## Abstract

We investigate the propagation properties and the radiation forces of Airy Gaussian vortex (AiGV) beams in a harmonic potential analytically and numerically in this paper. Obtaining the propagation expression of AiGV beams by solving the dimensionless linear (2+1) D Schrödinger equation in a harmonic potential, we perform the track, the intensity and phase distributions, the propagation shapes, the energy flow and the angular momentum of AiGV beams in a harmonic potential with the method of numerical simulations. The trajectory acting like a cosine curve is shown. Periodic inversion and phase oscillation are demonstrated during propagation. The influence of the distribution factor and the vortex factor on the propagation of AiGV beams in a harmonic potential are discussed. Likewise, the motion of the Poynting vector and the angular momentum is elucidated respectively. As for the radiation forces, we explore the gradient and scattering forces on Rayleigh dielectric particles induced by AiGV beams. In particular, it’s found that the value of the scattering force is approximately seven orders of magnitude larger than that of the gradient force during the propagation in a harmonic potential.

© 2017 Optical Society of America

## 1. Introduction

In 1979, by solving Schrödinger equation, Berry and Balazs introduced the nonspreading Airy wave packets [1] which bring many researchers’ interests due to their unique properties of non-spreading and constant acceleration in free space. In 2007, Siviloglou *et al.* obtained finite energy Airy beams by adding a decay factor, and investigated those beams in both one- and two-dimensional configurations theoretically [2] and experimentally [3]. After that, the finite-energy Airy beams have been the hot spots of researches [4–18] due to their unique properties, such as non-diffracting [4], self-healing [5] and self-acceleration [6]. The dynamics of Airy beams was considered in linear medium [4–6], nonlinear medium [7, 8], and media with external potentials [9–14]. Nowadays, Airy beams are involed in various fields such as atmospheric science [15], optical trapping particle [16–18], etc.

Optical vortices [19] are twisted lights which contain information on the phase and orbital angular momentum of lights [20], carrying phase singularities or phase-defects where both real and imaginary values of optical field go to zero [21]. They can be produced by utilizing spiral phase plates [22], synthetic holograms [23], liquid crystal cells [24], higher order laser mode separation [25], dielectric wedge [26], and interference patterns [27]. Vortex beams are widely used in many applications such as optical tweezers to manipulate micrometre-sized particles, micro-motors to provide angular momentum and improving channel capacity in optical and radio-wave information transfer [28].

On the other hand, ways to effectively modulate light beams have always been high on research agenda in optics. As an effective tool, a photonic potential embedded in the medium’s index of refraction is often used in linear optics and extensively referenced in the literature [29]. As exemplified by vastly different photonic crystal structures, it comes in different forms. A linear potential which affects the properties of an Airy plasmon beam and is used to control acceleration of Airy beams has been reported in [10,11]. An external longitudinally-dependent transverse potential will modulate the propagating trajectory of the light beam according to the form of the potential [12]. In addition, the propagation dynamics of optical beams with different external potential in the fractional Schrödinger equation (FSE), which is a generalization of the standard Schrödinger equation (SE) that contains fractional Laplacian, has attracted widespread attention recently. For instance, Zhang *et al.* reported the propagation of a chirped Gaussian beam in the (FSE) with a harmonic potential [30]. In the paper, it is found that the chirped Gaussian beam propagates along a zigzag trajectory in one dimension and evolves into a breathing ring structure in two dimensions. Also, the dynamics of waves in the FSE with PT-symmetric potential are investigated by Zhang and co-workers [31].

In this paper, based on the standard linear SE, we investigate the propagation properties and the radiation forces of the AiGV beams which are formed by multiplying a Gaussian factor and a vortex factor to the Airy beams with the modulation of a harmonic potential. Different from the previous researches of Airy beams with external potentials [9–14], the properties of Airy beams, Gaussian beams and vortex beams are synthetically indicated and further used to explore the optical trapping with radiation forces on a Rayleigh particle. To the best of our knowledge, the topic in our works has not been reported so far.

The paper is organized as follows. Firstly in Sec. 2, we start from the model with linear (2+1)D Schrödinger equation in a harmonic potential and obtain the solution of AiGV beams. Then in Sec. 3, different typical properties of 2D AiGV beams in a harmonic potential are analyzed. Additionally in Sec. 4, we go a step further to explore the radiation forces on a Rayleigh dielectric particles induced by AiGV beams. Finally in Sec. 5, the paper is concluded.

