Space-time Wave Packets Research Articles

Abrupt X-to-O-wave structural field transition in presence of anomalous dispersion

All linear, propagation-invariant, paraxial pulsed beams are spatiotemporally X-shaped (conical waves) in the absence of group-velocity dispersion (GVD) or in the presence of normal GVD. It is known, however, that such conical waves become O-shaped in the presence of anomalous GVD, resulting in a field profile that is circularly symmetric in space and time. To date, experiments generating conical waves in which the wavelength of a high-energy pump laser is tuned across the zero-dispersion wavelength of a nonlinear medium have not revealed the expected X-to-O-wave structural field transition. We report here an unambiguous observation of a fixed-central-wavelength X-to-O-wave structural field transition occurring in linear dispersion-free wave packets in the anomalous GVD regime, without needing to change the sign or magnitude of the GVD. Instead, by tuning the group velocity of a space–time wave packet (STWP) across a threshold value that we call the “escape velocity,” we observe an abrupt transition in the STWP from an O-shaped to an X-shaped spatiotemporal profile. This transition is associated with an abrupt change in the associated spatiotemporal spectrum of the STWP: from closed elliptical spatiotemporal spectra below the escape velocity to open hyperbolic spectra above it. These results may furnish new opportunities for engineering the phase-matching conditions in nonlinear and quantum optics.

Synthesis and characterization of space-time light sheets: a tutorial

Abstract Space-time wave packets (STWPs) are a new class of pulsed optical beams with many unique and intriguing attributes, including propagation invariance and tunable group velocity in linear optical media. STWPs are a form of spatiotemporally structured light, so their synthesis poses challenges that are not shared by conventional monochromatic structured light fields. We present here a detailed description of the synthesis of STWPs that are localized along one transverse dimension and uniform along the other; i.e., space-time light sheets. We also describe the main characterization schemes needed for benchmarking the unique properties of space-time light sheets.

Space-time dispersive symmetric Pearcey-Pearcey wave packets in the fractional Schrödinger equation.

This study presents an investigation into the propagation characteristics of a symmetric Pearcey-Pearcey space-time (SPPST) wave packet in a dispersive medium for the first time, to the best of our knowledge, in an optical system based on the fractional Schrödinger equation. Subsequently, the influence of the dispersion (normal and abnormal dispersion) on the SPPST packet is analyzed comprehensively. By manipulating the parameters of the SPPST wave packet including the parameters of the symmetric Pearcey beam, the value of the chirp, and the dispersion in the medium, it is possible to control its shape, orientation, and propagation dynamics. Simultaneously, the study delves into the effects of the combination of the dispersion and the second-order chirp on the evolution of SPPST wave packets and the associated intensity with these wave packets. Studying self-focusing wave packets with spatiotemporal symmetry provides new theoretical support for the development of quantum optics and optical communication.

Temporally and longitudinally tailored dynamic space-time wave packets.

In general, space-time wave packets with correlations between transverse spatial fields and temporal frequency spectra can lead to unique spatiotemporal dynamics, thus enabling control of the instantaneous light properties. However, spatiotemporal dynamics generated in previous approaches manifest themselves at a given propagation distance yet are not arbitrarily tailored longitudinally. Here, we propose and demonstrate a new versatile class of judiciously synthesized wave packets whose spatiotemporal evolution can be arbitrarily engineered to take place at various predesigned distances along the longitudinal propagation path. Spatiotemporal synthesis is achieved by introducing a 2-dimensional spectrum comprising both temporal and longitudinal wavenumbers associated with specific transverse Bessel-Gaussian fields. The resulting spectra are then employed to produce wave packets evolving in both time and axial distance - in full accord with the theoretical analysis. In this respect, various light degrees of freedom can be independently manipulated, such as intensity, polarization, and transverse spatial distribution (e.g., orbital angular momentum). Through a temporal-longitudinal frequency comb spectrum, we simulate the synthesis of the aforementioned wave packet properties, indicating a decrease in relative error compared to the desired phenomena as more spectral components are incorporated. Additionally, we experimentally demonstrate tailorable spatiotemporal fields carrying time- and longitudinal-varying orbital angular momentum, such that the local topological charge evolves every ∼1 ps in the time domain and 10 cm axially. We believe our space-time wave packets can significantly expand the exploration of spatiotemporal dynamics in the longitudinal dimension. Such wave packets might potentially enable novel applications in light-matter interactions and nonlinear optics.

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.

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.

Experimental realization of Lorentz boosts of space-time wave packets

Experimental realization of Lorentz boosts of space-time wave packets

(3+1)-dimensional Pearcey-Gaussian wave packet with arbitrary velocity driven by flying focus.

The group velocity (GV) modulation of space-time wave packets (STWPs) along the transverse and longitudinal directions in free space is constrained by various factors. To surmount this limitation, a technique called "flying focus" has been developed, which enables the generation of laser pulses with dynamic focal points that can propagate at arbitrary velocities independent of GV. In this Letter, we propose a (3+1)-dimensional Pearcey-Gauss wave packet based on the "flying focus" technique, which exhibits superluminal propagation, transverse focus oscillation, and longitudinal periodic autofocusing. By selecting appropriate parameters, we can flexibly manipulate the position, the size, and the number of focal points- or make the wave packet follow a desired trajectory. This work may pave the way for the advancement of space-time structured light fields.

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.

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.

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.

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.

