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Journal of the Optical Society of America B

Journal of the Optical Society of America B


  • Editor: Henry van Driel
  • Vol. 28, Iss. 8 — Aug. 1, 2011
  • pp: 1988–1993

Condition for canard explosion in a semiconductor optical amplifier

Elena Shchepakina and Olga Korotkova  »View Author Affiliations

JOSA B, Vol. 28, Issue 8, pp. 1988-1993 (2011)

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A model for the semiconductor optical amplifier (SOA) consisting of two coupled, nonlinear, first-order differential equations is analytically explored on the basis of the geometric theory of singularly perturbed dynamical systems. The value of the control parameter μ, accounting for the stimulated emission, for which the solution exhibits the phenomenon of “canard explosion” is determined depending on the rest of the SOA’s parameters. Such value is represented by a power series of a small parameter ε of the singularly perturbed system. An example is considered where the canard explosion is numerically evaluated for a typical set of the SOA’s parameters. The importance of rigorous determination of the critical regime in the SOA for optical synchronization and photonic clocking is outlined.

© 2011 Optical Society of America

OCIS Codes
(140.3280) Lasers and laser optics : Laser amplifiers
(140.3430) Lasers and laser optics : Laser theory
(140.4480) Lasers and laser optics : Optical amplifiers
(140.5960) Lasers and laser optics : Semiconductor lasers

ToC Category:
Lasers and Laser Optics

Original Manuscript: May 2, 2011
Revised Manuscript: June 23, 2011
Manuscript Accepted: June 28, 2011
Published: July 22, 2011

Elena Shchepakina and Olga Korotkova, "Condition for canard explosion in a semiconductor optical amplifier," J. Opt. Soc. Am. B 28, 1988-1993 (2011)

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  1. R. E. O’Malley, Jr., Introduction to Singular Perturbations (Academic, 1974).
  2. A. B. Vasil’eva, V. F. Butuzov, and L. V. Kalachev, The Boundary Function Method for Singular Perturbation Problems (SIAM, 1995), Vol.  14. [CrossRef]
  3. F. Marino, G. Catalán, P. Sánchez, S. Balle, and O. Piro, “Thermo-optical “canard orbits” and excitable limit cycles,” Phys. Rev. Lett. 92, 073901 (2004). [CrossRef] [PubMed]
  4. S. Baer and T. Erneux, “Singular Hopf bifurcation to relaxation oscillations,” SIAM J. Appl. Math. 46, 721–739 (1986). [CrossRef]
  5. B. Braaksma, “Critical phenomena in dynamical systems of van der Pol type,” Ph.D. thesis (University of Utrecht, 1993).
  6. E. F. Mishchenko, Y. S. Kolesov, A. Y. Kolesov, and N. K. Rozov, Asymptotic Methods in Singularly Perturbed Systems(Plenum, 1995).
  7. M.P.Mortell, R.E.O’Malley, A.Pokrovskii, and V.A.Sobolev, eds. Singular Perturbations and Hysteresis (SIAM, 2005). [CrossRef]
  8. A. W. L. Chan, K. L. Lee, and C. Shu, “Self-starting photonic clock using semiconductor optical amplifier based Mach-Zehnder interferometer,” Electron. Lett. 40, 827–828 (2004). [CrossRef]
  9. E. I. Volkov, E. Ullner, A. A. Zaikin, and J. Kurths, “Oscillatory amplification of stochastic resonance in excitable systems,” Phys. Rev. E 68, 026214 (2003). [CrossRef]
  10. M. Brøns and K. Bar-Eli, “Canard explosion and excitation in a model of the Belousov-Zhabotinsky reaction,” J. Phys. Chem. 95, 8706–8713 (1991). [CrossRef]
  11. M. Brøns and K. Bar-Eli, “Asymptotic analysis of canards in the EOE equations and the role of the inflection line,” Proc. R. Soc. A 445, 305–322 (1994). [CrossRef]
  12. M. Brøns and J. Sturis, “Explosion of limit cycles and chaotic waves in a simple nonlinear chemical system,” Phys. Rev. E 64, 026209 (2001). [CrossRef]
  13. J. Moehlis, “Canards in a surface oxidation reaction,” J. Nonlinear Sci. 12, 319–345 (2002). [CrossRef]
  14. M. Sekikawa, N. Inaba, and T. Tsubouchi, “Chaos via duck solution breakdown in a piecewise linear van der Pol oscillator driven by an extremely small periodic perturbation,” Physica D 194, 227–249 (2004). [CrossRef]
  15. M. Brøns, “Relaxation oscillations and canards in a nonlinear model of discontinuous plastic deformation in metals at very low temperatures,” Proc. R. Soc. A 461, 2289–2302 (2005). [CrossRef]
  16. E. F. Mishchenko and N. K. Rozov, Differential Equations with Small Parameters and Relaxation Oscillations (Plenum, 1980).
  17. J. D. Murray, Mathematical Biology (Springer-Verlag, 2003).
  18. J. Grasman, Asymptotic Methods for Relaxation Oscillations and Applications (Springer-Verlag, 1987). [CrossRef]
  19. M. Diener, Nessie et Les Canards (Publication IRMA, 1979).
  20. E. Benoit, J. L. Calot, F. Diener, and M. Diener, “Chasse au canard,” Collectanea Mathematica 31–32, 37–119 (1981).
  21. E. Benoit, “Systèmes lents-rapides dans R3 et leurs canards,” Astérisque 109–110, 159–191 (1983).
  22. W. Eckhaus, “Relaxation oscillations including a standart chase on French ducks,” Lect. Notes Math. 985, 449–494(1983). [CrossRef]
  23. A. K. Zvonkin and M. A. Shubin, “Non-standard analysis and singular perturbations of ordinary differential equations,” Russ. Math. Surv. 39, 69–131 (1984). [CrossRef]
  24. G. N. Gorelov and V. A. Sobolev, “Mathematical modeling of critical phenomena in thermal explosion theory,” Combust. Flame 87, 203–210 (1991). [CrossRef]
  25. G. N. Gorelov and V. A. Sobolev, “Duck-trajectories in a thermal explosion problem,” Appl. Math. Lett. 5, 3–6 (1992). [CrossRef]
  26. E. Shchepakina and V. Sobolev, “Black swans and canards in laser and combustion models,” in Singular Perturbations and Hysteresis, M.P.Mortell, R.E.O’Malley, A.Pokrovskii, and V.A.Sobolev, eds. (SIAM, 2005), pp. 207–255. [CrossRef]
  27. V. Sobolev and E. Shchepakina, “Duck trajectories in a problem of combustion theory,” Differ. Equ. 32, 1177–1186 (1996).
  28. F. Marino, F. Marino, S. Balle, and O. Piro, “Chaotically spiking canards in an excitable system with 2D inertial fast manifolds,” Phys. Rev. Lett. 98, 074104 (2007). [CrossRef] [PubMed]
  29. M. Desroches, B. Krauskopf, and H. M. Osinga, “Numerical continuation of canard orbits in slow-fast dynamical systems,” Nonlinearity 23, 739–765 (2010). [CrossRef]
  30. V. V. Strygin and V. A. Sobolev, “Effect of geometric and kinetic parameters and energy dissipation on orientation stability of satellites with double spin,” Cosmic Res. 14, 331–335 (1976).
  31. S. Balle, Institut Mediterrani d’Estudis Avançats, CSIC-UIB, E-07071, Palma de Mallorca, Spain (personal communication, 2011).

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