Abstract

Parametric devices based on four-wave mixing in fibers perform many signal-processing functions required by optical communication systems. In these devices, strong pumps drive weak signal and idler sidebands, which can have one or two polarization components, and one or many frequency components. The evolution of these components (modes) is governed by a system of coupled-mode equations. Schmidt decompositions of the associated transfer matrices determine the natural input and output mode vectors of such systems, and facilitate the optimization of device performance. In this paper, the basic properties of Schmidt decompositions are derived from first principles and are illustrated by two simple examples (one- and two-mode parametric amplification). In a forthcoming paper, several nontrivial examples relevant to current research (including four-mode parametric amplification) will be discussed.

© 2013 OSA

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    [CrossRef]
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    [CrossRef] [PubMed]
  32. 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]
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    [CrossRef]
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  35. A. I. Lvovsky, W. Wasilewski, and K. Banaszek, “Decomposing a pulsed optical parametric amplifier into independent squeezers,” J. Mod. Opt.54, 721–733 (2007).
    [CrossRef]
  36. C. J. McKinstrie, M. G. Raymer, and H. J. McGuinness, “Spatial-temporal evolution of asymmetrically-pumped phase conjugation I: General formalism,” Alcatel-Lucent ITD-09-48636Q, available upon request.
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    [CrossRef]
  42. Y. S. Kim and M. E. Noz, “Illustrative examples of the symplectic group,” Am. J. Phys.51, 368–375 (1983).
    [CrossRef]
  43. A. Mufti, H. A. Schmitt, and M. Sargent, “Finite-dimensional matrix representations as calculational tools in quantum optics,” Am. J. Phys.61, 729–733 (1993).
    [CrossRef]
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    [CrossRef] [PubMed]

2012 (4)

S. Radic, “Parametric signal processing,” IEEE J. Sel. Top. Quantum Electron.18, 670–680 (2012).
[CrossRef]

C. J. McKinstrie, L. Mejling, M. G. Raymer, and K. Rottwitt, “Quantum-state-preserving optical pulse reshaping and multiplexing by four-wave mixing in fibers,” Phys. Rev. A85, 053829 (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]

C. J. McKinstrie and N. Alic, “Information efficiencies of parametric devices,” J. Sel. Top. Quantum Electron.18, 794–811 (2012).
[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 (3)

O. Cohen, J. S. Lundeen, B. J. Smith, G. Puentes, P. J. Mosley, and I. A. Walmsley, “Tailored photon-pair generation in optical fibers,” Phys. Rev. Lett.102, 123603 (2009).
[CrossRef] [PubMed]

C. J. McKinstrie, “Unitary and singular value decompositions of parametric processes in fibers,” Opt. Commun.282, 583–593 (2009).
[CrossRef]

M. Halder, J. Fulconis, B. Cemlyn, A. Clark, C. Xiong, W. J. Wadsworth, and J. G. Rarity, “Nonclassical 2-photon interference with separate intrinsically narrowband fibre sources,” Opt. Express17, 4670–4676 (2009).
[CrossRef] [PubMed]

2008 (1)

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]

2007 (2)

A. I. Lvovsky, W. Wasilewski, and K. Banaszek, “Decomposing a pulsed optical parametric amplifier into independent squeezers,” J. Mod. Opt.54, 721–733 (2007).
[CrossRef]

P. A. Andrekson and M. Westlund, “Nonlinear optical fiber based high resolution all-optical waveform sampling,” Laser Photon. Rev.1, 231–248 (2007).
[CrossRef]

2006 (1)

2005 (4)

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

J. H. Lee, “All-optical signal processing devices based on holey fiber,” IEICE Trans. Electron.E88-C, 327–334 (2005).
[CrossRef]

S. Radic and C. J. McKinstrie, “Optical amplification and signal processing in highly nonlinear optical fiber,” IEICE Trans. Electron.E88-C, 859–869 (2005).
[CrossRef]

S. L. Braunstein, “Squeezing as an irreducible resource,” Phys. Rev. A71, 055801 (2005).
[CrossRef]

2004 (2)

2003 (1)

2002 (2)

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]

M. Fiorentino, P. L. Voss, J. E. Sharping, and P. Kumar, “All-fiber photon-pair source for quantum communications,” IEEE Photon. Technol. Lett.14, 983–985 (2002).
[CrossRef]

