Abstract

Using computer simulations, we examine the effects of higher-order dispersive and nonlinear propagation processes on the spectral and time development of ultrashort, high-intensity pulses propagating in single-mode optical fibers having normal dispersion. Our results indicate that both the cubic-dispersion term and the shock term of the nonlinear Schrödinger equation contribute to asymmetry in the pulse power spectrum and cause highly nonlinear chirp.

© 1987 Optical Society of America

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References

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  1. W. J. Tomlinson et al., J. Opt. Soc. Am. B 1, 139 (1984).
    [CrossRef]
  2. W. H. Knox et al., Appl. Phys. Lett. 46, 1120 (1985).
    [CrossRef]
  3. D. Marcuse, Appl. Opt. 19, 1653 (1980).
    [CrossRef] [PubMed]
  4. F. DeMartini et al., Phys. Rev. 164A, 312 (1967).
    [CrossRef]
  5. G. Yang, Y. R. Shen, Opt. Lett. 9, 510 (1984).
    [CrossRef] [PubMed]
  6. D. Anderson, M. Lisak, Phys. Rev. A 27, 1393 (1983).
    [CrossRef]
  7. E. Bourkoff, in Proceedings of the National Science Foundation Workshop on Optical Nonlinearities, Fast Phenomena and Signal Processing (University of Arizona, Tucson, Ariz., 1986), pp. 15–28.
  8. E. Bourkoff et al., Opt. News 12(9), 190 (1986); E. Bourkoff et al., J. Opt. Soc. Am. A 3(13), P90 (1986); “Intensity-dependent spectra of pulses propagating in optical fibers,” Opt. Commun. (to be published).
  9. Y. R. Shen, Principles of Nonlinear Optics (Wiley, New York, 1984).
  10. D. Yevick, B. Hermansson, Opt. Commun. 47, 101 (1983).
    [CrossRef]
  11. J. Satsuma, N. Yajima, Suppl. Prog. Theor. Phys. 55, 284 (1974).
    [CrossRef]
  12. D. N. Christodoulides, R. I. Joseph, Electron. Lett. 20, 659 (1984).
    [CrossRef]
  13. W. Zhao, E. Bourkoff, paper TuGG4 to be presented at the International Quantum Electronics Conference, April 27–May 3, 1987, Baltimore, Maryland.
  14. R. Y. Chiao, E. Garmire, C. H. Townes, Phys. Rev. Lett. 13, 479 (1964).
    [CrossRef]
  15. F. M. Mitschke, L. F. Mollenauer, Opt. Lett. 11, 659 (1986).
    [CrossRef] [PubMed]
  16. J. P. Gordon, Opt. Lett. 11, 662 (1986).
    [CrossRef] [PubMed]

1986 (3)

E. Bourkoff et al., Opt. News 12(9), 190 (1986); E. Bourkoff et al., J. Opt. Soc. Am. A 3(13), P90 (1986); “Intensity-dependent spectra of pulses propagating in optical fibers,” Opt. Commun. (to be published).

F. M. Mitschke, L. F. Mollenauer, Opt. Lett. 11, 659 (1986).
[CrossRef] [PubMed]

J. P. Gordon, Opt. Lett. 11, 662 (1986).
[CrossRef] [PubMed]

1985 (1)

W. H. Knox et al., Appl. Phys. Lett. 46, 1120 (1985).
[CrossRef]

1984 (3)

1983 (2)

D. Yevick, B. Hermansson, Opt. Commun. 47, 101 (1983).
[CrossRef]

D. Anderson, M. Lisak, Phys. Rev. A 27, 1393 (1983).
[CrossRef]

1980 (1)

1974 (1)

J. Satsuma, N. Yajima, Suppl. Prog. Theor. Phys. 55, 284 (1974).
[CrossRef]

1967 (1)

F. DeMartini et al., Phys. Rev. 164A, 312 (1967).
[CrossRef]

1964 (1)

R. Y. Chiao, E. Garmire, C. H. Townes, Phys. Rev. Lett. 13, 479 (1964).
[CrossRef]

Anderson, D.

