Abstract

Attosecond streaking is a powerful tool to investigate ultrafast electron dynamics on the attosecond time scale. To obtain the highest temporal resolution in a pump-probe experiment, soft-X-ray (SXR) and infrared (IR) pulses have to be carefully characterized. Here, we present a detailed description of our recent generalization of the Volkov-transform generalized projection algorithm (VTGPA) and its application to multiple overlapping photoelectron bands. This method allows for the complete temporal reconstruction of both IR and SXR pulses under the inclusion of accurate complex photoionization matrix elements (PMEs). In this article, we compare the performance of our new method with traditional algorithms. We particularly focus on the important role played by the photoelectron angular distribution (PAD) which needs to be taken into account for the highest fidelity of attosecond pulse reconstruction. For this purpose, we investigate numerically the influence of the finite collection angle of the electron spectrometer on the retrieval and the obtained pulse parameters. We further theoretically demonstrate the reliability of the reconstruction for pulse durations even shorter than the atomic unit of time.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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2017 (4)

Y. Pertot, C. Schmidt, M. Matthews, A. Chauvet, M. Huppert, V. Svoboda, A. von Conta, A. Tehlar, D. Baykusheva, J.-P. Wolf, and H. J. Wörner, “Time-resolved x-ray absorption spectroscopy with a water window high-harmonic source,” Science 355, 264–267 (2017).
[Crossref] [PubMed]

J. Li, X. Ren, Y. Yin, K. Zhao, A. Chew, Y. Cheng, E. Cunningham, Y. Wang, S. Hu, Y. Wu, M. Chini, and Z. Chang, “53-attosecond X-ray pulses reach the carbon K-edge,” Nat. Commun. 8, 186 (2017).
[Crossref] [PubMed]

H. Li, N. G. Kling, T. Gaumnitz, C. Burger, R. Siemering, J. Schötz, Q. Liu, L. Ban, Y. Pertot, J. Wu, A. M. Azzeer, R. de Vivie-Riedle, H. J. Wörner, and M. F. Kling, “Sub-cycle steering of the deprotonation of acetylene by intense few-cycle mid-infrared laser fields,” Opt. Express 25, 14192 (2017).
[Crossref] [PubMed]

T. Gaumnitz, A. Jain, Y. Pertot, M. Huppert, I. Jordan, F. Ardana-Lamas, and H. J. Wörner, “Streaking of 43-attosecond soft-X-ray pulses generated by a passively CEP-stable mid-infrared driver,” Opt. Express 25, 27506 (2017).
[Crossref] [PubMed]

2016 (4)

M. Huppert, I. Jordan, D. Baykusheva, A. von Conta, and H. J. Wörner, “Attosecond Delays in Molecular Photoionization,” Phys. Rev. Lett. 117, 093001 (2016).
[Crossref] [PubMed]

S. Pabst, M. Lein, and H. J. Wörner, “Preparing attosecond coherences by strong-field ionization,” Phys. Rev. A 93, 023412 (2016).
[Crossref]

P. D. Keathley, S. Bhardwaj, J. Moses, G. Laurent, and F. X. Kärtner, “Volkov transform generalized projection algorithm for attosecond pulse characterization,” New J. Phys. 18, 073009 (2016).
[Crossref]

H. Wei, T. Morishita, and C. D. Lin, “Critical evaluation of attosecond time delays retrieved from photoelectron streaking measurements,” Phys. Rev. A 93, 053412 (2016).
[Crossref]

2015 (2)

P. M. Kraus, B. Mignolet, D. Baykusheva, A. Rupenyan, L. Horny, E. F. Penka, G. Grassi, O. I. Tolstikhin, J. Schneider, F. Jensen, L. B. Madsen, A. D. Bandrauk, F. Remacle, and H. J. Wörner, “Measurement and laser control of attosecond charge migration in ionized iodoacetylene,” Science 350, 790–795 (2015).
[Crossref] [PubMed]

M. Lucchini, M. Brügmann, A. Ludwig, L. Gallmann, U. Keller, and T. Feurer, “Ptychographic reconstruction of attosecond pulses,” Opt. Express 23, 29502–29513 (2015).
[Crossref] [PubMed]

2014 (3)

S. L. Cousin, F. Silva, S. Teichmann, M. Hemmer, B. Buades, and J. Biegert, “High-flux table-top soft x-ray source driven by sub-2-cycle, CEP stable, 1.85-µ m 1-kHz pulses for carbon K-edge spectroscopy,” Opt. Lett. 39, 5383 (2014).
[Crossref]

E. P. Månsson, D. Guénot, C. L. Arnold, D. Kroon, S. Kasper, J. M. Dahlström, E. Lindroth, A. S. Kheifets, A. L’huillier, S. L. Sorensen, and M. Gisselbrecht, “Double ionization probed on the attosecond timescale,” Nat. Phys. 10, 207–211 (2014).
[Crossref]

F. Calegari, D. Ayuso, A. Trabattoni, L. Belshaw, S. De Camillis, S. Anumula, F. Frassetto, L. Poletto, A. Palacios, P. Decleva, J. B. Greenwood, F. Martin, and M. Nisoli, “Ultrafast electron dynamics in phenylalanine initiated by attosecond pulses,” Science 346, 336–339 (2014).
[Crossref] [PubMed]

2013 (3)

A. S. Kheifets, “Time delay in valence-shell photoionization of noble-gas atoms,” Phys. Rev. A 87, 1–9 (2013).
[Crossref]

S. Bhardwaj, S. K. Son, K. H. Hong, C. J. Lai, F. X. Kärtner, and R. Santra, “Recombination-amplitude calculations of noble gases, in both length and acceleration forms, beyond the strong-field approximation,” Phys. Rev. A 88, 1–7 (2013).
[Crossref]

G. Laurent, W. Cao, I. Ben-Itzhak, and C. L. Cocke, “Attosecond pulse characterization,” Opt. Express 21, 16914 (2013).
[Crossref] [PubMed]

2012 (2)

I. Grguraš, A. R. Maier, C. Behrens, T. Mazza, T. J. Kelly, P. Radcliffe, S. Düsterer, A. K. Kazansky, N. M. Kabachnik, T. Tschentscher, J. T. Costello, M. Meyer, M. C. Hoffmann, H. Schlarb, and A. L. Cavalieri, “Ultrafast X-ray pulse characterization at free-electron lasers,” Nat. Photonics 6, 852–857 (2012).
[Crossref]

T. Popmintchev, M.-C. Chen, D. Popmintchev, P. Arpin, S. Brown, S. Alisauskas, G. Andriukaitis, T. Balciunas, O. D. Mücke, A. Pugzlys, A. Baltuska, B. Shim, S. E. Schrauth, A. Gaeta, C. Hernandez-Garcia, L. Plaja, A. Becker, A. Jaron-Becker, M. M. Murnane, and H. C. Kapteyn, “Bright Coherent Ultrahigh Harmonics in the keV X-ray Regime from Mid-Infrared Femtosecond Lasers,” Science 336, 1287–1291 (2012).
[Crossref] [PubMed]

2010 (3)

G. Sansone, F. Kelkensberg, J. F. Pérez-Torres, F. Morales, M. F. Kling, W. Siu, O. Ghafur, P. Johnsson, M. Swoboda, E. Benedetti, F. Ferrari, F. Lépine, J. L. Sanz-Vicario, S. Zherebtsov, I. Znakovskaya, A. L’Huillier, M. Y. Ivanov, M. Nisoli, F. Martín, and M. J. J. Vrakking, “Electron localization following attosecond molecular photoionization,” Nature 465, 763–766 (2010).
[Crossref] [PubMed]

