Terawatt-scale optical half-cycle attosecond pulses

Extreme-ultravoilet (XUV) attosecond pulses with durations of a few tens of attosecond have been successfully applied for exploring ultrafast electron dynamics at the atomic scale. But their weak intensities limit the further application in demonstrating nonlinear responses of inner-shell electrons. Optical attosecond pulses will provide sufficient photon flux to initiate strong-field processes. Here we proposed a novel method to generate an ultra-intense isolated optical attosecond pulse through relativistic multi-cycle laser pulse interacting with a designed gas-foil target. The underdense gas target sharpens the multi-cycle laser pulse, producing a dense layer of relativistic electrons with a thickness of a few hundred nanometers. When the dense electron layer passes through an oblique foil, it emits single ultra-intense half-cycle attosecond pulse in the visible and ultraviolet spectral range. The emitted pulse has a peak intensity exceeding 1018 W/cm2 and full-width-half-maximum duration of 200 as. The peak power of this attosecond light source reaches 2 terawatt. The proposed method relaxes the single-cycle requirement on the driving pulse for isolated attosecond pulse generation and significantly boosts the peak power, thus it may open up the route to new experiments tracking the nonlinear response of inner-shell electrons as well as nonlinear attosecond phenomena investigation.

generated ultra-intense half-cycle pulse is present with the second low-density gas cell (n e2 region). The high-intensity unipolar half-cycle pulse maintains its temporal structure in low-density gas and accelerates background electrons to relativistic energies with asymmetric angular distribution. The half-cycle pulse can be verified through the energy and emitted angle measurement of accelerated electrons.
The fundamental scheme of the ultra-intense half-cycle attosecond pulse generation was studied using the two-dimensional particle-in-cell simulation code VORPAL 20 . Simulation results indicate that the optical attosecond pulses were generated with different laser-target parameters. Here we take one laser-target example to describe the generation of ultra-intense optical half-cycle pulse, and the simulation parameters are present here. A moving simulation window with a size of 50 μm × 60 μm in the x-y plane is divided into 15000 × 1200 cells, where the time step of . c 3 3 nm/ is fine enough to resolve the attosecond features of the radiated pulse. The employed underdense plasma has a homogenous electron distribution with a density of = × − n 1 10 cm e1 20 3 . A p-polarized laser pulse along the y-axis propagates from the left boundary of the simulation box along x-axis. The typical employed incident laser pulse has a temporal profile of . Here E 0 is the peak incident laser field, ω 0 is the central laser frequency, e and m e are the electron charge and mass, respectively, and c is the light speed in vacuum. The laser shaping process in the underdense plasma depends on the initial laser pulse duration, where it takes more time for the case of long pulse duration 18 . To minimize the computation time in the high-resolution runs, we chose a relatively small initial laser pulse duration of 8 fs. The laser pulse has a focal spot size of 15 μm, around 5 times larger than the plasma wavelength in the n e1 plasma. The oblique flat foil target is located at x = 45 μm with a thickness of d = 2 μm and bias angle of θ =°45 . It is fully ionized with an electron density of 3 shapes the initial multi-cycle laser pulse to a steep laser pulse and produces a dense layer of relativistic electrons, which is essential for isolated ultra-intense attosecond pulse generation. When the femtosecond laser pulse propagates in the n e1 plasma, the laser front is continuously etched backward and is steepened gradually, and this effect is referred to as the laser shaping effect 18 or self-etching effect 21 . Therefore, after propagating for a certain distance in the underdense plasma, the laser shaping effect will cut off the laser front, resulting in a step-like pulse. For our simulation case, the initial pulse duration of the employed laser pulse is 8 fs, the electric field of the resulted step-like pulse is plotted in Fig. 2a. Meanwhile, the laser pulse maintains its high intensity (Ẽ E / 1) y 0 because the self-focusing effect compensates the laser energy depletion, here E 0 is the initial peak laser amplitude. Such a step-like intense pulse poses a longitudinal ponderomotive force so strong in the first half laser cycle that it pushes up a huge number of background electrons and generates a density spike in that position, i.e. a highly-dense electron layer is formed 21 . Figure 2a displays the density plot of the electron layer along the longitudinal direction and it shows that the density spike has a peak value of 0.7 n c with a narrow full-width-half-maximum (FWHM) thickness of 320 nm. Moreover, this electron layer has good transverse uniformity over a length of more than 10 μm because of the large laser focal spot size. The electron layer carries a total charge of more than 3 nC assuming that the electron layer is symmetric in the transverse y-z plane. As the initial laser peak intensity and its focal spot size get matched in the underdense plasma, the formed dense electron layer keeps its narrow thickness and high-density peak for a propagation distance of a few tens of micrometers since self-focusing effect and energy depletion will maintain the laser peak intensity and transverse size for a relatively long distance 22 . The electron layer formation depends on the laser pulse evolution in the n e1 gas target and requires high-intensity of initial laser pulse.
