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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.

All linear, propagation-invariant, paraxial pulsed beams are spatiotemporally X-shaped (conical waves) in absence of group-velocity dispersion (GVD), or in presence of normal GVD. It is known, however, that such conical waves become O-shaped in 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 unambiguous observation of a fixed-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.

For partially coherent optical fields in which a single binary degree of freedom (DoF) is relevant, such as polarization, entropy uniquely identifies the class of optical fields that can be converted into each other via unitary transformations. However, when multiple DoFs are taken into consideration, entropy no longer serves this purpose. We investigate the structure of the family of iso-entropy partially coherent optical fields defined by two binary DoFs (polarization and two spatial modes) and described by a 4x4 coherence matrix G. We find that the rank of G (the number of its non-zero eigenvalues) plays a critical role in this context: whereby any pair of iso-entropy rank-2 fields can be converted into each other unitarily, this is not necessarily the case for a pair of rank-3 or rank-4 fields. Furthermore, unitary transformations between iso-entropy fields of different ranks are strictly forbidden. Instead, such conversions require entropy-maintaining non-unitary transformations that potentially combine filtering projections and randomizing operations. We experimentally synthesize partially coherent iso-entropy optical fields of all ranks, and tomographically reconstruct their coherence matrices. Moreover, we steer the coherence matrix over iso-entropy trajectories that maintain a fixed rank (intra-rank conversion) or that involve changes in the rank (inter-rank conversion). These findings offer a new perspective for the potential utility of partially coherent light in optical communications and sensing.

We introduce a taxonomy for partially coherent optical fields spanning multiple degrees of freedom (DoFs) based on the rank of the associated coherence matrix (the number of non-zero eigenvalues). When DoFs comprise two spatial modes and polarization, a fourfold classification emerges, with rank-1 fields corresponding to fully coherent fields. We demonstrate theoretically and confirm experimentally that these classes have heretofore unrecognized different properties. Specifically, whereas rank-2 fields can always be rendered separable with respect to its DoFs via a unitary transformation, rank-3 fields are always non-separable. Consequently, the entropy for a rank-2 field can always be concentrated into a single DoF (thus ridding the other DoF of statistical fluctuations), whereas some entropy is always 'locked' in one DoF of a rank-3 field.

It is now well-understood that a Lorentz boost of a spatially coherent monochromatic optical beam yields a so-called space-time wave packet (STWP): a propagation-invariant pulsed beam whose group velocity is determined by the relative velocity between the source and observer. Moreover, the Lorentz boost of an STWP is another STWP, whose group velocities are related by the relativistic law for addition of velocities typically associated with massive particles. We present an experimental procedure for testing this prediction in both the subluminal and superluminal regimes that makes use of spatio-temporal Fourier synthesis via a spatial light modulator. Our approach enables realizing the change in temporal bandwidth, the invariance of the spatial bandwidth, the concomitant change in the spatio-temporal wave-packet envelope, and the change in group velocity that all accompany a Lorentz boost of a monochromatic optical beam. The only consequence of the Lorentz boost not captured by this methodology is the Doppler shift in the optical carrier. This work may provide an avenue for further table-top demonstration of relativistic transformations of optical fields., Comment: 13 pages, 10 figures

Chirped Bragg volume gratings (CBGs) offer a useful alternative for spectral analysis, but increasing the bandwidth necessitates increasing the device area. In contrast, recently developed rotated CBGs (r-CBGs), in which the Bragg structure is rotated by $45^{\circ}$ with respect to the device facets, require increasing only the device length to extend the bandwidth, in addition to the convenience of resolving the spectrum at normal incidence. Here, we multiplex r-CBGs in the same device to enable spectral analysis in two independent spectral windows without increasing the system volume. This new device, which we term an X-CBG, allows for compact multi-band spectroscopy in contiguous or separated spectral windows in the visible and near-infrared for applications in nonlinear microscopy and materials identification and sensing., Comment: This is the 3rd paper in the series of papers on rotated chirped volume Bragg gratings (CBG). The previous two papers can be found in Optics Letters 48 (5), 1180-1183 and Optics Letters 48 (10), 2500-2503

