Shaped multi-cycle two-color laser field for generating an intense isolated XUV pulse toward 100 attoseconds

Qingbin Zhang; Lixin He; Pengfei Lan; Peixiang Lu

doi:10.1364/OE.22.013213

Shaped multi-cycle two-color laser field for generating an intense isolated XUV pulse toward 100 attoseconds

Qingbin Zhang, Lixin He, Pengfei Lan, and Peixiang Lu

Optics Express
Vol. 22,
Issue 11,
pp. 13213-13233
(2014)
•https://doi.org/10.1364/OE.22.013213

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Qingbin Zhang, Lixin He, Pengfei Lan, and Peixiang Lu, "Shaped multi-cycle two-color laser field for generating an intense isolated XUV pulse toward 100 attoseconds," Opt. Express 22, 13213-13233 (2014)

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Related Topics
Table of Contents Category
- Ultrafast Optics
Optics & Photonics Topics
?

The topics in this list come from the OSA Optics and Photonics Topics applied to this article.
Previously assigned OCIS codes
- Multiharmonic generation (190.4160)
- Ultrafast nonlinear optics (190.7110)
- Spectroscopy, x-ray (300.6560)

About this Article
History
- Original Manuscript: February 18, 2014
- Revised Manuscript: May 9, 2014
- Manuscript Accepted: May 16, 2014
- Published: May 23, 2014

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Open Access

Abstract

The isolated attosecond pulse (IAP) generated from high-order harmonic (HH) radiation has been established as an important technique for the ultrafast optics over past decade. The applications of IAP in ultrafast processes can be greatly extended by further developing the high-intensity IAP. Here, we theoretically propose to shape a two-color field by performing peak amplitude-wavelength analysis. It is found that a 240-as IAP can be generated even without carrier envelop phase (CEP) stabilization using a 25 fs/800 nm fundamental field and a relative weak 25 fs/1330 nm control field, which enables us to markedly relax the requirements of the driving laser fields both in pulse duration and CEP control. On the other hand, if the CEPs of driving laser fields are stabilized, a 65-eV broadband continual harmonic, supporting a 81-as IAP, can be directly produced with the optimized intensity ratio of 0.866 and control wavelength of 1400 nm. Moreover, the propagation effect of two-color field on the macroscopic build-up of HH for generating a high-energy IAP is discussed. We found that the method of phase match still works for the efficient continuous harmonic generation as long as the ionization level and the pressure of gas medium are kept low enough. Since the phase-matched short IAP can be generated with our shaped two-color scheme in combination with a relaxed requirement of driving laser fields, the commercial available high-energy laser source with a loosely focused geometry is promising for scaling up the energy of IAP, showing the potential for the realization of IAP with high focused intensity toward 100 attoseconds.

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|
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|
Author
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Publication

