## High-repetition-rate soliton-train generation using fiber Bragg gratings

Optics Express, Vol. 3, Issue 11, pp. 411-417 (1998)

http://dx.doi.org/10.1364/OE.3.000411

Acrobat PDF (135 KB)

### Abstract

We propose a high-repetition-rate soliton-train source based on adiabatic compression of a dual-frequency optical signal in nonuniform fiber Bragg gratings. As the signal propagates through the grating, it is reshaped into a train of Bragg solitons whose repetition rate is predetermined by the frequency of initial sinusoidal modulation. We develop an approximate analytical model to predict the width of compressed soliton-like pulses and to provide conditions for adiabatic compression. We demonstrate numerically the formation of a 40-GHz train of 2.6-ps pulses and find that the numerical results are in good agreement with the predictions of our analytical model. The scheme relies on the dispersion provided by the grating, which can be up to six orders of magnitude larger than of fiber and makes it possible to reduce the fiber length significantly.

© Optical Society of America

## 1. Introduction

1. J. J. Veselka and S. K. Korotky, “Pulse generation for soliton systems using lithium niobate modulators,” IEEE J. Sel. Top. Quantum Electron. **2**, 300–310 (1996). [CrossRef]

2. A. Hasegawa, “Generation of a train of soliton pulses by induced modulational instability in optical fibers,” Opt. Lett. **9**, 288–290 (1984). [CrossRef] [PubMed]

3. K. Tai, A. Tomita, J. L. Jewell, and A. Hasegawa, “Generation of subpicosecond solitonlike optical pulses at 0.3 THz repetition rate by induced modulational instability,” Appl. Phys. Lett. **49**, 236–238 (1986). [CrossRef]

4. E. M. Dianov, P. V. Mamyshev, A. M. Prokhorov, and S. V. Chernikov, “Generation of a train of fundamental solitons at a high repetition rate in optical fibers,” Opt. Lett. **14**, 1008–1010 (1989). [CrossRef] [PubMed]

5. V. A. Bogatyrev, M. M. Bubnov, E. M. Dianov, A. S. Kurkov, P. V. Mamyshev, A. M. Prokhorov, S. D. Rumyantsev, V. A. Semenov, S. L. Semenov, A. A. Sysoliatin, S. V. Chernikov, A. N. Gur’yanov, G. G. Devyatykh, and S. I. Miroshnichenko, “A single-mode fiber with chromatic dispersion varying along the length,” J. Lightwave Technol. **9**, 561–566 (1991). [CrossRef]

6. P. V. Mamyshev, S. V. Chernikov, and E. M. Dianov, “Generation of fundamental soliton trains for high-bit-rate optical fiber communication lines,” IEEE J. Quantum Electron. **27**, 2347–2355 (1991). [CrossRef]

6. P. V. Mamyshev, S. V. Chernikov, and E. M. Dianov, “Generation of fundamental soliton trains for high-bit-rate optical fiber communication lines,” IEEE J. Quantum Electron. **27**, 2347–2355 (1991). [CrossRef]

12. B. J. Eggleton, T. Stephens, P. A. Krug, G. Dhozi, Z. Brodzeli, and F. Ouellette, “Dispersion compensation over 100 km at 10 Gbit/s using a fiber grating in transmission,” Electron. Lett. **32**, 1610–1611 (1996). [CrossRef]

13. N. M. Litchinitser, B. J. Eggleton, and D. B. Patterson, “Fiber Bragg gratings for dispersion compensation in transmission: Theoretical model and design criterion for nearly ideal pulse compression,” J. Lightwave Technol. **15**, 1303–1313 (1997). [CrossRef]

15. A. Asseh, H. Storay, B. E. Sahlgren, S. Sandgren, and R. A. H. Stubbe, “A writing technique for long fiber Bragg gratings with complex reflectivity profiles,” J. Lightwave Technol. **15**, 1419–1423 (1997). [CrossRef]

17. R. E. Slusher, B. J. Eggleton, C. M. de Sterke, and T. A. Strasser, “Nonlinear pulse reflections from chirped fiber gratings,” Opt. Express **3**, (1998). http://epubs.osa.org/oearchive/source/6996.htm [CrossRef] [PubMed]

