Abstract

We consider the accuracy of modeling ultrashort pulse propagation and supercontinuum generation in optical fibers based on the assumption of a material Raman response that varies linearly with frequency. Numerical simulations in silica fiber using the linear Raman gain approximation are compared with simulations using the full Raman response, and differences in the spectral, temporal and stability characteristics are considered. A major finding is that for conditions typical of many experiments, although the input pulses may satisfy the criteria where the linear gain approximation is valid, the subsequent evolution and breakup of the input pulse can rapidly lead to a situation where the linear model leads to severe inaccuracies. Numerical artifacts within the linear model inducing unphysical pulse collapse are also identified.

©2010 Optical Society of America

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  17. A. V. Gorbach and D. V. Skryabin, “Light trapping in gravity-like potentials and expansion of supercontinuum spectra in photonic-crystal fibres,” Nat. Photonics 1(11), 653–657 (2007).
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  18. M. Facão, M. I. Carvalho, and D. F. Parker, “Soliton self-frequency shift: Self-similar solutions and their stability,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 81(4 ), 046604 (2010).
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  19. C. Conti, S. Stark, P. S. Russell, and F. Biancalana, “Multiple hydrodynamical shocks induced by Raman effect in photonic crystal fibres,” Phys. Rev. A 82(1), 013838 (2010).
    [Crossref]
  20. W. Hodel and H. P. Weber, “Decay of femtosecond higher-order solitons in an optical fiber induced by Raman self-pumping,” Opt. Lett. 12(11), 924–926 (1987).
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  21. M. N. Islam, G. Sucha, I. Bar-Joseph, M. Wegener, J. P. Gordon, and D. S. Chemla, “Broad bandwidths from frequency-shifting solitons in fibers,” Opt. Lett. 14(7), 370–372 (1989).
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  22. J. M. Harbold, F. Ö. Ilday, F. W. Wise, T. A. Birks, W. J. Wadsworth, and Z. Chen, “Long-wavelength continuum generation about the second dispersion zero of a tapered fiber,” Opt. Lett. 27(17), 1558–1560 (2002).
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  23. K. Saitoh and M. Koshiba, “Highly nonlinear dispersion-flattened photonic crystal fibers for supercontinuum generation in a telecommunication window,” Opt. Express 12(10), 2027–2032 (2004), http://www.opticsinfobase.org/abstract.cfm?URI=oe-12-10-2027 .
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  24. Y. Nan, J. Wang, C. Lou, and Y. Gao, “Performance analysis for a supercontinuum continuous-wave optical source for dense wavelength division multiplexed transmission,” J. Opt. A, Pure Appl. Opt. 7(3), 129–134 (2005).
    [Crossref]
  25. J. N. Kutz, C. Lyngå, and B. Eggleton, “Enhanced Supercontinuum Generation through Dispersion-Management,” Opt. Express 13(11), 3989–3998 (2005), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-13-11-3989 .
    [Crossref] [PubMed]
  26. A. Demircan and U. Bandelow, “Supercontinuum generation by the modulation instability,” Opt. Commun. 244(1-6), 181–185 (2005).
