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.

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References

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  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. 1, 257–279 (1876).
  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]
  4. P. G. Drazin and R. S. Johnson, Solitons: An Introduction, 2nd ed. (Cambridge University Press, 1989).
  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]
  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]
  9. 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.
  10. S. Chouli and P. Grelu, “Rains of solitons in a fiber laser,” Opt. Express 17(14), 11776–11781 (2009).
    [CrossRef] [PubMed]
  11. H. Statz and G. De Mars, “Transients and oscillation pulses in masers,” in Quantum Electronics (Columbia University Press, 1960), pp. 530–537.
  12. L. Tarassov, Physique des Processus dans les Générateurs de Rayonnement Optique Cohérent (Éditons MIR, 1981).
  13. M. Braun, Differential Equations and their Applications: An Introduction to Applied Mathematics (Springer, 1992).
  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 Cr4+:YAG passive switch under the weak resonant signal control,” Opt. Commun. 169(1-6), 309–316 (1999).
    [CrossRef]
  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]
  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]
  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. 367(1901), 3213–3226 (2009).
    [CrossRef] [PubMed]
  20. A. C. Newell and J. V. Moloney, Nonlinear Optics (Addison-Wesley Publishing Co., 1992).
  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 24(3), 592–605 (2007).
    [CrossRef]
  22. N. Akhmediev and A. Ankiewicz, Solitons Nonlinear Pulses and Beams (Chapman & Hall, 1997).

2010 (1)

2009 (2)

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]

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

2008 (1)

2007 (1)

2004 (1)

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]

2003 (1)

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]

1999 (1)

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]

1993 (1)

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]

1974 (1)

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]

1973 (2)

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]

1965 (1)

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]

1876 (1)

L. Rayleigh, “On waves,” Philos. Mag. 1, 257–279 (1876).

Aboites, V.

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]

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]

Ashkin, A. A.

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]

Baldwin, K. J.

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]

Bjorkholm, J. E.

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]

Chouli, S.

Clerc, M. G.

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]

Clowes, J.

Coulibaly, S.

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]

Crofts, G. J.

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]

Damzen, M. J.

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]

Dinda, P. T.

Dupriez, P.

Egorov, A. A.

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]

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]

Gèrôme, F.

Grelu, P.

Hasegawa, A.

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]

He, J.

Il’ichev, N. N.

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]

Kanna, T.

Kartashov, Y. A.

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]

Kartashov, Y. V.

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]

Kir’yanov, A.

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]

Knight, J. C.

Kruskal, M. D.

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]

Li, J.

Liang, X.

Mujica, N.

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]

Navarro, R.

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]

Radhakrishnan, R.

Rayleigh, L.

L. Rayleigh, “On waves,” Philos. Mag. 1, 257–279 (1876).

Sauma, T.

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]

Tappert, F.

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]

Torner, L.

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]

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]

Vysloukh, V. A.

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]

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]

Wadsworth, W. J.

Xu, J.

Zabusky, N. J.

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]

Zelenina, A. S.

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]

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]

Zhao, Z.

Zheng, L.

Appl. Phys. Lett. (2)

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]

J. Opt. Soc. Am. B (1)

Opt. Commun. (2)

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]

Opt. Express (3)

Philos. Mag. (1)

L. Rayleigh, “On waves,” Philos. Mag. 1, 257–279 (1876).

Philos. Transact. A Math. Phys. Eng. Sci. (1)

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]

Phys. Rev. E Stat. Nonlin. Soft Matter Phys. (1)

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]

Phys. Rev. Lett. (3)

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]

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]

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]

Other (8)

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

P. G. Drazin and R. S. Johnson, Solitons: An Introduction, 2nd ed. (Cambridge University Press, 1989).

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).

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.

A. C. Newell and J. V. Moloney, Nonlinear Optics (Addison-Wesley Publishing Co., 1992).

N. Akhmediev and A. Ankiewicz, Solitons Nonlinear Pulses and Beams (Chapman & Hall, 1997).

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Figures (5)

Fig. 1
Fig. 1

Optical scheme for cnoidal wave generation. AM and SA are active medium and saturable absorber, M 1 and M 2 are total reflected and semi-transparent laser mirrors, and EOM is an electro-optical modulator.

Fig. 2
Fig. 2

Stability condition given by the relation between α and αa .

Fig. 3
Fig. 3

Laser output intensity for α a = 15 and control frequencies (a) ω = 1, (b) 5, (c) 15, (d) 25, (e) 50, and (f) 75.

Fig. 4
Fig. 4

Overlapping of one pulse taken at αa = 15 and ω = 25 (solid line) with a sech 2 wave form (dashed line).

Fig. 5
Fig. 5

Modulation frequency ωs and absorption ratio of saturable absorber corresponding to soliton-shape pulses.

Equations (4)

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

d S d t = Γ v σ N S Γ v l a l k a S 1 T S , d N d t = β σ w N S + N 0 N τ , d k a d t = 2 σ a k a S w + k 0 a k a τ a ,
d m d t ' = G m ( n + n a 1 ) , d n d t ' = α n ( m + 1 ) , d n a d t ' = δ α a [ 1 + cos ( ω t ) 2 ] n a ( ρ m + δ ) ,
d m d t ' = G m ( n + n a 1 ) , d n d t ' = α n ( m + 1 ) , d n a d t ' = δ α a [ 1 + cos ( x ) 2 ] n a ( ρ m + δ ) , d x d t ' = ω .
[ G ( α + α a 1 ) λ ] [ 1 λ ] [ δ λ ] [ λ ] = 0 ,

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