Abstract

In vector four-wave mixing, one or two strong pump waves drive two weak signal and idler waves, each of which has two polarization components. In this paper, vector four-wave mixing processes in a randomly-birefringent fiber (modulation interaction, phase conjugation and Bragg scattering) are studied in detail. For each process, the Schmidt decompositions of the coupling matrices facilitate the solution of the signal–idler equations and the Schmidt decomposition of the associated transfer matrix. The results of this paper are valid for arbitrary pump polarizations.

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  1. J. Hansryd, P. A. Andrekson, M. Westlund, J. Li, and P. O. Hedekvist, “Fiber-based optical parametric amplifiers and their applications,” IEEE J. Sel. Top. Quantum Electron.8, 506–520 (2002).
    [CrossRef]
  2. C. J. McKinstrie, S. Radic, and A. H. Gnauck, “All-optical signal processing by fiber-based parametric devices,” Opt. Photon. News18(3), 34–40 (2007).
    [CrossRef]
  3. H. Cruz-Ramirez, R. Ramirez-Alarcon, M. Corona, K. Garay-Palmett, and A. B. U’Ren, “Spontaneous parametric processes in quantum optics,” Opt. Photon. News22(11), 37–41 (2011).
    [CrossRef]
  4. M. G. Raymer and K. Srinivasan, “Manipulating the color and shape of single photons,” Phys. Today65(11), 32–37 (2012).
    [CrossRef]
  5. C. J. McKinstrie and M. Karlsson, “Schmidt decompositions of parametric processes I: Basic theory and simple examples,” Opt. Express21, 1374–1394 (2013) and references therein.
    [CrossRef] [PubMed]
  6. H. P. Yuen, “Two-photon coherent states of the radiation field,” Phys. Rev. A13, 2226–2243 (1976).
    [CrossRef]
  7. C. M. Caves, “Quantum limits on noise in linear ampifiers,” Phys. Rev. D26, 1817–1839 (1982).
    [CrossRef]
  8. K. Inoue, “Polarization effect on four-wave mixing efficiency in a single-mode fiber,” IEEE J. Quantum Electron.28, 883–894 (1992).
    [CrossRef]
  9. C. J. McKinstrie, H. Kogelnik, R. M. Jopson, S. Radic, and A. V. Kanaev, “Four-wave mixing in fibers with random birefringence,” Opt. Express12, 2033–2055 (2004).
    [CrossRef] [PubMed]
  10. H. Kogelnik and C. J. McKinstrie, “Dynamic eigenstates of parametric interactions in randomly birefringent fibers,” IEEE Photon. Technol. Lett.21, 1036–1038 (2009).
    [CrossRef]
  11. S. V. Manakov, “On the theory of two-dimensional stationary self-focusing of electromagnetic waves,” Sov. Phys. JETP38, 248–253 (1974).
  12. P. K. A. Wai, C. R. Menyuk, and H. H. Chen, “Effects of randomly varying birefringence on soliton interactions in optical fibers,” Opt. Lett.16, 1735–1737 (1991).
    [CrossRef] [PubMed]
  13. S. G. Evangelides, L. F. Mollenauer, J. P. Gordon, and N. S. Bergano, “Polarization multiplexing with solitons,” J. Lightwave Technol.10, 28–35 (1992).
    [CrossRef]
  14. P. K. A. Wai and C. R. Menyuk, “Polarization mode dispersion, decorrelation and diffusion in optical fibers with randomly varying birefringence,” J. Lightwave Technol.14, 148–157 (1996).
    [CrossRef]
  15. T. I. Lakoba, “Concerning the equations governing nonlinear pulse propagation in randomly birefringent fibers,” J. Opt. Soc. Am. B13, 2006–2011 (1996).
    [CrossRef]
  16. D. Marcuse, C. R. Menyuk, and P. K. A. Wai, “Application of the Manakov-PMD equation to studies of signal propagation in optical fibers with randomly varying birefringence,” J. Lightwave Technol.15, 1735–1746 (1997).
    [CrossRef]
  17. C. J. McKinstrie, H. Kogelnik, G. G. Luther, and L. Schenato, “Stokes-space derivations of generalized Schrödinger equations for wave propagation in various fibers,” Opt. Express15, 10964–10983 (2007).
    [CrossRef] [PubMed]
  18. J. P. Gordon and H. Kogelnik, “PMD fundamentals: Polarization mode dispersion in optical fibers,” Proc. Nat. Acad. Sci.97, 4541–4550 (2000).
    [CrossRef] [PubMed]
  19. P. O. Hedekvist, M. Karlsson, and P. A. Andrekson, “Polarization dependence and efficiency in a fiber four-wave mixing phase conjugator with orthogonal pump waves,” IEEE Photon. Technol. Lett.8, 776–778 (1996).
    [CrossRef]
  20. F. Yaman, Q. Lin, and G. P. Agrawal, “Effects of polarization-mode dispersion in dual-pump fiber-optic parametric amplifiers,” IEEE Photon. Technol. Lett.16, 431–433 (2004).
    [CrossRef]
  21. M. Karlsson and H. Sunnerud, “Effects of nonlinearities on PMD-induced system impairments,” J. Lightwave Technol.24, 4127–4137 (2006).
    [CrossRef]
  22. C. J. McKinstrie and S. Radic, “Phase-sensitive amplification in a fiber,” Opt. Express12, 4973–4979 (2004).
    [CrossRef] [PubMed]
  23. 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 (2006).
    [CrossRef]
  24. M. G. Raymer, S. J. van Enk, C. J. McKinstrie, and H. J. McGuinness, “Interference of two photons of different color,” Opt. Commun.283, 747–752 (2010).
    [CrossRef]
  25. S. Prasad, M. O. Scully, and W. Martienssen, “A quantum description of the beam splitter,” Opt. Commun.62, 139–145 (1987).
    [CrossRef]
  26. H. Fearn and R. Loudon, “Quantum theory of the lossless beam splitter,” Opt. Commun.64, 485–490 (1987).
    [CrossRef]

