Abstract

The nonlinear physical processes influencing pulse propagation are emphasized by analyzing equations for the pulse spectrum amplitude and phase. A constant pulse is maintained when it has the correct form necessary to suppress the four-photon mixing processes, so that the amplitude is constant and the total optical Kerr effect counterbalances the frequency dependence of the linear dispersion, thus preventing variable frequency-dependent terms in the phase. The effects are demonstrated explicitly and offer an alternative to the self-phase-modulation picture based on the equivalent nonlinear Schrödinger equation. The fundamental soliton is immediately obtained. Constants associated with pulse propagation in a nonlinear medium are related to moments of the pulse spectrum. The constraints on pulse shape and magnitude are discussed, and the effects of power absorption are examined.

© 1986 Optical Society of America

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References

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  1. L. F. Mollenauer and R. H. Stolen, “Solitons in optical fibers,” Laser Focus 18(4), 193 (1982).
  2. R. H. Stolen, “Active fibers,” in New Directions in Guided Wave and Coherent Optics, D. B. Ostrowsky and E. Spitz, eds. (Nijhoff, The Hague, 1984), Vol. 1.
  3. R. P. Feynman, The Feyman Lectures on Physics (Addison-Wesley, Reading, Mass., 1964), Vol. II, Sec. 10-4.
  4. J. A. Armstrong, H. Bloembergen, J. Ducuing, and P. S. Pershan, “Interactions between light waves in a non-linear dielectric,” Phys. Rev. 127, 1918 (1962).
    [CrossRef]
  5. N. Bloembergen, Nonlinear Optics (Benjamin, New York, 1965).
  6. 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. A,  137, 801 (1965).
  7. R. H. Stolen and A. Ashkin, “Optical Kerr effect in glass waveguide,” Appl. Phys. Lett. 22, 294–296 (1973).
    [CrossRef]
  8. K. O. Hill, D. C. Johnson, B. S. Kawasaki, and R. I. MacDonald, “Cw three-wave mixing in single mode optical fibers,” J. Appl. Phys. 49, 5098 (1978).
    [CrossRef]
  9. R. H. Stolen and E. P. Ippen, “Raman gain in glass optical waveguide,” Appl. Phys. Lett. 22, 276 (1973).
    [CrossRef]
  10. D. Marcuse, “Gaussian approximation of the fundamental modes of graded index fibers,” J. Opt. Soc. Am. 68, 103 (1978).
    [CrossRef]
  11. L. F. Mollenauer, R. H. Stolen, and J. P. Gordon, “Experimental observation of picosecond pulse narrowing and soliton in optical fibers,” Phys. Rev. Lett. 15, 1095 (1980).
    [CrossRef]
  12. R. H. Stolen, L. F. Mollenauer, and W. J. Tomlinson, “Observation of pulse restoration at the soliton period in optical fibers,” Opt. Lett. 8, 186 (1983).
    [CrossRef] [PubMed]
  13. V. E. Zakharov and A. S. Shabat, “Exact theory of two-dimensional self focusing and one-dimensional self-modulation of waves in nonlinear media,” Sov. Phys. JETP 34, 62 (1972).
  14. A. Papoulis, Systems and Transforms with Applications in Optics (McGraw-Hill, New York, 1968); see problem 3-7.
  15. A. Hasegawa and Y. Kodoma, “Signal transmission by optical solitons in monomode fibers,” Proc. IEEE 69, 1145 (1981).
    [CrossRef]
  16. N. J. Doran and K. J. Blow, “Solitons in optical communications,” IEEE J. Quantum Electron. QE-19, 1883 (1983).
    [CrossRef]
  17. K. J. Blow and N. J. Doran, “The asymptotic dispersion of soliton pulses in lossy fibers,” Opt. Commun. 52, 367 (1985).
    [CrossRef]
  18. E. Shiojiri and Y. Fujii, “Transmission capability of an optical fiber communication system using index nonlinearity,” Appl. Opt. 24, 358 (1985).
    [CrossRef] [PubMed]
  19. For example, see G. A. Korn and T. M. Korn, Mathematical Handbook for Scientists and Engineers (McGraw-Hill, New York, 1968), Sec. 9.5-6.

