Eliminating the dipole phase in attosecond pulse characterization using Rydberg wave packets

We propose a technique to fully characterize the temporal structure of extreme ultraviolet pulses by ionizing a bound coherent electronic wave packet. The influence of the dipole phase, which is the main obstacle for state-of-the-art pulse characterization schemes, can be eliminated by angle integration of the photoelectron spectrum. We show that in particular, atomic Rydberg wave packets are ideal and that wave packets involving multiple electronic states provide redundant information that can be used to cross-check the consistency of the phase reconstruction.

Figure 1

Sketch of the method to characterize a uv pulse using a bound-electron wave packet. (a) Preparation of the wave packet with a laser pulse. (b) Field-free propagation of the electronic wave packet $\psi$ for the duration $\tau$. (c) Ionization of the wave packet by the uv pulse. Due to the coherent superposition of different electronic states, different spectral components of the uv pulse are absorbed to reach the same final continuum state with energy $\epsilon$. The interferences between the different spectral components contain the spectral phase information of the uv pulse that can be extracted by varying the delay $\tau$ and repeating the experiment.

Figure 2

(a)–(c) Photoelectron spectra of the $4p-5p$ Rydberg wave packet ionized by an attosecond pulse with (a) no chirp, (b) a linear chirp, and (c) a quadratic chirp ionizing as a function of the pulse delay. (d)–(f) Photoelectron spectra where the delay axis is Fourier-transformed. The spectrum contains a static signal and an oscillating component with frequency 1.36 eV. (g)–(i) Retrieved phase differences of the 1.36 eV component. Analytic results are shown as black dashed lines. The corresponding group delay of the phase differences is shown on the right.

Figure 3

(a) Spectra of the attosecond pulse, $E_{\mathrm{X}}\left({\omega }_{fj}\right)$, shifted by the Rydberg binding energies ${\epsilon }_{4p}, {\epsilon }_{7p}$, and ${\epsilon }_{8p}$ (indicated by arrows). (b) Photoelectron spectrum of the $4p-7p-8p$ Rydberg wave packet ionized by a quadratic-chirped single attosecond pulse. (c) The static and the 0.14, 2.12, and 2.26 eV oscillating signals of the photoelectron spectrum. (d) Retrieved phases of the three frequency components. The analytic results are shown as black dashed lines.

Figure 4

(a) Spectra of the attosecond pulse train, $E_{\mathrm{X}}\left({\omega }_{fj}\right)$, shifted by the Rydberg binding energies ${\epsilon }_{4p}, {\epsilon }_{7p}$, and ${\epsilon }_{8p}$ (indicated by arrows). (b) Photoelectron spectrum of the $4p-7p-8p$ Rydberg wave packet ionized by the attosecond pulse train. (c) The static and the 0.14, 2.12, and 2.26 eV oscillating components of the photoelectron spectrum. (d) Retrieved phases of the three frequency components. The analytic results are shown as black dashed lines. The highlighted phases (red circles) at 73.7 and 75.3 eV for the $4p-7p$ beating are used to determined the relative phases between harmonics.

Figure 5

(a) Dipole phases ${\phi }_{4p/5p}^z\left(\epsilon \right)=\text{arg}[\langle \epsilon ,{\vec{e}}_z\left|z\right|4p/5p\rangle ]$ for ionizing an electron in the $z$ direction with kinetic energy $\epsilon$. (b) Delay of the directional dipole phase ${\phi }_{4p}^z\left(\epsilon \right)$ (red dashed line) and the relative delay between $4p$ and $5p, {\varphi }_D^{(4p,5p)}\left(\epsilon \right)$, entering the directional (blue dashed line) and angle-integrated (blue solid line) photoelectron spectra. Data are produced by static HF calculations.

Figure 6

Dipole-induced delay for the directional (dashed lines) and for the angle-integrated (solid lines) photoelectron spectra of potassium treated in the HF approximation (blue lines) and RPAE (yellow lines).

Figure 7

Perturbation diagram for (a) HF, (b) direct-forward RPAE, (c) direct-backward RPAE, (d) exchange-forward RPAE, and (e) exchange-backward RPAE. Up (down) arrows label electron (hole) states. Further details are given in the main text.

Figure 8

(a) Angle-integrated photoelectron spectrum of the $2p^{-1}3s–2s^{-1}4s$ wave packet ionized by an isolated attosecond pulse. (b) The static (black dashed line) and oscillating components (solid red line) of the angle-integrated photoelectron spectrum. (c) Retrieved phase of the oscillating components of the angle-integrated (red solid line) and directional (green dashed-dotted line) photoelectron spectrum for the full TDCIS model. The intrachannel TDCIS results for the angle-integrated photoelectron spectrum are shown as a reference (blue dashed line). The corresponding group delay of the phase differences is shown on the right.