Extracting photoelectron spectra from the time-dependent wave function: Comparison of the projection onto continuum states and window-operator methods

Over the past three decades numerous numerical methods for solving the time-dependent Schrödinger equation within the single-active electron approximation have been developed for studying ionization of atomic targets exposed to an intense laser field. In addition, various numerical techniques for extracting the photoelectron spectra from the time-dependent wave function have emerged. In this paper we compare photoelectron spectra obtained by either projecting the time-dependent wave function at the end of the laser pulse onto the continuum state having the proper incoming boundary condition or by using the window-operator method. Our results for three different atomic targets show that the boundary condition imposed onto the continuum states plays a crucial role for obtaining correct spectra accurate enough to resolve fine details of the interference structures of the photoelectron angular distribution.

Figure 1

Differential detachment probabilities of ${\mathrm{F}}^{-}$ ions for emission of electrons in the directions ${\theta }_{\mathbf{k}}=0^{\circ }$, ${90}^{\circ }$, and ${180}^{\circ }$, as functions of the photoelectron energy in units of the ponderomotive energy $U_p$, for the following laser-field parameters: $I=1.3\times {10}^{13}\text{W}/{\text{cm}}^2$, $\lambda =1800\text{nm}$, and $N_c=6$. The results are obtained by projecting the time-dependent wave function $\mathrm{\Psi }\left(T_p\right)$ onto the ${\mathrm{\Phi }}_{\mathbf{k}}^{(-)}$ states (black solid line) and ${\mathrm{\Phi }}_{\mathbf{k}}^{(+)}$ states (green dot-dashed line) and using the WO method with $\gamma =2\times {10}^{-3}$ (red dashed line).

Figure 2

Full PADs for the same parameters as in Fig. 1. The upper panel shows the PAD obtained by projecting onto the continuum states ${\mathrm{\Phi }}_{\mathbf{k}}^{(-)}$, while the lower panel shows the PAD obtained by the WO method. The WO method gives additional structure for angles ${\theta }_{\mathbf{k}}\in ({30}^{\circ },{150}^{\circ })$ and energies $E_{\mathbf{k}}>3U_p$.

Figure 3

Differential ionization probabilities of H atoms for emission of electrons in the directions ${\theta }_{\mathbf{k}}=0^{\circ }$, ${90}^{\circ }$, and ${180}^{\circ }$, as functions of the photoelectron energy in units of the ponderomotive energy $U_p$, for the following laser-field parameters: $I={10}^{14}\text{W}/{\text{cm}}^2$, $\lambda =800\text{nm}$, and $N_c=6$. The results are obtained by projecting the time-dependent wave function $\mathrm{\Psi }\left(T_p\right)$ onto the ${\mathrm{\Phi }}_{\mathbf{k}}^{(-)}$ states (black solid line) and the ${\mathrm{\Phi }}_{\mathbf{k}}^{(+)}$ states (green dot-dashed line) and by using the WO method with $\gamma =6\times {10}^{-3}$ (red dashed line).

Figure 4

Full PADs for the H atom and laser-field parameters as in Fig. 3. The upper panel shows the PAD obtained by projecting onto the Coulomb wave for the free particle and the lower panel shows the PAD obtained by the WO method. The WO method gives additional interference structures for angles ${\theta }_{\mathbf{k}}\in ({30}^{\circ },{150}^{\circ })$ and $E_{\mathbf{k}}>4U_p$.

Figure 5

Differential ionization probabilities of the Ar atom for emission of electrons in the directions ${\theta }_{\mathbf{k}}=0^{\circ }$, ${90}^{\circ }$, and ${180}^{\circ }$, as functions of the photoelectron energy in units of the ponderomotive energy $U_p$, for the following laser-field parameters: $I=8\times {10}^{13}\text{W}/{\text{cm}}^2$, $\lambda =800\text{nm}$, and $N_c=6$. The results are obtained by projecting the time-dependent wave function $\mathrm{\Psi }\left(T_p\right)$ onto the ${\mathrm{\Phi }}_{\mathbf{k}}^{(-)}$ states (black solid line) and ${\mathrm{\Phi }}_{\mathbf{k}}^{(+)}$ states (green dot-dashed line) and by using the WO method with $\gamma =6\times {10}^{-3}$ (red dashed line).

Figure 6

Full PADs for the Ar atom and the same laser-field parameters as in Fig. 5. The upper panel shows the PAD obtained by projecting onto the continuum states ${\mathrm{\Phi }}_{\mathbf{k}}^{(-)}$ and the lower panel shows the PAD obtained by the WO method. The WO method gives additional structure for angles ${\theta }_{\mathbf{k}}\in ({30}^{\circ },{150}^{\circ })$ and $E_{\mathbf{k}}>4U_p$.

Figure 7

Angle-integrated spectra for the Ar atom and the same laser-field parameters as in Fig. 5. The spectra are obtained by projecting onto continuum states with incoming boundary condition (black solid line) and using the WO method (dashed red line).