时空光场调控以及时空光涡旋波包的研究进展（特邀）

曹前，詹其文

（上海理工大学光电信息与计算机工程学院，上海200093）

0 Introduction

The development of ultrafast laser has opened new avenues for physicists and scientists.Differed from Continuous Wave（CW）laser，ultrafast laser emits output in the form of optical pulses whose pulse duration is at 10～100 femtosecond scale（1 fs = 10-15s）［1-2］.Thanks to its short pulse duration，high pulse energy，high peak intensity，and a broadband spectral coverage，ultrafast laser has become an enabling tool in the research fields for probing ultrafast events［3］，strong field physics［4］，nonlinear optics［5-6］，and spectroscopic applications［7-8］.

In recent years，there is a growing research interest in studying complex Spatiotemporal coupled（STc）optical fields generated by ultrafast lasers.These complex STc optical fields exhibit unique photonic properties that are previously unavailable.For example，Spatiotemporal Optical Vortices（STOV）wavepackets that have a spiral phase in ST domain can carry photons with a pure transverse Orbital Angular Momentum（OAM）［9-19］.Space-time light sheets with a modulated field distribution in the spatial frequency and spectral domain（k-ωspace）can possess an arbitrary group velocity［20-25］，achieve negative refraction on interfaces［26］，propagate with self-accelerating in space-time［27］or propagate without spread in space［28-37］.These newly published research works have pushed the frontier of photonic researches and created new opportunities in the photonics research field，and hence we feel it is necessary to review the state-of-art research status of STc optical fields，especially for research works published in the last decade.

In this paper，we review the research progress of novel STc optical fields with a particular emphasis on the newly discovered STOV wavepackets.We review the origin and early study of STc optical fields.Then we introduce the theoretical model for describing STc optical field.It covers the mathematical equations for simulating STc optical field and provides an example of the evolution of STOV wavepackets under dispersive and non-dispersive propagation.Thirdly，we discuss the experimental realization of STc optical fields.In generating STc optical fields，the interplay between dispersion and diffraction is more subtle compared with conventional pulse shaping or beam shaping techniques.These will be discussed in this section.Finally，we give a detailed review of the newly discovered STOV wavepackets in the fourth section.We hope this review can serve as a mini guide book for more photonics researchers in studying novel STc optical fields.

1 Origin and early study of STc optical fields

Due to the time-frequency uncertainty principle，ultrafast laser has a significantly broader spectrum compared with CW laser and can be thus affected by chromatism.A typical source for chromatism is material dispersion，i.e.，broadband light propagating in bulky optical elements such as lenses and prisms.A simple example of chromatism is the purple fringes found in the high contrast boundary areas of a smartphone photo（see Fig.1（a））.Those purple fringes are caused by a radially varying dispersion of the imaging system that results in a color-dependent lateral shift at the image plane［38］.Such chromatic effect is considered detrimental to the image quality and optics scientists have invented achromatic lenses to prevent it.

It is noteworthy that smartphone photos are images illuminated by incoherent light sources，and，therefore，chromatism only affects the light propagation geometrically.For an ultrafast laser，the output pulse profile in the time domain is the result of constructive and destructive interference between a phase-coherent broadband spectrum in the frequency domain.In this case，chromatism leads to a spectrally varying phase and it can severely distort the temporal pulse profile.Fig.1 illustrates two examples for the influence of chromatism for an ultrafast laser，including pulse broadening［2］as shown in Fig.1（b）and pulse compression［39-42］as shown in Fig.1（c）.For almost any ultrafast laser oscillator，the dispersion management is of great importance［43-44］.External to the laser cavity，a proper management of dispersion has been widely used in Chirped Pulse Amplification（CPA）technique［45］，facilitating the construction of super-intense lasers［46］.

Fig.1 Chromatism on optical beams，pulses，and Spatiotemporal（ST）optical fields

There are already several review articles published on the theory，measurement technique and application for those STc optical fields［71-74］that mainly have a first-or second-order coupling between the spatial（or angular）and temporal（or spectral）coordinates.In recent years，there are more research works on studying STc optical field with a much more complex field structure.In the next section，we are to introduce the theoretical model for these complex STc optical fields，which provides the theoretical background for studying these fields.

