Abstract

In this paper a detailed analysis is made of the phase-sensitive amplification (PSA) of optical pulses by degenerate four-wave mixing (phase-conjugation) in fibers. Formulas are derived, which show that the amplification level and phase sensitivity depend strongly on the phase of the signal pulse, but weakly on its chirp, and the difference between its carrier frequency and the average pump frequency. Solitons, which are unchirped, and dispersion-managed solitons, which are weakly chirped, are suitable for in-line and post-transmission PSA. Pseudo-linear pulses, which are strongly chirped, are unsuitable for in-line PSA, but post-transmission dispersion compensation makes them suitable for PSA prior to detection.

©2007 Optical Society of America

Full Article  |  PDF Article
OSA Recommended Articles
Phase squeezing and dispersion tolerance of phase sensitive amplifier using periodically poled LiNbO3 waveguide

Masaki Asobe, Takeshi Umeki, Kouji Enbutsu, Osamu Tadanaga, and Hirokazu Takenouchi
J. Opt. Soc. Am. B 31(12) 3164-3169 (2014)

Phase sensitive amplification with noise figure below the 3 dB quantum limit using CW pumped PPLN waveguide

Masaki Asobe, Takeshi Umeki, and Osamu Tadanaga
Opt. Express 20(12) 13164-13172 (2012)

Pulse propagation in nonlinear optical fiber lines that employ phase-sensitive parametric amplifiers

J. Nathan Kutz, Cheryl V. Hile, William L. Kath, Ruo-Ding Li, and Prem Kumar
J. Opt. Soc. Am. B 11(10) 2112-2123 (1994)

References

  • View by:
  • |
  • |
  • |

  1. R. Loudon, “Theory of noise accumulation in linear optical-amplifier chains,” IEEE J. Quantum Electron. 21,766–773 (1985).
    [Crossref]
  2. H. P. Yuen, “Reduction of quantum fluctuation and suppression of the Gordon-Haus effect with phase-sensitive linear amplifiers,” Opt. Lett. 17,73–75 (1992).
    [Crossref] [PubMed]
  3. Y. Mu and C. M. Savage, “Parametric amplifiers in phase-noise-limited optical communications,” J. Opt. Soc. Am. B 9,65–70 (1992).
    [Crossref]
  4. R. D. Li, P. Kumar, W. L. Kath, and J. N. Kutz, “Combating dispersion with parametric amplifers,” IEEE Photon. Technol. Lett. 5,669–672 (1993).
    [Crossref]
  5. W. Imajuku and A. Takada, “Reduction of fiber-nonlinearity-enhanced amplifier noise by means of phase-sensitive amplifiers,” Opt. Lett. 22,31–33 (1997).
    [Crossref] [PubMed]
  6. R. O. Moore, W. L. Kath, B. Sandstede, C. K. R. T. Jones, and J. C. Alexander, “Stability of multiple pulses in optical fibers with phase-sensitive amplification and noise,” Opt. Commun. 195,127–139 (2001).
    [Crossref]
  7. C. J. McKinstrie and S. Radic, “Phase-sensitive amplification in a fiber,” Opt. Express 12,4973–4979 (2004).
    [Crossref] [PubMed]
  8. L. F. Mollenauer, J. P. Gordon, and P. V. Mamyshev, “Solitons in high bit-rate, long-distance transmission,” in Optical Fiber Telecommunications IIIA, edited by I. P. Kaminow and T. L. Koch (Academic Press, 1997), pp.373–460.
  9. S. K. Turitsyn, V. K. Mezentsev, and E. G. Shapiro, “Dispersion-managed solitons and optimization of the dispersion management,” Opt. Fiber. Technol. 4,384–452 (1998).
    [Crossref]
  10. R. J. Essiambre, G. Raybon, and B. Mikkelsen “Pseudo-linear transmission of high-speed TDM signals: 40 and 160 Gb/s,” in Optical Fiber Telecommunications IVB, edited by I. P. Kaminow and T. Li (Academic Press, 2002) pp.232–304.
  11. C. J. McKinstrie, H. Kogelnik, R. M. Jopson, S. Radic, and A. V. Kanaev, “Four-wave mixing in fibers with random birefringence,” Opt. Express 12,2033–2055 (2004).
    [Crossref] [PubMed]
  12. C. J. McKinstrie, H. Kogelnik, and L. Schenato, “Four-wave mixing in a rapidly-spun fiber,” Opt. Express 14,8516–8534 (2006).
    [Crossref] [PubMed]
  13. R. Loudon and P. L. Knight, “Squeezed light,” J. Mod. Opt. 34,709–759 (1987).
    [Crossref]
  14. R. Loudon, The Quantum Theory of Light, 3rd Ed. (Oxford University Press, 2000).
  15. J. P. Gordon and H. A. Haus, “Random walk of coherently amplified solitons in optical fiber transmission,” Opt. Lett. 11,665–667 (1986).
    [Crossref] [PubMed]
  16. C. J. McKinstrie, “Gordon-Haus timing jitter in dispersion-managed systems with distributed amplification,” Opt. Commun. 200,165–177 (2001).
    [Crossref]
  17. J. P. Gordon and L. F. Mollenauer, “Phase noise in photonic communications systems using linear amplifiers,” Opt. Lett. 15,1351–1353 (1990).
    [Crossref] [PubMed]
  18. C. J. McKinstrie and C. Xie, “Phase-jitter in single-channel soliton systems with constant dispersion,” IEEE J. Sel. Top. Quantum Electron. 8,616–625 (2002).
    [Crossref]
  19. C. J. McKinstrie, C. Xie, and T. I. Lakoba, “Efficient modeling of phase jitter in dispersion-managed soliton systems,” Opt. Lett. 27,1887–1889 (2002).
    [Crossref]
  20. M. Hanna, D. Boivin, P. A. Lacourt, and J. P. Goedgebuer, “Calculation of optical phase jitter in dispersion-managed systems by use of the moment method,” J. Opt. Soc. Am. B 21,24–28 (2004).
    [Crossref]
  21. A. H. Gnauck and P. J. Winzer, “Optical phase-shift-keyed transmission,” J. Lightwave Technol. 23,115–130 (2005).
    [Crossref]
  22. K. Croussore, I. Kim, Y. Han, C. Kim, G. Li, and S. Radic, “Demonstration of phase regeneration of DPSK signals based on phase-sensitive amplification,” Opt. Express 13,3945–3950 (2005).
    [Crossref] [PubMed]
  23. A. Mecozzi, C. B. Clausen, and M. Shtaif, “Analysis of intrachannel nonlinear effects in highly dispersed optical pulse transmission,” IEEE Photon. Technol. Lett 12,392–394 (2000).
    [Crossref]
  24. X. Wei and X. Liu, “Analysis of intrachannel four-wave mixing in differential phase-shift keying transmission with large dispersion,” Opt. Lett. 28,2300–2302 (2003).
    [Crossref] [PubMed]
  25. S. N. Vlasov, V. A. Petrishchev, and V. I. Talanov, “Averaged description of wave beams in linear and nonlinear media,” Radiophys. Quantum Electron. 14,1062–1070 (1971).
    [Crossref]
  26. W. J. Firth, “Propagation of laser beams through inhomogeneous media,” Opt. Commun. 22,226–230 (1977).
    [Crossref]
  27. W. L. Kath, “A modified conservation law for the phase of the nonlinear Schrödinger equation,” Methods and Applications of Analysis 4,141–155 (1997).
  28. M. V. Kozlov, C. J. McKinstrie, and C. Xie, “Moment equations for optical pulses in dispersive and dissipative systems,” Opt. Commun. 251,194–208 (2005).
    [Crossref]
  29. Subsets of the variation equations have been known since the 1980s. A complete set is derived in C. J. McKinstrie, “Frequency shifts caused by pulse collisions in wavelength-division-multiplexed systems,” Opt. Commun. 205,123–137 (2002).
    [Crossref]
  30. I. S. Gradsteyn and I. M. Ryzhik, Tables of Integrals, Series and Products, 5th Ed. (Academic Press, 1994), pp.98–101.

