Abstract

Short optical pulses in nonlinear fibers are susceptible to a variety of higher-order physical effects, including the Raman self-frequency shift and cubic and nonlinear dispersions. These effects directly modify pulse propagation and contribute to noise-induced phenomena such as the Gordon–Haus jitter. We show that phase-sensitive amplification, if used to compensate for loss, acts as a restoring force in frequency and compensates for the Raman self-frequency shift. Furthermore, phase-sensitive amplification controls the Gordon–Haus jitter, including the contributions of the Raman self-frequency shift and the third-order dispersion.

© 1997 Optical Society of America

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References

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  1. N. S. Bergano, “Undersea lightwave transmission systems using Er-doped fiber amplifiers,” Opt. Photonics News 4(1), 8–14 (1993).
    [CrossRef]
  2. N. Bergano, “Circulating loop transmission experiments for the study of long-haul transmission systems using erbium-doped fiber amplifiers,” J. Lightwave Technol. 13, 879–888 (1995).
    [CrossRef]
  3. A. Hasegawa and Y. Kodama, “Guiding-center soliton in optical fibers,” Opt. Lett. 15, 1443–1445 (1990).
    [CrossRef] [PubMed]
  4. L. F. Mollenauer, S. G. Evangelides, Jr., and H. A. Haus, “Long-distance soliton propagation using lumped amplifiers and dispersion shifted fiber,” J. Lightwave Technol. 9, 194–197 (1991).
    [CrossRef]
  5. L. F. Mollenauer, M. J. Neubelt, S. G. Evangelides, J. P. Gordon, J. R. Simpson, and L. G. Cohen, “Experimental study of soliton transmission over more than 10, 000 km in dispersion-shifted fiber,” Opt. Lett. 15, 1203–1205 (1990).
    [CrossRef] [PubMed]
  6. E. Desurvire, Erbium-Doped Fiber Amplifiers: Theory and Applications (Wiley, New York, 1994).
  7. J. P. Gordon and H. A. Haus, “Random walk of coherently amplified solitons in optical fiber transmission,” Opt. Lett. 11, 665–667 (1986).
    [CrossRef] [PubMed]
  8. A. Mecozzi, J. Moores, H. Haus, and Y. Lai, “Soliton transmission control,” Opt. Lett. 16, 1841–1843 (1991).
    [CrossRef] [PubMed]
  9. Y. Kodama and A. Hasegawa, “Generation of asymptotically stable optical solitons and suppression of the Gordon–Haus effect,” Opt. Lett. 17, 31–33 (1992).
    [CrossRef] [PubMed]
  10. L. F. Mollenauer, J. P. Gordon, and S. G. Evangelides, “The sliding-frequency guiding filter: an improved form of soliton jitter control,” Opt. Lett. 17, 1575–1577 (1992).
    [CrossRef] [PubMed]
  11. L. F. Mollenauer, P. V. Mamyshev, and M. J. Neubelt, “Measurement of timing jitter in filter-guided soliton transmission at 10 Gbits/s and achievement of 375 Gbit/s-Mm, error-free, at 12.5 and 15 Gbits/s,” Opt. Lett. 19, 704–706 (1994).
    [CrossRef] [PubMed]
  12. L. F. Mollenauer, E. Lichtman, M. J. Neubelt, and G. T. Harvey, “Demonstration, using sliding-frequency guiding filters, of error-free soliton transmission over more than 20 Mm at 10 Gbit/s, single channel, and over more than 13 Mm at 20 Gbit/s in a two-channel WDM,” Electron. Lett. 29, 910–911 (1993).
    [CrossRef]
  13. K. J. Blow, N. J. Doran, and D. Wood, “Suppression of the soliton self-frequency shift by bandwidth-limited amplification,” J. Opt. Soc. Am. B 5, 1301–1304 (1988).
    [CrossRef]
  14. M. Ding and K. Kikuchi, “Analysis of soliton transmission in optical fibers with the soliton self-frequency shift being compensated by distributed frequency dependent gain,” IEEE Photonics Technol. Lett. 4, 497–500 (1992).
    [CrossRef]
  15. G. P. Agrawal, Nonlinear Fiber Optics (Academic, Boston, Mass., 1989).
