Abstract

We investigate the use of nonlinear optical loop mirrors as saturable absorbers in picosecond soliton transmission systems. It is found that they allow short (1 – 5-ps) pulses to be propagated through chains of optical amplifiers spaced at intervals of typically 10 km. The loop mirror removes dispersive waves and stabilizes the peak amplitude of the soliton. An additional advantage is that the self-frequency shift of the soliton may be suppressed by bandwidth filtering without causing growth of dispersive waves at the center of the passband. The timing jitter and soliton interactions present in the scheme are also described.

© 1995 Optical Society of America

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  1. M. Nakazawa, E. Yamada, H. Kubota, and K. Suzuki, “10 Gb/s soliton data transmission over one million kilometers,” Electron. Lett. 27, 1270–1272 (1991).
    [CrossRef]
  2. N. J. Smith, K. J. Blow, W. J. Firth, and K. Smith, “Soliton dynamics in the presence of phase modulators,” Opt. Commun. 102, 324–328 (1993).
    [CrossRef]
  3. A. Mecozzi, J. D. Moores, H. A. Haus, and Y. Lai, “Modulation and filtering control of soliton transmission,” J. Opt. Soc. Am. B 9, 1350–1357 (1992).
    [CrossRef]
  4. L. F. Mollenauer, J. P. Gordon, and S. G. Evangelides, “The sliding frequency filter: an improved form of soliton jitter control,” Opt. Lett. 17, 1575–1577 (1992).
    [CrossRef] [PubMed]
  5. F. Fontana, L. Bossalini, P. Franco, M. Midrio, M. Romagnoli, and S. Wabnitz, “Self-starting sliding-frequency fiber soliton laser,” Electron. Lett. 30, 321–322 (1994).
    [CrossRef]
  6. Y. Kodama, M. Romagnoli, and S. Wabnitz, “Stabilization of optical solitons by an acoustooptic modulator and filter,” Electron. Lett. 30, 261–262 (1994).
    [CrossRef]
  7. I. N. Duling, “All-fiber ring soliton laser mode locked with a nonlinear mirror,” Opt. Lett. 16, 539–541 (1991).
    [CrossRef]
  8. Y. Kodama, H. Romagnoli, and S. Wabnitz, “Soliton stability and interactions in fiber lasers,” Electron. Lett. 28, 1981–1983 (1992).
    [CrossRef]
  9. M. Mastsumoto, H. Ikeda, and A. Hasegawa, “Suppression of noise accumulation in bandwidth-limited soliton transmission by means of nonlinear loop mirrors,” Opt. Lett. 19, 183–185 (1994).
    [CrossRef]
  10. K. J. Blow and N. J. Doran, “Average soliton dynamics and the operation of soliton systems with lumped amplifiers,” IEEE Photon. Technol. Lett. 3, 369–371 (1991).
    [CrossRef]
  11. A. Hasegawa and Y. Kodama, ”Guiding-center soliton in optical fibers,” Opt. Lett. 15, 1443–1445 (1990).
    [CrossRef] [PubMed]
  12. J. P. Gordon, “Dispersive perturbations of solitons of the nonlinear Schrödinger equation,” J. Opt. Soc. Am. B 9, 91–97 (1992).
    [CrossRef]
  13. N. J. Smith, K. J. Blow, and I. Andonovic, “Sideband generation through perturbations to the average soliton model,” J. Lightwave Technol. 10, 1329–1333 (1992).
    [CrossRef]
  14. D. U. Noske, N. Pandit, and J. R. Taylor, “Source of spectral and temporal instability in soliton fiber lasers,” Opt. Lett. 17, 1515–1517 (1992).
    [CrossRef] [PubMed]
  15. M. Nakazawa and K. Kurokawa, “Femtosecond soliton transmission in 18 km long dispersion shifted distributed erbium-doped fibre amplifier,” Electron. Lett. 27, 1369–1371 (1991).
    [CrossRef]
  16. N. J. Doran and D. Wood, “Nonlinear optical loop mirror,” Opt. Lett. 13, 56–58 (1988).
    [CrossRef] [PubMed]
  17. K. J. Blow, B. K. Nayar, and N. J. Doran, “Experimental demonstration of optical soliton switching in an all-fiber nonlinear Sagnac interferometer,” Opt. Lett. 14, 754–756, (1989).
    [CrossRef] [PubMed]
  18. M. N. Islam, E. R. Sunderman, R. H. Stolen, W. Pleibel, and J. R. Simpson, “Soliton switching in a fiber nonlinear loop mirror,” Opt. Lett. 14, 811–813 (1989).
    [CrossRef] [PubMed]
  19. M. E. Fermann, F. Haberl, M. Hofer, and H. Hochreiter, “Nonlinear amplifying loop mirror,” Opt. Lett. 15, 752–754 (1990).
    [CrossRef] [PubMed]
  20. K. Smith, N. J. Doran, and P. G. J. Wigley, “Pulse shaping, compression, and pedestal suppression employing a nonlinear-optical loop mirror,” Opt. Lett. 15, 1294–1296 (1990).
    [CrossRef] [PubMed]
  21. N. Finlayson, B. K. Nayar, and N. J. Doran, “Switch inversion and polarization sensitivity of the nonlinear-optical loop mirror,” Opt. Lett. 17, 112–114 (1992).
    [CrossRef] [PubMed]
  22. Y. Kodama and A. Hasegawa, “Generation of asymptotically stable solitons and suppression of the Gordon–Haus effect,” Opt. Lett. 17, 31–34 (1992).
    [CrossRef] [PubMed]
  23. 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]
  24. J. P. Gordon, “Theory of the soliton self-frequency shift,” Opt. Lett. 11, 662–664 (1986).
    [CrossRef] [PubMed]
  25. K. J. Blow and D. Wood, “Theoretical description of transient stimulated Raman scattering in optical fibers,” IEEE J. Quantum Electron. 25, 2665–2673 (1989).
    [CrossRef]
  26. 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]
  27. D. Wood, “Constraints on the bit rate in direct detection optical communication systems using linear or soliton pulses,” J. Lightwave Technol. 8, 1097–1106 (1990).
    [CrossRef]
  28. K. Smith and L. F. Mollenauer, “Experimental observation of soliton interaction over long fiber paths: discovery of a long-range interaction,” Opt. Lett. 14, 1284–1286 (1989).
    [CrossRef] [PubMed]
  29. E. M. Dianov, A. V. Luchikov, A. N. Pilipetskii, and A. N. Starodumov, “Electrostriction mechanism of soliton interaction in optical fibers,” Opt. Lett. 15, 314–316 (1990).
    [CrossRef] [PubMed]

