Abstract

A method is proposed to engineer the continuum fraction of the electron wave packet in high-order harmonic generation (HHG) such that a quasi-monochromatic recollision with the atomic core is rendered possible even for parts of the wave packet that were launched to the continuum at different laser phases. Because of this, the HHG spectrum is shown to be enhanced in a specified controllable spectral window. The electron wave-packet engineering is achieved via driving HHG by the combined fields of weak x rays and a strong shaped pulse of infrared radiation. Our calculations based on the strong field approximation show how the enhanced spectral window can be controlled by the shaping of the driving infrared pulse. The scheme is illustrated via a semiclassical trajectory-based analysis.

© 2012 Optical Society of America

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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  44. Note that in the employed saddle-point approximation, the Taylor expansion of the complex phase of Eq. (1) up to the second-order term is applied. When minimizing the functional determinant, the second-order expansion term is decreasing, increasing the role of the third-order term. However, we verified numerically the validity of excluding the third-order term in the applied parameter regime.

2012

M. C. Kohler, T. Pfeiffer, K. Z. Hatsagortsyan, and C. H. Keitel, “Frontiers of atomic high-harmonic generation,” Adv. At. Mol. Opt. Phys. 61, 159–208 (2012).
[CrossRef]

T. Popmintchev, M.-C. Chen, D. Popmintchev, P. Arpin, S. Brown, S. Ališauskas, G. Andriukaitis, T. Baliunas, O. Mücke, A. Pugzlys, Andrius Baltuška, B. Shim, S. Schrauth, A. Gaeta, C. Hernández-Garciá, L. Plaja, A. Becker, Agnieszka Jaron-Becker, M. M. Murnane, and H. C. Kapteyn, “Bright coherent ultrahigh harmonics in the keV x-ray regime from mid-infrared femtosecond lasers,” Science 336, 1287–1291 (2012).
[CrossRef]

M. C. Kohler and K. Z. Hatsagortsyan, “Macroscopic aspects of relativistic x-ray-assisted high-order-harmonic generation,” Phys. Rev. A 85, 023819 (2012).
[CrossRef]

2011

2010

H. Du, H. Wang, and B. Hu, “Isolated short attosecond pulse generated using a two-color laser and a high-order pulse,” Phys. Rev. A 81, 063813 (2010).
[CrossRef]

L. E. Chipperfield, J. W. G. Tisch, and J. P. Marangos, “Derivation of the perfect wave for optimising the strong-field driven electron recollision energy,” J. Mod. Opt. 57, 992–998 (2010).
[CrossRef]

M.-C. Chen, P. Arpin, T. Popmintchev, M. Gerrity, B. Zhang, M. Seaberg, D. Popmintchev, M. M. Murnane, and H. C. Kapteyn, “Bright, coherent, ultrafast soft x-ray harmonics spanning the water window from a tabletop light source,” Phys. Rev. Lett. 105, 173901 (2010).
[CrossRef]

2009

T. Popmintchev, M.-C. Chen, A. Bahabad, M. Gerrity, P. Sidorenko, O. Cohen, I. P. Christov, M. M. Murnane, and H. C. Kapteyn, “Phase matching of high harmonic generation in the soft and hard x-ray regions of the spectrum,” Proc. Natl. Acad. Sci. USA 106, 10516–10521 (2009).
[CrossRef]

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

L. E. Chipperfield, J. S. Robinson, J. W. G. Tisch, and J. P. Marangos, “Ideal waveform to generate the maximum possible electron recollision energy for any given oscillation period,” Phys. Rev. Lett. 102, 063003 (2009).
[CrossRef]

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]

K. Z. Hatsagortsyan, M. Klaiber, C. Müller, M. C. Kohler, and C. H. Keitel, “Laser-driven relativistic recollisions,” J. Opt. Soc. Am. B 25, B92–B103 (2008).
[CrossRef]

Ł. Kornaszewski, M. Kohler, U. K. Sapaev, and D. T. Reid, “Designer femtosecond pulse shaping using grating-engineered quasi-phase-matching in lithium niobate,” Opt. Lett. 33, 378–380 (2008).
[CrossRef]

