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Dispersive wave generation by solitons in microstructured optical fibers

Ilaria Cristiani, Riccardo Tediosi, Luca Tartara, and Vittorio Degiorgio

Open Access Open Access

Abstract

We study the nonlinear propagation of femtosecond pulses in the anomalous dispersion region of microstructured fibers, where soliton fission mechanisms play an important role. The experiment shows that the output spectrum contains, besides the infrared supercontinuum, a narrow-band 430-nm peak, carrying about one fourth of the input energy. By combining simulation and experiments, we explore the generation mechanism of the visible peak and describe its properties. The simulation demonstrates that the blue peak is generated only when the input pulse is so strongly compressed that the short-wavelength tail of the spectrum includes the wavelength predicted for the dispersive wave. In agreement with simulation, intensity-autocorrelation measurements show that the duration of the blue pulse is in the picosecond time range, and that, by increasing the input intensity, satellite pulses of lower intensity are generated.

©2004 Optical Society of America

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References

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  • Publication
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    [Crossref]
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    [Crossref] [PubMed]
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    [Crossref]
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    [Crossref]
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    [Crossref]
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2003 (5)

L. Tartara, I. Cristiani, and V. Degiorgio, “Blue light and infrared continuum generation by soliton fission in a microstructured fiber,” Appl. Phys. B 77, 307–311, (2003).
[Crossref]

X. Fang, N. Karasawa, R. Morita, R.S. Windeler, and M. Yamashita, “Nonlinear propagation of a-few-optical-cycle pulses in a photonic crystal fiber - Experimental and theoretical studies beyond the slowly varying - envelope approximation,” IEEE Photon. Technol. Lett. 15, 233–235, (2003)
[Crossref]

E. Sorokin, V.L. Kalashnikov, S. Naumov, J. Teipel, F. Warken, H. Giessen, and I. T. Sorokina, “Intra-and extra-cavityspectral broadening and continuum generation at 1.5 µm using compact low-energy femtosecond Cr:YAG laser,” Appl. Phys. B 77, 197–204, (2003).
[Crossref]

A. Efimov, A.J. Taylor, F.G. Omenetto, J.C. Knight, W.J. Wadsworth, and P.S. Russell, “Phase-matched third harmonic generation in microstructured fibers,” Opt. Express 11, 2567–2576 (2003), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-20-2567
[Crossref] [PubMed]

L. Tartara, I. Cristiani, V. Degiorgio, F. Carbone, D. Faccio, M. Romagnoli, and W Belardi, “Phase-matched nonlinear interactions in a holey fiber induced by infrared super-continuum generation,” Opt. Commun. 215, 191–197 (2003).
[Crossref]

2002 (5)

2001 (1)

A.V. Husakou and J. Herrmann, “Supercontinuum generation of higher - order solitons by fission in photonic crystal fibers,” Phys. Rev. Lett. 87, 203901, (2001).
[Crossref] [PubMed]

2000 (1)

1995 (1)

N. Akhmediev and M. Karlsson, “Cherenkov radiation emitted by solitons in optical fibers,” Phys. Rev. A 51, 2602–2607, (1995).
[Crossref] [PubMed]

1994 (1)

1993 (1)

1989 (1)

K. Blow and D. Wood, “Theoretical description of transient stimulated Raman scattering in fibers,” IEEE J. of Quantum Electronics 25, 2665–2673 (1989).
[Crossref]

1987 (1)

Y. Kodama and A. Hasegawa, “Nonlinear pulse propagation in a monomode dielectric waveguide,” IEEE J. Quantum Electron. QE-23, 510–524, (1987).
[Crossref]

1986 (1)

Agrawal, G.P.

G.P. Agrawal, Nonlinear Fiber Optics (Academic Press, San Diego, 1995).

Akhmediev, N.

N. Akhmediev and M. Karlsson, “Cherenkov radiation emitted by solitons in optical fibers,” Phys. Rev. A 51, 2602–2607, (1995).
[Crossref] [PubMed]

Belardi, W

L. Tartara, I. Cristiani, V. Degiorgio, F. Carbone, D. Faccio, M. Romagnoli, and W Belardi, “Phase-matched nonlinear interactions in a holey fiber induced by infrared super-continuum generation,” Opt. Commun. 215, 191–197 (2003).
[Crossref]

Belardi, W.

