Optical space-time wave packets having arbitrary group velocities in free space

Spatiotemporal hologram

Spatiotemporal structured light has opened up new avenues for optics and photonics. Current spatiotemporal manipulation of light mostly relies on phase-only devices such as liquid crystal spatial light modulators to generate spatiotemporal optical fields with unique photonic properties. However, simultaneous manipulation of both amplitude and phase of the complex field for the spatiotemporal light is still lacking, limiting the diversity and richness of achievable photonic properties. In this work, a simple and versatile spatiotemporal holographic method that can arbitrarily sculpt the spatiotemporal light is presented. The capabilities of this simple yet powerful method are demonstrated through the generation of fundamental and higher-order spatiotemporal Bessel wavepackets, spatiotemporal crystal-like and quasi-crystal-like structures, and spatiotemporal flat-top wavepackets. Fully customizable spatiotemporal wavepackets will find broader application in investigating the dynamics of spatiotemporal fields and interactions between ultrafast spatiotemporal pulses and matters, unveiling previously hidden light-matter interactions and unlocking breakthroughs in photonics and beyond.

Exciting space-time surface plasmon polaritons by irradiating a nanoslit structure

Space-time (ST) wave packets are propagation-invariant pulsed optical beams that travel freely in dielectrics at a tunable group velocity without diffraction or dispersion. Because ST wave packets maintain these characteristics even when only one transverse dimension is considered, they can realize surface-bound waves (e.g., surface plasmon polaritons at a metal-dielectric interface, which we call ST-SPPs) that have the same unique characteristics as their freely propagating counterparts. However, because the spatiotemporal spectral structure of ST-SPPs is key to their propagation invariance on the metal surface, their excitation methodology must be considered carefully. Using finite-difference time-domain simulations, we show that an appropriately synthesized ST wave packet in free space can be coupled to an ST-SPP via a single nanoscale slit inscribed in the metal surface. Our calculations confirm that this excitation methodology yields surface-bound ST-SPPs that are localized in all dimensions (and can thus be considered as plasmonic "bullets"), which travel rigidly at the metal-dielectric interface without diffraction or dispersion at a tunable group velocity.

Space-Time Structured Plasma Waves

Electrostatic waves play a critical role in nearly every branch of plasma physics from fusion to advanced accelerators, to astro, solar, and ionospheric physics. The properties of planar electrostatic waves are fully determined by the plasma conditions, such as density, temperature, ionization state, or details of the distribution functions. Here we demonstrate that electrostatic wave packets structured with space-time correlations can have properties that are independent of the plasma conditions. For instance, an appropriately structured electrostatic wave packet can travel at any group velocity, even backward with respect to its phase fronts, while maintaining a localized energy density. These linear, propagation-invariant wave packets can be constructed with or without orbital angular momentum by superposing natural modes of the plasma and can be ponderomotively excited by space-time structured laser pulses like the flying focus.

Propagation-Invariant Space-Time Plasmonic Pulse in Subwavelength MIM Waveguide

The metal-insulator-metal (MIM) plasmonic waveguide has been highly anticipated for confining and guiding surface plasmon polaritons (SPPs) on the subwavelength scale. However, perennial drawbacks such as a short propagation length and an unbounded transverse field have set limits on the use of the MIM waveguide in various applications. Herein, diffraction- and dispersion-free MIM modes are synthesized by using space-time wave packets (STWPs) and are therefore referred to as space-time MIM (ST-MIM) waveguide modes. Compared to a Gaussian pulse of the same duration and spectral bandwidth, the ST-MIM demonstrates enhanced propagation lengths of about 2.4 times for the symmetric mode and about 6.3 times for the antisymmetric mode. In the simulations, the ST-MIMs are confined in all transverse dimensions, thereby overriding the diffraction limits. In addition, the group velocities of the ST-MIMs can be arbitrarily designed, which makes it possible to synchronize the pulse propagation speeds of the symmetric and antisymmetric MIM modes.

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.