2001 (2)

W. P. Grice, A. B. U’Ren, and I. A. Walmsley, “Eliminating frequency and space-time correlations in multiphoton states,” Phys. Rev. A64, 063815 (2001).
[CrossRef]

C. C. Gerry, “Remarks on the use of group theory in quantum optics,” Opt. Express8, 76–85 (2001).
[CrossRef] [PubMed]

2000 (1)

C. K. Law, I. A. Walmsley, and J. H. Eberly, “Continuous frequency entanglement: Effective finite Hilbert space and entropy control,” Phys. Rev. Lett.84, 5304–5307 (2000).
[CrossRef] [PubMed]

1999 (1)

E. Telatar, “Capacity of multi-antenna Gaussian channels,” Euro. Trans. Telecom.10, 585–595 (1999).
[CrossRef]

1993 (2)

A. Mufti, H. A. Schmitt, and M. Sargent, “Finite-dimensional matrix representations as calculational tools in quantum optics,” Am. J. Phys.61, 729–733 (1993).
[CrossRef]

G. W. Stewart, “On the early history of the singular value decomposition,” SIAM Rev.35, 551–566 (1993).
[CrossRef]

1992 (1)

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

1991 (1)

A. K. Ekert and P. L. Knight, “Relationship between semiclassical and quantum-mechanical input-output theories of optical response,” Phys. Rev. A43, 3934–3938 (1991).
[CrossRef] [PubMed]

1983 (1)

Y. S. Kim and M. E. Noz, “Illustrative examples of the symplectic group,” Am. J. Phys.51, 368–375 (1983).
[CrossRef]

1982 (1)

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

1976 (1)

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

1973 (1)

H. Takenaka, “A unified formalism for polarization optics by using group theory,” Nou. Rev. Opt.4, 37–41 (1973).
[CrossRef]

1957 (1)

M. T. Weiss, “Quantum derivation of energy relations analogous to those for nonlinear reactances,” Proc. IRE45, 1012–1013 (1957).

1956 (1)

J. M. Manley and H. E. Rowe, “Some general properties of nonlinear elements—Part I. General energy relations,” Proc. IRE44, 904–913 (1956).
[CrossRef]

Alic, N.

C. J. McKinstrie and N. Alic, “Information efficiencies of parametric devices,” J. Sel. Top. Quantum Electron.18, 794–811 (2012).
[CrossRef]

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]

P. A. Andrekson and M. Westlund, “Nonlinear optical fiber based high resolution all-optical waveform sampling,” Laser Photon. Rev.1, 231–248 (2007).
[CrossRef]

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]

Arnold, V. I.

V. I. Arnold, Mathematical Methods of Classical Mechanics, 2nd Ed. (Springer, 2000).

Banaszek, K.

A. I. Lvovsky, W. Wasilewski, and K. Banaszek, “Decomposing a pulsed optical parametric amplifier into independent squeezers,” J. Mod. Opt.54, 721–733 (2007).
[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]

Braunstein, S. L.

S. L. Braunstein, “Squeezing as an irreducible resource,” Phys. Rev. A71, 055801 (2005).
[CrossRef]

Caves, C. M.

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

Cemlyn, B.

Clark, A.

Cohen, O.

O. Cohen, J. S. Lundeen, B. J. Smith, G. Puentes, P. J. Mosley, and I. A. Walmsley, “Tailored photon-pair generation in optical fibers,” Phys. Rev. Lett.102, 123603 (2009).
[CrossRef] [PubMed]

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]

Eberly, J. H.

C. K. Law, I. A. Walmsley, and J. H. Eberly, “Continuous frequency entanglement: Effective finite Hilbert space and entropy control,” Phys. Rev. Lett.84, 5304–5307 (2000).
[CrossRef] [PubMed]

Edwards, D. A.

D. A. Edwards, J. D. Fehribach, R. O. Moore, and C. J. McKinstrie, “An application of matrix theory to the evolution of coupled modes,” to appear in SIAM Rev.

Ekert, A. K.

A. K. Ekert and P. L. Knight, “Relationship between semiclassical and quantum-mechanical input-output theories of optical response,” Phys. Rev. A43, 3934–3938 (1991).
[CrossRef] [PubMed]

Fehribach, J. D.

