Volume 14, Issue 6 - Special Issue: Spatiotemporal Optical Fields

Shaping transverse degrees of freedom across different temporal frequency spectra has unlocked new possibilities for realizing a wide variety of novel spatiotemporal phenomena. In particular, using the discrete frequencies of optical frequency combs for spatiotemporal synthesis enables precise spatial separation and control of individual spectral lines, thereby facilitating the experimental generation of space-time wave packets (STWPs). This review explores the theoretical investigation and experimental demonstration of versatile STWPs synthesized using correlated frequency comb lines and spatial fields, including the following: (a) STWPs exhibiting dynamic evolution of spatial properties; (b) STWPs with customized group velocities; and (c) longitudinal control over the dynamic evolution of STWPs.

Ultrafast fibre lasers, characterized by ultrashort pulse duration and broad spectral bandwidth, have drawn significant attention due to their vast potential across a wide range of applications, from fundamental scientific to industrial processing and beyond. As dissipative nonlinear systems, ultrafast fibre lasers not only generate single solitons, but also exhibit various forms of spatiotemporal soliton bunching. Analogous to molecules composed of multiple atoms in chemistry, soliton molecules (SMs) – alias bound states – in ultrafast fibre lasers are a key concept for gaining a deeper understanding of nonlinear interaction and hold a promise for advancing high-capacity fibre-optic communications. SMs are particularly notable for their high degree of controllability, including their internal temporal separation, and relative phase differences, thereby suggesting new possibilities for manipulating multi-pulse systems. In this review, we provide a comprehensive overview of recent advancements in the studies of SMs with the multidimensional parameter space in ultrafast fibre lasers. Owing to the flexibility afforded by mode-locking techniques and dispersion management, various types of SMs – with diverse values of the soliton number, relative phase, pulse separation, carrier frequencies, and even modal dispersion – have been experimentally demonstrated. We also discuss other basic nonlinear optical phenomena observed in fibre lasers, including the formation, spatiotemporal pulsations, and interaction dynamics of SMs. Furthermore, we explore the multidimensional control of SMs through approaches such as gain modulation, polarization control, dispersion management, and photomechanical effects, along with their applications to optical data encoding. Finally, we discuss challenges and future development of multidimensional technologies for the manipulation of SMs.

Joint space-time modulation of light fields has recently garnered intense attention for enabling precise control over both spatial and temporal characteristics of light, leading to the creation of space-time beams with unique properties, such as diffraction-free propagation and transverse orbital angular momentum. Here, we theoretically propose and experimentally demonstrate spatiotemporal Moiré lattice light fields by controlling the discrete rotational symmetry of a pulse’s spatiotemporal spectrum. Using a 4 f pulse shaper and an x − ω modulation strategy, we generate tunable spatiotemporal Moiré patterns with varying sublattice sizes and confirm their diffraction-free behavior in time-averaged intensities. Additionally, we demonstrate spatiotemporal Moiré lattices carrying transverse orbital angular momentum. These findings provide a novel platform for studying spatiotemporal light–matter interactions and may open new possibilities for applications in other wave-based systems, such as acoustics and electron waves.

Dissipative optical solitons, i.e. packets of radiation localized not due to the presence of optical inhomogeneities of the scheme or medium, but due to the balance of energy inflow and outflow in a nonlinear medium, deserve special attention for a number of reasons. First, these solitons are “calibrated” with a discrete set of basic parameters. This will lead to their increased stability: dissipative solitons are attractors, they are not sensitive to small perturbations. Second, progress in laser technology and the emergence of new laser and nonlinear optical materials provides an opportunity not only to study the rich physics of dissipative solitons, but also to propose their promising applications. This paper, which combines both a review of the current level of theory and original results, is devoted mainly to new types of these solitons. These types exploit the topological features of structured radiation, characteristic of vector, polarization dissipative solitons, which have a nontrivial internal structure. We sequentially present one-dimensional (1D), two-dimensional (2D) and three-dimensional (3D) polarization solitons, identify limitations in the topological protection of the information that can be encoded by topological charges and indices and discuss development prospects in this area.

Reconnections of spatiotemporal optical vortices have been shown to occur between line vortices. Here, we show that reconnections also occur between spatiotemporal loop vortices in optical waves. As optical loop vortices propagate in a media with spatial diffraction and material group velocity dispersion, unique reconnections occur. The birth and death of loops can occur, with certain loop vortices emerging from or collapsing to a single point while interacting with others. As certain parameters are varied in the model, complex arrangements of loops form in space-time from simple initial fields.

