Abstract

A general theory of second-harmonic generation, including all the effects of group-velocity dispersion, is given for coherent ultrashort pulses with arbitrary shapes and carrier chirps. Ultrashort-pulse second-harmonic generation is analyzed for transform-limited fundamental pulses. The effects of intrapulse group-velocity dispersion (IGVD) on the second-harmonic (SH) pulse shape are investigated for parameters representative of popular phase-matchable crystals and wavelength, including Ti:sapphire lasers. In phase-matched structures IGVD at the SH cannot be neglected for pulses approaching 10 fs. It results in a spectral quadratic phase on the SH and in some cases can shorten the pulse. External dispersive shaping of the SH pulses distorted by group-velocity mismatch (GVM) is examined, and some pulse shortening is found possible. It is shown that the effect of IGVD at the SH wavelength on the pulse is similar to that of the spectral quadratic phase provided by an external pulse shaper. Group-velocity-matched configurations are also investigated. IGVD at both the fundamental and the SH wavelengths is found to limit the optimum thickness of the nonlinear medium. A measure of the interaction length in which the pulse width of the fundamental pulse is preserved in the SH is introduced. It is defined in terms of the GVM and the pulse bandwidth for phase-matched structures and in terms of the IGVD and pulse bandwidths for group-velocity-matched configurations.

© 1995 Optical Society of America

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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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  11. T. R. Zhang, H. R. Choo, and M. C. Downer, "Phase and group velocity matching for second harmonic generation of femtosecond pulses," Appl. Opt. 29, 3927 (1990).
    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef] [PubMed]
  14. E. Sidick, A. Knoesen, and A. Dienes, "Ultrashort pulse second harmonic generation in optimized nonlinear polymer thin film structures," Intern. J. Nonlin. Opt. Phys. 3, 543 (1994).
    [CrossRef]
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  16. G. Khanarian, M. A. Mortazavi, and A. J. East, "Phase-matched 2nd-harmonic generation from free-standing periodically stacked polymer films," Appl. Phys. Lett. 63, 1462 (1993).
    [CrossRef]
  17. M. A. Mortazavi and G. Khanarian, "Quasi-phase-matched frequency doubling in bulk periodic polymeric structures," Opt. Lett. 19, 1290 (1994).
    [CrossRef] [PubMed]
  18. D. R. Yankelevich, A. Dienes, A. Knoesen, R. W. Schoenlein, and C. V. Shank, "Generation of 312 nm, femtosecond pulses using a poled copolymer film," IEEE J. Quantum Electron. 28, 2398 (1992).
    [CrossRef]
  19. S. A. Akhmanov, A. S. Chirkin, K. N. Drabovich, A. I. Kovrigin, R. V. Khokhlov, and A. P. Sukhorukov, "Nonstationary nonlinear optical effects and ultrashort light pulse formation," IEEE J. Quantum Electron. QE-4, 598 (1968).
    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
  31. J. T. Manassah, "Effects of velocity dispersion on a generated second harmonic signal," Appl. Opt. 27, 4365 (1988).
    [CrossRef] [PubMed]
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    [CrossRef]
  33. A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986), Chap. 9.
  34. J. D. Harvey, J. M. Dudley, P. F. Curley, C. Spielmann, and F. Krausz, "Coherent effects in a self-mode-locked Ti:sapphire laser," Opt. Lett. 19, 972 (1994).
    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]

1995 (1)

1994 (7)

1993 (4)

K. Hayata and M. Koshiba, "Group-velocity-matched second-harmonic generation: an efficient scheme for femtosecond ultraviolet pulse generation in periodically domain-inverted β-BaB2O4," Appl. Phys. Lett. 62, 2188 (1993).
[CrossRef]

P. F. Curley, Ch. Spielmann, T. Brabec, F. Krausz, E. Wintner, and J. Schmidt, "Operation of a femtosecond Ti-sapphire solitary laser in the vicinity of zero group-delay dispersion," Opt. Lett. 18, 54 (1993).
[CrossRef] [PubMed]