## 2. The AiGV beam’s model and its solutions in a harmonic potential

Let us consider the AiGV beams propagate along the *z* axis in Cartesian coordinate system in a harmonic potential. The dimensionless electric field distribution of the AiGV beams in the input plane *z* = 0 can be written as

*E*(

*x*

_{0},

*y*

_{0}, 0) stands for the electric field distribution at the input plane,

*x*

_{0},

*y*

_{0}are the normalized transverse coordinates scaled by Airy characteristic transverse width

*w*

_{0},

*A*

_{0}is the complex amplitude of the beams,

*Ai*(·) is the Airy function,

*a*stands for the decay factor [2,3], which makes the energy of the AiGV beams be finite.

*χ*

_{0}is a distribution factor, [

*x*

_{0}−

*x*

_{1}+

*i*(

*y*

_{0}−

*y*

_{1})]

*and [*

^{m}*x*

_{0}−

*x*

_{2}−

*i*(

*y*

_{0}−

*y*

_{2})]

*respectively represent positive and negative vortex factor,*

^{l}*m*and

*l*are the orders of the vortex factor.

*x*(

_{i}*i*= 1, 2) and

*y*(

_{i}*i*= 1, 2) denote positions of the center of the positive and negative vortex factors, respectively. For the sake of simplicity, we will consider the case of

*m*= 1,

*l*= 0 in our next discussion.

Figure 1 depicts the intensity and phase distributions of the AiGV beams with *χ*_{0} = 0.18 at the initial plane (*z* = 0). It is clearly found that the value of the intensity distribution in the center of the AiGV beams is slightly smaller than that around its surroundings due to the effect of the vortex factor. In the meantime the phase distribution apparently appears a vortex distribution.

The AiGV beams propagating in the paraxial approximation and in the harmonic potential are governed by the dimensionless linear (2+1)D Schrödinger equation [13]

*α*stands for a factor controlling the potential width [13].

*E*(

*x*,

*y*,

*z*) is the electric field distribution at the

*z*plane, ${\nabla}_{\perp}^{2}={\partial}^{2}/\partial {x}^{2}+{\partial}^{2}/\partial {y}^{2}$ is the transversal Laplace operator, the variables

*x*and

*y*are the normalized transverse coordinates, which are scaled by some characteristic transverse width

*w*

_{0}, the variables

*z*is the propagation distance and ${Z}_{0}=k{w}_{0}^{2}$ is the corresponding Rayleigh range. Here,

*k*= 2

*πn/λ*is the wave number, and

*n*is the ambient index of refraction and

*λ*is the wavelength in free space. The nonlocal nonlinear Schrödinger equation [32] and the Eq. (2) are two questions which are similar in form but different in nature. The nonlocal nonlinear Schrödinger equation is the nonlinearity limit with the degrees of nonlocality approaching to infinity, while the Eq. (2) is standard linear Schrödinger equation. Together with Eqs. (1) and (2), the electric field distribution of the AiGV beams propagating a distance

*z*in a harmonic potential can be derived where

*Ai′*(·) is the derivative of the Airy function.

From above equations, we can find that the component of *F*_{1} shows us a description of a conventional AiGV beams propagating a distance *z*, *F*_{2} represents the profile of the AiGV beams and its *x* component is described by the combination of the Airy function and its derivative while *y* component is only about the Airy function, *F*_{3} is similar to *F*_{2} but the *x* and *y* coordinates are exchanged.

## 3. The propagation properties of AiGV beams in a harmonic potential

With the foregoing discussion and the analytical results of the propagation expression, we will further investigate the propagation properties of AiGV beams in a harmonic potential numerically.

