Obtaining Intense Attosecond Pulses in the Far Field from Relativistic Laser-Plasma Interactions

In this paper, we show that the Gouy phase shift plays a key role in the far-field waveform evolution of the reflected harmonic radiations from plasma surfaces driven by a relativistic Gaussian laser. With a proper adjustment of laser focal position away from the plasma surface, the inherent separations between the peaks of different harmonic carrier waves as well as the fundamental wave due to different wavelengths can be cleared away when they propagate from near to far field, since they experience the same Gouy phase shift of $\pi /2$. Using this method, intense attosecond pulses can be obtained in the far field with no need of any spectral filters. Three-dimensional particle-in-cell simulations show that far-field attosecond pulses with intensity of $4\times {10}^{15}\mathrm{W}/{\mathrm{cm}}^2$ ($2.56\times {10}^{17}\mathrm{W/sr}$; 65 times increase) and duration of $76\mathrm{as}$ (50% decrease) can be obtained by lasers at intensities of ${10}^{21}\mathrm{W}/{\mathrm{cm}}^2$. Such brilliant pulses with fully reserved spectra significantly benefit applications in attosecond science.

Figure 1

(a) Schematic for electric field evolution from the near to far field of the RP (green arrow) in RHHG, where a Gaussian laser (blue arrow) is incident on a solid target (gray block) and an optically curved PM (purple block) is formed. The temporal profiles of $E_y$ on the $x$ axis of the RP in the near (green) and far (red) fields, obtained from 3D PIC simulations with laser focal position $x_f=b{x}_R$ and preplasma length scale $L=0.2{\lambda }_L$, are plotted. (b) Temporal profiles of $E_y$ on the $x$ axis of the RP in the near (green) and far (red) fields for the case $x_f=0$ and $L=0$, where the inset (blue) shows $E_y$ in the far field after shifting back a phase $\pi /2$. (c) The harmonic spectra for the cases in (b). (d) Temporal profile of $E_y$ (red) of the far-field RP in the simulation of (a) but for $x_f=0$, where the black dashed line shows the fundamental component. Note that all values of $E_y$ are normalized by the maximum $E_y$ (red) in (d) and other parameters are described in the text.

Figure 2

(a) On-axis far-field electric field $E_y$ of high harmonics (solid, $n>1$) and the fundamental frequency (dashed) in RP in 3D simulations, where $x_f=0$ (green, for which the value is magnified by 8 times) and $x_f=b{x}_R\approx 49.4{\lambda }_L$ (red) are taken, respectively. The inset shows the near-field results. (b) On-axis far-field $|E_y{|}^2$ for $L=0.1{\lambda }_L$ (red) and $0.2{\lambda }_L$ (green) when $x_f=b{x}_R$ (solid) and $x_f=0$ (dashed). (c) Dependences of the Gouy phase shift $\mathrm{\Delta }{\varphi }_{Gn}$ and beam waist $w_{fn}$ on $x_f$ for the 5th-order (dashed when ${\beta }_5=0.8$, shading when $0.4<{\beta }_5<1$) and 31st-order (solid when ${\beta }_{31}=0.5$, shading when $0.4<{\beta }_{31}<1$) harmonics, where the curved lines are from Eqs. (4) and (5) and triangle and circle symbols represent simulation results. (d) Similar to (c), but for those depending on $L$, where $x_f=b{x}_R$.

Figure 3

(a) Divergence angles ${\theta }_n={\lambda }_L/n\pi w_{fn}$ of the reflected harmonic waves in the far field varying with different $n$ for the cases of $x_f=b{x}_R$ (red square) and $x_f=0$ (blue square), where the dashed lines are calculated from the diffraction limit ${\theta }_n={\theta }_1/n$. (b) The harmonic spectra on the $x$ axis in the far field, where all intensities are normalized by the reflected fundamental one for $x_f=0$. (c),(d) Spatial-temporal distributions of $E_y$ on the $t$-$z$ plane ($y=0$) in near field for the cases of $x_f=b{x}_R$ and $x_f=0$.

Figure 4

(a) Intensity distributions of the obtained attosecond pulses in the far field at distance $x_{\mathrm{far}}={10}^5{\lambda }_L$ for three cases of $x_f=b{x}_R\approx 24.7{\lambda }_L$ and $L=0.1{\lambda }_L$ (red), $x_f=0$ and $L=0.1{\lambda }_L$ (blue), and $x_f=0$ and $L=0$ (green). (b) Intensity distribution $|E_y{|}^2$ of the RP on the $x$ axis for three cycles around the peak laser intensity when $x_f=0.75b{x}_R$, $b{x}_R$, and $1.25b{x}_R$.

Figure 5

(a) Dependences of the coefficient $h$ on $a_{\mathrm{peak}}$ for $L=0.1{\lambda }_L$ (red) and $0.2{\lambda }_L$ (blue) when $t=t_{\max }$ (dashed) as well as $|t-t_{\max }|\le 0.5T$ (shading). (b) On-axis intensity distributions of the far-field attosecond pulses obtained from cases with movable (red dashed) or immovable (black solid) ions, where $x_{\mathrm{far}}={10}^5{\lambda }_L$, $x_f=b{x}_R\approx 24.7{\lambda }_L$, and $L=0.1{\lambda }_L$.