# Bunch–excited wakefield in dielectric waveguide with hollow plasma channel

K.V. Galaydych, P.I. Markov, G.V. Sotnikov Corresponding author:
<EMAIL_ADDRESS>
(National Science Center Kharkiv Institute of Physics and Technology 1,
Akademichna St., Kharkiv, 61108, Ukraine)

## Abstract

Wakefield excitation by a single relativistic electron bunch in a
plasma–dielectric accelerating structure has been studied both analytically
and numerically. The structure represents a dielectric–loaded cylindrical
metal waveguide, which has partially plasma–filled channel (the hollow plasma
channel) to transport charged particles. Assuming the linear regime of
excitation, analytical expressions have been derived for the longitudinal and
radial wakefields generated by a finite–size electron bunch. Axial profiles of
wakefield component amplitudes have been studied, and their mode and spectrum
analyses have been performed. Furthermore, the electron bunch–driven wakefield
excitation has been PIC–simulated numerically for the quasi–linear regime. The
comparative analysis of the data resulting from analytical studies and the
ones obtained by numerical simulation has demonstrated qualitative agreement
between the results.

##  1 Introduction

The wakefield acceleration of charged particles in dielectric structures [1]
ranges among new methods such as PWFA [2, 3], LWFA [4] and DLA [5, 6], and can
be considered as one of the potential alternatives to the classical methods.
Presently, all these methods continue to be currently central in the research,
and are actively developed showing steady progress. The papers [7, 8, 9, 10,
11, 12, 13] presents road maps of future studies in these directions for a
wide range of problems and challenges in the physics of accelerators and
charged particle beams. Thus, in dielectric structures, owing to the use of
ceramic and polycrystalline materials (artificial diamond, quartz) that
exhibit a high–level breakdown field, and also, due to advance in production
of short (about a few tens of microns) electron bunches with a high charge
value (up to a few tens of nC), it appears possible to attain high–amplitude
excited fields. The experiments [14] have demonstrated the possibility of
generating GV/m wakefields in the THz waveguide. In these structures, the
wakefield is excited by relativistic high–charge drive bunches. In its passage
through the structure, the drive bunch excites the Cherenkov radiation, which
being reflected from the metal coating of the structure, forms the
longitudinal field behind the drive bunch. In turn, this field can be used to
accelerate the test bunch having a lower charge as opposed to that of the
drive bunch. The delay time of the test bunch injection is matched so that in
the passage through the charged particles transport channel (being generally
vacuum), the test bunch could stay in the accelerating phase of the field as
long as possible. It has been proposed in [15] that the particle transport
channel should be filled with cold plasma. As demonstrated in [15], that gives
rise to the focusing properties of the wakefield excited in this
plasma–dielectric waveguide due to the Langmuir wave excitation. Previous
analytical studies of wakefields in the cylindrical plasma–dielectric
waveguide were carried out for the case, where homogeneous plasma fully filled
the charged particle transport channel. With that, consideration has been
given to two important cases, namely, i) when the drive bunch is injected
strictly along the waveguide axis [15], and ii) when the drive bunch is
injected with initial transverse offset [16]. It has been shown in [16] that
for the cylindrical plasma–dielectric accelerating structure the presence of
the initial transverse offset of the drive bunch does not lead to BBU as
opposed to the dielectric–loaded structure without plasma filling [17, 18,
19]. Such plasma–dielectric accelerating structures cannot fully replace the
classical focusing systems, yet, owing to the intrinsic focusing properties,
they can substantially reduce the requirements to the external magnetic
structures. By using PIC simulation in [20, 21], consideration has been given
to the non–linear regime of wakefield acceleration of test electron and
positron bunches for the cases of homogeneous and inhomogeneous plasmas, where
the electron bunch was considered to be the driver. However, despite the
possibility of investigating the processes of excitation and acceleration
through the use of the PIC simulation, a number of problems still remain to be
solved. The need to have a clearer interpretation of the simulation results
calls for more detailed analytical studies and development of the
corresponding theory. Among the questions that remain to be answered we
mention, for example, the question about the particular mechanism of formation
of the focusing transverse field in the plasma region for the linear regime.
The numerical analysis based on the analytical results makes it possible to
carry out scanning and optimization of the parameters for the dielectric
waveguide, plasma and drive bunch in an effort to determine the parameters
that would provide steady test bunch acceleration. Besides, the feasible
preliminary matching of the parameters appears to be a rather useful tool of
theoretical analysis, as it can reduce the amount of numerical simulation. It
is pertinent to note that the numerical simulation, being capable of taking
into account many factors simultaneously, provides, on the one hand, a more
comprehensive (when compared to the theory) description of the processes under
study, but on the other hand, presents a computational extremely resource-
intensive for the plasma systems comprised of a great number of particles. It
is therefore only natural in the analytical studies to turn to the case of
inhomogeneous plasma. As the inhomogeneous plasma, for which the analytical
theory can be built without using any approximation methods, we have chosen
the case, where the transport channel is filled with homogeneous plasma
incompletely, and along the waveguide axis there remains a cylindrical vacuum
region. Compared to the waveguide with the fully plasma-filled transport
channel, in the waveguide with incompletely filled channel one should expect
the occurrence of additional eigenwaves, which, in turn, can be excited by the
drive bunch, and thus can contribute to the excited wakefield. This profile of
plasma filling, known as the hollow plasma channel, may take place, for
example, at the capillary discharge [22, 23], or, when creating plasma by the
use of a laser [24] having the annular cross–section of its intensity profile.
It should be noted that by themselves, the hollow plasma channels as wakefield
acceleration structures show much promise and are of great interest because of
the possibility to accelerate positrons [25, 26, 27, 28]. For this reason,
their intensive studies as promising acceleration structures are being
continued, and the results obtained have been presented in numerous
publications [29, 30].

The present paper has been focused on building a linear theory of wakefield
excitation by a drive charged particle bunch in a cylindrical dielectric
waveguide with a hollow plasma channel.

The paper is organized as follows. The statement of problem is formulated in
Section 2. Section 3 describes analytical investigations. Section 4 deals with
numerical analysis based on the results described in Section 3. Section 5
gives the summary.

##  2 Statement of problem

The accelerating structure under consideration represents a cylindrical metal
waveguide of radius $b$, partially filled with homogeneous isotropic
dielectric having the relative permittivity $\varepsilon_{d}$. Inside the
dielectric, there is a vacuum channel of radius $a$, partially filled with
homogeneous isotropic plasma of density $n_{p}$. In turn, the plasma has a
vacuum channel of radius $R_{c}$, and forms in this way a hollow plasma
channel. Along the axis of the accelerating structure, at a constant velocity
$v$ there propagates a non-evolving drive electron bunch of charge $Q_{b}$,
length $L_{b}$ and radius $R_{b}$. In its passage the electron bunch excites
the wakefield. In the general case, the drive bunch radius is larger than the
vacuum channel radius. So, during its propagation, one part of the drive bunch
is localized within the vacuum region, whereas the other is in the plasma
region. Figure 1 schematically shows the waveguide, the dielectric, the hollow
plasma channel and the drive bunch as longitudinal and transverse cross
sections.

Figure 1: General view of the round plasma–dielectric waveguide. Metal
coating, dielectric, plasma and drive beam are shown schematically. The
on–axis drive beam moves along to the waveguide axis.

The plasma ions are assumed to be immobile, and the thermal motion of plasma
electrons is neglected in paper. The vacuum-plasma boundary is taken to be
sharp. It is also assumed that the dielectric has neither frequency nor
spatial dispersion. The material of the metal conductor is assumed to be the
ideal conductor. It is suggested that the density of the drive electron bunch
is low in comparison with the plasma density, enabling us to consider the
plasma as a linear medium. The external magnetic field is absent. The main
goal of the present paper is to construct a linear analytical theory of
wakefield excitation by a single electron bunch in the dielectric waveguide
with a hollow plasma channel. The electrodynamics problem of structure
excitation will be solved here in waveguide statement without taking into
account the effects related to the longitudinal boundedness of the structure.

##  3 Analytical studies

The theory of wakefield excitation by the drive electron bunch in the
waveguide under study will be constructed in cylindrical coordinates
$(r,\varphi,z)$ with the $z$–axis directed along the waveguide axis. According
to the problem description: (I) the drive bunch propagates in the accelerating
structure without any initial transverse offset, (II) only the longitudinal
component of the drive bunch particle velocity vector is nonzero, (III) the
charge distribution of the drive bunch is azimuthally homogeneous. Therefore,
the nonzero contribution to the total wakefield will come only from the eigen
$TM$–waves $(E_{r},\,E_{z},\,H_{\varphi})$ of the bunch–excited waveguide
under consideration.