1994 (3)

F. Fontana, L. Bossalini, P. Franco, M. Midrio, M. Romagnoli, and S. Wabnitz, “Self-starting sliding-frequency fiber soliton laser,” Electron. Lett. 30, 321–322 (1994).
[CrossRef]

Y. Kodama, M. Romagnoli, and S. Wabnitz, “Stabilization of optical solitons by an acoustooptic modulator and filter,” Electron. Lett. 30, 261–262 (1994).
[CrossRef]

M. Mastsumoto, H. Ikeda, and A. Hasegawa, “Suppression of noise accumulation in bandwidth-limited soliton transmission by means of nonlinear loop mirrors,” Opt. Lett. 19, 183–185 (1994).
[CrossRef]

1993 (1)

N. J. Smith, K. J. Blow, W. J. Firth, and K. Smith, “Soliton dynamics in the presence of phase modulators,” Opt. Commun. 102, 324–328 (1993).
[CrossRef]

1992 (8)

1991 (4)

M. Nakazawa and K. Kurokawa, “Femtosecond soliton transmission in 18 km long dispersion shifted distributed erbium-doped fibre amplifier,” Electron. Lett. 27, 1369–1371 (1991).
[CrossRef]

M. Nakazawa, E. Yamada, H. Kubota, and K. Suzuki, “10 Gb/s soliton data transmission over one million kilometers,” Electron. Lett. 27, 1270–1272 (1991).
[CrossRef]

K. J. Blow and N. J. Doran, “Average soliton dynamics and the operation of soliton systems with lumped amplifiers,” IEEE Photon. Technol. Lett. 3, 369–371 (1991).
[CrossRef]

I. N. Duling, “All-fiber ring soliton laser mode locked with a nonlinear mirror,” Opt. Lett. 16, 539–541 (1991).
[CrossRef]

1990 (5)

1989 (4)

1988 (2)

1986 (2)

Andonovic, I.

N. J. Smith, K. J. Blow, and I. Andonovic, “Sideband generation through perturbations to the average soliton model,” J. Lightwave Technol. 10, 1329–1333 (1992).
[CrossRef]

Blow, K. J.

