Abstract

We present a semi-classical approach for predicting the quantum noise properties of fiber optical parametric amplifiers. The unavoidable contributors of noise, vacuum fluctuations, loss-induced noise, and spontaneous Raman scattering, are included in the analysis of both phase-insensitive and phase-sensitive amplifiers. We show that the model agrees with earlier fully quantum approaches in the linear gain regime, whereas in the saturated gain regime, in which the classical equations are valid, we predict that the amplifier increases the signal-to-noise ratio by generating an amplitude-squeezed state of light. Also, in the same process, we analyze the quantum noise properties of the pump, which is difficult using standard quantum approaches, and we discover that the pump displays complicated dynamics in both the linear and the nonlinear gain regimes.

© 2013 Optical Society of America

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  1. J. Hansryd, P. A. Andrekson, M. Westlund, J. Li, and P. Hedekvist, “Fiber-based optical parametric amplifiers and their applications,” IEEE J. Sel. Top. Quantum Electron.8, 506–520 (2002).
    [CrossRef]
  2. J. A. Levenson, I. Abram, T. Rivera, and P. Grangier, “Reduction of quantum noise in optical parametric amplification,” J. Opt. Soc. Am. B10, 2233–2238 (1993).
    [CrossRef]
  3. C. M. Caves, “Quantum limits on noise in linear amplifiers,” Phys. Rev. D26, 1817–1839 (1982).
    [CrossRef]
  4. Z. Tong, C. Lundström, P. A. Andrekson, C. J. McKinstrie, M. Karlsson, D. J. Blessing, E. Tipsuwannakul, B. J. Puttnam, H. Toda, and L. Grüner-Nielsen, “Towards ultrasensitive optical links enabled by low-noise phase-sensitive amplifiers,” Nat. Photonics5, 340–436 (2011).
    [CrossRef]
  5. C. J. McKinstrie and J. P. Gordon, “Field fluctuations produced by parametric processes in fibers,” IEEE J. Sel. Top. Quantum Electron.18, 958–969 (2012).
    [CrossRef]
  6. C. J. McKinstrie and S. Radic, “Phase-sensitive amplification in a fiber,” Opt. Express12, 4973–4979 (2004).
    [CrossRef] [PubMed]
  7. C. J. McKinstrie, J. D. Harvey, S. Radic, and M. G. Raymer, “Translation of quantum states by four-wave mixing in fibers,” Opt. Express13, 9131–9142 (2005).
    [CrossRef] [PubMed]
  8. C. J. McKinstrie, M. Yu, M. G. Raymer, and S. Radic, “Quantum noise properties of parametric processes,” Opt. Express13, 4986–5012 (2005).
    [CrossRef] [PubMed]
  9. C. J. McKinstrie, M. G. Raymer, S. Radic, and M. V. Vasilyev, “Quantum mechanics of phase-sensitive amplification in a fiber,” Opt. Commun.257, 146–163 (2005).
    [CrossRef]
  10. P. L. Voss and P. Kumar, “Raman-effect induced noise limits on χ(3) parametric amplifiers and wavelength converters,” J. Opt. B Quantum Semiclassical Opt.6, 762–770 (2004).
    [CrossRef]
  11. P. L. Voss, K. G. Köprülü, and P. Kumar, “Raman-noise-induced quantum limits for χ(3) nondegenerate phase-sensitive amplification and quadrature squeezing,” J. Opt. Soc. Am. B23, 598–609 (2006).
    [CrossRef]
  12. G. P. Agrawal, Nonlinear Fiber Optics (Academic, 2007).
  13. H. Kidorf, K. Rottwitt, M. Nissov, M. Ma, and E. Rabarijaona, “Pump interactions in a 100-nm bandwidth Raman amplifier,” IEEE Photonics Technol. Lett.11, 530–532 (1999).
    [CrossRef]
  14. R. H. Stolen, J. P. Gordon, W. J. Tomlinson, and H. A. Haus, “Raman response function of silica-core fibers,” J. Opt. Soc. Am. B6, 1159–1166 (1989).
    [CrossRef]
  15. K. Rottwitt, J. Bromage, A. J. Stentz, L. Leng, M. E. Lines, and H. Smith, “Scaling of the Raman gain coefficient: Applications to germanosilicate fibers,” J. Lightwave Technol.21, 1652–1662 (2003).
    [CrossRef]
  16. A. S. Y. Hsieh, G. K. L. Wong, S. G. Murdoch, S. Coen, F. Vanholsbeeck, R. Leonhardt, and J. D. Harvey, “Combined effect of Raman and parametric gain on single-pump parametric amplifiers,” Opt. Express15, 8104–8114 (2007).
    [CrossRef] [PubMed]
  17. C. J. McKinstrie and M. G. Raymer, “Four-wave-mixing cascades near the zero-dispersion frequency,” Opt. Express14, 9600–9610 (2006).
    [CrossRef] [PubMed]
  18. Z. Tong, C. Lundström, P. A. Andrekson, M. Karlsson, and A. Bogris, “Ultralow noise, broadband phase-sensitive optical amplifiers, and their applications,” IEEE J. Sel. Top. Quantum Electron.18, 1016–1032 (2012).
    [CrossRef]
  19. M. Vasilyev, “Distributed phase-sensitive amplification,” Opt. Express13, 7563–7571 (2005).
    [CrossRef] [PubMed]
  20. C. C. Gerry and P. L. Knight, Introductory Quantum Optics (Cambridge University, 2005).
  21. C. J. McKinstrie, S. Radic, R. M. Jopson, and A. R. Chraplyvy, “Quantum noise limits on optical monitoring with parametric devices,” Opt. Commun.259, 309–320 (2006).
    [CrossRef]
  22. K. Rottwitt and A. J. Stentz, “Raman amplification in lightwave communication systems,” in Optical Fiber Telecommunications IV-A,I. Kaminow and T. Li, eds. (Academic, 2002), pp. 213–257.
    [CrossRef]
  23. Z. Tong, A. Bogris, M. Karlsson, and P. A. Andrekson, “Full characterization of the signal and idler noise figure spectra in single-pumped fiber optical parametric amplifiers,” Opt. Express18, 2884–2893 (2010).
    [CrossRef] [PubMed]
  24. K. Croussore and G. Li, “Phase and amplitude regeneration of differential phase-shift keyed signals using phase-sensitive amplification,” IEEE J. Sel. Top. Quantum Electron.14, 648–658 (2008).
    [CrossRef]
  25. M. Sköld, J. Yang, H. Sunnerud, M. Karlsson, S. Oda, and P. A. Andrekson, “Constellation diagram analysis of DPSK signal regeneration in a saturated parametric amplifier,” Opt. Express16, 5974–5982 (2008).
    [CrossRef] [PubMed]
  26. C. Peucheret, M. Lorenzen, J. Seoane, D. Noordegraaf, C. V. Nielsen, L. Grüner-Nielsen, and K. Rottwitt, “Amplitude regeneration of RZ-DPSK signals in single-pump fiber-optic parametric amplifiers,” IEEE Photonics Technol. Lett.21, 872–874 (2009).
    [CrossRef]
  27. M. Matsumoto, “Fiber-based all-optical signal regeneration,” IEEE J. Sel. Top. Quantum Electron.18, 738–752 (2012).
    [CrossRef]

