Abstract

In a previous paper [Opt. Express 13, 4986 (2005)], formulas were derived for the field-quadrature and photon-number variances produced by multiple-mode parametric processes. In this paper, formulas are derived for the quadrature and number correlations. The number formulas are used to analyze the properties of basic devices, such as two-mode amplifiers, attenuators and frequency convertors, and composite systems made from these devices, such as cascaded parametric amplifiers and communication links. Amplifiers generate idlers that are correlated with the amplified signals, or correlate pre-existing pairs of modes, whereas attenuators decorrelate pre-existing modes. Both types of device modify the signal-to-noise ratios (SNRs) of the modes on which they act. Amplifiers decrease or increase the mode SNRs, depending on whether they are operated in phase-insensitive (PI) or phase-sensitive (PS) manners, respectively, whereas attenuators always decrease these SNRs. Two-mode PS links are sequences of transmission fibers (attenuators) followed by two-mode PS amplifiers. Not only do these PS links have noise figures that are 6-dB lower than those of the corresponding PI links, they also produce idlers that are (almost) completely correlated with the signals. By detecting the signals and idlers, one can eliminate the effects of electronic noise in the detectors.

© 2010 Optical Society of America

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  1. J. Hansryd, P. A. Andrekson, M. Westland, 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. S. Radic, and C. J. McKinstrie, “Optical amplification and signal processing in highly nonlinear optical fiber,” IEICE Trans. Electron,” E 88-C, 859–869 (2005).
    [CrossRef]
  3. P. A. Andrekson, and M. Westlund, “Nonlinear optical fiber based all-optical waveform sampling,” Laser Photon. Rev. 1, 231–248 (2007).
    [CrossRef]
  4. S. Radic, “Parametric amplification and processing in optical fibers,” Laser Photon. Rev. 2, 489–513 (2008).
    [CrossRef]
  5. C. J. McKinstrie, S. Radic, and M. G. Raymer, “Quantum noise properties of parametric amplifiers driven by two pump waves,” Opt. Express 12, 5037–5066 (2004).
    [CrossRef] [PubMed]
  6. C. J. McKinstrie, M. Yu, M. G. Raymer, and S. Radic, “Quantum noise properties of parametric processes,” Opt. Express 13, 4986–5012 (2005).
    [CrossRef] [PubMed]
  7. R. Loudon, and P. L. Knight, “Squeezed light,” J. Mod. Opt. 34, 709–759 (1987).
    [CrossRef]
  8. S. M. Barnett, and P. M. Radmore, Methods in Theoretical Quantum Optics (Oxford University Press, 1997).
  9. N. Christensen, R. Leonhardt, and J. D. Harvey, “Noise characteristics of cross-phase modulation instability light,” Opt. Commun. 101, 205–212 (1993).
    [CrossRef]
  10. J. E. Sharping, M. Fiorentino, and P. Kumar, “Observation of twin-beam-type quantum correlation in optical fiber,” Opt. Lett. 26, 367–369 (2001).
    [CrossRef]
  11. R. Tang, P. Devgan, P. L. Voss, V. S. Grigoryan, and P. Kumar, “In-line frequency-nondegenerate phase-sensitive fiber-optical parametric amplifier,” IEEE Photon. Technol. Lett. 17, 1845–1847 (2005).
    [CrossRef]
  12. R. Tang, J. Lasri, P. S. Devgan, V. Grigoryan, and P. Kumar, “Gain characteristics of a frequency nondegenerate phase-sensitive fiber-optic parametric amplifier with phase self-stabilized input,” Opt. Express 13, 10483–10493 (2005).
    [CrossRef] [PubMed]
  13. C. Lundström, J. Kakande, P. A. Andrekson, Z. Tong, M. Karlsson, P. Petropoulos, F. Parmigiani and D. J. Richardson, “Experimental comparison of gain and saturation characteristics of a parametric amplifier in phase-sensitive and phase-insensitive mode,” ECOC 2009, paper 1.1.1.
  14. J. Kakande, C. Lundström, P. A. Andrekson, Z. Tong, M. Karlsson, P. Petropoulos, F. Parmigiani, and D. J. Richardson, “Detailed characterization of a fiber-optic parametric amplifier in phase-sensitive and phase-insensitive operation,” Opt. Express 18, 4130–4137 (2010).
    [CrossRef] [PubMed]
  15. R. Loudon, “Theory of noise accumulation in linear optical-amplifier chains,” J. Quantum Electron. 21, 766–773 (1985).
    [CrossRef]
  16. R. E. Slusher, and B. Yurke, “Squeezed light for coherent communications,” J. Lightwave Technol. 8, 466–477 (1990).
    [CrossRef]
  17. M. Vasilyev, “Distributed phase-sensitive amplification,” Opt. Express 13, 7563–7571 (2005).
    [CrossRef] [PubMed]
  18. 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]
  19. J. M. Manley, and H. E. Rowe, “Some general properties of nonlinear elements–Part I. General energy relations,” Proc. IRE 44, 904–913 (1956).
    [CrossRef]
  20. M. T. Weiss, “Quantum derivation of energy relations analogous to those for nonlinear reactances,” Proc. IRE 45, 1012–1013 (1957).
  21. C. J. McKinstrie, J. D. Harvey, S. Radic, and M. G. Raymer, “Translation of quantum states by four-wave mixing in fibers,” Opt. Express 13, 9131–9142 (2005).
    [CrossRef] [PubMed]
  22. 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]
  23. Z. Tong, A. Bogris, C. Lundström, C. J. McKinstrie, M. Vasilyev, M. Karlsson, and P. A. Andrekson, “Modeling and measurement of the noise figure of a cascaded non-degenerate phase-sensitive parametric amplifier,” Opt. Express 18, 14820–14835 (2010).
    [CrossRef] [PubMed]
  24. Z. Tong, C. J. McKinstrie, C. Lundström, M. Karlsson, and P. A. Andrekson, “Noise performance of optical fiber transmission links that use non-degenerate cascaded phase-sensative amplifiers,” Opt. Express 18, 15426–15439 (2010).
    [CrossRef] [PubMed]
  25. L. A. Krivitsky, U. L. Andersen, R. Dong, A. Huck, C. Wittmann, and G. Leuchs, “Electronic noise-free measurements of squeezed light,” Opt. Lett. 33, 2395–2397 (2008).
    [CrossRef] [PubMed]
  26. C. J. McKinstrie, S. Radic, M. G. Raymer, and M. V. Vasilyev, “Quantum mechanics of phase-sensitive amplification in fibers,” Opt. Commun. 257, 146–163 (2006).
    [CrossRef]

