 Review
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Optical spatiotemporal vortices
eLight volume 3, Article number: 11 (2023)
Abstract
Spatiotemporal vortices of light, featuring transverse orbital angular momentum (OAM) and energy circulation in the spatiotemporal domain, have received increasing attention recently. The experimental realization of the controllable generation of spatiotemporal vortices triggers a series of research in this field. This review article covers the latest developments of spatiotemporal vortices of light ranging from theoretical physics, experimental generation schemes, and characterization methods, to applications and future perspectives. This new degree of freedom in photonic OAM endowed by spatiotemporal vortices paves the way to the discovery of novel physical mechanisms and photonic applications in light science.
1 Main text
Vortices of light, null regions of intensity surrounded by spiral phases, constitute the skeleton of a wavefield and influence the properties of light. Since the discovery of the tight connection of optical vortices and orbital angular momentum (OAM) of light in the seminal paper in 1992 [1], considerable research progress has been made to unveil the beauty of vortices of light. Numerous applications of optical OAM have been discovered in both classic and quantum optics [2], including optical communication [3], quantum entanglement and cryptography [4,5,6], optical tweezing [7], driving torque for a micromachine [8], rotational Doppler shift [9], and imaging [10,11,12]. Optical vortices feature azimuthal phase dependence of exp(ilϕ) where l is an integer known as the topological charge. The OAM carried by each photon is proportional to the topological charge and quantized in units of ħ. The spin angular momentum (SAM), associated with circular polarization states, is limited to [− ħ, ħ] per photon. The OAM per photon, however, is theoretically unbounded. The direction of angular momentum of light is usually directed along the propagation direction. A decade ago, the interest in transverse SAM started to emerge and transversely spinning electric fields were discovered in strongly focused beams and evanescent waves [13,14,15,16,17,18,19,20,21,22,23,24]. The word “photonic wheel” is coined to describe the orthogonal relationship of the SAM direction and the propagation direction. Analogously, optical vortices carrying transverse OAM have attracted rapidly growing interest [25]. Tilted vortices of light were predicted using the special theory of relativity [26]. A spatial optical vortex is seen by a transversely moving observer near the speed of light as a tilted vortex. Spatiotemporal optical vortices (STOVs), occupying a small fraction of total energy, were observed in femtosecond filaments in air [27]. Controllable generation of STOVs with transverse OAM were demonstrated in a linear manner [28, 29]. Second harmonic generation (SHG) of transverse OAM were reported and the conservation of OAM was demonstrated [30, 31]. Rigorous calculation of transverse OAM and the coupling of transverse OAM and SAM were accomplished [32,33,34]. Schemes were designed to generate STOVs using metasurfaces and photonic crystals [35,36,37]. Undoubtedly, the experimental realization of STOVs has been driving the increasing interest in STOVs and spurring their potential applications in various optical phenomena. This review article is intended to cover the latest research progress in the field of spatiotemporal vortices in optics. The mathematical theory and physical interpretation are presented in Sect. 2. The generation, propagation and conversion of STOVs are discussed in Sect. 3. Section 4 reviews the current characterization methods and their limitations. Section 5 introduces different types of STOVs. The applications of STOVs are discussed in Sect. 6. Future perspectives on STOVs are concluded in the summary. It should be noted that this article focuses on the very recent advances related to STOVs. Readers interested in the latest developments of more general spatiotemporal wave packets and structured waves may refer to review articles [38, 39].
2 Mathematical expression and physical interpretation
2.1 Mathematical expression
The mathematical expression of an optical wave field is generally a function of the threedimensional (3D) space and the onedimensional (1D) time. The spatial and temporal dimensions are generally considered separable for simplification. In fact, the coupling of space and time is ubiquitous and becomes nonnegligible in cases of high intensity and tight focusing [40]. Spatiotemporal coupling is generally considered detrimental and regarded as side effects that should be avoided. Recently, interest has growing on exploiting the spatiotemporal coupling effect on purpose. Spatiotemporal light sheet is a typical example of spatiotemporal light field that has shown exotic features such as invariant propagation, controllable group velocity, and veiled Talbot effect [41,42,43].
