Radiative relaxation of relativistic electron beam in helical undulator
A process of radiative relaxation of the monoenergetic relativistic electron beam, moving through a static spatially periodic helical magnetic field is investigated theoretically. Interaction of electron with random incoherent electromagnetic field of spontaneous radiation is shown to lead to spre...
Збережено в:
| Опубліковано в: : | Вопросы атомной науки и техники |
|---|---|
| Дата: | 2006 |
| Автор: | |
| Формат: | Стаття |
| Мова: | Англійська |
| Опубліковано: |
Національний науковий центр «Харківський фізико-технічний інститут» НАН України
2006
|
| Теми: | |
| Онлайн доступ: | https://nasplib.isofts.kiev.ua/handle/123456789/80433 |
| Теги: |
Додати тег
Немає тегів, Будьте першим, хто поставить тег для цього запису!
|
| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Цитувати: | Radiative relaxation of relativistic electron beam in helical undulator / V.V. Ognivenko // Вопросы атомной науки и техники. — 2006. — № 5. — С. 7-9. — Бібліогр.: 6 назв. — англ. |
Репозитарії
Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1859836649876226048 |
|---|---|
| author | Ognivenko, V.V. |
| author_facet | Ognivenko, V.V. |
| citation_txt | Radiative relaxation of relativistic electron beam in helical undulator / V.V. Ognivenko // Вопросы атомной науки и техники. — 2006. — № 5. — С. 7-9. — Бібліогр.: 6 назв. — англ. |
| collection | DSpace DC |
| container_title | Вопросы атомной науки и техники |
| description | A process of radiative relaxation of the monoenergetic relativistic electron beam, moving through a static spatially
periodic helical magnetic field is investigated theoretically. Interaction of electron with random incoherent electromagnetic
field of spontaneous radiation is shown to lead to spread in the electron-momentum. The conditions
necessary for self-amplified spontaneous emission process realization in the short-wavelength region are derived.
Теоретически исследован процесс радиационной релаксации моноэнергетического релятивистского электронного пучка, движущегося в пространственно периодическом спиральном магнитном поле ондулятора. Показано, что взаимодействие электронов со случайным некогерентным электромагнитным полем спонтанного излучения приводит к разбросу по импульсам электронов. Сформулированы условия, необходимые для реализации процесса самопроизвольного усиления спонтанного излучения в ультракоротковолновой области.
Теоретично досліджено процес радіаційної релаксації моноенергетичного релятивістського електронного
пучка, що рухається в просторово-періодичному спіральному магнітному полі ондулятора. Показано, що
взаємодія електронів з випадковим некогерентним електромагнітним полем спонтанного випромінювання
призводить до розкиду по імпульсах електронів. Сформульовано умови, необхідні для реалізації процесу
самочинного підсилення спонтанного випромінювання в ультракороткохвильовий області.
|
| first_indexed | 2025-12-07T15:35:14Z |
| format | Article |
| fulltext |
RADIATIVE RELAXATION OF RELATIVISTIC ELECTRON BEAM
IN HELICAL UNDULATOR
V.V. Ognivenko
National Science Center “Kharkov Institute of Physics and Technology”, Kharkov, Ukraine
E-mail: ognivenko@kipt.kharkov.ua
A process of radiative relaxation of the monoenergetic relativistic electron beam, moving through a static spatial-
ly periodic helical magnetic field is investigated theoretically. Interaction of electron with random incoherent elec-
tromagnetic field of spontaneous radiation is shown to lead to spread in the electron-momentum. The conditions
necessary for self-amplified spontaneous emission process realization in the short-wavelength region are derived.
PACS: 41.60.Cr
1. INTRODUCTION
Relativistic electron beam, moving through a period-
ic magnetic field (undulator), is known to be frequency
tunable source of powerful short-wavelength electro-
magnetic radiation. Under certain conditions initial in-
coherent spontaneous radiation can be amplified by
rather dense monoenergetic electron beam at single un-
dulator passage (mode of self-amplified spontaneous
emission (SASE)) [1-3]. Determination of real prospects
of such process realization for the stimulated emission
from an ultra-relativistic electron beams is of consider-
able interest in connection with the researches, directed
on the development of X-ray free electron lasers (FEL).
In this work theoretical analysis of relativistic elec-
tron beam relaxation in the mode of incoherent sponta-
neous radiation in the undulator is presented. Interaction
of electrons with incoherent field of spontaneous radia-
tion is shown to lead to a growth of the electron mo-
mentum spread. The rate of change in the mean-square
value of the longitudinal momentum is found for initial-
ly monoenergetic electron beam and conditions neces-
sary for SASE process realization in the short-wave-
length region are also formulated.
