Abstract

We studied experimentally and theoretically the pulse compression using a zero-dispersion photonic crystal fiber in order to optimize the pulse duration and pulse shape. 20.3-fs pulses centered at 1070 nm have been produced using a diode-pumped system based on Yb:SYS crystal. The limitations such as pre-pulse amplitude or solitonic fission have also been studied.

© 2004 Optical Society of America

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    [CrossRef]
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  22. X. Gu, M. Kimmel, A. P. Shreenath, R. Trebino, J. M. Dudley, S. Coen, and R. S. Windeler, “Experimental studies of the coherence of microstructure-fiber supercontinuum,” Opt. Express 11, 2997–2703 (2003).
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2004 (1)

2003 (5)

2002 (4)

2001 (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]

B.R. Washburn, S.E. Ralph, P.A. Lacourt, J.M. Dudley, W.T. Rhodes, R.S. Windeler, and S. Coen, “Tunable near-infrared femtosecond soliton generation in photonic crystal fibres,” Electron. Lett. 37, 1510–1512 (2001).
[CrossRef]

1999 (2)

C. Hönninger, R. Paschotta, M. Graf, F. Morier-Genoud, G. Zhang, M. Moser, S. Biswal, J. Nees, A. Braun, G. Mourou, I. Johannsen, A. Giesen, W. Seeber, and U. Keller, “Ultrafast ytterbium-doped bulk laser amplifiers,” Appl. Phys B,  69, 3–17 (1999).
[CrossRef]

C. M.J. Gander, R. McBride, J.D.C. Jones, D. Mogilevstev, T.A. Birks, J.C. Knight, and P.St.J. Russel, “Experimental measurement of group velocity dispersion in photonic crystal fiber,” Electon. Lett. 35, 63–64 (1999).
[CrossRef]

1997 (2)

M.E. Fermann, A. Galvanauskas, G. Sucha, and D. Harter, “Ultrafast pulse sources based on multi-mode optical fibers,” Appl. Phys. B 65, 259–275 (1997).
[CrossRef]

R. Trebino, K. W. DeLong, D. N. Fittinghoff, J. N. Sweetser, M. A. Krumbugel, B. A. Richman, and D. J. Kane, “Measuring ultrashort laser pulses in the time-frequency domain using frequency-resolved optical gating,” Rev. Sci. Instrum. 68, 3277–3295 (1997).
[CrossRef]

1984 (2)

Agrawal, G. P.

G. P. Agrawal, Nonlinear fiber optics, (Academic press, Inc.1994).

Andersen, T. V.

Aschwanden, A.

Bagget, J.

Balembois, F.

Birks, T.A.

C. M.J. Gander, R. McBride, J.D.C. Jones, D. Mogilevstev, T.A. Birks, J.C. Knight, and P.St.J. Russel, “Experimental measurement of group velocity dispersion in photonic crystal fiber,” Electon. Lett. 35, 63–64 (1999).
[CrossRef]

Biswal, S.

C. Hönninger, R. Paschotta, M. Graf, F. Morier-Genoud, G. Zhang, M. Moser, S. Biswal, J. Nees, A. Braun, G. Mourou, I. Johannsen, A. Giesen, W. Seeber, and U. Keller, “Ultrafast ytterbium-doped bulk laser amplifiers,” Appl. Phys B,  69, 3–17 (1999).
[CrossRef]

Braun, A.

C. Hönninger, R. Paschotta, M. Graf, F. Morier-Genoud, G. Zhang, M. Moser, S. Biswal, J. Nees, A. Braun, G. Mourou, I. Johannsen, A. Giesen, W. Seeber, and U. Keller, “Ultrafast ytterbium-doped bulk laser amplifiers,” Appl. Phys B,  69, 3–17 (1999).
[CrossRef]

Brunner, F.

Chénais, S.

Coen, S.