2012 (3)

C. J. McKinstrie and J. P. Gordon, “Field fluctuations produced by parametric processes in fibers,” IEEE J. Sel. Top. Quantum Electron.18, 958–969 (2012).
[CrossRef]

Z. Tong, C. Lundström, P. A. Andrekson, M. Karlsson, and A. Bogris, “Ultralow noise, broadband phase-sensitive optical amplifiers, and their applications,” IEEE J. Sel. Top. Quantum Electron.18, 1016–1032 (2012).
[CrossRef]

M. Matsumoto, “Fiber-based all-optical signal regeneration,” IEEE J. Sel. Top. Quantum Electron.18, 738–752 (2012).
[CrossRef]

2011 (1)

Z. Tong, C. Lundström, P. A. Andrekson, C. J. McKinstrie, M. Karlsson, D. J. Blessing, E. Tipsuwannakul, B. J. Puttnam, H. Toda, and L. Grüner-Nielsen, “Towards ultrasensitive optical links enabled by low-noise phase-sensitive amplifiers,” Nat. Photonics5, 340–436 (2011).
[CrossRef]

2010 (1)

2009 (1)

C. Peucheret, M. Lorenzen, J. Seoane, D. Noordegraaf, C. V. Nielsen, L. Grüner-Nielsen, and K. Rottwitt, “Amplitude regeneration of RZ-DPSK signals in single-pump fiber-optic parametric amplifiers,” IEEE Photonics Technol. Lett.21, 872–874 (2009).
[CrossRef]

2008 (2)

K. Croussore and G. Li, “Phase and amplitude regeneration of differential phase-shift keyed signals using phase-sensitive amplification,” IEEE J. Sel. Top. Quantum Electron.14, 648–658 (2008).
[CrossRef]

