## Abstract

The dynamics of Airy beams modeled by the fractional Schrödinger equation (FSE) with the potential barrier are numerically investigated. Adjusting the Lévy index provides a convenient way to control the diffraction of Airy beams which presents the non-diffraction and splitting property. It has been found that the total reflection of beams occurs when the depth of the single potential barrier exceeds the threshold, and that the number of reflected waves is influenced by the Lévy index and the location of potential barrier. However, the periodic self-imaging phenomenon of Airy beams is shown under a symmetric potential barrier when the Lévy index is equal to one, and the self-imaging period of the asymmetric Airy beams is analytically demonstrated and is as twice as that of symmetric Gaussian beams, moreover, the chaoticon of light field is formed during propagation as the Lévy index increases. All the properties of Airy beams modeled by FSE confirm the potential application in optical manipulation.

© 2017 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

Fractional effects, including fractional quantum Hall effect [1], fractional Talbot effect [2] and fractional Josephson effect [3], have been found in various branch of physics [4, 5]. The fractional Schrödinger equation (FSE) formulated by Laskin is a generalization of the standard Schrödinger equation (SE) by replacing the second-order spatial derivative with a fractional Lévy index [6–8], and it can describe the fractional field theory and the behavior of particles with fractional spin. In the last decades, the research on FSE was intensive and mainly discussed the mathematical aspects [9–11]. Not until 2015 was the experimental scheme proposed in optics by Longhi [12]. In the paper, the model is based on transverse light dynamics in aspherical optical cavities, and the dual Airy function is of the eigenmode of a massless harmonic oscillator.

Longhi’s work paves the way for the experimental investigation of the fractional models developed in quantum physics and stimulates interest to investigate the beam dynamics in the FSE. In the linear region, the beam splitting and non-diffraction of sub-waves as well as the Talbot effect are presented [13], and the splitting property can be explained from the view of group delay [14, 15]; When the Gaussian beam is linearly chirped, the zigzag trajectory in the FSE with a harmonic potential is shown [16]; Under a periodic PT-symmetric potential, the linear and symmetric band structure results in a nondiffracting propagation and conical diffraction of input beams [17]. In the nonlinear cases, the breathing soliton and the soliton pair are generated with the super-Gaussian beams in the nonlinear FSE [15]. And a deeper insight into the soliton dynamics is provided as the optical lattice is considered, and the existence range of stable solitons and nonlinear bound states are uncovered [18]. Recently, the spatiotemporal modulational instability (MI) in a fractional nonlinear Schrödinger equation has been invesitgated, and the Lévy index affects the fastest growth frequencies and MI bandwidth and gain [19]. The relation between the fractional Schrödinger equation and light propagation in honeycomb lattice has been uncovered [20]. However, few results about the dynamics of Airy beams modeled by FSE have been reported, in fact, Airy beam has attracted considerable attention owing to the unique features, such as non-diffraction [21], self-acceleration [22] and self-healing [23] during propagation. These properties promote applications in particle manipulating [24], image transmission [25] and self-bending plasma channels [26, 27], as a result, controlling the dynamics of Airy beams is of great importance. Moreover, the propagation of beams under a refractive barrier present the fascinating properties, and the reflection and transmission have been presented [28–30].

Stimulating by the results, the potential barrier is introduced to control the propagation properties of Airy beams in FSE, and some novel properties have been discovered. Without the potential barrier, the symmetric beam splitting and non-diffraction of sub-waves are exhibited. When a single potential barrier is considered, the beam’s reflection dynamics closely relate to the Lévy index and the location of the potential barrier. However, with the symmetric potential barrier, the beam evolution changes from the periodic self-imaging phenomenon to the chaoticon when the Lévy index increases, also the period is analytically discussed. These results show that the dynamics of Airy beams can be well controlled by the FSE with the potential barriers. To the best of our knowledge, the topic in our works has not been reported so far.

The paper is organized as follows. In Sec. 2, the theoretical model is introduced, and the diffraction features are demonstrated. In Sec. 3, the dynamics of Airy beams in FSE with the single potential barrier are analyzed. In Sec. 4, we go a step further to explore the propagation of Airy beams under the symmetric potential barrier. Finally in Sec. 5, the paper is concluded.

## 2. Theoretical model and diffraction properties

The theoretical dimensionless model, governing the propagation of beams in FSE in the presence of a potential, is given by [11, 14, 31]

*φ*is the dimensionless field amplitude,

*x*and

*z*are the normalized transverse and longitudinal coordinates, respectively.

