Abstract

A theoretical treatment of the high-order harmonic generation by electrons undergoing collisions with ions in a plasma in the presence of two single-mode linearly polarized fields is presented. Except for polarization, the two radiation fields may have arbitrary parameters. The treatment is based on an approximate solution of the Fokker–Planck equation for the plasma electron velocity distribution function. A set of representative numerical calculations is carried out. It is found that the generation of odd harmonics of each field is followed by the generation as well of satellites at mixed frequencies. For a given odd harmonic of one field, the satellites are shifted from it by multiple numbers of twice the frequency of the other field. When there are no coincident frequencies among the generated ones, the intensity of the odd harmonics of one field is found to decrease owing to the action of the other field. When, instead, the satellite frequencies of low-order harmonics coincide with the frequencies of high-order harmonics, a considerable enhancement of the latter is predicted. It is shown also that the harmonic intensity and the polarization direction depend significantly on the angle between the polarization directions of the two mixing fields.

© 2000 Optical Society of America

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  1. J. L. Krause, K. J. Schafer, and K. C. Kulander, “High-order harmonic generation from atoms and ions in the high intensity regime,” Phys. Rev. Lett. 68, 3535–3538 (1992).
    [CrossRef] [PubMed]
  2. J. K. Crane, M. D. Perry, S. Herman, and R. W. Falcone, “High-order harmonic generation in helium,” Opt. Lett. 17, 1256–1258 (1992).
    [CrossRef] [PubMed]
  3. A. L’Huiller and Ph. Balcou, “High-order harmonic generation in rare gases with a 1-ps 1053-nm laser,” Phys. Rev. Lett. 70, 774–777 (1993).
    [CrossRef]
  4. K. Kondo, N. Sarukura, K. Sajiki, and S. Watanabe, “High-order harmonic generation by ultrashort KrF and Ti:sapphire lasers,” Phys. Rev. A 47, R2480–R2483 (1993).
    [CrossRef] [PubMed]
  5. S. G. Preston, A. Sanpera, M. Zepf, W. J. Blyth, C. G. Smith, J. S. Wark, M. H. Key, K. Burnett, M. Nakai, D. Neely, and A. A. Offenberger, “High-order harmonics of 248.6-nm KrF laser from helium and neon ions,” Phys. Rev. A 53, R31–R34 (1996).
    [CrossRef] [PubMed]
  6. I. P. Christov, J. Zhou, J. Peatross, A. Rumdequist, M. M. Murname, and H. C. Kapteyn, “Nonadiabatic effects in high-harmonic generation with ultrashort pulses,” Phys. Rev. Lett. 77, 1743–1746 (1996).
    [CrossRef] [PubMed]
  7. E. Charron, A. Giusti-Suzor, and F. H. Mies, “Two-color coherent control of H2+ photodissociation in intense laser fields,” Phys. Rev. Lett. 71, 692–695 (1993).
    [CrossRef] [PubMed]
  8. M. Protopapas, P. L. Knight, and K. Burnett, “Phase dependence in two-color excitation of a model atom by intense fields,” Phys. Rev. A 49, 1945–1949 (1994).
    [CrossRef] [PubMed]
  9. D. W. Schumacher, F. Weihe, H. G. Muller, and P. H. Bucksbaum, “Phase dependence of intense field ionization: a study using two colors,” Phys. Rev. Lett. 73, 1344–1347 (1994).
    [CrossRef] [PubMed]
  10. K. Kondo, T. Tamida, Y. Nabekawa, and S. Watanabe, “High-order harmonic generation and ionization using ultrashort KrF and Ti:sapphire lasers,” Phys. Rev. A 49, 3881–3889 (1994).
    [CrossRef] [PubMed]
  11. W.-C. Liu and C. W. Clark, “Intensity dependence of the phase of harmonics in one- and two-frequency laser fields,” Phys. Rev. A 53, 3582–3586 (1996).
    [CrossRef] [PubMed]
  12. D. B. Milosevic and B. Piraux, “High-order harmonic generation in a bichromatic elliptically polarized laser field,” Phys. Rev. A 54, 1522–1531 (1996).
    [CrossRef] [PubMed]
  13. M. B. Gaarde, A. L’Huillier, and M. Lewenstein, “Theory of high-order sum and difference frequency mixing in a strong bichromatic laser field,” Phys. Rev. A 54, 4236–4248 (1996).
    [CrossRef] [PubMed]
  14. S. Varro and F. Ehlotzky, “Higher harmonic generation at metal surfaces by powerful bichromatic laser fields,” Phys. Rev. A 56, 2439–2442 (1997).
    [CrossRef]
  15. J. Z. Kaminski and F. Ehlotzky, “Phase-dependent sidelobes in high-order above-threshold ionization in a bichromatic laser field,” J. Phys. B 30, 5729–5735 (1997).
    [CrossRef]
  16. V. T. Platonenko and V. V. Strelkov, “High-order harmonic generation in the field of intense laser radiation,” Quantum Electron. 28, 564–583 (1998).
    [CrossRef]
  17. X.-M. Tong and S.-I. Chu, “Generation of circularly polarized multiple high-order harmonic emission from two-color crossed laser beams,” Phys. Rev. A 58, R2656–R2659 (1998).
    [CrossRef]
  18. C. Figueira de Morisson Faria, M. Dorr, W. Becker, and W. Sandner, “Time-frequency analysis of two-color high-harmonic generation,” Phys. Rev. A 60, 1377–1384 (1999).
    [CrossRef]
  19. W. Becker, B. N. Chichkov, and B. Wellegehausen, “Schemes for the generation of circularly polarized high-order harmonics by two-color mixing,” Phys. Rev. A 60, 1721–1722 (1999).
    [CrossRef]
  20. V. P. Silin, “Nonlinear high-frequency plasma conductivity,” Sov. Phys. JETP 20, 1510–1516 (1965).
  21. G. Ferrante, S. A. Uryupin, M. Zarcone, and P. I. Porshnev, “Harmonic generation in a plasma with laser-heated electron velocity distributions,” J. Opt. Soc. Am. B 14, 1716–1722 (1997).
    [CrossRef]
  22. V. P. Silin, “On the coherent bremsstrahlung radiation of harmonics in a laser plasma,” JETP 87, 486–494 (1998).
    [CrossRef]
  23. V. P. Silin, “Coherent bremsstrahlung generation of harmonics in a laser-produced plasma,” Quantum Electron. 29, 11–18 (1999).
    [CrossRef]
  24. V. L. Ginzburg, The Propagation of Electromagnetic Waves in a Plasma (Pergamon, New York, 1970).
  25. S. Nuzzo, M. Zarcone, G. Ferrante, and S. Basile, “Elementary kinetic theory of harmonics generation and wave mixing,” Laser Phys. 10, 254–261 (2000).
  26. B. N. Chichkov, S. A. Shumsky, and S. A. Uryupin, “Nonstationary electron distribution functions in a laser field,” Phys. Rev. A 45, 7475–7479 (1992).
    [CrossRef] [PubMed]

