error of paraxial approximation t. agoh 2010.11.8, kek csr impedance below cutoff frequency not...
TRANSCRIPT
![Page 1: Error of paraxial approximation T. Agoh 2010.11.8, KEK CSR impedance below cutoff frequency Not important in practical applications](https://reader036.vdocument.in/reader036/viewer/2022062806/56649cba5503460f94981d5e/html5/thumbnails/1.jpg)
Error of paraxial approximation
T. Agoh
2010.11.8, KEK
• CSR impedance below cutoff frequency
• Not important in practical applications
![Page 2: Error of paraxial approximation T. Agoh 2010.11.8, KEK CSR impedance below cutoff frequency Not important in practical applications](https://reader036.vdocument.in/reader036/viewer/2022062806/56649cba5503460f94981d5e/html5/thumbnails/2.jpg)
Wave equation in the frequency domain
Curvature
2
Gauss’s law
Parabolic equation
cutoff frequency
Condition
![Page 3: Error of paraxial approximation T. Agoh 2010.11.8, KEK CSR impedance below cutoff frequency Not important in practical applications](https://reader036.vdocument.in/reader036/viewer/2022062806/56649cba5503460f94981d5e/html5/thumbnails/3.jpg)
3Transient field of CSR in a beam pipe
G. V. Stupakov and I. A. Kotelnikov, PRST-AB 12, 104401 (2009)
Eigenmode expansion
Laplace transform
Initial value problem with respect to s
![Page 4: Error of paraxial approximation T. Agoh 2010.11.8, KEK CSR impedance below cutoff frequency Not important in practical applications](https://reader036.vdocument.in/reader036/viewer/2022062806/56649cba5503460f94981d5e/html5/thumbnails/4.jpg)
Generally speaking about field analysis,
Eigenmode expansion Fourier, Laplace analysis
Algebraic
Analytic
easier to describe resonant field = eigenstate of the structure
resonance = singularity,δ-function
easier to examine the structure in non-resonant region
easier to deal with on computer = useful in practical applications
(special treatment needed)
concrete and clear picture to human brain
compact & beautiful formalism
straightforward formulation but often complex & intricate
(orthonormality, completeness)
<abstract, symbolic>
<concrete, graphic>
4
![Page 5: Error of paraxial approximation T. Agoh 2010.11.8, KEK CSR impedance below cutoff frequency Not important in practical applications](https://reader036.vdocument.in/reader036/viewer/2022062806/56649cba5503460f94981d5e/html5/thumbnails/5.jpg)
Field in front of the bunch
Regularity at the origin
Field structure of steady CSR in a perfectly conducting rectangular pipe
5
(byproduct) effect of sidewalls
No pole nor branch point
Imaginary poles
symmetrycondition
e.g.)
![Page 6: Error of paraxial approximation T. Agoh 2010.11.8, KEK CSR impedance below cutoff frequency Not important in practical applications](https://reader036.vdocument.in/reader036/viewer/2022062806/56649cba5503460f94981d5e/html5/thumbnails/6.jpg)
Transient field of CSR inside magnet
• Transverse field
• Longitudinal field
6
s-dependence is only here
![Page 7: Error of paraxial approximation T. Agoh 2010.11.8, KEK CSR impedance below cutoff frequency Not important in practical applications](https://reader036.vdocument.in/reader036/viewer/2022062806/56649cba5503460f94981d5e/html5/thumbnails/7.jpg)
s-dependence of transient CSR inside magnet
in the framework of the paraxial approximation.
7
Assumption: This fact must hold also in the exact Maxwell theory.
That is, froze
n
![Page 8: Error of paraxial approximation T. Agoh 2010.11.8, KEK CSR impedance below cutoff frequency Not important in practical applications](https://reader036.vdocument.in/reader036/viewer/2022062806/56649cba5503460f94981d5e/html5/thumbnails/8.jpg)
• Steady field a in perfectly conducting rectangular pipe
• Exact solution for periodic system
Debye expansion
Olver expansion
Asymptotic expansion
no longer periodic
8
(no periodicity)
contradiction
(para-approx.)