M. T. Asaki, C. P. Huang, D. Garvey, J. P. Zhou, H. C. Kapteyn, and M. M. Murnane, "Generation of 11-fs pulses from a self-mode-locked Ti-sapphire laser," Opt. Lett. 18, 977 (1993).
[CrossRef] [PubMed]

G. Khanarian, M. A. Mortazavi, and A. J. East, "Phase-matched 2nd-harmonic generation from free-standing periodically stacked polymer films," Appl. Phys. Lett. 63, 1462 (1993).
[CrossRef]

1992 (1)

D. R. Yankelevich, A. Dienes, A. Knoesen, R. W. Schoenlein, and C. V. Shank, "Generation of 312 nm, femtosecond pulses using a poled copolymer film," IEEE J. Quantum Electron. 28, 2398 (1992).
[CrossRef]

1990 (3)

G. Szabó and Z. Bor, "Broadband frequency doubler for femtosecond pulses," Appl. Phys. B 50, 51 (1990).
[CrossRef]

T. R. Zhang, H. R. Choo, and M. C. Downer, "Phase and group velocity matching for second harmonic generation of femtosecond pulses," Appl. Opt. 29, 3927 (1990).
[CrossRef] [PubMed]

S. Lin, Z. Sun, B. Wu, and C. Chen, "The nonlinear optical characteristics of a LiB3O5 crystal," J. Appl. Phys. 67, 634 (1990).
[CrossRef]

1989 (3)

1988 (3)

1987 (2)

D. Eimerl, L. Davis, S. Velsko, E. K. Graham, and A. Zalkin, "Optical, mechanical, and thermal properties of barium borate," J. Appl. Phys. 62, 1968 (1987).
[CrossRef]

J. T. Manassah and O. R. Cockings, "Induced phase modulation of a generated second-harmonic signal," Opt. Lett. 12, 1005 (1987).
[CrossRef] [PubMed]

1984 (2)

R. C. Eckardt and J. Reintjes, "Phase matching limitations of high efficiency second harmonic generation," IEEE J. Quantum Electron. QE-20, 1178 (1984).
[CrossRef]

W. H. Knox, M. C. Downer, R. L. Fork, and C. V. Shank, "Amplified femtosecond optical pulses and continuum generation at a 5-kHz repetition rate," Opt. Lett. 9, 552 (1984).
[CrossRef] [PubMed]

1983 (1)

A. M. Weiner, "Effect of group velocity mismatch on the measurement of ultrashort optical pulses via second harmonic generation," IEEE J. Quantum Electron. QE-19, 1276 (1983).
[CrossRef]

1975 (1)

R. A. Fisher and W. K. Bischel, "Numerical studies of the interplay between self-phase modulation and dispersion for intense plane-wave laser pulses," J. Appl. Phys. 46, 4921 (1975).
[CrossRef]

1969 (2)

S. A. Akhmanov, A. P. Sukhorukov, and A. S. Chirkin, "Non-stationary phenomena and space-time analogy in nonlinear optics," Sov. Phys. JETP 28, 748 (1969).

W. H. Glenn, "Second harmonic generation from picosecond pulses," IEEE J. Quantum Electron. QE-5, 284 (1969).
[CrossRef]

1968 (2)

J. Comly and E. Garmire, "Second harmonic generation from short pulses," Appl. Phys. Lett. 12, 7 (1968).
[CrossRef]

S. A. Akhmanov, A. S. Chirkin, K. N. Drabovich, A. I. Kovrigin, R. V. Khokhlov, and A. P. Sukhorukov, "Nonstationary nonlinear optical effects and ultrashort light pulse formation," IEEE J. Quantum Electron. QE-4, 598 (1968).
[CrossRef]

1964 (1)

Akhmanov, S. A.