M. Sköld, J. Yang, H. Sunnerud, M. Karlsson, S. Oda, and P. A. Andrekson, “Constellation diagram analysis of DPSK signal regeneration in a saturated parametric amplifier,” Opt. Express16, 5974–5982 (2008).
[CrossRef] [PubMed]

2007 (1)

2006 (3)

2005 (4)

2004 (2)

P. L. Voss and P. Kumar, “Raman-effect induced noise limits on χ(3) parametric amplifiers and wavelength converters,” J. Opt. B Quantum Semiclassical Opt.6, 762–770 (2004).
[CrossRef]

C. J. McKinstrie and S. Radic, “Phase-sensitive amplification in a fiber,” Opt. Express12, 4973–4979 (2004).
[CrossRef] [PubMed]

2003 (1)

2002 (1)

J. Hansryd, P. A. Andrekson, M. Westlund, J. Li, and P. Hedekvist, “Fiber-based optical parametric amplifiers and their applications,” IEEE J. Sel. Top. Quantum Electron.8, 506–520 (2002).
[CrossRef]

1999 (1)

H. Kidorf, K. Rottwitt, M. Nissov, M. Ma, and E. Rabarijaona, “Pump interactions in a 100-nm bandwidth Raman amplifier,” IEEE Photonics Technol. Lett.11, 530–532 (1999).
[CrossRef]

1993 (1)

1989 (1)

1982 (1)

C. M. Caves, “Quantum limits on noise in linear amplifiers,” Phys. Rev. D26, 1817–1839 (1982).
[CrossRef]

Abram, I.

Agrawal, G. P.

G. P. Agrawal, Nonlinear Fiber Optics (Academic, 2007).

Andrekson, P. A.

Z. Tong, C. Lundström, P. A. Andrekson, M. Karlsson, and A. Bogris, “Ultralow noise, broadband phase-sensitive optical amplifiers, and their applications,” IEEE J. Sel. Top. Quantum Electron.18, 1016–1032 (2012).
[CrossRef]

Z. Tong, C. Lundström, P. A. Andrekson, C. J. McKinstrie, M. Karlsson, D. J. Blessing, E. Tipsuwannakul, B. J. Puttnam, H. Toda, and L. Grüner-Nielsen, “Towards ultrasensitive optical links enabled by low-noise phase-sensitive amplifiers,” Nat. Photonics5, 340–436 (2011).
[CrossRef]

Z. Tong, A. Bogris, M. Karlsson, and P. A. Andrekson, “Full characterization of the signal and idler noise figure spectra in single-pumped fiber optical parametric amplifiers,” Opt. Express18, 2884–2893 (2010).
[CrossRef] [PubMed]

M. Sköld, J. Yang, H. Sunnerud, M. Karlsson, S. Oda, and P. A. Andrekson, “Constellation diagram analysis of DPSK signal regeneration in a saturated parametric amplifier,” Opt. Express16, 5974–5982 (2008).
[CrossRef] [PubMed]

J. Hansryd, P. A. Andrekson, M. Westlund, J. Li, and P. Hedekvist, “Fiber-based optical parametric amplifiers and their applications,” IEEE J. Sel. Top. Quantum Electron.8, 506–520 (2002).
[CrossRef]

Blessing, D. J.

Z. Tong, C. Lundström, P. A. Andrekson, C. J. McKinstrie, M. Karlsson, D. J. Blessing, E. Tipsuwannakul, B. J. Puttnam, H. Toda, and L. Grüner-Nielsen, “Towards ultrasensitive optical links enabled by low-noise phase-sensitive amplifiers,” Nat. Photonics5, 340–436 (2011).
[CrossRef]

Bogris, A.

Z. Tong, C. Lundström, P. A. Andrekson, M. Karlsson, and A. Bogris, “Ultralow noise, broadband phase-sensitive optical amplifiers, and their applications,” IEEE J. Sel. Top. Quantum Electron.18, 1016–1032 (2012).
[CrossRef]

Z. Tong, A. Bogris, M. Karlsson, and P. A. Andrekson, “Full characterization of the signal and idler noise figure spectra in single-pumped fiber optical parametric amplifiers,” Opt. Express18, 2884–2893 (2010).
[CrossRef] [PubMed]

Bromage, J.

Caves, C. M.

C. M. Caves, “Quantum limits on noise in linear amplifiers,” Phys. Rev. D26, 1817–1839 (1982).
[CrossRef]

Chraplyvy, A. R.

C. J. McKinstrie, S. Radic, R. M. Jopson, and A. R. Chraplyvy, “Quantum noise limits on optical monitoring with parametric devices,” Opt. Commun.259, 309–320 (2006).
[CrossRef]

Coen, S.

Croussore, K.

