Abstract

We discuss several methods to realize efficient rescattering of the ionized electron in various relativistic regimes with the recollision energy spanning from the kilo-electron-volt to the mega-electron-volt range. In an energy domain of several tens of kilo-electron-volts, the recollisions are achieved via assisting the infrared strong laser field with a weak attosecond pulse train (APT) of extreme ultraviolet photons. For recollisions in the mega-electron-volt domain, strong counterpropagating APTs are employed. High-order harmonic emission in these setups are enhanced by several orders of magnitude in comparison to the case of a single laser wave. This way hard x-ray harmonics and extreme short pulses can be feasible.

© 2008 Optical Society of America

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    [CrossRef] [PubMed]
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2008

M. Klaiber, K. Z. Hatsagortsyan, C. Müller, and C. H. Keitel, “Coherent hard x rays from attosecond pulse train-assisted harmonic generation,” Opt. Lett. 33, 411-413 (2008).
[CrossRef] [PubMed]

C. Müller, K. Z. Hatsagortsyan, and C. H. Keitel, “Particle physics with a laser-driven positronium atom,” Phys. Lett. B 659, 209-213 (2008).
[CrossRef]

2007

M. Verschl and C. H. Keitel, “Relativistic classical and quantum dynamics in intense crossed laser beams of various polarizations,” Phys. Rev. ST Accel. Beams 10, 024001 (2007).
[CrossRef]

M. Klaiber, K. Z. Hatsagortsyan, and C. H. Keitel, “Fully relativistic laser-induced ionization and recollision processes,” Phys. Rev. A 75, 063413 (2007).
[CrossRef]

M. Verschl and C. H. Keitel, “Relativistic recollisions with two consecutive laser pulses,” J. Phys. B 40, F69-F76 (2007).
[CrossRef]

M. Verschl and C. H. Keitel, “Refocussed relativistic recollisions,” Europhys. Lett. 77, 64004 (2007).
[CrossRef]

E. J. Takahashi, T. Kanai, K. L. Ishikawa, Y. Nabekawa, and K. Midorikawa, “Dramatic enhancement of high-order harmonic generation,” Phys. Rev. Lett. 99, 053904 (2007).
[CrossRef] [PubMed]

R. Fischer, M. Lein, and C. H. Keitel, “Strongly enhanced high-harmonic generation via antisymmetric ionic states,” J. Phys. B 40, F113-F119 (2007).
[CrossRef]

R. Fischer, M. Lein, and C. H. Keitel, “Enhanced recollision dynamics via the combination of antisymmetric wave functions and beyond-dipole effects,” J. Mod. Opt. 54, 1911-1926 (2007).
[CrossRef]

2006

Q. Lin, S. Li, and W. Becker, “High-order harmonic generation in a tightly focused laser beam,” Opt. Lett. 31, 2163-2165 (2006).
[CrossRef] [PubMed]

Y. I. Salamin, S. X. Hu, K. Z. Hatsagortsyan, and C. H. Keitel, “Relativistic high-power laser-matter interactions,” Phys. Rep. 427, 41-155 (2006).
[CrossRef]

M. Klaiber, K. Z. Hatsagortsyan, and C. H. Keitel, “Relativistic ionization rescattering with tailored laser pulses,” Phys. Rev. A 74, 051803 (2006).
[CrossRef]

R. Fischer, M. Lein, and C. H. Keitel, “Enhanced recollisions for antisymmetric molecular orbitals in intense laser fields,” Phys. Rev. Lett. 97, 143901 (2006).
[CrossRef] [PubMed]

S. Palaniyappan, I. Ghebregziabher, A. DiChiara, J. MacDonald, and B. C. Walker, “Emergence from nonrelativistic strong-field rescattering to ultrastrong-field laser-atom physics: a semiclassical analysis,” Phys. Rev. A 74, 033403 (2006).
[CrossRef]

K. Z. Hatsagortsyan, C. Müller, and C. H. Keitel, “Microscopic laser-driven high-energy colliders,” Europhys. Lett. 76, 29-35 (2006).
[CrossRef]

C. Müller, K. Z. Hatsagortsyan, and C. H. Keitel, “Muon pair creation from positronium in a circularly polarized laser field,” Phys. Rev. D 74, 074017 (2006).
[CrossRef]

A. Heinrich, W. Kornelis, M. P. Anscombe, C. P. Hauri, P. Schlup, J. Biegert, and U. Keller, “Enhanced VUV-assisted high harmonic generation,” J. Phys. B 39, S275-S281 (2006).
[CrossRef]

C. Figueira de Morisson Faria, P. Salières, P. Villain, and M. Lewenstein, “Controlling high-order harmonic generation and above-threshold ionization with an attosecond-pulse train,” Phys. Rev. A 74, 053416 (2006).
[CrossRef]

Strong attosecond pulses of up to kilo-electron-volt photon energy can be produced, e.g., via laser pulse reflection from overdense plasma, see G. Tsakiris, K. Eidmann, J. Meyer-ter-Vehn, and F. Krausz, “Route to intense single attosecond pulses,” New J. Phys. 8, 19 (2006).
[CrossRef]

M. Klaiber, K. Z. Hatsagortsyan, and C. H. Keitel, “Gauge-invariant relativistic strong-field approximation,” Phys. Rev. A 73, 053411 (2006).
[CrossRef]

2005

M. B. Gaarde, K. J. Schafer, A. Heinrich, J. Biegert, and U. Keller, “Large enhancement of macroscopic yield in attosecond pulse train-assisted harmonic generation,” Phys. Rev. A 72, 013411 (2005).
[CrossRef]

A. Bandrauk, S. Chelkowski, and S. Goudreau, “Control of harmonic generation using two-color femtosecond-attosecond laser fields: quantum and classical perspectives,” J. Mod. Phys. 52, 411-428 (2005).

2004

N. Milosevic, P. B. Corkum, and T. Brabec, “How to use lasers for imaging attosecond dynamics of nuclear processes,” Phys. Rev. Lett. 92, 013002 (2004).
[CrossRef] [PubMed]

M. Spanner, O. Smirnova, P. B. Corkum, and M. Y. Ivanov, “Reading diffraction images in strong field ionization of diatomic molecules,” J. Phys. B 37, L243-L250 (2004).
[CrossRef]

J. Itatani, J. Levesque, D. Zeidler, H. Niikura, H. Pepin, J. C. Kieffer, P. B. Corkum, and D. M. Villeneuve, “Tomographic imaging of molecular orbitals,” Nature 432, 867-871 (2004).
[CrossRef] [PubMed]

G. Mocken and C. H. Keitel, “Bound atomic dynamics in the MeV regime,” J. Phys. B 37, L275-L283 (2004).
[CrossRef]

C. C. Chirilă, C. J. Joachain, N. J. Kylstra, and R. M. Potvliege, “Interaction of superintense laser pulses with relativistic ions,” Phys. Rev. Lett. 93, 243603 (2004).
[CrossRef]

B. Henrich, K. Z. Hatsagortsyan, and C. H. Keitel, “Positronium in intense laser fields,” Phys. Rev. Lett. 93, 013601 (2004).
[CrossRef]

K. J. Schafer, M. B. Gaarde, A. Heinrich, J. Biegert, and U. Keller, “Strong field quantum path control using attosecond pulse trains,” Phys. Rev. Lett. 92, 023003 (2004).
[CrossRef] [PubMed]

2003

H. Niikura, F. Legare, R. Hasbani, M. Yu. Ivanov, D. M. Villeneuve, and P. B. Corkum, “Probing molecular dynamics with attosecond resolution using correlated wave packet pairs,” Nature 421, 826-829 (2003).
[CrossRef] [PubMed]

K. Ishikawa, “Photoemission and ionization of He+ under simultaneous irradiation of fundamental laser and high-order harmonic pulses,” Phys. Rev. Lett. 91, 043002 (2003).
[CrossRef] [PubMed]

2002

A. Bandrauk and N. H. Shon, “Attosecond control of ionization and high-order harmonic generation in molecules,” Phys. Rev. A 66, 031401 (2002).
[CrossRef]

C. C. Chirilă, N. J. Kylstra, R. M. Potvliege, and C. J. Joachain, “Nondipole effects in photon emission by laser-driven ions,” Phys. Rev. A 66, 063411 (2002).
[CrossRef]

V. D. Taranukhin and N. Y. Shubin, “High-order harmonic generation by atoms with traveling- and standing-wave pumps of relativistic intensity,” J. Opt. Soc. Am. B 19, 1132-1140 (2002).
[CrossRef]

H. Niikura, F. Legare, R. Hasbani, A. D. Bandrauk, M. Yu. Ivanov, D. M. Villeneuve, and P. B. Corkum, “Sub-laser-cycle electron pulses for probing molecular dynamics,” Nature 417, 917-922 (2002).
[CrossRef] [PubMed]

C. H. Keitel and S. X. Hu, “Coherent x-ray pulse generation in the subangstrøm regime,” Appl. Phys. Lett. 80, 541-543 (2002).
[CrossRef]

D. B. Milošević, S. Hu, and W. Becker, “Relativistic ultrahigh-order harmonic generation,” Laser Phys. 12, 389-397 (2002).

