Chaos in Free-Electron Lasers

  • H. P. Freund
  • T. M. AntonsenJr.


Researchers in the field of chaos are concerned with the basic properties of the solutions of systems of nonlinear equations. This interest stems from the fact that almost every physical system can be described at some level of approximation by a system of nonlinear equations. The development of this field has led to several general conclusions about nonlinear systems. On the one hand, even the simplest deterministic nonlinear systems can exhibit behavior that is complicated and appears to be random. This behavior has been termed chaos. On the other hand, the chaotic behavior of much more complicated systems often seems to follow the same rules as the simple systems. Thus, there is order in the chaos. In this chapter, we introduce some basic concepts regarding chaos and nonlinear dynamics before going on to a discussion of the application of these concepts to the physics of free-electron lasers.


Chaos Nonlinear dynamics Lyapunov exponent Integrable trajectories KAM surfaces Attractor Gyroresonance Arnold diffusion Return maps Slippage 


  1. 1.
    A. Lichtenburg, M. Liberman, Regular and Stochastic Motion (Springer, New York, 1983)CrossRefGoogle Scholar
  2. 2.
    R.Z. Sagdeev, D.A. Usikov, G.M. Zaslavky, Nonlinear Physics: From the Pendulum to Turbulence and Chaos (Harwood Academic Publishers, Chur, 1988)Google Scholar
  3. 3.
    E. Ott, Strange attractors and chaotic motions of dynamical systems. Rev. Mod. Phys. 53, 655 (1981)MathSciNetCrossRefGoogle Scholar
  4. 4.
    C. Chen, R.C. Davidson, Chaotic particle dynamics in free-electron lasers. Phys. Rev. A 43, 5541 (1991)CrossRefGoogle Scholar
  5. 5.
    L. Michel, A. Bourdier, J.M. Buzzi, Chaos electron trajectories in a free-electron laser. Nucl. Instr. Meth. A304, 465 (1991)CrossRefGoogle Scholar
  6. 6.
    G. Spindler, G. Renz, Chaotic behavior of electron orbits in a free-electron laser near magnetoresonance. Nucl. Instr. Meth. A304, 492 (1991)CrossRefGoogle Scholar
  7. 7.
    R.C. Davidson, Physics of Nonneutral Plasmas (Addison-Wesley, Reading, 1990)Google Scholar
  8. 8.
    T.M. Antonsen, Jr., Nonlinear dynamics of radiation in a free-electron laser, in Nonlinear Dynamics and Particle Acceleration, ed. Y.H. Ichikawa, T. Tajima (AIP Conference Proceedings No. 230, New York, 1991), p. 106Google Scholar
  9. 9.
    S. Riyopoulos, C.M. Tang, Chaotic electron motion caused by sidebands in free-electron lasers. Phys. Fluids 31, 3387 (1988)CrossRefGoogle Scholar
  10. 10.
    N.S. Ginzburg, M.I. Petelin, Multi-frequency generation in free-electron lasers with quasi-optical resonators. Int. J. Electronics 59, 291 (1985)CrossRefGoogle Scholar
  11. 11.
    M.J. Fiegenbaum, Quantitative universality for a class of nonlinear transformations. J. Stat. Phys. 19, 25 (1978)MathSciNetCrossRefGoogle Scholar
  12. 12.
    L.R. Elias, G.J. Ramian, J. Hu, A. Amir, Observation of single-mode operation in a free-electron laser. Phys. Rev. Lett. 75, 424 (1986)CrossRefGoogle Scholar
  13. 13.
    T.M. Antonsen, B. Levush, Mode competition and suppression in free-electron laser oscillators. Phys. Fluids B 1, 1097 (1989)CrossRefGoogle Scholar
  14. 14.
    T.M. Antonsen, B. Levush, Mode competition and control in free-electron laser oscillators. Phys. Rev. Lett. 62, 1488 (1989)CrossRefGoogle Scholar
  15. 15.
    B.G. Danly, S.G. Evangelides, T.S. Chu, R.J. Tempkin, G. Ramian, J. Hu, Direct spectral measurements of a quasi-cw free-electron laser. Phys. Rev. Lett. 65, 2251 (1990)CrossRefGoogle Scholar
  16. 16.
    Y.L. Bogomolov, V.L. Bratman, N.S. Ginzburg, M.I. Petelin, A.D. Yunakovsky, Nonstationary generation in free-electron lasers. Opt. Comm. 36, 209 (1981)CrossRefGoogle Scholar
  17. 17.
    W.B. Colson, R.A. Freedman, Synchrotron instability for long pulses in free-electron lasers. Opt. Commun. 46, 37 (1983)CrossRefGoogle Scholar
  18. 18.
    B. Levush, T.M. Antonsen Jr., Effect of nonlinear mode competition on the efficiency of low gain free-electron laser oscillator. SPIE 1061, 2 (1989)Google Scholar
  19. 19.
    R.W. Warren, J.E. Sollid, D.W. Feldman, W.E. Stein, W.J. Johnson, A.H. Lumpkin, J.C. Goldstein, Near-ideal lasing with a uniform wiggler. Nucl. Instr. Meth. A285, 1 (1989)CrossRefGoogle Scholar
  20. 20.
    R.W. Warren, J.C. Goldstein, The generation and suppression of synchrotron sidebands. Nucl. Instr. Meth. A272, 155 (1988)CrossRefGoogle Scholar
  21. 21.
    M. Billardon, Storage-ring free-electron laser and chaos. Phys. Rev. Lett. 65, 713 (1990)CrossRefGoogle Scholar
  22. 22.
    M. Billardon, Chaotic behavior of the storage-ring free-electron laser. Nucl. Instr. Meth. A304, 37 (1991)CrossRefGoogle Scholar
  23. 23.
    P. Ellaume, Macrotemporal structure of free-electron lasers. J. Phys. 45, 997 (1984)CrossRefGoogle Scholar
  24. 24.
    C. Grebogi, E. Ott, J.A. Yorke, Chaos, strange attractors, and fractal basin boundaries in nonlinear dynamics. Science 238, 585 (1987)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  • H. P. Freund
    • 1
  • T. M. AntonsenJr.
    • 2
  1. 1.University of Maryland, University of New MexicoViennaUSA
  2. 2.University of MarylandPotomacUSA

Personalised recommendations