Abstract

Exact solutions to the equations describing general nonlinear four-wave mixing (including parametric mixing and three-wave sum-frequency generation) in optical fibers are presented. This complements our earlier research on four-photon parametric mixing in optical fibers [ Opt. Lett. 14, 87 ( 1989)].

© 1989 Optical Society of America

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References

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  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]
  2. K. O. Hill, D. D. Johnson, B. C. Kawasaki, and R. I. MacDonald, “Cw three-wave mixing in single-mode optical fibers,” J. Appl. Phys. 49, 5098–5106 (1978).
    [Crossref]
  3. R. H. Stolen and J. E. Bjorkholm, “Parametric amplification and frequency conversion in optical fibers,” IEEE J. Quantum Electron. QE-18, 1062–1082 (1982).
    [Crossref]
  4. A. Vaterescu, “Light conversion in nonlinear monomode optical fibers,” IEEE J. Lightwave Technol. LT-5, 1652–1659 (1987).
    [Crossref]
  5. C. Lin, W. A. Reed, A. D. Pearson, and H. T. Shang, “Phase matching in the minimum-chromatic-dispersion region of single-mode fibers for stimulated four-photon mixing,” Opt. Lett. 6, 493–495 (1981).
    [Crossref] [PubMed]
  6. R. H. Stolen and W. N. Leibolt, “Optical fiber modes using stimulated four-photon mixing,” Appl. Opt. 15, 239–243 (1976).
    [Crossref] [PubMed]
  7. R. H. Stolen, “Phase-matched-stimulated four-photon mixing in silica-fiber waveguides,” IEEE J. Quantum Electron. QE-11, 100–103 (1975).
    [Crossref]
  8. R. H. Stolen, J. E. Bjorkholm, and A. Ashkin, “Phase-matched three-wave mixing in silica fiber optical waveguides,” Appl. Phys. Lett. 24, 308–310 (1974).
    [Crossref]
  9. J. M. Gabriagues, “Third-harmonic and three-wave sum-frequency light generation in an elliptical-core optical fiber,” Opt. Lett. 8, 183–185 (1983).
    [Crossref] [PubMed]
  10. J. H. Marbuger and J. F. Lam, “Nonlinear theory of degenerate four wave mixing,” Appl. Phys. Lett. 34, 389–391 (1979).
    [Crossref]
  11. A. W. Snyder and J. D. Love, Optical Waveguide Theory (Chapman & Hall, New York, 1983).
  12. R. W. Terhune and D. A. Weinberger, “Second-harmonic generation in fibers,” J. Opt. Soc. Am. B 4, 661–674 (1987).
    [Crossref]
  13. 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]
  14. B. Daino, G. Gregori, and S. Wabnitz, “Stability analysis of nonlinear coherent coupling,” J. Appl. Phys. 58, 4512–4514 (1985).
    [Crossref]
  15. Here the term “not very large” is a relative description comparing the case of an extremely large frequency shift discussed below. For instance, if one takes the usual fiber parameters, core radius ρ = 3 μm, relative refractive-index difference Δ = 0.0066, at the long-wavelength range λp = 1.32 μm, C02 will be greater than zero when the frequency shift is smaller than ∼2600 cm−1; with ρ = 3.5 μm, Δ = 0.006, at the short wavelength range λp = 0.532 μm, C02 will be greater than zero at a frequency shift of less than ∼9000 cm−1. These critical frequency shifts (2600, 9000 cm−1) are much larger than those within Raman frequency band.
  16. M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Function, Natl. Bur. Std. (U.S.) Appl. Math. Ser.55 (1964);P. F. Byrd and M. D. Friedman, Handbook of Elliptic Integrals for Engineers and Scientists (Springer-Verlag, Berlin, 1971).
    [Crossref]
  17. 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]

1989 (1)

1987 (2)

R. W. Terhune and D. A. Weinberger, “Second-harmonic generation in fibers,” J. Opt. Soc. Am. B 4, 661–674 (1987).
[Crossref]

