Abstract

We show analytically and numerically that parabolic pulses and similaritons are not always synonyms and that a self-phase modulation amplification regime can precede the self-similar evolution. The properties of the recompressed pulses after SPM amplification are investigated. We also demonstrate that negatively chirped parabolic pulses can exhibit a spectral recompression during amplification leading to high-power chirp-free parabolic pulses at the amplifier output.

© 2006 Optical Society of America

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References

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  1. 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]
  2. J.P. Limpert, T. Schreiber, T. Clausnitzer, K. Zöllner, H.J. Fuchs, E.B. Kley, H. Zellmer, and A. Tünnermann, "High-power femtosecond Yb-doped fiber amplifier," Opt. Express 10,628-638 (2002).
    [PubMed]
  3. A. Malinowski, A. Piper, J.H.V. Price, K. Furusawa, Y. Jeong, J. Nilsson, and D.J. Richardson, "Ultrashort-pulse Yb3+ fiber based laser and amplifier system producing > 25 W average power," Opt. Lett. 29,2073-2075 (2004).
    [CrossRef] [PubMed]
  4. 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]
  5. 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]
  6. 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]
  7. 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]
  8. C.K. Nielsen, B. Ortac, T. Schreiber, J.P. Limpert, R. Hohmuth, W. Richter, and A. Tünnermann, "Self-starting self-similar all-polarization maintaining Yb-doped fiber laser," Opt. Express 13,9346-9351 (2005).
    [CrossRef] [PubMed]
  9. 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]
  10. 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]
  11. G. Chang, A. Galvanauskas, H.G. Winful, and T.B. Norris, "Dependence of parabolic pulse amplification on stimulated Raman scattering and gain bandwith," Opt. Lett. 29,2647-2549 (2004).
    [CrossRef] [PubMed]
  12. G.P. Agrawal, Nonlinear Fiber Optics, Third Edition. (Academic Press, San Francisco, 2001).
  13. S. Boscolo, S.K. Turitsyn, V.Y. Novokshenov, and J.H.B. Nijhof, "Self-similar parabolic optical solitary waves," Theor. Math. Phys. 133,1647-1656 (2002).
    [CrossRef]
  14. V.I. Kruglov, A.C. Peacock, J.D. Harvey, and J.M. Dudley, "Self-similar propagation of parabolic pulses in normal-dispersion fiber amplifiers," J. Opt. Soc. Am. B 19,461-469 (2002).
    [CrossRef]
  15. Y. Ozeki, Y. Takushima, K. Taira, and K. Kikuchi. "Clean similariton generation from an initial pulse optimized by the backward propagation method," in Conference on Lasers and Electro-Optics/Quantum Electronics and Laser Science 2004 (Optical Society of America, Washington DC, 2004), paper CTUBB5, (2004).
  16. D. Anderson, M. Desaix, M. Karlson, M. Lisak, and M.L. Quiroga-Teixeiro, "Wave-breaking-free pulses in nonlinear optical fibers," J. Opt. Soc. Amer. B 10,1185-1190 (1993).
    [CrossRef]
  17. J.P. Limpert, A. Liem, T. Gabler, H. Zellmer, A. Tünnermann, S. Unger, S. Jetschke, and H.-R. Müller, "High-average-power picosecond Yb-doped fiber amplifier," Opt. Lett. 16,1849-1851 (2001).
    [CrossRef]
  18. A.C. Peacock, "Self-similar amplification and propagation of parabolic pulses in optical fibers," Master thesis (Auckland University, New-Zealand, 2001).
  19. V.I. Kruglov, A.C. Peacock, J.M. Dudley, and J.D. Harvey, "Self-similar propagation of high-power parabolic pulses in optical fiber amplifiers," Opt. Lett. 25,1753-1755 (2000).
    [CrossRef]
  20. B.R. Washburn, J.A. Buck, and S.E. Ralph, "Transform-limited spectral compression due to self-phase modulation in fibers," Opt. Lett. 25,445-447 (2000).
    [CrossRef]
  21. J.P. Limpert, N. Deguil-Robin, I. Manek-Hönninger, F. Salin, T. Schreiber, A. Liem, F. Röser, H. Zellmer, A. Tünnermann, A. Courjaud, C. Hönninger, and E. Mottay, "High-power picosecond fiber amplifer based on nonlinear spectral compression," Opt. Lett. 30,714-716 (2005).
    [CrossRef] [PubMed]
  22. M. Oberthaler and R.A. Höpfel, "Spectral narrowing of ultrashort laser pulses by self-phase modulation in optical fibers," Appl. Phys. Lett. 63,1017-1019 (1993).
    [CrossRef]

