NUMERICAL SIMULATION OF
NONLINEAR EFFECTS IN VOLUME
FREE ELECTRON LASER (VFEL)
K. Batrakov, S. SytovaResearch Institute for Nuclear
Problems,Belarusian State University
Free Electron Laser
Theoretical investigations show that it is one of the effective schemes with n-wave volume distributed feedback
(VDFB)
FEL lasing is aroused by different types of spontaneous radiation: undulator radiation, Smith-Purcell or Cherenkov radiation and so on. Using positive feedback in FELs reduces the working length and provides oscillation regime of generation. This feedback is usually one-dimensional and can be formed either by two parallel mirrors or by one-dimensional diffraction grating, in which incident and diffracted waves move along the electron beam.
What is volume distributed feedback ?
Volume (non-one-dimensional) multi-wave distributed
feedback is the distinctive feature of Volume Free Electron
Laser (VFEL)
one-dimensional distributed feedback
two-dimensional distributed feedback
Benefits provided by the volume distributed feedback
This new law provides for noticeable reduction of electron beam current density
(108 A/cm2 for LiH crystal against 1013 A/cm2 required in G.Kurizki, M.Strauss, I.Oreg,
N.Rostoker, Phys.Rev. A35 (1987) 3427) necessary for achievement the generation
threshold. In X-ray range this generation threshold can be reached for the
induced parametric X-ray radiation in crystals, i.e. to create X-ray laser
(V.G.Baryshevsky, K.G.Batrakov, I.Ya. Dubovskaya, J.Phys D24 (1991) 1250)
The new law of instability for an electron beam passing through a spatially-
periodic medium provides the increment of instability in degeneration points
proportional to 1/(3+s) , here s is the number of surplus waves appearing due to
diffraction. This increment differs from the conventional increment for single-wave
system (TWTA and FEL), which is proportional to 1/3.
(V.G.Baryshevsky, I.D.Feranchuk, Phys.Lett. 102A (1984) 141)
s23start LkkL1~j
This law is universal and valid for all wavelength ranges regardless the This law is universal and valid for all wavelength ranges regardless the
spontaneous radiation mechanismspontaneous radiation mechanism
volume diffraction grating
volume diffraction grating
What is Volume Free Electron Laser ? *
* Eurasian Patent no. 004665
volume “grid” resonator X-ray VFEL
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
Use of volume distributed feedback makes available:
VFEL experimental history1996
Experimental modeling of electrodynamic processes in the
volume diffraction grating made from dielectric threads
2001
First lasing of volume free electron laser in mm-wavelength range.
Demonstration of validity of VFEL principles. Demonstration of possibility for
frequency tuning at constant electron energy
2004
New VFEL prototype with volume “grid” resonator.
V.G.Baryshevsky, K.G.Batrakov,A.A.Gurinovich, I.I.Ilienko, A.S.Lobko,
V.I.Moroz, P.F.Sofronov, V.I.Stolyarsky, NIM 483 A (2002) 21
V.G.Baryshevsky, K.G.Batrakov, I.Ya. Dubovskaya, V.A.Karpovich, V.M.Rodionova, NIM 393A (1997) 71
Resonator (2001)side view
top view
diffraction gratings
The interaction of the exciting diffraction grating with the electron beam arouses Smith-Purcell radiation. The second resonant grating provides distributed feedback of generated radiation with electron beam by Bragg dynamical diffraction.
New VFEL generator (2004)Main features:
volume gratings (volume “grid”)
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
First observation of generation in the backward wave oscillator with a "grid" diffraction grating and lasing of the volume FEL with a "grid" volume resonator*
( V. Baryshevsky et al., LANL e-print archive: physics/0409125)
Electrodynamical properties of a volume resonator that is formed by a periodic structure built from the metallic threads inside a rectangular waveguide depend on diffraction conditions.
Electrodynamical properties of a "grid" volume resonator *
* V. Baryshevsky, A. Gurinovich, LANL e-print archive: physics/0409107
Then in this system the effect of anomalous transmission for electromagnetic waves could appear similarly to the Bormann effect well-known in the dynamical diffraction theory of X-rays.