de Broglie wave packets accompanying moving particles are dispersive and lack an intrinsic length scale dictated solely by the particle mass and velocity. Mackinnon proposed almost 45~years ago a localized non-dispersive wave packet constructed out of dispersive de Broglie phase waves via a Copernican inversion of the roles of particle and observer, whereupon an intrinsic length scale emerges by accounting for every possible observer -- rather than by introducing an \textit{ad hoc} uncertainty in the particle velocity. The de Broglie-Mackinnon (dBM) wave packet has nevertheless remained to date a theoretical entity. Here, we report the observation of optical dBM wave packets using paraxial space-time-coupled pulsed laser fields in presence of anomalous group-velocity dispersion. Crucially, the bandwidth of dBM wave packets has an upper limit that is compatible with the wave-packet group velocity and equivalent mass. In contrast to previously observed linear propagation-invariant wave packets whose spatio-temporal profiles at any axial plane are X-shaped, those for dBM wave packets are uniquely O-shaped (circularly symmetric with respect to space and time). By sculpting their spatio-temporal spectral structure, we produce dispersion-free dBM wave packets in the dispersive medium, observe their circularly symmetric spatio-temporal profiles, and tune the field parameters corresponding to particle mass and velocity that uniquely determine the wave-packet length scale.

In addition to longitudinal spin angular momentum (SAM) along the axis of propagation of light, spatially structured electromagnetic fields such as evanescent waves and focused beams have recently been found to possess transverse SAM in the direction perpendicular to the axis of propagation. In particular, the SAM of SPPs with spatial structure has been extensively studied in the last decade after it became clear that evanescent fields with spatially structured energy flow generate threedimensional spin texture. Here we present numerical calculations of the space-time surface plasmon polariton (ST-SPP) wave packet, a plasmonic bullet that propagates at an arbitrary group velocity while maintaining its spatial distribution. ST-SPP wave packets with complex spatial structure and energy flow density distribution determined by the group velocity are found to propagate with accompanying three-dimensional spin texture and finite topological charge density. Furthermore, the spatial distribution of the spin texture and topological charge density determined by the spatial structure of the SPP is controllable, and the deformation associated with propagation is negligible. ST-SPP wave packets, which can stably transport customizable three-dimensional spin textures and topological charge densities, can be excellent subjects of observation in studies of spinphotonics and optical topological materials., Comment: 15 pages, 6 figures

Resonant field enhancement in optical cavities is provided over only narrow linewidths and for specific spatial modes. Consequently, spectrally restrictive planar Fabry-P{\'e}rot cavities have not contributed to date to white-light imaging, which necessitates a highly multimoded broadband field to satisfy the resonance condition. Here we show that introducing judicious angular-dispersion circumvents the fundamental trade-off between cavity linewidth and finesse in a Fabry-P{\'e}rot cavity by exciting a 130-nm-bandwidth achromatic resonance across the visible spectrum, which far exceeds the finesse-limited linewidth (0.5~nm), and even exceeds the free spectral range (45~nm). This omni-resonant configuration enables broadband color-imaging over a 100-nm-bandwidth in the visible with minimal spherical and chromatic aberrations. We demonstrate omni-resonant imaging using coherent and incoherent light, and spatially extended and localized fields comprising stationary and moving objects. This work paves the way to harnessing broadband resonant enhancements for spatially structured fields, as needed for example in solar windows.

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 counterpart 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 (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 degrees 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 25x25x8 mm3, which is orders of magnitude smaller than previous arrangements., Comment: 5 pages, 5 figures

We introduce a new 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 Bragg volume 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 an 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 ultra-compact spectrometers and pulse modulators., Comment: 5-page letter

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 of their freely propagating counterparts. However, because the spatio-temporal spectral structure of ST-SPPs is key to their propagation invariance on the metal surface, their excitation methodology must be considered carefully. We show here using finite-difference time-domain (FDTD) simulations that an appropriately synthesized ST wave packet in free space can be couples to a ST-SPP via a single nano-scale slit inscribed in the metal surface. Our calculations confirm that this excitation methodology yields surface-bound ST-SPPs that are locarized 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., Comment: 10 pages, 6 figures

