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Abstract
We investigate the mutual interaction of two spatially-separated Airy beams in the nonlinear Schrödinger equation with the fractional Laplacian. Depending on the beam separation ($d$), relative phase and Lévy index ($\alpha $), we observed an anomalous attraction or repulsion between the Airy beams. Anomalous attraction leads to a single breather soliton with a period that grows exponentially as $\alpha $ increases. In this region of the parameter space, we identify a crossover between two asymmetric regimes: as the Lévy index exceeds a critical value ${\alpha _c}$, the period of breather soliton for $d > 0$ is orders of magnitude larger than for $d < 0$, while the opposite occurs as $\alpha < {\alpha _c}$. Our results reveal a novel scenario for Airy beams interaction in the framework of fractional nonlinear Schrödinger equation and provide an alternative mechanism to control breather soliton generation.
© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Fractional calculus is an extremely useful and advanced tool in mathematical analysis, which has been widely used for dealing with complex dynamics in diverse fields of science and engineering [1–3]. In quantum mechanics, the space-fractional Schrödinger equation (FSE) was proposed by Laskin as the Lévy trajectories was substituted by the Brownian trajectories in the path integral formalism [4–6]. Compared to the standard Schrödinger equation, FSE has the Riesz space-fractional derivative instead of the conventional Laplacian. By using paradigms of the fractional calculus, Gutiérrez-Vega found new fractional-order beam solutions to the paraxial wave equation [7,8]. In 2013, Stickler introduced one-dimensional Lévy crystal to experimentally realize FSE in condensed-matter environment [9]. More recently, Longhi put forward an optical experiment scheme for modeling the FSE [10]. By exploiting the Dirac-Weyl equation, Zhang et al. constructed the link between FSE and a real physical system as the honeycomb lattice [11]. These breakthroughs make the experimental investigation of unique behavior of FSE accessible in optics.
On the other hand, the fractional Laplacian derivative provides an additional degree of freedom to shape the dispersion relation of the physical system, leading to a profound change in the properties of the wave function. Consequently, considerable attention has been paid to investigate beam propagation dynamics in FSE from linear to nonlinear regimes. Diffraction-free or self-splitting beam [12,13], light propagations in FSE with different potentials [14–16] or variable coefficient [17], modulation instability [18], Rabi oscillations [19], Anderson delocalization [20] and rich families of solitons [21–27] in fractional dimensions have been reported.
Airy wave packet was first discovered in the quantum mechanics as a solution to the potential-free Schrödinger equation [28]. The experimental realization of optical Airy beams in 2007 [29,30] stimulated increasing investigations towards generation, applications and control of Airy wave packets [31], owing to its unique characteristics of non-spreading, acceleration, and self-healing [29,30,32]. Airy beams propagating in nonlinear Kerr media are able to shed solitons with controllable features [33]. Breather solitons and soliton pairs can be generated through Airy beam interaction, a phenomenon that has been investigated in Kerr and saturable nonlinear media [34–36], photorefractive media [37], nonlocal nonlinear media [38] and nematic liquid crystals [39]. However, so far all the interaction properties of these beams are based on the conventional nonlinear Schrödinger equation (CNLSE). How Airy wavepackets interact when governed by the fractional nonlinear Schrödinger equation (FNLSE) is an open question. In this case, a unique dynamics is expected to emerge from the interplay of the fractional Laplacian, which is inherently a nonlocal operator, and the extended and asymmetric character of Airy fields. However, to the best of our knowledge, the interactions between Airy beams in the FNLSE framework remains unexplored.
Here, we investigated Airy beam interaction in a FNLSE, revealing the impact of the Lévy index on the generation of nonlinear structures. We point out a novel scenario without any correspondence with the dynamics of continuous waves. The anomalous attractive behavior is modelled according to the complex structure generated by a general superposition of two Airy beams.
