S vetlana Sytova Research Institute for Nuclear Problems, Belarusian State University

1. Bifurcations and chaos in relativistic beams interacting with three-dimensional periodical structures Svetlana Sytova Research Institute for Nuclear Problems, Belarusian State University INP

2. Outline • What is new • What is Volume Free Electron Laser • VFEL physical and mathematical models in Bragg geometry • Code VOLC for VFEL simulation • Examples of different chaotic regimes of VFEL intensity • Sensibility to initial conditions – the “Butterfly” effect • The largest Lyapunov exponent • Two-parametric analysis of root to chaotic lasing in VFEL

3. What is new? Investigation of chaotic behaviour of relativistic beam in three-dimensional periodical structure under volume (non-one-dimensional) multi-wave distributed feedback (VDFB)

4. What is Volume Free Electron Laser ? * volume diffraction grating volume diffraction grating volume “grid” resonator X-ray VFEL * Eurasian Patent no. 004665

5. Benefits of volume distributed feedback* • frequency tuning at fixed energy of electron beam in significantly wider range than conventional systems can provide • more effective interaction of electron beam and electromagnetic wave allows significant reduction of threshold current of electron beam and, as a result, miniaturization of generator • reduction of limits for available output power by the use of wide electron beams and diffraction gratings of large volumes • simultaneous generation at several frequencies This law is universal and valid for all wavelength ranges regardless the spontaneous radiation mechanism The increment of instability for an electron beam passing through a spatially-periodic medium in degeneration points~1/(3+s)instead of~1/3for the single-wave systems (TWTA and FEL) ( s is the number of surplus waves appearing due to diffraction). The following estimation for threshold current in degeneration points is valid: *V.G.Baryshevsky, I.D.Feranchuk, Phys. Lett. 102A (1984) 141; V.G.Baryshevsky, K.G.Batrakov, I.Ya. Dubovskaya, J.Phys D24 (1991) 1250

6. VFEL generator with a "grid" volume resonator*: • Main features: • electron beam of large cross-section • electron beam energy 180-250 keV • possibility of gratings rotation • operation frequency 10 GHz • tungsten threads with diameter 100 m Resonant grating provides VDFB of generated radiation with electron beam * V.G. Baryshevsky et al., Nucl. Instr. Meth. B 252 (2006) 86 V.G.Baryshevskyet al., Proc. FEL06, Germany (2006), p.331

7. t k u kt L 0 VFEL in Bragg geometry

8. Equations for electron beam is an electron phase in a wave

9. System for two-wave VFEL: Right-hand side of this system is more complicated than usually used because it takes into account two-dimensional distributions with respect to spatial coordinate and electron phase p. So, they allow to simulate electron beam dynamics more precisely. This is very important when electron beam moves angularly to electromagnetic waves. are system parameters, - detuning from exact Cherenkov condition g0,1are direction cosines, b = g0 / g1 is an asymmetry factor χ0, χ±1are Fourier components of the dielectric susceptibility of the target

10. System of equations for BWT, TWT etc. * System is versatile in the sense that they remain the same within some normalization for a wide range of electronic devices (FEL, BWT, TWB etc). *N.S.Ginzburg, S.P.Kuznetsov, T.N.Fedoseeva. Izvestija VUZov - Radiophysics, 21 (1978), 1037 (in Russian).

11. Code VOLC (“VOLumeCode”),version 2.0 for VFEL simulation

12. j j L δ χ li βi Results of numerical simulation (2002-2007): Root to chaos (parameter planes: (j,β), (j,d),(j, L)) Laue- Laue VFEL Three-wave SASE Two-wave Amplification regime Bragg Some cases of bifurcation points Laue Generation regime Amplification regime Generation regime Bragg- Bragg SASE Amplification regime Synchronism of several modes: 1, 2, 3 Generation threshold Generation regime with external mirrors Generation threshold Generation regime with external mirrors Bragg- Laue All results obtained numerically are in good agreement with analytical predictions. β

13. Dynamical systems* Chaotic dynamics means the tendency of wide range of systems to transition into states with deterministic behavior and unpredictable behavior. We know a lot of such examples as turbulence in fluid, gas and plasma, lasers and nonlinear optical devices**, accelerators, chaos in biological and chemical systems and so on. Nonlinearity is necessary but non-sufficient condition for chaos in the system. The main origin of chaos is the exponential divergence of initially close trajectories in the nonlinear systems. This is so-called the “Butterfly effect”*** (the sensibility to initial conditions). Bifurcation is any qualitative changes of the system when control parameter mpasses through the bifurcational value m0. *H.-G. Schuster, "Deterministic Chaos" An Introduction, Physik Verlag, (1984) ** M.E.Couprie, Nucl. Instr. Meth. A507 (2003), 1 M.S.Hur, H.J. Lee, J.K.Lee., Phys. Rev. E58 (1998), 936 *** E.N.Lorenz, J. Atmos. Sci. 20 (1963), 130

14. Example of periodic regimes of VFEL intensity VFEL Intensity Intensity Fourier transform Attractor is a set of points in phase space of the dynamical system. System trajectories tend to this set. Attractors are constructed from continious time series with some delay d. Poincaré maps or sections are a signature of periodic regimes in phase space. Poincaré map is constructed as a section of attractor by a small plan area. Poincaré map Attractor

15. Quasiperiodic oscillations Quasiperiodicity is associated with the Hopf bifurcations which introduces a new frequency into the system. The ratios between the fundamental frequencies are incommensurate (Hahn, Lee, Phys. Rev.E(1993),48, 2162)

16. “Weak” chaotic regime Dependence of amplitude in time seems as approximate repetition of equitype spikes close in dimensions per approximately equal time space.