## 2. Analytical model

19. A. B. Aceves and S. Wabnitz, “Self-induced transparency solitons in nonlinear refractive periodic media,” Phys. Lett. A **141**, 37–42 (1989). [CrossRef]

*E*

_{+}and

*E*

_{-}are the slowly varying amplitudes of forward and backward propagating waves, respectively,

*V*=

*c*/

*n*is the velocity of light in fiber,

*n*is the average refractive index,

*κ*=

*πηΔn*/

*λ*

_{B}is the coupling coefficient,

*Δn*is the index modulation depth,

*η*is the fraction of the energy in the fiber core,

*λ*

_{B}is the Bragg wavelength. Γ

_{s}and Γ

_{×}are the nonlinear parameters responsible for self- and cross-phase modulation such that Γ

_{s}= Γ

_{×}≡ Γ

_{0}= 2

*π*

*n*

_{2}/

*λ*,

*n*

_{2}is the nonlinear refractive index, and

*λ*is the signal wavelength. The nonlinear coupled mode equations have solitary-wave solutions, called Bragg solitons [19

20. B. J. Eggleton, R. E. Slusher, C. M. de Sterke, P. A. Krug, and J. E. Sipe, “Bragg grating solitons,” Phys. Rev. Lett. **76**, 1627–1630 (1996). [CrossRef] [PubMed]

20. B. J. Eggleton, R. E. Slusher, C. M. de Sterke, P. A. Krug, and J. E. Sipe, “Bragg grating solitons,” Phys. Rev. Lett. **76**, 1627–1630 (1996). [CrossRef] [PubMed]

23. C. M. de Sterke and J. E. Sipe, “Coupled modes and the nonlinear Schrödinger equation,” Phys. Rev. A **42**, 550–555 (1990). [CrossRef]

*δ*=

*n*/

*c*(

*ω*-

*ω*

_{B}) represents detuning of the incident signal from the Bragg frequency,

*v*= [1-(

*κ*/

*δ*)

^{2}]

^{1/2}, and

*γ*= (1-

*v*

^{2})

^{-1/2}. We note that if the grating parameters

*κ*or

*δ*vary with

*z*, both β

_{2}and

*Γ*also vary along the grating length.

*L*

_{D}=

*T*

^{2}/|β

_{2}| is the dispersion length,

*L*

_{NL}= (Γ

*I*

_{in})

^{-1}is the nonlinear length,

*I*

_{in}= |

*A*|

^{2}is the peak intensity of the pulse inside the grating, and

*T*is a measure of the pulse width. For a pulse of the form sech(

*t*/

*T*),

*T*=

*T*

_{FWHM}/1.763, where

*T*

_{FWHM}is the intensity full width at half maximum. The soliton energy

*E*

_{S}= 2

*I*

_{in}

*Tσ*

_{eff}, where

*σ*

_{eff}is the effective core area. Note that the intensity inside the uniform fiber grating is enhanced with respect to that outside the grating by a factor 1/

*v*, i.e.

*I*

_{in}=

*I*

_{out}/

*v*[25

25. B. J. Eggleton, C. M. de Sterke, A. Aceves, J. E. Sipe, T. A. Strasser, and R. E. Slusher, “Modulational instabilities and tunable multiple soliton generation in apodized fiber gratings,” Opt. Commun. **149**, 267–271 (1998). [CrossRef]

_{2}and Γ are

*z*dependent. Combining equations (4

4. E. M. Dianov, P. V. Mamyshev, A. M. Prokhorov, and S. V. Chernikov, “Generation of a train of fundamental solitons at a high repetition rate in optical fibers,” Opt. Lett. **14**, 1008–1010 (1989). [CrossRef] [PubMed]

*N*=1 soliton in a nonuniform grating:

_{2}and Γ (see Eq. (4

4. E. M. Dianov, P. V. Mamyshev, A. M. Prokhorov, and S. V. Chernikov, “Generation of a train of fundamental solitons at a high repetition rate in optical fibers,” Opt. Lett. **14**, 1008–1010 (1989). [CrossRef] [PubMed]