2006

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

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]

S. Boscolo, S.K. Turitsyn, V.Y. Novokshenov, and J.H.B. Nijhof, "Self-similar parabolic optical solitary waves," Theor. Math. Phys. 133,1647-1656 (2002).
[CrossRef]

V.I. Kruglov, A.C. Peacock, J.D. Harvey, and J.M. Dudley, "Self-similar propagation of parabolic pulses in normal-dispersion fiber amplifiers," J. Opt. Soc. Am. B 19,461-469 (2002).
[CrossRef]

J.P. Limpert, T. Schreiber, T. Clausnitzer, K. Zöllner, H.J. Fuchs, E.B. Kley, H. Zellmer, and A. Tünnermann, "High-power femtosecond Yb-doped fiber amplifier," Opt. Express 10,628-638 (2002).
[PubMed]

2001

J.P. Limpert, A. Liem, T. Gabler, H. Zellmer, A. Tünnermann, S. Unger, S. Jetschke, and H.-R. Müller, "High-average-power picosecond Yb-doped fiber amplifier," Opt. Lett. 16,1849-1851 (2001).
[CrossRef]

2000

1993

M. Oberthaler and R.A. Höpfel, "Spectral narrowing of ultrashort laser pulses by self-phase modulation in optical fibers," Appl. Phys. Lett. 63,1017-1019 (1993).
[CrossRef]

D. Anderson, M. Desaix, M. Karlson, M. Lisak, and M.L. Quiroga-Teixeiro, "Wave-breaking-free pulses in nonlinear optical fibers," J. Opt. Soc. Amer. B 10,1185-1190 (1993).
[CrossRef]

Anderson, D.

D. Anderson, M. Desaix, M. Karlson, M. Lisak, and M.L. Quiroga-Teixeiro, "Wave-breaking-free pulses in nonlinear optical fibers," J. Opt. Soc. Amer. B 10,1185-1190 (1993).
[CrossRef]

Billet, C.

Boscolo, S.

S. Boscolo, S.K. Turitsyn, V.Y. Novokshenov, and J.H.B. Nijhof, "Self-similar parabolic optical solitary waves," Theor. Math. Phys. 133,1647-1656 (2002).
[CrossRef]

Buck, J.A.

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]

Chang, G.

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]

Clausnitzer, T.

Courjaud, A.

Deguil-Robin, N.

Desaix, M.

D. Anderson, M. Desaix, M. Karlson, M. Lisak, and M.L. Quiroga-Teixeiro, "Wave-breaking-free pulses in nonlinear optical fibers," J. Opt. Soc. Amer. B 10,1185-1190 (1993).
[CrossRef]

Dudley, J.M.

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]

Finot, C.

Fuchs, H.J.

Furusawa, K.

Gabler, T.

J.P. Limpert, A. Liem, T. Gabler, H. Zellmer, A. Tünnermann, S. Unger, S. Jetschke, and H.-R. Müller, "High-average-power picosecond Yb-doped fiber amplifier," Opt. Lett. 16,1849-1851 (2001).
[CrossRef]

Galvanauskas, A.

Harvey, J.D.

V.I. Kruglov, A.C. Peacock, J.D. Harvey, and J.M. Dudley, "Self-similar propagation of parabolic pulses in normal-dispersion fiber amplifiers," J. Opt. Soc. Am. B 19,461-469 (2002).
[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]

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]

V.I. Kruglov, A.C. Peacock, J.M. Dudley, and J.D. Harvey, "Self-similar propagation of high-power parabolic pulses in optical fiber amplifiers," Opt. Lett. 25,1753-1755 (2000).
[CrossRef]

Hohmuth, R.

Hönninger, C.

Höpfel, R.A.

M. Oberthaler and R.A. Höpfel, "Spectral narrowing of ultrashort laser pulses by self-phase modulation in optical fibers," Appl. Phys. Lett. 63,1017-1019 (1993).
[CrossRef]

Ibsen, M.

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]

Jeong, Y.

Jetschke, S.