Ultrabroadband flying-focus using an axiparabola-echelon pair

Flying-focus pulses promise to revolutionize laser-driven secondary sources by decoupling the trajectory of the peak intensity from the native group velocity of the medium over distances much longer than a Rayleigh range. Previous demonstrations of the flying focus have either produced an uncontrolled trajectory or a trajectory that is engineered using chromatic methods that limit the duration of the peak intensity to picosecond scales. Here we demonstrate a controllable ultrabroadband flying focus using a nearly achromatic axiparabola-echelon pair. Spectral interferometry using an ultrabroadband superluminescent diode was used to measure designed super- and subluminal flying-focus trajectories and the effective temporal pulse duration as inferred from the measured spectral phase. The measurements demonstrate that a nearly transform- and diffraction-limited moving focus can be created over a centimeter-scale-an extended focal region more than 50 Rayleigh ranges in length. This ultrabroadband flying-focus and the novel axiparabola-echelon configuration used to produce it are ideally suited for applications and scalable to >100 TW peak powers.

Space-time wave packets with reduced divergence and tunable group velocity generated in free space after multi-mode fiber propagation

Previously, space-time wave packets (STWPs) have been generated in free space with reduced diffraction and a tunable group velocity by combining multiple frequency comb lines each carrying a single Bessel mode with a unique wave number. It might be potentially desirable to propagate the STWP through fiber for reconfigurable positioning. However, fiber mode coupling might degrade the output STWP and distort its propagation characteristics. In this Letter, we experimentally demonstrate STWP generation and propagation over 1-m graded-index multi-mode fiber. Fiber mode coupling is mitigated by pre-distortion according to the inverse matrix of the fiber mode coupling matrix. Measurement of the STWP at the fiber output shows that its group velocity can vary from 1.0042c to 0.9967c by tuning the wave number of the Bessel mode on each frequency. The measured time-averaged intensity profiles show that the beam radius remains similar after 150-mm free-space propagation after exiting the fiber.

The propagation speed of optical speckle

That the speed of light in vacuum is constant is a cornerstone of modern physics. However, recent experiments have shown that when the light field is confined in the transverse plane, the observed propagation speed of the light is reduced. This effect is a consequence of the transverse structure which reduces the component of wavevector of the light in the direction of propagation, thereby modifying both the phase and group velocity. Here, we consider the case of optical speckle, which has a random transverse distribution and is ubiquitous with scales ranging from the microscopic to the astronomical. We numerically investigate the plane-to-plane propagation speed of the optical speckle by using the method of angular spectrum analysis. For a general diffuser with Gaussian scattering over an angular range of 5°, we calculate the slowing of the propagation speed of the optical speckle to be on the order of 1% of the free-space speed, resulting in a significantly higher temporal delay compared to the Bessel and Laguerre-Gaussian beams considered previously. Our results have implications for studying optical speckle in both laboratory and astronomical settings.

Theory of space-time supermodes in planar multimode waveguides

When an optical pulse is focused into a multimode waveguide or fiber, the energy is divided among the available guided modes. Consequently, the initially localized intensity spreads transversely, the spatial profile undergoes rapid variations with axial propagation, and the pulse disperses temporally. Space-time (ST) supermodes are pulsed guided field configurations that propagate invariantly in multimode waveguides by assigning each mode to a prescribed wavelength. ST supermodes can be thus viewed as spectrally discrete, guided-wave counterparts of the recently demonstrated propagation-invariant ST wave packets in free space. The group velocity of an ST supermode is tunable independently-in principle-of the waveguide structure, group-velocity dispersion is eliminated or dramatically curtailed, and the time-averaged intensity profile is axially invariant along the waveguide in absence of mode-coupling. We establish here a theoretical framework for studying ST supermodes in planar waveguides. Modal engineering allows sculpting this axially invariant transverse intensity profile from an on-axis peak or dip (dark beam) to a multi-peak or flat distribution. Moreover, ST supermodes can be synthesized using spectrally incoherent light, thus paving the way to potential applications in optical beam delivery for lighting applications.

Ultra-compact synthesis of space-time wave packets

Space-time wave packets (STWPs) are pulsed fields in which a strictly prescribed association between the spatial and temporal frequencies yields surprising and useful behavior. However, STWPs to date have been synthesized using bulky free-space optical systems that require precise alignment. We describe a compact system that makes use of a novel optical component: a chirped volume Bragg grating that is rotated by 45° with respect to the plane-parallel device facets. By virtue of this grating's unique structure, cascaded gratings resolve and recombine the spectrum without free-space propagation or collimation. We produce STWPs by placing a phase plate that spatially modulates the resolved spectrum between such cascaded gratings, with a device volume of 25 × 25 × 8 mm³, which is orders-of-magnitude smaller than previous arrangements.