The theory will be developed as follows. First, we determine the wakefield
excited by the point–like particle (Green’s function) of charge $q_{0}$ with
the transverse coordinates $(r_{0},\,\varphi_{0})$. The particle is injected
into the accelerating structure ($z=0$) at an arrival time $t_{0}$, and is
moving at a constant velocity $v$. With the knowledge of the point–like
particle–excited wakefield, we can derive the wakefield excited by the drive
bunch of finite longitudinal and transverse size values by integrating over
the injection time and transverse coordinate values of the particles.

Before proceeding to deriving the Green’s function, it is of importance to
note the following. The primary emphasis in the present study has been given
to the excited electromagnetic fields, the structure of which is specified
only by the external source, and no account has been taken of the effects that
are associated with the longitudinal boundedness of the waveguide structure.
When constructing the linear theory of excitation, we assume that the time and
longitudinal coordinate dependence of the wakefield is determined in the same
way as that for the source exciting the mentioned field.

The initial set of equations that describes the wakefield excitation
represents the system of Maxwell equations for the radial component of the
electric field $E_{r}$, the axial component of the electric field $E_{z}$, and
the azimuthal component of the magnetic field $H_{\varphi}$

$\begin{split}&-\frac{1}{v}\frac{\partial E_{r}}{\partial\xi}-\frac{\partial
E_{z}}{\partial r}=-\frac{1}{c}\frac{\partial H_{\varphi}}{\partial\xi},\\\
&\frac{1}{v}\frac{\partial H_{\varphi}}{\partial\xi}=\frac{1}{c}\frac{\partial
D_{r}}{\partial\xi},\\\ &\frac{1}{r}\frac{\partial}{\partial
r}\left(rH_{\varphi}\right)=\frac{4\pi}{c}j_{z}+\frac{1}{c}\frac{\partial
D_{z}}{\partial\xi}\end{split}$ (1)

where the densities of charge $\rho$ and of current $j_{z}$, created by the
point–like charged particle are, respectively, equal to

$\begin{split}\rho&=\frac{q_{0}}{v}\frac{{\delta(r-{r_{0}})}}{r}\delta(\varphi-{\varphi_{0}})\delta(\xi-{t_{0}}),\\\
j_{r}&=0,\,j_{\varphi}=0,\,j_{z}=\rho v,\end{split}$ (2)

where $c$ is the speed of light, and $\xi=t-z/v$. The formulated problem is
uniform in both the time and direction of the drive particle motion. The
excited wakefield, and also, the densities of charge and current can be
written as the Fourier integrals over the variable $\xi$

$\begin{split}\begin{pmatrix}\mathbf{E}(r,\xi)\\\ \mathbf{D}(r,\xi)\\\
\mathbf{H}(r,\xi)\\\
\end{pmatrix}&=\int\limits_{-\infty}^{+\infty}{d\omega\begin{pmatrix}\mathbf{E}^{\omega}(r,\omega)\\\
\varepsilon(\omega)\mathbf{E}^{\omega}(r,\omega)\\\
\mathbf{H}^{\omega}(r,\omega)\\\ \end{pmatrix}}{e^{-i\omega\xi}}{\text{,}}\\\
\begin{pmatrix}\rho(r,\xi)\\\ j_{z}(r,\xi)\\\
\end{pmatrix}&=\int\limits_{-\infty}^{+\infty}{d\omega\begin{pmatrix}\rho^{\omega}(r,\omega)\\\
j_{z}^{\omega}(r,\omega)\\\
\end{pmatrix}}{e^{-i\omega\xi}}{\text{.}}\end{split}$ (3)

The dielectric permittivity in the vacuum regions ($0\leq r\leq R_{c}$),
plasma ($R_{c}\leq r\leq a$) and dielectric ($a\leq r\leq b$) is,
respectively, equal to

$\begin{split}\varepsilon(\omega)=\begin{cases}1,&\text{$0\leq r\leq
R_{c}$;}\\\
\varepsilon_{p}(\omega)=1-\omega_{p}^{2}/{\omega^{2}},&\text{$R_{c}\leq r\leq
a$;}\\\ \varepsilon_{d},&\text{$a\leq r\leq b$.}\end{cases}\end{split}$ (4)

where $\omega_{p}$ is the plasma frequency.

From equations (1) we obtain the inhomogeneous wave equation for the Fourier
transform of the axial component of the electric field $E_{z}^{\omega}$

$\begin{split}\frac{1}{r}\frac{\partial}{{\partial r}}\left({r\frac{{\partial
E_{z}^{\omega}}}{{\partial
r}}}\right)-\frac{{{\omega^{2}}}}{{{v^{2}}}}(1-{\beta^{2}}\varepsilon(\omega))E_{z}^{\omega}=\frac{{4\pi\omega
i}}{{{v^{2}}}}\,\frac{{1-{\beta^{2}}\varepsilon(\omega)}}{{\varepsilon(\omega)}}j_{z}^{\omega}.\end{split}$
(5)

In the right–hand side of (5) we have the Fourier transform of the axial
component of the current density $j_{z}^{\omega}$, which can be written as

$\begin{split}j_{z}^{\omega}=\frac{q_{0}e^{i\omega
t_{0}}}{4\pi^{2}}\int\limits_{0}^{+\infty}J_{0}(\lambda r)J_{0}(\lambda
r_{0})\lambda d\lambda.\end{split}$ (6)

In the case that the radius of drive bunch is larger than the radius of vacuum
channel, the process of wakefield excitation is contributed by both the bunch
particles being in the vacuum region and by the ones being in the plasma
region. Below it will be shown that the frequency contents of the wakefield
(and consequently, its spatiotemporal structure) is dependent on the
excitation source location. Since the consideration is given here to the
linear regime of excitation, then the total wakefield may be represented as a
superposition of wakefields excited independently by the particles, being both
in the vacuum and in the plasma.

The right–hand side of the wave equation for the Fourier transform of the
axial component of the electric field
$E_{z}^{\omega(I)}(r,\omega,r_{0},t_{0})$, that is excited by the point–like
charged particle propagating in vacuum is nonzero only in the vacuum region
and is equal to $4\pi\omega i(1-{\beta^{2}})j_{z}^{\omega}/v^{2}$. In the case
when the source of wakefield excitation is a charged particle propagating in
the plasma, the right part of the equation for the Fourier transform of the
axial component of the electric field
$E_{z}^{\omega(II)}(r,\omega,r_{0},t_{0})$ is nonzero only in the plasma
region, and is equal to $4\pi\omega
i(1-{\beta^{2}\varepsilon_{p}(\omega)})j_{z}^{\omega}/v^{2}\varepsilon_{p}(\omega)$.

Equation (5) must be supplemented by the boundary conditions satisfied by the
electromagnetic field excited in the waveguide. For the above–defined problem,
these conditions are as follows. The amplitudes of wakefield components are
finite at $r=0$. On the surface of the perfectly conducting metal coating, the
longitudinal electric field component turns to zero
$E_{z}^{\omega(I,II)}(r=b)=0$. At vacuum–plasma and plasma–dielectric
boundaries, the axial electric field component and the azimuthal magnetic
field component are continuous
$E_{z}^{\omega(I,II)}(r=R_{c}-0)=E_{z}^{\omega(I,II)}(r=R_{c}+0)$,
$H_{\varphi}^{\omega(I,II)}(r=R_{c}-0)=H_{\varphi}^{\omega(I,II)}(r=R_{c}+0)$,
$E_{z}^{\omega(I,II)}(r=a-0)=E_{z}^{\omega(I,II)}(r=a+0)$,
$H_{\varphi}^{\omega(I,II)}(r=a-0)=H_{\varphi}^{\omega(I,II)}(r=a+0)$.

