Abstract

We numerically and experimentally demonstrate that pulses with a parabolic intensity profile can be formed by passive reshaping of more conventional laser pulses using nonlinear propagation in a length of normally dispersive nonlinear fibre. Moreover, we show that the parabolic shape can be stabilised by launching these pulses into a second length of fiber with suitably different nonlinear and dispersive characteristics relative to the initial reshaping fiber.

© 2007 Optical Society of America

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References

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  1. D. Anderson, M. Desaix, M. Karlson, M. Lisak, and M. L. Quiroga-Teixeiro, "Wave-breaking-free pulses in nonlinear optical fibers," J. Opt. Soc. Am. B 10, 1185-1190 (1993).
    [CrossRef]
  2. W. J. Tomlinson, R. H. Stolen, and A. M. Johnson, "Optical wave-breaking of pulses in nonlinear optical fibers," Opt. Lett. 10, 457-459 (1985).
    [CrossRef] [PubMed]
  3. D. Anderson, M. Desaix, M. Lisak, and M. L. Quiroga-Teixeiro, "Wave-breaking in nonlinear optical fibers," J. Opt. Soc. Am. B 9,1358-1361 (1992).
    [CrossRef]
  4. M. E. Fermann, V. I. Kruglov, B. C. Thomsen, J. M. Dudley, and J. D. Harvey, "Self-similar propagation and amplification of parabolic pulses in optical fibers," Phys. Rev. Lett. 84, 6010-6013 (2000).
    [CrossRef] [PubMed]
  5. P. Dupriez, C. Finot, A. Malinowski, J. K. Sahu, J. Nilsson, D. J. Richardson, K. G. Wilcox, H. D. Foreman, and A. C. Tropper, "High-power, high repetition rate picosecond and femtosecond sources based on Yb-doped fiber amplification of VECSELs," Opt. Express 14, 9611-9616 (2006).
    [CrossRef] [PubMed]
  6. C. Billet, J. M. Dudley, N. Joly, and J. C. Knight, "Intermediate asymptotic evolution and photonic bandgap fiber compression of optical similaritons around 1550 nm," Opt. Express 13, 3236-3241 (2005).
    [CrossRef] [PubMed]
  7. Y. Ozeki, Y. Takushima, K. Aiso, K. Taira, and K. Kikuchi, "Generation of 10 GHz similariton pulse trains from 1.2 km-long erbium-doped fibre amplifier for application to multi-wavelength pulse sources," Electron. Lett. 40, 1103-1104 (2004).
    [CrossRef]
  8. C. Finot, G. Millot, C. Billet, and J. M. Dudley, "Experimental generation of parabolic pulses via Raman amplification in optical fiber," Opt. Express 11, 1547-1552 (2003).
    [CrossRef] [PubMed]
  9. F. Ö. Ilday, J. R. Buckley, W. G. Clark, and F. W. Wise, "Self-similar evolution of parabolic pulses in a laser," Phys. Rev. Lett. 92, 213902 (2004).
    [CrossRef] [PubMed]
  10. A. Ruehl, O. Prochnow, D. Wandt, D. Kracht, B. Burgoyne, N. Godbout, and S. Lacroix, "Dynamics of parabolic pulses in an ultrafast fiber laser," Opt. Lett. 31, 2734-2736 (2006).
    [CrossRef] [PubMed]
  11. C. Finot, F. Parmigiani, P. Petropoulos, and D. J. Richardson, "Parabolic pulse evolution in normally dispersive fiber amplifiers preceding the similariton formation regime," Opt. Express 14, 3161-3170 (2006).
    [CrossRef] [PubMed]
  12. T. Hirooka, and M. Nakazawa, "Parabolic pulse generation by use of a dispersion-decreasing fiber with normal group-velocity dispersion," Opt. Lett. 29, 498-500 (2004).
    [CrossRef] [PubMed]
  13. A. Latkin, S. K. Turitsyn, and A. Sysoliatin, "On the theory of parabolic pulse generation in tapered fibre," Opt. Lett. (to be published).
    [PubMed]
  14. S. V. Chernikov, and P. V. Mamyshev, "Femtosecond soliton propagation in fibers with slowly decreasing dispersion," J. Opt. Soc. Am. B 8, 1633-1641 (1991).
    [CrossRef]
  15. B. Kibler, C. Billet, P. A. Lacourt, R. Ferrière, L. Larger, and J. M. Dudley, "Parabolic pulse generation in comb-like profiled dispersion decreasing fibre," Electron. Lett. 42,965-966 (2006).
    [CrossRef]
  16. F. Parmigiani, C. Finot, K. Mukasa, M. Ibsen, M. A. F. Roelens, P. Petropoulos, and D. J. Richardson, "Ultra-flat SPM-broadened spectra in a highly nonlinear fiber using parabolic pulses formed in a fiber Bragg grating," Opt. Express 14, 7617-7622 (2006).
    [CrossRef] [PubMed]
  17. S. Pitois, C. Finot, J. Fatome, and G. Millot, "Generation of 20-Ghz picosecond pulse trains in the normal and anomalous dispersion regimes of optical fibers," Opt. Commun. 260, 301-306 (2006).
    [CrossRef]
  18. G. P. Agrawal, Nonlinear Fiber Optics, Third Edition (San Francisco, CA : Academic Press, 2001).
  19. H. Nakatsuka, D. Grischkowsky, and A. C. Balant, "Nonlinear picosecond-pulse propagation through optical fibers with positive group velocity dispersion," Phys. Rev. Lett. 47, 910-913 (1981).
    [CrossRef]
  20. C. Jirauschek, F. Ö. Ilday, and F. X. Kärtner, "A Semi-Analytic Theory of the Self-Similar Laser Oscillator," in Non Linear Guided Waves and their Applications (NLGW)(Dresden, 2005).
  21. C. Finot, G. Millot, and J. M. Dudley, "Asymptotic characteristics of parabolic similariton pulses in optical fiber amplifiers," Opt. Lett. 29, 2533-2535 (2004).
    [CrossRef] [PubMed]
  22. R. Trebino, Frequency-Resolved Optical Gating : the measurement of ultrashort laser pulses (Norwell, MA : Kluwer Academic Publishers, 2000).
  23. F. Parmigiani, P. Petropoulos, M. Ibsen, and D. J. Richardson, "Pulse retiming based on XPM using parabolic pulses formed in a fiber Bragg grating," IEEE Photon. Technol. Lett. 18, 829-831 (2006).
    [CrossRef]
  24. A. C. Peacock, R. J. Kruhlak, J. D. Harvey, and J. M. Dudley, "Solitary pulse propagation in high gain optical fiber amplifiers with normal group velocity dispersion," Opt. Commun. 206, 171-177 (2002).
    [CrossRef]