N. J. Smith, K. J. Blow, W. J. Firth, and K. Smith, “Soliton dynamics in the presence of phase modulators,” Opt. Commun. 102, 324–328 (1993).
[CrossRef]

N. J. Smith, K. J. Blow, and I. Andonovic, “Sideband generation through perturbations to the average soliton model,” J. Lightwave Technol. 10, 1329–1333 (1992).
[CrossRef]

K. J. Blow and N. J. Doran, “Average soliton dynamics and the operation of soliton systems with lumped amplifiers,” IEEE Photon. Technol. Lett. 3, 369–371 (1991).
[CrossRef]

K. J. Blow, B. K. Nayar, and N. J. Doran, “Experimental demonstration of optical soliton switching in an all-fiber nonlinear Sagnac interferometer,” Opt. Lett. 14, 754–756, (1989).
[CrossRef] [PubMed]

K. J. Blow and D. Wood, “Theoretical description of transient stimulated Raman scattering in optical fibers,” IEEE J. Quantum Electron. 25, 2665–2673 (1989).
[CrossRef]

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]

Bossalini, L.

F. Fontana, L. Bossalini, P. Franco, M. Midrio, M. Romagnoli, and S. Wabnitz, “Self-starting sliding-frequency fiber soliton laser,” Electron. Lett. 30, 321–322 (1994).
[CrossRef]

Dianov, E. M.

Doran, N. J.

Duling, I. N.

Evangelides, S. G.

Fermann, M. E.

Finlayson, N.

Firth, W. J.

N. J. Smith, K. J. Blow, W. J. Firth, and K. Smith, “Soliton dynamics in the presence of phase modulators,” Opt. Commun. 102, 324–328 (1993).
[CrossRef]

Fontana, F.

F. Fontana, L. Bossalini, P. Franco, M. Midrio, M. Romagnoli, and S. Wabnitz, “Self-starting sliding-frequency fiber soliton laser,” Electron. Lett. 30, 321–322 (1994).
[CrossRef]

Franco, P.

F. Fontana, L. Bossalini, P. Franco, M. Midrio, M. Romagnoli, and S. Wabnitz, “Self-starting sliding-frequency fiber soliton laser,” Electron. Lett. 30, 321–322 (1994).
[CrossRef]

Gordon, J. P.

Haberl, F.

Hasegawa, A.

Haus, H. A.

Hochreiter, H.

Hofer, M.

Ikeda, H.

Islam, M. N.

Kodama, Y.

Y. Kodama, M. Romagnoli, and S. Wabnitz, “Stabilization of optical solitons by an acoustooptic modulator and filter,” Electron. Lett. 30, 261–262 (1994).
[CrossRef]

Y. Kodama, H. Romagnoli, and S. Wabnitz, “Soliton stability and interactions in fiber lasers,” Electron. Lett. 28, 1981–1983 (1992).
[CrossRef]

Y. Kodama and A. Hasegawa, “Generation of asymptotically stable solitons and suppression of the Gordon–Haus effect,” Opt. Lett. 17, 31–34 (1992).
[CrossRef] [PubMed]

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

Kubota, H.

M. Nakazawa, E. Yamada, H. Kubota, and K. Suzuki, “10 Gb/s soliton data transmission over one million kilometers,” Electron. Lett. 27, 1270–1272 (1991).
[CrossRef]

Kurokawa, K.

M. Nakazawa and K. Kurokawa, “Femtosecond soliton transmission in 18 km long dispersion shifted distributed erbium-doped fibre amplifier,” Electron. Lett. 27, 1369–1371 (1991).
[CrossRef]

Lai, Y.

Luchikov, A. V.

Mastsumoto, M.

Mecozzi, A.

Midrio, M.

F. Fontana, L. Bossalini, P. Franco, M. Midrio, M. Romagnoli, and S. Wabnitz, “Self-starting sliding-frequency fiber soliton laser,” Electron. Lett. 30, 321–322 (1994).
[CrossRef]

Mollenauer, L. F.

Moores, J. D.

Nakazawa, M.

M. Nakazawa, E. Yamada, H. Kubota, and K. Suzuki, “10 Gb/s soliton data transmission over one million kilometers,” Electron. Lett. 27, 1270–1272 (1991).
[CrossRef]

M. Nakazawa and K. Kurokawa, “Femtosecond soliton transmission in 18 km long dispersion shifted distributed erbium-doped fibre amplifier,” Electron. Lett. 27, 1369–1371 (1991).
[CrossRef]

Nayar, B. K.

