Bunching system based on the evanescent waves
To improve the beam bunching at the initial stage of acceleration it is necessary to create an increasing field distribution. Such distribution can be created in the ordinary disk-loaded waveguide in its stopbands. Generally, there are two eigen evanescent waves, one of which has increasing distribu...
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| Zitieren: | Bunching system based on the evanescent waves / M.I. Ayzatsky, K.Yu. Kramarenko, S.A. Perezhogin // Вопросы атомной науки и техники. — 2001. — № 3. — С. 83-85. — Бібліогр.: 4 назв. — англ. |
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| author | Ayzatsky, M.I. Kramarenko, K.Yu. Perezhogin, S.A. |
| author_facet | Ayzatsky, M.I. Kramarenko, K.Yu. Perezhogin, S.A. |
| citation_txt | Bunching system based on the evanescent waves / M.I. Ayzatsky, K.Yu. Kramarenko, S.A. Perezhogin // Вопросы атомной науки и техники. — 2001. — № 3. — С. 83-85. — Бібліогр.: 4 назв. — англ. |
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| container_title | Вопросы атомной науки и техники |
| description | To improve the beam bunching at the initial stage of acceleration it is necessary to create an increasing field distribution. Such distribution can be created in the ordinary disk-loaded waveguide in its stopbands. Generally, there are two eigen evanescent waves, one of which has increasing distribution along the longitudinal axes and another - decreasing one. The field structure in the bounded system is a superposition of these two evanescent waves whose amplitudes are determined by the operating frequency and by the geometry of the boundary cavities. The results of the simulation of the buncher for the case of the 25 keV injected electron energy are presented in the paper.
|
| first_indexed | 2025-12-07T18:24:12Z |
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BUNCHING SYSTEM BASED ON THE EVANESCENT WAVES
M.I. Ayzatsky, K.Yu. Kramarenko, S.A. Perezhogin
National Science Center Kharkov Institute of Physics&Technology
1 Academicheskaya Str., NSC KIPT, 61108 Kharkov, Ukraine
E-mail aizatsky@nik.kharkov.ua
To improve the beam bunching at the initial stage of acceleration it is necessary to create an increasing field distri-
bution. Such distribution can be created in the ordinary disk-loaded waveguide in its stopbands. Generally, there are
two eigen evanescent waves, one of which has increasing distribution along the longitudinal axes and another - de-
creasing one. The field structure in the bounded system is a superposition of these two evanescent waves whose am-
plitudes are determined by the operating frequency and by the geometry of the boundary cavities. The results of the
simulation of the buncher for the case of the 25 keV injected electron energy are presented in the paper.
PACS numbers: 29.27.-a
1 INTRODUCTION
To improve the bunching process at the initial stage
of acceleration it is necessary to create an increasing
field distribution. It was shown, that the amplitude dis-
tribution of the eigen oscillations in the bounded period-
ic structure within the stopband corresponds to the in-
creasing amplitude distribution [1]. It is known, that in
the boundless periodic structure two eigen electromag-
netic oscillations exist. In the passbands the eigen oscil-
lations represent travelling waves. In the stopbands the
eigen oscillations do not transfer energy in the direction
of periodicity and have either a decreasing or increasing
character. In the bounded periodic structure it is possi-
ble to create the field distribution corresponding to the
one (increasing or decreasing) eigen oscillation. The in-
creasing field distribution cannot be obtained in the
smooth waveguide. As a result of the boundary condi-
tions, the amplitude of the increasing solution in the
smooth waveguide is always less than the amplitude of
the decreasing one.
2 MATHEMATICAL MODEL
Bunching system based on the segment of cylindri-
cal disk-loaded waveguide is considered. To investigate
the field distribution in this structure we use the oscilla-
tion equations of the weakly coupled cavities. The cou-
pling between the neighboring cavities is taken into ac-
count [2].
