In recent decades, the field of ultrafast optics has significantly matured, leading to promising applications that rely on the design and generation of structured laser pulses. This progress has brought to light several underlying theoretical and practical challenges that must be addressed. A current challenge is the design of ultrashort structured laser pulses with precise control over their spatiotemporal properties, tailored to specific applications. Significant advancements on spatiotemporal control of laser pulses are documented in recent reviews and tutorials underlying its potential applications; see, for example, Refs. [1–3]. Nevertheless, achieving precise control over the spatiotemporal properties of pulses presents difficulties that have yet to be overcome. These difficulties arise primarily from the polychromatic bandwidth of ultrashort pulses, which leads to the emergence of spatiotemporal couplings (STCs). In ultrafast optics, a beam is said to exhibit STC when its temporal and spatial properties are interdependent, meaning the electric field cannot be described as a simple product of separate spatial and temporal components [1,2,4,5]. The polychromatic characteristics of ultrashort laser pulses endow them with unique properties, such as extremely short durations and exceptionally high peak powers. In practice, the inherent STCs of an ultrashort laser pulse have to be governed according to the specific application in order to harness its potential. Recent advancements show that controlled STCs could unlock a new range of experimental capabilities [4–15], which are of high interest in light-matter applications such as laser wakefield electron acceleration [16,17] and control of plasma wakefields [6,18].

A notable example of laser pulses with controlled STCs is the spatiotemporal light spring (LS) [6,18], named for its helical intensity and phase profiles. These pulses are obtained by superimposing Laguerre-Gauss (LG) beams whose azimuthal mode index is correlated to their frequency [6]. As is well known, its helical wavefront is responsible for the orbital angular momentum (OAM) carried by such LG beams [19]. While a single helical LG beam with azimuthal index $l$ (topological charge) can transfer its OAM ($l\hslash$ per photon) to micro- and nano-sized particles [20], it cannot transfer OAM to a plasma on a laser wakefield acceleration scenario due to the nature of stimulated Raman scattering [18], where absorbed and emitted photons carry the same OAM, resulting in no net transfer. In the case of stimulated Raman scattering, the OAM transfer to a plasma wakefield becomes possible when the OAM per photon in the laser beam is frequency-dependent. The spatiotemporal LS is designed as a collinear superposition of helical LG beams, each with a frequency-dependent topological charge $l(\omega )$, creating a spatio-spectrally coupled helical beam capable of transferring OAM to control plasma wakefields optically [18]. This is a clear example that illustrates the need in the state of the art to design structured pulses for the application under consideration. However, there exists another mechanism in which plasma electrons gain angular velocity by using LG beams with OAM thanks to the dephasing process induced by the combined action of the ponderomotive force and the laser-induced radial oscillation [21]. Despite the significant interest in creating spatiotemporal LS, current pulse shaping techniques are limited to narrow topological-spectral bands. This limitation arises from the technical challenges associated with imparting different OAM values to the multiple spectral LG-mode components of the pulse. This technical difficulty in the broadband control of topological-spectral correlations has recently been addressed by using helical Bessel-Gauss beams [22] instead of the LG ones, allowing for the experimental generation of a type of spatiotemporal light coil with circular geometry carrying OAM.

In parallel, the use of ultrashort vortex laser pulses carrying angular momentum has been the subject of intense research activity in the last two decades. Recent advancements in the synthesis of pulses with different types of angular momentum are particularly notable [1,2]. These innovations include the generation of pulses with toroidal topology [23,24], the creation of scalar optical hopfions [24], and the generation of ultrashort laser pulses with controllable spatiotemporal optical vortices [11,25–35]. Note that a spatiotemporal optical vortex is characterized by its transverse OAM, which refers to the component of angular momentum of a light beam that is perpendicular to the direction of propagation. This should not be confused with longitudinal OAM, which is the component of angular momentum aligned with the direction of propagation, as carried by the previously mentioned helical Bessel-Gauss and LG beams used to create spatiotemporal light coils [6,18,22]. Other sophisticated methods for synthesizing ultrafast wave packets with precisely tailored spatiotemporal properties have also been proposed in the last years [7,36].

The concept of curve-shaped laser (CSL) pulses has been recently introduced, whose intensity and phase distributions are independently tailored along any curve [12]. CSL pulses can be easily generated using a Fourier space-time (FST) pulse shaper that includes a conventional programmable spatial light modulator (SLM), as experimentally demonstrated in Ref. [12]. One of the degrees of freedom provided by the CSL pulse that facilitates controlling the pulse’s spatiotemporal dynamics is the ability to design its phase gradient on demand without altering the beam size and shape. This capability is not possible with other types of laser pulses, such as Bessel-Gauss and LG vortices or any of their superpositions. The inherent independent control of the intensity and phase of the CSL pulses allows for tuning their OAM, regardless of the curve geometry [12]. Moreover, the CSL pulse forms a diffraction-limited light curve, with its width constrained by the size of the point spread function of the optical focusing system [12]. Therefore, in an ultrashort CSL pulse, the pulse intensity can be precisely and strongly focused along any curve, with independent control over the spatial phase along it, which is advantageous for light-matter applications. Let us underline that these advantages are not achievable with conventional pulse shaping methods including the ones that rely on superposing helical LG [6,18] or Bessel-Gauss beams [22].

In this article, we present an alternative strategy for designing the spatiotemporal characteristics of structured ultrashort laser pulses. This is accomplished by precisely superimposing two scalar ultrashort laser (CSL) pulses, arranged in a suitable spatio-spectral coupling configuration. Specifically, the structured ultrashort laser pulse is obtained by interfering two pulse wavefields with different frequencies and spatial phases, which can be easily adjusted. This setup allows independent manipulation of the intensity and phase of the CSL pulses to drive the STCs in the resulting structured ultrashort laser pulse. Our approach leverages the design of both linear and non-linear phase gradients to advantageously exploit STCs. In this way, we have successfully generated novel types of flying structured ultrashort laser pulses, in the form of springs and multi-rings with different geometries, whose spatiotemporal behavior can directly be controlled. These pulses are referred to as flying structured ultrashort laser pulses because they propagate along the optical axis of the FST pulse shaper at the speed of light while preserving the three-dimensional shape of their intensity distribution. This causes the intensity peak of the resulting pulse to rotate at an extremely high frequency at the focal plane as the flying structured pulse crosses it. Moreover, these ultrashort laser pulses inherit the advantages of CSL pulses, including precise and strong light focusing along the trajectory traced by their intensity peak. Our theoretical framework unveils the physical mechanisms underlying this enhanced control over the spatiotemporal behavior of the pulse intensity peaks and wavefronts, demonstrating how it can be easily driven by the spatial phase design of the interfering CSL pulses. The presented results underscore the importance of designing the spatial phase, a capability that has not been fully exploited for controlling STCs until now due to the lack of efficient and versatile techniques for structuring the spatial phase on ultrashort laser pulses. The direct engineering of the spatiotemporal characteristics of flying spring and multi-ring pulses is demonstrated through numerical and analytical results obtained from the developed theoretical model, which incorporates the characteristics of the experimental setup (FST pulse shaper) required for their generation.

A conceptual sketch of the experimental setup required for the generation of the proposed flying structured laser pulses is displayed in Fig. 1(a). An input collimated ultrashort laser pulse illuminates at normal incidence a programmable SLM, which holographically encodes [37] the complex field amplitude ${\tilde{E}}_0({\mathbf{r}}_0)$ required to shape the proposed flying structured pulse, with ${\mathbf{r}}_0$ being the position vector at the SLM’s display plane. Specifically, the modulated input pulse is focused by a convergent lens and the structured laser pulse is obtained by an optical Fourier transformation as indicated in Fig. 1(a). Note that this type of optical setup corresponds to the same FST pulse shaper experimentally demonstrated in Ref. [12] for the generation of ultrashort CSL beams. Here, the FST pulse shaper allows the generation of a flying structured pulse, for instance, in the form of a flying helical spring as sketched in Fig. 1(a), that propagates along the optical axis direction at the speed of light $c$.