2000 (1)

S. Nuzzo, M. Zarcone, G. Ferrante, and S. Basile, “Elementary kinetic theory of harmonics generation and wave mixing,” Laser Phys. 10, 254–261 (2000).

1999 (3)

C. Figueira de Morisson Faria, M. Dorr, W. Becker, and W. Sandner, “Time-frequency analysis of two-color high-harmonic generation,” Phys. Rev. A 60, 1377–1384 (1999).
[CrossRef]

W. Becker, B. N. Chichkov, and B. Wellegehausen, “Schemes for the generation of circularly polarized high-order harmonics by two-color mixing,” Phys. Rev. A 60, 1721–1722 (1999).
[CrossRef]

V. P. Silin, “Coherent bremsstrahlung generation of harmonics in a laser-produced plasma,” Quantum Electron. 29, 11–18 (1999).
[CrossRef]

1998 (3)

V. P. Silin, “On the coherent bremsstrahlung radiation of harmonics in a laser plasma,” JETP 87, 486–494 (1998).
[CrossRef]

V. T. Platonenko and V. V. Strelkov, “High-order harmonic generation in the field of intense laser radiation,” Quantum Electron. 28, 564–583 (1998).
[CrossRef]

X.-M. Tong and S.-I. Chu, “Generation of circularly polarized multiple high-order harmonic emission from two-color crossed laser beams,” Phys. Rev. A 58, R2656–R2659 (1998).
[CrossRef]

1997 (3)

S. Varro and F. Ehlotzky, “Higher harmonic generation at metal surfaces by powerful bichromatic laser fields,” Phys. Rev. A 56, 2439–2442 (1997).
[CrossRef]

J. Z. Kaminski and F. Ehlotzky, “Phase-dependent sidelobes in high-order above-threshold ionization in a bichromatic laser field,” J. Phys. B 30, 5729–5735 (1997).
[CrossRef]

G. Ferrante, S. A. Uryupin, M. Zarcone, and P. I. Porshnev, “Harmonic generation in a plasma with laser-heated electron velocity distributions,” J. Opt. Soc. Am. B 14, 1716–1722 (1997).
[CrossRef]

1996 (5)

S. G. Preston, A. Sanpera, M. Zepf, W. J. Blyth, C. G. Smith, J. S. Wark, M. H. Key, K. Burnett, M. Nakai, D. Neely, and A. A. Offenberger, “High-order harmonics of 248.6-nm KrF laser from helium and neon ions,” Phys. Rev. A 53, R31–R34 (1996).
[CrossRef] [PubMed]

I. P. Christov, J. Zhou, J. Peatross, A. Rumdequist, M. M. Murname, and H. C. Kapteyn, “Nonadiabatic effects in high-harmonic generation with ultrashort pulses,” Phys. Rev. Lett. 77, 1743–1746 (1996).
[CrossRef] [PubMed]

W.-C. Liu and C. W. Clark, “Intensity dependence of the phase of harmonics in one- and two-frequency laser fields,” Phys. Rev. A 53, 3582–3586 (1996).
[CrossRef] [PubMed]

D. B. Milosevic and B. Piraux, “High-order harmonic generation in a bichromatic elliptically polarized laser field,” Phys. Rev. A 54, 1522–1531 (1996).
[CrossRef] [PubMed]

M. B. Gaarde, A. L’Huillier, and M. Lewenstein, “Theory of high-order sum and difference frequency mixing in a strong bichromatic laser field,” Phys. Rev. A 54, 4236–4248 (1996).
[CrossRef] [PubMed]

1994 (3)

M. Protopapas, P. L. Knight, and K. Burnett, “Phase dependence in two-color excitation of a model atom by intense fields,” Phys. Rev. A 49, 1945–1949 (1994).
[CrossRef] [PubMed]

D. W. Schumacher, F. Weihe, H. G. Muller, and P. H. Bucksbaum, “Phase dependence of intense field ionization: a study using two colors,” Phys. Rev. Lett. 73, 1344–1347 (1994).
[CrossRef] [PubMed]