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]

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

2001

D. B. Milošević, S. Hu, and W. Becker, “Quantum-mechanical model for ultrahigh-order harmonic generation in the moderately relativistic regime,” Phys. Rev. A 63, 011403 (2001).
[CrossRef]

C. H. Keitel, “Relativistic quantum optics,” Contemp. Phys. 42, 353-363 (2001).
[CrossRef]

S. X. Hu and C. H. Keitel, “Dynamics of multiply charged ions in intense laser fields,” Phys. Rev. A 63, 053402 (2001).
[CrossRef]

2000

N. J. Kylstra, R. A. Worthington, A. Patel, P. L. Knight, J. R. Vázquez de Aldana, and L. Roso, “Breakdown of stabilization of atoms interacting with intense, high-frequency laser pulses,” Phys. Rev. Lett. 85, 1835-1838 (2000).
[CrossRef] [PubMed]

V. D. Taranukhin, “Relativistic high-order harmonic generation,” Laser Phys. 10, 330-336 (2000).

V. D. Taranukhin and N. Yu. Shubin, “High-order harmonic generation by multielectron atoms in the field of a relativistic-intensity standing electromagnetic wave,” Quantum Electron. 31, 179-184 (2000).
[CrossRef]

1996

T. Zuo, A. D. Bandrauk, and P. B. Corkum, “Laser-induced electron diffraction: a new tool for probing ultrafast molecular dynamics,” Chem. Phys. Lett. 259, 313-320 (1996).
[CrossRef]

Y. I. Salamin and F. H. M. Faisal, “Harmonic generation by superintense light scattering from relativistic electrons,” Phys. Rev. A 54, 4383-4395 (1996).
[CrossRef] [PubMed]

1993

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

1989

S. Augst, D. Strickland, D. D. Meyerhofer, S. L. Chin, and J. H. Eberly, “Tunneling ionization of noble gases in a high-intensity laser field,” Phys. Rev. Lett. 63, 2212-2215 (1989).
[CrossRef] [PubMed]

1987

M. Yu. Kuchiev, “Atomic antenna,” JETP Lett. 45, 404-406 (1987).

Anscombe, M. P.

A. Heinrich, W. Kornelis, M. P. Anscombe, C. P. Hauri, P. Schlup, J. Biegert, and U. Keller, “Enhanced VUV-assisted high harmonic generation,” J. Phys. B 39, S275-S281 (2006).
[CrossRef]

Augst, S.

S. Augst, D. Strickland, D. D. Meyerhofer, S. L. Chin, and J. H. Eberly, “Tunneling ionization of noble gases in a high-intensity laser field,” Phys. Rev. Lett. 63, 2212-2215 (1989).
[CrossRef] [PubMed]

Bandrauk, A.

A. Bandrauk, S. Chelkowski, and S. Goudreau, “Control of harmonic generation using two-color femtosecond-attosecond laser fields: quantum and classical perspectives,” J. Mod. Phys. 52, 411-428 (2005).

A. Bandrauk and N. H. Shon, “Attosecond control of ionization and high-order harmonic generation in molecules,” Phys. Rev. A 66, 031401 (2002).
[CrossRef]

Bandrauk, A. D.

H. Niikura, F. Legare, R. Hasbani, A. D. Bandrauk, M. Yu. Ivanov, D. M. Villeneuve, and P. B. Corkum, “Sub-laser-cycle electron pulses for probing molecular dynamics,” Nature 417, 917-922 (2002).
[CrossRef] [PubMed]

T. Zuo, A. D. Bandrauk, and P. B. Corkum, “Laser-induced electron diffraction: a new tool for probing ultrafast molecular dynamics,” Chem. Phys. Lett. 259, 313-320 (1996).
[CrossRef]

Becker, W.

Q. Lin, S. Li, and W. Becker, “High-order harmonic generation in a tightly focused laser beam,” Opt. Lett. 31, 2163-2165 (2006).
[CrossRef] [PubMed]

D. B. Milošević, S. Hu, and W. Becker, “Relativistic ultrahigh-order harmonic generation,” Laser Phys. 12, 389-397 (2002).

D. B. Milošević, S. Hu, and W. Becker, “Quantum-mechanical model for ultrahigh-order harmonic generation in the moderately relativistic regime,” Phys. Rev. A 63, 011403 (2001).
[CrossRef]

Biegert, J.

A. Heinrich, W. Kornelis, M. P. Anscombe, C. P. Hauri, P. Schlup, J. Biegert, and U. Keller, “Enhanced VUV-assisted high harmonic generation,” J. Phys. B 39, S275-S281 (2006).
[CrossRef]

M. B. Gaarde, K. J. Schafer, A. Heinrich, J. Biegert, and U. Keller, “Large enhancement of macroscopic yield in attosecond pulse train-assisted harmonic generation,” Phys. Rev. A 72, 013411 (2005).
[CrossRef]

K. J. Schafer, M. B. Gaarde, A. Heinrich, J. Biegert, and U. Keller, “Strong field quantum path control using attosecond pulse trains,” Phys. Rev. Lett. 92, 023003 (2004).
[CrossRef] [PubMed]

Brabec, T.

N. Milosevic, P. B. Corkum, and T. Brabec, “How to use lasers for imaging attosecond dynamics of nuclear processes,” Phys. Rev. Lett. 92, 013002 (2004).
[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]

Chelkowski, S.

A. Bandrauk, S. Chelkowski, and S. Goudreau, “Control of harmonic generation using two-color femtosecond-attosecond laser fields: quantum and classical perspectives,” J. Mod. Phys. 52, 411-428 (2005).

Chin, S. L.

S. Augst, D. Strickland, D. D. Meyerhofer, S. L. Chin, and J. H. Eberly, “Tunneling ionization of noble gases in a high-intensity laser field,” Phys. Rev. Lett. 63, 2212-2215 (1989).
[CrossRef] [PubMed]

Chirila, C. C.

C. C. Chirilă, C. J. Joachain, N. J. Kylstra, and R. M. Potvliege, “Interaction of superintense laser pulses with relativistic ions,” Phys. Rev. Lett. 93, 243603 (2004).
[CrossRef]

C. C. Chirilă, N. J. Kylstra, R. M. Potvliege, and C. J. Joachain, “Nondipole effects in photon emission by laser-driven ions,” Phys. Rev. A 66, 063411 (2002).
[CrossRef]

Corkum, P. B.

N. Milosevic, P. B. Corkum, and T. Brabec, “How to use lasers for imaging attosecond dynamics of nuclear processes,” Phys. Rev. Lett. 92, 013002 (2004).
[CrossRef] [PubMed]

M. Spanner, O. Smirnova, P. B. Corkum, and M. Y. Ivanov, “Reading diffraction images in strong field ionization of diatomic molecules,” J. Phys. B 37, L243-L250 (2004).
[CrossRef]

J. Itatani, J. Levesque, D. Zeidler, H. Niikura, H. Pepin, J. C. Kieffer, P. B. Corkum, and D. M. Villeneuve, “Tomographic imaging of molecular orbitals,” Nature 432, 867-871 (2004).
[CrossRef] [PubMed]

H. Niikura, F. Legare, R. Hasbani, M. Yu. Ivanov, D. M. Villeneuve, and P. B. Corkum, “Probing molecular dynamics with attosecond resolution using correlated wave packet pairs,” Nature 421, 826-829 (2003).
[CrossRef] [PubMed]

H. Niikura, F. Legare, R. Hasbani, A. D. Bandrauk, M. Yu. Ivanov, D. M. Villeneuve, and P. B. Corkum, “Sub-laser-cycle electron pulses for probing molecular dynamics,” Nature 417, 917-922 (2002).
[CrossRef] [PubMed]

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

T. Zuo, A. D. Bandrauk, and P. B. Corkum, “Laser-induced electron diffraction: a new tool for probing ultrafast molecular dynamics,” Chem. Phys. Lett. 259, 313-320 (1996).
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P. B. Corkum, “Plasma perspective on strong field multiphoton ionization,” Phys. Rev. Lett. 71, 1994-1997 (1993).
[CrossRef] [PubMed]

DiChiara, A.

S. Palaniyappan, I. Ghebregziabher, A. DiChiara, J. MacDonald, and B. C. Walker, “Emergence from nonrelativistic strong-field rescattering to ultrastrong-field laser-atom physics: a semiclassical analysis,” Phys. Rev. A 74, 033403 (2006).
[CrossRef]

Eberly, J. H.

S. Augst, D. Strickland, D. D. Meyerhofer, S. L. Chin, and J. H. Eberly, “Tunneling ionization of noble gases in a high-intensity laser field,” Phys. Rev. Lett. 63, 2212-2215 (1989).
[CrossRef] [PubMed]

Eidmann, K.

Strong attosecond pulses of up to kilo-electron-volt photon energy can be produced, e.g., via laser pulse reflection from overdense plasma, see G. Tsakiris, K. Eidmann, J. Meyer-ter-Vehn, and F. Krausz, “Route to intense single attosecond pulses,” New J. Phys. 8, 19 (2006).
[CrossRef]

Faisal, F. H. M.

Y. I. Salamin and F. H. M. Faisal, “Harmonic generation by superintense light scattering from relativistic electrons,” Phys. Rev. A 54, 4383-4395 (1996).
[CrossRef] [PubMed]

Figueira de Morisson Faria, C.

