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