K. Kondo, T. Tamida, Y. Nabekawa, and S. Watanabe, “High-order harmonic generation and ionization using ultrashort KrF and Ti:sapphire lasers,” Phys. Rev. A 49, 3881–3889 (1994).
[CrossRef] [PubMed]

1993 (3)

E. Charron, A. Giusti-Suzor, and F. H. Mies, “Two-color coherent control of H2+ photodissociation in intense laser fields,” Phys. Rev. Lett. 71, 692–695 (1993).
[CrossRef] [PubMed]

A. L’Huiller and Ph. Balcou, “High-order harmonic generation in rare gases with a 1-ps 1053-nm laser,” Phys. Rev. Lett. 70, 774–777 (1993).
[CrossRef]

K. Kondo, N. Sarukura, K. Sajiki, and S. Watanabe, “High-order harmonic generation by ultrashort KrF and Ti:sapphire lasers,” Phys. Rev. A 47, R2480–R2483 (1993).
[CrossRef] [PubMed]

1992 (3)

B. N. Chichkov, S. A. Shumsky, and S. A. Uryupin, “Nonstationary electron distribution functions in a laser field,” Phys. Rev. A 45, 7475–7479 (1992).
[CrossRef] [PubMed]

J. L. Krause, K. J. Schafer, and K. C. Kulander, “High-order harmonic generation from atoms and ions in the high intensity regime,” Phys. Rev. Lett. 68, 3535–3538 (1992).
[CrossRef] [PubMed]

J. K. Crane, M. D. Perry, S. Herman, and R. W. Falcone, “High-order harmonic generation in helium,” Opt. Lett. 17, 1256–1258 (1992).
[CrossRef] [PubMed]

1965 (1)

V. P. Silin, “Nonlinear high-frequency plasma conductivity,” Sov. Phys. JETP 20, 1510–1516 (1965).

Balcou, Ph.

A. L’Huiller and Ph. Balcou, “High-order harmonic generation in rare gases with a 1-ps 1053-nm laser,” Phys. Rev. Lett. 70, 774–777 (1993).
[CrossRef]

Basile, S.

S. Nuzzo, M. Zarcone, G. Ferrante, and S. Basile, “Elementary kinetic theory of harmonics generation and wave mixing,” Laser Phys. 10, 254–261 (2000).

Becker, W.

C. Figueira de Morisson Faria, M. Dorr, W. Becker, and W. Sandner, “Time-frequency analysis of two-color high-harmonic generation,” Phys. Rev. A 60, 1377–1384 (1999).
[CrossRef]

W. Becker, B. N. Chichkov, and B. Wellegehausen, “Schemes for the generation of circularly polarized high-order harmonics by two-color mixing,” Phys. Rev. A 60, 1721–1722 (1999).
[CrossRef]

Blyth, W. J.

S. G. Preston, A. Sanpera, M. Zepf, W. J. Blyth, C. G. Smith, J. S. Wark, M. H. Key, K. Burnett, M. Nakai, D. Neely, and A. A. Offenberger, “High-order harmonics of 248.6-nm KrF laser from helium and neon ions,” Phys. Rev. A 53, R31–R34 (1996).
[CrossRef] [PubMed]

Bucksbaum, P. H.

D. W. Schumacher, F. Weihe, H. G. Muller, and P. H. Bucksbaum, “Phase dependence of intense field ionization: a study using two colors,” Phys. Rev. Lett. 73, 1344–1347 (1994).
[CrossRef] [PubMed]

Burnett, K.

S. G. Preston, A. Sanpera, M. Zepf, W. J. Blyth, C. G. Smith, J. S. Wark, M. H. Key, K. Burnett, M. Nakai, D. Neely, and A. A. Offenberger, “High-order harmonics of 248.6-nm KrF laser from helium and neon ions,” Phys. Rev. A 53, R31–R34 (1996).
[CrossRef] [PubMed]

M. Protopapas, P. L. Knight, and K. Burnett, “Phase dependence in two-color excitation of a model atom by intense fields,” Phys. Rev. A 49, 1945–1949 (1994).
[CrossRef] [PubMed]

Charron, E.

E. Charron, A. Giusti-Suzor, and F. H. Mies, “Two-color coherent control of H2+ photodissociation in intense laser fields,” Phys. Rev. Lett. 71, 692–695 (1993).
[CrossRef] [PubMed]

Chichkov, B. N.

W. Becker, B. N. Chichkov, and B. Wellegehausen, “Schemes for the generation of circularly polarized high-order harmonics by two-color mixing,” Phys. Rev. A 60, 1721–1722 (1999).
[CrossRef]

B. N. Chichkov, S. A. Shumsky, and S. A. Uryupin, “Nonstationary electron distribution functions in a laser field,” Phys. Rev. A 45, 7475–7479 (1992).
[CrossRef] [PubMed]

Christov, I. P.

I. P. Christov, J. Zhou, J. Peatross, A. Rumdequist, M. M. Murname, and H. C. Kapteyn, “Nonadiabatic effects in high-harmonic generation with ultrashort pulses,” Phys. Rev. Lett. 77, 1743–1746 (1996).
[CrossRef] [PubMed]

Chu, S.-I.

X.-M. Tong and S.-I. Chu, “Generation of circularly polarized multiple high-order harmonic emission from two-color crossed laser beams,” Phys. Rev. A 58, R2656–R2659 (1998).
[CrossRef]

Clark, C. W.

W.-C. Liu and C. W. Clark, “Intensity dependence of the phase of harmonics in one- and two-frequency laser fields,” Phys. Rev. A 53, 3582–3586 (1996).
[CrossRef] [PubMed]

Crane, J. K.

