1
Coupled cavity model for disc-loaded waveguides M.I. Ayzatsky 1 , V.V.Mytrochenko National Science Center Kharkov Institute of Physics and Technology (NSC KIPT), 610108, Kharkov, Ukraine The coupled cavity model for calculation characteristics of inhomogeneous disc-loaded waveguides with taking into account the rounding of the disk hole edges was developed. On the base of this model simulation of section tuning process with the field measurement method was conducted. Accuracy of this method under tuning of disc-loaded waveguides with β ph ∼ 1 was
evaluated. As it follows from our investigation one can use simple coefficients in tuning process, but the needed values of these coefficients must be obtained by calculation of DLW parameters. 1 Introduction Disc-loaded waveguides (DLW) have been heavily investigated both numerically and analytically over the past seven decades (see, for example, [1,2] and cited there literature). They have also been used, and continue to be used, in a variety of microwave devices such as linear accelerators [3,4], travelling-wave tube amplifiers, backward-wave oscillators [5], etc. Earlier we have developed approach (a coupled cavity model).that used the eigenmodes of circular cavities as the basic functions for calculation the properties of homogeneous [6] and inhomogeneous DLWs [7,8,9].with cylindrical openings in discs. Here we present the extension of that model for the case of the DLWs with the rounding of the disk hole edges. 2 Electromagnetic fields in nonuniform disk-loaded waveguide Let’s consider a cylindrical nonuniform DLW (Fig.1). We will consider only axially symmetric fields with Ez , Er , Hϕ components. Time dependence is exp(−i ω t ) .
Fig. 1 1
M.I. Aizatskyi, N.I.Aizatsky;
[email protected]
2 We can divide the DLW volume into infinite number of different cylindrical volumes that are contiguous with each other over circle area. In each large volume (cavity) we expand the electromagnetic field with the short-circuit resonant cavity modes E ( k ) = ∑ eq( k ) (t ) Eq( k ) (r ) , (1.1) q
H
(k )
= i ∑ hq( k ) (t ) H q( k ) (r ) ,
(1.2)
q
where q = {0, m, n} . Eq( k ) , H q( k ) are the solutions of homogenous Maxwell equations rot Eq( k ) = i ωq( k ) μ0 H q( k ) , rot H q( k ) = −i ωq( k )ε 0 Eq( k )
(1.3)
with boundary condition Eτ = 0 on the metal surface ⎛λ ⎞ ⎛π ⎞ E0( km)n , z = J 0 ⎜ m r ⎟ cos ⎜ n ( z − zk ) ⎟ , ⎝ bk ⎠ ⎝ dk ⎠ ⎞ ε b ⎛λ ⎞ ⎛ π H 0( km)n ,ϕ = −i ωn( k, m) 0 k J1 ⎜ m r ⎟ cos ⎜ n ( z − zk ) ⎟ , λm ⎝ bk ⎠ ⎝ dk ⎠ ⎞ b π n ⎛ λm ⎞ ⎛ π E0( km)n , r = k J1 ⎜ r ⎟ sin ⎜ n ( z − zk ) ⎟ , λm d k ⎝ bk ⎠ ⎝ d k ⎠
ω
( k )2 0m n
where J 0 (λm ) = 0 .
⎧⎪⎛ λ ⎞2 ⎛ π n ⎞ 2 ⎫⎪ = c ⎨⎜ m ⎟ + ⎜ ⎟ ⎬, ⎪⎩⎝ bk ⎠ ⎝ d k ⎠ ⎭⎪ 2
(1.4) (1.5) (1.6)
(1.7)
Under time dependence exp(−i ω t ) the equations for expansion amplitudes eq( k,ω) and hq( k,ω) can be written as
(ω
( k )2 q
−ω )e 2
(k ) q ,ω
⎞ i ωq( k ) ⎛ = ( k ) ⎜ ∫ [ Ec( k ) H q( k )∗ ] dS + ∫ [ Ec( k +1) H q( k )∗ ] dS ⎟ , ⎟ N q ⎜⎝ Sk Sk +1 ⎠ hq( k,ω) = − i
ω (k ) e , ωq( k ) q ,ω
(1.8) (1.9)
where Sk is the cross section of the waveguide at the left side of the k-volume, N 0 mn
ω02mn, m b 4 = 2 2 ε 0 π d J12 (λm ) δ n ,0 , c λm
(1.10)
n = 0, ⎧1 , ⎩ 0.5 n > 0.
(1.11)
δn = ⎨
Each small volume inside a disc (waveguide) we divide into a series of homogeneous waveguides (Fig. 2).
