## Abstract

Controlling the focal length and the intensity of the optical focus in the media is an important task. Here we investigate the propagation properties of the sharply autofocused ring Airy Gaussian vortex beams numerically and some numerical experiments are performed. We introduce the distribution factor *b* into the initial beams, and discuss the influences for the beams. With controlling the factor *b*, the beams that tend to a ring Airy vortex beam with the smaller value, or a hollow Gaussian vortex beam with the larger one. By a choice of initial launch condition, we find that the number of topological charge of the incident beams, as well as its size, greatly affect the focal intensity and the focal length of the autofocused ring Airy Gaussian vortex beams. Furthermore, we show that the off-axis autofocused ring Airy Gaussian beams with vortex pairs can be implemented.

© 2015 Optical Society of America

## 1. Introduction

Siviloglou et al. [1, 2] have first reported that an optical Airy beam can remain diffraction free within a certain distance, while its main intensity lobe freely self-bends during propagation in 2007. The Airy Gaussian beams carry finite power, retain the nondiffracting propagation properties within a finite propagation distance, and can be realized experimentally to a very good approximation [3]. For example, the propagation properties of Airy Gaussian beams in the strongly nonlocal nonlinear media [4] and in the Kerr medium [5] have been studied. And furthermore, the propagation dynamic of the vortex incorporated in various background beams and the partially coherent beams carrying optical vortices have been reported [6, 7]. More recently, the research on the propagation of an optical vortex superimposed on Airy beams and Airy Gaussian beams has been a highlight [8–11].

Ring Airy beams are a recently introduced type of abruptly autofocusing waves [12–18], which is cylindrically symmetric and can be able to abruptly autofocus in the linear media. Their maximum intensity can abruptly increase by orders of magnitude just at the focus. They have been introduced theoretically [12] and have been demonstrated experimentally [13]. In some cases, for example in biomedical treatment, it is extremely necessary for a beam to focus in a particular target, preserving low intensity without damaging the material before the focus. Up to now, the propagation of the circular Airy beams with optical vortices [19, 20], and the radially polarized circular Airy beams [21] are proposed and observed in experiment.

Here we will investigate the propagation properties of the sharply autofocused ring Airy Gaussian vortex (RAiGV) beams in the linear media. Using numerical simulations, we will study the intensity and phase distributions of the ring Airy Gaussian (RAiG) beams with one optical vortex. We also investigate the influences of different parameter *b* on the beams. By a choice of initial launch condition, we analyze the effects of the number of topological charge of the incident beams, as well as its size, on the focal length and the focal intensity. Finally, we show that the off-axis autofocused RAiG beams with vortex pairs can be implemented.

To the best of our knowledge, it is the first time to introduce the RAiGV beams. The RAiG beams describe in a more realistic way the propagation of the abruptly autofocusing circular Airy beams because RAiG beams retain the abruptly autofocusing properties within a certain propagation distance, and can be realized experimentally to a very good approximation. The circular Airy beams [12, 13] are special cases of the RAiG beams. Here, we will do some numerical experiments to show the properties and advantages of the RAiGV beams.

## 2. Paraxial propagation of ring Airy Gaussian beams with vortex

By considering the diffraction of a RAiG beam with vortex propagates in a linear medium. The RAiG beams with vortex are solutions of Maxwell’s equations that obey axial symmetry in amplitude. Within the framework of the paraxial approximation, we consider the propagation of the field envelope *u*(*x, y, z*) described by equation

*u*is the amplitude of the optical wave,

*x, y*are the scaled transverse coordinates, and

*z*is the propagation distance, where

*k*= 2

*π/λ*is the wave number in free space,

*λ*is the wavelength of the incident light. For paraxial solution in cylindrical coordinates, it is useful to express the wave Eq. (1) in cylindrical coordinates

*r*,

*φ*, and

*z*:

The axially symmetric solution of the electric field will have the form

where*m*is an integer,

*U*(

*r, z*) satisfies the following equation under the paraxial and slowly varying envelope approximation:

The electric fields of the initial RAiG beams superimposed by a spiral phase in cylindrical coordinate can be expressed as