When the dense electron layer forms, it is positioned in the laser front and propagates along with the laser pulse. These electrons inside the layer are directly accelerated by the driving laser pulse, where the electron motion follows the single-electron picture in the local laser electric field 23 . The longitudinal momentum is and transverse momentum is P m c a / y e . When arriving at the front side of the oblique foil, the laser pulse is reflected. There is an angle of θ 2 between the initial and reflected laser pulses. The reflected laser Figure 1. Physical scheme of half-cycle attosecond pulse emission and detection. A relativistic multi-cycle laser pulse shots into a gas-foil target (n e1 gas cell and an oblique foil) and produces isolated ultra-intense half-cycle attosecond pulse. The generated attosecond pulse will be detected based on the second gas cell (n e2 region) through the measurement of emitted electrons. pulse propagates along the −y axis for the case of θ =°45 . The foil causes the wave break and injection of this thin dense electron layer. At the same time, the laser pulse gets reflected by the foil surface and thus there is no laser-driven wakefield for the injected electron layer. Therefore the electron layer moves forward freely and passes through the foil target. As we consider one electron inside the relativistic electron layer, it interacts with the reflected laser for a distance of about λ /4 0 until it reaches the foil front surface. During the interaction, its transverse momentum decreases from P m c a / . Figure 2b shows the P y spatial distribution of the electrons inside the layer before and after the laser pulse is reflected. One can see the single electron layer as well as the oblique foil target, which is marked by the dash line. At that time, the upper-part (y < 1.7 μm) of laser pulse has already been reflected by the oblique foil while the down-part (y < 1.7 μm) is still approaching the foil. At the front surface of the foil, the transverse momentum P y of the electron inside the layer drops from , which is in good agreement with the calculated value. It should be also noticed that the electron's longitudinal momentum increases significantly from . P m c / 29 5 x e to P m c / 40 x e after interaction with the reflected laser pulse. When the dense electron layer finally reaches the foil, these electrons inside the layer have large longitudinal momenta and a transverse momenta of θ P m c / t an y e . Therefore it will trigger an intense transverse current 24 . The current is however screened inside the foil target. When it appears at the rear side of the foil, this transverse current of the electron layer emerges again and emits half-cycle pulse radiation 15 . The emitted pulse has only half cycle along the + y axis, as same as the electron transverse momentum  phase-stabilized because its amplitude has only a half cycle in + y direction, and thus does not require extra phase stabilizing processes in the potential applications 1 .