M. Schultze, M. Fieß, N. Karpowicz, J. Gagnon, M. Korbman, M. Hofstetter, S. Neppl, A. L. Cavalieri, Y. Komninos, T. Mercouris, C. A. Nicolaides, R. Pazourek, S. Nagele, J. Feist, J. Burgdörfer, A. M. Azzeer, R. Ernstorfer, R. Kienberger, U. Kleineberg, E. Goulielmakis, F. Krausz, and V. S. Yakovlev, “Delay in Photoemission,” Science 328, 1658–1662 (2010).
[Crossref] [PubMed]

M. Chini, S. Gilbertson, S. D. Khan, and Z. Chang, “Characterizing ultrabroadband attosecond lasers,” Opt. Express 18, 13006 (2010).
[Crossref] [PubMed]

2008 (3)

P. Eckle, A. N. Pfeiffer, C. Cirelli, A. Staudte, R. Dorner, H. G. Muller, M. Buttiker, and U. Keller, “Attosecond Ionization and Tunneling Delay Time Measurements in Helium,” Science 322, 1525–1529 (2008).
[Crossref] [PubMed]

E. J. Takahashi, T. Kanai, K. L. Ishikawa, Y. Nabekawa, and K. Midorikawa, “Coherent water window X ray by phase-matched high-order harmonic generation in neutral media,” Phys. Rev. Lett. 101, 1–4 (2008).
[Crossref]

J. Gagnon, E. Goulielmakis, and V. Yakovlev, “The accurate FROG characterization of attosecond pulses from streaking measurements,” Appl. Phys. B 92, 25–32 (2008).
[Crossref]

2007 (1)

A. L. Cavalieri, N. Müller, T. Uphues, V. S. Yakovlev, A. Baltuška, B. Horvath, B. Schmidt, L. Blümel, R. Holzwarth, S. Hendel, M. Drescher, U. Kleineberg, P. M. Echenique, R. Kienberger, F. Krausz, and U. Heinzmann, “Attosecond spectroscopy in condensed matter,” Nature 449, 1029–1032 (2007).
[Crossref] [PubMed]

2006 (1)

J. Mauritsson, P. Johnsson, E. Gustafsson, A. L’Huillier, K. J. Schafer, and M. B. Gaarde, “Attosecond Pulse Trains Generated Using Two Color Laser Fields,” Phys. Rev. Lett. 97, 013001 (2006).
[Crossref] [PubMed]

2005 (1)

Y. Mairesse and F. Quéré, “Frequency-resolved optical gating for complete reconstruction of attosecond bursts,” Phys. Rev. A 71, 1–4 (2005).
[Crossref]

2004 (1)

E. Seres, J. Seres, F. Krausz, and C. Spielmann, “Generation of Coherent Soft-X-Ray Radiation Extending Far Beyond the Titanium L-Edge,” Phys. Rev. Lett. 92, 163002 (2004).
[Crossref]

2002 (3)

J. Itatani, F. Quéré, G. L. Yudin, M. Y. Ivanov, F. Krausz, and P. B. Corkum, “Attosecond streak camera,” Phys. Rev. Lett. 88, 173903 (2002).
[Crossref] [PubMed]

M. Kitzler, N. Milosevic, A. Scrinzi, F. Krausz, and T. Brabec, “Quantum Theory of Attosecond XUV Pulse Measurement by Laser Dressed Photoionization,” Phys. Rev. Lett. 88, 173904 (2002).
[Crossref] [PubMed]

M. Drescher, M. Hentschel, R. Kienberger, M. Uiberacker, V. Yakovlev, A. Scrinzi, T. Westerwalbesloh, U. Kleineberg, U. Heinzmann, and F. Krausz, “Time-resolved atomic inner-shell spectroscopy,” Nature 419, 803–807 (2002).
[Crossref] [PubMed]

1997 (1)

C. Spielmann, “Generation of Coherent X-rays in the Water Window Using 5-Femtosecond Laser Pulses,” Science 278, 661–664 (1997).
[Crossref]

1977 (1)

G. C. King, M. Tronc, F. H. Read, and R. C. Bradford, “An investigation of the structure near the L 2,3 edges of argon, the M 4,5 edges of krypton and the N 4,5 edges of xenon, using electron impact with high resolution,” J. Phys. B 10, 2479–2495 (1977).
[Crossref]

1976 (1)

J. B. West and G. V. Marr, “The Absolute Photoionization Cross Sections of Helium, Neon, Argon and Krypton in the Extreme Vacuum Ultraviolet Region of the Spectrum,” Proc. Royal Soc. A 349, 397–421 (1976).
[Crossref]

Alisauskas, S.

T. Popmintchev, M.-C. Chen, D. Popmintchev, P. Arpin, S. Brown, S. Alisauskas, G. Andriukaitis, T. Balciunas, O. D. Mücke, A. Pugzlys, A. Baltuska, B. Shim, S. E. Schrauth, A. Gaeta, C. Hernandez-Garcia, L. Plaja, A. Becker, A. Jaron-Becker, M. M. Murnane, and H. C. Kapteyn, “Bright Coherent Ultrahigh Harmonics in the keV X-ray Regime from Mid-Infrared Femtosecond Lasers,” Science 336, 1287–1291 (2012).
[Crossref] [PubMed]

Andriukaitis, G.

T. Popmintchev, M.-C. Chen, D. Popmintchev, P. Arpin, S. Brown, S. Alisauskas, G. Andriukaitis, T. Balciunas, O. D. Mücke, A. Pugzlys, A. Baltuska, B. Shim, S. E. Schrauth, A. Gaeta, C. Hernandez-Garcia, L. Plaja, A. Becker, A. Jaron-Becker, M. M. Murnane, and H. C. Kapteyn, “Bright Coherent Ultrahigh Harmonics in the keV X-ray Regime from Mid-Infrared Femtosecond Lasers,” Science 336, 1287–1291 (2012).
[Crossref] [PubMed]

Anumula, S.

F. Calegari, D. Ayuso, A. Trabattoni, L. Belshaw, S. De Camillis, S. Anumula, F. Frassetto, L. Poletto, A. Palacios, P. Decleva, J. B. Greenwood, F. Martin, and M. Nisoli, “Ultrafast electron dynamics in phenylalanine initiated by attosecond pulses,” Science 346, 336–339 (2014).
[Crossref] [PubMed]

Ardana-Lamas, F.

Arnold, C. L.

E. P. Månsson, D. Guénot, C. L. Arnold, D. Kroon, S. Kasper, J. M. Dahlström, E. Lindroth, A. S. Kheifets, A. L’huillier, S. L. Sorensen, and M. Gisselbrecht, “Double ionization probed on the attosecond timescale,” Nat. Phys. 10, 207–211 (2014).
[Crossref]

Arpin, P.

T. Popmintchev, M.-C. Chen, D. Popmintchev, P. Arpin, S. Brown, S. Alisauskas, G. Andriukaitis, T. Balciunas, O. D. Mücke, A. Pugzlys, A. Baltuska, B. Shim, S. E. Schrauth, A. Gaeta, C. Hernandez-Garcia, L. Plaja, A. Becker, A. Jaron-Becker, M. M. Murnane, and H. C. Kapteyn, “Bright Coherent Ultrahigh Harmonics in the keV X-ray Regime from Mid-Infrared Femtosecond Lasers,” Science 336, 1287–1291 (2012).
[Crossref] [PubMed]

Ayuso, D.

