Abstract

We measure the degree of coherence of supercontinua generated in tapered fibers by subsequent fs pulses. By means of interference experiments we study its dependence on the input pulse duration and power. We also present numerical simulations that allow us to explain the experimental observations which show a decreasing degree of coherence with increasing input power. We attribute this loss of coherence to phase noise due to the cross-phase modulation by several solitons with randomly varying parameters due to quantum noise.

© 2007 Optical Society of America

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References

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  1. J. K. Ranka, R. S. Windeler, A. J. Stentz, "Visible continuum generation in air silica microstructure optical fibers with anomalous dispersion at 800nm," Opt. Lett. 25, 25-27 (2000).
    [CrossRef]
  2. T. A. Birks, W. J. Wadsworth, P. St. J. Russell, "Supercontinuum generation in tapered fibers," Opt. Lett. 25,1415-1417 (2000).
    [CrossRef]
  3. A. Husakou, J. Herrmann, "Supercontinuum Generation of Higher-Order Solitons by Fission in Photonic Crystal Fibers," Phys. Rev. Lett. 87, 203901 (2001).
    [CrossRef] [PubMed]
  4. J. Herrmann, U. Griebner, N. Zhavoronkov, A. Husakou, D. Nickel, J. C. Knight, W. J. Wadsworth, P. St. J. Russell, G. Korn, "Experimental Evidence for Supercontinuum Generation by Fission of Higher-Order Solitons in Photonic Fibers," Phys. Rev. Lett. 88, 173901 (2002).
    [CrossRef] [PubMed]
  5. J. M. Dudley, G. Genty, and S. Coen, "Supercontinuum generation in photonic crystal fibers," Rev. Mod. Phys. 78, 1135-1184 (2006).
    [CrossRef]
  6. B. Schenkel, R. Paschotta, U. Keller, "Pulse compression with supercontinuum generation in microstructure fibers," J. Opt. Soc. Am. B 22, 687-693 (2005).
    [CrossRef]
  7. R. Holzwarth, Th. Udem, T. W. Hänsch, J. C. Knight, W. J. Wadsworth, P. St. J. Russell, "Optical Frequency Synthesizer for Precision Spectroscopy," Phys. Rev. Lett. 85, 2264-2267 (2000).
    [CrossRef] [PubMed]
  8. J. M. Dudley, S. Coen, "Coherence properties of supercontinuum spectra generated in photonic crystal and tapered optical fibers," Opt. Lett. 27, 1180-1182 (2002).
    [CrossRef]
  9. K. L. Corwin, N. R. Newbury, J. M. Dudley, S. Coen, S. A. Diddams, K. Weber, R. S. Windeler, "Fundamental Noise Limitations to Supercontinuum Generation in Microstructure Fiber," Phys. Rev. Lett. 90, 113904 (2003).
    [CrossRef] [PubMed]
  10. T. M. Fortier, J. Ye, S. T. Cundiff, R. S. Windeler, "Nonlinear phase noise generated in air-silica microstructure fiber and its effect on carrier-envelope phase," Opt. Lett. 27, 445-447 (2002).
    [CrossRef]
  11. N. R. Newbury, B. R. Washburn, K. L. Corwin, R. S. Windeler, "Noise amplification during supercontinuum generation in microstructure fiber," Opt. Lett. 28, 944-946 (2003).
    [CrossRef] [PubMed]
  12. A. L. Gaeta, "Nonlinear propagation and continuum generation in microstructured optical fibers," Opt. Lett. 27, 924-926 (2002).
    [CrossRef]
  13. M. Bellini, T. W. H¨ansch, "Phase-locked white-light continuum pulses: toward a universal optical frequency-comb synthesizer," Opt. Lett. 25, 1049-1051 (2000).
    [CrossRef]
  14. X. Gu, M. Kimmel, A. P. Shreenath, R. Trebino, J. M. Dudley, S. Coen, R. S. Windeler, "Experimental studies of the coherence of microstructure-fiber supercontinuum," Opt. Express 11, 2697-2703 (2003).
    [CrossRef] [PubMed]
  15. F. Lu, W. H. Knox, "Generation of a broadband continuum with high spectral coherence in tapered single-mode optical fibers," Opt. Express 12, 347-353 (2004).
    [CrossRef] [PubMed]
  16. J. N. Ames, S. Ghosh, R. S. Windeler, A. L. Gaeta, S. T. Cundiff, "Excess noise generation during spectral broadening in a microstructured fiber," Appl. Phys. B 77, 279-284 (2003).
    [CrossRef]
  17. J. Teipel, K. Franke, D. T¨urke, F.Warken, D. Meiser, M. Leuschner, H. Giessen, "Characteristics of supercontinuum generation in tapered fibers using femtosecond laser pulses," Appl. Phys. B 77,245-251 (2003).
    [CrossRef]
  18. J. Stenger and H. R. Telle, "Kerr-lens mode-locked lasers for optical frequency measurements," in Laser Frequency Stabilization, Standards, Measurement, and Applications, John L. Hall and Jun Ye, eds., Proc. SPIE 4269, 72-76 (2001).
    [CrossRef]
  19. L. Mandel, E. Wolf, Optical coherence and quantum optics (Cambridge University Press, Cambridge, 1995).
  20. J. W. Nicholson, M. F. Yan, "Cross-coherence measurements of supercontinua generated in highly-nonlinear, dispersion shifted fiber at 1550 nm," Opt. Express 12, 679-688 (2004).
    [CrossRef] [PubMed]
  21. P. D. Drummond, J. F. Corney, "Quantum noise in optical fibers. I. Stochastic equations," J. Opt. Soc. Am. B 18, 139-152 (2001).
    [CrossRef]
  22. M. Bass (ed.), Handbook of Optics (McGraw-Hill, New York, 1995).
  23. A. Husakou and J. Herrmann, in preparation.