The STOV is another quintessential example that takes advantage of the spatiotemporal coupling effect. In physics, the STOV belongs to a category of optical field that carries a spatiotemporal spiral phase. In contrast to conventional optical vortex that has a spiral phase expressed with two spatial dimensions, the STOV carries a spatiotemporal spiral phase expressed with one spatial coordinate and one temporal coordinate:
Equation 1 expresses a STOV wave packet propagating in the z direction under paraxial and narrow band approximation, where \(A\left( {x,y,z,t} \right)\exp \left( {il\varphi_{st} } \right)\) is the complex envelope function, \(l\varphi_{st}\) denotes the spatiotemporal spiral phase, k is the wave number of the central frequency. The azimuthal angle in the spatiotemporal plane, e.g. the x–t plane, can be expressed as
where \(\tau = {\raise0.7ex\hbox{$z$} \!\mathord{\left/ {\vphantom {z v}}\right.\kern0pt} \!\lower0.7ex\hbox{$v$}}  t\) is the local time coordinate, x_{s} and τ_{s} are the widths of the wave packet along the x and τ directions. The normalization in Eq. (2) is necessary because the spatial coordinate and the temporal coordinate are in different units. The mathematical expression of a STOV shows that the spatial and temporal dimensions are coupled and nonseparable. The spatiotemporal coupling in the form of a vortex in the spatiotemporal domain is the most salient feature of a STOV where a number of novel physical phenomena originate from. It is worthy of noting that ultrashort pulses with other nontrivial spatiotemporal coupling and energy circulation in the transverse spatial dimensions are not classified as STOVs [44,45,46,47,48,49,50].
2.2 Physical interpretation: momentum density, angular momentum density, and transverse OAM
It is well known that light carries linear momentum. The momentum density (or optical current) can be decomposed into a dominant longitudinal component that represents the light propagation in the main direction, and a local component determined by the 3D phase structure of the wave packet given by [51],
where ψ is the envelope function, Im stands for imaginary part, * denotes complex conjugate, and \(\nabla\) is the nabla operator. The OAM density equals the cross product of the position vector r and the momentum density J,
The OAM per wave packet equals the volume integral of the OAM density L. The average OAM per photon is the total OAM of the wave packet divided by the total number of photons contained in the wave packet. It is worthy of noting that the OAM is tightly connected to the spiral phase structure. A spiral phase structure, no matter in the spatial domain or the spatiotemporal domain, leads to spacedependent and tilted wave vectors. These tilted wave vectors, excluding the dominant longitudinal component, form circulating momentum density. For a spatial vortex, the momentum density circulates in the x–y plane, leading to an OAM density parallel to the z axis; For a STOV, the momentum density circulates in a spatiotemporal plane, e.g. x–t plane, resulting in an OAM density orthogonal to the z axis. The Poynting vector is given by c^{2} times the momentum density. The circulation of momentum density in a STOV indicates that a small fraction of energy circulates in the spatiotemporal domain while advancing at the speed of light along the propagation direction. This behavior of light resembles the movement of a cyclone where the rotating axis and the advancing direction are perpendicular to each other. The OAM carried by STOVs is named transverse OAM to differ from the longitudinal OAM carried by a spatial vortex beam. It is worthy of noting that there is lateral shift between the energydensity and photon (probability) centroid of a STOV. Choosing the probability centroid as the origin is preferred as discussed in [52].
3 Generation, propagation and conversion of STOVs
One theoretical paper was published in 2012 discussing how to observe a spatial vortex as a STOV [26]. However, the special theory of relativity and the requirement of a fastmoving observer makes it extremely difficult to realize a STOV in a laboratory. The STOV phenomenon was observed in the nonlinear collapse and selfarrest of an extremely intense optical pulse in air [27]. Only a small fraction of energy contributes to the phenomenon and there is no way to readily control the properties. The first controllable generation of STOVs was achieved only recently [28, 29, 53]. It circumvents the necessity of a fastmoving observer and requires neither complex nonlinear configuration nor high pulse energy.
The idea originates from the conservation of OAM of light in the spatiotemporal domain and in the spatiotemporal frequency domain. The creation of a spiral phase in the spatiotemporal domain is controlled by manipulating the phase structure in the spatiotemporal frequency domain. This breakthrough leads to the experimental generation of STOVs in a linear and manageable manner. The exploration of novel physical properties and potential applications of STOVs has since then become experimentally plausible.
Higher order STOVs are immediately observed to be easily breakable upon propagation into multiple unitary STOVs [28, 29, 54]. This linear phenomenon attributes to the imbalance between dispersion and diffraction that affect the broadening in the temporal and spatial dimensions. Detection schemes of STOVs are developed based on their diffraction properties in free space [55]. Nonlinear conversion such as second harmonic generation (SHG) and high harmonic generation (HHG) are studied and the conservation of OAM is verified.
3.1 Spiral phase modulation using 2D pulse shaper
A 2D pulse shaper consists of two diffraction gratings, two cylindrical lenses and a 2D bespoke phase element as shown in Fig. 1a. The phase element can be a spiral phase plate or a spatial light modulator (SLM) that implements a spiral phase modulation. The optical setup is positioned in a 4f configuration that the phase element, the cylindrical lenses and the gratings separated by one focal length. The diffraction grating disperses different wavelengths along the horizontal direction. The vertical direction in the SLM plane is considered as the spatial frequency direction. The SLM applies different phase to different wavelengths as well as to different spatial positions to introduce controllable spatiotemporal phase coupling. For the generation of STOVs, the SLM imprints a spiral phase in the spatiotemporal frequency plane. The spiral phase can have arbitrary topological charge with either positive or negative signs. The aspect ratio of the spiral phase pattern and the phase singularity position are tuned to obtain an optimized experimental result. The spiral phase applied by the SLM in the spatiotemporal frequency domain is maintained in the spatiotemporal domain through a 2D spatiotemporal Fourier transform.
