Abstract

A model is constructed for the Raman effect as responsible for the self-frequency shift of solitons. The effect is related to and compared with the measured Raman gain in silica fibers. The model is quantized and includes the thermal and the quantum noise of the optical phonons generating the Raman effect. It is shown that the Raman effect causes excess noise for ultrashort pulses and limits the squeezing of subpicosecond solitons.

© 1994 Optical Society of America

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  1. R. M. Shelby, M. D. Levenson, and P. W. Bayer, “Guided acoustic-wave Brillouin scattering,” Phys. Rev. B 31, 5244–5252 (1985).
    [CrossRef]
  2. K. Bergman, H. A. Haus, and M. Shirasaki, “Analysis and measurement of GAWBS spectrum in a nonlinear fiber ring,” Appl. Phys. B 55, 242–249 (1992).
    [CrossRef]
  3. R. M. Shelby, M. D. Levenson, S. H. Perlmutter, R. G. DeVoe, and D. F. Walls, “Broadband parametric deamplification of quantum noise in an optical fiber,” Phys. Rev. Lett. 57, 691–694 (1986).
    [CrossRef] [PubMed]
  4. M. Shirasaki and H. A. Haus, “Squeezing of pulses in a nonlinear interferometer,” J. Opt. Soc. Am. B 7, 30–34 (1990).
    [CrossRef]
  5. M. Rosenbluh and R. M. Shelby, “Squeezing optical solitons,” Phys. Rev. Lett. 66, 153–156 (1991).
    [CrossRef] [PubMed]
  6. K. Bergman and H. A. Haus, “Squeezing in fibers with optical pulses,” Opt. Lett. 16, 663–665 (1991).
    [CrossRef] [PubMed]
  7. M. Shirasaki and H. A. Haus, “Reduction of guided-acoustic-wave Brillouin scattering noise in a squeezer,” Opt. Lett. 17, 1225–1227 (1992).
    [CrossRef] [PubMed]
  8. K. Bergman, C. R. Doerr, H. A. Haus, and M. Shirasaki, “Sub-shot-noise measurement with fiber-squeezed optical pulses,” Opt. Lett. 18, 643–645 (1993).
    [CrossRef] [PubMed]
  9. K. Bergman, H. A. Haus, and M. Shirasaki, “Squeezing and suppression of guided-acoustic-wave Brillouin scattering with 1-GHz pulses,” in Annual Meeting, Vol. 16 of 1993 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1993), paper WII1, p. 142.
  10. T. von Foerster and R. J. Glauber, “Quantum theory of light propagation in amplifying media,” Phys. Rev. A 3, 1484–1511 (1971).
    [CrossRef]
  11. D. F. Walls, “A master equation approach to the Raman effect,” J. Phys. A 6, 496–605 (1973).
    [CrossRef]
  12. Shen and N. Bloembergen, “Theory of stimulated Brillouin and Raman scattering,” Phys. Rev. 137, A1787–A1805 (1965).
    [CrossRef]
  13. S. J. Carter and P. D. Drummond, “Squeezed quantum solitons and Raman noise,” Phys. Rev. Lett. 67, 3757–3750 (1991).
    [CrossRef] [PubMed]
  14. P. D. Drummond and A. D. Hardman, “Simulation of quantum effects in Raman-active waveguides,” Europhys. Lett. 21, 279–284 (1993).
    [CrossRef]
  15. R. M. Shelby, E. H. Whittaker, P. D. Drummond, S. H. Carter, and M. Rosenbluh, “Quantum noise and thermal noise in optical soliton propagation,” in Quantum Electronics and Laser Science, Vol. 13 of 1992 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1992), paper QWA4, p. 112.
  16. D. Dougherty, F. X. Kärtner, I. P. Ippen, and H. A. Haus, “Low-frequency Raman gain measurements,” to be submitted to Opt. Lett.
  17. E. Marx, ed., G. Placek and Marx Handbuch der Radiologie, 2nd ed. (Academische Verlagsgesellschaft, Leipzig, Germany, 1934), Vol. 6, Part 2, pp. 209–374.
  18. R. H. Stolen and E. P. Ippen, “Raman gain in glass optical waveguides,” Appl. Phys. Lett. 22, 276–278 (1973).
    [CrossRef]
  19. R. W. Hellwarth, J. Cherlow, and T.-T. Yang, “Origin and frequency dependence of nonlinear optical susceptibilities of glasses,” Phys. Rev. B 11, 964–967 (1975).
    [CrossRef]
  20. R. H. Stolen and M. A. Bösch, “Low-frequency and low-temperature Raman scattering in silica fibers,” J. Opt. Soc. Am. B 10, 475–484 (1993).
  21. R. H. Stolen, J. P. Gordon, W. J. Tomlinson, and H. A. Haus, “Raman response function of silica-core fibers,” J. Opt. Soc. Am. B 6, 1159–1166 (1989).
    [CrossRef]
  22. G. P. Agarwal, Nonlinear Fiber Optics (Academic, San Diego, Calif., 1989).
  23. R. K. John, J. H. Shapiro, and P. Kumar, “Classical and quantum noise transformations produced by self-phase modulation,” in International Quantum Electronics Conference, Vol. 21 of 1987 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1987), p. 204.
  24. K. J. Blow, R. Loudon, and S. J. D. Phoenix, “Exact solution for quantum self-phase modulation,” J. Opt. Soc. Am. B 8, 1750–1756 (1991).
    [CrossRef]
  25. F. X. Kärtner, L. Joneckis, and H. A. Haus, “Classical and quantum dynamics of a pulse in a dispersionless nonlinear fiber,” Quantum Opt. 4, 379–396 (1992).
    [CrossRef]
  26. L. Joneckis and J. Shapiro, “Quantum propagation in a Kerr medium: lossless, dispersionless fiber,” J. Opt. Soc. Am. B 10, 1102–1120 (1993).
    [CrossRef]
  27. L. Boivin, F. X. Kärtner, and H. A. Haus, “Quantum theory of self-phase modulation with finite response time,” submitted to Phys. Rev. Lett.
  28. P. D. Drummond and S. J. Carter, “Quantum-field theory of squeezing in solitons,” J. Opt. Soc. Am. B 4, 1565–1573 (1987).
    [CrossRef]
  29. P. D. Drummond, “Electromagnetic quantization in dispersive inhomogeneous nonlinear dielectrics,” Phys. Rev. A 42, 6845–6857 (1990).
    [CrossRef] [PubMed]
  30. B. Yurke and S. Denker, “Quantum network theory,” Phys. Rev. A 29, 1419 (1984).
    [CrossRef]
  31. R. M. Mitschke and L. F. Mollenauer, “Discovery of the soliton self-frequency shift,” Opt. Lett. 11, 659–661 (1986).
    [CrossRef] [PubMed]
  32. J. P. Gordon, “Theory of the soliton self-frequency shift,” Opt. Lett. 11, 662–664 (1986).
    [CrossRef] [PubMed]
  33. H. A. Haus and Y. Lai, “Quantum theory of soliton squeezing: a linearized approach,” Opt. Soc. Am. B 7, 386–392 (1990).
    [CrossRef]
  34. D. J. Kaup, “Perturbation theory of solitons in optical fibers,” Phys. Rev. A 42, 5689–5694 (1990).
    [CrossRef] [PubMed]
  35. Y. Lai, “Quantum theory of soliton propagation: a unified approach based on the linearization approximation,” J. Opt. Soc. Am. B 10, 475–484 (1993).
    [CrossRef]
  36. H. A. Haus and F. X. Kärtner, “Quantization of the nonlinear Schrödinger equation,” Phys. Rev. A 46, R1175–R1176 (1992).
    [CrossRef]
  37. R. M. Shelby, P. D. Drummond, and S. J. Carter, “Phase noise scaling in quantum soliton propagation,” Phys. Rev. A 42, 2966–2976 (1990).
    [CrossRef] [PubMed]