The most of the accelerating structures operate in the
Е010 – mode. The field amplitude distribution of Е010 –
mode in the structure consisting of N cavities is deter-
mined by the set of N equations:
0~)1( 21
2
1
1
1
1
2
1
2
1 =+
−+− A
Q
iA εωω ωεωω , (1.1)
,0~)21( 11
2
03
2
0
0
02
0
2
2 =++
−+− AA
Q
iA εωεωω ωεωω (1.2)
,0)21( 1
2
01
2
0
0
02
0
2 =++
−+− +− nnn AA
Q
iA εωεωω ωεωω (1.3)
,0~)21( 2
02
2
0
0
02
0
2
1 =++
−+− −− NNNN AA
Q
i
A εωεω
ω ω
εωω (1.4)
0~)1( 1
222 =+
−+− −NNN
N
N
NNN A
Q
iA εωω ωεωω , (1.5)
where An is the field amplitude in n-th resonator,
n=1, 2,…N; )(3
2
1
2
1 aJπ
χ = ,
Db
a
2
3
χε = ,
NN
N Db
a
,1
2
,1
3
,1 χε = ,
NN
N DDbb
a
,1,1
3
,1
~ χε = , a is the radius of coupling aperture,
b is the radius of the cavity, D is the length of the cavi-
ty, ω0 is the resonant frequency of the n-th cavity (n≠
1,N), ω1 is the resonant frequency of the 1-st cavity, ωN
is the resonant frequency of the N-th cavity. Equations
(1.1-1.5) are true in the case of thin diaphragms. The set
of equations (1.1-1.5) is the set of homogeneous differ-
ence equations of the second order with constant coeffi-
cients. The field amplitude distribution in the boundless
structure is determined by the set of equivalent equa-
tions, which are similar to the equation (1.3). There are
two partial solutions of this equation: n
nA 1ρ= and
n
nA 2ρ= , where 12
2,1 −±= ββρ are the roots of the
characteristic equation 0122 =+− β ρρ . The value β is
defined as:
εωω ωωεωβ 2
0
0
022
0 2/)21(
+−+=
Q
i
. (2)
At first, the ideal structure ( ∞=0Q ) will be consid-
ered. In the passband, when the frequency changes from
ω0 ( 1=β ) up to ωπ ( 1−=β ), nie ψρ =1 and
nie ψρ −=1 . The phase shift per cell is defined from the
dispersion equation: βψ =cos . In the stopbands,
0ωω < and πωω > , the values 21 ρρ , correspond to
the evanescent oscillation: 121 ≠,ρ . At the frequencies
ω>ωπ 21, ρρ take the negative values: πρρ ie11 = ,
πρρ ie22 = .
The field amplitude distribution in the bounded
structure equals to the sum of two partial solutions with
constant coefficients – nn
n CСA 2211 ρρ += . Excluding
1A from (1.1, 1.2) and AN from (1.4, 1.5), we get the set
ВОПРОСЫ АТОМНОЙ НАУКИ И ТЕХНИКИ. 2001. №3.
Серия: Ядерно-физические исследования (38), с. 83-85.
83
mailto: aizatsky@nik.kharkov.ua
of equations to define C1, C2:
( ) ( ) 0)()( 21
2
2211
2
11 =+++ ρωρρωρ fCfС , (3.1)
( ) ( ) 0)()( 1
1
222
1
11 =+++ −− ρωρρωρ N
N
N
N fCfС , (3.2)
where ( )2
1
2
1
2
1
2
11 )1(/~2)( ωεωεεωβω −++−=f ,
( )2222 )1(/~2)( ωεωεεωβω −++−= NNNNNf .
The set of equations (3) has the non-trivial solution
if determinant is equal to zero:
( ) ( )
( ) ( ) .0)()(
)()(
2
1
121
2
2
1
1
211
2
1
=++−
−++
−
−
ρωρρωρ
ρωρρωρ
N
N
N
N
ff
ff
(4)
This equation determines the resonant frequencies of
the bounded structure. The structure consisting of N
cavities has N resonant frequencies. Resonant frequen-
cies of the structure with half boundary cells
(D1 = DN = D/2) lay within the interval ω0 ≤ ωn ≤ ωπ. In
this case, the resonant oscillation represents a standing
wave formed by two traveling waves – ni ne ψ and
ni ne ψ− , ψn = πn/(N-1), n=0, … N-1.
Let us consider the case, when the cavity chain has
arbitrary (not half) boundary cells. Suppose, that at the
some frequency 1ω ′ the following conditions are ful-
filled:
0f 211 =+′ ρω )( , 0f 111 ≠+′ ρω )( , (5.1)
0f 11N =+′ ρω )( , 0f 21N ≠+′ ρω )( . (5.2)
If boundary cells differ from other ones only in
length - D1 = ξ1D, DN = ξND, than conditions (5.1), (5.2)
will have the form:
1
1 1
1
ρ
ξ
−
= , (6.1)
21
1
ρ
ξ
−
=N . (6.2)
In this case, as it follows from equations (3.1, 3.2),
C1=0. The field amplitude distribution is as follows:
n
n СA 22 ρ= .
By similar way, one can obtain conditions for creat-
ing the field amplitude distribution of the form
n
n СA 11ρ= at the some frequency 2ω ′ .
