1. INTRODUCTION

Ultrashort laser pulses are used in multiple applications and research fields in physics, chemistry, photonics, microscopy, etc. Because of their ultrashort irradiation duration and high intensities, femtosecond laser pulses can serve as a probe to study light–matter interactions and measure fast dynamic processes (e.g., atomic and molecular dynamics and chemical bonds) [1,2] or be applied to trigger relevant processes such as material ablation (e.g., for laser micro-machining and material processing) [3,4] and laser-driven particle acceleration [5–7].

These relevant applications have stimulated the interest in the development of advanced techniques for shaping focused ultrashort laser beams to optimize and control their spatial and/or temporal characteristics. In the last years, several spatiotemporal control techniques have been proposed to create ultrashort pulses focused in the form of a straight line along the optical axis enabling tunable control of the velocity of the pulse intensity peak along the line [8–13]. Such a tunable control of the velocity of the pulse intensity peak is interesting for laser-driven particle acceleration, photon accelerators, and photon–electron light sources [9,10,14]. Another relevant case of structured laser pulses is the so-called “space-time” wave packets focused in the form of a light sheet [15–17]. This type of one-dimensional pulsed light sheets travel in free space at a fixed [15,16] or variable [17] velocity exhibiting an elongated transverse focused intensity profile. Structured laser pulses corresponding to femtosecond Bessel [10,18], Laguerre–Gaussian [19,20], and Airy [21] beams have been also studied in the context of spatiotemporal control and laser micro-machining [3,4,22,23]. However, the ability to control the group velocity in free space of these structured pulses would be very limited [16].

The spatiotemporal control of both the intensity peak and phase of focused ultrashort laser pulses along arbitrary trajectories is a long-standing problem. For instance, this is required to govern the pulse propagation dynamics including the time delay function and speed of the peak intensity traveling along the targeted trajectory. Here, to address this challenging problem, we present a technique that allows us to create a completely different type of structured ultrashort laser pulses strongly focused in form of any curve, which is further referred to as curve-shaped laser (CSL) pulses. Specifically, by using the proposed technique, any pulse phase profile can be prescribed along the target curve along which the intensity peak travels with controlled time delay and velocity, decoupled from the group velocity of the laser. This spatiotemporal control of both the intensity peak and phase of the pulse along the target curve is achieved in a programmable way. Moreover, the temporal duration of the input pulse is preserved, which is crucial for maintaining the properties of extreme ultrashort laser pulses such as the femtosecond ones. Since the ultrashort duration (i.e., femtosecond scale) of the CSL pulse is preserved, there is no need for pulse compression. Thus, this work advances in the understanding of how to achieve tailored propagation dynamics of laser pulses along target trajectories on demand preserving their ultrashort temporal character.

In addition, the strong focusing character of the CSL pulses creates a diffraction-limited light curve with high intensity gradients, which are known to be responsible for optical trapping forces exerted over micro- and nanoparticles [24–27]. The control of the prescribed phase profile allows us to design the spatial phase gradients of the beam, which are responsible for the optical forces able to transport particles along the light curve [28–30]. We also show that the ability to control the prescribed phase profile allows us to tailor the orbital angular momentum (OAM, [31–34]) carried by the beam, yielding CSL optical vortex pulses. All these advantages expand the flexibility of spatiotemporal pulse control for laser-based applications. In particular, these CSL pulses are promising for laser micro-machining [22,23], optical manipulation [24–27], and laser-driven particle acceleration [5–7] applications demanding ultrashort laser pulses focused in the form of arbitrary trajectory with controlled propagation dynamics.

In this paper, the theoretical framework that describes the design and properties of the CSL pulses in vacuum or any linear medium is presented. We have found a closed-form expression for the pulse wave field that predicts its propagation dynamics observed in the numerical simulations performed for distinct spatiotemporal configurations. To further illustrate the practical versatility of the proposed technique, we also show how contour-shaped pulses can be created to follow the outline of objects at micrometer scale enabling single-shot laser micro-machining. The proposed experimental system transforms input femtosecond laser pulses into CSL ones preserving the pulse duration by using a conventional spatial light modulator (SLM, liquid-crystal display-based technology). The required beam information is holographically encoded as a phase mask onto the SLM by using the direct (noniterative) algorithm reported in [35]. Let us mention that conventional beam shaping techniques often apply the well-known iterative Gerchberg–Saxton algorithm to compute a holographic phase mask for creating target intensity patterns of a monochromatic laser beam or laser pulses of narrow spectral bandwidths [36,37]. Such techniques are not suited for shaping both the intensity and phase of femtosecond laser pulses whose broader spectrum cannot be neglected. In contrast, the proposed technique considers the effects of the broader spectrum for the spatiotemporal control of femtosecond CSL pulses. It can be also used for controlling CSL pulses of narrow spectral linewidth if needed.

The paper is organized as follows. Section 2 introduces the theoretical framework required for the proposed spatiotemporal control of the laser pulse and the description of the propagation dynamics of the CSL pulse for any curve geometry. This allows us to find the role played by the phase prescribed along the curve and its geometry on the ultrashort pulse dynamics, establishing the basis for straightforward engineering of the CSL pulse. Section 3 presents representative examples of CSL pulses for different curve geometries, phase distributions, temporal delay profiles, and propagation dynamics. In Section 4, the experimental setup and results demonstrating the experimental generation of CSL femtosecond pulses are presented. The work ends with concluding remarks and discussions.

2. THEORETICAL FRAMEWORK

The design of the considered CSL pulse is inspired by the concept of the polymorphic laser beam (monochromatic), which can be focused in the form of an arbitrary light curve with independent control of its intensity and phase distributions prescribed along the curve [38]. Specifically, the complex field amplitude of a scalar polymorphic beam in the traverse input plane (i.e., the SLM display required for its experimental generation) is written in Cartesian coordinates ${{\textbf{r}}_0} = ({{x_{0,}}{y_0}})$ as follows:

To generate a CSL pulse, an input ultrashort laser pulse illuminating the SLM is used, and each spectral component of the pulse is focused in form of light curve, ${{\textbf{r}}_c}({\alpha ,\omega}) = ({\omega _0}/\omega) \cdot {{\textbf{r}}_c}(\alpha)$, where the frequency-scaling factor ${\omega _0}/\omega$ arises from the optical Fourier transformation performed by the focusing lens [40]. The superposition of these frequency-scaled light curves ${{\textbf{r}}_c}({\alpha ,\omega})$ yields the output CSL pulse. A conceptual sketch of the system comprising the SLM and the focusing lens, required for this holographic shaping of the input pulse, is shown in Fig. 1(a). Here, we assume a plane pulse linearly polarized that illuminates the SLM in normal incidence; therefore, each spectral component of the pulse has the same field amplitude ${E_0}({{x_0},{y_0}})$ at the screen plane $({{x_0},{y_0}})$ of the SLM.

