Abstract

We present a quantum theory of nondegenerate phase-sensitive parametric amplification in a χ(3) nonlinear medium. The nonzero response time of the Kerr (χ(3)) nonlinearity determines the quantum-limited noise figure of χ(3) parametric amplification, as well as the limit on quadrature squeezing. This nonzero response time of the nonlinearity requires coupling of the parametric process to a molecular vibration phonon bath, causing the addition of excess noise through spontaneous Raman scattering. We present analytical expressions for the quantum-limited noise figure of frequency nondegenerate and frequency degenerate χ(3) parametric amplifiers operated as phase-sensitive amplifiers. We also present results for frequency nondegenerate quadrature squeezing. We show that our nondegenerate squeezing theory agrees with the degenerate squeezing theory of Boivin and Shapiro as degeneracy is approached. We have also included the effect of linear loss on the phase-sensitive process.

© 2006 Optical Society of America

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  1. K. K. Y. Wong, K. Shimizu, M. E. Marhic, K. Uesaka, G. Kalogerakis, and L. G. Kazovsky, "Continuous-wave fiber optical parametric wavelength converter with 40-dB conversion efficiency and a 3.8-dB noise figure," Opt. Lett. 28, 692-694 (2003).
    [CrossRef] [PubMed]
  2. J. Hansryd, P. A. Andrekson, M. Westlund, J. Li, and P. Hedekvist, "Fiber-based optical parametric amplifiers and their applications," IEEE J. Sel. Top. Quantum Electron. 8, 506-520 (2002).
    [CrossRef]
  3. Y. Su, L. Wang, A. Agarwal, and P. Kumar, "Wavelength-tunable all-optical clock recovery using a fiber-optic parametric oscillator," Opt. Commun. 184, 151-156 (2000).
    [CrossRef]
  4. L. Wang, Y. Su, A. Agarwal, and P. Kumar, "Synchronously mode-locked fiber laser based on parametric gain modulation and soliton shaping," Opt. Commun. 194, 313-317 (2001).
    [CrossRef]
  5. L. Wang, A. Agarwal, Y. Su, and P. Kumar, "All-optical picosecond-pulse packet buffer based on four-wave mixing loading and intracavity soliton control," IEEE J. Quantum Electron. 38, 614-619 (2002).
    [CrossRef]
  6. J. L. Blows and S. E. French, "Low-noise-figure optical parametric amplifier with a continuous-wave frequency-modulated pump," Opt. Lett. 27, 491-493 (2002).
    [CrossRef]
  7. R. Tang, J. Lasri, P. Devgan, J. E. Sharping, and P. Kumar, "Microstructure fiber-based optical parametric amplifier mth gain slope of ~ 200 dB/W/km in the telecom range," Electron. Lett. 39, 195-196 (2003).
    [CrossRef]
  8. S. Radic, C. J. McKinstrie, R. M. Jopson, J. Centanni, Q. Lin, and G. Agrawal, "Record performance of parametric amplifier constructed with highly nonlinear fiber," Electron. Lett. 39, 838-839 (2003).
    [CrossRef]
  9. P. L. Voss, R. Y. Tang, and P. Kumar, "Measurement of the photon statistics and the noise figure of a fiber-optic parametric amplifier," Opt. Lett. 28, 549-551 (2003).
    [CrossRef] [PubMed]
  10. P. L. Voss and P. Kumar, "Raman-noise induced noise-figure limit for chi(3) parametric amplifiers," Opt. Lett. 29, 445-447 (2004).
    [CrossRef] [PubMed]
  11. P. Voss and P. Kumar, "Raman-effect induced noise limits on chi(3) parametric amplifiers and wavelength converters," J. Opt. 6, S762-S770 (2004).
    [CrossRef]
  12. R. Tang, P. L. Voss, J. Lasri, P. Devgan, and P. Kumar, "Noise-figure of a fiber parametric amplifier and wavelength converter: experimental investigation," Opt. Lett. 29, 2372-2374 (2004).
    [CrossRef] [PubMed]
  13. M. E. Marhic, C. H. Hsia, and J. M. Jeong, "Optical amplification in a nonlinear fiber interferometer," Electron. Lett. 27, 201-211 (1991).
    [CrossRef]
  14. G. Bartolini, R. D. Li, P. Kumar, W. Riha, and K. V. Reddy, "1.5-mm phase-sensitive amplifier for ultrahigh-speed communications," in Conference on Optical Fiber Communication, Vol. 4 of 1994 OSA Technical Digest Series (Optical Society of America, 1994), pp. 202-203.
  15. C. M. Caves, "Quantum limits on noise in limited amplifiers," Phys. Rev. D 26, 1817-1839 (1982).
    [CrossRef]
  16. W. Imajuku, A. Takada, and Y. Yamabayashi, "Inline coherent optical amplifier with noise figure lower than 3 dB quantum limit," Electron. Lett. 36, 63-64 (2000).
    [CrossRef]
  17. D. Levandovsky, M. Vasilyev, and P. Kumar, "Near-noiseless amplification of light by a phase-sensitive fibre amplifier," Pramana, J. Phys. 56, 281-285 (2001).
    [CrossRef]
  18. R. M. Shelby, M. D. Levenson, and P. W. Bayer, "Guided acoustic-wave Brillouin scattering," Phys. Rev. B 31, 5244-5252 (1985).
    [CrossRef]
  19. K. Bergman, H. A. Haus, E. P. Ippen, and M. Shirasaki, "Squeezing in a fiber interferometer with a gigahertz pump," Opt. Lett. 19, 290-292 (1994).
    [CrossRef] [PubMed]
  20. R. Tang, P. Devgan, P. L. Voss, V. Grigoryan, and P. Kumar, "An in-line frequency-non-degenerate phase-sensitive fiber-optical parametric amplifier," IEEE Photon. Technol. Lett. 17, 1845-1847 (2005).
    [CrossRef]
  21. R. Tang, P. Devgan, V. Grigoryan, and P. Kumar, "In-line frequency-non-degenerate phase-sensitive fiber parametric amplifier for fiber-optic communications," Electron. Lett. 41, 1072-1073 (2005).
    [CrossRef]
  22. M. Vasilyev, "Distributed phase-sensitive amplification," Opt. Express 13, 7563-7571 (2005).
    [CrossRef] [PubMed]
  23. H. P. Yuen, "2-photon coherent states of radiation-field," Phys. Rev. A 13, 2226-2243 (1976).
    [CrossRef]
  24. H. P. Yuen and J. H. Shapiro, "Optical communication with 2-photon coherent states. 1. Quantum-state propagation and quantum-noise reduction," IEEE Trans. Inf. Theory 24, 657-668 (1978).
    [CrossRef]
  25. J. H. Shapiro, H. P. Yuen, and J. A. M. Mata, "Optical communication with 2-photon coherent states. 2. Photoemissive detection and structured receiver performance," IEEE Trans. Inf. Theory 25, 179-192 (1979).
    [CrossRef]
  26. H. P. Yuen and J. H. Shapiro, "Optical communication with 2-photon coherent states. 3. Quantum measurements realizable with photo-emissive detectors," IEEE Trans. Inf. Theory 26, 78-82 (1980).
    [CrossRef]
  27. C. M. Caves, "Quantum-mechanical noise in an interferometer," Phys. Rev. D 23, 1693-1708 (1981).
    [CrossRef]
  28. N. Treps, N. Grosse, W. P. Bowen, C. Fabre, H. A. Bachor, and P. K. Lam, "A quantum laser pointer," Science 301, 940-943 (2003).
    [CrossRef] [PubMed]
  29. A. N. Boto, P. Kok, D. S. Abrams, S. L. Braunstein, C. P. Williams, and J. P. Dowling, "Quantum interferometric optical lithography: exploiting entanglement to beat the diffraction limit," Phys. Rev. Lett. 86, 2736-2739 (2000).
  30. J. H. Shapiro and L. Boivin, "Raman-noise limit on squeezing in continuous-wave four-wave mixing," Opt. Lett. 20, 925-927 (1995).
    [CrossRef] [PubMed]
  31. L. Boivin, F. X. Kärtner, and H. A. Haus, "Analytical solution to the quantum field theory of self-phase modulation with a finite response time," Phys. Rev. Lett. 73, 240-243 (1994).
    [CrossRef] [PubMed]
  32. F. X. Kärtner, D. J. Dougherty, H. A. Haus, and E. P. Ippen, "Raman noise and soliton squeezing," J. Opt. Soc. Am. B 11, 1267-1276 (1994).
    [CrossRef]
  33. P. D. Drummond and J. F. Corney, "Quantum noise in optical fibers: I. Stochastic equations," J. Opt. Soc. Am. B 18, 139-152 (2001).
    [CrossRef]
  34. N. R. Newbury, "Raman gain: pump-wavelength dependence in single-mode fiber," Opt. Lett. 27, 1232-1234 (2002).
    [CrossRef]
  35. N. R. Newbury, "Pump-wavelength dependence of Raman gain in single-mode optical fibers," J. Lightwave Technol. 21, 3364-3373 (2003).
    [CrossRef]
  36. G. P. Agrawal, Nonlinear Fiber Optics, 3rd ed. (Academic, 2001).
  37. E. Golovchenko, P. V. Mamyshev, A. N. Pilipetskii, and E. M. Dianov, "Mutual influence of the parametric effects and stimulated Raman scattering in optical fibers," IEEE J. Quantum Electron. 26, 1815-1820 (1990).
    [CrossRef]
  38. S. J. Carter, P. D. Drummond, M. D. Reid, and R. M. Shelby, "Squeezing of quantum solitons," Phys. Rev. Lett. 58, 1841-1843 (1987).
    [CrossRef] [PubMed]