1985 (2)

K. J. Blow and N. J. Doran, “The asymptotic dispersion of soliton pulses in lossy fibers,” Opt. Commun. 52, 367 (1985).
[CrossRef]

E. Shiojiri and Y. Fujii, “Transmission capability of an optical fiber communication system using index nonlinearity,” Appl. Opt. 24, 358 (1985).
[CrossRef] [PubMed]

1983 (2)

1982 (1)

L. F. Mollenauer and R. H. Stolen, “Solitons in optical fibers,” Laser Focus 18(4), 193 (1982).

1981 (1)

A. Hasegawa and Y. Kodoma, “Signal transmission by optical solitons in monomode fibers,” Proc. IEEE 69, 1145 (1981).
[CrossRef]

1980 (1)

L. F. Mollenauer, R. H. Stolen, and J. P. Gordon, “Experimental observation of picosecond pulse narrowing and soliton in optical fibers,” Phys. Rev. Lett. 15, 1095 (1980).
[CrossRef]

1978 (2)

D. Marcuse, “Gaussian approximation of the fundamental modes of graded index fibers,” J. Opt. Soc. Am. 68, 103 (1978).
[CrossRef]

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

1973 (2)

R. H. Stolen and E. P. Ippen, “Raman gain in glass optical waveguide,” Appl. Phys. Lett. 22, 276 (1973).
[CrossRef]

R. H. Stolen and A. Ashkin, “Optical Kerr effect in glass waveguide,” Appl. Phys. Lett. 22, 294–296 (1973).
[CrossRef]

1972 (1)

V. E. Zakharov and A. S. Shabat, “Exact theory of two-dimensional self focusing and one-dimensional self-modulation of waves in nonlinear media,” Sov. Phys. JETP 34, 62 (1972).

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. A,  137, 801 (1965).

1962 (1)

J. A. Armstrong, H. Bloembergen, J. Ducuing, and P. S. Pershan, “Interactions between light waves in a non-linear dielectric,” Phys. Rev. 127, 1918 (1962).
[CrossRef]

Armstrong, J. A.

J. A. Armstrong, H. Bloembergen, J. Ducuing, and P. S. Pershan, “Interactions between light waves in a non-linear dielectric,” Phys. Rev. 127, 1918 (1962).
[CrossRef]

Ashkin, A.

R. H. Stolen and A. Ashkin, “Optical Kerr effect in glass waveguide,” Appl. Phys. Lett. 22, 294–296 (1973).
[CrossRef]

Bloembergen, H.

J. A. Armstrong, H. Bloembergen, J. Ducuing, and P. S. Pershan, “Interactions between light waves in a non-linear dielectric,” Phys. Rev. 127, 1918 (1962).
[CrossRef]

Bloembergen, N.

N. Bloembergen, Nonlinear Optics (Benjamin, New York, 1965).

Blow, K. J.

K. J. Blow and N. J. Doran, “The asymptotic dispersion of soliton pulses in lossy fibers,” Opt. Commun. 52, 367 (1985).
[CrossRef]

N. J. Doran and K. J. Blow, “Solitons in optical communications,” IEEE J. Quantum Electron. QE-19, 1883 (1983).
[CrossRef]

Doran, N. J.

K. J. Blow and N. J. Doran, “The asymptotic dispersion of soliton pulses in lossy fibers,” Opt. Commun. 52, 367 (1985).
[CrossRef]

N. J. Doran and K. J. Blow, “Solitons in optical communications,” IEEE J. Quantum Electron. QE-19, 1883 (1983).
[CrossRef]

Ducuing, J.

J. A. Armstrong, H. Bloembergen, J. Ducuing, and P. S. Pershan, “Interactions between light waves in a non-linear dielectric,” Phys. Rev. 127, 1918 (1962).
[CrossRef]

Feynman, R. P.

R. P. Feynman, The Feyman Lectures on Physics (Addison-Wesley, Reading, Mass., 1964), Vol. II, Sec. 10-4.

Fujii, Y.