2006 (1)

J. M. Dudley, G. Genty, and S. Coen, "Supercontinuum generation in photonic crystal fibers," Rev. Mod. Phys. 78, 1135-1184 (2006).
[CrossRef]

2005 (1)

2004 (2)

2003 (5)

N. R. Newbury, B. R. Washburn, K. L. Corwin, R. S. Windeler, "Noise amplification during supercontinuum generation in microstructure fiber," Opt. Lett. 28, 944-946 (2003).
[CrossRef] [PubMed]

X. Gu, M. Kimmel, A. P. Shreenath, R. Trebino, J. M. Dudley, S. Coen, R. S. Windeler, "Experimental studies of the coherence of microstructure-fiber supercontinuum," Opt. Express 11, 2697-2703 (2003).
[CrossRef] [PubMed]

K. L. Corwin, N. R. Newbury, J. M. Dudley, S. Coen, S. A. Diddams, K. Weber, R. S. Windeler, "Fundamental Noise Limitations to Supercontinuum Generation in Microstructure Fiber," Phys. Rev. Lett. 90, 113904 (2003).
[CrossRef] [PubMed]

J. N. Ames, S. Ghosh, R. S. Windeler, A. L. Gaeta, S. T. Cundiff, "Excess noise generation during spectral broadening in a microstructured fiber," Appl. Phys. B 77, 279-284 (2003).
[CrossRef]

J. Teipel, K. Franke, D. T¨urke, F.Warken, D. Meiser, M. Leuschner, H. Giessen, "Characteristics of supercontinuum generation in tapered fibers using femtosecond laser pulses," Appl. Phys. B 77,245-251 (2003).
[CrossRef]

2002 (4)

2001 (2)

P. D. Drummond, J. F. Corney, "Quantum noise in optical fibers. I. Stochastic equations," J. Opt. Soc. Am. B 18, 139-152 (2001).
[CrossRef]

A. Husakou, J. Herrmann, "Supercontinuum Generation of Higher-Order Solitons by Fission in Photonic Crystal Fibers," Phys. Rev. Lett. 87, 203901 (2001).
[CrossRef] [PubMed]

2000 (4)

Ames, J. N.

