1. Electromagnetic fields in a resistive cylindrical beam pipe
3rd Impedance meeting - N. Mounet and E. Métral - CERN/BE-ABP-LIS - 10/07/2009

2. Electromagnetic fields in a resistive cylindrical beam pipe
Analytic calculation from Bruno Zotter’s formalism
Position of the problem
Maxwell’s equations
 the issue on permittivity
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
3rd Impedance meeting - N. Mounet and E. Métral - CERN/BE-ABP-LIS - 10/07/2009

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) offseted from center.
In time domain:
In frequency domain:
and in each case
3rd Impedance meeting - N. Mounet and E. Métral - CERN/BE-ABP-LIS - 10/07/2009

4. Position of the problem
Azimuthal mode decomposition
Consider for the computation one mode at a time
3rd Impedance meeting - N. Mounet and E. Métral - CERN/BE-ABP-LIS - 10/07/2009

5. Maxwell’s equations and the issue on permittivity
In a linear medium without losses, in time domain
In r and J we need to put the free charges and currents (= the beam, here) but also bound charges and currents, coming from charges displacement in the conductive medium, due to the fields. 
3rd Impedance meeting - N. Mounet and E. Métral - CERN/BE-ABP-LIS - 10/07/2009

6. The issue on permittivity
From the continuity equation
3rd Impedance meeting - N. Mounet and E. Métral - CERN/BE-ABP-LIS - 10/07/2009

7. The issue on permittivity
Now in frequency domain
 We can define
3rd Impedance meeting - N. Mounet and E. Métral - CERN/BE-ABP-LIS - 10/07/2009

8. The issue on permittivity: connection with boundary conditions
the radial boundary conditions become
We know (see. E. Métral et al, Resistive-wall impedance of an infinitely long multilayer cylindrical beam pipe, PAC 2007) that for e.g. the mode m=0
and the tangential components of are continuous, so this can work only with and not with
3rd Impedance meeting - N. Mounet and E. Métral - CERN/BE-ABP-LIS - 10/07/2009

9. Conclusion on the permittivity issue
We should write
but the referee of the PRST-AB paper does not seem to agree…
3rd Impedance meeting - N. Mounet and E. Métral - CERN/BE-ABP-LIS - 10/07/2009

10. 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)
3rd Impedance meeting - N. Mounet and E. Métral - CERN/BE-ABP-LIS - 10/07/2009

11. Matrix formalism
It is possible to relate the constants of a layer to those of the previous one, finally getting or
3rd Impedance meeting - N. Mounet and E. Métral - CERN/BE-ABP-LIS - 10/07/2009

12. 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
3rd Impedance meeting - N. Mounet and E. Métral - CERN/BE-ABP-LIS - 10/07/2009

13. 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)
Courtesy of B. Salvant (existing code)
3rd Impedance meeting - N. Mounet and E. Métral - CERN/BE-ABP-LIS - 10/07/2009

14. Matrix formalism: conclusion
We are no longer limited to 3 layers (I tried 4 and it works fine, probably more is possible).
Computation of the transverse impedance is a factor of twenty or so faster than previously (which is not negligible in the case of three layers).
We might find some useful approximations for some specific multilayer pipe, and compute the impedance analytically (idea from S. Fartoukh).
Hahn and Ivanyan already thought about a similar matrix formalism (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).
3rd Impedance meeting - N. Mounet and E. Métral - CERN/BE-ABP-LIS - 10/07/2009

15. A first result in the first frequency regime
LHC graphite collimator transverse impedance
3rd Impedance meeting - N. Mounet and E. Métral - CERN/BE-ABP-LIS - 10/07/2009

16. 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
3rd Impedance meeting - N. Mounet and E. Métral - CERN/BE-ABP-LIS - 10/07/2009