Abstract

In a parametric amplifier (PA) driven by two pump waves the signal sideband is coupled to three idler sidebands, all of which are frequency-converted (FC) images of the signal, and two of which are phase-conjugated (PC) images of the signal. If such a device is to be useful, the signal must be amplified, and the PC and FC idlers must be produced, with minimal noise. In this paper the quantum noise properties of two-sideband (TS) parametric devices are reviewed and the properties of many-sideband devices are determined. These results are applied to the study of two-pump PAs, which are based on the aforementioned four-sideband (FS) interaction. As a general guideline, the more sidebands that interact, the higher are the noise levels. However, if the pump frequencies are tuned to maximize the frequency bandwidth of the FS interaction, the signal and idler noise-figures are only slightly higher than the noise figures associated with the limiting TS interactions.

© 2004 Optical Society of America

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  1. K. O. Hill, D. C. Johnson, B. S. Kawasaki, and R. I. MacDonald, “CW three-wave mixing in single-mode optical fibers,” J. Appl. Phys. 49, 5098–5106 (1978).
    [Crossref]
  2. R. H. Stolen and J. E. Bjorkholm, “Parametric amplification and frequency conversion in optical fibers,” IEEE J. Quantum Electron. 18, 1062–1072 (1982).
    [Crossref]
  3. C. J. McKinstrie, S. Radic, and A. R. Chraplyvy, “Parametric amplifiers driven by two pump waves,” IEEE J. Sel. Top. Quantum Electron. 8, 538–547 and 956 (2002), and references therein.
    [Crossref]
  4. C. J. McKinstrie, S. Radic, and C. Xie, “Parametric instabilities driven by orthogonal pump waves in birefringent fibers,” Opt. Express 11, 2619–2633 (2003) and references therein.
    [Crossref] [PubMed]
  5. 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) and refrences therein.
    [Crossref] [PubMed]
  6. S. Radic, C. J. McKinstrie, R. M. Jopson, Q. Lin, and G. P. Agrawal, “Record performance of a parametric amplifier constructed with highly-nonlinear fiber,” Electron. Lett. 39, 838–839 (2003).
    [Crossref]
  7. T. Tanemura, C. S. Goh, K. Kikuchi, and S. Y. Set, “Highly efficient arbitrary wavelength conversion within entire C-band based on nondegenerate fiber four-wave mixing,” IEEE Photon. Tech. Lett. 16, 551–553 (2004).
    [Crossref]
  8. 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]
  9. M. T. Weiss, “Quantum derivation of energy relations analogous to those for nonlinear reactances,” Proc. IRE 45, 1012–1013 (1957).
  10. W. H. Louisell, A. Yariv, and A. E. Siegman, “Quantum fluctuations and noise in parametric processes I,” Phys. Rev. 124, 1646–1654 (1961).
    [Crossref]
  11. J. P. Gordon, W. H. Louisell, and L. R. Walker, “Quantum fluctuations and noise in parametric processes II,” Phys. Rev. 129, 481–485 (1963).
    [Crossref]
  12. W. H. Louisell, Radiation and Noise in Quantum Electronics (McGraw-Hill, New York, 1964).
  13. B. R. Mollow and R. J. Glauber, “Quantum theory of parametric amplification I,” Phys. Rev. 160, 1076–1096 (1967).
    [Crossref]
  14. B. R. Mollow and R. J. Glauber, “Quantum theory of parametric amplification II,” Phys. Rev. 160, 1097–1108 (1967).
    [Crossref]
  15. J. Mostowski and M. G. Raymer, “Quantum statistics in nonlinear optics,” in Contemporary Nonlinear Optics, edited by G. P. Agrawal and R. W. Boyd (Academic Press, San Diego, 1992), and references therein.
  16. R. Loudon, The Quantum Theory of Light, 3rd Ed. (Oxford University Press, Oxford, 2000).
  17. L. D. Landau, E. M. Lifshitz, and L. P. Pitaevskii, Electrodynamics of Continuous Media, 2nd Ed. (Butterworth-Heinemann, Oxford, 1984).
  18. T. H. Stix, Waves in Plasmas (American Institute of Physics, New York, 1992).
  19. R. J. Glauber, “Coherent and incoherent states of the radiation field,” Phys. Rev. 131, 2766–2788 (1963).
    [Crossref]
  20. F. T. Arecchi, “Measurement of the statistical distribution of Gaussian and laser sources,” Phys. Rev. Lett. 15, 912–916 (1965).
    [Crossref]
  21. C. Freed and H. A. Haus, “Photoelectron statistics produced by a laser operating below and above the threshold of oscillation,” IEEE J. Quantum Electron. 2, 190–195 (1966).
    [Crossref]
  22. W. Schottky, “On spontaneous current fluctuations in different electrical conductors,” Ann. Phys. (Leipzig) 57, 541–567 (1918).
  23. S. Radic and C. J. McKinstrie, “Two-pump fiber parametric amplifiers,” Opt. Fiber Technol. 9, 7–23 (2003).
    [Crossref]
  24. S. Radic, C. J. McKinstrie, R. M. Jopson, A. H. Gnauck, J. C. Centanni, and A. R. Chraplyvy, “Multi-band bit-level switching in two-pump fiber parametric devices,” IEEE Photon. Technol. Lett. 16, 852–854 (2004).
    [Crossref]
  25. S. Radic, C. J. McKinstrie, A. R. Chraplyvy, G. Raybon, J. C. Centanni, C. G. Jorgensen, K. Brar, and C. Headley, “Continuous-wave parametric-gain synthesis using nondegenerate-pump four-wave mixing,” IEEE Photon. Tech-nol. Lett. 14, 1406–1408 (2002).
    [Crossref]
  26. P. C. Becker, N. A. Olssen, and J. R. Simpson, Erbium-Doped Fiber Amplifiers (Academic Press, San Diego, 1999).
  27. K. Rottwitt and A. J. Stentz, “Raman amplification in lightwave communication systems,” in Optical Fiber Telecommunications IVA, edited by I. Kaminow and T. Li (Academic Press, San Diego, 2002), pp. 213–257.
    [Crossref]
  28. A. H. Nayfeh, Introduction to Perturbation Techniques (Wiley, New York, 1981).