  16. F. M. Mitschke and L. F. Mollenauer, “Discovery of the soliton self-frequency shift,” Opt. Lett. 11, 659–661 (1986).
    [CrossRef] [PubMed]
  17. J. P. Gordon, “Theory of the soliton self-frequency shift,” Opt. Lett. 11, 662–664 (1986).
    [CrossRef] [PubMed]
  18. C. G. Goedde, W. L. Kath, and P. Kumar, “Compensation of the soliton self-frequency shift with phase-sensitive amplifiers,” Opt. Lett. 19, 2077–2079 (1994).
    [CrossRef] [PubMed]
  19. H. P. Yuen, “Reduction of quantum fluctuation and suppression of the Gordon–Haus effect with phase-sensitive linear amplifiers,” Opt. Lett. 17, 73–75 (1992).
    [CrossRef] [PubMed]
  20. S.-H. Lee, Ph. D. dissertation (Northwestern University, Evanston, Ill., 1992).
  21. I. H. Deutsch and I. Abram, “Reduction of quantum noise in soliton propagation by using phase sensitive amplification,” J. Opt. Soc. Am. B 11, 2303–2313 (1994).
    [CrossRef]
  22. A. Mecozzi, W. L. Kath, P. Kumar, and C. G. Goedde, “Long-term storage of a soliton bit stream by use of phase-sensitive amplification,” Opt. Lett. 19, 2050–2052 (1994).
    [CrossRef] [PubMed]
  23. J. N. Kutz, C. V. Hile, W. L. Kath, R.-D. Li, and P. Kumar, “Pulse propagation in nonlinear optical fiber lines that employ phase-sensitive parametric amplification,” J. Opt. Soc. Am. B 11, 2112–2123 (1994).
    [CrossRef]
  24. C. Kim and P. Kumar, “Quadrature-squeezed light detection using a self-generated matched local oscillator,” Phys. Rev. Lett. 73, 1605–1608 (1994).
    [CrossRef] [PubMed]
  25. W. Sohler and H. Suche, “Optical parametric amplification in Ti-diffused LiNbO3 waveguides,” Appl. Phys. Lett. 37, 255–257 (1980).
    [CrossRef]
  26. S. Helmfrid, F. Laurell, and G. Arvidsson, “Optical parametric amplification of a 1.54 μm single-mode DFB laser in a Ti:LiNbO3 waveguide,” J. Lightwave Technol. 11, 1459–1469 (1993).
    [CrossRef]
  27. M. L. Bortz, M. A. Arbore, and M. M. Fejer, “Quasi-phase-matched optical parametric amplification and oscillation in periodically poled LiNbO3 waveguides,” Opt. Lett. 20, 49–51 (1995).
    [CrossRef] [PubMed]
  28. M. Shirasaki and H. A. Haus, “Squeezing of pulses in a nonlinear interferometer,” J. Opt. Soc. Am. B 7, 30–34 (1990).
    [CrossRef]
  29. M. E. Marhic, C. H. Hsia, and J.-M. Jeong, “Optical amplification in a nonlinear fibre interferometer,” Electron. Lett. 27, 210–211 (1991).
    [CrossRef]
  30. G. Bartolini, R.-D. Li, P. Kumar, W. Riha, and K. V. Reddy, “1.5 μm phase-sensitive amplifier for ultra-high speed communications,” in Optical Fiber Communication Conference, Vol. 4 of 1994 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1994), pp. 202–203.
  31. A. Takada and W. Imajuki, “Amplitude noise suppression by using high-gain phase sensitive amplifier as a limiting amplifier,” Electron. Lett. 32, 677–679 (1996).
    [CrossRef]
  32. H. A. Haus, “Quantum noise in a solitonlike repeater system,” J. Opt. Soc. Am. B 8, 1122–1126 (1991).
    [CrossRef]
  33. A. Mecozzi, “Long-distance soliton transmission with filtering,” J. Opt. Soc. Am. B 10, 2321–2330 (1993).
    [CrossRef]
  34. B. Friedman, Principles and Techniques of Applied Mathematics (Wiley, New York, 1956).
  35. J. D. Moores, W. S. Wong, and H. A. Haus, “Stability and timing maintenance in soliton transmission and storage rings,” Opt. Commun. 113, 153–175 (1994).
    [CrossRef]
  36. Y. Kodama and A. Hasegawa, “Theoretical foundation of optical-soliton concept in fibers,” Prog. Opt. 30, 205–259 (1992).
    [CrossRef]
  37. V. I. Karpman and E. M. Maslov, “Perturbation theory for solitons,” Sov. Phys. JETP 46, 281–291 (1977).
  38. H. A. Haus and Y. Lai, “Quantum theory of soliton squeezing: a linearized approach,” J. Opt. Soc. Am. B 7, 386–392 (1990).
    [CrossRef]