K. Croussore and G. Li, “Phase and amplitude regeneration of differential phase-shift keyed signals using phase-sensitive amplification,” IEEE J. Sel. Top. Quantum Electron.14, 648–658 (2008).
[CrossRef]

Gerry, C. C.

C. C. Gerry and P. L. Knight, Introductory Quantum Optics (Cambridge University, 2005).

Gordon, J. P.

C. J. McKinstrie and J. P. Gordon, “Field fluctuations produced by parametric processes in fibers,” IEEE J. Sel. Top. Quantum Electron.18, 958–969 (2012).
[CrossRef]

R. H. Stolen, J. P. Gordon, W. J. Tomlinson, and H. A. Haus, “Raman response function of silica-core fibers,” J. Opt. Soc. Am. B6, 1159–1166 (1989).
[CrossRef]

Grangier, P.

Grüner-Nielsen, L.

Z. Tong, C. Lundström, P. A. Andrekson, C. J. McKinstrie, M. Karlsson, D. J. Blessing, E. Tipsuwannakul, B. J. Puttnam, H. Toda, and L. Grüner-Nielsen, “Towards ultrasensitive optical links enabled by low-noise phase-sensitive amplifiers,” Nat. Photonics5, 340–436 (2011).
[CrossRef]

C. Peucheret, M. Lorenzen, J. Seoane, D. Noordegraaf, C. V. Nielsen, L. Grüner-Nielsen, and K. Rottwitt, “Amplitude regeneration of RZ-DPSK signals in single-pump fiber-optic parametric amplifiers,” IEEE Photonics Technol. Lett.21, 872–874 (2009).
[CrossRef]

Hansryd, J.

J. Hansryd, P. A. Andrekson, M. Westlund, J. Li, and P. Hedekvist, “Fiber-based optical parametric amplifiers and their applications,” IEEE J. Sel. Top. Quantum Electron.8, 506–520 (2002).
[CrossRef]

Harvey, J. D.

Haus, H. A.

Hedekvist, P.

J. Hansryd, P. A. Andrekson, M. Westlund, J. Li, and P. Hedekvist, “Fiber-based optical parametric amplifiers and their applications,” IEEE J. Sel. Top. Quantum Electron.8, 506–520 (2002).
[CrossRef]

Hsieh, A. S. Y.

Jopson, R. M.

C. J. McKinstrie, S. Radic, R. M. Jopson, and A. R. Chraplyvy, “Quantum noise limits on optical monitoring with parametric devices,” Opt. Commun.259, 309–320 (2006).
[CrossRef]

Karlsson, M.

Z. Tong, C. Lundström, P. A. Andrekson, M. Karlsson, and A. Bogris, “Ultralow noise, broadband phase-sensitive optical amplifiers, and their applications,” IEEE J. Sel. Top. Quantum Electron.18, 1016–1032 (2012).
[CrossRef]

Z. Tong, C. Lundström, P. A. Andrekson, C. J. McKinstrie, M. Karlsson, D. J. Blessing, E. Tipsuwannakul, B. J. Puttnam, H. Toda, and L. Grüner-Nielsen, “Towards ultrasensitive optical links enabled by low-noise phase-sensitive amplifiers,” Nat. Photonics5, 340–436 (2011).
[CrossRef]

Z. Tong, A. Bogris, M. Karlsson, and P. A. Andrekson, “Full characterization of the signal and idler noise figure spectra in single-pumped fiber optical parametric amplifiers,” Opt. Express18, 2884–2893 (2010).
[CrossRef] [PubMed]

M. Sköld, J. Yang, H. Sunnerud, M. Karlsson, S. Oda, and P. A. Andrekson, “Constellation diagram analysis of DPSK signal regeneration in a saturated parametric amplifier,” Opt. Express16, 5974–5982 (2008).
[CrossRef] [PubMed]

Kidorf, H.

H. Kidorf, K. Rottwitt, M. Nissov, M. Ma, and E. Rabarijaona, “Pump interactions in a 100-nm bandwidth Raman amplifier,” IEEE Photonics Technol. Lett.11, 530–532 (1999).
[CrossRef]

Knight, P. L.

C. C. Gerry and P. L. Knight, Introductory Quantum Optics (Cambridge University, 2005).

Köprülü, K. G.

Kumar, P.

P. L. Voss, K. G. Köprülü, and P. Kumar, “Raman-noise-induced quantum limits for χ(3) nondegenerate phase-sensitive amplification and quadrature squeezing,” J. Opt. Soc. Am. B23, 598–609 (2006).
[CrossRef]

P. L. Voss and P. Kumar, “Raman-effect induced noise limits on χ(3) parametric amplifiers and wavelength converters,” J. Opt. B Quantum Semiclassical Opt.6, 762–770 (2004).
[CrossRef]

Leng, L.

