Trapping of nonmonoenergetic electron bunches into a wake wave
The trapping of nonmonoenergetic electron bunches in a wake field wave excited by a laser pulse in a plasma channel is studied analytically. Electrons are injected into the region of the wake wave potential maximum at a ve-locity lower than the phase velocity of the wave. The formula for length of a...
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України
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irk-123456789-1121142017-01-18T03:03:17Z Trapping of nonmonoenergetic electron bunches into a wake wave Kuznetsov, S.V. Теория и техника ускорения частиц The trapping of nonmonoenergetic electron bunches in a wake field wave excited by a laser pulse in a plasma channel is studied analytically. Electrons are injected into the region of the wake wave potential maximum at a ve-locity lower than the phase velocity of the wave. The formula for length of a bunch in the accelerating stage as func-tions of its initial energy spread and initial sizes (length and cross-section radius) is derived. Проведено аналітичне дослідження процесу захоплення в кільватерній хвилі, збудженій лазерним імпульсом у плазмовому каналі, компактних немоноенергетичних електронних згустків. Електрони інжектуються біля максимуму потенціалу кільватерної хвилі зі швидкістю менше її фазової швидкості. Отримана формула, що визначає довжину згустка на прискорювальній стадії, як функцію його початкових розмірів (його довжини і радіуса поперечного перерізу) і початкового розкиду електронів по енергіям в ньому. Проведено аналитическое исследование процесса захвата в кильватерной волне, возбуждаемой лазерным импульсом в плазменном канале, компактных немоноэнергетических электронных сгустков. Электроны инжектируются в окрестность максимума потенциала кильватерной волны со скоростью меньше её фазовой скорости. Получена формула, определяющая длину сгустка на ускоряющей стадии, как функция его начальных размеров (его длины и радиуса поперечного сечения) и начального разброса электронов по энергиям в нем. 2015 Article Trapping of nonmonoenergetic electron bunches into a wake wave / S.V. Kuznetsov // Вопросы атомной науки и техники. — 2015. — № 3. — С. 127-131. — Бібліогр.: 18 назв. — англ. 1562-6016 PACS: 52.38.Kd; 41.75.Jv http://dspace.nbuv.gov.ua/handle/123456789/112114 en Вопросы атомной науки и техники Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
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Теория и техника ускорения частиц Теория и техника ускорения частиц |
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Теория и техника ускорения частиц Теория и техника ускорения частиц Kuznetsov, S.V. Trapping of nonmonoenergetic electron bunches into a wake wave Вопросы атомной науки и техники |
| description |
The trapping of nonmonoenergetic electron bunches in a wake field wave excited by a laser pulse in a plasma channel is studied analytically. Electrons are injected into the region of the wake wave potential maximum at a ve-locity lower than the phase velocity of the wave. The formula for length of a bunch in the accelerating stage as func-tions of its initial energy spread and initial sizes (length and cross-section radius) is derived. |
| format |
Article |
| author |
Kuznetsov, S.V. |
| author_facet |
Kuznetsov, S.V. |
| author_sort |
Kuznetsov, S.V. |
| title |
Trapping of nonmonoenergetic electron bunches into a wake wave |
| title_short |
Trapping of nonmonoenergetic electron bunches into a wake wave |
| title_full |
Trapping of nonmonoenergetic electron bunches into a wake wave |
| title_fullStr |
Trapping of nonmonoenergetic electron bunches into a wake wave |
| title_full_unstemmed |
Trapping of nonmonoenergetic electron bunches into a wake wave |
| title_sort |
trapping of nonmonoenergetic electron bunches into a wake wave |
| publisher |
Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| publishDate |
2015 |
| topic_facet |
Теория и техника ускорения частиц |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/112114 |
| citation_txt |
Trapping of nonmonoenergetic electron bunches into a wake wave / S.V. Kuznetsov // Вопросы атомной науки и техники. — 2015. — № 3. — С. 127-131. — Бібліогр.: 18 назв. — англ. |
| series |
Вопросы атомной науки и техники |
| work_keys_str_mv |
AT kuznetsovsv trappingofnonmonoenergeticelectronbunchesintoawakewave |
| first_indexed |
2025-07-08T03:25:00Z |
| last_indexed |
2025-07-08T03:25:00Z |
| _version_ |
1837047586830155776 |
| fulltext |
TRAPPING OF NONMONOENERGETIC ELECTRON
BUNCHES INTO A WAKE WAVE
S.V.Kuznetsov∗
Joint Institute for High Temperatures of RAS, Moscow, Russia
(Received September 02, 2013)
The trapping of nonmonoenergetic electron bunches in a wake field wave excited by a laser pulse in a plasma channel
is studied analytically. Electrons are injected into the region of the wake wave potential maximum at a ve-locity
lower than the phase velocity of the wave. The formula for length of a bunch in the accelerating stage as func-tions
of its initial energy spread and initial sizes (length and cross-section radius) is derived.
