Spatiotemporal sculpturing of light: a tutorial

Qiwen Zhan; Qiwen Zhan; Qiwen Zhan

doi:10.1364/AOP.507558

1. Introduction and Background

Controlling light in all its degrees of freedom and producing light fields with required properties represent one major milestone for fully harnessing the power of light. In the past three decades, spatial and temporal modulation of light with multiple degrees of freedom are steadily gaining traction, extending from the familiar one-dimensional (1D) temporal waveform shaping [1–3], two-dimensional (2D) transverse mode patterning [4–13] to three-dimensional (3D) spatial control [14–18], and to include spatiotemporal control for four-dimensional (4D) and even higher-dimensional forms [19–28].

On the temporal front, advances in pulse shaping have enabled optical pulses to be produced with nearly arbitrary waveforms [3]. Temporal shaping of light, especially in the ultrafast domain, has already benefited from diverse applications. For example, the locking of the carrier envelope phase of an optical pulse can result in a frequency comb as a powerful tool for ultraprecise metrology [29]. The shaping of an ultrafast laser pulse into burst profiles enables the improved effect of ablation-cooled material removal for advanced laser processing [30], for example.

On the other hand, spatial shaping of light has also attracted growing attention for arbitrary tailoring of light patterns for diverse purposes. Studies have extended our familiar 2D transverse forms of electromagnetic waves to include 3D control (for all three components of the electric field), to sophisticated topology of structured light with recent advances of exploiting of geometric transformation under general symmetry to reveal “hidden” degrees of freedom of light [31]. Two particular examples with broad interests are the optical vortex carrying orbital angular momentum (OAM) [12] and cylindrical vector beams [13].

Although photonic spin angular momentum (SAM) has been very well known since the early days of quantum mechanics [32,33], creating photons with OAM was restricted to unlikely quadrupole transitions in atoms, and remained the realm of atomic physics textbooks until only 30 years ago [34]. The seminal work by Allen et al. revealed the importance of the spatial structure of light, particularly its helical phase, and its connection to OAM. The explosion of activities that followed was fueled by the ease of which OAM light could be created, bringing photonic angular momentum tools into optical laboratories. Almost in parallel, vectorial optical fields with structured polarization, particularly those with cylindrical polarization symmetry and polarization singularities received lots of attention mainly driven by their focusing properties [35–37], and their intrinsic links to SAM and OAM have been investigated through spin–orbital conversion, coupling, and interactions [38,39].

Research into these spatially structured optical fields have come quite a long way and already produced practical applications with economical impacts. For examples, optical communication links with data rates as high as 1,000 Tbit/s have been demonstrated recently using OAM multiplexing [40–42]. Polarization engineered optical fields have been employed in heat-assisted magnetic recording [43] that is available on the market and azimuthal polarization illumination has found use in deep ultraviolet (DUV) immersion lithography system. These applications were not imaginable at all when research in this field started about 30 years ago.

Despite the significant progress, there remains much to do with open challenges and opportunities. A very natural extension is to combine both spatial and temporal modulations together and to have the spatial and temporal structures entangled simultaneously. Traditionally, spatiotemporal light pulses are treated as spatiotemporally separable wave packet as a solution to Maxwell’s equations. When temporal shaping meets spatial shaping with complex space–time nonseparability under control, spatiotemporally structured light comes into play, injecting new vitality into photonic science and applications [44].

Adding temporal control to the spatially complex optical as the fifth degree of freedom has already seen very recent breakthroughs [45]. More generalized forms of spatiotemporally nonseparable solution started to emerge with growing importance for their striking physical effects [27,46–49]. Most recently, optical toroidal vortex, a unique topology never previously conceived in optics, was successfully demonstrated through the combination of spatiotemporal modulation of light and spatial conformal mapping [23]. Meanwhile, the generation of toroidal space–time nonseparable pulses has promised to excite new forms of multi-pole moments in matter and free space [24]. A simple illustration of the evolution in complexity of light fields is summarized in Fig. 1.

Figure 1. Illustration of the evolution of complexity of light fields. Simple (top row): Gaussian → spherical wavefront→ spatially homogeneous polarization → continuous wave → pulsed Gaussian; and Complex (bottom row): LG modes → optical vortex → cylindrical vector modes [13] → spatiotemporal wave packets (spatiotemporal optical vortex [48], toroidal vortex [23], toroidal pulse [24], and optical hopfion [27]). Reprinted with permission from [13]. © The Optical Society. Reprinted from [23,27] under a Creative Commons license.

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The large emergent community has steadily uncovered further connections among multiple degrees of freedom in structured light [10,50], spatial and temporal, and its angular momentum, including spin–orbit coupling [51–54], geometric representations of photonic angular momentum states [55], transverse angular momentum and photonic wheels [56–58], and spatiotemporal vortices [46–49]. The ultrafast vortex or vector pulses can be used to manufacture helical micro- or nano-structure for producing chiral functional materials [59], emulate topological textures of quasiparticles in condensed matters [60], and to excite new nonlinear effect in light–matter interactions [61]. Therefore, sculpturing light in the spatiotemporal domain with increasingly complex topological structures represents a cutting-edge frontier for both fundamental physics and applied optics.

This tutorial aims to highlight the rapidly growing body of works on the generation and characterization of various spatiotemporally structured light pulses as well as the associated novel physics that mainly happened in the past 5 years. Different from other previous reviews, this work focuses on the “sculpturing” capability to produce spatiotemporal structure on demand, rather than finding the propagation invariant or eigenmodes of propagation through certain media. In some sense, this is the most generalized spatiotemporal distribution of light that would exhibit complex propagation, focusing, and interaction properties. From a linear system point of view, the modulation to optical fields at one plane is related to the optical fields at another designated plane through a linear transformation. Thus, essentially the problem of sculpturing light into desired spatiotemporal distribution can be treated as a “spatiotemporal hologram” [62]. The benefit of adopting this point of view allows us to consider true 3D + 1D structuring of light fields. In the previously reported cases, including four-wave mixing (FWM) [63,64], X-wave [65,66], O-wave [67–69], flying focus [70], tilted-pulse-front pulses [71], abrupt focusing needles of light [72], and those propagation invariant spatiotemporal wave packets (STWPs) discussed in a recent review article [45], only 1D space + 1D temporal could be imagined and formulated.

We will first make an attempt to quickly go through the mathematics for solutions to Maxwell’s equations where the spatiotemporal coupling was traditionally considered. Then fundamentals of the modulation and characterization of the optical field in spatial, temporal, and spatiotemporal domains are briefly summarized along with various novel STWPs with designed spatiotemporal coupling that have been studied recently. The widespread and potential applications of spatiotemporal light pulses along with new opportunities for both fundamental and applied sciences will be reviewed. Finally, speculations on the near future trends and key challenges that need to be addressed will be presented in the perspective section.

2. Maxwell’s Equations and Solutions

Let’s start with Maxwell’s equations in the differential form:

\nabla \times \vec{E} ={-} {\mu _r}{\mu _0}\frac{{\partial \vec{H}}}{{\partial t}},\nabla \cdot \vec{E} = 0,

(1)$$\nabla \times \vec{H} = {\epsilon _r}{\epsilon _0}\frac{{\partial \vec{E}}}{{\partial t}},\nabla \cdot \vec{H} = 0, $$

which lead to the wave equation for the electric field component as

(2)$${\nabla ^2}\vec{E} + \nabla \left( {\frac{1}{{{\epsilon_r}}}\nabla {\epsilon_r} \cdot \vec{E}} \right) - \frac{{{\epsilon _r}}}{{{c^2}}}\frac{{{\partial ^2}\vec{E}}}{{\partial {t^2}}} = 0, $$

where ${\epsilon _r} = {n^2}$ is the dielectric constant of the medium and n is the index of refraction. In general, the solutions to this wave equation subject to given boundary conditions are vectorial in nature. To make the discussion more accessible, we will restrict ourselves to the cases that the vectorial field is transverse so that the vectorial nature can be considered separately. This is valid for light fields that are paraxial and not too short in the temporal domain. This assumption, as we shall see in a moment, is the same as the cases for spatiotemporal sculpturing of light that we will focus on in this tutorial. Consequently, Eq. (2) can be reduced to a scalar form as

(3)$${\nabla ^2}U - \frac{1}{{{c^2}}}\frac{{{\partial ^2}U}}{{\partial {t^2}}} = 0,$$

where $U({\vec{r},t} )$ represents a scalar solution for one of the field components. The commonly known solutions to this scalar wave equation include pulsed plane wave $U({r,t} )= A\left( {t - \frac{z}{c}} \right)\textrm{exp}\left[ {i{\omega_0}\left( {t - \frac{z}{c}} \right)} \right]$ and pulsed spherical wave $U({r,t} )= \left( {\frac{1}{r}} \right)g\left( {t - \frac{r}{c}} \right)\textrm{exp}\left[ {i{\omega_0}\left( {t - \frac{r}{c}} \right)} \right]$. The characteristic of these solutions is that the pulse width does not change upon propagation and the spatial distribution of the wave is not “constrained.” In other words, no diffraction effect needs to be considered.

When spatial constraints are applied to the field, the situation will get more complicated. In general, the spatial and temporal evolution are intricately coupled together under this situation. Under the slow varying envelope (SVE) approximation $\frac{{{\partial ^2}A}}{{\partial {t^2}}} \ll \omega _0^2A$, the solution can be regarded as a quasi-continuous wave (quasi-CW) pulsed wave with a center frequency ${\omega _0}$ expressed as

(4)$$U({\vec{r},t} )= A({\vec{r},t} ){e^{ - i{k_0}z}}{e^{i{\omega _0}t}}. $$

If we further impose the paraxial condition $\frac{{{\partial ^2}A}}{{\partial {z^2}}} \ll k_0^2A$, the scalar wave Eq. (3) leads to the equation for the envelope as

(5)$$\nabla _T^2A - i\frac{{4\pi }}{{{\lambda _0}}}\left( {\frac{{\partial A}}{{\partial z}} + \frac{1}{c}\frac{{\partial A}}{{\partial t}}} \right) = 0, $$

where $\nabla _T^2 = \frac{{{\partial ^2}}}{{\partial {x^2}}} + \frac{{{\partial ^2}}}{{\partial {y^2}}}$ is the transverse Laplacian operator. This equation may be satisfied by $A({\rho ,z,t} )= g({t - z/c} ){A_0}({\vec{r}} )$, where ${A_0}({\vec{r}} )$ satisfies the paraxial Helmholtz equation:

(6)$$\nabla _T^2{A_0} - i\frac{{4\pi }}{{{\lambda _0}}}\frac{{\partial {A_0}}}{{\partial z}} = 0.$$

Under these conditions, the spatial and temporal parts are now separated. In other words, the paraxial spatial content is modulated by a slowly varying pulse. In this case, the propagation does not alter the temporal waveform in nondispersive medium. One example of these solutions is the familiar pulsed Gaussian Beam that can be expressed as [73]

(7)$$A({\rho ,z,t} )= g\left( {t - \frac{z}{c}} \right)\frac{{i{z_0}}}{{z + i{z_0}}}\textrm{exp}\left( { - i\frac{\pi }{{{\lambda_0}}}\frac{{{\rho^2}}}{{z + i{z_0}}}} \right), $$

where g(t) is an arbitrary slowly varying function and z₀ is the Rayleigh range.

These space–time separable solutions allow us to treat the temporal and spatial behaviors of light fields separately. However, in general, the optical field distribution does not have to be separable in space and time. The question is whether the space–time distribution can be maintained through propagation. The spatial propagation is affected by diffraction, while the temporal wave evolution does not “suffer” this in nondispersive medium or free space. Thus, there is unbalance between the spatial diffraction and temporal evolution. Even if a STWP is prepared in the paraxial domain at the beginning, the spatiotemporal distribution will be distorted as it propagates unless proper dispersion is introduced.

For a dispersive medium, if we perform the Taylor expansion of the propagation constant up to the second-order term,

(8)$$\beta ^{\prime} = \frac{{d\beta }}{{d\omega }}{|_{{\omega _0}}} = \frac{1}{v}, $$

(9)$$\beta ^{\prime\prime} = \frac{{{d^2}\beta }}{{d{\omega ^2}}}{|_{{\omega _0}}} = \frac{{{D_v}}}{{2\pi }}, $$

where $\beta = \omega n(\omega )/{c_0}$ is the propagation constant, $v = {c_0}/N$ is the group velocity, $N = n - {\lambda _0}\frac{{dn}}{{d{\lambda _0}}}\; $is the group index, and ${D_v} = 2\pi \beta ^{\prime\prime}$ is the dispersion coefficient. Under such a situation, it can be shown that the envelope function (ignoring the spatial distribution) satisfies the partial differential equation:

(10)$$\frac{{{\partial ^2}A}}{{\partial {t^2}}} + i\frac{{4\pi }}{{{D_v}}}\left( {\frac{{\partial A}}{{\partial z}} + \frac{1}{v}\frac{{\partial A}}{{\partial t}}} \right) = 0. $$

If we consider a coordinate system moving with the pulse group velocity v, the equation is reduced to the SVE diffusion equation as

(11)$$\frac{{{\partial ^2}A}}{{\partial {t^2}}} + i\frac{{4\pi }}{{{D_v}}}\frac{{\partial A}}{{\partial z}} = 0. $$

A comparison between this equation and the paraxial Helmholtz Eq. (6) reveals the analogy between dispersion and diffraction. Such similarity between the spatial and temporal evolutions offers the possibility to engineer space–time propagation invariant solutions via properly chosen temporal and spatial dispersion.