2006

P. Dupriez, C. Finot, A. Malinowski, J. K. Sahu, J. Nilsson, D. J. Richardson, K. G. Wilcox, H. D. Foreman, and A. C. Tropper, "High-power, high repetition rate picosecond and femtosecond sources based on Yb-doped fiber amplification of VECSELs," Opt. Express 14, 9611-9616 (2006).
[CrossRef] [PubMed]

A. Ruehl, O. Prochnow, D. Wandt, D. Kracht, B. Burgoyne, N. Godbout, and S. Lacroix, "Dynamics of parabolic pulses in an ultrafast fiber laser," Opt. Lett. 31, 2734-2736 (2006).
[CrossRef] [PubMed]

C. Finot, F. Parmigiani, P. Petropoulos, and D. J. Richardson, "Parabolic pulse evolution in normally dispersive fiber amplifiers preceding the similariton formation regime," Opt. Express 14, 3161-3170 (2006).
[CrossRef] [PubMed]

B. Kibler, C. Billet, P. A. Lacourt, R. Ferrière, L. Larger, and J. M. Dudley, "Parabolic pulse generation in comb-like profiled dispersion decreasing fibre," Electron. Lett. 42,965-966 (2006).
[CrossRef]

F. Parmigiani, C. Finot, K. Mukasa, M. Ibsen, M. A. F. Roelens, P. Petropoulos, and D. J. Richardson, "Ultra-flat SPM-broadened spectra in a highly nonlinear fiber using parabolic pulses formed in a fiber Bragg grating," Opt. Express 14, 7617-7622 (2006).
[CrossRef] [PubMed]

S. Pitois, C. Finot, J. Fatome, and G. Millot, "Generation of 20-Ghz picosecond pulse trains in the normal and anomalous dispersion regimes of optical fibers," Opt. Commun. 260, 301-306 (2006).
[CrossRef]