 

Fig. 1. (a) Conceptual sketch of the system required for generation of CSL pulses. An input laser pulse (e.g., 50 fs, Fourier-limited) illuminates the SLM where a hologram [encoding the field Eq. (1)] is displayed for spatiotemporal control of the pulse. The CSL pulse in form of arbitrary 3D curve is created in the focal region of the focusing lens. (b) Sketch of the laser curve shaping for the target curve ${{\textbf{r}}_c}(\alpha)$ corresponding to the central frequency ${\omega _0}$ of the pulse. The light focusing extends over a region around the curve (e.g., see green curve) due to the PSF extension. The field $E(t)$ and its spectral amplitude $\tilde G(\omega)$ of the input pulse are indicated as an example. (c) The CSL pulse results from the superposition of all the spectral components for each curve ${{\textbf{r}}_c}(\alpha ,\omega)$, according to Eq. (2), sketched in 2D ($x - y$ plane) as an example.

Download Full Size | PDF

The spatiotemporal evolution of the pulse can be found by propagating each spectral component from the SLM to the focal region of the focusing lens, obtaining the following wave field:

The complexity of the expression Eq. (2), however, makes it difficult to find analytical solutions for $E({{\textbf{r}},t})$; therefore, its numerical computation is required. Moreover, from Eq. (2), it is not intuitive to describe the role played by the curve geometry ${{\textbf{r}}_c}(\alpha)$ and the prescribed phase $\Psi (\alpha)$ along the curve on the pulse propagation dynamics. Nevertheless, it can be easily described by analyzing the CSL pulse formation in the focal region of the focusing lens as follows.

The representation of the field $E({{\textbf{r}},t})$ as a convolution operation in Eq. (2) directly shows the independent role played by the curve geometry encoded into the source function $H({{\textbf{r}},\omega})$ and the characteristics of the optical system given by ${\rm PSF}({{\textbf{r}},\omega})$. This operation can be explicitly written as

A. CSL Pulses for 2D Curves

The function ${\cal F}({\alpha ,\omega})$ in the integral Eq. (2) behaves as a complex-valued spectral filter acting on the spectral amplitude $\tilde G(\omega)$ of the input pulse. This spectral filtering varies along the curve position, ${{\textbf{r}}_c}(\alpha)$, and also depends on the particular curve geometry. To find a general solution, it is convenient to simplify the analysis by describing ${{\textbf{r}}_c}(\alpha)$ and the prescribed phase $\Psi (\alpha)$ as Taylor series up to first order in the parameter $\alpha$ for each point of the curve. Therefore, the pulse wave field at $\alpha$ is obtained by using this series expansion centered at $\alpha$. This is the key to the high robustness of this local approximation. The accuracy of this approximation is discussed in detail in Supplement 1, Section 6. Under this approximation, the phase of the filter ${\cal F}({\alpha ,\omega})$ is a linear function in the frequency $\omega$ (see Supplement 1, Section 2) while its modulus $| {{\cal F}({\alpha ,\omega})} |$ is a symmetric function for $\omega$ (with respect the central frequency ${\omega _0}$), which filters the wings of the input spectrum distribution $\tilde G(\omega)$ (see Supplement 1, Section 3). Thus, the phase of ${\cal F}({\alpha ,\omega})$ governs both the temporal evolution of the pulse phase and the peak intensity of the pulse, whereas its modulus sets the pulse duration along the curve. In ultrashort pulse applications, it is often convenient to preserve the input pulse duration as much as possible along the whole target trajectory ${{\textbf{r}}_c}(\alpha)$; therefore, a spectral filter is needed whose modulus $| {{\cal F}({\alpha ,\omega})} |$ is approximately constant. This requirement is fulfilled when $| {{{\textbf{r}}_c}({\alpha ,\omega}) - {{\textbf{r}}_c}(\alpha)} |$ is much smaller than the size of the mainlobe of the transverse section of the PSF for the whole light spectrum of the input pulse. Thus, the spectral phase filter ${\cal F}({\alpha ,\omega})$ is given by

This spectral phase filter Eq. (5) plays a fundamental role in the pulse dynamics. Indeed, a closed-form expression for the pulse wave field is found by introducing Eq. (5) in Eq. (2), obtaining

Let us note that the complex field amplitude ${E_c}({\alpha ,t})$ given by Eq. (6) fully describes the dynamics of the CSL pulse at each position ${{\textbf{r}}_c}(\alpha)$. Interestingly, the temporal dependence of the CSL pulse Eq. (6) at every point ${{\textbf{r}}_c}(\alpha)$ is proportional to ${\partial _t}G({t - \tau (\alpha)})$, which is the temporal derivative of the input pulse delayed by $\tau (\alpha)$. Note that the operation ${\partial _t}G(t)$ changes the input $G(t)$ (except for a constant) very little in the case of a Fourier-limited input pulse longer than a few femtoseconds. For instance, an input pulse with central wavelength in the near-infrared range (e.g., ${\lambda _0} = 750 \;{\rm{nm}}$) and with a duration longer than 10 fs fulfills ${\partial _t}G(t) \propto G(t)$; therefore, the temporal dependence of ${E_c}({\alpha ,t})$ is given by $G({t - \tau (\alpha)})$. Note that the time delay $\tau (\alpha)$ given by Eq. (8) shows the role played by the prescribed phase and the curve geometry on the temporal behavior of the peak intensity of the pulse.

In the derivation of Eq. (5), it has been considered that the modulus of the spectral filter is approximately constant, so the input spectrum of the pulse is fully or largely preserved at each point of the curve. Let us note that the amplitude spectral filtering relies on the limited extension of the frequency-scaled transverse section of ${\rm PSF}({{\textbf{r}},\omega})$. This filters the wings of the input spectrum distribution and stretches the pulse in time. The increase of the pulse duration can be quantified by assuming a Fourier-limited input pulse satisfying ${\partial _t}G(t) \propto G(t)$, as has been previously mentioned. It allows us to determine the region that limits the maximum temporal stretching of the pulse due to the spectral filtering. Specifically, this region is a circle embedded in the focal plane and centered at the focal point $(x = 0,y = 0,z = 0)$ (see Supplement 1, Section 3). Therefore, any CSL pulse created inside this reference circle has a temporal duration less than $\beta T$, where $T$ is the input pulse duration and $\beta \gt 1$ is the ratio between the maximum output pulse stretching and $T$. The radius of the reference circle is given by

In this paper, we will study in detail an interesting CSL pulse given by

In general, the spatial phase prescribed along a curve can be easily defined as [38]

Let us note that the pulse wavefronts for a uniform prescribed phase move along the curve at constant velocity ${v_w}$. The time it takes for a wavefront to travel the full curve is given by $\Delta t = [{\Psi ({{\alpha _2}}) - \Psi ({{\alpha _1}})}]/{\omega _0} = 2\pi m/{\omega _0}$; therefore, the wavefront velocity can be calculated as ${v_w} = S({\alpha _2})/\Delta t = {\omega _0}S({\alpha _2})/({2\pi m})$. The temporal evolution of the wavefront is not dependent on the curve shape for this important case.