2004 (3)

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) and refrences therein.
[Crossref] [PubMed]

T. Tanemura, C. S. Goh, K. Kikuchi, and S. Y. Set, “Highly efficient arbitrary wavelength conversion within entire C-band based on nondegenerate fiber four-wave mixing,” IEEE Photon. Tech. Lett. 16, 551–553 (2004).
[Crossref]

S. Radic, C. J. McKinstrie, R. M. Jopson, A. H. Gnauck, J. C. Centanni, and A. R. Chraplyvy, “Multi-band bit-level switching in two-pump fiber parametric devices,” IEEE Photon. Technol. Lett. 16, 852–854 (2004).
[Crossref]

2003 (3)

S. Radic and C. J. McKinstrie, “Two-pump fiber parametric amplifiers,” Opt. Fiber Technol. 9, 7–23 (2003).
[Crossref]

S. Radic, C. J. McKinstrie, R. M. Jopson, Q. Lin, and G. P. Agrawal, “Record performance of a parametric amplifier constructed with highly-nonlinear fiber,” Electron. Lett. 39, 838–839 (2003).
[Crossref]

C. J. McKinstrie, S. Radic, and C. Xie, “Parametric instabilities driven by orthogonal pump waves in birefringent fibers,” Opt. Express 11, 2619–2633 (2003) and references therein.
[Crossref] [PubMed]

2002 (2)

C. J. McKinstrie, S. Radic, and A. R. Chraplyvy, “Parametric amplifiers driven by two pump waves,” IEEE J. Sel. Top. Quantum Electron. 8, 538–547 and 956 (2002), and references therein.
[Crossref]

S. Radic, C. J. McKinstrie, A. R. Chraplyvy, G. Raybon, J. C. Centanni, C. G. Jorgensen, K. Brar, and C. Headley, “Continuous-wave parametric-gain synthesis using nondegenerate-pump four-wave mixing,” IEEE Photon. Tech-nol. Lett. 14, 1406–1408 (2002).
[Crossref]

1982 (1)

R. H. Stolen and J. E. Bjorkholm, “Parametric amplification and frequency conversion in optical fibers,” IEEE J. Quantum Electron. 18, 1062–1072 (1982).
[Crossref]

1978 (1)

K. O. Hill, D. C. Johnson, B. S. Kawasaki, and R. I. MacDonald, “CW three-wave mixing in single-mode optical fibers,” J. Appl. Phys. 49, 5098–5106 (1978).
[Crossref]

1967 (2)

B. R. Mollow and R. J. Glauber, “Quantum theory of parametric amplification I,” Phys. Rev. 160, 1076–1096 (1967).
[Crossref]

B. R. Mollow and R. J. Glauber, “Quantum theory of parametric amplification II,” Phys. Rev. 160, 1097–1108 (1967).
[Crossref]

1966 (1)

C. Freed and H. A. Haus, “Photoelectron statistics produced by a laser operating below and above the threshold of oscillation,” IEEE J. Quantum Electron. 2, 190–195 (1966).
[Crossref]

1965 (1)

F. T. Arecchi, “Measurement of the statistical distribution of Gaussian and laser sources,” Phys. Rev. Lett. 15, 912–916 (1965).
[Crossref]

1963 (2)

J. P. Gordon, W. H. Louisell, and L. R. Walker, “Quantum fluctuations and noise in parametric processes II,” Phys. Rev. 129, 481–485 (1963).
[Crossref]

R. J. Glauber, “Coherent and incoherent states of the radiation field,” Phys. Rev. 131, 2766–2788 (1963).
[Crossref]

1961 (1)

W. H. Louisell, A. Yariv, and A. E. Siegman, “Quantum fluctuations and noise in parametric processes I,” Phys. Rev. 124, 1646–1654 (1961).
[Crossref]

1957 (1)

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

1956 (1)

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]

1918 (1)

W. Schottky, “On spontaneous current fluctuations in different electrical conductors,” Ann. Phys. (Leipzig) 57, 541–567 (1918).

Agrawal, G. P.

S. Radic, C. J. McKinstrie, R. M. Jopson, Q. Lin, and G. P. Agrawal, “Record performance of a parametric amplifier constructed with highly-nonlinear fiber,” Electron. Lett. 39, 838–839 (2003).
[Crossref]

Arecchi, F. T.

F. T. Arecchi, “Measurement of the statistical distribution of Gaussian and laser sources,” Phys. Rev. Lett. 15, 912–916 (1965).
[Crossref]

Becker, P. C.

P. C. Becker, N. A. Olssen, and J. R. Simpson, Erbium-Doped Fiber Amplifiers (Academic Press, San Diego, 1999).

Bjorkholm, J. E.

R. H. Stolen and J. E. Bjorkholm, “Parametric amplification and frequency conversion in optical fibers,” IEEE J. Quantum Electron. 18, 1062–1072 (1982).
[Crossref]

Brar, K.

S. Radic, C. J. McKinstrie, A. R. Chraplyvy, G. Raybon, J. C. Centanni, C. G. Jorgensen, K. Brar, and C. Headley, “Continuous-wave parametric-gain synthesis using nondegenerate-pump four-wave mixing,” IEEE Photon. Tech-nol. Lett. 14, 1406–1408 (2002).
[Crossref]

Centanni, J. C.

S. Radic, C. J. McKinstrie, R. M. Jopson, A. H. Gnauck, J. C. Centanni, and A. R. Chraplyvy, “Multi-band bit-level switching in two-pump fiber parametric devices,” IEEE Photon. Technol. Lett. 16, 852–854 (2004).
[Crossref]

S. Radic, C. J. McKinstrie, A. R. Chraplyvy, G. Raybon, J. C. Centanni, C. G. Jorgensen, K. Brar, and C. Headley, “Continuous-wave parametric-gain synthesis using nondegenerate-pump four-wave mixing,” IEEE Photon. Tech-nol. Lett. 14, 1406–1408 (2002).
[Crossref]

Chraplyvy, A. R.

S. Radic, C. J. McKinstrie, R. M. Jopson, A. H. Gnauck, J. C. Centanni, and A. R. Chraplyvy, “Multi-band bit-level switching in two-pump fiber parametric devices,” IEEE Photon. Technol. Lett. 16, 852–854 (2004).
[Crossref]

S. Radic, C. J. McKinstrie, A. R. Chraplyvy, G. Raybon, J. C. Centanni, C. G. Jorgensen, K. Brar, and C. Headley, “Continuous-wave parametric-gain synthesis using nondegenerate-pump four-wave mixing,” IEEE Photon. Tech-nol. Lett. 14, 1406–1408 (2002).
[Crossref]

C. J. McKinstrie, S. Radic, and A. R. Chraplyvy, “Parametric amplifiers driven by two pump waves,” IEEE J. Sel. Top. Quantum Electron. 8, 538–547 and 956 (2002), and references therein.
[Crossref]

Freed, C.

C. Freed and H. A. Haus, “Photoelectron statistics produced by a laser operating below and above the threshold of oscillation,” IEEE J. Quantum Electron. 2, 190–195 (1966).
[Crossref]

Glauber, R. J.