J.P. Limpert, A. Liem, T. Gabler, H. Zellmer, A. Tünnermann, S. Unger, S. Jetschke, and H.-R. Müller, "High-average-power picosecond Yb-doped fiber amplifier," Opt. Lett. 16,1849-1851 (2001).
[CrossRef]

Joly, N.

Karlson, M.

D. Anderson, M. Desaix, M. Karlson, M. Lisak, and M.L. Quiroga-Teixeiro, "Wave-breaking-free pulses in nonlinear optical fibers," J. Opt. Soc. Amer. B 10,1185-1190 (1993).
[CrossRef]

Kley, E.B.

Knight, J.C.

Kruglov, V.I.

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]

Liem, A.

Limpert, J.P.

Lisak, M.

D. Anderson, M. Desaix, M. Karlson, M. Lisak, and M.L. Quiroga-Teixeiro, "Wave-breaking-free pulses in nonlinear optical fibers," J. Opt. Soc. Amer. B 10,1185-1190 (1993).
[CrossRef]

Malinowski, A.

Manek-Hönninger, I.

Millot, G.

Mottay, E.

Müller, H.-R.

J.P. Limpert, A. Liem, T. Gabler, H. Zellmer, A. Tünnermann, S. Unger, S. Jetschke, and H.-R. Müller, "High-average-power picosecond Yb-doped fiber amplifier," Opt. Lett. 16,1849-1851 (2001).
[CrossRef]

Nielsen, C.K.

Nijhof, J.H.B.

S. Boscolo, S.K. Turitsyn, V.Y. Novokshenov, and J.H.B. Nijhof, "Self-similar parabolic optical solitary waves," Theor. Math. Phys. 133,1647-1656 (2002).
[CrossRef]

Nilsson, J.

Norris, T.B.

Novokshenov, V.Y.

S. Boscolo, S.K. Turitsyn, V.Y. Novokshenov, and J.H.B. Nijhof, "Self-similar parabolic optical solitary waves," Theor. Math. Phys. 133,1647-1656 (2002).
[CrossRef]

Oberthaler, M.

M. Oberthaler and R.A. Höpfel, "Spectral narrowing of ultrashort laser pulses by self-phase modulation in optical fibers," Appl. Phys. Lett. 63,1017-1019 (1993).
[CrossRef]

Ortac, B.

Parmigiani, F.

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]

Peacock, A.C.

Petropoulos, P.

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]

Piper, A.

Price, J.H.V.

Quiroga-Teixeiro, M.L.

D. Anderson, M. Desaix, M. Karlson, M. Lisak, and M.L. Quiroga-Teixeiro, "Wave-breaking-free pulses in nonlinear optical fibers," J. Opt. Soc. Amer. B 10,1185-1190 (1993).
[CrossRef]

Ralph, S.E.

Richardson, D.J.

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]

A. Malinowski, A. Piper, J.H.V. Price, K. Furusawa, Y. Jeong, J. Nilsson, and D.J. Richardson, "Ultrashort-pulse Yb3+ fiber based laser and amplifier system producing > 25 W average power," Opt. Lett. 29,2073-2075 (2004).
[CrossRef] [PubMed]

Richter, W.

Röser, F.

Salin, F.

Schreiber, T.

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]

Tünnermann, A.

Turitsyn, S.K.

S. Boscolo, S.K. Turitsyn, V.Y. Novokshenov, and J.H.B. Nijhof, "Self-similar parabolic optical solitary waves," Theor. Math. Phys. 133,1647-1656 (2002).
[CrossRef]

Unger, S.

J.P. Limpert, A. Liem, T. Gabler, H. Zellmer, A. Tünnermann, S. Unger, S. Jetschke, and H.-R. Müller, "High-average-power picosecond Yb-doped fiber amplifier," Opt. Lett. 16,1849-1851 (2001).
[CrossRef]

Washburn, B.R.

Winful, H.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]

Zellmer, H.

Zöllner, K.

Appl. Phys. Lett.

M. Oberthaler and R.A. Höpfel, "Spectral narrowing of ultrashort laser pulses by self-phase modulation in optical fibers," Appl. Phys. Lett. 63,1017-1019 (1993).
[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

J. Opt. Soc. Amer. B

D. Anderson, M. Desaix, M. Karlson, M. Lisak, and M.L. Quiroga-Teixeiro, "Wave-breaking-free pulses in nonlinear optical fibers," J. Opt. Soc. Amer. B 10,1185-1190 (1993).
[CrossRef]

Opt. Commun.

