Abstract

In cases in which the ponderomotive energy is much larger than the ionization energy, which correspond to the highintensity limit or to the low-frequency limit, harmonic generation in gases near the ionization threshold can be understood from a plasma-physics point of view. Multiphoton ionization, which can be described by using the dc tunneling limit, takes place in a time interval localized around the maximum of the electric field. The plasma current thus generated varies on the same fast time scale and is responsible for harmonic generation. A model using those properties that has been developed shows the presence of a large number of odd harmonics.

© 1990 Optical Society of America

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  1. M. Ferray, A. L’Huillier, X. F. Li, L. A. Lompré, G. Mainfray, and C. Manus, “Multiple-harmonic conversion of 1064 nm radiation in rare gases,” J. Phys. B 21, L31 (1988);X. F. Li, A. L’Huillier, M. Ferray, L. A. Lompre, and G. Mainfray, “Multiple-harmonic generation in rare gases at high laser intensity,” Phys. Rev. A 39, 5751 (1989).
    [Crossref] [PubMed]
  2. A. McPherson, G. Gibson, H. Jara, U. Johan, T. S. Luk, I. A. Mclntyre, K. Boyer, and C. K. Rhodes, “Studies of multiphoton production of vacuum-ultraviolet radiation in the rare gases,” J. Opt. Soc. Am. B 4, 595 (1987).
    [Crossref]
  3. K. C. Kulander and B. W. Shore, “Calculation of multiple-harmonic conversion of 1064 nm radiation in Xe,” Phys. Rev. Lett. 62, 524 (1989).
    [Crossref] [PubMed]
  4. J. H. Eberly, Q. Su, and J. Javanainen, “Nonlinear light scattering accompanying multiphoton ionization,” Phys. Rev. Lett. 62, 881 (1989);“High-order harmonics in light scattering by atomic electrons above threshold,” in Short Wavelength Coherent Radiation: Generation and Application, R. W. Falcone and J. Kirz, eds. Proceedings of the topical meeting held September 26–29, 1988 (Optical Society of America, Washington, D.C., to be published).
    [Crossref] [PubMed]
  5. P. B. Corkum, N. H. Burnett, and F. Brunei, “Above-threshold ionization in the long wavelength limit,” Phys. Rev. Lett. 62, 1259 (1989).
    [Crossref] [PubMed]
  6. N. H. Burnett and P. B. Corkum, “Cold plasma production for recombination extreme ultraviolet lasers by optical field induced ionization,” J. Opt. Soc. Am. B 6, 1195 (1989).
    [Crossref]
  7. L. D. Landau and E. M. Lifshitz, Quantum Mechanics, 2nd ed. (Pergamon, New York, 1965), p. 276.
  8. T. J. Englert and E. A. Rinehart, “Second-harmonic photons from the interaction of free electrons with intense laser radiation,” Phys. Rev. A 28, 1539 (1983).
    [Crossref]
  9. P. Sprangle, A. Ting, and E. Esarey, “The nonlinear interaction of intense laser pulses in plasmas,” NRL Memo. Rep. 6545 (Naval Research Laboratory, Washington, D.C., 1989).
  10. T. Tajima and Y. C. Lee, “Absorbing boundary condition and Budden turning point technique for electromagnetic plasma simulations,” J. Comput. Phys. 42, 406 (1981).
    [Crossref]
  11. S. A. Akhmanov, S. Gladkov, and N. Koroteev, “New mechanism of higher-order optical harmonic generation in super-strong laser fields,” presented at the Conference on Super-Intense Laser–Atom Physics,University of Rochester, Rochester, New York, June 28–30, 1989.
  12. L. D. Landau and E. M. Lipshitz, Electrodynamics (Addison-Wesley, Reading, Mass., 1960), Sec. 70;R. Z. Sagdeev, D. A. Usikov, and G. M. Zaslavsky, Nonlinear Physics: From the Pendulum to Turbulence and Chaos (Harwood Academic, New York, 1988), p. 594.