B. R. Mollow and R. J. Glauber, “Quantum theory of parametric amplification I,” Phys. Rev. 160, 1076–1096 (1967).
[Crossref]

B. R. Mollow and R. J. Glauber, “Quantum theory of parametric amplification II,” Phys. Rev. 160, 1097–1108 (1967).
[Crossref]

R. J. Glauber, “Coherent and incoherent states of the radiation field,” Phys. Rev. 131, 2766–2788 (1963).
[Crossref]

Gnauck, A. H.

S. Radic, C. J. McKinstrie, R. M. Jopson, A. H. Gnauck, J. C. Centanni, and A. R. Chraplyvy, “Multi-band bit-level switching in two-pump fiber parametric devices,” IEEE Photon. Technol. Lett. 16, 852–854 (2004).
[Crossref]

Goh, C. S.

T. Tanemura, C. S. Goh, K. Kikuchi, and S. Y. Set, “Highly efficient arbitrary wavelength conversion within entire C-band based on nondegenerate fiber four-wave mixing,” IEEE Photon. Tech. Lett. 16, 551–553 (2004).
[Crossref]

Gordon, J. P.

J. P. Gordon, W. H. Louisell, and L. R. Walker, “Quantum fluctuations and noise in parametric processes II,” Phys. Rev. 129, 481–485 (1963).
[Crossref]

Haus, H. A.

C. Freed and H. A. Haus, “Photoelectron statistics produced by a laser operating below and above the threshold of oscillation,” IEEE J. Quantum Electron. 2, 190–195 (1966).
[Crossref]

Headley, C.

S. Radic, C. J. McKinstrie, A. R. Chraplyvy, G. Raybon, J. C. Centanni, C. G. Jorgensen, K. Brar, and C. Headley, “Continuous-wave parametric-gain synthesis using nondegenerate-pump four-wave mixing,” IEEE Photon. Tech-nol. Lett. 14, 1406–1408 (2002).
[Crossref]

Hill, K. O.

K. O. Hill, D. C. Johnson, B. S. Kawasaki, and R. I. MacDonald, “CW three-wave mixing in single-mode optical fibers,” J. Appl. Phys. 49, 5098–5106 (1978).
[Crossref]

Johnson, D. C.

K. O. Hill, D. C. Johnson, B. S. Kawasaki, and R. I. MacDonald, “CW three-wave mixing in single-mode optical fibers,” J. Appl. Phys. 49, 5098–5106 (1978).
[Crossref]

Jopson, R. M.

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) and refrences therein.
[Crossref] [PubMed]

S. Radic, C. J. McKinstrie, R. M. Jopson, A. H. Gnauck, J. C. Centanni, and A. R. Chraplyvy, “Multi-band bit-level switching in two-pump fiber parametric devices,” IEEE Photon. Technol. Lett. 16, 852–854 (2004).
[Crossref]

S. Radic, C. J. McKinstrie, R. M. Jopson, Q. Lin, and G. P. Agrawal, “Record performance of a parametric amplifier constructed with highly-nonlinear fiber,” Electron. Lett. 39, 838–839 (2003).
[Crossref]

Jorgensen, C. G.

S. Radic, C. J. McKinstrie, A. R. Chraplyvy, G. Raybon, J. C. Centanni, C. G. Jorgensen, K. Brar, and C. Headley, “Continuous-wave parametric-gain synthesis using nondegenerate-pump four-wave mixing,” IEEE Photon. Tech-nol. Lett. 14, 1406–1408 (2002).
[Crossref]

Kanaev, A. V.

Kawasaki, B. S.

K. O. Hill, D. C. Johnson, B. S. Kawasaki, and R. I. MacDonald, “CW three-wave mixing in single-mode optical fibers,” J. Appl. Phys. 49, 5098–5106 (1978).
[Crossref]

Kikuchi, K.

T. Tanemura, C. S. Goh, K. Kikuchi, and S. Y. Set, “Highly efficient arbitrary wavelength conversion within entire C-band based on nondegenerate fiber four-wave mixing,” IEEE Photon. Tech. Lett. 16, 551–553 (2004).
[Crossref]

Kogelnik, H.

Landau, L. D.

L. D. Landau, E. M. Lifshitz, and L. P. Pitaevskii, Electrodynamics of Continuous Media, 2nd Ed. (Butterworth-Heinemann, Oxford, 1984).

Lifshitz, E. M.

L. D. Landau, E. M. Lifshitz, and L. P. Pitaevskii, Electrodynamics of Continuous Media, 2nd Ed. (Butterworth-Heinemann, Oxford, 1984).

Lin, Q.

S. Radic, C. J. McKinstrie, R. M. Jopson, Q. Lin, and G. P. Agrawal, “Record performance of a parametric amplifier constructed with highly-nonlinear fiber,” Electron. Lett. 39, 838–839 (2003).
[Crossref]

Loudon, R.

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

Louisell, W. H.

J. P. Gordon, W. H. Louisell, and L. R. Walker, “Quantum fluctuations and noise in parametric processes II,” Phys. Rev. 129, 481–485 (1963).
[Crossref]

W. H. Louisell, A. Yariv, and A. E. Siegman, “Quantum fluctuations and noise in parametric processes I,” Phys. Rev. 124, 1646–1654 (1961).
[Crossref]

W. H. Louisell, Radiation and Noise in Quantum Electronics (McGraw-Hill, New York, 1964).

MacDonald, R. I.

K. O. Hill, D. C. Johnson, B. S. Kawasaki, and R. I. MacDonald, “CW three-wave mixing in single-mode optical fibers,” J. Appl. Phys. 49, 5098–5106 (1978).
[Crossref]

Manley, J. M.

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]

McKinstrie, C. J.

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) and refrences therein.
[Crossref] [PubMed]

S. Radic, C. J. McKinstrie, R. M. Jopson, A. H. Gnauck, J. C. Centanni, and A. R. Chraplyvy, “Multi-band bit-level switching in two-pump fiber parametric devices,” IEEE Photon. Technol. Lett. 16, 852–854 (2004).
[Crossref]

S. Radic and C. J. McKinstrie, “Two-pump fiber parametric amplifiers,” Opt. Fiber Technol. 9, 7–23 (2003).
[Crossref]

C. J. McKinstrie, S. Radic, and C. Xie, “Parametric instabilities driven by orthogonal pump waves in birefringent fibers,” Opt. Express 11, 2619–2633 (2003) and references therein.
[Crossref] [PubMed]

S. Radic, C. J. McKinstrie, R. M. Jopson, Q. Lin, and G. P. Agrawal, “Record performance of a parametric amplifier constructed with highly-nonlinear fiber,” Electron. Lett. 39, 838–839 (2003).
[Crossref]

C. J. McKinstrie, S. Radic, and A. R. Chraplyvy, “Parametric amplifiers driven by two pump waves,” IEEE J. Sel. Top. Quantum Electron. 8, 538–547 and 956 (2002), and references therein.
[Crossref]

S. Radic, C. J. McKinstrie, A. R. Chraplyvy, G. Raybon, J. C. Centanni, C. G. Jorgensen, K. Brar, and C. Headley, “Continuous-wave parametric-gain synthesis using nondegenerate-pump four-wave mixing,” IEEE Photon. Tech-nol. Lett. 14, 1406–1408 (2002).
[Crossref]

Mollow, B. R.

