Abstract

Supercontinuum (SC) generation in all-normal dispersion photonic crystal fiber under high energy femtosecond pumping is numerically investigated. It is shown that coherent octave spanning SC spectra with flatness of better than ±1 dB can be achieved over the entire bandwidth. A single pulse is maintained in the time domain, which may be externally compressed to the sub-10 fs regime even by simple linear chirp elimination. The single optical cycle limit is approached for full phase compensation, leading to peak power spectral densities of multiple kilowatts/nanometer. The generated SC is therefore ideal for applications which require high broadband spectral power densities as well as a defined pulse profile in the time domain. The properties of the generated SC are shown to be independent of the input pulse duration.

© 2010 Optical Society of America

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2009 (3)

2008 (1)

2007 (4)

2006 (3)

M.-L. V. Tse, P. Horak, F. Poletti, N. G. Broderick, J. H. Price, J. R. Hayes, and D. J. Richardson, “Supercontinuum generation at 1.06 μm in holey fibers with dispersion flattened profiles,” Opt. Express 14, 4445-4451 (2006).
[CrossRef] [PubMed]

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

K. Chow, Y. Takushima, C. Lin, C. Shu, and A. Bjarklev, “Flat super-continuum generation based on normal dispersion nonlinear photonic crystal fibre,” Electron. Lett. 42, 989-991 (2006).
[CrossRef]

2005 (4)

2004 (2)

2003 (2)

X. Gu, M. Kimmel, A. Shreenath, R. Trebino, J. Dudley, S. Coen, and R. 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, and R. S. Windeler, “Fundamental noise limitations to supercontinuum generation in microstructure fiber,” Phys. Rev. Lett. 90, 113904 (2003).
[CrossRef] [PubMed]

2002 (3)

1997 (1)

K. Mori, H. Takara, S. Kawanishi, M. Saruwatari, and T. Morioka, “Flatly broadened supercontinuum spectrum generated in a dispersion decreasing fibre with convex dispersion profile,” Electron. Lett. 33, 1806-1808 (1997).
[CrossRef]

1992 (1)

1989 (1)

Agrawal, G. P.

G. P. Agrawal, Nonlinear Fiber Optics, 3rd ed. (Academic, 2001).

Alfano, R. R.

R. R. Alfano, The Supercontinuum Laser Source (Springer, 2006).
[CrossRef]

Andersen, T.

Anderson, D.

Andresen, E. R.

Bang, O.

Bartelt, H.

A. M. Heidt, G. Bosman, M. Becker, M. Rothhardt, K. Schuster, J. Kobelke, and H. Bartelt, “Prospects of high energy ultrashort pulse generation with frequency shifted feedback fiber oscillators,” in European Conference on Lasers and Electro-Optics 2009 and the European Quantum Electronics Conference (IEEE, 2009), paper CJ.P. 37.
[CrossRef] [PubMed]

Baumgartl, M.

Becker, M.

A. M. Heidt, G. Bosman, M. Becker, M. Rothhardt, K. Schuster, J. Kobelke, and H. Bartelt, “Prospects of high energy ultrashort pulse generation with frequency shifted feedback fiber oscillators,” in European Conference on Lasers and Electro-Optics 2009 and the European Quantum Electronics Conference (IEEE, 2009), paper CJ.P. 37.
[CrossRef] [PubMed]

Birkedal, V.

Bjarklev, A.

K. Chow, Y. Takushima, C. Lin, C. Shu, and A. Bjarklev, “Flat super-continuum generation based on normal dispersion nonlinear photonic crystal fibre,” Electron. Lett. 42, 989-991 (2006).
[CrossRef]

Bosman, G.

A. M. Heidt, G. Bosman, M. Becker, M. Rothhardt, K. Schuster, J. Kobelke, and H. Bartelt, “Prospects of high energy ultrashort pulse generation with frequency shifted feedback fiber oscillators,” in European Conference on Lasers and Electro-Optics 2009 and the European Quantum Electronics Conference (IEEE, 2009), paper CJ.P. 37.
[CrossRef] [PubMed]

Botten, L. C.

Broderick, N. G.

Burger, J. P.

Chong, A.

Chow, K.