S. B. P. Radnor, L. E. Chipperfield, P. Kinsler, and G. H. C. New, “Carrier-wave steepened pulses and gradient-gated high-order harmonic generation,” Phys. Rev. A 77, 033806 (2008).
[CrossRef]

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]

2007

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]

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

C. Figueira de Morisson Faria and P. Salières, “High-order harmonic generation with a strong laser field and an attosecond-pulse train: the Dirac-Delta comb and monochromatic limits,” Laser Phys. 17, 390–400 (2007).
[CrossRef]

2006

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

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

E. Seres, J. Seres, and C. Spielmann, “X-ray absorption spectroscopy in the keV range with laser generated high harmonic radiation,” Appl. Phys. Lett 89, 181919 (2006).
[CrossRef]

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]

2005

T. Pfeifer, D. Walter, C. Winterfeldt, C. Spielmann, and G. Gerber, “Controlling the spectral shape of coherent soft x-rays,” Appl. Phys. B 80, 277–280 (2005).
[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]

2004

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]

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]

D. H. Reitze, S. Kazamias, F. Weihe, G. Mullot, D. Douillet, F. Aug, O. Albert, V. Ramanathan, J. P. Chambaret, D. Hulin, and P. Balcou, “Enhancement of high-order harmonic generation at tuned wavelengths through adaptive control,” Opt. Lett. 29, 86–88 (2004).
[CrossRef]

P. Agostini and L. F. D. Mauro, “The physics of attosecond light pulses,” Rep. Prog. Phys. 67, 813–855 (2004).
[CrossRef]

2003

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]

2002

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

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

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

2000

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 (2000).
[CrossRef]

R. Bartels, S. Backus, E. Zeek, L. Misoguti, G. Vdovin, I. P. Christov, M. M. Murnane, and H. C. Kapteyn, “Shaped-pulse optimization of coherent emission of high-harmonic soft x-rays,” Nature 406, 164–166 (2000).
[CrossRef]

1999

1996

M. Y. Ivanov, T. Brabec, and N. Burnett, “Coulomb corrections and polarization effects in high-intensity high-harmonic emission,” Phys. Rev. A 54, 742–745 (1996).
[CrossRef]

1993

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

1990

1980

H. R. Reiss, “Effect of an intense electromagnetic field on a weakly bound system,” Phys. Rev. A 22, 1786–1813 (1980).
[CrossRef]

1973

F. H. M. Faisal, “Multiple absorption of laser photons by atoms,” J. Phys. B: At. Mol. Phys. 6, L89–L92 (1973).
[CrossRef]

1964

L. V. Keldysh, “Ionization in the field of a strong electromagnetic wave,” Zh. Eksp. Teor. Fiz. 47, 1945–1957 (1964); Sov. Phys. JETP 20, 1307–1319 (1965).

Agostini, P.

P. Agostini and L. F. D. Mauro, “The physics of attosecond light pulses,” Rep. Prog. Phys. 67, 813–855 (2004).
[CrossRef]

Albert, O.

Ališauskas, S.

T. Popmintchev, M.-C. Chen, D. Popmintchev, P. Arpin, S. Brown, S. Ališauskas, G. Andriukaitis, T. Baliunas, O. Mücke, A. Pugzlys, Andrius Baltuška, B. Shim, S. Schrauth, A. Gaeta, C. Hernández-Garciá, L. Plaja, A. Becker, Agnieszka Jaron-Becker, M. M. Murnane, and H. C. Kapteyn, “Bright coherent ultrahigh harmonics in the keV x-ray regime from mid-infrared femtosecond lasers,” Science 336, 1287–1291 (2012).
[CrossRef]

Altucci, C.

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]

Andriukaitis, G.

T. Popmintchev, M.-C. Chen, D. Popmintchev, P. Arpin, S. Brown, S. Ališauskas, G. Andriukaitis, T. Baliunas, O. Mücke, A. Pugzlys, Andrius Baltuška, B. Shim, S. Schrauth, A. Gaeta, C. Hernández-Garciá, L. Plaja, A. Becker, Agnieszka Jaron-Becker, M. M. Murnane, and H. C. Kapteyn, “Bright coherent ultrahigh harmonics in the keV x-ray regime from mid-infrared femtosecond lasers,” Science 336, 1287–1291 (2012).
[CrossRef]

Aquila, A. L.