2005 (3)

R. Tang, P. Devgan, P. L. Voss, V. Grigoryan, and P. Kumar, "An in-line frequency-non-degenerate phase-sensitive fiber-optical parametric amplifier," IEEE Photon. Technol. Lett. 17, 1845-1847 (2005).
[CrossRef]

R. Tang, P. Devgan, V. Grigoryan, and P. Kumar, "In-line frequency-non-degenerate phase-sensitive fiber parametric amplifier for fiber-optic communications," Electron. Lett. 41, 1072-1073 (2005).
[CrossRef]

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

2004 (3)

2003 (6)

K. K. Y. Wong, K. Shimizu, M. E. Marhic, K. Uesaka, G. Kalogerakis, and L. G. Kazovsky, "Continuous-wave fiber optical parametric wavelength converter with 40-dB conversion efficiency and a 3.8-dB noise figure," Opt. Lett. 28, 692-694 (2003).
[CrossRef] [PubMed]

R. Tang, J. Lasri, P. Devgan, J. E. Sharping, and P. Kumar, "Microstructure fiber-based optical parametric amplifier mth gain slope of ~ 200 dB/W/km in the telecom range," Electron. Lett. 39, 195-196 (2003).
[CrossRef]

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

P. L. Voss, R. Y. Tang, and P. Kumar, "Measurement of the photon statistics and the noise figure of a fiber-optic parametric amplifier," Opt. Lett. 28, 549-551 (2003).
[CrossRef] [PubMed]

N. Treps, N. Grosse, W. P. Bowen, C. Fabre, H. A. Bachor, and P. K. Lam, "A quantum laser pointer," Science 301, 940-943 (2003).
[CrossRef] [PubMed]

N. R. Newbury, "Pump-wavelength dependence of Raman gain in single-mode optical fibers," J. Lightwave Technol. 21, 3364-3373 (2003).
[CrossRef]

2002 (4)

N. R. Newbury, "Raman gain: pump-wavelength dependence in single-mode fiber," Opt. Lett. 27, 1232-1234 (2002).
[CrossRef]

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

L. Wang, A. Agarwal, Y. Su, and P. Kumar, "All-optical picosecond-pulse packet buffer based on four-wave mixing loading and intracavity soliton control," IEEE J. Quantum Electron. 38, 614-619 (2002).
[CrossRef]

J. L. Blows and S. E. French, "Low-noise-figure optical parametric amplifier with a continuous-wave frequency-modulated pump," Opt. Lett. 27, 491-493 (2002).
[CrossRef]

2001 (3)

L. Wang, Y. Su, A. Agarwal, and P. Kumar, "Synchronously mode-locked fiber laser based on parametric gain modulation and soliton shaping," Opt. Commun. 194, 313-317 (2001).
[CrossRef]

P. D. Drummond and J. F. Corney, "Quantum noise in optical fibers: I. Stochastic equations," J. Opt. Soc. Am. B 18, 139-152 (2001).
[CrossRef]

D. Levandovsky, M. Vasilyev, and P. Kumar, "Near-noiseless amplification of light by a phase-sensitive fibre amplifier," Pramana, J. Phys. 56, 281-285 (2001).
[CrossRef]

2000 (3)

A. N. Boto, P. Kok, D. S. Abrams, S. L. Braunstein, C. P. Williams, and J. P. Dowling, "Quantum interferometric optical lithography: exploiting entanglement to beat the diffraction limit," Phys. Rev. Lett. 86, 2736-2739 (2000).

W. Imajuku, A. Takada, and Y. Yamabayashi, "Inline coherent optical amplifier with noise figure lower than 3 dB quantum limit," Electron. Lett. 36, 63-64 (2000).
[CrossRef]

Y. Su, L. Wang, A. Agarwal, and P. Kumar, "Wavelength-tunable all-optical clock recovery using a fiber-optic parametric oscillator," Opt. Commun. 184, 151-156 (2000).
[CrossRef]

1995 (1)

1994 (3)

1991 (1)

M. E. Marhic, C. H. Hsia, and J. M. Jeong, "Optical amplification in a nonlinear fiber interferometer," Electron. Lett. 27, 201-211 (1991).
[CrossRef]

1990 (1)

E. Golovchenko, P. V. Mamyshev, A. N. Pilipetskii, and E. M. Dianov, "Mutual influence of the parametric effects and stimulated Raman scattering in optical fibers," IEEE J. Quantum Electron. 26, 1815-1820 (1990).
[CrossRef]

1987 (1)

S. J. Carter, P. D. Drummond, M. D. Reid, and R. M. Shelby, "Squeezing of quantum solitons," Phys. Rev. Lett. 58, 1841-1843 (1987).
[CrossRef] [PubMed]

1985 (1)

R. M. Shelby, M. D. Levenson, and P. W. Bayer, "Guided acoustic-wave Brillouin scattering," Phys. Rev. B 31, 5244-5252 (1985).
[CrossRef]

1982 (1)

C. M. Caves, "Quantum limits on noise in limited amplifiers," Phys. Rev. D 26, 1817-1839 (1982).
[CrossRef]

1981 (1)

C. M. Caves, "Quantum-mechanical noise in an interferometer," Phys. Rev. D 23, 1693-1708 (1981).
[CrossRef]

1980 (1)

H. P. Yuen and J. H. Shapiro, "Optical communication with 2-photon coherent states. 3. Quantum measurements realizable with photo-emissive detectors," IEEE Trans. Inf. Theory 26, 78-82 (1980).
[CrossRef]

1979 (1)

J. H. Shapiro, H. P. Yuen, and J. A. M. Mata, "Optical communication with 2-photon coherent states. 2. Photoemissive detection and structured receiver performance," IEEE Trans. Inf. Theory 25, 179-192 (1979).
[CrossRef]

1978 (1)

H. P. Yuen and J. H. Shapiro, "Optical communication with 2-photon coherent states. 1. Quantum-state propagation and quantum-noise reduction," IEEE Trans. Inf. Theory 24, 657-668 (1978).
[CrossRef]

1976 (1)

H. P. Yuen, "2-photon coherent states of radiation-field," Phys. Rev. A 13, 2226-2243 (1976).
[CrossRef]

Abrams, D. S.

A. N. Boto, P. Kok, D. S. Abrams, S. L. Braunstein, C. P. Williams, and J. P. Dowling, "Quantum interferometric optical lithography: exploiting entanglement to beat the diffraction limit," Phys. Rev. Lett. 86, 2736-2739 (2000).

Agarwal, A.

L. Wang, A. Agarwal, Y. Su, and P. Kumar, "All-optical picosecond-pulse packet buffer based on four-wave mixing loading and intracavity soliton control," IEEE J. Quantum Electron. 38, 614-619 (2002).
[CrossRef]

L. Wang, Y. Su, A. Agarwal, and P. Kumar, "Synchronously mode-locked fiber laser based on parametric gain modulation and soliton shaping," Opt. Commun. 194, 313-317 (2001).
[CrossRef]

Y. Su, L. Wang, A. Agarwal, and P. Kumar, "Wavelength-tunable all-optical clock recovery using a fiber-optic parametric oscillator," Opt. Commun. 184, 151-156 (2000).
[CrossRef]

Agrawal, G.

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

Agrawal, G. P.

G. P. Agrawal, Nonlinear Fiber Optics, 3rd ed. (Academic, 2001).

Andrekson, P. A.

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

Bachor, H. A.

N. Treps, N. Grosse, W. P. Bowen, C. Fabre, H. A. Bachor, and P. K. Lam, "A quantum laser pointer," Science 301, 940-943 (2003).
[CrossRef] [PubMed]

Bartolini, G.

G. Bartolini, R. D. Li, P. Kumar, W. Riha, and K. V. Reddy, "1.5-mm phase-sensitive amplifier for ultrahigh-speed communications," in Conference on Optical Fiber Communication, Vol. 4 of 1994 OSA Technical Digest Series (Optical Society of America, 1994), pp. 202-203.

Bayer, P. W.

R. M. Shelby, M. D. Levenson, and P. W. Bayer, "Guided acoustic-wave Brillouin scattering," Phys. Rev. B 31, 5244-5252 (1985).
[CrossRef]

Bergman, K.

Blows, J. L.

