Abstract

Second-harmonic generation (SHG) is investigated for a dispersive medium with quadratic and cubic nonlinearities. All possible solutions assuming a zero initial value for the second harmonic are obtained and tabulated. Based on the analysis, the optimal condition for SHG is deduced and the dependence of the SHG conversion efficiency upon the nonlinearity, the dispersion, and the input field intensity is discussed. Also, a thorough numerical study of the behavior of SHG efficiency is performed and the results are reported. A technique for estimation of the unknown third-order nonlinearity coefficients of a material from SHG observation is proposed.

© 1991 Optical Society of America

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References

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  1. P. A. Franken, A. E. Hill, C. W. Peters, and G. W. Weinreigh, “Generation of optical harmonics,” Phys. Rev. Lett. 7, 118–119 (1961).
    [Crossref]
  2. J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interactions between light waves in a nonlinear dielectric,” Phys. Rev. 127, 1918–1939 (1962).
    [Crossref]
  3. P. D. Maker and R. W. Terhune, “Study of optical effects due to an induced polarization third order in the electric field strength,” Phys. Rev. 137, A801–A818 (1965).
    [Crossref]
  4. G. D. Boyd and D. A. Kleinman, “Parametric interaction of focused Gaussian light beams,” J. Appl. Phys. 39, 3597–3639 (1968).
    [Crossref]
  5. M. M. T. Loy and Y. R. Shen, “Small-scale filaments in liquids and tracks of moving foci,” Phys. Rev. Lett. 22, 994–997 (1969).
    [Crossref]
  6. Y. R. Shen, “Self-focusing: experimental,” Prog. Quantum Electron. 4, 1–34 (1975).
    [Crossref]
  7. J. H. Marburger, “Self-focusing: theory,” Prog. Quantum Electron. 4, 35–110 (1975).
    [Crossref]
  8. S. A. Magnitskii, V. I. Pryalkin, V. G. Tunkin, and A. I. Kholodnykh, “Influence of optical inhomogeneity on the parametric amplification of picosecond light pulses in lithium niobate crystals,” Sov. J. Quantum Electron. 12, 900–903 (1982).
    [Crossref]
  9. R. S. Adhav and R. W. Wallace, “Second harmonic generation in 90° phase-matched KDP isomorphs,” IEEE J. Quantum Electron. QE-9, 855–856 (1973).
    [Crossref]
  10. N. C. Kothari and X. Carlotti, “Transient second-harmonic generation: influence of effective group velocity dispersion,” J. Opt. Soc. Am. B 5, 756–764 (1988).
    [Crossref]
  11. R. Danelyus, G. Dikchyus, V. Kabelka, A. Piskarskas, A. Stabinis, and Ya. Yasevichyute, “Parametric excitation of light in the picosecond range,” Sov. J. Quantum Electron. 7, 1360–1368 (1977).
    [Crossref]
  12. R. Yu. Orlov, I. B. Skidan, and L. S. Telegin, “Investigation of breakdown produced in dielectrics by ultrashort laser pulses,” Sov. Phys. JETP 34, 418–421 (1972).
  13. S. A. Akhmanov and R. V. Khokhlov, Nonlinear Optics (Gordon & Breach, New York, 1972).
  14. D. J. Harter and D. C. Brown, “Effects of higher order nonlinearities on second-order frequency mixing,” IEEE J. Quantum Electron. QE-18, 1146–1151 (1982).
    [Crossref]
  15. L. S. Telegin and A. S. Chirkin, “Interaction in frequency doubling of ultrashort laser pulses,” Sov. J. Quantum Electron. 12, 1354–1356 (1982).
    [Crossref]
  16. T. B. Razumikhina, L. S. Telegin, A. I. Kholodnykh, and A. S. Chirkin, “Three-frequency interactions of high-intensity light waves in media with quadratic and cubic nonlinearities,” Sov. J. Quantum Electron. 14, 1358–1363 (1984).
    [Crossref]
  17. F. Zernike and J. E. Midwinter, Applied Nonlinear Optics (Wiley, New York, 1973), Chap. 3.
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    [Crossref]
  19. A. Yariv and P. Yeh, Optical Waves in Crystals (Wiley, New York, 1984), Chap. 12.
  20. Y. R. Shen, The Principles of Nonlinear Optics (Wiley, New York, 1984), Chaps. 1–3, 6, 7.
  21. G. P. Agrawal, Nonlinear Fiber Optics (Academic, New York, 1989), Chap. 2.
  22. V. I. Karpman, Nonlinear Waves in Dispersive Media (Pergamon, Oxford, 1975), Chap. 5.
  23. A. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1965).
  24. I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, New York, 1965).
  25. E. W. Van Stryland, W. E. Williams, M. J. Soileau, and A. L. Smirl, “Laser induced damage, nonlinear absorption, and doubling efficiency of LiIO3,” IEEE J. Quantum Electron. QE-20, 434–439 (1984).
    [Crossref]
  26. C. J. McKinstrie, G. G. Luther, and S. H. Batha, “Signal enhancement in collinear four-wave mixing,” J. Opt. Soc. Am. B 7, 340–344 (1990).
    [Crossref]
  27. Y. Chen and A. W. Snyder, “Four-photon parametric mixing in optical fibers: effect of pump depletion,” Opt. Lett. 14, 87–89 (1989).
    [Crossref] [PubMed]
  28. Y. Chen, “Four-wave mixing in optical fibers: exact solution,” J. Opt. Soc. Am. B 6, 1986–1993 (1989).
    [Crossref]