Space-time wave packets with both arbitrary transverse and longitudinal accelerations

The group velocity in the free space of space-time wave packets (STWPs) and light bullets can be flexibly regulated by many advanced strategies; however, these regulations are restricted to only the longitudinal group velocity. In this work, a computational model based on catastrophe theory is proposed, to devise STWPs with both arbitrary transverse and longitudinal accelerations. In particular, we investigate the attenuation-free Pearcey-Gauss STWP, which enriches the family of non-diffracting STWPs. This work may advance the development of space-time structured light fields.

Rotated chirped volume Bragg gratings for compact spectral analysis

We introduce a new, to the best of our knowledge, optical component-a rotated chirped volume Bragg grating (r-CBG)-that spatially resolves the spectrum of a normally incident light beam in a compact footprint and without the need for subsequent free-space propagation or collimation. Unlike conventional chirped volume Bragg gratings in which both the length and width of the device must be increased to increase the bandwidth, by rotating the Bragg structure we sever the link between the length and width of a r-CBG, leading to a significantly reduced device footprint for the same bandwidth. We fabricate and characterize such a device in multiple spectral windows, we study its spectral resolution, and confirm that a pair of cascaded r-CBGs can resolve and then recombine the spectrum. Such a device can lead to ultracompact spectrometers and pulse modulators.

Generation of OAM-carrying space-time wave packets with time-dependent beam radii using a coherent combination of multiple LG modes on multiple frequencies

Space-time (ST) wave packets, in which spatial and temporal characteristics are coupled, have gained attention due to their unique propagation characteristics, such as propagation invariance and tunable group velocity in addition to their potential ability to carry orbital angular momentum (OAM). Through experiment and simulation, we explore the generation of OAM-carrying ST wave packets, with the unique property of a time-dependent beam radius at various ranges of propagation distances. To achieve this, we synthesize multiple frequency comb lines, each assigned to a coherent combination of multiple Laguerre-Gaussian (LG_ℓ,p) modes with the same azimuthal index but different radial indices. The time-dependent interference among the spatial modes at the different frequencies leads to the generation of the desired OAM-carrying ST wave packet with dynamically varying radii. The simulation results indicate that the dynamic range of beam radius oscillations increases with the number of modes and frequency lines. The simulated ST wave packet for OAM of orders +1 or +3 has an OAM purity of >95%. In addition, we experimentally generate and measure the OAM-carrying ST wave packets with time-dependent beam radii. In the experiment, several lines of a Kerr frequency comb are spatially modulated with the superposition of multiple LG modes and combined to generate such an ST wave packet. In the experiment, ST wave packets for OAM of orders +1 or +3 have an OAM purity of >64%. In simulation and experiment, OAM purity decreases and beam radius becomes larger over the propagation.

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.

Excitation of surface plasmon polaritons by diffraction-free and vector beams

Surface plasmon polaritons (SPPs) are traditionally excited by plane waves within the Rayleigh range of a focused transverse-magnetic (TM) Gaussian beam. Here we investigate and confirm the coupling between SPPs and two-dimensional Gaussian and Bessel-Gauss wave packets, as well as one-dimensional light sheets and space-time wave packets. We encode the incoming wavefronts with spatially varying states of polarization; then we couple the respective TM components of radial and azimuthal vector beam profiles to confirm polarization-correlation and spatial-mode selectivity. Our results do not require material optimization or multi-dimensional confinement via periodically corrugated metal surfaces to achieve coupling at a greater extent, hereby outlining a pivotal, yet commonly overlooked, path towards the development of long-range biosensors and all-optical integrated plasmonic circuits.

Demonstration of speckle resistance using space-time light sheets

The capacity of self-healing fields to reconstruct after passing through scattering media may prove useful in reducing speckle formation. Here, we study the speckle response of the space-time (ST) light sheet compared to a Gaussian wave packet, Airy beam, and Bessel Gauss beam. We find that the Pearson's correlation coefficient for the ST light sheet is 50%, 48% and 40% larger than that of the Gaussian, Airy beam and Bessel Gauss beams, respectively, demonstrating a strong correlation to an input beam that has not been speckled. These results suggest that the ST light sheet exhibits considerable resistance to speckle generation. We also investigate the speckle response of the ST light sheet at its second-harmonic frequency and observe a mean Pearson's correlation coefficient close to 0.6, comparable to the second-harmonic Bessel Gauss beam, and 2.8 × the value obtained for the second-harmonic Gaussian beam. Our results lend themselves to a variety of applications including bioimaging, communications, and optical tweezers.

