Higher-order spatiotemporal wave packets with Gouy phase dynamics.

Spatiotemporal (ST) wave packets constitute a broad class of optical pulses whose spatial and temporal degrees of freedom cannot be treated independently. Such space-time non-separability can induce exotic physical effects such as non-diffraction, non-transverse waves, and sub or superluminal propagation. Here, a higher-order generalised family of ST modes is presented, where modal orders are proposed to enrich their ST structural complexity, analogous to spatial higher-order Gaussian modes. This framework also incorporates spatial eigenmodes and typical ST pulses (e.g., toroidal light pulses) as elementary members. The modal orders are strongly coupled to the Gouy phase, which can unveil anomalous ST Gouy-phase dynamics, including ultrafast cycle-switching evolution, ST self-healing, and sub/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/group-velocity laws. Moreover, an approach to analysing the phase velocity and group velocity of the higher-order ST modes is proposed to quantitatively characterise the sub/super-luminal effects. The method is universal for a larger group of complex structured pulses, laying the basis for both fundamental physics and advanced applications in ultrafast optics and structured light.