D. A. Edwards, J. D. Fehribach, R. O. Moore, and C. J. McKinstrie, “An application of matrix theory to the evolution of coupled modes,” to appear in SIAM Rev.

Fiorentino, M.

M. Fiorentino, P. L. Voss, J. E. Sharping, and P. Kumar, “All-fiber photon-pair source for quantum communications,” IEEE Photon. Technol. Lett.14, 983–985 (2002).
[CrossRef]

Fulconis, J.

Gbur, G. J.

G. J. Gbur, Mathematical Methods for Optical Physics and Engineering (Cambridge, 2011), Sec. 5.4.

Gerry, C. C.

Goldstein, H.

H. Goldstein, Classical Mechanics, 2nd Ed. (Addison-Wesley, 1980).

Grice, W. P.

W. P. Grice, A. B. U’Ren, and I. A. Walmsley, “Eliminating frequency and space-time correlations in multiphoton states,” Phys. Rev. A64, 063815 (2001).
[CrossRef]

Halder, M.

Hamermesh, M.

M. Hamermesh, Group Theory and its Application to Physical Problems (Dover, 1989).

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]

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.

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]

Kazovsky, L. G.

Kim, Y. S.

Y. S. Kim and M. E. Noz, “Illustrative examples of the symplectic group,” Am. J. Phys.51, 368–375 (1983).
[CrossRef]

Knight, P. L.

A. K. Ekert and P. L. Knight, “Relationship between semiclassical and quantum-mechanical input-output theories of optical response,” Phys. Rev. A43, 3934–3938 (1991).
[CrossRef] [PubMed]

Kogelnik, H.

Kumar, P.

M. Fiorentino, P. L. Voss, J. E. Sharping, and P. Kumar, “All-fiber photon-pair source for quantum communications,” IEEE Photon. Technol. Lett.14, 983–985 (2002).
[CrossRef]

Law, C. K.

C. K. Law, I. A. Walmsley, and J. H. Eberly, “Continuous frequency entanglement: Effective finite Hilbert space and entropy control,” Phys. Rev. Lett.84, 5304–5307 (2000).
[CrossRef] [PubMed]

Lee, J. H.

J. H. Lee, “All-optical signal processing devices based on holey fiber,” IEICE Trans. Electron.E88-C, 327–334 (2005).
[CrossRef]

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

Loudon, R.

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

Lundeen, J. S.

O. Cohen, J. S. Lundeen, B. J. Smith, G. Puentes, P. J. Mosley, and I. A. Walmsley, “Tailored photon-pair generation in optical fibers,” Phys. Rev. Lett.102, 123603 (2009).
[CrossRef] [PubMed]

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]

Lvovsky, A. I.

A. I. Lvovsky, W. Wasilewski, and K. Banaszek, “Decomposing a pulsed optical parametric amplifier into independent squeezers,” J. Mod. Opt.54, 721–733 (2007).
[CrossRef]

Manley, J. M.

J. M. Manley and H. E. Rowe, “Some general properties of nonlinear elements—Part I. General energy relations,” Proc. IRE44, 904–913 (1956).
[CrossRef]

Marhic, M. E.

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]

C. J. McKinstrie, M. G. Raymer, and H. J. McGuinness, “Spatial-temporal evolution of asymmetrically-pumped phase conjugation I: General formalism,” Alcatel-Lucent ITD-09-48636Q, available upon request.

McKinstrie, C. J.

C. J. McKinstrie and N. Alic, “Information efficiencies of parametric devices,” J. Sel. Top. Quantum Electron.18, 794–811 (2012).
[CrossRef]

C. J. McKinstrie, L. Mejling, M. G. Raymer, and K. Rottwitt, “Quantum-state-preserving optical pulse reshaping and multiplexing by four-wave mixing in fibers,” Phys. Rev. A85, 053829 (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, “Unitary and singular value decompositions of parametric processes in fibers,” Opt. Commun.282, 583–593 (2009).
[CrossRef]

C. J. McKinstrie, H. Kogelnik, and L. Schenato, “Four-wave mixing in a rapidly-spun fiber,” Opt. Express14, 8516–8534 (2006).
[CrossRef] [PubMed]

S. Radic and C. J. McKinstrie, “Optical amplification and signal processing in highly nonlinear optical fiber,” IEICE Trans. Electron.E88-C, 859–869 (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, 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]

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

D. A. Edwards, J. D. Fehribach, R. O. Moore, and C. J. McKinstrie, “An application of matrix theory to the evolution of coupled modes,” to appear in SIAM Rev.

