Extreme-ultraviolet coherent pulse amplification in argon

The amplification of ultrashort extreme-ultraviolet (XUV) pulses in argon in high-order harmonic generation processes is studied by using the time-dependent Schrödinger equation in the spin-free one-active-electron and single-atom approximation. We consider a neutral argon atom initially in the valence 3p state and a sufficiently intense two-cycle driving infrared (IR) pulse for the atom to be mainly ionized after the first laser cycle. The correlated dynamics and transitions from the valence 3p to a virtual subvalence 3s state and in the ionized regions are examined by synchronizing a 1.5-fs XUV pulse to the IR pulse. The calculated single-atom gain spectrum (26–45 eV) agrees with recent experimental measurements. We discuss different channels that can be present in the gain process as a function of pulse parameters and through an analysis of the dynamics of the populations in terms of field-free eigenstates. When the XUV pulse is considered at the end of the driving IR field the amplification is due to contributions of stimulated recombination from excited Rydberg and low energy continuum states to the 3s and 3p states of argon. In regions where the IR field is intense, high energy and angular momentum states are populated and the medium can interact with the pulses through bound and continuum states involving parametric transitions, which is further confirmed by studying classical electron trajectories. We discuss how these parametric interactions might be suitable for amplification of photon energies far from the ionization threshold as observed in the experiments.

Figure 1

We consider the IR field resulting from a 800-nm two-cycle ${\cos }^2$ vector potential interacting with the argon atom together with a XUV weak pulse synchronized with the IR field at different delays. (a) The IR field with peak intensity of $5\times {10}^{15}$ W/${\mathrm{cm}}^2$ is shown to produce a large ionization of the model atom, which is considered initially in the ground 3p state. Examples of two weak XUV pulses at two different delays are also shown. (b) The population in the 3s, 3p, and 3d field-free states is plotted in the time interval the XUV pulse interacts.

Figure 2

Spectrally integrated XUV probe absorption signal $S_a$ [see Eq. (2)] as a function of the delay of the XUV pulse with respect to the IR pulse. Negative values show the total XUV gain regions while positive values show absorption regions. The peak IR field intensities in terms of ${10}^{15}$ W/${\mathrm{cm}}^2$ (4.0, 5.0, and 6.0) are indicated in the subplots and by the different styles and colors of the lines. The central frequencies of the XUV pulse considered are shown as H13-H33 in each subplot, Hn meaning the corresponding n harmonic of an 800-nm field. The light gray-dashed line is shown for clarity and corresponds to an example of the IR field, with $3\times {10}^{15}$ W/${\mathrm{cm}}^2$ (see values in the right axis in a.u.).

Figure 3

(a) Spectrally integrated single-atom XUV absorption signal $S_a$ (in barns) as a function of harmonic order and the IR peak intensity (in terms of ${10}^{15}$ W/${\mathrm{cm}}^2$). The delay of the XUV pulse is in this case $t=3$ fs. (b) $S_a$ for $5\times {10}^{15}$ W/${\mathrm{cm}}^2$, as obtained from a cut of (a). Note that the vertical axis is reversed for a clearer comparison with the measurements. (c) Experimental results obtained with an IR peak intensity of $\sim 5\times {10}^{14}$ W/${\mathrm{cm}}^2$ and of 26-fs duration (FWHM).

Figure 4

Ten-cycle 800-nm flat-top IR pulse with intensity of ${10}^{15}$ W/${\mathrm{cm}}^2$ which produces amplification of an H21 XUV pulse synchronized at $t=8$ fs with integrated probe absorption of $-9.5$ b, as indicated. The population in the bound states after the IR pulse is $\sim 0.006$%.

Figure 5

(a) Frequency resolved absorption signal $S\left(\omega \right)$ of H21 as a function of delay for an IR peak intensity of $6.0\times {10}^{15}$ W/${\mathrm{cm}}^2$. (b) Some absorption spectra $S\left(\omega \right)$ of H21 at different delays from (a), as indicated, are shown. The integrated value $S_a$ is also given.

Figure 6

Weighted histograms of the population differences $\mathrm{\Delta }\rho$ in the case of H21 at a delay of $t=3.0$ fs and for an IR peak intensity is of $6.0\times {10}^{15}$ W/${\mathrm{cm}}^2$. As indicated, the histograms are determined by the total number of states in time (a), and by the quantum number $n$ (b)–(d), the angular momentum $L$ (e)–(g), and the energies of the field-free states $E$ (h)–(j) in particular XUV interaction time windows. In (h)–(j) $E=0$ denotes the transition from bound states to the continuum. The threshold for $\mathrm{\Delta }\rho$ is ${10}^{-8}$ and all vertical axes are divided by this number.

Figure 7

Weighted histograms of the population differences $\mathrm{\Delta }\rho$ in the case of H21 at a delay of $t=1.0$ fs and for an IR peak intensity is of $6.0\times {10}^{15}$ W/${\mathrm{cm}}^2$. As in Fig. 6, the histograms are determined by the total number of states in time (a), and by the quantum number $n$ (b)–(d), the angular momentum $L$ (e)–(g), and the energies of the field-free states $E$ (h)–(j) in particular XUV interaction time windows. In (h)–(j) $E=0$ denotes the transition from bound states to the continuum. The threshold for $\mathrm{\Delta }\rho$ is $5\times {10}^{-8}$ and all vertical axes are divided by this number.

Figure 8

Weighted histograms of the population differences $\mathrm{\Delta }\rho$ in the case of H21 at a delay of $t=1.0$ fs and for an IR peak intensity is of $6.0\times {10}^{15}$ W/${\mathrm{cm}}^2$ in the XUV interaction time window $0.66<t<0.8$ fs. The shown histograms are determined by the total number of states in time (a), and by the energies of the field-free states $E$ (b) and (c) in the particular XUV interaction time windows, as indicated. In (b) and (c) $E=0$ denotes the transition from bound states to the continuum. The threshold for $\mathrm{\Delta }\rho$ is $2\times {10}^{-9}$ and all vertical axes are divided by this number.

Figure 9

(Thick full red line) Classical trajectory of an electron ionized from the 3s state of argon at $t=0.981$ fs by a photon of energy 1.114 a.u. (H19.56) in a two-cycle IR pulse of $6.0\times {10}^{15}$ W/${\mathrm{cm}}^2$. At $t=1.426$ fs the electron can recombine with the 3s state with an energy of 1.244 a.u. (H21.84), as indicated, opening the channel for IXPA parametric amplification. (Different styles and colors, thin lines) Electron trajectories of tunnel-ionized electrons that open the channel for XPA parametric amplification. The recombination energies are indicated in terms of (not integer) harmonics.