X. Gu, M. Kimmel, A. P. Shreenath, R. Trebino, J. M. Dudley, S. Coen, and R. S. Windeler, “Experimental studies of the coherence of microstructure-fiber supercontinuum,” Opt. Express 11, 2997–2703 (2003).
[CrossRef]

J. M. Dudley, X. Gu, L. Xu, M. Kimmel, E. Zeek, P. O’Shea, R. Trebino, S. Coen, and R. S. Windeler, “Cross-correlation frequency resolved optical gating analysis of broadband continuum generation in photonic crystal fiber: simulations and experiments,” Opt. Express 10,. 1215–1221 (2002).
[CrossRef] [PubMed]

B.R. Washburn, S.E. Ralph, P.A. Lacourt, J.M. Dudley, W.T. Rhodes, R.S. Windeler, and S. Coen, “Tunable near-infrared femtosecond soliton generation in photonic crystal fibres,” Electron. Lett. 37, 1510–1512 (2001).
[CrossRef]

J. Dudley and S. Coen “Coherence properties of supercontinuum spectra generated in photonic crystal and tapered optical fibers,” Opt. Lett.27, 1180–82 (2002).J. Dudley and S. Coen, “Coherence properties of supercontinuum spectra generated in photonic crystal and tapered optical fibers,” Opt. Lett.27, 1180–82 (2002).
[CrossRef]

J. Dudley and S. Coen “Coherence properties of supercontinuum spectra generated in photonic crystal and tapered optical fibers,” Opt. Lett.27, 1180–82 (2002).J. Dudley and S. Coen, “Coherence properties of supercontinuum spectra generated in photonic crystal and tapered optical fibers,” Opt. Lett.27, 1180–82 (2002).
[CrossRef]

Cormack, I.G.

I.G. Cormack, D.T. Reid, W.J. Wadsworth, J.C. Knight, and P.ST.J. Russel, “Observation of soliton self-frequency shift in photonic crystal fibre,” Electron. Lett. 38, 167–168 (2002).
[CrossRef]

DeLong, K. W.

R. Trebino, K. W. DeLong, D. N. Fittinghoff, J. N. Sweetser, M. A. Krumbugel, B. A. Richman, and D. J. Kane, “Measuring ultrashort laser pulses in the time-frequency domain using frequency-resolved optical gating,” Rev. Sci. Instrum. 68, 3277–3295 (1997).
[CrossRef]

Dhellemmes, S.

Druon, F.

Dudley, J.

J. Dudley and S. Coen “Coherence properties of supercontinuum spectra generated in photonic crystal and tapered optical fibers,” Opt. Lett.27, 1180–82 (2002).J. Dudley and S. Coen, “Coherence properties of supercontinuum spectra generated in photonic crystal and tapered optical fibers,” Opt. Lett.27, 1180–82 (2002).
[CrossRef]

J. Dudley and S. Coen “Coherence properties of supercontinuum spectra generated in photonic crystal and tapered optical fibers,” Opt. Lett.27, 1180–82 (2002).J. Dudley and S. Coen, “Coherence properties of supercontinuum spectra generated in photonic crystal and tapered optical fibers,” Opt. Lett.27, 1180–82 (2002).
[CrossRef]

Dudley, J. M.

Dudley, J.M.

B.R. Washburn, S.E. Ralph, P.A. Lacourt, J.M. Dudley, W.T. Rhodes, R.S. Windeler, and S. Coen, “Tunable near-infrared femtosecond soliton generation in photonic crystal fibres,” Electron. Lett. 37, 1510–1512 (2001).
[CrossRef]

Fermann, M.E.

M.E. Fermann, A. Galvanauskas, G. Sucha, and D. Harter, “Ultrafast pulse sources based on multi-mode optical fibers,” Appl. Phys. B 65, 259–275 (1997).
[CrossRef]

Fittinghoff, D. N.

R. Trebino, K. W. DeLong, D. N. Fittinghoff, J. N. Sweetser, M. A. Krumbugel, B. A. Richman, and D. J. Kane, “Measuring ultrashort laser pulses in the time-frequency domain using frequency-resolved optical gating,” Rev. Sci. Instrum. 68, 3277–3295 (1997).
[CrossRef]

Fork, R. L.

Furasawa, K.

Furusawa, K.

Galvanauskas, A.

M.E. Fermann, A. Galvanauskas, G. Sucha, and D. Harter, “Ultrafast pulse sources based on multi-mode optical fibers,” Appl. Phys. B 65, 259–275 (1997).
[CrossRef]

Gander, C. M.J.