2. FORMULATION OF THE PROBLEM
Let us consider the beam of relativistic electrons
with uniform particle density n0, moving in the positive
direction of z axis in the helical periodic magnetic field
Hu
( ) ( )[ ]zkzkH uyuxu sincos0 eeH += , z>0, (1)
where uuk λπ= 2 , H0 and uλ are the amplitude and
period of magnetic field, ex, ey are unit vectors along the
x and y axes.
The electron beam is monoenergetic at z=0 the un-
dulator entrance and unbounded on radius.
Taking into account a discrete structure of the beam,
we will derive the mean-square longitudinal electron-
momentum spread due to interaction of electrons with
each other via incoherent electromagnetic field, pro-
duced by the same electrons moving through the undu-
lator.
3. BASIC EQUATIONS
The evolution in time of mean-square value of elec-
tron-momentum spread one can be found directly by
considering motion of some individual (test) electron in
the undulator field and in the fields produced by all oth-
er electrons of beam, moving in the undulator. The
equations of motion for i-th electron in the magnetic
field of undulator and in the field, produced by other in-
dividual electrons of beam, can be written in the form:
( )[ ] ( ) ( )[ ]∑==
s
si
s
ziz
zi xttFtt
dt
d
;,,F
p
rr , (2)
( )
( )tm
t
dt
d
i
ii
γ
=
pr
, (3)
( ) ( ) ( ) ( )( )[ ]
+= zszss
s
z t
c
tEextF ,1,;, rHvrr , (4)
where ri , pi are the position and momentum of i-th elec-
tron at time t; Fz
(s)(r,t;xs) is the longitudinal force of two
electrons interaction via the fields of s-th electron,
xs(t)={rs(t), ps(t)} means the Cartesian coordinates and
momentum of s-th electron, ( ) ( )tt ss ,,, rHrE are the
strength of electric and magnetic fields, produced by the
s-th electron at time t in r coordinate; e, m are the
charge and the mass of electron, dtdrv = ;
2221 cmp+=γ .
The electric and magnetic fields are expressed via
the retarded potentials of electromagnetic field for the
individual electrons moving in an undulator [4]:
( ) ( )
( )
( )[ ][ ]
( )
′′−′
′−′′
+
′′−′
β ′−′−′
=
3232
2
11
1
iii
iiii
iii
iii
i
RcR
e
βn
vβnn
βn
βn
E
, (5)
[ ]iii EnH ′= , (6)
where iii RRn = , ( )tii rrR −= , cvβ = , dtdvv = ,
a prime denotes, that a value is taken in the retarded mo-
ment at time t′, determined by the following relation
( ) ctRtt i ′−=′ .
The value of field at observation point r at time t is
determined by only those electrons, initial coordinates
of which satisfy condition:
( ) cttt iii 00 rr −>− , (7)
where r0i={x0i, y0i, 0} is the coordinate of the i-th elec-
tron at time t0i.
Since coordinates of electron at the undulator en-
trance are random values, the electromagnetic field pro-
duced by these electrons at the coordinate r at time t,
will be random too. This electromagnetic field gives rise
to random deviation from mean value of electron mo-
mentum. The deviation of longitudinal momentum from
the mean value ∆pzi for the i-th particle will be obtained
by integrating Eq. (2):
___________________________________________________________
ВОПРОСЫ АТОМНОЙ НАУКИ И ТЕХНИКИ. 2006. № 5.
Серия: Плазменная электроника и новые методы ускорения (5), с.7-9. 7
( )[ ]∫ ′′′δ=∆
t
t
izzi
i
tdttFp
0
,r , (8)
where ∆pzi=pzi-<pzi>; zzz FFF −=δ ; <pzi>, <Fz> are
the mean value of longitudinal momentum and force.
Using Eqs. (2) and (8) of a test particle motion we
obtain the expression for rate of change of mean-square
spread in the longitudinal momentum of particles
( ) ( )[ ] ( )[ ]∫ ′δ′′δ=∆
t
t
izizzi
i
tdttFttFp
dt
d
0
,,22 rr , (9)
where angular brackets indicate average values.