Let us now introduce the fundamentals of the underlying physical mechanism used for the generation of this type of ultrashort laser pulse. To create a flying structured laser pulse, a coherent superposition (interference) of the spectral wavefield components of the input beam, reshaped similarly to CSL pulses, is required. The FST pulse shaper enables a versatile manipulation of this interference mechanism. Note that all the spectral components of the laser pulse modulated by the SLM have the same complex field amplitude ${\tilde{E}}_0({\mathbf{r}}_0)$ at the SLM’s display plane. The wavefield propagated at the Fourier plane (focal plane of the FST pulse shaper) is given by the optical Fourier transformation for each spectral component of frequency $\omega$. Due to this transformation, a frequency-scaled wavefield for each spectral component is obtained at the focal plane. For example, the FST pulse shaper can be configured to focus the input laser pulse in the form of a ring of radius $R$ (for the central frequency ${\omega }_0$ of the input pulse) at the focal plane [12]. At this plane, a series of concentric ring-shaped wavefield components is obtained, each corresponding to a spectral component of the input pulse with a radius $r(\omega )={\omega }_{0}R/\omega$ given by the frequency scaling factor (${\omega }_0/\omega$). The interference between these ring-shaped spectral wavefield components, with a tunable prescribed spatial phase along them, gives rise to the output ring-shaped laser pulse, which belongs to the family of CSL pulses experimentally demonstrated in Ref. [12]. This frequency-dependent spatial scaling factor of the wavefield’s spectral components does not pose a disadvantage; in fact, here we show how it can be exploited to generate more sophisticated structured laser pulses (with tunable shape and temporal evolution) on the example of flying spring and multi-ring laser pulses. Notably, the geometry of these ring-shaped wavefield components (pulses) extends beyond the circular configuration described in Ref. [22]. Indeed, they can be configured with any polygonal geometry if needed. Therefore, the proposed approach allows for tuning the geometry of the spring and multi-ring pulse, such as circular or polygonal shapes, with the capability to prescribe the wavefield’s phase distribution along it.

To create a flying spring pulse with circular geometry, as shown in Fig. 1(a), the FST pulse shaper is configured to simultaneously focus two ring-shaped laser pulses with similar radii $R_1$ and $R_2$ for the central frequency ${\omega }_0$ (with $R_2>R_1)$, at the focal plane. Each pulse has a different prescribed spatial phase profile, ${\mathrm{\Psi }}_1(\alpha )$ and ${\mathrm{\Psi }}_2(\alpha )$, where $\alpha$ is the polar angle. Thus, the superposition of their corresponding concentric ring-shaped wavefield components (further referred to as spectral-ring sets) results in the interference of two pulse wavefields with different frequencies (${\omega }_1$ and ${\omega }_2$) and spatial phases [${\mathrm{\Psi }}_1(\alpha )$ and ${\mathrm{\Psi }}_2(\alpha )$] for each radial position $r$ at the focal plane. The value of each ring radius $R_{j=1,2}$ is chosen to ensure the spatial overlap between the corresponding two spectral-ring sets, whose spectral components have radii $r_{j=1,2}(\omega )={\omega }_0{R}_j/\omega$. This overlap is necessary for the required interference. To illustrate these concepts, the intensity and phase distributions of each spectral-ring set are displayed in Fig. 1(b) (see spectral-ring sets 1 and 2). Their interference yields the spring pulse whose intensity and phase at the focal plane (at time $t=0\mathrm{fs}$) are also displayed in Fig. 1(b). In this example, the prescribed phase is ${\mathrm{\Psi }}_1(\alpha )=7\alpha$ and ${\mathrm{\Psi }}_2(\alpha )=6\alpha$. Thus, for the time $t=0\mathrm{fs}$ the constructive interference yields a spring pulse whose intensity peak crosses the focal plane at the point $(R_{\mathrm{avg}}=(R_1+R_2)/2,\alpha =0)$, where these interfering spectral-ring sets are in phase [i.e., ${\mathrm{\Psi }}_2(0)-{\mathrm{\Psi }}_1(0)=0$]. This transverse section of the pulse intensity peak trajectory is further referred to as intensity peak point. Due to the radially varying frequencies of the spectral-ring components, their phase values evolve at different rates over time. This results in a time-varying interference that drives the rotation of the intensity peak, positioned at $(R_{\mathrm{avg}},\alpha (t))$, as the flying spring pulse traverses the focal plane. The spectral distribution of each spectral-ring set is sketched in Fig. 1(c) to help the visualization as well. Specifically, in this spatio-spectral coupling configuration the two spectral-ring sets overlap at the radial positions $r_1({\omega }_1)=r_2({\omega }_2)$, which in turns requires ${\omega }_1/{\omega }_2=R_1/R_2$ resulting in a frequency difference $\mathrm{\Delta }\omega ={\omega }_2-{\omega }_1={\omega }_1\mathrm{\Delta }R/R_1$, where $\mathrm{\Delta }R=R_2-R_1$. As demonstrated in the next subsection, the interference of these two spectral-ring sets governs the spatiotemporal dynamics of the pulse. This is achieved by adjusting the phase difference $\mathrm{\Delta }\mathrm{\Psi }(\alpha )={\mathrm{\Psi }}_2(\alpha )-{\mathrm{\Psi }}_1(\alpha )$ and the frequency difference $\mathrm{\Delta }\omega$ through $\mathrm{\Delta }R$ (i.e., by selecting $R_1$ and $R_2$). In fact, both $\mathrm{\Delta }R$ and $\mathrm{\Delta }\mathrm{\Psi }(\alpha )$ act as control knobs, enabling the generation of different types of flying spring pulses and multi-ring pulses with tunable spatiotemporal behavior, as explained below.

The complex field amplitude ${\tilde{E}}_0({\mathbf{r}}_0)$, encoded onto the SLM, is given by $${\tilde{E}}_0({\mathbf{r}}_0)\propto \sum _{j=1}^2{\int }_0^{2\pi }{\int }_0^{\infty }|{\partial }_{\alpha }{\mathbf{c}}_j(\alpha )|F_j(\alpha )\delta [r-R_j(\alpha )]\exp [\mathrm{i}\frac{{\omega }_0}{c\mathrm{f}}{r}_{0}r\cos (\varphi -\alpha )]\mathrm{d}r\mathrm{d}\alpha ,$$(1)where ${\mathbf{r}}_0(r_0,\varphi )$ is the position vector (in polar coordinates) at the display plane of the SLM and $|{\partial }_{\alpha }{\mathbf{c}}_j(\alpha )|=[{({\partial }_{\alpha }{R}_j(\alpha ))}^2+R_j^2(\alpha ){]}^{1/2}$ is the speed of the curve ${\mathbf{c}}_j(\alpha )$. The expression Eq. (1) has been calculated by performing the inverse Fourier transformation of the target wavefield proportional to $\sum _{j=1}^2{F}_j(\alpha )\delta [r-R_j(\alpha )]/r$, at the Fourier plane. Note that at this plane the position vector is $\mathbf{r}=r(\cos \alpha ,\sin \alpha )$, given by polar coordinates $(r,\alpha )$. Specifically, ${\tilde{E}}_0({\mathbf{r}}_0)$ results from the superposition of the two interfering curve-shaped spectral components associated to the curves ${\mathbf{c}}_{j=1,2}(\alpha )=R_j(\alpha )\cdot (\cos \alpha ,\sin \alpha )$, for the central frequency ${\omega }_0$ of the input pulse. Let us underline that the radii $R_1(\alpha )$ and $R_2(\alpha )$ are now given as a function of the polar angle $\alpha$ (see Appendix A), enabling the creation of flying structured pulses of any geometry (e.g., polygonal rings) apart from the circular one (constants $R_1$ and $R_2$). Note that the function $F_j(\alpha )=|F_j(\alpha )|\exp [\mathrm{i}{\mathrm{\Psi }}_j(\alpha )]$ allows for prescribing the intensity and spatial phase along the curve ${\mathbf{c}}_j(\alpha )$. In addition, the factor $|{\partial }_{\alpha }{\mathbf{c}}_j(\alpha )|$ in Eq. (1) provides the appropriate weighting to the prescribed amplitude, $F_j(\alpha )$, to compensate for the arc length differences measured along the curve for the same angular differential $\mathrm{d}\alpha$.

