1. Introduction

In recent years, catastrophe optical fields have attracted extensive attention due to their oscillating structures with catastrophe functions, where the oscillating structure in the spatial domain generally shows the combination of the main lobe and the side lobe. Seven catastrophe functions can be described by fold, cusp, swallowtail, butterfly, hyperbolic umbilic, elliptic umbilic, and parabolic umbilic catastrophes, respectively^[1-2]. The fold and cusp cases correspond to the well-known Airy and Pearcey functions, respectively^[3-5]. Airy beams in the optical field have been found to exhibit intriguing properties and applications such as self-healing, diffraction-free, self-bends, optical cleaning, and trapping^[4]. The form-invariance, auto-focusing, and self-healing of Pearcey beams on propagation have also been found by Ring et al.^[5].

After the fold and cusp catastrophes, much effort has been devoted to investigating the optical swallowtail, butterfly, and hyperbolic umbilic beams in theoretical and experimental aspects. For example, the optical swallowtail and butterfly beams were reported by Zannotti et al. in 2017, which demonstrated the evolution behavior of particular swallowtail beams decaying to lower-order cusp catastrophe during propagation^[6-7]. The Poynting vector and angular momentum density of swallowtail and butterfly beams modulated by Gaussian factors have been studied and the polarization states and Stokes vortices of dual butterfly-Gauss vortex beams have been explored in previous work by the authors of Refs. [8-10]. Subsequently, the caustics, wavefront, and auto-focusing of Swallowtail-Gauss have been further reported by Qian et al.^[11-13]. The tornado Swallowtail wave superimposed by two pairs of ring Swallowtail vortex beams has been proposed by Deng et al., and it presents a high value in angular acceleration^[14]. The multi-focus and autofocusing ability of circular hyperbolic umbilic beams were first investigated experimentally and theoretically by Zhang et al. in 2022^[15].

On the other hand, the Fractional Schrödinger equation (FSE) can be considered as the extension of the standard Schrödinger equation in quantum mechanics by replacing Brownian trajectory with Lévy flight for Lévy index 1<α≤2^[16], and its experimental design was realized in aspherical optical cavities by Longhi in 2015^[17]. When a Gaussian beam propagates in the FSE, the periodically oscillating behavior and split phenomenon can be found^[18]. Since then, the optical propagation dynamics of different types of beams with external potential in FSE systems have attracted much attention owing to their zigzag, funnel-like trajectories or more interesting phenomena^[19]. For example, Zhang et al. have systematically studied the propagation behaviors of Airy, Bessel-Gauss, Laguerre-Gauss, and Hermite-Gauss beams^[20-22] with different potentials in FSE, where their propagation trajectories and focus positions are flexibly controlled by potential functions and initial input pulses. Remarkably, the study of optical solitons in nonlinear regimes based on nonlinear fractional Schrödinger equations (NLFSEs) has become a topic of interest in optics^[23-28]. For example, Zeng et al. have studied two-dimensional trapped solitons in NLFSEs with attractive-repulsive cubic-quintic nonlinearity and an optical lattice^[23] and have also further numerically investigated the existence, shapes, and stabilities of various localized modes^[27]. Yan et al. have explored a spontaneous symmetry-breaking phenomenon of solitons supported by the two-dimensional NLFSEs with focusing and defocusing Kerr media under non-Hermitian potential^[28]. Furthermore, the accelerating trajectory and controllable focusing behavior of optical Pearcey-related beams in different potentials have been investigated by introducing cusp catastrophe or Pearcey function to the optical field^[29-33]. In addition, Liu et al. have started experimental realizations of FSE in the temporal domain and observed temporal dynamics behavior such as solitary, splitting, and merging pulses, as well as double-Airy and “rain-like” multi-pulse structures^[34].

It is natural to ask whether the optical Swallowtail beams modulated by the Gaussian factor, i.e., Swallowtail-Gaussian (SG) beams, show distinctive properties, especially regarding the evolution dynamics of inversion and focusing behaviors in FSE. This paper aims to explore the periodic evolution dynamics of SG beams with different potentials in the FSE by a split-step Fourier method, which stresses their periodic inversion and focusing behaviors in the linear, parabolic, Gaussian potentials, and non-potential cases. The results obtained in this paper differ from the periodic evolution of the Pearcey beams in FSE. The properties of SG beams are expected to control inversion and focusing trajectory in FSE, which should help design optical modulators and switches.

