Abstract

We report experimental observation and characterization of rogue wave-like extreme value statistics arising from pump-signal noise transfer in a fiber Raman amplifier. Specifically, by exploiting Raman amplification with an incoherent pump, the amplified signal is shown to develop a series of temporal intensity spikes whose peak power follows a power-law probability distribution. The results are interpreted using a numerical model of the Raman gain process using coupled nonlinear Schrödinger equations, and the numerical model predicts results in good agreement with experiment.

© 2008 Optical Society of America

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  1. K. Tai, A. Hasegawa, and A. Tomita, "Observation of modulational instability in optical fibers," Phys. Rev. Lett. 56, 135-138 (1986).
    [CrossRef] [PubMed]
  2. L. F. Mollenauer, R. H. Stolen, and J. P. Gordon, "Experimental observation of picosecond pulse narrowing and solitons in optical fibers," Phys. Rev. Lett. 45, 1095-1098 (1980).
    [CrossRef]
  3. A. Picozzi, M. Haelterman, S. Pitois, and G. Millot, "Incoherent solitons in instantaneous response nonlinear media," Phys. Rev. Lett. 92, 143906 (2004).
    [CrossRef] [PubMed]
  4. J. M. Dudley, C. Finot, G. Millot, and D. J. Richardson, "Self-similarity in ultrafast nonlinear optics," Nat. Phys. 3, 597-603 (2007).
    [CrossRef]
  5. D. R. Solli, C. Ropers, P. Koonath, and B. Jalali, "Optical rogue waves," Nature 450, 1054 (2007).
    [CrossRef] [PubMed]
  6. 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]
  7. J. R. Thompson and R. Roy, "Statistical fluctuations in multiple four-wave mixing in a single-mode optical fiber," Phys. Rev. A 44, 7605-7614 (1991).
    [CrossRef] [PubMed]
  8. E. Huynh, E. Manzano, A. Medrano, T. Thorn, A. Zavala, C. G. Goedde, and J. R. Thompson, "The interplay of thermal and pump fluctuations in stimulated Brillouin scattering," Opt. Commun. 281, 836-845 (2008).
    [CrossRef]
  9. L. Garcia, J. Jenkins, Y. Lee, N. Poole, K. Salit, P. Sidereas, C. G. Goedde, and J. R. Thompson, "Influence of classical pump noise on long-pulse multiorder stimulated Raman scattering in optical fiber," J. Opt. Soc. Am. B 19, 2727-2736 (2002).
    [CrossRef]
  10. A. Betlej, P. Schmitt, P. Sidereas, R. Tracy, C. G. Goedde, and J. R. Thompson, "Increased Stokes pulse energy variation from amplified classical noise in a fiber Raman generator," Opt. Express 13, 2948-2960 (2005).
    [CrossRef] [PubMed]
  11. E. Landahl, D. Baiocchi, and J. R. Thompson, "A simple analytic model for noise shaping by an optical fiber Raman generator," Opt. Commun. 150, 339-347 (1998).
    [CrossRef]
  12. C. Headley and G. P. Agrawal, Raman amplification in fiber optical communications (Academic Press, 2005).
  13. A. S. Grabtchikov, A. I. Vodtchits, and V. A. Orlovich, "Pulse-energy statistics in the linear regime of stimulated Raman scattering with a broad-band pump," Phys. Rev. A 56, 1666-1669 (1997).
    [CrossRef]
  14. F. Vanholsbeeck, S. Martin-Lopez, M. Gonzalez-Herraez, and S. Coen, "The role of pump incoherence in continuous-wvae supercontinuum generation," Opt. Express 13, 6615-6625 (2005).
    [CrossRef] [PubMed]
  15. C. Fludger, V. Handerek, and R. J. Mears, "Pump to signal RIN transfer in Raman fiber amplifiers," J. Lightwave Technol. 19, 1140-1148 (2001).
    [CrossRef]
  16. G. Ravet, A. A. Fotidai, and P. Mégret, "Spectral broadening in Raman fiber amplifier pumped by partially coherent wave," in CLEO Europe, (2007).
  17. B. E. Olsson, P. �?hlen, L. Rau, and D. J. Blumenthal, "A simple and robust 40-Gb/s wavelength converter using fiber cross-phase modulation and optical filtering," IEEE Photon. Technol. Lett. 12, 846-848 (2000).
    [CrossRef]
  18. D. Borlaug and B. Jalali, "Extreme value statistics in silicon photonics," to be presented at the 21st Annual Meeting of The IEEE Lasers & Electro-Optics Society, Newport Beach, United-States, 9-13 Nov. 2008. http://arxiv.org/abs/0809.0152v1