The input collimated laser pulse (with central frequency ${\omega }_0$) is modulated by the encoded signal ${\tilde{E}}_0({\mathbf{r}}_0)$ given by Eq. (1). Therefore, the output structured wavefield (focused by a lens of focal length $f$) at the Fourier plane is given by $$E(\mathbf{r},t)\propto \sum _{j=1}^2{\partial }_t{\int }_0^{\infty }{\mathcal{F}}_j(\mathbf{r},\omega )\tilde{G}(\omega )\exp (-\mathrm{i}\omega t)\mathrm{d}\omega ,$$(2)where the term $\tilde{G}(\omega )$ is the spectral amplitude of the input laser pulse. Let us recall that the conjugate function $G(t)={(2\pi )}^{-1}\int \tilde{G}(\omega )\exp (-\mathrm{i}\omega t)\mathrm{d}\omega$ describes the temporal evolution of the input pulse. The function $${\mathcal{F}}_j(\mathbf{r},\omega )={\int }_0^{2\pi }\mathrm{PSF}(|\mathbf{r}-{\mathbf{c}}_j(\alpha ,\omega )|,\omega )|{\partial }_{\alpha }{\mathbf{c}}_j(\alpha )|F_j(\alpha )\mathrm{d}\alpha$$(3)behaves as a complex-valued spatial-spectral filter acting on the spectral amplitude $\tilde{G}(\omega )$. The vectors ${\mathbf{c}}_j(\alpha ,\omega )=R_j(\alpha )\frac{{\omega }_0}{\omega }(\cos \alpha ,\sin \alpha )$ define the set of concentric curve-shaped spectral components [with frequency-scaled radii $R_j(\alpha ){\omega }_0/\omega$] comprising the output structured wavefield. Thus, for the central frequency ${\omega }_0$ two spectral curves ${\mathbf{c}}_{j=1,2}(\alpha ,{\omega }_0)$ of radii $R_1(\alpha )$ and $R_2(\alpha )$ are obtained at the Fourier plane as expected.

The finite numerical aperture (NA) of the focusing lens is considered in Eq. (3) through the transverse section of the point spread function (PSF) given by $$\mathrm{PSF}(r,\omega )\propto \frac{J_1\left(\frac{\mathrm{NA}}{c}\omega r\right)}{\frac{\mathrm{NA}}{c}\omega r},$$(4)where $J_1(·)$ is the Bessel function of the first kind of order one. As it is well known, the PSF describes the optical response of the focusing system to a light point source of frequency $\omega$, located at the Fourier plane of the focusing lens. For further details on the role of the PSF in the generation of CSL pulses, see Ref. [12].

Let us now explain in detail the wavefield characteristics of the flying structured pulse with circular geometry. The wavefield of the pulse is described by Eq. (2) and it can be understood as the interference of two wavefields each corresponding to a set of spectral-ring components [e.g., see Fig. 1(b)]. According to Eq. (2), each spectral-ring set comprises frequency-scaled 2D light curves described by $\mathrm{PSF}(|\mathbf{r}-{\mathbf{c}}_j(\alpha ,\omega )|,\omega )$, weighted by both the input spectral amplitude $\tilde{G}(\omega )$ and the complex amplitude $F_j(\alpha )$, which is prescribed along each spectral curve ${\mathbf{c}}_j(\alpha ,\omega )$. Since the function $F_j(\alpha )$ is slower than the $\mathrm{PSF}(|\mathbf{r}-{\mathbf{c}}_j(\alpha ,\omega )|,\omega )$ with respect to the variable $\alpha$, the spatial-spectral filter function Eq. (3) is simplified as follows: $${\mathcal{F}}_j(r,\alpha ,\omega )\simeq \mathrm{sinc}\left[(r-R_j\frac{{\omega }_0}{\omega })\frac{\mathrm{NA}}{c}\omega \right]|F_j(\alpha )|\exp [\mathrm{i}{\mathrm{\Psi }}_j(\alpha )],$$(5)with the function $\mathrm{sinc}(x)=\sin (x)/x$. In practice, the input spectral amplitude function $\tilde{G}(\omega )$ is also slowly varying in the frequency $\omega$ compared to the function ${\mathcal{F}}_j(r,\alpha ,\omega )$, especially for sufficiently large radii $R_1$ and $R_2$. This facilitates the integration of Eq. (2) for obtaining the expression of the output pulse wavefield. In the case of a flying structured pulse with circular geometry, the output pulse wavefield is given by $$E(r,\alpha ,t)\propto \mathrm{rect}\left(\frac{ct}{\mathrm{NA}r}\right)\sum _{j=1}^2\frac{|F_j(\alpha )|R_j}{r^2}\tilde{G}\left({\omega }_0\frac{R_j}{r}\right)\exp [\mathrm{i}{\mathrm{\Psi }}_j(\alpha )]\exp (-\mathrm{i}\frac{R_j}{r}{\omega }_{0}t),$$(6)where the rectangle function, defined as $\mathrm{rect}(\tau )=\{1,\mathrm{if}|\tau |\le 1;0,\mathrm{otherwise}\}$, windows the temporal duration of the output pulse. In this work, without loss of generality, we assume a spectral amplitude function $\tilde{G}(\omega )$ symmetric with respect to the central pulse frequency ${\omega }_0$. This is the case of the considered Gaussian Fourier-limited input pulse, $G(t)$, whose spectral amplitude function is $\tilde{G}(\omega )=\sqrt{\pi }{T}_{\mathrm{input}}\cdot \exp [-T_{\mathrm{input}}^2{(\omega -{\omega }_0)}^2/4]$, with $T_{\mathrm{input}}$ being the input pulse duration (e.g., $T_{\mathrm{input}}=15\mathrm{fs}$). Thus, from Eq. (6), the spatiotemporal evolution of the output peak intensity is described by the wavefield $$E(R_{\mathrm{avg}},\alpha ,t)\propto \mathrm{rect}\left(\frac{ct}{\mathrm{NA}{R}_{\mathrm{avg}}}\right)\cos [\frac{\mathrm{\Delta }\mathrm{\Psi }(\alpha )}{2}-\frac{{\mathrm{\Omega }}_0}{2}t]\exp \{\mathrm{i}[{\mathrm{\Psi }}_{\mathrm{avg}}(\alpha )-{\omega }_{0}t]\},$$(7)evaluated at the averaged radial position $R_{\mathrm{avg}}$ for the intensity peak, where ${\mathrm{\Omega }}_0={\omega }_0\mathrm{\Delta }R/R_{\mathrm{avg}}$, $\mathrm{\Delta }\mathrm{\Psi }(\alpha )={\mathrm{\Psi }}_2(\alpha )-{\mathrm{\Psi }}_1(\alpha )$, and ${\mathrm{\Psi }}_{\mathrm{avg}}(\alpha )=[{\mathrm{\Psi }}_1(\alpha )+{\mathrm{\Psi }}_2(\alpha )]/2$. Note that the expression Eq. (7) is derived from Eq. (6) with $|F_j(\alpha )|\propto R_{\mathrm{avg}}/R_j$. The output pulse duration $T_{\mathrm{output}}=2\mathrm{NA}{R}_{\mathrm{avg}}/c$ is easily determined from the expression Eq. (7), given the rectangle function $\mathrm{rect}(ct/\mathrm{NA}{R}_{\mathrm{avg}})$.

