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Bifurcations and chaos in relativistic beams interacting with threedimensional periodical structures. S vetlana Sytova Research Institute for Nuclear Problems, Belarusian State University. INP. Outline. What is new What is Volume Free Electron Laser
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Svetlana Sytova
Research Institute for Nuclear Problems,
Belarusian State University
INP
Investigation of chaotic behaviour of relativistic beam in threedimensional periodical structure under volume (nononedimensional) multiwave distributed feedback (VDFB)
volume diffraction grating
volume diffraction grating
volume “grid” resonator
Xray VFEL
* Eurasian Patent no. 004665
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 spatiallyperiodic medium in degeneration points~1/(3+s)instead of~1/3for the singlewave 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
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
is an electron phase in a wave
Righthand side of this system is more complicated than usually used because it takes into account twodimensional 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
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).
j
L
δ
χ
li
βi
Results of numerical simulation (20022007):Root to chaos
(parameter planes:
(j,β), (j,d),(j, L))
Laue
Laue
VFEL
Threewave
SASE
Twowave
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
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 nonsufficient 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 socalled 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
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
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)
Dependence of amplitude in time seems as approximate
repetition of equitype spikes close in dimensions per approximately equal time space.
Intermittency is closely related to saddlenode bifurcations. This means the collision between stable and unstable points, that then disappears.
(Hahn, Lee, Phys. Rev.E(1993),48, 2162)
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 pointsFirst 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
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.
transmitted wave
diffracted wave
under threshold length
periodic regime
quasiperiodic regime
chaotic regime
2
0
0
0
A
/
c
m
2
Periodic regime
2
0
0
0
+
108
.
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
Е  = 1
Е  = 1.1
j = 2300 А/ сm2
Е  = 1
Е  = 1+1015
Sensibility to initial conditions for amplification regimePeriodic regime
Chaotic regime
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), 117134
with largescale and smallscale amplitude regimes
*S.P.Kuznetsov. Izvestia Vuzov “Applied Nonlinear Dynamics”, 2006, v.14, 335
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 lasing0 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.
0 depicts a domain under beam current threshold. P – periodic regimes, Q –quasiperiodicity, A – domains with transition between largescale and smallscale amplitudes, I – intermittency, C – chaos.