Post on 31-Dec-2015
SSvetlana vetlana SytovaSytova
Research Institute for Nuclear Research Institute for Nuclear Problems,Problems,
Belarusian State UniversityBelarusian State University
Bifurcations and Bifurcations and chaos in relativistic chaos in relativistic beams interacting beams interacting
with with three-dimensional three-dimensional
periodical structuresperiodical structures
INP
OutlineOutline
What is newWhat is new What is Volume Free Electron LaserWhat is Volume Free Electron Laser VFEL physical and mathematical models in VFEL physical and mathematical models in
Bragg geometryBragg geometry Code VOLC Code VOLC forfor VFEL simulation VFEL simulation Examples of different chaotic regimes of Examples of different chaotic regimes of
VFEL intensityVFEL intensity Sensibility to initial conditions – the Sensibility to initial conditions – the
“Butterfly” effect“Butterfly” effect The largest Lyapunov exponentThe largest Lyapunov exponent Two-parametric analysis of root to chaotic Two-parametric analysis of root to chaotic
lasing in VFELlasing in VFEL
What is new?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)
volume diffraction grating
volume diffraction grating
WhatWhat is Volume Free Electron Laser is Volume Free Electron Laser ? *? *
* Eurasian Patent no. 004665
volume “grid” resonator X-ray VFEL
Benefits of volume distributed Benefits of volume distributed feedback*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:
s23start LkkL1~j
*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
VFELVFEL generator 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
VFEL in Bragg geometryVFEL in Bragg geometry
k
0
u
L
k
0conditionsCherenkov
,02
conditionsBragg2
ku
τkτ
Equations for electron beamEquations for electron beam
]2,2[],,0[,0
,),0,(
,/),0,(
)),,,(exp(
),/(Re),,(),,(
3
232
2
pLzt
ppt
ukdz
ptd
pzti
zuztEdz
pztdk
m
e
dz
pztd
),,( pzt is an electron phase in a wave
200 )//(1 cul
System for tSystem for twowo-wave VFEL-wave VFEL::
05.05.0
,8
22
5.05.0
11
),,(),,(2
02
0
EliEiz
Ec
t
E
dpeep
j
EilEiz
Ec
t
E
pztipzti
2 2 20
2
0
, 0,1,
,
ii
k cl i
l l
- detuning from exact Cherenkov condition
0,1 are direction cosines, is an asymmetry factor
χ0, 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.
System of equations for BWT, System of equations for BWT, TWT etc. *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 Code VOLCVOLC (“ (“VOLVOLumeume CCode”),ode”),version 2.0 forversion 2.0 for VFEL simulation VFEL simulation
Some cases of bifurcation
points
Results of Results of numericalnumerical simulation (2002- simulation (2002-2007):2007):
Bragg
Laue
SASE
Laue-Laue
Bragg-Laue
Synchronism of several
modes: 1, 2, 3
VFEL
Two-wave
Three-waveAmplification
regime
j δ
βili
Bragg-BraggSASE
Generationregime with
external mirrors
Amplificationregime
Amplificationregime
Generationregime with
external mirrors
Generationregime
j L
χ
Generation threshold
Generation threshold
Generationregime
Root to chaos (parameter planes:
(j, β), (j, ), (j, L))
β
All results obtained numerically are in good agreement with analytical predictions.
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*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
Example of periodic regimes of Example of periodic regimes of VFEL intensityVFEL intensity
1800 1850 1900 1950 2000 20501800
1820
1840
1860
1880
1900
1920
1940
1960
1980
2000
E(t
+d
)
E(t)
17501800
18501900
19502000
2050 17501800
18501900
19502000
2050
1800
1850
1900
1950
2000
2050
Z A
xis
E(t+d)
E(t)
0 100 200 300
0
4
8
IFT
, a .u .
0 200 400 600 800t, ns
0
500
1E+3
1.5E+3
2E+3
2.5E+3|E |
Intensity Fourier transform
Poincaré 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.
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.
Quasiperiodic oscillationsQuasiperiodic oscillations
1400 1600 1800 20001100
1200
1300
1400
1500
1600
1700
1800
1900
2000
E(t
+d
)
E(t)
0 100 200 300
0
5
10
15
IFT
, a . u .
0 100 200 300 400t, ns
0.1
1
10
1E+2
1E+3
|E |
1000120014001600
18002000
22001000 1200 1400 1600 1800 2000 2200
1200
1400
1600
1800
2000
2200
E(t
+2d
)
E(t+d)
E(t)
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)
““Weak” chaotic regimeWeak” chaotic regime
2900 3000 3100 3200 3300 3400 3500 3600 37003100
3200
3300
3400
3500
3600
3700E
(t+
d)
E(t)
0 200 400 600 800t, ns
0
1E+3
2E+3
3E+3
4E +3|E |
0 200 400 600
0
2
4
6
8
10
IFT
, a .u .
3000
3200
3400
3600
3800
30003200
34003600
38003000
3200
3400
3600
3800
E(t
+2d
)
E(t+d)
E(t)
Dependence of amplitude in time seems as approximaterepetition of equitype spikes close in dimensions per approximately equal time space.
