Abstract

The dynamics of a soliton propagating in a single-mode optical fiber with gain, loss, and Raman coupling to thermal phonons is analyzed. Using both soliton perturbation theory and exact numerical techniques, we propose that intrinsic thermal quantum noise from the phonon reservoirs is a larger source of jitter and other perturbations than the gain-related Gordon–Haus noise for short pulses (≲1 ps), assuming typical fiber parameters. The size of the Raman timing jitter is evaluated for both bright and dark (topological) solitons and is larger for bright solitons. Because Raman thermal quantum noise is a nonlinear, multiplicative noise source, these effects are stronger for the more intense pulses that are needed to propagate as solitons in the short-pulse regime. Thus Raman noise may place additional limitations on fiber-optical communications and networking by use of ultrafast (subpicosecond) pulses.

© 2001 Optical Society of America

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References

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  1. J. F. Corney, P. D. Drummond, and A. Liebman, “Quantum noise limits to terabaud communications,” Opt. Commun. 140, 211–215 (1997).
    [CrossRef]
  2. 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]
  3. 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]
  4. L. F. Mollenauer, J. P. Gordon, and M. N. Islam, “Soliton propagation in long fibers with periodically compensated loss,” IEEE J. Quantum Electron. 22, 157–173 (1986).
    [CrossRef]
  5. 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]
  6. A. K. Atieh, P. Myslinski, J. Chrostowski, and P. Galko, “Measuring the Raman time constant (TR) for soliton pulses in standard single-mode fiber,” J. Lightwave Technol. 17, 216–221 (1999).
    [CrossRef]
  7. D.-M. Baboiu, D. Mihalache, and N.-C. Panoiu, “Combined influence of amplifier noise and intrapulse Raman scattering on the bit-rate limit of optical fiber communication systems,” Opt. Lett. 20, 1865–1867 (1995); D. Mihalache, L.-C. Crasovan, N.-C. Panoiu, F. Moldoveanu, and D.-M. Baboiu, “Timing jitter of femtosecond solitons in monomode optical fibers,” Opt. Eng. 35, 1611–1615 (1996); D. Wood, “Constraints on the bit rates in direct detection optical communication systems using linear or soliton pulses,” J. Lightwave Technol. JLTEDG 8, 1097–1106 (1990).
    [CrossRef] [PubMed]
  8. D. Shenoy and A. Puri, “Compensation for the soliton self-frequency shift and the third-order dispersion using bandwidth-limited optical gain,” Opt. Commun. 113, 410–406 (1995); S. V. Chernikov and S. M. J. Kelly, “Stability of femtosecond solitons in optical fibres influenced by optical attenuation and bandwidth limited gain,” Electron. Lett. 28, 238–240 (1992).
    [CrossRef]
  9. G. P. Agrawal, Nonlinear Fiber Optics, 2nd ed. (Academic, San Diego, Calif. 1995), p. 475.
  10. P. D. Drummond and W. Man, “Quantum noise in reversible soliton logic,” Opt. Commun. 105, 99–103 (1994).
    [CrossRef]
  11. H. A. Haus and W. S. Wong, “Solitons in optical communications,” Rev. Mod. Phys. 68, 423–444 (1996).