Topological structures reveal the hidden secrets and beauty in nature, such as the double helix in DNA, whilst, the manipulation of which in physical fields, especially in ultrafast structured light, draw booming attention. Here we introduce a new family of spatiotemporal light fields, i.e. helical pulses, carrying sophisticated double-helix singularities in its electromagnetic topological structures. The helical pulses were solved from Maxwell’s equation as chiral extensions of toroidal light pulses but with controlled angular momentum dependence. We unveil that the double helix singularities can maintain their topological invariance during propagation and the field exhibits paired generation and annihilation of vortices and antivortices in ultrafast space-time, so as to be potential information carriers beating previous conventional vortex structured light.

When light propagates through a randomly correlated, slowly varying medium, it generates optical branched flow. Previous studies have demonstrated that the self-focusing effect in optical media can accelerate the appearance of the first branching points and sharpen the filaments of branched flow. In this study, we investigate the influence of the nonlocality of the nonlinear response on branched flow. We find that, due to its averaging effect, as the range of nonlocality increases, the first branching point shifts to a greater distance, and the flow structures broaden, thus nonlocality ultimately restores the branched flow to its linear condition. We have developed a semi-analytical formula and confirmed the screening of the self-focusing effect on branching flow by nonlocality.

We report the experimental observation and characterization of Replica Symmetry Breaking (RSB) manifestation while analyzing the transverse intensity profile of laser pulses in filamentation experiments using sapphire crystal and distilled water, excited by a femtosecond laser centered at 800 nm. The RSB arises from the competition between self-focusing and plasma defocusing, subject to local fluctuations in the nonlinear refractive index generated by plasma via multiphoton excitation, which subsequently promotes frustration among modes. Our results confirm the existence of glassy-like photonic states not only in multifilamentation, as previously reported [ W. Ettoumi, J. Kasparian, and J. Wolf, “Spin-glass model governs laser multiple filamentation,” Phys. Rev. Lett., vol. 115, no. 3, pp. 033902, 2015 ], but also in the generation of a single filament and in filamentation accompanied by conical emission. These findings improve the understanding of statistical nonlinear optics by establishing connections with magnetism and highlighting the glassy-like behavior of light in the context of ultrafast optical phenomena.

Discrete spatial solitons are self-consistent solutions of the discrete nonlinear Schrödinger equation that maintain their shape during propagation. Here we show, using a pump-probe technique, that soliton formation can be used to optically induce and control a linear topological end state in the bulk of a Su–Schrieffer–Heeger lattice, using evanescently-coupled waveguide arrays. Specifically, we observe an abrupt nonlinearly-induced transition above a certain power threshold due to an inversion symmetry-breaking nonlinear bifurcation. Our results demonstrate all-optical active control of topological states.

We measure the change in transverse orbital angular momentum (tOAM) per photon, Δ⟨ L y ⟩, applied to an optical pulse by a pure amplitude perturbation. The results are in excellent agreement with calculations and simulations of the spatiotemporal torque based on our tOAM theory [Phys. Rev. Lett. 127 , 193901 (2021)]. The crucial factor in determining Δ⟨ L y ⟩ is the spatiotemporal distribution of tOAM density in the pulse. We show that even Gaussian pulses with zero total tOAM can have net tOAM induced by an amplitude perturbation stationary in the lab frame. As a prelude to the paper, we review and clarify several recent theoretical approaches to tOAM and reemphasize several fundamental principles needed for the correct analysis of experiments and simulations.

When an excited electromagnetically open optical waveguide goes through a temporal transition of its material properties, it radiates to the ambient surroundings. In this paper, we explore this radiation and reveal, using asymptotic evaluation of path integral in the complex frequency (Laplace) plane, a peculiar space-time dependence of its frequency. Specifically, we derive an asymptotically exact formula (Eq. (11)) for the instantaneous radiation frequency, which exhibits a chirp behavior with respect to time. This simple formula depends on the ambient properties and on the longitudinal wavenumber β of the guided mode before the temporal transition but not on the specific waveguide structure or materials. In addition, we derive a t −3/2 decay rate of the radiative field amplitude on time. We verify our analytic results using full-wave simulations of a dispersive and lossy indium tin oxide waveguide that undergoes smooth temporal long transitions over up to ∼200 cycles at the initially guided mode frequency. Thus, these theoretical findings offer valuable insights into the behavior of general optical waveguides experiencing temporal transitions and provide a powerful tool for analyzing and designing such THz and optical setups, with potential use in sensing and imaging.

