Scaling Laws of the Two-Electron Sum-Energy Spectrum in Strong-Field Double Ionization

The sum-energy spectrum of two correlated electrons emitted in nonsequential strong-field double ionization (SFDI) of Ar was studied for intensities of 0.3 to $2\times {10}^{14}\mathrm{W}/{\mathrm{cm}}^2$. We find the mean sum energy, the maximum of the distributions as well as the high-energy tail of the scaled (to the ponderomotive energy) spectra increase with decreasing intensity below the recollision threshold (BRT). At higher intensities the spectra collapse into a single distribution. This behavior can be well explained within a semiclassical model providing clear evidence of the importance of multiple recollisions in the BRT regime. Here, ultrafast thermalization between both electrons is found occurring within three optical cycles only and leaving its clear footprint in the sum-energy spectra.

Figure 1

(a) The measured sum-energy distribution for SFDI at 800 nm and intensities of $4\times {10}^{13}$, $7\times {10}^{13}$, and $1.2\times {10}^{14}\mathrm{W}/{\mathrm{cm}}^2$. The sum energy is scaled with respect to $U_p$ at each intensity and the maxima of the distributions are scaled to unity. In (b) and (c), the sum-energy spectra are fitted with the function of $f(E)=c{E}^{\alpha }{e}^{-\beta E}$ at $7\times {10}^{13}$ and $4\times {10}^{13}\mathrm{W}/{\mathrm{cm}}^2$, respectively.

Figure 2

(a) The calculated sum-energy distribution for SFDI at 800 nm in the intensity region from $4\times {10}^{13}$ to $2\times {10}^{14}\mathrm{W}/{\mathrm{cm}}^2$. In (b) and (c), the sum-energy spectra are fitted with the function $f(E)=c{E}^{\alpha }{e}^{-\beta E}$ at $7\times {10}^{13}$ and $4\times {10}^{13}\mathrm{W}/{\mathrm{cm}}^2$, respectively.

Figure 3

(a) Intensity dependence of the peak position of the theoretical sum-energy spectra (i.e., the parameter $\alpha /\beta$, blue circles) and the arithmetic mean value of the sum energy (black squares). (b) The percentages of multiple-recollision trajectories (MRT). $I_{\mathrm{th}}=6\times {10}^{13}\mathrm{W}/{\mathrm{cm}}^2$ is the threshold intensity.

Figure 4

(a)–(j) Snapshots of the compensated-velocity distribution as time evolves from 0.75 o.c. to 13 o.c.For clear visualization, we only show those DI events triggered by tunneled electrons released at 17° of the cosine-shaped field. The laser intensity is $4\times {10}^{13}\mathrm{W}/{\mathrm{cm}}^2$. The blue curve in (i) is a Maxwell velocity distribution at the temperature of $4.5\times {10}^4\mathrm{K}$ (3.9 eV), added by hand to guide the eye.