K. Chow, Y. Takushima, C. Lin, C. Shu, and A. Bjarklev, “Flat super-continuum generation based on normal dispersion nonlinear photonic crystal fibre,” Electron. Lett. 42, 989-991 (2006).
[CrossRef]

Coen, S.

G. Genty, S. Coen, and J. M. Dudley, “Fiber supercontinuum sources (Invited),” J. Opt. Soc. Am. B 24, 1771-1785 (2007).
[CrossRef]

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

K. L. Corwin, N. R. Newbury, J. M. Dudley, S. Coen, S. A. Diddams, K. Weber, and 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. Shreenath, R. Trebino, J. Dudley, S. Coen, and R. Windeler, “Experimental studies of the coherence of microstructure-fiber supercontinuum,” Opt. Express 11, 2697-2703 (2003).
[CrossRef] [PubMed]

Corwin, K. L.

K. L. Corwin, N. R. Newbury, J. M. Dudley, S. Coen, S. A. Diddams, K. Weber, and R. S. Windeler, “Fundamental noise limitations to supercontinuum generation in microstructure fiber,” Phys. Rev. Lett. 90, 113904 (2003).
[CrossRef] [PubMed]

de Sterke, C. M.

Desaix, M.

Diddams, S. A.

K. L. Corwin, N. R. Newbury, J. M. Dudley, S. Coen, S. A. Diddams, K. Weber, and R. S. Windeler, “Fundamental noise limitations to supercontinuum generation in microstructure fiber,” Phys. Rev. Lett. 90, 113904 (2003).
[CrossRef] [PubMed]

Dudley, J.

Dudley, J. M.

G. Genty, S. Coen, and J. M. Dudley, “Fiber supercontinuum sources (Invited),” J. Opt. Soc. Am. B 24, 1771-1785 (2007).
[CrossRef]

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

K. L. Corwin, N. R. Newbury, J. M. Dudley, S. Coen, S. A. Diddams, K. Weber, and R. S. Windeler, “Fundamental noise limitations to supercontinuum generation in microstructure fiber,” Phys. Rev. Lett. 90, 113904 (2003).
[CrossRef] [PubMed]

Falk, P.

Finot, C.

Frosz, M.

Frosz, M. H.

Genty, G.

G. Genty, S. Coen, and J. M. Dudley, “Fiber supercontinuum sources (Invited),” J. Opt. Soc. Am. B 24, 1771-1785 (2007).
[CrossRef]

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

Gordon, J. P.

Griebner, U.

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

Gu, X.

Hansen, K.

Haus, H. A.

Hayes, J. R.

Heidt, A. M.

A. M. Heidt, “Efficient adaptive step size method for the simulation of supercontinuum generation in optical fibers,” J. Lightwave Technol. 27, 3984-3991 (2009).
[CrossRef]

A. M. Heidt, J. P. Burger, J.-N. Maran, and N. Traynor, “High power and high energy ultrashort pulse generation with a frequency shifted feedback fiber laser,” Opt. Express 15, 15892-15897 (2007).
[CrossRef] [PubMed]

A. M. Heidt, G. Bosman, M. Becker, M. Rothhardt, K. Schuster, J. Kobelke, and H. Bartelt, “Prospects of high energy ultrashort pulse generation with frequency shifted feedback fiber oscillators,” in European Conference on Lasers and Electro-Optics 2009 and the European Quantum Electronics Conference (IEEE, 2009), paper CJ.P. 37.
[CrossRef] [PubMed]

Herrmann, J.

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

Hilligsøe, K. M.

Horak, P.

Hult, J.

Husakou, A.

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

Kawanishi, S.

K. Mori, H. Takara, S. Kawanishi, M. Saruwatari, and T. Morioka, “Flatly broadened supercontinuum spectrum generated in a dispersion decreasing fibre with convex dispersion profile,” Electron. Lett. 33, 1806-1808 (1997).
[CrossRef]

Keiding, S.

Keiding, S. R.

Kibler, B.

Kieu, K.

Kimmel, M.

Knight, J.

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

Kobelke, J.