F. Parmigiani, P. Petropoulos, M. Ibsen, and D. J. Richardson, "Pulse retiming based on XPM using parabolic pulses formed in a fiber Bragg grating," IEEE Photon. Technol. Lett. 18, 829-831 (2006).
[CrossRef]

2005

2004

Y. Ozeki, Y. Takushima, K. Aiso, K. Taira, and K. Kikuchi, "Generation of 10 GHz similariton pulse trains from 1.2 km-long erbium-doped fibre amplifier for application to multi-wavelength pulse sources," Electron. Lett. 40, 1103-1104 (2004).
[CrossRef]

T. Hirooka, and M. Nakazawa, "Parabolic pulse generation by use of a dispersion-decreasing fiber with normal group-velocity dispersion," Opt. Lett. 29, 498-500 (2004).
[CrossRef] [PubMed]

F. Ö. Ilday, J. R. Buckley, W. G. Clark, and F. W. Wise, "Self-similar evolution of parabolic pulses in a laser," Phys. Rev. Lett. 92, 213902 (2004).
[CrossRef] [PubMed]

C. Finot, G. Millot, and J. M. Dudley, "Asymptotic characteristics of parabolic similariton pulses in optical fiber amplifiers," Opt. Lett. 29, 2533-2535 (2004).
[CrossRef] [PubMed]

2003

2002

A. C. Peacock, R. J. Kruhlak, J. D. Harvey, and J. M. Dudley, "Solitary pulse propagation in high gain optical fiber amplifiers with normal group velocity dispersion," Opt. Commun. 206, 171-177 (2002).
[CrossRef]

2000

M. E. Fermann, V. I. Kruglov, B. C. Thomsen, J. M. Dudley, and J. D. Harvey, "Self-similar propagation and amplification of parabolic pulses in optical fibers," Phys. Rev. Lett. 84, 6010-6013 (2000).
[CrossRef] [PubMed]

1993

1992

1991

1985

1981

H. Nakatsuka, D. Grischkowsky, and A. C. Balant, "Nonlinear picosecond-pulse propagation through optical fibers with positive group velocity dispersion," Phys. Rev. Lett. 47, 910-913 (1981).
[CrossRef]

Aiso, K.

Y. Ozeki, Y. Takushima, K. Aiso, K. Taira, and K. Kikuchi, "Generation of 10 GHz similariton pulse trains from 1.2 km-long erbium-doped fibre amplifier for application to multi-wavelength pulse sources," Electron. Lett. 40, 1103-1104 (2004).
[CrossRef]

Anderson, D.

Balant, A. C.

H. Nakatsuka, D. Grischkowsky, and A. C. Balant, "Nonlinear picosecond-pulse propagation through optical fibers with positive group velocity dispersion," Phys. Rev. Lett. 47, 910-913 (1981).
[CrossRef]

Billet, C.

Buckley, J. R.

F. Ö. Ilday, J. R. Buckley, W. G. Clark, and F. W. Wise, "Self-similar evolution of parabolic pulses in a laser," Phys. Rev. Lett. 92, 213902 (2004).
[CrossRef] [PubMed]

Burgoyne, B.

Chernikov, S. V.

Clark, W. G.

F. Ö. Ilday, J. R. Buckley, W. G. Clark, and F. W. Wise, "Self-similar evolution of parabolic pulses in a laser," Phys. Rev. Lett. 92, 213902 (2004).
[CrossRef] [PubMed]

Desaix, M.

Dudley, J. M.

B. Kibler, C. Billet, P. A. Lacourt, R. Ferrière, L. Larger, and J. M. Dudley, "Parabolic pulse generation in comb-like profiled dispersion decreasing fibre," Electron. Lett. 42,965-966 (2006).
[CrossRef]

C. Billet, J. M. Dudley, N. Joly, and J. C. Knight, "Intermediate asymptotic evolution and photonic bandgap fiber compression of optical similaritons around 1550 nm," Opt. Express 13, 3236-3241 (2005).
[CrossRef] [PubMed]

C. Finot, G. Millot, and J. M. Dudley, "Asymptotic characteristics of parabolic similariton pulses in optical fiber amplifiers," Opt. Lett. 29, 2533-2535 (2004).
[CrossRef] [PubMed]

