Abstract

A new model of pump noise in supercontinuum and rogue wave generation is presented. Simulations are compared with experiments and show that the new model provides significantly better agreement than the currently ubiquitously used one-photon-per-mode model. The new model also allows for a study of the influence of the pump spectral line width on the spectral broadening mechanisms. Specifically, it is found that for four-wave mixing (FWM) a narrow spectral line width (≲ 0.1 nm) initially leads to a build-up of FWM from quantum noise, whereas a broad spectral line width (≳ 1 nm) initially leads to a gradual broadening of the pump spectrum. Since the new model provides better agreement with experiments and is still simple to implement, it is particularly important that it is used for future studies of the statistical properties of nonlinear spectral broadening, such as the formation of rogue waves.

© 2010 Optical Society of America

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  1. M. N. Islam, G. Sucha, I. Bar-Joseph, M. Wegener, J. P. Gordon, and D. S. Chemla, “Femtosecond distributed soliton spectrum in fibers,” J. Opt. Soc. Am. B 6, 1149–1158 (1989), http://www.opticsinfobase.org/abstract.cfm?URI=josab-6-6-1149.
    [CrossRef]
  2. M. H. Frosz, O. Bang, and A. Bjarklev, “Soliton collision and Raman gain regimes in continuous-wave pumped supercontinuum generation,” Opt. Express 14, 9391–9407 (2006), http://www.opticsinfobase.org/abstract.cfm?URI=oe-14-20-9391.
    [CrossRef] [PubMed]
  3. D. R. Solli, C. Ropers, P. Koonath, and B. Jalali, “Optical rogue waves,” Nature 450, 1054–1057 (2007).
    [CrossRef] [PubMed]
  4. D. R. Solli, C. Ropers, B. Jalali, and C. Ropers, “Active control of rogue waves for stimulated supercontinuum generation,” Phys. Rev. Lett. 101, 233902 (2008).
    [CrossRef] [PubMed]
  5. J. M. Dudley, G. Genty, and B. J. Eggleton, “Harnessing and control of optical rogue waves in supercontinuum generation,” Opt. Express 16, 3644–3651 (2008).
    [CrossRef] [PubMed]
  6. A. Mussot, A. Kudlinski, M. Kolobov, E. Louvergneaux, M. Douay, and M. Taki, “Observation of extreme temporal events in CW-pumped supercontinuum,” Opt. Express 17, 17010–17015 (2009).
    [CrossRef] [PubMed]
  7. J. M. Dudley, G. Genty, and S. Coen, “Supercontinuum generation in photonic crystal fiber,” Rev. Mod. Phys. 78, 1135–1184 (2006), http://link.aps.org/abstract/RMP/v78/p1135.
    [CrossRef]
  8. J. C. Travers, A. B. Rulkov, B. A. Cumberland, S. V. Popov, and J. R. Taylor, “Visible supercontinuum generation in photonic crystal fibers with a 400 W continuous wave fiber laser,” Opt. Express 16, 14435–14447 (2008).
    [CrossRef] [PubMed]
  9. J. C. Travers, “Continuous wave supercontinuum generation” Chap. 8 in Supercontinuum generation in optical fibers, J. M. Dudley and J. R. Taylor, eds. (Cambridge University Press, 2010), ISBN 978–0-521–51480–4.
    [CrossRef]
  10. J. Lægsgaard, “Mode profile dispersion in the generalised nonlinear Schrödinger equation,” Opt. Express 15, 16110–16123 (2007).
    [CrossRef] [PubMed]
  11. J. C. Travers, M. H. Frosz, and J. M. Dudley, “Nonlinear fibre optics overview” Chap. 3 in Supercontinuum generation in optical fibers, J. M. Dudley and J. R. Taylor, eds. (Cambridge University Press, 2010), ISBN 978–0-521–51480–4.
  12. G. P. Agrawal, “Nonlinear Fiber Optics (Academic Press, Burlington, MA, USA, 2007), 4th ed.
  13. K. J. Blow, and D. Wood, “Theoretical description of transient stimulated Raman scattering in optical fibers,” IEEE J. Quantum Electron. 25, 2665–2673 (1989).
    [CrossRef]
  14. S. B. Cavalcanti, G. P. Agrawal, and M. Yu, “Noise amplification in dispersive nonlinear media,” Phys. Rev. A 51, 4086–4092 (1995), http://dx.doi.org/10.1103/PhysRevA.51.4086.
    [CrossRef] [PubMed]
  15. A. Mussot, E. Lantz, H. Maillotte, T. Sylvestre, C. Finot, and S. Pitois, “Spectral broadening of a partially coherent CW laser beam in single-mode optical fibers,” Opt. Express 12, 2838–2843 (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-13-2838.
    [CrossRef] [PubMed]
  16. J. W. Goodman, Statistical Optics (John Wiley & Sons Inc., 2000). ISBN 0471399167.
  17. J. Schröder, and S. Coen, “Observation of high-contrast, fast intensity noise of a continuous wave Raman fiber laser,” Opt. Express 17, 16444–16449 (2009).
    [CrossRef] [PubMed]
  18. B. E. A. Saleh, and M. C. Teich, Fundamentals of Photonics (John Wiley & Sons, Inc., New York, 1991). ISBN 0–471–83965–5.
    [CrossRef]
  19. P. M. Moselund, “Long-pulse supercontinuum light sources,” Ph.D. thesis, DTU Fotonik, Department of Photonics Engineering, Technical University of Denmark, Oersteds Plads 343, DK-2800 Kgs. Lyngby, Denmark (2009), ISBN: 87–02062–31–8, http://orbit.dtu.dk/getResource?recordId=251910&objectId=1&versionId=1.
  20. M. H. Frosz, P. M. Moselund, P. D. Rasmussen, C. L. Thomsen, and O. Bang, “Increasing the blue-shift of a picosecond pumped supercontinuum” Chap. 7 in Supercontinuum generation in optical fibers, J. M. Dudley and J. R. Taylor, eds. (Cambridge University Press, 2010), ISBN 978–0-521–51480–4.
  21. A. Kudlinski, Personal communication (2010).
  22. J. M. Dudley, G. Genty, F. Dias, B. Kibler, and N. Akhmediev, “Modulation instability, Akhmediev Breathers and continuous wave supercontinuum generation,” Opt. Express 17, 21497–21508 (2009).
    [CrossRef] [PubMed]

