Abstract

We consider phase matching in the generation of very short-wavelength coherent radiation by nonlinear frequency upconversion in plasma and suggest some ways to improve phase matching in high-order harmonic generation. We obtain what are to our knowledge the first simple analytical expressions for a phase-matching factor in multiphoton mixing of an arbitrary order and demonstrate theoretically that high-order difference-frequency mixing in plasma could be, owing to its potential for phase-matching optimization, a more promising method of large-scale frequency upconversion than high-order harmonic generation.

© 1993 Optical Society of America

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References

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  1. A. L’Huillier, Ph. Balcou, Phys. Rev. Lett. 70, 774 (1993); J. J. Macklin, J. D. Kmetec, C. L. Gordon, Phys. Rev. Lett. 70, 776 (1993).
    [CrossRef]
  2. Ph. Balcou, A. L’Huillier, Phys. Rev A 47, 1447 (1993).
    [CrossRef] [PubMed]
  3. Y. Akiyama, K. Midorikawa, Y. Nagata, M. Obara, H. Tashiro, K. Toyoda, Phys. Rev. Lett. 69, 2176 (1992); J. L. Krause, K. J. Schafer, K. C. Kulander, Phys. Rev. Lett. 68, 3535 (1992).
    [CrossRef] [PubMed]
  4. P. W. Millony, J. H. Eberly, Lasers (Wiley, New York, 1988), Chap. 3.
  5. D. H. Close, Phys. Rev. 153, 360 (1967); P. L. Shkolnikov, A. E. Kaplan, Phys. Rev. A 44, 6951 (1991).
    [CrossRef] [PubMed]
  6. J. F. Reintjes, Nonlinear Optical Processes in Liquids and Gases (Academic, New York, 1984), Chap. 3.
  7. P. L. Shkolnikov, A. E. Kaplan, Opt. Lett. 16, 1973 (1991).
    [CrossRef] [PubMed]
  8. A. Lago, G. Hilber, R. Wallenstein, Phys. Rev. A 36, 3827 (1987); G. Hilber, D. J. Brink, A. Lago, R. Wallenstein, Phys. Rev. A 38, 6231 (1988).
    [CrossRef] [PubMed]
  9. G. Bjorklund, IEEE J. Quantum Electron. QE-11, 287 (1975).
    [CrossRef]
  10. I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, New York, 1980), Tables 3.384.9, 9.237.2, 8.97.

1993 (2)

A. L’Huillier, Ph. Balcou, Phys. Rev. Lett. 70, 774 (1993); J. J. Macklin, J. D. Kmetec, C. L. Gordon, Phys. Rev. Lett. 70, 776 (1993).
[CrossRef]

Ph. Balcou, A. L’Huillier, Phys. Rev A 47, 1447 (1993).
[CrossRef] [PubMed]

1992 (1)

Y. Akiyama, K. Midorikawa, Y. Nagata, M. Obara, H. Tashiro, K. Toyoda, Phys. Rev. Lett. 69, 2176 (1992); J. L. Krause, K. J. Schafer, K. C. Kulander, Phys. Rev. Lett. 68, 3535 (1992).
[CrossRef] [PubMed]

1991 (1)

1987 (1)

A. Lago, G. Hilber, R. Wallenstein, Phys. Rev. A 36, 3827 (1987); G. Hilber, D. J. Brink, A. Lago, R. Wallenstein, Phys. Rev. A 38, 6231 (1988).
[CrossRef] [PubMed]

1975 (1)

G. Bjorklund, IEEE J. Quantum Electron. QE-11, 287 (1975).
[CrossRef]

1967 (1)

D. H. Close, Phys. Rev. 153, 360 (1967); P. L. Shkolnikov, A. E. Kaplan, Phys. Rev. A 44, 6951 (1991).
[CrossRef] [PubMed]

Akiyama, Y.

Y. Akiyama, K. Midorikawa, Y. Nagata, M. Obara, H. Tashiro, K. Toyoda, Phys. Rev. Lett. 69, 2176 (1992); J. L. Krause, K. J. Schafer, K. C. Kulander, Phys. Rev. Lett. 68, 3535 (1992).
[CrossRef] [PubMed]

Balcou, Ph.

