candle.amcandle.am/wp-content/uploads/2017/04/samvel-zakaryan...- 2 - contents introduction
TRANSCRIPT
Center for the Advancement of Natural Discoveries using Light
Emission
Samvel Zakaryan
Electromagnetic waves in laminated structures and particle
bunching
Thesis
to take a candidate degree in physical and mathematical sciences 01.04.20
โCharged Particle Beam Physics and Accelerator Technologyโ
Scientific supervisor:
Doctor of Physical and Mathematical Sciences
Mikayel Ivanyan
Yerevan 2017
- 2 -
Contents
Introduction ............................................................................................. - 4 -
Chapter 1: Slow waves in laminated structures .................................. - 10 -
1.1 Introduction ..................................................................................................................... - 10 -
1.2 Slow waves in cylindrical structures ........................................................................ - 10 -
1.3 Attenuation parameter .................................................................................................. - 13 -
1.4 Longitudinal impedance and radiation pattern ...................................................... - 14 -
1.4. Slow waves in flat structures .................................................................................... - 16 -
1.5 Summary .......................................................................................................................... - 24 -
Chapter 2: Accelerating structures with flat geometry ....................... - 26 -
2.1 Introduction ..................................................................................................................... - 26 -
2.2 Multilayer flat structure ................................................................................................ - 27 -
2.3 Two-layer flat structure ................................................................................................ - 35 -
2.4 Impedance of two-layer symmetrical flat structure .............................................. - 37 -
2.5 Approximate analytical representation of impedance ......................................... - 47 -
2.6 Summary .......................................................................................................................... - 53 -
Chapter 3: Rectangular resonator with two-layer vertical walls ........ - 55 -
3.1 Introduction ..................................................................................................................... - 55 -
3.2 IHLCC cavity with ideally conductive walls ............................................................ - 56 -
3.3 Pure copper cavity. Measurements and interpretation ....................................... - 60 -
3.4 IHLCC cavity resonance frequency comparison with experimental results .. - 62 -
3.5 Summary .......................................................................................................................... - 67 -
Chapter 4: Electron bunch compression in single-mode structure ... - 68 -
4.1 Introduction ..................................................................................................................... - 68 -
4.2 Wakefields and Impedances ....................................................................................... - 69 -
4.3 Ballistic bunching .......................................................................................................... - 71 -
4.4 Energy modulation, bunch compression and microbunching in cold neutral
plasma ..................................................................................................................................... - 73 -
4.5 Energy modulation, bunch compression and microbunching in internally
coated metallic tube ............................................................................................................. - 81 -
- 3 -
4.6 Summary .......................................................................................................................... - 88 -
Summary ................................................................................................ - 90 -
Acknowledgements ............................................................................... - 93 -
References ............................................................................................. - 94 -
- 4 -
Introduction
The investigation of new acceleration structures and high frequency, high brightness
coherent radiation sources is an important research area in modern accelerator physics [1-
7]. High brightness and high intensity radiation in ๐๐ป๐ง region opens opportunities of
researches in wide range of areas [8-11]. European XFEL, SwissFEL accelerators are
constructed to obtain such radiation [12, 13]. To emit the ๐๐ป๐ง radiation, one needs to
generate ultrashort, intensive bunches [12, 14-21]. When lengths of microbunches become
smaller than the synchronous mode wavelength (๐๐ง < ๐), the large number of electrons
starts to radiate coherently and the intensity of the radiation field grows quadratically with
the number of particles [22].
The direct generation of sub-ps bunches in RF guns is limited due to technical
characteristics of photocathodes and lasers. Because of that reason, sub-ps bunches are
usually formed by initial long bunches shortening. One of the main principles to obtain
ultrashort bunches is to use SASE process [23, 24], which is widely used in modern
accelerators (FELs) [12, 13]. Also it is possible to use additional external laser field to have
a stronger microbunching []. The main principle to generate ultrashort bunches is to obtain
energy modulation within the bunch and convert it into charge density modulation [19]. For
the high energy bunches magnetic chicanes and for low energy bunches velocity bunching
are used to generate sub-ps bunches [25-29]. Energy modulation within the bunch can be
obtained via bunch interaction with its own radiated wakefield or wakefield emitted from
bunches in front of it []. For such a process disc loaded or dielectric loaded structures are
widely used [30-35]. Both structure types are characterized by the high order modes,
excited during the beam passage through them [35-46]. These high order modes play
parasitic role in particle acceleration processes.
Unlike disc and dielectric loaded structures, in the single mode structures like
plasma and circular waveguide with two-layer metallic walls (ICMT) there are no parasitic
high order modes [47-50]. In the dissertation the microbunching processes are studied in
single mode structure.
- 5 -
Theoretically is shown that ICMT has a single slowly propagating high frequency
mode [40]. At appropriate structure geometry, a charged particle moving along the
structure axis radiates in the terahertz frequency region [50]. While the theoretical study of
ICMT shows that for certain conditions it is a single-mode structure, the experiment on it is
related to some technical difficulties. To serve as a high frequency single-mode structure,
ICMT requires high accuracy, ๐๐ โ sub-๐๐ low conductivity internal coating, and an inner
diameter in the order of several ๐๐ [50], which is technically complex problem. Because
of these problems, it is preferable to do an experiment on a structure which is similar to
ICMT and is more simple in mechanical means. As such a structure, multilayer flat metallic
structure is considered. For appropriate parameters these two structures are similar, and it
is expected that two-layer flat structure should be single-mode, high frequency too.
The thesis consists of introduction, four chapters, summary and bibliography.
In introduction the short review of the actuality of the problems and the main results of the
thesis are given.
In the first chapter of the dissertation is devoted to the study of slow waves in
laminated structures. The study of high frequency accelerating structures is an important
issue for the development of future compact accelerator concepts [51, 52]. The laminated
structures are widely used in advanced accelerators to meet the technical specifications
like high vacuum performance, cure of static charge, reduction of the impedance, etc. [53].
The electrodynamic properties of two-layer structures, based on field matching technique,
have been studied in Refs. [54โ57]. For the internally coated metallic tube (ICMT), with
coating thickness smaller than the skin depth, the longitudinal impedance has a narrow
band resonance at high frequency [57], which is conditioned by the synchronous ๐๐01
fundamental mode [50]. In the first chapter, the peculiarities of high frequency ๐๐01 mode,
slowly propagating in ICMT structure, is analyzed both numerically and analytically. The
TM modes dispersion curves and the ๐๐01 mode attenuation constant are given.
Dispersion relations of flat two-layer structure is investigated. The longitudinal impedance
and the radiation patterns for large aperture ICMT structures are discussed.
The main results of the first chapter are
- 6 -
The single slow propagating mode properties in laminated structures are
studied. It is shown that two-layer cylindrical structure has single slowly traveling
mode (๐๐01) in ๐๐ป๐ง frequency range.
Explicit expression for attenuation constant of two-layer structureโs ๐๐01 mode
is obtained.
Dispersion relations of flat two-layer structure are derived.
The second chapter of the thesis is devoted to study of electrodynamic properties
of multilayer two infinite metallic parallel plates. A bunch traveling through the structure
excites wakefields. The longitudinal component of the wakefield produces extra voltage for
the trailing particles in the bunch. The Fourier transformation of the wake potential for the
point driving charge is an impedance of the structure, which presents the excited
electromagnetic field in frequency domain [37]. For ultrarelativistic particles the impedance
is independent of the beam parameters and can describe a structure in frequency domain.
The field matching technique, based on matching the tangential components of the
electromagnetic fields at the borders of layers, is used to calculate electromagnetic fields.
Matrix formalism is developed to couple electromagnetic field tangential components in
two borders of layers. Fourier transformed electromagnetic fields excited by a relativistic
point charge are obtained analytically for two-layer unbounded two parallel infinite plates.
As a special cases of the above, electromagnetic fields of single-layer unbounded
structure and two-layer structure with perfectly conducting outer layer are obtained.
Longitudinal impedance of two-layer structure with outer perfectly conducting layer, excited
by an ultrarelativistic particle is calculated numerically. It is shown that for low conductivity
inner layer the driving particle radiation has a narrow-band resonance. The resonance
frequency is well described by the formula ๐๐๐๐ =๐
2๐โ
1
๐ ๐, where ๐ is a half-height of a
structure, ๐ is a layer thickness and ๐ is the velocity of light in vacuum. The longitudinal
wake potential is calculated and it is shown that it is a quasi-periodic function with the
period ฮs given by the resonant frequency ฮs =c
๐๐๐๐ .
The main results of the second chapter are
- 7 -
Method of calculating point charge excited electromagnetic fields in multilayer
infinite parallel plates is developed.
Expressions of radiation fields in two-layer parallel plates are derived analytically.
Longitudinal impedance and dispersion relations of symmetrical two-layer
parallel plates with perfectly conducting outer layer is derived.
Comparison of two-layer parallel platesโ electro-dynamic properties and ICMTโs
electro-dynamic properties are performed.
Conditions in which the point charge radiation in two-layer parallel plates has a
narrow-band resonance are obtained.
Wakefields generated by a point charge traveling through the center of the two-
layer parallel plates with outer perfectly conducting material are calculated.
In the third chapter theoretical study of rectangular cavity with laminated walls and
comparison with experimental results is presented. On the second chapter it is shown that
for appropriate parameters two-layered parallel infinite plates have a distinct, narrow band
resonance. In this chapter, a real, limited structure, which is similar to the parallel plates, is
studied. As such a structure a rectangular cavity with laminated horizontal walls is
considered. For the cavity with vertical dimensions much bigger than horizontals, the
electrodynamic characteristics are expected to be similar to the ones for parallel plates. In
the case of resonator, unlike the parallel platesโ case, one has a discrete set of
eigenfrequencies in all three degrees of freedom. To be able to distinguish the main
resonance from secondary ones, finite wall resonator model (perfectly conducting
rectangular cavity with inner low conductivity layers at the top and bottom walls) is
developed and resonances of the model are studied.
Comparison of resonant frequencies of resonator model with resonances of the test
copper cavity with inner germanium layers and parallel two-layer plates is done. Good
agreement between these results is obtained (ฮ๐ ๐๐๐๐ง < 0.03โ ). Experimental results show
the presence of the fundamental resonance which is caused by the two-layer horizontal
cavity walls. The measured fundamental frequencies are close to those calculated for the
- 8 -
two-layer parallel plates, their values are subjected to the laws specific to this kind of
structures: they are decreasing with cavity height increase.
The main results of the third chapter are
The resonance properties of a copper rectangular resonator and a copper
resonator with an internal germanium coating are experimentally investigated
and theoretically substantiated.
Theoretical model of a rectangular cavity with horizontal laminated walls is
created and electro-dynamical properties of the model are studied.
The good correlation between resonance frequencies of the model and test
facility is fixed: all resonances of the test structure corresponds to the modes of
the model.
It is shown that for a resonator with vertical dimensions much bigger than
horizontal ones, the resonance frequencies are in good agreement with
resonances of two-layer infinite parallel plates.
The forth chapter of the dissertation is devoted to the generation of sub-ps bunches
via bunch compression and microbunching processes in single-mode structures.
Wakefields, generated by electron bunch, propagating through a structure, act back on the
bunch particles or the particles traveling behind it producing energy modulation or
transverse kick [36-38, 59-61]. For appropriate parameters, this process can lead to
energy modulation within the bunch, which than can be transformed into charge density
modulation. For non-ultrarelativistic (10 ๐๐๐) electron bunches, the energy modulation
causes velocity modulation and for proper ballistic distance leads to bunch compression or
microbunching.
Single-mode structures, due to absence of high order parasitic modes, are ideal
candidates for this process. Plasma channel and internally coated metallic tube are
observed as an example of single-mode structures. Ballistic bunching method is used: the
bunch is considered rigid in a structure and only energy modulations occur, than in the drift
space the energy modulation leads to charge redistribution.
The main results of the forth chapter are
- 9 -
It is shown that the Gaussian distributed bunch interaction with single mode
structure leads to its compression. For appropriate bunch and structure
parameters 8 and 12 times bunch compression is reached for plasma and ICMT,
respectively.
It is shown that microbunching of non-Gaussian bunches is possible in single
mode structures. For parabolic distributed bunches 20 ๐๐ and 3 ๐๐
microbunches and for rectangular distributed bunches 40 ๐๐ and 8 ๐๐
microbunches can be formed for plasma and ICMT, respectively.
The relations between the bunch length, structures parameters and microbunch
formation distance are analyzed for various bunch shapes. The optimal
parameters for structures and drift space distances are obtained.
The main results of the dissertation are the following:
The single slowly propagating mode properties in laminated structures are
studied.
The electromagnetic fields of multilayer parallel plates are analytically derived.
Dispersion relations of two-layer parallel plates is derived and single resonance
frequency is obtained.
The narrow-band high frequency longitudinal impedance and longitudinal wake
function of two-layer parallel plates are calculated.
Resonance frequencies of rectangular cavity with horizontal double-layer
metallic walls are derived.
Good agreement between theoretical results and experiment is fixed.
It is shown that generation of ultrashort bunches (~10 ๐๐ ) in single-mode
structures is possible via ballistic bunching.
The results of the study have been reported at international conference, during the
seminars at CANDLE Synchrotron Research Institute, Yerevan State University, DESY,
Paul Scherrer Institute and are published in the scientific journals.
- 10 -
Chapter 1: Slow waves in laminated structures
1.1 Introduction
In this chapter the conditions in which slowly propagating free oscillations are
formed in cylindrical and flat two-layer metallic structures are studied. The importance of
establishing the presence of such oscillations is due to the fact that the radiation of a
moving particle in accelerating structures is synchronous (or is synphase) with its motion.
In particular, for an ultrarelativistic particle moving at the speed of light, the phase velocity
of the emitted wave is equal to the speed of light.
As is known, an arbitrary field, generated or propagating in a closed structure can
be expressed with the help of a superposition of the eigen-oscillations of a given structure.
The field generated by a particle must contain only synchronous components of its own
oscillations.
In this chapter, we consider mechanisms of the formation of a synchronous mode in
two-layer cylindrical and planar structures and demonstrate their connection with
longitudinal impedances. In particular, for a plane structure, a relationship between the
eigenvalues of the synchronous mode and the poles of the impedance integral is
established.
1.2 Slow waves in cylindrical structures
Consider a round metallic two-layer hollow pipe of inner radius ๐ with perfectly
conducting outer layer and the metallic inner layer of conductivity ๐ and thickness ๐ = ๐ โ
๐ (Figure 1.1).
- 11 -
Figure 1.1: The geometry of the internally coated metallic tube.
In axially symmetric geometry, the frequency domain non zero electromagnetic field
components (๐ธ๐ง, ๐ธ๐, ๐ป๐) of the axially symmetric monopole TM modes are proportional to
๐๐ฅ๐(๐๐ฝ0๐๐ง โ ๐๐๐ก) , where ๐ฝ0๐ = โ๐2 โ ๐0๐
2 and ๐๐ are the longitudinal and radial
propagation constants, respectively, ๐ = ๐ ๐โ is the wave number, ๐ is the frequency and ๐
is the velocity of light. The dispersion relation for the modes is derived by the matching of
electromagnetic field tangential components (๐ธ๐ง, ๐ป๐) at the layers boundaries. Omitting the
factor ๐๐ฅ๐(๐๐ฝ0๐๐ง โ ๐๐๐ก), the tangential components of electromagnetic fields in vacuum
(๐ < ๐) and metallic wall (๐ โค ๐ โค ๐) regions are given as
๐ธ๐ง(๐, ๐ง) = ๐ด๐ฝ0(๐0๐๐)
๐ป๐(๐, ๐ง) = ๐ด๐๐๐0
๐0๐๐ฝ1(๐0๐๐)
๐ธ๐ง(๐, ๐ง) = ๐ต๐ฝ0(๐0๐๐) + ๐ถ๐ป0(1)(๐0๐๐)
๐ป๐(๐, ๐ง) =๐๐๐1
๐0๐[๐ต๐ฝ1(๐0๐๐) + ๐ถ๐ป1
(1)(๐0๐๐)]
for
๐ < ๐
๐ โค ๐ โค ๐
(1.1)
where ๐0๐ = โ๐0๐2 โ ๐2, ๐ = ๐ โ2 ๐ฟ
โ is the radial propagation constant in metallic internal
layer, ๐ฟ = ๐ (2ํ0
๐๐โ ) is the skin depth, ํ0 and ํ1 = ํ0 +๐๐๐โ are the vacuum and metal
dielectric constants, respectively, ๐ฝ0,1(๐ฅ), ๐ป0,1(1)(๐ฅ) are the zero and first order Bessel and
Hankel functions respectively.
The unknown coefficients ๐ด, ๐ต, ๐ถ are defined by the matching of tangential
components (๐ธ๐ง , ๐ป๐ ) at (๐ = ๐) and the vanishing of the electric field ๐ธ๐ง at the perfect
- 12 -
conducting outer wall (๐ = ๐). Non-zero solution is provided by zero determinant of three
linear algebraic equations system, which in its turn defines the eigenvalue equation.
๐0๐ฝ1(๐0๐๐)
๐1๐ฝ0(๐0๐๐)=๐0๐๐ฝ1(๐0๐๐)๐ป0
(1)(๐0๐๐)โ๐ฝ0(๐0๐๐)๐ป1
(1)(๐0๐๐)
๐0๐๐ฝ0(๐0๐๐)๐ป0(1)(๐0๐๐)โ๐ฝ0(๐0๐๐)๐ป0
(1)(๐0๐๐)
(1.2)
In high frequency range, where the inequalities |๐| โซ |๐0๐| (|๐0๐| โโ2
๐ฟโ ), |๐0๐|๐ โซ
1 and ๐ฟ โช ๐ are hold, Eq. 1.2 is modified to
๐0๐ฝ1(๐0๐๐)
๐0๐๐ฝ0(๐0๐๐)=
๐1
๐0๐cot(๐0๐๐) (1.3)
For the inner metallic layer thickness ๐, less than the skin depth ๐ฟ (๐ โช ๐ฟ), and not
very high frequencies ๐ โช ๐ํ0โ (ํ1 โ
๐๐๐โ ), the Eq. 1.3 is expressed as
1
๐0๐๐
๐ฝ1(๐0๐๐)
๐ฝ0(๐0๐๐)=
1
๐2๐๐ (1.4)
In the frequency range of interest (THz), the inequality ๐ โช ๐ํ0โ holds well for
practically all metals (for copper, e.g. ๐ ํ0โ โ 105 ๐๐ป). The frequency range of applicability
for expression (1.4) is given as ๐ โช ๐ฟ โช ๐.
The analysis of the Eq. 1.4 shows that the radial propagation constants ๐0๐ of high
๐๐0๐ (๐ > 1) modes are purely real and nonzero. Therefore, the high order modes phase
velocity ๐๐โ =๐๐๐ฝ0๐โ is higher than velocity of light. For fundamental ๐๐01 mode, the
parameter ๐01 vanishes (๐01 = 0) at frequency ๐0 = โ2/๐๐ (๐๐โ = ๐), ๐01 is purely real
for frequencies ๐ < ๐0 (๐๐โ > ๐) and ๐01 is purely imaginary for frequencies ๐ > ๐0 (๐๐โ <
๐). Thus, the structure under consideration is characterized by a single slowly propagating
๐๐01mode at high frequency. Figure 1.2 shows the dispersion curves for the fundamental
๐๐01 and high order ๐๐0๐ (๐ > 1) modes. As is seen, only the fundamental mode crosses
the synchronous line ๐ฝ = ๐ (๐๐โ = ๐) at a resonant frequency of ๐ = ๐0 = โ2/๐๐.
- 13 -
Figure 1.2: Dispersion curves for fundamental ๐๐01 and high order ๐๐0๐ (๐ > 1) modes.
1.3 Attenuation parameter
To evaluate the ๐๐01 mode attenuation constant ๐ผ01 = ๐ผ๐(๐ฝ01) for thin inner layer
(๐ โช ๐ฟ), the second term in expansion of formula 1.3 should be taken into account, i.e.
cot(๐) โ 1 ๐โ โ๐3โ ,
๐ฝ1(๐01๐)
๐01๐๐ฝ0(๐01๐)=
๐02
2๐2โ ๐
๐
๐๐ (1.5)
where ํ = (๐ + ๐โ1)/โ3 , ๐ = ๐0๐๐/โ3 . The numerical solutions for real and
imaginary parts of ๐๐01 mode transverse propagation constant are presented on Figure
1.3 (๐ ๐โ = 10โ3 , ๐ = 1 ). For small arguments of Bessel functions |๐01|๐ โช 1 one can
obtain the following approximate expression
๐012 โ
16
๐2(โ
1
2+
๐02
2๐2โ ๐
๐
๐๐) (1.6)
Note, that the real and imaginary parts of transverse propagation constant ๐01 at
the resonance frequency ๐ = ๐0 = โ2๐๐โ are given as
๐ ๐(๐01) = โ๐ผ๐(๐01) =1
๐โ8๐
๐0๐ (1.7)
- 14 -
and the relation |๐01|๐ โช 1 well hold around the resonance frequency. For high frequency
๐ โซ |๐01|, the longitudinal propagation constant ๐ฝ01 = โ๐2 โ ๐012 is given as
๐ฝ โ ๐ +4
๐3๐2(๐2 โ ๐0
2) +8๐
๐(๐๐)2 (1.8)
The attenuation constant at resonance frequency (๐ = ๐0) is then given as ๐ผ01 =
4๐ํ/๐2.
Figure 1.3: Real (solid) and imaginary (dashed) parts of ๐๐01 mode transverse eigenvalue ๐ =
๐01๐.
1.4 Longitudinal impedance and radiation pattern
Consider a relativistic charge traveling along the axis of a round metallic two-layer
hollow pipe (Fig. 1.1). The excited nonzero electromagnetic field components (๐ธ๐ง, ๐ธ๐, ๐ป๐)
of the axially symmetric monopole mode in the vacuum region can be derived based on
the field matching technique [53]. A good analytical approximation for metallic type two-
layer tube longitudinal impedance can be obtained in high frequency range, when the skin
depths ๐ฟ are much smaller than the tube radius a, i.e. ๐ฟ โช ๐ [57].
๐โฅ0 = ๐
๐0
๐๐๐2(1 +
2
๐
๐
๐0๐๐๐กโ(๐๐))
โ1
(1.9)
- 15 -
where ๐0 = 120๐ ฮฉ is the impedance of free space. For the layer thickness ๐ smaller with
respect to skin depth ๐ฟ (๐ โช ๐ฟ) and not very high frequencies (๐ โช ๐/ํ0) the impedance
can be presented in a form of the impedance of parallel resonance circuit
๐โฅ0 = ๐ [1 + ๐๐ (
๐0
๐โ
๐
๐0)]โ1
(1.10)
where ๐0 = ๐โ2/๐๐ is the resonant frequency, ๐ = ๐0(2๐๐ํ)โ1 is the shunt impedance,
๐ = ๐0๐(2๐ํ)โ1 is the quality factor.
Figure 1.4: Longitudinal impedance for Cu, LCM (solid) and Cu-LCM (dashed) tubes.
Figure 1.4 presents the field matching based exact numerical simulations of
longitudinal impedance for copper (Cu, ๐1 = 5.8 ร 107 ฮฉโ1๐โ1) tube internally coated by
the low conductivity metal (LCM, ๐1 = 5 ร 103 ฮฉโ1๐โ1 ) of ๐ = 1 ๐๐ and ๐ = 0.25 ๐๐
thickness. The radius of tube is ๐ = 1 ๐๐. As is seen, in the transition region for thin LCM
layer the impedance modified to narrow band resonance at the frequency ๐0 = ๐0/2๐,
corresponding to ๐0 = 0.675 ๐๐ป๐ง for ๐ = 1 ๐๐ and ๐0 = 1.35 ๐๐ป๐ง for ๐ = 0.25 ๐๐ . The
impedance has a different nature for single and two-layer tubes. It has a broadband
maximum for the single layer tube and narrow band resonance at high frequency ฯ for
two-layer tube. An important feature is that the resonance is observed when the inner layer
thickness is less than the layer skin depth. Thus, the resonance is conditioned by the
interference of scattered electromagnetic fields from the inner and outer layers.
- 16 -
The radiation pattern at the open end of the waveguide can be evaluated by using
from near to far fields transformation technique.
Figure 1.5: Radiation pattern for frequencies ๐ = 1.35 ๐๐ป๐ง (solid) and ๐ = 4.7 ๐๐ป๐ง (dashed).
Fig. 1.5 presents the angular distribution of the normalized radiation intensity with
respect to waveguide axis ํ = 0 for the internal layer thickness of ๐ = 0.25 ๐๐ and
resonant frequencies ๐ = 4.7 ๐๐ป๐ง (๐ = 1 ๐๐) and ๐ = 1.35 ๐๐ป๐ง (๐ = 1 ๐๐). The radiation
pattern is zero at ํ = 0 and reaches its maximum value at ํ = ๐๐๐๐ ๐๐(2.3โ๐/2๐).