Chaotic self-oscillations Chaotic self-oscillations (hyperchaos)(hyperchaos)
0 200 400 600 800t, ns
0
5E+2
1E+3
1.5E+3
2E+3
2.5E+3
3E+3
3.5E+3
4E+3
4.5E+3
5E+3
5.5E+3|E |
0 100 200 300
0
1
2
3
4
5
IFT
, a .u.
42004300
44004500
46004700
48004900
5000 42004300
44004500
46004700
48004900
50004200
4300
4400
4500
4600
4700
4800
4900
5000
E(t
+2d
)
E(t+d)E(t)
4400 4500 4600 4700 4800 49004300
4400
4500
4600
4700
4800
E(t
+d
)
E(t)
Quasiperiodicity and intermittencyQuasiperiodicity 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/ см2 2) 1950 А/ см2 3) 2150 А/ см2
4) 2220 А/ см2
5) 2300 А/ см2
6) 2340 А/ см2 7) 2350 А/ см2
500
1000
1500
2000
2500
3000
3500
5001000
15002000
25003000
3500
1000
1500
2000
2500
3000
3500
E(t+d)
E(t+
2d)
E(t)
Quasiperiodicity and intermittencyQuasiperiodicity and intermittency
1000
1500
2000
2500
3000
3500
5001000
15002000
25003000
3500
500
1000
1500
2000
2500
3000
3500
E(t
+2d
)
E(t+
d)
E(t)
1) 1750 А/ см2 2) 1950 А/ см2 3) 2150 А/ см2
4) 2300 А/ см2
5) 2340 А/ см2 6) 2350 А/ см2
Amplification and generation Amplification and generation regimes are first and second regimes are first and second
bifurcation pointsbifurcation points
180 200 220 240 260j, A/cm1E-9
1E-81E-71E-61E-5
0.00010.0010.010.1
110
1001000
| E |
E = 0
E =1
E
=
0.01
E = 0.0001
2
A B
L
=
20
cm
0
0
0
0
A BFirst 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.
0 200 400 600 800t, ns
0
200
400
600
800
|E |
1
2
3
4
67
5
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.
Amplification regime Amplification regime
0 200 400 600 800t, ns
0
40
80
120|E |
1
2
34
6
7
5
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.
0 200 400 600 800t, ns
0
40
80
120|E |
1
2
3
4
0 200 400 600 800t, ns
0
300
600
900|E |
1
2
3
4
j is equal to:1) 490 A/ cm2, 2) 505 A/cm2, 3) 530 A/cm2, 4) 550 A/cm2
Generation regimeGeneration regime
Initiation of quasiperiodic regimes Initiation of quasiperiodic regimes at relatively small resonator at relatively small resonator length near threshold pointlength near threshold point
10 15 20 25 30 35 40
L, cm
transmitted wave
diffracted wave
under threshold lengthperiodic regimequasiperiodic regimechaotic regime
Sensibility to initial conditionsSensibility to initial conditions for for generator regimegenerator regime
0 100 200 300 400t, ns
0.1
1
10
1E+2
1E +3
|E |
0 100 200 300 400t, ns
0.1
1
10
1E+2
1E +3
|E |
0 200 400 600t, ns
0
1E+3
2E+3
3E+3
4E +3|E |
“Weak” chaos
Quasiperiodic oscillations
j = 1950 A/cm2
j = 1960 A/cm2
0 200 400 600t, ns
0
1E+3
2E+3
3E+3
4E+3
5E+3
6E+3
7E+3
8E+3
9E+3
1E+4
1.1E+4
1.2E+4|E |
Periodic regime
2000 A/cm2
2010 A/cm2000.1 A/cm2000+10-8. A/cm
2
2
2
0 200 400 600 800t, ns
0
30
60
90|E |
Sensibility to initial conditionsSensibility to initial conditions for for amplification regimeamplification regime
0 200 400 600t, ns
0.1
1
10
1E+2
1E+3
1E +4|E |
j = 2300 А/ сm2
|Е | = 1|Е | = 1+10-15
j = 500 А/ сm2
|Е | = 1|Е | = 1.1
Periodic regime Chaotic regime
0 20 40 60 80 100t, ns
-6
-5
-4
ln (
d )
1
jThe largest Lyapunov exponentThe largest Lyapunov exponent
reconstructed with Rosenstein reconstructed with Rosenstein approach*approach*
0 20 40 60 80 100t, ns
-4
-2
0
ln(d
) j
1
**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
Map of BWT dynamical regimes Map of BWT dynamical regimes with strong with strong reflections*reflections*
*S.P.Kuznetsov. Izvestia Vuzov “Applied Nonlinear Dynamics”, 2006, v.14, 3-35
with large-scale with large-scale and small-scale and small-scale amplitude amplitude regimesregimes
Domains with transition between Domains with transition between large-scale and small-scale large-scale and small-scale
amplitudesamplitudes
Root to chaotic lasingRoot 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
/cm
2
I
C
0
P
Q
2j,
A/c
m
500
1000
1500
2000
2500
-20 -15 -10 -5
Another root to chaotic lasingAnother 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.
-20 -10 0 10 20
k L
500
1000
1500
2000
2500
3000
j, A
/cm
2
-20 -10 0 10 20
k L
500
1000
1500
2000
2500
3000
j, A
/cm
2
0
P
Q
A
I
C
ReferencesReferences
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