B. CSL Pulses for 3D Curves

To illustrate the generation of CSL pulses extended along a 3D trajectory, let us consider the important case of a straight line ${{\textbf{r}}_l}(\alpha) = ({\cos \phi ,0,\sin \phi})\alpha$ tilted by an angle $\phi$ with respect the focal plane. We recall that it has been reported that laser pulses can be created with tunable control of the velocity of the pulse intensity peak along a straight line in the $z$ axis (i.e., $\phi = \pi /2$) [8–13], which is interesting for laser-driven particle acceleration, photon accelerators, and photon–electron light sources [9,10,14]. Here, we show that ultrashort needle-shaped laser pulses whose intensity peak propagates along a tilted line ${{\textbf{r}}_l}(\alpha)$ with tunable velocity control can be also generated.

The spatiotemporal evolution of the needle-shaped laser pulse along the target line is obtained by following the same procedure as for 2D curves embedded in the focal plane. Specifically, in this case, the spectral phase filter is given by (see Supplement 1, Section 5)

There are several families of 3D curve geometries that present a closed-form expression for the pulse wave field evaluated along the curve, beyond the case of the tilted 3D straight line studied in this section (see Supplement 1, Section 7). We conclude that it is possible to tailor the time delay of the intensity peak and the phase of the CSL pulse even for complex 3D geometries.

3. PROPAGATION DYNAMICS OF CSL PULSES

A. CSL Pulse Dynamics for 2D Curves

In this section, we analyze the dynamics of the CSL pulse for the case of 2D curves ${{\textbf{r}}_c}(\alpha) = ({{\textbf{R}}(\alpha),0})$, embedded in the focal plane (located at $z = 0$), and for uniform as well as non-uniform prescribed phase $\Psi (\alpha)$ [Eq. (13)]. In particular, we study the propagation behavior of both the peak intensity and phase $\Phi ({\alpha ,t})$ [Eq. (12)] of the pulse. The dynamics of the pulse peak intensity is governed by the time delay $\tau (\alpha)$, Eq. (8), which is given by the product of the phase derivative $\dot \Psi (\alpha)$ and the geometric factor ${\tau _g}(\alpha)$, as we have previously mentioned. On the other hand, the pulse phase given by Eq. (12) depends on the prescribed spatial phase $\Psi (\alpha)$; therefore, it is not dependent on the curve geometry. Thus, it is convenient to study the pulse dynamics in two steps: first by analyzing the effect of the geometric factor ${\tau _g}(\alpha)$ on the time delay, and second by analyzing the effect of the prescribed spatial phase $\Psi (\alpha)$ on both the time delay $\tau (\alpha)$ and the pulse phase $\Phi ({\alpha ,t})$.

 

Fig. 2. (a) Ring-shaped vortex pulse (at time $t = 0\;{\rm{fs}}$) with radius $R = 50\;\unicode{x00B5}{\rm m}$ and uniform phase distribution (linear phase $\Psi (\alpha) = m\alpha$ prescribed along the circle) with charge $m = 4$ (see Visualization 1). (b) Ring-shaped vortex pulse (at time $t = 0\;{\rm{fs}}$) with non-uniform phase distribution (quadratic phase $\Psi (\alpha) = m{\alpha ^2}/2\pi$ prescribed along the circle) with $m = 4$ (see Visualization 2). The shape of the ring-shaped pulse is preserved despite the change of the prescribed phase $\Psi (\alpha)$, as expected.

Download Full Size | PDF

To evaluate the pulse dynamics, here we consider an achromatic focusing lens with ${\rm{NA}} = 0.15$ and a Fourier-limited input femtosecond pulse with a duration of $T = 50\;{\rm{fs}}$ (central wavelength ${\lambda _0} = 750 \;{\rm{nm}}$ and spectral width $\Delta \lambda = 30\; {\rm{nm}}$ FWHM, according to our experimental setup). As it has been previously mentioned, to preserve the pulse duration, we consider a pulse stretching ratio of $\beta = 1.1$. For these system parameters, the reference circle where the pulse duration is largely preserved [thus, Eq. (11) accurately describes the pulse wave field] has a radius of $B \simeq 70\;\unicode{x00B5}{\rm m}$. We have performed a numerical simulation of the pulse propagation in the focal region in order to analyze the spatiotemporal behavior of the CSL pulse for different configurations of the prescribed phase and curve geometries. This numerical simulation implements the well-known computation algorithm reported in [44,45] for monochromatic scalar beams that we have adapted for the case of a Fourier-limited input femtosecond pulse, which is modulated by the signal Eq. (1) encoded onto the SLM and then transformed into the CSL pulse by the focusing lens (Fig. 1). We have used a fast computation device (GPU, NVIDIA TITAN X, providing 11 TFLOPS) enabling a fast numerical simulation of the considered pulses in a time of ${\sim}5\;{\rm{s}}$, which is ${\sim}20 \times$ faster than a conventional PC-CPU device. Here, the numerical simulation results are compared with the theoretical ones obtained from the expression Eq. (11) describing the considered CSL pulses.

Let us first analyze the case corresponding to a circle of radius $R$ centered at the origin (the focal point), which corresponds to ${\textbf{R}}(\alpha) = R({\cos \alpha ,\sin \alpha})$. For this geometry, the vector ${{\textbf{r}}_c}(\alpha)$ and the tangent vector ${\textbf{u}}(\alpha)$ to the circle are orthogonal for all the values of the angle $\alpha$. Thus, the geometric factor is ${\tau _g}(\alpha) = 0$; therefore, the time delay is $\tau (\alpha) = 0$ for any type of spatial phase $\Psi (\alpha)$ prescribed along the circle. This means that the pulse simultaneously focuses along the whole circle, reaching its peak intensity at time $t = 0\;{\rm{s}}$, irrespective of the prescribed phase $\Psi (\alpha)$. Consequently, the pulse peak intensity and its phase Eq. (12) are independently controlled in the case of the centered circle.