Boivin, L.

J. H. Shapiro and L. Boivin, "Raman-noise limit on squeezing in continuous-wave four-wave mixing," Opt. Lett. 20, 925-927 (1995).
[CrossRef] [PubMed]

L. Boivin, F. X. Kärtner, and H. A. Haus, "Analytical solution to the quantum field theory of self-phase modulation with a finite response time," Phys. Rev. Lett. 73, 240-243 (1994).
[CrossRef] [PubMed]

Boto, A. N.

A. N. Boto, P. Kok, D. S. Abrams, S. L. Braunstein, C. P. Williams, and J. P. Dowling, "Quantum interferometric optical lithography: exploiting entanglement to beat the diffraction limit," Phys. Rev. Lett. 86, 2736-2739 (2000).

Bowen, W. P.

N. Treps, N. Grosse, W. P. Bowen, C. Fabre, H. A. Bachor, and P. K. Lam, "A quantum laser pointer," Science 301, 940-943 (2003).
[CrossRef] [PubMed]

Braunstein, S. L.

A. N. Boto, P. Kok, D. S. Abrams, S. L. Braunstein, C. P. Williams, and J. P. Dowling, "Quantum interferometric optical lithography: exploiting entanglement to beat the diffraction limit," Phys. Rev. Lett. 86, 2736-2739 (2000).

Carter, S. J.

S. J. Carter, P. D. Drummond, M. D. Reid, and R. M. Shelby, "Squeezing of quantum solitons," Phys. Rev. Lett. 58, 1841-1843 (1987).
[CrossRef] [PubMed]

Caves, C. M.

C. M. Caves, "Quantum limits on noise in limited amplifiers," Phys. Rev. D 26, 1817-1839 (1982).
[CrossRef]

C. M. Caves, "Quantum-mechanical noise in an interferometer," Phys. Rev. D 23, 1693-1708 (1981).
[CrossRef]

Centanni, J.

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

Corney, J. F.

Devgan, P.

R. Tang, P. Devgan, P. L. Voss, V. Grigoryan, and P. Kumar, "An in-line frequency-non-degenerate phase-sensitive fiber-optical parametric amplifier," IEEE Photon. Technol. Lett. 17, 1845-1847 (2005).
[CrossRef]

R. Tang, P. Devgan, V. Grigoryan, and P. Kumar, "In-line frequency-non-degenerate phase-sensitive fiber parametric amplifier for fiber-optic communications," Electron. Lett. 41, 1072-1073 (2005).
[CrossRef]

R. Tang, P. L. Voss, J. Lasri, P. Devgan, and P. Kumar, "Noise-figure of a fiber parametric amplifier and wavelength converter: experimental investigation," Opt. Lett. 29, 2372-2374 (2004).
[CrossRef] [PubMed]

R. Tang, J. Lasri, P. Devgan, J. E. Sharping, and P. Kumar, "Microstructure fiber-based optical parametric amplifier mth gain slope of ~ 200 dB/W/km in the telecom range," Electron. Lett. 39, 195-196 (2003).
[CrossRef]

Dianov, E. M.

E. Golovchenko, P. V. Mamyshev, A. N. Pilipetskii, and E. M. Dianov, "Mutual influence of the parametric effects and stimulated Raman scattering in optical fibers," IEEE J. Quantum Electron. 26, 1815-1820 (1990).
[CrossRef]

Dougherty, D. J.

Dowling, J. P.

A. N. Boto, P. Kok, D. S. Abrams, S. L. Braunstein, C. P. Williams, and J. P. Dowling, "Quantum interferometric optical lithography: exploiting entanglement to beat the diffraction limit," Phys. Rev. Lett. 86, 2736-2739 (2000).

Drummond, P. D.

P. D. Drummond and J. F. Corney, "Quantum noise in optical fibers: I. Stochastic equations," J. Opt. Soc. Am. B 18, 139-152 (2001).
[CrossRef]

S. J. Carter, P. D. Drummond, M. D. Reid, and R. M. Shelby, "Squeezing of quantum solitons," Phys. Rev. Lett. 58, 1841-1843 (1987).
[CrossRef] [PubMed]

Fabre, C.

N. Treps, N. Grosse, W. P. Bowen, C. Fabre, H. A. Bachor, and P. K. Lam, "A quantum laser pointer," Science 301, 940-943 (2003).
[CrossRef] [PubMed]

French, S. E.

Golovchenko, E.

E. Golovchenko, P. V. Mamyshev, A. N. Pilipetskii, and E. M. Dianov, "Mutual influence of the parametric effects and stimulated Raman scattering in optical fibers," IEEE J. Quantum Electron. 26, 1815-1820 (1990).
[CrossRef]

Grigoryan, V.

R. Tang, P. Devgan, P. L. Voss, V. Grigoryan, and P. Kumar, "An in-line frequency-non-degenerate phase-sensitive fiber-optical parametric amplifier," IEEE Photon. Technol. Lett. 17, 1845-1847 (2005).
[CrossRef]

R. Tang, P. Devgan, V. Grigoryan, and P. Kumar, "In-line frequency-non-degenerate phase-sensitive fiber parametric amplifier for fiber-optic communications," Electron. Lett. 41, 1072-1073 (2005).
[CrossRef]

Grosse, N.

N. Treps, N. Grosse, W. P. Bowen, C. Fabre, H. A. Bachor, and P. K. Lam, "A quantum laser pointer," Science 301, 940-943 (2003).
[CrossRef] [PubMed]

Hansryd, J.

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

Haus, H. A.

Hedekvist, P.

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

Hsia, C. H.

M. E. Marhic, C. H. Hsia, and J. M. Jeong, "Optical amplification in a nonlinear fiber interferometer," Electron. Lett. 27, 201-211 (1991).
[CrossRef]

Imajuku, W.

W. Imajuku, A. Takada, and Y. Yamabayashi, "Inline coherent optical amplifier with noise figure lower than 3 dB quantum limit," Electron. Lett. 36, 63-64 (2000).
[CrossRef]

Ippen, E. P.

Jeong, J. M.

M. E. Marhic, C. H. Hsia, and J. M. Jeong, "Optical amplification in a nonlinear fiber interferometer," Electron. Lett. 27, 201-211 (1991).
[CrossRef]

Jopson, R. M.

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

Kalogerakis, G.

Kärtner, F. X.

L. Boivin, F. X. Kärtner, and H. A. Haus, "Analytical solution to the quantum field theory of self-phase modulation with a finite response time," Phys. Rev. Lett. 73, 240-243 (1994).
[CrossRef] [PubMed]

F. X. Kärtner, D. J. Dougherty, H. A. Haus, and E. P. Ippen, "Raman noise and soliton squeezing," J. Opt. Soc. Am. B 11, 1267-1276 (1994).
[CrossRef]

Kazovsky, L. G.

Kok, P.

A. N. Boto, P. Kok, D. S. Abrams, S. L. Braunstein, C. P. Williams, and J. P. Dowling, "Quantum interferometric optical lithography: exploiting entanglement to beat the diffraction limit," Phys. Rev. Lett. 86, 2736-2739 (2000).

Kumar, P.

R. Tang, P. Devgan, P. L. Voss, V. Grigoryan, and P. Kumar, "An in-line frequency-non-degenerate phase-sensitive fiber-optical parametric amplifier," IEEE Photon. Technol. Lett. 17, 1845-1847 (2005).
[CrossRef]

R. Tang, P. Devgan, V. Grigoryan, and P. Kumar, "In-line frequency-non-degenerate phase-sensitive fiber parametric amplifier for fiber-optic communications," Electron. Lett. 41, 1072-1073 (2005).
[CrossRef]

P. Voss and P. Kumar, "Raman-effect induced noise limits on chi(3) parametric amplifiers and wavelength converters," J. Opt. 6, S762-S770 (2004).
[CrossRef]

R. Tang, P. L. Voss, J. Lasri, P. Devgan, and P. Kumar, "Noise-figure of a fiber parametric amplifier and wavelength converter: experimental investigation," Opt. Lett. 29, 2372-2374 (2004).
[CrossRef] [PubMed]

P. L. Voss and P. Kumar, "Raman-noise induced noise-figure limit for chi(3) parametric amplifiers," Opt. Lett. 29, 445-447 (2004).
[CrossRef] [PubMed]

P. L. Voss, R. Y. Tang, and P. Kumar, "Measurement of the photon statistics and the noise figure of a fiber-optic parametric amplifier," Opt. Lett. 28, 549-551 (2003).
[CrossRef] [PubMed]

R. Tang, J. Lasri, P. Devgan, J. E. Sharping, and P. Kumar, "Microstructure fiber-based optical parametric amplifier mth gain slope of ~ 200 dB/W/km in the telecom range," Electron. Lett. 39, 195-196 (2003).
[CrossRef]

L. Wang, A. Agarwal, Y. Su, and P. Kumar, "All-optical picosecond-pulse packet buffer based on four-wave mixing loading and intracavity soliton control," IEEE J. Quantum Electron. 38, 614-619 (2002).
[CrossRef]

L. Wang, Y. Su, A. Agarwal, and P. Kumar, "Synchronously mode-locked fiber laser based on parametric gain modulation and soliton shaping," Opt. Commun. 194, 313-317 (2001).
[CrossRef]

D. Levandovsky, M. Vasilyev, and P. Kumar, "Near-noiseless amplification of light by a phase-sensitive fibre amplifier," Pramana, J. Phys. 56, 281-285 (2001).
[CrossRef]

Y. Su, L. Wang, A. Agarwal, and P. Kumar, "Wavelength-tunable all-optical clock recovery using a fiber-optic parametric oscillator," Opt. Commun. 184, 151-156 (2000).
[CrossRef]

G. Bartolini, R. D. Li, P. Kumar, W. Riha, and K. V. Reddy, "1.5-mm phase-sensitive amplifier for ultrahigh-speed communications," in Conference on Optical Fiber Communication, Vol. 4 of 1994 OSA Technical Digest Series (Optical Society of America, 1994), pp. 202-203.

