## Generation of cnoidal waves by a laser system with a controllable saturable absorber |

Optics Express, Vol. 19, Issue 15, pp. 14210-14216 (2011)

http://dx.doi.org/10.1364/OE.19.014210

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### Abstract

We demonstrate the cnoidal wave formation in a two-laser system with a saturable absorber in the cavity of one of the lasers. Another laser is used to activate the saturable absorber in order to control the pulse shape, width, intensity and frequency. Using the three-level laser model based on the Statz - De Mars equations, we show that for any value of the saturable absorber parameter there exists a certain modulation frequency for which the pulse shape is very close to a soliton shape with less than 5% error at the pulse base. Such a device may be prominent for optical communication and laser engineering applications.

© 2011 OSA

## 1. Introduction

3. N. J. Zabusky and M. D. Kruskal, “Interaction of ‘solitons’ in a collisionless plasma and the recurrence of initial states,” Phys. Rev. Lett. **15**(6), 240–243 (1965). [CrossRef]

5. J. E. Bjorkholm and A. A. Ashkin, “Cw self-focusing and self-trapping of light in sodium vapor,” Phys. Rev. Lett. **32**(4), 129–132 (1974). [CrossRef]

6. A. Hasegawa and F. Tappert, “Transmission of stationary nonlinear optical physics in dispersive dielectric fibers I: Anomalous dispersion,” Appl. Phys. Lett. **23**(3), 142–144 (1973). [CrossRef]

7. A. Hasegawa and F. Tappert, “Transmission of stationary nonlinear optical physics in dispersive dielectric fibers II: Normal dispersion,” Appl. Phys. Lett. **23**(4), 171–172 (1973). [CrossRef]

7. A. Hasegawa and F. Tappert, “Transmission of stationary nonlinear optical physics in dispersive dielectric fibers II: Normal dispersion,” Appl. Phys. Lett. **23**(4), 171–172 (1973). [CrossRef]

8. F. Gèrôme, P. Dupriez, J. Clowes, J. C. Knight, and W. J. Wadsworth, “High power tunable femtosecond soliton source using hollow-core photonic bandgap fiber, and its use for frequency doubling,” Opt. Express **16**(4), 2381–2386 (2008). [CrossRef] [PubMed]

10. S. Chouli and P. Grelu, “Rains of solitons in a fiber laser,” Opt. Express **17**(14), 11776–11781 (2009). [CrossRef] [PubMed]

## 2. Theoretical model

*S*is the emitted photon density,

*N*is the population inversion of the active medium, and

*k*is the resonant absorption of the saturable absorber. Γ,

_{a}*ν*,

*σ*, and

*T*stand, respectively, for cavity filling coefficient, optical frequency, active medium cross-section, and photon lifetime in the cavity,

*β*is the coefficient which accounts for the difference in population inversion coursed by lasing,

*l*and

*l*are, respectively, the active medium and the SA lengths,

_{a}*k*

_{0}

*is the linear resonant SA absorption coefficient without lasing,*

_{a}*σ*is the SA cross-section,

_{a}*N*

_{0}is the population inversion in the active medium without radiation,

*τ*and

*τ*stand for relaxation time in the active medium and in the SA, respectively, and finally

_{a}*t*´ =

*t*/

*τ*,

*G*=

*τ*/

*T*,

*δ*=

*τ*/

*τ*,

_{a}*ρ*= 2

*σ*/

_{a}*βσ*,

*α*=

*ΓνσTN*,

*α*= -

_{a}*ΓνTk*

_{0}

*(*

_{a}*l*/

_{a}*l*),

*n*(

*t*´) =

*ΓνσTN*(

*t*´),

*n*(

_{a}*t´*) = -

*Γν*(

*l*/

_{a}*l*)

*Tk*(

_{a}*t*´), and

*m*(

*t*´) = 2

*πβστS*(

*t*´)/

*hw*, and including the normalized control harmonic modulation (1 + cos(

*ωt*))/2 to

*α*in the third equation which describes the SA:

_{a}14. V. Aboites, K. J. Baldwin, G. J. Crofts, and M. J. Damzen, “Fast high power optical switch,” Opt. Commun. **98**(4-6), 298–302 (1993). [CrossRef]

15. A. Kir’yanov, V. Aboites, and N. N. Il’ichev, “A polarisation-bistable neodymium laser with a Cr^{4+}:YAG passive switch under the weak resonant signal control,” Opt. Commun. **169**(1-6), 309–316 (1999). [CrossRef]