1990 (1)

1989 (2)

1988 (1)

1984 (2)

T. B. Razumikhina, L. S. Telegin, A. I. Kholodnykh, and A. S. Chirkin, “Three-frequency interactions of high-intensity light waves in media with quadratic and cubic nonlinearities,” Sov. J. Quantum Electron. 14, 1358–1363 (1984).
[Crossref]

E. W. Van Stryland, W. E. Williams, M. J. Soileau, and A. L. Smirl, “Laser induced damage, nonlinear absorption, and doubling efficiency of LiIO3,” IEEE J. Quantum Electron. QE-20, 434–439 (1984).
[Crossref]

1982 (3)

D. J. Harter and D. C. Brown, “Effects of higher order nonlinearities on second-order frequency mixing,” IEEE J. Quantum Electron. QE-18, 1146–1151 (1982).
[Crossref]

L. S. Telegin and A. S. Chirkin, “Interaction in frequency doubling of ultrashort laser pulses,” Sov. J. Quantum Electron. 12, 1354–1356 (1982).
[Crossref]

S. A. Magnitskii, V. I. Pryalkin, V. G. Tunkin, and A. I. Kholodnykh, “Influence of optical inhomogeneity on the parametric amplification of picosecond light pulses in lithium niobate crystals,” Sov. J. Quantum Electron. 12, 900–903 (1982).
[Crossref]

1981 (1)

R. S. Craxton, “High efficiency frequency tripling schemes for high-power Nd:glass lasers,” IEEE J. Quantum Electron. QE-17, 1771–1782 (1981), App. I.
[Crossref]

1977 (1)

R. Danelyus, G. Dikchyus, V. Kabelka, A. Piskarskas, A. Stabinis, and Ya. Yasevichyute, “Parametric excitation of light in the picosecond range,” Sov. J. Quantum Electron. 7, 1360–1368 (1977).
[Crossref]

1975 (2)

Y. R. Shen, “Self-focusing: experimental,” Prog. Quantum Electron. 4, 1–34 (1975).
[Crossref]

J. H. Marburger, “Self-focusing: theory,” Prog. Quantum Electron. 4, 35–110 (1975).
[Crossref]

1973 (1)

R. S. Adhav and R. W. Wallace, “Second harmonic generation in 90° phase-matched KDP isomorphs,” IEEE J. Quantum Electron. QE-9, 855–856 (1973).
[Crossref]

1972 (1)

R. Yu. Orlov, I. B. Skidan, and L. S. Telegin, “Investigation of breakdown produced in dielectrics by ultrashort laser pulses,” Sov. Phys. JETP 34, 418–421 (1972).

1969 (1)

M. M. T. Loy and Y. R. Shen, “Small-scale filaments in liquids and tracks of moving foci,” Phys. Rev. Lett. 22, 994–997 (1969).
[Crossref]

1968 (1)

G. D. Boyd and D. A. Kleinman, “Parametric interaction of focused Gaussian light beams,” J. Appl. Phys. 39, 3597–3639 (1968).
[Crossref]

1965 (1)

P. D. Maker and R. W. Terhune, “Study of optical effects due to an induced polarization third order in the electric field strength,” Phys. Rev. 137, A801–A818 (1965).
[Crossref]

1962 (1)

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interactions between light waves in a nonlinear dielectric,” Phys. Rev. 127, 1918–1939 (1962).
[Crossref]

1961 (1)

P. A. Franken, A. E. Hill, C. W. Peters, and G. W. Weinreigh, “Generation of optical harmonics,” Phys. Rev. Lett. 7, 118–119 (1961).
[Crossref]

Abramowitz, A.

A. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1965).

Adhav, R. S.

R. S. Adhav and R. W. Wallace, “Second harmonic generation in 90° phase-matched KDP isomorphs,” IEEE J. Quantum Electron. QE-9, 855–856 (1973).
[Crossref]

Agrawal, G. P.