Lam, P. K.

N. Treps, N. Grosse, W. P. Bowen, C. Fabre, H. A. Bachor, and P. K. Lam, "A quantum laser pointer," Science 301, 940-943 (2003).
[CrossRef] [PubMed]

Lasri, J.

R. Tang, P. L. Voss, J. Lasri, P. Devgan, and P. Kumar, "Noise-figure of a fiber parametric amplifier and wavelength converter: experimental investigation," Opt. Lett. 29, 2372-2374 (2004).
[CrossRef] [PubMed]

R. Tang, J. Lasri, P. Devgan, J. E. Sharping, and P. Kumar, "Microstructure fiber-based optical parametric amplifier mth gain slope of ~ 200 dB/W/km in the telecom range," Electron. Lett. 39, 195-196 (2003).
[CrossRef]

Levandovsky, D.

D. Levandovsky, M. Vasilyev, and P. Kumar, "Near-noiseless amplification of light by a phase-sensitive fibre amplifier," Pramana, J. Phys. 56, 281-285 (2001).
[CrossRef]

Levenson, M. D.

R. M. Shelby, M. D. Levenson, and P. W. Bayer, "Guided acoustic-wave Brillouin scattering," Phys. Rev. B 31, 5244-5252 (1985).
[CrossRef]

Li, J.

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

Li, R. D.

G. Bartolini, R. D. Li, P. Kumar, W. Riha, and K. V. Reddy, "1.5-mm phase-sensitive amplifier for ultrahigh-speed communications," in Conference on Optical Fiber Communication, Vol. 4 of 1994 OSA Technical Digest Series (Optical Society of America, 1994), pp. 202-203.

Lin, Q.

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

Mamyshev, P. V.

E. Golovchenko, P. V. Mamyshev, A. N. Pilipetskii, and E. M. Dianov, "Mutual influence of the parametric effects and stimulated Raman scattering in optical fibers," IEEE J. Quantum Electron. 26, 1815-1820 (1990).
[CrossRef]

Marhic, M. E.

Mata, J. A. M.

J. H. Shapiro, H. P. Yuen, and J. A. M. Mata, "Optical communication with 2-photon coherent states. 2. Photoemissive detection and structured receiver performance," IEEE Trans. Inf. Theory 25, 179-192 (1979).
[CrossRef]

McKinstrie, C. J.

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

Newbury, N. R.

Pilipetskii, A. N.

E. Golovchenko, P. V. Mamyshev, A. N. Pilipetskii, and E. M. Dianov, "Mutual influence of the parametric effects and stimulated Raman scattering in optical fibers," IEEE J. Quantum Electron. 26, 1815-1820 (1990).
[CrossRef]

Radic, S.

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

Reddy, K. V.

G. Bartolini, R. D. Li, P. Kumar, W. Riha, and K. V. Reddy, "1.5-mm phase-sensitive amplifier for ultrahigh-speed communications," in Conference on Optical Fiber Communication, Vol. 4 of 1994 OSA Technical Digest Series (Optical Society of America, 1994), pp. 202-203.

Reid, M. D.

S. J. Carter, P. D. Drummond, M. D. Reid, and R. M. Shelby, "Squeezing of quantum solitons," Phys. Rev. Lett. 58, 1841-1843 (1987).
[CrossRef] [PubMed]

Riha, W.

G. Bartolini, R. D. Li, P. Kumar, W. Riha, and K. V. Reddy, "1.5-mm phase-sensitive amplifier for ultrahigh-speed communications," in Conference on Optical Fiber Communication, Vol. 4 of 1994 OSA Technical Digest Series (Optical Society of America, 1994), pp. 202-203.

Shapiro, J. H.

J. H. Shapiro and L. Boivin, "Raman-noise limit on squeezing in continuous-wave four-wave mixing," Opt. Lett. 20, 925-927 (1995).
[CrossRef] [PubMed]

H. P. Yuen and J. H. Shapiro, "Optical communication with 2-photon coherent states. 3. Quantum measurements realizable with photo-emissive detectors," IEEE Trans. Inf. Theory 26, 78-82 (1980).
[CrossRef]

J. H. Shapiro, H. P. Yuen, and J. A. M. Mata, "Optical communication with 2-photon coherent states. 2. Photoemissive detection and structured receiver performance," IEEE Trans. Inf. Theory 25, 179-192 (1979).
[CrossRef]

H. P. Yuen and J. H. Shapiro, "Optical communication with 2-photon coherent states. 1. Quantum-state propagation and quantum-noise reduction," IEEE Trans. Inf. Theory 24, 657-668 (1978).
[CrossRef]

Sharping, J. E.

R. Tang, J. Lasri, P. Devgan, J. E. Sharping, and P. Kumar, "Microstructure fiber-based optical parametric amplifier mth gain slope of ~ 200 dB/W/km in the telecom range," Electron. Lett. 39, 195-196 (2003).
[CrossRef]

Shelby, R. M.

S. J. Carter, P. D. Drummond, M. D. Reid, and R. M. Shelby, "Squeezing of quantum solitons," Phys. Rev. Lett. 58, 1841-1843 (1987).
[CrossRef] [PubMed]

R. M. Shelby, M. D. Levenson, and P. W. Bayer, "Guided acoustic-wave Brillouin scattering," Phys. Rev. B 31, 5244-5252 (1985).
[CrossRef]

Shimizu, K.

Shirasaki, M.

Su, Y.

L. Wang, A. Agarwal, Y. Su, and P. Kumar, "All-optical picosecond-pulse packet buffer based on four-wave mixing loading and intracavity soliton control," IEEE J. Quantum Electron. 38, 614-619 (2002).
[CrossRef]

L. Wang, Y. Su, A. Agarwal, and P. Kumar, "Synchronously mode-locked fiber laser based on parametric gain modulation and soliton shaping," Opt. Commun. 194, 313-317 (2001).
[CrossRef]

Y. Su, L. Wang, A. Agarwal, and P. Kumar, "Wavelength-tunable all-optical clock recovery using a fiber-optic parametric oscillator," Opt. Commun. 184, 151-156 (2000).
[CrossRef]

Takada, A.

W. Imajuku, A. Takada, and Y. Yamabayashi, "Inline coherent optical amplifier with noise figure lower than 3 dB quantum limit," Electron. Lett. 36, 63-64 (2000).
[CrossRef]

Tang, R.

R. Tang, P. Devgan, P. L. Voss, V. Grigoryan, and P. Kumar, "An in-line frequency-non-degenerate phase-sensitive fiber-optical parametric amplifier," IEEE Photon. Technol. Lett. 17, 1845-1847 (2005).
[CrossRef]

R. Tang, P. Devgan, V. Grigoryan, and P. Kumar, "In-line frequency-non-degenerate phase-sensitive fiber parametric amplifier for fiber-optic communications," Electron. Lett. 41, 1072-1073 (2005).
[CrossRef]

R. Tang, P. L. Voss, J. Lasri, P. Devgan, and P. Kumar, "Noise-figure of a fiber parametric amplifier and wavelength converter: experimental investigation," Opt. Lett. 29, 2372-2374 (2004).
[CrossRef] [PubMed]

R. Tang, J. Lasri, P. Devgan, J. E. Sharping, and P. Kumar, "Microstructure fiber-based optical parametric amplifier mth gain slope of ~ 200 dB/W/km in the telecom range," Electron. Lett. 39, 195-196 (2003).
[CrossRef]

Tang, R. Y.

Treps, N.

N. Treps, N. Grosse, W. P. Bowen, C. Fabre, H. A. Bachor, and P. K. Lam, "A quantum laser pointer," Science 301, 940-943 (2003).
[CrossRef] [PubMed]

Uesaka, K.

Vasilyev, M.

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

D. Levandovsky, M. Vasilyev, and P. Kumar, "Near-noiseless amplification of light by a phase-sensitive fibre amplifier," Pramana, J. Phys. 56, 281-285 (2001).
[CrossRef]

Voss, P.