F. Calegari, D. Ayuso, A. Trabattoni, L. Belshaw, S. De Camillis, S. Anumula, F. Frassetto, L. Poletto, A. Palacios, P. Decleva, J. B. Greenwood, F. Martin, and M. Nisoli, “Ultrafast electron dynamics in phenylalanine initiated by attosecond pulses,” Science 346, 336–339 (2014).
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Azzeer, A. M.

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Y. Pertot, C. Schmidt, M. Matthews, A. Chauvet, M. Huppert, V. Svoboda, A. von Conta, A. Tehlar, D. Baykusheva, J.-P. Wolf, and H. J. Wörner, “Time-resolved x-ray absorption spectroscopy with a water window high-harmonic source,” Science 355, 264–267 (2017).
[Crossref] [PubMed]

Schneider, J.

P. M. Kraus, B. Mignolet, D. Baykusheva, A. Rupenyan, L. Horny, E. F. Penka, G. Grassi, O. I. Tolstikhin, J. Schneider, F. Jensen, L. B. Madsen, A. D. Bandrauk, F. Remacle, and H. J. Wörner, “Measurement and laser control of attosecond charge migration in ionized iodoacetylene,” Science 350, 790–795 (2015).
[Crossref] [PubMed]

Schötz, J.

Schrauth, S. E.

T. Popmintchev, M.-C. Chen, D. Popmintchev, P. Arpin, S. Brown, S. Alisauskas, G. Andriukaitis, T. Balciunas, O. D. Mücke, A. Pugzlys, A. Baltuska, B. Shim, S. E. Schrauth, A. Gaeta, C. Hernandez-Garcia, L. Plaja, A. Becker, A. Jaron-Becker, M. M. Murnane, and H. C. Kapteyn, “Bright Coherent Ultrahigh Harmonics in the keV X-ray Regime from Mid-Infrared Femtosecond Lasers,” Science 336, 1287–1291 (2012).
[Crossref] [PubMed]

Schultze, M.

M. Schultze, M. Fieß, N. Karpowicz, J. Gagnon, M. Korbman, M. Hofstetter, S. Neppl, A. L. Cavalieri, Y. Komninos, T. Mercouris, C. A. Nicolaides, R. Pazourek, S. Nagele, J. Feist, J. Burgdörfer, A. M. Azzeer, R. Ernstorfer, R. Kienberger, U. Kleineberg, E. Goulielmakis, F. Krausz, and V. S. Yakovlev, “Delay in Photoemission,” Science 328, 1658–1662 (2010).
[Crossref] [PubMed]

Scrinzi, A.

M. Kitzler, N. Milosevic, A. Scrinzi, F. Krausz, and T. Brabec, “Quantum Theory of Attosecond XUV Pulse Measurement by Laser Dressed Photoionization,” Phys. Rev. Lett. 88, 173904 (2002).
[Crossref] [PubMed]

M. Drescher, M. Hentschel, R. Kienberger, M. Uiberacker, V. Yakovlev, A. Scrinzi, T. Westerwalbesloh, U. Kleineberg, U. Heinzmann, and F. Krausz, “Time-resolved atomic inner-shell spectroscopy,” Nature 419, 803–807 (2002).
[Crossref] [PubMed]

Seres, E.

E. Seres, J. Seres, F. Krausz, and C. Spielmann, “Generation of Coherent Soft-X-Ray Radiation Extending Far Beyond the Titanium L-Edge,” Phys. Rev. Lett. 92, 163002 (2004).
[Crossref]

Seres, J.

E. Seres, J. Seres, F. Krausz, and C. Spielmann, “Generation of Coherent Soft-X-Ray Radiation Extending Far Beyond the Titanium L-Edge,” Phys. Rev. Lett. 92, 163002 (2004).
[Crossref]

Shim, B.

T. Popmintchev, M.-C. Chen, D. Popmintchev, P. Arpin, S. Brown, S. Alisauskas, G. Andriukaitis, T. Balciunas, O. D. Mücke, A. Pugzlys, A. Baltuska, B. Shim, S. E. Schrauth, A. Gaeta, C. Hernandez-Garcia, L. Plaja, A. Becker, A. Jaron-Becker, M. M. Murnane, and H. C. Kapteyn, “Bright Coherent Ultrahigh Harmonics in the keV X-ray Regime from Mid-Infrared Femtosecond Lasers,” Science 336, 1287–1291 (2012).
[Crossref] [PubMed]

Siemering, R.

Silva, F.

Siu, W.

G. Sansone, F. Kelkensberg, J. F. Pérez-Torres, F. Morales, M. F. Kling, W. Siu, O. Ghafur, P. Johnsson, M. Swoboda, E. Benedetti, F. Ferrari, F. Lépine, J. L. Sanz-Vicario, S. Zherebtsov, I. Znakovskaya, A. L’Huillier, M. Y. Ivanov, M. Nisoli, F. Martín, and M. J. J. Vrakking, “Electron localization following attosecond molecular photoionization,” Nature 465, 763–766 (2010).
[Crossref] [PubMed]

Son, S. K.

S. Bhardwaj, S. K. Son, K. H. Hong, C. J. Lai, F. X. Kärtner, and R. Santra, “Recombination-amplitude calculations of noble gases, in both length and acceleration forms, beyond the strong-field approximation,” Phys. Rev. A 88, 1–7 (2013).
[Crossref]

Sorensen, S. L.

E. P. Månsson, D. Guénot, C. L. Arnold, D. Kroon, S. Kasper, J. M. Dahlström, E. Lindroth, A. S. Kheifets, A. L’huillier, S. L. Sorensen, and M. Gisselbrecht, “Double ionization probed on the attosecond timescale,” Nat. Phys. 10, 207–211 (2014).
[Crossref]

Spielmann, C.

E. Seres, J. Seres, F. Krausz, and C. Spielmann, “Generation of Coherent Soft-X-Ray Radiation Extending Far Beyond the Titanium L-Edge,” Phys. Rev. Lett. 92, 163002 (2004).
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C. Spielmann, “Generation of Coherent X-rays in the Water Window Using 5-Femtosecond Laser Pulses,” Science 278, 661–664 (1997).
[Crossref]

Staudte, A.

P. Eckle, A. N. Pfeiffer, C. Cirelli, A. Staudte, R. Dorner, H. G. Muller, M. Buttiker, and U. Keller, “Attosecond Ionization and Tunneling Delay Time Measurements in Helium,” Science 322, 1525–1529 (2008).
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Svoboda, V.

Y. Pertot, C. Schmidt, M. Matthews, A. Chauvet, M. Huppert, V. Svoboda, A. von Conta, A. Tehlar, D. Baykusheva, J.-P. Wolf, and H. J. Wörner, “Time-resolved x-ray absorption spectroscopy with a water window high-harmonic source,” Science 355, 264–267 (2017).
[Crossref] [PubMed]

Swoboda, M.