1.4. Slow waves in flat structures
In this section, we determine the dispersion relations that determine the
eigenfrequencies of the electromagnetic oscillations of the structure consisting of two
infinite parallel planes, each of which is covered from the inside by a thin conducting layer.
The metal layers are identical: with a thickness ๐ and a conductivity ๐. The permittivity of
the layers is characterized by the expression ํ = ํโฒ + ๐ ๐ ๐โ , where ํโฒ the static permittivity
of the metal (for most metals ํโฒ = ํ0, where ํ0 is the permittivity of vacuum, for germanium
ํโฒ = 16ํ0) and ๐ is a frequency. The distance between the inner surfaces of the plates 2๐.
The solution for the natural electromagnetic oscillations (eigen-oscillations) of the
structure is sought by the method of partial regions. In the case under consideration, we
can distinguish three regions: the region of the upper (๐ โค ๐ฆ โค ๐ + ๐, ๐ = 1) and lower
(โ๐ โ ๐ โค ๐ฆ โค ๐, ๐ = โ1) metal layer and the vacuum region between the plates (โ๐ โค
- 17 -
๐ฆ โค ๐, ๐ = 0). In the outer ideally conducting layers, the fields and currents are absent.
Since all three regions have finite dimensions along the transverse direction, in each of
them can propagate quasi-plane waves with both damped and increasing amplitudes.
Thus, in each of the three regions, the fields can be written in an identical form. In
particular, electrical components in each of the three regions (๐ = 0, ยฑ1) are written as:
๐ธ๐ฅ,๐ฆ,๐ง(๐)
= ๐ธ1,๐ฅ,๐ฆ,๐ง(๐)
+ ๐ธ2,๐ฅ,๐ฆ,๐ง(๐)
(1.11)
with
๐ธ1,2,๐ฅ,๐ฆ,๐ง(๐) (๐) = โซ โซ ๏ฟฝฬ๏ฟฝ1,2,๐ฅ,๐ฆ,๐ง
(๐)๐๐๐ฅ๐๐๐ง๐๐
โ
โโ
โ
โโ (1.12)
where
๏ฟฝฬ๏ฟฝ1,2,๐ฅ(๐)
= ๐ด1,2,๐ฅ {๐โ (๐๐ฆ
(๐)๐ฆ)
๐ถโ (๐๐ฆ(๐)๐ฆ)} ๐๐(๐๐ฅ๐ฅ+๐๐ง๐งโ๐๐ก)
๏ฟฝฬ๏ฟฝ1,2,๐ฆ(๐)
= ๐ด1,2,๐ฆ {๐ถโ (๐๐ฆ
(๐)๐ฆ)
๐โ (๐๐ฆ(๐)๐ฆ)} ๐๐(๐๐ฅ๐ฅ+๐๐ง๐งโ๐๐ก)
๏ฟฝฬ๏ฟฝ1,2,๐ง(๐)
= ๐ด1,2,๐ง {๐โ (๐๐ฆ
(๐)๐ฆ)
๐ถโ (๐๐ฆ(๐)๐ฆ)} ๐๐(๐๐ฅ๐ฅ+๐๐ง๐งโ๐๐ก)
(1.13)
Here ๐ด1,2,๐ฅ,๐ฆ,๐ง are arbitrary weight coefficients (amplitudes) and ๐ = ๐๐ . The
magnetic field components are determined from electrical field components with the help
of Maxwell equations:
๏ฟฝโโ๏ฟฝ1,2(๐)(๐) =
1
๐๐๐๐๐ก ๏ฟฝโโ๏ฟฝยฑ
(๐)(๐) (1.14)
whence
- 18 -
๏ฟฝฬ๏ฟฝ1,2,๐ฅ(๐) =
1
๐๐(๐๐ฆ
(๐)๐ด1,2,๐ง(๐) โ ๐๐๐ง๐ด1,2,๐ฆ
(๐) ) {๐ถโ(๐๐ฆ
(๐)๐ฆ)
๐โ(๐๐ฆ(๐)๐ฆ)} ๐๐(๐๐ฅ๐ฅ+๐๐ง๐งโ๐๐ก)
๏ฟฝฬ๏ฟฝ1,2,๐ฆ(๐) = โ
1
๐(๐๐ฅ๐ด1,2,๐ง
(๐) โ ๐๐ง๐ด1,2,๐ฅ(๐) ) {
๐โ(๐๐ฆ(๐)๐ฆ)
๐ถโ(๐๐ฆ(๐)๐ฆ)
} ๐๐(๐๐ฅ๐ฅ+๐๐ง๐งโ๐๐ก)
๏ฟฝฬ๏ฟฝ1,2,๐ฅ(๐)
= โ1
๐๐(๐๐ฆ
(๐)๐ด1,2,๐ฅ(๐)
โ ๐๐๐ง๐ด1,2,๐ฆ(๐) ) {
๐ถโ(๐๐ฆ(๐)๐ฆ)
๐โ(๐๐ฆ(๐)๐ฆ)
} ๐๐(๐๐ฅ๐ฅ+๐๐ง๐งโ๐๐ก)
(1.15)
The relationship between the amplitudes ๐ด1,2,๐ง(๐)
and ๐ด1,2,๐ฅ(๐) , ๐ด1,2,๐ฆ
(๐) is determined from
the Maxwell equation
๐๐๐ฃ๏ฟฝโโ๏ฟฝ1,2(๐)= 0 (1.16)
whence
๐ด1,2,๐ง(๐)
=1
๐๐ง(๐๐๐ฆ
(๐)๐ด1,2,๐ฅ(๐)
โ ๐๐ฅ๐ด1,2,๐ฅ(๐)
) (1.17)
To determine the relationship between wave numbers and frequency we use the
identity, which follows from Eq. 1.13
๐๐๐ก ๐๐๐ก๏ฟฝโโ๏ฟฝ1,2(๐)= (๐๐ฅ
2 โ (๐๐ฆ(๐))2+ ๐๐ง
2) ๏ฟฝโโ๏ฟฝ1,2(๐)(๐๐ฅ, ๐๐ง, ๐, ๐) (1.18)
and the equation, which is a consequence of Maxwell's equations:
๐๐๐ก ๐๐๐ก๏ฟฝโโ๏ฟฝยฑ(๐)= ๐2ํ๐
โฒ๐๐โฒ๏ฟฝโโ๏ฟฝยฑ(๐)
(1.19)
with ํ๐โฒ, ๐๐
โฒ relative electric and magnetic permeability in the respective regions (ํยฑ1โฒ = 1 +
๐ ๐ ํ0๐, ๐ยฑ1โฒ = 1โ ; ํ0
โฒ = ๐0โฒ = 1), and ๐ = ๐ ๐โ .
Comparing Eq. 1.18 and Eq. 1.19, we obtain
๐๐ฆ(๐)= โโ๐2ํ๐
โฒ๐๐โฒ + ๐๐ฅ
2 + ๐๐ง2 (1.20)
For inner (vacuum) part of structure
๐๐ฆ(0)= โโ๐2 + ๐๐ฅ
2 + ๐๐ง2 (1.21)
- 19 -
To construct the dispersion relations (relationship between ๐๐ฅ and k) one should set
up a system of linear equations based on fields matching at the boundaries of the structure
areas:
1. ๐ธ๐ฅ(1)= 0, ๐ธ๐ง
(1)= 0 at ๐ฆ = ๐ + ๐
(1.22)
2. ๐ธ๐ฅ(1) = ๐ธ๐ฅ
(0), ๐ธ๐ง(1) = ๐ธ๐ง
(0); ๐ต๐ฅ(1) = ๐ต๐ฅ
(0), ๐ต๐ง(1) = ๐ต๐ง
(0) at ๐ฆ = ๐
3. ๐ธ๐ฅ(โ1) = ๐ธ๐ฅ
(0), ๐ธ๐ง(โ1) = ๐ธ๐ง
(0); ๐ต๐ฅ(โ1) = ๐ต๐ฅ
(0), ๐ต๐ง(โ1) = ๐ต๐ง
(0) at ๐ฆ = โ๐
4. ๐ธ๐ฅ(โ1)
= 0, ๐ธ๐ง(โ1)
= 0 at ๐ฆ = โ๐ โ ๐
As a result, a system of 12 equations containing 12 unknown amplitudes is obtained.
Dispersion relations are determined by equating to zero the determinant of this system,
which takes the form consisting of the product of four factors:
๐ท = โ ๐๐4๐=1 = 0 (1.23)
where
๐1 = ๐๐ฆ(1)๐ถโ (๐๐๐ฆ
(1))๐โ (๐๐๐ฆ(0)) + ๐๐ฆ
(0)๐โ (๐๐๐ฆ(1))๐ถโ (๐๐๐ฆ
(0))
๐2 = ๐๐ฆ(1)๐ถโ (๐๐๐ฆ
(1))๐ถโ (๐๐๐ฆ(0)) + ๐๐ฆ
(0)๐โ (๐๐๐ฆ(1))๐โ (๐๐๐ฆ
(0))
๐3 = ๐๐ฆ(0)(๐๐ฆ(1)2
โ ๐๐ฅ2 โ ๐๐ง
2)๐ถโ (๐๐๐ฆ(1)) ๐โ (๐๐๐ฆ
(0)) + ๐๐ฆ
(1)(๐๐ฆ(0)2
โ ๐๐ฅ2 โ ๐๐ง
2) ๐โ (๐๐๐ฆ(1))๐ถโ (๐๐๐ฆ
(0))
๐4 = ๐๐ฆ(0) (๐๐ฆ
(1)2 โ ๐๐ฅ2 โ ๐๐ง
2)๐ถโ (๐๐๐ฆ(1))๐ถโ (๐๐๐ฆ
(0)) + ๐๐ฆ(1) (๐๐ฆ
(0)2 โ ๐๐ฅ2 โ ๐๐ง
2)๐โ (๐๐๐ฆ(1)) ๐โ (๐๐๐ฆ
(0))
(1.24)
Taking into account ๐๐ฆ(๐) = โโ๐2ํ๐
โฒ๐๐โฒ + ๐๐ฅ2 + ๐๐ง2 from Eq. 1.18, instead of ๐3 and ๐4
we can write
๐3 = โ๐๐ฆ(0)๐2ํ๐
โฒ๐๐โฒ๐ถโ(๐๐๐ฆ
(1))๐โ(๐๐๐ฆ(0)) โ ๐๐ฆ
(1)๐2๐โ(๐๐๐ฆ
(1))๐ถโ(๐๐๐ฆ(0))
๐4 = โ๐๐ฆ(0)๐2ํ๐
โฒ๐๐โฒ๐ถโ(๐๐๐ฆ
(1))๐ถโ(๐๐๐ฆ(0)) โ ๐๐ฆ
(1)๐2๐โ(๐๐๐ฆ(1))๐โ(๐๐๐ฆ
(0)) (1.25)
The equality to zero of each of the factors (๐๐ = 0, ๐ โ 1,2,3,4) determines an
independent system of natural oscillations of the structure.
The radiation generated by a particle is synchronous with its motion. The spectral
components of its radiation field have a phase velocity equal to the speed of light ๐ฃ๐โ =
๐ ๐โ = ๐. In this case, the direction of motion of the particle (direction along the z axis) is
- 20 -
distinguished. In the case of natural electromagnetic oscillations, determined in the
absence of a particle or some other source of radiation, the directions along both
horizontal axes ๏ฟฝโ๏ฟฝ and ๐ง are equivalent. This circumstance manifests itself in a symmetrical
form of the dependence of the frequency ๐ = ๐ ๐โ on the wave numbers ๐๐ฅ and ๐๐ง (1.21):
๐ = โ๐๐ฅ2 + ๐๐ง
2 โ ๐๐ฆ(0)2
(1.26)
The phase velocity of the eigenfunctions of the structure has the form:
๐ฃ๐โ = ๐ ๐๐งโ =๐
๐๐ง๐ =
โ๐๐ฅ2+๐๐ง
2โ๐๐ฆ(0)2
๐๐ง๐ (1.27)
As can be seen from Eq. 1.27, the phase velocity of the natural oscillations is equal
to the velocity of the particle (the speed of light in a vacuum) under condition ๐ = ๐๐ง. The
amplitudes ๐ด1,2,๐ฅ,๐ฆ,๐ง are then written in a form containing the Dirac delta function: ๐ด1,2,๐ฅ,๐ฆ,๐ง =
๏ฟฝฬ๏ฟฝ1,2,๐ฅ,๐ฆ,๐ง๐ฟ(๐ โ ๐๐ง). Taking into account ๐ = ๐๐ง the vertical wave numbers ๐๐ฆ(๐)
will take the
form ๐๐ฆ(๐) = โ๐2(1 โ ํ๐
โฒ๐๐โฒ) + ๐๐ฅ2 + ๐๐ง2 for ๐ = ยฑ1 (in layers) and ๐๐ฆ
(0) = ๐๐ฅ at ๐ = 0 (in the
inner vacuum region)
Thus, for synchronous eigenmodes generated by a particle, instead of (1.24, 1.25),
one can write (by replacing ๐๐ฆ(0)
with ๐๐ฅ)
๐1 = ๐๐ฆ(1)๐ถโ (๐๐๐ฆ
(1)) ๐โ(๐๐๐ฅ) + ๐๐ฅ๐โ (๐๐๐ฆ
(1)) ๐ถโ(๐๐๐ฅ)
๐2 = ๐๐ฆ(1)๐ถโ (๐๐๐ฆ
(1))๐ถโ(๐๐๐ฅ) + ๐๐ฅ๐โ (๐๐๐ฆ
(1)) ๐โ(๐๐๐ฅ)
๐3 = โ๐๐ฅ๐2ํ๐โฒ๐๐โฒ๐ถโ (๐๐๐ฆ
(1)) ๐โ(๐๐๐ฅ) โ ๐๐ฆ
(1)๐2๐โ (๐๐๐ฆ
(1)) ๐ถโ(๐๐๐ฅ)
๐4 = โ๐๐ฅ๐2ํ๐โฒ๐๐โฒ๐ถโ (๐๐๐ฆ
(1))๐ถโ(๐๐๐ฅ) โ ๐๐ฆ
(1)๐2๐โ (๐๐๐ฆ
(1)) ๐โ(๐๐๐ฅ)
(1.28)
Let us compare the obtained expressions 1.28 with the denominators of the
impedance (Eq. 2.45) obtained in Chapter 2 writing them in a form convenient for
comparison
- 21 -
๐ 1๐ถ
๐ 2๐ถ} = ยฑ{
๐๐ฆ(1)
๐๐ฅ๐2ํ๐โฒ๐๐โฒ}๐ถโ(๐๐ฆ
(1)๐)๐ถโ(๐๐ฅ๐1) ยฑ {๐๐ฅ
๐๐ฆ(1)๐2} ๐โ(๐๐ฆ
(1)๐)๐โ(๐๐ฅ๐1)
๐ 1๐
๐ 2๐} = ยฑ{
๐๐ฆ(1)
๐๐ฅ๐2ํ๐โฒ๐๐โฒ}๐ถโ(๐๐ฆ
(1)๐)๐โ(๐๐ฅ๐1) ยฑ {๐๐ฅ
๐๐ฆ(1)๐2} ๐โ(๐๐ฆ
(1)๐)๐ถโ(๐๐ฅ๐1)
(1.29)
The comparison gives a complete coincidence of the expressions 1.28 and 1.29
๐1 = ๐ 1๐, ๐2 = ๐ 1
๐ถ , ๐3 = ๐ 2๐, ๐4 = ๐ 2
๐ถ (1.30)
Thus, the complex roots of the dispersion equations (๐๐ = 0, ๐ = 1,2,3,4) are the
poles of impedance function on the complex plane ๐๐ฅ. When the particle moves along the
symmetry plane of the structure (ฮ๐ฆ = 0), only two dispersion equations are realized: ๐2 =
0 and ๐4 = 0. With a parallel offset of the particle from the horizontal plane of symmetry
(ฮ๐ฆ โ 0), all four dispersion equations are realized: ๐๐ = 0, ๐ = 1,2,3,4.
In the approximation ๐๐ฅ2 โช ๐0๐๐๐ง or ๐๐ฆ
(1) โ โโ๐๐0๐๐๐ง , (in the cylindrical case this
approximation is equivalent to approximation for the permittivity of the metal ํ โ ๐ ๐ ๐โ ,
which is valid for not very large frequencies), the expressions (1.28) or (1.29) become
analytic on the complex plane ๐๐ฅ , which allows us to calculate directly the integral
functions through which the impedance is expressed using the residue theorem [62].
The final form of the second and fourth equations (1.28) is
๐๐ฅ๐โ(๐๐๐ฅ) = โโโ๐๐0๐๐๐ถ๐กโ(๐โโ๐๐0๐๐)
๐โ(๐๐๐ฅ)
๐๐ฅ=โโ๐๐๐0๐
๐2๐ถ๐กโ(๐โโ๐๐๐0๐)
(1.31)
At a fixed frequency, the right-hand sides of Eq. 1.31 are constant (independent of
the wave number ๐๐ฅ to be determined). Equations (1.31) can be expressed through
dimensionless parameters ๏ฟฝฬ๏ฟฝ๐ฅ = ๐๐๐ฅ , ๏ฟฝฬ๏ฟฝ = ๐๐, ๏ฟฝฬ๏ฟฝ = ๐ ๐โ and ๐ = ๐0๐๐:
๐ถ๐กโ(๏ฟฝฬ๏ฟฝ๐ฅ)
๏ฟฝฬ๏ฟฝ๐ฅ= โ๏ฟฝฬ๏ฟฝ
๐กโ(โโ๐๐๏ฟฝฬ๏ฟฝ)
โโ๐๐๏ฟฝฬ๏ฟฝ (1.32a)
๐โ(๏ฟฝฬ๏ฟฝ๐ฅ)
๏ฟฝฬ๏ฟฝ๐ฅ= ๏ฟฝฬ๏ฟฝโโ๐๐๏ฟฝฬ๏ฟฝ ๐ถ๐กโ (โโ๐๐๏ฟฝฬ๏ฟฝ) ๏ฟฝฬ๏ฟฝ2โ (1.32b)
- 22 -
Fig.1.6. Eigenvalues of a planar two-layer structure. Solutions of equation 1.32a (left) and 1.32b
(right) for the different values of parameter ๐ (๐ = 1, 2 ๐๐๐ 5) at a resonance frequency ๏ฟฝฬ๏ฟฝ =
๐๐๐๐ง๏ฟฝฬ๏ฟฝ = โ๐ ๐โ , ๏ฟฝฬ๏ฟฝ = 10โ4.
Fig.1.7. Eigenvalues of a planar two-layer structure. Solutions of equation 1.32a (left) and 1.32b
(right) in the case of ๐ = 1 for the several values of frequency: ๏ฟฝฬ๏ฟฝ = ๐๐๐๐ง๏ฟฝฬ๏ฟฝ = โ๐ ๐โ , ๏ฟฝฬ๏ฟฝ = 0.5โ๐ ๐โ <
๐๐๐๐ง๏ฟฝฬ๏ฟฝ and ๏ฟฝฬ๏ฟฝ = 1.5โ๐ ๐โ > ๐๐๐๐ง๏ฟฝฬ๏ฟฝ; ๏ฟฝฬ๏ฟฝ = 10โ4.
Figures 1.6 and 1.7 show the eigenvalue distributions of the synchronous mode of a
two-layered planar structure. On the horizontal and vertical axes, respectively, the
imaginary and real components of the eigenvalues are plotted on the graphs. The graphs
are the result of the exact solution of Equations (1.32a, b). For small right-hand sides of
equations (1.32), their asymptotic solution can be represented as the sum of the solution
for ideally conducting plates (๐๐ฅ0๐ = ๐ ๐ 2โ + ๐๐(๐ โ 1) for Equation (1.32a) and ๐๐ฅ0๐ = ๐๐๐
for Equation (1.32b), ๐ = 1,2,4, โฆ ) and a small additive ๐ผ๐(๐) , taking into account the
presence of the inner coating:
- 23 -
๐๐ฅ๐(๐) = ๐๐๐ฅ0๐(๐) + ๐ผ๐(๐), ๐ผ๐(๐) = โ๐๐๐(๐)๐0๐(๐)
1ยฑ๐๐(๐) (1.33)
where
๐๐ = ๏ฟฝฬ๏ฟฝ ๐กโ (โโ๐๐๏ฟฝฬ๏ฟฝ) โโ๐๐๏ฟฝฬ๏ฟฝโ , ๐๐ = ๏ฟฝฬ๏ฟฝโโ๐๐๏ฟฝฬ๏ฟฝ ๐ถ๐กโ (โโ๐๐๏ฟฝฬ๏ฟฝ) ๏ฟฝฬ๏ฟฝ2โ (1.34)
The smallness of the parameters ๐๐(๐) (1.34) is conditioned by not only due to the
smallness of the relative thickness of the layer ๏ฟฝฬ๏ฟฝ, but also to the smallness of the other
factors included in this parameter. Thus, for ๏ฟฝฬ๏ฟฝ = 10โ4 and ๐ = 1 at the resonant frequency
๏ฟฝฬ๏ฟฝ = โ๐ ๐โ , the parameter ๐๐ is of the order of unity (โ๐๐โ โ 1).
The eigenvalues ๐๐ฅ๐ (solutions of the equation (1.32a)) within the parameters
shown in Figures 1.6, 1.7 (left), are approximated rather well by the formula (1.33). They
are characterized by imaginary components closed enough to the eigenvalues of the
structure consisting of two parallel ideally conducting planes (๐ผ๐๐๐ฅ๐ โ ๐๐ฅ๐0), and by small
real additions, whose values decrease with increasing parameter ๐ (Fig.1.6, left). We note
that for fixed values of the parameter ๐ , small real corrections increase linearly with
increasing imaginary components (Fig. 1.6, 1,7, left) on x axis and decreases with
increasing frequency (Fig. 1,7, left).
The behavior of sequences of eigenvalues ๐๐ (solutions of equation (1.32ะฑ)) is
much more complicated. Their real components are two or three orders of magnitude
smaller than the corresponding roots of equation (1.32a). For small ๐ (๐ = 1 and ๐ = 2 on
Figure 1.6, right) they decrease exponentially with increasing imaginary components,
deposited on the horizontal axis. With an increase of the parameter ๐, the lower values
begin to increase to a certain limit, and then the increase goes down (๐ = 5 on Figure 1.6,
right). In contrast to the eigenvalues of ๐๐ (solutions of equation (1.32a)), there is a
general increase of the real components of eigenvalues with increasing of parameter ๐
(Figure 1.6, right). The reverse order is also valid for fixed values of ๐: lower frequencies
correspond to smaller values of the real components of the eigenvalues ๐๐. In contrast to
the real values of ๐๐, they increase with increasing frequency (Fig. 1.7, right).
- 24 -
The essential difference between the imaginary components of the eigenvalues ๐๐
and the corresponding values for ideally conducting parallel planes is explained by the
magnitude of the parameter ๐๐ (1.34): its smallness is ensured at very low relative
thicknesses of the metallic layer ๏ฟฝฬ๏ฟฝ (๏ฟฝฬ๏ฟฝ โช 1)or at sufficiently high frequencies (๏ฟฝฬ๏ฟฝ โซ ๏ฟฝฬ๏ฟฝ๐๐๐ง).
1.5 Summary
The first chapter establishes the connection between the radiation of a charged
particle and the synchronous electromagnetic eigenmodes of two-layer cylindrical and flat
metal structures.
For a two-layered metal cylindrical waveguide, it is shown, that a single slowly-
propagated (with phase velocity equal to the speed of light) ๐๐01 mode at certain
frequency (resonant frequency ๐ = โ1 ๐0๐โ , ๐0 inner radius of waveguide, ๐ inner layer
thickness) is in phase with a radiation field of a charged particle and synchronized with the
ultra-relativistic particle, moving along the waveguide axis. This circumstance is clearly
demonstrated in Figure 1.2, where only one dispersion curve corresponding to the above-
mentioned mode intersects a line describing the phase velocity equal to the speed of light.
The important results of this Chapter also include the calculation of the attenuation
coefficient of the synchronous mode for the resonance frequency (Eq. 1.8), which is
defined as the real component of its phase.
The remarkable properties of the phase of the synchronous mode, presented on
Figure 1.3, are also determined.
The second main direction of this chapter is to investigate the mechanism of
synchronization of the electromagnetic eigen-oscillations of a two-layer flat metallic
structure.
Equations for the eigenvalue numbers of free vibrations in the structure under
consideration are obtained. A subsystem of free oscillations, synchronous with a particle,
moving along a rectilinear trajectory parallel to the symmetry plane of structure, is
distinguished from the complete set of free oscillations. Equations for eigenvalues are
obtained both for a complete system of free oscillations and for their synchronous
- 25 -
subsystem. A unique relationship is established between the roots of the equations for
synchronous oscillations and the complex poles of the integrand for the longitudinal
impedance. Based on graphic constructions, the regularities of the behavior of eigenvalues
are analyzed.
