Abstract

A numerical study is presented on the frequency chirp and spectra of light pulses affected by both self-phase modulation and stimulated Raman scattering in optical fibers. The asymmetric distortion of pump and Stokes spectra is attributed to walk-off between the pulses and found to be in good agreement with recent experimental results. The stationary-phase method is employed to explain the relation between the structure of the spectra and the form of the pulse chirp. An analytic expression describing the chirp of a depletable pump and of the first Stokes pulse is given for negligible walk-off. The dominant influence of self-phase modulation over the cross phase modulation on the buildup of the chirp in the case of large walk-off and severe pump depletion is shown.

© 1987 Optical Society of America

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References

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  1. B. Nikolaus and D. Grischkowsky, “12× pulse compression using optical fibers,” Appl. Phys. Lett. 42, 1–2 (1983).
    [CrossRef]
  2. J. D. Kafka, B. H. Kolner, T. Baer, and D. M. Bloom, “Compression of pulses from a continuous-wave mode-locked Nd:YAG laser,” Opt. Lett. 9, 505–506 (1984).
    [CrossRef] [PubMed]
  3. A. S. L. Gomes, U. Österberg, W. Sibbett, and J. R. Taylor, “An experimental study of the primary parameters that determine the temporal compression of cw Nd:YAG laser pulses,” Opt. Commun. 54, 377–382 (1985).
    [CrossRef]
  4. A. S. L. Gomes, W. Sibbett, and J. R. Taylor, “Spectral and temporal study of picosecond-pulse propagation in a single-mode optical fibre,” Appl. Phys. B 39, 44–46 (1986).
    [CrossRef]
  5. A. M. Weiner, J. P. Heritage, and R. H. Stolen, “Effect of stimulated Raman scattering and pulse walk off on self-phase modulation in optical fibers,” in Digest of the Conference on Lasers and Electro-Optics (Optical Society of America, Washington, D.C., 1986), p. 246.
  6. B. Zysset and H. P. Weber, “Temporal and spectral investigation of Nd:YAG pulse propagation in optical fibers and its application to pulse compression,” in Digest of the Conference on Lasers and Electro-Optics (Optical Society of America, Washington, D.C., 1986), p. 182.
  7. Lu Hian-Hua, Li Yu-Lin, and Jiang Jia-Lin, “On combined self-phase modulation and stimulated Raman scattering in optical fibres,” Opt. Quantum Electron. 17, 187–193 (1985).
    [CrossRef]
  8. E. M. Dianov, A. Ya. Karasik, P. G. Mamyshev, A. M. Prokhorov, and V. N. Serkin, “Generation of ultrashort pulses by spectral filtering during stimulated Raman scattering in an optical fiber,” Sov. Phys. JETP 62, 448–455 (1985).
  9. D. Schadt, B. Jaskorzynska, and U. Österberg, “Numerical study on stimulated Raman scattering and self phase modulation in optical fibers influenced by walk-off between the pulses,” J. Opt. Soc. Am. B 3, 1257–1262 (1986).
    [CrossRef]
  10. V. A. Vysloukh and V. N. Serkin, “Nonlinear transformation of solitons in optical fiber lightguides,” Izv. Akad. Nauk. SSSR Ser. Fiz. 48, 1777–1781 (1984).
  11. R. H. Stolen and E. P. Ippen, “Raman gain in glass optical waveguides,” Appl. Phys. Lett. 22, 276–278 (1973).
    [CrossRef]
  12. J. Auyeung and A. Yariv, “Spontaneous and stimulated Raman scattering in low loss fibers,” IEEE J. Quantum Electron. QE-14, 347–352 (1978).
    [CrossRef]
  13. R. H. Stolen and C. Lin, “Self-phase-modulation in silica optical fibers,” Phys. Rev. A 17, 1448–1453 (1977).
    [CrossRef]
  14. Fujio Shimizu, “Frequency broadening in liquids by a short light pulse,” Phys. Rev. Lett. 19, 1097–1100 (1967).
    [CrossRef]
  15. A. Papoulis, Systems and Transforms with Applications in Optics (McGraw-Hill, New York, 1968).
  16. T. K. Gustafson, J. P. Taran, H. A. Haus, J. R. Lifsitz, and P. L. Kelley, “Self-modulation, self-steepening, and spectral development of light in small-scale trapped filaments,” Phys. Rev. 177, 306–313 (1969).
    [CrossRef]
  17. E. M. Dianov, A. Y. Karasik, P. V. Mamyshev, G. I. Onishchukov, A. M. Prokhorov, M. F. Stel’makh, and A. A. Formichev, “Picosecond structure of the pump pulse in stimulated Raman scattering in a single-mode optical fiber,” JETP Lett. 39, 691–695 (1984).

