Ultra-intense lasers facilitate studies of matter and particle dynamics at unprecedented electromagnetic field strengths. In order to quantify these studies, precise knowledge of the laser’s spatiotemporal shape is required. Due to material damage, however, conventional metrology devices are inapplicable at highest intensities, limiting laser metrology there to indirect schemes at attenuated intensities. Direct metrology, capable of benchmarking these methods, thus far only provides static properties of short-pulsed lasers with no scheme suggested to extract dynamical laser properties. Most notably, this leaves an ultra-intense laser pulse’s duration in its focus unknown at full intensity. Here we demonstrate how the electromagnetic radiation pattern emitted by an electron bunch with a temporal energy chirp colliding with the laser pulse depends on the laser’s pulse duration. This could eventually facilitate to determine the pulse’s temporal duration directly in its focus at full intensity, in an example case to an accuracy of order 10% for fs-pulses, indicating the possibility of an order-of magnitude estimation of this previously inaccessible parameter.

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## Introduction

## Ultra-intense Laser Metrology

The development of lasers towards ever higher intensities, motivated by their utility as tools for studies of fundamental physics1,2 and applications such as particle acceleration3 or high-energy photon sources4,5, is driven by compressing their energy to ever shorter pulse durations. Several high-power facilities are already operating pulse durations of less than 30 fs6,7,8,9, with facilities under construction, aiming at pulse energies exceeding 100 J at less than 10 fs pulse durations10,11,12,13. In this parameter regime, novel physics are predicted or were readily observed, ranging from the broadening of emission spectra14 to the sub-cycle dependence of atomic ionization15 or even electron-positron pair production16,17. All these effects feature a delicate dependence on the laser’s spatial and temporal profiles. These, however, may distort unpredictably upon amplification and propagation18,19. Consequently, a thorough characterization of the laser field inside its focus, where the interaction takes place, is required. Due to material damage, on the other hand20, ultra-intense laser pulses cannot be sent directly through solid state devices, conventionally used to measure pulse durations21,22,23,24, energies25, spot sizes26 and sub-cycle field structures15,27,28,29,30,31,32 at lower intensities. The strong electromagnetic fields of ultra-intense lasers as considered in this work would immediately disintegrate such devices whence metrology has to be performed either far from focus or at attenuated intensities. On the other hand, attenuating the laser’s full intensity or using a secondary laser beam line of reduced intensity would amount to using indirect pulse characterization schemes. Such indirect schemes, however, should, at least in principle, be benchmarked by direct measurements, providing a signal originating directly from the unattenuated and fully focused laser pulse. The desire for direct laser characterization techniques, benchmarking the well-established indirect methods was consequently expressed in community meetings33 as well as by experts on experimental high-power laser science34. One possible solution is quantifying the emission patterns of electrons scattered from the laser focus, giving access to a direct determination of its carrier-envelope phase35,36 and intensity18, with the latter already implemented experimentally37. This latter experiment is in line with a series of recent experiments on laser-electron accelerators38,39,40 and all-optical radiation sources4,5,41,42,43, for which there exists an abundance of refined detectors for both the electrons and emitted radiation44. These schemes are based on the interaction of electrons (mass and charge m and e \({p}_{i}^{\mu }=({\varepsilon }_{i}/c,{{\boldsymbol{p}}}_{i})\), where c is the speed of light, with an ultra-intense laser pulse of peak electric field E0 and central frequency ω0, corresponding to a wavelength \({\lambda }_{0}=2\pi c/{\omega }_{0}\). Here, ultra-intense refers to lasers for which the dimensionless amplitude

exceeds unity, indicating that the electrons will be accelerated to relativistic velocities within one field oscillation. A relativistic electron \({\varepsilon }_{i}\gg m{c}^{2}\), scattered from an ultra-intense laser pulse \(\xi \gg 1\), emits radiation into a cone around its or the laser’s initial propagation direction with opening angle35,45

$$\delta \zeta \sim \{\begin{array}{cc}\frac{m{c}^{2}\xi }{{\varepsilon }_{i}} & {\rm{i}}{\rm{f}}\,{\varepsilon }_{i}\gg m{c}^{2}\xi \\ \frac{{\varepsilon }_{i}}{m{c}^{2}\xi } & {\rm{i}}{\rm{f}}\,{\varepsilon }_{i}\ll m{c}^{2}\xi \\ 1 & {\rm{i}}{\rm{f}}\,{\varepsilon }_{i}\approx m{c}^{2}\xi .\end{array}$$

On the other hand, the pulse duration, a highly relevant pulse parameter, cannot be determined at full intensity by either solid state devices or the electron scattering schemes mentioned above, indicating the lack of a possibility to benchmark indirect pulse duration measurements in ultra-intense lasers’ foci at full intensity.

Here we present a fundamentally novel approach to amend well-developed pulse metrology techniques operating at attenuated intensities. We show how an ultra-intense laser’s pulse duration can in principle be measured directly in its focus at ultra-high field strengths, providing an order-of-magnitude benchmark for conventional indirect pulse metrology. Its basic working principle is to imprint a temporal structure onto the electron bunch to be scattered from the ultra-intense laser pulse, in the form of an energy chirp as are common in laser-accelerated electron bunches and need to be mitigated with great effort46,47. Keeping the electron bunch chirped, the particles’ changing energy will lead to a temporal evolution of the angular range into which the electrons emit radiation. Here we quantify this temporal change by means of a simplified analytical model which is then benchmarked by numerical simulations, demonstrating that the pulse’s duration can be quantitatively inferred. To assess how feasible the proposed scheme is we corroborate our analysis by a thorough analysis of the relevant error sources affecting its experimental implementation.