Leonhardt, R.

Levenson, J. A.

Li, G.

K. Croussore and G. Li, “Phase and amplitude regeneration of differential phase-shift keyed signals using phase-sensitive amplification,” IEEE J. Sel. Top. Quantum Electron.14, 648–658 (2008).
[CrossRef]

Li, J.

J. Hansryd, P. A. Andrekson, M. Westlund, J. Li, and P. Hedekvist, “Fiber-based optical parametric amplifiers and their applications,” IEEE J. Sel. Top. Quantum Electron.8, 506–520 (2002).
[CrossRef]

Lines, M. E.

Lorenzen, M.

C. Peucheret, M. Lorenzen, J. Seoane, D. Noordegraaf, C. V. Nielsen, L. Grüner-Nielsen, and K. Rottwitt, “Amplitude regeneration of RZ-DPSK signals in single-pump fiber-optic parametric amplifiers,” IEEE Photonics Technol. Lett.21, 872–874 (2009).
[CrossRef]

Lundström, C.

Z. Tong, C. Lundström, P. A. Andrekson, M. Karlsson, and A. Bogris, “Ultralow noise, broadband phase-sensitive optical amplifiers, and their applications,” IEEE J. Sel. Top. Quantum Electron.18, 1016–1032 (2012).
[CrossRef]

Z. Tong, C. Lundström, P. A. Andrekson, C. J. McKinstrie, M. Karlsson, D. J. Blessing, E. Tipsuwannakul, B. J. Puttnam, H. Toda, and L. Grüner-Nielsen, “Towards ultrasensitive optical links enabled by low-noise phase-sensitive amplifiers,” Nat. Photonics5, 340–436 (2011).
[CrossRef]

Ma, M.

H. Kidorf, K. Rottwitt, M. Nissov, M. Ma, and E. Rabarijaona, “Pump interactions in a 100-nm bandwidth Raman amplifier,” IEEE Photonics Technol. Lett.11, 530–532 (1999).
[CrossRef]

Matsumoto, M.

M. Matsumoto, “Fiber-based all-optical signal regeneration,” IEEE J. Sel. Top. Quantum Electron.18, 738–752 (2012).
[CrossRef]

McKinstrie, C. J.

C. J. McKinstrie and J. P. Gordon, “Field fluctuations produced by parametric processes in fibers,” IEEE J. Sel. Top. Quantum Electron.18, 958–969 (2012).
[CrossRef]

Z. Tong, C. Lundström, P. A. Andrekson, C. J. McKinstrie, M. Karlsson, D. J. Blessing, E. Tipsuwannakul, B. J. Puttnam, H. Toda, and L. Grüner-Nielsen, “Towards ultrasensitive optical links enabled by low-noise phase-sensitive amplifiers,” Nat. Photonics5, 340–436 (2011).
[CrossRef]

C. J. McKinstrie and M. G. Raymer, “Four-wave-mixing cascades near the zero-dispersion frequency,” Opt. Express14, 9600–9610 (2006).
[CrossRef] [PubMed]

C. J. McKinstrie, S. Radic, R. M. Jopson, and A. R. Chraplyvy, “Quantum noise limits on optical monitoring with parametric devices,” Opt. Commun.259, 309–320 (2006).
[CrossRef]

C. J. McKinstrie, M. G. Raymer, S. Radic, and M. V. Vasilyev, “Quantum mechanics of phase-sensitive amplification in a fiber,” Opt. Commun.257, 146–163 (2005).
[CrossRef]

C. J. McKinstrie, J. D. Harvey, S. Radic, and M. G. Raymer, “Translation of quantum states by four-wave mixing in fibers,” Opt. Express13, 9131–9142 (2005).
[CrossRef] [PubMed]

C. J. McKinstrie, M. Yu, M. G. Raymer, and S. Radic, “Quantum noise properties of parametric processes,” Opt. Express13, 4986–5012 (2005).
[CrossRef] [PubMed]

C. J. McKinstrie and S. Radic, “Phase-sensitive amplification in a fiber,” Opt. Express12, 4973–4979 (2004).
[CrossRef] [PubMed]

Murdoch, S. G.

Nielsen, C. V.

C. Peucheret, M. Lorenzen, J. Seoane, D. Noordegraaf, C. V. Nielsen, L. Grüner-Nielsen, and K. Rottwitt, “Amplitude regeneration of RZ-DPSK signals in single-pump fiber-optic parametric amplifiers,” IEEE Photonics Technol. Lett.21, 872–874 (2009).
[CrossRef]

Nissov, M.