*A*

_{0}is the constant amplitude of the electric field,

*Ai*(·) corresponds to the Airy function,

*r*is the radial coordinate,

*r*

_{0}represents the radius of the primary Airy ring, and

*w*is a scaling factor, 0 ≤

*a*< 1 is the exponential truncation factor which determines the propagation distance, and

*m*is the topological charge of the optical vortex. Here,

*b*is a distribution factor parameter. We first introduce the factor

*b*into the RAiGV beams, which can adjust the scale between the ring Airy factor and the hollow Gaussian factor, and it describes in a more realistic way to adjust the initial beams. The choice of proper distribution factor

*b*can make the initial beams tend to a ring Airy vortex beam with the smaller value, such as

*b*= 0.1 in Fig. 1, or a hollow Gaussian vortex beam with the larger one, such as

*b*= 0.4 in Fig. 1. By comparing to the circular Airy beams with optical vortices (see [20]), we can find that the circular Airy vortex beam is one special case of the RAiGV beams. The influences of the parameter

*b*will be explored in more detail in a later section.

With the initial electric fields *u*(*r, φ, z* = 0), we can not get the *u*(*r, φ, z*) analytically. To explore the propagation properties of the sharply autofocused RAiGV beams, we perform the propagation numerical calculation and show the simulation findings of RAiGV beams. From the theoretical part has been discussed above, by solving the Eq. (1) or Eq. (2) with the initial input electric fields *u*(*r, φ, z* = 0), we find some interesting and useful properties. By taking Eq. (5) as an initial electric field, here we use the Fast Fourier transform method to solve Eq. (1) and obtain the numerical electric field *u*(*r, φ, z*) of the propagation. In the simulations, we assume that *A*_{0} = 1, *λ* = 532 × 10^{−}^{6}*mm*, *a* = 0.05, *b* = 0.1, *r*_{0} = 1*mm*, *w* = 0.5*mm*. The Rayleigh distance of the beam is *Z _{R}* =

*kw*

^{2}/2. Using these parameters, the RAiGV beams can be autofocusing at a certain position. The focal length of the case

*b*= 0.1 is about

*z*= 0.21

*Z*in the simulation.

_{R}We first discuss the properties of the RAiGV beams with the topological charge *m* = 1 and the distribution factor *b* = 0.1. Figure 2 depicts the numerical results of RAiGV beams propagating on-axis in the linear media. As shown in Figs. 2(a) and 2(b), the intensity peak follows a curved trajectory as the beams propagate toward the autofocusing occurs. It is different from the circular Airy beams with optical vortices [20], the peak intensity of the RAiGV beams decreases first. Close to the focal point, the power of the RAiGV rings is concentrated in a small area and the maximum intensity near the center increases sharply. The lateral acceleration of the RAiGV beams is attained and energy rushes in an accelerated fashion toward the focus. However, the center intensity of the beams persists zero due to the vortex on axis. After the focal plane, the maximum intensity begins to reduce. As can be seen in Fig. 2(a), the decrease is not monotonic, but it slight oscillates, which is mainly generated by the subsequent Airy rings. Due to the parameter *b* is small, the Gaussian factor has only minor effects on the beams. The properties of the beams are similar to the ring Airy vortex beams.

In Fig. 3(a), which simulated side-view propagation of the RAiGV beams, it is quite clear that the beams sharply autofocus after a certain distance of propagation. Before the intensity increases sharply, the rings of the inward accelerating Airy Gaussian vortex beams move on a parabolic trajectory, and as the propagation distance increases, the radius of the Airy Gaussian beams decreases. After autofocusing occurs, we can see that the intensity of the profile keeps the dark hollow rings, and a dark hollow channel forms because of the vortex on axis along the propagation. The focal point is located at about the distance *z* = 0.21*Z _{R}* in the simulation, the intensity patterns and phase distribution are shown in Fig. 3(b3) and Fig. 3(c3). We can see that the intensity patterns are similar to the vortex Bessel beams [22] after the focal plane.

In Figs. 3(c1)–3(c4), the phase of the beams carrying the optical vortex on-axis along the propagation can clearly display the distribution of defects. When the vortex travels on axis, it exhibits a clockwise screw before the autofocusing occurs. After the beams passing through the point where autofocusing occurs, the rotation of the optical vortex presents an anticlockwise screw. In fact, the relative positions of the vortex relative to the host beam are preserved but the pattern is rotated.