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.

Spatial resolution of omni-resonant imaging

Omni-resonance refers to the broadening of the spectral transmission through a planar cavity, not by changing the cavity structure, but by preconditioning the incident optical field. As such, broadband imaging can be performed through such a cavity with all the wavelengths simultaneously resonating. We examine here the spatial resolution of omni-resonant imaging and find that the spectral linewidth of the cavity resonance determines the spatial resolution. Surprisingly, the spatial resolution improves at longer wavelengths because of the negative angular dispersion intrinsic to Fabry-Pérot resonances, in contrast to conventional diffraction-limited optical imaging systems where the spatial resolution improves at shorter wavelengths. These results are important for applications ranging from transparent solar windows to nonlinear resonant image processing.

Time diffraction-free transverse orbital angular momentum beams

The discovery of optical transverse orbital angular momentum (OAM) has broadened our understanding of light and is expected to promote optics and other physics. However, some fundamental questions concerning the nature of such OAM remain, particularly whether they can survive from observed mode degradation and hold OAM values higher than 1. Here, we show that the strong degradation actually origins from inappropriate time-delayed kx-ω modulation, instead, for transverse OAM having inherent space-time coupling, immediate modulation is necessary. Thus, using immediate x-ω modulation, we demonstrate theoretically and experimentally degradation-free spatiotemporal Bessel (STB) vortices with transverse OAM even beyond 102. Remarkably, we observe a time-symmetrical evolution, verifying pure time diffraction on transverse OAM beams. More importantly, we quantify such nontrivial evolution as an intrinsic dispersion factor, opening the door towards time diffraction-free STB vortices via dispersion engineering. Our results may find analogues in other physical systems, such as surface plasmon-polaritons, superfluids, and Bose-Einstein condensates.

Prism-based approach to create intensity-interferometric non-diffractive cw light sheets

Light sheets are optical beam-like fields with one-dimensional intensity localization. Ideally, the field intensity should be independent of the longitudinal and one of the transverse coordinates, which is difficult to achieve even for truncated light sheets. In this work, we present a general theoretical framework for intensity-interferometric continuous wave (cw) light sheets formed by overlapping the interference fringe patterns of mutually uncorrelated frequency components of the field. We show that the key parameters of the light sheets can be calculated using simple analytical expressions. We propose a practical way to generate such light sheets with the help of prisms and demonstrate numerically the abilities of the method. Both bright and dark light sheets with an exceptionally small thickness and long divergence-free propagation distance are possible to generate. We also show that the transverse profile of the generated light sheets can be shaped by modifying the spectrum of the light. We believe our findings advance the beam-engineering technology and its applications.

Synthesis of near-diffraction-free orbital-angular-momentum space-time wave packets having a controllable group velocity using a frequency comb

Novel forms of light beams carrying orbital angular momentum (OAM) have recently gained interest, especially due to some of their intriguing propagation features. Here, we experimentally demonstrate the generation of near-diffraction-free two-dimensional (2D) space-time (ST) OAM wave packets (ℓ = +1, +2, or +3) with variable group velocities in free space by coherently combining multiple frequency comb lines, each carrying a unique Bessel mode. Introducing a controllable specific correlation between temporal frequencies and spatial frequencies of these Bessel modes, we experimentally generate and detect near-diffraction-free OAM wave packets with high mode purities (>86%). Moreover, the group velocity can be controlled from 0.9933c to 1.0069c (c is the speed of light in vacuum). These ST OAM wave packets might find applications in imaging, nonlinear optics, and optical communications. In addition, our approach might also provide some insights for generating other interesting ST beams.

Ultrabright Electron Bunch Injection in a Plasma Wakefield Driven by a Superluminal Flying Focus Electron Beam

We propose a new method for self-injection of high-quality electron bunches in the plasma wakefield structure in the blowout regime utilizing a "flying focus" produced by a drive beam with an energy chirp. In a flying focus the speed of the density centroid of the drive bunch can be superluminal or subluminal by utilizing the chromatic dependence of the focusing optics. We first derive the focal velocity and the characteristic length of the focal spot in terms of the focal length and an energy chirp. We then demonstrate using multidimensional particle-in-cell simulations that a wake driven by a superluminally propagating flying focus of an electron beam can generate GeV-level electron bunches with ultralow normalized slice emittance (∼30  nm rad), high current (∼17  kA), low slice energy spread (∼0.1%), and therefore high normalized brightness (>10^{19}  A/m^{2}/rad^{2}) in a plasma of density ∼10^{19}  cm^{-3}. The injection process is highly controllable and tunable by changing the focal velocity and shaping the drive beam current. Near-term experiments at FACET II where the capabilities to generate tens of kA, <10  fs drivers are planned, could potentially produce beams with brightness near 10^{20}  A/m^{2}/rad^{2}.