    [CrossRef]
  12. F. X. Kartner, 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]
  13. D. J. Kaup, “Perturbation theory for solitons in optical fibers,” Phys. Rev. A 42, 5689–5694 (1990).
    [CrossRef] [PubMed]
  14. Y. S. Kivshar, M. Haelterman, P. Emplit, and J. P. Hamaide, “Gordon–Haus effect on dark solitons,” Opt. Lett. 19, 19–21 (1994); Y. S. Kivshar, “Dark solitons in nonlinear optics,” IEEE J. Quantum Electron. 29, 250–264 (1993).
    [CrossRef] [PubMed]
  15. I. M. Uzunov and V. S. Gerdjikov, “Self-frequency shift of dark solitons in optical fibers,” Phys. Rev. A 47, 1582–1585 (1993).
    [CrossRef] [PubMed]
  16. A. Hasegawa and F. Tappert, “Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers. II. Normal dispersion,” Appl. Phys. 23, 171–172 (1973).
  17. J. Hamaide, P. Emplit, and M. Haelterman, “Dark-soliton jitter in amplified optical transmission systems,” Opt. Lett. 16, 1578–1580 (1991).
    [CrossRef] [PubMed]
  18. R. H. Stolen, C. Lee, and R. K. Jain, “Development of the stimulated Raman spectrum in single-mode silica fibers,” J. Opt. Soc. Am. B 1, 652–657 (1984); D. J. Dougherty, F. X. Kartner, H. A. Haus, and E. P. Ippen, “Measurement of the Raman gain spectrum of optical fibers,” Opt. Lett. 20, 31–33 (1995); 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 JOBPDE 6, 1159–1166 (1989).
    [CrossRef] [PubMed]
  19. Y. Lai and S.-S. Yu, “General quantum theory of nonlinear optical-pulse propagation,” Phys. Rev. A 51, 817–829 (1995); S.-S. Yu and Y. Lai, “Impacts of self-Raman effect and third-order dispersion on pulse squeezed state generation using optical fibers,” J. Opt. Soc. Am. B 12, 2340–2346 (1995).
    [CrossRef] [PubMed]
  20. A. Mecozzi, M. Midrio, and M. Romagnoli, “Timing jitter in soliton transmission with sliding filters,” Opt. Lett. 21, 402–404 (1996); 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 Gbits/s-Mm, error free, at 12.5 and 15 Gbits/s,” Opt. Lett. 19, 704–706 (1994); 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 10000 km in dispersion-shifted fiber,” Opt. Lett. OPLEDP 15, 1203–1205 (1990).
    [CrossRef] [PubMed]
  21. P. D. Drummond, “Central partial difference propagation algorithms,” Comput. Phys. Commun. 29, 211–225 (1983).
    [CrossRef]
  22. P. D. Drummond and A. D. Hardman, “Simulation of quantum effects in Raman-active waveguides,” Europhys. Lett. 21, 279–284 (1993).
    [CrossRef]
  23. P. D. Drummond and I. K. Mortimer, “Computer simulations of multiplicative stochastic differential equations,” J. Comput. Phys. 93, 144–170 (1991).
    [CrossRef]
  24. M. J. Werner and P. D. Drummond, “Robust algorithms for solving stochastic partial differential equations,” J. Comput. Phys. 132, 312–326 (1997).
    [CrossRef]
  25. C. X. Yu, S. Namiki, and H. A. Haus, “Noise of the stretched pulse fiber laser. II. Experiments,” IEEE J. Quantum Electron. 33, 660–668 (1997).
    [CrossRef]
  26. S. Namiki, C. X. Yu, and H. A. Haus, “Observation of nearly quantum-limited timing jitter in an all-fiber ring laser,” J. Opt. Soc. Am. B 13, 2817–2823 (1996).
    [CrossRef]