1996 (1)

A. Takada and W. Imajuki, “Amplitude noise suppression by using high-gain phase sensitive amplifier as a limiting amplifier,” Electron. Lett. 32, 677–679 (1996).
[CrossRef]

1995 (2)

M. L. Bortz, M. A. Arbore, and M. M. Fejer, “Quasi-phase-matched optical parametric amplification and oscillation in periodically poled LiNbO3 waveguides,” Opt. Lett. 20, 49–51 (1995).
[CrossRef] [PubMed]

N. Bergano, “Circulating loop transmission experiments for the study of long-haul transmission systems using erbium-doped fiber amplifiers,” J. Lightwave Technol. 13, 879–888 (1995).
[CrossRef]

1994 (7)

1993 (4)

A. Mecozzi, “Long-distance soliton transmission with filtering,” J. Opt. Soc. Am. B 10, 2321–2330 (1993).
[CrossRef]

S. Helmfrid, F. Laurell, and G. Arvidsson, “Optical parametric amplification of a 1.54 μm single-mode DFB laser in a Ti:LiNbO3 waveguide,” J. Lightwave Technol. 11, 1459–1469 (1993).
[CrossRef]

L. F. Mollenauer, E. Lichtman, M. J. Neubelt, and G. T. Harvey, “Demonstration, using sliding-frequency guiding filters, of error-free soliton transmission over more than 20 Mm at 10 Gbit/s, single channel, and over more than 13 Mm at 20 Gbit/s in a two-channel WDM,” Electron. Lett. 29, 910–911 (1993).
[CrossRef]

N. S. Bergano, “Undersea lightwave transmission systems using Er-doped fiber amplifiers,” Opt. Photonics News 4(1), 8–14 (1993).
[CrossRef]

1992 (5)

1991 (4)

M. E. Marhic, C. H. Hsia, and J.-M. Jeong, “Optical amplification in a nonlinear fibre interferometer,” Electron. Lett. 27, 210–211 (1991).
[CrossRef]

H. A. Haus, “Quantum noise in a solitonlike repeater system,” J. Opt. Soc. Am. B 8, 1122–1126 (1991).
[CrossRef]

A. Mecozzi, J. Moores, H. Haus, and Y. Lai, “Soliton transmission control,” Opt. Lett. 16, 1841–1843 (1991).
[CrossRef] [PubMed]

L. F. Mollenauer, S. G. Evangelides, Jr., and H. A. Haus, “Long-distance soliton propagation using lumped amplifiers and dispersion shifted fiber,” J. Lightwave Technol. 9, 194–197 (1991).
[CrossRef]

1990 (4)

1988 (1)

1986 (3)

1980 (1)

W. Sohler and H. Suche, “Optical parametric amplification in Ti-diffused LiNbO3 waveguides,” Appl. Phys. Lett. 37, 255–257 (1980).
[CrossRef]

1977 (1)

V. I. Karpman and E. M. Maslov, “Perturbation theory for solitons,” Sov. Phys. JETP 46, 281–291 (1977).

Abram, I.