2. Propagation model
Light beam propagation described by the dimensionless FNLSE can be written as [13]:
We use the standard split-step Fourier method to numerically integrate Eq. (1). The Fourier transform of Eq. (1) can be written as [13]:
For Airy beam,
where $Ai(X )= \frac{1}{\pi }\int_0^\infty {\cos \left( {Xt + \frac{{{t^3}}}{3}} \right)dt}$ and A are Airy function and beam amplitude respectively. Therefore, the general expressions for the in-phase and out-of-phase fields are,3. Numerical results
We first study the interaction of two in-phase beams in the FNLSE with Lévy index $\alpha = 1.5$ and $N = 1$ for different beam intervals d. Without losing generality, we fix $A = 2.25$ so that the amplitude of Airy beams with $a = 0.2$ is equal to one. In order to achieve a clear visualization of the beam dynamics in the FNLSE, hereafter, we plot the evolution of ${|U |^{0.5}}$, instead of the intensity ${|U |^2}$, as a function of propagation distance. Figures 1(a)–1(c) show the spatial evolution of the interaction of Airy beams for relative separation varying from −2 to 2. For comparison, Figs. 1(d)–1(f) show an example of the interaction process obtained when sech beams are used. We can clearly observe that, for $d = 0$, the evolution of Airy beam interaction shown in Fig. 1(b) is very similar to that occurring for sech beams [Fig. 1(e)]. After an initial shaping stage, the combined beams first attract each other and then evolve into a breathing soliton with a very small period. Weak soliton pairs with different angels also coexist with the process of breathing soliton formation. However, as the value of relative separation between the Airy beams is varied from negative to positive, i.e., the left and right beams swap suits, the interacting dynamics presents a radically different significantly behavior. For sech beams, the evolution depends only on the absolute values of d, irrespectively of the sign of the beam interval parameter. Differently, in the case of Airy beams, the interaction processes display evident differences as the beam interval is varied from negative to positive. The propagation distance on which the breathing soliton forms for $d = - 2$ is always shorter than those for $d = 2$, whereas the period of the breathing soliton is smaller for positive d values. Moreover, the widths of breathing solitons formed from Airy beam interaction are wider than those generated from sech beams.
Fig. 1. Spatial evolution of the interaction of two in-phase Airy (top row) and sech beams (bottom row) with different initial interval d in the FNLSE.
To further disclose and understand the different behaviors between the two interacting scenarios, Figs. 2(a) and 2(c) display the maximal value of ${|U |^{0.5}}$ as a function of propagation distance. It can be clearly observed from Figs. 2(a) and 2(c) that the peak intensity increases rapidly as a result of the attractive force between the two in-phase beams. At this point, a single breathing soliton forms. Its period, measured at the location $X = 0$, strongly depends on the input condition. For three cases presented in Figs. 2(a) and 2(c), the periods of breathing soliton generated by Airy beam interaction are larger than that for sech beams interaction. The smaller the interval is, the stronger the attraction will be, thus leading to breathing solitons with smaller periods. Specifically, the period of breather soliton decreases as the beam interval increases. For symmetric sech beams, as shown in Fig. 2(c), the curves of maximum $\sqrt {|U |} $ overlap entirely for both positive and negative values of the relative interval. This indicates that the period of the breathing soliton remains constant if the absolute values of d is kept fixed. However, for Airy beam interaction an anomalous behavior emerges. The breathing solitons have different peak intensity and period when the relative interval parameter d changes sign. The breathing soliton has an increased stability and a larger period in the case of a negative spatial interval $d = - 2$ than those for a positive interval $d = 2$.
Fig. 2. (a, c) The evolution of maximum values of ${|U |^{0.5}}$ versus propagation distance and (b, d) the incident beam profiles, which consist in the superposition of two in-phase beams with different initial intervals for (a, b) Airy and (c, d) sech beams.
The qualitative differences between the Airy and sech beam interaction can be understood by noting that the input beam is composed by a coherent superposition of two beams with relative separation and phase delay. We plot in Figs. 2(b) and 2(d) the shape of incident beams by calculating Eq. (4) directly. As evident from Fig. 1, all breathing solitons are located at $X = 0$, indicating that the central part of the incident beams would play an important role in the process of breathing soliton formation. In the case of two in-phase sech beams, their superposition only changes from a single peak to double peak structure as the value of relative separation becomes different from zero [Fig. 2(d)]. However, due to the asymmetric oscillatory structure of the Airy beams, the distributions resulting from their coherent superposition exhibit a more complex structure that depends on both their initial relative spacing and phase difference [see Fig. 2(b)]. The shape of region between the two main peaks changes from a valley to a peak as d is varied from 2 to −2. This unusual variation of intensity in the center region directly alters the corresponding refractive index. As a result, Airy beam interaction can be strongly affected by the specific input condition in the case of negative relative separations.
To get a deep insight on the complex interaction between counterpropagating Airy beams, Fig. 3 shows the interference pattern produced by the superposition of two beams as a function of relative separation d for both the in-phase and out-of-phase case. As reported in Fig. 3, an Airy beam superposition gives a rich and complex dynamics within its central zone when the value of relative interval is negative and crosses the peak maxima. There are many secondary peaks between the outermost two dominant peaks, their number and intensity being strongly dependent on the value of relative interval and phase difference. Interestingly, there are even and odd numbers of secondary peaks within the region between the two strongest peaks, respectively for in phase and out of phase input beams. These peaks increase two-by-two as the negative d decreases. For the out of phase case, a gap appears in Fig. 3(d) for $X = 0$ due to the symmetric $d = 0$ shape sech beams, while it disappears for asymmetric Airy beams, as reported in Fig. 3(c). In addition, the superimposed beam profile still has a symmetric multipeak structure with respect to $X = 0$, with the exception of the corresponding on-axis amplitude that is always zero due to destructive interference.
