1. Page 1 J-M Lagniel ESS-Lund Feb 03, 2015 Resonances in linac beams Jean-Michel Lagniel (CEA/GANIL) Resonances are the main source of emittance growths and halo formation in linac beams. After a description of the tools used for their study a detailed analysis of the uncoupling and coupling resonances potentially excited in matched and mismatched linac beams will be presented.

2. Page 2 Menu 1- Introduction - Motivations - Basics - Tools for resonance studies 2- Resonances in linac beams 3- Matched beam resonances - Matched envelopes - Uncoupling resonances - Coupling resonances 4- Mismatched beam resonances - Mismatched envelopes (New, unpublished) - Uncoupling resonances (New, unpublished) - Coupling resonances (New, unpublished) 5- Summary J-M Lagniel ESS-Lund Feb 03, 2015

3. Page 3 1- Motivations J-M Lagniel ESS-Lund Feb 03, 2015 Resonances = Emittance growths and Emittance exchanges  Beam quality degradations  An issue for any accelerator, in particular for S3 @ SPIRAL 2 and injection in the ESSνSB ring Resonances = the single source of diffusion at large amplitudes  Halo formation  Beam losses  Activation  An issue for any high-power accelerator < 1 W/m for ESS and SPIRAL 2 < 1 nA/m for SARAF

4. Page 4 1- Basics 1/2 J-M Lagniel ESS-Lund Feb 03, 2015 Resonances = Resonances between single particle oscillations and an oscillating perturbing force Particles considered as linear oscillators Resonance studies restricted to 2 degrees of freedom 2-D continuous beams : Horizontal (x) / Vertical (y) 2.5-D bunched beams (equivalent circular cross section) Transverse (x or r) / Longitudinal (y or z) Resonances characterized by 3 integers : n, m = perturbing-force polynomial series coefficients k = perturbing-force Fourier spectrum harmonic number Excitation = f (a x,y_n_m_k ) a x,y_n_m_k = 0 no excitation

5. Page 5 J-M Lagniel ESS-Lund Feb 03, 2015 1- Basics 2/2 Resonances conditions analyzed introducing the solutions of the homogeneous equations in the perturbing terms Resonance condition fulfilled when a perturbing force component (n, m, k) has a frequency component equal to the particle oscillation “natural frequency” i x + j y = k x =  x /  p y =  y /  p i related to n j related m of the x n y m cos(k  p s) perturbing force component  i  +  j  = Order of the resonance i or j = 0  uncoupling resonance i and j  0  coupling resonance

6. Page 6 J-M Lagniel ESS-Lund Feb 03, 2015 1- Tool #1 Tune diagram Synchrotrons : “Wave number” = “Tune” ( x, y ) = “Working point” Direct link between Tunes and Phase advances per focusing system period Perturbing force period = One turn

7. Page 7 J-M Lagniel ESS-Lund Feb 03, 2015 1- Tool #1 Tune diagram x y Beam “footprint” ( x, y ) surface occupied by the particles i x + j y = k resonance lines Direct view of “all” the resonances which can affect the beam Knowledge of the part of the beam (core or halo) which can be affected Major tool in the accelerator physicist toolbox ! 2 x = 1 x - y = 0

8. Page 8 J-M Lagniel ESS-Lund Feb 03, 2015 1- Tool #2 [ , ,  ] charts    y /  x  x   x /  0x   E y /E x  Constant One point in the chart = One beam footprint in the tune diagram when  0x is known and as long as  stays constant [ , ,  ] chart Tune diagram WARNING : The [ , ,  ] charts cannot be used to study beam-footprint-position dependent phenomena when  0x is not known

9. Page 9 J-M Lagniel ESS-Lund Feb 03, 2015 1- Tool #2 [ , ,  ] charts Beam footprint position as a function of  0x for a given point [  2.2  x  0.5   1.5]  0x = 100°  0x = 80°  0x = 60°  0x = 40°  0x = 20° -1- The uncoupling resonance (x, y) and portion of the beam affected by the resonances is highly dependent of the value of  0x  The resonance effects cannot be estimated without the knowledge of  0x -2- We know that the space-charge resonance excitation Is directly related to the beam density profile (a n,m,k )  The resonance effects cannot be estimated without the knowledge of the beam distribution DONT TRUST in [ , ,  ] charts pretending to predict the resonance strengths ! JML Space-charge workshop, Oxford