*z*such that the soliton energy

*E*

_{s}remains constant,

*T*must follow changes in

*v*increases with

*z*. The soliton width at the end of the grating is given by

*κ*and

*L*, for adiabatic compression of a short pulse inside the grating. Following Refs. [5

5. V. A. Bogatyrev, M. M. Bubnov, E. M. Dianov, A. S. Kurkov, P. V. Mamyshev, A. M. Prokhorov, S. D. Rumyantsev, V. A. Semenov, S. L. Semenov, A. A. Sysoliatin, S. V. Chernikov, A. N. Gur’yanov, G. G. Devyatykh, and S. I. Miroshnichenko, “A single-mode fiber with chromatic dispersion varying along the length,” J. Lightwave Technol. **9**, 561–566 (1991). [CrossRef]

6. P. V. Mamyshev, S. V. Chernikov, and E. M. Dianov, “Generation of fundamental soliton trains for high-bit-rate optical fiber communication lines,” IEEE J. Quantum Electron. **27**, 2347–2355 (1991). [CrossRef]

3. K. Tai, A. Tomita, J. L. Jewell, and A. Hasegawa, “Generation of subpicosecond solitonlike optical pulses at 0.3 THz repetition rate by induced modulational instability,” Appl. Phys. Lett. **49**, 236–238 (1986). [CrossRef]

_{2}and Γ to a standard NLSE with an effective gain. With the transformation

*τ*=

*t*/

*T*(0) and assume β

_{2}is negative,

*g*(

*ξ*) takes into account axial variations of both β

_{2}and Γ.

*G*

_{eff}represents the factor by which the input pulse is compressed. Equations (12) and (13) show that the compression factor is determined by the ratio of the effective dispersion at the end points of the grating.

*g*(ξ) varies slowly with ξ, it can be averaged over the grating length. From Eq. (11),

*G*

_{eff}(

*L*) = exp(2

*gL*) for

*z*=

*L*. The condition (14) can thus be written as ln(

*G*

_{eff}) ≪

*L*/

*L*

_{D}or, using Eq.(12), as

13. N. M. Litchinitser, B. J. Eggleton, and D. B. Patterson, “Fiber Bragg gratings for dispersion compensation in transmission: Theoretical model and design criterion for nearly ideal pulse compression,” J. Lightwave Technol. **15**, 1303–1313 (1997). [CrossRef]

*n*decreases along its length (see Fig. 1). Although the theory developed in Section 2 assumes that the input pulses are solitons, it allows to predict the width of compressed soliton-like pulses even if the input signal is sinusoidal and to provide conditions for adiabatic compression. As Δ

*n*decreases, grating stop band narrows, as shown in Fig. 1. The dual-frequency optical signal can be obtained from a laser operating simultaneously in two longitudinal modes or by using two distributed feedback semiconductor lasers, each operating in a single longitudinal mode. Coherent beating between the two modes generates a sinusoidally modulated optical signal such that

*T*

_{S}= (Δ

*v*)

^{-1}is the modulation period for a mode spacing of Δ

_{ν}. We consider a fiber grating whose coupling coefficient is nonuniform with a functional form

*κ*

_{0}is maximum coupling coefficient, and C is a constant. The effective dispersion parameter corresponding to this profile is given by

## 4. Conclusions

^{2}). The required intensity can be scaled down using a medium with high nonlinearity. For example, the nonlinear coefficient of As

_{2}S

_{3}chalcogenide fibers is 100 times larger than that of standard silica fibers [28

28. M. Asobe, “Nonlinear optical properties of chalcogenide glass fibers and their application to all-optical switching,” Opt. Fiber Technol. **3**, 142–148 (1997). [CrossRef]

^{2}used in our numerical example in Section III will scale down to 70 MW/cm

^{2}(peak power ~ 10 W) if such fibers are used.

