Open Access
Add to BibSonomyAbstract
Based on a geometric caustic argument and diffraction catastrophe theory, we generate a novel form of accelerating beams using a symmetric 3/2 phase-only pattern. Such beams can be called accelerating quad Airy beams (AQABs) because they look very much like four face-to-face combined Airy beams. Optical characteristics of AQABs are subsequently investigated. The research results show that the beams have axial-symmetrical and centrosymmetrical transverse intensity patterns and quasi-diffraction-free propagation features for their four main lobes while undergoing transverse shift along parabolic trajectories. Moreover, we also demonstrate that AQABs possess self-construction ability when local areas are blocked. The unique optical properties of these beams will make them useful tools for future scientific applications.
© 2014 Optical Society of America
1. Introduction
Accelerating Airy beams were first experimentally generated in 2007 [1]. Subsequently, accelerating Airy beams have been used as unique light sources of energy and as optical probes in various research fields. For example, because accelerating Airy beams follow precise parabolic trajectories, they can be used as optical spanners to trap, sort, transport, manipulate, guide and mix particles in the areas where blank light cannot arrive [2]. Airy beams can also be used in light–matter interactions to generate curved filamentation [3], curved plasma channels [4], curved electron acceleration [5], etc. Moreover, due to the gradient forces originating from the unique intensity pattern of Airy beams, researchers can use Airy beams to ‘snowblow’, a process that optically clears particles through colloidal suspensions [6].
To meet various experimental needs, research teams have produced multiple kinds of accelerating beams with different optical structure and acceleration [7–23]. Although each accelerating beam has different topological structure and propagation characteristic, all accelerating beams possess some common optical characteristics. First, all accelerating beams must have one or several narrow main lobes that concentrate optical energy. This can be regarded as pseudo-localized wave packets to be used in experiment. Such optical characteristics are similar to those of Bessel beams [24]. Second, the main lobes of accelerating beams can propagate along curved ballistic trajectories in free space.
Moreover, accelerating beams can generally be divided into two classes by optical structures. One class includes accelerating beams with a single main lobe, such as Airy beams [1], Pearcey beams [7], and other accelerating caustic beams [8–13]. The other class includes accelerating beams with several main lobes, such as accelerating regular polygon beams [14,15,22]. Note that all accelerating regular polygon beams have an odd number (three, five, or seven …) of main lobes—researchers have not yet been able to generate accelerating beams with even number (two, four, or six ...) of main lobes. Of course, some researchers have combined several accelerating beams to produce accelerating beams with several main lobes (including even number of main lobes). For example, combined Airy beams are used for purposes such as colloidal suspensions [6], as well as multiple optical tweezers setups [16]. In [17], Li et al. has even claimed that two back-to-back combined and crossed Airy beams are more beneficial to the energy gain for an electron than the acceleration scheme in a single Airy beam. However, it need to be pointed out that, combining several Airy beams cannot be considered as a true accelerating beam with several main lobes, but as groups of Airy beams with a single main lobe each.
From the perspective of a generating method, all accelerating beams can basically be divided into two categories based on whether a Fourier transform is performed. Fourier transform accelerating beams are in canonical catastrophe polynomial forms in Fourier space (so a Fourier transform must be performed). Canonical catastrophe polynomial forms are shown to exist for classes of standard exponential expressions [18]. Fourier transform accelerating beams include Airy beams [1], Pearcey beams [7], Mathieu beams [11,12], parabolic beams [13], and regular polygon beams [14,15,22], etc. Note that generation of Airy beams using the nonlinear effect of photonic crystals also requires performing a Fourier transform [23]. The second category of accelerating beams can be named as phase-only directly modulated accelerating beams. These beams can be directly produced using the phase-only modulation method, so no Fourier transform is required [8,9,20]. The original work of generating accelerating beams using the phase-only directly modulated method has been introduced by Cottrell et al. in [20]. In nature, phase-only directly modulated accelerating beams are exact rays from the aperture distribution that evolves into longitudinal caustic in finite distance [18–21].
