Abstract

The parametric decay and sum-frequency generation of intense optical radiation, in a medium with both second- and third-order nonlinearities, is considered. In steady state, nonlinear wave-number shifts usually detune these interactions before a complete exchange of energy between the pump and product waves can be achieved. Fortunately, however, it is possible to compensate for this nonlinear detuning by imposing a linear wave-number mismatch on each type of interaction. Simple criteria are given for the conditions under which a complete exchange of energy is possible, and a simple expression is obtained for the required medium length. Some exact solutions of the three-wave equations are used to illustrate the underlying wave physics, and some useful approximate solutions are described. As an application of the general theory, Type II frequency doubling in a KDP crystal is discussed briefly.

© 1993 Optical Society of America

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  1. A. Jurkus and P. N. Robson, “Saturation effects in a traveling-wave parametric amplifier,” Proc. Inst. Electr. Eng. Part B 107, 119–122 (1960).
  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. J. M. Manley and H. E. Rowe, “Some general properties of nonlinear elements—Part I. General energy relations,” Proc. IRE 44, 904–913 (1956).
    [CrossRef]
  4. M. T. Weiss, “Quantum derivation of energy relations analogous to those for nonlinear reactances,” Proc. IRE 45, 1012–1013 (1957).
  5. V. V. Berezovskii, A. V. Lebedev, A. I. Maimistov, and Z. A. Manykin, “Second harmonic generation under two-photon resonance conditions,” Sov. J. Quantum Electron. 10, 270–275 (1980).
    [CrossRef]
  6. L. S. Telegin and A. S. Chirkin, “Interaction in frequency doubling of ultrashort laser pulses,” Sov. J. Quantum Electron. 12, 1354–1356 (1982).
    [CrossRef]
  7. C. E. Clayton, C. Joshi, C. Darrow, and D. Umstadter, “Relativistic plasma-wave excitation by collinear optical mixing,” Phys. Rev. Lett. 54, 2343–2346 (1985).
    [CrossRef] [PubMed]
  8. J. Fukai, S. Krishan, and E. G. Harris, “Explosive plasma instabilities,” Phys. Rev. Lett. 23, 910–912 (1969).
    [CrossRef]
  9. M. N. Rosenbluth and C. S. Liu, “Excitation of plasma waves by two laser beams,” Phys. Rev. Lett. 29, 701–705 (1972).
    [CrossRef]
  10. S. A. Akhmanov and R. V. Khokhlov, Problems of Nonlinear Optics (Gordon & Breach, New York, 1972), pp. 169–174, 213–218.
  11. 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]
  12. C. J. McKinstrie and D. W. Forslund, “The detuning of relativistic Langmuir waves in the beat-wave accelerator,” Phys. Fluids 30, 904–908 (1987), and references therein.
    [CrossRef]
  13. V. N. Oraevskii, H. Wilhelmsson, E. Ya. Kogan, and V. P. Pavlenko, “On the stabilization of explosive instabilities by nonlinear frequency shifts,” Phys. Scr. 7, 217–221 (1973).
    [CrossRef]
  14. J. Weiland and H. Wilhelmsson, “Repetitive explosive instabilities,” Phys. Scr. 7, 222–229 (1973).
    [CrossRef]
  15. T. B. Razumikhina, L. S. Telegin, A. I. Kolodnykh, 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]
  16. C. J. McKinstrie and D. F. DuBois, “Relativistic solitary-wave solutions of the beat-wave equations,” Phys. Rev. Lett. 57, 2022–2025 (1986); erratum 58, 286 (1987).
    [CrossRef] [PubMed]
  17. C. J. McKinstrie, “Relativistic solitary-wave solutions of the beat-wave equations,” Phys. Fluids 31, 288–297 (1988).
    [CrossRef]
  18. C. J. McKinstrie and G. G. Luther, “Solitary-wave solutions of the generalised three-wave and four-wave equations,” Phys. Lett. A 127, 14–18 (1988).
    [CrossRef]
  19. X. D. Cao and C. J. McKinstrie, “The optimization of sum-frequency generation,” presented at the 21st Annual Anomalous Absorption Conference, Banff, Alberta, April 15–19, 1991. This work was completed before our discovery of Refs. 13–15.
  20. W. Choe, P. P. Banerjee, and F. C. Caimi, “Second-harmonic generation in an optical medium with second- and third-order nonlinear susceptibilities,” J. Opt. Soc. Am. B 8, 1013–1022 (1991).
    [CrossRef]
  21. R. Y. Chiao, P. L. Kelley, and E. Garmire, “Stimulated four-photon interaction and its influence on stimulated Rayleigh-wing scattering,” Phys. Rev. Lett. 17, 1158–1161 (1966).
    [CrossRef]
  22. J. H. Marburger and J. F. Lam, “Effect of nonlinear index changes on degenerate four-wave mixing,” Appl. Phys. Lett. 35, 249–251 (1979).
    [CrossRef]
  23. R. W. Boyd, Nonlinear Optics (Academic, Boston, Mass., 1992), pp. 85–90.
  24. M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions (National Bureau of Standards, Washington, D.C., 1964), pp. 17, 591.
  25. R. S. Craxton, “Theory of high-efficiency third-harmonic generation of high-power Nd:glass laser radiation,” Opt. Commun. 34, 474–478 (1980).
    [CrossRef]
  26. R. S. Craxton, “High-efficiency frequency tripling schemes for high-power Nd:glass lasers,” IEEE J. Quantum Electron. QE-17, 1771–1782 (1981).
    [CrossRef]
  27. R. S. Craxton, S. D. Jacobs, J. E. Rizzo, and R. Boni, “Basic properties of KDP related to the frequency conversion of 1 μ m laser radiation,” IEEE J. Quantum Electron. QE-17, 1782–1786 (1981).
    [CrossRef]
  28. C. J. McKinstrie, X. D. Cao, and J. S. Li, “Nonlinear detuning of four-wave interactions,” submitted to J. Opt. Soc. Am. B.
  29. I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products (Academic, New York, 1980), p. 84.
  30. P. F. Byrd and M. D. Friedman, Handbook of Elliptic Integrals for Engineers and Scientists, 2nd ed. (Springer-Verlag, New York, 1971), primarily pp. 18–21, 120, 133.