2 Theoretical model for STc optical field

Spatiotemporal （ST） optical field is characterized in a three-dimensional (x，y，t) space.For a spatiotemporally uncoupled field，the field can be written as the product of its transverse field（beam）and its temporal field （pulse），E(x，y，t) =E(x，y) ·E( )t.Therefore，the ST field can be analytically or experimentally studied separately in the spatial domain and time domain［75-77］.For Spatiotemporally Coupled（STc）optical field，it cannot be written as the space-time field product and it is thus mandatory to use a theoretical model that accounts physical effects in the spatial domain and time domain simultaneously.In this section，we will introduce the theoretical model used for studying STc optical fields and how to use the model to simulate the evolution of STc optical fields under dispersive or non-dispersive propagation.

In most scenarios，STc optical fields are investigated in the paraxial regime，meaning the dimension of the STc fields in the transverse domain is much larger than the optical wavelength，wb≫λ0.This assumption can greatly simplify the calculation process for propagation a STc optical field.Another assumption that is always taken is the Slowly Varying Envelope Approximation（SVEA）.It equivalently means the spectral bandwidth of the field is much smaller than the optical center frequency，wΩ≪ω0.Besides，we assume the field is linearly polarized at this stage.The field can be thus written in a scalar form aswhereA(x，y，t) is the field amplitude andφ(x，y，t) is the field phase.STc optical field can be thus related to its spatial frequency and spectral counterparts using the following Fourier transformations，

here，(kx，ky，ω) is the spatial frequency and spectral coordinate which is reciprocal to(x，y，t).In this Fourier transformation expression，normalization factors are neglected because we are more interested in the relative distribution of the STc field.Using these equations，the propagation of a STc optical field in a dispersive medium can be written as

Fig.2 Evolution of STOV wavepacket with a topological charge ofl=+1

wherek0is the averaged propagation constant of the field，β2is the Group Velocity Dispersion（GVD）coefficient of the medium，Ω=ω−ω0is the relative optical angular frequency，andLis the propagation distance.For a non-dispersive propagation，β2is set as zero.

Using Eq.（3），the evolution of STc optical fields can be both analytically and numerically calculated.To give an example，Fig.2 shows the evolution of a Spatiotemporal Optical Vortices（STOV）wavepacket［9，10，18］under dispersive and non-dispersive propagation.The 3D plots show the iso-surface shell of the wavepacket where the localized intensity of the field drops to 10% of the peak intensity and the surface color is registered to the phase of the wavepacket.The initial STOV wavepacket shown in the leftmost figure has a spiral phase of exp(ilθx−t)，wherelstands for the topological charge andθx−tis the polar angle in(x，t)space.The topological chargelis +1.

Fig.2 Evolution of STOV wavepacket with a topological charge of l=+1

During dispersive propagation，STc optical fields such as STOV wavepacket will spread in all directions in(x，y，t) space.During non-dispersive propagation，STc fields only spread in the spatial domain.Consider thex-direction diffractive phase term and the dispersive phase term in Eq.（3），it can be noticed that the evolution of STc fields under normal dispersion resembles an optical beam propagation process except that theycoordinate in beam propagation is now replaced byt-coordinate.Therefore，STOV wavepacket can maintain its ring-like shape under normal dispersive propagation while it splits into a diagonal shape under anomalous dispersive propagation due to a strong astigmatism between the accumulated dispersive phase and diffractive phase［18］.Meanwhile，it is also noteworthy that STOV wavepacket has a spiral phase reversal during anomalous dispersive propagation（see the rightmost figure in Fig.2（c））.It means the photon within the STOV field now carries a transverse OAM with a topological charge of−1 instead of+1.