Important difference between the mechanisms of formation of synchronous
oscillations for two-layered cylindrical and planar structures should be mentioned. In the
first case, we have a single synchronous mode, with synchronization being achieved only
at the resonant frequency ๐ = โ2 ๐0๐โ . In the second case, there is an infinite set of eigen-
oscillations, synchronous on the entire infinite frequency range. The radiation field of a
particle is formed by the total contribution of the entire infinite sequence of natural
synchronous oscillations of the structure.
As will be shown in the Second Chapter, the result of the total contribution of
synchronous oscillations is radiation with a narrow-band frequency spectrum and a single
resonant frequency ๐ = โ1 ๐๐โ
- 26 -
Chapter 2: Accelerating structures with flat geometry
2.1 Introduction
Impedances and wake functions in multilayer cylindrical structure are studied in [55].
It is shown that under certain conditions there is a narrow-band longitudinal impedance in
the structure. The study of dispersive relations of a two-layer cylindrical structure shows
that in the same conditions the structure becomes a high frequency single slowly (๐ฃ < ๐)
traveling wave structure. Single-mode structures are good candidates both for novel
accelerating structures and as a source of high frequency monochromatic radiation. As it
is shown in previous chapter, a single-mode structure can be used also for bunch
compression and for microbunching. To be a high frequency single-mode structure, it
requires a high accuracy ๐๐ โ sub-๐๐ low conductivity internal coating, and an inner
diameter of order of several ๐๐, which is technically complex problem for structures with
๐๐ diameters.
In this chapter multilayer flat metallic structure is studied as an alternative of the
abovementioned cylindrical structure. Longitudinal impedance and dispersive relations of
the structure are presented, wake fields generated by a particle travelling through the
structure are observed. Analysis of symmetrical and asymmetrical multilayer flat
waveguides are performed. Conditions in which the longitudinal impedance has a narrow-
band resonance are obtained. Comparison of properties of multilayer flat metallic structure
with cylindrical structure are done.
To calculate electromagnetic fields the field matching technique that implies the
continuity of the tangential components of electric and magnetic fields at the borders of
the layers is used for solving Maxwell equation. The matrix formalism described in [55] is
used to couple electromagnetic field tangential components in two borders of a layer.
Longitudinal impedance of the structure and wake fields in it are obtained via Fourier
inverse transformation of electromagnetic fields.
- 27 -
2.2 Multilayer flat structure
We will start with the most general case: structure consists of two parallel multilayer
infinite plates (Figure 2.1). The boundaries between layers are infinite parallel planes. The
number of layers in the top ๐(๐ก) and bottom ๐(๐) plates can generally be unequal (๐(๐ก) โ
๐(๐)). The filling of layers may be both dielectric or metallic and in general is characterized
by a static complex dielectric permittivity ํ๐(๐ข), ๐ = 1,2, . . ๐(๐ข), ๐ข = ๐ก, ๐, magnetic permeability
๐๐(๐ข)
and conductivity ๐๐(๐ข)
. The dielectric constant material layer can be written as ํ๏ฟฝฬ๏ฟฝ(๐ข) =
ํ๐(๐ข) + ๐ ๐๐
(๐ข) ๐โ , where ๐ is frequency. For the majority of highly conductive metal generally
accepted ํ๐(๐ข) = ํ0; for non-magnetic materials ๐๐
(๐ข) = ๐0 (ํ0, ๐0 permittivity and magnetic
permeability of vacuum). The thickness of ๐-th layer is ๐๐(๐ข)
. The distance from the plane of
symmetry of the vacuum chamber to the boundary planes, separating layers are equal to
๐๐(๐ข)
; ๐๐+1(๐ข) = ๐๐
(๐ข) + ๐๐(๐ข), ๐ = 1,2,3, โฆ๐(๐ข) (for upper half-space).The upper and lower
unbounded regions can be filled by an arbitrary medium, electromagnetic properties of
which are characterized by parameters ํ๏ฟฝฬ๏ฟฝ(๐ข)+1
(๐ข)= ํ
๐(๐ข)+1
(๐ข)+ ๐ ๐
๐(๐ข)+1
(๐ข)๐โ , ๐
๐(๐ข)+1
(๐ข)= ๐0.
The point-like charge is traveling through layered metallic tube with velocity ๐
parallel to the ๐ง-axis at vertical offset ฮ.
Figure 2.1: Geometry of multilayer flat metallic structure.
- 28 -
Maxwellโs equations must be solved to calculate EM fields excited by the point
charge. Cartesian coordinates ๐ = (๐ฅ, ๐ฆ, ๐ง) are used, where ๐ฆ is a direction of vertical axis.
For (๐(๐ก), ๐(๐)) layered tube there are ๐(๐ก) + ๐(๐) + 3 regions (including inner region and
two outer regions of the structure) and ๐(๐ก) + ๐(๐) + 2 borders. The complete number of
matching equations is 4(๐(๐ก) + ๐(๐) + 2) with 4(๐(๐ก) + ๐(๐) + 2) unknown coefficients as in
each border there are four boundary conditions that match tangential (๐ธ๐ฅ, ๐ธ๐ง , ๐ต๐ฅ, ๐ต๐ง)
components of electromagnetic fields.
Radiation fields are searched by the method of partial areas [64]. The geometry of
the problem divides into three regions where the fields are presented in different ways:
charge existing region โ๐โ1 โค ๐ฆ โค ๐1, metallic layers of the structure โ๐โn(๐) < ๐ฆ < โ๐โ1
and ๐1 < ๐ฆ < ๐n(๐ก) and outer vacuum region ๐ฆ < โaโn(๐) and ๐ฆ > an(๐ก). In all regions the
solution is sought in the form of the general solution of the homogeneous Maxwell
equations except for the area between the plates, which, to the general solution of the
homogeneous Maxwell equations is added a particular solution of inhomogeneous
Maxwell's equations.
Electromagnetic fields and point charge are presented via Fourier transformation as
๐ธ(๐ฅ, ๐ฆ, ๐ง) = โซ ๏ฟฝฬ๏ฟฝ(๐๐ฅ, ๐ฆ, ๐๐ง)๐๐๐๐ฅ๐ฅ๐๐๐(๐งโ๐๐ก)๐๐๐ฅ๐๐๐ง
โ
โโ
๐(๐ฅ, ๐ฆ, ๐ง) = โซ ๏ฟฝฬ๏ฟฝ(๐๐ฅ, ๐ฆ, ๐๐ง)๐๐๐๐ฅ๐ฅ๐๐๐๐ง๐ง๐๐๐ฅ๐๐๐ง
โ
โโ
(2.1)
where ๐ = ๐ ๐กโ is the wave number.
Boundary conditions states that tangential components of full electric and magnetic
fields should be continues at layers boundaries.
E๐ฅ๐ (ยฑ๐๐) = E๐ฅ
๐+1(ยฑ๐๐)
E๐ง๐ (ยฑ๐๐) = E๐ง
๐+1(ยฑ๐๐)
B๐ฅ๐ (ยฑ๐๐) = B๐ฅ
๐+1(ยฑ๐๐)
B๐ง๐ (ยฑ๐๐) = B๐ง
๐+1(ยฑ๐๐)
(2.2)
where + and โ signs corresponds for layer boundaries in the upper and lower regions,
respectively.
- 29 -
An electric charge traveling in the structure produces electromagnetic field which
can be presented as a sum of two parts
๐ฌ๐๐๐ = ๐ฌ๐ + ๐ฌ, ๐ฉ๐๐๐ = ๐ฉ๐ +๐ฉ (2.3)
where ๐ฌ๐ฉ, ๐ฉ๐ฉ are electric and magnetic fields in the case of perfectly conducting plates and
๐ฌ,๐ฉ are solutions of source free Maxwellโs equations with a metallic current ๐ = ๐๐ฌ. The
last onesโ dependence on the source particles comes only with boundary conditions.
Firstly electric ๐ฌ๐ฉ and magnetic ๐ฉ๐ฉ fields in the perfectly conducting parallel plates
with a static charge in the structure are calculated, then fields when there is a charge
traveling in the direction of structure axis can be obtained via Lorentz transformation [64].
Maxwellโs equations for the structure with a static charge ๐ in it are
โ โ ๐ฌ๐ =๐
๐0
โ โ ๐ฉ๐ = 0
โ ร ๐ฌ๐ = 0
โ ร ๐ฉ๐ = 0
(2.4)
where โ is a nabla operator.
Using Fourier transformation in Maxwellโs equations and expressing electric field ๐ฅ
and ๐ฆ components via ๐ง component, the wave equation for a point charge can be derived
๐2๐ธ๐๐ง
๐๐ฆ2โ ๐ผ2๐ธ๐๐ง =
๐
4๐2๐0๐๐๐ง๐ฟ(๐ฆ โ ฮ) (2.5)
where ๐ผ = โ๐๐ฅ2 + ๐๐ง
2 and ฮ is the point charge deviation from the center of ๐ฆ axis.
The solution of Eq. 2.5 is a sum of homogenous equation solution (๏ฟฝฬ๏ฟฝ๐ง,โ) and a
particular solution (๏ฟฝฬ๏ฟฝ๐ง,๐).
Presenting the solution of homogenous part as a sum of hyperbolic cosine and sine
with unknown coefficients and finding a particular solution by integrating two parts of the
wave equation, its full solution can be obtained.
๐ธ๐๐งยฑ(๐๐ฅ, ๐ฆ, ๐๐ง) = โ
๐๐๐๐ง
4๐2๐ผ๐0
sinh(๐ผ(๐ยฑฮ)) sinh(๐ผ(๐โ๐ฆ))
sinh(2๐ผ๐) (2.6)
- 30 -
where + sign stand for an electric field in upper part of the structure (๐ฆ > ฮ ) and โ for an
electric field in lower part (๐ฆ < ฮ ).
๐ฅ and ๐ฆ component of the electric field can be obtained via Faradayโs low and
magnetic field is zero for a static charge.
๐ธ๐๐ฅยฑ(๐๐ฅ, ๐ฆ, ๐๐ง) = โ
๐๐๐๐ฅ
4๐2๐ผ๐0
๐ ๐๐โ(๐ผ(๐โ๐ฆ)) ๐ ๐๐โ(๐ผ(๐ยฑ๐ฅ))
๐ ๐๐โ(2๐ผ๐)
๐ธ๐๐ฆยฑ(๐๐ฅ, ๐ฆ, ๐๐ง) = ยฑ
๐
4๐2๐0
๐๐๐ (๐ผ(๐โ๐ฆ)) ๐ ๐๐โ(๐ผ(๐ยฑ๐ฅ))
๐ ๐๐โ(2๐ผ๐)
(2.7)
The Fourier components of the electromagnetic fields of a moving charge can be
obtained via Lorentz transformation
๐ธ๐๐งยฑ(๐๐ฅ, ๐ฆ, ๐๐ง) = โ
1
๐พ2๐๐๐๐ง
4๐2๐๐ฅ๐0
sinh(๐๐ฅ(๐ยฑฮ)) sinh(๐๐ฅ(๐โ๐ฆ))
sinh(2๐๐ฅ๐)
๐ธ๐๐ฅยฑ(๐๐ฅ, ๐ฆ, ๐๐ง) = ๐พ
2 ๐๐ฅ
๐๐ง๐ธ๐๐งยฑ (๐๐ฅ, ๐ฆ, ๐๐ง)
๐ธ๐๐ฆยฑ(๐๐ฅ, ๐ฆ, ๐๐ง) = ยฑ๐พ
2๐๐๐ฅ
๐๐งcoth(๐๐ฅ(๐ โ ๐ฆ))๐ธ๐๐ง
ยฑ (๐๐ฅ, ๐ฆ, ๐๐ง)
๐ต๐๐งยฑ(๐ฅ, ๐ฆ, ๐ง) = 0
๐ต๐๐ฅยฑ(๐๐ฅ, ๐ฆ, ๐๐ง) =
๐ฃ
๐2๐ธ๐๐ฆยฑ (๐๐ฅ, ๐ฆ, ๐๐ง)
๐ต๐๐ฆยฑ(๐๐ฅ, ๐ฆ, ๐๐ง) = โ
๐ฃ
๐2๐ธ๐๐ฅยฑ (๐๐ฅ, ๐ฆ, ๐๐ง)
(2.8)
note that variable of integration is replaced by ๐๐ง โ ๐๐งโฒ ๐พโ and ๐ผ = โ๐๐ฅ
2 + (๐๐งโฒ
๐พ)2
โ ๐๐ฅ.
We write fields in Eq. 2.8 in the form of plane waves
๐ธ๐๐ฅ,๐ฆ,๐ง = ๐ธ1,๐๐ฅ,๐ฆ,๐ง + ๐ธ2,๐๐ฅ,๐ฆ,๐ง = ๐ด1,๐ฅ,๐ฆ,๐ง(ยฑ)
๐(โ)๐๐๐โ๐๐ฆ + ๐ด2,๐ฅ,๐ฆ,๐ง(ยฑ)
๐(ยฑ)๐๐๐โ๐๐ฆ
๐ต๐๐ฅ,๐ฆ,๐ง = ๐ต1,๐๐ฅ,๐ฆ,๐ง + ๐ต2,๐๐ฅ,๐ฆ,๐ง = ๐ต1,๐ฅ,๐ฆ,๐ง(ยฑ)
๐(โ)๐๐๐โ๐๐ฆ + ๐ต2,๐ฅ,๐ฆ,๐ง(ยฑ)
๐(ยฑ)๐๐๐โ๐๐ฆ (2.9)
where
๐ด1,๐ง(ยฑ)= โ๐๐พโ2
๐๐๐ง
8๐2๐0๐cosh(2๐๐) sinh(๐(๐ ยฑ โ)) ๐ด2,๐ง
(ยฑ)= โ๐ด1,,๐ง
(ยฑ)
๐ด1,๐ฅ(ยฑ)= โ๐
๐๐๐ฅ
8๐2๐0๐๐ถ๐ โ(2๐๐)๐โ{๐(๐ ยฑ โ)} = ๐พ2๐ด1,๐ง
(ยฑ)๐ด2,๐ง(ยฑ)= โ๐ด2,๐ง
(ยฑ)
๐ด1,๐ฆ(ยฑ)= ยฑ
๐
8๐2๐0๐ถ๐ โ(2๐๐)๐โ{๐(๐ ยฑ โ)} ๐ด2,๐ฆ
(ยฑ)= ๐ด2,,๐ฆ
(ยฑ)
(2.10)
On the metallic surfaces the electromagnetic field components are
- 31 -
๐ธ๐๐ฅ(๐๐ฅ, ยฑ๐, ๐๐ง) = ๐ธ๐๐ง
(๐๐ฅ, ยฑ๐, ๐๐ง) = 0
๐ธ๐๐ฆ(๐๐ฅ, ยฑ๐, ๐๐ง) =
๐
4๐2๐0
sinh(๐๐ฅ(ฮยฑ๐))
sinh(2๐๐ฅ๐)
๐ต๐๐ฅ(๐๐ฅ, ยฑ๐, ๐๐ง) =
๐
4๐2๐0
๐ฃ
๐2
sinh(๐๐ฅ(ฮยฑ๐))
sinh(2๐๐ฅ๐)
๐ต๐๐ฆ(๐๐ฅ, ยฑ๐, ๐๐ง) = ๐ต๐๐ง
(๐๐ฅ, ยฑ๐, ๐๐ง) = 0
(2.11)
The only non-zero tangential component at the boundaries ๐ฆ = ยฑ๐ is a component
๐ต๐๐ฅ, which in ultrarelativistic approximation (๐ฃ โ ๐) is
๐ต๐๐ฅ(๐๐ฅ, ยฑ๐, ๐๐ง) =
๐
4๐2๐0๐
sinh(๐๐ฅ(ฮยฑ๐))
sinh(2๐๐ฅ๐) (2.12)
As it was mentioned, the total electromagnetic field excited by a point charge
traveling in the multilayer structure was presented as a sum of two parts. Now the second
part of total electromagnetic field will be calculated, i.e. solutions of the source free
Maxwellโs equations with a metallic current ๐ = ๐๐ฌ. The solution inside layers is sought in
the form of the general solution of the homogenous Maxwellโs equations.
๐ฌ(๐)(๐ฅ, ๐ฆ, ๐ง) = ๐ฌ1(๐)(๐ฅ, ๐ฆ, ๐ง) + ๐ฌ2
(๐)(๐ฅ, ๐ฆ, ๐ง)
๐ฉ(๐)(๐ฅ, ๐ฆ, ๐ง) = ๐ฉ1(๐)(๐ฅ, ๐ฆ, ๐ง) + ๐ฉ2
(๐)(๐ฅ, ๐ฆ, ๐ง) (2.13)
where
๐ฌ1,2(๐)(๐ฅ, ๐ฆ, ๐ง) = โซ โซ ๐ฌ1,2
(๐)(๐๐ฅ, ๐ฆ, ๐๐ง)๐๐(๐๐ฅ๐ฅ+๐๐ง(๐งโ๐ฃ๐ก))
โ
โโ๐๐๐ฅ๐๐
โ
โโ
๐ฉ1,2(๐)(๐ฅ, ๐ฆ, ๐ง) = โซ โซ ๐ฉ1,2
(๐)(๐๐ฅ, ๐ฆ, ๐๐ง)๐๐(๐๐ฅ๐ฅ+๐๐ง(๐งโ๐ฃ๐ก))
โ
โโ๐๐๐ฅ๐๐
โ
โโ
(2.14)
with
๐ฌ1,2(๐)(๐๐ฅ, ๐ฆ, ๐๐ง) = ๐จ1,2
(๐)๐โ๐๐ฆ
(๐)๐ฆ
๐ฉ1,2(๐)(๐๐ฅ, ๐ฆ, ๐๐ง) =
๐โ๐(๐๐ฅ๐ฅ+๐๐ง(๐งโ๐ฃ๐ก))
๐๐๐ง๐ฃ๐๐๐ก [๐ฌ1,2
(๐)(๐๐ฅ, ๐ฆ, ๐๐ง)๐๐(๐๐ฅ๐ฅ+๐๐ง(๐งโ๐ฃ๐ก))]
(2.15)
For the inner region (โ๐โ1 < ๐ฆ < ๐1) we should add the partial solution (๐ฌ๐, ๐ฉ๐) of
inhomogeneous Maxwellโs equation (Eq. 2.8) to the general solution of homogeneous
Maxwellโs equation
- 32 -
๐ฌ(0)(๐ฅ, ๐ฆ, ๐ง) = ๐ฌ1(๐)(๐ฅ, ๐ฆ, ๐ง) + ๐ฌ2
(๐)(๐ฅ, ๐ฆ, ๐ง) + ๐ฌ๐(๐ฅ, ๐ฆ, ๐ง)
๐ฉ(0)(๐ฅ, ๐ฆ, ๐ง) = ๐ฉ1(๐)(๐ฅ, ๐ฆ, ๐ง) + ๐ฉ2
(๐)(๐ฅ, ๐ฆ, ๐ง) + ๐ฉ๐(๐ฅ, ๐ฆ, ๐ง) (2.16)
Finally, for the upper and lower outer regions fields are sought in the form of
vanishing at infinity (at ๐ฆ โ ยฑโ)
๐ฌ(๐(๐ก)+1)(๐ฅ, ๐ฆ, ๐ง) = ๐ฌ1
(๐(๐ก)+1)(๐ฅ, ๐ฆ, ๐ง)
๐ฉ(๐(๐ก)+1)(๐ฅ, ๐ฆ, ๐ง) = ๐ฉ1
(๐(๐ก)+1)(๐ฅ, ๐ฆ, ๐ง)
๐ฌ(โ๐(๐)โ1)(๐ฅ, ๐ฆ, ๐ง) = ๐ฌ2
(โ๐(๐)โ1)(๐ฅ, ๐ฆ, ๐ง)
๐ฉ(โ๐(๐)โ1)(๐ฅ, ๐ฆ, ๐ง) = ๐ฉ2
(โ๐(๐)โ1)(๐ฅ, ๐ฆ, ๐ง)
(2.17)
for top (๐ฆ > ๐๐(๐ก)+1) and bottom (๐ฆ < โ๐โ๐(๐)โ1) regions, respectively.
As it is mentioned, we obtain a system of 4(๐ +๐ + 2) equations for the same
number of unknown amplitudes. Amplitudes ๐ด๐ง,1๐ and ๐ด๐ง,2
๐ can be presented by ๐ด๐ฅ,1๐ , ๐ด๐ฆ,1
๐
and ๐ด๐ฅ,2๐ , ๐ด๐ฆ,2
๐ ๐ด๐ฅ,1 via Gaussโs low
๐ด๐ง,1๐ = โ
๐๐ฅ๐ด๐ฅ,1๐ +๐๐๐ฆ
๐ ๐ด๐ฆ,1๐
๐๐ง
๐ด๐ง,2๐ = โ
๐๐ฅ๐ด๐ฅ,2๐ โ๐๐๐ฆ
๐ ๐ด๐ฆ,2๐
๐๐ง
(2.18)
and magnetic field components can be derived via Faradayโs low.
To determine the electromagnetic field we will use the field matching technique that
implies the continuity of the tangential components of electromagnetic fields at the borders
of layers (๐ฆ = ยฑ๐๐). We are interested in tangential components of fields, so we will
introduce the tangential field vector ๏ฟฝฬ๏ฟฝ๐(๐ธ๐ฅ๐, ๐ธ๐ง
๐ , c๐ต๐ฅ๐ , c๐ต๐ง
๐). We will use the matrix approach
introduced in [63] to obtain electromagnetic fields in the vacuum region.
Connection between the components of the fields and amplitudes is realized by
using the basic transformation matrixes
- 33 -
๏ฟฝฬ๏ฟฝ1(๐)(๐ฆ) =
๐โ๐๐ฆ๐ ๐ฆ
๐๐ง2
(
๐๐ง2 0 0 0
โ๐๐ฅ๐๐ง โ๐๐๐ฆ๐ ๐๐ง 0 0
โ๐๐๐ฆ๐ ๐๐ฅ ๐๐ฆ
๐ 2 โ ๐๐ง2 0 0
โ๐๐๐ฆ๐ ๐๐ง ๐๐ฅ๐๐ง 0 0)
(2.19)
๏ฟฝฬ๏ฟฝ2(๐)(๐ฆ) =
๐๐๐ฆ๐ ๐ฆ
๐๐ง2
(
0 0 ๐๐ง2 0
0 0 โ๐๐ฅ๐๐ง โ๐๐๐ฆ๐ ๐๐ง
0 0 ๐๐๐ฆ๐ ๐๐ฅ ๐๐ฆ
๐ 2 โ ๐๐ง2
0 0 โ๐๐๐ฆ๐ ๐๐ง ๐๐ฅ๐๐ง )
(2.20)
๏ฟฝฬ๏ฟฝ(๐)(๐ฆ) = ๏ฟฝฬ๏ฟฝ1(๐)(๐ฆ)+๏ฟฝฬ๏ฟฝ2
(๐)(๐ฆ) (2.21)
where ๐๐ฆ(๐) = ๐๐ฅ
2 + (1 โ ํ(๐)๐(๐)๐ฃ2)๐๐ง2.
Let's establish a connection between the values of the fields on the lower borders of
the neighboring layers. The fields on the upper and lower borders of the ๐-th layer of the
upper half-space can be expressed in terms of the values of the same amplitudes:
๏ฟฝฬ๏ฟฝ๐(๐๐) = ๏ฟฝฬ๏ฟฝ1๐(๐๐)๏ฟฝฬ๏ฟฝ
๐
๏ฟฝฬ๏ฟฝ๐(๐๐+1) = ๏ฟฝฬ๏ฟฝ1๐(๐๐+1)๏ฟฝฬ๏ฟฝ
๐โ ๏ฟฝฬ๏ฟฝ๐(๐๐+1) = ๏ฟฝฬ๏ฟฝ1
๐(๐๐+1)๏ฟฝฬ๏ฟฝ1๐(๐๐)
โ1๏ฟฝฬ๏ฟฝ๐(๐๐) (2.22)
From field tangential components equality on layer borders (๏ฟฝฬ๏ฟฝ๐+1(๐๐+1) = ๏ฟฝฬ๏ฟฝ๐(๐๐+1))
one obtains
๏ฟฝฬ๏ฟฝ๐+1(๐๐+1) = ๏ฟฝฬ๏ฟฝ1๐(๐๐+1)๏ฟฝฬ๏ฟฝ1
๐(๐๐)โ1๏ฟฝฬ๏ฟฝ๐(๐๐) (2.23)
Consistent application of (2.23) for upper half-spaces gives
๏ฟฝฬ๏ฟฝ๐(๐ก)+1(๐๐(๐ก)+1) = ๐
(๐ก)๏ฟฝฬ๏ฟฝ0(๐1) (2.24)
where ๐(๐ก) = โ ๐๐(๐ก)๐(๐ก)
๐=1 and ๐๐(๐ก) = ๏ฟฝฬ๏ฟฝ(๐)(๐๐+1)๏ฟฝฬ๏ฟฝ
(๐)(๐๐)โ1.