F. Krausz and M. Ivanov, “Attosecond physics,” Rev. Mod. Phys. 81, 163–234 (2009).
[Crossref]
M. Drescher, M. Hentschel, R. Kienberger, M. Uiberacker, V. Yakovlev, A. Scrinzi, Th. Westerwalbesloh, U. Kleineberg, U. Heinzmann, and F. Krausz, “Time-resolved atomic inner-shell spectroscopy,” Nature 419, 803–807 (2002).
[Crossref] [PubMed]
M. Uiberacker, Th. Uphues, M. Schultze, A. J. Verhoef, V. Yakovlev, M. F. Kling, J. Rauschenberger, N. M. Kabachnik, H. Schröder, M. Lezius, K. L. Kompa, H.-G. Muller, M. J. J. Vrakking, S. Hendel, U. Kleineberg, U. Heinzmann, M. Drescher, and F. Krausz, “Attosecond real-time observation of electron tunnelling inatoms,” Nature 446, 627–632 (2007).
[Crossref] [PubMed]
Th. Weber, H. Giessen, M. Weckenbrock, G. Urbasch, A. Staudte, L. Spielberger, O. Jagutzki, V. Mergel, M. Vollmer, and R. Dörner, “Correlated electron emission in multiphoton double ionization,” Nature 405, 658–661 (2000).
[Crossref] [PubMed]
L. Miaja-Avila, G. Saathoff, S. Mathias, J. Yin, C. La-o-vorakiat, M. Bauer, M. Aeschlimann, M. M. Murnane, and H. C. Kapteyn, “Direct measurement of core-level relaxation dynamics on a surface-adsorbate system,” Phys. Rev. Lett. 101, 046101 (2008).
[Crossref] [PubMed]
A. D. Shiner, B. E. Schmidt, C. Trallero-Herrero, H. J. Wörner, S. Patchkovskii, P. B. Corkum, J-C. Kieffer, F. Lègarè, and D. M. Villeneuve, “Probing collective multi-electron dynamics in xenon with high-harmonic spectroscopy,” Nat. Phys. 7, 464–467 (2011).
[Crossref]
Y. Zhou, C. Huang, Q. Liao, and P. Lu, “Classical Simulations Including Electron Correlations for Sequential Double Ionization,” Phys. Rev. Lett. 109, 053004 (2012).
[Crossref] [PubMed]
Y. Zhou, Q. Zhang, C. Huang, and P. Lu, “Classical description of strong-field double ionization by elliptical laser pulses,” Phys. Rev. A 86, 043427 (2012).
[Crossref]
E. Gagnon, P. Ranitovic, X. M. Tong, C. L. Cocke, M. M. Murnane, H. C. Kapteyn, and A. S. Sandhu, “Soft X-ray-driven femtosecond molecular dynamics,” Science 317, 1374–1378 (2007).
[Crossref] [PubMed]
K. T. Kim, D. H. Ko, J. Park, N. N. Choi, C. M. Kim, K. L. Ishikawa, J. Lee, and C. H. Nam, “Amplitude and phase reconstruction of electron wave packets for probing ultrafast photoionization dynamics,” Phys. Rev. Lett. 108, 093001 (2012).
[Crossref] [PubMed]
T. Rohwer, S. Hellmann, M. Wiesenmayer, C. Sohrt, A. Stange, B. Slomski, A. Carr, Y. Liu, L. M. Avila, M. Kallane, S. Mathias, L. Kipp, K. Rossnagel, and M. Bauer, “Collapse of long-range charge order tracked by time-resolved photoemission at high momenta,” Nature 471, 490–493 (2011).
[Crossref] [PubMed]
R. E. F. Silva, F. Catoire, P. Rivière, H. Bachau, and F. Martín, “Correlated electron and nuclear dynamics in strong field photoionization of H2+,” Phys. Rev. Lett. 110, 113001 (2013).
[Crossref]
K. Liu, Q. Zhang, and P. Lu, “Enhancing electron localization in molecular dissociation by two-color mid- and near-infrared laser fields,” Phys. Rev. A 86, 033410 (2012).
[Crossref]
M. I. Stockman, M. F. Klin, U. Kleineberg, and F. Krausz, “Attosecond nanoplasmonic-field microscope,” Nat. Photonics 1, 539–544 (2007).
[Crossref]
M. E. Siemens, Q. Li, R. Yang, K. A. Nelson, E. H. Anderson, M. M. Murnane, and H. C. Kapteyn, “Quasi-ballistic thermal transport from nanoscale interfaces observed using ultrafast coherent soft X-ray beams,” Nature Mater. 9, 26–30 (2010).
[Crossref]
E. Turgut, C. La-o-vorakiat, J. M. Shaw., P. Grychtol, H. T. Nembach, D. Rudolf, R. Adam, M. Aeschlimann, C. M. Schneider, T. J. Silva, M. M. Murnane, H. C. Kapteyn, and S. Mathias, “Controlling the competition between optically induced ultrafast spin-flip scattering and spin transport in magnetic multilayers,” Phys. Rev. Lett. 110, 197201 (2013).
[Crossref] [PubMed]
P. B. Corkum, “Plasma perspective on strong field multiphoton ionization,” Phys. Rev. Lett. 71, 1994–1997 (1993).
[Crossref] [PubMed]
R. Kienberger, E. Goulielmakis, M. Uiberacker, A. Baltuska, V. Yakovlev, F. Bammer, A. Scrinzi, Th. Westerwalbesloh, U. Kleineberg, U. Heinzmann, M. Drescher, and F. Krausz, “Atomic transient recorder,” Nature 427, 817 (2004).
[Crossref] [PubMed]
S. Svensson, B. Eriksson, N. Martensson, G. Wendin, and U. J. Gelius, “Electron shake-up and correlation satellites and continuum shake-off distributions in X-Ray photoelectron spectra of the rare gas atoms,” J. Electron Spectrosc. Relat. Phenom. 47, 327–384 (1988).
[Crossref]
J. Breidbach and L. S. Cederbaum, “Universal attosecond response to the removal of an electron,” Physical review letters 94, 033901 (2005).
[Crossref] [PubMed]
F. Remacle and R. D. Levine, “An electronic time scale in chemistry,” Proc. Natl. Acad. Sci. USA 103, 6793–6798 (2006).
[Crossref] [PubMed]
G. Sansone, E. Benedetti, F. Calegari, C. Vozzi, L. Avaldi, R. Flammini, L. Poletto, P. Villoresi, C. Altucci, R. Velotta, S. Stagira, S. De Silvestri, and M. Nisoli, Isolated single-cycle attosecond pulses,” Science 314, 443–446 (2006).
[Crossref] [PubMed]
E. Goulielmakis, M. Schultze, M. Hofstetter, V. S. Yakovlev, J. Gagnon, M. Uiberacker, A. L. Aquila, E. M. Gullikson, D. T. Attwood, R. Kienberger, F. Krausz, and U. Kleineberg, “Single-cycle nonlinear optics,” Science 320, 1614–1617 (2008).
[Crossref] [PubMed]
K. Zhao, Q. Zhang, M. Chini, Y. Wu, X. Wang, and Z. Chang, “Tailoring a 67 attosecond pulse through advantageous phase-mismatch,” Opt. Lett. 37, 3891–3893 (2012).
[Crossref] [PubMed]
V. S. Yakovlev, M. Ivanov, and F. Krausz, “Enhanced phase-matching for generation of soft X-ray harmonics and attosecond pulses in atomic gases,” Opt. Express 15, 15351–15364 (2007).
[Crossref] [PubMed]
J. Tate, T. Auguste, H. G. Muller, P. Salieres, P. Agostini, and L. F. DiMauro, “Scaling of wave-packet dynamics in an intense midinfrared field,” Phys. Rev. Lett. 98, 013901 (2007).
[Crossref] [PubMed]
K. Schiessl, K. L. Ishikawa, E. Persson, and J. Burgdorfer, “Quantum path interference in the wavelength dependence of high-harmonic generation,” Phys. Rev. Lett. 99, 253903 (2007).
[Crossref]
M. V. Frolov, N. L. Manakov, and A. F. Starace, “Wavelength scaling of high-harmonic yield: threshold phenomena and bound state symmetry dependence,” Phys. Rev. Lett. 100, 173001 (2008).
[Crossref] [PubMed]
P. Colosimo, G. Doumy, C. I. Blaga, J. Wheeler, C. Hauri, F. Catoire, J. Tate, R. Chirla, A. M. March, G. G. Paulus, H. G. Muller, P. Agostini, and L. F. DiMauro, “Scaling strong-field interactions towards the classical limit,” Nat. Physics 4, 386–389 (2008).
[Crossref]
A. D. Shiner, C. Trallero-Herrero, N. Kajumba, H. C. Bandulet, D. Comtois, F. Legare, M. Giguere, J. C. Kieffer, P. B. Corkum, and D. M. Villeneuve, “Wavelength scaling of high harmonic generation efficiency,” Phys. Rev. Lett. 103, 073902 (2009).
[Crossref] [PubMed]
A. Ravasio, D. Gauthier, F. R. N. C. Maia, M. Billon, J.-P. Caumes, D. Garzella, M. Géléoc, O. Gobert, J.-F. Hergott, A.-M. Pena, H. Perez, B. Carré, E. Bourhis, J. Gierak, A. Madouri, D. Mailly, B. Schiedt, M. Fajardo, J. Gautier, P. Zeitoun, P. H. Bucksbaum, J. Hajdu, and H. Merdji, ”Single-Shot Diffractive Imaging with a Table-Top Femtosecond Soft X-Ray Laser-Harmonics Source,” Phys. Rev. Lett. 103, 028104 (2009).
[Crossref] [PubMed]
Y. Nabekawa, H. Hasegawa, E. J. Takahashi, and K. Midorikawa, “Production of Doubly Charged Helium Ions by Two-Photon Absorption of an Intense Sub-10-fs Soft X-Ray Pulse at 42 eV Photon Energy,” Phys. Rev. Lett. 94, 043001 (2005).
[Crossref] [PubMed]
K. Hoshina, A. Hishikawa, K. Kato, T. Sako, K. Yamanouchi, E. J. Takahashi, Y. Nabekawa, and K. Midorikawa, “Dissociative ATI of H2 and D2 in intense soft x-ray laser fields,” J. Phys. B 39, 813–829 (2006).
[Crossref]
Y. Nabekawa, T. Shimizu, Y. Furukawa, E. J. Takahashi, and K. Midorikawa, “Interferometry of Attosecond Pulse Trains in the Extreme Ultraviolet Wavelength Region,” Phys. Rev. Lett. 102, 213904 (2009).
[Crossref] [PubMed]
P. Tzallas, E. Skantzakis, L. A. A. Nikolopoulos, G. D. Tsakiris, and D. Charalambidis, “Extreme-ultraviolet pumpCprobe studies of one-femtosecond-scale electron dynamics,” Nat. Physics 7, 781–784 (2011).
[Crossref]
E. Allaria, F. Bencivenga, R. Borghes, F. Capotondi, D. Castronovo, P. Charalambous, P. Cinquegrana, M. B. Danailov, G. De Ninno, A. Demidovich, S. Di Mitri, B. Diviacco, D. Fausti, W. M. Fawley, E. Ferrari, L. Froehlich, D. Gauthier, A. Gessini, L. Giannessi, R. Ivanov, M. Kiskinova, G. Kurdi, B. Mahieu, N. Mahne, I. Nikolov, C. Masciovecchio, E. Pedersoli, G. Penco, L. Raimondi, C. Serpico, P. Sigalotti, S. Spampinati, C. Spezzani, C. Svetina, M. Trovo, and M. Zangrando, “Two-colour pump-probe experiments with a twin-pulse-seed extreme ultraviolet free-electron laser,” Nat. Commun. 4, 2476 (2013).
[Crossref] [PubMed]
E. J. Takahashi, Y. Nabekawa, and K. Midorikawa, “Low-divergence coherent soft x-ray source at 13 nm by high-order harmonics,” Appl. Phys. Lett. 84, 4–6 (2004).
[Crossref]
E. J. Takahashi, Y. Nabekawa, and K. Midorikawa, “Generation of 10- microJ coherent extreme-ultraviolet light by use of high-order harmonics,” Opt. Lett. 27, 1920–1922 (2002).
[Crossref]
E. J. Takahashi, Y. Nabekawa, T. Otsuka, M. Obara, and K. Midorikawa, “Generation of highly coherent submicrojoule soft x rays by high-orderharmonics,” Phys. Rev. A 66, 021802(R) (2002).
[Crossref]
J.-F. Hergott, M. Kovacev, H. Merdji, C. Hubert, Y. Mairesse, E. Jean, P. Breger, P. Agostini, B. Carr, and P. Saliéres, “Extreme-ultraviolet high-order harmonic pulses in the microjoule range,” Phys. Rev. A 66, 021801(R) (2002).
[Crossref]
Y. Wu, E. Cunningham, H. Zang, J. Li, M. Chini, X. Wang, Y. Wang, K. Zhao, and Z. Chang, “Generation of high-flux attosecond extreme ultraviolet continuum with a 10 TW laser,” Appl. Phys. Lett. 102, 201104 (2013).
[Crossref]
P. Rudawski, C. M. Heyl, F. Brizuela, J. Schwenke, A. Persson, E. Mansten, R. Rakowski, L. Rading, F. Campi, B. Kim, P. Johnsson, and A. L’Huillier, “A high-flux high-order harmonic source,” Rev. Sci. Instrum. 84, 073103 (2013).
[Crossref] [PubMed]
P. Tzallas, E. Skantzakis, C. Kalpouzos, E. P. Bents, G. D. Tsakiris, and D. Charalambidis, “Generation of intense continuum extreme-ultraviolet radiation by many-cycle laser fields,” Nat. Physics 3, 846–850 (2007).
[Crossref]
X. Feng, S. Gilbertson, H. Mashiko, H. Wang, S. D. Khan, M. Chini, Y. Wu, K. Zhao, and Z. Chang, “Generation of Isolated Attosecond Pulses with 20 to 28 Femtosecond Lasers,” Phys. Rev. Lett. 103, 183903 (2009).
[Crossref]
P. Lan, P. Lu, W. Cao, Y. Li, and X. Wang, “Isolated sub-100-as pulse generation via controlling electron dynamics,” Phys. Rev. A 76, 011402(R) (2007).
[Crossref]
P. Lan, P. X. Lu, W. Cao, Y. H. Li, and X. L. Wang, “Attosecond ionization gating for isolated attosecond electron wave packet and broadband attosecond xuv pulses,” Phys. Rev. A 76, 051801(R) (2007).
[Crossref]
Q. Zhang, P. Lan, W. Hong, Q. Liao, Z. Yang, and P. Lu, “The effect of controlling laser field on broadband suppercontinuum generation,” Acta Phys Sin-Ch Ed 58, 4908–4913 (2009).
W. X. Chen, G. L. Chen, and D. E. Kim, “Two-color field for the generation of an isolated attosecond pulse in water-window region,” Opt. Express 19, 20610–20615 (2011).
[Crossref] [PubMed]
B. Kim, J. Ahn, Y. L. Yu, Y. Cheng, Z. Z. Xu, and D. E. Kim, “Optimization of multi-cycle two-color laser fields for the generation of an isolated attosecond pulse,” Opt Express 16, 10331–10340 (2008).
[Crossref] [PubMed]
E. J. Takahashi, P. Lan, Oliver D. Muöcke, Y. Nabekawa, and K. Midorikawa, “Attosecond nonlinear optics using gigawatt-scale isolated attosecond pulses,” Nat. Commun. 4, 2691 (2013).
[Crossref] [PubMed]
Q. Zhang, E. J. Takahashi, O. D. Mücke, P. Lu, and K. Midorikawa, “Dual-chirped optical parametric amplification for generating few hundred mJ infrared pulses,” Opt. Express 19, 7190–7212 (2011).
[Crossref] [PubMed]
Z. Hong, Q. Zhang, and P. Lu, “Compact dual-crystal optical parametric amplification for broadband IR pulse generation using a collinear geometry,” Opt. Express 21, 9491–9504 (2013).
[Crossref] [PubMed]
S. Fang, G. Cirmi, S.-H. Chia, O. D. Mücke, F. X. Kärtner, C. Manzoni, P. Farinello, and G. Cerullo, “Multi-mJ parametric synthesizer generating two-octave-wide optical waveforms,” Conference on Lasers and Electro-Optics Pacific Rim, 1–2 (CLEO-PR, 2013).
M. Lewenstein, Ph. Balcou, M. Yu. Ivanov, A. L’Huillier, and P. B. Corkum, “Theory of high-harmonic generation by low-frequency laser fields,” Phys. Rev. A 49, 2117–2132 (1994).
[Crossref] [PubMed]
Q. Zhang, P. Lu, W. Hong, Q. Liao, P. Lan, and X. B. Wang, “Enhanced high-order harmonic generation via controlling ionization in spatially extended systems,” Phys. Rev. A 79, 053406 (2009).
[Crossref]
R. Lopez-Martens, K. Varju, P. Johnsson, J. Mauritsson, Y. Mairesse, P. Salieres, M. B. Gaarde, K. J. Schafer, A. Persson, S. Svanberg, C. G. Wahlstrom, and A. L’Huillier, “Amplitude and phase control of attosecond light pulses,” Phys. Rev. Lett. 94, 033001 (2005).
[Crossref] [PubMed]
N. Dudovich, O. Smirnova, J. Levesque, Y. Mairesse, M. Y. Ivanov, D. M. Villeneuve, and P. B. Corkum, “Measuring and controlling the birth of attosecond XUV pulses,” Nat. Phys. 2, 781–786 (2006).
[Crossref]
Y. Zheng, Z. Zeng, P. Zou, L. Zhang, X. Li, P. Liu, R. Li, and Z. Xu, “Dynamic Chirp Control and Pulse Compression for Attosecond High-Order Harmonic Emission,” Phys. Rev. Lett 103, 043904 (2009).
[Crossref] [PubMed]
C. Winterfeldt, C. Spielmann, and G. Gerber, “Colloquium: Optimal control of high-harmonic generation,” Rev. Mod. Phys. 80,, 117–140 (2008).
[Crossref]
D. B. Milosević and W. Becker, “Role of long quantum orbits in high-order harmonic generation,” Phys. Rev. A 66, 063417 (2002).
[Crossref]
P. Lan, E. J. Takahashi, and K. Midorikawa, “Wavelength scaling of efficient high-order harmonic generation by two-color infrared laser fields,” Phys. Rev. A 81, 061802(R) (2010).
[Crossref]
H. Mashiko, A. Suda, and K. Midorikawa, “Focusing coherent soft-x-ray radiation to a micrometer spot size with an intensity of 1014 W/cm2,” Opt. Lett. 29, 1927–1929 (2004).
[Crossref] [PubMed]