The generated half-cycle attosecond pulse has a broad spectrum, plotted in Fig. 2d, which mainly covers the visible and part of the ultraviolet range. The radiation spectrum extends up to ω 35 0 , limited by the longitudinal resolution and its amplitude has a slow decreasing until the rollover frequency ω rs of approximately ω 5 0 . The high-frequency radiation above ω rs is not coherent because the electron layer has the longitudinal thickness of a few hundred nanometers. Therefore, the radiation intensity above ω rs decreases rapidly with increasing frequency. The half-cycle attosecond pulse radiation is mostly in the visible and nearby spectra range, and thus, it is referred to as an optical attosecond pulse. In our method, the peak intensity of the optical attosecond pulse is several orders of magnitude more intense than that generated in the attosecond light-field synthesizer 16 . Such an intense attosecond pulse will make it possible to study the ultrafast electronic quantum dynamics on atomic scale and other attosecond high-field physics.
The radiation intensity of optical attosecond pulse strongly depends on the foil angle θ as well as peak electron density and the high energy of the dense electron layer ( γ θn sin e ). In the normal incidence case of θ = 0, the electrons inside the layer have zero transverse momenta when passing through the oblique foil and thus there is no radiation 25 . The simulation also confirms that there is no visible radiation signal at the case of θ = 0, which indicates that the transition radiation of electron layer plays ignorable role in radiation of attosecond pulse. As the foil angle increasing, the transverse momentum of relativistic electrons inside the layer increases and thus the half-cycle pulse intensity increases. The electron layer density plays a critical role in the radiation intensity, where the dense electron layer is resulted from the large longitudinal pondermotive force of the steep laser pulse. Both the initial laser intensity and focal spot size have to be matched to underdense plasma to optimize the electron layer formation. The radiation amplitude of emitted attosecond pulse also relies on the relativistic energy of the electron layer. During momentum modulation by the reflected laser, the electron longitudinal momentum P m c / x e increases significantly for the p-polarized incident laser case while it is constant for the s-polarized laser case (See electron momentum evolution of electron layer in the supplementary materials). Employing the p-polarized incident laser will enhance the radiation intensity. Additionally, the p-polarized laser could produce a smaller pulse duration of the attosecond pulse owing to the higher longitudinal momenta of the electrons inside the layer. We also learned from the simulation that the intense half-cycle attosecond pulse is generated with well-structure shape and short pulse duration at the case of foil thickness increasing to 15 μm. Therefore the foil target can be very thick in our proposed mechanism, where the foil is strong enough to handle the large gas pressure in experiments. Moreover, such a thick foil will not allow to transmit the initial laser pulse and the radiation of laser-foil interaction is not able to pass through the thick foil. The emitted half-cycle pulse gradually separates from the electron layer when they exits the foil as Coulomb expansion of the electron layer will cause a fast decline of the electron density. The half-cycle attosecond maintains its temporal structure in vacuum for a long distance. A remarkable feature of the half-cycle pulse in our method is that the optical half-cycle pulse is highly-collimated with a large transverse size. The generated half-cycle pulse has a transverse size of around 8.6 µm since the electron layer keeps its electron density and narrow thickness in a large transverse range of 10 µm. Figure 3a presents the spatial distribution of the half-cycle pulse at different time steps, the oblique target is also plotted. The half-cycle pulse detaches from the foil target at t = 70 μm/c with a peak field of 4.4 × 10 12 V/m and a FWHM transverse size of 8.6 µm. After a propagation distance of 140 µm, the transverse size slowly increases to 10.6 µm, as plotted in Fig. 3b. Additionally, the peak amplitude of the attosecond pulse decreases by only 27% and its pulse duration slightly decreases and then keeps stable at 200 as. It suggests that the half-cycle pulse is highly collimated with a full divergence angle as small as 0.95°, which is crucial for providing intense on-target beam intensities of the attosecond pulse in future applications. The emitted half-cycle pulse carries a total energy of 0.6 mJ, which corresponds to . × − 3 3 10 5 of the incident laser energy. Therefore, it reaches a peak power of 2.1 TW. High-intensity attosecond pulse with a small duration of 200 as is suitable for new experiments demonstrating nonlinear time-resolved attosecond dynamical process.