At the frequency 1ω ′ ( 2ω ′ ) the resonant oscillation
of the structure consisting from N cavities is based on
the one eigen oscillation of the boundless structure. As
ξ1, ξN > 0, than πωωω >′′ 21 , . Hence, the eigen oscilla-
tion is based on evanescent oscillation of the boundless
structure. Let us suppose, that at the frequency ω > ωπ
1ρ is less than one ( 11 <ρ ) and 2ρ is grater than
one ( 12 >ρ ). So, at the frequency 1ω ′ the amplitude
distribution in the structure ( n
n СA 22 ρ= ) is an increas-
ing one: nn AА >+ 1 ; at the frequency 2ω ′ the ampli-
tude distribution in the structure ( n
n СA 11ρ= ) is a de-
creasing one: nn AА <+ 1 . It is easy to show that both
for increasing and decreasing amplitude distributions
the following condition is satisfied:
D1 + DN = D. (7)
The resonant oscillation of the structure with
D1 > D/2, DN < D/2, D1 + DN = D at the frequency
ω > ωπ corresponds to the increasing eigen oscillation
(Fig. 1, curve b); as for the decreasing one, the length of
the boundary cavities must fulfill such conditions:
D1 < D/2, DN > D/2, D1 + DN = D (Fig. 1, curve а).
Consider the case, when the structure is exited at the
frequency 1ω ′ . Taking into account losses in the struc-
ture, one can obtain:
)()1(2
2
2
1 Q
С
С N δρ −≈ . (8)
As one can see, the amplitude distribution depends on
the number of cavities. When 1)()1(2
2 < <− QN δρ , than
C1 << C2. In this case, the amplitude distribution in the
structure is increasing. If 1)()1(2
2 ≈− QN δρ , then the
amplitude distribution represents a superposition of two
eigen oscillations. When N → ∞, than C1 >> C2 and the
amplitude distribution is decreasing. It satisfies the con-
dition that in the half-bounded structure the increasing
distribution cannot be realized.
One resonant frequency of the structure, the bound-
ary cells of which differ from half ones, lays outside the
passband: ω > ωπ. If condition (7) is not fulfilled, the
resonant oscillation represents the sum of two evanes-
cent oscillations (Fig. 1, curve c).
1 2 3 4 5 6 7
0
1
2
3
4
5
6
n
A
,
a
rb
it
.
u
n
it
s
a
b
c
Fig. 1. Amplitude distribution in the structure con-
sisting of 7 cavities at the frequency ω=1.004ωπ:
a=1.5 см, b=4.4026 см, D =2.438 см;
curve а – D1 =0.9698 cm, DN = 1.4682 cm;
curve b – D1 =1.4682 cm, DN= 0.9698 cm;
curve c – D1 =1.4682 cm, DN =1.219 cm.
We considered the situation, when we changed only
the length of the boundary cells. Similar effect can be
achieved by changing the eigen frequencies of boundary
cavities. Resonant frequency of the Е010 - mode is in in-
verse proportion to the radius of the cavity. Let us des-
ignate – b1 = ζ1b, bN = ζNb. We shall suppose that D1 =
DN = D.
For the realization of the increasing field amplitude
distribution n
n СA 22 ρ= , С1 = 0 (Fig. 2, curve b) it is
necessary to select the resonant frequency of the bound-
ed structure laying above ωπ , and to satisfy conditions
(5.1, 5.2). Setting πωω >′1 , we define the values ρ1, ρ2.
Coefficients ζ1, ζN, which determine the radii of bound-
ary cavities with respect to other ones, are defined from
two nonlinear equations:
( )
−++−+= 4
1
4
1
2
11114
1
1 14)()(
2
1 ζζγαγα
ζ
ρ , (9.1)
84
( ) ( )
−++−+= 442
42 14)(
2
1
NNNNNN
N
ζζγαγα
ζ
ρ . (9.2)
To obtain the decreasing amplitude distribution one can
reverse boundary cells (Fig. 2, curve a).
1 2 3 4 5 6 7
0
2
4
6
8
n
A
,a
rb
it
.
u
n
it
s
b
a
c
Fig. 2. Amplitude distribution in the structure con-
sisting of 7 cavities at the frequency ω=1.004ωπ:
a=1.5 cm, b=4.4026 cm, D =2.438 cm;
curve а – b1 =4.1636 cm, bN = 4.2516 cm;
curve b – b1 =4.2516 cm, bN = 4.1636 cm;
curve c – b1 = bN =4.1825 cm.
3 SIMULATION
Proceeding from the above-presented theory, the in-
jector system based on evanescent oscillation was simu-
lated using SUPERFISH [3] and PARMELA [4] codes.
The simulation was held under the electron beam initial
energy W0=25 keV and current 50 mA with space
charge forces taken into account. Peak value of on-axis
electric field is 30 MV/m.