Next, we perform some numerical simulations and numerical experiments by changing the distribution factor *b* for RAiGV beams. Figure 4 shows a comparison of the RAiGV beams for different parameters *b*. Figures. 4(a)–4(c) display the initial intensity distribution of the 1D and 2D RAiGV beams with *b* = 0.1, 0.2, 0.4, respectively. As expected, the RAiGV beams tend to be a ring Airy vortex beam with the parameter *b* = 0.1, and a hollow Gaussian vortex beam with the larger one, such as *b* = 0.4, the beams become a hollow Gaussian vortex beam, more or less. Figures. 4(a1)–4(c1) demonstrate the numerically simulated side-view propagation of the RAiGV beams for various parameters *b* = 0.1, 0.2, 0.4, and Figs. 4(a2)–4(c2) show the corresponding peak intensity distribution. We can find that the distribution factor *b* greatly affects the focal length and the focal intensity of the autofocusing beams. The focal length of the case *b* = 0.1 is about *z* = 0.21*Z _{R}*, while the focal length of the case

*b*= 0.4 is about

*z*= 1.25

*Z*. The focal intensity of the former case is more than four times of the latter.

_{R}By a suitable choice of distribution factor *b*, the focal length and the focal intensity of the RAiGV beams can be easily modulated, which was shown in the numerical experiment for different distribution factor *b* in Fig. 5. We can choose different *b* to control the initial beam patterns. For numerical experimental generation, we launch an initial beam to reconstruct the off-axis computer-generated holograms [see Figs. 5(a1)–5(c1)] of the desired beam profiles [14]. The holograms are obtained by computing the off-axis interference patterns between the complex amplitude profile of the RAiGV beams at the *z* = 0 plane and a plane wave [see Figs. 5(a2)–5(c2)]. The transverse intensity patterns taken at input plane (Figs. 5(a3)–5(c3)) and focal plane (Figs. 5(a4)–5(c4)) indicate clearly that the RAiGV beams autofocus along the propagation direction. When the factor *b* is smaller (*b* = 0.1), the intensity pattern is sharply autofocused. On the contrary, the spot is not too abrupt. These observations are in good agreement with numerical simulations.

The number of topological charge of the incident beams, as well as its size, greatly affect the focal intensity and the focal length of the autofocused RAiGV beams as well. Figure 6 shows the propagation dynamics of the autofocused RAiGV beams on-axis. Figures 6(a)–6(c) compare the intensity of the initial incident RAiGV beams for topological charge *m* = 0, 1, 2 (*m* = 0, namely ring Airy Gaussian beams), respectively, which shows the more topological charge, the more rings exist. Figures 6(a1)–6(c1) demonstrate the numerically simulated side-view propagation of the RAiGV beams for *m* = 0, 1, 2, respectively, the dashed line indicates the focal plane. Comparing to the RAiG beams (Fig. 6(a1)), interestingly, for the vortex exists, a dark hollow channel forms on axis along the propagation. The beam characterization can be demonstrated by the beam width [23], which are shown in Figs. 7(a)–7(c). The full width at half maximum intensity for the different cases clearly show the properties of the beams. For *m* = 0, the beams autofocus sharply in a certain distance, and then diffract slightly. For *m* = 1 and *m* = 2, the beams also autofocus abruptly, while the beams keep a hollow channel along the propagation. The diameter of the hollow channel for *m* = 2 is larger and diffracts faster than the case *m* = 1.

We can also find that the focal plane of the autofocused beams is tiny shifted towards the laser source as the topological charge is increased. The focus position versus the different topological charge is shown in Fig. 8. Figures 6(a2)–6(c2) show the intensity of the focal plane respectively. The intensity of RAiG beams becomes Bessel beams, the center intensity of the RAiG beams with one or two vortex persists zero, and rings remain. With more topological charge, the main ring has a larger core diameter in the focal plane. More precise contrast is described in Fig. 9(a). On the other hand, in Fig. 9(b), comparing to the RAiG beams, the results show that with the number of topological charge increase, the maximum intensity of RAiG beams with vortices decreases first and then increases greatly at the focal plane. The axial intensity changes smoothly because of the Gaussian factor exists in the intensity distribution, the oscillation caused by Airy factor is weakened. It is different from the ring Airy beams [12] or the circular Airy vortex beams [20], which have several oscillations.