C. M.J. Gander, R. McBride, J.D.C. Jones, D. Mogilevstev, T.A. Birks, J.C. Knight, and P.St.J. Russel, “Experimental measurement of group velocity dispersion in photonic crystal fiber,” Electon. Lett. 35, 63–64 (1999).
[CrossRef]

Gaumé, R.

Georges, P.

Giesen, A.

C. Hönninger, R. Paschotta, M. Graf, F. Morier-Genoud, G. Zhang, M. Moser, S. Biswal, J. Nees, A. Braun, G. Mourou, I. Johannsen, A. Giesen, W. Seeber, and U. Keller, “Ultrafast ytterbium-doped bulk laser amplifiers,” Appl. Phys B,  69, 3–17 (1999).
[CrossRef]

Gordon, J. P.

Graf, M.

C. Hönninger, R. Paschotta, M. Graf, F. Morier-Genoud, G. Zhang, M. Moser, S. Biswal, J. Nees, A. Braun, G. Mourou, I. Johannsen, A. Giesen, W. Seeber, and U. Keller, “Ultrafast ytterbium-doped bulk laser amplifiers,” Appl. Phys B,  69, 3–17 (1999).
[CrossRef]

Gu, X.

Hansen, K. P.

Hansen, K.P.

Harter, D.

M.E. Fermann, A. Galvanauskas, G. Sucha, and D. Harter, “Ultrafast pulse sources based on multi-mode optical fibers,” Appl. Phys. B 65, 259–275 (1997).
[CrossRef]

Haumesser, P.H.

Herrmann, J.

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]

Hilligsøe, K.M.

Honninger, C.

Hönninger, C.

C. Hönninger, R. Paschotta, M. Graf, F. Morier-Genoud, G. Zhang, M. Moser, S. Biswal, J. Nees, A. Braun, G. Mourou, I. Johannsen, A. Giesen, W. Seeber, and U. Keller, “Ultrafast ytterbium-doped bulk laser amplifiers,” Appl. Phys B,  69, 3–17 (1999).
[CrossRef]

Hring, R.

Husakou, A. V.

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]

Innerhofer, E.

Jasapara, J.

J.W. Nicholson, J. Jasapara, W. Rudolph, F.G. Omenetto, and A.J. Taylor, “Full-f ield characterization of femtosecond pulses by spectrum and cross-correlation measurements,” Opt. Lett.1774–76 (1999) and 138 (2000).
[CrossRef]

Johannsen, I.

C. Hönninger, R. Paschotta, M. Graf, F. Morier-Genoud, G. Zhang, M. Moser, S. Biswal, J. Nees, A. Braun, G. Mourou, I. Johannsen, A. Giesen, W. Seeber, and U. Keller, “Ultrafast ytterbium-doped bulk laser amplifiers,” Appl. Phys B,  69, 3–17 (1999).
[CrossRef]

Jones, J.D.C.

C. M.J. Gander, R. McBride, J.D.C. Jones, D. Mogilevstev, T.A. Birks, J.C. Knight, and P.St.J. Russel, “Experimental measurement of group velocity dispersion in photonic crystal fiber,” Electon. Lett. 35, 63–64 (1999).
[CrossRef]

Kane, D. J.

R. Trebino, K. W. DeLong, D. N. Fittinghoff, J. N. Sweetser, M. A. Krumbugel, B. A. Richman, and D. J. Kane, “Measuring ultrashort laser pulses in the time-frequency domain using frequency-resolved optical gating,” Rev. Sci. Instrum. 68, 3277–3295 (1997).
[CrossRef]

Keiding, S.

Keller, U.

Kimmel, M.

Knight, J.C.

I.G. Cormack, D.T. Reid, W.J. Wadsworth, J.C. Knight, and P.ST.J. Russel, “Observation of soliton self-frequency shift in photonic crystal fibre,” Electron. Lett. 38, 167–168 (2002).
[CrossRef]

C. M.J. Gander, R. McBride, J.D.C. Jones, D. Mogilevstev, T.A. Birks, J.C. Knight, and P.St.J. Russel, “Experimental measurement of group velocity dispersion in photonic crystal fiber,” Electon. Lett. 35, 63–64 (1999).
[CrossRef]

Kristiansen, R.