H. Kidorf, K. Rottwitt, M. Nissov, M. Ma, and E. Rabarijaona, “Pump interactions in a 100-nm bandwidth Raman amplifier,” IEEE Photonics Technol. Lett.11, 530–532 (1999).
[CrossRef]

Noordegraaf, D.

C. Peucheret, M. Lorenzen, J. Seoane, D. Noordegraaf, C. V. Nielsen, L. Grüner-Nielsen, and K. Rottwitt, “Amplitude regeneration of RZ-DPSK signals in single-pump fiber-optic parametric amplifiers,” IEEE Photonics Technol. Lett.21, 872–874 (2009).
[CrossRef]

Oda, S.

Peucheret, C.

C. Peucheret, M. Lorenzen, J. Seoane, D. Noordegraaf, C. V. Nielsen, L. Grüner-Nielsen, and K. Rottwitt, “Amplitude regeneration of RZ-DPSK signals in single-pump fiber-optic parametric amplifiers,” IEEE Photonics Technol. Lett.21, 872–874 (2009).
[CrossRef]

Puttnam, B. J.

Z. Tong, C. Lundström, P. A. Andrekson, C. J. McKinstrie, M. Karlsson, D. J. Blessing, E. Tipsuwannakul, B. J. Puttnam, H. Toda, and L. Grüner-Nielsen, “Towards ultrasensitive optical links enabled by low-noise phase-sensitive amplifiers,” Nat. Photonics5, 340–436 (2011).
[CrossRef]

Rabarijaona, E.

H. Kidorf, K. Rottwitt, M. Nissov, M. Ma, and E. Rabarijaona, “Pump interactions in a 100-nm bandwidth Raman amplifier,” IEEE Photonics Technol. Lett.11, 530–532 (1999).
[CrossRef]

Radic, S.

C. J. McKinstrie, S. Radic, R. M. Jopson, and A. R. Chraplyvy, “Quantum noise limits on optical monitoring with parametric devices,” Opt. Commun.259, 309–320 (2006).
[CrossRef]

C. J. McKinstrie, J. D. Harvey, S. Radic, and M. G. Raymer, “Translation of quantum states by four-wave mixing in fibers,” Opt. Express13, 9131–9142 (2005).
[CrossRef] [PubMed]

C. J. McKinstrie, M. G. Raymer, S. Radic, and M. V. Vasilyev, “Quantum mechanics of phase-sensitive amplification in a fiber,” Opt. Commun.257, 146–163 (2005).
[CrossRef]

C. J. McKinstrie, M. Yu, M. G. Raymer, and S. Radic, “Quantum noise properties of parametric processes,” Opt. Express13, 4986–5012 (2005).
[CrossRef] [PubMed]

C. J. McKinstrie and S. Radic, “Phase-sensitive amplification in a fiber,” Opt. Express12, 4973–4979 (2004).
[CrossRef] [PubMed]

Raymer, M. G.

Rivera, T.

Rottwitt, K.

C. Peucheret, M. Lorenzen, J. Seoane, D. Noordegraaf, C. V. Nielsen, L. Grüner-Nielsen, and K. Rottwitt, “Amplitude regeneration of RZ-DPSK signals in single-pump fiber-optic parametric amplifiers,” IEEE Photonics Technol. Lett.21, 872–874 (2009).
[CrossRef]

K. Rottwitt, J. Bromage, A. J. Stentz, L. Leng, M. E. Lines, and H. Smith, “Scaling of the Raman gain coefficient: Applications to germanosilicate fibers,” J. Lightwave Technol.21, 1652–1662 (2003).
[CrossRef]

H. Kidorf, K. Rottwitt, M. Nissov, M. Ma, and E. Rabarijaona, “Pump interactions in a 100-nm bandwidth Raman amplifier,” IEEE Photonics Technol. Lett.11, 530–532 (1999).
[CrossRef]

K. Rottwitt and A. J. Stentz, “Raman amplification in lightwave communication systems,” in Optical Fiber Telecommunications IV-A,I. Kaminow and T. Li, eds. (Academic, 2002), pp. 213–257.
[CrossRef]

Seoane, J.

C. Peucheret, M. Lorenzen, J. Seoane, D. Noordegraaf, C. V. Nielsen, L. Grüner-Nielsen, and K. Rottwitt, “Amplitude regeneration of RZ-DPSK signals in single-pump fiber-optic parametric amplifiers,” IEEE Photonics Technol. Lett.21, 872–874 (2009).
[CrossRef]

Sköld, M.

Smith, H.

Stentz, A. J.