G. P. Agrawal, Nonlinear Fiber Optics (Academic, New York, 1989), Chap. 2.

Akhmanov, S. A.

S. A. Akhmanov and R. V. Khokhlov, Nonlinear Optics (Gordon & Breach, New York, 1972).

Armstrong, J. A.

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interactions between light waves in a nonlinear dielectric,” Phys. Rev. 127, 1918–1939 (1962).
[Crossref]

Batha, S. H.

Bloembergen, N.

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interactions between light waves in a nonlinear dielectric,” Phys. Rev. 127, 1918–1939 (1962).
[Crossref]

Boyd, G. D.

G. D. Boyd and D. A. Kleinman, “Parametric interaction of focused Gaussian light beams,” J. Appl. Phys. 39, 3597–3639 (1968).
[Crossref]

Brown, D. C.

D. J. Harter and D. C. Brown, “Effects of higher order nonlinearities on second-order frequency mixing,” IEEE J. Quantum Electron. QE-18, 1146–1151 (1982).
[Crossref]

Carlotti, X.

Chen, Y.

Chirkin, A. S.

T. B. Razumikhina, L. S. Telegin, A. I. Kholodnykh, and A. S. Chirkin, “Three-frequency interactions of high-intensity light waves in media with quadratic and cubic nonlinearities,” Sov. J. Quantum Electron. 14, 1358–1363 (1984).
[Crossref]

L. S. Telegin and A. S. Chirkin, “Interaction in frequency doubling of ultrashort laser pulses,” Sov. J. Quantum Electron. 12, 1354–1356 (1982).
[Crossref]

Craxton, R. S.

R. S. Craxton, “High efficiency frequency tripling schemes for high-power Nd:glass lasers,” IEEE J. Quantum Electron. QE-17, 1771–1782 (1981), App. I.
[Crossref]

Danelyus, R.

R. Danelyus, G. Dikchyus, V. Kabelka, A. Piskarskas, A. Stabinis, and Ya. Yasevichyute, “Parametric excitation of light in the picosecond range,” Sov. J. Quantum Electron. 7, 1360–1368 (1977).
[Crossref]

Dikchyus, G.

R. Danelyus, G. Dikchyus, V. Kabelka, A. Piskarskas, A. Stabinis, and Ya. Yasevichyute, “Parametric excitation of light in the picosecond range,” Sov. J. Quantum Electron. 7, 1360–1368 (1977).
[Crossref]

Ducuing, J.

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interactions between light waves in a nonlinear dielectric,” Phys. Rev. 127, 1918–1939 (1962).
[Crossref]

Franken, P. A.

P. A. Franken, A. E. Hill, C. W. Peters, and G. W. Weinreigh, “Generation of optical harmonics,” Phys. Rev. Lett. 7, 118–119 (1961).
[Crossref]

Gradshteyn, I. S.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, New York, 1965).

Harter, D. J.

D. J. Harter and D. C. Brown, “Effects of higher order nonlinearities on second-order frequency mixing,” IEEE J. Quantum Electron. QE-18, 1146–1151 (1982).
[Crossref]

Hill, A. E.

P. A. Franken, A. E. Hill, C. W. Peters, and G. W. Weinreigh, “Generation of optical harmonics,” Phys. Rev. Lett. 7, 118–119 (1961).
[Crossref]

Kabelka, V.

R. Danelyus, G. Dikchyus, V. Kabelka, A. Piskarskas, A. Stabinis, and Ya. Yasevichyute, “Parametric excitation of light in the picosecond range,” Sov. J. Quantum Electron. 7, 1360–1368 (1977).
[Crossref]

Karpman, V. I.

V. I. Karpman, Nonlinear Waves in Dispersive Media (Pergamon, Oxford, 1975), Chap. 5.

Khokhlov, R. V.

S. A. Akhmanov and R. V. Khokhlov, Nonlinear Optics (Gordon & Breach, New York, 1972).

Kholodnykh, A. I.

T. B. Razumikhina, L. S. Telegin, A. I. Kholodnykh, and A. S. Chirkin, “Three-frequency interactions of high-intensity light waves in media with quadratic and cubic nonlinearities,” Sov. J. Quantum Electron. 14, 1358–1363 (1984).
[Crossref]

S. A. Magnitskii, V. I. Pryalkin, V. G. Tunkin, and A. I. Kholodnykh, “Influence of optical inhomogeneity on the parametric amplification of picosecond light pulses in lithium niobate crystals,” Sov. J. Quantum Electron. 12, 900–903 (1982).
[Crossref]

Kleinman, D. A.

G. D. Boyd and D. A. Kleinman, “Parametric interaction of focused Gaussian light beams,” J. Appl. Phys. 39, 3597–3639 (1968).
[Crossref]

Kothari, N. C.

Loy, M. M. T.