3
Fig. 2
In each homogeneous region we expand the electromagnetic field with the waveguide modes
(
)
H ( k , p ) = ∑ Cs( k , p )H s( k , p ) + C−( ks , p )H−(sk , p ) , s
(
)
E ( k , p ) = ∑ Cs( k , p )E s ( k , p ) + C−( ks , p )E −(sk , p ) ,
(1.12) (1.13)
s
where ⎛ λ ⎞ E±( ks ,,zp ) = J 0 ⎜ ( sp ) r ⎟ exp {±γ s( k , p ) ( z − zk( p ) )} , ⎝ ak ⎠ ε ε a( p) ⎛ λ ⎞ H±( ks ,,ϕp ) = −i ω 0 k J1 ⎜ ( sp ) r ⎟ exp {±γ s( k , p ) ( z − zk( p ) )} , λs ⎝ ak ⎠ ( p) ⎛ λ ⎞ a E±( ks ,,rp ) = ∓ k γ s( k , p ) J1 ⎜ ( sp ) r ⎟ exp {±γ s( k ) ( z − zk( p ) )} , λs ⎝ ak ⎠
(1.14) (1.15) (1.16)
2
⎛ λ ⎞ ω2 a ( p )2ω 2 ⎞ 1 1 ⎛ 1 γ = ⎜ ( sp ) ⎟ − 2 = ( p )2 ⎜ λs2 − k 2 ⎟ = ( p )2 ( λs2 − Ω (k p )2 ) = ( p )2 γ s( k , p )2 , (1.17) ak ⎝ c ak ⎠ ak ⎝ ak ⎠ c ( p) ( p) a ω ak 2π f Ω (k p ) = k = , (1.18) c c p = 0,1,..., 2 P . (1.19) Magnetic field boundary conditions must be satisfied at the interface between the cavities and waveguides: (1.20) ∑ ( Cs( k ,0)Hs(,ϕk ,0) + C−( ks )H−(sk,ϕ) ) = i∑ hq( k −1) H q( k,ϕ−1) ( k , p )2 s
s
q
at the plane located at z = zk −1 + d k −1 , and
∑ (C s
H s(,ϕk ,0) + C−( ks )H−(sk,ϕ) ) = i ∑ hq( k ) H q( k,ϕ)
( k ,0) s
q
(1.21)
4 at z = zk . Since the magnetic waveguide modes are orthogonal, from the previous equations we can obtain
{C
{exp (γ
( k ,2 P ) (2 P ) k
t
( k ,0) s
)C
+ C−( ks ,0) } = ∑ (−1) n e0( kmn−1) Γ (mk ,,sk −1) , n ,m
( k ,2 P ) s
}
(1.22)
+ exp ( −γ ( k ,2 P )tk(2 P ) ) C−( ks ,2 P ) = ∑ e0( kmn) Γ (mk ,k ) ,
(1.23)
n,m
where ⎛ a (2 P ) λm ⎞ Fs ⎜ k ⎟, J1 ( λs ) ⎝ bk ⎠ ⎛ a (0) λ ⎞ 2λs =− Fs ⎜ k m ⎟ , J1 ( λs ) ⎝ bk −1 ⎠
Γ (mk ,,sk ) = − Γ (mk,,sk −1)
2λs
Fs ( x ) =
J0 ( x)
( x ) − ( λs ) 2
2
(1.24) (1.25)
.
(1.26)
The equations (1.8) we can rewrite as
(ω
( k )2 0 mn
−ω )e 2
(k ) 0 mn
2ω0( km)20 =− 2 × bk d k J12 (λm ) δ n ,0
⎡ J1 ( λs′ ) ⎧⎪ ( k ,2 P ) (2 P ) 3 ⎛ ak(2 P ) λm ⎞ ( k ) ⎛ ak(0)+1λm ⎞ ( k +1) ⎫⎪⎤ n ( k +1,0) (0) 3 × ⎢∑ ( ak ) Fs′ ⎜ b ⎟ Cr ,s′ − (−1) γ s′ ( ak +1 ) Fs′ ⎜ b ⎟ Cl ,s′ ⎬⎥ ⎨γ s′ ⎪⎭⎥⎦ ⎢⎣ s′ λs′ ⎪⎩ k ⎝ ⎠ ⎝ k ⎠ where Cr(,ks) = ⎡⎣ −Cs( k ,2 P ) exp γ s( k ,2 P )tk(2 P ) + C−( ks ,2 P ) exp −γ s( k ,2 P )tk(2 P ) ⎤⎦
(
)
(
,
(1.27)
)
(1.28) Cl(,ks ) = ⎡⎣ −Cs( k ,0) + C−( ks ,0) ⎤⎦ Indexes l (left) and r (right) refer to the coefficients on the left and right sides of the disk. We will suppose that for 1 ≤ p ≤ P : ak( p ) > ak( p +1) and for P + 1 ≤ p ≤ 2 P : ak( p ) < ak( p +1) . The magnetic field boundary conditions inside a disc at z = zk( p ) are
∑ (C s
Hs(,ϕk , p −1) + C−( ks , p −1)H−(sk,ϕ, p −1) ) = ∑ ( Cs( k , p )Hs(,ϕk , p ) + C−( ks , p )H−(sk,ϕ, p ) ) ,
( k , p −1) s
(1.29)
s
1≤ p ≤ P , 0 ≤ r < ak( p ) ,
(1.30)
P +1 ≤ p ≤ 2P , 0 ≤ r < ak( p −1) . The electric field boundary conditions inside a disc at z = z 1≤ p ≤ P ,
(1.31) ( p) k
are
⎧ 0, ak( p ) ≤ r < ak( p −1) ⎪ ∑s ( Cs(k , p −1)Es(,kr, p −1) + C−( ks , p−1)E−( ks,,rp−1) ) = ⎨∑ ( Cs( k , p )Es(,kr, p ) + C−( ks , p )E−( ks,,rp ) ), r < ak( p ) , ⎪⎩ s
(1.32)