## Acknowledgments

## References and links

1. | J. J. Veselka and S. K. Korotky, “Pulse generation for soliton systems using lithium niobate modulators,” IEEE J. Sel. Top. Quantum Electron. |

2. | A. Hasegawa, “Generation of a train of soliton pulses by induced modulational instability in optical fibers,” Opt. Lett. |

3. | K. Tai, A. Tomita, J. L. Jewell, and A. Hasegawa, “Generation of subpicosecond solitonlike optical pulses at 0.3 THz repetition rate by induced modulational instability,” Appl. Phys. Lett. |

4. | E. M. Dianov, P. V. Mamyshev, A. M. Prokhorov, and S. V. Chernikov, “Generation of a train of fundamental solitons at a high repetition rate in optical fibers,” Opt. Lett. |

5. | V. A. Bogatyrev, M. M. Bubnov, E. M. Dianov, A. S. Kurkov, P. V. Mamyshev, A. M. Prokhorov, S. D. Rumyantsev, V. A. Semenov, S. L. Semenov, A. A. Sysoliatin, S. V. Chernikov, A. N. Gur’yanov, G. G. Devyatykh, and S. I. Miroshnichenko, “A single-mode fiber with chromatic dispersion varying along the length,” J. Lightwave Technol. |

6. | P. V. Mamyshev, S. V. Chernikov, and E. M. Dianov, “Generation of fundamental soliton trains for high-bit-rate optical fiber communication lines,” IEEE J. Quantum Electron. |

7. | S.V. Chernikov, D.J. Richardson, R.I. Laming, E.M. Dianov, and D.N. Payne, “70 Gbit/s fiber based source of fundamental solitons at 1550nm,” Electron. Lett. |

8. | M. Romagnoli, S. Trillo, and S. Wabnitz, “Soliton switching in nonlinear couplers,” Optical and Quantum Electon. |

9. | S. V. Chernikov, J. R. Taylor, and R. Kashyap, “Comblike dispersion-profiled fiber for soliton pulse train generation,” Opt. Lett. |

10. | E. A. Swanson and S. R. Chinn, “23-GHz and 123-GHz soliton pulse generation using two cw lasers and standard single-mode fiber,” IEEE Photon. Technol. Lett. |

11. | N. Akhmediev and A. Ankiewicz, “Generation of a train of solitons with arbitrary phase difference between neighboring solitons,” Opt. Lett. |

12. | B. J. Eggleton, T. Stephens, P. A. Krug, G. Dhozi, Z. Brodzeli, and F. Ouellette, “Dispersion compensation over 100 km at 10 Gbit/s using a fiber grating in transmission,” Electron. Lett. |

13. | N. M. Litchinitser, B. J. Eggleton, and D. B. Patterson, “Fiber Bragg gratings for dispersion compensation in transmission: Theoretical model and design criterion for nearly ideal pulse compression,” J. Lightwave Technol. |

14. | G. Lenz and B. J. Eggleton, “Adiabatic Bragg soliton compression in nonuniform grating structures,” J. Opt. Soc. Am. B, in press. |

15. | A. Asseh, H. Storay, B. E. Sahlgren, S. Sandgren, and R. A. H. Stubbe, “A writing technique for long fiber Bragg gratings with complex reflectivity profiles,” J. Lightwave Technol. |

16. | L. Dong, M. J. Cole, M. Durkin, M. Ibsen, and R. I. Laming, “40Gbit/s 1.55um transmission over 109km of non-dispersion-shifted fiber with long continuously chirped fiber gratings,” in 1996Opt. Fiber Commun. Conf. (OFC'96), postdeadline paper PD6. |

17. | R. E. Slusher, B. J. Eggleton, C. M. de Sterke, and T. A. Strasser, “Nonlinear pulse reflections from chirped fiber gratings,” Opt. Express |

18. | C. M. de Sterke and J. E. Sipe, in |

19. | A. B. Aceves and S. Wabnitz, “Self-induced transparency solitons in nonlinear refractive periodic media,” Phys. Lett. A |

20. | B. J. Eggleton, R. E. Slusher, C. M. de Sterke, P. A. Krug, and J. E. Sipe, “Bragg grating solitons,” Phys. Rev. Lett. |

21. | D. Taverner, N.G.R. Broderick, D.T. Richardson, R.I. Laming, and M. Ibsen, “Nonlinear self-switching and multiple gap-soliton formation in a fiber Bragg grating,” Opt. Lett. |

22. | B. J. Eggleton, C. M. de Sterke, and R. E. Slusher, “Bragg solitons in the nonlinear Schrödinger limit: theory and experiment,” submitted to J. Opt. Soc. Am. B (1998). |