    [Crossref]
  27. D. R. Solli, C. Ropers, and B. Jalali, “Active control of optical rogue waves for stimulated supercontinuum generation,” Phys. Rev. Lett. 101(23), 233902 (2008).
    [Crossref] [PubMed]
  28. H. Lu, X. Liu, Y. Gong, X. Hu, and X. Li, “Optimization of supercontinuum generation in air-silica nanowires,” J. Opt. Soc. Am. B 27(5), 904–908 (2010).
    [Crossref]
  29. D. R. Solli, C. Ropers, P. Koonath, and B. Jalali, “Optical rogue waves,” Nature 450(7172), 1054–1057 (2007).
    [Crossref] [PubMed]
  30. G. Genty, J. M. Dudley, and B. J. Eggleton, “Modulation control and spectral shaping of optical fiber supercontinuum generation in the picosecond regime,” Appl. Phys. B 94(2), 187–194 (2009).
    [Crossref]
  31. M. Erkintalo, G. Genty, and J. M. Dudley, “On the statistical interpretation of optical rogue waves,” Eur. Phys. J. Spec. Top. 185(1), 135–144 (2010).
    [Crossref]
  32. Z. Chen, A. J. Taylor, and A. Efimov, “Soliton dynamics in non-uniform fiber tapers: analytical description through an improved moment method,” J. Opt. Soc. Am. B 27(5), 1022–1030 (2010).
    [Crossref]
  33. D. J. Dougherty, F. X. Kärtner, H. A. Haus, and E. P. Ippen, “Measurement of the Raman gain spectrum of optical fibers,” Opt. Lett. 20(1), 31–33 (1995).
    [Crossref] [PubMed]
  34. A. K. Atieh, P. Myslinski, J. Chrostowski, and P. Galko, “Measuring the Raman Time Constant for Soliton Pulses in Standard Single-Mode Fiber,” J. Lightwave Technol. 17(2), 216–221 (1999).
    [Crossref]
  35. There is considerable uncertainty about the slope of the Raman gain slope at zero frequencies. Some measurements of Raman gain for small wavelength shifts suggest that a value of TR = 3 fs when fitting to the gain peak (Fig. 1) is also consistent with a good fit to the slope near ω = 0. See e.g. A. Dogariu and D. Hagan, “Low frequency Raman gain measurements using chirped pulses,” Opt. Express 1, 73–76 (1997). G. Shaulov, V. J. Mazurczyk, and E. A. Golovchenko, “Measurement of Raman gain coefficient for small wavelength shifts,” in Optical Fiber Communication Conference, Paper TuA4 (2000).
  36. J. M. Dudley and S. Coen, “Coherence properties of supercontinuum spectra generated in photonic crystal and tapered optical fibers,” Opt. Lett. 27(13), 1180–1182 (2002).
    [Crossref]
  37. M. H. Frosz, “Validation of input-noise model for simulations of supercontinuum generation and rogue waves,” Opt. Express 18(14), 14778–14787 (2010), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-14-14778 .
    [Crossref] [PubMed]
  38. G. Genty, M. Lehtonen, and H. Ludvigsen, “Effect of cross-phase modulation on supercontinuum generated in microstructured fibers with sub-30 fs pulses,” Opt. Express 12(19), 4614–4624 (2004), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-12-19-4614 .
    [Crossref] [PubMed]
  39. J. M. Dudley, G. Genty, F. Dias, B. Kibler, and N. Akhmediev, “Modulation instability, Akhmediev Breathers and continuous wave supercontinuum generation,” Opt. Express 17(24), 21497–21508 (2009), http://www.opticsinfobase.org/abstract.cfm?URI=oe-17-24-21497 .
    [Crossref] [PubMed]
  40. K. Hammani, B. Kibler, C. Finot, and A. Picozzi, “Emergence of rogue waves from optical turbulence,” Phys. Lett. A 374(34), 3585–3589 (2010).
    [Crossref]