Arbore, M. A.

Arvidsson, G.

S. Helmfrid, F. Laurell, and G. Arvidsson, “Optical parametric amplification of a 1.54 μm single-mode DFB laser in a Ti:LiNbO3 waveguide,” J. Lightwave Technol. 11, 1459–1469 (1993).
[CrossRef]

Bergano, N.

N. Bergano, “Circulating loop transmission experiments for the study of long-haul transmission systems using erbium-doped fiber amplifiers,” J. Lightwave Technol. 13, 879–888 (1995).
[CrossRef]

Bergano, N. S.

N. S. Bergano, “Undersea lightwave transmission systems using Er-doped fiber amplifiers,” Opt. Photonics News 4(1), 8–14 (1993).
[CrossRef]

Blow, K. J.

Bortz, M. L.

Cohen, L. G.

Deutsch, I. H.

Ding, M.

M. Ding and K. Kikuchi, “Analysis of soliton transmission in optical fibers with the soliton self-frequency shift being compensated by distributed frequency dependent gain,” IEEE Photonics Technol. Lett. 4, 497–500 (1992).
[CrossRef]

Doran, N. J.

Evangelides, S. G.

Evangelides , Jr., S. G.

L. F. Mollenauer, S. G. Evangelides, Jr., and H. A. Haus, “Long-distance soliton propagation using lumped amplifiers and dispersion shifted fiber,” J. Lightwave Technol. 9, 194–197 (1991).
[CrossRef]

Fejer, M. M.

Goedde, C. G.

Gordon, J. P.

Harvey, G. T.

L. F. Mollenauer, E. Lichtman, M. J. Neubelt, and G. T. Harvey, “Demonstration, using sliding-frequency guiding filters, of error-free soliton transmission over more than 20 Mm at 10 Gbit/s, single channel, and over more than 13 Mm at 20 Gbit/s in a two-channel WDM,” Electron. Lett. 29, 910–911 (1993).
[CrossRef]

Hasegawa, A.

Haus, H.

Haus, H. A.

Helmfrid, S.

S. Helmfrid, F. Laurell, and G. Arvidsson, “Optical parametric amplification of a 1.54 μm single-mode DFB laser in a Ti:LiNbO3 waveguide,” J. Lightwave Technol. 11, 1459–1469 (1993).
[CrossRef]

Hile, C. V.

Hsia, C. H.

M. E. Marhic, C. H. Hsia, and J.-M. Jeong, “Optical amplification in a nonlinear fibre interferometer,” Electron. Lett. 27, 210–211 (1991).
[CrossRef]

Imajuki, W.

A. Takada and W. Imajuki, “Amplitude noise suppression by using high-gain phase sensitive amplifier as a limiting amplifier,” Electron. Lett. 32, 677–679 (1996).
[CrossRef]

Jeong, J.-M.

M. E. Marhic, C. H. Hsia, and J.-M. Jeong, “Optical amplification in a nonlinear fibre interferometer,” Electron. Lett. 27, 210–211 (1991).
[CrossRef]

Karpman, V. I.

V. I. Karpman and E. M. Maslov, “Perturbation theory for solitons,” Sov. Phys. JETP 46, 281–291 (1977).

Kath, W. L.

Kikuchi, K.

M. Ding and K. Kikuchi, “Analysis of soliton transmission in optical fibers with the soliton self-frequency shift being compensated by distributed frequency dependent gain,” IEEE Photonics Technol. Lett. 4, 497–500 (1992).
[CrossRef]

Kim, C.

C. Kim and P. Kumar, “Quadrature-squeezed light detection using a self-generated matched local oscillator,” Phys. Rev. Lett. 73, 1605–1608 (1994).
[CrossRef] [PubMed]

Kodama, Y.

Kumar, P.

Kutz, J. N.

Lai, Y.

Laurell, F.