C. J. McKinstrie, M. G. Raymer, and H. J. McGuinness, “Spatial-temporal evolution of asymmetrically-pumped phase conjugation I: General formalism,” Alcatel-Lucent ITD-09-48636Q, available upon request.

Mejling, L.

C. J. McKinstrie, L. Mejling, M. G. Raymer, and K. Rottwitt, “Quantum-state-preserving optical pulse reshaping and multiplexing by four-wave mixing in fibers,” Phys. Rev. A85, 053829 (2012).
[CrossRef]

Moore, R. O.

D. A. Edwards, J. D. Fehribach, R. O. Moore, and C. J. McKinstrie, “An application of matrix theory to the evolution of coupled modes,” to appear in SIAM Rev.

Mosley, P. J.

O. Cohen, J. S. Lundeen, B. J. Smith, G. Puentes, P. J. Mosley, and I. A. Walmsley, “Tailored photon-pair generation in optical fibers,” Phys. Rev. Lett.102, 123603 (2009).
[CrossRef] [PubMed]

Mufti, A.

A. Mufti, H. A. Schmitt, and M. Sargent, “Finite-dimensional matrix representations as calculational tools in quantum optics,” Am. J. Phys.61, 729–733 (1993).
[CrossRef]

Noz, M. E.

Y. S. Kim and M. E. Noz, “Illustrative examples of the symplectic group,” Am. J. Phys.51, 368–375 (1983).
[CrossRef]

Puentes, G.

O. Cohen, J. S. Lundeen, B. J. Smith, G. Puentes, P. J. Mosley, and I. A. Walmsley, “Tailored photon-pair generation in optical fibers,” Phys. Rev. Lett.102, 123603 (2009).
[CrossRef] [PubMed]

Radic, S.

Rarity, J. G.

Raymer, M. G.

C. J. McKinstrie, L. Mejling, M. G. Raymer, and K. Rottwitt, “Quantum-state-preserving optical pulse reshaping and multiplexing by four-wave mixing in fibers,” Phys. Rev. A85, 053829 (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. Yu, M. G. Raymer, and S. Radic, “Quantum noise properties of parametric processes,” Opt. Express13, 4986–5012 (2005).
[CrossRef] [PubMed]

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Rottwitt, K.

C. J. McKinstrie, L. Mejling, M. G. Raymer, and K. Rottwitt, “Quantum-state-preserving optical pulse reshaping and multiplexing by four-wave mixing in fibers,” Phys. Rev. A85, 053829 (2012).
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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).
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O. Cohen, J. S. Lundeen, B. J. Smith, G. Puentes, P. J. Mosley, and I. A. Walmsley, “Tailored photon-pair generation in optical fibers,” Phys. Rev. Lett.102, 123603 (2009).
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W. P. Grice, A. B. U’Ren, and I. A. Walmsley, “Eliminating frequency and space-time correlations in multiphoton states,” Phys. Rev. A64, 063815 (2001).
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C. K. Law, I. A. Walmsley, and J. H. Eberly, “Continuous frequency entanglement: Effective finite Hilbert space and entropy control,” Phys. Rev. Lett.84, 5304–5307 (2000).
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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).
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[CrossRef]

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[CrossRef]

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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).
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J. H. Lee, “All-optical signal processing devices based on holey fiber,” IEICE Trans. Electron.E88-C, 327–334 (2005).
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[CrossRef]

J. Mod. Opt. (1)

A. I. Lvovsky, W. Wasilewski, and K. Banaszek, “Decomposing a pulsed optical parametric amplifier into independent squeezers,” J. Mod. Opt.54, 721–733 (2007).
[CrossRef]

J. Opt. Soc. Am. B (1)

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

C. J. McKinstrie and N. Alic, “Information efficiencies of parametric devices,” J. Sel. Top. Quantum Electron.18, 794–811 (2012).
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P. A. Andrekson and M. Westlund, “Nonlinear optical fiber based high resolution all-optical waveform sampling,” Laser Photon. Rev.1, 231–248 (2007).
[CrossRef]