J. N. Ames, S. Ghosh, R. S. Windeler, A. L. Gaeta, S. T. Cundiff, "Excess noise generation during spectral broadening in a microstructured fiber," Appl. Phys. B 77, 279-284 (2003).
[CrossRef]

Bellini, M.

Birks, T. A.

Coen, S.

J. M. Dudley, G. Genty, and S. Coen, "Supercontinuum generation in photonic crystal fibers," Rev. Mod. Phys. 78, 1135-1184 (2006).
[CrossRef]

K. L. Corwin, N. R. Newbury, J. M. Dudley, S. Coen, S. A. Diddams, K. Weber, R. S. Windeler, "Fundamental Noise Limitations to Supercontinuum Generation in Microstructure Fiber," Phys. Rev. Lett. 90, 113904 (2003).
[CrossRef] [PubMed]

X. Gu, M. Kimmel, A. P. Shreenath, R. Trebino, J. M. Dudley, S. Coen, R. S. Windeler, "Experimental studies of the coherence of microstructure-fiber supercontinuum," Opt. Express 11, 2697-2703 (2003).
[CrossRef] [PubMed]

J. M. Dudley, S. Coen, "Coherence properties of supercontinuum spectra generated in photonic crystal and tapered optical fibers," Opt. Lett. 27, 1180-1182 (2002).
[CrossRef]

Corney, J. F.

Corwin, K. L.

N. R. Newbury, B. R. Washburn, K. L. Corwin, R. S. Windeler, "Noise amplification during supercontinuum generation in microstructure fiber," Opt. Lett. 28, 944-946 (2003).
[CrossRef] [PubMed]

K. L. Corwin, N. R. Newbury, J. M. Dudley, S. Coen, S. A. Diddams, K. Weber, R. S. Windeler, "Fundamental Noise Limitations to Supercontinuum Generation in Microstructure Fiber," Phys. Rev. Lett. 90, 113904 (2003).
[CrossRef] [PubMed]

Cundiff, S. T.

J. N. Ames, S. Ghosh, R. S. Windeler, A. L. Gaeta, S. T. Cundiff, "Excess noise generation during spectral broadening in a microstructured fiber," Appl. Phys. B 77, 279-284 (2003).
[CrossRef]

T. M. Fortier, J. Ye, S. T. Cundiff, R. S. Windeler, "Nonlinear phase noise generated in air-silica microstructure fiber and its effect on carrier-envelope phase," Opt. Lett. 27, 445-447 (2002).
[CrossRef]

Diddams, S. A.

K. L. Corwin, N. R. Newbury, J. M. Dudley, S. Coen, S. A. Diddams, K. Weber, R. S. Windeler, "Fundamental Noise Limitations to Supercontinuum Generation in Microstructure Fiber," Phys. Rev. Lett. 90, 113904 (2003).
[CrossRef] [PubMed]

Drummond, P. D.

Dudley, J. M.

J. M. Dudley, G. Genty, and S. Coen, "Supercontinuum generation in photonic crystal fibers," Rev. Mod. Phys. 78, 1135-1184 (2006).
[CrossRef]

K. L. Corwin, N. R. Newbury, J. M. Dudley, S. Coen, S. A. Diddams, K. Weber, R. S. Windeler, "Fundamental Noise Limitations to Supercontinuum Generation in Microstructure Fiber," Phys. Rev. Lett. 90, 113904 (2003).
[CrossRef] [PubMed]

X. Gu, M. Kimmel, A. P. Shreenath, R. Trebino, J. M. Dudley, S. Coen, R. S. Windeler, "Experimental studies of the coherence of microstructure-fiber supercontinuum," Opt. Express 11, 2697-2703 (2003).
[CrossRef] [PubMed]

J. M. Dudley, S. Coen, "Coherence properties of supercontinuum spectra generated in photonic crystal and tapered optical fibers," Opt. Lett. 27, 1180-1182 (2002).
[CrossRef]

Fortier, T. M.