Dorr, M.

C. Figueira de Morisson Faria, M. Dorr, W. Becker, and W. Sandner, “Time-frequency analysis of two-color high-harmonic generation,” Phys. Rev. A 60, 1377–1384 (1999).
[CrossRef]

Ehlotzky, F.

J. Z. Kaminski and F. Ehlotzky, “Phase-dependent sidelobes in high-order above-threshold ionization in a bichromatic laser field,” J. Phys. B 30, 5729–5735 (1997).
[CrossRef]

S. Varro and F. Ehlotzky, “Higher harmonic generation at metal surfaces by powerful bichromatic laser fields,” Phys. Rev. A 56, 2439–2442 (1997).
[CrossRef]

Falcone, R. W.

Ferrante, G.

S. Nuzzo, M. Zarcone, G. Ferrante, and S. Basile, “Elementary kinetic theory of harmonics generation and wave mixing,” Laser Phys. 10, 254–261 (2000).

G. Ferrante, S. A. Uryupin, M. Zarcone, and P. I. Porshnev, “Harmonic generation in a plasma with laser-heated electron velocity distributions,” J. Opt. Soc. Am. B 14, 1716–1722 (1997).
[CrossRef]

Figueira de Morisson Faria, C.

C. Figueira de Morisson Faria, M. Dorr, W. Becker, and W. Sandner, “Time-frequency analysis of two-color high-harmonic generation,” Phys. Rev. A 60, 1377–1384 (1999).
[CrossRef]

Gaarde, M. B.

M. B. Gaarde, A. L’Huillier, and M. Lewenstein, “Theory of high-order sum and difference frequency mixing in a strong bichromatic laser field,” Phys. Rev. A 54, 4236–4248 (1996).
[CrossRef] [PubMed]

Giusti-Suzor, A.

E. Charron, A. Giusti-Suzor, and F. H. Mies, “Two-color coherent control of H2+ photodissociation in intense laser fields,” Phys. Rev. Lett. 71, 692–695 (1993).
[CrossRef] [PubMed]

Herman, S.

Kaminski, J. Z.

J. Z. Kaminski and F. Ehlotzky, “Phase-dependent sidelobes in high-order above-threshold ionization in a bichromatic laser field,” J. Phys. B 30, 5729–5735 (1997).
[CrossRef]

Kapteyn, H. C.

I. P. Christov, J. Zhou, J. Peatross, A. Rumdequist, M. M. Murname, and H. C. Kapteyn, “Nonadiabatic effects in high-harmonic generation with ultrashort pulses,” Phys. Rev. Lett. 77, 1743–1746 (1996).
[CrossRef] [PubMed]

Key, M. H.

S. G. Preston, A. Sanpera, M. Zepf, W. J. Blyth, C. G. Smith, J. S. Wark, M. H. Key, K. Burnett, M. Nakai, D. Neely, and A. A. Offenberger, “High-order harmonics of 248.6-nm KrF laser from helium and neon ions,” Phys. Rev. A 53, R31–R34 (1996).
[CrossRef] [PubMed]

Knight, P. L.

M. Protopapas, P. L. Knight, and K. Burnett, “Phase dependence in two-color excitation of a model atom by intense fields,” Phys. Rev. A 49, 1945–1949 (1994).
[CrossRef] [PubMed]

Kondo, K.

K. Kondo, T. Tamida, Y. Nabekawa, and S. Watanabe, “High-order harmonic generation and ionization using ultrashort KrF and Ti:sapphire lasers,” Phys. Rev. A 49, 3881–3889 (1994).
[CrossRef] [PubMed]

K. Kondo, N. Sarukura, K. Sajiki, and S. Watanabe, “High-order harmonic generation by ultrashort KrF and Ti:sapphire lasers,” Phys. Rev. A 47, R2480–R2483 (1993).
[CrossRef] [PubMed]

Krause, J. L.

J. L. Krause, K. J. Schafer, and K. C. Kulander, “High-order harmonic generation from atoms and ions in the high intensity regime,” Phys. Rev. Lett. 68, 3535–3538 (1992).
[CrossRef] [PubMed]

Kulander, K. C.

J. L. Krause, K. J. Schafer, and K. C. Kulander, “High-order harmonic generation from atoms and ions in the high intensity regime,” Phys. Rev. Lett. 68, 3535–3538 (1992).
[CrossRef] [PubMed]

L’Huiller, A.

A. L’Huiller and Ph. Balcou, “High-order harmonic generation in rare gases with a 1-ps 1053-nm laser,” Phys. Rev. Lett. 70, 774–777 (1993).
[CrossRef]

L’Huillier, A.

M. B. Gaarde, A. L’Huillier, and M. Lewenstein, “Theory of high-order sum and difference frequency mixing in a strong bichromatic laser field,” Phys. Rev. A 54, 4236–4248 (1996).
[CrossRef] [PubMed]

Lewenstein, M.

M. B. Gaarde, A. L’Huillier, and M. Lewenstein, “Theory of high-order sum and difference frequency mixing in a strong bichromatic laser field,” Phys. Rev. A 54, 4236–4248 (1996).
[CrossRef] [PubMed]

Liu, W.-C.

W.-C. Liu and C. W. Clark, “Intensity dependence of the phase of harmonics in one- and two-frequency laser fields,” Phys. Rev. A 53, 3582–3586 (1996).
[CrossRef] [PubMed]

Mies, F. H.

