Calculation of the dispersion performances of slow wave structures with artificial anisotropical dielectrics
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| Дата: | 1999 |
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| Мова: | English |
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України
1999
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| Назва видання: | Вопросы атомной науки и техники |
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| Цитувати: | Calculation of the dispersion performances of slow wave structures with artificial anisotropical dielectrics / V.G. Papkovich, N.A. Khizhnyak // Вопросы атомной науки и техники. — 1999. — № 4. — С. 31-33. — Бібліогр.: 5 назв. — англ. |
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irk-123456789-815212015-05-18T03:02:17Z Calculation of the dispersion performances of slow wave structures with artificial anisotropical dielectrics Papkovich, V.G. Khizhnyak, N.A. 1999 Article Calculation of the dispersion performances of slow wave structures with artificial anisotropical dielectrics / V.G. Papkovich, N.A. Khizhnyak // Вопросы атомной науки и техники. — 1999. — № 4. — С. 31-33. — Бібліогр.: 5 назв. — англ. 1562-6016 http://dspace.nbuv.gov.ua/handle/123456789/81521 en Вопросы атомной науки и техники Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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DSpace DC |
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English |
| format |
Article |
| author |
Papkovich, V.G. Khizhnyak, N.A. |
| spellingShingle |
Papkovich, V.G. Khizhnyak, N.A. Calculation of the dispersion performances of slow wave structures with artificial anisotropical dielectrics Вопросы атомной науки и техники |
| author_facet |
Papkovich, V.G. Khizhnyak, N.A. |
| author_sort |
Papkovich, V.G. |
| title |
Calculation of the dispersion performances of slow wave structures with artificial anisotropical dielectrics |
| title_short |
Calculation of the dispersion performances of slow wave structures with artificial anisotropical dielectrics |
| title_full |
Calculation of the dispersion performances of slow wave structures with artificial anisotropical dielectrics |
| title_fullStr |
Calculation of the dispersion performances of slow wave structures with artificial anisotropical dielectrics |
| title_full_unstemmed |
Calculation of the dispersion performances of slow wave structures with artificial anisotropical dielectrics |
| title_sort |
calculation of the dispersion performances of slow wave structures with artificial anisotropical dielectrics |
| publisher |
Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| publishDate |
1999 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/81521 |
| citation_txt |
Calculation of the dispersion performances of slow wave structures with artificial anisotropical dielectrics / V.G. Papkovich, N.A. Khizhnyak // Вопросы атомной науки и техники. — 1999. — № 4. — С. 31-33. — Бібліогр.: 5 назв. — англ. |
| series |
Вопросы атомной науки и техники |
| work_keys_str_mv |
AT papkovichvg calculationofthedispersionperformancesofslowwavestructureswithartificialanisotropicaldielectrics AT khizhnyakna calculationofthedispersionperformancesofslowwavestructureswithartificialanisotropicaldielectrics |
| first_indexed |
2025-07-06T06:31:26Z |
| last_indexed |
2025-07-06T06:31:26Z |
| _version_ |
1836878127246082048 |
| fulltext |
CALCULATION OF THE DISPERSION PERFORMANCES OF SLOW
WAVE STRUCTURES WITH ARTIFICIAL ANISOTROPICAL
DIELECTRICS
V.G.Papkovich, N.A.Khizhnyak
NSC KIPT, Kharkov
The cylindrical waveguide periodically loaded
with dielectric disks [1] represents the waveguide of
slow waves which can find application in the technique
of amplification and generation of electromagnetic
waves, as well as in accelerating technique. The
waveguides with a dielectric load are effective in a wide
range of phase velocities vph (β = vph/c ~ 0,1-1,0) and
have extra properties favourably distinguishing them
from structures with an all metal load. And though they
have the some limitations caused by the known
behaviour of dielectrics in high-frequency fields of the
high strength, the structures with such advantages can
attract attention of the developers of new high-
frequency technique.
In development of such an electrophysical
equipment one of basic problems is to satisfy the
requirements of synchronism between the velocity of
charged particle beam and the phase velocity of
electromagnetic wave propagating in the structure. The
aim of the present paper is to study the dispersion
characteristics of the waveguide loaded with dielectric
disks with central holes for passing the particle beam.
Let us consider the metal waveguide of circular
section of radius R, periodically loaded with dielectric
disks of a thickness b, distance between disks is equal to
a, L = a+b, period of disk arrangement. The permittivity
of disk material is identical and equal to ε, between
disks ε = 1. The radius of the pass channel is designated
as r0. The losses of electromagnetic energy in a metal
wall of the waveguide and in a dielectric volume in
sectional operation are not taken into account. In the
waveguide the symmetric wave of E-type propagates
(below the time factor exp( )i tω is omitted). The
solution of this problem will be carried out in a general
form where the structure period is comparable with the
wavelength in the waveguide (L ~ λg).