The interplay between diffraction and dispersion can make the process quite complex. For pulses with relatively narrow spectra, approximation similar to those that led to paraxial Helmholtz equation and SVE equation can be used to derive a partial differential equation for the envelope as the generalized paraxial wave equation:

(12)$$- {\lambda _0}\nabla _T^2A + {D_v}\frac{{{\partial ^2}A}}{{\partial {t^2}}} + i\left( {\frac{{\partial A}}{{\partial z}} + \frac{1}{v}\frac{{\partial A}}{{\partial t}}} \right) = 0. $$

Under these approximations, the solution may still be separable such as the space–time Gaussian pulse beam [69]:

(13)$$A({x,y,z,t} )= {A_0}{\left( {\frac{{ - iz_0^{\prime}}}{{z - iz_0^{\prime}}}} \right)^{1/2}}\textrm{exp}\left( { - i\frac{\pi }{{{D_v}}}\frac{{t - z/v}}{{z - iz_0^{\prime}}}} \right) \cdot \left( {\frac{{iz_0^{}}}{{z + iz_0^{}}}} \right)\textrm{exp}\left( { - i\frac{\pi }{\lambda }\frac{{{\rho^2}}}{{z + iz_0^{}}}} \right). $$

However, when these conditions are not met, this will lead to a more general partial differential equation for the envelope without the paraxial and weak dispersion approximations [69]:

(14)$$- {\lambda _0}\nabla _T^2A + {D_v}\frac{{{\partial ^2}A}}{{\partial {t^2}}} + i4\pi \left( {\frac{\partial }{{\partial z}} + \frac{1}{v}\frac{\partial }{{\partial t}}} \right)A - {\lambda _0}\left( {\frac{{{\partial^2}}}{{\partial {z^2}}} - \frac{1}{v}\frac{{{\partial^2}}}{{\partial {t^2}}}} \right)A = 0. $$

When expressed in the moving coordinates with the pulse, with $t^{\prime} = t - \frac{z}{c}\; \;\textrm{and}\;\; z^{\prime} = z$ the equation can be rewritten as [73]:

(15)$$- {\lambda _0}\nabla _T^2A + {D_v}\frac{{{\partial ^2}A}}{{\partial t{^{\prime 2}}}} + i4\pi \frac{{\partial A}}{{\partial z^{\prime}}} - {\lambda _0}\left( {\frac{{{\partial^2}A}}{{\partial z{^{\prime 2}}}} - \frac{2}{v}\frac{{{\partial^2}A}}{{\partial t^{\prime}\partial z^{\prime}}}} \right) = 0. $$

This equation clearly exhibits spatiotemporal coupling effects that have been studied extensively in the past.

Thus, in general, the spatial and temporal characteristics of pulsed light waves are inherently coupled together. In other words, the spatial spreading depends on the initial temporal profile and the temporal waveform is influenced by the initial spatial pattern and diffraction. Under paraxial and SVE approximations, the temporal and spatial evolutions may be effectively decoupled. These conditions would be violated under highly focusing and ultrashort pulses situations, where the spatiotemporal coupling must be considered.

This coupling has long been considered as a nuisance that needs to be avoided or combatted. The balance and interplay between diffraction and dispersion play important roles in the spatiotemporal coupling. In the past four decades, attention has been focused on finding solutions for propagation invariant STWPs such as such as FWM, X-waves, O-waves, Bessel-X waves [74], and lately Airy–Bessel linear light bullets [75], and other localized waves with spherical harmonic symmetries [76,77]. The difficulties in the generation of many of these concepts in the optical regime severely hampered the interests and almost “doomed” this field. Until very recently, new tools for STWP synthesis and developments of theoretical pictures in the parameter space injected fresh life into this field. A recent review article by Yessenov et al. [45] provides an overview of propagation invariant STWPs and the latest developments in the optical generation and characterization. However, this review focused on the propagation invariant characteristics explained with a k–ω spectral support description and the spatiotemporal structuring is mostly limited to one space dimension and one temporal dimension, which does not have to be the case.

Rather than disentangling the spatial and temporal characteristics of pulsed light fields, the focus of this tutorial is to intentionally introduce spatiotemporal coupling under the paraxial and SVE conditions through various modulation schemes to produce STWPs with specific distributions at a designated time and location, not necessarily propagation invariant. This is what we meant by “spatiotemporal sculpturing” of light. In an analogy to the optical spatial diffraction which can be classified into the near field, the far field, and the intermediate Fresnel regime, “spatiotemporal sculpturing” of light sits in the much more complicated yet phenomena-rich regime that is between the space–spectral domain and the far field space–time domain. There are much more in the middle.

Another important aspect of optical field that has long been ignored in the STWP synthesis is the is the vectorial nature of light. Part of the reason for this ignorance is due to the extreme difficulty in control or tailoring spatiotemporal distribution of polarization within one pulse. Under paraxial and SVE conditions, vectorial property of light will be transverse in nature. For free space or linear homogeneous nondispersive medium, this allows the vectorial nature to be considered separately. However, it is worth pointing out that as the pulse approaches single cycle and/or the size of beam approaches wavelength, the vectorial nature and particularly the non-transverse components of the fields will be prominent or even dominant [78], which adds another dimensional complexity as well as more opportunity to construct light fields with much more exquisite topological features.

The context of discussions in this article are mainly restricted to the linear regime. Within this regime, Fourier analysis can be applied to the most general situations, including single cycle, highly focusing, and with polarization considered [79,80]. Such a general approach can be employed to deal with propagation/focusing and later in their applications. It also allows us to handle much more complicated situations including ultrashort pulse duration, tight focusing, and even with the vectorial nature also involved.

3. The Basics of Light Modulation

3.1 Spatial Modulation of Light

In principle, anything that can affect the 2D profile of light field can be regarded as spatial modulation. Here we restrict ourself to optical devices that can perform precisely defined spatial modification to one or multiple parameters of the optical field. This can range from simple transmission masks/gratings, to deformable mirror devices (DMDs), liquid crystal spatial light modulators (LC-SLMs), to the recently developed metasurfaces, and so on. Generally speaking, all pixelated spatial light modulation necessarily undergoes diffraction related processes, thus they all can be regarded as diffractive optical elements (DOEs). Traditionally, the word modulation sometime also implies some sort of dynamically updating capabilities, through effects such as electro-optic (EO), acousto-optic (AO), and magneto-optic (MO) effects [73]. However, fixed devices can be used for static applications that does not require dynamic refreshing after optimization is done. Owing to space limitations, we briefly present three classes of spatial light modulation techniques that are widely used in the technical areas that we are interested in.

3.1a Liquid Crystal Spatial Light Modulator

A LC-SLM utilizes the EO effect of liquid crystal (LC) materials. Through applying an electric field, the orientation of LC molecules can be adjusted and consequently the birefringence is continuously tuned by the applied voltage. Using a linearly polarized input aligned with the LC birefringent axis, phase modulation can be produced. Pixelated devices can be made with microfabrication tools, offering spatial light modulation capabilities. In principle, LC-SLM can be made for both amplitude and phase modulation. However, they are often intertwined. The most popular version is the reflection type pure phase liquid crystal on silicon (LCOS) device. This is a mature technique and there are many commercial products available on the market nowadays. In practice, one needs to choose the proper products according to the wavelength, power, phase modulation depth requirements, filling factor, and refreshing rates, etc.

One of the key features of LC-SLM is its polarization dependence. It only responds to one specific input linear polarization while leaving the other orthogonal polarization component intact. This feature makes it possible to expand its modulation capabilities to include both amplitude and polarization (with two degrees of freedom to control) through the combination with linear polarizers and λ/4 plate. For example, a spatially variant polarization rotator can be constructed with a λ/4 plate and a reflective LC-SLM (Fig. 2) [81]. A full degree of freedom (phase, amplitude, and polarization) modulation has been realized with the use of software segmented LC-SLMs through 4f relay systems in a configuration called VOF-Gen illustrated in Fig. 3 [81,82].

Figure 2. Illustration of a spatially variant polarization rotation operation using a reflection-type liquid crystal spatial light modulator. Such a configuration allows the realization of amplitude (combined with linear polarizer) and polarization modulation using a pure phase modulator. Reprinted with permission from [81]. © The Optical Society.

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Figure 3. Vectorial optical field generator (VOF-Gen) that is capable of generating arbitrarily complex optical fields. Its capability of producing phase, amplitude, polarization, and polarization ellipticity within the beam cross section given a Gaussian input beam is illustrated by the patterns shown in the top row as insets. Reprinted with permission from [81]. © The Optical Society.

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In certain sense LC-SLM can be regarded as DOEs with dynamic refreshing capabilities. However, it should be noticed that in many cases, such as the VOF-Gen, the “diffraction” feature is suppressed by the 4f relay systems. These devices produce structured optical field at designated plane and the diffraction (propagation) phenomenon will be considered later. This is also the same way we treat the spatiotemporal sculpturing of the optical field in this tutorial, meaning that these devices provide a “on-spot” structuring to the STWP with the propagation concerned later.

3.1b Multi-Plane Light Conversion

However, the diffraction and propagation can be utilized in the shaping of transverse profiles of optical fields. A good example is the multi-plane light conversion (MPLC) illustrated in Fig. 4 [83,84]. Multiple DOE elements can be cascaded together to realize certain spatial shaping functionality utilizing multiple reflections/transmissions and light propagation/diffraction through the DOE elements. Specifically designed far-field patterns can be synthesized in general using modal analysis. Compared with the LC-SLM devices, MPLC typically can be designed with higher efficiency for light throughput. However, one drawback of this approach clearly is that the “resolution” will be reduced due to the reliance on the diffraction/propagation. Current MPLC designs use phase-only devices while amplitude and polarization are generally not considered. However, these functions can be incorporated through the use of LC-SLM or metasurfaces as part of the implementing system. The incorporation of dynamic elements into MPLC design will also enable certain level of reconfigurability.

Figure 4. Schematic for multi-plane light conversion (MPLC). Designed phase masks are placed at multiple designated locations along the path of free-space propagation to convert one set of orthogonal modes to the desired output modes. Reprinted with permission from [84].

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3.1c Metasurfaces

Recent rapid development on metasurfaces deserve special attention as they offer optical modulation capabilities with distinctive features. Metasurface performs wavefront modulation based on the geometric phase introduced by the orientation of individual subwavelength building blocks (Fig. 5) [85,86]. Thus, the device thickness can be very thin as it no longer needs the thickness to build up the phase profile. Metasurface offers miniaturized and low profile solutions for light spatial modulation. Taking advantages of the modern nanofabrication technology, the pixel size (spatial resolution) of metasurface devices can be made much smaller than that of the LC-SLM or traditional DOEs. In many cases, the boundary between metasurface and DOE is kind of vague and the advantages of metasurface over DOE in some cases have been challenged. However, in terms of optical spatial modulation, metasurface devices offer one distinctive advantage over traditional DOEs: the capability of modulation multiple parameters (phase, amplitude, and polarization) with the same integrated device [87,88]. In addition, there are ongoing efforts to make metasurface dynamic/reconfigurable by embedding proper materials (such as LC or phase change materials) into the architect [89,90]. It is conceivable that further combination of metasurface with MPLC or 4f relay system will potentially offer much more powerful tools for light modulation.

Figure 5. Metasurface device that utilizes the geometrical phase to produce wavefront modulation. (A) Scanning electron micrograph of the metasurface device with spatially variant elemental orientation to produce different phase. (B) Enlarged view of (A). (C, E, G) and (D, F, H) Experimental and theoretical intensity and interferograms to demonstrate the generation of optical vortex with this device. From Yu et al., Science 334, 333–337 (2011). Reprinted with permission from AAAS.

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3.2 Temporal Waveform Shaping with a 1D Pulse Shaper

Modulators based on EO, AO, or thermo-optic (TO) effects, etc., typically produce envelope modulations that are much slower than the electrical field oscillation of light field. However, for temporal waveform shaping we refer to methods that can be used to reshape the temporal waveform from tens of cycles to potentially single cycle. Temporal control of a light field has advanced significantly as demonstrated by pulse shaping techniques and the frequency comb. For the completeness of this article, a pulse shaper using 4f optical relay is presented as one of the most widely used and relevant pulse shaper techniques (Fig. 6). A typical pulse shaper using 4f optical system consists of a pair of gratings, a pair of cylindrical lenses, and a mask at the frequency plane that can modify the amplitude and/or phase of the spectral content. The output waveform is determined by the input pulse spectral and the frequency plane modulation through Fourier transform. Such a pulse shaper architect is very mature and commercial products are available on the market. It should be noted that other pulse shaper techniques are also available and may be further explored in the future for spatiotemporal sculpturing of light as well. More comprehensive discussions on pulse shaping can be found in [3,73].

Figure 6. Schematic of a typical 4f pulse shaper for temporal waveform synthesis. The 4f configuration is used to ensure the system is dispersion free. Reprinted with permission from Weiner, Rev. Sci. Instrum. 71, 1929–1960 (2000), [3]. Copyright 2000, AIP Publishing LLC.

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3.3 Spatiotemporal Modulation Using a 2D Pulse Shaper

In the traditional 4f pulse shaper architecture, one dimensional amplitude or phase mask is used as the frequency plane modulator, only modulating the frequency contents while leaving the other dimension (space) untouched. However, this modulator does not have to be 1D. The modulation is typically done with LC-SLM, but can also be replaced with other spatial modulation devices such as metasurfaces, DOEs, DMDs, and so on. These modulation devices are naturally 2D. Thus, the 4f pulse shaper architect can be very easily extended to not only reshape the temporal waveforms but also the spatial profile (Fig. 7). Furthermore, mathematically the frequency dependence and spatial dependence can be made nonseparable, leading to a 2D pulse shaper that is capable of spatiotemporal modulation.

Figure 7. Schematic of a 2D pulse shaper as a STWP generator. The generator has the same configuration as a standard zero-dispersion 4f pulse shaper, except the modulator has a 2D format. A reflection-type configuration is shown here.