P +1 ≤ p ≤ 2P , ⎧ 0, ak( p −1) ≤ r < ak( p ) ⎪ ∑s ( Cs( k , p )Es(,kr, p ) + C−( ks, p )E−( ks ,,rp ) ) = ⎨∑ ( Cs( k , p −1)Es(,kr, p −1) + C−( ks , p−1)E−( ks ,,rp−1) ), r < ak( p−1) . ⎪⎩ s
(1.33)
5 We arrange the amplitudes C±( ks , p ) ( s = 1, 2,3....∞ ) in the column vectors C±( k , p )
C±( k , p )
⎛ C±( 1k , p ) ⎞ ⎜ (k , p) ⎟ C = ⎜ ±( 2k , p ) ⎟ , ⎜ C±3 ⎟ ⎜⎜ ⎟⎟ ⎝ .... ⎠
(1.34)
Using the orthogonal properties of electric field we transform equalities (1.29),(1.32), (1.33) into matrix ones 1≤ p ≤ P
⎡⎣C+( k , p ) + C−( k , p ) ⎤⎦ = H ( ak( p −1) , ak( p ) ) ⎡exp {γ ( k , p −1)tk( p −1) } C+( k , p −1) + exp {−γ ( k , p −1)tk( p −1) } C−( k , p −1) ⎤ ,(1.35) ⎣ ⎦ ⎡exp {γ ( k , p −1)tk( p −1) } C+( k , p −1) − exp {−γ ( k , p −1)tk( p −1) } C−( k , p −1) ⎤ = E ( ak( p −1) , ak( p ) ) ⎡C+( k , p ) − C−( k , p ) ⎤ ,(1.36) ⎣ ⎦ ⎣ ⎦
P +1 ≤ p ≤ 2P ( k , p −1) ( p −1) ⎡ tk } C+( k , p −1) + exp {−γ ( k , p −1)tk( p −1) } C−( k , p −1) ⎦⎤ = H ( ak( p ) , ak( p −1) ) ⎣⎡C+( k , p ) + C−( k , p ) ⎦⎤ ,(1.37) ⎣ exp {γ
⎡⎣C+( k , p ) − C−( k , p ) ⎤⎦ = E (ak( p ) , ak( p −1) ) ⎡exp {γ ( k , p −1)tk( p −1) } C+( k , p −1) − exp {−γ ( k , p −1)tk( p −1) } C−( k , p −1) ⎤ ,(1.38) ⎣ ⎦
where the elements of matrices H , E and exp {±γ ( k , p )tk( p ) } are 3
⎛ ak( p −1) λs′ ⎞ 2λs ⎛ ak( p ) ⎞ H s , s′ ( a , a ) = − J ( λ ) ⎜ a( p−1)λ ⎟ Fs ⎜ a( p ) ⎟ , s ⎝ k s′ ⎠ 1 ⎝ k ⎠ ( k , p −1) ( p −1) ( p −1) 2J (λ ) γ ⎛a λ ⎞ a Es , s′ (ak( p ) , ak( p −1) ) = − k ( p ) 2 1 s′ s′ ( k , p ) Fs′ ⎜ k ( p ) s ⎟ , ak J1 ( λs ) λs′λsγ s ⎝ ak ⎠ ( p) k
( p −1) k
(1.39) (1.40)
(k , p) ( p) ⎡ tk }⎦⎤ = δ s , s′ exp {±γ s( k , p )tk( p ) } . ⎣ exp {±γ s , s′ The matrix equalities (1.35),(1.37),(1.38) we can rewrite using Scattering Matrix (GSM) of the waveguide step 1≤ p ≤ P
⎡ exp {γ ( k , p −1)tk( p −1) } C+( k , p −1) ⎤ ⎡ S11( k , p −1: p ,l) ⎢ ⎥ = ⎢ ( k , p −1: p ,) (k , p) C − ⎣⎢ ⎦⎥ ⎣ S21
⎡ S11( k , p −1: p ,l) ⎢ ( k , p −1: p ,) ⎣ S21
S12( k , p −1: p ,l) ⎤ ⎡ exp {−γ ⎢ ( k , p −1: p ,l) ⎥ S22 ⎦ ⎣⎢
⎡ − I + 2 H −1 ( E + H −1 )−1 S12( k , p −1: p ,l) ⎤ == ⎢ ( k , p −1: p ,l) ⎥ −1 ⎢ S22 ⎦ 2 ( E + H −1 ) ⎣⎢
E = E ( ak( p −1) , ak( p ) ) P +1 ≤ p ≤ 2P
⎡ exp {γ ( k , p −1)tk( p −1) } C+( k , p −1) ⎤ ⎡ S11( k , p −1: p ,r ) ⎢ ⎥ = ⎢ ( k , p −1: p ,r ) (k , p) C − ⎣⎢ ⎦⎥ ⎣ S21
the Generalized
( k , p −1) ( p −1) k
t
}C
( k , p −1) −
C+( k , p )
H −1 + H −1 ( E + H −1 )
H = H ( ak( p −1) , ak( p ) )
(1.41)
(E + H
(1.42)
( E − H )⎥⎤ ,(1.43) ⎥ ) ( E − H ) ⎦⎥ −1
−1 −1
−1
−1
,
S12( k , p −1: p ,r ) ⎤ ⎡ exp {−γ ⎥⎢ S22( k , p −1: p ,r ) ⎦ ⎣⎢
⎤ ⎥, ⎥⎦
(1.44)
( k , p −1) ( p −1) k
t
C+( k , p )
}C
( k , p −1) −
⎤ ⎥ , (1.45) ⎦⎥
6 ⎡ S11( k , p −1: p ,r ) ⎢ ( k , p −1: p ,r ) ⎣ S 21
−1 S12( k , p −1: p , r ) ⎤ ⎡ − I + 2 H ( I + EH ) E ⎥=⎢ S 22( k , p −1: p , r ) ⎦ ⎢ 2 ( I + EH )−1 E ⎣
H = H ( ak( p ) , ak( p −1) ) E = E (ak( p ) , ak( p −1) ) Here, I is a unit matrix and
( )
−1
H + H ( I + EH ) ( I − EH ) ⎤ ⎥, −1 ( I + EH ) ( I − EH ) ⎥⎦ −1
.