1993 (5)

1992 (4)

H. A. Haus and F. X. Kärtner, “Quantization of the nonlinear Schrödinger equation,” Phys. Rev. A 46, R1175–R1176 (1992).
[CrossRef]

F. X. Kärtner, L. Joneckis, and H. A. Haus, “Classical and quantum dynamics of a pulse in a dispersionless nonlinear fiber,” Quantum Opt. 4, 379–396 (1992).
[CrossRef]

M. Shirasaki and H. A. Haus, “Reduction of guided-acoustic-wave Brillouin scattering noise in a squeezer,” Opt. Lett. 17, 1225–1227 (1992).
[CrossRef] [PubMed]

K. Bergman, H. A. Haus, and M. Shirasaki, “Analysis and measurement of GAWBS spectrum in a nonlinear fiber ring,” Appl. Phys. B 55, 242–249 (1992).
[CrossRef]

1991 (4)

M. Rosenbluh and R. M. Shelby, “Squeezing optical solitons,” Phys. Rev. Lett. 66, 153–156 (1991).
[CrossRef] [PubMed]

K. Bergman and H. A. Haus, “Squeezing in fibers with optical pulses,” Opt. Lett. 16, 663–665 (1991).
[CrossRef] [PubMed]

S. J. Carter and P. D. Drummond, “Squeezed quantum solitons and Raman noise,” Phys. Rev. Lett. 67, 3757–3750 (1991).
[CrossRef] [PubMed]

K. J. Blow, R. Loudon, and S. J. D. Phoenix, “Exact solution for quantum self-phase modulation,” J. Opt. Soc. Am. B 8, 1750–1756 (1991).
[CrossRef]

1990 (5)

P. D. Drummond, “Electromagnetic quantization in dispersive inhomogeneous nonlinear dielectrics,” Phys. Rev. A 42, 6845–6857 (1990).
[CrossRef] [PubMed]

R. M. Shelby, P. D. Drummond, and S. J. Carter, “Phase noise scaling in quantum soliton propagation,” Phys. Rev. A 42, 2966–2976 (1990).
[CrossRef] [PubMed]

H. A. Haus and Y. Lai, “Quantum theory of soliton squeezing: a linearized approach,” Opt. Soc. Am. B 7, 386–392 (1990).
[CrossRef]

D. J. Kaup, “Perturbation theory of solitons in optical fibers,” Phys. Rev. A 42, 5689–5694 (1990).
[CrossRef] [PubMed]

M. Shirasaki and H. A. Haus, “Squeezing of pulses in a nonlinear interferometer,” J. Opt. Soc. Am. B 7, 30–34 (1990).
[CrossRef]

1989 (1)

1987 (1)

1986 (3)

R. M. Mitschke and L. F. Mollenauer, “Discovery of the soliton self-frequency shift,” Opt. Lett. 11, 659–661 (1986).
[CrossRef] [PubMed]

J. P. Gordon, “Theory of the soliton self-frequency shift,” Opt. Lett. 11, 662–664 (1986).
[CrossRef] [PubMed]

R. M. Shelby, M. D. Levenson, S. H. Perlmutter, R. G. DeVoe, and D. F. Walls, “Broadband parametric deamplification of quantum noise in an optical fiber,” Phys. Rev. Lett. 57, 691–694 (1986).
[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]

1984 (1)

B. Yurke and S. Denker, “Quantum network theory,” Phys. Rev. A 29, 1419 (1984).
[CrossRef]

1975 (1)