S. A. Akhmanov, A. P. Sukhorukov, and A. S. Chirkin, "Non-stationary phenomena and space-time analogy in nonlinear optics," Sov. Phys. JETP 28, 748 (1969).

S. A. Akhmanov, A. S. Chirkin, K. N. Drabovich, A. I. Kovrigin, R. V. Khokhlov, and A. P. Sukhorukov, "Nonstationary nonlinear optical effects and ultrashort light pulse formation," IEEE J. Quantum Electron. QE-4, 598 (1968).
[CrossRef]

Asaki, M. T.

Bakker, H. J.

Bischel, W. K.

R. A. Fisher and W. K. Bischel, "Numerical studies of the interplay between self-phase modulation and dispersion for intense plane-wave laser pulses," J. Appl. Phys. 46, 4921 (1975).
[CrossRef]

Bor, Z.

G. Szabó and Z. Bor, "Broadband frequency doubler for femtosecond pulses," Appl. Phys. B 50, 51 (1990).
[CrossRef]

Boyd, R. W.

R. W. Boyd, Nonlinear Optics (Academic, San Diego, Calif., 1992), p. 52.

Brabec, T.

Carlotti, X.

Chen, C.

S. Lin, Z. Sun, B. Wu, and C. Chen, "The nonlinear optical characteristics of a LiB3O5 crystal," J. Appl. Phys. 67, 634 (1990).
[CrossRef]

B. Wu, N. Chen, C. Chen, D. Deng, and Z. Xu, "Highly efficient ultraviolet generation at 355 nm in LiB3O5," Opt. Lett. 14, 1080 (1989).
[CrossRef] [PubMed]

Chen, N.

Chirkin, A. S.

S. A. Akhmanov, A. P. Sukhorukov, and A. S. Chirkin, "Non-stationary phenomena and space-time analogy in nonlinear optics," Sov. Phys. JETP 28, 748 (1969).

S. A. Akhmanov, A. S. Chirkin, K. N. Drabovich, A. I. Kovrigin, R. V. Khokhlov, and A. P. Sukhorukov, "Nonstationary nonlinear optical effects and ultrashort light pulse formation," IEEE J. Quantum Electron. QE-4, 598 (1968).
[CrossRef]

Choo, H. R.

Christov, I. P.

Cockings, O. R.

Comly, J.

J. Comly and E. Garmire, "Second harmonic generation from short pulses," Appl. Phys. Lett. 12, 7 (1968).
[CrossRef]

Curley, P. F.

Davis, L.

D. Eimerl, L. Davis, S. Velsko, E. K. Graham, and A. Zalkin, "Optical, mechanical, and thermal properties of barium borate," J. Appl. Phys. 62, 1968 (1987).
[CrossRef]

Deng, D.

Dienes, A.

E. Sidick, A. Dienes, and A. Knoesen, "Ultrashort-pulse second-harmonic generation. II. Non-transform-limited fundamental pulses," J. Opt. Soc. Am. B 12, 1713 (1995).
[CrossRef]

E. Sidick, A. Knoesen, and A. Dienes, "Ultrashort pulse second harmonic generation in optimized nonlinear polymer thin film structures," Intern. J. Nonlin. Opt. Phys. 3, 543 (1994).
[CrossRef]

E. Sidick, A. Knoesen, and A. Dienes, "Ultrashort pulse second-harmonic generation in quasi-phase-matched dispersive media," Opt. Lett. 19, 266 (1994).
[CrossRef] [PubMed]

D. R. Yankelevich, A. Dienes, A. Knoesen, R. W. Schoenlein, and C. V. Shank, "Generation of 312 nm, femtosecond pulses using a poled copolymer film," IEEE J. Quantum Electron. 28, 2398 (1992).
[CrossRef]

Downer, M. C.

Drabovich, K. N.

