# Bayesian Analysis of Femtosecond Pump-Probe Photoelectron-Photoion Coincidence Spectra with Fluctuating Laser Intensities

^{1}

^{2}

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

## 2. Experiment

## 3. Bayesian Data Analysis

#### 3.1. Preliminary Considerations

#### 3.2. The Posterior PDF

#### 3.3. The Prior PDF

#### 3.4. The Likelihood

#### 3.5. Remarks on the Posterior Sampling

## 4. Mock Data Analysis

#### 4.1. False Coincidences

#### 4.2. Background Subtraction

## 5. Application to Experimental Data

## 6. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A. Solution of the 〈λ^{n} 〉-Integral

## Appendix B. Channel-Resolved Single Coincidences

## Appendix C. Transformation of the Dirac Distribution

## Appendix D. The Jacobian Determinant

## Appendix E. Probabilities of the Count-Pairs (N_{e}, N_{i})

## References

- Brehm, B.; von Puttkamer, E. Koinzidenzmessung von Photoionen und Photoelektronen bei Methan. Z. Naturforsch. A
**1967**, 22, 8–10. [Google Scholar] [CrossRef] - Boguslavskiy, A.E.; Mikosch, J.; Gijsbertsen, A.; Spanner, M.; Patchkovskii, S.; Gador, N.; Vrakking, M.J.J.; Stolow, A. The multielectron ionization dynamics underlying attosecond strong-field spectroscopies. Science
**2012**, 335, 1336–1340. [Google Scholar] [CrossRef] [PubMed] - Sándor, P.; Zhao, A.; Rozgonyi, T.; Weinacht, T. Strong field molecular ionization to multiple ionic states: Direct versus indirect pathways. J. Phys. B At. Mol. Opt. Phys.
**2014**, 47, 124021. [Google Scholar] [CrossRef] - Koch, M.; Heim, P.; Thaler, B.; Kitzler, M.; Ernst, W.E. Direct observation of a photochemical activation energy: A Case study of acetone photodissociation. J. Phys. B At. Mol. Opt. Phys.
**2017**, 50, 125102. [Google Scholar] [CrossRef] - Arion, T.; Hergenhahn, U. Coincidence spectroscopy: Past, present and perspectives. J. Electron Spectrosc. Relat. Phenom.
**2015**, 200, 222–231. [Google Scholar] [CrossRef][Green Version] - Continetti, R.E. Coincidence Spectroscopy. Ann. Rev. Phys. Chem.
**2001**, 52, 165–192. [Google Scholar] [CrossRef] [PubMed] - Maierhofer, P.; Bainschab, M.; Thaler, B.; Heim, P.; Ernst, W.E.; Koch, M. Disentangling multichannel photodissociation dynamics in acetone by time-resolved photoelectron-photoion coincidence spectroscopy. J. Phys. Chem. A
**2016**, 120, 6418–6423. [Google Scholar] [CrossRef] [PubMed] - Couch, D.E.; Kapteyn, H.C.; Murnane, M.M.; Peters, W.K. Uncovering highly-excited state mixing in acetone using ultrafast VUV pulses and coincidence imaging techniques. J. Phys. Chem. A
**2017**, 121, 2361–2366. [Google Scholar] [CrossRef] [PubMed] - Wilkinson, I.; Boguslavskiy, A.E.; Mikosch, J.; Bertrand, J.B.; Wörner, H.J.; Villeneuve, D.M.; Spanner, M.; Patchkovskii, S.; Stolow, A. Excited state dynamics in SO2. I. Bound state relaxation studied by time-resolved photoelectron-photoion coincidence spectroscopy. J. Chem. Phys.
**2014**, 140, 204301. [Google Scholar] [CrossRef][Green Version] - Hertel, I.V.; Radloff, W. Ultrafast dynamics in isolated molecules and molecular clusters. Rep. Prog. Phys.
**2006**, 69, 1897–2003. [Google Scholar] [CrossRef] - Stert, V.; Radloff, W.; Schulz, C.P.; Hertel, I.V. Ultrafast photoelectron spectroscopy: Femtosecond pump-probe coincidence detection of ammonia cluster ions and electrons. Eur. Phys. J. D
**1999**, 5, 97–106. [Google Scholar] [CrossRef] - Stolow, A.; Bragg, A.E.; Neumark, D.M. Femtosecond time-resolved photoelectron spectroscopy. Chem. Rev.
**2004**, 104, 1719–1757. [Google Scholar] [CrossRef] [PubMed] - Nugent-Glandorf, L.; Scheer, M.; Samuels, D.A.; Mulhisen, A.M.; Grant, E.R.; Yang, X.; Bierbaum, V.M.; Leone, S.R. Ultrafast time-resolved soft X-ray photoelectron spectroscopy of dissociating Br2. Phys. Rev. Lett.
**2001**, 87, 193002. [Google Scholar] [CrossRef] - Gregory, P. Bayesian Logical Data Analysis for the Physical Sciences: A Comparative Approach with Mathematica
^{®}Support; Cambridge University Press: Cambridge, UK, 2005. [Google Scholar] - Loredo, T.J. The promise of bayesian inference for astrophysics. In Statistical Challenges in Modern Astronomy; Feigelson, E., Babu, G., Eds.; Springer: New York, NY, USA, 1992; pp. 275–297. [Google Scholar]
