Space-time wave packets violate the universal relationship between angular dispersion and pulse-front tilt

Universal angular-dispersion synthesizer

We uncover a surprising gap in optics with regards to angular dispersion (AD). A systematic examination of pulsed optical field configurations classified according to their three lowest dispersion orders resulting from AD (the axial phase velocity, group velocity, and group-velocity dispersion) reveals that the majority of possible classes of fields have eluded optics thus far. This gap is due in part to the limited technical reach of the standard components that provide AD such as gratings and prisms, but due in part also to misconceptions regarding the set of physically admissible field configurations that can be accessed via AD. For example, it has long been thought that AD cannot yield normal group-velocity dispersion in free space. We introduce a "universal AD synthesizer": a pulsed-beam shaper that produces a wavelength-dependent propagation angle with arbitrary spectral profile, thereby enabling access to all physically admissible field configurations realizable via AD. This universal AD synthesizer is a versatile tool for preparing pulsed optical fields for dispersion cancellation, optical signal processing, and nonlinear optics.

Non-differentiable angular dispersion as an optical resource

Introducing angular dispersion into a pulsed field associates each frequency with a particular angle with respect to the propagation axis. A perennial yet implicit assumption is that the propagation angle is differentiable with respect to the frequency. Recent work on space-time wave packets has shown that the existence of a frequency at which the derivative of the propagation angle does not exist-which we refer to as non-differentiable angular dispersion-allows for the optical field to exhibit unique and useful characteristics that are unattainable by endowing optical fields with conventional angular dispersion. Because these novel, to the best of our knowledge, features are retained in principle even when the specific non-differentiable frequency is not part of the selected spectrum, the question arises as to the impact of the proximity of the spectrum to this frequency. We show here that operating in the vicinity of the non-differentiable frequency is imperative to reduce the deleterious impact of (1) errors in implementing the angular-dispersion profile and (2) the spectral uncertainty intrinsic to finite-energy wave packets in any realistic system. Non-differential angular dispersion can then be viewed as a resource-quantified by a Schmidt number-that is maximized in the vicinity of the non-differentiable frequency. These results will be useful in designing novel phase-matching of nonlinear interactions in dispersive media.

Investigating group-velocity-tunable propagation-invariant optical wave-packets

The group-velocity of the propagation-invariant optical wave-packet generated by the conical superposition can be controlled by introducing well-designed arbitrarily-axisymmetric pulse-front deformation, which permits realizing superluminal, subluminal, accelerating, decelerating, and even nearly-programmable group-velocities. To better understand the tunability of the group-velocity, the generation methods of this propagation-invariant optical wave-packet and the mechanisms of the tunable group-velocity in both the physical and Fourier spaces are investigated. We also have studied the relationship with the recently-reported space–time wave-packet, and this group-velocity-tunable propagation-invariant optical wave-packet should be a subset of the space–time wave-packet.

Space-time wave packets localized in all dimensions

Optical wave packets that are localized in space and time, but nevertheless overcome diffraction and travel rigidly in free space, are a long sought-after field structure with applications ranging from microscopy and remote sensing, to nonlinear and quantum optics. However, synthesizing such wave packets requires introducing non-differentiable angular dispersion with high spectral precision in two transverse dimensions, a capability that has eluded optics to date. Here, we describe an experimental strategy capable of sculpting the spatio-temporal spectrum of a generic pulsed beam by introducing arbitrary radial chirp via two-dimensional conformal coordinate transformations of the spectrally resolved field. This procedure yields propagation-invariant 'space-time' wave packets localized in all dimensions, with tunable group velocity in the range from 0.7c to 1.8c in free space, and endowed with prescribed orbital angular momentum. By providing unprecedented flexibility in sculpting the three-dimensional structure of pulsed optical fields, our experimental strategy promises to be a versatile platform for the emerging enterprise of space-time optics.

Consequences of non-differentiable angular dispersion in optics: tilted pulse fronts versus space-time wave packets

Conventional diffractive and dispersive devices introduce angular dispersion (AD) into pulsed optical fields, thus producing so-called 'tilted pulse fronts'. Naturally, it is always assumed that the functional form of the wavelength-dependent propagation angle[s] associated with AD is differentiable with respect to wavelength. Recent developments in the study of space-time wave packets - pulsed beams in which the spatial and temporal degrees of freedom are inextricably intertwined - have pointed to the existence of non-differentiable AD: field configurations in which the propagation angle does not possess a derivative at some wavelength. Here we investigate the consequences of introducing non-differentiable AD into a pulsed field and show that it is the crucial ingredient required to realize group velocities that deviate from c (the speed of light in vacuum) along the propagation axis in free space. In contrast, the on-axis group velocity for conventional pulsed fields in free space is always equal to c. Furthermore, we show that non-differentiable AD is needed for realizing anomalous or normal group-velocity dispersion along the propagation axis, while simultaneously suppressing all higher-order dispersion terms. We experimentally verify these and several other consequences of non-differentiable AD using a pulsed-beam shaper capable of introducing AD with arbitrary spectral profile. Non-differentiable AD is not an exotic phenomenon, but is rather an accessible, robust, and versatile resource for sculpting pulsed optical fields.

