Comparative analysis of excitation of LSM and lSE waves by a bunch train in dielectric loaded rectangular resonator
The results of a three-dimensional analysis of wake field excitation in a slab-symmetric dielectric-loaded resonator
 by rigid electron bunches are presented. The complete set of solutions, including the solenoidal and potential parts of
 the electromagnetic field, consists of LSM an...
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| Zitieren: | Comparative analysis of excitation of LSM and lSE waves by a bunch train in dielectric loaded rectangular resonator / T.C. Marshall, I.N. Onishchenko, G.V Sotnikov // Вопросы атомной науки и техники. — 2006. — № 6. — С. 172-174. — Бібліогр.: 3 назв. — англ. |
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| author | Marshal, T.C. Onishchenko, I.N. Sotnikov, G.V |
| author_facet | Marshal, T.C. Onishchenko, I.N. Sotnikov, G.V |
| citation_txt | Comparative analysis of excitation of LSM and lSE waves by a bunch train in dielectric loaded rectangular resonator / T.C. Marshall, I.N. Onishchenko, G.V Sotnikov // Вопросы атомной науки и техники. — 2006. — № 6. — С. 172-174. — Бібліогр.: 3 назв. — англ. |
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| container_title | Вопросы атомной науки и техники |
| description | The results of a three-dimensional analysis of wake field excitation in a slab-symmetric dielectric-loaded resonator
by rigid electron bunches are presented. The complete set of solutions, including the solenoidal and potential parts of
the electromagnetic field, consists of LSM and LSE modes. Each of the LSM and LSE modes contains odd and even
waves. A numerical analysis of wake field excitation by symmetric electron bunches is carried out. The threedimensional
spatial structure of the longitudinal electric field is investigated. The influence of the drift vacuum channel
on the wake field amplitude and on the coherent summation of wakefields for a regular sequence of bunches is studied.
|
| first_indexed | 2025-12-07T17:26:16Z |
| format | Article |
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172 Problems of Atomic Science and Technology. 2006, 6. Series: Plasma Physics (12), p. 172-174
COMPARATIVE ANALYSIS OF EXCITATION
OF LSM AND LSE WAVES BY A BUNCH TRAIN
IN DIELECTRIC LOADED RECTANGULAR RESONATOR
T.C. Marshall1, I.N. Onishchenko2, G.V Sotnikov2
1Columbia University, New York City, USA;
2NSC “Kharkov Institute of Physics and Technology”, Kharkov, Ukraine
The results of a three-dimensional analysis of wake field excitation in a slab-symmetric dielectric-loaded resonator
by rigid electron bunches are presented. The complete set of solutions, including the solenoidal and potential parts of
the electromagnetic field, consists of LSM and LSE modes. Each of the LSM and LSE modes contains odd and even
waves. A numerical analysis of wake field excitation by symmetric electron bunches is carried out. The three-
dimensional spatial structure of the longitudinal electric field is investigated. The influence of the drift vacuum channel
on the wake field amplitude and on the coherent summation of wakefields for a regular sequence of bunches is studied.
PACS: 41.75.Jv, 41.75.Lx, 41.75.Ht, 96.50.Pw
1. INTRODUCTION
Dielectric-lined structures show promise for genera-
tion of strong accelerating fields by relativistic electron
bunches. Recently attention of specialists in the area of
accelerating techniques has been directed to dielectric-
lined waveguides with rectangular configuration. Simplic-
ity of manufacturing, possibility of realizing a multimode
regime of excitation with equally-spaced frequencies re-
sulting in a significant increase of accelerating field am-
plitude, easy fine tuning of working frequency, additional
intrinsic focusing, and other advantages make dielectric-
lined structures in a rectangular configuration attractive
for excitation of accelerating fields by a laser pulse or
electron bunches.
In a dielectric loaded waveguide of finite length that
is excited by a train of electron bunches, the important
factors restricting the summation of Cherenkov wake-
fields of bunches are the transition radiation and the
“quenching wave”. With the purpose of eliminating these
factors, we proposed to use the dielectric resonator [1].
The principles of a resonator concept for the planar
dielectric wakefield accelerator have been reported before
[2]. They were based on a two-dimensional analysis, ne-
glecting the influence of the bunch vacuum channel upon
the eigen-frequencies of the dielectric resonator.
In this work we investigate the excitation of the rec-
tangular resonator, loaded with two symmetric dielectric
slabs (DLRR). The separation of waves into LSM and
LSE modes [3] is very effective at studying such prob-
lems. A numerical analysis of wakefield excitation by
symmetric electron bunches of LSM and LSE mode is
carried out.
