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Abstract
Described by the fractional Schrödinger equation (FSE) with the parabolic potential, the periodic evolution of the astigmatic chirped symmetric Pearcey Gaussian vortex beams (SPGVBs) is exhibited numerically and some interesting behaviors are found. The beams show stable oscillation and autofocus effect periodically during the propagation for a larger Lévy index (0 < α ≤ 2). With the augment of the α, the focal intensity is enhanced and the focal length becomes shorter when 0 < α ≤ 1. However, for a larger α, the autofocusing effect gets weaker, and the focal length monotonously reduces, when 1 < α ≤ 2. Moreover, the symmetry of the intensity distribution, the shape of the light spot and the focal length of the beams can be controlled by the second-order chirped factor, the potential depth, as well as the order of the topological charge. Finally, the Poynting vector and the angular momentum of the beams prove the autofocusing and diffraction behaviors. These unique properties open more opportunities of developing applications to optical switch and optical manipulation.
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1. Introduction
Based on the path-integral approach, a fractional generalization of the Schrödinger equation, developed in the framework of the fractional quantum and statistical mechanics, was proposed by Laskin [1]. Subsequently, fractional Schrödinger equations (FSEs) have drawn much interest in various areas of physics [2–6]. In optics, the fractional diffraction effect also has practical significance in physics [7,8]. The realization of the FSE theory in optical fields provides abundant possibilities for studies of the fractional order beam-propagation dynamics. Subsequently, the propagation of beams in FSE with different external potentials and nonlinear terms was investigated [9–12]. This breakthrough laid a foundation for the experimental study of fractional model developed in quantum physics and aroused great interest in the research of laser beams propagation deduced by the FSEs. For instance, chirped Gaussian beams propagating in the FSE with an external harmonic potential were studied by Zhang et al. [6]. The results show that the trajectories of chirped Gaussian beams can evolve into single splitting, diffraction free or zigzag trajectories. Furthermore, circular Airy Gaussian vortex beams [13] for a FSE have also been discussed. Most recently, dynamics of a Pearcey-Gaussian beam (PGB) with Gaussian potential in the FSE is investigated, varying the Levy index offers a convenient way to control the splitting and bending angle of the beam [14]. Spatial optical solitons have been widely investigated in FSEs with different nonlinearties and potentials [15–17], in which various stable spatial solitons are obtained and controlled by external potentials. Photon potentials are commonly used in linear optics, and have been widely cited in the literature. Efremidis found that the Airy wave trajectory is controlled by adjusting the refractive index gradient in the refractive index potential [18]. As one of the potential modes, different Airy beams propagating in the parabolic potential were widely researched, such as circular Airy beams [19], Cosh-Airy and Cos-Airy beams [20,21], hypergeometric laser beams [22], etc. A wide application of parabolic potentials in optics is the propagation of pulses in gradient-exponential fibers. If a Gaussian beam was fired into such a potential, it would oscillate back and forth in a perfectly harmonic fashion without changing its profile. However, if the Airy beam is emitted with a parabolic potential, its propagation is very different. Abruptly autofocusing optical beams are subjects of broad interest in theoretical [23] and experimental [24–26] studies. Such beams can sharply concentrate their energy at focal points, which has potential applications in biomedical therapy [23], generation of “light bullets” [27], design of precise optical tweezers [25,28], stimulating photo-polymerization [29] and controlling matter waves in Bose-Einstein condensates [30]. The finite energy PGBs were generated by modulating Gaussian factor generated by spatial light modulator. The Pearcey beams have autofocusing, self-healing and inversion properties shown by Ring et al. [31]. Then, Kovalev et al. verified that the superposition of two paraxial two-dimensional half Pearcey laser beams can form the whole Pearcey beams, in which the half Pearcey beams show acceleration and deceleration trajectories before and after autofocusing respectively [32]. Recently, in 2017, Zhang et al. [33] reported Hermit-Gauss, Bessel-Gauss, and finite-energy Airy beams transmitting in a parabolic potential. Then, in 2020, the PGBs were used in a medium carrying parabolic refractive index [34]. Additionally, the propagation properties of the PGBs with vortex in a harmonic potential [35] and the propagation dynamics of the odd-Pearcey Gaussian beams and Pearcey Gaussian vortex beams in a parabolic potential were explored [36,37]. A new type of Pearcey-like vortex beam generated by the appropriate phase modulation of the optical wavefront propagates along a parabolic trajectory in the free space is presented [38]. Recently, the symmetric PGB can be formed by two quartic spectral phases in theory and experiment [39], processes a focusing intensity approximately 1.32 times stronger than that of the symmetric Airy beam.