2009

2008

2007

2006

2004

1995

S. B. Cavalcanti, G. P. Agrawal, and M. Yu, “Noise amplification in dispersive nonlinear media,” Phys. Rev. A 51, 4086–4092 (1995), http://dx.doi.org/10.1103/PhysRevA.51.4086.
[CrossRef] [PubMed]

1989

Agrawal, G. P.

S. B. Cavalcanti, G. P. Agrawal, and M. Yu, “Noise amplification in dispersive nonlinear media,” Phys. Rev. A 51, 4086–4092 (1995), http://dx.doi.org/10.1103/PhysRevA.51.4086.
[CrossRef] [PubMed]

Akhmediev, N.

Bang, O.

Bar-Joseph, I.

Bjarklev, 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, 2665–2673 (1989).
[CrossRef]

Cavalcanti, S. B.

S. B. Cavalcanti, G. P. Agrawal, and M. Yu, “Noise amplification in dispersive nonlinear media,” Phys. Rev. A 51, 4086–4092 (1995), http://dx.doi.org/10.1103/PhysRevA.51.4086.
[CrossRef] [PubMed]

Chemla, D. S.

Coen, S.

Cumberland, B. A.

Dias, F.

Douay, M.

Dudley, J. M.

Eggleton, B. J.

Finot, C.

Frosz, M. H.

Genty, G.

Gordon, J. P.

Islam, M. N.

Jalali, B.

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

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

Kibler, B.

Kolobov, M.

Koonath, P.

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

Kudlinski, A.

Lægsgaard, J.

Lantz, E.

Louvergneaux, E.

Maillotte, H.

Mussot, A.

Pitois, S.

Popov, S. V.

Ropers, C.

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

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

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

Rulkov, A. B.

Schröder, J.

Solli, D. R.

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

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

Sucha, G.

Sylvestre, T.

Taki, M.

Taylor, J. R.

Travers, J. C.

Wegener, M.

Wood, D.

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

Yu, M.

S. B. Cavalcanti, G. P. Agrawal, and M. Yu, “Noise amplification in dispersive nonlinear media,” Phys. Rev. A 51, 4086–4092 (1995), http://dx.doi.org/10.1103/PhysRevA.51.4086.
[CrossRef] [PubMed]

IEEE J. Quantum Electron.

