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.

© 2013 OSA

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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. News 18(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. News 22(11), 37–41 (2011).
    [CrossRef]
  4. M. G. Raymer and K. Srinivasan, “Manipulating the color and shape of single photons,” Phys. Today 65(11), 32–37 (2012).
    [CrossRef]
  5. C. J. McKinstrie and M. Karlsson, “Schmidt decompositions of parametric processes I: Basic theory and simple examples,” Opt. Express 21, 1374–1394 (2013) and references therein.
    [CrossRef] [PubMed]
  6. H. P. Yuen, “Two-photon coherent states of the radiation field,” Phys. Rev. A 13, 2226–2243 (1976).
    [CrossRef]
  7. C. M. Caves, “Quantum limits on noise in linear ampifiers,” Phys. Rev. D 26, 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. Express 12, 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. JETP 38, 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. B 13, 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. Express 15, 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. Express 12, 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 (1)

2012 (1)

M. G. Raymer and K. Srinivasan, “Manipulating the color and shape of single photons,” Phys. Today 65(11), 32–37 (2012).
[CrossRef]

2011 (1)

H. Cruz-Ramirez, R. Ramirez-Alarcon, M. Corona, K. Garay-Palmett, and A. B. U’Ren, “Spontaneous parametric processes in quantum optics,” Opt. Photon. News 22(11), 37–41 (2011).
[CrossRef]

2010 (1)

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 (1)

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 (2)

C. J. McKinstrie, S. Radic, and A. H. Gnauck, “All-optical signal processing by fiber-based parametric devices,” Opt. Photon. News 18(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. Express 15, 10964–10983 (2007).
[CrossRef] [PubMed]

2006 (2)

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 (3)

2002 (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]

2000 (1)

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 (1)

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 (3)

T. I. Lakoba, “Concerning the equations governing nonlinear pulse propagation in randomly birefringent fibers,” J. Opt. Soc. Am. B 13, 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]

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]

1992 (2)

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

1991 (1)

1987 (2)

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 (1)

C. M. Caves, “Quantum limits on noise in linear ampifiers,” Phys. Rev. D 26, 1817–1839 (1982).
[CrossRef]

1976 (1)

H. P. Yuen, “Two-photon coherent states of the radiation field,” Phys. Rev. A 13, 2226–2243 (1976).
[CrossRef]

1974 (1)

S. V. Manakov, “On the theory of two-dimensional stationary self-focusing of electromagnetic waves,” Sov. Phys. JETP 38, 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. D 26, 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. News 22(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. News 22(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. News 22(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. News 18(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. Express 15, 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. Express 12, 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. JETP 38, 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. Express 21, 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. News 18(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. Express 15, 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. 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]

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. News 18(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. 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]

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. News 22(11), 37–41 (2011).
[CrossRef]

Raymer, M. G.

M. G. Raymer and K. Srinivasan, “Manipulating the color and shape of single photons,” Phys. Today 65(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. Today 65(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. News 22(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. A 13, 2226–2243 (1976).
[CrossRef]

IEEE J. Quantum Electron. (1)

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. (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]

IEEE Photon. Technol. Lett. (3)

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. (4)

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 (1)

Opt. Commun. (4)

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 (4)

Opt. Lett. (1)

Opt. Photon. News (2)

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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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