2001

1999

1997

J. F. Corney, P. D. Drummond, and A. Liebman, “Quantum noise limits to terabaud communications,” Opt. Commun. 140, 211–215 (1997).
[CrossRef]

M. J. Werner and P. D. Drummond, “Robust algorithms for solving stochastic partial differential equations,” J. Comput. Phys. 132, 312–326 (1997).
[CrossRef]

C. X. Yu, S. Namiki, and H. A. Haus, “Noise of the stretched pulse fiber laser. II. Experiments,” IEEE J. Quantum Electron. 33, 660–668 (1997).
[CrossRef]

1996

1994

F. X. Kartner, 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]

P. D. Drummond and W. Man, “Quantum noise in reversible soliton logic,” Opt. Commun. 105, 99–103 (1994).
[CrossRef]

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]

1993

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

I. M. Uzunov and V. S. Gerdjikov, “Self-frequency shift of dark solitons in optical fibers,” Phys. Rev. A 47, 1582–1585 (1993).
[CrossRef] [PubMed]

1991

J. Hamaide, P. Emplit, and M. Haelterman, “Dark-soliton jitter in amplified optical transmission systems,” Opt. Lett. 16, 1578–1580 (1991).
[CrossRef] [PubMed]

P. D. Drummond and I. K. Mortimer, “Computer simulations of multiplicative stochastic differential equations,” J. Comput. Phys. 93, 144–170 (1991).
[CrossRef]

1990

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

1986

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, J. P. Gordon, and M. N. Islam, “Soliton propagation in long fibers with periodically compensated loss,” IEEE J. Quantum Electron. 22, 157–173 (1986).
[CrossRef]

1983

P. D. Drummond, “Central partial difference propagation algorithms,” Comput. Phys. Commun. 29, 211–225 (1983).
[CrossRef]

1973

A. Hasegawa and F. Tappert, “Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers. II. Normal dispersion,” Appl. Phys. 23, 171–172 (1973).

Atieh, A. K.

Chrostowski, J.

Corney, J. F.

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]

J. F. Corney, P. D. Drummond, and A. Liebman, “Quantum noise limits to terabaud communications,” Opt. Commun. 140, 211–215 (1997).
[CrossRef]

Dougherty, D. J.

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]

M. J. Werner and P. D. Drummond, “Robust algorithms for solving stochastic partial differential equations,” J. Comput. Phys. 132, 312–326 (1997).
[CrossRef]

J. F. Corney, P. D. Drummond, and A. Liebman, “Quantum noise limits to terabaud communications,” Opt. Commun. 140, 211–215 (1997).
[CrossRef]

P. D. Drummond and W. Man, “Quantum noise in reversible soliton logic,” Opt. Commun. 105, 99–103 (1994).
[CrossRef]

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

P. D. Drummond and I. K. Mortimer, “Computer simulations of multiplicative stochastic differential equations,” J. Comput. Phys. 93, 144–170 (1991).
[CrossRef]

P. D. Drummond, “Central partial difference propagation algorithms,” Comput. Phys. Commun. 29, 211–225 (1983).
[CrossRef]

Emplit, P.

Galko, P.

Gerdjikov, V. S.

I. M. Uzunov and V. S. Gerdjikov, “Self-frequency shift of dark solitons in optical fibers,” Phys. Rev. A 47, 1582–1585 (1993).
[CrossRef] [PubMed]

Gordon, J. P.

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, J. P. Gordon, and M. N. Islam, “Soliton propagation in long fibers with periodically compensated loss,” IEEE J. Quantum Electron. 22, 157–173 (1986).
[CrossRef]

Haelterman, M.

Hamaide, J.

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]

Hasegawa, A.

A. Hasegawa and F. Tappert, “Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers. II. Normal dispersion,” Appl. Phys. 23, 171–172 (1973).

Haus, H. A.

C. X. Yu, S. Namiki, and H. A. Haus, “Noise of the stretched pulse fiber laser. II. Experiments,” IEEE J. Quantum Electron. 33, 660–668 (1997).
[CrossRef]

S. Namiki, C. X. Yu, and H. A. Haus, “Observation of nearly quantum-limited timing jitter in an all-fiber ring laser,” J. Opt. Soc. Am. B 13, 2817–2823 (1996).
[CrossRef]

H. A. Haus and W. S. Wong, “Solitons in optical communications,” Rev. Mod. Phys. 68, 423–444 (1996).
[CrossRef]

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]

F. X. Kartner, 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]

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]

Ippen, E. P.

Islam, M. N.

L. F. Mollenauer, J. P. Gordon, and M. N. Islam, “Soliton propagation in long fibers with periodically compensated loss,” IEEE J. Quantum Electron. 22, 157–173 (1986).
[CrossRef]

Kartner, F. X.

Kaup, D. J.

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

Liebman, A.

J. F. Corney, P. D. Drummond, and A. Liebman, “Quantum noise limits to terabaud communications,” Opt. Commun. 140, 211–215 (1997).
[CrossRef]

Man, W.

P. D. Drummond and W. Man, “Quantum noise in reversible soliton logic,” Opt. Commun. 105, 99–103 (1994).
[CrossRef]

Mollenauer, L. F.