With the use of the Maxwell equations, the Fourier transforms of the radial
electric field $E_{r}^{\omega}$, the axial magnetic field
$H_{\varphi}^{\omega}$, and as a result, their combination
$W_{r}^{\omega}=E_{r}^{\omega}-\beta H_{\varphi}^{\omega}$, which is involved
in the radial force, can be expressed in terms of the Fourier transform of the
axial electric field $E_{z}^{\omega}$

$\begin{split}&E_{r}^{\omega}=-i\frac{v/\omega}{{(1-{\beta^{2}}\varepsilon(\omega))}}{\frac{{\partial
E_{z}^{\omega}}}{{\partial r}}},\\\
&H_{\varphi}^{\omega}=-i\beta\varepsilon(\omega)\frac{v/\omega}{{(1-{\beta^{2}}\varepsilon(\omega))}}{\frac{{\partial
E_{z}^{\omega}}}{{\partial r}}},\\\ &W_{r}^{\omega}=E_{r}^{\omega}-\beta
H_{\varphi}^{\omega}=-i\frac{v}{\omega}\frac{\partial E_{z}^{\omega}}{\partial
r}\end{split}$ (7)

Equation (5) can be solved by the superposition a particular solution of the
inhomogeneous equation and the general solution of the corresponding
homogeneous equation

$\begin{split}E_{z}^{\omega(I)}(0\leq
r<r_{0})=-i\frac{q_{0}\varkappa_{v}^{2}}{\pi\omega}\frac{I_{0}(\varkappa_{v}r)}{I_{0}(\varkappa_{v}R_{c})}\Delta_{0}(\varkappa_{v}R_{c},\varkappa_{v}r_{0})e^{i\omega
t_{0}}-\\\ i\frac{q_{0}}{\pi\omega
R_{c}}\frac{I_{0}(\varkappa_{v}r_{0})I_{0}(\varkappa_{v}r)}{I_{0}^{2}(\varkappa_{v}R_{c})}\frac{D_{2}(\omega)}{D(\omega)}e^{i\omega
t_{0}}\end{split}$ (8)

$\begin{split}E_{z}^{\omega(I)}(r_{0}<r\leq
R_{c})=-i\frac{q_{0}\varkappa_{v}^{2}}{\pi\omega}\frac{I_{0}(\varkappa_{v}r_{0})}{I_{0}(\varkappa_{v}R_{c})}\Delta_{0}(\varkappa_{v}R_{c},\varkappa_{v}r)e^{i\omega
t_{0}}-\\\ i\frac{q_{0}}{\pi\omega
R_{c}}\frac{I_{0}(\varkappa_{v}r_{0})I_{0}(\varkappa_{v}r)}{I_{0}^{2}(\varkappa_{v}R_{c})}\frac{D_{2}(\omega)}{D(\omega)}e^{i\omega
t_{0}}\end{split}$ (9)

$\begin{split}E_{z}^{\omega(I)}(R_{c}\leq r\leq a)=-i\frac{q_{0}}{\pi\omega
R_{c}}\frac{I_{0}(\varkappa_{v}r_{0})}{I_{0}(\varkappa_{v}R_{c})}\frac{D_{2}(\omega)}{D(\omega)}\frac{\Delta_{0}(\varkappa_{v}r,\varkappa_{v}a)}{\Delta_{0}(\varkappa_{v}R_{c},\varkappa_{v}a)}e^{i\omega
t_{0}}-\\\
i\frac{q_{0}\varepsilon_{p}(\omega)}{\pi\omega\varkappa_{p}^{2}aR_{c}D(\omega)}\frac{I_{0}(\varkappa_{v}r_{0})}{I_{0}(\varkappa_{v}R_{c})}\frac{\Delta_{0}(\varkappa_{p}r,\varkappa_{p}R_{c})}{\Delta_{0}^{2}(\varkappa_{p}a,\varkappa_{p}R_{c})}e^{i\omega
t_{0}}\end{split}$ (10)

$\begin{split}E_{z}^{\omega(II)}(0\leq r\leq R_{c})=\frac{ic}{\omega
D(\omega)}\frac{I_{0}(\varkappa r)}{I_{0}(\varkappa
R_{c})}\left(f_{1}D_{2}-\frac{f_{2}\varepsilon_{p}}{\varkappa_{p}^{2}R_{c}\Delta_{0}(\varkappa_{p}a,\varkappa_{p}R_{c})}\right)\end{split}$
(11)

$\begin{split}E_{z}^{\omega(II)}(R_{c}\leq r\leq
r_{0})=\frac{iq_{0}\varkappa_{p}^{2}}{\pi\omega\varepsilon_{p}}G_{p}(r,r_{0},\omega)e^{i\omega
t_{0}}+\\\ \frac{ic}{\omega
D(\omega)}\left(f_{1}D_{2}-\frac{f_{2}\varepsilon_{p}}{\varkappa_{p}^{2}R_{c}\Delta_{0}(\varkappa_{p}a,\varkappa_{p}R_{c})}\right)\frac{\Delta_{0}(\varkappa_{p}r,\varkappa_{p}a)}{\Delta_{0}(\varkappa_{p}R_{c},\varkappa_{p}a)}-\\\
\frac{ic}{\omega
D(\omega)}\left(f_{2}D_{1}+\frac{f_{1}\varepsilon_{p}}{\varkappa_{p}^{2}a\Delta_{0}(\varkappa_{p}R_{c},\varkappa_{p}a)}\right)\frac{\Delta_{0}(\varkappa_{p}r,\varkappa_{p}R_{c})}{\Delta_{0}(\varkappa_{p}a,\varkappa_{p}R_{c})}\end{split}$
(12)

$\begin{split}E_{z}^{\omega(II)}(r_{0}\leq r\leq
a)=\frac{iq_{0}\varkappa_{p}^{2}}{\pi\omega\varepsilon_{p}}H_{p}(r,r_{0},\omega)e^{i\omega
t_{0}}+\\\ \frac{ic}{\omega
D(\omega)}\left(f_{1}D_{2}-\frac{f_{2}\varepsilon_{p}}{\varkappa_{p}^{2}R_{c}\Delta_{0}(\varkappa_{p}a,\varkappa_{p}R_{c})}\right)\frac{\Delta_{0}(\varkappa_{p}r,\varkappa_{p}a)}{\Delta_{0}(\varkappa_{p}R_{c},\varkappa_{p}a)}-\\\
\frac{ic}{\omega
D(\omega)}\left(f_{2}D_{1}+\frac{f_{1}\varepsilon_{p}}{\varkappa_{p}^{2}a\Delta_{0}(\varkappa_{p}R_{c},\varkappa_{p}a)}\right)\frac{\Delta_{0}(\varkappa_{p}r,\varkappa_{p}R_{c})}{\Delta_{0}(\varkappa_{p}a,\varkappa_{p}R_{c})},\end{split}$
(13)

In expressions (8)– (13) the following designations are used:

$\begin{split}\varkappa_{v}^{2}=(\omega/v)^{2}(1-{\beta^{2}}),\,\varkappa_{p}^{2}=(\omega/v)^{2}(1-{\beta^{2}}{\varepsilon_{p}}({\omega})),\end{split}$
(14)
$\begin{split}\Delta_{0}(x,y)=I_{0}(x)K_{0}(y)-K_{0}(x)I_{0}(y),\,\Delta_{1}(x,y)=I_{1}(x)K_{0}(y)+K_{1}(x)I_{0}(y),\end{split}$
(15)

where $I_{0,1}$ and $K_{0,1}$ are the modified Bessel and MacDonald functions
of the zero and first order, and the functions $f_{1}(\omega)$,
$f_{2}(\omega)$, $G_{p}(r,r_{0},\omega)$, $H_{p}(r,r_{0},\omega)$, $D(\omega)$
take the following forms:

$\begin{split}f_{1}(\omega)=-\frac{q_{0}e^{i\omega t_{0}}}{\pi
cR_{c}}\frac{\Delta_{0}(\varkappa_{p}r_{0},\varkappa_{p}a)}{\Delta_{0}(\varkappa_{ps}R_{c},\varkappa_{ps}a)},\,f_{2}(\omega)=\frac{q_{0}e^{i\omega
t_{0}}}{\pi
ca}\frac{\Delta_{0}(\varkappa_{p}r_{0},\varkappa_{p}R_{c})}{\Delta_{0}(\varkappa_{ps}a,\varkappa_{ps}R_{c})},\end{split}$
(16)

$\begin{split}G_{p}(r,r_{0},\omega)=\frac{\Delta_{0}(\varkappa_{p}r,\varkappa_{p}R_{c})\Delta_{0}(\varkappa_{p}a,\varkappa_{p}r_{0})}{\Delta_{0}(\varkappa_{p}R_{c},\varkappa_{p}a)},\,H_{p}(r,r_{0},\omega)=\frac{\Delta_{0}(\varkappa_{p}r,\varkappa_{p}a)\Delta_{0}(\varkappa_{p}r_{0},\varkappa_{p}R_{c})}{\Delta_{0}(\varkappa_{p}a,\varkappa_{p}R_{c})},\end{split}$
(17)

