Single-Shot Carrier-Envelope Phase Determination of Long Superintense Laser Pulses

The impact of the carrier-envelope phase (CEP) of an intense multicycle laser pulse on the radiation of an electron beam during nonlinear Compton scattering is investigated. We have identified a CEP effect specific to the ultrarelativistic regime. When the electron beam counterpropagates with the laser pulse, pronounced high-energy x-ray double peaks emerge near the backward direction relative to the initial electron motion. This is achieved in the relativistic interaction domain, where both the electron energy is required to be lower than for the electron reflection condition at the laser peak and the stochasticity effects in the photon emission need to be weak. The asymmetry parameter of the double peaks in the angular radiation distribution is shown to serve as a sensitive measure for the CEP of up to 10-cycle long laser pulses and can be applied for the characterization of extremely strong laser pulses in present and near future laser facilities.

Figure 1

Interaction scheme where the focused laser and the electron beam counterpropagate. For $\gamma \ll \xi$, parts of the electron beam are reflected, escape from the laser beam in different laser cycles, and emit x rays at different angles in backward direction. Two backward x-ray peaks are sensitive to the CEP of the long driving laser pulse, and forward emission arises before the electron reflection. Only the emission at azimuthal angle $\phi =0$ is shown.

Figure 2

(a),(b) Angle-resolved radiation energy ${\log }_{10}[\mathrm{d}{ϵ}_R/\mathrm{d}\mathrm{\Omega }]{\mathrm{rad}}^{-2}$ in units of $m$ vs the emission polar angle $\theta$ and the azimuthal angle $\varphi$ ($\mathrm{\Omega }$ is the emission solid angle) in a 6-cycle laser pulse; the CEP ${\psi }_{\mathrm{CEP}}=0°$ and 180°, respectively. (c),(d) $\mathrm{d}{\widetilde{ϵ}}_R/[\mathrm{d}\theta \sin (\theta )]$ vs $\theta$; ${\psi }_{\mathrm{CEP}}=0°$ in (a),(c), and ${\psi }_{\mathrm{CEP}}=180°$ in (b),(d). $P_1$ and $P_2$ correspond to the two main peaks from left to right. All other parameters are given in the text.

Figure 3

(a) The asymmetry parameter $\mathcal{A}$ of the backward radiation peaks $P_1$ and $P_2$ vs CEP. The polar angles (b) ${\theta }_{P_1}$ and (c) ${\theta }_{P_2}$ vs CEP. (red-solid lines) $\xi =600$, ${ϵ}_0=10\mathrm{MeV}$, (blue-dotted lines) $\xi =100$ and ${ϵ}_0=3\mathrm{MeV}$. The periodic variation of $\mathcal{A}$ for $\xi =600$ is shown in (a), and is omitted for other cases. Other parameters are the same as in Fig. 2.

Figure 4

Emergence of the radiation peaks $P_1$, $P_2$ (see Fig. 2): Left column is for ${\psi }_{\mathrm{CEP}}=0°$, and right column for ${\psi }_{\mathrm{CEP}}=180°$. (a),(b) Radiation intensity integrated over the azimuthal angle of $-15°\le \varphi \le +15°$, ${\log }_{10}\{d^2{\widetilde{ϵ}}_R/[d\overline{\eta }d\theta \sin (\theta )]\}$, vs emission phase $\overline{\eta }$, with $\overline{\eta }=({\omega }_{0}t-k_{0}z)/2\pi$. (c),(d) The transverse component of electric fields $E_x$ at the focus scaled by the laser amplitude $E_0$ vs $\overline{\eta }$. (e),(f) The electron’s initial spatial distribution in the cross section of the electron beam, which contributes to the spectral peak $P_1$ (yellow only), and to the spectral peak $P_2$ (black and yellow). The red circles show the boundary of the electron beam. (g),(h) Example trajectories of the electrons initially in the yellow region (yellow curves), and in the black region (outside of the yellow part, black curves), respectively. The photon emission is indicated by circles, and the dashed line shows the laser beam radius $w_z=w_0\sqrt{1+{(z/z_r)}^2}$. Other parameters are the same as in Fig. 2.

Figure 5

The variations of (a) $\mathcal{A}$ between $P_1$ and $P_2$ and the polar angles (b) ${\theta }_{P_1}$ and (c) ${\theta }_{P_2}$ with respect to the CEP. The black-solid, blue-dotted, red-solid, green-dash-dotted, and cyan-solid curves represent the cases of $\tau =2$, 4, 6, 8, and 10 $T_0$, respectively. Other parameters are the same as in Fig. 2.