L. F. Mollenauer, J. P. Gordon, and M. N. Islam, “Soliton propagation in long fibers with periodically compensated loss,” IEEE J. Quantum Electron. 22, 157–173 (1986).
[CrossRef]

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]

Mortimer, I. K.

P. D. Drummond and I. K. Mortimer, “Computer simulations of multiplicative stochastic differential equations,” J. Comput. Phys. 93, 144–170 (1991).
[CrossRef]

Myslinski, P.

Namiki, S.

C. X. Yu, S. Namiki, and H. A. Haus, “Noise of the stretched pulse fiber laser. II. Experiments,” IEEE J. Quantum Electron. 33, 660–668 (1997).
[CrossRef]

S. Namiki, C. X. Yu, and H. A. Haus, “Observation of nearly quantum-limited timing jitter in an all-fiber ring laser,” J. Opt. Soc. Am. B 13, 2817–2823 (1996).
[CrossRef]

Tappert, F.

A. Hasegawa and F. Tappert, “Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers. II. Normal dispersion,” Appl. Phys. 23, 171–172 (1973).

Uzunov, I. M.

I. M. Uzunov and V. S. Gerdjikov, “Self-frequency shift of dark solitons in optical fibers,” Phys. Rev. A 47, 1582–1585 (1993).
[CrossRef] [PubMed]

Werner, M. J.

M. J. Werner and P. D. Drummond, “Robust algorithms for solving stochastic partial differential equations,” J. Comput. Phys. 132, 312–326 (1997).
[CrossRef]

Wong, W. S.

H. A. Haus and W. S. Wong, “Solitons in optical communications,” Rev. Mod. Phys. 68, 423–444 (1996).
[CrossRef]

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]

Yu, C. X.

C. X. Yu, S. Namiki, and H. A. Haus, “Noise of the stretched pulse fiber laser. II. Experiments,” IEEE J. Quantum Electron. 33, 660–668 (1997).
[CrossRef]

S. Namiki, C. X. Yu, and H. A. Haus, “Observation of nearly quantum-limited timing jitter in an all-fiber ring laser,” J. Opt. Soc. Am. B 13, 2817–2823 (1996).
[CrossRef]

Appl. Phys.

A. Hasegawa and F. Tappert, “Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers. II. Normal dispersion,” Appl. Phys. 23, 171–172 (1973).

Comput. Phys. Commun.

P. D. Drummond, “Central partial difference propagation algorithms,” Comput. Phys. Commun. 29, 211–225 (1983).
[CrossRef]

Europhys. Lett.

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

IEEE J. Quantum Electron.

C. X. Yu, S. Namiki, and H. A. Haus, “Noise of the stretched pulse fiber laser. II. Experiments,” IEEE J. Quantum Electron. 33, 660–668 (1997).
[CrossRef]

L. F. Mollenauer, J. P. Gordon, and M. N. Islam, “Soliton propagation in long fibers with periodically compensated loss,” IEEE J. Quantum Electron. 22, 157–173 (1986).
[CrossRef]

J. Comput. Phys.

P. D. Drummond and I. K. Mortimer, “Computer simulations of multiplicative stochastic differential equations,” J. Comput. Phys. 93, 144–170 (1991).
[CrossRef]

M. J. Werner and P. D. Drummond, “Robust algorithms for solving stochastic partial differential equations,” J. Comput. Phys. 132, 312–326 (1997).
[CrossRef]

J. Lightwave Technol.

J. Opt. Soc. Am. B

Opt. Commun.

J. F. Corney, P. D. Drummond, and A. Liebman, “Quantum noise limits to terabaud communications,” Opt. Commun. 140, 211–215 (1997).
[CrossRef]

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]

P. D. Drummond and W. Man, “Quantum noise in reversible soliton logic,” Opt. Commun. 105, 99–103 (1994).
[CrossRef]

Opt. Lett.

Phys. Rev. A

I. M. Uzunov and V. S. Gerdjikov, “Self-frequency shift of dark solitons in optical fibers,” Phys. Rev. A 47, 1582–1585 (1993).
[CrossRef] [PubMed]

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

Rev. Mod. Phys.