2010 (4)

2008 (2)

2007 (1)

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

2006 (2)

C. J. McKinstrie, S. Radic, M. G. Raymer, and M. V. Vasilyev, “Quantum mechanics of phase-sensitive amplification in fibers,” Opt. Commun. 257, 146–163 (2006).
[CrossRef]

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]

2005 (6)

2004 (1)

2002 (1)

J. Hansryd, P. A. Andrekson, M. Westland, 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]

2001 (1)

1993 (1)

N. Christensen, R. Leonhardt, and J. D. Harvey, “Noise characteristics of cross-phase modulation instability light,” Opt. Commun. 101, 205–212 (1993).
[CrossRef]

1990 (1)

R. E. Slusher, and B. Yurke, “Squeezed light for coherent communications,” J. Lightwave Technol. 8, 466–477 (1990).
[CrossRef]

1987 (1)

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

1985 (1)

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

Andersen, U. L.

Andrekson, P. A.

Bogris, A.

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]

Christensen, N.

N. Christensen, R. Leonhardt, and J. D. Harvey, “Noise characteristics of cross-phase modulation instability light,” Opt. Commun. 101, 205–212 (1993).
[CrossRef]

Devgan, P.

R. Tang, P. Devgan, P. L. Voss, V. S. Grigoryan, and P. Kumar, “In-line frequency-nondegenerate phase-sensitive fiber-optical parametric amplifier,” IEEE Photon. Technol. Lett. 17, 1845–1847 (2005).
[CrossRef]

Devgan, P. S.

Dong, R.

Fiorentino, M.

Grigoryan, V.

Grigoryan, V. S.

R. Tang, P. Devgan, P. L. Voss, V. S. Grigoryan, and P. Kumar, “In-line frequency-nondegenerate phase-sensitive fiber-optical parametric amplifier,” IEEE Photon. Technol. Lett. 17, 1845–1847 (2005).
[CrossRef]

Hansryd, J.

J. Hansryd, P. A. Andrekson, M. Westland, 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]

Harvey, J. D.

C. J. McKinstrie, J. D. Harvey, S. Radic, and M. G. Raymer, “Translation of quantum states by four-wave mixing in fibers,” Opt. Express 13, 9131–9142 (2005).
[CrossRef] [PubMed]

N. Christensen, R. Leonhardt, and J. D. Harvey, “Noise characteristics of cross-phase modulation instability light,” Opt. Commun. 101, 205–212 (1993).
[CrossRef]

Hedekvist, P. O.

J. Hansryd, P. A. Andrekson, M. Westland, 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]

Huck, A.

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]

Kakande, J.

Karlsson, M.

Knight, P. L.

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

Krivitsky, L. A.

Kumar, P.

Lasri, J.

Leonhardt, R.

N. Christensen, R. Leonhardt, and J. D. Harvey, “Noise characteristics of cross-phase modulation instability light,” Opt. Commun. 101, 205–212 (1993).
[CrossRef]

Leuchs, G.

Li, J.

J. Hansryd, P. A. Andrekson, M. Westland, 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, and P. L. Knight, “Squeezed light,” J. Mod. Opt. 34, 709–759 (1987).
[CrossRef]

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

Lundström, C.

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.

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]

Z. Tong, A. Bogris, C. Lundström, C. J. McKinstrie, M. Vasilyev, M. Karlsson, and P. A. Andrekson, “Modeling and measurement of the noise figure of a cascaded non-degenerate phase-sensitive parametric amplifier,” Opt. Express 18, 14820–14835 (2010).
[CrossRef] [PubMed]

Z. Tong, C. J. McKinstrie, C. Lundström, M. Karlsson, and P. A. Andrekson, “Noise performance of optical fiber transmission links that use non-degenerate cascaded phase-sensative amplifiers,” Opt. Express 18, 15426–15439 (2010).
[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, S. Radic, M. G. Raymer, and M. V. Vasilyev, “Quantum mechanics of phase-sensitive amplification in fibers,” Opt. Commun. 257, 146–163 (2006).
[CrossRef]

S. Radic, and C. J. McKinstrie, “Optical amplification and signal processing in highly nonlinear optical fiber,” IEICE Trans. Electron,” E 88-C, 859–869 (2005).
[CrossRef]

C. J. McKinstrie, M. Yu, M. G. Raymer, and S. Radic, “Quantum noise properties of parametric processes,” Opt. Express 13, 4986–5012 (2005).
[CrossRef] [PubMed]

C. J. McKinstrie, J. D. Harvey, S. Radic, and M. G. Raymer, “Translation of quantum states by four-wave mixing in fibers,” Opt. Express 13, 9131–9142 (2005).
[CrossRef] [PubMed]

C. J. McKinstrie, S. Radic, and M. G. Raymer, “Quantum noise properties of parametric amplifiers driven by two pump waves,” Opt. Express 12, 5037–5066 (2004).
[CrossRef] [PubMed]

Parmigiani, F.

Petropoulos, P.

Radic, S.

S. Radic, “Parametric amplification and processing in optical fibers,” Laser Photon. Rev. 2, 489–513 (2008).
[CrossRef]

C. J. McKinstrie, S. Radic, M. G. Raymer, and M. V. Vasilyev, “Quantum mechanics of phase-sensitive amplification in fibers,” Opt. Commun. 257, 146–163 (2006).
[CrossRef]

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. Express 13, 9131–9142 (2005).
[CrossRef] [PubMed]

C. J. McKinstrie, M. Yu, M. G. Raymer, and S. Radic, “Quantum noise properties of parametric processes,” Opt. Express 13, 4986–5012 (2005).
[CrossRef] [PubMed]

S. Radic, and C. J. McKinstrie, “Optical amplification and signal processing in highly nonlinear optical fiber,” IEICE Trans. Electron,” E 88-C, 859–869 (2005).
[CrossRef]

C. J. McKinstrie, S. Radic, and M. G. Raymer, “Quantum noise properties of parametric amplifiers driven by two pump waves,” Opt. Express 12, 5037–5066 (2004).
[CrossRef] [PubMed]

Raymer, M. G.