We first deeply research the generation mechanism for accelerating Airy beams using the phase-only directly modulated method. Then, on the basis of a geometrical longitudinal caustic model and diffraction catastrophe theory, we provide new phase-modulated masks for directly generating new accelerating beams. We then construct novel finite-energy versions of accelerating beams with four symmetrical accelerating intensity maxima. Our research results show that these beams have a particular topology structure and propagating feature. It is well-known that laser beams (especially when in focus mode), are good light sources for energy and optical probes. Since different kinds of laser beams can play different roles in scientific fields, the new type of beams can be conceived to play a new role in the optical community.
The paper is presented as follows: first, an introduction followed by theoretical framework and formulae used to analyze phase masks for generation of new accelerating beams. In the third section experimental and numerical results of new accelerating beams are given, which includes analysis and discussion of optical characteristics. The fourth section covers the self-construction properties. Finally, main conclusions are drawn briefly.
2. Theory
In the past, Airy beams were usually generated after the Fourier plane of lens by imposing a cubic phase on the input beams. Recently, based on the approximate expression of the Airy function Ai(x) ≈x–1/4exp(iCx3/2), Airy beams have been directly generated by encoding the appropriate phase onto a phase modulation element. This method eliminates the need for an optical Fourier transform system and thus reduces the system length [20]. Nonbroading or weak-diffracting accelerating beams along arbitrary convex trajectories have also been generated using phase-only modulated patterns [8,9]. Abruptly focusing vortex beams are also generated using this method [21]. The principles of generating accelerating beams that use phase-only methods can be understood using the paraxial caustic theory of geometrical arguments [18–21,25].
Wavefront distribution φ(x0, y0) in the surface of the phase modulation element can be specified by the height of the wavefront above a reference plane, which is also called the geometrical optics wavefront [18]. Assuming that the coordinates of the observation point are (x, y, z), the distance between the modulated wavefront and observation point is:
In the paraxial approximation condition, for which z >> φ(x0, y0), z >> |x – x0| and z >> |y – y0|, Eq. (1) can be written as:Based on Fermat’s principle, r is stationary. The optical path should be a maximum, minimum, or saddle. Therefore there are:Substituting Eq. (2) into Eq. (3), there are:Based on Eq. (4), the relationship between phase distribution φ(x0, y0) and geometrical caustic argument can be derived. Moreover, in [18], Nye have pointed out that caustics are associated with diffraction patterns. In fact, all caustics (including folds, cusps, swallowtails, and elliptic and hyperbolic umbilics) are phenomena of diffraction catastrophes [18]. Since the directly generated accelerating beams using phase-only pattern are that modulated beams of finite size propagate in a finite distance, these diffraction catastrophes can essentially be regarded as Fresnel diffraction [8,9,20,21]. As a result, the 3D point spread function in free space with phase mask φ(x0, y0) can be written by an integral representation [26]:
whereNote that the initial phase distribution φ(x0, y0) can be generated using phase modulation elements such as spatial light modulators (SLMs).
In [20], Cottrell et al. write the phase function as:
where w is a parameter controlling the peak-to-valley aberration amount over the SLM and σ is a positive value. Apparently, Airy beams are generated using 1/4 quadrant phase-only patterns in reference [20], for which the phase distribution shows axial symmetry (with an axis of symmetry along the x0 = y0 direction) because there are φ(x0, y0) = φ(y0, x0) in Eq. (7). This generating mechanism is distinct because the corresponding relationship between phase distribution and the geometrical caustic has been given in Fig. 1 of reference [20]. Based on the geometrical model, centrosymmetrical phase distribution in the whole quadrant should be able to produce four accelerating Airy beams. Such a phase pattern can be written as:Apparently, there are:
Fig. 1 2D geometrical caustic model for accelerating quad Airy beams. The dashed lines represent the caustic lines (rainbow lines), and the solid lines represent their generating rays. The z–axis is optical axis, and the phase–modulated plane of SLM is overlapping with the x–axis.