H. A. Haus and W. S. Wong, “Solitons in optical communications,” Rev. Mod. Phys. 68, 423–444 (1996).
[CrossRef]

Other

R. H. Stolen, C. Lee, and R. K. Jain, “Development of the stimulated Raman spectrum in single-mode silica fibers,” J. Opt. Soc. Am. B 1, 652–657 (1984); D. J. Dougherty, F. X. Kartner, H. A. Haus, and E. P. Ippen, “Measurement of the Raman gain spectrum of optical fibers,” Opt. Lett. 20, 31–33 (1995); 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 JOBPDE 6, 1159–1166 (1989).
[CrossRef] [PubMed]

Y. Lai and S.-S. Yu, “General quantum theory of nonlinear optical-pulse propagation,” Phys. Rev. A 51, 817–829 (1995); S.-S. Yu and Y. Lai, “Impacts of self-Raman effect and third-order dispersion on pulse squeezed state generation using optical fibers,” J. Opt. Soc. Am. B 12, 2340–2346 (1995).
[CrossRef] [PubMed]

A. Mecozzi, M. Midrio, and M. Romagnoli, “Timing jitter in soliton transmission with sliding filters,” Opt. Lett. 21, 402–404 (1996); 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 Gbits/s-Mm, error free, at 12.5 and 15 Gbits/s,” Opt. Lett. 19, 704–706 (1994); 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 10000 km in dispersion-shifted fiber,” Opt. Lett. OPLEDP 15, 1203–1205 (1990).
[CrossRef] [PubMed]

D.-M. Baboiu, D. Mihalache, and N.-C. Panoiu, “Combined influence of amplifier noise and intrapulse Raman scattering on the bit-rate limit of optical fiber communication systems,” Opt. Lett. 20, 1865–1867 (1995); D. Mihalache, L.-C. Crasovan, N.-C. Panoiu, F. Moldoveanu, and D.-M. Baboiu, “Timing jitter of femtosecond solitons in monomode optical fibers,” Opt. Eng. 35, 1611–1615 (1996); D. Wood, “Constraints on the bit rates in direct detection optical communication systems using linear or soliton pulses,” J. Lightwave Technol. JLTEDG 8, 1097–1106 (1990).
[CrossRef] [PubMed]

D. Shenoy and A. Puri, “Compensation for the soliton self-frequency shift and the third-order dispersion using bandwidth-limited optical gain,” Opt. Commun. 113, 410–406 (1995); S. V. Chernikov and S. M. J. Kelly, “Stability of femtosecond solitons in optical fibres influenced by optical attenuation and bandwidth limited gain,” Electron. Lett. 28, 238–240 (1992).
[CrossRef]

G. P. Agrawal, Nonlinear Fiber Optics, 2nd ed. (Academic, San Diego, Calif. 1995), p. 475.

Y. S. Kivshar, M. Haelterman, P. Emplit, and J. P. Hamaide, “Gordon–Haus effect on dark solitons,” Opt. Lett. 19, 19–21 (1994); Y. S. Kivshar, “Dark solitons in nonlinear optics,” IEEE J. Quantum Electron. 29, 250–264 (1993).
[CrossRef] [PubMed]

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Figures (2)

Fig. 1
Fig. 1

Spectrum of the fluorescence function F(ω) for the 11-Lorentzian model (solid curve) and the single-Lorentzian model (dashed curve) for a temperature of T=300 K. Also shown is the spectrum of a t0=1 ps soliton.

Fig. 2
Fig. 2

Timing jitter in t0=500 fs (a) bright and (b) dark solitons that is due to initial quantum fluctuations, the Gordon-Haus effect, and Raman noise. The asterisks give the total jitter, and the continuous curve gives the approximate analytic results for the Raman jitter.