2013 (7)

R. E. F. Silva, F. Catoire, P. Rivière, H. Bachau, and F. Martín, “Correlated electron and nuclear dynamics in strong field photoionization of H2+,” Phys. Rev. Lett. 110, 113001 (2013).
[Crossref]

E. Turgut, C. La-o-vorakiat, J. M. Shaw., P. Grychtol, H. T. Nembach, D. Rudolf, R. Adam, M. Aeschlimann, C. M. Schneider, T. J. Silva, M. M. Murnane, H. C. Kapteyn, and S. Mathias, “Controlling the competition between optically induced ultrafast spin-flip scattering and spin transport in magnetic multilayers,” Phys. Rev. Lett. 110, 197201 (2013).
[Crossref] [PubMed]

E. Allaria, F. Bencivenga, R. Borghes, F. Capotondi, D. Castronovo, P. Charalambous, P. Cinquegrana, M. B. Danailov, G. De Ninno, A. Demidovich, S. Di Mitri, B. Diviacco, D. Fausti, W. M. Fawley, E. Ferrari, L. Froehlich, D. Gauthier, A. Gessini, L. Giannessi, R. Ivanov, M. Kiskinova, G. Kurdi, B. Mahieu, N. Mahne, I. Nikolov, C. Masciovecchio, E. Pedersoli, G. Penco, L. Raimondi, C. Serpico, P. Sigalotti, S. Spampinati, C. Spezzani, C. Svetina, M. Trovo, and M. Zangrando, “Two-colour pump-probe experiments with a twin-pulse-seed extreme ultraviolet free-electron laser,” Nat. Commun. 4, 2476 (2013).
[Crossref] [PubMed]

Y. Wu, E. Cunningham, H. Zang, J. Li, M. Chini, X. Wang, Y. Wang, K. Zhao, and Z. Chang, “Generation of high-flux attosecond extreme ultraviolet continuum with a 10 TW laser,” Appl. Phys. Lett. 102, 201104 (2013).
[Crossref]

P. Rudawski, C. M. Heyl, F. Brizuela, J. Schwenke, A. Persson, E. Mansten, R. Rakowski, L. Rading, F. Campi, B. Kim, P. Johnsson, and A. L’Huillier, “A high-flux high-order harmonic source,” Rev. Sci. Instrum. 84, 073103 (2013).
[Crossref] [PubMed]

E. J. Takahashi, P. Lan, Oliver D. Muöcke, Y. Nabekawa, and K. Midorikawa, “Attosecond nonlinear optics using gigawatt-scale isolated attosecond pulses,” Nat. Commun. 4, 2691 (2013).
[Crossref] [PubMed]

Z. Hong, Q. Zhang, and P. Lu, “Compact dual-crystal optical parametric amplification for broadband IR pulse generation using a collinear geometry,” Opt. Express 21, 9491–9504 (2013).
[Crossref] [PubMed]

2012 (5)

K. Zhao, Q. Zhang, M. Chini, Y. Wu, X. Wang, and Z. Chang, “Tailoring a 67 attosecond pulse through advantageous phase-mismatch,” Opt. Lett. 37, 3891–3893 (2012).
[Crossref] [PubMed]

K. Liu, Q. Zhang, and P. Lu, “Enhancing electron localization in molecular dissociation by two-color mid- and near-infrared laser fields,” Phys. Rev. A 86, 033410 (2012).
[Crossref]

K. T. Kim, D. H. Ko, J. Park, N. N. Choi, C. M. Kim, K. L. Ishikawa, J. Lee, and C. H. Nam, “Amplitude and phase reconstruction of electron wave packets for probing ultrafast photoionization dynamics,” Phys. Rev. Lett. 108, 093001 (2012).
[Crossref] [PubMed]

Y. Zhou, C. Huang, Q. Liao, and P. Lu, “Classical Simulations Including Electron Correlations for Sequential Double Ionization,” Phys. Rev. Lett. 109, 053004 (2012).
[Crossref] [PubMed]

Y. Zhou, Q. Zhang, C. Huang, and P. Lu, “Classical description of strong-field double ionization by elliptical laser pulses,” Phys. Rev. A 86, 043427 (2012).
[Crossref]

2011 (5)

A. D. Shiner, B. E. Schmidt, C. Trallero-Herrero, H. J. Wörner, S. Patchkovskii, P. B. Corkum, J-C. Kieffer, F. Lègarè, and D. M. Villeneuve, “Probing collective multi-electron dynamics in xenon with high-harmonic spectroscopy,” Nat. Phys. 7, 464–467 (2011).
[Crossref]

T. Rohwer, S. Hellmann, M. Wiesenmayer, C. Sohrt, A. Stange, B. Slomski, A. Carr, Y. Liu, L. M. Avila, M. Kallane, S. Mathias, L. Kipp, K. Rossnagel, and M. Bauer, “Collapse of long-range charge order tracked by time-resolved photoemission at high momenta,” Nature 471, 490–493 (2011).
[Crossref] [PubMed]

Q. Zhang, E. J. Takahashi, O. D. Mücke, P. Lu, and K. Midorikawa, “Dual-chirped optical parametric amplification for generating few hundred mJ infrared pulses,” Opt. Express 19, 7190–7212 (2011).
[Crossref] [PubMed]

W. X. Chen, G. L. Chen, and D. E. Kim, “Two-color field for the generation of an isolated attosecond pulse in water-window region,” Opt. Express 19, 20610–20615 (2011).
[Crossref] [PubMed]

P. Tzallas, E. Skantzakis, L. A. A. Nikolopoulos, G. D. Tsakiris, and D. Charalambidis, “Extreme-ultraviolet pumpCprobe studies of one-femtosecond-scale electron dynamics,” Nat. Physics 7, 781–784 (2011).
[Crossref]

2010 (2)

P. Lan, E. J. Takahashi, and K. Midorikawa, “Wavelength scaling of efficient high-order harmonic generation by two-color infrared laser fields,” Phys. Rev. A 81, 061802(R) (2010).
[Crossref]

M. E. Siemens, Q. Li, R. Yang, K. A. Nelson, E. H. Anderson, M. M. Murnane, and H. C. Kapteyn, “Quasi-ballistic thermal transport from nanoscale interfaces observed using ultrafast coherent soft X-ray beams,” Nature Mater. 9, 26–30 (2010).
[Crossref]

2009 (8)

A. D. Shiner, C. Trallero-Herrero, N. Kajumba, H. C. Bandulet, D. Comtois, F. Legare, M. Giguere, J. C. Kieffer, P. B. Corkum, and D. M. Villeneuve, “Wavelength scaling of high harmonic generation efficiency,” Phys. Rev. Lett. 103, 073902 (2009).
[Crossref] [PubMed]