2010 (7)

M. Facão, M. I. Carvalho, and D. F. Parker, “Soliton self-frequency shift: Self-similar solutions and their stability,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 81(4 ), 046604 (2010).
[Crossref] [PubMed]

C. Conti, S. Stark, P. S. Russell, and F. Biancalana, “Multiple hydrodynamical shocks induced by Raman effect in photonic crystal fibres,” Phys. Rev. A 82(1), 013838 (2010).
[Crossref]

M. Erkintalo, G. Genty, and J. M. Dudley, “On the statistical interpretation of optical rogue waves,” Eur. Phys. J. Spec. Top. 185(1), 135–144 (2010).
[Crossref]

K. Hammani, B. Kibler, C. Finot, and A. Picozzi, “Emergence of rogue waves from optical turbulence,” Phys. Lett. A 374(34), 3585–3589 (2010).
[Crossref]

H. Lu, X. Liu, Y. Gong, X. Hu, and X. Li, “Optimization of supercontinuum generation in air-silica nanowires,” J. Opt. Soc. Am. B 27(5), 904–908 (2010).
[Crossref]

Z. Chen, A. J. Taylor, and A. Efimov, “Soliton dynamics in non-uniform fiber tapers: analytical description through an improved moment method,” J. Opt. Soc. Am. B 27(5), 1022–1030 (2010).
[Crossref]

M. H. Frosz, “Validation of input-noise model for simulations of supercontinuum generation and rogue waves,” Opt. Express 18(14), 14778–14787 (2010), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-14-14778 .
[Crossref] [PubMed]

2009 (2)

G. Genty, J. M. Dudley, and B. J. Eggleton, “Modulation control and spectral shaping of optical fiber supercontinuum generation in the picosecond regime,” Appl. Phys. B 94(2), 187–194 (2009).
[Crossref]

J. M. Dudley, G. Genty, F. Dias, B. Kibler, and N. Akhmediev, “Modulation instability, Akhmediev Breathers and continuous wave supercontinuum generation,” Opt. Express 17(24), 21497–21508 (2009), http://www.opticsinfobase.org/abstract.cfm?URI=oe-17-24-21497 .
[Crossref] [PubMed]

2008 (1)

D. R. Solli, C. Ropers, and B. Jalali, “Active control of optical rogue waves for stimulated supercontinuum generation,” Phys. Rev. Lett. 101(23), 233902 (2008).
[Crossref] [PubMed]

2007 (2)

D. R. Solli, C. Ropers, P. Koonath, and B. Jalali, “Optical rogue waves,” Nature 450(7172), 1054–1057 (2007).
[Crossref] [PubMed]

A. V. Gorbach and D. V. Skryabin, “Light trapping in gravity-like potentials and expansion of supercontinuum spectra in photonic-crystal fibres,” Nat. Photonics 1(11), 653–657 (2007).
[Crossref]

2006 (3)

J. M. Dudley, G. Genty, and S. Coen, “Supercontinuum generation in photonic crystal fiber,” Rev. Mod. Phys. 78(4), 1135–1184 (2006).
[Crossref]

S. V. Smirnov, J. D. Ania-Castanon, T. J. Ellingham, S. M. Kobtsev, S. V. Kukarin, and S. K. Turitsyn, “Optical spectral broadening and supercontinuum generation in telecom applications,” Opt. Fiber Technol. 12(2), 122–147 (2006).
[Crossref]

Q. Lin and G. P. Agrawal, “Raman response function for silica fibers,” Opt. Lett. 31(21), 3086–3088 (2006).
[Crossref] [PubMed]

2005 (3)

J. N. Kutz, C. Lyngå, and B. Eggleton, “Enhanced Supercontinuum Generation through Dispersion-Management,” Opt. Express 13(11), 3989–3998 (2005), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-13-11-3989 .
[Crossref] [PubMed]

Y. Nan, J. Wang, C. Lou, and Y. Gao, “Performance analysis for a supercontinuum continuous-wave optical source for dense wavelength division multiplexed transmission,” J. Opt. A, Pure Appl. Opt. 7(3), 129–134 (2005).
[Crossref]

A. Demircan and U. Bandelow, “Supercontinuum generation by the modulation instability,” Opt. Commun. 244(1-6), 181–185 (2005).
[Crossref]

2004 (2)

2002 (3)

1999 (1)

1996 (1)

N. Akhmediev, W. Krolikowski, and A. J. Lowery, “Influence of the Raman-effect on solitons in optical fibers,” Opt. Commun. 131(4-6), 260–266 (1996).
[Crossref]

1995 (1)

1990 (1)

1989 (3)

1988 (1)

1987 (2)

W. Hodel and H. P. Weber, “Decay of femtosecond higher-order solitons in an optical fiber induced by Raman self-pumping,” Opt. Lett. 12(11), 924–926 (1987).
[Crossref] [PubMed]

Y. Kodama and A. Hasegawa, “Nonlinear pulse propagation in a monomode dielectric guide,” IEEE J. Quantum Electron. 23(5), 510–524 (1987).
[Crossref]

1986 (2)

1972 (2)