Averaging in the right-hand side of Eq. (9) is per-
formed with the distribution function in phase space of
coordinates and momentum of all particles, being in the
undulator at time t. Using the principle of the phase vol-
ume conservation the averaging in Eq.(9) over the dis-
tribution at time t will be replaced by averaging over
initial distribution. Take account that particles are iden-
tical and neglecting initial correlation between them it
can be shown that the ensemble average of δFz(r,t)δFz(r
′,t′) can be expressed as:
( ) ( ) ( )
( ) ( )[ ] ( ) ( )[ ] ,,;,;
1
oss
s
zoss
s
z
ososzz
xtxtFxtxtF
xfdxtFtF
′′×
×=′ ∫δδ
(10)
where f1 is the single-particle distribution function,
x0s=xs(0).
Choosing the time τ=t-t0i as quite small in compari-
son with tr time (τ<tr) in which essential electron move-
ment change occurs, in the right-hand side of Eq. (9) ri
(t) can be replaced by unperturbed trajectory ( ) ( )ti
0r of
electron in the undulator. Then the Eq. (9) for initially
monoenergetic electron beam takes the form:
( ) ( ) ( ) ( ) ( )[ ]∫∫ ′=∆ ssi
s
zs
t
t
zi qtxttFndqtdp
dt
d
i
0
0
00
2 ,;,2
0
r ×
× ( ) ( ) ( ) ( )[ ]ssi
s
z qtxttF 0
0 ,';','r , (11)
( ) ( ) ( ) szsuuysuuxss zzkrzkr eeerr ++−= cossin0
0 ,
( ) ( ) ( ) ozzsuysuxs zkzk vsinvcosvv 0 eee +−−= ⊥⊥ ,
where ( ) ( )szs tttz 00v −= , q0s={x0s, y0s, v0zt0s}, v0z is the
axial velocity of electrons at z=0, 00vK γ= zuu kcr ,
0K/v γ=⊥ c , ukmcHe 2
0K = .
4. RESULTS AND DISCUSSION
Analytical expressions for the force, acting on the
individual electron in the electromagnetic field, pro-
duced by other particle, is derived assuming that an un-
dulator parameter is small: K2<<1. Expanding expres-
sions (5) and (6) in a power series of this parameter up
to linear terms in K we substitute these expressions into
Eq. (4). Considering interaction of electrons only via the
fields of electromagnetic radiation the following expres-
sion for longitudinal components of the force, acting on
the i-th electron in the field of the s-th one is obtained:
( ) ( )[ ] ( ) ( )γρ∆Φβγ= sisiusizs zkeqttF ,K;, 2
000
0r , (12)
( )ψΦ−=Φ i
reRe ,
( )
,
2
11,
2
*
2
0
2
*
2
0
0
*
0
*
0
*0*0
−−+−
−
+=Φ
R
y
RkR
xi
R
x
RkRk
yxr
βββ
β
where ( ) ( )*0
2
0, Rxkyx u β+γ=ψ ,
( ) ( ) 2122
* , yxyxR += , ( )isossi ttz 00v −=∆ ,
( ) ( )[ ] 2122
oiosoiossi yyxx −+−=ρ , 0
2
00 βγ= ukk .
In the K2<<1 approximation considered the inequali-
ty (7) can be written in the form of:
( )*0
2
0 Rzz si β+∆γ≥ . (13)
Substituting expression (12) into Eq. (11), the fol-
lowing expression for the rate of change of mean-square
longitudinal electron-momentum deviation can be ob-
tained:
( ) ( )
( )[ ]
( )
,,
K2
'0V
0
2
0
4
00
2
∫∫ ∆Φ′×
×=∆
t
ossisi
uzi
dnztd
kep
dt
d
qρ
βγ
τ (14)
where V0 is the integration region determined by condi-
tion (13).
In approximation considered of a homogeneous and
unbounded in radius electron beam the average beam
density is independent on the initial transversal coordi-
nates x0s, y0s and time t0s. Therefore in the right-hand
side of Eq. (14) integration over these coordinates will
be replaced by integration over r′, θ, ϕ coordinates
which is determined as: ϕθ′γ=− cossin000 rxx is ,
ϕθ′γ=− sinsin000 ryy is , ( ) θ′=− cosv 000 rtt isz .
Integrating right-hand side of Eq. (14) by taking into
account the fields radiating in forward direction only (z
≥zs(t′)), the expression
( ) 222
00
2442 K cznkep
dz
d
uz βπ=∆ (15)
can be obtained.
Here the distance z from the undulator entrance is
chosen as the independent variable.
Integrating in Eq. (15) over z we obtain the follow-
ing expression for the mean-square longitudinal elec-
tron-momentum deviation from mean value
( )[ ] ( ) 23
0
2120
0
212 K
3
2 zkN
r
pp u
u
zz γ
λ
=∆ , (16)
where 43
0 8γλ= unN .