1991 (1)

1988 (2)

C. J. McKinstrie, “Relativistic solitary-wave solutions of the beat-wave equations,” Phys. Fluids 31, 288–297 (1988).
[CrossRef]

C. J. McKinstrie and G. G. Luther, “Solitary-wave solutions of the generalised three-wave and four-wave equations,” Phys. Lett. A 127, 14–18 (1988).
[CrossRef]

1987 (1)

C. J. McKinstrie and D. W. Forslund, “The detuning of relativistic Langmuir waves in the beat-wave accelerator,” Phys. Fluids 30, 904–908 (1987), and references therein.
[CrossRef]

1986 (1)

C. J. McKinstrie and D. F. DuBois, “Relativistic solitary-wave solutions of the beat-wave equations,” Phys. Rev. Lett. 57, 2022–2025 (1986); erratum 58, 286 (1987).
[CrossRef] [PubMed]

1985 (1)

C. E. Clayton, C. Joshi, C. Darrow, and D. Umstadter, “Relativistic plasma-wave excitation by collinear optical mixing,” Phys. Rev. Lett. 54, 2343–2346 (1985).
[CrossRef] [PubMed]

1984 (1)

T. B. Razumikhina, L. S. Telegin, A. I. Kolodnykh, 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]

1982 (2)

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]

1981 (2)

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

R. S. Craxton, S. D. Jacobs, J. E. Rizzo, and R. Boni, “Basic properties of KDP related to the frequency conversion of 1 μ m laser radiation,” IEEE J. Quantum Electron. QE-17, 1782–1786 (1981).
[CrossRef]

1980 (2)

R. S. Craxton, “Theory of high-efficiency third-harmonic generation of high-power Nd:glass laser radiation,” Opt. Commun. 34, 474–478 (1980).
[CrossRef]

V. V. Berezovskii, A. V. Lebedev, A. I. Maimistov, and Z. A. Manykin, “Second harmonic generation under two-photon resonance conditions,” Sov. J. Quantum Electron. 10, 270–275 (1980).
[CrossRef]

1979 (1)

J. H. Marburger and J. F. Lam, “Effect of nonlinear index changes on degenerate four-wave mixing,” Appl. Phys. Lett. 35, 249–251 (1979).
[CrossRef]

1973 (2)

V. N. Oraevskii, H. Wilhelmsson, E. Ya. Kogan, and V. P. Pavlenko, “On the stabilization of explosive instabilities by nonlinear frequency shifts,” Phys. Scr. 7, 217–221 (1973).
[CrossRef]

J. Weiland and H. Wilhelmsson, “Repetitive explosive instabilities,” Phys. Scr. 7, 222–229 (1973).
[CrossRef]

1972 (1)

M. N. Rosenbluth and C. S. Liu, “Excitation of plasma waves by two laser beams,” Phys. Rev. Lett. 29, 701–705 (1972).
[CrossRef]

1969 (1)

J. Fukai, S. Krishan, and E. G. Harris, “Explosive plasma instabilities,” Phys. Rev. Lett. 23, 910–912 (1969).
[CrossRef]

1966 (1)

R. Y. Chiao, P. L. Kelley, and E. Garmire, “Stimulated four-photon interaction and its influence on stimulated Rayleigh-wing scattering,” Phys. Rev. Lett. 17, 1158–1161 (1966).
[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]

1960 (1)

A. Jurkus and P. N. Robson, “Saturation effects in a traveling-wave parametric amplifier,” Proc. Inst. Electr. Eng. Part B 107, 119–122 (1960).

1957 (1)

M. T. Weiss, “Quantum derivation of energy relations analogous to those for nonlinear reactances,” Proc. IRE 45, 1012–1013 (1957).

1956 (1)

J. M. Manley and H. E. Rowe, “Some general properties of nonlinear elements—Part I. General energy relations,” Proc. IRE 44, 904–913 (1956).
[CrossRef]

Akhmanov, S. A.

S. A. Akhmanov and R. V. Khokhlov, Problems of Nonlinear Optics (Gordon & Breach, New York, 1972), pp. 169–174, 213–218.

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]

Banerjee, P. P.

Berezovskii, V. V.

V. V. Berezovskii, A. V. Lebedev, A. I. Maimistov, and Z. A. Manykin, “Second harmonic generation under two-photon resonance conditions,” Sov. J. Quantum Electron. 10, 270–275 (1980).
[CrossRef]

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]

Boni, R.

R. S. Craxton, S. D. Jacobs, J. E. Rizzo, and R. Boni, “Basic properties of KDP related to the frequency conversion of 1 μ m laser radiation,” IEEE J. Quantum Electron. QE-17, 1782–1786 (1981).
[CrossRef]

Boyd, R. W.

R. W. Boyd, Nonlinear Optics (Academic, Boston, Mass., 1992), pp. 85–90.

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]

Byrd, P. F.

P. F. Byrd and M. D. Friedman, Handbook of Elliptic Integrals for Engineers and Scientists, 2nd ed. (Springer-Verlag, New York, 1971), primarily pp. 18–21, 120, 133.

Caimi, F. C.

Cao, X. D.

X. D. Cao and C. J. McKinstrie, “The optimization of sum-frequency generation,” presented at the 21st Annual Anomalous Absorption Conference, Banff, Alberta, April 15–19, 1991. This work was completed before our discovery of Refs. 13–15.