Refraction of space-time wave packets in a dispersive medium

Space-time (ST) wave packets are a class of pulsed optical beams whose spatiotemporal spectral structure results in propagation invariance, tunable group velocity, and anomalous refractive phenomena. Here, we investigate the refraction of ST wave packets normally incident onto a planar interface between two dispersive, homogeneous, isotropic media. We formulate a new, to the best of our knowledge, refractive invariant for ST wave packets in this configuration, from which we obtain a law of refraction that determines the change in their group velocity across the interface. We verify this new refraction law in ZnSe and CdSe, both of which manifest large chromatic dispersion at near-infrared frequencies in the vicinity of their band edges. ST wave packets can thus be utilized in nonlinear optics for bridging large group-velocity mismatches in highly dispersive scenarios.

Spatiotemporal control of laser intensity through cross-phase modulation

Spatiotemporal pulse shaping provides control over the trajectory and range of an intensity peak. While this control can enhance laser-based applications, the optical configurations required for shaping the pulse can constrain the transverse or temporal profile, duration, or orbital angular momentum (OAM). Here we present a novel technique for spatiotemporal control that mitigates these constraints by using a "stencil" pulse to spatiotemporally structure a second, primary pulse through cross-phase modulation (XPM) in a Kerr lens. The temporally shaped stencil pulse induces a time-dependent focusing phase within the primary pulse. This technique, the "flying focus X," allows the primary pulse to have any profile or OAM, expanding the flexibility of spatiotemporal pulse shaping for laser-based applications. As an example, simulations show that the flying focus X can deliver an arbitrary-velocity, variable-duration intensity peak with OAM over distances much longer than a Rayleigh range.

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.

Arbitrarily accelerating space-time wave packets

All known realizations of optical wave packets that accelerate along their propagation axis, such as Airy wave packets in dispersive media or wave-front-modulated X-waves, exhibit a constant acceleration; that is, the group velocity varies linearly with propagation. Here we synthesize space-time wave packets that travel in free space with arbitrary axial acceleration profiles, including group velocities that change with integer or fractional exponents of the distance. Furthermore, we realize a composite acceleration profile: the wave packet accelerates from an initial to a terminal group velocity, before decelerating back to the initial value. These never-before-seen optical-acceleration phenomena are produced using the same experimental arrangement that precisely sculpts the wave packet's spatio-temporal spectral structure.

Space-time vector light sheets

We introduce the space-time (ST) vector light sheet. This unique one-dimensional ST wave packet is characterized by classical entanglement (CE), a correlation between at least two non-separable intrinsic degrees-of-freedom (DoFs), which in this case are the spatiotemporal DoFs in parallel with the spatial-polarization DoFs. We experimentally confirm that the ST vector light sheet maintains the intrinsic features of the uniformly polarized ST light sheet, such as near-diffraction-free propagation and self-healing, while also maintaining the intrinsic polarization structure of common vector beams, such as those that are radially polarized and azimuthally polarized. We also show that the vector beam structure of the ST vector light sheet is maintained in the subluminal and superluminal regimes.

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.

Space-Time Wave Packets from Smith-Purcell Radiation

Space-time wave packets are electromagnetic waves with strong correlations between their spatial and temporal degrees of freedom. These wave packets have gained much attention for fundamental properties like propagation invariance and user-designed group velocities, and for potential applications like optical microscopy, micromanipulation, and laser micromachining. Here, free-electron radiation is presented as a natural and versatile source of space-time wave packets that are ultra-broadband and highly tunable in frequency. For instance, ab initio theory and numerical simulations show that the intensity profile of space-time wave packets from Smith-Purcell radiation can be directly tailored through the grating properties, as well as the velocity and shape of the electron bunches. The result of this work indicates a viable way of generating space-time wave packets at exotic frequencies such as the terahertz and X-ray regimes, potentially paving the way toward new methods of shaping electromagnetic wave packets through free-electron radiation.

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.