2013

2012

M. G. Raymer and K. Srinivasan, “Manipulating the color and shape of single photons,” Phys. Today65(11), 32–37 (2012).
[CrossRef]

2011

H. Cruz-Ramirez, R. Ramirez-Alarcon, M. Corona, K. Garay-Palmett, and A. B. U’Ren, “Spontaneous parametric processes in quantum optics,” Opt. Photon. News22(11), 37–41 (2011).
[CrossRef]

2010

M. G. Raymer, S. J. van Enk, C. J. McKinstrie, and H. J. McGuinness, “Interference of two photons of different color,” Opt. Commun.283, 747–752 (2010).
[CrossRef]

2009

H. Kogelnik and C. J. McKinstrie, “Dynamic eigenstates of parametric interactions in randomly birefringent fibers,” IEEE Photon. Technol. Lett.21, 1036–1038 (2009).
[CrossRef]

2007

2006

M. Karlsson and H. Sunnerud, “Effects of nonlinearities on PMD-induced system impairments,” J. Lightwave Technol.24, 4127–4137 (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 (2006).
[CrossRef]

2004

2002

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

2000

J. P. Gordon and H. Kogelnik, “PMD fundamentals: Polarization mode dispersion in optical fibers,” Proc. Nat. Acad. Sci.97, 4541–4550 (2000).
[CrossRef] [PubMed]

1997

D. Marcuse, C. R. Menyuk, and P. K. A. Wai, “Application of the Manakov-PMD equation to studies of signal propagation in optical fibers with randomly varying birefringence,” J. Lightwave Technol.15, 1735–1746 (1997).
[CrossRef]

1996

P. K. A. Wai and C. R. Menyuk, “Polarization mode dispersion, decorrelation and diffusion in optical fibers with randomly varying birefringence,” J. Lightwave Technol.14, 148–157 (1996).
[CrossRef]

T. I. Lakoba, “Concerning the equations governing nonlinear pulse propagation in randomly birefringent fibers,” J. Opt. Soc. Am. B13, 2006–2011 (1996).
[CrossRef]

P. O. Hedekvist, M. Karlsson, and P. A. Andrekson, “Polarization dependence and efficiency in a fiber four-wave mixing phase conjugator with orthogonal pump waves,” IEEE Photon. Technol. Lett.8, 776–778 (1996).
[CrossRef]

1992

S. G. Evangelides, L. F. Mollenauer, J. P. Gordon, and N. S. Bergano, “Polarization multiplexing with solitons,” J. Lightwave Technol.10, 28–35 (1992).
[CrossRef]

K. Inoue, “Polarization effect on four-wave mixing efficiency in a single-mode fiber,” IEEE J. Quantum Electron.28, 883–894 (1992).
[CrossRef]

1991

1987

S. Prasad, M. O. Scully, and W. Martienssen, “A quantum description of the beam splitter,” Opt. Commun.62, 139–145 (1987).
[CrossRef]

H. Fearn and R. Loudon, “Quantum theory of the lossless beam splitter,” Opt. Commun.64, 485–490 (1987).
[CrossRef]

1982

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

1976

H. P. Yuen, “Two-photon coherent states of the radiation field,” Phys. Rev. A13, 2226–2243 (1976).
[CrossRef]

1974

S. V. Manakov, “On the theory of two-dimensional stationary self-focusing of electromagnetic waves,” Sov. Phys. JETP38, 248–253 (1974).