PACS: 52.38.Kd; 41.75.Jv
INTRODUCTION
A number of sufficient successful experiments on
laser-plasma acceleration of electron bunches to ener-
gies of a few GeV while in relative energy spread of
the accelerated monoenergetic bunch ∼ 5...10% [1, 2]
was carried out in recent years. However, for many
practical applications, the energy spread of the elec-
tron bunch after acceleration must be tenths of a per-
cent [3, 4]. In this regard, the theoretical study of the
influence of various factors on the energy spread of
the accelerated electron bunch is very important to
identify ways to get in the experiment bunches of high
energy electrons with a small energy spread between
them.
It is known that the energy spread of the acceler-
ated electron bunch largely depends from its length
in the accelerating stage [5, 6]. In laser-plasma accel-
erators using an external electron injector the length
of accelerated electron bunch is determined by the
regimen of the injection bunch into a wake field, its
parameters at the instant of injection and the pa-
rameters of the wake field [7, 9]. In this paper we
theoretically investigate the influence of the trans-
verse dimensions of the injected bunch on its length
in the accelerating stage in the scheme of injection,
when a bunch is injected into the vicinity of the wake
wave potential maximum with a velocity less than
the phase velocity of the wake wave [9, 10]. Group-
ing phenomenon is observed in this injection scheme,
owing to which electrons are arranged in the trap-
ping area along the longitudinal axis more densely
than at the instant of injection. The phenomenon of
electron grouping allows to relax the requirements to
the original length of the injected bunch. In previ-
ous studies the phenomenon of electron bunching by
wake wave was studied in one dimensional formula-
tion [11, 12]. The aim of this work is an analytical
study of this phenomenon in two-dimensional formu-
lation and definition of the corrections to the trapped
electrons bunch length, resulting from transverse di-
mensions of the injected bunch.
1. TRAPPING OF A SINGLE ELECTRON
BY A WAKE WAVE
To describe the motion of relativistic electrons of
a bunch accelerated in the wake field generated by an
axisymmetric laser pulse propagating along the OZ
axis, we use the motion equations in the form:
dpz
dτ
= Fz(ξ, ρ), (1)
dpr
dτ
= Fr(ξ,ρ), (2)
dξ
dτ
=
pz√
1 + p2z + p2r
− β, (3)
dρ
dτ
=
pr√
1 + p2z + p2r
, (4)
where pz and pr = {px, py} are normalized to mc
the longitudinal and transverse components of the
momentum of a bunch electron, perpendicular to
the OZ axis; kp(z − Vpht), ρ = kpr = kp{x, y} =
kpr{cosϕ, sinϕ}, ϕ = arctan(y/x) are its dimen-
sionless coordinates; τ = ωp t; electron plasma fre-
quency ωp =
√
4πe2n0/m is determined from the
background plasma density n0; kp = ωp/c; Vph is
the phase velocity of the wake wave; and β = Vph/c.
The axial and radial components of the normalized
force acting on an electron moving along the OZ axis
with a velocity close to the velocity of light c can be
expressed in terms of potential φ of the wake field as
(Fr = {Fx, Fy} = Fr{cosϕ, sinϕ}:
Fz ≡ eEz
mcωp
=
|e|
mc2
∂φ
∂ ξ
, (5)
Fr ≡ eEr
mcωp
− eBϕ
mcωp
=
|e|
mc2
∂φ
∂ρ
, (6)
∗Corresponding author E-mail address: shenau@rambler.ru
ISSN 1562-6016. PROBLEMS OF ATOMIC SCIENCE AND TECHNOLOGY, 2015, N3(97).