#### 3.1. The propagation path, the intensity distribution and phase evolution

Figures 2(a1)–2(a8) show the longitudinal normalized intensity of the AiGV beams with the different propagation distance in a harmonic potential, and the centers of optical vortex are marked by the red arrows. From Figs. 2(a1)–2(a8), one can see that the optical vortex keeps moving as the propagation distance increases. In addition, when the AiGV beams propagate at the positions 0.25*π*, 1.75*π*, 2*π*, shown in Figs. 2(a1), 2(a7), 2(a8) respectively, the light intensities of the AiGV beams mainly locate in the top left, while at the positions 0.75*π*, 1*π*, 1.25*π*, shown in Figs. 2(a3), 2(a4), 2(a5) respectively, the light intensities of the AiGV beams locate in the bottom right. During the propagation, when 0 < *z* < 0.5*π*, the intensities locate in the top left, and the energy of the AiGV beams gradually converges to the center (showed in Fig. 1(a) and Fig. 2(a1)) due to the influence of the vortex factor. Next, the envelope of beam gradually becomes a ring structure(showed in Fig. 2(a2)) due to the modulation of the harmonic potential. When 0.5*π* < *z* < *π*, symmetrically and inversely the intensities locate in the bottom right, and the envelope of the beam gradually approaches to the initial one. But the envelope in 0 < *z* < 0.5*π* is centrosymmetric with the one in 0.5*π* < *z* < *π*. During the propagation from *z* = *π* to *z* = 2*π*, it inverts from the intensity distribution *z* = *π* to *z* = 0.

Figure 2(b) shows the propagation trajectory along *z* axis in the observation plane *x* = *y*. It shows the oscillating period is *D* = 2*π/α* [14]. Due to the properties of self-acceleration of AiGV beams and the modulation of the harmonic potential, the trajectory looks like a cosine curve. On the other hand, for the phase evolution (see in Fig. 2(c)), it can be clearly seen the beam also undergoes phase oscillation. In a conclusion, the AiGV beam undergoes periodic inversion with the energy periodically converging to the center and then dispersing.

#### 3.2. The influence of the distribution factor *χ*_{0}

Next, we perform numerical simulations by changing the distribution factor *χ*_{0} of the AiGV beams and something interesting can be found. Figures 3(a1), 3(b1), and 3(c1) depict the initial intensity distribution with *χ*_{0} = 0.01, 0.18, 1, respectively. It is interesting to find that as *χ*_{0} increases, the value of the intensity distribution in the center of the AiGV beams at the input plane gradually becomes larger than that around its surroundings. The smaller *χ*_{0} is, the more strongly the Airy factor affects, and on the contrary, the more strongly the Gaussian factor affects. Gaussian factor can strengthen main lobes and weaken side lobes, while the vortex factor can weaken main lobes. Figures 3(a2), 3(b2), and 3(c2) show the 3D propagation of the AiGV beams in a harmonic potential with *χ*_{0} = 0.01, 0.18, 1, respectively. In Figs. 3(a2)–3(c2), the values of intensity are not only showed in different color, but also showed in different transparencies. The smaller the values are, the higher their transparencies are. In Fig. 3(a2), it can be clearly seen that with a smaller *χ*_{0}, the AiGV beams tend to preserve the properties of the Airy beams with non-diffraction and self-accelerating. As can be seen in Fig. 3(c2), with a larger *χ*_{0}, the beams do diffract and converge periodically in a harmonic potential, which miss most of the properties of the Airy beams. Hence, it indicates that as *χ*_{0} increases, the Airy component gradually is replaced by the Gaussian component. In Figs. 4(a), 4(b) and 4(c), we display respectively the iso-surface plot of the beam during propagation with *χ*_{0} = 0.01, 0.18, 1, which can further demonstrate the effect of *χ*_{0}. On the other hand, the twisted shapes of the AiGV beams can be clearly seen from the iso-surface plot with *χ*_{0} = 0.01, 0.18. During the propagation from 0 to 2*π*, the positions of twist appear twice, one is nearby the plane *z* = 0.5*π*, the other is nearby the plane *z* = 1.5*π*. The role of the latter twist is to make the AiGV beams be self-healing. Additionally, the motion of the two twists will act more strongly with *χ*_{0} decreasing.

In order to understand the influence of *χ*_{0} better, it is instructive to look at the dynamics of the “center of mass” [13,14,33], coming from the intensity distribution

Figure 5(a) demonstrates the variation of the center of mass related to Eq. (11) as a varies from 1 to 20 at the planes *z* = 1.25*Z*_{0} and *z* = *Z*_{0}. We can see that the center of mass fluctuates in an approximate cosine mode as *α* increases, which further demonstrates that the oscillating period is related to *α*. Besides, in Fig. 5(b), the variation of the center of mass versus the axial propagation distance is depicted with the different distribution factor *χ*_{0}. It is interesting to see that it also fluctuates in an approximate cosine mode. Additionally when *χ*_{0} is valued smaller, the center of mass will fluctuate more intensively due to the Airy factor.