Spatiotemporal sculpturing of light pulse with ultimately sophisticated structures represents the holy grail of the human everlasting pursue of ultrafast information transmission and processing as well as ultra-intense energy concentration and extraction. It also holds the key to unlock new extraordinary fundamental physical effects. Traditionally, spatiotemporal light pulses are always treated as spatiotemporally separable wave packet as solution of the Maxwell's equations. In the past decade, however, more generalized forms of spatiotemporally nonseparable solution started to emerge with growing importance for their striking physical effects. This roadmap intends to highlight the recent advances in the creation and control of increasingly complex spatiotemporally sculptured pulses, from spatiotemporally separable to complex nonseparable states, with diverse geometric and topological structures, presenting a bird's eye viewpoint on the zoology of spatiotemporal light fields and the outlook of future trends and open challenges., Comment: This is the version of the article before peer review or editing, as submitted by an author to Journal of Optics. IOP Publishing Ltd is not responsible for any errors or omissions in this version of the manuscript or any version derived from it

We report on the diffraction-free propagation of space-time wave packets (STWPs) -- a class of propagation-invariant pulsed beams -- for $\sim\!1$ km in an open-air laser range in a low-turbulence scenario. Making use of $\approx\!100$-fs pulses (bandwidth $\sim\!25$ nm) at a wavelength of $\approx\!1$ $\mu$m, we construct an STWP with a transverse width of $\approx\!2$ mm that expands to $\approx\!3$ mm after $\sim\!500$ m, and another that expands from $\approx\!8$ mm to $\approx\!10$ mm after 1 km. The propagation of the STWPs is compared to Gaussian wave packets of the same transverse spatial width and bandwidth. We establish a theoretical model that accounts for the significant factors limiting the STWP propagation distance and suggests the path to further extending this distance.

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 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 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.

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 quasi-monochromatic optical \textit{beam} (of finite transverse width) when recorded by a moving detector. Because of the space-time coupling engendered by 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 family of `space-time wave packets'. We show that the predicted transformation from a quasi-monochromatic beam to a pulsed wave packet can be observed even at terrestrial speeds., Comment: 5 pages, 6 figures

Space-time wave packets (STWPs) are propagation-invariant pulsed beams whose characteristics stem from the tight association between their spatial and temporal degrees of freedom. Until recently, only scalar STWPs have been synthesized in the form of light sheets. Here we synthesize vector STWPs that are localized in all dimensions by preparing polarization-structured spatio-temporal spectra and unveil the polarization distribution over the STWP volume via time-resolved complex field measurements. Such vector STWPs are endowed with cylindrically symmetric polarization vector structures, which require joint manipulation of the spatial, temporal, and polarization degrees-of-freedom of the optical field. These results may be useful in particle manipulation, and in nonlinear and quantum optics., Comment: 5 pages, 6 figures

Space-time wave packets (STWPs) are pulsed beams that propagate invariantly (without diffraction or dispersion) in linear media. The behavior of STWPs in free space is now well-established, and recently their propagation invariance was confirmed in both the normal and anomalous dispersion regimes. However, yet-to-be-observed rich dynamics of spectral reorganization have been predicted to occur in the presence of normal group-velocity dispersion (GVD). Indeed, propagation invariance in the normal-GVD regime is compatible with spatio-temporal spectra that are X-shaped, hyperbolic, parabolic, elliptical, or even separable along the spatial and temporal degrees-of-freedom. These broad varieties of field structures can be classified in a two-dimensional space parameterized by the group velocity of the STWP and its central axial wave number. Here we observe the entire span of spectral reorganization for STWPs in the paraxial regime in normally dispersive ZnSe at a wavelength $\sim1$~$\mu$m with STWPs of on-axis pulse width of $\sim\!200$~fs. By tuning the group velocity and central axial wave number of the STWP, we observe transitions in the structure of the spatio-temporal spectrum and verify the associated change in its intensity profile. These results lay the foundation for initiating novel phase-matched nonlinear processes using STWPs in dispersive media.

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 couple the respective TM components of radial and azimuthal vector beam profiles to confirm SPP 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 greater extents. Hereby, outlining a pivotal, yet commonly overlooked, path towards the development of long-range biosensors and all-optical integrated plasmonic circuits., Comment: 15 pages and 5 figures

Omni-resonance refers to the broadening of the spectral transmission through a planar cavity, not by changing the cavity structure, but by judiciously 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-Perot 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.