C. Finot, G. Millot, C. Billet, and J. M. Dudley, "Experimental generation of parabolic pulses via Raman amplification in optical fiber," Opt. Express 11, 1547-1552 (2003).
[CrossRef] [PubMed]

A. C. Peacock, R. J. Kruhlak, J. D. Harvey, and J. M. Dudley, "Solitary pulse propagation in high gain optical fiber amplifiers with normal group velocity dispersion," Opt. Commun. 206, 171-177 (2002).
[CrossRef]

M. E. Fermann, V. I. Kruglov, B. C. Thomsen, J. M. Dudley, and J. D. Harvey, "Self-similar propagation and amplification of parabolic pulses in optical fibers," Phys. Rev. Lett. 84, 6010-6013 (2000).
[CrossRef] [PubMed]

Dupriez, P.

Fatome, J.

S. Pitois, C. Finot, J. Fatome, and G. Millot, "Generation of 20-Ghz picosecond pulse trains in the normal and anomalous dispersion regimes of optical fibers," Opt. Commun. 260, 301-306 (2006).
[CrossRef]

Fermann, M. E.

M. E. Fermann, V. I. Kruglov, B. C. Thomsen, J. M. Dudley, and J. D. Harvey, "Self-similar propagation and amplification of parabolic pulses in optical fibers," Phys. Rev. Lett. 84, 6010-6013 (2000).
[CrossRef] [PubMed]

Ferrière, R.

B. Kibler, C. Billet, P. A. Lacourt, R. Ferrière, L. Larger, and J. M. Dudley, "Parabolic pulse generation in comb-like profiled dispersion decreasing fibre," Electron. Lett. 42,965-966 (2006).
[CrossRef]

Finot, C.

Foreman, H. D.

Godbout, N.

Grischkowsky, D.

H. Nakatsuka, D. Grischkowsky, and A. C. Balant, "Nonlinear picosecond-pulse propagation through optical fibers with positive group velocity dispersion," Phys. Rev. Lett. 47, 910-913 (1981).
[CrossRef]

Harvey, J. D.

A. C. Peacock, R. J. Kruhlak, J. D. Harvey, and J. M. Dudley, "Solitary pulse propagation in high gain optical fiber amplifiers with normal group velocity dispersion," Opt. Commun. 206, 171-177 (2002).
[CrossRef]

M. E. Fermann, V. I. Kruglov, B. C. Thomsen, J. M. Dudley, and J. D. Harvey, "Self-similar propagation and amplification of parabolic pulses in optical fibers," Phys. Rev. Lett. 84, 6010-6013 (2000).
[CrossRef] [PubMed]

Hirooka, T.

Ibsen, M.

F. Parmigiani, C. Finot, K. Mukasa, M. Ibsen, M. A. F. Roelens, P. Petropoulos, and D. J. Richardson, "Ultra-flat SPM-broadened spectra in a highly nonlinear fiber using parabolic pulses formed in a fiber Bragg grating," Opt. Express 14, 7617-7622 (2006).
[CrossRef] [PubMed]

F. Parmigiani, P. Petropoulos, M. Ibsen, and D. J. Richardson, "Pulse retiming based on XPM using parabolic pulses formed in a fiber Bragg grating," IEEE Photon. Technol. Lett. 18, 829-831 (2006).
[CrossRef]

Ilday, F. Ö.

F. Ö. Ilday, J. R. Buckley, W. G. Clark, and F. W. Wise, "Self-similar evolution of parabolic pulses in a laser," Phys. Rev. Lett. 92, 213902 (2004).
[CrossRef] [PubMed]

Johnson, A. M.

Joly, N.

Karlson, M.

Kibler, B.

B. Kibler, C. Billet, P. A. Lacourt, R. Ferrière, L. Larger, and J. M. Dudley, "Parabolic pulse generation in comb-like profiled dispersion decreasing fibre," Electron. Lett. 42,965-966 (2006).
[CrossRef]

Kikuchi, K.

Y. Ozeki, Y. Takushima, K. Aiso, K. Taira, and K. Kikuchi, "Generation of 10 GHz similariton pulse trains from 1.2 km-long erbium-doped fibre amplifier for application to multi-wavelength pulse sources," Electron. Lett. 40, 1103-1104 (2004).
[CrossRef]

Knight, J. C.

Kracht, D.

Kruglov, V. I.