From Eq. (7), the instantaneous peak intensity of the output pulse is directly obtained: $$I(R_{\mathrm{avg}},\alpha ,t)\propto \mathrm{rect}\left(\frac{2t}{T_{\mathrm{output}}}\right)\{1+\cos [\mathrm{\Delta }\mathrm{\Psi }(\alpha )-{\mathrm{\Omega }}_{0}t]\}.$$(8)

The latter expression, Eq. (8), directly reveals the spatial (angular) and temporal behavior of the intensity peak of the flying structured pulse, in terms of the difference $\mathrm{\Delta }R$ on the radii (through ${\mathrm{\Omega }}_0$) and the difference $\mathrm{\Delta }\mathrm{\Psi }(\alpha )$ on the prescribed phases. Indeed, this demonstrates that the action of the angular-time coupling can be controlled by adjusting two parameters, which function as control knobs in Eq. (8): the angular frequency ${\mathrm{\Omega }}_0={\omega }_0\mathrm{\Delta }R/R_{\mathrm{avg}}$ and the phase difference $\mathrm{\Delta }\mathrm{\Psi }(\alpha )$. This result allows establishing the design rules for governing the spatiotemporal behavior of the structured pulse as a function of physical meaningful parameters: the radii $R_1$ and $R_2$ and prescribed phases ${\mathrm{\Psi }}_1(\alpha )$ and ${\mathrm{\Psi }}_2(\alpha )$.

As previously mentioned, Fig. 1(b) displays the intensity and phase distributions of each spectral-ring set for the case of a flying spring pulse in the form of a helix. Specifically, in this example we have considered a typical input ultrashort laser pulse with central wavelength ${\lambda }_0=785\mathrm{nm}$ and input pulse duration $T_{\mathrm{input}}=15\mathrm{fs}$. The radii of the spectral-ring sets are $R_1=529\mathrm{\mu m}$ and $R_2=561\mathrm{\mu m}$ for the central frequency ${\omega }_0$, while the phases prescribed along them are ${\mathrm{\Psi }}_1(\alpha )=Q_1\alpha$ and ${\mathrm{\Psi }}_2(\alpha )=Q_2\alpha$, with topological charges $Q_1=7$ and $Q_2=6$. Let us recall that the topological charge gives the number of times the beam phase passes through the interval $[0,2\pi ]$ along the ring. According to Eq. (7), the phase distribution ${\mathrm{\Psi }}_{\mathrm{avg}}(\alpha )=Q_{\mathrm{avg}}\alpha$ holds in this case with $Q_{\mathrm{avg}}=(Q_1+Q_2)/2=6.5$. The numerical aperture of the focusing lens is $\mathrm{NA}=0.062$; thus, the output pulse has a temporal duration $T_{\mathrm{output}}=227\mathrm{fs}$. As observed in Fig. 1(b), the interference yields a well defined intensity maximum located at $r=R_{\mathrm{avg}}$ (with $\alpha =0$, for time $t=0\mathrm{fs}$) according to Eq. (8). Note that when the flying spring pulse [see right panel of Fig. 1(a)] reaches the Fourier plane, its intensity peak crosses it, describing a rotatory movement on this plane around the focal point ($r=0$). The temporal evolution of this rotatory peak intensity, described by Eq. (8), traces a space-time circular helix of radius $R_{\mathrm{avg}}$ as observed in the time-lapse image displayed in the right panel of Fig. 1(a).

To facilitate the analysis of the total phase distribution $\mathrm{\Psi }(r,\alpha ,t)$ of the pulse wavefield Eq. (6), it is convenient to rewrite it as follows: $$E(r,\alpha ,t)\propto \mathrm{rect}\left(\frac{ct}{\mathrm{NA}r}\right)\frac{R_{\mathrm{avg}}}{r^2}{(A_1^2+A_2^2+2A_1{A}_2\cos \mathrm{\Delta }\psi )}^{1/2}\exp [\mathrm{i}\mathrm{\Psi }(r,\alpha ,t)],$$(9)where the total phase distribution of the flying structured pulse is given by $$\mathrm{\Psi }(r,\alpha ,t)={\psi }_{\mathrm{avg}}(r,\alpha ,t)+\delta (r,\alpha ,t)+\sigma +\frac{\pi }{2}.$$(10)

For simplicity, in Eq. (9), the phase and amplitude of the interfering pulses are defined as ${\psi }_{1,2}(r,\alpha ,t)={\mathrm{\Psi }}_{1,2}(\alpha )-R_{1,2}{\omega }_{0}t/r$ and $A_{1,2}(r)=\tilde{G}({\omega }_0{R}_{1,2}/r)$, respectively, and the phase difference of the interfering pulses as $\mathrm{\Delta }\psi (r,\alpha ,t)={\psi }_2(r,\alpha ,t)-{\psi }_1(r,\alpha ,t)$.

Expression Eq. (10) reveals that the total phase distribution consists of three distinct contributions. The first term, $${\psi }_{\mathrm{avg}}(r,\alpha ,t)={\mathrm{\Psi }}_{\mathrm{avg}}(\alpha )-\frac{R_{\mathrm{avg}}}{r}{\omega }_{0}t,$$(11)incorporates the average prescribed phase ${\mathrm{\Psi }}_{\mathrm{avg}}(\alpha )$. The second term, $\delta (r,\alpha ,t)$, accounts for the prescribed phase difference $\mathrm{\Delta }\mathrm{\Psi }(\alpha )$ and the amplitude functions $A_{1,2}(r)$ of the interfering pulses via $$\tan \delta (r,\alpha ,t)=\frac{A_1(r)+A_2(r)}{A_1(r)-A_2(r)}\cot [\frac{\mathrm{\Delta }\mathrm{\Psi }(\alpha )}{2}-{\omega }_0\frac{\mathrm{\Delta }R}{2r}t].$$(12)

The third term, $$\sigma =\frac{\pi }{2}\left\{\mathrm{sign}\right\{[A_2(r)-A_1(r)]\sin [\frac{\mathrm{\Delta }\mathrm{\Psi }(\alpha )}{2}-{\omega }_0\frac{\mathrm{\Delta }R}{2r}t]\}-1\},$$(13)is a phase constant (i.e., $0,-\pi$); thus, $\exp (\mathrm{i}\sigma )=\pm 1$. Therefore, in the total phase function $\mathrm{\Psi }(r,\alpha ,t)$, the interplay between spatiotemporal couplings, developed along both the radial $r$ and angular $\alpha$ spatial coordinates, and the prescribed spatial phase design, ${\mathrm{\Psi }}_{1,2}(\alpha )$, is clearly distinguishable. This combined action governs the spatiotemporal evolution of both the pulse intensity peaks and the wavefronts.

The topological charge of the beam along a closed trajectory (curve C) is defined as $Q_T={(2\pi )}^{-1}{\oint }_C\nabla \mathrm{\Psi }\mathrm{d}\mathbf{c}$, where $\nabla \mathrm{\Psi }$ is the phase gradient of the beam and $\mathrm{d}\mathbf{c}$ is an infinitesimal displacement along C [38]. In the case of flying structured ultrashort laser pulses with circular geometry, this expression simplifies to $$Q_T(r,t)=\frac{1}{2\pi }{\int }_0^{2\pi }{\partial }_{\alpha }\mathrm{\Psi }(r,\alpha ,t)\mathrm{d}\alpha =\frac{1}{2\pi }[\mathrm{\Psi }(r,2\pi ,t)-\mathrm{\Psi }(r,0,t)],$$(14)which gives the phase accumulation $2\pi Q_T(r,t)$ along a ring of radius $r$ at time $t$. For instance, in the case of the considered flying helical spring with a linear prescribed phase $\mathrm{\Delta }\mathrm{\Psi }(\alpha )=\alpha \mathrm{\Delta }Q$ with $\mathrm{\Delta }Q=Q_2-Q_1$, the beam’s topological charge $Q_T(r,t)$ evaluated at radii $r<R_{\mathrm{avg}}$, $r=R_{\mathrm{avg}}$, and $r>R_{\mathrm{avg}}$ is given by $Q_1$, $Q_{\mathrm{avg}}$, and $Q_2$, respectively, at any time. This result highlights how the local topological charge of the flying spring pulse can exhibit spatial variations even in the case of a linear prescribed phase. As illustrated in Fig. 1(b), the phase distribution, $\mathrm{\Psi }(r,\alpha ,t)$, of this flying spring pulse varies in both radial and angular coordinates while the phase of the spectral-ring sets only varies azimuthally.