Figure 2(a) shows the intensity and phase distributions of a ring-shaped vortex pulse (at time $t = 0\;{\rm{s}}$) with linear prescribed phase $\Psi (\alpha) = m\alpha$, charge $m = 4$, and radius $R = 50 \;{\rm{\unicode{x00B5}{\rm m}}}$. Note that this case corresponds to the familiar helical (vortex) Bessel pulse with phase $\Phi ({\alpha ,t}) = m\alpha - {\omega _0}t$ that exhibits a constant phase velocity and constant phase gradient $\xi = m/R$. Specifically, the first and second row of Fig. 2(a) correspond to the numerically propagated pulse and to the theoretically predicted one Eq. (11), respectively. These results are in good agreement supporting the developed theoretical framework. The temporal evolution of this ring-shaped vortex pulse can be observed in Visualization 1.

This ring-shaped vortex pulse as well as other optical vortex beams carry an OAM $m\hbar$ per photon [34,46]. Previous works on the generation of broadband beams carrying OAM (see, e.g., [31,33,34]), have been only focused on tuning the OAM value of a ring-like optical vortex (e.g., Laguerre–Gaussian and Bessel–Gaussian vortices) with constant phase gradient. It is well known that a monochromatic optical vortex beam carrying OAM can exert transverse optical forces able to propel particles at micro/nanoscale and set them into fast rotation [28,46]. These optical propulsion forces indeed arise from the phase gradient of the beam that redirects part of the light radiation pressure to transport the particles [28,47,48]. Specifically, the propulsion force is proportional to the product of the phase gradient vector and the intensity of the beam $I({\textbf{r}})$ [28,47], which in the case of a monochromatic laser curve corresponds to the force ${\textbf{j}}(\alpha) \propto \dot \Psi (\alpha)I({{\textbf{r}}_c}(\alpha)){\textbf{u}}(\alpha)$ [30]. The phase gradient also plays a crucial role in the control of the local OAM density of a monochromatic optical vortex, defined by ${L_z}({\alpha ,{\textbf{r}}}) \propto \omega \dot \Psi (\alpha)I({\textbf{r}})$ [46], which is attractive in quantum optics and laser free-space communications [49]. Therefore, the ability to design the phase gradient prescribed in the CSL pulse is crucial for relevant applications such as optical manipulation of particles [28,30,47,48], as well as other light–matter interactions [34,50], and laser material ablation [43] based on optical vortex pulses.

 

Fig. 3. (a) The intensity and phase distributions of a line-shaped pulse with quadratic time delay $\tau (\alpha) = b{\alpha ^2}$ ($n = 2$ and $b = - 300\,{c}^{-1}{{\rm{m}}^{- {{2}}}}\,{\rm{s}}$) are displayed for time $t = - 24\;{\rm{fs}}$ and $t = 0\;{\rm{fs}}$ (see full temporal sequence in Visualization 3). This quadratic time delay function is represented in (b) along with linear time delay $\tau (\alpha) = b\alpha$ ($n = 1$ and $b = 0.033\,{c}^{-1}{{\rm{m}}^{- {{1}}}}\,{\rm{s}}$). A time lapse intensity profile of the line-shaped pulse is shown in (c) and (d) for the case of quadratic and linear prescribed phase, respectively.

Download Full Size | PDF

Thus, let us illustrate the ability to design the phase gradient independently of the curve geometry. For instance, Fig. 2(b) shows the intensity and phase distributions of a ring-shaped vortex pulse (at time $t = 0\;{\rm{s}}$) with quadratic phase $\Psi (\alpha) = m{\alpha ^2}/2\pi$ (with charge $m = 4$) yielding a non-constant phase gradient, $\xi (\alpha) = m\alpha /\pi R$. Specifically, the first and second rows of Fig. 2(b) correspond to the numerically propagated pulse and to the theoretically predicted one, Eq. (11), respectively. These results are also in good agreement supporting the developed theoretical framework and proving that a ring-shaped vortex pulse with non-constant phase gradient (corresponding to a non-linear phase distribution) can be created preserving its shape (same radius $R$) and the value of the topological charge $m$. The temporal evolution of the intensity and phase distributions of this type of ring-shaped vortex pulse is provided in Visualization 2. The dynamics of the peak intensity is preserved (the pulse simultaneously focuses on the whole circle), as expected, because the time delay is $\tau (\alpha) = 0$ irrespective of the non-constant phase gradient $\xi (\alpha) = m\alpha /\pi R$ prescribed along the circle. A non-constant phase gradient is interesting, for example, in optical manipulation applications to tailor the speed of optically transported particles, as has been experimentally demonstrated in [28,30] for the case of monochromatic laser traps.

Another interesting case is a line-shaped pulse corresponding to ${{\textbf{r}}_c}(\alpha) = \alpha {{\textbf{e}}_x}$ (a straight line crossing the origin of coordinates located at the focal point), where ${{\textbf{e}}_x}$ is the unitary vector for the $x$ axis and $\alpha$ has length units. For this case, ${{\textbf{r}}_c}(\alpha)$ and ${\textbf{u}}(\alpha)$ are parallel vectors for all the values of $\alpha$ resulting in the geometric factor ${\tau _g}(\alpha) = \alpha /{\omega _0}$ and in the time delay $\tau (\alpha) = ({\dot \Psi (\alpha)/{\omega _0}})\alpha$. Since the dynamics of the pulse peak intensity is governed by $\tau (\alpha)$ then it can be controlled by tuning $\dot \Psi (\alpha)$ (i.e., the phase gradient) as required. To illustrate the control of the peak intensity dynamics, let us consider as an example the power law dependence $\tau (\alpha) = b{\alpha ^n}$ obtained by using the phase $\Psi (\alpha) = b{\omega _0}{\alpha ^n}/n$ with $n$ and $b$ being an integer and a real number, respectively. Taking into account the expression Eq. (11), in this case the pulse propagation along the line is described by

 

Fig. 4. (a) Square-shaped vortex pulse with prescribed uniform phase distribution (topological charge $m = 45$) displayed at time $t = - 8.7\;{\rm{fs}}$ and $t = 8.7\;{\rm{fs}}$. The pulse peak intensity propagates following the curve between the indicated points ${\sigma _{1 - 8}}$ (see Visualization 4). The intensity time-lapse profiles along the four sides of the square are shown in (b). The time delay $\tau (\alpha)$ displayed in (c) governs the peak intensity propagation along the square. (d) Starfish-shaped vortex pulse with prescribed uniform phase distribution (topological charge $m = 30$) displayed at time $t = - 8.3\;{\rm{fs}}$ and $t = 8.3\;{\rm{fs}}$. The peak intensity also propagates along the curve (see Visualization 5), according to the corresponding time delay $\tau (\alpha)$ displayed in (e).