Series: Nuclear Physics Investigations (64), p.127-131.
127
where Er, Ez, Bϕ - electric and magnetic fields of the
wake wave.
To simplify theoretical analysis of the motion of
electrons in the wake field, we assume here that
the wake field potential depends, apart from radius
ρ, only on the concomitant variable ξ (and it does
not depend explicitly on time); this implies steady-
state waveguide propagation of the laser pulse in the
plasma channel. It should also be noted that expres-
sion (6) for the radial force taking into account mag-
netic field Bϕ is correct and quite accurate only for
electrons with radial velocities much smaller than the
velocity of light c. However, in the case of not very
strong nonlinearity of the wake field (|eφ|/mc2 ≤ 1)
and its characteristic transverse scale much exceed-
ing skin depth k−1
p , the magnetic field of the wake
wave is much weaker than its electric field, and both
force components, (5) and (6), are independent of the
radial velocities of electrons being accelerated.
The effect of the laser pulse on the subsequent
slow (low-frequency) motion of electrons can be dis-
regarded because the ponderomotive potential of the
laser field (which is proportional to ∼ a20) in the co-
ordinate system associated with the wave is small as
compared to the wake field potential proportional to
∼ γphφ (a0 = |e|EL,0/mcω, EL,0 being the amplitude
of the high-frequency laser pulse and ω being its fre-
quency, γph = 1/
√
1− β2) provided that a0 ≤ 1.
The effect of the intrinsic charge of the bunch on
the process of its acceleration is also disregarded; this
is justified for not very narrow bunches with a char-
acteristic transverse size of 10−4 < kpRb0, where Rb0
is the radius of the injected electron bunch, which
satisfy the condition [13]:
Ne << n0(c/ωp)
3
= 4 · 106λp, (7)
where Ne is the number of electrons in the bunch and
λp is the wavelength of plasma oscillations. Condition
(7) indicates that the fields induced by the charge of
the bunch are much weaker than the wake field in
which electrons are accelerated.
We assume that a short relativistic nonmonoener-
getic electron bunch is injected into a stationary wake
wave propagating without changing its shape so that
the region of bunch injection includes coordinate (or
phase) ξm at which the potential of the wake wave has
the maximal value φmax = φ (ξm, ρ = 0). The elec-
tron bunch injection scheme is shown in Fig.1, where
along the axis OZ dotted line shows the laser pulse
generating wake field, the solid line is the wake po-
tential, dash-dot line shows the boundary of focusing
and defocusing areas of wake field on the axis.
The direction of bunch injection coincides
with the direction of phase velocity Vph of the
wake wave, the energy of electron injection be-
ing Einj = Eres = γphmc2. It is believed
that the electron bunch injection conditions are
such that all injected electrons are trapped by
wake field and grouped during their trapping near
the boundary of focusing and defocusing areas.
Fig.1. Injection scheme. Mutual location along the
OZ axis of the laser pulse, wake wave and bunch at
the instant of injection
The time-independence of the wake field in the
coordinate system associated with the wave (i.e., in
the concomitant system of coordinates moving with
velocity Vph along the OZ axis) makes it possible to
write the energy conservation law in this coordinate
system for an arbitrary electron bunch in the 2D for-
mulation in the form
E′ − |e|φ′(ξ, ρ) = E′
inj − |e|φ′(ξinj , ρinj), (8)
here and below, primes indicate that the primed
quantities are taken in the wave frame of reference,
ξinj , ρinj and Einj are the coordinate and energy of
electron injection into the wake wave.
We assume that the electron bunch is quite com-
pact at the instant of injection and in the subse-
quent trapping and acceleration process, and that
the electron energies differ insignificantly (∆ξinj =
kp∆zinj << 1, ρinj << 1, ∆Einj << Einj); i.e.,
electrons of the bunch in phase space (ξ, E) are quite
close to one another at any instant. Then the analy-
sis of the spatial structure of the bunch along the OZ
axis in the region of trapping can be done as follows.
