Radiation and beam energy spread in a channel with random rougness
The energy spread of particles inside a short bunch propagating along the co-axial channel with small random axially symmetric wall perturbations is calculated. Although the radiation is small, its reaction on particles distributed along a bunch can create a spread of particle energy and leads to an...
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| Опубліковано в: : | Вопросы атомной науки и техники |
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| Дата: | 2001 |
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| Формат: | Стаття |
| Мова: | Англійська |
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України
2001
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| Цитувати: | Radiation and beam energy spread in a channel with random rougness / A.V. Agafonov, A.N. Lebedev // Вопросы атомной науки и техники. — 2001. — № 3. — С. 161-163. — Бібліогр.: 4 назв. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1859649609860644864 |
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| author | Agafonov, A.V. Lebedev, A.N. |
| author_facet | Agafonov, A.V. Lebedev, A.N. |
| citation_txt | Radiation and beam energy spread in a channel with random rougness / A.V. Agafonov, A.N. Lebedev // Вопросы атомной науки и техники. — 2001. — № 3. — С. 161-163. — Бібліогр.: 4 назв. — англ. |
| collection | DSpace DC |
| container_title | Вопросы атомной науки и техники |
| description | The energy spread of particles inside a short bunch propagating along the co-axial channel with small random axially symmetric wall perturbations is calculated. Although the radiation is small, its reaction on particles distributed along a bunch can create a spread of particle energy and leads to an uncontrolled growth on a transverse emittance of the transported beam.
|
| first_indexed | 2025-12-07T13:32:09Z |
| format | Article |
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RADIATION AND BEAM ENERGY SPREAD IN A CHANNEL
WITH RANDOM ROUGNESS
A.V. Agafonov, A.N. Lebedev
Lebedev Physical Institute
Leninsky Pr. 53, Moscow, Russia, 119991
agafonov@sci.lebedv.ru, lebedev@sci.lebedv.ru
The energy spread of particles inside a short bunch propagating along the co-axial channel with small random axi-
ally symmetric wall perturbations is calculated. Although the radiation is small, its reaction on particles distributed
along a bunch can create a spread of particle energy and leads to an uncontrolled growth on a transverse emittance
of the transported beam.
PACS numbers: 41.60.Cr
1 INTRODUCTION
Short-wave free electron lasers are very sensitive to
transverse and longitudinal emittances of an electron
beam. The problem of radiation of electron bunches in-
side a channel with small random perturbations of the
wall has recently been discussed. Although the radiation
is small, its reaction on particles distributed along a
bunch creates a spread of particle energy and could lead
to an uncontrolled growth of transverse emittance in the
transported beam. Estimations [1 – 3] of the effect show
its possible danger for practically reached uniformity of
the wall surface. These estimations are based on the
concept of effective beam-channel coupling impedance
of a beam and a channel. Real part of the impedance is
interpreted as the losses due to diffraction radiation, and
imaginary part is calculated using Kronig-Kramers rela-
tions (see, for example, [4]). Radiation losses them-
selves are estimated for a statistical model of the sur-
face. It seems that this approach is not quite correct, or,
at any case, has a vague area of applicability. Actually,
the use of coupling impedance as a coefficient of pro-
portionality between a harmonic of the current of fre-
quency ω and of wave number k and a harmonic of elec-
tric field with the same characteristics does mean the
equality of phase velocities of these two wave pro-
cesses. Then, the field acting on a test particle of the
mono-energetic beam depends only on the position of
the particle relative to the bunch and does not depend
explicitly on time. Consequently, it results in linear
growth of energy deflexion and transverse pulse with
the distance z.
Waves propagating in the system with non-regular
(random) pertuberations of the wall have continuous
spectrum with respect to k, because reflections from in-
dividual perturbations are non-coherent. It means the
absence of slow eigenwaves of the channel and non-sta-
tionary behaviour of the radiation field acting on the test
particle even for a harmonically space-modulated beam.
Hence, the variations of particle energy are non-regular
and the conception of impedance looses its physical
meaning. Therefore, we calculate it using the straight
way of classical electrodynamics.
