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Deep learning in attosecond metrology

Christian Brunner, Andreas Duensing, Christian Schröder, Michael Mittermair, Vladimir Golkov, Maximilian Pollanka, Daniel Cremers, and Reinhard Kienberger

Open Access Open Access

Abstract

Time-resolved photoelectron spectroscopy provides a versatile tool for investigating electron dynamics in gaseous, liquid, and solid samples on sub-femtosecond time scales. The extraction of information from spectrograms recorded with the attosecond streak camera remains a difficult challenge. Common algorithms are highly specialized and typically computationally heavy. In this work, we apply deep neural networks to map from streaking traces to near-infrared pulses as well as electron wavepackets and extensively benchmark our results on simulated data. Additionally, we illustrate domain-shift to real-world data. We also attempt to quantify the model predictive uncertainty. Our deep neural networks display competitive retrieval quality and superior tolerance against noisy data conditions, while reducing the computational time by orders of magnitude.

© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement

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Supplementary Material (1)

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Supplement 1       Supplemental Document

Data availability

Data underlying the results presented in this paper as well as the source code and trained neural networks may be obtained from the authors upon reasonable request.

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Figures (7)
Fig. 1.
Fig. 1. Visual impression of the range of simulated signal-to-noise ratios ($\mathrm {SNR}$). Pseudo-color plots of a unique WPA-based spectrogram without background subtraction (left column) and with differential background subtraction (right column). Applying our pre-processing pipeline we generate data that resemble noisy real-world conditions.

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Fig. 2.
Fig. 2. Comparison of different methods for single-trace retrieval. (a) Pseudo-color plot of the simulated test set spectrogram with $\mathrm {SNR} \approx 12~\mathrm {dB}$. (b), (d), (f) NIR vector potential retrievals compared to the label (simulation input). (c), (e), (g) Electron wavepacket retrievals compared to label. Our deep learning approaches better match the ground-truth labels, especially regarding the wavepacket spectral phase.

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Fig. 3.
Fig. 3. Single-trace retrieval quality as function of input noise. Box plots depicting the distribution of the respective quality criterion over the evaluated samples concerning: (a) Wavepacket spectrum, (b) Wavepacket spectral phase, (c) NIR vector potential and (d) Spectrogram reconstruction. Lower values are better, showing that our deep learning approaches significantly outperform ePIE at high noise levels.

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Fig. 4.
Fig. 4. Real-world single-trace retrieval using a deep ensemble. (a) Pseudo-color plot of the experimentally recorded spectrogram with Helium as a target. (b) Pseudo-color plot of the attained spectrogram reconstruction. (c) NIR vector potential retrieval with uncertainty estimate. (d) Electron wavepacket retrieval with uncertainty estimate. We find excellent agreement between recorded and reconstructed streaking trace.

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Fig. 5.
Fig. 5. Deep learning approaches for dual-trace retrieval. (a) Pseudo-color plot of the simulated test set spectrogram with $\mathrm {SNR} \approx 12~\mathrm {dB}$. (b) Pseudo-color plot of the spectrogram after differential background subtraction. (c), (e) NIR vector potential retrievals compared to label. (d), (f) Electron wavepacket retrievals compared to labels. Our deep learning approaches nicely match the ground-truth labels, managing to reconstruct a first-order phase term for the lower energetic wavepacket.

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Fig. 6.
Fig. 6. Central-energy photoemission time delay retrieval quality as function of input noise. (a) Time delay retrieval $\mathrm {RMSE}$. (b) Estimated bias of rTDSE methods. (c) Estimated bias of deep learning methods. (d) Percentage of plausible predictions for the respective methods on the benchmark dataset. Our deep learning approaches yield unbiased predictions and competitive $\mathrm {RMSE}$ with respect to rTDSE methods.

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Fig. 7.
Fig. 7. Real-world dual trace retrieval using a deep ensemble. (a) Pseudo-color plot of the experimentally recorded spectrogram with Argon as a target. (b) Pseudo-color plot of the attained spectrogram reconstruction. (c) NIR vector potential retrieval with uncertainty estimate. (d) Electron wavepacket retrievals with uncertainty estimate. We observe good agreement between recorded and reconstructed spectrogram. More data will be required to extract a statistically significant photoemission time delay value.

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Equations (11)

Equations on this page are rendered with MathJax. Learn more.

S W P A ( E f , τ ) = | ȷ ∫ − ∞ ∞ d t χ ( t + τ ) e ȷ ( E f + I p ) t e ȷ Φ V ( E f , t ) | 2 ,
Φ V ( E , t ) = − ∫ t ∞ d t ′ ( 2 E A N I R ( t ′ ) + 1 2 A N I R 2 ( t ′ ) ) ,
χ ~ ( E ) = I W P ( E ) e ȷ ϕ W P ( E ) ,
S C M A ( E f , τ ) ∝ | F − 1 [ χ ( t + τ ) G ~ ( t ) ] | 2 ,
S N R = 10 log 10 ⁡ [ ∑ i ∑ j S [ i , j ] ∑ i ∑ j | S [ i , j ] − S ′ [ i , j ] | ] ,
J ( θ → ) = 1 N ∑ n = 0 N − 1 L ( y ( n ) , f ( x ( n ) ; θ → ) ) ,
L A ( A → N I R , A → ^ N I R ) = ‖ A → N I R − A → ^ N I R ‖ 2 ‖ A → N I R ‖ 2 .
L ϕ ( ϕ → W P , ϕ → ^ W P ) = ∑ i = 0 I − 1 [ w [ i ] ( ϕ W P [ i ] − ϕ ^ W P [ i ] ) 2 + η ( I W P [ i ] − I ^ W P [ i ] ) 2 ] .
w [ i ] = 1 ( I W P [ i ] > t h ) ,
Q ϕ ( ϕ → W P , ϕ → ^ W P ) = [ ∑ i w [ i ] ( ϕ W P [ i ] − ϕ ^ W P [ i ] ) 2 ∑ i w [ i ] ] 1 2 .
Q S ( S , S ^ ) = [ ∑ i ∑ j ( S [ i , j ] − S ^ [ i , j ] ) 2 ∑ i ∑ j ( S [ i , j ] ) 2 ] 1 2 .
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