Refraction of space-time wave packets: II. experiments at normal incidence

The refraction of space-time (ST) wave packets offers many fascinating surprises with respect to conventional pulsed beams. In the first paper in this sequence [J. Opt. Soc. Am. A38, 1409 (2021)10.1364/JOSAA.430105], we theoretically described the refraction of all families of ST wave packets at normal and oblique incidence at a planar interface between two nondispersive, homogeneous, isotropic dielectrics. Here, in this second paper in the sequence, we present experimental verification of the refractive phenomena predicted for baseband ST wave packets upon normal incidence on a planar interface. Specifically, we observe group velocity invariance, normal and anomalous refraction, and group velocity inversion leading to group delay cancellation. These phenomena are verified in a set of optical materials with refractive indices ranging from 1.38 to 1.76, including MgF₂, fused silica, BK7 glass, and sapphire. We also provide a geometrical representation of the physics associated with anomalous refraction in terms of the dynamics of the spectral support domain for ST wave packets on the surface of the light cone.

Simulation of near-diffraction- and near-dispersion-free OAM pulses with controllable group velocity by combining multiple frequencies, each carrying a Bessel mode

Optical pulses carrying orbital angular momentum (OAM) have recently gained interest. In general, it might be beneficial to simultaneously achieve: (i) minimum diffraction, (ii) minimum dispersion, and (iii) controllable group velocity. Here, we explore via simulation the generation of near-diffraction-free and near-dispersion-free OAM pulses with arbitrary group velocities by coherently combining multiple frequencies. Each frequency carries a specific Bessel mode with the same topological charge (ℓ) but different k_r (spatial frequency) values based on space-time correlations. Moreover, we also find that (i) both positive and negative group velocities could be achieved and continuously controlled from the subluminal to superluminal values and (ii) when the ℓ is varied from 0 to 10, the simulated value of the group velocity remains the same. However, as the ℓ value increases, the pulse duration becomes longer for a given number of frequency lines.

Propagation-invariant space-time caustics of light

Caustics are responsible for a wide range of natural phenomena, from rainbows and mirages to sparkling seas. Here, we present caustics in space-time wavepackets, a class of pulsed beams featuring strong coupling between spatial and temporal frequencies. Space-time wavepackets have attracted much attention with their propagation-invariant intensity profiles that travel at tunable superluminal and subluminal group velocities. These intensity profiles, however, have been largely restricted to an X-shape or similar pattern. We show that space-time caustics combine the propagation invariance of space-time wavepackets with the flexible design of caustics, allowing for customizable intensity patterns in space-time wavepackets. Our method directly provides the phase distribution needed to realize user-designed caustic patterns in space-time wavepackets. We show that space-time caustics can feature in a broad range of intriguing optical phenomena, including backward traveling caustics formed from purely forward propagating waves, and nondiffracting beams that evolve with time. Our findings should open the doors to an even wider range of structured light with spatiotemporal coupling.

Structured 3D linear space-time light bullets by nonlocal nanophotonics

We propose the generation of 3D linear light bullets propagating in free space using a single passive nonlocal optical surface. The nonlocal nanophotonics can generate space-time coupling without any need for bulky pulse-shaping and spatial modulation techniques. Our approach provides simultaneous control of various properties of the light bullets, including the external properties such as the group velocity and the propagation distance, and internal degrees of freedom such as the spin angular momentum and the orbital angular momentum.

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.

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

Introducing angular dispersion into a pulsed field tilts the pulse front with respect to the phase front. There exists between the angular dispersion and pulse-front tilt a universal relationship that is device-independent, and also independent of the pulse shape and bandwidth. We show here that this relationship is violated by propagation-invariant space-time (ST) wave packets, which are pulsed beams endowed with precise spatiotemporal structure corresponding to a particular form of angular dispersion. We demonstrate theoretically and experimentally that ST wave packets represent, to the best of our knowledge, the first example in optics of non-differentiable angular dispersion, resulting in pulse-front tilt that depends on the square-root of the pulse bandwidth.

