Spatiotemporal sculpturing of light pulse with ultimately sophisticated structures represents a major goal of the everlasting pursue of ultra-fast information transmission and processing as well as ultra-intense energy concentration and extraction. It also holds the key to unlock new extraordinary fundamental physical effects. Traditionally, spatiotemporal light pulses are always treated as spatiotemporally separable wave packet as solution of the Maxwell’s equations. In the past decade, however, more generalized forms of spatiotemporally nonseparable solution started to emerge with growing importance for their striking physical effects. This roadmap intends to highlight the recent advances in the creation and control of increasingly complex spatiotemporally sculptured pulses, from spatiotemporally separable to complex nonseparable states, with diverse geometric and topological structures, presenting a bird’s eye viewpoint on the zoology of spatiotemporal light fields and the outlook of future trends and open challenges.

The precise observation of the angle-frequency spectrum of light filaments in water reveals a scenario incompatible with current models of conical emission (CE). Its description in terms of linear X-wave modes leads us to understand filamentation dynamics requiring a phase- and group-matched, Kerr-driven four-wave-mixing process that involves two highly localized pumps and two X waves. CE and temporal splitting arise naturally as two manifestations of this process.

We find a family of solutions of the paraxial wave equation that represents ultrashort pulsed light beams propagating in free space. These pulsed beams have an arbitrary temporal form and a nearly Gaussian cross section, while modeling for the pulses emitted by mode-locked lasers with stable two-mirror resonators. We also study the effects arising from their spatiotemporal coupled behavior, such as pulse time delay, distortion, and a frequency shift toward the beam periphery. Time-varying diffraction (with diffraction reduction at the first instants of arrival of the pulsed beam at a given distance) and the dependence of the spatial distribution of energy on the pulse form are also described. These effects become important for pulsed beams with a few optical oscillations within the pulse envelope.

Basic concepts of three-dimensional wave packets are applied to the description of transverse effects on the propagation of ultrashort (femtosecond) pulses. The frequency-dependent nature of diffraction acts as a kind of dispersion that modifies the pulse front surface, its group velocity, the envelope form, and the carrier frequency. If the diffracted field in the monochromatic case is known, these changes can be straightforwardly quantified. Finding the propagated pulsed beam field reduces to a well-known and simpler problem of one-dimensional pulse propagation with group velocity dispersion. The method is applied to pulsed Gaussian beams and pulsed Bessel beams. Anomalous pulse front behavior, including superluminality in pulsed Gaussian beams is found. The carrier phase at any point of space is calculated.

Nonlinear losses accompanying self-focusing substantially impact the dynamic balance of diffraction and nonlinearity, permitting the existence of localized and stationary solutions of the $2\mathrm{D}+1$ nonlinear Schrödinger equation, which are stable against radial collapse. These are featured by linear, conical tails that continually refill the nonlinear, central spot. An experiment shows that the discovered solution behaves as a strong attractor for the self-focusing dynamics in Kerr media.

In recent experiments, localized and stationary optical wave packets have been generated in second-order nonlinear processes with femtosecond pulses, whose asymptotic features relate to those of nondiffracting and nondispersing polychromatic Bessel beams in linear dispersive media. We investigate the nature of these linear waves and show that they can be identified with the X-shaped (O-shaped) modes of the hyperbolic (elliptic) wave equation in media with normal (anomalous) dispersion. Depending on the relative strengths of mode phase mismatch, group velocity mismatch with respect to a plane pulse, and the defeated group velocity dispersion, these modes can adopt the form of pulsed Bessel beams, focus wave modes, and X waves (O waves), respectively.