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This simple 2D pulse shaper is the core of many discussions in this article. In some sense, it is this device that has been driving the resurgence of STWP synthesis in the past several years. A k–ω spectral support domain picture has been advanced to understand the synthesis of some of the propagation invariant STWPs. However, its impact goes far beyond when the propagation invariant requirement is removed as evidenced by the recent developments such as spatiotemporal optical vortices (STOVs). In addition, the modulation applied in the space–frequency plane does not need to be phase only. It can contain all degrees of freedom of light: phase, amplitude, and polarization. Generally speaking, the optical field at the spectral–space plane and the optical field in the far field or any other planes outside the pulse shaper is connected through a linear transformation. As long as there are a sufficient number of degrees of freedom within the spectral–space modulation, a linear mapping could be established to produce desired spatiotemporal distribution at a designated plane outside of the pulse shaper. The design process can be treated as analogy to the iterative Fourier-transform algorithm (IFTA) or Gerchberg–Saxton algorithm that is widely adopted in computer-generated holography and laser shaping techniques [91–93]. Along this analogy, a spatiotemporal “hologram” concept can be established that would greatly expand the classes of STWPs with specifically designed spatiotemporal structures to meet the needs of certain applications [62].

3.4 Spatiotemporal Modulation Enhanced with Additional Spatial Modulation

A very natural thought to further expand the spatiotemporal modulation capability is to combine the recently developed spatiotemporal modulation techniques such as the 2D pulse shaper along with those already matured spatial modulation capabilities. In general, there are two approaches to carry out this combination (Fig. 8). The first is to incorporate the spatial modulation capabilities within the 2D pulse shaper. In other words, more comprehensive spatial modulation (amplitude, phase, and polarization as opposed to phase only) is performed in the space–frequency domain [22]. The second approach is to cascade spatiotemporal modulation with spatial modulation. The spatial modulation that follows the spatiotemporal modulation unit could be direct spatial modulation in amplitude, phase, and/or polarization [21,94], or spatial shaping devices such as MPLC [25] or other more general spatial operation such as a conformal mapper [23]. Essentially, this is to combine the techniques described in Sections 3.1 and 3.3 to offer much more sophisticated spatiotemporal sculpturing capabilities with almost unlimited possibilities.

Figure 8. Schematic of approaches to generate more complex spatiotemporal optical fields. (A) Replacing the pure phase or amplitude modulator with any general 2D spatial operation or modulation. (B) Cascading any general 2D spatial operation or modulation after the spatiotemporal modulation. The general 2D spatial operation or modulation can be as simple as a DOE such as a vortex plate, vectorial vortex plate, metasurface, or SLM, or can be more complicated optical system that does spatial transformation such as MPLC or conformal mapper.

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4. Basics of the Characterization of Light Fields

4.1 Spatial Characterization

Spatial characterization of light fields in general refers to measurements of field distributions within the beam cross section. Although early techniques ranging from photographic plates to scanning point/pinhole/slit, rotating drum, knife edge, and tomographic constructions are still used in certain situations and mostly for unusual wavelengths, they are mostly replaced with 2D array detectors such as charged coupled device (CCD), complementary metal-oxide-semiconductor (CMOS), and bolometer array. Combined with data collection and beam diagnostic software, intensity profiling of light beams has become a very mature technology and many commercial products are available nowadays.

These beam profilers measure the intensity or amplitude distributions of optical beams. However, to fully characterize optical beams, it is also necessary to retrieve other information such as phase and polarization. Spatially resolved phase information can be obtained with various interferometric setup in combination with array detectors or through different wavefront sensing techniques such as Shack–Hartmann, curvature sensing, or phase retrieval techniques developed by the digital holography and computational imaging community [95–97]. On top of the amplitude and phase information, optical vectorial fields contain spatially variant polarization distributions within the beam cross section. Thus, it is also of interest to be able to spatially resolve the state of polarization. This is typically done with polarimetric imaging settings through measuring the spatially distributed Stokes parameters. For general information regarding polarimetric imaging, readers may refer to textbooks on optical polarization such as [98].

The techniques described above are, in general, suitable for monochromatic CW or quasi-monochromatic quasi-CW optical fields. For broadband, polychromatic fields or ultrashort pulses, dispersive devices or spectral filters may be required to separate the spectral contents and then perform spatial characterization for individual spectral components. Otherwise, the recorded results would be an average over the spectral range. Full spatial characterization of light fields requires the proper combinations of these techniques, consequently leading of serial measurements that is time consuming. When temporal resolution is required for rapidly changing dynamic light fields, the speed of measurements would be the bottleneck that needs to be addressed.

4.2 Temporal Characterization: Pulse Waveform Measurement

The development of mode-locking lasers raises the problem of ultrashort pulse measurement since these optical pulses are significantly shorter than any photodetector response time. The need for temporal waveform characterization has increased along with the advances of sources producing shorter and more complicated waveforms and their continuously expanding applications. Many ultrashort pulse measurement techniques have been developed over the years. Two major considerations in ultrashort pulse characterization are the physical arrangement of the linear and nonlinear optical components and the inversion procedure. Most of the techniques utilize nonlinear optical component to circumvent the obstacle of lack of detectors with high enough speed. However, recent developments also show that pure linear optical techniques can also be employed to characterize ultrashort pulse waveforms. The early technique that dominated the field for many years measures the intensity autocorrelation of a pulse using nonlinear optical processes giving an estimation of the pulse duration rather than the actually resolved waveform. Recently developed ultrashort pulse measurement techniques to resolve the waveform in general can be categorized into three types: spectrography, tomography, and interferometry. The temporal information can often be dispersed into the spectral information and measured with the spatial 2D measurements techniques in terms of the spectral amplitude and phase information. Once these are determined, its corresponding temporal information can be reconstructed via inversion or filtering algorithms. Popular diagnostic techniques developed to characterize the electric field of a pulse in time domain include frequency-resolved optical gating (FROG) [99–102], spectral phase interferometry for direct electric-field reconstruction (SPIDER) [103–107], self-referenced spectral interferometry (SRSI) [108–110], and D-Scan [111,112]. Details of these and other ultrashort pulse characterization techniques can be found in an excellent review article published recently [113].

In these temporal waveform characterization techniques, spatiotemporal coupling typically was not considered while the temporal waveform is considered ideally independent of the spatial location in the beam. This property may not be preserved due to spatiotemporal coupling raised by certain physical processes or by some optical pulse generation mechanisms themselves. Spatiotemporal characterization of optical waveforms brings in the spatial dependence of the temporal waveforms of optical pulses, which creates further complexity and technical challenges.

4.3 Spatiotemporal Characterization

Spatiotemporal coupling is significant for strongly focused and ultrashort pulses and intentionally generated STWPs. Characterization of spatiotemporal optical fields is challenging owing to the amount of information involved. It may require 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 main categories: (1) spectrally resolved spectral/temporal measurement techniques; (2) spatially resolved spectral measurement techniques [114]. Scanning spectral/temporal measurement over space usually is unable to provide a full spatiotemporal measurement because these techniques are blind to the carrier-envelope relative phase and the pulse arrival time across space. A compact twofold spectral interferometer, based on in-line bulk interferometry and fiber-optic coupler-assisted interferometry, has been reported to measure infrared femtosecond vector beams with polarization evolving at the micrometer and femtosecond scales [115]. Digital holography has been demonstrated for spatiotemporal metrology in the terahertz spectrum [116,117]. Techniques such as the total electrical field reconstruction using a Michelson interferometer temporal scan (TERMITES) and INSIGHT are promising to become available products [118–120]. Considering the vector nature of electric fields, spatiotemporal vector beams provide additional challenge for characterization especially when the longitudinal electric components are prominent [24].

4.3a Self-Referenced Interferometry

The characterization of some of STWPs such as STOVs, however, is a less challenging task because only 2D information needs to be measured and the wave packet is considered scalar with the electric field oscillating along the direction of the spatiotemporal phase singularity line. One easy-to-implement method is the self-referenced interferometry technique that only uses linear optics [48,121]. As shown in Fig. 9, a STOV pulse is generated from a chirped wave packet. A reference pulse is split from the same light source and compressed through a grating pair. The reference pulse is transform-limited and considerably shorter than the STOV wave packet. The reference is brought to interfere with temporal slices of the STOV wave packet using a motorized precision stage. Each interference pattern contains the information of a temporal slice of the STOV. 2D amplitude and phase information can be extracted through a Fourier filtering algorithm and, finally, the 3D field information is reconstructed based on hundreds of 2D temporal slices.

Figure 9. Self-reference interferometry setup that only uses linear optics developed for STWP characterization. A transform-limited reference pulse is brought to interfere with temporal slices of the generated STWP. A linear translation stage is used to take snapshots of different temporal slices. Full spatiotemporal structure of the STWP can be reconstructed through Fourier filtering. Reprinted from [121] under a Creative Commons license.

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4.3b Single-Shot STWP Characterization

The self-referenced interferometry technique described in Section 4.3a requires high repetition rate, thus average over many pulses are performed. Spatiotemporal characterization of single-shot or low repetition rate (of the order of hertz) source remains to be a challenging task that needs solution. However, for STWPs with relatively simple structures or certain known characteristics, both linear and nonlinear single-shot characterization has been demonstrated. For example, STOVs can be characterized using single-shot supercontinuum spectral interferometry (Fig. 10) [47]. 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 (the witness plate). The spatiotemporal phase imposed on the supercontinuum probe pulse is extracted in an imaging spectrometer using a supercontinuum reference pulse. This nonlinear characterization technique requires fairly high peak intensity in order for the scheme to work. For weak STWPs, characterization techniques using linear optics have also been demonstrated recently by Chen et al. and Liu et al. (as shown in Fig. 11) [122,123].

Figure 10. Supercontinuum spectral interferometry setup that has been employed to perform single-shot STOV characterization. This nonlinear characterization technique requires high peak intensity. Reprinted with permission from [47]. © The Optical Society.

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Figure 11. Spatially resolved spectral interferometer setup that has been employed to perform single-frame STOV characterization. Since only linear optics is used, this technique is also applicable for weak optical fields. Reprinted from [122] under a Creative Commons license.

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5. Latest Developments in Spatiotemporal Sculpturing of Light

First, we would like to point out that many of the earlier developments in STWPs, including FWM, X-waves, O-waves, and the so-called localized waves, are covered in this review. This is partly due to the lack of experimental realization or observation of the predicted characteristics. For a historical overview of these earlier developments, please refer to the excellent recent review article [45]. The research field has been more or less dormant for a couple of decades and the linear light bullet work reported by Wise et al. may be regarded as a precursor to the revitalization of the field [75]. With the availability of novel STWP synthesizer and reports of new STWPs, particularly the implementation of 2D pulse shaper and the realization of STOVs, we have seen a rapid increase of research activities in this research area. With the recent demonstration of STOVs, toroidal vortices, toroidal pulses and hopfions, and beyond, this research field is at its cusp of taking off. Some of the latest developments in the spatiotemporal sculpturing of light reported within the past several years are summarized in this section.

5.1 Propagation-Invariant STWPs

First, we examine the freely propagating STWPs in this section. Searching for propagation invariant solutions to the Maxwell’s equations remains one the major objectives in spatiotemporal optics. The field structure of these STWPs typically exhibits propagation-invariant and self-healing properties observed within certain diffraction-free propagation distance that is given by the spectral uncertainty. Recent successes mostly are based on the 2D 4f pulse shaper or STWP synthesizer. To synthesize a propagation-invariant STWP, the phase pattern on the SLM for each wavelength takes the form of a linearly varying phase grating, which imparts a particular spatial frequency onto each wavelength. Whereas a traditional periodic grating provides a well-defined change in the transverse wavenumber to the entire spectrum, the linearly varying grating provides a different change in the transverse wavenumber to each wavelength, creating the necessary space–frequency (k–ω) correlation for STWP synthesis.

This principle is illustrated in Fig. 12, where the light cone is given by

(16)$$k_x^2 + k_z^2 = \frac{{{\omega ^2}}}{{{c^2}}},\; \; {k_y} = 0.$$

Clearly the wave packet constructed on this light cone assumes the optical field is constant along the y direction. Hence, the spatiotemporal structuring is limited to the x–t plane. Precise association of the spatial frequency and spectral frequency, and consequently space and time, can be understood with a variety of spectral support domain corresponding to conic sections at the interception of the light cone with a plane. Different types of STWP can be viewed as different conic sections produced by planes with different orientations, each of which describes precise correlation between temporal frequency and spatial frequency and consequently different types of STWPs can be constructed depending on the slicing angle. For examples, spatiotemporal Bessel wave packets with nondiffraction propagation properties (Fig. 13) and spatiotemporal Airy wave packet that travels along curved spatiotemporal trajectory have been demonstrated.

Figure 12. Concept of a diffraction-free space–time sheet explained with the k–ω light cone. (a, d, g, j) Intersection of the light cone for a monochromatic beam, pulsed Gaussian beam, hyperbolic diffraction-free beam, and elliptic diffraction-free beam. (b, e, h, k) The corresponding spatial frequencies to points in (a, d, g, j). (c) Calculated spatiotemporal intensity profiles at z = 0, 10z₀, and 20z₀ (z₀ is the Rayleigh range) for (a, d, g, j), showing diffractive spreading of monochromatic beam and pulsed Gaussian and the nonspreading feature for a hyperbolic and elliptic diffraction-free beam. The top panel shows the phase patterns and the corresponding k–ω light cone pictures for a beam with uncorrelated degrees of freedom, diffraction-free light sheets with different hyperbolic spectral correlation, and diffraction-free light sheets with elliptic spectral correlation. Reprinted by permission from Springer Nature: Kondakci and Abouraddy, Nat. Photonics 11, 733–740, Copyright 2017.