Similar relation can be obtained for lower half-space
๏ฟฝฬ๏ฟฝ(โ(๐(๐)+1))
(โ๐โ(๐(๐)+1)) = ๐(๐)๏ฟฝฬ๏ฟฝ(0)(โ๐โ1) (2.25)
where ๐(๐) = โ ๐๐(๐)๐(๐)
๐=1 , ๐๐(๐) = ๏ฟฝฬ๏ฟฝ(โ๐)(โ๐โ(๐+1))๏ฟฝฬ๏ฟฝ
(โ๐)(โ๐โ๐)โ1.
- 34 -
For the symmetrical case ๐๐(๐) = ๐๐
(๐ก)โ1 and, respectively, ๐(๐) = โ ๐๐
(๐ก)โ1๐๐=1 .
Given the matching conditions on the borders of the plates, we have finally
๏ฟฝฬ๏ฟฝ1(๐(๐ก)+1)
(๐๐(๐ก)+1)๏ฟฝฬ๏ฟฝ(๐ก๐) = ๐๐ก[๏ฟฝฬ๏ฟฝ(0)(๐1)๏ฟฝฬ๏ฟฝ
(0) + ๏ฟฝฬ๏ฟฝ(๐ก)] (2.26)
๏ฟฝฬ๏ฟฝ2(โ(๐(๐)+1))
(โ๐โ(๐(๐)+1)) ๏ฟฝฬ๏ฟฝ(๐ก๐) = ๐๐[๏ฟฝฬ๏ฟฝ(0)(โ๐โ1)๏ฟฝฬ๏ฟฝ
(0) + ๏ฟฝฬ๏ฟฝ(๐)] (2.27)
where ๏ฟฝฬ๏ฟฝ(๐ก,๐) and ๏ฟฝฬ๏ฟฝ(๐ก,๐) are the single-column matrixes
๏ฟฝฬ๏ฟฝ(๐ก,๐) = {{0}, {0}, {2๐๐ฆ(0)๐ฃ csch (2๐๐ฆ
(0)๐) sinh [๐๐ฆ
(0)(โ ยฑ ๐1)]} , {0}}
๏ฟฝฬ๏ฟฝ(๐ก๐) = {{๐ด๐ฅ,1(๐(๐ก)+1)
} , {๐ด๐ฆ,1(๐(๐ก)+1)
} , {๐ด๐ฅ,2(โ(๐(๐)+1))
} , {๐ด๐ฆ,2(โ(๐(๐)+1))
}} (2.28)
On the Eq. 2.9 and Eq. 2.10 condition that fields are vanishing on ๐ฆ = ยฑโ is taken
into account.
The amplitudes ๏ฟฝฬ๏ฟฝ(0) may be expressed in terms of amplitudes ๏ฟฝฬ๏ฟฝ(๐ก๐) using Eq. 2.26
and Eq. 2.27
๏ฟฝฬ๏ฟฝ(0) = ๏ฟฝฬ๏ฟฝ๐ก๏ฟฝฬ๏ฟฝ(๐ก๐) โ ๏ฟฝฬ๏ฟฝ(0)(๐1)โ1๏ฟฝฬ๏ฟฝ(๐ก)
๏ฟฝฬ๏ฟฝ(0) = ๏ฟฝฬ๏ฟฝ๐๏ฟฝฬ๏ฟฝ(๐ก๐) โ ๏ฟฝฬ๏ฟฝ(0)(โ๐โ1)โ1๏ฟฝฬ๏ฟฝ(๐)
(2.29)
where
๏ฟฝฬ๏ฟฝ๐ก = ๏ฟฝฬ๏ฟฝ(0)(๐1)โ1๐๐ก
โ1๏ฟฝฬ๏ฟฝ1(๐(๐ก)+1)
(๐๐(๐ก)+1)
๏ฟฝฬ๏ฟฝ๐ = ๏ฟฝฬ๏ฟฝ(0)(โ๐โ1)โ1๐๐
โ1๏ฟฝฬ๏ฟฝ2(โ(๐(๐)+1))
(โ๐โ(๐(๐)+1)) (2.30)
Eliminating the outer layer coefficients we obtain explicit expressions for the
amplitudes of the radiation field of the particle in the space between the plates
๏ฟฝฬ๏ฟฝ(0) =1
2(๏ฟฝฬ๏ฟฝ(+)๏ฟฝฬ๏ฟฝ(โ)
โ1๏ฟฝฬ๏ฟฝ(โ) โ ๏ฟฝฬ๏ฟฝ(+)) (2.31)
with
๏ฟฝฬ๏ฟฝ(ยฑ) = ๏ฟฝฬ๏ฟฝ๐ก ยฑ ๏ฟฝฬ๏ฟฝ๐, ๏ฟฝฬ๏ฟฝ(ยฑ) = ๏ฟฝฬ๏ฟฝ(0)(๐1)โ1๏ฟฝฬ๏ฟฝ(๐ก) ยฑ ๏ฟฝฬ๏ฟฝ(0)(โ๐โ1)
โ1๏ฟฝฬ๏ฟฝ(๐) (2.32)
- 35 -
Amplitudes of the radiation field for the structure with an arbitrary number of layers
in each of the plates can be obtained by Eq. 2.31 numerically. The particle velocity can
also be arbitrary.
2.3 Two-layer flat structure
Note that in the symmetric case (๐๐ = ๐โ๐, ๐๐ฆ(๐) = ๐๐ฆ
(โ๐), ๐(๐ก) = ๐(๐)) ๐๐(๐) = ๐๐
(๐ก)โ1 and
๐(๐) = ๐(๐ก)โ1
. For some important special cases, it is possible to obtain explicit
expressions.
Figure 2.2: Two-layer flat structure. Geometry and electromagnetic parameters.
An important special case is a symmetrical two-layer structure with unbounded
external walls (Figure 2.2). We write the amplitudes of the longitudinal electric component
of the radiation field of the particles in the vacuum region between the plates (corresponds
to Eq. 2.15 for ๐ = 0)
๐ด๐ง,1,2(0)
= ๐ด๐ถ cosh (๐๐ฆ(0)ฮ) โ ๐ด๐ sinh (๐๐ฆ
(0)ฮ) (2.33)
where
๐ด๐ถ = ๐๐๐๐ง
16๐2๐0
๐ฃ2
๐2 ๐ด+๐ต tanh(๐๐ฆ
(0)๐1)
๐ท1๐ถ๐ท2
๐ถ , ๐ด๐ = ๐๐๐๐ง
16๐2๐0
๐ฃ2
๐2 ๐ด+๐ต coth(๐๐ฆ
(0)๐1)
๐ท1๐๐ท2
๐ (2.34)
The following notations are made in Eq. 2.34
๐ด = ๐๐ฆ(0)[(๐๐ฆ
(1)2โ ๐๐ฆ
(2)2) (๐๐ฅ
2๐พ12 โ ๐๐ฆ
(1)2๐๐ง2) + (๐๐ฆ
(1)2โ ๐๐ฅ
2) ๐]
ํ1, ๐1
ํ1, ๐1
ํ2, ๐2
ํ2, ๐2
- 36 -
๐ต = 2๐๐ฆ(1)(๐๐ฅ
2 โ ๐๐ฆ(0)2) ๐1๏ฟฝฬ๏ฟฝ2
๐ท1๐ถ = ๐๐ฆ
(1)๐2 cosh (๐๐ฆ
(0)๐1) + ๐๐ฆ
(0)๐1 sinh (๐๐ฆ
(0)๐1)
๐ท1๐ = ๐๐ฆ
(1)๐2 sinh (๐๐ฆ
(0)๐1) + ๐๐ฆ
(0)๐1 cosh (๐๐ฆ
(0)๐1)
๐ท2๐ถ = ๐๐ฆ
(0)๐พ12๏ฟฝฬ๏ฟฝ1 cosh (๐๐ฆ
(0)๐1) + ๐๐ฆ
(1)๐พ02๏ฟฝฬ๏ฟฝ2 sinh (๐๐ฆ
(0)๐1)
๐ท2๐ = ๐๐ฆ
(0)๐พ12๏ฟฝฬ๏ฟฝ1 sinh (๐๐ฆ
(0)๐1) + ๐๐ฆ
(1)๐พ02๏ฟฝฬ๏ฟฝ2 cosh (๐๐ฆ
(0)๐1) (2.35)
๐1 = ๐๐ฆ(1) cosh(๐๐ฆ
(1)๐) + ๐๐ฆ(2) sinh(๐๐ฆ
(1)๐) ๐2 = ๐๐ฆ(2) cosh(๐๐ฆ
(1)๐) + ๐๐ฆ(1) sinh(๐๐ฆ
(1)๐)
๏ฟฝฬ๏ฟฝ1 = ๐๐ฆ(1)๐พ2
2 cosh(๐๐ฆ(1)๐) + ๐๐ฆ
(2)๐พ12 sinh(๐๐ฆ
(1)๐) ๏ฟฝฬ๏ฟฝ2 = ๐๐ฆ(2)๐พ1
2 cosh(๐๐ฆ(1)๐) + ๐๐ฆ
(1)๐พ22 sinh(๐๐ฆ
(1)๐)
๐ = (๐๐ฆ(2)2๐พ1
2 + ๐๐ฆ(1)2๐พ2
2) cosh(2๐๐ฆ(1)๐) + ๐๐ฆ
(1)๐๐ฆ(2)(๐พ1
2 + ๐พ22) sinh(2๐๐ฆ
(1)๐)
๐พ๐2 = ๐๐ฆ
(๐)2โ ๐๐ฅ
2 โ ๐๐ง2, ๐ = 0,1,2
Result of Eq. 2.33 โ Eq. 2.35 is a generalization of the well-known result [64] on the
two-layer structure for an arbitrary velocity of the particle. In the transition to single-layer
unbounded plates (๐๐ฆ(1) = ๐๐ฆ
(2)) the first term in ๐ด vanishes. In the transition to ultra-
relativistic approximation (๐๐ฆ(0) = ๐๐ฅ) factor ๐ต in vanishes. The following are the values of
the field amplitudes for the structure, consisting of two unbounded parallel plates and the
arbitrary particle velocity.
๐ด๐ถ = ๐๐๐๐ง
8๐2๐0
๐ฃ2
๐2
(๐๐ฅ2โ๐๐ฆ
(1)2)๐๐ฆ(0)+๐๐ฆ
(1)(๐๐ฅ2โ๐๐ฆ
(0)2) tanh(๐๐ฆ
(0)๐1)
(๐๐ฆ(1)cosh(๐๐ฆ
(0)๐1)+๐๐ฆ
(0)sinh(๐๐ฆ
(0)๐1))(๐๐ฆ
(0)๐พ12 cosh(๐๐ฆ
(0)๐1)+๐๐ฆ
(1)๐พ02 sinh(๐๐ฆ
(0)๐1))
๐ด๐ = ๐๐๐๐ง
8๐2๐0
๐ฃ2
๐2
(๐๐ฅ2โ๐๐ฆ
(1)2)๐๐ฆ(0)+๐๐ฆ
(1)(๐๐ฅ2โ๐๐ฆ
(0)2) coth(๐๐ฆ
(0)๐1)
(๐๐ฆ(0)cosh(๐๐ฆ
(0)๐1)+๐๐ฆ
(1)sinh(๐๐ฆ
(0)๐1))(๐๐ฆ
(1)๐พ02 cosh(๐๐ฆ
(0)๐1)+๐๐ฆ
(0)๐พ12 sinh(๐๐ฆ
(0)๐1))
(2.36)
At ๐๐ฆ(0) = ๐๐ฅ and ๐ฃ = ๐ amplitudes Eq. 2.36 coincide with the corresponding
amplitudes in [64], obtained in the ultra-relativistic approximation.
- 37 -
In the case of the high conductivity of outer layer, it can be considered as a perfect
conductor. Transition to this case is made by means of equating to infinity in Eq. 2.34 the
wave number ๐๐ฆ(2)
. In this case the field amplitudes are
๐ด1๐ถ = ๐๐๐๐ง
16๐2๐0
๐ฃ2
๐2
(๐๐ฅ2โ๐๐ฆ
(1)2)๐๐ฆ(0)sinh(2 ๐๐ฆ
(1)๐)+2๐๐ฆ
(1)(๐๐ฆ(0)2
โ๐๐ฅ2) sinh2(๐๐ฆ
(1)๐) tanh(๐๐ฆ
(0)๐1)
๐ 1๐ถ๐ 2
๐ถ
๐ด1๐ = ๐๐๐๐ง
16๐2๐0
๐ฃ2
๐2
(๐๐ฅ2โ๐๐ฆ
(1)2)๐๐ฆ(0)sinh(2 ๐๐ฆ
(1)๐)+2๐๐ฆ
(1)(๐๐ฆ(0)2
โ๐๐ฅ2) sinh2(๐๐ฆ
(1)๐) coth(๐๐ฆ
(0)๐1)
๐ 1๐๐ 2๐
(2.37)
where the following notations are made
๐ 1๐ถ = ๐๐ฆ
(1) cosh(๐๐ฆ(1)๐) cosh(๐๐ฆ
(0)๐1) + ๐๐ฆ(0) sinh(๐๐ฆ
(1)๐) sinh(๐๐ฆ(0)๐1)
๐ 1๐ = ๐๐ฆ
(1)cosh(๐๐ฆ
(1)๐) sinh(๐๐ฆ
(0)๐1) + ๐๐ฆ
(0)sinh(๐๐ฆ
(1)๐) cosh(๐๐ฆ
(0)๐1)
๐ 2๐ถ = ๐๐ฆ
(0)๐พ12 cosh(๐๐ฆ
(1)๐) cosh(๐๐ฆ(0)๐1) + ๐๐ฆ
(1)๐พ02 sinh(๐๐ฆ
(1)๐) sinh(๐๐ฆ(0)๐1)
๐ 2๐ = ๐๐ฆ
(0)๐พ12 cosh(๐๐ฆ
(1)๐) sinh(๐๐ฆ(0)๐1) + ๐๐ฆ
(1)๐พ02 sinh(๐๐ฆ
(1)๐) cosh(๐๐ฆ(0)๐1)
(2.38)
Below the longitudinal electric field in case of asymmetric filling of plates is
presented. A structure with perfectly conducting plates, and only one of them covered with
a layer of finite conductivity material is considered.
๐ธ๐ง = ๐
๐๐๐ง
16๐2๐0{๐ด1๐ง๐
โ๐๐ฅ๐1 + ๐ด2๐ง๐๐๐ฅ๐1} = ๐
๐๐๐ง๐ด0
8๐2๐0๐ถโ(2๐๐ฅ๐1)
๐ด1,2๐ง = โ๐ด0๐โ{๐๐ฅ(๐ + โ)}๐โ๐๐ฅ๐1
(2.39)
with
๐ด0 = ๐๐ฅ (๐๐ฆ(1)2 โ ๐๐ฅ
2) ๐โ(2๐๐y(1)) {(๐๐ฅ๐ถโ(2๐1๐๐ฅ)๐โ(๐๐๐ฆ
(1)) + ๐๐ฆ(1)๐โ(2๐1๐๐ฅ)๐ถโ(๐๐๐ฆ
(1)))
(๐๐ฆ(1)๐๐ง
2๐ถโ(2๐1๐๐ฅ)๐โ(๐๐๐ฆ(1)) โ ๐๐ฅ๐พ1
2๐โ(2๐1๐๐ฅ)๐ถโ(๐๐๐ฆ(1)))}
(2.40)
2.4 Impedance of two-layer symmetrical flat structure
As it is mentioned in the first chapter, the longitudinal impedance is a Fourier
transformation of normalized longitudinal component of Lorentz force. General expression
of Lorentz force is
๐ญ(๐ฅ, ๐ฆ, ๐ง) = ๐[๐ฌ(๐ฅ, ๐ฆ, ๐ง) + ๐ ร ๐ฉ(๐ฅ, ๐ฆ, ๐ง)] (2.41)
- 38 -
The longitudinal (๐ง direction) impedance ๐โฅ(๐ฅ, ๐ฆ, ๐๐ง) is a Fourier transformation of
normalized longitudinal component of Lorenz force ๐น๐ง(๐๐ฅ, ๐ฆ, ๐๐ง) . In the case, when a
charge ๐ is traveling along ๐ง axis, the longitudinal component of Lorentz force becomes
๐น๐ง(๐๐ฅ, ๐ฆ, ๐๐ง) = ๐๐ธ๐ง(๐๐ฅ, ๐ฆ, ๐๐ง) (2.42)
and consequently the longitudinal impedance becomes
๐โฅ(๐ฅ, ๐ฆ, ๐๐ง) = โ1
๐๐โซ ๐ธ๐ง(๐๐ฅ, ๐ฆ, ๐๐ง)๐
โ๐๐๐ฅ๐ฅ๐๐๐ฅโ
โโ (2.43)
Consider the most important and at the same time quite simple special case:
symmetrical two-layer flat structure. In the case of the perfectly conductive outer layer
(๐๐ฆ(2) โ โ) and the particle velocity equal to the velocity of light (๐๐ฆ
(0) = ๐๐ฅ, ๐ฃ = ๐), the
formulae Eq. 2.37 are simplified to the form
๐ด1๐ถ(๐) = ๐๐๐๐ง๐๐ฆ
(1)
16๐2๐0
(๐๐ฅ2โ๐๐ฆ
(1)2) sinh(2๐๐ฆ
(1)๐)
๐ 1๐ถ(๐)๐ 2
๐ถ(๐) (2.44)
where
๐ 1๐ถ
๐ 2๐ถ} = {
๐๐ฆ(1)
๐๐ฅ๐พ12} ๐ถโ(๐๐ฆ
(1)๐)๐ถโ(๐๐ฅ๐1) ยฑ {๐๐ฅ
๐๐ฆ(1)๐๐ฅ2} ๐โ(๐๐ฆ
(1)๐)๐โ(๐๐ฅ๐1)
๐ 1๐
๐ 2๐} = {
๐๐ฆ(1)
๐๐ฅ๐พ12} ๐ถโ(๐๐ฆ
(1)๐)๐โ(๐๐ฅ๐1) ยฑ {๐๐ฅ
๐๐ฆ(1)๐๐ฅ
2} ๐โ(๐๐ฆ(1)๐)๐ถโ(๐๐ฅ๐1)
(2.45)
This case is fairly well simulates the highly conducting outer wall and is a two-
dimensional analogue of a two-layer cylindrical tube, the unique resonant properties were
discovered earlier. In the cylindrical waveguide with two-layer metal walls under high
conductivity of the thick outer wall and the relatively low conductivity of the thin inner layer,
the radiation of the driving particles has a narrow-band resonance. Resonance is the sole,
and formed by single slowly propagating waveguide mode. We show that the same is true
of the two-dimensional two-layer metal structure.
In the cylindrical case, the resonance frequency is equal to ๐๐ง,๐๐๐ง = โ2 ๐1๐โ (๐1
waveguide inner radius, ๐ inner layer thickness). Minimum damping (in the case of perfect
- 39 -
conductor outer layer) is provided by the condition ํ =1
โ3๐0๐1๐ (๐0 = 120๐ ฮฉ, ๐1 -inner
layer conductivity).
Let us demonstrate in the first step the single narrow-band extremal nature of the
longitudinal impedance of the two-dimensional two-layer structure.
Figure 2.3: Real (left) and imaginary (right) parts of the longitudinal impedance of two-layer flat
structure. On the graphs, the thicknesses of the inner layer in micrometers are shown.
Figure 2.3 shows a series of distributions of real and imaginary components of the
longitudinal impedance of a two-layer planar structure with a perfectly conductive outer
layer (Figure4). The distance between the plates is 2 ๐๐ (half-height ๐1 = 1 ๐๐) and the
conductivity of the inner layer is 2 ร 103 ฮฉโ1๐โ1 The numbers on the curves in the graph
indicate the thickness of the inner layer (๐๐): "1" (1๐๐), "2" (2๐๐) and so on. As we see,
the longitudinal impedance has a single narrow-band resonance with a resonance
frequency dependent on the thickness of the inner layer. As is shown in Table 1, it is quite
well defined by the formula
๐๐๐๐ง =๐
2๐โ
1
๐1๐ (2.46)
where ๐1 = ๐ 2โ is a half-height of a structure.
- 40 -
๐ (๐๐) 1 2 3 4 5 6 7 8 10 16
๐๐๐๐ง (๐๐ป๐ง) 0.477 0.338 0.276 0.239 0.214 0.195 0.180 0.169 0.151 0.119
๐๐๐๐ฅ (๐๐ป๐ง) 0.479 0.339 0.277 0.241 0.215 0.197 0.183 0.171 0.154 0.124
๐0 (๐๐ป๐ง) 0.479 0.503 0.277 0.241 0.215 0.197 0.183 0.171 0.154 0.124
๐๐ 0.251 0.339 0.754 1.001 1.257 1.508 1.760 2.011 2.513 4.021
Table 2.1: Resonance frequencies of two-layer flat structure calculated analytically (๐๐๐๐ง),
numerically (๐๐๐๐ฅ), impedance imaginary component vanishing frequency (๐0) and a
parameter ๐๐.
In Table 2.1: line ๐๐๐๐ง shows the resonance frequency calculated by Eq. 2.46; in line
๐๐๐๐ฅ the resonance frequencies calculated numerically by the exact formulae are given;
line ๐0 shows the frequency values in which the imaginary components of the impedance
vanish; in the last row given the values of the parameter ๐๐ =1
3๐0๐1๐๐. As the table shows,
there is a good coincidence between the frequency values ๐๐๐๐ง and ๐๐๐๐ฅ . Thus, the
resonant frequency in a flat two-layer metal structure, in contrast to the corresponding
cylindrical structure, is defined by Eq. 2.46. In cylindrical case ๐๐๐๐ง =๐
2๐โ
2
๐1๐., i.e. in โ2
times higher. There is also a complete coincidence of frequencies ๐๐๐๐ฅ and ๐0, i.e. the
maximum value of the impedance is purely real. Comparing the maximum values of the
real component of impedance distributions (Figure 2.3, left) with the corresponding
parameter ๐๐ values, we note the following regularities: impedance reaches the highest
value at ๐ = 4 ๐๐ , which corresponds to value of parameter ๐๐ equal to unity. The
maximum values of the other curves (Figure 2.3, left) on either side of it, decrease as the
distance.
As it is shown in Figure 2.3 (left), there are a pair of curves, maximum values of
which are located on a same level. These curves correspond to a pair of mutually ๐๐
parameter reverse values. Thus, as it follows from Figure 2.3 and Table 1: ๐1 โ 1 ๐16โ and
๐2 โ 1 ๐8โ . Similar phenomena occur also in a two-layer cylindrical waveguide. The
difference is in the value of the parameter ๐๐: ๐ =1
3๐0๐1๐ for planar and ๐ =
1
โ3๐0๐1๐ for
cylindrical cases, i.e. โ3 times less. Recall that the resonance frequencies in the planar
case to โ2 times less than in the corresponding cylindrical: ๐๐๐๐ง = ๐โ1 ๐1๐โ (๐1 inner half
- 41 -
height of planar structure) instead of ๐๐๐๐ง = ๐โ2 ๐1๐โ (๐1 inner radius of cylinder). Such
relations for broadband resonance frequencies have been identified in [65] for the single-
layer resistive two-dimensional flat and cylindrical structures (Figure 2.4). For comparison,
Figure 2.4 shows the longitudinal impedance distributions for a two-layer metal tube with
an ideally conducting outer layer. Its geometric and electromagnetic parameters are
brought into correspondence with the parameters of the flat two-layer metal structure
considered above. The internal radius of the cylinder ๐0 = 1 ๐๐, the conductivity of the
inner layer is ๐ = 2 ร 103 ฮฉโ1๐โ1, and its thickness varies between 1 ๐๐ and 5 ๐๐.
Figure 2.4: Real (left) and imaginary (right) parts of the longitudinal impedance of two-layer
cylindrical structure. On the graphs, the thicknesses of the inner layer in microns are shown.
The wake potential is defined as the inverse Fourier transform of impedance (2.43)
๐โฅ(๐ฅ, ๐ฆ, ๐ ) = โซ ๐โฅ(๐ฅ, ๐ฆ, ๐)โ
โโ๐โ๐๐๐ ๐๐ , ๐ = ๐ง โ ๐ฃ๐ก (2.47)
Figure 2.5 shows the distribution of the longitudinal wake potential for three different
thicknesses of the inner layer at the fixed conductivity. In all three cases the wake
potentials are quasi-periodic functions with the period ฮ๐ given by the resonant frequency
ฮ๐ = ๐ ๐๐๐๐งโ , ๐๐๐๐ง =๐
2๐โ1
๐๐. The latter indicates the particle radiation monochromaticity
similar to two-layer cylindrical metal waveguide [50].
- 42 -
Figure 2.5: Longitudinal wake potential of the two-layer flat structure for different values of
the thickness of the inner layer and the fixed conductivity ๐ = 2 ร 103 ฮฉโ1๐โ1; half-height of
structure ๐ = 1 ๐๐. For comparison, the wake function for two unbounded infinite single-layer
plates with halt-width ๐ = 1 ๐๐ and ๐ = 2 ร 103 ฮฉโ1๐โ1 is given (dashed curve).