A. L’Huillier, Ph. Balcou, Phys. Rev. Lett. 70, 774 (1993); J. J. Macklin, J. D. Kmetec, C. L. Gordon, Phys. Rev. Lett. 70, 776 (1993).
[CrossRef]

Ph. Balcou, A. L’Huillier, Phys. Rev A 47, 1447 (1993).
[CrossRef] [PubMed]

Bjorklund, G.

G. Bjorklund, IEEE J. Quantum Electron. QE-11, 287 (1975).
[CrossRef]

Close, D. H.

D. H. Close, Phys. Rev. 153, 360 (1967); P. L. Shkolnikov, A. E. Kaplan, Phys. Rev. A 44, 6951 (1991).
[CrossRef] [PubMed]

Eberly, J. H.

P. W. Millony, J. H. Eberly, Lasers (Wiley, New York, 1988), Chap. 3.

Gradshteyn, I. S.

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, New York, 1980), Tables 3.384.9, 9.237.2, 8.97.

Hilber, G.

A. Lago, G. Hilber, R. Wallenstein, Phys. Rev. A 36, 3827 (1987); G. Hilber, D. J. Brink, A. Lago, R. Wallenstein, Phys. Rev. A 38, 6231 (1988).
[CrossRef] [PubMed]

Kaplan, A. E.

L’Huillier, A.

Ph. Balcou, A. L’Huillier, Phys. Rev A 47, 1447 (1993).
[CrossRef] [PubMed]

A. L’Huillier, Ph. Balcou, Phys. Rev. Lett. 70, 774 (1993); J. J. Macklin, J. D. Kmetec, C. L. Gordon, Phys. Rev. Lett. 70, 776 (1993).
[CrossRef]

Lago, A.

A. Lago, G. Hilber, R. Wallenstein, Phys. Rev. A 36, 3827 (1987); G. Hilber, D. J. Brink, A. Lago, R. Wallenstein, Phys. Rev. A 38, 6231 (1988).
[CrossRef] [PubMed]

Midorikawa, K.

Y. Akiyama, K. Midorikawa, Y. Nagata, M. Obara, H. Tashiro, K. Toyoda, Phys. Rev. Lett. 69, 2176 (1992); J. L. Krause, K. J. Schafer, K. C. Kulander, Phys. Rev. Lett. 68, 3535 (1992).
[CrossRef] [PubMed]

Millony, P. W.

P. W. Millony, J. H. Eberly, Lasers (Wiley, New York, 1988), Chap. 3.

Nagata, Y.

Y. Akiyama, K. Midorikawa, Y. Nagata, M. Obara, H. Tashiro, K. Toyoda, Phys. Rev. Lett. 69, 2176 (1992); J. L. Krause, K. J. Schafer, K. C. Kulander, Phys. Rev. Lett. 68, 3535 (1992).
[CrossRef] [PubMed]

Obara, M.

Y. Akiyama, K. Midorikawa, Y. Nagata, M. Obara, H. Tashiro, K. Toyoda, Phys. Rev. Lett. 69, 2176 (1992); J. L. Krause, K. J. Schafer, K. C. Kulander, Phys. Rev. Lett. 68, 3535 (1992).
[CrossRef] [PubMed]

Reintjes, J. F.

J. F. Reintjes, Nonlinear Optical Processes in Liquids and Gases (Academic, New York, 1984), Chap. 3.

Ryzhik, I. M.

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, New York, 1980), Tables 3.384.9, 9.237.2, 8.97.

Shkolnikov, P. L.

Tashiro, H.

Y. Akiyama, K. Midorikawa, Y. Nagata, M. Obara, H. Tashiro, K. Toyoda, Phys. Rev. Lett. 69, 2176 (1992); J. L. Krause, K. J. Schafer, K. C. Kulander, Phys. Rev. Lett. 68, 3535 (1992).
[CrossRef] [PubMed]

Toyoda, K.

Y. Akiyama, K. Midorikawa, Y. Nagata, M. Obara, H. Tashiro, K. Toyoda, Phys. Rev. Lett. 69, 2176 (1992); J. L. Krause, K. J. Schafer, K. C. Kulander, Phys. Rev. Lett. 68, 3535 (1992).
[CrossRef] [PubMed]

Wallenstein, R.