Gordon, J. P.

L. F. Mollenauer, R. H. Stolen, and J. P. Gordon, “Experimental observation of picosecond pulse narrowing and soliton in optical fibers,” Phys. Rev. Lett. 15, 1095 (1980).
[CrossRef]

Hasegawa, A.

A. Hasegawa and Y. Kodoma, “Signal transmission by optical solitons in monomode fibers,” Proc. IEEE 69, 1145 (1981).
[CrossRef]

Hill, K. O.

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

Ippen, E. P.

R. H. Stolen and E. P. Ippen, “Raman gain in glass optical waveguide,” Appl. Phys. Lett. 22, 276 (1973).
[CrossRef]

Johnson, D. C.

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

Kawasaki, B. S.

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

Kodoma, Y.

A. Hasegawa and Y. Kodoma, “Signal transmission by optical solitons in monomode fibers,” Proc. IEEE 69, 1145 (1981).
[CrossRef]

Korn, G. A.

For example, see G. A. Korn and T. M. Korn, Mathematical Handbook for Scientists and Engineers (McGraw-Hill, New York, 1968), Sec. 9.5-6.

Korn, T. M.

For example, see G. A. Korn and T. M. Korn, Mathematical Handbook for Scientists and Engineers (McGraw-Hill, New York, 1968), Sec. 9.5-6.

MacDonald, R. I.

K. O. Hill, D. C. Johnson, B. S. Kawasaki, and R. I. MacDonald, “Cw three-wave mixing in single mode optical fibers,” J. Appl. Phys. 49, 5098 (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. A,  137, 801 (1965).

Marcuse, D.

Mollenauer, L. F.

R. H. Stolen, L. F. Mollenauer, and W. J. Tomlinson, “Observation of pulse restoration at the soliton period in optical fibers,” Opt. Lett. 8, 186 (1983).
[CrossRef] [PubMed]

L. F. Mollenauer and R. H. Stolen, “Solitons in optical fibers,” Laser Focus 18(4), 193 (1982).

L. F. Mollenauer, R. H. Stolen, and J. P. Gordon, “Experimental observation of picosecond pulse narrowing and soliton in optical fibers,” Phys. Rev. Lett. 15, 1095 (1980).
[CrossRef]

Papoulis, A.

A. Papoulis, Systems and Transforms with Applications in Optics (McGraw-Hill, New York, 1968); see problem 3-7.

Pershan, P. S.

J. A. Armstrong, H. Bloembergen, J. Ducuing, and P. S. Pershan, “Interactions between light waves in a non-linear dielectric,” Phys. Rev. 127, 1918 (1962).
[CrossRef]

Shabat, A. S.

V. E. Zakharov and A. S. Shabat, “Exact theory of two-dimensional self focusing and one-dimensional self-modulation of waves in nonlinear media,” Sov. Phys. JETP 34, 62 (1972).

Shiojiri, E.

Stolen, R. H.

R. H. Stolen, L. F. Mollenauer, and W. J. Tomlinson, “Observation of pulse restoration at the soliton period in optical fibers,” Opt. Lett. 8, 186 (1983).
[CrossRef] [PubMed]

L. F. Mollenauer and R. H. Stolen, “Solitons in optical fibers,” Laser Focus 18(4), 193 (1982).

L. F. Mollenauer, R. H. Stolen, and J. P. Gordon, “Experimental observation of picosecond pulse narrowing and soliton in optical fibers,” Phys. Rev. Lett. 15, 1095 (1980).
[CrossRef]

R. H. Stolen and E. P. Ippen, “Raman gain in glass optical waveguide,” Appl. Phys. Lett. 22, 276 (1973).
[CrossRef]

R. H. Stolen and A. Ashkin, “Optical Kerr effect in glass waveguide,” Appl. Phys. Lett. 22, 294–296 (1973).
[CrossRef]

R. H. Stolen, “Active fibers,” in New Directions in Guided Wave and Coherent Optics, D. B. Ostrowsky and E. Spitz, eds. (Nijhoff, The Hague, 1984), Vol. 1.

