Analytical solutions in the two-cavity coupling problem
Analytical solutions of precise equations that describe the rf-coupling of two cavities through a coaxial cylindrical hole are given for various limited cases. For their derivation we have used the method of solution of an infinite set of linear algebraic equations, based on its transformation into...
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України
2000
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| Cite this: | Analytical solutions in the two-cavity coupling problem / N.I. Ayzatsky // Вопросы атомной науки и техники. — 2000. — № 2. — С. 66-68. — Бібліогр.: 8 назв. — англ. |
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Ayzatsky, N.I. 2015-05-27T13:59:40Z 2015-05-27T13:59:40Z 2000 Analytical solutions in the two-cavity coupling problem / N.I. Ayzatsky // Вопросы атомной науки и техники. — 2000. — № 2. — С. 66-68. — Бібліогр.: 8 назв. — англ. 1562-6016 PACS: 84.40.Cb https://nasplib.isofts.kiev.ua/handle/123456789/82286 Analytical solutions of precise equations that describe the rf-coupling of two cavities through a coaxial cylindrical hole are given for various limited cases. For their derivation we have used the method of solution of an infinite set of linear algebraic equations, based on its transformation into dual integral equations. The author wishes to thank Academician V.A. Marchenko, Professor R.L. Gluckstern and Professor V.I. Kurilko for discussions relating results of this work. en Національний науковий центр «Харківський фізико-технічний інститут» НАН України Вопросы атомной науки и техники Theory and technics of particle acceleration Analytical solutions in the two-cavity coupling problem Аналитические решения в проблеме взаимодействия двух резонаторов Article published earlier |
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Analytical solutions in the two-cavity coupling problem Ayzatsky, N.I. Theory and technics of particle acceleration |
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Analytical solutions in the two-cavity coupling problem |
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Analytical solutions in the two-cavity coupling problem |
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Analytical solutions in the two-cavity coupling problem |
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analytical solutions in the two-cavity coupling problem |
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Ayzatsky, N.I. |
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Ayzatsky, N.I. |
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Theory and technics of particle acceleration |
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Theory and technics of particle acceleration |
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Вопросы атомной науки и техники |
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Аналитические решения в проблеме взаимодействия двух резонаторов |
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Analytical solutions of precise equations that describe the rf-coupling of two cavities through a coaxial cylindrical hole are given for various limited cases. For their derivation we have used the method of solution of an infinite set of linear algebraic equations, based on its transformation into dual integral equations.
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Analytical solutions in the two-cavity coupling problem / N.I. Ayzatsky // Вопросы атомной науки и техники. — 2000. — № 2. — С. 66-68. — Бібліогр.: 8 назв. — англ. |
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T H E O R Y A N D T E C H N I C S O F P A R T I C L E A C C E L E R A T I O N
ANALYTICAL SOLUTIONS IN THE TWO-CAVITY COUPLING
PROBLEM
N.I. Ayzatsky
National Science Center “Kharkov Institute of Physics and Technology”, Kharkov, Ukraine
Analytical solutions of precise equations that describe the rf-coupling of two cavities through a co-axial
cylindrical hole are given for various limited cases. For their derivation we have used the method of solution of an
infinite set of linear algebraic equations, based on its transformation into dual integral equations.
PACS: 84.40.Cb
1. INTRODUCTION
Authors of the basic works on RF-coupling [1-3]
made a suggestion that in the case, when the hole
dimensions are much smaller than the wavelength, the
coupling of two resonant volumes can be described with
some accuracy by a system of equations of two resonant
circuits. They also assumed that coupling coefficients
could be calculated within the same accuracy by solving
the static problem (а/λ→0). In our papers [4-5] we
showed that RF-coupling of two resonators through a
center hole of arbitrary dimensions in the frequency
domain is strictly described by the system of equations
of two resonant circuits with the coupling coefficients
that depend on the frequency. On the base of these
equations we calculated numerically the relationship of
coupling coefficients versus different parameters
(frequency, hole radius, etc.). But there is a question on
confirming the analytical results of a static approach by
obtaining the static coupling coefficients from precise
equations in the limit а/λ→0. As we know, this problem
was not solved up to this time. For dealing this problem
we developed a new mathematical method for solving an
infinite set of linear algebraic equations that is based on
its transformation into dual integral equations. Besides,
expressions are derived for coupling coefficients which
are valid up to the second order of the ratio between the
hole dimension and the free-space wavelength (а/λ).
