Spatiotemporal amplitude and phase retrieval of Bessel-X pulses using a Hartmann-Shack sensor

The propagation speed of optical speckle

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

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.

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.

Phase front retrieval and correction of Bessel beams

Bessel beams generated with non-ideal axicons are affected by aberrations. We introduce a method to retrieve the complex amplitude of a Bessel beam from intensity measurements alone, and then use this information to correct the wavefront and intensity profile using a deformable mirror.

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.

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

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

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

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

What is the maximum differential group delay achievable by a space-time wave packet in free space?

The group velocity of 'space-time' wave packets - propagation-invariant pulsed beams endowed with tight spatio-temporal spectral correlations - can take on arbitrary values in free space. Here we investigate theoretically and experimentally the maximum achievable group delay that realistic finite-energy space-time wave packets can achieve with respect to a reference pulse traveling at the speed of light. We find that this delay is determined solely by the spectral uncertainty in the association between the spatial frequencies and wavelengths underlying the wave packet spatio-temporal spectrum - and not by the beam size, bandwidth, or pulse width. We show experimentally that the propagation of space-time wave packets is delimited by a spectral-uncertainty-induced 'pilot envelope' that travels at a group velocity equal to the speed of light in vacuum. Temporal walk-off between the space-time wave packet and the pilot envelope limits the maximum achievable differential group delay to the width of the pilot envelope. Within this pilot envelope the space-time wave packet can locally travel at an arbitrary group velocity and yet not violate relativistic causality because the leading or trailing edge of superluminal and subluminal space-time wave packets, respectively, are suppressed once they reach the envelope edge. Using pulses of width ∼ 4 ps and a spectral uncertainty of ∼ 20 pm, we measure maximum differential group delays of approximately ±150 ps, which exceed previously reported measurements by at least three orders of magnitude.

Broadband space-time wave packets propagating 70 m

The propagation distance of a pulsed beam in free space is ultimately limited by diffraction and space-time coupling. "Space-time" (ST) wave packets are pulsed beams endowed with tight spatio-temporal spectral correlations that render them propagation-invariant. Here we explore the limits of the propagation distance for ST wave packets. Making use of a specially designed phase plate inscribed by gray-scale lithography and having a laser-damage threshold of ∼0.5  J/cm², we synthesize a ST light sheet of width ≈700  μm and bandwidth ∼20  nm, and confirm a propagation distance of ≈70  m.

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

Controlling the group velocity of an optical pulse typically requires traversing a material or structure whose dispersion is judiciously crafted. Alternatively, the group velocity can be modified in free space by spatially structuring the beam profile, but the realizable deviation from the speed of light in vacuum is small. Here we demonstrate precise and versatile control over the group velocity of a propagation-invariant optical wave packet in free space through sculpting its spatio-temporal spectrum. By jointly modulating the spatial and temporal degrees of freedom, arbitrary group velocities are unambiguously observed in free space above or below the speed of light in vacuum, whether in the forward direction propagating away from the source or even traveling backwards towards it.

Controlling the group velocity of light in free space has been limited to small deviations so far. Here, the authors present a method to control the spatio-temporal spectrum and allow arbitrary group velocities of a wave packet in free space both above and below the speed of light.

A trillion frames per second: the techniques and applications of light-in-flight photography

Cameras capable of capturing videos at a trillion frames per second allow to freeze light in motion, a very counterintuitive capability when related to our everyday experience in which light appears to travel instantaneously. By combining this capability with computational imaging techniques, new imaging opportunities emerge such as 3D imaging of scenes that are hidden behind a corner, the study of relativistic distortion effects, imaging through diffusive media and imaging of ultrafast optical processes such as laser ablation, supercontinuum and plasma generation. We provide an overview of the main techniques that have been developed for ultra-high speed photography with a particular focus on 'light-in-flight' imaging, i.e. applications where the key element is the imaging of light itself at frame rates that allow to freeze its motion and therefore extract information that would otherwise be blurred out and lost.