Agrawal, G. P.

F. Yaman, Q. Lin, and G. P. Agrawal, “Effects of polarization-mode dispersion in dual-pump fiber-optic parametric amplifiers,” IEEE Photon. Technol. Lett.16, 431–433 (2004).
[CrossRef]

Andrekson, P. A.

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

P. O. Hedekvist, M. Karlsson, and P. A. Andrekson, “Polarization dependence and efficiency in a fiber four-wave mixing phase conjugator with orthogonal pump waves,” IEEE Photon. Technol. Lett.8, 776–778 (1996).
[CrossRef]

Bergano, N. S.

S. G. Evangelides, L. F. Mollenauer, J. P. Gordon, and N. S. Bergano, “Polarization multiplexing with solitons,” J. Lightwave Technol.10, 28–35 (1992).
[CrossRef]

Caves, C. M.

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

Chen, H. H.

Corona, M.

H. Cruz-Ramirez, R. Ramirez-Alarcon, M. Corona, K. Garay-Palmett, and A. B. U’Ren, “Spontaneous parametric processes in quantum optics,” Opt. Photon. News22(11), 37–41 (2011).
[CrossRef]

Cruz-Ramirez, H.

H. Cruz-Ramirez, R. Ramirez-Alarcon, M. Corona, K. Garay-Palmett, and A. B. U’Ren, “Spontaneous parametric processes in quantum optics,” Opt. Photon. News22(11), 37–41 (2011).
[CrossRef]

Evangelides, S. G.

S. G. Evangelides, L. F. Mollenauer, J. P. Gordon, and N. S. Bergano, “Polarization multiplexing with solitons,” J. Lightwave Technol.10, 28–35 (1992).
[CrossRef]

Fearn, H.

H. Fearn and R. Loudon, “Quantum theory of the lossless beam splitter,” Opt. Commun.64, 485–490 (1987).
[CrossRef]

Garay-Palmett, K.

H. Cruz-Ramirez, R. Ramirez-Alarcon, M. Corona, K. Garay-Palmett, and A. B. U’Ren, “Spontaneous parametric processes in quantum optics,” Opt. Photon. News22(11), 37–41 (2011).
[CrossRef]

Gnauck, A. H.

C. J. McKinstrie, S. Radic, and A. H. Gnauck, “All-optical signal processing by fiber-based parametric devices,” Opt. Photon. News18(3), 34–40 (2007).
[CrossRef]

Gordon, J. P.

J. P. Gordon and H. Kogelnik, “PMD fundamentals: Polarization mode dispersion in optical fibers,” Proc. Nat. Acad. Sci.97, 4541–4550 (2000).
[CrossRef] [PubMed]

S. G. Evangelides, L. F. Mollenauer, J. P. Gordon, and N. S. Bergano, “Polarization multiplexing with solitons,” J. Lightwave Technol.10, 28–35 (1992).
[CrossRef]

Hansryd, J.

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

Hedekvist, P. O.

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

P. O. Hedekvist, M. Karlsson, and P. A. Andrekson, “Polarization dependence and efficiency in a fiber four-wave mixing phase conjugator with orthogonal pump waves,” IEEE Photon. Technol. Lett.8, 776–778 (1996).
[CrossRef]

Inoue, K.

K. Inoue, “Polarization effect on four-wave mixing efficiency in a single-mode fiber,” IEEE J. Quantum Electron.28, 883–894 (1992).
[CrossRef]

Jopson, R. M.

Kanaev, A. V.

Karlsson, M.

Kogelnik, H.