Figure 2.6 shows the wake potential distributions for a two-layer cylindrical structure.
Figure 2.6: Longitudinal wake potential of the two-layer cylindrical waveguide for different
values of the thickness of the inner layer and the fixed conductivity ๐ = 2 ร 103 ฮฉโ1๐โ1;
inner radius of cylinder is equal to half height of flat structure (Figure 2.5) ๐0 = ๐ = 1 ๐๐. The
designations of the curves correspond to the thicknesses of the inner coating in microns. The
wake potential for unbounded single-layer resistive cylinder with radius ๐0 = 1 ๐๐ and ๐ =
2 ร 103 ฮฉโ1๐โ1 is given in addition (dashed curve).
- 43 -
As in the plane case, the wake function of a cylindrical structure is a quasi-
monochromatic oscillating damped function with a single oscillation frequency ๐๐๐๐ง =๐
2๐โ2
๐๐.
The period of the oscillations is ฮ๐ = ๐ ๐๐๐๐งโ (โ2 times less of one of flat case).
Comparing the curves for impedances and wake potentials for a two-layered
cylindrical structure, we notice from Figure 2.4 (curve with label โ2.5โ), that in the
frequency representation the maximum quality factor (maximum amplitude) of the
impedance reaches at the thickness of the inner coating equal to 2.5 ๐๐. In the space-time
representation this correlates with the minimum attenuation of the wake (Figure 2.6, curve
with label โ2.5โ). In the cylindrical case, the attenuation coefficient is proportional to the
value of ํ = ๐ + ๐โ1, with ๐ =1
โ3๐0๐๐ and attains its minimum value at ๐ = 1 [50].
The above patterns (Figure 2.3 and Figure 2.5) are revealed in the case of fixed
value of the inner layer conductivity. The regularities of the structure with a fixed thickness
of the inner layer, are denoted in Figure 2.7. The Figure shows the distribution of the
amplitude values of the impedance curve at ๐ = 1๐๐ and 1500 ฮฉโ1๐โ1 < ๐ <
11000 ฮฉโ1๐โ1, which corresponds to the range of variation of the parameter ๐ =1
โ3๐๐๐0
within 0.3 < ๐ < 2.5.
- 44 -
Figure 2.7: Distribution of resonance values of the two-layer flat structure impedance as a function
of the conductivity ๐ of the inner layer for fixed geometric parameters of the structure (๐ =
1๐๐, ๐ = 1 ๐๐). The horizontal dashed lines indicate the mutually conjugate values of the
conductivities ๐๐ and ๐๐ for which ํ๐ = 1 ํ๐โ with ํ๐(๐) = 3โ1 2โ ๐๐0๐๐(๐). Vertical dashed line
indicates the maximum possible level of impedance for a given thickness of the inner layer ๐ =
1๐๐.
A significant difference between the cylindrical and flat cases is that in the first case
the optimal condition (in the sense of the maximum Q of the system) is the fulfillment of
equality ๐ =1
โ3๐0๐๐ = 1. In the second case, when fixing the conductivity value of the
inner layer ฯ, the condition for optimizing the structure is the equality ๐ =1
3๐0๐๐ = 1, i.e.
the optimum layer thickness in this case is determined through its conductivity ๐ by the
expression ๐ =3
๐0๐. In other hand, fixation of the thickness of the inner layer ๐ leads to the
optimization condition ๐ =1
โ3๐0๐๐ = 1 and to the optimal conductivity of the layer ๐ =
โ3
๐0๐.
Thus, for a given conductivity of the inner layer, in order to select the optimum thickness of
the layer, it is necessary to use the relation ๐ =3
๐0๐, and for a given layer thickness d, its
optimal conductivity is determined by the expression ๐ =โ3
๐0๐.
- 45 -
Figure 2.8: Optimization of the parameters of a two-dimensional two-layer metal structure with the
initial fixation of the thickness ๐ = 1๐๐ of the inner layer.
Figure 2.8 shows the results of the sequential optimization process with the initial
fixation value of the thickness of the inner layer at the level of ๐ = 1๐๐. The figure shows
5 curves depicting the distribution of longitudinal impedances for the half-height of
structure ๐ = 1๐๐. Curve 1 (black solid curve) is plotted for the thickness of the inner layer
๐ = 1 ๐๐ with a conductivity ๐ = 4594 ฮฉโ1๐โ1 , determined by the formula ๐ =โ3
๐0๐, i.e.
curve 1 represents the impedance with the highest possible amplitude level, achievable at
๐ = 1 ๐๐ by means of the corresponding selection of the conductivity of the inner layer.
Fixing, then, the received value of conductivity, by the formula ๐2 =3
๐0๐ we determine for it
the optimum value of layer thickness ๐2 = 1.732 ๐๐ , wich is optimal for this value of
conductivity. As a result, we obtain curve 2 (red solid curve). Optimization leads to an
increase in the thickness of the inner layer and, consequently, to a decrease in the
resonant frequency. Further optimization again assumes the use of the formula ๐2 =โ3
๐0๐.
As a result we have curve 3 with reduced conductivity ๐3 = 2653 ฮฉโ1๐โ1 and a resonance
conserved frequency. This procedure, consisting in the successive application of the
formulas ๐๐ =3
๐0๐๐ and ๐๐+1 =
โ3
๐0๐๐, can be repeated many times, successively increasing
the thickness of the inner layer and its conductivity and decreasing the resonance
frequency (Figure 2.8).
- 46 -
An alternative way to perform a series of optimizations is shown in Figure 2.9. It is
obtained by first fixing the conductivity of the inner layer of the structure. In the figure,
curve โ1โ corresponds to the originally selected conductivity of the inner layer ๐ = 2 ร
103 ฮฉโ1๐โ1. Its thickness is optimized using the formula ๐ =3
๐0๐= 3.98 ๐๐. Curve โ2โ (red,
dashed) is obtained by optimizing the conductivity for the resulting length: ๐2 =โ3
๐0๐=
1154 ฮฉโ1๐โ1 . The resonant frequency does not change in this case. The further
optimization step, corresponding to curve โ3โ, is carried out using the formula ๐2 =3
๐0๐2=
6.89 ๐๐. This optimization stage is accompanied by an increase in the thickness of the
inner layer and by a decrease of resonant frequency. At the next stage, the optimum
conductivity value is determined for the layer new thickness ๐3 =โ3
๐0๐2= 385 ฮฉโ1๐โ1 (curve
โ3โ) and so on.
Figure 2.9: Optimization of the parameters of a two-dimensional two-layer metal structure with the
initial fixation of the inner layer conductivity ๐ = 2 ร 103 ฮฉโ1๐โ1.
Thus, there are two different possibilities to optimize a two-dimensional two-layer
metal structure: the first (Figure 2.8) is accompanied by an increase of the conductivity of
the material of the inner layer, while the second (Figure 2.9) lowers it.
The practical significance of the above study is the ability to correctly select the
thickness of the inner layer and the electromagnetic characteristics of the material filling it.
- 47 -
This, for example, was successfully done during of experimentally testing the resonant
properties of a two-layer metal structure [67] with severe limitations on the measuring
equipment, whose frequency range was limited by 14 GHz bandwidth.
Figure: 2.10: Longitudinal impedance of cylindrical pipe with optimized parameters: optimal inner
layer width ๐ = 2.3๐๐ for given conductivity ๐ = 2000 ฮฉโ1๐โ1 (red); optimal inner layer
conductivity ๐ = 4600 ฮฉโ1๐โ1 for given layer width ๐ = 1๐๐ (black).
In the case of a two-layered cylindrical structure, the picture is much simpler. In this
case, the narrow-band resonance character of the impedance curve is completely due to
the proximity to the unity of the parameter ๐ =1
โ3๐0๐๐ [50]. The impedance curve is
completely determined by setting two parameters (๐, ๐; ๐, ๐ or ๐, ๐) of three (๐, ๐, ๐).
2.5 Approximate analytical representation of impedance
The integral representation of the longitudinal impedance (Eq. 2.43 โ Eq. 2.45) is
accurate (exact) and, as was shown above, it is possible in principle to calculate both the
impedance and the wake potential (represented in the form of a double integral (Eq. 2.47)).
For calculation, a corresponding software package has been developed by using the
software package Mathematica8. The integral form of the expressions for the impedance
and the wake function, as well as the complexity of the integrand expression, leads,
however, to some difficulties in interpreting them. Integration by means of deformation of
the contour of integration and its transfer to the complex plane is associated with taking
- 48 -
into account, in addition to complex poles of denominator, also integration along the cuts
on the Riemann surface. The latter circumstance is connected with the non-analyticity of
the function ๐๐ฆ(1) = โ๐๐ฅ2 + (1 โ ํ
(1)๐(1)๐2)๐2, which leads to the non-analyticity of the
denominator of the integrand as a whole.
The analytizing of the integrand in (Eq. 2.43) is performed taking into account the
smallness of the parameter ๐ (๐ โช ๐), characterized the thickness of the inner layer. For
this purpose, we rewrite the function ๐ด1๐ถ (Eq. 2.44) in the form
๐ด1๐ถ =๐๐0๐๐ฅ
8๐2๐0๐๐
๏ฟฝฬ๏ฟฝ, ๏ฟฝฬ๏ฟฝ = ๏ฟฝฬ๏ฟฝ1
๐ถ๏ฟฝฬ๏ฟฝ2๐ถ (2.48)
where now
๏ฟฝฬ๏ฟฝ1๐ถ = ๐๐ฆ
(1)๐ถโ(๐๐ฅ๐)๐ถ๐กโ (๐๐ฆ
(1)๐) + ๐๐ฅ๐โ(๐๐ฅ๐)
๏ฟฝฬ๏ฟฝ2๐ถ = ๐๐ฅ(๐ + ๐ ๐0๐)๐ถโ(๐๐ฅ๐) + ๐๐๐ฆ
(1)๐โ(๐๐ฅ๐)๐โ (๐๐ฆ
(1)๐)
(2.49)
After multiplication, the denominator in Eq. 2.48 takes the form
๏ฟฝฬ๏ฟฝ = ๐๐ฆ(1)๐๐ฅ(๐+ ๐ ๐0๐)๐ถโ
2(๐๐ฅ๐)๐ถ๐กโ (๐๐ฆ
(1)๐)+
1
2(๐๐ฆ
(1)2๐+ ๐๐ฅ
2(๐+ ๐ ๐0๐))๐โ(2๐๐ฅ๐)+๐๐ฆ
(1)๐๐ฅ๐๐โ
2(๐๐ฅ๐)๐โ (๐๐ฆ
(1)๐)
(2.50)
Further, we expand ๏ฟฝฬ๏ฟฝ in a series with respect to the small parameter ๐ and leave
the first three terms of the expansion (๐โ1, ๐0 and ๐). As a result, we get
๏ฟฝฬ๏ฟฝ โ ๐ขโ1 ๐โ + ๐ข0 + ๐ข1๐ (2.51)
where
๐ข1 =1
3๐๐ฆ(1)2๐๐ฅ ((๐ + ๐ ๐0๐)๐ถโ
2(๐๐ฅ๐) + 3๐๐โ2(๐๐ฅ๐))
๐ข0 =1
2(๐๐ฆ
(1)2๐ + ๐๐ฅ
2(๐ + ๐ ๐0๐))๐โ(2๐๐ฅ๐)
๐ขโ1 = ๐๐ฅ(๐ + ๐ ๐0๐)๐ถโ2(๐๐ฅ๐)
(2.52)
Expression 2.51 with taking into account Eq. 2.52 no longer contains square roots
of the integration variables and is analytical.
- 49 -
Believing, further ๐๐ฅ2 โช ๐0๐๐, (in the cylindrical case this approximation is equivalent
to approximation for the permittivity of the metal ํ โ ๐ ๐ ๐โ , which is valid for not very large
frequencies) we have:
๐๐ฆ(1)2
โ โ๐๐0๐๐ (2.53)
and instead of Eq. 2.52, we can write
๐ข1 = โ1
3๐๐0๐๐๐๐ฅ((๐ + ๐ ๐0๐)๐ถโ
2(๐๐ฅ๐) + 3๐๐โ2(๐๐ฅ๐))
๐ข0 =1
2(๐๐๐ฅ
2 + ๐ ๐0๐(๐๐ฅ2 โ ๐2))๐โ(2๐๐ฅ๐)
๐ขโ1 = ๐๐ฅ(๐ + ๐ ๐0๐)๐ถโ2(๐๐ฅ๐)
(2.54)
Figures (2.11โ2.15) show comparison of calculations for exact formulae and in
approximation (Eq. 2.51 โ Eq. 2.54) for three different half-widths of a plane structure: ๐ =
10 ๐๐ (Figure 2.11 and Figure 2.12), ๐ = 1๐๐ (Figure 2.13 and Figure 2.14) and ๐ = 1 ๐๐
(Figure 2.15). The thickness of the inner layer is chosen equal to 7 ๐๐. Dashed red curves
represent the results obtained from exact formulae. The approximate results are
represented by solid black curves.
Fig. 2.11. Real (left) and imaginary (right) components of the longitudinal impedance of a
two-layer metal structure, calculated by exact formulae (red, dashed) and using the approximation
(2.48, 2.51) (black, solid): 1 - ๐ = 2 ร 104 ฮฉโ1๐โ1, ํ = 30.5; 2 - ๐ = 2 ร 103 ฮฉโ1๐โ1, ํ = 3.05; 3 -
๐ = 656 ฮฉโ1๐โ1, ํ = 1; ๐ = 10 ๐๐, ๐ = 7 ๐๐.
- 50 -
Fig. 2.13. Real (left) and imaginary (right) components of the longitudinal impedance of a two-layer
metal structure, calculated by exact formulae (red, dashed) and using the approximation (2.48,
2.51) (black, solid): 1 - ๐ = 2 ร 104 ฮฉโ1๐โ1, ํ = 30.5; 2 - ๐ = 2 ร 103 ฮฉโ1๐โ1, ํ = 3.05; 3 - ๐ =
656 ฮฉโ1๐โ1, ํ = 1; ๐ = 1 ๐๐, ๐ = 7 ๐๐.
As can be seen from Figure 2.11 and Figure 2.13, for relatively large values of the
half-widths of the structure (1 ๐๐ โค ๐ โค 10 ๐๐) there is a complete coincidence of the
approximation calculations with the exact ones even for much larger (in comparison with
unity) values of the parameter ํ =1
โ3๐0๐ (up to ํ~30). Significant discrepancies occurs at
large values of this parameter ํ~100 (Figure 2.12 and Figure 2.14).
Fig. 2.12. Real (left) and imaginary (right) components of the longitudinal impedance of a two-layer
metal structure, calculated by exact formulae (red, dashed) and using the approximation (2.48,
2.51) (black, solid): 1 - ๐ = 2 ร 105 ฮฉโ1๐โ1, ํ = 135 ; 2 - ๐ = 2 ร 106 ฮฉโ1๐โ1, ํ = 1523 ; ๐ =
10 ๐๐, ๐ = 7 ๐๐.
- 51 -
Fig. 2.14. Real (left) and imaginary (right) components of the longitudinal impedance of a two-layer
metal structure, calculated by exact formulae (red, dashed) and using the approximation (2.48,
2.51) (black, solid): 1 - ๐ = 2 ร 105 ฮฉโ1๐โ1, ํ = 135 ; 2 - ๐ = 2 ร 106 ฮฉโ1๐โ1, ํ = 1523 ; ๐ =
1 ๐๐, ๐ = 7 ๐๐.
As follows from the Figure 2.15, a decrease in the gap between the plates leads to
a toughening of the requirement of ํ be close to unity. Thus, for ํ = 30 (curve โ1โ on
Figure 2.15) already there is an essential discrepancy between the exact and approximate
solution, whereas for ํ = 7.6 (curve โ2โ on Figure 2.15) they practically coincide.
Fig. 2.15. Real (left) and imaginary (right) components of the longitudinal impedance of a two-layer
metal structure, calculated by exact formulae (red, dashed) and using the approximation (2.48,
2.51) (black, solid): 1 - ๐ = 2 ร 104 ฮฉโ1๐โ1, ํ = 30.5; 2 - ๐ = 5 ร 103 ฮฉโ1๐โ1, ํ = 7.6. ๐ = 1 ๐๐,
๐ = 7 ๐๐.
Further simplification of the integrand (Eq. 2.50 โ Eq. 2.53) is achieved by replacing
the hypergeometric functions by their decomposition. The expansions in a series of
hypergeometric functions ๐โ(๐ฅ) and ๐ถโ(๐ฅ) have the form [68]
- 52 -
๐โ(๐ฅ) = โ๐ฅ2๐โ1
(2๐โ1)!โ๐=1 , ๐ถโ(๐ฅ) = โ
(๐ฅ)2๐โ2
(2๐)!โ๐=1 (2.55)
Substituting Eq. 2.55 in Eq. 2.53 and leaving the first N terms of the expansion in
them, we have instead of Eq. 2.48
๐ด1๐ถ =๐๐0
8๐2๐0๐๐
๏ฟฝฬ๏ฟฝโฒ (2.56)
where
๏ฟฝฬ๏ฟฝโฒ โ ๐ขโ1โฒ ๐โ + ๐ข0
โฒ + ๐ข1โฒ๐ (2.57)
and
๐ข1โฒ =
1
6๐๐๐๐0((2๐ โ ๐๐๐0) โ (4๐ + ๐๐๐0)๐ถโ๐(2๐๐๐ฅ))
๐ขโ1โฒ =
1
2(๐ + ๐ ๐0๐)(1 + ๐ถโ๐(2๐๐๐ฅ)
๐ข0โฒ =
1
2(๐๐๐ฅ
2 + ๐ ๐0๐(๐๐ฅ2 โ ๐2))
๐โ๐(2๐๐ฅ๐)
๐๐ฅ
(2.58)
with
๐โ๐(๐ฅ) = โ๐ฅ2๐โ1
(2๐โ1)!๐๐=1 , ๐ถโ๐(๐ฅ) = โ
๐ฅ2๐โ2
(2๐)!๐๐=1 (2.59)
In this approximation, the denominator (Eq. 2.57) is represented in the form of an
algebraic polynomial of 2N-th order in powers of ๐๐ฅ
๏ฟฝฬ๏ฟฝโฒ = โ ๐๐๐๐ฅ๐2๐
๐=0 (2.60)
and can be written in a form containing its complex roots ๐ฅ๐ , ๐ = 1,2, โฆ 2๐
๏ฟฝฬ๏ฟฝโฒ = ๐2๐โ (๐๐ฅ โ ๐ฅ๐)2๐๐=1 (2.61)
Thus, Eq. 2.56 can be rewritten as follows
๐ด๐ถ =๐๐0
8๐2๐0๐๐
๐2๐โ (๐๐ฅโ๐ฅ๐)2๐๐=1
(2.62)
The longitudinal impedance, which is an integral of this expression over ๐๐ฅ, can be
expressed explicitly in terms of the roots ๐ฅ๐ of the equation ๏ฟฝฬ๏ฟฝโฒ(๐๐ฅ) = 0
- 53 -
๐|| = โซ ๐ด๐ถ๐๐๐ฅโ
โโ= โ
๐02๐๐
4๐2๐2๐โ
๐๐{โ๐ฅ๐}
โ (๐ฅ๐โ๐ฅ๐)2๐๐=1๐โ ๐
2๐๐=1 (2.63)
The roots ๐ฅ๐ of the equation ๏ฟฝฬ๏ฟฝโฒ(๐๐ฅ) = 0 are determined numerically for an arbitrary
value of N with the help of the software package Mathematica.
Longitudinal impedance of a two-layer metal structure calculated by exact and
approximate formulas formulae is presented in Figure 2.16. In the figure black solid
impedance curves are calculated using the approximate equations 2.51 and 2.53, blue
dashed ones are from Eq. 2.63 for ๐ = 5 and red dashed ones are for calculated by exact
formula.
Fig. 2.16. Real (left) and imaginary (right) components of the longitudinal impedance of a two-layer
metal structure, calculated by exact formulae (red, dashed), using the approximation (2.48, 2.51)
(black, solid) and using approximation (2.60) for n=5 (blue, dashed): 1 - ๐ = 2 ร 104 ฮฉโ1๐โ1, ํ =
30.5; 2 - ๐ = 5 ร 103 ฮฉโ1๐โ1, ํ = 7.6; ๐ = 1 ๐๐, ๐ = 7 ๐๐.
In Figure 2.16, the impedance curves calculated using the asymptotic (Eq. 2.62) are
superimposed on the impedance curves shown in Fig. 2.15.
2.6 Summary
Method of calculating the longitudinal impedance of a multilayer two-dimensional
flat structure is presented. The matrix formalism that allows to couple electromagnetic field
components in the inner and outer regions of the structure has been developed. Non-
ultrarelativistic particle radiation fields in the multilayer flat symmetric and asymmetric
structure are obtained. Explicit expressions of the radiation fields in two-layer structure
- 54 -
with unbounded external walls and in single-layer unbounded structure are derived as a
special case of fields in multilayer structure.
An ultrarelativistic particle radiation in a symmetrical two-layer flat structure with
perfectly conducting outer layer is derived. Longitudinal impedance of this structure with
low conductivity thin inner layer and high conductivity thick outer layer wall is obtained
numerically. It is shown that the impedance has a narrow-band resonance. The resonance
frequency dependence of a structure parameters is obtained. Parameter ๐๐ =1
3๐0๐1๐๐ is
introduced, which describes maximum values of impedance. It is shown that impedances
of structures with reverse ๐ parameters (๐๐ โ ๐๐ = 1) has the same maximum values and it
reaches the highest maximum when ๐๐ is equal to unity.
- 55 -
Chapter 3: Rectangular resonator with two-layer vertical walls
3.1 Introduction
In this chapter, dispersion relations are obtained for the electromagnetic eigen-
oscillations in a rectangular resonator with ideally conducting walls. The horizontal (top
and bottom) walls of this resonator are covered from the inside by a thin low-conductive
layer.
On the second Chapter it is shown that double-layered parallel plates for appropriate
parameters have a distinct, narrow band resonance. In this Chapter rectangular copper
cavity with double-layered horizontal walls is observed. Here, unlike of the parallel plates,
one has a discrete set of wavenumbers in all three degree of freedom. Compared with
two-dimensional structure, the pattern of modal frequency distribution in such a resonator
is distorted: in addition to the fundamental resonance frequency, a set of secondary
resonances caused by side walls exists. It is important to note the presence of the
fundamental resonance, fixed by measurements, and caused by the two-layer horizontal
cavity walls. To be able to distinguish the main mode from secondary resonances, finite
wall resonator model is developed and resonance frequencies of the resonator are
calculated.
In this Chapter secondary resonances caused by side walls are obtained. Criterions
of resonator materials choice are interpreted and resonance frequencies are predicted via
the resonator model with ideal conducting walls. The choice is conditioned by the
frequency bandwidth of the experimental set-up ( 14 ๐บ๐ป๐ง ) and the predicted single
resonance for two parallel laminated plates (chapter 2). The electro-dynamical properties
of test resonator model (perfectly conducting rectangular cavity with inner low conducting
layers) are studied. Resonance frequencies of the test structure are measured
experimentally and comparison with predicted resonant frequencies of resonator model
and two parallel double-layer plates is done. Good agreement between these results is
obtained.
- 56 -
Experimentally obtained resonance frequencies are subjected to the laws specific to
this kind of structures: their value decreases with increasing height of the cavity. At the
same time, the difference between the measured and calculated values increases.
Decreasing a structure height, the contribution of its sidewalls also decrease, thatโs why for
a structures with small height there is a good agreement between experimental and
theoretical resonant frequencies.
3.2 IHLCC cavity with ideally conductive walls
To obtain dispersion relations analytically for a tested cavity, a simplified resonator
model is studied. It is assumed that all four sidewalls of IHLCC cavity are ideally
conductive materials and upper and lower ideally conductive walls are covered with a thin
layer of low-conductive material (Fig 3.1).
Figure 3.1: IHLCC cavity shape.