*α*(1 <

*α*≤ 2) is the Lévy index which describes the fractional-order diffraction effect, and the model changes into the standard SE with

*α*= 2. From [32], the Fourier transform of ${\left(-\frac{{\partial}^{2}}{\partial {x}^{2}}\right)}^{\alpha /2}\phi $ is Ψ|

*k*|

_{x}*with Ψ being the Fourier transform of*

^{α}*φ*. In fact, the FSE optical model is physically realizable by using the scheme in [12], and the phase mask

*t*

_{1}(

*x*) and

*t*

_{2}(

*x*) have the transmission function

*t*

_{1}(

*x*) = exp (−

*i*|

*k*|

_{x}*) and*

^{α}*t*

_{2}(

*x*) = exp (−

*iV*(

*x*)) respectively. Numerically, Eq. (1) can be solved by using the standard split-step Fourier method In this paper, the propagation properties of Airy beams modeled by FSE are investigated, and the expression of input beams is: In Eq. (2), $\text{Ai}(x)=1/\pi {\displaystyle {\int}_{0}^{\infty}\mathrm{cos}({t}^{3}/3+xt)\mathrm{d}t}$ represents the Airy function,

*a*(0 <

*a*< 1) is the truncation coefficient that decreases the tails of Airy function, ensuring the finite energy of Airy beams, and

*F*(

*a*) normalizes the peak intensity of input field to 1. According to [33], the spectrum of the input Airy beams is given by:

In the following discussions, the truncation coefficient *a* = 0.1 is fixed. At first, the diffraction properties of Airy beams in FSE are demonstrated in Fig. 1, and the direct numerical simulation results of Eq. (1) (first row) and the numerical integration of Eq. (4) (second row) matches well with each other. With *α* ≠ 2, the beam splitting occurs, and this phenomenon can be explained from the group delay which is expressed as: $\partial \left(-{k}_{x}^{3}/3+{a}_{0}^{2}{k}_{x}+|{k}_{x}{|}^{\alpha}z/2\right)/\partial {k}_{x}$ [14]. One can find the opposite value and abrupt jump of the group delay at *k _{x}* = 0, thus it results in the reverse consequence in real space. That is, the one part corresponding to negative frequency is decelerating, while the other part relating to positive frequency is accelerating. Moreover, the two waves propagate to the opposite direction, leading to the beam splitting. As the Lévy index decreases, the splitting trend get enhanced with smaller bending angle of the main lobe, and the parabolic trajectory of Airy beams under

*α*= 2 is distorted in Fig. 1(e), the center of gravity gradually decreases and even retains invariant, simultaneously, the diffraction of Airy beams becomes weak, and it can be concluded from Fig. 1(f) where the peak intensity decreases slower. More interesting is that the peak intensity gets enhanced comparing with the case of

*α*= 2, and this property can be utilized to the autofocusing enhancement of ring Airy beams with adjusting the Lévy index [34]. From the peak intensity and the center of gravity under

*α*= 1 in Figs. 1(e) and 1(f), the non-diffracting and symmetric beam splitting can be clearly indicated. Also, in Fig. 1(a), the lateral shift of one sub-wave almost present the linear relation with the propagation distance. The next, the dynamics of finite energy Airy beams with the potential barrier are to be investigated based on its diffraction and the splitting property.

## 3. Reflection of Airy beams with the right side potential barrier

Here we focus on the dynamics of Airy beams with a single potential barrier, and *V*(*x*) is set to be a barrier at the right side:

*V*

_{0}is the depth of the potential barrier, and

*L*is its location. With the potential barrier under

*α*= 1, the transmission and reflection of Airy beams are shown in Figs. 2(a) and 2(b). For

*α*= 1, the non-diffraction of the reflected wave and the transmitted wave resulting from the non-diffraction property of sub-waves is shown, and the intensity shape of the reflected wave is generally identical to the shedding wave at the left side due to the interference between the main lobe and the side lobes. From Fig. 2(d), the reflectivity

*R*and the transmissivity

*T*always meet the condition:

*R*= 1 −

*T*. As the potential depth increases, more energy is reflected owing to the stronger potential barrier, resulting in the increase of the reflectivity, and then the transmissivity decreases, moreover, this regularity also suits for the case of

*α*> 1. And one can find that a threshold value of the potential depth

*V*is necessary to realize the total reflection, and that this threshold value gradually increases with the Lévy index in Fig. 2(c). For instance, the threshold