S. Helmfrid, F. Laurell, and G. Arvidsson, “Optical parametric amplification of a 1.54 μm single-mode DFB laser in a Ti:LiNbO3 waveguide,” J. Lightwave Technol. 11, 1459–1469 (1993).
[CrossRef]

Li, R.-D.

Lichtman, E.

L. F. Mollenauer, E. Lichtman, M. J. Neubelt, and G. T. Harvey, “Demonstration, using sliding-frequency guiding filters, of error-free soliton transmission over more than 20 Mm at 10 Gbit/s, single channel, and over more than 13 Mm at 20 Gbit/s in a two-channel WDM,” Electron. Lett. 29, 910–911 (1993).
[CrossRef]

Mamyshev, P. V.

Marhic, M. E.

M. E. Marhic, C. H. Hsia, and J.-M. Jeong, “Optical amplification in a nonlinear fibre interferometer,” Electron. Lett. 27, 210–211 (1991).
[CrossRef]

Maslov, E. M.

V. I. Karpman and E. M. Maslov, “Perturbation theory for solitons,” Sov. Phys. JETP 46, 281–291 (1977).

Mecozzi, A.

Mitschke, F. M.

Mollenauer, L. F.

Moores, J.

Moores, J. D.

J. D. Moores, W. S. Wong, and H. A. Haus, “Stability and timing maintenance in soliton transmission and storage rings,” Opt. Commun. 113, 153–175 (1994).
[CrossRef]

Neubelt, M. J.

Shirasaki, M.

Simpson, J. R.

Sohler, W.

W. Sohler and H. Suche, “Optical parametric amplification in Ti-diffused LiNbO3 waveguides,” Appl. Phys. Lett. 37, 255–257 (1980).
[CrossRef]

Suche, H.

W. Sohler and H. Suche, “Optical parametric amplification in Ti-diffused LiNbO3 waveguides,” Appl. Phys. Lett. 37, 255–257 (1980).
[CrossRef]

Takada, A.

A. Takada and W. Imajuki, “Amplitude noise suppression by using high-gain phase sensitive amplifier as a limiting amplifier,” Electron. Lett. 32, 677–679 (1996).
[CrossRef]

Wong, W. S.

J. D. Moores, W. S. Wong, and H. A. Haus, “Stability and timing maintenance in soliton transmission and storage rings,” Opt. Commun. 113, 153–175 (1994).
[CrossRef]

Wood, D.

Yuen, H. P.

Appl. Phys. Lett. (1)

W. Sohler and H. Suche, “Optical parametric amplification in Ti-diffused LiNbO3 waveguides,” Appl. Phys. Lett. 37, 255–257 (1980).
[CrossRef]

Electron. Lett. (3)

M. E. Marhic, C. H. Hsia, and J.-M. Jeong, “Optical amplification in a nonlinear fibre interferometer,” Electron. Lett. 27, 210–211 (1991).
[CrossRef]

A. Takada and W. Imajuki, “Amplitude noise suppression by using high-gain phase sensitive amplifier as a limiting amplifier,” Electron. Lett. 32, 677–679 (1996).
[CrossRef]

L. F. Mollenauer, E. Lichtman, M. J. Neubelt, and G. T. Harvey, “Demonstration, using sliding-frequency guiding filters, of error-free soliton transmission over more than 20 Mm at 10 Gbit/s, single channel, and over more than 13 Mm at 20 Gbit/s in a two-channel WDM,” Electron. Lett. 29, 910–911 (1993).
[CrossRef]

IEEE Photonics Technol. Lett. (1)

M. Ding and K. Kikuchi, “Analysis of soliton transmission in optical fibers with the soliton self-frequency shift being compensated by distributed frequency dependent gain,” IEEE Photonics Technol. Lett. 4, 497–500 (1992).
[CrossRef]

J. Lightwave Technol. (3)

N. Bergano, “Circulating loop transmission experiments for the study of long-haul transmission systems using erbium-doped fiber amplifiers,” J. Lightwave Technol. 13, 879–888 (1995).
[CrossRef]