23. | C. M. de Sterke and J. E. Sipe, “Coupled modes and the nonlinear Schrödinger equation,” Phys. Rev. A |

24. | C. M. de Sterke and B. J. Eggleton, “Bragg solitons and the nonlinear Schrödinger equation,” Phys. Rev. E. , in press. |

25. | B. J. Eggleton, C. M. de Sterke, A. Aceves, J. E. Sipe, T. A. Strasser, and R. E. Slusher, “Modulational instabilities and tunable multiple soliton generation in apodized fiber gratings,” Opt. Commun. |

26. | G. P. Agrawal, |

27. | J. P. Gordon, “Interaction forces among solitons in optical fibers,” Opt. Lett. |

28. | M. Asobe, “Nonlinear optical properties of chalcogenide glass fibers and their application to all-optical switching,” Opt. Fiber Technol. |

**OCIS Codes**

(060.4370) Fiber optics and optical communications : Nonlinear optics, fibers

(230.6080) Optical devices : Sources

(270.5530) Quantum optics : Pulse propagation and temporal solitons

(320.5520) Ultrafast optics : Pulse compression

(350.2770) Other areas of optics : Gratings

**ToC Category:**

Focus Issue: Bragg solitons and nonlinear optics of periodic structures

**History**

Published: November 23, 1998

**Citation**

Natalia Litchinitser, Govind Agrawal, Benjamin Eggleton, and Gadi Lenz, "High-repetition-rate soliton-train generation using fiber Bragg gratings," Opt. Express **3**, 411-417 (1998)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-3-11-411

