Nonlinear dynamics of relativistic electron bunch in an undulator
The dynamics of electron bunch motion in a helical undulator for the regime of self–amplification spontaneous undulator radiation is investigated theoretically using the method of self-consistent modeling the motion of electrons in the total field of their undulator radiation. The effects of bunch e...
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України
2001
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| Цитувати: | Nonlinear dynamics of relativistic electron bunch in an undulator / V.V. Ognivenko // Вопросы атомной науки и техники. — 2001. — № 1. — С. 107-108. — Бібліогр.: 5 назв. — англ. |
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irk-123456789-785242015-03-19T03:02:51Z Nonlinear dynamics of relativistic electron bunch in an undulator Ognivenko, V.V. Theory and technics of particle acceleration The dynamics of electron bunch motion in a helical undulator for the regime of self–amplification spontaneous undulator radiation is investigated theoretically using the method of self-consistent modeling the motion of electrons in the total field of their undulator radiation. The effects of bunch edge on start up processes are studied. The bunching of electrons within the radiation wavelength and within the full length of the beam is investigated. The efficiency of free–electron lasers using such beam is determined. 2001 Article Nonlinear dynamics of relativistic electron bunch in an undulator / V.V. Ognivenko // Вопросы атомной науки и техники. — 2001. — № 1. — С. 107-108. — Бібліогр.: 5 назв. — англ. 1562-6016 PACS: 41.60Cr. http://dspace.nbuv.gov.ua/handle/123456789/78524 en Вопросы атомной науки и техники Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
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English |
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Theory and technics of particle acceleration Theory and technics of particle acceleration |
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Theory and technics of particle acceleration Theory and technics of particle acceleration Ognivenko, V.V. Nonlinear dynamics of relativistic electron bunch in an undulator Вопросы атомной науки и техники |
| description |
The dynamics of electron bunch motion in a helical undulator for the regime of self–amplification spontaneous undulator radiation is investigated theoretically using the method of self-consistent modeling the motion of electrons in the total field of their undulator radiation. The effects of bunch edge on start up processes are studied. The bunching of electrons within the radiation wavelength and within the full length of the beam is investigated. The efficiency of free–electron lasers using such beam is determined. |
| format |
Article |
| author |
Ognivenko, V.V. |
| author_facet |
Ognivenko, V.V. |
| author_sort |
Ognivenko, V.V. |
| title |
Nonlinear dynamics of relativistic electron bunch in an undulator |
| title_short |
Nonlinear dynamics of relativistic electron bunch in an undulator |
| title_full |
Nonlinear dynamics of relativistic electron bunch in an undulator |
| title_fullStr |
Nonlinear dynamics of relativistic electron bunch in an undulator |
| title_full_unstemmed |
Nonlinear dynamics of relativistic electron bunch in an undulator |
| title_sort |
nonlinear dynamics of relativistic electron bunch in an undulator |
| publisher |
Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| publishDate |
2001 |
| topic_facet |
Theory and technics of particle acceleration |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/78524 |
| citation_txt |
Nonlinear dynamics of relativistic electron bunch in an undulator / V.V. Ognivenko // Вопросы атомной науки и техники. — 2001. — № 1. — С. 107-108. — Бібліогр.: 5 назв. — англ. |
| series |
Вопросы атомной науки и техники |
| work_keys_str_mv |
AT ognivenkovv nonlineardynamicsofrelativisticelectronbunchinanundulator |
| first_indexed |
2025-07-06T02:35:40Z |
| last_indexed |
2025-07-06T02:35:40Z |
| _version_ |
1836863288411947008 |
| fulltext |
NONLINEAR DYNAMICS OF RELATIVISTIC ELECTRON BUNCH IN
AN UNDULATOR
V.V. Ognivenko
National Science Center “Kharkov Institute of Physics and Technology”, Kharkov, Ukraine
The dynamics of electron bunch motion in a helical undulator for the regime of self–amplification spontaneous
undulator radiation is investigated theoretically using the method of self-consistent modeling the motion of electrons in
the total field of their undulator radiation. The effects of bunch edge on start up processes are studied. The bunching of
electrons within the radiation wavelength and within the full length of the beam is investigated. The efficiency of free–
electron lasers using such beam is determined.
PACS: 41.60Cr.
As known [1,2], the regime of self–amplification
spontaneous radiation from relativistic electron beams
in free–electron lasers (FEL) allows one to obtain
frequency tunable coherent ultrashort wavelength
radiation by the use of a sufficiently intense relativistic
electron beam. For shortening the wavelength of
coherent radiation and generation high intensity
electromagnetic radiation in FELs the more impotence
takes investigations of self–amplification undulator
radiation process by finite length relativistic electron
beam, in particular, the electron beam the length of
which is order of several wavelength of electromagnetic
radiation. In this paper the dynamics of the relativistic
electrons motion in the field of helical magnetic
undulator and the total field of an undulator radiation of
these electrons is investigated. The effects of bunch
edge are studied and it is shown that self–amplification
process start up from the spontaneous coherent
undulator radiation of electrons moving at the trailing
edge of the bunch.