Fig. 3. Beam shapes from the coherent superposition of two Airy (a, c) and sech (b, d) beams as a function of relative separation for the cases of (a, b) in phase and (c, d) out of phase. Black dashed lines denote the position of peak intensity.
Figures 4(a) and 4(b) show the maximum amplitudes of the two-beam superposition as a function of the values of the relative spacing for in phase and out of phase, respectively. The variation of the maximum amplitude for sech beams is smooth: it increases (decreases) from 1 to 2 (0), then decreases (increases) to 1 for in (out of) phase as the value of relative spacing d is varied from positive to negative. This behavior is fundamentally altered when Airy beams are combined. In this case, the corresponding curves not only have an asymmetric shape but they can also have a multi-peak structure near the leading edge of their shapes. Even more surprisingly, the minimum (maximum) amplitude is about 0.5 (1.5) rather than being zero (one) for out of phase case. The oscillatory structure directly reflects the complex shape of the two superimposed Airy beams.
Fig. 4. (a, b) Maximum amplitude and (c) on-axis amplitude of coherent superposition of two in-phase (a, c) and out-of-phase (b) beams as a function of their relative separation. (d) The period of breathing soliton formed from in-phase beams interaction.
The on-axis amplitude is a crucial factor in the formation of breathing soliton. As seen from Fig. 4(c), the on-axis amplitude for two in-phase superimposed Airy beams shows a strong oscillating and multi-peaked structure, while that for sech beams has only a single peak. In fact, the former curve can be expressed as an Airy function, while the latter one is described by hyperbolic secant function [see Eq. (8)]. The complex waveforms generated by Airy beam superposition lead to the formation of a breathing soliton formation with variable properties. Figure 4(d) shows the comparison between the breathing soliton period generated from Airy and sech beam interactions. We observe that Airy beams give a larger period for the breather soliton than sech beams. However, the Airy behavior exhibits an oscillatory structure as $d < 0$. This change is consistent with the variation of the on-axis amplitude shown in Fig. 4(c). In addition, comparing Figs 4(a) and 4(d) we can find that, for $d < 0$, the period of the breather soliton exhibits multipeak values at the points of minimum values of max amplitude. In other words, the smaller the max amplitude is, the bigger the period becomes.
Next, we focus on the impact of the Lévy index on Airy beam interaction. Figure 5 displays the FNLSE propagation dynamics of two in-phase Airy beams with different intervals for different Lévy index. Since a large interval between the two Airy beams leads to weak interaction, here we only consider the case where such interval is small. For a Lévy index $\alpha = 2$, Eq. (1) reduced to the CNLSE, and the corresponding interactions are shown in Figs. 2(a3)–2(e3). The nonlinear effect would be enhanced as the Lévy index decreases. We observe the effect in all the simulated dynamics reported in Fig. 5. Here, Airy beam’s amplitude is chosen as $A{\ =\ }3$ for better displaying the evolution of breather soliton generated from their interaction. As observed, the first lobes of two Airy beams form two solitons after shedding some radiation, then the two solitons attract each other and keep a certain period along the propagation distances. When the interval $d = - 1$, the Airy beam superposition exhibits a single maximum peak located at $X = 0$ [see black dashed line in Fig. 3(a)], which leads to the strongest interaction and smallest breathing soliton period. When the Lévy index decreases gradually, the propagation features of the two in-phase Airy beams can be effectively controlled. We can see from Figs. 5(e1) and 5(e3) that, with a decreasing Lévy index, the period of the breathing soliton becomes smaller. In other words, small Lévy indices are able to enhance attractive interaction, an effect leading to a stronger gravitational pull between the Airy beams. It is worth noting that, when the Lévy index is small enough, the periodically breathing soliton will disappear and stable brothering pairs generated from individual Airy beam would appear. The reason is that nonlinearity fully dominates over fractional diffraction effects.
Fig. 5. The interactions of two in-phase Airy beams with $A = 3$ for different initial interval and different Lévy index $\alpha $ in the FNLSE.