1989 (4)

K. C. Kulander and B. W. Shore, “Calculation of multiple-harmonic conversion of 1064 nm radiation in Xe,” Phys. Rev. Lett. 62, 524 (1989).
[Crossref] [PubMed]

J. H. Eberly, Q. Su, and J. Javanainen, “Nonlinear light scattering accompanying multiphoton ionization,” Phys. Rev. Lett. 62, 881 (1989);“High-order harmonics in light scattering by atomic electrons above threshold,” in Short Wavelength Coherent Radiation: Generation and Application, R. W. Falcone and J. Kirz, eds. Proceedings of the topical meeting held September 26–29, 1988 (Optical Society of America, Washington, D.C., to be published).
[Crossref] [PubMed]

P. B. Corkum, N. H. Burnett, and F. Brunei, “Above-threshold ionization in the long wavelength limit,” Phys. Rev. Lett. 62, 1259 (1989).
[Crossref] [PubMed]

N. H. Burnett and P. B. Corkum, “Cold plasma production for recombination extreme ultraviolet lasers by optical field induced ionization,” J. Opt. Soc. Am. B 6, 1195 (1989).
[Crossref]

1988 (1)

M. Ferray, A. L’Huillier, X. F. Li, L. A. Lompré, G. Mainfray, and C. Manus, “Multiple-harmonic conversion of 1064 nm radiation in rare gases,” J. Phys. B 21, L31 (1988);X. F. Li, A. L’Huillier, M. Ferray, L. A. Lompre, and G. Mainfray, “Multiple-harmonic generation in rare gases at high laser intensity,” Phys. Rev. A 39, 5751 (1989).
[Crossref] [PubMed]

1987 (1)

1983 (1)

T. J. Englert and E. A. Rinehart, “Second-harmonic photons from the interaction of free electrons with intense laser radiation,” Phys. Rev. A 28, 1539 (1983).
[Crossref]

1981 (1)

T. Tajima and Y. C. Lee, “Absorbing boundary condition and Budden turning point technique for electromagnetic plasma simulations,” J. Comput. Phys. 42, 406 (1981).
[Crossref]

Akhmanov, S. A.

S. A. Akhmanov, S. Gladkov, and N. Koroteev, “New mechanism of higher-order optical harmonic generation in super-strong laser fields,” presented at the Conference on Super-Intense Laser–Atom Physics,University of Rochester, Rochester, New York, June 28–30, 1989.

Boyer, K.

Brunei, F.

P. B. Corkum, N. H. Burnett, and F. Brunei, “Above-threshold ionization in the long wavelength limit,” Phys. Rev. Lett. 62, 1259 (1989).
[Crossref] [PubMed]

Burnett, N. H.

N. H. Burnett and P. B. Corkum, “Cold plasma production for recombination extreme ultraviolet lasers by optical field induced ionization,” J. Opt. Soc. Am. B 6, 1195 (1989).
[Crossref]

P. B. Corkum, N. H. Burnett, and F. Brunei, “Above-threshold ionization in the long wavelength limit,” Phys. Rev. Lett. 62, 1259 (1989).
[Crossref] [PubMed]

Corkum, P. B.

N. H. Burnett and P. B. Corkum, “Cold plasma production for recombination extreme ultraviolet lasers by optical field induced ionization,” J. Opt. Soc. Am. B 6, 1195 (1989).
[Crossref]

P. B. Corkum, N. H. Burnett, and F. Brunei, “Above-threshold ionization in the long wavelength limit,” Phys. Rev. Lett. 62, 1259 (1989).
[Crossref] [PubMed]

Eberly, J. H.

J. H. Eberly, Q. Su, and J. Javanainen, “Nonlinear light scattering accompanying multiphoton ionization,” Phys. Rev. Lett. 62, 881 (1989);“High-order harmonics in light scattering by atomic electrons above threshold,” in Short Wavelength Coherent Radiation: Generation and Application, R. W. Falcone and J. Kirz, eds. Proceedings of the topical meeting held September 26–29, 1988 (Optical Society of America, Washington, D.C., to be published).
[Crossref] [PubMed]

Englert, T. J.