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H. Takenaka, “A unified formalism for polarization optics by using group theory,” Nou. Rev. Opt.4, 37–41 (1973).
[CrossRef]

Opt. Commun. (2)

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, “Unitary and singular value decompositions of parametric processes in fibers,” Opt. Commun.282, 583–593 (2009).
[CrossRef]

Opt. Express (6)

Phys. Rev. A (5)

C. J. McKinstrie, L. Mejling, M. G. Raymer, and K. Rottwitt, “Quantum-state-preserving optical pulse reshaping and multiplexing by four-wave mixing in fibers,” Phys. Rev. A85, 053829 (2012).
[CrossRef]

W. P. Grice, A. B. U’Ren, and I. A. Walmsley, “Eliminating frequency and space-time correlations in multiphoton states,” Phys. Rev. A64, 063815 (2001).
[CrossRef]

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

Phys. Rev. Lett. (2)

C. K. Law, I. A. Walmsley, and J. H. Eberly, “Continuous frequency entanglement: Effective finite Hilbert space and entropy control,” Phys. Rev. Lett.84, 5304–5307 (2000).
[CrossRef] [PubMed]

O. Cohen, J. S. Lundeen, B. J. Smith, G. Puentes, P. J. Mosley, and I. A. Walmsley, “Tailored photon-pair generation in optical fibers,” Phys. Rev. Lett.102, 123603 (2009).
[CrossRef] [PubMed]

Proc. IRE (2)

J. M. Manley and H. E. Rowe, “Some general properties of nonlinear elements—Part I. General energy relations,” Proc. IRE44, 904–913 (1956).
[CrossRef]

M. T. Weiss, “Quantum derivation of energy relations analogous to those for nonlinear reactances,” Proc. IRE45, 1012–1013 (1957).

SIAM Rev. (1)

G. W. Stewart, “On the early history of the singular value decomposition,” SIAM Rev.35, 551–566 (1993).
[CrossRef]

Other (9)

G. J. Gbur, Mathematical Methods for Optical Physics and Engineering (Cambridge, 2011), Sec. 5.4.

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

M. E. Marhic, Fiber Optical Parametric Amplifiers, Oscillators and Related Devices (Cambridge, 2007).
[CrossRef]

D. A. Edwards, J. D. Fehribach, R. O. Moore, and C. J. McKinstrie, “An application of matrix theory to the evolution of coupled modes,” to appear in SIAM Rev.

C. J. McKinstrie, M. G. Raymer, and H. J. McGuinness, “Spatial-temporal evolution of asymmetrically-pumped phase conjugation I: General formalism,” Alcatel-Lucent ITD-09-48636Q, available upon request.

H. Goldstein, Classical Mechanics, 2nd Ed. (Addison-Wesley, 1980).

V. I. Arnold, Mathematical Methods of Classical Mechanics, 2nd Ed. (Springer, 2000).

D. H. Sattinger and O. L. Weaver, Lie groups and Algebras with Applications to Physics, Geometry and Mechanics (Springer, 1986).
[CrossRef]

M. Hamermesh, Group Theory and its Application to Physical Problems (Dover, 1989).

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Figures (1)

Fig. 1
Fig. 1

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

Equations (128)

