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