(1.47)
represents the inverse matrix.
Then the GSM of the whole aperture ⎤ ⎡ S (k ) ⎡ C+( k ,0) ⎢ (k , p) ⎥ = ⎢ 11( k ) ( k ,2 P ) (2 P ) − γ exp C t { }⎦⎥ ⎣ S21 k ⎣⎢ − we can find as
⎤ C−( k ,0) S12( k ) ⎤ ⎡ ⎢ ⎥ ⎥ S 22( k ) ⎦ ⎣⎢C+( k ,2 P ) exp {γ ( k ,2 P )tk(2 P ) }⎦⎥
S ( k ) = Sw( k ,0) ⊗ S ( k ,0,1,l ) ⊗ Sw( k ,1) ⊗ S ( k ,1,2,l ) ⊗ Sw( k ,2) ............. ⊗ S ( k , P −1, P ,l ) ⊗ Sw( k , P ) ⊗ S ( k , P , P +1, r ) ⊗ Sw( k , P +1) ........ ⊗ S ( k ,2 P −1,2 P , r ) ⊗ Sw( k ,2 P ) where the star product S ( c ) = S ( b ) ⊗ S ( a ) is determined as follows: −1 S11( c ) = ⎡⎢ S11( a ) + S12( a ) S11(b ) ( I − S22( a ) S11(b ) ) S21( a ) ⎤⎥ ⎣ ⎦ −1 ( a ) ( b ) S12( c ) = S12( a ) ⎡⎢ S11(b ) ( I − S22( a ) S11( b ) ) S22 S12 + S12(b ) ⎤⎥ ⎣ ⎦.
S
Matrix Sw( k , p )
=S
(c) 21
(b) 21
(1.46)
(I − S (I − S
)
( a ) ( b ) −1 22 11
S
S
(1.48)
, (1.49)
(1.50)
(a) 21
( a ) ( b ) −1 ( a ) ( b ) (b ) ⎤ S 22( c ) = ⎡⎢ S21(b ) 22 S11 ) S 22 S12 + S 22 ⎥ ⎣ ⎦ is the S matrix of the uniform section of waveguide
0 exp {−γ ( k , p )tk( p ) }⎤ Sw12( k , p ) ⎤ ⎡ ⎥. =⎢ (1.51) (k , p) ⎥ (k , p) ( p) Sw22 ⎢ ⎥ 0 tk } ⎦ ⎣exp {−γ ⎦ Using(1.48), from (1.22) and (1.23) we can find the amplitudes in the end waveguides and auxiliary coefficients (1.28) P ,k ,k ) P , k , k −1) + ∑ (−1) n e0( kmn−1) Ξ (2 Cr(,ks) = ∑ e0( kmn) Ξ (2 m,s m,s ⎡ Sw11( k , p ) ⎢ w( k , p ) ⎣ Sw21
n,m
(k ) l ,s
C
n,m
= ∑e
(k ) 0 mn
n,m
Ξ
(0, k .k ) m,s
k . k −1) + ∑ (−1) n e0( kmn−1) Ξ (0, m,s
,
(1.52)
n,m
where (k ) k ,k ) ⎤⎦ Γ (mk ,k ) Ξ (0, = ⎡⎣Θ(21k ) − S11( k ) Θ(21k ) − S12( k ) Θ11 m, s (k ) k , k −1) ⎤⎦ Γ (mk ,k −1) Ξ (0, = ⎡⎣Θ (22k ) − S11( k ) Θ(22k ) − S12( k ) Θ12 m,s , (k ) (k ) (k ) (k ) ⎤⎦ Γ (mk ,k ) Ξ (2m ,Ps, k , k ) = ⎡⎣ S 21 Θ 21 + S 22( k ) Θ11 − Θ11 (k ) (k ) P , k , k −1) ⎤⎦ Γ (mk ,k −1) − Θ12 Ξ (2 = ⎡⎣ S 21( k ) Θ(22k ) + S 2(2k ) Θ12 m,s
(1.53)
7 (k ) ⎡ Θ11 ⎢ (k ) ⎣ Θ21
(k ) ⎤ ⎡ ( I + S 22( k ) ) S 21( k ) ⎤ Θ12 ⎥=⎢ ⎥ ( I + S11( k ) ) ⎦ Θ(22k ) ⎦ ⎣ S12( k )
−1
−1
(k ) ( k ) −1 ( k ) ( k ) −1 ) S 21 ⎤⎦ S12( k ) ( I + S 22 ) Θ11 = ( I + S 22( k ) ) −1 + ( I + S 22( k ) ) −1 S 21( k ) ⎡⎣ ( I + S11( k ) ) − S12( k ) ( I + S 22 −1
Θ(21k ) = − ⎡⎣( I + S11( k ) ) − S12( k ) ( I + S 22( k ) ) −1 S 21( k ) ⎤⎦ S12( k ) ( I + S 22( k ) ) −1 (k ) (k ) ⎤⎦ Θ12 = −( I + S 22( k ) ) −1 S 21( k ) ⎡⎣( I + S11( k ) ) − S12( k ) ( I + S 22( k ) ) −1 S 21
. (1.54)
−1
−1
Θ(22k ) = ⎡⎣( I + S11( k ) ) − S12( k ) ( I + S 22( k ) ) −1 S 21( k ) ⎤⎦ Lets us chose the E010 modes of the cavities as the basic modes (oscillators) [6,10]. In this case the equations (1.52) we can write in the form P ,k ,k ) P , k , k −1) ( k ) (2 P , k , k ) ( k −1) (2 P , k , k −1) − ∑ ' (−1) n e0( kmn−1) Ξ (2 = e010 Ξ1, s + e010 Ξ1, s Cr(,ks) − ∑ ' e0( kmn) Ξ (2 m,s m,s n,m
(k ) l ,s
C
n,m
−∑ e
' (k ) 0 mn
Ξ
(0, k . k ) m,s
n,m
where
∑
'
=
∑
k .k −1) ( k ) (0, k .k ) ( k −1) (0, k .k −1) Ξ1, s − ∑ ' (−1) n e0( kmn−1) Ξ (0, = e010 Ξ1, s + e010 m, s
(1.55)
n,m
.