Krumbugel, M. A.

R. Trebino, K. W. DeLong, D. N. Fittinghoff, J. N. Sweetser, M. A. Krumbugel, B. A. Richman, and D. J. Kane, “Measuring ultrashort laser pulses in the time-frequency domain using frequency-resolved optical gating,” Rev. Sci. Instrum. 68, 3277–3295 (1997).
[CrossRef]

Kumkar, M.

Lacourt, P.A.

B.R. Washburn, S.E. Ralph, P.A. Lacourt, J.M. Dudley, W.T. Rhodes, R.S. Windeler, and S. Coen, “Tunable near-infrared femtosecond soliton generation in photonic crystal fibres,” Electron. Lett. 37, 1510–1512 (2001).
[CrossRef]

Larat, C.

Lefort, L.

Lucas-Leclin, G.

Martinez, O. E.

McBride, R.

C. M.J. Gander, R. McBride, J.D.C. Jones, D. Mogilevstev, T.A. Birks, J.C. Knight, and P.St.J. Russel, “Experimental measurement of group velocity dispersion in photonic crystal fiber,” Electon. Lett. 35, 63–64 (1999).
[CrossRef]

McConnell, G.

G. McConnell and E. Riis, “Ultrashort pulse compression using photonic crystal fiber,” Appl. Phys. B557–563 (2004)
[CrossRef]

Mogilevstev, D.

C. M.J. Gander, R. McBride, J.D.C. Jones, D. Mogilevstev, T.A. Birks, J.C. Knight, and P.St.J. Russel, “Experimental measurement of group velocity dispersion in photonic crystal fiber,” Electon. Lett. 35, 63–64 (1999).
[CrossRef]

Mølmer, K.

Monro, T.

Monro, T.M.

Morier-Genoud, F.

C. Hönninger, R. Paschotta, M. Graf, F. Morier-Genoud, G. Zhang, M. Moser, S. Biswal, J. Nees, A. Braun, G. Mourou, I. Johannsen, A. Giesen, W. Seeber, and U. Keller, “Ultrafast ytterbium-doped bulk laser amplifiers,” Appl. Phys B,  69, 3–17 (1999).
[CrossRef]

Moser, M.

C. Hönninger, R. Paschotta, M. Graf, F. Morier-Genoud, G. Zhang, M. Moser, S. Biswal, J. Nees, A. Braun, G. Mourou, I. Johannsen, A. Giesen, W. Seeber, and U. Keller, “Ultrafast ytterbium-doped bulk laser amplifiers,” Appl. Phys B,  69, 3–17 (1999).
[CrossRef]

Mourou, G.

C. Hönninger, R. Paschotta, M. Graf, F. Morier-Genoud, G. Zhang, M. Moser, S. Biswal, J. Nees, A. Braun, G. Mourou, I. Johannsen, A. Giesen, W. Seeber, and U. Keller, “Ultrafast ytterbium-doped bulk laser amplifiers,” Appl. Phys B,  69, 3–17 (1999).
[CrossRef]

Nees, J.

C. Hönninger, R. Paschotta, M. Graf, F. Morier-Genoud, G. Zhang, M. Moser, S. Biswal, J. Nees, A. Braun, G. Mourou, I. Johannsen, A. Giesen, W. Seeber, and U. Keller, “Ultrafast ytterbium-doped bulk laser amplifiers,” Appl. Phys B,  69, 3–17 (1999).
[CrossRef]

Nicholson, J.W.

J.W. Nicholson, J. Jasapara, W. Rudolph, F.G. Omenetto, and A.J. Taylor, “Full-f ield characterization of femtosecond pulses by spectrum and cross-correlation measurements,” Opt. Lett.1774–76 (1999) and 138 (2000).
[CrossRef]

Nielsen, C. K.

O’Shea, P.

Omenetto, F.G.

J.W. Nicholson, J. Jasapara, W. Rudolph, F.G. Omenetto, and A.J. Taylor, “Full-f ield characterization of femtosecond pulses by spectrum and cross-correlation measurements,” Opt. Lett.1774–76 (1999) and 138 (2000).
[CrossRef]

Ortiz, V.

Paschotta, R.