A. Lago, G. Hilber, R. Wallenstein, Phys. Rev. A 36, 3827 (1987); G. Hilber, D. J. Brink, A. Lago, R. Wallenstein, Phys. Rev. A 38, 6231 (1988).
[CrossRef] [PubMed]

IEEE J. Quantum Electron. (1)

G. Bjorklund, IEEE J. Quantum Electron. QE-11, 287 (1975).
[CrossRef]

Opt. Lett. (1)

Phys. Rev A (1)

Ph. Balcou, A. L’Huillier, Phys. Rev A 47, 1447 (1993).
[CrossRef] [PubMed]

Phys. Rev. (1)

D. H. Close, Phys. Rev. 153, 360 (1967); P. L. Shkolnikov, A. E. Kaplan, Phys. Rev. A 44, 6951 (1991).
[CrossRef] [PubMed]

Phys. Rev. A (1)

A. Lago, G. Hilber, R. Wallenstein, Phys. Rev. A 36, 3827 (1987); G. Hilber, D. J. Brink, A. Lago, R. Wallenstein, Phys. Rev. A 38, 6231 (1988).
[CrossRef] [PubMed]

Phys. Rev. Lett. (2)

Y. Akiyama, K. Midorikawa, Y. Nagata, M. Obara, H. Tashiro, K. Toyoda, Phys. Rev. Lett. 69, 2176 (1992); J. L. Krause, K. J. Schafer, K. C. Kulander, Phys. Rev. Lett. 68, 3535 (1992).
[CrossRef] [PubMed]

A. L’Huillier, Ph. Balcou, Phys. Rev. Lett. 70, 774 (1993); J. J. Macklin, J. D. Kmetec, C. L. Gordon, Phys. Rev. Lett. 70, 776 (1993).
[CrossRef]

Other (3)

P. W. Millony, J. H. Eberly, Lasers (Wiley, New York, 1988), Chap. 3.

J. F. Reintjes, Nonlinear Optical Processes in Liquids and Gases (Academic, New York, 1984), Chap. 3.

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, New York, 1980), Tables 3.384.9, 9.237.2, 8.97.

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Equations (16)

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Δ k HHG r e q λ N e [ 1 2 1 q 2 ( N i / N e ) ( g 1 f 12 / g 2 ) 1 λ q / λ 0 ] .
q 2 ( N i / N e ) ( g 1 f 12 / g 2 ) ( 1 λ q / λ 0 ) > 2 ,
2 1 q 2 ( N i / N e ) ( g 1 f 12 / g 2 ) 1 λ q / λ 0 = 1 + 2 ( q 2 ) b r e q λ N e .
ω = m ω 1 l ω 2 ,
Δ k r e N e m [ λ 1 ( l / m ) λ 2 λ / m ] , λ = λ 1 λ 2 / ( m λ 2 l λ 1 ) .
ω = j = 1 m ω j j = 1 l ω ˜ j ,
E j ( x , y , z ) = E j 0 exp ( i z k j ) ( 1 + i j ) 1 × exp [ k j R 2 / b j ( 1 + i j ) ] .
P ( x , y , z ) = S χ N i j = 1 m E j j = 1 l E ˜ j * ,
E ( x , y , z ) = S χ N i j = 1 m E j 0 j = 1 m E ˜ j 0 * ( i k 0 2 k 1 b ) exp ( i z k ) × ζ d exp [ ( b Δ k / 2 ) ( ) ] ( 1 + i ) m 1 ( 1 + i ) l 1 ( k i k ) H × exp ( R 2 b H ) .
k k .
F = | ζ d exp ( b Δ k / 2 ) ( 1 + i ) m 1 ( 1 i ) l | 2 .
F r , s = u r , s 2 ( b Δ k ) , if Δ k > 0 , = u s , r 2 ( b | Δ k | ) , if Δ k < 0 ,
u r , s ( x ) exp ( x / 2 ) π r ! 2 r + s k = 1 s ( r + s k ) ! ( s k ) ! k ! x k , r m 2 , s l 1 .
F r , 0 = π 2 2 2 r exp ( b Δ k ) , Δ k > 0 , = π 2 2 2 r exp ( b | Δ k | ) [ k = 0 r ( b | Δ k | ) k / k ! ] 2 , Δ k < 0 .
( b Δ k ) opt 2 r 2 / ( r + 1 ) , r = m 2 .
( b Δ k ) opt 2 ( r s ) .

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