H. Kogelnik and C. J. McKinstrie, “Dynamic eigenstates of parametric interactions in randomly birefringent fibers,” IEEE Photon. Technol. Lett.21, 1036–1038 (2009).
[CrossRef]

C. J. McKinstrie, H. Kogelnik, G. G. Luther, and L. Schenato, “Stokes-space derivations of generalized Schrödinger equations for wave propagation in various fibers,” Opt. Express15, 10964–10983 (2007).
[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. Express12, 2033–2055 (2004).
[CrossRef] [PubMed]

J. P. Gordon and H. Kogelnik, “PMD fundamentals: Polarization mode dispersion in optical fibers,” Proc. Nat. Acad. Sci.97, 4541–4550 (2000).
[CrossRef] [PubMed]

Lakoba, T. I.

Li, J.

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

Lin, Q.

F. Yaman, Q. Lin, and G. P. Agrawal, “Effects of polarization-mode dispersion in dual-pump fiber-optic parametric amplifiers,” IEEE Photon. Technol. Lett.16, 431–433 (2004).
[CrossRef]

Loudon, R.

H. Fearn and R. Loudon, “Quantum theory of the lossless beam splitter,” Opt. Commun.64, 485–490 (1987).
[CrossRef]

Luther, G. G.

Manakov, S. V.

S. V. Manakov, “On the theory of two-dimensional stationary self-focusing of electromagnetic waves,” Sov. Phys. JETP38, 248–253 (1974).

Marcuse, D.

D. Marcuse, C. R. Menyuk, and P. K. A. Wai, “Application of the Manakov-PMD equation to studies of signal propagation in optical fibers with randomly varying birefringence,” J. Lightwave Technol.15, 1735–1746 (1997).
[CrossRef]

Martienssen, W.

S. Prasad, M. O. Scully, and W. Martienssen, “A quantum description of the beam splitter,” Opt. Commun.62, 139–145 (1987).
[CrossRef]

McGuinness, H. J.

M. G. Raymer, S. J. van Enk, C. J. McKinstrie, and H. J. McGuinness, “Interference of two photons of different color,” Opt. Commun.283, 747–752 (2010).
[CrossRef]

McKinstrie, C. J.

C. J. McKinstrie and M. Karlsson, “Schmidt decompositions of parametric processes I: Basic theory and simple examples,” Opt. Express21, 1374–1394 (2013) and references therein.
[CrossRef] [PubMed]

M. G. Raymer, S. J. van Enk, C. J. McKinstrie, and H. J. McGuinness, “Interference of two photons of different color,” Opt. Commun.283, 747–752 (2010).
[CrossRef]

H. Kogelnik and C. J. McKinstrie, “Dynamic eigenstates of parametric interactions in randomly birefringent fibers,” IEEE Photon. Technol. Lett.21, 1036–1038 (2009).
[CrossRef]

C. J. McKinstrie, S. Radic, and A. H. Gnauck, “All-optical signal processing by fiber-based parametric devices,” Opt. Photon. News18(3), 34–40 (2007).
[CrossRef]

C. J. McKinstrie, H. Kogelnik, G. G. Luther, and L. Schenato, “Stokes-space derivations of generalized Schrödinger equations for wave propagation in various fibers,” Opt. Express15, 10964–10983 (2007).
[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 (2006).
[CrossRef]

C. J. McKinstrie and S. Radic, “Phase-sensitive amplification in a fiber,” Opt. Express12, 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. Express12, 2033–2055 (2004).
[CrossRef] [PubMed]

Menyuk, C. R.

D. Marcuse, C. R. Menyuk, and P. K. A. Wai, “Application of the Manakov-PMD equation to studies of signal propagation in optical fibers with randomly varying birefringence,” J. Lightwave Technol.15, 1735–1746 (1997).
[CrossRef]

P. K. A. Wai and C. R. Menyuk, “Polarization mode dispersion, decorrelation and diffusion in optical fibers with randomly varying birefringence,” J. Lightwave Technol.14, 148–157 (1996).
[CrossRef]

P. K. A. Wai, C. R. Menyuk, and H. H. Chen, “Effects of randomly varying birefringence on soliton interactions in optical fibers,” Opt. Lett.16, 1735–1737 (1991).
[CrossRef] [PubMed]

Mollenauer, L. F.

S. G. Evangelides, L. F. Mollenauer, J. P. Gordon, and N. S. Bergano, “Polarization multiplexing with solitons,” J. Lightwave Technol.10, 28–35 (1992).
[CrossRef]

Prasad, S.

S. Prasad, M. O. Scully, and W. Martienssen, “A quantum description of the beam splitter,” Opt. Commun.62, 139–145 (1987).
[CrossRef]

Radic, S.