Franke, K.

J. Teipel, K. Franke, D. T¨urke, F.Warken, D. Meiser, M. Leuschner, H. Giessen, "Characteristics of supercontinuum generation in tapered fibers using femtosecond laser pulses," Appl. Phys. B 77,245-251 (2003).
[CrossRef]

Gaeta, A. L.

J. N. Ames, S. Ghosh, R. S. Windeler, A. L. Gaeta, S. T. Cundiff, "Excess noise generation during spectral broadening in a microstructured fiber," Appl. Phys. B 77, 279-284 (2003).
[CrossRef]

A. L. Gaeta, "Nonlinear propagation and continuum generation in microstructured optical fibers," Opt. Lett. 27, 924-926 (2002).
[CrossRef]

Genty, G.

J. M. Dudley, G. Genty, and S. Coen, "Supercontinuum generation in photonic crystal fibers," Rev. Mod. Phys. 78, 1135-1184 (2006).
[CrossRef]

Ghosh, S.

J. N. Ames, S. Ghosh, R. S. Windeler, A. L. Gaeta, S. T. Cundiff, "Excess noise generation during spectral broadening in a microstructured fiber," Appl. Phys. B 77, 279-284 (2003).
[CrossRef]

Griebner, U.

J. Herrmann, U. Griebner, N. Zhavoronkov, A. Husakou, D. Nickel, J. C. Knight, W. J. Wadsworth, P. St. J. Russell, G. Korn, "Experimental Evidence for Supercontinuum Generation by Fission of Higher-Order Solitons in Photonic Fibers," Phys. Rev. Lett. 88, 173901 (2002).
[CrossRef] [PubMed]

Gu, X.

Hänsch, T. W.

R. Holzwarth, Th. Udem, T. W. Hänsch, J. C. Knight, W. J. Wadsworth, P. St. J. Russell, "Optical Frequency Synthesizer for Precision Spectroscopy," Phys. Rev. Lett. 85, 2264-2267 (2000).
[CrossRef] [PubMed]

Herrmann, J.

J. Herrmann, U. Griebner, N. Zhavoronkov, A. Husakou, D. Nickel, J. C. Knight, W. J. Wadsworth, P. St. J. Russell, G. Korn, "Experimental Evidence for Supercontinuum Generation by Fission of Higher-Order Solitons in Photonic Fibers," Phys. Rev. Lett. 88, 173901 (2002).
[CrossRef] [PubMed]

A. Husakou, J. Herrmann, "Supercontinuum Generation of Higher-Order Solitons by Fission in Photonic Crystal Fibers," Phys. Rev. Lett. 87, 203901 (2001).
[CrossRef] [PubMed]

Holzwarth, R.

R. Holzwarth, Th. Udem, T. W. Hänsch, J. C. Knight, W. J. Wadsworth, P. St. J. Russell, "Optical Frequency Synthesizer for Precision Spectroscopy," Phys. Rev. Lett. 85, 2264-2267 (2000).
[CrossRef] [PubMed]

Husakou, A.

J. Herrmann, U. Griebner, N. Zhavoronkov, A. Husakou, D. Nickel, J. C. Knight, W. J. Wadsworth, P. St. J. Russell, G. Korn, "Experimental Evidence for Supercontinuum Generation by Fission of Higher-Order Solitons in Photonic Fibers," Phys. Rev. Lett. 88, 173901 (2002).
[CrossRef] [PubMed]

A. Husakou, J. Herrmann, "Supercontinuum Generation of Higher-Order Solitons by Fission in Photonic Crystal Fibers," Phys. Rev. Lett. 87, 203901 (2001).
[CrossRef] [PubMed]

Keller, U.

Kimmel, M.

Knight, J. C.