10. Page 10 J-M Lagniel ESS-Lund Feb 03, 2015 2- Resonances in linac beams -Type 1- Periodic perturbing force @ periodic or quasi-periodic setting of the radial and longitudinal focusing elements ( = synchrotron case ) - Systematic “errors” on all the elements along the linac (eg. quad multipolar component) - Systematic “unavoidable” perturbing forces induced by the nonlinear field of the accelerating cavities (see my 2014 ESS seminar “Zero-current longitudinal beam dynamics”)  Space-charges forces which have the periodicity of the focusing systems when the beam is matched -Type 2- Periodic perturbing force when the beam is mismatched  Space-charge forces which have the periodicity of the mismatched beam envelope oscillations

11. Page 11 J-M Lagniel ESS-Lund Feb 03, 2015 2- Resonances in matched linac beams Matched beam uncoupling resonances  0x* and  0z* < 90° ( 0x and 0z < 0.25) to avoid the strong =1/4 (4 =1) 4 th order res  Deal with =1/6 (  * = 60°) =1/8 (  * = 45°) … Matched beam coupling resonances  0x* and  0z* < 90° ↔ 0x and 0z < 0.25  Only difference coupling resonances  x -  z  0 2 nd order  x - 2  z  0 2  x -  z  0 3 rd order  x - 3  z  0 3  x -  z  0 4 th order 2  x - 2  z  0 (strong excitation)

12. Page 12 J-M Lagniel ESS-Lund Feb 03, 2015 2- Resonances in mismatched linac beams Envelope oscillation period (fundamental period of the space-charge perturbing forces) no longer equal to the period of the radial and longitudinal focusing systems Mismatched beam resonances studies starting from the equations governing the mismatched beam envelope oscillations  4 th part of this seminar Tune diagram Area in which the mismatched beam footprint can be located Area in which the matched beam footprint can be located Half-integer / integer resonances always / sometimes present in mismatched beams

13. Page 13 J-M Lagniel ESS-Lund Feb 03, 2015 3- Matched beam resonances Perturbing force = space-charge force oscillations @ matched beam envelope oscillations 2-D horizontal / vertical case FDO FODO FDO vs FODO : same peak-to-peak amplitudes of the space-charge perturbing force Higher harmonics (k = 2, 3) in the case of FDO

14. Page 14 J-M Lagniel ESS-Lund Feb 03, 2015 3- Matched beam resonances Matched beam envelope oscillations : 2.5-D transverse / longitudinal case Longitudinal space-charge perturbing force oscillations  radial envelope mean value FDO FODO FODO : Smaller amplitude of the radial envelope mean value (magenta)  Weaker coupling induced by the coupling resonances than in the FDO case

15. Page 15 J-M Lagniel ESS-Lund Feb 03, 2015 3- Uncoupling Matched beam resonances 2-D example with  0x* 90°  0x* = 80° ( =0.222)  x* = 40° ( =0.111)  0y* = 97.4° ( =0.27)  y* = 59° ( =0.164) 90° 1/4 60° 1/6 Uncouling resonance predictions : (x, x’) 60° (1/6) + 45° (1/8) (y, y’) 90° (1/4) + 60° (1/6) Phase advances / lattice f (particle amplitude)

16. Page 16 J-M Lagniel ESS-Lund Feb 03, 2015 2-D example  0x* = 80°  x* = 40°  0y* = 97.4°  y* = 59° 3- Uncoupling Matched beam resonances (x, x’) Phase-space portraits (y, y’) Uncoupled particles with initial conditions (x, x’, y=y’=0) and (y, y’, x=x’=0) (x, x’) : 1/6 and 1/8 present with thin chaotic layers, no resonance overlap (y, y’) : resonance overlap up to the 1/4 around the beam core  large amplitude diffusion Diffusion characteristic time = one focusing system period

17. Page 17 J-M Lagniel ESS-Lund Feb 03, 2015 3- Uncoupling Matched beam resonances 2-D example  0x* = 80°  x* = 40°  0y* = 97.4°  y* = 59°  x* = f(x i, y i ) Phase-advances / lattice  y* = f(x i, y i ) Coupled particles with initial conditions (x, y, x’=y’=0)  (x, x’, y, y’) motions The  x = 60° (1/6) affect also particles with small x oscillation amplitudes The  y = 90° (1/4) affect also particles with small y oscillation amplitudes