2 EVOLUTION OF FIELD ALONG NON-
REGULAR SYSTEM
We consider longitudinally modulated electron beam
of velocity βc and current density j(r, z/βc-t) which is
assumed to have multiplicative distribution:
( , , ) ( ) ( ) exp[ ( - )]zj r z t cq r i t
c
β ω ψ ω
β
= , (1)
propagating co-axially along a round quasi-regular
waveguide (see Fig. 1). Here, q(ω) is the Fourier-amp-
litude of the linear charge density, ψ(r) is a normalised
to unity transverse density distribution.
Fig. 1. Geometry and notations.
We limit ourselves the case of axially symmetric
perturbations of the wall surface when only TM-waves
can be excited. They influence the particle energy only
and don't lead to transverse displacements. The devi-
ations of the real radius of the channel r0(z) from an ap-
proximating radius a of a coaxial smooth cylinder are
ВОПРОСЫ АТОМНОЙ НАУКИ И ТЕХНИКИ. 2001. №3.
Серия: Ядерно-физические исследования (38), с. 161-163.
161
assumed small enough (in the sense discussed further).
Non-uniform wave equations must be completed by
the following boundary condition:
0
0 0( ( ), ) ( ( ), ).z r
drE r z z E r z z
dz
≡ − (2)
It means vanishing of a tangential component of the
electric field on the perfectly conducting wall of the
waveguide.
Following methodically [1, 2] we approximate the
boundary condition by its expansion at r = a using the
smallness of δ(z) = a - r0(z):
,),(),( +
∂
∂+′≈ = ar
z
rz r
EzaEzaE δδ (3)
where prime denotes a total derivative with respect to z.
This approximation is reasonable if δ' << 1, i.e., if
perturbations at the surface are smooth enough. The
area of its applicability was discussed in [1, 2]. Under
the condition, the fields excited by the beam inside a
regular wavegude of radius a can be substituted into the
right part of (3). Because the radiation can't exist in a
regular waveguide (except of transition radiation at ends
of the waveguide) then the zero order field propagates
with the velocity of the beam having periodicity ∝
exp(ik0z/β), where k0 = ω/c, and represents just a Cou-
lomb's field in the laboratory frame. The method of its
calculations is a standard one. It is necessary to use the
expansion the field over the full system of eigenfunc-
tions of the transverse Laplacian's operator (for the case,
Bessel's functions of zero order with roots λn).
Taking into account this expansion the boundary
condition (3) can be written as
),/exp()()(),( 00 βzikzfkAqzaEz = (4)
where
.)(4,)(
22
0
222
1
22
2
0 ∑ +
=+′=
n n
nn
ak
JAikzf
γβλ
λψγπ β
β γ
δδ (5)
The solution of the non-uniform equation for Ez with the
non-uniform boundary condition (4) can be represented
in the following form:
0 10
0( , ) ( , ) ( ) ( ) exp ( , ),z z z
ik zE r z E r z Aq k f z E r z
β
= + +
where E1
z(r,z) satisfies the non-uniform equation with
zero boundary condition.
Note once more that E0
z(r,z) and E0
z(a,z) have no
poles for real k0 and, consequently, can't describe free
propagating radiation field. The last is described by the
term E1
z(r,z) and have to meet the demands of the Som-
merfeld's principle (principle of causality). For the case:
at the input of the waveguide (z=0) waves exist and
propagate only in negative direction of z-axes; and at
the output of the waveguide (z = L) waves propagate
only in positive direction. This rule of the solution
choice is valid and for L → ∞ as well.
As far as E1
z(a,z)= 0 we can find the solution of non-
uniform equation for complex amplitudes of the radi-
ation field with zero boundary condition for E1
z(r,z) in
the form of the expansion over the same eigenfunctions
J0(λmr/a).
,exp
)(
)(2
)exp()()exp()()(
0
1
0
βλλ
κκ
zik
J
fkAq
zizAzizAzE
ss
sssss
−
−+= −+
where
- 2 2
0( ) exp( ) exp(- ) 0, - s s s s s sA z i z A i z kκ κ κ λ+ + = = .
Taking into account above-mentioned principle of
causality we can find the amplitudes of forward and
backdirected waves.
