Abstract

The Fokker–Planck approach is used to study the statistics of an optical pulse propagating in semiconductor optical amplifiers in which the amplified spontaneous emission noise dominates. The probability density functions (pdfs) of the pulse peak sample are calculated for different saturation levels. Analytical results are possible for an ideal linear amplifier and when the simplest first-order approximation for nonlinear amplification is used. In other cases, we solve numerically the Fokker–Planck equation and the appropriate Langevin equation. Multicanonical Monte Carlo simulations ensure efficient calculations of the pdfs whose high- and low-power tails deviate from noncentral chi-square statistics for moderate and deep saturation levels, thus implying that the electric field process is not Gaussian.

© 2005 Optical Society of America

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  4. E. Desurvire, Erbium-Doped Fiber Amplifiers (Wiley, 1994).
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    [CrossRef]
  6. M. L. Dakss and P. Melman, "Amplified spontaneous Raman scattering and gain in fiber Raman amplifiers," J. Lightwave Technol. 3, 806-813 (1985).
    [CrossRef]
  7. D. Dahan and G. Eisenstein, "The properties of amplified spontaneous emission noise in saturated fiber Raman amplifiers operating with cw signals," Opt. Commun. 236, 279-288 (2004).
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  9. P. O. Hedekvist and P. A. Andrekson, "Noise characteristics of fiber-based optical phase conjugators," J. Lightwave Technol. 17, 74-79 (1999).
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  11. Y. Yamamoto, "Noise and error rate performance of semiconductor laser amplifiers in PCM-IM optical transmission systems," IEEE J. Quantum Electron. 16, 1073-1081 (1980).
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    [CrossRef]
  13. M. Shtaif and G. Eisenstein, "Calculation of bit error rates in all-optical signal processing applications exploiting nondegenerate four-wave mixing in semiconductor optical amplifiers," J. Lightwave Technol. 14, 2069-2077 (1996).
    [CrossRef]
  14. K. Obermann, I. Koltchanov, K. Petermann, S. Diez, R. Ludwig, and H. G. Weber, "Noise analysis of frequency converters utilizing semiconductor-laser amplifiers," IEEE J. Quantum Electron. 33, 81-88 (1997).
    [CrossRef]
  15. M. Shtaif, B. Tromborg, and G. Eisenstein, "Noise spectra of semiconductor optical amplifiers: relation between semiclassical and quantum descriptions," IEEE J. Quantum Electron. 34, 869-878 (1998).
    [CrossRef]
  16. M. Shtaif and G. Eisenstein, "Noise properties of nonlinear semiconductor optical amplifiers," Opt. Lett. 21, 1851-1853 (1996).
    [CrossRef] [PubMed]
  17. A. Bilenca and G. Eisenstein, "On the noise properties of linear and nonlinear quantum-dot semiconductor optical amplifiers: the impact of inhomogeneously broadened gain and fast carrier dynamics," IEEE J. Quantum Electron. 40, 690-702 (2004).
    [CrossRef]
  18. D. Hadass, A. Bilenca, R. Alizon, H. Dery, V. Mikhelashvili, G. Eisenstein, R. Schwertberger, A. Somers, J. P. Reithmaier, A. Forchel, M. Calligaro, S. Bansropun, and M. Krakowski, "Gain and noise saturation spectra of wide band InAs/InP quantum dash optical amplifiers: model and experiments," IEEE J. Sel. Top. Quantum Electron., submitted for publication.
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  27. H. Risken, The Fokker-Planck Equation, 2nd ed. (Springer, 1989).
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    [CrossRef] [PubMed]
  29. R. Holzlhner and C. R. Menyuk, "Use of multicanonical Monte Carlo simulations to obtain accurate bit error rates in optical communications systems," Opt. Lett. 28, 1894-1896 (2003).
    [CrossRef]
  30. A. Bilenca and G. Eisenstein, "Statistical noise properties of an optical pulse propagating in a nonlinear semiconductor optical amplifier," IEEE J. Quantum Electron. 41, 36-44 (2005)
    [CrossRef]
  31. T. C. Gard, Introduction to Stochastic Differential Equations (Marcel Dekker, 1988).
  32. A. Papoulis, Probability, Random Variables, and Stochastic Processes, 3rd ed. (McGraw-Hill, 1991).
  33. G. A. Korn and T. M. Korn, Mathematical Handbook for Scientists and Engineers (Dover, 2000).
  34. C. Andrieu, N. De Freitas, A. Doucet, and M. I. Jordan, "An introduction to MCMC for machine learning," Mach. Learn. 50, 5-43 (2003).
    [CrossRef]
  35. P. Duchateau and D. Zachmann, Applied Partial Differential Equations (Dover2002).