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.

B.R. Washburn, J.A. Buck, and S.E. Ralph, "Transform-limited spectral compression due to self-phase modulation in fibers," Opt. Lett. 25,445-447 (2000).
[CrossRef]

V.I. Kruglov, A.C. Peacock, J.M. Dudley, and J.D. Harvey, "Self-similar propagation of high-power parabolic pulses in optical fiber amplifiers," Opt. Lett. 25,1753-1755 (2000).
[CrossRef]

A. Malinowski, A. Piper, J.H.V. Price, K. Furusawa, Y. Jeong, J. Nilsson, and D.J. Richardson, "Ultrashort-pulse Yb3+ fiber based laser and amplifier system producing > 25 W average power," Opt. Lett. 29,2073-2075 (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]

G. Chang, A. Galvanauskas, H.G. Winful, and T.B. Norris, "Dependence of parabolic pulse amplification on stimulated Raman scattering and gain bandwith," Opt. Lett. 29,2647-2549 (2004).
[CrossRef] [PubMed]

J.P. Limpert, N. Deguil-Robin, I. Manek-Hönninger, F. Salin, T. Schreiber, A. Liem, F. Röser, H. Zellmer, A. Tünnermann, A. Courjaud, C. Hönninger, and E. Mottay, "High-power picosecond fiber amplifer based on nonlinear spectral compression," Opt. Lett. 30,714-716 (2005).
[CrossRef] [PubMed]

J.P. Limpert, A. Liem, T. Gabler, H. Zellmer, A. Tünnermann, S. Unger, S. Jetschke, and H.-R. Müller, "High-average-power picosecond Yb-doped fiber amplifier," Opt. Lett. 16,1849-1851 (2001).
[CrossRef]

Phys. Rev. Lett.

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]

Theor. Math. Phys.

S. Boscolo, S.K. Turitsyn, V.Y. Novokshenov, and J.H.B. Nijhof, "Self-similar parabolic optical solitary waves," Theor. Math. Phys. 133,1647-1656 (2002).
[CrossRef]

Other

Y. Ozeki, Y. Takushima, K. Taira, and K. Kikuchi. "Clean similariton generation from an initial pulse optimized by the backward propagation method," in Conference on Lasers and Electro-Optics/Quantum Electronics and Laser Science 2004 (Optical Society of America, Washington DC, 2004), paper CTUBB5, (2004).

A.C. Peacock, "Self-similar amplification and propagation of parabolic pulses in optical fibers," Master thesis (Auckland University, New-Zealand, 2001).

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

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Figures (6)

Fig. 1.
Fig. 1.

Longitudinal evolution of different initial chirp-free Gaussian pulses with initial temporal widths T0 = 0.8, 2, 4, 8 and 16 ps (solid black, dotted black, mixed black, solid grey and dotted grey lines respectively). (a) Evolution of the FWHM temporal width compared with the asymptotic SS solution (circles, Eq. (3)). (b) Evolution of the misfit function M (see Eq.5).

Fig. 2.
Fig. 2.

(a) Longitudinal evolution of the misfit function M for different initial pulse shapes (parabolic, Gaussian and sech, solid, dotted and mixed lines respectively) and different temporal widths (0.8 ps and 8 ps, grey and black respectively). (b) Normalised intensity profile of a 0.8 ps parabolic pulse evolving in a purely dispersive media for different distance of propagation (0, 0.5 and 2 m, solid, dotted and mixed lines respectively).

Fig. 3.
Fig. 3.

Longitudinal evolution of different initial chirp-free pulses with an initial energy U0 = 50 pJ. Different initial shapes (parabolic, Gaussian, and sech, solid, dotted and mixed lines respectively) and different initial FWHM temporal widths (800 fs and 8 ps, grey and black lines respectively) are considered. Analytical results in the SS regime (red circles, Eq. 3) and in the SPM regime (blue diamonds, 8 ps parabolic initial pulse, Eq. (10) and (11)) are compared with the numerical integration of Eq. (2). (a) Evolution of the linear chirp parameter C. (b) Longitudinal evolution of the spectral width. (c) Spectrum after 8 m of amplification.

Fig. 4.
Fig. 4.

