Controlling high-order harmonic generation and above-threshold ionization with an attosecond-pulse train

We perform a detailed analysis of how high-order harmonic generation (HHG) and above-threshold ionization (ATI) can be controlled by a time-delayed attosecond-pulse train superposed to a strong, near-infrared laser field. In particular we show that the high-order harmonic and photoelectron intensities, the high-order harmonic plateau structure and cutoff energies, and the ATI angular distributions can be manipulated by changing this delay. This is a direct consequence of the fact that the attosecond pulse train can be employed as a tool for constraining the instant an electronic wave packet is ejected in the continuum. A change in such initial conditions strongly affects its subsequent motion in the laser field, and thus HHG and ATI. In our studies, we employ the strong-field approximation and explain the features observed in terms of interference effects between various electron quantum orbits. Our results are in agreement with recent experimental findings and theoretical studies employing purely numerical methods.

Figure 1

(Color online) Direct above-threshold ionization spectra, integrated over the solid angle and in linear scale [Eq. (10)], for argon $(I_p=0.58\mathrm{a.u.})$ interacting with a monochromatic field of frequency $\omega =0.057\mathrm{a.u.}$ and an attosecond-pulse train [Eq. (4)], as functions of the time delay between both driving fields. Panels (a) and (b) have been computed for an intensity of $I=3.3\times {10}^{13}$ and $5\times {10}^{14}\mathrm{W}∕{\mathrm{cm}}^2$ for the low-frequency laser field, respectively. In panels (a) and (b), the ATI orders (labeled with the number of photons from the ground state) correspond to photoelectron energies roughly between 0 and $36.5\mathrm{eV}$ and 0 and $47.4\mathrm{eV}$, respectively, due to the ponderomotive shift. The yield has been computed from Eq. (6), with $\zeta =0.04$. The intensities of the attosecond pulse trains have been chosen as $I_h=I∕10$, i.e., one tenth of those of the low-frequency field.

Figure 2

Upper panels: modulus squared of the transition amplitudes (8, 9), integrated over the solid angle, for $1s$ initial states (black line) and within the broad Gaussian approximation (gray line), as functions of the time delay between the attosecond-pulse train and the IR field. Lower panels: ${\mid d_{+}\mid }^2$ and ${\mid d_{-}\mid }^2$ integrated over the solid angle, for $1s$ initial states. We considered argon under the influence of an IR field of $I=3.3\times {10}^{13}\mathrm{W}∕{\mathrm{cm}}^2$ superposed to an attosecond-pulse train of $I_h=I∕10$ [panel (a) in Fig. 2]. The curves in the upper panels have been normalized so that their maximum values are unity, and those in the lower panel to the maximum value of $D_I^{(-)}$. The numbers $N$ in the figure denote the ATI orders.

Figure 3

(Color online) Angular distributions for the ATI direct electrons, in a logarithmic scale, for the same field and atomic parameters as in Fig. 1. The upper and lower panels have been computed for an intensity of $I=3.3\times {10}^{13}$ and $5\times {10}^{14}\mathrm{W}∕{\mathrm{cm}}^2$ for the low-frequency laser field, respectively. The yields are displayed as functions of the time delay $\varphi =\omega t_d$ and orientation angle $\theta$ between the laser field and the final electron momenta. The ATI orders $N$ are indicated on the upper parts of the panels, and all contours have been normalized to the highest yield in the figure (left panels).

Figure 4

High-order harmonic spectra for neon $(I_p=0.79\mathrm{a.u.})$ interacting with a monochromatic field of intensity $I=5\times {10}^{14}\mathrm{W}∕{\mathrm{cm}}^2$, and frequency $\omega =0.057\mathrm{a.u.}$, and a train of attosecond pulses of intensity $I_h={10}^{13}\mathrm{W}∕{\mathrm{cm}}^2$ [Eq. (4)], and several time delays between the attosecond pulse train and the IR field. We considered the orbits with the five shortest pairs of excursion times $t-t^{\prime }$. In the figure, we only show the harmonic orders for which $\Omega >I_p$.

Figure 5

(Color online) Real part of the recombination times as a function of the high-order harmonic order, for neon $(I_p=0.79\mathrm{a.u.})$ interacting with a monochromatic field of intensity $I=5\times {10}^{14}\mathrm{W}∕{\mathrm{cm}}^2$ and frequency $\omega =0.057\mathrm{a.u.}$ and an attosecond-pulse train of intensity $I_h={10}^{13}\mathrm{W}∕{\mathrm{cm}}^2$, composed of an infinite number of harmonics [Eq. (4)]. Parts (a), (b), (c), and (d) correspond to delays $\varphi =0$, $\varphi =0.25\pi$, $\varphi =0.5\pi$, and $\varphi =0.75\pi$ between the attosecond-pulse train and the low-frequency driving wave, respectively. In the figure, we only show the harmonic orders for which $\Omega >I_p$, and the pairs of orbits are indicated by the natural numbers $(i,j)$, which increase according to the electron excursion time in the continuum. The ionization times are indicated by the blue dashed lines, and the numbers in the upper part of the figure give the approximate kinetic energy of the electron upon return, in units of the ponderomotive energy $U_p$.

Figure 6

(Color online) Contributions of isolated pairs of orbits to the high-order harmonic spectra for the same atomic and external-field parameters in Figs. 4, 5, and delays $\varphi =0$, $\varphi =0.25\pi$, $\varphi =0.5\pi$, and $\varphi =0.75\pi$ between the attosecond-pulse train and the low-frequency driving wave [(a), (b), (c), and (d), respectively]. In the figure, we only show the harmonic orders for which $\Omega >I_p$. The contributions from specific pairs of orbits are depicted in the same colors as those employed in Fig. 5, for depicting each corresponding pair. For comparison, the spectra computed with the ten shortest orbits and with orbit 3 to 10 are given as the thick lines in the figure. The numbers on the upper parts of the figure give the approximate kinetic energy of the electron upon return, in units of the ponderomotive energy.