Equations (69)

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

ζϕ(τ, ζ)=-0dτg(τ-τ)ϕ(τ, ζ)+Γ(τ, ζ)+i±12 2ϕτ2+0dτh(τ-τ)[ϕ(τ,ζ)]*×ϕ(τ, ζ)+iΓR(τ, ζ)ϕ(τ, ζ).
ζϕ(τ, ζ)=±i2 2τ2+iϕ*(τ, ζ)ϕ(τ, ζ)×ϕ(τ, ζ)+ΓC(τ, ζ),
ΓC(τ, ζ)=Γ(τ, ζ)+iΓR(τ, ζ)ϕ(τ, ζ).
Δϕ(τ, 0)Δϕ*(τ, 0)=12n¯δ(τ-τ).
Γ(Ω, ζ)Γ*(Ω, ζ)=(αG+αA)2n¯δ(ζ-ζ)δ(Ω+Ω),
Γ(Ω, ζ)=12π - dτΓ(τ, ζ)exp(iΩτ).
ΓR(Ω, ζ)ΓR(Ω, ζ)=1n¯δ(ζ-ζ)δ(Ω+Ω)×nth(Ω)+12αR(Ω),
ϕbright(τ, ζ)=A sech[Aτ-q(ζ)]exp[iVτ+iθ(ζ)],
ϕ(τ, ζ)=ϕ¯(τ, ζ)+Δϕ(τ, ζ),
ϕ¯(τ, ζ)=A(ζ)sech[A(ζ)τ-qζ)]exp[iV(ζ)τ+iθ(ζ)]
ζΔϕ(τ, ζ)=±i2 2τ2+i2ϕ¯*(τ, ζ)ϕ¯(τ, ζ)×Δϕ(τ, ζ)+iϕ¯(τ, ζ)2Δϕ*(τ, ζ)+Γ¯(τ, ζ),
Γ¯(τ, ζ)=Γ(τ, ζ)+iΓR(τ, ζ)ϕ¯(τ, ζ).
Δϕ(τ, ζ)=i ϕ¯(τ, ζ)PiΔPi+Δϕc(τ, ζ)
=i fPiΔPi+Δϕc(τ, ζ),
fA=[(1/A)-τ tanh(Aτ-q)]ϕ¯,
fq=tanh(Aτ-q)ϕ¯,
fV=iτϕ¯,
fθ=iϕ¯.
fA¯=ϕ¯,
fq¯=τϕ¯,
fV¯=i tanh(Aτ-q)ϕ¯,
fθ¯=iτ tanh(Aτ-q)ϕ¯.
R- dτfPi fPj¯*=δi, j.
ζΔq(ζ)=AΔV(ζ)+Γq(ζ),
ζΔV(ζ)=ΓV(ζ),
ΓPi(ζ)=R-dτfp*¯(ζ)Γ¯(τ, ζ).
Γq(ζ)=R-dτAτ sech(Aτ-q)×exp(-iVτ-iθ)Γ¯(τ, ζ)
=-dτAτ sech(Aτ-q)×R{exp(-iVτ-iθ)Γ},
ΓV(ζ)=R-dτA(-i)sech(Aτ-q)×tanh(Aτ-q)exp(-iVτ-iθ)Γ¯(τ, ζ)
=-dτA sech(Aτ-q)tanh(Aτ-q)×{A sech(Aτ-q)ΓR+I[exp(-iVτ-iθ)Γ]}.
ΔV(ζ)=ΔV(0)+0ζdζΓV(ζ)=R-Δϕ(τ, ζ)fV*¯dτ+0ζdζΓV(ζ).
ΔV(ζ)ΔV*(ζ)
=ΔV(0)ΔV*(0)+0ζ0ζdζdζΓV(ζ)ΓV*(ζ)
=A6n¯+αGA3n¯+2A2I(t0)n¯ζ,ζ<ζ,
I(t0)=--dτdτ×tanh(τ)sech2(τ)tanh(τ)sech2(τ)F˜(τ/A-τ/A).
Δq(ζ)Δq*(ζ)=(Δq(0)Δq*(0)+0ζ0ζdζdζ[A2ΔV(ζ)ΔV*(ζ)+Γq(ζ)Γq*(ζ)],ζ<ζ.
[Δτ(ζ)]2=Δq(ζ)Δq*(ζ)=π224n¯+π2αG12n¯ζ+A36n¯ζ2+αGA39n¯+2A4I(t0)3n¯ζ3,
