt} ,\,{C_3} = {a_3} - 10{m^2}{\xi ^2},\,\psi = mX\xi + {m^2}Y{\xi ^2} + {m^2}({{a_3}m - 1} ){\xi ^3} - 4{m^5}{\xi ^5}{\kern 1pt}$, Note that the expressions for the coefficients are true only for one single value $m = 1/5$.
The evolution of the $Sw(X,Y,{a_3})$ beam can be described again with an optical swallowtail catastrophe beam $Sw({C_1},{C_2},{C_3})$ where the control parameters are functions of X, Y, ${a_3}$, and $\xi$. By combining Eqs. (5) and (10), we can derive the following parameter equations that describe the propagating caustic:
Figure 6(a) exhibits the caustic surface of the $Sw(X,Y,{a_3})$ beam during propagation in real 3D space $(X,Y,\xi )$, where $\xi$ denotes the propagation distance. The red line in the $\xi = 0$ plane represents the initial caustic line, which matches the numerical and experimental results displayed in Figs. 5(a1) and 5(a2). The yellow and blue lines represent the propagating caustic lines, which are in good agreement with the results shown in Figs. 5(b1) and 5(c1). The blue line represents the Pearcey caustic structure, which validates our theoretical prediction in Eq. (8) and the experimental shown in Fig. 5(c2). As seen in Fig. 6(b), the surface includes two intertwined cross-sections, which gradually separate and evolve into a cusp and a flat structure. During propagation, these structures become increasingly smooth and curvilinear. Consequently, the caustic structure also clearly exhibits accelerating property. With these structures, it is possible for caustic beams to propagate along a given path, a feature that will prove valuable in the design and construction of artificial accelerating beams.
Fig. 6. Two different perspectives of the visual caustic surface of $Sw(X,Y,{a_3})$ beam during propagation for ${a_3} = 0$.
As discussed above, the frequency spectrum for a $Sw(X,Y,{a_3}){\kern 1pt}$ beam is $\widetilde {Sw}({K_X},{K_Y}) = {\kern 1pt} \delta ({K_X}^2 - {K_Y}) \cdot {e^{i({K_X}^5 + {a_3}{K_X}^3)}}{\kern 1pt} {\kern 1pt}$, where the parabolic spectral amplitude $\delta ({K_X}^2 - {K_Y})$ is satisfied when ${K_Y} = {K_X}^2$. Similarly, the even-symmetrical spectrum for a $Sw({a_1},Y,Z)$ beam is ${\kern 1pt} \delta ({|{{K_X}} |^{\frac{3}{2}}} - {K_Y}) \cdot {e^{i{{|{{K_X}} |}^{\frac{5}{2}}}}}{e^{i{a_1}{{|{{K_X}} |}^{\frac{1}{2}}}}} + \delta ( - {|{{K_X}} |^{\frac{3}{2}}} - {K_Y}) \cdot {e^{ - i{{|{{K_X}} |}^{\frac{5}{2}}}}}{e^{ - i{a_1}{{|{{K_X}} |}^{\frac{1}{2}}}}}$ and the cubic spectrum for a $Sw(X,{a_2},Z)$ beam is $\delta ({K_X}^3 - {K_Y}) \cdot {e^{i({K_X}^5 + {a_2}{K_X}^2) }}$. For ${a_1} = {a_2} = {a_3} = 0$, we see that the numerical intensity distributions (Figs. 7(a1)–7(c1)) for these spectra are consistent with our experimental results, as recorded by the CCD placed in the Fourier plane (Figs. 7(a3)–7(c3)). The corresponding spectral phases are displayed in Figs. 7(a2)–7(c2). The spectral phases of $Sw(X,Y,{a_3})$ and $Sw(X,{a_2},Z)$ are identical to one another because they both evaluate to ${K_X}^5{\kern 1pt}$ when ${a_2} = {a_3} = 0$. Here, the spectrum can be described by polynomials, which form the swallowtail structures.
Fig. 7. (a1–c1) Numerical spectral intensity distributions, (a2–c2) phase structures, and (a3–c3) experimental intensity results in Fourier space for the $Sw(X,Y,{a_3}),\,Sw(X,{a_2},Z)$, and $Sw({a_1},Y,Z)$ beams, respectively, for ${a_1} = {a_2} = {a_3} = 0$.