2008 (2)

E. Huynh, E. Manzano, A. Medrano, T. Thorn, A. Zavala, C. G. Goedde, and J. R. Thompson, "The interplay of thermal and pump fluctuations in stimulated Brillouin scattering," Opt. Commun. 281, 836-845 (2008).
[CrossRef]

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]

2007 (2)

J. M. Dudley, C. Finot, G. Millot, and D. J. Richardson, "Self-similarity in ultrafast nonlinear optics," Nat. Phys. 3, 597-603 (2007).
[CrossRef]

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

2005 (2)

2004 (1)

A. Picozzi, M. Haelterman, S. Pitois, and G. Millot, "Incoherent solitons in instantaneous response nonlinear media," Phys. Rev. Lett. 92, 143906 (2004).
[CrossRef] [PubMed]

2002 (1)

2001 (1)

2000 (1)

B. E. Olsson, P. �?hlen, L. Rau, and D. J. Blumenthal, "A simple and robust 40-Gb/s wavelength converter using fiber cross-phase modulation and optical filtering," IEEE Photon. Technol. Lett. 12, 846-848 (2000).
[CrossRef]

1998 (1)

E. Landahl, D. Baiocchi, and J. R. Thompson, "A simple analytic model for noise shaping by an optical fiber Raman generator," Opt. Commun. 150, 339-347 (1998).
[CrossRef]

1997 (1)

A. S. Grabtchikov, A. I. Vodtchits, and V. A. Orlovich, "Pulse-energy statistics in the linear regime of stimulated Raman scattering with a broad-band pump," Phys. Rev. A 56, 1666-1669 (1997).
[CrossRef]

1991 (1)

J. R. Thompson and R. Roy, "Statistical fluctuations in multiple four-wave mixing in a single-mode optical fiber," Phys. Rev. A 44, 7605-7614 (1991).
[CrossRef] [PubMed]

1986 (1)

K. Tai, A. Hasegawa, and A. Tomita, "Observation of modulational instability in optical fibers," Phys. Rev. Lett. 56, 135-138 (1986).
[CrossRef] [PubMed]

1980 (1)

L. F. Mollenauer, R. H. Stolen, and J. P. Gordon, "Experimental observation of picosecond pulse narrowing and solitons in optical fibers," Phys. Rev. Lett. 45, 1095-1098 (1980).
[CrossRef]

??hlen, P.

B. E. Olsson, P. �?hlen, L. Rau, and D. J. Blumenthal, "A simple and robust 40-Gb/s wavelength converter using fiber cross-phase modulation and optical filtering," IEEE Photon. Technol. Lett. 12, 846-848 (2000).
[CrossRef]

Baiocchi, D.

E. Landahl, D. Baiocchi, and J. R. Thompson, "A simple analytic model for noise shaping by an optical fiber Raman generator," Opt. Commun. 150, 339-347 (1998).
[CrossRef]

Betlej, A.

Blumenthal, D. J.

B. E. Olsson, P. �?hlen, L. Rau, and D. J. Blumenthal, "A simple and robust 40-Gb/s wavelength converter using fiber cross-phase modulation and optical filtering," IEEE Photon. Technol. Lett. 12, 846-848 (2000).
[CrossRef]

Coen, S.

Dudley, J. M.

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]

J. M. Dudley, C. Finot, G. Millot, and D. J. Richardson, "Self-similarity in ultrafast nonlinear optics," Nat. Phys. 3, 597-603 (2007).
[CrossRef]

Eggleton, B. J.

Finot, C.

J. M. Dudley, C. Finot, G. Millot, and D. J. Richardson, "Self-similarity in ultrafast nonlinear optics," Nat. Phys. 3, 597-603 (2007).
[CrossRef]

Fludger, C.

Garcia, L.

Genty, G.

Goedde, C. G.

Gonzalez-Herraez, M.

Gordon, J. P.

L. F. Mollenauer, R. H. Stolen, and J. P. Gordon, "Experimental observation of picosecond pulse narrowing and solitons in optical fibers," Phys. Rev. Lett. 45, 1095-1098 (1980).
[CrossRef]

Grabtchikov, A. S.