P. Voss and P. Kumar, "Raman-effect induced noise limits on chi(3) parametric amplifiers and wavelength converters," J. Opt. 6, S762-S770 (2004).
[CrossRef]

Voss, P. L.

Wang, L.

L. Wang, A. Agarwal, Y. Su, and P. Kumar, "All-optical picosecond-pulse packet buffer based on four-wave mixing loading and intracavity soliton control," IEEE J. Quantum Electron. 38, 614-619 (2002).
[CrossRef]

L. Wang, Y. Su, A. Agarwal, and P. Kumar, "Synchronously mode-locked fiber laser based on parametric gain modulation and soliton shaping," Opt. Commun. 194, 313-317 (2001).
[CrossRef]

Y. Su, L. Wang, A. Agarwal, and P. Kumar, "Wavelength-tunable all-optical clock recovery using a fiber-optic parametric oscillator," Opt. Commun. 184, 151-156 (2000).
[CrossRef]

Westlund, M.

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

Williams, C. P.

A. N. Boto, P. Kok, D. S. Abrams, S. L. Braunstein, C. P. Williams, and J. P. Dowling, "Quantum interferometric optical lithography: exploiting entanglement to beat the diffraction limit," Phys. Rev. Lett. 86, 2736-2739 (2000).

Wong, K. K. Y.

Yamabayashi, Y.

W. Imajuku, A. Takada, and Y. Yamabayashi, "Inline coherent optical amplifier with noise figure lower than 3 dB quantum limit," Electron. Lett. 36, 63-64 (2000).
[CrossRef]

Yuen, H. P.

H. P. Yuen and J. H. Shapiro, "Optical communication with 2-photon coherent states. 3. Quantum measurements realizable with photo-emissive detectors," IEEE Trans. Inf. Theory 26, 78-82 (1980).
[CrossRef]

J. H. Shapiro, H. P. Yuen, and J. A. M. Mata, "Optical communication with 2-photon coherent states. 2. Photoemissive detection and structured receiver performance," IEEE Trans. Inf. Theory 25, 179-192 (1979).
[CrossRef]

H. P. Yuen and J. H. Shapiro, "Optical communication with 2-photon coherent states. 1. Quantum-state propagation and quantum-noise reduction," IEEE Trans. Inf. Theory 24, 657-668 (1978).
[CrossRef]

H. P. Yuen, "2-photon coherent states of radiation-field," Phys. Rev. A 13, 2226-2243 (1976).
[CrossRef]

Electron. Lett. (5)

R. Tang, J. Lasri, P. Devgan, J. E. Sharping, and P. Kumar, "Microstructure fiber-based optical parametric amplifier mth gain slope of ~ 200 dB/W/km in the telecom range," Electron. Lett. 39, 195-196 (2003).
[CrossRef]

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

M. E. Marhic, C. H. Hsia, and J. M. Jeong, "Optical amplification in a nonlinear fiber interferometer," Electron. Lett. 27, 201-211 (1991).
[CrossRef]

W. Imajuku, A. Takada, and Y. Yamabayashi, "Inline coherent optical amplifier with noise figure lower than 3 dB quantum limit," Electron. Lett. 36, 63-64 (2000).
[CrossRef]

R. Tang, P. Devgan, V. Grigoryan, and P. Kumar, "In-line frequency-non-degenerate phase-sensitive fiber parametric amplifier for fiber-optic communications," Electron. Lett. 41, 1072-1073 (2005).
[CrossRef]

IEEE J. Quantum Electron. (2)

E. Golovchenko, P. V. Mamyshev, A. N. Pilipetskii, and E. M. Dianov, "Mutual influence of the parametric effects and stimulated Raman scattering in optical fibers," IEEE J. Quantum Electron. 26, 1815-1820 (1990).
[CrossRef]

L. Wang, A. Agarwal, Y. Su, and P. Kumar, "All-optical picosecond-pulse packet buffer based on four-wave mixing loading and intracavity soliton control," IEEE J. Quantum Electron. 38, 614-619 (2002).
[CrossRef]

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

J. Hansryd, P. A. Andrekson, M. Westlund, J. Li, and P. 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. Grigoryan, and P. Kumar, "An in-line frequency-non-degenerate phase-sensitive fiber-optical parametric amplifier," IEEE Photon. Technol. Lett. 17, 1845-1847 (2005).
[CrossRef]

IEEE Trans. Inf. Theory (3)

H. P. Yuen and J. H. Shapiro, "Optical communication with 2-photon coherent states. 1. Quantum-state propagation and quantum-noise reduction," IEEE Trans. Inf. Theory 24, 657-668 (1978).
[CrossRef]

J. H. Shapiro, H. P. Yuen, and J. A. M. Mata, "Optical communication with 2-photon coherent states. 2. Photoemissive detection and structured receiver performance," IEEE Trans. Inf. Theory 25, 179-192 (1979).
[CrossRef]

H. P. Yuen and J. H. Shapiro, "Optical communication with 2-photon coherent states. 3. Quantum measurements realizable with photo-emissive detectors," IEEE Trans. Inf. Theory 26, 78-82 (1980).
[CrossRef]

J. Lightwave Technol. (1)

J. Opt. (1)

P. Voss and P. Kumar, "Raman-effect induced noise limits on chi(3) parametric amplifiers and wavelength converters," J. Opt. 6, S762-S770 (2004).
[CrossRef]

J. Opt. Soc. Am. B (2)

Opt. Commun. (2)

Y. Su, L. Wang, A. Agarwal, and P. Kumar, "Wavelength-tunable all-optical clock recovery using a fiber-optic parametric oscillator," Opt. Commun. 184, 151-156 (2000).
[CrossRef]

L. Wang, Y. Su, A. Agarwal, and P. Kumar, "Synchronously mode-locked fiber laser based on parametric gain modulation and soliton shaping," Opt. Commun. 194, 313-317 (2001).
[CrossRef]

Opt. Express (1)

Opt. Lett. (8)

Phys. Rev. A (1)

H. P. Yuen, "2-photon coherent states of radiation-field," Phys. Rev. A 13, 2226-2243 (1976).
[CrossRef]

Phys. Rev. B (1)

R. M. Shelby, M. D. Levenson, and P. W. Bayer, "Guided acoustic-wave Brillouin scattering," Phys. Rev. B 31, 5244-5252 (1985).
[CrossRef]

Phys. Rev. D (2)

C. M. Caves, "Quantum-mechanical noise in an interferometer," Phys. Rev. D 23, 1693-1708 (1981).
[CrossRef]

C. M. Caves, "Quantum limits on noise in limited amplifiers," Phys. Rev. D 26, 1817-1839 (1982).
[CrossRef]

Phys. Rev. Lett. (3)

A. N. Boto, P. Kok, D. S. Abrams, S. L. Braunstein, C. P. Williams, and J. P. Dowling, "Quantum interferometric optical lithography: exploiting entanglement to beat the diffraction limit," Phys. Rev. Lett. 86, 2736-2739 (2000).

L. Boivin, F. X. Kärtner, and H. A. Haus, "Analytical solution to the quantum field theory of self-phase modulation with a finite response time," Phys. Rev. Lett. 73, 240-243 (1994).
[CrossRef] [PubMed]

S. J. Carter, P. D. Drummond, M. D. Reid, and R. M. Shelby, "Squeezing of quantum solitons," Phys. Rev. Lett. 58, 1841-1843 (1987).
[CrossRef] [PubMed]

Pramana, J. Phys. (1)

D. Levandovsky, M. Vasilyev, and P. Kumar, "Near-noiseless amplification of light by a phase-sensitive fibre amplifier," Pramana, J. Phys. 56, 281-285 (2001).
[CrossRef]

Science (1)

N. Treps, N. Grosse, W. P. Bowen, C. Fabre, H. A. Bachor, and P. K. Lam, "A quantum laser pointer," Science 301, 940-943 (2003).
[CrossRef] [PubMed]

Other (2)

G. Bartolini, R. D. Li, P. Kumar, W. Riha, and K. V. Reddy, "1.5-mm phase-sensitive amplifier for ultrahigh-speed communications," in Conference on Optical Fiber Communication, Vol. 4 of 1994 OSA Technical Digest Series (Optical Society of America, 1994), pp. 202-203.

G. P. Agrawal, Nonlinear Fiber Optics, 3rd ed. (Academic, 2001).

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Figures (5)

Fig. 1
Fig. 1

Gain spectra versus detuning for optimized PSAs made from (a) lossless fiber of length L = 3.63 km and Im { γ Ω } = 0 (solid curve); (b) same as (a), but Im { γ Ω } calculated for dispersion-shifted fiber (DSF) as explained in the text (circles); (c) 4.44 km of DSF with L eff = 3.63 km for α a = α s = α p = 0.41 dB km and Im { γ Ω } calculated for DSF as explained in the text (dotted curve). Gain spectra versus detuning for optimized PSDs made from (d) lossless fiber of length L = 3.63 km and Im { γ Ω } = 0 (dashed curve); (b) same as (a), but Im { γ Ω } calculated for DSF as explained in the text (squares); (c) 4.44 km of DSF with L eff = 3.63 km for α a = α s = α p = 0.41 dB km and Im { γ Ω } calculated for DSF as explained in the text (dashed–dotted curve). Input pump power is 0.33 W , λ 0 = 1551.16 nm , pump wavelength is 1551.5 nm , and the dispersion slope is 57 ps ( nm 2 km ) .

