Betatron Motion with Coupling of Horizontal and Vertical

Betatron Motion with Coupling of Horizontal and Vertical Degrees of Freedom – Part I Alex Bogacz, Geoff Krafft and Timofey Zolkin Thomas Jefferson National Accelerator Facility Operated by JSA for the U. S. Department of Energy Lecture 7 - Coupled Betatron Motion I USPAS, Fort Collins, CO, June 10 -21, 2013 1

Outline Introduction Equations of Motion, Symplecticity and Eigen-vectors and Particle Ellipsoid in 4 D Phase Space Generalized Twiss Functions Derivatives of Tunes and Beta-Functions – 4 D 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, http: //dx. doi. org/10. 1088/1748 -0221/5/10/P 10010 Thomas Jefferson National Accelerator Facility Operated by JSA for the U. S. Department of Energy Lecture 7 - Coupled Betatron Motion I USPAS, Fort Collins, CO, June 10 -21, 2013 2

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) Thomas Jefferson National Accelerator Facility Operated by JSA for the U. S. Department of Energy Lecture 7 - Coupled Betatron Motion I USPAS, Fort Collins, CO, June 10 -21, 2013 3

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 2 D betatron motion) Thomas Jefferson National Accelerator Facility Operated by JSA for the U. S. Department of Energy Lecture 7 - Coupled Betatron Motion I USPAS, Fort Collins, CO, June 10 -21, 2013 4

Unresolved issues for both parametrizations Quest for versatile representation conveniently describing both storage rings and transfer lines 2 D emittances - how are they related to the 4 D beam emittance? How to determine the beam emittances and the generalized Twiss parameters from the particle beam ellipsoid (bilinear form), and from the secondorder moments of the particle distribution? Thomas Jefferson National Accelerator Facility Operated by JSA for the U. S. Department of Energy Lecture 7 - Coupled Betatron Motion I USPAS, Fort Collins, CO, June 10 -21, 2013 5

Equations of Motion and Symplecticity Condition Thomas Jefferson National Accelerator Facility Operated by JSA for the U. S. Department of Energy Lecture 7 - Coupled Betatron Motion I USPAS, Fort Collins, CO, June 10 -21, 2013 6

Hamiltonian formulation - equations of motion Hamiltonian matrix: Unit symplectic matrix : Thomas Jefferson National Accelerator Facility Operated by JSA for the U. S. Department of Energy Lecture 7 - Coupled Betatron Motion I USPAS, Fort Collins, CO, June 10 -21, 2013 7

Hamiltonian formulation - equations of motion Thomas Jefferson National Accelerator Facility Operated by JSA for the U. S. Department of Energy Lecture 7 - Coupled Betatron Motion I USPAS, Fort Collins, CO, June 10 -21, 2013 8

Hamiltonian formulation - equations of motion Thomas Jefferson National Accelerator Facility Operated by JSA for the U. S. Department of Energy Lecture 7 - Coupled Betatron Motion I USPAS, Fort Collins, CO, June 10 -21, 2013 9

Hamiltonian formulation - Sympecticity Thomas Jefferson National Accelerator Facility Operated by JSA for the U. S. Department of Energy Lecture 7 - Coupled Betatron Motion I USPAS, Fort Collins, CO, June 10 -21, 2013 10

Eigen-vectors Thomas Jefferson National Accelerator Facility Operated by JSA for the U. S. Department of Energy Lecture 7 - Coupled Betatron Motion I USPAS, Fort Collins, CO, June 10 -21, 2013 11

Eigen-vectors Thomas Jefferson National Accelerator Facility Operated by JSA for the U. S. Department of Energy Lecture 7 - Coupled Betatron Motion I USPAS, Fort Collins, CO, June 10 -21, 12

Eigen-vectors Thomas Jefferson National Accelerator Facility Operated by JSA for the U. S. Department of Energy Lecture 7 - Coupled Betatron Motion I USPAS, Fort Collins, CO, June 10 -21, 2013 13