A. S. Grabtchikov, A. I. Vodtchits, and V. A. Orlovich, "Pulse-energy statistics in the linear regime of stimulated Raman scattering with a broad-band pump," Phys. Rev. A 56, 1666-1669 (1997).
[CrossRef]

Haelterman, M.

A. Picozzi, M. Haelterman, S. Pitois, and G. Millot, "Incoherent solitons in instantaneous response nonlinear media," Phys. Rev. Lett. 92, 143906 (2004).
[CrossRef] [PubMed]

Handerek, V.

Hasegawa, A.

K. Tai, A. Hasegawa, and A. Tomita, "Observation of modulational instability in optical fibers," Phys. Rev. Lett. 56, 135-138 (1986).
[CrossRef] [PubMed]

Huynh, E.

E. Huynh, E. Manzano, A. Medrano, T. Thorn, A. Zavala, C. G. Goedde, and J. R. Thompson, "The interplay of thermal and pump fluctuations in stimulated Brillouin scattering," Opt. Commun. 281, 836-845 (2008).
[CrossRef]

Jalali, B.

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

Jenkins, J.

Koonath, P.

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

Landahl, E.

E. Landahl, D. Baiocchi, and J. R. Thompson, "A simple analytic model for noise shaping by an optical fiber Raman generator," Opt. Commun. 150, 339-347 (1998).
[CrossRef]

Lee, Y.

Manzano, E.

E. Huynh, E. Manzano, A. Medrano, T. Thorn, A. Zavala, C. G. Goedde, and J. R. Thompson, "The interplay of thermal and pump fluctuations in stimulated Brillouin scattering," Opt. Commun. 281, 836-845 (2008).
[CrossRef]

Martin-Lopez, S.

Mears, R. J.

Medrano, A.

E. Huynh, E. Manzano, A. Medrano, T. Thorn, A. Zavala, C. G. Goedde, and J. R. Thompson, "The interplay of thermal and pump fluctuations in stimulated Brillouin scattering," Opt. Commun. 281, 836-845 (2008).
[CrossRef]

Millot, G.

J. M. Dudley, C. Finot, G. Millot, and D. J. Richardson, "Self-similarity in ultrafast nonlinear optics," Nat. Phys. 3, 597-603 (2007).
[CrossRef]

A. Picozzi, M. Haelterman, S. Pitois, and G. Millot, "Incoherent solitons in instantaneous response nonlinear media," Phys. Rev. Lett. 92, 143906 (2004).
[CrossRef] [PubMed]

Mollenauer, L. F.

L. F. Mollenauer, R. H. Stolen, and J. P. Gordon, "Experimental observation of picosecond pulse narrowing and solitons in optical fibers," Phys. Rev. Lett. 45, 1095-1098 (1980).
[CrossRef]

Olsson, B. E.

B. E. Olsson, P. �?hlen, L. Rau, and D. J. Blumenthal, "A simple and robust 40-Gb/s wavelength converter using fiber cross-phase modulation and optical filtering," IEEE Photon. Technol. Lett. 12, 846-848 (2000).
[CrossRef]

Orlovich, V. A.

A. S. Grabtchikov, A. I. Vodtchits, and V. A. Orlovich, "Pulse-energy statistics in the linear regime of stimulated Raman scattering with a broad-band pump," Phys. Rev. A 56, 1666-1669 (1997).
[CrossRef]

Picozzi, A.

A. Picozzi, M. Haelterman, S. Pitois, and G. Millot, "Incoherent solitons in instantaneous response nonlinear media," Phys. Rev. Lett. 92, 143906 (2004).
[CrossRef] [PubMed]

Pitois, S.

A. Picozzi, M. Haelterman, S. Pitois, and G. Millot, "Incoherent solitons in instantaneous response nonlinear media," Phys. Rev. Lett. 92, 143906 (2004).
[CrossRef] [PubMed]

Poole, N.

Rau, L.

B. E. Olsson, P. �?hlen, L. Rau, and D. J. Blumenthal, "A simple and robust 40-Gb/s wavelength converter using fiber cross-phase modulation and optical filtering," IEEE Photon. Technol. Lett. 12, 846-848 (2000).
[CrossRef]

Richardson, D. J.

J. M. Dudley, C. Finot, G. Millot, and D. J. Richardson, "Self-similarity in ultrafast nonlinear optics," Nat. Phys. 3, 597-603 (2007).
[CrossRef]

Ropers, C.

