1. Ivan Bazarov, Single HOM two-pass analysis, SRF mtg, 4 June 2003 1 CHESS / LEPP Single HOM, two-pass analysis Motivation: explain the plot produced by BBU code bi R/Q = 100 Ohm Q = 10000 m 12 =  10 –6 m(c/eV)  = 2  2 GHz t 0 = (1.3 GHz) –1

2. Ivan Bazarov, Single HOM two-pass analysis, SRF mtg, 4 June 2003 2 CHESS / LEPP Frequency of the instability Dipole mode is excited by first current moment thru interaction with longitudinal field of the mode Infinite number of bunches with finite number of passes (as opposed to finite number of bunches with infinite number of passes for BBU in storage rings) Potentially, any frequency can be present in FT of the current moment for infinite delta-function current train Instability occurs with frequency close to that of HOM, where impedance is maximal FT

3. Ivan Bazarov, Single HOM two-pass analysis, SRF mtg, 4 June 2003 3 CHESS / LEPP Getting the master equation Sample solution: Summing geometric series

4. Ivan Bazarov, Single HOM two-pass analysis, SRF mtg, 4 June 2003 4 CHESS / LEPP Perturbative approach Solve the master equation for instability frequency treating K as small parameter: The frequency up to the first order in K: Requiring Im(  ) = 0 yields famous Problem 1: Half solution is missing Problem 2: Unphysical exponent

5. Ivan Bazarov, Single HOM two-pass analysis, SRF mtg, 4 June 2003 5 CHESS / LEPP Second order perturbative term The second order term is found to be: Im(  ) = 0 yields quadratic equation for the threshold current 1 st order 2 nd order Observation: Clearly, the other half of the solution is not a 2 nd order effect

6. Ivan Bazarov, Single HOM two-pass analysis, SRF mtg, 4 June 2003 6 CHESS / LEPP Complex current approach Solving master equation directly for current gives the following: Re(I 0 ) Im(I 0 ) solution space

7. Ivan Bazarov, Single HOM two-pass analysis, SRF mtg, 4 June 2003 7 CHESS / LEPP Max and min currents

8. Ivan Bazarov, Single HOM two-pass analysis, SRF mtg, 4 June 2003 8 CHESS / LEPP Obtaining complete first order solution In the following limit (HOM damping is small of timescale of t 0 ) (instability frequency shift is small compared to bunching frequency, or as seen later, equivalent to number of bunches in recirculating loop >> 1) Solving Im(I 0 ) = 0, yields the threshold and instability frequency Note:It’s sin(  t r ) not sin(  t r ) Unphysical exponent is gone.

9. Ivan Bazarov, Single HOM two-pass analysis, SRF mtg, 4 June 2003 9 CHESS / LEPP Linearized solutions for instability frequency Transcendental equation for instability frequency can be linearized for two important cases: look at solutions closest to 

10. Ivan Bazarov, Single HOM two-pass analysis, SRF mtg, 4 June 2003 10 CHESS / LEPP Comparison with tracking

11. Ivan Bazarov, Single HOM two-pass analysis, SRF mtg, 4 June 2003 11 CHESS / LEPP zoomed in  Solving Im(I 0 )=0 numerically with increasing t r

12. Ivan Bazarov, Single HOM two-pass analysis, SRF mtg, 4 June 2003 12 CHESS / LEPP Comparison of tracking with numeric solution of Im(I 0 )=0

13. Ivan Bazarov, Single HOM two-pass analysis, SRF mtg, 4 June 2003 13 CHESS / LEPP Large t r (  r <<  /2Q) t r does not matter as opposed to small accelerators case, threshold is approx. given by I in

14. Ivan Bazarov, Single HOM two-pass analysis, SRF mtg, 4 June 2003 14 CHESS / LEPP A word on quad HOM BBU coupling term Wake functions are identical in the form, except for the loss factor difference In the approximation that alignment error of cavity transverse position dominates and causes dipole-like BBU (b is beam pipe radius), i.e. ~ 2 orders of magnitude bigger