Realizing normal group-velocity dispersion in free space via angular dispersion

It has long been thought that normal group-velocity dispersion (GVD) cannot be produced in free space via angular dispersion. Indeed, conventional diffractive or dispersive components such as gratings or prisms produce only anomalous GVD. We identify the conditions that must be fulfilled by the angular dispersion introduced into a plane-wave pulse to yield normal GVD. We then utilize a pulsed-beam shaper capable of introducing arbitrary angular-dispersion profiles to symmetrically produce normal and anomalous GVD in free space, which are realized here on the same footing for the first time, to our knowledge.

Refraction of space-time wave packets: I. theoretical principles

Space-time (ST) wave packets are pulsed optical beams endowed with precise spatio-temporal structure by virtue of which they exhibit unique and useful characteristics such as propagation invariance and tunable group velocity. We study in detail here, and in two accompanying papers, the refraction of ST wave packets at planar interfaces between non-dispersive, homogeneous, and isotropic dielectrics. We formulate a law of refraction that determines the change in the ST wave-packet group velocity across such an interface as a consequence of a newly identified optical refractive invariant that we call the "spectral curvature". Because the spectral curvature vanishes in conventional optical fields where the spatial and temporal degrees of freedom are separable, these phenomena have not been observed to date. We derive the laws of refraction for baseband, X wave, and sideband ST wave packets that reveal fascinating refractive phenomena, especially for the former class of wave packets. We predict theoretically, and confirm experimentally in the accompanying papers, refractive phenomena such as group-velocity invariance (ST wave packets whose group velocity does not change across the interface), anomalous refraction (group-velocity increase in higher-index media), group-velocity inversion (change in the sign of the group velocity upon refraction but not its magnitude), and the dependence of the group velocity of the refracted ST wave packet on the angle of incidence.

Refraction of space-time wave packets: III. experiments at oblique incidence

The refraction of space-time (ST) wave packets at planar interfaces between non-dispersive, homogeneous, isotropic dielectrics exhibits fascinating phenomena, even at normal incidence. Examples of such refractive phenomena include group-velocity invariance across the interface, anomalous refraction, and group-velocity inversion. Crucial differences emerge at oblique incidence with respect to the results established at normal incidence. For example, the group velocity of the refracted ST wave packet can be tuned simply by changing the angle of incidence. In the third paper, we present experimental verification of the refractive phenomena exhibited by ST wave packets at oblique incidence that were in the first paper of this sequence [J. Opt. Soc. Am. A38, 1409 (2021)10.1364/JOSAA.430105]. We also examine a proposal for "blind synchronization," whereby identical ST wave packets arrive simultaneously at different receivers without a priori knowledge of their locations except that they are all located at the same depth beyond an interface between two media. A first proof-of-principle experimental demonstration of this effect is provided.

Temporal Talbot effect in free space

The temporal Talbot effect refers to the periodic revivals of a pulse train propagating in a dispersive medium and is a temporal analog of the spatial Talbot effect with group-velocity dispersion in time replacing diffraction in space. Because of typically large temporal Talbot lengths, this effect has been observed to date in only single-mode fibers, rather than with freely propagating fields in bulk dispersive media. Here we demonstrate for the first time, to the best of our knowledge, the temporal Talbot effect in free space by employing dispersive space-time wave packets, whose spatiotemporal structure induces group-velocity dispersion of controllable magnitude and sign in free space.

Isochronous space-time wave packets

The group delay incurred by an optical wave packet depends on its path length. Therefore, when a wave packet is obliquely incident on a planar homogeneous slab, the group delay upon traversing it inevitably increases with the angle of incidence. Here, we confirm the existence of isochronous "space-time" (ST) wave packets: pulsed beams whose spatiotemporal structure enables them to traverse the layer with a fixed group delay over a wide span of incident angles. This unique behavior stems from the dependence of the group velocity of a refracted ST wave packet on its angle of incidence. Isochronous ST wave packets are observed in slabs of optical materials with indices ranging from 1.38 to 2.5 for angles up to 50° away from normal incidence.