The formula (16) describes the mean square devia-
tion from the mean value of longitudinal electron-mo-
mentum in the initially monoenergetic stream of the rel-
ativistic electrons, moving through the spatially periodic
magnetic field (1) of undulator. From Eq. (16) follows,
that longitudinal momentum spread increases as beam
moving along z axis in undulator. Interaction of elec-
trons via the incoherent electromagnetic field is shown
to result in momentum spread, therefore on the whole a
process can be considered as the initial before-kinetic
stage of particles diffusion in momentum space at the
incoherent spontaneous emission from the relativistic
electron beam in undulator.
8
The expression for the momentum spread can be
written in the form:
( )[ ] ( )zNFzp cohR
z
z π
=∆
2
3
v0
212 , (17)
where Ncoh=Nz/λu , ( )( ) 3
0
2
00032 βγ= HrFR is the force
of radiative deceleration for pointed electron, r0=e2/mc2
is the classical electron radius.
It should be noted that the relativistic electron beams
must be rather dense for self-amplification of sponta-
neous emission [5]: N>>1, therefore Ncoh>>1 for such
beams.
The condition for collective instability and exponen-
tial growth of electromagnetic radiation intensity at the
self-amplified spontaneous emission is a small momen-
tum spread in the electron beam [2,3,6]:
ρ<∆ zz pp 0 ,
where ( ) 0
312
00
2 16 γπλ=ρ urnK .
Instability saturation takes place at the distance Lsat
[2,3,6]: Lsat=λu/ρ.
Consequently, the mean-square momentum spread
due to the effect of radiative relaxation of electrons in
incoherent electromagnetic field produced by them, is
possible to be neglect when
21
0
02
3
K8
γ
λ
π> >ρ
u
r
. (18)
If a condition opposite (18) is fulfilled and momen-
tum spread brought in by the incoherent spontaneous
undulator radiation will be considerably larger than pa-
rameter ρ, the mode of the intensive amplification of in-
coherent spontaneous radiation will not be able to be re-
alized.
The author thanks Prof. K.N. Stepanov for fruitful
discussion.
REFERENCES
1. R. Bonifachio, C. Pellegrini., L.M. Narducci. Col-
lective instabilities and high-gain regime in a free-
electron laser // Opt. Commun. 1984, v.50, №6,
p.373-379.
2. С. Pellegrini. Summary of the Working Group on
FEL theory // Nucl. Instr. and Meth. A. 1985, v.239,
№1, p.127-129.
3. K.J. Kim. An analysis of self-amplified sponta-
neous emission // Nucl. Instr. and Meth. A. 1986,
v.250, №1,2, p.396-403.
4. L.D. Landau, E.M. Lifshits. Theory of field.
M.: “Nauka”, 1967, p.460 (in Russian).
5. V.I. Kurilko, V.V. Ognivenko. Dynamics of mo-
noenergetic stream of point electrons in spiral un-
dulator // Physics of plasma. 1994, v.20, p.634-639
(in Russian).
6. P. Sprangle, R.A. Smith. Theory of free-electron
lasers // Phys. Rev. A. 1980, v.21, №1, p.293-301.
РАДИАЦИОННАЯ РЕЛАКСАЦИЯ РЕЛЯТИВИСТСКОГО ЭЛЕКТРОННОГО ПУЧКА В СПИРАЛЬ-
НОМ ОНДУЛЯТОРЕ
В.В. Огнивенко
Теоретически исследован процесс радиационной релаксации моноэнергетического релятивистского элек-
тронного пучка, движущегося в пространственно периодическом спиральном магнитном поле ондулятора.
Показано, что взаимодействие электронов со случайным некогерентным электромагнитным полем спонтан-
ного излучения приводит к разбросу по импульсам электронов. Сформулированы условия, необходимые для
реализации процесса самопроизвольного усиления спонтанного излучения в ультракоротковолновой обла-
сти.
РАДІАЦІЙНА РЕЛАКСАЦІЯ РЕЛЯТИВІСТСЬКОГО ЕЛЕКТРОННОГО ПУЧКА В СПІРАЛЬНОМУ
ОНДУЛЯТОРІ
В.В. Огнівенко
Теоретично досліджено процес радіаційної релаксації моноенергетичного релятивістського електронного
пучка, що рухається в просторово-періодичному спіральному магнітному полі ондулятора. Показано, що
взаємодія електронів з випадковим некогерентним електромагнітним полем спонтанного випромінювання
призводить до розкиду по імпульсах електронів. Сформульовано умови, необхідні для реалізації процесу
самочинного підсилення спонтанного випромінювання в ультракороткохвильовий області.
