Abstract

The nonlinear propagation of femtosecond pulses in photonic-crystal fibers is investigated theoretically without the use of the slowly varying envelope approximation. Low-intensity supercontinuum generation caused by fission of higher-order solitons into red-shifted fundamental solitons and blue-shifted nonsolitonic radiation is studied in a large range of fiber and pulse parameters. It is shown that phase matching of degenerate four-wave mixing can be achieved in an extremely broad frequency range from the IR to the UV. Spontaneous generation of new frequency components and parametric amplification by four-wave mixing as well as its possible overlap with soliton fission are studied in detail.

© 2002 Optical Society of America

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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef]
  23. M. Geissler, G. Tempea, A. Scrinzi, F. Krausz, and T. Brabec, “Light propagation in field-ionizing media: extreme nonlinear optics,” Phys. Rev. Lett. 83, 2930–2933 (1999).
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]

2001 (6)

2000 (7)

T. A. Birks, W. J. Wadsworth, and P. St. J. Russel, “Supercontinuum generation in tapered fibers,” Opt. Lett. 25, 1415–1417 (2000).
[CrossRef]

J. K. Ranka, R. S. Windeler, and A. J. Steinz, “Visible supercontinuum generation in an air–silica microstructure optical fiber with anomalous dispersion at 800 nm,” Opt. Lett. 25, 25–27 (2000).
[CrossRef]

J. C. Knight, J. Arriaga, and T. A. Birks, “Anomalous dispersion in photonic crystal fiber,” IEEE Photon. Technol. Lett. 12, 807–809 (2000).
[CrossRef]

V. P. Kalosha and J. Herrmann, “Self-phase modulation and compression of few-optical-cycle pulses,” Phys. Rev. A 62, 011804(R) (2000).
[CrossRef]

D. A. Jones, S. A. Diddams, J. K. Ranka, A. Stentz, R. S. Windeler, J. L. Hall, and S. T. Cundiff, “Carrier-envelope phase control of femtosecond mode-locked lasers and direct optical frequency synthesis,” Science 288, 635–639 (2000).
[CrossRef] [PubMed]

R. Holzwarth, Th. Udem, T. W. Hänsch, J. C. Knight, W. J. Wadsworth, and P. St. J. Russel, “Optical frequency synthesizer for precision spectroscopy,” Phys. Rev. Lett. 85, 2264 (2000).
[CrossRef] [PubMed]

V. P. Kalosha and J. Herrmann, “Phase relation, quasicontinuous spectra and subfemtosecond pulses in high-order stimulated Raman scattering with short-pulse excitation,” Phys. Rev. Lett. 85, 1226–1229 (2000).
[CrossRef] [PubMed]

1999 (2)

M. Geissler, G. Tempea, A. Scrinzi, F. Krausz, and T. Brabec, “Light propagation in field-ionizing media: extreme nonlinear optics,” Phys. Rev. Lett. 83, 2930–2933 (1999).
[CrossRef]

V. P. Kalosha and J. Herrmann, “Formation of optical subcycle pulses and full Maxwell–Bloch solitary waves by coherent propagation effects,” Phys. Rev. Lett. 83, 544–547 (1999).
[CrossRef]

1998 (1)

1997 (2)

T. A. Birks, J. C. Knight, and P. St. J. Russel, “Endlessly single-mode photonic-crystal fiber,” Opt. Lett. 22, 961–963 (1997).
[CrossRef] [PubMed]

T. Brabec and F. Krausz, “Nonlinear pulse propagation in the single-pulse regime,” Phys. Rev. Lett. 78, 3282–3285 (1997).
[CrossRef]

1996 (1)

1995 (2)

S. G. Murdoch, R. Leonhardt, and J. D. Harvey, “Polarization modulational instability in weakly birefringent fibers,” Opt. Lett. 20, 866–868 (1995).
[CrossRef] [PubMed]

J. N. Elgin, T. Brabec, and S. M. J. Kelly, “A perturbative theory of soliton propagation in the presence of third-order dispersion,” Opt. Commun. 114, 321–328 (1995).
[CrossRef]

1994 (1)

1992 (1)

1991 (1)

M. S. Sheik-Bahae, D. C. Hutchings, D. J. Hagan, and E. W. van Stryland, “Dispersion of bound electronic nonlinear refractive index,” IEEE J. Quantum Electron. 27, 1296–1309 (1991).
[CrossRef]