M. M. T. Loy and Y. R. Shen, “Small-scale filaments in liquids and tracks of moving foci,” Phys. Rev. Lett. 22, 994–997 (1969).
[Crossref]

Luther, G. G.

Magnitskii, S. A.

S. A. Magnitskii, V. I. Pryalkin, V. G. Tunkin, and A. I. Kholodnykh, “Influence of optical inhomogeneity on the parametric amplification of picosecond light pulses in lithium niobate crystals,” Sov. J. Quantum Electron. 12, 900–903 (1982).
[Crossref]

Maker, P. D.

P. D. Maker and R. W. Terhune, “Study of optical effects due to an induced polarization third order in the electric field strength,” Phys. Rev. 137, A801–A818 (1965).
[Crossref]

Marburger, J. H.

J. H. Marburger, “Self-focusing: theory,” Prog. Quantum Electron. 4, 35–110 (1975).
[Crossref]

McKinstrie, C. J.

Midwinter, J. E.

F. Zernike and J. E. Midwinter, Applied Nonlinear Optics (Wiley, New York, 1973), Chap. 3.

Orlov, R. Yu.

R. Yu. Orlov, I. B. Skidan, and L. S. Telegin, “Investigation of breakdown produced in dielectrics by ultrashort laser pulses,” Sov. Phys. JETP 34, 418–421 (1972).

Pershan, P. S.

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interactions between light waves in a nonlinear dielectric,” Phys. Rev. 127, 1918–1939 (1962).
[Crossref]

Peters, C. W.

P. A. Franken, A. E. Hill, C. W. Peters, and G. W. Weinreigh, “Generation of optical harmonics,” Phys. Rev. Lett. 7, 118–119 (1961).
[Crossref]

Piskarskas, A.

R. Danelyus, G. Dikchyus, V. Kabelka, A. Piskarskas, A. Stabinis, and Ya. Yasevichyute, “Parametric excitation of light in the picosecond range,” Sov. J. Quantum Electron. 7, 1360–1368 (1977).
[Crossref]

Pryalkin, V. I.

S. A. Magnitskii, V. I. Pryalkin, V. G. Tunkin, and A. I. Kholodnykh, “Influence of optical inhomogeneity on the parametric amplification of picosecond light pulses in lithium niobate crystals,” Sov. J. Quantum Electron. 12, 900–903 (1982).
[Crossref]

Razumikhina, T. B.

T. B. Razumikhina, L. S. Telegin, A. I. Kholodnykh, and A. S. Chirkin, “Three-frequency interactions of high-intensity light waves in media with quadratic and cubic nonlinearities,” Sov. J. Quantum Electron. 14, 1358–1363 (1984).
[Crossref]

Ryzhik, I. M.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, New York, 1965).

Shen, Y. R.

Y. R. Shen, “Self-focusing: experimental,” Prog. Quantum Electron. 4, 1–34 (1975).
[Crossref]

M. M. T. Loy and Y. R. Shen, “Small-scale filaments in liquids and tracks of moving foci,” Phys. Rev. Lett. 22, 994–997 (1969).
[Crossref]

Y. R. Shen, The Principles of Nonlinear Optics (Wiley, New York, 1984), Chaps. 1–3, 6, 7.

Skidan, I. B.

R. Yu. Orlov, I. B. Skidan, and L. S. Telegin, “Investigation of breakdown produced in dielectrics by ultrashort laser pulses,” Sov. Phys. JETP 34, 418–421 (1972).

Smirl, A. L.

E. W. Van Stryland, W. E. Williams, M. J. Soileau, and A. L. Smirl, “Laser induced damage, nonlinear absorption, and doubling efficiency of LiIO3,” IEEE J. Quantum Electron. QE-20, 434–439 (1984).
[Crossref]

Snyder, A. W.

Soileau, M. J.

E. W. Van Stryland, W. E. Williams, M. J. Soileau, and A. L. Smirl, “Laser induced damage, nonlinear absorption, and doubling efficiency of LiIO3,” IEEE J. Quantum Electron. QE-20, 434–439 (1984).
[Crossref]

Stabinis, A.

R. Danelyus, G. Dikchyus, V. Kabelka, A. Piskarskas, A. Stabinis, and Ya. Yasevichyute, “Parametric excitation of light in the picosecond range,” Sov. J. Quantum Electron. 7, 1360–1368 (1977).
[Crossref]

Stegun, I. A.

A. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1965).

Telegin, L. S.