C. J. McKinstrie, X. D. Cao, and J. S. Li, “Nonlinear detuning of four-wave interactions,” submitted to J. Opt. Soc. Am. B.

Chiao, R. Y.

R. Y. Chiao, P. L. Kelley, and E. Garmire, “Stimulated four-photon interaction and its influence on stimulated Rayleigh-wing scattering,” Phys. Rev. Lett. 17, 1158–1161 (1966).
[CrossRef]

Chirkin, A. S.

T. B. Razumikhina, L. S. Telegin, A. I. Kolodnykh, 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]

Choe, W.

Clayton, C. E.

C. E. Clayton, C. Joshi, C. Darrow, and D. Umstadter, “Relativistic plasma-wave excitation by collinear optical mixing,” Phys. Rev. Lett. 54, 2343–2346 (1985).
[CrossRef] [PubMed]

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).
[CrossRef]

R. S. Craxton, S. D. Jacobs, J. E. Rizzo, and R. Boni, “Basic properties of KDP related to the frequency conversion of 1 μ m laser radiation,” IEEE J. Quantum Electron. QE-17, 1782–1786 (1981).
[CrossRef]

R. S. Craxton, “Theory of high-efficiency third-harmonic generation of high-power Nd:glass laser radiation,” Opt. Commun. 34, 474–478 (1980).
[CrossRef]

Darrow, C.

C. E. Clayton, C. Joshi, C. Darrow, and D. Umstadter, “Relativistic plasma-wave excitation by collinear optical mixing,” Phys. Rev. Lett. 54, 2343–2346 (1985).
[CrossRef] [PubMed]

DuBois, D. F.

C. J. McKinstrie and D. F. DuBois, “Relativistic solitary-wave solutions of the beat-wave equations,” Phys. Rev. Lett. 57, 2022–2025 (1986); erratum 58, 286 (1987).
[CrossRef] [PubMed]

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]

Forslund, D. W.

C. J. McKinstrie and D. W. Forslund, “The detuning of relativistic Langmuir waves in the beat-wave accelerator,” Phys. Fluids 30, 904–908 (1987), and references therein.
[CrossRef]

Friedman, M. D.

P. F. Byrd and M. D. Friedman, Handbook of Elliptic Integrals for Engineers and Scientists, 2nd ed. (Springer-Verlag, New York, 1971), primarily pp. 18–21, 120, 133.

Fukai, J.

J. Fukai, S. Krishan, and E. G. Harris, “Explosive plasma instabilities,” Phys. Rev. Lett. 23, 910–912 (1969).
[CrossRef]

Garmire, E.

R. Y. Chiao, P. L. Kelley, and E. Garmire, “Stimulated four-photon interaction and its influence on stimulated Rayleigh-wing scattering,” Phys. Rev. Lett. 17, 1158–1161 (1966).
[CrossRef]

Gradshteyn, I. S.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products (Academic, New York, 1980), p. 84.

Harris, E. G.

J. Fukai, S. Krishan, and E. G. Harris, “Explosive plasma instabilities,” Phys. Rev. Lett. 23, 910–912 (1969).
[CrossRef]

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]

Jacobs, S. D.

R. S. Craxton, S. D. Jacobs, J. E. Rizzo, and R. Boni, “Basic properties of KDP related to the frequency conversion of 1 μ m laser radiation,” IEEE J. Quantum Electron. QE-17, 1782–1786 (1981).
[CrossRef]

Joshi, C.

C. E. Clayton, C. Joshi, C. Darrow, and D. Umstadter, “Relativistic plasma-wave excitation by collinear optical mixing,” Phys. Rev. Lett. 54, 2343–2346 (1985).
[CrossRef] [PubMed]

Jurkus, A.

A. Jurkus and P. N. Robson, “Saturation effects in a traveling-wave parametric amplifier,” Proc. Inst. Electr. Eng. Part B 107, 119–122 (1960).

Kelley, P. L.

R. Y. Chiao, P. L. Kelley, and E. Garmire, “Stimulated four-photon interaction and its influence on stimulated Rayleigh-wing scattering,” Phys. Rev. Lett. 17, 1158–1161 (1966).
[CrossRef]

Khokhlov, R. V.

S. A. Akhmanov and R. V. Khokhlov, Problems of Nonlinear Optics (Gordon & Breach, New York, 1972), pp. 169–174, 213–218.

Kogan, E. Ya.

V. N. Oraevskii, H. Wilhelmsson, E. Ya. Kogan, and V. P. Pavlenko, “On the stabilization of explosive instabilities by nonlinear frequency shifts,” Phys. Scr. 7, 217–221 (1973).
[CrossRef]

Kolodnykh, A. I.

T. B. Razumikhina, L. S. Telegin, A. I. Kolodnykh, 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]

Krishan, S.

J. Fukai, S. Krishan, and E. G. Harris, “Explosive plasma instabilities,” Phys. Rev. Lett. 23, 910–912 (1969).
[CrossRef]

Lam, J. F.

J. H. Marburger and J. F. Lam, “Effect of nonlinear index changes on degenerate four-wave mixing,” Appl. Phys. Lett. 35, 249–251 (1979).
[CrossRef]

Lebedev, A. V.

V. V. Berezovskii, A. V. Lebedev, A. I. Maimistov, and Z. A. Manykin, “Second harmonic generation under two-photon resonance conditions,” Sov. J. Quantum Electron. 10, 270–275 (1980).
[CrossRef]

Li, J. S.

C. J. McKinstrie, X. D. Cao, and J. S. Li, “Nonlinear detuning of four-wave interactions,” submitted to J. Opt. Soc. Am. B.

Liu, C. S.

M. N. Rosenbluth and C. S. Liu, “Excitation of plasma waves by two laser beams,” Phys. Rev. Lett. 29, 701–705 (1972).
[CrossRef]

Luther, G. G.