- Gertner, I.; Heber, O.; Zajfman, J.; Zajfman, D.; Rosner, B. Comparison between two computer codes for PIXE studies applied to trace element analysis in amniotic fluid. Nucl. Instrum. Methods Phys. Res. Sect. B Beam Interact. Mater. Atoms
**1989**, 36, 74–81. [Google Scholar] [CrossRef] - Von der Linden, W.; Dose, V.; Padayachee, J.; Prozesky, V. Signal and background separation. Phys. Rev. E
**1999**, 59, 6527–6534. [Google Scholar] [CrossRef][Green Version] - Von der Linden, W.; Dose, V.; Fischer, R. How to separate the signal from the background. In Proceedings of the MAXENT96—Maximum Entropy Conference, Berg-en-Dal, South Africa, 12–17 August 1996; p. 146. [Google Scholar]
- Prozesky, V.M.; Padayachee, J.; Fischer, R.; von der Linden, W.; Dose, V.; Ryan, C.G. The use of maximum entropy and Bayesian techniques in nuclear microprobe applications. Nucl. Instrum. Methods Phys. Res. Sect. B Beam Interact. Mater. Atoms
**1997**, 130, 113–117. [Google Scholar] [CrossRef] - Padayachee, J.; Prozesky, V.; von der Linden, W.; Nkwinika, M.S.; Dose, V. Bayesian PIXE background subtraction. Nucl. Instrum. Methods Phys. Res. Sect. B Beam Interact. Mater. Atoms
**1999**, 150, 129–135. [Google Scholar] [CrossRef] - Fischer, R.; Hanson, K.M.; Dose, V.; von der Linden, W. Background estimation in experimental spectra. Phys. Rev. E
**2000**, 61, 1152–1160. [Google Scholar] [CrossRef] - Rumetshofer, M.; Heim, P.; Thaler, B.; Ernst, W.E.; Koch, M.; von der Linden, W. Analysis of femtosecond pump-probe photoelectron-photoion coincidence measurements applying Bayesian probability theory. Phys. Rev. A
**2018**, 97, 062503. [Google Scholar] [CrossRef][Green Version] - Dörner, R.; Mergel, V.; Jagutzki, O.; Spielberger, L.; Ullrich, J.; Moshammer, R.; Schmidt-Böcking, H. Cold target recoil ion momentum spectroscopy: A momentum microscope to view atomic collision dynamics. Phys. Rep.
**2000**, 330, 95–192. [Google Scholar] [CrossRef] - Ullrich, J.; Moshammer, R.; Dorn, A.; Dörner, R.; Schmidt, L.P.H.; Schmidt-Böcking, H. Recoil-ion and electron momentum spectroscopy: Reaction-microscopes. Rep. Prog. Phys.
**2003**, 66, 1463–1545. [Google Scholar] [CrossRef] - Frasinski, L.J.; Codling, K.; Hatherly, P.A. Covariance mapping: A correlation method applied to multiphoton multiple ionization. Science
**1989**, 246, 1029–1031. [Google Scholar] [CrossRef] [PubMed] - Mikosch, J.; Patchkovskii, S. Coincidence and covariance data acquisition in photoelectron and -ion spectroscopy. I. Formal theory. J. Mod. Opt.
**2013**, 60, 1426–1438. [Google Scholar] [CrossRef] - Mikosch, J.; Patchkovskii, S. Coincidence and covariance data acquisition in photoelectron and -ion spectroscopy. II. Analysis and applications. J. Mod. Opt.
**2013**, 60, 1439–1451. [Google Scholar] [CrossRef] - Koch, M.; Thaler, B.; Heim, P.; Ernst, W.E. The Role of Rydberg–Valence Coupling in the Ultrafast Relaxation Dynamics of Acetone. J. Phys. Chem. A
**2017**, 121, 6398–6404. [Google Scholar] [CrossRef] [PubMed][Green Version] - Heim, P.; Rumetshofer, M.; Ranftl, S.; Thaler, B.; Ernst, W.E.; Koch, M.; von der Linden, W. Bayesian Analysis of Femtosecond Pump-Probe Photoelectron-Photoion Coincidence Spectra. In Proceedings of the 38th International Workshop on Bayesian Inference and Maximum Entropy Methods in Science and Engineering, London, UK, 2–6 July 2018; Volume 38. [Google Scholar]
- Von der Linden, W.; Dose, V.; von Toussaint, U. Bayesian Probability Theory: Applications in the Physical Sciences; Cambridge University Press: Cambridge, UK, 2014. [Google Scholar]

