Near-Forward Rescattering Photoelectron Holography in Strong-Field Ionization: Extraction of the Phase of the Scattering Amplitude

We revisit the concept of near-forward rescattering strong-field photoelectron holography introduced by Y. Huismans et al. [Science 331, 61 (2011)]. The recently developed adiabatic theory is used to show how the phase of the scattering amplitude for near-forward rescattering of an ionized electron by the parent ion is encoded in and can be read out from the corresponding interference pattern in photoelectron momentum distributions (PEMDs) produced in the ionization of atoms and molecules by intense laser pulses. A procedure to extract the phase is proposed. Its application to PEMDs obtained by solving the time-dependent Schrödinger equation for a model atom yields results in good agreement with scattering calculations. This establishes a novel general approach to extracting structural information from strong-field observables capable of providing time-resolved imaging of ultrafast processes.

Figure 1

(a) TDSE results for the PEMD produced in the ionization of a model atom described by the potential (9) with $a=10$ by a one-cycle pulse with $\lambda \approx 800\mathrm{nm}$ and the intensity of $3.5\times {10}^{14}\mathrm{W}/{\mathrm{cm}}^2$. (b) Open circles: interference minima of the near-forward rescattering SFPEH extracted from the TDSE results using a procedure described in the text. Solid lines: positions of the minima in the adiabatic approximation found from $\mathrm{\Delta }{\varphi }_{\mathrm{AA}}=(2n+1)\pi$, with $n=0,1,\dots ,4$ and $\mathrm{\Delta }{\varphi }_{\mathrm{AA}}$ given by Eq. (8).

Figure 2

(a) Cuts of the PEMDs calculated for the potential (9) with $a=10$ (red circles) and $a=20$ (blue squares) and for the Coulomb potential (black diamonds) and averaged over $k_z$ in the interval $k_z=2.2\pm 0.1$ near the vertical dashed line in Fig. 1 as functions of $k_{\perp }$. (b) The function $\cos \mathrm{\Delta }{\varphi }_{\mathrm{TDSE}}$ extracted by fitting the results in panel (a) by Eq. (10). (c) The phase $\alpha$ of the scattering amplitude extracted from the results in panel (b) using $\mathrm{\Delta }{\varphi }_{\mathrm{TDSE}}=\mathrm{\Delta }{\varphi }_{\mathrm{AA}}$ and Eq. (8) (symbols) and obtained from scattering calculations (solid lines).

Figure 3

(a) The electric field of a few-cycle pulse with $\lambda \approx 1200\mathrm{nm}$ and the intensity of $3.5\times {10}^{14}\mathrm{W}/{\mathrm{cm}}^2$. (b) The PEMD produced by this pulse calculated for the potential (9) with $a=10$. (c) The phase $\alpha$ (in units of $\pi$) of the scattering amplitude extracted from the results in panel (b). (d) The same phase obtained from scattering calculations. (e) The extracted (symbols) and calculated (solid lines) phase $\alpha (p,\theta )$ at a fixed incident momentum $p=2.1$ as a function of the scattering angle $\theta$ taken along the white dashed lines in panels (c) and (d), respectively, for the potentials with $a=10$ and 20.