Download Full Size | PDF

To easily create 2D curves ${{\textbf{r}}_c}(\alpha) = R(\alpha)({\cos \alpha ,\sin \alpha ,0})$ of any symmetry, the so-called Superformula

For instance, here we consider a square and starfish curve given by the set ${Q_\square} = (1, 1, 7, 7, 7, 4)$ and ${Q_ \star} = (10, 10, 2, 7, 7, 5)$ with a base radius $\rho = 8\;\unicode{x00B5}{\rm m}$, respectively. In Fig. 4(a), the intensity and phase distributions of the numerically simulated square-shaped vortex pulse are displayed along with the corresponding theoretical prediction for the case of the uniform prescribed phase given by Eqs. (13) and (14) with topological charge $m = 45$. These numerical and theoretical results are in good agreement, again supporting the proposed technique. We have considered these polygonal curves as an example to illustrate the rich interplay between the curve geometry and the prescribed phase that governs the temporal evolution of the pulse peak intensity. Interestingly, the pulse peak intensity propagates along the square following a specific sequence: Its propagation simultaneously starts at time $t \simeq - 8.7\;{\rm{fs}}$ close to the vertices, at the four points ${\sigma _{1 - 4}}$ indicated in Fig. 4(a), and then it propagates toward the four points ${\sigma _{5 - 8}}$ at which the peak intensity arrives at $t = 8.7\;{\rm{fs}}$ (see Visualization 4). The whole square is illuminated at $t = 0\;{\rm{fs}}$ as observed in Fig. 4(b), where the corresponding time lapse profiles are also displayed to illustrate the peak intensity propagation along the four sides of the square. The positions of the points ${\sigma _{1 - 8}}$ are given by the time delay function Eq. (8) displayed in Fig. 4(c) for this square-shaped pulse. Indeed, the positions of the points ${\sigma _{1 - 8}}$ are defined by the maximum and minimum values of the time delay function, which is proportional to the topological charge ($m$) as $\tau (\alpha) = \dot \Psi (\alpha){\tau _g}(\alpha) \propto m{\tau _g}(\alpha)$. For instance, the peak intensity at the point ${\sigma _1}$ propagates following the paths ${\sigma _1} \to {\sigma _6}$ and ${\sigma _1} \to {\sigma _5}$ (along the square), and it simultaneously arrives at the point ${\sigma _5}$ and ${\sigma _6}$ ($t = 8.7\;{\rm{fs}}$). The peak intensity propagates much faster in the path ${\sigma _1} \to {\sigma _6}$ than in the path ${\sigma _1} \to {\sigma _5}$ because the former one is much longer. Since the time delay function $\tau (\alpha)$ is proportional to the topological charge $m$, then the propagation velocity of the pulse peak intensity can be easily changed, and its propagation direction is reversed just by switching the sign of $m$ if needed. For instance, the propagation ${\sigma _1} \to {\sigma _6}$ for $m = 45$ is reversed to ${\sigma _6} \to {\sigma _1}$ by using $m = - 45$, and correspondingly for the rest of propagation paths. This behavior of the pulse dynamics can be also obtained for more complex CSL pulses as, for example, the starfish one displayed in Fig. 4(d) (see also Visualization 5). Indeed, the pulse peak intensity simultaneously starts (time $t = - 8.3\;{\rm{fs}}$) at the points ${\eta _{1 - 5}}$ located close to the vertices of the starfish curve and then it propagates toward the points ${\eta _{6 - 10}}$, at which it arrives at time $t = 8.3\;{\rm{fs}}$. The whole curve is illuminated at $t = 0\;{\rm{fs}}$ (see Visualization 5). The corresponding time delay function $\tau (\alpha)$ displayed in Fig. 4(e) is also proportional to the charge $m$ because the prescribed phase $\Psi (\alpha)$ is uniform along the starfish curve. Again, the peak intensity propagates much faster in the longer paths. For instance, the peak intensity propagates faster in the path ${\eta _1} \to {\eta _7}$ than in short path ${\eta _1} \to {\eta _6}$. Let us note that for the uniform prescribed phase considered in these examples the wavefronts of the square and starfish-shaped vortex pulse move along the curve at constant velocity ${v_w} = 3.5\;\unicode{x00B5}{\rm m}/{\rm{fs}}$ and ${v_w} = 6.7\;\unicode{x00B5}{\rm m}/{\rm{fs}}$ (see Visualization 4 and Visualization 5), respectively.

 

Fig. 5. (a) A contour-shaped vortex pulse can be generated to follow the outline of an object. The bitmap image of the object (shown in step 1) is processed by using a conventional contour-tracing algorithm providing the raster contour trajectory shown in the step 2. Then it is represented as a piece-wise defined curve ${{\textbf{r}}_c}(\alpha) = \{{{\textbf{b}}_1}(\alpha),...,{{\textbf{b}}_N}(\alpha)\}$ comprising a set of $N$ parametric Bézier splines ${{\textbf{b}}_n}(\alpha)$ as indicated in step 3. The corresponding contour-shaped pulse (intensity and phase for the topological charge $m = 5$) is shown for the time $t = 0$ fs in step 4 (see also Visualization 6). (b) The intensity and phase of the signal Eq. (1) required to create this contour-shaped pulse are also displayed as an example. (c) The theoretical results for the intensity and phase distributions of the considered contour-shaped pulse are also shown for $t = 0\;{\rm{fs}}$.