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. A,  137, 801 (1965).

Tomlinson, W. J.

Zakharov, V. E.

V. E. Zakharov and A. S. Shabat, “Exact theory of two-dimensional self focusing and one-dimensional self-modulation of waves in nonlinear media,” Sov. Phys. JETP 34, 62 (1972).

Appl. Opt. (1)

Appl. Phys. Lett. (2)

R. H. Stolen and A. Ashkin, “Optical Kerr effect in glass waveguide,” Appl. Phys. Lett. 22, 294–296 (1973).
[CrossRef]

R. H. Stolen and E. P. Ippen, “Raman gain in glass optical waveguide,” Appl. Phys. Lett. 22, 276 (1973).
[CrossRef]

IEEE J. Quantum Electron. (1)

N. J. Doran and K. J. Blow, “Solitons in optical communications,” IEEE J. Quantum Electron. QE-19, 1883 (1983).
[CrossRef]

J. Appl. Phys. (1)

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

J. Opt. Soc. Am. (1)

Laser Focus (1)

L. F. Mollenauer and R. H. Stolen, “Solitons in optical fibers,” Laser Focus 18(4), 193 (1982).

Opt. Commun. (1)

K. J. Blow and N. J. Doran, “The asymptotic dispersion of soliton pulses in lossy fibers,” Opt. Commun. 52, 367 (1985).
[CrossRef]

Opt. Lett. (1)

Phys. Rev. (1)

J. A. Armstrong, H. Bloembergen, J. Ducuing, and P. S. Pershan, “Interactions between light waves in a non-linear dielectric,” Phys. Rev. 127, 1918 (1962).
[CrossRef]

Phys. Rev. A (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. A,  137, 801 (1965).

Phys. Rev. Lett. (1)

L. F. Mollenauer, R. H. Stolen, and J. P. Gordon, “Experimental observation of picosecond pulse narrowing and soliton in optical fibers,” Phys. Rev. Lett. 15, 1095 (1980).
[CrossRef]

Proc. IEEE (1)

A. Hasegawa and Y. Kodoma, “Signal transmission by optical solitons in monomode fibers,” Proc. IEEE 69, 1145 (1981).
[CrossRef]

Sov. Phys. JETP (1)

V. E. Zakharov and A. S. Shabat, “Exact theory of two-dimensional self focusing and one-dimensional self-modulation of waves in nonlinear media,” Sov. Phys. JETP 34, 62 (1972).

Other (5)

A. Papoulis, Systems and Transforms with Applications in Optics (McGraw-Hill, New York, 1968); see problem 3-7.

For example, see G. A. Korn and T. M. Korn, Mathematical Handbook for Scientists and Engineers (McGraw-Hill, New York, 1968), Sec. 9.5-6.

N. Bloembergen, Nonlinear Optics (Benjamin, New York, 1965).

R. H. Stolen, “Active fibers,” in New Directions in Guided Wave and Coherent Optics, D. B. Ostrowsky and E. Spitz, eds. (Nijhoff, The Hague, 1984), Vol. 1.

R. P. Feynman, The Feyman Lectures on Physics (Addison-Wesley, Reading, Mass., 1964), Vol. II, Sec. 10-4.

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Figures (2)

Fig. 1
Fig. 1

Phase factor g for (a) hyperbolic secant spectrum shape, Eqs. (41) and (44); (b) Gaussian spectrum shape, Eqs. (45) and (48). In each case g(−υ) = g(υ) and the dashed curve gives the spectrum shape A(υ)/A0. For the sech case, g = 0 when α = 1 and g = |g| if α > 1, whereas g = −|g| if α < 1.

Fig. 2
Fig. 2

Pulse-width factor W as in Eq. (58) for various values of the matching fraction α, Eq. (43). The distance is z and γ is the power-attenuation coefficient. The dashed curve, exp(−γz), represents the fraction of power remaining.