ϕdark(τ, ζ)=ϕ0{1-A2 sech2[ϕ0Aτ-q(ζ)]}1/2 exp[iθ(ζ)]exp[iσ(ζ, τ)],
σ(ζ, τ)=arcsinA tanh[ϕ0Aτ-q(ζ)]{1-A2 sech2[ϕ0Aτ-q(ζ)]}1/2,
ϕdark(τ, ζ)=ϕ0 exp[iθ(ζ)-iκτ]cos ψ2+i sin ψ2 tanhϕ0τ sin ψ2-q(ζ),
fθ=-ϕ0 tanh(ϕ0τ-q)exp(iθ-iκτ),
fϕ0=i[tanh(ϕ0τ-q)+ϕ0τ sech2(ϕ0τ-q)]×exp(iθ-iκτ),
fq=-iϕ0 sech2(ϕ0τ-q)exp(iθ-iκτ),
fψ=ϕ0[β1ϕ0τ tanh(ϕ0τ-q)-(1/2)]exp(iθ-iκτ),
fq¯=-i3γq4 sech2(ϕ0τ-q)exp(iθ-iκτ),
fψ¯=γψβ1-1 sech2(ϕ0τ-q)exp(iθ-iκτ),
R-τl+q/ϕ0τl+q/ϕ0dτfPi fPj*¯=δi, j.
ζΔq(ζ)=ϕ0β22Δψ(ζ)+Γq(ζ),
ζΔψ(ζ)=Γψ(ζ),
Γψ(ζ)=R-τl+q/ϕ0τl+q/ϕ0dτfψ*¯(ζ)Γ¯(τ, ζ)=-τl+q/ϕ0τl+q/ϕ0dτ γψβ1-1 sech2(ϕ0τ-q)×{R[exp(-iϕ+iκτ)Γ]-ϕ0 tanh(ϕ0τ-q)ΓR},
Δψ(ζ)Δψ*(ζ)
=Δψ(0)Δψ*(0)+0ζ0ζdζdζΓψ(ζ)Γψ*(ζ)
=γψ23n¯γqϕ0+2αGγψ23n¯γq(β1-1)2ϕ0+2γψ2Iτl(t0)n¯(β1-1)2ζ,
ζ<ζ,
Iτl(t0)=-ϕ0τlϕ0τl-ϕ0τlϕ0τldτdτ tanh(τ)sech2(τ)tanh(τ)×sech2(τ)F˜(τ/ϕ0-τ/ϕ0).
Δq(ζ)Δq*(ζ)
=ϕ02β224 0ζ0ζdζdζΔψ(ζ)Δψ*(ζ)=ϕ0β22γψ212n¯γqz2+αGϕ0β22γψ218n¯γq(β1-1)2+Iτl(t0)ϕ02β22γψ26n¯ζ3.
[Δτ(ζ)]2=ϕ0312n¯ζ2+αGϕ0318n¯+I(t0)ϕ046n¯ζ3,
[Δτ(ζ)]2I=π224n¯+16n¯ζ2(bright).
[Δτ(ζ)]2I=π248n¯+112n¯ζ2(dark).
[Δτ(ζ)]2GHαG9n¯ζ3(bright),
[Δτ(ζ)]2GHαG18n¯ζ3(dark),
[Δτ(ζ)]2R=2I(t0)3n¯ζ3(bright),
[Δτ(ζ)]2R=I(t0)6n¯ζ3(dark),
F(Ω)=12 nth(Ω)+12αR(Ω)2F1Ω1δ12kBT(Ω12+δ12)2=F(0),
[Δt(x)]2R8|k|2n2ω02F(0)45Act03x3=8t02F(0)45n¯ xx03(bright),
[Δt(x)]2R2|k|2n2ω02F(0)45Act03x3=2t02F(0)45n¯ xx03(dark).
[Δt(x)]2R[Δt(x)]2GH=6I(t0)|k|Gt02=6I(t0)Gx08F(0)5Gx0(bright),
[Δt(x)]2R[Δt(x)]2GH=3I(t0)|k|Gt02=3I(t0)Gx04F(0)5Gx0 (dark).

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