Download Full Size | PDF

Relevant applications, such as femtosecond laser material processing including two-photon polymerization, optical waveguide fabrication, and laser-assisted cutting and grooving of materials [23], often require high-speed scanning of a focused laser spot or line-shaped beam along a scanning trajectory enabling both macro- and micro-machining. For instance, the scanning trajectory describes the contour of an object shape applied for laser cutting or engraving of the material. The proposed technique is also promising for these applications because it allows both shaping the laser pulse along the contour trajectory and the control of the pulse dynamics along it. In particular, it can be used for single-shot machining instead of the conventional scanning-based one. Let us illustrate the pulse shaping process for an arbitrary trajectory that cannot be described by an analytical parametric curve ${{\textbf{r}}_c}(\alpha)$, as for example the one displayed in Fig. 5(a) described in four steps. In the first step, the contour trajectory is automatically traced (in a few milliseconds) from a given image of an object [52], which is a standard task in laser material processing applications [23]. In particular, we have considered a bitmap (raster) image of a letter “t” (cooper black font, or any other) as an object whose complex contour is similar to a circuit. This image has been processed by using a well-known contour-tracing algorithm [52] to trace its contour displayed in the second step depicted in Fig. 5(a). The traced contour defines the trajectory as a set of points (i.e., a raster contour), and it can be represented by a piece-wise defined curve ${{\textbf{r}}_c}(\alpha) = \{{{\textbf{b}}_1}(\alpha),...,{{\textbf{b}}_N}(\alpha)\}$ [see step 3 in Fig. 5(a)] comprising a set of parametric Bézier splines ${{\textbf{b}}_n}(\alpha)$ (e.g., cubic polynomials), which are joined to each other in the so-called knot points [39,53] that belong to the traced contour. This piece-wise curve (created in a few milliseconds) represents the contour as scalable vector graphics (instead of a raster one) enabling fast and accurate transformations such as scaling, rotation, and dynamic morphing of laser curves as reported in [39] for monochromatic light. The piece-wise contour ${{\textbf{r}}_c}(\alpha) = \{{{\textbf{b}}_1}(\alpha),...,{{\textbf{b}}_N}(\alpha)\}$ of the object is used in Eq. (1) to create the CSL pulse [see the fourth step in Fig. 5(a)] as it has been previously described. The intensity and phase distributions (at $t = 0\;{\rm{fs}}$) of the numerically simulated contour-shaped pulse, shown in Fig. 5(a), indeed follow the outline of the object (letter “t”) as expected. The corresponding signal Eq. (1) to be encoded into the SLM is also shown in Fig. 5(b). As in the previous examples, the prescribed phase $\Psi (\alpha)$ is uniform for this contour-shaped pulse (with topological charge $m = 5$), and the corresponding time delay function $\tau (\alpha) = \dot \Psi (\alpha){\tau _g}(\alpha) \propto m{\tau _g}(\alpha)$ is displayed in Fig. 5(c). We have considered this configuration to show that a small time delay (up to 2.5 fs) can be also obtained spite of the complex shape of the curve. The theoretical intensity and phase distributions of the contour-shaped pulse are also displayed in Fig. 5(c), and they are in good agreement with the numerical simulation ones [Fig. 5(a)] (see also Visualization 6). Note that the wavefronts of the contour-shaped vortex pulse move at constant velocity ${v_w} = 206.9\;\unicode{x00B5}{\rm m}/{\rm{fs}}$ for the considered uniform prescribed phase.

These results underline the ability to create CSL pulses on demand for practical applications. Let us recall that a non-linear phase distribution can be also prescribed along any curve, including these piece-wise contour curves as well as the ones given by the expression Eq. (21), yielding more sophisticated pulse propagation dynamics if needed.

B. CSL Pulse Dynamics for 3D Curves

Here, we study the dynamics of the needle-shaped laser pulse, Eq. (18), as a representative example of CSL extended in the 3D space (around the focal plane). To show the control of its dynamics, let us consider a linear phase $\Psi (\alpha) = b{\omega _0}\alpha /c$ (with $b$ being a real number) prescribed along the target propagation line ${{\textbf{r}}_l}(\alpha) = ({\cos \phi ,0,\sin \phi})\alpha$, tilted by an angle $\phi$ with respect to the focal plane. Note that wave field Eq. (18) for the case $\phi = 0$ collapses to the one corresponding to the line-shaped laser pulse extended along the $x$-axis [see Eq. (20) and Fig. 3], as expected.

 

Fig. 6. (a) Instantaneous intensity distribution $(x - z$ section) of a needle-shaped laser pulse (with $b = 0$) for time $t = - 8500\;{\rm{fs}}$, $t = 0$, and $t = 8500\;{\rm{fs}}$. The pulse propagates like a 3D light bullet traveling along a target line tilted by an angle $\phi ={ 89.6^ \circ}$ (with respect the focal plane). (b) The average intensity distribution displayed in this panel reveals the resulting needle-shaped laser pulse. (c) Needle-shaped laser pulse (ratio $v/c$ versus $b$): The theoretical velocity ratio $| {\textbf{v}} |/c = 1/({\sin \phi + b})$ of the pulse peak intensity is plotted as a function of the parameter $b$, for angles: $\phi ={ 3^ \circ}$ (blue line), $\phi ={ 20^ \circ}$ (red line), and $\phi ={ 87^ \circ}$ (green line), along with the speed ratios obtained from the numerical simulation of the pulse propagation (see colored dots).

Download Full Size | PDF

The time delay for the needle-shaped laser pulse is $\tau (\alpha) = ({\sin \phi + b})\alpha /c$, and its phase is given by $\Phi ({\alpha ,t}) = ({\tau (\alpha) - t}){\omega _0}$. Thus, the spatiotemporal evolution of the pulse wavefront and of the peak intensity match, and they are driven by the temporal dependence $\alpha (t)$ obtained by solving $\tau (\alpha) - t = 0$. It is interesting to parameterize the curve in terms of the time $t$ just by introducing $\alpha (t)$ on ${{\textbf{r}}_l}(\alpha)$ yielding ${{\textbf{r}}_l}(t) = ({\cos \phi ,0,\sin \phi})c t/({\sin \phi + b})$. Hence, the pulse peak intensity and the pulse wavefront propagate at a constant velocity ${\textbf{v}} = {\partial _t}{{\textbf{r}}_l}(t) = ({\cos \phi ,0,\sin \phi})c/({\sin \phi + b})$. Thus, the dynamics of the needle-shaped pulse relies on tuning the value of $b$ (for fixed angle $\phi$). Note that the sign of the term $\sin \phi + b$ in the latter velocity expression allows for a co-propagating or counter-propagating pulse peak intensity along the target line ${{\textbf{r}}_l}(\alpha)$, illustrating the versatile control of the pulse peak intensity velocity.