C. Figueira de Morisson Faria, P. Salières, P. Villain, and M. Lewenstein, “Controlling high-order harmonic generation and above-threshold ionization with an attosecond-pulse train,” Phys. Rev. A 74, 053416 (2006).
[CrossRef]

Fischer, R.

R. Fischer, M. Lein, and C. H. Keitel, “Strongly enhanced high-harmonic generation via antisymmetric ionic states,” J. Phys. B 40, F113-F119 (2007).
[CrossRef]

R. Fischer, M. Lein, and C. H. Keitel, “Enhanced recollision dynamics via the combination of antisymmetric wave functions and beyond-dipole effects,” J. Mod. Opt. 54, 1911-1926 (2007).
[CrossRef]

R. Fischer, M. Lein, and C. H. Keitel, “Enhanced recollisions for antisymmetric molecular orbitals in intense laser fields,” Phys. Rev. Lett. 97, 143901 (2006).
[CrossRef] [PubMed]

Gaarde, M. B.

M. B. Gaarde, K. J. Schafer, A. Heinrich, J. Biegert, and U. Keller, “Large enhancement of macroscopic yield in attosecond pulse train-assisted harmonic generation,” Phys. Rev. A 72, 013411 (2005).
[CrossRef]

K. J. Schafer, M. B. Gaarde, A. Heinrich, J. Biegert, and U. Keller, “Strong field quantum path control using attosecond pulse trains,” Phys. Rev. Lett. 92, 023003 (2004).
[CrossRef] [PubMed]

Ghebregziabher, I.

S. Palaniyappan, I. Ghebregziabher, A. DiChiara, J. MacDonald, and B. C. Walker, “Emergence from nonrelativistic strong-field rescattering to ultrastrong-field laser-atom physics: a semiclassical analysis,” Phys. Rev. A 74, 033403 (2006).
[CrossRef]

Goudreau, S.

A. Bandrauk, S. Chelkowski, and S. Goudreau, “Control of harmonic generation using two-color femtosecond-attosecond laser fields: quantum and classical perspectives,” J. Mod. Phys. 52, 411-428 (2005).

Hasbani, R.

H. Niikura, F. Legare, R. Hasbani, M. Yu. Ivanov, D. M. Villeneuve, and P. B. Corkum, “Probing molecular dynamics with attosecond resolution using correlated wave packet pairs,” Nature 421, 826-829 (2003).
[CrossRef] [PubMed]

H. Niikura, F. Legare, R. Hasbani, A. D. Bandrauk, M. Yu. Ivanov, D. M. Villeneuve, and P. B. Corkum, “Sub-laser-cycle electron pulses for probing molecular dynamics,” Nature 417, 917-922 (2002).
[CrossRef] [PubMed]

Hatsagortsyan, K. Z.

M. Klaiber, K. Z. Hatsagortsyan, C. Müller, and C. H. Keitel, “Coherent hard x rays from attosecond pulse train-assisted harmonic generation,” Opt. Lett. 33, 411-413 (2008).
[CrossRef] [PubMed]

C. Müller, K. Z. Hatsagortsyan, and C. H. Keitel, “Particle physics with a laser-driven positronium atom,” Phys. Lett. B 659, 209-213 (2008).
[CrossRef]

M. Klaiber, K. Z. Hatsagortsyan, and C. H. Keitel, “Fully relativistic laser-induced ionization and recollision processes,” Phys. Rev. A 75, 063413 (2007).
[CrossRef]

M. Klaiber, K. Z. Hatsagortsyan, and C. H. Keitel, “Relativistic ionization rescattering with tailored laser pulses,” Phys. Rev. A 74, 051803 (2006).
[CrossRef]

Y. I. Salamin, S. X. Hu, K. Z. Hatsagortsyan, and C. H. Keitel, “Relativistic high-power laser-matter interactions,” Phys. Rep. 427, 41-155 (2006).
[CrossRef]

M. Klaiber, K. Z. Hatsagortsyan, and C. H. Keitel, “Gauge-invariant relativistic strong-field approximation,” Phys. Rev. A 73, 053411 (2006).
[CrossRef]

K. Z. Hatsagortsyan, C. Müller, and C. H. Keitel, “Microscopic laser-driven high-energy colliders,” Europhys. Lett. 76, 29-35 (2006).
[CrossRef]

C. Müller, K. Z. Hatsagortsyan, and C. H. Keitel, “Muon pair creation from positronium in a circularly polarized laser field,” Phys. Rev. D 74, 074017 (2006).
[CrossRef]

B. Henrich, K. Z. Hatsagortsyan, and C. H. Keitel, “Positronium in intense laser fields,” Phys. Rev. Lett. 93, 013601 (2004).
[CrossRef]

M. Klaiber, K. Z. Hatsagortsyan, and C. H. Keitel, “Zeptosecond γ-ray pulses,” arXiv.org e-print archive, physics.atom-ph/0707.2900, July 19, 2007, http://arXiv.org/abs/0707.2900.

Hauri, C. P.

A. Heinrich, W. Kornelis, M. P. Anscombe, C. P. Hauri, P. Schlup, J. Biegert, and U. Keller, “Enhanced VUV-assisted high harmonic generation,” J. Phys. B 39, S275-S281 (2006).
[CrossRef]

Heinrich, A.

A. Heinrich, W. Kornelis, M. P. Anscombe, C. P. Hauri, P. Schlup, J. Biegert, and U. Keller, “Enhanced VUV-assisted high harmonic generation,” J. Phys. B 39, S275-S281 (2006).
[CrossRef]

M. B. Gaarde, K. J. Schafer, A. Heinrich, J. Biegert, and U. Keller, “Large enhancement of macroscopic yield in attosecond pulse train-assisted harmonic generation,” Phys. Rev. A 72, 013411 (2005).
[CrossRef]

K. J. Schafer, M. B. Gaarde, A. Heinrich, J. Biegert, and U. Keller, “Strong field quantum path control using attosecond pulse trains,” Phys. Rev. Lett. 92, 023003 (2004).
[CrossRef] [PubMed]

Henrich, B.

B. Henrich, K. Z. Hatsagortsyan, and C. H. Keitel, “Positronium in intense laser fields,” Phys. Rev. Lett. 93, 013601 (2004).
[CrossRef]

Hu, S.

D. B. Milošević, S. Hu, and W. Becker, “Relativistic ultrahigh-order harmonic generation,” Laser Phys. 12, 389-397 (2002).

D. B. Milošević, S. Hu, and W. Becker, “Quantum-mechanical model for ultrahigh-order harmonic generation in the moderately relativistic regime,” Phys. Rev. A 63, 011403 (2001).
[CrossRef]

Hu, S. X.

Y. I. Salamin, S. X. Hu, K. Z. Hatsagortsyan, and C. H. Keitel, “Relativistic high-power laser-matter interactions,” Phys. Rep. 427, 41-155 (2006).
[CrossRef]

C. H. Keitel and S. X. Hu, “Coherent x-ray pulse generation in the subangstrøm regime,” Appl. Phys. Lett. 80, 541-543 (2002).
[CrossRef]

S. X. Hu and C. H. Keitel, “Dynamics of multiply charged ions in intense laser fields,” Phys. Rev. A 63, 053402 (2001).
[CrossRef]

Ishikawa, K.

K. Ishikawa, “Photoemission and ionization of He+ under simultaneous irradiation of fundamental laser and high-order harmonic pulses,” Phys. Rev. Lett. 91, 043002 (2003).
[CrossRef] [PubMed]

Ishikawa, K. L.

E. J. Takahashi, T. Kanai, K. L. Ishikawa, Y. Nabekawa, and K. Midorikawa, “Dramatic enhancement of high-order harmonic generation,” Phys. Rev. Lett. 99, 053904 (2007).
[CrossRef] [PubMed]

Itatani, J.

J. Itatani, J. Levesque, D. Zeidler, H. Niikura, H. Pepin, J. C. Kieffer, P. B. Corkum, and D. M. Villeneuve, “Tomographic imaging of molecular orbitals,” Nature 432, 867-871 (2004).
[CrossRef] [PubMed]

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

Ivanov, M. Y.

M. Spanner, O. Smirnova, P. B. Corkum, and M. Y. Ivanov, “Reading diffraction images in strong field ionization of diatomic molecules,” J. Phys. B 37, L243-L250 (2004).
[CrossRef]

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

Ivanov, M. Yu.

H. Niikura, F. Legare, R. Hasbani, M. Yu. Ivanov, D. M. Villeneuve, and P. B. Corkum, “Probing molecular dynamics with attosecond resolution using correlated wave packet pairs,” Nature 421, 826-829 (2003).
[CrossRef] [PubMed]

H. Niikura, F. Legare, R. Hasbani, A. D. Bandrauk, M. Yu. Ivanov, D. M. Villeneuve, and P. B. Corkum, “Sub-laser-cycle electron pulses for probing molecular dynamics,” Nature 417, 917-922 (2002).
[CrossRef] [PubMed]

Joachain, C. J.