2. FIELD EXPRESSIONS
IN THE DIELECTRIC RESONATOR
Let’s consider a rectangular metal resonator loaded
with oppositely placed dielectric slabs of permittivityε .
The transverse size of a resonator in y-direction is b and
in x-direction is a . The transverse size of the vacuum
channel in the y-direction is 1b (the thickness of slabs is
1 2d ( b b ) /= − ). The length of the resonator is L . Par-
allel to the dielectric slabs along the z-axis a regular se-
quence of electron bunches is injected at the plane z=0,
which travel through the resonator.
We shall proceed from the following equations for
electromangnetic field components transverse to the slabs:
2
2 2
2
2 2
1 4
4
y y y
y y
y y
y y z
y
E
E E ,
y y c t y
H jH ,
c t c x
ε ε ρ
π
ε ε
ε π
∂ ∂∂ ∂
∆ + − = ∂ ∂ ∂ ∂
∂ ∂
∆ − =
∂ ∂
(1)
where: 0zj vρ= , ρ is charge density and yε ε= for
1b | y | b≤ ≤ or 1yε = for 1| y | b< .
Having executed a time Fourier transformation in
time of the equations (1), (2) and having expanded into a
series in harmonic functions with respect to x-,y-
coordinates, we shall obtain the equations in ordinary de-
rivatives with respect to the y-coordinate. Solutions of
each of the equations jointly contain two sets of inde-
pendent eigenfunctions. The component yH contains odd
eigenfunctions (symmetric with respect to plane 0y = )
and even eigenfunctions (antisymmetric). The component
yE contains odd eigenfunctions (antisymmetric) and even
eigenfunctions (symmetric). All these eigenfunctions are
orthogonal with respect to a certain weight factor among
themselves; eigenvalues for them are determined from the
corresponding four dispersion equations. Having ex-
panded yE and yH in a series on eigenfunctions and
having executed the inverse Fourier transformation, we
can obtain final expressions. Other components of the
field can be obtained from the Maxwell equations through
yE and yH . To obtain expressions for the fields of a
bunch having finite transverse size it is necessary to inte-
grate these expressions over the transverse locations of
the composing point bunches. We consider symmetric
(with respect to planes 0x = and 0y = ) electron
bunches, therefore upon integration over the locations
0 0i ix , y of bunches there will remain only the symmetric
solutions. For different components of field the symmetric
solutions correspond both to even and odd LSE and LSM
173
modes. The longitudinal component zE of the electric field contains only odd LSE and LSM modes:
2 2
0
0 0 0 0 02 2 2
1
0 0 0 0
1 1 1
11
n l
v l z
z l i mnl i i l iml
l ,m ,n mnl l || l mnl l
l m LSM
mnl i i x b y ,n b
mnl
v ( p c ) cos( k z )E E sin ( t t ) sin ( t t ) ( t t ) sin ( t t )
L ( )
( ) sin ( t t L / v ) ( t t L / v ) G ( x )G ( y )cos( k
ω
ω ω θ ω
ω ω ω ω ω ω
ω θ
ω
∞
=
= − − − − − − −
− − − − − −
∑
2
2 22 2
0 0
0 0 0 02 2 2 2 2
1
0
0
1 1
1
n
||om
x o
mnl
ml mll m
|| ||l z mnl x
l l i mnl i i lml
m,n mnl l || mnl mnl mnl
l
l i
e ( y )
x )
|| D ||
v cos( k z ) ( k v )E sin ( t t ) sin ( t t ) ( t t )
L ( )
sin ( t t ) ( )
ω ωδ ϖ
ω ω ϖ θ ω
ϖ ω ω ϖ ϖ ϖ
ω
∞
=
=
− − − − − − − − −
× − − −
∑
2
0
0 0 0 0 2
nm
yol m LSE mx
mnl i i x b y ,n b x o
mnl mnl
h ( y )( k v ) sin ( t t L / v ) ( t t L / v ) G ( x )G ( y )cos( k x )
|| H ||
ϖ θ
ϖ
− − − −
(2)
a b
Fig.1. Wakefield in the DLRR, shown in the z-y plane (x=0): a) after injection of 5-th bunch, b) after injection of 61-th
bunch. Crossline line shows sections of image/contour plot. The corresponding 1D structures of wakefield are pre-
sented at the top and at the right of images. Rectangles show bunch locations
where: 0 32 bE Q / abπ= ; bQ is bunch charge;
2 2 2 2n ml
v mnl ||( p ) / c ( k )ω= − , 2 2 2 2n ml
d mnl ||( p ) / c ( k )ω ε= − ,
2 2 2 2n ml
v mnl ||( q ) / c ( k )ϖ= − , 2 2 2 2n ml
d mnl ||( q ) / c ( k )ϖ ε= − ;
2 2 2 2m
x b b bG ( x ) sin ( m / ) sin( mx / a ) /( mx / a )π π π= ,
2 2LSM n n
y ,n b v b v bG ( y ) sin( p y / ) /( p y / )= , 2 2 2ml m l
|| x zk k k= + ,
2 2LSE n n
y ,n b v b v bG ( y ) sin( q y / ) /( q y / )= ; lδ is equal 1 if
0l = and is equal 2 if 0l ≠ ; 0
l
l zk vω = , 0
ml ml
|| ||k vω = .