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]

Arpin, P.

T. Popmintchev, M.-C. Chen, D. Popmintchev, P. Arpin, S. Brown, S. Ališauskas, G. Andriukaitis, T. Baliunas, O. Mücke, A. Pugzlys, Andrius Baltuška, B. Shim, S. Schrauth, A. Gaeta, C. Hernández-Garciá, L. Plaja, A. Becker, Agnieszka Jaron-Becker, M. M. Murnane, and H. C. Kapteyn, “Bright coherent ultrahigh harmonics in the keV x-ray regime from mid-infrared femtosecond lasers,” Science 336, 1287–1291 (2012).
[CrossRef]

M.-C. Chen, P. Arpin, T. Popmintchev, M. Gerrity, B. Zhang, M. Seaberg, D. Popmintchev, M. M. Murnane, and H. C. Kapteyn, “Bright, coherent, ultrafast soft x-ray harmonics spanning the water window from a tabletop light source,” Phys. Rev. Lett. 105, 173901 (2010).
[CrossRef]

Attwood, D. T.

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]

Aug, F.

Avaldi, L.

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]

Backus, S.

R. Bartels, S. Backus, E. Zeek, L. Misoguti, G. Vdovin, I. P. Christov, M. M. Murnane, and H. C. Kapteyn, “Shaped-pulse optimization of coherent emission of high-harmonic soft x-rays,” Nature 406, 164–166 (2000).
[CrossRef]

E. Zeek, K. Maginnis, S. Backus, U. Russek, M. Murnane, G. Mourou, H. Kapteyn, and G. Vdovin, “Pulse compression by use of deformable mirrors,” Opt. Lett. 24, 493–495 (1999).
[CrossRef]

Bahabad, A.

T. Popmintchev, M.-C. Chen, A. Bahabad, M. Gerrity, P. Sidorenko, O. Cohen, I. P. Christov, M. M. Murnane, and H. C. Kapteyn, “Phase matching of high harmonic generation in the soft and hard x-ray regions of the spectrum,” Proc. Natl. Acad. Sci. USA 106, 10516–10521 (2009).
[CrossRef]

Balcou, P.

Baliunas, T.

T. Popmintchev, M.-C. Chen, D. Popmintchev, P. Arpin, S. Brown, S. Ališauskas, G. Andriukaitis, T. Baliunas, O. Mücke, A. Pugzlys, Andrius Baltuška, B. Shim, S. Schrauth, A. Gaeta, C. Hernández-Garciá, L. Plaja, A. Becker, Agnieszka Jaron-Becker, M. M. Murnane, and H. C. Kapteyn, “Bright coherent ultrahigh harmonics in the keV x-ray regime from mid-infrared femtosecond lasers,” Science 336, 1287–1291 (2012).
[CrossRef]

Baltuška, Andrius

T. Popmintchev, M.-C. Chen, D. Popmintchev, P. Arpin, S. Brown, S. Ališauskas, G. Andriukaitis, T. Baliunas, O. Mücke, A. Pugzlys, Andrius Baltuška, B. Shim, S. Schrauth, A. Gaeta, C. Hernández-Garciá, L. Plaja, A. Becker, Agnieszka Jaron-Becker, M. M. Murnane, and H. C. Kapteyn, “Bright coherent ultrahigh harmonics in the keV x-ray regime from mid-infrared femtosecond lasers,” Science 336, 1287–1291 (2012).
[CrossRef]

Bartels, R.

R. Bartels, S. Backus, E. Zeek, L. Misoguti, G. Vdovin, I. P. Christov, M. M. Murnane, and H. C. Kapteyn, “Shaped-pulse optimization of coherent emission of high-harmonic soft x-rays,” Nature 406, 164–166 (2000).
[CrossRef]

Becker, A.