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

Roy, R.

J. R. Thompson and R. Roy, "Statistical fluctuations in multiple four-wave mixing in a single-mode optical fiber," Phys. Rev. A 44, 7605-7614 (1991).
[CrossRef] [PubMed]

Salit, K.

Schmitt, P.

Sidereas, P.

Solli, D. R.

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

Stolen, R. H.

L. F. Mollenauer, R. H. Stolen, and J. P. Gordon, "Experimental observation of picosecond pulse narrowing and solitons in optical fibers," Phys. Rev. Lett. 45, 1095-1098 (1980).
[CrossRef]

Tai, K.

K. Tai, A. Hasegawa, and A. Tomita, "Observation of modulational instability in optical fibers," Phys. Rev. Lett. 56, 135-138 (1986).
[CrossRef] [PubMed]

Thompson, J. R.

E. Huynh, E. Manzano, A. Medrano, T. Thorn, A. Zavala, C. G. Goedde, and J. R. Thompson, "The interplay of thermal and pump fluctuations in stimulated Brillouin scattering," Opt. Commun. 281, 836-845 (2008).
[CrossRef]

A. Betlej, P. Schmitt, P. Sidereas, R. Tracy, C. G. Goedde, and J. R. Thompson, "Increased Stokes pulse energy variation from amplified classical noise in a fiber Raman generator," Opt. Express 13, 2948-2960 (2005).
[CrossRef] [PubMed]

L. Garcia, J. Jenkins, Y. Lee, N. Poole, K. Salit, P. Sidereas, C. G. Goedde, and J. R. Thompson, "Influence of classical pump noise on long-pulse multiorder stimulated Raman scattering in optical fiber," J. Opt. Soc. Am. B 19, 2727-2736 (2002).
[CrossRef]

E. Landahl, D. Baiocchi, and J. R. Thompson, "A simple analytic model for noise shaping by an optical fiber Raman generator," Opt. Commun. 150, 339-347 (1998).
[CrossRef]

J. R. Thompson and R. Roy, "Statistical fluctuations in multiple four-wave mixing in a single-mode optical fiber," Phys. Rev. A 44, 7605-7614 (1991).
[CrossRef] [PubMed]

Thorn, T.

E. Huynh, E. Manzano, A. Medrano, T. Thorn, A. Zavala, C. G. Goedde, and J. R. Thompson, "The interplay of thermal and pump fluctuations in stimulated Brillouin scattering," Opt. Commun. 281, 836-845 (2008).
[CrossRef]

Tomita, A.

K. Tai, A. Hasegawa, and A. Tomita, "Observation of modulational instability in optical fibers," Phys. Rev. Lett. 56, 135-138 (1986).
[CrossRef] [PubMed]

Tracy, R.

Vanholsbeeck, F.

Vodtchits, A. I.

A. S. Grabtchikov, A. I. Vodtchits, and V. A. Orlovich, "Pulse-energy statistics in the linear regime of stimulated Raman scattering with a broad-band pump," Phys. Rev. A 56, 1666-1669 (1997).
[CrossRef]

Zavala, A.

E. Huynh, E. Manzano, A. Medrano, T. Thorn, A. Zavala, C. G. Goedde, and J. R. Thompson, "The interplay of thermal and pump fluctuations in stimulated Brillouin scattering," Opt. Commun. 281, 836-845 (2008).
[CrossRef]

IEEE Photon. Technol. Lett. (1)

B. E. Olsson, P. �?hlen, L. Rau, and D. J. Blumenthal, "A simple and robust 40-Gb/s wavelength converter using fiber cross-phase modulation and optical filtering," IEEE Photon. Technol. Lett. 12, 846-848 (2000).
[CrossRef]

J. Lightwave Technol. (1)

J. Opt. Soc. Am. B (1)

Nat. Phys. (1)

J. M. Dudley, C. Finot, G. Millot, and D. J. Richardson, "Self-similarity in ultrafast nonlinear optics," Nat. Phys. 3, 597-603 (2007).
[CrossRef]

Nature (1)

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

Opt. Commun. (2)

E. Huynh, E. Manzano, A. Medrano, T. Thorn, A. Zavala, C. G. Goedde, and J. R. Thompson, "The interplay of thermal and pump fluctuations in stimulated Brillouin scattering," Opt. Commun. 281, 836-845 (2008).
[CrossRef]