C. C. Chirilă, C. J. Joachain, N. J. Kylstra, and R. M. Potvliege, “Interaction of superintense laser pulses with relativistic ions,” Phys. Rev. Lett. 93, 243603 (2004).
[CrossRef]

C. C. Chirilă, N. J. Kylstra, R. M. Potvliege, and C. J. Joachain, “Nondipole effects in photon emission by laser-driven ions,” Phys. Rev. A 66, 063411 (2002).
[CrossRef]

Kanai, T.

E. J. Takahashi, T. Kanai, K. L. Ishikawa, Y. Nabekawa, and K. Midorikawa, “Dramatic enhancement of high-order harmonic generation,” Phys. Rev. Lett. 99, 053904 (2007).
[CrossRef] [PubMed]

Keitel, C. H.

M. Klaiber, K. Z. Hatsagortsyan, C. Müller, and C. H. Keitel, “Coherent hard x rays from attosecond pulse train-assisted harmonic generation,” Opt. Lett. 33, 411-413 (2008).
[CrossRef] [PubMed]

C. Müller, K. Z. Hatsagortsyan, and C. H. Keitel, “Particle physics with a laser-driven positronium atom,” Phys. Lett. B 659, 209-213 (2008).
[CrossRef]

M. Verschl and C. H. Keitel, “Relativistic recollisions with two consecutive laser pulses,” J. Phys. B 40, F69-F76 (2007).
[CrossRef]

M. Verschl and C. H. Keitel, “Refocussed relativistic recollisions,” Europhys. Lett. 77, 64004 (2007).
[CrossRef]

M. Verschl and C. H. Keitel, “Relativistic classical and quantum dynamics in intense crossed laser beams of various polarizations,” Phys. Rev. ST Accel. Beams 10, 024001 (2007).
[CrossRef]

M. Klaiber, K. Z. Hatsagortsyan, and C. H. Keitel, “Fully relativistic laser-induced ionization and recollision processes,” Phys. Rev. A 75, 063413 (2007).
[CrossRef]

R. Fischer, M. Lein, and C. H. Keitel, “Enhanced recollision dynamics via the combination of antisymmetric wave functions and beyond-dipole effects,” J. Mod. Opt. 54, 1911-1926 (2007).
[CrossRef]

R. Fischer, M. Lein, and C. H. Keitel, “Strongly enhanced high-harmonic generation via antisymmetric ionic states,” J. Phys. B 40, F113-F119 (2007).
[CrossRef]

R. Fischer, M. Lein, and C. H. Keitel, “Enhanced recollisions for antisymmetric molecular orbitals in intense laser fields,” Phys. Rev. Lett. 97, 143901 (2006).
[CrossRef] [PubMed]

Y. I. Salamin, S. X. Hu, K. Z. Hatsagortsyan, and C. H. Keitel, “Relativistic high-power laser-matter interactions,” Phys. Rep. 427, 41-155 (2006).
[CrossRef]

M. Klaiber, K. Z. Hatsagortsyan, and C. H. Keitel, “Relativistic ionization rescattering with tailored laser pulses,” Phys. Rev. A 74, 051803 (2006).
[CrossRef]

C. Müller, K. Z. Hatsagortsyan, and C. H. Keitel, “Muon pair creation from positronium in a circularly polarized laser field,” Phys. Rev. D 74, 074017 (2006).
[CrossRef]

K. Z. Hatsagortsyan, C. Müller, and C. H. Keitel, “Microscopic laser-driven high-energy colliders,” Europhys. Lett. 76, 29-35 (2006).
[CrossRef]

M. Klaiber, K. Z. Hatsagortsyan, and C. H. Keitel, “Gauge-invariant relativistic strong-field approximation,” Phys. Rev. A 73, 053411 (2006).
[CrossRef]

G. Mocken and C. H. Keitel, “Bound atomic dynamics in the MeV regime,” J. Phys. B 37, L275-L283 (2004).
[CrossRef]

B. Henrich, K. Z. Hatsagortsyan, and C. H. Keitel, “Positronium in intense laser fields,” Phys. Rev. Lett. 93, 013601 (2004).
[CrossRef]

C. H. Keitel and S. X. Hu, “Coherent x-ray pulse generation in the subangstrøm regime,” Appl. Phys. Lett. 80, 541-543 (2002).
[CrossRef]

S. X. Hu and C. H. Keitel, “Dynamics of multiply charged ions in intense laser fields,” Phys. Rev. A 63, 053402 (2001).
[CrossRef]

C. H. Keitel, “Relativistic quantum optics,” Contemp. Phys. 42, 353-363 (2001).
[CrossRef]

M. Klaiber, K. Z. Hatsagortsyan, and C. H. Keitel, “Zeptosecond γ-ray pulses,” arXiv.org e-print archive, physics.atom-ph/0707.2900, July 19, 2007, http://arXiv.org/abs/0707.2900.

Keller, U.

A. Heinrich, W. Kornelis, M. P. Anscombe, C. P. Hauri, P. Schlup, J. Biegert, and U. Keller, “Enhanced VUV-assisted high harmonic generation,” J. Phys. B 39, S275-S281 (2006).
[CrossRef]

M. B. Gaarde, K. J. Schafer, A. Heinrich, J. Biegert, and U. Keller, “Large enhancement of macroscopic yield in attosecond pulse train-assisted harmonic generation,” Phys. Rev. A 72, 013411 (2005).
[CrossRef]

K. J. Schafer, M. B. Gaarde, A. Heinrich, J. Biegert, and U. Keller, “Strong field quantum path control using attosecond pulse trains,” Phys. Rev. Lett. 92, 023003 (2004).
[CrossRef] [PubMed]

Kieffer, J. C.

J. Itatani, J. Levesque, D. Zeidler, H. Niikura, H. Pepin, J. C. Kieffer, P. B. Corkum, and D. M. Villeneuve, “Tomographic imaging of molecular orbitals,” Nature 432, 867-871 (2004).
[CrossRef] [PubMed]

Kitzler, M.

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]

Klaiber, M.

M. Klaiber, K. Z. Hatsagortsyan, C. Müller, and C. H. Keitel, “Coherent hard x rays from attosecond pulse train-assisted harmonic generation,” Opt. Lett. 33, 411-413 (2008).
[CrossRef] [PubMed]

M. Klaiber, K. Z. Hatsagortsyan, and C. H. Keitel, “Fully relativistic laser-induced ionization and recollision processes,” Phys. Rev. A 75, 063413 (2007).
[CrossRef]

M. Klaiber, K. Z. Hatsagortsyan, and C. H. Keitel, “Relativistic ionization rescattering with tailored laser pulses,” Phys. Rev. A 74, 051803 (2006).
[CrossRef]

M. Klaiber, K. Z. Hatsagortsyan, and C. H. Keitel, “Gauge-invariant relativistic strong-field approximation,” Phys. Rev. A 73, 053411 (2006).
[CrossRef]

M. Klaiber, K. Z. Hatsagortsyan, and C. H. Keitel, “Zeptosecond γ-ray pulses,” arXiv.org e-print archive, physics.atom-ph/0707.2900, July 19, 2007, http://arXiv.org/abs/0707.2900.

Knight, P. L.

N. J. Kylstra, R. A. Worthington, A. Patel, P. L. Knight, J. R. Vázquez de Aldana, and L. Roso, “Breakdown of stabilization of atoms interacting with intense, high-frequency laser pulses,” Phys. Rev. Lett. 85, 1835-1838 (2000).
[CrossRef] [PubMed]

Kornelis, W.

A. Heinrich, W. Kornelis, M. P. Anscombe, C. P. Hauri, P. Schlup, J. Biegert, and U. Keller, “Enhanced VUV-assisted high harmonic generation,” J. Phys. B 39, S275-S281 (2006).
[CrossRef]

Krause, J. L.

K. C. Kulander, K. J. Schafer, and J. L. Krause, “Dynamics of short-pulse excitation and ionization and harmonic generation,” in Super-Intense Laser-Atom Physics, B.Piraux, A.L'Huillier, and K.Rzażewski, eds. (Plenum, 1993), pp. 95-110.

Krausz, F.

Strong attosecond pulses of up to kilo-electron-volt photon energy can be produced, e.g., via laser pulse reflection from overdense plasma, see G. Tsakiris, K. Eidmann, J. Meyer-ter-Vehn, and F. Krausz, “Route to intense single attosecond pulses,” New J. Phys. 8, 19 (2006).
[CrossRef]

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]

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

Kuchiev, M. Yu.

M. Yu. Kuchiev, “Atomic antenna,” JETP Lett. 45, 404-406 (1987).

Kulander, K. C.

K. C. Kulander, K. J. Schafer, and J. L. Krause, “Dynamics of short-pulse excitation and ionization and harmonic generation,” in Super-Intense Laser-Atom Physics, B.Piraux, A.L'Huillier, and K.Rzażewski, eds. (Plenum, 1993), pp. 95-110.

Kylstra, N. J.

C. C. Chirilă, C. J. Joachain, N. J. Kylstra, and R. M. Potvliege, “Interaction of superintense laser pulses with relativistic ions,” Phys. Rev. Lett. 93, 243603 (2004).
[CrossRef]

C. C. Chirilă, N. J. Kylstra, R. M. Potvliege, and C. J. Joachain, “Nondipole effects in photon emission by laser-driven ions,” Phys. Rev. A 66, 063411 (2002).
[CrossRef]

N. J. Kylstra, R. A. Worthington, A. Patel, P. L. Knight, J. R. Vázquez de Aldana, and L. Roso, “Breakdown of stabilization of atoms interacting with intense, high-frequency laser pulses,” Phys. Rev. Lett. 85, 1835-1838 (2000).
[CrossRef] [PubMed]

Legare, F.

