Error of paraxial approximation

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

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

3. 3 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

4. 4 Generally speaking about field analysis, Eigenmode expansion Fourier, Laplace analysis Algebraic Analytic <abstract, symbolic> <concrete, graphic> compact & beautiful formalism straightforward formulation but often complex & intricate (orthonormality, completeness) 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

5. 5 e.g.) Field structure of steady CSR in a perfectly conducting rectangular pipe Imaginary poles Field in front of the bunch Regularity at the origin symmetry condition No pole nor branch point (byproduct) effect of sidewalls

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

7. 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.

8. 8 • Steady field a in perfectly conducting rectangular pipe (no periodicity) Asymptotic expansion contradiction Olver expansion (para-approx.) • Exact solution for periodic system Debye expansion no longer periodic

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

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

11. 11 Fluctuation of impedance at low frequency agreement with grid calculation

12. 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. 13 Conventional parabolic equation Modified parabolic equation radiation condition Curvature factor

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

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

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

17. 17 Summary • Modified parabolic equation • 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.