T. J. Englert and E. A. Rinehart, “Second-harmonic photons from the interaction of free electrons with intense laser radiation,” Phys. Rev. A 28, 1539 (1983).
[Crossref]

Esarey, E.

P. Sprangle, A. Ting, and E. Esarey, “The nonlinear interaction of intense laser pulses in plasmas,” NRL Memo. Rep. 6545 (Naval Research Laboratory, Washington, D.C., 1989).

Ferray, M.

M. Ferray, A. L’Huillier, X. F. Li, L. A. Lompré, G. Mainfray, and C. Manus, “Multiple-harmonic conversion of 1064 nm radiation in rare gases,” J. Phys. B 21, L31 (1988);X. F. Li, A. L’Huillier, M. Ferray, L. A. Lompre, and G. Mainfray, “Multiple-harmonic generation in rare gases at high laser intensity,” Phys. Rev. A 39, 5751 (1989).
[Crossref] [PubMed]

Gibson, G.

Gladkov, S.

S. A. Akhmanov, S. Gladkov, and N. Koroteev, “New mechanism of higher-order optical harmonic generation in super-strong laser fields,” presented at the Conference on Super-Intense Laser–Atom Physics,University of Rochester, Rochester, New York, June 28–30, 1989.

Jara, H.

Javanainen, J.

J. H. Eberly, Q. Su, and J. Javanainen, “Nonlinear light scattering accompanying multiphoton ionization,” Phys. Rev. Lett. 62, 881 (1989);“High-order harmonics in light scattering by atomic electrons above threshold,” in Short Wavelength Coherent Radiation: Generation and Application, R. W. Falcone and J. Kirz, eds. Proceedings of the topical meeting held September 26–29, 1988 (Optical Society of America, Washington, D.C., to be published).
[Crossref] [PubMed]

Johan, U.

Koroteev, N.

S. A. Akhmanov, S. Gladkov, and N. Koroteev, “New mechanism of higher-order optical harmonic generation in super-strong laser fields,” presented at the Conference on Super-Intense Laser–Atom Physics,University of Rochester, Rochester, New York, June 28–30, 1989.

Kulander, K. C.

K. C. Kulander and B. W. Shore, “Calculation of multiple-harmonic conversion of 1064 nm radiation in Xe,” Phys. Rev. Lett. 62, 524 (1989).
[Crossref] [PubMed]

L’Huillier, A.

M. Ferray, A. L’Huillier, X. F. Li, L. A. Lompré, G. Mainfray, and C. Manus, “Multiple-harmonic conversion of 1064 nm radiation in rare gases,” J. Phys. B 21, L31 (1988);X. F. Li, A. L’Huillier, M. Ferray, L. A. Lompre, and G. Mainfray, “Multiple-harmonic generation in rare gases at high laser intensity,” Phys. Rev. A 39, 5751 (1989).
[Crossref] [PubMed]

Landau, L. D.

L. D. Landau and E. M. Lifshitz, Quantum Mechanics, 2nd ed. (Pergamon, New York, 1965), p. 276.

L. D. Landau and E. M. Lipshitz, Electrodynamics (Addison-Wesley, Reading, Mass., 1960), Sec. 70;R. Z. Sagdeev, D. A. Usikov, and G. M. Zaslavsky, Nonlinear Physics: From the Pendulum to Turbulence and Chaos (Harwood Academic, New York, 1988), p. 594.

Lee, Y. C.

T. Tajima and Y. C. Lee, “Absorbing boundary condition and Budden turning point technique for electromagnetic plasma simulations,” J. Comput. Phys. 42, 406 (1981).
[Crossref]

Li, X. F.

M. Ferray, A. L’Huillier, X. F. Li, L. A. Lompré, G. Mainfray, and C. Manus, “Multiple-harmonic conversion of 1064 nm radiation in rare gases,” J. Phys. B 21, L31 (1988);X. F. Li, A. L’Huillier, M. Ferray, L. A. Lompre, and G. Mainfray, “Multiple-harmonic generation in rare gases at high laser intensity,” Phys. Rev. A 39, 5751 (1989).
[Crossref] [PubMed]

Lifshitz, E. M.

