Download Impedance and Wake of Two-Layer Metallic Flat Structure

Armenian Journal of Physics, 2017, vol. 10, issue 1, pp. 36-41
Impedance and Wake of Two-Layer Metallic Flat Structure
M. I. Ivanyan1*, A. Grigoryan1, S. Zakaryan1, A.V. Tsakanian2
1
CANDLE Synchrotron Research Institute, 0040 Yerevan, Armenia
2
Helmholtz-Zentrum Berlin, D-12489 Berlin, Germany
*
Email: [email protected]
Received 07 March 2017
Abstract. Electrodynamic properties of two-layer two infinite metallic parallel plates are studied. Explicit
expressions for the impedance and wake potential are obtained. Their properties are studied. It is shown
that at low conductivity of the inner layer, the frequency distribution of particle radiation in such a
structure has a narrow-band resonance nature and concentrated in a vicinity of the resonant frequency
determined by the geometrical parameters of the structure.
Keywords: metallic flat structure, impedance and wake potential
1. Introduction
For the acceleration of charged particles mainly disc-loaded accelerating structures are used [1,2].
Recently, much attention is also paid to the dielectric–loaded ones [2]. Both structure types are characterized
by the high order modes, excited during the charge particles beam passage through the structure. In [4, 5] an
alternative structure is proposed: circular waveguide with two-layer metallic walls. A charged particle
moving along the axis of such structure, under certain conditions, only fundamental TM01 modes are
generated. This mode is slow propagating with the phase velocity less than the speed of light. At appropriate
structure geometry, this radiation reaches the terahertz frequency range [6].
This paper deals with the particle radiation in two-layer parallel metallic plates (flat structure). The study
of such structures is important not only in terms of its resonance properties, but also for modeling the NEG
(Non-Evaporable Gutter) coated rectangular or elliptic beam pipes with comparatively wide transverse
dimensions. The impedance and wake potential of single-layer resistive flat structure are considered in [7]
(exact solution for ultra-relativistic case) and [8] (solution in surface impedance approximation). General
problem of particle radiation in the multilayer flat structure is considered in [9].
In this paper, the explicit expression for the radiation field of ultra-relativistic particle in two-layer
metallic flat structure with perfectly conducting outer layer is obtained. The properties of the longitudinal
impedance and wake functions are studied.
2. Statement of the problem and the basic equations
Considering the structure, which consists of two parallel perfectly conducting surfaces, each coated with
an isotropic metal layer of thickness from the inside with dielectric permittivity = + ⁄ ( metal
dielectric constant, metal conductivity, freguency) and magnetic permeability . The distance between
the plates is 2 . The point particle with charge moves parallel to the axis z of the Cartesian coordinate
frame , , with the constant velocity . Distance between the axis z and the trajectory of the particle equals
to Δ < (Fig.1). The solution is constructed by the method of partial areas. In this case, there are three
regions: the upper ( = 1)and bottom ( = −1) layers and the vacuum area ( = 0) between the plates. In the
layer and inner vacuum regions ( = 0, ±1) the solution is sought in the form of the general solution of the
homogeneous Maxwell equations. For the area between the plates, to the general solution of the
homogeneous Maxwell equations a particular solution of inhomogeneous equations is added.
Impedance and Wake of Two-Layer Metallic Flat Structure || Armenian Journal of Physics, 2017, vol. 10, issue 1
Figure 1. Geometry of the problem
In all three areas, homogeneous solution of Maxwell's equations for electric and magnetic
field components can be written in the form:
( )(
, , )=
()
( , , )+
()
( )(
, , )=
()
( , , )+
()
( , , ),
( , , ),
()
,
()
radiation
(1)
= 0, ±1
where
()
, (
, , )=
()
, (
, , )=
()
, (
,
()
, (
()
, (
(
, )
,
,
, )
)
(
)
(2)
with
, )=
() ∓ ()
,
(3)
In the ultra-relativistic case ( → , is a speed of light):
()
=
=
+ (1 −
)
,( = ±1),
( )
=
.