A. M. Heidt, G. Bosman, M. Becker, M. Rothhardt, K. Schuster, J. Kobelke, and H. Bartelt, “Prospects of high energy ultrashort pulse generation with frequency shifted feedback fiber oscillators,” in European Conference on Lasers and Electro-Optics 2009 and the European Quantum Electronics Conference (IEEE, 2009), paper CJ.P. 37.
[CrossRef] [PubMed]

Korn, G.

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

Koshiba, M.

Kristiansen, R.

Kuhlmey, B. T.

Larsen, J.

Limpert, J.

Lin, C.

K. Chow, Y. Takushima, C. Lin, C. Shu, and A. Bjarklev, “Flat super-continuum generation based on normal dispersion nonlinear photonic crystal fibre,” Electron. Lett. 42, 989-991 (2006).
[CrossRef]

Lisak, M.

Maran, J. -N.

Maystre, D.

McPhedran, R. C.

Mølmer, K.

Mori, K.

K. Mori, H. Takara, S. Kawanishi, M. Saruwatari, and T. Morioka, “Flatly broadened supercontinuum spectrum generated in a dispersion decreasing fibre with convex dispersion profile,” Electron. Lett. 33, 1806-1808 (1997).
[CrossRef]

Morioka, T.

K. Mori, H. Takara, S. Kawanishi, M. Saruwatari, and T. Morioka, “Flatly broadened supercontinuum spectrum generated in a dispersion decreasing fibre with convex dispersion profile,” Electron. Lett. 33, 1806-1808 (1997).
[CrossRef]

Newbury, N. R.

K. L. Corwin, N. R. Newbury, J. M. Dudley, S. Coen, S. A. Diddams, K. Weber, and R. S. Windeler, “Fundamental noise limitations to supercontinuum generation in microstructure fiber,” Phys. Rev. Lett. 90, 113904 (2003).
[CrossRef] [PubMed]

Nickel, D.

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

Nielsen, C.

Nishizawa, N.

Ortaç, B.

O'Shea, P.

Paulsen, H.

Paulsen, H. N.

Poletti, F.

Price, J. H.

Provost, L.

Quiroga-Teixeiro, M. L.

Renninger, W. H.

Renversez, G.

Richardson, D. J.

Rothhardt, M.

A. M. Heidt, G. Bosman, M. Becker, M. Rothhardt, K. Schuster, J. Kobelke, and H. Bartelt, “Prospects of high energy ultrashort pulse generation with frequency shifted feedback fiber oscillators,” in European Conference on Lasers and Electro-Optics 2009 and the European Quantum Electronics Conference (IEEE, 2009), paper CJ.P. 37.
[CrossRef] [PubMed]

Russell, P. S. J.

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

Saitoh, K.

Saruwatari, M.

K. Mori, H. Takara, S. Kawanishi, M. Saruwatari, and T. Morioka, “Flatly broadened supercontinuum spectrum generated in a dispersion decreasing fibre with convex dispersion profile,” Electron. Lett. 33, 1806-1808 (1997).
[CrossRef]

Schuster, K.

A. M. Heidt, G. Bosman, M. Becker, M. Rothhardt, K. Schuster, J. Kobelke, and H. Bartelt, “Prospects of high energy ultrashort pulse generation with frequency shifted feedback fiber oscillators,” in European Conference on Lasers and Electro-Optics 2009 and the European Quantum Electronics Conference (IEEE, 2009), paper CJ.P. 37.
[CrossRef] [PubMed]

Shreenath, A.

Shreenath, A. P.

Shu, C.

K. Chow, Y. Takushima, C. Lin, C. Shu, and A. Bjarklev, “Flat super-continuum generation based on normal dispersion nonlinear photonic crystal fibre,” Electron. Lett. 42, 989-991 (2006).
[CrossRef]

Stolen, R. H.

Takara, H.

K. Mori, H. Takara, S. Kawanishi, M. Saruwatari, and T. Morioka, “Flatly broadened supercontinuum spectrum generated in a dispersion decreasing fibre with convex dispersion profile,” Electron. Lett. 33, 1806-1808 (1997).
[CrossRef]

Takayanagi, J.

Takushima, Y.

K. Chow, Y. Takushima, C. Lin, C. Shu, and A. Bjarklev, “Flat super-continuum generation based on normal dispersion nonlinear photonic crystal fibre,” Electron. Lett. 42, 989-991 (2006).
[CrossRef]

Thøgersen, J.