n , m ; n ≠ 0, m ≠1
n,m
Substituting (1.27) into (1.55) gives a set of equations ( k ) ( k ,2) ( k +1) ( k ,3) ( k −1) ( k ,4) Cl(,ks ) + ∑ Cr(,ks)′Tl ,(sk,,1) s ′ + ∑ Cl , s ′ Tl , s , s′ − ∑ Cl , s′ Tl , s , s′ − ∑ Cr , s′ Tl , s , s′ = s′
=R
s′
( k ,1) ( k ) l ,s 010
e
( k ,2) ( k −1) l ,s 010
−R
e
= ∑ δ j ,k {R
s′
s′
( k ,1) ( j ) s 010
e
( k ,2) ( j −1) s 010
−R
e
j
} = ∑ {δ
j ,k
( j) Rl(,ks ,1) − δ j +1,k Rl(,ks ,2) } e010
j
,(1.56)
( k +1) ( k ,3) ( k −1) ( k ,4) Cr(,ks) + ∑ Cr(,ks)′Tr(,ks ,,1)s′ + ∑ Cl(,ks′)Tr(,ks ,,2) s ′ − ∑ Cl , s ′ Tr , s , s ′ − ∑ Cr , s ′ Tr , s , s ′ = s′
=R
s′
( k ,1) ( k ) r ,s 010
e
( k ,2) ( k −1) r ,s 010
−R
e
= ∑ δ j ,k {R
s′
s′
( k ,1) ( j ) r ,s 010
j
e
( k ,2) ( j −1) r ,s 010
−R
e
} = ∑ {δ
j ,k
( j) Rr(,ks,1) − δ j −1, k Rr(,ks,2) } e010
j
where
C
γ s(′k ,2 P ) ( k ) = Cr , s′ J1 (λs′ ) λs′
Cl(,ks′)
γ ( k ,0) = s′ Cl(,ks′) J1 (λs′ ) λs′
(k ) r , s′
,
( k ,1) l , s , s′
γ s( k ,0) J1 ( λs ) 2ak(2 P )3 ⎛ ak(2 P ) λm ⎞ (0,k , k ) λm2 (1, k ) = Λ F ∑m m J 2 (λ ) s′ ⎜ b ⎟ Ξ m,s λs bk3 m k 1 ⎝ ⎠
( k ,3) l , s , s′
γ s( k ,0) J1 ( λs ) 2ak(0)3 ⎛ ak(0)+1λm ⎞ (0,k ,k ) λm2 (2, k ) +1 = Λ F ⎟ Ξ m, s ′ ∑ m s ⎜ bk3 m J12 (λm ) ⎝ bk ⎠ λs
( k ,4) l , s , s′
γ s( k ,0) J1 ( λs ) 2ak(2−1P )3 ⎛ ak(2−1P ) λm ⎞ (0,k ,k −1) λm2 (2, k −1) =− ∑ Λ m J 2 (λ ) Fs′ ⎜ b ⎟ Ξ m,s bk3−1 m λs 1 m ⎝ k −1 ⎠
T
T T
Tl (, sk,,2) s′ = −
γ s( k ,0) J1 ( λs ) 2ak(0)3 ⎛ ak(0) λm ⎞ (0, k ,k −1) λm2 (1, k −1) F Λ ∑ m J 2 (λ ) s ′ ⎜ b ⎟ Ξ m , s λs bk3−1 n , m 1 m ⎝ k −1 ⎠
(1.57)
,
(1.58)
8
γ s( k ,0) J1 ( λs ) (0,k ,k ) Ξ1, s λs
Rl(,ks ,1) = ( k ,2) l ,s
R
Tr(,ks ,,1)s′ = ( k ,3) r , s , s′
T
( k ,4) r , s , s′
T
Tr(,ks ,,2) s′
γ s( k ,0) J1 ( λs ) (0,k ,k −1) =− Ξ1, s λs
,
(1.59)
k) 2 γ s( k ,2 P ) J1 ( λs ) 2ak(2 P )3 Λ (1, ⎛ ak(2 P ) λm ⎞ (2 P ,k ,k ) m λm F ∑m J 2 (λ ) s′ ⎜ b ⎟ Ξ m,s bk3 λs 1 m k ⎝ ⎠