Resonant grating provides VDFB of generated radiation with electron beam
“Grid” volume resonator*
V.G.Baryshevsky, N.A.Belous, A.Gurinovich et al., Abstracts of FEL06, (2006), TUPPH012
VFEL in Bragg geometry
k
0
u
L
k
0conditionsCherenkov
,02
conditionsBragg2
ku
τkτ
Laue geometry
u
Three-wave VFEL, Bragg-Bragg geometry
k2
k1
0
u
L
k
Equations for electron beam
2
2 3( ) Re exp ( ) ,
( , , ) electron phase in a wave
z
d eE i k z t
dt mt z p
e n k r
]2,2[],,0[,0
,),0,(,/),0,(
pLzt
pptukdz
ptd
Main equations
2
2
1( ) ,b
c t t
jE
E E
( )( )0 1( )
1( )
0
( ),
In the common wave case:
i
i ti t
i tb
ni t
ii
E e E eje
n
E e
k rkr
kr
k r
E ej e
E e
System for two-wave VFEL:
05.05.0
,8
22
5.05.0
11011
11
),,(),,(2
02
1100
00
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
γj are direction cosines,
χ±1 are Fourier components of the dielectric susceptibility of the target
System parametersThe integral form of current is obtained by averaging over two initial phases: entrance time of electron in interaction zone and transverse coordinate of entrance point in interaction zone. In the mean field approximation double integration over two initial phases can be reduced
to single integration.
Code VOLC - VFEL simulation
Some cases of bifurcation
points
Results of numerical simulation (2002-2006):
BraggLaue
SASE
Laue-Laue
Bragg-Laue
Synchronism of several
modes: 1, 2, 3
VFEL
Two-wave
Three-waveAmplification
regime
j δ
βili
Bragg-Bragg
SASE
Generationregime with
external mirrors
Amplificationregime
Amplificationregime
Generationregime with
external mirrors
Generationregime
j L
βχ
Generation threshold
Generation threshold
Generationregime
Dependence of the threshold current on assymetry factor of VDFB
j,
A/c
m2
with absorptionwithout absorption
-20 -18 -16 -14 -12 -10 -8 -6 -4 -2
0
30
60
90
120
150
180
210
240
270
300
This relation confirms that VDFB
allows to control threshold current.
Current threshold for two- and three-wave geometry in dependence on L
L, cm10 15 20 25 30 35 40 45 50 55 60 65 70
0.01
0.1
1
10
1E+2
1E+3
1E+4
1E+5
|E|
two-wave geometry,one mode in synchronism2-root degeneration point in three-wave geometry3-root degeneration point
3 2( 1)
1~th nj
kL
For n modes in synchronism*:
*V.Baryshevsky, K.Batrakov, I.Dubovskaya, NIM 358A (1995) 493
So, threshold current can be significantly decreased when modes are degenerated in multiwave diffraction geometry
References (VFEL theory and experiment) V.G.Baryshevsky, I.D.Feranchuk, Phys.Lett. 102A (1984) 141-144
V.G.Baryshevsky, Doklady Akademii Nauk USSR 299 (1988) 19
V.G.Baryshevsky, K.G.Batrakov, I.Ya. Dubovskaya, Journ.Phys D24 (1991) 1250-1257
V.G.Baryshevsky, NIM 445A (2000) 281-283
Belarus Patent no. 20010235 (09.10.2001)
Russian Patent no. 2001126186/20 (04.10.2001)
Eurasian Patent no. 004665 (25.09.2001)
V.G.Baryshevsky et al., NIM A 483 (2002) 21-24
V.G.Baryshevsky et al., NIM A 507 (2003) 137-140
V. Baryshevsky et al., LANL e-print archive: physics/ 0409107, physics/0409125, physics/0605122, physics/0608068
V.G.Baryshevsky et al., Abstracts of FEL06, (2006), TUPPH012, TUPPH013
References (VFEL simulation)
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. Abstracts of FEL06, (2006), MOPPH005
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