## Abstract

By using the diffractive optical elements written onto a spatial light modulator, we experimentally obtain optical regular triple-cusp beams. Their propagation characteristics and topological structures are subsequently investigated. The experimental results demonstrate that each cusp of an optical regular triple-cusp beam, similar to the main lobe of an Airy beam, propagates along curved paths in free space, hence tends to adopt the “transverse acceleration” property. Moreover, we experimentally prove that optical regular triple-cusp beams can resist local distorted deformation. Such beams can thus be applied in adverse optical environments, such as a probe for the exploration of microscopic world and as an energy source for research on high-field laser–matter interactions.

©2012 Optical Society of America

## 1. Introduction

Airy beams have recently attracted tremendous research interests. Perhaps the most important reason is because the main lobe of Airy beams tends to transversely accelerate along parabolic ballistic trajectories during propagation, just similar to the shapes of rainbows [1–5]. Even in nonlinear media, a nonlinear accelerating Airy-like beam can still travel along the parabolic trajectory just as it propagates in free space [6–8]. Due to their unusual optical propagation characteristics, Airy beams play a novel role in a wide range of applications as new-style probes or energy sources. For example, an Airy beam can be exploited to trap, sort, mix, and measure particles by precisely following parabolic trajectories [2,3]. This means that Airy beams can be used as control and transport micro particles in areas where an ordinary point-blank beam cannot arrive. Airy beams are also used in the field of light–matter interactions to generate curved filaments and curved plasma channels, as well as to accelerate electrons along a curved trajectory based on their self-bending property during propagation [9–15].

In 2010, another family of finite-energy accelerating beams was theoretically introduced, which were named as “accelerating regular polygon beams”. Such beams possess an odd number of high-intensity peaks that follow curved paths during propagation. The findings paved the way for innovating optical techniques [16]. However, accelerating regular polygon beams presented in reference [16] were only studied theoretically. In order to apply accelerating regular polygon beams in practical scientific researches, we must generate in advance regular polygon beams in experiments. Moreover, understanding the basic propagation properties of accelerating regular polygon beams is also of fundamental importance for various optical applications. This paper reports on the observation of optical regular triple-cusp beams (RTBs) in free space, which belong to one case of the regular polygon beams, by deriving the necessary phase patterns. The topological structures of optical RTBs are investigated by experiments and numerical propagation simulations. Our experimental results are in excellent agreements with theoretical predictions. The ballistic dynamics of optical RTBs are also studied experimentally. The research results are expected to lay the foundation for the future applications of optical RTBs.

## 2. Diffraction integrals and catastrophes

By associating diffraction integrals and the catastrophe polynomials [16–19], the three-dimensional (3D) intensity distribution function of optical RTBs can be written in dimensionless Cartesian coordinates:

where the amplitude distribution*E*(

*x*,

*y*,

*z*) is written as:

*x*,

*y*) are the transversal coordinates and

*z*is the longitudinal coordinate along the optical axis. (

*ξ*,

*η*) are the transversal spectral variables associating with (

*x*,

*y*).

*A*is a constant number and does not influence the amplitude distribution. According to Eqs. (1) and (2), a finite optical 3D amplitude distribution of optical RTBs can be theoretically predicted.

## 3. Experimental setup

The experimental schematic diagram used to generate the optical RTBs is shown in Fig. 1
. The phase patterns are displayed on a spatial light modulator (SLM, Holoeye PLUTO, 1080 × 1920 pixels with a pitch of 8.3um). A collimated He-Ne laser beam with an approximate width of 1 cm (full–width at half–maximum) used to cover the effective working area of the computer-controlled SLM. This SLM is used to impose the phase modulation, which is necessary to produce optical RTBs. The gray level image representation of phase mask to generate optical RTBs is shown in Fig. 2
. After the SLM, the phase-modulated beam is Fourier–transformed by using a lens with a 40-cm focal length to generate optical RTBs. Then, a well-aligned CCD camera used to view the beam. This CCD camera can slide along the optical axis away from the focal plane of lens to record optical RTBs at different observation planes. The observed distances are referred from the focal plane (*z* = 0 cm) of lens.

## 4. Results and discussions

To better applying optical RTBs in scientific researches, we hope to understand beforehand their optical properties. Hence, we studied first the internal topological structures of optical RTBs. In order to show more details in every figure, the scale changes between the panels in Fig. 3
since optical RTBs are actually expanding. Figures 3(a) to 3(e) display the recorded two-dimensional (2D) intensity photographs of the optical RTB in a series of sections corresponding to different propagation distances. The images reveal the rich structure of the optical RTB. Based on the wave propagation theory, a series of numerical propagation simulations of optical RTBs is performed based on Eqs. (1) and (2) for comparison with experimental observation results. Apparently, the experimental photographs fairly well coincide with the simulation configurations of optical RTBs shown in Figs. 3(f) to 3(j), even in the finest details. Based on Fig. 3, the entire 3D propagating sketch configuration of an optical RTB is shown in Fig. 4
. Figures 3(a) and 3(f) illustrate that the optical RTB is an on-axis singular point at *z* = 0 (the focal plane of the lens), which is a focus with elliptic umbilical structure. When *z*≠0, Figs. 3(b) to 3(e) and 3(g) to 3(j) show that this singular point subsequently splits into three interconnecting cusp lines, which is just the optical RTBs. Such beams are one case of accelerating regular polygon beams introduced by Barwick [16].