E. Charron, A. Giusti-Suzor, and F. H. Mies, “Two-color coherent control of H2+ photodissociation in intense laser fields,” Phys. Rev. Lett. 71, 692–695 (1993).
[CrossRef] [PubMed]

Milosevic, D. B.

D. B. Milosevic and B. Piraux, “High-order harmonic generation in a bichromatic elliptically polarized laser field,” Phys. Rev. A 54, 1522–1531 (1996).
[CrossRef] [PubMed]

Muller, H. G.

D. W. Schumacher, F. Weihe, H. G. Muller, and P. H. Bucksbaum, “Phase dependence of intense field ionization: a study using two colors,” Phys. Rev. Lett. 73, 1344–1347 (1994).
[CrossRef] [PubMed]

Murname, M. M.

I. P. Christov, J. Zhou, J. Peatross, A. Rumdequist, M. M. Murname, and H. C. Kapteyn, “Nonadiabatic effects in high-harmonic generation with ultrashort pulses,” Phys. Rev. Lett. 77, 1743–1746 (1996).
[CrossRef] [PubMed]

Nabekawa, Y.

K. Kondo, T. Tamida, Y. Nabekawa, and S. Watanabe, “High-order harmonic generation and ionization using ultrashort KrF and Ti:sapphire lasers,” Phys. Rev. A 49, 3881–3889 (1994).
[CrossRef] [PubMed]

Nakai, M.

S. G. Preston, A. Sanpera, M. Zepf, W. J. Blyth, C. G. Smith, J. S. Wark, M. H. Key, K. Burnett, M. Nakai, D. Neely, and A. A. Offenberger, “High-order harmonics of 248.6-nm KrF laser from helium and neon ions,” Phys. Rev. A 53, R31–R34 (1996).
[CrossRef] [PubMed]

Neely, D.

S. G. Preston, A. Sanpera, M. Zepf, W. J. Blyth, C. G. Smith, J. S. Wark, M. H. Key, K. Burnett, M. Nakai, D. Neely, and A. A. Offenberger, “High-order harmonics of 248.6-nm KrF laser from helium and neon ions,” Phys. Rev. A 53, R31–R34 (1996).
[CrossRef] [PubMed]

Nuzzo, S.

S. Nuzzo, M. Zarcone, G. Ferrante, and S. Basile, “Elementary kinetic theory of harmonics generation and wave mixing,” Laser Phys. 10, 254–261 (2000).

Offenberger, A. A.

S. G. Preston, A. Sanpera, M. Zepf, W. J. Blyth, C. G. Smith, J. S. Wark, M. H. Key, K. Burnett, M. Nakai, D. Neely, and A. A. Offenberger, “High-order harmonics of 248.6-nm KrF laser from helium and neon ions,” Phys. Rev. A 53, R31–R34 (1996).
[CrossRef] [PubMed]

Peatross, J.

I. P. Christov, J. Zhou, J. Peatross, A. Rumdequist, M. M. Murname, and H. C. Kapteyn, “Nonadiabatic effects in high-harmonic generation with ultrashort pulses,” Phys. Rev. Lett. 77, 1743–1746 (1996).
[CrossRef] [PubMed]

Perry, M. D.

Piraux, B.

D. B. Milosevic and B. Piraux, “High-order harmonic generation in a bichromatic elliptically polarized laser field,” Phys. Rev. A 54, 1522–1531 (1996).
[CrossRef] [PubMed]

Platonenko, V. T.

V. T. Platonenko and V. V. Strelkov, “High-order harmonic generation in the field of intense laser radiation,” Quantum Electron. 28, 564–583 (1998).
[CrossRef]

Porshnev, P. I.

Preston, S. G.

S. G. Preston, A. Sanpera, M. Zepf, W. J. Blyth, C. G. Smith, J. S. Wark, M. H. Key, K. Burnett, M. Nakai, D. Neely, and A. A. Offenberger, “High-order harmonics of 248.6-nm KrF laser from helium and neon ions,” Phys. Rev. A 53, R31–R34 (1996).
[CrossRef] [PubMed]

Protopapas, M.

M. Protopapas, P. L. Knight, and K. Burnett, “Phase dependence in two-color excitation of a model atom by intense fields,” Phys. Rev. A 49, 1945–1949 (1994).
[CrossRef] [PubMed]

Rumdequist, A.

I. P. Christov, J. Zhou, J. Peatross, A. Rumdequist, M. M. Murname, and H. C. Kapteyn, “Nonadiabatic effects in high-harmonic generation with ultrashort pulses,” Phys. Rev. Lett. 77, 1743–1746 (1996).
[CrossRef] [PubMed]

Sajiki, K.

K. Kondo, N. Sarukura, K. Sajiki, and S. Watanabe, “High-order harmonic generation by ultrashort KrF and Ti:sapphire lasers,” Phys. Rev. A 47, R2480–R2483 (1993).
[CrossRef] [PubMed]

Sandner, W.

C. Figueira de Morisson Faria, M. Dorr, W. Becker, and W. Sandner, “Time-frequency analysis of two-color high-harmonic generation,” Phys. Rev. A 60, 1377–1384 (1999).
[CrossRef]

Sanpera, A.

S. G. Preston, A. Sanpera, M. Zepf, W. J. Blyth, C. G. Smith, J. S. Wark, M. H. Key, K. Burnett, M. Nakai, D. Neely, and A. A. Offenberger, “High-order harmonics of 248.6-nm KrF laser from helium and neon ions,” Phys. Rev. A 53, R31–R34 (1996).
[CrossRef] [PubMed]

Sarukura, N.