When an optical pulse is spatially localized in a highly multimoded waveguide, its energy is typically distributed among a multiplicity of modes, thus giving rise to a speckled transverse spatial profile that undergoes erratic changes with propagation. It has been suggested theoretically that pulsed multimode fields in which each wavelength is locked to an individual mode at a prescribed axial wave number will propagate invariantly along the waveguide at a tunable group velocity. In this conception, an initially localized field remains localized along the waveguide. Here, we provide proof-of-principle experimental confirmation for the existence of this new class of pulsed guided fields, which we denote space-time supermodes, and verify their propagation invariance in a planar waveguide. By superposing up to 21 modes, each assigned to a prescribed wavelength, we construct space-time supermodes in a 170-micron-thick planar glass waveguide with group indices extending from 1 to 2. The initial transverse width of the field is 6 microns, and the waveguide length is 9.1 mm, which is 257x the associated Rayleigh range. A variety of axially invariant transverse spatial profiles are produced by judicious selection of the modes contributing to the ST supermode, including single-peak and multi-peak fields, dark fields (containing a spatial dip), and even flat uniform intensity profiles.

Because surface plasmon polaritons (SPPs) are surface waves characterized by one free transverse dimension, the only monochromatic diffraction-free spatial profiles for SPPs are cosine and Airy waves. Pulsed SPP wave packets have been recently formulated that are propagation-invariant and localized in the in-plane dimensions by virtue of a tight spectral association between their spatial and temporal frequencies, which have thus been dubbed `space-time' (ST) SPPs. Because of the spatio-temporal spectral structure unique to ST-SPPs, the optimal launching strategy of such novel plasmonic field configurations remains an open question. We present here a critical step towards realizing ST-SPPs by reporting observations of ultrabroadband striped ST-SPPs. These are SPPs in which each wavelength travels at a prescribed angle with respect to the propagation axis to produce a periodic (striped) transverse spatial profile that is diffraction-free. We start with a free-space ST wave packet that is coupled to a ST-SPP at a gold-dielectric interface, and unambiguously identify the ST-SPP via an axial beating detected in two-photon fluorescence produced by the superposition of incident ST wave packet and the excited surface-bound ST-SPP. These results highlight a viable approach for efficient and reliable coupling to ST-SPPs, and thus represent the first crucial step towards realization of the full potential of ST-SPPs for plasmonic sensing and imaging., Comment: 9 pages, 8 figures

Angular dispersion can counterbalance normal group-velocity dispersion (GVD) that increases the wave-vector length in a dispersive medium. By tilting the wave vector, angular dispersion reduces the axial wave number in this case to match the pre-GVD value. By the same token, however, angular dispersion fails to counterbalance anomalous GVD, which in contrast reduces the wave-vector length. Consequently, GVD-cancellation via angular dispersion has not been demonstrated to date in the anomalous dispersion regime. We synthesize here structured femtosecond pulsed beams known as `space-time' wave packets designed to realize dispersion-cancellation symmetrically in either the normal- or anomalous-GVD regimes by virtue of non-differentiable angular dispersion inculcated into the pulsed field. Furthermore, we also verify GVD-inversion: reversing the GVD sign experienced by the field with respect to that dictated by the chromatic dispersion of the medium itself.

"Space-time" (ST) wave packets 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 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 ST wave packets: their spectra are not separable with respect to the spatial and temporal degrees of freedom. Indeed, the spatio-temporal structure is endowed with non-differentiable angular dispersion, in which each spatial frequency is associated with a single prescribed wavelength. Furthermore, deviation from this particular spatio-temporal 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 ST wave packets has been known since the 1980's, only very recently has rapid experimental development emerged. These advances are made possible by innovations in spatio-temporal Fourier synthesis, thereby opening a new frontier for structured light at the intersection of beam optics and ultrafast optics., Comment: This is a review paper on 'Space-time' wave packets. It contains 103 pages, 57 figures

Space-time (ST) wave packets are a class of pulsed optical beams whose spatio-temporal spectral structure results in propagation invariance, tunable group velocity, and fascinating refractive phenomena. Here, we investigate the refraction of ST wave packets at a planar interface between two dispersive, homogeneous, isotropic media. We formulate a new 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 a tool in nonlinear optics for bridging large group-velocity mismatches in highly dispersive scenarios., Comment: 5 pages, 4 figures