Spatially and temporally modulated current of large
enough frequency (higher than the cutoff frequency λ
sc/a) excites forward and backdirected modes of radi-
ation. Low frequency components of the current (ima-
ginary κs) do not excite radiation, and corresponding
fields propagate together with the beam. These fields
represent corrections of Coulomb's field more or less
locally related to the current density. Note, that the cor-
rection themselves are calculated without taking into ac-
count boundary effects at z=0 and z=L. However this
can be of importance only if k0a ≈ λs, when Coulomb's
field is not yet screened by the walls.
The radiation field as distinct from Coulomb's field
is defined by overall beam inside the waveguide, and
grow (for forward wave) along it. However, because of
random positions of small perturbations of the wall sur-
face the phase of the field is random as well, and the ra-
diation can be only partially coherent even a perfect reg-
ular arrangement of primary radiators i.e., of beam
particles. The degree of the coherency is defined by cor-
relation properties of a non-uniform surface.
3 PARTICLE ENERGY VARIATIONS DUE
TO RADIATION FIELD
We find the energy variation ∆sγ(ζ) of a test particle
placed at a lag distance ζ (ζ= 0 at the centre of a bunch)
in longitudinal direction and at radial position r under
the action of s-mode. We restore omitted term exp(-
ik0ct) in the field expression putting there t=(z+ζ)/βc,
and integrate J0(λsr/a)Es(z)exp(-ik0z/β) along the wave-
guide (the influence of Coulomb's field of zero order
and backdirected wave is neglected) and over k0, i.e.,
the inverse Fourier transformation is to be performed to
restore a real spatial/temporal structure of the bunched
beam field:
∫ −=∆
L
ss
s
s dGL
Jmc
arieJ
01
2
0 ,)()(
)(
)/(
ξξξδ
λλ
λ
γ
where
[ ]
0 0
0
0 0
0
( ) ( ) exp( / )
1 exp ( / ) 1 .
/
s
s
s
s
G Aq k ik
k i k dk
k
ξ κ ζ β
κ β κ β ξ
κ β
+ ∞
− ∞
= − − ×
− + − − −
∫
(6)
The behaviour of G(ξ) is determined by an hierarchy of
three characteristic lengths - the channel transverse radi-
us a, the channel length L (or the distance ξ = L-z), and
the bunch longitudinal size in the lab frame lb. Apart
from pure geometry this hierarchy depends essentially
on the Lorentz factor γ. We can consider two limiting
162
cases.
1) If the independent variable ξ satisfies the inequal-
ity aγ/λsξ >> 1 then for the majority of the harmonics
(with wave-numbers essentially lesser than λsγ2/a the
phase slip at the total length is small. The corresponding
exponent in (6) may be changed then for unity. Then G(
ξ) ≈ G(0) is independent of ξ and the expression for the
energy gain is simplified. Thus case can be considered
as the “short channel” solution if the length of the chan-
nel L << L* = aγ/λs.
2) If ξ >> L* even a smooth minimum of the func-
tion κs – k0/β plays certain selective role and quasi-res-
onant harmonic can be of importance. It can be showed
that asymptotic behaviour of G(ξ) for large ξ does not
depend on ξ and for this case G(ξ) =G(∞).
4 RESULTS AND ESTIMATIONS
It follows from the consideration above that the total
intensity and amplitude of the most effective harmonic
depends on a spatial distribution of the bunch current.
We shall restrict ourselves by the case of a transversally
uniform (0 < r < b) single bunch of charge Q with Gaus-
sian longitudinal distribution of half-width lb
The estimates obtained above are valid for an arbit-
rary realisation of the perturbation δ(z) and contain cer-
tain information on the asymptotic phase of the radi-
ation field. In the case of random perturbation this in-
formation is not of importance. Let us suppose that the
mathematical expectation δ = 0. As far G(ξ) does not
change its value at the correlation length lc the mathem-
atical expectation ∆s vanishes as well. The role of a
real characteristic of the energy diffusion is played by
the diffusion coefficient <∆2>, which is a sum over
transverse harmonics squared and averaged:
.