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

J. Opt. Soc. Am. B

Nature

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

Opt. Express

M. H. Frosz, O. Bang, and A. Bjarklev, “Soliton collision and Raman gain regimes in continuous-wave pumped supercontinuum generation,” Opt. Express 14, 9391–9407 (2006), http://www.opticsinfobase.org/abstract.cfm?URI=oe-14-20-9391.
[CrossRef] [PubMed]

J. M. Dudley, G. Genty, and B. J. Eggleton, “Harnessing and control of optical rogue waves in supercontinuum generation,” Opt. Express 16, 3644–3651 (2008).
[CrossRef] [PubMed]

A. Mussot, A. Kudlinski, M. Kolobov, E. Louvergneaux, M. Douay, and M. Taki, “Observation of extreme temporal events in CW-pumped supercontinuum,” Opt. Express 17, 17010–17015 (2009).
[CrossRef] [PubMed]

J. C. Travers, A. B. Rulkov, B. A. Cumberland, S. V. Popov, and J. R. Taylor, “Visible supercontinuum generation in photonic crystal fibers with a 400 W continuous wave fiber laser,” Opt. Express 16, 14435–14447 (2008).
[CrossRef] [PubMed]

J. Lægsgaard, “Mode profile dispersion in the generalised nonlinear Schrödinger equation,” Opt. Express 15, 16110–16123 (2007).
[CrossRef] [PubMed]

J. Schröder, and S. Coen, “Observation of high-contrast, fast intensity noise of a continuous wave Raman fiber laser,” Opt. Express 17, 16444–16449 (2009).
[CrossRef] [PubMed]

A. Mussot, E. Lantz, H. Maillotte, T. Sylvestre, C. Finot, and S. Pitois, “Spectral broadening of a partially coherent CW laser beam in single-mode optical fibers,” Opt. Express 12, 2838–2843 (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-13-2838.
[CrossRef] [PubMed]

J. M. Dudley, G. Genty, F. Dias, B. Kibler, and N. Akhmediev, “Modulation instability, Akhmediev Breathers and continuous wave supercontinuum generation,” Opt. Express 17, 21497–21508 (2009).
[CrossRef] [PubMed]

Phys. Rev. A

S. B. Cavalcanti, G. P. Agrawal, and M. Yu, “Noise amplification in dispersive nonlinear media,” Phys. Rev. A 51, 4086–4092 (1995), http://dx.doi.org/10.1103/PhysRevA.51.4086.
[CrossRef] [PubMed]

Phys. Rev. Lett.

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

Rev. Mod. Phys.

J. M. Dudley, G. Genty, and S. Coen, “Supercontinuum generation in photonic crystal fiber,” Rev. Mod. Phys. 78, 1135–1184 (2006), http://link.aps.org/abstract/RMP/v78/p1135.
[CrossRef]

Other

J. C. Travers, “Continuous wave supercontinuum generation” Chap. 8 in Supercontinuum generation in optical fibers, J. M. Dudley and J. R. Taylor, eds. (Cambridge University Press, 2010), ISBN 978–0-521–51480–4.
[CrossRef]

J. C. Travers, M. H. Frosz, and J. M. Dudley, “Nonlinear fibre optics overview” Chap. 3 in Supercontinuum generation in optical fibers, J. M. Dudley and J. R. Taylor, eds. (Cambridge University Press, 2010), ISBN 978–0-521–51480–4.

G. P. Agrawal, “Nonlinear Fiber Optics (Academic Press, Burlington, MA, USA, 2007), 4th ed.

B. E. A. Saleh, and M. C. Teich, Fundamentals of Photonics (John Wiley & Sons, Inc., New York, 1991). ISBN 0–471–83965–5.
[CrossRef]

P. M. Moselund, “Long-pulse supercontinuum light sources,” Ph.D. thesis, DTU Fotonik, Department of Photonics Engineering, Technical University of Denmark, Oersteds Plads 343, DK-2800 Kgs. Lyngby, Denmark (2009), ISBN: 87–02062–31–8, http://orbit.dtu.dk/getResource?recordId=251910&objectId=1&versionId=1.

M. H. Frosz, P. M. Moselund, P. D. Rasmussen, C. L. Thomsen, and O. Bang, “Increasing the blue-shift of a picosecond pumped supercontinuum” Chap. 7 in Supercontinuum generation in optical fibers, J. M. Dudley and J. R. Taylor, eds. (Cambridge University Press, 2010), ISBN 978–0-521–51480–4.

A. Kudlinski, Personal communication (2010).

J. W. Goodman, Statistical Optics (John Wiley & Sons Inc., 2000). ISBN 0471399167.

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Figures (5)

Fig. 1.
Fig. 1.