Noske, D. U.

Pandit, N.

Pilipetskii, A. N.

Pleibel, W.

Romagnoli, H.

Y. Kodama, H. Romagnoli, and S. Wabnitz, “Soliton stability and interactions in fiber lasers,” Electron. Lett. 28, 1981–1983 (1992).
[CrossRef]

Romagnoli, M.

Y. Kodama, M. Romagnoli, and S. Wabnitz, “Stabilization of optical solitons by an acoustooptic modulator and filter,” Electron. Lett. 30, 261–262 (1994).
[CrossRef]

F. Fontana, L. Bossalini, P. Franco, M. Midrio, M. Romagnoli, and S. Wabnitz, “Self-starting sliding-frequency fiber soliton laser,” Electron. Lett. 30, 321–322 (1994).
[CrossRef]

Simpson, J. R.

Smith, K.

Smith, N. J.

N. J. Smith, K. J. Blow, W. J. Firth, and K. Smith, “Soliton dynamics in the presence of phase modulators,” Opt. Commun. 102, 324–328 (1993).
[CrossRef]

N. J. Smith, K. J. Blow, and I. Andonovic, “Sideband generation through perturbations to the average soliton model,” J. Lightwave Technol. 10, 1329–1333 (1992).
[CrossRef]

Starodumov, A. N.

Stolen, R. H.

Sunderman, E. R.

Suzuki, K.

M. Nakazawa, E. Yamada, H. Kubota, and K. Suzuki, “10 Gb/s soliton data transmission over one million kilometers,” Electron. Lett. 27, 1270–1272 (1991).
[CrossRef]

Taylor, J. R.

Wabnitz, S.

F. Fontana, L. Bossalini, P. Franco, M. Midrio, M. Romagnoli, and S. Wabnitz, “Self-starting sliding-frequency fiber soliton laser,” Electron. Lett. 30, 321–322 (1994).
[CrossRef]

Y. Kodama, M. Romagnoli, and S. Wabnitz, “Stabilization of optical solitons by an acoustooptic modulator and filter,” Electron. Lett. 30, 261–262 (1994).
[CrossRef]

Y. Kodama, H. Romagnoli, and S. Wabnitz, “Soliton stability and interactions in fiber lasers,” Electron. Lett. 28, 1981–1983 (1992).
[CrossRef]

Wigley, P. G. J.

Wood, D.

D. Wood, “Constraints on the bit rate in direct detection optical communication systems using linear or soliton pulses,” J. Lightwave Technol. 8, 1097–1106 (1990).
[CrossRef]

K. J. Blow and D. Wood, “Theoretical description of transient stimulated Raman scattering in optical fibers,” IEEE J. Quantum Electron. 25, 2665–2673 (1989).
[CrossRef]

N. J. Doran and D. Wood, “Nonlinear optical loop mirror,” Opt. Lett. 13, 56–58 (1988).
[CrossRef] [PubMed]

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]

Yamada, E.

M. Nakazawa, E. Yamada, H. Kubota, and K. Suzuki, “10 Gb/s soliton data transmission over one million kilometers,” Electron. Lett. 27, 1270–1272 (1991).
[CrossRef]

Electron. Lett. (5)

M. Nakazawa, E. Yamada, H. Kubota, and K. Suzuki, “10 Gb/s soliton data transmission over one million kilometers,” Electron. Lett. 27, 1270–1272 (1991).
[CrossRef]

F. Fontana, L. Bossalini, P. Franco, M. Midrio, M. Romagnoli, and S. Wabnitz, “Self-starting sliding-frequency fiber soliton laser,” Electron. Lett. 30, 321–322 (1994).
[CrossRef]

Y. Kodama, M. Romagnoli, and S. Wabnitz, “Stabilization of optical solitons by an acoustooptic modulator and filter,” Electron. Lett. 30, 261–262 (1994).
[CrossRef]

Y. Kodama, H. Romagnoli, and S. Wabnitz, “Soliton stability and interactions in fiber lasers,” Electron. Lett. 28, 1981–1983 (1992).
[CrossRef]

M. Nakazawa and K. Kurokawa, “Femtosecond soliton transmission in 18 km long dispersion shifted distributed erbium-doped fibre amplifier,” Electron. Lett. 27, 1369–1371 (1991).
[CrossRef]

IEEE J. Quantum Electron. (1)