S. A. Akhmanov, A. S. Chirkin, K. N. Drabovich, A. I. Kovrigin, R. V. Khokhlov, and A. P. Sukhorukov, "Nonstationary nonlinear optical effects and ultrashort light pulse formation," IEEE J. Quantum Electron. QE-4, 598 (1968).
[CrossRef]

Dudley, J. M.

East, A. J.

G. Khanarian, M. A. Mortazavi, and A. J. East, "Phase-matched 2nd-harmonic generation from free-standing periodically stacked polymer films," Appl. Phys. Lett. 63, 1462 (1993).
[CrossRef]

Eckardt, R. C.

R. C. Eckardt and J. Reintjes, "Phase matching limitations of high efficiency second harmonic generation," IEEE J. Quantum Electron. QE-20, 1178 (1984).
[CrossRef]

Eimerl, D.

D. Eimerl, L. Davis, S. Velsko, E. K. Graham, and A. Zalkin, "Optical, mechanical, and thermal properties of barium borate," J. Appl. Phys. 62, 1968 (1987).
[CrossRef]

Fisher, R. A.

R. A. Fisher and W. K. Bischel, "Numerical studies of the interplay between self-phase modulation and dispersion for intense plane-wave laser pulses," J. Appl. Phys. 46, 4921 (1975).
[CrossRef]

Fork, R. L.

Garmire, E.

J. Comly and E. Garmire, "Second harmonic generation from short pulses," Appl. Phys. Lett. 12, 7 (1968).
[CrossRef]

Garvey, D.

Glenn, W. H.

W. H. Glenn, "Second harmonic generation from picosecond pulses," IEEE J. Quantum Electron. QE-5, 284 (1969).
[CrossRef]

Graham, E. K.

D. Eimerl, L. Davis, S. Velsko, E. K. Graham, and A. Zalkin, "Optical, mechanical, and thermal properties of barium borate," J. Appl. Phys. 62, 1968 (1987).
[CrossRef]

Harvey, J. D.

Hayata, K.

K. Hayata and M. Koshiba, "Group-velocity-matched second-harmonic generation: an efficient scheme for femtosecond ultraviolet pulse generation in periodically domain-inverted β-BaB2O4," Appl. Phys. Lett. 62, 2188 (1993).
[CrossRef]

Heritage, J. P.

A. M. Weiner, J. P. Heritage, and E. M. Kirschner, "High-resolution femtosecond pulse shaping," J. Opt. Soc. Am. B 8, 1563 (1988).
[CrossRef]

Huang, C. P.

Kapteyn, H. C.

Khanarian, G.

M. A. Mortazavi and G. Khanarian, "Quasi-phase-matched frequency doubling in bulk periodic polymeric structures," Opt. Lett. 19, 1290 (1994).
[CrossRef] [PubMed]

G. Khanarian, M. A. Mortazavi, and A. J. East, "Phase-matched 2nd-harmonic generation from free-standing periodically stacked polymer films," Appl. Phys. Lett. 63, 1462 (1993).
[CrossRef]

Khokhlov, R. V.

S. A. Akhmanov, A. S. Chirkin, K. N. Drabovich, A. I. Kovrigin, R. V. Khokhlov, and A. P. Sukhorukov, "Nonstationary nonlinear optical effects and ultrashort light pulse formation," IEEE J. Quantum Electron. QE-4, 598 (1968).
[CrossRef]

Kirschner, E. M.

A. M. Weiner, J. P. Heritage, and E. M. Kirschner, "High-resolution femtosecond pulse shaping," J. Opt. Soc. Am. B 8, 1563 (1988).
[CrossRef]

Knoesen, A.