T. B. Razumikhina, L. S. Telegin, A. I. Kholodnykh, and A. S. Chirkin, “Three-frequency interactions of high-intensity light waves in media with quadratic and cubic nonlinearities,” Sov. J. Quantum Electron. 14, 1358–1363 (1984).
[Crossref]

L. S. Telegin and A. S. Chirkin, “Interaction in frequency doubling of ultrashort laser pulses,” Sov. J. Quantum Electron. 12, 1354–1356 (1982).
[Crossref]

R. Yu. Orlov, I. B. Skidan, and L. S. Telegin, “Investigation of breakdown produced in dielectrics by ultrashort laser pulses,” Sov. Phys. JETP 34, 418–421 (1972).

Terhune, R. W.

P. D. Maker and R. W. Terhune, “Study of optical effects due to an induced polarization third order in the electric field strength,” Phys. Rev. 137, A801–A818 (1965).
[Crossref]

Tunkin, V. G.

S. A. Magnitskii, V. I. Pryalkin, V. G. Tunkin, and A. I. Kholodnykh, “Influence of optical inhomogeneity on the parametric amplification of picosecond light pulses in lithium niobate crystals,” Sov. J. Quantum Electron. 12, 900–903 (1982).
[Crossref]

Van Stryland, E. W.

E. W. Van Stryland, W. E. Williams, M. J. Soileau, and A. L. Smirl, “Laser induced damage, nonlinear absorption, and doubling efficiency of LiIO3,” IEEE J. Quantum Electron. QE-20, 434–439 (1984).
[Crossref]

Wallace, R. W.

R. S. Adhav and R. W. Wallace, “Second harmonic generation in 90° phase-matched KDP isomorphs,” IEEE J. Quantum Electron. QE-9, 855–856 (1973).
[Crossref]

Weinreigh, G. W.

P. A. Franken, A. E. Hill, C. W. Peters, and G. W. Weinreigh, “Generation of optical harmonics,” Phys. Rev. Lett. 7, 118–119 (1961).
[Crossref]

Williams, W. E.

E. W. Van Stryland, W. E. Williams, M. J. Soileau, and A. L. Smirl, “Laser induced damage, nonlinear absorption, and doubling efficiency of LiIO3,” IEEE J. Quantum Electron. QE-20, 434–439 (1984).
[Crossref]

Yariv, A.

A. Yariv and P. Yeh, Optical Waves in Crystals (Wiley, New York, 1984), Chap. 12.

Yasevichyute, Ya.

R. Danelyus, G. Dikchyus, V. Kabelka, A. Piskarskas, A. Stabinis, and Ya. Yasevichyute, “Parametric excitation of light in the picosecond range,” Sov. J. Quantum Electron. 7, 1360–1368 (1977).
[Crossref]

Yeh, P.

A. Yariv and P. Yeh, Optical Waves in Crystals (Wiley, New York, 1984), Chap. 12.

Zernike, F.

F. Zernike and J. E. Midwinter, Applied Nonlinear Optics (Wiley, New York, 1973), Chap. 3.

IEEE J. Quantum Electron. (4)

R. S. Adhav and R. W. Wallace, “Second harmonic generation in 90° phase-matched KDP isomorphs,” IEEE J. Quantum Electron. QE-9, 855–856 (1973).
[Crossref]

D. J. Harter and D. C. Brown, “Effects of higher order nonlinearities on second-order frequency mixing,” IEEE J. Quantum Electron. QE-18, 1146–1151 (1982).
[Crossref]

R. S. Craxton, “High efficiency frequency tripling schemes for high-power Nd:glass lasers,” IEEE J. Quantum Electron. QE-17, 1771–1782 (1981), App. I.
[Crossref]

E. W. Van Stryland, W. E. Williams, M. J. Soileau, and A. L. Smirl, “Laser induced damage, nonlinear absorption, and doubling efficiency of LiIO3,” IEEE J. Quantum Electron. QE-20, 434–439 (1984).
[Crossref]

J. Appl. Phys. (1)

G. D. Boyd and D. A. Kleinman, “Parametric interaction of focused Gaussian light beams,” J. Appl. Phys. 39, 3597–3639 (1968).
[Crossref]

J. Opt. Soc. Am. B (3)

Opt. Lett. (1)

Phys. Rev. (2)

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interactions between light waves in a nonlinear dielectric,” Phys. Rev. 127, 1918–1939 (1962).
[Crossref]

P. D. Maker and R. W. Terhune, “Study of optical effects due to an induced polarization third order in the electric field strength,” Phys. Rev. 137, A801–A818 (1965).
[Crossref]

Phys. Rev. Lett. (2)

P. A. Franken, A. E. Hill, C. W. Peters, and G. W. Weinreigh, “Generation of optical harmonics,” Phys. Rev. Lett. 7, 118–119 (1961).
[Crossref]

M. M. T. Loy and Y. R. Shen, “Small-scale filaments in liquids and tracks of moving foci,” Phys. Rev. Lett. 22, 994–997 (1969).
[Crossref]

Prog. Quantum Electron. (2)