R. H. Stolen, E. P. Ippen, and A. R. Tynes, “Raman oscillation in glass optical waveguides,” Appl. Phys. Lett. 20(2), 62–64 (1972).
[Crossref]

R. G. Smith, “Optical power handling capacity of low loss optical fibers as determined by stimulated Raman and brillouin scattering,” Appl. Opt. 11(11), 2489–2494 (1972).
[Crossref] [PubMed]

1970 (1)

E. P. Ippen, “Low-power quasi-cw Raman oscillator,” Appl. Phys. Lett. 16(8), 303–305 (1970).
[Crossref]

Agrawal, G. P.

Akhmediev, N.

Ania-Castanon, J. D.

S. V. Smirnov, J. D. Ania-Castanon, T. J. Ellingham, S. M. Kobtsev, S. V. Kukarin, and S. K. Turitsyn, “Optical spectral broadening and supercontinuum generation in telecom applications,” Opt. Fiber Technol. 12(2), 122–147 (2006).
[Crossref]

Atieh, A. K.

Bandelow, U.

A. Demircan and U. Bandelow, “Supercontinuum generation by the modulation instability,” Opt. Commun. 244(1-6), 181–185 (2005).
[Crossref]

Bar-Joseph, I.

Bekki, N.

Biancalana, F.

C. Conti, S. Stark, P. S. Russell, and F. Biancalana, “Multiple hydrodynamical shocks induced by Raman effect in photonic crystal fibres,” Phys. Rev. A 82(1), 013838 (2010).
[Crossref]

Birks, T. A.

Blow, K. J.

K. J. Blow and D. Wood, “Theoretical description of transient stimulated Raman scattering in optical fibers,” IEEE J. Quantum Electron. 25(12), 2665–2673 (1989).
[Crossref]

Cantrell, C. D.

Carvalho, M. I.

M. Facão, M. I. Carvalho, and D. F. Parker, “Soliton self-frequency shift: Self-similar solutions and their stability,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 81(4 ), 046604 (2010).
[Crossref] [PubMed]

Chemla, D. S.

Chen, Z.

Chernikov, S. V.

Chrostowski, J.

Coen, S.

J. M. Dudley, G. Genty, and S. Coen, “Supercontinuum generation in photonic crystal fiber,” Rev. Mod. Phys. 78(4), 1135–1184 (2006).
[Crossref]

J. M. Dudley and S. Coen, “Coherence properties of supercontinuum spectra generated in photonic crystal and tapered optical fibers,” Opt. Lett. 27(13), 1180–1182 (2002).
[Crossref]

Conti, C.

C. Conti, S. Stark, P. S. Russell, and F. Biancalana, “Multiple hydrodynamical shocks induced by Raman effect in photonic crystal fibres,” Phys. Rev. A 82(1), 013838 (2010).
[Crossref]

Demircan, A.

A. Demircan and U. Bandelow, “Supercontinuum generation by the modulation instability,” Opt. Commun. 244(1-6), 181–185 (2005).
[Crossref]

Dias, F.

Dougherty, D. J.

Dudley, J. M.

M. Erkintalo, G. Genty, and J. M. Dudley, “On the statistical interpretation of optical rogue waves,” Eur. Phys. J. Spec. Top. 185(1), 135–144 (2010).
[Crossref]

G. Genty, J. M. Dudley, and B. J. Eggleton, “Modulation control and spectral shaping of optical fiber supercontinuum generation in the picosecond regime,” Appl. Phys. B 94(2), 187–194 (2009).
[Crossref]

J. M. Dudley, G. Genty, F. Dias, B. Kibler, and N. Akhmediev, “Modulation instability, Akhmediev Breathers and continuous wave supercontinuum generation,” Opt. Express 17(24), 21497–21508 (2009), http://www.opticsinfobase.org/abstract.cfm?URI=oe-17-24-21497 .
[Crossref] [PubMed]

J. M. Dudley, G. Genty, and S. Coen, “Supercontinuum generation in photonic crystal fiber,” Rev. Mod. Phys. 78(4), 1135–1184 (2006).
[Crossref]

J. M. Dudley and S. Coen, “Coherence properties of supercontinuum spectra generated in photonic crystal and tapered optical fibers,” Opt. Lett. 27(13), 1180–1182 (2002).
[Crossref]

Efimov, A.

