Ultrafast XUV emission from laser wakefields in underdense plasma

First, let us start by presenting the results on XUV attosecond emission from one-dimensional (1D) PIC simulations with the code KLAP-1D [31–32]. In the PIC simulations, we take a laser pulse with an amplitude profile a = a_L sin² (π t/τ ) , where a_L = 10 is the normalized peak amplitude corresponding to a laser intensity of 1.37 × 10²⁰ W cm^{- 2} at a laser wavelength of λ_L = 1 μ m , and τ = 10T_L is the full laser period with T_L being the oscillating period of the incident laser. We assume that the laser pulse is linearly polarized along the z-direction and propagates along the x-direction in uniform plasma with a density of n_e /n_c = 0.04 , where n_c is the critical density. The left plasma boundary is located at x = 20λ_L and the plasma length is 200λ_L . We chose the cell size of λ_L /200 to ensure enough resolutions in time and space under which we can observe the attosecond pulse emission properly. This is due to the fact plasma wave-breaking is a highly nonlinear process and the trapped electron bunch structure is sensitive to the space and time resolution in the simulation.

Figure 1 illustrates the evolution of the laser pulse and its wakefield (shown by the electron density), as well as the subsequent attosecond pulse generation. In figure 1(a), a high electron density spike is shown behind the laser pulse, which is due to the excitation of the high-amplitude laser wakefield by the ponderomotive force and subsequent electron trapping. At a later time t = 100T_L , the laser pulse front steepens, which also drives a density peak at the pulse front, as shown in figure 1(b). At t = 150T_L , the laser pulse shortens significantly and the density peak at the front of the laser pulse is still there. But the density spike behind the laser pulse disappears, as shown in figure 1(c). During the laser pulse evolution, one may note that an extremely short pulse emerges near the density spike behind the driving laser pulse. Figures 1(d)–(f) display zoomed-in plots of the extremely short pulses from figures 1(a)–(c). One may note that its amplitude seems to increase with time and its duration also increases from sub-cycle to about one and a half cycles from t = 50T_L to 150T_L . At the latter time, the pulse has a duration of about one-seventh of the incident laser period or about 460 attoseconds.

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In figure 2, we plot the attosecond pulse in vacuum after passing through the plasma region of about 200λ_L and its frequency spectrum (shown by black solid lines) under the same laser and plasma conditions as in figure 1. Even though the whole duration of the attosecond pulse is about T_L /7 , its frequency spectrum extends to over 70ω_L . This implies that one may get attosecond pulses with the duration around T_L /70 when the low-frequency components are filtered out. Note that the normalized peak field is close to 2.0, corresponding to an intensity over 4 × 10¹⁸ W cm^{- 2} .

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We have conducted a series of simulations by changing the plasma density, the laser duration and the laser intensity. It is found that the attosecond pulse generation is a robust process as long as highly nonlinear wakefields are excited and the driving laser is self-modulated and steepened significantly. Therefore, it usually requires a laser intensity in excess of 10¹⁹ W cm^{- 2} , while ours is 4.2 × 10¹⁹ W cm^{- 2} for a_L = 10 and n = 0.04 . In this case, the produced laser wakefields just break after the first wave bucket and the attosecond pulse emerges in the first wave bucket. In figure 2, the attosecond pulses produced with different pulse durations at different plasma densities are shown together with their spectra. It shows that as long as the attosecond pulses are produced, their peak amplitudes do not change significantly with the laser and plasma parameters and their durations are typically around a few hundreds of attoseconds. The pulses have ultrabroad spectra in the XUV range with the spectrum widths, decreasing slightly with increasing plasma densities. One also expects that the duration of the attosecond pulses depends upon the pump laser intensity and plasma density. For example, for the laser parameters a_L = 10 and τ = 10T_L and the plasma density n_e /n_c = 0.04 , one observes the attosecond pulse roughly with 3/2 cycles as shown in figure 2. However, one observes an attosecond pulse with nearly five cycles when taking a_L = 40 , apparently due to the fact that the driving laser can propagate a longer distance before it is depleted.