1986 (2)

A. S. L. Gomes, W. Sibbett, and J. R. Taylor, “Spectral and temporal study of picosecond-pulse propagation in a single-mode optical fibre,” Appl. Phys. B 39, 44–46 (1986).
[CrossRef]

D. Schadt, B. Jaskorzynska, and U. Österberg, “Numerical study on stimulated Raman scattering and self phase modulation in optical fibers influenced by walk-off between the pulses,” J. Opt. Soc. Am. B 3, 1257–1262 (1986).
[CrossRef]

1985 (3)

Lu Hian-Hua, Li Yu-Lin, and Jiang Jia-Lin, “On combined self-phase modulation and stimulated Raman scattering in optical fibres,” Opt. Quantum Electron. 17, 187–193 (1985).
[CrossRef]

E. M. Dianov, A. Ya. Karasik, P. G. Mamyshev, A. M. Prokhorov, and V. N. Serkin, “Generation of ultrashort pulses by spectral filtering during stimulated Raman scattering in an optical fiber,” Sov. Phys. JETP 62, 448–455 (1985).

A. S. L. Gomes, U. Österberg, W. Sibbett, and J. R. Taylor, “An experimental study of the primary parameters that determine the temporal compression of cw Nd:YAG laser pulses,” Opt. Commun. 54, 377–382 (1985).
[CrossRef]

1984 (3)

E. M. Dianov, A. Y. Karasik, P. V. Mamyshev, G. I. Onishchukov, A. M. Prokhorov, M. F. Stel’makh, and A. A. Formichev, “Picosecond structure of the pump pulse in stimulated Raman scattering in a single-mode optical fiber,” JETP Lett. 39, 691–695 (1984).

J. D. Kafka, B. H. Kolner, T. Baer, and D. M. Bloom, “Compression of pulses from a continuous-wave mode-locked Nd:YAG laser,” Opt. Lett. 9, 505–506 (1984).
[CrossRef] [PubMed]

V. A. Vysloukh and V. N. Serkin, “Nonlinear transformation of solitons in optical fiber lightguides,” Izv. Akad. Nauk. SSSR Ser. Fiz. 48, 1777–1781 (1984).

1983 (1)

B. Nikolaus and D. Grischkowsky, “12× pulse compression using optical fibers,” Appl. Phys. Lett. 42, 1–2 (1983).
[CrossRef]

1978 (1)

J. Auyeung and A. Yariv, “Spontaneous and stimulated Raman scattering in low loss fibers,” IEEE J. Quantum Electron. QE-14, 347–352 (1978).
[CrossRef]

1977 (1)

R. H. Stolen and C. Lin, “Self-phase-modulation in silica optical fibers,” Phys. Rev. A 17, 1448–1453 (1977).
[CrossRef]

1973 (1)

R. H. Stolen and E. P. Ippen, “Raman gain in glass optical waveguides,” Appl. Phys. Lett. 22, 276–278 (1973).
[CrossRef]

1969 (1)

T. K. Gustafson, J. P. Taran, H. A. Haus, J. R. Lifsitz, and P. L. Kelley, “Self-modulation, self-steepening, and spectral development of light in small-scale trapped filaments,” Phys. Rev. 177, 306–313 (1969).
[CrossRef]

1967 (1)

Fujio Shimizu, “Frequency broadening in liquids by a short light pulse,” Phys. Rev. Lett. 19, 1097–1100 (1967).
[CrossRef]

Auyeung, J.