Let us consider a finite length relativistic electron
beam passing through a helical magnetic field of
undulator: ( )[ ]H eu uz H ik z( ) Re exp= −0 , where
e e e− = −x yi , ku=2π/λu, λu is period of undulator, ex ey
are the unit vectors along the axes ОХ, ОY. The initial
velocity of the electron beam is in the axial direction
ezv0.
We consider one-dimentional model assuming that
in x, y directions the bunch is infinite. The pulse
duration of the electron beam prior to entering the
undulator field is tb.
The electric and magnetic fields of the electron in
the undulator derived from formulae for the field of
charge particle moving with an acceleration [3], take the
form as given in [4,5]. Assuming the beam density is
independent of initial transverse coordinates (x0, y0) at
the undulator entrance z=0 the expressions for the total
field of radiation take the form:
( ) ( )
( )E e=
−
− ∫2
10 0 0
0
π β
β δ
ev dt n t
i
zu bRe
exp
sgn
ϑ
, (1)
( ) ( ) ( )
( )H e= −
−
− ∫2
10 0 0
0
π β
δ
β δ
ev dt n t
z i
zu bIm
sgn exp
sgn
ϑ
,(2)
where ( ) ( ) ( )
ϑ
∆
=
−
− − −
k v
z
t z
c
z t
z t
v
u 0
0
0
0
01 β δ
δ
sgn
sgn
,
,
( ) ( ) ( )δ z z v t t z t= − − −0 0 0∆ , , ∆ – is the longitudinal
displacement of radiating electron relative of its
equilibrium trajectory, β γu K= / 0 , ( )γ β0 0
2 1 2
1= −
− /
,
K e H mc ku= 0
2/ ; e, m are the charge and the rest
mass of electron.
The leading edge of the bunch cross the z=0 plane at
time t0=0 and t0 integration in Eqs. (1) (2) is over region
from 0 to t z c− .
The axial dynamics of electrons in the undulator
region z>0 determine the corresponding component of
Lorentz force: ( )F e H k z H k zz
tot
u x
tot
u y
tot
u= −β sin cos .
For the simplification of the formulae we neglect the
space–charge force.
The self–consistent nonlinear equations of particle
motion are:
( ) ( )
( )v
dp
dz
e v dt n t
z
zz
z
u b= −
−∫2
1
2 2
0 0 0
0
π β
δ ψ
β δ
sgn cos
sgn
, (3)
dt
dz v
L
z
= 1
, (4)
where
( )
( )ψ
δ
β δ
=
−
k z
z
u
1 0 sgn
.
The efficiency of particle kinetic energy conversion
into electromagnetic energy will be defined as the ratio
of kinetic beam energy losses to its initial energy (<...>
denotes an average over the beam pulse duration):
( )[ ] ( )η γ γ γ= − −0 0 01z t, . (5)
Below will be considered the motion of electrons
under force due to forward traveling radiation field,
since it is this force that is responsible for the axial
bunching of electrons and leads to the growth of
coherent short wavelength radiation.
The density of the electrons at the entrance to the
undulator is assumed to be uniform over the beam pulse
duration 0<t0<tb . The electrons are monoenergetic
( )d dτ ξ τ τξ = = =0 00 0, and no input signal in the
undulator, where ( )[ ]τ ω= −t z t z vL , 0 .
In the linear regime (ξ<<ξsat=1 3ρ τ b ) when
longitudinal displacement of electrons relative its
equilibrium trajectory is less then wavelength of
radiation, from Eqs. (3) (4) can be obtained following
PROBLEMS OF ATOMIC SCIENCE AND TECHNOLOGY. 2001, № 1.
Series: Nuclear Physics Investigations (37), p. 107-108.
107
expression for the amplitude of this displacement:
( ) ( ) ( )τ ξ τ τ ξ τ τ τ, , sin0 0 0 0= + −R b , (6)
( ) ( ) ( ) ( )R b bξ τ ρ ξ τ τ θ ξ τ τ, 0
3
0
2
01 2= + − + − , (7)
where ρ
γ
ω
=
1
0
2 3
b
uk c
K
/
, ω πb e n m= 4 2
0 , ξ = k zu ,
τ ω0 0= t , n0 is the unperturbed beam density,
( )ω β= −k vu 0 01 , λ π ω λ γ= ≈2 2 0
2c u are the
frequency and wavelength of the field radiated along the
axis OZ in the direction of bunch motion,
τ ω π λb b bt l= = 2 , lb is bunch length, θ(x) is the
Heaviside unit step function.
