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

2. Outline What is new What is new What is Volume Free Electron Laser What is Volume Free Electron Laser VFEL physical and mathematical models in Bragg geometry VFEL physical and mathematical models in Bragg geometry Code VOLC for VFEL simulation Code VOLC for VFEL simulation Examples of different chaotic regimes of VFEL intensity Examples of different chaotic regimes of VFEL intensity Sensibility to initial conditions – the “Butterfly” effect Sensibility to initial conditions – the “Butterfly” effect The largest Lyapunov exponent The largest Lyapunov exponent Two-parametric analysis of root to chaotic lasing in VFEL 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. volume diffraction grating What is Volume Free Electron Laser ? * * Eurasian Patent no. 004665 volume “grid” resonator X-ray VFEL

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 The increment of instability for an electron beam passing through a spatially- periodic medium in degeneration points ~  1/(3+s) instead of ~  1/3 for 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 This law is universal and valid for all wavelength ranges regardless the spontaneous radiation mechanism

6. VFEL generator : 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 * V.G. Baryshevsky et al., Nucl. Instr. Meth. B 252 (2006) 86 V.G.Baryshevsky et al., Proc. FEL06, Germany (2006), p.331 Resonant grating provides VDFB of generated radiation with electron beam

7. VFEL in Bragg geometry k k 0 u L k 

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

9. System for two-wave VFEL: - detuning from exact Cherenkov condition  0,1 are direction cosines,     is an asymmetry factor χ 0, χ ±1 are Fourier components of the dielectric susceptibility of the target are system parameters, 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.

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 (“VOLume Code”), version 2.0 for VFEL simulation

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

13. 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  passes through the bifurcational value  0. Dynamical systems* * 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 Intensity Fourier transform Poincar é map Attractor VFEL Intensity 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. Poincar é maps or sections are a signature of periodic regimes in phase space. Poincar é map is constructed as a section of attractor by a small plan area.

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 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) 1)1750 А/ см 2 2)1950 А/ см 2 3)2150 А/ см 2 4)2220 А/ см 2 5)2300 А/ см 2 6)2340 А/ см 2 7)2350 А/ см 2

19. Quasiperiodicity and intermittency 1)1750 А/ см 2 2)1950 А/ см 2 3)2150 А/ см 2 4)2300 А/ см 2 5)2340 А/ см 2 6)2350 А/ см 2

20. Amplification and generation regimes are first and second bifurcation points 180200220240260 j, A/cmcm 1E-9 1E-8 1E-7 1E-6 1E-5 0.0001 0.001 0.01 0.1 1 10 100 1000 | E | E = 0 E=1 E = 0.01 E = 0.0001 2 A B L = 20 cm 0 0 0 0 A B 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.

21. j is equal to:1) 350 A/ cm 2, 2) 450 A/cm 2, 3) 470 A/cm 2, 4) 515 A/cm 2, 5) 525 A/cm 2, 6) 528 A/cm 2, 7) 550 A/cm 2. Amplification regime 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. j is equal to: 1) 490 A/ cm 2, 2) 505 A/cm 2, 3) 530 A/cm 2, 4) 550 A/cm 2 Generation regime

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. Sensibility to initial conditions for generator regime “Weak” chaos Quasiperiodic oscillations j = 1950 A/cm 2 j = 1960 A/cm 2 Periodic regime 2000 A/cm 2 2010 A/cm 2000.1 A/cm 2000+10 -8. A/cm 2 2 2

25. Sensibility to initial conditions for amplification regime j = 2300 А/ сm 2 |Е | = 1 |Е | = 1+10 -15 j = 500 А/ сm 2 |Е | = 1 |Е | = 1.1 Periodic regimeChaotic regime

26. The largest Lyapunov exponent reconstructed with Rosenstein approach* * * M.T.Rosenstein et al. Physica D65 (1993), 117-134 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

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

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

29. Root to chaotic lasing 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. 0 depicts a domain under beam current threshold. P – periodic regimes, Q – quasiperiodicity, I – intermittency, C – chaos. -20-15-10-5  500 1000 1500 2000 2500 j, A / c m 2 I C 0 P Q 2 j, A / c m 500 1000 1500 2000 2500

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