**Figure 1.**Utterly simplified sketch of a time-resolved photoionization study carried out with a pump-probe setup and a time-of-flight spectrometer. A commercial Ti:sapphire laser system delivers pulses of $800$ nm in center wavelength and $25$ fs in temporal length at a repetition rate of $3$ kHz. The delay stage is used to control the length of the optical path, and hence the time delay. The energy level diagram shows how the electron kinetic energy, given the energy of the states and the photons, identifies the state the system was in at the moment of ionization. A detailed description of the setup can be found in our previous publications [7,28].

**Figure 3.**Simulation with mock data for studying the influence of $\lambda $-fluctuations on false coincidences. The black lines are the spectra used to generate the data; the green (blue) lines including $\pm \sigma $ error bands are the reconstructed spectra (not) including $\lambda $-fluctuations in the reconstruction. The parameters are ${\xi}_{i}={\xi}_{e}=0.5$ and ${\mathcal{N}}_{p}={10}^{7}$. For ${\underline{\lambda}}_{2}=1.5$, differences between the algorithms are negligible even at relatively high $\lambda $-fluctuations with ${\sigma}_{2}=0.5$; see spectra (

**a**,

**b**). When choosing ${\underline{\lambda}}_{2}=0.5$ (

**c**,

**d**), the algorithm not including $\lambda $-fluctuations produces small deviations, e.g., underestimation of the false coincidences at the first Gaussian in the fragment spectrum.