As can be seen from Fig. 5, the breather solitons become stable after propagating a certain distance. The period and amplitude of the stable breather solitons from Fig. 5 were extracted and shown in Figs. 6(a) and 6(b), respectively. As evident from Fig. 6(a), in the five cases, the periods of the breathing solitons increase with increasing Lévy index $\alpha $. Comparing with the presented cases, for $d = 0$ the period is the shortest and the corresponding amplitude is the biggest. It clearly seen from Fig. 6(a) that the period changes for $d = 2$ is very sensitive to the values of Lévy index, which increases steeply as the Lévy index reaches a critical value ${\alpha _c} \approx 1.81$ for the Lévy index below ${\alpha _c}$ the period for $d = - 2$ is larger than that of the period for $d = 2$ while the opposite occurs for the Lévy index above this value. But for the case of $d = \pm 1$, the critical value of the Lévy index does not exist. This is very interesting, by comparing Fig. 4(a) and 4(d), we can find that the period of the breathing soliton near the minimum point ($d < 0$) of max amplitude is larger than that at the corresponding point ($d > 0$). Therefore, we show the evolution diagram of the breathing soliton period with an initial interval of $d = \pm 3.5$ in Fig. 6(a). Interestingly, the critical value of the Lévy index appears which further confirms our conclusion. The behind reason is that, when the Lévy index decreases, the diffraction term changes from conventional to fractional, making the beam spreading weaken. Consequently, the dominated two peaks generated soliton individually. The overlapping part between two peaks for $d = - 2$ advances the breathing formation, see first column of Fig. 5. For the case of $d = 2$ shown in last column of Fig. 5, the overlapping part disappear, making the positions for the first collision almost unchanged. But the individual solitons become stronger as the Lévy index decreases, making the period of breathing soliton small. It can be seen from Fig. 6(b) that in all three cases, the amplitude of breathing soliton decreases with increasing Lévy index. The smaller the relative spacing, the greater the amplitude. This indicates that the period and amplitude of the breathing solitons can be manipulated by changing the Lévy index and relative spacing in the FNLSE.
Fig. 6. (a) Period and (b) maximal values of ${|U |^{0.5}}$ of the breathing soliton for varying $\alpha $ Dashed green line in (a) denotes the critical point of the Lévy index. Inset in (a) a zoomed range.
The propagation dynamics of the out-of-phase Airy beams are shown in Fig. 7, in which the numerical parameters are same as that used in the in-phase case shown in Fig. 5. As seen from Fig. 7, the two out-of-phase Airy beams repel each other and form soliton pairs, and a repulsive force between the two generated solitons would be decreased with decreasing Lévy index. In general, it is commonly accepted that the smaller the interval, the greater the repulsive force. But some anomalous behavior happens for two out-of-phase Airy beam interaction.
Fig. 7. The interactions of two out-of-phase Airy beams with $A = 3$ for different initial interval d and different Lévy index $\alpha $ in the FNLSE.
By comparing the peak intensity [double peaks with same intensity, see Fig. 3(c)] and their interval resulted from the superposition of two Airy beams with $\pi $ phase difference listed in Table 1, we find the Airy beams interaction for $d = 0$ breaks the conventional picture. In this case, the separation angle is the largest, although the intensity and interval are both the smallest. The reason behind this anomalous phenomenon is related to the intrinsic behaviors of Airy beam. For $d = 0$ the dominant peak intensity of a superposition of two Airy beam is cut off at a level of 37% of the peak value. As a result, the peak intensity below one, leading the diffraction effect dominates over the nonlinearity effect. On the other hand, the double Airy beams are able to self-accelerating propagation owing to their self-healing capability, manifested as widen separation angle. It also should be pointed that, for $d = - 1$, the main lobe of two out of phase Airy beams is almost completely offset and their second peaks become the dominate lobe. The separation angle is much bigger compared with the case of $d = 1$. In other words, the soliton pair shown in Figs. 7(b) is generated from the second lobes, while the others are generated by the main lobes.
Table 1. Peak intensity and its interval for different initial relative spacing
4. Conclusion
In conclusion, we have investigated the interaction of both in-phase and out-of-phase Airy beams with a different Lévy index and spatial interval based on the FNLSE. The complex interference pattern generated from a superposition of two Airy beams and tunable Lévy index leads to a series of novel propagation dynamics. In the in-phase case, when the value of beam separation changes from positive to negative, the distribution in the vicinity of $X = 0$ becomes complex, leading to oscillatory variation of on-axis intensity. As a result, anomalous mutual interaction of Airy beams was observed. In addition, the interaction dynamics of Airy beams with positive value of beam separation is much more sensitive to the varying Lévy index, a phenomenon manifested through a period of generated breathing soliton that grows exponentially increasing Lévy index. On the contrary, it varies linearly with the Lévy index for the case of Airy beams with negative beam separations. In the out-of-phase scenario, two Airy beams always experience repulsion. This nonlinear effect is gradually enhanced as the Lévy index is decreased. Therefore, our results not only shed new light on the mutual interaction of two beams with asymmetric profile but also suggest an efficient strategy to generate and control breather solitons.
Funding
National Natural Science Foundation of China (61975130, 61505116); Natural Science Foundation of Guangdong Province (2016A030313049); China Postdoctoral Science Foundation (2019T120748, 2018M630978); Natural Science Foundation of SZU (000053); Science and Technology Planning Project of Shenzhen Municipality (ZDSYS20170727104468); Department of Education of Guangdong Province (2016KCXTD006).
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