H. Niikura, F. Legare, R. Hasbani, M. Yu. Ivanov, D. M. Villeneuve, and P. B. Corkum, “Probing molecular dynamics with attosecond resolution using correlated wave packet pairs,” Nature 421, 826-829 (2003).
[CrossRef] [PubMed]

H. Niikura, F. Legare, R. Hasbani, A. D. Bandrauk, M. Yu. Ivanov, D. M. Villeneuve, and P. B. Corkum, “Sub-laser-cycle electron pulses for probing molecular dynamics,” Nature 417, 917-922 (2002).
[CrossRef] [PubMed]

Lein, M.

R. Fischer, M. Lein, and C. H. Keitel, “Strongly enhanced high-harmonic generation via antisymmetric ionic states,” J. Phys. B 40, F113-F119 (2007).
[CrossRef]

R. Fischer, M. Lein, and C. H. Keitel, “Enhanced recollision dynamics via the combination of antisymmetric wave functions and beyond-dipole effects,” J. Mod. Opt. 54, 1911-1926 (2007).
[CrossRef]

R. Fischer, M. Lein, and C. H. Keitel, “Enhanced recollisions for antisymmetric molecular orbitals in intense laser fields,” Phys. Rev. Lett. 97, 143901 (2006).
[CrossRef] [PubMed]

Levesque, J.

J. Itatani, J. Levesque, D. Zeidler, H. Niikura, H. Pepin, J. C. Kieffer, P. B. Corkum, and D. M. Villeneuve, “Tomographic imaging of molecular orbitals,” Nature 432, 867-871 (2004).
[CrossRef] [PubMed]

Lewenstein, M.

C. Figueira de Morisson Faria, P. Salières, P. Villain, and M. Lewenstein, “Controlling high-order harmonic generation and above-threshold ionization with an attosecond-pulse train,” Phys. Rev. A 74, 053416 (2006).
[CrossRef]

Li, S.

Lin, Q.

MacDonald, J.

S. Palaniyappan, I. Ghebregziabher, A. DiChiara, J. MacDonald, and B. C. Walker, “Emergence from nonrelativistic strong-field rescattering to ultrastrong-field laser-atom physics: a semiclassical analysis,” Phys. Rev. A 74, 033403 (2006).
[CrossRef]

Meyerhofer, D. D.

S. Augst, D. Strickland, D. D. Meyerhofer, S. L. Chin, and J. H. Eberly, “Tunneling ionization of noble gases in a high-intensity laser field,” Phys. Rev. Lett. 63, 2212-2215 (1989).
[CrossRef] [PubMed]

Meyer-ter-Vehn, J.

Strong attosecond pulses of up to kilo-electron-volt photon energy can be produced, e.g., via laser pulse reflection from overdense plasma, see G. Tsakiris, K. Eidmann, J. Meyer-ter-Vehn, and F. Krausz, “Route to intense single attosecond pulses,” New J. Phys. 8, 19 (2006).
[CrossRef]

Midorikawa, K.

E. J. Takahashi, T. Kanai, K. L. Ishikawa, Y. Nabekawa, and K. Midorikawa, “Dramatic enhancement of high-order harmonic generation,” Phys. Rev. Lett. 99, 053904 (2007).
[CrossRef] [PubMed]

Milosevic, N.

N. Milosevic, P. B. Corkum, and T. Brabec, “How to use lasers for imaging attosecond dynamics of nuclear processes,” Phys. Rev. Lett. 92, 013002 (2004).
[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]

Miloševic, D. B.

D. B. Milošević, S. Hu, and W. Becker, “Relativistic ultrahigh-order harmonic generation,” Laser Phys. 12, 389-397 (2002).

D. B. Milošević, S. Hu, and W. Becker, “Quantum-mechanical model for ultrahigh-order harmonic generation in the moderately relativistic regime,” Phys. Rev. A 63, 011403 (2001).
[CrossRef]

Mocken, G.

G. Mocken and C. H. Keitel, “Bound atomic dynamics in the MeV regime,” J. Phys. B 37, L275-L283 (2004).
[CrossRef]

Müller, C.

M. Klaiber, K. Z. Hatsagortsyan, C. Müller, and C. H. Keitel, “Coherent hard x rays from attosecond pulse train-assisted harmonic generation,” Opt. Lett. 33, 411-413 (2008).
[CrossRef] [PubMed]

C. Müller, K. Z. Hatsagortsyan, and C. H. Keitel, “Particle physics with a laser-driven positronium atom,” Phys. Lett. B 659, 209-213 (2008).
[CrossRef]

C. Müller, K. Z. Hatsagortsyan, and C. H. Keitel, “Muon pair creation from positronium in a circularly polarized laser field,” Phys. Rev. D 74, 074017 (2006).
[CrossRef]

K. Z. Hatsagortsyan, C. Müller, and C. H. Keitel, “Microscopic laser-driven high-energy colliders,” Europhys. Lett. 76, 29-35 (2006).
[CrossRef]

Nabekawa, Y.

E. J. Takahashi, T. Kanai, K. L. Ishikawa, Y. Nabekawa, and K. Midorikawa, “Dramatic enhancement of high-order harmonic generation,” Phys. Rev. Lett. 99, 053904 (2007).
[CrossRef] [PubMed]

Niikura, H.

J. Itatani, J. Levesque, D. Zeidler, H. Niikura, H. Pepin, J. C. Kieffer, P. B. Corkum, and D. M. Villeneuve, “Tomographic imaging of molecular orbitals,” Nature 432, 867-871 (2004).
[CrossRef] [PubMed]

H. Niikura, F. Legare, R. Hasbani, M. Yu. Ivanov, D. M. Villeneuve, and P. B. Corkum, “Probing molecular dynamics with attosecond resolution using correlated wave packet pairs,” Nature 421, 826-829 (2003).
[CrossRef] [PubMed]

H. Niikura, F. Legare, R. Hasbani, A. D. Bandrauk, M. Yu. Ivanov, D. M. Villeneuve, and P. B. Corkum, “Sub-laser-cycle electron pulses for probing molecular dynamics,” Nature 417, 917-922 (2002).
[CrossRef] [PubMed]

Palaniyappan, S.

S. Palaniyappan, I. Ghebregziabher, A. DiChiara, J. MacDonald, and B. C. Walker, “Emergence from nonrelativistic strong-field rescattering to ultrastrong-field laser-atom physics: a semiclassical analysis,” Phys. Rev. A 74, 033403 (2006).
[CrossRef]

Patel, A.

N. J. Kylstra, R. A. Worthington, A. Patel, P. L. Knight, J. R. Vázquez de Aldana, and L. Roso, “Breakdown of stabilization of atoms interacting with intense, high-frequency laser pulses,” Phys. Rev. Lett. 85, 1835-1838 (2000).
[CrossRef] [PubMed]

Pepin, H.

J. Itatani, J. Levesque, D. Zeidler, H. Niikura, H. Pepin, J. C. Kieffer, P. B. Corkum, and D. M. Villeneuve, “Tomographic imaging of molecular orbitals,” Nature 432, 867-871 (2004).
[CrossRef] [PubMed]

Potvliege, R. M.

C. C. Chirilă, C. J. Joachain, N. J. Kylstra, and R. M. Potvliege, “Interaction of superintense laser pulses with relativistic ions,” Phys. Rev. Lett. 93, 243603 (2004).
[CrossRef]

C. C. Chirilă, N. J. Kylstra, R. M. Potvliege, and C. J. Joachain, “Nondipole effects in photon emission by laser-driven ions,” Phys. Rev. A 66, 063411 (2002).
[CrossRef]

Quere, F.

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

Roso, L.

N. J. Kylstra, R. A. Worthington, A. Patel, P. L. Knight, J. R. Vázquez de Aldana, and L. Roso, “Breakdown of stabilization of atoms interacting with intense, high-frequency laser pulses,” Phys. Rev. Lett. 85, 1835-1838 (2000).
[CrossRef] [PubMed]

Salamin, Y. I.

Y. I. Salamin, S. X. Hu, K. Z. Hatsagortsyan, and C. H. Keitel, “Relativistic high-power laser-matter interactions,” Phys. Rep. 427, 41-155 (2006).
[CrossRef]

Y. I. Salamin and F. H. M. Faisal, “Harmonic generation by superintense light scattering from relativistic electrons,” Phys. Rev. A 54, 4383-4395 (1996).
[CrossRef] [PubMed]

Salières, P.

C. Figueira de Morisson Faria, P. Salières, P. Villain, and M. Lewenstein, “Controlling high-order harmonic generation and above-threshold ionization with an attosecond-pulse train,” Phys. Rev. A 74, 053416 (2006).
[CrossRef]

Schafer, K. J.