(4)
As a partial solution of the inhomogeneous Maxwell's equations, similar to [7], the field of ultra-relativistic
charge, which moving between two parallel ideally conducting plates is taken as:
( ,
, )=
( ,
, )=±
ℎ(2
ℎ(2
) ℎ
) ℎ
( ± ∆) ℎ
( ± ∆) ℎ
37
( ∓ )
( ∓ )
Ivanyan et al. || Armenian Journal of Physics, 2017, vol. 10, issue 1
=−
=
,
,
=
=0
(5)
The upper and lower signs (±) in (5) refer, two cases > Δ and < Δ, respectively. Thus, the full field of
the particle in the vacuum region between the plates can be written as:
( )(
,
( )(
, )=
( )
, )=
( )
,
( ,
( ,
, )+
( )
, )+
( )
( ,
( ,
, )+
( ,
, )
, )+
( ,
, )
(6)
3. Field matching
()
The unknown weight coefficients
, , , , are determined by the requirement of the electric and
magnetic tangential field components continuity at the boundaries of the inner vacuum region ( = ± ) and
the requirements of the electric tangential field components vanishing on the border with a perfect conductor
= ±( + ) . The relevant equations are:
( )
, (
( )
, (
,
, )=
,
, )=
( )
, (−
( )
, (−
( )
, ,
( )
, ,
,
, )=
,
, )=
( )
, , ( + ,
( )
, , (− −
( ,
, )+
( ,
, )+
(
)
, , (−
( )
, , (−
,
, )+
,
, )+
( )
, , ( +
( )
+ , , (−
, )+
,
, )
( )
, ,
( )
, ,
( ,
, )
( ,
, )
(
)
, , (−
( )
, , (−
,
, )
,
, )
, )=0
,
− ,
, )=0
Given that the longitudinal components of the amplitudes
()
, , ,
()
, (
using Maxwell's equations
containing 12 unknown amplitudes
presented below:
( )
, ,
=
ℎ(
Δ) ∓
ℎ(
()
, , ,
(7)
()
, ,
can be expressed in terms of amplitudes
, , ) , from (7) we obtain the system of 12 equations,
( = 0, ±1). The explicit forms of the amplitudes
Δ)
()
з, ,
are
(8)
where
( )
(
=
−
)
( ( )
)
(9)
( )
with
=
ℎ(
) ℎ(
)±
ℎ(
) ℎ(
)
=
ℎ(
) ℎ(
)±
ℎ(
) ℎ(
)
and
38
(10)
Impedance and Wake of Two-Layer Metallic Flat Structure || Armenian Journal of Physics, 2017, vol. 10, issue 1
=
−
−
(11)
Obviously, as in case of the resistive single-layer structure [7], the components of Lorentz force, ( ) =
( )
+ × ( ) acting on the test particle are expressed in terms of the above-determined longitudinal
amplitudes (8)-(11):
( )
where
( )
,
=
,
,
+
( )
,
( )
,
+
−
( )
,
+
( )
,
+
( )
,
(12)
are corresponding unit vectors of Cartesian frame axes (Fig.1).
4. Impedance and wake potential
The vector wake function is determined by double integration of the Lorentz force:
( )
=
( )
(
)
,
=
−
(13)
Vector impedance is determined by the inverse Fourier transform of the wake function (13)
( )
=
( )
=
(14)
The correspondence between (12-14) and a single-layer resistive structure [7] is reached by striving to
infinity thickness of the metal layer ( → ∞).
At an axial motion of the particle (Δ = 0), the longitudinal impedance (frequency distribution of the
longitudinal electric field component) gets the following form on the axis ( = = 0):
( )(
)=
(
−
)
( )
(15)
The impedance (15) is a function of frequency ( = ⁄ ) and depends on the geometric ( , ) and
electromagnetic ( , , ) parameters of structure. We restrict ourselves to the case of non-magnetic metals
vacuum magnetic permeability) with the dielectric constant =
of vacuum dielectric
( = , where
permittivity.