J. Auyeung and A. Yariv, “Spontaneous and stimulated Raman scattering in low loss fibers,” IEEE J. Quantum Electron. QE-14, 347–352 (1978).
[CrossRef]

Baer, T.

Bloom, D. M.

Dianov, E. M.

E. M. Dianov, A. Ya. Karasik, P. G. Mamyshev, A. M. Prokhorov, and V. N. Serkin, “Generation of ultrashort pulses by spectral filtering during stimulated Raman scattering in an optical fiber,” Sov. Phys. JETP 62, 448–455 (1985).

E. M. Dianov, A. Y. Karasik, P. V. Mamyshev, G. I. Onishchukov, A. M. Prokhorov, M. F. Stel’makh, and A. A. Formichev, “Picosecond structure of the pump pulse in stimulated Raman scattering in a single-mode optical fiber,” JETP Lett. 39, 691–695 (1984).

Formichev, A. A.

E. M. Dianov, A. Y. Karasik, P. V. Mamyshev, G. I. Onishchukov, A. M. Prokhorov, M. F. Stel’makh, and A. A. Formichev, “Picosecond structure of the pump pulse in stimulated Raman scattering in a single-mode optical fiber,” JETP Lett. 39, 691–695 (1984).

Gomes, A. S. L.

A. S. L. Gomes, W. Sibbett, and J. R. Taylor, “Spectral and temporal study of picosecond-pulse propagation in a single-mode optical fibre,” Appl. Phys. B 39, 44–46 (1986).
[CrossRef]

A. S. L. Gomes, U. Österberg, W. Sibbett, and J. R. Taylor, “An experimental study of the primary parameters that determine the temporal compression of cw Nd:YAG laser pulses,” Opt. Commun. 54, 377–382 (1985).
[CrossRef]

Grischkowsky, D.

B. Nikolaus and D. Grischkowsky, “12× pulse compression using optical fibers,” Appl. Phys. Lett. 42, 1–2 (1983).
[CrossRef]

Gustafson, T. K.

T. K. Gustafson, J. P. Taran, H. A. Haus, J. R. Lifsitz, and P. L. Kelley, “Self-modulation, self-steepening, and spectral development of light in small-scale trapped filaments,” Phys. Rev. 177, 306–313 (1969).
[CrossRef]

Haus, H. A.

T. K. Gustafson, J. P. Taran, H. A. Haus, J. R. Lifsitz, and P. L. Kelley, “Self-modulation, self-steepening, and spectral development of light in small-scale trapped filaments,” Phys. Rev. 177, 306–313 (1969).
[CrossRef]

Heritage, J. P.

A. M. Weiner, J. P. Heritage, and R. H. Stolen, “Effect of stimulated Raman scattering and pulse walk off on self-phase modulation in optical fibers,” in Digest of the Conference on Lasers and Electro-Optics (Optical Society of America, Washington, D.C., 1986), p. 246.

Hian-Hua, Lu

Lu Hian-Hua, Li Yu-Lin, and Jiang Jia-Lin, “On combined self-phase modulation and stimulated Raman scattering in optical fibres,” Opt. Quantum Electron. 17, 187–193 (1985).
[CrossRef]

Ippen, E. P.

R. H. Stolen and E. P. Ippen, “Raman gain in glass optical waveguides,” Appl. Phys. Lett. 22, 276–278 (1973).
[CrossRef]

Jaskorzynska, B.

Jia-Lin, Jiang

Lu Hian-Hua, Li Yu-Lin, and Jiang Jia-Lin, “On combined self-phase modulation and stimulated Raman scattering in optical fibres,” Opt. Quantum Electron. 17, 187–193 (1985).
[CrossRef]

Kafka, J. D.

Karasik, A. Y.

E. M. Dianov, A. Y. Karasik, P. V. Mamyshev, G. I. Onishchukov, A. M. Prokhorov, M. F. Stel’makh, and A. A. Formichev, “Picosecond structure of the pump pulse in stimulated Raman scattering in a single-mode optical fiber,” JETP Lett. 39, 691–695 (1984).

Karasik, A. Ya.

E. M. Dianov, A. Ya. Karasik, P. G. Mamyshev, A. M. Prokhorov, and V. N. Serkin, “Generation of ultrashort pulses by spectral filtering during stimulated Raman scattering in an optical fiber,” Sov. Phys. JETP 62, 448–455 (1985).