M. B. Gaarde, K. J. Schafer, A. Heinrich, J. Biegert, and U. Keller, “Large enhancement of macroscopic yield in attosecond pulse train-assisted harmonic generation,” Phys. Rev. A 72, 013411 (2005).
[CrossRef]

K. J. Schafer, M. B. Gaarde, A. Heinrich, J. Biegert, and U. Keller, “Strong field quantum path control using attosecond pulse trains,” Phys. Rev. Lett. 92, 023003 (2004).
[CrossRef] [PubMed]

K. C. Kulander, K. J. Schafer, and J. L. Krause, “Dynamics of short-pulse excitation and ionization and harmonic generation,” in Super-Intense Laser-Atom Physics, B.Piraux, A.L'Huillier, and K.Rzażewski, eds. (Plenum, 1993), pp. 95-110.

Schlup, P.

A. Heinrich, W. Kornelis, M. P. Anscombe, C. P. Hauri, P. Schlup, J. Biegert, and U. Keller, “Enhanced VUV-assisted high harmonic generation,” J. Phys. B 39, S275-S281 (2006).
[CrossRef]

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]

Shon, N. H.

A. Bandrauk and N. H. Shon, “Attosecond control of ionization and high-order harmonic generation in molecules,” Phys. Rev. A 66, 031401 (2002).
[CrossRef]

Shubin, N. Y.

Shubin, N. Yu.

V. D. Taranukhin and N. Yu. Shubin, “High-order harmonic generation by multielectron atoms in the field of a relativistic-intensity standing electromagnetic wave,” Quantum Electron. 31, 179-184 (2000).
[CrossRef]

Smirnova, O.

M. Spanner, O. Smirnova, P. B. Corkum, and M. Y. Ivanov, “Reading diffraction images in strong field ionization of diatomic molecules,” J. Phys. B 37, L243-L250 (2004).
[CrossRef]

Spanner, M.

M. Spanner, O. Smirnova, P. B. Corkum, and M. Y. Ivanov, “Reading diffraction images in strong field ionization of diatomic molecules,” J. Phys. B 37, L243-L250 (2004).
[CrossRef]

Strickland, D.

S. Augst, D. Strickland, D. D. Meyerhofer, S. L. Chin, and J. H. Eberly, “Tunneling ionization of noble gases in a high-intensity laser field,” Phys. Rev. Lett. 63, 2212-2215 (1989).
[CrossRef] [PubMed]

Takahashi, E. J.

E. J. Takahashi, T. Kanai, K. L. Ishikawa, Y. Nabekawa, and K. Midorikawa, “Dramatic enhancement of high-order harmonic generation,” Phys. Rev. Lett. 99, 053904 (2007).
[CrossRef] [PubMed]

Taranukhin, V. D.

V. D. Taranukhin and N. Y. Shubin, “High-order harmonic generation by atoms with traveling- and standing-wave pumps of relativistic intensity,” J. Opt. Soc. Am. B 19, 1132-1140 (2002).
[CrossRef]

V. D. Taranukhin, “Relativistic high-order harmonic generation,” Laser Phys. 10, 330-336 (2000).

V. D. Taranukhin and N. Yu. Shubin, “High-order harmonic generation by multielectron atoms in the field of a relativistic-intensity standing electromagnetic wave,” Quantum Electron. 31, 179-184 (2000).
[CrossRef]

Tsakiris, G.

Strong attosecond pulses of up to kilo-electron-volt photon energy can be produced, e.g., via laser pulse reflection from overdense plasma, see G. Tsakiris, K. Eidmann, J. Meyer-ter-Vehn, and F. Krausz, “Route to intense single attosecond pulses,” New J. Phys. 8, 19 (2006).
[CrossRef]

Vázquez de Aldana, J. R.

N. J. Kylstra, R. A. Worthington, A. Patel, P. L. Knight, J. R. Vázquez de Aldana, and L. Roso, “Breakdown of stabilization of atoms interacting with intense, high-frequency laser pulses,” Phys. Rev. Lett. 85, 1835-1838 (2000).
[CrossRef] [PubMed]

Verschl, M.

M. Verschl and C. H. Keitel, “Relativistic classical and quantum dynamics in intense crossed laser beams of various polarizations,” Phys. Rev. ST Accel. Beams 10, 024001 (2007).
[CrossRef]

M. Verschl and C. H. Keitel, “Relativistic recollisions with two consecutive laser pulses,” J. Phys. B 40, F69-F76 (2007).
[CrossRef]

M. Verschl and C. H. Keitel, “Refocussed relativistic recollisions,” Europhys. Lett. 77, 64004 (2007).
[CrossRef]

Villain, P.

C. Figueira de Morisson Faria, P. Salières, P. Villain, and M. Lewenstein, “Controlling high-order harmonic generation and above-threshold ionization with an attosecond-pulse train,” Phys. Rev. A 74, 053416 (2006).
[CrossRef]

Villeneuve, D. M.

J. Itatani, J. Levesque, D. Zeidler, H. Niikura, H. Pepin, J. C. Kieffer, P. B. Corkum, and D. M. Villeneuve, “Tomographic imaging of molecular orbitals,” Nature 432, 867-871 (2004).
[CrossRef] [PubMed]

H. Niikura, F. Legare, R. Hasbani, M. Yu. Ivanov, D. M. Villeneuve, and P. B. Corkum, “Probing molecular dynamics with attosecond resolution using correlated wave packet pairs,” Nature 421, 826-829 (2003).
[CrossRef] [PubMed]

H. Niikura, F. Legare, R. Hasbani, A. D. Bandrauk, M. Yu. Ivanov, D. M. Villeneuve, and P. B. Corkum, “Sub-laser-cycle electron pulses for probing molecular dynamics,” Nature 417, 917-922 (2002).
[CrossRef] [PubMed]

Walker, B. C.

S. Palaniyappan, I. Ghebregziabher, A. DiChiara, J. MacDonald, and B. C. Walker, “Emergence from nonrelativistic strong-field rescattering to ultrastrong-field laser-atom physics: a semiclassical analysis,” Phys. Rev. A 74, 033403 (2006).
[CrossRef]

Worthington, R. A.

N. J. Kylstra, R. A. Worthington, A. Patel, P. L. Knight, J. R. Vázquez de Aldana, and L. Roso, “Breakdown of stabilization of atoms interacting with intense, high-frequency laser pulses,” Phys. Rev. Lett. 85, 1835-1838 (2000).
[CrossRef] [PubMed]

Yudin, G. L.

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

Zeidler, D.

J. Itatani, J. Levesque, D. Zeidler, H. Niikura, H. Pepin, J. C. Kieffer, P. B. Corkum, and D. M. Villeneuve, “Tomographic imaging of molecular orbitals,” Nature 432, 867-871 (2004).
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Appl. Phys. Lett.

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Chem. Phys. Lett.

T. Zuo, A. D. Bandrauk, and P. B. Corkum, “Laser-induced electron diffraction: a new tool for probing ultrafast molecular dynamics,” Chem. Phys. Lett. 259, 313-320 (1996).
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C. H. Keitel, “Relativistic quantum optics,” Contemp. Phys. 42, 353-363 (2001).
[CrossRef]

Europhys. Lett.

M. Verschl and C. H. Keitel, “Refocussed relativistic recollisions,” Europhys. Lett. 77, 64004 (2007).
[CrossRef]

K. Z. Hatsagortsyan, C. Müller, and C. H. Keitel, “Microscopic laser-driven high-energy colliders,” Europhys. Lett. 76, 29-35 (2006).
[CrossRef]

J. Mod. Opt.

R. Fischer, M. Lein, and C. H. Keitel, “Enhanced recollision dynamics via the combination of antisymmetric wave functions and beyond-dipole effects,” J. Mod. Opt. 54, 1911-1926 (2007).
[CrossRef]

J. Mod. Phys.

A. Bandrauk, S. Chelkowski, and S. Goudreau, “Control of harmonic generation using two-color femtosecond-attosecond laser fields: quantum and classical perspectives,” J. Mod. Phys. 52, 411-428 (2005).

J. Opt. Soc. Am. B

J. Phys. B

R. Fischer, M. Lein, and C. H. Keitel, “Strongly enhanced high-harmonic generation via antisymmetric ionic states,” J. Phys. B 40, F113-F119 (2007).
[CrossRef]

A. Heinrich, W. Kornelis, M. P. Anscombe, C. P. Hauri, P. Schlup, J. Biegert, and U. Keller, “Enhanced VUV-assisted high harmonic generation,” J. Phys. B 39, S275-S281 (2006).
[CrossRef]

M. Verschl and C. H. Keitel, “Relativistic recollisions with two consecutive laser pulses,” J. Phys. B 40, F69-F76 (2007).
[CrossRef]

G. Mocken and C. H. Keitel, “Bound atomic dynamics in the MeV regime,” J. Phys. B 37, L275-L283 (2004).
[CrossRef]

M. Spanner, O. Smirnova, P. B. Corkum, and M. Y. Ivanov, “Reading diffraction images in strong field ionization of diatomic molecules,” J. Phys. B 37, L243-L250 (2004).
[CrossRef]

JETP Lett.

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Laser Phys.

V. D. Taranukhin, “Relativistic high-order harmonic generation,” Laser Phys. 10, 330-336 (2000).

D. B. Milošević, S. Hu, and W. Becker, “Relativistic ultrahigh-order harmonic generation,” Laser Phys. 12, 389-397 (2002).