*m*= 0,

^{s}*n*=

^{s}*α*,

*n*=

_{a}^{s}*α*, and

_{a}*x*= 0) is:where

^{s}*λ*

_{1}=

*G*(

*α*+

*α*– 1),

_{a}*λ*

_{2}= −1,

*λ*

_{3}= -

*δ*, and

*λ*

_{4}= 0 are eigenvalues which are all real, being

*λ*

_{2}and

*λ*

_{3}always negative. Therefore, the stability condition is defined only by the sign of

*λ*

_{1}, i.e. the fixed point is a source when

*α*+

_{a}*α*> 1, as shown in Fig. 2 . Thus, to obtain oscillations in the laser with SA [12],

*α*and

*α*should be chosen inside the dashed region shown in Fig. 2.

_{a}14. V. Aboites, K. J. Baldwin, G. J. Crofts, and M. J. Damzen, “Fast high power optical switch,” Opt. Commun. **98**(4-6), 298–302 (1993). [CrossRef]

*G*= 200,

*α*= 4,

*δ*= 1,

*ρ*= 0.001, and the initial conditions near one of the critical stable points of Eqs. (2), i.e.

*m*

_{0}= 0.25,

*n*

_{0}= 0, and

*n*

_{a}_{0}= 0.152. Since the parameter

*α*depends on geometrical values and on the absorbent centers density in the SA (the dye concentration in a dye SA cell), we use it as the SA defining parameter and call it

_{a}*absorption ratio*.

## 3. Results

*α*= 15 and different modulation frequencies

_{a}*ω*. For small

*ω*(lower than the laser relaxation oscillation frequency), the laser generates pulse trains with localized undulation windows, which are the damped relaxation oscillations (Figs. 3(a)–3(c)). For higher

*ω*, only one frequency remains, i.e. the laser oscillates with the modulation frequency (Figs. 3(d)–3(f)), and the pulse shape strongly depends on

*ω*. One important aspect is that as

*ω*is increased; the peak amplitude first increases, reaches a maximum, and then decreases, thus going from a

*sech*

^{2}(when the amplitude is maximum, Fig. 3(d)) to almost harmonic oscillations (Fig. 3(f)). While the peak amplitude is decreasing, the laser intensity never falls down to zero again; the continuous background appears because the frequency applied to the SA is so high that it has not enough time, neither to relax to its ground state nor to saturate. As

*ω*further increases, the signal behavior becomes more and more sinusoidal with relatively small amplitude. We repeat the simulations for different

*α*with a step of 5 and find that the results shown in Fig. 3 for

_{a}*α*= 15 follow exactly the same qualitative pattern for any other

_{a}*α*∈ [5, 60].

_{a}*sech*

^{2}. Note, that the generated cnoidal waves are asymptotically stable due to quadratic nonlinearities of the SA [16

16. Y. A. Kartashov, A. A. Egorov, A. S. Zelenina, V. A. Vysloukh, and L. Torner, “Stabilization of one-dimensional periodic waves by saturation of the nonlinear response,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. **68**(6), 065605 (2003). [CrossRef]

17. Y. V. Kartashov, A. A. Egorov, A. S. Zelenina, V. A. Vysloukh, and L. Torner, “Stable multicolor periodic-wave arrays,” Phys. Rev. Lett. **92**(3), 033901 (2004). [CrossRef] [PubMed]

*sech*

^{2}(soliton-like) shape approximates the pulse shape with a very good precision as demonstrated in Fig. 4 . This is confirmed by overlapping one pulse with a

*sech*

^{2}waveform; the difference that appears on the base right hand side is very small (in the order of 2%, and always less than 5%). We find that for every saturable absorber coefficient

*α*there is an optimal modulation frequency

_{a}*ω*for which this soliton-shape approximation has better precision than for other frequencies. As seen from Fig. 5 ,

_{s}*ω*increases approximately linearly with

_{s}*α*with two jumps at

_{a}*α*= 5 and

_{a}*α*= 30. We should note that several theoretical and experimental works report the existence of solitons [18

_{a}18. J. Li, X. Liang, J. He, L. Zheng, Z. Zhao, and J. Xu, “Diode pumped passively mode-locked Yb:SSO laser with 2.3 ps duration,” Opt. Express **18**(17), 18354–18359 (2010). [CrossRef] [PubMed]

*sech*and

*sech*

^{2}). While in the cited works, the difference between the reported pulses and the soliton shape is larger than 5%; our system allows the soliton generation with a higher precision, which makes it prominent for optical communication purposes.