1988 (1)

1987 (3)

1979 (1)

R. K. Bulough, P. M. Jack, P. W. Kitchenside, and R. Saunders, “Solitons in laser physics,” Phys. Scr. 20, 364–381 (1979).
[CrossRef]

1970 (1)

R. R. Alfano and S. L. Shapiro, “Observation of self-phase modulation and small-scale filaments in crystal and glasses,” Phys. Rev. Lett. 24, 592–594 (1970).
[CrossRef]

1966 (1)

K. S. Yee, “Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media,” IEEE Trans. Antennas Propag. 14, 302–307 (1966).
[CrossRef]

1955 (1)

N. Akhmediev and M. Karlsson, “Cherenkov radiation emitted by solitons in optical fibers,” Phys. Rev. A 51, 2602–2607 (1955).
[CrossRef]

Akhmediev, N.

N. Akhmediev and M. Karlsson, “Cherenkov radiation emitted by solitons in optical fibers,” Phys. Rev. A 51, 2602–2607 (1955).
[CrossRef]

Alfano, R. R.

R. R. Alfano and S. L. Shapiro, “Observation of self-phase modulation and small-scale filaments in crystal and glasses,” Phys. Rev. Lett. 24, 592–594 (1970).
[CrossRef]

Arriaga, J.

J. C. Knight, J. Arriaga, and T. A. Birks, “Anomalous dispersion in photonic crystal fiber,” IEEE Photon. Technol. Lett. 12, 807–809 (2000).
[CrossRef]

Atkin, D. M.

Becker, P. C.

Birks, T. A.

Brabec, T.

M. Geissler, G. Tempea, A. Scrinzi, F. Krausz, and T. Brabec, “Light propagation in field-ionizing media: extreme nonlinear optics,” Phys. Rev. Lett. 83, 2930–2933 (1999).
[CrossRef]

T. Brabec and F. Krausz, “Nonlinear pulse propagation in the single-pulse regime,” Phys. Rev. Lett. 78, 3282–3285 (1997).
[CrossRef]

J. N. Elgin, T. Brabec, and S. M. J. Kelly, “A perturbative theory of soliton propagation in the presence of third-order dispersion,” Opt. Commun. 114, 321–328 (1995).
[CrossRef]

Brito Cruz, C. H.

Bulough, R. K.

R. K. Bulough, P. M. Jack, P. W. Kitchenside, and R. Saunders, “Solitons in laser physics,” Phys. Scr. 20, 364–381 (1979).
[CrossRef]

Chen, H. H.

Chen, Y. L.

Chudoba, C.

Coker, A.

Cundiff, S. T.

D. A. Jones, S. A. Diddams, J. K. Ranka, A. Stentz, R. S. Windeler, J. L. Hall, and S. T. Cundiff, “Carrier-envelope phase control of femtosecond mode-locked lasers and direct optical frequency synthesis,” Science 288, 635–639 (2000).
[CrossRef] [PubMed]

de Boeij, W.

Diddams, S. A.

D. A. Jones, S. A. Diddams, J. K. Ranka, A. Stentz, R. S. Windeler, J. L. Hall, and S. T. Cundiff, “Carrier-envelope phase control of femtosecond mode-locked lasers and direct optical frequency synthesis,” Science 288, 635–639 (2000).
[CrossRef] [PubMed]

Elgin, J. N.

J. N. Elgin, T. Brabec, and S. M. J. Kelly, “A perturbative theory of soliton propagation in the presence of third-order dispersion,” Opt. Commun. 114, 321–328 (1995).
[CrossRef]

Faldon, M. E.

Fiorentino, M.

Fork, R. L.

Fujimoto, J. G.

Geissler, M.

M. Geissler, G. Tempea, A. Scrinzi, F. Krausz, and T. Brabec, “Light propagation in field-ionizing media: extreme nonlinear optics,” Phys. Rev. Lett. 83, 2930–2933 (1999).
[CrossRef]

Ghanta, R. K.

Gouveia-Neto, A. S.

Hagan, D. J.

M. S. Sheik-Bahae, D. C. Hutchings, D. J. Hagan, and E. W. van Stryland, “Dispersion of bound electronic nonlinear refractive index,” IEEE J. Quantum Electron. 27, 1296–1309 (1991).
[CrossRef]

Hall, J. L.

