Imaging electron angular distributions to assess a full-power petawatt-class laser focus

We present a novel technique to assess the focal volume of petawatt-class lasers at full power. Our approach exploits quantitative measurement of the angular distribution of electrons born in the focus via ionization of rarefied gas, which are accelerated forward and ejected ponderomotively by the field. We show that a bivariate ($\theta, \phi$) angular distribution, which was obtained with image plates, not only enables the peak intensity to be extracted, but also reflects nonideality of the focal-spot intensity distribution. In our prototype demonstration at intensities of a few $\times 10^{19}$ to a few $\times 10^{20}$ $\mathrm{W/cm^2}$, an f/10 optic produced a focal spot in the paraxial regime. This allows a plane-wave parameterization of the peak intensity given by $\tan{\theta_c} = 2/a_0$ ($a_0$ being the normalized vector potential and $\theta_c$ the minimum ejection angle) to be compared with our measurements. Qualitative agreement was found using an $a_0$ inferred from the pulse energy, pulse duration and the focal spot distribution with a modified parameterization, $\tan{\theta_c} = 2\eta/a_0$ ($\eta = 2.02^{+0.26}_{-0.22}$). This highlights the need for (i) better understanding of intensity degradation due to focal-spot distortions and (ii) more robust modeling of the ejection dynamics. Using single-shot detection of electrons, we showed that while there is significant shot-to-shot variation in the number of electrons ejected at a given angular position, the average distribution scales with the pulse energy in a way that is consistent with that seen with the image plates. Finally, we note that the asymptotic behavior as $\theta \to 0^{\circ}$ limits the usability of angular measurement. For 800 nm, this limit is at an intensity $\sim 10^{21}\ \mathrm{W/cm^2}$.