To analyze the dynamics of the needle-shaped laser pulse, we consider the system parameters previously used in the case of 2D curves and the same Fourier-limited input femtosecond pulse. As an example, the instantaneous intensity distributions of the needle-shaped pulse for $b = 0$ and tilting angle $\phi ={ 89.6^ \circ}$ are shown in Fig. 6(a) for time $t = - 8500\;{\rm{fs}}$, $t = 0\;{\rm{fs}}$, and $t = 8500\;{\rm{fs}}$. The pulse peak intensity propagates preserving its spatial shape while it travels at constant velocity $| {\textbf{v}} | = 1.00002c$ along the target tilted line of length 10 mm . Interestingly, this type of pulse behaves like the light bullet reported in [12,54], but in our case the target propagation trajectory (a straight line ${{\textbf{r}}_l}(\alpha)$ extended in 3D space) is not restricted to the optical axis ($\phi ={ 90^ \circ}$). Figure 6(b) also displays the average intensity revealing the needle-shaped pulse tilted by an angle $\phi ={ 89.6^ \circ}$ (line of length 10 mm), as expected. The theoretical and numerical simulation results corresponding to the instantaneous intensity distribution and average intensity of the needle-shaped pulse are in good agreement [see Figs. 6(a) and 6(b), respectively]. In Fig. 6(c), the values of $| {\textbf{v}} |/c = 1/({\sin \phi + b})$ are plotted as a function of the parameter $b$ for different values of the tilting angle: small angle $\phi ={ 3^ \circ}$ (blue line), medium $\phi ={ 20^ \circ}$ (red line), and also large angle $\phi ={ 87^ \circ}$ (green line). For small values of the tilting angle $\phi$ [see Fig. 6(c), blue lines, $\phi ={ 3^ \circ}$], the interval of velocity values is wider than in the case of medium and large tilting angles. Indeed, for the case $\phi ={ 3^ \circ}$, both co-propagating ($58 \le | {\textbf{v}} |/c \le \infty$) and counter-propagating (${-}\infty \le | {\textbf{v}} |/c \le - 58$) pulse peak intensities can be obtained along the target line ${{\textbf{r}}_l}(\alpha)$. The velocity ratios $| {\textbf{v}} |/c$ obtained from the numerical simulation of the pulse propagation are also plotted in Fig. 6(c) (see blue, red, and green dots) and fit well to the theoretical velocity ratios. These findings illustrate the versatility of the proposed technique to create light bullets [12,54] with tailored velocity along a target 3D propagation trajectory.

4. EXPERIMENTAL RESULTS

A schematic diagram of the experimental setup used for femtosecond laser pulse shaping along arbitrary 3D curves is shown in Fig. 7. It comprises a programmable SLM (reflective phase-only LCOS, Holoeye Pluto, 8-bit phase level, pixel size of 8 µm) device for holographic encoding of the signal Eq. (1) required for pulse shaping and an optical parametric amplifier (TOPAS Prime-U, Spectra Physics) pumped by an amplified Ti:sapphire femtosecond laser (3.6 mJ per pulse at 1 kHz repetition rate) providing Fourier-limited input pulses of 50 fs with a central wavelength of ${\lambda _0} = 750 \;{\rm{nm}}$ (spectral width $\Delta \lambda = 30 \;{\rm{nm}}$ FWHM). The input laser pulse has been collimated (beam diameter of 11 mm) to illuminate the SLM displaying a computer generated hologram (CGH) [35] working as a phase-only diffraction grating (DG) that encodes the complex field amplitude of the polymorphic beam Eq. (1). Specifically, the encoded beam is reconstructed at the first diffraction order of the CGH, which is obtained at the focal plane of the focusing parabolic mirror as indicated in Fig. 7. This off-axis parabolic mirror works as a reflective lens suited for use with femtosecond pulsed lasers because it enables achromatic focusing of the laser without pulse stretching. Nevertheless, since the CGH is a type of DG, then the focused pulse suffers from angular spectral dispersion, which can be compensated by using an additional phase-only DG with the same grating period of the CGH. This compensation can be straightforwardly achieved by projecting back the modulated pulse (by using mirror M1) onto the SLM where the DG is displayed (see Fig. 7).

 

Fig. 7. Sketch of the experimental setup for femtosecond laser pulse shaping along arbitrary 3D curves. The collimated input pulse (50 fs, Fourier-limited) illuminates the screen of the SLM where the CGH is displayed for shaping of the input pulse. A knife prism mirror (KERAP, Knife-Edge Right-Angle Prism Mirror, MRAK25-P01 Thorlabs), mirror M1, and M2 (D-Shaped Mirrors, PFD10-03-AG Thorlabs) enable redirection of the modulated pulse. The parabolic mirror (focal length of 200 mm, MPD189-P01, Thorlabs) and M1 are used to project the spatially modulated beam onto DG for compensation of the angular spectral dispersion of the pulse. The achromatic lens FL1 (focal length of 200 mm, AC254-200-B-ML, Thorlabs) focuses the pulse onto the sCMOS camera. The intensity distributions of the first-order pulse (without angular spectral dispersion) measured at the focal plane of FL1 are displayed for a ring, square, and star-like curve (vortices with $m = 10$).

Download Full Size | PDF

In our case, the CGH and DG are displayed on the left and right sides of the SLM screen (see Fig. 7), respectively. This split-screen method is cost-effective because it requires just one SLM for angular spectral dispersion compensation instead of two paired SLMs, which are often used in holographic shaping of femtosecond lasers [23]. Note that the use of the mirror M1 allows us to redirect the pulse toward the DG while the parabolic mirror enables the optical projection of the pulse onto DG as a ${\rm{4 -}}f$ system (Keplerian ${{1}} \times$ telescope). All the optical elements of the proposed pulse shaping setup operate in reflection so that the pulse at the output of the system remains Fourier-limited (as the input pulse). This can be also achieved by using two SLMs instead of the proposed split-screen method by displaying the CGH onto the first SLM and the DG onto the second one. In such a case, the illuminated CGH is optically projected onto the second SLM by using the same ${\rm{4 -}}f$ system comprising two parabolic mirrors, which preserve the temporal duration of the input pulse as in the case of the split-screen method. The implementation of two SLMs is more expensive; however, it allows full screen display of the CGH and DG if needed.

The pulse modulated by the DG is finally redirected (by using the mirror M2) toward the achromatic lens (FL1, focal length of 200 mm), which focuses the pulse onto the sCMOS camera (Thorlabs CS505CU, 12-bit level, pixel size of 3.45 µm). Note that the angular spectral dispersion is compensated at the first diffraction order of the DG (see Supplement 1, Section 4). Thus, the CSL pulse is created at this first diffraction order in the focal region of the focusing lens FL1, without angular spectral dispersion. This is also referred to as the first-order pulse to distinguish it from the uncompensated zeroth-order counterpart.

This experimental configuration allows the generation of CSL pulses with a temporal duration close to the one of the input pulse ($T = 50\;{\rm{fs}}$), inside a reference circle of radius $B \simeq 300\;\unicode{x00B5}{\rm m}$ corresponding to a stretching ratio $\beta \simeq 1.025$ [and ${\rm{NA}} = 0.015$, see Eq. (10)]. We have chosen this configuration as an example to illustrate the experimental generation of femtosecond CSL pulses (first-order pulse), whose intensity distributions are shown in Fig. 7 for the case of a circle (radius $R = 230\;\unicode{x00B5}{\rm m}$), square, and starfish curves (embedded in the focal plane). A uniform phase distribution with a topological charge $m = 10$ has been prescribed along these curves.