Tomlinson, W. J.

Traynor, N.

Trebino, R.

Tse, M. -L. V.

Tünnermann, A.

Wabnitz, S.

Wadsworth, W.

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

Weber, K.

K. L. Corwin, N. R. Newbury, J. M. Dudley, S. Coen, S. A. Diddams, K. Weber, and R. S. Windeler, “Fundamental noise limitations to supercontinuum generation in microstructure fiber,” Phys. Rev. Lett. 90, 113904 (2003).
[CrossRef] [PubMed]

White, T. P.

Windeler, R.

Windeler, R. S.

K. L. Corwin, N. R. Newbury, J. M. Dudley, S. Coen, S. A. Diddams, K. Weber, and R. S. Windeler, “Fundamental noise limitations to supercontinuum generation in microstructure fiber,” Phys. Rev. Lett. 90, 113904 (2003).
[CrossRef] [PubMed]

X. Gu, L. Xu, M. Kimmel, E. Zeek, P. O'Shea, A. P. Shreenath, R. Trebino, and R. S. Windeler, “Frequency-resolved optical gating and single-shot spectral measurements reveal fine structure in microstructure-fiber continuum,” Opt. Lett. 27, 1174-1176 (2002).
[CrossRef]

Wise, F. W.

Xu, L.

Zeek, E.

Zhavoronkov, N.

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

Electron. Lett. (2)

K. Mori, H. Takara, S. Kawanishi, M. Saruwatari, and T. Morioka, “Flatly broadened supercontinuum spectrum generated in a dispersion decreasing fibre with convex dispersion profile,” Electron. Lett. 33, 1806-1808 (1997).
[CrossRef]

K. Chow, Y. Takushima, C. Lin, C. Shu, and A. Bjarklev, “Flat super-continuum generation based on normal dispersion nonlinear photonic crystal fibre,” Electron. Lett. 42, 989-991 (2006).
[CrossRef]

J. Lightwave Technol. (2)

J. Opt. Soc. Am. B (7)

Opt. Express (7)

M. Frosz, P. Falk, and O. Bang, “The role of the second zero-dispersion wavelength in generation of supercontinua and bright-bright soliton-pairs across the zero-dispersion wavelength,” Opt. Express 13, 6181-6192 (2005).
[CrossRef] [PubMed]

P. Falk, M. H. Frosz, and O. Bang, “Supercontinuum generation in a photonic crystal fiber with two zero-dispersion wavelengths tapered to normal dispersion at all wavelengths,” Opt. Express 13, 7535-7540 (2005).
[CrossRef] [PubMed]

M.-L. V. Tse, P. Horak, F. Poletti, N. G. Broderick, J. H. Price, J. R. Hayes, and D. J. Richardson, “Supercontinuum generation at 1.06 μm in holey fibers with dispersion flattened profiles,” Opt. Express 14, 4445-4451 (2006).
[CrossRef] [PubMed]

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

K. M. Hilligsøe, T. Andersen, H. Paulsen, C. Nielsen, K. Mølmer, S. Keiding, R. Kristiansen, K. Hansen, and J. Larsen, “Supercontinuum generation in a photonic crystal fiber with two zero dispersion wavelengths,” Opt. Express 12, 1045-1054 (2004).
[CrossRef] [PubMed]

A. M. Heidt, J. P. Burger, J.-N. Maran, and N. Traynor, “High power and high energy ultrashort pulse generation with a frequency shifted feedback fiber laser,” Opt. Express 15, 15892-15897 (2007).
[CrossRef] [PubMed]

K. Saitoh and M. Koshiba, “Empirical relations for simple design of photonic crystal fibers,” Opt. Express 13, 267-274 (2005).
[CrossRef] [PubMed]

Opt. Lett. (4)

Phys. Rev. Lett. (2)

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

K. L. Corwin, N. R. Newbury, J. M. Dudley, S. Coen, S. A. Diddams, K. Weber, and 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 fiber,” Rev. Mod. Phys. 78, 1135-1184 (2006).
[CrossRef]

Other (3)