___________________________________________________________
ВОПРОСЫ АТОМНОЙ НАУКИ И ТЕХНИКИ. 2006. № 5.
Серия: Плазменная электроника и новые методы ускорения (5), с.7-9. 9
National Science Center “Kharkov Institute of Physics and Technology”, Kharkov, Ukraine
PACS: 41.60.Cr
3. Basic equAtions
4. results and Discussion
В.В. Огнивенко
В.В. Огнівенко
|
| id | nasplib_isofts_kiev_ua-123456789-80433 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1562-6016 |
| language | English |
| last_indexed | 2025-12-07T15:35:14Z |
| publishDate | 2006 |
| publisher | Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| record_format | dspace |
| spelling | Ognivenko, V.V. 2015-04-17T19:21:33Z 2015-04-17T19:21:33Z 2006 Radiative relaxation of relativistic electron beam in helical undulator / V.V. Ognivenko // Вопросы атомной науки и техники. — 2006. — № 5. — С. 7-9. — Бібліогр.: 6 назв. — англ. 1562-6016 PACS: 41.60.Cr https://nasplib.isofts.kiev.ua/handle/123456789/80433 A process of radiative relaxation of the monoenergetic relativistic electron beam, moving through a static spatially periodic helical magnetic field is investigated theoretically. Interaction of electron with random incoherent electromagnetic field of spontaneous radiation is shown to lead to spread in the electron-momentum. The conditions necessary for self-amplified spontaneous emission process realization in the short-wavelength region are derived. Теоретически исследован процесс радиационной релаксации моноэнергетического релятивистского электронного пучка, движущегося в пространственно периодическом спиральном магнитном поле ондулятора. Показано, что взаимодействие электронов со случайным некогерентным электромагнитным полем спонтанного излучения приводит к разбросу по импульсам электронов. Сформулированы условия, необходимые для реализации процесса самопроизвольного усиления спонтанного излучения в ультракоротковолновой области. Теоретично досліджено процес радіаційної релаксації моноенергетичного релятивістського електронного пучка, що рухається в просторово-періодичному спіральному магнітному полі ондулятора. Показано, що взаємодія електронів з випадковим некогерентним електромагнітним полем спонтанного випромінювання призводить до розкиду по імпульсах електронів. Сформульовано умови, необхідні для реалізації процесу самочинного підсилення спонтанного випромінювання в ультракороткохвильовий області. The author thanks Prof. K.N. Stepanov for fruitful discussion. en Національний науковий центр «Харківський фізико-технічний інститут» НАН України Вопросы атомной науки и техники Релятивистская и нерелятивистская плазменная СВЧ-электроника Radiative relaxation of relativistic electron beam in helical undulator Радиационная релаксация релятивистского электронного пучка в спиральном ондуляторе Радіаційна релаксація релятивістського електронного пучка в спіральному ондуляторі Article published earlier |
| spellingShingle | Radiative relaxation of relativistic electron beam in helical undulator Ognivenko, V.V. Релятивистская и нерелятивистская плазменная СВЧ-электроника |
| title | Radiative relaxation of relativistic electron beam in helical undulator |
| title_alt | Радиационная релаксация релятивистского электронного пучка в спиральном ондуляторе Радіаційна релаксація релятивістського електронного пучка в спіральному ондуляторі |
| title_full | Radiative relaxation of relativistic electron beam in helical undulator |
| title_fullStr | Radiative relaxation of relativistic electron beam in helical undulator |
| title_full_unstemmed | Radiative relaxation of relativistic electron beam in helical undulator |
| title_short | Radiative relaxation of relativistic electron beam in helical undulator |
| title_sort | radiative relaxation of relativistic electron beam in helical undulator |
| topic | Релятивистская и нерелятивистская плазменная СВЧ-электроника |
| topic_facet | Релятивистская и нерелятивистская плазменная СВЧ-электроника |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/80433 |
| work_keys_str_mv | AT ognivenkovv radiativerelaxationofrelativisticelectronbeaminhelicalundulator AT ognivenkovv radiacionnaârelaksaciârelâtivistskogoélektronnogopučkavspiralʹnomondulâtore AT ognivenkovv radíacíinarelaksacíârelâtivístsʹkogoelektronnogopučkavspíralʹnomuondulâtorí |