J. Herrmann, U. Griebner, N. Zhavoronkov, A. Husakou, D. Nickel, J. C. Knight, W. J. Wadsworth, P. St. J. Russell, G. Korn, "Experimental Evidence for Supercontinuum Generation by Fission of Higher-Order Solitons in Photonic Fibers," Phys. Rev. Lett. 88, 173901 (2002).
[CrossRef] [PubMed]

R. Holzwarth, Th. Udem, T. W. Hänsch, J. C. Knight, W. J. Wadsworth, P. St. J. Russell, "Optical Frequency Synthesizer for Precision Spectroscopy," Phys. Rev. Lett. 85, 2264-2267 (2000).
[CrossRef] [PubMed]

Knox, W. H.

Korn, G.

J. Herrmann, U. Griebner, N. Zhavoronkov, A. Husakou, D. Nickel, J. C. Knight, W. J. Wadsworth, P. St. J. Russell, G. Korn, "Experimental Evidence for Supercontinuum Generation by Fission of Higher-Order Solitons in Photonic Fibers," Phys. Rev. Lett. 88, 173901 (2002).
[CrossRef] [PubMed]

Lu, F.

Newbury, N. R.

N. R. Newbury, B. R. Washburn, K. L. Corwin, R. S. Windeler, "Noise amplification during supercontinuum generation in microstructure fiber," Opt. Lett. 28, 944-946 (2003).
[CrossRef] [PubMed]

K. L. Corwin, N. R. Newbury, J. M. Dudley, S. Coen, S. A. Diddams, K. Weber, R. S. Windeler, "Fundamental Noise Limitations to Supercontinuum Generation in Microstructure Fiber," Phys. Rev. Lett. 90, 113904 (2003).
[CrossRef] [PubMed]

Nicholson, J. W.

Nickel, D.

J. Herrmann, U. Griebner, N. Zhavoronkov, A. Husakou, D. Nickel, J. C. Knight, W. J. Wadsworth, P. St. J. Russell, G. Korn, "Experimental Evidence for Supercontinuum Generation by Fission of Higher-Order Solitons in Photonic Fibers," Phys. Rev. Lett. 88, 173901 (2002).
[CrossRef] [PubMed]

Paschotta, R.

Ranka, J. K.

Russell, P. St. J.

J. Herrmann, U. Griebner, N. Zhavoronkov, A. Husakou, D. Nickel, J. C. Knight, W. J. Wadsworth, P. St. J. Russell, G. Korn, "Experimental Evidence for Supercontinuum Generation by Fission of Higher-Order Solitons in Photonic Fibers," Phys. Rev. Lett. 88, 173901 (2002).
[CrossRef] [PubMed]

R. Holzwarth, Th. Udem, T. W. Hänsch, J. C. Knight, W. J. Wadsworth, P. St. J. Russell, "Optical Frequency Synthesizer for Precision Spectroscopy," Phys. Rev. Lett. 85, 2264-2267 (2000).
[CrossRef] [PubMed]

T. A. Birks, W. J. Wadsworth, P. St. J. Russell, "Supercontinuum generation in tapered fibers," Opt. Lett. 25,1415-1417 (2000).
[CrossRef]

Schenkel, B.

Shreenath, A. P.

Stentz, A. J.

Teipel, J.

J. Teipel, K. Franke, D. T¨urke, F.Warken, D. Meiser, M. Leuschner, H. Giessen, "Characteristics of supercontinuum generation in tapered fibers using femtosecond laser pulses," Appl. Phys. B 77,245-251 (2003).
[CrossRef]

Trebino, R.

Udem, Th.

R. Holzwarth, Th. Udem, T. W. Hänsch, J. C. Knight, W. J. Wadsworth, P. St. J. Russell, "Optical Frequency Synthesizer for Precision Spectroscopy," Phys. Rev. Lett. 85, 2264-2267 (2000).
[CrossRef] [PubMed]

Wadsworth, W. J.

