Intra- and intercycle interference of electron emissions in laser-assisted XUV atomic ionization

We study the ionization of atomic hydrogen in the direction of polarization due to a linearly polarized XUV pulse in the presence of a strong IR laser. We describe the photoelectron spectra as an interference problem in the time domain. Electron trajectories stemming from different optical laser cycles give rise to intercycle interference energy peaks known as sidebands. These sidebands are modulated by a coarse-grained structure coming from the intracycle interference of the two electron trajectories born during the same optical cycle. We make use of a simple semiclassical model that offers the possibility to establish a connection between emission times and the photoelectron kinetic energy. We analyze such interference pattern as a function of the time delay between the IR and the XUV pulses and also as a function of the laser intensity. We compare the semiclassical predictions with the continuum-distorted-wave strong-field approximation and the ab initio solution of the time-dependent Schrödinger equation.

Figure 1

(a) and (c) Total electric field $F\left(t\right)=F_L\left(t\right)+F_X\left(t\right)$ and (b) and (d) vector potential $A\left(t\right)=A_L\left(t\right)+A_X\left(t\right)$ as a function of time. The IR laser parameters are $F_{L0}=0.02$, ${\omega }_L=0.05$, ${\tau }_L=7T_L$, and the XUV pulse with the parameters $F_{X0}=0.01$ and ${\omega }_X=1.5$. The duration of the XUV pulse is (a) and (b) ${\tau }_X=T_L$ and (c) and (d) ${\tau }_X=3T_L$. In (b) and (d) the electron emission early (late) times for a given final momentum $k_z$ are marked by circles (triangles). The shaded area denotes the active time window when XUV ionization takes place.

Figure 2

(a) Buildup of the interference pattern following the SCM. The intracycle pattern is given by the structure pattern $F\left(k_z\right)$ in a red thick line and intercycle interference is given by the function $B\left(k_z\right)$ with $N=3$ [Eq. (23)]. (b) Total interference pattern $F\left(k_z\right)B\left(k_z\right)$ with $N=3$. The IR laser parameters are $F_{L0}=0.02$, ${\omega }_L=0.05$, the XUV frequency ${\omega }_X=1.5$, and delay time $t_{12}=0$. Both fields are cosinelike. For the sake of comparison, the intracycle pattern $F\left(k_z\right)$ of (a) is shown in light gray. Vertical lines depict the positions of the sidebands $E_n$ of Eq. (24).

Figure 3

Energy distribution in the forward direction for the same laser parameters and XUV frequency as in Fig. 1 calculated within (a) the SFA and (b) the TDSE. The laser duration is ${\tau }_L=7T_L=879.65$ and the XUV field amplitude $F_{X0}=0.01$. Vertical lines depict the positions of the sidebands $E_n$ in Eq. (24).

Figure 4

Photoelectron spectra in the forward direction (in arbitrary units) for time delay $t_{12}=0$ for the SFA (black dashed line), the approximation of Bivona et al. [13] (green solid line), the approximation of Kazansky et al. [11, 12] (orange solid line), the semiclassical model (purple solid line), and the soft-photon approximation (thin red solid line with circles). The IR laser parameters are $F_{L0}=0.02$, ${\omega }_L=0.05$, ${\tau }_L=7T_L$, and the XUV pulse with the parameters $F_{X0}=0.01$, ${\omega }_X=1.5$, and duration ${\tau }_X=T_L$. Both fields are cosinelike.

Figure 5

Photoelectron spectra in the forward direction (in arbitrary units) calculated at different laser field strengths within (a) the SCM, (b) the SFA, and (c) the TDSE. The IR laser frequency is ${\omega }_L=0.05$ and the XUV pulse parameters are $F_{X0}=0.01$, ${\omega }_X=1.5$, and ${\tau }_X=T_L$. Both XUV and IR fields are cosinelike. The classical boundaries of laser-assisted photoemission are shown with dashed lines. The high-intensity spot in the right bottom corner of (c) corresponds to ionization by the IR laser pulse alone.

Figure 6

Photoelectron spectra in the forward direction (in arbitrary units) as a function of the optical phase (time delay $t_{12}$) within (a) and (d) the SCM, (b) and (e) the SFA, and (c) and (f) the TDSE. The IR laser frequency is ${\omega }_L=0.05$, ${\tau }_L=7T_L$, and the XUV pulse has $F_{X0}=0.01$, ${\omega }_X=1.5$, XUV pulse duration (a)–(c) ${\tau }_X=T_L$ and (d)–(f) ${\tau }_X=2T_L$.

Figure 7

Photoelectron spectra in the forward direction (in arbitrary units) for time delays $t_{12}$ corresponding to (a)–(c) $\varphi =\pi /2$ and (d)–(f) $\varphi =3\pi /2$ within (a) and (d) the SCM, (b) and (e) the SFA, and (c) and (f) the TDSE. The IR laser frequency is ${\omega }_L=0.05$, ${\tau }_L=7T_L$, and the XUV pulse parameters are $F_{X0}=0.01$, ${\omega }_X=1.5$, and ${\tau }_X=T_L$ (thick dark gray curve) and ${\tau }_X=2T_L$ (thick light gray curve).

Figure 8

Energy distribution in the forward direction calculated within (a) the SFA and (b) the TDSE. The laser duration is ${\tau }_L=9T_L=1130.97$ and the envelope function of the XUV pulse is $F_{X0}\left(t\right)={sin}^2(\pi t/{\tau }_X)$ with ${\tau }_X=345.2$ and $690.4$, which correspond to a FWHM duration of ${\tau }_X^{\mathrm{FWHM}}=T_L$ and $2T_L$, respectively. For completeness, we also show the intracycle momentum for the flattop XUV pulse of Fig. 3 (thick gray curve). Vertical lines depict the positions of the sidebands $E_n$ according to Eq. (24).