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d z X = A X + B X * ,
X ( z ) = M ( z ) X ( 0 ) + N ( z ) X * ( 0 ) ,
X ( z ) = V D μ U X ( 0 ) + V D ν U t X * ( 0 ) ,
x ¯ j ( z ) = μ j ( z ) x ¯ j ( 0 ) + ν j ( z ) x ¯ j * ( 0 ) .
d z x = i δ x + i γ x * ,
x ( z ) = μ ( z ) x ( 0 ) + ν ( z ) x * ( 0 ) .
μ ( z ) = cos ( k z ) + i δ sin ( k z ) / k , ν ( z ) = i γ sin ( k z ) / k ,
| μ ( z ) | 2 | ν ( z ) | 2 = 1 .
μ ( z ) = μ * ( z ) , ν ( z ) = ν ( z ) .
x ( z ) = v | μ | u * x ( 0 ) + v | ν | u x * ( 0 ) ,
d z Y = i L y Y ,
L y = [ δ γ γ * δ ] .
Y ( z ) = T y ( z ) Y ( 0 ) ,
T y ( z ) = exp ( i L y z ) .
T y ( z ) = [ μ ( z ) ν ( z ) ν * ( z ) μ * ( z ) ] .
T y ( z ) = [ μ ( z ) ν ( z ) ν * ( z ) μ * ( z ) ] = T y 1 ( z ) = [ μ * ( z ) ν ( z ) ν * ( z ) μ ( z ) ] .
T y ( z ) = 1 2 1 / 2 [ e i ϕ s e i ϕ s e i ϕ s e i ϕ s ] [ | μ | + | ν | 0 0 | μ | | ν | ] 1 2 1 / 2 [ e i ϕ d e i ϕ d e i ϕ d e i ϕ d ] ,
T y 1 ( z ) = 1 2 1 / 2 [ i e i ϕ d i e i ϕ d i e i ϕ d i e i ϕ d ] [ | μ | + | ν | 0 0 | μ | | ν | ] 1 2 1 / 2 [ i e i ϕ s i e i ϕ s i e i ϕ s i e i ϕ s ] .
d z x 1 = i δ 1 x 1 + i γ x 2 * , d z x 2 = i δ 2 x 2 + i γ x 1 * ,
x 1 ( z ) = μ 11 ( z ) x 1 ( 0 ) + ν 12 ( z ) x 2 * ( 0 ) , x 2 ( z ) = μ 22 ( z ) x 2 ( 0 ) + ν 21 ( z ) x 1 * ( 0 ) .
| μ 11 ( z ) | 2 | ν 12 ( z ) | 2 = 1 , | μ 11 ( z ) | 2 | ν 21 ( z ) | 2 = 1 ,
μ 11 ( z ) = μ 11 * ( z ) , ν 12 ( z ) = ν 21 ( z ) .
d z X = i L x X ,
L x = [ δ 1 γ γ * δ 2 ] .
X ( z ) = T x ( z ) X ( 0 ) ,
T x ( z ) = exp ( i L x z ) .
T x ( z ) = [ μ 11 ( z ) ν 12 ( z ) ν 21 * ( z ) μ 22 * ( z ) ] = e ( z ) [ μ ( z ) ν ( z ) ν * ( z ) μ * ( z ) ] .
T x ( z ) = e ( z ) [ μ ( z ) ν ( z ) ν * ( z ) μ * ( z ) ] = T x 1 ( z ) = e * ( z ) [ μ * ( z ) ν ( z ) ν * ( z ) μ ( z ) ] .
T x ( z ) = e i ϕ δ 2 1 / 2 [ e i ϕ s e i ϕ s e i ϕ s e i ϕ s ] [ | μ | + | ν | 0 0 | μ | | ν | ] 1 2 1 / 2 [ e i ϕ d e i ϕ d e i ϕ d e i ϕ d ] ,
T x 1 ( z ) = 1 2 1 / 2 [ i e i ϕ d i e i ϕ d i e i ϕ d i e i ϕ d ] [ | μ | + | ν | 0 0 | μ | | ν | ] e i ϕ δ 2 1 / 2 [ i e i ϕ s i e i ϕ s i e i ϕ s i e i ϕ s ] .
M = [ μ 11 0 0 μ 22 ] = [ e μ 0 0 e * μ ] ,
N = [ 0 ν 12 ν 21 0 ] = [ 0 e ν e * ν 0 ] .
M = 1 2 1 / 2 [ e i ϕ δ i e i ϕ δ e i ϕ δ i e i ϕ δ ] [ μ 0 0 μ ] 1 2 1 / 2 [ 1 1 i i ] ,
N = 1 2 1 / 2 [ e i ϕ δ i e i ϕ δ e i ϕ δ i e i ϕ δ ] [ ν 0 0 ν ] 1 2 1 / 2 [ 1 1 i i ] .
M = e i ϕ s 2 1 / 2 [ e i ϕ δ i e i ϕ δ e i ϕ δ i e i ϕ δ ] [ | μ | 0 0 | μ | ] e i ϕ d 2 1 / 2 [ 1 1 i i ] ,