Richardson, D. J.

Sharping, J. E.

Slusher, R. E.

R. E. Slusher, and B. Yurke, “Squeezed light for coherent communications,” J. Lightwave Technol. 8, 466–477 (1990).
[CrossRef]

Tang, R.

R. Tang, J. Lasri, P. S. Devgan, V. Grigoryan, and P. Kumar, “Gain characteristics of a frequency nondegenerate phase-sensitive fiber-optic parametric amplifier with phase self-stabilized input,” Opt. Express 13, 10483–10493 (2005).
[CrossRef] [PubMed]

R. Tang, P. Devgan, P. L. Voss, V. S. Grigoryan, and P. Kumar, “In-line frequency-nondegenerate phase-sensitive fiber-optical parametric amplifier,” IEEE Photon. Technol. Lett. 17, 1845–1847 (2005).
[CrossRef]

Tong, Z.

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.

Vasilyev, M. V.

C. J. McKinstrie, S. Radic, M. G. Raymer, and M. V. Vasilyev, “Quantum mechanics of phase-sensitive amplification in fibers,” Opt. Commun. 257, 146–163 (2006).
[CrossRef]

Voss, P. L.

R. Tang, P. Devgan, P. L. Voss, V. S. Grigoryan, and P. Kumar, “In-line frequency-nondegenerate phase-sensitive fiber-optical parametric amplifier,” IEEE Photon. Technol. Lett. 17, 1845–1847 (2005).
[CrossRef]

Westland, M.

J. Hansryd, P. A. Andrekson, M. Westland, 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]

Westlund, M.

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

Wittmann, C.

Yu, M.

Yurke, B.

R. E. Slusher, and B. Yurke, “Squeezed light for coherent communications,” J. Lightwave Technol. 8, 466–477 (1990).
[CrossRef]

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

J. Hansryd, P. A. Andrekson, M. Westland, 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. (1)

R. Tang, P. Devgan, P. L. Voss, V. S. Grigoryan, and P. Kumar, “In-line frequency-nondegenerate phase-sensitive fiber-optical parametric amplifier,” IEEE Photon. Technol. Lett. 17, 1845–1847 (2005).
[CrossRef]

IEICE Trans. Electron. E (1)

S. Radic, and C. J. McKinstrie, “Optical amplification and signal processing in highly nonlinear optical fiber,” IEICE Trans. Electron,” E 88-C, 859–869 (2005).
[CrossRef]

J. Lightwave Technol. (1)

R. E. Slusher, and B. Yurke, “Squeezed light for coherent communications,” J. Lightwave Technol. 8, 466–477 (1990).
[CrossRef]

J. Mod. Opt. (1)

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

J. Quantum Electron. (1)

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

Laser Photon. Rev. (2)

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

S. Radic, “Parametric amplification and processing in optical fibers,” Laser Photon. Rev. 2, 489–513 (2008).
[CrossRef]

Opt. Commun. (4)

N. Christensen, R. Leonhardt, and J. D. Harvey, “Noise characteristics of cross-phase modulation instability light,” Opt. Commun. 101, 205–212 (1993).
[CrossRef]

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]

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, S. Radic, M. G. Raymer, and M. V. Vasilyev, “Quantum mechanics of phase-sensitive amplification in fibers,” Opt. Commun. 257, 146–163 (2006).
[CrossRef]

Opt. Express (8)

C. J. McKinstrie, S. Radic, and M. G. Raymer, “Quantum noise properties of parametric amplifiers driven by two pump waves,” Opt. Express 12, 5037–5066 (2004).
[CrossRef] [PubMed]

C. J. McKinstrie, M. Yu, M. G. Raymer, and S. Radic, “Quantum noise properties of parametric processes,” Opt. Express 13, 4986–5012 (2005).
[CrossRef] [PubMed]

M. Vasilyev, “Distributed phase-sensitive amplification,” Opt. Express 13, 7563–7571 (2005).
[CrossRef] [PubMed]

C. J. McKinstrie, J. D. Harvey, S. Radic, and M. G. Raymer, “Translation of quantum states by four-wave mixing in fibers,” Opt. Express 13, 9131–9142 (2005).
[CrossRef] [PubMed]

R. Tang, J. Lasri, P. S. Devgan, V. Grigoryan, and P. Kumar, “Gain characteristics of a frequency nondegenerate phase-sensitive fiber-optic parametric amplifier with phase self-stabilized input,” Opt. Express 13, 10483–10493 (2005).
[CrossRef] [PubMed]

J. Kakande, C. Lundström, P. A. Andrekson, Z. Tong, M. Karlsson, P. Petropoulos, F. Parmigiani, and D. J. Richardson, “Detailed characterization of a fiber-optic parametric amplifier in phase-sensitive and phase-insensitive operation,” Opt. Express 18, 4130–4137 (2010).
[CrossRef] [PubMed]

Z. Tong, A. Bogris, C. Lundström, C. J. McKinstrie, M. Vasilyev, M. Karlsson, and P. A. Andrekson, “Modeling and measurement of the noise figure of a cascaded non-degenerate phase-sensitive parametric amplifier,” Opt. Express 18, 14820–14835 (2010).
[CrossRef] [PubMed]

Z. Tong, C. J. McKinstrie, C. Lundström, M. Karlsson, and P. A. Andrekson, “Noise performance of optical fiber transmission links that use non-degenerate cascaded phase-sensative amplifiers,” Opt. Express 18, 15426–15439 (2010).
[CrossRef] [PubMed]

Opt. Lett. (2)

Other (4)

J. M. Manley, and H. E. Rowe, “Some general properties of nonlinear elements–Part I. General energy relations,” Proc. IRE 44, 904–913 (1956).
[CrossRef]

M. T. Weiss, “Quantum derivation of energy relations analogous to those for nonlinear reactances,” Proc. IRE 45, 1012–1013 (1957).

S. M. Barnett, and P. M. Radmore, Methods in Theoretical Quantum Optics (Oxford University Press, 1997).

C. Lundström, J. Kakande, P. A. Andrekson, Z. Tong, M. Karlsson, P. Petropoulos, F. Parmigiani and D. J. Richardson, “Experimental comparison of gain and saturation characteristics of a parametric amplifier in phase-sensitive and phase-insensitive mode,” ECOC 2009, paper 1.1.1.