C. J. McKinstrie and G. G. Luther, “Solitary-wave solutions of the generalised three-wave and four-wave equations,” Phys. Lett. A 127, 14–18 (1988).
[CrossRef]

Maimistov, A. I.

V. V. Berezovskii, A. V. Lebedev, A. I. Maimistov, and Z. A. Manykin, “Second harmonic generation under two-photon resonance conditions,” Sov. J. Quantum Electron. 10, 270–275 (1980).
[CrossRef]

Manley, J. M.

J. M. Manley and H. E. Rowe, “Some general properties of nonlinear elements—Part I. General energy relations,” Proc. IRE 44, 904–913 (1956).
[CrossRef]

Manykin, Z. A.

V. V. Berezovskii, A. V. Lebedev, A. I. Maimistov, and Z. A. Manykin, “Second harmonic generation under two-photon resonance conditions,” Sov. J. Quantum Electron. 10, 270–275 (1980).
[CrossRef]

Marburger, J. H.

J. H. Marburger and J. F. Lam, “Effect of nonlinear index changes on degenerate four-wave mixing,” Appl. Phys. Lett. 35, 249–251 (1979).
[CrossRef]

McKinstrie, C. J.

C. J. McKinstrie, “Relativistic solitary-wave solutions of the beat-wave equations,” Phys. Fluids 31, 288–297 (1988).
[CrossRef]

C. J. McKinstrie and G. G. Luther, “Solitary-wave solutions of the generalised three-wave and four-wave equations,” Phys. Lett. A 127, 14–18 (1988).
[CrossRef]

C. J. McKinstrie and D. W. Forslund, “The detuning of relativistic Langmuir waves in the beat-wave accelerator,” Phys. Fluids 30, 904–908 (1987), and references therein.
[CrossRef]

C. J. McKinstrie and D. F. DuBois, “Relativistic solitary-wave solutions of the beat-wave equations,” Phys. Rev. Lett. 57, 2022–2025 (1986); erratum 58, 286 (1987).
[CrossRef] [PubMed]

X. D. Cao and C. J. McKinstrie, “The optimization of sum-frequency generation,” presented at the 21st Annual Anomalous Absorption Conference, Banff, Alberta, April 15–19, 1991. This work was completed before our discovery of Refs. 13–15.

C. J. McKinstrie, X. D. Cao, and J. S. Li, “Nonlinear detuning of four-wave interactions,” submitted to J. Opt. Soc. Am. B.

Oraevskii, V. N.

V. N. Oraevskii, H. Wilhelmsson, E. Ya. Kogan, and V. P. Pavlenko, “On the stabilization of explosive instabilities by nonlinear frequency shifts,” Phys. Scr. 7, 217–221 (1973).
[CrossRef]

Pavlenko, V. P.

V. N. Oraevskii, H. Wilhelmsson, E. Ya. Kogan, and V. P. Pavlenko, “On the stabilization of explosive instabilities by nonlinear frequency shifts,” Phys. Scr. 7, 217–221 (1973).
[CrossRef]

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]

Razumikhina, T. B.

T. B. Razumikhina, L. S. Telegin, A. I. Kolodnykh, 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]

Rizzo, J. E.

R. S. Craxton, S. D. Jacobs, J. E. Rizzo, and R. Boni, “Basic properties of KDP related to the frequency conversion of 1 μ m laser radiation,” IEEE J. Quantum Electron. QE-17, 1782–1786 (1981).
[CrossRef]

Robson, P. N.

A. Jurkus and P. N. Robson, “Saturation effects in a traveling-wave parametric amplifier,” Proc. Inst. Electr. Eng. Part B 107, 119–122 (1960).

Rosenbluth, M. N.

M. N. Rosenbluth and C. S. Liu, “Excitation of plasma waves by two laser beams,” Phys. Rev. Lett. 29, 701–705 (1972).
[CrossRef]

Rowe, H. E.

J. M. Manley and H. E. Rowe, “Some general properties of nonlinear elements—Part I. General energy relations,” Proc. IRE 44, 904–913 (1956).
[CrossRef]

Ryzhik, I. M.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products (Academic, New York, 1980), p. 84.

Telegin, L. S.

T. B. Razumikhina, L. S. Telegin, A. I. Kolodnykh, 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]

Umstadter, D.

C. E. Clayton, C. Joshi, C. Darrow, and D. Umstadter, “Relativistic plasma-wave excitation by collinear optical mixing,” Phys. Rev. Lett. 54, 2343–2346 (1985).
[CrossRef] [PubMed]

Weiland, J.

J. Weiland and H. Wilhelmsson, “Repetitive explosive instabilities,” Phys. Scr. 7, 222–229 (1973).
[CrossRef]

Weiss, M. T.

M. T. Weiss, “Quantum derivation of energy relations analogous to those for nonlinear reactances,” Proc. IRE 45, 1012–1013 (1957).

Wilhelmsson, H.