E. Sidick, A. Dienes, and A. Knoesen, "Ultrashort-pulse second-harmonic generation. II. Non-transform-limited fundamental pulses," J. Opt. Soc. Am. B 12, 1713 (1995).
[CrossRef]

E. Sidick, A. Knoesen, and A. Dienes, "Ultrashort pulse second-harmonic generation in quasi-phase-matched dispersive media," Opt. Lett. 19, 266 (1994).
[CrossRef] [PubMed]

E. Sidick, A. Knoesen, and A. Dienes, "Ultrashort pulse second harmonic generation in optimized nonlinear polymer thin film structures," Intern. J. Nonlin. Opt. Phys. 3, 543 (1994).
[CrossRef]

D. R. Yankelevich, A. Dienes, A. Knoesen, R. W. Schoenlein, and C. V. Shank, "Generation of 312 nm, femtosecond pulses using a poled copolymer film," IEEE J. Quantum Electron. 28, 2398 (1992).
[CrossRef]

Knox, W. H.

Koshiba, M.

K. Hayata and M. Koshiba, "Group-velocity-matched second-harmonic generation: an efficient scheme for femtosecond ultraviolet pulse generation in periodically domain-inverted β-BaB2O4," Appl. Phys. Lett. 62, 2188 (1993).
[CrossRef]

Kothari, N. C.

Kovrigin, A. I.

S. A. Akhmanov, A. S. Chirkin, K. N. Drabovich, A. I. Kovrigin, R. V. Khokhlov, and A. P. Sukhorukov, "Nonstationary nonlinear optical effects and ultrashort light pulse formation," IEEE J. Quantum Electron. QE-4, 598 (1968).
[CrossRef]

Krausz, F.

Lin, S.

S. Lin, Z. Sun, B. Wu, and C. Chen, "The nonlinear optical characteristics of a LiB3O5 crystal," J. Appl. Phys. 67, 634 (1990).
[CrossRef]

Manassah, J. T.

Martinez, O. E.

O. E. Martinez, "Achromatic phase matching for second harmonic generation of femtosecond pulses," IEEE J. Quantum Electron. 25, 2464 (1989).
[CrossRef]

Mortazavi, M. A.

M. A. Mortazavi and G. Khanarian, "Quasi-phase-matched frequency doubling in bulk periodic polymeric structures," Opt. Lett. 19, 1290 (1994).
[CrossRef] [PubMed]

G. Khanarian, M. A. Mortazavi, and A. J. East, "Phase-matched 2nd-harmonic generation from free-standing periodically stacked polymer films," Appl. Phys. Lett. 63, 1462 (1993).
[CrossRef]

Muller, H. G.

Murnane, M. M.

Planken, P. C. M.

Reintjes, J.

R. C. Eckardt and J. Reintjes, "Phase matching limitations of high efficiency second harmonic generation," IEEE J. Quantum Electron. QE-20, 1178 (1984).
[CrossRef]

Schmidt, J.

Schoenlein, R. W.

D. R. Yankelevich, A. Dienes, A. Knoesen, R. W. Schoenlein, and C. V. Shank, "Generation of 312 nm, femtosecond pulses using a poled copolymer film," IEEE J. Quantum Electron. 28, 2398 (1992).
[CrossRef]

Shank, C. V.

D. R. Yankelevich, A. Dienes, A. Knoesen, R. W. Schoenlein, and C. V. Shank, "Generation of 312 nm, femtosecond pulses using a poled copolymer film," IEEE J. Quantum Electron. 28, 2398 (1992).
[CrossRef]

W. H. Knox, M. C. Downer, R. L. Fork, and C. V. Shank, "Amplified femtosecond optical pulses and continuum generation at a 5-kHz repetition rate," Opt. Lett. 9, 552 (1984).
[CrossRef] [PubMed]

Sidick, E.

Siegman, A. E.

A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986), Chap. 9.

Spielmann, C.

Spielmann, Ch.

Stingl, A.

Sukhorukov, A. P.

S. A. Akhmanov, A. P. Sukhorukov, and A. S. Chirkin, "Non-stationary phenomena and space-time analogy in nonlinear optics," Sov. Phys. JETP 28, 748 (1969).