_{cr}*V*increases from 1.5 to 4.2 with

_{cr}*α*ranging from 1 to 2. The more interesting properties are the products of the reflection of Airy beams under the effects of the Lévy index as shown in Fig. 3, where the depth of the potential barrier is set to be over the threshold so that the wave at the right side can be completely reflected. Obviously, more numbers of the reflected waves are presented for increasing the Lévy index, and the diffraction property of the reflected waves gets enhanced, this phenomenon mainly attributes to the effects of the Lévy index on the diffraction of Airy beams, also the location of beams’ reflection decreases because of the larger bending angle of main lobe. For more details of the reflected waves, the width between the two adjacent waves firstly increases with the Lévy index in Figs. 3(b) and 3(c) and almost retain invariant when the Lévy index goes beyond 1.6 comparing Figs. 3(d) and 3(e); Obviously, the width between the two adjacent waves gradully decreases from the right side to the left side under a certain Lévy index, however, the peak of the reflected waves gradully shifts to the right side for a larger Lévy index as Figs. 3(b) and 3(d) shown.

The above results are discussed under a fixed *L*, and the influences of the location of the barrier are presented in Fig. 4. Due to the non-diffracting property of the sub-wave under *α* = 1, and the potential barrier do not change the diffraction of sub-waves but its propagation direction, then the reflected wave still retains the non-diffracting property, resulting in the fact that altering the location of the barrier *L* do not impact the products of the beams’ reflection in Figs. 4(a0)–4(a3). However, when *α* > 1, as an example of *α* = 1.6, the reflected waves are greatly influenced, especially, the number of reflected waves gradually increases for a larger *L* comparing Figs. 4(b0) and 4(b2) with the decreased width of single relfected wave. For a small *L*, more energy from the main lobe of Airy beams is reflected, and the energy forms a new barrier to impede the reflection of the side lobes as Fig. 4(b3) shows, as a result, the reflected waves get reduced comparing with the case of *L* = 20 under *α* = 1.6 in Fig. 4(b1).

## 4. Self-imaging of Airy beams with the symmetric potential barriers

Stimulating by the fact that the reflected wave still remains non-diffracting property under *α* = 1, in this section, the products of the reflection of Airy beams with the symmetric potential barrier will be investigated, and the potential is set to be as follows:

We mainly focus on the dynamics when the Lévy index is equal to 1 due to the symmetric splitting of energy, and the results are presented in Fig. 5 where the potential depth is larger than the threshold value of total reflection. With the symmetric potential barrier, the two sub-waves are reflected and then concentrate into the reversed Airy shape at the center, simultaneously, the splitting and reflection occurs again, while the waves evolve into the Airy shape at the center, as a result, the periodic self-imaging property can be clearly seen. The more details are shown in Figs. 5(e) and 5(f), for a large *L* (*L* = 20), the shape of the reconstructed wave is identical to the input Airy beam, while only a truncated Airy shape is reconstructed under a small *L* (*L* = 10) because a fraction of initial Airy tails is separated by the potential barrier, thus these results lead to the interesting point that the arbitrary part of the input beam can be reproduced by altering *L*. This intriguing feature may be potential in optical controlling and optical signal transmission. It should be noted additionally that the light field at the center also presents the Airy shape with its intensity gradually decreasing during propagation when the potential depth is lower than the threshold value. One can find that the period decreases for a larger *L*, as Figs. 5(a)–5(c) show, for comparison, the propagation of Gaussian beams under the same case is shown in Fig. 5(d), the fascinating feature is suggested that the period of self-imaging of Gaussian beams is half of Airy beams. The next, we will give a clarification of the evolution period of Airy beams in FSE with a symmetric potential barrier. The spatial light field during propagation without a potential under *α* = 1 can be obtained:

_{0}(

*k*, 0) is the spatial spectrum of the input light field. To give a qualitative analysis, Airy beams are composed by multiple off-axis Gaussian functions [35]:

_{x}*φ*(

*x*, 0) = exp(−

*x*

^{2}), the light field changes into the form:

*∂*

^{2}

*φ*(

*x, z*)/

*∂z*

^{2}−

*∂*

^{2}

*φ*(

*x, z*)/4

*∂x*

^{2}= 0, which gives the two sub-waves solution with the identical peak intensity. Eq. (6) clearly means that the lateral shift of the two waves is