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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).
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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).
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T. Popmintchev, M.-C. Chen, D. Popmintchev, P. Arpin, S. Brown, S. Ališauskas, G. Andriukaitis, T. Baliunas, O. Mücke, A. Pugzlys, Andrius Baltuška, B. Shim, S. Schrauth, A. Gaeta, C. Hernández-Garciá, L. Plaja, A. Becker, Agnieszka Jaron-Becker, M. M. Murnane, and H. C. Kapteyn, “Bright coherent ultrahigh harmonics in the keV x-ray regime from mid-infrared femtosecond lasers,” Science 336, 1287–1291 (2012).
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H. Du, H. Wang, and B. Hu, “Isolated short attosecond pulse generated using a two-color laser and a high-order pulse,” Phys. Rev. A 81, 063813 (2010).
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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 (2000).
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Kapteyn, H. C.

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[CrossRef]

M.-C. Chen, P. Arpin, T. Popmintchev, M. Gerrity, B. Zhang, M. Seaberg, D. Popmintchev, M. M. Murnane, and H. C. Kapteyn, “Bright, coherent, ultrafast soft x-ray harmonics spanning the water window from a tabletop light source,” Phys. Rev. Lett. 105, 173901 (2010).
[CrossRef]

T. Popmintchev, M.-C. Chen, A. Bahabad, M. Gerrity, P. Sidorenko, O. Cohen, I. P. Christov, M. M. Murnane, and H. C. Kapteyn, “Phase matching of high harmonic generation in the soft and hard x-ray regions of the spectrum,” Proc. Natl. Acad. Sci. USA 106, 10516–10521 (2009).
[CrossRef]

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Keitel, C. H.

M. C. Kohler, T. Pfeiffer, K. Z. Hatsagortsyan, and C. H. Keitel, “Frontiers of atomic high-harmonic generation,” Adv. At. Mol. Opt. Phys. 61, 159–208 (2012).
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M. C. Kohler, C. H. Keitel, and K. Z. Hatsagortsyan, “Attochirp-free high-order harmonic generation,” Opt. Express 19, 4411–4420 (2011).
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M. C. Kohler, M. Klaiber, K. Z. Hatsagortsyan, and C. H. Keitel, “Phase-matched coherent hard x-rays from relativistic high-order harmonic generation,” Europhys. Lett. 94, 14002 (2011).
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M. Klaiber, K. Z. Hatsagortsyan, and C. H. Keitel, “Fully relativistic laser-induced ionization and recollision processes,” Phys. Rev. A 75, 063413 (2007).
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M. Klaiber, K. Z. Hatsagortsyan, and C. H. Keitel, “Relativistic ionization rescattering with tailored laser pulses,” Phys. Rev. A 74, 051803 (2006).
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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]

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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).
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S. B. P. Radnor, L. E. Chipperfield, P. Kinsler, and G. H. C. New, “Carrier-wave steepened pulses and gradient-gated high-order harmonic generation,” Phys. Rev. A 77, 033806 (2008).
[CrossRef]

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M. C. Kohler, M. Klaiber, K. Z. Hatsagortsyan, and C. H. Keitel, “Phase-matched coherent hard x-rays from relativistic high-order harmonic generation,” Europhys. Lett. 94, 14002 (2011).
[CrossRef]

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]

K. Z. Hatsagortsyan, M. Klaiber, C. Müller, M. C. Kohler, and C. H. Keitel, “Laser-driven relativistic recollisions,” J. Opt. Soc. Am. B 25, B92–B103 (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]

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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]

Kohler, M.

Kohler, M. C.

M. C. Kohler, T. Pfeiffer, K. Z. Hatsagortsyan, and C. H. Keitel, “Frontiers of atomic high-harmonic generation,” Adv. At. Mol. Opt. Phys. 61, 159–208 (2012).
[CrossRef]

M. C. Kohler and K. Z. Hatsagortsyan, “Macroscopic aspects of relativistic x-ray-assisted high-order-harmonic generation,” Phys. Rev. A 85, 023819 (2012).
[CrossRef]

M. C. Kohler, C. H. Keitel, and K. Z. Hatsagortsyan, “Attochirp-free high-order harmonic generation,” Opt. Express 19, 4411–4420 (2011).
[CrossRef]