To further characterize the beam’s phase topology, the local topological charge, defined by the phase derivative ${\partial }_{\alpha }\mathrm{\Psi }(r,\alpha ,t)$, is often analyzed [38]. If necessary, the global OAM carried by the beam can be determined by integrating the OAM density, which in this case is given by ${|E(r,\alpha ,t)|}^2{\partial }_{\alpha }\mathrm{\Psi }(r,\alpha ,t)$, across all spatial coordinates [19,38–40]. The OAM density essentially represents the product of the beam intensity and the phase gradient strength [38–40]. This product is also known as optical current [41], which is responsible for optical forces harnessed to optically manipulate micro- and nanoparticles [42,43]. Before analyzing the pulse spatiotemporal dynamics in detail, it is necessary to identify the main properties and design rules of the developed flying structured laser pulses, as done in the next subsection.

The spatiotemporal behavior of the intensity peak of the flying structured pulses is governed by Eq. (8). Specifically, the pulse’s peak intensity is reached when the following constructive interference condition is fulfilled: $$\mathrm{\Delta }\mathrm{\Psi }[\alpha -2\pi n(\alpha )]-{\mathrm{\Omega }}_{0}t=-2\pi n(\alpha ),$$(15)where $n(\alpha )=\lfloor \alpha /2\pi \rfloor$, with $\lfloor ·\rfloor$ being the floor function giving as output the greatest integer number less than or equal to $\alpha /2\pi$. The time delay in which a peak intensity is obtained is given as follows: $$\tau (\alpha )=\frac{1}{{\mathrm{\Omega }}_0}\{\mathrm{\Delta }\mathrm{\Psi }[\alpha -2\pi n(\alpha )]+2\pi n(\alpha )\},$$(16)which is derived from Eq. (15). When the flying structured pulse (of circular geometry) reaches the focal plane, its intensity peaks cross it, describing a trajectory (i.e., a time lapse curve) around the focal point given by $$\mathbf{h}(\alpha )=(R_{\mathrm{avg}}\cos \alpha ,R_{\mathrm{avg}}\sin \alpha ,\tau (\alpha )).$$(17)

Specifically, $\mathbf{h}(\alpha )$ is a piecewise curve, given as a set of time-shifted copies of a master curve segment defined in the interval $\alpha \in (0,2\pi ]$, which in turn corresponds to a constant value $n(\alpha )$. Indeed, this piecewise behavior arises from the staircase function $n(\alpha )$. This can be interpreted as an intrinsic replication mechanism that forms the intensity peak trajectory along the pulse duration by inserting the master curve cyclically in each period [obtained from Eq. (16)]: $$\delta t=\frac{1}{{\mathrm{\Omega }}_0}2\pi ;$$(18)thus, ${\mathrm{\Omega }}_0$ is revealed to be a characteristic angular frequency. Specifically, its value is given by the design of the spectral-ring sets (through $\mathrm{\Delta }R$ and $R_{\mathrm{avg}}$) and the central frequency ${\omega }_0$ of the input pulse. Let us also underline that ${\mathrm{\Omega }}_0\delta t=2\pi$ is a fundamental condition arising from the interference of the pulse spectral-ring components. The angular frequency ${\mathrm{\Omega }}_0={\omega }_2-{\omega }_1$ is indeed given by the frequency difference between the corresponding spectral components of each spectral-ring set evaluated at the radial position $R_{\mathrm{avg}}$. The period $\delta t$ is also the time in which the value $n(\alpha )$ increases in one unit (for any angular position $\alpha$). Thus, the motion of the pulse’s intensity peak is cyclically reproduced every period $\delta t$. As previously mentioned, the angular frequency ${\mathrm{\Omega }}_0$ is one of the control knobs that allows for adjusting the pulse dynamics. In this context, the period $\delta t$ (which, for example, is associated with the helix pitch in the flying spring pulse) is another fundamental parameter that can be controlled through ${\mathrm{\Omega }}_0$.

From expression Eq. (16) the time can also be obtained: $$\delta t_{\mathrm{loop}}=\frac{1}{{\mathrm{\Omega }}_0}[\mathrm{\Delta }\mathrm{\Psi }(2\pi )-\mathrm{\Delta }\mathrm{\Psi }(0)],$$(19)corresponding to the time interval in which the peak intensity is located at $\alpha =0$ and $\alpha =2\pi$ [for the same value $n(\alpha )$]. Here, it is important to note that ${\mathrm{\Omega }}_0\delta t_{\mathrm{loop}}=\mathrm{\Delta }\mathrm{\Psi }(2\pi )-\mathrm{\Delta }\mathrm{\Psi }(0)$ can take any real value; in particular it can be different from $2\pi$. The ratio between these two characteristic times, $\delta t_{\mathrm{loop}}/\delta t=[\mathrm{\Delta }\mathrm{\Psi }(2\pi )-\mathrm{\Delta }\mathrm{\Psi }(0)]/2\pi$, is crucial in determining the spatiotemporal behavior of the pulse’s intensity peak during its propagation. As we will show below, the value $\delta t_{\mathrm{loop}}/\delta t$ defines three families of flying structured pulses with distinct spatiotemporal behavior. Moreover, the phase difference function $\mathrm{\Delta }\mathrm{\Psi }(\alpha )$ is confirmed as a control knob for governing the motion of the intensity peak, enabling different pulse dynamics in each family.

The first family is the flying spring pulse whose intensity peak can describe different types of helical trajectories. This family is obtained when $\delta t_{\mathrm{loop}}/\delta t=N$, with $N$ being a non-zero integer number; see Fig. 2(a) corresponding to $N=-1$. The second family corresponds to $N=0$, which yields a flying closed multi-ring pulse whose intensity peak space-time trajectory $\mathbf{h}(\alpha )$ [see Eq. (17)] is shown in Fig. 2(b). In this case the intensity peak trajectory is a closed curve because $\delta t_{\mathrm{loop}}=0$. In other words, $\mathrm{\Delta }\mathrm{\Psi }(2\pi )=\mathrm{\Delta }\mathrm{\Psi }(0)$ holds, indicating that the intensity peak simultaneously arrives at both the end ($\alpha =2\pi$) and start ($\alpha =0$) angular positions, yielding a closed light curve. This condition is not met for the third family, which is obtained for fractional number $N$. This results in a flying open multi-ring pulse with intensity peak trajectory $\mathbf{h}(\alpha )$ as shown in Fig. 2(c).

In each of these families, the prescribed phase difference $\mathrm{\Delta }\mathrm{\Psi }(\alpha )$ can be designed to govern the spatiotemporal behavior of the peak intensity [i.e., the shape of $\mathbf{h}(\alpha )$], which is cyclically repeated over the period $\delta t$. For example, in the case of a linear phase $\mathrm{\Delta }{\mathrm{\Psi }}_{\mathrm{L}}(\alpha )=N\alpha$ a flying spring pulse comprising a set of $|N|$ interleaved helices, $${\mathbf{h}}_N(\alpha )=R_{\mathrm{avg}}(\cos \alpha ,\sin \alpha ,\frac{\delta t}{2\pi R_{\mathrm{avg}}}N\alpha ),$$(20)of radius $R_{\mathrm{avg}}$ and pitch $\delta t$, is obtained. Let us underline that in this case $N=\mathrm{\Delta }Q$ holds, which is the difference $Q_2-Q_1$ of topological charges of the spectral-ring sets. The simple yet important case $|N|=1$ corresponds to a single space-time helix coil of radius $R_{\mathrm{avg}}$ and pitch $\delta t$. The revolution time of the peak intensity around the focal point coincides with this period (i.e., $\delta t_{\mathrm{loop}}=\delta t$). The number of helix turns, comprising the flying spring pulse, is given by the ratio $T_{\mathrm{output}}/\delta t$. More sophisticated pulse dynamics and helical shapes can be obtained using a phase function $\mathrm{\Delta }{\mathrm{\Psi }}_{\mathrm{NL}}(\alpha )$ with non-linear dependence on $\alpha$, for example, as indicated in Fig. 2(a). These results illustrate the important role played by the difference $\mathrm{\Delta }\mathrm{\Psi }(\alpha )$ of the prescribed phase on the spatiotemporal behavior of the pulse dynamic.