A. Ravasio, D. Gauthier, F. R. N. C. Maia, M. Billon, J.-P. Caumes, D. Garzella, M. Géléoc, O. Gobert, J.-F. Hergott, A.-M. Pena, H. Perez, B. Carré, E. Bourhis, J. Gierak, A. Madouri, D. Mailly, B. Schiedt, M. Fajardo, J. Gautier, P. Zeitoun, P. H. Bucksbaum, J. Hajdu, and H. Merdji, ”Single-Shot Diffractive Imaging with a Table-Top Femtosecond Soft X-Ray Laser-Harmonics Source,” Phys. Rev. Lett. 103, 028104 (2009).
[Crossref] [PubMed]

Y. Nabekawa, T. Shimizu, Y. Furukawa, E. J. Takahashi, and K. Midorikawa, “Interferometry of Attosecond Pulse Trains in the Extreme Ultraviolet Wavelength Region,” Phys. Rev. Lett. 102, 213904 (2009).
[Crossref] [PubMed]

F. Krausz and M. Ivanov, “Attosecond physics,” Rev. Mod. Phys. 81, 163–234 (2009).
[Crossref]

Y. Zheng, Z. Zeng, P. Zou, L. Zhang, X. Li, P. Liu, R. Li, and Z. Xu, “Dynamic Chirp Control and Pulse Compression for Attosecond High-Order Harmonic Emission,” Phys. Rev. Lett 103, 043904 (2009).
[Crossref] [PubMed]

X. Feng, S. Gilbertson, H. Mashiko, H. Wang, S. D. Khan, M. Chini, Y. Wu, K. Zhao, and Z. Chang, “Generation of Isolated Attosecond Pulses with 20 to 28 Femtosecond Lasers,” Phys. Rev. Lett. 103, 183903 (2009).
[Crossref]

Q. Zhang, P. Lan, W. Hong, Q. Liao, Z. Yang, and P. Lu, “The effect of controlling laser field on broadband suppercontinuum generation,” Acta Phys Sin-Ch Ed 58, 4908–4913 (2009).

Q. Zhang, P. Lu, W. Hong, Q. Liao, P. Lan, and X. B. Wang, “Enhanced high-order harmonic generation via controlling ionization in spatially extended systems,” Phys. Rev. A 79, 053406 (2009).
[Crossref]

2008 (6)

C. Winterfeldt, C. Spielmann, and G. Gerber, “Colloquium: Optimal control of high-harmonic generation,” Rev. Mod. Phys. 80,, 117–140 (2008).
[Crossref]

B. Kim, J. Ahn, Y. L. Yu, Y. Cheng, Z. Z. Xu, and D. E. Kim, “Optimization of multi-cycle two-color laser fields for the generation of an isolated attosecond pulse,” Opt Express 16, 10331–10340 (2008).
[Crossref] [PubMed]

L. Miaja-Avila, G. Saathoff, S. Mathias, J. Yin, C. La-o-vorakiat, M. Bauer, M. Aeschlimann, M. M. Murnane, and H. C. Kapteyn, “Direct measurement of core-level relaxation dynamics on a surface-adsorbate system,” Phys. Rev. Lett. 101, 046101 (2008).
[Crossref] [PubMed]

E. Goulielmakis, M. Schultze, M. Hofstetter, V. S. Yakovlev, J. Gagnon, M. Uiberacker, A. L. Aquila, E. M. Gullikson, D. T. Attwood, R. Kienberger, F. Krausz, and U. Kleineberg, “Single-cycle nonlinear optics,” Science 320, 1614–1617 (2008).
[Crossref] [PubMed]

M. V. Frolov, N. L. Manakov, and A. F. Starace, “Wavelength scaling of high-harmonic yield: threshold phenomena and bound state symmetry dependence,” Phys. Rev. Lett. 100, 173001 (2008).
[Crossref] [PubMed]

P. Colosimo, G. Doumy, C. I. Blaga, J. Wheeler, C. Hauri, F. Catoire, J. Tate, R. Chirla, A. M. March, G. G. Paulus, H. G. Muller, P. Agostini, and L. F. DiMauro, “Scaling strong-field interactions towards the classical limit,” Nat. Physics 4, 386–389 (2008).
[Crossref]

2007 (9)

J. Tate, T. Auguste, H. G. Muller, P. Salieres, P. Agostini, and L. F. DiMauro, “Scaling of wave-packet dynamics in an intense midinfrared field,” Phys. Rev. Lett. 98, 013901 (2007).
[Crossref] [PubMed]

K. Schiessl, K. L. Ishikawa, E. Persson, and J. Burgdorfer, “Quantum path interference in the wavelength dependence of high-harmonic generation,” Phys. Rev. Lett. 99, 253903 (2007).
[Crossref]

E. Gagnon, P. Ranitovic, X. M. Tong, C. L. Cocke, M. M. Murnane, H. C. Kapteyn, and A. S. Sandhu, “Soft X-ray-driven femtosecond molecular dynamics,” Science 317, 1374–1378 (2007).
[Crossref] [PubMed]

M. Uiberacker, Th. Uphues, M. Schultze, A. J. Verhoef, V. Yakovlev, M. F. Kling, J. Rauschenberger, N. M. Kabachnik, H. Schröder, M. Lezius, K. L. Kompa, H.-G. Muller, M. J. J. Vrakking, S. Hendel, U. Kleineberg, U. Heinzmann, M. Drescher, and F. Krausz, “Attosecond real-time observation of electron tunnelling inatoms,” Nature 446, 627–632 (2007).
[Crossref] [PubMed]

M. I. Stockman, M. F. Klin, U. Kleineberg, and F. Krausz, “Attosecond nanoplasmonic-field microscope,” Nat. Photonics 1, 539–544 (2007).
[Crossref]

P. Lan, P. Lu, W. Cao, Y. Li, and X. Wang, “Isolated sub-100-as pulse generation via controlling electron dynamics,” Phys. Rev. A 76, 011402(R) (2007).
[Crossref]

P. Lan, P. X. Lu, W. Cao, Y. H. Li, and X. L. Wang, “Attosecond ionization gating for isolated attosecond electron wave packet and broadband attosecond xuv pulses,” Phys. Rev. A 76, 051801(R) (2007).
[Crossref]

P. Tzallas, E. Skantzakis, C. Kalpouzos, E. P. Bents, G. D. Tsakiris, and D. Charalambidis, “Generation of intense continuum extreme-ultraviolet radiation by many-cycle laser fields,” Nat. Physics 3, 846–850 (2007).
[Crossref]

V. S. Yakovlev, M. Ivanov, and F. Krausz, “Enhanced phase-matching for generation of soft X-ray harmonics and attosecond pulses in atomic gases,” Opt. Express 15, 15351–15364 (2007).
[Crossref] [PubMed]

2006 (4)

N. Dudovich, O. Smirnova, J. Levesque, Y. Mairesse, M. Y. Ivanov, D. M. Villeneuve, and P. B. Corkum, “Measuring and controlling the birth of attosecond XUV pulses,” Nat. Phys. 2, 781–786 (2006).
[Crossref]

F. Remacle and R. D. Levine, “An electronic time scale in chemistry,” Proc. Natl. Acad. Sci. USA 103, 6793–6798 (2006).
[Crossref] [PubMed]

G. Sansone, E. Benedetti, F. Calegari, C. Vozzi, L. Avaldi, R. Flammini, L. Poletto, P. Villoresi, C. Altucci, R. Velotta, S. Stagira, S. De Silvestri, and M. Nisoli, Isolated single-cycle attosecond pulses,” Science 314, 443–446 (2006).
[Crossref] [PubMed]

K. Hoshina, A. Hishikawa, K. Kato, T. Sako, K. Yamanouchi, E. J. Takahashi, Y. Nabekawa, and K. Midorikawa, “Dissociative ATI of H2 and D2 in intense soft x-ray laser fields,” J. Phys. B 39, 813–829 (2006).
[Crossref]

2005 (3)

Y. Nabekawa, H. Hasegawa, E. J. Takahashi, and K. Midorikawa, “Production of Doubly Charged Helium Ions by Two-Photon Absorption of an Intense Sub-10-fs Soft X-Ray Pulse at 42 eV Photon Energy,” Phys. Rev. Lett. 94, 043001 (2005).
[Crossref] [PubMed]

J. Breidbach and L. S. Cederbaum, “Universal attosecond response to the removal of an electron,” Physical review letters 94, 033901 (2005).
[Crossref] [PubMed]

R. Lopez-Martens, K. Varju, P. Johnsson, J. Mauritsson, Y. Mairesse, P. Salieres, M. B. Gaarde, K. J. Schafer, A. Persson, S. Svanberg, C. G. Wahlstrom, and A. L’Huillier, “Amplitude and phase control of attosecond light pulses,” Phys. Rev. Lett. 94, 033001 (2005).
[Crossref] [PubMed]

2004 (3)

H. Mashiko, A. Suda, and K. Midorikawa, “Focusing coherent soft-x-ray radiation to a micrometer spot size with an intensity of 1014 W/cm2,” Opt. Lett. 29, 1927–1929 (2004).
[Crossref] [PubMed]

E. J. Takahashi, Y. Nabekawa, and K. Midorikawa, “Low-divergence coherent soft x-ray source at 13 nm by high-order harmonics,” Appl. Phys. Lett. 84, 4–6 (2004).
[Crossref]

R. Kienberger, E. Goulielmakis, M. Uiberacker, A. Baltuska, V. Yakovlev, F. Bammer, A. Scrinzi, Th. Westerwalbesloh, U. Kleineberg, U. Heinzmann, M. Drescher, and F. Krausz, “Atomic transient recorder,” Nature 427, 817 (2004).
[Crossref] [PubMed]

2002 (5)