J. Herrmann, U. Griebner, N. Zhavoronkov, A. Husakou, D. Nickel, J. C. Knight, W. J. Wadsworth, P. St. J. Russell, G. Korn, "Experimental Evidence for Supercontinuum Generation by Fission of Higher-Order Solitons in Photonic Fibers," Phys. Rev. Lett. 88, 173901 (2002).
[CrossRef] [PubMed]

R. Holzwarth, Th. Udem, T. W. Hänsch, J. C. Knight, W. J. Wadsworth, P. St. J. Russell, "Optical Frequency Synthesizer for Precision Spectroscopy," Phys. Rev. Lett. 85, 2264-2267 (2000).
[CrossRef] [PubMed]

T. A. Birks, W. J. Wadsworth, P. St. J. Russell, "Supercontinuum generation in tapered fibers," Opt. Lett. 25,1415-1417 (2000).
[CrossRef]

Washburn, B. R.

Weber, K.

K. L. Corwin, N. R. Newbury, J. M. Dudley, S. Coen, S. A. Diddams, K. Weber, R. S. Windeler, "Fundamental Noise Limitations to Supercontinuum Generation in Microstructure Fiber," Phys. Rev. Lett. 90, 113904 (2003).
[CrossRef] [PubMed]

Windeler, R. S.

Yan, M. F.

Ye, J.

Zhavoronkov, N.

J. Herrmann, U. Griebner, N. Zhavoronkov, A. Husakou, D. Nickel, J. C. Knight, W. J. Wadsworth, P. St. J. Russell, G. Korn, "Experimental Evidence for Supercontinuum Generation by Fission of Higher-Order Solitons in Photonic Fibers," Phys. Rev. Lett. 88, 173901 (2002).
[CrossRef] [PubMed]

Appl. Phys. B (2)

J. N. Ames, S. Ghosh, R. S. Windeler, A. L. Gaeta, S. T. Cundiff, "Excess noise generation during spectral broadening in a microstructured fiber," Appl. Phys. B 77, 279-284 (2003).
[CrossRef]

J. Teipel, K. Franke, D. T¨urke, F.Warken, D. Meiser, M. Leuschner, H. Giessen, "Characteristics of supercontinuum generation in tapered fibers using femtosecond laser pulses," Appl. Phys. B 77,245-251 (2003).
[CrossRef]

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

Opt. Express (3)

Opt. Lett. (7)

Phys. Rev. Lett. (4)

A. Husakou, 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. Husakou, D. Nickel, J. C. Knight, W. J. Wadsworth, P. St. J. Russell, G. Korn, "Experimental Evidence for Supercontinuum Generation by Fission of Higher-Order Solitons in Photonic Fibers," Phys. Rev. Lett. 88, 173901 (2002).
[CrossRef] [PubMed]

R. Holzwarth, Th. Udem, T. W. Hänsch, J. C. Knight, W. J. Wadsworth, P. St. J. Russell, "Optical Frequency Synthesizer for Precision Spectroscopy," Phys. Rev. Lett. 85, 2264-2267 (2000).
[CrossRef] [PubMed]

K. L. Corwin, N. R. Newbury, J. M. Dudley, S. Coen, S. A. Diddams, K. Weber, R. S. Windeler, "Fundamental Noise Limitations to Supercontinuum Generation in Microstructure Fiber," Phys. Rev. Lett. 90, 113904 (2003).
[CrossRef] [PubMed]

Rev. Mod. Phys. (1)

J. M. Dudley, G. Genty, and S. Coen, "Supercontinuum generation in photonic crystal fibers," Rev. Mod. Phys. 78, 1135-1184 (2006).
[CrossRef]

Other (4)

J. Stenger and H. R. Telle, "Kerr-lens mode-locked lasers for optical frequency measurements," in Laser Frequency Stabilization, Standards, Measurement, and Applications, John L. Hall and Jun Ye, eds., Proc. SPIE 4269, 72-76 (2001).
[CrossRef]

L. Mandel, E. Wolf, Optical coherence and quantum optics (Cambridge University Press, Cambridge, 1995).

M. Bass (ed.), Handbook of Optics (McGraw-Hill, New York, 1995).

A. Husakou and J. Herrmann, in preparation.

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Figures (8)

Fig. 1.
Fig. 1.