Y. R. Shen, “Self-focusing: experimental,” Prog. Quantum Electron. 4, 1–34 (1975).
[Crossref]

J. H. Marburger, “Self-focusing: theory,” Prog. Quantum Electron. 4, 35–110 (1975).
[Crossref]

Sov. J. Quantum Electron. (4)

S. A. Magnitskii, V. I. Pryalkin, V. G. Tunkin, and A. I. Kholodnykh, “Influence of optical inhomogeneity on the parametric amplification of picosecond light pulses in lithium niobate crystals,” Sov. J. Quantum Electron. 12, 900–903 (1982).
[Crossref]

R. Danelyus, G. Dikchyus, V. Kabelka, A. Piskarskas, A. Stabinis, and Ya. Yasevichyute, “Parametric excitation of light in the picosecond range,” Sov. J. Quantum Electron. 7, 1360–1368 (1977).
[Crossref]

L. S. Telegin and A. S. Chirkin, “Interaction in frequency doubling of ultrashort laser pulses,” Sov. J. Quantum Electron. 12, 1354–1356 (1982).
[Crossref]

T. B. Razumikhina, L. S. Telegin, A. I. Kholodnykh, and A. S. Chirkin, “Three-frequency interactions of high-intensity light waves in media with quadratic and cubic nonlinearities,” Sov. J. Quantum Electron. 14, 1358–1363 (1984).
[Crossref]

Sov. Phys. JETP (1)

R. Yu. Orlov, I. B. Skidan, and L. S. Telegin, “Investigation of breakdown produced in dielectrics by ultrashort laser pulses,” Sov. Phys. JETP 34, 418–421 (1972).

Other (8)

S. A. Akhmanov and R. V. Khokhlov, Nonlinear Optics (Gordon & Breach, New York, 1972).

F. Zernike and J. E. Midwinter, Applied Nonlinear Optics (Wiley, New York, 1973), Chap. 3.

A. Yariv and P. Yeh, Optical Waves in Crystals (Wiley, New York, 1984), Chap. 12.

Y. R. Shen, The Principles of Nonlinear Optics (Wiley, New York, 1984), Chaps. 1–3, 6, 7.

G. P. Agrawal, Nonlinear Fiber Optics (Academic, New York, 1989), Chap. 2.

V. I. Karpman, Nonlinear Waves in Dispersive Media (Pergamon, Oxford, 1975), Chap. 5.

A. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1965).

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, New York, 1965).

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Figures (3)

Fig. 1
Fig. 1

Computer estimation of J versus T3 and T4, which shows the dependence of solutions for SHG on the parameters T3 = (T2T1)/4 = R(2γ12γ22 + γ21 − 2γ11)/4β and T4 = (ΔP + T1)/2 = Δk/2βR + R(2γ11γ21)/2β.

Fig. 2
Fig. 2

Numerical evaluation of the changes in (a) the maximum conversion efficiency and (b) the medium length required to achieve the maximum in a dispersive medium with quadratic and cubic nonlinearities.

Fig. 3
Fig. 3

Contour maps for (a) the maximum conversion efficiency and (b) the length of the medium required to achieve the maximum. Each contour line is plotted at intervals of 5% for the efficiency and at intervals of 0.2 for the normalized medium length. One can also use these maps when T4 has a negative value by changing the signs of T3 and T4 simultaneously (i.e., T3T3 and T4 → −T4), since the maximum efficiency and the medium length are invariant to changes in sign of T3 and T4.

Tables (1)

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Table 1 Types of Solution for SHG

Equations (62)

