Abstract

An analytical theory of the generation of high-order harmonics of laser radiation has been developed on the basis of a two-level model atom with degenerate levels. Among other parameters, onset, width, and cutoff of the plateau in the harmonic spectrum are obtained in simple analytical forms that connect the basic problem parameters and permit a transparent interpretation of the mechanism underlying the spectrum formation for this specific case. Selected numerical calculations are reported to corroborate the analytical findings and to investigate other harmonic-spectrum features.

© 1996 Optical Society of America

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References

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  1. M. Gavrila, ed., Advances in Atomic, Molecular, and Optical Physics, Supplement (Academic, New York, 1992).
  2. N. B. Delone and M. V. Fedorov, Sov. Phys. Usp. 32, 500 (1990).
    [Crossref]
  3. J. Javanainen, J. H. Eberly, and K. Rzazewski, Phys. Rep. 204, 331 (1991).
    [Crossref]
  4. G. Mainfray and C. Manus, Rep. Prog. Phys. 54, 1333 (1991).
    [Crossref]
  5. R. R. Freeman and P. H. Bucksbaum, J. Phys. B. 24, 325 (1991).
    [Crossref]
  6. A. L’Huillier, K. J. Schafer, and K. C. Kulander, J. Phys. B.: At. Mol. Opt. Phys. 24, 3315 (1991).
    [Crossref]
  7. A. L’Huillier, L. A. Lompré, G. Mainfray, and C. Manus, in Advances in Atomic, Molecular, and Optical Physics, Supplement (Academic, New York, 1992), p. 139.
  8. H. R. Reiss, Prog. Quantum Electron. 16, 1 (1992).
    [Crossref]
  9. K. Burnett, V. C. Reed, and P. L. K. night, J. Phys. B 26, 561 (1993).
    [Crossref]
  10. A. E. Kaplan and P. L. Shkolnikov, Phys. Rev. A 49, 1275 (1994).
    [Crossref] [PubMed]
  11. W. Becker, S. Long, and J. K. McIver, Phys. Rev. A 50, 1540 (1994).
    [Crossref] [PubMed]
  12. J. K. Krause, K. J. Schafer, and K. C. Kulander, Phys. Rev. Lett. 68, 3535 (1992).
    [Crossref] [PubMed]
  13. M. Lewenstein, Ph. Balcou, M. Yu. Ivanov, A. L’Huillier, and P. B. Corkum, Phys. Rev. A 49, 2117 (1994).
    [Crossref] [PubMed]
  14. J. K. Krause, K. J. Schafer, and K. C. Kulander, Phys. Rev. A 45, 4998 (1992).
    [Crossref] [PubMed]
  15. See, for instance, A. M. Zheltikov, G. Ferrante, and N. I. Koroteev, Laser Phys. 5, 126 (1995).
  16. M. D. Perry and J. K. Crane, Phys. Rev. A 48, R4051 (1993).
    [Crossref]
  17. J. B. Peatross, “The far-field angular distribution of high-order harmonics produced in light scattering from a thin low-density gas target,” Ph.D. dissertation (University of Rochester, Rochester, New York, 1993).

1995 (1)

See, for instance, A. M. Zheltikov, G. Ferrante, and N. I. Koroteev, Laser Phys. 5, 126 (1995).

1994 (3)

M. Lewenstein, Ph. Balcou, M. Yu. Ivanov, A. L’Huillier, and P. B. Corkum, Phys. Rev. A 49, 2117 (1994).
[Crossref] [PubMed]

A. E. Kaplan and P. L. Shkolnikov, Phys. Rev. A 49, 1275 (1994).
[Crossref] [PubMed]

W. Becker, S. Long, and J. K. McIver, Phys. Rev. A 50, 1540 (1994).
[Crossref] [PubMed]

1993 (2)

M. D. Perry and J. K. Crane, Phys. Rev. A 48, R4051 (1993).
[Crossref]

K. Burnett, V. C. Reed, and P. L. K. night, J. Phys. B 26, 561 (1993).
[Crossref]

1992 (3)

J. K. Krause, K. J. Schafer, and K. C. Kulander, Phys. Rev. A 45, 4998 (1992).
[Crossref] [PubMed]

J. K. Krause, K. J. Schafer, and K. C. Kulander, Phys. Rev. Lett. 68, 3535 (1992).
[Crossref] [PubMed]

H. R. Reiss, Prog. Quantum Electron. 16, 1 (1992).
[Crossref]

1991 (4)