Research on time crystals concerns the spontaneous breaking of translational symmetry in time, as well as the realization of phenomena and phases known from solid-state physics in the time domain. Periodically driven systems of massive particles are widely used in these studies. In the present work, we consider a photonic system and demonstrate that stable nonlinear propagation of a strong optical wave in a fiber with the third-order dispersion may lead to the establishment of quasi-periodic oscillations in the electromagnetic field intensity. A second, weaker signal optical wave propagating in the fiber senses these oscillations and, as a result, undergoes exponential localization in time. This is a temporal analog of Aubry–André localization. If an optical detector is placed at a certain position in the fiber, the temporal localization of the probe wave will be observed in the form of the signal which emerges and then decays as a function of time.

We analyze the spatiotemporal solitary waves of a graded-index multimode optical fiber with a parabolic transverse index profile. Using the nonpolynomial Schrödinger equation approach, we derive an effective one-dimensional Lagrangian associated with the Laguerre–Gauss modes with a generic radial mode number p and azimuthal index m . We show that the form of the equations of motion for any Laguerre–Gauss mode is particularly simple, and we derive the critical power for the collapse for every mode. By solving the nonpolynomial Schrödinger equation, we provide a comparison of the stationary mode profiles in the radial and temporal coordinates.

Space-time separability is commonly assumed in the theoretical description of laser beams. However, recent progresses have demonstrated that this assumption often breaks down for ultrashort realistic pulses, giving rise to spatio-temporal effects that modify both the spatial and temporal characteristics of the laser field. In this work, we introduce semi-analytical and numerical diffraction integral models to investigate these spatio-temporal effects in tightly focused configurations. In particular, we investigate how the TM 01 beam mode is modified at the focus by chromatic angular dispersion, curvature, and spatial chirp. We compare the two formalisms, thus creating a toolset for modeling extreme localization of structured electromagnetic beams in time and space.

Spin–orbit coupling (SOC) in tightly focused optical fields offers a powerful mechanism for manipulating the complex motion of particles. However, to date, such a mechanism has only been applied to the single-orbit motion for particles, while multi-orbital dynamics have not yet been experimentally demonstrated. Here, the theoretical and experimental realization of dual-orbit rotational dynamics of nanoparticles in a tightly focused circularly polarized Laguerre-Gaussian beam is reported. Analyses reveal that the dual-orbit rotation of nanoparticles originates from SOC in a tightly focused vortex beam, with the motion velocity and direction determined by the topological charge of the beam. Experimentally, the dual-orbit rotation of polystyrene nanoparticles was observed for the first time using an inverted optical tweezer. In addition, the rotation velocity showed a clear linear dependence on the topological charge of the incident beam. This work reveals the pivotal role of SOC in enabling precise dual-orbit control at the nanoscale, paving the way for applications in optical sorting, grinding and delivery of microparticles.

Understanding the intricate properties of spatiotemporal optical vortices (STOVs) is crucial for their growing number of applications, particularly where they drive light–matter interactions that can create up-converted STOV-like structures at attosecond and nanometer scales. We derive closed-form expressions for the propagation of STOVs, and spatiotemporal (ST) tilted Hermite-lobed (THL) pulses forming STOVs, through cascaded optical systems represented by ABCD matrices such as free space propagation, lenses, mirrors, similar in simplicity as those in Gaussian beam optics. STOVs and ST THL pulses are found to be spatial and temporal Fourier transform pairs at the same time, so that a STOV is a spatiospectral (SS) THL pulse in SS domain and a ST THL pulse is a SS vortex. This duality allow us to find expressions for the transformation of their spatiospectra through arbitrary optics, which is particularly important at the attosecond and nanometer scales, where ST characterization techniques are limited.

Photocurrent-induced harmonics appear in gases and solids due to tunnel ionization of electrons in strong fields and subsequent acceleration. In contrast to three-step harmonic emission, no return to the parent ions is necessary. Here we show that the same mechanism produces harmonics in metallic nanostructures in strong fields. Furthermore, we demonstrate how strong local field gradient, appearing as a consequence of the field enhancement, affects photocurrent-induced harmonics. This influence can shed light at the state of electron as it appears in the continuum, in particular, to its initial velocity.

Intra-system entanglement occurs between non-separable modes within the same system. For optical systems, the various degrees of freedom of light represent different modes, and the potential use of light to create higher dimensional classical entangle states offers a promising potential to drive new technological developments. In this work, we present experimental results demonstrating the orthogonality between transverse orbital angular momentum (t-OAM) of different spatiotemporal topological charges, a previously unverified property of t-OAM. Based on those results, we developed methods to create and characterize a novel family of t-OAM and polarization entangled spatiotemporal structured light. We further provide theoretical analysis to support our study of the entanglement between those modes. By demonstrating the feasibility of leveraging t-OAM as a new family of modes for classical entanglement, our work represents a new advancement towards higher dimensional classical entanglement strategies.