Blow, K.

K. Blow and D. Wood, “Theoretical description of transient stimulated Raman scattering in fibers,” IEEE J. of Quantum Electronics 25, 2665–2673 (1989).
[Crossref]

Broeng, J.

G. Genty, M. Lehtonen, H. Ludvigsen, J. Broeng, and M. Kaivola, “Spectral broadening of femtosecond pulses into continuum radiation in microstructured fibers,” Opt.Express 10, 1083–1098, (2002), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-20-1083.
[Crossref] [PubMed]

Carbone, F.

L. Tartara, I. Cristiani, V. Degiorgio, F. Carbone, D. Faccio, M. Romagnoli, and W Belardi, “Phase-matched nonlinear interactions in a holey fiber induced by infrared super-continuum generation,” Opt. Commun. 215, 191–197 (2003).
[Crossref]

Chau, A.H. L.

Coen, S.

Cristiani, I.

L. Tartara, I. Cristiani, V. Degiorgio, F. Carbone, D. Faccio, M. Romagnoli, and W Belardi, “Phase-matched nonlinear interactions in a holey fiber induced by infrared super-continuum generation,” Opt. Commun. 215, 191–197 (2003).
[Crossref]

L. Tartara, I. Cristiani, and V. Degiorgio, “Blue light and infrared continuum generation by soliton fission in a microstructured fiber,” Appl. Phys. B 77, 307–311, (2003).
[Crossref]

Degiorgio, V.

L. Tartara, I. Cristiani, and V. Degiorgio, “Blue light and infrared continuum generation by soliton fission in a microstructured fiber,” Appl. Phys. B 77, 307–311, (2003).
[Crossref]

L. Tartara, I. Cristiani, V. Degiorgio, F. Carbone, D. Faccio, M. Romagnoli, and W Belardi, “Phase-matched nonlinear interactions in a holey fiber induced by infrared super-continuum generation,” Opt. Commun. 215, 191–197 (2003).
[Crossref]

Dudley, J.M.

Efimov, A.

Eggleton, B.J.

Faccio, D.

L. Tartara, I. Cristiani, V. Degiorgio, F. Carbone, D. Faccio, M. Romagnoli, and W Belardi, “Phase-matched nonlinear interactions in a holey fiber induced by infrared super-continuum generation,” Opt. Commun. 215, 191–197 (2003).
[Crossref]

Fang, X.

X. Fang, N. Karasawa, R. Morita, R.S. Windeler, and M. Yamashita, “Nonlinear propagation of a-few-optical-cycle pulses in a photonic crystal fiber - Experimental and theoretical studies beyond the slowly varying - envelope approximation,” IEEE Photon. Technol. Lett. 15, 233–235, (2003)
[Crossref]

Genty, G.

G. Genty, M. Lehtonen, H. Ludvigsen, J. Broeng, and M. Kaivola, “Spectral broadening of femtosecond pulses into continuum radiation in microstructured fibers,” Opt.Express 10, 1083–1098, (2002), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-20-1083.
[Crossref] [PubMed]

Giessen, H.

E. Sorokin, V.L. Kalashnikov, S. Naumov, J. Teipel, F. Warken, H. Giessen, and I. T. Sorokina, “Intra-and extra-cavityspectral broadening and continuum generation at 1.5 µm using compact low-energy femtosecond Cr:YAG laser,” Appl. Phys. B 77, 197–204, (2003).
[Crossref]

Gordon, J.P.

Griebner, U.

J. Herrmann, U. Griebner, N. Zhavoronkov, A.V. Husakou, D. Nickel, J.C. Knight, W.J. Wadsworth, P.St.J. Russell, and J. Korn, “Experimental evidence for supercontinuum generation by fission of higher - order solitons in photonic crystal fibers,” Phys. Rev. Lett. 88, 173901, (2002).
[Crossref] [PubMed]

Grossard, N.

Harvey, J.D.

Hasegawa, A.

Y. Kodama and A. Hasegawa, “Nonlinear pulse propagation in a monomode dielectric waveguide,” IEEE J. Quantum Electron. QE-23, 510–524, (1987).
[Crossref]

Herrmann, J.