2005 (1)

A. Bilenca and G. Eisenstein, "Statistical noise properties of an optical pulse propagating in a nonlinear semiconductor optical amplifier," IEEE J. Quantum Electron. 41, 36-44 (2005)
[CrossRef]

2004 (2)

D. Dahan and G. Eisenstein, "The properties of amplified spontaneous emission noise in saturated fiber Raman amplifiers operating with cw signals," Opt. Commun. 236, 279-288 (2004).
[CrossRef]

A. Bilenca and G. Eisenstein, "On the noise properties of linear and nonlinear quantum-dot semiconductor optical amplifiers: the impact of inhomogeneously broadened gain and fast carrier dynamics," IEEE J. Quantum Electron. 40, 690-702 (2004).
[CrossRef]

2003 (3)

2002 (1)

2001 (1)

1999 (2)

1998 (1)

M. Shtaif, B. Tromborg, and G. Eisenstein, "Noise spectra of semiconductor optical amplifiers: relation between semiclassical and quantum descriptions," IEEE J. Quantum Electron. 34, 869-878 (1998).
[CrossRef]

1997 (4)

A. Mecozzi and J. Mørk, "Saturation effects in nondegenerate four-wave mixing between short optical pulses in semiconductor laser amplifiers," IEEE J. Sel. Top. Quantum Electron. 3, 1190-1207 (1997).
[CrossRef]

K. Obermann, I. Koltchanov, K. Petermann, S. Diez, R. Ludwig, and H. G. Weber, "Noise analysis of frequency converters utilizing semiconductor-laser amplifiers," IEEE J. Quantum Electron. 33, 81-88 (1997).
[CrossRef]

M. Shtaif and G. Eisenstein, "Experimental study of the statistical properties of nonlinearly amplified signals in semiconductor optical amplifiers," IEEE Photonics Technol. Lett. 9, 904-906 (1997).
[CrossRef]

B. Chan and J. Conradi, "On the non-Gaussian noise in erbium-doped fiber amplifiers," J. Lightwave Technol. 15, 680-687 (1997).
[CrossRef]

1996 (4)

M. Shtaif and G. Eisenstein, "Noise characteristics of nonlinear semiconductor optical amplifiers in the Gaussian limit," IEEE J. Quantum Electron. 32, 1801-1809 (1996).
[CrossRef]

M. Shtaif and G. Eisenstein, "Calculation of bit error rates in all-optical signal processing applications exploiting nondegenerate four-wave mixing in semiconductor optical amplifiers," J. Lightwave Technol. 14, 2069-2077 (1996).
[CrossRef]

C. H. Henry and R. Kazarinov, "Quantum noise in photonics," Rev. Mod. Phys. 68, 801-853 (1996).
[CrossRef]

M. Shtaif and G. Eisenstein, "Noise properties of nonlinear semiconductor optical amplifiers," Opt. Lett. 21, 1851-1853 (1996).
[CrossRef] [PubMed]

1995 (1)

A. D'Ottavi, E. Iannone, A. Mecozzi, S. Scotti, P. Spano, R. Dall'Ara, J. Eckner, and G. Guekos, "Efficiency and noise performance of wavelength converters based on FWM in semiconductor optical amplifiers," IEEE Photonics Technol. Lett. 7, 357-359 (1995).
[CrossRef]

1992 (1)

B. A. Berg and T. Neuhaus, "Multicanonical ensemble: a new approach to simulate first-order phase transitions," Phys. Rev. Lett. 68, 9-12 (1992).
[CrossRef] [PubMed]

1989 (3)

G. P. Agrawal, "Self-phase modulation and spectral broadening of optical pulses in semiconductor laser amplifiers," IEEE J. Quantum Electron. 25, 2297-2306 (1989).
[CrossRef]

Y. Yamamoto and T. Mukai, "Fundamentals of optical amplifiers," Opt. Quantum Electron. 21, S1-S14 (1989).
[CrossRef]

N. A. Olsson, "Lightwave systems with optical amplifiers," J. Lightwave Technol. 7, 1071-1082 (1989).
[CrossRef]

1985 (1)

M. L. Dakss and P. Melman, "Amplified spontaneous Raman scattering and gain in fiber Raman amplifiers," J. Lightwave Technol. 3, 806-813 (1985).
[CrossRef]

1982 (1)

T. Mukai and Y. Yamamoto, "Noise in an AlGaAs semiconductor laser amplifier," IEEE J. Quantum Electron. 18, 564-575 (1982).
[CrossRef]

1980 (1)

Y. Yamamoto, "Noise and error rate performance of semiconductor laser amplifiers in PCM-IM optical transmission systems," IEEE J. Quantum Electron. 16, 1073-1081 (1980).
[CrossRef]

Agrawal, G. P.

G. P. Agrawal, "Self-phase modulation and spectral broadening of optical pulses in semiconductor laser amplifiers," IEEE J. Quantum Electron. 25, 2297-2306 (1989).
[CrossRef]

Alizon, R.