C. J. McKinstrie, S. Radic, and A. H. Gnauck, “All-optical signal processing by fiber-based parametric devices,” Opt. Photon. News18(3), 34–40 (2007).
[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 (2006).
[CrossRef]

C. J. McKinstrie and S. Radic, “Phase-sensitive amplification in a fiber,” Opt. Express12, 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. Express12, 2033–2055 (2004).
[CrossRef] [PubMed]

Ramirez-Alarcon, R.

H. Cruz-Ramirez, R. Ramirez-Alarcon, M. Corona, K. Garay-Palmett, and A. B. U’Ren, “Spontaneous parametric processes in quantum optics,” Opt. Photon. News22(11), 37–41 (2011).
[CrossRef]

Raymer, M. G.

M. G. Raymer and K. Srinivasan, “Manipulating the color and shape of single photons,” Phys. Today65(11), 32–37 (2012).
[CrossRef]

M. G. Raymer, S. J. van Enk, C. J. McKinstrie, and H. J. McGuinness, “Interference of two photons of different color,” Opt. Commun.283, 747–752 (2010).
[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 (2006).
[CrossRef]

Schenato, L.

Scully, M. O.

S. Prasad, M. O. Scully, and W. Martienssen, “A quantum description of the beam splitter,” Opt. Commun.62, 139–145 (1987).
[CrossRef]

Srinivasan, K.

M. G. Raymer and K. Srinivasan, “Manipulating the color and shape of single photons,” Phys. Today65(11), 32–37 (2012).
[CrossRef]

Sunnerud, H.

U’Ren, A. B.

H. Cruz-Ramirez, R. Ramirez-Alarcon, M. Corona, K. Garay-Palmett, and A. B. U’Ren, “Spontaneous parametric processes in quantum optics,” Opt. Photon. News22(11), 37–41 (2011).
[CrossRef]

van Enk, S. J.

M. G. Raymer, S. J. van Enk, C. J. McKinstrie, and H. J. McGuinness, “Interference of two photons of different color,” Opt. Commun.283, 747–752 (2010).
[CrossRef]

Vasilyev, M. V.

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 (2006).
[CrossRef]

Wai, P. K. A.

D. Marcuse, C. R. Menyuk, and P. K. A. Wai, “Application of the Manakov-PMD equation to studies of signal propagation in optical fibers with randomly varying birefringence,” J. Lightwave Technol.15, 1735–1746 (1997).
[CrossRef]

P. K. A. Wai and C. R. Menyuk, “Polarization mode dispersion, decorrelation and diffusion in optical fibers with randomly varying birefringence,” J. Lightwave Technol.14, 148–157 (1996).
[CrossRef]

P. K. A. Wai, C. R. Menyuk, and H. H. Chen, “Effects of randomly varying birefringence on soliton interactions in optical fibers,” Opt. Lett.16, 1735–1737 (1991).
[CrossRef] [PubMed]

Westlund, M.

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

Yaman, F.

F. Yaman, Q. Lin, and G. P. Agrawal, “Effects of polarization-mode dispersion in dual-pump fiber-optic parametric amplifiers,” IEEE Photon. Technol. Lett.16, 431–433 (2004).
[CrossRef]

Yuen, H. P.

H. P. Yuen, “Two-photon coherent states of the radiation field,” Phys. Rev. A13, 2226–2243 (1976).
[CrossRef]

IEEE J. Quantum Electron.

K. Inoue, “Polarization effect on four-wave mixing efficiency in a single-mode fiber,” IEEE J. Quantum Electron.28, 883–894 (1992).
[CrossRef]

IEEE J. Sel. Top. Quantum Electron.

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

IEEE Photon. Technol. Lett.