Since the attosecond pulse observed in the simulation is coherent emission, it is related with a transverse electron current. Simulation indicates that it is produced by the current produced near the density spikes of the wakefield behind the laser pulse. In figures 1(d)–(f), we plot snapshots of the transverse current distributions, which are located near the attosecond pulse. Actually, they co-move with the electron density spike. Firstly, according to the continuity equation of the electron fluid, the electron density can be described as n = n₀ /(1 -v/v_g ) , where v_g is the group velocity of the laser radiation in the ambient plasma. A density spike appears around the positions where electrons move with a velocity close to v_g . If one assumes that the electron density spike can be described by n_spike = δ (x - v_g t) , then the transverse current can be written as

$$\begin{equation}j_{z,{\rm{spike}}} = - ev_z \delta (x - v_{\rm{g}} t),\end{equation} \tag{ 1 }$$

where v_z is the slowly varying transverse electron velocity near the density spike along the laser polarization direction to be discussed later. As shown in figure 1(c), the electron density spike disappears at t = 150T_L , because the driving laser pulse has been depleted significantly and the laser wakefield decreases. Then the transverse current disappears correspondingly as shown in figure 1(f). Afterwards, the attosecond pulse is found to propagate in underdense plasma without visible changes for a long distance. It is thus clear that the transverse currents related with the electron density spikes are responsible for the attosecond pulse generation. In the following, we shall show how the transverse electron velocity v_z is produced in the z-direction or the laser polarization direction.

Figure 3(a) displays snapshots of longitudinal distributions of the laser field and the transverse kinetic momentum of electrons p_z in the laser polarization direction. Before t = 50T_L , one observes that p_z has not vanished only inside the laser pulse, which are basically described by $p_z = a_z = - \int_0^t {E_z \,{\rm{d}}t}$ following the canonical momentum conservation, where a_z is the normalized vector potential. Behind the laser pulse, one finds that p_z = a_z ≈ 0 . At later times, one finds that p_z and a_z do not vanish behind the laser pulse. Instead, they take a single finite value at a given longitudinal position and are distributed in a wide space behind the laser pulse. In the following, we call the finite kinetic momentum of electrons behind the laser pulse as the residual momentum. Note that the change in the residual momentum with the space coordinate is much slower than that found inside the laser pulse; the latter changes in a period of the laser wavelength. The occurrence of this residual momentum is related with the violent evolution of the laser pulse front with time, which includes the changes in the carrier phase of the laser pulse and the pulse front steepening. These are caused by two well-known nonlinear effects, namely the relativistic nonlinearity and the ponderomotive force of the laser pulse. According to the refractive index of a homogeneous plasma, η = (1 - γ^{- 1} ω_p² /ω_L² )1/2 where γ is the relativistic factor and ω_p = (4π n₀ e² /m_e )^1/2 is the electron plasma frequency; the relativistic nonlinearity changes the refractive index in such a way that the high-intensity part of the laser can propagate at a high speed. The ponderomotive force at the pulse front drives a density peak, which induces a frequency down-shift of the pulse front [15]. Both effects tend to produce pulse front steepening and carrier phase changes. At t = 150T_L , the laser pulse is depleted to nearly a single cycle, and the residual momentum becomes negative behind the pulse.

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Figure 3(b) illustrates the evolution of the residual electron momentum at a distance about 6λ_L behind the laser pulse. It shows that the residual momentum begins to build up after t = 50T_L due to strong pulse evolution. They even change with time periodically and the time scale for this periodic change is roughly given by 2π ω_L /ω_p² , which is the typical time scale of self-modulation. It is clear that the slowly varying residual transverse momentum and the electron density spikes behind the laser pulses are responsible for the formation of the fast-moving narrow current sheet, which emits attosecond pulses.

