Photoelectron holography of atomic targets

We study the spatial interference effects appearing during the ionization of atoms (H, He, Ne, and Ar) by few-cycle laser pulses using single-electron ab initio calculations. The spatial interference is the result of the coherent superposition of the electronic wave packets created during one half cycle of the driving field following different spatial paths. This spatial interference pattern may be interpreted as the hologram of the target atom. With the help of a wave-function analysis (splitting) technique and approximate (strong-field and Coulomb-Volkov) calculations, we directly show that the hologram is the result of the electronic-wave-packet scattering on the parent ion. On the He target we demonstrate the usefulness of the wave-function splitting technique in the disentanglement of different interference patterns. Further, by performing calculations for the different targets, we show that the pattern of the hologram does not depend on the angular symmetry of the initial state and it is strongly influenced by the atomic species of the target: A deeper bounding potential leads to a denser pattern.

Figure 1

Shape of the two-cycle laser pulse used in the present calculations. Here $E_0$ is the amplitude, $T=2\pi /\omega$ is the period of the carrier wave, $t_i$ stands for the time moments when the electric field is zero, and ${\mathrm{\Psi }}_{Ci}$ indicates the continuum electron wave packets created during each time interval of the laser field. The pulse duration is $\tau =2T$.

Figure 2

Illustration of the wave splitting technique. Solid red arrow indicate the time propagation and the dotted blue arrows the wave function splitting. Here ${\mathrm{\Psi }}_B$ is the bound part of the wave function, while ${\mathrm{\Psi }}_{Ci}$ are the continuum partial wave functions created during different time intervals of the laser field.

Figure 3

Distribution of the continuum electrons as a function of the parallel ($k_{\text{par}}$) and perpendicular ($k_{\text{per}}$) momentum components for a H atom, which interacts with a two-cycle laser pulse [Eq. (3)] with the following parameters: $\omega =0.4445\mathrm{a}.\mathrm{u}., E_0=1\mathrm{a}.\mathrm{u}.$, and $\tau =28.26\mathrm{a}.\mathrm{u}$. (a)–(e) Momentum distributions for the electronic wave packets created during each half cycle of the driving laser field: (a) ${\mathrm{\Psi }}_{C1}$, (b) ${\mathrm{\Psi }}_{C2}$, (c) ${\mathrm{\Psi }}_{C3}$, (d) ${\mathrm{\Psi }}_{C4}$, and (e) ${\mathrm{\Psi }}_{C5}$. (f)–(i) Results of the gradual coherent summation of these wave packets: (f) ${\mathrm{\Psi }}_{C1}+{\mathrm{\Psi }}_{C2}$, (g) ${\mathrm{\Psi }}_{C1}+{\mathrm{\Psi }}_{C2}+{\mathrm{\Psi }}_{C3}$, (h) ${\mathrm{\Psi }}_{C1}+{\mathrm{\Psi }}_{C2}+{\mathrm{\Psi }}_{C3}+{\mathrm{\Psi }}_{C4}$, and (i) ${\mathrm{\Psi }}_{C1}+{\mathrm{\Psi }}_{C2}+{\mathrm{\Psi }}_{C3}+{\mathrm{\Psi }}_{C4}+{\mathrm{\Psi }}_{C5}$.

Figure 4

Total ionization probability (i.e., the norm of the continuum electronic wave packets) as a function of time. Shown at the top of the figure is the electric component of the driving laser field. The target atom is H and the laser pulse parameters are the same as in Fig. 3.

Figure 5

Ionization probability density of the H atom as a function of electron momentum components. Calculations were performed in the framework of (a) the strong-field approximation, (b) the Coulomb-Volkov approximation, and (c) the time-dependent close coupling model. The laser pulse parameters are the same as in Fig. 3.

Figure 6

Model potential $V_{\text{eff}}\left(r\right)$ as a function of the radial coordinate $r$ for the H, He, Ne, and Ar targets. For each potential curve (thick lines) the ground-state energy of the active electron is shown (thin horizontal lines), while the ionization energy $I_p$ for each target is specified in the legend.

Figure 7

Ionization probability density as a function of electron momentum components parallel and perpendicular to the laser polarization axis. The TDCC results for the H, He, Ne, and Ar targets are compared. The laser pulse parameters are the same as in Fig. 3.

Figure 8

(a) Laser field, (b) total ionization probability, and (c) expectation value of the $z$ coordinate $\langle z\left(t\right)\rangle$ as a function of time for the H, He, Ne, and Ar targets. The laser pulse parameters are the same as in Fig. 3.

Figure 9

Angular distribution of the continuum electrons at fixed $k=2.6\mathrm{a}.\mathrm{u}.$ electron momentum for different atomic targets: (a) H and Ar and (b) He and Ne. For an easier comparison the angular distributions are normalized. The electron ejection angle is measured from the laser polarization axis. The laser pulse parameters are the same as in Fig. 3.

Figure 10

Ionization probability density as a function of electron momentum components parallel and perpendicular to the laser polarization axis. The TDCC results for a H-like target started from the $2s$ and $2p$ states are compared. The effective charge of the H-like core is set to $Z_{\text{eff}}=2.152$. The laser pulse parameters are the same as in Fig. 3.

Figure 11

Angular distribution of the continuum electrons at fixed $k=2.2\mathrm{a}.\mathrm{u}.$ electron momentum for H-like targets ($2s$ and $2p$). For an easier comparison the angular distributions are normalized. The electron ejection angle is measured from the laser polarization axis. The laser pulse parameters are the same as in Fig. 3.

Figure 12

Distribution of the continuum electrons as a function of the parallel ($k_{\text{par}}$) and perpendicular ($k_{\text{per}}$) momentum components for a He atom interacting with a two-cycle laser pulse with the following parameters: $\omega =0.4445\mathrm{a}.\mathrm{u}., E_0=1\mathrm{a}.\mathrm{u}.$, and $\tau =28.26\mathrm{a}.\mathrm{u}$. (a)–(e) Momentum distribution for the electronic wave packets created during each half cycle of the driving laser field: (a) ${\mathrm{\Psi }}_{C1}$, (b) ${\mathrm{\Psi }}_{C2}$, (c) ${\mathrm{\Psi }}_{C3}$, (d) ${\mathrm{\Psi }}_{C4}$, and (e) ${\mathrm{\Psi }}_{C5}$. (f)–(i) Results of the gradual coherent summation of these wave packets: (f) ${\mathrm{\Psi }}_{C1}+{\mathrm{\Psi }}_{C2}$, (g) ${\mathrm{\Psi }}_{C1}+{\mathrm{\Psi }}_{C2}+{\mathrm{\Psi }}_{C3}$, (h) ${\mathrm{\Psi }}_{C1}+{\mathrm{\Psi }}_{C2}+{\mathrm{\Psi }}_{C3}+{\mathrm{\Psi }}_{C4}$, and (i) ${\mathrm{\Psi }}_{C1}+{\mathrm{\Psi }}_{C2}+{\mathrm{\Psi }}_{C3}+{\mathrm{\Psi }}_{C4}+{\mathrm{\Psi }}_{C5}$.