In a case, when the load consists of solid
dielectric disks (r0 = 0), the dispersion properties of the
waveguide are featured by the well- known equation [2]
cos cos( ) cos( )
sin( ) sin( )
ψ
ε
ε
= −
− +
p a p b
p
p
p
p
p a p b
1 2
1
2
2
1
1 2
1
2
, (1)
where p k k p k k1
2 2
1
2
2
2 2
1
2= − = −, ε is the propagation
constants in separate layers of the waveguide load,
k c= ω / - wave number, k R1 0= σ / - transversal
wave number, σ0 - first root of cylindrical functions of
the first order, ψ ω β= =L v kLph/ / - shift of the
phase in the structure period. From this equation for
given values of geometrical and electrotechnical
parameters it is possible to determine a phase velocity
of a wave in the waveguide.
The situation varies radically, when in the
waveguide it is necessary to have the channel for
passing particles interacting with a field of a slow wave.
The presence of holes in disks leads to decrease of a
total dielectric load of the waveguide and to
redistribution of a field existing in the waveguide at
r0 = 0. To solve the dispersion equation at r0 ≠ 0 we shall
take advantage of a procedure formulated in [3]. The
dispersion equation obtained has, as in the case of the
waveguide loaded with metal diaphragms, a form of an
infinite determinant but the form of the field expansion
in separate regions and character of convergence of the
solution obtained are completely various.
The field in the pass channel is represented as an
infinite decomposition on space harmonics
E a J r eя
I
n n n
i z
n
n= −
= − ∞
∞
∑ ( )χ β ,
(2)
H a
ik
J r eI
n
n
n
i z
n
n
ϕ
β
χ
χ= −
= − ∞
∞
∑ 1( ) ,
where χ βn nk2 2 2= − , and the values of βn are
interrelated to a stationary value of distribution of the
first harmonic β ω0 0= / v by a Floquet’s relation
β β πn n L= +0 2 / . A field in the annular region is
represented as the expansion in terms of space
harmonics of the cylindrical waveguide periodically
loaded with solid dielectric disks
E c Z r
W z
zz
II
m m
m
m
=
= − ∞
∞
∑ 0 ( )
( )
( )Γ ε
,
(3)
H c
ik
Z r W zII
m
m
m m
m
ϕ =
= − ∞
∞
∑ Γ
Γ1( ) ( ) .
Here Ae
mn and Am
mn are coefficients of the expansion
A
L
e
W z
z
dzmn
e i z m
L
n= ∫
1
0
β
ε
( )
( )
,
A
L
e W z dzmn
m i z
m
L
n= ∫
1
0
β ( ) . (4)
Wm(z) - periodic function with a period L, describing
the dependence of a field in the waveguide on the
longitudinal coordinate. For the wave, propagating in
positive direction of then axis z, the function Wm(z) with
unit starting conditions can be written as
W z u z
u L e
u L
u zm
i
( ) ( )
( )
( )
( )= −
− −
1
1
2
2
ψ
. (5)
where the functions u1(z) and u2(z) represent the
fundamental solutions of the equation
ВОПРОСЫ АТОМНОЙ НАУКИ И ТЕХНИКИ. 1999. № 4.
Серия: Ядерно-физические исследования (35), с. 31-33.
31
ε ε ε( ) ( ) [ ( ) ]z
d
dz z
dW
dz k z Wm
m
1
02 2+ − =Γ . (6)
The radial part in the expansion (3) is determined with a
combination of cylindrical functions of the zero and first
orders
Z0(Γmr) = N0(ΓmR)J0(Γmr) - J0(ΓmR)N0(Γmr),
Z1(Γmr) = N0(ΓmR)J1(Γmr) - J0(ΓmR)N1(Γmr).
Equating the corresponding harmonics at r = r0
we come to the following set of equations concerning
required coefficients an and cm:
a J r c Z r An n m m mn
e
m
0 0 0 0( ) ( )χ =
= − ∞
∞
∑ Γ ,
a
r J r c r Z r An
n
n m
m
m mn
m
mχ χ
0
1 0
0
1 0
1
( ) ( )=
= − ∞
∞
∑ Γ Γ . (7)
The condition for solving the obtained homogeneous set
of algebraic equations is the equality to zero of its
determinant
1 1
0
0
1 0
0 0 0
1 0
0 0χ
χ
χn
n
n m
m
m
mn
m
mn
er
J r
J r r
Z r
Z r
A
A
( )
( )
( )
( )
− =
Γ
Γ
Γ
. (8)
which is the required dispersion equation of the
waveguide loaded with dielectric disks with central
holes. Similarly to the case of the waveguide loaded
with metal diaphragms, it looks like an infinite
determinant. Note, that the convergence of the
determinant obtained will be determined by a behavior
of functions, on which the expansion is carried out [4].