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

E. J. Takahashi, Y. Nabekawa, T. Otsuka, M. Obara, and K. Midorikawa, “Generation of highly coherent submicrojoule soft x rays by high-orderharmonics,” Phys. Rev. A 66, 021802(R) (2002).
[Crossref]

J.-F. Hergott, M. Kovacev, H. Merdji, C. Hubert, Y. Mairesse, E. Jean, P. Breger, P. Agostini, B. Carr, and P. Saliéres, “Extreme-ultraviolet high-order harmonic pulses in the microjoule range,” Phys. Rev. A 66, 021801(R) (2002).
[Crossref]

D. B. Milosević and W. Becker, “Role of long quantum orbits in high-order harmonic generation,” Phys. Rev. A 66, 063417 (2002).
[Crossref]

E. J. Takahashi, Y. Nabekawa, and K. Midorikawa, “Generation of 10- microJ coherent extreme-ultraviolet light by use of high-order harmonics,” Opt. Lett. 27, 1920–1922 (2002).
[Crossref]

2000 (1)

Th. Weber, H. Giessen, M. Weckenbrock, G. Urbasch, A. Staudte, L. Spielberger, O. Jagutzki, V. Mergel, M. Vollmer, and R. Dörner, “Correlated electron emission in multiphoton double ionization,” Nature 405, 658–661 (2000).
[Crossref] [PubMed]

1994 (1)

M. Lewenstein, Ph. Balcou, M. Yu. Ivanov, A. L’Huillier, and P. B. Corkum, “Theory of high-harmonic generation by low-frequency laser fields,” Phys. Rev. A 49, 2117–2132 (1994).
[Crossref] [PubMed]

1993 (1)

P. B. Corkum, “Plasma perspective on strong field multiphoton ionization,” Phys. Rev. Lett. 71, 1994–1997 (1993).
[Crossref] [PubMed]

1988 (1)

S. Svensson, B. Eriksson, N. Martensson, G. Wendin, and U. J. Gelius, “Electron shake-up and correlation satellites and continuum shake-off distributions in X-Ray photoelectron spectra of the rare gas atoms,” J. Electron Spectrosc. Relat. Phenom. 47, 327–384 (1988).
[Crossref]

Adam, R.

Aeschlimann, M.

Agostini, P.

Ahn, J.

Allaria, E.

Altucci, C.

Anderson, E. H.

Aquila, A. L.

Attwood, D. T.

Auguste, T.

Avaldi, L.

Avila, L. M.

Bachau, H.

Balcou, Ph.

Baltuska, A.

Bammer, F.

Bandulet, H. C.

Bauer, M.

Becker, W.

D. B. Milosević and W. Becker, “Role of long quantum orbits in high-order harmonic generation,” Phys. Rev. A 66, 063417 (2002).
[Crossref]

Bencivenga, F.

Benedetti, E.

Bents, E. P.

Billon, M.

Blaga, C. I.

Borghes, R.

Bourhis, E.

Breger, P.

Breidbach, J.

J. Breidbach and L. S. Cederbaum, “Universal attosecond response to the removal of an electron,” Physical review letters 94, 033901 (2005).
[Crossref] [PubMed]

Brizuela, F.

Bucksbaum, P. H.

Burgdorfer, J.

K. Schiessl, K. L. Ishikawa, E. Persson, and J. Burgdorfer, “Quantum path interference in the wavelength dependence of high-harmonic generation,” Phys. Rev. Lett. 99, 253903 (2007).
[Crossref]

Calegari, F.

Campi, F.

Cao, W.

P. Lan, P. Lu, W. Cao, Y. Li, and X. Wang, “Isolated sub-100-as pulse generation via controlling electron dynamics,” Phys. Rev. A 76, 011402(R) (2007).
[Crossref]

Capotondi, F.

Carr, A.

Carr, B.

Carré, B.

Castronovo, D.

Catoire, F.

Caumes, J.-P.

Cederbaum, L. S.

J. Breidbach and L. S. Cederbaum, “Universal attosecond response to the removal of an electron,” Physical review letters 94, 033901 (2005).
[Crossref] [PubMed]

Cerullo, G.

S. Fang, G. Cirmi, S.-H. Chia, O. D. Mücke, F. X. Kärtner, C. Manzoni, P. Farinello, and G. Cerullo, “Multi-mJ parametric synthesizer generating two-octave-wide optical waveforms,” Conference on Lasers and Electro-Optics Pacific Rim, 1–2 (CLEO-PR, 2013).

Chang, Z.

K. Zhao, Q. Zhang, M. Chini, Y. Wu, X. Wang, and Z. Chang, “Tailoring a 67 attosecond pulse through advantageous phase-mismatch,” Opt. Lett. 37, 3891–3893 (2012).
[Crossref] [PubMed]

Charalambidis, D.

Charalambous, P.

Chen, G. L.

W. X. Chen, G. L. Chen, and D. E. Kim, “Two-color field for the generation of an isolated attosecond pulse in water-window region,” Opt. Express 19, 20610–20615 (2011).
[Crossref] [PubMed]

Chen, W. X.

W. X. Chen, G. L. Chen, and D. E. Kim, “Two-color field for the generation of an isolated attosecond pulse in water-window region,” Opt. Express 19, 20610–20615 (2011).
[Crossref] [PubMed]

Cheng, Y.

Chia, S.-H.

Chini, M.

K. Zhao, Q. Zhang, M. Chini, Y. Wu, X. Wang, and Z. Chang, “Tailoring a 67 attosecond pulse through advantageous phase-mismatch,” Opt. Lett. 37, 3891–3893 (2012).
[Crossref] [PubMed]

Chirla, R.

Choi, N. N.

Cinquegrana, P.

Cirmi, G.

Cocke, C. L.

Colosimo, P.

Comtois, D.

Corkum, P. B.

P. B. Corkum, “Plasma perspective on strong field multiphoton ionization,” Phys. Rev. Lett. 71, 1994–1997 (1993).
[Crossref] [PubMed]

Cunningham, E.

Danailov, M. B.

De Ninno, G.

De Silvestri, S.

Demidovich, A.

Di Mitri, S.

DiMauro, L. F.

Diviacco, B.

Dörner, R.

Doumy, G.

Drescher, M.

Dudovich, N.

Eriksson, B.

Fajardo, M.

Fang, S.

Farinello, P.

Fausti, D.

Fawley, W. M.

Feng, X.

Ferrari, E.

Flammini, R.

Froehlich, L.

Frolov, M. V.

Furukawa, Y.

Gaarde, M. B.

Gagnon, E.

Gagnon, J.

Garzella, D.

Gauthier, D.

Gautier, J.

Géléoc, M.

Gelius, U. J.

Gerber, G.

C. Winterfeldt, C. Spielmann, and G. Gerber, “Colloquium: Optimal control of high-harmonic generation,” Rev. Mod. Phys. 80,, 117–140 (2008).
[Crossref]

Gessini, A.

Giannessi, L.

Gierak, J.

Giessen, H.

Giguere, M.

Gilbertson, S.

Gobert, O.

Goulielmakis, E.

Grychtol, P.

Gullikson, E. M.

Hajdu, J.

Hasegawa, H.

Hauri, C.

Heinzmann, U.

Hellmann, S.

Hendel, S.

Hentschel, M.

Hergott, J.-F.

Heyl, C. M.

Hishikawa, A.

Hofstetter, M.

Hong, W.

Q. Zhang, P. Lan, W. Hong, Q. Liao, Z. Yang, and P. Lu, “The effect of controlling laser field on broadband suppercontinuum generation,” Acta Phys Sin-Ch Ed 58, 4908–4913 (2009).

Hong, Z.

Hoshina, K.

Huang, C.

Y. Zhou, Q. Zhang, C. Huang, and P. Lu, “Classical description of strong-field double ionization by elliptical laser pulses,” Phys. Rev. A 86, 043427 (2012).
[Crossref]

Y. Zhou, C. Huang, Q. Liao, and P. Lu, “Classical Simulations Including Electron Correlations for Sequential Double Ionization,” Phys. Rev. Lett. 109, 053004 (2012).
[Crossref] [PubMed]

Hubert, C.

Ishikawa, K. L.

K. Schiessl, K. L. Ishikawa, E. Persson, and J. Burgdorfer, “Quantum path interference in the wavelength dependence of high-harmonic generation,” Phys. Rev. Lett. 99, 253903 (2007).
[Crossref]

Ivanov, M.

F. Krausz and M. Ivanov, “Attosecond physics,” Rev. Mod. Phys. 81, 163–234 (2009).
[Crossref]

Ivanov, M. Y.

Ivanov, R.

Jagutzki, O.

Jean, E.

Johnsson, P.

Kabachnik, N. M.

Kajumba, N.

Kallane, M.

Kalpouzos, C.

Kapteyn, H. C.

Kärtner, F. X.

Kato, K.

Khan, S. D.

Kieffer, J. C.

Kieffer, J-C.

Kienberger, R.

Kim, B.

Kim, C. M.

Kim, D. E.

W. X. Chen, G. L. Chen, and D. E. Kim, “Two-color field for the generation of an isolated attosecond pulse in water-window region,” Opt. Express 19, 20610–20615 (2011).
[Crossref] [PubMed]

Kim, K. T.

Kipp, L.

Kiskinova, M.

Kleineberg, U.

M. I. Stockman, M. F. Klin, U. Kleineberg, and F. Krausz, “Attosecond nanoplasmonic-field microscope,” Nat. Photonics 1, 539–544 (2007).
[Crossref]

Klin, M. F.