D. A. Jones, S. A. Diddams, J. K. Ranka, A. Stentz, R. S. Windeler, J. L. Hall, and S. T. Cundiff, “Carrier-envelope phase control of femtosecond mode-locked lasers and direct optical frequency synthesis,” Science 288, 635–639 (2000).
[CrossRef] [PubMed]

Hänsch, T. W.

R. Holzwarth, Th. Udem, T. W. Hänsch, J. C. Knight, W. J. Wadsworth, and P. St. J. Russel, “Optical frequency synthesizer for precision spectroscopy,” Phys. Rev. Lett. 85, 2264 (2000).
[CrossRef] [PubMed]

Hartl, J.

Harvey, J. D.

Hasegawa, A.

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

Herrmann, J.

V. P. Kalosha and J. Herrmann, “Pulse compression without chirp control and frequency detuning by high-order coherent Raman scattering in impulsively excited media,” Opt. Lett. 26, 456–458 (2001).
[CrossRef]

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]

A. V. Husakou, V. P. Kalosha, and J. Herrmann, “Supercontinuum generation and subcycle pulse compression in hollow waveguides,” Opt. Lett. 26, 1022–1024 (2001).
[CrossRef]

V. P. Kalosha and J. Herrmann, “Phase relation, quasicontinuous spectra and subfemtosecond pulses in high-order stimulated Raman scattering with short-pulse excitation,” Phys. Rev. Lett. 85, 1226–1229 (2000).
[CrossRef] [PubMed]

V. P. Kalosha and J. Herrmann, “Self-phase modulation and compression of few-optical-cycle pulses,” Phys. Rev. A 62, 011804(R) (2000).
[CrossRef]

V. P. Kalosha and J. Herrmann, “Formation of optical subcycle pulses and full Maxwell–Bloch solitary waves by coherent propagation effects,” Phys. Rev. Lett. 83, 544–547 (1999).
[CrossRef]

Holzwarth, R.

R. Holzwarth, Th. Udem, T. W. Hänsch, J. C. Knight, W. J. Wadsworth, and P. St. J. Russel, “Optical frequency synthesizer for precision spectroscopy,” Phys. Rev. Lett. 85, 2264 (2000).
[CrossRef] [PubMed]

Husakou, A. V.

A. V. Husakou, V. P. Kalosha, and J. Herrmann, “Supercontinuum generation and subcycle pulse compression in hollow waveguides,” Opt. Lett. 26, 1022–1024 (2001).
[CrossRef]

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]

Hutchings, D. C.

M. S. Sheik-Bahae, D. C. Hutchings, D. J. Hagan, and E. W. van Stryland, “Dispersion of bound electronic nonlinear refractive index,” IEEE J. Quantum Electron. 27, 1296–1309 (1991).
[CrossRef]

Jack, P. M.

R. K. Bulough, P. M. Jack, P. W. Kitchenside, and R. Saunders, “Solitons in laser physics,” Phys. Scr. 20, 364–381 (1979).
[CrossRef]

Jones, D. A.

D. A. Jones, S. A. Diddams, J. K. Ranka, A. Stentz, R. S. Windeler, J. L. Hall, and S. T. Cundiff, “Carrier-envelope phase control of femtosecond mode-locked lasers and direct optical frequency synthesis,” Science 288, 635–639 (2000).
[CrossRef] [PubMed]

Kalosha, V. P.

A. V. Husakou, V. P. Kalosha, and J. Herrmann, “Supercontinuum generation and subcycle pulse compression in hollow waveguides,” Opt. Lett. 26, 1022–1024 (2001).
[CrossRef]

V. P. Kalosha and J. Herrmann, “Pulse compression without chirp control and frequency detuning by high-order coherent Raman scattering in impulsively excited media,” Opt. Lett. 26, 456–458 (2001).
[CrossRef]

V. P. Kalosha and J. Herrmann, “Self-phase modulation and compression of few-optical-cycle pulses,” Phys. Rev. A 62, 011804(R) (2000).
[CrossRef]

V. P. Kalosha and J. Herrmann, “Phase relation, quasicontinuous spectra and subfemtosecond pulses in high-order stimulated Raman scattering with short-pulse excitation,” Phys. Rev. Lett. 85, 1226–1229 (2000).
[CrossRef] [PubMed]

V. P. Kalosha and J. Herrmann, “Formation of optical subcycle pulses and full Maxwell–Bloch solitary waves by coherent propagation effects,” Phys. Rev. Lett. 83, 544–547 (1999).
[CrossRef]

Karlsson, M.

