## Enhanced mutual capture of colored solitons by matched modulator

Optics Express, Vol. 12, Issue 16, pp. 3759-3764 (2004)

http://dx.doi.org/10.1364/OPEX.12.003759

Acrobat PDF (424 KB)

### Abstract

The mutual capture of two colored solitons is enhanced by a modulator, to a level which enables its practical exploitation, e.g., for a read- write mechanism in a soliton buffer. The enhanced capture was analyzed using closed form particle-like soliton perturbation, and verified by numerical simulations. Optimal modulator frequency and modulation depth are obtained. This mutual capture can be utilized for all-optical soliton logic and memory.

© 2004 Optical Society of America

## 1. Introduction

1. R.A. Barry, V.W.S. Chan, K.L. Hall, E.S. Kintzer, J.D. Moors, K.A. Rauschenbach, E.A. Swanson, L.E. Adams, C.R. Doerr, S.G. Finn, H.A. Haus, E.P. Ippen, W.S. Wong, and M. Haner, “All-optical network consortium - ultrafast TDM netwarks,” J. Sel. Areas In Com. **14**, 999–1012 (1996). [CrossRef]

2. E. Feigenbaum and M. Orenstein, “Colored solitons interactions: particle-like and beyond,” Opt. Express **12**, 2193–2206 (2004). http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-10-2193. [CrossRef] [PubMed]

3. M. Nakazawa, H. Kubota, E. Yamada, and K. Suzuki, “Infinite-distance soliton transmission with soliton controls in time and frequency domains,” Elec. Lett. **28**, p. 1099–1100 (1992). [CrossRef]

2. E. Feigenbaum and M. Orenstein, “Colored solitons interactions: particle-like and beyond,” Opt. Express **12**, 2193–2206 (2004). http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-10-2193. [CrossRef] [PubMed]

6. M. Karlsson, D Anderson, A Höök, and M. Lisak, “A variational approach to optical soliton collisions,” Phys. Scripta **50**, 265–270 (1994). [CrossRef]

11. N.C. Panoiu, I.V. Mel’nikov, D. Mihalache, C. Etrich, and F. Lederer, “Soliton generation from a multi-frequency optical signal,” J. Opt. B: Quantum Semiclass. Opt. **4**, R53–R68 (2002). [CrossRef]

2. E. Feigenbaum and M. Orenstein, “Colored solitons interactions: particle-like and beyond,” Opt. Express **12**, 2193–2206 (2004). http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-10-2193. [CrossRef] [PubMed]

**12**, 2193–2206 (2004). http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-10-2193. [CrossRef] [PubMed]

9. N.C. Panoiu, D. Mihalache, D. Mazilu, L.C. Crasovan, and I.V. Mel’nikov, “Soliton dynamics of symmetry-endowed two-soiton solutions of the nonlinear Schrodinger equation,” Chaos **10**, 625–640 (2000). [CrossRef]

## 2. Particle-like description

**12**, 2193–2206 (2004). http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-10-2193. [CrossRef] [PubMed]

13. V.I. Karpman and V.V. Solov’ev, “A perturbational approach to the two-soliton systems,” Physica D **3**, 487–502 (1981). [CrossRef]

*z*denotes the propagation distance,

*β*” the group velocity dispersion coefficient,

*δ*the Kerr constant.

*W, τ, p*and

*θ*stand for the solitons peak amplitude, temporal center, central frequency and phase.