Equation (9) indicates that phase function given in Eq. (8) is a completely mirror-symmetrical and centrosymmetrical.
Substituting Eq. (8) into Eq. (4), we can also provide the corresponding geometrical caustic model (see Fig. 1). We know that a caustic is an envelope of a family of rays of geometrical optics, which are associated with critical lines of accelerating beams. Unlike Fig. 1 of reference [20], Fig. 1 shows that four curved caustics of rays (there are two rays in the 2D cross-section) can be generated based on the phase distribution of Eq. (8). Using Eqs. (5) and (9), we can also prove that:
This equation tells us that the intensity distributions of modulated beams are axis-symmetrical and centrosymmetrical. Such results can be verified by experiment and numerical simulation in the following.
3. Experiment
Accelerating beam can be generated by modulating an incident beam using phase modulation elements. We constructed an experimental device, as shown in Fig. 2. By reflecting a collimated linearly polarized He-Ne beam on the effective working area of an optically-addressed liquid crystal SLM programmed with a given phase distribution, the phase-only modulated beam subsequently propagates in free space, but a Fourier transform optical system is not required. Then experimental results can be recorded by a properly aligned charge-coupled device (CCD) camera [8,9,20,21].
Fig. 2 Schematic of principle for directly generating accelerating beams using phase-only patterns encoded onto SLM. BS, beam splitter; SLM, Spatial light modulation; PMM, Phase modulated mask; CCD, Charge-coupled device.
Based on Eq. (8), the discrete phase distribution function encoding on SLM is given by:
where Δ is the pixel size of the SLM. There are M ∈ (–M0/2, M0/2–1), N ∈ (–N0/2, N0/2–1). M0 (N0) is relied on the total number of pixels of SLM in the transverse (longitudinal) direction. We define a 1080 × 1080 phase pattern (i.e. M0 = N0 = 1080) and use only the central portion of our SLM (1080 × 1920 pixels with a pixel spacing of 8 µm), since the phase mask is square in our experiment.We construct the phase mask based on Eq. (11), with a typically w of 0.0018 mm–0.5. Then this new phase-modulated mask (see Fig. 2) is written into a SLM to generate accelerating beams. The recorded results at different propagation distances are shown in Figs. 3(a) to 3(f). Each photo shows an area of 5 × 5 mm. Based on the wave propagation theory of Eq. (5), we numerically simulate intensity distribution of such beams during propagation, as shown in Figs. 3(g) to 3(l). The experimental results match well with the theoretical predictions.
Fig. 3 Experimental results for the 2D optical intensity distribution of an AQAB at distances of z = {30, 40, 50, 60, 70, 80} cm (a)–(f), and corresponding numerical simulation results at these same distances (g)–(l).
Figures 3(a) to 3(d) show how the modulated beam gradually evolves as four symmetrical Airy beams during propagation after the SLM. By observing Figs. 3(d) to 3(f), we find the whole beam looks like four face-to-face combined Airy beams. Therefore we call the beams accelerating ‘quad’ Airy beams (AQABs).