E. Landahl, D. Baiocchi, and J. R. Thompson, "A simple analytic model for noise shaping by an optical fiber Raman generator," Opt. Commun. 150, 339-347 (1998).
[CrossRef]

Opt. Express (3)

Phys. Rev. A (2)

A. S. Grabtchikov, A. I. Vodtchits, and V. A. Orlovich, "Pulse-energy statistics in the linear regime of stimulated Raman scattering with a broad-band pump," Phys. Rev. A 56, 1666-1669 (1997).
[CrossRef]

J. R. Thompson and R. Roy, "Statistical fluctuations in multiple four-wave mixing in a single-mode optical fiber," Phys. Rev. A 44, 7605-7614 (1991).
[CrossRef] [PubMed]

Phys. Rev. Lett. (3)

K. Tai, A. Hasegawa, and A. Tomita, "Observation of modulational instability in optical fibers," Phys. Rev. Lett. 56, 135-138 (1986).
[CrossRef] [PubMed]

L. F. Mollenauer, R. H. Stolen, and J. P. Gordon, "Experimental observation of picosecond pulse narrowing and solitons in optical fibers," Phys. Rev. Lett. 45, 1095-1098 (1980).
[CrossRef]

A. Picozzi, M. Haelterman, S. Pitois, and G. Millot, "Incoherent solitons in instantaneous response nonlinear media," Phys. Rev. Lett. 92, 143906 (2004).
[CrossRef] [PubMed]

Other (3)

G. Ravet, A. A. Fotidai, and P. Mégret, "Spectral broadening in Raman fiber amplifier pumped by partially coherent wave," in CLEO Europe, (2007).

C. Headley and G. P. Agrawal, Raman amplification in fiber optical communications (Academic Press, 2005).

D. Borlaug and B. Jalali, "Extreme value statistics in silicon photonics," to be presented at the 21st Annual Meeting of The IEEE Lasers & Electro-Optics Society, Newport Beach, United-States, 9-13 Nov. 2008. http://arxiv.org/abs/0809.0152v1

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Figures (5)

Fig. 1.
Fig. 1.

(a) Experimental set-up (b) Autocorrelation of the pump at 1455 nm, experimental results (solid blue line) compared to a Gaussian distribution with a coherence time of 25 ps (circles). (c) Corresponding intensity fluctuations normalized according to the median value.

Fig. 2.
Fig. 2.

(a). Amplified CW signal in a HNLF amplifier at 12 dB gain. The intensity is normalized relative to the median value. (b). Autocorrelation of pump and amplified signal in HNLF and DSF amplifiers at gains indicated.

Fig. 3.
Fig. 3.

(a). Eye-diagrams of: (a1) the initial ps pulse train compared with (a2-a5) amplified signals in DSF and HNLF for 3-dB and 12-dB on-off gains (b) Probability of the peak powers for both fibers and different gains. Peak-power values are normalized compared to the median value. Experimental results on a log-scale (lines) are compared with an exponential-decreasing fit (grey and black circles for HNLF amplifier) or with a linear fit (red circles).

Fig. 4.
Fig. 4.

Numerical results based on the integration of Eq. (1). (a) 3D representation of the longitudinal evolution of the temporal intensity profile of a rogue intensity spike. (b) Amplified continuous signal observed at the amplifier output (c) Autocorrelation signal of the pump and output amplified signal after propagation in HNLF and DSF for gain values as indicated (d) Statistics of the peak powers for DSF and HNLF-based amplifiers (subplots (d1) and (d2) respectively) and for different gains. Results on a logarithmic scale are compared with a linear fit (red dots) or an exponential decreasing fit (grey and black circles).

Fig. 5.
Fig. 5.

(a). Optical spectra of the amplified signal at different gains. Results for the HNLF amplifier are compared with the intial seed and the DSF results. Experimental and simulated results are plotted in (a1) and (a2), respectively. (b). Spectro-temporal plot of a rogue wave-type fluctuation (c). Experimental temporal signal obtained after spectral slicing by a 9 GHz filter and for different spectral offsets of (c1) 0 GHz, (c2) 150 GHz and (c3) 350 GHz.

Equations (1)

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{ ψ s z = g r 2 ψ p 2 ψ s α 2 ψ s + i γ [ ψ s 2 + 2 ψ p 2 ] ψ s ψ p z = g r 2 ψ s 2 ψ p α 2 ψ p + i γ [ ψ p 2 + 2 ψ s 2 ] ψ p δ ψ p T

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