*z*/2 when propagating to the distance

*z*, and the linear relationship between the lateral shift of one sub-wave and the propagation distance is confirmed. From the above results, the trajectory of sub-waves can be approximately considered as:

*z*= ±2

*x*, thus the angle

*θ*(see Fig. 6(a)) between the right sub-wave and the

*x*-axis is easily found to be

*θ*= arctan(2). Fig. 6(a) gives the schematics of Airy beams in FSE with the symmetric potential, the symbols

*φ*

_{1}and

*φ*

_{2}indicate the two sub-waves. When the sub-wave of Airy beams propagates to the edge of the potential barrier, the reflection occurs, and the lateral shift value is

*L*, then the propagation distance

*Z*

_{0}in Fig. 6(a) can be expressed as:

*Z*

_{0}=

*L*tan(

*θ*) = 2

*L*. As a result, the period of self-imaging of Airy beams is:

*T*= 8

*L*, but for Gaussian beams, the period decreases to

*T*= 4

*L*. The difference attributes to the asymmetric shape of initial Airy beams, when the sub-waves are reflected by the potential, the intensity of one sub-wave is similar to the sub-wave at the opposite side due to the interference between the main lobe and the side lobes, and the intensity distribution of the two sub-waves differs from each other although presenting the identical peak intensity value, thus two reflected waves firstly construct the reversed Airy shape and then changes into the Airy shape after the second reflection. However, for the symmetric Gaussian beams, the intensity of two sub-waves is identical to each other, as a result, the Gaussian shape is formed again after the first reflection, and the period is only half of Airy beams. From this result, a general conclusion is easy to be achieved: in the FSE optical system with the symmetric potential barrier, the period of self-imaging of the input asymmetric beams is half of the input symmetric beams. One importance needs to be pointed out that this result also suits for the case of small

*L*, for example, part of the Airy beams is self-imaged with

*L*= 10, and the period for the self-imaging of truncated Airy shape is also

*T*= 8

*L*as Fig. 5(f) shows. Moreover, the analytical result and numerical result of the self-imaging period for Airy beams and Gaussian beams are given in Fig. 6(b), and it confirms that the analytical result agrees well with the numerical result.

Finally, the propagation dynamics of Airy beams with *α* > 1 are also been given in Fig. 7. It can be clearly seen that the beams eventually evolve into the chaoticons, and that the chaoticons are constructed faster with increase of the Lévy index comparing Fig. 7(a) with Fig. 7(c). This phenomenon is easy to be clarified, the non-symmetric and weak beam splitting is confirmed for *α* > 1, and the diffraction of the sub-waves and the reflected waves gets enhanced, for this reason, as the two sub-waves are reflected and interacts with each other, the beams evolve into the chaoticons at a long propagation distance because of the interference of the reflected waves. And the larger Lévy index leads to the larger bending angle of the main lobe, therefore, the waves are reflected at the smaller propagation distance, and then the construction of chaoticons is faster. Although the light field presents the random distribution that may be potential in optical imaging, it should be noted that the chaoticons here only describes the disordered state of the light field and is completely different from the spatiotemporal chaotic localized structures in dissipative systems. It should be noted that the solution of the Eq. (1) under three dimensions *φ(x*, y, *z*) can be treated as *φ*(*x, z*)*φ*(y, *z*) owing to the linear property of the equation, therefore, these results can be scaled to full 3D propagation although this paper only gives the one-dimensional results.

## 5. Conclusion

The propagation properties of Airy beams modeled by the fractional Schrödinger equation(FSE) with the potential barrier are investigated. When the potential is not considered, the beams diffraction and splitting can be controlled by the Lévy index, and the non-diffraction and symmetric splitting property are demonstrated. With a single potential barrier, the beams’ reflection and transmission are presented, and the total reflection occurs as the depth of the potential is larger than the threshold value, also the products of the total reflection show interesting property, the number of reflected waves gradually increases for increasing the Lévy index. Under the symmetric potential barrier, the periodic self-imaging phenomenon of Airy beams is shown due to the non-diffracting and splitting property when the Lévy index is equal to one, the periodic evolution is analytically discussed, and the period of asymmetric Airy beams is doubled compared with the symmetric Gaussian beams, however, a chaoticon can be constructed during propagation as the Lévy index increases. These unique features show the possibility in controlling the dynamics of Airy beams with the FSE system, and potential applications can be found in optical controlling and optical imaging.

## Funding

National Natural Science Foundation of China (61571183); Hunan Provincial Natural Science Foundation of China (2017JJ1014).

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