K. Rottwitt, J. Bromage, A. J. Stentz, L. Leng, M. E. Lines, and H. Smith, “Scaling of the Raman gain coefficient: Applications to germanosilicate fibers,” J. Lightwave Technol.21, 1652–1662 (2003).
[CrossRef]

K. Rottwitt and A. J. Stentz, “Raman amplification in lightwave communication systems,” in Optical Fiber Telecommunications IV-A,I. Kaminow and T. Li, eds. (Academic, 2002), pp. 213–257.
[CrossRef]

Stolen, R. H.

Sunnerud, H.

Tipsuwannakul, E.

Z. Tong, C. Lundström, P. A. Andrekson, C. J. McKinstrie, M. Karlsson, D. J. Blessing, E. Tipsuwannakul, B. J. Puttnam, H. Toda, and L. Grüner-Nielsen, “Towards ultrasensitive optical links enabled by low-noise phase-sensitive amplifiers,” Nat. Photonics5, 340–436 (2011).
[CrossRef]

Toda, H.

Z. Tong, C. Lundström, P. A. Andrekson, C. J. McKinstrie, M. Karlsson, D. J. Blessing, E. Tipsuwannakul, B. J. Puttnam, H. Toda, and L. Grüner-Nielsen, “Towards ultrasensitive optical links enabled by low-noise phase-sensitive amplifiers,” Nat. Photonics5, 340–436 (2011).
[CrossRef]

Tomlinson, W. J.

Tong, Z.

Z. Tong, C. Lundström, P. A. Andrekson, M. Karlsson, and A. Bogris, “Ultralow noise, broadband phase-sensitive optical amplifiers, and their applications,” IEEE J. Sel. Top. Quantum Electron.18, 1016–1032 (2012).
[CrossRef]

Z. Tong, C. Lundström, P. A. Andrekson, C. J. McKinstrie, M. Karlsson, D. J. Blessing, E. Tipsuwannakul, B. J. Puttnam, H. Toda, and L. Grüner-Nielsen, “Towards ultrasensitive optical links enabled by low-noise phase-sensitive amplifiers,” Nat. Photonics5, 340–436 (2011).
[CrossRef]

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J. Lightwave Technol. (1)

J. Opt. B Quantum Semiclassical Opt. (1)

P. L. Voss and P. Kumar, “Raman-effect induced noise limits on χ(3) parametric amplifiers and wavelength converters,” J. Opt. B Quantum Semiclassical Opt.6, 762–770 (2004).
[CrossRef]

J. Opt. Soc. Am. B (3)

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[CrossRef]

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Figures (4)

Fig. 1
Fig. 1

Schematics of six different configurations of parametric processes based on four-wave mixing, {p, s, i} denote pump, signal and idler, respectively.

Fig. 2
Fig. 2

(a) PIA (gray) and PSA (black) gains and NFs versus fiber length with and without the Raman effect and loss (the gain with Raman and loss are not shown; line styles apply to both PIA and PSA curves), (b) Gain and NF spetrca at the amplifier output (z = 300 m), which show the effect of SRS and LIN on the NF. Parameters: Pp,0 = 1.4 W, Ps,0 = 10−7 W, λp = 1560.7 nm, λ0 = 1559 nm, γ = 11 (Wkm)−1, ∂D/∂λ = 0.03 ps/(nm2 km), α = 0.4 dB/km, T = 300 K, and Δz = 1 m.

Fig. 3
Fig. 3

(a) Signal gain (black) and NF (gray) of a depleted PIA without loss and Raman scattering (solid lines), and with loss and Raman scattering (dashed lines). (b) The top plots show constellation diagrams of the signal at the positions i)–iii) in (a), where loss and Raman scattering are included in all of them; the scalings in the three diagrams are not equal, so the sizes of the ensembles cannot be compared, only their shapes (the dotted lines point towards the origins of the phase-space diagrams). The bottom plot shows the degree of amplitude squeezing through the amplifiers with and without loss and Raman scattering. All plots indicate that neglecting Raman scattering beyond full pump depletion leads to a significant error. The parameters are the same as in Fig. 2, but with Ps,0 = 10−4 W.

Fig. 4
Fig. 4

(a) Pump power and NF through a PIA with and without loss and Raman scattering. The dots i)–vii) denote the positions to which the constellation diagrams in (b) belong, where i) z = 0 m, ii) z = 67 m, iii) z = 133 m, iv) z = 200 m, v) z = 269 m, vi) z = 332 m and vii) z = 369 m; loss and Raman scattering are included in all of them. The scaling on the axes of all the diagrams are equal except for the denoted zooms, so the sizes and shapes of the ensembles can be compared; the diagrams show the case with loss and Raman scattering. The parameters are the same as in Fig. 3.