3.2 Generation of STOV with nanostructures
Nanostructures of subwavelength feature size are versatile in controlling the amplitude, phase, and polarization of a light beam by choosing the material type, shape, and formation of subwavelength structures. The spacedependent and wavelengthdependent response makes nanostructures suitable for applying spatiotemporal modulation to a wave packet. Nanophotonic platforms have been successfully demonstrated for spatial OAM multiplexing, chiral engineering in isotropic materials, creating vortices in real space, momentum space, and the spatiotemporal domain [56,57,58,59]. Compared with the 2D pulse shaper, nanostructures are considerably compact and easily integratable. A spatiotemporal differentiator that breaks spatial mirror symmetry is designed and demonstrated to generate STOVs carrying transverse OAM without spatial or temporal Fourier transform [35]. As shown in Fig. 1b, the nanostructure is 1D periodic silicon structure with two rods per period of different heights and widths. The period of the structure is smaller than the wavelength, therefore, only the zeroth order light will be transmitted.
The parameters of the nanostructure are optimized to obtain a desired transmission spectrum function that is valid around k_{x} = 0 and Ω = 0 and is given by,
where Ω = ω − ω_{0} is the angular frequency, C_{x} and C_{t} are two complex numbers that has a phase difference. The phase singularity in the spatiotemporal spectrum plane results in a spiral phase in the spatiotemporal plane. The presented nanostructures are expected to be fabricated through standard lithography and etching processing [60, 61].
The topological property of nonlocal spatialmirrorsymmetrybreaking metasurfaces has also been investigated, adding a topological perspective for STOV generation using nanostructures [62]. All parameters in Fig. 1b are fixed except for the rod height h_{1}. For h_{1} < 230 nm, no phase singularity is observed in the transmission spectrum function. For h_{1} between 238.5 nm and 388.7 nm, the mirror symmetry is broken and phase singularities appear in the transmission map in pairs. For h_{1} > 388.7 nm, h_{1} is significantly taller than h_{2} (160 nm), the phase singularity disappears. The transition from trivial case to topologically protected STOV and back to trivial case indicates the region where the STOV pulse can be generated and immune to fabrication imperfections. Randomly generated defects such as partial convexities, concavities, and various kinds of chamfers are introduced to verify the robustness of the structure for STOV generation. The topological robustness for STOV generation is crucial for the fabrication and integration of such nanostructures with other optoelectronic devices.
Another example is the design of a photonic crystal slab structure as shown in Fig. 1c that possesses a corresponding nodal line in its transmission spectrum function [37]. The transmission spectrum function is given by,
where C_{y} is real, C_{x} is complex, and b characterizes the direction of the nodal line in the spatiotemporal frequency domain. An incident Gaussian pulse travels through the device and transforms into a STOV with its nodal line and OAM in arbitrary 3D direction.
3.3 Second harmonic generation (SHG)
The behavior of STOV pulses during frequency upconversion of SHG has been investigated [30, 31]. Using an experimental setup as shown in Fig. 1d, a fundamental STOV pulse of l = 1 is generated. Afterwards, thin beta barium borate (BBO) crystal plates are placed in both arms of a Mach–Zehnder setup. The reference pulse is frequencydoubled Gaussian pulse. The STOV pulse after nonlinear conversion, is experimentally verified to be frequencydoubled with twice the transverse OAM per photon. The law of OAM conservation is valid if no SAM coupling is involved. The thickness of the BBO crystal affects the distortion caused by spatiotemporal astigmatism. Upconverted STOV pulse of l = 2 displays a single hole and a 4π spiral phase for a thin (20 μm) BBO crystal, and two holes of 2π spiral phase for a thick (1 mm) BBO crystal. The SHG experiment with STOV pulses spurs interest of the investigation of nonlinear process with STOVs, such as stimulated Raman scattering, harmonic spin–orbit angular momentum conversion, and photonic entanglement generated by parametric process.
3.4 High harmonic generation (HHG)
High harmonic generation (HHG) is an extreme nonlinear process in lightmatter interaction. HHG with transverse OAM driven by STOV pulses shows distinct features [63]. As shown in Fig. 1e, the spatial chirp of the fundamental STOV results in a significant tilt in HHG spectrum. The spatiotemporal singularity in STOV has an impact on electron subcycle dynamics leading to fringe patterns in the spectrum. The conservation of transverse OAM holds true for HHG if no spin–orbit coupling is involved. The spectral feature can be effectively controlled by twocolor counterspin. The microscopic and macroscopic characteristics of HHG driven by transverse OAM provide a promising tool for attosecond science and extreme optics.