L. F. Mollenauer, S. G. Evangelides, Jr., and H. A. Haus, “Long-distance soliton propagation using lumped amplifiers and dispersion shifted fiber,” J. Lightwave Technol. 9, 194–197 (1991).
[CrossRef]

S. Helmfrid, F. Laurell, and G. Arvidsson, “Optical parametric amplification of a 1.54 μm single-mode DFB laser in a Ti:LiNbO3 waveguide,” J. Lightwave Technol. 11, 1459–1469 (1993).
[CrossRef]

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

Opt. Commun. (1)

J. D. Moores, W. S. Wong, and H. A. Haus, “Stability and timing maintenance in soliton transmission and storage rings,” Opt. Commun. 113, 153–175 (1994).
[CrossRef]

Opt. Lett. (13)

F. 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]

J. P. Gordon and H. A. Haus, “Random walk of coherently amplified solitons in optical fiber transmission,” Opt. Lett. 11, 665–667 (1986).
[CrossRef] [PubMed]

L. F. Mollenauer, M. J. Neubelt, S. G. Evangelides, J. P. Gordon, J. R. Simpson, and L. G. Cohen, “Experimental study of soliton transmission over more than 10, 000 km in dispersion-shifted fiber,” Opt. Lett. 15, 1203–1205 (1990).
[CrossRef] [PubMed]

A. Hasegawa and Y. Kodama, “Guiding-center soliton in optical fibers,” Opt. Lett. 15, 1443–1445 (1990).
[CrossRef] [PubMed]

A. Mecozzi, J. Moores, H. Haus, and Y. Lai, “Soliton transmission control,” Opt. Lett. 16, 1841–1843 (1991).
[CrossRef] [PubMed]

Y. Kodama and A. Hasegawa, “Generation of asymptotically stable optical solitons and suppression of the Gordon–Haus effect,” Opt. Lett. 17, 31–33 (1992).
[CrossRef] [PubMed]

H. P. Yuen, “Reduction of quantum fluctuation and suppression of the Gordon–Haus effect with phase-sensitive linear amplifiers,” Opt. Lett. 17, 73–75 (1992).
[CrossRef] [PubMed]

L. F. Mollenauer, J. P. Gordon, and S. G. Evangelides, “The sliding-frequency guiding filter: an improved form of soliton jitter control,” Opt. Lett. 17, 1575–1577 (1992).
[CrossRef] [PubMed]

L. F. Mollenauer, P. V. Mamyshev, and M. J. Neubelt, “Measurement of timing jitter in filter-guided soliton transmission at 10 Gbits/s and achievement of 375 Gbit/s-Mm, error-free, at 12.5 and 15 Gbits/s,” Opt. Lett. 19, 704–706 (1994).
[CrossRef] [PubMed]

A. Mecozzi, W. L. Kath, P. Kumar, and C. G. Goedde, “Long-term storage of a soliton bit stream by use of phase-sensitive amplification,” Opt. Lett. 19, 2050–2052 (1994).
[CrossRef] [PubMed]

C. G. Goedde, W. L. Kath, and P. Kumar, “Compensation of the soliton self-frequency shift with phase-sensitive amplifiers,” Opt. Lett. 19, 2077–2079 (1994).
[CrossRef] [PubMed]

M. L. Bortz, M. A. Arbore, and M. M. Fejer, “Quasi-phase-matched optical parametric amplification and oscillation in periodically poled LiNbO3 waveguides,” Opt. Lett. 20, 49–51 (1995).
[CrossRef] [PubMed]

Opt. Photonics News (1)

N. S. Bergano, “Undersea lightwave transmission systems using Er-doped fiber amplifiers,” Opt. Photonics News 4(1), 8–14 (1993).
[CrossRef]

Phys. Rev. Lett. (1)

C. Kim and P. Kumar, “Quadrature-squeezed light detection using a self-generated matched local oscillator,” Phys. Rev. Lett. 73, 1605–1608 (1994).
[CrossRef] [PubMed]

Prog. Opt. (1)