L. D. Landau and E. M. Lifshitz, Quantum Mechanics, 2nd ed. (Pergamon, New York, 1965), p. 276.

Lipshitz, E. M.

L. D. Landau and E. M. Lipshitz, Electrodynamics (Addison-Wesley, Reading, Mass., 1960), Sec. 70;R. Z. Sagdeev, D. A. Usikov, and G. M. Zaslavsky, Nonlinear Physics: From the Pendulum to Turbulence and Chaos (Harwood Academic, New York, 1988), p. 594.

Lompré, L. A.

M. Ferray, A. L’Huillier, X. F. Li, L. A. Lompré, G. Mainfray, and C. Manus, “Multiple-harmonic conversion of 1064 nm radiation in rare gases,” J. Phys. B 21, L31 (1988);X. F. Li, A. L’Huillier, M. Ferray, L. A. Lompre, and G. Mainfray, “Multiple-harmonic generation in rare gases at high laser intensity,” Phys. Rev. A 39, 5751 (1989).
[Crossref] [PubMed]

Luk, T. S.

Mainfray, G.

M. Ferray, A. L’Huillier, X. F. Li, L. A. Lompré, G. Mainfray, and C. Manus, “Multiple-harmonic conversion of 1064 nm radiation in rare gases,” J. Phys. B 21, L31 (1988);X. F. Li, A. L’Huillier, M. Ferray, L. A. Lompre, and G. Mainfray, “Multiple-harmonic generation in rare gases at high laser intensity,” Phys. Rev. A 39, 5751 (1989).
[Crossref] [PubMed]

Manus, C.

M. Ferray, A. L’Huillier, X. F. Li, L. A. Lompré, G. Mainfray, and C. Manus, “Multiple-harmonic conversion of 1064 nm radiation in rare gases,” J. Phys. B 21, L31 (1988);X. F. Li, A. L’Huillier, M. Ferray, L. A. Lompre, and G. Mainfray, “Multiple-harmonic generation in rare gases at high laser intensity,” Phys. Rev. A 39, 5751 (1989).
[Crossref] [PubMed]

Mclntyre, I. A.

McPherson, A.

Rhodes, C. K.

Rinehart, E. A.

T. J. Englert and E. A. Rinehart, “Second-harmonic photons from the interaction of free electrons with intense laser radiation,” Phys. Rev. A 28, 1539 (1983).
[Crossref]

Shore, B. W.

K. C. Kulander and B. W. Shore, “Calculation of multiple-harmonic conversion of 1064 nm radiation in Xe,” Phys. Rev. Lett. 62, 524 (1989).
[Crossref] [PubMed]

Sprangle, P.

P. Sprangle, A. Ting, and E. Esarey, “The nonlinear interaction of intense laser pulses in plasmas,” NRL Memo. Rep. 6545 (Naval Research Laboratory, Washington, D.C., 1989).

Su, Q.

J. H. Eberly, Q. Su, and J. Javanainen, “Nonlinear light scattering accompanying multiphoton ionization,” Phys. Rev. Lett. 62, 881 (1989);“High-order harmonics in light scattering by atomic electrons above threshold,” in Short Wavelength Coherent Radiation: Generation and Application, R. W. Falcone and J. Kirz, eds. Proceedings of the topical meeting held September 26–29, 1988 (Optical Society of America, Washington, D.C., to be published).
[Crossref] [PubMed]

Tajima, T.

T. Tajima and Y. C. Lee, “Absorbing boundary condition and Budden turning point technique for electromagnetic plasma simulations,” J. Comput. Phys. 42, 406 (1981).
[Crossref]

Ting, A.

P. Sprangle, A. Ting, and E. Esarey, “The nonlinear interaction of intense laser pulses in plasmas,” NRL Memo. Rep. 6545 (Naval Research Laboratory, Washington, D.C., 1989).