K. Kondo, N. Sarukura, K. Sajiki, and S. Watanabe, “High-order harmonic generation by ultrashort KrF and Ti:sapphire lasers,” Phys. Rev. A 47, R2480–R2483 (1993).
[CrossRef] [PubMed]

Schafer, K. J.

J. L. Krause, K. J. Schafer, and K. C. Kulander, “High-order harmonic generation from atoms and ions in the high intensity regime,” Phys. Rev. Lett. 68, 3535–3538 (1992).
[CrossRef] [PubMed]

Schumacher, D. W.

D. W. Schumacher, F. Weihe, H. G. Muller, and P. H. Bucksbaum, “Phase dependence of intense field ionization: a study using two colors,” Phys. Rev. Lett. 73, 1344–1347 (1994).
[CrossRef] [PubMed]

Shumsky, S. A.

B. N. Chichkov, S. A. Shumsky, and S. A. Uryupin, “Nonstationary electron distribution functions in a laser field,” Phys. Rev. A 45, 7475–7479 (1992).
[CrossRef] [PubMed]

Silin, V. P.

V. P. Silin, “Coherent bremsstrahlung generation of harmonics in a laser-produced plasma,” Quantum Electron. 29, 11–18 (1999).
[CrossRef]

V. P. Silin, “On the coherent bremsstrahlung radiation of harmonics in a laser plasma,” JETP 87, 486–494 (1998).
[CrossRef]

V. P. Silin, “Nonlinear high-frequency plasma conductivity,” Sov. Phys. JETP 20, 1510–1516 (1965).

Smith, C. G.

S. G. Preston, A. Sanpera, M. Zepf, W. J. Blyth, C. G. Smith, J. S. Wark, M. H. Key, K. Burnett, M. Nakai, D. Neely, and A. A. Offenberger, “High-order harmonics of 248.6-nm KrF laser from helium and neon ions,” Phys. Rev. A 53, R31–R34 (1996).
[CrossRef] [PubMed]

Strelkov, V. V.

V. T. Platonenko and V. V. Strelkov, “High-order harmonic generation in the field of intense laser radiation,” Quantum Electron. 28, 564–583 (1998).
[CrossRef]

Tamida, T.

K. Kondo, T. Tamida, Y. Nabekawa, and S. Watanabe, “High-order harmonic generation and ionization using ultrashort KrF and Ti:sapphire lasers,” Phys. Rev. A 49, 3881–3889 (1994).
[CrossRef] [PubMed]

Tong, X.-M.

X.-M. Tong and S.-I. Chu, “Generation of circularly polarized multiple high-order harmonic emission from two-color crossed laser beams,” Phys. Rev. A 58, R2656–R2659 (1998).
[CrossRef]

Uryupin, S. A.

G. Ferrante, S. A. Uryupin, M. Zarcone, and P. I. Porshnev, “Harmonic generation in a plasma with laser-heated electron velocity distributions,” J. Opt. Soc. Am. B 14, 1716–1722 (1997).
[CrossRef]

B. N. Chichkov, S. A. Shumsky, and S. A. Uryupin, “Nonstationary electron distribution functions in a laser field,” Phys. Rev. A 45, 7475–7479 (1992).
[CrossRef] [PubMed]

Varro, S.

S. Varro and F. Ehlotzky, “Higher harmonic generation at metal surfaces by powerful bichromatic laser fields,” Phys. Rev. A 56, 2439–2442 (1997).
[CrossRef]

Wark, J. S.

S. G. Preston, A. Sanpera, M. Zepf, W. J. Blyth, C. G. Smith, J. S. Wark, M. H. Key, K. Burnett, M. Nakai, D. Neely, and A. A. Offenberger, “High-order harmonics of 248.6-nm KrF laser from helium and neon ions,” Phys. Rev. A 53, R31–R34 (1996).
[CrossRef] [PubMed]

Watanabe, S.

K. Kondo, T. Tamida, Y. Nabekawa, and S. Watanabe, “High-order harmonic generation and ionization using ultrashort KrF and Ti:sapphire lasers,” Phys. Rev. A 49, 3881–3889 (1994).
[CrossRef] [PubMed]

K. Kondo, N. Sarukura, K. Sajiki, and S. Watanabe, “High-order harmonic generation by ultrashort KrF and Ti:sapphire lasers,” Phys. Rev. A 47, R2480–R2483 (1993).
[CrossRef] [PubMed]

Weihe, F.

D. W. Schumacher, F. Weihe, H. G. Muller, and P. H. Bucksbaum, “Phase dependence of intense field ionization: a study using two colors,” Phys. Rev. Lett. 73, 1344–1347 (1994).
[CrossRef] [PubMed]

Wellegehausen, B.

W. Becker, B. N. Chichkov, and B. Wellegehausen, “Schemes for the generation of circularly polarized high-order harmonics by two-color mixing,” Phys. Rev. A 60, 1721–1722 (1999).
[CrossRef]

Zarcone, M.

S. Nuzzo, M. Zarcone, G. Ferrante, and S. Basile, “Elementary kinetic theory of harmonics generation and wave mixing,” Laser Phys. 10, 254–261 (2000).

G. Ferrante, S. A. Uryupin, M. Zarcone, and P. I. Porshnev, “Harmonic generation in a plasma with laser-heated electron velocity distributions,” J. Opt. Soc. Am. B 14, 1716–1722 (1997).
[CrossRef]

Zepf, M.