Waveguide section composed from five accelerating
cells was taken for simulations of bunching system
based on disk-loaded waveguide. It is well known that
the disk-loaded waveguide has many stopbands. As a
working stopband we have chosen the second stopband
of the symmetric wave. If we want to work in the stop-
band, the conditions for the eigen frequency of the sys-
tem to lay in the stopband must be created. The simplest
way of creating such situation is shifting the frequency
of the last cell. In the second stopband the phase shift
per cell equals π (ρ<0). The time-transit angle for rela-
tivistic particle was chosen equal to π30. per period. As
a result of simulations, the on-axis increasing field dis-
tribution was obtained. The results are shown in Fig. 3.
Fig. 3. Geometry of bunching system base on disk-
loaded waveguide and corresponding on-axis elec-
tric field distribution.
Electrodynamic performances of the simulated sys-
tem are the follows: the quality factor Q=11163, shunt
impedance Rsh=39.8 MOhm/m. Simulation of particle
dynamics in the system has shown that the maximum
energy is 0.779 MeV, average energy is 0.697 MeV, en-
ergy spectrum is 9% (70% of particle), phase length is
32°, normalize emittance is 28 mm⋅mrad and capture is
91.2%.
4 CONCLUSION
The increasing amplitude distribution necessary for
the effective bunching process can be obtained in the
regular disk-loaded waveguide. Results of simulation of
the bunching process show the efficiency of using the
bunching system on evanescent oscillations. It was also
shown that in the bounded structure with losses ampli-
tude distribution depends on the number of resonators.
The authors express gratitude to V.A. Kushnir and
V.V. Mitrochenko for the participation in discussing the
results.
REFERENCES
1. M.I.Ayzatsky. Electromagnetic oscillations in peri-
odic mediums and waveguides outside the passband
// Problems of Atomic Science and Technology. Is-
sue: Nuclear-Physics Research (34). 1999, N 3,
p. 6-9.
2. H.A.Bethe Theory of diffraction by small holes //
Phys. Rev. 1944, 66, N7-8, p. 163-182.
3. L.M.Young. “PARMELA”, Los Alamos National
Laboratory, LA-UR-96-1835, 1996.
4. J.H.Billen and L.M.Young. POISSON/SUPER-
FISH on PC compatibles // Proc. 1993 Particle Ac-
celerator Conf., Washington (USA), 1993, p. 790–
792.
ВОПРОСЫ АТОМНОЙ НАУКИ И ТЕХНИКИ. 2001. №3.
Серия: Ядерно-физические исследования (38), с. 85-85.
85
|
| id | nasplib_isofts_kiev_ua-123456789-79206 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1562-6016 |
| language | English |
| last_indexed | 2025-12-07T18:24:12Z |
| publishDate | 2001 |
| publisher | Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| record_format | dspace |
| spelling | Ayzatsky, M.I. Kramarenko, K.Yu. Perezhogin, S.A. 2015-03-29T18:23:38Z 2015-03-29T18:23:38Z 2001 Bunching system based on the evanescent waves / M.I. Ayzatsky, K.Yu. Kramarenko, S.A. Perezhogin // Вопросы атомной науки и техники. — 2001. — № 3. — С. 83-85. — Бібліогр.: 4 назв. — англ. 1562-6016 PACS numbers: 29.27.-a https://nasplib.isofts.kiev.ua/handle/123456789/79206 To improve the beam bunching at the initial stage of acceleration it is necessary to create an increasing field distribution. Such distribution can be created in the ordinary disk-loaded waveguide in its stopbands. Generally, there are two eigen evanescent waves, one of which has increasing distribution along the longitudinal axes and another - decreasing one. The field structure in the bounded system is a superposition of these two evanescent waves whose amplitudes are determined by the operating frequency and by the geometry of the boundary cavities. The results of the simulation of the buncher for the case of the 25 keV injected electron energy are presented in the paper. The authors express gratitude to V.A. Kushnir and V.V. Mitrochenko for the participation in discussing the results. en Національний науковий центр «Харківський фізико-технічний інститут» НАН України Вопросы атомной науки и техники Bunching system based on the evanescent waves Бипериодический группирователь пучка электронов на нераспространяющемся колебании Article published earlier |
| spellingShingle | Bunching system based on the evanescent waves Ayzatsky, M.I. Kramarenko, K.Yu. Perezhogin, S.A. |
| title | Bunching system based on the evanescent waves |
| title_alt | Бипериодический группирователь пучка электронов на нераспространяющемся колебании |
| title_full | Bunching system based on the evanescent waves |
| title_fullStr | Bunching system based on the evanescent waves |
| title_full_unstemmed | Bunching system based on the evanescent waves |
| title_short | Bunching system based on the evanescent waves |
| title_sort | bunching system based on the evanescent waves |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/79206 |
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