Such coupling coefficients were also obtained in [6] by
using a variation method, but our approach gives the
possibility of finding the electrical field distribution on
the coupling area.
2. PROBLEM DEFINITION ORIGINAL
EQUATIONS.
Let us consider the coupling of two cavities through
a circular hole with the radius a in a separating wall of a
thickness t. For simplicity sake, we will consider the
case of two identical cavities, with b being the cavity
radii and d their length. In [4,5] it was demonstrated that
if the field is expanded with the short-circuit resonant
cavity modes and E 0 1 0 -modes are selected as
fundamental, the precise set of equations will consist of
two equations for the amplitudes of E 0 1 0 -modes, where
coupling coefficients are defined by soluting the infinite
set of linear algebraic equations. Let us generalize the
case considered in [4,5], choosing as fundamental the
E p q0 -modes of closed cavities (q is the number of field
variations across the radius, p is the number of field
variations along the longitudinal coordinates). Using the
method similar to that one in [4,5], one can show that
the set of equations describing the system under
consideration has the form:
Λ−−Λ−=
=Ζ
2
)1;2(
,)1(1
)2;1(
,2
3
)(2
13
42
0,
)2;1(
,,
pqap
pqa
db
a
qJq
pqapqp
λπ
ω
ε
, (1)
where
( )[ ]Ζ q p q p q p q
p s
c b p d
p
p p J s
, , ,, / / ,
,
, , , . . . , ( ) , , . . . ,
= − = +
=
=
≠
= ∞ = = ∞
ω ω ω λ π
ε λ
2 2 2 2 2 2 2
0
2 0
1 0 0 1 0 1 2
a q p
i
,
( ) is the amplitude of E q p0 -mode in the i-th cavity
(i=1,2). The normalized coupling coefficients Λ i are
determined by the expression:
,22/
1
)()(2
0)(
−
∞
=
=Λ=Λ ∑ qs
s
i
swqJii θλθω (2)
where w s
i( ) ’s are the solution of the following set of
linear equations:
,22/)1(3
)2()1;2()1()2;1(
,
)2;1(
∗Ω−=
=
++ ∑
mmf
s
mfswmfswsmGmw
λπ
(3)
( ) ( ) ( / ) , 1
(2 / ) 1 , 2 ,( )
− =µ= − =
m mj m
m
mm
ch q ch q t L j
f
ch q d L jsh q
2 2
2 2 2
/ , 2 , ,
/ , ( / ) .∗
= µ = + µ = λ − Ω
Ω = ω Ω = Ω − π
m m m mq L a L d t
a c a p d
,
, ,
2 3 2
0
2 2 2 2 2 2
( / )
2
2 ( )
,
( ) ( ) [ ( / ) ]
δ
= − µ +
µ
π θ θ
+
ε χ λ − θ λ − θ µ + π
m s
m s m s m
m
q q
p q m q s q m
G B cth d a
a J
d b a p d
2 2
0
, 2 2 2 2
1
( )
,
( ) ( )
∞
=
θ θ
= π
χ λ − θ λ − θ∑ l l l
l l l l
m s
m s
J RaB
b
66 ВОПРОСЫ АТОМНОЙ НАУКИ И ТЕХНИКИ. 2000, № 2.
Серия: Ядерно-физические исследования (36), c. 66-68.
2 2 2
1/ , ( ) / 2 , ,θ = λ χ = π λ λ ν = θ − Ωl l l l l l la b J
2 2
( / ) /
2 /{ ( ( / ) } , ,
( / ) / , .
θ ν ν −
= − θ ε ν + π =
θ ν ν ≠
l l
l
q q q
q p q
q q q
cth d a
R a d ap d q
cth d a q
The coefficients w s
i( ) have a simple physical sense.
Really, it is easy to show that the tangential electric field
component in the left cross-section of the coupling hole
E rr
( ) ( )− has the form:
),()2()2(
,.0
~)()1()1(
,.0
~)2()1()()( rQpqErQpqEEErrE −=−=−
i n di n d (4)
where Q r
J r a
J
wi s
ss
s
i( ) ( )( )
( / )
( )
,= ∑
1
3
1
1π
λ
λ
~
, ,
( )E q p0
1 is the
value of the longitudinal (perpendicular to the hole)
electric field of (0, q, p)-mode in the first cavity on the
left coupling hole cross-section for r = a, while
~
, ,
( )E q p0
2 is
the same value for the right-hand cavity on the right
coupling hole cross-section for the same radius. From
the expression (4) it follows that the tangential electric
field component on the left coupling hole cross-section1
is equal to the difference of two induced fields, each of
which is proportional to the perpendicular electric field
components of E q p0 , , -modes taken to be fundamental.