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Figure 13. Experimental setup for the generation of diffraction-free space–time light sheets using a 4f 2D pulse shaper. The phase pattern on the SLM is designed to produce wave packets corresponding to different intersections on the k–ω light cone. (a) Measured propagation of a generated diffraction-free space–time sheet. (b) Measured propagation of a generated diffraction-free space–time hollow sheet. Reprinted by permission from Springer Nature: Kondakci and Abouraddy, Nat. Photonics 11, 733–740, Copyright 2017.

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This k–ω spectral support picture is also instrumental to understand the group velocity tunability of STWPs. Through varying the intersection angle, it can be shown that the group velocity of the so-called baseband STWPs can be widely tuned into either subliminal, superluminal, or even negative regimes. As opposed to the minuscule deviations from speed of light c in previous works, the group velocity tunability range of these baseband STWPs can be very wide. For example, group velocity tunable range from 30c to –4c has been reported in [124]. Of course, group velocity tunability, particularly for superluminal cases, would inevitably raise the question of violation of causality and special relativity. For details, please refer to the recent review on space–time wave packets [45].

The theoretical model assumes infinite spectral resolution. In practice, this is impossible to realize due to the limited dispersive power of optical devices, spectral content of the source and pixelation of the SLM. Hence, there is always a finite spectral uncertainty associated with these STWP synthesizer. This spectral uncertainty determines the characteristics and quality of the generated propagation invariant STWPs through the capability of controlling their group velocities and diffraction-free propagation distance.

Such a k–ω light cone picture definitely is very useful in understanding the synthesis and properties/characteristics of the propagation invariant STWPs. Despite the success in generating propagation-invariant STWPs, the k–ω picture also limited the scope of STWPs and our imagination to some extent. Lifting the propagation-invariant requirement would provide much broader space for STWP synthesis and lead to findings of novel STWPs with rather peculiar physical properties, inevitably deviating from the k–ω space picture above. This is best illustrated with the recent developments in STOVs.

5.2 Spatiotemporal Optical Vortices

Photons can carry angular momentum, which consists of SAM and OAM. Photonic SAM is associated with the circular polarization state of the optical field, while the OAM is linked to the spatial helical phase structure of light. Under paraxial conditions, both the SAM associated with the circular polarization state and the OAM associated with the spatial helical phase patterns are in parallel with the Poynting vector, thus they can be termed as longitudinal SAM and longitudinal OAM.

Under situations that involve strong focusing or evanescent waves, the situation can change dramatically, and SAM with a component that is orthogonal to the propagation direction can be generated, which will be referred to as transverse SAM [56–58]. The transverse SAM of light has gained increasing attention because of its unique physical characteristics. The spin axis of the optical beam with purely transverse SAM is orthogonal to its propagation direction because of the existence of a longitudinal electric field component that is in the quadrature (π∕2 phase difference) with respect to its transverse field components.

Meanwhile, optical vortices carrying transverse OAM remained elusive until very recently. A light field that may carry transverse OAM was first proposed in [125]. Theoretical analysis of STOVs with transverse OAM component was developed using the special theory of relativity, while a spatial optical vortex is seen by a transversely moving observer near the speed of light as a tilted vortex [126]. Experimental wise, STOVs accounting for a very small fraction of the total energy were observed in femtosecond filaments in air [46]. It was not until very recently that controllable generation of STOVs with transverse OAM with linear optics were achieved using the 2D 4f pulse shaper (Fig. 14) [48]. A helical phase pattern is loaded to the SLM within the pulse shaper. Such a singularity in the ω–k domain is then transformed into a singularity in the x–t domain, creating a STWP containing OAM that is purely transverse.

Figure 14. Experimental generation and characterization of a STOV. A vortex phase is loaded onto the SLM in the 2D pulse shaper. (a) Illustration of the temporal slicing. (b) Predicted and measured interferograms at different temporal slices. (c) Measured spatiotemporal phase map showing the vortex phase in space and time. Reprinted by permission from Springer Nature: Chong et al., Nat. Photonics 14, 350–354, Copyright 2020.

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The success generated a flurry of interests, leading to a series of studies on STOVs. Second-harmonic generation (SHG) (Fig. 15) [127,128] and high-harmonic generation (HHG) [129] of transverse OAM were reported and the conservation of transverse OAM was examined. Rigorous calculation of transverse OAM and the coupling of transverse OAM and SAM were studied [130–132]. Schemes were designed to generate STOVs using metasurfaces and photonic crystals [133,134]. To prevent the collapsing of STOVs due to the spatiotemporal astigmatism through propagation and generation stable transverse OAM with high topological charge, Bessel STOV (Fig. 16) and high-order Bessel STOVs (Fig. 17) have been demonstrated [135,136].

Figure 15. SHG for STOV and transverse OAM conservation. (a) Conceptual illustration of STOV SHG. (b) Experimental setup for generation and characterization. (c) Measured STOV for fundamental. (d) Measured STOV for SHG STOV. Reprinted by permission from Springer Nature: Gui et al., Nat. Photonics 15, 608–613, Copyright 2021.

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Figure 16. Generation of spatiotemporal Bessel wave packets carrying STOVs. The top row shows the measured Bessel STOVs with different topological charge and their propagation stability. Reprinted from [135] under a Creative Commons license.

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Figure 17. Generation of spatiotemporal Bessel STOVs with extremely high topological charge. (a)–(d) Phase patterns for the generation of Bessel STOVs with topological charges of 10, 25, 50, and 100. (e) The experimental setup for the generation and characterization of high-order Bessel STOVs. Top left panel: The k–ω pictures explain the generation of STOV and Bessel STOV with a topological charge of 5. Top right panel: measured Bessel STOVs with topological charges of 10, 25, 50, and 100. Reprinted from [136] under a Creative Commons license.

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Undoubtedly, the experimental realization of the controllable generation of STOVs triggered a series of research in this field and in some sense stimulated the research interests in the general spatiotemporal optics. Interested readers may refer to a recent review article that covers the latest developments of spatiotemporal vortices of light ranging from theoretical physics, experimental generation schemes, and characterization methods, to applications and future perspectives [49]. It is believed that this new degree of freedom in photonic OAM endowed by STOVs may pave the way to the discovery of novel physical mechanisms and photonic applications in light sciences.

5.3 Combination of Spatiotemporal and Spatial Modulation

A natural thought to further expand the classes of STWP is to combine spatiotemporal modulation with spatial modulation. This has been implemented in several different ways. In terms of optical singularities, a spatial vortex and a STOV can be combined and the interactions between these two types of phase singularities were observed experimentally (Fig. 18). With this method, one can synthesize the OAM orientation and quantity in a way that is similar to the recently developed techniques to control photonic SAM [94,137].

Figure 18. Collision between a STOV and a spatial optical vortex. This is an example of cascading spatiotemporal modulation with spatial modulation. (a)–(d) Interferograms at temporal slices shown in (e). (e), (h) Cut-through views of the spatiotemporal and spatial vortices, respectively. Clear twisting interactions between these two types of optical vortices can be observed. Reprinted from [94] under a Creative Commons license.

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A similar two-stage configuration using MPLC as the spatial mode shaping module has also been demonstrated (Fig. 19) [25]. With this method, STWP synthesis is understood with spectral–spatial coherent modal decomposition. In the pulse shaper module, one dimension of the SLM is responsible for imprinting the desired spectral phase profiles, whereas the other dimension is utilized to controllably steer the ultrashort pulse to a specific output position by applying a linear phase. By encoding a superposition of multiple holograms onto the SLM, the 2D pulse shaper generates an array of spectrally modulated beams in the vertical direction. The MPLC as the second stage converts the linear array of Gaussian beams into a set of co-propagating 2D spatial modes, forming the desired STWP. Both separable and nonseparable trains of ultrafast wave packets with time-varying dynamic angular momentum and tailored spectral characteristics have been demonstrated. To characterize the spatiotemporal composition of the synthesized STWPs, a spectrally resolved holographic technique that combines off-axis Mach–Zehnder interferometry with a controllably tilted narrow bandpass filter is developed to retrieve the complex (amplitude/phase) spatial composition of each spectral component at different time locations.

Figure 19. Generation of spatially and temporally correlation wave packets using the combination of 2D pulse shaper and MPLC. The MPLC cascaded after the 2D pulse shaper enabling the precise engineering of the space–time correlation. The top panel illustrates an example wave packet with temporally varying spatial modes. Reprinted by permission from Springer Nature: Cruz-Delgado et al., Nat. Photonics 16, 686–691, Copyright 2022.

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The strategy to cascade spatiotemporal modulation with different types of spatial modulation/operation has been employed to generate other novel STWPs, such as vectorial STWPs [21], optical toroidal vortices [23], optical hopfions [27], and so on, which are discussed in later sections. These cascaded methods use spatial modulation outside of the pulse shaper such that the spatiotemporal modulation and the spatial modulation are performed sequentially. However, these already matured spatial modulation/operation techniques can be incorporated inside the 2D pulse shaper as well, which is illustrated in the vectorial pulse shaping in Section 5.5 [22]. It is anticipated that various combinations and permutations of spatiotemporal and spatial modulation techniques will enable the synthesis of STWPs with nearly unlimited possibilities [62].

5.4 Linear Mapping with Chirped Pulses

One can further take advantage of the chirping property of ultrashort pulses to sculpt various spatiotemporal optical structures in space–time. It is well known that for a chirped pulse, the wavefront is tilted in space–time. Thus, different spectral content within the pulse actually arrives at the modulator at different time. Based on this observation, a linear mapping relationship between the spectral–space modulation loaded on the SLM and the spatiotemporal distribution after the pulse shaper can be derived and utilized to perform spatiotemporal structuring. The concept is experimentally verified through the generation of multiple STOVs and STOV lattice within the same STWP (Fig. 20) [138].

Figure 20. Generation of STWPs through linear space-to-time mapping utilizing the wavefront tilt of a chirped pulse. The top panel figures in (a)–(f) show exemplary wave packets with spatiotemporally distributed STOVs with different topological charges. Figures on the right-hand side of the top panel demonstrate the capabilities of adding additional GDDs to different portions of the wave packet to simulate acceleration and deacceleration of different parts of the wave packet. Reprinted with permission from [138]. © The Optical Society.

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This work also demonstrated another important advantage of such a simple 2D 4f pulse shaper architect in terms of its flexibility and versatility. It is very straightforward to load different phase modulation both along the spectral direction and the spatial direction for segments within the SLM to perform different spatial and temporal operations, such as focusing/defocusing and accelerating/deaccelerating to different parts of the spatiotemporal pulse, opening a whole new avenue for STWP manipulation and operation. The power of this method is demonstrated by STOV polarity reversal, STOV collision, STOV annihilation, and overlapping (see Fig. 20 inset). Such a direct mapping technique opens tremendous potential opportunities for sculpturing complex spatiotemporal waveforms.

5.5 Vectorial and Amplitude Spatiotemporal Modulation

Polarization is an important vectorial characteristic of an optical field. A natural way to add polarization structures to STWP is to cascade STWP with spatial polarization modulation device. For example, cylindrically polarized STOV have been generated and characterized successfully by simply adding a vector vortex plate to the STOV (Fig. 21) [21]. This example is also a demonstration of the coexistence of two types of singularities within the same STWP: phase singularity in the spatiotemporal domain and polarization singularity in the spatial domain. One thing to notice, however, is the different behavior between this wave packet and the wave packet reported in [94], which also contains two types of singularities: phase singularity in the spatiotemporal domain and phase singularity in the spatial domain. The two types of phase singularities strongly interact and twist each other, while the two types of singularities in the cylindrical vector STOV do not interact with each other at least under the paraxial condition.

Figure 21. Cylindrically polarized STOV generated by cascading a vectorial vortex plate after the 2D pulse shaper. (a), (f) Illustrations of the x- and y-polarization components of the wave packet. (b),(c),(d),(e) and (g),(h),(i),(j) Interferograms for both polarization components, respectively. The reconstructed wave packet is shown in (k), and (l) is a cut-through view of (k). Note that no twisting effect is observed, which is very different from the phenomenon shown in Fig. 18. Reprinted from [21] under a Creative Commons license.

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A more challenging vectorial modulation is to rapidly change polarization in both space and time. Spatiotemporal vectorial pulse shaping can produce ultrafast optical pulses that enable unprecedented coherent control for light–matter interactions. A traditional pulse shaper produces ultrafast pulses with scalar programmable waveforms for various quantum control applications. However, quantum systems are 3D in nature. Thus, the interactions are inevitability vectorial. Polarization shaping of ultrafast laser has been pursued for quantum control and realized in different ways [139]. However, existing polarization pulse shaping techniques, which are often difficult to align and cumbersome to handle, can only produce dynamic polarization modulation in the temporal domain [140–143]. Spatiotemporal vectorial structuring of ultrashort pulses was realized recently through simply introducing a λ/4 plate in a pulse shaper using a 2D LC-SLM (Fig. 22) [22]. STWPs with sophisticated spatiotemporal vectorial structures such as spatiotemporal spin grating, spatiotemporal spin lattice, and spatiotemporally twisting polarization have been demonstrated (Fig. 23). The generated vectorial STWPs were characterized with polarization selective interferometry. This development also offers a straightforward way to perform spatiotemporal amplitude modulation by following a linearly polarizer after the vectorial spatiotemporal modulation. Such a spatiotemporal amplitude modulation capability was demonstrated with the generation of a “USST” logo in the space–time domain as shown in Fig. 24.

Figure 22. Producing spatiotemporal polarization modulation with a vectorial 2D pulse shaper. The vectorial 2D pulse shaper is realized by simply inserting a λ/4 plate into the regular 2D pulse shaper. The specially oriented λ/4 plate and the reflection type LC-SLM form a polarization rotator illustrated in Fig. 2. (a), (c) Measured 3D and 2D spatiotemporal polarization distribution when the phase pattern (b) is loaded. (d), (f) Measured 3D and 2D spatiotemporal polarization distribution when the phase pattern (e) is loaded. Reprinted from [22] under a Creative Commons license.

