1. Integrable Optics Test Accelerator (IOTA) physics goals S. Nagaitsev (FNAL) February 23, 2012

2. Motivations  Physics (academic interest)  General physics  Accelerators  Practical (potential outcomes)  improved collider schemes  increased Landau damping  Educational  many PhD and postdoc research topics  opportunities to collaborate with Universities  Playing from Fermilab strength  we have unique expertise in beam manipulations and beam cooling  no competition in this area of physics, a lot of interest…  NML capabilities IOTA Physics Goals - Nagaitsev 2

3. IOTA goals  Construct and commission an electron storage ring, IOTA, designed to conduct accelerator research.  Using the IOTA beam:  Advance understanding of strong nonlinear effects generated by an electron lens  Demonstrate large (~1) tune shift with external nonlinear magnets  Demonstrate the feasibility of optical stochastic cooling IOTA Physics Goals - Nagaitsev 3

4. 4 Original NML building Photoinjector and low energy test beamlines 1 to 6 cryomodules High energy test beamlines New tunnel extension IOTA storage ring 75 meters

5. Beam lines layout IOTA Physics Goals - Nagaitsev 5

6. IOTA layout  pc = 150 MeV, electrons (single bunch, 10^9)  ~36 m circumference  50 quadrupoles, 8 dipoles, 50-mm diam vac chamber  hor and vert kickers, 16 BPMs IOTA Physics Goals - Nagaitsev 6 Nonlinear inserts Injection, rf cavity Stochastic cooling insert

7. Why electrons?  Small size (~50 um), pencil beam  Reasonable damping time (~1 sec)  No space charge IOTA Physics Goals - Nagaitsev 7

8. Prior Art  IUCF Cooler Ring: experiments CE-22 and CE-48 (DOE-funded, 1991-1998)  Nonlinear beam dynamics, study of 1D and 2D resonances and chaos  45 MeV protons (pc = 300 MeV) pencil beam (electron cooled)  using kickers and BPMs studied phase-space trajectories IOTA Physics Goals - Nagaitsev 8

9. What’s new?  IUCF Cooler experiments studied “natural” ring nonlinearites.  Mostly linear system and a single isolated resonance  No attempt was made to make the system highly nonlinear and integrable  We are proposing to construct a dedicated ring to study nonlinear focusing with an aim to develop a system either without or with very weak resonances.  By design, the ring has 1-4 locations for nonlinear elements, separated by regular (linear) focusing elements providing special optical transfer matrices. IOTA Physics Goals - Nagaitsev 9

10. Proposed experiments  We are proposing 4 different experiments with nonlinear lenses  2 with an electron lens  2 with special electromagnets  In all experiments the electron bunch is kicked transversely to “sample” nonlinearities. We intend to measure the turn-by-turn BPM positions as well as synch light to obtain information about phase space trajectories.  We are proposing 2 experiments with optical stochastic cooling (OSC)  2-um optical band -- without an optical amplifier  6-um optical band – with and without an optical amplifier (future)  Both OSC experiments use same hardware IOTA Physics Goals - Nagaitsev 10

11. Experiments with electron lens  For IOTA ring, we would need a 5-kG, ~1-m long solenoid  Electron beam: ~0.5 A, ~5 keV, ~1 mm radius IOTA Physics Goals - Nagaitsev 11 Example: Tevatron electron lens 150 MeV beam solenoid

12. Experiment 1: thin electron lens  The system consists of a thin nonlinear lens (electron beam) and a linear focusing ring  Axially-symmetric thin lens:  Electron lens with a special density profile  The ring has the following transfer matrix IOTA Physics Goals - Nagaitsev 12 electron lens

13. Experiment 1:  The system is integrable. Two integrals of motion (transverse):  Angular momentum:  McMillan-type integral, quadratic in momentum  For large amplitudes, the fractional tune is 0.25  For small amplitude, the electron (defocusing) lens can give a tune shift of ~-0.3  Potentially, can cross an integer resonance IOTA Physics Goals - Nagaitsev 13

14. Experiment 1  Goal 1: demonstrate a tune shift of 0.3  Goal 2: study effects of integer resonance crossing  Goal 3: quantify effects of a non-ideal lens IOTA Physics Goals - Nagaitsev 14

15. Experiment 2: thick electron lens  Suppose that the linear optical system consists of two portions:  A section with a constant beta function. For a 150-MeV electron beam, a solenoid 5-kG gives 2-m beta functions (constant over the solenoid length, L)  The rest of the ring has a phase advance of nπ (+ arb. rotation)  The fractional tunes are: IOTA Physics Goals - Nagaitsev 15