2006 (1)

2005 (3)

2004 (3)

2003 (1)

2002 (3)

Subsets of the variation equations have been known since the 1980s. A complete set is derived in C. J. McKinstrie, “Frequency shifts caused by pulse collisions in wavelength-division-multiplexed systems,” Opt. Commun. 205,123–137 (2002).
[Crossref]

C. J. McKinstrie and C. Xie, “Phase-jitter in single-channel soliton systems with constant dispersion,” IEEE J. Sel. Top. Quantum Electron. 8,616–625 (2002).
[Crossref]

C. J. McKinstrie, C. Xie, and T. I. Lakoba, “Efficient modeling of phase jitter in dispersion-managed soliton systems,” Opt. Lett. 27,1887–1889 (2002).
[Crossref]

2001 (2)

C. J. McKinstrie, “Gordon-Haus timing jitter in dispersion-managed systems with distributed amplification,” Opt. Commun. 200,165–177 (2001).
[Crossref]

R. O. Moore, W. L. Kath, B. Sandstede, C. K. R. T. Jones, and J. C. Alexander, “Stability of multiple pulses in optical fibers with phase-sensitive amplification and noise,” Opt. Commun. 195,127–139 (2001).
[Crossref]

2000 (1)

A. Mecozzi, C. B. Clausen, and M. Shtaif, “Analysis of intrachannel nonlinear effects in highly dispersed optical pulse transmission,” IEEE Photon. Technol. Lett 12,392–394 (2000).
[Crossref]

1998 (1)

S. K. Turitsyn, V. K. Mezentsev, and E. G. Shapiro, “Dispersion-managed solitons and optimization of the dispersion management,” Opt. Fiber. Technol. 4,384–452 (1998).
[Crossref]

1997 (2)

W. Imajuku and A. Takada, “Reduction of fiber-nonlinearity-enhanced amplifier noise by means of phase-sensitive amplifiers,” Opt. Lett. 22,31–33 (1997).
[Crossref] [PubMed]

W. L. Kath, “A modified conservation law for the phase of the nonlinear Schrödinger equation,” Methods and Applications of Analysis 4,141–155 (1997).