To show the effect of the angular spectral dispersion, the measured intensity distribution of the first-order pulse for each curve is displayed in Figs. 8(a)–8(c) along with the corresponding one of the zeroth-order pulse. As it is observed, the zeroth-order pulse exhibits remarkably angular spectral dispersion arising from the DG (CGH) operating in the $x$ axis direction. In contrast, the first-order pulse is fully compensated, and it reproduces the CSL pulse as expected. Specifically, the first-order pulse has a stretching ratio $\beta = 1.025$ (a ratio of ${\sim}2.5 \%$ supporting its dispersion-free character) while the zeroth-order pulse has ${\beta _0} = 1.767$ (a ratio of ${\sim}77 \%$) that confirms its remarkably angular spectral dispersion (see Supplement 1, Section 4 for further details).

 

Fig. 8. Experimental results. Intensity distributions of the first-order pulse (without angular spectral dispersion) and zeroth-order pulse (with angular spectral dispersion) measured at the focal plane ($z = 0$) and corresponding to the following 2D curves: (a) ring, (b) square, and (c) starfish curve. The volumetric representations displayed in (d) and (e) correspond to the measured and numerically simulated intensity distributions for the square-shaped laser pulse (1st-order pulse) and starfish-shaped laser pulse (first-order pulse) propagated in the focal region of the focusing lens, respectively. The volumetric representation of the measured intensity of the needle-shaped laser pulse (first-order pulse) is displayed in (f) along with its $x - z$ section and the numerically simulated one.

Download Full Size | PDF

In the previous discussion, we have omitted the pulse stretching due to the considered refractive focusing lens (FL1) since this optical element does not form part of the proposed pulse shaping setup. Let us recall that a focusing lens with a proper NA should be carefully chosen to preserve the temporal duration of the CSL pulse inside the reference circle, as has been previously explained [see Eq. (10)]. For instance, an objective focusing lens with NA ${\sim}0.55$ (a typical NA value for long working distance dry objective lenses) is often required for laser micro-machining applications. Let us note that the pulse stretching due to the focusing lens FL1 can be easily pre-compensated by using a pulse compressor placed before the shaping system, or alternatively FL1 can be replaced by a reflective parabolic mirror so that the pulse duration remains unchanged under focusing. Nevertheless, the experimental results shown in Figs. 8(a)–8(c) for the first-order pulse are in good agreement with theoretical ones analyzed in the previous section.

The measured intensity distribution of the CSL pulse (first-order pulse) propagated in the focal region of the focusing lens FL1 is also displayed in Fig. 8(d) and 8(e) for the case of the square and starfish, respectively. This volumetric representation of the time-averaged intensity distribution of the pulse is also displayed for the corresponding numerically propagated pulses. The volumetric representation of the measured intensity distribution of the needle-shaped pulse (line of length ${\sim}9.5 \;{\rm{mm}}$) is displayed in Fig. 8(f) along with its $x - z$ section and the corresponding numerically simulated one (line of length ${\sim}10 \;{\rm{mm}}$). All these experimental results are in good agreement with the numerical/theoretical ones and show the strong focusing of a CSL pulse in distinct geometries in 3D, which is a heritage of the diffraction-limited focusing in the form of the light curve provided by the polymorphic beam in Eq. (1) [38].

A specifically designed interferometric setup is required to experimentally characterize the temporal evolution of the peak intensity and phase of the pulse along the target curve. This is a challenging technical task that is out of the scope of this work, and it will be a subject of our future research. Nevertheless, the good agreement between the theoretical results and the ones corresponding to the numerical simulation of the proposed experimental setup supports the spatiotemporal control of the peak intensity and phase of the pulse along the target curve.

5. DISCUSSION

Here, we have demonstrated that an ultrashort CSL pulse can be straightforwardly designed with a versatile control of the dynamics of its intensity and phase distributions along an arbitrary target trajectory ${{\textbf{r}}_c}(\alpha)$. We have found a closed-form expression that describes the spatiotemporal behavior of the pulse wave field and clearly shows the role played by the curve geometry and spatial phase $\Psi (\alpha)$ prescribed along the curve ${{\textbf{r}}_c}(\alpha)$. The control of the temporal evolution of both the peak intensity and phase of the pulse along the curve has been demonstrated by considered realistic examples including contour-shaped pulses created to follow the outline of objects. Interestingly, the spatiotemporal behavior of the pulse depends on a well-defined interplay between the prescribed spatial phase $\Psi (\alpha)$ and the curve geometry, which in turns allows us to increase the degrees of freedom in the control of the pulse dynamics. These findings pave a promising way to create sophisticated structured ultrashort laser pulses with easily engineered propagation dynamics tailored to the standing application. The peak intensity of the CSL pulse can interact with the material or particles along the target curve during the pulse duration. Thus, to the benefits provided by the ultrashort pulse duration, we add the programmable control of the pulse phase, time delay, and velocity of the peak intensity traveling along the curve. This provides new degrees of freedom of interest in the study of light–matter interactions such as electron acceleration, ultrafast laser machining, microscopy, and attosecond spectroscopy.

We have also demonstrated that CSL vortex pulses can be created for any curve geometry and phase gradient to obtain rich pulse dynamics along the target trajectory. In addition, a needle-shaped laser pulse with tunable titling angle, length, and velocity of its peak intensity has been demonstrated as an example of CSL extended in 3D. Interestingly, the behavior of the needle-shaped laser pulse is similar to 3D linear light bullets [12,54], which are also interesting as probes of light–matter interactions.

Another relevant advantage is that the developed technique allows the generation of a strongly focused CSL pulse exhibiting high intensity and phase gradients. This advantage is particularly attractive for applications such as laser micro-machining and material ablation. This diffraction-limited character of the laser curve and the control of the pulse dynamics along it open up new perspectives for the study of light–matter interactions including laser-driven particle acceleration. For instance, it has been reported that the phase $\Psi (\alpha)$ prescribed into a monochromatic CSL beam (a continuous wave diffraction-limited laser curve) is responsible for optical propulsion forces driving the transport of particles along the curve [28,30,47,48]. Thus, we may expect the development of more sophisticated laser-driven particle dynamics by using the proposed ultrashort CSL pulse.

The proposed experimental setup is compact and cost-effective because it only requires a single SLM for the holographic shaping of an input femtosecond laser pulse as a CSL pulse. The experimental results demonstrate the generation of femtosecond CSL pulses of different geometries that are in good agreement with the numerically simulated and theoretical predictions.

Both the developed theoretical framework and experimental technique have been explained in detail to provide a practical guide for easy engineering of the ultrashort CSL pulse with controlled pulse dynamics. We envision that the reported achievements and findings can prompt the future development of multiple applications demanding tailored structured laser pulses.