J. Herrmann, U. Griebner, N. Zhavoronkov, A.V. Husakou, D. Nickel, J.C. Knight, W.J. Wadsworth, P.St.J. Russell, and J. Korn, “Experimental evidence for supercontinuum generation by fission of higher - order solitons in photonic crystal fibers,” Phys. Rev. Lett. 88, 173901, (2002).
[Crossref] [PubMed]

A.V. Husakou and J. Herrmann, “Supercontinuum generation of higher - order solitons by fission in photonic crystal fibers,” Phys. Rev. Lett. 87, 203901, (2001).
[Crossref] [PubMed]

Husakou, A.V.

J. Herrmann, U. Griebner, N. Zhavoronkov, A.V. Husakou, D. Nickel, J.C. Knight, W.J. Wadsworth, P.St.J. Russell, and J. Korn, “Experimental evidence for supercontinuum generation by fission of higher - order solitons in photonic crystal fibers,” Phys. Rev. Lett. 88, 173901, (2002).
[Crossref] [PubMed]

A.V. Husakou and J. Herrmann, “Supercontinuum generation of higher - order solitons by fission in photonic crystal fibers,” Phys. Rev. Lett. 87, 203901, (2001).
[Crossref] [PubMed]

Kaivola, M.

G. Genty, M. Lehtonen, H. Ludvigsen, J. Broeng, and M. Kaivola, “Spectral broadening of femtosecond pulses into continuum radiation in microstructured fibers,” Opt.Express 10, 1083–1098, (2002), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-20-1083.
[Crossref] [PubMed]

Kalashnikov, V.L.

E. Sorokin, V.L. Kalashnikov, S. Naumov, J. Teipel, F. Warken, H. Giessen, and I. T. Sorokina, “Intra-and extra-cavityspectral broadening and continuum generation at 1.5 µm using compact low-energy femtosecond Cr:YAG laser,” Appl. Phys. B 77, 197–204, (2003).
[Crossref]

Karasawa, N.

X. Fang, N. Karasawa, R. Morita, R.S. Windeler, and M. Yamashita, “Nonlinear propagation of a-few-optical-cycle pulses in a photonic crystal fiber - Experimental and theoretical studies beyond the slowly varying - envelope approximation,” IEEE Photon. Technol. Lett. 15, 233–235, (2003)
[Crossref]

Karlsson, M.

N. Akhmediev and M. Karlsson, “Cherenkov radiation emitted by solitons in optical fibers,” Phys. Rev. A 51, 2602–2607, (1995).
[Crossref] [PubMed]

Knight, J.C.

Kodama, Y.

Y. Kodama, M. Romagnoli, S. Wabnitz, and M. Midrio, “Role of third-order dispersion on soliton instabilities and interactions in optical fibers,” Opt. Lett. 19, 165–167, (1994).
[Crossref] [PubMed]

Y. Kodama and A. Hasegawa, “Nonlinear pulse propagation in a monomode dielectric waveguide,” IEEE J. Quantum Electron. QE-23, 510–524, (1987).
[Crossref]

Korn, J.

J. Herrmann, U. Griebner, N. Zhavoronkov, A.V. Husakou, D. Nickel, J.C. Knight, W.J. Wadsworth, P.St.J. Russell, and J. Korn, “Experimental evidence for supercontinuum generation by fission of higher - order solitons in photonic crystal fibers,” Phys. Rev. Lett. 88, 173901, (2002).
[Crossref] [PubMed]

Lehtonen, M.

G. Genty, M. Lehtonen, H. Ludvigsen, J. Broeng, and M. Kaivola, “Spectral broadening of femtosecond pulses into continuum radiation in microstructured fibers,” Opt.Express 10, 1083–1098, (2002), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-20-1083.
[Crossref] [PubMed]

Leonhardt, R.

Ludvigsen, H.

G. Genty, M. Lehtonen, H. Ludvigsen, J. Broeng, and M. Kaivola, “Spectral broadening of femtosecond pulses into continuum radiation in microstructured fibers,” Opt.Express 10, 1083–1098, (2002), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-20-1083.
[Crossref] [PubMed]

Maillotte, H.

Malinowski, A.

Matera, F.

Mecozzi, A.

Midrio, M.

Monro, T.M.

Morita, R.