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Figures (15)

Fig. 1.
Fig. 1.

In a two-mode amplifier (▷), the signal-mode operator (a 1) is coupled to the hermitian conjugate of the idler-mode operator (a 2). Amplifiers are characterized by their transfer coefficients μ and ν.

Fig. 2.
Fig. 2.

In a two-mode attenuator (◁), the signal-mode operator (a 1) is coupled to the loss-mode operator (a 3). Attenuators are characterized by their transfer coefficients τ and ρ.

Fig. 3.
Fig. 3.

Architecture of a copier. A two-mode amplifier (▷) is followed by two attenuators (◁) in parallel. Mode 1 is the signal, mode 2 is the idler (copied signal), and modes 3 and 4 are the loss modes.

Fig. 4.
Fig. 4.

Architecture of an idealized one-stage link. Two attenuators (◁) in parallel are followed by a two-mode amplifier (▷). Mode 1 is the signal, mode 2 is the idler, and modes 3 and 4 are the loss modes.

Fig. 5.
Fig. 5.

Architecture of a cascaded phase-sensitive amplifier. A two-mode amplifier (▷) is followed by two attenuators (◁) in parallel and another two-mode amplifier. The signal, idler and loss modes are labeled 1, 2, 3 and 4, respectively.

Fig. 6.
Fig. 6.

Architecture of a multiple-stage link. The copier consists of a two-mode amplifier (▷) followed by two attenuators (◁) in parallel, whereas stage r of the link consists of two attenuators in parallel followed by a two-mode amplifier. Mode 1 is the signal, mode 2 is the idler, modes −1 and 0 are the loss modes of the copier, and modes 2r + 1 and 2r + 2 are the loss modes of the stage.

Fig. 7.
Fig. 7.

Properties of a two-mode PI amplifier. (a) Noise figures plotted as functions of gain. The solid and dashed curves represent the signal and idler, respectively. (b) Correlation coefficient plotted as a function of gain.

Fig. 8.
Fig. 8.

Properties of a two-mode PS amplifier. (a) Noise figures and (b) correlation coefficients plotted as functions of gain. The solid, dot-dashed and dashed curves represent the phase differences 0 (in-phase), π/2 and π (out-of-phase), respectively.

Fig. 9.
Fig. 9.

Noise figures plotted as functions of loss for a two-mode attenuator. The solid and dashed curves represent the signal and loss-mode, respectively.

Fig. 10.
Fig. 10.

Properties of a balanced copier. (a) Noise figures plotted as functions of gain. The solid and dashed curves represent the signal and idler, respectively. (b) Correlation coefficient plotted as a function of gain.

Fig. 11.
Fig. 11.

Properties of a balanced one-stage link. (a) Noise figures and (b) correlation coefficients plotted as functions of loss. The solid and dashed curves represent PS and PI links, respectively.

Fig. 12.
Fig. 12.

Properties of a cascaded PS amplifier with a balanced copier. (a) Noise figure and (b) correlation coefficient plotted as functions of the gain G 2, for the case in which G 1 = 10 and the transmission T = 0.1.

Fig. 13.
Fig. 13.

Properties of a cascaded PS amplifier with high loss. (a) Noise figure and (b) correlation coefficient plotted as functions of loss, for the case in which the gains G 1 = 10 and G 2 = 10. The solid and dashed curves represent the PS amplifier and an unbalanced copier, respectively.

Fig. 14.
Fig. 14.

Properties of an idealized 3-stage PS link. (a) Noise figure and (b) correlation coefficient plotted as functions of the stage loss. The solid curves represent the exact results [Eqs. (92) and (93)], whereas the dashed curves represent the approximate results [Eqs. (121) and (122)].

Fig. 15.
Fig. 15.

Properties of a realistic 3-stage PS link with a balanced copier. (a) Noise figure and (b) correlation coefficient plotted as functions of the stage loss. The solid curves represent the exact results [Eqs. (107) and (108)], whereas the dashed curves represent the approximate results [Eqs. (123) and (124)].

Tables (2)

Tables Icon

Table 1. Commutators for effective modes 1 and 2

Tables Icon

Table 2. Commutators for effective modes 1 and 3

Equations (200)