2. Swallowtail-Gaussian beams with non-potential in FSE

The evolution of optical field propagation in the fractional system can be described by the normalized FSE^[19]

where ψ(x, z) is the optical field amplitude, the x and z are normalized and dimensionless transversal and longitudinal scales, respectively. $- {\partial ^2}/\partial {x^2}$ is the Laplace operator, α is the Lévy index (1<α≤2), and V(x) represents the potential function. If α=2, Eq. (1) is reduced to the standard Schrödinger equation. When the potential function V(x)=0, one can obtain the solution of Eq. (1) expressed as

where $\varphi (k,{\textit{z}}) = \displaystyle\int_{ - \infty }^{ + \infty } {\psi (x,{\textit{z}})\exp ( - ikx){\mathrm{d}}x}$ is the Fourier transform of ψ(x, z), k represents spatial frequency.

Assume that the incident beam in Eq. (1) is a profile of a one-dimensional optical Swallowtail-Gaussian field described by

where the Swallowtail catastrophe in one dimension is^[6-7]

The distribution factor of the Gaussian function σ>0 in Eq. (3) ensures its finite energy.

The spectrum of the input one-dimensional optical Swallowtail-Gaussian field of ψ(x, 0) is obtained by Fourier transform

On substituting from Eq. (5) into Eq. (2), one can find its propagating optical field expressed by

where the group delay is given by

With the spectral phase of ϕ(Sw) corresponding to the Swallowtail function in Eq. (6). One can find that the factor of $|k{|^\alpha }$ plays an important role in the intensity evolution. More specifically, the propagating optical field and group delay in Eqs. (6) and (7) can expand to ±k^α for Lévy index α≠2, which results in the reverse optical field in the opposite direction in real space.

On the other hand, the analytical propagation expressions for the Lévy indexes of α=1 and 2 can be obtained by the direct integration method as

with

and

For the case of 1<α<2, it is difficult to obtain a complete analytical expression for Eq. (6) due to the fractional Lévy indexes. Still, their intensity evolutions in free space can be calculated numerically using the split-step Fourier method.

Fig. 1 (color online) shows the group delay and intensity split of the SG beams with non-potential in the FSE for different Lévy indexes. It shows that the SG beams without potential present split behaviors along +x and −x directions for α≠2 in the propagation evolution, with the splitting becoming weaker as α increases. The split phenomenon can be further explained by the group delay, as shown in Figs. 1(a)−1(d), where a distinct spectrum jump is observed at k=0 for α=1, with the jump gradually decreasing as α increases. In addition, for the case of α=1, the evolution of the SG beams follows the straight line of x=z/2 determined by Eq. (9), which is also depicted by the main lobe marked by white dotted lines in Fig. 1(e). The splitting trajectories of SG beams for α=2 follow hyperbolic curves rather than the parabola of Airy beams due to their intrinsic properties in catastrophe functions^[35]. As the Lévy index α increases, the evolution of the main lobe gradually changes from a straight line into a curve, as shown in Eq. (10) and Figs. 1(f)−1(h).

Figure  1.  Group delay and intensity split of one-dimensional Swallowtail-Gaussian beam without potentials for different Lévy indexes of α=1, 1.4, 1.8, and 2. White dotted lines: the evolutions of the main lobe

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3. Swallowtail-Gaussian beams with different potentials in FSE

In this section, we focus on the periodic evolution of SG beams with linear, parabolic, and Gaussian potentials in FSE by using split-step Fourier method.

3.1 Linear potential

When the potential function is V(x)=ax (linear potential coefficient a≠0) in Eq. (1), the Fourier transform on both sides of Eq. (1) can be expressed as^[36]

where the wave function of the Fourier transform is given by

On substituting Eq. (12) into Eq. (11), the solution of Eq. (11) in the Fourier frequency domain is

The intensity evolution of the SG beams can be simulated by the split-step Fourier method, where the beam in the x-direction boundary is set as T_max=80, the number of points is N=1024, the horizontal step is Δx=2T_max/N, the propagation distance step is set as Δz=0.01, and the frequency domain step is Δk=π/T_max.

Fig. 2 gives the intensity and spectral distributions of the SG beams with a linear potential for different Lévy indexes α and linear potential coefficients a, where the split behaviors are not found for their continuous group delays. The beams present periodic oscillations in linear potentials, and their intensities focus on the cusp marked by white dotted circles in Figs. 2(h) and 2(i). The beams gradually change from periodic lines into curves as α increases. Their beam profiles can reverse sharply near focal points, and the focal points also get larger with greater α. Their periodic evolutions in positive linear potential are mirrored by those in negative cases. For example, for the case of linear potential coefficient a=8, the beams begin to deflect along −x direction, whereas for a=−8, their deflections are inverse, as shown in Figs. 2(g) and 2(h). The periods in intensity and spectral evolutions show good consistencies, as shown by the white dotted lines in Figs. 2(f)、2(i)、2(l) and 2(o). Although the Lévy index α can affect the deflection range and direction, it contributes nothing to periodic intensity distribution. The periodic oscillations in propagation, i.e., inversion and focusing behaviors, can be explained by k space in Figs. 2(m)−2(o). When the value of k is negative, the beam deflects along the −x direction. However, the spectrum jumps rapidly from −k to +k, as shown by the white dotted line in Fig. 2(o); the beam will be pulled from −x to +x direction, which leads to the focusing phenomenon. The inversion behavior is due to the beam’s capability of being reflected at the places of x=0 for k=0. The underlying mechanisms result from the intrinsic attributes of the fractional Schrödinger equation and inherent autofocusing properties of SG beams.