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d u / d ξ = u v sin θ ,
d v / d ξ = - u 2 sin θ ,
d θ / d ξ = Δ P + ( 2 v - u 2 / v ) cos θ + T 1 u 2 + T 2 v 2 .
u = A 1 / R ,
v = A 2 / R ,
A 1 = a ^ 1 A 1 ,
A 2 = a ^ 2 A 2 ,
E ( z , t ) = 1 2 n = - 2 0 2 A n ( z ) exp [ j ( n ω 0 t - k n z ) ] ,
R = ( A 1 2 + A 2 2 ) 1 / 2 .
θ = ( k 2 - 2 k 1 ) z + ϕ 2 - 2 ϕ 1 ,
A 1 = A 1 exp ( - j ϕ 1 ) = A - 1 * ,
A 2 = A 2 exp ( - j ϕ 2 ) = A - 2 * ,
T 1 = R ( 2 γ 11 - γ 21 ) / β ,
T 2 = R ( 2 γ 12 - γ 22 ) / β ,
Δ P = ( k 2 - 2 k 1 ) / β R = Δ k / β R ,
ξ = β R z ,
β = ( ω 0 / 2 c n 1 ) a ^ 1 · χ ˜ ( 2 ) ( 2 ω 0 , - ω 0 ) : a ^ 1 a ^ 2 = ( ω 0 / 2 c n 2 ) a ^ 2 · χ ˜ ( 2 ) ( ω 0 , ω 0 ) : a ^ 1 a ^ 1 ,
γ m m = ( 3 m ω 0 / 8 c n m ) a ^ m · χ ˜ ( 3 ) ( m ω 0 , m ω 0 , - m ω 0 ) : a ^ m a ^ m a ^ m ,         m = 1 , 2 ,
γ m m = ( 3 m ω 0 / 4 c n m ) a ^ m · χ ˜ ( 3 ) ( m ω 0 , n ω 0 , - n ω 0 ) : a ^ m a ^ n a ^ n ,         m , n = 1 , 2 ,             m n .
χ ˜ ( 2 ) ( ω l , ω m ) = - d τ 1 d τ 2 × exp [ - j ( ω l τ 1 + ω m τ 2 ) ] χ ( 2 ) ( τ 1 , τ 2 ) ,
χ ˜ ( 3 ) ( ω l , ω m , ω s ) = - d τ 1 d τ 2 d τ 3 × exp [ - j ( ω l τ 1 + ω m τ 2 + ω s τ 3 ) ] χ ( 3 ) ( τ 1 , τ 2 , τ 3 ) .
P NL ( z , t ) = ɛ 0 - χ ( 2 ) ( t - t 1 , t - t 2 ) : E ( z , t 1 ) E ( z , t 2 ) d t 1 d t 2 + ɛ 0 - χ ( 3 ) ( t - t 1 , t - t 2 , t - t 3 ) E ( z , t 1 ) E ( z , t 2 ) × E ( z , t 3 ) d t 1 d t 2 d t 3 ,
( 2 t 2 - c 2 2 ) E ( z , t ) = - 1 ɛ 0 2 t 2 [ P ( z , t ) ] ,
P = P L + P NL
P L = ɛ 0 - χ ( 1 ) ( t - t ) · E ( z , t ) d t .
u 2 + v 2 = 1 ,
ξ = ± 1 2 h d h / D ( h ) ,
D ( h ) = h 3 - ( T 4 2 + 2 ) h 2 + ( 1 - 2 T 3 T 4 ) h - T 3 2 = ( h - h 1 ) ( h - h 2 ) ( h - h 3 ) ,
h = 1 v 2 ,
T 3 = T 2 - T 1 4 = R ( 2 γ 12 - γ 22 + γ 21 - 2 γ 11 ) 4 β ,
T 4 = Δ P + T 1 2 = Δ k 2 β R + R ( γ 11 - γ 21 ) 2 β .
J = ( 1 / 108 ) [ 27 T 3 4 + 4 T 3 3 ( T 4 3 + 18 T 4 ) + 2 T 3 2 ( 4 T 4 4 + 31 T 4 2 - 2 ) + 4 T 3 ( T 4 5 + 4 T 4 3 - 2 T 4 ) - ( T 4 4 + 4 T 4 2 ) ] .
h 1 ,             h 2 = h 3 * = h r + j h i ,
D ( h ) = h 3 - ( h 1 + 2 h r ) h 2 + ( 2 h 1 h r + h r 2 + h i 2 ) h - h 1 ( h r 2 + h i 2 ) .
v 2 = 1 / h ( ξ ) = [ 1 - cn ( 2 λ ξ , m ) ] / [ ( h 1 + λ 2 ) - ( h 1 - λ 2 ) cn ( 2 λ ξ , m ) ] ,
λ 2 = ( { d [ D ( h ) ] / d h } h = h 1 ) 1 / 2 = [ ( h 1 - h r ) 2 + h i 2 ] 1 / 2 .
m = sin 2 α = 1 2 - 1 8 { d 2 [ D ( h ) ] / d h 2 d [ D ( h ) ] / d h } h = h 1 .
h 1 = h 2 .
v 2 ( ξ ) = 1 / [ h 3 + ( h 3 - h 1 ) cot 2 ( h 3 - h 1 ξ ) ] .
v 2 ( ξ ) = ξ 2 / ( 1 + ξ 2 ) .
v 2 ( ξ ) = 1 / [ h 3 + ( h 1 - h 3 ) coth 2 ( h 1 - h 3 ξ ) ] ,