Nature

H. Niikura, F. Legare, R. Hasbani, A. D. Bandrauk, M. Yu. Ivanov, D. M. Villeneuve, and P. B. Corkum, “Sub-laser-cycle electron pulses for probing molecular dynamics,” Nature 417, 917-922 (2002).
[CrossRef] [PubMed]

H. Niikura, F. Legare, R. Hasbani, M. Yu. Ivanov, D. M. Villeneuve, and P. B. Corkum, “Probing molecular dynamics with attosecond resolution using correlated wave packet pairs,” Nature 421, 826-829 (2003).
[CrossRef] [PubMed]

J. Itatani, J. Levesque, D. Zeidler, H. Niikura, H. Pepin, J. C. Kieffer, P. B. Corkum, and D. M. Villeneuve, “Tomographic imaging of molecular orbitals,” Nature 432, 867-871 (2004).
[CrossRef] [PubMed]

New J. Phys.

Strong attosecond pulses of up to kilo-electron-volt photon energy can be produced, e.g., via laser pulse reflection from overdense plasma, see G. Tsakiris, K. Eidmann, J. Meyer-ter-Vehn, and F. Krausz, “Route to intense single attosecond pulses,” New J. Phys. 8, 19 (2006).
[CrossRef]

Opt. Lett.

Phys. Lett. B

C. Müller, K. Z. Hatsagortsyan, and C. H. Keitel, “Particle physics with a laser-driven positronium atom,” Phys. Lett. B 659, 209-213 (2008).
[CrossRef]

Phys. Rep.

Y. I. Salamin, S. X. Hu, K. Z. Hatsagortsyan, and C. H. Keitel, “Relativistic high-power laser-matter interactions,” Phys. Rep. 427, 41-155 (2006).
[CrossRef]

Phys. Rev. A

S. X. Hu and C. H. Keitel, “Dynamics of multiply charged ions in intense laser fields,” Phys. Rev. A 63, 053402 (2001).
[CrossRef]

S. Palaniyappan, I. Ghebregziabher, A. DiChiara, J. MacDonald, and B. C. Walker, “Emergence from nonrelativistic strong-field rescattering to ultrastrong-field laser-atom physics: a semiclassical analysis,” Phys. Rev. A 74, 033403 (2006).
[CrossRef]

Y. I. Salamin and F. H. M. Faisal, “Harmonic generation by superintense light scattering from relativistic electrons,” Phys. Rev. A 54, 4383-4395 (1996).
[CrossRef] [PubMed]

M. B. Gaarde, K. J. Schafer, A. Heinrich, J. Biegert, and U. Keller, “Large enhancement of macroscopic yield in attosecond pulse train-assisted harmonic generation,” Phys. Rev. A 72, 013411 (2005).
[CrossRef]

C. Figueira de Morisson Faria, P. Salières, P. Villain, and M. Lewenstein, “Controlling high-order harmonic generation and above-threshold ionization with an attosecond-pulse train,” Phys. Rev. A 74, 053416 (2006).
[CrossRef]

C. C. Chirilă, N. J. Kylstra, R. M. Potvliege, and C. J. Joachain, “Nondipole effects in photon emission by laser-driven ions,” Phys. Rev. A 66, 063411 (2002).
[CrossRef]

M. Klaiber, K. Z. Hatsagortsyan, and C. H. Keitel, “Relativistic ionization rescattering with tailored laser pulses,” Phys. Rev. A 74, 051803 (2006).
[CrossRef]

M. Klaiber, K. Z. Hatsagortsyan, and C. H. Keitel, “Fully relativistic laser-induced ionization and recollision processes,” Phys. Rev. A 75, 063413 (2007).
[CrossRef]

A. Bandrauk and N. H. Shon, “Attosecond control of ionization and high-order harmonic generation in molecules,” Phys. Rev. A 66, 031401 (2002).
[CrossRef]

D. B. Milošević, S. Hu, and W. Becker, “Quantum-mechanical model for ultrahigh-order harmonic generation in the moderately relativistic regime,” Phys. Rev. A 63, 011403 (2001).
[CrossRef]

M. Klaiber, K. Z. Hatsagortsyan, and C. H. Keitel, “Gauge-invariant relativistic strong-field approximation,” Phys. Rev. A 73, 053411 (2006).
[CrossRef]

Phys. Rev. D

C. Müller, K. Z. Hatsagortsyan, and C. H. Keitel, “Muon pair creation from positronium in a circularly polarized laser field,” Phys. Rev. D 74, 074017 (2006).
[CrossRef]

Phys. Rev. Lett.

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]

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

K. J. Schafer, M. B. Gaarde, A. Heinrich, J. Biegert, and U. Keller, “Strong field quantum path control using attosecond pulse trains,” Phys. Rev. Lett. 92, 023003 (2004).
[CrossRef] [PubMed]

S. Augst, D. Strickland, D. D. Meyerhofer, S. L. Chin, and J. H. Eberly, “Tunneling ionization of noble gases in a high-intensity laser field,” Phys. Rev. Lett. 63, 2212-2215 (1989).
[CrossRef] [PubMed]

K. Ishikawa, “Photoemission and ionization of He+ under simultaneous irradiation of fundamental laser and high-order harmonic pulses,” Phys. Rev. Lett. 91, 043002 (2003).
[CrossRef] [PubMed]

E. J. Takahashi, T. Kanai, K. L. Ishikawa, Y. Nabekawa, and K. Midorikawa, “Dramatic enhancement of high-order harmonic generation,” Phys. Rev. Lett. 99, 053904 (2007).
[CrossRef] [PubMed]

N. Milosevic, P. B. Corkum, and T. Brabec, “How to use lasers for imaging attosecond dynamics of nuclear processes,” Phys. Rev. Lett. 92, 013002 (2004).
[CrossRef] [PubMed]

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

N. J. Kylstra, R. A. Worthington, A. Patel, P. L. Knight, J. R. Vázquez de Aldana, and L. Roso, “Breakdown of stabilization of atoms interacting with intense, high-frequency laser pulses,” Phys. Rev. Lett. 85, 1835-1838 (2000).
[CrossRef] [PubMed]

C. C. Chirilă, C. J. Joachain, N. J. Kylstra, and R. M. Potvliege, “Interaction of superintense laser pulses with relativistic ions,” Phys. Rev. Lett. 93, 243603 (2004).
[CrossRef]

B. Henrich, K. Z. Hatsagortsyan, and C. H. Keitel, “Positronium in intense laser fields,” Phys. Rev. Lett. 93, 013601 (2004).
[CrossRef]

R. Fischer, M. Lein, and C. H. Keitel, “Enhanced recollisions for antisymmetric molecular orbitals in intense laser fields,” Phys. Rev. Lett. 97, 143901 (2006).
[CrossRef] [PubMed]

Phys. Rev. ST Accel. Beams

M. Verschl and C. H. Keitel, “Relativistic classical and quantum dynamics in intense crossed laser beams of various polarizations,” Phys. Rev. ST Accel. Beams 10, 024001 (2007).
[CrossRef]

Quantum Electron.

V. D. Taranukhin and N. Yu. Shubin, “High-order harmonic generation by multielectron atoms in the field of a relativistic-intensity standing electromagnetic wave,” Quantum Electron. 31, 179-184 (2000).
[CrossRef]

Other

K. C. Kulander, K. J. Schafer, and J. L. Krause, “Dynamics of short-pulse excitation and ionization and harmonic generation,” in Super-Intense Laser-Atom Physics, B.Piraux, A.L'Huillier, and K.Rzażewski, eds. (Plenum, 1993), pp. 95-110.

M. Klaiber, K. Z. Hatsagortsyan, and C. H. Keitel, “Zeptosecond γ-ray pulses,” arXiv.org e-print archive, physics.atom-ph/0707.2900, July 19, 2007, http://arXiv.org/abs/0707.2900.

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Figures (16)

Fig. 1
Fig. 1

Conditions for TI and for a significant relativistic drift in the case of an IR laser ω = 0.05 a.u. In the region of the parameters ξ and I p below the thick curve, the ionization occurs via tunneling [condition of Eq. (4)]. In the region of the parameters above the thin curve, the drift plays a significant role [condition of Eq. (2)]. In the overlap of these regions, the drift plays a significant role and ionization occurs via tunneling. For these parameters HHG can be enhanced when the drift is eliminated.

Fig. 2
Fig. 2

Dependence of the initial electron energy ε d = ω a I p on the ionization phase η 2 of the IR laser field for different rescattering energies of 20.5, 21.5, 22.5, 23.5, and 24.3 keV (for the first pair of trajectories, from left to center; for the second pair of trajectories, from right to center). The laser electric field is E 0 = 1.69 a.u. , the angular frequency is 0.05 a.u. , and I p = 4.51 a.u. ( Li 2 + ions).

Fig. 3
Fig. 3

HHG spectra d w n d Ω via Eq. (14) in a laser field with E 0 = 1.69 a.u. and ω = 0.05 a.u. for I p = 4.51 a.u. ( Li 2 + ) : (a) Klein–Gordon equation (red curve), (b) additionally assisted by an APT of Gaussian shape with E 0 a = 0.03 a.u. , a carrier frequency ω a = 5.8 a.u. , a pulse length τ = 4 a.u. , and a phase delay of 1.2 rad with respect to the laser wave (black curve), (c) additionally assisted by an APT of a δ function shape with identical field strength and phase delay (blue curve), and (d) DA (gray curve).