N. Akhmediev and M. Karlsson, “Cherenkov radiation emitted by solitons in optical fibers,” Phys. Rev. A 51, 2602–2607 (1955).
[CrossRef]

Kelly, S. M. J.

J. N. Elgin, T. Brabec, and S. M. J. Kelly, “A perturbative theory of soliton propagation in the presence of third-order dispersion,” Opt. Commun. 114, 321–328 (1995).
[CrossRef]

Kitchenside, P. W.

R. K. Bulough, P. M. Jack, P. W. Kitchenside, and R. Saunders, “Solitons in laser physics,” Phys. Scr. 20, 364–381 (1979).
[CrossRef]

Knight, J. C.

R. Holzwarth, Th. Udem, T. W. Hänsch, J. C. Knight, W. J. Wadsworth, and P. St. J. Russel, “Optical frequency synthesizer for precision spectroscopy,” Phys. Rev. Lett. 85, 2264 (2000).
[CrossRef] [PubMed]

J. C. Knight, J. Arriaga, and T. A. Birks, “Anomalous dispersion in photonic crystal fiber,” IEEE Photon. Technol. Lett. 12, 807–809 (2000).
[CrossRef]

T. A. Birks, J. C. Knight, and P. St. J. Russel, “Endlessly single-mode photonic-crystal fiber,” Opt. Lett. 22, 961–963 (1997).
[CrossRef] [PubMed]

J. C. Knight, T. A. Birks, P. St. J. Russel, and D. M. Atkin, “All-silica single-mode optical fiber with photonic-crystal cladding,” Opt. Lett. 21, 1547–1549 (1996).
[CrossRef] [PubMed]

Ko, T. H.

Kodama, Y.

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

Krausz, F.

M. Geissler, G. Tempea, A. Scrinzi, F. Krausz, and T. Brabec, “Light propagation in field-ionizing media: extreme nonlinear optics,” Phys. Rev. Lett. 83, 2930–2933 (1999).
[CrossRef]

T. Brabec and F. Krausz, “Nonlinear pulse propagation in the single-pulse regime,” Phys. Rev. Lett. 78, 3282–3285 (1997).
[CrossRef]

Kumar, P.

Lee, Y. C.

Leonhardt, R.

Li, Q.

Li, X. D.

Luo, S. Y.

Menyuk, C. R.

Mogilevtsev, D.

Murdoch, S. G.

Pan, W.

Pshenichnikov, M.

Ranka, J. K.

Rothenberg, J. E.

Russel, P. St. J.

Saunders, R.

R. K. Bulough, P. M. Jack, P. W. Kitchenside, and R. Saunders, “Solitons in laser physics,” Phys. Scr. 20, 364–381 (1979).
[CrossRef]

Scrinzi, A.

M. Geissler, G. Tempea, A. Scrinzi, F. Krausz, and T. Brabec, “Light propagation in field-ionizing media: extreme nonlinear optics,” Phys. Rev. Lett. 83, 2930–2933 (1999).
[CrossRef]

Shank, C. V.

Shapiro, S. L.

R. R. Alfano and S. L. Shapiro, “Observation of self-phase modulation and small-scale filaments in crystal and glasses,” Phys. Rev. Lett. 24, 592–594 (1970).
[CrossRef]

Sharping, J. E.

Sheik-Bahae, M. S.

M. S. Sheik-Bahae, D. C. Hutchings, D. J. Hagan, and E. W. van Stryland, “Dispersion of bound electronic nonlinear refractive index,” IEEE J. Quantum Electron. 27, 1296–1309 (1991).
[CrossRef]

Steinz, A. J.

Stentz, A.

D. A. Jones, S. A. Diddams, J. K. Ranka, A. Stentz, R. S. Windeler, J. L. Hall, and S. T. Cundiff, “Carrier-envelope phase control of femtosecond mode-locked lasers and direct optical frequency synthesis,” Science 288, 635–639 (2000).
[CrossRef] [PubMed]

Taylor, J. R.

Tempea, G.