We choose an electron that is exactly at the potential
maximum of the wake field at the instant of injection
and has injection energy equal to the mean electron
injection energy of the bunch; we will henceforth refer
to this electron as the central electron. The energy
conservation law (8) at a known potential φ(ξ , ρ) and
known energy injection Einj can determine the phase
of the wake wave its trapping ξtr on the axis OZ, in
which the velocity of the central electron is equal to
the phase velocity of the wake wave (respectively, the
electron energy will be equal E = γphmc2).
Earlier it was shown [11] that the greatest degree
of grouping of 1D bunch along the axis OZ is realized
when the trapping area of the injected electrons is lo-
cated in the vicinity of the phase of the wake wave,
in which its accelerating field is maximum. Since for
the process of laser-plasma acceleration with minimal
energy spread is preferred minimum length of accel-
erated bunch, then we assume that this condition is
satisfied. Since the properties of linear and weakly
nonlinear wake waves and look close enough, we can
assume that for a weakly nonlinear wake wave the
boundary of focusing and defocusing areas of wake
128
field ρ = ρdf (ξ) close to the plane, as well as in the
linear wake wave. On this boundary it is executed
∂φ
∂ρ = 0 by definition, and the accelerating field near
the border is close to the maximum.
All the trapped electrons of the bunch come to
trapping area, i.e. acquire energy γphmc2, at differ-
ent instants. But as in the trapping area the velocity
of electrons close to the speed of light, then later in
the process of the acceleration their mutual arrange-
ment practically is unchanged. Therefore, the length
of electron bunch in accelerating stage is determined
by the length in the trapping region, which depends
on the electron distribution along the axis OZ in this
area.
Deviation of the phase of trapping along the axis
OZ any trapped electron relatively the central elec-
tron ξtr can be found by varying the integral (8) in
the small deviations δξinj , ρinj , δE
′
inj of the electron
from the central electron at the instant of injection.
Neglecting in the trapping region the terms contain-
ing derivatives of wake potential along the radius,
which are small due to the fact that the trapping
area is located near the boundary of focusing and
defocusing wake wave phase, we obtain the relation:
δξtr =
{
1
2
(
∂2φ
∂ξ2m
δξ2inj +
∂2φ
∂ρ2m
ρ2inj
)
+
+
(
Vph
uinj
− 1
)
δEinj
|e|
}[
∂φ
∂ξtr
]−1
,
(9)
where ∂2φ
∂ξ2m
and ∂2φ
∂ρ2
m
are determined in the phase of
wake field ξm, in which the wake potential is maxi-
mum, ∂φ
∂ξm
is calculated in the trapping point of cen-
tral electron.
2. TRAPPING OF ELECTRON BUNCH
The relation (9) makes it possible to analytically
evaluate the length of the bunch in the trapping area
and in the accelerating stage, if the spatial distribu-
tion of electrons in the bunch and their energy dis-
tribution in it at the instant of injection are known.
If the distributions are normal (Gaussian), in which
σz,inj , σγ,inj , σE,inj are the corresponding standard
deviations of the bunch electrons from the injection
coordinate and energy of the central electron, we ob-
tain the following expression for the bunch length in
the accelerating stage:
kpLb = 2
{
1
2
(
∂2φ
∂ξ2m
)2
(kpσz,inj)
4 +
(
Vph
uinj
− 1
)2
×
×
σ2
E,inj
|e|2
+
(
∂2φ
∂ρ2m
)2
(kpσr,inj)
4
}1/2 [
∂φ
∂ξtr
]−1
.
(10)
In this formula it is assumed that all the electrons
injected into the wake field, were trapped for subse-
quent acceleration.
To check the formula (10) we numerically simu-
lated the formation by wake field of electron bunch
during their trapping. The simulation were per-
formed for wake field that was established at the
end of transition process after injecting into a plasma
channel of short (τFWHM ≈ 100 fs) pulse Ti:Sa laser
with a wavelength of λ0 = 0.8 µm, a power 78 TW
and the size of the focal spot rL ≈ 68 µm.