*ε*is the first-order soliton bandwidth to peak-amplitude ratio (

*ε*=(δ/|β”|)

^{1/2}). The coordinate system is symmetrical in respect to the initial conditions: τ

_{1}=-τ

_{2}=τ, p

_{1}=-p

_{2}=p. The LHS of Eq. (1a) is an equivalent force term for a particle-like model, which asserts an incoherent interaction. Even though the frequency difference varies along capture process, and hence the interaction coherency does vary too, the particle-like model describes reasonably well this process [2

**12**, 2193–2206 (2004). http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-10-2193. [CrossRef] [PubMed]

**12**, 2193–2206 (2004). http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-10-2193. [CrossRef] [PubMed]

_{τ}) is introduced into Eq. (1b), to yield:

*S*, are projected out of the NLSE perturbation source, using the adjoint perturbation functions (f

_{τ}_{m}) given in [14

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

*M, ω*and

_{m}*t*are the depth, frequency, and temporal center of the modulator respectively and

_{m}*t*is the

*z*dependent time coordinate (traveling with the carrier). The NLSE perturbation source is obtained from the transmission function:

*u*stands for the slowly varying amplitude of the soliton. The modulator is uniformly distributed along the fiber, which is a good approximation, disregarding lumped periodic effects such as Kelly’s sidebands [15

15. S.M.J. Kelly, “Characteristic sideband instability of periodically amplified average soliton,” Elec. Lett. **28**, 806–807 (1992). [CrossRef]

16. H.A. Haus, W.S. Wong, and F.I. Kharti, “Continuum generation by perturbation of soliton,” J. Opt. Soc. Am. B. **14**, 304–313(1997) [CrossRef]

_{m}) is smaller than the soliton period (z

_{0}). Furthermore, the actual modulator period should be negligible in terms of the colored solitons walk-off due to dispersion. This dictates an accumulated solitons temporal shift along the modulator period (2β”pz

_{m}) smaller than soliton effective width (~10/εW):

^{2}/Km), δ=1.3(1/Watt Km), W=1(Watt

^{1/2}), z

_{m}=10(m), the soliton frequency difference should be smaller than 600(THz), whereas we discuss a much smaller difference of the order of 1(THz). A detailed comparison to lumped modulator is given at the end.

_{m}=t

_{m}-τ and using Eq. (2) and the perturbation source of Eq. (5) the damping coefficient (V) is:

*V*is positive when the soliton peak coincides with the modulator transmission peak (|τ-t

_{m}|<π/(2ω

_{m})) and hence, the modulator attracts the soliton to this peak. Deviation of the soliton center generates asymmetric Kerr refractive index, pulling the soliton back to the transmission peak (t

_{m}). The damping coefficient (V) is enhanced with the modulation depth, and has a maximal value for an optimal modulation bandwidth.

*f*

_{V}(ω

_{m}) vanishes both for zero and infinity

*ω*, with a maximum for ω

_{m}_{m}~1.6εW. As

*ω*is raised, the variations of the solitons around

_{m}*t*, in terms of the modulator temporal period, become larger and cos(ω

_{m}_{m}Δτ

_{m}) may accept negative values. Consequently the

*ω*value, for optimal

_{m}*V*is lower than

*f*optimal value (~1.6εW), and is similar of the soliton bandwidth (εW).

_{v}_{0}). These results were obtained from direct calculations of the modified particle-like equations (Eq. (1),(2),(5)). The value of

*ω*in Fig. 2(a) is 0.9(THz) - similar to soliton bandwidth (εW~0.8(THz), ω

_{m}_{m}~1.1εW). In Fig. 2(b), the soliton bandwidths (εW) are smaller, resulting in the decrease of optimal modulator frequency at the same ratio.

## 3. Simulation results

^{1/2}). These solitons have a capture threshold of 2p

_{0-TH}=2×0.19×2π and therefore for initial frequencies difference 2p

_{0-TH}=2×0.20×2π, the solitons escape as depicted in Fig. 3(a). In Fig. 3(b), using the same initial frequencies but with a modulator in the loop, the capture of the data soliton to the control soliton, as well as the post capture oscillations damping are evident. In Fig. 3(c) it is clear that for the same initial conditions when no control soliton is applied, but with a modulator in the loop, the data soliton propagates virtually undisturbed by the modulator. To compensate for the losses of the modulator, a distributed gain of 5% per soliton period (z

_{0}) was applied.

