Nonlinear Optics as a Path to High-Intensity Circular Machines S. Nagaitsev, A. Valishev (Fermilab), and V. Danilov (ORNL) Draft talk for HB2010 Sep 23, 2010 Report at HEAC 1971 1965, Priceton-Stanford CBX: First mention of an 8-pole CBX layout (1962) magnet Observed vertical resistive wall instability With octupoles, increased beam current from ~5 to 500 mA CERN PS: In 1959 had 10 octupoles; not used until 1968 At 1012 protons/pulse observed (1st time) head-tail instability. Octupoles helped. Once understood, chromaticity jump at transition was developed using sextupoles. More instabilities were discovered; helped by octupoles and by feedback. 2 How to make a high-intensity machine? (OR, how to make a high-intensity beam stable?) 1. Landau damping the beams immune system. It is related to the spread of betatron oscillation frequencies. The larger the spread, the more stable the beam is against collective instabilities. 2. External damping (feed-back) system

presently the most commonly used mechanism to keep the beam stable. Can not be used for some instabilities (headtail) Noise 3 Most accelerators rely on both LHC Has a transverse feedback system Has 336 Landau Damping Octupoles Provide tune spread of 0.001 at 1-sigma at injection In all machines there is a trade-off between Landau damping and dynamic aperture. But it does not have to be. 4 Todays talk will be about How to improve beams immune system (Landau damping through betatron frequency spread) Tune spread not ~0.001 but 10-50% What can be wrong with the

immune system? The main feature of all present accelerators particles have nearly identical betatron frequencies (tunes) by design. This results in two problems: I. II. Single particle motion can be unstable due to resonant perturbations (magnet imperfections and non-linear elements); Landau damping of instabilities is suppressed because the frequency spread is small. 5 Courant-Snyder Invariant z" K ( s ) z 0, z x or y Equation of motion for betatron oscillations Courant and Snyder found a conserved quantity: J where 2 1 2 ( s) z

z ( s ) z 2 ( s) 2 K ( s) 1 3 -- auxiliary equation H ( J x , J y ) x J x y J y J x , J y-- are Courant-Snyder integrals of motion H x J x y H -- betatron frequencies J y 6 Linear function of actions: good or bad? H ( J x , J y ) x J x y J y It is convenient (to have linear optics), easy to model,

but it is NOT good for stability. We did not know (until now) how to make it any other way! To create the tune spread, we add non-linear elements (octupoles) as best we can Destroys octupole integrability! Tune spread depends on a linear tune location 1-D system: Theoretical max. spread is 0.125 2-D system: Max. spread < 0.05 7 Long-term stability The first paper on the subject was written by Nikolay Nekhoroshev in 1971 He proved that for sufficiently small provided that H0(I) meets certain conditions know as steepness Convex and quasi-convex functions H0(I) are the steepest An example of a NON-STEEP function is a linear function H 0 ( I1 , I 2 ) 1I1 2 I 2 Another example of a NON-STEEP function is H 0 ( I1 , I 2 ) I12 I 22

8 Non-linear Hamiltonians We were looking for (and found) non-linear 2-D steep Hamiltonians that can be implemented in an accelerator Other authors worked on this subject: Yu. Orlov (1962-65), E. McMillan (1967-71), and recently, J. Carry et al., S. Danilov, E. Perevedentsev The problem in 2-D is that the fields of non-linear elements are coupled by the Laplace equation. An example H 0 (of I1 , Ia2 )steep 1 I12 (convex) 2 I 22 , 0 Hamiltonian is but we DO NOT know how to implement it with magnetic fields 9 What are we looking for? We are looking for a 2-D integrable convex H 0 ( I1 , I 2 ) h( I1 , I 2 ) non-linear Hamiltonian, -- convex curves h( I1 , I 2 ) const I2 I1 10

Our approach See: Phys. Rev. ST Accel. Beams 13, 084002 Start with a round axially-symmetric LINEAR focusing lattice (FOFO) V ( x, y , s ) potential V ( x, y ) V(x,y,s) 0 Add specialnon-linear such that V(x,y,s) V(x,y,s) V(x,y,s) 0 V(x,y,s) 0 BETA_X&Y[m] V(x,y,s) DISP_X&Y[m]

20 5 Sun Apr 25 20:48:31 2010 OptiM - MAIN: - C:\Documents and Settings\nsergei\My Documents\Papers\Invariants\Round 0 BETA_X BETA_Y DISP_X DISP_Y 40 11 Special time-dependent potential Lets consider a Hamiltonian 2 x2 y2 p x2 p y H K s V ( x, y, s ) 2 2 2 2 where V(x,y,s) satisfies the Laplace equation in 2d:

V ( x, y , s ) V ( x, y ) 0 z zN ( s) In normalized variables we will have: p p HN p 2 xN p 2 2 yN N , ( s) ( s ) z , 2 ( s) x N2 y N2 ( )V x N ( ) , y N ( ) , s ( ) 2

s ds Where new time variable is ( s ) (s) 0 12 Four main ideas 1. Chose the potential to be time-independent 2 2 p xN p yN in new variables HN 2 x N2 y N2 U (xN , yN ) 2 (s) 2. Element of periodicity ( s) L sk L s

Lk 1 1 2 2 T insert 1 k 0 0 0 0 0 1 0 0 0 1 0 0 k 1 L the second 3. Find potentials U(x, y) with s integral of motion 4. Convert Hamiltonian to action variables