R. W. Hellwarth, J. Cherlow, and T.-T. Yang, “Origin and frequency dependence of nonlinear optical susceptibilities of glasses,” Phys. Rev. B 11, 964–967 (1975).
[CrossRef]

1973 (2)

R. H. Stolen and E. P. Ippen, “Raman gain in glass optical waveguides,” Appl. Phys. Lett. 22, 276–278 (1973).
[CrossRef]

D. F. Walls, “A master equation approach to the Raman effect,” J. Phys. A 6, 496–605 (1973).
[CrossRef]

1971 (1)

T. von Foerster and R. J. Glauber, “Quantum theory of light propagation in amplifying media,” Phys. Rev. A 3, 1484–1511 (1971).
[CrossRef]

1965 (1)

Shen and N. Bloembergen, “Theory of stimulated Brillouin and Raman scattering,” Phys. Rev. 137, A1787–A1805 (1965).
[CrossRef]

Agarwal, G. P.

G. P. Agarwal, Nonlinear Fiber Optics (Academic, San Diego, Calif., 1989).

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.

K. Bergman, C. R. Doerr, H. A. Haus, and M. Shirasaki, “Sub-shot-noise measurement with fiber-squeezed optical pulses,” Opt. Lett. 18, 643–645 (1993).
[CrossRef] [PubMed]

K. Bergman, H. A. Haus, and M. Shirasaki, “Analysis and measurement of GAWBS spectrum in a nonlinear fiber ring,” Appl. Phys. B 55, 242–249 (1992).
[CrossRef]

K. Bergman and H. A. Haus, “Squeezing in fibers with optical pulses,” Opt. Lett. 16, 663–665 (1991).
[CrossRef] [PubMed]

K. Bergman, H. A. Haus, and M. Shirasaki, “Squeezing and suppression of guided-acoustic-wave Brillouin scattering with 1-GHz pulses,” in Annual Meeting, Vol. 16 of 1993 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1993), paper WII1, p. 142.

Bloembergen, N.

Shen and N. Bloembergen, “Theory of stimulated Brillouin and Raman scattering,” Phys. Rev. 137, A1787–A1805 (1965).
[CrossRef]

Blow, K. J.

Boivin, L.

L. Boivin, F. X. Kärtner, and H. A. Haus, “Quantum theory of self-phase modulation with finite response time,” submitted to Phys. Rev. Lett.

Bösch, M. A.

Carter, S. H.

R. M. Shelby, E. H. Whittaker, P. D. Drummond, S. H. Carter, and M. Rosenbluh, “Quantum noise and thermal noise in optical soliton propagation,” in Quantum Electronics and Laser Science, Vol. 13 of 1992 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1992), paper QWA4, p. 112.

Carter, S. J.

S. J. Carter and P. D. Drummond, “Squeezed quantum solitons and Raman noise,” Phys. Rev. Lett. 67, 3757–3750 (1991).
[CrossRef] [PubMed]

R. M. Shelby, P. D. Drummond, and S. J. Carter, “Phase noise scaling in quantum soliton propagation,” Phys. Rev. A 42, 2966–2976 (1990).
[CrossRef] [PubMed]

P. D. Drummond and S. J. Carter, “Quantum-field theory of squeezing in solitons,” J. Opt. Soc. Am. B 4, 1565–1573 (1987).
[CrossRef]

Cherlow, J.

R. W. Hellwarth, J. Cherlow, and T.-T. Yang, “Origin and frequency dependence of nonlinear optical susceptibilities of glasses,” Phys. Rev. B 11, 964–967 (1975).
[CrossRef]

Denker, S.

B. Yurke and S. Denker, “Quantum network theory,” Phys. Rev. A 29, 1419 (1984).
[CrossRef]

DeVoe, R. G.

R. M. Shelby, M. D. Levenson, S. H. Perlmutter, R. G. DeVoe, and D. F. Walls, “Broadband parametric deamplification of quantum noise in an optical fiber,” Phys. Rev. Lett. 57, 691–694 (1986).
[CrossRef] [PubMed]

Doerr, C. R.

Dougherty, D.

D. Dougherty, F. X. Kärtner, I. P. Ippen, and H. A. Haus, “Low-frequency Raman gain measurements,” to be submitted to Opt. Lett.

Drummond, P. D.

P. D. Drummond and A. D. Hardman, “Simulation of quantum effects in Raman-active waveguides,” Europhys. Lett. 21, 279–284 (1993).
[CrossRef]

S. J. Carter and P. D. Drummond, “Squeezed quantum solitons and Raman noise,” Phys. Rev. Lett. 67, 3757–3750 (1991).
[CrossRef] [PubMed]

R. M. Shelby, P. D. Drummond, and S. J. Carter, “Phase noise scaling in quantum soliton propagation,” Phys. Rev. A 42, 2966–2976 (1990).
[CrossRef] [PubMed]

P. D. Drummond, “Electromagnetic quantization in dispersive inhomogeneous nonlinear dielectrics,” Phys. Rev. A 42, 6845–6857 (1990).
[CrossRef] [PubMed]

P. D. Drummond and S. J. Carter, “Quantum-field theory of squeezing in solitons,” J. Opt. Soc. Am. B 4, 1565–1573 (1987).
[CrossRef]

R. M. Shelby, E. H. Whittaker, P. D. Drummond, S. H. Carter, and M. Rosenbluh, “Quantum noise and thermal noise in optical soliton propagation,” in Quantum Electronics and Laser Science, Vol. 13 of 1992 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1992), paper QWA4, p. 112.

Glauber, R. J.

T. von Foerster and R. J. Glauber, “Quantum theory of light propagation in amplifying media,” Phys. Rev. A 3, 1484–1511 (1971).
[CrossRef]

Gordon, J. P.