In the used approach the decomposition of fields is
carried out on eigenfunctions of the inhomogeneous
cylindrical waveguide, therefore convergence of the
dispersion equation will be the better, the smaller radius
of a central hole in disks is.
To demonstrate peculiarities of calculations for
dispersion properties of such waveguides we consider,
as an example, the calculation of the dielectric disk
thickness as a function of the radius value of a central
hole in the waveguide with the given value of the phase
velocity (β = const). The waveguide is specified by the
following parameters: ε = 90, R = 4,2 cm, working
wavelength λ = 10 cm. In the waveguide the travelling
wave propagates with a phase velocity β = 0,4 and
phase shift ψ = π/2 in a structure period.
0 0 . 2 5 0 . 5
0 . 2
0 . 2 5
0 . 3
0 . 3 5
r / R
b
/ L
1
2
3
Fig. 1.
In Fig. 1 the change of a fill factor of a structure
period with the dielectric b/L is shown depending on the
radius of a central hole r0/R. The curve 1 corresponds to
the change b/L at loading the waveguide with solid
dielectric disks, 2 - calculation for the zero term of the
equation (8) and 3 - calculation for the complete
dispersion equation (8). From this consideration it
follows, that at r0/R < 0,1 dispersion properties of
structure do not differ from properties of waveguide
loaded by solid disks, but at r0/R > 0,15 it is necessary
to take into account the changed structure of the field,
i.e. to take into account the presence of the spectrum of
space harmonics.
0 1 3 5 7 9 1 1
0 . 2
0 . 3
0 . 4
m
b
/ L
0 . 1
0 . 3
0 . 4
0 . 5
Fig. 2.
In Fig. 2 the character of convergence of the
solution of the dispersion equation (8) is shown
depending on the order of a computed determinant. In
this figure the reduced radius of the central hole is taken
as a parameter. From theconsideration it follows, that
with calculations being sufficiently accurate for practice
(~ 1 micron) one may restrict itself to the solution of the
determinant with m = 5.
ВОПРОСЫ АТОМНОЙ НАУКИ И ТЕХНИКИ. 1999. № 4.
Серия: Ядерно-физические исследования (35), с. 31-33.
31
2 . 8 2 . 9 3 3 . 1 3 . 2 3 . 3 3 . 4 3 . 5
0 . 3
0 . 4
0 . 5
0 . 6
0 . 7
0 . 8
0 . 9
1
f , G H z
v
/ c
1 a
2 a
1 b 2 b
Fig. 3.
Fig. 3 represents the results of numerical
calculations of dispersion curve for the waveguide
(solid) in comparison with these experimentally
obtained (points) taken from the paper [5] For this case
r0/R = 1/3,6: ε = 80. The curve 1b corresponds to the
waveguide loaded with dielectric disks of a thickness b
= 3 mm with a period of arrangement L = 1,2 cm; a
curve 2b - b = 4 mm and L = 2,2 cm. The dotted lines 1a
and 2a show dispersion characteristics for the
corresponding waveguides loaded by solid dielectric
disks. From the above dependence it follows that the
data obtained correspond each other within the limits of
an experimental error. That proves the chosen approach
to the solution of a problem in distribution of
electromagnetic waves in the circular metal waveguide
loaded with dielectric disks having central holes.
REFERENCES
1. Fainberg Ya.B., Khizhnyak N.A. Artificial
anisotropic mediums // ZTF. 1955. Vol. 25. No.4. P.
711-719. (in Russian)
2. Malkin I.G. A stability theory of motion. M.-L.:
GITTL, 1952, 432 p. (in Russian)
3. Khizhnyak N.A. The theory of waveguides loaded
with dielectric disks. Radio engineering and
Electronics. 1960. Vol.5. No.3. P. 413-421. (in
Russian)
4. Bulyak Ye.V., Kurilko V.I., Papkovich V.G., The
dispersion theory of a circular disk waveguide. //
VANT. Ser. Nuclear and physical investigations.
1989. No.6 (6). ZNIIAtominform, KIPT. P. 37-43.
(in Russian)
5. Brizgalov G.A., Papkovich V.G., Khizhnyak N.A.,
Shulika N.G. Examination of dispersion and
electrodynamic properties of waveguides loaded
with dielectric fillers. // Accelerators. 1975. No.14.
M.: Atomizdat. P.15-17. (in Russian).
ВОПРОСЫ АТОМНОЙ НАУКИ И ТЕХНИКИ. 1999. № 4.
Серия: Ядерно-физические исследования (35), с. 31-33.
31
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