H 0 ( I1 , I 2 ) h( I1 , I 2 ) and check it for steepness 13 Integrable 2-D Hamiltonians 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) p x2 p y2 x 2 y 2 H U ( x, y ) So, we are looking 2 for2integrable potentials Second integral: such that I Ap x2 Bp x p y Cp y2 D( x, y ) A ay 2 c 2 , B 2axy, C ax 2 , 14 Darboux equation (1901)

Let a 0 and c 0, then we will take a = 1 xy U xx U yy y 2 x 2 c 2 U xy 3 yU x 3xU y 0 General solution f ( ) g ( ) U ( x, y ) 2 2 x c 2 y 2 x c 2 y 2 2c x c 2 y 2 2c x c 2 y 2 : [1, ], : [-1, 1], f and g arbitrary functions 15 The second integral The 2nd integral I x, y, p x , p y xp y yp x Example: 2 2 2 f

( ) g ( ) c 2 p x2 2c 2 2 2 1 2 U ( x, y ) x y 2 2 c2 2 2 f 1 ( ) 1 2 c2 2 g1 ( ) 1 2 2 I x, y, p x , p y xp y yp x c 2 p x2 c 2 x 2 2

16 Laplace equation Now we look for potentials that also satisfy the Laplace equation (in addition to the Darboux equation): U xx U yy 0 We found a family with 4 free parameters (b, 2 f 2 ( ) 1 d t acosh c, d, t): g 2 ( ) 1 2 b t acos f ( ) g ( ) U ( x, y ) 2 2 The most interesting: c=1, t=1, d=0,b 2 17 The integrable Hamiltonian H p x2 p y2 2 x2 y2 kU ( x, y )

2 Multipole expansion 2 8 16 128 2 4 6 8 x iy 10 ... U ( x, y ) Re x iy x iy x iy x iy 3 15 35 315 |k| < 0.5 to provide linear stability for small amplitudes For k > 0 adds focusing in x Small-amplitude tune s: 1 1 2k 2 1 2k B 18 Convex Hamiltonian This Hamiltonian is convex (steep) Example of tunes for k = 0.4

1(J1, 0) 2(0, J2) 0.8 2 1 2k 1j w2j 0.6 1 2k 1 2t 0.4 1 2t 1 0.2 0 0 0.2 0.4 0.6 0.8

1 0 0 0.2 J1j For k -> 0.5 tune spreads of ~ 100% is possible 0.4 0.6 0.8 J2j 19 How to realize it? (s) T insert Need to create an element 1 0 0 0 k 1 0 0 0 0 1 0

0 0 k 1 of periodicity. The T-insert can also be 1 0 0 0 k 1 0 0 0 0 1 0 0 0 k 1 s L which results in a phase advance 0.5 (180 degrees) for the T-insert. The drift space L can give the phase advance of at most 0.5 (180 degrees). 20 Location: New Muon Lab New cryoplant and horizontal cryomodule test stands Existing NML building

Photoinjector and low energy test beamlines up to 6 cryomodules High energy test beamlines Possible 10m storage ring 75 New tunnel extension 21 Nonlinear Lens Block e- Energy 150 MeV Circumference 38 m Dipole field

0.5 T Betatron tunes Qx=Qy=3. 2 (2.4 to 3.6) Radiation damping time 1-2 s (107 turns) Equilibrium emittance, rms, non-norm 0.06 mm Nonlinear lens block mx,my=0. 5 x y Dx Length 2.5 m Number of elements 20

Element length 0.1 m Max. gradient 1 T/m Pole-to-pole ~ 2 cm 22 Current and Proposed Studies Numerical Simulations Nonlinear lenses implemented in a multi-particle tracking code (thin kick) Studied particle stability Effects of imperfections (phase advance, beta-functions, etc.) acceptable Synchrotron motion - acceptable Number of nonlinear lenses - 20 Simulated observable tune spread To Do: Possible Experiments at Test Ring Demonstrate large betatron tune spread

Demonstrate part of the beam crossing integer resonance Map phase space with pencil beam by varying an injection error All particles are stable! Ring nonlinearities Chromaticity Spectrum of horizontal dipole moment Q00=0.9x4=3.6 5000 particles 8000 revolutions 6 6 23 Conclusions We found first examples of completely integrable non-linear optics. Tune spreads of 50% are possible. In our Test Ring simulation we achieved tune spread of about 1.5 (out of 3.6); Nonlinear integrable accelerator optics has advanced to possible practical implementations Provides infinite Landau damping Potential to make an order of magnitude jump in

beam brightness and intensity Fermilab is in a good position to use of all these developments for next accelerator projects Rings or linacs 24 Extra slides 25 Examples of time-independent Hamiltonians q V x , y , s x y Quadrupole s U ( x , y ) q x y 2 2 2 N 2 N

N HN 2 1 (s) 0 0.2 0.4 0.6 s x N2 y N2 q x N2 y N2 2 x2 02 (1 2q) y2 02 (1 2q) Tune spread: zero 0.5 0 2

Tunes: ( s) q( s ) 2 2 p xN p yN Integrable but still linear quadrupole amplitude 1.5 2 N 0.8 L 26 Examples of time-independent Hamiltonians Octupole x 4 y 4 3x 2 y 2

V x, y , s 3 4 2 s 4 x N4 y N4 3 y N2 x N2 U 4 2 4 1 2 1 2 k 4 2 2 H ( px p y ) ( x y ) x y 4 6 x 2 y 2 2 2 4 This Hamiltonian is NOT integrable

Tune spread (in both x and y) is limited to ~12% B 27 Spectrum of vertical dipole moment. Q00=0.905x4=3.62 28