Furthermore, the dynamics of the pulse peak intensity is described by its angular position as a function of time, $\alpha (\tau )$, which is obtained from expression Eq. (16). Specifically, the angular velocity $\mathrm{\Omega }={\partial }_{\tau }\alpha (\tau )$ of the intensity peak observed in the Fourier plane is given by $$\mathrm{\Omega }(\alpha )=\frac{{\mathrm{\Omega }}_0}{{\partial }_{\alpha }[\mathrm{\Delta }\mathrm{\Psi }(\alpha )]}.$$(21)

This expression also highlights the crucial joint role played by the prescribed phase gradient, represented by the term ${\partial }_{\alpha }[\mathrm{\Delta }\mathrm{\Psi }(\alpha )]$, and the geometry of the spatial-spectral configuration represented here by the angular frequency ${\mathrm{\Omega }}_0$. When $\mathrm{\Delta }\mathrm{\Psi }(\alpha )$ exhibits a linear dependence on $\alpha$, the angular velocity remains constant: $\mathrm{\Omega }(\alpha )\propto {\mathrm{\Omega }}_0$. In contrast, a non-linear dependence on $\alpha$ introduces a variable angular velocity $\mathrm{\Omega }(\alpha )$ that changes at each angular position. Therefore, a proper design of $\mathrm{\Delta }\mathrm{\Psi }(\alpha )$ also enables tailoring the behavior of the intensity peak in terms of its angular velocity and acceleration, as we will show in the next section. These results demonstrate that the phase difference $\mathrm{\Delta }\mathrm{\Psi }(\alpha )$ acts as a control knob for easily adjusting the pulse’s spatiotemporal behavior and, therefore, the shape of the intensity peak trajectory.

Let us first analyze the spatiotemporal behavior of these three families of flying structured pulses for the case of circular geometry, as previously outlined in Fig. 2. Here we consider a realistic numerical simulation of the free-space propagation of the flying structured pulse at the output of the FST pulse shaper, as outlined in Fig. 1(a). The free-space propagation of the encoded wavefield, as described by Eq. (1), is computed for each spectral component of the input pulse. The corresponding wavefields are coherently superposed to obtain the output flying structured pulse at any time and at any propagation distance around the Fourier plane (i.e., the focal plane of the FST pulse shaper).

The intensity and phase distributions of a flying spring pulse at the focal plane, at times $t=0\mathrm{fs}$ and $t=\delta t/2=22\mathrm{fs}$, are displayed in Fig. 3(a) for the case of a single left-handed helix ${\mathbf{h}}_{N=-1}(\alpha )$ with linear phase difference $\mathrm{\Delta }{\mathrm{\Psi }}_{\mathrm{L}1}(\alpha )=-\alpha$ (corresponding to $Q_2=Q_1-1$). The intensity distribution displayed in the first row of Fig. 3(a) is the same for the two considered phase functions: ${\mathrm{\Psi }}_{\mathrm{avg}}(\alpha )=19.5\alpha$ (for $Q_1=20$ and $Q_2=19$) and ${\mathrm{\Psi }}_{\mathrm{avg}}(\alpha )=3.5\alpha$ (for $Q_1=4$ and $Q_2=3$) as seen in the second and third rows, respectively. Since in this example the sign of the function $\mathrm{\Delta }{\mathrm{\Psi }}_{\mathrm{L}1}(\alpha )=-\alpha$ is negative the intensity peak exhibits the same clockwise rotatory motion (with constant angular velocity ${\mathrm{\Omega }}_{\mathrm{L}1}=-{\mathrm{\Omega }}_0$) regardless of the value of $Q_{\mathrm{avg}}$; see Visualization 1. This result evidences that the value of the phase index $Q_{\mathrm{avg}}$ can be changed without altering the shape and size of the spring pulse. Note that the pulse’s wavefronts exhibit a counter-clockwise rotatory motion for $Q_{\mathrm{avg}}>0$ (see Visualization 1) and clockwise rotation for $Q_{\mathrm{avg}}<0$. The complete temporal evolution (over a duration of $T_{\mathrm{output}}=227\mathrm{fs}$) of the intensity and phase distributions of this type of flying spring pulse is provided in Visualization 1 [see also Fig. 3(a)], for both numerical and analytical results [obtained from Eq. (6)]. The good agreement between these results supports the developed theoretical model. The volumetric $xyz$-plot displayed in Fig. 3(b) reveals the spatial shape of this single-helix flying spring pulse at $t=0\mathrm{fs}$ (an intensity peak in the form of a right-handed $xyz$-helix with pitch $\delta z$). Let us recall that the flying structured pulse developed in this work preserves its spatial shape (i.e., its intensity peak in the form of an $xyz$-curve) as it propagates at speed $\delta z/\delta t=c$ along the optical axis. Consequently, as the flying spring pulse crosses the focal plane, the intensity peak section observed at a transverse plane rotates at a high frequency: $\delta t^{-1}\sim 23\mathrm{THz}$ in this example. Note that $\delta t^{-1}$ can be increased by increasing $\mathrm{\Delta }R$ while keeping $R_1$ constant. This rotating intensity peak section describes a clockwise space-time helical trajectory ${\mathbf{h}}_{N=-1}(\alpha )$, a left-handed $xyt$–helix as observed in Fig. 3(c), which is endowed by the rectangular function over time: $\mathrm{rect}(2t/T_{\mathrm{output}})$, with duration of $T_{\mathrm{output}}=227\mathrm{fs}$, as predicted by expression Eq. (8). Therefore, in this example, the extension of the single-helix flying spring pulse is limited to a number $T_{\mathrm{output}}/\delta t\approx 5$ of helix loops; see Figs. 3(b) and 3(c).

The ability to prescribe the spatial phase $\mathrm{\Delta }\mathrm{\Psi }(\alpha )$ provides relevant degrees of freedom and advantages for designing pulse dynamics in a straightforward manner. For instance, a non-linear prescribed phase can be applied to easily achieve sophisticated propagation dynamics of the pulse intensity peak. To illustrate this fact, Figs. 4(a) and 4(b) display the intensity and phase distributions corresponding to two different types of single-helix flying spring pulses obtained using the following non-linear prescribed phase functions: $\mathrm{\Delta }{\mathrm{\Psi }}_{\mathrm{NL}1}(\alpha )=-{\alpha }^4/{(2\pi )}^3$ and $\mathrm{\Delta }{\mathrm{\Psi }}_{\mathrm{NL}2}(\alpha )=-\alpha -4\cos (2\alpha )$, as in Fig. 2(a). The complete time evolution sequence is provided in Visualization 2 and Visualization 3 for both cases. To better understand the complex motion of the observed intensity peak sections, the angular position dynamics $\alpha (t)$ is displayed in Figs. 4(c) and 4(d) for the case of $\mathrm{\Delta }{\mathrm{\Psi }}_{\mathrm{NL}1}(\alpha )$ and $\mathrm{\Delta }{\mathrm{\Psi }}_{\mathrm{NL}2}(\alpha )$, respectively. Note that in one period (e.g., interval $[0,\delta t]$) several master curve segments [$n(\alpha )$] can be involved in the intensity peak dynamics $\alpha (t)$ of the flying spring pulse, as indicated Figs. 4(c) and 4(d). Specifically, for $\mathrm{\Delta }{\mathrm{\Psi }}_{\mathrm{NL}1}(\alpha )$ the intensity peak at $t=0$ arrives at the focal plane illuminating an arc defined between the angular position 0 and $\approx 3\pi /4$; see Fig. 4(a). The analytical expression of the intensity peak dynamics $\alpha (t)$, represented in Fig. 4(b), predicts this behavior observed at $t=0$ as well as for any time value. In the case of the prescribed phase $\mathrm{\Delta }{\mathrm{\Psi }}_{\mathrm{NL}2}(\alpha )$, the intensity peak exhibits a more sophisticated motion; see Fig. 4(d). At $t=0\mathrm{fs}$ five intensity peak points are observed at the five angular positions indicated by gray and red points in Figs. 4(b) and 4(d). These points belong to different parts of the space-time helix curve $\mathbf{h}(\alpha (t=0))$, corresponding to $n(\alpha )=0$ (two gray points) and $n(\alpha )=1$ (three red points) as depicted in Fig. 4(d). At time $t=\delta t/2=22\mathrm{fs}$ another five points are observed but now corresponding to other parts of the space-time helix [i.e., $\mathbf{h}(\alpha (t=\delta t/2))$]: four red points [belonging to $n(\alpha )=1$] and one green point [belonging to $n(\alpha )=2$]. Note that these intensity peak sections (located at the colored points) exhibit a complex temporal dynamics $\alpha (t)$ dictated by the transverse cross-section of the flying spring pulse with a non-linear prescribed phase. These results once again show a good agreement between numerical simulation and the analytical model developed to design and characterize the behavior of the flying structured pulse.