k) 2 γ s( k ,2 P ) J1 ( λs ) 2ak(0)3 Λ (2, λm ⎛ ak(0)+1λm ⎞ (2 P ,k ,k ) +1 m = ∑m J 2 (λ ) Fs′ ⎜ b ⎟ Ξ m,s bk3 λs 1 m ⎝ k ⎠ k −1) 2 γ s( k ,2 P ) J1 ( λs ) 2ak(2−1P )3 Λ (2, λm ⎛ ak(2−1P ) λm ⎞ (2 P ,k ,k −1) m Fs′ ⎜ =− ⎟ Ξ m,s ∑ bk3−1 m J12 (λm ) λs ⎝ bk −1 ⎠ k −1) 2 γ s( k ,2 P ) J1 ( λs ) 2ak(0)3 Λ (1, λm ⎛ ak(0) λm ⎞ (2 P ,k ,k −1) m Fs′ ⎜ =− ⎟ Ξ m,s ∑ bk3−1 m J12 (λm ) λs ⎝ bk −1 ⎠ γ s( k ,2 P ) J1 ( λs ) (2 P ,k ,k ) ( k ,1) Ξ1, s Rr , s = λs ( k ,2) r ,s
R
γ ( k ,2 P ) J1 ( λs ) (2 P ,k ,k −1) =− s Ξ1, s λs
,
,
(1.60)
(1.61)
k) Λ (1, m
⎧ 1 ⎡ 1 ⎤ (k ) ⎪ ( k ) ⎢cth(d k h1 ) − ⎥, m =1 d k h1( k ) ⎦ ⎪ bk h1 ⎣ =⎨ , 1 ⎪ cth(d k hm( k ) ), m >1 ⎪ b h( k ) ⎩ k m
(1.62)
k) Λ (2, m
⎧ 1 ⎡ 1 1 ⎤ , m =1 − ⎪ (k ) ⎢ (k ) (k ) ⎥ ⎪ bk h1 ⎣ sh(d k h1 ) d k h1 ⎦ =⎨ , 1 ⎪ , m >1 ⎪ bk hm( k ) sh(d k hm( k ) ) ⎩
(1.63)
λm2
hm( k ) =
−
ω2
. (1.64). bk2 c 2 The systems (1.56) are the infinite linear coupled systems of equations, so one can represent its solution in the form ( j) k, j) k, j) Cl(,ks ) = ∑ (Yl (1, − Yl (2, ) e010 ,s ,s j
C
(k ) r ,s
k, j) k, j) ( j) = ∑ (Yr(1, − Yr(2, ) e010 ,s ,s
,
(1.65)
j
(1, k , j ) l ,s
where the new unknowns Y
(2, k , j ) l ,s
,Y
k, j) k, j) , Yr(1, , Yr(2, are the solutions of such sets 2 of ,s ,s
equations
2
equations
Instead two sets of unknowns we introduce four sets of unknowns, so we can add two new sets of
9 k +1, j ) k +1, j ) ( k +1,1) k +1, j ) ( k +1,2) k + 2, j ) ( k +1,3) k , j ) ( k +1,4) + ∑ Yr(1, Yl (1, Tl , s ,s′ + ∑ Yl (1, Tl , s , s′ − ∑ Yl (1, Tl , s , s′ − ∑ Yr(1, Tl , s , s′ ,s , s′ , s′ , s′ , s′ s′
s′
s′
s′
δ
( k +1,1) l ,s k +1, j
=R
(1, k +1, j ) ( k ,3) k −1, j ) ( k ,4) k, j) k , j ) ( k ,1) + ∑ Yr(1, Yr(1, Tr , s , s′ + ∑ Yl ,(1,s′ k , j )Tr(,ks ,,2) Tr , s ,s′ − ∑ Yr(1, Tr , s , s′ = , s′ s ′ − ∑ Yl , s ′ ,s , s′ s′
s′
=R
s′
,
(1.66)
s′
δ
( k ,1) r ,s k, j
k +1, j ) k +1, j ) ( k +1,1) k +1, j ) ( k +1,2) k + 2, j ) ( k +1,3) k , j ) ( k +1,4) + ∑ Yr(2, Yl (2, Tl , s , s′ + ∑ Yl (2, Tl , s , s′ − ∑ Yl (2, Tl , s , s′ − ∑ Yr(2, Tl , s , s′ = ,s , s′ , s′ , s′ , s′ s′
s′
s′
s′
δ
( k +1,2) l ,s k, j
=R
k , j ) ( k ,2) k +1, j ) ( k ,3) k −1, j ) ( k ,4) k, j) k , j ) ( k ,1) + ∑ Yr(2, Yr(2, Tr , s , s′ + ∑ Yl (2, Tr , s , s′ − ∑ Yl ,(2, Tr , s , s′ − ∑ Yr(2, Tr ,s ,s′ = , s′ s′ ,s , s′ , s′ s′
=R
s′
s′
.