J. Weiland and H. Wilhelmsson, “Repetitive explosive instabilities,” Phys. Scr. 7, 222–229 (1973).
[CrossRef]

V. N. Oraevskii, H. Wilhelmsson, E. Ya. Kogan, and V. P. Pavlenko, “On the stabilization of explosive instabilities by nonlinear frequency shifts,” Phys. Scr. 7, 217–221 (1973).
[CrossRef]

Appl. Phys. Lett. (1)

J. H. Marburger and J. F. Lam, “Effect of nonlinear index changes on degenerate four-wave mixing,” Appl. Phys. Lett. 35, 249–251 (1979).
[CrossRef]

IEEE J. Quantum Electron. (3)

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

R. S. Craxton, S. D. Jacobs, J. E. Rizzo, and R. Boni, “Basic properties of KDP related to the frequency conversion of 1 μ m laser radiation,” IEEE J. Quantum Electron. QE-17, 1782–1786 (1981).
[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]

J. Opt. Soc. Am. B (1)

Opt. Commun. (1)

R. S. Craxton, “Theory of high-efficiency third-harmonic generation of high-power Nd:glass laser radiation,” Opt. Commun. 34, 474–478 (1980).
[CrossRef]

Phys. Fluids (2)

C. J. McKinstrie and D. W. Forslund, “The detuning of relativistic Langmuir waves in the beat-wave accelerator,” Phys. Fluids 30, 904–908 (1987), and references therein.
[CrossRef]

C. J. McKinstrie, “Relativistic solitary-wave solutions of the beat-wave equations,” Phys. Fluids 31, 288–297 (1988).
[CrossRef]

Phys. Lett. A (1)

C. J. McKinstrie and G. G. Luther, “Solitary-wave solutions of the generalised three-wave and four-wave equations,” Phys. Lett. A 127, 14–18 (1988).
[CrossRef]

Phys. Rev. (1)

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Figures (6)

Fig. 1
Fig. 1

(a) Action flux densities of the pump and probe waves and (b) the relative phase of the three waves, plotted as functions of position for the case in which λ = 0.1 and = 0.01. For these parameters, which correspond to a moderate-intensity pump wave and a weak probe wave, solution (4.9) is relevant. The action flux densities are normalized to the initial action flux density of the pump wave, and the relative phase is measured in units of π. Distance is measured in units of the growth length of the linear three-wave instability.

Fig. 2
Fig. 2

Exact solution (4.9) and approximate solution (3.12) superimposed for the case in which λ = 0.1 and = 0.01. (a) Normalized action flux density of the probe wave. (b) Relative phase of the three waves, in units of π.

Fig. 3
Fig. 3

(a) Normalized action flux densities of the pump and probe waves and (b) the relative phase of the three waves, in units of π, plotted as functions of position for the case in which λ = 2 and = 1. For these parameters, which correspond to a high-intensity pump wave and a strong probe wave, solution (4.15) is relevant. Notice that the variation in relative phase is much larger than for the parameters of Fig. 1.

Fig. 4
Fig. 4

(a) Normalized action flux densities of the pump and probe waves and (b) the relative phase of the three waves, in units of π, plotted as functions of position for the case in which λ = 2.00 and = 0.563. For these parameters, is only marginally in excess of the value required by the second of conditions (4.8). Solution (4.15) is relevant.

Fig. 5
Fig. 5

Potential Q(F) associated with Eq. (4.3). (a) λ = 2.00 and = 0.563, as in Fig. 4. (b) λ = 2 and = 1, as in Fig. 3. (c) λ = 0.1 and = 0.01, as in Fig. 1.

Fig. 6
Fig. 6

(a) Normalized action flux densities of the fundamental and second-harmonic waves and (b) their relative phase, in units of π, plotted as functions of position for the case in which λ = 1. For this value of λ, which corresponds to a high-intensity fundamental wave, the second of solutions (5.15) is relevant.

Equations (162)