J. Javanainen, J. H. Eberly, and K. Rzazewski, Phys. Rep. 204, 331 (1991).
[Crossref]

G. Mainfray and C. Manus, Rep. Prog. Phys. 54, 1333 (1991).
[Crossref]

R. R. Freeman and P. H. Bucksbaum, J. Phys. B. 24, 325 (1991).
[Crossref]

A. L’Huillier, K. J. Schafer, and K. C. Kulander, J. Phys. B.: At. Mol. Opt. Phys. 24, 3315 (1991).
[Crossref]

1990 (1)

N. B. Delone and M. V. Fedorov, Sov. Phys. Usp. 32, 500 (1990).
[Crossref]

Balcou, Ph.

M. Lewenstein, Ph. Balcou, M. Yu. Ivanov, A. L’Huillier, and P. B. Corkum, Phys. Rev. A 49, 2117 (1994).
[Crossref] [PubMed]

Becker, W.

W. Becker, S. Long, and J. K. McIver, Phys. Rev. A 50, 1540 (1994).
[Crossref] [PubMed]

Bucksbaum, P. H.

R. R. Freeman and P. H. Bucksbaum, J. Phys. B. 24, 325 (1991).
[Crossref]

Burnett, K.

K. Burnett, V. C. Reed, and P. L. K. night, J. Phys. B 26, 561 (1993).
[Crossref]

Corkum, P. B.

M. Lewenstein, Ph. Balcou, M. Yu. Ivanov, A. L’Huillier, and P. B. Corkum, Phys. Rev. A 49, 2117 (1994).
[Crossref] [PubMed]

Crane, J. K.

M. D. Perry and J. K. Crane, Phys. Rev. A 48, R4051 (1993).
[Crossref]

Delone, N. B.

N. B. Delone and M. V. Fedorov, Sov. Phys. Usp. 32, 500 (1990).
[Crossref]

Eberly, J. H.

J. Javanainen, J. H. Eberly, and K. Rzazewski, Phys. Rep. 204, 331 (1991).
[Crossref]

Fedorov, M. V.

N. B. Delone and M. V. Fedorov, Sov. Phys. Usp. 32, 500 (1990).
[Crossref]

Ferrante, G.

See, for instance, A. M. Zheltikov, G. Ferrante, and N. I. Koroteev, Laser Phys. 5, 126 (1995).

Freeman, R. R.

R. R. Freeman and P. H. Bucksbaum, J. Phys. B. 24, 325 (1991).
[Crossref]

Ivanov, M. Yu.

M. Lewenstein, Ph. Balcou, M. Yu. Ivanov, A. L’Huillier, and P. B. Corkum, Phys. Rev. A 49, 2117 (1994).
[Crossref] [PubMed]

Javanainen, J.

J. Javanainen, J. H. Eberly, and K. Rzazewski, Phys. Rep. 204, 331 (1991).
[Crossref]

Kaplan, A. E.

A. E. Kaplan and P. L. Shkolnikov, Phys. Rev. A 49, 1275 (1994).
[Crossref] [PubMed]

Koroteev, N. I.

See, for instance, A. M. Zheltikov, G. Ferrante, and N. I. Koroteev, Laser Phys. 5, 126 (1995).

Krause, J. K.

J. K. Krause, K. J. Schafer, and K. C. Kulander, Phys. Rev. A 45, 4998 (1992).
[Crossref] [PubMed]

J. K. Krause, K. J. Schafer, and K. C. Kulander, Phys. Rev. Lett. 68, 3535 (1992).
[Crossref] [PubMed]

Kulander, K. C.

J. K. Krause, K. J. Schafer, and K. C. Kulander, Phys. Rev. Lett. 68, 3535 (1992).
[Crossref] [PubMed]

J. K. Krause, K. J. Schafer, and K. C. Kulander, Phys. Rev. A 45, 4998 (1992).
[Crossref] [PubMed]

A. L’Huillier, K. J. Schafer, and K. C. Kulander, J. Phys. B.: At. Mol. Opt. Phys. 24, 3315 (1991).
[Crossref]

L’Huillier, A.