Analysis of the nonlinear structure of a wake wave
used in our study for simulating the trapping of elec-
trons is based on the equations described in [14-16],
where the generation of the wake field excited by
a laser pulse propagating in the preliminarily pre-
pared plasma channel was studied. We assume that
the density n(r) of plasma electrons in the chan-
nel varies in the direction perpendicular to chan-
nel axis OZ in accordance with the parabolic law
n(r) = n(r = 0)
[
1 + r2/R2
ch
]
, where channel radius
Rch is related to radius rL of the focal spot of laser
radiation by the condition Rch = kpr
2
L/2. The den-
sity n(r) = n(r = 0) of the plasma on the channel
axis was chosen such that the resonance condition
ωpτFWHM = 2
√
2 ln 2 was satisfied for the excitation
of a wake wave by the laser pulse of duration τFWHM .
The envelope of electric field EL of the laser pulse
generating the wake wave and focused at the input of
the plasma channel can be written in the form:
a =
|e|EL
mcω
= a0 exp
[
− r2
r2L
− 2 ln 2
(ξ − ξ0)
2
(ωpτFWHM )2
]
,
(11)
where τFWHM is the total duration of the laser pulse
at half the maximum intensity (FWHM).
The resonance density of plasma electrons on the
channel axis is n0 = n(r = 0) ≈ 1.75 · 1017 cm−3 in
this case and the matched channel radius is Rch ≈
180 µm, which corresponds to the following values
of dimensionless parameters: kprL = 5.35, kpRch =
14.3, a0 = 0.71. The gamma factor determined by
the phase velocity of the wake wave being excited is
γph = 1/
√
1− β2 = 100.
Injected electron bunch were characterized by an
average energy of electron injection Einj = 1.8 mc2.
Distribution of the injected electrons in the bunch,
as all spatial axes, and in the energy space was con-
sidered Gaussian. Initial bunch length was 30.2 fs,
which corresponds to the standard deviation in the
distribution of the electron bunch length σz,inj ≈
3.2 µm (or kpσz,inj ≈ 0.25). The bunches of such
longitudinal dimension can be generated by the best
modern injectors [17, 18]. Original size of the bunch
in the longitudinal direction was chosen on the basis
of that in the case of one-dimensional bunch (bunch
of infinitely small radius) all of its electrons do not go
during the trapping process in the defocusing region
of wake wave. Formula (10) in the limit (σz,inj → 0)
gives the result obtained earlier in [9-12], which in-
dicates a significant contraction in the longitudinal
direction of the injected bunch. Thus, it is expected
that the application of the above expansion in the
small parameters in the derivation of (10) is an ade-
quate representation, and the length of the trapped
bunch in the form (10) is right for compact injected
129
electron bunch of not too large transverse dimension.
Fig.2 compares the results of numerical simulation
(squares) with the results of a calculation, using for-
mula (10) for the electron bunch length (dashed line),
trapped from the initial nonmonoenergetic bunch
with the initial energy spread between the electrons
σE,inj = 0.00825 mc2 which corresponds to a rela-
tive energy spread 1 %. The axis of abscissa shows
the characteristic transverse dimension of the in-
jected bunch, on the left y-axis it is shown the bunch
length in the trapping region. In the calculations we
used the values |e|
mc2
dφ
dξtr
= 0.22, |e|
mc2
d2φ
dξ2m
= −0.16,
|e|
mc2
d2φ
dρ2
m
= −0.055, that characterize this wakefield.
Fig.2. Length of the trapped bunch and the propor-
tion of trapped electrons (circles) as a function of
the transverse dimension of the bunch at the instant
of injection. For length: squares -simulation, dash -
calculation by formula (10)
From Fig. 2 it is seen that the bunch length, cal-
culated from the analytical formula (10), sufficiently
coincides exactly with the simulation results up to the
value of the radius of the injected bunch, comparable
to the longitudinal dimension of the injected bunch
kpσz,inj ≈ 0.3. Then there is a growing difference
between the graphs.
The reason is that with the growth of kpσz,inj
an increasing number of electrons is not trapped by
wake field, as electrons pass the boundary of focus-
ing and defocusing regions of the wake wave. This
clearly shows the graph (dashed line marked by cir-
cles) of the proportion of trapped electrons (right y-
axis) in Fig. 2. Electrons scattered by wave wake
not participate in the formation of a trapped electron
bunch and, therefore, does not determine its length.
Therefore, the formula (10) for the injected electron
bunches of large transversal radius gives an inaccu-
rate result.