M. Geissler, G. Tempea, A. Scrinzi, F. Krausz, and T. Brabec, “Light propagation in field-ionizing media: extreme nonlinear optics,” Phys. Rev. Lett. 83, 2930–2933 (1999).
[CrossRef]

Udem, Th.

R. Holzwarth, Th. Udem, T. W. Hänsch, J. C. Knight, W. J. Wadsworth, and P. St. J. Russel, “Optical frequency synthesizer for precision spectroscopy,” Phys. Rev. Lett. 85, 2264 (2000).
[CrossRef] [PubMed]

van Stryland, E. W.

M. S. Sheik-Bahae, D. C. Hutchings, D. J. Hagan, and E. W. van Stryland, “Dispersion of bound electronic nonlinear refractive index,” IEEE J. Quantum Electron. 27, 1296–1309 (1991).
[CrossRef]

Wadsworth, W. J.

T. A. Birks, W. J. Wadsworth, and P. St. J. Russel, “Supercontinuum generation in tapered fibers,” Opt. Lett. 25, 1415–1417 (2000).
[CrossRef]

R. Holzwarth, Th. Udem, T. W. Hänsch, J. C. Knight, W. J. Wadsworth, and P. St. J. Russel, “Optical frequency synthesizer for precision spectroscopy,” Phys. Rev. Lett. 85, 2264 (2000).
[CrossRef] [PubMed]

Wai, P. K. A.

Wiersma, D.

Windeler, R. S.

Yee, K. S.

K. S. Yee, “Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media,” IEEE Trans. Antennas Propag. 14, 302–307 (1966).
[CrossRef]

Zhang, J.

IEEE J. Quantum Electron. (2)

M. S. Sheik-Bahae, D. C. Hutchings, D. J. Hagan, and E. W. van Stryland, “Dispersion of bound electronic nonlinear refractive index,” IEEE J. Quantum Electron. 27, 1296–1309 (1991).
[CrossRef]

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

IEEE Photon. Technol. Lett. (1)

J. C. Knight, J. Arriaga, and T. A. Birks, “Anomalous dispersion in photonic crystal fiber,” IEEE Photon. Technol. Lett. 12, 807–809 (2000).
[CrossRef]

IEEE Trans. Antennas Propag. (1)

K. S. Yee, “Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media,” IEEE Trans. Antennas Propag. 14, 302–307 (1966).
[CrossRef]

Opt. Commun. (1)

J. N. Elgin, T. Brabec, and S. M. J. Kelly, “A perturbative theory of soliton propagation in the presence of third-order dispersion,” Opt. Commun. 114, 321–328 (1995).
[CrossRef]

Opt. Lett. (16)

J. K. Ranka, R. S. Windeler, and A. J. Steinz, “Visible supercontinuum generation in an air–silica microstructure optical fiber with anomalous dispersion at 800 nm,” Opt. Lett. 25, 25–27 (2000).
[CrossRef]

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Science (1)

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Other (3)

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R. Alfano, ed., The Supercontinuum Laser Source (Springer Verlag, New York, 1989).

G. P. Agrawal, Nonlinear Fiber Optics (Academic Press, New York, 1994).

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Figures (13)

Fig. 1
Fig. 1

Cross section of the PCF (a) with small air holes, (b) with large air holes, (c) a schematic of a tapered fiber.

Fig. 2
Fig. 2

Comparison of spectra by use of different propagation equations and PCF dispersion parameters: Spectra of a 15-fs, 40-TW/cm2 pulse after propagation of 0.5 mm in standard fiber as calculated by (a) the FME and (b) the reduced Maxwell equation are compared with the results from the full wave equation (dashed curves). (c) The effective mode area S1 (solid curve) and the nonlinear reduction factor α(ω) (dashed curve). (d) Dispersion parameter D(ω) for a PCF with Λ=1.5 µm, d=1.3 µm (curve 1) and for bulk silica (curve 2) together with phase matching for the weakest (curve 3) and the strongest (curve 4) solitons for Fig. 4.

Fig. 3
Fig. 3

Output pulse (a) shapes and (b) spectra for L=15 mm, I0=0.6 TW/cm2, τ0=100 fs, and different input frequencies as indicated. Initial spectra (dashed curves; scaled for clarity here and hereafter) are also presented.

Fig. 4
Fig. 4

Evolution of the output pulse (a) shape and (b) spectrum for ω0=0.85ωZD, I0=0.6 TW/cm2, and τ0=100 fs. Spectral phases are shown by the dotted curves.