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To extract the electrons’ radiation signal from the laser’s focal region and to decouple it from the strong optical laser radiation background, it is favorable to let the electrons collide with the laser pulse perpendicularly. We thus analyze the emission from an electron bunch propagating along the y-axis colliding with a laser pulse propagating along the positive z-axis and polarized along the x-axis (s. Fig. 1). In order to quantify the overall angular distribution of the bunch’s emitted radiation, we observe that at each time instant the radiation emitted by an ultra-relativistic electron is confined to a narrow cone around its instantaneous direction of propagation. Thus, we approximate the time-dependent direction into which radiation is emitted as the time-dependent electron propagation direction. Naturally, this approximation is limited for low-energy electrons, but from a numerical simulation, taking into account the exact angular distribution of an electron’s emission, we find it to be still reliable for energies as low as 5 MeV (s. below). We thus need to solve an electron’s equation of motion, which is generally involved in a laser field of arbitrary focusing. On the other hand, it was demonstrated that for schemes utilizing the emission of a laser-driven electron bunch a clear detection signal is provided by the boundaries of the angular radiation distribution18,35,37, determined by the strongest electron deflection, i.e., at the regions of highest field strength. Consequently, we focus on the electron dynamics close to the laser’s focus, which we assume to lie in the origin \((x,y,z)\equiv {\bf{0}}\) in the following. In this focal region the laser field is well approximated by a plane wave, provided the transit time of the electrons through the focal volume τt is longer than the pulse duration τL, so that they will not experience non-plane wave field contributions far away from the laser focus. The ratio of these quantities, which has to exceed unity, can be found as \({\tau }_{t}/{\tau }_{L}=2{w}_{0}/(c{\tau }_{L})\), where w0 is the laser’s spot radius and we assumed the electrons to propagate with the speed of light. We thus see that the plane wave approximation is valid only for not too tight focusing and correspondingly short laser pulses. It will be demonstrated, however, that for FWHM pulse durations down to τL ~ 10 fs the analytical model provides good agreement with a full numerical simulation. We assume the above condition to be satisfied and model the laser field as a plane wave with four-potential \({A}^{\mu }(\eta )=(m{c}^{2}\xi /|e|){\varepsilon }_{0}^{\mu }g(\eta )\), with \(\eta =t-z/c\), the pulse’s polarization vector \({\varepsilon }_{0}^{\mu }\) and temporal shape g(η). We quantify the electrons’ emission direction by the spherical coordinate angles θ and ϕ, with respect to the z-axis as polar axis and the x-axis as azimuthal axis. As we are interested in the boundaries of the angular region into which the electrons emit radiation, we only consider the deflection of the electrons from their initial propagation direction \(\delta \zeta ={\zeta }_{i}-\zeta \) with \(\zeta \in <\theta ,\varphi >\) and where the electrons’ initial propagation direction in the considered setup is given by \({\theta }_{i}={\varphi }_{i}=\pi \mathrm{/2}\). It can be shown that in the ϕ-θ plane the emission is confined to a rectangular emission box in the plane spanned by the emission angles \((\theta ,\phi )\in (<{\phi }_{i}-|\delta \phi ({t}_{0})|,{\phi }_{i}+|\delta \phi ({t}_{0})|>,<{\theta }_{i}-\delta \theta ({t}_{0}),{\theta }_{i}>))\), where t0 is the time at which the angular deviations are maximal (s. Methods). The azimuthal deflection δϕ points into both halfspaces δϕ > 0 and δϕ δθ in a plane wave, on the other hand, always points towards the laser’s propagation direction, resulting in δθ > 0. The advantage of analyzing the emission box rather than only one of the angles is that the emission box is sensitive to the electron dynamics in a broader parameter range than just one of the angles. For a laser pulse with a typical Gaussian shape in time \(g(\eta )=\exp <\,-\,2{(\eta /{\tilde{\tau }}_{L})}^{2}>\) with the physical and scaled FWHM pulse durations τL and \({\tilde{\tau }}_{L}={\tau }_{L}/\sqrt{\log \,\mathrm{(2)}}\), respectively, an electron bunch with constant energy \({\varepsilon }_{i}\) yields \({t}_{0}\equiv 0\), i.e., emission towards the maximal emission angles originates from the regions of highest field strength, i.e., the pulse’s temporal and spatial center. For an electron bunch with a linear energy chirp, on the other hand, due to the time dependence of the electrons’ energy, contrary to the naive guess, t0 instead increases for increasing pulse durations τL. Consequently, as \(\delta \phi ({t}_{0}),\delta \theta ({t}_{0})\) explicitly depend on τL, the emission box’s boundaries facilitate to determine the laser’s pulse duration. To quantitatively exemplify the dependency of the emission box’s boundary angles on τL we consider the interaction of a moderately relativistic electron bunch \({\varepsilon }_{i}/m{c}^{2}=10\). We assume the bunch to have a large energy spread \(\Delta \varepsilon ={\varepsilon }_{i}\) and to be short \({\tau }_{E}=55\) fs, as can be typical for an electron bunch from a laser-accelerator not optimized for monochromaticity38,39,48. The laser is assumed to have a dimensionless amplitude of ξ = 10 at a central frequency \(\hslash {\omega }_{0}=1.55\) eV. For these parameters quantum electrodynamical (QED) effects like single-photon recoil of the electron are suppressed by a factor \(\chi ={\varepsilon }_{i}\xi \hslash {\omega }_{0}/{m}^{2}{c}^{4} \sim {10}^{-4}\) 35,49 and are hence neglected. Plotting the resulting changes of the emission box’s cutoff angles (s. Methods) we find a clear dependence on the laser’s pulse duration (s. Fig. 2). Thus, determining the boundary angles of the emission box emitted by a chirped electron bunch scattering from a relativistically intense laser pulse allows to determine the pulse’s duration.