N = e i ϕ s 2 1 / 2 [ e i ϕ δ i e i ϕ δ e i ϕ δ i e i ϕ δ ] [ | ν | 0 0 | ν | ] e i ϕ d 2 1 / 2 [ 1 1 i i ] .
x ± = ( x 1 ± x 2 ) / 2 1 / 2 .
d z x ± = i δ s x ± + i δ d x ± i γ x ± * ,
H = X J X + X K X * + X t K * X ,
d z X = i H / X
d z X = i J X + i K X * .
X ( z ) = M ( z ) X ( 0 ) + N ( z ) X * ( 0 ) ,
d z Y = i L y Y ,
Y = [ X X * ] , L y = [ J K K * J * ] ,
Y ( z ) = T y ( z ) Y ( 0 ) ,
T y ( z ) = exp ( i L y z ) .
H = X 1 J 1 X 1 + X 2 J 2 X 2 + X 1 K X 2 * + X 1 t K * X 2 ,
d z X j = i H / X j
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 * .
X 1 ( z ) = M 11 ( z ) X 1 ( 0 ) + N 12 ( z ) X 2 * ( 0 ) , X 2 ( z ) = M 22 ( z ) X 2 ( 0 ) + N 21 ( z ) X 1 * ( 0 ) ,
d z X = i L x X ,
X = [ X 1 X 2 * ] , L x = [ J 1 K K J 2 * ] ,
X ( z ) = T x ( z ) X ( 0 ) ,
T x ( z ) = exp ( i L x z ) .
S = [ I 0 0 I ] .
S L x = L x S .
d z C = i X ( S L x L x S ) X .
S L x n = ( L x ) n S .
S T x ( z ) = T x ( z ) S .
T x ( z ) = T x 1 ( z ) ,
det ( λ I A ) = det [ S ( λ I A ) S ] = det ( λ I S A S ) .
det ( λ I A B ) = det [ A 1 ( λ I A B ) A ] = det ( λ I B A ) .
( T x T x ) 1 = T x 1 ( T x ) 1 = ( S T x S ) ( S T x S ) = S ( T x T x ) S ,
( T x T x ) 1 = ( T x ) 1 T x 1 = ( S T x S ) ( S T x S ) = S ( T x T x ) S .
T x ( z ) = [ M 11 ( z ) N 12 ( z ) N 21 * ( z ) M 22 * ( z ) ] = S T x ( z ) S = [ M 11 ( z ) N 21 t ( z ) N 12 ( z ) M 22 t ( z ) ] .
M 11 ( z ) = M 11 ( z ) , N 12 ( z ) = N 21 t ( z ) , M 22 ( z ) = M 22 ( z ) , N 21 ( z ) = N 12 t ( z ) .
T x 1 ( z ) T x ( z ) = [ M 11 N 21 t N 12 M 22 t ] [ M 11 N 12 N 21 * M 22 * ] = [ M 11 M 11 N 21 t N 21 * M 11 N 12 N 21 t M 22 * M 22 t N 21 * N 12 M 11 M 22 t M 22 * N 12 N 12 ] = [ I 0 0 I ] ,
T x ( z ) T x 1 ( z ) = [ M 11 N 12 N 21 * M 22 * ] [ M 11 N 21 t N 12 M 22 t ] = [ M 11 M 11 N 12 N 12 N 12 M 22 t M 11 N 21 t N 21 * M 11 M 22 * N 12 M 22 * M 22 t N 21 * N 21 t ] = [ I 0 0 I ] .
U 11 D 11 2 U 11 U 21 * D 21 2 U 21 t = I ,
U 22 * D 22 2 U 11 t U 12 D 12 2 U 12 = I ,
V 11 D 11 2 V 11 V 12 D 12 2 V 12 = I ,
V 22 * D 22 2 V 22 t V 21 * D 21 2 V 21 t = I .
T x ( 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 ] .
T x T x = [ U 1 ( D ν 2 + D ν 2 ) U 1 U 1 ( 2 D μ D ν ) U 2 t U 2 * ( 2 D μ D ν ) U 1 U 2 * ( D μ 2 + D ν 2 ) U 2 t ] .
D ± = ( D μ ± D ν ) 2 , E ± = [ U 1 t , ± U 2 ] t / 2 1 / 2 .
T x T x = [ V 1 ( D ν 2 + D ν 2 ) V 1 V 1 ( 2 D μ D ν ) V 2 t V 2 * ( 2 D μ D ν ) V 1 V 2 * ( D μ 2 + D ν 2 ) V 2 t ] .