Equations (68)

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E ( z , t ) = x ˆ ( 1 2 ) E ( ω ) exp [ i [ β ( ω ) z ω t ] ] d ω ,
E ( z , t ) = x ˆ ( 1 2 ) 0 E ( ω ) exp { i [ β ( ω ) z ω t ] } d ω + c . c . ,
β ( ω ) = β ( ω 0 ) + β ( ω 0 ) ( ω ω 0 ) + 1 2 β ( ω 0 ) ( ω ω 0 ) 2 + β 0 + β 0 υ + 1 2 β 0 υ 2 + ,
E ( z , t ) = x ˆ ( 1 2 ) exp [ i ( β 0 z ω 0 t ) ] × S ( υ , z ) exp [ i ( β 0 z t ) υ ] d υ + c . c . ,
S ( υ , z ) = E ( υ + ω 0 ) exp ( i β 0 υ 2 z / 2 ) , υ > ω 0 , = 0 , υ < ω 0 .
S ( υ , z ) = S ( υ , 0 ) exp [ i θ ( z ) ] ,
Λ Λ E + 1 c 2 2 E t 2 = 4 π c 2 2 P t 2 ,
d E ( ω ) d z = 2 π i ω 2 β ( ω ) c 2 P NL ( ω ) exp [ i β ( ω ) z ] ,
P NL ( ω ) = d ω 1 d ω 2 χ E * ( ω 1 ) E ( ω 2 ) E ( ω 3 ) exp { i [ β ( ω 2 ) + β ( ω 3 ) β ( ω 1 ) β ( ω ) ] z }
ω + ω 1 = ω 2 + ω 3 .
d S ( υ ) d z = ( i β 0 υ 2 / 2 ) S ( υ ) + i C NL d υ 1 d υ 2 S * ( υ 1 ) S ( υ 2 ) S ( υ 3 ) .
C NL = 6 π χ ω 0 2 β 0 c 2 ,
E E ( ω ) ψ ( ω , r ) / N ,
C NL = 48 π 2 ω 0 χ c 2 n 2 a eff ,
a eff = 2 π ( 0 ψ 2 r d r ) 2 / 0 ψ 4 r d r .
( c n / 4 ) 0 ψ 2 ( ω , r ) r d r = N 2 .
ψ ( ω , r ) = exp [ r 2 / r 0 2 ( ω ) ] ,
C NL = 24 π c 2 ω 0 χ n 2 r 0 2 .
S ( υ , z ) = A ( υ , z ) exp [ i ϕ ( υ , z ) ] .
d A ( υ ) d z = C NL d υ 1 d υ 2 A ( υ 1 ) A ( υ 2 ) A ( υ 3 ) sin Φ
d ϕ ( υ ) d z = β 0 υ 2 2 + C NL A ( υ ) d υ 1 d υ 2 A ( υ 1 ) A ( υ 2 ) A ( υ 3 ) cos Φ ,
Φ = ϕ ( υ 2 ) + ϕ ( υ 3 ) ϕ ( υ 1 ) ϕ ( υ ) ,
υ + υ 1 = υ 2 + υ 3 .
sin Φ = 0 , all z .
ϕ ( υ , z ) = g 1 + g 2 z ,
g 2 = β 0 υ 2 2 + C NL A ( υ ) A ( υ 1 ) A ( υ 2 ) A ( υ 3 ) d υ 1 d υ 2 .
A ( υ ) = A 0 sech ( υ / υ ¯ ) .
g 2 = β 0 2 υ 2 + 2 A 0 2 C NL [ υ 2 + ( π υ ¯ / 2 ) 2 ] .
A 0 2 = β 0 / 4 C NL .
g 2 = π 2 β 0 υ ¯ 2 / 8 .
E ( z , t ) = ( A 0 / 2 ) exp { i [ ( β 0 + g 2 ) z ω 0 t ] } × sech ( υ / υ ¯ ) exp [ i ( β 0 z t ) υ ] d υ + c . c .
= ( π υ ¯ A 0 / 2 ) exp { i [ ( β 0 + g 2 ) z ω 0 t } × sech [ π υ ¯ ( β 0 z t ) / 2 ] + c . c .