Figure  2.  (Color online) Intensity and spectral distributions of SG beams with a linear potential for different Lévy indexes α and linear potential coefficients a.

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Furthermore, the period of intensity in the spatial domain is determined by T=2π/(|a|Δx), and its period increases as linear potential |a| decreases. Moreover, the peak intensities and their positions of the SG beams for different Lévy indexes and linear potentials are also further shown in Fig. 3 (color online). For linear potential coefficient a=8, their peak intensities only appear at z=2.5, 7.5, 12.5, 17.5, 22.5 and 27.5 for different Lévy indexes in Fig. 3 (a), which also indicate that Lévy indexes can vary in peak intensity. Fig. 3 (c) shows that a higher peak value is accompanied by a larger Lévy index. If the peak intensity position is assumed as the shift distance in the x direction, a higher α and smaller a can lead to a larger shift distance.

Figure  3.  Peak intensities and their positions of SG beams with a linear potential for different Lévy indexes α and linear potential coefficients a during propagation. (a) a = 8, (b) a = 4, (c) the peak intensity (Max) of the longitudinal coordinate in transmission varies with the change of the Lévy indexes α(a = −4, +4, −8, +8)

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3.2 Parabolic potential

The potential function is V(x)=b²x²/2 in Eq. (1), with b being the parabolic potential coefficient. For α=2, the solution to Eq. (1) can be expressed as^{[19, 21]}

where B=bcot(bz)/2, K=bxcsc(bz), $f(x,\textit{z}) = \exp \left( {iB{x^2}} \right)\sqrt { - iK/\left( {2{\text{π}} x} \right)}$. It can be seen that the integral part of Eq. (14) is equivalent to the Fourier transform of ψ(x, 0)exp(iBx²), i.e., the beam in a parabolic potential propagates periodically from it to the Fourier transform of itself in a duplicate mode; therefore, the propagation in a parabolic potential is also known as self-Fourier propagation^[21]. The solution to SG beam propagation in parabolic potentials for α=2 is obtained by

with η=σ–ibcot(bz)/2. One can see that the SG beam presents a periodic inversion and focusing, whose main lobe follows the evolution trajectory expressed by

where x_± represents the upper and lower boundaries of the evolution trajectory.

Fig. 4 (color online) shows the intensity evolution of the SG beams with parabolic potential for different Lévy indexes and parabolic potential coefficients. One can see that the evolutions of the beams for α=1, 1.4, and 1.8 are chaotic and their periods are also indistinct. However, the evolution of the beam becomes gradually apparent to a complete period as the Lévy index increases to α=2 in Figs. 4(j)−4(e), where the beam is inverse and focused in one evolution period, and their inversion and focusing positions in the z direction are determined by

Figure  4.  Intensity evolution of the SG beams with parabolic potential for different Lévy indexes and parabolic potential coefficients. White solid lines in (j)−(l): evolution trajectories of main lobes; Red and white dotted circles: inversion and focus positions of main lobes

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With positive integer n, respectively, while for a smaller parabolic potential, e.g., b=0.1, the reversal and focus positions coincide to 2πn/|α|b. For example, for the case of α=2 and b=4 in Fig. 4(k) the reversal positions are π/4, π/2, 3π/4, π, and the focus points are located at π/8, 3π/8, 5π/8, 7π/8, 9π/8, etc. Whereas for α=2 and b=0.1 in Fig. 4(l), their locations appear in the same positions as 10π, 20π, 30π, 40π, etc. The evolution of the peak intensities of the SG beams for different parabolic potentials is further given in Fig. 5(a) (color online). For a smaller b, e.g., b=0.1, the evolution profile in peak intensity is funnel-shaped, and its value decreases before gradually increasing during propagation. Their funnel-shaped evolution profiles become sharper, and their bottoms (see green dotted circles in Fig. 5(a)) are gradually elevated with a greater b, e.g., b=0.2 and 0.4 in Fig. 5(a). If the bottom of the funnel shape is treated as a reversal position, the peak intensity in the reversal position gets larger with bigger b, and the value equals the initial intensity of the beam up to b=0.1832, as shown in Fig. 5(b).