This model is very close to the cavity with copper walls, so it is expected that
analytical results would be in a good agreement with tested cavity results. Exact dispersion
equations may be obtained for a similar structure by the method of partial areas. In this
case, three regions may be distinguished:
1. 0 โค ๐ฅ โค ๐, 0 โค ๐ง โค ๐ top low conductivity material plate
(3.1) 2. 0 โค ๐ฅ โค ๐, 0 โค ๐ง โค ๐ vacuum chamber of cavity
3. 0 โค ๐ฅ โค ๐, 0 โค ๐ง โค ๐ lower low conductivity material plate
- 57 -
Electric and magnetic fields of cavity in each of the three regions are sought in the
following form
๏ฟฝโโ๏ฟฝ๐ = ๏ฟฝโโ๏ฟฝ๐(1)+ ๏ฟฝโโ๏ฟฝ๐
(2), ๏ฟฝโโ๏ฟฝ๐ = (๐๐๐)
โ1๐๐๐ก๏ฟฝโโ๏ฟฝ๐, ๐ = 1,2,3 (3.2)
with
๐ธ๐ฅ,๐(1)
๐ธ๐ฅ,๐(2)} = {
๐ด๐ฅ,๐(1)๐ถโ(๐๐๐ฆ)
๐ด๐ฅ,๐(2)๐โ(๐๐๐ฆ)
} ๐ถ๐๐ (๐๐ฅ๐ฅ)๐๐๐(๐๐ง๐ง)
๐ธ๐ฆ,๐(1)
๐ธ๐ฆ,๐(2)} = {
๐ด๐ฆ,๐(1)๐โ(๐๐๐ฆ)
๐ด๐ฆ,๐(2)๐ถโ(๐๐๐ฆ)
} ๐๐๐(๐๐ฅ๐ฅ)๐๐๐(๐๐ง๐ง)
๐ธ๐ง,๐(1)
๐ธ๐ง,๐(2)} = {
๐ด๐ง,๐(1)๐ถโ(๐๐๐ฆ)
๐ด๐ง,๐(2)๐โ(๐๐๐ฆ)
} ๐๐๐(๐๐ฅ๐ฅ)๐ถ๐๐ (๐๐ง๐ง)
(3.3)
where ๐ด๐ฅ,๐ฆ,๐ง,๐(1,2)
are arbitrary amplitudes, ๐1,3 = ๐พ,๐2 = ๐๐ฆ and
๐พ = โ๐2(1 โ ํโฒ๐โฒ) + ๐๐ฆ2, ๐ =
๐
๐= โ๐๐ฅ
2 + ๐๐ง2 โ ๐๐ฆ
2, (3.4)
The boundary conditions on the perfectly conducting sidewalls are
๐ธ๐ฅ,๐ = ๐ธ๐ฆ,๐ = 0
๐ธ๐ฆ,๐ = ๐ธ๐ง,๐ = 0 at
๐ง = 0, ๐ง = ๐๐ฅ = 0, ๐ฅ = ๐
, ๐ = 1,2,3 (3.5)
To satisfy these conditions horizontal wavenumbers should be in the form of ๐๐ฅ =
(๐ โ 1)๐
๐, ๐๐ง = (๐ โ 1)
๐
๐, ๐, ๐ = 1,2,3โฆ. ๐๐ฆ transverse vertical wavenumber should be
determined from boundary condition on low-conductive layer surfaces and vertical walls.
On vacuum-low conductive layer borders, from the continuity of tangential electric
and magnetic field components
๐ธ๐ฅ,1 = ๐ธ๐ฅ,2, ๐ธ๐ง,1 = ๐ธ๐ง,2๐ต๐ฅ,1 = ๐ต๐ฅ,2, ๐ต๐ง,1 = ๐ต๐ง,2
at ๐ฆ = ๐/2 (3.6)
๐ธ๐ฅ,2 = ๐ธ๐ฅ,3, ๐ธ๐ง,2 = ๐ธ๐ง,3๐ต๐ฅ,2 = ๐ต๐ฅ,3, ๐ต๐ง,2 = ๐ต๐ง,3
at ๐ฆ = โ๐/2 (3.7)
- 58 -
On the outer boundaries between low conductive layers and ideal conductors,
electrical tangential components of fields must vanish
๐ธ๐ฅ,1 = 0, ๐ธ๐ง,1 = 0
๐ธ๐ฅ,3 = 0, ๐ธ๐ง,3 = 0 at
๐ฆ = ๐โฒ/2
๐ฆ = โ๐โฒ/2 (3.8)
Using Maxwell's equation ๐๐๐ฃ๏ฟฝโโ๏ฟฝ๐(1,2) = 0, the amplitudes ๐ด๐ง,๐
(1,2) can be expressed by
the amplitudes ๐ด๐ฅ,๐(1,2)
and ๐ด๐ฆ,๐(1,2)
. Thus, we have a linear homogeneous system consisting
of 12 equations (Eq. 3.6 - Eq. 3.8) and containing the same number of indeterminate
coefficients: ๐ด๐ฅ,๐(1,2)
, ๐ด๐ฆ,๐(1,2)
with ๐ = 1,2,3. Condition of existence of non-trivial solutions of this
system is the vanishing of its determinant
๐ท = ๐1๐2๐3๐4 = 0 (3.9)
where
๐1 = ๐พ๐๐๐ถโ(๐๐พ)๐โ(๐๐๐ฆ๐) + ๐๐ฆ๐๐โ(๐๐พ๐๐)๐ถโ(๐๐๐ฆ๐) (3.10a)
๐2 = ๐พ๐๐๐ถโ(๐๐พ๐๐)๐ถโ(๐1๐๐ฆ๐) + ๐๐ฆ๐๐โ(๐๐พ๐๐)๐โ(๐1๐๐ฆ๐) (3.10b)
๐3 = โ๐พ๐๐๐2๐โ(๐๐พ๐๐)๐ถโ(๐๐๐ฆ๐) โ ๐๐ฆ๐๐
2ํโฒ๐โฒ๐ถโ(๐๐พ๐๐)๐โ(๐๐๐ฆ๐) (3.10c)
๐4 = โ๐๐ฆ๐๐2ํโฒ๐โฒ๐ถโ(๐๐พ๐๐)๐ถโ(๐๐๐ฆ๐) โ๐พ๐๐๐
2๐โ(๐๐พ๐๐)๐โ(๐๐๐ฆ๐) (3.10d)
where ๐พ๐๐ = โ๐๐ฅ๐2 + ๐๐ง๐
2 โ ๐2ํโฒ๐โฒ, ๐ =๐
๐= โ๐๐ฅ๐
2 + ๐๐ง๐2 โ ๐๐ฆ๐
2 , ๐,๐, ๐ =
1,2,3,โฆ.
The result is four independent dispersion equations:
๐๐ = 0, ๐ = 1,2,3,4 (3.11)
Each of these four equations includes three eigenvalues: ๐๐ฅ๐, ๐๐ง๐ and ๐๐ฆ๐ .
Moreover, the horizontal transverse ๐๐ฅ๐ and longitudinal ๐๐ง๐ wave numbers are fixed for
given numbers ๐ and ๐. They form two infinite sequences generated by integer indices ๐
and ๐ . Thus, each of equations (Eq. 3.10.a -3.10d) defines the sequence of ๐๐ฆ๐ , (๐ =
1,2,3, โฆ ) values for each fixed combination of indices ๐ and ๐. As a result, from these
- 59 -
equations we get dependent on ๐ and ๐ sequences of eigenvalues ๐๐ฆ = ๐๐๐๐(๐) (๐ = 1,2,3,4)
for = 1,2,3โฆ .
Let us compare the obtained expressions with the corresponding expressions,
obtained in Chapter 1, for the two-layer flat structure (1.24):
๐1 = ๐๐ฆ(1)๐ถโ(๐๐๐ฆ
(1))๐โ(๐๐๐ฆ(0)) + ๐๐ฆ
(0)๐โ(๐๐๐ฆ(1))๐ถโ(๐๐๐ฆ
(0))
๐2 = ๐๐ฆ(1)๐ถโ(๐๐๐ฆ
(1))๐ถโ(๐๐๐ฆ(0)) + ๐๐ฆ
(0)๐โ(๐๐๐ฆ(1))๐โ(๐๐๐ฆ
(0))
๐3 = โ๐๐ฆ(0)๐2ํ๐
โฒ๐๐โฒ๐ถโ(๐๐๐ฆ
(1))๐โ(๐๐๐ฆ(0)) โ ๐๐ฆ
(1)๐2๐โ(๐๐๐ฆ(1))๐ถโ(๐๐๐ฆ
(0))
๐4 = โ๐๐ฆ(0)๐2ํ๐
โฒ๐๐โฒ๐ถโ(๐๐๐ฆ
(1))๐ถโ(๐๐๐ฆ(0)) โ ๐๐ฆ
(1)๐2๐โ(๐๐๐ฆ(1))๐โ(๐๐๐ฆ
(0))
(3.12)
Obviously, there is a correspondence: ๐๐ โ ๐๐ , ๐ = 1,2,3,4 . A complete coincidence
cannot take place, since in the case of an infinite two-dimensional structure, the
variables ๐๐ฅ and ๐๐ง are integration variables, but in the case of a resonator, the
wave numbers ๐๐ฅ๐ and ๐๐ง๐ are discrete and fixed for specified values by the index
n and m.
The standing wave excited in the resonator consists of the imposition of two
traveling waves propagating in opposite directions: a wave incident on the front wall of the
resonator (๏ฟฝโโ๏ฟฝ1) and reflected from it (๏ฟฝโโ๏ฟฝ2):
๐ธ๐ฅ,๐ฆ,๐ง = ๐ธ1,๐ฅ,๐ฆ,๐ง ยฑ ๐ธ2,๐ฅ,๐ฆ,๐ง, ๐ธ1,2,๐ฅ,๐ฆ,๐ง = ๐ด1,2,๐ฅ,๐ฆ,๐ง๐๐(ยฑ๐๐ง๐งโ๐๐ก) (3.13)
The wave ๏ฟฝโโ๏ฟฝ1 can serve as an analog of a wave propagating in an infinite planar two-layer
structure, generated by a particle. Since its frequency ๐ in this case is complex, it can be
represented in the form:
๐ = ๐โ๐ด + ๐๐ต = ๐โโ๐ด2+๐ต2+๐ต
2+ ๐๐ ๐๐๐๐ ๐ตโ
โ๐ด2+๐ต2โ๐ต
2 (3.14)
with ๐๐ฆ๐ = ๐๐ฆ๐โฒ + ๐๐๐ฆ๐
โฒโฒ , ๐ด = ๐๐ฅ๐2 + ๐๐ง๐
2 โ ๐๐ฆ๐โฒ2 + ๐๐ฆ๐
โฒโฒ2, ๐ต = 2๐๐ฆ๐โฒ ๐๐ฆ๐
โฒโฒ
The real part of (3.14) determines the phase velocity of the wave, and the imaginary part
characterizes its damping. The phase velocity of the wave, incident on the front wall of the
resonator, is determined by the following relation:
- 60 -
๐ฃ๐โ =๐ ๐{๐}
๐๐ง (3.15)
This speed can be either more (at ๐๐ฅ๐2 > ๐๐ฅ0๐
2 ) or less (at ๐๐ฅ๐2 < ๐๐ฅ0๐
2 ) than the speed of
light, where
๐๐ฅ0๐2 = ๐๐ฆ๐
โฒ2 โ ๐๐ฆ๐โฒโฒ2 โ ๐๐ง๐
2 ยฑ 2๐๐ง๐โ2๐๐ฆ๐โฒ ๐๐ฆ๐
โฒโฒ + ๐๐ง๐2 (3.16)
Equality ๐๐ฅ๐2 = ๐๐ฅ0๐
2 (at which the wave has a phase velocity equal to the speed of light)
cannot be fulfilled exactly because ๐๐ฆ๐โฒ2 , ๐๐ฆ๐
โฒโฒ2 and ๐๐ง๐2 are discrete and fixed. The condition
for the greatest closeness of the phase velocity of a wave to the velocity of light, thus, is
the requirement ๐๐๐|๐๐ฅ๐2 โ๐๐ฅ0๐
2 |. The mode (or set of modes) that satisfies this requirement
forms the main resonance arising as a result of the interactions in it of elementary free
oscillations.
3.3 Pure copper cavity. Measurements and interpretation
To check the accuracy of manufacturing, firstly an experimental resonances of pure
copper cavity are compared with theoretical results for the cavity.
The resonant behavior of pure copper cavity has been investigated experimentally
in CANDLE SRI by measuring the testing device (cavity) transmission S-parameter ๐บ๐๐
(forward voltage gain) [69] using Network Analyzer ZVB14 [70]. The measurements are
performed for the rectangular cross-section copper cavities.
Measured narrow-band extremes are compared with the standing wave
eigenfrequencies of rectangular cavity with perfectly conducting walls [72] with same
dimensions (๐โฒ = 22, 32, 42 ๐๐, ๐ = 65๐๐, ๐ = 200 ๐๐):
๐๐๐๐ =๐
2โ(
๐โ1
๐โฒ)2
+(๐โ1
๐)2
+ (๐โ1
๐)2
๐, ๐, ๐ = 1,2,3โฆ {
๐๐ ๐ = 1, ๐, ๐ > 1 ๐๐ ๐ = 1, ๐, ๐ > 1๐๐ ๐ = 1, ๐, ๐ > 1
(3.17)
Note that in IHLCC cavity when ๐ = 0, both equations (๐1 and ๐2) are transformed
to equation for the vertical wave number of the ideal resonator (Eq. 3.12): ๐๐๐โ(๐๐๐ฆ) = 0
with solution ๐๐ฆ = ๐ ๐(๐ โ 1) ๐โ . Similarly, in the limiting case of ๐พ = ๐๐ฆ (ํโฒ๐โฒ = 1) we have
- 61 -
both from (Eq. 3.10.a) and (Eq. 3.10.b): ๐๐๐โ(๐โฒ๐๐ฆ) = 0 with solution ๐๐ฆ = ๐ ๐(๐ โ 1) ๐โฒโ .
In these extreme cases, all three eigenvalues are independent from each other.
๐โฒ = 22 ๐๐ ๐โฒ = 32 ๐๐
N m n q Exp. Calc. N m n q Exp. Calc.
1 2 2 6 8.04 8.116 1 1 1 12 8.25 8.25
2 2 2 6 8.12 8.116 2 1 4 8 8.7 8.69
3 2 2 7 8.48 8.489 3 1 1 13 9.06 9
4 2 2 8 8.86 8.909 4 3 1 2 9.4 9.405
5 1 2 13 9.31 9.291 5 1 3 12 9.46 9.453
6 2 2 9 9.39 9.371 6 3 2 2 9.69 9.684
7 2 1 10 9.56 9.594 7 2 3 11 9.97 9.976
8 1 4 10 9.65 9.670 8 3 2 6 10.36 10.358
9 1 5 6 9.92 9.963 9 3 3 5 10.88 10.872
10 1 2 14 10.04 10.019 10 3 2 8 10.97 10.99
11 2 4 5 10.15 10.169 11 1 6 1 11.58 11.539
12 1 5 7 10.28 10.269 12 1 6 3 11.65 11.636
13 1 1 15 10.51 10.50 13 3 4 3 11.76 11.750
14 1 5 8 10.6 10.619 14 1 6 5 11.91 11.922
15 1 5 9 11.01 11.009 15 1 4 14 11.96 11.960
16 1 3 15 11.46 11.470 16 3 3 9 12.05 12.050
17 1 6 5 11.93 11.922 17 1 3 16 12.16 12.160
18 1 5 12 12.3 12.38 18 3 3 10 12.43 12.440
19 2 1 15 12.53 12.519 19 3 2 12 12.7 12.700
20 2 5 8 12.605 12.62 20 3 1 13 13.0 12.996
21 2 2 15 12.71 12.73 21 2 2 17 13.11 13.088
22 1 5 13 12.9 12.892 22 3 4 10 13.48 13.468
23 2 1 16 13.14 13.155
24 2 3 15 13.33 13.343
25 2 6 3 13.47 13.486
26 2 4 6 13.57 13.59
27 2 5 11 13.7 13.709
28 2 1 17 13.8 13.802
29 2 3 16 13.94 13.941
30 1 5 15 13.98 13.980
Table 3.1.a: Comparison of the measured (Exp., GHz) and calculated (Calc., GHz) values of the
eigenfrequencies of a rectangular cavity with a height of 22 ๐๐ and 32 ๐๐.
Table 3.1 shows the correlation of eigenfrequencies of the cavity with the perfectly
conducting walls, calculated by Eq. 3.12, with measured peaks.
- 62 -
๐โฒ = 42 ๐๐
N m n q Exp. Calc. N m n q Exp. Calc.
1 1 4 7 4.12 4.129 10 1 3 10 8.18 8.177
2 1 4 8 4.33 4.344 11 1 2 12 8.56 8.5666
3 1 5 2 4.65 4.63 12 2 1 12 8.98 8.9899
4 1 2 7 5.04 5.057 13 1 4 9 9.17 9.1601
5 1 3 5 5.52 5.505 14 1 3 12 9.45 9.4533
6 1 3 16 6.08 6.080 15 2 5 2 9.93 9.9260
7 2 3 5 6.66 6.562 16 3 3 9 10.42 10.408
1 1 10 6.75 17 3 3 10 10.84 10.857
8 1 4 5 7.55 7.545 18 3 4 7 10.90 10.918
9 2 3 8 7.85 7.850
Table 3.1.b: Comparison of the measured (Exp., GHz) and calculated (Calc., GHz) values of the
eigenfrequencies of a rectangular cavity with 42 ๐๐ height.
As can be seen from Table 3.1, there is a good matching (up to the second decimal)
between experimentally obtained and calculated eigenvalues. This is achieved due to the
high precision machining of internal surfaces of the cavity with keeping dimensional
accuracy.
The main result of the above research is the establishment of agreement of
experimentally obtained narrowband peaks with resonant frequencies of the rectangular
copper cavities. This means that the accuracy of manufacturing of copper resonator
component allow to obtain the corresponding frequency distribution for the IHLCC cavity
with acceptable distortions.
3.4 IHLCC cavity resonance frequency comparison with experimental results
The transmission ๐บ๐๐ parameters of IHLCC cavity has been measured to perform a
comparison between resonant behaviors of real structure and IHLCC.
The defining parameters of the cavity are the following: the height of the inner
chamber ๐, the thickness of the low conductive layer ๐ =๐โฒโ๐
2, the width ๐ and length ๐ of
the chamber, conductivity ๐1 and relative dielectric constant ํ1 of low conductive layer.
The choice of material and the thickness of a low-conductive metallic layer,
covering the top and bottom walls of cavity is conditioned by following two factors:
- 63 -
1. network analyzer ZVB14 provides spectral distribution in the range of 0 โ 14 ๐บ๐ป๐ง.
The resonance frequency is determined roughly by the geometrical parameters of
the cross-section of the cavity: ๐๐๐๐ง โ๐
2๐โ2/๐๐, which depends on the thickness of
the inner layer ๐ and structure height ๐. Having a bunch in a structure imposes the
lower limit on its height (๐~1 ๐๐). Also with the increase of the height the impedance
decreases, so the height should be near to its lower limit. For a structure with 2 โ
4 ๐๐ height the thickness of the coating should be about 1 โ 0.5 ๐๐ to have a
resonance frequency in 0 โ 14 ๐บ๐ป๐ง range, given by measuring devise.
2. for the double-layered parallel plates the condition of resonance ํ โ ๐0๐1๐~1 is
derived in chapter 2. Assuming that for the resonator the condition should be
approximately the same, the low conducting material is chosen to have ๐1~2.65 ๐๐โ
conductivity for a layer with ๐ = 1 ๐๐ thickness.
As a result, germanium (Ge) crystal plates with sizes ๐ = 30 c๐, ๐ = 6 c๐, ๐ =
1 ๐๐ with conductivity ๐1 = 2 ฮฉโ1๐โ1 [73] and the relative dielectric constant ํ1 = 16,
were selected. Measurements were carried out for three different heights: ๐ = 2 ๐๐, ๐ =
3 ๐๐ and ๐ = 4 ๐๐ (๐โฒ = 2.2, ๐โฒ = 3.2 and ๐โฒ = 4.2 ๐๐, respectively).
For the above-given vertical apertures and the Ge layer thickness, the skin depth at
resonant frequency ๐ฟ is larger than the Ge layer thickness by factor of 3 (๐ฟ/๐ > 3). The
predicted resonant frequencies of the corresponding two parallel plates lie in the X-Band
region between 9 โ 15 ๐บ๐ป๐ง.
3.4.1 Comparison with flat two-layer structure
The measured resonant frequencies of the test cavity with laminated horizontal
walls have been compared with resonant frequencies of the corresponding two-
dimensional two-layered structure (outer layer is assumed as a perfect conductor) with
inner 1 ๐๐ Ge layers.
The longitudinal impedance of two parallel infinite laminated plates is derived in the
second chapter of the dissertation. For an ultrarelativistic particle moving parallel to the
plates along geometrical axis of the structure with perfectly conducting outer and low
- 64 -
conducting inner metallic layers the longitudinal impedance can be presented as (Eq. 2.15,
Eq. 2.33, Eq. 2.42 and Eq. 2.43)
๐โฅ =๐0
16๐2โซ
๐๐ข(๐2โ๐ข2)๐โ(2๐๐)
๐1๐2
โ
โโ๐๐ข (3.18)
where
๐1 = ๐๐ถโ(๐๐)๐ถโ(๐ข๐) + ๐ข๐โ(๐๐)๐โ(๐ข๐)
๐2 = (๐ข2 + ๐2 โ ๐2)๐ถโ(๐๐)๐ถโ(๐ข๐) +
๐พ๐2
๐๐โ(๐๐)๐โ(๐ข๐)
(3.19)
with ๐1 = ๐ 2โ , ๐ = ๐ ๐โ longitudinal wavenumber and
๐พ = โ๐ข2 + ๐2(1 โ ํ1) โ ๐๐๐0๐ (3.20)
Figure 3.2 presents the longitudinal impedance curves for such parallel laminated
plates with 2 ๐๐, 3 ๐๐ and 4 ๐๐ vertical apertures. The distances between the plates and
material of the inner layers (germanium) are the same as in the tested cavities. The
dashed lines indicate the measured main resonant frequencies of the corresponding
copper cavities with laminated top and bottom walls.
Figure 3.2: The longitudinal impedance of perfect conductor-Ge infinite parallel plates
(solid lines) and measured resonant frequencies of copper cavity with Ge inner layers at
the top and bottom walls (dashed lines). The Ge layer thickness is 1 ๐๐, the vertical
apertures are 2 ๐๐, 3 ๐๐ and 4 ๐๐.
- 65 -
Figure 3.2 shows that there is a good agreement between measured and calculated
resonant frequencies. The relative difference in resonant frequency is 2.6 % for 2 ๐๐ ,
3.5 % for 3 ๐๐ and 7 % for 4 ๐๐ aperture cavities. The difference is caused mainly due to
the limited volume of the cavity: the presence of highly conductive sidewalls. The presence
of additional bursts is the result of the display of modal structure of the test cavity. Their
contribution is weakened by decreasing of the distance between plates. Thus, by
decreasing the height of cavity, its resonant characteristics approach to the characteristics
of a flat structure consisting of two identical two-layer parallel plates.
3.4.2 Comparison with IHLCC resonator theoretical results
Resonant frequencies of the cavity with outer perfectly conducting walls and inner
germanium layers at the top and bottom walls can be obtained from Eq. 3.11.
Experimentally fixed resonances (Figure 3.5) can be identified as concrete solutions
(modes) of dispersion relations given by Eq. 3.11. Choosing the needed mode (๐๐ฅ, ๐๐ง) the
main resonances frequencies can be obtained for our IHLCC cavity with ideally conductive
walls. Table. 3.2 shows comparison of experimentally observed resonance frequencies of
the test cavity (โExperimentโ in table) and analytically calculated resonances of cavity with
outer perfectly conducting walls (โCalculatedโ in table).
Aperture
2
Experiment
4
5
Calculated Rel. dif.
2 ๐๐
12.32 ๐บ๐ป๐ง
12.29 ๐บ๐ป๐ง
0.2 %
3 ๐๐ 11.42 ๐บ๐ป๐ง
11.36 ๐บ๐ป๐ง
0.5 %
4 ๐๐
10.6 ๐บ๐ป๐ง
10.49 ๐บ๐ป๐ง
1 %
Table. 3.2: The main resonances obtained experimentally and calculated for a cavity with
perfectly conducting walls and inner Ge layers at the top and bottom walls.
0.251
0.339
0.754
1.001
1.257
1.508
1.760
As can be seen from Table. 3.2, there is a good agreement on measured and
calculated results. The relative differences between the calculated and measured
frequencies (โRel. dif.โ in table) are below 1 %.
As it was noted earlier (see section 3.2), free electromagnetic waves formed in the
resonator form standing waves, which can be represented as the superimposition of two
oppositely directed traveling waves: incident on the front wall of the resonator and going in
- 66 -
the opposite direction (see Eq.3.13). The phase velocities of these waves are equal in
absolute value and opposite in sign (since they propagate in opposite directions): ๐ฃ๐โ =
๐ ๐ ๐{๐} ๐๐งโ with ๐ =๐
๐= โ๐๐ฅ2 + ๐๐ง2 โ ๐๐ฆ2 ; ๐๐ฅ and ๐๐ง are real numbers determined by the
indices m and n and ๐๐ฆ is determined by the equations (3.10) and is a set of complex
numbers. Thus, k is a complex number, the real part of which determines a discrete series
of eigenfrequencies of the resonator, and its imaginary part, caused by the finite
conductivity of the inner layers covering the inside walls, determines the finite Q of the
resonator at its eigen frequencies. Determining k from the equations (3.10), we can
determine the dependence of the phase velocity of the partial traveling waves in the cavity,
depending on the frequency. As noted earlier, the exact equality of the phase velocity of
the partial wave in the resonator with the speed of light is unlikely. Investigations of the
solutions of the equations (3.10) show that in the frequency range 0-14 GHz with the
corresponding cavity parameters, only equation ๐4 = 0 (3.10d) give solutions for slow
partial waves. The corresponding graphs are shown in the Figure 3.3.