Kelley, P. L.

T. K. Gustafson, J. P. Taran, H. A. Haus, J. R. Lifsitz, and P. L. Kelley, “Self-modulation, self-steepening, and spectral development of light in small-scale trapped filaments,” Phys. Rev. 177, 306–313 (1969).
[CrossRef]

Kolner, B. H.

Lifsitz, J. R.

T. K. Gustafson, J. P. Taran, H. A. Haus, J. R. Lifsitz, and P. L. Kelley, “Self-modulation, self-steepening, and spectral development of light in small-scale trapped filaments,” Phys. Rev. 177, 306–313 (1969).
[CrossRef]

Lin, C.

R. H. Stolen and C. Lin, “Self-phase-modulation in silica optical fibers,” Phys. Rev. A 17, 1448–1453 (1977).
[CrossRef]

Mamyshev, P. G.

E. M. Dianov, A. Ya. Karasik, P. G. Mamyshev, A. M. Prokhorov, and V. N. Serkin, “Generation of ultrashort pulses by spectral filtering during stimulated Raman scattering in an optical fiber,” Sov. Phys. JETP 62, 448–455 (1985).

Mamyshev, P. V.

E. M. Dianov, A. Y. Karasik, P. V. Mamyshev, G. I. Onishchukov, A. M. Prokhorov, M. F. Stel’makh, and A. A. Formichev, “Picosecond structure of the pump pulse in stimulated Raman scattering in a single-mode optical fiber,” JETP Lett. 39, 691–695 (1984).

Nikolaus, B.

B. Nikolaus and D. Grischkowsky, “12× pulse compression using optical fibers,” Appl. Phys. Lett. 42, 1–2 (1983).
[CrossRef]

Onishchukov, G. I.

E. M. Dianov, A. Y. Karasik, P. V. Mamyshev, G. I. Onishchukov, A. M. Prokhorov, M. F. Stel’makh, and A. A. Formichev, “Picosecond structure of the pump pulse in stimulated Raman scattering in a single-mode optical fiber,” JETP Lett. 39, 691–695 (1984).

Österberg, U.

D. Schadt, B. Jaskorzynska, and U. Österberg, “Numerical study on stimulated Raman scattering and self phase modulation in optical fibers influenced by walk-off between the pulses,” J. Opt. Soc. Am. B 3, 1257–1262 (1986).
[CrossRef]

A. S. L. Gomes, U. Österberg, W. Sibbett, and J. R. Taylor, “An experimental study of the primary parameters that determine the temporal compression of cw Nd:YAG laser pulses,” Opt. Commun. 54, 377–382 (1985).
[CrossRef]

Papoulis, A.

A. Papoulis, Systems and Transforms with Applications in Optics (McGraw-Hill, New York, 1968).

Prokhorov, A. M.

E. M. Dianov, A. Ya. Karasik, P. G. Mamyshev, A. M. Prokhorov, and V. N. Serkin, “Generation of ultrashort pulses by spectral filtering during stimulated Raman scattering in an optical fiber,” Sov. Phys. JETP 62, 448–455 (1985).

E. M. Dianov, A. Y. Karasik, P. V. Mamyshev, G. I. Onishchukov, A. M. Prokhorov, M. F. Stel’makh, and A. A. Formichev, “Picosecond structure of the pump pulse in stimulated Raman scattering in a single-mode optical fiber,” JETP Lett. 39, 691–695 (1984).

Schadt, D.

Serkin, V. N.

E. M. Dianov, A. Ya. Karasik, P. G. Mamyshev, A. M. Prokhorov, and V. N. Serkin, “Generation of ultrashort pulses by spectral filtering during stimulated Raman scattering in an optical fiber,” Sov. Phys. JETP 62, 448–455 (1985).

V. A. Vysloukh and V. N. Serkin, “Nonlinear transformation of solitons in optical fiber lightguides,” Izv. Akad. Nauk. SSSR Ser. Fiz. 48, 1777–1781 (1984).

