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

For citation:

Krenc A. A., Molevich N. E. Birth of a stable torus from the critical closed curve and its bifurcations in a laser system with frequency detuning. Izvestiya VUZ. Applied Nonlinear Dynamics, 2010, vol. 18, iss. 5, pp. 67-80. DOI: 10.18500/0869-6632-2010-18-5-67-80

This is an open access article distributed under the terms of Creative Commons Attribution 4.0 International License (CC-BY 4.0).
Full text PDF(Ru):
PDF icon 2010no5p068.pdf
(downloads: 174)
Language: 
Russian
Heading: 
Bifurcations in Dynamical Systems
Article type: 
Article
UDC: 
535.374:621.375.8
DOI: 
10.18500/0869-6632-2010-18-5-67-80

Birth of a stable torus from the critical closed curve and its bifurcations in a laser system with frequency detuning

Autors: 
Krenc Anton Anatolevich, Samara branch of Physical Institute. P. N. Lebedev Of The Russian Academy Of Sciences
Molevich Nonna Evgenevna, Samara branch of Physical Institute. P. N. Lebedev Of The Russian Academy Of Sciences
Abstract: 

Realization of stable two­frequency oscillations is shown in the Maxwell–Bloch model. Birth of a stable ergodic two­dimensional torus from the critical closed curve is observed. The conditions of the passage to chaos via a cascade of torus doubling bifurcations are obtained. It is established that at bifurcations points a structurally unstable three­dimensional torus is produced, which gives rise to a stable doubled ergodic torus. Analytical approximation describing dynamics of the system near a point of torus birth is found.