A. Vaterescu, “Light conversion in nonlinear monomode optical fibers,” IEEE J. Lightwave Technol. LT-5, 1652–1659 (1987).
[Crossref]

1985 (1)

B. Daino, G. Gregori, and S. Wabnitz, “Stability analysis of nonlinear coherent coupling,” J. Appl. Phys. 58, 4512–4514 (1985).
[Crossref]

1983 (1)

1982 (1)

R. H. Stolen and J. E. Bjorkholm, “Parametric amplification and frequency conversion in optical fibers,” IEEE J. Quantum Electron. QE-18, 1062–1082 (1982).
[Crossref]

1981 (1)

1979 (1)

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

1978 (1)

K. O. Hill, D. D. Johnson, B. C. Kawasaki, and R. I. MacDonald, “Cw three-wave mixing in single-mode optical fibers,” J. Appl. Phys. 49, 5098–5106 (1978).
[Crossref]

1976 (1)

1975 (1)

R. H. Stolen, “Phase-matched-stimulated four-photon mixing in silica-fiber waveguides,” IEEE J. Quantum Electron. QE-11, 100–103 (1975).
[Crossref]

1974 (1)

R. H. Stolen, J. E. Bjorkholm, and A. Ashkin, “Phase-matched three-wave mixing in silica fiber optical waveguides,” Appl. Phys. Lett. 24, 308–310 (1974).
[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]

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]

Ashkin, A.

R. H. Stolen, J. E. Bjorkholm, and A. Ashkin, “Phase-matched three-wave mixing in silica fiber optical waveguides,” Appl. Phys. Lett. 24, 308–310 (1974).
[Crossref]

Bjorkholm, J. E.

R. H. Stolen and J. E. Bjorkholm, “Parametric amplification and frequency conversion in optical fibers,” IEEE J. Quantum Electron. QE-18, 1062–1082 (1982).
[Crossref]

R. H. Stolen, J. E. Bjorkholm, and A. Ashkin, “Phase-matched three-wave mixing in silica fiber optical waveguides,” Appl. Phys. Lett. 24, 308–310 (1974).
[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]

Chen, Y.

Daino, B.

B. Daino, G. Gregori, and S. Wabnitz, “Stability analysis of nonlinear coherent coupling,” J. Appl. Phys. 58, 4512–4514 (1985).
[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]

Gabriagues, J. M.

Gregori, G.

B. Daino, G. Gregori, and S. Wabnitz, “Stability analysis of nonlinear coherent coupling,” J. Appl. Phys. 58, 4512–4514 (1985).
[Crossref]

Hill, K. O.

K. O. Hill, D. D. Johnson, B. C. Kawasaki, and R. I. MacDonald, “Cw three-wave mixing in single-mode optical fibers,” J. Appl. Phys. 49, 5098–5106 (1978).
[Crossref]

Johnson, D. D.

K. O. Hill, D. D. Johnson, B. C. Kawasaki, and R. I. MacDonald, “Cw three-wave mixing in single-mode optical fibers,” J. Appl. Phys. 49, 5098–5106 (1978).
[Crossref]

Kawasaki, B. C.

K. O. Hill, D. D. Johnson, B. C. Kawasaki, and R. I. MacDonald, “Cw three-wave mixing in single-mode optical fibers,” J. Appl. Phys. 49, 5098–5106 (1978).
[Crossref]

Lam, J. F.

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

Leibolt, W. N.

Lin, C.

Love, J. D.

A. W. Snyder and J. D. Love, Optical Waveguide Theory (Chapman & Hall, New York, 1983).

MacDonald, R. I.

K. O. Hill, D. D. Johnson, B. C. Kawasaki, and R. I. MacDonald, “Cw three-wave mixing in single-mode optical fibers,” J. Appl. Phys. 49, 5098–5106 (1978).
[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]

Marbuger, J. H.

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

Pearson, A. D.

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]

Reed, W. A.

Shang, H. T.