Superluminal X-waves in a polariton quantum fluid

In this work, we experimentally demonstrate for the first time the spontaneous generation of two-dimensional exciton-polariton X-waves. X-waves belong to the family of localized packets that can sustain their shape without spreading, even in the linear regime. This allows the wavepacket to maintain its shape and size for very low densities and very long times compared to soliton waves, which always necessitate a nonlinearity to compensate the diffusion. Here, we exploit the polariton nonlinearity and uniquely structured dispersion, comprising both positive- and negative-mass curvatures, to trigger an asymmetric four-wave mixing in momentum space. This ultimately enables the self-formation of a spatial X-wave front. Using ultrafast imaging experiments, we observe the early reshaping of the initial Gaussian packet into the X-pulse and its propagation, even for vanishingly small densities. This allows us to outline the crucial effects and parameters that drive the phenomena and to tune the degree of superluminal propagation, which we found to be in close agreement with numerical simulations.

Spatiotemporal evolutions of ultrashort vortex pulses generated by spiral multi-pinhole plate

We use the spiral multi-pinhole plate to generate ultrashort vortex pulses and study their spatiotemporal evolutions involving intensity, phase, orbital angular momentum, and energy current. In the experiment, a Mach-Zehnder-type interferometer is employed to perform the investigation of ultrashort vortices. Combining the experimental results and the theoretical analyses, we discuss the spatiotemporal evolutions of ultrashort vortex pulses in femtosecond regime. The results show that the distribution of orbital angular momentum in the cross-section of the vortex pulse is maintained almost invariable in the pulse duration, while both the intensity and the energy current obey a Gaussian-like distribution. With time evolution, the phase contour lines of such vortex pulses rotate around the propagation axis.

Complete measurement of spatiotemporally complex multi-spatial-mode ultrashort pulses from multimode optical fibers using delay-scanned wavelength-multiplexed holography

We introduce a simple delay-scanned complete spatiotemporal intensity-and-phase measurement technique based on wavelength-multiplexed holography to characterize long, complex pulses in space and time. We demonstrate it using pulses emerging from multi-mode fiber. This technique extends the temporal range and spectral resolution of the single-frame STRIPED FISH technique without using an otherwise-required expensive ultranarrow-bandpass filter. With this technique, we measured the complete intensity and phase of up to ten fiber modes from a multi-mode fiber (normalized frequency V ≈10) over a ~3ps time range. Spatiotemporal complexities such as intermodal delay, modal dispersion, and material dispersion were also intuitively displayed by the retrieved results. Agreement between the reconstructed color movies and the monitored time-averaged spatial profiles confirms the validity to this delay-scanned STRIPED FISH method.

Observation of image pair creation and annihilation from superluminal scattering sources

The invariance of the speed of light is one of the foundational pillars of our current understanding of the universe. It implies a series of consequences related to our perception of simultaneity and, ultimately, of time itself. Whereas these consequences are experimentally well studied in the case of subluminal motion, the kinematics of superluminal motion lack direct evidence or even a clear experimental approach. We investigate kinematic effects associated with the superluminal motion of a light source. By using high-temporal-resolution imaging techniques, we directly demonstrate that if the source approaches an observer at superluminal speeds, the temporal ordering of events is inverted and its image appears to propagate backward. Moreover, for a source changing its speed and crossing the interface between subluminal and superluminal propagation regions, we observe image pair annihilation and creation, depending on the crossing direction. These results are very general and show that, regardless of the emitter speed, it is not possible to unambiguously determine the kinematics of an event from imaging and time-resolved measurements alone. This has implications not only for light, but also, for example, for sound and other wave phenomena.

