Ionization by few-cycle pulses: Tracing the electron orbits

High-order above-threshold ionization by few-cycle laser pulses is analyzed in terms of quantum orbits. For a given carrier-envelope phase, the number of contributing orbits and their ionization and rescattering times determine the shape of the angle-resolved spectrum in all detail. Conversely, analysis of a given spectrum reveals the carrier-envelope phase and the various interfering pathways from which the electron could choose.

Figure 1

(Color online) Differential ionization probabilities of argon $(I_p=15.76\mathrm{eV})$ as functions of the electron energy $E_{\mathbf{p}}$, for emission in the direction $\theta =0°$ (black curves) and $\theta =180°$ [red (gray) curves], for the linearly polarized seven-cycle sine-square pulse (5) with CE phase $\varphi =0$. The laser-field intensity is ${10}^{14}\mathrm{W}{\mathrm{cm}}^{-2}$ and the wavelength $800\mathrm{nm}$. The results were obtained by numerical integration, which is responsible for the jitter above $80\mathrm{eV}$.

Figure 2

(Color online) Quantum-orbit analysis of the spectra of Fig. 1. For $\theta =0°$ (upper panel), the contributions of the dominant three pairs of quantum orbits are presented, as well as the direct-ATI probability $\left({D}\right)$. The highly structured (dark blue) curve that extends from the upper left to the lower right gives the coherent sum of all these contributions $({D}+\mathrm{SP})$. For $\theta =180°$ (lower panel), there are five dominant pairs as well as the direct electrons. Contributions represented by dashed lines must be excluded from the coherent sum (1) after their respective cutoffs.

Figure 3

(Color online) Real parts of the quantum orbits (3) investigated in Fig. 2, for the same parameters as in Fig. 1. Their distance from the ion (in multiples of the excursion amplitude $A_0∕\omega$) is plotted as a function of time (in multiples of the carrier-wave period $T$). The black dashed curve shows the seven-cycle electric field (5), which underlies the calculation. The upper and middle panels correspond to the directions $\theta =0°$ and $\theta =180°$, respectively. Only the orbits most important at $56\mathrm{eV}$ are depicted. These are, using the nomenclature of Fig. 2, $a$, $b$, $b^{\prime }$, and $c,c^{\prime }$ for $\theta =0°$, and $e$, $e^{\prime }$, and $d$ for $\theta =180°$. The lowest panel displays the probabilities $w_{x+y}$ defined in the text for the orbits $x$ and $y$ specified in the inset.