# Equivalence of RABBITT and Streaking Delays in Attosecond-Time-Resolved Photoemission Spectroscopy at Solid Surfaces

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## Abstract

**:**

## 1. Introduction

## 2. Methods

## 3. Results

#### 3.1. From RABBITT Regime to Streaking Regime

#### 3.2. Comparison of Streaking and RABBITT Delays with Classical Photoemission Times

## 4. Conclusions

_{2}photoemission spectra were successfully explained by the model applied here using jellium-type potential and including electron–hole interaction for electron transport while treating angular momentum influences as intra-atomic effects. Additionally, they exhibited stable surface conditions, which is helpful for such a challenging experiment [10].

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A. Extraction of Delays

**Figure A1.**Analysis of simulated photoelectron spectra for a 50 eV photoelectron interacting with an IR pulse with a center wavelength of 800 nm, pulse duration of 5 fs, and maximum electric field amplitude ${E}_{0}$ = $1\mathrm{V}/\mathrm{nm}$. (

**a**) Photoelectron is generated by a single-attosecond pulse (streaking). IR vector potential (red line) is fitted to the simulated data points. The residuum between model function and data points (blue line) exhibits systematic deviations in the order of ${10}^{-3}$ with respect to the streaking amplitude. (

**b**) Photoelectron is generated by an attosecond pulse train (RABBITT). Equation (9) (red line) is fitted to the simulated data points. Here, the residuum between model function and data points (blue line) shows systematic deviations in the order of a few percent of maximum side-band intensity.

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**Figure 1.**Principle of (

**a**) as-streaking and (

**b**) reconstruction of attosecond beating by interference of two-photon transitions (RABBITT) at solid surfaces: Initial state ${\Phi}_{0}\left(z\right)$ is excited into the continuum by a single-attosecond extreme-ultraviolet (EUV) pulse (green) or by an attosecond pulse train (red), respectively. The time–energy bandwidth product results in broad energy distribution in the continuum in the case of a single-attosecond pulse (SAP, green) and two narrow lines in the case of an attosecond pulse train (APT, red). The continuum wave packet $\Phi \left(z\right)$, i.e., a single electron wave packet (streaking) or an electron pulse train (RABBITT), travels toward the surface and is emitted into the vacuum (schematic wave packet and laser-pulse plots are not true to scale). As soon as the photoelectron leaves the surface, it starts interacting with the infrared (IR) pulse (violet). In the case of a single electron wave packet, the energy distribution accumulates a shift determined by IR vector potential. This results in a streaking spectrogram (upper-right panel). Here, the white solid line corresponds to the streaked momentum expectation value, while the white dashed line corresponds to the momentum expectation value without IR interaction. White vertical lines indicate the delay between streaking trace and zero IR–EUV pulse delay. If it is excited by an APT (RABBITT), at each half-cycle of the IR field, a photoelectron wave packet is emitted. Absorption and emission of IR photons leads to the formation of side bands in the RABBITT spectrum (lower-right panel). Intensity oscillation in the central side band (between white dashed lines) is shifted with respect to zero IR–EUV pulse delay (white vertical lines).

**Figure 2.**Schematic high harmonic spectrum of a few-cycle IR pulse. In the intermediate energy regime, separated harmonic lines of equal height form a plateau. Spectral filtering in this regime leads to EUV APTs. In the high-energy regime, a cutoff continuum is formed. To obtain SAPs, this regime needs to be spectrally filtered.

**Figure 3.**Normalized intensity of EUV excitation pulses (solid line) in the time domain corresponding to the EUV spectra shown in Figure 4a–d. Pulse train envelopes are also shown (dotted lines). The envelope full width at half maximum (FWHM) pulse durations are (

**a**) 6.7 fs, (

**b**) 1.7 fs, (

**c**) 1.1 fs, and (

**d**) 474 as.

**Figure 4.**Simulated photoelectron spectra (color plots) corresponding to four different EUV excitation spectra (black lines). All spectra were individually normalized. EUV pulse (train) was given by the Fourier transform of the EUV spectra. IR pulse was treated independently, with a center wavelength of 800 nm, pulse duration of 5 fs, and maximum electric field amplitude ${E}_{0}=1\mathrm{V}/\mathrm{nm}$. Effective propagation distance was 5 Å, and initial-state binding energy was ${\epsilon}_{0}=-52.9\mathrm{eV}$.

**Figure 5.**Comparison of streaking and RABBITT delays with classical photoemission transport times ${t}_{\mathrm{PE}}$. (

**a**) Variation of streaking (green crosses) and RABBITT delays (red dots), as well as classical photoemission times (black) as a function of (vacuum) kinetic energy. For the streaking data, error bars are not shown because they are smaller than the symbol size. Streaking and RABBITT delays were extracted from spectrograms, calculated by solving the one-dimensional time-dependent Schrödinger equation (TDSE) (Equation (6)) for potential properties described in the main text. (

**b**) Difference between RABBITT and streaking delays and classical photoemission times. (

**c**) Difference between RABBITT and streaking delays.

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**MDPI and ACS Style**

Gebauer, A.; Neb, S.; Enns, W.; Stadtmüller, B.; Aeschlimann, M.; Pfeiffer, W.
Equivalence of RABBITT and Streaking Delays in Attosecond-Time-Resolved Photoemission Spectroscopy at Solid Surfaces. *Appl. Sci.* **2019**, *9*, 592.
https://doi.org/10.3390/app9030592

**AMA Style**

Gebauer A, Neb S, Enns W, Stadtmüller B, Aeschlimann M, Pfeiffer W.
Equivalence of RABBITT and Streaking Delays in Attosecond-Time-Resolved Photoemission Spectroscopy at Solid Surfaces. *Applied Sciences*. 2019; 9(3):592.
https://doi.org/10.3390/app9030592

**Chicago/Turabian Style**

Gebauer, Andreas, Sergej Neb, Walter Enns, Benjamin Stadtmüller, Martin Aeschlimann, and Walter Pfeiffer.
2019. "Equivalence of RABBITT and Streaking Delays in Attosecond-Time-Resolved Photoemission Spectroscopy at Solid Surfaces" *Applied Sciences* 9, no. 3: 592.
https://doi.org/10.3390/app9030592