Phase retrieval from angular streaking of XUV atomic ionization

This paper demonstrates and validates a technique for accurately retrieving atomic ionization phases and time delays from XUV photoelectrons streaked by a circularly polarized IR field, confirming its equivalence to RABBITT simulations via numerical solutions of the time-dependent Schrödinger equation for hydrogen.

The Gist

Imagine you are trying to take a photograph of a lightning bolt. The problem is that lightning happens so fast that your camera shutter is too slow to capture it clearly. In the world of atoms, electrons move even faster than lightning. To "see" them, scientists use ultra-short flashes of light called attosecond pulses (one attosecond is to a second what a second is to the age of the universe).

However, there's a catch. The machines that create these flashes (called Free-Electron Lasers) are a bit chaotic. They don't fire perfectly on a schedule; the timing of each flash is slightly random, like a drummer who keeps a great beat but occasionally hits the snare a millisecond early or late.

This paper introduces a clever new way to measure these lightning-fast electrons, even when the "drummer" is unpredictable.

The Problem: The Jittery Drummer

Traditionally, to measure how fast an electron leaves an atom, scientists used a technique called RABBITT. Think of this like a dance routine where two partners (the XUV light flash and a laser field) must move in perfect, synchronized steps. To get the measurement, you have to carefully adjust the timing between the partners step-by-step.

But with Free-Electron Lasers, you can't do this step-by-step dance. The "drummer" (the laser) is too jittery. You can't predict exactly when the next flash will hit, so you can't set up a perfect, controlled dance.

The Solution: The Spinning Top (Angular Streaking)

The authors propose a new method called Angular Streaking. Instead of a synchronized dance, imagine a spinning top.

1. The Setup: You have an atom (like a hydrogen atom). You hit it with a super-fast flash of light (the XUV pulse) to knock an electron out.
2. The Spin: At the same time, you blast it with a circularly polarized laser. This laser acts like a giant, invisible spinning top or a merry-go-round.
3. The Result: When the electron is knocked out, it doesn't fly straight. The spinning laser field grabs it and twists its path, like a wind blowing a kite. The direction the electron flies depends on exactly when it was knocked out relative to the spin of the laser.

The Magic Trick: Random Shots Work Too

Here is the genius part of this paper. Because the laser is spinning, the electron's final direction tells you the time it was knocked out, even if you don't know exactly when the flash happened.

• Old Way: You need to know the exact time difference between the flash and the laser to calculate the speed. (Requires a perfect, predictable schedule).
• New Way: You just take a bunch of random snapshots. Some electrons fly left, some right, some up, some down. By looking at the pattern of where all the electrons land after thousands of random shots, you can mathematically reconstruct the timing.

It's like trying to figure out the speed of a car by watching where it lands after hitting a spinning fan. Even if you don't know exactly when the car hit the fan, if you see where the car parts land, you can figure out the speed and the timing of the impact.

Why This Matters

The authors tested this idea using a computer simulation (solving complex math equations called the Schrödinger equation) on a hydrogen atom. They found that:

1. It's Accurate: The new method gives the same precise timing information as the old, difficult methods.
2. It's Robust: It works even when the laser shots are completely random and unpredictable.
3. It's Versatile: It works for electrons with different energies, from very slow to very fast.

The Big Picture

This research is a breakthrough for attosecond science. It means scientists can finally use the powerful, chaotic Free-Electron Lasers to study the fastest processes in nature (like electrons moving in atoms or molecules) without needing to perfectly synchronize the equipment.

In short: They found a way to take a clear, high-speed photo of an electron's journey using a "spinning laser" camera, even when the camera's shutter is firing at random times. This opens the door to understanding the fundamental timing of how atoms react to light, which could lead to faster electronics and new materials.

Technical Summary

1. Problem Statement

The characterization of isolated attosecond pulses (IAPs) from Free-Electron Laser (FEL) sources is critical for attosecond science. However, existing interferometric techniques (such as RABBITT or standard streaking) typically require a systematic, controllable scan of the time delay between the ionizing XUV pulse and the streaking IR laser pulse.

This requirement presents a significant bottleneck for FEL sources because:

• Stochastic Nature: Self-amplified spontaneous emission (SASE) in FELs produces radiation with inherent timing jitter.
• Random Arrival: The arrival time of XUV pulses relative to the IR laser varies randomly from shot to shot.
• Impossibility of Scanning: The inherent jitter makes it impossible to perform the precise, incremental delay scans required by traditional methods in a single measurement set.

The paper addresses the need for a phase retrieval technique that can work in a "random shot mode," where the time delay ($\tau$) between XUV and IR pulses is unknown and varies stochastically for every shot.