$\begin{split}D(\omega)&=D_{1}(\omega)D_{2}(\omega)+D_{3}(\omega),\\\
D_{1}(\omega)&=\frac{1}{\varkappa_{v}}\frac{I_{1}(\varkappa_{v}R_{c})}{I_{0}(\varkappa_{v}R_{c})}-\frac{\varepsilon_{p}}{\varkappa_{p}}\frac{\Delta_{1}(\varkappa_{p}R_{c},\varkappa_{p}a)}{\Delta_{0}(\varkappa_{p}R_{c},\varkappa_{p}a)},\\\
D_{2}(\omega)&=\frac{\varepsilon_{p}}{\varkappa_{p}}\frac{\Delta_{1}(\varkappa_{p}a,\varkappa_{p}R_{c})}{\Delta_{0}(\varkappa_{p}a,\varkappa_{p}R_{c})}+\frac{\varepsilon_{d}}{\varkappa_{d}}\frac{F_{1}(\varkappa_{d}a,\varkappa_{d}b)}{F_{0}(\varkappa_{d}a,\varkappa_{d}b)},\\\
D_{3}(\omega)&=\frac{\varepsilon_{p}^{2}}{\varkappa_{p}^{2}}\frac{\Delta_{1}(\varkappa_{p}R_{c},\varkappa_{p}R_{c})}{\Delta_{0}(\varkappa_{p}a,\varkappa_{p}R_{c})}\frac{\Delta_{1}(\varkappa_{p}a,\varkappa_{p}a)}{\Delta_{0}(\varkappa_{p}R_{c},\varkappa_{p}a)},\end{split}$
(18)

and $\varkappa_{d}^{2}=(\omega/v)^{2}(\beta^{2}\varepsilon_{d}-1)$.

By integrating over the frequency (3) we obtain the expressions for the
Green’s functions of the longitudinal component of the electric field excited
by point–like charged particles, which are propagating in the vacuum channel
$E_{Gz}^{(I)}(r,\xi,r_{0},t_{0})$, and in the plasma
$E_{Gz}^{(II)}(r,\xi,r_{0},t_{0})$

$\begin{split}E_{Gz}^{(I)}(0\leq r\leq
R_{c})=-\sum\limits_{s=1}^{+\infty}\frac{4q_{0}\varkappa_{vs}}{\omega_{s}R_{c}}\frac{I_{0}(\varkappa_{vs}r)I_{0}(\varkappa_{vs}r_{0})}{I_{0}^{2}(\varkappa_{vs}R_{c})}\frac{D_{2s}}{D^{\prime}({\omega_{s}})}\times\\\
cos\omega_{s}(\xi-t_{0})\theta(\xi-t_{0})\end{split}$ (19)

$\begin{split}E_{Gz}^{(I)}(R_{c}\leq r\leq
a)=-\sum\limits_{s=1}^{+\infty}\frac{4q_{0}}{\omega_{s}D^{\prime}({\omega_{s}})R_{c}}\frac{I_{0}(\varkappa_{vs}r_{0})}{I_{0}(\varkappa_{vs}R_{c})}cos\omega_{s}(\xi-
t_{0})\theta(\xi-t_{0})\times\\\
\left(D_{2s}\frac{\Delta_{0}(\varkappa_{ps}r,\varkappa_{ps}a)}{\Delta_{0}(\varkappa_{ps}R_{c},\varkappa_{ps}a)}-\frac{\varepsilon_{ps}}{\varkappa_{ps}^{2}a}\frac{\Delta_{0}(\varkappa_{ps}r,\varkappa_{ps}R_{c})}{\Delta_{0}(\varkappa_{ps}a,\varkappa_{ps}R_{c})}\right)\end{split}$
(20)

$\begin{split}W_{Gr}^{(I)}(0\leq r\leq
R_{c})=\sum\limits_{s=1}^{+\infty}\frac{4q_{0}v\varkappa_{vs}}{\omega_{s}^{2}R_{c}}\frac{I_{1}(\varkappa_{vs}r)I_{0}(\varkappa_{vs}r_{0})}{I_{0}^{2}(\varkappa_{vs}R_{c})}\times\\\
\frac{D_{2s}}{D^{\prime}({\omega_{s}})}sin\omega_{s}(\xi-t_{0})\theta(\xi-
t_{0})\end{split}$ (21)

$\begin{split}W_{Gr}^{(I)}(R_{c}\leq r\leq
a)=\sum\limits_{s=1}^{+\infty}\frac{4q_{0}v\varkappa_{ps}}{\omega_{s}^{2}D^{\prime}(\omega_{s})}\frac{I_{0}(\varkappa_{vs}r_{0})}{I_{0}(\varkappa_{vs}R_{c})}\times\\\
\left(\frac{D_{2s}}{R_{c}}\frac{\Delta_{1}(\varkappa_{ps}r,\varkappa_{ps}a)}{\Delta_{0}(\varkappa_{ps}R_{c},\varkappa_{ps}a)}-\frac{\varepsilon_{ps}}{\varkappa_{ps}^{2}aR_{c}\Delta_{0}(\varkappa_{ps}R_{c},\varkappa_{ps}a)}\frac{\Delta_{1}(\varkappa_{ps}r,\varkappa_{ps}R_{c})}{\Delta_{0}(\varkappa_{ps}a,\varkappa_{ps}R_{c})}\right)\times\\\
sin\omega_{s}(\xi-t_{0})\theta(\xi-t_{0})\end{split}$ (22)

$\begin{split}E_{Gz}^{(II)}(0\leq r\leq
R_{c})=-\sum\limits_{s=1}^{+\infty}\frac{4q_{0}}{\omega_{s}D^{\prime}({\omega_{s}})}\frac{I_{0}(\varkappa_{vs}r)}{I_{0}(\varkappa_{vs}R_{c})}cos\omega_{s}(\xi-
t_{0})\theta(\xi-t_{0})\times\\\
\left(\frac{D_{2s}\Delta_{0}(\varkappa_{ps}r_{0},\varkappa_{ps}a)}{R_{c}\Delta_{0}(\varkappa_{ps}R_{c},\varkappa_{ps}a)}-\frac{\varkappa_{ps}^{2}D_{3s}}{\varepsilon_{ps}}\Delta_{0}(\varkappa_{ps}r_{0},\varkappa_{ps}R_{c})\right)\end{split}$
(23)

$\begin{split}E_{Gz}^{(II)}(R_{c}\leq r\leq
r_{0})=2q_{0}k_{p}^{2}G_{p}(r,r_{0},\omega_{p})cos\omega_{p}(\xi-
t_{0})\theta(\xi-t_{0})-\\\
\sum\limits_{s=1}^{+\infty}\frac{4q_{0}}{\omega_{s}D^{\prime}({\omega_{s}})}\frac{\Delta_{0}(\varkappa_{ps}r,\varkappa_{ps}a)}{\Delta_{0}(\varkappa_{ps}R_{c},\varkappa_{ps}a)}cos\omega_{s}(\xi-
t_{0})\theta(\xi-t_{0})\times\\\
\left(\frac{D_{2s}\Delta_{0}(\varkappa_{ps}r_{0},\varkappa_{ps}a)}{R_{c}\Delta_{0}(\varkappa_{ps}R_{c},\varkappa_{ps}a)}-\frac{\varkappa_{ps}^{2}D_{3s}}{\varepsilon_{ps}}\Delta_{0}(\varkappa_{ps}r_{0},\varkappa_{ps}R_{c})\right)-\\\
\sum\limits_{s=1}^{+\infty}\frac{4q_{0}}{\omega_{s}D^{\prime}(\omega_{s})}\frac{\Delta_{0}(\varkappa_{ps}r,\varkappa_{ps}R_{c})}{\Delta_{0}(\varkappa_{ps}a,\varkappa_{ps}R_{c})}cos\omega_{s}(\xi-
t_{0})\theta(\xi-t_{0})\times\\\
\left(\frac{D_{1s}\Delta_{0}(\varkappa_{ps}r_{0},\varkappa_{ps}R_{c})}{a\Delta_{0}(\varkappa_{ps}a,\varkappa_{ps}R_{c})}+\frac{\varkappa_{ps}^{2}D_{3s}}{\varepsilon_{ps}}\Delta_{0}(\varkappa_{ps}r_{0},\varkappa_{ps}a)\right)\end{split}$
(24)