D. Hadass, A. Bilenca, R. Alizon, H. Dery, V. Mikhelashvili, G. Eisenstein, R. Schwertberger, A. Somers, J. P. Reithmaier, A. Forchel, M. Calligaro, S. Bansropun, and M. Krakowski, "Gain and noise saturation spectra of wide band InAs/InP quantum dash optical amplifiers: model and experiments," IEEE J. Sel. Top. Quantum Electron., submitted for publication.

Andrekson, P. A.

Andrieu, C.

C. Andrieu, N. De Freitas, A. Doucet, and M. I. Jordan, "An introduction to MCMC for machine learning," Mach. Learn. 50, 5-43 (2003).
[CrossRef]

Aurelius, A.

F. Öhman, B. Tromborg, J. Mørk, A. Aurelius, A. Djupsjöbacka, and A. Berntson, "Measurements of non-linear noise re-distribution in an SOA," in Conference of Lasers and Electro-Optics, Vol. 96 of OSA Trends in Optics and Photonics Series (Optical Society of America, 2004).

Bansropun, S.

D. Hadass, A. Bilenca, R. Alizon, H. Dery, V. Mikhelashvili, G. Eisenstein, R. Schwertberger, A. Somers, J. P. Reithmaier, A. Forchel, M. Calligaro, S. Bansropun, and M. Krakowski, "Gain and noise saturation spectra of wide band InAs/InP quantum dash optical amplifiers: model and experiments," IEEE J. Sel. Top. Quantum Electron., submitted for publication.

Berg, B. A.

B. A. Berg and T. Neuhaus, "Multicanonical ensemble: a new approach to simulate first-order phase transitions," Phys. Rev. Lett. 68, 9-12 (1992).
[CrossRef] [PubMed]

Berntson, A.

F. Öhman, B. Tromborg, J. Mørk, A. Aurelius, A. Djupsjöbacka, and A. Berntson, "Measurements of non-linear noise re-distribution in an SOA," in Conference of Lasers and Electro-Optics, Vol. 96 of OSA Trends in Optics and Photonics Series (Optical Society of America, 2004).

Bilenca, A.

A. Bilenca and G. Eisenstein, "Statistical noise properties of an optical pulse propagating in a nonlinear semiconductor optical amplifier," IEEE J. Quantum Electron. 41, 36-44 (2005)
[CrossRef]

A. Bilenca and G. Eisenstein, "On the noise properties of linear and nonlinear quantum-dot semiconductor optical amplifiers: the impact of inhomogeneously broadened gain and fast carrier dynamics," IEEE J. Quantum Electron. 40, 690-702 (2004).
[CrossRef]

D. Hadass, A. Bilenca, R. Alizon, H. Dery, V. Mikhelashvili, G. Eisenstein, R. Schwertberger, A. Somers, J. P. Reithmaier, A. Forchel, M. Calligaro, S. Bansropun, and M. Krakowski, "Gain and noise saturation spectra of wide band InAs/InP quantum dash optical amplifiers: model and experiments," IEEE J. Sel. Top. Quantum Electron., submitted for publication.

Calligaro, M.

D. Hadass, A. Bilenca, R. Alizon, H. Dery, V. Mikhelashvili, G. Eisenstein, R. Schwertberger, A. Somers, J. P. Reithmaier, A. Forchel, M. Calligaro, S. Bansropun, and M. Krakowski, "Gain and noise saturation spectra of wide band InAs/InP quantum dash optical amplifiers: model and experiments," IEEE J. Sel. Top. Quantum Electron., submitted for publication.

Chan, B.

B. Chan and J. Conradi, "On the non-Gaussian noise in erbium-doped fiber amplifiers," J. Lightwave Technol. 15, 680-687 (1997).
[CrossRef]

Conradi, J.

B. Chan and J. Conradi, "On the non-Gaussian noise in erbium-doped fiber amplifiers," J. Lightwave Technol. 15, 680-687 (1997).
[CrossRef]

Dahan, D.

D. Dahan and G. Eisenstein, "The properties of amplified spontaneous emission noise in saturated fiber Raman amplifiers operating with cw signals," Opt. Commun. 236, 279-288 (2004).
[CrossRef]

Dakss, M. L.

M. L. Dakss and P. Melman, "Amplified spontaneous Raman scattering and gain in fiber Raman amplifiers," J. Lightwave Technol. 3, 806-813 (1985).
[CrossRef]

Dall'Ara, R.

A. D'Ottavi, E. Iannone, A. Mecozzi, S. Scotti, P. Spano, R. Dall'Ara, J. Eckner, and G. Guekos, "Efficiency and noise performance of wavelength converters based on FWM in semiconductor optical amplifiers," IEEE Photonics Technol. Lett. 7, 357-359 (1995).
[CrossRef]

De Freitas, N.

C. Andrieu, N. De Freitas, A. Doucet, and M. I. Jordan, "An introduction to MCMC for machine learning," Mach. Learn. 50, 5-43 (2003).
[CrossRef]

Dery, H.