4. Conclusions
In conclusion, we developed an approach for generating high-order catastrophe beams, specifically for Swallowtail beams, using diffraction catastrophe integral. These beams are realized experimentally by projecting the cross sections of the 3D light field onto their corresponding transverse planes. The caustic surfaces of catastrophe beams depend solely on their potential functions. Owing to the flexibility of high-order diffraction catastrophes, Swallowtail beams are highly tunable and can be used to produce a diverse array of light field structures. These fantastic swallowtail caustics result in the beams with singular behaviors such as the ability to propagate along curved trajectories. These unique propagation dynamics will be vital in developing new applications in the fields of micromachining, optical manipulation, and the tailoring of arbitrary accelerating caustic beams.
Funding
Key Laboratory of Optical Field Manipulation of Zhejiang Province (ZJOFM2019004); National Natural Science Foundation of China (11974314).
Disclosures
The authors declare no conflicts of interest.
References
1. G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Observation of accelerating Airy beams,” Phys. Rev. Lett. 99(21), 213901 (2007). [CrossRef]
2. T. Melamed and A. Shlivinski, “Practical algorithm for custom-made caustic beams,” Opt. Lett. 42(13), 2499–2502 (2017). [CrossRef]
3. M. Goutsoulas, R. S. Penciu, and N. K. Efremidis, “Independent amplitude and trajectory/beam-width control of nonparaxial beams,” Opt. Express 26(15), 18969–18974 (2018). [CrossRef]
4. Y. Lan, F. Hu, and Y. Qian, “Generation of spirally accelerating optical beams,” Opt. Lett. 44(8), 1968–1971 (2019). [CrossRef]
5. L. Froehly, F. Courvoisier, A. Mathis, M. Jacquot, L. Furfaro, R. Giust, P. A. Lacourt, and J. M. Dudley, “Arbitrary accelerating micron-scale caustic beams in two and three dimensions,” Opt. Express 19(17), 16455–16465 (2011). [CrossRef]
6. Y. Kaganovsky and E. Heyman, “Wave analysis of Airy beams,” Opt. Express 18(8), 8440–8452 (2010). [CrossRef]
7. Y. Hu, D. Bongiovanni, Z. Chen, and R. Morandotti, “Periodic self-accelerating beams by combined phase and amplitude modulation in the Fourier space,” Opt. Lett. 38(17), 3387–3389 (2013). [CrossRef]
8. M. A. Bandres, “Accelerating beams,” Opt. Lett. 34(24), 3791–3793 (2009). [CrossRef]
9. I. Kaminer, R. Bekenstein, J. Nemirovsky, and M. Segev, “Nondiffracting accelerating wave packets of Maxwell’s equations,” Phys. Rev. Lett. 108(16), 163901 (2012). [CrossRef]
10. J. Zhao, I. D. Chremmos, D. Song, D. N. Christodoulides, N. K. Efremidis, and Z. Chen, “Curved singular beams for three-dimensional particle manipulation,” Sci. Rep. 5(1), 12086 (2015). [CrossRef]
11. J. A. Rodrigo, T. Alieva, E. Abramochkin, and I. Castro, “Shaping of light beams along curves in three dimensions,” Opt. Express 21(18), 20544–20555 (2013). [CrossRef]
12. J. C. Gutiérrez-Vega and M. A. Bandres, “Helmholtz–Gauss waves,” J. Opt. Soc. Am. A 22(2), 289–298 (2005). [CrossRef]
13. S. N. Khonina, “Specular and vortical Airy beams,” Opt. Commun. 284(19), 4263–4271 (2011). [CrossRef]
14. S. N. Khonina, A. V. Ustinov, and S. Chávez-Cerda, “Generalized parabolic nondiffracting beams of two orders,” J. Opt. Soc. Am. A 35(9), 1511–1517 (2018). [CrossRef]