G. Sansone, F. Kelkensberg, J. F. Pérez-Torres, F. Morales, M. F. Kling, W. Siu, O. Ghafur, P. Johnsson, M. Swoboda, E. Benedetti, F. Ferrari, F. Lépine, J. L. Sanz-Vicario, S. Zherebtsov, I. Znakovskaya, A. L’Huillier, M. Y. Ivanov, M. Nisoli, F. Martín, and M. J. J. Vrakking, “Electron localization following attosecond molecular photoionization,” Nature 465, 763–766 (2010).
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Takahashi, E. J.

E. J. Takahashi, T. Kanai, K. L. Ishikawa, Y. Nabekawa, and K. Midorikawa, “Coherent water window X ray by phase-matched high-order harmonic generation in neutral media,” Phys. Rev. Lett. 101, 1–4 (2008).
[Crossref]

Tehlar, A.

Y. Pertot, C. Schmidt, M. Matthews, A. Chauvet, M. Huppert, V. Svoboda, A. von Conta, A. Tehlar, D. Baykusheva, J.-P. Wolf, and H. J. Wörner, “Time-resolved x-ray absorption spectroscopy with a water window high-harmonic source,” Science 355, 264–267 (2017).
[Crossref] [PubMed]

Teichmann, S.

Tolstikhin, O. I.

P. M. Kraus, B. Mignolet, D. Baykusheva, A. Rupenyan, L. Horny, E. F. Penka, G. Grassi, O. I. Tolstikhin, J. Schneider, F. Jensen, L. B. Madsen, A. D. Bandrauk, F. Remacle, and H. J. Wörner, “Measurement and laser control of attosecond charge migration in ionized iodoacetylene,” Science 350, 790–795 (2015).
[Crossref] [PubMed]

Trabattoni, A.

F. Calegari, D. Ayuso, A. Trabattoni, L. Belshaw, S. De Camillis, S. Anumula, F. Frassetto, L. Poletto, A. Palacios, P. Decleva, J. B. Greenwood, F. Martin, and M. Nisoli, “Ultrafast electron dynamics in phenylalanine initiated by attosecond pulses,” Science 346, 336–339 (2014).
[Crossref] [PubMed]

Tronc, M.

G. C. King, M. Tronc, F. H. Read, and R. C. Bradford, “An investigation of the structure near the L 2,3 edges of argon, the M 4,5 edges of krypton and the N 4,5 edges of xenon, using electron impact with high resolution,” J. Phys. B 10, 2479–2495 (1977).
[Crossref]

Tschentscher, T.

I. Grguraš, A. R. Maier, C. Behrens, T. Mazza, T. J. Kelly, P. Radcliffe, S. Düsterer, A. K. Kazansky, N. M. Kabachnik, T. Tschentscher, J. T. Costello, M. Meyer, M. C. Hoffmann, H. Schlarb, and A. L. Cavalieri, “Ultrafast X-ray pulse characterization at free-electron lasers,” Nat. Photonics 6, 852–857 (2012).
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Uiberacker, M.

M. Drescher, M. Hentschel, R. Kienberger, M. Uiberacker, V. Yakovlev, A. Scrinzi, T. Westerwalbesloh, U. Kleineberg, U. Heinzmann, and F. Krausz, “Time-resolved atomic inner-shell spectroscopy,” Nature 419, 803–807 (2002).
[Crossref] [PubMed]

Uphues, T.

A. L. Cavalieri, N. Müller, T. Uphues, V. S. Yakovlev, A. Baltuška, B. Horvath, B. Schmidt, L. Blümel, R. Holzwarth, S. Hendel, M. Drescher, U. Kleineberg, P. M. Echenique, R. Kienberger, F. Krausz, and U. Heinzmann, “Attosecond spectroscopy in condensed matter,” Nature 449, 1029–1032 (2007).
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von Conta, A.

Y. Pertot, C. Schmidt, M. Matthews, A. Chauvet, M. Huppert, V. Svoboda, A. von Conta, A. Tehlar, D. Baykusheva, J.-P. Wolf, and H. J. Wörner, “Time-resolved x-ray absorption spectroscopy with a water window high-harmonic source,” Science 355, 264–267 (2017).
[Crossref] [PubMed]

M. Huppert, I. Jordan, D. Baykusheva, A. von Conta, and H. J. Wörner, “Attosecond Delays in Molecular Photoionization,” Phys. Rev. Lett. 117, 093001 (2016).
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Vrakking, M. J. J.

G. Sansone, F. Kelkensberg, J. F. Pérez-Torres, F. Morales, M. F. Kling, W. Siu, O. Ghafur, P. Johnsson, M. Swoboda, E. Benedetti, F. Ferrari, F. Lépine, J. L. Sanz-Vicario, S. Zherebtsov, I. Znakovskaya, A. L’Huillier, M. Y. Ivanov, M. Nisoli, F. Martín, and M. J. J. Vrakking, “Electron localization following attosecond molecular photoionization,” Nature 465, 763–766 (2010).
[Crossref] [PubMed]

Wang, Y.

J. Li, X. Ren, Y. Yin, K. Zhao, A. Chew, Y. Cheng, E. Cunningham, Y. Wang, S. Hu, Y. Wu, M. Chini, and Z. Chang, “53-attosecond X-ray pulses reach the carbon K-edge,” Nat. Commun. 8, 186 (2017).
[Crossref] [PubMed]

Wei, H.

H. Wei, T. Morishita, and C. D. Lin, “Critical evaluation of attosecond time delays retrieved from photoelectron streaking measurements,” Phys. Rev. A 93, 053412 (2016).
[Crossref]

West, J. B.

J. B. West and G. V. Marr, “The Absolute Photoionization Cross Sections of Helium, Neon, Argon and Krypton in the Extreme Vacuum Ultraviolet Region of the Spectrum,” Proc. Royal Soc. A 349, 397–421 (1976).
[Crossref]

Westerwalbesloh, T.

M. Drescher, M. Hentschel, R. Kienberger, M. Uiberacker, V. Yakovlev, A. Scrinzi, T. Westerwalbesloh, U. Kleineberg, U. Heinzmann, and F. Krausz, “Time-resolved atomic inner-shell spectroscopy,” Nature 419, 803–807 (2002).
[Crossref] [PubMed]

Wolf, J.-P.

Y. Pertot, C. Schmidt, M. Matthews, A. Chauvet, M. Huppert, V. Svoboda, A. von Conta, A. Tehlar, D. Baykusheva, J.-P. Wolf, and H. J. Wörner, “Time-resolved x-ray absorption spectroscopy with a water window high-harmonic source,” Science 355, 264–267 (2017).
[Crossref] [PubMed]

Wörner, H. J.