B. R. Mollow and R. J. Glauber, “Quantum theory of parametric amplification I,” Phys. Rev. 160, 1076–1096 (1967).
[Crossref]

B. R. Mollow and R. J. Glauber, “Quantum theory of parametric amplification II,” Phys. Rev. 160, 1097–1108 (1967).
[Crossref]

Mostowski, J.

J. Mostowski and M. G. Raymer, “Quantum statistics in nonlinear optics,” in Contemporary Nonlinear Optics, edited by G. P. Agrawal and R. W. Boyd (Academic Press, San Diego, 1992), and references therein.

Nayfeh, A. H.

A. H. Nayfeh, Introduction to Perturbation Techniques (Wiley, New York, 1981).

Olssen, N. A.

P. C. Becker, N. A. Olssen, and J. R. Simpson, Erbium-Doped Fiber Amplifiers (Academic Press, San Diego, 1999).

Pitaevskii, L. P.

L. D. Landau, E. M. Lifshitz, and L. P. Pitaevskii, Electrodynamics of Continuous Media, 2nd Ed. (Butterworth-Heinemann, Oxford, 1984).

Radic, S.

S. Radic, C. J. McKinstrie, R. M. Jopson, A. H. Gnauck, J. C. Centanni, and A. R. Chraplyvy, “Multi-band bit-level switching in two-pump fiber parametric devices,” IEEE Photon. Technol. Lett. 16, 852–854 (2004).
[Crossref]

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) and refrences therein.
[Crossref] [PubMed]

C. J. McKinstrie, S. Radic, and C. Xie, “Parametric instabilities driven by orthogonal pump waves in birefringent fibers,” Opt. Express 11, 2619–2633 (2003) and references therein.
[Crossref] [PubMed]

S. Radic and C. J. McKinstrie, “Two-pump fiber parametric amplifiers,” Opt. Fiber Technol. 9, 7–23 (2003).
[Crossref]

S. Radic, C. J. McKinstrie, R. M. Jopson, Q. Lin, and G. P. Agrawal, “Record performance of a parametric amplifier constructed with highly-nonlinear fiber,” Electron. Lett. 39, 838–839 (2003).
[Crossref]

C. J. McKinstrie, S. Radic, and A. R. Chraplyvy, “Parametric amplifiers driven by two pump waves,” IEEE J. Sel. Top. Quantum Electron. 8, 538–547 and 956 (2002), and references therein.
[Crossref]

S. Radic, C. J. McKinstrie, A. R. Chraplyvy, G. Raybon, J. C. Centanni, C. G. Jorgensen, K. Brar, and C. Headley, “Continuous-wave parametric-gain synthesis using nondegenerate-pump four-wave mixing,” IEEE Photon. Tech-nol. Lett. 14, 1406–1408 (2002).
[Crossref]

Raybon, G.

S. Radic, C. J. McKinstrie, A. R. Chraplyvy, G. Raybon, J. C. Centanni, C. G. Jorgensen, K. Brar, and C. Headley, “Continuous-wave parametric-gain synthesis using nondegenerate-pump four-wave mixing,” IEEE Photon. Tech-nol. Lett. 14, 1406–1408 (2002).
[Crossref]

Raymer, M. G.

J. Mostowski and M. G. Raymer, “Quantum statistics in nonlinear optics,” in Contemporary Nonlinear Optics, edited by G. P. Agrawal and R. W. Boyd (Academic Press, San Diego, 1992), and references therein.

Rottwitt, K.

K. Rottwitt and A. J. Stentz, “Raman amplification in lightwave communication systems,” in Optical Fiber Telecommunications IVA, edited by I. Kaminow and T. Li (Academic Press, San Diego, 2002), pp. 213–257.
[Crossref]

Rowe, H. E.

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]

Schottky, W.

W. Schottky, “On spontaneous current fluctuations in different electrical conductors,” Ann. Phys. (Leipzig) 57, 541–567 (1918).

Set, S. Y.

T. Tanemura, C. S. Goh, K. Kikuchi, and S. Y. Set, “Highly efficient arbitrary wavelength conversion within entire C-band based on nondegenerate fiber four-wave mixing,” IEEE Photon. Tech. Lett. 16, 551–553 (2004).
[Crossref]

Siegman, A. E.

W. H. Louisell, A. Yariv, and A. E. Siegman, “Quantum fluctuations and noise in parametric processes I,” Phys. Rev. 124, 1646–1654 (1961).
[Crossref]

Simpson, J. R.

P. C. Becker, N. A. Olssen, and J. R. Simpson, Erbium-Doped Fiber Amplifiers (Academic Press, San Diego, 1999).

Stentz, A. J.

K. Rottwitt and A. J. Stentz, “Raman amplification in lightwave communication systems,” in Optical Fiber Telecommunications IVA, edited by I. Kaminow and T. Li (Academic Press, San Diego, 2002), pp. 213–257.
[Crossref]

Stix, T. H.

T. H. Stix, Waves in Plasmas (American Institute of Physics, New York, 1992).

Stolen, R. H.

R. H. Stolen and J. E. Bjorkholm, “Parametric amplification and frequency conversion in optical fibers,” IEEE J. Quantum Electron. 18, 1062–1072 (1982).
[Crossref]

Tanemura, T.

T. Tanemura, C. S. Goh, K. Kikuchi, and S. Y. Set, “Highly efficient arbitrary wavelength conversion within entire C-band based on nondegenerate fiber four-wave mixing,” IEEE Photon. Tech. Lett. 16, 551–553 (2004).
[Crossref]

Walker, L. R.

J. P. Gordon, W. H. Louisell, and L. R. Walker, “Quantum fluctuations and noise in parametric processes II,” Phys. Rev. 129, 481–485 (1963).
[Crossref]

Weiss, M. T.

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

Xie, C.

Yariv, A.

W. H. Louisell, A. Yariv, and A. E. Siegman, “Quantum fluctuations and noise in parametric processes I,” Phys. Rev. 124, 1646–1654 (1961).
[Crossref]

Ann. Phys. (Leipzig) (1)

W. Schottky, “On spontaneous current fluctuations in different electrical conductors,” Ann. Phys. (Leipzig) 57, 541–567 (1918).