Figure 8 depicts the focus position versus the different topological charge in the medium. By increasing the number of topological charge, we can find that the focal point is tiny shifted towards the laser source, but it tends to move less and less as the topological charge is increased.

The size of the incident beams also affects the focal intensity and the focal length of the RAiGV beams. In Fig. 10, we can find that as the radius of the primary Airy ring increases, the focal plane is shifted outwards the laser source no matter what the topological charge is. In addition, the maximum intensity of RAiG beams decreases, while the RAiG beams with vortices increases greatly at the focal plane.

From the above analysis we can see that one can modify the focal length and the intensity of the optical focus by adjusting the distribution factor, the number of topological charge and the size of the incident beams in the media.

## 3. Off-axis propagation of ring Airy Gaussian beams with vortex pairs

To investigate the propagation properties of the off-axis autofocused RAiGV beams, we show the RAiG beams with positive vortex pairs in our case. We assume that the vortex pairs distribute symmetrical about the origin in x-axis, and the phase *φ _{k}* = 0. So the fields in the initial plane can be expressed as:

*r*) denotes the location of the optical vortex. In the simulation, we assume that the positive vortex pairs are off-axis with

_{k}, φ_{k}*r*= 0.6

_{k}*mm*, and the other parameters are the same to Fig. 3. We show the properties of the off-axis RAiGV beams numerically and experimentally.

Figure 11 depicts the off-axis propagation of RAiG beams with positive vortex pairs. From the side-view propagation (Fig. 11(a)) and the transverse normalized intensity patterns (Figs. 11(b1)–11(b4)), we can see that the beams also autofocus, but the intensity patterns are influenced by the vortex pairs. After autofocusing occurs, there are two dark hollow channels formed, and the transverse intensity patterns in the *x*-direction are weakened. The evolution of off-axis phase pattern is shown in Figs. 11(c1)–11(c4). The off-axis optical vortex pairs are rotated relative to the propagation axis, interestingly, they are also forced to move to near the center. The distance between the optical vortex pairs and the beam center is 0.6*mm* initially, but it becomes about 0.02*mm* at the focal plane. Nevertheless, the vortex pairs can not be at the beam center and they can not get the same point. After the focal plane, the distance between the vortex pairs increases gradually, and the beams propagate according to a constant angle of divergence. At the same time, the intensity becomes weaker. We demonstrated the propagation properties of the off-axis RAiGV beams numerical experimentally. By using the computer-generated hologram (Fig. 12(a)) of the off-axis RAiGV beam profile, it is clear that the results (Figs. 12(a1)–12(a4)) are in good agreement with the numerical simulations (Figs. 11(b1)–11(b4)).

## 4. Conclusion

In summary, we have presented the propagation properties of the sharply autofocused RAiGV beams numerically. Through numerical calculations, we can find that one can modify the focal length and the intensity of the optical focus by appropriately selecting the distribution factor, the number of topological charge and the size of the incident beams in the media. We first introduce the distribution factor parameter *b* into the RAiGV beams, which can describe in a more realistic way to adjust the initial beams. The choice of proper distribution factor *b* can make the initial beams tend to a ring Airy vortex beam with the smaller value, or a hollow Gaussian vortex beam with the larger one. The results have been shown by the numerical experiments. In addition, we have shown that the off-axis propagation of autofocused RAiG beams with vortex pairs. With controlling the focal length and the intensity of the optical focus in the media is an important task. We believe the sharply autofocused RAiGV beams can be applied to various applications such as medical treatment, particle manipulation, and optical communication.

## Acknowledgments

This research was supported by the National Natural Science Foundation of China ( 11374108, 10904041, 11374107), and the Foundation of Cultivating Outstanding Young Scholars (Thousand, Hundred, Ten Program) of Guangdong Province in China. CAS Key Laboratory of Geospace Environment, University of Science and Technology of China.

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