Sort: Journal | Reset

### References

- J. J. Veselka and S. K. Korotky, "Pulse generation for soliton systems using lithium niobate modulators," IEEE J. Sel. Top. Quantum Electron. 2, 300-310 (1996). [CrossRef]
- A. Hasegawa, "Generation of a train of soliton pulses by induced modulational instability in optical fibers," Opt. Lett. 9, 288-290 (1984). [CrossRef] [PubMed]
- K. Tai, A. Tomita, J. L. Jewell and A. Hasegawa, "Generation of subpicosecond solitonlike optical pulses at 0.3 THz repetition rate by induced modulational instability," Appl. Phys. Lett. 49, 236-238 (1986). [CrossRef]
- E. M. Dianov, P. V. Mamyshev, A. M. Prokhorov and S. V. Chernikov, "Generation of a train of fundamental solitons at a high repetition rate in optical fibers," Opt. Lett. 14, 1008-1010 (1989). [CrossRef] [PubMed]
- V. A. Bogatyrev, M. M. Bubnov, E. M. Dianov, A. S. Kurkov, P. V. Mamyshev, A. M. Prokhorov, S. D. Rumyantsev, V. A. Semenov, S. L. Semenov, A. A. Sysoliatin, S. V. Chernikov, A. N. Guryanov, G. G. Devyatykh and S. I. Miroshnichenko, "A single-mode fiber with chromatic dispersion varying along the length," J. Lightwave Technol. 9, 561-566 (1991). [CrossRef]
- P. V. Mamyshev, S. V. Chernikov and E. M. Dianov, "Generation of fundamental soliton trains for high-bit- rate optical fiber communication lines," IEEE J. Quantum Electron. 27, 2347-2355 (1991). [CrossRef]
- S.V. Chernikov, D.J. Richardson, R.I. Laming, E.M. Dianov and D.N. Payne, "70 Gbit/s fiber based source of fundamental solitons at 1550nm," Electron. Lett. 28, 1210-1212 (1992). [CrossRef]
- M. Romagnoli, S. Trillo and S. Wabnitz, "Soliton switching in nonlinear couplers," Optical and Quantum Electon. 24, S1237-S1267 (1992). [CrossRef]
- S. V. Chernikov, J. R. Taylor and R. Kashyap, "Comblike dispersion-profiled fiber for soliton pulse train generation," Opt. Lett. 19, 539-541 (1994). [CrossRef] [PubMed]
- E. A. Swanson and S. R. Chinn, "23-GHz and 123-GHz soliton pulse generation using two cw lasers and standard single-mode fiber," IEEE Photon. Technol. Lett. 6, 796-799 (1994). [CrossRef]
- N. Akhmediev and A. Ankiewicz, "Generation of a train of solitons with arbitrary phase difference between neighboring solitons," Opt. Lett. 19, 545-547 (1994). [CrossRef] [PubMed]
- B. J. Eggleton, T. Stephens, P. A. Krug, G. Dhozi, Z. Brodzeli and F. Ouellette, "Dispersion compensation over 100 km at 10 Gbit/s using a fiber grating in transmission," Electron. Lett. 32, 1610-1611 (1996). [CrossRef]
- N. M. Litchinitser, B. J. Eggleton and D. B. Patterson, "Fiber Bragg gratings for dispersion compensation in transmission: Theoretical model and design criterion for nearly ideal pulse compression," J. Lightwave Technol. 15, 1303-1313 (1997). [CrossRef]
- G. Lenz and B. J. Eggleton, "Adiabatic Bragg soliton compression in nonuniform grating structures," J. Opt. Soc. Am. B, in press.
- A. Asseh, H. Storoy, B. E. Sahlgren, S. Sandgren and R. A. H. Stubbe, "A writing technique for long fiber Bragg gratings with complex reflectivity profiles," J. Lightwave Technol. 15, 1419-1423 (1997). [CrossRef]
- L. Dong, M. J. Cole, M. Durkin, M. Ibsen and R. I. Laming, "40Gbit/s 1.55um transmission over 109km of non-dispersion-shifted fiber with long continuously chirped fiber gratings," in 1996 Opt. Fiber Commun. Conf. (OFC96), postdeadline paper PD6.
- R. E. Slusher, B. J. Eggleton, C. M. de Sterke and T. A. Strasser, "Nonlinear pulse reflections from chirped fiber gratings," Opt. Express 3, (1998). http://epubs.osa.org/oearchive/source/6996.htm [CrossRef] [PubMed]
- C. M. de Sterke and J. E. Sipe, in Progress in Optics XXXIII, ed. by E. Wolf (North-Holand, Amsterdam, 1994), pp. 203-260.
- A. B. Aceves and S. Wabnitz, "Self-induced transparency solitons in nonlinear refractive periodic media," Phys. Lett. A 141, 37-42 (1989). [CrossRef]
- B. J. Eggleton, R. E. Slusher, C. M. de Sterke, P. A. Krug and J. E. Sipe, "Bragg grating solitons," Phys. Rev. Lett. 76, 1627-1630 (1996). [CrossRef] [PubMed]
- D. Taverner, N. G. R. Broderick, D. T. Richardson , R. I. Laming and M. Ibsen, "Nonlinear self-switching and multiple gap-soliton formation in a fiber Bragg grating," Opt. Lett. 23, 328-330 (1998). [CrossRef]
- B. J. Eggleton, C. M. de Sterke and R. E. Slusher, "Bragg solitons in the nonlinear Schrödinger limit: theory and experiment," submitted to J. Opt. Soc. Am. B (1998).
- C. M. de Sterke and J. E. Sipe, "Coupled modes and the nonlinear Schrödinger equation," Phys. Rev. A 42, 550-555 (1990). [CrossRef]
- C. M. de Sterke and B. J. Eggleton, "Bragg solitons and the nonlinear Schrödinger equation," Phys. Rev. E., in press.
- B. J. Eggleton, C. M. de Sterke, A. Aceves, J. E. Sipe, T. A. Strasser and R. E. Slusher, "Modulational instabilities and tunable multiple soliton generation in apodized fiber gratings," Opt. Commun. 149, 267-271 (1998). [CrossRef]
- G. P. Agrawal, Nonlinear Fiber Optics, (Academic, New York, 1995).
- J. P. Gordon, "Interaction forces among solitons in optical fibers," Opt. Lett. 8, 596-598 (1983). [CrossRef] [PubMed]
- M. Asobe, "Nonlinear optical properties of chalcogenide glass fibers and their application to all-optical switching," Opt. Fiber Technol. 3, 142-148 (1997). [CrossRef]

## Cited By |
Alert me when this paper is cited |

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

« Previous Article | Next Article »

OSA is a member of CrossRef.