Snyder, A. W.

Stolen, R. H.

R. H. Stolen and J. E. Bjorkholm, “Parametric amplification and frequency conversion in optical fibers,” IEEE J. Quantum Electron. QE-18, 1062–1082 (1982).
[Crossref]

R. H. Stolen and W. N. Leibolt, “Optical fiber modes using stimulated four-photon mixing,” Appl. Opt. 15, 239–243 (1976).
[Crossref] [PubMed]

R. H. Stolen, “Phase-matched-stimulated four-photon mixing in silica-fiber waveguides,” IEEE J. Quantum Electron. QE-11, 100–103 (1975).
[Crossref]

R. H. Stolen, J. E. Bjorkholm, and A. Ashkin, “Phase-matched three-wave mixing in silica fiber optical waveguides,” Appl. Phys. Lett. 24, 308–310 (1974).
[Crossref]

Terhune, R. W.

R. W. Terhune and D. A. Weinberger, “Second-harmonic generation in fibers,” J. Opt. Soc. Am. B 4, 661–674 (1987).
[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]

Vaterescu, A.

A. Vaterescu, “Light conversion in nonlinear monomode optical fibers,” IEEE J. Lightwave Technol. LT-5, 1652–1659 (1987).
[Crossref]

Wabnitz, S.

B. Daino, G. Gregori, and S. Wabnitz, “Stability analysis of nonlinear coherent coupling,” J. Appl. Phys. 58, 4512–4514 (1985).
[Crossref]

Weinberger, D. A.

Appl. Opt. (1)

Appl. Phys. Lett. (2)

R. H. Stolen, J. E. Bjorkholm, and A. Ashkin, “Phase-matched three-wave mixing in silica fiber optical waveguides,” Appl. Phys. Lett. 24, 308–310 (1974).
[Crossref]

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

IEEE J. Lightwave Technol. (1)

A. Vaterescu, “Light conversion in nonlinear monomode optical fibers,” IEEE J. Lightwave Technol. LT-5, 1652–1659 (1987).
[Crossref]

IEEE J. Quantum Electron. (2)

R. H. Stolen, “Phase-matched-stimulated four-photon mixing in silica-fiber waveguides,” IEEE J. Quantum Electron. QE-11, 100–103 (1975).
[Crossref]

R. H. Stolen and J. E. Bjorkholm, “Parametric amplification and frequency conversion in optical fibers,” IEEE J. Quantum Electron. QE-18, 1062–1082 (1982).
[Crossref]

J. Appl. Phys. (2)

B. Daino, G. Gregori, and S. Wabnitz, “Stability analysis of nonlinear coherent coupling,” J. Appl. Phys. 58, 4512–4514 (1985).
[Crossref]

K. O. Hill, D. D. Johnson, B. C. Kawasaki, and R. I. MacDonald, “Cw three-wave mixing in single-mode optical fibers,” J. Appl. Phys. 49, 5098–5106 (1978).
[Crossref]

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

Opt. Lett. (3)

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]

Other (3)

A. W. Snyder and J. D. Love, Optical Waveguide Theory (Chapman & Hall, New York, 1983).

Here the term “not very large” is a relative description comparing the case of an extremely large frequency shift discussed below. For instance, if one takes the usual fiber parameters, core radius ρ = 3 μm, relative refractive-index difference Δ = 0.0066, at the long-wavelength range λp = 1.32 μm, C02 will be greater than zero when the frequency shift is smaller than ∼2600 cm−1; with ρ = 3.5 μm, Δ = 0.006, at the short wavelength range λp = 0.532 μm, C02 will be greater than zero at a frequency shift of less than ∼9000 cm−1. These critical frequency shifts (2600, 9000 cm−1) are much larger than those within Raman frequency band.