![Page 9: Error of paraxial approximation T. Agoh 2010.11.8, KEK CSR impedance below cutoff frequency Not important in practical applications](https://reader036.vdocument.in/reader036/viewer/2022062806/56649cba5503460f94981d5e/html5/thumbnails/9.jpg)
(+)
( C )
( C )
(+)
Imaginary impedance
9
exact limit
eq.
paraxial approximation
resistive
walltransient, fr
ee
steady, free
cutoff frequency
shielding threshold
transient,
copper
perfectly conducting pipe
( C ) transient,
![Page 10: Error of paraxial approximation T. Agoh 2010.11.8, KEK CSR impedance below cutoff frequency Not important in practical applications](https://reader036.vdocument.in/reader036/viewer/2022062806/56649cba5503460f94981d5e/html5/thumbnails/10.jpg)
Real impedance
10
resistive
wall
transie
nt,
free
steady, free
cutoff frequency
shielding threshold
transient,
copper
transient, perfectly conducting pipe
The lowest resonance
strange structure
![Page 11: Error of paraxial approximation T. Agoh 2010.11.8, KEK CSR impedance below cutoff frequency Not important in practical applications](https://reader036.vdocument.in/reader036/viewer/2022062806/56649cba5503460f94981d5e/html5/thumbnails/11.jpg)
Fluctuation of impedance at low frequency
11
agreement with grid calculation
![Page 12: Error of paraxial approximation T. Agoh 2010.11.8, KEK CSR impedance below cutoff frequency Not important in practical applications](https://reader036.vdocument.in/reader036/viewer/2022062806/56649cba5503460f94981d5e/html5/thumbnails/12.jpg)
Q) What’s the physical sense of this structure?or, error of the paraxial
approximation?
12
Higher order modes
or, due to some assumption?
![Page 13: Error of paraxial approximation T. Agoh 2010.11.8, KEK CSR impedance below cutoff frequency Not important in practical applications](https://reader036.vdocument.in/reader036/viewer/2022062806/56649cba5503460f94981d5e/html5/thumbnails/13.jpg)
Curvature factor
Conventional parabolic equation
Modified parabolic equation
radiation condition
13
![Page 14: Error of paraxial approximation T. Agoh 2010.11.8, KEK CSR impedance below cutoff frequency Not important in practical applications](https://reader036.vdocument.in/reader036/viewer/2022062806/56649cba5503460f94981d5e/html5/thumbnails/14.jpg)
(+)
( C )
36% error
14Imaginary impedance in a perfectly conducting pipe
conventional equation
modified equation
resistive
wall
( C )
exact limit expression
(+)
![Page 15: Error of paraxial approximation T. Agoh 2010.11.8, KEK CSR impedance below cutoff frequency Not important in practical applications](https://reader036.vdocument.in/reader036/viewer/2022062806/56649cba5503460f94981d5e/html5/thumbnails/15.jpg)
15Real impedance in a perfectly conducting pipe
conventional
modified
resistive
wall
![Page 16: Error of paraxial approximation T. Agoh 2010.11.8, KEK CSR impedance below cutoff frequency Not important in practical applications](https://reader036.vdocument.in/reader036/viewer/2022062806/56649cba5503460f94981d5e/html5/thumbnails/16.jpg)
CSR in a resistive pipe16
Imaginary part
Real part
The difference is almost buried in the resistive wall impedance.
somewhat different (~20%) around the cutoff wavenumber
conventional
modified
modified
conventional
resistive
wallresistive wall
![Page 17: Error of paraxial approximation T. Agoh 2010.11.8, KEK CSR impedance below cutoff frequency Not important in practical applications](https://reader036.vdocument.in/reader036/viewer/2022062806/56649cba5503460f94981d5e/html5/thumbnails/17.jpg)
Summary
The imaginary impedance for this parabolic equation has better behavior than the conventional equation.
However, the error is still large (~40%) below the cutoff wavenumber, compared to the exact equation.
The impedance has strange structure in transient state, though the picture is not clear.
17
Modified parabolic equation