Bifurcations and chaos in relativistic beams interacting with three-dimensional periodical structuresSvetlana SytovaResearch Institute for Nuclear Problems,Belarusian State UniversityINP

OutlineWhat is newWhat is Volume Free Electron LaserVFEL physical and mathematical models in Bragg geometryCode VOLC for VFEL simulationExamples of different chaotic regimes of VFEL intensitySensibility to initial conditions the Butterfly effectThe largest Lyapunov exponentTwo-parametric analysis of root to chaotic lasing in VFEL

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)

What is Volume Free Electron Laser ? ** Eurasian Patent no. 004665volume grid resonatorX-ray VFEL

Benefits of volume distributed feedback*frequency tuning at fixed energy of electron beam in significantly wider range than conventional systems can providemore 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 volumessimultaneous generation at several frequenciesThe 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) 1250This law is universal and valid for all wavelength ranges regardless the spontaneous radiation mechanism

VFEL generator with a "grid" volume resonator*:Main features:electron beam of large cross-section electron beam energy 180-250 keVpossibility of gratings rotationoperation 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.331Resonant grating provides VDFB of generated radiation with electron beam

VFEL in Bragg geometry

Equations for electron beam is an electron phase in a wave

System for two-wave VFEL:

- detuning from exact Cherenkov conditiong0,1 are direction cosines, b = g0 / g1 is an asymmetry factor0, 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.

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).

Code VOLC (VOLume Code),version 2.0 for VFEL simulation

Results of numerical simulation (2002-2007):Some cases of bifurcation pointsBraggLaueSASELaue-LaueBragg-LaueSynchronism of several modes: 1, 2, 3VFELTwo-waveThree-waveAmplificationregimeBragg-BraggSASEGenerationregime with external mirrorsAmplificationregimeAmplificationregimeGenerationregime with external mirrorsGenerationregimeGeneration thresholdGeneration thresholdGenerationregimeRoot to chaos (parameter planes: (j, ), (j, d), (j, L))

All results obtained numerically are in good agreement with analytical predictions.

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 m passes 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

Example of periodic regimes of VFEL intensityIntensity Fourier transformPoincar mapAttractorVFEL 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 maps or sections are a signature of periodic regimes in phase space. Poincar map is constructed as a section of attractor by a small plan area.

Quasiperiodic oscillationsQuasiperiodicity 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)

Weak chaotic regimeDependence of amplitude in time seems as approximaterepetition of equitype spikes close in dimensions per approximately equal time space.

Chaotic self-oscillations (hyperchaos)

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)1750 / 2 1950 / 2 2150 / 22220 / 22300 / 22340 / 2 2350 / 2

Quasiperiodicity and intermittency 1750 / 2 1950 / 2 2150 / 22300 / 22340 / 2 2350 / 2

Amplification and generation regimes are first and second bifurcation pointsABFirst 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.

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.

Generation regimej is equal to:1) 490 A/ cm2, 2) 505 A/cm2, 3) 530 A/cm2, 4) 550 A/cm2

Initiation of quasiperiodic regimes at relatively small resonator length near threshold pointtransmitted wavediffracted waveunder threshold lengthperiodic regimequasiperiodic regimechaotic regime

Sensibility to initial conditions for generator regimeWeak chaosQuasiperiodic oscillationsj = 1960 A/cm2

Sensibility to initial conditions for amplification regime

Periodic regimeChaotic regime

The largest Lyapunov exponent reconstructed with Rosenstein approach*

*M.T.Rosenstein et al. Physica D65 (1993), 117-134The 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 regimeWeak chaos

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

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

Root to chaotic lasingLarger 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.

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.

ReferencesBatrakov K., Sytova S. Mathematical Modelling and Analysis, 9 (2004) 1-8.Batrakov K., Sytova S. Computational Mathematics and Mathematical Physics 45: 4 (2005) 666676 Batrakov K., Sytova S.Mathematical Modelling and Analysis. 10: 1 (2005) 18Batrakov 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) 4248Batrakov K., Sytova S. Mathematical Modelling and Analysis. 11: 1 (2006) 1322Batrakov K., Sytova S. Proceedings of FEL06, Germany (2006), p.41Batrakov K., Sytova S. Proceedings of RUPAC, Novosibirsk, Russia, (2006), p.141

MULTIWAVE DIFFRACTION - mention the number of the diffracted waves it will be important later