## 4. Conclusions

*sech*

^{2}), we obtained less than 5% error at the pulse base. The proposed system can be a base for building a reliable and cheap device to generate cnoidal waves as efficient information carriers for optical communication; the proposed system is economic due to the elements involved, at using general purpose laser elements and not ultra-fast optics elements, the experimental implementation of the presented scheme results less expensive.

## Acknowledgments

## References and links

1. | J. S. Russell, “Report on waves,” Fourteenth Meeting of the British Association for the Advancement of Science (1844). |

2. | L. Rayleigh, “On waves,” Philos. Mag. |

3. | N. J. Zabusky and M. D. Kruskal, “Interaction of ‘solitons’ in a collisionless plasma and the recurrence of initial states,” Phys. Rev. Lett. |

4. | P. G. Drazin and R. S. Johnson, |

5. | J. E. Bjorkholm and A. A. Ashkin, “Cw self-focusing and self-trapping of light in sodium vapor,” Phys. Rev. Lett. |

6. | A. Hasegawa and F. Tappert, “Transmission of stationary nonlinear optical physics in dispersive dielectric fibers I: Anomalous dispersion,” Appl. Phys. Lett. |

7. | A. Hasegawa and F. Tappert, “Transmission of stationary nonlinear optical physics in dispersive dielectric fibers II: Normal dispersion,” Appl. Phys. Lett. |

8. | F. Gèrôme, P. Dupriez, J. Clowes, J. C. Knight, and W. J. Wadsworth, “High power tunable femtosecond soliton source using hollow-core photonic bandgap fiber, and its use for frequency doubling,” Opt. Express |

9. | R. Herda and O. G. Okhotnikov, “All-fiber soliton source tunable over 500 nm,” in |

10. | S. Chouli and P. Grelu, “Rains of solitons in a fiber laser,” Opt. Express |

11. | H. Statz and G. De Mars, “Transients and oscillation pulses in masers,” in |

12. | L. Tarassov, |

13. | M. Braun, |

14. | V. Aboites, K. J. Baldwin, G. J. Crofts, and M. J. Damzen, “Fast high power optical switch,” Opt. Commun. |

15. | A. Kir’yanov, V. Aboites, and N. N. Il’ichev, “A polarisation-bistable neodymium laser with a Cr |

16. | Y. A. Kartashov, A. A. Egorov, A. S. Zelenina, V. A. Vysloukh, and L. Torner, “Stabilization of one-dimensional periodic waves by saturation of the nonlinear response,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. |

17. | Y. V. Kartashov, A. A. Egorov, A. S. Zelenina, V. A. Vysloukh, and L. Torner, “Stable multicolor periodic-wave arrays,” Phys. Rev. Lett. |

18. | J. Li, X. Liang, J. He, L. Zheng, Z. Zhao, and J. Xu, “Diode pumped passively mode-locked Yb:SSO laser with 2.3 ps duration,” Opt. Express |

19. | M. G. Clerc, S. Coulibaly, N. Mujica, R. Navarro, and T. Sauma, “Soliton pair interaction law in parametrically driven Newtonian fluid,” Philos. Transact. A Math. Phys. Eng. Sci. |

20. | A. C. Newell and J. V. Moloney, |

21. | P. T. Dinda, R. Radhakrishnan, and T. Kanna, “Energy-exchange collision of the Manakov vector solitons under strong environmental perturbations,” J. Opt. Soc. Am. B |

22. | N. Akhmediev and A. Ankiewicz, |

**OCIS Codes**

(140.0140) Lasers and laser optics : Lasers and laser optics

(190.0190) Nonlinear optics : Nonlinear optics

(190.6135) Nonlinear optics : Spatial solitons

**ToC Category:**

Nonlinear Optics

**History**

Original Manuscript: May 5, 2011

Revised Manuscript: June 17, 2011

Manuscript Accepted: June 17, 2011

Published: July 11, 2011

**Citation**

M. Wilson, V. Aboites, A. N. Pisarchik, V. Pinto, and M. Taki, "Generation of cnoidal waves by a laser system with a controllable saturable absorber," Opt. Express **19**, 14210-14216 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-15-14210