D. Hadass, A. Bilenca, R. Alizon, H. Dery, V. Mikhelashvili, G. Eisenstein, R. Schwertberger, A. Somers, J. P. Reithmaier, A. Forchel, M. Calligaro, S. Bansropun, and M. Krakowski, "Gain and noise saturation spectra of wide band InAs/InP quantum dash optical amplifiers: model and experiments," IEEE J. Sel. Top. Quantum Electron., submitted for publication.

Desurvire, E.

E. Desurvire, Erbium-Doped Fiber Amplifiers (Wiley, 1994).

Diez, S.

K. Obermann, I. Koltchanov, K. Petermann, S. Diez, R. Ludwig, and H. G. Weber, "Noise analysis of frequency converters utilizing semiconductor-laser amplifiers," IEEE J. Quantum Electron. 33, 81-88 (1997).
[CrossRef]

Djupsjöbacka, A.

F. Öhman, B. Tromborg, J. Mørk, A. Aurelius, A. Djupsjöbacka, and A. Berntson, "Measurements of non-linear noise re-distribution in an SOA," in Conference of Lasers and Electro-Optics, Vol. 96 of OSA Trends in Optics and Photonics Series (Optical Society of America, 2004).

D'Ottavi, A.

A. D'Ottavi, E. Iannone, A. Mecozzi, S. Scotti, P. Spano, R. Dall'Ara, J. Eckner, and G. Guekos, "Efficiency and noise performance of wavelength converters based on FWM in semiconductor optical amplifiers," IEEE Photonics Technol. Lett. 7, 357-359 (1995).
[CrossRef]

Doucet, A.

C. Andrieu, N. De Freitas, A. Doucet, and M. I. Jordan, "An introduction to MCMC for machine learning," Mach. Learn. 50, 5-43 (2003).
[CrossRef]

Duchateau, P.

P. Duchateau and D. Zachmann, Applied Partial Differential Equations (Dover2002).

Eckner, J.

A. D'Ottavi, E. Iannone, A. Mecozzi, S. Scotti, P. Spano, R. Dall'Ara, J. Eckner, and G. Guekos, "Efficiency and noise performance of wavelength converters based on FWM in semiconductor optical amplifiers," IEEE Photonics Technol. Lett. 7, 357-359 (1995).
[CrossRef]

Eisenstein, G.

A. Bilenca and G. Eisenstein, "Statistical noise properties of an optical pulse propagating in a nonlinear semiconductor optical amplifier," IEEE J. Quantum Electron. 41, 36-44 (2005)
[CrossRef]

D. Dahan and G. Eisenstein, "The properties of amplified spontaneous emission noise in saturated fiber Raman amplifiers operating with cw signals," Opt. Commun. 236, 279-288 (2004).
[CrossRef]

A. Bilenca and G. Eisenstein, "On the noise properties of linear and nonlinear quantum-dot semiconductor optical amplifiers: the impact of inhomogeneously broadened gain and fast carrier dynamics," IEEE J. Quantum Electron. 40, 690-702 (2004).
[CrossRef]

M. Shtaif, B. Tromborg, and G. Eisenstein, "Noise spectra of semiconductor optical amplifiers: relation between semiclassical and quantum descriptions," IEEE J. Quantum Electron. 34, 869-878 (1998).
[CrossRef]

M. Shtaif and G. Eisenstein, "Experimental study of the statistical properties of nonlinearly amplified signals in semiconductor optical amplifiers," IEEE Photonics Technol. Lett. 9, 904-906 (1997).
[CrossRef]

M. Shtaif and G. Eisenstein, "Calculation of bit error rates in all-optical signal processing applications exploiting nondegenerate four-wave mixing in semiconductor optical amplifiers," J. Lightwave Technol. 14, 2069-2077 (1996).
[CrossRef]

M. Shtaif and G. Eisenstein, "Noise properties of nonlinear semiconductor optical amplifiers," Opt. Lett. 21, 1851-1853 (1996).
[CrossRef] [PubMed]

M. Shtaif and G. Eisenstein, "Noise characteristics of nonlinear semiconductor optical amplifiers in the Gaussian limit," IEEE J. Quantum Electron. 32, 1801-1809 (1996).
[CrossRef]

D. Hadass, A. Bilenca, R. Alizon, H. Dery, V. Mikhelashvili, G. Eisenstein, R. Schwertberger, A. Somers, J. P. Reithmaier, A. Forchel, M. Calligaro, S. Bansropun, and M. Krakowski, "Gain and noise saturation spectra of wide band InAs/InP quantum dash optical amplifiers: model and experiments," IEEE J. Sel. Top. Quantum Electron., submitted for publication.

Forchel, A.

D. Hadass, A. Bilenca, R. Alizon, H. Dery, V. Mikhelashvili, G. Eisenstein, R. Schwertberger, A. Somers, J. P. Reithmaier, A. Forchel, M. Calligaro, S. Bansropun, and M. Krakowski, "Gain and noise saturation spectra of wide band InAs/InP quantum dash optical amplifiers: model and experiments," IEEE J. Sel. Top. Quantum Electron., submitted for publication.