Relativistic transformations of quasi-monochromatic paraxial optical beams

A monochromatic plane wave recorded by an observer moving with respect to the source undergoes a Doppler shift and spatial aberration. We investigate here the transformation undergone by a generic, paraxial, spectrally coherent quasimonochromatic optical beam (of finite transverse width) when recorded by a moving detector. Because of the space-time coupling intrinsic to the Lorentz transformation, the monochromatic beam is converted into a propagation-invariant pulsed beam traveling at a group velocity equal to that of the relative motion and which belongs to the recently studied class of ``space-time wave packets.'' We show that the predicted transformation from a quasimonochromatic beam to a pulsed wave packet can be observed even at terrestrial speeds.

Experimental Observation of the Spontaneous Emission of a Space–Time Wavepacket in a Multimode Optical Fiber

We provide a complete analysis, from theory to experiment, of the spontaneous emergence of a discretized conical wave of X-type (i.e., a localized 2D+1 space-time wavepacket) when an intense ultrashort pulse nonlinearly propagates in a multimode fiber. In particular, we reveal that this spatiotemporal phenomenon corresponds to broadband intermodal dispersive wave emission from an unsteady localized wave structure formed during nonlinear propagation. Theoretical phase-matching predictions are experimentally and numerically confirmed in a commercially available step-index multimode fiber. Our results provide a general understanding of phase-matched radiations emitted by nonlinear waves in multidimensional dispersive optical system.

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 (LGℓ,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.

Strong double space-time wave packets using optical parametric amplification

The space-time (ST) wave packet, a type of light source with many anomalous behaviors, has already found application in weak-field optics, however its generation method limits its energy and therefore its application in strong-field optics. Here we show that the type-I collinear optical parametric amplification (OPA) is a natural amplifier for a ST wave packet, because in it, the ST spectrum (i.e., angle-dependent phase-matching spectrum) of the signal and idler has the same analytical expression as that of a ST wave packet under the narrowband approximation. The high gain in a thin-crystal OPA allows for large energy amplification while ensuring unchanged ST correlation. Meanwhile, a double ST wave packet containing two collinear-propagating ones of the amplified signal and the generated idler via the nondegenerate OPA is reported, which with short- and long-wavelengths (relative to the degenerate-wavelength) have superluminal and subluminal velocities, respectively. This study realizes energy amplification, wavelength conversion, and velocity switching of a ST wave packet and will expand its application in many fields.

Tunability of space-time wave packet carrying tunable and dynamically changing OAM value.

Space-time (ST) wave packets have gained much interest due to their dynamic optical properties. Such wave packets can be generated by synthesizing frequency comb lines, each having multiple complex-weighted spatial modes, to carry dynamically changing orbital angular momentum (OAM) values. Here, we investigate the tunability of such ST wave packets by varying the number of frequency comb lines and the combinations of spatial modes on each frequency. We experimentally generate and measure the wave packets with tunable OAM values from +1 to +6 or from +1 to +4 during a ∼5.2-ps period. We also investigate, in simulation, the temporal pulse width of the ST wave packet and the nonlinear variation of the OAM values. The simulation results show that: (i) a pulse width can be narrower for the ST wave packet carrying dynamically changing OAM values using more frequency lines; and (ii) the nonlinearly varying OAM value can result in different frequency chirps along the azimuthal direction at different time instants.

Synthesizing ultrafast optical pulses with arbitrary spatiotemporal control.

The ability to control the instantaneous state of light, from high-energy pulses down to the single-photon level, is an indispensable requirement in photonics. This has, for example, facilitated spatiotemporal probing and coherent control of ultrafast light-matter interactions, and enabled capabilities such as generation of exotic states of light with complexity, or at wavelengths, that are not easily accessible. Here, by leveraging the multifunctional control of light at the nanoscale offered by metasurfaces embedded in a Fourier transform setup, we present a versatile approach to synthesize ultrafast optical transients with arbitrary control over its complete spatiotemporal evolution. Our approach, supporting an ultrawide bandwidth with simultaneously high spectral and spatial resolution, enables ready synthesis of complex states of structured space-time wave packets. We expect our results to offer unique capabilities in coherent ultrafast light-matter interactions and facilitate applications in microscopy, communications, and nonlinear optics.

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.

Space-time wave packets

Space-time wave packets (STWPs) constitute a broad class of pulsed optical fields that are rigidly transported in linear media without diffraction or dispersion, and are therefore propagation-invariant in the absence of optical nonlinearities or waveguiding structures. Such wave packets exhibit unique characteristics, such as controllable group velocities in free space and exotic refractive phenomena. At the root of these behaviors is a fundamental feature underpinning STWPs: their spectra are not separable with respect to the spatial and temporal degrees of freedom. Indeed, the spatiotemporal structure is endowed with non-differentiable angular dispersion, in which each spatial frequency is associated with a single prescribed wavelength. Furthermore, controlled deviation from this particular spatiotemporal structure yields novel behaviors that depart from propagation-invariance in a precise manner, such as acceleration with an arbitrary axial distribution of the group velocity, tunable dispersion profiles, and Talbot effects in space–time. Although the basic concept of STWPs has been known since the 1980s, only very recently has rapid experimental development emerged. These advances are made possible by innovations in spatiotemporal Fourier synthesis, thereby opening a new frontier for structured light at the intersection of beam optics and ultrafast optics. Furthermore, a plethora of novel spatiotemporally structured optical fields (such as flying-focus wave packets, toroidal pulses, and spatiotemporal optical vortices) are now providing a swath of surprising characteristics, ranging from tunable group velocities to transverse orbital angular momentum. We review the historical development of STWPs, describe the new experimental approaches for their efficient synthesis, and enumerate the various new results and potential applications for STWPs and other spatiotemporally structured fields, before casting an eye on a future roadmap for this field.