S. G. Preston, A. Sanpera, M. Zepf, W. J. Blyth, C. G. Smith, J. S. Wark, M. H. Key, K. Burnett, M. Nakai, D. Neely, and A. A. Offenberger, “High-order harmonics of 248.6-nm KrF laser from helium and neon ions,” Phys. Rev. A 53, R31–R34 (1996).
[CrossRef] [PubMed]

Zhou, J.

I. P. Christov, J. Zhou, J. Peatross, A. Rumdequist, M. M. Murname, and H. C. Kapteyn, “Nonadiabatic effects in high-harmonic generation with ultrashort pulses,” Phys. Rev. Lett. 77, 1743–1746 (1996).
[CrossRef] [PubMed]

J. Opt. Soc. Am. B (1)

J. Phys. B (1)

J. Z. Kaminski and F. Ehlotzky, “Phase-dependent sidelobes in high-order above-threshold ionization in a bichromatic laser field,” J. Phys. B 30, 5729–5735 (1997).
[CrossRef]

JETP (1)

V. P. Silin, “On the coherent bremsstrahlung radiation of harmonics in a laser plasma,” JETP 87, 486–494 (1998).
[CrossRef]

Laser Phys. (1)

S. Nuzzo, M. Zarcone, G. Ferrante, and S. Basile, “Elementary kinetic theory of harmonics generation and wave mixing,” Laser Phys. 10, 254–261 (2000).

Opt. Lett. (1)

Phys. Rev. A (12)

B. N. Chichkov, S. A. Shumsky, and S. A. Uryupin, “Nonstationary electron distribution functions in a laser field,” Phys. Rev. A 45, 7475–7479 (1992).
[CrossRef] [PubMed]

M. Protopapas, P. L. Knight, and K. Burnett, “Phase dependence in two-color excitation of a model atom by intense fields,” Phys. Rev. A 49, 1945–1949 (1994).
[CrossRef] [PubMed]

X.-M. Tong and S.-I. Chu, “Generation of circularly polarized multiple high-order harmonic emission from two-color crossed laser beams,” Phys. Rev. A 58, R2656–R2659 (1998).
[CrossRef]

C. Figueira de Morisson Faria, M. Dorr, W. Becker, and W. Sandner, “Time-frequency analysis of two-color high-harmonic generation,” Phys. Rev. A 60, 1377–1384 (1999).
[CrossRef]

W. Becker, B. N. Chichkov, and B. Wellegehausen, “Schemes for the generation of circularly polarized high-order harmonics by two-color mixing,” Phys. Rev. A 60, 1721–1722 (1999).
[CrossRef]

K. Kondo, N. Sarukura, K. Sajiki, and S. Watanabe, “High-order harmonic generation by ultrashort KrF and Ti:sapphire lasers,” Phys. Rev. A 47, R2480–R2483 (1993).
[CrossRef] [PubMed]

S. G. Preston, A. Sanpera, M. Zepf, W. J. Blyth, C. G. Smith, J. S. Wark, M. H. Key, K. Burnett, M. Nakai, D. Neely, and A. A. Offenberger, “High-order harmonics of 248.6-nm KrF laser from helium and neon ions,” Phys. Rev. A 53, R31–R34 (1996).
[CrossRef] [PubMed]

K. Kondo, T. Tamida, Y. Nabekawa, and S. Watanabe, “High-order harmonic generation and ionization using ultrashort KrF and Ti:sapphire lasers,” Phys. Rev. A 49, 3881–3889 (1994).
[CrossRef] [PubMed]

W.-C. Liu and C. W. Clark, “Intensity dependence of the phase of harmonics in one- and two-frequency laser fields,” Phys. Rev. A 53, 3582–3586 (1996).
[CrossRef] [PubMed]

D. B. Milosevic and B. Piraux, “High-order harmonic generation in a bichromatic elliptically polarized laser field,” Phys. Rev. A 54, 1522–1531 (1996).
[CrossRef] [PubMed]

M. B. Gaarde, A. L’Huillier, and M. Lewenstein, “Theory of high-order sum and difference frequency mixing in a strong bichromatic laser field,” Phys. Rev. A 54, 4236–4248 (1996).
[CrossRef] [PubMed]

S. Varro and F. Ehlotzky, “Higher harmonic generation at metal surfaces by powerful bichromatic laser fields,” Phys. Rev. A 56, 2439–2442 (1997).
[CrossRef]

Phys. Rev. Lett. (5)

I. P. Christov, J. Zhou, J. Peatross, A. Rumdequist, M. M. Murname, and H. C. Kapteyn, “Nonadiabatic effects in high-harmonic generation with ultrashort pulses,” Phys. Rev. Lett. 77, 1743–1746 (1996).
[CrossRef] [PubMed]

E. Charron, A. Giusti-Suzor, and F. H. Mies, “Two-color coherent control of H2+ photodissociation in intense laser fields,” Phys. Rev. Lett. 71, 692–695 (1993).
[CrossRef] [PubMed]

D. W. Schumacher, F. Weihe, H. G. Muller, and P. H. Bucksbaum, “Phase dependence of intense field ionization: a study using two colors,” Phys. Rev. Lett. 73, 1344–1347 (1994).
[CrossRef] [PubMed]

J. L. Krause, K. J. Schafer, and K. C. Kulander, “High-order harmonic generation from atoms and ions in the high intensity regime,” Phys. Rev. Lett. 68, 3535–3538 (1992).
[CrossRef] [PubMed]

A. L’Huiller and Ph. Balcou, “High-order harmonic generation in rare gases with a 1-ps 1053-nm laser,” Phys. Rev. Lett. 70, 774–777 (1993).
[CrossRef]

