Abstract

The solitary-wave solutions of the four-wave equations are studied, and their relevance to four-wave mixing in finite media is discussed. In general, the transfer of action from the pump waves to the probe and signal waves is limited by nonlinear phase shifts that detune the interaction. However, by controlling the linear phase mismatch judiciously, it is often possible to effect a complete transfer of action from the pump waves to the probe and signal waves.

© 1990 Optical Society of America

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References

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  1. R. Y. Chiao, P. L. Kelley, and E. Garmire, “Stimulated fourphoton interaction and its influence on stimulated Rayleigh–Wing scattering,” Phys. Rev. Lett. 17, 1158–1161 (1966).
    [CrossRef]
  2. R. L. Carman, R. Y. Chiao, and P. L. Kelley, “Observation of degenerate stimulated four-photon interaction and four-wave parametric amplification,” Phys. Rev. Lett. 17, 1281–1283 (1966).
    [CrossRef]
  3. 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]
  4. T. Tajima and J. M. Dawson, “Laser electron accelerator,” Phys. Rev. Lett. 43, 267–270 (1979).
    [CrossRef]
  5. C. Joshi, W. B. Mori, T. Katsouleas, J. M. Dawson, J. M. Kindel, and D. W. Forslund, “Ultrahigh-gradient particle acceleration by intense laser-driven plasma density waves,” Nature 311, 525–529 (1984).
    [CrossRef]
  6. C. M. Tang, P. Sprangle, and R. N. Sudan, “Excitation of the plasma wave in the laser beat wave accelerator,” Appl. Phys. Lett. 45, 375–377 (1984).
    [CrossRef]
  7. C. M. Tang, P. Sprangle, and R. N. Sudan, “Dynamics of spacecharge waves in the laser beat wave accelerator,” Phys. Fluids 28, 1974–1983 (1985).
    [CrossRef]
  8. C. J. McKinstrie and D. W. Forslund, “The detuning of relativistic Langmuir waves in the beat-wave accelerator,” Phys. Fluids 30, 904–908 (1987).
    [CrossRef]
  9. C. J. McKinstrie and D. F. DuBois, “Relativistic solitary-wave solutions of the beat-wave equations,” Phys. Rev. Lett. 57, 2022–2025 (1986);Phys. Rev. Lett. 58, 286 (1987).
    [CrossRef] [PubMed]
  10. C. J. McKinstrie, “Relativistic solitary-wave solutions of the beat-wave equations,” Phys. Fluids 31, 288–297 (1988).
    [CrossRef]
  11. 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]
  12. C. J. McKinstrie, G. G. Luther, and S. H. Batha, “Enhancement of the conjugate signal in nondegenerate collinear four-wave mixing,” presented at the 18th Anomalous Absorption Conference, (L’Estérel, Quebec, June 27– July 1, 1988.
  13. 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]
  14. 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]
  15. Y. Inoue, “Resonant four-wave interaction in a dispersive medium,” J. Phys. Soc. Jpn. 39, 1092–1099 (1975).
    [CrossRef]
  16. T. J. M. Boyd and J. G. Turner, “Four-wave interactions in plasmas,” Lett. Math. Phys. 1, 477–484 (1977).
    [CrossRef]
  17. J. H. Marburger and J. F. Lam, “Nonlinear theory of degenerate four-wave mixing,” Appl. Phys. Lett. 34, 389–391 (1979).
    [CrossRef]
  18. 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]
  19. R. Lytel, “Pump-depletion effects in noncollinear degenerate four-wave mixing in Kerr media,” J. Opt. Soc. Am. B 3, 1580–1584 (1986), and references therein.
    [CrossRef]

1989 (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).
[CrossRef]

1986 (2)

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

R. Lytel, “Pump-depletion effects in noncollinear degenerate four-wave mixing in Kerr media,” J. Opt. Soc. Am. B 3, 1580–1584 (1986), and references therein.
[CrossRef]

1985 (1)

C. M. Tang, P. Sprangle, and R. N. Sudan, “Dynamics of spacecharge waves in the laser beat wave accelerator,” Phys. Fluids 28, 1974–1983 (1985).
[CrossRef]

1984 (2)

C. Joshi, W. B. Mori, T. Katsouleas, J. M. Dawson, J. M. Kindel, and D. W. Forslund, “Ultrahigh-gradient particle acceleration by intense laser-driven plasma density waves,” Nature 311, 525–529 (1984).
[CrossRef]

C. M. Tang, P. Sprangle, and R. N. Sudan, “Excitation of the plasma wave in the laser beat wave accelerator,” Appl. Phys. Lett. 45, 375–377 (1984).
[CrossRef]

1979 (3)

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]