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., Comment: 12 pages of Main text with 7 figures. 37 pages of Supplementary document with 18 figures

The structure of an optical waveguide determines the characteristics of its guided modes, such as their spatial profile and group index. General features are shared by modes regardless of the waveguiding structure; for example, modal dispersion is inevitable in multimode waveguides, every mode experiences group-velocity dispersion, and higher-order modes are usually slower than their lower-order counterparts. We show here that such trends can be fundamentally altered -- altogether severing the link between modal order and group index hybrid and eliminating dispersion -- by exploiting hybrid guided space-time modes in a planar multimode waveguide. Such modes are confined in one-dimension by the waveguide and in the other by the spatio-temporal spectral structure of the field itself. Direct measurements of the modal group delays confirm that the group index for low-loss, dispersion-free, hybrid space-time modes can be each tuned away from the group index of the conventional mode of same order, and that the transverse size of these hybrid modes can be varied independently of the modal order and group index. These findings are verified in a few-mode planar waveguide consisting of a 25.5-mm-long, 4-$\mu$m-thick silica film deposited on a MgF$_2$ substrate.

We uncover a surprising gap in optics with regards to angular dispersion (AD) that has persisted for decades. 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. To rectify this state of affairs, 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 compensation, optical signal processing, and nonlinear optics.

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 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 phase and group velocities are always equal in conventional scenarios. 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. These and several other consequences of non-differentiable AD are verified experimentally using a pulsed-beam shaper capable of introducing AD with arbitrary spectral profile. Rather than being an exotic phenomenon, non-differentiable AD is an accessible, robust, and versatile resource for sculpting pulsed optical fields.

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 first accelerates from an initial to a terminal group velocity, decelerates back to the initial value, and then travels at a fixed group velocity. These never-before-seen optical-acceleration phenomena are all produced using the same experimental arrangement that precisely sculpts the wave packet's spatio-temporal spectral structure.

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.

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, 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.

The refraction of space-time (ST) wave packets at planar interfaces between non-dispersive, homogeneous, isotropic dielectrics exhibit 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 paper (III) of this sequence, we present experimental verification of the refractive phenomena exhibited by ST wave packets at oblique incidence that were predicted in paper (I). We also examine a proposal for 'blind synchronization' whereby identical ST wave packets arrive simultaneously at different receivers without \textit{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.

The refraction of space-time (ST) wave packets offers many fascinating surprises with respect to conventional pulsed beams. In paper (I) of this sequence, we described theoretically the refraction of all families of ST wave packets at normal and oblique incidence at a planar interface between two non-dispersive, homogeneous, isotropic dielectrics. Here, in paper (II) of this sequence, we present experimental verification of the novel 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 MgF$_2$, 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.

Coherent perfect absorption (CPA) refers to interferometrically induced complete absorption of incident light by a partial absorber independently of its intrinsic absorption (which may be vanishingly small) or its thickness. CPA is typically realized in a resonant device, and thus cannot be achieved over a broad continuous spectrum, which thwarts its applicability to photodetectors and solar cells, for example. Here, we demonstrate broadband omni-resonant CPA by placing a thin weak absorber in a planar cavity and pre-conditioning the incident optical field by introducing judicious angular dispersion. We make use of monolayer graphene embedded in silica as the absorber and boost its optical absorption from ~1.6% to ~60% over a bandwidth of ~70 nm in the visible. Crucially, an analytical model demonstrates that placement of the graphene monolayer at a peak in the cavity standing-wave field is not necessary to achieve CPA, contrary to conventional wisdom.

Introducing precise spatio-temporal structure into a pulsed optical field can lead to remarkable changes with its free propagation. `Space-time' (ST) wave packets, for example, propagate rigidly at a tunable group velocity in free space by inculcating a one-to-one relationship between the axial wave numbers and the temporal frequencies. Here we introduce a new class of ST wave packets that we call V-waves (so named because of their characteristically V-shaped spatio-temporal spectrum) in which a linear one-to-one relationship is introduced between the temporal frequencies and the transverse wave numbers (or spatial frequencies). We confirm experimentally that V-waves experience anomalous group velocity dispersion in free space, all the while maintaining the group velocity fixed at the speed of light in vacuum. Extremely large values of group velocity dispersion can be easily realized, which are not accessible with traditional optical materials or photonic structures. Moreover, V-waves are the unique optical wave packets whose diffraction and dispersion lengths are intrinsically equal by virtue of the spatio-temporal structure of the field itself. These results are of interest for optical signal processing and nonlinear optics.