M. I. Stockman, M. F. Klin, U. Kleineberg, and F. Krausz, “Attosecond nanoplasmonic-field microscope,” Nat. Photonics 1, 539–544 (2007).
[Crossref]

Kling, M. F.

Ko, D. H.

Kompa, K. L.

Kovacev, M.

Krausz, F.

F. Krausz and M. Ivanov, “Attosecond physics,” Rev. Mod. Phys. 81, 163–234 (2009).
[Crossref]

M. I. Stockman, M. F. Klin, U. Kleineberg, and F. Krausz, “Attosecond nanoplasmonic-field microscope,” Nat. Photonics 1, 539–544 (2007).
[Crossref]

Kurdi, G.

L’Huillier, A.

Lan, P.

P. Lan, E. J. Takahashi, and K. Midorikawa, “Wavelength scaling of efficient high-order harmonic generation by two-color infrared laser fields,” Phys. Rev. A 81, 061802(R) (2010).
[Crossref]

Q. Zhang, P. Lan, W. Hong, Q. Liao, Z. Yang, and P. Lu, “The effect of controlling laser field on broadband suppercontinuum generation,” Acta Phys Sin-Ch Ed 58, 4908–4913 (2009).

P. Lan, P. Lu, W. Cao, Y. Li, and X. Wang, “Isolated sub-100-as pulse generation via controlling electron dynamics,” Phys. Rev. A 76, 011402(R) (2007).
[Crossref]

La-o-vorakiat, C.

Lee, J.

Legare, F.

Lègarè, F.

Levesque, J.

Levine, R. D.

F. Remacle and R. D. Levine, “An electronic time scale in chemistry,” Proc. Natl. Acad. Sci. USA 103, 6793–6798 (2006).
[Crossref] [PubMed]

Lewenstein, M.

Lezius, M.

Li, J.

Li, Q.

Li, R.

Li, X.

Li, Y.

P. Lan, P. Lu, W. Cao, Y. Li, and X. Wang, “Isolated sub-100-as pulse generation via controlling electron dynamics,” Phys. Rev. A 76, 011402(R) (2007).
[Crossref]

Li, Y. H.

Liao, Q.

Y. Zhou, C. Huang, Q. Liao, and P. Lu, “Classical Simulations Including Electron Correlations for Sequential Double Ionization,” Phys. Rev. Lett. 109, 053004 (2012).
[Crossref] [PubMed]

Q. Zhang, P. Lan, W. Hong, Q. Liao, Z. Yang, and P. Lu, “The effect of controlling laser field on broadband suppercontinuum generation,” Acta Phys Sin-Ch Ed 58, 4908–4913 (2009).

Liu, K.

K. Liu, Q. Zhang, and P. Lu, “Enhancing electron localization in molecular dissociation by two-color mid- and near-infrared laser fields,” Phys. Rev. A 86, 033410 (2012).
[Crossref]

Liu, P.

Liu, Y.

Lopez-Martens, R.

Lu, P.

K. Liu, Q. Zhang, and P. Lu, “Enhancing electron localization in molecular dissociation by two-color mid- and near-infrared laser fields,” Phys. Rev. A 86, 033410 (2012).
[Crossref]

Y. Zhou, C. Huang, Q. Liao, and P. Lu, “Classical Simulations Including Electron Correlations for Sequential Double Ionization,” Phys. Rev. Lett. 109, 053004 (2012).
[Crossref] [PubMed]

Y. Zhou, Q. Zhang, C. Huang, and P. Lu, “Classical description of strong-field double ionization by elliptical laser pulses,” Phys. Rev. A 86, 043427 (2012).
[Crossref]

Q. Zhang, P. Lan, W. Hong, Q. Liao, Z. Yang, and P. Lu, “The effect of controlling laser field on broadband suppercontinuum generation,” Acta Phys Sin-Ch Ed 58, 4908–4913 (2009).

P. Lan, P. Lu, W. Cao, Y. Li, and X. Wang, “Isolated sub-100-as pulse generation via controlling electron dynamics,” Phys. Rev. A 76, 011402(R) (2007).
[Crossref]

Lu, P. X.

Madouri, A.

Mahieu, B.

Mahne, N.

Maia, F. R. N. C.

Mailly, D.

Mairesse, Y.

Manakov, N. L.

Mansten, E.

Manzoni, C.

March, A. M.

Martensson, N.

Martín, F.

Masciovecchio, C.

Mashiko, H.

H. Mashiko, A. Suda, and K. Midorikawa, “Focusing coherent soft-x-ray radiation to a micrometer spot size with an intensity of 1014 W/cm2,” Opt. Lett. 29, 1927–1929 (2004).
[Crossref] [PubMed]

Mathias, S.

Mauritsson, J.

Merdji, H.

Mergel, V.

Miaja-Avila, L.

Midorikawa, K.

P. Lan, E. J. Takahashi, and K. Midorikawa, “Wavelength scaling of efficient high-order harmonic generation by two-color infrared laser fields,” Phys. Rev. A 81, 061802(R) (2010).
[Crossref]

E. J. Takahashi, Y. Nabekawa, and K. Midorikawa, “Low-divergence coherent soft x-ray source at 13 nm by high-order harmonics,” Appl. Phys. Lett. 84, 4–6 (2004).
[Crossref]

H. Mashiko, A. Suda, and K. Midorikawa, “Focusing coherent soft-x-ray radiation to a micrometer spot size with an intensity of 1014 W/cm2,” Opt. Lett. 29, 1927–1929 (2004).
[Crossref] [PubMed]

E. J. Takahashi, Y. Nabekawa, and K. Midorikawa, “Generation of 10- microJ coherent extreme-ultraviolet light by use of high-order harmonics,” Opt. Lett. 27, 1920–1922 (2002).
[Crossref]

Milosevic, D. B.

D. B. Milosević and W. Becker, “Role of long quantum orbits in high-order harmonic generation,” Phys. Rev. A 66, 063417 (2002).
[Crossref]

Mücke, O. D.

Muller, H. G.

Muller, H.-G.

Muöcke, Oliver D.

Murnane, M. M.

Nabekawa, Y.

E. J. Takahashi, Y. Nabekawa, and K. Midorikawa, “Low-divergence coherent soft x-ray source at 13 nm by high-order harmonics,” Appl. Phys. Lett. 84, 4–6 (2004).
[Crossref]

E. J. Takahashi, Y. Nabekawa, and K. Midorikawa, “Generation of 10- microJ coherent extreme-ultraviolet light by use of high-order harmonics,” Opt. Lett. 27, 1920–1922 (2002).
[Crossref]

Nam, C. H.

Nelson, K. A.

Nembach, H. T.

Nikolopoulos, L. A. A.

Nikolov, I.

Nisoli, M.

Obara, M.

Otsuka, T.

Park, J.

Patchkovskii, S.

Paulus, G. G.

Pedersoli, E.

Pena, A.-M.

Penco, G.

Perez, H.

Persson, A.

Persson, E.

K. Schiessl, K. L. Ishikawa, E. Persson, and J. Burgdorfer, “Quantum path interference in the wavelength dependence of high-harmonic generation,” Phys. Rev. Lett. 99, 253903 (2007).
[Crossref]

Poletto, L.

Rading, L.

Raimondi, L.

Rakowski, R.

Ranitovic, P.

Rauschenberger, J.

Ravasio, A.

Remacle, F.

F. Remacle and R. D. Levine, “An electronic time scale in chemistry,” Proc. Natl. Acad. Sci. USA 103, 6793–6798 (2006).
[Crossref] [PubMed]

Rivière, P.

Rohwer, T.

Rossnagel, K.

Rudawski, P.

Rudolf, D.

Saathoff, G.

Sako, T.

Salieres, P.

Saliéres, P.

Sandhu, A. S.

Sansone, G.

Schafer, K. J.

Schiedt, B.

Schiessl, K.

K. Schiessl, K. L. Ishikawa, E. Persson, and J. Burgdorfer, “Quantum path interference in the wavelength dependence of high-harmonic generation,” Phys. Rev. Lett. 99, 253903 (2007).
[Crossref]

Schmidt, B. E.

Schneider, C. M.

Schröder, H.

Schultze, M.

Schwenke, J.

Scrinzi, A.

Serpico, C.

Shaw., J. M.

Shimizu, T.

Shiner, A. D.

Siemens, M. E.

Sigalotti, P.

Silva, R. E. F.

Silva, T. J.

Skantzakis, E.

Slomski, B.

Smirnova, O.

Sohrt, C.

Spampinati, S.

Spezzani, C.

Spielberger, L.

Spielmann, C.

C. Winterfeldt, C. Spielmann, and G. Gerber, “Colloquium: Optimal control of high-harmonic generation,” Rev. Mod. Phys. 80,, 117–140 (2008).
[Crossref]

Stagira, S.

Stange, A.

Starace, A. F.

Staudte, A.

Stockman, M. I.

M. I. Stockman, M. F. Klin, U. Kleineberg, and F. Krausz, “Attosecond nanoplasmonic-field microscope,” Nat. Photonics 1, 539–544 (2007).
[Crossref]

Suda, A.

H. Mashiko, A. Suda, and K. Midorikawa, “Focusing coherent soft-x-ray radiation to a micrometer spot size with an intensity of 1014 W/cm2,” Opt. Lett. 29, 1927–1929 (2004).
[Crossref] [PubMed]

Svanberg, S.

Svensson, S.

Svetina, C.

Takahashi, E. J.