Eigen-vectors and Particle Ellipsoid in 4 D Space Thomas Jefferson National Accelerator Facility Operated by JSA for the U. S. Department of Energy Lecture 7 - Coupled Betatron Motion I USPAS, Fort Collins, CO, June 10 -21, 2013 14

Eigen-vectors and Particle Ellipsoid in 4 D Space Thomas Jefferson National Accelerator Facility Operated by JSA for the U. S. Department of Energy Lecture 7 - Coupled Betatron Motion I USPAS, Fort Collins, CO, June 10 -21, 2013 15

Eigen-vectors and Particle Ellipsoid in 4 D Space Thomas Jefferson National Accelerator Facility Operated by JSA for the U. S. Department of Energy Lecture 7 - Coupled Betatron Motion I USPAS, Fort Collins, CO, June 10 -21, 2013 16

Beam emittance 4 -D Thomas Jefferson National Accelerator Facility Operated by JSA for the U. S. Department of Energy Lecture 7 - Coupled Betatron Motion I USPAS, Fort Collins, CO, June 10 -21, 2013 17

Beam emittance 4 -D Thomas Jefferson National Accelerator Facility Operated by JSA for the U. S. Department of Energy Lecture 7 - Coupled Betatron Motion I USPAS, Fort Collins, CO, June 10 -21, 2013 18

Beam emittance 4 -D Thomas Jefferson National Accelerator Facility Operated by JSA for the U. S. Department of Energy Lecture 7 - Coupled Betatron Motion I USPAS, Fort Collins, CO, June 10 -21, 2013 19

Second order moments of the particle distribution Thomas Jefferson National Accelerator Facility Operated by JSA for the U. S. Department of Energy Lecture 7 - Coupled Betatron Motion I USPAS, Fort Collins, CO, June 10 -21, 2013 20

Beam emittance 4 -D Thomas Jefferson National Accelerator Facility Operated by JSA for the U. S. Department of Energy Lecture 7 - Coupled Betatron Motion I USPAS, Fort Collins, CO, June 10 -21, 2013 21

Beam emittance 4 -D Thomas Jefferson National Accelerator Facility Operated by JSA for the U. S. Department of Energy Lecture 7 - Coupled Betatron Motion I USPAS, Fort Collins, CO, June 10 -21, 2013 22

Beam emittance 4 -D Thomas Jefferson National Accelerator Facility Operated by JSA for the U. S. Department of Energy Lecture 7 - Coupled Betatron Motion I USPAS, Fort Collins, CO, June 10 -21, 2013 23

Beam emittance 4 -D Thomas Jefferson National Accelerator Facility Operated by JSA for the U. S. Department of Energy Lecture 7 - Coupled Betatron Motion I USPAS, Fort Collins, CO, June 10 -21, 2013 24

Twiss Functions for Coupled 2 D Motion Thomas Jefferson National Accelerator Facility Operated by JSA for the U. S. Department of Energy Lecture 7 - Coupled Betatron Motion I USPAS, Fort Collins, CO, June 10 -21, 2013 25

Twiss Functions for Coupled 2 D Motion Thomas Jefferson National Accelerator Facility Operated by JSA for the U. S. Department of Energy Lecture 7 - Coupled Betatron Motion I USPAS, Fort Collins, CO, June 10 -21, 2013 26

Twiss Functions for Coupled 2 D Motion Thomas Jefferson National Accelerator Facility Operated by JSA for the U. S. Department of Energy Lecture 7 - Coupled Betatron Motion I USPAS, Fort Collins, CO, June 10 -21, 2013 27

Twiss Functions for Coupled 2 D Motion Thomas Jefferson National Accelerator Facility Operated by JSA for the U. S. Department of Energy Lecture 7 - Coupled Betatron Motion I USPAS, Fort Collins, CO, June 10 -21, 2013 28

Twiss Functions for Coupled 2 D Motion Thomas Jefferson National Accelerator Facility Operated by JSA for the U. S. Department of Energy Lecture 7 - Coupled Betatron Motion I USPAS, Fort Collins, CO, June 10 -21, 2013 29