P = ( π υ ¯ A 0 ) 2 10 7 ( W ) = ( 3.525 A 0 / T ) 2 10 7 ( W ) ,
u ( ξ , z ) = S ( υ , z ) e i ξ υ d υ = F ( S ) ,
ξ = t β 0 z
d u d z = u z + u ξ d ξ d z = F ( d S / d z ) β 0 u / ξ .
i u z β 0 2 2 u ξ 2 + C NL u | u | 2 = 0.
A 2 ( υ ) d υ = K 1 ,
υ A 2 ( υ ) d υ = K 2 ,
υ 2 A 2 ( υ ) d υ = K 3 1 ( 1 + 8 π 2 ) A 2 d ϕ d z d υ ,
υ m A 2 ( υ ) d υ = constant , all m .
i N i = constant ,
i ω i N i = constant .
S ( υ , 0 ) = A ( υ , 0 ) = A 0 sech ( υ / υ ¯ ) .
d ϕ / d z = g 2 ,
A 0 2 = α ( β 0 / 4 C NL ) ,
d ϕ d z | 0 = β 0 2 ( 1 α ) υ 2 + 2 A 0 2 C NL ( π υ ¯ 2 ) 2 , = g ( υ ) ( β 0 υ ¯ 2 / 2 ) + g 0 .
S ( υ , 0 ) = A ( υ , 0 ) = A 0 exp ( υ 2 / υ ¯ 2 ) .
d ϕ d z | 0 = β 0 υ 2 / 2 + C NL A 0 2 υ ¯ 2 ( π / 3 ) exp ( 2 υ 2 / 3 υ ¯ 2 ) .
A 0 2 = ( β 0 3 3 / 4 π C NL ) .
d ϕ d z | 0 = ( β 0 υ ¯ 2 / 2 ) { ( υ / υ ¯ 2 ) ( 3 α / 2 ) exp ( 2 υ 2 / 3 υ ¯ 2 ) + 3 α / 2 } 3 α β 0 υ ¯ 2 / 4 , = g ( υ ) { β 0 υ ¯ 2 / 2 } + g 0 ,
ϕ ( υ , z ) g ( υ ) { β 0 υ ¯ 2 / 2 } z + g 0 z , small z .
τ 2 = | u | 2 ξ 2 d ξ / | u | 2 d ξ ,
τ 2 = [ ( d A d υ ) 2 + A 2 ( d ϕ d υ ) 2 ] d υ / A 2 d υ .
A 2 ( υ , z ) = A 2 ( υ , 0 ) exp ( γ z ) .
A ( υ , z ) = A 0 exp ( γ z / 2 ) sech ( υ / υ ¯ ) .
d ϕ d z = β 0 2 υ 2 + 2 A 0 2 + C NL [ υ 2 + ( π υ ¯ / 2 ) 2 ] exp ( γ z ) ,
ϕ ( υ , z ) = β 0 2 υ 2 z α β 0 2 [ υ 2 + ( π υ ¯ / 2 ) 2 ] [ 1 exp ( γ z ) ] / γ .
τ 2 τ 0 2 = 1 + ( π 6 τ 0 2 ) 2 ( β z ) 2 ,
β eff = β 0 { 1 α [ 1 exp ( γ z ) ] γ z } .
τ 2 τ 0 2 = 1 + ( π β 0 6 τ 0 2 γ ) 2 W 2 ( γ z , α ) ,
W ( γ z , α ) = γ z α [ 1 exp ( γ z ) ] .
F ( k ) = F ( f ) = f ( t ) exp ( i k t ) d t .
d 2 F d k 2 = F ( t 2 f ) ,
F [ f 1 ( t p ) f 2 ( p ) d p ] = F 1 ( k ) F 2 ( k ) ,
F [ f ( t + p ) f * ( p ) d p ] = | F ( k ) | 2 .
F [ f * ( t 1 ) f ( t 2 ) f ( t + t 1 t 2 ) d t 1 d t 2 ] = F ( k ) | F ( k ) | 2 .
d 2 A ˜ d k 2 = ( 2 C NL β 0 ) A ˜ 3 ( 2 g 2 β 0 ) A ˜ .

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