调查结果表明，在广东省青云山自然保护区通过红外相机调查野生动物的分布是有效的检测手段之一，为地面和林下活动警惕的物种监测提供了有效手段，值得长期和扩大范围应用。但由于各种条件的限制，本次调查仅限于森林动态监测样地范围内，且野外调查和相机实际工作时间较短，投入的相机数量有限，尚难以全面准确地评估该地区鸟类和兽类的多样性格局及重要物种的分布和种群状况，特别是斑灵狸、豹猫、白鹇等关键物种的分布和密度，为进一步掌握野生动物的资源现状，亟需在大范围内开展长期的调查和监测工作，并合理增加红外相机的数量(按公里网格法)，扩大监测区域和延长监测时间，从而更全面地掌握该地区野生动物现状及分布格局。

3 Experimental realization of STc optical field

The experimental realization of STc optical field is normally achieved by a phase and/or amplitude modulation in the spatial and spectral domain of the input optical field.One typical setup for generating STc optical field has the same configuration as a zero-dispersion 4-fpulse shaper［78-80］.Fig.3 illustrates the schematic of a STc field generator.The generator has the same configuration as a standard zero-dispersion 4-fpulse shaper［78-80］except that the 1D Liquid-Crystal Spatial Light Modulator（LC-SLM）in a pulse shaper is now replaced by a 2D SLM so that it can modulate the spatial and spectral phase of the input field［9，17］.The generator constitutes a pair of diffraction gratings，a pair of cylindrical lenses，and a transmissive 2D LC-SLM.In the setup，each component is separated by the focal length of the cylindrical lens.In this way，the spatial-spectral（x-ω）component of the input field can be projected at the LC-SLM plane.The spatial-spectral field at LCSLM planeESLM(x，Ω) can be related with the input fieldEin(x′，t) via a one-dimensional Fourier transformation overtwhich can be written as

whereΩ=ω−ω0.In the setup，the gratings disperse or re-collimate the light field alongy′-direction（see Fig.3 for the exact geometry）and we assume the field is uniformly distributed alongy′-direction.Using Eq.（4），STc optical field after the generatorEout(x′，t) can be expressed as

Fig.3 Schematic of a spatiotemporal coupled（STc）optical field generator

whereφSLM(x，Ω) stands for the 2D phase applied on LC-SLM and we assume there is negligible diffraction effect alongx-direction when the light field propagates inside the generator.

STc field generator has been extensively used for generating and studying STc optical fields such as STOV wavepackets［9］，self-accelerating ST Airy wavepacket［27］，and space-time light sheets［31］.Besides，it is also possible to replace the programmable LC-SLM device in the setup with a transmission optical mask so that the spatial-spectral transmittance is modulated.Some specific STc optical fields such as ST Bessel wavepackets can be generated using this non-programmable setup［81］.

4 Regimes of STc optical field generation

It should be noted that although 4-fpulse shaper setup for STc optical field generation appear to be simple，the subtle interplay between dispersion and diffraction during the process makes the generation of STc optical fields more sophisticated.Among the research works that utilize STc field generator，there are distinct differences in those realizations according to the distance between the exit plane of the generator and the observation plane.Here we can roughly categorize them into three main different operation regimes：

1）The“far-field”regime.In this approach，the observation plane is set at several Rayleigh ranges after the generator［9，14］，or，alternatively，an additional lens or an additional imaging system can be inserted after the generator so that the“far-field”position is now relocated to the back focal plane of the lens or the imaging system［31］［81］.In both scenarios，the observed STc optical fieldEob(x，t) becomes mathematically the Fourier transform of the STc field at the exit plane of the generator，which transform between thex′-direction coordinate of the exiting STc field andx-direction coordinate of the observed STc field［82］.That is equivalent to say，the generator now modulates spatial frequency and spectral phaseφ(kx，ω) of the observed field，or it changes the spatial frequency and spectral field distributionof the observed field.To give an example，in the experimental realization of STOV wavepacket with a controllable transverse Orbital Angular Momentum（OAM）（see Fig.4（a）for the visualization of a simulated STOV wavepacket），a spiral phase of e−ilθis applied on the LC-SLM in the generator.The observation plane is set at several Rayleigh ranges after the generator，and，therefore，a 2D Fourier transform can give the observed STc optical field inx-tspace via［83］

Fig.4 STOV wavepackets generation using different operation regime of the spatiotemporal coupled optical field generator［9，17，18］

here，(ρ，φ) is the polar coordinate in (x，t) space defined byandφ=arctan (x/t)，FT is the Fourier transform，andHlis the Hankel transform of orderl.In a Fourier transformation，the spiral phase and its related optical OAM are conserved.Fig.4（b）plots the theoretical prediction of a generated STOV wavepacket and Fig.4（c）shows the measured ST phase，indicating a STOV wavepacket can be generated by applying a spiral phase in its(kx，ω)space［9］.