Gard, T. C.

T. C. Gard, Introduction to Stochastic Differential Equations (Marcel Dekker, 1988).

Guekos, G.

A. D'Ottavi, E. Iannone, A. Mecozzi, S. Scotti, P. Spano, R. Dall'Ara, J. Eckner, and G. Guekos, "Efficiency and noise performance of wavelength converters based on FWM in semiconductor optical amplifiers," IEEE Photonics Technol. Lett. 7, 357-359 (1995).
[CrossRef]

Hadass, D.

D. Hadass, A. Bilenca, R. Alizon, H. Dery, V. Mikhelashvili, G. Eisenstein, R. Schwertberger, A. Somers, J. P. Reithmaier, A. Forchel, M. Calligaro, S. Bansropun, and M. Krakowski, "Gain and noise saturation spectra of wide band InAs/InP quantum dash optical amplifiers: model and experiments," IEEE J. Sel. Top. Quantum Electron., submitted for publication.

Hedekvist, P. O.

Henry, C. H.

C. H. Henry and R. Kazarinov, "Quantum noise in photonics," Rev. Mod. Phys. 68, 801-853 (1996).
[CrossRef]

Holzlhner, R.

Iannone, E.

A. D'Ottavi, E. Iannone, A. Mecozzi, S. Scotti, P. Spano, R. Dall'Ara, J. Eckner, and G. Guekos, "Efficiency and noise performance of wavelength converters based on FWM in semiconductor optical amplifiers," IEEE Photonics Technol. Lett. 7, 357-359 (1995).
[CrossRef]

Inoue, K.

Jordan, M. I.

C. Andrieu, N. De Freitas, A. Doucet, and M. I. Jordan, "An introduction to MCMC for machine learning," Mach. Learn. 50, 5-43 (2003).
[CrossRef]

Kazarinov, R.

C. H. Henry and R. Kazarinov, "Quantum noise in photonics," Rev. Mod. Phys. 68, 801-853 (1996).
[CrossRef]

Koltchanov, I.

K. Obermann, I. Koltchanov, K. Petermann, S. Diez, R. Ludwig, and H. G. Weber, "Noise analysis of frequency converters utilizing semiconductor-laser amplifiers," IEEE J. Quantum Electron. 33, 81-88 (1997).
[CrossRef]

Korn, G. A.

G. A. Korn and T. M. Korn, Mathematical Handbook for Scientists and Engineers (Dover, 2000).

Korn, T. M.

G. A. Korn and T. M. Korn, Mathematical Handbook for Scientists and Engineers (Dover, 2000).

Krakowski, M.

D. Hadass, A. Bilenca, R. Alizon, H. Dery, V. Mikhelashvili, G. Eisenstein, R. Schwertberger, A. Somers, J. P. Reithmaier, A. Forchel, M. Calligaro, S. Bansropun, and M. Krakowski, "Gain and noise saturation spectra of wide band InAs/InP quantum dash optical amplifiers: model and experiments," IEEE J. Sel. Top. Quantum Electron., submitted for publication.

Ludwig, R.

K. Obermann, I. Koltchanov, K. Petermann, S. Diez, R. Ludwig, and H. G. Weber, "Noise analysis of frequency converters utilizing semiconductor-laser amplifiers," IEEE J. Quantum Electron. 33, 81-88 (1997).
[CrossRef]

Mecozzi, A.

A. Mecozzi and J. Mørk, "Saturation effects in nondegenerate four-wave mixing between short optical pulses in semiconductor laser amplifiers," IEEE J. Sel. Top. Quantum Electron. 3, 1190-1207 (1997).
[CrossRef]

A. D'Ottavi, E. Iannone, A. Mecozzi, S. Scotti, P. Spano, R. Dall'Ara, J. Eckner, and G. Guekos, "Efficiency and noise performance of wavelength converters based on FWM in semiconductor optical amplifiers," IEEE Photonics Technol. Lett. 7, 357-359 (1995).
[CrossRef]

Melman, P.

M. L. Dakss and P. Melman, "Amplified spontaneous Raman scattering and gain in fiber Raman amplifiers," J. Lightwave Technol. 3, 806-813 (1985).
[CrossRef]

Menyuk, C. R.

Mikhelashvili, V.

D. Hadass, A. Bilenca, R. Alizon, H. Dery, V. Mikhelashvili, G. Eisenstein, R. Schwertberger, A. Somers, J. P. Reithmaier, A. Forchel, M. Calligaro, S. Bansropun, and M. Krakowski, "Gain and noise saturation spectra of wide band InAs/InP quantum dash optical amplifiers: model and experiments," IEEE J. Sel. Top. Quantum Electron., submitted for publication.

Mørk, J.