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Function, Natl. Bur. Std. (U.S.) Appl. Math. Ser.55 (1964);P. F. Byrd and M. D. Friedman, Handbook of Elliptic Integrals for Engineers and Scientists (Springer-Verlag, Berlin, 1971).
[Crossref]

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Figures (2)

Fig. 1
Fig. 1

Optimum index mismatch ΔSmax versus frequency shift Δf for P4(0) = 0. P3(0) = 0.001P1(0), P2(0) = 1.5P1(0), fiber core radius ρ = 3.5 μm, relative refractive-index difference Δ = 0.006, pump wavelength λp1 = λp2 = λp = 0.532 μm.

Fig. 2
Fig. 2

Normalized power P ¯ i = P i / P versus normalized distance Γ for (a) ΔS = ΔSmax = 0.149, (b) ΔS = 0.6, (c) ΔS = −0.15.

Equations (106)

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E t = i A i ( z ) e i ( r , θ ) exp [ j ( β i z ω i t ) ] ,
d A 1 d z = j o 1 ω 1 A 2 * A 3 A 4 exp ( j Δ β z ) + j o 2 ω 1 A 1 i = 1 4 I 1 i | A i | 2 ,
d A 2 d z = j o 1 ω 2 A 1 * A 3 A 4 exp ( j Δ β z ) + j o 2 ω 2 A 2 i = 1 4 I 2 i | A i | 2 ,
d A 3 d z = j o 1 ω 3 A 1 A 2 A 4 * exp ( j Δ β z ) + j o 2 ω 3 A 3 i = 1 4 I 3 i | A i | 2 ,
d A 4 d z = j o 1 ω 4 A 1 A 2 A 3 * exp ( j Δ β z ) + j o 2 ω 4 A 4 i = 1 4 I 4 i | A i | 2 ,
o 1 = 3 2 0 Re ( χ 3 ) ψ 1 ψ 2 ψ 3 ψ 4 d A ,
I m i = ψ m 2 ψ i 2 d A , ( m i ) ,
I i i = 1 2 ψ i 4 d A ,
ω 1 + ω 2 + ω 3 = ω 4 ,
Δ β = β 1 + β 2 + β 3 β 4 .
d A 1 d z = j o 1 ω 1 A 2 * A 3 * A 4 exp ( j Δ β z ) + j o 2 ω 1 A 1 i = 1 4 I 1 i | A i | 2 ,
d A 2 d z = j o 1 ω 2 A 1 * A 3 * A 4 exp ( j Δ β z ) + j o 2 ω 2 A 2 i = 1 4 I 2 i | A i | 2 ,
d A 3 d z = j o 1 ω 3 A 1 * A 2 * A 4 exp ( j Δ β z ) + j o 2 ω 3 A 3 i = 1 4 I 3 i | A i | 2 ,
d A 4 d z = j o 1 ω 4 A 1 A 2 A 3 exp ( j Δ β z ) + j o 2 ω 4 A 4 i = 1 4 I 4 i | A i | 2 .