2）The“near-field”regime.In contrast to the“far-field”regime，the“near-field”regime means the observation plane is placed within the Rayleigh range after the generator.In this case，there is negligible diffraction and the observed optical field can be expressed by Eq.（5）.One interesting application of this“nearfield”approach is to use a linearly chirped pulse as the input to seed the generator.Since for a linearly chirped pulse，its instantaneous frequency is linearly dispersed in time，the spatial phase modulation applied on the LCSLM inside the generator now translates to a direct ST phase modulation for the generated field.This allows a direct sculpturing of the STc optical field in the ST domain，permitting the generation and manipulation of more complicated STc optical fields in an intuitive fashion.Fig.4（d）plots the measurement results for the ST intensity and phase of two STOV lattices［18］.Each lattice contains four STOV wavepackets with different topological charge.Using this“near-field”operation of the generator，STOV collision and STOV annihilation experiments are demonstrated.

3）The“intermediate”regime.In this operation regime，the interplay between dispersion and diffraction for STc field propagation is more subtle.Once the dispersive phase and the diffractive phase are perfectly balanced，the evolution of a STc optical field has the exact same mathematical expression of a Fresnel diffraction integral，except that they-direction coordinate in a Fresnel diffraction is now replaced by time［17］.It is well known that an optical axicon can be used to generate a Bessel beam［84］.The STc field generator can also apply a spatial-spectral conical phase for generating a ST Bessel wavepacket.If an additional helical phase is also applied，the generator can generate Bessel STOV（BeSTOV）wavepackets with a higher-order Bessel function distribution in ST domain.The results for BeSTOV wavepackets with a charge of +1，+2 and +3 are shown in Fig.4（e）and Fig.4（f）.

5 Recent developments in STOV wavepackets

One of the most recent examples of STc optical fields that attracted significant attention is the Spatiotemporal Optical Vortices（STOV）wavepacket.The STOV wavepackets have unique photonic property that the wavepacket can carry pure transverse Orbital Angular Momentum（OAM）.Over the last three decades，photonics research community has been extensively studying optical fields that carry a longitudinal OAM［85］or a longitudinal Spin

Angular Momentum（SAM）［86］，i.e.，vortex beam or left/right circular polarized（LCP/RCP）beam.The angular momentum such field carries has an orientation that is in parallel with the propagation direction of the light field.It was later discovered that transverse SAM can occur in special cases such as a tightly focused beam［87］and evanescent waves from waveguides［88］.On the other hand，only limited research works have theoretically predicted the possibility of photonic transverse OAM in the form of a STOV wavepacket［89-90］.The first demonstration of STOV wavepacket is achieved using the nonlinear collapse and self-arrest of an extremely intense optical pulse in air［91］.The resulting STOV wavepacket has limited pulse energy and this realization approach lacks the control over STOV.

Recently，scientists have successfully generated a STOV wavepacket using a STc optical field generator setup（see Fig.3）based on a 2D phase modulation device［9-10］.The generated STOV wavepacket can have a controllable ST field structure with a spiral phase of exp(ilθx−t)in the spatiotemporal domain（see Fig.5（a）for the experimental results of STOV wavepackets with a topological chargelof +1 and +2）.The creation of a controllable photonic transverse OAM has granted scientists a new degree of freedom for manipulating the photons.It also means the control over photonic OAM is extended from previous 1D to 3D.