Asymmetric Michelson interferometer. V-10: Verdi V-10 pump laser (532nm, 10W), Ti:Sa: Ti:sapphire laser, MF: Microstructure fiber, PD: Photodiode for measuring the amplitude noise, ISO: Faraday isolator, IF: Interference band pass filter, DS: Delay stage, Spec: Spectrometer, S: Screen for observing the interference fringes.

Fig. 2.
Fig. 2.

(a) Typical SC output spectrum for 148 fs pumping. (b) Normalized interference pattern I(λ)/2I 0(λ).

Fig. 3.
Fig. 3.

Decrease of the amplitude of the visibility depending on the amplitude noise for a rectangular distribution (red dashed line) and a Gaussian distribution (black solid line).

Fig. 4.
Fig. 4.

Experimental results of the visibility V=〈I〉/(2(〈0〉)- 1 (red, left scale) corrected by removing the amplitude noise for different input peak powers (kW) and pulse length [(a) 148 fs, (b) 410 fs]. Additionally the SC spectrum is plotted in black (right scale). The spectral resolution of the visibility measurement is about 10 nm.

Fig. 5.
Fig. 5.

Output spectra (black dashed curves) and coherence (red solid curves) for 148-fs input pulses for 2.5 kW (a) and 5.9 kW (b) input peak power. The input wavelength is 775 nm, the fiber length is 9 cm, the fiber diameter is 2.1 μm.

Fig. 6.
Fig. 6.

Output spectra (black dashed curves) and coherence (red solid curves) for 410-fs input pulses for 0.09 kW (a), 0.9 kW (b), and 5.9 kW (c) input peak power. The input wavelength is 775 nm, the fiber length is 9 cm, the fiber diameter is 2.1 μm.

Fig. 7.
Fig. 7.

Output spectra (black dashed curve) and coherence (red solid curve) for 15-fs input pulses for 5.9 kW input peak power. The input wavelength is 775 nm, the fiber length is 9 cm, the fiber diameter is 2.1 μm.

Fig. 8.
Fig. 8.

Simulation of the output temporal shape for input peak power of 5.9 kW and 410 fs (a) and 15 fs (b) duration. The input wavelength is 775 nm, the fiber length is 9 cm, the fiber diameter is 2.1 μm.

Equations (14)

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

E ( t ) = ( 1 + Δ E ) E exp iωt = ( 1 + Δ P ) P exp iωt ,
I 0 = E 0 2 = 1 ρ σ ( Δ P ) (1+ΔP) P d Δ P 1 ρ σ ( Δ P ) d Δ P
V ( τ ) = E i ( t ) + E j ( t + τ ) 2 2 I 0 1
= ( 1 ρ σ ( x ) 1 + x dx ) 2 1 ρ σ ( x ) ( 1 + x ) dx 1 ρ σ ( x ) dx cos ωτ
= V ( σ ) cos ωτ ,
E z ( z , ω ) = i ( β ( ω ) ω n g / c ) E ( z , ω ) + i ω 2 μ 0 2 β ( ω ) P NL ( z , ω ) .
P NL ( z , t ) = ( 1 f ) ε 0 χ ( 3 ) E ( z , t ) 3
+ f ε 0 χ ( 3 ) 2 π T R τ R 2 T R 2 + 4 π 2 τ R 2 E ( z , τ ) t E ( z , t ) 2
× exp ( ( t τ ) / τ R ) sin ( 2 π ( t τ ) / T R ) ,
< Δ E ( t 1 ) Δ E ( t 2 ) > = δ ( t 1 t 2 ) 1 2 n ¯
g ( ω ) = [ < E b ( L , ω ) E a * ( L , ω ) > ab , a b < E a ( L , ω ) E a * ( L , ω ) > a ]
f i ( z , t ) z = i Δ β i ˜ f i ( z , t )
+ f i ( z , t ) Σ j A j ( z , t ) e i ω i t i k i z 2
+ ( ( 3 ) 3 / t 3 β ( 4 ) 4 / t 4 + . . . ) A i ( z , t ) ,

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