By carefully observing the topological structures and propagation characteristics of AQABs based on Figs. 3(d) to 3(f), we obtain some important optical characteristics of the AQABs. The most important fact is that, unlike accelerating Airy beams with only a single main lobe or accelerating regular polygon beams with odd main lobes, the AQAB has “quadruplet” main lobes. This means that an AQAB has an even number of main lobes. The internal margin of the four main lobes is fringe structures similar to classical Airy beams. Compared with the peripheral fringe area, the four main lobes have higher optical intensity (intensity maximum of beams). Therefore the main lobes of AQABs can be regarded as focal beams to be used in science research. We also find that each lobe of an AQAB is self-bending, i.e. it gradually deviates from the optical axis during propagation just like classical Airy beams. In other words, AQABs exhibit obvious transversal acceleration feature. Therefore, there is no doubt that AQABs are accelerating beams. But the maximum intensity cusp points of an AQAB exhibit four curved trajectories, and the four main lobe of an AQAB propagate in mutually opposite directions, with the centroid of the beam stayed on the optical axis. The sketch of propagating structure of the four main lobes of an AQAB is given in Fig. 4. Clearly there are essential links among Figs. 1, 3 and 4. The contour lines of Figs. 3(a) to 3(f) correspond to the A to F planes of Fig. 4, respectively. The A to F planes of Fig. 4 correspond to the sections A to F of Fig. 1. The trajectories of the high-intensity main lobes of Fig. 4 exactly correspond to the lines of the caustic lines of Fig. 1. Apparently, the optical caustic phenomena are of no doubt the foundation of generating accelerating beams. It is already well understood why each accelerating beams have their own topology structures and propagation characteristics according to their caustic phenomena. Therefore, the shape evolution of the AQAB given in Figs. 3(a) to 3(f) can readily be understood from Fig. 1. It is not difficult to know why the sectional photographs of the AQAB given in Figs. 3(a) to 3(c) are the beam distribution with weak fringe in the peripheral areas, but those in Figs. 3(d) to 3(f) are hollow beam distribution.
Fig. 4 Three-dimensional skeleton frame of optical AQABs. Black lines correspond to propagating trajectory of four main lobes of an AQAB. Blue lines are 2D cross-section outline of AQABs in given position.
Moreover, AQABs can resist diffraction, so their main lobes can propagate over a large range of z with minimal shape distortion. In the past, we knew that diffraction results in inevitably strong spreads of small localized beams (especially being focal beams). The spreads restrict optical micromanipulation to the vicinity of the focal plane. By virtue of their Airy transverse amplitude profile, the main lobes of AQABs are intrinsically diffraction-resistant. Therefore the AQAB keep its peak intensity and FWHM (the full width at half the maximum) area of the four maximum spots approximately constant in a transmission range of about 49 cm (from 55 cm to 104 cm away from the SLM) in our experiment. Numerical results show that the lobe’s FWHM diameter of Airy beams only increases from about 142 µm to 203 µm in this about 49 cm range, which means that the size of the main lobes only increases by 1.429 times. Apparently, the four main lobes of the AQAB are very stable during propagation. Experimental results well match the theoretical predictions.
After propagating 104 cm from the SLM, the AQAB begin to gradually deform. It is readily understandable why AQABs deform and even disappear, i.e. finite-energy Airy beams can also remain quasi-invariant up to a finite propagation distance until diffraction eventually takes over [1]. Although not being absolutely diffraction-free beams, AQABs can basically maintain stable propagation over extended distances—just similar to finite-energy Airy beams, in other words, they are weak diffraction beams [1]. Although the accelerating property of weak-diffracting main lobes can also be found in classical Airy beams, AQABs are more meaningful, because AQABs have transversely-symmetrical intensity patterns. The four main lobes of an optical AQAB also allow four symmetrical accelerating sampling densities to coexist, which is beneficial for some applications.
At the same time, note that the optical structure of AQABs differs from that of regular polygon beams of references [14,15,22], in which the regular polygon beam has an odd number of main lobes (three, five, or seven …). The main lobes of regular polygon beams become more and more flat, i.e. the shape of the main lobe gradually changes during propagation [14,15,22]. On the contrary, the shapes of main lobes of AQABs are always nearly round until the basic structure of AQABs has disappeared after long distance propagation. Therefore, the accelerating sampling points (main lobes) of AQABs are of higher quality than those of regular polygon beams.
