Classification of (3+1)D propagation-invariant space-time wave packets in free space: extension of the 10-class scheme and emergence of new classes

A 2019 experimental and theoretical study organized (2+1)D propagation-invariant 'space-time' (ST) wave packets — pulsed optical light sheets whose spatio-temporal spectra lie on the intersection of the light cone with tilted spectral hyperplanes — into 10 classes indexed by the magnitude and sign of their freely tunable group velocity and by whether low spatial frequencies are allowed (baseband) or forbidden (sideband). That study left open whether this classification extends to (3+1)D volumetric wave packets, whether genuinely new classes emerge in higher dimensions, and how such packets could be synthesized. The sibling open questions raised alongside it — the Lorentz-boost interpretation of class transitions, classical entanglement of continuous degrees of freedom, and synthesis with broadband incoherent light including a tunable 'speed of coherence' — have since been substantially settled by follow-up work. The surviving open core is the classification problem itself: establish the criteria under which a (3+1)D spatio-temporal spectral support on the light cone yields propagation invariance, enumerate the resulting classes (including any with no (2+1)D counterpart, arising for example from non-planar spectral loci, azimuthal/orbital-angular-momentum structure, or polarization structure), settle the conjecture that the 10-class scheme is exhaustive under the replacement of the transverse spatial frequency by its radial magnitude, and experimentally synthesize and characterize a representative of every class.

Background

A pulsed optical beam normally spreads as it propagates: diffraction widens its transverse profile and dispersion lengthens its pulse. 'Space-time' (ST) wave packets evade both by introducing tight, engineered correlations between the spatial and temporal degrees of freedom of the field. In the spatio-temporal spectral domain, every plane-wave component of a free-space field lies on the light cone, the dispersion surface $k_x^2+k_z^2=(\omega/c)^2$ relating transverse spatial frequency, axial spatial frequency $k_z$, and temporal frequency $\omega$. If each temporal frequency $\omega$ is paired with one prescribed spatial frequency so that the axial wave number is linear in frequency, $k_z=k_0+\Omega/\tilde{v}$ (with $\Omega=\omega-\omega_0$), all components stay in phase and the packet propagates rigidly — it is propagation-invariant — at a group velocity $\tilde{v}$ that can be tuned at will: subluminal, superluminal, or even negative (traveling backwards). Geometrically, such spectra occupy the intersection of the light cone with a spectral hyperplane tilted by an angle $\theta$ relative to the $k_z$ axis, and this tilt is a form of non-differentiable angular dispersion — each wavelength propagates at its own angle, with no common pulse front, so the wavelength-to-angle mapping cannot be smoothed into an ordinary differentiable dispersion law. Building on the diffraction-free ST light sheets of Kondakci and Abouraddy (Nat. Photon. 11, 733 (2017)), Yessenov, Bhaduri, Kondakci, and Abouraddy (Phys. Rev. A 99, 023856 (2019)) gave a complete classification of $(2+1)$D ST wave packets (one transverse dimension plus time, i.e. light sheets): 10 classes determined by three criteria — the magnitude of the group velocity (subluminal, luminal, superluminal), its sign (forward or backward), and whether the low spatial frequencies are physically allowed ('baseband', whose spectral hyperplane passes through the carrier point $(k_x,k_z,\omega/c)=(0,k_0,\omega_0/c)$ on the light cone, so that $k_x=0$ is admitted) or forbidden ('sideband', whose hyperplane misses the low-frequency region, as in X-shaped waves — conical pulsed fields whose spectral hyperplane passes through the origin rather than through the carrier point) — and they experimentally synthesized all 10 classes using a spatial-light-modulator-based spatio-temporal Fourier synthesis system.

The same paper asked whether this classification extends to $(3+1)$D volumetric wave packets (two transverse dimensions plus time), where the spectral constraint requires each temporal frequency to be paired with a single radial spatial frequency $k_{\mathrm{T}}=(k_x^2+k_y^2)^{1/2}$ — non-differentiable radial angular dispersion in two transverse dimensions, a capability that did not then exist. The authors conjectured that the classification is in fact exhaustive, since the extension formally amounts to exchanging $|k_x|$ for $k_{\mathrm{T}}$, but they could not see how to generalize their synthesis strategy. Since then, the experimental barrier has been broken for one class: Yessenov et al. (Nat. Commun. 13, 4573 (2022)) synthesized propagation-invariant wave packets localized in all dimensions with tunable group velocity from $0.7c$ to $1.8c$ and endowed with orbital angular momentum (OAM, the helical phase structure associated with Laguerre-Gaussian-type modes), using two-dimensional conformal coordinate transformations to imprint arbitrary radial chirp; Pang et al. (Opt. Lett. 46, 4678 (2021)) simulated OAM-carrying near-diffraction-free pulses from comb sources. At the same time, results have accumulated that sit outside the tilted-plane taxonomy: Béjot and Kibler (ACS Photonics 9, 2066 (2022)) produced invariant ST wave packets whose spectral loci are quadric surfaces rather than planes; vector (polarization-structured) ST wave packets have been synthesized (Opt. Lett. 47, 4131 (2022); Opt. Express 29, 37225 (2021)); propagation-invariant spatiotemporal vortices were reported in 2025; and Almeida et al. (Opt. Lett. 50, 3393 (2025)) showed that ST wave packets and the 'flying focus' share one universal propagation-invariant spatiotemporal structure. No published work, however, enumerates the $(3+1)$D classes or tests the exhaustiveness conjecture, so the classification of volumetric propagation-invariant packets remains an open question with one class experimentally realized.

Problem Statement

Determine whether the classification of $(2+1)$D propagation-invariant space-time wave packets in free space into 10 classes extends to $(3+1)$D volumetric wave packets, and whether genuinely new classes emerge in higher dimensions. Concretely, with the same theoretical and experimental completeness standard that the source paper achieved for $(2+1)$D: (i) establish the criteria under which a $(3+1)$D spatio-temporal spectral support on the light cone — each temporal frequency paired with a prescribed radial spatial frequency $k_{\mathrm{T}}=(k_x^2+k_y^2)^{1/2}$ — yields a propagation-invariant wave packet, including the magnitude and sign of the group velocity, whether low transverse spatial frequencies are allowed (baseband) or forbidden (sideband), and any criteria that are genuinely three-dimensional, such as the geometry of the spectral locus (planar versus non-planar, e.g. quadric) and azimuthal (orbital-angular-momentum) or polarization structure of the spectrum; (ii) enumerate the resulting classes, show they are distinct, and settle the source's conjecture that replacing $|k_x|$ by $k_{\mathrm{T}}$ renders the $(2+1)$D classification exhaustive in $(3+1)$D — or else exhibit classes with no $(2+1)$D counterpart; and (iii) experimentally synthesize and characterize a representative of every class, verifying propagation invariance and the prescribed group velocity. These three requested results jointly support one overarching objective — a complete classification of volumetric propagation-invariant space-time wave packets in free space — and the problem is complete when that classification is established, since each component (criteria, enumeration with exhaustiveness, and experimental realization of the classes) is a load-bearing part of what 'classification' meant in the source work.

References

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