However, there is no report on the propagation properties of the astigmatic chirped symmetric Pearcey Gaussian beams (SPGVBs) in the FSE with parabolic potential so far. Hence, in the paper, we study the focusing characteristics of the beams under the fractional parabolic Schrödinger equation by using split-step Fourier method. Due to the parabolic potential, the beams undergo a profound change, including the periodic autofocus during propagation, and the periodic inversion. This behavior raises a lot of questions. Most obviously, why and how does periodic autofocus occur? How to understand and describe this process? Do the beams maintain diffraction-free? These issues will be resolved in this article.
The setup of the article is as follows. In Sec. 2, we introduce the numerically model; in Sec. 3, different classic focusing properties of the beams in the parabolic potential are investigated. Then, we explore the Poynting vector and the angular momentum induced by astigmatic chirped SPGVBs in Sec. 4. Eventually, we summarize the paper in Sec. 5.
2. Model
In the paraxial optical system, the beams in propagation obey the normalized dimensionless fractional parabolic (Schrödinger) equation
The model’s Hamiltonian is $i \frac {\partial u}{\partial z}=H_\alpha u$, here Hamiltonian operator is
The initial astigmatic chirped SPGVBs can be written in terms of the rectangular coordinates:
It is difficult to get the analytically result of the $u(x, y, z)$ with the initial electric fields $u(x, y, z=0)$ in the optical system. Fortunately, numerical simulations are carried out to perform the propagation numerical calculation and show the simulation findings of the astigmatic chirped SPGVBs. By taking Eq. (3) as an initial electric field, we solve Eq. (1) by using the fast Fourier transform method and obtain the numerical electric field $u(x, y, z)$ of the propagation.
3. Numerical results
3.1 Periodic evolution and Inversion
With the numerical solution obtained by the split-step Fourier method, the propagation properties of the astigmatic chirped SPGVBs will be further studied in the Eq. (1), and the calculation parameters are chosen as: $\gamma =1, \kappa =0.1, x_0=-0.2$, $y_0=-0.2, p=1, g=2.25, \beta =5.3, v=1.5, x_{\mathrm {t}}=0.05, y_{\mathrm {t}}=0.05, m=1$ and $R=2$.
The intensity distributions, phase distributions and the side-view distribution of the astigmatic chirped SPGVBs at different propagation distances anp shown in the Fig. 1. From Fig. 1(a), the periodic evolution of the beams in the process of propagation is clearly shown. The chirp modulation, the parabolic potential and the fractional diffraction effect work together when $\alpha =1.8$.
Fig. 1. Propagation of astigmatic chirped SPGVBs in the FSE with parabolic potential within $z=4\mathrm {L}$ for $\alpha =1.8$. (a) is the numerically simulated side view of astigmatic chirped SPGVBs propagation; (b1)-(b5) snapshots of transverse intensity patterns at planes 1-4 marked in (a).