1993 (1)

R. D. Li, P. Kumar, W. L. Kath, and J. N. Kutz, “Combating dispersion with parametric amplifers,” IEEE Photon. Technol. Lett. 5,669–672 (1993).
[Crossref]

1992 (2)

1990 (1)

1987 (1)

R. Loudon and P. L. Knight, “Squeezed light,” J. Mod. Opt. 34,709–759 (1987).
[Crossref]

1986 (1)

1985 (1)

R. Loudon, “Theory of noise accumulation in linear optical-amplifier chains,” IEEE J. Quantum Electron. 21,766–773 (1985).
[Crossref]

1977 (1)

W. J. Firth, “Propagation of laser beams through inhomogeneous media,” Opt. Commun. 22,226–230 (1977).
[Crossref]

1971 (1)

S. N. Vlasov, V. A. Petrishchev, and V. I. Talanov, “Averaged description of wave beams in linear and nonlinear media,” Radiophys. Quantum Electron. 14,1062–1070 (1971).
[Crossref]

Alexander, J. C.

R. O. Moore, W. L. Kath, B. Sandstede, C. K. R. T. Jones, and J. C. Alexander, “Stability of multiple pulses in optical fibers with phase-sensitive amplification and noise,” Opt. Commun. 195,127–139 (2001).
[Crossref]

Boivin, D.

Clausen, C. B.

A. Mecozzi, C. B. Clausen, and M. Shtaif, “Analysis of intrachannel nonlinear effects in highly dispersed optical pulse transmission,” IEEE Photon. Technol. Lett 12,392–394 (2000).
[Crossref]

Croussore, K.

Essiambre, R. J.

R. J. Essiambre, G. Raybon, and B. Mikkelsen “Pseudo-linear transmission of high-speed TDM signals: 40 and 160 Gb/s,” in Optical Fiber Telecommunications IVB, edited by I. P. Kaminow and T. Li (Academic Press, 2002) pp.232–304.

Firth, W. J.

W. J. Firth, “Propagation of laser beams through inhomogeneous media,” Opt. Commun. 22,226–230 (1977).
[Crossref]

Gnauck, A. H.

Goedgebuer, J. P.

Gordon, J. P.

J. P. Gordon and L. F. Mollenauer, “Phase noise in photonic communications systems using linear amplifiers,” Opt. Lett. 15,1351–1353 (1990).
[Crossref] [PubMed]

J. P. Gordon and H. A. Haus, “Random walk of coherently amplified solitons in optical fiber transmission,” Opt. Lett. 11,665–667 (1986).
[Crossref] [PubMed]

L. F. Mollenauer, J. P. Gordon, and P. V. Mamyshev, “Solitons in high bit-rate, long-distance transmission,” in Optical Fiber Telecommunications IIIA, edited by I. P. Kaminow and T. L. Koch (Academic Press, 1997), pp.373–460.

Gradsteyn, I. S.

I. S. Gradsteyn and I. M. Ryzhik, Tables of Integrals, Series and Products, 5th Ed. (Academic Press, 1994), pp.98–101.

Han, Y.

Hanna, M.

Haus, H. A.

Imajuku, W.

Jones, C. K. R. T.

R. O. Moore, W. L. Kath, B. Sandstede, C. K. R. T. Jones, and J. C. Alexander, “Stability of multiple pulses in optical fibers with phase-sensitive amplification and noise,” Opt. Commun. 195,127–139 (2001).
[Crossref]

Jopson, R. M.

Kanaev, A. V.

Kath, W. L.

R. O. Moore, W. L. Kath, B. Sandstede, C. K. R. T. Jones, and J. C. Alexander, “Stability of multiple pulses in optical fibers with phase-sensitive amplification and noise,” Opt. Commun. 195,127–139 (2001).
[Crossref]

W. L. Kath, “A modified conservation law for the phase of the nonlinear Schrödinger equation,” Methods and Applications of Analysis 4,141–155 (1997).

R. D. Li, P. Kumar, W. L. Kath, and J. N. Kutz, “Combating dispersion with parametric amplifers,” IEEE Photon. Technol. Lett. 5,669–672 (1993).
[Crossref]

Kim, C.

Kim, I.

Knight, P. L.

R. Loudon and P. L. Knight, “Squeezed light,” J. Mod. Opt. 34,709–759 (1987).
[Crossref]

Kogelnik, H.

Kozlov, M. V.

M. V. Kozlov, C. J. McKinstrie, and C. Xie, “Moment equations for optical pulses in dispersive and dissipative systems,” Opt. Commun. 251,194–208 (2005).
[Crossref]

Kumar, P.

R. D. Li, P. Kumar, W. L. Kath, and J. N. Kutz, “Combating dispersion with parametric amplifers,” IEEE Photon. Technol. Lett. 5,669–672 (1993).
[Crossref]

Kutz, J. N.

R. D. Li, P. Kumar, W. L. Kath, and J. N. Kutz, “Combating dispersion with parametric amplifers,” IEEE Photon. Technol. Lett. 5,669–672 (1993).
[Crossref]

Lacourt, P. A.

Lakoba, T. I.

Li, G.

Li, R. D.

R. D. Li, P. Kumar, W. L. Kath, and J. N. Kutz, “Combating dispersion with parametric amplifers,” IEEE Photon. Technol. Lett. 5,669–672 (1993).
[Crossref]

Liu, X.