M. C. Kohler, M. Klaiber, K. Z. Hatsagortsyan, and C. H. Keitel, “Phase-matched coherent hard x-rays from relativistic high-order harmonic generation,” Europhys. Lett. 94, 14002 (2011).
[CrossRef]

C. Buth, M. C. Kohler, J. Ullrich, and C. H. Keitel, “High-order harmonic generation enhanced by XUV light,” Opt. Lett. 36, 3530–3532 (2011).
[CrossRef]

K. Z. Hatsagortsyan, M. Klaiber, C. Müller, M. C. Kohler, and C. H. Keitel, “Laser-driven relativistic recollisions,” J. Opt. Soc. Am. B 25, B92–B103 (2008).
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Krausz, F.

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

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]

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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 ion,” Phys. Rev. A 66, 063411 (2002).
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Li, S.

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L. E. Chipperfield, J. W. G. Tisch, and J. P. Marangos, “Derivation of the perfect wave for optimising the strong-field driven electron recollision energy,” J. Mod. Opt. 57, 992–998 (2010).
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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).
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D. B. Milošević, S. X. Hu, and W. Becker, “Relativistic ultrahigh-order harmonic generation,” Laser Phys. 12, 389–397 (2002).

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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 (2000).
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R. Bartels, S. Backus, E. Zeek, L. Misoguti, G. Vdovin, I. P. Christov, M. M. Murnane, and H. C. Kapteyn, “Shaped-pulse optimization of coherent emission of high-harmonic soft x-rays,” Nature 406, 164–166 (2000).
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T. Popmintchev, M.-C. Chen, D. Popmintchev, P. Arpin, S. Brown, S. Ališauskas, G. Andriukaitis, T. Baliunas, O. Mücke, A. Pugzlys, Andrius Baltuška, B. Shim, S. Schrauth, A. Gaeta, C. Hernández-Garciá, L. Plaja, A. Becker, Agnieszka Jaron-Becker, M. M. Murnane, and H. C. Kapteyn, “Bright coherent ultrahigh harmonics in the keV x-ray regime from mid-infrared femtosecond lasers,” Science 336, 1287–1291 (2012).
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M.-C. Chen, P. Arpin, T. Popmintchev, M. Gerrity, B. Zhang, M. Seaberg, D. Popmintchev, M. M. Murnane, and H. C. Kapteyn, “Bright, coherent, ultrafast soft x-ray harmonics spanning the water window from a tabletop light source,” Phys. Rev. Lett. 105, 173901 (2010).
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T. Popmintchev, M.-C. Chen, A. Bahabad, M. Gerrity, P. Sidorenko, O. Cohen, I. P. Christov, M. M. Murnane, and H. C. Kapteyn, “Phase matching of high harmonic generation in the soft and hard x-ray regions of the spectrum,” Proc. Natl. Acad. Sci. USA 106, 10516–10521 (2009).
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R. Bartels, S. Backus, E. Zeek, L. Misoguti, G. Vdovin, I. P. Christov, M. M. Murnane, and H. C. Kapteyn, “Shaped-pulse optimization of coherent emission of high-harmonic soft x-rays,” Nature 406, 164–166 (2000).
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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).
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S. B. P. Radnor, L. E. Chipperfield, P. Kinsler, and G. H. C. New, “Carrier-wave steepened pulses and gradient-gated high-order harmonic generation,” Phys. Rev. A 77, 033806 (2008).
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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).
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T. Pfeifer, D. Walter, C. Winterfeldt, C. Spielmann, and G. Gerber, “Controlling the spectral shape of coherent soft x-rays,” Appl. Phys. B 80, 277–280 (2005).
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M. C. Kohler, T. Pfeiffer, K. Z. Hatsagortsyan, and C. H. Keitel, “Frontiers of atomic high-harmonic generation,” Adv. At. Mol. Opt. Phys. 61, 159–208 (2012).
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T. Popmintchev, M.-C. Chen, D. Popmintchev, P. Arpin, S. Brown, S. Ališauskas, G. Andriukaitis, T. Baliunas, O. Mücke, A. Pugzlys, Andrius Baltuška, B. Shim, S. Schrauth, A. Gaeta, C. Hernández-Garciá, L. Plaja, A. Becker, Agnieszka Jaron-Becker, M. M. Murnane, and H. C. Kapteyn, “Bright coherent ultrahigh harmonics in the keV x-ray regime from mid-infrared femtosecond lasers,” Science 336, 1287–1291 (2012).
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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).
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T. Popmintchev, M.-C. Chen, D. Popmintchev, P. Arpin, S. Brown, S. Ališauskas, G. Andriukaitis, T. Baliunas, O. Mücke, A. Pugzlys, Andrius Baltuška, B. Shim, S. Schrauth, A. Gaeta, C. Hernández-Garciá, L. Plaja, A. Becker, Agnieszka Jaron-Becker, M. M. Murnane, and H. C. Kapteyn, “Bright coherent ultrahigh harmonics in the keV x-ray regime from mid-infrared femtosecond lasers,” Science 336, 1287–1291 (2012).
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M.-C. Chen, P. Arpin, T. Popmintchev, M. Gerrity, B. Zhang, M. Seaberg, D. Popmintchev, M. M. Murnane, and H. C. Kapteyn, “Bright, coherent, ultrafast soft x-ray harmonics spanning the water window from a tabletop light source,” Phys. Rev. Lett. 105, 173901 (2010).
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T. Popmintchev, M.-C. Chen, D. Popmintchev, P. Arpin, S. Brown, S. Ališauskas, G. Andriukaitis, T. Baliunas, O. Mücke, A. Pugzlys, Andrius Baltuška, B. Shim, S. Schrauth, A. Gaeta, C. Hernández-Garciá, L. Plaja, A. Becker, Agnieszka Jaron-Becker, M. M. Murnane, and H. C. Kapteyn, “Bright coherent ultrahigh harmonics in the keV x-ray regime from mid-infrared femtosecond lasers,” Science 336, 1287–1291 (2012).
[CrossRef]