3.5 Dipole radiation theory
An infinitesimal dipole can be considered as an ideal point light source. Dipole models have been utilized in unidirectional excitation of electromagnetic guided modes and spin–orbit interaction via chiroptical responses [21, 64]. Focusing the complex conjugate of the radiation field emitted from an infinitesimal dipole results in a diffractionlimited spot. An array of infinitesimal dipoles with controllable relative amplitude, phase and oscillation direction digitize an optical focal field pixel by pixel (Fig. 1f). Both transversely oriented radially polarized and azimuthally polarized optical fields are demonstrated based on the timereversal method and the vectorial diffraction theory. The 4Pi optical configuration is utilized to collect dipole radiation in all directions [65]. The superposition of transversely oriented radially polarized and azimuthally polarized leads to a controllable optical field carrying controllable transverse SAM and transverse OAM. It is worthy of noting that the generated field is not a propagating wave packet, but is stationary in the focal region.
4 Characterization of STOVs
Characterization of spatiotemporal optical fields is crucial for implementing structured ultrafast pulse as a key tool for microscopy, ultrafast manipulation, and attosecond science. Spatiotemporal metrology requires accurate 3D measurement of the amplitude, phase and polarization that change rapidly both in space and time. Spatiotemporal metrology measures both spatial and spectral information, and can be categorized into two methodologies: (1) spectrally resolved spectral/temporal measurement; (2) spatiallyresolved spectral measurement techniques [66]. Scanning spectral/temporal measurement over space usually is unable to provide a full spatiotemporal measurement because these techniques are blind to the carrierenvelope relative phase and the pulse arrival time across space. A compact twofold spectral interferometer, based on inline bulk interferometry and fiberoptic coupler assisted interferometry, is reported to measure infrared femtosecond vector beams with polarization evolving at the micrometer and femtosecond scales [67]. Digital holography has been demonstrated for spatiotemporal metrology in the terahertz spectrum [68, 69]. Techniques such as the total Efield reconstruction using a Michelson interferometer temporal scan (TERMITES) and INSIGHT are promising to become available products [70,71,72]. Considering the vector nature of electric fields, spatiotemporal vector beams provide additional challenge for characterization especially when the longitudinal electric components are prominent. The characterization of STOVs, however, is a less challenging task because only 2D information is required to measure and the wave packet is considered scalar with the electric field oscillating along the direction of the spatiotemporal phase singularity line. Differentiating STOVs of different topological charge usually does not require spatiotemporal metrology, and can be accomplished based on their diffraction properties [55].
4.1 Selfreferenced interferometry technique
One easytoimplement method is the selfreferenced interferometry technique [29]. As shown in Fig. 2a, a STOV pulse is generated from a chirped wave packet. A reference pulse is split from the light source and dechirped through a grating pair. The reference pulse is transformlimited and considerably shorter than the STOV wave packet. The reference pulse is interfered with each temporal slice of the STOV wave packet using a motorized precision stage. Each interference pattern contains the information of a temporal slice of the STOV. The twodimensional amplitude and phase information can be extracted through a Fourier filtering algorithm. The threedimensional field information is reconstructed based on hundreds of twodimensional temporal slices. The interference patterns are vertical fringes at the beginning. As the slicing process moves toward the center of STOV, the vertical fringes begin to bend in the middle. As the reference pulse coincides with the spatiotemporal phase singularity of the STOV, the fringe pattern becomes two rows of fringes with a halfperiod shift that indicates a π phase difference between the upper and lower halves of the STOV. As the slicing process continues, the fringes start to bend in the other direction. The sequence of fringe bending directions is directly related to the sign of the topological charge of the STOV.
4.2 Singleshot supercontinuum spectral interferometry
STOVs are reported to be measured using singleshot supercontinuum spectral interferometry (Fig. 2b). Three beams are involved in the measurement: a STOV pulse, a supercontinuum probe pulse and a supercontinuum reference pulse. The STOV pulse causes phase modulation to the spatially and temporally overlapped chirped supercontinuum probe pulse in a thin fused silica plate (witness plate). The spatiotemporal phase imposed on the supercontinuum probe pulse is extracted in an imaging spectrometer using a supercontinuum reference pulse.
4.3 Spatially resolved spectral interferometry
Singleframe characterization of STOV pulses is accomplished using spatially resolved spectral interferometry [73]. The main characteristics such as the topological charge and helicity can be identified from the raw data. Other information including the pulse dispersion and beam divergence can be recovered after data processing and reconstruction. This characterization method enables singleshot measurements of ultrafast STOV pulses without requiring high pulse intensity.