Figure  5.  (a) Peak intensities of SG beams for different parabolic potential coefficients during propagation; (b) peak intensity in reversal positions versus parabolic potentials

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3.3 Gaussian potential

The potential function is a Gaussian form in Eq. (1), and it follows the expression of^[31-32]

where x₀ is the central location of Gaussian potential, A and w₀ are the potential height and width, respectively. If the parameter is fixed to A=10, the potential barriers are combined and determined by x₀ and w₀. This section of the paper focuses on the influence of the Gaussian potential and Lévy indexes on the SG beams. The evolution dynamics of SG beams with Gaussian potential for different Gaussian potentials and Lévy indexes are plotted in Fig. 6 (color online). It is obvious that the SG beams are restrained in potential barriers. One can regard these barriers as mirrors reflecting the main and side lobes. For example, the split beams propagate from +x and −x directions and are reflected by potential barriers after they travel a certain distance in Fig. 6(b), where the barriers and reflection points are marked by white dotted lines and red circles, respectively. The inversion focusing behaviors are periodically reimaged in a period length marked by yellow dotted lines in Figs. 6(b), and its length increases with larger x₀. As seen in Figs. 6(d)−(f), the potential barriers can also be modulated by w₀ for a fixed x₀=20. The barrier width being compressed by a larger w₀ results in the gradual breaking of the evolution period. Figs. 6(g)−(i) show that the interference between the traveling and reflected beams becomes chaotic after the beams propagate a certain distance, and the distance gets smaller as α increases.

Figure  6.  The evolution dynamics of SG beams for different Gaussian potentials and Lévy indexes. (a)−(c): α=1, w₀=1; (d)−(f): α=1, x₀=20; (g)−(i): x₀=20, w₀=1

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Fig. 7 (color online) describes the evolution dynamics of SG beams in a larger Gaussian potential width of w₀=10 for different Lévy indexes. As the Lévy index increases, the periodic dynamics become clear, and the beam gradually evolves from chaos into inversion and self-imaging. This chaos phenomenon can be explained by the interference of the reflected main and side lobes with potential barriers. The inversion and self-imaging behaviors are attributed to the suppressed side-lobes and strengthened main-lobes by a larger Lévy index, e.g., α=1.8 or 2. The results indicate that the periodic evolution dynamics of the SG beams make it easier to control inversion focusing behaviors by varying Gaussian potential and Lévy index.

Figure  7.  The evolution dynamics of SG beams in a larger Gaussian potential width of w₀=10 for different Lévy indexes (x₀=20)

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4. Conclusion

The evolution dynamics of the Swallowtail-Gaussian beams with different potentials in FSE were investigated using the split-step Fourier method, where the linear, parabolic, and Gaussian potential and non-potential cases are discussed. In a non-potential case, the SG beams split into two sub-beams from +x and −x directions, and their splitting behavior can be changed from a straight line into a curve trajectory by increasing the Lévy index in FSE. In a parabolic potential case, the SG beams evolve from chaotic deflection into periodic inversion and focusing of main and side lobes with an increase of the Lévy index. The inversion and focus position are combined and determined by a parabolic potential and a Lévy index of πn/|α|b and π(2n−1)/|α|b, respectively. Whereas for a smaller parabolic potential, their positions are equivalent to 2πn/|α|b. In a linear potential case, the periodic inversion and focusing behaviors are found, and their period distances are only affected by linear potentials; however, a larger Lévy index can strengthen their periodic evolutions in curve trajectories. In contrast with Ref. [36], where the center of gravity and the beam width of Airy beams were analyzed, this paper focuses on SG beams’ controllable inversion and focusing behaviors.

In a Gaussian potential, the evolution dynamics of the SG beams are evidently constrained within potential barriers, where the barriers can be regarded as reflection mirrors. For a smaller potential width w_0, the periodic inversion and focusing behaviors become clear with a smaller Lévy index α and larger central location x₀. However, for a larger w_0, the cases are opposite to the Lévy index. The inversion and focusing behaviors grow stronger because of the suppression of the side-lobes by a larger Lévy index.

Although it is challenging to design or manufacture the nonlinear optical media corresponding to the fractional Schrödinger equation with a variable Lévy index and potential function, the results obtained may offer potential benefits for the fabrication of optical modulators and switches by controlling the periodic inversion and focusing behavior of SG beams in FSE, and extend the higher Swallowtail catastrophe wavefield to the fractional Schrödinger optical system.