There, the coefficients w s
i( ) are the ones in the
expansion of appropriate functions with the complete set
of functions { }J r as1 ( / )λ . Thus, the two-cavity
coupling problem, rigorously formulated on the base of
the electric field expansion with the short-circuit
resonant cavity mode, is reduced to the induced field
definition on the right and left cylindrical hole cross-
sections.
3. INFINITELY THIN WALL CASE
An important role in the problem of cavity coupling
plays the case of infinitely thin wall dividing the cavities
(t = 0). In this case, from Eq (3) it follows that
w w wm m m
( ) ( )1 2= = . Here the set of equations for w m will
take the form2:
∑
∞
=
∗Ω−=
1
.222/3,
s
msmBsw λπ (5)
For the case t=0 Λ Λ Λ1 1= = , where
.22/)(2
0 ∑
−=Λ
s
qsswqJ θλθ (6)
3.1. SMALL COUPLING HOLE CASE ( a → 0 )
If in Eqs. (5,6) the hole radius tends to zero3, then
Eq. (5) will become:
1The same is true for the right cross section
2We have neglected terms of order a 5 in the expression
for G m s,
3In this case, as follows from Eqs.(1), the coupling
coefficients will be proportional to a 3
.
22
3
1 0 2222
)(2
0
2
ms ms
J
dsw
λ
π
θλθλ
θθ
θ =
∞
=
∞
−
−
∑ ∫ (7)
In order to get the solution for Eq. (7) we will introduce
an integer odd function f z1 ( ) the values of which in the
points z s= λ are equal to ( ) ( )f w Js s s1 1λ λ= . Let us
assume that at ( )z f z→ ∞ 1 grows not faster than
exp(z), then, in accordance with the Cauchy theorem,
the function ( )( )f z J z( ) / 0 can be expanded into the
series over mere fractions
( ) ( ) .
1
22/20/1 ∑
∞
=
−=
n
znnwzzJzf λ (8)
Using (8), and, also, multiplying Eq. (7) by
( ) ( )J x Jm m1 1λ λ/ , where 0< x <1, and doing
summation over sub-index m, we will get
( )∫
∞
<<=
0
.10,2/31)(1 xxdzzxJzf π (9)
By multiplying (8) by ( )z J x z1 and integrating over z
from 0 to ∞, we will obtain at x >1:
( ) ( ) .1,00 11 >∞ =∫ xdzzxJzfz (10)
In this way, the set of linear algebraic equations (7) with
a complicated matrix coefficients that cannot be
expressed via elementary functions and can be
calculated only numerically, has been reduced to two
integral equations (9,10). Having determined the kind of
the function ( )f z1 , there is no need in calculating the
sum (6), since
( ) ( ){ }[ ]∑
→
==Λ
s
zJzzf
z
ssw .02/1
0
lim2/ λ (11)
The method of solving the dual integral equations of
the type (9,10) on the base of the Mellin transformation,
as well as the property of Cauchy-type integrals, can be
found in [7]. The brief summary of their solutions is
given in [8]. We shall dwell briefly on a simpler method
of resolving this system.
Since ( )f z1 is the odd function it can be represented
in the form ( ) ( ) ( )f z z t t d t1 0= ∫ ∞ s i n .η Substituting this
expression in Eq. (10) we obtain the following integral
equation for ( )η t : ( )d t t t x xx η / , .2 2 0 1− = >∫ ∞ The
solution of this equation is ( )η t = 0 for t >1.
Consequently, any function of the type
( ) ( ) ( )∫= 1
0sin1 dtttzzf η (12)
satisfies Eq. (10). Substituting (12) into (9), we obtain
the first kind Volterra equation of Abelian type
∫ <<=−x xxtxttdt0 ,10,2/2322/)( πη (13)
the solution for which can be found in the analytical
form. Omitting the intermediate formulae, we shall give
the final expression for the function ( )f z1 :
67
( )f z j z1 16= ( ) , where j zn ( ) is the spherical Bessel
function of order n.