**Figure 4.**Simulated test spectra for studying the influence of $\lambda $-fluctuations on the background subtraction. The parameters are ${\underline{\lambda}}_{1}={\underline{\lambda}}_{2}=0.5$, ${\xi}_{i}={\xi}_{e}=0.5$, and ${\mathcal{N}}_{p}={10}^{7}$. ${\sigma}_{1}$ and ${\sigma}_{2}$ are different for every sub-figure. If ${\sigma}_{1}={\sigma}_{2}=0.1$ (

**a**) or ${\sigma}_{1}={\sigma}_{2}=0.5$ (

**b**), both algorithms (with (green line) and without (blue line) including $\lambda $-fluctuations) reconstruct the spectra correctly. ${\sigma}_{1}=0.1$ and ${\sigma}_{2}=0.5$ lead to an underestimation of the background when neglecting $\lambda $-fluctuations (

**c**). Overestimation of the background happens in the case of ${\sigma}_{1}=0.5$ and ${\sigma}_{2}=0.1$ (

**d**).

**Table 1.**Estimated parameters ${\underline{\lambda}}_{2}$, ${\sigma}_{2}$, ${\xi}_{i}$, and ${\xi}_{e}$. In the lines showing the results of the algorithm presented in [22], ${\lambda}_{2}$ is shown instead of ${\underline{\lambda}}_{2}$. Each value denotes the mean and standard deviation of the parameter’s distribution.

λ${}_{2}$ | ${\mathit{\sigma}}_{2}$ | ${\mathit{\xi}}_{\mathit{i}}$ | ${\mathit{\xi}}_{\mathit{e}}$ | |
---|---|---|---|---|

Parameters (Figure 3a,b) | $1.5$ | $0.5$ | $0.5$ | $0.5$ |

Algorithm in [22] | $1.4177\pm 0.0008$ | - | $0.5235\pm 0.0003$ | $0.5235\pm 0.0003$ |

Algorithm with $\lambda $-fluctuations | $1.499\pm 0.001$ | $0.501\pm 0.002$ | $0.4998\pm 0.0003$ | $0.5000\pm 0.0003$ |

Parameters (Figure 3c,d) | $0.5$ | $0.5$ | $0.5$ | $0.5$ |

Algorithm in [22] | $0.4365\pm 0.0003$ | - | $0.5618\pm 0.0004$ | $0.5621\pm 0.0004$ |

Algorithm with $\lambda $-fluctuations | $0.4992\pm 0.0005$ | $0.499\pm 0.001$ | $0.5000\pm 0.0004$ | $0.5004\pm 0.0004$ |

**Table 2.**Estimated parameters ${\underline{\lambda}}_{1}$, ${\underline{\lambda}}_{2}$, ${\sigma}_{1}$, ${\sigma}_{2}$, ${\xi}_{i}$, and ${\xi}_{e}$. The parameter regimes denoted by the identifications (a–d) are according to Figure 4. For each parameter set, the first line denotes the true value, while Line 2 (3) contains the parameter estimation performed with the algorithm without (with) $\lambda $-fluctuations, respectively. In the lines showing the results of the algorithm ignoring $\lambda $-fluctuations, ${\lambda}_{j}$ is shown instead of ${\underline{\lambda}}_{j}$. Each value denotes the mean and standard deviation of the parameter’s distribution.

λ${}_{1}$ | λ${}_{2}$ | ${\mathit{\sigma}}_{1}$ | ${\mathit{\sigma}}_{2}$ | ${\mathit{\xi}}_{\mathit{i}}$ | ${\mathit{\xi}}_{\mathit{e}}$ | |
---|---|---|---|---|---|---|

(a) | $0.5$ | $0.5$ | $0.1$ | $0.1$ | $0.5$ | $0.5$ |

$0.4970\pm 0.0003$ | $0.4972\pm 0.0005$ | - | - | $0.5032\pm 0.0002$ | $0.5029\pm 0.0002$ | |

$0.5000\pm 0.0003$ | $0.5007\pm 0.0005$ | $0.098\pm 0.002$ | $0.101\pm 0.005$ | $0.5003\pm 0.0003$ | $0.5000\pm 0.0003$ | |

(b) | $0.5$ | $0.5$ | $0.5$ | $0.5$ | $0.5$ | $0.5$ |

$0.4367\pm 0.0003$ | $0.4317\pm 0.0004$ | - | - | $0.5618\pm 0.0002$ | $0.5620\pm 0.0002$ | |

