† Corresponding author. E-mail:
Project supported by the National Natural Science Foundation of China (Grant No. 11604058), the Natural Science Foundation of Ningbo City, China (Grant No. ZX2015000617), the K C Wong Magna Fund in Ningbo University, China, and the Natural Science Foundation of Guangxi Zhuang Autonomous Region, China (Grant Nos. 2016GXNSFBA380244 and 2015GXNSFBA139011).
We numerically and experimentally demonstrate that a three-Airy autofocusing beam can be generated by superposing three deformed two-dimensional (2D) Airy beams with a triangle symmetry. When the initial angle between two wings of the deformed 2D Airy beams increases, such a three-Airy autofocusing beam exhibits that the focusing length decreases and the intensity contrast at the focal point changes. Moreover, after introducing an optical vortex phase, this three-Airy autofocusing beam displays a transverse rotation in propagation. The rotation angle is determined by the topological charge of the vortex and the initial wing angle. Our results may have some potential applications in optical manipulation.
Accelerating optical beams, represented by Airy beams, have attracted a lot of attention due to their unique properties including self-acceleration, self-reconstruction, and non-diffraction.[1–3] They have been used in various areas including optical micromanipulation,[4,5] filamentation,[6] light bullets,[7] optical routing,[8] imaging,[9–13] guiding discharges,[14] and material processing.[15] Triggered by the prosperity in optics, accelerating wave-packets were continuously realized in other wave systems such as plasmons,[16–18] acoustic waves,[19] electronic beams.[20]
In these different classes of applications, a special family of accelerating beams, namely, abruptly autofocusing (AAF) beams, plays an important role because of their autofocusing property.[21–27] The peak intensity for each of these beams almost is maintained at a constant level during early propagation, yet at a particular distance it abruptly increases by several orders of magnitude.[23–27] In general, the AAF beams are produced from radially symmetric Airy beams. In order to better control the autofocusing properties, many other methods were developed.[22–27] One can synthesize the AAF via a superposition of multiple one-dimensional (1D) or two-dimensional (2D) Airy beams distributed around a circle. Their autofocusing propagation is thus readily changed by engineering the lateral accelerations of the combining Airy beams. In these approaches, standard 2D Airy beams, for each of which the tangle between their two wings is the same, are employed. Recently, it was shown that the acceleration of 2D Airy beam changes with the angle between their two wings,[29–31] thus offering us an additional degree of freedom to control the AAF. Here, we propose a new kind of AAF beam via superposing three deformed 2D Airy beams with a triangle symmetry. Propagation dynamics of this new kind of AAF beam, namely, three-Airy autofocusing beam is investigated theoretically and experimentally. It is found that the three-Airy autofocusing beam definitely shows a focusing propagation and its focusing length varies with the angle between two wings of the superposing 2D Airy beam. Moreover, by engineering the phase profile of three-Airy autofocusing beam via a vortex structure, we can make the beams have a rotation. This rotation can be controlled by varying the topological charge of vortex structure and the initial wing angle. Our results will be very valuable for the optical manipulation.
In the paraxial condition, the intensity distribution of a three-Airy autofocusing beam at different propagation positions can be defined as a sum of three 2D Airy beams
In experiment, we employed a setup similar to those in our previous work[29–31] to generate a three-Airy autofocusing beam (i.e., via Fourier transforming a proper phase), as schematically shown in Fig.
Figures
In contrast, for the cases of θ ≠ 90°, the superposed three 2D Airy beams cannot maintain their shapes, particularly for a long distance propagation, because of a “hyperbolic umbilic” catastrophe as mentioned in Ref. [29]. Instead, their profiles tend to evolve into the 1D or 2D standard Airy patterns (see Figs.
Also, from Fig.
In theory, the initial position of the upper superposed Airy beam (see Figs.
So, let yd = 0, i.e., three superposed Airy beams meet together at the focal point, then we will have a relation to calculate the focusing length Fz of three Airy autofocusing beams as follows:
Similarly, we also analyze the effects of initial displacement (c0) of the 2D Airy beams on their propagation as shown in Figs.
Figures
Next, the propagation of three-Airy autofocusing beam embedded an optical vortex phase with different topological charges L is also studied in simulation and experiments. The left three columns in Figs.
It should be noted that the rotation of three-Airy autofocusing beam is enhanced for larger topological charge that brings about larger orbital angular momentum. Specially, the rotation angle is proportional to the topological charge as shown in Fig.
In this work, we investigate the propagations of three-Airy autofocusing beam with different initial wing angles theoretically and experimentally. Our results show that the focusing length of the beam increases with initial wing angle turning small, due to the smaller transverse self-acceleration of the main lobes. However, the main lobes of three-Airy autofocusing beam with a non-90° wing angle shows a power separation in propagation, which is caused by “hyperbolic umbilic” catastrophe. Finally, the beam presents a strongest intensity contrast only in the case of 90°. Furthermore, when a vortex is imposed onto three-Airy autofocusing beams, a rotation proportional to topological charge appears. This rotation becomes weaker as the initial wing angle increases. Our result may be useful for designing and controlling the dynamical autofocusing beams for various applications such as optical manipulation.
[1] | |
[2] | |
[3] | |
[4] | |
[5] | |
[6] | |
[7] | |
[8] | |
[9] | |
[10] | |
[11] | |
[12] | |
[13] | |
[14] | |
[15] | |
[16] | |
[17] | |
[18] | |
[19] | |
[20] | |
[21] | |
[22] | |
[24] | |
[25] | |
[26] | |
[27] | |
[28] | |
[29] | |
[30] | |
[31] | |
[32] |