Abstract

With a simple model, scaling laws are obtained for self-compression in the limit of a large soliton number. Numerical results for Gaussian and hyperbolic-secant initial pulse shapes and various initial amplitudes are used to verify the approximate analytic result and to determine accurate scaling constants. Self-decompression (self-dispersion) is also treated in the same fashion.

© 2002 Optical Society of America

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References

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  1. For a review of this topic, see G. P. Agrawal, Nonlinear Fiber Optics, 2nd ed. (Academic, San Diego, 1995).
  2. L. F. Mollenauer, R. H. Stolen, J. P. Gordon, and W. J. Tomlinson, “Extreme picosecond pulse narrowing by means of the soliton effect in single-mode optical fibers,” Opt. Lett. 8, 289–291 (1983).
    [CrossRef] [PubMed]
  3. E. M. Dianov, Z. S. Nikonova, A. M. Prokhorov, and A. A. Podshivalov, “Optimal compression of multi-soliton pulses,” Sov. Tech. Phys. Lett. 12, 311–313 (1986).
  4. M. J. Potasek, G. P. Agrawal, and S. C. Pinault, “Analytic and numerical study of pulse broadening in nonlinear dispersive fibers,” J. Opt. Soc. Am. B 3, 205–211 (1986).
    [CrossRef]
  5. P. L. Kelley, “The nonlinear index of refraction and self-action effects in optical propagation,” IEEE J. Sel. Top. Quantum Electron. 6, 1259–1264 (2000).
    [CrossRef]
  6. S. A. Akhmanov, R. V. Khokhlov, and A. P. Sukhorukov, “Self-focusing, self-defocusing, and self-modulation of laser beams,” in Laser Handbook, F. T. Arecchi and E. O. Schulz-Dubois, eds. (North-Holland, Amsterdam, 1972), Vol. 2, pp. 1151–1228.
  7. O. Svelto, “Self-focusing, self-trapping, and self-phase-modulation of light beams,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1972), Vol. 12, pp. 1–51.
  8. J. H. Marburger, “Self-focusing: theory,” in Progress in Quantum Electronics, J. H. Sanders and S. Stenholm, eds. (Pergamon, New York, 1974), Vol. 4, pp. 35–110.
  9. Y. Silberberg, “Collapse of optical pulses,” Opt. Lett. 15, 1282–1285 (1990).
    [CrossRef] [PubMed]
  10. G. P. Agrawal, “Effect of intrapulse stimulated Raman scattering on soliton-effect pulse compression in optical fibers,” Opt. Lett. 15, 224–226 (1990).
    [CrossRef] [PubMed]
  11. K. T. Chan and W. H. Cao, “Improved soliton-effect pulse compression by combined action of negative third-order dispersion and Raman self-scattering in optical fibers,” J. Opt. Soc. Am. B 15, 2371–2375 (1998).
    [CrossRef]
  12. R. A. Fisher, P. L. Kelley, and T. K. Gustafson, “Subpicosecond pulse generation using the optical Kerr effect,” Appl. Phys. Lett. 14, 140–143 (1969).
    [CrossRef]
  13. N. V. Akhmediev and N. V. Mitzkevich, “Extremely high degree of N-soliton pulse compression in optical fiber,” IEEE J. Quantum Electron. 27, 849–857 (1991).
    [CrossRef]
  14. P. L. Kelley, “Self-focusing of optical beams,” Phys. Rev. Lett. 15, 1005–1008 (1965).
    [CrossRef]
  15. V. E. Zakharov and A. B. Shabat, “Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media,” Sov. Phys. JETP 34, 62–69 (1972).
  16. J. Satsuma and N. Yajima, “Initial value problems of one-dimensional self-modulation of nonlinear waves in dispersive media,” Suppl. Prog. Theor. Phys. 55, 284–306 (1974).
    [CrossRef]
  17. M. J. Ablowitz and H. Segur, Solitons and the Inverse Scattering Transform (Society for Industrial and Applied Mathematics, Philadelphia, 1981).
  18. G. L. Lamb, Jr., Elements of Soliton Theory (Wiley, New York, 1981).
  19. J. A. C. Weideman and B. M. Herbst, “Split-step methods for the solution of the nonlinear Schrödinger equation,” SIAM (Soc. Ind. Appl. Math.) J. Numer. Anal. 23, 485–507 (1986).
    [CrossRef]

