Correcting Coupling in the PEP-II Low Energy Ring with Model Independent Analysis

1. Correcting Coupling in the PEP-II Low Energy Ring with Model Independent Analysis Lacey Kitch Mentor: Yiton Yan Beam Physics SULI 2007

2. SLAC’s PEP-II • HER (High Energy Ring) carries electrons at 9 GeV • LER (Low Energy Ring) carries positrons at 3.1 GeV • Luminosity: related to likelihood of collisions at IP (Interaction Point, BaBar) • Want to increase luminosity Luminosity • On the way to increasing luminosity, we want to decrease emittance (how well the particles are clumped in the beam) in LER • We decrease emittance by decreasing coupling in LER

3. y s x Storage Ring Optics • Dipole magnets bend the beam, but they bend particles of different energies by different amounts, and thus spread out the beam • Need beam focusing: Quadrupole magnets • focus in one transverse direction (x or y), defocus in the other • alternate focusing and defocusing in each direction Quadrupole configuration (1-D): Coordinate system: x Betatron motion s

4. Matrix Formulation of Betatron Motion • Each magnet and drift space can be represented by a matrix operation on a particle’s state in phase space Focusing magnet Defocusing magnet Outgoing particle Incoming particle Drift spaces length d

5. Linear Maps Matrices for each magnet and drift space can be combined to create a Linear Map, Mab, that describes the transition between any two points on the ring

6. Two-Dimensional Linear Maps • In reality, the quadrupole magnets are not perfect and there is coupling between x and y (x and y do not have independent linear maps) • M must actually be 4x4. It’s still factorable. The ideal, uncoupled parts are captured in A and R, which are block diagonal. The coupling is captured in C.

7. The Important Parts • Quadrupole (beam focusing) magnets cause Betatron motion, where the particles move kind of sinusoidally in x and y • This motion can be represented by linear maps, or matrix operators which transform the phase state vector at one point to the phase state vector at another point • The matrices can be factored into a coupling part and an uncoupled part phase space state vector Linear Map (4x4)

8. Emittance: Transverse rms beam size Beta function (from earlier) gives information about amplitude and phase of Betatron oscillations Emittance Coupling, Emittance, and Luminosity • Our ultimate goal: Increase PEP-II’s luminosity • Increase luminosity by making the particles in the beam clump together more, this making collisions more likely. This means decreasing emittance. • Emittance: constant area in phase space that beam occupies • x emittance and y emittance • x and x’ vary a lot (bending!), so x emittance is large • When x and y are coupled, this means y emittance is large Reducing coupling will reduce emittance, which will increase luminosity. If the beams match up….

9. We know about linear maps, and we know that we want to reduce coupling in order to reduce emittance in order to increase luminosity. How? • Well, we: • Study the current accelerator • Make a model of it • Tweak the model to find better magnet settings • Apply the better settings to the real machine

10. Step One: Take Data, Form Orbits • Take a DFT of the position data to find dominant (non-zero) mode • Each mode gives a sine-like and a cosine-like orbit • Two dominant modes (horizontal and vertical excitation) and two orbits per mode gives four orbits • Excite the beam along horizontal and vertical eigentunes • Take 1024 turns worth of Beam Position Monitor (BPM) data from the 319 BPM sites in the LER

11. The Four Orbits

12. Remember: So if we have: Then Mab is completely determined: Why Four Orbits? • M is 4x4 • x,y phase space vector is 4x1 • We need four orbits to determine M

13. Step Two: Calculate Green’s Functions and Phase Advances • In order to find the best-fitting Virtual Model, we calculate some quantities from the orbit data and use them to fit a virtual model. The quantities are: Green’s Functions • Four matrix elements of M • Exist between any two BPM pairs (a,b), just like M • Green’s functions in BPM frame (ie, as measured) have measurement errors (BPM gain and cross-coupling) • Use BPM gain and cross-coupling to calculate real Green’s functions

14. Betatron phase = advance Step Two: Calculate Green’s Functions and Phase Advances • In order to find the best-fitting Virtual Model, we calculate some quantities from the orbit data and use them to fit a virtual model. The quantities are: Phase Advances • Phase advance describes where in the Betatron oscillation the particle lies

15. Step Three: Fit a Virtual Accelerator to the Data Phase advances and Green’s functions of Virtual Machine BPM gain and cross-coupling Phase Advances and Green’s functions as derived from measurement • Initialize x to some previously measured machine values • Calculate what change in x would bring Y closer to Ym • Update x, repeat • The end result: a set of machine parameters (BPM gain and cross-coupling, from which we can calculate other things we want) which fits the data well. This fitted model is the Virtual Machine.

16. Betatron phase = advance Step Two: Fit a Virtual Accelerator to the Data Phase advances and Green’s functions of Virtual Machine BPM gain and cross-coupling Phase Advances and Green’s functions as derived from measurement Linear Green’s Functions Phase Advances

17. Results: The Virtual Accelerator Our target for improvement: coupling Blue is the Ideal Accelerator Lattice; Red is the Virtual Accelerator. We want to reduce coupling and make the accelerator closer to the ideal lattice.

18. How we know that our Virtual Accelerator is a good model of the real accelerator Red: Data Blue: Our model We didn’t use coupling as a fitting parameter, but it fits. So our accelerator must be a good model.

19. A Strange Result: High BPM gains in a particular region of ring BPM gain: abnormal BPM cross-coupling: normal

20. Step Four: Find a solution that will reduce coupling • A solution is a set of magnet strength adjustments • We wanted a solution which would reduce coupling • We pick some magnets to use as variables, assign weights to various parameters (beta functions, coupling, tune, etc.), and then try to fit a solution • We found three solutions (red: virtual machine; dark blue: solution we picked)

21. Sidenote: Magnets for Adjustment • Global and Local Skew Quadrupoles • Trombone Quadrupoles • Not before messed with: Symmetric Y sextupole bumps • Sextupoles provide second order corrections, so when the beam is separated from the sextupoles by a distance x0, an effect that is linear in x is produced. We adjust the sextupoles to get at this linear effect in order to fix coupling. Cancel by design Linear!

22. Step Four: Find a solution that will reduce coupling • A solution is a set of magnet strength adjustments • We wanted a solution which would reduce coupling • We pick some magnets to use as variables, assign weights to various parameters (beta functions, coupling, tune, etc.), and then try to fit a solution. GOAL: reduce coupling while leaving parameters at IP the same so that LER will still match HER • We found three solutions (red: virtual machine; dark blue: solution we picked)

23. Predicted Improvement in Coupling

24. Step Five: Dial Solution into Real Accelerator • Turn our magnet strength adjustments into a digital knob that will slowly adjust the strengths as desired • Partway through, we decided to stop at 50% of our calculated solution

25. Coupling Improvement After Dial-In

26. Conclusion • Summary of what we did: • Took BPM data; Fourier Transformed it to get modes and orbits • Calculated Green’s functions and phase advances from the orbits and used them to fit a Virtual Model which matched the real machine • Found a solution which adjusted magnet strengths in the virtual machine to improve coupling • Dialed solution into the real machine • We succeeded in improving coupling! • Luminosity increase may come after lots of adjusting to make the HER match up with the LER (the complicated job of the operators, or another MIA analysis on the HER)

27. Thanks! • Yiton Yan • Tom Knight • Franz-Josef Decker, Uli Wienands, Gerald Yocky, Martin Donald, Walter Wittmer, Michael Sullivan • Steve Rock, Mike Woods, and Farah Rahbar