M. Lewenstein, Ph. Balcou, M. Yu. Ivanov, A. L’Huillier, and P. B. Corkum, Phys. Rev. A 49, 2117 (1994).
[Crossref] [PubMed]

A. L’Huillier, K. J. Schafer, and K. C. Kulander, J. Phys. B.: At. Mol. Opt. Phys. 24, 3315 (1991).
[Crossref]

A. L’Huillier, L. A. Lompré, G. Mainfray, and C. Manus, in Advances in Atomic, Molecular, and Optical Physics, Supplement (Academic, New York, 1992), p. 139.

Lewenstein, M.

M. Lewenstein, Ph. Balcou, M. Yu. Ivanov, A. L’Huillier, and P. B. Corkum, Phys. Rev. A 49, 2117 (1994).
[Crossref] [PubMed]

Lompré, L. A.

A. L’Huillier, L. A. Lompré, G. Mainfray, and C. Manus, in Advances in Atomic, Molecular, and Optical Physics, Supplement (Academic, New York, 1992), p. 139.

Long, S.

W. Becker, S. Long, and J. K. McIver, Phys. Rev. A 50, 1540 (1994).
[Crossref] [PubMed]

Mainfray, G.

G. Mainfray and C. Manus, Rep. Prog. Phys. 54, 1333 (1991).
[Crossref]

A. L’Huillier, L. A. Lompré, G. Mainfray, and C. Manus, in Advances in Atomic, Molecular, and Optical Physics, Supplement (Academic, New York, 1992), p. 139.

Manus, C.

G. Mainfray and C. Manus, Rep. Prog. Phys. 54, 1333 (1991).
[Crossref]

A. L’Huillier, L. A. Lompré, G. Mainfray, and C. Manus, in Advances in Atomic, Molecular, and Optical Physics, Supplement (Academic, New York, 1992), p. 139.

McIver, J. K.

W. Becker, S. Long, and J. K. McIver, Phys. Rev. A 50, 1540 (1994).
[Crossref] [PubMed]

night, P. L. K.

K. Burnett, V. C. Reed, and P. L. K. night, J. Phys. B 26, 561 (1993).
[Crossref]

Peatross, J. B.

J. B. Peatross, “The far-field angular distribution of high-order harmonics produced in light scattering from a thin low-density gas target,” Ph.D. dissertation (University of Rochester, Rochester, New York, 1993).

Perry, M. D.

M. D. Perry and J. K. Crane, Phys. Rev. A 48, R4051 (1993).
[Crossref]

Reed, V. C.

K. Burnett, V. C. Reed, and P. L. K. night, J. Phys. B 26, 561 (1993).
[Crossref]

Reiss, H. R.

H. R. Reiss, Prog. Quantum Electron. 16, 1 (1992).
[Crossref]

Rzazewski, K.

J. Javanainen, J. H. Eberly, and K. Rzazewski, Phys. Rep. 204, 331 (1991).
[Crossref]

Schafer, K. J.

J. K. Krause, K. J. Schafer, and K. C. Kulander, Phys. Rev. A 45, 4998 (1992).
[Crossref] [PubMed]

J. K. Krause, K. J. Schafer, and K. C. Kulander, Phys. Rev. Lett. 68, 3535 (1992).
[Crossref] [PubMed]

A. L’Huillier, K. J. Schafer, and K. C. Kulander, J. Phys. B.: At. Mol. Opt. Phys. 24, 3315 (1991).
[Crossref]

Shkolnikov, P. L.

A. E. Kaplan and P. L. Shkolnikov, Phys. Rev. A 49, 1275 (1994).
[Crossref] [PubMed]

Zheltikov, A. M.

See, for instance, A. M. Zheltikov, G. Ferrante, and N. I. Koroteev, Laser Phys. 5, 126 (1995).

J. Phys. B (1)

K. Burnett, V. C. Reed, and P. L. K. night, J. Phys. B 26, 561 (1993).
[Crossref]

J. Phys. B. (1)

R. R. Freeman and P. H. Bucksbaum, J. Phys. B. 24, 325 (1991).
[Crossref]

J. Phys. B.: At. Mol. Opt. Phys. (1)

A. L’Huillier, K. J. Schafer, and K. C. Kulander, J. Phys. B.: At. Mol. Opt. Phys. 24, 3315 (1991).
[Crossref]

Laser Phys. (1)

See, for instance, A. M. Zheltikov, G. Ferrante, and N. I. Koroteev, Laser Phys. 5, 126 (1995).