T. Tajima and J. M. Dawson, “Laser electron accelerator,” Phys. Rev. Lett. 43, 267–270 (1979).
[CrossRef]

J. H. Marburger and J. F. Lam, “Nonlinear theory of degenerate four-wave mixing,” Appl. Phys. Lett. 34, 389–391 (1979).
[CrossRef]

1977 (1)

T. J. M. Boyd and J. G. Turner, “Four-wave interactions in plasmas,” Lett. Math. Phys. 1, 477–484 (1977).
[CrossRef]

1975 (1)

Y. Inoue, “Resonant four-wave interaction in a dispersive medium,” J. Phys. Soc. Jpn. 39, 1092–1099 (1975).
[CrossRef]

1966 (2)

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

R. L. Carman, R. Y. Chiao, and P. L. Kelley, “Observation of degenerate stimulated four-photon interaction and four-wave parametric amplification,” Phys. Rev. Lett. 17, 1281–1283 (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]

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]

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.

C. J. McKinstrie, G. G. Luther, and S. H. Batha, “Enhancement of the conjugate signal in nondegenerate collinear four-wave mixing,” presented at the 18th Anomalous Absorption Conference, (L’Estérel, Quebec, June 27– July 1, 1988.

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, T. J. M.

T. J. M. Boyd and J. G. Turner, “Four-wave interactions in plasmas,” Lett. Math. Phys. 1, 477–484 (1977).
[CrossRef]

Carman, R. L.

R. L. Carman, R. Y. Chiao, and P. L. Kelley, “Observation of degenerate stimulated four-photon interaction and four-wave parametric amplification,” Phys. Rev. Lett. 17, 1281–1283 (1966).
[CrossRef]

Chen, Y.

Chiao, R. Y.

R. L. Carman, R. Y. Chiao, and P. L. Kelley, “Observation of degenerate stimulated four-photon interaction and four-wave parametric amplification,” Phys. Rev. Lett. 17, 1281–1283 (1966).
[CrossRef]

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

Dawson, J. M.

C. Joshi, W. B. Mori, T. Katsouleas, J. M. Dawson, J. M. Kindel, and D. W. Forslund, “Ultrahigh-gradient particle acceleration by intense laser-driven plasma density waves,” Nature 311, 525–529 (1984).
[CrossRef]

T. Tajima and J. M. Dawson, “Laser electron accelerator,” Phys. Rev. Lett. 43, 267–270 (1979).
[CrossRef]

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);Phys. Rev. Lett. 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).
[CrossRef]

C. Joshi, W. B. Mori, T. Katsouleas, J. M. Dawson, J. M. Kindel, and D. W. Forslund, “Ultrahigh-gradient particle acceleration by intense laser-driven plasma density waves,” Nature 311, 525–529 (1984).
[CrossRef]

Garmire, E.

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

Inoue, Y.

Y. Inoue, “Resonant four-wave interaction in a dispersive medium,” J. Phys. Soc. Jpn. 39, 1092–1099 (1975).
[CrossRef]

Joshi, C.

C. Joshi, W. B. Mori, T. Katsouleas, J. M. Dawson, J. M. Kindel, and D. W. Forslund, “Ultrahigh-gradient particle acceleration by intense laser-driven plasma density waves,” Nature 311, 525–529 (1984).
[CrossRef]

Katsouleas, T.

C. Joshi, W. B. Mori, T. Katsouleas, J. M. Dawson, J. M. Kindel, and D. W. Forslund, “Ultrahigh-gradient particle acceleration by intense laser-driven plasma density waves,” Nature 311, 525–529 (1984).
[CrossRef]

Kelley, P. L.

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

R. L. Carman, R. Y. Chiao, and P. L. Kelley, “Observation of degenerate stimulated four-photon interaction and four-wave parametric amplification,” Phys. Rev. Lett. 17, 1281–1283 (1966).
[CrossRef]

Kindel, J. M.

C. Joshi, W. B. Mori, T. Katsouleas, J. M. Dawson, J. M. Kindel, and D. W. Forslund, “Ultrahigh-gradient particle acceleration by intense laser-driven plasma density waves,” Nature 311, 525–529 (1984).
[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]

J. H. Marburger and J. F. Lam, “Nonlinear theory of degenerate four-wave mixing,” Appl. Phys. Lett. 34, 389–391 (1979).
[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]

C. J. McKinstrie, G. G. Luther, and S. H. Batha, “Enhancement of the conjugate signal in nondegenerate collinear four-wave mixing,” presented at the 18th Anomalous Absorption Conference, (L’Estérel, Quebec, June 27– July 1, 1988.

Lytel, R.

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]

Marburger, J. H.