Figure 3.3: The phase velocities of the partial waves in the resonator for modes ๐4,๐๐๐ (solutions of
equation ๐4 = 0) for ๐ = 2, ๐ = 1,2, ๐ = 1,2,3,4,โฆ; ๐ = 1 ๐๐ (left), 1.5 ๐๐ (middle), ๐ = 2 ๐๐ (right).
Each of the graphs shows two sequences (discrete curves) of phase velocities of
the modes: ๐4;21๐ and ๐4;2,2๐ with monotonically decreasing phase velocities. On all three
graphs, a discrete curve depicting a sequence of modes ๐4;2,2๐ intersects the
synchronization line ๐ฃ๐โ ๐โ = 1. The second discrete curve containing the phase velocities
of the modes ๐4;21๐ is located above the line ๐ฃ๐โ ๐โ = 1, gradually approaching it as the
frequency increases and intersecting with the first line. In the first and second cases
(for ๐ = 1๐๐ and ๐ = 1.5๐๐ ), the measured fundamental resonance frequency
corresponds to the intersection point of two discrete lines adjacent to the synchronization
- 67 -
line and calculated (for two โ dimensional planar case) is closed to intersection point of
first curve and line of synchronization. In the third case, the situation is reversed: the
measured fundamental resonance frequency corresponds to intersection point of first
curve and line of synchronization and calculated is closed to the intersection point of two
discrete lines.
It is obvious that in all three cases the main resonance is formed by several modes
with closed frequencies and close in phase velocity: modes ๐4;2,1,15,
๐4;2,1,16, ๐4;2,2,15, ๐4;2,2,16, ๐4;2,2,17 for ๐ = 1๐ , ๐4;2,1,13, ๐4;2,1,14, ๐4;2,1,15, ๐4;2,213, ๐4;2,2,14, ๐4;2,2,15
for ๐ = 1.5๐๐ and ๐4;2,1,13, ๐4;2,1,14, ๐4;2,1,15, ๐4;2,213, ๐4;2,2,14, ๐4;2,2,15 for ๐ = 2๐๐.
3.5 Summary
The chapter is the logical follow-up of the study of high frequency single-mode
accelerating structures with laminated metallic walls (flat and cylindrical) [50, 75, 76].
Theoretical model of a rectangular cavity with horizontal double layer metallic walls
is created and electro-dynamic properties of the model are studied. The model with
sufficient accuracy displays experimental tested device and enables clearly interpret the
measurement results. Extreme frequencies on the experimental curves have been
correlated with the respective modes of resonator model. The experimental results are
also compared with theoretical results for two infinite parallel plates. A good agreement
between them stimulates the experimental study of beam radiation and acceleration in
laminated structures. The corresponding experimental program at AREAL test facility is
foreseen.
- 68 -
Chapter 4: Electron bunch compression in single-mode
structure
4.1 Introduction
The generation and acceleration of ultrashort electron bunches is an important
issue for generation of coherent radiation in THz and Infrared regions, driving the free
electron lasers (FEL) and direct applications for ultrafast electron diffraction [14-16, 77-79].
Length of bunches generated in RF guns is limited due to technical characteristics
of photocathode and lasers. So for having a sub-ps bunches one need to shorten ps
bunches generated in RF gun. There are several methods for obtaining sub-ps bunches,
e.g. magnetic chicanes are used for high energy beams shortening [80], velocity and
ballistic bunching are for low energy beams [26-29]. For all these methods one needs to
have an energy modulated beam.
It is well known that the relativistic charge, moving along the structure, interacts with
the surrounding environment and excites the electromagnetic field known as wakefield
[36-38, 59-61]. Structures are usually the disk, plasma or dielectric loaded channels that
drive the corresponding wakefield accelerator (WFA) concepts [25, 34, 35, 37, 55, 57], the
performance of which can be improved by using the asymmetric driving beam to achieve
high transformer ratio [45, 83]. For the charge distribution the wakefield acts back to
bunch particles producing an energy modulation within the bunch by longitudinal wake
potential. The form of longitudinal wake potential depends both on the bunch charge
distribution and the surrounding structure [36]. Although for Gaussian bunch the energy
modulation is driven by the bunch head-tail energy exchange, for rectangular and
parabolic bunch shapes the energy modulation at high frequency can be obtained.
To have an energy modulation disc and dielectric loaded structures are used.
These kind of structures have high order modes excited by a charge when it passes the
structure, which leads to energy storage in that parasitic modes and beam instabilities [35-
46]. To avoid such effects one needs a single-mode structure. In this chapter wakefield
induced bunch energy modulation during bunch interaction with the single-mode
- 69 -
structures is studied. As a single mode structure cold plasma and recently proposed
internally coated metallic tube (ICMT) are considered [47-50]. In a cold plasma purely
sine-like wakefield without damping are generated, while in ICMT wakefields are sine-like
with damping factor [50].
The direct application of the holding integral for bunch rectangular or parabolic
shape and point wake potential lead to energy modulation within the bunch at the excited
mode frequency. To obtain sub-๐๐ bunches, a structure with excited frequency higher
than ๐๐ป๐ง is needed. In cold plasma and ICMT, for appropriate parameters, the excited
mode is within sub-๐๐ป๐ง to ๐๐ป๐ง frequency region.
For comparatively low energy bunches wakefield generated energy modulation
causes velocity modulation and for proper ballistic distance leads to density modulation or
microbunching. To have that effect in proper distance, energy modulation in a bunch after
the structure should be greater than several ๐๐๐, which can be achieved with 100 ๐๐ โ
500 ฮผ๐ bunches. Electron bunches with 100 ๐๐ถ charge and 10 ๐๐๐ energy are observed,
which corresponds to usual values in modern linear accelerators [84].
4.2 Wakefields and Impedances
A relativistic charge ๐ , passing throw a structure interacts with it. Due to this
interaction the particle radiates electromagnetic fields, known as wakefields, which then
can act to the test charge ๐. From the causality principle, wakefields generated by an
ultrarelativistic driving charge should vanish in front of it [37]. The longitudinal component
of the wakefield produces energy gain (or loss) of the test particle, while the transverse
component produces the transverse kick of the test particle.
- 70 -
Figure 4.1: Driving ๐ and test ๐ charge moving in a structure
The longitudinal wake function (point wake potential) is the integrated longitudinal
Lorentz force excited in a structure by a point charge ๐ experienced by the test particle
traveling on an ๐ distance behind it [36]
๐ค๐ง(๐, ๐๐ธ, ๐ ) = โ1
๐โซ ๐ธ๐ง (๐, ๐๐ธ, ๐ง, ๐ก =
๐ง+๐
๐)๐๐ง
โ
โโ (4.1)
where ๐ธ๐ง is the longitudinal component of excited electric field, ๐ is a transverse offset of
test particle, ๐๐ธ is a transverse offset of source particle and ๐ is the speed of light in
vacuum (Figure 4.1). In the definition ๐ > 0 corresponds to the distance behind the driving
point charge, ๐ = 0 to the position of the charge and ๐ < 0 to the distance in front of the
charge.
The transverse wake function is defined as an integrated transverse Lorentz force
acting on a test particle
๐โฅ(๐, ๐๐ธ, ๐ ) = โ1
๐โซ (๐ฌ + ๐ ร ๐ฉ)โฅ (๐, ๐๐ธ, ๐ง, ๐ก =
๐ง+๐
๐) ๐๐ง
โ
โโ (4.2)
For a longitudinal distributed ๐(๐ ) driving bunch, the wake potentials are defined as
an integrated longitudinal and transverse Lorentz forces excited by a bunch at the
position of the test particle traveling on an ๐ distance behind it. Here distance ๐ is
- 71 -
measured from the center of driving bunch. The longitudinal wake potential could be
obtained from the wake functions using the convolution integral
๐โฅ(๐ ) = โซ ๐ค๐ง(๐ โ ๐ โฒ)๐(๐ ) ๐๐ โฒ๐
โโ (4.3)
here integration is from โโ to ๐ , because there is no field in front of driving bunch.
The total energy loss of the driving charge in a structure due to its radiation is
proportional to the square of the charge
ฮ๐ = ๐2๐พ๐๐๐ ๐ (4.4)
The proportionality factor ๐พ๐๐๐ ๐ is a longitudinal loss factor and can be presented by
wake potentials
๐๐๐๐ ๐ = โซ ๐โฅ(๐)๐(๐)๐๐โ
โโ (4.5)
The wake potential is a function of time, while Fourier transformation of the wake
potential gives an information in frequency domain and is called the impedance [36, 47]
๐โฅ(๐) = โซ ๐๐ง(๐)๐โ๐๐๐๐๐
โ
โโ=1
๐โซ ๐๐ง(๐ง)๐
โ๐๐๐ง๐โ ๐๐ง
โ
โโ
๐๐ง(๐) =1
2๐โซ ๐โฅ(๐)๐
๐๐๐๐๐โ
โโ
(4.6)
4.3 Ballistic bunching
Traveling in the structure bunch particles gain (loss) energy due to interaction with
wakefields generated by themselves and particles in front of them. Ballistic bunching
method of studying microbunching and bunch compression is used. In ballistic method it is
assumed that in the structure charge is rigid and only energy modulations occur. The
energy modulation leads to particles velocity change in the structure and for non-
ultrarelativistic bunches charge density redistribution occurs in drift space after the
structure (Figure 4.2). For appropriate initial bunch energy and energy spread after the
structure, bunches with several ๐๐ length can be formed.
- 72 -
Figure 4.2: Geometry of ballistic bunching (blue line illustrates the wakefield)
After bunch passes the structure energy gain (loss) of ๐-th particle with initial ๐ ๐
coordinate is determined by
ฮ๐(๐ ๐) = ๐๐๐๐ง(๐ ๐)๐ฟ (4.7)
where ๐ is a charge of test particle, ๐ is a total charge of a source bunch, ๐๐ง(๐ ๐) is a wake
potential in a structure and ๐ฟ is a structure length.
Velocity of ๐-th particle on the entrance of drift space can be presented as
๐ฝ(๐ ๐) = โ1 โ (๐๐๐๐ ๐ก
๐๐(๐ ๐))2
(4.8)
where ๐๐๐๐ ๐ก is a particle rest energy and ๐๐(๐ ๐) = ๐0 + ฮ๐(๐ ๐) is an energy of ๐-th particle
after the structure. Here cold bunch approximation is assumed, which means that there is
no energy spread in the initial bunch. Having a velocity of each particle at the end of the
structure, the coordinate ๐ ๐,๐ of ๐-th particle after passing ๐๐๐๐๐๐ก distance in a drift space can
be calculating by
๐ ๐,๐ = ๐ ๐ + ฮ๐ ๐ = ๐ ๐ + ๐๐๐๐๐๐ก (1 โ๐ฝ(๐ ๐)
๐ฝ0) (4.9)
- 73 -
where ๐ ๐ is the initial coordinate of ๐ -th particle, ๐ฝ0 = โ1 โ (๐๐๐๐ ๐ก
๐0)2
is the bunch initial
velocity, ๐ฝ(๐ ๐) is the velocity of ๐-th particle after passing the structure. With the coordinate
of each particle in a bunch after a drift space, one can obtain the bunchโs shape.
Numerical simulations are done for observing wakefield effects on various line
charge distributions. The bunch is presented as an ensemble of ๐ particles that
experience the energy deviations due to longitudinal wakefield. For simulations ๐ = 106
particles have been used.
4.4 Energy modulation, bunch compression and microbunching in cold
neutral plasma
Bunch charge density modulation in infinite cold neutral plasma is studied via
ballistic bunching method. It is assumed that the electrons temperature is much higher
than the ions temperature (cold plasma). Under this assumption the plasma can be
considered as a medium with ํ = 1 โ๐๐2
๐2โ macroscopic dielectric constant, where the
plasma frequency is ๐๐ = 56.4 โ 103โ๐๐ ๐ป๐ง, with ๐๐ being the plasma electron density (in
๐๐โ3 units) [47]. The wake function of the relativistic point charge, passing the cold neutral
plasma, is given by [47, 48]
๐ค๐ง(๐ ) = โ2๐ป(๐ )๐พ๐๐๐ ๐ cos(๐๐๐ ) (4.10)
where ๐ป(๐ ) is the Heaviside step function, ๐๐ =๐๐
๐โ is the wave number, ๐พ๐๐๐ ๐ is the
longitudinal loss factor of plasma channel.
- 74 -
Figure 4.3: Point charge wake function in cold neutral plasma (๐๐ = 56.4 โ 103โ๐๐ ๐ป๐ง and
๐พ๐๐๐ ๐ = 45๐๐
๐/๐๐ถ)
Numerical simulations of bunch compression and microbunching have been
performed for the driving bunch with 100 ๐๐ถ charge and 10 ๐๐๐ initial energy.
Wake potential generated by a Gaussian bunch in plasma channel can be obtained
by Eq. 4.2
๐โฅ(๐ ) =1
โ2๐๐โซ ๐ค๐ง(๐ โ ๐ โฒ)๐
โ๐ 2
2๐2 ๐๐ โฒ๐
โโ (4.11)
Figure 4.4 presents the wake potential of the Gaussian charge distribution with
0.1 ๐๐ rms bunch length in plasma channel with ๐๐ =๐๐
2๐โ = 1.43 ๐๐ป๐ง plasma frequency
(wavelength ๐๐ = 0.2 ๐๐).
- 75 -
Figure 4.4: Gaussian bunch wake potential.
Figure 4.4 shows that the wake potential is decelerating at bunchโs head (๐ < 0)
and accelerating at bunchโs tail (๐ > 0). This leads to bunch compression till some critical
drift space distance, after which particles with coordinates ๐ > 0 overtake particles with
coordinates ๐ < 0. For distances bigger than the critical distance bunch starts to expand
and debunching process starts. From Figure 4.4 it can be seen that the central part
(~70%) of the bunch sees focusing wake, while bunchโs tail sees a decelerating wake.
Compressed bunch shape at various drift space distances after 2 ๐ plasma channel
is shown in Figure 4.5. The dashed line represents the initial Gaussian bunch distribution.
It can be seen, that till 3.5 ๐ drift space length bunch compresses, after which it starts to
expand. On 3.5 ๐ drift space length the rms length of bunchโs central part that contains
70 % of the particles is 15 ๐๐.
- 76 -
a
b
c
d
Figure 4.5: Line charge distribution after bunch passes ๐๐๐๐๐๐ก distance in drift space
๐) ๐๐๐๐๐๐ก = 1.5 ๐, b) ๐๐๐๐๐๐ก = 3.5 ๐, c) ๐๐๐๐๐๐ก = 4 ๐, d) ๐๐๐๐๐๐ก = 4.5 ๐
To find the maximum compression, rms length dependence of distance is
calculated. rms length of the Gaussian bunch 70 % (central part) and 100 % particles
versus bunch travel distance in drift space is shown in Figure 4.6. As it was mentioned,
bunchโs tail sees defocusing wakefield. This defocused part artificially increases rms
length of a whole bunch and it always stays above 70 ๐๐ . For 70 % bunch particles
maximum bunch compression is reached at 4 ๐ distance (bunch rms length is ~ 12 ๐๐)
after which the debunching process starts.
- 77 -
Figure 4.6: The Gaussian bunch longitudinal rms length for 70 % (left) and 100 % (right)
versus drift distance.
To have a microbunching, one needs a wakefield with several times smaller wave
length than bunchโs length. High frequency fields generated by a Gaussian distribution
are very low, because of exponential damping factor in distribution. While for Gaussian
bunch it is not possible to have a microbunching in appropriate distances, for some non-
Gaussian distributed bunches it is possible.
For the rectangular uniform charge distribution the wake potential in plasma
channel produces bunch energy modulation at the excited mode wavelength.
Rectangular uniform line charge distribution is in the form of
๐(๐ ) = { 1
2๐, |๐ | โค ๐
0 , |๐ | > ๐ (4.12)
and the rms length of the bunch is given by ๐ =๐
โ3.
Wake potential of such a bunch can be derived from Eq. 4.2
๐๐ง(๐ ) = โซ ๐ค๐ง (๐๐(๐ โ ๐ โฒ)) ๐(๐ โฒ)๐๐ โฒ
โ
โโ= {
0 , ๐ < โ๐
๐๐๐๐ ๐
2๐๐sin (๐๐(๐ + ๐)) , |๐ | โค ๐
๐๐๐๐ ๐
๐๐sin(๐๐๐) cos(๐๐๐) , ๐ > ๐
(4.13)
Figure 4.7 shows the wake potential generated by a rectangular uniform distributed
bunch of 0.58 ๐๐ rms length (2 ๐๐ total length) in a plasma channel with 0.6 ๐๐ (๐๐ โ
0.5 ๐๐ป๐ง) excited mode wavelength.
- 78 -
Figure 4.7: Uniform bunch wake potential.
The energy modulation at the wakefield excited mode frequency leads to the charge
density modulation as the beam travels in free space.
a
b
c
d
Figure 4.8: Line charge distribution after passing ๐๐๐๐๐๐ก distance in a drift space
๐) ๐๐๐๐๐๐ก = 1.5 ๐, b) ๐๐๐๐๐๐ก = 2.5 ๐, c) ๐๐๐๐๐๐ก = 3.5 ๐, d) ๐๐๐๐๐๐ก = 5 ๐
- 79 -
Charge density modulation at various drift space distances after 0.3 ๐ plasma
channel is shown in Figure 4.8. As it can be seen, the density modulation leads to three
microbunches formation.
The best bunching is at the 3.5 ๐ drift space distance, where the initial rectangular
bunch is transformed into three microbunches with 14 ๐๐ถ charge and 40 ๐๐ full width at
half maximum (FWHM). After that distance debunching process starts (Figure 4.8(d)).
Parabolic line charge distribution is taken in the form of
๐(๐ ) = { 1
2๐3(๐2 โ ๐ 2), |๐ | โค ๐
0 , |๐ | > ๐ (4.14)
and rms length of the bunch is determined by ๐ =๐
โ5.
The longitudinal wake potential of parabolic charge can be derived from Eq. 1.2
๐๐ง(๐ ) = โซ ๐ค๐ง(๐(๐ โ ๐ โฒ))๐(๐ โฒ)๐๐ โฒ
โ
โโ=
{
0 , ๐ < โ๐
๐๐๐๐ ๐
๐3๐3[sin(๐(๐ + ๐)) โ ๐๐ cos(๐(๐ + ๐)) โ ๐๐ ] , |๐ | โค ๐
๐๐๐๐ ๐
๐3๐3[sin(๐(๐ + ๐ )) + sin(๐(๐ โ ๐ )) โ ๐๐(cos(๐(๐ + ๐ )) + cos(๐(๐ โ ๐ )))] , ๐ > ๐
(4.15)
The wake potential for parabolic charge distribution of 0.45 ๐๐ rms length (2 ๐๐
full length) is shown in Figure 4.9. Dashed line represents the bunch shape. To satisfy a
microbunching criterion a plasma channel with 0.6 ๐๐ ( ๐๐ โ 0.5 ๐๐ป๐ง ) excited mode
wavelength is taken.
- 80 -
Figure 4.9: Parabolic bunch wake potential.
Particles energy modulation is observed with linear ramp of the average energy
deviation.
Figure 4.10 shows the parabolic charge density modulation at various drift space
distances after 1 ๐ plasma channel length. The density modulation leads to microbunching
and initial bunch transforms into three microbunches.
a
b
c
d
Figure 4.10: Line charge distribution after passing ๐๐๐๐๐๐ก distance in a drift space:
๐) ๐๐๐๐๐๐ก = 1 ๐, b) ๐๐๐๐๐๐ก = 3 ๐, c) ๐๐๐๐๐๐ก = 5 ๐, d) ๐๐๐๐๐๐ก = 7 ๐
The best bunching is observed on a 5 ๐ drift space length, where microbunches
with 10 ๐๐ถ charge 20 ๐๐ length (FWHM) are formed. It can be seen that at 7 ๐ drift space
length debunching of all three microbunches occurs.
- 81 -
4.5 Energy modulation, bunch compression and microbunching in internally
coated metallic tube
Energy and charge density modulation of Gaussian, rectangular uniform and
parabolic charge distributions in ICMT are observed. The geometry of ICMT is shown in
Figure 4.11.
Figure 4.11: Geometry of the ICMT
In a high frequency range, for a thin, low conductivity inner layer and infinite thick,
perfectly conducting outer layer the ICMT is a single-mode structure with ๐0 =๐
2๐โ2 ๐๐โ
resonant frequency, where ๐ is the inner radii of a layer and ๐ is its thickness.
Wake function generated in ICMT by an ultrarelativistic point charge moving along
the structure axis can be presented as [50]
๐คโฅ(๐ ) = โ๐0๐
๐๐2๐โ๐ผ๐ [cos(๐๐๐ ) โ
๐ผ
๐๐sin(๐๐๐ )] (4.17)
where, ๐๐ = โ๐02 โ ๐ผ2 , ๐0 =
2๐
๐๐0 and ๐ผ damping factor is a function of layer thickness
and conductivity.
- 82 -
For non-ultrarelativistic particle (10 ๐๐๐) the longitudinal wake function calculated in
[85] is used. Wake functions of a point charge with various energies are shown in Figure
4.12.
a
b
c
d
Figure 4.12: Wake functions of a point charge with gamma factor ๐พ a) ๐พ = 10, b) ๐พ = 20, c) ๐พ = 40, d) ๐พ = 80
Matching the inner radii of the internal cover and its thickness one can obtain a
resonant frequency in ๐๐ป๐ง region in the ICMT.
Numerical simulations are performed to determine bunch shape after drift space for
driving bunch charge of ๐ = 100 ๐๐ถ with ๐0 = 10 ๐๐๐ initial energy. The inner diameter of
a structure is 0.5 ๐๐, electric conductivity of the first layer is 104 ๐
๐, thickness of the first
layer is 1 ๐๐. Excited mode frequency in ICMT with these parameters is approximately
๐0 =๐
2๐โ2 ๐๐โ = 3 ๐๐ป๐ง , which corresponds to 0.1 ๐๐ wavelength. Point wake function of
such a structure is shown in Figure 4.12.b.
- 83 -
Figure 4.13 presents the wake potential of the Gaussian charge distribution with
0.1 ๐๐ rms length (dashed line shows bunch shape).
Figure 4.13: 100 ๐๐ Gaussian bunch wake potential.
The wake potential is decelerating at bunchโs head (๐ < 0) and accelerating at
bunchโs tail (๐ > 0). This energy modulation can lead to bunch compression for appropriate
drift space length.
Compressed bunch shape at various drift space distances after 1 ๐ ICMT is shown
in Figure 4.14. Dashed line presents the initial Gaussian bunch distribution. The bunch
compresses till about 4 ๐ drift space length after which tail of the bunch passes its head
and bunch starts to expand.
- 84 -
a
b
c
d
Figure 4.14: Line charge distribution at ๐๐๐๐๐๐ก drift space distances.
๐) ๐๐๐๐๐๐ก = 1 ๐, b) ๐๐๐๐๐๐ก = 2 ๐, c) ๐๐๐๐๐๐ก = 4 ๐, d) ๐๐๐๐๐๐ก = 6 ๐
In ICMT, like in plasma, the Gaussian bunch sees strong focusing field on its central
part, while in its head and tail there is a weak field. Note that in ICMT there is no
debunching field on bunchโs tail. rms lengths of 70 % (central part) and 100 % particles
versus bunch travel distance in drift space is shown in Figure 4.15. The best compression
is reached at 3.8 ๐ drift space distance, where a bunch with 70 ๐๐ถ charge and 8.02 ๐๐
rms length is formed.
- 85 -
Figure 4.15: rms of 70% (left) and 100% (right) of particles in a bunch versus drift space
distance.
To have a microbunching process in a structure one need a bunch several times
longer than the excited wavelength (0.1 ๐๐ ). This criterion is taken into account for
studying microbunching of uniform rectangular and parabolic bunches.
Microbunching of rectangular uniform charge distribution with 0.17 ๐๐ rms length
(0.6 ๐๐ total length) in ICMT is studied. Wake potential of the bunch is presented in
Figure 4.16 (dashed line shows bunch shape). This wake potential generates an energy
modulation within the bunch at excited mode wavelength.
Figure 4.16: Uniform bunch wake potential.
The excited energy modulation leads to charge density modulation while it travels in
a free space. Figure 4.17 shows charge density modulations in various drift space
distances after 0.2 ๐ ICMT. The initial uniform bunch transforms into several
microbunches.
- 86 -
a
b
c
d
Figure 4.17: Line charge distribution after passing ๐๐๐๐๐๐ก distance in a drift space
๐) ๐๐๐๐๐๐ก = 1 ๐, b) ๐๐๐๐๐๐ก = 2 ๐, c) ๐๐๐๐๐๐ก = 3 ๐, d) ๐๐๐๐๐๐ก = 4 ๐
The best bunching is on 3 ๐ drift space length, after which a debunching process
starts. Full width at half maximum (FWHM) of the first microbunch is 8 ๐๐, which is a good
length for generating THz radiation. 7.8 % of total charge is concentrated in a full width
region of the first microbunch, and 10 % in three full width region. Full width of the second
microbunch is 16 ๐๐ . 5.94 % of total charge is concentrated in a full width region and
10.5 % in three full width region.