$0.5002\pm 0.0004$ | $0.5009\pm 0.0006$ | $0.5026\pm 0.0008$ | $0.496\pm 0.002$ | $0.4996\pm 0.0003$ | $0.4998\pm 0.0003$ | |

(c) | $0.5$ | $0.5$ | $0.1$ | $0.5$ | $0.5$ | $0.5$ |

$0.4840\pm 0.0003$ | $0.4608\pm 0.0005$ | - | - | $0.5201\pm 0.0002$ | $0.5201\pm 0.0002$ | |

$0.4991\pm 0.0003$ | $0.5013\pm 0.0006$ | $0.098\pm 0.002$ | $0.500\pm 0.002$ | $0.5001\pm 0.0002$ | $0.5001\pm 0.0003$ | |

(d) | $0.5$ | $0.5$ | $0.5$ | $0.1$ | $0.5$ | $0.5$ |

$0.4452\pm 0.0003$ | $0.4675\pm 0.0004$ | - | - | $0.5441\pm 0.0002$ | $0.5440\pm 0.0002$ | |

$0.4998\pm 0.0004$ | $0.5005\pm 0.0006$ | $0.5002\pm 0.0009$ | $0.102\pm 0.004$ | $0.4998\pm 0.0003$ | $0.4997\pm 0.0003$ |

**Table 3.**Estimated parameters ${\underline{\lambda}}_{1}$, ${\underline{\lambda}}_{2}$, ${\sigma}_{1}$, ${\sigma}_{2}$, ${\xi}_{i}$, and ${\xi}_{e}$. Line 1 (2) contains the parameter estimations performed with the algorithm without (with) $\lambda $-fluctuations, respectively. In the line showing the results of the algorithm ignoring $\lambda $-fluctuations, ${\lambda}_{j}$ is shown instead of ${\underline{\lambda}}_{j}$. Each value denotes the mean and standard deviation of the parameter’s distribution.

λ${}_{1}$ | λ${}_{2}$ | ${\mathit{\sigma}}_{1}$ | ${\mathit{\sigma}}_{2}$ | ${\mathit{\xi}}_{\mathit{i}}$ | ${\mathit{\xi}}_{\mathit{e}}$ |
---|---|---|---|---|---|

$0.3328\pm 0.0009$ | $0.132\pm 0.001$ | - | - | $0.3247\pm 0.0008$ | $0.542\pm 0.001$ |

$0.3328\pm 0.0009$ | $0.1335\pm 0.0003$ | $0.0004\pm 0.0010$ | $0.0004\pm 0.0012$ | $0.3245\pm 0.0008$ | $0.541\pm 0.001$ |

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Heim, P.; Rumetshofer, M.; Ranftl, S.; Thaler, B.; Ernst, W.E.; Koch, M.; von der Linden, W.
Bayesian Analysis of Femtosecond Pump-Probe Photoelectron-Photoion Coincidence Spectra with Fluctuating Laser Intensities. *Entropy* **2019**, *21*, 93.
https://doi.org/10.3390/e21010093

**AMA Style**

Heim P, Rumetshofer M, Ranftl S, Thaler B, Ernst WE, Koch M, von der Linden W.
Bayesian Analysis of Femtosecond Pump-Probe Photoelectron-Photoion Coincidence Spectra with Fluctuating Laser Intensities. *Entropy*. 2019; 21(1):93.
https://doi.org/10.3390/e21010093

**Chicago/Turabian Style**

Heim, Pascal, Michael Rumetshofer, Sascha Ranftl, Bernhard Thaler, Wolfgang E. Ernst, Markus Koch, and Wolfgang von der Linden.
2019. "Bayesian Analysis of Femtosecond Pump-Probe Photoelectron-Photoion Coincidence Spectra with Fluctuating Laser Intensities" *Entropy* 21, no. 1: 93.
https://doi.org/10.3390/e21010093