Our ability to generate new distributions of light has been remarkably enhanced in recent years. At the most fundamental level, these light patterns are obtained by ingeniously combining different electromagnetic modes. Interestingly, the modal superposition occurs in the spatial, temporal as well as spatio-temporal domain. This generalized concept of structured light is being applied across the entire spectrum of optics: generating classical and quantum states of light, harnessing linear and nonlinear light-matter interactions, and advancing applications in microscopy, spectroscopy, holography, communication, and synchronization. This Roadmap highlights the common roots of these different techniques and thus establishes links between research areas that complement each other seamlessly. We provide an overview of all these areas, their backgrounds, current research, and future developments. We highlight the power of multimodal light manipulation and want to inspire new eclectic approaches in this vibrant research community., Comment: Under review in J. Opt

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 the temporal Talbot effect in free space by employing dispersive space-time wave packets, whose spatio-temporal structure induces group-velocity dispersion of controllable magnitude and sign in free space.

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 spatio-temporal 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 in the range from 1.38 to 2.5 for angles up to $50^{\circ}$ away from normal incidence.

Spatial structuring of an optical pulse can lead in some cases upon free propagation to changes in its temporal profile. For example, introducing conventional angular dispersion into the field results in the pulse encountering group-velocity dispersion in free space. However, only limited control is accessible via this strategy. Here we show that precise and versatile control can be exercised in free space over the dispersion profile of so-called `space-time' wave packets: a class of pulsed beams undergirded by non-differentiable angular dispersion. This abstract mathematical feature allows us to tune the magnitude and sign of the different dispersion orders without introducing losses, thereby realizing arbitrary dispersion profiles, and achieving dispersion values unattainable in optical materials away from resonance. Unlike optical materials and photonic structures in which the values of the different dispersion orders are not independent of each other, these orders are addressable separately using our strategy. These results demonstrate the versatility of space-time wave packets as a platform for structured light and point towards their utility in nonlinear and quantum optics.

The Talbot effect, epitomized by periodic revivals of a freely evolving periodic field structure, has been observed with waves of diverse physical nature in space and separately in time, whereby diffraction underlies the former and dispersion the latter. To date, a combined spatio-temporal Talbot effect has not been realized in any wave field because diffraction and dispersion are independent physical phenomena, typically unfolding at incommensurable length scales. Here we report the observation of an optical 'space-time' Talbot effect, whereby a spatio-temporal optical lattice structure undergoes periodic revivals after suffering the impact of both diffraction and dispersion. The discovered space-time revivals are governed by a single self-imaging length scale, which encompasses both spatial and temporal degrees of freedom. Key to this effect is the identification of a unique pulsed optical field structure, which we refer to as a V-wave, that is endowed with intrinsically equal diffraction and dispersion lengths in free space, thereby enabling self-imaging to proceed in lockstep in space and time.

Introducing angular dispersion into a pulsed field tilts the pulse front with respect to the phase front. There exists between the angular dispersion and the 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 spatio-temporal structure corresponding to a particular form of angular dispersion. We demonstrate theoretically and experimentally that underlying ST wave packets is -- 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 pulse bandwidth even at fixed angular dispersion.

Coherent perfect absorption (CPA) is an interferometric effect that guarantees full absorption in a lossy layer independently of its intrinsic losses. To date, it has been observed only at a single wavelength or narrow bandwidths, whereupon wavelength-dependent absorption can be ignored. Here we produce CPA over a bandwidth of ~ 60 nm in a 2-um-thick polymer film with a low-doping concentration of an organic laser dye. A planar cavity is designed with a spectral `dip' to accommodate the dye resonant linewidth, and CPA is thus achieved even at its absorption edges. This approach allows realizing strong absorption in laser dyes - and resonant materials in general - independently of the intrinsic absorption levels, with a at spectral profile and without suffering absorption quenching due to high doping levels.