H. Kogelnik and C. J. McKinstrie, “Dynamic eigenstates of parametric interactions in randomly birefringent fibers,” IEEE Photon. Technol. Lett.21, 1036–1038 (2009).
[CrossRef]

P. O. Hedekvist, M. Karlsson, and P. A. Andrekson, “Polarization dependence and efficiency in a fiber four-wave mixing phase conjugator with orthogonal pump waves,” IEEE Photon. Technol. Lett.8, 776–778 (1996).
[CrossRef]

F. Yaman, Q. Lin, and G. P. Agrawal, “Effects of polarization-mode dispersion in dual-pump fiber-optic parametric amplifiers,” IEEE Photon. Technol. Lett.16, 431–433 (2004).
[CrossRef]

J. Lightwave Technol.

M. Karlsson and H. Sunnerud, “Effects of nonlinearities on PMD-induced system impairments,” J. Lightwave Technol.24, 4127–4137 (2006).
[CrossRef]

S. G. Evangelides, L. F. Mollenauer, J. P. Gordon, and N. S. Bergano, “Polarization multiplexing with solitons,” J. Lightwave Technol.10, 28–35 (1992).
[CrossRef]

P. K. A. Wai and C. R. Menyuk, “Polarization mode dispersion, decorrelation and diffusion in optical fibers with randomly varying birefringence,” J. Lightwave Technol.14, 148–157 (1996).
[CrossRef]

D. Marcuse, C. R. Menyuk, and P. K. A. Wai, “Application of the Manakov-PMD equation to studies of signal propagation in optical fibers with randomly varying birefringence,” J. Lightwave Technol.15, 1735–1746 (1997).
[CrossRef]

J. Opt. Soc. Am. B

Opt. Commun.

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 (2006).
[CrossRef]

M. G. Raymer, S. J. van Enk, C. J. McKinstrie, and H. J. McGuinness, “Interference of two photons of different color,” Opt. Commun.283, 747–752 (2010).
[CrossRef]

S. Prasad, M. O. Scully, and W. Martienssen, “A quantum description of the beam splitter,” Opt. Commun.62, 139–145 (1987).
[CrossRef]

H. Fearn and R. Loudon, “Quantum theory of the lossless beam splitter,” Opt. Commun.64, 485–490 (1987).
[CrossRef]

Opt. Express

Opt. Lett.

Opt. Photon. News

C. J. McKinstrie, S. Radic, and A. H. Gnauck, “All-optical signal processing by fiber-based parametric devices,” Opt. Photon. News18(3), 34–40 (2007).
[CrossRef]

H. Cruz-Ramirez, R. Ramirez-Alarcon, M. Corona, K. Garay-Palmett, and A. B. U’Ren, “Spontaneous parametric processes in quantum optics,” Opt. Photon. News22(11), 37–41 (2011).
[CrossRef]

Phys. Rev. A

H. P. Yuen, “Two-photon coherent states of the radiation field,” Phys. Rev. A13, 2226–2243 (1976).
[CrossRef]

Phys. Rev. D

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

Phys. Today

M. G. Raymer and K. Srinivasan, “Manipulating the color and shape of single photons,” Phys. Today65(11), 32–37 (2012).
[CrossRef]

Proc. Nat. Acad. Sci.

J. P. Gordon and H. Kogelnik, “PMD fundamentals: Polarization mode dispersion in optical fibers,” Proc. Nat. Acad. Sci.97, 4541–4550 (2000).
[CrossRef] [PubMed]

Sov. Phys. JETP

S. V. Manakov, “On the theory of two-dimensional stationary self-focusing of electromagnetic waves,” Sov. Phys. JETP38, 248–253 (1974).

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Figures (5)

Fig. 1
Fig. 1

Frequency diagrams for (a) modulation interaction and (b) inverse modulation interaction. Long arrows denote pumps (p and q), whereas short arrows denote sidebands (r and s). Downward arrows denote modes that lose photons, whereas upward arrows denote modes that gain photons.

Fig. 2
Fig. 2

Frequency diagrams for (a) outer-band and (b) inner-band phase conjugation. Long arrows denote pumps (p and q), whereas short arrows denote sidebands (r and s). Downward arrows denote modes that lose photons, whereas upward arrows denote modes that gain photons.

Fig. 3
Fig. 3

Normalized Schmidt coefficients (γ±/|BpBq|) plotted as functions of the pump-polarization alignment (p⃗ ·q⃗). The solid and dashed curves represent γ+ and γ, respectively.

Fig. 4
Fig. 4

Frequency diagrams for (a) distant and (b) nearby Bragg scattering. Long arrows denote pumps (p and q), whereas short arrows denote sidebands (r and s). Downward arrows denote modes that lose photons, whereas upward arrows denote modes that gain photons. The directions of the arrows are reversible.

Fig. 5
Fig. 5

Normalized Schmidt coefficients (γ±/|BpBq|) plotted as functions of the pump-polarization alignment (p⃗ ·q⃗). The solid and dashed curves represent γ+ and γ, respectively.