Quantum Electron. (2)

V. P. Silin, “Coherent bremsstrahlung generation of harmonics in a laser-produced plasma,” Quantum Electron. 29, 11–18 (1999).
[CrossRef]

V. T. Platonenko and V. V. Strelkov, “High-order harmonic generation in the field of intense laser radiation,” Quantum Electron. 28, 564–583 (1998).
[CrossRef]

Sov. Phys. JETP (1)

V. P. Silin, “Nonlinear high-frequency plasma conductivity,” Sov. Phys. JETP 20, 1510–1516 (1965).

Other (1)

V. L. Ginzburg, The Propagation of Electromagnetic Waves in a Plasma (Pergamon, New York, 1970).

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Figures (6)

Fig. 1
Fig. 1

(a) Relative intensity of the harmonic 3ω1 for γ1=3 as a function of the angle α between the fields E1 and E2 for four values of the parameter γ2: 1-γ2=0, 2-γ2=0.5, 3-γ2=1, and 4-γ2=2. (b) The angle between the field E(1, 0) of the harmonic with frequency 3ω1 and the field E1 for γ1=3 as a function of the angle α between the fields E1 and E2, for four values of the parameter γ2: 1-γ2=0, 2-γ2=0.5, 3-γ2=1, and 4-γ2=2.

Fig. 2
Fig. 2

Relative intensity of the harmonic 5ω1 for γ1=3 as a function of the angle α between the fields E1 and E2, for the same values of γ2 as in Fig. 1(a). (b) The angle between the field E(2, 0) of the harmonic with frequency 5ω1 and the field E1 for γ1=3 as a function of the angle α between the fields E1 and E2, for the same values of γ2 as in Fig. 1(a).

Fig. 3
Fig. 3

Angle between the field E(n, 0) of the harmonic with frequency (2n+1)ω1 and the field E1 for γ1=3 and γ2=2 as a function of the harmonic order n, for two values of α: (triangles) α=π/18 and (squares) α=π/6.

Fig. 4
Fig. 4

(a) Intensity of the satellites with frequencies |ω1+2mω2| for γ1=3, α=π/2, and for three values of the parameter γ2: (solid circles) γ2=1, (squares) γ2=2, and (hollow circles) γ2=4. (b) The intensity of the satellites with frequencies |ω1+2mω2| for γ1=3, γ2=2, and for three values of the angle α: (triangles) α=π/18, (squares) α=π/6, and (hollow circles) α=π/2.

Fig. 5
Fig. 5

Intensity of the satellites with frequencies |3ω1+2mω2| for γ1=3, α=π/2, and for three values of the parameter γ2: (solid circles) γ2=1, (squares) γ2=2, and (hollow circles) γ2=4.

Fig. 6
Fig. 6

Comparison of the intensities of the odd harmonics (2n+1)ω1 [one field only, γ1=3, and γ2=0 (squares)] with the intensities of the satellites with combined frequencies Ω(0,-1)=3ω1, Ω(0, 1)=5ω1, Ω(0,-2)=7ω1, Ω(0, 2)=9ω1, and Ω(0,-3)=11ω1 (γ1=3, γ2=4) for two values of the angle α: (triangles) α=π/18 and (hollow circles) α=π/2.

Equations (26)

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E(t)=E1 cos(ω1t+δ1)+E2 cos(ω2t+δ2),
ft+em E(t) fv=12 ν(v) vi(v2δij-vivj) fvj,
ν(v)=4πZe4nm2v3Λ,
vE=vE(t)=eE1mω1 sin(ω1t+δ1)+eE2mω2 sin(ω2t+δ2)
fm(u)=n2π2πvT3 exp-u22vT2,
f=fm(v-vE)+δf
tδf+em E(t) vδf
=12 ν(v) vi(v2δij-vivj) vjfm(v-vE).
t j=en t vE+tδj,
tδj=12eduu ui{ν(|u+vE|)[(u+vE)2δij-(ui+vE, i)(uj+vE, j)]} ujfm(u).
tδj=envTνn=0m=-I(2n+1,2m,γ1,γ2,α)×sin[(2n+1)(ω1t+δ1)+2m(ω2t+δ2)]+m=0n=-I(2n,2m+1,γ1,γ2,α)×sin[2n(ω1t+δ1)+(2m+1)(ω2t+δ2)],
I(2n+1,2m,γ1,γ2,α)
=1π 0du02πdφu1-erfu2×J2n+1(uγ1 cos φ)J2m[uγ2 cos(φ-α)],
|(2n+1)ω1+2mω2|=Ω(n, m),
E(n, m)sin[(2n+1)(ω1t+δ1)+2m(ω2t+δ2)],
E(n, m)E1=νω1 G(n, m),
G(n, m)=-1γ1Δ(n, m) I(n, m),
Δ(n, m)=4n(n+1)+4m2+4m(2n+1) ω1ω2-k1k2c2ωL2.
ψ=ψ(n, m)=arctanGy(n, m)Gx(n, m)
Gx(α)=Gx(π-α)=Gx(π+α)=Gx(-α),
Gy(α)=-Gy(π-α)=Gy(π+α)=-Gy(-α).
E2(n, m)E12=νω12G2(n, m)1.5×10-4G2(n, m).
m=-[n/2]E(n-2m, m)sin[(2n+1)ω1t
+(2n+1-4m)δ1+2mδ2]
-m=-[-(n+1)/2]E(-1-n-2m, m)
×sin[(2n+1)ω1t+(2n+1+4m)δ1-2mδ2],

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