P. Lan, E. J. Takahashi, and K. Midorikawa, “Wavelength scaling of efficient high-order harmonic generation by two-color infrared laser fields,” Phys. Rev. A 81, 061802(R) (2010).
[Crossref]

E. J. Takahashi, Y. Nabekawa, and K. Midorikawa, “Low-divergence coherent soft x-ray source at 13 nm by high-order harmonics,” Appl. Phys. Lett. 84, 4–6 (2004).
[Crossref]

E. J. Takahashi, Y. Nabekawa, and K. Midorikawa, “Generation of 10- microJ coherent extreme-ultraviolet light by use of high-order harmonics,” Opt. Lett. 27, 1920–1922 (2002).
[Crossref]

Tate, J.

Tong, X. M.

Trallero-Herrero, C.

Trovo, M.

Tsakiris, G. D.

Turgut, E.

Tzallas, P.

Uiberacker, M.

Uphues, Th.

Urbasch, G.

Varju, K.

Velotta, R.

Verhoef, A. J.

Villeneuve, D. M.

Villoresi, P.

Vollmer, M.

Vozzi, C.

Vrakking, M. J. J.

Wahlstrom, C. G.

Wang, H.

Wang, X.

K. Zhao, Q. Zhang, M. Chini, Y. Wu, X. Wang, and Z. Chang, “Tailoring a 67 attosecond pulse through advantageous phase-mismatch,” Opt. Lett. 37, 3891–3893 (2012).
[Crossref] [PubMed]

P. Lan, P. Lu, W. Cao, Y. Li, and X. Wang, “Isolated sub-100-as pulse generation via controlling electron dynamics,” Phys. Rev. A 76, 011402(R) (2007).
[Crossref]

Wang, X. B.

Wang, X. L.

Wang, Y.

Weber, Th.

Weckenbrock, M.

Wendin, G.

Westerwalbesloh, Th.

Wheeler, J.

Wiesenmayer, M.

Winterfeldt, C.

C. Winterfeldt, C. Spielmann, and G. Gerber, “Colloquium: Optimal control of high-harmonic generation,” Rev. Mod. Phys. 80,, 117–140 (2008).
[Crossref]

Wörner, H. J.

Wu, Y.

K. Zhao, Q. Zhang, M. Chini, Y. Wu, X. Wang, and Z. Chang, “Tailoring a 67 attosecond pulse through advantageous phase-mismatch,” Opt. Lett. 37, 3891–3893 (2012).
[Crossref] [PubMed]

Xu, Z.

Xu, Z. Z.

Yakovlev, V.

Yakovlev, V. S.

Yamanouchi, K.

Yang, R.

Yang, Z.

Q. Zhang, P. Lan, W. Hong, Q. Liao, Z. Yang, and P. Lu, “The effect of controlling laser field on broadband suppercontinuum generation,” Acta Phys Sin-Ch Ed 58, 4908–4913 (2009).

Yin, J.

Yu, Y. L.

Yu. Ivanov, M.

Zang, H.

Zangrando, M.

Zeitoun, P.

Zeng, Z.

Zhang, L.

Zhang, Q.

K. Zhao, Q. Zhang, M. Chini, Y. Wu, X. Wang, and Z. Chang, “Tailoring a 67 attosecond pulse through advantageous phase-mismatch,” Opt. Lett. 37, 3891–3893 (2012).
[Crossref] [PubMed]

Y. Zhou, Q. Zhang, C. Huang, and P. Lu, “Classical description of strong-field double ionization by elliptical laser pulses,” Phys. Rev. A 86, 043427 (2012).
[Crossref]

K. Liu, Q. Zhang, and P. Lu, “Enhancing electron localization in molecular dissociation by two-color mid- and near-infrared laser fields,” Phys. Rev. A 86, 033410 (2012).
[Crossref]

Q. Zhang, P. Lan, W. Hong, Q. Liao, Z. Yang, and P. Lu, “The effect of controlling laser field on broadband suppercontinuum generation,” Acta Phys Sin-Ch Ed 58, 4908–4913 (2009).

Zhao, K.

K. Zhao, Q. Zhang, M. Chini, Y. Wu, X. Wang, and Z. Chang, “Tailoring a 67 attosecond pulse through advantageous phase-mismatch,” Opt. Lett. 37, 3891–3893 (2012).
[Crossref] [PubMed]

Zheng, Y.

Zhou, Y.

Y. Zhou, Q. Zhang, C. Huang, and P. Lu, “Classical description of strong-field double ionization by elliptical laser pulses,” Phys. Rev. A 86, 043427 (2012).
[Crossref]

Y. Zhou, C. Huang, Q. Liao, and P. Lu, “Classical Simulations Including Electron Correlations for Sequential Double Ionization,” Phys. Rev. Lett. 109, 053004 (2012).
[Crossref] [PubMed]

Zou, P.

Acta Phys Sin-Ch Ed (1)

Q. Zhang, P. Lan, W. Hong, Q. Liao, Z. Yang, and P. Lu, “The effect of controlling laser field on broadband suppercontinuum generation,” Acta Phys Sin-Ch Ed 58, 4908–4913 (2009).

Appl. Phys. Lett. (2)

E. J. Takahashi, Y. Nabekawa, and K. Midorikawa, “Low-divergence coherent soft x-ray source at 13 nm by high-order harmonics,” Appl. Phys. Lett. 84, 4–6 (2004).
[Crossref]

J. Electron Spectrosc. Relat. Phenom. (1)

J. Phys. B (1)

Nat. Commun. (2)

Nat. Photonics (1)

M. I. Stockman, M. F. Klin, U. Kleineberg, and F. Krausz, “Attosecond nanoplasmonic-field microscope,” Nat. Photonics 1, 539–544 (2007).
[Crossref]

Nat. Phys. (2)

Nat. Physics (3)

Nature (5)

Nature Mater. (1)

Opt Express (1)

Opt. Express (4)

W. X. Chen, G. L. Chen, and D. E. Kim, “Two-color field for the generation of an isolated attosecond pulse in water-window region,” Opt. Express 19, 20610–20615 (2011).
[Crossref] [PubMed]

Opt. Lett. (3)

H. Mashiko, A. Suda, and K. Midorikawa, “Focusing coherent soft-x-ray radiation to a micrometer spot size with an intensity of 1014 W/cm2,” Opt. Lett. 29, 1927–1929 (2004).
[Crossref] [PubMed]

K. Zhao, Q. Zhang, M. Chini, Y. Wu, X. Wang, and Z. Chang, “Tailoring a 67 attosecond pulse through advantageous phase-mismatch,” Opt. Lett. 37, 3891–3893 (2012).
[Crossref] [PubMed]

E. J. Takahashi, Y. Nabekawa, and K. Midorikawa, “Generation of 10- microJ coherent extreme-ultraviolet light by use of high-order harmonics,” Opt. Lett. 27, 1920–1922 (2002).
[Crossref]

Phys. Rev. A (10)

P. Lan, P. Lu, W. Cao, Y. Li, and X. Wang, “Isolated sub-100-as pulse generation via controlling electron dynamics,” Phys. Rev. A 76, 011402(R) (2007).
[Crossref]

D. B. Milosević and W. Becker, “Role of long quantum orbits in high-order harmonic generation,” Phys. Rev. A 66, 063417 (2002).
[Crossref]

P. Lan, E. J. Takahashi, and K. Midorikawa, “Wavelength scaling of efficient high-order harmonic generation by two-color infrared laser fields,” Phys. Rev. A 81, 061802(R) (2010).
[Crossref]

K. Liu, Q. Zhang, and P. Lu, “Enhancing electron localization in molecular dissociation by two-color mid- and near-infrared laser fields,” Phys. Rev. A 86, 033410 (2012).
[Crossref]

Y. Zhou, Q. Zhang, C. Huang, and P. Lu, “Classical description of strong-field double ionization by elliptical laser pulses,” Phys. Rev. A 86, 043427 (2012).
[Crossref]

Phys. Rev. Lett (1)

Phys. Rev. Lett. (15)

Y. Zhou, C. Huang, Q. Liao, and P. Lu, “Classical Simulations Including Electron Correlations for Sequential Double Ionization,” Phys. Rev. Lett. 109, 053004 (2012).
[Crossref] [PubMed]

P. B. Corkum, “Plasma perspective on strong field multiphoton ionization,” Phys. Rev. Lett. 71, 1994–1997 (1993).
[Crossref] [PubMed]

K. Schiessl, K. L. Ishikawa, E. Persson, and J. Burgdorfer, “Quantum path interference in the wavelength dependence of high-harmonic generation,” Phys. Rev. Lett. 99, 253903 (2007).
[Crossref]

Physical review letters (1)

J. Breidbach and L. S. Cederbaum, “Universal attosecond response to the removal of an electron,” Physical review letters 94, 033901 (2005).
[Crossref] [PubMed]

Proc. Natl. Acad. Sci. USA (1)

F. Remacle and R. D. Levine, “An electronic time scale in chemistry,” Proc. Natl. Acad. Sci. USA 103, 6793–6798 (2006).
[Crossref] [PubMed]

Rev. Mod. Phys. (2)

F. Krausz and M. Ivanov, “Attosecond physics,” Rev. Mod. Phys. 81, 163–234 (2009).
[Crossref]

C. Winterfeldt, C. Spielmann, and G. Gerber, “Colloquium: Optimal control of high-harmonic generation,” Rev. Mod. Phys. 80,, 117–140 (2008).
[Crossref]

Rev. Sci. Instrum. (1)

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Fig. 1 The variation of the ratio R as a function of the control field wavelength using 25-fs two-color fields with different intensity ratios: (E₁/E₀)²=0.05 (blue solid line), (E₁/E₀)²=0.1 (green dashed line), and (E₁/E₀)²=0.15 (red chain line). Here, ϕ₀ and ϕ₁ are both set to π.