#### 3.3. The Poynting vector and the angular momentum

In the end of Sec. 3 we investigate the Poynting vector and angular momentum of the AiGV beam as it propagates in a harmonic potential. The rate of electromagnetic energy flow per unit area, which is usually expressed in terms of the Poynting vector, is a commonly known quantity in electrodynamics [34, 35]. The Poynting vector is defined as
$\overrightarrow{S}=\frac{c}{4\pi}\overrightarrow{E}\times \overrightarrow{B}$, where *c* is the speed of light in vacuum, *E⃗* and *B⃗* are the electric and magnetic fields, respectively. Besides, we can give a vector potential *A⃗* = *ζ̂E*(*x*, *y*, *z*) exp(−*ikz*), where *ζ̂* is an arbitrary polarization. In the Lorenz gauge, assuming an *x̂*-polarized field, the time-averaged Poynting vector can be expressed as [36]

*x*,

*y*and

*z*directions, respectively, and * denotes the complex conjugate,

*ω*is the angular frequency. It is easy to find from Eq. (12) that the energy flowing in the

*z*direction is proportional to the light intensity.

For observing the rotation of the energy flow as the AiGV beams propagate in a harmonic potential, we perform numerical simulations of the Poynting vector around the optical vortex corresponding to the positions marked by red arrows in Figs. 2(a)–2(h). Figures 6(a)–6(h) show the Poynting vector of the AiGV beams with the different propagation distance in a harmonic potential. In Fig. 6, the trend of the Poynting vector is clearly seen. From Figs. 6(a)–6(h), one can see that the transverse energy flow appears to rotate clockwise or counterclockwise around the positive optical vortex due to the topological charge of the AiGV beams. Observing in more detail, we can find that the direction of rotation is clockwise firstly (0 < *z* < *π*), in the next place counterclockwise (*π* < *z* < 1.5*π*), once again clockwise (1.5*π* < *z* < 1.75*π*), and counterclockwise in the end (1.75*π* < *z* < 2*π*). Likewise, it is interesting to note that these properties are obviously not in accordance with the properties in Ref. [37] that the Poynting vector will point towards the direction of the energy flow of the main Airy peak (45°). Therefore, we could draw a conclusion that the topological charge of the positive optical vortex can change the trend of the Poynting vector.

As in mechanics, the time-averaged angular momentum density for the electromagnetic field is the angular momentum per unit area (per unit time) [38], obtained by forming the cross product of the position vector with the time-averaged momentum density, which can be written as [37]

*π*, 1.75

*π*, shown in Figs. 7(a), 7(g) respectively, the angular momentum mainly concentrates on the main lobe of the AiGV beams, while at the positions 1

*π*, 2

*π*, shown in Figs. 7(d), 7(h) respectively, the angular momentum mainly concentrates on the side lobe of the AiGV beams.

## 4. The radiation force of AiGV beams in a harmonic potential

In this section, we investigate the radiation forces of the AiGV beams in a harmonic potential. The gradient force and the scattering force are deemed as two kinds of the radiation forces [39]. By employing the vector identity and the solution of the Maxwell equations, the gradient force can be written as [40]

*n*

_{1}and

*n*

_{2}are the refractive index of the nano-particles and the surrounding medium respectively,

*r*

_{0}is the radius of the nano-particles,

*∊*

_{0}is the permittivity of vacuum. We asumme the particle is in stable state, its gradient force is the time average [40],

*c*is the light velocity, $I(x,y,z)=c{n}_{2}{\u220a}_{0}\frac{{\left|E(x,y,z)\right|}^{2}}{2}$.

The change of the electromagnetic momentum will influence the scattering of light, with which the scattering force is associated. The scattering force can be expressed as [40]

*C*

_{pr0}is the radiation pressure section of the particle. Considering the isotropy of the particle, the radiation pressure section of the particle is equal to the scattering force section of the scatterer:

Comparing Eq. (15) with Eq. (16), one can see that the gradient force and the scattering force are proportional to the intensity gradient and the intensity, respectively. From the vector $\overrightarrow{{e}_{z}}$ in Eq. (16), the scattering force aims at the direction of the AiGV beam’s propagation and drives the particles to move along the optical axis when propagating in a harmonic potential.