J. Comput. Phys. (1)

T. Tajima and Y. C. Lee, “Absorbing boundary condition and Budden turning point technique for electromagnetic plasma simulations,” J. Comput. Phys. 42, 406 (1981).
[Crossref]

J. Opt. Soc. Am. B (2)

J. Phys. B (1)

M. Ferray, A. L’Huillier, X. F. Li, L. A. Lompré, G. Mainfray, and C. Manus, “Multiple-harmonic conversion of 1064 nm radiation in rare gases,” J. Phys. B 21, L31 (1988);X. F. Li, A. L’Huillier, M. Ferray, L. A. Lompre, and G. Mainfray, “Multiple-harmonic generation in rare gases at high laser intensity,” Phys. Rev. A 39, 5751 (1989).
[Crossref] [PubMed]

Phys. Rev. A (1)

T. J. Englert and E. A. Rinehart, “Second-harmonic photons from the interaction of free electrons with intense laser radiation,” Phys. Rev. A 28, 1539 (1983).
[Crossref]

Phys. Rev. Lett. (3)

K. C. Kulander and B. W. Shore, “Calculation of multiple-harmonic conversion of 1064 nm radiation in Xe,” Phys. Rev. Lett. 62, 524 (1989).
[Crossref] [PubMed]

J. H. Eberly, Q. Su, and J. Javanainen, “Nonlinear light scattering accompanying multiphoton ionization,” Phys. Rev. Lett. 62, 881 (1989);“High-order harmonics in light scattering by atomic electrons above threshold,” in Short Wavelength Coherent Radiation: Generation and Application, R. W. Falcone and J. Kirz, eds. Proceedings of the topical meeting held September 26–29, 1988 (Optical Society of America, Washington, D.C., to be published).
[Crossref] [PubMed]

P. B. Corkum, N. H. Burnett, and F. Brunei, “Above-threshold ionization in the long wavelength limit,” Phys. Rev. Lett. 62, 1259 (1989).
[Crossref] [PubMed]

Other (4)

L. D. Landau and E. M. Lifshitz, Quantum Mechanics, 2nd ed. (Pergamon, New York, 1965), p. 276.

P. Sprangle, A. Ting, and E. Esarey, “The nonlinear interaction of intense laser pulses in plasmas,” NRL Memo. Rep. 6545 (Naval Research Laboratory, Washington, D.C., 1989).

S. A. Akhmanov, S. Gladkov, and N. Koroteev, “New mechanism of higher-order optical harmonic generation in super-strong laser fields,” presented at the Conference on Super-Intense Laser–Atom Physics,University of Rochester, Rochester, New York, June 28–30, 1989.

L. D. Landau and E. M. Lipshitz, Electrodynamics (Addison-Wesley, Reading, Mass., 1960), Sec. 70;R. Z. Sagdeev, D. A. Usikov, and G. M. Zaslavsky, Nonlinear Physics: From the Pendulum to Turbulence and Chaos (Harwood Academic, New York, 1988), p. 594.

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Figures (3)

Fig. 1
Fig. 1

Density profile and power spectrum in the case when the laser pulse has a flat part that is 10 laser cyles long with ramped parts at both ends that are 5 cyles long, (a) Density profile produced by the ionization process as the laser pulse is halfway through the gas slab and is propagating toward the right, (b) The power spectrum of the electric field signal that is collected on the right-hand side of the gas slab. The signal has been multiplied by a masking function fM = exp[−(ttc)2/tM2] before being Fourier analyzed in order to reduce the background or the continuous noise that is due to the shape of the pulse envelope. tc corresponds to the center of the pulse, and tM equals four laser periods.

Fig. 2
Fig. 2

Power spectrum and density profile in the case of a laser pulse with a Gaussian envelope, (a) Density profile as the laser signal is halfway through the gas slab. Notice by the quantity of the steps observed that only few central cycles participate in the ionization. (b) Power spectrum of the electric field signal that is collected on the right-hand side; in this case no masking function has been applied.

Fig. 3
Fig. 3

Same as Fig. 2(a) but here the laser electric field signal has been multiplied by a masking function fM = exp[−(ttc)2/tM2], with tM = 1.35 laser cycles before Fourier analysis.