Fig. 2
Fig. 2

Gain versus fiber length for a PSA made from dispersion-shifted fiber for (a) phase-sensitive deamplification with optimum power splitting in lossless fiber ( α a = α s = α p = 0 ) (thick solid line), (b) phase-sensitive deamplification with optimum power splitting in a lossy fiber ( α a = α s = α p = 0.25 dB km ) (dotted curve), (c) phase-sensitive deamplification in a lossless fiber with ζ a 2 = ζ s 2 (dashed–dotted curve), (d) phase-sensitive deamplification in a lossy fiber with ζ a 2 = ζ s 2 (dashed curve), (e) phase-sensitive amplification in a lossless fiber with ζ a 2 = ζ s 2 (squares) and optimum input power splitting (solid line under the squares), and (f) phase-sensitive amplification in a lossy fiber with ζ a 2 = ζ s 2 (circles) and optimum input power splitting (solid line under the circles). Input pump power is 4 W , pump–signal detuning is 1 THz , and phase matching is achieved at the input [ Δ k = 2 Re { γ Ω } I p ( 0 ) ] .

Fig. 3
Fig. 3

PSA NF versus gain for various detunings for a highly nonlinear fiber. For thick curves, the fiber attenuation is 0.75 dB km at pump, Stokes, and anti-Stokes wavelengths; for thin curves, the fiber is lossless. Ω 2 π = 13.8 THz (dashed curves); Ω 2 π = 1.38 THz (dashed–dotted curves); Ω 2 π = 40 GHz (solid curves); Ω 2 π = 0 Hz (dotted curves). Except for the dotted curves, phase matching at the input [ Δ k = 2 Re { γ Ω } I p ( 0 ) ] is achieved. For dotted curves, Δ k = 0 . The anti-Stokes and Stokes relative phase and power splitting at the input is for optimal classical gain. Initial pump power is 340 mW , γ ( 0 ) = 9 × 10 3 W 1 m 1 , the peak imaginary part of γ Ω is 3.5 × 10 3 W 1 m 1 . Fiber length is 1 km .

Fig. 4
Fig. 4

PSA gain and NF spectrum versus detuning. Anti-Stokes and Stokes relative phase and input power splitting is for optimal classical gain. Initial pump power is 300 mW , γ 0 = 2 × 10 3 W 1 m 1 , the peak imaginary part of γ Ω is 0.75 × 10 3 W 1 m 1 . Attenuation is 0.41 dB km at the pump, Stokes, and anti-Stokes wavelengths. Fiber length is 4 km . λ 0 = 1551.15 nm , λ p = 1555.5 nm , and the dispersion slope is 57 ps ( nm km 2 ) .

Fig. 5
Fig. 5

Squeezing versus nonlinear phase shift in a lossless fiber. Lower curves, without Raman effect. Upper curves, with Raman effect. Dashed curves signify that phase matching is achieved at the input [ Δ k = 2 Re { γ Ω } A p 2 ] . The thick dashed curve is for optimal LO power splitting; thin dashed curves are for equal LO power splitting. Dashed–dotted curves are for Δ k = ( 2 3 ) Re { γ Ω } A p 2 . Thick dashed–dotted curve is for optimal LO power splitting; thin dashed curves are for equal LO power splitting. In all other curves, Δ k = 0 . Raman effect neglected, dotted curve; Raman effect included and equal LO power splitting are shown with crosses; Raman effect included and optimal LO power splitting are shown with circles; cw limit of Eq. (99) with γ ¯ i ( 0 ) k T γ ¯ 0 = 0.026 , thin solid curve. Pump–signal detuning is 40 GHz .

Equations (99)