2. Methodology

The authors propose and validate a method based on Angular Streaking of XUV Ionization (ASX) using a circularly polarized IR laser field. The methodology relies on the following theoretical and numerical framework:

• Physical Mechanism: The technique combines elements of the "attoclock" (circular polarization rotating the photoelectron momentum) and the "attosecond streak camera." The IR field rotates the photoelectron momentum distribution (PMD) in the polarization plane.
• Theoretical Basis:
  • The authors utilize the Strong Field Approximation (SFA) and Lowest Order Perturbation Theory (LOPT).
  • They derive a modified isochrone equation (Eq. 8) that relates the photoelectron momentum ($k$), emission angle ($\phi$), and time delay ($\tau$).
  • Crucially, they introduce a streaking phase ($\Phi_S$) into this equation. This phase is composed of the XUV ionization (Wigner) phase and the laser-induced continuum-continuum (CC) phase.
  • The relationship is approximated as: $k^2/2 - E_0 \approx k A_0 \cos[\phi - \omega\tau + \omega\alpha]$, where $\omega\alpha$ represents the streaking phase $\Phi_S$.
• Numerical Approach:
  • The authors solve the Time-Dependent Schrödinger Equation (TDSE) numerically for the hydrogen atom.
  • They simulate a series of "random shots" where the XUV/IR delay $\tau$ is varied stochastically rather than incrementally.
  • Data Extraction: Instead of scanning $\tau$, they analyze the radial momentum profiles ($P_{\pm}$) integrated over specific angular lobes ($\pm 90^\circ$ around $\phi=0$ and $\phi=180^\circ$).
  • By fitting the displacement of these momentum lobes ($k_{\pm}$) against the known vector potential $A_0$, they extract the average streaking phase $\Phi_S$ and the effective vector potential magnitude, even without knowing the exact $\tau$ for each individual shot.

3. Key Contributions

1. Random Shot Phase Retrieval: The primary contribution is demonstrating that accurate phase retrieval is possible without a controlled delay scan. The method extracts the streaking phase from a statistical ensemble of random shots, overcoming the timing jitter limitations of FELs.
2. Validation via Hydrogen Atom: The technique is rigorously tested on the hydrogen atom across a wide range of photon energies (from the ionization threshold to several times higher).
3. Benchmarking against RABBITT: The authors compare their ASX results with the well-established RABBITT (Reconstruction of Attosecond Beating by Interference of Two-photon Transitions) technique. They show that the atomic time delays derived from ASX ($\tau_S = \Phi_S/\omega$) are comparable to those from RABBITT ($\tau_R = \Phi_R/2\omega$).
4. Robustness Analysis: The study demonstrates that the retrieved phase is:
   • Insensitive to IR Intensity: Variations in IR intensity (by 50%) result in only minor changes (~10%) in the retrieved phase.
   • Insensitive to Ellipticity: The method remains robust even if the IR polarization deviates slightly from perfect circularity.
   • Calibration Capable: The maximum vertical displacement of the PMD lobes allows for the determination of the effective vector potential $A_0$ and the zero-delay point ($\tau=0$) purely from the data statistics.

4. Key Results

• Accuracy: The method achieves an accuracy of approximately 20% in retrieving the streaking phase, which is comparable to other streaking measurements.
• Phase Agreement: Figure 4 in the paper shows a close agreement between the RABBITT phase ($\Phi_R$) and twice the streaking phase ($2\Phi_S$) across a wide energy range.
• Time Delay Consistency: The extracted atomic time delays ($\tau_S$) align well with analytical Coulombic Wigner time delays (augmented by CC corrections) for the hydrogen atom.
• Yukawa Atom Test: Simulations on a Yukawa atom (screened Coulomb potential) showed that the streaking phase vanishes, confirming that the observed time delay in hydrogen is indeed of Coulombic origin.
• Mid-IR Extension: The method was successfully extended to mid-IR laser wavelengths ($\lambda = 10.6 \, \mu m$), demonstrating stability and the ability to measure longer time delays (e.g., 260 as) compared to near-IR (125 as).

5. Significance

• Enabling FEL Characterization: This work provides a practical solution for characterizing attosecond pulses at X-ray Free-Electron Lasers (XFELs), where timing jitter is unavoidable. It removes the need for complex, slow delay-scanning hardware.
• Single-Shot Potential: While the current demonstration uses an ensemble average, the methodology suggests that with sufficient statistics, single-shot or few-shot phase retrieval is feasible, which is crucial for studying non-reproducible or transient samples.
• Broad Applicability: The technique is not limited to hydrogen; the authors note its extension to the $H_2$ molecule, where it successfully reproduces orientation and two-center interference effects.
• Fundamental Physics: It reinforces the interpretation of atomic time delays as the sum of the photoionization time and the measurement process time, validating theoretical models of electron dynamics in strong fields.

In summary, the paper establishes Angular Streaking from Random Shots as a viable, robust, and accurate alternative to traditional interferometric methods for characterizing attosecond pulses in stochastic FEL environments.