A. Mecozzi and J. Mørk, "Saturation effects in nondegenerate four-wave mixing between short optical pulses in semiconductor laser amplifiers," IEEE J. Sel. Top. Quantum Electron. 3, 1190-1207 (1997).
[CrossRef]

F. Öhman, B. Tromborg, J. Mørk, A. Aurelius, A. Djupsjöbacka, and A. Berntson, "Measurements of non-linear noise re-distribution in an SOA," in Conference of Lasers and Electro-Optics, Vol. 96 of OSA Trends in Optics and Photonics Series (Optical Society of America, 2004).

Mukai, T.

Neuhaus, T.

B. A. Berg and T. Neuhaus, "Multicanonical ensemble: a new approach to simulate first-order phase transitions," Phys. Rev. Lett. 68, 9-12 (1992).
[CrossRef] [PubMed]

Obermann, K.

K. Obermann, I. Koltchanov, K. Petermann, S. Diez, R. Ludwig, and H. G. Weber, "Noise analysis of frequency converters utilizing semiconductor-laser amplifiers," IEEE J. Quantum Electron. 33, 81-88 (1997).
[CrossRef]

Öhman, F.

F. Öhman, B. Tromborg, J. Mørk, A. Aurelius, A. Djupsjöbacka, and A. Berntson, "Measurements of non-linear noise re-distribution in an SOA," in Conference of Lasers and Electro-Optics, Vol. 96 of OSA Trends in Optics and Photonics Series (Optical Society of America, 2004).

Olsson, N. A.

N. A. Olsson, "Lightwave systems with optical amplifiers," J. Lightwave Technol. 7, 1071-1082 (1989).
[CrossRef]

Papoulis, A.

A. Papoulis, Probability, Random Variables, and Stochastic Processes, 3rd ed. (McGraw-Hill, 1991).

Petermann, K.

K. Obermann, I. Koltchanov, K. Petermann, S. Diez, R. Ludwig, and H. G. Weber, "Noise analysis of frequency converters utilizing semiconductor-laser amplifiers," IEEE J. Quantum Electron. 33, 81-88 (1997).
[CrossRef]

Reithmaier, J. P.

D. Hadass, A. Bilenca, R. Alizon, H. Dery, V. Mikhelashvili, G. Eisenstein, R. Schwertberger, A. Somers, J. P. Reithmaier, A. Forchel, M. Calligaro, S. Bansropun, and M. Krakowski, "Gain and noise saturation spectra of wide band InAs/InP quantum dash optical amplifiers: model and experiments," IEEE J. Sel. Top. Quantum Electron., submitted for publication.

Risken, H.

H. Risken, The Fokker-Planck Equation, 2nd ed. (Springer, 1989).

Schwertberger, R.

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A. D'Ottavi, E. Iannone, A. Mecozzi, S. Scotti, P. Spano, R. Dall'Ara, J. Eckner, and G. Guekos, "Efficiency and noise performance of wavelength converters based on FWM in semiconductor optical amplifiers," IEEE Photonics Technol. Lett. 7, 357-359 (1995).
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M. Shtaif and G. Eisenstein, "Noise characteristics of nonlinear semiconductor optical amplifiers in the Gaussian limit," IEEE J. Quantum Electron. 32, 1801-1809 (1996).
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D. Hadass, A. Bilenca, R. Alizon, H. Dery, V. Mikhelashvili, G. Eisenstein, R. Schwertberger, A. Somers, J. P. Reithmaier, A. Forchel, M. Calligaro, S. Bansropun, and M. Krakowski, "Gain and noise saturation spectra of wide band InAs/InP quantum dash optical amplifiers: model and experiments," IEEE J. Sel. Top. Quantum Electron., submitted for publication.

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A. D'Ottavi, E. Iannone, A. Mecozzi, S. Scotti, P. Spano, R. Dall'Ara, J. Eckner, and G. Guekos, "Efficiency and noise performance of wavelength converters based on FWM in semiconductor optical amplifiers," IEEE Photonics Technol. Lett. 7, 357-359 (1995).
[CrossRef]

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M. Shtaif, B. Tromborg, and G. Eisenstein, "Noise spectra of semiconductor optical amplifiers: relation between semiclassical and quantum descriptions," IEEE J. Quantum Electron. 34, 869-878 (1998).
[CrossRef]

F. Öhman, B. Tromborg, J. Mørk, A. Aurelius, A. Djupsjöbacka, and A. Berntson, "Measurements of non-linear noise re-distribution in an SOA," in Conference of Lasers and Electro-Optics, Vol. 96 of OSA Trends in Optics and Photonics Series (Optical Society of America, 2004).