Twiss Functions for Coupled 2 D Motion Thomas Jefferson National Accelerator Facility Operated by JSA for the U. S. Department of Energy Lecture 7 - Coupled Betatron Motion I USPAS, Fort Collins, CO, June 10 -21, 2013 30

Twiss Functions for Coupled 2 D Motion Thomas Jefferson National Accelerator Facility Operated by JSA for the U. S. Department of Energy Lecture 7 - Coupled Betatron Motion I USPAS, Fort Collins, CO, June 10 -21, 2013 31

Beam sizes Thomas Jefferson National Accelerator Facility Operated by JSA for the U. S. Department of Energy Lecture 7 - Coupled Betatron Motion I USPAS, Fort Collins, CO, June 10 -21, 2013 32

Derivatives of Tunes and Beta-Functions Thomas Jefferson National Accelerator Facility Operated by JSA for the U. S. Department of Energy Lecture 7 - Coupled Betatron Motion I USPAS, Fort Collins, CO, June 10 -21, 2013 33

Derivatives of Tunes and Beta-Functions Thomas Jefferson National Accelerator Facility Operated by JSA for the U. S. Department of Energy Lecture 7 - Coupled Betatron Motion I USPAS, Fort Collins, CO, June 10 -21, 2013 34

Transfer Matrix in terms of Twiss Functions Thomas Jefferson National Accelerator Facility Operated by JSA for the U. S. Department of Energy Lecture 7 - Coupled Betatron Motion I USPAS, Fort Collins, CO, June 10 -21, 2013 35

Transfer Matrix in terms of Twiss Functions Thomas Jefferson National Accelerator Facility Operated by JSA for the U. S. Department of Energy Lecture 7 - Coupled Betatron Motion I USPAS, Fort Collins, CO, June 10 -21, 2013 36

Transfer Matrix in terms of Twiss Functions Thomas Jefferson National Accelerator Facility Operated by JSA for the U. S. Department of Energy Lecture 7 - Coupled Betatron Motion I USPAS, Fort Collins, CO, June 10 -21, 2013 37

Transfer Matrix in terms of Twiss Functions Thomas Jefferson National Accelerator Facility Operated by JSA for the U. S. Department of Energy Lecture 7 - Coupled Betatron Motion I USPAS, Fort Collins, CO, June 10 -21, 2013 38

Beam ellipsoid in 4 D space - bilinear form Thomas Jefferson National Accelerator Facility Operated by JSA for the U. S. Department of Energy Lecture 7 - Coupled Betatron Motion I USPAS, Fort Collins, CO, June 10 -21, 2013 39

Beam ellipsoid in 4 D space - bilinear form Thomas Jefferson National Accelerator Facility Operated by JSA for the U. S. Department of Energy Lecture 7 - Coupled Betatron Motion I USPAS, Fort Collins, CO, June 10 -21, 2013 40

Beam ellipsoid in 4 D space - bilinear form Thomas Jefferson National Accelerator Facility Operated by JSA for the U. S. Department of Energy Lecture 7 - Coupled Betatron Motion I USPAS, Fort Collins, CO, June 10 -21, 2013 41

Second order moments in terms of Twiss functions Thomas Jefferson National Accelerator Facility Operated by JSA for the U. S. Department of Energy Lecture 7 - Coupled Betatron Motion I USPAS, Fort Collins, CO, June 10 -21, 2013 42

Second order moments in terms of Twiss functions Thomas Jefferson National Accelerator Facility Operated by JSA for the U. S. Department of Energy Lecture 7 - Coupled Betatron Motion I USPAS, Fort Collins, CO, June 10 -21, 2013 43

Summary Thomas Jefferson National Accelerator Facility Operated by JSA for the U. S. Department of Energy Lecture 7 - Coupled Betatron Motion I USPAS, Fort Collins, CO, June 10 -21, 2013 44