Phys. Rep. (1)

J. Javanainen, J. H. Eberly, and K. Rzazewski, Phys. Rep. 204, 331 (1991).
[Crossref]

Phys. Rev. A (5)

A. E. Kaplan and P. L. Shkolnikov, Phys. Rev. A 49, 1275 (1994).
[Crossref] [PubMed]

W. Becker, S. Long, and J. K. McIver, Phys. Rev. A 50, 1540 (1994).
[Crossref] [PubMed]

M. D. Perry and J. K. Crane, Phys. Rev. A 48, R4051 (1993).
[Crossref]

M. Lewenstein, Ph. Balcou, M. Yu. Ivanov, A. L’Huillier, and P. B. Corkum, Phys. Rev. A 49, 2117 (1994).
[Crossref] [PubMed]

J. K. Krause, K. J. Schafer, and K. C. Kulander, Phys. Rev. A 45, 4998 (1992).
[Crossref] [PubMed]

Phys. Rev. Lett. (1)

J. K. Krause, K. J. Schafer, and K. C. Kulander, Phys. Rev. Lett. 68, 3535 (1992).
[Crossref] [PubMed]

Prog. Quantum Electron. (1)

H. R. Reiss, Prog. Quantum Electron. 16, 1 (1992).
[Crossref]

Rep. Prog. Phys. (1)

G. Mainfray and C. Manus, Rep. Prog. Phys. 54, 1333 (1991).
[Crossref]

Sov. Phys. Usp. (1)

N. B. Delone and M. V. Fedorov, Sov. Phys. Usp. 32, 500 (1990).
[Crossref]

Other (3)

M. Gavrila, ed., Advances in Atomic, Molecular, and Optical Physics, Supplement (Academic, New York, 1992).

J. B. Peatross, “The far-field angular distribution of high-order harmonics produced in light scattering from a thin low-density gas target,” Ph.D. dissertation (University of Rochester, Rochester, New York, 1993).

A. L’Huillier, L. A. Lompré, G. Mainfray, and C. Manus, in Advances in Atomic, Molecular, and Optical Physics, Supplement (Academic, New York, 1992), p. 139.

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Figures (7)

Fig. 1
Fig. 1

Schematic of a two-level atom with a twofold degeneracy of the upper level: (a) without the external radiation field, (b) with the external radiation field. ћΩ11 is the linear Stark shift.

Fig. 2
Fig. 2

Connection between the field-modified (Stark-split) energy spectrum of a two-level atom and the expected shape of the harmonic spectrum. The upper level is Stark split by the amount 2ћΩ11 = 2d12E. All the transitions that originate inside the band of width 2ћΩ11 back to the ground state will form the plateau, with the onset corresponding to transitions from the bottom of the band and the cutoff corresponding to transitions from the top. The bottom of the band is located at ћ(ω10 − Ω11), the top, at ћ(ω10 + Ω11). Transitions from outside the band correspond to the decreasing parts of the harmonic spectrum.

Fig. 3
Fig. 3

Sn2 versus the harmonic order n =2N + 1. The harmonic spectra in the quasi-resonant and the off-resonant cases at Ω11 = 15 ω are compared. Solid curve, ω10 = 25.1ω; dotted curve, ω10 = 25.5ω.

Fig. 4
Fig. 4

Same as Fig. 3, but with Ω11 = 25 ω. Solid curve, ω10 = 35.1ω; dotted curve, ω10 = 35.5ω.

Fig. 5
Fig. 5

Behavior of three selected harmonics (n = 9, 35, 63) versus Ω11/ω, a quantity dependent linearly on the laser electric field E. ω10 = 35.5ω. The arrow marks the point at which ω10 ≈ Ω11.

Fig. 6
Fig. 6

Sn2 versus the harmonic order n = 2N + 1, with Ω11 = 5ω. Solid curve, ω10 = 15.5ω; dotted curve, ω10 = 35.5ω. The arrow marks the point at which n = (ω10 + Ω11)/ω.

Fig. 7
Fig. 7

Sn2 versus the harmonic order n = 2N + 1, with Ω11 = 15ω. Upper solid curve, ω10 = 25.5ω; dotted curve, ω10 = 35.5ω; lower solid curve, ω10 = 45.5ω. The arrows mark the point at which n = (ω10 + Ω11)/ω.