(1.67)
s′
δ Using (1.65) we can rewrite (1.27) as
( k ,2) r ,s k , j +1
(ω
( k )2 0 mn
− ω 2 ) e0( kmn) = ω0( kmn)2
∞
∑e
j =−∞
(k , j ) α mn ,
( j) 010
(1.68)
where
α
(k , j ) mn
⎛a ⎞ 2π ε 0bk3 ⎡ k, j) k, j) = ( k ) 3 ⎢ −∑ (Yr(1, − Yr(2, Fs ⎜ k λm ⎟ + ) ,s ,s N m , n λm ⎣ s ⎝ bk ⎠
. (1.69) ⎤ ⎛ ⎞ a k +1, j ) k +1, j ) + (−1) n ∑ (Yl (1, − Yl ,(2, ) Fs ⎜ bk +1 λm ⎟ ⎥ s ,s s ⎝ k ⎠⎦ Let’s note that relations (1.69) are the exact ones. It follows from (1.68) that for finding the amplitudes of the main E010 mode we have to solve a system of coupled equations ∞
(k ) ( k )2 ( j) (k , j) α10 . = ω010 (ω010( k )2 − ω 2 ) e010 ∑ e010
(1.70)
j =−∞
Amplitudes of other modes ( (m, n) ≠ (1, 0) ) can be found by summing the relevant series (k ) 0 mn
e
ω0( kmn)2 = ( k )2 ω0 mn (1 − α m( k,n,k ) ) − ω 2
{
∞
}
∑e
j =−∞
(k , j) α mn .
( j) 010
(1.71)
The system of coupled equations (1.70) is very similar to that one that can be constructed on the base of equivalent circuits approach. But in our approach the coefficients α m( k,n, j ) are electrodynamically strictly defined and can be calculated with necessary accuracy. They depend on both the frequency ω and geometrical sizes of all volumes. As it follows from physics of couplings, the contribution of ”long range” couplings is small 3 and one can confine oneself to consideration a finite number of adjacent couplings. In this case the system of coupled equations (1.70) transforms into such one ( k )2 (k ) ( k )2 − ω 2 ) e010 = ω010 (ω010
k+N
∑
j =k − N
( j ) (k , j ) α10 . e010
.
(1.72)
For dealing with this system we have to develop a procedure for α10( k , j ) calculation. From (1.69) it follows that we have to know the coefficients (1.73) (Yl (1,,s k +1, j ) , Yr(1,,s k , j ) ) , (Yl ,(2,s k +1, j ) , Yr(2,,s k , j ) )
3
In the case under consideration (when
through evanescent fields
E010 modes are the basic ones) ”long range” coupling is realized
10 for k − N ≤ j ≤ k + N . Coefficients Y are the solutions of the infinite set of infinite systems of linear equations. We suppose that this set can be truncated and the solution of such truncated set of equations converge under increasing the number of equations. Besides, we have to make restriction on the number of waveguide modes ( M ) in the field expansion inside apertures and on the number of radial modes ( L ) in the field expansion inside resonators. Used truncation method is explained in [7]. In addition to the case without rounding we have additional parameter 2 P +1 - number of homogeneous waveguides that represent the real opening in the disk. It is necessary to introduce some criterion for choosing the values of M , L, P that can give correct simulation results. As such criterion we chose the frequency calculation accuracy for the given phase shift per cell in the case of homogeneous DLW. Results of our previous investigations have shown [7,9] that N ≤ 4 (number of “side couplings”) gives good results in (k ) all cases that are interesting for accelerator physics. The amplitudes e010 of homogeneous DLW obey the Floquet theorem (k ) (0) e010 exp(ikϕ ) = e010 (1.74) and the dispersive equation has such form [6,7,9]
{ω (1 + α ) − ω } = 2ω ∑ α N
(0)2 010
(0,0) 1,0
2
(0)2 010
j =1
(0, j ) 10
(ω ) cos( jϕ ) .
(1.75)
For given ϕ = ϕ∗ , N and geometrical sizes we can find from this equation corresponding frequency ω = ω∗ .
Fig. 3 Dependence of frequency of 2π / 3 mode on number of expansion modes M for homogeneous DLW with d = 3.099 cm, t = 0.4 cm, b = 4.134 cm, a = 1.2 cm
11
Fig. 4 Dependence of frequency of 2π / 3 mode on number of homogeneous waveguides that represent the real opening in the disk ( 2 P + 1 ) for homogeneous DLW with d = 3.099 cm, t = 0.4 cm, b = 4.134 cm, a = 1.2 cm
In Fig. 3 and Fig. 4 calculation results of frequency are presented with taking into account a coupling of nine cavities ( N = 4) for homogeneous DLW with d = 3.099 cm, t = 0.4 cm, b = 4.134 cm, a = 1.2 cm, R = t / 2 = 0.2 cm (radius of rounding), SUPERFISH frequency of 2π / 3 mode is f = 2855.06 MHz. We can see that solutions of the truncated set of equations converge. Two curves correspond to different laws of changing of lengths l p of homogeneous waveguides inside the opening. The first curve was obtained under the law l p = l p −1 + Δl ( l0 = R /(10 P) ) and the second one under the law l p = const ( ∑ l p = R = t / 2 ). Radii a ( p ) of homogeneous waveguides inside the opening were chosen from such condition