Y. Pertot, C. Schmidt, M. Matthews, A. Chauvet, M. Huppert, V. Svoboda, A. von Conta, A. Tehlar, D. Baykusheva, J.-P. Wolf, and H. J. Wörner, “Time-resolved x-ray absorption spectroscopy with a water window high-harmonic source,” Science 355, 264–267 (2017).
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H. Li, N. G. Kling, T. Gaumnitz, C. Burger, R. Siemering, J. Schötz, Q. Liu, L. Ban, Y. Pertot, J. Wu, A. M. Azzeer, R. de Vivie-Riedle, H. J. Wörner, and M. F. Kling, “Sub-cycle steering of the deprotonation of acetylene by intense few-cycle mid-infrared laser fields,” Opt. Express 25, 14192 (2017).
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T. Gaumnitz, A. Jain, Y. Pertot, M. Huppert, I. Jordan, F. Ardana-Lamas, and H. J. Wörner, “Streaking of 43-attosecond soft-X-ray pulses generated by a passively CEP-stable mid-infrared driver,” Opt. Express 25, 27506 (2017).
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M. Huppert, I. Jordan, D. Baykusheva, A. von Conta, and H. J. Wörner, “Attosecond Delays in Molecular Photoionization,” Phys. Rev. Lett. 117, 093001 (2016).
[Crossref] [PubMed]

S. Pabst, M. Lein, and H. J. Wörner, “Preparing attosecond coherences by strong-field ionization,” Phys. Rev. A 93, 023412 (2016).
[Crossref]

P. M. Kraus, B. Mignolet, D. Baykusheva, A. Rupenyan, L. Horny, E. F. Penka, G. Grassi, O. I. Tolstikhin, J. Schneider, F. Jensen, L. B. Madsen, A. D. Bandrauk, F. Remacle, and H. J. Wörner, “Measurement and laser control of attosecond charge migration in ionized iodoacetylene,” Science 350, 790–795 (2015).
[Crossref] [PubMed]

Wu, J.

Wu, Y.

J. Li, X. Ren, Y. Yin, K. Zhao, A. Chew, Y. Cheng, E. Cunningham, Y. Wang, S. Hu, Y. Wu, M. Chini, and Z. Chang, “53-attosecond X-ray pulses reach the carbon K-edge,” Nat. Commun. 8, 186 (2017).
[Crossref] [PubMed]

Yakovlev, V.

J. Gagnon, E. Goulielmakis, and V. Yakovlev, “The accurate FROG characterization of attosecond pulses from streaking measurements,” Appl. Phys. B 92, 25–32 (2008).
[Crossref]

M. Drescher, M. Hentschel, R. Kienberger, M. Uiberacker, V. Yakovlev, A. Scrinzi, T. Westerwalbesloh, U. Kleineberg, U. Heinzmann, and F. Krausz, “Time-resolved atomic inner-shell spectroscopy,” Nature 419, 803–807 (2002).
[Crossref] [PubMed]

Yakovlev, V. S.

M. Schultze, M. Fieß, N. Karpowicz, J. Gagnon, M. Korbman, M. Hofstetter, S. Neppl, A. L. Cavalieri, Y. Komninos, T. Mercouris, C. A. Nicolaides, R. Pazourek, S. Nagele, J. Feist, J. Burgdörfer, A. M. Azzeer, R. Ernstorfer, R. Kienberger, U. Kleineberg, E. Goulielmakis, F. Krausz, and V. S. Yakovlev, “Delay in Photoemission,” Science 328, 1658–1662 (2010).
[Crossref] [PubMed]

A. L. Cavalieri, N. Müller, T. Uphues, V. S. Yakovlev, A. Baltuška, B. Horvath, B. Schmidt, L. Blümel, R. Holzwarth, S. Hendel, M. Drescher, U. Kleineberg, P. M. Echenique, R. Kienberger, F. Krausz, and U. Heinzmann, “Attosecond spectroscopy in condensed matter,” Nature 449, 1029–1032 (2007).
[Crossref] [PubMed]

Yin, Y.

J. Li, X. Ren, Y. Yin, K. Zhao, A. Chew, Y. Cheng, E. Cunningham, Y. Wang, S. Hu, Y. Wu, M. Chini, and Z. Chang, “53-attosecond X-ray pulses reach the carbon K-edge,” Nat. Commun. 8, 186 (2017).
[Crossref] [PubMed]

Yudin, G. L.

J. Itatani, F. Quéré, G. L. Yudin, M. Y. Ivanov, F. Krausz, and P. B. Corkum, “Attosecond streak camera,” Phys. Rev. Lett. 88, 173903 (2002).
[Crossref] [PubMed]

Zhao, K.

J. Li, X. Ren, Y. Yin, K. Zhao, A. Chew, Y. Cheng, E. Cunningham, Y. Wang, S. Hu, Y. Wu, M. Chini, and Z. Chang, “53-attosecond X-ray pulses reach the carbon K-edge,” Nat. Commun. 8, 186 (2017).
[Crossref] [PubMed]

Zherebtsov, S.

G. Sansone, F. Kelkensberg, J. F. Pérez-Torres, F. Morales, M. F. Kling, W. Siu, O. Ghafur, P. Johnsson, M. Swoboda, E. Benedetti, F. Ferrari, F. Lépine, J. L. Sanz-Vicario, S. Zherebtsov, I. Znakovskaya, A. L’Huillier, M. Y. Ivanov, M. Nisoli, F. Martín, and M. J. J. Vrakking, “Electron localization following attosecond molecular photoionization,” Nature 465, 763–766 (2010).
[Crossref] [PubMed]

Znakovskaya, I.

G. Sansone, F. Kelkensberg, J. F. Pérez-Torres, F. Morales, M. F. Kling, W. Siu, O. Ghafur, P. Johnsson, M. Swoboda, E. Benedetti, F. Ferrari, F. Lépine, J. L. Sanz-Vicario, S. Zherebtsov, I. Znakovskaya, A. L’Huillier, M. Y. Ivanov, M. Nisoli, F. Martín, and M. J. J. Vrakking, “Electron localization following attosecond molecular photoionization,” Nature 465, 763–766 (2010).
[Crossref] [PubMed]

Appl. Phys. B (1)

J. Gagnon, E. Goulielmakis, and V. Yakovlev, “The accurate FROG characterization of attosecond pulses from streaking measurements,” Appl. Phys. B 92, 25–32 (2008).
[Crossref]

J. Phys. B (1)

G. C. King, M. Tronc, F. H. Read, and R. C. Bradford, “An investigation of the structure near the L 2,3 edges of argon, the M 4,5 edges of krypton and the N 4,5 edges of xenon, using electron impact with high resolution,” J. Phys. B 10, 2479–2495 (1977).
[Crossref]

Nat. Commun. (1)

J. Li, X. Ren, Y. Yin, K. Zhao, A. Chew, Y. Cheng, E. Cunningham, Y. Wang, S. Hu, Y. Wu, M. Chini, and Z. Chang, “53-attosecond X-ray pulses reach the carbon K-edge,” Nat. Commun. 8, 186 (2017).
[Crossref] [PubMed]

Nat. Photonics (1)

I. Grguraš, A. R. Maier, C. Behrens, T. Mazza, T. J. Kelly, P. Radcliffe, S. Düsterer, A. K. Kazansky, N. M. Kabachnik, T. Tschentscher, J. T. Costello, M. Meyer, M. C. Hoffmann, H. Schlarb, and A. L. Cavalieri, “Ultrafast X-ray pulse characterization at free-electron lasers,” Nat. Photonics 6, 852–857 (2012).
[Crossref]

Nat. Phys. (1)

E. P. Månsson, D. Guénot, C. L. Arnold, D. Kroon, S. Kasper, J. M. Dahlström, E. Lindroth, A. S. Kheifets, A. L’huillier, S. L. Sorensen, and M. Gisselbrecht, “Double ionization probed on the attosecond timescale,” Nat. Phys. 10, 207–211 (2014).
[Crossref]

Nature (3)

M. Drescher, M. Hentschel, R. Kienberger, M. Uiberacker, V. Yakovlev, A. Scrinzi, T. Westerwalbesloh, U. Kleineberg, U. Heinzmann, and F. Krausz, “Time-resolved atomic inner-shell spectroscopy,” Nature 419, 803–807 (2002).
[Crossref] [PubMed]