K. J. Blow and D. Wood, “Theoretical description of transient stimulated Raman scattering in optical fibers,” IEEE J. Quantum Electron. 25, 2665–2673 (1989).
[CrossRef]

IEEE Photon. Technol. Lett. (1)

K. J. Blow and N. J. Doran, “Average soliton dynamics and the operation of soliton systems with lumped amplifiers,” IEEE Photon. Technol. Lett. 3, 369–371 (1991).
[CrossRef]

J. Lightwave Technol. (2)

N. J. Smith, K. J. Blow, and I. Andonovic, “Sideband generation through perturbations to the average soliton model,” J. Lightwave Technol. 10, 1329–1333 (1992).
[CrossRef]

D. Wood, “Constraints on the bit rate in direct detection optical communication systems using linear or soliton pulses,” J. Lightwave Technol. 8, 1097–1106 (1990).
[CrossRef]

J. Opt. Soc. Am. B (3)

Opt. Commun. (1)

N. J. Smith, K. J. Blow, W. J. Firth, and K. Smith, “Soliton dynamics in the presence of phase modulators,” Opt. Commun. 102, 324–328 (1993).
[CrossRef]

Opt. Lett. (16)

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

M. Mastsumoto, H. Ikeda, and A. Hasegawa, “Suppression of noise accumulation in bandwidth-limited soliton transmission by means of nonlinear loop mirrors,” Opt. Lett. 19, 183–185 (1994).
[CrossRef]

I. N. Duling, “All-fiber ring soliton laser mode locked with a nonlinear mirror,” Opt. Lett. 16, 539–541 (1991).
[CrossRef]

N. J. Doran and D. Wood, “Nonlinear optical loop mirror,” Opt. Lett. 13, 56–58 (1988).
[CrossRef] [PubMed]

K. J. Blow, B. K. Nayar, and N. J. Doran, “Experimental demonstration of optical soliton switching in an all-fiber nonlinear Sagnac interferometer,” Opt. Lett. 14, 754–756, (1989).
[CrossRef] [PubMed]

M. N. Islam, E. R. Sunderman, R. H. Stolen, W. Pleibel, and J. R. Simpson, “Soliton switching in a fiber nonlinear loop mirror,” Opt. Lett. 14, 811–813 (1989).
[CrossRef] [PubMed]

M. E. Fermann, F. Haberl, M. Hofer, and H. Hochreiter, “Nonlinear amplifying loop mirror,” Opt. Lett. 15, 752–754 (1990).
[CrossRef] [PubMed]

K. Smith, N. J. Doran, and P. G. J. Wigley, “Pulse shaping, compression, and pedestal suppression employing a nonlinear-optical loop mirror,” Opt. Lett. 15, 1294–1296 (1990).
[CrossRef] [PubMed]

N. Finlayson, B. K. Nayar, and N. J. Doran, “Switch inversion and polarization sensitivity of the nonlinear-optical loop mirror,” Opt. Lett. 17, 112–114 (1992).
[CrossRef] [PubMed]

Y. Kodama and A. Hasegawa, “Generation of asymptotically stable solitons and suppression of the Gordon–Haus effect,” Opt. Lett. 17, 31–34 (1992).
[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]

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

K. Smith and L. F. Mollenauer, “Experimental observation of soliton interaction over long fiber paths: discovery of a long-range interaction,” Opt. Lett. 14, 1284–1286 (1989).
[CrossRef] [PubMed]

E. M. Dianov, A. V. Luchikov, A. N. Pilipetskii, and A. N. Starodumov, “Electrostriction mechanism of soliton interaction in optical fibers,” Opt. Lett. 15, 314–316 (1990).
[CrossRef] [PubMed]

D. U. Noske, N. Pandit, and J. R. Taylor, “Source of spectral and temporal instability in soliton fiber lasers,” Opt. Lett. 17, 1515–1517 (1992).
[CrossRef] [PubMed]

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

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Figures (12)

Fig. 1
Fig. 1

Propagation of 1.5-ps FWHM solitons in fiber without loop mirror filters. Amplifier spacing, 10 km.

Fig. 2
Fig. 2

Schematic diagram of NOLM’s inserted into a soliton transmission line.

Fig. 3
Fig. 3

Pulse intensity and spectral profiles (as measured at amplifier output) for propagation through concatenated NOLM’s. Loop-mirror spacing, 10 km; gain per amplifier, 2.7 dB. The dispersion is 1 (ps/nm)/km.