D. Anderson, M. Lisak, Phys. Rev. A 27, 1393 (1983).
[CrossRef]

Bourkoff, E.

E. Bourkoff et al., Opt. News 12(9), 190 (1986); E. Bourkoff et al., J. Opt. Soc. Am. A 3(13), P90 (1986); “Intensity-dependent spectra of pulses propagating in optical fibers,” Opt. Commun. (to be published).

W. Zhao, E. Bourkoff, paper TuGG4 to be presented at the International Quantum Electronics Conference, April 27–May 3, 1987, Baltimore, Maryland.

E. Bourkoff, in Proceedings of the National Science Foundation Workshop on Optical Nonlinearities, Fast Phenomena and Signal Processing (University of Arizona, Tucson, Ariz., 1986), pp. 15–28.

Chiao, R. Y.

R. Y. Chiao, E. Garmire, C. H. Townes, Phys. Rev. Lett. 13, 479 (1964).
[CrossRef]

Christodoulides, D. N.

D. N. Christodoulides, R. I. Joseph, Electron. Lett. 20, 659 (1984).
[CrossRef]

DeMartini, F.

F. DeMartini et al., Phys. Rev. 164A, 312 (1967).
[CrossRef]

Garmire, E.

R. Y. Chiao, E. Garmire, C. H. Townes, Phys. Rev. Lett. 13, 479 (1964).
[CrossRef]

Gordon, J. P.

Hermansson, B.

D. Yevick, B. Hermansson, Opt. Commun. 47, 101 (1983).
[CrossRef]

Joseph, R. I.

D. N. Christodoulides, R. I. Joseph, Electron. Lett. 20, 659 (1984).
[CrossRef]

Knox, W. H.

W. H. Knox et al., Appl. Phys. Lett. 46, 1120 (1985).
[CrossRef]

Lisak, M.

D. Anderson, M. Lisak, Phys. Rev. A 27, 1393 (1983).
[CrossRef]

Marcuse, D.

Mitschke, F. M.

Mollenauer, L. F.

Satsuma, J.

J. Satsuma, N. Yajima, Suppl. Prog. Theor. Phys. 55, 284 (1974).
[CrossRef]

Shen, Y. R.

G. Yang, Y. R. Shen, Opt. Lett. 9, 510 (1984).
[CrossRef] [PubMed]

Y. R. Shen, Principles of Nonlinear Optics (Wiley, New York, 1984).

Tomlinson, W. J.

Townes, C. H.

R. Y. Chiao, E. Garmire, C. H. Townes, Phys. Rev. Lett. 13, 479 (1964).
[CrossRef]

Yajima, N.

J. Satsuma, N. Yajima, Suppl. Prog. Theor. Phys. 55, 284 (1974).
[CrossRef]

Yang, G.

Yevick, D.

D. Yevick, B. Hermansson, Opt. Commun. 47, 101 (1983).
[CrossRef]

Zhao, W.

W. Zhao, E. Bourkoff, paper TuGG4 to be presented at the International Quantum Electronics Conference, April 27–May 3, 1987, Baltimore, Maryland.

Appl. Opt. (1)

Appl. Phys. Lett. (1)

W. H. Knox et al., Appl. Phys. Lett. 46, 1120 (1985).
[CrossRef]

Electron. Lett. (1)

D. N. Christodoulides, R. I. Joseph, Electron. Lett. 20, 659 (1984).
[CrossRef]

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

Opt. Commun. (1)

D. Yevick, B. Hermansson, Opt. Commun. 47, 101 (1983).
[CrossRef]

Opt. Lett. (3)

Opt. News (1)

E. Bourkoff et al., Opt. News 12(9), 190 (1986); E. Bourkoff et al., J. Opt. Soc. Am. A 3(13), P90 (1986); “Intensity-dependent spectra of pulses propagating in optical fibers,” Opt. Commun. (to be published).