Space-time (ST) wave packets are propagation-invariant pulsed optical beams whose group velocity can be tuned in free space by tailoring their spatio-temporal spectral structure. To date, efforts on synthesizing ST wave packets have striven to maintain their propagation invariance. Here, we demonstrate that one degree of freedom of a ST wave packet -- its on-axis spectrum -- can be isolated and purposefully controlled independently of the others. Judicious spatio-temporal spectral amplitude and phase modulation yields ST wave packets with programmable spectral changes along the propagation axis; including red-shifting or blue-shifting spectra, or more sophisticated axial spectral encoding including bidirectional spectral shifts and accelerating spectra. In all cases, the spectral shift can be larger than the initial on-axis bandwidth, while preserving the propagation-invariance of the other degrees of freedom, including the wave packet spatio-temporal profile. These results may be useful in range-finding in microscopy or remote sensing via spectral stamping., Comment: one main article and one supplementary

'Space-time' (ST) wave packets are propagation-invariant pulsed optical beams that travel rigidly in linear media without diffraction or dispersion at a potentially arbitrary group velocity. These unique characteristics are a result of spatio-temporal spectral correlations introduced into the field; specifically, each spatial frequency is associated with a single temporal frequency (or wavelength). Consequently, the spatial and temporal bandwidths of ST wave packets are correlated, so that exploiting an optical source with a large temporal bandwidth or achieving an ultralow group velocity necessitate an exorbitantly large numerical aperture. Here we show that `spectral recycling' can help overcome these challenges. 'Recycling' or `reusing' each spatial frequency by associating it with multiple distinct but widely separated temporal frequencies allows one to circumvent the proportionality between the spatial and temporal bandwidths of ST wave packets, which has been one of their permanent characteristics since their inception. We demonstrate experimentally that the propagation invariance, maximum propagation distance, and group velocity of ST wave packets are unaffected by spectral recycling. We also synthesize a ST wave packet with group velocity c/14.3 (c is the speed of light in vacuum) with a reasonable numerical aperture made possible by spectral recycling., Comment: 9 pages, 7 figures

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 spatio-temporal 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 space-time 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.

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 spatio-temporal structure leads to group-velocity changes as high as $\sim c$ observed over a distance of $\sim20$ mm in free space, which represents a boost of at least $\sim4$ 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.

We introduce the unique class of propagation-invariant surface plasmon polaritons (SPPs) representing pulsed surface wave packets propagating along unpatterned metal-dielectric interfaces and are localized in all dimensions - with potentially subwavelength transverse spatial widths. The characteristic features of such linear diffraction-free, dispersion-free `plasmonic bullets' stem from tight spatio-temporal correlations incorporated into the SPP spectral support domain, and we thus call them `space-time' SPPs. We show that the group velocity of space-time SPP wave packets can be readily tuned to subluminal, superluminal, and even negative values by tailoring the spatio-temporal field structure independently of any material properties. We present an analytical framework and numerical simulations for the propagation of space-time SPPs in comparison with traditional pulsed SPPs whose spatial and temporal degrees of freedom are separable, thereby verifying the propagation-invariance of the former.

Light can be confined transversely and delivered axially in a waveguide. However, waveguides are lossy static structures whose modal characteristics are fundamentally determined by the boundary conditions, and thus cannot be readily changed post-fabrication. Here we show that unpatterned planar optical films can be exploited for low-loss two-dimensional waveguiding by using `space-time' wave packets, which are the unique family of one-dimensional propagation-invariant pulsed optical beams. We observe `hybrid guided' space-time modes that are index-guided in one transverse dimension in the film and localized along the unbounded transverse dimension via the intrinsic spatio-temporal structure of the field. We demonstrate that these field configurations enable overriding the boundary conditions by varying post-fabrication the group index of the fundamental mode in a 2-$\mu$m-thick, 25-mm-long silica film, which is achieved by modifying the field's spatio-temporal structure along the unbounded dimension. Tunability of the group index over an unprecedented range from 1.26 to 1.77 around the planar-waveguide value of 1.47 is verified - while maintaining a spectrally flat zero-dispersion profile. Our work paves the way to to the utilization of space-time wave packets in on-chip photonic platforms, and may enable new phase-matching strategies that circumvent the restrictions due to intrinsic material properties.