A. L. Cavalieri, N. Müller, T. Uphues, V. S. Yakovlev, A. Baltuška, B. Horvath, B. Schmidt, L. Blümel, R. Holzwarth, S. Hendel, M. Drescher, U. Kleineberg, P. M. Echenique, R. Kienberger, F. Krausz, and U. Heinzmann, “Attosecond spectroscopy in condensed matter,” Nature 449, 1029–1032 (2007).
[Crossref] [PubMed]

G. Sansone, F. Kelkensberg, J. F. Pérez-Torres, F. Morales, M. F. Kling, W. Siu, O. Ghafur, P. Johnsson, M. Swoboda, E. Benedetti, F. Ferrari, F. Lépine, J. L. Sanz-Vicario, S. Zherebtsov, I. Znakovskaya, A. L’Huillier, M. Y. Ivanov, M. Nisoli, F. Martín, and M. J. J. Vrakking, “Electron localization following attosecond molecular photoionization,” Nature 465, 763–766 (2010).
[Crossref] [PubMed]

New J. Phys. (1)

P. D. Keathley, S. Bhardwaj, J. Moses, G. Laurent, and F. X. Kärtner, “Volkov transform generalized projection algorithm for attosecond pulse characterization,” New J. Phys. 18, 073009 (2016).
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Opt. Express (5)

Opt. Lett. (1)

Phys. Rev. A (5)

S. Pabst, M. Lein, and H. J. Wörner, “Preparing attosecond coherences by strong-field ionization,” Phys. Rev. A 93, 023412 (2016).
[Crossref]

H. Wei, T. Morishita, and C. D. Lin, “Critical evaluation of attosecond time delays retrieved from photoelectron streaking measurements,” Phys. Rev. A 93, 053412 (2016).
[Crossref]

Y. Mairesse and F. Quéré, “Frequency-resolved optical gating for complete reconstruction of attosecond bursts,” Phys. Rev. A 71, 1–4 (2005).
[Crossref]

A. S. Kheifets, “Time delay in valence-shell photoionization of noble-gas atoms,” Phys. Rev. A 87, 1–9 (2013).
[Crossref]

S. Bhardwaj, S. K. Son, K. H. Hong, C. J. Lai, F. X. Kärtner, and R. Santra, “Recombination-amplitude calculations of noble gases, in both length and acceleration forms, beyond the strong-field approximation,” Phys. Rev. A 88, 1–7 (2013).
[Crossref]

Phys. Rev. Lett. (6)

M. Huppert, I. Jordan, D. Baykusheva, A. von Conta, and H. J. Wörner, “Attosecond Delays in Molecular Photoionization,” Phys. Rev. Lett. 117, 093001 (2016).
[Crossref] [PubMed]

J. Mauritsson, P. Johnsson, E. Gustafsson, A. L’Huillier, K. J. Schafer, and M. B. Gaarde, “Attosecond Pulse Trains Generated Using Two Color Laser Fields,” Phys. Rev. Lett. 97, 013001 (2006).
[Crossref] [PubMed]

E. Seres, J. Seres, F. Krausz, and C. Spielmann, “Generation of Coherent Soft-X-Ray Radiation Extending Far Beyond the Titanium L-Edge,” Phys. Rev. Lett. 92, 163002 (2004).
[Crossref]

E. J. Takahashi, T. Kanai, K. L. Ishikawa, Y. Nabekawa, and K. Midorikawa, “Coherent water window X ray by phase-matched high-order harmonic generation in neutral media,” Phys. Rev. Lett. 101, 1–4 (2008).
[Crossref]

J. Itatani, F. Quéré, G. L. Yudin, M. Y. Ivanov, F. Krausz, and P. B. Corkum, “Attosecond streak camera,” Phys. Rev. Lett. 88, 173903 (2002).
[Crossref] [PubMed]

M. Kitzler, N. Milosevic, A. Scrinzi, F. Krausz, and T. Brabec, “Quantum Theory of Attosecond XUV Pulse Measurement by Laser Dressed Photoionization,” Phys. Rev. Lett. 88, 173904 (2002).
[Crossref] [PubMed]

Proc. Royal Soc. A (1)

J. B. West and G. V. Marr, “The Absolute Photoionization Cross Sections of Helium, Neon, Argon and Krypton in the Extreme Vacuum Ultraviolet Region of the Spectrum,” Proc. Royal Soc. A 349, 397–421 (1976).
[Crossref]

Science (7)

Y. Pertot, C. Schmidt, M. Matthews, A. Chauvet, M. Huppert, V. Svoboda, A. von Conta, A. Tehlar, D. Baykusheva, J.-P. Wolf, and H. J. Wörner, “Time-resolved x-ray absorption spectroscopy with a water window high-harmonic source,” Science 355, 264–267 (2017).
[Crossref] [PubMed]

T. Popmintchev, M.-C. Chen, D. Popmintchev, P. Arpin, S. Brown, S. Alisauskas, G. Andriukaitis, T. Balciunas, O. D. Mücke, A. Pugzlys, A. Baltuska, B. Shim, S. E. Schrauth, A. Gaeta, C. Hernandez-Garcia, L. Plaja, A. Becker, A. Jaron-Becker, M. M. Murnane, and H. C. Kapteyn, “Bright Coherent Ultrahigh Harmonics in the keV X-ray Regime from Mid-Infrared Femtosecond Lasers,” Science 336, 1287–1291 (2012).
[Crossref] [PubMed]

C. Spielmann, “Generation of Coherent X-rays in the Water Window Using 5-Femtosecond Laser Pulses,” Science 278, 661–664 (1997).
[Crossref]

F. Calegari, D. Ayuso, A. Trabattoni, L. Belshaw, S. De Camillis, S. Anumula, F. Frassetto, L. Poletto, A. Palacios, P. Decleva, J. B. Greenwood, F. Martin, and M. Nisoli, “Ultrafast electron dynamics in phenylalanine initiated by attosecond pulses,” Science 346, 336–339 (2014).
[Crossref] [PubMed]

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Other (5)

A. Kramida, Yu. Ralchenko, J. Reader, and NIST ASD Team, NIST Atomic Spectra Database (ver. 5.5.1), [Online]. National Institute of Standards and Technology, Gaithersburg, MD. (2017).

J. Gagnon, “Attosecond Electron Spectroscopy - Theory and its Applications,” Ph.D. thesis, LMUMünchen (2010).

S. Bhardwaj, “Modeling generation and characterization of attosecond pulses,” Ph.D. thesis, Massachusetts Institute of Technology (2014).