The transverse gradient force of AiGV beams is numerically simulated based on Eqs. (1) and (3). Figures 8(a)–8(j) show the section distributions of the transverse gradient force of AiGV beams for the particles with *n*_{1} = 1.50, *r*_{0} = 60*nm* when the AiGV beams propagating at every single oscillating period (2*π*) in a harmonic potential.

From the results we can quantificationally observe and qualitatively analyze the gradient force of the AiGV beams. As is seen in Figs. 8(a) and 8(b), the gradient force mainly concentrates on side-lobe of the AiGV beams when the beams propagate from 0 to 2*π*. Then due to the effect of the vortex factor, the gradient force gradually converges to the main-lobe of the AiGV beams when the beams propagate from 4*π* to 8*π*, which is shown in Figs. 8(c)–8(e). Comparing Figs. 8(a)–8(e) with Figs. 8(f)–8(j), one can see that the self-healing process, one of the charming properties of the AiGV beams, will finish as the beams propagate the distance about 20*π*. During the propagation from 12*π* to 20*π*, the gradient force mainly transfers from the top left to the bottom right with the effect of optical vortex and the modulation of a harmonic potential. This behavior indicates that the AiGV beams can rotate 180° clockwise or counterclockwise with dragging some nano-particles.

Figure 9 shows the scattering force of the AiGV beams propagating at the same plane as those in Fig. 8. It is easy to find that there are many similar behaviors between the scattering force and the gradient force. For instance, when 0 < *z* < 8*π*, both of the radiation forces mainly locate in the top left and they transfer to the bottom right symmetrically and inversely when 12 < *z* < 20*π*. However, making a comparison between Fig. 8 and Fig. 9, we find that as the AiGV beams propagate in a harmonic potential, the gradient force mainly converges to the both side of the main-lobe or the side-lobe, acting like a letter “*U*”, while the scattering force mainly converges to the center of the main-lobe or the side-lobe acting like a focal spot.

The ability of optical trapping particles is related to whether the gradient force overwhelms the scattering force or not [43]. Figure 10 indicates distribution and evolution of the gradient force and the scattering force at three different *z* planes, showed in Figs. 10(a) and 10(b) respectively. From this result, on the one hand, it is obvious that both of the radiation forces will climb up and then decline as the propagation distance increases. On the other hand, specifically both of the radiation forces become relatively low value and tortuous distribution at the plane 10*π*. In addition, noting that the difference of the unit of *y* axis between Figs. 10(a) and 10(b), one can find that the value of the scattering force is approximately seven orders of magnitude larger than that of the gradient force.

## 5. Conclusion

In conclusion, we investigate the propagation properties and the radiation forces of AiGV beams in a harmonic potential. In the part about the propagation properties, the beams undergoing periodic inversion and phase oscillation are shown. Then, the numerical simulation is performed with the different distribution factor *χ*_{0} and one can modify the AiGV beams to Airy vortex beams with a smaller *χ*_{0} while to Gaussian vortex beam with a large one. Further the variation of the center of mass with different *χ*_{0} at some fixed planes is calculated and it fluctuates in an approximate cosine mode. Moreover, the Poynting vector and the angular momentum are derived. One can see that the transverse energy flow appears to rotate inversely and constantly. As for the angular momentum, the spatial distributions are more coherent and the positions of optical vortex become unconspicuous. In the part about the radiation forces, we analyze the gradient force and the scattering force respectively. It is found that both of the radiation forces exerted on the nano-particles will transfer from the side-lobe to the main-lobe of the the AiGV beams in a harmonic potential. But the gradient force mainly converges to the two sides of the beams while the scattering force converges to the center. Additionally, the value of the scattering force is approximately seven orders of magnitude larger than that of the gradient force at the same plane. We believe the theoretical results presented in this paper would be useful for understanding about AiGV beams in a harmonic potential and applications in particle trapping.

## Funding

National Natural Science Foundation of China (NSFC) (11374108); CAS Key Laboratory of Geospace Environment, University of Science and Technology of China; Special Funds for the Cultivation of Guangdong College Students’ Scientific and Technological Innovation (pdjh2017b0137).