Loudon, R.

R. Loudon and P. L. Knight, “Squeezed light,” J. Mod. Opt. 34,709–759 (1987).
[Crossref]

R. Loudon, “Theory of noise accumulation in linear optical-amplifier chains,” IEEE J. Quantum Electron. 21,766–773 (1985).
[Crossref]

R. Loudon, The Quantum Theory of Light, 3rd Ed. (Oxford University Press, 2000).

Mamyshev, P. V.

L. F. Mollenauer, J. P. Gordon, and P. V. Mamyshev, “Solitons in high bit-rate, long-distance transmission,” in Optical Fiber Telecommunications IIIA, edited by I. P. Kaminow and T. L. Koch (Academic Press, 1997), pp.373–460.

McKinstrie, C. J.

C. J. McKinstrie, H. Kogelnik, and L. Schenato, “Four-wave mixing in a rapidly-spun fiber,” Opt. Express 14,8516–8534 (2006).
[Crossref] [PubMed]

M. V. Kozlov, C. J. McKinstrie, and C. Xie, “Moment equations for optical pulses in dispersive and dissipative systems,” Opt. Commun. 251,194–208 (2005).
[Crossref]

C. J. McKinstrie and S. Radic, “Phase-sensitive amplification in a fiber,” Opt. Express 12,4973–4979 (2004).
[Crossref] [PubMed]

C. J. McKinstrie, H. Kogelnik, R. M. Jopson, S. Radic, and A. V. Kanaev, “Four-wave mixing in fibers with random birefringence,” Opt. Express 12,2033–2055 (2004).
[Crossref] [PubMed]

C. J. McKinstrie, C. Xie, and T. I. Lakoba, “Efficient modeling of phase jitter in dispersion-managed soliton systems,” Opt. Lett. 27,1887–1889 (2002).
[Crossref]

Subsets of the variation equations have been known since the 1980s. A complete set is derived in C. J. McKinstrie, “Frequency shifts caused by pulse collisions in wavelength-division-multiplexed systems,” Opt. Commun. 205,123–137 (2002).
[Crossref]

C. J. McKinstrie and C. Xie, “Phase-jitter in single-channel soliton systems with constant dispersion,” IEEE J. Sel. Top. Quantum Electron. 8,616–625 (2002).
[Crossref]

C. J. McKinstrie, “Gordon-Haus timing jitter in dispersion-managed systems with distributed amplification,” Opt. Commun. 200,165–177 (2001).
[Crossref]

Mecozzi, A.

A. Mecozzi, C. B. Clausen, and M. Shtaif, “Analysis of intrachannel nonlinear effects in highly dispersed optical pulse transmission,” IEEE Photon. Technol. Lett 12,392–394 (2000).
[Crossref]

Mezentsev, V. K.

S. K. Turitsyn, V. K. Mezentsev, and E. G. Shapiro, “Dispersion-managed solitons and optimization of the dispersion management,” Opt. Fiber. Technol. 4,384–452 (1998).
[Crossref]

Mikkelsen, B.

R. J. Essiambre, G. Raybon, and B. Mikkelsen “Pseudo-linear transmission of high-speed TDM signals: 40 and 160 Gb/s,” in Optical Fiber Telecommunications IVB, edited by I. P. Kaminow and T. Li (Academic Press, 2002) pp.232–304.

Mollenauer, L. F.

J. P. Gordon and L. F. Mollenauer, “Phase noise in photonic communications systems using linear amplifiers,” Opt. Lett. 15,1351–1353 (1990).
[Crossref] [PubMed]

L. F. Mollenauer, J. P. Gordon, and P. V. Mamyshev, “Solitons in high bit-rate, long-distance transmission,” in Optical Fiber Telecommunications IIIA, edited by I. P. Kaminow and T. L. Koch (Academic Press, 1997), pp.373–460.

Moore, R. O.

R. O. Moore, W. L. Kath, B. Sandstede, C. K. R. T. Jones, and J. C. Alexander, “Stability of multiple pulses in optical fibers with phase-sensitive amplification and noise,” Opt. Commun. 195,127–139 (2001).
[Crossref]

Mu, Y.

Petrishchev, V. A.

S. N. Vlasov, V. A. Petrishchev, and V. I. Talanov, “Averaged description of wave beams in linear and nonlinear media,” Radiophys. Quantum Electron. 14,1062–1070 (1971).
[Crossref]

Radic, S.

Raybon, G.

R. J. Essiambre, G. Raybon, and B. Mikkelsen “Pseudo-linear transmission of high-speed TDM signals: 40 and 160 Gb/s,” in Optical Fiber Telecommunications IVB, edited by I. P. Kaminow and T. Li (Academic Press, 2002) pp.232–304.

Ryzhik, I. M.

I. S. Gradsteyn and I. M. Ryzhik, Tables of Integrals, Series and Products, 5th Ed. (Academic Press, 1994), pp.98–101.

Sandstede, B.

R. O. Moore, W. L. Kath, B. Sandstede, C. K. R. T. Jones, and J. C. Alexander, “Stability of multiple pulses in optical fibers with phase-sensitive amplification and noise,” Opt. Commun. 195,127–139 (2001).
[Crossref]

Savage, C. M.