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b i = Σ k ( μ ik a k + ν ik a k ) ,
Σ k ( μ ik ν jk ν ik μ jk ) = 0 ,
Σ k ( μ ik μ jk * ν ik ν jk * ) = δ ij .
β i = Σ k ( μ ik α k + ν ik α k * ) .
q i ( θ i ) = ( β i e i θ i + β i * e i θ i ) 2 1 2
a i = α i + v i , b i = β i + w i ,
δ q i ( θ i ) = q i ( θ i ) q j ( θ j ) = ( w i e i θ i + w i e i θ i ) 2 1 2 .
δ q i ( θ i ) δ q j ( θ j ) = ( w i e i θ i + w i e i θ i ) ( w j e i θ j + w j e i θ j ) 2 .
w i w j = Σ k μ ik ν jk , w i w j = Σ k μ ik μ jk * ,
w i w j = Σ k ν ik * ν jk , w i w j = Σ k ν ik * μ jk * ,
δ q i ( θ i ) δ q j ( θ j ) = Σ k ( μ ik e i θ i + ν ik * e i θ i ) ( μ jk * e i θ j + ν jk e i θ j ) 2 .
n i = β i 2 + β i * w i + β i w i + w i w i .
n i = β i 2 + w i w i ,
δ n i = n i n i = β i * w i + β i w i + w i w i w i w i .
δ n i δ n j = ( β i * w i + β i w i ) ( β j * w j + β j w j ) + w i w i w j w j w i w i w j w j .
δ n i δ n j sn = 2 ρ i ρ j δ q i ( ϕ i ) δ q j ( ϕ j ) ,
δ n i δ n j sn = Σ k ( β i * μ ik + β i ν ik * ) ( β j μ jk * + β j * ν jk ) .
( w i w i ) 1 = Σ k Σ l ( v ik * μ il v k v l + v ik * v il v k v l ) ,
( w j w j ) r = Σ k Σ l ( μ jk * v jl v k v l + v jk * v jl v k v l ) ,
δ n i δ n j nn = 2 Σ k ( μ ik ν ik * ) ( μ jk * ν jk ) + Σ k Σ l > k ( μ ik ν il * + μ il ν ik * ) ( μ jk * ν jl + μ jl * ν jk ) .
δ n i δ n j = Σ k ( β i * μ ik + β i ν ik * ) ( β j μ jk * + β j * ν jk ) + 2 Σ k ( μ ik ν ik * ) ( μ jk * ν jk )
+ Σ k Σ l > k ( μ ik ν il * + μ il ν ik * ) ( μ jk * ν jl + μ jl * ν jk ) .
( δ n i δ n j ) 2 = Σ k β i * μ ik + β i ν ik * β j * μ jk β j ν jk * 2 + 2 Σ k μ ik ν ik * μ jk ν jk * 2
+ Σ k Σ l > k μ ik ν il * + μ il ν ik * μ jk ν jl * μ jl ν jk * 2
δ n i 2 sn = β i 2 Σ k λ ik 2 ,
δ n i δ n j sn = β i * β j Σ k μ ik μ jk * + β i β j * Σ l ν il * ν jl ,
δ n i δ n j sn = Re ( β i * β j Σ k λ ik λ jk * ) ,
δ n i δ n j sn = β i * β j * Σ k μ ik ν jk + β i β j Σ l ν il * μ jl * ,
δ n i δ n j sn = Re ( β i * β j * Σ k λ ik λ jk ) ,
b 1 = μ a 1 + ν a 2 ,
b 2 = ν a 1 + μ a 2 ,
β 1 2 = μ α 1 2 + ν α 2 2 + 2 μ ν α 1 α 2 cos θ ,
β 2 2 = μ α 2 2 + ν α 1 2 + 2 μ ν α 1 α 2 cos θ ,
δ n 1 2 = ( μ 2 + ν 2 ) ( μ α 1 2 + ν α 2 2 + 2 μ ν α 1 α 2 cos θ ) ,
δ n 2 2 = ( μ 2 + ν 2 ) ( μ α 2 2 + ν α 1 2 + 2 μ ν α 1 α 2 cos θ ) ,
δ n 1 δ n 2 = 2 μ ν [ μ ν ( α 1 2 + α 2 2 ) + ( μ 2 + ν 2 ) α 1 α 2 cos θ ] ,
( δ n 1 δ n 2 ) 2 = α 1 2 + α 2 2
δ n 1 2 = ( μ 2 + ν 2 ) μ α 1 2 ,
δ n 2 2 = ( μ 2 + ν 2 ) ν α 1 2 ,
δ n 1 δ n 2 = 2 μ ν α 1 2
( δ n 1 δ n 2 ) 2 = α 1 2 .
δ n i 2 = ( μ 2 + ν 2 ) [ ( μ 2 + ν 2 ) + 2 μ ν cos θ ] α 2 ,
δ n 1 δ n 2 = 2 μ ν [ 2 μ ν + ( μ 2 + ν 2 ) cos θ ] α 2 .
b 1 = τ a 1 + ρ a 3 ,
b 3 = ρ * a 1 + τ * a 3 ,
β 1 2 = τ α 1 2 + ρ α 3 2 + 2 τ ρ α 1 α 3 cos θ ,
β 3 2 = τ α 3 2 + ρ α 1 2 2 τ ρ α 1 α 3 cos θ ,
δ n i 2 = ( τ 2 + ρ 2 ) β i 2 ,
δ n 1 δ n 3 = 0 ,
( δ n 1 δ n 3 ) 2 = α 1 2 + α 3 2
b 1 = ( τ 1 μ ) a 1 + ( τ 1 ν ) a 2 + ρ 1 a 3 ,
b 2 = ( τ 2 ν ) a 1 + ( τ 2 μ ) a 2 + ρ 2 a 4 ,
β 1 2 = τ 1 2 ( μ α 1 2 + ν α 2 2 + 2 μ ν α 1 α 2 cos θ ) ,