M.-C. Chen, P. Arpin, T. Popmintchev, M. Gerrity, B. Zhang, M. Seaberg, D. Popmintchev, M. M. Murnane, and H. C. Kapteyn, “Bright, coherent, ultrafast soft x-ray harmonics spanning the water window from a tabletop light source,” Phys. Rev. Lett. 105, 173901 (2010).
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T. Popmintchev, M.-C. Chen, A. Bahabad, M. Gerrity, P. Sidorenko, O. Cohen, I. P. Christov, M. M. Murnane, and H. C. Kapteyn, “Phase matching of high harmonic generation in the soft and hard x-ray regions of the spectrum,” Proc. Natl. Acad. Sci. USA 106, 10516–10521 (2009).
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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 ion,” Phys. Rev. A 66, 063411 (2002).
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S. B. P. Radnor, L. E. Chipperfield, P. Kinsler, and G. H. C. New, “Carrier-wave steepened pulses and gradient-gated high-order harmonic generation,” Phys. Rev. A 77, 033806 (2008).
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C. Figueira de Morisson Faria and P. Salières, “High-order harmonic generation with a strong laser field and an attosecond-pulse train: the Dirac-Delta comb and monochromatic limits,” Laser Phys. 17, 390–400 (2007).
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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).
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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).
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T. Popmintchev, M.-C. Chen, D. Popmintchev, P. Arpin, S. Brown, S. Ališauskas, G. Andriukaitis, T. Baliunas, O. Mücke, A. Pugzlys, Andrius Baltuška, B. Shim, S. Schrauth, A. Gaeta, C. Hernández-Garciá, L. Plaja, A. Becker, Agnieszka Jaron-Becker, M. M. Murnane, and H. C. Kapteyn, “Bright coherent ultrahigh harmonics in the keV x-ray regime from mid-infrared femtosecond lasers,” Science 336, 1287–1291 (2012).
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T. Popmintchev, M.-C. Chen, D. Popmintchev, P. Arpin, S. Brown, S. Ališauskas, G. Andriukaitis, T. Baliunas, O. Mücke, A. Pugzlys, Andrius Baltuška, B. Shim, S. Schrauth, A. Gaeta, C. Hernández-Garciá, L. Plaja, A. Becker, Agnieszka Jaron-Becker, M. M. Murnane, and H. C. Kapteyn, “Bright coherent ultrahigh harmonics in the keV x-ray regime from mid-infrared femtosecond lasers,” Science 336, 1287–1291 (2012).
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T. Popmintchev, M.-C. Chen, A. Bahabad, M. Gerrity, P. Sidorenko, O. Cohen, I. P. Christov, M. M. Murnane, and H. C. Kapteyn, “Phase matching of high harmonic generation in the soft and hard x-ray regions of the spectrum,” Proc. Natl. Acad. Sci. USA 106, 10516–10521 (2009).
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E. Seres, J. Seres, and C. Spielmann, “X-ray absorption spectroscopy in the keV range with laser generated high harmonic radiation,” Appl. Phys. Lett 89, 181919 (2006).
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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).
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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).
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L. E. Chipperfield, J. W. G. Tisch, and J. P. Marangos, “Derivation of the perfect wave for optimising the strong-field driven electron recollision energy,” J. Mod. Opt. 57, 992–998 (2010).
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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).
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Other