Equations (72)

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d z X 1 = i J 1 X 1 + i K X 2 * , d z X 2 = i J 2 X 2 + i K t X 1 * ,
d z X = i L X ,
X = [ X 1 X 2 * ] , L = [ J 1 K K J 2 * ] ,
X ( z ) = T ( z ) X ( 0 ) ,
T ( z ) = [ V 1 D μ U 1 V 1 D ν U 2 t V 2 * D ν U 1 V 2 * D μ U 2 t ] ,
x ¯ 1 j ( 0 ) = μ j ( z ) x ¯ 1 j ( 0 ) + ν j ( z ) x ¯ 2 j * ( 0 ) , x ¯ 2 j * ( 0 ) = ν j ( z ) x ¯ 1 j ( 0 ) + μ j ( z ) x ¯ 2 j * ( 0 ) ,
z A = i β ( i t ) A + i γ ( A A ) A ,
A ( z , t ) = A p ( z ) exp ( i ω p t ) + A r ( z ) exp ( i ω r t ) + A s ( z ) exp ( i ω s t )
d z A p = i β p A p + i γ ( A p A p ) A p ,
d z A r = i β r A r + i γ ( A p A p + A p A p ) A r + i γ ( A p A p t ) A s * ,
d z A s = i β s A s + i γ ( A p A p + A p A p ) A s + i γ ( A p A p t ) A r * ,
d z O p = i ( β p + γ A p A p ) O p
O p ( z ) = exp [ i ( β p + γ A p A p ) z ] ,
A j ( z ) = O p ( z ) B j ( z ) .
d z B r = i ( β r β p + γ | B p | 2 ) B r + i γ ( B p B p t ) B s * ,
d z B s = i ( β s β p + γ | B p | 2 ) B s + i γ ( B p B p t ) B r * .
B r = j b r j V j B s = j b s j V j ,
d z b r j = i δ r b r j + i γ j b s j * , d z b s j = i δ s b s j + i γ j b r j * ,
b r j ( z ) = e ( z ) μ j ( z ) b r j ( 0 ) + e ( z ) ν j ( z ) b s j * ( 0 ) ,
b s j ( z ) = e * ( z ) μ j ( z ) b s j ( 0 ) + e * ( z ) ν j ( z ) b r j * ( 0 ) ,
μ j ( z ) = cos ( k j z ) + i δ a sin ( k j z ) / k j ,
ν j ( z ) = i γ j sin ( k j z ) / k j ,
e ( z ) = exp ( i δ d z ) ,
b r j = V j B r , b s j = V j B s ,
B r ( z ) = j V j e ( z ) μ j ( z ) V j B r ( 0 ) + j V j e ( z ) ν j ( z ) V j t B s * ( 0 ) ,
B s * ( z ) = j V j * e ( z ) ν j * ( z ) V j B r ( 0 ) + j V j * e ( z ) μ j * ( z ) V j t B s * ( 0 ) .
[ B r ( z ) B s * ( z ) ] = [ V e D μ V V e D ν V t V * e D ν * V V * e D μ * V t ] [ B r ( 0 ) B s * ( 0 ) ] .
[ B r ( z ) B s * ( z ) ] = [ V r | D μ | U V r | D ν | U t V s * | D ν | U V s * | D μ | U t ] [ B r ( 0 ) B s * ( 0 ) ] ,
A ( z , t ) = A p ( z ) exp ( i ω p t ) + A q ( z ) exp ( i ω q t ) + A r ( z ) exp ( i ω r t ) + A s ( z ) exp ( i ω s t )
d z A p = i β p A p + i γ ( A p A p + A q A q + A q A q ) A p ,
d z A q = i β q A q + i γ ( A q A q + A p A p + A p A p ) A q ,
d z A r = i β r A r + i γ ( A p A p + A p A p + A q A q + A q A q ) A r + i γ ( A p A q t + A q A p t ) A s * ,
d z A s = i β s A s + i γ ( A p A p + A p A p + A q A q + A q A q ) A s + i γ ( A p A q t + A q A p t ) A r * .
d z O p = i [ β p + γ | A q | 2 + γ ( A p A p + A q A q ) ] O p ,
d z O q = i [ β q + γ | A p | 2 + γ ( A p A p + A q A q ) ] O q ,
O p ( z ) = exp { i [ β p + γ | A q | 2 + γ ( A p A p + A q A q ) ] z } ,