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Figures (10)

Fig. 1.
Fig. 1.

Illustration of the constituent FWM processes.

Fig. 2.
Fig. 2.

Gains and noise figures of a two-mode parametric amplifier. The solid curves represent the signal, whereas the dashed curves represent the idler. Distance is normalized to the gain length [γε(P 1 P 2)1/2]-1.

Fig. 3.
Fig. 3.

Transmissions and noise figures of a two-mode frequency converter. The solid curves represent the signal, whereas the dashed curves represent the idler. Distance is normalized to the interaction length [γϕ(P 1 P 2)1/2]-1.

Fig. 4.
Fig. 4.

Gains and noise figures of a four-mode process with quadratic gain, for co-polarized pumps (ε = 2). The solid and long-dashed curves represent the 1- signal and 1+ idler, respectively, and the medium-dashed curves represent the 2- and 2+ idlers. Distance is normalized to the characteristic length (γP)-1.

Fig. 5.
Fig. 5.

Gains and noise figures of a four-mode process with quadratic gain, for cross-polarized pumps in a fiber with constant birefringence (ε = 2/3). The solid and long-dashed curves represent the 1- signal and 1+ idler, respectively, and the medium-dashed curves represent the 2- and 2+ idlers. Distance is normalized to the characteristic length (γP)-1.

Fig. 6.
Fig. 6.

Gains and noise figures of a four-mode process with quadratic gain, for cross-polarized pumps in a fiber with random birefringence (ε = 1). The solid curve represents the 1- signal, and the long-dashed curves represent the 1+, 2- and 2+ idlers. Distance is normalized to the characteristic length (γP)-1.

Fig. 7.
Fig. 7.

Gains and noise figures of a four-mode process with exponential gain, for cross-polarized pumps in a fiber with random birefringence (ε = 1). The solid curves represent the 1- signal, the long-dashed curves represent the 1+ and 2+ idlers, and the medium-dashed curves represent the 2- idler. Distance is normalized to the gain length (2γP)-1.

Fig. 8.
Fig. 8.

Gains and noise figures of a four-mode process driven by cross-polarized pumps in a fiber with random birefringence (ε= 1). The solid curves represent the 1- signal, and the long-, medium- and short-dashed curves represent the 1+, 2- and 2+ idlers, respectively. Frequencies are measured relative to the zero-dispersion frequency of the fiber. The vertical lines denote the pump frequencies.

Fig. 9.
Fig. 9.

Gains and noise figures of a four-mode process driven by cross-polarized pumps in a fiber with random birefringence (ε= 1). The long-dashed curves represent the 1+ signal, and the solid, medium- and short-dashed curves represent the 1-, 2- and 2+ idlers, respectively. Frequencies are measured relative to the zero-dispersion frequency of the fiber. The vertical lines denote the pump frequencies.

Fig. 10.
Fig. 10.

Transmissions and noise figures of a four-mode process driven by cross-polarized pumps in a fiber with random birefringence (ε = 1), plotted as functions of the pump frequency ω 1. The long-dashed curves represent the 1+ signal, and the solid, medium-and short-dashed curves represent the 1-, 2- and 2+ idlers, respectively. Frequencies are measured relative to the zero-dispersion frequency of the fiber.

Equations (202)

