Self-accelerating beam dynamics in the space fractional Schrödinger equation

Self-accelerating beams are fascinating solutions of the Schrödinger equation. Thanks to their particular phase engineering, they can accelerate without the need for external potentials or applied forces. Finite-energy approximations of these beams have led to many applications, spanning from particle manipulation to robust in vivo imaging. The most studied and emblematic beam, the Airy beam, has been recently investigated in the context of the fractional Schrödinger equation. It was notably found that the packet acceleration decreased with a reduction in the fractional order. Here, I study the case of a general $n\mathrm{th}$-order self-accelerating caustic beam in the fractional Schrödinger equation. Using a Madelung decomposition combined with the wavelet transform, I derive the analytical expression of the beam's acceleration. I show that the nonaccelerating limit is reached for infinite phase order or when the fractional order is reduced to 1. This work provides a quantitative description of self-accelerating caustic beams' properties.

Figure 1

Diffusion of a Gaussian wave packet in the SFSE of order $\alpha =1.5$ and with initial width ${\sigma }_k=1/1.5$. (a) Wave function density ${\left|\psi (x,t)\right|}^2$. (b–d) Wavelet energy density ${\left|\mathbb{W}(x,k)\right|}^2$ computed at selected times and with $w_{\mathcal{G}}=8$. Dashed green lines represent the mode displacement $d^{\left(\alpha \right)}(k,t)$ computed from Eq. (8) and derived from the dispersion relation only. Supplemental Movie S1 provides an animation of the Gaussian wave-packet dynamics with its WT [39].

Figure 2

Airy beam dynamics for different fractional orders $\alpha$. Each row corresponds to a different value of $\alpha$. (a)–(c) Wave function density $|{\psi }_{\mathrm{Ai}}^{\left(\alpha \right)}{(x,t)|}^2$. The dashed blue line shows the wave-packet front trajectory, computed from the extremum mode displacement in Eq. (25). (d)–(l) Corresponding wavelet energy density ${\left|\mathbb{W}(x,k)\right|}^2$ computed at selected times and with $w_{\mathcal{G}}=8$. Dashed purple and orange lines are the mode displacement $d_{\mathrm{Ai}}^{\left(\alpha \right)}(k,t)$ computed from Eq. (23). The point $P_I$, around which the self-interference occurs, is shown as a red circle, and the green line represents its trajectory in the $x\text{−}k$ phase space. (m)–(o) Far-field density $|{\psi }_{\mathrm{Ai}}^{\left(\alpha \right)}{(k,E)|}^2$. The field's density follows the dispersion relation $E^{\left(\alpha \right)}\left(k\right)$, shown as the solid red line. The parabolic dispersion $E^{(\alpha =2)}\left(k\right)$ is plotted as the dashed blue line for comparison. Parameters are as follows: $a=0.01$, $b=0.3$, $D_{\alpha }=0.5$. Supplemental Movie S2 provides an animation of the different Airy beam dynamics with their WT [39].

Figure 3

Exponent of the wave-front acceleration $t^{\beta }$ as a function of the fractional order $\alpha$ for self-accelerating beams of different orders $n$. The three cases with $n=3$, 4, and 5 correspond to an Airy, a Pearcey, and a Swallowtail beam, respectively. These beams can reach the same acceleration for a different fractional order $\alpha$, for example, $\beta =4/3$. The dots at $\alpha =2$ indicate the acceleration obtained with the conventional Schrödinger equation. The nonaccelerating limit $(\beta \to 1)$ is reached either when $\alpha \to 1$ or when $n\to \infty$.

Figure 4

High-order accelerating beam dynamics. The first column corresponds to the case of a Pearcey beam with $n=4$ and $\alpha =1.75$, and the second column to a Swallowtail beam with $n=5$ and $\alpha =2$. (a), (b) Wave-function density $|{\psi }_n^{\left(\alpha \right)}{(x,t)|}^2$. Dashed blue lines indicate the trajectory of the extremum mode, derived from Eq. (31). (c), (d) Wavelet energy density ${\left|\mathbb{W}(x,k)\right|}^2$ computed at $t=150$ and with $w_{\mathcal{G}}=8$. Dashed orange and purple lines represent the mode displacement $d_n^{\left(\alpha \right)}(k,t)$ derived from Eq. (29). The point $P_I$, around which the self-interference occurs, is shown as a red circle, and the green line shows its trajectory in the $x\text{−}k$ phase space. Both packets accelerate as $t^{4/3}$. Supplemental Movie S3 provides an animation of the Pearcey and Swallowtail beam dynamics with their WT [39].