Functions n n
||oe ( y ) ( p y )ϕ= and n n
yoh ( y ) ( q y )ϕ= de-
scribe the transverse structure of wakefield
1
1
2 2
2 2
n n n
v d d
n
n
v
cos( g b / ) sin g ( b / | y |) / sin g d ,
( g y ) if b / | y | b / ,
cos( g y ), else
ϕ
−
= ≤ ≤
and norms 2o
mnl|| D || , 2o
mnl|| H || of odd LSM and LSE
modes are defined in accordance with:
2 2 2
2 1 11
2 2
1
2 11
1
2
1
2
1
21 1
2
22 1
2
n n n
o v v v
mnl n n n
v d d
n n
od v
mnln n
d v
n n
v d
n n
d d
sin( p b ) ( p ) cos ( p b )b|| D || d
b p b ( p ) sin ( p d )
sin( p d ) sin( q b )b; || H ||
p d ) b q b
cos ( q b ) sin( q d )
d sin ( q d ) q d )
ε
ε
= − +
× + = +
+ −
Eigenfrequencies mnlω of odd LSM modes are deter-
mined from the dispersion equation
1 2 0n n n n
d v v dp tg( p b / ) p ctg( p d )ε− = , (3)
frequencies mnlϖ of odd LSE modes from the equation
1 2 0n n n n
d d v vq ctg( q d ) q tg( q b / )− = (4)
It should be noted, that when
mnl l mnl l,ω ω ϖ ω= = (5)
the relevant items in the expression (2) have removable
singularities. The conditions (5) are nothing else than the
conditions of Cherenkov radiation in a slowing medium.
3. NUMERICAL RESULTS
For a numerical analysis of wakefield excitation, a
metal resonator with the sizes 1 6a . cm= , 7 4b . cm= was
chosen. Parameters of a sequence of bunches are: repeti-
tion frequency 0 2805f MHz= , energy of electrons
4 5. MeV , macropulse current is 1 A , bunch length is
1 6bL . cm= , transverse sizes 21 1b bx y cm× = × . For a
coherent summation of bunch wakefields, at using a
bunch multiplicity 10N = , it is necessary [3] to choose
the length of the resonator 53 16L . cm= . The thickness of
the slab ( 8 2.ε = ) slabs was chosen from the condition (5)
For the excitation of the 1110, ,LSM -mode with frequency
0f from the equation (3) it follows 1 6 41b . mm= .
In Fig.1 the structure of the excited wave is presented.
During the time before the bunches reach the exit end of
the resonator, a wakefield is formed which is similar to
the wakefield in a semi-infinite waveguide. The field
grows from the head of a bunch, and Cherenkov cones
174
with reflections from the metal walls of the resonator are
easily seen. The presence of the Cherenkov cone is the
typical sign of multimode radiation in a slow-wave me-
dium. Near the resonator entrance the field of the transi-
tion radiation is appreciable. In the direction traverse to
the slabs the field is practically homogeneous.
As the next new bunches are injected, the field in the
resonator builds up and its amplitude becomes more ho-
mogeneous in the longitudinal direction, oscillating in
time with the frequency of the 1110, ,LSM -mode. The dy-
namics of the longitudinal electric field at the entrance
and exit of the resonator is given in Fig.2,a. It is seen that
the field in the resonator grows in time on both ends of
the resonator and the field of the transition radiation is no
longer significant.
In Fig.2,b the axial distribution of wakefields at time
t=32.11 ns, after injection of 91 bunches, is shown. From
these plots it follows that the dominant contribution to the
total field is given by LSM modes.
a b
Fig. 2. a) Dynamics of the zE at exit (left) and at entrance (right) of DLRR axis (x=y=0); b) Electric field zE at time
t=32.11ns along the axis of resonator ( 0x y= = ) after injection of 91 bunches. Solid line (1) - total field, dashed line
(2) - of all LSM modes, dash-dot line (3) - all LSE modes, dotted line (4) - the shape and position of bunches
We also carried out calculations of DLRR excita-
tion in the case where the size of the slabs is chosen so
that the frequency of the 1110, ,LSE -mode is equal to the
frequency of bunch repetition ( 1 5 24b . mm= ). Qualita-
tively, the structure of the longitudinal electric field is
similar to the one presented above. However, the rate
of increase is a little smaller than in the case of the
LSM-modes. Again the field of the transition radiation
does not appreciably affect the rate of increase of the
wakefield.