16. Optics example IOTA Physics Goals - Nagaitsev 16 Solenoid “Ring” “Ring” has the matrix: One turn

17. Experiment 2  Now, add an axially symmetric electron lens (length L)  Achievable linear tune shift for small amplitudes:  At large amplitudes (larger than electron lens radius): IOTA Physics Goals - Nagaitsev 17 Electron lens

18. Experiment 2  The system is integrable. Two integrals of motion (transverse):  Angular momentum:  The total transverse energy (Hamiltonian) in the solenoid section.  A very interesting case is near (just above) the integer resonance  The system can lose linear (small amplitude) stability but retain the large amplitude stability IOTA Physics Goals - Nagaitsev 18

19. Experiment 2  Goal 1: demonstrate a tune shift of 0.3  Goal 2: study effects of integer resonance crossing  Goal 3: quantify effects of a non-ideal lens IOTA Physics Goals - Nagaitsev 19

20. Nonlinear magnets See: Phys. Rev. ST Accel. Beams 13, 084002 Start with a round axially-symmetric LINEAR focusing lattice (FOFO) Add special non-linear potential V(x,y,s) such that IOTA Physics Goals - Nagaitsev 20 V(x,y,s) βx = βy

21. Fake thin lens inserts IOTA Physics Goals - Nagaitsev 21 V(x,y,s)

22. Time-dependent system IOTA Physics Goals - Nagaitsev 22 Let’s consider a Hamiltonian of this FOFO system: where V(x,y,s) satisfies the Laplace equation in 2d: In normalized variables we will have: Where new “time” variable is, s is “time” variable

23. Main ideas IOTA Physics Goals - Nagaitsev 23 1.Start with a time-dependent Hamiltonian: 2.Chose the potential to be time-independent in new variables 3.Find potentials U(x, y) with the second integral of motion and such that ΔU(x, y) = 0

24. Example with quadrupoles IOTA Physics Goals - Nagaitsev 24 quadrupoles: β(s) quadrupole amplitude L Tunes: Tune spread: zero Integrable but still linear…

25. Example with quadrupoles IOTA Physics Goals - Nagaitsev 25 Quadrupole strength (T/m) Quadrupoles set to 0 βx = βy βxβx βyβy q = 0 q = 0.45

26. Experiment 3: octupoles IOTA Physics Goals - Nagaitsev 26 This Hamiltonian is NOT integrable Tune spread (in both x and y) is limited to ~ 12% Octupoles:

27. Experiment 4: Integrable nonlinear focusing IOTA Physics Goals - Nagaitsev 27  Look for second integrals quadratic in momentum  All such potentials are separable in some variables (cartesian, polar, elliptic, parabolic)  First comprehensive study by Gaston Darboux (1901)  So, we are looking for integrable potentials such that Second integral:

28. Darboux equation IOTA Physics Goals - Nagaitsev 28  Let a ≠ 0 and c ≠ 0, then we will take a = 1  General solution ξ : [1, ∞], η : [-1, 1], f and g arbitrary functions

29. Laplace equation IOTA Physics Goals - Nagaitsev 29  Now we look for potentials that also satisfy the Laplace equation (in addition to the Darboux equation):  We found a family with 4 free parameters (b, c, d, t):

30. Examples of potentials IOTA Physics Goals - Nagaitsev 30  c – location of singularities on x-axis (x = +/- c)  t, b and d define the type of the potential  There are 3 possible types and (their combinations) t = 0, d =0t = 0, b =0 d =0 and Dipole-like Quad-like

31. Experiment 4: nonlinear lens IOTA Physics Goals - Nagaitsev 31 Multipole expansion (electrostatic case): For c = 1 |t| < 0.5 to provide linear stability for small amplitudes For t > 0 adds focusing in x Small-amplitude tune s: This potential has two adjustable parameters: t – strength and c – location of singularities For |z| < c

32. Transverse forces IOTA Physics Goals - Nagaitsev 32 FxFy Focusing Defocusing

33. Examples of trajectories IOTA Physics Goals - Nagaitsev 33

34. Experimets 3 & 4: Goals  Goal 1: demonstrate a tune shift of ~1  Goal 2: study effects of integer resonance crossing  Goal 3: quantify effects of a non-ideal lens  Goal 4: Develop a practical lens design for proton machines IOTA Physics Goals - Nagaitsev 34

35. Summary  We have found first (practical) examples of completely integrable non-linear optics.  We have explored these ideas with modeling and tracking simulations.  We are now ready for a practical demonstration of these ideas. IOTA Physics Goals - Nagaitsev 35