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K. Obermann, I. Koltchanov, K. Petermann, S. Diez, R. Ludwig, and H. G. Weber, "Noise analysis of frequency converters utilizing semiconductor-laser amplifiers," IEEE J. Quantum Electron. 33, 81-88 (1997).
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Y. Yamamoto and K. Inoue, "Noise in amplifiers," J. Lightwave Technol. 21, 2895-2915 (2003).
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[CrossRef]

Y. Yamamoto, "Noise and error rate performance of semiconductor laser amplifiers in PCM-IM optical transmission systems," IEEE J. Quantum Electron. 16, 1073-1081 (1980).
[CrossRef]

K. Obermann, I. Koltchanov, K. Petermann, S. Diez, R. Ludwig, and H. G. Weber, "Noise analysis of frequency converters utilizing semiconductor-laser amplifiers," IEEE J. Quantum Electron. 33, 81-88 (1997).
[CrossRef]

M. Shtaif, B. Tromborg, and G. Eisenstein, "Noise spectra of semiconductor optical amplifiers: relation between semiclassical and quantum descriptions," IEEE J. Quantum Electron. 34, 869-878 (1998).
[CrossRef]

M. Shtaif and G. Eisenstein, "Noise characteristics of nonlinear semiconductor optical amplifiers in the Gaussian limit," IEEE J. Quantum Electron. 32, 1801-1809 (1996).
[CrossRef]

G. P. Agrawal, "Self-phase modulation and spectral broadening of optical pulses in semiconductor laser amplifiers," IEEE J. Quantum Electron. 25, 2297-2306 (1989).
[CrossRef]

A. Bilenca and G. Eisenstein, "Statistical noise properties of an optical pulse propagating in a nonlinear semiconductor optical amplifier," IEEE J. Quantum Electron. 41, 36-44 (2005)
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[CrossRef]

IEEE J. Sel. Top. Quantum Electron. (1)

A. Mecozzi and J. Mørk, "Saturation effects in nondegenerate four-wave mixing between short optical pulses in semiconductor laser amplifiers," IEEE J. Sel. Top. Quantum Electron. 3, 1190-1207 (1997).
[CrossRef]

IEEE Photonics Technol. Lett. (2)

M. Shtaif and G. Eisenstein, "Experimental study of the statistical properties of nonlinearly amplified signals in semiconductor optical amplifiers," IEEE Photonics Technol. Lett. 9, 904-906 (1997).
[CrossRef]

A. D'Ottavi, E. Iannone, A. Mecozzi, S. Scotti, P. Spano, R. Dall'Ara, J. Eckner, and G. Guekos, "Efficiency and noise performance of wavelength converters based on FWM in semiconductor optical amplifiers," IEEE Photonics Technol. Lett. 7, 357-359 (1995).
[CrossRef]

J. Lightwave Technol. (7)

M. Shtaif and G. Eisenstein, "Calculation of bit error rates in all-optical signal processing applications exploiting nondegenerate four-wave mixing in semiconductor optical amplifiers," J. Lightwave Technol. 14, 2069-2077 (1996).
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D. Dahan and G. Eisenstein, "The properties of amplified spontaneous emission noise in saturated fiber Raman amplifiers operating with cw signals," Opt. Commun. 236, 279-288 (2004).
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F. Öhman, B. Tromborg, J. Mørk, A. Aurelius, A. Djupsjöbacka, and A. Berntson, "Measurements of non-linear noise re-distribution in an SOA," in Conference of Lasers and Electro-Optics, Vol. 96 of OSA Trends in Optics and Photonics Series (Optical Society of America, 2004).

H. Risken, The Fokker-Planck Equation, 2nd ed. (Springer, 1989).

P. Duchateau and D. Zachmann, Applied Partial Differential Equations (Dover2002).

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D. Hadass, A. Bilenca, R. Alizon, H. Dery, V. Mikhelashvili, G. Eisenstein, R. Schwertberger, A. Somers, J. P. Reithmaier, A. Forchel, M. Calligaro, S. Bansropun, and M. Krakowski, "Gain and noise saturation spectra of wide band InAs/InP quantum dash optical amplifiers: model and experiments," IEEE J. Sel. Top. Quantum Electron., submitted for publication.

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Figures (2)

Fig. 1
Fig. 1

Pdfs of the peak of an optical pulse emerging at the output of a SOA computed for three levels of gain saturation. (a) Low saturation (0.8 dB), (b) moderate saturation (3.5 dB), and (c) deep saturation (9 dB). The noncentral chi-square ( n c - χ 2 ) function is shown in a solid black curve; the approximated pdf computed with MMC simulations is indicated by a dashed black curve; and the complete pdf calculated through MMC simulations is described by a solid gray curve.

Fig. 2
Fig. 2

Calculated pdf of the peak of an optical pulse emerging at the output of a deeply saturated SOA ( gain saturation = 9 dB ) . The noncentral chi-square ( n c - χ 2 ) function is shown in a solid gray curve; the complete pdf computed through MMC simulations is indicated by a dashed gray curve; and the FPE solution is described by a solid black curve.