Properties of the recompressed pulse after amplification for different 8 ps input pulse shapes (same conventions as in Fig. 3(b)). The analytical predictions (Eq. (13) and (14)) obtained in the SPM regime are compared to the numerical simulations. (a) Intensity profiles of the recompressed pulses after 8 meters of amplification (peak intensity normalized to 1) (b) Evolution of the FWHM temporal width TC of the recompressed pulses. (c) Evolution of the peak-power PC of the recompressed pulse.

Fig. 5.
Fig. 5.

Longitudinal evolution of parabolic pulses with different initial chirp coefficients CP0 (zero, positive and negative, blue, red and green respectively). (a) Evolution of the chirp coefficient. Numerical integration of Eq. (2) (solid line) is compared to the integration of the system (18) (colored diamonds) and to expression in the SS regime (black circles) and SPM regime (black diamonds). (b) Evolution of spectral FWHM width (note that the horizontal axis is different from Fig. 5a).

Fig. 6.
Fig. 6.

Spectral recompression of pulses with an initial negative linear chirp coefficient. The results are displayed at optimal compression distance. The pulses have the same initial energy, temporal width and linear chirp, but different shapes (parabolic, Gaussian and sech, solid, dotted and mixed lines respectively) (a) The spectrum after amplification in 6.6 m of fiber (black line) is compared to the initial parabolic spectrum (grey line) and to the spectrum of a transformed limited parabolic pulse (blue diamonds). (b) Intensity and phase profiles after spectral recompression.

Equations (19)

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

{ ψ P ( t ) = P P 1 2 t 2 T P 2 exp ( i C P 2 t 2 ) if t T P 2 ψ P ( t ) = 0 otherwise ,
i ψ z = i g 2 ψ + β 2 2 2 ψ t 2 γ ψ 2 ψ .
{ P P _ SS = 1 2 ( g U 0 β 2 γ 2 ) 1 3 exp ( g 3 z ) C P _ SS = g 3 β 2 T P _ SS = 3 2 ( U 0 β 2 γ 2 g 2 ) 1 3 exp ( g 3 z )
L D L NL = γ U 0 T 0 2 β 2 π ln 2
M 2 = [ ψ 2 ψ P _ FIT 2 ] 2 dt ψ 4 dt
U P 0 _ SS = g 2 T P 0 _ SS 3 3 3 γ β 2 2 ,
ψ ( t ) = P P 0 exp ( 1 2 g z ) ψ ̂ ( t ) and γ ̂ = γ P P 0 exp ( g z ) ,
i ψ ̂ z = β 2 2 2 ψ ̂ t 2 γ ̂ ψ ̂ 2 ψ ̂ .
ψ ̂ P _ SPM = ψ ̂ P 0 exp ( i ψ ̂ P 0 2 0 z γ ̂ dz ) = ψ ̂ P 0 exp ( i ψ ̂ P 0 2 γ P P 0 exp ( g z ) 1 g )
C P _ SPM = 4 0 z γ ̂ dz T P 0 2 + C P 0 = 3 2 γ U P 0 T P 0 3 exp ( g z ) 1 g + C P 0
T P _ SPM = T P 0 and P P _ SPM = 3 2 4 U P 0 T P 0 exp ( g z )
f P C P T P 2 π .
ψ CP ( t ) J 1 ( 2 π f P t ) ( 2 π f P t ) ,
T CP _ SPM = 1 1.33 f P _ SPM and P PC _ SPM = 3 π 4 2 U P 0 Γ 2 ( 3 2 ) f P _ SPM exp ( g z )
A ̂ z = β 2 A ̂ t φ t + β 2 2 A ̂ 2 φ t 2 and φ z = β 2 2 { ( φ t ) 2 1 A ̂ 2 A ̂ t 2 } + γ ̂ A ̂ 2
P ̂ z = β 2 t ( P ̂ δω ) and δω z = β 2 2 δ ω 2 t γ ̂ P ̂ t
P ̂ P z = β 2 P ̂ P C P , 1 P ̂ P P ̂ P z + 1 T P T P z = 0 and C P z + β 2 C P 2 = 4 γ ̂ P ̂ P T P 2
2 T P z 2 = 3 2 U P 0 γ β 2 T P 2 exp ( g z ) , C P = 1 β 2 1 T P T P z and P P = 3 2 4 U P 0 T P exp ( g z )
f P 1 2 π β 2 T P z

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