Shimizu, Fujio

Fujio Shimizu, “Frequency broadening in liquids by a short light pulse,” Phys. Rev. Lett. 19, 1097–1100 (1967).
[CrossRef]

Sibbett, W.

A. S. L. Gomes, W. Sibbett, and J. R. Taylor, “Spectral and temporal study of picosecond-pulse propagation in a single-mode optical fibre,” Appl. Phys. B 39, 44–46 (1986).
[CrossRef]

A. S. L. Gomes, U. Österberg, W. Sibbett, and J. R. Taylor, “An experimental study of the primary parameters that determine the temporal compression of cw Nd:YAG laser pulses,” Opt. Commun. 54, 377–382 (1985).
[CrossRef]

Stel’makh, M. F.

E. M. Dianov, A. Y. Karasik, P. V. Mamyshev, G. I. Onishchukov, A. M. Prokhorov, M. F. Stel’makh, and A. A. Formichev, “Picosecond structure of the pump pulse in stimulated Raman scattering in a single-mode optical fiber,” JETP Lett. 39, 691–695 (1984).

Stolen, R. H.

R. H. Stolen and C. Lin, “Self-phase-modulation in silica optical fibers,” Phys. Rev. A 17, 1448–1453 (1977).
[CrossRef]

R. H. Stolen and E. P. Ippen, “Raman gain in glass optical waveguides,” Appl. Phys. Lett. 22, 276–278 (1973).
[CrossRef]

A. M. Weiner, J. P. Heritage, and R. H. Stolen, “Effect of stimulated Raman scattering and pulse walk off on self-phase modulation in optical fibers,” in Digest of the Conference on Lasers and Electro-Optics (Optical Society of America, Washington, D.C., 1986), p. 246.

Taran, J. P.

T. K. Gustafson, J. P. Taran, H. A. Haus, J. R. Lifsitz, and P. L. Kelley, “Self-modulation, self-steepening, and spectral development of light in small-scale trapped filaments,” Phys. Rev. 177, 306–313 (1969).
[CrossRef]

Taylor, J. R.

A. S. L. Gomes, W. Sibbett, and J. R. Taylor, “Spectral and temporal study of picosecond-pulse propagation in a single-mode optical fibre,” Appl. Phys. B 39, 44–46 (1986).
[CrossRef]

A. S. L. Gomes, U. Österberg, W. Sibbett, and J. R. Taylor, “An experimental study of the primary parameters that determine the temporal compression of cw Nd:YAG laser pulses,” Opt. Commun. 54, 377–382 (1985).
[CrossRef]

Vysloukh, V. A.

V. A. Vysloukh and V. N. Serkin, “Nonlinear transformation of solitons in optical fiber lightguides,” Izv. Akad. Nauk. SSSR Ser. Fiz. 48, 1777–1781 (1984).

Weber, H. P.

B. Zysset and H. P. Weber, “Temporal and spectral investigation of Nd:YAG pulse propagation in optical fibers and its application to pulse compression,” in Digest of the Conference on Lasers and Electro-Optics (Optical Society of America, Washington, D.C., 1986), p. 182.

Weiner, A. M.

A. M. Weiner, J. P. Heritage, and R. H. Stolen, “Effect of stimulated Raman scattering and pulse walk off on self-phase modulation in optical fibers,” in Digest of the Conference on Lasers and Electro-Optics (Optical Society of America, Washington, D.C., 1986), p. 246.

Yariv, A.

J. Auyeung and A. Yariv, “Spontaneous and stimulated Raman scattering in low loss fibers,” IEEE J. Quantum Electron. QE-14, 347–352 (1978).
[CrossRef]

Yu-Lin, Li

Lu Hian-Hua, Li Yu-Lin, and Jiang Jia-Lin, “On combined self-phase modulation and stimulated Raman scattering in optical fibres,” Opt. Quantum Electron. 17, 187–193 (1985).
[CrossRef]

Zysset, B.

B. Zysset and H. P. Weber, “Temporal and spectral investigation of Nd:YAG pulse propagation in optical fibers and its application to pulse compression,” in Digest of the Conference on Lasers and Electro-Optics (Optical Society of America, Washington, D.C., 1986), p. 182.