Hardman, A. D.

P. D. Drummond and A. D. Hardman, “Simulation of quantum effects in Raman-active waveguides,” Europhys. Lett. 21, 279–284 (1993).
[CrossRef]

Haus, H. A.

K. Bergman, C. R. Doerr, H. A. Haus, and M. Shirasaki, “Sub-shot-noise measurement with fiber-squeezed optical pulses,” Opt. Lett. 18, 643–645 (1993).
[CrossRef] [PubMed]

M. Shirasaki and H. A. Haus, “Reduction of guided-acoustic-wave Brillouin scattering noise in a squeezer,” Opt. Lett. 17, 1225–1227 (1992).
[CrossRef] [PubMed]

K. Bergman, H. A. Haus, and M. Shirasaki, “Analysis and measurement of GAWBS spectrum in a nonlinear fiber ring,” Appl. Phys. B 55, 242–249 (1992).
[CrossRef]

H. A. Haus and F. X. Kärtner, “Quantization of the nonlinear Schrödinger equation,” Phys. Rev. A 46, R1175–R1176 (1992).
[CrossRef]

F. X. Kärtner, L. Joneckis, and H. A. Haus, “Classical and quantum dynamics of a pulse in a dispersionless nonlinear fiber,” Quantum Opt. 4, 379–396 (1992).
[CrossRef]

K. Bergman and H. A. Haus, “Squeezing in fibers with optical pulses,” Opt. Lett. 16, 663–665 (1991).
[CrossRef] [PubMed]

H. A. Haus and Y. Lai, “Quantum theory of soliton squeezing: a linearized approach,” Opt. Soc. Am. B 7, 386–392 (1990).
[CrossRef]

M. Shirasaki and H. A. Haus, “Squeezing of pulses in a nonlinear interferometer,” J. Opt. Soc. Am. B 7, 30–34 (1990).
[CrossRef]

R. H. Stolen, J. P. Gordon, W. J. Tomlinson, and H. A. Haus, “Raman response function of silica-core fibers,” J. Opt. Soc. Am. B 6, 1159–1166 (1989).
[CrossRef]

L. Boivin, F. X. Kärtner, and H. A. Haus, “Quantum theory of self-phase modulation with finite response time,” submitted to Phys. Rev. Lett.

D. Dougherty, F. X. Kärtner, I. P. Ippen, and H. A. Haus, “Low-frequency Raman gain measurements,” to be submitted to Opt. Lett.

K. Bergman, H. A. Haus, and M. Shirasaki, “Squeezing and suppression of guided-acoustic-wave Brillouin scattering with 1-GHz pulses,” in Annual Meeting, Vol. 16 of 1993 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1993), paper WII1, p. 142.

Hellwarth, R. W.

R. W. Hellwarth, J. Cherlow, and T.-T. Yang, “Origin and frequency dependence of nonlinear optical susceptibilities of glasses,” Phys. Rev. B 11, 964–967 (1975).
[CrossRef]

Ippen, E. P.

R. H. Stolen and E. P. Ippen, “Raman gain in glass optical waveguides,” Appl. Phys. Lett. 22, 276–278 (1973).
[CrossRef]

Ippen, I. P.

D. Dougherty, F. X. Kärtner, I. P. Ippen, and H. A. Haus, “Low-frequency Raman gain measurements,” to be submitted to Opt. Lett.

John, R. K.

R. K. John, J. H. Shapiro, and P. Kumar, “Classical and quantum noise transformations produced by self-phase modulation,” in International Quantum Electronics Conference, Vol. 21 of 1987 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1987), p. 204.

Joneckis, L.

L. Joneckis and J. Shapiro, “Quantum propagation in a Kerr medium: lossless, dispersionless fiber,” J. Opt. Soc. Am. B 10, 1102–1120 (1993).
[CrossRef]

F. X. Kärtner, L. Joneckis, and H. A. Haus, “Classical and quantum dynamics of a pulse in a dispersionless nonlinear fiber,” Quantum Opt. 4, 379–396 (1992).
[CrossRef]

Kärtner, F. X.

F. X. Kärtner, L. Joneckis, and H. A. Haus, “Classical and quantum dynamics of a pulse in a dispersionless nonlinear fiber,” Quantum Opt. 4, 379–396 (1992).
[CrossRef]

H. A. Haus and F. X. Kärtner, “Quantization of the nonlinear Schrödinger equation,” Phys. Rev. A 46, R1175–R1176 (1992).
[CrossRef]

L. Boivin, F. X. Kärtner, and H. A. Haus, “Quantum theory of self-phase modulation with finite response time,” submitted to Phys. Rev. Lett.

D. Dougherty, F. X. Kärtner, I. P. Ippen, and H. A. Haus, “Low-frequency Raman gain measurements,” to be submitted to Opt. Lett.

Kaup, D. J.

D. J. Kaup, “Perturbation theory of solitons in optical fibers,” Phys. Rev. A 42, 5689–5694 (1990).
[CrossRef] [PubMed]

Kumar, P.

R. K. John, J. H. Shapiro, and P. Kumar, “Classical and quantum noise transformations produced by self-phase modulation,” in International Quantum Electronics Conference, Vol. 21 of 1987 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1987), p. 204.

Lai, Y.

Y. Lai, “Quantum theory of soliton propagation: a unified approach based on the linearization approximation,” J. Opt. Soc. Am. B 10, 475–484 (1993).
[CrossRef]

H. A. Haus and Y. Lai, “Quantum theory of soliton squeezing: a linearized approach,” Opt. Soc. Am. B 7, 386–392 (1990).
[CrossRef]

Levenson, M. D.