Figures 1(b1)–1(b4) clearly show the transverse intensity distributions of the beams at different propagation distances. The beams travel periodically focusing along the $z$-axis, due to the parabolic potential. In addition, there are three times autofocusing effects whithin $z=4\mathrm {L}$, and the beams autofocus first, then divergent and focus second. Figure 1(b1) shows the initial field of the beams is composed of a crescent-shaped light spot with zero intensity dark in the center due to the optical vortex embedded. Most of the energy is concentrated on the outer ring of the spot, and the vortex pair is located in a hollow region. Before the first autofocus generated, the propagation characteristics of the beams are similar to that of the free space, the crescent shaped main lobe gradually focuses on the center, and the light spot continuously rotates and shrinks with the increasing propagation distance to complete the first automatic focusing along the $y=x$ direction, as shown in Figs. 1(b2) and 1(b3). The first autofocusing effect occurs at $z=0.7\mathrm {~L}$, and the focal intensity contrast is 42. Interestingly, we find that an off axis open optical bottle is generated at the automatic focusing plane of the beam. In general, the off axis optics vortex superimposed on the astigmatic chirped SPGVBs can not only form an off axis open end optical bottle, but also change the position and size of the optical bottle. Then, the beams undergo a slight diffraction, which are suppressed by chirp modulation and nonlinear effect, results in the second autofocusing effect, and finally the energy of the beams concentrates at the second focus. The second focus at $z=2.2\,\mathrm{L}$, and the focal intensity contrast reaches 32, which is smaller than that of the first focusing. The transverse intensity modes from Figs. 1(b1)–1(b5) are taken on the planes marked with dotted lines in Fig. 1(a). As shown in Figs. 1(b1)–1(b5), the transverse intensities of the beams in the parabolic potential show centrosymmetric distributions by choosing $R=2$.
In order to directly observe the formation process of the focus of the beams during the propagation, the peak intensity curve and the three-dimensional schematic diagram are given in Fig. 2.
Fig. 2. (a) is the transverse intensity of astigmatic chirped SPGVBs in the FSE with parabolic potential vs. the propagation distance. (b) is the three-dimensional propagation diagram of the beams whose intensity is relative to that of the (a).
3.2 Influences of the parameters of astigmatic chirped SPGVBs and the parabolic potential
The numerically simulated side views and intensity contrasts of astigmatic chirped SPGVBs in the FSE with parabolic potential within $z=8\,\mathrm{L}$ from left to right respectively are depicted in Fig. 3. The values of Lévy index are selected as 1.2, 1.5 and 2, respectively. In Figs. 3(a1)–3(c1) and 3(a2)–3(c2), as $\alpha$ increases, the beams autofocus sharply at a certain distance firstly, then diffract and autofocus for the next time. It can be seen from Figs. 3(a1)–3(c1), the focal intensities appear at the center of the transmission axis $y=0$ for $m=0$. The times of the "defocusing to focusing" effect gradually increase, the number of the focal point becomes fewer, the positions of the focal point gradually approach the initial position, and the autofocusing effect gradually weakens. That is, as $\alpha$ decreases, the focal intensity is enhanced and the focal length becomes longer by comparing Fig. 3(a1) with Fig. 3(a2) or Fig. 3(a3). More important is that the focal intensity for the case of $\alpha <2$ is always greater than that for $\alpha =2$. This means that the focal intensity of the beams in the FSE optical system with parabolic potential is always larger than that in the standard Schrödinger equation with parabolic potential.Compared with Fig. 3(c2), Figs. 3(a2)–3(b2) no longer show periodic auto-focusing phenomenon, but slight oscillates, generated by the fractional diffraction effect.
Fig. 3. (a1)-(c1) are the numerically simulated side-views and (a2)-(c2) are the intensity contrasts of astigmatic chirped SPGBs in the FSE with parabolic potential within different distances for different Lévy indexes, $\alpha =1.2, \alpha =1.5$, and $\alpha =2$, respectively. Other parameters are the same as those in Fig. 1 except $\beta =5, \kappa =1, g=1, v=1, m=0,$ and $R=5$
The three-dimensional isosurfaces of astigmatic chirped SPGVBs propagating in the FSE with parabolic potential are presented in a systematic form in Fig. 4. In agreement with Fig. 3, the focal intensity monotonously decreases and focal length decreases with the increase of $\alpha$. Therefore, a larger $\alpha$ naturally causes faster autofocusing effect and more focal points, as the diffraction term in Eq. (1).
Fig. 4. (a)-(c) are three-dimensional isosurfaces of astigmatic chirped SPGBs propagating in the FSE with parabolic potential vs. the different Lévy indexes. All parameters are the same as those in Fig. 3.