Appl. Phys. B (1)

A. S. L. Gomes, W. Sibbett, and J. R. Taylor, “Spectral and temporal study of picosecond-pulse propagation in a single-mode optical fibre,” Appl. Phys. B 39, 44–46 (1986).
[CrossRef]

Appl. Phys. Lett. (2)

B. Nikolaus and D. Grischkowsky, “12× pulse compression using optical fibers,” Appl. Phys. Lett. 42, 1–2 (1983).
[CrossRef]

R. H. Stolen and E. P. Ippen, “Raman gain in glass optical waveguides,” Appl. Phys. Lett. 22, 276–278 (1973).
[CrossRef]

IEEE J. Quantum Electron. (1)

J. Auyeung and A. Yariv, “Spontaneous and stimulated Raman scattering in low loss fibers,” IEEE J. Quantum Electron. QE-14, 347–352 (1978).
[CrossRef]

Izv. Akad. Nauk. SSSR Ser. Fiz. (1)

V. A. Vysloukh and V. N. Serkin, “Nonlinear transformation of solitons in optical fiber lightguides,” Izv. Akad. Nauk. SSSR Ser. Fiz. 48, 1777–1781 (1984).

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

JETP Lett. (1)

E. M. Dianov, A. Y. Karasik, P. V. Mamyshev, G. I. Onishchukov, A. M. Prokhorov, M. F. Stel’makh, and A. A. Formichev, “Picosecond structure of the pump pulse in stimulated Raman scattering in a single-mode optical fiber,” JETP Lett. 39, 691–695 (1984).

Opt. Commun. (1)

A. S. L. Gomes, U. Österberg, W. Sibbett, and J. R. Taylor, “An experimental study of the primary parameters that determine the temporal compression of cw Nd:YAG laser pulses,” Opt. Commun. 54, 377–382 (1985).
[CrossRef]

Opt. Lett. (1)

Opt. Quantum Electron. (1)

Lu Hian-Hua, Li Yu-Lin, and Jiang Jia-Lin, “On combined self-phase modulation and stimulated Raman scattering in optical fibres,” Opt. Quantum Electron. 17, 187–193 (1985).
[CrossRef]

Phys. Rev. (1)

T. K. Gustafson, J. P. Taran, H. A. Haus, J. R. Lifsitz, and P. L. Kelley, “Self-modulation, self-steepening, and spectral development of light in small-scale trapped filaments,” Phys. Rev. 177, 306–313 (1969).
[CrossRef]

Phys. Rev. A (1)

R. H. Stolen and C. Lin, “Self-phase-modulation in silica optical fibers,” Phys. Rev. A 17, 1448–1453 (1977).
[CrossRef]

Phys. Rev. Lett. (1)

Fujio Shimizu, “Frequency broadening in liquids by a short light pulse,” Phys. Rev. Lett. 19, 1097–1100 (1967).
[CrossRef]

Sov. Phys. JETP (1)

E. M. Dianov, A. Ya. Karasik, P. G. Mamyshev, A. M. Prokhorov, and V. N. Serkin, “Generation of ultrashort pulses by spectral filtering during stimulated Raman scattering in an optical fiber,” Sov. Phys. JETP 62, 448–455 (1985).

Other (3)

A. M. Weiner, J. P. Heritage, and R. H. Stolen, “Effect of stimulated Raman scattering and pulse walk off on self-phase modulation in optical fibers,” in Digest of the Conference on Lasers and Electro-Optics (Optical Society of America, Washington, D.C., 1986), p. 246.

B. Zysset and H. P. Weber, “Temporal and spectral investigation of Nd:YAG pulse propagation in optical fibers and its application to pulse compression,” in Digest of the Conference on Lasers and Electro-Optics (Optical Society of America, Washington, D.C., 1986), p. 182.

A. Papoulis, Systems and Transforms with Applications in Optics (McGraw-Hill, New York, 1968).

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Figures (6)

Fig. 1
Fig. 1

Spectra and chirps for the data of row a of Table 1 (walk-off neglected). a, Spectrum of the pump pulse; b, spectral broadening of the Stokes pulse because of PM; c, chirp (solid line) and shape (dashed line) of the pump pulse; d, chirp (solid line) and shape (dashed line) of the Stokes pulse.

