1. Alex Bogacz, Geoff Krafft and Timofey Zolkin
Betatron Motion with Coupling of Horizontal and Vertical Degrees of Freedom – Part I
Alex Bogacz, Geoff Krafft and Timofey Zolkin
USPAS, Fort Collins, CO, June 10-21, 2013

2. Outline
Introduction
Equations of Motion, Symplecticity and Eigen-vectors
Eigen-vectors and Particle Ellipsoid in 4D Phase Space
Generalized Twiss Functions
Derivatives of Tunes and Beta-Functions – 4D Floque formulae
Second order moments in terms of generalized Twiss functions
V. Lebedev, A. Bogacz, ‘Betatron Motion with Coupling of Horizontal and Vertical Degrees of Freedom’, 2000,
USPAS, Fort Collins, CO, June 10-21, 2013

3. Introduction
Courant-Snyder representation for one-dimensional betatron motion
Simple relations between Twiss parameters, eigen-vectors and bilinear form for the particle ellipsoid
Symplecticity  22 – 1 = 3 parameters
From uncoupled to strongly coupled motion by design
“Moebius Twist Accelerator” to create round beams (Cornell)
Ionization cooling channel for Neutrino Factory and Muon Collider
Vertex to plane adapter for electron cooling (Fermilab)
USPAS, Fort Collins, CO, June 10-21, 2013

4. Two dimensional coupled betatron motion
Symplecticity  44 – 6 = 10 parameters
Effective parameterization in terms of generalized Twiss functions
Shortcomings of the existing representations
Edwards and Teng, Fermilab (1973)
Ambiguity of the rotation angle
Physical meaning of the betatron phase advance?
G. Ripken, et al., DESY (1987)
Oriented for circular accelerators
Incomplete parametrization (one needs 10 independent parameters to fully describe 2D betatron motion)
USPAS, Fort Collins, CO, June 10-21, 2013

5. Unresolved issues for both parametrizations
Quest for versatile representation conveniently describing both storage rings and transfer lines
2D emittances - how are they related to the 4D beam emittance?
How to determine the beam emittances and the generalized Twiss parameters from the particle beam ellipsoid (bilinear form), and from the second-order moments of the particle distribution?
USPAS, Fort Collins, CO, June 10-21, 2013

6. Equations of Motion and Symplecticity Condition
USPAS, Fort Collins, CO, June 10-21, 2013

7. Hamiltonian formulation - equations of motion
Hamiltonian matrix:
Unit symplectic matrix :
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8. Hamiltonian formulation - equations of motion
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9. Hamiltonian formulation - equations of motion
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10. Hamiltonian formulation - Sympecticity
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11. Eigen-vectors
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12. Eigen-vectors
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13. Eigen-vectors
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14. Eigen-vectors and Particle Ellipsoid in 4D Space
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15. Eigen-vectors and Particle Ellipsoid in 4D Space
USPAS, Fort Collins, CO, June 10-21, 2013

16. Eigen-vectors and Particle Ellipsoid in 4D Space
USPAS, Fort Collins, CO, June 10-21, 2013

17. Beam emittance 4-D
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18. Beam emittance 4-D
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19. Beam emittance 4-D
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20. Second order moments of the particle distribution
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21. Beam emittance 4-D
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22. Beam emittance 4-D
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23. Beam emittance 4-D
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24. Beam emittance 4-D
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25. Twiss Functions for Coupled 2D Motion
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26. Twiss Functions for Coupled 2D Motion
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27. Twiss Functions for Coupled 2D Motion
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28. Twiss Functions for Coupled 2D Motion
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29. Twiss Functions for Coupled 2D Motion
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30. Twiss Functions for Coupled 2D Motion
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31. Twiss Functions for Coupled 2D Motion
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32. Beam sizes
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33. Derivatives of Tunes and Beta-Functions
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34. Derivatives of Tunes and Beta-Functions
USPAS, Fort Collins, CO, June 10-21, 2013

35. Transfer Matrix in terms of Twiss Functions
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36. Transfer Matrix in terms of Twiss Functions
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37. Transfer Matrix in terms of Twiss Functions
USPAS, Fort Collins, CO, June 10-21, 2013

38. Transfer Matrix in terms of Twiss Functions
USPAS, Fort Collins, CO, June 10-21, 2013

39. Beam ellipsoid in 4D space - bilinear form
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40. Beam ellipsoid in 4D space - bilinear form
USPAS, Fort Collins, CO, June 10-21, 2013

41. Beam ellipsoid in 4D space - bilinear form
USPAS, Fort Collins, CO, June 10-21, 2013

42. Second order moments in terms of Twiss functions
USPAS, Fort Collins, CO, June 10-21, 2013

43. Second order moments in terms of Twiss functions
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44. Summary
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