CONCLUSIONS
The process of trapping of nonmonoenergetic
compact electron bunch injected into the vicinity of
the maximum potential of the wake wave generated
by the laser pulse is investigated analytically and by
numerical simulation. The phenomenon of electron
grouping arising in the process of the trapping of elec-
trons in the wake wave, if the initial velocity of the
injected electrons is much smaller than its phase ve-
locity, is studied in two-dimensional formulation of
the problem. A simple formula, quite accurately pre-
dicts the longitudinal size of the bunch in the accel-
erating stage when all electrons of injected bunch are
trapped. It is shown that the trapped electron bunch
length in an accelerating stage is determined addi-
tively by its initial energy spread and spatial trans-
verse and longitudinal dimensions. It is found that
for the wake field in which the characteristic longi-
tudinal and transverse dimensions are approximately
equal, the initial transverse characteristic size of the
bunch affects the length of the trapped bunch to the
same extent as its initial longitudinal characteristic
size at the instant of injection. Analytical results are
confirmed by the results of simulation.
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ÇÀÕÂÀÒ ÍÅÌÎÍÎÝÍÅÐÃÅÒÈ×ÅÑÊÈÕ ÝËÅÊÒÐÎÍÍÛÕ ÑÃÓÑÒÊÎÂ
 ÊÈËÜÂÀÒÅÐÍÎÉ ÂÎËÍÅ
Ñ.Â.Êóçíåöîâ
Ïðîâåäåíî àíàëèòè÷åñêîå èññëåäîâàíèå ïðîöåññà çàõâàòà â êèëüâàòåðíîé âîëíå, âîçáóæäàåìîé ëà-
çåðíûì èìïóëüñîì â ïëàçìåííîì êàíàëå, êîìïàêòíûõ íåìîíîýíåðãåòè÷åñêèõ ýëåêòðîííûõ ñãóñòêîâ.
Ýëåêòðîíû èíæåêòèðóþòñÿ â îêðåñòíîñòü ìàêñèìóìà ïîòåíöèàëà êèëüâàòåðíîé âîëíû ñî ñêîðîñòüþ
ìåíüøå å¼ ôàçîâîé ñêîðîñòè. Ïîëó÷åíà ôîðìóëà, îïðåäåëÿþùàÿ äëèíó ñãóñòêà íà óñêîðÿþùåé ñòà-
äèè, êàê ôóíêöèÿ åãî íà÷àëüíûõ ðàçìåðîâ (åãî äëèíû è ðàäèóñà ïîïåðå÷íîãî ñå÷åíèÿ) è íà÷àëüíîãî
ðàçáðîñà ýëåêòðîíîâ ïî ýíåðãèÿì â íåì.
ÇÀÕÎÏËÅÍÍß ÍÅÌÎÍÎÅÍÅÐÃÅÒÈ×ÍÈÕ ÅËÅÊÒÐÎÍÍÈÕ ÇÃÓÑÒÊIÂ
Ó ÊIËÜÂÀÒÅÐÍIÉ ÕÂÈËI
Ñ.Â.Êóçíåöîâ
Ïðîâåäåíî àíàëiòè÷íå äîñëiäæåííÿ ïðîöåñó çàõîïëåííÿ â êiëüâàòåðíié õâèëi, çáóäæåíié ëàçåðíèì iì-
ïóëüñîì ó ïëàçìîâîìó êàíàëi, êîìïàêòíèõ íåìîíîåíåðãåòè÷íèõ åëåêòðîííèõ çãóñòêiâ. Åëåêòðîíè ií-
æåêòóþòüñÿ áiëÿ ìàêñèìóìó ïîòåíöiàëó êiëüâàòåðíî¨ õâèëi çi øâèäêiñòþ ìåíøå ¨¨ ôàçîâî¨ øâèäêîñòi.
Îòðèìàíà ôîðìóëà, ùî âèçíà÷๠äîâæèíó çãóñòêà íà ïðèñêîðþâàëüíié ñòàäi¨, ÿê ôóíêöiÿ éîãî ïî÷àò-
êîâèõ ðîçìiðiâ (éîãî äîâæèíè i ðàäióñà ïîïåðå÷íîãî ïåðåðiçó) i ïî÷àòêîâîãî ðîçêèäó åëåêòðîíiâ ïî
åíåðãiÿì â íüîìó.
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