Fig. 2
Fig. 2

Spectra and chirps for the data of row b of Table 1. (Row b differs from row a by the introduction of walk-off and a longer propagation distance.) a, Spectrum of the pump pulse; b, spectral broadening of the Stokes pulse because of PM; c, chirp (solid line) and shape (dashed line) of the pump pulse; d, chirp (solid line) and shape (dashed line) of the Stokes pulse.

Fig. 3
Fig. 3

Spectra and chirps for the data of row c of Table 1 (The data of row c correspond to the same physical situation as those of row b but with 50% higher input power.) a, Spectrum of the pump; b, spectral broadening of the Stokes pulse because of PM; c, chirp (solid line) and shape (dashed line) of the pump pulse; d, chirp (solid line) and shape (dashed line) of the Stokes pulse.

Fig. 4
Fig. 4

a, Pump spectrum and b, chirp for the data as in row c of Table 1 but without walk-off.

Fig. 5
Fig. 5

Chirp components that are due to SPM and cross PM for the data of row b of Table 1. a, Pump chirp because of SPM only; b, Stokes chirp because of SPM only; c, pump chirp because cross PM only; d, Stokes chirp because of cross PM only.

Fig. 6
Fig. 6

As in Fig. 5 but for the data of row c of Table 1.

Tables (1)

Tables Icon

Table 1 Values of Parameters Used in the Calculations

Equations (8)

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

A p Z + z K z W A p T + i 2 z K z D 2 A p T 2 = - 1 2 Ω p Ω s z K z A A s 2 A p + i 2 ( A p 2 + 2 A s 2 ) A p - z K z L A p , A s Z + i 2 k s k p z K z D 2 A s T 2 = 1 2 z K z A A p 2 A s + i 2 Ω s Ω p ( A s 2 + 2 A p 2 ) A s - Γ s Γ p z K z L A s ,
- δ Ω p = { 1 2 ( ln N 0 p / T ) Ω p N Z 1 + ( N 0 s / N 0 p ) exp [ ( z K / z A ) Ω p N Z ] } SPM + { ( ln N o s / T ) Ω s N Z 1 + ( N 0 p + N 0 s ) exp [ - ( z K / z A ) Ω p N Z ] } CROSS PM , - δ Ω s = { 1 2 Ω s Ω p ( ln N 0 s / T ) Ω s N Z 1 + ( N 0 p / N 0 s ) exp [ - ( z K / z A ) Ω p N Z ] } SPM + { Ω s Ω p ( ln N 0 p / T ) Ω p N Z 1 + ( N 0 s / N 0 p ) exp [ ( z K / z A ) Ω p N Z ] } CROSS PM ,
N 0 l = A l ( 0 , T ) 2 / Ω l ,             l = p , s ,             N = N 0 p + N 0 s .
δ Ω l = - ϕ l / T .
F τ Ω + Ω 0 2 ~ 2 π δ 0 2 W δ Ω { 1 - V δ Ω [ 1 - cos ( ψ 2 δ Ω - ψ 1 δ Ω ) ] } , ϕ ( T 1 δ Ω ) 0 ,             ϕ ( T 2 δ Ω ) 0 ,
W δ Ω = [ a ˜ ( T 1 δ Ω ) + a ˜ ( T 2 δ Ω ) ] 2 , V δ Ω = 2 a ˜ ( T 2 δ Ω ) / a ˜ ( T 1 δ Ω ) [ 1 + a ˜ ( T 2 δ Ω ) / a ˜ ( T 1 δ Ω ) ] 2 , Ψ n δ Ω = ϕ ( T n δ Ω ) + Ω · T n δ Ω ,             n = 1 , 2 , - ϕ ( T 1 δ Ω ) = - ϕ ( T 2 δ Ω ) = δ Ω ,
( ψ 2 δ Ω - ψ 1 δ Ω ) ( δ Ω ) = T 2 δ Ω - T 1 δ Ω
[ δ Ω s ] cross = Ω s Ω p z W z K [ | A s ( T - z K z W Z ) | 2 - A s ( T ) 2 ] .

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