## Abstract

We have investigated the propagation dynamics of super-Gaussian optical beams in fractional Schrödinger equation. We have identified the difference between the propagation dynamics of super-Gaussian beams and that of Gaussian beams. We show that, the linear propagation dynamics of the super-Gaussian beams with order *m* > 1 undergo an initial compression phase before they split into two sub-beams. The sub-beams with saddle shape separate each other and their interval increases linearly with propagation distance. In the nonlinear regime, the super-Gaussian beams evolve to become a single soliton, breathing soliton or soliton pair depending on the order of super-Gaussian beams, nonlinearity, as well as the Lévy index. In two dimensions, the linear evolution of super-Gaussian beams is similar to that for one dimension case, but the initial compression of the input super-Gaussian beams and the diffraction of the splitting beams are much stronger than that for one dimension case. While the nonlinear propagation of the super-Gaussian beams becomes much more unstable compared with that for the case of one dimension. Our results show the nonlinear effects can be tuned by varying the Lévy index in the fractional Schrödinger equation for a fixed input power.

© 2016 Optical Society of America

## 1. Introduction

Fractional effects have been found in various branch of physics [1, 2]. In quantum physics, Mandelbrot first introduced the concept of fractality into the Feynman path integral approach to quantum mechanics [3]. Laskin formulated the fractional Schrödinger equation (SE) to describe the fractional quantum mechanics by applying the path integral over Lévy trajectories instead of Brownian trajectories [4–6]. The fractional SE has the spatial derivative of order $\alpha $ instead of the second ($\alpha \text{=}2$) order spatial derivative in the standard SE.

Recently, Longhi proposed an optical experiment scheme to realize the fractional SE, which is based on transverse light dynamics in aspherical optical cavities [7]. He found that the dual Airy function was of the eigenmode of a massless harmonic oscillator. This breakthrough paved the way for the experimental investigation of the fractional models developed in quantum physics and stimulated an increasing interest on the study of the laser beam propagation governed by the fractional SE. Zhang *et al*. investigated a chirped Gaussian beam propagation in the fractional SE with or without an external harmonic potential [8, 9]. It is shown that the evolution of a chirped Gaussian beam can be characterized as single splitting, non-diffracting or zigzag trajectories. The fractional SE with a linear potential taken into accout has been investigated recently [10]. Those fascinating behaviors are quite different from those observed in a standard SE. In addition, the fractional SE with PT-symmetric potential has been reported [11]. There are also some literatures on the fractional nonlinear SE [12–17], in which the attentions are mainly focused on the mathematical issues of the solitary waves solutions and the numerical solving methods.

A beam with super-Gaussian (SG) profile, a well experimental approximation of flat-top beam, has been used in a wide variety of laser application, such as optical processing [18, 19], laser-driven particle acceleration [20, 21], optical trapping [22], gravitational wave detectors [23] as well as multi-filamentation generation and control [24, 25]. On the other hand, the propagation dynamics of SG beams in Kerr medium is different from those of Gaussian beam [26–28]. These investigations are, however, based on the standard nonlinear SE. A natural question one may ask is how the SG beams propagation governed by the fractional SE both in the linear and nonlinear regimes. What is the difference between the SG beams propagation in the fractional SE and the standard one? Does its evolution in the fractional SE is the same as that of Gaussian beam reported in Ref. 9? And what would happen if the SG beams propagation is decribed by the fractional nonlinear SE? These questions will be addressed in this paper.

The aim of our work is to present a detailed investigation of the SG beams propagation in the fractional linear and nonlinear SEs. In the linear regime, the SG beam goes through an initial compression phase before it splits into two sub-beams with saddle hat worn. The two sub-beams separate each other and their interval increases linearly with propagation distance. In the nonlinear regime, single soliton, breathing soliton and soliton pair will be formed depending on the order of SG beam, nonlinearity, as well as the Lévy index. Those novel behaviors can be explained by both analytically investigating the change of sepectal phase and using the spatial-frequency analysis.