R. P. Brent, Algorithms for minimization without derivatives (Prentice-Hall, Englewood Cliffs, New Jersey, 1973).

J. Burkardt and R. P. Brent, “BRENT, available at http://people.sc.fsu.edu/∼jburkardt/cpp_src/brent/brent.html ;.

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Figures (11)

Fig. 1
Fig. 1 Simulated traces for a few-cycle near- or mid-IR pulse streaking photoelectrons emitted from xenon by narrow- or broad-band SXR pulses. Panel (A) shows a simulated streaking spectrogram S (E, τ) for the xenon 5p, 5s, and 4d shells of a few-cycle pulse centered at λc = 650 nm for a τSXR = 420 as SXR pulse centered at W0 = 110 eV. The difference in intensity is due to the included PMEs. The spectrum is sufficiently narrow that the three (5s, 5p, 4d) photoelectron bands are separated from each other. In panel (B) the spectrum of the τSXR = 42 as pulse is so broad that the individual photoelectron bands from 5p and 5s cannot be separated anymore and a single photoelectron band is observed. The ionization potential of the 4d shell is sufficiently large to be separated from the two other lines, even the wings of the photoelectron lines start to overlap. In panel (C) a mid-IR streaking pulse (λc = 1650 nm) with the same intensity was used for comparison with the result in panel (B), where the streaking amplitude has increased due to the longer wavelength of the streaking pulse. All spectrograms are normalized to the peak intensity and plotted on the same false-color representation. The 5s, 5p lines have been scaled by a factor of ≈3.3 in intensity, due to the lower cross section.
Fig. 2
Fig. 2 Simulated streaking traces in helium with a 350 as SXR pulse centered at the photon energy 110 eV for different polar collection angles of the spectrometer. Simulated streaking spectrogram S (E, τ) in helium for a τSXR = 350 as SXR pulse for a spectrometer integrating over a collection polar angle of θmax = 90° (A), θmax = 120° (B), and θmax = 150° (C) normalized on the peak count rate assuming an isotropic photoelectron angular distribution, corresponding to an anisotropy parameter β = 0.
Fig. 3
Fig. 3 SXR- and IR-field polarization and the time-of-flight detector. (A) SXR (blue) and IR (red) pulse with linear polarizations, the SXR pulse is described by E SXR ( t ) = E ˜ SXR ( t ) e i W 0 t / e x, where E ˜ SXR ( t ) is the time-dependent SXR envelope and W0 the central photon energy, while the IR streaking pulse is described by E ( t ) = E ( t ) e x. The polarization of both pulses is parallel to the TOF axis. (B) The SXR pulse E ˜ SXR ( t ) leads to photoelectron emission, where | k 0 | is the absolute value of the initial momentum of the photoelectrons. The angular intensity distribution is included in the PME d ( k + A ( t ) , θ ) = d ( k 0 , θ ). The streaking field E ( t ) changes the momentum by Δ k and the photoelectron with final momentum k = k 0 + Δ k is detected by the TOF if its angle θ (angle between polarization of the streaking field and the detection direction) is smaller than the maximum photoemission polar acceptance angle θmax of the TOF.
Fig. 4
Fig. 4 Flow-chart of the multi-line VTGPA including photoelectron angular distributions and photoionization cross sections of xenon. (A) Flow-chart of the multi-line VTGPA including photoelectron angular distribution (PAD). Starting guesses for E ˜ SXR ( t ) and A(t) are used together with the PMEs d(i,k,θ) for the numerical integration of the streaking Eq. (1) to obtain S ( | k | , τ ) from the Ns contributing streaking spectra S i from the individual shells integrated over the acceptance polar angle θ of the photoelectron spectrometer. This is compared with the measured (input) spectrogram S ( | k | , τ ) to minimize the error using the merit function in Eq. (6) with the least-squares (LS) method and Brent’s minimization. (B) Cross sections for the 5s, 5p, and 4d shells calculated from the complex photoionization matrix elements (PMEs) [36]. The red box indicates the photon energy range, where the numerical experiments presented in section 4.1 are performed, which also corresponds to the energy range typically covered by Ti:Sa driven attosecond sources. The strong energy dependence of the cross sections has to be accounted for in pulse-retrieval methods for broadband pulses.
Fig. 5
Fig. 5 Multi-line streaking spectrogram in xenon. Simulated (A) and reconstructed (B) streaking spectrogram to test the ML-VTGPA algorithm with an attosecond pulse duration of τSXR = 350 as in xenon including the PMEs for the 5p, 5s (both multiplied by 3.3, for better visibility of all photoelectron bands) and 4d photoelectron bands. The input vector potential A(t) (blue line) and the reconstructed vector potential (black crosses) are scaled and plotted on top of the 5p photoelectron line for comparison. (C) False color representation of the difference between S and S′ multiplied by three orders of magnitude. The residual mean square error between measured and reconstructed spectrogram was found to be 1.25 · 10−7 after 350 iterations of the algorithm. The unscaled vector potentials (input and retrieved) are also shown as an inset to the panel. (D) Reconstructed SXR pulse parameters amplitude Ienv(t) and phase Φ(t) (black circles and crosses) for detailed comparison with the input SXR pulse parameters (blue, red lines).
Fig. 6
Fig. 6 SXR input pulse and retrieved vector potential. (A) Chirped (blue) and unchirped (orange) SXR input pulse in spectral and temporal (inset) domain. The τSXR = 350 as used in Fig. 5 centered around W0 = 110 eV with a bandwidth of a ΔE = 5.2 eV. For the corresponding chirped SXR pulse from Fig. 7, the bandwidth increases to ΔE = 6.9 eV, and the spectral phase has a quadratic energy dependence (blue dash-dotted line), compared to the flat spectral phase for the previous case (orange dash-dotted). The inset also shows the FTL pulse duration for the increased bandwidth case (yellow), that corresponds to a τSXR,FTL = 265 as pulse. (B) LSGPA (orange crosses) and ML-VTGPA (blue circles) retrieved vector potential for the case presented in Fig. 7, in comparison with the input vector potential (black line).
Fig. 7
Fig. 7 Reconstruction of a multi-line streaking spectrogram in xenon for a chirped 350 as SXR pulse, using the ML-VTGPA and LSGPA methods. (A) Simulated input streaking spectrogram for xenon including the PMEs for the 5p, 5s (both multiplied by 3.3, for better visibility) and 4d photoelectron bands. The input parameters are the same as for Fig. 5, except for a second order SXR chirp of −1.00 · 10−5 as−2, leading to larger bandwidth and broader lines that partially overlap. For reconstruction the ML-VTGPA (C, D) and the LSGPA (E, F) methods have been used. The retrieved attosecond SXR pulses are shown in panel (B) together with the obtained pulse duration, while the obtained vector potentials are shown on top of the retrieved spectrogram S′ for both methods. The input pulses are given by blue lines, while the reconstructed vector potentials are given by the black crosses (Every 40th data point is plotted for better visibility.). It has to be noted that the panels (D) and (F) are using the same color scale, for a better comparison between the two methods. In panel (D) the upper part of the plot (Ekin > 65 eV) has been amplified by x10 for a better visibility of the residuals. Panel (A), (C), and (E) use the same color scale as used before for input and reconstructed spectrogram, c.f. Fig. 5.
Fig. 8
Fig. 8 Multi-line streaking spectrogram in xenon for a chirped 55 as SXR pulse with overlapping 5s, 5p photoelectron band. Retrieved (A) streaking spectrogram for a chirped τSXR = 55 as SXR pulse with a spectrum supporting τSXR = 50 as at a center energy of W0 = 110 eV in the region of overlapping xenon 5p, 5s photoelectron band including the PMEs. The input parameters are the same as for Fig. 5. The blue solid lines represent the vector potential of the IR pulse A(t) used for the simulation at the central kinetic energies (W0Ip,i) of the xenon lines, while the reconstructed vector potential of the streaking pulse is given by the black crosses. The difference between calculated and reconstructed spectrogram is given in panel (B), where the maximum deviation is ≪ 2.5 · 10−3, while the overall MSE of the reconstruction is already below 1.5 · 10−6 after 12000 iterations (inset to panel (D)). In panel (C) every 40th point of the retrieved vector potential A(t) is plotted together with the input pulse (blue) while in the inset every 10th retrieved point is shown. Panel (D) shows the retrieved SXR pulse parameter Ienv(t) and Φ(t) together with the input pulse. Panel (A) uses the same color scale as used before for input and reconstructed spectrogram, c.f. Fig. 5.
Fig. 9
Fig. 9 Streaking spectrogram in helium for a 20 as SXR pulse centered at 260 eV. Reconstructed (A) streaking spectrogram for an SXR pulse duration τSXR = 20 as with a central photon energy of W0 = 260 eV in He including the PMEs calculated by Bhardwaj et al. [37]. The input parameters for the vector potential of the IR streaking pulse A(t) are the same as in example 1. The difference between calculated and reconstructed spectrogram is given in panel (B), where the maximum deviation is ≪ 2 · 10−2, while the overall error (MSE) of the reconstruction is after approx. 300 iterations already below ≪ 2 · 10−6 (inset to panel (D)). The scaled reconstructed (black crosses) and the input (blue line) vector potential are given together at the initial kinetic energy of the photoelectrons (W0Ip) in panel (A). Panel (C) shows A(t) on the atomic unit scale. Every 40th point of the retrieved A(t) is shown, while in the inset every 10th point around the maximum of A(t) is shown. In panel (D), the retrieved SXR pulse envelope and phase are shown together with the input pulse. As inset to the panel, the evolution of the MSE during the reconstruction is plotted. Panel (A) uses the same color scale as used before for input and reconstructed spectrogram, c.f. Fig. 5.
Fig. 10
Fig. 10 Angle integrated streaking spectrogram in helium. Simulated input spectrogram S(A) and reconstructed (B) streaking spectrogram S′ to test the ML-VTGPA including photoelectron angular distribution with an attosecond pulse duration of τSXR = 350 as in helium using the PMEs. For the reconstruction an acceptance angle of θmax = 120° has been assumed. Therefore the number of reconstruction angles Nθ was set to 5 with an equidistant distribution between 0 ≤ θ ≤ 120°. The vector potential A(t) used for the simulation (blue line) reconstructed vector potential (black crosses, every tenth data point) are plotted on top for comparison. (C) False color representation of the difference between simulated and reconstructed streaking trace multiplied by 100, for a better comparison. (D) Simulated SXR pulse (intensity envelope and phase: red and blue lines) together with the corresponding reconstructed quantities. The mean squared error for the reconstruction is presented as an inset to (D), showing a residual MSE ≪ 2 · 10−5.
Fig. 11
Fig. 11 Reconstructed vector potential and SXR intensity envelope for different acceptance angles of the MB-TOF. Retrieval results for the streaking vector potential A(t) (A) and the attosecond SXR pulse Ienv(t) (B) as a function of the acceptance angle of the spectrometer. The input pulses are given by black crosses for comparison. The angle integration was performed over 5 angles (Nθ = 5) with a variation of maximum acceptance angle θmax between 90° and 140°. (C) shows the residual MSE after 45 iterations of the algorithm, where the case of θmax = 90° already starts to diverge. (D) SXR- and IR-pulse as a function of maximum acceptance angle after convergence of the reconstruction. The input pulse durations (thick blue lines) are shown together with a ±5 % deviation (thin blue lines).