ACKNOWLEDGEMENTS
This study was supported by CRDF Grant No.
UP2-2569-KH-04 and Ukr. DFFD 02.07/325.
REFERENCES
1. T.C. Marshall, J-.M. Fang, J.L. Hirshfield, S.J. Park.
Multi-mode, Multi-bunch Dielectric Wake Field Reso-
nator Accelerator// AIP Conference Proceedings 569,
American Institute of Physics, New York. 2001, p. 316.
2. T.C. Marshall, I.N. Onishchenko, N.I. Onishchenko,
G.V. Sotnikov. Mode-Locking In A Dielectric Wake
Field Resonator Accelerator//Proc. of VI Int. Workshop
Strong Microwaves in Plasma, Inst. of Appl. Phys.
RAS, Nizhny Novgorod, 2006. V.1, p.277.
3. L. Pincherle. Electromagnetic Waves in Metal
Tubes Filled Longitudinally with Two Dielectrics//
Phys.Rev. 1944, v. 66, N5-6, p. 118-130.
LSM LSE
. , . , .
.
LSM LSE
. , . , .
.
|
| id | nasplib_isofts_kiev_ua-123456789-82297 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1562-6016 |
| language | English |
| last_indexed | 2025-12-07T17:26:16Z |
| publishDate | 2006 |
| publisher | Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| record_format | dspace |
| spelling | Marshal, T.C. Onishchenko, I.N. Sotnikov, G.V 2015-05-27T14:59:52Z 2015-05-27T14:59:52Z 2006 Comparative analysis of excitation of LSM and lSE waves by a bunch train in dielectric loaded rectangular resonator / T.C. Marshall, I.N. Onishchenko, G.V Sotnikov // Вопросы атомной науки и техники. — 2006. — № 6. — С. 172-174. — Бібліогр.: 3 назв. — англ. 1562-6016 PACS: 41.75.Jv, 41.75.Lx, 41.75.Ht, 96.50.Pw https://nasplib.isofts.kiev.ua/handle/123456789/82297 The results of a three-dimensional analysis of wake field excitation in a slab-symmetric dielectric-loaded resonator
 by rigid electron bunches are presented. The complete set of solutions, including the solenoidal and potential parts of
 the electromagnetic field, consists of LSM and LSE modes. Each of the LSM and LSE modes contains odd and even
 waves. A numerical analysis of wake field excitation by symmetric electron bunches is carried out. The threedimensional
 spatial structure of the longitudinal electric field is investigated. The influence of the drift vacuum channel
 on the wake field amplitude and on the coherent summation of wakefields for a regular sequence of bunches is studied. This study was supported by CRDF Grant No.
 UP2-2569-KH-04 and Ukr. DFFD 02.07/325. en Національний науковий центр «Харківський фізико-технічний інститут» НАН України Вопросы атомной науки и техники Plasma electronics Comparative analysis of excitation of LSM and lSE waves by a bunch train in dielectric loaded rectangular resonator Article published earlier |
| spellingShingle | Comparative analysis of excitation of LSM and lSE waves by a bunch train in dielectric loaded rectangular resonator Marshal, T.C. Onishchenko, I.N. Sotnikov, G.V Plasma electronics |
| title | Comparative analysis of excitation of LSM and lSE waves by a bunch train in dielectric loaded rectangular resonator |
| title_full | Comparative analysis of excitation of LSM and lSE waves by a bunch train in dielectric loaded rectangular resonator |
| title_fullStr | Comparative analysis of excitation of LSM and lSE waves by a bunch train in dielectric loaded rectangular resonator |
| title_full_unstemmed | Comparative analysis of excitation of LSM and lSE waves by a bunch train in dielectric loaded rectangular resonator |
| title_short | Comparative analysis of excitation of LSM and lSE waves by a bunch train in dielectric loaded rectangular resonator |
| title_sort | comparative analysis of excitation of lsm and lse waves by a bunch train in dielectric loaded rectangular resonator |
| topic | Plasma electronics |
| topic_facet | Plasma electronics |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/82297 |
| work_keys_str_mv | AT marshaltc comparativeanalysisofexcitationoflsmandlsewavesbyabunchtrainindielectricloadedrectangularresonator AT onishchenkoin comparativeanalysisofexcitationoflsmandlsewavesbyabunchtrainindielectricloadedrectangularresonator AT sotnikovgv comparativeanalysisofexcitationoflsmandlsewavesbyabunchtrainindielectricloadedrectangularresonator |