Equations (40)

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× ( × E ) + 1 c 2 2 E t 2 + 4 π c 2 J t = 0 ,
J = n q e v f
v f = 1 N α = 1 N v α
J t = q e 2 m e n E ,
× ( × E ) + 1 c 2 2 E t 2 + ω pe 2 ( t ) c 2 E = 0 ,
ω pe 2 = 4 π n q e 2 m e
d n d t = w ( n 0 n ) ,
w = 4 ω a E a E exp ( 2 3 E a E ) ,
E = r E r ( x , t ) exp [ i ( k r x ω r t ) ] , ω pe 2 = r ω pe r 2 ( x , t ) exp [ i ( r k F x r ω F t ) ] .
k r 2 E r + 2 i k r E r x + ω r 2 c 2 E r + 2 i ω r c 2 E r t ω pe 0 2 c 2 E r = r , r r ω pe r 2 c 2 E r exp ( i Δ k r x i Δ ω r t ) ,
Δ k r = r k F + k r k r
Δ ω r = r ω F + ω r ω r .
ω r 2 = ω pe 0 2 ( x , t ) + k r 2 c 2 ,
x E r + 1 υ gr t E r = i 2 k r r , r r ω pe r 2 c 2 E r exp ( i Δ k r x i Δ ω r t ) ,
δ n = n g t 0 t w d t ,
w = w 0 2 + r = 1 w r cos r ω F t
δ n = δ n 0 2 t + r = 1 δ n r sin r ω F t ,
w r = ω F π π / ω f π / ω f d t w ( t ) cos r ω F t .
w ( t ) 4 ω a ζ exp ( ζ ω L 2 t 2 / 3 ) ,
w r = P ( ω F 2 π ) exp ( 3 r 2 / ζ ) ,
P = 8 3 π ω a ω L ζ 1 / 2 exp ( 2 3 ζ )
P 0 2 π / ω L w ( t ) d t .
δ n r = n g P 2 π r exp ( 3 r 2 / ζ ) ,
ω pe r 2 = i ω p g 2 P 4 π r exp ( 3 r 2 / ζ ) ,
x E r = i 2 k r r , r ω pe r 2 c 2 E r exp ( i Δ k r x ) .
Δ k r k L ω pe 0 2 ω L 2 ,
x E 2 r + 1 = k L P 8 π r ( 2 r + 1 ) ω p g 2 ω L 2 exp ( i Δ k x ) × { exp ( 3 r 2 / ζ ) + r r + 1 exp [ 3 ( r + 1 ) 2 / ζ ] } E 1 .
E 2 r + 1 = P k L 8 π r ( 2 r + 1 ) ω p g 2 ω L 2 [ exp ( i Δ k L ) 1 i Δ k ] × { exp ( 3 r 2 / ζ ) + r r + 1 exp [ 3 ( r + 1 ) 2 / ζ ] } E 1 .
| E 2 r + 1 | | E 1 | = P 4 π r 2 ( 2 r + 1 ) n g n p × { exp ( 3 r 2 / ζ ) + r ( r + 1 ) exp [ 3 ( r + 1 ) 2 / ζ ] } ,
E t = 4 π J + c × B
B t = c × E
t ( n υ f ) = n q e m e E .
J ( t ) = q e Δ V α = 1 N ( t ) v α ( t ) ,
J ( t + Δ t ) = q Δ V α = 1 N ( t ) v α ( t + Δ t ) + q Δ V α = N ( t ) + 1 N ( t + Δ t ) v α ( t + Δ t ) .
J t = q e Δ V α = 1 N ( t ) d v α d t .
d v α d t = q e m e E ,
1 4 π v d V [ c · ( E × B ) + 1 2 E 2 t + 1 2 B 2 t ] + V d V J · E = 0 ,
J t = t n q e v f = q e 2 m e n E ,
m e v f · t n v f = n q e v f · E = J · E ,
1 2 t m e n υ f 2 + m e 2 υ f 2 n t = J · E .

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