Higher-order spatiotemporal wave packets with Gouy phase dynamics

Abstract

Spatiotemporal (ST) wave packets refer to a broad class of optical pulses whose spatial and temporal dependence cannot be treated separately. Such space time non-separability can induce exotic physical effects such as non-diffraction, non-transverse waves, and sub or superluminal propagation. Here, a family of ST non-separable pulses is presented, where a modal order is proposed to extend their spatiotemporal structural complexity, analogous to the spatial higher-order Gaussian modes. The modal order is strongly coupled to the Gouy phase, which can unveil anomalous spatiotemporal dynamics, including ultrafast cycle-switching evolution, ST self-healing, and sub- or super-luminal propagation. We further introduce a stretch parameter that stretches the temporal envelope while keeping the Gouy-phase coefficient unchanged. This stretch invariance decouples pulse duration from modal order, allowing us to tune the few-cycle width without shifting temporal-revival positions or altering the phase or group-velocity laws. Moreover, an approach to analyzing the phase velocity and group velocity of the higher-order ST modes is proposed to quantitatively characterize the sub- or supe-luminal effects. The method is universal for a larger group of complex structured ultrafast pulses, laying the basis for both fundamental physics and advanced applications in ultrafast optics and structured light.

Figures (5)

• Figure 1: Spatiotemporal evolutions of fundamental FP and FD pulses:a,b, The isosurfaces for the electric fields of (a) $E^{\text{(FP)}}(\mathbf{r},t)$ at amplitude levels of $E=\pm0.08$, and (b) $E^{\text{(FD)}}(\mathbf{r},t)$ at amplitude levels of $E=\pm0.22$, with $q_2=100q_1$, at different times of $t=0$, $\pm z_0/(2c)$, and $\pm z_0/c$, inserted with temporal profiles at various spatial positions, $z=0$, $\pm z_0$, and $\pm\infty$, and the $x$-$z$ map plotting the distributions of corresponding instantaneous electric fields. Spatiotemporal evolutions of higher-order FP and FD pulses:c,d, The $r$-$z$ distributions for the electric fields of (c) $E^{\text{(FP)}}_\alpha(\mathbf{r},t)$ with $\alpha=4$, $9$, and $16$, and (d) $E^{\text{(FD)}}_\alpha(\mathbf{r},t)$ with $\alpha=3$, $6$, and $10$, at different times of $t=0$, $\pm z_0/(2c)$, and $\pm z_0/c$, with emerged multi-cycle structures highlighted in dashed lines. The unit for the spatial coordinate is $q_1$. See the detailed dynamic evolutions of various high-order pulses versus index $\alpha$ and time $t$ in Video 1 and Video 2 in Supplementary Materials.
• Figure 2: Iso‑amplitude morphology of FP/FD pulses under the $p$‑stretch and approach to the beam limit. Left column: FP; right column: FD. (a1,b1) Three‑dimensional Equipotential surface of an electric field for a representative stretch $p=12$; arrows indicate the local polarization (FP: linear $y$; FD: azimuthal). (a2,b2) Insets: time traces for the same parameters [instantaneous $\mathrm{Re}\,E$ together with $\pm|E|$]. (a3,a4) Three-dimensional isosurfaces of the electric field are shown for FP at representative stretch parameters of $p=70$ and $p=500$. As $p$ increases, a progressively denser stack of half-cycle sheets appears along the local-time direction, leading to an extended temporal envelope. (a5) In the large‑$p$ limit the FP tends to a linearly polarized Gaussian beam; (b3,b4) Three-dimensional isosurfaces of the electric field are shown for FD at representative stretch parameters of $p=300$ and $p=1000$. As $p$ increases, a progressively denser stack of half-cycle sheets appears along the local-time direction, leading to an extended temporal envelope. (b5) the FD tends to a cylindrical‑vector (azimuthally polarized) beam. Red/blue encode positive/negative instantaneous electric field; color bars indicate the normalized iso‑amplitude levels used in the renderings.
• Figure 3: Temporal revivals and $p$-stretch invariance. (a) Fundamental FP; (b) fundamental FD; (c) 3rd‑order FD ($\alpha=3$); (d) a superposed state $E^{(\mathrm{FD})}_{\alpha=3}+E^{(\mathrm{FD})}_{\alpha=7}+E^{(\mathrm{FD})}_{\alpha=11}$. Vertical dashed lines indicate sampling positions $\theta=\tan^{-1}(z/z_0)$. In (c) we plot two rows labelled $p=1$ (top) and $p=9$ (bottom). The red dashed boxes mark the profile‑revival positions at $\theta=\pm\pi/4$; the boxed waveforms for $p=1$ and $p=9$ coincide, demonstrating that while $p$ stretches the temporal envelope, the Gouy‑phase coefficient and the corresponding revival locations are unaffected by $p$ ($p$-stretch invariance).
• Figure 4: Phase velocity of HOSTP:a, The on-axis phase velocity distributions versus propagation distance of the higher-order FP (solid lines) and FD (dashed lines) pulses with various order indices $\alpha$ from 1 to 6; b, The phase velocity distributions on various off-axis positions from $r=0$ to $r=w_0$ versus propagation distance of the 2nd-order ($\alpha=2$) FP (solid lines) and FD (dashed lines) pulses.
• Figure 5: Group velocity of HOSTP:a, The group velocity distributions on various off-axis position from $r=0$ to $r=w_0$ versus propagation distance of the fundamental FP and FD pulses; b, The group velocity distributions on off-axis position $r=w_0$ of the HOSTP with various order indices $\alpha$ from 1 to 6. The gray and yellow regions mark the regions of subluminal and superluminal group velocity.