S. A. Akhmanov, A. S. Chirkin, K. N. Drabovich, A. I. Kovrigin, R. V. Khokhlov, and A. P. Sukhorukov, "Nonstationary nonlinear optical effects and ultrashort light pulse formation," IEEE J. Quantum Electron. QE-4, 598 (1968).
[CrossRef]

Sun, Z.

S. Lin, Z. Sun, B. Wu, and C. Chen, "The nonlinear optical characteristics of a LiB3O5 crystal," J. Appl. Phys. 67, 634 (1990).
[CrossRef]

Szabó, G.

G. Szabó and Z. Bor, "Broadband frequency doubler for femtosecond pulses," Appl. Phys. B 50, 51 (1990).
[CrossRef]

Szipöcs, R.

Taft, G.

Velsko, S.

D. Eimerl, L. Davis, S. Velsko, E. K. Graham, and A. Zalkin, "Optical, mechanical, and thermal properties of barium borate," J. Appl. Phys. 62, 1968 (1987).
[CrossRef]

Weiner, A. M.

A. M. Weiner, J. P. Heritage, and E. M. Kirschner, "High-resolution femtosecond pulse shaping," J. Opt. Soc. Am. B 8, 1563 (1988).
[CrossRef]

A. M. Weiner, "Effect of group velocity mismatch on the measurement of ultrashort optical pulses via second harmonic generation," IEEE J. Quantum Electron. QE-19, 1276 (1983).
[CrossRef]

Wintner, E.

Wu, B.

S. Lin, Z. Sun, B. Wu, and C. Chen, "The nonlinear optical characteristics of a LiB3O5 crystal," J. Appl. Phys. 67, 634 (1990).
[CrossRef]

B. Wu, N. Chen, C. Chen, D. Deng, and Z. Xu, "Highly efficient ultraviolet generation at 355 nm in LiB3O5," Opt. Lett. 14, 1080 (1989).
[CrossRef] [PubMed]

Xu, Z.

Yankelevich, D. R.

D. R. Yankelevich, A. Dienes, A. Knoesen, R. W. Schoenlein, and C. V. Shank, "Generation of 312 nm, femtosecond pulses using a poled copolymer film," IEEE J. Quantum Electron. 28, 2398 (1992).
[CrossRef]

Zalkin, A.

D. Eimerl, L. Davis, S. Velsko, E. K. Graham, and A. Zalkin, "Optical, mechanical, and thermal properties of barium borate," J. Appl. Phys. 62, 1968 (1987).
[CrossRef]

Zernike, F.

Zhang, T. R.

Zhou, J. P.

Appl. Opt. (2)

Appl. Phys. B (1)

G. Szabó and Z. Bor, "Broadband frequency doubler for femtosecond pulses," Appl. Phys. B 50, 51 (1990).
[CrossRef]

Appl. Phys. Lett. (3)

K. Hayata and M. Koshiba, "Group-velocity-matched second-harmonic generation: an efficient scheme for femtosecond ultraviolet pulse generation in periodically domain-inverted β-BaB2O4," Appl. Phys. Lett. 62, 2188 (1993).
[CrossRef]

G. Khanarian, M. A. Mortazavi, and A. J. East, "Phase-matched 2nd-harmonic generation from free-standing periodically stacked polymer films," Appl. Phys. Lett. 63, 1462 (1993).
[CrossRef]

J. Comly and E. Garmire, "Second harmonic generation from short pulses," Appl. Phys. Lett. 12, 7 (1968).
[CrossRef]

IEEE J. Quantum Electron. (6)

W. H. Glenn, "Second harmonic generation from picosecond pulses," IEEE J. Quantum Electron. QE-5, 284 (1969).
[CrossRef]

A. M. Weiner, "Effect of group velocity mismatch on the measurement of ultrashort optical pulses via second harmonic generation," IEEE J. Quantum Electron. QE-19, 1276 (1983).
[CrossRef]