M. E. Fermann, V. I. Kruglov, B. C. Thomsen, J. M. Dudley, and J. D. Harvey, "Self-similar propagation and amplification of parabolic pulses in optical fibers," Phys. Rev. Lett. 84, 6010-6013 (2000).
[CrossRef] [PubMed]

Kruhlak, R. J.

A. C. Peacock, R. J. Kruhlak, J. D. Harvey, and J. M. Dudley, "Solitary pulse propagation in high gain optical fiber amplifiers with normal group velocity dispersion," Opt. Commun. 206, 171-177 (2002).
[CrossRef]

Lacourt, P. A.

B. Kibler, C. Billet, P. A. Lacourt, R. Ferrière, L. Larger, and J. M. Dudley, "Parabolic pulse generation in comb-like profiled dispersion decreasing fibre," Electron. Lett. 42,965-966 (2006).
[CrossRef]

Lacroix, S.

Larger, L.

B. Kibler, C. Billet, P. A. Lacourt, R. Ferrière, L. Larger, and J. M. Dudley, "Parabolic pulse generation in comb-like profiled dispersion decreasing fibre," Electron. Lett. 42,965-966 (2006).
[CrossRef]

Latkin, A.

A. Latkin, S. K. Turitsyn, and A. Sysoliatin, "On the theory of parabolic pulse generation in tapered fibre," Opt. Lett. (to be published).
[PubMed]

Lisak, M.

Malinowski, A.

Mamyshev, P. V.

Millot, G.

Mukasa, K.

Nakatsuka, H.

H. Nakatsuka, D. Grischkowsky, and A. C. Balant, "Nonlinear picosecond-pulse propagation through optical fibers with positive group velocity dispersion," Phys. Rev. Lett. 47, 910-913 (1981).
[CrossRef]

Nakazawa, M.

Nilsson, J.

Ozeki, Y.

Y. Ozeki, Y. Takushima, K. Aiso, K. Taira, and K. Kikuchi, "Generation of 10 GHz similariton pulse trains from 1.2 km-long erbium-doped fibre amplifier for application to multi-wavelength pulse sources," Electron. Lett. 40, 1103-1104 (2004).
[CrossRef]

Parmigiani, F.

Peacock, A. C.

A. C. Peacock, R. J. Kruhlak, J. D. Harvey, and J. M. Dudley, "Solitary pulse propagation in high gain optical fiber amplifiers with normal group velocity dispersion," Opt. Commun. 206, 171-177 (2002).
[CrossRef]

Petropoulos, P.

Pitois, S.

S. Pitois, C. Finot, J. Fatome, and G. Millot, "Generation of 20-Ghz picosecond pulse trains in the normal and anomalous dispersion regimes of optical fibers," Opt. Commun. 260, 301-306 (2006).
[CrossRef]

Prochnow, O.

Quiroga-Teixeiro, M. L.

Richardson, D. J.

Roelens, M. A. F.

Ruehl, A.

Sahu, J. K.

Stolen, R. H.

Sysoliatin, A.

A. Latkin, S. K. Turitsyn, and A. Sysoliatin, "On the theory of parabolic pulse generation in tapered fibre," Opt. Lett. (to be published).
[PubMed]

Taira, K.

Y. Ozeki, Y. Takushima, K. Aiso, K. Taira, and K. Kikuchi, "Generation of 10 GHz similariton pulse trains from 1.2 km-long erbium-doped fibre amplifier for application to multi-wavelength pulse sources," Electron. Lett. 40, 1103-1104 (2004).
[CrossRef]

Takushima, Y.

Y. Ozeki, Y. Takushima, K. Aiso, K. Taira, and K. Kikuchi, "Generation of 10 GHz similariton pulse trains from 1.2 km-long erbium-doped fibre amplifier for application to multi-wavelength pulse sources," Electron. Lett. 40, 1103-1104 (2004).
[CrossRef]

Thomsen, B. C.

M. E. Fermann, V. I. Kruglov, B. C. Thomsen, J. M. Dudley, and J. D. Harvey, "Self-similar propagation and amplification of parabolic pulses in optical fibers," Phys. Rev. Lett. 84, 6010-6013 (2000).
[CrossRef] [PubMed]

Tomlinson, W. J.

Tropper, A. C.

Turitsyn, S. K.