X. Fang, N. Karasawa, R. Morita, R.S. Windeler, and M. Yamashita, “Nonlinear propagation of a-few-optical-cycle pulses in a photonic crystal fiber - Experimental and theoretical studies beyond the slowly varying - envelope approximation,” IEEE Photon. Technol. Lett. 15, 233–235, (2003)
[Crossref]

Naumov, S.

E. Sorokin, V.L. Kalashnikov, S. Naumov, J. Teipel, F. Warken, H. Giessen, and I. T. Sorokina, “Intra-and extra-cavityspectral broadening and continuum generation at 1.5 µm using compact low-energy femtosecond Cr:YAG laser,” Appl. Phys. B 77, 197–204, (2003).
[Crossref]

Nickel, D.

J. Herrmann, U. Griebner, N. Zhavoronkov, A.V. Husakou, D. Nickel, J.C. Knight, W.J. Wadsworth, P.St.J. Russell, and J. Korn, “Experimental evidence for supercontinuum generation by fission of higher - order solitons in photonic crystal fibers,” Phys. Rev. Lett. 88, 173901, (2002).
[Crossref] [PubMed]

Omenetto, F.G.

Piper, A.

Price, J.H.V.

Provino, L.

Ranka, J. K.

Richardson, D. J.

Romagnoli, M.

Russell, P.S.

Russell, P.St.J.

S. Coen, A.H. L. Chau, R. Leonhardt, J.D. Harvey, J.C. Knight, W.J. Wadsworth, and P.St.J. Russell, “Supercontinuum generation by stimulated Raman scattering and parametric four-wave-mixing in photonic crystal fibers,” J. Opt. Soc. Am. B 19, 753–763, (2002).
[Crossref]

J. Herrmann, U. Griebner, N. Zhavoronkov, A.V. Husakou, D. Nickel, J.C. Knight, W.J. Wadsworth, P.St.J. Russell, and J. Korn, “Experimental evidence for supercontinuum generation by fission of higher - order solitons in photonic crystal fibers,” Phys. Rev. Lett. 88, 173901, (2002).
[Crossref] [PubMed]

Settembre, M.

Sorokin, E.

E. Sorokin, V.L. Kalashnikov, S. Naumov, J. Teipel, F. Warken, H. Giessen, and I. T. Sorokina, “Intra-and extra-cavityspectral broadening and continuum generation at 1.5 µm using compact low-energy femtosecond Cr:YAG laser,” Appl. Phys. B 77, 197–204, (2003).
[Crossref]

Sorokina, I. T.

E. Sorokin, V.L. Kalashnikov, S. Naumov, J. Teipel, F. Warken, H. Giessen, and I. T. Sorokina, “Intra-and extra-cavityspectral broadening and continuum generation at 1.5 µm using compact low-energy femtosecond Cr:YAG laser,” Appl. Phys. B 77, 197–204, (2003).
[Crossref]

Stentz, A. J.

Tartara, L.

L. Tartara, I. Cristiani, and V. Degiorgio, “Blue light and infrared continuum generation by soliton fission in a microstructured fiber,” Appl. Phys. B 77, 307–311, (2003).
[Crossref]

L. Tartara, I. Cristiani, V. Degiorgio, F. Carbone, D. Faccio, M. Romagnoli, and W Belardi, “Phase-matched nonlinear interactions in a holey fiber induced by infrared super-continuum generation,” Opt. Commun. 215, 191–197 (2003).
[Crossref]

Taylor, A.J.

Teipel, J.

E. Sorokin, V.L. Kalashnikov, S. Naumov, J. Teipel, F. Warken, H. Giessen, and I. T. Sorokina, “Intra-and extra-cavityspectral broadening and continuum generation at 1.5 µm using compact low-energy femtosecond Cr:YAG laser,” Appl. Phys. B 77, 197–204, (2003).
[Crossref]

Wabnitz, S.

Wadsworth, W.J.

Warken, F.

E. Sorokin, V.L. Kalashnikov, S. Naumov, J. Teipel, F. Warken, H. Giessen, and I. T. Sorokina, “Intra-and extra-cavityspectral broadening and continuum generation at 1.5 µm using compact low-energy femtosecond Cr:YAG laser,” Appl. Phys. B 77, 197–204, (2003).
[Crossref]

Windeler, R. S.

Windeler, R.S.

