ISSN 0869-6632 (Print)
ISSN 2542-1905 (Online)

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Grigorieva E. V., Kashchenko S. A. Local dynamics of laser chain model with optoelectronic delayed unidirectional coupling. Izvestiya VUZ. Applied Nonlinear Dynamics, 2022, vol. 30, iss. 2, pp. 189-207. DOI: 10.18500/0869-6632-2022-30-2-189-207

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Local dynamics of laser chain model with optoelectronic delayed unidirectional coupling

Grigorieva Elena Viktorovna, Belarus State Economic University (BSEU)
Kashchenko Sergej Aleksandrovich, P. G. Demidov Yaroslavl State University

Purpose. The local dynamics of the laser chain model with optoelectronic delayed unidirectional coupling is investigated. A system of equations is considered that describes the dynamics of a closed chain of a large number of lasers with optoelectronic delayed coupling between elements. An equivalent distributed integro-differential model with a small parameter inversely proportional to the number of lasers in the chain is proposed. For a distributed model with periodic edge conditions, the critical value of the coupling coefficient is obtained, at which the stationary state in the chain becomes unstable. It is shown that in a certain neighborhood of the bifurcation point, the number of roots of the characteristic equation with a real part close to zero increases indefinitely when the small parameter decreases. In this case, a two-dimensional complex Ginzburg–Landau equation with convection is constructed as a normal form. Its nonlocal dynamics determines the behavior of the solutions of the original boundary value problem. Research methods. Methods for studying local dynamics based on the construction of normal forms on central manifolds are used as applied to critical cases of (asymptotically) infinite dimension. An algorithm for reducing the original boundary value problem to the equation for slowly varying amplitudes is proposed. Results. The simplest homogeneous periodic solutions of Ginzburg–Landau equation and corresponding to them inhomogeneous solutions in the form of traveling waves in a distributed model are obtained. Such solutions can be interpreted as phase locking regimes in the chain of coupled lasers. The frequencies and amplitudes of oscillations of the radiation intensity of each laser and the phase difference between adjacent oscillators are determined.