Equations (31)

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φ 0 = φ 0 0 φ 1 , 2 = ( φ 1 0 ± φ 2 0 ) / ( 2 ) 1 / 2 ,
d 11 = d 22 = d 12 0 , d 12 = 0 , d 01 = ( d 02 0 + d 01 0 ) / ( 2 ) 1 / 2 , d 02 = ( d 02 0 d 01 0 ) / ( 2 ) 1 / 2 .
Ψ = a j exp ( i E j t / ћ ) φ j
a ˙ j = i Ω ˜ jj a j + i Ω ˜ j 0 exp ( i ω 10 t ) a 0 j 0 ,
a ˙ 0 = i Ω ˜ 0 j exp ( i ω 10 t ) a j ,
a 0 = 1 , a j = t i Ω ˜ j 0 ( τ ) exp ( i ω 10 τ ) d τ exp [ i τ t Ω ˜ jj ( t ) d t ] .
t Ω ˜ jj ( t ) d t Ω ω cos ω t
t Ω exp [ i ( ω 10 ± l ω ) t ] d t Ω exp [ i ( ω 10 ± l ω ) t ] i ( ω 10 ± l ω ) .
exp ( iz cos φ ) = + i k J k ( z ) exp ( ik φ )
d ¯ = j , k a j * a k d jk exp ( i ω jk t )
d ¯ = n = d n exp ( in ω t ) ,
d n = ½ ( i ) n S n α n E ,
S n = l = 1 ( ω 10 2 ω 2 ) ( 1 ) 1 ( ω 10 2 l 2 ω 2 ) ( J l 1 + J l + 1 ) ( J l n + J l + n ) = l = 1 σ l ( n ) ,
α 2 N + 1 = α = 2 ω 01 [ ( d 01 0 ) 2 + ( d 02 0 ) 2 ] ћ ( ω 01 2 ω 2 ) ,
α 2 N = 4 ω d 01 0 d 02 0 ћ ( ω 01 2 ω 2 ) .
d 01 0 = 0 .
d 2 N + 1 = i 2 ( 1 ) N S 2 N + 1 α E [ | a 0 ( ) | 2 | a 1 0 ( ) | 2 + | a 2 0 ( ) | 2 2 ] d 2 N = ( 1 ) N 2 S 2 N α E [ a 1 0 ( ) a 2 0 * ( ) + c . c . ] ,
| l = l ω ω 10 + l ω J l exp ( il ω t ) | 2 ( d 12 0 d 02 0 ) 2 .
Ω 11 = d 11 E ћ = d 12 0 E ћ
φ 1 = 1 2 ( φ 1 0 + φ 2 0 ) exp ( i ћ E 1 t ) s ( 1 ) s × J s ( Ω 11 ω ) exp ( is ω t ) ,
φ 2 = 1 2 ( φ 1 0 + φ 2 0 ) exp ( i ћ E 1 t ) s ( 1 ) s × J s ( Ω 11 ω ) exp ( is ω t ) ,
σ l ( n ) = ε ( l ) R ( l , n ) ,
ε ( l ) = 2 ( ω 10 2 ω 2 ) ( ω 10 2 l 2 ω 2 ) ( l ω Ω 11 ) J l ( Ω 11 ω ) ,
R ( l , n ) = J l n ( Ω 11 ω ) + J l + n ( Ω 11 ω ) .
S 2 N + 1 = l σ l ( 2 N + 1 ) σ l 0 ( 2 N + 1 ) = ε ( l 0 ) R ( l 0 , 2 N + 1 ) ,
l 0 ( 2 N + 1 ) Ω 11 / ω ,
( 2 N + 1 ) ω l 0 ω Ω 11 ω 10 Ω 11 ,
| l 0 ( 2 N + 1 ) | = ( 2 N + 1 ) l 0 Ω 11 / ω .
( 2 N + 1 ) ω Ω 11 + l 0 ω ω 10 + Ω 11 ,
Δ ω 10 + Ω 11 ( ω 10 Ω 11 ) 2 Ω 11 .
ε ( l 0 ) = 2 ( ω 10 2 ω 2 ) ( ω 10 2 l 0 2 ω 2 ) ( l 0 ω Ω 11 ) J l 0 ( Ω 11 ω )

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