Equations (15)

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a i ( | k | , θ , τ ) = i E ˜ SXR ( t τ ) d ( i , k + A ( t ) , θ ) e i Φ ( k , θ , t ) e i ( k 2 / 2 + I p , i ) t d t Φ ( k , θ , t ) = t ( k A ( t ) + A 2 ( t ) / 2 ) d t .
S i ( | k | , τ ) = θ = 0 θ max | a i ( | k | , θ , τ ) | 2 sin ( θ ) d θ .
S ( | k | , τ ) = i = 1 N s S i ( | k | , τ ) ,
M = l = 1 N W Δ W [ l ] m = 1 N τ Δ τ ( a [ l , m ] a [ l , m ] ) 2
with a 2 = | a | 2 = S = i = 1 N s θ = 0 θ max | a i | 2 sin ( θ ) d θ .
M = l = 1 N W Δ W [ l ] m = 1 N τ Δ τ ( a [ l , m ] 2 + a [ l , m ] 2 2 a [ l , m ] a [ l , m ] ) .
M α [ n ] = 0 and M ϕ [ n ] = 0 .
M α [ n ] = l = 1 N W Δ W [ l ] m = 1 N τ Δ τ ( | a [ l , m ] | 2 α [ n ] + | a [ l , m ] | 2 α [ n ] 2 a [ l , m ] a [ l , m ] α [ n ] ) = i = 1 N s j = 1 N θ Δ θ [ j ] l = 1 N W Δ W [ l ] m = 1 N τ Δ τ ( | a i [ l , m ] | 2 sin ( θ j ) α [ n ] a [ l , m ] a [ l , m ] | a i [ l , m ] | 2 sin ( θ j ) α [ n ] )
M ϕ [ n ] = i = 1 N s j = 1 N θ Δ θ [ j ] l = 1 N W Δ W [ l ] m = 1 N τ Δ τ ( | a i [ l , m ] | 2 sin ( θ j ) ϕ [ n ] a [ l , m ] a [ l , m ] | a i [ l , m ] | 2 sin ( θ j ) ϕ [ n ] ) .
a α = 0 .
0 = i = 1 N s j = 1 N θ Δ θ [ j ] Im ( e i ϕ [ c ] ( Γ ˜ i [ c ] + β ˜ i [ c ] ) ) sin ( θ j )
0 = α [ c ] i = 1 N s j = 1 N θ Δ θ [ j ] l = 1 N W Δ W [ l ] m = 1 N τ Δ τ | d ˜ ( i , | k [ l ] + A [ n + L m ] | , θ j ) | 2 sin ( θ j ) + i = 1 N s j = 1 N θ Δ θ [ j ] Re ( e i ϕ [ c ] ( Γ ˜ i [ c ] + β ˜ i [ c ] ) ) sin ( θ j ) ,
Γ ˜ i ( c ) = l = 1 N W Δ W [ l ] m = 1 N τ Δ τ n = 1 , n c N E Δ t 2 d ˜ ( i , | k [ l ] + A [ n + L m ] | , θ j ) d ˜ ( i , | k [ l ] + A [ n + L m ] | , θ j ) * E ˜ SXR * [ n ] e { i ( I p , i + k [ l ] 2 / 2 ) Δ t ( c n ) } e { i Δ t b = n + L m N A Φ ( l , b ) } e { i Δ t b = c + L m N A Φ ( l , b ) }
β ˜ i ( c ) = i l = 1 N W Δ W [ l ] m = 1 N τ Δ τ Δ t d ( i , | k [ l ] + A [ n + L m ] | , θ j ) e { i Δ t ( I p , i + k [ l ] 2 ) Δ t c } e { i Δ t b = c + L m N A Φ ( l , b ) } a i * [ l , m ] a [ l , m ] a [ l , m ] .
E ˜ SXR [ c ] = i = 1 N s j = 1 N θ Δ θ [ j ] ( Γ ˜ i [ c ] + β ˜ i [ c ] ) * sin ( θ j ) i = 1 N s j = 1 N θ Δ θ [ j ] l = 1 N W Δ W [ l ] m = 1 N τ Δ τ | d i , | k [ l ] + A [ c + L m ] | , θ j | 2 sin ( θ j ) .

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