5 Various types of spatiotemporal vortices
STOV wave packets exist in various forms that have additional spatiotemporal characteristics. The associated OAM density can be timevarying, oriented in arbitrary three dimensions. The STOV wave packets can be engineered to be propagationinvariant, or be shaped into a toroidal vortex form.
5.1 Timevarying STOV
Recently, Rego et al. demonstrated selftorqued beams carrying timevarying longitudinal OAM in extremeultraviolet spectrum [74]. Realizing wave packets carrying timevarying transverse OAM is also possible and in fact more straightforward [75]. Creating multiple spatially separated spatial vortices in the spatiotemporal frequency domain lead to multiple temporally separated STOVs within one wave packet. Each STOV can have different signs and values of topological charge. The OAM of the wave packet varies rapidly from head to tail. Experimental demonstration shows rapid transverse OAM alternation in the range from 0.5 to 1 ps.
5.2 Tilted STOV
Each photon within a STOV wave packet carries orbital angular momentum. The direction of the OAM is parallel to the phase singularity line and perpendicular to the light propagation direction. Since the discovery of transverse OAM, it is intriguing to explore plausible schemes to realize tilted STOVs with a tilted phase singularity line and OAM in arbitrary 3D directions.
The concept of a tilted STOV was first predicted using the special theory of relativity. A monochromatic spatial vortex beam emitted from a fastmoving source is seen by an observer as a tilted STOV [26]. This concept is interesting but difficult to experimentally realize it. Since the experimental realization of STOVs with transverse OAM, various schemes have been proposed and demonstrated to generate tilted STOVs. Photonic crystals with custom nodal lines are theoretically proposed to be capable of creating tilted STOVs [37].
Cylindrical lens system is commonly used as an astigmatic mode converter for transformation between Hermite Gaussian and Laguerre Gaussian modes [76]. Cylindrical lens system has been proven capable of coupling longitudinal OAM component to a STOV wave packet carrying pure transverse OAM to obtain a tilted STOV with a tilted OAM [77]. Figure 3a shows the experimental setup that utilizes cylindrical lens mode converter to create tilted STOVs. The tilt angle of the STOV is directed related to the rotation angle of the cylindrical lens pair. The tilt angle is limited to around 40° as the rotation angle of the mode converter increases to 80°. Further increase in the rotation angle causes the wave packet to break apart due to the asigmatic focusing effect.
Tilted STOVs carrying 3D oriented OAM have potential applications in highspeed optical communication [3, 78], generation of entangled tilted OAM states [79, 80], and lightmatter interaction such as sculpting photocurrent distribution in semiconductors [81, 82].
5.3 Besseltype STOV
STOV wave packets are 2D solutions to the Maxwell’s equations that have spatiotemporal spiral phase. The expression is a function of 1D spatial coordinate and 1D temporal coordinate. These solutions are analogues of wellstudied spatial modes that are dependent on two spatial coordinates. A Bessel beam is a propagationinvariant with selfhealing ability. A Bessel beam corresponds to a ring shape in the spatial frequency domain. An axicon can apply a conical phase to a plane wave and turns it into a Bessel beam. Similarly, a spatiotemporal Bessel wave packet can be generated by applying a spatiospectral conical phase [83]. A Bessel STOV is a spatiotemporal Bessel wave packet that has a spatiotemporal spiral phase and multiplering structure for the intensity distribution in the spatiotemporal domain. If the dispersion and diffraction effects can be balanced using materials of custom dispersion parameters or using nonlinear process, the Bessel STOV is expected to have the central lobe propagation invariant both spatially and temporally for a long distance.
5.4 Degradationfree STOV
Higherorder STOVs are unstable and tend to split into a number of elemental STOVs upon free space propagation [54]. This phenomenon is attributed to spatiotemporal astigmatism, i.e. the wave packet broadens in the spatial dimension due to diffraction and undergoes no dispersion in the temporal dimension. The free space propagation is necessary for spatiotemporal Fourier transform and the degradation of highorder STOVs seems unavoidable. One solution to the problem is to view the SLM plane in a 2D pulse shaper (Fig. 1a) as the x–ω plane instead of k_{x}–ω plane and generate STOVs without free space propagation [84]. The generated STOVs corresponds to a circle on the light cone in contrast to a patch that is associated with a conventional STOV (Fig. 3b). The topological charge of the degradationfree STOVs is experimentally achieved up to 100.
5.5 Toroidal vortex
STOVs are spatiotemporal wave packets with 2D degrees of freedom: one spatial coordinate and one temporal coordinate. Transformation optics can be utilized to take a leap forward to the customization of 3D spatiotemporal wave packets. One interesting example is the demonstration of optical toroidal vortex [85, 86]. A toroidal vortex of light is a ringshaped wave packet with poloidal spatiotemporal spiral phase. A STOV wave packet is elongated to a STOV tube and then conformally mapped to a toroidal vortex using transformation optics that completes logpolar to Cartesian mapping. The toroidal vortex is an approximate solution to the Maxwell’s equations. The hollow core of the toroidal vortex corresponds to a circular phase singularity line (Fig. 3c).