2000

P. L. Kelley, “The nonlinear index of refraction and self-action effects in optical propagation,” IEEE J. Sel. Top. Quantum Electron. 6, 1259–1264 (2000).
[CrossRef]

1998

1991

N. V. Akhmediev and N. V. Mitzkevich, “Extremely high degree of N-soliton pulse compression in optical fiber,” IEEE J. Quantum Electron. 27, 849–857 (1991).
[CrossRef]

1990

1986

E. M. Dianov, Z. S. Nikonova, A. M. Prokhorov, and A. A. Podshivalov, “Optimal compression of multi-soliton pulses,” Sov. Tech. Phys. Lett. 12, 311–313 (1986).

M. J. Potasek, G. P. Agrawal, and S. C. Pinault, “Analytic and numerical study of pulse broadening in nonlinear dispersive fibers,” J. Opt. Soc. Am. B 3, 205–211 (1986).
[CrossRef]

J. A. C. Weideman and B. M. Herbst, “Split-step methods for the solution of the nonlinear Schrödinger equation,” SIAM (Soc. Ind. Appl. Math.) J. Numer. Anal. 23, 485–507 (1986).
[CrossRef]

1983

1974

J. Satsuma and N. Yajima, “Initial value problems of one-dimensional self-modulation of nonlinear waves in dispersive media,” Suppl. Prog. Theor. Phys. 55, 284–306 (1974).
[CrossRef]

1972

V. E. Zakharov and A. B. Shabat, “Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media,” Sov. Phys. JETP 34, 62–69 (1972).

1969

R. A. Fisher, P. L. Kelley, and T. K. Gustafson, “Subpicosecond pulse generation using the optical Kerr effect,” Appl. Phys. Lett. 14, 140–143 (1969).
[CrossRef]

1965

P. L. Kelley, “Self-focusing of optical beams,” Phys. Rev. Lett. 15, 1005–1008 (1965).
[CrossRef]

Agrawal, G. P.

Akhmediev, N. V.

N. V. Akhmediev and N. V. Mitzkevich, “Extremely high degree of N-soliton pulse compression in optical fiber,” IEEE J. Quantum Electron. 27, 849–857 (1991).
[CrossRef]

Cao, W. H.

Chan, K. T.

Dianov, E. M.

E. M. Dianov, Z. S. Nikonova, A. M. Prokhorov, and A. A. Podshivalov, “Optimal compression of multi-soliton pulses,” Sov. Tech. Phys. Lett. 12, 311–313 (1986).

Fisher, R. A.

R. A. Fisher, P. L. Kelley, and T. K. Gustafson, “Subpicosecond pulse generation using the optical Kerr effect,” Appl. Phys. Lett. 14, 140–143 (1969).
[CrossRef]

Gordon, J. P.

Gustafson, T. K.

R. A. Fisher, P. L. Kelley, and T. K. Gustafson, “Subpicosecond pulse generation using the optical Kerr effect,” Appl. Phys. Lett. 14, 140–143 (1969).
[CrossRef]

Herbst, B. M.

J. A. C. Weideman and B. M. Herbst, “Split-step methods for the solution of the nonlinear Schrödinger equation,” SIAM (Soc. Ind. Appl. Math.) J. Numer. Anal. 23, 485–507 (1986).
[CrossRef]

Kelley, P. L.

P. L. Kelley, “The nonlinear index of refraction and self-action effects in optical propagation,” IEEE J. Sel. Top. Quantum Electron. 6, 1259–1264 (2000).
[CrossRef]

R. A. Fisher, P. L. Kelley, and T. K. Gustafson, “Subpicosecond pulse generation using the optical Kerr effect,” Appl. Phys. Lett. 14, 140–143 (1969).
[CrossRef]

P. L. Kelley, “Self-focusing of optical beams,” Phys. Rev. Lett. 15, 1005–1008 (1965).
[CrossRef]

Mitzkevich, N. V.

N. V. Akhmediev and N. V. Mitzkevich, “Extremely high degree of N-soliton pulse compression in optical fiber,” IEEE J. Quantum Electron. 27, 849–857 (1991).
[CrossRef]

Mollenauer, L. F.

Nikonova, Z. S.