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ω 1 = ω 2 + ω 3 ,             k 1 = k 2 + k 3 .
s 1 d z A 1 = - i A 2 A 3 + i ( δ 1 + λ 1 β A β 2 ) A 1 , s 2 d z A 2 = - i A 3 * A 1 + i ( δ 2 + λ 2 β A β 2 ) A 2 , s 3 d z A 3 = - i A 1 A 2 * + i ( δ 3 + λ 3 β A β 2 ) A 3 ,
A α = F α 1 / 2 exp ( i ϕ α ) .
s 1 d z F 1 = - 2 ( F 1 F 2 F 3 ) 1 / 2 sin ϕ , s 2 d z F 2 = 2 ( F 1 F 2 F 3 ) 1 / 2 sin ϕ , s 3 d z F 3 = 2 ( F 1 F 2 F 3 ) 1 / 2 sin ϕ
s 1 d z ϕ 1 = - ( F 2 F 3 / F 1 ) 1 / 2 cos ϕ + ( δ 1 + λ 1 β F β ) , s 2 d z ϕ 2 = - ( F 3 F 1 / F 2 ) 1 / 2 cos ϕ + ( δ 2 + λ 2 β F β ) , s 3 d z ϕ 3 = - ( F 1 F 2 / F 3 ) 1 / 2 cos ϕ + ( δ 3 + λ 3 β F β ) ,
ϕ = ϕ 1 - ϕ 2 - ϕ 3
H = 2 ( F 1 F 2 F 3 ) 1 / 2 cos ϕ - ( δ α + 1 / 2 λ α β F β ) F α ,
s α d F α d z = H ϕ α ,             s α d ϕ α d z = - H F α .
( d z F 3 ) 2 = 4 F 1 F 2 F 3 - [ H + ( δ α + 1 / 2 λ α β F β ) F α ] 2 .
d z ( s 1 F 1 + s 2 F 2 ) = 0 ,             d z ( s 2 F 2 - s 3 F 3 ) = 0.
F 1 ( - ) = 1 ,             F 2 ( - ) = 0 ,             F 3 ( - ) = 0.
F 2 ( z ) = F 3 ( z ) = F ( z ) ,             F 1 ( z ) = 1 - F ( z ) .
δ ¯ = 1 / 2 ( δ 1 - δ 2 - δ 3 ) ,             λ ¯ α = 1 / 4 ( λ 1 α - λ 2 α - λ 3 α ) ,
δ = δ ¯ + 2 λ ¯ 1 ,             λ = λ ¯ 1 - λ ¯ 2 - λ ¯ 3 ,
H + ( δ α + 1 / 2 λ α β F β ) F α = - 2 ( δ - λ F ) F .
( d z F ) 2 = 4 F 2 [ ( 1 - F ) - ( δ - λ F ) 2 ] .
a = 4 ( 1 - δ 2 ) ,             b = 4 ( 2 δ λ - 1 ) ,             c = - 4 λ 2 ,
F ( z ) = { 2 a - b + ( b 2 - 4 a c ) 1 / 2 cosh ( a 1 / 2 z ) if a > 0 4 b - 4 c + b 2 z 2 if a = 0             and             b > 0
cos ϕ = - ( δ - λ F ) ( 1 - F ) 1 / 2 .
δ = λ .
λ 1.
F ( z ) = { 1 - λ 2 1 - λ 2 + sinh 2 [ ( 1 - λ 2 ) 1 / 2 z ] if λ < 1 1 1 + z 2 if λ = 1 .
F 1 ( 0 ) = 1 ,             F 2 ( 0 ) = ,             F 3 ( 0 ) = 0.
F 3 ( z ) = F ( z ) ,             F 1 ( z ) = 1 - F ( z ) ,             F 2 ( z ) = + F ( z ) .
( d z F ) 2 = 4 F [ ( + F ) ( 1 - F ) - F ( δ - λ F ) 2 ] ,
δ = δ ¯ + 2 λ ¯ 1 + 2 λ ¯ 2
cos ϕ = - ( δ - λ F ) F [ ( 1 - F ) ( + F ) F ] 1 / 2 .
δ = λ .
f ± = - ( 1 - λ 2 ) ± [ ( 1 - λ 2 ) 2 - 4 λ 2 ] 1 / 2 2 λ 2 .
λ 1 ,             or λ > 1             and             > ( λ 2 - 1 ) 2 / 4 λ 2 .
F ( z ) = - f + sn 2 ( k z , m ) 1 - f + - sn 2 ( k z , m ) ,
k 2 = - λ 2 ( 1 - f + ) f - , m 2 = - ( f + - f - ) / [ ( 1 - f + ) f - ] .
k l = K ( m ) ,
F ( l + z ) = - f - cn 2 ( k z , m ) - f - + sn 2 ( k z , m ) ,
F ( l + z ) = cn 2 ( k z , m ) ,
k 2 = 1 + ,             m 2 = 1 / ( 1 + ) .
F ( z ) = β [ 1 - cn ( k z , m ) ] ( α + β ) + ( α - β ) cn ( k z , m ) ,
k 2 = 4 λ 2 α β ,             m 2 = [ 1 - ( α - β ) 2 ] / 4 α β ,
α 2 = ( 1 + ) / λ 2 ,             β 2 = / λ 2 .
k l = 2 K ( m ) .
F ( l + z ) = β [ 1 + cn ( k z , m ) ] ( α + β ) - ( α - β ) cn ( k z , m ) .
F ( z ) = f sin 2 ( k z ) 1 + f - sin 2 ( k z ) .
d ϕ d z = δ - 3 λ F + [ δ - 2 λ ( 1 - ) ] F 2 + λ F 3 ( 1 - F ) ( + F ) ,
d ϕ d z = λ ( - 2 F - F 2 ) ( + F )
l log [ 4 ( 1 - λ 2 ) / 1 / 2 ] ( 1 - λ 2 ) 1 / 2
F ( z ) sinh 2 [ ( 1 - λ 2 ) 1 / 2 z ] ( 1 - λ 2 ) .
l 1 ~ log [ 4 ( 1 - λ 2 ) / ] 2 ( 1 - λ 2 ) 1 / 2 .
l 2 ~ log [ 4 ( 1 - λ 2 ) ] 2 ( 1 - λ 2 ) 1 / 2 .
F 1 ( 0 ) = 0 ,             F 2 ( 0 ) = 1 + ,             F 3 ( 0 ) = 1.
F 1 ( z ) = F ( z ) ,             F 2 ( z ) = 1 + - F ( z ) , F 3 ( z ) = 1 - F ( z ) .