Velocity property of an optical chain generated by the tightly focused femtosecond radially polarization pulse

Based on the Richards-Wolf vector diffraction integration, we obtained the expressions of the intensity and velocity of femtosecond radially polarized pulses at the focus near a dielectric interface, and the pulses are modulated by an optical system consisting of diffractive optical elements (DOEs) and a high numerical aperture (NA) lens. The factors that affected the intensity distribution and velocity evolution of the three-dimensional optical capture structural pulse (optical chain) are also analyzed. These factors include the DOE structural parameters (bandwidth, phase difference between the rings), the interception ratio of incident beam, the NA, the central wavelength of pulses, and the refractive index of exiting medium. The results show that the velocity of the optical chain will increase with an increase in the DOE bandwidth or a decrease in the refractive index of the exiting medium, and the maximum of the optical chain velocity will decrease versus the NA. Furthermore, the dependence of the optical chain velocity on its intensity distribution is also revealed. The superluminal and subluminal can also be found during the propagation of the optical chain. The velocity distribution difference between bright and dark areas along the z axis is more conducive to distinguishing the trapping of the Rayleigh particles. We believe these interesting results have great potential to improve the space-time resolution to detect particle positions during high-speed optical trapping.

Discretized Conical Waves in Multimode Optical Fibers

Multimode optical fibers are essential in bridging the gap between nonlinear optics in bulk media and single-mode fibers. The understanding of the transition between the two fields remains complex due to intermodal nonlinear processes and spatiotemporal couplings, e.g., some striking phenomena observed in bulk media with ultrashort pulses have not yet been unveiled in such waveguides. Here we generalize the concept of conical waves described in bulk media towards structured media, such as multimode optical fibers, in which only a discrete and finite number of modes can propagate. Such propagation-invariant optical wave packets can be linearly generated, in the limit of superposed monochromatic fields, by shaping their spatiotemporal spectrum, whatever the dispersion regime and waveguide geometry. Moreover, they can also spontaneously emerge when a rather intense short pulse propagates nonlinearly in a multimode waveguide, their finite energy is also associated with temporal dispersion. The modal distribution of optical fibers then provides a discretization of conical emission (e.g., discretized X waves). Future experiments in multimode fibers could reveal different forms of dispersion-engineered conical emission and supercontinuum light bullets.

Radial Structure of OAM-Carrying Fundamental X-Waves

We investigate the spectral degree of freedom of OAM-carrying localized waves and its influence on their transverse intensity distribution. In particular, we focus our attention on exponentially decaying spectra, which are very tightly connected to fundamental X-waves; we then show how it is possible to structure their transverse intensity distribution, thus creating a radial structure similar to that of Bessel beams.

Veiled Talbot Effect

A freely propagating optical field having a periodic transverse spatial profile undergoes periodic axial revivals-a well-known phenomenon known as the Talbot effect or self-imaging. We show here that introducing tight spatiotemporal spectral correlations into an ultrafast pulsed optical field with a periodic transverse spatial profile eliminates all axial dynamics in physical space, while revealing a novel veiled Talbot effect that can be observed only when carrying out time-resolved measurements. Indeed, "time diffraction" is observed, whereupon the temporal profile of the field envelope at a fixed axial plane corresponds to a segment of the spatial propagation profile of a monochromatic field sharing the initial spatial profile and observed at the same axial plane. Time averaging, which is intrinsic to observing the intensity, altogether veils this effect.

Hybrid guided space-time optical modes in unpatterned films

Light is confined transversely and delivered axially in a waveguide. However, waveguides are lossy static structures whose modal characteristics are fundamentally determined by their boundary conditions. Here we show that unpatterned planar waveguides can provide low-loss two-dimensional waveguiding by using space-time wave packets, which are unique one-dimensional propagation-invariant pulsed optical beams. We observe hybrid guided space-time modes that are index-guided in one transverse dimension and localized along the unbounded dimension. We confirm that these fields enable overriding the boundary conditions by varying post-fabrication the group index of the fundamental mode in a 2-μm-thick, 25-mm-long silica film, achieved by modifying the field’s spatio-temporal structure. Tunability of the group index over an unprecedented range from 1.26 to 1.77 is verified while maintaining a spectrally flat zero-dispersion profile. Our work paves the way to utilizing space-time wave packets in on-chip platforms, and enable phase-matching strategies that circumvent restrictions due to intrinsic material properties.

Waveguides typically function by using boundary conditions to contain light. Here, the authors show that by using space-time wavepackets, light can be guided in an unpatterned planar waveguide as the field remains localized along the unbounded dimension.