R. M. Shelby, M. D. Levenson, S. H. Perlmutter, R. G. DeVoe, and D. F. Walls, “Broadband parametric deamplification of quantum noise in an optical fiber,” Phys. Rev. Lett. 57, 691–694 (1986).
[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]

Loudon, R.

Mitschke, R. M.

Mollenauer, L. F.

Perlmutter, S. H.

R. M. Shelby, M. D. Levenson, S. H. Perlmutter, R. G. DeVoe, and D. F. Walls, “Broadband parametric deamplification of quantum noise in an optical fiber,” Phys. Rev. Lett. 57, 691–694 (1986).
[CrossRef] [PubMed]

Phoenix, S. J. D.

Rosenbluh, M.

M. Rosenbluh and R. M. Shelby, “Squeezing optical solitons,” Phys. Rev. Lett. 66, 153–156 (1991).
[CrossRef] [PubMed]

R. M. Shelby, E. H. Whittaker, P. D. Drummond, S. H. Carter, and M. Rosenbluh, “Quantum noise and thermal noise in optical soliton propagation,” in Quantum Electronics and Laser Science, Vol. 13 of 1992 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1992), paper QWA4, p. 112.

Shapiro, J.

Shapiro, J. H.

R. K. John, J. H. Shapiro, and P. Kumar, “Classical and quantum noise transformations produced by self-phase modulation,” in International Quantum Electronics Conference, Vol. 21 of 1987 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1987), p. 204.

Shelby, R. M.

M. Rosenbluh and R. M. Shelby, “Squeezing optical solitons,” Phys. Rev. Lett. 66, 153–156 (1991).
[CrossRef] [PubMed]

R. M. Shelby, P. D. Drummond, and S. J. Carter, “Phase noise scaling in quantum soliton propagation,” Phys. Rev. A 42, 2966–2976 (1990).
[CrossRef] [PubMed]

R. M. Shelby, M. D. Levenson, S. H. Perlmutter, R. G. DeVoe, and D. F. Walls, “Broadband parametric deamplification of quantum noise in an optical fiber,” Phys. Rev. Lett. 57, 691–694 (1986).
[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]

R. M. Shelby, E. H. Whittaker, P. D. Drummond, S. H. Carter, and M. Rosenbluh, “Quantum noise and thermal noise in optical soliton propagation,” in Quantum Electronics and Laser Science, Vol. 13 of 1992 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1992), paper QWA4, p. 112.

Shen,

Shen and N. Bloembergen, “Theory of stimulated Brillouin and Raman scattering,” Phys. Rev. 137, A1787–A1805 (1965).
[CrossRef]

Shirasaki, M.

K. Bergman, C. R. Doerr, H. A. Haus, and M. Shirasaki, “Sub-shot-noise measurement with fiber-squeezed optical pulses,” Opt. Lett. 18, 643–645 (1993).
[CrossRef] [PubMed]

M. Shirasaki and H. A. Haus, “Reduction of guided-acoustic-wave Brillouin scattering noise in a squeezer,” Opt. Lett. 17, 1225–1227 (1992).
[CrossRef] [PubMed]

K. Bergman, H. A. Haus, and M. Shirasaki, “Analysis and measurement of GAWBS spectrum in a nonlinear fiber ring,” Appl. Phys. B 55, 242–249 (1992).
[CrossRef]

M. Shirasaki and H. A. Haus, “Squeezing of pulses in a nonlinear interferometer,” J. Opt. Soc. Am. B 7, 30–34 (1990).
[CrossRef]

K. Bergman, H. A. Haus, and M. Shirasaki, “Squeezing and suppression of guided-acoustic-wave Brillouin scattering with 1-GHz pulses,” in Annual Meeting, Vol. 16 of 1993 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1993), paper WII1, p. 142.

Stolen, R. H.

Tomlinson, W. J.

von Foerster, T.

T. von Foerster and R. J. Glauber, “Quantum theory of light propagation in amplifying media,” Phys. Rev. A 3, 1484–1511 (1971).
[CrossRef]

Walls, D. F.

R. M. Shelby, M. D. Levenson, S. H. Perlmutter, R. G. DeVoe, and D. F. Walls, “Broadband parametric deamplification of quantum noise in an optical fiber,” Phys. Rev. Lett. 57, 691–694 (1986).
[CrossRef] [PubMed]

D. F. Walls, “A master equation approach to the Raman effect,” J. Phys. A 6, 496–605 (1973).
[CrossRef]

Whittaker, E. H.

R. M. Shelby, E. H. Whittaker, P. D. Drummond, S. H. Carter, and M. Rosenbluh, “Quantum noise and thermal noise in optical soliton propagation,” in Quantum Electronics and Laser Science, Vol. 13 of 1992 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1992), paper QWA4, p. 112.

Yang, T.-T.

R. W. Hellwarth, J. Cherlow, and T.-T. Yang, “Origin and frequency dependence of nonlinear optical susceptibilities of glasses,” Phys. Rev. B 11, 964–967 (1975).
[CrossRef]

Yurke, B.

