Towards a theory of an attosecond transient recorder

Laser-assisted photoemission by a chirped subfemtosecond extreme ultraviolet (xuv) pulse is considered within an exactly solvable quantum-mechanical model. Special emphasis is given to the energy dependence of photoexcitation cross section. The streaked spectra are analyzed within the classical picture of initial time-momentum distribution $r_{\text{ini}}\left(p,t\right)$ of photoelectrons mapped to the final energy scale. The actual time-momentum distribution in the absence of the probe laser field is shown to be a poor choice for $r_{\text{ini}}$, and a more adequate ansatz is suggested. The semiclassical theory offers a simple practically useful approximation for streaked spectra. Its limitations for sufficiently long chirped xuv pulses are established.

Figure 1

(a) Potential $V\left(z\right)$ of the model 1D system. The atom is represented by a 6 a.u. wide potential well centered at $z=-8\text{a}\text{.u}\text{.}$ relative to the surface barrier. (b) Density of states. (c) Superposition of the electric field of the xuv pulse of duration $D_{\text{X}}$ and the laser pulse of duration $D_{\text{L}}$.

Figure 2

(Color online) Time-dependent occupation of final states $J\left(ϵ,t\right)$ for the duration $D_{\text{X}}=2000$ as, ${\Gamma }_{\text{X}}^0=3\text{eV}$ (left column), and 400 as, ${\Gamma }_{\text{X}}^0=15\text{eV}$ (right column). The chirp rate $b$ is zero in graphs (a) and (d) and (b) $b=6$, (c) 12, (e) 96, and (f) 240 eV/fs. Energy is given relative to ${ϵ}_{\text{i}}+{\omega }_{\text{X}}^0$ [the frequency at $t=0$ is ${\omega }_{\text{X}}^0=91\text{eV}$, see Eq. (2)] and time relative to $t_{\text{X}}$. The maximum intensity is set to 1. The vertical cross section at the right border of the map is the final intensity distribution $J\left(ϵ\right)$.

Figure 3

(Color online) Time dependence of the photoelectron spectrum center of gravity ${ϵ}_g\left(t\right)$ for the chirp rate $b=\pm 24\text{eV}/\text{fs}$ for three pulse durations: $D_{\text{X}}=400$, 800, and 1200 as. Positive (negative) chirp is shown by full (dashed) lines. Right panel shows the function $S\left(ϵ\right)$ [see Eq. (5)]. Photoelectron energy $ϵ$ is given relative to ${ϵ}_{\text{i}}+{\omega }_{\text{X}}^0$.

Figure 4

(Color online) (a) Temporal intensity profile of the xuv pulse envelope $U\left(t\right)$ [see Eq. (1)], (solid line) compared to the laser-free transition rate $r_0\left(p,t\right)$ for the central momentum, $p_0^2={ϵ}_{\text{i}}+{\omega }_{\text{X}}-V_0$ (dashed line), and for a momentum far from the center $p_1$ (dot-dashed line) obtained from the TDSE. (b) Electric field $E_{\text{L}}\left(t\right)$ of the laser pulse (dashed line) and the transferred momentum $\Delta p\left(t\right)$. The circles indicate the release points $t_{\text{X}}$ of the xuv pulses studied in Fig. 5.

Figure 5

(Color online) Streaked photoelectron EDCs for xuv pulse durations [(a) and (b)] $D_{\text{X}}=1000$ as and [(c) and (d)] 1600 as and release points [(a) and (c)] $t_{\text{X}}=600$ as and [(b) and (d)] $-400$ as indicated in Fig. 4b. Dotted lines are the full solution of the TDSE, and solid lines are obtained from SSA Eq. (6) with the function $r_{\text{ini}}$ from Eq. (7). Dashed curves result from using the laser-free function $r_0$ in Eq. (6).

Figure 6

(Color online) Graphs (a) and (b) show the dependence of the streaking broadening $\Delta {\Gamma }_{\text{e}}$ on the duration $D_{\text{X}}$ of the xuv pump pulse for two release points (a) $t_{\text{X}}=840$ as and (b) 600 as by the full solution of the TDSE (solid lines) and by the SSA (dashed lines). The curves are presented for three values of the laser field amplitude: $E_0=6\times {10}^7\text{V}/\text{cm}$, as well as $2E_0$ and $4E_0$. Graph (c) shows the dependence of $\Delta {\Gamma }_{\text{e}}$ on release time for $D_{\text{X}}=300$, 400, 600, and 800 as.

Figure 7

(Color online) Dependence of the streaking broadening $\Delta {\Gamma }_{\text{e}}$ of the photoelectron spectra on the release point $t_{\text{X}}$ for four values of chirp rate $b=10$, 25, 50, and 100 eV/fs and four pulse durations $D_{\text{X}}=400$, 600, 800, and 1000 as. Solid lines are the exact solution of the TDSE and dashed lines are obtained from the SSA Eqs. (6, 10).

Figure 8

(Color online) Dependence of the laser-free spectral FWHM ${\Gamma }_{\text{e}}^0$ and streaking broadening $\Delta {\Gamma }_{\text{e}}$ on the duration $D_{\text{X}}$ and chirp $b$. Positive $\Delta {\Gamma }_{\text{e}}$ corresponds to the release point $t_{\text{X}}=-840$ as and negative to $t_{\text{X}}=840$ as. The $b=\text{const}$ lines (solid lines) are labeled by the chirp rate in the units of eV/fs. The dots in the lines are placed at equal intervals of 50 as between $D_{\text{X}}=300$ and 1000 as. Points of equal duration are connected by dotted lines.