Photodetachment microscopy in time-dependent fields

Photodetachment of negative ions in combined laser and low-frequency fields is investigated. The time-dependent Green's function method is used for calculation of electron flux at a macroscopic distance from the photodetachment source, typical for a photodetachment microscopy experiment. In calculating the electron flux, we use the stationary phase method for the time integral, equivalent to the semiclassical approximation, to compute the time-dependent wave function. The stationary points $t_1^{\left(i\right)}, i=1,...,n$ correspond to time instances of launching of classical trajectories arriving at the detector at a given spacetime point $(\mathbf{r},t)$. The number of trajectories $n$ contributing to the electron flux at any point in the classically allowed spacetime domain can be controlled by varying the switching interval of the high-frequency laser which initiates the photodetachment process. The divergences inherent in the electron flux in the semiclassical treatment are removed by using the uniform Airy approximation near the caustics.

Figure 1

Graphical solution of the equation $\partial S/\partial t^{\prime }=0$ for $t^{\prime }$ at different detection times $t$. For $t=351.00$ ns no real roots exist. This means that there are no real trajectories arriving at the center of the detector. For $t=357.00$ ns there are two roots corresponding to two electron trajectories arriving at the center of the detector at the same time. The amplitude and frequency of the field are 50 V/cm and 100 MHz, respectively. The detector is 0.5 m from the source.

Figure 2

Electron trajectories (thin lines) and projections of the caustic surfaces on the $x-z$ plane (thick lines). Panels (a)–(c) show the variation of the caustic surface for different initial times.

Figure 3

Electron trajectories (thin lines) and the caustic surface (thick solid line) for the case of a static field. $F=50$ V/cm, $E=0.5$ meV.

Figure 4

Profile of $\mathrm{\Lambda }(t,t_0,t_2)$ for the laser interval $I_2$.

Figure 5

Relation between the detection time $t$ and the launching time $t^{\prime }$. (a) The switching interval is $I_2$. Maximum of eight trajectories grouped as $G_1$ and $G_2$ are possible. (b) Switching interval $I_1$. Maximum of four trajectories grouped as $A_1$ and $A_2$ are possible. $C_1$ and $C_2$ indicate the points of the caustic condition (16).

Figure 6

Electron flux at the center of the detector is shown as a function of time for $F_0=50$ V/cm and $\omega =100$ MHz. Divergence in the semiclassical flux (red dashed line) is corrected by using the uniform Airy approximation (thick solid line) near the caustic.

Figure 7

Same as Fig. 6, but the electron flux near the transition region from two to four trajectories is shown.

Figure 8

Calculations for $F_0=50$ V/cm and rf 200 MHz. (a) Electron-arrival time plot, where $C_1$ and $C_2$ indicate the points which satisfy the caustic conditions (16). (b) Semiclassical flux (red dashed line) and the uniform Airy function approximation (thick solid line) when the number of electron trajectories increases from 0 to 2. (c) The same as panel (b), but the number of trajectories increases from two to four.

Figure 9

Initial electron launching time as a function of the $x$ coordinate on the detector plane for different final times $t$.

Figure 10

(a) Variation of the flux along $x$ at $t=360$ ns. (b) Interference pattern in the detector plane. The field amplitude $F_0=50$ V/cm and frequency 100 MHz.

Figure 11

(a), (c) Variation of the spatial interference pattern along the $x$ coordinate for $t=358$ ns and $t=359$ ns, respectively. (b), (d) Corresponding spatial distribution of the electron flux in the detector plane.