d A 1 d z = j o 3 ω 1 A 1 * A 1 * A 3 exp ( j Δ β z ) + j o 2 ω 1 A 1 ( I 11 | A 1 | 2 + I 13 | A 3 | 2 ) ,
d A 3 d z = j 1 3 o 3 ω 3 A 1 3 exp ( j Δ β z ) + j o 2 ω 3 A 3 ( I 13 | A 1 | 2 + I 33 | A 3 | 2 ) ,
o 3 = 3 4 0 Re ( χ 3 ) I 13 .
d ξ 1 d z = o 1 ω 1 ξ 2 ξ 3 ξ 4 sin θ ,
d ξ 2 d z = o 1 ω 2 ξ 1 ξ 3 ξ 4 sin θ ,
d ξ 3 d z = o 1 ω 3 ξ 1 ξ 2 ξ 4 sin θ ,
d ξ 4 d z = o 1 ω 4 ξ 1 ξ 2 ξ 3 sin θ ,
d θ d z = Δ β + o 1 ( ω 1 ξ 2 ξ 3 ξ 4 ξ 1 + ω 2 ξ 1 ξ 3 ξ 4 ξ 2 ω 3 ξ 1 ξ 2 ξ 3 ξ 4 ω 4 ξ 1 ξ 2 ξ 3 ξ 4 ) cos θ + o 2 ( ω 1 i = 1 4 I 1 i ξ i 2 + ω 2 i = 1 4 I 2 i ξ i 2 ω 3 i = 1 4 I 3 i ξ i 2 ω 4 i = 1 4 I 4 i ξ i 2 ) .
P = ξ 1 2 + ξ 2 2 + ξ 3 2 + ξ 4 2 .
ξ i = ω i P ζ i ( i = 1 , 4 ) ,
Γ = o 1 P ( ω 1 ω 2 ω 3 ω 4 ) 1 / 2 z
d ζ 1 d Γ = ζ 2 ζ 3 ζ 4 sin θ ,
d ζ 2 d Γ = ζ 1 ζ 3 ζ 4 sin θ ,
d ζ 3 d Γ = ζ 1 ζ 2 ζ 4 sin θ ,
d ζ 4 d Γ = ζ 1 ζ 2 ζ 3 sin θ ,
d θ d Γ = Δ S + cos θ sin θ d d Γ ln ( ζ 1 ζ 2 ζ 3 ζ 4 ) + Q 1 ζ 1 2 + Q 2 ζ 2 2 + Q 3 ζ 3 2 + Q 4 ζ 4 2 ,
Δ S = Δ β o 1 P ( ω 1 ω 2 ω 3 ω 4 ) 1 / 2 ,
Q i = o 2 m m ω m I m i ω i o 1 ( ω 1 ω 2 ω 3 ω 4 ) 1 / 2 ,
m = { 1 m = 1 , 2 1 m = 3 , 4 .
x 1 = ζ 1 2 + ζ 4 2 ,
x 2 = ζ 2 2 + ζ 4 2 ,
x 3 = ζ 3 2 ζ 4 2 ,
C = Δ S 2 ζ 4 2 + ζ 1 ζ 2 ζ 3 ζ 4 cos θ + 1 4 ( Q 1 ζ 1 4 + Q 2 ζ 2 4 Q 3 ζ 3 4 Q 4 ζ 4 4 ) .
Γ = ± 1 2 x ( 0 ) x ( Γ ) d x f ( x ) ,
f ( x ) = x ( x 1 x ) ( x 2 x ) ( x 3 + x ) [ C + Δ S 2 x Q 1 4 ( x 1 x ) 2 Q 2 4 ( x 2 x ) 2 + Q 3 4 ( x 3 + x ) 2 + Q 4 4 x 2 ] 2 .
f ( x ) = 0
C 0 2 = 1 1 16 ( Q 1 + Q 2 Q 3 Q 4 ) 2 ,
η 2 x η . 3
y 2 = x η 2 x η 1 η 3 η 1 η 3 η 2
Γ = ± 1 C 0 [ ( η 3 η 1 ) ( η 4 η 2 ) ] 1 / 2 y ( 0 ) y ( Γ ) d y [ ( 1 y 2 ) ( 1 k 2 y 2 ) ] 1 / 2 ,
k = [ ( η 3 η 2 ) ( η 4 η 1 ) / ( η 3 η 1 ) ( η 4 η 2 ) ] 1 / 2 .
P 4 ( z ) = ω 4 P η 2 η η 1 s n 2 ( z + z 0 z c , k ) 1 η s n 2 ( z + z 0 z c , k ) ,
P 3 ( z ) = P 3 ( 0 ) ω 3 ω 4 [ P 4 ( 0 ) P 4 ( z ) ] ,