Note that in the employed saddle-point approximation, the Taylor expansion of the complex phase of Eq. (1) up to the second-order term is applied. When minimizing the functional determinant, the second-order expansion term is decreasing, increasing the role of the third-order term. However, we verified numerically the validity of excluding the third-order term in the applied parameter regime.

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Figures (3)

Fig. 1.
Fig. 1.

Classical electron positions (black circles) are displayed (a) at the initial moment ti and (b) at the final moment te for two possible trajectories, one starting earlier than the second. Their spatial separation at the ionization moment of the second electron is denoted by δxi, and that at the recombination moment of the first electron by δxe. The position of the parent ion is the gray circle.

Fig. 2.
Fig. 2.

(a) displays the optimized tailored laser field (blue, black) and sinusoidal field (red, gray) with the same period. Both fields are assisted by a weak x-ray field. The optimized field is determined by the procedure described in the text and (b) shows the recollision energies for electrons depending on the ionization phase calculated from the saddle points with the parameter ωxIp=0.04 a.u. for the fields in (a): the optimized tailored laser field (blue, black) and sinusoidal field (red, gray). The downhill and uphill trajectories are distinguished by a solid or dashed plot style.

Fig. 3.
Fig. 3.

Single-atom HHG rate with a hydrogen-like lithium (Ip=4.5 a.u.) by the shaped pulse (blue, black) of Fig. 2(a) and a conventional sinusoidal pulse (red, gray) both assisted by an x-ray field with the photon energy of ωx=4.54 a.u. and the field amplitude EX=0.3 a.u. The sinusoidal laser field has a field amplitude of E=0.6 a.u. and a frequency of ω=0.05 a.u.

Equations (10)

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d˜q=idttdtd3qΦ0|x|p+A(t)×EX(t)p+A(t)|x|Φ0eiSq(p,t,t),
d˜q=is(2πi)5det(S˜s)Φ0|x|ps+A(te)×EX(ti)ps+A(ti)|x|Φ0eiSq(ps,te,ti),
tite[ps+A(t)]dt=0,
[ps+A(ti)]2/2=ωxIp,
[ps+A(te)]2/2+Ip=qω
S˜s=(px,pxSqpx,pySqpx,pzSqpx,tSqpx,tSqpy,pxSqpy,pySqpy,pzSqpy,tSqpy,tSqpz,pxSqpz,pySqpz,pzSqpz,tSqpz,tSqt,pxSqt,pySqt,pzSqt,tSqt,tSqt,pxSqt,pySqt,pzSqt,tSqt,tSq).
S˜s=(τ00pK(te)pK(ti)0τ00000τ00pK(te)00E(te)pK(te)0pK(ti)000E(ti)pK(ti)),
detS˜s=τ2pK,ipK,e[EipK,e+Ee(Eiτ+pK,i)].
ωHtipK,eδpK,e/δti=EipK,e+Ee(Eiτ+pK,i).
detS˜s=τ2pK,ipK,eωHti.

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