1. Electromagnetic fields in a resistive cylindrical beam pipe
4th Impedance meeting - N. Mounet and E. Métral - CERN/BE-ABP-LIS - 26/08/2009

2. Electromagnetic fields in a resistive cylindrical beam pipe
Analytic calculation from Zotter & Métral’s formalism
Position of the problem
Maxwell’s equations
General calculation of the field components in frequency domain
Field matching to get the constants
 multilayer case (matrix formalism)
Force and impedance computation (summing all the azimuthal modes)
Approximations in the single-layer/good conductor case
 impedance in the first frequency regime
Back to the time domain
 Wake field computation (new FT technique): answer to V. Kornilov question on causality violiation for the wake before the bunch
4th Impedance meeting - N. Mounet and E. Métral - CERN/BE-ABP-LIS - 26/08/2009

3. Position of the problem
Beam circulating at speed v in a cylindrical pipe made of any multilayer linear medium.
The beam is a macroparticle (charge Q) offsetted from center.
In time domain:
In frequency domain:
and in each case
4th Impedance meeting - N. Mounet and E. Métral - CERN/BE-ABP-LIS - 26/08/2009

4. Position of the problem
Azimuthal mode decomposition
Consider for the computation one mode at a time
4th Impedance meeting - N. Mounet and E. Métral - CERN/BE-ABP-LIS - 26/08/2009

5. Multilayer field matching: Matrix formalism
Longitudinal field components in each layer
We want to determine the constants in front of the Bessel’s functions from field matching, going as far as possible in the analytic process (to avoid numerical problems later)
4th Impedance meeting - N. Mounet and E. Métral - CERN/BE-ABP-LIS - 26/08/2009

6. Matrix formalism
It is possible to relate the constants of a layer to those of the previous one, finally getting or
4th Impedance meeting - N. Mounet and E. Métral - CERN/BE-ABP-LIS - 26/08/2009

7. Matrix formalism In the end (and something similar for aTE)
 We get everything from a 4x4 matrix which is the product of N-1 (relatively) simple 4x4 matrices
4th Impedance meeting - N. Mounet and E. Métral - CERN/BE-ABP-LIS - 26/08/2009

8. Comparison with existing exact 2-layers formula and L
Comparison with existing exact 2-layers formula and L. Vos’s thin wall formula
For 2 layers we can compare with Elias’s formula, and the thin wall with inductive bypass one, when the pipe thickness is very small (with respect to its radius)
450 Gev, stainless steel, radius = 20 mm, thickness = 0.02mm
4th Impedance meeting - N. Mounet and E. Métral - CERN/BE-ABP-LIS - 26/08/2009

9. Comparison with existing multilayer code (from Benoît Salvant)
For 3 layers (the code is still Mathematica® in symbolic, the only difference is in the way to compute aTM)
Copper coated LHC graphite collimator (450 GeV)
Courtesy of B. Salvant (existing code)
4th Impedance meeting - N. Mounet and E. Métral - CERN/BE-ABP-LIS - 26/08/2009

10. Comparison with other multilayer formalisms
For 3 layers (copper coated LHC graphite collimator at 450 GeV)
Close agreement, except:
At very high frequency with BL
At low frequency (real part) with LV (at high freq., numerical pb – we did not use Mathematica)
4th Impedance meeting - N. Mounet and E. Métral - CERN/BE-ABP-LIS - 26/08/2009

11. Some multilayered examples
Calculation from 1 up to 5 layers (LHC graphite collimator at 450 GeV)
The biggest difference is between graphite alone and copper coated graphite
4th Impedance meeting - N. Mounet and E. Métral - CERN/BE-ABP-LIS - 26/08/2009

12. Matrix formalism: conclusion
We are no longer limited to 3 layers (we computed up to 5 layers).
Computation of the transverse impedance is about 130 faster than previously, in the case of three layers (computation time of a few hours is replaced by a few minutes).
We might find some useful approximations for some specific multilayer pipe, and derive simple analytical formula for the impedance (idea from S. Fartoukh).
Hahn and Ivanyan already thought about a similar matrix method (in longitudinal, see H. Hahn, Matrix solution to longitudinal impedance of multi-layer circular structures, BNL, C-A/AP/#336, more generally see M. Ivanyan et al, Multilayer tube impedance and external radiation, PRST-AB 2008), but not in the same formalism.
4th Impedance meeting - N. Mounet and E. Métral - CERN/BE-ABP-LIS - 26/08/2009

13. A first result in the first frequency regime
LHC graphite collimator transverse impedance
4th Impedance meeting - N. Mounet and E. Métral - CERN/BE-ABP-LIS - 26/08/2009

14. Real part of the transverse impedance in the first frequency regime
From Elias Métral’s transverse impedance formula (developing the Bessel’s function up to second order – the first order giving the inductive bypass limit) we get at low frequencies
d : skin depth
 proportional to w ln(w), and to s ,
 what matters is d instead of b
4th Impedance meeting - N. Mounet and E. Métral - CERN/BE-ABP-LIS - 26/08/2009

15. Answer to V. Kornilov question
“Where is the limit due to causality on a wake field plot before the bunch, i.e. where is the point after which it must be zero?”
4th Impedance meeting - N. Mounet and E. Métral - CERN/BE-ABP-LIS - 26/08/2009

16. Answer to V. Kornilov question
Actually, nowhere !
Assuming infinite length, there always exists a time interval Dt such that at t-Dt the travelling source has emitted light that can arrive (after interaction with the wall) at the test point at t, no matter how v is near the speed of light and how far the test point is before the bunch. Only for v=c the light cannot catch up with the beam and then there cannot be any EM field before the bunch.
Pipe wall v
Source at t
Source at t-Dt
Light (speed c>v)
Test point (at t)
4th Impedance meeting - N. Mounet and E. Métral - CERN/BE-ABP-LIS - 26/08/2009

17. Answer to V. Kornilov question
From Physics of Intensity Dependent Instabilities, Lecture Notes by K. Y. Ng USPAS, Los Angeles, January 2002
4th Impedance meeting - N. Mounet and E. Métral - CERN/BE-ABP-LIS - 26/08/2009