Schenato, L.

Shapiro, E. G.

S. K. Turitsyn, V. K. Mezentsev, and E. G. Shapiro, “Dispersion-managed solitons and optimization of the dispersion management,” Opt. Fiber. Technol. 4,384–452 (1998).
[Crossref]

Shtaif, M.

A. Mecozzi, C. B. Clausen, and M. Shtaif, “Analysis of intrachannel nonlinear effects in highly dispersed optical pulse transmission,” IEEE Photon. Technol. Lett 12,392–394 (2000).
[Crossref]

Takada, A.

Talanov, V. I.

S. N. Vlasov, V. A. Petrishchev, and V. I. Talanov, “Averaged description of wave beams in linear and nonlinear media,” Radiophys. Quantum Electron. 14,1062–1070 (1971).
[Crossref]

Turitsyn, S. K.

S. K. Turitsyn, V. K. Mezentsev, and E. G. Shapiro, “Dispersion-managed solitons and optimization of the dispersion management,” Opt. Fiber. Technol. 4,384–452 (1998).
[Crossref]

Vlasov, S. N.

S. N. Vlasov, V. A. Petrishchev, and V. I. Talanov, “Averaged description of wave beams in linear and nonlinear media,” Radiophys. Quantum Electron. 14,1062–1070 (1971).
[Crossref]

Wei, X.

Winzer, P. J.

Xie, C.

M. V. Kozlov, C. J. McKinstrie, and C. Xie, “Moment equations for optical pulses in dispersive and dissipative systems,” Opt. Commun. 251,194–208 (2005).
[Crossref]

C. J. McKinstrie and C. Xie, “Phase-jitter in single-channel soliton systems with constant dispersion,” IEEE J. Sel. Top. Quantum Electron. 8,616–625 (2002).
[Crossref]

C. J. McKinstrie, C. Xie, and T. I. Lakoba, “Efficient modeling of phase jitter in dispersion-managed soliton systems,” Opt. Lett. 27,1887–1889 (2002).
[Crossref]

Yuen, H. P.

IEEE J. Quantum Electron. (1)

R. Loudon, “Theory of noise accumulation in linear optical-amplifier chains,” IEEE J. Quantum Electron. 21,766–773 (1985).
[Crossref]

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

C. J. McKinstrie and C. Xie, “Phase-jitter in single-channel soliton systems with constant dispersion,” IEEE J. Sel. Top. Quantum Electron. 8,616–625 (2002).
[Crossref]

IEEE Photon. Technol. Lett (1)

A. Mecozzi, C. B. Clausen, and M. Shtaif, “Analysis of intrachannel nonlinear effects in highly dispersed optical pulse transmission,” IEEE Photon. Technol. Lett 12,392–394 (2000).
[Crossref]

IEEE Photon. Technol. Lett. (1)

R. D. Li, P. Kumar, W. L. Kath, and J. N. Kutz, “Combating dispersion with parametric amplifers,” IEEE Photon. Technol. Lett. 5,669–672 (1993).
[Crossref]

J. Lightwave Technol. (1)

J. Mod. Opt. (1)

R. Loudon and P. L. Knight, “Squeezed light,” J. Mod. Opt. 34,709–759 (1987).
[Crossref]

J. Opt. Soc. Am. B (2)

Methods and Applications of Analysis (1)

W. L. Kath, “A modified conservation law for the phase of the nonlinear Schrödinger equation,” Methods and Applications of Analysis 4,141–155 (1997).

Opt. Commun. (5)

M. V. Kozlov, C. J. McKinstrie, and C. Xie, “Moment equations for optical pulses in dispersive and dissipative systems,” Opt. Commun. 251,194–208 (2005).
[Crossref]

Subsets of the variation equations have been known since the 1980s. A complete set is derived in C. J. McKinstrie, “Frequency shifts caused by pulse collisions in wavelength-division-multiplexed systems,” Opt. Commun. 205,123–137 (2002).
[Crossref]

W. J. Firth, “Propagation of laser beams through inhomogeneous media,” Opt. Commun. 22,226–230 (1977).
[Crossref]

C. J. McKinstrie, “Gordon-Haus timing jitter in dispersion-managed systems with distributed amplification,” Opt. Commun. 200,165–177 (2001).
[Crossref]

R. O. Moore, W. L. Kath, B. Sandstede, C. K. R. T. Jones, and J. C. Alexander, “Stability of multiple pulses in optical fibers with phase-sensitive amplification and noise,” Opt. Commun. 195,127–139 (2001).
[Crossref]

Opt. Express (4)

Opt. Fiber. Technol. (1)

S. K. Turitsyn, V. K. Mezentsev, and E. G. Shapiro, “Dispersion-managed solitons and optimization of the dispersion management,” Opt. Fiber. Technol. 4,384–452 (1998).
[Crossref]

Opt. Lett. (6)

Radiophys. Quantum Electron. (1)

S. N. Vlasov, V. A. Petrishchev, and V. I. Talanov, “Averaged description of wave beams in linear and nonlinear media,” Radiophys. Quantum Electron. 14,1062–1070 (1971).
[Crossref]

Other (4)

I. S. Gradsteyn and I. M. Ryzhik, Tables of Integrals, Series and Products, 5th Ed. (Academic Press, 1994), pp.98–101.

R. Loudon, The Quantum Theory of Light, 3rd Ed. (Oxford University Press, 2000).

R. J. Essiambre, G. Raybon, and B. Mikkelsen “Pseudo-linear transmission of high-speed TDM signals: 40 and 160 Gb/s,” in Optical Fiber Telecommunications IVB, edited by I. P. Kaminow and T. Li (Academic Press, 2002) pp.232–304.