$\begin{split}E_{Gz}^{(II)}(r_{0}\leq r\leq
a)=2q_{0}k_{p}^{2}H_{p}(r,r_{0},\omega_{p})cos\omega_{p}(\xi-t_{0})\theta(\xi-
t_{0})-\\\
\sum\limits_{s=1}^{+\infty}\frac{4q_{0}}{\omega_{s}D^{\prime}({\omega_{s}})}\frac{\Delta_{0}(\varkappa_{ps}r,\varkappa_{ps}a)}{\Delta_{0}(\varkappa_{ps}R_{c},\varkappa_{ps}a)}cos\omega_{s}(\xi-
t_{0})\theta(\xi-t_{0})\times\\\
\left(\frac{D_{2s}\Delta_{0}(\varkappa_{ps}r_{0},\varkappa_{ps}a)}{R_{c}\Delta_{0}(\varkappa_{ps}R_{c},\varkappa_{ps}a)}-\frac{\varkappa_{ps}^{2}D_{3s}}{\varepsilon_{ps}}\Delta_{0}(\varkappa_{ps}r_{0},\varkappa_{ps}R_{c})\right)-\\\
\sum\limits_{s=1}^{+\infty}\frac{4q_{0}}{\omega_{s}D^{\prime}(\omega_{s})}\frac{\Delta_{0}(\varkappa_{ps}r,\varkappa_{ps}R_{c})}{\Delta_{0}(\varkappa_{ps}a,\varkappa_{ps}R_{c})}cos\omega_{s}(\xi-
t_{0})\theta(\xi-t_{0})\times\\\
\left(\frac{D_{1s}\Delta_{0}(\varkappa_{ps}r_{0},\varkappa_{ps}R_{c})}{a\Delta_{0}(\varkappa_{ps}a,\varkappa_{ps}R_{c})}+\frac{\varkappa_{ps}^{2}D_{3s}}{\varepsilon_{ps}}\Delta_{0}(\varkappa_{ps}r_{0},\varkappa_{ps}a)\right)\end{split}$
(25)

$\begin{split}W_{Gr}^{(II)}(0\leq r\leq
R_{c})=-\sum\limits_{s=1}^{+\infty}\frac{4\pi
cv\varkappa_{vs}}{\omega_{s}^{2}D^{\prime}(\omega_{s})}\frac{I_{1}(\varkappa_{vs}r)}{I_{0}(\varkappa_{vs}R_{c})}\times\\\
\left(f_{1s}D_{2s}-\frac{f_{2s}\varepsilon_{ps}}{\varkappa_{ps}^{2}R_{c}\Delta_{0}(\varkappa_{p}a,\varkappa_{p}R_{c})}\right)sin\omega_{s}(\xi-
t_{0})\theta(\xi-t_{0})\end{split}$ (26)

$\begin{split}W_{Gr}^{(II)}(R_{c}\leq r\leq
r_{0})=-2q_{0}k_{p}^{2}G^{\prime}_{p}(\omega_{p})\times\\\ sin\omega_{p}(\xi-
t_{0})\theta(\xi-t_{0})-\sum\limits_{s=1}^{+\infty}\frac{4\pi
cv\varkappa_{ps}}{\omega_{s}^{2}D^{\prime}(\omega_{s})}sin\omega_{s}(\xi-
t_{0})\theta(\xi-t_{0})\times\\\
\left(\frac{\Delta_{1}(\varkappa_{ps}r,\varkappa_{ps}a)}{\Delta_{0}(\varkappa_{ps}R_{c},\varkappa_{ps}a)}\left\\{f_{1s}D_{2s}-\frac{f_{2s}\varepsilon_{ps}}{\varkappa_{ps}^{2}R_{c}\Delta_{0}(\varkappa_{p}a,\varkappa_{p}R_{c})}\right\\}-\right.\\\
\left.\frac{\Delta_{1}(\varkappa_{ps}r,\varkappa_{ps}R_{c})}{\Delta_{0}(\varkappa_{ps}a,\varkappa_{ps}R_{c})}\left\\{f_{2s}D_{1s}+\frac{f_{1s}\varepsilon_{ps}}{\varkappa_{ps}^{2}a\Delta_{0}(\varkappa_{ps}R_{c},\varkappa_{ps}a)}\right\\}\right)\end{split}$
(27)

$\begin{split}W_{Gr}^{(II)}(r_{0}\leq r\leq
a)=-2q_{0}k_{p}^{2}H^{\prime}_{p}(\omega_{p})\times\\\ sin\omega_{p}(\xi-
t_{0})\theta(\xi-t_{0})-\sum\limits_{s=1}^{+\infty}\frac{4\pi
cv\varkappa_{ps}}{\omega_{s}^{2}D^{\prime}(\omega_{s})}sin\omega_{s}(\xi-
t_{0})\theta(\xi-t_{0})\times\\\
\left(\frac{\Delta_{1}(\varkappa_{ps}r,\varkappa_{ps}a)}{\Delta_{0}(\varkappa_{ps}R_{c},\varkappa_{ps}a)}\left\\{f_{1s}D_{2s}-\frac{f_{2s}\varepsilon_{ps}}{\varkappa_{ps}^{2}R_{c}\Delta_{0}(\varkappa_{p}a,\varkappa_{p}R_{c})}\right\\}-\right.\\\
\left.\frac{\Delta_{1}(\varkappa_{ps}r,\varkappa_{ps}R_{c})}{\Delta_{0}(\varkappa_{ps}a,\varkappa_{ps}R_{c})}\left\\{f_{2s}D_{1s}+\frac{f_{1s}\varepsilon_{ps}}{\varkappa_{ps}^{2}a\Delta_{0}(\varkappa_{ps}R_{c},\varkappa_{ps}a)}\right\\}\right),\end{split}$
(28)

where the index $s$ denotes that the corresponding quantities were calculated
at $\omega=\omega_{s}$, while
$D^{\prime}(\omega_{s})=\left(\frac{dD(\omega)}{d\omega}\right)_{\omega=\omega_{s}}.$

Expressions (19)– (28) demonstrate the fact that the frequency content of the
excited wakefield depends on the region of the waveguide in which the charged
particle, which is its source, propagates. Namely, if the exciting particle is
in the vacuum channel, then the expressions $E_{Gz}^{(I)}(r,\xi,r_{0},t_{0})$
for the regions of both the vacuum and the plasma, and
$E_{Gz}^{(II)}(r,\xi,r_{0},t_{0})$ for the vacuum region, contain only the
frequencies $\omega_{s}$, which are defined by the pole $D(\omega_{s})=0$ in
the integrands. The frequencies $\omega_{s}$ are the fundamental frequencies
of the waveguide under study, which are in resonance with the bunch (Cherenkov
synchronism). This exhibits the fact that on moving in vacuum, the charged
particle does not excite the plasma Langmuir wave. Furthermore, the plasma
Langmuir wave, excited at the frequency $\omega_{p}$ by the charged particle
propagating in the plasma, is localized in the plasma. And in this case, the
structure of the total wakefield is determined by the wakefields excited at
the frequencies $\omega_{s}$ being synchronous with the drive bunch. While the
expressions $E_{Gz}^{(II)}(r,\xi,r_{0},t_{0})$ in the plasma region are
determined by both the resonant frequencies $\omega_{s}$, and the plasma
frequency $\omega_{p}$, which is defined by the pole
$\varepsilon_{p}(\omega_{p})=0$. And in this case, the total wakefield is
formed by both the wakefields at the frequencies $\omega_{s}$, being in
synchronism with the bunch, and the wakefield of the plasma wave at the
frequency $\omega_{p}$.