X. Fang, N. Karasawa, R. Morita, R.S. Windeler, and M. Yamashita, “Nonlinear propagation of a-few-optical-cycle pulses in a photonic crystal fiber - Experimental and theoretical studies beyond the slowly varying - envelope approximation,” IEEE Photon. Technol. Lett. 15, 233–235, (2003)
[Crossref]

J.M. Dudley, L. Provino, N. Grossard, H. Maillotte, R.S. Windeler, B.J. Eggleton, and S. Coen, “Supercontinuum generation in air-silica microstructured fibers with nanosecond and femtosecond pulse pumping,” J. Opt. Soc. Am. B 19, 765–771, (2002).
[Crossref]

Wood, D.

K. Blow and D. Wood, “Theoretical description of transient stimulated Raman scattering in fibers,” IEEE J. of Quantum Electronics 25, 2665–2673 (1989).
[Crossref]

Yamashita, M.

X. Fang, N. Karasawa, R. Morita, R.S. Windeler, and M. Yamashita, “Nonlinear propagation of a-few-optical-cycle pulses in a photonic crystal fiber - Experimental and theoretical studies beyond the slowly varying - envelope approximation,” IEEE Photon. Technol. Lett. 15, 233–235, (2003)
[Crossref]

Zhavoronkov, N.

J. Herrmann, U. Griebner, N. Zhavoronkov, A.V. Husakou, D. Nickel, J.C. Knight, W.J. Wadsworth, P.St.J. Russell, and J. Korn, “Experimental evidence for supercontinuum generation by fission of higher - order solitons in photonic crystal fibers,” Phys. Rev. Lett. 88, 173901, (2002).
[Crossref] [PubMed]

Appl. Phys. B (2)

L. Tartara, I. Cristiani, and V. Degiorgio, “Blue light and infrared continuum generation by soliton fission in a microstructured fiber,” Appl. Phys. B 77, 307–311, (2003).
[Crossref]

E. Sorokin, V.L. Kalashnikov, S. Naumov, J. Teipel, F. Warken, H. Giessen, and I. T. Sorokina, “Intra-and extra-cavityspectral broadening and continuum generation at 1.5 µm using compact low-energy femtosecond Cr:YAG laser,” Appl. Phys. B 77, 197–204, (2003).
[Crossref]

IEEE J. of Quantum Electronics (1)

K. Blow and D. Wood, “Theoretical description of transient stimulated Raman scattering in fibers,” IEEE J. of Quantum Electronics 25, 2665–2673 (1989).
[Crossref]

IEEE J. Quantum Electron. (1)

Y. Kodama and A. Hasegawa, “Nonlinear pulse propagation in a monomode dielectric waveguide,” IEEE J. Quantum Electron. QE-23, 510–524, (1987).
[Crossref]

IEEE Photon. Technol. Lett. (1)

X. Fang, N. Karasawa, R. Morita, R.S. Windeler, and M. Yamashita, “Nonlinear propagation of a-few-optical-cycle pulses in a photonic crystal fiber - Experimental and theoretical studies beyond the slowly varying - envelope approximation,” IEEE Photon. Technol. Lett. 15, 233–235, (2003)
[Crossref]

J. Opt. Soc. Am. B (2)

Opt. Commun. (1)

L. Tartara, I. Cristiani, V. Degiorgio, F. Carbone, D. Faccio, M. Romagnoli, and W Belardi, “Phase-matched nonlinear interactions in a holey fiber induced by infrared super-continuum generation,” Opt. Commun. 215, 191–197 (2003).
[Crossref]

Opt. Express (2)

Opt. Lett. (4)

Opt.Express (1)

G. Genty, M. Lehtonen, H. Ludvigsen, J. Broeng, and M. Kaivola, “Spectral broadening of femtosecond pulses into continuum radiation in microstructured fibers,” Opt.Express 10, 1083–1098, (2002), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-20-1083.
[Crossref] [PubMed]

Phys. Rev. A (1)

N. Akhmediev and M. Karlsson, “Cherenkov radiation emitted by solitons in optical fibers,” Phys. Rev. A 51, 2602–2607, (1995).
[Crossref] [PubMed]

Phys. Rev. Lett. (2)

A.V. Husakou and J. Herrmann, “Supercontinuum generation of higher - order solitons by fission in photonic crystal fibers,” Phys. Rev. Lett. 87, 203901, (2001).
[Crossref] [PubMed]

J. Herrmann, U. Griebner, N. Zhavoronkov, A.V. Husakou, D. Nickel, J.C. Knight, W.J. Wadsworth, P.St.J. Russell, and J. Korn, “Experimental evidence for supercontinuum generation by fission of higher - order solitons in photonic crystal fibers,” Phys. Rev. Lett. 88, 173901, (2002).
[Crossref] [PubMed]

Other (1)

G.P. Agrawal, Nonlinear Fiber Optics (Academic Press, San Diego, 1995).