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Figure 23. Experimentally generated vectorial STWPs obtained by a loading vortex phase into the vectorial 2D pulse shaper shown in Fig. 22. The state of polarization continuously evolved with respect to time, and the evolution is spatially variant. Reprinted from [22] under a Creative Commons license.

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Figure 24. Demonstration of spatiotemporal amplitude modulation by cascading a linear polarizer after the vectorial 2D pulse shaper shown in Fig. 22. The logo of “USST” is generated and measured in the spatiotemporal domain. Reprinted from [22] under a Creative Commons license.

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5.6 Spatiotemporal Modulation with Metasurface Devices

As discussed in Section 3.1c, the functionality of SLM can be realized with metasurfaces to reduce the cost/size/weight of the pulse shaper as long as reconfigurability is not required. As shown in Fig. 25, a metasurface is used in the 4f pulse shaper configuration for pulse shaping [144]. With the modern nanofabrication capabilities, the use of metasurface as SLM offers the benefit of higher spatiotemporal resolution (smaller pixel size). In addition, metasurface devices also provides multidimensional modulation (phase, polarization, and amplitude) capability within the same device, representing another advantage over the traditional phase only LC-SLM-based pulse shaper configurations. For example, spatiotemporal polarization modulation with metasurface element has been reported recently (Fig. 26) [145].

Figure 25. Ultrafast optical pulse shaping using dielectric metasurfaces. In a 4f configuration, a metasurface device is used to perform the 2D phase modulation at the frequency plane (A). The phase modulation is realized by the geometric phase controlled by the orientation of individual elements (B, C). From Divitt et al., Science 364, 890–894 (2019) Reprinted with permission from AAAS.

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Figure 26. A 4f 2D pulse shaper setup using a metasurface that is capable of synthesizing ultrafast optical pulses with sophisticated spatiotemporal control. The metasurface device performs various modulations to different spectral contents of the pulse and simultaneously transforms them into targeted waveforms with the requisite amplitude, phase, polarization, and wavefront distribution and is coherently recombined. The top panel illustrates an example of an OAM-carrying, polarization-swept ultrafast pulse. Reprinted from [145] under a Creative Commons license.

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In addition, if LC materials or phase change materials are incorporated within the metasurface configuration, it is conceivable that a certain level of reconfigurability or dynamic modulation may be realized [89,90,146]. For metasurfaces made with phase change materials, both high spatiotemporal resolution and high refresh rate can be implemented simultaneously. Thus, the developments in this area and their applications in STWP synthesis definitely worth of future attention.

5.7 Integrated Devices for STWP Synthesis

Despite the use of metasurface reducing the cost/size/weight for experimental STWP synthesis, a 4f pulse shaper configuration is still needed. Although this 4f pulse shaper configuration offers great flexibility and reconfigurability, it is bulky and takes a good chunk of space. Several recent reports showed that the pulse shaper may be replaced with single nanophotonic integrated device. For example, photonic crystal devices for STOV generation and orientation control have been reported (Fig. 27) [134]. With the recent development of ultrafast laser on a chip [147], this type of devices may enable the development of integrated on-chip devices/systems for STWP synthesis.

Figure 27. Generation of STOV with different orientations with a photonic crystal. (a) Input Gaussian pulse. (b)–(d) Various STOVs with different orientations. (e) Device for generating transmission functions shown in (f)–(h) that correspond to the transmission functions needed to convert the incident Gaussian pulse into the STOVs shown in (b)–(d). (i) The geometry of the photonic crystal slab used in the study. (j), (k) The amplitude and phase for the STOV generated by passing a Gaussian pulse through the photonic crystal slab. The state of polarization continuously evolve with respect to time and the evolution is spatially variant. From Divitt et al., Science 364, 890–894 (2019). Reprinted with permission from AAAS.

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5.8 Self-Torque Pulse and Light Spring

An interesting kind of STWP is the so-called helical beam or light spring with self-torque, which essentially is STWP containing time-varying OAM. This type of STWP can be synthesized with OAM pulses with different center frequency and different topological charge [148]. Such self-torque pulse has been demonstrated with HHG at extreme ultraviolet (EUV) wavelength (Fig. 28) [149]. However, the setup is very bulky and expensive. The synthesis is complex and characterization is challenging as well. A relatively compact setup using diffractive axicon with circular geometry was employed by Piccardo et al. recently to create broadband topological correlations in space–time beams and produce light springs [150].

Figure 28. Light springs and optical pulse with self-torque. (A) Experimental scheme for generating and (B) measuring light beams with self-torque [149]. The top left panels show examples of the spectral intensity and variation of the azimuthal mode index for a single-coil light spring and double-coil light spring [148]. The top right panel shows the spatial HHG azimuthal spectrum for the experimental generation EUV self-torque pulse [149]. Reprinted with permission from [148]. © The Optical Society and from Rego et al., Science 364, eaaw9486 (2019). Reprinted with permission from AAAS.

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Multiple optical-frequency-comb lines with superposition of LG_lp modes for each line was proposed and numerically studied to produce spatiotemporal beam with dynamic rotation (OAM) revolving along a spring-like trajectory [151]. Microcomb platform emitting different OAM orders was experimentally demonstrated to produce such helical beam [152]. The microcomb device can generate a frequency comb with different OAM order between two adjacent frequencies. When these frequencies are phase locked together, self-torque pulses with adjustable OAM order difference and hence helical, double-helical, and triple-helical spatiotemporal structures etc. can be synthesized, offering a much more compact and flexible platform for light spring generation.

Using the 2D pulse shaper technique described above, STWP with rapidly varying OAM may be generated with a much simpler method. For example, as reported in [153], pulses with changing transverse OAM can be realized through a simple phase pattern on the SLM (Fig. 29). If this is cascaded with another SLM, pulses with variable OAM orientation and magnitude can be easily realized as well. Beyond light pulses with temporally varying OAM, STWPs in the form of STOV lattice containing spatiotemporally varying OAM have also been realized [138].

Figure 29. Generation of STWPs with rapidly changing transverse OAM using 2D pulse shaper: (a) interferograms measured at different temporal slices of (b) the corresponding wave packet; and (c) the phase map loaded onto the SLM. Reprinted from Sci. Bull. 65, Wan et al., “Generation of ultrafast spatiotemporal wave packet embedded with time-varying orbital angular momentum,” pp. 1334–1336, Copyright 2020, with permission from Elsevier.

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5.9 Toroidal Vortices of Light

The rapid developments of spatial and spatiotemporal structuring of light enable the generation of optical fields with a broad variety of topological structures that was not possible or not even conceived before. One such example is the toroidal vortex of light or optical vortex rings [23]. Toroidal vortices are intriguing propagating ring-shaped structures with whirling disturbances rotating about the ring. Toroidal vortices are very common in nature and science, such as smoke rings puffed by smokers and bubble rings produced by dolphins. The scientific investigation of vortex rings can even dates back to Lord Kelvin’s vortex atom model [154]. Research into vortex rings remains active in a variety of disciplines. However, its optical realization has never been reported or even considered until very recently.

In [23], a photonic toroidal vortex is shown to be an approximate solution to Maxwell’s equations that can propagate without distortion in a uniform medium with anomalous group velocity dispersion (GVD). The experimental generation of toroidal vortex builds upon the success of STOV generation. A STOV is stretched with a pair of cylindrical lenses and then a Cartesian-to-polar coordinates conformal mapping technique is utilized to wrap the stretched STOV into a ring shape, forming toroidal vortex of light (Fig. 30). The generation of toroidal vortex is confirmed with the same interference technique used in the STOV characterization reported in [48,121]. It should be pointed out that the vortex phase structures of the generated optical toroidal vortex were examined at several cross sections of the ring. Full characterization of the field distribution of the entire structure without stitching remain to be a challenging task.

Figure 30. Generation and characterization of optical toroidal vortex. A 2D pulse shaper is used to produce STOV first. The STOV is elongated by a pair of cylindrical lenses. Conformal mapping is then performed to wrap the elongated STOV into a vortex ring: optical toroidal vortex. The top panel shows the experimentally reconstructed intensity distributions and the vortex phase at different locations of the ring. Reprinted from [23] under aCreative Commons license.

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5.10 Toroidal Pulse

The optical toroidal vortex described in Section 5.9 is a scalar solution. Almost at the same time, toroidal light pulses with vectorial structure were also reported [24]. In 1996, Hellwarth and Nouchi theoretically identified a radically different, non-transverse electromagnetic pulse with a toroidal topology [155]. These pulses are propagating counterparts of localized toroidal dipole excitations in matter with unique properties such as nontransversal, vectorial, space–time nonseparable, and single cycle. However, these exact same features also make it extremely difficult to realize and characterize experimentally. Hence, the actual generation and observation of toroidal pulses remained elusive until the recent report on experimental observation of toroidal pulses in the optical and terahertz regions using metasurfaces with spatially tailored dispersion (Fig. 31) [24].

Figure 31. Generation and characterization of toroidal pulse. (a), (b) Spatiotemporal and spatiospectral structure of the toroidal pulse. (c) Schematic of the generation of an optical toroidal pulse. (d) Schematic of the generation of terahertz toroidal pulses using plasmonic metasurfaces. (e), (f) Experimentally measured spatiotemporal and spatial structures of the transverse field component of the generated optical toroidal pulse. (g) Calculated spatiotemporal structure of the longitudinal field component of the generated optical toroidal pulse. Reprinted by permission from Springer Nature: Zdagkas et al., Nat. Photonics 16, 523–528, Copyright 2022.

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The generation scheme for optical toroidal pulses comprises a linear-to-radial polarization converter and a nanostructured metasurface that are driven by ultrashort linearly polarized laser pulses. The metasurface consists of concentric gold rings that is designed with the transmission resonance wavelength varying along the radial direction, providing the necessary radial chirp for the generation of toroidal pulses. The spatiotemporal and spatiospectral structures of the generated toroidal pulses were characterized by hyperspectral imaging of their transverse profiles and spatially resolved Fourier transform interferometry. The latter is based on interference of the generated toroidal pulse with a known reference linearly polarized pulse, allowing retrieval of the spectral amplitude and phase for all transverse polarization components of the electric field at any point of the toroidal pulse.

It should be pointed out that strictly speaking the optical pulses generated above were not toroidal pulses as the single-cycle requirement was not satisfied. In addition, the characterization techniques used cannot directly measure the nontransverse component of the field. To complement the optical results, single-cycle toroidal pulses in the terahertz spectral region was demonstrated and characterized through optical rectification of femtosecond near-infrared pulses on a Pancharatnam–Berry phase metasurface. Despite the further improvement still needed in the optical regime to truly demonstrate an optical toroidal pulse, this work nevertheless paves the way for experimental studies of energy and information transfer with toroidal light pulses, their spatiotemporal coupling and light–matter interactions.

5.11 Optical Hopfions and Topological Perspectives

The realization of optical toroidal vortices and toroidal pulses opens new opportunities for topological photonics. Much more complicated optical topology can be generated with the techniques used for the creation of a toroidal vortex. One such recent example is the optical hopfion. Hopfions are 3D topological states discovered in the field theory, magnetics, and hydrodynamics that resemble particle-like objects in the physical space [156]. Hopfions inherit the topological features of the Hopf fibration, a homotopic mapping from unit sphere in 4D space to unit sphere in 3D space. Formation of optical hopfion structures has been demonstrated using vectorial light with the 3D polarization plus phase as the fourth dimension [26]. Nevertheless, previously reported formations of optical hopfions are static and not complete.

Building upon the optical toroidal vortex, dynamic scalar optical hopfions in the shape of a toroidal vortex can be created simply by adding a spatial vortex phase, which can be shown as a solution to the Maxwell’s equations under the paraxial approximation (Fig. 32) [27]. The equiphase lines corresponding to disjoint and interlinked loops can form complete ring tori in 3D space. The Hopf invariant, the product of two winding numbers, is given by the topological charge of the poloidal spatiotemporal vortices and the spatial vortices in toroidal coordinates. It should be emphasized that such a dynamic scalar hopfionic structure is formed with equal phase lines of the optical field as opposed to those optical hopfionic structures defined with 3D vector textures [26]. The realization of scalar optical hopfions provides a photonic platform for topological studies and may be utilized as high-dimensional information carriers.

Figure 32. Experimental setup for the generation and characterization of scalar optical hopfions. The setup is essentially the same as that for the optical toroidal vortex generation. An additional spatial vortex phase can be loaded to the last SLM. The phase of the electrical field can be mapped into the optical hopfions. The top panel shows various optical hopfion structures with different winding numbers. Reprinted from [27] under a Creative Commons license.

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Mathematically, topology can be understood as projection from higher dimension to lower dimension. The toroidal vortex, vortex pulse and hopfions can be considered as special members of knot, a subfield of topology that traces back to Lord Kelvin. In principle, the high-dimensional parameters offered by STWPs creates unprecedented opportunities for topological studies. In addition to scalar optical hopfions, several other interesting developments including supertoroidal pulses as free-space electromagnetic skyrmions [157], optical twisted phase stripes [158], topological transformation and free-space transport of optical hopfion [28] have also been studied very recently (Fig. 33).