( d z F ) 2 = 4 F [ ( 1 + - F ) ( 1 - F ) - F ( δ + λ F ) 2 ] ,
δ = δ ¯ + 2 λ ¯ 2 ( 1 + ) + 2 λ ¯ 3
cos ϕ = ( δ + λ F ) F [ ( 1 - F ) ( 1 + - F ) F ] 1 / 2 .
δ = - λ .
λ 1 ,             or λ > 1             and             > ( λ 2 - 1 ) 2 / 4 λ 2 .
F ( z ) = ( 1 - f - ) sn 2 ( k z , m ) - f - + sn 2 ( k z , m ) ,
F ( z ) = sn 2 ( k z , m ) ,
F ( z ) = α [ 1 - cn ( k z , m ) ] ( α + β ) - ( α - β ) cn ( k z , m ) ,
F ( z ) = ( 1 + f ) sin 2 ( k z ) f + sin 2 ( k z ) .
( d z F ) 2 = 4 F [ ( 1 - F ) 2 - F ( δ + λ F ) 2 ] ,
cos ϕ = ( δ + λ F ) F 1 / 2 ( 1 - F ) .
δ = - λ             and             λ 1.
F ( z ) = { sinh 2 [ ( 1 - λ 2 ) 1 / 2 z ] 1 - λ 2 + sinh 2 [ ( 1 - λ 2 ) 1 / 2 z ] if λ < 1 z 2 1 + z 2 if λ = 1 ,
F ( z ) { sin 2 ( δ z ) / δ 2 if δ 0 z 2 if δ = 0 .
Δ k [ mm - 1 ] - 0.26 Δ θ [ mrad ]
l g [ mm ] 7.2 ( I [ GW cm - 2 ] ) - 1 / 2
δ ¯ 0.92 ( Δ θ [ mrad ] ) ( I [ GW cm - 2 ] ) - 1 / 2 .
f ± = 1 + δ 2 / 2 ± [ δ 2 ( 1 + δ 2 / 4 ) ] 1 / 2 .
f + { 1 + δ if δ 1 δ 2 + 2 if δ 1 ,
f - { 1 - δ if δ 1 1 / δ 2 if δ 1 .
F ( z ) = f - sn 2 ( k z , m ) ,
k 2 = f + ,             m 2 = f - / f + .
m 2 { 1 - 2 δ if δ 1 1 / δ 4 if δ 1 ,
l { log [ ( 8 / δ ) 1 / 2 ] if δ 1 π / 4 if δ 1 .
F ( z ) = tanh 2 ( z ) ,
Δ k nl [ mm - 1 ] ~ 2.5 × 10 - 3 I [ GW cm - 2 ] .
λ 33 ~ 0.018 ( I [ GW cm - 2 ] ) 1 / 2 .
Δ ϕ 3 [ rad ] ~ 0.054 ( I [ GW cm - 2 ] ) 1 / 2 .
F max 1 - δ + λ .
A α = B α exp ( i ψ α ) ,
ψ α = [ s α ( λ α β - λ β α ) / 2 ] 0 z B β ( z ) 2 d z ,
s 1 d z B 1 = - i B 2 B 3 exp ( - i ψ ) + i ( δ 1 + λ 1 β + B β 2 ) B 1 , s 2 d z B 2 = - i B 3 * B 1 exp ( i ψ ) + i ( δ 2 + λ 2 β + B β 2 ) B 2 , s 3 d z B 3 = - i B 1 B 2 * exp ( i ψ ) + i ( δ 3 + λ 3 β + B β 2 ) B 3 ,
λ α β + = 1 / 2 ( λ α β + λ β α ) ,
ψ = ψ 1 - ψ 2 - ψ 3 .
B α = F α 1 / 2 exp ( i ϕ α ) ,
ϕ = ϕ 1 - ϕ 2 - ϕ 3 + ψ ( z ) .
d 2 ( s 1 s 2 s 3 ψ ) / d z 2 = 0
ψ ( z ) = ψ z .
d z H = - 2 ψ ( F 1 F 2 F 3 ) 1 / 2 sin ϕ = 1 / 2 ( λ α β - λ β α ) F β d z F α = λ α β - F β d z F α .
d z [ 2 ( F 1 F 2 F 3 ) 1 / 2 cos ϕ ] = [ δ α + λ α β F β ] d z F α = [ δ α + ( λ α β + + λ α β - ) F β ] d z F α .
2 ( F 1 F 2 F 3 ) 1 / 2 cos ϕ = K + ( δ α + 1 / 2 λ α β + F β ) F α - ψ s 3 F 3 ,
K = H + ψ s 3 F 3
s 3 d F 3 d z = - K ϕ ,             s 3 d ϕ d z = K F 3 ,
ψ = 1 / 2 ( λ 12 - λ 21 ) + 1 / 2 ( λ 13 - λ 31 ) .
2 δ = 2 δ ¯ + ( λ 11 + - λ 21 + - λ 31 + ) + ψ = 2 δ ¯ + ( λ 11 - λ 21 - λ 31 ) = 2 ( δ ¯ + 2 λ ¯ 1 ) ,
d z A 1 = i ( δ 1 + λ 11 A 1 2 ) A 1 , d z A 2 = - i A 3 * A 1 + i ( δ 2 + λ 21 A 1 2 ) A 2 , d z A 3 = - i A 1 A 2 * + i ( δ 3 + λ 31 A 1 2 ) A 3 .
A 1 ( z ) = exp [ i ( δ 1 + λ 11 ) z ] .
A α = C α exp ( i θ α ) ,             θ α = ( δ α + λ α 1 ) z ,
d z C 2 = - i C 3 * exp ( i θ ) ,             d z C 3 = - i C 2 * exp ( i θ ) ,
θ = [ ( δ 1 - δ 2 - δ 3 ) + ( λ 11 - λ 21 - λ 31 ) ] z = 2 δ z
C α = D α exp ( i δ z ) ,
( d z + i δ ) D 2 = - i D 3 * ,             ( d z + i δ ) D 3 = - i D 2 * .
D 2 ( 0 ) = Δ ,             D 3 ( 0 ) = 0 ,
D 2 = Δ { cosh [ ( 1 - δ 2 ) 1 / 2 z ] - i δ sinh [ ( 1 - δ 2 ) 1 / 2 z ] ( 1 - δ 2 ) 1 / 2 } , D 3 = - i Δ * sinh [ ( 1 - δ 2 ) 1 / 2 z ] ( 1 - δ 2 ) 1 / 2 .
D 2 2 = D 3 2 + Δ 2
cos ϕ = Re ( D 2 * D 3 * / D 2 D 3 ) , sin ϕ = Im ( D 2 * D 3 * / D 2 D 3 ) .
cos ϕ 0 ,             sin ϕ 1
cos ϕ - δ ,             sin ϕ ( 1 - δ 2 ) 1 / 2