L. F. Mollenauer, J. P. Gordon, and P. V. Mamyshev, “Solitons in high bit-rate, long-distance transmission,” in Optical Fiber Telecommunications IIIA, edited by I. P. Kaminow and T. L. Koch (Academic Press, 1997), pp.373–460.

Cited By

OSA participates in Crossref's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (11)

Fig. 1.
Fig. 1. Polarization diagrams for degenerate PC driven by parallel LP pumps.
Fig. 2.
Fig. 2. Polarization diagrams for degenerate PC driven by co-rotating CP pumps. The dashed line denotes a signal that is not coupled to the pumps.
Fig. 3.
Fig. 3. Polarization diagrams for degenerate PC driven by orthogonal pumps.
Fig. 4.
Fig. 4. Energy-amplification factor plotted as a function of the chirp cs . The solid and dashed curves represent the frequency offsets ωs = 0 and ωs = 0.5/π s , respectively, where 1/π s is the frequency bandwidth. The upper curves represent maximal gain, whereas the lower curves represent minimal gain.
Fig. 5.
Fig. 5. Pulse power and spectrum at various points in a system with anomalous dispersion. The input pulse frequency equals the average pump frequency. Because the pulse propagates without distorting, the curves are identical.
Fig. 6.
Fig. 6. Pulse power and spectrum in a system with anomalous dispersion. The input pulse frequency does not equal the average pump frequency. Dotted curves represent z = 0 Mm (input), dot-dashed curves represent z = 1 (before the PSA), dashed curves represent z = 1 (after the PSA) and solid curves represent z = 5 (output). Because the pulse propagates without distorting before the PSA, the dotted and dot-dashed curves are identical.
Fig. 7.
Fig. 7. (a) Width, (b) chirp and (c) phase plotted as functions of distance for a DM system with average dispersion 〈β〉 = -0.15 ps2/Km. (d) Power plotted as a function of time. The solid, dot-dashed and dashed curves represent the beginning of the normal half-section, end of the normal half-section and midpoint of the anomalous section, respectively.
Fig. 8.
Fig. 8. Pulse power and spectrum at various points in a DM system. The input pulse frequency equals the average pump frequency. Because the pulse evolution is periodic, and the data sets pertain to equivalent points in the dispersion map, the curves are identical.
Fig. 9.
Fig. 9. Pulse power and spectrum in a DM system. The input pulse frequency does not equal the average pump frequency. Dotted curves represent z = 0 Mm (input), dot-dashed curves represent z = 1 (before the PSA), dashed curves represent z = 1 (after the PSA) and solid curves represent z = 5 (output). Because the pulse evolution is periodic before the PSA, and the data sets pertain to equivalent points in the dispersion map, the dotted and dot-dashed curves are identical.
Fig. 10.
Fig. 10. Pulse power and spectrum in a DM system. The PSA is located at the beginning of a normal section. Dotted curves represent z = 0 Mm (input), dot-dashed curves represent z ≈ 1 (before the PSA), dashed curves represent z ≈ 1 (after the PSA) and solid curves represent z = 5 (output).
Fig. 11.
Fig. 11. Pulse power and spectrum in a DM system. The PSA is located at the end of a normal section. Dotted curves represent z = 0 Mm (input), dot-dashed curves represent z ≈ 1 (before the PSA), dashed curves represent z ≈ 1 (after the PSA) and solid curves represent z = 5 (output).

Tables (2)

Tables Icon

Table 1. Degenerate PC driven by aligned pumps (1 and 3)

Tables Icon

Table 2. Degenerate PC driven by orthogonal pumps (1 and 3)

Equations (100)

Equations on this page are rendered with MathJax. Learn more.