Eggleton, B.

Eggleton, B. J.

G. Genty, J. M. Dudley, and B. J. Eggleton, “Modulation control and spectral shaping of optical fiber supercontinuum generation in the picosecond regime,” Appl. Phys. B 94(2), 187–194 (2009).
[Crossref]

Ellingham, T. J.

S. V. Smirnov, J. D. Ania-Castanon, T. J. Ellingham, S. M. Kobtsev, S. V. Kukarin, and S. K. Turitsyn, “Optical spectral broadening and supercontinuum generation in telecom applications,” Opt. Fiber Technol. 12(2), 122–147 (2006).
[Crossref]

Erkintalo, M.

M. Erkintalo, G. Genty, and J. M. Dudley, “On the statistical interpretation of optical rogue waves,” Eur. Phys. J. Spec. Top. 185(1), 135–144 (2010).
[Crossref]

Facão, M.

M. Facão, M. I. Carvalho, and D. F. Parker, “Soliton self-frequency shift: Self-similar solutions and their stability,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 81(4 ), 046604 (2010).
[Crossref] [PubMed]

Finot, C.

K. Hammani, B. Kibler, C. Finot, and A. Picozzi, “Emergence of rogue waves from optical turbulence,” Phys. Lett. A 374(34), 3585–3589 (2010).
[Crossref]

Frosz, M. H.

Galko, P.

Gao, Y.

Y. Nan, J. Wang, C. Lou, and Y. Gao, “Performance analysis for a supercontinuum continuous-wave optical source for dense wavelength division multiplexed transmission,” J. Opt. A, Pure Appl. Opt. 7(3), 129–134 (2005).
[Crossref]

Genty, G.

M. Erkintalo, G. Genty, and J. M. Dudley, “On the statistical interpretation of optical rogue waves,” Eur. Phys. J. Spec. Top. 185(1), 135–144 (2010).
[Crossref]

G. Genty, J. M. Dudley, and B. J. Eggleton, “Modulation control and spectral shaping of optical fiber supercontinuum generation in the picosecond regime,” Appl. Phys. B 94(2), 187–194 (2009).
[Crossref]

J. M. Dudley, G. Genty, F. Dias, B. Kibler, and N. Akhmediev, “Modulation instability, Akhmediev Breathers and continuous wave supercontinuum generation,” Opt. Express 17(24), 21497–21508 (2009), http://www.opticsinfobase.org/abstract.cfm?URI=oe-17-24-21497 .
[Crossref] [PubMed]

J. M. Dudley, G. Genty, and S. Coen, “Supercontinuum generation in photonic crystal fiber,” Rev. Mod. Phys. 78(4), 1135–1184 (2006).
[Crossref]

G. Genty, M. Lehtonen, and H. Ludvigsen, “Effect of cross-phase modulation on supercontinuum generated in microstructured fibers with sub-30 fs pulses,” Opt. Express 12(19), 4614–4624 (2004), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-12-19-4614 .
[Crossref] [PubMed]

Gong, Y.

Gorbach, A. V.

A. V. Gorbach and D. V. Skryabin, “Light trapping in gravity-like potentials and expansion of supercontinuum spectra in photonic-crystal fibres,” Nat. Photonics 1(11), 653–657 (2007).
[Crossref]

Gordon, J. P.

Hammani, K.

K. Hammani, B. Kibler, C. Finot, and A. Picozzi, “Emergence of rogue waves from optical turbulence,” Phys. Lett. A 374(34), 3585–3589 (2010).
[Crossref]

Harbold, J. M.

Hasegawa, A.

K. Tai, A. Hasegawa, and N. Bekki, “Fission of optical solitons induced by stimulated Raman effect,” Opt. Lett. 13(5), 392–394 (1988).
[Crossref] [PubMed]

Y. Kodama and A. Hasegawa, “Nonlinear pulse propagation in a monomode dielectric guide,” IEEE J. Quantum Electron. 23(5), 510–524 (1987).
[Crossref]

Haus, H. A.