Y. Kodama and A. Hasegawa, “Theoretical foundation of optical-soliton concept in fibers,” Prog. Opt. 30, 205–259 (1992).
[CrossRef]

Sov. Phys. JETP (1)

V. I. Karpman and E. M. Maslov, “Perturbation theory for solitons,” Sov. Phys. JETP 46, 281–291 (1977).

Other (5)

B. Friedman, Principles and Techniques of Applied Mathematics (Wiley, New York, 1956).

G. Bartolini, R.-D. Li, P. Kumar, W. Riha, and K. V. Reddy, “1.5 μm phase-sensitive amplifier for ultra-high speed communications,” in Optical Fiber Communication Conference, Vol. 4 of 1994 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1994), pp. 202–203.

S.-H. Lee, Ph. D. dissertation (Northwestern University, Evanston, Ill., 1992).

E. Desurvire, Erbium-Doped Fiber Amplifiers: Theory and Applications (Wiley, New York, 1994).

G. P. Agrawal, Nonlinear Fiber Optics (Academic, Boston, Mass., 1989).

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Figures (9)

Fig. 1
Fig. 1

Evolution of the intensity |q|2 (arbitrary units) for a 3.5-ps pulse in a transmission line with PSA’s; (a) and (b) are two perspectives of the same pulse. See text for parameters.

Fig. 2
Fig. 2

Total change in frequency of a 3.5-ps pulse when PSA’s are used to control the Raman self-frequency shift.

Fig. 3
Fig. 3

Effect of the Raman self-frequency shift on the evolution of the intensity |q|2 of a pulse in a transmission line with PIA’s for the same parameters as in Fig. 1.

Fig. 4
Fig. 4

Comparison between the solution of the exact equations [Eqs. (1) and (2)] and the solution of the averaged equation [Eqs. (13) and (11b)] at z/Z0=150; solid curves denote the in-phase component, and dashed curves denote the quadrature component. (a) Amplitude of the solution. (b) Difference between the exact and the averaged solutions.

Fig. 5
Fig. 5

Comparison between the perturbation theory and the averaged solutions at z/Z0=150. (a) Amplitude of the solution. (b) Difference between the exact and the averaged solutions.

Fig. 6
Fig. 6

Variance of the timing jitter with PIA’s and linear fiber loss; solid curves denote variance of 400 numerical runs, and dashed curves denote the variance calculated from Eq. (40). Upper curves show the variance when the higher-order terms are included.

Fig. 7
Fig. 7

Variance of the timing jitter with PIA’s and both line and lumped loss; solid curves denote variance of 400 numerical runs, and dashed curves denote the variance calculated from Eq. (40). Upper curves show the variance when the higher-order terms are included.

Fig. 8
Fig. 8

Variance of the timing jitter with PSA’s and linear line loss. The solid curves denote the variance of 400 numerical runs, both with and without the higher-order terms; the dashed line indicates the variance calculated from Eq. (43).

Fig. 9
Fig. 9

Variance of the timing jitter with PSA’s and both line and lumped loss as calculated from 400 numerical runs. The upper curve shows the variance when the higher-order terms are included.

Equations (66)

Equations on this page are rendered with MathJax. Learn more.