E. M. Dianov, Z. S. Nikonova, A. M. Prokhorov, and A. A. Podshivalov, “Optimal compression of multi-soliton pulses,” Sov. Tech. Phys. Lett. 12, 311–313 (1986).

Pinault, S. C.

Podshivalov, A. A.

E. M. Dianov, Z. S. Nikonova, A. M. Prokhorov, and A. A. Podshivalov, “Optimal compression of multi-soliton pulses,” Sov. Tech. Phys. Lett. 12, 311–313 (1986).

Potasek, M. J.

Prokhorov, A. M.

E. M. Dianov, Z. S. Nikonova, A. M. Prokhorov, and A. A. Podshivalov, “Optimal compression of multi-soliton pulses,” Sov. Tech. Phys. Lett. 12, 311–313 (1986).

Satsuma, J.

J. Satsuma and N. Yajima, “Initial value problems of one-dimensional self-modulation of nonlinear waves in dispersive media,” Suppl. Prog. Theor. Phys. 55, 284–306 (1974).
[CrossRef]

Shabat, A. B.

V. E. Zakharov and A. B. Shabat, “Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media,” Sov. Phys. JETP 34, 62–69 (1972).

Silberberg, Y.

Stolen, R. H.

Tomlinson, W. J.

Weideman, J. A. C.

J. A. C. Weideman and B. M. Herbst, “Split-step methods for the solution of the nonlinear Schrödinger equation,” SIAM (Soc. Ind. Appl. Math.) J. Numer. Anal. 23, 485–507 (1986).
[CrossRef]

Yajima, N.

J. Satsuma and N. Yajima, “Initial value problems of one-dimensional self-modulation of nonlinear waves in dispersive media,” Suppl. Prog. Theor. Phys. 55, 284–306 (1974).
[CrossRef]

Zakharov, V. E.

V. E. Zakharov and A. B. Shabat, “Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media,” Sov. Phys. JETP 34, 62–69 (1972).

Appl. Phys. Lett.

R. A. Fisher, P. L. Kelley, and T. K. Gustafson, “Subpicosecond pulse generation using the optical Kerr effect,” Appl. Phys. Lett. 14, 140–143 (1969).
[CrossRef]

IEEE J. Quantum Electron.

N. V. Akhmediev and N. V. Mitzkevich, “Extremely high degree of N-soliton pulse compression in optical fiber,” IEEE J. Quantum Electron. 27, 849–857 (1991).
[CrossRef]

IEEE J. Sel. Top. Quantum Electron.

P. L. Kelley, “The nonlinear index of refraction and self-action effects in optical propagation,” IEEE J. Sel. Top. Quantum Electron. 6, 1259–1264 (2000).
[CrossRef]

J. Opt. Soc. Am. B

Opt. Lett.

Phys. Rev. Lett.

P. L. Kelley, “Self-focusing of optical beams,” Phys. Rev. Lett. 15, 1005–1008 (1965).
[CrossRef]

SIAM (Soc. Ind. Appl. Math.) J. Numer. Anal.

J. A. C. Weideman and B. M. Herbst, “Split-step methods for the solution of the nonlinear Schrödinger equation,” SIAM (Soc. Ind. Appl. Math.) J. Numer. Anal. 23, 485–507 (1986).
[CrossRef]

Sov. Phys. JETP

V. E. Zakharov and A. B. Shabat, “Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media,” Sov. Phys. JETP 34, 62–69 (1972).

Sov. Tech. Phys. Lett.

E. M. Dianov, Z. S. Nikonova, A. M. Prokhorov, and A. A. Podshivalov, “Optimal compression of multi-soliton pulses,” Sov. Tech. Phys. Lett. 12, 311–313 (1986).

Suppl. Prog. Theor. Phys.

J. Satsuma and N. Yajima, “Initial value problems of one-dimensional self-modulation of nonlinear waves in dispersive media,” Suppl. Prog. Theor. Phys. 55, 284–306 (1974).
[CrossRef]

Other

M. J. Ablowitz and H. Segur, Solitons and the Inverse Scattering Transform (Society for Industrial and Applied Mathematics, Philadelphia, 1981).

G. L. Lamb, Jr., Elements of Soliton Theory (Wiley, New York, 1981).

For a review of this topic, see G. P. Agrawal, Nonlinear Fiber Optics, 2nd ed. (Academic, San Diego, 1995).