Figure 33. Various topological structures realized with STWPs. Top panel: skyrmionic structure embedded as the vectorial features of toroidal and supertoroidal pulses [157]. Middle panel: hopfion structure established with the phase of toroidal vortex [27]. Bottom panel: twisted stripes constructed with phase structure of toroidal vortex [158]. Top and middle sections reprinted from [27,157] under Creative Commons licenses. Bottom panel reprinted with permission from Zhong et al., ACS Photonics 10, 3384–3389 (2023) [158]. Copyright 2023 American Chemical Society, https://doi.org/10.1021/acsphotonics.3c00881

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5.12 STOV with Partially Coherent Light

One other interesting but perhaps somehow counterintuitive development is the spatiotemporal structure in (quasi-CW) partially coherent light. Because of the partial coherent nature, the phases of different frequency and modes are not completely correlated, thus the spatiotemporal structure would be random and no interesting spatiotemporal structures would have been anticipated. However, singularities in the spatiotemporal domain may provide certain topological stability or invariants that may be maintained, making the feasibility of observing STOVs with partially coherent light very intriguing. Theoretical descriptions of the formation and propagation of partially coherent vortex beams in the space–frequency and spatiotemporal domain was studied theoretically by Hyde in [159]. Meanwhile, numerical and experimental demonstration of STOVs generated using a superluminescent diode (SLD) as a light source with partial temporal coherence and fluctuating temporal structures was reported in [160,161] (Fig. 34). The existence of spatiotemporal phase singularity structures can be understood through longitudinal modes with random phase fluctuations under a global spiral phase envelope in the space–frequency domain. If the phase randomness is completely eliminated, coherent STOV is formed as discussed in Section 5.2. As the phase randomness increases, the shape of the STOV is severely distorted from the ring-shaped profile with multiple singularities occurring at various temporal locations. The study shows that cheap broadband partially coherent sources instead of mode-locked lasers can be used for generating STOVs, offering a low-cost option for spatiotemporal vortex generation. It should, however, be pointed out that this could be just the beginning of the investigation of partially coherent light with spatiotemporal structures and lots of rich phenomena remain to be explored. As evidenced by the history of their spatial counterparts, partially coherent spatially structured light with or without singularities have been shown to offer certain advantages compared with coherent sources for specific applications such as imaging or propagation through turbulent media [162]. In analog, similar and new benefits may be waiting to be discovered for partially coherent optical fields with embedded and designed spatiotemporal structures.

Figure 34. Generation of STOVs with partially coherent light: (a),(b) simulated intensity and phase distribution of partially coherent STOV with Poynting vector superimposed; (c) experimental setup using a 2D pulse shaper configuration with an incoherent light source; (d) spectrum of the incoherent source. Reprinted with permission from [160]. © The Optical Society.

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6. Manipulation of STWPs and Applications

6.1 Propagation of STWPs

In order to make use of these spatiotemporally sculptured optical fields, it is necessary to understand the propagation properties and develop methods to manipulate these STWPs, including reflection, refraction, and focusing. It has been shown that propagation of STWPs in free space and nondispersive homogeneous media exhibit unique properties such as wave front rotation [163] and space–time Talbot effect [164] with various applications including attosecond pulse generation [165].

As pointed out earlier, the STWPs discussed in this tutorial do not focus on the propagation invariant properties. In general, due to the dispersion and diffraction mismatch, if the STWP propagates in free space or a homogeneous medium, the mismatch will cause the spatiotemporal distributions and structures to change and eventually stabilize at a certain spatiotemporal profile that may be very different from the initial spatiotemporal distributions (Fig. 35). However, under paraxial and SVE approximations, the STWPs can be assumed to be stable within the Rayleigh range ${z_0} = \frac{{\pi w_0^2}}{\lambda }$ and dispersion length ${z_d} = \frac{{\pi \tau _0^2}}{{{D_v}}}$.

Figure 35. (a) Free-space propagation of STOV [47]. (b) The propagation properties can be readily simulated by adding different amounts of GDD onto the SLM in the 4f 2D pulse shaper configuration [138]. Reprinted with permission from [47,138]. © The Optical Society.

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The mismatch is mainly caused by the lack of dispersion for free space propagation. This can be pre-compensated with loaded GDD to the pulse in order to produce a specific STWP structure at a fixed location (as shown in the bottom panel of Fig. 35). Dispersion can also be introduced to balance the diffraction. Under this situation, STWPs can be regarded as eigenmodes of the medium and remain unchanged during the entire propagation through such a dispersive medium.

Mathematically, the propagation of general STWPs can be handled with plane wave decomposition. The initial STWP is decomposed into monochromatic plane waves through Fourier transform. Each plane wave component can be treated individually as it propagates through the space. At the observation plane, inverse Fourier transform is then applied to synthesize output. Using this approach, propagation of various STWPs including STOVs and STOV modulated with spatial OAM have been studied [47,94].

6.2 Focusing of STWPs

To apply STWPs in light–matter interactions, often time it is necessary to confine the interaction within a small volume. In other words, we need to strongly focus STWPs and produce a focused STWP with prescribed characteristics within the focal volume. Hence, it is necessary to study the focusing properties of various STWPs. In general, this can be done through plane wave decomposition, where the STWP is Fourier transformed and decomposed into monochromatic plane waves and each individual plane wave is dealt with separately with the existing methods such as vectorial Debye integral or Richard–Wolf vectorial diffraction theory [79,80]. Then the focused STWP is synthesized from these individual plane wave contributions through inverse Fourier transform. This strategy can, in principle, be applied to deal with the focusing problems for arbitrarily complex STWPs.

However, in practice the computation load could potentially be insurmountable. A simpler model has been developed to study some of the characteristics of STWP focusing (Fig. 36) [166]. In this model, the STWP source is sliced in the temporal domain and each spatial slice is treated with a traditional focusing model such as Debye integral and the eventual focused STWP at the focal plane (or any other planes near the focus) is repackaged by stitching these focused slices together using the following formula:

(17)$${E_f}({{r_f},\phi ,{z_f},t} )= \mathop \smallint \limits_0^\alpha \mathop \smallint \limits_0^{2\pi } {E_\mathrm{\Omega }}({\theta ,\varphi ,t} )\times {e^{ - jk[{{r_f}\sin \theta \cos ({\varphi - \phi } )+ {z_f}\cos \theta } ]}}\sin \theta d\theta d\varphi , $$

where $\alpha $ is the convergence semi-angle determined by the numerical aperture (NA) of the lens, ${r_f} = \sqrt {x_f^2 + y_f^2} $, $\phi = {\tan ^{ - 1}}({{y_f}/{x_f}} )$, ${E_\mathrm{\Omega }}({\theta ,\varphi ,t} )$ is the incident optical field on the spherical surface $\mathrm{\Omega }$. A planatic objective lens that obeys the sine condition is used, which gives the pupil apodization function $\sqrt {\cos \theta } $.

Figure 36. Configuration for a simple numerical model for the calculation of highly focused STWPs. The assumption is that each temporal slice of the incident STWP is mapped on the spherical surface Ω and then focused toward the focal plane. The focusing process is performed with the Richard–Wolf vectorial diffraction method. Reprinted with permission from Chen et al., ACS Photonics 9, 793–799 (2022) [132]. Copyright 2022 American Chemical Society, https://doi.org/10.1021/acsphotonics.1c01190

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This simple treatment allows us to appreciate a distinctively different perspective of spatiotemporal focusing, which is the unbalance between the spatial diffraction and temporal dispersion. This unbalance between diffraction and dispersion is similar to the astigmatism induced by cylindrical lens in 2D spatial focusing [167,168]. Such a spatiotemporal astigmatism needs to be pre-compensated in order to produce the desired STWP at the focal plane. Using this strategy, subwavelength STOV focusing can be realized with high-NA objective lens (Fig. 37). The scalar Eq. (17) can be easily extended to include polarization into consideration. The spin–orbital coupling (SOC) within the focal volume for polarized STOVs has been successfully investigated [132] and very distinctive topological structures have been observed numerically [169]. This simplified model was tested with a more rigorous study with the general plane wave decomposition method [170]:

(18)$${E_f}({{r_f},\phi ,{z_f},\omega } )= \mathop \smallint \limits_0^\alpha \mathop \smallint \limits_0^{2\pi } {E_\mathrm{\Omega }}({\theta ,\varphi ,\omega } )\times {e^{ - jk[{{r_f}\sin \theta \cos ({\varphi - \phi } )+ {z_f}\cos \theta } ]}}\sin \theta d\theta d\varphi , $$

where ${E_\mathrm{\Omega }}({\theta ,\varphi ,\omega } )$ is the frequency component of the incident optical field on the spherical surface $\mathrm{\Omega }$. The focused STWP can be calculated with inverse temporal Fourier transform of Eq. (18) back into the spatiotemporal domain. It has been shown that as long as the pulse is not too short (e.g., longer than 100 fs at 1 µm central wavelength), the simplified method remains to be an excellent approximation. As the pulse width gets shorter, the simplified model results will deviate and a more comprehensive model is necessary, especially as the pulse width approaches single cycle of the light wave.

Figure 37. Top panel: amplitude and phase of the incident STOV and the corresponding focused wave packet, showing the collapsing of the STOV structure at the focus. Bottom panel: With precondition STWP, a subwavelength STOV structure can be produced at the focus. Reprinted with permission from [166]. © The Optical Society.

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However, it should be kept in mind that in order to fully appreciate the focusing behavior of STWPs, material properties of the objective lens and various type of aberrations also need to be considered. This is especially critical for cases where extremely short pulse width and, hence, very broad spectral width are involved. On the other hand, it is also worth pointing out that although these factors will in most cases be nuisances, they may offer interesting means of producing unexpected STWP structures as well.

6.3 Reflection and Refraction of STWP

Perhaps the simplest types of light–matter interactions are reflection/refraction at interfaces. In principle, plane wave decomposition can be used to study the behavior of any beams or wave packets at interfaces using Snell’s law. The collective behavior resulted from the interferences of all the constituting plane wave components may lead to unexpected phenomena different from individual plane wave component. One such example similar to optical beams in the spatial domain is the Goos–Hänchen shift and the spin-Hall effect [171]. Small shifts and time delays of STWPs such as Goos–Hänchen shift, Wigner time delay, and spin-Hall effect have attracted increasing attention. 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 techniques.

Vortex beams carrying longitudinal OAM have been shown to experience beam shifts increased by the factor of their topological charge. In parallel, STOVs carrying transverse OAM exhibit peculiar properties of beam shifts and time delays depending on the value and orientation of the OAM. Because of the transverse nature of the OAM carried by STOVs, reflection/refraction of STOVs can be classified into two categories, with the OAM either perpendicular or parallel to the incident plane (Fig. 38), similar to the s- and p-polarization for plane wave reflection/refraction at interfaces. The normalized integral OAM of the incident pulse can be shown to be $\left\langle {{L^i}} \right\rangle = \frac{{\gamma + {\gamma ^{ - 1}}}}{2}l$, where γ is the shape ellipticity of STOV and l is the topological charge. The topological charge of the STOV is inverted in the reflected pulse for transverse OAM perpendicular to the incident plane and remains unchanged for the other situation. For the refracted pulse, a 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, leading to a distortion to the shape (ellipticity) of transmitted STOV.

Figure 38. Schematics of the reflection and refraction of STOV at the planar interface. The incident, reflected, and transmitted pulses, together with their accompanying coordinate frames and intrinsic OAM, are shown. The longitudinal and angular shifts are shown for the reflected and transmitted pulses in (A), whereas the transverse and angular shifts are shown for the transmitted and reflected pulses in (B). Reprinted from [171] under a Creative Commons license.

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Paraxial polarized STOVs will experience all shifts known for the Gaussian beams such as Goos–Hänchen shift and spin-Hall effect. On top of these common shifts, STOVs experience additional types of shifts that depend on the topological charge. The first type is related to the conservation of the z component of the total angular momentum. The orbital-Hall effect requires the transverse y-shift 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 spin-Hall shifts with an additional factor of (1 +|l|). The typical scale is independent of the ellipticity and is of the order of the inverse Rayleigh range. Thus, tighter focusing of STOVs would lead to larger shifts of this type. The third type of shift is the longitudinal shift of STOVs reflected/refracted by a planar interface.

Clearly, STWPs such as STOVs render much richer phenomena at interfaces compared with their spatial counterparts. 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 independent of the frequency, which in turn enable subluminal and superluminal pulse propagation without the need to rely on medium dispersion.

6.4 Spin–Orbital Interaction and Coupling

Spin–orbital interaction (SOI) and SOC have attracted significant interests for spatially structured light [53]. These coupling and interaction phenomena have both fundamental and application significance. Owing to their unique propagation and focusing properties described in Sections 6.1 and 6.2, STWPs exhibit SOI and SOC effects that are distinctively different from their spatial counterparts [131]. In particular, for STOVs and other related spatiotemporal vortices that carry transverse intrinsic OAM and OAM with arbitrarily controllable orientations, these new degrees of freedom will inevitably render novel SOI and SOC phenomena.

For paraxial propagation, it has been shown that a Bessel-type STOV can be constructed using a superposition of plane waves with wave vectors distributed over a circle within the k_x–k_z plane. These wave vectors have a spiral phase difference proportional to the topological charge l. Two polarization situations, i.e., out-of-plane and in-plane cases exist for Bessel-type STOVs. The out-of-plane polarization case is equivalent to the scalar case, while the in-plane 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. The interference of plane waves with different phase, spatial frequency, and linear polarizations results in nonzero spin density indicating the presence of transverse spin. Despite such local spin–orbit interaction, the integral SAM vanishes and the integral OAM is equal to lℏ per photon as long as the intensity distribution is circularly symmetric. However, if the intensity profiles deviate from circular symmetry, the OAM per photon is larger than lℏ.