The second-order chirped factor also has a significant impact on the propagation dynamics of the beams, as presented in Fig. 5. In fact, multiple focal points produced by the "defocusing to focusing" effect with $y=0$ on the propagation axis, which can be explained that the first focus due to the auto-focusing and the Fourier transform of the beams, and the other focuses are mainly caused by $\beta$, which makes the beams seem to pass through a "focusing lens" and converge multiple times. With the attenuation of the $\beta$, we can see that the peak intensity of the beams becomes stronger in Figs. 5(a) and 5(b) in the case of the same $\alpha$. Before the first autofocusing effect occurs, the diffraction is restrained by the chirp modulation and the parabolic potential, which make the beams’ focus at a certain distance. After the first autofocusing effect, the beams diffract slightly, and then the next autofocusing effect occurs. In addition, with the change of the second-order chirped factor, the distance from the each focus to the initial plane during the propagation of the beams is not significantly affected.
Fig. 5. Intensity ratios as a function of the normalized propagation distance $z$ with different $\beta$, (a) $\alpha =1.5$ and (b) $\alpha =1.8$. Other parameters are the same as those in Fig. 1 except $\kappa =1, g=1, v=1, m=0,$ and $R=5$.
To further research the influences of the parameters $\gamma$ and $\alpha$ on the intensity ratios, focal positions, and the intensity ratios of astigmatic chirped SPGVBs in the FSE with parabolic potential for different potential depths are presented in Fig. 6. In Fig. 6(a), we show the dependence of intensity on propagation distance versus different $\gamma$ for $\alpha =1.6$. As the propagation distance increases, the first autofocusing effect occurs at 0.64$\,\mathrm{L}$ for $\gamma =2$, and the focal intensity ratio reaches 147 in the first autofocusing while the ratio drops to 46 in the second autofocusing effect. As shown in Fig. 6(a), the beams undergo three times "defocusing-to-focusing" effects. We find that the ratio of the focal intensity goes down successively with the increase of the transmission distance for the case of $\alpha <2$, which is different from the case of $\alpha =2$ in Fig. 6(b). The same phenomenon occurs in that of $\gamma =1$ and $\gamma =1.5$ in Fig. 6(a). Interestingly, a larger $\gamma$ can increase the times of the autofocusing effect, and strengthen the peak intensity ratios while shorten the focal length for the fixed Lévy index in Figs. 6(a) and 6(b). Therefore, learning from Fig. 6 , we could make a conclusion that the bigger potential depth is, the shorter the focal length becomes, the stronger the focal intensity ratio becomes for the fixed Lévy index.
Fig. 6. The intensity ratios of astigmatic chirped SPGVBs in the FSE with parabolic potential for different potential depths. (a) $\alpha =1.6$, and (b) $\alpha =2$, respectively. Other parameters are the same as those in Fig. 1 except $\kappa =1, g=1, \beta =5, v=1, m=0, R=5$.
The topological charge of the vortex on the astigmatic chirped SPGVBs is also an important factor controlling the propagating motion of the beams, as shown in Fig. 6. It is evident that the amount of $l$ can affect the propagation properties of the beams. Compared with Fig. 7(b), Fig. 7(a) shows that when $l$ increases, the focal intensity ratio becomes higher while the focal length gets longer within $z=2 \,\mathrm{L}$ when $\alpha <2$. Moreover, we find that the lower $\alpha$ is, the longer the focal length becomes, the stronger the focal intensity ratio becomes.
Fig. 7. Numerically simulated side views of the astigmatic chirped SPGVBs propagating in the FSE with parabolic potential for different topological charges. Intensity as a function of the normalized propagation distance $z$ with different topological charges, (a) $\alpha =1.6$ and (b) $\alpha =2$. Other parameters are the same as those in Fig. 1 except $\kappa =1,\beta =5, v=1,$ $R=5$.