15. A. Mathis, F. Courvoisier, L. Froehly, L. Furfaro, M. Jacquot, P. A. Lacourt, and J. M. Dudley, “Micromachining along a curve: Femtosecond laser micromachining of curved profiles in diamond and silicon using accelerating beams,” Appl. Phys. Lett. 101(7), 071110 (2012). [CrossRef]
16. 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]
17. T. Vettenburg, H. I. C. Dalgarno, J. Nylk, C. Coll Lladó, D. E. K. Ferrier, T. Čižmár, F. J. Gunn-Moore, and K. Dholakia, “Light-sheet microscopy using an Airy beam,” Nat. Methods 11(5), 541–544 (2014). [CrossRef]
18. A. Chong, W. H. Renninger, D. N. Christodoulides, and F. W. Wise, “Airy–Bessel wave packets as versatile linear light bullets,” Nat. Photonics 4(2), 103–106 (2010). [CrossRef]
19. L. Li, T. Li, S. Wang, C. Zhang, and S. Zhu, “Plasmonic Airy beam generated by in-plane diffraction,” Phys. Rev. Lett. 107(12), 126804 (2011). [CrossRef]
20. P. Zhang, Y. Hu, T. Li, D. Cannan, X. Yin, R. Morandotti, Z. Chen, and X. Zhang, “Nonparaxial Mathieu and Weber accelerating beams,” Phys. Rev. Lett. 109(19), 193901 (2012). [CrossRef]
21. Y. A. Kravtsov and Y. I. Orlov, “Caustics, catastrophes and wave fields,” Springer (2012).
22. V. I. Arnol’d, “Catastrophe theory,” Springer (2003).
23. E. Greenfield, M. Segev, W. Walasik, and O. Raz, “Accelerating light beams along arbitrary convex trajectories,” Phys. Rev. Lett. 106(21), 213902 (2011). [CrossRef]
24. G. Lasry, T. Melamed, and Y. Brick, “Manipulation and control of 3-D caustic beams over an arbitrary trajectory,” Opt. Express 28(14), 20645–20659 (2020). [CrossRef]
25. A. Zannotti, C. Denz, M. A. Alonso, and M. R. Dennis, “Shaping caustics into propagation-invariant light,” Nat. Commun. 11(1), 1–7 (2020). [CrossRef]
26. I. D. Chremmos, Z. Chen, D. N. Christodoulides, and N. K. Efremidis, “Abruptly autofocusing and autodefocusing optical beams with arbitrary caustics,” Phys. Rev. A 85(2), 023828 (2012). [CrossRef]
27. 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]
28. 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]
29. S. N. Khonina, A. V. Ustinov, and A. P. Porfirev, “Aberration laser beams with autofocusing properties,” Appl. Opt. 57(6), 1410–1416 (2018). [CrossRef]
30. A. Brimis, K. G. Makris, and D. G. Papazoglou, “Tornado waves,” Opt. Lett. 45(2), 280–283 (2020). [CrossRef]
31. 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]
32. P. Vaveliuk, A. Lencina, J. A. Rodrigo, and O. M. Matos, “Symmetric airy beams,” Opt. Lett. 39(8), 2370–2373 (2014). [CrossRef]
33. P. Zhang, Z. Zhang, J. Prakash, S. Huang, D. Hernandez, M. Salazar, D. N. Christodoulides, and Z. Chen, “Trapping and transporting aerosols with a single optical bottle beam generated by moiré techniques,” Opt. Lett. 36(8), 1491–1493 (2011). [CrossRef]
34. T. Poston and I. Stewart, “Catastrophe theory and its applications,” Pitman, Dover (1996).
35. 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]
36. J. F. Nye, “From Airy rings to the elliptic umbilic diffraction catastrophe,” J. Opt. A: Pure Appl. Opt. 5(5), 503–510 (2003). [CrossRef]
37. M. V. Berry and C. Upstill, “IV catastrophe optics: morphologies of caustics and their diffraction patterns,” Prog. Optics 18, 257–346 (1980). [CrossRef]
38. M. V. Berry, J. F. Nye, and F. Wright, “The elliptic umbilic diffraction catastrophe,” Philos. T. R. Soc A 291(1382), 453–484 (1979). [CrossRef]
39. T. Needham, “Visual complex analysis,” Oxford University Press (1998).