B. Yurke and S. Denker, “Quantum network theory,” Phys. Rev. A 29, 1419 (1984).
[CrossRef]

Appl. Phys. B (1)

K. Bergman, H. A. Haus, and M. Shirasaki, “Analysis and measurement of GAWBS spectrum in a nonlinear fiber ring,” Appl. Phys. B 55, 242–249 (1992).
[CrossRef]

Appl. Phys. Lett. (1)

R. H. Stolen and E. P. Ippen, “Raman gain in glass optical waveguides,” Appl. Phys. Lett. 22, 276–278 (1973).
[CrossRef]

Europhys. Lett. (1)

P. D. Drummond and A. D. Hardman, “Simulation of quantum effects in Raman-active waveguides,” Europhys. Lett. 21, 279–284 (1993).
[CrossRef]

J. Opt. Soc. Am. B (7)

J. Phys. A (1)

D. F. Walls, “A master equation approach to the Raman effect,” J. Phys. A 6, 496–605 (1973).
[CrossRef]

Opt. Lett. (5)

Opt. Soc. Am. B (1)

H. A. Haus and Y. Lai, “Quantum theory of soliton squeezing: a linearized approach,” Opt. Soc. Am. B 7, 386–392 (1990).
[CrossRef]

Phys. Rev. (1)

Shen and N. Bloembergen, “Theory of stimulated Brillouin and Raman scattering,” Phys. Rev. 137, A1787–A1805 (1965).
[CrossRef]

Phys. Rev. A (6)

T. von Foerster and R. J. Glauber, “Quantum theory of light propagation in amplifying media,” Phys. Rev. A 3, 1484–1511 (1971).
[CrossRef]

D. J. Kaup, “Perturbation theory of solitons in optical fibers,” Phys. Rev. A 42, 5689–5694 (1990).
[CrossRef] [PubMed]

P. D. Drummond, “Electromagnetic quantization in dispersive inhomogeneous nonlinear dielectrics,” Phys. Rev. A 42, 6845–6857 (1990).
[CrossRef] [PubMed]

B. Yurke and S. Denker, “Quantum network theory,” Phys. Rev. A 29, 1419 (1984).
[CrossRef]

H. A. Haus and F. X. Kärtner, “Quantization of the nonlinear Schrödinger equation,” Phys. Rev. A 46, R1175–R1176 (1992).
[CrossRef]

R. M. Shelby, P. D. Drummond, and S. J. Carter, “Phase noise scaling in quantum soliton propagation,” Phys. Rev. A 42, 2966–2976 (1990).
[CrossRef] [PubMed]

Phys. Rev. B (2)

R. W. Hellwarth, J. Cherlow, and T.-T. Yang, “Origin and frequency dependence of nonlinear optical susceptibilities of glasses,” Phys. Rev. B 11, 964–967 (1975).
[CrossRef]

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. Lett. (3)

S. J. Carter and P. D. Drummond, “Squeezed quantum solitons and Raman noise,” Phys. Rev. Lett. 67, 3757–3750 (1991).
[CrossRef] [PubMed]

M. Rosenbluh and R. M. Shelby, “Squeezing optical solitons,” Phys. Rev. Lett. 66, 153–156 (1991).
[CrossRef] [PubMed]

R. M. Shelby, M. D. Levenson, S. H. Perlmutter, R. G. DeVoe, and D. F. Walls, “Broadband parametric deamplification of quantum noise in an optical fiber,” Phys. Rev. Lett. 57, 691–694 (1986).
[CrossRef] [PubMed]

Quantum Opt. (1)

F. X. Kärtner, L. Joneckis, and H. A. Haus, “Classical and quantum dynamics of a pulse in a dispersionless nonlinear fiber,” Quantum Opt. 4, 379–396 (1992).
[CrossRef]

Other (7)

G. P. Agarwal, Nonlinear Fiber Optics (Academic, San Diego, Calif., 1989).

R. K. John, J. H. Shapiro, and P. Kumar, “Classical and quantum noise transformations produced by self-phase modulation,” in International Quantum Electronics Conference, Vol. 21 of 1987 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1987), p. 204.

L. Boivin, F. X. Kärtner, and H. A. Haus, “Quantum theory of self-phase modulation with finite response time,” submitted to Phys. Rev. Lett.

K. Bergman, H. A. Haus, and M. Shirasaki, “Squeezing and suppression of guided-acoustic-wave Brillouin scattering with 1-GHz pulses,” in Annual Meeting, Vol. 16 of 1993 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1993), paper WII1, p. 142.

R. M. Shelby, E. H. Whittaker, P. D. Drummond, S. H. Carter, and M. Rosenbluh, “Quantum noise and thermal noise in optical soliton propagation,” in Quantum Electronics and Laser Science, Vol. 13 of 1992 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1992), paper QWA4, p. 112.

D. Dougherty, F. X. Kärtner, I. P. Ippen, and H. A. Haus, “Low-frequency Raman gain measurements,” to be submitted to Opt. Lett.

E. Marx, ed., G. Placek and Marx Handbuch der Radiologie, 2nd ed. (Academische Verlagsgesellschaft, Leipzig, Germany, 1934), Vol. 6, Part 2, pp. 209–374.

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Figures (5)

Fig. 1
Fig. 1

Solid curve, normalized measured Raman gain according to Ref. 16 for frequencies less than 100 GHz, extrapolated to zero detuning by f3. Dashed curve, linear approximation. Note that positive and negative frequencies are interchanged to conform with the literature.

Fig. 2
Fig. 2

Raman-scattering noise SR(ω) for room temperature T = 300 K.

Fig. 3
Fig. 3

(a) Noise reduction for soliton squeezing Rmin as a function of the phase shift of the soliton Φ and the POPS parameter C. (b) Fluctuations in the antisqueezing component Rmax as a function of the phase shift of the soliton Φ and the POPS parameter C.

Fig. 4
Fig. 4

(a) Normalized slope of the Raman gain as it contributes to the Kerr effect. b) Spectral distribution of the weight function |h(f)|2, which projects the phase noise for a 100-fs soliton. (c) Thermal enhancement of the Raman noise for different temperatures.

Fig. 5
Fig. 5

POPS parameter for solitons with pulse widths from 100 fs to 10 ps for different temperatures.