## 2. Propagation model

The theoretical model for describing a light beam propagation is based on the following fractional nonlinear SE [13–17],

## 3. Results

#### 3.1 Linear propagation of the SG beams in fractional SE

In the linear case, by taking Fourier transform of Eq. (1) with $\gamma =0$, one can obtain

Figure 1 demonstrates the linear propagation of the SG beams with different values of $m$ in the fractional linear SE with $\alpha =1$. Figures 1(a)-1(d) show the numerical results of Eq. (1) with $\gamma \text{=}0$, while the corresponding analytical results according to Eq. (6) are shown in Figs. 1(e)-1(h). The analytical and numerical results shown in Fig. 1 clearly indicate the incident SG beams with different order $m$ split into two beams after a short propagation distances. These sub-beams are located at $\pm \zeta /2$ and symmetrical with respect to $\mu =0$. They separate from each other and the interval increases linearly with the propagation distance. This process is similar to the case $m=1$, which corresponds to Gaussian beam. In this case, the intensity distribution of sub-beams is relatively smooth and has single peak. However, these behaviors change drastically for $m>1$. Under such condition, the sub-beams become unsmooth, and exhibit two peaks. The similarity and difference between the evolution of Gaussian and SG beams are clearly seen from Fig. 1. The analytical results by performing the numerical integration of Eq. (6) are in excellent agreement with numerical simulation of Eq. (1).

To intuitively display the dependence of the sub-beam shapes on the order of the incident SG beams, Fig. 2 shows the beam shapes at four propagation distances for different values of $m$. The splitting process of the input beams can be clearly seen in Fig. 2. It is interesting to note that, if $m>1$, the splitting beams have two peaks whose location near the central region being more intense. The shapes of the splitting beams are similar the word “U”, and look like a saddle. While this behavior does not occur for the case $m=1$, where the sub-beams shapes are very smooth. When the splitting beams completely separate each other, the minimum intensity of the shape “U” is 0.25 for $m>1$. This value is equal to the peak intensity for the case $m=1$. The splitting-beam shapes make a transition from symmetry to asymmetry with the increase of the order. The characteristic nonsymmetry faded away gradually with the increase of propagation distance.

Figure 3(a) shows the peak intensity as a function of propagation distances for several values of $m$. In the case $m=1$, the peak intensity always decreases, and tends to a fixed value of 0.25. But for $m>1$, the peak intensity initially increases with an increasing propagation distance, reaches a maximum value, and then decreases quickly toward a fixed value with further increase in $\zeta $. These fixed values are larger than 0.25 and depend on the parameter $m$. This feature indicates that the propagation of the splitting beams are quasi-non-diffraction after they separate fundamentally each other. Figure 3(b) shows the peak intensity and the corresponding position as a function of the parameter $m$ for the case of $\alpha \text{=}1$. The peak intensity first increases gradually with increasing $m$, and then becomes saturable. The evolution of the position of peak intensity has the same tendency, but its rise speed is much faster. It means that the linear evolution of the SG beams with larger order $m$ are nearly the same.

Now we consider how the Lévy index ($1<\alpha \le 2$) affects the linear propagation dynamics of the SG beam. Figure 4 shows spatial evolution of the SG beam with $m=4$ as a function of propagation distance in the fractional linear SE with the values of $\alpha $ raging from 1.2 to 2. It can be clearly seen form Fig. 5 that, as the Lévy index $\alpha $ increases, the initial compression always occurs, but the symmetrical splitting process disappears gradually. The reason of disappearance can be attributed to the increase of the diffraction effects. One can find from Fig. 5 that, as the Lévy index increases, the peak intensity first increases, reaches to a maximium value, and then decreases. While the position of peak intensity always decreases with increase in $\alpha $.

#### 3.2 Nonlinear propagation of SG beams in fractional SE

Next, we focus our attention on the SG beams propagation in the fractional SE under the action of Kerr nonlinearity. The splitting behavior disappears when the action of nonlinear self-focusing is taken into account, which disrupts the phase evolution that determined the observed results in the linear setting. Figure 6 shows the spatial evolution of SG beams with different order $m$ for the case of $\alpha \text{=}1$ and $\gamma =1$. The corresponding input power is 0.925 times the self-focusing critical power ${P}_{cr}$ of Gaussian beam. When $m=1$, the Gaussian beam undergoes an slightly initial narrowing stage, and then diffracts with the propagation distance further increased. The initial compression of the SG beams with $m\ge 2$ have been seen apparently. All the beam’s energy is concentrated in a very short spatial slot, featuring *tightly focusing* in the space domain, a property that may lead to the realization of novel beam-focusing technique. After the initial compression, the SG beams of order $m\ge 2$ are transformed into a breathing soliton during propagation. This process is always accompanied with the radiation shedding. The periods of breathing soliton are nearly the same. The soliton, as well as the radiation shedding tend to travel along straight lines.