It follows from Eqs. (6), (7) that at the beginning of
selfbunching process there is self–modulation of
longitudinal displacement (and longitudinal momentum)
of electrons moving at the trailing edge of the beam.
The front of modulation propagates from the trailing
edge of the beam to its leading edge as beam moves in
the undulator.
The results from computer simulation of the electron
bunch motion in the undulator are shown in Figs. 1 and
2. Fig. 1 shows the efficiency as a function of axial
distance (ζ=ρξ) for different bunch length lb λ =5 (1),
10 (2), 20 (3) and for ρ=0.02.
0 5 10 15 20 25
0,0
0,1
0,2
0,3
0,4
0,5
3
2
1
η / ρ
ζ
Fig. 1. The efficiency η as a function of normalized
axial distance ζ for several value of bunch length
lb=5λ (1), 10λ (2), 20λ (3) and for ρ=0.02.
It is seen from this figure that the spatial growth rate
increases and the saturation length decreases with an
increase of the beam length. The maximum value of the
efficiency grows with an increase of the beam length
too.
Fig. 2 shows the phase space ( τ τ ξ'≡ d d , τ) of
electron beam for the case of lb=10λ. The longitudinal
momentum as a function of relative time the particles
cross the axial positions ζ=1, 9, 12, 15 are plots in this
figure. At the initial position ζ=0 the particles have
uniform axial velocity ( )τ τ' ,0 00 = .
It can see from this figure that axial velocity of
electrons is modulated with spatial period of
ponderomotive force, approximately equal λ. The
perturbation of axial velocity arises at the trailing edge
of the bunch and propagates in forward direction to its
leading edge. Physically in considered case only
coherent spontaneous undulator radiation of electrons
moving at the trailing edge of the bunch is the source of
initial perturbation of axial momentum of electrons.
Notice that in Fig. 2 the particles at trailing edge have
maximum value of entry time. Some electrons have
gained energy while others have lost energy depending
on their phase relation with ponderomotive force.
0 2 4 6 8 10
-0,1
0,0
0,1
a)
τ '
0 2 4 6 8 10
-2
-1
0
1
2
b)
0 2 4 6 8 10
-2
-1
0
1
2
c)
τ /2 π
τ '
0 2 4 6 8 10
-2
-1
0
1
2
d)
τ /2 π
Fig. 2. The phase space of electron bunch crossing
the axial position: ζ=1 (a), 9 (b), 12 (c), 15 (d)
At the ξ=τ the front of axial momentum perturbation
reaches the leading edge of the bunch. Since the
electrons undergo grouping force due to the radiation of
electrons moving behind of the considered, the particles
placed at the head of bunch have amplitude of
displacement in axial momentum larger than the
particles at the tail of the bunch. Fig. 2 c, 2 d depict the
particles at saturation and show trapping and spatial
bunching of electrons. Many of particles within each of
wavelength cross the ζ=12 plane (and also ζ=15 plane)
at about the same time. Fig. 2d, at ζ=15, shows the
formation of microbunches practically within full length
of beam. Thus, the performed analytical and numerical
simulations show the possibility of development of
collective selfbunching process in short pulse relativistic
electron beam moving through an undulator.
REFERENCES
1. J.B. Murphy, С. Pellegrini. Free–electron lasers for
the XUV spectral region // Nucl. Instr. and Meth.
1985, v. A237, N 1–2, p. 159-167.
2. R. Bonifacio, P. Pierini, C. Pellegrini, at al.
Slippage, noise and superradiant effects in the
UCLA FEL experiment // N M, 1994, A341, p. 285–
288.
3. L.D. Landau, E.M. Lifshits, Teoriya Polya. 1967, M:
Nauka, 460 p. (in Russian).
4. V.I. Kurilko, V.V. Ognivenko. Dynamics of mo-
noenergetic point electron flow in a helix undulator
// Plasma Physics Report. 1994, v. 20. №7, p. 568–
573.
5. V.V. Ognivenko. The dynamics of finite-length
electron beam motion in an undulator // PAST,
series: Plasma Physics. 1999, Т3(3), 4(4), p. 230-
232.
ВОПРОСЫ АТОМНОЙ НАУКИ И ТЕХНИКИ. 2000, №2.
Серия: Ядерно-физические исследования (36), с. 3-6.
4
V.V. Ognivenko
National Science Center “Kharkov Institute of Physics and Technology”, Kharkov, Ukraine
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