Compared with the relatively weak SOI and SOC during propagation and reflection/refraction at interfaces, tight focusing of spatiotemporally structured wave packets produces much stronger coupling between the transverse OAM and SAM in a way similar to the highly focused spatially structured light. Using the Debye focusing method, numerical modeling has been performed to study the conversion from longitudinal SAM of the incident light to transverse OAM in the focused light occurs during the focusing process. With the focusing model for STWP developed in Section 6.2, SOI and SOC for highly focused circularly polarized STOVs (Fig. 39) and cylindrically polarized STOVs (Fig. 40) have been performed and very complicated SOI and SOC phenomena have been observed [131,169]. It should be noted that these models are still rather over-simplified. With rigorous model, even more sophisticated SOI and SOC effects are expected. Connected and knotted topological structures may be possible through these SOI and SOC processes. These spin–orbit couplings and interactions involving transverse OAM are expected to give rise to new phenomena in light–matter interactions leading to novel applications.

Figure 39. Strong focusing of circularly polarized STOV and spin orbital coupling. (a)–(e),(g) Spatiotemporal structures of the intensity and phase for x, y, and z components of the field. (f) Extracted vortex core to illustrate the evolution of the phase singularities within the wave packet for the z component. Reprinted with permission from Chen et al., ACS Photonics 9, 793–799 (2022) [132]. Copyright 2022 American Chemical Society, https://doi.org/10.1021/acsphotonics.1c01190

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Figure 40. Creation of optical toroidal wave packet through tight focusing of radially polarized 2D STOV. (a)–(d) The spatiotemporal intensity and phase structures for the x and y components of a radially polarized 2D STOV. (e) The spatiotemporal structures for the total intensity of a radially polarized 2D STOV. (f) The polarization distribution of the radially polarized 2D STOV. (g) The spatiotemporal intensity distribution at the focus for highly focused radially polarized 2D STOV showing toroidal distribution. (h),(i),(j) The intensity distribution combined with polarization map in various x–y planes sliced through the wave packet at focus. Reprinted with permission from [169]. © Optica Publishing Group.

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6.5 STWPs in Optical Fiber

Optical communication relies on the transmission of optical pulses to convey digital information. The information can be encoded in different physical quantities, such as intensity, wavelength, phase and polarization, spatial modes, angular momentum, and so on. To expand the communication capacity of a fixed physical channel, different physical quantities can be multiplexed and transmitted through the same channel simultaneously. Although optical communications can be conducted via free-space or optical fiber, major interests remain in fiber-optic solutions. In the past decade, significant efforts have been invested in the utilization of vortex beams carrying longitudinal OAM. The transverse OAM offered by STOVs represent an additional degree of freedom that could be exploited to further expand the optical communication capacity. However, transmission of STWPs such as STOVs through optical fiber or waveguide is rather challenging, due to the spatiotemporal coupling within the STWP and the modal/spectral dispersion from the fiber. Very recently, the first experimental demonstration of transmission of STOVs through a commercial few-mode optical fiber have been reported (Fig. 41) [172]. This also opens the possibility of generating STOVs and other more complicated STWPs in a laser cavity, rather than externally. Nevertheless, much remains to be done in order to test the use of transverse OAM as an information carrier.

Figure 41. Experimental demonstration of transmitting STOVs through few mode optical fiber. The top panel shows the wave packets with topological charge ±1 STOV at the output of the fiber. Reprinted from [172] under a Creative Commons license.

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6.6 STWPs as Information Carriers

Optical communication relies on the coding, transmission, routing, and decoding of optical pulses to convey digital information. To further expand optical communication bandwidth, multi-dimensional information such as intensity, wavelength, phase, spatial modes, polarization, and angular momentum can be incorporated into the coding and decoding schemes simultaneously. In the past couple of decades, increasing interest has been paid to the utilization of vortex beams carrying longitudinal OAM [173]. Very recently, a transmission rate of 1,000 TB/s has been reported using OAM multiplexing with a 7-core optical fiber link [42].

Transverse OAM carried by STWPs adds an additional degree of freedom to OAM-based optical communication. Furthermore, toroidal vortices are closely related to particle-like waves such as hopfions, which are regarded as high-dimensional particle-like waves that contain the information of stereographic projection from three-sphere (S³) to two-sphere (S²) (see Fig. 33) [27]. Using hopfions and other related topological structures of light pulses as information carriers may increase the information dimension per pulse for optical communication as long as appropriate coding/decoding schemes can be engineered [174,175].

6.7 Information Processing with STWPs

Photonic structures with spatial dispersion have been designed to perform analog computing. For example, spatial differentiation can be performed with these devices for edge enhancement in imaging applications. Similar operation can be realized in the spatiotemporal domain. As discussed in Section 5.7, 1D periodic structure with broken mirror symmetry has been designed to generate STOVs. This same structure can also be utilized as a spatiotemporal differentiator, as demonstrated by Ruan et al. [176]. The resolution of detecting sharp changes of 18 µm in the spatial coordinate and 182 fs in the temporal coordinate have been reported (Fig. 42).

Figure 42. Illustration of spatiotemporal differentiator. (a) With mirror symmetry, for an incident pulse with both Gaussian envelopes in both spatial and temporal domains, the phase distribution of the transmitted pulse has the same symmetry about the mirror plane. (b) Breaking mirror symmetry of a spatiotemporal differentiator is necessary for generating the phase singularity, where the transmitted pulse corresponds to a STOV carrying transverse OAM. The top panel shows the spatiotemporal amplitude and phase of a “ZJU” logo pattern and the corresponding amplitude and phase after passing through the spatiotemporal differentiator. Huang et al., Laser Photonics Rev. 16, 2100357 (2022) [176]. Copyright Wiley-VCH Verlag GmbH & Co. KGaA. Reproduced with permission.

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6.8 Polarization Spectroscopy and Quantum Control

Spatiotemporal vectorial pulse shaping can produce ultrafast optical pulses that enable unprecedented coherent control for light–matter interactions. Quantum systems are 3D in nature. Thus, the interactions are inevitability vectorial. Preliminary polarization pulse shaping techniques have already been utilized to study nuclear motion, molecular rotation, momentum of magnetization, etc. [139,177]. With the vectorial spatiotemporal sculpturing technique described above, much more sophisticated spatiotemporal vectorial structures such as spatiotemporal spin grating, spatiotemporal spin lattice, and spatiotemporally twisting polarization can be generated [22]. This greatly expands the scope of spatiotemporal structuring of the wave packet and will find broad applications that require vectorial coherent control of light–matter interactions ranging from multiphoton absorption and photoionization [178], coherent Raman scattering [179], optical control of lattice vibrations [180], selective production of enantiomers [181], magnetization [142], nonlinear optical wavelength conversion [182], to vectorized optoelectronic control within semiconductors (Fig. 43) [183].

Figure 43. Vectorized coherent optoelectronic control in a semiconductor. (a) Schematic of the experimental configuration used to generate vector beams at ω and 2ω and to synthesize their waveform. (b) Typical intensity profile of an azimuthally polarized beam. (c) Detected y component of current as a function of Δφ_ω_,2ω measured at points P₁ and P₂ in the beam cross section (shown in (b)). The top panel shows the electronic band structure of GaAs and various current distributions excited by different polarization patterns. Reproduced with permission from Springer Nature: Sederberg et al., Nat. Photonics 14, 680–685, Copyright 2020.

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6.9 Quantum Optics

Semi-classical theories break down when it comes to studying structured single-photon pulses [184,185]. The description of single photons with Stokes parameters and the Poincaré sphere ignores the vectorial nature of spatiotemporally localized spin and OAM densities. Recently, a framework based on quantum field theory has been established to engineer the local angular momentum densities of quantum structured light pulse, especially for STOVs [186]. A 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 long-range 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. It is demonstrated that the spin texture of a single-photon pulse can exhibit a very rich and interesting structure in the vectorial case (Fig. 44) [186]. Using free-electron interactions, quantum photonic states with unique quantum properties can be shaped through judicious choice of the input light and electron states [187]. The additional degrees of freedom brought in by STOV and the associated new coupling mechanisms open new avenues in the quantum optics aspects.

Figure 44. Helical spin texture of a twisted quantum pulse where the spin texture of a single-photon left-handed circularly polarized Bessel pulse on the pulse center plane is shown. Reprinted from [186] under a Creative Commons license.

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6.10 Optical Manipulation

The conservation of linear and angular momenta is a fundamental law that governs light–matter interaction. Transfer of momentum from light to particles exerts force and torques onto the particles, enabling optical microtools for particle manipulation. These phenomena have been exploited extensively in optical tweezers, trapping, spanners, sweepers, and so on. Since the discovery of the connection of spatial spiral phase with longitudinal OAM, vortex beams have been exploited extensively in optical confinement and manipulation. The interaction between small particles and light carrying OAM generates light-induced torque and leads to rotation of particles. In most previous studies, the transfer of angular momentum is limited to the direction parallel to the beam propagation. The mechanical motion induced for nanoparticles thus is limited to rotations around its own axis or the singularity line of a spatial vortex.

Recently, observation and measurement of transverse SAM and OAM transfer to particles of several micrometers in size are reported (Fig. 45) [188]. Transverse angular momentum transfer is especially advantageous in creating shear stress for studying mechanotransduction within cellular structures. Thus, STOVs carrying transverse OAM may be utilized to rotate particles about an axis that is perpendicular to the propagation direction of the wave packet, offering a versatile tool for transferring transverse OAM to small particles. In addition, the enriched SOC and conversion discussed above and various novel wave packets including self-torque pulses and light springs will undoubtedly further expand our optical micromanipulation toolbox.

Figure 45. Demonstration of controlled transverse OAM transfer to optically trapped birefringent microparticles. (a) Configuration of inducing transverse rotation of a trapped particle. (b)–(d) Diagram of momentum of two crossed focused beams. Experimental verification is shown in the right panel. Reproduced with permission from Springer Nature: Stilgoe et al., Nat. Photonics 16, 346–351, Copyright 2022.

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6.11 Particle Acceleration and Novel Radiation Sources

Spatially structured light has been studied in particle acceleration in the development of new table-top radiation sources for extreme short wavelengths. The spatial phase and polarization distributions have been shown to be beneficial for the improvement of radiation quality such as divergence angle and brightness. It is natural to extend this to the STWPs, although this is still in its infancy. For electron acceleration and new radiation sources development, in a recent study particle-in-cell (PIC) numerical modeling is employed to study the use of relativistic STOV to produce isolated attosecond (∼600 as) electron sheets and showed encouraging results, although experimental confirmation is still needed [189].

7. Future Perspectives and Speculations

Despite the rapid developments made in spatiotemporal sculpturing of light in the past several years, this is still an emerging research area and there are much more to be explored. In this section, we discuss a few potential future directions of this field, some of which are what we would like to see happen, some are drawn in parallel from the past development in spatially structured light, and some may be mere speculations.

7.1 STWP On-Demand

Until recently, the parameter of light fields under spatiotemporal modulation is typically limited to either amplitude or phase. Now polarization has been added to the list of spatiotemporal modulations to form spatiotemporally vectorial wave packets. With each of the parameters added to the modulation list, the dimensions of the mathematical description increase. Ultimately, how many parameters or dimensions can we modulate simultaneously? With more dimensions involved, the math (topology), physics, and information description of these STWPs will be the subjects of investigation.

Spatiotemporally sculpturing light fields requires modulation of the light field parameters spatially and temporally. Several fundamental questions to be answered are as follows. What is the smallest spatial dimension one can modulate? Is wavelength the limit? In parallel, what is the shortest time frame one can modulate? Is single cycle the limit? Combined together, what is the smallest spatiotemporal volume one can modulate? What are the light field parameters that can be modulated simultaneously within such volume?

Once these questions are addressed, it would be interesting to find out what new and how complicated spatiotemporal optical topologies can be formed, what the math would be needed to describe them, how robust the topology is under propagation (linear and nonlinear, dispersive and nondispersive) and focusing, and whether these topologies can be made invariant (soliton or soliton-like).

7.2 High-Speed Modulation for STWP Synthesis

Metasurface and MPLC-based STWP synthesis schemes in general lack the capability of tuning. On the other hand, the LC-SLM used in the STWP synthesis offers flexibility of tuning with low (∼100 Hz) refresh rate. In applications that require dynamically adjusting the STWP, high-speed reconfigurable modulators would be desirable. DMD-based devices can operate at higher speed. However, the functionality is limited to amplitude and/or phase only. Recently, developments in modulators based on phase change materials deserve special attention [141]. Spatial light modulators built with this type of material potentially could lead to gigahertz or even higher-speed modulation that can be incorporated into the spatiotemporal pulse shaper and SLM for spatial modulation, offering a much faster refresh rate for STWP generation.

7.3 Single-Shot Real-Time Characterization

Currently one of the biggest challenges in spatiotemporal photonics is in characterization. In many senses, the characterization capability is lagging significantly behind the generation method. With the current generation toolkits, many exotic STWPs could be generated or actually have been generated. However, full characterization of the amplitude, phase, and polarization in the spatiotemporal domain is sometimes impossible with the existing techniques. The interference technique is one of the most capable and versatile. However, when polarization, especially the longitudinal component, or single/few cycle, or structures that lack symmetry are involved, things become much more challenging. A direct and full characterization toolkit is needed in order to further advance this field.

Perhaps an even more challenging task is to perform single-shot real-time characterization of STWPs. This is particularly relevant when high-power STWPs are desired. Typically for high-power systems, the repetition rate is much lower (∼1 Hz), making the use of the fringe averaging technique in the interference method impractical. Such a single-shot full parameter measurement capability represents the tantamount challenge for STWP characterization.

7.4 Closed-Loop Adaptive Control

If high-speed modulation and single-shot real-time characterization are available, it is then possible to establish a closed-loop adaptive control system for STWP synthesis. Such a closed-loop adaptive wave packet sculpturing system could be developed to deliver optimized STWPs at the target that can produce the desired light–matter interaction results (as illustrated in Fig. 46). Ultimately, this could be a dream system that would allow us to harness the full power of light for light–matter interactions.

Figure 46. Illustration of a closed-loop adaptive system that is capable of characterizing the generated STWP, assessing the involved light–matter interaction, and subsequently calculating and updating the spatiotemporal modulation unit to produce the optimally desired output.