Equations (85)

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z A s = i μ v g Q ( ω q ) A p ,
2 t 2 Q + Γ t Q + Ω 2 Q = μ Ω [ A s * A p exp ( i ω q t ) + A p * A s exp ( - i ω q t ) ] .
A s z = i A p 2 κ R Ω 2 Ω 2 - ω q 2 - i ω q Γ A s ,
κ R μ 2 v g Ω ,
g R ( ω q ) = - κ R 2 Ω 2 Γ ω q ( Ω 2 - ω q 2 ) 2 + ( ω q Γ ) 2 .
A z = i κ R A 2 A .
A s ( ω ) z = i κ R ( ω ) A p 2 A s ( ω ) + 1 2 g R ( ω ) A p 2 A s ( ω ) ,
g R , tot ( ω ) = - 0 D ( Ω ) 2 κ R ( Ω ) Ω 2 Γ ( Ω ) ω ( Ω 2 - ω 2 ) 2 + [ ω Γ ( Ω ) ] 2 d Ω ,
z A ( z , t ) = i μ v g Q ( z , t ) A ( z , t ) ,
2 t 2 Q ( z , t ) + Γ t Q ( z , t ) + Ω 2 Q ( z , t ) = μ Ω A * ( z , t ) A ( z , t ) .
G ( ω ) = 1 Ω 2 - ω 2 - i ω Γ ,
G ( t - t ) = u ( t - t ) exp [ ( - Γ / 2 ) ( t - t ) ] [ Ω 2 - ( Γ / 2 ) 2 ] 1 / 2 × sin { [ Ω 2 - ( Γ 2 ) 2 ] 1 / 2 ( t - t ) } ,
u ( t ) = { 1 t 0 0 t < 0 .
f ( t - t ) = κ R Ω 2 G ( t - t ) ,
A ( z , t ) z = i - t f ( t - t ) A ( z , t ) 2 d t A ( z , t ) .
- t f ( t - t ) A ( z , t ) 2 d t = n = 0 1 n ! F n n A ( z , t ) 2 t n ,
F n = ( - 1 ) n 0 t n f ( t ) d t
F n = 2 π i n d n d ω n f ( ω ) ω = 0 ,
f ( t ) = - f ( ω ) exp ( - i ω t ) d ω , f ( ω ) = - f ( t ) exp ( i ω t ) d t 2 π .
A z = i κ R ( A 2 A - T R A 2 t A ) ,
T R Γ / Ω 2 .
2 κ R T R - g R ( ω ) ω | ω = 0 .
A z = i κ ( A 2 A - T res t A 2 A ) ,
T res = η T R .
κ R = - 1 π 0 g R ( ω ) ω d ω .
η = κ R / κ = 18.6 % ,
H ^ = - d z { ω 0 A ^ ( z ) A ^ ( z ) + i v g [ A ^ ( z ) z A ^ ( z ) - A ^ ( z ) A ^ ( z ) z ] + Ω 2 [ P ^ 2 ( z ) + Q ^ 2 ( z ) ] - μ A ^ ( z ) A ^ ( z ) Q ^ ( z ) + 0 ω b ^ ω ( z ) b ^ ω ( z ) d ω + i Q ^ ( z ) 0 ( ω Γ π Ω ) 1 / 2 [ b ^ ω ( z ) - b ^ ω ( z ) ] d ω } .
[ A ^ ( z ) , A ^ ( z ) ] = δ ( z - z ) , [ b ^ ω ( z ) , b ^ ω ( z ) ] = δ ( ω - ω ) δ ( z - z ) , [ Q ^ ( z ) , P ^ ( z ) ] = i δ ( z - z ) ,
t A ^ = - i ω 0 A ^ - v g z A ^ + i μ Q ^ A ^ ,
t Q ^ = Ω P ^ ,
t P ^ = - Ω Q ^ + μ A ^ A ^ + i 0 ( ω Γ π Ω ) 1 / 2 ( b ^ ω - b ^ ω ) d ω ,
t b ^ ω = - i ω b ^ ω - ( ω Γ π Ω ) 1 / 2 Q ^ .
b ^ ω ( z , t ) = b ^ ω ( z , 0 ) exp ( - i ω t ) - 0 t ( ω Γ π Ω ) 1 / 2 Q ^ ( z , t ) × exp [ - i ω ( t - t ) ] d t ,
b ^ ω ( z , 0 ) b ^ ω ( z , 0 ) = n th ( ω ) δ ( z - z ) δ ( ω - ω ) , b ^ ω ( z , 0 ) b ^ ω ( z , 0 ) = [ n th ( ω ) + 1 ] δ ( z - z ) δ ( ω - ω ) ,
n th ( ω ) = 1 exp ( ω / k T ) - 1 .
t P ^ ( z , t ) = - Ω Q ^ ( z , t ) + μ A ^ ( z , t ) A ^ ( z , t ) + 2 Γ 0 t Q ^ ( z , t ) t δ ( t - t ) d t + ( 2 Γ Ω ) 1 / 2 ξ ^ ( z , t ) ,
ξ ^ ( z , t ) = i 0 ( ω 2 π ) 1 / 2 × [ b ^ ω ( z , 0 ) exp ( i ω t ) - b ^ ω ( z , 0 ) exp ( - i ω t ) ] d ω
ξ ^ ( z , ω ) = i ( ω 2 ω ) 1 / 2 { - b ^ ω ( z , 0 ) ω > 0 b ^ ω ( z , 0 ) ω < 0 .
2 t 2 Q ^ ( z , t ) + Γ t Q ^ ( z , t ) + Ω 2 Q ^ ( z , t ) = μ Ω A ^ ( z , t ) A ^ ( z , t ) + ( 2 Γ ) 1 / 2 Ω ξ ^ ( z , t ) .