O q ( z ) = exp { i [ β q + γ | A p | 2 + γ ( A p A p + A q A q ) ] z } .
A p ( z ) = O p ( z ) B p ( z ) , A q ( z ) = O q ( z ) B q ( z ) ,
A r ( z ) = O p ( z ) B r ( z ) , A s ( z ) = O q ( z ) B s ( z ) .
d z B r = i ( β r β p + γ | B p | 2 ) B r + i γ ( B p B q t + B q B p t ) B s * ,
d z B s = i ( β s β q + γ | B q | 2 ) B s + i γ ( B p B q t + B q B p t ) B r * .
r = s = ( p + q ) / ( 2 + 2 p q ) 1 / 2 .
γ ± 2 = [ 3 + p q ± 2 ( 2 + 2 p q ) 1 / 2 ] | B p B q | 2 / 2 ,
B r ( z ) = C r ( z ) exp ( i δ d z ) , B s ( z ) = C s ( z ) exp ( i δ d z ) ,
d z C r = i δ a C r + i γ ( B p B q t + B q B p t ) C s * ,
d z C s = i δ a C s + i γ ( B p B q t + B q B p t ) C r * ,
d z A s = i β s A s + i γ ( A p A p + A p A p + A q A q + A q A q ) A s + i γ ( A p A q t + A q A p t ) A s * .
O s ( z ) = exp { i [ ( β p + β q ) / 2 + γ ( | A p | 2 + | A q | 2 ) / 2 + γ ( A p A p + A q A q ) ] z } ,
d z B s = i δ s B s + i γ ( B p B q t + B q B p t ) B s * ,
d z A p = i β p A p + i γ ( A p A p + A q A q + A q A q ) A p ,
d z A q = i β q A q + i γ ( A q A q + A p A p + A p A p ) A q ,
d z A r = i β r A r + i γ ( A p A p + A p A p + A q A q + A q A q ) A r + i γ ( A p A q + A q A p ) A s ,
d z A s = i β s A s + i γ ( A p A p + A p A p + A q A q + A q A q ) A s + i γ ( A p A q + A q A p ) A r .
d z X = i H X ,
X = [ A r A s ] , H = [ J r K K J s ] ,
T ( z ) = [ V 1 D τ U 1 V 1 D ρ U 2 V 2 D ρ U 1 V 2 D τ U 2 ] ,
d z B r = i ( β r β p + γ | B p | 2 ) B r + i γ ( B p B q + B q B p ) B s ,
d z B s = i ( β s β q + γ | B q | 2 ) B s + i γ ( B p B q + B q B p ) B r .
r = ( 2 p + q ) / ( 5 + 4 p q ) 1 / 2 , s = ( p + 2 q ) / ( 5 + 4 p q ) 1 / 2 ,
γ ± 2 = [ 3 + 2 p q ± ( 5 + 4 p q ) 1 / 2 ] | B p B q | 2 / 2.
B r = j b r j U j , B s = j b s j V j ,
d z b r j = i δ r b r j + i γ j b s j , d z b s j = i δ s b s j + i γ j b r j ,
b r j ( z ) = e ( z ) τ j ( z ) b r j ( 0 ) + e ( z ) ρ j ( z ) b s j ( 0 ) ,
b s j ( z ) = e ( z ) ρ j * ( z ) b r j ( 0 ) + e ( z ) τ j * ( z ) b s j ( 0 ) ,
τ j ( z ) = cos ( k j z ) + i δ d sin ( k j z ) / k j ,
ρ j ( z ) = i γ j sin ( k j z ) / k j ,
e ( z ) = exp ( i δ a z ) ,
b r j = U j B r , b s j = V j B s ,
B r ( z ) = j U j e ( z ) τ j ( z ) U j B r ( 0 ) + j U j e ( z ) ρ j ( z ) V j B s ( 0 ) ,
B s ( z ) = j V j e ( z ) ρ j * ( z ) U j B r ( 0 ) + j V j e ( z ) τ j * ( z ) V j B s ( 0 ) .
[ B r ( z ) B s ( z ) ] = [ U e D τ U U e D ρ V V e D ρ * U V e D τ * V ] [ B r ( 0 ) B s ( 0 ) ] .
[ B r ( z ) B s ( z ) ] = [ V r | D τ | U r V r | D ρ | U s V s | D ρ | U r V s | D τ | U s ] [ B r ( 0 ) B s ( 0 ) ] ,

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