π (a
( p )2
− a ) l p = Vp = 2
z p +1
∫ dz
zp
a + R − R 2 −( z − R )
∫
2
rdrdϕ
(1.76)
R0
Table 1 a = 1.2 cm, b = 4.134 cm, d = 3.099 cm, t = 2 R = 0.4 cm (0,0) α1,0
0.04706432
α
(0,1) 1,0
0.01091948
(0,2) 1,0
α
-1.9937E-05
(0,3) 1,0
α
3.7465E-08
(0,4) α1,0
-7.0364E-11
Taking into account a coupling of nine cavities ( N = 4) for the homogeneous DLW with (0,4) considered geometry 4 is excessive. Indeed, from Table 1 it follows that coefficient α1,0 is less
Taking into account a coupling of nine cavities ( N = 4) can be necessary for a DLW with greater coupling (small d or large a). 4
12 than the needed accuracy. So, in our calculations we shall use N = 3 approach for the DLWs with geometry close to considered one above. In paper [9] we presented some results that clarify some aspects of the tuning methods that are based on the field measurements and were used for tuning nonuniform DLWs [11,12,13,14,15,16,17,18,19] . These results concerned the possibility of using some parameters that characterize the detuning of cell without taking into account the parameters of neighboring cells. Using the above coupled cavity model we can make calculation of characteristics of the inhomogeneous disc-loaded waveguides that are used in real linacs with taking into account the rounding of the disk hole edges. In paper [9] we proposed to characterize the sell detuning in the case of small coupling by such parameter Z ( m,3) Fk(3) = k( c ,3) − 1 , (1.77) Zk where 5 ( k −1, m ) ( k +1, m ) α ( c ,k ,k −1) e010 + α10( c , k ,k +1) e010 Z k( m,3) = 10 , (1.78) ( k ,m) e010
Z k( c ,3) =
( k −1, c ) ( k +1, c ) α10( c ,k ,k −1) e010 + α10( c ,k ,k +1) e010 ( k ,c ) e010
,
(1.79)
( k ,c ) - calculated complex amplitudes α10( c ,k ,k −1) , α10( c ,k ,k +1) - calculated coupling coefficients, e010 ( k ,m ) = of E010 mode, e010
1 dk
∫ E (r = 0, z)dz k
- measured amplitudes of E010 mode, Ek - measured
dk
longitudinal electric field of the k -cavity. In the case of moderate coupling we can use such parameter Z ( m,5) Fk(5) = k( c ,5) − 1 . Zk
Z k( m ,5) =
( k − 2, m ) ( k −1, m ) ( k +1, m ) ( k + 2, m ) + α10( c , k , k −1) e010 + α10( c , k ,k +1) e010 + α10( c , k , k + 2) e010 α10( c ,k ,k − 2) e010 ( k ,m ) e010
(1.80) , (1.81)
Z k(5) =
( k − 2, c ) ( k −1, c ) ( k +1, c ) ( k + 2, c ) α10( c ,k ,k − 2) e010 + α10( c , k , k −1) e010 + α10( c , k , k +1) e010 + α10( c , k ,k + 2) e010 ( k ,c ) e010
. (1.82)
There also was proposed to use such parameter [11,12,13,14,15,16,17,18,19] E ( m ) + Ek( +m1) − 2 cos ϕ Ek( m ) Fk(3,0) = k −1 , Ek( m ) 2 cos ϕ
(1.83)
where Ek( m ) = Ek (r = 0, z = d k / 2) measured electric field in the center of the k -cavity. We also introduced an additional parameter that can be obtained from (1.77) under assumption that Z k( c ,3) ≈ 2α10( c , k ,k −1) cos ϕ , α10( c , k ,k −1) ≈ α10( c ,k , k +1) [9] (compare with (1.83))
Fk(3,00) =
5
( k −1, m ) ( k +1, m ) ( k ,m) + e010 − 2 cos ϕ e010 e010 . ( k ,m ) 2 cos ϕ e010
c- calculated values, m – measured ones
(1.84)
13 To study quality of introduced parameters we calculated these parameters for a homogeneous DLW 6 with β ph ≈ 1 ( d k(0) = 3.099 cm, tk(0) = 0.4 cm, bk(0) = 4.1334 cm,
ak(0) = 1.2056 cm, Rk(0) = t / 2 = 0.2 cm, SUPERFISH frequency of 2π / 3 mode f = 2856.04 MHz, frequency of 2π / 3 mode under calculation with P = 50, L = 250, M = 50 - f = 2856.38 MHz) that has a local “defect”. We used the developed coupled cavity model with N = 3 for calculation the necessary coefficients.
Fig. 5 Modules of amplitudes of E010 mode in the case of the “b-defect”: b24 = b24(0) − 10μ m
(0) Fig. 6 Phase shifts ( ϕk − 120 ) in the case of the “b-defect”: b24 = b24 − 10μ m
Calculations were carried out for 25 cavities with assumption that at frequency f = 2856.38 MHz there is a full matching at the ends [8]. Amplitudes and phase shifts ( ϕk − 120 ) ( k ,c ) and δϕ ( k ,c ) ). for the DLW without defects are presented in Fig. 5 and Fig. 6 (coefficients e010
Reflection 6
coefficient
equals
8E-5;
real
parts
of
Fk(3,00) and
Fk(3,0)
equal:
The last cell of an accelerating structure for industrial linac ( M.I. Ayzatskiy, A.N. Dovbnya, V.F. Ziglo et al. Accelerating system for an industrial linac. PAST, 2012, N4, pp.24-28)
14 Real( Fk(3,00) ) = Real( Fk(3,0) ) = -1.39E-4; modules of real parts of Fk(3,m) and Fk(5,m) are less than E-14; imaginary parts are very small for all coefficients (modules
