1. LCLS II Subharmonic Beam Loading G. A. Krafft

2. Subharmonic Beam Loading Under condition of constant incident RF power, there is a voltage fluctuation in the fundamental accelerating mode when the beam load is sub harmonically related to the cavity frequency Have some old results, from the days when we investigated FELs in the CEBAF accelerator These results can be used to quantify the voltage fluctuations expected from the subharmonic beam load in LCLS II

3. CEBAF FEL Results Krafft and Laubach, CEBAF-TN-0153 (1989) Bunch repetition time τ 1 chosen to emphasize physics

4. Model Single standing wave accelerating mode. Reflected power absorbed by matched circulator. Beam current (Constant) Incident RF (β coupler coupling)

5. Analytic Method of Solution Green function Geometric series summation Excellent approximation

6. Phasor Diagram of Solution

7. Single Subharmonic Beam

8. Beam Cases Case 1 Case 2 Case 3 BeamBeam Pulse Rep. RateBunch Charge (pC)Average Current (µA) HXR1 MHz145 Straight10 kHz1451.45 SXR1 MHz145 BeamBeam Pulse Rep. RateBunch Charge (pC)Average Current (µA) HXR1 MHz295 Straight10 kHz2952.95 SXR100 kHz202 BeamBeam Pulse Rep. RateBunch Charge (pC)Average Current (µA) HXR100 kHz295 Straight10 kHz2952.95 SXR100 kHz202

9. Case 1 Straight Current HXR Current SXR Current Total Voltage 2 µsec 100 µsec

10. Case 1 Volts t/20 nsec... ΔE/E=10 -4

11. Case 2 Volts t/20 nsec... ΔE/E=10 -4

12. Case 3 t/20 nsec Volts ΔE/E=10 -4

13. Summaries of Beam Energies Case 1 Case 2 Case 3 BeamMinimum (kV)Maximum (kV)Form HXR0.6901.814Linear 10 Straight1.574 Constant SXR0.7531.876Linear 10 BeamMinimum (kV)Maximum (kV)Form HXR0.6311.943Linear 100 Straight1.550 Constant SXR0.7881.911Linear 10 BeamMinimum (kV)Maximum (kV)Form HXR0.6151.222Linear 100 Straight0.908 Constant SXR0.6181.225Linear 100 For off-crest cavities, multiply by cos φ

14. Summary Fluctuations in voltage from constant intensity subharmonic beams can be computed analytically Basic character is a series of steps at bunch arrival, the step magnitude being (R/Q)πf c q Energy offsets were evaluated for some potential operating scenarios. Spread sheet provided that can be used to investigate differing current choices

15. Case I’ Zero Crossing Left Deflection Right Deflection Total Voltage 20 µsec 100 µsec

16. Case 2 (100 kHz contribution minor) Zero Crossing Left Deflection Right Deflection Total Voltage 2 µsec 100 µsec

17. Case 3 Zero Crossing Left Deflection Right Deflection Total Voltage 20 µsec 100 µsec