Tables (1)

Tables Icon

Table 1 Physical Parameters Used in the Calculations

Equations (31)

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A ( τ , z ) z = 1 2 ( 1 i α ) g A ( τ , z ) + g + Γ a N T n ( τ , z ) ,
E [ n c ( τ , z ) n c ( τ , z ) ] = E [ n s ( τ , z ) n s ( τ , z ) ] = q ( τ τ ) δ ( z z ) ,
E n c = E n s = 0 .
P ( τ , z ) z = g P ( τ , z ) + 2 P ( τ , z ) ( g + Γ a N T ) [ n c ( τ , z ) cos ϕ ( τ , z ) + n s ( τ , z ) sin ϕ ( τ , z ) ] ,
ϕ ( τ , z ) z = α 2 g [ g + Γ a N T P ( τ , z ) ] 1 2 [ n c ( τ , z ) sin ϕ ( τ , z ) n s ( τ , z ) cos ϕ ( τ , z ) ] .
d P i d z = g i P i + 2 P i ( g i + Γ a N T ) ( n c i cos ϕ i + n s i sin ϕ i ) ,
d ϕ i d z = α 2 g i ( g i + Γ a N T P i ) 1 2 ( n c i sin ϕ i n s i cos ϕ i ) ,
D P i = h i + 1 2 k = 1 N q ( τ k τ i ) ( f c i x k f c k + f c i y k r c k + f s i x k f s k + f s i y k r s k ) ,
D ϕ i = α 2 g i + 1 2 k = 1 N q ( τ k τ i ) ( r c i x k f c k + r c i y k r c k + r s i x k f s k + r s i y k r s k ) ,
D P i P j = 1 2 q ( τ i τ j ) ( f c i f c j + f s i f s j ) ,
D ϕ i ϕ j = 1 2 q ( τ i τ j ) ( r c i r c j + r s i r s j ) ,
D P i ϕ j = D ϕ j P i = 1 2 q ( τ i τ j ) ( f c i r c j + f s i r s j ) .
W z = i = 1 N [ P i ( D P i W ) + ϕ i ( D ϕ i W ) ] + i = 1 N j = 1 N [ 2 P i P j ( D P i P j W ) + 2 ϕ i ϕ j ( D ϕ i ϕ j W ) + 2 2 P i ϕ j ( D P i ϕ j W ) ] ,
D P i = g i P i + σ 2 [ 2 ( g i + Γ a N T ) + P i g i P i ] ,
D ϕ i = ( α 2 ) g i ,
D P i P j = 2 σ 2 P i ( g i + Γ a N T ) δ i j ,
D ϕ i ϕ j = 0.5 σ 2 g i + Γ a N T P i δ i j ,
D P i ϕ j = D ϕ i P j = 0 ,
P z = g P + σ 2 ( g + Γ a N T ) + 2 σ P ( g + Γ a N T ) n P ,
ϕ z = α 2 g σ ( g + Γ a N T P ) 1 2 n ϕ ,
E [ n P ( τ , z ) n P ( τ , z ) ] = E [ n ϕ ( τ , z ) n ϕ ( τ , z ) ] = δ ( τ τ ) δ ( z z ) ,
E [ n P ( τ , z ) n ϕ ( τ , z ) ] = 0 ,
E n P = E n ϕ = 0 ,
W z = i = 1 N P i ( g i P i W ) + σ 2 P i { [ 2 ( g i + Γ a N T ) + P i g i P i ] W } α 2 g i W ϕ i + σ 2 i = 1 N 2 P i 2 [ P i ( g i + Γ a N T ) W ] + g i + Γ a N T 2 P i 2 W ϕ i 2 .
W z = P [ P λ 1 ( P ; z ) W ] σ 2 P ( { 2 [ λ 1 ( P ; z ) + Γ a N T ] + P λ 2 ( P ; z ) } W ) + 2 σ 2 2 P 2 { P [ λ 1 ( P ; z ) + Γ a N T ] W } ,
W ( P , z = L P ( z = 0 ) ) = 1 2 σ n 2 exp [ ( s 2 + P ) 2 σ n 2 ] I 0 ( P s σ n 2 ) ,
s 2 = G P = G ss P ( z = 0 ) , σ n 2 = 0.5 ω 0 B opt n sp ( G ss 1 ) .
λ 1 ( P ; z ) = E ( g P ; z ) = E ( g ) g 0 ( τ q , z ) = g ss exp [ τ q P 0 ( τ , z ) E sat d τ ]
P 0 ( τ , z ) z = g 0 ( τ , z ) P 0 ( τ , z ) + 2 σ 2 [ g 0 ( τ , z ) + Γ a N T ] .
s 2 = G 0 ( τ q , z = L ) P ( z = 0 ) = exp [ 0 L g 0 ( τ q , z ) d z ] P ( z = 0 ) ,
σ n 2 = 0.5 ω 0 B opt 0 L n sp , 0 ( τ q , z ) g 0 ( τ q , z ) exp [ z L g 0 ( τ q , z ) d z ] d z ,

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