Figure 7 shows the propagation dynamics of the SG beams under the action of an even stronger nonlinearity. It is quite different compared to the case $\gamma \text{=1}$ shown in Fig. 6. When $\gamma \text{=}2$ (corresponding to 1.85${P}_{cr}$), after the initial compression, the SG beams form a stable soliton for $m=1$ and $m=2$, see Figs. 7(a1)-7(b1). A part of the beam energy is inevitably shed away in the form of dispersive waves during the soliton formation. When $m=3$, see Fig. 7(c1), soliton pair is formed out of the compression beam in the first 2 propagation distances. But, the soliton pair attracts each other and fusion into one channel. With increasing the order of beam $m=4$, Fig. 7(d1) shows the soliton pair is attractive as well, then fusion into breathing soliton, and finally generates a stable soilton. When we further increase the order of SG beams ($m=5$ or 6), a soliton pair or breathing soliton pair would be produced with stable propagation.

Figures 7(a2)-7(f2) show the intensity evolution of the SG beams with different order $m$ propagating in the fractional nonlinear SE at even high nonlinear regime ($\gamma =3$, equivalent to 2.725${P}_{cr}$). In this case, for $m\text{=}1$, there still only one stable solition formation, which is similar to the preceding case ($\gamma \text{=}2$ and $m\text{=}1$) shown in Fig. 7(a1) except that a shorter distance is required for the soliton formation. While stable soliton pair can be generated for $m>2$. They just propagate individually and do not interact each other any more. It can also be found from Fig. 7 that the interval between the generated soliton pair is determined by both the order of SG beam $m$ and the nonlinearity $\gamma $. The interval increases with increasing $m$ and $\gamma $. But the increasing amount of the interval is more sensitive to the change of $\gamma $ than that of $m$. For the large values of $m$ or $\gamma $, see Figs. 7(b2)-7(f2), the incident SG beams are much easier to generate the soliton pair. At the same time, the shedding radiations carry a relatively big amount of energy. When the values of $m$ increases, the shapes of the incident SG beams become nearly rectangular with increasingly steeper both leading and trailing edges, which is beefit for self-focusing effects.

Once the soliton pair is formed, one half of the shedding radiation with most intense goes inside a cage formed by the soliton pair. The shedding radiation collides with the one of the soliton and is almost completely reflected. It then goes toward the other soliton, and the same collision occurs again and again. This periodical process leads to generation of “fish net” pattern, see typical case in Fig. 7(d2). When the shedding radiations become stronger, they can form an intense soliton located at $\mu =0$, known as central soliton. But such soliton is unstable and quickly split into two sub-beams. The sub-beams collide with the solitons pair, and go outside the solitonic cage. These collisions lead to a transverse displacement of soliton pair toward to the center. If the shedding radiations are much stronger, the central soliton can experience stable propagation within 4 propagation distances, see Fig. 7(f2).

#### 3.3 Linear and nonlinear propagation of SG beams in fractional SE with two dimensions

The fractional nonlinear SE with two dimensions can be written as

Figure 8(a) and Visualization 1 show the linear propagation of the SG beam with $m=4$ as a function of propagation distance under the condition of two dimensions. It should be compared with Fig. 1(g) where the case of one dimension is displayed. One can obtain the linear evolution of the SG beam still undergoes the compression and split stages, which is similar as that appears for one dimension case, but the SG beam is compressed tightly, and evolves into a ring with saddle shape, and then the ring beam diffracts quickly. When the Kerr nonlinear effects are taken into account, the SG beams form a ring shape in the beginning. However, the ring shape is unstable and will break apart into filaments. These filaments interact each other and continue to collapse with increasing propagation distance. This behavior can be clearly seen from the Fig. 8(b) and Visualization 2. It is quite different from that for one dimensional case shown in Fig. 7(d1).