R. R. Alfano, The Supercontinuum Laser Source (Springer, 2006).
[CrossRef]

G. P. Agrawal, Nonlinear Fiber Optics, 3rd ed. (Academic, 2001).

A. M. Heidt, G. Bosman, M. Becker, M. Rothhardt, K. Schuster, J. Kobelke, and H. Bartelt, “Prospects of high energy ultrashort pulse generation with frequency shifted feedback fiber oscillators,” in European Conference on Lasers and Electro-Optics 2009 and the European Quantum Electronics Conference (IEEE, 2009), paper CJ.P. 37.
[CrossRef] [PubMed]

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Figures (9)

Fig. 1
Fig. 1

Calculated dispersion parameter and effective MFD of PCF with Λ = 1.55 μ m and values of d / Λ ranging between 0.3 and 0.42 in steps of 0.02.

Fig. 2
Fig. 2

(a) Dependence of the generated SC spectrum on the relative air hole diameter d / Λ in PCF with pitch Λ = 1.55 μ m in a logarithmic density plot. An input pulse of 50 fs duration with 5 nJ energy propagating through a 10 cm piece of fiber is considered. The blue dotted line indicates the ZDWs: the dispersion is anomalous in the enclosed region and normal elsewhere. (b) Corresponding pulse profiles at the end of the fiber for selected d / Λ values in linear scale.

Fig. 3
Fig. 3

Spectral evolution with propagation distance z in a PCF with Λ = 1.55 μ m and d / Λ = 0.39 in a logarithmic density plot. The input pulse parameters are identical to those used in Fig. 2.

Fig. 4
Fig. 4

(a) Pulse profiles at the input and after z = 6.5   mm of propagation. (b) Spectrogram of the pulse at the onset of optical WB.

Fig. 5
Fig. 5

Recalculation of the spectral evolution with input pulse and fiber parameters identical to those used in Fig. 3, but only certain effects are included into the calculations: (a) SPM and GVD; (b) SPM, GVD, and self-steepening; (c) SPM, GVD, and frequency dependence of γ ( ω ) ; and (d) all of the above.

Fig. 6
Fig. 6

(a) Spectrum and corresponding degree of coherence after 10 cm propagation of a 5 nJ 50 fs pulse in the PCF with Λ = 1.55 μ m and d / Λ = 0.37 . (b) Pulse profiles after full phase compensation (gray shading) and after compensation of only linear chirp.

Fig. 7
Fig. 7

(a) Calculated dispersion parameter of PCF with d / Λ = 0.37 and selected pitch values between Λ = 1.45 and 1.75 μ m in 0.05 μ m steps. (b) Dependence of the generated SC spectrum on fiber pitch in a logarithmic density plot. Input pulse parameters are identical to Fig. 2. The blue dotted line indicates the position of the ZDWs.

Fig. 8
Fig. 8

(a) Generated SC spectra for different input pulse energies on linear scale. The pulse duration of 50 fs and propagation distance of 10 cm remain unchanged. (b) Spectrogram of the SC generated with the 15 nJ input pulse. The initial 50 fs pulse was used as the gate function.

Fig. 9
Fig. 9

(a) SC spectra generated with different input pulse durations. The pulse energies are adjusted in order to keep the peak power constant at about 90 kW. The fiber length is chosen such that the spectrum does not change anymore with further propagation. On the top, the degree of coherence is shown for the 300 fs input pulse. (b) Temporal profile and phase of the 200 fs 20 nJ pulse after 20 cm propagation through the fiber.

Equations (4)

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A z = α 2 A + k 2 i k + 1 k ! β k k A T k + i γ ( 1 + i τ shock T ) [ A ( z , t ) ( + R ( T ) | A ( z , T T ) | 2 d T + i Γ R ( z , T ) ) ] ,
Γ R ( Ω , z ) Γ R ( Ω , z ) = 2 f R ω 0 γ | Im [ h R ( Ω ) ] | [ n th ( | Ω | ) + U ( Ω ) ] δ ( z z ) δ ( Ω Ω ) ,
| g 12 ( 1 ) ( λ , t 1 t 2 ) | = | E 1 ( λ , t 1 ) E 2 ( λ , t 2 ) | E 1 ( λ , t 1 ) | 2 | E 2 ( λ , t 2 ) | 2 | .
ω FWM = 2 ω pump ω seed

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