To further analyze the intensity peak dynamics, the plot for the corresponding angular velocity functions, calculated from Eq. (21), is displayed in Fig. 4(e) for each case: ${\mathrm{\Omega }}_{\mathrm{NL}1}(\alpha )=-{\mathrm{\Omega }}_0{\pi }^3/(2{\alpha }^3)$ (orange plot line), ${\mathrm{\Omega }}_{\mathrm{NL}2}(\alpha )=-{\mathrm{\Omega }}_0/[-1+8\sin (2\alpha )]$ (violet plot line), as well as for the case of the linear prescribed phase (${\mathrm{\Omega }}_{\mathrm{L}1}=-{\mathrm{\Omega }}_0$, blue plot line) considered here for direct comparison. The angular velocity ${\mathrm{\Omega }}_{\mathrm{NL}1}(\alpha )$ diverges approaching $\alpha =0$ while it converges to a constant value at $\alpha =2\pi$. In contrast, for the linear phase difference $\mathrm{\Delta }{\mathrm{\Psi }}_{\mathrm{L}1}(\alpha )$ the angular velocity of the intensity peak is constant (${\mathrm{\Omega }}_{\mathrm{L}1}=-{\mathrm{\Omega }}_0$) at any time and angular position $\alpha$; see blue line in Fig. 4(e). In the case of $\mathrm{\Delta }{\mathrm{\Psi }}_{\mathrm{NL}2}(\alpha )$ the behavior of the intensity peak is even more complex, as observed in Figs. 4(d) and 4(e). Specifically, the angular velocity ${\mathrm{\Omega }}_{\mathrm{NL}2}(\alpha )$ of the intensity peak changes its sign (i.e., its apparent rotation direction); see violet line in Fig. 4(e). This behavior contrasts with the other two cases, ${\mathrm{\Omega }}_{\mathrm{L}1}(\alpha )$ and ${\mathrm{\Omega }}_{\mathrm{NL}1}(\alpha )$, for which the angular speed preserves their signs at any time and angular position. These examples underscore the simplicity of the proposed technique in designing sophisticated spatiotemporal behavior of structured ultrashort pulses. This is confirmed in the following examples for the remaining families.

The intensity and phase distributions calculated at the focal plane for different times (within the period $\delta t$) are displayed in Fig. 5 for the second family: the flying closed multi-ring pulse (with $\delta t_{\mathrm{loop}}=0$). The first case, displayed in Fig. 5(a), corresponds to the trivial phase difference $\mathrm{\Delta }{\mathrm{\Psi }}_{\mathrm{L}2}(\alpha )=0$ that yields a ring-shaped intensity peak of radius $R_{\mathrm{avg}}$. Since the whole ring-shaped intensity peak simultaneously arrives at the transverse plane its angular speed ${\mathrm{\Omega }}_{\mathrm{L}2}(\alpha )$ is not defined, as expected. Note that the whole ring-shaped intensity peak appears in each period $\delta t=44\mathrm{fs}$, as seen in the corresponding plot for the pulse’s dynamics $\alpha (t)$ and in the volumetric $xyt$–plot (time lapse) displayed in the second and third rows of Fig. 5(a), respectively. The complete temporal evolution of this flying multi-ring pulse is also provided in Visualization 4. Let us underline that this example of a flying multi-ring pulse consists of five ring-shaped vortices, as seen in the phase distribution displayed in Fig. 5(a) for $t=0$. These ring-shaped optical vortices illuminate the focal plane at a rate of $\delta t^{-1}\sim 23\mathrm{THz}$. Conventional techniques have proved the application of ultra-intense optical vortices for femtosecond laser material ablation and micro-machining at a rate limited by the repetition rate of laser emission, typically in the kHz or MHz range. Recently, a technique that could create a pulse burst at a repetition rate of around 0.36 THz has been proposed [44]. In contrast, the developed flying multi-ring pulse allows for illuminating a material with a set of ultra-intense optical vortices at a significantly higher rate $\delta t^{-1}$ (of the order of several THz) than the conventional techniques. At this point, it is relevant to recall that the value of this rate $\delta t^{-1}={\mathrm{\Omega }}_0/2\pi$ can be easily tuned through the angular frequency ${\mathrm{\Omega }}_0={\omega }_0\mathrm{\Delta }R/R_{\mathrm{avg}}$. As previously explained, this frequency acts as a control knob by selecting $\mathrm{\Delta }R$ (i.e., the difference between the radii $R_1$ and $R_2$ of the spectral-ring sets). Moreover, any value of the topological charge $Q_{\mathrm{avg}}$ (and therefore of the OAM) can be chosen without altering the repetition rate $\delta t^{-1}$ or the shape of the vortex flying structured pulse.

The second case of a flying closed multi-ring pulse is created by using a non-linear prescribed phase. For instance, Fig. 5(b) shows the intensity and phase distributions obtained using the prescribed phase difference $\mathrm{\Delta }{\mathrm{\Psi }}_{\mathrm{NL}3}(\alpha )=-4{(\alpha -\pi )}^2/2\pi$. In this case the angular speed is ${\mathrm{\Omega }}_{\mathrm{NL}3}(\alpha )=-{\mathrm{\Omega }}_0\pi /[4(\alpha -\pi )]$, which is positive (counterclockwise propagation) for $\alpha \in (0,\pi ]$ and negative (clockwise propagation) for $\alpha \in (\pi ,2\pi ]$. Indeed, the intensity peak at the focal plane splits into several points propagating in opposite angular directions according to the pulse’s dynamics function $\alpha (t)$ displayed in the second row of Fig. 5(b). This interesting behavior arises from the spatiotemporal structure of the flying multi-ring pulse, which exhibits a pointed vertex located at $\alpha =0$ as observed in the time lapse displayed in the third row of Fig. 5(b). As in previous cases, the theoretical model accurately describes the evolution of the intensity and phase of the pulse throughout its duration (see also Visualization 5). The third example of a flying multi-ring pulse corresponds to the following non-linear phase difference: $\mathrm{\Delta }{\mathrm{\Psi }}_{\mathrm{NL}4}(\alpha )=-4\cos (2\alpha )$; see Fig. 5(c) and Visualization 6. It yields a nonuniform angular speed ${\mathrm{\Omega }}_{\mathrm{NL}4}(\alpha )=-{\mathrm{\Omega }}_0/[8\sin (2\alpha )]$. Again a closed multi-ring structure is obtained but it is more waved than the previous example [with $\mathrm{\Delta }{\mathrm{\Psi }}_{\mathrm{NL}3}(\alpha )$]. Consecutively, four intensity peak points are observed at $t=0$ (gray points at $\alpha =\pm \pi /4$ and $\pm 3\pi /4$) that propagate and split into eight points at time $t=\delta t/2=22\mathrm{fs}$ (four red and four gray points), according to the expected intensity peak dynamics $\alpha (t)$ displayed in the second row of Fig. 5(c). This motion shows that several pairs of intensity peak points can propagate in opposite angular directions to finally merge into one point. A similar behavior is observed for the pulse wavefront (phase distribution), but with the propagation in the opposite angular directions as observed in Visualization 5.