Fig. 4
Fig. 4

Cw switching curves for a NOLM with various values of the coupling ratio. The bold curves indicate those regions that permit operating points that are stable against small changes in the input power (see the analysis of Section 4).

Fig. 5
Fig. 5

Input–output mapping relationship for the NOLM, showing how the system will iterate to a fixed point beyond the peak of the switching curve.

Fig. 6
Fig. 6

Mean pulse frequencies observed in a concatenated loop-mirror chain. NOLM spacing, 10 km; 20% couplers; gain per amplifier, 3 dB; amplifier bandwidth, 1 THz.

Fig. 7
Fig. 7

Variance in pulse positions observed when a sliding filter is added to the simulations. The rate of sliding is −1 GHz per amplifier.

Fig. 8
Fig. 8

Pulse energies and frequencies when a narrow filter at each amplifier is used to suppress the soliton self-frequency shift. Amplifier spacing, 10 km; gain per amplifier, 4.6 dB; 20% couplers; amplifier bandwidth, 0.25 THz.

Fig. 9
Fig. 9

Spectral evolution when a narrow filter at each amplifier is used to suppress the soliton self-frequency shift. Amplifier spacing, 10 km; gain per amplifier, 4.6 dB; 20% couplers; amplifier bandwidth, 0.25 THz.

Fig. 10
Fig. 10

Root-mean-square statistics for the system in which the filtering suppressed the soliton self-frequency shift. Amplifier spacing, 10 km; gain per amplifier, 4.6 dB; 20% couplers; amplifier bandwidth, 0.25 THz.

Fig. 11
Fig. 11

Mutual repulsion of two in-phase pulses initially separated by 40 ps. Loop-mirror spacing, 15 km; amplifier gain, 3.8 dB; 10% coupler; amplifier bandwidth, 1 THz.

Fig. 12
Fig. 12

Mutual attraction of two in-phase pulses initially separated by 36 ps. The loop mirrors and amplifiers are spaced at 15 km. The coupler ratio, 10% amplifier gain, 4 dB; bandwidth, 1.0 THz.

Equations (22)

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

T = P o P i = { 1 2 r ( 1 r ) [ 1 + cos ( Φ + π P i P switch ) ] } ,
P switch = λ A eff 2 π n 2 L ( 1 2 r ) .
i u z = β ¨ 2 2 u t 2 + γ NL u 2 u * + i Δ u .
u out = u in g exp ( ω 2 / Δ ω 2 ) .
P o = P i g 2 { 1 2 r ( 1 r ) [ 1 + cos ( Φ + π g 2 P i P switch ) ] } .
P o = P i ,
1 < d P o d P i < + 1 .
d P o d P i = P o P i + 2 P i g 2 r ( 1 r ) ( π g 2 P sw ) sin ( Φ + π g 2 P i P switch ) .
d P o d P i = 1 + 2 P i g 2 r ( 1 r ) ( π g 2 P sw ) sin ( Φ + π g 2 P i P switch ) .
2 < 2 P i g 2 r ( 1 r ) ( π g 2 P sw ) sin ( Φ + π g 2 P i P switch ) < 0 .
1 g 2 = 1 2 r ( 1 r ) [ 1 + cos ( Φ + π g 2 P i P switch ) ] .
u s = η sech ( η t ξ ) exp ( i κ t ) .
d κ dz = 4 15 τ R τ η 4 ,
d ξ d z = κ .
Δ η = η η 0 ,
Δ κ = κ κ 0 + 4 15 τ R τ η 0 4 z ,
Δ ξ = ξ 0 + 2 15 τ R τ η 0 4 z 2 .
Δ η 2 ( n ) = n δ η 2 ,
Δ κ 2 ( n ) = n δ κ 2 + ( 16 15 τ R τ η 0 3 z a ) 2 1 2 n 2 δ η 2 ,
Δ ξ 2 ( n ) = 1 6 n 3 z a 2 n δ κ 2 + ( 16 15 τ R τ η 0 3 ) 2 1 24 n 4 z a 3 δ η 2 .
δ η 2 = η ( G 1 ) / N 0 .
Δ T 2 = [ η sp h ν ( G 1 ) γ NL 3 ] × [ 1 6 β ¨ τ n 3 Z a 2 + 1 8 β ¨ 2 3 τ 3 ( 16 τ R 15 τ ) 2 n 4 Z a 3 ] .

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