O. E. Martinez, "Achromatic phase matching for second harmonic generation of femtosecond pulses," IEEE J. Quantum Electron. 25, 2464 (1989).
[CrossRef]

D. R. Yankelevich, A. Dienes, A. Knoesen, R. W. Schoenlein, and C. V. Shank, "Generation of 312 nm, femtosecond pulses using a poled copolymer film," IEEE J. Quantum Electron. 28, 2398 (1992).
[CrossRef]

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[CrossRef] [PubMed]

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[CrossRef] [PubMed]

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[CrossRef] [PubMed]

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Other (2)

R. W. Boyd, Nonlinear Optics (Academic, San Diego, Calif., 1992), p. 52.

A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986), Chap. 9.

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Figures (9)

Fig. 1
Fig. 1

Schematic of a nonlinear medium used for ultra-short-pulse SHG. The fundamental and the SH pulses propagate obliquely in the xz plane inside the medium, making angles of θ1 and θ2 from the positive z direction.

Fig. 2
Fig. 2

(a) Pulse-width-preservation length Lτ, (b) effective in-trapulse GVD parameters ξ1Lτ and ξ2Lτ, (c) normalized coupling coefficient ρLτ of KDP, LBO, and BBO as functions of the fundamental wavelength λ1 for a fundamental pulse width of 10 fs.

Fig. 3
Fig. 3

(a) Peak intensity I2 and (b) pulsewidth τp2 (in units of the fundamental pulse width τp1) of the SH versus the crystal thickness L (in units of the pulse-width-preservation length Lτ) for I0 = 100 GW/cm2 (solid curves) and I0 = 1000 GW/cm2 (dashed curves). Other parameters are ρLτ = 200 pm/V and ξ1Lτ = 0. In (a) the I2 corresponding to I0 = 100 GW/cm2 is magnified by a factor of 100.

Fig. 4
Fig. 4

SH pulse profile corresponding to the case of I0 = 1000 GW/cm2 in Fig. 2. (a) ξ2Lτ = 0, (b) ξ2Lτ = 0.5. T/1.763 is the time normalized by the fundamental pulse width τp1.

Fig. 5
Fig. 5

(a) SH pulse spectral amplitude sinc(ΩηL/2)/sinhc(πΩ/2) and quadratic spectral phase b2Ω2 versus the normalized frequency Ω = (ω − 2ω0)τ. (b) SH pulse profiles: solid curves, original pulse (τp2/τp1 = 2.9); dashed curve, pulse obtained after the replacement sinc(ΩηL/2) → |sinc(ΩηL/2)| is made (τp2/τp1 = 1.5); dotted curve, pulse shaped by a quadratic phase compensator with b2 = 1.19 (τp2/τp1 = 1.8). L = 3Lτ.

Fig. 6
Fig. 6

Normalized SH pulse width τp2/τp1 versus the phase coefficient b2 for a transform-limited fundamental pulse.

Fig. 7
Fig. 7

Same as Fig. (6), except that ξ2Lτ = 0.5.

Fig. 8
Fig. 8

SH pulse profiles before (solid curve) and after (dashed curve) shaping. The SH pulses were calculated for ξ2Lτ = 0.5 and L = 3Lτ. The value of the shaping parameter b2 used is 0.98, at which τp2 is a minimum.

Fig. 9
Fig. 9

(a) Conversion efficiency and (b) the widths of the fundamental and the SH pulses as functions of the number of periods in a periodically domain-inverted BBO crystal. In (b) ξ1Lc = 0.0032 and ξ2Lc = 0.0058. Other parameters are initial fundamental pulse width τp1 = 10 fs, nonlinear coherence length Lc = 2 μm, I0 = 500 GW/cm2, and ρ1Lc = ρ2Lc = 0.59 pm/V.