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n 2 = 3 χ ( 3 ) 4 ϵ 0 n 0 2 c ,
χ ( 3 ) χ 1111 ( 3 ) [ m 2 V 2 ] .
N 2 ( Ω ) = n 2 e + n 2 r F ( Ω ) .
γ Ω = 2 π N 2 ( Ω ) λ A eff [ 1 Wm ] ,
A ( t ) = A p + A s exp ( i Ω t ) + A a exp ( i Ω t )
A ( t ) z = i [ d τ γ ( t τ ) A * ( τ ) A ( τ ) ] A ( t ) α ( Ω ) 2 A ̃ ( Ω ) exp ( i Ω t ) d Ω ,
d A p d z = i γ 0 A p 2 A p α p 2 A p ,
d A a d z = i ( γ 0 + γ Ω ) A p 2 A a + i γ Ω A p 2 A s * exp ( i Δ k z ) α a 2 A a ,
d A s d z = i ( γ 0 + γ Ω ) A p 2 A a + i γ Ω A p 2 A a * exp ( i Δ k z ) α a 2 A s .
A a ( z , L ) = μ a ( z , L ) A a ( z ) + ν a ( z , L ) A s * ( z ) ,
A s ( z , L ) = μ s ( z , L ) A s ( z ) + ν s ( z , L ) A a * ( z ) ,
A p ( z ) = exp [ i γ 0 I p ( 0 ) z eff α p z 2 ] A p ( 0 ) ,
A a = B a exp [ i ( γ 0 + γ Ω ) I p ( 0 ) z eff α a z 2 ] ,
A s = B s exp [ i ( γ 0 + γ Ω ) I p ( 0 ) z eff α s z 2 ]
d B a d z eff = i γ Ω I p ( 0 ) exp [ f ( z eff ) ] B s * ,
d B s * d z eff = i γ Ω * I p ( 0 ) exp [ f ( z eff ) ] B a ,
f ( z eff ) = i [ γ Ω + γ Ω * ] I p ( 0 ) z eff ( α s α a + 2 i Δ k ) ln ( 1 α p z eff ) 2 α p .
F a = B a exp [ f ( z eff ) ] ,
F s * = B s * exp [ f ( z eff ) ] ,
d F a d z eff ( Γ + Λ 1 α p z eff ) F a = ξ 1 F s * ,
d F s * d z eff + ( Γ + Λ 1 α p z eff ) F s * = ξ 2 F a ,
1 1 α p z eff = n = 0 α p n z eff n
μ a ( z , L ) = exp [ p ( z , L ) ] n = 0 a n L eff n ( a 0 = 1 , s 0 * = 0 ) ,
μ s ( z , L ) = exp [ p ( z , L ) ] n = 0 s n L eff n ( a 0 = 0 , s 0 * = 1 ) ,
ν a ( z , L ) = exp [ p ( z , L ) ] n = 0 a n L eff n ( a 0 = 0 , s 0 * = 1 ) ,
ν s ( z , L ) = exp [ p ( z , L ) ] n = 0 s n L eff n ( a 0 = 1 , s 0 * = 0 ) ,
p ( z , L ) = i [ γ 0 + ( γ Ω γ Ω * ) 2 ] I p ( z ) L eff i Δ k ( L z ) 2 α a ( L z ) 4 α s ( L z ) 4 ,
a n = Γ a n 1 + ξ 1 s n 1 * + Λ j = 0 n 1 α p j a n 1 j n ,
s n * = Γ s n 1 * + ξ 2 a n 1 Λ j = 0 n 1 α p j s n 1 j * n .
μ a ( z , L ) = exp { i [ Δ k ( 2 γ 0 + γ Ω γ Ω * ) I p ] ( L z ) 2 } × { i κ 2 g sinh [ g ( L z ) ] + cosh [ g ( L z ) ] } ,
μ a ( z , L ) = exp { i [ Δ k ( 2 γ 0 + γ Ω γ Ω * ) I p ] ( L z ) 2 } { i κ * 2 g * sinh [ g * ( L z ) ] + cosh [ g * ( L z ) ] } ,
ν a ( z , L ) = exp { i [ Δ k ( 2 γ 0 + γ Ω γ Ω * ) I p ] ( L z ) 2 } i γ Ω A p ( z ) 2 g sinh [ g ( L z ) ] ,
ν s ( z , L ) = exp { i [ Δ k ( 2 γ 0 + γ Ω γ Ω * ) I p ] ( L z ) 2 } i γ Ω A p ( z ) 2 g * sinh [ g * ( L z ) ] .
μ a ( z , L ) = exp [ i ( 2 γ 0 + γ Ω γ Ω * 2 ) I p ( L z ) ] [ 1 + i γ Ω ( L z ) I p ] ,
μ s ( z , L ) = exp [ i ( 2 γ 0 + γ Ω γ Ω * 2 ) I p ( L z ) ] [ 1 + i γ Ω ( L z ) I p ] ,
ν a ( z , L ) = exp [ i ( 2 γ 0 + γ Ω γ Ω * 2 ) I p ( L z ) ] i γ Ω ( L z ) A p ( z ) 2 ,
ν s ( z , L ) = exp [ i ( 2 γ 0 + γ Ω γ Ω * 2 ) I p ( L z ) ] i γ Ω ( L z ) A p ( z ) 2 .
G = A a ( L ) 2 + A s ( L ) 2 A a ( 0 ) 2 + A s ( 0 ) 2 = μ a ζ a + ν a ζ s * 2 + μ s ζ s + ν s ζ a * 2 ζ a 2 + ζ s 2 = ( μ a 2 + ν s 2 ) ζ a 2 + ( ν a 2 + μ s 2 ) ζ s 2 + ( { ( μ s ν s * + μ a ν a * ) ζ a ζ s exp [ i ( θ a + θ s ) ] } + c.c. ) ζ a 2 + ζ s 2 .
θ PSA , opt = arg ( μ s ν s * + μ a ν a * ) ,
θ PSD , opt = π arg ( μ s ν s * + μ a ν a * )
ζ a 2 ζ a 2 + ζ s 2 = 1 2 [ 1 ± μ s 2 μ a 2 4 μ a ν a * + μ s ν s * 2 + ( μ s 2 μ a 2 ) 2 ] ,
G PSA = ( μ a 2 + μ s 2 + ν a 2 + ν s 2 ) 2 + 4 μ a ν a * + μ s ν s * 2 + ( μ s 2 μ a 2 ) 2 2 + ( μ s 2 μ a 2 ) ( ν a 2 ν s 2 ) 4 μ a ν a * + μ s ν s * 2 + ( μ s 2 μ a 2 ) 2 .
A ̂ ( t ) z = α ( Ω ) 2 A ̂ ̃ ( Ω ) exp ( i Ω t ) d Ω + i [ d τ γ ¯ ( t τ ) A ̂ ( τ ) A ̂ ( τ ) ] A ̂ ( t ) + i m ̂ ( z , t ) A ̂ ( t ) + l ̂ ( z , t ) ,
m ̂ ( z , t ) = 0 d Ω 2 Im { γ ¯ Ω } 2 π [ i d ̂ Ω ( z ) exp ( i Ω t ) i d ̂ Ω ( z ) exp ( i Ω t ) ] ,
l ̂ ( t ) = 0 d Ω α Ω 2 π exp ( i Ω t ) v ̂ Ω ( t )
[ A ̂ ( t ) , A ̂ ( t ) ] = δ ( t t ) ,
[ A ̂ ( t ) , A ̂ ( t ) ] = 0 .
[ d ̂ Ω ( z ) , d ̂ Ω ( z ) ] = δ ( Ω Ω ) δ ( z z ) ,
d ̂ Ω ( z ) d ̂ Ω ( z ) = δ ( Ω Ω ) δ ( z z ) n th
[ v ̂ ± Ω ( z ) , v ̂ ± Ω ( z ) ] = δ ( ± Ω ± Ω ) δ ( z z )
A ̂ ( t ) = A ̂ p + A ̂ s exp ( i Ω t ) + A ̂ a exp ( i Ω t ) .
d A ̂ p d z = i γ ¯ 0 A ̂ p A ̂ p A ̂ p + i ( γ ¯ 0 + γ ¯ Ω ) A ̂ s A ̂ s A ̂ p + i ( γ ¯ 0 + γ ¯ Ω ) A ̂ a A ̂ a A ̂ p + i ( γ ¯ Ω + γ ¯ Ω ) A ̂ p A ̂ s A ̂ a exp ( i Δ k z ) 2 Im { γ ¯ Ω } A ̂ a d ̂ Ω ( z ) exp [ i ( k p k a ) k z ] + 2 Im { γ ¯ Ω } A ̂ s d ̂ Ω ( z ) exp [ i ( k p k s ) k z ] + 2 Im { γ Ω 0 } A ̂ p [ d ̂ Ω 0 + ( z ) + d ̂ Ω 0 ( z ) ] α p 2 A ̂ p + α p v ̂ p ( z ) ,
d A ̂ a d z = i ( γ ¯ 0 + γ ¯ Ω ) A ̂ p A ̂ p A ̂ a + i γ ¯ 0 A ̂ a A ̂ a A ̂ a + i ( γ ¯ 0 + γ ¯ 2 Ω ) A ̂ s A ̂ s A ̂ a + i γ ¯ Ω A ̂ p 2 A ̂ s exp ( i Δ k z ) 2 Im { γ ¯ Ω } A ̂ p exp [ i ( k p k a ) z ] d ̂ Ω ( z ) α a 2 A ̂ a + α a v ̂ a ( z ) ,
d A ̂ s d z = i ( γ ¯ 0 + γ ¯ Ω ) A ̂ p A ̂ p A ̂ s + i γ ¯ 0 A ̂ s A ̂ s A ̂ s + i ( γ ¯ 0 + γ ¯ 2 Ω ) A ̂ a A ̂ a A ̂ s + i γ ¯ Ω A ̂ p 2 A ̂ a exp ( i Δ k z ) + 2 Im { γ ¯ Ω } A ̂ p exp [ i ( k p k s ) z ] d ̂ Ω ( z ) α s 2 A ̂ s + α s v ̂ s ( z ) .
A ¯ j = A ¯ j + a ̂ j ,
d A ¯ p d z = i γ ¯ 0 A ¯ p * A ¯ p A ¯ p + i ( γ ¯ 0 + γ ¯ Ω ) A ¯ s * A ¯ s A ¯ p + i ( γ ¯ 0 + γ ¯ Ω ) A ¯ a * A ¯ a A ¯ p + i ( γ ¯ Ω + γ ¯ Ω ) A ¯ p * A ¯ s A ¯ a exp ( i Δ k z ) α p 2 A ¯ p ,
d A ¯ a d z = i ( γ ¯ 0 + γ ¯ Ω ) A ¯ p * A ¯ p A ¯ a + i γ ¯ 0 A ¯ a * A ¯ a A ¯ a + i ( γ ¯ 0 + γ ¯ 2 Ω ) A ¯ s * A ¯ s A ¯ a + i γ ¯ Ω A ¯ p 2 A ¯ s * exp ( i Δ k z ) α a 2 , A ¯ a