**12**, 2193–2206 (2004). http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-10-2193. [CrossRef] [PubMed]

6. M. Karlsson, D Anderson, A Höök, and M. Lisak, “A variational approach to optical soliton collisions,” Phys. Scripta **50**, 265–270 (1994). [CrossRef]

_{0}, (~ capture oscillation period), the decay coefficient is depicted versus the M-phase (the relative shift in the modulator location, in units of modulator periods). The maximal decay coefficient was obtained for M-phase=0.5, which is the location where the solitons are most apart. The minimal decay coefficient was for overlapping solitons (M-phase=0). The distributed modulator decay coefficient is about the mean value obtained for the lumped modulators.

_{m}≪z

_{0}which usually coincides with typical system parameters, e.g soliton period of kilometers and storage ring meters long. Since M-phase is a fraction of the modulator period, which is negligible relative to the capture oscillation period, the modulator is effectively distributed. However for larger storage rings, M-phase design may improve the capture process.

## 4. Conclusions

## References and Links

1. | R.A. Barry, V.W.S. Chan, K.L. Hall, E.S. Kintzer, J.D. Moors, K.A. Rauschenbach, E.A. Swanson, L.E. Adams, C.R. Doerr, S.G. Finn, H.A. Haus, E.P. Ippen, W.S. Wong, and M. Haner, “All-optical network consortium - ultrafast TDM netwarks,” J. Sel. Areas In Com. |

2. | E. Feigenbaum and M. Orenstein, “Colored solitons interactions: particle-like and beyond,” Opt. Express |

3. | M. Nakazawa, H. Kubota, E. Yamada, and K. Suzuki, “Infinite-distance soliton transmission with soliton controls in time and frequency domains,” Elec. Lett. |

4. | J.D. Moors, W.S. Wong, and H.A. Haus, “Stability and timing maintenance in soliton transmission and storage rings,” Opt. Comm. |

5. | H.A. Haus, “Lecture 11” in |

6. | M. Karlsson, D Anderson, A Höök, and M. Lisak, “A variational approach to optical soliton collisions,” Phys. Scripta |

7. | N. C. Panoiu, I. V. Melǹikov, D. Mihalache, C. Etrich, and F. Lederer, “Soliton generation in optical fibers for dual-frequency input,” Phys. Rev. E |

8. | V. V. Afanasjev and V. A. Vysloukh, “Interaction of initially overlapping solitons with different frequencies,” J. Opt. Soc. Am. B |

9. | N.C. Panoiu, D. Mihalache, D. Mazilu, L.C. Crasovan, and I.V. Mel’nikov, “Soliton dynamics of symmetry-endowed two-soiton solutions of the nonlinear Schrodinger equation,” Chaos |

10. | C. Etrich, N.C. Panoiu, D. Mihalache, and F. Lederer, “Limits for interchanel frequency separation in a soliton wavelength-division multiplexing system,” Phys. Rev. E |

11. | N.C. Panoiu, I.V. Mel’nikov, D. Mihalache, C. Etrich, and F. Lederer, “Soliton generation from a multi-frequency optical signal,” J. Opt. B: Quantum Semiclass. Opt. |

12. | G.P. Agrawal, |

13. | V.I. Karpman and V.V. Solov’ev, “A perturbational approach to the two-soliton systems,” Physica D |

14. | H.A. Haus and W.S. Wong, “Solitons in optical communications,” Rev. of Mod. Phys. |

15. | S.M.J. Kelly, “Characteristic sideband instability of periodically amplified average soliton,” Elec. Lett. |

16. | H.A. Haus, W.S. Wong, and F.I. Kharti, “Continuum generation by perturbation of soliton,” J. Opt. Soc. Am. B. |

**OCIS Codes**

(060.5530) Fiber optics and optical communications : Pulse propagation and temporal solitons

(190.2640) Nonlinear optics : Stimulated scattering, modulation, etc.