## References and links

**1. **M. V. Berry and N. L. Balazs, “Nonspreading wave packets,” Am. J. Phys. **47**, 264–267 (1979). [CrossRef]

**2. **G. A. Siviloglou and D. N. Christodoulides, “Accelerating finite energy Airy beams,” Opt. Lett. **32**(8), 979–981 (2007). [CrossRef] [PubMed]

**3. **G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Observation of accelerating Airy beams,” Phys. Rev. Lett. **99**(21), 213901 (2007). [CrossRef]

**4. **G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Ballistic dynamics of Airy beams,” Opt. Lett. **33**(3), 207–209 (2008). [CrossRef] [PubMed]

**5. **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]

**6. **Y. Hu, G. A. Siviloglou, P. Zhang, N. K. Efremidis, D. N. Christodoulides, and Z. G. Chen, *Self-accelerating Airy Beams: Generation, Control, and Applications* (Springer, 2012) Vol. 170, pp. 1–46.

**7. **L. Zhang, K. Liu, H. Zhong, J. Zhang, Y. Li, and D. Fan, “Effect of initial frequency chirp on Airy pulse propagation in an optical fiber,” Opt. Express **23**(3), 2566–2576 (2015). [CrossRef] [PubMed]

**8. **R. Driben, V. V. Konotop, and T. Meier, “Coupled Airy breathers,” Opt. Lett. **39**(19), 5523–5526 (2014). [CrossRef] [PubMed]

**9. **N. K. Efremidis, “Airy trajectory engineering in dynamic linear index potentials,” Opt. Lett. **36**(15), 3006–3008 (2011). [CrossRef] [PubMed]

**10. **W. Liu, D. N. Neshev, I. V. Shadrivov, A. E. Miroshnichenko, and Y. S. Kivshar, “Plasmonic airy beam manipulation in linear optical potentials,” Opt. Lett. **36**(7), 1164–1166 (2011). [CrossRef] [PubMed]

**11. **Z. Ye, S. Liu, C. Lou, P. Zhang, Y. Hu, D. Song, J. Zhao, and Z. Chen, “Acceleration control of airy beams with optically induced refractiveindex gradient,” Opt. Lett. **36**(16), 3230–3232 (2011). [CrossRef] [PubMed]

**12. **N. K. Efremidis, “Accelerating beam propagation in refractive-index potentials,” Phys. Rev. A **89**(2), 023841 (2014). [CrossRef]

**13. **Y. Zhang, M. R. Belic, L. Zhang, W. Zhong, D. Zhu, R. Wang, and Y. Zhang, “Periodic inversion and phase transition of finite energy airy beams in a medium with parabolic potential,” Opt. Express **23**(8), 10467–10480 (2015). [CrossRef] [PubMed]

**14. **Y. Zhang, X. Liu, M. R. Belic, W. Zhong, F. Wen, and Y. Zhang, “Anharmonic propagation of two-dimensional beams carrying orbital angular momentum in a harmonic potential,” Opt. Lett. **40**(16), 3786–3789 (2015). [CrossRef] [PubMed]

**15. **Y. Gu and G. Gbur, “Scintillation of airy beam arrays in atmospheric turbulence,” Opt. Lett. **35**(20), 3456–3458 (2010). [CrossRef] [PubMed]

**16. **D. N. Christodoulides, “Optical trapping riding along an airy beam,” Nat. Photonics **2**(11), 652–653 (2008). [CrossRef]

**17. **J. Baumgartl, M. Mazilu, and K. Dholakia, “Optically mediated particle clearing using Airy wavepackets,” Nat. Photonics **2**(11), 675–678 (2008). [CrossRef]

**18. **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] [PubMed]

**19. **P. Coulet, L. Gil, and F. Rocca, “Optical vortices,” Opt. Commun. **73**(5), 403–408 (1989). [CrossRef]

**20. **L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of laguerre-gaussian laser modes,” Phys. Rev. A **45**(11), 8185–8189 (1992). [CrossRef] [PubMed]

**21. **J. F. Nye and M. V. Berry, “Dislocations in wave trains,” P. Roy. Soc. Lon. A Mat. **336**(1605), 165–190 (1974). [CrossRef]

**22. **V. V. Kotlyar, A. A. Almazov, S. N. Khonina, V. A. Soifer, H. Elfstrom, and J. Turunen, “Generation of phase singularity through diffracting a plane or gaussian beam by a spiral phase plate,” J. Opt. Soc. Am. A **22**(5), 849–861 (2005). [CrossRef]

**23. **A. Vasara, J. Turunen, and A. T. Friberg, “Realization of general nondiffracting beams with computer-generated holograms,” J. Opt. Soc. Am. A **6**(11), 1748–1754 (1989). [CrossRef] [PubMed]