A. Latkin, S. K. Turitsyn, and A. Sysoliatin, "On the theory of parabolic pulse generation in tapered fibre," Opt. Lett. (to be published).
[PubMed]

Wandt, D.

Wilcox, K. G.

Wise, F. W.

F. Ö. Ilday, J. R. Buckley, W. G. Clark, and F. W. Wise, "Self-similar evolution of parabolic pulses in a laser," Phys. Rev. Lett. 92, 213902 (2004).
[CrossRef] [PubMed]

Electron. Lett.

Y. Ozeki, Y. Takushima, K. Aiso, K. Taira, and K. Kikuchi, "Generation of 10 GHz similariton pulse trains from 1.2 km-long erbium-doped fibre amplifier for application to multi-wavelength pulse sources," Electron. Lett. 40, 1103-1104 (2004).
[CrossRef]

B. Kibler, C. Billet, P. A. Lacourt, R. Ferrière, L. Larger, and J. M. Dudley, "Parabolic pulse generation in comb-like profiled dispersion decreasing fibre," Electron. Lett. 42,965-966 (2006).
[CrossRef]

IEEE Photon. Technol. Lett.

F. Parmigiani, P. Petropoulos, M. Ibsen, and D. J. Richardson, "Pulse retiming based on XPM using parabolic pulses formed in a fiber Bragg grating," IEEE Photon. Technol. Lett. 18, 829-831 (2006).
[CrossRef]

J. Opt. Soc. Am. B

Opt. Commun.

S. Pitois, C. Finot, J. Fatome, and G. Millot, "Generation of 20-Ghz picosecond pulse trains in the normal and anomalous dispersion regimes of optical fibers," Opt. Commun. 260, 301-306 (2006).
[CrossRef]

A. C. Peacock, R. J. Kruhlak, J. D. Harvey, and J. M. Dudley, "Solitary pulse propagation in high gain optical fiber amplifiers with normal group velocity dispersion," Opt. Commun. 206, 171-177 (2002).
[CrossRef]

Opt. Express

Opt. Lett.

Phys. Rev. Lett.

H. Nakatsuka, D. Grischkowsky, and A. C. Balant, "Nonlinear picosecond-pulse propagation through optical fibers with positive group velocity dispersion," Phys. Rev. Lett. 47, 910-913 (1981).
[CrossRef]

M. E. Fermann, V. I. Kruglov, B. C. Thomsen, J. M. Dudley, and J. D. Harvey, "Self-similar propagation and amplification of parabolic pulses in optical fibers," Phys. Rev. Lett. 84, 6010-6013 (2000).
[CrossRef] [PubMed]

F. Ö. Ilday, J. R. Buckley, W. G. Clark, and F. W. Wise, "Self-similar evolution of parabolic pulses in a laser," Phys. Rev. Lett. 92, 213902 (2004).
[CrossRef] [PubMed]

Other

G. P. Agrawal, Nonlinear Fiber Optics, Third Edition (San Francisco, CA : Academic Press, 2001).

C. Jirauschek, F. Ö. Ilday, and F. X. Kärtner, "A Semi-Analytic Theory of the Self-Similar Laser Oscillator," in Non Linear Guided Waves and their Applications (NLGW)(Dresden, 2005).

R. Trebino, Frequency-Resolved Optical Gating : the measurement of ultrashort laser pulses (Norwell, MA : Kluwer Academic Publishers, 2000).

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Figures (12)

Fig. 1.
Fig. 1.

(a) Evolution of the temporal intensity profile of an initial Gaussian pulse in a normally dispersive fiber for N = 4 and for different propagation distance ξ: 0, 0.1, 0.2 and 0.6 (Fig. a1, a2, a3 and a4 respectively). Results from numerical integration of Eq. (2) (solid lines) are compared to parabolic fits (blue circles). The discrepancy between the two is highlighted by the yellow shaded areas. (b) Evolution of the misfit parameter M versus the normalized propagation distance ξ for N = 4.

Fig. 2.
Fig. 2.

(a) Evolution of the misfit parameter M versus N and ξ. The wave-breaking condition according to Eq. (6) is given by the white solid line. (b) Closer view of Fig. a. Black solid line joins the points where the optimal parabolic shape is reached. Points 1, 2, 3 4 (circles) represent the points used in Fig. 1(a), points A, B, C, D (crosses) are used in Fig 3(a) and points I and II (yellow diamonds) correspond to the experimental conditions used in Fig 8 and 12 respectively.