Paulsen, H. N.

Peterson, A.

Price, J.H.V.

Ralph, S.E.

B.R. Washburn, S.E. Ralph, P.A. Lacourt, J.M. Dudley, W.T. Rhodes, R.S. Windeler, and S. Coen, “Tunable near-infrared femtosecond soliton generation in photonic crystal fibres,” Electron. Lett. 37, 1510–1512 (2001).
[CrossRef]

Raybaut, P.

Reid, D.T.

I.G. Cormack, D.T. Reid, W.J. Wadsworth, J.C. Knight, and P.ST.J. Russel, “Observation of soliton self-frequency shift in photonic crystal fibre,” Electron. Lett. 38, 167–168 (2002).
[CrossRef]

Rhodes, W.T.

B.R. Washburn, S.E. Ralph, P.A. Lacourt, J.M. Dudley, W.T. Rhodes, R.S. Windeler, and S. Coen, “Tunable near-infrared femtosecond soliton generation in photonic crystal fibres,” Electron. Lett. 37, 1510–1512 (2001).
[CrossRef]

Richardson, D.

Richardson, D.J.

Richman, B. A.

R. Trebino, K. W. DeLong, D. N. Fittinghoff, J. N. Sweetser, M. A. Krumbugel, B. A. Richman, and D. J. Kane, “Measuring ultrashort laser pulses in the time-frequency domain using frequency-resolved optical gating,” Rev. Sci. Instrum. 68, 3277–3295 (1997).
[CrossRef]

Riis, E.

G. McConnell and E. Riis, “Ultrashort pulse compression using photonic crystal fiber,” Appl. Phys. B557–563 (2004)
[CrossRef]

Rudolph, W.

J.W. Nicholson, J. Jasapara, W. Rudolph, F.G. Omenetto, and A.J. Taylor, “Full-f ield characterization of femtosecond pulses by spectrum and cross-correlation measurements,” Opt. Lett.1774–76 (1999) and 138 (2000).
[CrossRef]

Russel, P.ST.J.

I.G. Cormack, D.T. Reid, W.J. Wadsworth, J.C. Knight, and P.ST.J. Russel, “Observation of soliton self-frequency shift in photonic crystal fibre,” Electron. Lett. 38, 167–168 (2002).
[CrossRef]

C. M.J. Gander, R. McBride, J.D.C. Jones, D. Mogilevstev, T.A. Birks, J.C. Knight, and P.St.J. Russel, “Experimental measurement of group velocity dispersion in photonic crystal fiber,” Electon. Lett. 35, 63–64 (1999).
[CrossRef]

Sanner, N.

Seeber, W.

C. Hönninger, R. Paschotta, M. Graf, F. Morier-Genoud, G. Zhang, M. Moser, S. Biswal, J. Nees, A. Braun, G. Mourou, I. Johannsen, A. Giesen, W. Seeber, and U. Keller, “Ultrafast ytterbium-doped bulk laser amplifiers,” Appl. Phys B,  69, 3–17 (1999).
[CrossRef]

Shank, C. V.

Shreenath, A. P.

Stolen, R. H.

Sucha, G.

M.E. Fermann, A. Galvanauskas, G. Sucha, and D. Harter, “Ultrafast pulse sources based on multi-mode optical fibers,” Appl. Phys. B 65, 259–275 (1997).
[CrossRef]

Sudmeyer, T.

Sweetser, J. N.

R. Trebino, K. W. DeLong, D. N. Fittinghoff, J. N. Sweetser, M. A. Krumbugel, B. A. Richman, and D. J. Kane, “Measuring ultrashort laser pulses in the time-frequency domain using frequency-resolved optical gating,” Rev. Sci. Instrum. 68, 3277–3295 (1997).
[CrossRef]

Taylor, A.J.

J.W. Nicholson, J. Jasapara, W. Rudolph, F.G. Omenetto, and A.J. Taylor, “Full-f ield characterization of femtosecond pulses by spectrum and cross-correlation measurements,” Opt. Lett.1774–76 (1999) and 138 (2000).
[CrossRef]

Tomlinson, W. J.

Trebino, R.

Viana, B.

Vivien, D.

Wadsworth, W.J.