A distinct type of toroidal vortex has a circular polarization singularity line [87]. The vectorial vortex ring has circulating electric (or magnetic) field lines surrounding the polarization singularity circle. Metasurfaces are fabricated to generate vectorial vortex rings in the terahertz domain. The longitudinal size of these vortex rings is wavelengthlong, considerably shorter the transverse dimensions (Fig. 3d).
Numerical simulation has shown that tight focusing of cylindrical vector 2DSTOV wave packets lead to peculiar Yo–Yo ball like spatiotemporal distributions. Particularly, highly focused radially polarized 2DSTOV wave packet creates a transverse magnetic toroidal topology whereas highly focused azimuthally polarized 2DSTOV wave packet leads to transverse electric toroidal topology [88].
Although toroidal vortices of light are new, their analogues in fluid dynamics have been studied for a long time, such as the bubble rings, microbursts, and dandelion flying. Comparison of different types of toroidal vortices may bring insights to the study of photonic toroidal vortex in different aspects such as highdimensional information carrier, toroidal mode excitation, and stable and highenergy pulses.
5.6 Collision of STOV and spatial vortex
A STOV wave packet can also embed an additional spatial vortex [89]. The phase singular lines of spatiotemporal and spatial vortices are crossed and normal to each other as shown by the isointensity plot (Fig. 3e). At the intersection, the two vortex tunnels intertwine with each other and both of them are bent with a bending direction dependent on the sign of the topological charge. The complex threedimensional phase structure results in spacedependent angular momentum density. The average OAM per photon within the wave packet is determined by the topological charge of the spatiotemporal and spatial vortices as well as the intensity distribution of the wave packet. Both the value and the 3D orientation of average OAM per photon are fully controlled.
5.7 Partially coherent STOV
STOVs can be generated from a light source with partial temporal coherence with fluctuating temporal structures. Without the need of modelocked laser sources, the partially coherent STOVs provide a costeffective light source carrying transverse OAM.
Partially coherent vortex beams carrying longitudinal OAM have received much attention and found their applications in beam shaping, beam rotation and selfreconstruction. The description and propagation of STOVs with partial temporal coherence has been proposed by Hyde [90]. The experimental realization has also been accomplished recently [91]. The amplified spontaneous emission (ASE) is used as a partially coherent source and modelled by applying a randomly distributed spectral phase and a Gaussian spectrum profile. As the phase randomness increases, the shape of the STOVs deviate from the ringshaped profile with singularities occurring at various temporal locations. Amplitude fluctuations randomly spread throughout the whole temporal domain with multiple amplitude peaks. It is found that higherorder partially coherent STOVs break up into unitary vortices even without propagation. The instability of higher order partially coherent STOVs is an analog to the spatial vortex splitting caused by atmospheric turbulence [92]. An ytterbiumdoped singlemode fiber laser with a central wavelength of 1030 nm is driven below the lasing threshold and the ASE is used as the light source for the experiment. The ASE from the singlemode fiber has a high degree of spatial coherence because only the fundamental transverse mode is lasing. The ASE state is temporally incoherent. Besides the ASE, a noiselike pulse (NLP) state from the fiber laser has also been investigated. The NLP state provides partial temporal coherence where the longitudinal modes are not strictly phase locked. The NLP state is generated through adjusting the nonlinear polarization evolution based saturable absorber. The spectrum width of the NLP state is 52 nm. STOVs generated from NLP states are useful for highpower applications because NLP states usually have higher peak power than ASE.
Both the spatial and temporal coherence of STOVs are reported to be controllable. It is found that the spatial and temporal coherence parameters play a vital role in controlling the phase distribution and position of a partially coherent STOV [93].
6 Applications of STOVs
STOV wave packets have spatiotemporal vortex shape and carry transverse OAM. New effects and phenomena are expected when this new degree of freedom for photons interacts with matter. Thus, quickly after its laboratory realization, the applications of STOVs in both fundamental physics and photonic engineering are being actively investigated.
6.1 Optical manipulation
Since the discovery of the connection of spatial spiral phase with longitudinal OAM, vortex beams have been vastly utilized in optical manipulation. The interaction between small particles and light carrying OAM generates lightinduced toque and leads to rotation of particles. The conservation of angular momentum is a fundamental law that governs lightmatter interaction. The transfer of angular momentum is generally limited to the direction parallel to the beam propagation direction. Nanoparticles can be rotated about the singularity line of a spatial vortex beam. Recently, the observation and measurement of transferring transverse spin and orbital angular momentum to micro particles of several micrometers in size are reported [94] (Fig. 4a). Transverse angular momentum transfer is especially advantageous in creating shear stress for studying mechanotransduction within the cell. STOVs carrying transverse OAM are expected to rotate particles about an axis that is perpendicular to the propagation direction of the wave packet and provide a versatile light source for transferring transverse OAM to small particles.