Hodel, W.

Hollenbeck, D.

Hu, X.

Ilday, F. Ö.

Ippen, E. P.

D. J. Dougherty, F. X. Kärtner, H. A. Haus, and E. P. Ippen, “Measurement of the Raman gain spectrum of optical fibers,” Opt. Lett. 20(1), 31–33 (1995).
[Crossref] [PubMed]

R. H. Stolen, E. P. Ippen, and A. R. Tynes, “Raman oscillation in glass optical waveguides,” Appl. Phys. Lett. 20(2), 62–64 (1972).
[Crossref]

E. P. Ippen, “Low-power quasi-cw Raman oscillator,” Appl. Phys. Lett. 16(8), 303–305 (1970).
[Crossref]

Islam, M. N.

Jalali, B.

D. R. Solli, C. Ropers, and B. Jalali, “Active control of optical rogue waves for stimulated supercontinuum generation,” Phys. Rev. Lett. 101(23), 233902 (2008).
[Crossref] [PubMed]

D. R. Solli, C. Ropers, P. Koonath, and B. Jalali, “Optical rogue waves,” Nature 450(7172), 1054–1057 (2007).
[Crossref] [PubMed]

Kärtner, F. X.

Kibler, B.

Kobtsev, S. M.

S. V. Smirnov, J. D. Ania-Castanon, T. J. Ellingham, S. M. Kobtsev, S. V. Kukarin, and S. K. Turitsyn, “Optical spectral broadening and supercontinuum generation in telecom applications,” Opt. Fiber Technol. 12(2), 122–147 (2006).
[Crossref]

Kodama, Y.

Y. Kodama and A. Hasegawa, “Nonlinear pulse propagation in a monomode dielectric guide,” IEEE J. Quantum Electron. 23(5), 510–524 (1987).
[Crossref]

Koonath, P.

D. R. Solli, C. Ropers, P. Koonath, and B. Jalali, “Optical rogue waves,” Nature 450(7172), 1054–1057 (2007).
[Crossref] [PubMed]

Koshiba, M.

Krolikowski, W.

N. Akhmediev, W. Krolikowski, and A. J. Lowery, “Influence of the Raman-effect on solitons in optical fibers,” Opt. Commun. 131(4-6), 260–266 (1996).
[Crossref]

Kukarin, S. V.

S. V. Smirnov, J. D. Ania-Castanon, T. J. Ellingham, S. M. Kobtsev, S. V. Kukarin, and S. K. Turitsyn, “Optical spectral broadening and supercontinuum generation in telecom applications,” Opt. Fiber Technol. 12(2), 122–147 (2006).
[Crossref]

Kutz, J. N.

Lehtonen, M.

Li, X.

Lin, Q.

Liu, X.

Lou, C.

Y. Nan, J. Wang, C. Lou, and Y. Gao, “Performance analysis for a supercontinuum continuous-wave optical source for dense wavelength division multiplexed transmission,” J. Opt. A, Pure Appl. Opt. 7(3), 129–134 (2005).
[Crossref]

Lowery, A. J.

N. Akhmediev, W. Krolikowski, and A. J. Lowery, “Influence of the Raman-effect on solitons in optical fibers,” Opt. Commun. 131(4-6), 260–266 (1996).
[Crossref]

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Phys. Rev. A (1)

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Other (2)

There is considerable uncertainty about the slope of the Raman gain slope at zero frequencies. Some measurements of Raman gain for small wavelength shifts suggest that a value of TR = 3 fs when fitting to the gain peak (Fig. 1) is also consistent with a good fit to the slope near ω = 0. See e.g. A. Dogariu and D. Hagan, “Low frequency Raman gain measurements using chirped pulses,” Opt. Express 1, 73–76 (1997). G. Shaulov, V. J. Mazurczyk, and E. A. Golovchenko, “Measurement of Raman gain coefficient for small wavelength shifts,” in Optical Fiber Communication Conference, Paper TuA4 (2000).