In the past, people have thought that Airy beams are somewhat improper in some application fields, because of their asymmetrical transverse patterns. Therefore, generating accelerating beams with symmetrical structure is important in some application fields [16]. Note that the AQABs differ completely from the quad Airy beams reported in reference [16]. In this theoretical work, they reported a quad Airy beam generated by directly combining four classical Airy beams. The purpose of generating these beams is to compensate for this weakness of single classical Airy beam (with asymmetrical transverse intensity patterns). Moreover, although theoretically combining four same Airy beams into a symmetrical quad Airy beams is feasible, it is not easy in an experiment to ensure the absolute symmetry not only of optical structure but also of spatial position. The AQABs that we generated are mirror-symmetrical accelerating beams not only along the x–axis and y–axis but also along two oblique diagonal lines (x = y and x = –y). Such results match the theoretical prediction of Eq. (10).
Different scientific researchers need accelerating beams with different acceleration. Experimental and theoretical results show that we can also control the acceleration of the main lobes of AQABs by designing a phase mask with a different controlling parameter w based on Eq. (11). Making reference to the optical axis, the lateral displacement of each main lobe of an AQAB is directly proportional to w2. Since the acceleration of main lobes of AQABs also varies as the square of the distance z from the SLM, the distance △L of two adjacent main lobes in the x or y axis of an AQAB can be written as:
This expression provides a convenient way to predict the acceleration trajectory of the main lobes of AQABs.4. Self-construction propagation of AQABs
When people research the propagation properties of accelerating beams, the structural stability of the beams is an important subject [7,15,22,27]. Structural stability means that a beam can retain its intensity distribution when its local area is perturbed or damaged. An important and exotic optical characteristic of Airy beams is that they have the ability to ‘heal’ themselves and retain their intensity profiles after propagation at a distance when the beams are partially blocked by a small obstacle [27]. In this way, the beams act like biological tissue that can repair damaged areas. Such unusual optical properties allow the beams to work in adverse conditions for optical manipulating and other applications.
Until now, we have known that all classical structurally stable beams in free space (Such as Bessel beams, regular polygon beams, and Pearcey beams) have self-construction ability; i.e. they can survive severe changes in the shape of the local area [7,15,22,27,28]. As a kind of new accelerating beams with four main lobes, an AQAB might look like the combination of four classical Airy beams, judging by appearance. It is worthy of our attention whether each part of Airy beam of an AQAB also has the ability to repair itself similar to classical Airy beams.
The main lobes are the most important area of accelerating beams. When the AQABs are used in the field of light–matter or manipulating microparticles, their main lobes can be damaged for reasons such as light absorption and scattering. Therefore we mainly probe whether the damaged main lobes of AQABs can reconstruct during propagation. Figures 5(a) to 5(d) show that one blocked main lobe of an AQAB is gradually reconstructed during propagation (see the top left corner of Figs. 5(a) to 5(d)). The experimental results are excellently verified by the numerical results based on Fresnel diffraction theory, as is shown in Figs. 5(e) to 5(h). Such results demonstrate that the AQABs provide structural stability similar to that of the classical Airy beams. At the same time, the results also indicate that the main lobes of AQABs indeed are the caustics associated with the diffraction pattern.
Fig. 5 Self-construction of an AQAB when the main lobe of the top left corner is blocked. Observed intensity profile at Δz = {0, 14, 28, 42} mm from the obstacle.
Further, we also need to verify whether the four Airy beams (its four parts) of an AQAB act as a dependent whole, or are a simple combination of four independent Airy beams as in reference [16]. We block one complete Airy beam of an AQAB (in other words, we block 1/4 area of an AQAB). And, we find that the 1/4 blocked area of the AQAB gradually recovers during propagation, as shown in Figs. 6(a) to 6(d). Such results verify that the four Airy beam patterns of an AQAB act as a dependent whole, rather than as a simple combination of four independent Airy beams. We also need to point out that, this moment, the recovery rate is much slower than the case that one main lobe of an AQAB is blocked. In the process of self-recovery, there is a serious deformation of the surviving part (the surviving three Airy beams) of an AQAB. Because 1/4 of the AQAB area is blocked and thus 25% of the light energy of an AQAB is lost, the three surviving Airy beams must be severely deformed as power flows from the surviving area toward the blocked area. Figures 6(e) to 6(h) show numerical simulation results are basically match our experimental observations.