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Fig. 2 (a) The distribution of R values as a function of ϕ₀ and ϕ₁, (b) ϕ₁-dependent R values with ϕ₀=π and (c) ϕ₀-dependent R values with ϕ₁=π in the mixed field with a control field of 1330-nm. (d)–(f) Same to (a)–(c), but for a 1250-nm control field. Here, the intensity ratios (E₁/E₀)² of the driving and control fields are chosen as 0.1.

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Fig. 3 (a) Calculated harmonic spectra for different CEP combinations with the 1330-nm control field. (b) Same to (a), but for the 1250-nm control field. Here, the intensity ratios

E_{1}^{2} / E_{0}^{2}

are fixed to 0.1.

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Fig. 4 (a) The CEP dependence of HHG with single-atom response in the optimized 25-fs two-color field. (b) Temporal profiles of attosecond pulses generated by selecting continuous harmonics in (a).

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Fig. 5 The dependence of the R value on both the CEPs of the fundamental and control laser fields. The wavelength of control field is chosen as (a) 1200 nm, (b) 1600 nm, (c) 2000 nm, and (d) 2400 nm, respectively

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Fig. 6 (a) The harmonic spectra calculated with different CEP combinations (ϕ₀, ϕ₁) of the fundamental and control fields: (π, π) (blue line), (π, 1.1π) (green line), and (π, 1.2π) (red line). The wavelength and intensity of the control field are 1600 nm and 1.1 × 10¹³ W/cm², respectively. (b) Same to (a), but for 2400-nm control field.

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Fig. 7 (a) The variation of the R value as a function of the wavelength of control field. The R values at 1500 nm, 1776 nm and 2400 nm are indicated by green, blue and red lines, respectively. (b) the harmonic spectra calculated with different wavelengths of control field. Here, the CEP combination (ϕ₀, ϕ₁) is set to be (π, π) and the laser intensity of the control field is 1.1 × 10¹³ W/cm².

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Fig. 8 The dependence of the ratio R on both the laser intensity ratio

E_{1}^{2} / E_{0}^{2}

and wavelength of the control field with the optimum CEP combination (π, π).

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Fig. 9 (a) The electric field of the optimum two-color field, (b) calculated harmonic spectra for single-atom response, (c) time-frequency diagram of the spectrum, and (d) the temporal profiles of the attosecond pulses by superposing the continuous harmonics. Inset: the TL pulse of the IAP shown in (d).

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Fig. 10 (a)–(e) The birth moment t_i and recollision moment t_r contributed to the 30^th–120^th harmonic with intensity ratios of 0, 0.2, 0.4, 0.6 and 0.8, respectively. The blue solid and red solid line denote the long and short trajectories, respectively. The marked blue squares and diamonds represent the t_i and t_r of 80^th harmonic for long trajectory, the marked red circles and triangles correspond the t_i and t_r of 80^th harmonic for short trajectory. (f)–(j) The calculated phases of 75^th–85^th harmonics for short trajectory, the intensity ratios are the same as (a)–(e). The marked stars represent the phases of 80^th harmonic.

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Fig. 11 (a) The dependence of intensity ratio on different focal geometries across the focal region of the two-color field. (b) The phase Φ_q(z) of the generated 80^th harmonic (contributed by short trajectory) for these three focal geometries along propagation direction. (c) The dispersion ∂Φ_q(z)/∂z. In (a), (b) and (c) the intensity ratio is set to be 0.86 at the waist of the focused gaussian laser beam. (d). The comparison of calculated dispersion with f_#= 300 for a constant intensity ratio along propagation direction (brown line) and varied intensity ratio same as that in (c) (green line).

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Fig. 12 (a) The CEP dependence of on-axis harmonic spectrum with collective response in the optimized 25 fs two-color field. (b)Temporal profiles of attosecond pulses generated by selecting continuous harmonics in (a). The focused beam waists are 35 μm and 45 μm for the fundamental and control fields, respectively. Other laser parameters are the same as Fig. 4.

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Fig. 13 (a) The produced IAPs with different CEPs. (b) The relationship between obtained pulse duration and CEP. The parameters are the same as Fig. 12.

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Fig. 14 The spatial distribution of harmonic radiation at the exit of gas medium. The thin white line denote the on-axis harmonic spectrum, and the single-atom response is plotted with cyan line for comparison. Inset: the obtained IAP by superposing the 105^th to 135^th harmonics in the continuous part. The focused beam waists are 35 μm and 45 μm for the fundamental and control fields, respectively. Other parameters are the same as Fig. 9.

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Fig. 15 Same as Fig. 14, but for loosely focused geometry. f_0# = 600 and f_1# = 464 correspond to the beam waists of 140 μm and 180 μm for the fundamental and control fields.

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Equations (17)

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\begin{array}{l} d_{n l} (t) & = & i \int_{- \infty}^{t} d t^{'} {[\frac{π}{ε + i (t - t^{'}) / 2}]}^{3 / 2} \\ \times d_{rec} [p_{s t} (t^{'}, t) - A (t)] d_{ion} [p_{s t} (t^{'}, t) - A (t^{'})] \\ \times exp [- i S_{s t} (t^{'}, t)] E (t^{'}) g (t^{'}) + c . c . \end{array}

a_{q} = {| \frac{1}{T} \int_{0}^{T} \vec{a} (t) exp (- i q ω t) |}^{2},

R = E_{1 m} / E_{2 m},

\nabla^{2} E_{l} (r, z, t) - \frac{1}{c^{2}} \frac{\partial^{2} E_{l} (r, z, t)}{\partial t^{2}} = \frac{ω^{2}}{c^{2}} (1 - n_{eff}^{2}) E_{l} (r, z, t),

η_{eff} = η_{0} (r, z, t) + η_{2} I (r, z, t) - \frac{ω_{p}^{2} (r, z, t)}{2 ω} .

\nabla^{2} E_{h} (r, z, t) - \frac{1}{c^{2}} \frac{\partial^{2} E_{h} (r, z, t)}{\partial t^{2}} = \frac{ω_{p}^{2} (r, z, t)}{c^{2}} E_{h} (r, z, t) + μ_{0} \frac{\partial^{2} P_{n l} (r, z, t)}{\partial t^{2}},

E (t) = E_{0} exp (- 2 ln 2 t^{2} / {τ_{0}}^{2}) cos (ω_{0} t + ϕ_{0}) + E_{1} exp (- 2 ln 2 t^{2} / {τ_{1}}^{2}) cos (ω_{1} t + ϕ_{1}),

Δ k = Δ k_{n} + Δ k_{p} + Δ k_{g} + Δ k_{p l} .

\begin{array}{l} Δ k_{i} & = q \frac{\partial ϕ_{f, i} (z)}{\partial z} - \frac{\partial ϕ_{q, i} (z)}{\partial z} \\ = q k_{f, i} - k_{q, i}, (i = n, p, g, p l), \end{array}

Δ k = \frac{\partial Φ_{q} (z)}{\partial z} - \frac{\partial Φ_{q, pro} (z)}{\partial z},

Φ_{q, pro} (z) = Φ_{q} (z_{0}) + [\frac{2 π}{λ} + \frac{2 π N_{a} n (λ_{q})}{λ} - N_{e} r_{e} λ_{q}] (z - z_{0}),

\begin{array}{l} ϕ (S_{0}, S_{1}) = & E_{0}^{2} {\frac{1}{2 ω_{0}^{2}} S_{0}^{2} (t_{r} - t_{i}) + \frac{E_{1}^{2}}{E_{0}^{2}} \frac{1}{2 ω_{1}^{2}} S_{1}^{2} (t_{r} - t_{i}) + \frac{E_{1}}{E_{0}} \frac{1}{ω_{0} ω_{1}} S_{0} S_{1} \\ + \frac{1}{8 ω_{0}^{3}} [sin (2 ω_{0} t_{r}) - sin (2 ω_{0} t_{i})] + \frac{E_{1}^{2}}{E_{0}^{2}} \frac{1}{8 ω_{1}^{3}} [sin (2 ω_{1} t_{r} + 2 Δ ϕ) - sin (2 ω_{1} t_{i} + 2 Δ ϕ)] \\ + \frac{E_{1}}{E_{0}} \frac{1}{ω_{0} ω_{1}} {\frac{sin [(ω_{0} + ω_{1}) t_{r} + Δ ϕ]}{ω_{0} + ω_{1}} - \frac{sin [(ω_{0} - ω_{1}) t_{r} - Δ ϕ]}{ω_{0} - ω_{1}}} \\ - \frac{1}{4 ω_{0}^{2}} (t_{r} - t_{i}) - \frac{E_{1}^{2}}{E_{0}^{2}} \frac{1}{4 ω_{1}^{2}} (t_{r} - t_{i})} \\ + q ω_{0} t_{r}, \end{array}

S_{0} = \frac{cos (ω_{0} t_{r}) - cos (ω_{0} t_{i})}{ω_{0} (t_{i} - t_{r})},

S_{1} = \frac{cos (ω_{1} t_{r} + Δ ϕ) - cos (ω_{1} t_{i} + Δ ϕ)}{ω_{1} (t_{i} - t_{r})},

P_{s} = \frac{1}{t_{i} - t_{r}} \int_{t_{r}}^{t_{i}} d t^{″} A (t^{″}),

\frac{{[P_{s} + A (t_{i})]}^{2}}{2} + I_{p},

{\frac{[P_{s} + A (t_{r})]}{2}}^{2} + I_{p} = q ω .