0 y d u u ( a + b u + c u 2 ) 1 / 2 = - 1 a 1 / 2 tanh - 1 [ 2 a 1 / 2 ( a + b y + c y 2 ) 1 / 2 ( 2 a + b y ) ] ,
F = 2 a b ( 1 - t 2 ) + 2 a [ ( b 2 - 4 a c ) ( 1 - t 2 ) ] 1 / 2 b 2 t 2 - 4 a c ,
a = 4 ( 1 - δ 2 ) ,             b = 4 ( 2 δ λ - 1 ) ,             c = - 4 λ 2 ,
t = tanh ( a 1 / 2 z ) .
F ( z ) = 2 a - b + ( b 2 - 4 a c ) 1 / 2 cosh ( a 1 / 2 z ) .
F max = 2 δ λ - 1 + [ 1 + 4 λ ( λ - δ ) ] 1 / 2 2 λ 2 ,
F ( z ) = 4 b - 4 c + b 2 z 2 .
d F [ F ( 1 - F ) ( F - f + ) ( F - f - ) ] 1 / 2 = ± 2 λ d z
b y d u [ ( a - u ) ( u - b ) ( u - c ) ( u - d ) ] 1 / 2 ,
a = 1 ,             b = 0 ,             c = f + ,             d = f - .
v 2 = ( a - c ) ( u - b ) ( a - b ) ( u - c ) ,
2 [ ( a - c ) ( b - d ) ] 1 / 2 0 v ( y ) d v [ ( 1 - v 2 ) ( 1 - m 2 v 2 ) ] 1 / 2 ,
m 2 = ( a - b ) ( c - d ) ( a - c ) ( b - d ) .
F ( z ) = b ( a - c ) - c ( a - b ) sn 2 ( k z , m ) ( a - c ) - ( a - b ) sn 2 ( k z , m ) ,
k 2 = λ 2 ( a - c ) ( b - d ) .
k 2 = 1 / 2 { ( 1 - λ 2 ) + 2 + [ ( 1 - λ 2 ) 2 - 4 λ 2 ] 1 / 2 } ,
m 2 = 2 [ ( 1 - λ 2 ) 2 - 4 λ 2 ] 1 / 2 ( 1 - λ 2 ) + 2 + [ ( 1 - λ 2 ) 2 - 4 λ 2 ] 1 / 2 .
sn ( K + x ) = cn ( x ) / dn ( x ) ,
dn ( x ) = [ 1 - m 2 sn 2 ( x ) ] 1 / 2 ,
F ( l + z ) = a ( b - d ) + d ( a - b ) sn 2 ( k z , m ) ( b - d ) + ( a - b ) sn 2 ( k z , m ) .
k 2 ( 1 - λ 2 ) + ( 1 - 2 λ 2 ) / ( 1 - λ 2 ) ,
m 2 1 - / ( 1 - λ 2 ) 2 .
K ( m ) log [ 4 / ( 1 - m 2 ) 1 / 2 ]
sn ( x ) tanh ( x ) ,             cn ( x ) sech ( x ) ,
F ( l + z ) 1 - λ 2 1 - λ 2 + sinh 2 [ ( 1 - λ 2 ) 1 / 2 z ] ,
b y d u { ( a - u ) ( u - b ) [ ( u - c ) 2 + d 2 ] } 1 / 2 ,
a = 1 ,             b = 0 ,             c = ( f + + f - ) / 2 ,             d = ( f + - f - ) / 2 i .
v = β ( a - u ) - α ( u - b ) β ( a - u ) + α ( u - b ) ,
α 2 = ( a - c ) 2 + d 2 ,             β 2 = ( b - c ) 2 + d 2 ,
1 ( α β ) 1 / 2 v ( y ) 1 d v [ ( 1 - v 2 ) ( 1 - m 2 + m 2 v 2 ) ] 1 / 2 ,
m 2 = ( a - b ) 2 - ( α - β ) 2 4 α β .
F ( z ) = ( α b + β a ) + ( α b - β a ) cn ( k z , m ) ( α + β ) + ( α - β ) cn ( k z , m ) ,
k 2 = 4 λ 2 α β .
k 2 = 4 [ ( 1 + ) ] 1 / 2 ,
m 2 = λ 2 - [ ( 1 + ) 1 / 2 - 1 / 2 ] 2 4 1 / 2 ( 1 + ) 1 / 2 .
cn ( 2 K + x ) = - cn ( x ) ,
F ( l + z ) = β [ 1 + cn ( k z , m ) ] ( α + β ) - ( α - β ) cn ( k z , m ) .
k 2 4 1 / 2 ,             m 2 1 / 2.
cn ( x ) 1 - x 2 / 2 ,
F ( l + z ) 1 1 + ( z ) 2 ,
d z A 1 = - i A 2 2 + i ( δ 1 + λ 1 β A β 2 ) A 1 , d 2 A 2 = - 2 i A 1 A 2 * + i ( δ 2 + λ 2 β A β 2 ) A 2 ,
A α = F α 1 / 2 exp ( i ϕ α )
d z F 1 = - 2 ( F 1 F 2 2 ) 1 / 2 sin ϕ , d 2 F 2 = 4 ( F 1 F 2 2 ) 1 / 2 sin ϕ
d z ϕ 1 = - ( F 2 2 / F 1 ) 1 / 2 cos ϕ + ( δ 1 + λ 1 β F β ) , d z ϕ 2 = - 2 F 1 1 / 2 cos ϕ + ( δ 2 + λ 2 β F β ) ,
ϕ = ϕ 1 - 2 ϕ 2
H = 2 ( F 1 F 2 2 ) 1 / 2 cos ϕ - ( δ α + 1 / 2 λ α β F β ) F α ,
F 1 ( 0 ) = 0 ,             F 2 ( 0 ) = 1.
F 1 ( z ) = F ( z ) ,             F 2 ( z ) = 1 - 2 F ( z ) .
δ ¯ = 1 / 2 ( δ 1 - 2 δ 2 ) ,             λ ¯ α = 1 / 4 ( λ 1 α - 2 λ 2 α ) ,
δ = δ ¯ + 2 λ ¯ 2 ,             λ = λ ¯ 1 - 2 λ ¯ 2 .
H + ( δ α + 1 / 2 λ α β F β ) F α = 2 ( δ + λ F ) F .
( d z F ) 2 = 4 F [ ( 1 - 2 F ) 2 - ( δ + λ F ) 2 F ] .
F = 2 F ,             z = 2 z ,             δ = δ / 2 ,             λ = λ / 2 2 ,
( d z F ) = 4 F [ ( 1 - F ) 2 - ( δ + λ F ) 2 F ] ,

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