D ± = ( D μ ± D ν ) 2 , E ± = [ V 1 t , ± V 2 ] t / 2 1 / 2 .
T x ( z ) = 1 2 1 / 2 [ V 1 V 1 V 2 * V 2 * ] [ D μ + D ν 0 0 D μ D ν ] 1 2 1 / 2 [ U 1 U 2 t U 1 U 2 ] ,
T x 1 ( z ) = [ U 1 D μ V 1 U 1 D ν V 2 t U 2 * D ν V 1 U 2 * D μ V 2 t ] .
T x 1 ( z ) = 1 2 1 / 2 [ U 1 U 1 U 2 * U 2 * ] [ D μ D ν 0 0 D μ + D ν ] 1 2 1 / 2 [ V 1 V 2 t V 1 V 2 t ] .
T x 1 ( z ) = 1 2 1 / 2 [ U 1 U 1 U 2 * U 2 * ] [ D μ + D ν 0 0 D μ D ν ] 1 2 1 / 2 [ V 1 V 2 t V 1 V 2 t ] .
P = [ 0 I I 0 ] .
D 1 = P D P .
S U = U P , V S = P V .
D μ ( z ) D μ ( z ) , D ν ( z ) D ν ( z ) , U j ( z ) i V j ( z ) , V j ( z ) i U j ( z )
M = [ M 11 0 0 M 22 ] , N = [ 0 N 12 N 21 0 ] ,
M = 1 2 1 / 2 [ V 1 i V 1 V 2 i V 2 ] [ D μ 0 0 D μ ] 1 2 1 / 2 [ U 1 U 2 i U 1 i U 2 ] ,
N = 1 2 1 / 2 [ V 1 i V 1 V 2 i V 2 ] [ D ν 0 0 D ν ] 1 2 1 / 2 [ U 1 t U 2 t i U 1 t i U 2 t ] .
H = X 1 J 1 X 1 + X 2 t J 2 * X 2 * + X 1 K X 2 * + X 2 t K X 1 ,
d z X 1 = i H / X 1 , d z X 2 * = i H / X 2 t
H = X G X
d z X = i S H / X ,
X = [ X 1 X 2 * ] , G = [ J 1 K K J 2 * ] ,
d z x i = j T i j d z x j = i j k T i j S j k H / x k * = i j k i T i j S j k T l k * H / x l * .
j k l T i j S j k T l k * = S i l .
T S T = S .
d z T = i S G T ,
d z ( T S T ) = i ( T S 2 G T T G S 2 T ) = 0 ,
T S T = S .
( T X ) S ( T X ) = X ( T S T ) X = X S X .
( T T ) S E = S ( S T S ) ( S T S ) E = S ( T T ) 1 E = λ 1 S E .
T 1 = S ( U D V ) S = ( S U ) D ( S V ) .
d z p = H / q , d z q = H / p .
d z X = J H / X ,
J = [ 0 1 1 0 ]
H = α ¯ p 2 / 2 + β ¯ p q + γ ¯ q 2 / 2 ,
d z p = β ¯ p + γ ¯ q , d z q = α ¯ p β ¯ q .
H = X t G X / 2 ,
G = [ α ¯ β ¯ β ¯ γ ¯ ] .
d z X = J G X ,
d z x i = j T i j d z x j = j k T i j J j k H / x k = j k l T i j J j k T l k H / x l .
j k l T i j J j k T l k = J i l .
T J T t = J .
d z T = J G T ,
d z ( T t J T ) = T t J 2 G T + T t G t J t J T = 0 ,
T t J T = J .
a = ( q + i p ) / 2 1 / 2 , a * = ( q i p ) / 2 1 / 2 ,
p = i ( a * a ) / 2 1 / 2 , q = ( a * + a ) / 2 1 / 2 .
d z a = i H / a * , d z a * = i H / a .
d z X = i S H / X * ,
H = δ | a | 2 + γ ( a * ) 2 / 2 + γ * a 2 / 2 ,
d z a = i δ a + i γ a * , d z a * = i δ a * i γ * a .
α ¯ = δ γ r , β ¯ = γ i , γ ¯ = δ + γ r .
H = X G X / 2 ,
G = [ δ γ γ * δ ] .
d z X = i S G X .
U = 1 2 1 / 2 [ i 1 i 1 ] .
U ( T r J T r t ) U = ( U T r U ) ( U J U ) ( U T r t U ) = U J U .

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