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i z A 1 = β 1 ( i t ) A 1 + γ ( A 1 2 + ε A 2 2 ) A 1 ,
i z A 2 = β 2 ( i t ) A 2 + γ ( ε A 1 2 + A 2 2 ) A 2 ,
A 1 ( z ) = P 1 1 2 exp [ i ϕ 1 ( z ) ] ,
A 2 ( z ) = P 1 1 2 exp [ i ϕ 2 ( z ) ] ,
A 1 z t = [ P 1 1 2 + B 1 + ( z ) exp ( iωt ) + B 1 ( z ) exp ( iωt ) ] exp [ i ϕ 1 ( z ) ] ,
A 2 z t = [ P 2 1 2 + B 2 + ( z ) exp ( iωt ) + B 2 ( z ) exp ( iωt ) ] exp [ i ϕ 2 ( z ) ] ,
d z B 1 * = i ( β 1 + γ P 1 ) B 1 * P 1 B 1 +
iγε ( P 1 P 2 ) 1 2 B 2 * iγε ( P 1 P 2 ) 1 2 B 2 + ,
d z B 1 + = P 1 B 1 * + i ( β 1 + + γ P 1 ) B 1 +
+ iγε ( P 1 P 2 ) 1 2 B 2 * + iγε ( P 1 P 2 ) 1 2 B 2 + ,
d z B 2 * = ε ( P 1 P 2 ) 1 2 B 1 * ε ( P 1 P 2 ) 1 2 B 1 +
i ( β 2 + γ P 2 ) B 2 * P 2 B 2 + ,
d z B 2 + = ε ( P 1 P 2 ) 1 2 B 1 * + ε ( P 1 P 2 ) 1 2 B 1 +
+ P 2 B 2 * + i ( β 2 + + γ P 2 ) B 2 + ,
d z ( P 1 P 1 + P 2 P 2 + ) = 0 .
B 1 * ( z ) = C 1 * ( z ) exp [ i ( β 1 + β 1 ) z 2 ] ,
B 1 + ( z ) = C 1 + ( z ) exp [ i ( β 1 + β 1 ) z 2 ]
d z C 1 * = C 1 * C 1 + ,
d z C 1 + = C 1 * + C 1 + ,
C ( z ) = M ( z ) C ( 0 ) ,
M ( z ) = [ cos ( kz ) sin ( kz ) k sin ( kz ) k sin ( kz ) k cos ( kz ) + sin ( kz ) k ]
d z ( P 1 P 1 + ) = 0
B 1 * ( z ) = C 1 * ( z ) exp { i [ β 2 + β 1 + γ ( P 2 + P 1 ) ] z 2 } ,
B 2 * ( z ) = C 2 * ( z ) exp { i [ β 2 + β 1 + γ ( P 2 + P 1 ) ] z 2 }
d z C 1 = C 1 + C 2 ,
d z C 2 = C 1 + C 2 ,
M ( z ) = [ cos ( kz ) sin ( kz ) k sin ( kz ) k sin ( kz ) k cos ( kz ) + sin ( kz ) k ]
d z ( P 1 + P 2 ) = 0
B 1 * ( z ) = C 1 * ( z ) exp { i [ β 2 + β 1 + γ ( P 2 P 1 ) ] z 2 } ,
B 2 + ( z ) = C 2 + ( z ) exp { i [ β 2 + β 1 + γ ( P 2 P 1 ) ] z 2 }
d z C 1 * = C 1 * C 2 + ,
d z C 2 + = C 1 * + C 2 + ,
M ( z ) = [ cos ( kz ) sin ( kz ) k sin ( kz ) k sin ( kz ) k cos ( kz ) + sin ( kz ) k ]
d z ( P 1 P 2 + ) = 0
d z B = LB ,
B j ( z ) B j ( 0 ) exp [ λ j ( 0 ) z + l jj ( 1 ) z ] + k j B k ( 0 ) l jk ( 1 ) exp [ λ k ( 0 ) z ] exp [ λ j ( 0 ) z ] λ k ( 0 ) λ j ( 0 ) .
B ( z ) = M ( z ) B ( 0 ) ,
μ jk ( z ) { exp [ λ j ( 0 ) z + l jj ( 1 ) z ] if k = j , l jk ( 1 ) exp [ λ k ( 0 ) z ] exp [ λ j ( 0 ) z ] λ k ( 0 ) λ j ( 0 ) if k j .
M ( z ) [ 1 iγPz iγPz iγεPz iγεPz iγPz 1 + iγPz iγεPz iγεPz iγεPz iγεPz 1 iγPz iγPz iγεPz iγεPz iγPz 1 + iγPz ] .
M ( z ) [ μ ( z ) ( z ) μ ( z ) ( z ) ( z ) μ ( z ) ( z ) μ ( z ) μ ( z ) ( z ) μ ( z ) ( z ) ( z ) μ ( z ) ( z ) μ ( z ) ] ,
[ a ̂ j , a ̂ k ] = δ jk ,
D ω k = ω 2 n 2 ( ω ) c 2 k 2 ,
E + t z = E 0 t z exp [ i ϕ 0 t z ] ,
B + t z = B 0 t z exp [ i ϕ 0 t z ]
t T 0 + z S 0 =0,
T 0 = ( D ω ) 0 E 0 2 4 π ω 0 ,
S 0 = ( D k ) 0 E 0 2 4 π ω 0
E ̂ 0 t z = ( 2 π h ̄ ω 0 n 0 c ) 1 2 a ̂ ( ω ) exp [ ( ω ) z iωt ] ( 2 π ) 1 2 ,
B ̂ 0 t z = ( 2 π h ̄ ω 0 n 0 c ) 1 2 a ̂ ( ω ) exp [ ( ω ) z iωt ] ( 2 π ) 1 2 ,
[ a ̂ ( ω ) , a ̂ ( ω ) ] = δ ( ω ω ) ,
T ̂ 0 t z dz = h ̄ ω 0 a ̂ ( ω ) a ̂ ( ω ) .
a ̂ ω z = a ̂ ( ω ) exp [ ( ω ) z ]
a ̂ t z = a ̂ ( ω ) exp [ ( ω ) z iωt ] ( 2 π ) 1 2 .
a ̂ t z a ̂ t z dt = a ̂ ω z a ̂ ω z
[ a ̂ ( ω ) , a ̂ ( ω , z ) ] = δ ( ω ω ) ,
[ a ̂ t z , a ̂ ( t , z ) ] = δ ( t t ) .
i z a ̂ = β ( i t ) a ̂ ,
S ̂ 0 t z = h ̄ ω 0 a ̂ t z a ̂ t z
m ̂ T t z = t T 2 t + T 2 a ̂ ( t , z ) a ̂ ( t , z ) dt .
{ α } = exp ( a ̂ α a ̂ α ) { 0 } ,
a α = α ( ω ) a ̂ ( ω )
a ̂ α = α ω z a ̂ ω z ,
α ω z = α ( ω ) exp [ ( w ) z ]
a ̂ ω z { α } = α ω z { α } ,
a ̂ t z { α } = α t z { α } ,
m ̄ T t z = t T 2 t + T 2 α ( t , z ) 2 dt .
δ m T 2 t z = m ̄ T t z .
a ̂ k = ω k Δ 2 ω k + Δ 2 a ̂ ( ω ) Δ 1 2 .
[ a k , a l ] = δ kl ,
a ̂ j ( t , z ) = ( v L ) 1 2 k a ̂ k exp [ ( ω k ) z i ω k t ] .
m ̂ T ( t , z ) = ( Tv / L ) k , l sinc ( ω lk T / 2 ) a ̂ k a ̂ l exp ( i β lk z i ω lk t ) ,