The third family is the flying open multi-ring pulse, which is obtained when the ratio $\delta t_{\mathrm{loop}}/\delta t$ is a fractional (non-integer) number. The intensity and phase distributions shown in Fig. 6(a) correspond to an open multi-ring pulse ($\delta t_{\mathrm{loop}}/\delta t=0.5$) obtained using a linear prescribed phase $\mathrm{\Delta }{\mathrm{\Psi }}_{\mathrm{L}3}(\alpha )=-0.5\alpha$, while those shown in Fig. 6(b) correspond to the case of the non-linear phase $\mathrm{\Delta }{\mathrm{\Psi }}_{\mathrm{NL}5}(\alpha )=-0.5\alpha -\cos (2\alpha +\pi /2)$. The volumetric time lapse plot, displayed in Figs. 6(a) and 6(b) for each case, reveals the structure of the flying open multi-ring pulse as a set of five consecutive helices. The corresponding intensity peak dynamics $\alpha (t)$ and its angular speed are shown in Figs. 6(c) and 6(d) for each case [blue plot line for $\mathrm{\Delta }{\mathrm{\Psi }}_{\mathrm{L}3}(\alpha )$ and orange plot line for $\mathrm{\Delta }{\mathrm{\Psi }}_{\mathrm{NL}5}(\alpha )$], respectively. Since $\delta t_{\mathrm{loop}}/\delta t=0.5$ holds, the angular speed of the rotating intensity peak is twice (i.e., ${\mathrm{\Omega }}_{\mathrm{L}3}=-2{\mathrm{\Omega }}_0$) compared with the case $\delta t_{\mathrm{loop}}/\delta t=1$ (i.e., ${\mathrm{\Omega }}_{\mathrm{L}1}=-{\mathrm{\Omega }}_0$). Therefore, the rotating intensity peak completes one loop in half time ($t=\delta t/2$) in both cases $\mathrm{\Delta }{\mathrm{\Psi }}_{\mathrm{L}2}(\alpha )$ and $\mathrm{\Delta }{\mathrm{\Psi }}_{\mathrm{NL}5}(\alpha )$, as seen in Fig. 6(c) (see also Visualization 7 and Visualization 8). The non-linear phase function $\mathrm{\Delta }{\mathrm{\Psi }}_{\mathrm{NL}5}(\alpha )$ has been chosen to achieve an angular speed ${\mathrm{\Omega }}_{\mathrm{NL}5}(\alpha )=-{\mathrm{\Omega }}_0/[2\sin (2\alpha +\pi /2)-1/2]$ [see orange plot line in Fig. 6(d)] whose acceleration is more complex than in the previous examples.

To conclude this study, we also examine a flying structured pulse with non-circular geometry to demonstrate the versatility of the developed spatiotemporal pulse shaping technique. In Fig. 7 the intensity and phase [with ${\mathrm{\Psi }}_{\mathrm{avg}}(\alpha )=14.5\alpha$] distributions of a flying triangular-spring pulse (e.g., a Reuleaux triangle) are shown for the three time points at which the peak intensity reaches each corner of the triangle (see also Visualization 9). As in the previous examples, the repetition rate is $\delta t^{-1}\sim 23\mathrm{THz}$ and the phase ${\mathrm{\Psi }}_{\mathrm{avg}}(\alpha )$ can be adjusted without altering the shape of the flying triangular-spring pulse. The time lapse volumetric $xyt$–plot of the rotatory motion of the pulse’s peak intensity, displayed in Fig. 7, reveals a well defined triangular helix trajectory as expected. To help the comparison between the studied families of flying structured pulses, in Fig. 8 the volumetric $xyz$–plot of the pulse’s intensity peak and spatial phase is also displayed as an example.

This article introduced the flying structured laser pulse, a novel ultrashort laser pulse with controllable spatiotemporal behavior. It includes flying spring and vortex multi-ring pulses, which can carry orbital angular momentum (OAM). The closed-form wavefield expression of the flying structured pulse has been derived, revealing the combined effect of spatiotemporal couplings (STCs) and the prescribed phase in controlling the spatiotemporal behavior of their intensity peaks and wavefronts. This control is achieved by adjusting two spatial parameters of the wavefield’s spectral components (i.e., CSL pulses shaped as rings of any geometry) that comprise these flying structured pulses. These parameters can be understood as two control knobs: the phase gradient difference [phase-control knob, $\mathrm{\Delta }\mathrm{\Psi }(\alpha )$] and the radius difference (frequency-control knob, ${\mathrm{\Omega }}_0$) of such interfering ring-shaped spectral components. The frequency-control knob tunes the frequency difference of the interfering waves, while the phase-control knob leverages the resulting STCs to govern the pulse wavefield dynamics directly. Specifically, the phase-control knob allows the creation of three families of flying structured laser pulses with distinct spatiotemporal characteristics: spring and closed and open multi-ring. The intensity peak of the first family can follow tailored helical trajectories, including intertwined helices if needed. In contrast, the intensity peak in the second and third families repeats over time as a closed or open curve, respectively. Additionally, the phase-control knob can tune the OAM density and locally vary the slope of the intensity peak trajectory for each family in a programmable way, allowing on-demand changes in the trajectory shape and pulse dynamics, while the frequency-control knob adjusts the pitch of the flying spring and the repetition rate of the intensity peak for the flying closed and open multi-ring pulses, achieving values of several tens of THz. Together, these two control knobs enable the generation of flying structured laser pulses with advanced spatiotemporal behavior. The experimental setup for generating these pulses is the FST shaper developed in Ref. [12], which easily allows for practical utilization of these pulse control knobs.

The proposed technique facilitates the creation of structured laser pulses with precise control over their spatiotemporal properties, tailored to specific applications. Let us highlight some relevant specific applications. The achieved control over the laser pulse front, facilitated by the locally varying slope of the intensity peak trajectory, enables the design of pulse front-tilt, which can be tailored to be either uniform or non-uniform as required. This capability is particularly useful for studying physical phenomena involving light-matter interactions, such as in helical-beam plasma wakefield accelerators [18], where ultrashort structured laser pulses can create wakefield topologies with novel configurations. In dielectric laser accelerators, flying structured pulses can adapt their pulse front-tilt in two transverse coordinates to prevent the walk-off between the pulse intensity peak and the electron being accelerated. This mechanism has been shown to extend electron acceleration lengths along one transverse direction [17,45,46] to a sub-MeV level within a single acceleration structure. The enhanced control over the wavefield dynamics available in flying structured pulses could facilitate electron acceleration over shorter distances along two-dimensional trajectories. A more challenging application is the synthesis of extreme-ultraviolet spring beams [47,48] with variable phase gradients in high-harmonic generation, driven by flying structured pulses exhibiting intensity peak trajectories with locally varying slopes. Fundamental questions arise regarding the stability in space-time of these extreme beams and their capability of possessing self-torque, similar to the pulses created in Ref. [49]. Femtosecond laser material processing applications can greatly benefit from the flexibility provided by flying structured pulses carrying OAM, whose value can be adjusted without changing the beam shape. These pulses can be tightly focused, with control over the prescribed phase gradient, which can be set uniformly or varied as required by the application. Moreover, flying vortex multi-ring pulses can be delivered at THz frequencies, making them promising for advancing laser-induced micro/nano-structure formation on dielectric and metallic materials. Typical laser-induced structures include helical micro-needles, helical surface reliefs, and branched fibers, which can be precisely engineered on materials such as metal, silicon, polymer, and nano-crystalline needles as demonstrated in Refs. [50,51] using laser pulses with constant phase gradients (e.g., LG vortex). In this context, the ability to design a non-uniform phase gradient enabled by our technique not only allows for sophisticated spatiotemporal behaviors but also can improve the design of ultrashort pulses for laser-induced material structuring, a capability that has not been considered until now.

Acknowledgment. We gratefully acknowledge the support of NVIDIA Corporation with the donation of the TITAN X GPU device (NVIDIA Academic Hardware Grant Program) used to compute the numerical simulations.