2
2 ∑ ∆=∆
s
sγ (7)
The energy gain at the output of the channel can be
written as
,22*22 LlGC cδ=∆
where C = r0N0/ba2, r0 is the electron classical radius, N0
is the number of electrons per bunch, and G* = G(0) or
G(∞) are dimensionless functions (see Fig. 2) of the lag
ζ (measured in channel radius a units) describing the
distribution of energy gain along the bunch. Note that
for the case <∆2> ∝ lcL as it should be for a diffusion
process.
Calculations were carried out for the set of paramet-
ers of LCLS: γ = 30000; lb/a = 0.002, (for radius of the
channel a = 0.5 cm the half-length of the bunch is
lb = 10 µm). The coefficient C = 1.4×103 m-2 for
b/a = 0.1 and bunch charge Q= 1 nC (N0 = 6.25×109).
For the set of the parameters it follows that L* ≈ 50 m.
Taking maximum values of dimensionless energy gains
from Fig. 2 and choosing rather pessimistic parameters
(<δ2>)1/2 ≈ 5 µm, lc = a and L >> L* we find the amp-
litude of energy oscillations ∆γ/γ ≈ 5×10-4.
Fig. 2. Dimensionless part of particle energy vari-
ation. The dependencies are slightly asymmetric
with respect to the centre of the bunch.
5 CONCLUSION
We consider the energy spread inside a bunch
propagating along the channel with small random per-
turbations of the wall as not dangerous for the X-ray
FEL's. From this point of view a more vital problems
are the energy spread forced by self space charge of the
bunch and angular spread forced by the radiation.
Work was supported by RFFI under grant
01-02-17519.
REFERENCES
1. G.V.Stupakov // Phys. Rev. Special Topics - Accel-
erators and Beams. 1998, v. 1, p. 064401.
2. G.Stupakov, R.E.Thompson, D.Walz, R.Carr //
Phys. Rev. Special Topics - Accelerators and
Beams. 1999, v. 2, p. 64401.
3. A.Novokhatski, A.Mosnier // Proc. of the 1997
Particle Accelerator Conference, Vancouver, B.C.,
1997. v. 2, p. 1661.
4. L.Landau and E.Lifshits. Electrodynamics of Con-
tinuous Media, Moscow: Nauka, 1982.
163
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| id | nasplib_isofts_kiev_ua-123456789-79267 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1562-6016 |
| language | English |
| last_indexed | 2025-12-07T13:32:09Z |
| publishDate | 2001 |
| publisher | Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| record_format | dspace |
| spelling | Agafonov, A.V. Lebedev, A.N. 2015-03-30T08:29:01Z 2015-03-30T08:29:01Z 2001 Radiation and beam energy spread in a channel with random rougness / A.V. Agafonov, A.N. Lebedev // Вопросы атомной науки и техники. — 2001. — № 3. — С. 161-163. — Бібліогр.: 4 назв. — англ. 1562-6016 PACS numbers: 41.60.Cr https://nasplib.isofts.kiev.ua/handle/123456789/79267 The energy spread of particles inside a short bunch propagating along the co-axial channel with small random axially symmetric wall perturbations is calculated. Although the radiation is small, its reaction on particles distributed along a bunch can create a spread of particle energy and leads to an uncontrolled growth on a transverse emittance of the transported beam. Work was supported by RFFI under grant 01-02-17519. en Національний науковий центр «Харківський фізико-технічний інститут» НАН України Вопросы атомной науки и техники Radiation and beam energy spread in a channel with random rougness Излучение и энергетический разброс в пучке, распространяющемся в канале со случайными неоднородностями стенки Article published earlier |
| spellingShingle | Radiation and beam energy spread in a channel with random rougness Agafonov, A.V. Lebedev, A.N. |
| title | Radiation and beam energy spread in a channel with random rougness |
| title_alt | Излучение и энергетический разброс в пучке, распространяющемся в канале со случайными неоднородностями стенки |
| title_full | Radiation and beam energy spread in a channel with random rougness |
| title_fullStr | Radiation and beam energy spread in a channel with random rougness |
| title_full_unstemmed | Radiation and beam energy spread in a channel with random rougness |
| title_short | Radiation and beam energy spread in a channel with random rougness |
| title_sort | radiation and beam energy spread in a channel with random rougness |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/79267 |
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