Wake potential of parabolic charge distribution with 0.13 ๐๐ rms length (0.6 ๐๐
total length) is presented in Figure 4.18 (dashed line shows bunch shape). Bunch damping
energy modulation with linear ramp of the average energy deviation is observed.
- 87 -
Figure 4.18: Parabolic bunch wake potential.
Bunch shape change in various drift space lengths after 1 ๐ ICMT is shown in
Figure19. The initial parabolic bunch transforms into several microbunches.
a
b
c
d
Figure 4.19: Line charge distribution after passing ๐๐๐๐๐๐ก distance in a drift space
๐) ๐๐๐๐๐๐ก = 3 ๐, b) ๐๐๐๐๐๐ก = 5 ๐, c) ๐๐๐๐๐๐ก = 7 ๐, d) ๐๐๐๐๐๐ก = 9 ๐
- 88 -
The best bunching is on 7 ๐ drift space length. On that distance FWHM of the first
microbunch is 7.5 ๐๐. 1.8 % of total charge is concentrated in a full width region of the first
microbunch, and 2.2 % in three full width region. FWHM of the second microbunch is 3 ๐๐.
1.54 % of total charge is concentrated in a full width region and 2.42 % in three full width
region.
4.6 Summary
Energy and charge density modulation of relativistic electron bunch after its
interaction with cold plasma and ICMT have been studied. The bunching and
microbunching processes of cold, low energy (10 ๐๐๐) electron bunches of Gaussian,
rectangular and parabolic charge distributions are researched by ballistic method. It is
shown, that the Gaussian bunch interaction with these structures leads to its compression.
Bunch shape ๐0 ๐0 ๐0 ๐๐, ๐๐ ๐๐ก, ๐๐ก
Gaussian 100 ๐๐ถ 10 ๐๐๐ 100 ๐๐ 12 ๐๐, 70 ๐๐ถ 8.02 ๐๐, 70 ๐๐ถ
Table 1.1: Gaussian bunch parameters before and after interaction with plasma and ICMT.
Table 1.1 shows Gaussian bunch compression after plasma channel and ICMT. In
the table ๐0 is for bunch initial charge, ๐0 for initial energy, ๐0 is initial rms length, ๐๐ and
๐๐ are compressed bunch rms length and charge for plasma channel and finally ๐๐ก and ๐๐ก
are rms length and charge for ICMT.
In a cold plasma a Gaussian bunch with 0.1 ๐๐ rms length and 100 ๐๐ถ charge can
be compressed by factor of 8, while in ICMT it can be compressed by factor of 12. For the
rectangular and parabolic bunches several times longer than the structure excited
frequency the microbunching process can be obtained.
Plasma ICMT
Bunch shape ๐0,๐ ๐๐, ๐๐ ๐0,๐ก ๐๐ก, ๐๐ก ๐๐ก, ๐๐ก
Rectangular 2 ๐๐ 40 ๐๐, 14 ๐๐ถ 0.6 ๐๐ 8 ๐๐, 7.8 ๐๐ถ 16 ๐๐, 6 ๐๐ถ
Parabolic 2 ๐๐ 20 ๐๐, 10 ๐๐ถ 0.6 ๐๐ 7.5 ๐๐, 1.8 ๐๐ถ 3 ๐๐, 1.5 ๐๐ถ
Table 1.2: Rectangular and parabolic bunchesโ parameters before and after interaction with plasma
and ICMT.
- 89 -
Table 1.2 shows microbunching results for rectangular and parabolic bunches. In
the table indexes ๐ and ๐ก are for plasma channel and ICMT, respectively. ๐0 is for bunchโs
initial length (FWHM), ๐and ๐ represents microbunchesโ lengths and charges.
It is shown that in a cold plasma a uniform bunch with 2 ๐๐ total length (0.58 ๐๐
rms) and 100 ๐๐ถ charge can be splitted into three microbunches with 40 ๐๐ FWHM length
and 14 ๐๐ถ charge. Parabolic bunch with 2 ๐๐ full length ( 0.45 ๐๐ rms) and 100 ๐๐ถ
charge forms three microbunches of 20 ๐๐ full width and 10 ๐๐ถ charges. Study of
microbunching process of rectangular bunch in ICMT shows that it is possible to generate
microbunches of 8 ๐๐ FWHM length with 7.8 ๐๐ถ charge and 16 ๐๐ FWHM length with
5.94 ๐๐ถ charge. Having an initial parabolic charge distribution, it is possible to generate
microbunches of 7.5 ๐๐ FWHM length with 1.8 ๐๐ถ charge and 3 ๐๐ FWHM length with
1.54 ๐๐ถ charge.
- 90 -
Summary
It is shown that in single-mode structures sub-ps bunches can be formed from
longer ones. The sub-ps bunch formation is researched in plasma channels and internally
coated metallic tubes. The ballistic bunching method is used in research. Bunch shape is
reconstructed numerically.
It is shown that two-layer structures with inner low conductivity material is a high
frequency single-mode structures. The theory of flat two-layer structure is developed. It is
shown that electrodynamic properties of the flat structure are similar to properties of the
cylindrical one. Rectangular cavity with horizontal two-layer metallic walls is studied and
correlation between modes obtained experimentally and theoretically is performed.
The main results of the dissertation are the following:
Matrix formalism has been developed which allows to couple point charge
radiated electromagnetic fields in the inner and outer regions of multilayer
parallel infinite plates.
Non-ultrarelativistic point charge excited electromagnetic fields in multilayer
parallel infinite plates are obtained analytically.
Explicit expression of non-ultrarelativistic point charge radiation fields in two-
layer parallel infinite plates with unbounded external walls is derived analytically.
Also point charge radiation fields in single-layer unbounded structure is derived
as a special case of two-layer structure.
Longitudinal impedance and dispersion relations of symmetrical two-layer
parallel infinite plates are derived. It is shown that, for low-conductivity thin inner
layer and high conductivity thick outer layer, the longitudinal impedance has a
narrow-band resonance in high frequency region.
- 91 -
The resonant frequency dependence of two-layer parallel infinite platesโ
parameters is studied and resonance frequency dependence on structure
parameters is derived empirically.
Longitudinal wakefields generated by a point charge traveling through the center
of the two-layer parallel infinite plates with outer perfectly conducting material
are calculated. The longitudinal wake potential is a quasi-periodic function with
period given by the resonant frequency as c
๐๐๐๐ .
Resonance frequencies of rectangular cavity with horizontal laminated walls are
obtained analytically.
It is shown that for rectangular cavity with vertical dimensions much bigger then
horizontal ones, the resonance frequencies are in good agreement with
frequencies of two-layer parallel infinite plates.
The measured resonances of copper cavity with horizontal walls internally
covered by germanium thin layer are correlated with analytically obtained
resonance modes of rectangular cavity. All resonances of the test cavity are in
conformity with analytically obtained modes.
It is shown that for appropriate structure parameters the rectangular cavity with
internally covered horizontal walls is a good candidate for bunch acceleration
and sub-ps (micro) bunch generation.
Compression of Gaussian bunches and microbunching of rectangular and
parabolic bunches is studied numerically in plasma channels and internally
coated metallic structures, based on the ballistic bunching method.
It is shown that Gaussian distributed bunches with initial 100 ๐๐ rms length can
be compressed to 12 ๐๐ and 8 ๐๐ rms length in plasma channel and ICMT,
respectively.
- 92 -
It is shown that bunches with 20 ๐๐ and 3 ๐๐ full length can be formed due to
microbunching of parabolic bunches in plasma channel and ICMT, respectively.
It is shown that bunches with 40 ๐๐ and 8 ๐๐ full length can be formed due to
microbunching of rectangular bunches in plasma channel and ICMT,
respectively.
The results of the study can be useful for the development of new accelerating
structures for particle acceleration, monochromatic coherent radiation sources, ultrashort
bunch generation.
- 93 -
Acknowledgements
I express sincere gratitude to the director of CANDLE prof. Tsakanov, the head of the
doctor's work. M.I. Ivanyan and all the staff of CANDLE for their constant support and
assistance.
- 94 -
References
[1] A. H. Zewail, โFemtochemistry: Atomic-Scale Dynamics of the Chemical Bondโ, J.
Phys. Chem. A 104, 5660โ5694 (2000)
[2] R. Kuroda,, H. Ogawa, N. Sei, H. Toyokawa, K. Yagi-Watanabe, M. Yasumoto, M.
Koike, K. Yamada, T. Yanagida, T. Nakajyo and F. Sakai, โDevelopment of Cs2Te
photocathode rf gun system for compact THz SASE-FELโ, Nucl. Instr. and Meth. A 593,
91-93 (2008).
[3] C. Yim, S. Noh and S. Ko, โBeam Parameter measurements of rf-THz linac at PALโ,
Proceedings of the International Particle Accelerator Conference, 1059-1061 (2010).
[4] J.B. Hasting, F. M. Rudakov, D. H. Dowell, J. F. Schmerge, J. D. Cardoza, J. M. Castro,
S. M. Gierman,, H. Loos and P. M. Weber, โUltrafast time-resolved electron diffraction with
megavolt electron beamsโ, Appl. Phys. Lett. 89, 184109 (2006).
[5] P. Musumeci, J. T. Moody, C. M. Scoby, M. S. Gutierrez, H. A. Bender, N. S. Wilcox,
โHigh quality single shot diffraction patterns using ultrashort megaelectron volt electron
beams from a radio frequency photoinjectorโ, Rew. Sci. Instrum. 81, 013306 (2010).
[6] T. Shintake et al., โA compact free-electron laser for generating coherent radiation in
the extreme ultraviolet regionโ, Nature Photonics 2, 555-559 (2008)
[7] J.-H. Han, โProduction of a sub-10 fs electron beam with 107 electronsโ Phys. Rev. ST
Accel. Beams 14, 050101 (2011).
[8] G. P. Williams, โFilling the THz gap-high power sources and applicationsโ, Rep. Prog.
Phys. 69, 301-326 (2006).
[9] M. Tonouchi, โCutting-edge terahertz technologyโ, Nat. Photonics 1, 97-105 (2007).
[10] P. H. Siegel, โTerahertz technology in biology and medicineโ, IEEE Trans. Microwave
Theory Tech. 52, 2438โ2446 (2004).
[11] H. T. Chen, R. Kersting, G. C. Cho, โTerahertz imaging with nanometer resolutionโ,
Appl. Phys. Lett. 83, 3009โ3011 (2003)
[12] http://www.xfel.eu/en/.
- 95 -
[13] https://www.psi.ch/swissfel/swissfel-accelerator.
[14] A. Gover, โSuperradiant and stimulated-superradiant emission in prebunched
electron-beam radiators. I. Formulationโ, Phys. Rev. ST Accel. Beams 8, 030701 (2005).
[15] A. S. Muller, โAccelerator-Based Sources of Infrared and Terahertz Radiationโ, Rev.
Accl. Sci. Tech. 03, 165-183 (2010).
[16] D.F. Gordon, A. Ting, T. Jones, B. Hazi, R.F. Hubbard and P. Sprangle, โParticle-in-
cell simulation of optical injector for plasma acceleratorsโ, in Proceedings of the Particle
Accelerator Conference (PACโ03), 1846-1848 (2003).
[17] G. L. Carr, M. C. Martin, W. R. McKinney, K. Jordan, G. R. Neil, and G. P. Williams,
โHigh-power terahertz radiation from relativistic electronsโ, Nature (London) 420, 153-156
(2002).
[18] M. Abo-Bakr, et al, Brilliant, โCoherent Far-Infrared (THz) Synchrotron Radiationโ,
Phys. Rev. Lett. 90, 094801 (2003).
[19] K. L. F. Bane and G. Stupakov, โTerahertz radiation from a pipe with small
corrugationsโ, Nucl. Instrum. Methods Phys. Res., Sect. A 677, 67-73 (2012).
[20] V. Ayvazyan et al., โFirst operation of a free-electron laser generating GW power
radiation at 32 nm wavelengthโ, Eur. Phys. J. D 37, 297-303 (2006).
[21] http://www-ssrl.slac.stanford.edu/lcls/.
[22] J. Nodvick and D. Saxon, โSuppression of Coherent Radiation by Electrons in a
Synchrotronโ, Phys. Rev. 96 (1), 180 (1954).
[23] S. V. Milton et al., โExponential gain and saturation of a self-amplified spontaneous
emission free-electron laserโ, Science 292, 2037-2041 (2001).
[24] A. M. Kondratenko, E. L. Saldin, โGeneration of coherent radiation by a relativistic
electron beam in an undulatorโ, Particle Accelerators 10, 207-216 (1980).
[25] M. Ivanyan, V. Tsakanov, A. Grigoryan, A. Tsakanian, โThe narrow-band resonance in
two-layer metallic round tubeโ, http://arxiv.org/abs/1301.7729, (2013).
- 96 -
[26] X. J. Wang, X. Qiu, and I. Ben-Zvi, โExperimental observation of high-brightness
microbunching in a photocathode rf electron gunโ, Phys. Rev. E 54 (4), R3121 (1996).
[27] X. J. Wang, X. Y. Chang, โFemto-seconds kilo-ampere electron beam generationโ,
Nucl. Instr. and Meth. A 507, 310-313 (2003).
[28] P. Piot, L. Carr, W. S. Graves, and H. Loos, โSubpicosecond compression by velocity
bunching in a photoinjectorโ, Phys. Rev. ST Accel. Beams 6, 033503 (2003).
[29] M. Ferrario, et al., โExperimental Demonstration of Emittance Compensation with
Velocity Bunchingโ, Phys. Rev. Lett. 104, 054801 (2010).
[30] A. M. Cook, R. Tikhoplav, S. Y. Tochitsky, G. Travish, O. B. Williams and J. B.
Rosenzweig, โObservation of Narrow-Band Terahertz Coherent Cherenkov Radiation from
a Cylindrical Dielectric-Lined Waveguideโ, Phys. Rev. Lett. 103, 095003 (2009).
[31] ะ. ะ. ะะฐะปัะดะฝะตั, ะ. ะ. ะจะฐะปัะฝะพะฒ, โะญะปะตะบััะพะผะฐะณะฝะธัะฝัะต ะฟะพะปั ะฒ ะดะธะฐััะฐะณะผะธัะพะฒะฐะฝะฝัั
ะฒะพะปะฝะพะฒะพะดะฐั ะปะธะฝะตะนะฝัั ัะปะตะบััะพะฝะฝัั ััะบะพัะธัะตะปะตะนโ, ะะพัะบะฒะฐ, ะะพัะฐัะพะผะธะทะดะฐั, (1963).
[32] ะญ. ะ. ะะฐะทะฐะทัะฝ, ะญ. ะ. ะะฐะทะธะตะฒ, โะ ัะตัะตะฝะบะพะฒัะบะพะผ ะธะทะปััะตะฝะธะธ ะฒ ะฒะพะปะฝะพะฒะพะดะตโ, ะะทะฒ. ะะะ
ะัะผ. ะกะกะ , ัะธะท.-ะผะฐั. ะฝะฐัะบะธ, 16, N2, 79, (1963).
[33] M. Ivanyan, V. Tsakanov, โCoupling Impedance of Rough Resistive Pipeโ, IPAC 2011,
MOPS045, Spain, 700-702, (2011).
[34] M. Ivanyan, A. Tsakanian, โImpedances and Wakes in Round Three-layer Ceramic
Waveguideโ, IPAC2011, MOPS046, Spain, pp. 703-705, (2011).
[35] G. Stupakov, K.L.F. Bane, โSurface impedance formalism for a metallic beam pipe
with small corrugationsโ, Phys. Rev. ST Accel. Beams 15, 124401, (2012).
[36] A. W. Chao, โPhysics of Collective Beam Instabilities in High Energy Acceleratorsโ,
New York, John Willey & Sons, Inc., (1993).
[37] B. W. Zotter and S.A. Kheifetz, โImpedances and Wakes in High-Energy Particle
Acceleratorsโ, Singapore, World Scientific, (1997).
[38] H. Wiedemann, โParticle Accelerator Physicsโ, Springer, Nederland, vol. 2, (1999).
- 97 -
[39] T.O. Raubenheimer, โElectron Beam Acceleration and Compression for Short
Wavelength FELsโ, Nucl. Instr. and Meth. A 40, 358-369 (1995). A 358 (1995) 40-43
[40] A. Hofmann, โBeam Instabilitiesโ, CAS Proceedings, Rhodes, Greece, CERN โ 95-06,
307-326 (1995).
[41] G. A. Amatuni, V. M. Tsakanov et al, โDynamic aperture, impedances and instabilities
in CANDLE light sourceโ, ICFA Beam Dyn. Newslett 44, pp. 176-183, (2007).
[42] J. L. Laclare, โCoasting Beam Longitudinal Coherent Instabilitiesโ, CAS Proc., Finland,
CERN โ 94-01, 345-405, (1994).
[43] K.Y. Ng, โPhysics of Intensity Dependent Beam Instabilitiesโ, World Scientific, (2005).
[44] Y. Martirosyan, M. Ivanyan, V. Tsakanov, โBeam Current Limitations Study for
CANDLE Light Sourceโ, Proc. of EPAC-2002, France, 715-717 (2002).
[45] K. L. Bane, P. B. Wilson, and T. Weiland, โWake fields and wake field accelerationโ,
AIP Conf. Proc. 127, 875-928 (1985).
[46] B. M. Bolotovskii, โTHEORY OF CERENKOV RADIATION (III)โ, Sov. Phys. Usp. 4,
781-811 (1962).
[47] ะฏ.ะ. ะคะฐะนะฝะฑะตัะณ, โะฃัะบะพัะตะฝะธะต ัะฐััะธั ะฒ ะฟะปะฐะทะผะตโ, ะัะพะผะฝะฐั ัะฝะตัะณะธั, 6 (4), 431-446
(1959).
[48] P. Chen, J. M. Dawson, โThe plasma wake field acceleratorโ, AIP Conf. Proc. 130,
201 (1985).
[49] R. Fitzpatrick. โClassical electromagnetismโ, The University of Texas at Austin, (2006).
[50] M. Ivanyan, A. Grigoryan, A. Tsakanian, and V. Tsakanov, โNarrow-band impedance
of a round metallic pipe with a low conductive thin layerโ, Phys. Rev. ST Accel. Beams 17,
021302 (2014).
[51] R.J. England et al., โDielectric laser acceleratorsโ, Rev. Mod. Phys. 86, 1337-1389
(2014).
- 98 -
[52] E. A. Nanni, W. R. Huang, K.-H. Hong, K. Ravi, A. Fallahi, G. Moriena, R. J. Dwayne
Miller and F. X. Kartner, โTerahertz-driven linear electron accelerationโ, Nat. Commun. 6,
8486 (2015).
[53] B. W. Zotter, S. A. Kheifetz, โImpedances and wakes in high-energy particle
acceleratorsโ, World Scientific, Singapore (1997).
[54] H. Hahn, โMatrix solution for the wall impedance of infinitely long multilayer circular
beam tubesโ, Phys. Rev. ST Accel. Beams 13, 012002 (2010).
[55] M. Ivanyan, E. Laziev, V. Tsakanov and A. Vardanyan, S. Heifets and A. Tsakanian,
โMultilayer tube impedance and external radiationโ, Phys. Rev. ST Accel. Beams 11,
084001 (2008).
[56] N. Wang and Q. Qin, โResistive-wall impedance of two-layer tubeโ, Phys. Rev. ST
Accel. Beams 10, 111003 (2007).
[57] M. Ivanyan, V. Tsakanov, โLongitudinal impedance of two-layer tubeโ, Phys. Rev. ST
Accel. Beams 7, 114402 (2004).
[59] W. Gai, M. Conde, J. G. Power, โConsiderations for a dielectric-based two-beam-
accelerator linear colliderโ, Proc. of IPAC10, Kyoto, Japan, 3428-3430 (2010).
[60] J. B. Rosenzweig, G. Andonian, P. Muggle, P. Niknejadi, G. Travish, O. Williams, K.
Xuan and V. Yakimenko, โHigh Frequency, High Gradient Dielectric Wakefield
Acceleration Experiments at SLAC and BNLโ, AIP Conf. Proc. 1299, 364-369 (2010).
[61] K.L.F. Bane and G. Stupakov, โTerahertz radiation from a pipe with small
corrugationsโ, Nucl. Instr. and Meth. A 677, 67-73 (2012).
[62] G. Korn and T. Korn, โMathematical handbook for scientists and engineersโ, Dover
Publications, Mineola, New York (1968).
[64] H. Henke and O. Napoly, โWake Fields Between Two Parallel Resistive Platesโ, in
Proc. of the 2nd European Particle Accel. Conf., France, 1046-1048 (1990).
[65] K. Bane and G. Stupakov, โUsing surface impedance for calculating wakefields in flat
geometryโ, Phys. Rev. ST Accel. Beams 18, 034401 (2015).
- 99 -
[67] M. I. Ivanyan, V. Sh. Avagyan, V. A. Danielyan, A. V. Tsakanian, A. S. Vardanyan and
V. S. Zakaryan, โOn the resonant behavior of laminated accelerating structuresโ, Journal of
Instrumentation 12, P03019 (2017).
[68] Abramowitz and Stegun, โHandbook of Mathematical Functions with formulas, graphs,
and mathematical tablesโ, National Bureau of Standards, (1964)
[69] David M. Pozar, โMicrowave Engineeringโ, 3rd edition, ISBN 10:471448788, John
Wiley & Sons Inc., 170-174 (2005).
[70] R&S ZVB Vector Network Analyzer, http:/www.rohde-schwarz.com.
[71] R. E. Collin, โFoundations for Microwave Engineeringโ, 2nd edition, John Wiley & Sons,
Inc., Hoboken, New Jersey (2001).
[72] ะ. ะะฐะนะฝััะตะนะฝ, โะญะปะตะบััะพะผะฐะณะฝะธัะฝัะต ะฒะพะปะฝัโ, 2-ะต ะธะทะดะฐะฝะธะต, ะะพัะบะฒะฐ, ะ ะฐะดะธะพ ะธ ัะฒัะทั,
(1988).
[73] S.M. Sze, Physics of Semiconductor Devices, John Wiley and Sons, New York (1981).
[75] M. I. Ivanyan, V. A. Danielyan, B. A. Grigoryan, A. H. Grigoryan, A. V. Tsakanian, V.
M. Tsakanov, A. S. Vardanyan and S. V. Zakaryan, โHigh frequency single mode traveling
wave structure for particle accelerationโ, Nuclear Instruments and Methods A 829, 187-189
(2016).
[76] M. Ivanyan, A. Grigoryan, A. Tsakanian and V. Tsakanov, โWakefield radiation from
the open end of an internally coated metallic tubeโ, Phys. Rev. ST Accel. Beams 17,
074701 (2014).
[77] A. H. Zewail and J. M. Thomas, โ4D Electron Microscopy: Imaging in Space and
Timeโ, Imperial College Press, London (2010).
[78] R. K. Li, P. Musumeci, H. A. Bender, N. S. Wilcox and M. Wu, โImaging single
electrons to enable the generation of ultrashort beams for single-shot femtosecond
relativistic electron diffractionโ, Journal of Appl. Phys. 110, 074512 (2011).
[79] A. Modena, Z. Najmudin, A. E. Dangor, C. E. Clayton, K. A. Marsh, C. Joshi, V. Malka,
C. B. Darrow, C. Danson, D. Neely & F. N. Walsh, โElectron acceleration from the breaking
of relativistic plasma wavesโ, Nature 377, 606-608 (1995)
- 100 -
[80] B. E. Carlsten and S. M. Russel, โSub-picosecond compression of 0.1 - 1 nC electron
bunches with a magnetic chicane at 8 MeVโ, Phys. Rev. E 53 3, R2072-R2075 (1996).
[83] J.G. Power, W. Gai, X. Sun, A. Kanareykin, โTransformer Ratio Enhancement Using a
Ramped Bunch Train in a Collinear Wakefield Acceleratorโ, PAC 2001 Proceedings, 114-
116 (2002).
[84] V. M. Tsakanov, et al., โAREAL test facility for advanced accelerator and radiation
source conceptsโ, Nucl. Instr. and Meth. A 829, 284-290 (2016).
[85] ะ. ะัะธะณะพััะฝ, โะ ะตะทะพะฝะฐะฝัะฝัะต ัะฒะพะนััะฒะฐ ะธะทะปััะตะฝะธั ะฝะธะทะบะพัะฝะตัะณะตัะธัะตัะบะธั
ัะปะตะบััะพะฝะฝัั ะฟััะบะพะฒ ะฒ ะดะฒัั ัะปะพะนะฝะพะผ ะฒะพะปะฝะพะฒะพะดะตโ, ะะทะฒ. ะะะ ะัะผะตะฝะธะธ, ะคะธะทะธะบะฐ 48, 383-
393 (2013).