Fig. 6 Self-construction of an AQAB when one of its four Airy beams (1/4 of the total area of an AQAB) is blocked. Observed intensity profile at Δz = {0, 6, 12, 18} cm from the obstacle.
In short, AQABs have particular optical features such as transverse acceleration, weak diffraction, and self-construction properties. These optical properties of AQABs are mainly derived from their particular topology structures. In fact, there is no doubt that all non-diffraction or weak diffraction beams in free space are beams with special spatial distribution, which is also one of the reasons that such beams have self-construction properties. We also notice that none of current accelerating beams are rotationally-symmetric pattern [1,2,7–15,22]. On the contrary, rotationally symmetrical non-diffracting Bessel beams are not accelerating beams [24,28]. In [7], Ring et al. also pointed out that all accelerating beams along arbitrary convex trajectories are always shaped like the squares of Airy functions. Apparently, the optical structures of AQABs are not an exception, because each 1/4 area of an AQAB is the distribution of the square of the Airy function. But the complete transverse profiles are both axis-symmetrical and centrosymmetrical. Therefore AQABs are unique accelerating beams, because the section profiles of other accelerating beams (including well-known Airy beams, Pearcey beams, parabolic beams, regular polygon beams, and elliptic Mathieu beams) only are axial symmetry, not centrosymmetry.
5. Conclusions
We have reported the first observation of AQABs based on both an axial-symmetric and centrosymmetric 3/2 phase-only pattern. AQABs are novel accelerating beams with several noteworthy optical topology structures and propagation features. First, an AQAB is accelerating beams with four main lobes; this is the first time, to our knowledge, that accelerating beams with an even number of accelerating main lobes have been generated. Second, an AQAB looks like four face-to-face Airy beams. Their whole transverse profiles are both axis-symmetrical and centrosymmetrical. Third, because their optical topological structures resemble that of classical finite-energy Airy beams, the main lobes of AQABs show weak diffraction and also provide accelerating (self-bending) properties in the transverse direction under free-space propagation. Unlike a classical Airy beam with one main lobe, an AQAB has four high-intensity sampling points, and each point tends to transversely accelerate in the reverse direction during propagation. Fourth, similar to Airy and Bessel beams, the AQABs also demonstrate self-construction properties when their main lobes are blocked. Finally, we also found that the four Airy beams of an AQAB act as a dependent whole. In short, we are confident that this kind of unique accelerating beams with four main lobes and a symmetrical structure can offer new opportunities for various research fields.
Acknowledgments
This work was supported by the National Natural Science Foundation of China under Grant No 11274278, and the program for Innovative Research Team, Zhejiang Normal University.
References and links
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] [PubMed]
2. D. N. Christodoulides, “Optical trapping riding along an Airy beam,” Nat. Photonics 2(11), 652–653 (2008). [CrossRef]
3. P. Polynkin, M. Kolesik, E. M. Wright, and J. V. Moloney, “Experimental tests of the new paradigm for laser filamentation in gases,” Phys. Rev. Lett. 106(15), 153902 (2011). [CrossRef] [PubMed]
4. P. Polynkin, M. Kolesik, J. V. Moloney, G. A. Siviloglou, and D. N. Christodoulides, “Curved plasma channel generation using ultraintense Airy beams,” Science 324(5924), 229–232 (2009). [CrossRef] [PubMed]
