Finite-energy accelerating beam dynamics in wavelet-based representations

Accelerating beams are wave packets that appear to spontaneously accelerate without external potentials or applied forces. Since their first physical realization in the form of Airy beams, they have found applications on various platforms, spanning from optics to plasma physics. We investigate the dynamics of examples of finite-energy accelerating beams derived from catastrophe theory. We use a Madelung transformation in momentum space, combined with a wavelet transform analysis, to demonstrate that the beams' properties arise from a type of vanishing self-interference. We identify the modes responsible for the wave packet's acceleration, and we derive the general acceleration for higher-order cupsoid-related beams. We also demonstrate how bright solitons resulting from nonlinear Airy beams can be unambiguously detected using the wavelet transform. This methodology will allow for a better understanding of special wave packet dynamics.

Figure 1

Airy beam propagation. (a) Wave function density ${\left|\psi (x,t)\right|}^2$. The peaks initially move towards the left until $t=t_{\mathrm{rev}}$. The dashed red, blue, and horizontal black lines indicate the wave front trajectory, the packet's center of mass $\overline{x}$, and the reversing time $t_{\mathrm{rev}}$, respectively. (b) Wave function density ${\left|\psi \left(x\right)\right|}^2$ at selected times. (c)–(f) Corresponding wavelet energy densities ${\left|\mathbb{W}(x,k)\right|}^2$. The dashed purple and orange lines are the mode displacements $d_{\text{Ai}}(k,t)$ derived from the FEAB solution's phase [see Eq. (9)]. The red dot indicates the position of the branch extremum $d_{\text{Ai}}\left(k_{\mathrm{ext}}\right)$ around which the self-interference occurs and the solid red line shows its trajectory. Parameters are as follows: $a=0.01$, $b=0.3$, $k_0=-1$, $w_{\mathcal{G}}=8$. Supplemental Movie S1 provides an animation of the FEAB dynamics with its WT [40].

Figure 2

Airy beam propagation with self-focusing nonlinearity $g=-0.5$ and initial momentum $k_0=-1$. (a) Wave function density ${\left|\psi (x,t)\right|}^2$. (b) Wave function density ${\left|\psi \left(x\right)\right|}^2$ at selected times. (c) Spectral density ${\left|\psi (k,E)\right|}^2$ lying on the parabolic dispersion $E\left(k\right)$ (blue line), and the bright soliton dispersion $E_{\mathrm{BS}}\left(k\right)$ (dashed red line). (Inset) Close up of the bright soliton dispersion. (d) Wavelet energy density ${\left|\mathbb{W}(x,k)\right|}^2$ at $t=200$. The vertical dashed red line stands for the mode displacement associated with the bright soliton dispersion $E_{\mathrm{BS}}\left(k\right)$. Supplemental Movie S2 provides an animation of the nonlinear FEAB dynamics with its WT [40].

Figure 3

Pearcey beam propagation. (a) Wave function density ${\left|\psi (x,t)\right|}^2$. (b) Wave function density ${\left|\psi \left(x\right)\right|}^2$ at selected times. (c)–(f) Corresponding wavelet energy densities ${\left|\mathbb{W}(x,k)\right|}^2$. The dashed purple and orange lines are the mode displacements $d_{\text{Pe}}(k,t)$ derived from Eq. (9). The red dot indicates the position of one branch extremum $d_{\text{Pe}}\left(k_{\mathrm{ext}}\right)$ around which the self-interference occurs, and the solid red line shows its trajectory. Parameters are as follows: $\sigma =1/1000$. Supplemental Movie S3 provides an animation of the FEPB dynamics with its WT [40].

Figure 4

Diffusion of an initially sharp ($\sigma =1$) Gaussian wave packet. (a) Wave function density ${\left|\psi (x,t)\right|}^2$. The dashed blue line indicates the packet's center of mass $\overline{x}$. (b) Density profile at selected times. (c)–(f) Corresponding wavelet energy density ${\left|\mathbb{W}(x,k)\right|}^2$. The dashed green lines indicate the displacement $d\left(k\right)$ of mode $k$ derived from the parabolic dispersion relation. Supplemental Video S4 provides an animation of the Gaussian packet dynamics with its WT [40].

Figure 5

Airy-engineered wave front. (a) Wave function density ${\left|\psi (x,t)\right|}^2$. (b) Wave function density ${\left|\psi \left(x\right)\right|}^2$ at selected times (top) and evolution of the total normalized population (bottom). (c) and (d) Wavelet energy density ${\left|\mathbb{W}(x,k)\right|}^2$ at two selected times. The left branch is fully damped from $t=0$ and the right branch is progressively damped, following the point $\{d_{\mathrm{Ai}}\left(k_{\mathrm{ext}}\right),k_{\mathrm{ext}}\}$, while the rest of the signal is amplified. Supplemental Video S5 provides an animation of the engineered-FEAB dynamics with its WT [40].