The normalized coupling coefficients, as follows
from (11), is equal to Λ =1. Since
( ) ( )w f Js s s= 1 1λ λ/ , then from (4) we will obtain
.
22
)0()2(
,,0)0()1(
,,0)()(
ra
rrpqErpqE
rrE
−
=−=
=−
π
Thus, based on rigorous electrodynamics description of
the two-cavity coupling system we are the first to prove,
by the way of the limit transition a → 0 , the correctness
of the equations formulated in [1-3] using the quasi-
static approximation, and to obtain the expression for
the tangential electric field on the hole.
3.2. THE CASE OF SMALL FINITE VALUES OF
COUPLING HOLE RADIUS
The above method presents the opportunity to obtain
analytical expressions for the normalized coupling
coefficients with accuracy of the order of ( / )a λ 2 . If
a / λ is small, though finite, then, the coefficients w s in
(5) will be dependent on the hole radius value a:
w w as s= ( ) . Let us introduce the function of two
variables: ( )ψ λ( , ) ( ) ( ) / .a z z J z w a zn n
n
= −∑
=
∞
2 0
2 2
1
We
assume that relative to the variable z the function
ψ ( , )a z obeys the conditions formulated in Subsec.3.1.
Using the technique similar to that described in
Subsec.3.1, the set (5) can be reduced to:
,/1,0/),()
1
(1 abxllal
l
xJl <<=
∞
=
∑ χθψθθ (14)
( )
( ) .10,
0
13
1
),()(1
<<
∗Ω∗Ω
∗Ω
=
∞
=
∑ x
J
xJ
l l
lRlalxJ
b
a π
χ
θψθ
π (15)
Letting a → 0 in Eqs. (14,15), we derive a set of
equations (9,10), and, consequently, ψ ( , ) ( )0 1z f z= .
Then we represent ψ ( , )a z in the form
ψ ψ ϕ( , ) ( , ) ( , )a z z a a z= +0 2 . From (14,15) it follows
that ϕ ( , )0 z satisfies the following equations:
,1,00 ),0()(1 >=∞
∫ xdxJ θθϕθθ (16)
,10),(0 ),0()(1 <<=∞
∫ xxFdxJ θθϕθ (17)
where ( )F x
x
a
x
( ) .= − − −
∗ ∗
3
8 4
22
2 2
2
2 2π
Ω Ω Ω Ω .
The solution of Eqs.(16,17) has the form
),(222
222
)(124
22
),0( zf
a
zf
a
z
Ω−∗Ω
−
Ω−∗Ω
=ϕ (18)
where f z j z j z z2 3 1
22 3 2( ) ( ) / ( ) / ( ) .= −
The normalized coupling coefficients Λ, accurate to
the order of ( / )a λ 2 , are determined by the relationship:
.
2
20
1
2
,
20
3
2
5
11
−
−
−≈Λ
c
a
c
pqa
b
qa ωωλ
(19)
For the case ω ω≈ q p, the expression (19) agrees
with that for the generalized polarizability, obtained in
[6] at b → ∞ via the variation technique. Note that the
expression (19) is true for the frequency ω that is not
close to the resonant frequencies of the non-fundamental
modes of closed cavities: ω ω≠ n m, if ( , ) ( , )m n q p≠ .
Knowing ψ ( , )a z , and, consequently, w as ( ) , the form
of the tangential electric field around the hole can be
reconstructed:
.
2
1
6
222
2212
2222
4
1
1
)0()2(
,,0)0()1(
,,0)(
−
Ω−∗Ω
+
−
Ω−∗Ω
+
−
=−=
=−
a
r
a
r
ra
r
qb
a
rpqErpqE
rE
λ
π
(20)
4. CONCLUSION
Based on our method of reducing the infinite linear
algebraic equation set to dual integral equations, we
obtained, in different limited cases, the rigorous
analytical solutions regarding the two-cavity coupling
problem. Along with the general theory significance, the
obtained solutions are of applied interest, since they can
be used for a better convergence of the original equation
solution (3), which are true for arbitrary dimensions of
the coupling hole.
The author wishes to thank Academician
V.A. Marchenko, Professor R.L. Gluckstern and
Professor V.I. Kurilko for discussions relating results of
this work.
REFERENCES
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Frequency dependence of the penetration of
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68
1. INTRODUCTION
3. INFINITELY THIN WALL CASE
REFERENCES
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