#### 3.4 Physical explanation of the novel behaviors

The SG beams propagating in fractional media exhibit novel behaviors. What is the physical mechanism behind such unusual propagation process, such as splitting and asymmetric shape? We will reveal the behind reasons in the following analysis. The physical interpretation can be easily acquired by analyzing the evolution of the spectral phase of beam in momentum space [31].

In the linear regime, we recall the general solution of Eq. (2) in $\kappa $ space, $\widehat{\psi}\left(\kappa ,\zeta \right)$, which can be recognized as a summation of the negative and positive frequency components. So, $\widehat{\psi}\left(\kappa ,\zeta \right)$ can be written as

Figure 9 displays the evolution of the spectral phase and the group delay as a function of the spatial frequency $\kappa $ and propagation distance $\zeta $ for the case of $\alpha =1$ and $\alpha =2$. It is clearly seen from Figs. 9(a) and 9(b) that, for $\alpha =2$, the spectral phase has a parabolic profile during the propagation, and the group delay increases linearly with the frequency $\kappa $ at a fixed rate $\zeta $. It contributes to the beam broadening in real space. But for the case of $\alpha =1$, see Figs. 9(c) and 9(d), the spectral phase distribution evolves into a “V” shape. Not only is the group delay independence of the spatial frequency $\kappa $ but the values of group delay of the negative and positive $\kappa $ are quite opposite. The former is equal to $0.5\zeta $ while the latter is $-0.5\zeta $. Thus it would result in a reverse consequence in real space. That is, the part of the negative frequency components experience a shift towards trailing space, but the part of positive frequency components is shifted to leading space. As a result, the part of ${\psi}^{-}$ corresponding to the negative $\kappa $ is decelerating with respect to $\mu =0$, while the opposite occurs for the part of ${\psi}^{\text{+}}$. Moreover, ${\psi}^{\text{+}}$ and ${\psi}^{-}$ propagate in opposite directions, leading to beam splitting in real space.

To obtain a better understanding and visualization of the differences between the linear propagation of the SG beams in the fractional SE with different the Lévy index, we track the SG beams in the spatial and spectral domains simultaneously by using the spatial-frequency analysis (also known as windowed Fourier transform) [32]. Intuitive images of the spatial-frequency distributions are displayed by plotting the spectrograms as the SG beams propagate in fractional media. The spatial-frequency distributions (*Sf*) are numerically calculated with a windowed Fourier transform, resulting in the following expression,

Figure 10 shows the spatial-frequency distributions of the SG beams propagation in the fractional SE with $\alpha \text{=1}$ and $\alpha \text{=2}$ at six different propagation distances. It can be clearly see from the top panel of Fig. 10 and Visualization 3 that, as the propagation distance increases, the left half part and the right half part of frequency components separate from each other and their spacing increases with distance monotonically. The larger the propagation distance the stronger the repulsion. The bottom panel of Fig. 10 and Visualization 4 show the spatial-frequency distributions of the SG beams for the case of $\alpha \text{=2}$, which is fundamentally different from those shown in the top panel of Fig. 10. The spatial-frequency distributions rotate clockwise around the center point during propagation. But the negative and positive spectra components do not separate each other. The SG beam is expected to broaden due to the effects of diffraction, which is inexistence for the case of $\alpha =1$.

The “U” shapes of the splitting beams can also be understood through the evolution of the spatial-frequency distributions shown in the Figs. 10(a1)-10(f1) and the group delay shown in Fig. 9(d). The reason can be ascribed to the spectral profiles of the SG beams with oscillatory talils in both negative and positive $\kappa $. Equation (2) is a linear system, and satisfies the linear superposition principle. In real space, moreover, the output is a superposition of ${\psi}^{-}$ and ${\psi}^{\text{+}}$. By performing a series of numerical simulations of individual propagation of ${\psi}^{-}$(or ${\psi}^{\text{+}}$), we find ${\psi}^{-}$ (or ${\psi}^{\text{+}}$) has a symmetric profile with “U” shape. So, what caused the asymmetrical shape? It can be interpreted as follows: although both ${\psi}^{-}$ and ${\psi}^{\text{+}}$ separate each other, there is still a connection through the pedastral between them. At $\zeta =0$, the spectral phase of the the left side tails are the same as those of the right side tails, their spectral phases are zero. Once the beam experiences propagation, the left pedestal of ${\psi}^{\text{+}}$ with $-\pi $ phase shift overlaps with the ${\psi}^{-}$, whose phase shift changes from $\pi $ to $-\pi $ across its whole space. The result of superposition leads to the left peak of “U” decreases and the right one increases, showing asymmetric “U” shapes duiring the propagation. And the oppositie occurs in the superposition of ${\psi}^{\text{+}}$ and the right pedestal of ${\psi}^{-}$. The values of pesdestal become smaller with long propagation distance, the asymmetry of the saddle shape tends to be unobvious.