I.G. Cormack, D.T. Reid, W.J. Wadsworth, J.C. Knight, and P.ST.J. Russel, “Observation of soliton self-frequency shift in photonic crystal fibre,” Electron. Lett. 38, 167–168 (2002).
[CrossRef]

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B.R. Washburn, S.E. Ralph, P.A. Lacourt, J.M. Dudley, W.T. Rhodes, R.S. Windeler, and S. Coen, “Tunable near-infrared femtosecond soliton generation in photonic crystal fibres,” Electron. Lett. 37, 1510–1512 (2001).
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B.R. Washburn, S.E. Ralph, P.A. Lacourt, J.M. Dudley, W.T. Rhodes, R.S. Windeler, and S. Coen, “Tunable near-infrared femtosecond soliton generation in photonic crystal fibres,” Electron. Lett. 37, 1510–1512 (2001).
[CrossRef]

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[CrossRef]

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Figures (15)

Fig. 1.
Fig. 1.

Experimental setup.

Fig. 2.
Fig. 2.

Evolution of spectra versus the coupled power in the fiber (in log scale).

Fig. 3.
Fig. 3.

Experimental and theoretical spectra (due to SPM, SS and TOD) for a coupled average power about 45 mW (Φ SPM ≈2π).

Fig. 4.
Fig. 4.

Theoretical, retrieved and experimental non-collinear autocorrelation traces.

Fig. 5.
Fig. 5.

Theoretical and experimental retrieved pulse shape and phase.

Fig. 6.
Fig. 6.

Example of interferometric autocorrelations for Φ SPM ≈2π. The fitting assuming a pulse shape given by the model allows a fairly accurate estimation of the pulse duration.

Fig. 7.
Fig. 7.

Pulse duration of compressed pulses after the prism-compressor and influence of the pre-pulse amplitude.

Fig. 8.
Fig. 8.

Experimental and theoretical spectra for Φ SPM ≈2.5π(≈57 mW) just below the SRS splitting.

Fig. 9.
Fig. 9.

XFROG traces for different non-linear phase shifts: the first column represents XFROG[Efiber,Einput] which puts the emphasis on the spectrum generation by SPM in the PCF fiber; the second column represents XFROG[Ecompressed,ETF] which puts the emphasis on the compression efficiency and the third column represents XFROG[Ecompressed,ETF]-XFROG[ETF,ETF] which puts the emphasis on the compressed-pulse quality.

Fig. 10.
Fig. 10.

Evolution of spectra versus the coupled power in the fiber (in log scale): evidence of the stimulated Raman scattering splitting and soliton self-frequency shift.

Fig. 11.
Fig. 11.

Pulse duration of compressed pulses after the prism-compressor and influence of the SRS splitting see also Fig.10.

Fig. 12.
Fig. 12.

Autocorrelation trace for Φ SPM ≈4π. The experimental trace shows 7 cycles at FWHM and the theory 3 cycles both are demonstrating important satellite pulses.

Fig. 13.
Fig. 13.

XFROG traces for incident pulse duration: the first column represents XFROG[Efiber,Einput] which puts the emphasis on the spectrum generation by SPM in the PCF fiber; the second column represents XFROG[Ecompressed,ETF] which puts the emphasis on the compression efficiency and the third column represents XFROG[Ecompressed,ETF]-XFROG[ETF,ETF] which puts the emphasis on the compressed-pulse quality.

Fig. 14.
Fig. 14.

Second pulse amplitude versus incident pulse duration for fixed compressed pulse durations Δτf ∈[15fs,20fs,30fs].

Fig. 15.
Fig. 15.

Time-bandwidth product at FWHM versus incident pulse duration for fixed compressed pulse durations Δτf ∈[15fs,20fs,30fs].

Equations (4)

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Φ SPM = γ P 0 L
Δ ( f ( x ) , g ( x ) ) = 1 N i = 1 N ( f ( x i ) g ( x i ) ) 2
XFROG [ E 1 , E 2 ] ( λ , τ ) = E 1 ( t ) E 2 ( τ t ) e i 2 π ct λ dt 2
Φ SPM ( Δ τ i ) = γ L P ¯ F . Δ τ i a Δ τ f Δ τ i a

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