E ( t , z ) = A ( t , z ) exp [ i ( k r z ω r t ) ] + c . c .
i z A = β ( i t ) A + γ a ( A A ) A + γ b ( A t A ) A * ,
d z B 2 = B 2 + B 2 * ,
B 2 ( z ) = μ ( z ) B 2 ( 0 ) + ν ( z ) B 2 * ( 0 ) ,
μ ( z ) = cos ( kz ) + sin ( kz ) k ,
ν ( z ) = sin ( kz ) k
d z B = B + B * ,
d z B * = i κ * B B * ,
B ( z ) = μ ( z ) B ( 0 ) + ν ( z ) B * ( 0 ) ,
B * ( z ) = ν * ( z ) B ( 0 ) + μ * ( z ) B * ( 0 ) ,
B 2 ( ω , z ) = μ ( z ) B 2 ( ω , 0 ) + ν ( z ) B 2 * ( ω , 0 ) ,
B ( ω , z ) = μ ( z ) B ( ω , 0 ) + ν ( z ) B * ( ω , 0 ) ,
B * ( ω , z ) = ν * ( z ) B ( ω , 0 ) + μ * ( z ) B * ( ω , 0 ) ,
i z A = β r tt 2 A 2 .
A ( t , 0 ) = f s 1 2 exp ( i ω s t t 2 2 σ s 2 ) ,
A ( ω ) = A ( t ) exp ( iωt ) dt 2 π 1 2 ,
A ( t ) = A ( ω ) exp ( iωt ) 2 π 1 2 ,
A ( t ) 2 dt = A ( ω ) 2 .
A ( ω , 0 ) = f s 1 2 σ s exp [ σ s 2 ( ω ω s ) 2 2 ] ,
A ( ω , z ) = f s 1 2 σ s exp ( i β r ω 2 z 2 ) exp [ σ s 2 ( ω ω s ) 2 2 ] ,
A ( t , z ) = f s 1 2 exp ( i β r ω s 2 z 2 ) ( 1 i β r z σ s 2 ) 1 2 exp [ s t ( t ) 2 2 ( σ s 2 i β r z ) ] ,
A ( ω , z o ) = μ ( z o z i ) A ( ω , z i ) + ν ¯ ( z o z i ) A * ( ω , z i ) ,
A ( ω , z o ) = μ f s 1 2 σ s exp ( i β r ω 2 z 2 ) exp [ σ s 2 ( ω ω s ) 2 2 ] + νf s 1 2 σ s exp ( r ω 2 z 2 ) exp [ σ s 2 ( ω + ω s ) 2 2 ] .
N o = [ μ 2 + μν * ρ exp ( ) + μ * νρ exp ( ) + ν 2 ] N i ,
ρ = exp ( σ s 2 ω s 2 ) ( 1 + c s 2 ) 1 4 ,
θ = ( tan 1 c s ) 2 ,
N o = ( μ 2 ± 2 μν ρ + ν 2 ) N i .
A ( ω , z + z ) = μ f s 1 2 σ s exp ( i β r ω 2 z s 2 ) exp [ σ s 2 ( ω ω s ) 2 2 ] + νf s 1 2 σ s exp ( r ω 2 z d 2 ) exp [ σ s 2 ( ω + ω s ) 2 2 ] ,
A ( t , z + z ) = μf s 1 2 exp ( i β r ω s 2 z s 2 ) ( 1 i β r z s σ s 2 ) 1 2 exp [ s t ( t ) 2 2 ( σ s 2 i β r z s ) ] + νf s 1 2 exp ( i β r ω s 2 z d 2 ) ( 1 + i β r z d σ s 2 ) 1 2 exp [ s t ( t ) 2 2 ( σ s 2 i β r z d ) ] ,
A ( t , z ) = f s 1 2 exp [ s s ( t τ s ) ( 1 + ic s ) ( t τ s ) 2 2 σ s 2 ] ,
d z e s = 0 ,
d z τ s = β r ω s ,
d z σ s = β r c s σ s ,
d z ω s = 0 ,
d z c s = β r ( 1 + c s 2 ) σ s 2 + γe s ( 2 π ) 1 2 σ s ,
d z ϕ s = β r ( 1 σ s 2 ω s 2 ) 2 + 5 γe s 4 ( 2 π ) 1 2 σ s ,
A ( ω , z ) = [ f s σ s 2 ( 1 + ic s ) ] 1 2 exp [ s + s ω σ s 2 ( ω ω s ) 2 2 ( 1 + ic s ) ] .
ρ = exp [ σ s 2 ω s 2 ( 1 + c s 2 ) ] ( 1 + c s 2 ) 1 4 ,
θ = 2 ϕ s ( tan 1 c s ) 2 + c s σ s 2 ω s 2 ( 1 + c s 2 ) .
ϕ s ( z ) = 3 γe s z 4 ( 2 π ) 1 2 σ s .
c n ( σ s ) + γ n κ n 1 2 log [ κ n 1 2 c n ( σ s ) + κ n σ s γ n κ n γ n ] = κ n z ,
ϕ s ( σ s ) = π 4 1 2 tan 1 [ γ n σ s + 1 c n ( σ s ) ] + 5 γ n 4 κ n 1 2 log [ κ n 1 2 c n ( σ s ) + κ n σ s γ n κ n γ n ] .
c a ( σ s ) γ a κ a 1 2 log [ κ a 1 2 c a ( σ s ) + κ a σ s + γ a κ a σ a + γ a ] = κ a z ,
