Angular distributions of atomic photoelectrons produced in the uv and xuv regimes

We present angular distributions of photoelectrons of atomic model systems excited by intense linearly polarized laser pulses in the vuv and xuv regimes. We solve the multidimensional time-dependent Schrödinger equation for one particle on large spatial grids and investigate the direction dependence of the emitted electrons for isotropic $s$ states as well as $p$ states. Although the ponderomotive potential is small compared to the binding energy of the initially bound electron and the photon energy of the exciting laser field, richly structured photoelectron angular distributions are found which sensitively depend on the laser frequency and intensity as well as on the number of absorbed photons. The occurring shapes are explained in terms of scattering mechanisms.

Figure 1

Schematic drawing of the used detector implementation. The wave function is saved at a large fixed radius $r$ after which it is damped by an imaginary absorbing potential. Typical detector parameters are $r⩾25.0\mathrm{a.u.}$ and more than 1500 detector points. The atom is located in the center region at $r=0$.

Figure 2

(Color online) Density of the initial states of the regularized Coulomb potential, Eq. (3), with $\kappa =0.1$, in the $x\text{−}y$ plane calculated via TDSE propagation in the imaginary time direction. The lower numbers give the energy eigenvalue of the state. The nomenclature of the states is chosen in analogy to the orbitals of the hydrogen atom.

Figure 3

(Color online) Angle- and energy-resolved spectra of the ionization of the $2s$ state for two photon energies: (a) $\omega =0.5\mathrm{a.u.}$ and (b) $\omega =1.0\mathrm{a.u.}$ The electrical field strength is fixed at $E_0=0.1\mathrm{a.u.}$ corresponding to a laser intensity of $I=3.5\times {10}^{14}\mathrm{W}∕{\mathrm{cm}}^2$. The direction of the linearly polarized laser field is indicated by the (red) dashed line.

Figure 4

(Color online) The same as Fig. 3, but for the (a) $p_x$ and (b) $p_y$ states, excited with a laser frequency of $\omega =0.5\mathrm{a.u}$.

Figure 5

Angular dependence of photoelectrons for different photon orders $N_{\mathit{Ph}}$ (columns) and different photon energies $\omega$ (rows, given in atomic units) for the $2s$ state at an intensity of $I=3.5\times {10}^{14}\mathrm{W}∕{\mathrm{cm}}^2$. The laser field is polarized in the $x$ direction (horizontally).

Figure 6

(Color) Angular dependence of different photon orders $N_{\mathit{Ph}}$ (columns) for different photon energies $\omega$ (rows, given in atomic units) investigated for the $2p$ states at a laser intensity of $I=3.5\times {10}^{14}\mathrm{W}∕{\mathrm{cm}}^2$. The colored lines indicate the angular distribution of photoelectrons of the single $2p_x$ (red lines) and $2p_y$ (blue lines) states. The laser field is polarized in the $x$ direction (horizontally).

Figure 7

(Color online) Comparison of the photoelectron angular distributions for different orientations of the initial states. The upper figures give the densities of the rotated states (a) $2p_x^{\prime }$ and (b) $2p_y^{\prime }$. The PADs of the nonrotated $2p$ states (cf. Fig. 2) are shown in (c) and for the rotated states in (d). The thin lines indicate the data for each individual orientation, whereas the bold line displays the total intensity $I\left(2p_x\right)+I\left(2p_y\right)$ in (c) and $I\left(2p_x^{\prime }\right)+I\left(2p_y^{\prime }\right)$ in (d), respectively.

Figure 8

Photoelectron angular dependence for different photon orders $N_{\mathit{Ph}}$ (columns) and different intensities (rows, electrical field amplitude $E_0$ given in atomic units) for the $2s$ state at a fixed photon energy of $\omega =1.69\mathrm{a.u}$. The laser field is polarized along the $x$ direction (horizontally).

Figure 9

(Color online) Parameters ${\beta }_l$ of Eq. (4) calculated via a least-squares fit to our numerical data for the first (b) and second (a), respectively, photon peaks of the PADs of the $2s$ state vs the photon energy $\omega$ for a fixed intensity of $I=3.5\times {10}^{10}\mathrm{W}∕{\mathrm{cm}}^2$.

Figure 10

(Color online) Intensity dependence of ${\beta }_l$ parameters in Eq. (4) for the $2s$ state at a fixed photon energy of $\omega =1.69\mathrm{a.u.}$ Again (b) gives the result for the first photon peak (single-photon absorption) and (a) the data for the second photon line.

Figure 11

(Color online) Comparison of PAD fitted according to Eq. (4) (black solid lines), with the accurate numerical data [(red) dashed lines] for some selected typical cases of $\omega$; see Fig. 5.