Equations (30)

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2 E x 2 + 2 E z 2 - μ 0 0 2 E t 2 - μ 0 2 P L t 2 = μ 0 2 P NL t 2 ,
E = E 1 + E 2 ,
E 1 = ( 1 / 2 ) A ( z , t ) exp [ i ( ω 0 t - k 1 x sin θ 1 - k 1 z cos θ 1 ) ] + c . c . ,
E 2 = ( 1 / 2 ) B ( z , t ) exp [ i ( 2 ω 0 t - k 2 x sin θ 2 - k 2 z cos θ 2 ) ] + c . c . ,
P NL = 0 d eff E 2 ,
2 A z 2 - i 2 k 1 cos θ 1 A z - i 2 k 1 k ˙ 1 A t - ( k ˙ 1 2 + k 1 k ¨ 1 ) 2 A t 2 = i ω 0 μ 0 σ 1 A - ( ω 0 / c ) 2 d eff A * B exp ( - i Δ k z ) ,
2 B z 2 - i 2 k 2 cos θ 2 B z - i 2 k 2 k ˙ 2 B t - ( k ˙ 2 2 + k 2 k ¨ 2 ) 2 B t 2 = i 2 ω 0 μ 0 σ 2 B - 2 ( ω 0 / c ) 2 d eff A 2 exp ( i Δ k z ) ,
Δ k = k 2 cos θ 2 - 2 k 1 cos θ 1 = ( 4 π / λ 1 ) ( n 2 cos θ 2 - n 1 cos θ 1 ) .
A z = η 2 A T + i ξ 1 2 A T 2 - i ρ 1 A * B ,
B z = - η 2 B T + i ξ 2 2 B T 2 - ( α 2 + i Δ k ) B - i ρ 2 A 2 ,
η = ( γ 2 - γ 1 ) / τ , ξ i = ( k ˙ i 2 - γ 2 + k i k ¨ i ) / 2 k i τ 2 cos     θ i , ρ i = ω 0 d eff / 2 n i c cos θ i , α 2 = σ 2 ω 0 μ 0 / k 2 cos θ 2 .
A z = η 2 A T + i ξ 1 2 A T 2 ,
B z = - η 2 B T + i ξ 2 2 B T 2 - ( α 2 + i Δ k ) B ,
A z = - i ρ 1 A * B ,
B z = - i ρ 2 A 2 .
A z = η 2 A T ,
B z = - η 2 B T - ( α 2 + i Δ k ) B - i ρ 2 A 2 .
A ( z , T ) = A 0 f ( T + η z / 2 ) ,
B ( L , T ) = - i B 0 exp ( - α 2 L ) 0 1 d ζ f 2 [ T + η L ( ζ - 1 / 2 ) ] × exp [ ( α 2 + i Δ k ) L ζ ] ,
B ( L , T ) = - i ρ A 0 2 [ tanh ( T + η L / 2 ) - tanh ( T - η L / 2 ) ] / η .
L τ = 2.02 / η ,
L τ = 0.36 Δ f B 1 γ 2 - γ 1 .
L τ = 0.54 Δ f B 1 γ 2 - γ 1 .
B ˜ ( L , ω ) = - i ( B 0 τ / 2 π ) exp ( Φ - α 2 L / 2 ) × sinhc ( Φ ) [ F ˜ ( ω ) F ˜ ( ω ) ] ,
B ˜ ( L , ω ) = - i 2 B 0 τ exp ( Φ - α 2 L / 2 ) sinhc ( Φ ) sinhc ( π Ω / 2 ) .
B ˜ ( L , ω ) = - i 2 τ B 0 exp ( i Ω η L / 2 ) sinhc ( Ω η L / 2 ) sinhc ( π Ω / 2 ) .
B ˜ c ( L , ω ) = B ˜ ( L , ω ) exp ( - i b 2 Ω 2 ) ,
b 2 = ξ 2 L τ / 2.
L τ 0.4 L D 1 + 0.8 L D 2
L τ 0.2 L D 1 + 0.5 L D 2

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