Fig. 5
Fig. 5

Evolution of the output pulse (a) shape and (b) spectrum for ω=0.85ωZD, I0=0.6 TW/cm2, and τ0=17.5 fs.

Fig. 6
Fig. 6

Scheme of SC generation by fission of higher-order solitons.

Fig. 7
Fig. 7

(a), (b) Spectra, (c), (d) pulse shapes, and (e), (f) phase differences for (a), (c), (e) τ0=29 fs and (b), (d), (f) τ0=100 fs in a PCF with a 2.5-µm core diameter. The input wavelength is 850 nm.

Fig. 8
Fig. 8

Output pulse (a) shape and (b) spectrum for L=15 mm, I0=3.3 TW/cm2, τ0=10 fs, and ω0=0.92ωZD.

Fig. 9
Fig. 9

Phase matching for a 2.95-µm-diameter tapered fiber and different input frequencies ω0=2.03 fs-1 (curve 1), 2.26 fs-1 (curve 2), 2.5 fs-1 (curve 3). Phase-matched frequencies are determined as crossing points of the corresponding curve and the horizontal line at I=Ip. See text for explanations of curves.

Fig. 10
Fig. 10

Evolution of the output pulse (a) shape and (b) spectrum for ω0=ωZD, I0=2 TW/cm2, and τ0=200 fs in a 2.95-µm-diameter tapered fiber.

Fig. 11
Fig. 11

Output (a) spectra and (b) pulse shapes for L=3 mm, I0=8.7 TW/cm2, τ0=200 fs, and different input frequencies as indicated.

Fig. 12
Fig. 12

Output (a) spectra and (b) pulse shapes for L=10 mm, I0=8.7 TW/cm2, τ0=200 fs, and different input frequencies as indicated.

Fig. 13
Fig. 13

Output spectra for L=1 mm and (a) I0=27 TW/cm2, τ0=200 fs, and ω0=ωZD and (b) two pulses with I0,1=I0,2=12.4 TW/cm2, τ0,1=τ0,2=100 fs, ω0,1=1.1ωZD, and ω0,2=1.84ωZD.

Equations (21)

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E(r, t)=12 A(r, t)exp[iω0t-k(ω0)z]+c.c.,
Atω0|A|,
2z2+ΔE-1c2 2z2 E=-μ0 2t2 P,
2E(z, ω, k)z2+βNL2(ω)E(z, ω, k)=0,
βNL(ω, k)=ω2c2[1+χ(ω)]-k2+μ0ω2BNL(z, ω, k)1/2,
BNL=PNL(z, ω, k)E(z, ω, k).
E+z(z, k, ω)=iβNL(z, k, ω)E+(z, k, ω)
E-z(z, k, ω)=-iβNL(z, k, ω)E-(z, k, ω).
β(ω, k)k(ω)-k22k(ω)+μ0ω22k(ω)BNL.
E(r, ω)ξ=i ωc[n(ω)-1]E(r, ω)+i2k(ω)ΔE(r, ω)+iμ0ωc2n(ω) PNL(r, ω).
Eξ=-120c Pη.
Az+i2β 2Aη2-16β Aη3=iγ|A|2A+iω0 |A|2Aη,
ΔF+k(ω)2F=β(ω)2F
E˜(ξ, ω)ξ=i [n(ω)-1]ωcE˜(ξ, ω)+iμ0c ωα(ω)2n(ω)PNL(ξ, ω).
F(ω, ρ)J0[γ(ω)ρ]-N0[γ(ω)ρ] J1[γ(ω)R]N1[γ(ω)R].
κ(ω) I1[κ(ω)r]I0[κ(ω)r] J0[γ(ω)r]-N0[γ(ω)r] J1[γ(ω)R]N1[γ(ω)R]
=-γ(ω)J1[γ(ω)r]-N1[γ(ω)r] J1[γ(ω)R]N1[γ(ω)R].
χ3=χ3(0)1+2.8 ω2ωg2+.
ϕs(ωs)=n(ωs)ωsL/c+n2IωsL/(2c)-ωsL/vs,
ϕr(ω)=n(ω)ωL/c-ωL/vs.
Δkm+ΔkWG=β(ωs)ωs+β(ωI)ωI-2β(ωp)ωp

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