1. with Model Independent
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. 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
Coordinate system:
Quadrupole configuration (1-D): s y x 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. 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
Emittance:
Transverse rms beam size
Beta function (from earlier) gives information about amplitude and phase of Betatron oscillations
Emittance
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. Why Four Orbits? Remember: So if we have:
M is 4x4
x,y phase space vector is 4x1
We need four orbits to determine M
So if we have:
Then Mab is completely determined:

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. 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
Betatron
phase =
advance
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. Linear Green’s Functions
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
Betatron phase = advance

17. The Virtual Accelerator
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
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