(190.3270) Nonlinear optics : Kerr effect

(190.4370) Nonlinear optics : Nonlinear optics, fibers

(200.4740) Optics in computing : Optical processing

(210.4680) Optical data storage : Optical memories

**ToC Category:**

Research Papers

**History**

Original Manuscript: June 1, 2004

Revised Manuscript: July 12, 2004

Published: August 9, 2004

**Citation**

Eyal Feigenbaum and Meir Orenstein, "Enhanced mutual capture of colored solitons by matched modulator," Opt. Express **12**, 3759-3764 (2004)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-12-16-3759

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### References

- R.A. Barry, V.W.S. Chan, K.L. Hall, E.S. Kintzer, J.D. Moors, K.A. Rauschenbach, E.A. Swanson, L.E. Adams, C.R. Doerr, S.G. Finn, H.A. Haus, E.P. Ippen, W.S. Wong, M. Haner, ???All-optical network consortium ??? ultrafast TDM netwarks,??? J. Sel. Areas In Com. 14, 999-1012 (1996). [CrossRef]
- E. Feigenbaum, M. Orenstein, ???Colored solitons interactions: particle-like and beyond,??? Opt. Express 12, 2193-2206 (2004),<a href=" http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-10-2193">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-10-2193</a> [CrossRef] [PubMed]
- M. Nakazawa, H. Kubota, E. Yamada, K. Suzuki, ???Infinite-distance soliton transmission with soliton controls in time and frequency domains,??? Elec. Lett. 28, p. 1099-1100 (1992). [CrossRef]
- J.D. Moors, W.S. Wong, H.A. Haus, ???Stability and timing maintenance in soliton transmission and storage rings,??? Opt. Comm. 113, p. 153-175 (1994). [CrossRef]
- H.A. Haus, ???Lecture 11??? in Optical Solitons: Theoretical Challenges and Industrial Perspectives, V.E. Zakarov and S. Wabnitz Ed. (Springer, NY,1999).
- M. Karlsson, D Anderson, A Höök, M. Lisak, "A variational approach to optical soliton collisions," Phys. Scripta 50, 265-270 (1994). [CrossRef]
- N. C. Panoiu, I. V. Mel`nikov, D. Mihalache, C. Etrich, F. Lederer, ???Soliton generation in optical fibers for dual-frequency input,??? Phys. Rev. E 60, 4868-4876 (1999). [CrossRef]
- V. V. Afanasjev, V. A. Vysloukh, ???Interaction of initially overlapping solitons with different frequencies,??? J. Opt. Soc. Am. B 11, 2385-2393 (1994). [CrossRef]
- N.C. Panoiu, D. Mihalache, D. Mazilu, L.C. Crasovan, I.V. Mel???nikov, ???Soliton dynamics of symmetry-endowed two-soiton solutions of the nonlinear Schrodinger equation,??? Chaos 10, 625-640 (2000). [CrossRef]
- C. Etrich, N.C. Panoiu, D. Mihalache, F. Lederer, ???Limits for interchanel frequency separation in a soliton wavelength-division multiplexing system,??? Phys. Rev. E 63, 016609 (2001). [CrossRef]
- N.C. Panoiu, I.V. Mel???nikov, D. Mihalache, C. Etrich, F. Lederer, ???Soliton generation from a multi-frequency optical signal,??? J. Opt. B: Quantum Semiclass. Opt. 4, R53-R68 (2002). [CrossRef]
- G.P. Agrawal, Nonlinear Fiber Optics, 2nd ed.(Academic,NY, 1995).
- 13. V.I. Karpman, V.V. Solov???ev, ???A perturbational approach to the two-soliton systems,??? Physica D 3, 487-502 (1981). [CrossRef]
- H.A. Haus, W.S. Wong, ???Solitons in optical communications,??? Rev. of Mod. Phys. 68, 423-444 (1996). [CrossRef]
- S.M.J. Kelly, ???Characteristic sideband instability of periodically amplified average soliton,??? Elec. Lett. 28, 806-807 (1992). [CrossRef]
- H.A. Haus, W.S. Wong, F.I. Kharti, ???Continuum generation by perturbation of soliton,??? J. Opt. Soc. Am. B. 14, 304 -313(1997) [CrossRef]

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