The work of S. A. Kashchenko was supported by the Russian Science Foundation (project No. 21-71-30011).
  1. Pikovsky A, Rosenblum M, Kurths J. Synchronization: A Universal Concept in Nonlinear Sciences. Cambridge: Cambridge University Press; 2001. 411 p. DOI: 10.1017/CBO9780511755743.
  2. Stankovski T, Pereira T, McClintock PVE, Stefanovska A. Coupling functions: Universal insights into dynamical interaction mechanisms. Rev. Mod. Phys. 2017;89(4):045001. DOI: 10.1103/RevModPhys.89.045001.
  3. Klinshov VV, Nekorkin VI. Synchronization of delay-coupled oscillator networks. Phys. Usp. 2013;56(12):1217–1229. DOI: 10.3367/UFNe.0183.201312c.1323.
  4. Kuramoto Y. Chemical Oscillations, Waves, and Turbulence. Berlin: Springer-Verlag; 1984. 158 p. DOI: 10.1007/978-3-642-69689-3.
  5. Schuster HG, Wagner P. Mutual entrainment of two limit cycle oscillators with time delayed coupling. Progress of Theoretical Physics. 1989;81(5):939–945. DOI: 10.1143/PTP.81.939.
  6. Perlikowski P, Yanchuk S, Popovych OV, Tass PA. Periodic patterns in a ring of delay-coupled oscillators. Phys. Rev. E. 2010;82(3):036208. DOI: 10.1103/PhysRevE.82.036208.
  7. Klinshov V, Shchapin D, Yanchuk S, Wolfrum M, D’Huys O, Nekorkin V. Embedding the dynamics of a single delay system into a feed-forward ring. Phys. Rev. E. 2017;96(4):042217. DOI: 10.1103/PhysRevE.96.042217.
  8. Dahms T, Lehnert J, Scholl E. Cluster and group synchronization in delay-coupled networks. Phys. Rev. E. 2012;86(1):016202. DOI: 10.1103/PhysRevE.86.016202.
  9. Ramana Reddy DV, Sen A, Johnston GL. Experimental evidence of time-delay induced death in coupled limit-cycle oscillators. Phys. Rev. Lett. 2000;85(16):3381–3384. DOI: 10.1103/PhysRevLett.85.3381.
  10. Soriano MC, Garcia-Ojalvo J, Mirasso CR, Fischer I. Complex photonics: Dynamics and applications of delay-coupled semiconductors lasers. Rev. Mod. Phys. 2013;85(1):421–470. DOI: 10.1103/RevModPhys.85.421.
  11. Hohl A, Gavrielides A, Erneux T, Kovanis V. Localized synchronization in two coupled nonidentical semiconductor lasers. Phys. Rev. Lett. 1997;78(25):4745–4748. DOI: 10.1103/PhysRevLett.78.4745.
  12. Wunsche HJ, Bauer S, Kreissl J, Ushakov O, Korneyev N, Henneberger F, Wille E, Erzgraber H, Peil M, Elsaßer W, Fischer I. Synchronization of delay-coupled oscillators: A study of semiconductor lasers. Phys. Rev. Lett. 2005;94(16):163901. DOI: 10.1103/PhysRevLett.94.163901.
  13. Otten J, Muller J, Monnigmann M. Bifurcation-aware optimization and robust synchronization of coupled laser diodes. Phys. Rev. E. 2018;98(6):062212. DOI: 10.1103/PhysRevE.98.062212.
  14. Carra TW, Taylor ML, Schwartz IB. Negative-coupling resonances in pump-coupled lasers. Physica D. 2006;213(2):152–163. DOI: 10.1016/j.physd.2005.10.015.
  15. Uchida A, Matsuura T, Kinugawa S, Yoshimori S. Synchronization of chaos in microchip lasers by using incoherent feedback. Phys. Rev. E. 2002;65(6):066212. DOI: 10.1103/PhysRevE.65.066212.
  16. Uchida A, Mizumura K, Yoshimori S. Chaotic dynamics and synchronization in microchip solid state lasers with optoelectronic feedback. Phys. Rev. E. 2006;74(6):066206. DOI: 10.1103/PhysRevE.74.066206.
  17. Kim MY, Roy R, Aron JL, Carr TW, Schwartz IB. Scaling behavior of laser population dynamics with time-delayed coupling: Theory and experiment. Phys. Rev. Lett. 2005;94(8):088101. DOI: 10.1103/PhysRevLett.94.088101.
  18. Vicente R, Tang S, Mulet J, Mirasso CR, Liu JM. Dynamics of semiconductor lasers with bidirectional optoelectronic coupling: Stability, route to chaos, and entrainment. Phys. Rev. E. 2004;70(4):046216. DOI: 10.1103/PhysRevE.70.046216.
  19. Vicente R, Tang S, Mulet J, Mirasso CR, Liu JM. Synchronization properties of two self-oscillating semiconductor lasers subject to delayed optoelectronic mutual coupling. Phys. Rev. E. 2006;73(4):047201. DOI: 10.1103/PhysRevE.73.047201.
  20. Schwartz IB, Shaw LB. Isochronal synchronization of delay-coupled systems. Phys. Rev. E. 2007;75(4):046207. DOI: 10.1103/PhysRevE.75.046207.
  21. Perego AM, Lamperti M. Collective excitability, synchronization, and array-enhanced coherence resonance in a population of lasers with a saturable absorber. Phys. Rev. A. 2016;94(3):033839. DOI: 10.1103/PhysRevA.94.033839.
  22. Kashchenko SA. On quasinormal forms for parabolic equations with small diffusion. Soviet Mathematics. Doklady. 1988;37(2):510–513.
  23. Kaschenko SA. Normalization in the systems with small diffusion. International Journal of Bifurcation and Chaos. 1996;6(6):1093–1109. DOI: 10.1142/S021812749600059X.
  24. Kashchenko SA. Asymptotic form of spatially non-uniform structures in coherent nonlinear optical systems. USSR Computational Mathematics and Mathematical Physics. 1991;31(3):97–102.
  25. Grigorieva EV, Haken H, Kaschenko SA. Theory of quasiperiodicity in model of lasers with delayed optoelectronic feedback. Optics Communications. 1999;165(4–6):279–292. DOI: 10.1016/S0030-4018(99)00236-9.
  26. Kashchenko SA. Dynamics of advectively coupled Van der Pol equations chain. Chaos. 2021;31(3): 033147. DOI: 10.1063/5.0040689.
  27. Khanin YI. Fundamentals of Laser Dynamics. Cambridge: Cambridge International Science Publishing; 2006. 361 p.
  28. Akhromeyeva TS, Kurdyumov SP, Malinetskii GG, Samarskii AA. Nonstationary dissipative structures and diffusion-induced chaos in nonlinear media. Phys. Rep. 1989;176(5–6):189–370. DOI: 10.1016/0370-1573(89)90001-X.