S. A. Akhmanov, R. V. Khokhlov, and A. P. Sukhorukov, “Self-focusing, self-defocusing, and self-modulation of laser beams,” in Laser Handbook, F. T. Arecchi and E. O. Schulz-Dubois, eds. (North-Holland, Amsterdam, 1972), Vol. 2, pp. 1151–1228.

O. Svelto, “Self-focusing, self-trapping, and self-phase-modulation of light beams,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1972), Vol. 12, pp. 1–51.

J. H. Marburger, “Self-focusing: theory,” in Progress in Quantum Electronics, J. H. Sanders and S. Stenholm, eds. (Pergamon, New York, 1974), Vol. 4, pp. 35–110.

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Figures (5)

Fig. 1
Fig. 1

Inverse of the self-compression distance, z SC , as a function of the soliton number, N, for Gaussian (open circles) and hyperbolic-secant (open triangles) input pulses. The distance is in units of z DIS .

Fig. 2
Fig. 2

Self-compression factor, F SC , as function of the soliton number, N, for Gaussian (open circles) and hyperbolic-secant (open triangles) input pulses.

Fig. 3
Fig. 3

Inverse of the quality factor, Q SC , as function of the soliton number, N, for Gaussian (open circles) and hyperbolic-secant (open triangles) input pulses.

Fig. 4
Fig. 4

Inverse of the self-decompression distance, z SD , as a function of the soliton number, N, for Gaussian (open circles) and hyperbolic-secant (open triangles) input pulses. The distance is in units of z DIS .

Fig. 5
Fig. 5

Power of the nonlinear time lens, p, as a function of the modified soliton number, N * , for Gaussian input pulses. The power is in units of 1 / z DIS .

Equations (30)

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i A z - β 2 2 2 A τ 2 + k n 2 n 0 | A | 2 A = 0 ,
i u z - 1 2 β 2 | β 2 | 2 u τ 2 + | u | 2 u = 0 ,
z NL = 1 k n 2 n 0 | A 0 | 2 ,
z DIS = τ 0 2 | β 2 | .
N = z DIS z NL .
A ( z ,   τ ) = A ( 0 ,   τ ) exp [ i Φ NL ( z ,   τ ) ] ,
Φ NL ( z ,   τ ) = kz n 2 n 0 | A ( 0 ,   τ ) | 2 .
Ω ( z ,   τ ) = - Φ NL τ = - kz n 2 n 0 | A ( 0 ,   τ ) | 2 τ .
z ( τ ) = ν g ( Ω ( τ ) ) ( t - τ ) ,
z SC ν g ( 0 ) = t = z SC ν g ( Ω ( τ ) ) + τ ,
z SC 1 ν g ( 0 ) - 1 ν g ( Ω ( τ ) ) = τ .
z SC = - τ β 2 Ω ( τ )
z SC = n 0 τ k β 2 n 2 | A | 2 / τ τ 0 .
A ( 0 ,   τ ) = A 0   sech ( τ / τ 0 ) .
A ( 0 ,   τ ) = A 0   exp ( - τ 2 / 2 τ 0 2 ) ,
1 τ | A | 2 τ τ 0 - 2 τ 0 2 | A 0 | 2 .
z SC z NL z DIS 2 = z DIS 2 N .
| A ( z SC ,   τ ) | = A 0 ( τ 0 / τ SC ) 1 / 2   exp ( - τ 2 / 2 τ SC 2 ) ,
τ SC z SC | β 2 | τ 0 = τ 0 2 N ,
F SC = τ 0 τ SC 2 N .
z SD ν g ( 0 ) = t
t + 2 τ = z SD ν g ( Ω ( τ ) ) + τ
z SD 1 ν g ( 0 ) - 1 ν g ( Ω ( τ ) ) = ( 1 - 2 ) τ .
z SD = ( 2 - 1 ) τ β 2 Ω ( τ ) ,
z SD = ( 1 - 2 ) n 0 τ k β 2 n 2 | A | 2 / τ τ 0 .
z SD ( 2 - 1 ) z NL z DIS 2 = ( 2 - 1 ) 2 z DIS N .
N * = N ,
N * = - N ,
p = 1 z SC   when N * > 0 ,
p = - 1 z SD   when N * < 0 .

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