J. H. Marburger and J. F. Lam, “Nonlinear theory of degenerate four-wave mixing,” Appl. Phys. Lett. 34, 389–391 (1979).
[CrossRef]

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).
[CrossRef]

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

C. J. McKinstrie, G. G. Luther, and S. H. Batha, “Enhancement of the conjugate signal in nondegenerate collinear four-wave mixing,” presented at the 18th Anomalous Absorption Conference, (L’Estérel, Quebec, June 27– July 1, 1988.

Mori, W. B.

C. Joshi, W. B. Mori, T. Katsouleas, J. M. Dawson, J. M. Kindel, and D. W. Forslund, “Ultrahigh-gradient particle acceleration by intense laser-driven plasma density waves,” Nature 311, 525–529 (1984).
[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]

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]

Snyder, A. W.

Sprangle, P.

C. M. Tang, P. Sprangle, and R. N. Sudan, “Dynamics of spacecharge waves in the laser beat wave accelerator,” Phys. Fluids 28, 1974–1983 (1985).
[CrossRef]

C. M. Tang, P. Sprangle, and R. N. Sudan, “Excitation of the plasma wave in the laser beat wave accelerator,” Appl. Phys. Lett. 45, 375–377 (1984).
[CrossRef]

Sudan, R. N.

C. M. Tang, P. Sprangle, and R. N. Sudan, “Dynamics of spacecharge waves in the laser beat wave accelerator,” Phys. Fluids 28, 1974–1983 (1985).
[CrossRef]

C. M. Tang, P. Sprangle, and R. N. Sudan, “Excitation of the plasma wave in the laser beat wave accelerator,” Appl. Phys. Lett. 45, 375–377 (1984).
[CrossRef]

Tajima, T.

T. Tajima and J. M. Dawson, “Laser electron accelerator,” Phys. Rev. Lett. 43, 267–270 (1979).
[CrossRef]

Tang, C. M.

C. M. Tang, P. Sprangle, and R. N. Sudan, “Dynamics of spacecharge waves in the laser beat wave accelerator,” Phys. Fluids 28, 1974–1983 (1985).
[CrossRef]

C. M. Tang, P. Sprangle, and R. N. Sudan, “Excitation of the plasma wave in the laser beat wave accelerator,” Appl. Phys. Lett. 45, 375–377 (1984).
[CrossRef]

Turner, J. G.

T. J. M. Boyd and J. G. Turner, “Four-wave interactions in plasmas,” Lett. Math. Phys. 1, 477–484 (1977).
[CrossRef]

Appl. Phys. Lett. (3)

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]

C. M. Tang, P. Sprangle, and R. N. Sudan, “Excitation of the plasma wave in the laser beat wave accelerator,” Appl. Phys. Lett. 45, 375–377 (1984).
[CrossRef]

J. H. Marburger and J. F. Lam, “Nonlinear theory of degenerate four-wave mixing,” Appl. Phys. Lett. 34, 389–391 (1979).
[CrossRef]

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

J. Phys. Soc. Jpn. (1)

Y. Inoue, “Resonant four-wave interaction in a dispersive medium,” J. Phys. Soc. Jpn. 39, 1092–1099 (1975).
[CrossRef]

Lett. Math. Phys. (1)

T. J. M. Boyd and J. G. Turner, “Four-wave interactions in plasmas,” Lett. Math. Phys. 1, 477–484 (1977).
[CrossRef]

Nature (1)

C. Joshi, W. B. Mori, T. Katsouleas, J. M. Dawson, J. M. Kindel, and D. W. Forslund, “Ultrahigh-gradient particle acceleration by intense laser-driven plasma density waves,” Nature 311, 525–529 (1984).
[CrossRef]

Opt. Lett. (1)

Phys. Fluids (3)

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

C. M. Tang, P. Sprangle, and R. N. Sudan, “Dynamics of spacecharge waves in the laser beat wave accelerator,” Phys. Fluids 28, 1974–1983 (1985).
[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).
[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)

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]

Phys. Rev. Lett. (4)

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

T. Tajima and J. M. Dawson, “Laser electron accelerator,” Phys. Rev. Lett. 43, 267–270 (1979).
[CrossRef]

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

R. L. Carman, R. Y. Chiao, and P. L. Kelley, “Observation of degenerate stimulated four-photon interaction and four-wave parametric amplification,” Phys. Rev. Lett. 17, 1281–1283 (1966).
[CrossRef]

Proc. IRE (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]

Other (1)

C. J. McKinstrie, G. G. Luther, and S. H. Batha, “Enhancement of the conjugate signal in nondegenerate collinear four-wave mixing,” presented at the 18th Anomalous Absorption Conference, (L’Estérel, Quebec, June 27– July 1, 1988.