6.2 Interaction of transverse OAM with SAM
STOVs carry pure transverse intrinsic OAM. A Besseltype STOV can be constructed using a superposition of plane waves with wave vectors distributed over a circle within the k_{x}–k_{z} plane [32]. These wave vectors have a spiral phase difference proportional to the topological charge l. The spectrum curve on the light cone of nondiffracting Besseltype STOVs is an ellipse with the major axis parallel to k_{z}. The ellipse is Lorentz transform related to a circle. The electric field of each plane wave constituting Besseltype STOVs must be orthogonal to its individual wave vector k. Two polarization situations, i.e. out of plane and in plane cases exist for Besseltype STOVs. The outofplane polarization case is equivalent to the scalar case. The inplane polarization case is less trivial because the electric field has both x and z components and is dependent on the wave vector of each plane wave (Fig. 4b). The interference of plane waves with different phase, spatial frequency and linear polarizations result in nonzero spin density indicating the presence of transverse spin. Despite the local spin–orbit interaction, the integral SAM vanishes and the integral OAM equals to lℏ per photon as long as the intensity distribution is circularly symmetric. If the intensity profiles deviate from circular symmetry, the OAM per photon is larger than lℏ.
Tight focusing structured light and structuring STOVs in the focal region also involves the coupling of transverse OAM and SAM [34]. Using the Debye integral, the focal spot is calculated to have an intensity distribution in the shape of an elongated ring with a hollow core along the yaxis. The conversion from longitudinal SAM of the incident light to transverse OAM in the focused light occurs during the focusing process. The spin–orbit couplings involving transverse OAM are expected to be exploited in lightmatter interactions.
6.3 Quantum optics
Semiclassical and approximate quantum theories break down when it comes to study structured singlephoton pulses. The description of single photons with Stokes parameters and the Poincaré sphere ignores the vector nature of local spin and orbital AM densities. Recently, a framework is established based on quantum field theory to engineer the local angular momentum densities of quantum structured light pulse, especially for STOVs [95]. The quantum correlator of photonic spin density is introduced to characterize the nonlocal spin noise in light, facilitating the process of exploring exotic phases of light with longrange spin order. It is predicted that large fluctuations in the OAM along orthogonal directions exist in Bessel pulses with large OAM. This quantum noise is claimed to be veritable in metrology experiments with OAM laser beams. The quantum uncertainties \(\hat{L}_{x}^{obs}\) and \(\hat{L}_{y}^{obs}\) are linearly proportional to the photon number and proportional to the square of the helical phase index m in a Bessel pulse. The OAM fluctuations in the transverse plane are dependent on the polar angle of the Bessel pulse. It is demonstrated that the spin texture of a singlephoton pulse can exhibit a very rich and interesting structure in the vectorial case (Fig. 4c). The spin density of a coherent pulse exhibit several features: the projection in the x–y plane is symmetric around the zaxis, leading the spatial integral to vanish; the projection in the x–y plane is either parallel or antiparallel to the azimuthal unit vector, resulting in the helical spin texture; the projection in the x–y plane is described by the product of two Bessel functions and the clockwise and anticlockwise structures oscillate in the spin texture; the sign of the zcomponent of spin is always positive (negative) for LCP (RCP) pulse, leading to the nonvanishing global spin.
6.4 Information carrier
Optical communication relies on the transmission of optical pulses to convey digital information. To expand the communication bandwidth, multiple physical quantities are incorporated, such as intensity, wavelength, phase and polarization. In the past couple of decades, increasing interest has been paid on the utilization of vortex beams carrying longitudinal OAM. The realization of transverse OAM undoubtedly adds an additional degree of freedom to OAMbased optical communication. More importantly, toroidal vortices are closely related to particlelike waves such as hopfions [96]. Hopfions are high dimensional particlelike waves that contain the information of stereographic projection from threesphere (S^{3}) to twosphere (S^{2}) (Fig. 4d). Using hopfions as information carrier will increase the information dimension per pulse for optical communication.
6.5 Spatiotemporal differentiator
As discussed in Sect. 3.2, 1D periodic structure that breaks the mirror symmetry has been designed to generate STOVs [35]. Similar to topological spatial differentiator, this structure can be utilized as a spatiotemporal differentiator. The resolution of detecting sharp changes of 18 μm in the spatial coordinate and 182 fs in the temporal coordinate have been reported (Fig. 4e).