qz=i|β|22qt2+iδ|q|2q-αq-iδTRqt|q|2+β63qt3-2δω0t|q|2q.
q+=(cosh g)exp(-F)q-+(sinh g)exp(iϕ-F)q-*.
q+=exp(g-F)q-.
q+=(A++iB+)exp(iϕ/2)=[A- exp(g-F)+iB- exp(-g-F)]exp(iϕ/2).
q=q0(ζ, z, ξ, t)+q1(ζ, z, ξ, t)+
=(A0+iB0)+(A1+iB1)+.
A0ζ+ΓZ0A0=0,
B0ζ+ΓZ0B0=0,
A0=Aˆ(z, ξ, t)exp(-Γζ/Z0),
B0=Bˆ(z, ξ, t)exp(-Γζ/Z0).
A0+=exp(g-F)A0-,
B0+=exp(-g-F)B0-.
A1ζ+ΓZ0A1=-A0z,
B1ζ+ΓZ0B1=[L+N(A0)]A0,
L=|β|22t2-ϕ(z)2
N(A)=δA2-2TRAAt
A0=Aˆ(ξ, t)exp(-Γζ/Z0),
B1=Z0exp(-2Γ-2F)1-exp(-2Γ-2F)+ζZ0LA0-Z02Γ1-exp(2Γζ/Z0)1-exp(-4Γ-2F)1-exp(-2Γ-2F)×N(A0)A0,
A2ζ+ΓZ0A2=-[L+N(A0)]B1-A0ξ+ΔgZ0A0+β63A0t3-2δω0A03t.
uξ+Z0C12[L+N(u)]2u+Z0C22[LN(u)-N(u)L]u
-β63ut3+2δω0u3t-ΔgZ0u=0,
u2=1-exp(-2Γ)2ΓAˆ2r2Aˆ2,
C1=coth(Γ+F),C2=coth(Γ)-1Γ.
Ωz=-8|β|TR15τ4.
u(ξ, t)=u0(ξ, t)+μu1(ξ, t)+ ,
u0ξ+Z0C12(L+δu02)2-2ΔgZ02C1u0=0.
u0ξ+Z0C12[(L-+δu02)(L++δu02)]u0=0,
L±=|β|22t2-ϕ2±2ΔgZ02C11/2
u0(ξ, t)=U0 sech(t/τ)
τ2=|β|ϕ+(8Δg/Z02C1)1/2,
U02=|β|δτ2.
ϕ=|β|τ021-8ΔgC11/2,
u1ξ+Z0C12N-(u0)N+(u0)u1
=Z0C1δTRN-(u0)u02u0T+β63u0T3
-6δω0u02u0T-λu0T,
N-(u)=L-+δu02,
N+(u)=L++3δu02.
[N-(u0)N+(u0)]v=0,
u, v=-uvdT.
u0T, vλ=β63u0T3, v-6δω0u02u0T, v+Z0C1δTRN-(u0)u02u0T, v.
λ=β6τ2I1-6|β|ω0τ2I2+C1|β|TRτ2I3,
λPIA=-8|β|2TR15τ4z.
f1(t)f1(t)=f2(t)f2(t)=G-12ηspδ(t-t)
f1(t)f1(t)=G-14δ(t-t),
f2(t)f2(t)=G-14G2δ(t-t)
a(t)=U0 sech[(t-T0)/τ],
U0=|β|δτ2,ψ=|β|2τ2,
q(z, t)=[a(t)+Δq(z, t)]exp(iψz+iθ0).
Δq(z, t)=fn(t)Δn(z)+fθ(t)Δθ(z)+fΩ(t)ΔΩ(z)+fT(t)ΔT(z)+Δqc,
zΔnn=Snn,
z(Δθ)=1LdΔnn+Sθ,
z(ΔΩτ)=-1LRΔnn+SΩτ,
zΔTτ=1LhoΔnn-1LdΔΩτ+STτ.
1LR=32|β|TR15τ3
1Lho=4|β|ω0τ3+β3τ3
Nn=2n(G-1)r2ηspZa,
Nθ=(G-1)r2ηspnZaπ218+23,
NΩ=2(G-1)r2ηsp3nZaτ2,
NT=π2(G-1)r2τ2ηsp6nZa.
ΔT2=τ2Nn20LR2Ld2n2z5+τ23Lho2Nnn2+Lho2Ld2τ2NΩ×z3+NTz.
zΔnn=-4LgΔθ+Snn,
z(Δθ)=1LdΔnn-2αΔθ+Sθ,
z(ΔΩτ)=-1LRΔnn-2αΔΩτ+SΩτ,
zΔTτ=1LhoΔnn+1μΔΩτ+STτ.
1μ=π23Lg-1Ld,
ΔT2=τ2Ld2Lho21+Lho2LRμα2Nθ+αLg2n2Nn+τ22µα2NΩ+NTz.

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