Fig. 3.
Fig. 3.

(a) Temporal intensity and chirp profiles at ξopt for different N values (N = 2, 2.6, 3 and 4, curves A, B, C, D, mixed, solid, dotted and dashed lines respectively). The initial Gaussian pulse is presented in blue. (b) Evolution of τext (solid line) versus N compared to the temporal widths of the pulse at various intensity points (mixed lines)

Fig. 4.
Fig. 4.

Evolution of Mopt versus N for different initial pulse shapes: Gaussian, sech, truncated-cosine or supergaussian (blue, red, green and purple solid lines respectively). Results are obtained at the optimum distance ξopt adapted for each initial condition.

Fig 5.
Fig 5.

(a) Comparison of intensity profiles of different pulse shape with the same FWHM and the same energy (parabolic pulse are presented in black solid line, other pulses are plotted with the same convention as Fig. 4). (b) Evolution of the misfit parameter M versus N and ξ for a sech pulse.

Fig. 6.
Fig. 6.

Evolution of the misfit parameter M versus N and ξ for a truncated cosine pulse (a) and for a supergaussian pulse (b).

Fig. 7 :
Fig. 7 :

(a) Evolution of M for N = 16.5 in the case of a supergaussian pulse. (b) Temporal intensity and chirp profiles obtained at the different propagation distances ξ (positions 1, 2 and 3 on Fig a). Initial supergaussian pulse is plotted in purple solid line.

Fig. 8.
Fig. 8.

(a) Experimental set-up used to passively generate parabolic pulses. The initial experimental pulses characterized by FROG are plotted in the inset (circles) and compared with a Gaussian fit (solid black line). (b) Temporal chirp (Fig. b1) and intensity (Fig. b2) profiles. The results are for initial pulse energies of 11, 47 and 115 pJ (repetition rate of 10, 5 and 2.5 GHz respectively). Experimental results obtained by the FROG technique (blue circles) are compared to numerical simulations (solid black line) and to Gaussian or parabolic fits (green dashed and red mixed lines respectively). The misfit area is shaded in yellow.

Fig. 9.
Fig. 9.

(a) Evolution of the misfit parameter M versus N and ξ for an initial Gaussian pulse. The change from N to N’=8 is done when Mopt is reached in the first segment. (b) Evolution of M in the second stage for a parabolic pulse generated in the second stage (solid black line). The results are compared with the evolution a Gaussian (blue line) or sech (red line) pulse of the same FWHM temporal width, same linear chirp and same energy launched in the second fiber. Inset: intensity profile at the output of the second fiber.

Fig 10.
Fig 10.

(a) Spectral intensity profiles of the initial Gaussian pulse (grey dotted line) and of the parabolic pulse obtained by numerical simulations at the output of the first segment (N = 2.6, mixed line). Results at the output of the second segment (N’ = 8, ξ = 4) are compared to a parabolic fit. (b) Longitudinal evolution of the FWHM spectral width of the pulse. Numerical results (solid line) are compared with analytical results (blue diamonds).

Fig 11.
Fig 11.

(a) Influence of the N’ value on the evolution of the misfit parameter Mout at the output of the second segment and on the evolution of the output temporal FWHM. (b) Map similar to the one presented Fig 9(a) but for a sech initial pulse.

Fig. 12.
Fig. 12.

Temporal chirp (top) and intensity (bottom) profiles of the pulses characterized by FROG after the first stage (pink diamonds) and after the second stage (blue circles). Experimental results are compared with parabolic fits (red mixed lines) and with numerical simulations (solid black line).

Tables (1)

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Table 1. Comparison of optimal parameters for different initial pulse shapes

Equations (7)

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i ψ z = β 2 2 2 ψ T 2 γ ψ 2 ψ ,
i u ξ = 1 2 2 u τ 2 u 2 u ,
u ξ τ = N U , U ξ τ = ψ P C , τ = T T 0 , ξ = z L D .
L D = T 0 2 β 2 , L NL = 1 γ P C , N = L D L NL .
M 2 = [ u 2 p 2 ] 2 u 4
ξ wb = 1 4 exp ( 3 2 ) N 2 1 .
N ' 2 = T 0 2 γ ' G P C β' 2 .

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