6.6 Slow light and fast light
Small shifts and time delays of wave packets such as Goos–Hänchen shift, Wigner time delay and spinHall effect have attracted increasing attention in nanophotonics. These shifts and time delays are typically on the scale of a wave period and can be extended to beam/pulse width through weak measurement technique. Vortex beams carrying longitudinal OAM have shown increased beam shifts by the factor of the topological charge. STOVs carry intrinsic transverse OAM exhibit peculiar properties of beam shifts and time delays depending on the value and orientation of the OAM [97]. Time delays become independent of the frequency enabling subluminal and superluminal pulse propagation without medium dispersion. The reflection/refraction of STOVs can be categorized into two cases: (A) the OAM is perpendicular to the incident plane; (B) the OAM lies within the incident plane (Fig. 4f). The normalized integral OAM of the incident pulse is given by,
where γ is the ellipticity of STOV and l is the topological charge. The STOV is inverted in the reflected pulse for case A and remains unchanged for case B. For the transmitted pulse, the STOV is stretched by a factor of \(\frac{{\cos \theta^{\prime}}}{\cos \theta }\) in the transverse direction and squeezed by a factor of 1/n in the longitudinal direction, where n is the relative refractive index. Paraxial polarized STOVs are assumed to experience all shifts known for Gaussian wave packets such as Goos–Hänchen and spinHall shifts. In addition, STOVs experience three additional shifts that are ldependent. The first type is related to the conservation of the zcomponent of the total angular momentum. The orbitalHall effect requires the transverse yshift of the refracted pulse that generates an extrinsic OAM to compensate the imbalance between the z component of the intrinsic OAM between the incident and transmitted pulses. The typical scale of this shift is on the order of the wavelength and dependent on the topological charge l. The second type of shift is the angular Goos–Hänchen and spinHall shifts with an additional factor of (1 + l). The typical scale is independent of the ellipticity γ and is on the order of the inverse Rayleigh range. The third type of ldependent shifts are the longitudinal shifts of STOVs reflected/refracted by a planar interface. These shifts are equivalent to time delays. However, unlike the Wigner time delays produced by temporal dispersion, these time delays originate from the spatial dispersion and thus are pure geometric phenomena. These time delays are be exploited to achieve subluminal and superluminal pulse propagation based upon Fresnel reflection/refraction instead of using dispersive media.
7 Summary and perspectives
Vortices are ubiquitous in nature. STOVs represent a novel type of optical vortices that have topological and conservative properties similar to spatial vortices as well as novel properties such as carrying transverse OAM. STOVs and their variants have exhibited peculiar photonic properties in various optical phenomena and become applicable in optical manipulation, spatiotemporal differentiator, subluminal and superluminal pulse propagation, and free space optical communication. Research remains to be done to fully understand the physical characteristics of these spatiotemporal phase singularities and explore their applications. Potential interesting topics include but certainly not limited to: propagation of STOVs via optical fibers and slab waveguides; reflection, refraction, and scattering of STOVs in nonuniform, anisotropic, or nonlinear materials; modeexcitation and light manipulation in nanostructures using STOVs; STOVassisted dichroism for probing molecular chirality [98–100], optical parametric chirpedpulse amplification (OPCPA) of STOVs, characterization and applications of ultrafast and ultraintense STOV pulses in lightmatter interaction. It is always hard to predict the future of such a nascent research area that is rapidly developing. However, it is well expected that an increasing amount of physical mechanisms associated with STOVs are to be discovered and a wide range of applications are to be found in the near future.
Data availability
All data presented in the report are available upon reasonable request.
Abbreviations
 OAM:

Orbital angular momentum
 STOV:

Spatiotemporal optical vortex
 1D/2D/3D:

One/two/three dimension
 SAM:

Spin angular momentum
 SHG:

Second harmonic generation
 HHG:

High harmonic generation
 BBO:

Beta barium borate
 TERMITES:

Total Efield reconstruction using a Michelson interferometer temporal scan
 SLM:

Spatial light modulator
 OPCPA:

Optical parametric chirpedpulse amplification
 ASE:

Amplified spontaneous emission
 NLP:

Noiselike pulse
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We acknowledge the support from the National Natural Science Foundation of China (NSFC) [92050202 (Q.Z.), 61875245 (C.W.)], Shanghai Science and Technology Committee [19060502500 (Q.Z.)], Wuhan Science and Technology Bureau [2020010601012169 (C.W.)], and the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) [2022R1A2C1091890 (A.C.)].
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Wan, C., Chong, A. & Zhan, Q. Optical spatiotemporal vortices. eLight 3, 11 (2023). https://doi.org/10.1186/s43593023000426
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DOI: https://doi.org/10.1186/s43593023000426
Keywords
 Spatiotemporal optical vortex
 Orbital angular momentum
 Spin–orbit interaction
 Linear momentum density
 Toroidal vortex
 Topological charge