The strong electric field of the ultra-intense attosecond pulse can immediately ionize light atoms (for example, helium gas) and accelerate electrons to relativistic energies. The second gas cell, as shown in Fig. 1, produces a low-density n e2 gas target for ultra-intense half-cycle pulse detection. The dispersion effect of low-density plasma onto the half-cycle pulse is negligible, and thus its temporal structure remains. Since the unipolar half-cycle attosecond pulse is always positive in the time domain ( > E 0 y ), the transverse momenta of excited electrons should be accelerated along the −y direction, and therefore the emitted energetic electrons will have a certain angle with the attosecond pulse propagation direction (x-axis). The energetic electrons will pass through the thin gas cell wall and be detected, therefore the half-cycle structure of optical attosecond pulse will be verified through the emitted direction measurement of accelerated electrons. We have two gas plasmas of different densities (n e1 and n e2 ) and a tilting plasma foil (n e,f ) in single simulation for attosecond pulse detection, as shown in Fig. 1. The parameters of plasma n e1 and the foil are taken as same as the case of θ =°45 . The low-density plasma is distributed at − μ 60 160 m along the x-axis with a homogenous density of = × − n 1 10 cm e2 17 3 . In this simulation, the three plasmas are recorded in three different data files so that we can know that which plasma the energetic electrons are coming from. When the half-cycle pulse propagates at x = 95 μm, the maximum transverse momentum of accelerated electrons reach − P m c / 2 0 y e , and the longitudinal momentum P m c / 15 x e . Figure 4a displays that the accelerated electrons have a thermal distribution of = .
T 3 3 MeV e with the cut-off energy of 13 MeV, proving again the extremely high intensity of the attosecond pulse. These high-energy electrons will easily come out of the low-density plasma and be detected. The generation of ultra-intense attosecond pulse makes the study of attosecond relativistic high-field physics possible. An important feature of these energetic electrons is that the P y distribution is asymmetric. It is restricted to the −y domain. The angular distribution of electrons with the energy above 1 MeV in Fig. 4b shows the accelerated electrons are mainly emitted at ϕ = − .°54 2 , where ϕ =°0 is the attosecond pulse propagation direction (x-axis). This specific emission angle relies exclusively on the field structure of the half-cycle attosecond pulse, thus hinting at a new method to experimentally verify the proposed mechanism. The emitted angle ϕ strongly relies on the peak amplitude of half-cycle attosecond pulse E y ,max and weakly depends on the half-cycle pulse duration. Therefore, the second low-density gas cell with helium gas is present for the detection of the generated ultra-intense optical attosecond pulse. Such high-intensity attosecond pulse provides sufficient photon flux to ionize inner-shell electrons of heavy atoms, offering the ability to observe relativistic inner-shell process. The second gas cell can be also used to study the inner-shell electron dynamics when it is filled with heavy-atom gases.
In summary, a novel method is proposed to generate an isolated ultra-intense half-cycle attosecond pulse with a pulse duration of 200 as based on a relativistic multi-cycle laser pulse irradiating a gas-foil target. The generated attosecond pulse in the visible and near-visible spectral ranges reaches a peak power of 2.1 TW. The method fundamentally releases the strict requirement for the high-contrast laser pulse in the laser-solid interaction 15,26,27 , and the relativistic multi-cycle laser pulse is currently available. Here we have verified the physical scheme by taking one laser-target example through the simulation work and the next step is the experimental demonstration. The gas-foil target in proposed method can be realized based on the specially-designed gas cells and replaceable foils in future experiments. Moreover, a simple way is present to measure the produced half-cycle pulse structure. The high-intensity optical attosecond pulse with a small divergence of 0.95° will provide sufficient photon flux on target to ionize heavy atoms and initiate strong field process. Therefore, this intense optical attosecond pulse generation method offers the ability to investigate nonlinear attosecond optics such as electron correlation as well as ultrafast dynamics inside the atom and molecules 5,28 .