**24. **D. Ganic, X. S. Gan, M. Gu, M. Hain, S. Somalingam, S. Stankovic, and T. Tschudi, “Generation of doughnut laser beams by use of a liquid-crystal cell with a conversion efficiency near 100,” Opt. Lett. **27**(15), 1351–1353 (2002). [CrossRef]

**25. **E. Abramochkin, N. Losevsky, and V. Volostnikov, “Generation of spiral-type laser beams,” Opt. Commun. **141**(12), 59–64 (1997). [CrossRef]

**26. **Y. Izdebskaya, V. Shvedov, and A. Volyar, “Generation of higher-order optical vortices by a dielectric wedge,” Opt. Lett. **30**(18), 2472–2474 (2005). [CrossRef] [PubMed]

**27. **Y. Zhang, Z. Nie, Y. Zhao, C. Li, R. Wang, J. Si, and M. Xiao, “Modulated vortex solitons of four-wave mixing,” Opt. Express **18**(11), 10963–10972 (2010). [CrossRef] [PubMed]

**28. **J. Verbeeck, H. Tian, and P. Schattschneider, “Production and application of electron vortex beams,” Nature **467**(7313), 301–304 (2010). [CrossRef] [PubMed]

**29. **Y. Zhang, X Liu, M. R. Belic, W. Zhong, M. S. Petrovic, and Y. Zhang, “Automatic Fourier transform and self-Fourier beams due to parabolic potential,” Ann. Phys. **363**, 305–315 (2015). [CrossRef]

**30. **Y. Zhang, X. Liu, M. R. Belic, W. Zhong, Y. Zhang, and M. Xiao, “Propagation Dynamics of a Light Beam in a Fractional Schrödinger Equation,” Phys. Rev. Lett. **115**(18), 180403 (2015). [CrossRef] [PubMed]

**31. **Y. Zhang, H. Zhong, M. R. Belic, Y. Zhu, W. Zhong, Y. Zhang, D. N. Christodoulides, and M. Xiao, “PT symmetry in a fractional Schrödinger equation,” Laser Photon. Rev. **10**(3), 526–531 (2016). [CrossRef]

**32. **A. W. Snyder and D. J. Mitchell, “Accessible solitons,” Science **276**(5318), 1538–1541 (1997). [CrossRef]

**33. **A. Bekshaev, M. S. Soskin, and M. Vasnetsov, *Paraxial Light Beams with Angular Momentum*, (Nova Science, 2008).

**34. **J. D. Jackson, *Classical Electrodynamics*, (Wiley, 1962).

**35. **M. Born and E. Wolf, *Principles of Optics*, 7th Edition, (Cambridge University Press, Cambridge, 1999). [CrossRef]

**36. **L. Allen, M. J. Padgett, and M. Babiker, “The orbital angular momentum of light,” Prog. Opt. **39**, 291–372 (1999). [CrossRef]

**37. **H.I. Sztul and R.R. Alfano, “The Poynting vector and angular momentum of Airy beams,” Opt. Express **16**(13), 9411–9416 (2008). [CrossRef] [PubMed]

**38. **B. Chen, C. Chen, X. Peng, and D. Deng, “Propagation of airy gaussian vortex beams through slabs of right-handed materials and left-handed materials,” J. Opt. Soc. Am. B **32**(1), 173–178 (2015). [CrossRef]

**39. **J. Huang, Z. Liang, F. Deng, W. Yu, R. Zhao, B. Chen, X. Yang, and D. Deng, “Propagation properties of right-hand circularly polarized Airy-Gaussian beams through slabs of right-handed materials and left-handed materials,” J. Opt. Soc. Am. A **32**(11), 2104–2109 (2015). [CrossRef]

**40. **Y. Harada and T. Asakura, “Radiation forces on a dielectric sphere in the Rayleigh scattering regime,” Opt. Commun. **124**, 529–541 (1996). [CrossRef]

**41. **J. A. Stratton, *Electromagnetic Theory*, (McGraw-Hill, 1941).

**42. **M. Kerker, *The Scattering of Light and Other Electromagnetic Radiation*, (Academic, 1969).

**43. **Z. Zheng, B. Zhang, H Chen, J. Ding, and H. Wang, “Optical trapping with focused Airy beams,” Appl. Opt. **50**(1), 43–49 (2011). [CrossRef] [PubMed]