Phys. Rev. (1)

F. DeMartini et al., Phys. Rev. 164A, 312 (1967).
[CrossRef]

Phys. Rev. A (1)

D. Anderson, M. Lisak, Phys. Rev. A 27, 1393 (1983).
[CrossRef]

Phys. Rev. Lett. (1)

R. Y. Chiao, E. Garmire, C. H. Townes, Phys. Rev. Lett. 13, 479 (1964).
[CrossRef]

Suppl. Prog. Theor. Phys. (1)

J. Satsuma, N. Yajima, Suppl. Prog. Theor. Phys. 55, 284 (1974).
[CrossRef]

Other (3)

W. Zhao, E. Bourkoff, paper TuGG4 to be presented at the International Quantum Electronics Conference, April 27–May 3, 1987, Baltimore, Maryland.

E. Bourkoff, in Proceedings of the National Science Foundation Workshop on Optical Nonlinearities, Fast Phenomena and Signal Processing (University of Arizona, Tucson, Ariz., 1986), pp. 15–28.

Y. R. Shen, Principles of Nonlinear Optics (Wiley, New York, 1984).

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Figures (4)

Fig. 1
Fig. 1

Power spectrum of 1.06-μm pulses exiting from a single-mode fiber having length l = 0.5z0 = 3.62 cm. q(0, τ) = N sech(τ), with N = 7. (ωω0)τ0 is a normalized frequency deviation from the pulse-center frequency, with ω0 corresponding to λ0 = 1.06 μm and τ0 = 28.41 fsec. The solid curve is a solution of the conventional NLS equation (no cubic dispersion and no shock term). The dashed line corresponds to a solution with the higher-order terms, as given by Eq. (3) in the text.

Fig. 2
Fig. 2

Phase spectrum of the pulse leaving the fiber. N = 7 and fiber length l = 0.5z0, as in Fig. 1. The solution to the conventional NLS equation is shown by the (quadratic) solid curve. The dashed curve shows the effect of the higher-order terms on the phase spectrum, indicating an asymmetric deviation from quadratic behavior.

Fig. 3
Fig. 3

The optical pulse leaving the single-mode fiber shown in the time domain. Same conditions as for Figs. 1 and 2. Arrows pointing to the left refer to plots of instantaneous frequency versus normalized time, whereas arrows pointing to the right refer to plots of pulse intensity versus normalized time. The dashed curves represent a solution to the conventional NLS equation, whereas the solid curves represent a solution to the more exact Eq. (3), which takes into account third-order dispersion and the shock term.

Fig. 4
Fig. 4

Autocorrelation intensity versus normalized delay time, τd/τ0. The (numerically calculated) dashed curve (FWHM = 15.6 fsec) corresponds to the nonlinearly chirped pulses (of Fig. 3) passing through a quadratic grating-pair compressor. The solid curve (FWHM = 12.9 fsec) corresponds to the same conditions as for the dashed curve, except that an optical element that compensates for the cubic dispersion of the fiber has been inserted before the grating-pair compressor. The two curves correspond to pulses whose intensities FWHM are 10.4 and 8.6 fsec, respectively.

Equations (4)

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i ( ϕ z + k 0 ϕ t ) - k 0 2 2 ϕ t 2 + ω 0 n 2 α c ϕ 2 ϕ - i k 0 6 3 ϕ t 3 + i n 2 α c t ( ϕ 2 ϕ ) = 0 ,
q ( ξ , τ ) ( k 0 z 0 n 2 π ) 1 / 2 ϕ ( z , t ) ,
i q ξ ± 1 2 2 q τ 2 + q 2 q = i β 3 q τ 3 - i ω 0 τ 0 τ ( q 2 q ) ,
N = τ 0 ( μ 0 0 ) 1 / 4 [ ω 0 n 2 n c k 0 ( P max S ) ] 1 / 2 .

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