Key words: 
Wide-aperture lasers
torus doubling bifurcation
ergodic torus
chaos
Reference: 
  1. Lorenz EN. Deterministic nonperiodic flow. J. Atm. Sci. 1963;20(2):130–141. DOI: 10.1175/1520-0469(1963)020<0130:DNF>2.0.CO;2.
  2. Grasiuk AZ, Oraevsky AN. Proceedings of the Fourth International Congress on Microwave Tubes Sheveningen. Holland; 1962, 446 p. Paper at Course 31, Enrico Fermi Summer School. Varenna, Italy; 1963. (in Russian).
  3. Oraevsky AN. Masers, lasers, and strange attractors. Sov. J. Quantum. Electron. 1981;11(1):71–78.
  4. Haken H. Analogy between higher instabilities in fluids and lasers. Phys. Lett. A. 1975;53(1):77–78. DOI: 10.1016/0375-9601(75)90353-9.
  5. Letellier C. Modding out a continuous rotation symmetry for disentangling a laser dynamics. International Journal of Bifurcation and Chaos. 2003;13(6):1573–1577. DOI: 10.1142/S0218127403007424.
  6. Weiss CO, Larionova Ye. Pattern formation in optical resonators. Rep. Prog. Phys. 2007;70(2):255–335. DOI: 10.1088/0034-4885/70/2/R03.
  7. Hollinger F, Jung Chr, Weber H. Simple mathematical model describing multitransversal solid-state lasers. J. Opt. Soc. Am. B. 1990;7(6):1013–1018. DOI: 10.1364/JOSAB.7.001013.
  8. Hollinger F, Jung Chr. Single-longitudinal-mode laser as a discrete dynamical system. J. Opt. Soc. Am. B. 1985;2(1):218–225. DOI: 10.1364/JOSAB.2.000218.
  9. Cabrera E, Calderon OG, Melle S, Guerra JM. Development of spatial turbulence from boundary-controlled patterns in class-B lasers. Phys. Rev. A. 2006;73(5):053820. DOI: 10.1103/PhysRevA.73.053820.
  10. Huyet G, Tredicce JR. Spatio-temporal chaos in the transverse section of lasers. Physica D. 1996;96:209–214. DOI: 10.1016/0167-2789(96)00021-8.
  11. Huyet G, Martinoni MC, Tredicce JR, Rica S. Spatiotemporal dynamics of lasers with a large Fresnel number. Phys. Rev. Lett. 1995;75(22):4027–4030. DOI: 10.1103/PhysRevLett.75.4027.
  12. O‘Neil E, Houlihan J, McInerney JG, Huyet G. Dynamics of traveling waves in the transverse section of a laser. Phys. Rev. Lett. 2005;94(14):143901. DOI: 10.1103/PhysRevLett.94.143901.
  13. Jacobsen PK, Moloney JV, Newell AC, Indik R. Space-time dynamics of wide-gain-section lasers. Phys. Rev. A. 1992;45(11):8129–8137. DOI: 10.1103/physreva.45.8129.
  14. Jacobsen PK, Lega J, Feng Q, Staley M, Moloney JV, Newell AC. Nonlinear transverse modes of large-aspect-ratio homogeneously broadened lasers: I. Analysis and numerical simulation. Phys. Rev. A. 1994;49(5):4189–4200. DOI: 10.1103/physreva.49.4189.
  15. Jacobsen PK, Lega J, Feng Q, Staley M, Moloney JV, Newell AC. Nonlinear transverse modes of large-aspect-ratio homogeneously broadened lasers: II. Pattern analysis near and beyond threshold. Phys. Rev. A. 1994;49(5):4201–4212. DOI: 10.1103/physreva.49.4201.
  16. Zaikin AP, Molevich NE. Effect of the cross-relaxation rate on the transverse radiation dynamics of a wide-aperture laser. Quantum Electron. 2004;34(8):731–735.
  17. Zaikin AP, Kurguzkin AA, Molevich NE. Periodic self-wave structures in a wide-aperture laser with frequency detuning. I. Bifurcation analysis. Quantum Electron. 1999;29(6):523–525.
  18. Zaikin AP, Kurguzkin AA, Molevich NE. Periodic self-wave structures in a wide-aperture laser with frequency detuning. II. Distributed model. Quantum Electron. 1999;29(6):526–529.
  19. Zaikin AP, Kurguzkin AA, Molevich NE. Influence of frequency tuning on the space-time structure of the optical field of a wide-aperture laser. Izvestiya VUZ. Applied Nonlinear Dynamics. 1999;7(5):87–96 (in Russian).
  20. Krents AA, Molevich NE. Cascade of torus doubling bifurcations in a detuned laser. Quantum Electron. 2009;39(8):751–756.
  21. Amroun D, Brunel M, Letellier C, Leblond H, Sanchez F. Complex intermittent dynamics in large-aspect-ratio homogeneously broadened single-mode lasers. Physica D. 2005;203:185–197. DOI: 10.1016/j.physd.2005.03.015.
  22. Lugiato LA, Oldano C, Narducci LM. Cooperative frequency locking and stationary spatial structures in lasers. J. Opt. Soc. Am. B. 1988;5:879–888. DOI: 10.1364/JOSAB.5.000879.
  23. Arrowsmith D, Place C. Ordinary differential equations : a qualitative approach with applications. Moscow: Mir; 1986. (in Russian).
  24. Kuznetsov SP. Dynamical Chaos: Course of Lectures. Moscow: Fizmatlit; 2001. 296 p. (in Russian).
  25. Schuster G. Deterministic Chaos. An Introduction. Moscow: Mir; 1988. (in Russian).
  26. Anishchenko VS, Nikolaev SM. Generator of quasi-periodic oscillations featuring two-dimensional torus doubling bifurcations. Tech. Phys. Lett. 2005;31(10):853–855. DOI: 10.1134/1.2121837.
  27. Anishchenko VS. Complex Oscillations in Simple Systems. Moscow: Nauka; 1990. (in Russian).
  28. Zeghlache H., Mandel P. Influence of detuning on properties of laser equations. J. Opt. Soc. Am. B. 1985;2(1):18–22. DOI: 10.1364/JOSAB.2.000018.
Received: 
08.02.2010
Accepted: 
25.05.2010
Published: 
31.12.2010
Short text (in English):
PDF icon a2010no5p068.doc_.pdf
(downloads: 118)
  • 1650 reads