P 2 ( z ) = P 2 ( 0 ) + ω 2 ω 4 [ P 4 ( 0 ) P 4 ( z ) ] ,
P 1 ( z ) = P 1 ( 0 ) + ω 1 ω 4 [ P 4 ( 0 ) P 4 ( z ) ] ,
z c 1 = o 1 P | C 0 | [ ( η 3 η 1 ) ( η 4 η 2 ) ω 1 ω 2 ω 3 ω 4 ] 1 / 2 ,
η = ( η 3 η 2 ) / ( η 3 η 1 ) ,
z o = z c F ( sin 1 { P 4 ( 0 ) ω 4 P η 2 η [ P 4 ( 0 ) ω 4 P η 1 ] } 1 / 2 , k ) ,
k = [ ( η 4 η 3 ) ( η 2 η 1 ) / ( η 3 η 1 ) ( η 4 η 2 ) ] 1 / 2 .
P 4 ( z ) = ω 4 P η 3 η η 2 s n 2 ( z + z 0 z c , k ) 1 η s n 2 ( z + z 0 z c , k ) ,
η = ( η 4 η 3 ) / ( η 4 η 2 ) ,
z 0 = z c F ( sin 1 { P 4 ( 0 ) ω 4 P η 3 η [ P 4 ( 0 ) ω 4 P η 2 ] } 1 / 2 , k ) ,
P 4 ( z ) = ω 4 P η 1 η η 4 s n 2 ( z + z o z c , k ) 1 η s n 2 ( z + z 0 z c , k ) ,
η = ( η 2 η 1 ) / ( η 4 η 2 ) ,
z 0 = z c F ( sin 1 { P 4 ( 0 ) ω 4 P η 1 η [ P 4 ( 0 ) + ω 4 P η 4 ] } 1 / 2 , k )
P 3 , P 4 P 1 , P 2 ,
m = { 1 m = 1 , 2 , 3 1 m = 4 ,
x 1 = ζ 1 2 + ζ 4 2 ,
x 2 = ζ 2 2 + ζ 4 2 ,
x 3 = ζ 3 2 + ζ 4 2 ,
C = Δ S 2 ζ 4 2 + ζ 1 ζ 2 ζ 3 ζ 4 cos θ + 1 4 ( Q 1 ζ 1 4 + Q 2 ζ 2 4 + Q 3 ζ 3 4 Q 4 ζ 4 4 ) .
Γ = ± 1 2 x ( o ) x ( r ) d x f 1 ( x ) ,
f 1 ( x ) = x ( x 1 x ) ( x 2 x ) ( x 3 x ) [ C + Δ S 2 x Q 1 4 ( x 1 x ) 2 Q 2 4 ( x 2 x ) 2 Q 3 4 ( x 3 x ) 2 + Q 4 4 x 2 ] 2 .
η 4 > η 3 > η 2 > η 1 0
η 4 > η 3 0 , η 1 < η 2 < 0 .
y 2 = ( η 4 η 2 ) ( η 2 η 1 ) ( x η 1 ) ( η 4 x )
Γ = ± 1 C 0 [ ( η 3 η 1 ) ( η 4 η 2 ) ] 1 / 2 y ( 0 ) y ( Γ ) d y [ ( 1 y 2 ) ( 1 k 2 y 2 ) ] 1 / 2 ,
C 0 = [ 1 + 1 / 16 ( Q 1 + Q 2 + Q 3 Q 4 ) 2 ] 1 / 2 ,
k = [ ( η 4 η 3 ) ( η 2 η 1 ) / ( η 3 η 1 ) ( η 4 η 2 ) ] 1 / 2 .
y 2 = ( η 4 η 2 ) ( x η 3 ) ( η 4 η 3 ) ( x η 2 )
P 4 ( z ) = ω 4 P η 1 + η η 4 s n 2 ( z + z o z c , k ) 1 + η s n 2 ( z + z o z c , k ) ,
P 3 ( z ) = P 3 ( 0 ) + ω 3 ω 4 [ P 4 ( 0 ) P 4 ( z ) ] ,
P 2 ( z ) = P 2 ( 0 ) + ω 2 ω 4 [ P 4 ( 0 ) P 4 ( z ) ] ,
P 1 ( z ) = P 1 ( 0 ) + ω 1 ω 4 [ P 4 ( 0 ) P 4 ( z ) ] ,