α j = exp ( α j a ̂ j α j * a ̂ j ) 0 .
a ̂ j α j = α j α j ,
m ̄ T = ( Tv / L ) m ̄ j ,
δ m T 2 = m ¯ T .
B = MA ,
M ( z ) = [ μ * ( z ) v * ( z ) v ( z ) μ ( z ) ]
μ 2 v 2 = 1 .
n 1 = μ 2 a 1 a 1 + v 2 a 2 a 2 + μ * v a 1 a 2 + μ v * a 2 a 1 ,
n 2 = v 2 a 1 a 1 + μ 2 a 2 a 2 + μ v * a 1 a 2 + μ * v a 2 a 1 .
n 1 = μ 2 m 1 + v 2 ( m 2 + 1 ) ,
n 2 = v 2 ( m 1 + 1 ) + μ 2 m 2 .
n 1 2 = ( μ 2 a 1 a 1 + v 2 a 2 a 2 ) 2 + ( μ * v a 1 a 2 + μ v * a 2 a 1 ) 2
+ ( μ 2 a 1 a 1 + v 2 a 2 a 2 ) ( μ * v a 1 a 2 + μ v * a 2 a 1 )
+ ( μ * v a 1 a 2 + μ v * a 2 a 1 ) ( μ 2 a 1 a 1 + v 2 a 2 a 2 ) .
δ n 1 2 = μ 2 v 2 ( 2 m 1 m 2 + m 1 + m 2 + 1 ) .
ψ = m 1 = 0 [ p ( m 1 ) ] 1 2 exp ( i m 1 ϕ ) m 1 , 0 ,
n 1 = μ 2 m ̄ 1 + v 2 ,
n 2 = v 2 ( m ̄ 1 + 1 ) .
δ n 1 2 = μ 4 m ̄ 1 + μ 2 v 2 ( m ̄ 1 + 1 ) ,
δ n 2 2 = v 4 m ̄ 1 + μ 2 v 2 ( m ̄ 1 + 1 ) .
F 1 = m ̄ 1 [ G 2 m ̄ 1 + G ( G 1 ) ( m ̄ 1 + 1 ) ] [ G m ̄ 1 + ( G 1 ) ] 2 ,
F 2 = m ̄ 1 [ ( G 1 ) 2 m ̄ 1 + G ( G 1 ) ( m ̄ 1 + 1 ) ] ( G 1 ) 2 + ( m ̄ 1 + 1 ) 2 .
M ( z ) = [ μ ( z ) v ( z ) v * ( z ) μ * ( z ) ] .
μ 2 + v 2 = 1 .
n 1 = μ 2 a 1 a 1 + v 2 a 2 a 2 + μ * v a 1 a 2 + μ v * a 2 a 1 ,
n 2 = v 2 a 1 a 1 + μ 2 a 2 a 2 μ * v a 1 a 2 μ v * a 2 a 1 .
n 1 = μ 2 m 1 + v 2 m 2 ,
n 2 = v 2 m 1 + μ 2 m 2 .
n 1 2 = ( μ 2 a 1 a 1 + v 2 a 2 a 2 ) 2 + ( μ * v a 1 a 2 + μ v * a 2 a 1 ) 2
+ ( μ 2 a 1 a 1 + v 2 a 2 a 2 ) ( μ * v a 1 a 2 + μ v * a 2 a 1 )
+ ( μ * v a 1 a 2 + μ v * a 2 a 1 ) ( μ 2 a 1 a 1 + v 2 a 2 a 2 ) .
δ n 1 2 = μ 2 v 2 ( 2 m 1 m 2 + m 1 + m 2 ) .
n 1 = μ 2 m ̄ 1 ,
n 2 = v 2 m ̅ 1 .
δ n 1 2 = μ 4 m ̄ 1 + μ 2 v 2 m ̅ 1 ,
δ n 2 2 = v 4 m ̅ 1 + μ 2 v 2 m ̅ 1 .
F 1 = 1 T ,
F 2 = 1 ( 1 T ) .
B = MA ,
μ 11 2 μ 12 2 + μ 13 2 = 1 .
μ 21 2 + μ 22 2 μ 23 2 = 1 .
n 1 = μ 11 2 a 1 a 1 + μ 12 2 a 2 a 2 + μ 13 2 a 3 a 3 + μ 11 μ 12 * a 1 a 2 + μ 12 μ 11 * a 2 a 1
+ μ 12 μ 13 * a 2 a 3 + μ 13 μ 12 * a 3 a 2 + μ 13 μ 11 * a 3 a 1 + μ 11 μ 13 * a 1 a 3 ,
n 2 = μ 21 2 a 1 a 1 + μ 22 2 a 2 a 2 + μ 23 2 a 3 a 3 + μ 21 * μ 22 a 1 a 2 + μ 22 * μ 21 a 2 a 1
+ μ 22 * μ 23 a 2 a 3 + μ 23 * μ 22 a 3 a 2 + μ 23 * μ 21 a 3 a 1 + μ 21 * μ 13 a 1 a 3 .
n 1 = μ 11 2 m 1 + μ 12 2 ( m 2 + 1 ) + μ 13 2 m 3 ,
n 2 = μ 21 2 ( m 1 + 1 ) + μ 22 2 m 2 + μ 23 2 ( m 3 + 1 ) .
δ n 1 2 = μ 11 μ 12 2 ( 2 m 1 m 2 + m 1 + m 2 + 1 )
+ μ 12 μ 13 2 ( 2 m 2 m 3 + m 2 + m 3 + 1 )
+ μ 13 μ 11 2 ( 2 m 3 m 1 + m 3 + m 1 ) ,
δ n 2 2 = μ 21 μ 22 2 ( 2 m 1 m 2 + m 1 + m 2 + 1 )
+ μ 22 μ 23 2 ( 2 m 2 m 3 + m 2 + m 3 + 1 )
+ μ 23 μ 21 2 ( 2 m 3 m 1 + m 3 + m 1 ) .
n 1 = μ 11 2 m ̄ 1 + μ 12 2 ,
n 2 = μ 21 2 ( m ̄ 1 + 1 ) + μ 23 2 .
δ n 1 2 = μ 11 4 m ¯ 1 + μ 11 μ 12 2 ( m ¯ 1 + 1 ) + μ 12 μ 13 2 + μ 13 μ 11 2 m ¯ 1 ,
δ n 2 2 = μ 21 4 m ¯ 1 + μ 21 μ 22 2 ( m ¯ 1 + 1 ) + μ 22 μ 23 2 + μ 23 μ 21 2 m ¯ 1 .
F 1 = m ¯ 1 [ μ 11 4 m ¯ 1 + μ 11 μ 12 2 ( m ¯ 1 + 1 ) + μ 12 μ 13 2 + μ 13 μ 11 2 m ¯ 1 ] [ μ 11 2 m ¯ 1 + μ 12 2 ] 2 ,
F 2 = m ¯ 1 [ μ 21 4 m ¯ 1 + μ 21 μ 22 2 ( m ¯ 1 + 1 ) + μ 22 μ 23 2 + μ 23 μ 21 2 m ¯ 1 ] [ μ 21 2 ( m ¯ 1 + 1 ) + μ 23 2 ] 2 .
F 1 1 + ( μ 12 2 + μ 13 2 ) μ 11 2 ,
F 2 1 + ( μ 22 2 + μ 23 2 ) μ 21 2 .
k μ jk 2 s jk = 1 ,
n j = k μ jk 2 ( m k + σ jk ) ,
δ n j 2 = k , l > k μ jk μ jl 2 ( 2 m k m l + m k + m l + σ kl ) ,
n j = μ ji 2 m ̄ i + k μ jk 2 σ jk
δ n j 2 = μ ji 4 m ¯ i + k i μ ji μ jk 2 m ¯ i + k , l > k μ jk μ jl 2 σ kl ,
F j = m ¯ i ( μ ji 4 m ¯ i + k i μ ji μ jk 2 m ¯ i + k , l > k μ jk μ jl 2 σ kl ) ( μ ji 2 m ¯ i + k μ jk 2 σ jk ) 2 .
F j 1 + k i μ jk 2 μ ji 2 .
F 1 ± = 2 + 2 ε 2 ,
F 2 ± = 2 + 2 ε 2 .
F 1 ± = F 2 ± = 4 .
d z B = LB ,
d z 0 B ( 0 ) L ( 0 ) B ( 0 ) = 0 ,
d z 0 B ( 1 ) L ( 0 ) B ( 1 ) = d z 1 B ( 0 ) + L ( 1 ) B ( 0 ) ,
B j ( 0 ) ( z 0 ) = C j ( 0 ) exp [ λ j ( 0 ) z 0 ] ,