Equations (29)

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E ( r , t ) = 1 2 x j = 1 , 2 , 3 F j ( x , y ) A j ( z ) exp ( i β j z i ω j t ) + c . c . ,
A 1 z = i γ ( [ | A 1 | 2 + 2 ( | A 2 | 2 + | A 3 | 2 ) ] A 1 + A 2 2 A 3 * exp ( i Δ β z ) ) ,
A 2 z = i γ ( [ | A 2 | 2 + 2 ( | A 1 | 2 + | A 3 | 2 ) ] A 2 + 2 A 1 A 3 A 2 * exp ( i Δ β z ) ) ,
A 3 z = i γ ( [ | A 3 | 2 + 2 ( | A 1 | 2 + | A 2 | 2 ) ] A 3 + A 2 2 A 1 * exp ( i Δ β z ) ) ,
g R 21 2 | A 2 | 2 ( n T 21 + 1 ) A 1 + g R 31 2 | A 3 | 2 ( n T 31 + 1 ) A 1 ,
g R 21 2 | A 2 | 2 n T 21 A 1 g R 31 2 | A 3 | 2 n T 31 A 1 .
g R 21 2 | A 1 | 2 ( n T 21 + 1 ) ω 1 ω 2 A 2 + g R 32 2 | A 3 | 2 ( n T 32 + 1 ) A 2 ,
g R 21 2 | A 1 | 2 n T 21 ω 1 ω 2 A 2 g R 32 2 | A 3 | 2 n T 32 A 2 .
A 1 z = i γ ( [ | A 1 | 2 + 2 ( | A 2 | 2 + | A 3 | 2 ) ] A 1 + A 2 2 A 3 * exp ( i Δ β z ) ) α 2 A 1 + g R 21 2 | A 2 | 2 A 1 + g R 31 2 | A 3 | 2 A 1 ,
A 2 z = i γ ( [ | A 2 | 2 + 2 ( | A 1 | 2 + | A 3 | 2 ) ] A 2 + 2 A 1 A 3 A 2 * exp ( i Δ β z ) ) α 2 A 2 g R 21 2 ω 2 ω 1 | A 1 | 2 A 2 + g R 32 2 | A 3 | 2 A 2 ,
A 3 z = i γ ( [ | A 3 | 2 + 2 ( | A 1 | 2 + | A 2 | 2 ) ] A 3 + A 2 2 A 1 * exp ( i Δ β z ) ) α 2 A 3 g R 31 2 ω 3 ω 1 | A 1 | 2 A 3 g R 32 2 ω 3 ω 2 | A 2 | 2 A 3 .
h R ( t ) = exp ( t / τ 2 ) sin ( t / τ 1 ) Θ ( t ) ,
G ( z ) = | A s ( z ) | 2 | A s ( 0 ) | 2 ,
SNR P I = | A s ( z ) | 2 2 Var ( | A s ( z ) | 2 ) .
SNR P S = | A s ( z ) | 2 + | A i ( z ) | 2 2 Var ( | A s ( z ) | 2 + | A i ( z ) | 2 ) .
A j = x ¯ j + δ x j + i ( p ¯ j + δ p j ) ,
n j = | A j | 2 = x ¯ j 2 + p ¯ j 2 + 1 / 2 ,
Var ( n j ) = | A j | 4 | A j | 2 2 = x ¯ j 2 + p ¯ j 2 + 1 / 4 .
δ a LIN = 0 ,
Var ( δ a LIN ) = α Δ z / 4 ,
δ a S = 0 ,
Var ( δ a S ) = ( 1 + n T ( Ω j k ) ) g R ( Ω j k ) | A j | 2 Δ z / 2 ,
δ a A S = 0 ,
Var ( δ a A S ) = n T ( Ω j k ) g R ( Ω j k ) | A j | 2 Δ z / 2 .
G P I ( z ) = 1 + [ γ P p g sinh ( g z ) ] 2 ,
NF P I ( z ) = 1 + G P I ( z ) 1 G P I ( z ) ,
P s ( z ) = ( G P I ( z ) P s , 0 + ( G P I ( z ) 1 ) P i , 0 ) 2 ,
P i ( z ) = ( G P I ( z ) P i , 0 + ( G P I ( z ) 1 ) P s , 0 ) 2 ,
NF P S ( z ) = ( P s , 0 + P i , 0 ) ( G P I P s + ( G P I 1 ) P i ) 2 + ( G P I P i + ( G P I 1 ) P s ) 2 ( P s + P i ) 2 .

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