h 1 > h 2 > h 3 ,
v 2 ( ξ ) = [ cn 2 ( h 1 - h 3 ξ , m ) - 1 ] / [ h 3 cn 2 ( h 1 - h 3 ξ , m ) - h 1 ] ,
m = sin 2 α = ( h 2 - h 3 ) / ( h 1 - h 3 ) .
D ( h ) = ( h - 1 ) 2 ( h - T 3 2 ) ,
R 2 Γ = Δ k ,
Γ = ( γ 22 - 2 γ 11 + γ 21 - 2 γ 12 ) / 2 ,
R Γ β ,
Γ = ( 2 γ 12 - γ 22 - 2 γ 11 + γ 21 ) / 4.
Δ T = T 3 - ( - T 4 ) = ( 1 / 2 β ) Δ k / R - Γ R .
P NL ( z , t ) = ɛ 0 - d t 1 d t 2 χ ( 2 ) ( t - t 1 , t - t 2 ) : 1 4 l A l exp [ j ( ω l t 1 - k l z ) ] m A m exp [ j ( ω m t 2 - k m z ) ] + ɛ 0 - d t 1 d t 2 d t 3 χ ( 3 ) ( t - t 1 , t - t 2 , t - t 3 ) : 1 8 l A l exp [ j ( ω l t 1 - k l z ) ] m A m exp [ j ( ω m t 2 - k m z ) ] s A s exp [ j ( ω s t s - k s z ) ] .
P NL ( z , t ) = ɛ 0 4 - d τ 1 d τ 2 χ ( 2 ) ( τ 1 , τ 2 ) : l A l exp { j [ ω l ( t - τ 1 ) - k l z ] } m A m exp { j [ ω m ( t - τ 2 ) - k m z ] } + ɛ 0 8 - d τ 1 d τ 2 d τ 3 χ ( 3 ) ( τ 1 , τ 2 , τ 3 ) : l A l exp { j [ ω l ( t - τ 1 ) - k l z ] } × m A m exp { j [ ω m ( t - τ 2 ) - k m z ] } s A s exp { j [ ω s ( t - τ s ) - k s z ] } .
P NL ( z , t ) = ɛ 0 4 l m - d τ 1 d τ 2 exp [ - j ( ω l τ 1 + ω m τ 2 ) ] χ ( 2 ) ( τ 1 , τ 2 ) : A l A m exp { j [ ( ω l + ω m ) t - ( k l + k m ) z ] } × ɛ 0 8 l m s - d τ 1 d τ 2 d τ 3 exp [ - j ( ω l τ 1 + ω m τ 2 + ω s τ 3 ) ] × χ ( 3 ) ( τ 1 , τ 2 , τ 3 ) : A l A m A s exp { j [ ( ω l + ω m + ω s ) t - ( k l + k m + k s ) z ] } .
P NL ( z , t ) = ɛ 0 4 l m χ ˜ ( 2 ) ( ω l , ω m ) : A l A m × exp { j [ ( ω l + ω m ) t - ( k l + k m ) z ] } + ɛ 0 8 l m s 0 χ ˜ ( 3 ) ( ω l , ω m , ω s ) : A l A m A s × exp { j [ ( ω l + ω m + ω s ) t - ( k l + k m + k s ) z ] } .
P NL ( z , t ) = ɛ 0 4 n p χ ˜ ( 2 ) ( p ω 0 , [ n - p ] ω 0 ) : A p A n - p × exp { j [ n ω 0 t - ( k p + k n - p ) z ] } + ɛ 0 8 n p q χ ˜ ( 3 ) ( p ω 0 , [ q - p ] ω 0 , [ n - q ] ω 0 ) : A p A q - p A n - q × exp { j [ n ω 0 t - ( k p + k q - p + k n - p ) z ] } .
d d ξ ( u 2 v cos θ ) + d d ξ [ ( Δ P + T 2 ) u 2 / 2 + ( T 1 - T 2 ) u 4 / 4 ] = 0.
u 2 v cos θ + ( Δ P + T 2 ) u 2 / 2 + ( T 1 - T 2 ) u 4 / 4 = C 0 ,
d v 2 / d ξ = 2 v ( d v / d ξ ) = - 2 u 2 v sin θ .
d v 2 d ξ = ± 2 ( v 2 ( 1 - v 2 ) 2 - { C 0 - Δ P ( 1 - v 2 ) 2 - [ T 2 2 + ( T 1 - T 2 ) ( 1 - v 2 ) 4 ] ( 1 - v 2 ) } 2 ) 1 / 2 .
C 0 [ v 0 = v ( ξ = 0 ) = 0 ] = Δ P / 2 + ( T 1 + T 2 ) / 4.
ξ - ξ 0 ( = 0 ) = ± 1 2 v 0 2 v 2 d v 2 [ D ( v 2 ) ] 1 / 2 ,
D ( v 2 ) = v 2 { 1 - [ 2 + ( Δ P + T 1 2 ) 2 ] v 2 + [ 1 + 2 ( Δ P + T 1 2 ) ( T 1 - T 2 4 ) ] v 4 - ( T 1 - T 2 4 ) 2 v 6 } .

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