β 2 2 = τ 2 2 ( μ α 2 2 + ν α 1 2 + 2 μ ν α 1 α 2 cos θ ) ,
δ n 1 2 = τ 1 2 ( τ 1 μ 2 + τ 1 v 2 + ρ 1 2 ) ( μ α 1 2 + v α 2 2 + 2 μ v α 1 α 2 cos θ ) ,
δ n 2 2 = τ 2 2 ( τ 2 μ 2 + τ 2 v 2 + ρ 2 2 ) ( μ α 2 2 + v α 1 2 + 2 μ v α 1 α 2 cos θ ) ,
δ n 1 δ n 2 = 2 τ 1 τ 2 2 μ v [ μ v ( α 1 2 + α 2 2 ) + ( μ 2 + v 2 ) α 1 α 2 cos θ ] ,
δ n 1 2 = ( τ μ 2 + τ v 2 + ρ 2 ) τ μ α 1 2 ,
δ n 2 2 = ( τ μ 2 + τ v 2 + ρ 2 ) τv α 1 2 ,
δ n 1 δ n 2 = 2 τ μ τ v α 1 2 .
( δ n 1 δ n 2 ) 2 = ( 1 + 2 ρ v 2 ) τ α 1 2 .
b 1 = ( μ τ 1 ) a 1 + ( v τ 2 * ) a 2 + ( μ ρ 1 ) a 3 + ( v ρ 2 * ) a 4 ,
b 2 = ( v τ 1 * ) a 1 + ( μ τ 2 ) a 2 + ( v ρ 1 * ) a 3 + ( μ ρ 2 ) a 4 .
β 1 2 = μ τ 1 α 1 2 + v τ 2 α 2 2 + 2 μ v τ 1 τ 2 α 1 α 2 cos θ ,
β 2 2 = μ τ 2 α 2 2 + v τ 1 α 1 2 + 2 μ v τ 1 τ 2 α 1 α 2 cos θ ,
δ n 1 2 = ( μ 2 + v 2 ) ( μ τ 1 α 1 2 + v τ 2 α 2 2 + 2 μ v τ 1 τ 2 α 1 α 2 cos θ ) ,
δ n 2 2 = ( μ 2 + v 2 ) ( μ τ 2 α 2 2 + v τ 1 α 1 2 + 2 μ v τ 1 τ 2 α 1 α 2 cos θ ) ,
δ n 1 δ n 2 = 2 μ v [ μ v ( τ 1 α 1 2 + τ 2 α 2 2 ) + ( μ 2 + v 2 ) τ 1 τ 2 α 1 α 2 cos θ ] ,
( δ n 1 δ n 2 ) 2 = τ 1 α 1 2 + τ 2 α 2 2 .
δ n i 2 = ( μ 2 + v 2 ) [ ( μ 2 + v 2 ) + 2 μ v cos θ ] τ α 2 ,
δ n 1 δ n 2 = 2 μ v [ 2 μ v + ( μ 2 + v 2 ) cos θ ] τ α 2 .
b 1 = ( μ τ 1 μ c + v τ 2 v c ) a 1 + ( μ τ 1 v c + v τ 2 μ c ) a 2 + ( μ ρ 1 ) a 3 + ( v ρ 2 ) a 4 ,
b 2 = ( μ τ 2 v c + v τ 1 μ c ) a 1 + ( μ τ 2 μ c + v τ 1 v c ) a 2 + ( v ρ 1 ) a 3 + ( μ ρ 2 ) a 4 ,
δ n 1 2 = [ 2 σ 2 ( μ + v ) 2 + ( μ 2 + v 2 ) 2 μ 2 τ 1 2 ] β 2 ,
δ n 2 2 = [ 2 σ 2 ( μ + v ) 2 + ( μ 2 + v 2 ) 2 v 2 τ 1 2 ] β 2 ,
δ n 1 δ n 2 = [ 2 σ 2 ( μ + v ) 2 + 2 μ v ( 1 τ 1 2 ) ] β 2 ,
( δ n 1 δ n 2 ) 2 = 2 ( 1 τ 1 2 ) ( σ α ) 2 .
δ n i 2 [ 2 G 0 + ( G 0 + 1 G 0 ) 2 ] β 2 ,
δ n 1 δ n 2 [ 2 G 0 + ( G 0 1 G 0 ) 2 ] β 2 .
δ n i 2 = [ τ c 2 ( γ 1 2 + γ 2 2 ) + ρ c 2 ( μ 2 + v 2 ) ] β i 2 ,
δ n 1 δ n 2 = [ τ c 2 ( 2 γ 1 γ 2 ) + ρ c 2 ( 2 μ v ) ] β 1 β 2 .
( δ n 1 δ n 2 ) 2 = [ 1 + 2 ρ c 2 ( μ γ 2 v γ 1 ) 2 ] ( τ c α ) 2 .
b 1 ( 2 ) = τ 2 ( μ 2 + ν 2 ) a 1 + τ 2 ( 2 μ v ) a 2 + ρ τ ( μ 2 + v 2 ) a 3
+ ρ τ ( 2 μ v ) a 4 + ( ρ μ ) a 5 + ( ρ v ) a 6 ,
b 2 ( 2 ) = τ 2 ( 2 μ v ) a 1 + τ 2 ( μ 2 + v 2 ) a 2 + ρ τ ( 2 μ v ) a 3
+ ρ τ ( μ 2 + v 2 ) a 4 + ( ρ v ) a 5 + ( ρ μ ) a 6 ,
b 1 ( 3 ) = τ 3 [ μ ( μ 2 + 3 v 2 ) a 1 + v ( 3 μ 2 + v 2 ) a 2 ] + ρ τ 2 [ μ ( μ 2 + 3 v 2 ) a 3 + v ( 3 μ 2 + v 2 ) a 4 ]
+ ρ τ [ ( μ 2 + v 2 ) a 5 + ( 2 μ v ) a 6 ] + ρ ( μ a 7 + v a 8 ) ,
b 2 ( 3 ) = τ 3 [ v ( 3 μ 2 + v 2 ) a 1 + μ ( μ 2 + 3 v 2 ) a 2 ] + ρ τ 2 [ v ( 3 μ 2 + v 2 ) a 3 + μ ( μ 2 + 3 v 2 ) a 4 ]
+ ρ τ [ ( 2 μ v ) a 5 + ( μ 2 + v 2 ) a 6 ] + ρ ( v a 7 + μ a 8 ) ,
b 1 ( n ) = τ n ( p n a 1 + q n a 2 ) + ρ Σ r = 1 n τ n r ( p n r + 1 a 2 r + 1 + q n r + 1 a 2 r + 2 ) ,
b 2 ( n ) = τ n ( q n a 1 + p n a 2 ) + ρ Σ r = 1 n τ n r ( q n r + 1 a 2 r + 1 + p n r + 1 a 2 r + 2 ) ,
p n 2 + q n 2 = [ ( μ + v ) 2 n + 1 ( μ + v ) 2 n ] 2 ,
2 p n q n = [ ( μ + v ) 2 n 1 ( μ + v ) 2 n ] 2 .
β 1 = τ n ( p n α 1 + q n α 2 * ) ,
β 2 = τ n ( q n α 1 * + p n α 2 ) .
δ n i 2 β 2 = τ 2 n ( p n 2 + q n 2 ) + ρ 2 Σ r = 1 n τ 2 ( n r ) ( p n r + 1 2 + q n r + 1 2 ) ,