d A ¯ s d z = i ( γ ¯ 0 + γ ¯ Ω ) A ¯ p * A ¯ p A ¯ s + i γ ¯ 0 A ¯ s * A ¯ s A ¯ s + i ( γ ¯ 0 + γ ¯ 2 Ω ) A ¯ a * A ¯ a A ¯ s + i γ ¯ Ω A ¯ p 2 A ¯ a * exp ( i Δ k z ) α s 2 A ¯ s .
d a ̂ p d z = i γ ¯ 0 ( 2 A ¯ p 2 a ̂ p + A ¯ p 2 a ̂ p ) + i ( γ ¯ 0 + γ ¯ Ω ) ( A ¯ s 2 a ̂ p + A ¯ s * A ¯ p a ̂ s + A ¯ s A ¯ p a ̂ s ) + i ( γ ¯ 0 + γ ¯ Ω ) ( A ¯ a 2 a ̂ p + A ¯ a * A ¯ p a ̂ a + A ¯ a A ¯ p a ̂ a ) + i ( γ ¯ Ω + γ ¯ Ω ) ( A ¯ p * A ¯ s a ̂ a + A ¯ p * A ¯ a a ̂ s + A ¯ s A ¯ a a ̂ p ) exp ( i Δ k z ) 2 Im { γ ¯ Ω } A ¯ a d ̂ Ω ( z ) exp [ i ( k p k a ) k z ] + 2 Im { γ ¯ Ω } A ¯ s d ̂ Ω ( z ) exp [ i ( k p k s ) k z ] + 2 Im { γ Ω 0 } A ¯ p [ d ̂ Ω 0 + ( z ) + d ̂ Ω 0 ( z ) ] α p 2 a ̂ p + α p v ̂ p ( z ) ,
d a ̂ a d z = i ( γ ¯ 0 + γ ¯ Ω ) ( A ¯ p 2 a ̂ a + A ¯ p * A ¯ a a ̂ p + A ¯ p A ¯ a a ̂ p ) + i γ ¯ 0 ( 2 A ¯ a 2 a ̂ a + A ¯ a 2 a ̂ a ) + i ( γ ¯ 0 + γ ¯ 2 Ω ) ( A ¯ s 2 a ̂ a + A ¯ s * A ¯ s a ̂ s + A ¯ s A ¯ a a ̂ s ) + i γ ¯ Ω ( A ¯ p 2 a ̂ s + 2 A ¯ p A ¯ s * a ̂ p ) exp ( i Δ k z ) + 2 Im { γ ¯ Ω } A ̂ p exp [ i ( k p k a ) z ] d ̂ Ω ( z ) α a 2 a ̂ a + α a v ̂ a ( z ) ,
d a ̂ s d z = i ( γ ¯ 0 + γ ¯ Ω ) ( A ¯ p 2 a ̂ s + A ¯ p * A ¯ s a ̂ p + A ¯ p A ¯ s a ̂ p ) + i γ ¯ 0 ( 2 A s 2 a ̂ s + A ¯ s 2 a ̂ s ) + i ( γ ¯ 0 + γ ¯ 2 Ω ) ( A ¯ a 2 a ̂ s + A ¯ a * A ¯ s a ̂ a + A ¯ a A ¯ s a ̂ a ) + i γ ¯ Ω ( A ¯ p 2 a ̂ a + 2 A ¯ p A ¯ a * a ̂ p ) exp ( i Δ k z ) 2 Im { γ ¯ Ω } A ̂ p exp [ i ( k p k s ) z ] d ̂ Ω ( z ) α s 2 α ̂ s + α s v ̂ s ( z ) .
d A ¯ p d z = i γ ¯ 0 A ¯ p 2 A ¯ p α p 2 A ¯ p ,
d A ¯ a d z = i ( γ ¯ 0 + γ ¯ Ω ) A ¯ p 2 A ¯ a + i γ ¯ Ω A ¯ p 2 A ¯ s * exp ( i Δ k z ) α a 2 A ¯ a ,
d A ¯ s d z = i ( γ ¯ 0 + γ ¯ Ω ) A ¯ p 2 A ¯ s + i γ ¯ Ω A ¯ p 2 A ¯ a * exp ( i Δ k z ) α s 2 A ¯ s .
2 Im { γ ¯ Ω } A ¯ p 2 ( d ̂ Ω d ̂ Ω + d ̂ Ω d ̂ Ω ) 2 4 γ ¯ Ω 2 A ¯ a 2 A ¯ p 2 ( d ̂ p d ̂ p + d ̂ p d ̂ p ) 2 = Im { γ ¯ Ω } ( n th + 1 ) 2 γ ¯ Ω 2 n ̂ a .
d a ̂ p d z = 0 .
d a ̂ a d z = i ( γ ¯ 0 + γ ¯ Ω ) A ¯ p 2 a ̂ a + i γ ¯ Ω A ¯ p 2 a ̂ s exp ( i Δ k z ) + 2 Im { γ ¯ Ω } A ̂ p exp [ i ( k p k a ) z ] d ̂ Ω ( z ) α a 2 a ̂ a + α a v ̂ a ( z ) ,
d a ̂ s d z = i ( γ ¯ 0 + γ ¯ Ω ) A ¯ p 2 a ̂ s + i γ ¯ Ω A ¯ p 2 a ̂ a exp ( i Δ k z ) 2 Im { γ ¯ Ω } A ̂ p exp [ i ( k p k s ) z ] d ̂ Ω ( z ) α s 2 a ̂ s + α s v ̂ s ( z ) .
d A ¯ p d z = i γ ¯ 0 A ¯ p 2 A ¯ p α p 2 A ¯ p ,
d A ̂ a d z = i ( γ ¯ 0 + γ ¯ Ω ) A ¯ p 2 A ̂ a + i γ ¯ Ω A ¯ p 2 A ̂ s exp ( i Δ k z ) α a 2 A ̂ a + 2 Im { γ ¯ Ω } A ¯ p exp [ i ( k p k a ) z ] d ̂ Ω ( z ) + α a v ̂ a ( z ) ,
d A ̂ s d z = i ( γ ¯ 0 + γ ¯ Ω ) A ¯ p 2 A ̂ s + i γ ¯ Ω A ¯ p 2 A ̂ a exp ( i Δ k z ) α s 2 A ̂ s 2 Im { γ ¯ Ω } A ¯ p exp [ i ( k p k s ) z ] d ̂ Ω ( z ) + α s v ̂ s ( z ) .
A ̂ s ( L ) = μ a ( 0 , L ) A ̂ a ( 0 ) + ν a ( 0 , L ) A ̂ s ( 0 ) + 2 Im { γ ¯ Ω } 0 L d z A ¯ p ( z ) exp [ i ( k p k a ) z ] [ μ a ( z , L ) ν a ( z , L ) ] d ̂ Ω ( z ) + 0 L d z [ α a μ a ( z , L ) v ̂ a ( z ) + α s ν a ( z , L ) v ̂ s ( z ) ] ,
A ̂ s ( L ) = μ s ( 0 , L ) A ̂ s ( 0 ) + ν s ( 0 , L ) A ̂ a ( 0 ) + 2 Im { γ ¯ Ω } 0 L d z A ¯ p ( z ) exp [ i ( k p k s ) z ] [ μ s ( z , L ) + ν s ( z , L ) ] d ̂ Ω ( z ) + 0 L d z [ α s μ s ( z , L ) v ̂ s ( z ) + α a ν s ( z , L ) v ̂ a ( z ) ] .
NF = SNR in SNR out .
SNR in = ( n ̂ a + n ̂ s ) 2 Δ n ̂ a 2 + Δ n ̂ s 2 = ( ζ a 2 + ζ s 2 ) 2 ζ a 2 + ζ s 2 = ζ a 2 + ζ s 2 .
NF = ( ζ a 2 + ζ s 2 ) ( Δ n ̂ PI 2 + Δ n ̂ PS 2 ) ( P a + P s ) 2 ,
P a = μ a 2 ζ a 2 + ν a 2 ζ s 2 + ( ζ a ζ s μ a ν a * + c.c. ) ,
P s = μ s 2 ζ s 2 + ν s 2 ζ a 2 + ( ζ a ζ s μ s ν s * + c.c. ) .
Δ n ̂ PI 2 = P a B a + P s B s ,
Δ n ̂ PS 2 = 2 Q * B 1 + 2 Q B 2 ,
B j = μ j 2 + ν j 2 + ( 2 n th + 1 ) r j 2 + c j 1 2 + c j 2 2 , ( j = a , s ) ,
Q = ( μ a ζ a + ν a ζ s * ) ( μ s ζ s + ν s ζ a * ) ,
B 1 = c x 1 + r x ( n th + 1 ) + μ a ν s ,
B 2 = c x 2 * + r x * n th + μ s * ν a *
r a 2 = 2 Im { γ ¯ Ω } 0 L d z A ¯ p ( z ) 2 μ a ( z , L ) ν a ( z , L ) 2 ,
r s 2 = 2 Im { γ ¯ Ω } 0 L d z A ¯ p ( z ) 2 μ s ( z , L ) ν s ( z , L ) 2 ,
c a ( s ) 1 2 = 0 L d z α a ( s ) μ a ( s ) ( z , L ) 2 ,
c a ( s ) 2 2 = 0 L d z α s ( a ) ν a ( s ) ( z , L ) 2 ,
r x = 2 Im { γ ¯ Ω } 0 L d z A ¯ p 2 ( z ) exp ( i Δ k z ) [ μ a ( z , L ) ν a ( z , L ) ] [ μ s ( z , L ) + ν s ( z , L ) ] ,
c x 1 = α a 0 L d z μ a ( z , L ) ν s ( z , L ) ,
c x 2 * = α s 0 L d z ν a ( z , L ) * μ s ( z , L ) * .
NF PSA , Ω 0 = 1 + 4 k T γ ¯ i ( 0 ) γ ¯ 0 [ 1 ϕ NL 1 + ϕ NL 2 ] 1 + 2 ϕ NL 2 + 2 ϕ NL 1 + ϕ NL 2 ,
I ̂ = b ̂ a q ̂ a + b ̂ s q ̂ s + H . c . ,
y j = α LO , j 2 α LO , a 2 + α LO , s 2 ,
S = Δ I ̂ 2 Δ I ̂ 2 vac = [ 1 + 2 ( ν a 2 + r a 2 n th ) ] y a + { 1 + 2 [ ν s 2 + r s 2 ( 1 + n th ) ] } y s + 2 { [ μ s ν a ( 1 + n th ) μ a ν s n th ] exp [ i ( θ a + θ s ) ] + c.c. } y a y s .
θ a + θ s = π + arg [ μ s ν a ( 1 + n th ) μ a ν a n th ] .
y a = 1 2 { 1 + r s 2 ( 1 + n th ) r a 2 n th 4 μ s ν a ( 1 + n th ) μ a ν s n th 2 + [ r s 2 ( n th + 1 ) r a 2 n th ] 2 } .
S opt = 1 + ν a 2 + ν s 2 + r a 2 n th + r s 2 ( n th + 1 ) 4 μ s ν a ( 1 + n th ) μ a ν s n th 2 + [ r s 2 ( 1 + n th ) r a 2 n th ] 2 .
S opt ( Ω 0 ) = 1 + 2 ϕ NL [ ϕ NL + 2 k T γ ¯ i ( 0 ) γ ¯ 0 ] 2 ϕ NL { 1 + [ ϕ NL + 2 k T γ ¯ i ( 0 ) γ ¯ 0 ] 2 } 1 2 ,

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