The main structure of an optical RTB consists of three cusp points and three concave sides. Hence, as viewed from the external features, the structure of optical RTBs has elastic stability (i.e., stretching or compression without tearing). The distance between vertices of an optical RTB is identical to each other. Accordingly, a circle can be inscribed within the regular concave side of the regular triangle. Three concave surfaces are simultaneously joined along three parabolic cusped skeletons, which touch at a singular on-axis point. The centroid of the optical RTB coincides with the optical axis throughout its propagation. Figures 3(b) to 3(e) show that the edge waves of the optical RTB are continuously distributed inside the three skeletons. At the fold point where the equilibrium breaks down, a catastrophic jump that forms a cusp point can be observed, which corresponds to peaks with high optical intensity. This type of abrupt change in the optical configuration of the critical points is so-called “diffraction catastrophe” [17–19]. Apparently, the optical intensity of the cusp is far larger than that of edge waves. Although the elementary contour of the optical RTB gradually expands during propagation, the intensity maxima pattern of the cusp point is basically stable over a large propagation distance. Thus, every cusp can actually be regarded as a focal beam with a very long focal depth or light bullets and can be used in the field of physical research. Three peaks of an optical RTB indicate that the coexisting of three sampling densities is allowed in certain applications. During propagation, the cusp points move away from the optic axis quadratically with increase of the propagation distance *z*. This indicates that each cusp point can transversely accelerate during propagation. It is necessary to note that the centre of gravity of the optical RTB remains invariant with *z* and constantly coincides with the straight optical axis, although each high-optical-intensity peak transversely accelerates.

A well-known optical property of an Airy beam is that its main lobe can transversely accelerate during propagation in free space [1–5]. Thus, it is reasonable to assume that each high optical intensity peak of optical RTBs has an optical property analogous to the main lobe of Airy beams. Both beams can be regarded as self-bending (or accelerating) beams. Unlike the accelerating Airy beam with one main peak [2,3], an optical accelerating RTB with three same main peaks belongs to another new family of accelerating beams. As displayed in Fig. 5 , the experimentally measured results show that each cusp point of an optical RTB we produced deflects by approximately 1.16 mm from the optic axis when the output is at a distance of 6 cm from the focal plane of lens.

The goal of generating any exotic laser beams, such as Bessel beams or Airy beams, is to use them as information carriers, energy sources or probes in scientific research. Optical RTBs are a new member of a general family of accelerating beams. Based on the propagation characteristics of optical RTBs mentioned above, we believe that the generation of optical RTBs has new scientific application values aside from the intrinsic scientific values being similar to Airy beams [2,3, 9–15]. For example, optical RTBs can be used to separate a mixture of particles along three parabolic paths because an on–axis singular point at *z* = 0 can be split into three gradual separation cusp points, as shown in Fig. 4. This property indicates that optical RTBs can be used as a new tool for sorting particles. Moreover, another potential application of optical RTBs should be used to realize three-channel synchronous signal transmission by modulating an on-axis singular point at *z* = 0. Optical RTBs can also be used in the field of light–matter interactions to generate curved filaments and curved plasma channels. In contrast to Airy beams, we believe that an optical RTB can generate three filaments or plasma channels simultaneously.

Finally, we address experimentally another intriguing property of optical RTBs. Catastrophe theory is known to mainly study the structurally stable singularities of gradient maps [16–19]. According to catastrophe theory, the geometrical structure of the regular triple-cusp deltoid with three concave surfaces, which has the distribution of the minimum potential energy, is more stable against certain perturbations than others. To experimentally confirm structural stability of optical RTBs, we monitor the self-reconstruction of such beams during the process of propagation. The scale changes between the panels in Fig. 6
. As shown in Fig. 6(a), we employed a small opaque obstacle to block one cusp point (including a few edge waves near the cusp point) of an optical RTB. Figures 6(b) to 6(d) depict the reformation of the damaged optical RTB after certain propagation distances. Just similar to the self-healing ability of Airy beam [20,21], the obstructing cusp point of an optical RTB is gradually reborn with the increase of propagation distance *z*. The self-healing ability of optical RTBs makes them very useful in optical manipulation and many other areas. We also note that almost no impact on the steady propagation of the other cusp points can be observed even when one cusp point is completely obstructed.

Similar to the theoretical predictions given in reference [16–19], we prove experimentally that optical RTBs are remarkably resilient to local intensity deformations. In other words, optical RTBs can survive perturbations to the medium through which the waves propagate, which is an important optical property. Intuitively, in some research areas, optical RTBs can potentially be applied in unfavorable optical environments. For instances, optical RTBs can be used as an effective probe in exploration of microscopic world and energy sources on researching high field laser–matter interactions, since optical RTBs can self-repair the damaged intensity pattern as a result of absorption or scattering. It also means that the completely damaged cusp point of an RTB does not influence the normal work of other two cusp points of this RTB. Apparently, such optical property is also advantageous when optical RTBs are used as information carriers in three-channel synchronous signal transmission.

## 5. Conclusions

In conclusion, optical accelerating RTBs were produced, and their intriguing properties were directly observed based on their 2D configurations. The novel beams exhibits three key characteristics. First, similar to the main lobe of Airy beams, every cusp of optical RTBs can be approximately regarded as focal beams with an over–extended depth of focus for some types of physical applications. However, an optical RTB we produced has three accelerating peaks while an Airy beam has only one single accelerating peak. Second, the intensity of the cusps of optical RTBs tends to transversely accelerate (or self-bend) during propagation in free space. Third, the structure of optical RTBs is stable over a large range of propagation distance, and the cusp can self-construct after perturbations during propagation. Apparently, optical RTBs are a new member of a general family of accelerating beams. Thus, the use of optical accelerating RTBs such as particle manipulation, signal transmission, and light–matter interactions may be another productive research direction. We believed that this investigation on optical RTBs will provide useful information for their future potential applications.

## Acknowledgments

We acknowledge financial support from the program for Innovative Research Team, Zhejiang Normal University, Jinhua, Zhejiang Province, P. R. China, NSFC under grant of 11274278 and 11074221.

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