δ n 1 δ n 2 β 2 = τ 2 n ( 2 p n q n ) + ρ 2 Σ r = 1 n τ 2 ( n r ) ( 2 p n r + 1 q n r + 1 ) ,
δ n i 2 = [ 1 + 1 L 2 n + n ( L 1 ) + ( 1 1 L 2 n ) ( L + 1 ) ] α 2 2 ,
= [ 1 + n ( L 1 ) + ( 1 + 1 L 2 n 1 ) ( L + 1 ) ] α 2 2 ,
δ n 1 δ n 2 = [ 1 1 L 2 n + n ( L 1 ) ( 1 1 L 2 n ) ( L + 1 ) ] α 2 2 ,
= [ 1 + n ( L 1 ) ( 1 + 1 L 2 n 1 ) ( L + 1 ) ] α 2 2 .
( δ n 1 δ n 2 ) 2 β 2 = 2 τ 2 n ( p n q n ) 2 + 2 ρ 2 Σ r = 1 n τ 2 ( n r ) ( p n r + 1 q n r + 1 ) 2 .
( δ n 1 δ n 2 ) 2 = 2 [ 1 L 2 n + ( 1 1 L 2 n ) ( L + 1 ) ] α 2
= 2 ( 1 + 1 L 2 n 1 ) α 2 ( L + 1 ) .
b 1 = τ n [ ( p n τ 1 μ c + q n τ 2 v c ) a 1 + ( p n τ 1 v c + q n τ 2 μ c ) a 2 + ( p n ρ 1 ) a 1
+ ( q n ρ 2 ) a 0 ] + ρ Σ r = 1 n τ n r ( p n r + 1 a 2 r + 1 + q n r + 1 a 2 r + 2 ) .
b 2 = τ n [ ( p n τ 2 v c + q n τ 1 μ c ) a 1 + ( p n τ 2 μ c + q n τ 1 v c ) a 2 + ( q n ρ 1 ) a 1
+ ( p n ρ 2 ) a 0 ] + ρ Σ r = 1 n τ n r ( p n r + 1 a 2 r + 1 + q n r + 1 a 2 r + 2 ) ,
δ n 1 2 β 2 = τ 2 n [ 2 σ 2 ( p n + q n ) 2 + p n 2 + q n 2 2 p n 2 τ 1 2 ] ,
δ n 2 2 β 2 = τ 2 n [ 2 σ 2 ( p n + q n ) 2 + p n 2 + q n 2 2 q n 2 τ 1 2 ] ,
δ n 1 δ n 2 β 2 = τ 2 n [ 2 σ 2 ( p n + q n ) 2 + ( 2 p n q n ) ( 1 τ 1 2 ) ] ,
( δ n 1 δ n 2 ) 2 β 2 = 2 τ 2 n ( p n q n ) 2 ( 1 τ 1 2 ) .
γ 1 2 + γ 2 2 = [ ( μ c + ν c ) 2 ( μ + ν ) 2 n + 1 ( μ c + ν c ) 2 ( μ + ν ) 2 n ] 2 ,
2 γ 1 γ 2 = [ ( μ c + ν c ) 2 ( μ + ν ) 2 n 1 ( μ c + ν c ) 2 ( μ + ν ) 2 n ] 2 .
δ n i 2 β i 2 = τ 2 n [ τ c 2 ( γ 1 2 + γ 2 2 ) + ρ c 2 ( p n 2 + q n 2 ) ] ,
δ n 1 δ n 2 β 1 β 2 = τ 2 n [ τ c 2 ( 2 γ 1 γ 2 ) + ρ c 2 ( 2 p n q n ) ] ,
( δ n 1 δ n 2 ) 2 ( τ n τ c α ) 2 = τ 2 n [ 1 + 2 ρ c 2 ( p n γ 2 q n γ 1 ) 2 ] ,
δ n i 2 β i 2 = [ T c ( G c 0 + 1 G c 0 L 2 n ) + ( 1 T c ) ( 1 + 1 L 2 n ) ] 2 ,
δ n 1 δ n 2 2 β 1 β 2 = [ T c ( G c 0 1 G c 0 L 2 n ) + ( 1 T c ) ( 1 1 L 2 n ) ] 2 ,
F i = ( 2 G 1 ) G i ,
C 12 = 2 [ G ( G 1 ) ] 1 2 ( 2 G 1 ) .
F = ( 2 G 1 ) G θ ,
C 12 = 2 [ G ( G 1 ) ] 1 2 C θ ( 2 G 1 ) G θ ,
F i = 1 T i ,
C 13 = 0 ,
F i = [ T ( 2 G 1 ) + ( 1 T ) ] T G i
C 12 = 2 T [ G ( G 1 ) ] 1 2 T ( 2 G 1 ) + ( 1 T ) .
F = ( 2 G 1 ) T G θ ,
C 12 = 2 [ G ( G 1 ) ] 1 2 C θ ( 2 G 1 ) G θ ,
F 1 = [ T ( 2 H 0 1 ) + ( 1 T ) ( 2 G 2 1 ) ] T H 0 ,
C 12 = 2 T [ H 0 ( H 0 1 ) ] 1 2 + 2 ( 1 T ) [ G 2 ( G 2 1 ) ] 1 2 T ( 2 H 0 1 ) + ( 1 T ) ( 2 G 2 1 ) .
F [ 1 + n ( L 1 ) ] 2 ,
C 12 1 2 ( L + 1 ) [ 1 + n ( L 1 ) ] .
F 2 [ T c G c 0 + ( 1 T c ) + n ( L 1 ) ] T c G c 0 ,
C 12 1 2 ( L + 1 ) [ T c G c 0 + ( 1 T c ) + n ( L 1 ) ] .
m = α 2 + α * v + α v + v v .
m = α 2 + v v .
m 2 α 4 + 2 α 2 v v + [ ( α * ) 2 v 2 + α 2 v v + α 2 v v + α 2 ( v ) 2 ] ,
δ m 2 ( α * ) 2 v 2 + α 2 v v + α 2 v v + α 2 ( v ) 2 .
m 1 m 2 α 1 α 2 2 + α 1 2 v 2 v 2 + α 2 2 v 1 v 1
+ α 1 * α 2 * v 1 v 2 + α 1 * α 2 v 1 v 2 + α 1 α 2 * v 1 v 2 + α 1 α 2 v 1 v 2 .
δ m 1 δ m 2 α 1 * α 2 * v 1 v 2 + α 1 * α 2 v 1 v 2 + α 1 α 2 * v 1 v 2 + α 1 α 2 v 1 v 2 .
w 1 2 = μ 2 v 1 2 + 2 μ ν v 1 v 2 + ν 2 ( v 2 ) 2 ,
w 1 w 1 = μ 2 v 1 v 1 + μ * ν v 1 v 2 + μ ν * v 1 v 2 + ν 2 v 2 v 2 ,