By integration over the times of arrival $t_{0}$, and also over the transverse
coordinates of the particles $(r_{0},\varphi_{0})$ we obtain the final
expressions for the longitudinal and transverse components of the wakefield
excited by the drive bunch, that has the radius larger than the vacuum channel
radius ($R_{b}>R_{c}$):

vacuum channel region $(0\leq r\leq R_{c})$

$\begin{split}E_{z}(r,\xi)=-\sum\limits_{s=1}^{+\infty}\frac{8Q_{b}}{\omega_{s}D^{\prime}(\omega_{s})R_{b}^{2}}\frac{I_{0}(\varkappa_{vs}r)}{I_{0}(\varkappa_{vs}R_{c})}\Psi_{\parallel}(\xi,\omega_{s})\times\\\
\left(\frac{D_{2s}I_{1}(\varkappa_{vs}R_{c})}{\varkappa_{vs}I_{0}(\varkappa_{vs}R_{c})}+\frac{D_{2s}\Phi_{1}(\varkappa_{ps}R_{c})}{\varkappa_{ps}^{2}R_{c}\Delta_{0}(\varkappa_{ps}R_{c},\varkappa_{ps}a)}-\frac{D_{3s}\Phi_{2}(\varkappa_{ps}R_{b})}{\varepsilon_{ps}}\right),\end{split}$
(29)

$\begin{split}W_{r}(r,\xi)=\sum\limits_{s=1}^{+\infty}\frac{8Q_{b}v}{\omega_{s}^{2}D^{\prime}(\omega_{s})R_{b}^{2}}\frac{I_{1}(\varkappa_{vs}r)}{I_{0}(\varkappa_{vs}R_{c})}\Psi_{\perp}(\xi,\omega_{s})\times\\\
\left(D_{2s}\frac{I_{1}(\varkappa_{vs}R_{c})}{I_{0}(\varkappa_{vs}R_{c})}+\frac{\varkappa_{vs}}{\varkappa_{ps}^{2}R_{c}}\left\\{\frac{D_{2s}\Phi_{1}(\varkappa_{ps}R_{c})}{\Delta_{0}(\varkappa_{ps}R_{c},\varkappa_{ps}a)}+\frac{\varepsilon_{ps}\Phi_{2}(\varkappa_{ps}R_{b})}{\varkappa_{ps}^{2}a\Delta_{0}^{2}(\varkappa_{ps}a,\varkappa_{ps}R_{c})}\right\\}\right)\end{split}$
(30)

the plasma region, which has the drive bunch in $(R_{c}\leq r\leq R_{b})$

$\begin{split}E_{z}(r,\xi)=\frac{4Q_{b}}{R_{b}^{2}}\Psi_{\parallel}(\xi,\omega_{p})\left(\frac{\Delta_{0}(k_{p}r,k_{p}a)}{\Delta_{0}(k_{p}a,k_{p}R_{c})}\Phi_{2}(k_{p}r)+\frac{\Delta_{0}(k_{p}r,k_{p}R_{c})}{\Delta_{0}(k_{p}a,k_{p}R_{c})}\Phi_{1}(k_{p}r)\right)\\\
-\sum\limits_{s=1}^{+\infty}\frac{8Q_{b}}{\omega_{s}D^{\prime}(\omega_{s})R_{b}^{2}}\Psi_{\parallel}(\xi,\omega_{s})\times\\\
\left(\frac{I_{1}(\varkappa_{vs}R_{b})}{\varkappa_{vs}I_{0}(\varkappa_{vs}R_{c})}\left\\{D_{2s}\frac{\Delta_{0}(\varkappa_{ps}r,\varkappa_{ps}a)}{\Delta_{0}(\varkappa_{ps}R_{c},\varkappa_{ps}a)}-\frac{\varepsilon_{ps}}{\varkappa_{ps}^{2}a}\frac{\Delta_{0}(\varkappa_{ps}r,\varkappa_{ps}R_{c})}{\Delta_{0}(\varkappa_{ps}a,\varkappa_{ps}R_{c})}\right\\}+\right.\\\
\left.\frac{\Delta_{0}(\varkappa_{ps}r,\varkappa_{ps}a)}{\Delta_{0}(\varkappa_{ps}R_{c},\varkappa_{ps}a)}\left\\{\frac{D_{2s}\Phi_{1}(\varkappa_{ps}R_{c})}{\varkappa_{ps}^{2}R_{c}\Delta_{0}(\varkappa_{ps}R_{c},\varkappa_{ps}a)}-\frac{D_{3s}}{\varepsilon_{ps}}\Phi_{2}(\varkappa_{ps}R_{b})\right\\}+\right.\\\
\left.\frac{\Delta_{0}(\varkappa_{ps}r,\varkappa_{ps}R_{c})}{\Delta_{0}(\varkappa_{ps}a,\varkappa_{ps}R_{c})}\left\\{\frac{D_{1s}\Phi_{2}(\varkappa_{ps}R_{b})}{\varkappa_{ps}^{2}a\Delta_{0}(\varkappa_{ps}a,\varkappa_{ps}R_{c})}+\frac{D_{3s}}{\varepsilon_{ps}}\Phi_{1}(\varkappa_{ps}R_{c})\right\\}\right)\end{split}$
(31)

$\begin{split}W_{r}(r,\xi)=-\frac{4Q_{b}}{R_{b}^{2}}\Psi_{\perp}(\xi,\omega_{p})\left(\frac{\Delta_{1}(k_{p}r,k_{p}a)}{\Delta_{0}(k_{p}a,k_{p}R_{c})}\Phi_{2}(k_{p}r)+\frac{\Delta_{1}(k_{p}r,k_{p}R_{c})}{\Delta_{0}(k_{p}a,k_{p}R_{c})}\Phi_{1}(k_{p}r)\right)+\\\
\sum\limits_{s=1}^{+\infty}\frac{8Q_{b}v}{\omega_{s}^{2}D^{\prime}(\omega_{s})R_{b}^{2}}\Psi_{\perp}(\xi,\omega_{s})\times\\\
\left(\frac{\varkappa_{ps}I_{1}(\varkappa_{vs}R_{c})}{\varkappa_{vs}I_{0}(\varkappa_{vs}R_{c})}\left\\{D_{2s}\frac{\Delta_{1}(\varkappa_{ps}r,\varkappa_{ps}a)}{\Delta_{0}(\varkappa_{ps}R_{c},\varkappa_{ps}a)}-\frac{\varepsilon_{ps}}{\varkappa_{ps}^{2}a\Delta_{0}(\varkappa_{ps}R_{c},\varkappa_{ps}a)}\frac{\Delta_{1}(\varkappa_{ps}r,\varkappa_{ps}R_{c})}{\Delta_{0}(\varkappa_{ps}a,\varkappa_{ps}R_{c})}\right\\}+\right.\\\
\left.\frac{\Delta_{1}(\varkappa_{ps}r,\varkappa_{ps}a)}{\Delta_{0}(\varkappa_{ps}R_{c},\varkappa_{ps}a)}\left\\{\frac{D_{2s}\Phi_{1}(\varkappa_{ps}R_{c})}{\varkappa_{ps}R_{c}\Delta_{0}(\varkappa_{ps}R_{c},\varkappa_{ps}a)}+\frac{\varepsilon_{ps}\Phi_{2}(\varkappa_{ps}R_{b})}{\varkappa_{ps}^{3}aR_{c}\Delta_{0}^{2}(\varkappa_{ps}a,\varkappa_{ps}R_{c})}\right\\}+\right.\\\
\left.\frac{\Delta_{1}(\varkappa_{ps}r,\varkappa_{ps}R_{c})}{\Delta_{0}(\varkappa_{ps}a,\varkappa_{ps}R_{c})}\left\\{\frac{D_{1s}\Phi_{2}(\varkappa_{ps}R_{b})}{\varkappa_{ps}a\Delta_{0}(\varkappa_{ps}a,\varkappa_{ps}R_{c})}-\frac{\varepsilon_{ps}\Phi_{1}(\varkappa_{ps}R_{c})}{\varkappa_{ps}^{3}aR_{c}\Delta_{0}^{2}(\varkappa_{ps}R_{c},\varkappa_{ps}a)}\right\\}\right),\end{split}$
(32)

where

$\begin{split}\Phi_{1}(\alpha r)&=\alpha R_{b}\Delta_{1}(\alpha R_{b},\alpha
a)-\alpha r\Delta_{1}(\alpha r,\alpha a),\\\ \Phi_{2}(\alpha r)&=\alpha
r\Delta_{1}(\alpha r,\alpha R_{c})-\alpha R_{c}\Delta_{1}(\alpha R_{c},\alpha
R_{c})\end{split}$ (33)

The functions $\Psi_{\parallel}(\xi,\omega)$ and $\Psi_{\perp}(\xi,\omega)$,
that describe the longitudinal structure of the axial and transverse wakefield
components take the following forms:

$\begin{split}\Psi_{\parallel}(\xi,\omega)=\frac{1}{\omega
T_{b}}\bigg{(}\theta(\xi)sin\omega\xi-\theta(\xi-T_{b})sin\omega(\xi-
T_{b})\bigg{)}-\\\
\frac{1}{4\pi^{2}-\omega^{2}T_{b}^{2}}\left(\theta(\xi)\left\\{2\pi
sin\frac{2\pi}{T_{b}}\xi-\omega T_{b}sin\omega\xi\right\\}-\right.\\\
\left.\theta(\xi-T_{b})\left\\{2\pi sin\frac{2\pi}{T_{b}}(\xi-T_{b})-\omega
T_{b}sin\omega(\xi-T_{b})\right\\}\right),\\\
\Psi_{\perp}(\xi,\omega)=\frac{1}{\omega
T_{b}}\bigg{(}\theta(\xi)\Big{(}1-cos\omega\xi\Big{)}-\theta(\xi-
T_{b})\Big{(}1-cos\omega(\xi-T_{b})\Big{)}\bigg{)}-\\\ \frac{\omega
T_{b}}{\omega^{2}T_{b}^{2}-4\pi^{2}}\left(\theta(\xi)\left\\{cos\frac{2\pi}{T_{b}}\xi-
cos\omega\xi\right\\}-\right.\\\ \left.\theta(\xi-
T_{b})\left\\{cos\frac{2\pi}{T_{b}}(\xi-T_{b})-cos\omega(\xi-
T_{b})\right\\}\right),\end{split}$ (34)

where $T_{b}=L_{b}/v$ is the temporal length of the drive bunch.

##  4 Numerical analysis

The analytical expressions derived for the longitudinal and transverse
components of the azimuthally homogeneous wakefield, excited by a single
relativistic drive electron bunch, make it possible to analyze the amplitude
profiles distribution of these components, as well as to perform their mode
and spectrum analyses. For numerical analysis, we have chosen the following
parameters: inner dielectric radius $a=500\,\mu m$, outer dielectric radius
$b=600\,\mu m$, vacuum channel radius $R_{c}=200\,\mu m$, permittivity of
dielectric $\varepsilon_{d}=3.75$ (quartz), plasma density $n_{p}=2.0\cdot
10^{14}\,cm^{-3}$ (plasma frequency $f_{p}=127.3\,GHz$), energy of electron
drive bunch $5\,GeV$, charge of the drive bunch $Q_{b}=3.0\,nC$, length of the
drive bunch $L_{b}=250\,\mu m$, radius of the drive bunch $R_{b}=450\,\mu m$.

Considering that in our case the wakefield excitation is based on the
Cherenkov interaction of the relativistic drive bunch of charged particles
with the eigenwaves of the waveguide under consideration, we have first of all
investigated numerically its dispersion. Figure 2 shows the first six
dispersion curves for the dielectric waveguide with the hollow plasma channel.

Figure 2: The dispersion of the first six TM eigenwaves of the dielectric
waveguide with hollow plasma channel.

When compared to the dielectric waveguide with the vacuum channel for charged
particles, the presence of the hollow plasma channel in the dielectric
waveguide gives appearance to two additional eigenwaves. These are the surface
waves, which can propagate along two vacuum–plasma and plasma–dielectric
boundaries. The lower-frequency surface wave, having the asymptotics
$f\rightarrow f_{p}\left/\sqrt{1+\varepsilon_{d}}\right.$ at
$k_{z}\rightarrow\infty$ , exhibits the normal dispersion law and is the
forward wave. The higher-frequency surface wave having the asymptotics
$f\rightarrow f_{p}\left/\sqrt{2}\right.$ at $k_{z}\rightarrow\infty$, shows
the anomalous dispersion law, and is the backward wave. The beam line with the
dispersion $\omega=k_{z}v$, crosses the waveguide modes at definite
frequencies. These are the resonant frequencies, which are determined by
solving numerically the equation $D(\omega_{s})=0$. The first four resonant
frequencies are found to be 120.2 GHz, 327.6 GHz, 1040.0 GHz and 1885.5 GHz.
It should be noted that for the parameters chosen for the numerical analysis,
the lowest–frequency mode does not cross the beam line, and therefore, will
not contribute to the excited wakefield. In the special case that the circular
waveguide comprises a hollow plasma channel without any dielectric, there
exist only two resonance frequencies. These are the frequencies of the surface
waves that exist on the inner and outer boundaries of the plasma [31]. The
presence of dielectric gives rise to additional resonant frequencies, which
belong to the waves having the frequency regions with phase velocities less
than the speed of light.

With knowledge of the resonant frequencies $\omega_{s}$ one can now turn to
the numerical analysis of the excited wakefield. Figure 3 shows the
longitudinal distributions of the axial wakefield $E_{z}$ and the radial
wakefield $E_{r}-\beta H_{\phi}$ in the vacuum channel region and the plasma
region, calculated at $r=100\,\mu m$ and $r=350\,\mu m$ respectively. It is
obvious that the transverse wakefield amplitude distributions in the vacuum
and plasma regions are essentially different, both qualitatively and
quantitatively. In a vacuum channel the amplitude of the transverse wakefield
is almost zero. This effect has been analytically and numerically demonstrated
in [25, 29], which are dedicated to the wakefield excitation in the hollow
plasma channel. The transverse wakefield amplitude has been shown to decrease
with increase in the drive bunch energy, and tends to zero in vacuum region
for GeV energies. In plasma, the transverse wakefield amplitude is nonzero,
and exhibits nearly a monochromatic character with weak shape distortions.

Figure 3: Longitudinal distributions of axial $E_{z}$ and radial $E_{r}-\beta
H_{\phi}$ fields, both excited by the drive electron bunch in vacuum (left)
and in plasma (right), at $r=100\,\mu m$ and at $r=350\,\mu m$ respectively.
The drive bunch propagates from right to left along the waveguide axis with
constant velocity. The position of the drive bunch head corresponds to
$vt-z=0$.

At the same time, the longitudinal wakefield amplitude is nonzero in both the
plasma and vacuum regions, and has a multiwave structure in the both regions.
Moreover, it is obvious that the characteristic spatial periods of
longitudinal and transverse wakefields considerably differ from each other.
The given structure of the drive bunch–excited wakefield components permits
making some preliminary conclusions as to the transverse dynamics of the test
bunch. The test bunch particles moving in vacuum will be subjected to neither
focusing nor defocusing force on the side of the wakefield under acceleration.
Unlike the vacuum region, in the plasma region behind the drive electron
bunch, responsible for the wakefield excitation, there exist, respectively,
the regions, where $E_{z}<0$, $E_{r}-\beta H_{\phi}>0$ (for the electron test
bunch), and $E_{z}>0$, $E_{r}-\beta H_{\phi}<0$ (for the positron test bunch).
And hence, the electron test bunch and the positron test bunch following the
drive bunch can be concurrently accelerated and focused in the radial
direction.

The reason for nonmonochromatic amplitudes is due to the excitation of a great
number of resonant frequencies of waveguide eigenwaves. To demonstrate the
effect, Fig. 4 and Fig. 5 show the longitudinal distribution of the axial
$E_{z,s}$ and radial $E_{r,s}-\beta H_{\phi,s}$ fields of the first four
resonant with the drive bunch modes ($s=1-4$), and also of the drive electron
bunch-excited plasma wave in vacuum at $r=100\,\mu m$, and in the plasma at
$r=350\,\mu m$. The mode analysis shows that the main contribution to the
formation of the total wakefield comes from the first three resonant modes,
and also from the Langmuir wave, which is localized in the plasma region. The
contribution of higher–frequency resonant modes to the excited wakefield is
negligibly small.

Figure 4: Longitudinal distribution of the axial wakefield of the first four
eigenmodes resonant with the drive bunch $E_{z,s}$, excited in the vacuum at
$r=100\,\mu m$ . The position of the drive bunch head corresponds to $vt-z=0$.
The axial wakefield spectrum $E_{z}$, excited in the vacuum at $r=100\,\mu m$.

It is obvious that in vacuum, the amplitude of the total longitudinal field is
almost equally contributed by the first two resonant modes. Their spatial
periods differ from one another, and the addition results in a complex
longitudinal structure of the total longitudinal field.

Figure 5: Longitudinal distributions of axial $E_{z,s}$ and radial
$E_{r,s}-\beta H_{\phi,s}$ fields of the first four resonant modes and the
ones of the plasma wave, excited in the plasma region at $r=350\,\mu m$. The
position of the drive bunch head corresponds to $vt-z=0$.