Spatio-temporal coherence mapping of few-cycle vortex pulses

Light carrying an orbital angular momentum (OAM) displays an optical phase front rotating in space and time and a vanishing intensity, a so-called vortex, in the center. Beyond continuous-wave vortex beams, optical pulses with a finite OAM are important for many areas of science and technology, ranging from the selective manipulation and excitation of matter to telecommunications. Generation of vortex pulses with a duration of few optical cycles requires new methods for characterising their coherence properties in space and time. Here we report a novel approach for flexibly shaping and characterising few-cycle vortex pulses of tunable topological charge with two sequentially arranged spatial light modulators. The reconfigurable optical arrangement combines interferometry, wavefront sensing, time-of-flight and nonlinear correlation techniques in a very compact setup, providing complete spatio-temporal coherence maps at minimum pulse distortions. Sub-7 fs pulses carrying different optical angular momenta are generated in single and multichannel geometries and characterised in comparison to zero-order Laguerre-Gaussian beams. To the best of our knowledge, this represents the shortest pulse durations reported for direct vortex shaping and detection with spatial light modulators. This access to space-time coupling effects with sub-femtosecond time resolution opens new prospects for tailored twisted light transients of extremely short duration.

Imaging cross-correlation FROG: measuring ultrashort, complex, spatiotemporal fields

We present imaging cross-correlation frequency-resolved optical gating (ImXFROG), a new method for the spatiotemporal phase retrieval of ultrashort pulses. It is demonstrated that ImXFROG can measure phase and intensity of arbitrary, spatiotemporally distorted pulses with femtosecond resolution and up to 10(7) independent variables. ImXFROG is implemented as a plug-in upgrade to an existing correlator and used to demonstrate the reconstruction of highly complex, optical pulses with femtosecond features and massive spatiotemporal distortion.

Spatio-temporal characterization of ultrashort pulses diffracted by circularly symmetric hard-edge apertures: theory and experiment

We carry out a complete spatio-temporal characterization of the electric field of an ultrashort laser pulse after passing through a diffractive optical element composed of several binary amplitude concentric rings. Analytical expressions for the total diffraction field in the time and spectral domain are provided, using the Rayleigh-Sommerfeld formulation of the diffraction. These expressions are experimentally validated. The spatio-temporal amplitude and phase structure of the pulse are measured at different planes beyond the diffractive optical element using spatially-resolved spectral interferometry assisted by an optical fiber coupler (STARFISH). Our results allow corroborating theoretical predictions on the presence of multiple pulses or complex spectral distributions due to the diffraction-induced effects by the hard-edge ring apertures.

Quantum radiation from superluminal refractive-index perturbations

We analyze in detail photon production induced by a superluminal refractive-index perturbation in realistic experimental operating conditions. The interaction between the refractive-index perturbation and the quantum vacuum fluctuations of the electromagnetic field leads to the production of photon pairs.

Spatiotemporal amplitude and phase retrieval of space-time coupled ultrashort pulses using the Shackled-FROG technique

We demonstrate the validity of the Shackled-frequency-resolved-optical-gating technique for the complete characterization, both in space and in time, of ultrashort optical pulses that present strong angular dispersion. Combining a simple imaging grating with a Hartmann-Shack sensor and standard frequency-resolved-optical-gating detection at a single spatial position, we are able to retrieve the full spatiotemporal structure of a tilted pulse.

Direct spatiotemporal measurements of accelerating ultrashort Bessel-type light bullets

We measure the spatiotemporal field of ultrashort pulses with complex spatiotemporal profiles using the linear-optical, interferometric pulse-measurement technique SEA TADPOLE. Accelerating and decelerating ultrashort, localized, nonspreading Bessel-X wavepackets were generated from a approximately 27 fs duration Ti:Sapphire oscillator pulse using a combination of an axicon and a convex or concave lens. The wavefields are measured with approximately 5 microm spatial and approximately 15 fs temporal resolutions. Our experimental results are in good agreement with theoretical calculations and numerical simulations.

Measuring the spatiotemporal field of ultrashort Bessel-X pulses

We present direct measurements of the spatiotemporal electric field of an ultrashort Bessel-X pulse generated using a conical lens (axicon). These measurements were made using the linear-optical interferometric technique SEA TADPOLE, which has micrometer spatial resolution and femtosecond temporal resolution. From our measurements, both the superluminal velocity of the Bessel pulse and the propagation invariance of the central spot are apparent. We verified our measurements with simulations.