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### References

- J. S. Russell, “Report on waves,” Fourteenth Meeting of the British Association for the Advancement of Science (1844).
- L. Rayleigh, “On waves,” Philos. Mag. 1, 257–279 (1876).
- N. J. Zabusky and M. D. Kruskal, “Interaction of ‘solitons’ in a collisionless plasma and the recurrence of initial states,” Phys. Rev. Lett. 15(6), 240–243 (1965). [CrossRef]
- P. G. Drazin and R. S. Johnson, Solitons: An Introduction, 2nd ed. (Cambridge University Press, 1989).
- J. E. Bjorkholm and A. A. Ashkin, “Cw self-focusing and self-trapping of light in sodium vapor,” Phys. Rev. Lett. 32(4), 129–132 (1974). [CrossRef]
- A. Hasegawa and F. Tappert, “Transmission of stationary nonlinear optical physics in dispersive dielectric fibers I: Anomalous dispersion,” Appl. Phys. Lett. 23(3), 142–144 (1973). [CrossRef]
- A. Hasegawa and F. Tappert, “Transmission of stationary nonlinear optical physics in dispersive dielectric fibers II: Normal dispersion,” Appl. Phys. Lett. 23(4), 171–172 (1973). [CrossRef]
- F. Gèrôme, P. Dupriez, J. Clowes, J. C. Knight, and W. J. Wadsworth, “High power tunable femtosecond soliton source using hollow-core photonic bandgap fiber, and its use for frequency doubling,” Opt. Express 16(4), 2381–2386 (2008). [CrossRef] [PubMed]
- R. Herda and O. G. Okhotnikov, “All-fiber soliton source tunable over 500 nm,” in Conference on Lasers and Electro-Optics/Quantum Electronics and Laser Science and Photonic Applications Systems Technologies, Technical Digest (CD) (Optical Society of America, 2005), paper JWB39.
- S. Chouli and P. Grelu, “Rains of solitons in a fiber laser,” Opt. Express 17(14), 11776–11781 (2009). [CrossRef] [PubMed]
- H. Statz and G. De Mars, “Transients and oscillation pulses in masers,” in Quantum Electronics (Columbia University Press, 1960), pp. 530–537.
- L. Tarassov, Physique des Processus dans les Générateurs de Rayonnement Optique Cohérent (Éditons MIR, 1981).
- M. Braun, Differential Equations and their Applications: An Introduction to Applied Mathematics (Springer, 1992).
- V. Aboites, K. J. Baldwin, G. J. Crofts, and M. J. Damzen, “Fast high power optical switch,” Opt. Commun. 98(4-6), 298–302 (1993). [CrossRef]
- A. Kir’yanov, V. Aboites, and N. N. Il’ichev, “A polarisation-bistable neodymium laser with a Cr4+:YAG passive switch under the weak resonant signal control,” Opt. Commun. 169(1-6), 309–316 (1999). [CrossRef]
- Y. A. Kartashov, A. A. Egorov, A. S. Zelenina, V. A. Vysloukh, and L. Torner, “Stabilization of one-dimensional periodic waves by saturation of the nonlinear response,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 68(6), 065605 (2003). [CrossRef]
- Y. V. Kartashov, A. A. Egorov, A. S. Zelenina, V. A. Vysloukh, and L. Torner, “Stable multicolor periodic-wave arrays,” Phys. Rev. Lett. 92(3), 033901 (2004). [CrossRef] [PubMed]
- J. Li, X. Liang, J. He, L. Zheng, Z. Zhao, and J. Xu, “Diode pumped passively mode-locked Yb:SSO laser with 2.3 ps duration,” Opt. Express 18(17), 18354–18359 (2010). [CrossRef] [PubMed]
- M. G. Clerc, S. Coulibaly, N. Mujica, R. Navarro, and T. Sauma, “Soliton pair interaction law in parametrically driven Newtonian fluid,” Philos. Transact. A Math. Phys. Eng. Sci. 367(1901), 3213–3226 (2009). [CrossRef] [PubMed]
- A. C. Newell and J. V. Moloney, Nonlinear Optics (Addison-Wesley Publishing Co., 1992).
- P. T. Dinda, R. Radhakrishnan, and T. Kanna, “Energy-exchange collision of the Manakov vector solitons under strong environmental perturbations,” J. Opt. Soc. Am. B 24(3), 592–605 (2007). [CrossRef]
- N. Akhmediev and A. Ankiewicz, Solitons Nonlinear Pulses and Beams (Chapman & Hall, 1997).

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