Now, let us move to the nonlinear propagation dynamics. In this case, the novel behaviors can be understood by recalling the linear propagation shown in section 3.1. It is indicated that, for $\alpha \text{=}1$, the SG beams split into two sub-beams with quasi-nondiffraction during the propagation, while the SG beams do not split and diffract for $\alpha \text{=2}$. For the intermediate case $1<\alpha \text{<2}$ shown in Fig. 4, therefore, the SG beams not only split but also spread out owing to the diffraction effects. When the nonlinearity is taken into account, however, the nonlinear phase is dominant and able to annihilate the linear phase. In addition, with the decrease of $\alpha $, the effects of diffraction become weaker and the nonlinear effects get stronger. As a result, the bigger the Lévy index, the higher the nonlinear $\gamma $ required to the solition, solitn breathing and soliton pair generation, and vice verse. This means that, while keeping the value of $\gamma $ fixed, the nonlinear effects can be tuned by the change of the Lévy index.

For eventual physical implication of these theoretical and numerical findings, we can used a modified schematic setup, which is based on the method shown in Fig. 1(a) of Ref. 7. The phase masks ${t}_{1}\left(x\right)$ and ${t}_{2}\left(x\right)$ have the transmission functions ${t}_{1}\left(x\right)={e}^{-0.5i{\left|\kappa \right|}^{\alpha}}$ and ${t}_{2}\left(x\right)={e}^{i\gamma {\left|\psi \right|}^{2}}$, respectively. The axicon plays a role as ${t}_{1}\left(x\right)$ for the linear propagation. The Kerr nonlinear medium can generate the function of ${t}_{2}\left(x\right)$.

## 4. Conclusion

In summary, the linear and nonlinear propagations of the SG beams in the fractional SE have been investigated and compared. We characterized the propagation process of the SG beams under different cases. In the linear regime, the SG beams with the order $m>1$ experience an initial compression phase before they are divided into two sub-beams with the shape of saddle. The sub-beams separate each other and their interval increases linearly with propagation distance. While $m=1$, the Gaussian beam have a similar evolution except that the sub beams have smooth shapes. In the nonlinear regime, the SG beams are transformed into a single soliton, breathing soliton or soliton pair, which are determined by the order of SG beam, nonlinearity, as well as the Lévy index. In two dimensions, the evolution of SG beams is similar to the case of one dimension during the linear propagation, but the initial compression of incident beam and the diffraction of the sub-beam are much stronger than that in one dimension case. While the nonlinear volution of the SG beams becomes much unstable compared with that for one dimension case. The physical interpretation of those novel behaviors are presented by analyzing the evolution of spectral phase and the spatial-frequency distributions. Our investigations show the nonlinear effects can be tuned by changing the value of the Lévy index and the fractional SE provides another freedom for beam control. The results not only further deepen understanting the fracrional SE, but also provide some potential applications, such as beam splitter, ring beam generation, and other.

## Acknowledgments

This work was supported by the National Natural Science Foundation of China (Grant Nos. 61505116, 61505113, 61575127), the Natural Science Foundation of Guangdong Province (Grant No. 2016A030313049), the Natural Science Foundation of SZU (Grant No. 000053), the Program of Fundamental Research of Shenzhen Science and Technology Plan (Grant Nos. JCYJ20140828163634005, JCYJ20140418095735599, JCYJ20140828163633984), the Science and Technology Planning Project of Guangdong Province (Grant No. 2016B050501005), and the Aid Program for Science and Technology Innovative Research Team in Higher Educational Institutions of Hunan Province.

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