Fig. 4
Fig. 4

Dependence of the HHG yield at 24.3 keV photons on the phase delay between the IR laser and APT fields. The IR laser field is E 0 = 1.69 a.u. , the angular frequency is ω = 0.05 a.u. , the APT field is E 0 a = 0.03 a.u. , the carrier frequency is ω a = 5.8 a.u. , the pulse duration is τ = 4 a.u. , and the ionization potential is I p = 4.51 a.u. ( Li 2 + ) .

Fig. 5
Fig. 5

Dependence of the HHG yield at 24.3 keV photons on the XUV photon energy. The IR laser field is E 0 = 1.69 a.u. , the angular frequency is ω = 0.05 a.u. , the APT field is E 0 a = 0.03 a.u. , the carrier frequency is ω a = 5.8 a.u. , the pulse duration is τ = 4 a.u. , and the phase delay between the APT and IR fields 1.2 and the ionization potential I p = 4.51 a.u. ( Li 2 + ) .

Fig. 6
Fig. 6

Two trajectories in the laser field (with parameters as in Fig. 3) with a final energy of 24.3 keV are displayed. The ionization is induced by the APT with the same parameters as in Fig. 5.

Fig. 7
Fig. 7

Pulse shape of the counterpropagating APTs. The propagation directions of the pulses are shown by the arrows.

Fig. 8
Fig. 8

Trajectory of the ionized cutoff electron in the crossed laser beam setup. The maximal electric field strength is 70 a.u. , the main angular frequency is ω = 0.05 a.u. , and the ionization potential is I p = 54 a.u. ( Na 9 + ) . The laser pulses are counterpropagating along the z axis and polarized along the x axis.

Fig. 9
Fig. 9

Harmonic emission rate in the laser propagation directions as a function of the harmonic energy ( d w n d Ω is the emission probability of the nth harmonic per unit time and per unit solid angle Ω). The laser field amplitude is E 0 = 70 a.u. , ω = 0.05 a.u. , and I p = 54 a.u. ( Na 9 + ) , two counterpropagating APTs via the Klein–Gordon equation, the time delay is 1.5 fs (solid black curve) two counterpropagating APTs within the DA (solid gray curve), and tailored APT as in [27] via the Klein–Gordon equation (dashed curve).

Fig. 10
Fig. 10

Harmonic emission rate in the laser propagation direction via log 10 ( d w n d Ω ) in Eq. (13) as a function of the harmonic energy with a maximal laser electric field strength of 70 a.u. The pulses are separated by 0.9, 1.2, 1.5, 2.2, and 3.8 fs , respectively; the ionization potential is 54 a.u.

Fig. 11
Fig. 11

Dependence on the time delay (a) of initial kinetic momentum in the laser propagation direction p z ( η ) , (b) of the time interval between the pulse peak and the ionization time, and (c) of the cutoff energy of HHG in the counterpropagating APT setup. In all cases the laser electric field strength is 70 a.u. and the ionization potential is 54 a.u.

Fig. 12
Fig. 12

(a) Kinetic energy of the ionized electron during the excursion in the laser pulses with a time delay between the pulses of 0.95 fs (black curve) and 3.8 fs (red/faint curve). The laser electric field strength is 70 a.u. and the ionization potential is 54 a.u. (b) Ionization and excursion phases of the electron (see details in the text). The corresponding energies of the electron are indicated in (a).

Fig. 13
Fig. 13

Harmonic emission rate in the laser propagation direction via log 10 ( d w n d Ω ) in Eq. (13) as a function of the harmonic energy with a maximal laser electric field strength of 250 a.u. for counterpropagating ATPs with different time delays compared with the spectrum in the DA (gray curve). The pulses are separated by 4.3, 5.1, 7.6, and 15 fs , respectively (from the highest to the lowest curve). The ionization potential is 129 a.u. ( S 15 + ) .

Fig. 14
Fig. 14

Dependence on the time delay of (a) initial kinetic momentum in the laser propagation direction p z ( η ) , (b) the time interval between the pulse peak and the ionization time, and (c) the cutoff energy of HHG in the counterpropagating APT setup. The laser electric field strength is 250 a.u. and the ionization potential is 129 a.u.

Fig. 15
Fig. 15

Kinetic energy of the ionized electron during the excursion in laser pulses with a time delay between the pulses of 3.8 fs (black curve) and 15 fs (red/faint curve). The laser electric field strength is 250 a.u.

Fig. 16
Fig. 16

Electron kinetic energy change per c 2 (solid curves) and the parameter p x ( c Λ ( p ) ) (dashed curves) during the electron excursion up to the recollision for moderate relativistic (left) and ultrarelativistic (right) regimes in the case of a short (thick curve) and a long time delay (thin curve).

Equations (29)

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

Δ z 1 ω 2 3 2 e E m 3 2 I p 1 2 .
η = 2 I p m c 2 ξ 3 16 m c 2 ω > 1 .
W ( p d ) exp ( 2 ( 2 m I p + p d 2 ) 3 2 3 m e E ) exp ( 2 ( 2 m I p ) 3 2 3 m e E 2 m I p 3 m e E p d 2 ) ,
ξ < ( I p m c 2 ) 3 2 m c 2 4 2 ω .
A ( η ) = e x ( E 0 ω c ) cos η ; η = ω ( t z c ) ,
E a ( y , t ) = e z E 0 a g ( y , t ) sin ( ω a ( t y c ) ) ,
g ( y , t ) = exp [ 2 ( t y c t 0 ) 2 τ 2 ] ,
( μ μ + c 2 ) Ψ ( x ) = ( V L + V X + V A + V H ) Ψ ( x ) ,
V A = 2 i V c 2 t + V 2 c 2 ,
V L = 2 i A ( η ) c A ( η ) 2 c 2 ,
V X = 2 i A a ( η ) c A a ( η ) 2 c 2 ,
V H ( x ) = 2 1 H A H ( x ) c ( p ̂ + A tot ( η ) c ) 0 H ,
d w = M a 2 d 3 k ( 2 π ) 3 .
M a = i d 4 x d 4 x { Φ ( x ) * V H ( x ) G V ( x , x ) [ x ( E + E a ) ( x ) ] Φ ( x ) } .
G V ( x , x ) = i θ ( t t ) c d 3 p 2 ε p ( 2 π ) 3 exp [ i p ( x x ) i η η d η ̃ ( ( p + A ( η ̃ ) 2 c ) A ( η ̃ ) c ω Λ ( p ) ) ] ,
d w n d Ω = n ω ( 2 π ) 3 ω 2 c 3 M n a 2 ,
M n a = π π d η η d η d 3 p m ̃ H a ( p , η , η ) exp ( i ( S ( p , η , η ) + ω a η ω n η ) ) ,
m ̃ H a ( p , η , η ) = c 2 ( p x + A ( η ) c ) ε p ω 2 0 p + A ( η ) c k ω ( ε p + I p c 2 ) p + A ( η ) c k ω ( ε p + I p c 2 ) x E a 0 ,
ε p ( η ) c 2 = ω a I p .
M n = i d 4 x d 4 x { Φ ( x ) * V H ( x ) G ( x , x ) V A ( x ) Φ ( x ) } ,
G ( x , x ) i d 3 x B G 2 V ( x , x B ) c t B G 1 V ( x B , x ) ,
G ( x , x ) i θ ( t t ) d 3 p 2 ( 2 π ) 3 ε p exp [ i p ( x x ) i t B + z 0 B c η 2 d η ̃ ( ( p + A 2 ( η ̃ ) 2 c ) A 2 ( η ̃ ) c Λ 2 ( p ) ) i η 1 t B z 0 B c d η ̃ ( ( p + A 1 ( η ̃ ) 2 c ) A 1 ( η ̃ ) c Λ 1 ( p ) ) ] ,
z 0 B = η 1 t B z 0 B c d η ̃ ( p z Λ 1 ( p ) + ( p + A 1 ( η ̃ ) 2 c ) A 1 ( η ̃ ) c c Λ 1 ( p ) 2 ) .
M n = d η η d η d 3 p m H ( p , η , η ) exp { i [ S 2 ( p , η , t B + z 0 B c ) + S 1 ( p , t B z 0 B c , η ) + 2 ( ε p c 2 + I p ) z 0 B c n ω η ] } ,
m H ( p , η , η ) = c 2 ( p + A 2 ( η ) c ) e ̂ H ( 2 π ) 3 2 ε p ϕ ̃ 0 * ( p + A 2 ( η ) c k ̂ 2 c ( ε p + I p c 2 ) ) p + A 1 ( η ) c k ̂ 1 c ( ε p + I p c 2 ) V 0 ,
p ( S ( p , η , η ) + 2 ( ε p c 2 + I p ) z 0 B c ) = 0 ,
ε ̃ p , 2 ( η ) = c 2 I p + n ω ,
ε ̃ p , 1 ( η ) = c 2 I p ,
θ 0 < 2 5 4 γ ( ω τ ) 2 ( c 2 I p ) 1 4 π λ c ξ λ ,

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