Coulomb-Volkov approximation for near-threshold ionization by short laser pulses

The formation of near-threshold structures in the angle-resolved photoelectron spectrum, the cusp in the transverse electron momentum distribution, and the forward-backward asymmetry is influenced by the residual Coulomb potential of the remaining atomic core acting on the electron detached by strong short-laser pulses. We show that an improvement beyond the strong-field approximation—the Coulomb-Volkov distorted wave approximation—is capable to reproduce these features found in the exact solution of the time-dependent Schrödinger equation in the multiphoton regime and the transition from the multiphoton to the tunneling regime.

Figure 1

(Color online) Doubly-differential electron momentum distributions (logarithmic scale) in cylindrical coordinates $\left(k_z,k_{\rho }\right)$ using (a) the SFA, (b) the CVA, and (c) the exact TDSE calculation: The parameters of the field are $F_0=0.05\text{a}\text{.u}\text{.}$, $\omega =0.25\text{a}\text{.u}\text{.}$, and $\tau =151\text{a}\text{.u}\text{.}$ which comprises six complete optical cycles.

Figure 2

(Color online) Doubly-differential electron momentum distributions (logarithmic scale) in cylindrical coordinates $\left(k_z,k_{\rho }\right)$ using (a) only the Coulomb normalization factor $N_T\left(k\right)$ with the SFA, (b) only the hypergeometric function ${{}_{1}F}_1$ with the SFA [see Eq. (11)], and (c) the EVA [see Eq. (12)]. The parameters of the field are the same as in Fig. 1.

Figure 3

(Color online) Doubly-differential electron momentum distributions (logarithmic scale) in cylindrical coordinates $\left(k_z,k_{\rho }\right)$ using (a) the SFA, (b) the CVA, and (c) exact TDSE calculation: The parameters of the field are $F_0=0.0377\text{a}\text{.u}\text{.}$, $\omega =0.05\text{a}\text{.u}\text{.}$, and $\tau =1005\text{a}\text{.u}\text{.}$ (eight complete cycles).

Figure 4

(Color online) As Fig. 3 but with peak electric field $F_0=0.0533\text{a}\text{.u}\text{.}$

Figure 5

(Color online) Normalized doubly-differential momentum distribution as a function of the cosine of the ejection angle of the electron for a fixed energy $\epsilon$ and different laser parameters: (a) $F_0=0.05\text{a}\text{.u}\text{.}$, $\omega =0.25\text{a}\text{.u}\text{.}$, and $\tau =151\text{a}\text{.u}\text{.}$, (b) $F_0=0.0377\text{a}\text{.u}\text{.}$, $\omega =0.05\text{a}\text{.u}\text{.}$, and $\tau =1005\text{a}\text{.u}\text{.}$, and (c) $F_0=0.053\text{a}\text{.u}\text{.}$, $\omega =0.05\text{a}\text{.u}\text{.}$, and $\tau =1005\text{a}\text{.u}\text{.}$. The corresponding Keldysh parameters are indicated in the figures. (Blue) Dashed line, SFA; (red) solid line, CVA; (black) thick solid line, exact ab initio results.

Figure 6

(Color online) Normalized transversal momentum distributions $dP/d{k}_{\perp }$. (Blue) Dashed line, SFA; (red) solid line, CVA; black thick solid line, exact TDSE results. The parameters of the laser pulse are (a) $F_0=0.05\text{a}\text{.u}\text{.}$, $\omega =0.25\text{a}\text{.u}\text{.}$, and $\tau =151\text{a}\text{.u}\text{.}$, (b) $F_0=0.0377\text{a}\text{.u}\text{.}$, $\omega =0.05\text{a}\text{.u}\text{.}$, and $\tau =1005\text{a}\text{.u}\text{.}$, and (c) $F_0=0.053\text{a}\text{.u}\text{.}$, $\omega =0.05\text{a}\text{.u}\text{.}$, and $\tau =1005\text{a}\text{.u}\text{.}$ The corresponding Keldysh parameters are indicated in the figures.

Figure 7

(Color online) Asymmetry coefficient $A\left({\varphi }_{\text{CE}}\right)$ as a function of the carrier-envelope phase: (Blue) Dashed line, SFA; (red) solid line, CVA; and thick black line, exact TDSE results. The parameters of the field are $\omega =0.05\text{a}\text{.u}\text{.}$, $\tau =503\text{a}\text{.u}\text{.}$ (four complete cycles) and (a) $F_0=0.0377\text{a}\text{.u}\text{.}$ and (b) $F_0=0.0533\text{a}\text{.u}\text{.}$