w 1 w 2 = μ 2 v 1 v 2 + μ ν v 1 v 1 + μ ν v 2 v 2 + ν 2 v 1 v 2 ,
w 1 w 2 = μ 2 v 1 v 2 + μ ν * v 1 2 + μ * ν ( v 2 ) 2 + ν 2 v 1 v 2 ,
w i w i = ν 2 , w 1 w 2 = μ ν .
δ n i 2 = β i 2 ( μ 2 + ν 2 ) ,
δ n 1 δ n 2 = β 1 * β 2 * μ ν + β 1 β 2 μ * ν * ,
μ ¯ 2 = μ 2 μ 1 2 + 2 Re ( μ 2 ν 2 * μ 1 ν 1 ) + ν 2 ν 1 2 ,
ν ¯ 2 = μ 2 ν 1 2 + 2 Re ( μ 2 ν 2 * μ 1 ν 1 ) + ν 2 μ 1 2 ,
w i w i = μ 2 2 v 1 2 + 2 Re ( μ 2 v 2 * μ 1 v 1 ) + v 2 2 μ 1 2 ,
w 1 2 = τ 2 v 1 2 + 2 τ ρ v 1 v 3 + ρ 2 v 3 2 ,
w 1 w 1 = τ 2 v 1 v 1 + τ * ρ v 1 v 3 + τ ρ * v 1 v 3 + ρ 2 v 3 v 3 ,
w 1 w 3 = τ ρ * v 1 2 + ( τ 2 ρ 2 ) v 1 v 3 + τ * ρ v 3 2 ,
w 1 w 3 = τ ρ v 1 v 1 + τ 2 v 1 v 3 ρ 2 v 1 v 3 + τ ρ v 3 v 3 ,
δ n i 2 = β i 2 ,
δ n 1 δ n 3 = 0 ,
w 1 w 1 = τ v 2 , w 3 w 3 = ρ v 2 , w 1 w 3 = τ ρ v 2 .
δ n 1 2 = ( τ μ 2 + τ v 2 + ρ 2 ) τ μ α 1 2 ,
δ n 3 2 = ( ρμ 2 + ρv 2 + τ 2 ) ρμ α 1 2 ,
δ n 1 δ n 3 = 2 τ ρ μ v α 1 2 ,
( δ n 1 + δ n 3 ) 2 = ( μ 2 + ν 2 ) μ α 1 2 .
b 1 = ( τ μ ) a 1 + ( ρ ν ) a 1 + ( ρ μ ) a 2 + ( τ ν ) a 2 ,
b 2 = ( ρ * μ ) a 1 + ( τ * ν ) a 1 + ( τ * μ ) a 2 ( ρ * ν ) a 2 .
b 1 = μ a + + ν a + ,
b 2 = μ a + ν a ,
w i 2 = ± μ ν , w i w i = ν 2 ,
δ n i 2 = ± β * μ + β ν * 2 ,
δ n 1 δ n 2 = 0 .
δ n i 2 nn = 2 Σ k μ i k ν i k 2 + Σ k Σ l > k μ i k ν i l * + μ i l ν i k * 2 ,
δ n i δ n j nn = 2 Σ k ( μ i k ν i k * ) ( μ j k * ν j k ) + Σ k Σ l > k ( μ i k ν i l * + μ i l ν i k * ) ( μ j k * ν j l + μ j l * ν j k ) ,
δ n i 2 = ( Σ k μ i k 2 ) ( Σ l ν i l 2 ) ,
δ n i δ n j = ( Σ k μ i k μ j k * ) ( Σ l ν i l * ν j l ) ,
δ n i δ n j = ( Σ k μ i k ν j k ) ( Σ l μ j l * ν i l * ) ,
δ n i 2 = μ ν 2 = δ n 1 δ n 2 .
δ n i 2 = 0 = δ n 1 δ n 3 .
δ n 1 2 = ( μ 11 2 + μ 13 2 ) ( ν 12 2 + ν 14 2 ) ,
δ n 2 2 = ( μ 22 2 + μ 24 2 ) ( ν 21 2 + ν 23 2 ) ,
δ n 1 δ n 2 = ( μ 11 ν 21 + μ 13 ν 23 ) ( μ 22 * ν 12 * + μ 24 * ν 14 * ) ,
δ n 1 δ n 3 = ( μ 11 μ 31 * + μ 13 μ 33 * ) ( ν 12 * ν 32 + ν 14 * ν 34 ) .
δ n 1 2 = [ T 1 G + ( 1 T 1 ) ] T 1 ( G 1 ) ,
δ n 2 2 = [ T 2 G + ( 1 T 2 ) ] T 2 ( G 1 ) ,
δ n 1 δ n 2 = T 1 T 2 G ( G 1 ) ,
δ n i 2 = G ( G 1 ) = δ n 1 δ n 2 .
δ n 1 2 = [ T H 0 + ( 1 T ) G 2 ] [ T ( H 0 1 ) + ( 1 T ) ( G 2 1 ) ] ,
δ n 2 2 = [ T H 0 + ( 1 T ) G 2 ] [ T ( H 0 1 ) + ( 1 T ) ( G 2 1 ) ] ,
δ n 1 δ n 2 = { T [ H 0 ( H 0 1 ) ] 1 2 + ( 1 T ) [ G 2 ( G 2 1 ) ] 1 2 } 2 .
b 1 = c 1 o + c 1 e , b 2 = c 2 o + c 2 e ,
n 1 2 = c 1 o c 1 e c 1 o c 1 e + c 1 o c 1 e c 1 e c 1 e + c 1 e c 1 e c 1 o c 1 e + c 1 e c 1 e c 1 e c 1 e
= c 1 o c 1 o c 1 e c 1 e + c 1 e c 1 e 2 .
δ n 1 2 = c 1 o c 1 o c 1 e c 1 e .
δ n 2 2 = c 2 o c 2 o c 2 e c 2 e .
n 1 n 2 = c 1 e c 1 o c 2 e c 2 o + c 1 e c 1 o c 2 o c 2 o + c 1 e c 1 e c 2 e c 2 o + c 1 e c 1 e c 2 o c 2 o
= c 1 o c 2 o c 1 e c 2 e + c 1 e c 1 e c 2 o c 2 o ,
δ n 1 δ n 2 = c 1 o c 2 o c 1 e c 2 e .
n 1 n 3 = c 1 e c 1 o c 3 o c 3 e + c 1 e c 1 o c 3 e c 3 e + c 1 e c 1 e c 3 o c 3 e + c 1 e c 1 e c 3 e c 3 e
= c 1 o c 3 o c 1 e c 3 e + c 1 e c 1 e c 3 e c 3 e .
δ n 1 δ n 3 = c 1 o c 3 o c 1 e c 3 e .

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