Accelerating and Decelerating Space-Time Optical Wave Packets in Free Space

Although a plethora of techniques are now available for controlling the group velocity of an optical wave packet, there are very few options for creating accelerating or decelerating wave packets whose group velocity varies controllably along the propagation axis. Here we show that "space-time" wave packets in which each wavelength is associated with a prescribed spatial bandwidth enable the realization of optical acceleration and deceleration in free space. Endowing the field with precise spatiotemporal structure leads to group-velocity changes as high as ∼c observed over a distance of ∼20  mm in free space, which represents a boost of at least ∼4 orders of magnitude over X waves and Airy pulses. The acceleration implemented is, in principle, independent of the initial group velocity, and we have verified this effect in both the subluminal and superluminal regimes.

Free-space optical delay line using space-time wave packets

An optical buffer featuring a large delay-bandwidth-product-a critical component for future all-optical communications networks-remains elusive. Central to its realization is a controllable inline optical delay line, previously accomplished via engineered dispersion in optical materials or photonic structures constrained by a low delay-bandwidth product. Here we show that space-time wave packets whose group velocity is continuously tunable in free space provide a versatile platform for constructing inline optical delay lines. By spatio-temporal spectral-phase-modulation, wave packets in the same or in different spectral windows that initially overlap in space and time subsequently separate by multiple pulse widths upon free propagation by virtue of their different group velocities. Delay-bandwidth products of  ~100 for pulses of width  ~1 ps are observed, with no fundamental limit on the system bandwidth.

Impact of frequency-dependent spherical aberration in the focusing of ultrashort pulses

In this paper, the temporal and spatial intensity pulse distributions are calculated around the focal region of an optical system using a combination of ray tracing and a wave propagation method. We analyze how to measure the width of the intensity pulse distributions to estimate pulse duration and spot size in order to study the impact of the variation of spherical aberration with frequency in a pulse on the intensity distributions. Two experimental techniques used in the laboratory are also modeled: the knife-edge test to measure spatial distribution and the intensity autocorrelation technique to measure the temporal distribution. We use two measuring criteria, the full-width half-maximum (FWHM) and standard deviation (σ), to compare the spatial and temporal intensity distributions of the calculated diffraction patterns and those obtained from the simulated experimental techniques. We show that the FWHM is not a good criterion, since it gives different results in the measured intensity distributions in time and space when they are measured directly from the theoretical modeling and when they are measured from the modeled experimental techniques used in the laboratory. The standard deviation, however, is a consistent criterion, giving the same results for the calculated intensity distributions and the modeled experiments.

Velocity and acceleration freely tunable straight-line propagation light bullet

Three-dimensional (3-D) light solitons in space–time, referred to as light bullets, have many novel properties and wide applications. Here we theoretically show how the combination of diffraction-free beam and ultrashort pulse spatiotemporal-coupling enables the creation of a straight-line propagation light bullet with freely tunable velocity and acceleration. This light bullet could propagate with a constant superluminal or subluminal velocity, and it could also counter-propagate with a very fast superluminal velocity (e.g., − 35.6c). Apart from uniform motion, an acceleration or deceleration straight-line propagation light bullet with a tunable instantaneous acceleration could also be produced. The high controllability of the velocity and the acceleration of a straight-line propagation light bullet would enable very specific applications, such as velocity and/or acceleration matched micromanipulation, microscopy, particle acceleration, radiation generation, and so on.

Omni-resonant space-time wave packets

We describe theoretically and verify experimentally a novel, to the best of our knowledge, class of diffraction-free pulsed optical beams that are "omni-resonant": they have the remarkable property of transmission through planar Fabry-Perot resonators without spectral filtering, even if their bandwidth far exceeds the cavity linewidth. Ultrashort wave packets endowed with a specific spatiotemporal structure couple to a single resonant mode independent of its linewidth. We confirm that such "space-time" omni-resonant wave packets retain their bandwidth (1.6 nm), spatiotemporal profile (1.3-ps pulse width, 4-µm beam width), and diffraction-free behavior upon transmission through cavities with resonant linewidths of 0.3 nm and 0.15 nm.

Controlling the velocity of a femtosecond laser pulse using refractive lenses

The combination of temporal chirp with a simple chromatic aberration known as longitudinal chromatism leads to extensive control over the velocity of laser intensity in the focal region of an ultrashort laser beam. We present the first implementation of this effect on a femtosecond laser. We demonstrate that by using a specially designed and characterized lens doublet to induce longitudinal chromatism, this velocity control can be implemented independent of the parameters of the focusing optic, thus allowing for great flexibility in experimental applications. Finally, we explain and demonstrate how this spatiotemporal phenomenon evolves when imaging the ultrashort pulse focus with a magnification different from unity.