17. Chaotic self-oscillations (hyperchaos)

18. Quasiperiodicity and intermittency • 1750 А/ см2 • 1950 А/ см2 • 2150 А/ см2 • 2220 А/ см2 • 2300 А/ см2 • 2340 А/ см2 • 2350 А/ см2 Intermittency is closely related to saddle-node bifurcations. This means the collision between stable and unstable points, that then disappears. (Hahn, Lee, Phys. Rev.E(1993),48, 2162)

19. Quasiperiodicity and intermittency • 1750 А/ см2 • 1950 А/ см2 • 2150 А/ см2 • 2300 А/ см2 • 2340 А/ см2 • 2350 А/ см2

20. B A | E | 1 0 0 0 1 0 0 E = 1 1 0 0 1 E = 0 . 0 1 0 . 1 0 0 . 0 1 0 . 0 0 1 E = 0 . 0 0 01 0 0 . 0 0 0 1 1 E - 5 cm L = 2 0 1 E - 6 1 E - 7 E = 0 0 1 E - 8 1 E - 9 2 j , A / cm 1 8 0 2 0 0 2 2 0 2 4 0 2 6 0 Amplification and generation regimes are first and second bifurcation points First threshold point corresponds to beginning of electron beam instability. Here regenerative amplification starts while the radiation gain of generating mode is less than radiation losses. Parameters at which radiation gain becomes equal to absorption correspond to the second threshold point after that generation progresses actively. B A

21. Amplification regime j is equal to:1) 350 A/cm2, 2) 450 A/cm2, 3) 470 A/cm2, 4) 515 A/cm2, 5) 525 A/cm2,6) 528 A/cm2, 7) 550 A/cm2. In simulations an important VFEL feature due to VDFB was shown. This is the initiation of quasiperiodic regimes at relatively small current near firsts threshold points.

22. Generation regime j is equal to: 1) 490A/cm2,2) 505A/cm2, 3) 530 A/cm2, 4) 550A/cm2

23. Initiation of quasiperiodic regimes at relatively small resonator length near threshold point transmitted wave diffracted wave under threshold length periodic regime quasiperiodic regime chaotic regime

24. 2 2 0 0 0 A / c m 2 Periodic regime 2 0 0 0 + 10-8 . A / c m 2 2 0 0 0 . 1 A / c m 2 2 0 1 0 A / c m j = 1950 A/cm2 Sensibility to initial conditions for generator regime “Weak” chaos Quasiperiodic oscillations j = 1960 A/cm2

25. j = 500 А/ сm2 |Е | = 1 |Е | = 1.1 j = 2300 А/ сm2 |Е | = 1 |Е | = 1+10-15 Sensibility to initial conditions for amplification regime Periodic regime Chaotic regime

26. The largest Lyapunov exponentreconstructed with Rosenstein approach* The largest Lyapunov exponent is a measurement of the stability of the underlying dynamics of time series. It specifies the mean velocity of divergence of neighboring points. Periodic regime “Weak” chaos *M.T.Rosenstein et al. Physica D65 (1993), 117-134

27. Map of BWT dynamical regimes with strong reflections* with large-scale and small-scale amplitude regimes *S.P.Kuznetsov. Izvestia Vuzov “Applied Nonlinear Dynamics”, 2006, v.14, 3-35

28. Domains with transition between large-scale and small-scale amplitudes

29. 2500 2500 I 2000 2000 C 2 2 m m c c / 1500 1500 / A A Q , j , j P 1000 1000 0 500 500 -20 -15 -10 -5 b Root to chaotic lasing 0 depicts a domain under beam current threshold. P – periodic regimes, Q – quasiperiodicity, I – intermittency, C – chaos. Larger number of principle frequencies for transmitted wave can be explained the fact that in VFEL simultaneous generation at several frequencies is available. Here electrons emit radiation namely in the direction of transmitted wave.

30. Another root to chaotic lasing 0 depicts a domain under beam current threshold. P – periodic regimes, Q –quasiperiodicity, A – domains with transition between large-scale and small-scale amplitudes, I – intermittency, C – chaos.

31. References • Batrakov K., Sytova S.Mathematical Modelling and Analysis,9 (2004) 1-8. • Batrakov K., Sytova S. Computational Mathematics and Mathematical Physics 45: 4 (2005) 666–676 • Batrakov K., Sytova S.Mathematical Modelling and Analysis. 10: 1 (2005) 1–8 • Batrakov K., Sytova S. Nonlinear Phenomena in Complex Systems,8 : 4(2005) 359-365 • Batrakov K., Sytova S. Nonlinear Phenomena in Complex Systems,8 : 1(2005) 42–48 • Batrakov K., Sytova S. Mathematical Modelling and Analysis. 11: 1 (2006) 13–22 • Batrakov K., Sytova S.Proceedings of FEL06, Germany (2006), p.41 • Batrakov K., Sytova S. Proceedings of RUPAC, Novosibirsk, Russia, (2006), p.141