In the following, we give a simple estimation of the attosecond pulse amplitude in terms of the electron current. For the moment, one can neglect the oscillation of the transverse velocity v_z because it varies slowly as compared to the density spike. Then according to equation (1), the transverse current can be considered as a function of τ = t-x/v_g only. Substituting it into the wave equation

$$\begin{eqnarray*} &&({{\partial ^2 }/{\partial x^2 }} - {{c^{ - 2} \partial ^2 }/{\partial t^2 }})E_z = ({{4\pi }/{c^2 }}){{\partial j_{z,{\rm{spike}}} }/{\partial t}},\end{eqnarray*}$$

the emitted pulse is given by

$$\begin{equation}E_z = 4\pi \gamma _{\rm{g}}^2 \frac{{v_g^2 }}{{c^2 }}\int_{ - \infty }^\tau {j_{z,{\rm{spike}}} (\tau ')\,{\rm{d}}\tau '} ,\end{equation} \tag{ 2 }$$

where r_g = (1 - v_g² /c² )^{- 1/2}. Further assuming that the density spike is described by δ = n_max  sin [(ω_atto (t -x/v_g )] with 0 ⩽ ω_atto (t -x/v_g ) ⩽ π , where n_max is the peak of the density perturbation, ω_atto is the frequency of the attosecond pulse, whose estimation will be given in the following, and v_g /ω_atto is the full-width of the density spike, the current is given by

$$\begin{eqnarray*}&&j_{z,{\rm{spike}}} = - j_0 \sin [\omega _{{\rm{atto}}} (t - {x/{v_{\rm{g}} }})]\end{eqnarray*}$$

with j₀ = ev_z n_max . From equation (2), it gives

$$\begin{eqnarray*}&&E_z = - ({{4\pi j_0 }/{\omega _{{\rm{atto}}} }})({{\gamma _{\rm{g}}^2 v_{\rm{g}}^2 }/{c^2 }})\{ \cos [\omega _{{\rm{atto}}} (t - {x/{v_{\rm{g}} )] - 1\} }}.\end{eqnarray*}$$

Normalizing E_z by m_e ω_Lc/e and normalizing the current j₀ by en_c c , one finds that the normalized field amplitude of the attosecond pulse is given by

$$\begin{equation}\frac{{E_{z0} }}{{{{m_{\rm{e}} \omega _{\rm{L}} c}/e}}} = \frac{{\omega _{\rm{L}} }}{{\omega _{{\rm{atto}}} }}\frac{{\gamma _{\rm{g}}^2 v_{\rm{g}}^2 }}{c}\frac{{j_0 }}{{en_{\rm{c}} c}} \approx \frac{{\omega _{\rm{L}} }}{{\omega _{{\rm{atto}}} }}\frac{{\omega _{\rm{L}}^2 }}{{\omega _{\rm{p}}^2 }}\frac{{j_0 }}{{en_{\rm{c}} c}},\end{equation} \tag{ 3 }$$

where we have assumed v_g = (1 - ω_p² /ω_L² ) . The estimation given in equation (3) agrees qualitatively with the numerical results shown in figures 1(d) and (e). According to j₀ = ev_z n_max , it shows that attosecond pulse amplitude depends mainly upon the density spike value n_max , and depends weakly upon the driving laser intensity since v_z ∼ 1 at maximum for most electrons as long as the wakefield is driven in the highly nonlinear regime or the wave-breaking regime.

The above simple theoretical model is subjected to a few assumptions: the density spike with the fixed width v_g /ω_atto and the constant transverse velocity v_z . In the fluid model, the density spike height and its width change with the longitudinal velocity v_x of electrons according to n/n₀ = 1/(1 - v_x /v_g ) , following the continuity equation. Then the higher the density spike n_max , the narrower its width v_g /ω_atto should be. However, the real situation is more complicated in the highly nonlinear or wave-breaking regime, where the fluid model is not applicable and particle trapping appears. Furthermore, since the transverse velocity changes sign continuously with time according to figure 3(b) and the corresponding transverse current also changes sign with time, this results in attosecond emission in a few cycles. When the density spike disappears after about t = 130T_L , the transverse current also disappears and the emission no longer grows, although the residual transverse momenta continue to change sign with time before the laser energy is fully depleted. Thus, one can understand that the front part of the attosecond emission shown in figure 1(f) is from a high density spike with a small width and the trailing part is from a relatively low density spike with a large width.