Figure 2. Longitudinal impedance of the two-layer flat structure for different values of the thickness of the
; half-height of structure = 1 .
inner layer and the fixed conductivity = 2 × 10 Ω
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Ivanyan et al. || Armenian Journal of Physics, 2017, vol. 10, issue 1
Figure 2 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 (Fig.1). The distance between the plates
is 2
(half-height = 1 ) and the conductivity of the inner layer 2 × 10 Ω
. The numbers on the
curves in the graph indicate the thickness of the inner layer ( ): "1"(1 ), "2"(2 ) 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 evident from Table I (compare rows and
), it is quite well defined by the formula:
=
(16)
Table I. Main parameters of longitudinal impedance of the two-layer flat structure for different values of
inner layer thickness presented in Fig. 2
(
) 1
2
3
4
5
6
7
8
10
16
0.477 0.338
0.276
0.239
0.214
0.195
0.180
0.169
0.151
0.119
0.479 0.339
0.277
0.241
0.215
0.197
0.183
0.171
0.154
0.124
0.479 0.503
0.277
0.241
0.215
0.197
0.183
0.171
0.154
0.124
0.251 0.339
0.754
1.001
1.257
1.508
1.760
2.011
2.513
4.021
(THz) shows the resonance frequency calculated by equation (16); in line
(THz),
In Table I, line
shows the
the resonance frequencies calculated numerically by the exact formulae are given; line
frequency values in which the imaginary components of the impedance vanish; in the last row, the values of
the parameter
values
and
=
are given. As the table shows, there is a good coincidence between the frequency
. Thus, the resonant frequency in a flat two-layer metal structure, in contrast to the
corresponding cylindrical structure, is defined by the formula (16). In cylindrical case,
=
(
and , i.e. the
tube radius) is √2 times larger. There is also a complete coincidence of frequencies
resonance value of the impedance is purely real. By comparing the maximum values of the real component of
impedance distributions (Figure 2, left) with the corresponding parameter values, we note the following
regularities: impedance reaches the highest value at = 4 , which corresponds to value of parameter
equal to unity ( = 1).The maximum values of other curves (Fig.2, left) on either side of it, decrease on the
distance.
As shown in Figure 2 (left), there are the pairs of curves, maximum values of which are located on a
same level. These curves correspond to a pair of mutually parameter reverse values. Thus, as it follows
≈ 1⁄ and
≈ 1⁄ . Similar phenomena occur also in a two-layer
from Figure 2 and Table 1:
cylindrical waveguide [4]. The difference is in the value of the parameter
√
:
=
for planar and
=
for cylindrical cases, i.e. √3 times less. Recall that the resonance frequency in the planar case
=
1⁄ is less by factor of √2, with respect to resonance frequency of cylindrical geometry (
=
2⁄ ) in the case of = . The last relation for broadband resonance frequency has been identified in
[8] for the single-layer resistive flat and cylindrical structures.
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Impedance and Wake of Two-Layer Metallic Flat Structure || Armenian Journal of Physics, 2017, vol. 10, issue 1
Figure 3. Longitudinal wake potential of the two-layer flat structure for different values of the thickness of
the inner layer and the fixed conductivity = 2 × 10 Ω
; half-height of structure = 1 .
Numbering corresponds to Table I
Figure 3 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 Δ given by the resonant frequency Δ = ⁄
. The latter indicates the monochromaticity of
particle radiation similar to two-layer cylindrical metal waveguide [4].
5. Conclusion
The frequency distribution of particle radiation in a flat two-layer metal structure has a narrow-band
resonant character. Under certain conditions, it is concentrated in the vicinity of a single resonant frequency
determined by the geometrical parameters of the structure. This is the consequence of the longitudinal
impedance resonant behavior. As in the case of a two-layer metal cylindrical waveguide, the radiation is
conditioned by the slow-propagating wave.
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