P 3 ( z ) = P η 3 η η 2 s n 2 ( z + z 0 z c , k ) 1 η s n 2 ( z + z o z c , k ) ,
z 0 = z c F ( sin 1 { P 3 ( 0 ) P η 3 η [ P 3 ( 0 ) P η 2 ] } 1 / 2 , k ) ,
η = η 4 η 3 η 4 η 2
P 3 ( z ) = P η 1 + η η 4 s n 2 ( z + z 0 z c , k ) 1 + η s n 2 ( z + z o z c , k ) ,
z o = z c F ( sin 1 { P 3 ( 0 ) P η 1 η [ P η 4 P 3 ( 0 ) ] } 1 / 2 , k ) ,
η = η 2 η 1 η 4 η 2
x ( 1 x ) 3 [ C + Δ S 2 x Q 1 4 ( 1 x ) 2 + Q 3 4 x 2 ] 2 = 0 ,
k = [ ( η 4 η 3 ) ( η 2 η 1 ) ( η 3 η 1 ) ( η 4 η 2 ) ] 1 / 2 ,
z c 1 = o 3 P C 0 [ ( η 3 η 1 ) ( η 4 η 2 ) ] 1 / 2 ,
C 0 = [ 1 + ( Q 1 Q 3 ) 2 / 16 ] 1 / 2 .
P 1 ( z ) = P 1 ( 0 ) + P 3 ( 0 ) P 3 ( z ) .
η slp ( Δ S ) | Δ S = Δ S max = 0 .
η slp = min [ 1 ω 1 P 1 ( 0 ) P , 1 ω 2 P 2 ( 0 ) P ]
P 3 ( z max ) P 3 ( 0 ) = P η slp P 3 ( 0 ) .
η slp = min [ 1 ω 1 P 1 ( 0 ) P , 1 ω 2 P 2 ( 0 ) P , 1 ω 3 P 3 ( 0 ) P ]
C 1 2 = x 3 x 1 x 2 + Δ S + x 1 Q 1 + x 2 Q 2 + x 3 Q 3 .
η 3 > η 2 > η 1 ,
P 4 ( z ) = ω 4 P η 2 k η 1 s n 2 ( z + z o z c , k ) 1 k s n 2 ( z + z o z c , k ) ,
z o = z c F ( sin 1 { P 4 ( 0 ) ω 4 P η 2 k [ P 4 ( 0 ) ω 4 P η 1 ] } 1 / 2 , k ) ,
k = ( η 3 η 2 ) / ( η 3 η 1 )
p 4 ( z ) = ω 4 P [ η 1 + ( η 2 η 1 ) s n 2 ( z + z o z c , k ) ] ,
z o = z c F { sin 1 [ P 4 ( 0 ) ω 4 P η 1 ω 4 P ( η 2 η 1 ) ] 1 / 2 , k }
k = ( η 2 η 1 ) / ( η 3 η 1 )
z c 1 = o 1 P | C 1 | [ ( η 3 η 1 ) ω 1 ω 2 ω 3 ω 4 ] 1 / 2 .
C 2 2 = 1 4 ( x 1 + x 2 + x 3 ) 2 + x 2 x 3 + 2 ( C Q 1 4 x 1 2 Q 2 4 x 2 2 + Q 3 4 x 3 2 ) .
P 4 ( z ) = ω 4 P ( η 1 η 2 2 sin { 2 | C 2 | o 1 P ( ω 1 ω 2 ω 3 ω 4 ) 1 / 2 z + sin 1 [ 2 P 4 ( 0 ) ( η 1 + η 2 ) ω 4 P ω 4 P ( η 1 η 2 ) ] } + η 1 + η 2 2 ) .
f 1 ( x ) = ( x η 1 ) ( η 2 x ) ( η 3 x ) ( η 4 x ) C 0 2
f 1 ( x ) = ( x η 1 ) ( x η 2 ) ( x η 3 ) ( η 4 x ) C 0 2

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