1. Numerical Simulations for IOTA Dmitry Shatilov BINP & FNAL IOTA Meeting, FNAL, 23 February 2012

2. Elliptic Lens Modulated by  -function  The model Nonlinear lens (2 m long) is represented by 20 thin lenses. The total strength (dimensionless) is 0.45 Linear -I maps between nonlinear lenses Ring consists of 4 blocks (nonlin. lens & linear map), betatron tune advance between linear maps is 0.3 Emittances were chosen so that 10  x corresponds to the pole, and the real working area in normalized betatron amplitudes is 5A x  15A y  Simulations (single particle dynamics) Long-term multi-particle tracking (no space charge!) Frequency Map Analysis

3. Frequency Map Analysis for Integrable Optics FMA can be useful to detect all working resonances, their widths and strengths When the dynamic system is (or very close to) integrable, nonlinear resonances disappear – this must be clearly seen in FMA plot Various imperfections and misalignments violate integrability. Then FMA easily allows to determine the regular and stochastic areas in the phase space In addition, we obtain a tune-amplitude dependence (footprint)

4. FMA Integrals of motion Nonlinear lens: 10 slices Replacing the long nonlinear lens by a number of thin slices violates the integrability. The more number of slices – the better. This is illustrated in the following 4 pages.

5. FMA Integrals of motion Nonlinear lens: 20 slices (nominal case)

6. FMA Integrals of motion Nonlinear lens: 50 slices

7. FMA Integrals of motion Nonlinear lens: 100 slices

8. Footprint (4 lenses, 20 slices each one) A x ≤ 5, A y ≤ 15 (b) (a) (d)(c) (a) (b) (c) (d)

9. Octupoles & Quadrupoles (4 lenses, 20 slices each one)

10. Octupoles only (4 lenses, 20 slices each one)

11. Misalignments and Imperfections Random shifts (horizontal and vertical) of individual lenses Random rotations of lenses in X-Y plane (tilt) Errors in lens geometry: distance between poles (parameter c1) Errors in strength of individual lenses (parameter t) Errors in betatron phase advances for MAPQ blocks (linear lattice between nonlinear lenses) Errors in beta-functions at the edges Dispersion at the straight sections

12. Misalignments: r.m.s. = 100 μ AxAx AyAy Initial coordinates of lost particles Turn Number of lost particles (from 10,000)

13. Misalignments: r.m.s. = 10 μ X Y X Y Initial distribution Distribution after 10 6 turns No lost particles! Almost no diffusion!

14. All Misalignments and Errors but Dispersion Shifts of individual lenses: 10 μ Rotations of lenses in X-Y plane: 1 mrad Variations in c1: 10 -3 Variations in strengths: 10 -3 Errors in betatron tune advances: 10 -3 Errors in beta-functions at the edges: 1% Number of lost particles (from 10,000) Turn

15. Dispersion, case 1 Synchrotron amplitude for FMA: A s = 1 Constant dispersion at the lens:  x = 50 cm, σ E = 10 -4 No misalignments and other errors Number of lost particles (from 10,000) Turn

16. Dispersion, case 2 Synchrotron amplitude for FMA: A s = 1 Dispersion changes sign at the center of lens. At the lens edges  x =  50 cm  x =  0.5, σ E = 10 -4 No misalignments and other errors Only 6 particles of 10,000 were lost after 1,000,000 turns

17. All Misalignments and Errors, plus Dispersion (case 2) Number of lost particles (from 10,000) Turn Synchrotron amplitude for FMA: A s = 1 Dispersion changes sign at the center of lens. At the lens edges  x =  50 cm  x =  0.5, σ E = 10 -4

18. One lens: misalignments and errors without dispersion Number of lost particles (from 10,000) Turn With one nonlinear lens the tune spread  y  0.2 can be achieved

19. Round Lens (McMillan case) Assuming the lens strength is zero, the revolution transport matrix at the center of lens has the form: A B 0 c∙  0 s∙  -B A -c/  0 -s/  0 Sum of betatron tune advances (linear map) is 0.5, beta-function 1 m, emittance 10 -6 m∙rad,  = 1 mm Nonlinear e-lens (1 m long) is represented by 50 thin slises. Electron beam radius is 1 mm. The total lens strength (tune shift) is 0.3 The angular momentum is preserved, the 2 nd integral of motion – not, as the lens is not thin. where A =, B =

20. All excited resonances have the form k ∙ ( x + y ) = m They do not cross each other, so there are no stochastic layers and diffusion Thick round lens, drifts between slices

21. Thick round lens, solenoids between slices The background color has shifted from blue to yellow-green (the reason is not clear yet), but no additional resonances appear.

22. Round Lens (Danilov-Nagaitsev case) Beta-function in solenoid (e-lens) must be constant Transport matrix outside the lens is I (identity matrix) In our first simulations the e-lens profile was the same as for McMillan case. However, it can be arbitrary now In order to get valuable tune spread, length of the lens must be much larger than  The lens was represented by 100 thin slices and solenoids between them Without nonlinear field, one of the tunes is exactly integer, another one is shifted by solenoid

23. Thick round lens,  = const The footprint with one defocusing lens crosses integer resonance

24. What else should be done in simulations  Elliptic Lens Realistic lattice for “linear map”: include nonparaxial approximation, chromaticity, etc. Try different options: lens strength, beta-function, etc. Other sources of imperfections (which ones ?)  Round Lenses Introduce imperfections: e-lens beam profile, orbit, … Try different options: other e-beam profiles, lens strength, beta-function, etc.

25. Congratulation with the Defender of the Fatherland Day! (especially those who served in the Soviet Army)