G. P. Agrawal, Nonlinear Fiber Optics, 4th Edition, (Academic Press, Boston, 2007)

Supplementary Material (3)

» Media 1: MOV (1485 KB)     
» Media 2: MOV (1507 KB)     
» Media 3: MOV (1496 KB)     

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Figures (6)

Fig. 1
Fig. 1 (a) Comparison of the two forms of the Raman response function: the experimental response (solid line) is adapted from Ref. 13. The linear response (dashed line) is calculated as explained in the text. (b) As a function of initial pulse bandwidth, we compare the rate of Raman self-frequency shift experienced by an N = 1 soliton for: the experimental (full) model (black) and the linear model (red).

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Fig. 2
Fig. 2 Comparison of output spectra for different input pulse parameters as shown, using the realistic finite bandwidth Raman response (black) and the linear Raman model (red). The propagation distance in each case is z = 10 L fiss. Dimensional parameters are: (i) P 0 = 5.2 W, z = 900 m; (ii) P 0 = 129.4 W, z = 36 m; (iii) P 0 = 2.07 kW, z = 2.26 m; (iv) P 0 = 20.7 W, z = 450 m; (v) P 0 = 517.5 W, z = 18 m; (vi) P 0 = 8.28 kW, z = 1.13 m. Note the change in wavelength span in figures (iv)-(vi).

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Fig. 3
Fig. 3 Simulated propagation dynamics and output characteristics for a 50 fs input pulse at 1035 nm with N = 10 using (a) the linear Raman model and (b) the full Raman response. For propagation distance of z = 20 L fiss subfigures (a) and (b) show the output temporal profile (top) and the spectral evolution dynamics (bottom). Results in (c) show the output spectrograms for both cases as indicated. Spectrogram movies of the results in Fig. 3(a) and 3(b) are at (Media 1) and (Media 2) respectively. The spectrograms are calculated using a 200 fs temporal gate function to sample the evolving field (see e.g. Ref. 11 for more details on the spectrogram representation of supercontinuum generation.)

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Fig. 4
Fig. 4 Simulated propagation dynamics and output characteristics for a 4 ps pulse at 1035 nm of 150 W peak power such that N ~107 and propagation is highly incoherent with significant shot to shot fluctuations. Results from simulations using the linear Raman gain and full Raman gain models are compared (a) at fixed propagation distance of 15 m and (b) for comparable mean spectral broadening on the long wavelength side of ~200 nm at the −20 dB level relative to the residual pump.

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Fig. 5
Fig. 5 Supercontinuum coherence as a function of soliton number for (a) 100 fs and (b) 200 fs pulses. Results from the full model (open circles) and linear model (closed red circles) are compared. The predicted coherence properties from the two models differ significantly in the partially coherent regime.

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Fig. 6
Fig. 6 Simulated propagation dynamics of soliton collapse observed in the linear model. (a) Soliton energy vs. distance (b) temporal profile evolution of the soltion plotted in a frame of reference moving at the soliton velocity. (c) and (d) show the output spectrograms before and after collapse at z = 16 L fiss and z = 20 L fiss, respectively. The spectrograms were calculated using a 200 fs gate function. Input pulse parameters are identical to that of Fig. 3. A movie of the spectrogram evolution in (c) which shows also the projected temporal and spectral evolution plotted on log scales is at (Media 3).

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Equations (2)

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∂ A ∂ z − ∑ k ≥ 2 i k + 1 k ! β k ∂ k A ∂ t k = i γ ( 1 + i τ s h o c k ∂ ∂ t ) ( A ( z , t ) ∫ − ∞ + ∞ R ( t ' ) | A ( z , t − t ' ) | 2 d t ' ) .
∂ A ∂ z − ∑ k ≥ 2 i k + 1 k ! β k ∂ k A ∂ t k = i γ ( A | A | 2 + i τ s h o c k ∂ ∂ t ( A | A | 2 ) − T R A ∂ | A | 2 ∂ t ) .

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