A ^ z = i μ Q ^ A ^ ,
2 t 2 Q ^ + Γ t Q ^ + Ω 2 Q ^ = μ Ω A ^ A ^ + ( 2 Γ ) 1 / 2 Ω ξ ^ .
z A ^ ( z , t ) = i - t f ( t - t ) A ^ ( z , t ) A ^ ( z , t ) d t A ^ ( z , t ) + i A ^ ( z , t ) m ^ ( z , t )
m ^ ( z , t ) = ( 2 T R κ R ) 1 / 2 - t f ( t - t ) ξ ^ ( z , t ) d t .
ξ ^ ( z , ω ) ξ ^ ( z , ω ) = ω 2 π [ n th ( ω ) + u ( - ω ) ] × δ ( z - z ) δ ( ω - ω )
ξ ^ ( z , ω ) ξ ^ ( z , ω ) = ω 4 π [ coth ( ω 2 k T ) - 1 ] × δ ( z - z ) δ ( ω - ω ) .
Δ A ^ ( z = l ) = i A p 0 l m ^ ( z , t ) d z ,
S R ( ω ) = 2 π l A p 2 d ω Δ A ^ ( l , ω ) Δ A ^ ( l , ω ) = g R ( ω ) [ n th ( ω ) + u ( - ω ) ] ,
A ^ z = - i β 2 2 A ^ t 2 + i ( κ - κ R ) A ^ A ^ A ^ + i - t f ( t - t ) A ^ ( z , t ) A ^ ( z , t ) d t A ^ ( z , t ) + i A ^ ( z , t ) m ^ ( z , t ) .
A ( z , t ) = A 0 ( t ) exp [ i Φ ( z ) ] ,
A 0 ( t ) = A 0 sech ( t / τ )
Φ ( z ) = β z 2 τ 2 .
κ A 0 2 = β τ 2
w 0 = 2 τ A 0 2 .
d d z Δ w ^ ( z ) = N ^ w ( z ) ,
d d z Δ θ ^ ( z ) = - β τ 2 Δ w ^ ( z ) w 0 + N ^ θ ( z ) ,
d d z Δ p ^ ( z ) = N ^ p ( z ) ,
d d z Δ t ^ ( z ) = β Δ p ^ ( z ) + N ^ t ( z ) ,
N ^ s ( z ) = i 2 d t [ f - s * ( t ) A 0 ( t ) m ^ ( z , t ) - f - s ( t ) A 0 * ( t ) m ^ ( z , t ) ] .
f - w ( t ) = 2 A 0 ( t ) ,
f - θ ( t ) = - i 2 w 0 [ 1 - t τ tanh ( t τ ) ] A 0 ( t ) ,
f - p ( t ) = i 2 w 0 τ tanh ( t τ ) A 0 ( t ) ,
f - t ( t ) = 2 w 0 t A 0 ( t ) .
N ^ s ( z ) = h s ( t ) m ^ ( z , t ) d t ,
h s ( t ) = i 2 [ f s * ( t ) - f s ( t ) ] A 0 ( t ) .
N ^ ω ( z ) N ^ ω ( z ) = 0 ,
N ^ t ( z ) N ^ t ( z ) = 0.
N ^ s ( z ) N ^ s ( z ) = h s ( t ) m ^ ( z , t ) m ^ ( z , t ) h s ( t ) d t d t .
N ^ s ( z ) N ^ s ( z ) = N s δ ( z - z ) ,
N s = 2 π h s ( ω ) 2 S R ( ω ) d ω .
Δ w 2 ( z ) = Δ w 2 ( 0 ) = w 0 ,
Δ θ ^ 2 ( z ) = Δ θ ^ 2 ( 0 ) + 4 Φ ( z ) 2 Δ w ^ 2 ( 0 ) w 0 2 + N θ z ,
Δ θ ^ ( z ) Δ w ( z ) = - 2 Φ ( z ) Δ w ^ 2 ( 0 ) w 0 ,
Δ θ ^ 2 ( 0 ) = 0.6075 / w 0 .
M ^ ( z ) = [ c n Δ n ^ ( z ) + c θ Δ θ ^ ( z ) ] ,
c n 2 Δ w 2 ( 0 ) + c θ 2 Δ θ 2 ( 0 ) = 1 / 4.
M ^ 2 ( z ) = [ c n + 2 c θ w 0 Φ ( z ) ] 2 Δ w ^ 2 ( 0 ) + c θ 2 Δ θ ^ 2 ( 0 ) + c θ 2 4 N θ τ w 0 κ Φ ( z ) ,
κ τ β = 2 w 0 .
R ( z ) = M ^ 2 ( z ) M ^ 2 ( 0 ) = 4 M ^ 2 ( z ) .
R max / min ( z ) = 1 + Φ ˜ ( z ) ( C + Φ ˜ ( z ) 2 ± { 1 + [ C + Φ ˜ ( z ) 2 ] 2 } 1 / 2 ) ,
Φ ˜ ( z ) = 2 w 0 [ Δ w 2 ( 0 ) Δ θ ^ 2 ( 0 ) ] 1 / 2 2.566 × Φ ( z )
C = 2 N θ τ [ Δ w 2 ( 0 ) Δ θ ^ 2 ( 0 ) κ ] 1 / 2 = 2 N θ τ 0.6075 κ .
C = 2 τ 2 π h θ ( ω ) 2 S R ( ω ) d w 0.6075 κ .
C = 2 π τ 2.01 0.6075 - h θ ( ω ) 2 g R ( ω ) g R , max coth ( w 2 k T ) d w .
2 π h θ ( ω ) 2 d ω = h θ ( t ) 2 d ω = 0.866 τ ,
C = 4.549 T res k T ,

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