ϕ s ( σ s ) ϕ s ( σ a ) = 1 2 tan 1 [ γ a σ s 1 c a ( σ s ) ] π 4 + 5 γ a 4 κ a 1 2 log [ κ a 1 2 c a ( σ s ) + κ a σ s + γ a κ a σ a + γ a ] .
σ s ( ζ ) = ( 1 + ζ 2 ) 1 2 ,
c s ( ζ ) = ζ ,
ϕ s ( ζ ) = ( tan 1 ζ ) 2 .
A z = μ g A 2 + ( ν f i β r ) A tt 2 + i γ a A 2 A ,
E = A 2 dt ,
T = t A 2 dt E ,
V = ( t T ) 2 A 2 dt E ,
Ω = Im ( A * A t ) dt E ,
C = 2 Im [ ( t T ) A * A t ] dt E ,
Φ z = Im ( A * A z ) dt E ,
E z = μE ν A t 2 dt ,
( ET ) z = μET + βEΩ + ν Re ( t A * A tt ) dt ,
( EV ) z = μEV + βEC + ν Re [ ( t T ) 2 A * A tt ] dt ,
( E Ω ) z = μE Ω + ν Im ( A t * A tt ) dt ,
( EC ) z = μEC + 2 β A t 2 dt 2 E Ω T z + 2 ν Im [ ( t T ) A t * A tt ] dt + γ A 4 dt ,
E Φ z = β A t 2 dt 2 + γ A 4 dt .
B = A t 2 dt E .
A ( t , z ) = a exp [ ( t τ ) ( 1 + ic ) ( t τ ) 2 2 σ 2 ] ,
e z = { μ ν [ ω 2 + ( 1 + c 2 ) 2 σ 2 ] } e ,
τ z = ( β νc ) ω ,
σ z = βc σ + ν ( 1 + c 2 ) 2 σ ,
ω z = ν ( 1 + c 2 ) ω σ 2 ,
c z = ( β νc ) ( 1 + c 2 ) σ 2 + γe ( 2 π ) 1 2 σ .
ϕ z = β ( 1 σ 2 ω 2 ) 2 + νc ( ω 2 1 2 σ 2 ) + 5 γe 4 ( 2 π ) 1 2 σ .
A ( t , z ) = a exp [ ( t τ ) ] { sech [ ( t τ ) σ ] } 1 + ic ,
e z = { μ ν [ ω 2 + ( 1 + c 2 ) 3 σ 2 ] } e ,
τ z = ( β νc ) ω ,
π 2 σ z 6 = βc σ + ν ( 2 c 2 ) 3 σ ,
ω z = 2 ν ( 1 + c 2 ) ω 3 σ 2
c z = ( 2 β νc ) ( 1 + c 2 ) 3 σ 2 + γe 3 σ .
ϕ z = β [ ω 2 + ( 1 + c 2 ) 3 σ 2 ] 2 + γe 3 σ ω τ z c σ z 2 σ + κ c z .
σ z = c σ ,
c z = ( 1 + c 2 ) σ 2 + γ n σ ,
ϕ z = 1 2 σ 2 + 5 γ n 4 σ ,
σ zz = 1 σ 3 + γ n σ 2 .
σ z 2 = Q n ( σ ) ,
Q n ( σ ) = 1 σ 2 + 2 γ n σ κ n
κ n σdσ c n ( σ ) = ± κ n dz ,
c n ( σ ) = κ n σ 2 2 γ n σ 1 1 2 .
c n ( σ ) + γ n κ n 1 2 log [ κ n 1 2 c n ( σ ) + κ n σ γ n κ n γ n ] = κ n z .
ϕ σ = 1 2 σ c n ( σ ) + 5 γ n 4 c n ( σ ) .
ϕ ( σ ) = π 4 1 2 tan 1 [ γ n σ + 1 c n ( σ ) ] + 5 γ n 4 κ n 1 2 log [ κ n 1 2 c n ( σ ) + κ n σ γ n κ n γ n ] .
σ z = c σ ,
c z = ( 1 + c 2 ) σ 2 + γ a σ ,
ϕ z = 1 2 σ 2 + 5 γ a 4 σ ,
Q a ( σ ) = 1 σ 2 2 γ a σ κ a .
c a ( σ ) = κ a σ 2 + 2 γ a σ 1 1 2 .
c a ( σ ) γ a κ a 1 2 log [ κ a 1 2 c a ( σ ) + κ a σ + γ a κ a + γ a ] = κ a z ,
( σ + 2 ) ( σ 1 ) 1 2 = 3 z 2 ,
c a ( σ ) + γ a κ a 1 2 [ π 2 sin 1 ( γ a κ a σ 1 γ a ) ] = κ a z ,
ϕ σ = 1 2 σ c a ( σ ) + 5 γ a 4 c a ( σ ) .
ϕ ( σ ) = { 1 2 tan 1 [ γ a σ 1 c a ( σ ) ] π 4 + 5 γ a 4 κ a 1 2 log [ κ a 1 2 c a ( σ ) + κ a σ + γ a κ a + γ a ] , tan 1 ( σ 1 ) 1 2 + 5 ( σ 1 ) 1 2 4 , 1 2 tan 1 [ γ a σ 1 c a ( σ ) ] π 4 + 5 γ a 4 κ a 1 2 [ π 2 sin 1 ( γ a κ a σ 1 γ a ) ] ,
κ a = 1 2 γ a + 2 ( γ a + γ n ) ( 1 1 σ n ) .
κ a ( σ a σ ) 2 + 2 ( γ a + κ a σ ) ( σ a σ ) + c a 2 ( σ ) = 0 ,
κ a = [ c n 2 ( σ n ) 2 γ a σ n + 1 ] σ n 2 ,
σ a = [ γ a + ( γ a 2 + κ a ) 1 2 ] κ a .

Metrics