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Figures (5)

Fig. 1
Fig. 1

Wave-vector matching diagrams for frequency-degenerate four-wave mixing. The pump waves are denoted by the subscripts 1 and 2, while the probe and signal waves are denoted by the subscripts 3 and 4, respectively. (a) For general angles of inclination, |k4| is not equal to |k3|. (b) At the optimal angle of inclination, |k4| is equal to |k3| and the signal wave is driven resonantly.

Fig. 2
Fig. 2

Action flux densities of the pump waves and daughter waves plotted as functions of position for the case in which the nonlinear phase-mismatch parameter λ is equal to 0.5 and the linear phase-mismatch parameter δ is equal to 0.0. For this nonoptimal choice of δ, the transfer of action from the pump waves to the daughter waves is incomplete.

Fig. 3
Fig. 3

Action flux densities of the pump waves and daughter waves plotted as functions of position for the case in which λ is equal to 0.5 and δ is equal to 0.495. For this near-optimal choice of δ, the transfer of action from the pump waves to the daughter waves is, for all practical purposes, complete. Notice that the region in which the signal waves has a large amplitude is much broader than the corresponding region in Fig. 1.

Fig. 4
Fig. 4

Maximal action flux density of the signal wave plotted as a function of δ for the case in which λ is equal to 0.5. Notice that symmetric shifts of δ from its optimal value do not lead to symmetric reductions in the maximal action flux density of the signal wave.

Fig. 5
Fig. 5

Full width of the signal wave at half of its maximal action flux density plotted as a function of δ for the case in which λ is equal to 0.5. Notice that the full width becomes infinite as δ tends to its optimal value of 0.5. This signifies that the transfer of action from the pump waves to the daughter waves is irreversible in this limit.

Equations (29)

Equations on this page are rendered with MathJax. Learn more.

ω 1 + ω 2 = ω 3 + ω 4 , k 1 + k 2 = k 3 + k 4 .
s 1 d d z A 1 = i A 2 * A 3 A 4 + i ( δ 1 + λ 1 β | A β | 2 ) A 1 ,
s 3 d d z A 3 = i A 4 * 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 d z F 1 = 2 ( F 1 F 2 F 3 F 4 ) 1 / 2 sin ϕ ,
s 3 d d z F 3 = 2 ( F 1 F 2 F 3 F 4 ) 1 / 2 sin ϕ
s 1 d d z ϕ 1 = ( F 2 F 3 F 4 / F 1 ) 1 / 2 cos ϕ + ( δ 1 + λ 1 β F β ) ,
s 3 d d z ϕ 3 = ( F 4 F 1 F 2 / F 3 ) 1 / 2 cos ϕ + ( δ 3 + λ 3 β F β ) ,
ϕ = ϕ 1 + ϕ 2 ϕ 3 ϕ 4
H = 2 ( F 1 F 2 F 3 F 4 ) 1 / 2 cos ϕ ( δ α + ½ 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 d z F 4 ) 2 = 4 F 1 F 2 F 3 F 4 [ H + ( δ α + ½ λ α β F β ) F α ] 2 .
d d z ( s 1 F 1 s 2 F 2 ) = 0 , d d z ( s 3 F 3 s 4 F 4 ) = 0 .
d d z ( s 1 F 1 + s 2 F 2 + s 3 F 3 + s 4 F 4 ) = 0 .
F 1 ( ) = F 2 ( ) = 1 , F 3 ( ) = F 4 ( ) = 0 .
F 3 ( z ) = F 4 ( z ) = F ( z ) , F 1 ( z ) = F 2 ( z ) = 1 F ( z ) .
δ ¯ = ½ ( δ 1 + δ 2 δ 2 δ 4 ) , λ ¯ α = ¼ ( λ α 1 + λ α 2 λ α 3 λ α 4 )
δ = δ ¯ + 2 ( λ ¯ 1 + λ ¯ 2 ) , λ = λ ¯ 1 + λ ¯ 2 λ ¯ 3 λ ¯ 4 .
( d d z F ) 2 = 4 F 2 [ ( 1 F ) 2 ( δ λ F ) 2 ] ,
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 = 8 ( 1 δ λ ) , c = 4 ( 1 λ 2 ) ,
t = tanh ( a 1 / 2 z ) .
δ opt = λ .
δ = λ + ,
F max ( ) = 1 | | 2 | | λ .
l log F 3 ( 0 ) ( 1 δ 2 ) 1 / 2 .
F 1 ( ) = F 4 ( ) = 0 , F 2 ( ) = F 3 ( ) = 1 .
F 1 ( z ) = F 4 ( z ) = F ( z ) , F 2 ( z ) = F 3 ( z ) = 1 F ( z ) .
F 1 ( 0 ) = 1 , F 2 ( l ) = 1 , F 3 ( 0 ) 0 .

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