Subsequently, in order to better observe the on-axis astigmatic chirped SPGVBs, we plot the intensity distributions and side views of the beams in Fig. 8. In Figs. 8(a1), 8(b1) and 8(d1), with the number of the topological charge increasing, the hollow area of the transverse intensity pattern becomes larger, and the energy of the beams is mainly concentrated on the outside rings of the beams. Moreover, by comparing Figs. 8(b1) and 8(c1), it is found that the shape and the size of the light spot displayed on the initial plane are not significantly affected with the increase of the Lévy index. Along the per-designed vortex track, the beams autofocus at a specific propagation distance within $z=2 \,\mathrm{L}$. In addition, by observing Figs. 8(a3), 8(b3) and 8(d3), the discontinuous zero intensity dark nuclei surrounded by higher intensity regions is generated at the focus, and the amount of dark nuclei generated on the axis is consistent with the number of topological charge embedded on the axis. Besides, the intensity at the focal point increases with the augment of the topological charge. Interestingly, comparing Figs. 8(b2), 8(c2), 8(b3) and $8(\mathrm {c} 3)$, we find that when the Lévy index is larger, the areas of discontinuous zero intensity dark nuclei become larger, which are surrounded by the lower intensity. Besides, the energy of the beams diffuses from the center to the outside after the focal position.
Fig. 8. Demonstrations of the astigmatic chirped SPGVBs propagating in the FSE with different topological charges. (a1)-(d1) are initial intensities of the beams, (a2)-(d2) are side-views of the beams, (a3)-(d3) are intensities of the beams at the focal points. (a4)-(d4) are the intensities of the beams at $z=2 \,\mathrm{L}$. (a1)-(a4) $m=1, \alpha =1.6$; (b1)-(b4) $m=2, \alpha =1.6$; (c1)-(c4) $m=2$, $\alpha =1.8$;(d1)-(d4) $m=3$, $\alpha =1.6$. Other parameters are the same as those in Fig. 7.
Beams with an on-axis intensity distribution are known to be important in fields such as optical micro-manipulation, and the above studies have predicted the dependence of the on-axis transverse intensity profile of such beams during transmission on the topological charge. The results provide great inspiration for the development of capturing particles along the desired shape, processing of precise laser patterns, and detection of topological charges.
In order to investigate the propagation characteristics of the astigmatic chirped SPGVBs in the FSE with parabolic potential, we introduce the Poynting vector and the angular momentum of the beams. As far as we know, the Poynting vector is a vector of the energy flux density, which is used to directive the size and the direction of the energy flow. Therefore, the Poynting vector reflects the corresponding trend of the energy flow in the side view. It is a well-known that the relationship between fractional evolution and momentum has been a hot topic. For example, the conservation law of angular momentum in the nonlocal Lagrange formula was confirmed in 2013 [40]. Subsequently, conservation laws for different spatiotemporal fractional order nonlinear evolution equations were proposed in 2018 [41]. The beams carry the orbital angular momentum, whose magnitude and orientation reflect the variation in the vortex field.
When we assume that there is a vector potential $\vec {A}=\overrightarrow {e_x} u(x, y, z) \exp \left (i \frac {z}{2 \sigma ^2}-2 \pi t\right )$ along the $x$-polarized direction, the normalized time-averaged Poynting vector of the $x$-polarized field can be expressed as $[13, 42]$
To explain the variation of the transverse energy distribution of the beams, the transverse energy flow density of the beams within the $z=2 \,\mathrm{L}$, namely the Poynting vector, is calculated and presented, as shown in Fig. 9. Using the normalized transverse intensity distribution of the beams as the background, the blue arrow indicates the energy flow density vector, where the arrow length is the size of the energy flow density, and the direction of arrow indicates the direction of the energy flow. Meanwhile, the green arrow denotes the angular momentum density vector, in which the direction of the green arrow indicates the direction of the angular momentum density, the size of the green arrow corresponds to the size of the angular momentum density.
Fig. 9. Transverse energy flow and transverse angular momentum density distribution for astigmatic chirped SPGVBs in the FSE with parabolic potential at (a1) and (b1) $z=0$, (a2) and (b2) $z=0.4 \,\mathrm{L}$, (a3) and (b3) $z=0.5 \,\mathrm{L}$, (a4) and (b4) $z=0.6 \,\mathrm{L}$, (a5) and (b5) $z=\mathrm {L}$, (a6) and (b6) $z=1.2 \,\mathrm{L}$. Other parameters are the same as those in Fig. 1 except $\alpha =1.96,\beta =2.25$, and $R=15$.