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7.5 Incorporation of Artificial Intelligence

In the past decade, great advances have been made in artificial intelligence (AI) photonics with regards to two aspects: AI for photonics and photonics for AI. Since we can treat the spatiotemporal sculpturing of light as a linear system, the transmission matrix concept developed for imaging through scattering media could be borrowed and applied. Advanced computing power as well as AI algorithms could be used to facilitate the design of optimal STWPs. With the availability of closed-loop adaptive control, AI photonics and algorithms can be extended to make the system converge faster in order to tailor light–matter interactions for the realization of given functionalities.

7.6 Scale Up to High Power

The other limit that needs us to push is the field strength of these wave packets. Currently most STWPs are generated in the weak light regime. Is it possible to increase the field strength of such complex wave packets into the relativistic regime (i.e., >10¹⁸ W/cm²) in a controllable manner with high efficiency and high quality? One possible approach is to generate such wave packets first in the wake field regime and then boost up the peak intensity with cascaded amplifiers. The challenge is that spatiotemporal structure of the wave packet inevitably will deteriorate due to the aberration and dispersion of the bulk media the pulse will experience through propagation. These negative effects could be pre-compensated at the generation stage in order to preserve the desired spatiotemporal structure at the target plane. These could also be mitigated with proper topological design of the STWP, which deserves special attention. Another approach is to perform the spatiotemporal structuring after the amplification. This could be done by constructing the pulse shaper with a large-aperture grating, reflective cylindrical mirror, and programable reflective-type devices such as a MEMS mirror array as the spatial light modulator.

7.7 Transmitting Through Complex Media

As mentioned above, the linear system approach we present in this tutorial allows one to connect the spatiotemporal problem with spatial modes propagation through complex/scattering media. Those media that are particularly relevant to STWPs are few mode and multimode fibers, turbulence, underwater environments, scattering media such as tissues and skin, and even plasma. Transmission through complex media such as turbulent atmosphere and multimode fiber with structured light has seen great advancements in the past decade (Fig. 47) [190–196]. It is conceivable that through matching the capabilities offered by STWP sculpturing techniques with material/medium spatial and temporal dispersion properties, many interesting and likely useful phenomena are awaiting to be discovered.

Figure 47. Demonstration of the invariance hidden in the vectorial structure through various complex media. (a) Propagation through turbulent air. (b) Focusing. (c) Transmission through multimode fiber. (d) Propagation through an underwater environment. Reproduced with permission from Springer Nature: Nape et al., Nat. Photonics 16, 538–546, Copyright 2022.

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7.8 Integrated Spatiotemporal Photonic Devices

Another emerging subfield that deserves special attention is integrated spatiotemporal photonic devices. The motivation for us to pursue the limit of spatiotemporal light modulation, of course, is not just for curiosity. Such complex wave packets may find direct applications in information carrying, transmission, and processing capabilities. It would be desirable to develop proper information encoding and decoding schemes using these topological modes. As we have already seen that spatial OAM modes have been exploited in spatial mode multiplexing for optical communications, it is possible that much higher transmission capacity can be realized with wave packets of much higher dimensionality. However, in order to make these applications practical, it is necessary to developed integrated spatiotemporal photonic devices with miniaturized sizes. Platforms such as photonic crystals, micro-ring resonators, micro-combs, and femtosecond light sources integrated on chip have been developed and are candidate components. With further developments and particularly in the area of integration of various components onto the same platform, there is a good chance that these devices may find practical applications in various areas, particularly in information storage, transmission, processing, and computation, as well as optical sensing.

7.9 Interaction with Matter

Even more interesting applications would come from light–matter interactions. Different topologies of light may render distinctively different interaction mechanisms. This has already been demonstrated for spatially structured light in the past. For example, structured light can be used to access different modes within atoms, molecules as well as nanostructures, to enhance certain transitions that are normally not accessible, and to enhance weak interactions such as chirality, Raman, chiral Raman, and so on [197–199]. With the availability of the ultimate degree of freedom of spatiotemporal sculpturing of light, it can be expected that much richer phenomena will be observed. The spatiotemporal topology may also provide means to excite or transfer these topologies into other fields or matter [200]. For example, optical skyrmions may be imprinted into magnetic domains, electron gas, or soft matter systems. Matching the light fields to shaped electron energy transition of target material excitations may offer access to states that are forbidden with optical excitations [201–203]. The new topology offered by recent STWPs will pave the way to excite a whole new zoology of topological structures such as torus, toroidal links, knots, and hopfions in various material systems.

7.10 Space–Time Entanglement

Research in spatiotemporally coupled optical fields may offer a new resource: space–time entanglement. Quantum entanglement represents one of the most striking outcomes of quantum mechanics and serves as the most fundamental quantum mechanical resource for many quantum computing and quantum information applications. Entanglement used to be discussed strictly in the quantum context. Recently, there has been much interest in constructing entanglement states using classical optical fields in the hope of retaining some of the features that are analogs to quantum entanglement while avoiding the drawbacks and pushing the quantum–classical boundary [204–208]. Entanglement prepared with classical electromagnetic fields is consequently referred to as classical entanglement. Despite the debates on the nomenclature regarding whether we should associate entanglement with classical fields, the consensus is that mathematically, they share the same “nonseparable” characteristic [209–211]. This common nonseparable feature offers the possibilities of new type of resources for both classical and quantum entanglement. Frameworks developed in quantum computing, quantum information, etc., can still be applied to classically entangled fields and find applications in, e.g., imaging, computing, information processing, and metrology, despite the lack of a quantum nonlocality feature [212–214]. This mathematical nonseparable feature also exists in spatiotemporally coupled optical fields, hence space–time entanglement could be further developed as new resources for novel high-dimensional entanglements [215]. The main challenge probably is in the detection, which on the other hand may offer another level of control and security.

8. Summary and Prospects

In the past several decades, our capabilities in modulating and structuring light fields in either spatial or temporal domains have seen unprecedented levels of development. In both the spatial domain and the temporal domain, we have seen a trend of moving “from marching band to orchestra.” In some sense, the evolution of technology for sculpturing light fields is analogous to how music evolved from the simple beats of a marching band to a contemporary complex symphony by spatially distributed instruments recorded by a complex full score. At their early stages, techniques were developed to mainly “synchronize” the phase of different spatial or spectral content of the lights fields to obtain transformation limited focusing in space or compressing in time. Then the goals of light field structuring become more diverse and complex in either the spatial or temporal domain. This trend toward complexity can be generally associated to collectively match to or resonant with certain structures or material systems. For example, complicated wavefronts can be designed to assist light to transmit through a scattering medium more efficiently [190,216,217]; the vectorial modes are used to assess the molecule orientation or excite typically hidden modes of materials or nanostructures [218,219]; shaped waveforms are created to excite certain transitions or initiate reactions more effectively [220].

However, structuring light in both the spatial and temporal domains simultaneously and jointly became feasible only very recently. In the early days, spatiotemporal coupling was regarded as a nuisance that should be avoided or compensated. Studies in X-waves, O-waves, etc., did not gain traction due to the lack of viable experimental tools. Recent advances in optics enable us to spatiotemporally sculpture light pulses and correspondingly characterize them with novel spectroscopic technology in the spatiofrequency domain. The combination of spatial and temporal structuring capabilities led to the renaissance of research interests in STWP, opening a whole new chapter of modern optics.

Specifically, the field of STWP has seen a dramatic increase in interest with the generation and observation of STOVs and other associated singularities. This is similar to the history of spatial structuring of light, which was largely driven by interests in phase singularities and polarization singularities embedded in the spatial structure. These novel STOVs have topological and conservative properties similar to spatial vortices as well as novel properties and distinctively different spatiotemporal evolution dynamics. STOVs and their variants, such as toroidal vortex, hopfions, and broader context in knotted optical fields have exhibited peculiar photonic properties in various optical phenomena and become applicable in optical manipulation, spatiotemporal differentiators, subluminal and superluminal pulse propagation, and free-space optical communication. Nevertheless, research remains to be done to fully understand the physics characteristics of these spatiotemporal singularities and explore their applications. It is always hard to predict the future of such a nascent research area that is rapidly developing. However, if the evolution of studies in spatial singularities can serve as a historical precursor, it is well expected that increasing physical mechanisms associated with spatiotemporal vortices are to be discovered and a wider range of applications are to be found in the near future.

More generally, the capability to produce a light field with desired spatiotemporal structures on demand will give us the ultimate harnessing power over light. It may unlock unprecedented information transmission, processing, and computing capacities, from classical to quantum, enshrined in light. The rapid advances made in nanomaterials and nanostructures offer strong synergy with the high-dimensional degree of freedom provided by these spatiotemporally sculptured fields. Advanced computing power as well as AI algorithms will further facilitate of the tailoring of light–matter interactions. Practical applications of these complex fields in intelligent manufacturing and high-capacity data communication as well as transmission through complex turbulent and dispersive media may become increasingly feasible. With these developments and the availability of relativistic high-power wave packets with optimal spatiotemporal structures through a closed-loop control in real time, we may start to contemplate novel and compact laser accelerators, new radiation sources, and laser-driven fusion reactions in ways that were unthinkable previously. In short, successes in this research area may have long-lasting impacts and offer numerous new possibilities in many fields, further pushing the endless frontier of light science and applications.

Funding

National Natural Science Foundation of China (92050202).

Acknowledgments

I would like extend my sincere thanks to my current and former group members and many collaborators and colleagues for collaborations and enlightening discussions over the past 20 years or so. I also would like to acknowledge financial support from the National Natural Science Foundation of China (NSFC 92050202), Zhangjiang Laboratory, Shanghai Municipal Science and Technology Commission, Shanghai Municipal Education Commission, and the Visiting Faculty Program from the King Abdullah University of Science and Technology.

Disclosures

The author declare no conflicts of interest.

Data availability

No data were generated or analyzed in this tutorial.

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Dr. Qiwen Zhan is Distinguished Chair Professor in Nanophotonics at the University of Shanghai for Science and Technology (USST) and founder of the Center for Complex Optical FieldS and Meta-Optics Structures (COSMOS). He also holds joint appointment with Zhangjiang Laboratory, China, as well as affiliate membership with the International Institute for Sustainability with Knotted Chiral Meta Matter (WPI-SKCM2) at the Hiroshima University, Japan. He received a B.S. in Physics (optoelectronic) from the University of Science and Technology of China (USTC) in 1996 and a Ph.D. in Electrical Engineering from the University of Minnesota in 2002. From 2002 to 2020, he held a tenured faculty position (Assistant Professor in 2002, Associate Professor in 2008, and Full Professor in 2012) in the Department of Electro-Optics and Photonics at the University of Dayton, USA. He has published 1 book, 9 book chapters, and more than 300 journal and conference publications, delivered many conference presentations and invited talks/lectures/seminars, and possessed more than 20 patents (US and China). He is an Associate Editor for Science Bulletin, Associate Editor-in-Chief for PhotoniX, Editorial Board Member for Scientific Reports and Chinese Optics Letters, Senior Member of the IEEE, elected Fellow of Optica (formerly OSA), and Fellow of the SPIE.

Abstract

1. Introduction and Background

2. Maxwell’s Equations and Solutions

3. The Basics of Light Modulation

3.1 Spatial Modulation of Light

3.1a Liquid Crystal Spatial Light Modulator

3.1b Multi-Plane Light Conversion

3.1c Metasurfaces

3.2 Temporal Waveform Shaping with a 1D Pulse Shaper

3.3 Spatiotemporal Modulation Using a 2D Pulse Shaper

3.4 Spatiotemporal Modulation Enhanced with Additional Spatial Modulation

4. Basics of the Characterization of Light Fields

4.1 Spatial Characterization

4.2 Temporal Characterization: Pulse Waveform Measurement

4.3 Spatiotemporal Characterization

4.3a Self-Referenced Interferometry

4.3b Single-Shot STWP Characterization

5. Latest Developments in Spatiotemporal Sculpturing of Light

5.1 Propagation-Invariant STWPs

5.2 Spatiotemporal Optical Vortices

5.3 Combination of Spatiotemporal and Spatial Modulation

5.4 Linear Mapping with Chirped Pulses

5.5 Vectorial and Amplitude Spatiotemporal Modulation

5.6 Spatiotemporal Modulation with Metasurface Devices

5.7 Integrated Devices for STWP Synthesis

5.8 Self-Torque Pulse and Light Spring

5.9 Toroidal Vortices of Light

5.10 Toroidal Pulse

5.11 Optical Hopfions and Topological Perspectives

5.12 STOV with Partially Coherent Light

6. Manipulation of STWPs and Applications

6.1 Propagation of STWPs

6.2 Focusing of STWPs

6.3 Reflection and Refraction of STWP

6.4 Spin–Orbital Interaction and Coupling

6.5 STWPs in Optical Fiber

6.6 STWPs as Information Carriers

6.7 Information Processing with STWPs

6.8 Polarization Spectroscopy and Quantum Control

6.9 Quantum Optics

6.10 Optical Manipulation

6.11 Particle Acceleration and Novel Radiation Sources

7. Future Perspectives and Speculations

7.1 STWP On-Demand

7.2 High-Speed Modulation for STWP Synthesis

7.3 Single-Shot Real-Time Characterization

7.4 Closed-Loop Adaptive Control

7.5 Incorporation of Artificial Intelligence

7.6 Scale Up to High Power

7.7 Transmitting Through Complex Media

7.8 Integrated Spatiotemporal Photonic Devices

7.9 Interaction with Matter

7.10 Space–Time Entanglement

8. Summary and Prospects

Funding

Acknowledgments

Disclosures

Data availability

References

Data availability

Cited By

Figures (47)

Equations (19)

Advances in Optics and Photonics