Comparison of an experimentally measured spectrum from a Cheos Oy SNP-13E laser emitting 690 ps (FWHM) pulses at 1064 nm with ~ 0.1 nm FWHM linewidth (blue, solid); spectrum obtained from one-photon-per-mode model (green, dashed); Lorentzian power spectrum (red, dotted); Gaussian power spectrum (cyan, dash-dotted). All spectra have the same FWHM width and are normalized to have the same total power.

Fig. 2.
Fig. 2.

Simulations of spectra after 36 cm of propagation (green, dashed) for the same parameters as used in Section 3.2. The input spectra are shown as blue, solid lines. Left: using the one-photon-per-mode model. Right: using the Gaussian spectrum phase-diffusion model.

Fig. 3.
Fig. 3.

Experimental measurement by P. M. Moselund [19] obtained with a 22 cm photonic crystal fiber pumped by 14.6 ps pulses with 6.0 kW peak power (blue, solid). Corresponding simulations using either one-photon-per-mode (green, dashed), phase noise with Gaussian spectrum (red, dotted), or a combination of the two noise models (cyan, dash-dotted). The simulated spectra are obtained by averaging over 5 simulations for each type of input noise, and afterwards smoothed by convolution with a 2 nm wide Gaussian function.

Fig. 4.
Fig. 4.

Experimental measurement of output spectrum containing rogue waves, with 10 W CW pump and other parameters as presented in Ref. [6] (blue, solid). Simulated output spectra using either one-photon-per-mode input noise (green, dashed), Gaussian spectrum with phase noise (red, dotted), or a combination of the two noise models (cyan, dashdotted). The simulated spectra are obtained by averaging over 50 simulations for all types of input noise, and afterwards smoothed by convolution with a 1 nmwide Gaussian function and down-sampling to 1001 points.

Fig. 5.
Fig. 5.

Experimental measurement of output spectrum for parameters as presented in Ref. [22] (blue, solid). Simulated output spectra using either one-photon-per-mode input noise (green, dashed), Gaussian spectrum with phase noise (red, dotted), or a combination of the two noise models (cyan, dash-dotted). The simulated spectra are obtained by averaging over 10 simulations for all types of input noise, and afterwards smoothed by convolution with a 0.4 nm wide Gaussian function and down-sampling to 1001 points.

Equations (14)

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C ˜ z i { β ( ω ) β ( ω 0 ) β 1 ( ω 0 ) [ ω ω 0 ] } C ˜ ( z , ω ) + α ( ω ) 2 C ˜ ( z , ω ) =
i γ ( ω ) [ 1 + ω ω 0 ω 0 ] { C ( z , t ) R ( T ) C ( z , T T ) 2 d T } ,
{ C ( z , t ) } = C ˜ ( z , ω ) = [ A eff ( ω ) A eff ( ω 0 ) ] 1 4 A ˜ ( z , ω ) ,
γ ( ω ) = n 2 n 0 ω 0 cn eff ( ω ) A eff ( ω ) A eff ( ω 0 ) ,
R ( t ) = ( 1 f R ) δ ( t ) + f R τ 1 2 + τ 2 2 τ 1 τ 2 2 exp ( t τ 2 ) sin ( t τ 1 ) Θ ( t ) ,
A ( 0 , T ) = P ( T ) exp [ i δ ϕ ( T ) ]
v i = v 0 + 1 2 π d ( δ ϕ ) d t = v 0 + v R ( T ) .
δ ϕ ( T ) = 2 π t v R ( ξ ) d ξ .
σ v R 2 = Δ v FWHM B 2 π ,
A ˜ L ( ω ) 2 = P av Δ v FWHM 2 π 1 ( v v 0 ) 2 + ( Δ v FWHM 2 ) 2 ,
A ˜ G ( ω ) 2 = P av 1 Δ v π exp [ ( v v 0 ) 2 Δ v 2 ] ,
A ˜ G ( ω ) A ˜ L ( ω ) =
2 [ ln ( 2 ) π ] 1 4 Δ v FWHM exp [ 1 2 ( v v 0 ) 2 [ Δ v FWHM ( 2 ln ( 2 ) ) ] 2 ] ( v v 0 ) 2 + ( Δ v FWHM 2 ) 2 .
d A ( z , λ s ) 2 d z A ( z , λ s ) ) 2 A ( z , λ p ) ) 2 ,

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