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.

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.

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.

Compact dual-band spectral analysis via multiplexed rotated chirped volume Bragg gratings.

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° 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, to the best of our knowledge, device, which we term an X-CBG, allows for compact multi-band spectroscopy in contiguous or separated spectral windows for the visible and near-infrared applications in nonlinear microscopy and material identification and sensing.

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.

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.

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.

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.

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.

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.

Reversible inter-degree-of-freedom optical-coherence conversion via entropy swapping.

The entropy associated with an optical field quantifies the field fluctuations and thus its coherence. Any binary optical degree-of-freedom (DoF) - such as polarization or the field at a pair of points in space - can each carry up to one bit of entropy. We demonstrate here that entropy can be reversibly swapped between different DoFs, such that coherence is converted back and forth between them without loss of energy. Specifically, starting with a spatially coherent but unpolarized field carrying one bit of entropy, we unitarily convert the coherence from the spatial DoF to polarization to produce a spatially incoherent but polarized field by swapping the entropy between the two DoFs. Next, we implement the inverse unitary operator, thus converting the coherence back to yield once again a spatially coherent yet unpolarized field. We exploit the intermediate stage between the two coherence conversions - where the spatial coherence has been converted to the polarization DoF - to verify that the field has become immune to the deleterious impact of spatial phase scrambling. Maximizing the spatial entropy protects the spatial DoF by preventing it from taking on any additional fluctuations. After the second coherence conversion, spatial coherence is readily retrieved, and the effect of spatial phase scrambling circumvented.

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.

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.

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.

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.

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 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.

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.