5. J. X. Li, W. P. Zang, and J. G. Tian, “Vacuum laser-driven acceleration by Airy beams,” Opt. Express 18(7), 7300–7306 (2010). [CrossRef] [PubMed]
6. J. Baumgartl, T. Cizmár, M. Mazilu, V. C. Chan, A. E. Carruthers, B. A. Capron, W. McNeely, E. M. Wright, and K. Dholakia, “Optical path clearing and enhanced transmission through colloidal suspensions,” Opt. Express 18(16), 17130–17140 (2010). [CrossRef] [PubMed]
7. 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] [PubMed]
8. E. Greenfield, M. Segev, W. Walasik, and O. Raz, “Accelerating light beams along arbitrary convex trajectories,” Phys. Rev. Lett. 106(21), 213902 (2011). [CrossRef] [PubMed]
9. 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] [PubMed]
10. 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] [PubMed]
11. P. Aleahmad, M. A. Miri, M. S. Mills, I. Kaminer, M. Segev, and D. N. Christodoulides, “Fully vectorial accelerating diffraction-free Helmholtz beams,” Phys. Rev. Lett. 109(20), 203902 (2012). [CrossRef] [PubMed]
12. P. Zhang, Y. Hu, T. C. Li, D. Cannan, X. B. Yin, R. Morandotti, Z. G. Chen, and X. Zhang, “Nonparaxial Mathieu and Weber accelerating beams,” Phys. Rev. Lett. 109(19), 193901 (2012). [CrossRef] [PubMed]
13. M. A. Alonso and M. A. Bandres, “Spherical fields as nonparaxial accelerating waves,” Opt. Lett. 37(24), 5175–5177 (2012). [CrossRef] [PubMed]
14. S. Barwick, “Accelerating regular polygon beams,” Opt. Lett. 35(24), 4118–4120 (2010). [CrossRef] [PubMed]
15. Z. J. Ren, L. W. Dong, C. F. Ying, and C. J. Fan, “Generation of optical accelerating regular triple-cusp beams and their topological structures,” Opt. Express 20(28), 29276–29283 (2012). [CrossRef] [PubMed]
16. C.-Y. Hwang, D. Choi, K.-Y. Kim, and B. Lee, “Dual Airy beam,” Opt. Express 18(22), 23504–23516 (2010). [CrossRef] [PubMed]
17. J. X. Li, X. L. Fan, W. P. Zang, and J. G. Tian, “Vacuum electron acceleration driven by two crossed Airy beams,” Opt. Lett. 36(5), 648–650 (2011). [CrossRef] [PubMed]
18. J. F. Nye, Natural Focusing and Fine Structure of Light: Caustics and Wave Dislocations (Institute of Physics Publishing, 1999).
19. Y. Kaganovsky and E. Heyman, “Nonparaxial wave analysis of three-dimensional Airy beams,” J. Opt. Soc. Am. A 29(5), 671–688 (2012). [CrossRef] [PubMed]
20. D. M. Cottrell, J. A. Davis, and T. M. Hazard, “Direct generation of accelerating Airy beams using a 3/2 phase-only pattern,” Opt. Lett. 34(17), 2634–2636 (2009). [CrossRef] [PubMed]
21. J. A. Davis, D. M. Cottrell, and J. M. Zinn, “Direct generation of abruptly focusing vortex beams using a 3/2 radial phase-only pattern,” Appl. Opt. 52(9), 1888–1891 (2013). [CrossRef] [PubMed]
22. Z. J. Ren, X. D. Li, C. J. Fan, and Z. Q. Xu, “Generation of optical Accelerating quinary-cusp beams and their optical characteristics,” Chin. Phys. Lett. 30(11), 114208 (2013). [CrossRef]
23. T. Ellenbogen, N. Voloch-Bloch, A. Ganany-Padowicz, and A. Arie, “Nonlinear generation and manipulation of Airy beams,” Nat. Photonics 3(7), 395–398 (2009). [CrossRef]
24. J. Durnin, J. J. Miceli Jr, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58(15), 1499–1501 (1987). [CrossRef] [PubMed]
25. M. V. Berry, “Looking at coalescing images and poorly resolved caustics,” J. Opt. A, Pure Appl. Opt. 9(7), 649–657 (2007). [CrossRef]
26. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1968).
27. 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] [PubMed]
28. Z. Bouchal, J. Wagner, and M. Chlup, “Self-reconstruction of a distorted nondiffracting beam,” Opt. Commun. 151(4-6), 207–211 (1998). [CrossRef]