d z 0 B j ( 1 ) λ j ( 0 ) B j ( 1 ) = d z 1 C j ( 0 ) exp [ λ j ( 0 ) z 0 ] + k l jk ( 1 ) C k ( 0 ) exp [ λ k ( 0 ) z 0 ] .
d z 1 C j ( 0 ) = l jj ( 1 ) C j ( 0 ) ,
C j ( 0 ) ( z 1 ) = C j ( 0 ) ( 0 ) exp [ l jj ( 1 ) z 1 ] .
B j ( 1 ) ( z 0 ) = k j l jk ( 1 ) C k ( 0 ) [ λ k ( 0 ) z 0 ] exp [ λ j ( 0 ) z 0 ] λ k ( 0 ) λ j ( 0 ) .
B ( z ) = M ( z ) B ( 0 ) ,
M ( z ) 1 + Lz .
d z G j = i β e H j ,
d z H j = i ( β e + 2 γP ) G j + i 2 γεP G k ,
[ d zz 2 + β e ( β e + 2 γ P ) ] G 1 + 2 β e γε P G 2 = 0 ,
2 β e γε P G 1 + [ d zz 2 + β e ( β e + 2 γ P ) ] G 2 = 0 .
k ± 2 = β e [ β e + 2 γ ( 1 ± ε ) P ] .
G 1 = 1 2 cos ( k + z ) + i σ β e 2 k + sin ( k + z ) + 1 2 cos ( k z ) i σ β e 2 k sin ( k z ) ,
H 1 = σ 2 cos ( k + z ) + i k + 2 β e sin ( k + z ) + σ 2 cos ( k z ) i k 2 β e sin ( k z ) ,
G 2 = 1 2 cos ( k + z ) + i σ β e 2 k + sin ( k + z ) 1 2 cos ( k z ) i σ β e 2 k sin ( k z ) ,
H 2 = σ 2 cos ( k + z ) + i k + 2 β e sin ( k + z ) σ 2 cos ( k z ) i k 2 β e sin ( k z ) .
C 1 * = 1 σ 4 cos ( k + z ) + i 4 ( σ β e k + k + β e ) sin ( k + z )
+ 1 σ 4 cos ( k z ) + i 4 ( σ β e k k β e ) sin ( k z ) ,
C 1 + = 1 + σ 4 cos ( k + z ) + i 4 ( σ β e k + + k + β e ) sin ( k + z )
+ 1 + σ 4 cos ( k z ) + i 4 ( σ β e k + k β e ) sin ( k z ) ,
C 2 * = 1 σ 4 cos ( k + z ) + i 4 ( σ β e k + k + β e ) sin ( k + z )
1 σ 4 cos ( k z ) i 4 ( σ β e k k β e ) sin ( k z ) ,
C 2 + = 1 + σ 4 cos ( k + z ) + i 4 ( σ β e k + + k + β e ) sin ( k + z )
1 + σ 4 cos ( k z ) i 4 ( σ β e k + k + β e ) sin ( k z ) .
d z G j = i β je H j ,
d z H j = i ( β je + 2 γ P j ) G j + i 2 γε ( P j P k ) 1 2 G k ,
[ d zz 2 + β 1 e ( β 1 e + 2 γ P 1 ) ] G 1 + 2 β 1 e γε ( P 1 P 2 ) 1 2 G 2 = 0 ,
2 β 2 e γε ( P 1 P 2 ) 1 2 G 1 + [ d zz 2 + β 2 e ( β 2 e + 2 γ P 2 ) ] G 2 = 0 .
2 k ± 2 = β 1 e ( β 1 e + 2 γ P 1 ) + β 2 e ( β 2 e + 2 γ P 2 )
+ { [ β 1 e ( β 1 e + 2 γ P 1 ) β 2 e ( β 2 e + 2 γ P 2 ) ] 2
+ 4 [ 4 β 1 e β 2 e ( γε ) 2 P 1 P 2 ] } 1 2 .
G 1 = cos ( k + z ) 1 α 1 + α 1 + i σ β 1 e sin ( k + z ) k + ( 1 α 1 + α 1 )
+ cos ( k z ) 1 α 1 α 1 + + i σ β 1 e sin ( k z ) k ( 1 α 1 α 1 + ) ,
H 1 = σ cos ( k + z ) 1 α 1 + α 1 + i k + sin ( k + z ) β 1 e ( 1 α 1 + α 1 )
+ σ cos ( k z ) 1 α 1 α 1 + + i k sin ( k z ) β 1 e ( 1 α 1 α 1 + ) ,
G 2 = α 1 + cos ( k + z ) 1 α 1 + α 1 + i σ α 1 + β 1 e sin ( k + z ) k + ( 1 α 1 + α 1 )
+ α 1 cos ( k z ) 1 α 1 α 1 + + i σ α 1 β 1 e sin ( k z ) k ( 1 α 1 α 1 + ) ,
H 2 = σ α 1 + β 1 e cos ( k + z ) β 2 e ( 1 α 1 + α 1 ) + i α 1 + k + sin ( k + z ) β 2 e ( 1 α 1 + α 1 )
+ σ α 1 β 1 e cos ( k z ) β 2 e ( 1 α 1 α 1 + ) + i α 1 k sin ( k z ) β 2 e ( 1 α 1 α 1 + ) ,
α 1 ± = [ k ± 2 β 1 e ( β 1 e + 2 γ P 1 ) ] 2 β 1 e γε ( P 1 P 2 ) 1 2 .
C 1 * = ( 1 σ ) cos ( k + z ) 2 ( 1 α 1 + α 1 ) + ( σ β 1 e k + k + β 1 e ) i sin ( k + z ) 2 ( 1 α 1 + α 1 ) + ( + ) ,
C 1 + = ( 1 + σ ) cos ( k + z ) 2 ( 1 α 1 + α 1 ) + ( σ β 1 e k + + k + β 1 e ) i sin ( k + z ) 2 ( 1 α 1 + α 1 ) + ( + ) ,
C 2 * = α 1 + ( 1 σ β 1 e β 2 e ) cos ( k + z ) 2 ( 1 α 1 + α 1 ) + ( σ β 1 e k + k + β 2 e ) i α 1 + sin ( k + z ) 2 ( 1 α 1 + α 1 ) ,
C 2 + = α 1 + ( 1 + σ β 1 e β 2 e ) cos ( k + z ) 2 ( 1 α 1 + α 1 ) + ( σ β 1 e k + + k + β 2 e ) i α 1 + sin ( k + z ) 2 ( 1 α 1 + α 1 ) .
B = MA ,
k μ jk 2 s jk = 1 ,
j μ jk 2 s jk = 1 ,
d z B = LB ,
b j ( z ) = n e j ( n ) f k ( n ) * exp [ λ ( n ) z ] b k ( 0 ) ,
μ jk = n e j ( n ) f k ( n ) * exp [ λ ( n ) z ] .
L = [ i δ 1 i δ 2 i δ 3 i δ 4 ] ,
l kj = l jk s jk .
l ki ( n + 1 ) = j l kj l ji ( n )
= j l jk s jk l ij ( n ) s ij
= j l ij ( n ) l jk s ij s jk .
l ki ( n + 1 ) = l ik ( n + 1 ) s ik ,
μ kj = μ jk s jk .

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