Since the first experimental demonstration of generating STOV wavepackets［9-10］，many research groups have studied STOV-related physical experiments.In a nonlinear optical experiment of a second harmonic generation （SHG） process， fundamental STOV wavepackets are converted into SHG STOV wavepackets［12-13］.The resulting SHG STOV wavepacket can maintain its ring-like field distribution in ST domain（see Fig.5（b） for the iso-surface plot for the measurement results）.Besides，the ST phase measurement has verified the conservation of transverse OAM during nonlinear optical processes（see Fig.5（c）for the ST phase measurement results）.Besides，STOV wavepacket can combined with spatial vortices whose OAM is in the longitudinal direction.The resulting STOV wavepacket can have a photonic OAM with a controllable orientation［16］（see Fig.6（a）for the 3D intensity measurement results for wavepackets whose photonic OAM has a controllable orientation）.Furthermore，if polarization control is introduced to modulate a STOV wavepacket，it was found that a ST phase singularity can co-exist with a spatial polarization singularity［19］（see Fig.6（b）for the 3D intensity measurement for a cylindrical vector STOV wavepackets）.The resulting complex STc optical field can possess a complex local density of photonic angular momentum in space and time，which is of great interest for researchers who study the Spin-Orbit Interaction（SOI）of light［15，95］.

Fig.5 STOV wavepacket and SHG STOV wavepacket

Fig.6 STOV wavepackets superposed with additional photonic singularities

6 Summary and future perspectives

In the past decade or so，there have been a dramatic increase in research interests in Spatiotemporal Coupled（STc）optical fields.These fields feature unique photonic properties that are previously unavailable from conventional optical fields.Considering the rapid developing nature，it is inevitable that such a brief review will leave many other important aspects and seminal works out.A special focus is given to the recent example of Spatiotemporal Optical Vortices （STOV） wavepackets that carry photonic transverse Orbital Angular Momentum（OAM）.The use of 4-fpulse shaper as an STc optical field generator offers great flexibility to create a wide variety of novel fields.Although the realization of these novel STc optical fields utilizes an experimental setup that is almost exact the same as a standard 4-fpulse shaper，the subtle interplay between dispersion and diffraction during the process makes the generation of STc optical fields more complicated compared with pulse shaping or beam shaping.Meanwhile，it is exactly such a complexity that will offer much richer physics deserving future explorations.During the formation of complicated STc optical field，there are also new physical phenomena yet to be investigated in the future，for example，the reversal of STOV polarity during a dispersive propagation of STOV wavepackets.

The pulse shaper based STc optical field generation method has seen great success and received tremendous interests.It should be pointed out that the current experimental setup can be easily modified to simultaneous modulate multiple parameters over the input light field so that the device can generate much more ST vectorial field.In addition，the generated STc optical fields can be further structured in the spatial domain using the very mature spatial modulation techniques to produce much more sophisticated STc fields such as photonic toroidal vortex［96］.It is fairly certain to conclude that unprecedented level of control for photons is now already on the horizon.

Despite the great success already achieved，there are much more need to be studied，understood and developed in STc optical fields.For example，new generation methods that utilize metasurfaces or other nanophotonic structures are highly desirable in order to miniaturize the generation system and incorporate multiple parameters modulation into the system.With multiple parameters modulated STc fields，new characterization method will be needed.Consequently，a more complete set of toolboxes for the generation and characterization of complex STc optical fields shall be developed.With such a toolbox available，many other STc fields that are highly interesting but was not possible to generate，such as those localized waves［29］will become feasible.Another important future direction for STc optical fields is to study novel light matter interaction phenomena with the generated STc fields.And finally，how to transfer the current STc optical field technique to super-intense and ultrafast high-power laser deserves special attention as it may hold the key to unlock the full potential of these large laser facilities.It is never wise to predict exactly what would come out from a nascent research field such as STc optical field.However，with the unprecedented level of control of the spatiotemporal degree of freedoms of light，it is safe to say that spatiotemporally sculptured optical fields purposefully will significantly enrich the photonics arsenal for scientists in broad research fields ranging from quantum optics， nanophotonics， spin-photonics and spintronics， optical information transmission and processing，optical spectroscopy，laser driven particle acceleration，and much more beyond.