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Error of paraxial approximation T. Agoh 2010.11.8, KEK CSR impedance below cutoff frequency Not important in practical applications

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Wave equation in the frequency domain Curvature 2 Gauss’s law Parabolic equation cutoff frequency Condition

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3 Transient 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

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Generally speaking about field analysis, Eigenmode expansionFourier, Laplace analysis AlgebraicAnalytic 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 formalismstraightforward formulation but often complex & intricate (orthonormality, completeness) 4

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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 symmetry condition e.g.)

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Transient field of CSR inside magnet Transverse field Longitudinal field 6 s-dependence is only here

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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, frozen

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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.)

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(+) ( ̶ ) (+) Imaginary impedance 9 exact limit eq. paraxial approximation resistive wall transient, free steady, free cutoff frequencyshielding threshold transient, copper perfectly conducting pipe ( ̶ ) transient,

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Real impedance 10 resistive wall transient, free steady, free cutoff frequencyshielding threshold transient, copper transient, perfectly conducting pipe The lowest resonance strange structure

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Fluctuation of impedance at low frequency 11 agreement with grid calculation

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Q) What’s the physical sense of this structure? or, error of the paraxial approximation? 12 Higher order modes or, due to some assumption?

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Curvature factor Conventional parabolic equation Modified parabolic equation radiation condition 13

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(+) ( ̶ ) 36% error 14 Imaginary impedance in a perfectly conducting pipe conventional equation modified equation resistive wall ( ̶ ) exact limit expression (+)

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15 Real impedance in a perfectly conducting pipe conventional modified resistive wall

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CSR in a resistive pipe 16 Imaginary partReal part The difference is almost buried in the resistive wall impedance. somewhat different (~20%) around the cutoff wavenumber conventional modified conventional resistive wall

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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

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