Multipath multicomponent self-accelerating beams through spectrum-engineered position mapping

We introduce the concept of spatial spectral phase gradient, and demonstrate, both theoretically and experimentally, how this concept could be employed for generating single- and multipath self-accelerating beams. In particular, we show that the trajectories of the accelerating beams are determined a priori by different key spatial frequencies through direct spectrum-to-distance mapping. In the nonparaxial regime, our results clearly illustrate the breakup of Airy beams from a different perspective, and demonstrate how circular, elliptic, or hyperbolic accelerating beams can be created by judiciously engineering the spectral phase. Furthermore, we found that the accelerating beams still follow the predicted trajectory also for vectorial wave fronts. Our approach not only generalizes the idea of Fourier-space beam engineering along arbitrary convex trajectories, but also offers possibilities for beam or pulse manipulation not achievable through standard direct real-space approaches or by way of time-domain phase modulation.

Figure 1

(a) Sketch of the experimental geometry used for the generation of self-accelerating beams. (b)–(e) Calculated spectrum (b), (d) and beam propagation (c), (e) with two typical examples of imposed cubic (second row) and sinusoidal (third row) phase under the paraxial approximation. White lines in (b), (d) mark the key frequency corresponding to the main lobe of the accelerating beams in (c), (e), while the predicted beam trajectories are traced by the dashed white curves in (c), (e). The inset in (c) shows the transverse intensity (arbitrary units) profiles at $z$ $=$ 0 as calculated from the frequency range under the constrains |$k_x$| ≤ 0.002$k$ (upper panel) and |$k_x$| ≤ 0.043$k$ (bottom panel). The numbers in (d), (e) indicate the correspondence between the key spectral ranges and the resulting trajectories. (All calculated trajectories are slightly shifted with respect to the main lobes associated to the beam evolutions).

Figure 2

Experimental results of self-accelerating beams obtained with a cubic (top panels) and a sinusoidal (bottom panels) phase similar to that presented in Fig. 1a and 1d. Spectral phase distribution: (b), (e) single and triple trajectories of the accelerating beams resulting from (a) and (d), respectively; (c) the residual spectrum of the filtered main hump in (b); (f) different spectral ranges responsible for different trajectories in (e).

Figure 3

Self-accelerating beams resulting from direct spectrum-to-distance mapping under the nonparaxial condition. (a) and (b) show the half-Bessel and Airy-like beam trajectories obtained by applying an inverse sinusoidal and a cubic phase, respectively; (d) and (e) show the elliptic and hyperbolic beam trajectories obtained with the spectral phases plotted in (c).

Figure 4

Vector self-accelerating beams obtained under the nonparaxial condition. (a) Shows the key frequency distribution, (b) the total intensity of the beam, (c) and (d) the intensity patterns for the $x$ and $z$ components, respectively. The numbers in (a), (b) indicate the correspondence between the key spectral ranges and the resulting trajectories. Note that (b) is given by the sum of (c) and (d).