Note that energy flow density of the beams rotates clockwise as a whole within $z=\mathrm {L}$, the phenomenon can be attributed to the helical phase shift of the beams, which make the intensity of the beams rotate clockwise with increasing transmission distance within $z=\mathrm {L}$. However, the magnitude and the direction of the flow energy change significantly at different focal planes. For example, from Fig. 9(a5), after the beams going through $z=\mathrm {L}$ plane, the direction of the energy flow of the beams is opposite to that of the initial plane. That is, the energy flux density of the beams rotates counterclockwise with the increase of the transmission distance in the range of $z=\mathrm {L}$ to 1.2L. From Figs. 9(b1)–9(b6), green arrows represent the direction of the angular momentum of the x-y cross-section. The overall angular momentum is distributed clockwise on each annular light spot or elliptical light spot, which is more concentrated in the high intensity area.
In the above study, we solely researched the auto-focusing properties when $1<\alpha \leq 2$ during the propagation process. Because there is no possibility for the onset of collapse for the linear beams propagation, the case $\alpha \leq 1$ also needed to be discussed too. Therefore, the numerically simulated side views of astigmatic chirped SPGVBs in the FSE with parabolic potential within $z=32 \,\mathrm{L}$ for different Lévy indexes $\alpha =0.8, \alpha =0.9$, and $\alpha =1$, from top to bottom respectively are depicted in Figs. 10(a)–10(c). Axial intensity distribution curves correspond to the white dashed lines are shown in Figs. 10(a)–10(c). The results are interesting, showing multifocal propagation characteristics. We find that the focal positions as well as the number of focal points of the beams in focal planes can be adjusted by setting $\alpha \leq 1$. As $\alpha$ increases, the focal intensities are enhanced and the focal lengths become shorter by comparing Fig. 10(a) with Fig. 10(b) or Fig. 10(c), which are different from the the case when $1<\alpha \leq 2$.
Fig. 10. (a)-(c) are the numerically simulated side-views of astigmatic chirped SPGVBs in the FSE with parabolic potential within different distances $z=32 \,\mathrm{L}$ for different Lévy indexes $\alpha =0.8, \alpha =0.9$, and $\alpha =1$, respectively. Other parameters are the same as those in Fig. 3.
These unique properties indicate the possibility of using the FSE system with parabolic potential to control astigmatic chirped SPGVBs dynamics, and have potential applications in optical control, optical splitting and optical switching.
4. Conclusion
In summary, based on the FSE with the parabolic potential, we report propagation behaviors as well as the effects of the parameters on the propagation of the astigmatic chirped SPGVBs. The focal intensity, the focal length, the phase evolution, the 3D propagation, the Poynting vector as well as the angular momentum are discussed. These propagation properties rely on selecting the initial parameters of the beams and $\alpha$ reasonably. The numerical results prove that the proposed beams perform periodic evolution and inversion. As $\alpha$ increases, the focal intensity is enhanced and the focal length becomes shorter when $\alpha \leq 1$. However, with the increase of $\alpha$, the autofocusing effect becomes weaker, and the focal length becomes shorter, when $1<\alpha \leq 2$. As $\beta$ increases, there is no observable change in the position of focal point while the focal intensity is enhanced for the fixed $\alpha$. Moreover, the whole autofocusing effect is strengthened, and the focal length increases slightly with the increase of $m$ for the fixed $\alpha$. Furthermore, the parabolic parameter affects the period, the intensity and the focal length of the beams. When $\gamma$ increases, the focal length decreases, the focus ability of the beams is enhanced, and the focal depth decreases, indicating that autofocus performs is more frequent for fixed $\alpha$. At the same time, the focus intensity gradually decreases with the increase of the propagation distance. Finally, the Poynting vector and the angular momentum of the beams further confirm the behaviors of the periodic Fourier transform and autofocus effect. In a word, the research results are of great help to expand the application of laser in micro particle control, laser processing, medical and military fields.
Funding
Guangdong Department of Education Projects of Improving Scientific Research Capabilities of Key Subjects Construction (2022ZDJS016); Natural Science Foundation of Guangdong Province (2022A1515011482); National Natural Science Foundation of China (12004081, 12174122, 62175042).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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