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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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