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Figures (9)
Fig. 1.
Fig. 1. Output spectrum showing highly efficient visible light generation in a 40-cm MF. See Ref. 9 for experimental details.

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Fig. 2.
Fig. 2. Phase matching condition for the dispersive wave. The red line refers to the case in which the nonlinear dephasing is taken into account (Po =5 kW)

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Fig. 3.
Fig. 3. Experimental output spectrum obtained using different pump wavelengths λ0 =810 nm (blue line), λ0 =890 nm (green line) and λ0 =760 nm (red line)

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Fig. 4.
Fig. 4. Comparison between the experimental output spectrum (pump wavelengths λ0=810 nm, fiber length L=40 cm) and the simulation results

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Fig. 5.
Fig. 5. Picture of the first centimeters of fiber. The input pulse generates blue light after a very short propagation distance

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Fig. 6.
Fig. 6. Spectrum evolution of a fourth-order-soliton on half-soliton period. Coupling between dispersive radiation and soliton occurs in correspondence with the first temporal contraction.

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Fig. 7.
Fig. 7. (a) Temporal evolution of the launched pulse on half soliton period; (b) Corresponding spectral evolution; (c) Temporal behaviour of the dispersive wave, by filtering the complete spectrum 150 <Ω< 200 THz.

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Fig. 8.
Fig. 8. Temporal behaviour of the dispersive radiation obtained by numerically filtering the output spectrum of a fifth order soliton at different fiber length. The plots are presented in log-scale to highlight the presence of the satellite pulses that have an intensity much lower than that of the main pulse, because the simulation is performed at low intensity.

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Fig. 9.
Fig. 9. Measured intensity autocorrelation of the dispersive radiation. a) L=33 cm and the input power P=90 mW, b) L=33 cm and P=50 mW, c) L=18 cm and P=50 mW.

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Equations (13)

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Δ κ = β ( ω ) − β ( ω o ) − ( ω − ω o ) ⁄ u g − γ P o = 0
β ( ω ) = ∑ k = 0 ∞ β k k ! Ω k
β ( ω ) = ∑ n ≥ 2 ∞ β k n ! Ω n − 2 ≅ 0
∂ A ∂ z + α 2 A − ∑ n ≥ 2 i n + 1 n ! β n ∂ n A ∂ T n = i γ ( 1 + i ω 0 ∂ ∂ T ) A · ∫ − ∞ + ∞ R ( T ' ) ∣ A ( z , T − T ' ) ∣ 2 s T '
A ( 0 , T ) = A o sech ⁡ ( T T o )
T = t − z u g ( ω o )
R ( T ) = ( 1 − f r ) δ ( T ) + f r h r ( T )
∂ A ∂ z = i γ ( 1 + i ω o ∂ ∂ T ) A · [ R ( T ) * ∣ A o ( z , T ) ∣ 2 ]
A ( z , T ) = A o exp [ i γ ( R ( T ) * ∣ A o ( z , T ) ∣ 2 ) z ]
V ( z , T ) = A ( z , T ) exp [ i γ ( R ( T ) * ∣ A o ∣ 2 ) ( z − z o ) ]
∂ V ∂ z = i γ V · R ( T ) * ( ∣ V o ∣ 2 + ∣ V ∣ 2 ) − γ 1 ω o ∂ ∂ T ( V · R ( T ) * ∣ V ∣ 2 )
∂ V ∂ z = i γ V · ℑ − 1 { R ( Ω ) · ℑ ( ∣ V o ∣ 2 + ∣ V ∣ 2 ) } − γ 1 ω o ∂ ∂ T ( V · ℑ − 1 { R ( Ω ) · ℑ ( ∣ V ∣ 2 ) } )
N 2 = γ P o T o 2 ∣ β 2 ∣

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