In the plasma region, for the longitudinal field, the amplitude is most
greatly contributed by the second resonant mode ($s=2$). The amplitudes of the
first ($s=1$), third ($s=3$) resonant modes, and also, the Langmuir wave
amplitude are nearly the same in value, which differs from the maximum
amplitude by a factor of about 4. A similar situation with the mode content is
observed for the transverse field, as well. Namely, the most substantial
contribution to the amplitude comes from one resonant mode, this being the
first resonant mode ($s=1$). The amplitudes of the rest transverse field
components (the second resonant mode ($s=2$) and the Langmuir wave) have close
values, and similarly differ from the maximum amplitude value by a factor of
about 4.

We have also performed the Fourier analysis of the spectrum for longitudinal
and transverse wakefields excited in the plasma region. The results of the
analysis are presented in Fig. 5 and Fig. 6, which show the corresponding
spectra.

Figure 6: Spectra of axial $E_{z}$ and radial $E_{r}-\beta H_{\phi}$
wakefields excited in the plasma region at $r=350\,\mu m$.

The frequencies, at which the spectral amplitudes are nonzero, represent the
resonant mode frequencies, and also, the Langmuir wave frequency. The Fourier
analysis data are in complete agreement with the results of the mode analysis,
and taken together, they give us an insight into the internal structure of the
excited wakefield as far as the amplitude-frequency content is concerned.
Besides, the numerical analysis has demonstrated that a further increase in
the number of resonant modes considered causes no changes in the profiles of
either longitudinal or transverse wakefields. In turn, this permits us to
estimate the maximal frequency value, which should be taken into account in
the computational codes, where to find the excited electromagnetic field, use
is made of the Fourier expansion (for example CST Studio). It should be noted
that in previous numerical studies based on the PIC–simulation [21], the
spectrum analysis was not performed. According to the results of those
studies, it was concluded that the excited transverse wakefield (and hence,
the focusing force for the test bunch) was mainly contributed by the plasma
Langmuir wave at the plasma frequency $\omega_{p}$. The spectrum analysis made
on the basis of the developed linear theory has demonstrated that the main
contribution to the transverse wakefield comes not from the plasma Langmuir
wave but from the backward plasma surface wave with its resonant frequency
close to the frequency of the plasma Langmuir wave (for the given parameters
of both the plasma–dielectric waveguide and the drive bunch).

The derived expressions make it possible to obtain analytical estimation of
the longitudinal and transverse wakefields, and that, in turn, permits
estimation the energy gain of the test bunch, the drive bunch energy losses,
and also, the focusing of the both bunches (in the approximation of neglecting
the inverse effect of the excited wakefield on bunch particles). Using the
obtained Green’s functions we can derive the expressions for the wakefields
excited by solid and annular drive bunches that propagate either through the
vacuum channel, or through the plasma, or, in the general case, by overlapping
completely with the cross–section of the charged particle transport channel.

The data obtained on a basis of the presented linear theory were compared with
the data of simulation of quasi–linear regime ($n_{b}/n_{p}=0.075$) of the
process of wakefield excitation by the drive electron bunch. To get this
regime the charge of the drive bunch has been reduced by ten time. For
simulation, we have used the native PIC–based code, which was earlier verified
with both the widely used code XOOPIC [32], and the data of the previous
analytical studies [15, 16]. Figure 7 demonstrates the resulting from
simulation longitudinal distributions of the axial field $E_{z}$ and the
radial field $E_{r}-\beta H_{\phi}$ in the vacuum channel region and in the
plasma region, calculated at $r=100\,\mu m$, and $r=350\,\mu m$ respectively.

Figure 7: Resulting–from–simulation longitudinal profiles of the electron
bunch–excited axial $E_{z}$ and radial $E_{r}-\beta H_{\phi}$ wakefields in
vacuum (left) and plasma (right) regions at $r=100\,\mu m$ and $r=350\,\mu m$
respectively. The drive bunch travels from right to left along the waveguide
axis with constant velocity. The position of the drive bunch head corresponds
to the longitudinal coordinate $z=8$.

The analytical theory data (Fig. 3) and the simulation results (Fig. 7) show
good qualitative agreement between longitudinal axial and radial wakefield
profiles in both the vacuum and plasma regions. It should be noted that this
agreement is observed for the $vt-z$ values from the drive bunch tail position
up to the position of the group front of the excited wakefield [33, 34]. For
$vt-z$ values from the position of the group front of wakefield up to the
position of the accelerating structure input, the profiles under consideration
differ significantly. Among the main reasons for the difference between the
results gained with the developed linear theory and the ones obtained by
simulation we may mention the following: (I) the linear theory assumes the
absence of plasma density perturbation and, consequently, the plasma electrons
falling into the vacuum channel. That takes place (though slightly) in
simulation even for the linear regime; (II) the presented theory has been
constructed in the approximation of the longitudinal unboundenness of the
accelerating structure, and this, in turn, may cause the difference in the
frequency spectrum of the excited wakefield in comparison with the field
spectrum estimated in the simulation. The results of the spectrum analysis for
the longitudinal and transverse wakefields excited in the plasma region at
$r=350\,\mu m$ and estimated by the numerical simulation are presented in Fig.
8.

Figure 8: Simulation spectra for the axial $E_{z}$ and radial $E_{r}-\beta
H_{\phi}$ wakefields excited in the plasma region at $r=350\,\mu m$.

A comparison between the spectrum analysis data obtained from the analytical
theory (Fig. 6) and from simulation (Fig. 8) shows that there is some
difference in the frequency content of the excited wakefields. The simulation
displays a more multifrequency spectrum , but the frequencies, to which the
major maxima of the spectrum correspond, are almost the same. A great many
nonresonant frequencies can be treated as a transient radiation field, which
arises at the drive bunch injection into the accelerating structure, and which
is not taken into account for the longitudinally unbounded waveguide statement
of the problem. And yet, considering that the numerically simulated
longitudinal amplitude profiles of the axial and transvers wakefields, where
these frequencies get excited and are present in the corresponding spectra,
are not too different in quality from the corresponding amplitude profiles
derived from the analytical theory, it can be concluded that the contribution
of these frequencies to the total wakefield is not essential (for the $vt-z$
values from the position of the drive bunch tail to the group front position
of the excited wakefield). Surely, these frequencies do contribute to the
total amplitude of the wakefield components, but they do not essentially
affect the longitudinal profile of these amplitudes.

The PIC–based numerical calculation of the drive bunch–excited wakefields
distribution in the plasma–dielectric wakefield structure is a very time-
consuming procedure. To provide good resolution in the excited oscillation
spectra, one must simulate lengthy systems with a great number of
macroparticles. For this reason, the above-given analytical results can be
used for a rapid choice of the parameters to perform further more exact
numerical computations.

##  5 Conclusions

In paper we have developed a linear theory of wakefield excitation by a drive
charged particle bunch in a cylindrical dielectric waveguide with a hollow
plasma channel. Green’s functions have been built for the cases, where the
point–like particle exciting the wakefield travels in both the vacuum channel
and the plasma region. Based on the constructed theory, we have performed
numerical studies on the amplitude structure of wakefield components excited
by the finite–size drive bunch. The studies have demonstrated the possibility
of radially-stable acceleration of test bunches of both types (electron and
positron). The mode and spectrum analyses of the excited wakefield have shown
that to obtain analytical estimations for the field amplitudes and their
space–time structure, it will suffice to take into account several first
resonant eigenmodes together with the plasma Langmuir wave (in the plasma
region). For the linear regime, expressions for the longitudinal and
transverse wakefields were earlier derived for two transversely unlimited
cases. One case is in [25], where the radius of the vacuum region is finite,
while the plasma thickness is infinite; the second case is in [27] for the
hollow plasma channel of finite thickness surrounded by vacuum, which extends
to infinity. Unlike the mentioned cases, the present study is concerned with
the accelerating structure of finite transverse size, having an additional
dielectric layer. It has been demonstrated that in the vacuum region the
transverse wakefield is almost absent (for high bunch energies), whereas in
the plasma region the transverse wakefield is nonzero and can serve as
focusing for the electron and positron test bunches. The simulation data are
in good agreement with the data of the developed linear analytical theory.

## Acknowledgments

The study is supported by the National Research Foundation of Ukraine under
the program “Leading and Young Scientists Research Support” (project #
2020.02/0299).

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