18. Page 18 J-M Lagniel ESS-Lund Feb 03, 2015 3- Coupling Matched beam resonances Coupling resonances  0x* and  0z* < 90° ↔ 0x and 0z < 0.25 Only difference coupling resonances  At the opposite of the sum coupling resonances, these difference coupling resonances produce emittance exchanges but not instabilities  No « stability charts » for such resonances !  These coupling resonances can produce emittance exchanges even if the initial emittances are equal Sum coupling resonance Difference coupling resonance

19. Page 19 J-M Lagniel ESS-Lund Feb 03, 2015 3- Coupling Matched beam resonances Construction of [ , ,  ] charts with the difference coupling resonance positions 1- Built a (x,y) coordinate system matched to the beam footprint f(  0x ) 2- Compute 0 < X cross < 1 = distance at which the resonance crosses the beam footprint diagonal X cross Core affected X cross > 0.25 => Halo affected 3- Compute 0 < dX p < 1 = distance at which the resonance crosses the beam footprint perpendicular X cross and dX p are geometrical factors independant of  0x

20. Page 20 J-M Lagniel ESS-Lund Feb 03, 2015 3- Coupling Matched beam resonances [ , ,  =1] chart 1 = core affected 0 = halo affected 1 - X cross 1 - dX p < 1 Sum The [ , ,  =1] charts allow immediately to see if a “working point” corresponds to a beam footprint affected by a resonance or not and, if yes, to determine if the resonance can affect core or halo particles WARNING : Difference coupling resonances only NO excitation level, stability criteria…. Simple C++ code, easy to understand, well defined limits of use, please ask…

21. Page 21 J-M Lagniel ESS-Lund Feb 03, 2015 3- Coupling Matched beam resonances Difference coupling resonances : How does it work ? Example [  = 2.1,  = 0.5,  = 0.4762]  0x* = 40° 2 x - y = 0  0x* = 40°  0x* = 20°  0x* = 65.5°  0x* = 42°  y /  x =  x /  y = 2.1  EQP To understand emittance transfers, think coupling resonances not EQP !

22. Page 22 J-M Lagniel ESS-Lund Feb 03, 2015 3- Coupling Matched beam resonances How does it work ? Example [  = 2.1,  = 0.5,  = 0.5762]  0x* = 40° y / x = f ( x i, y i ) Single particle x(s) and y(s) oscillations Very slow coupling in this case y / x = 2 Difference coupling resonances Read Montague CERN 68-38 yellow report 1968

23. Page 23 J-M Lagniel ESS-Lund Feb 03, 2015 Matched  Mismatched beam resonances Matched beam resonances Mismatched beam resonances

24. Page 24 J-M Lagniel ESS-Lund Feb 03, 2015 4- Mismatched beam resonances Perturbing force = space-charge force oscillations @ mismatched beam envelope oscillations 2-D 2.5-D Envelope equations Oscillation frequencies = f (  0 x,y, K sc,  x,y ) Oscillation eigen-mode (even, odd) and amplitude = f ( initial mismatch values ) Matched beam envelope frequency

25. Page 25 J-M Lagniel ESS-Lund Feb 03, 2015 4- Mismatched beam resonances Resonances in mismatched beam present knowledge mainly limited to 2-D CW round beams 2.5-D : only analytical studies of the envelope oscillation eigen-modes with rough (unacceptable) approximations (Pabst-Bongart, ESS note 1997) or with numerical integration to compute the space-charge potential (Pichoff, Saclay note, 1998) 2-D CW round beam Lagniel 1995 Demonstration that the half-integer resonance Is always present

26. Page 26 J-M Lagniel ESS-Lund Feb 03, 2015 4- Mismatched beam resonances New (unpublished) « Resonances in mismatched beam theory » for 2-D continuous and 2.5-D bunched beams  Analytical expression of the mismatched beam envelope oscillations f ( accelerator and beam parameters + mismatch initial conditions)  Analytical expression of the “mismatched beam tunes”  Mismatched-beam uncoupling and coupling resonance studies using “Mismatched tune diagrams” “Mismatched [ , ,  ] charts ”

27. Page 27 J-M Lagniel ESS-Lund Feb 03, 2015 4- Mismatched beam resonances Analytical expression of the mismatched beam envelope oscillations Starting point : Nonlinear envelope equations of motion  Mismatched beam envelope linearized equations of motion Mismatched beam envelope motions are determined by 4 coefficients (2.5-D) 3 coefficients (2-D) which are function of [ , ,  ] and propotional to  0t

28. Page 28 J-M Lagniel ESS-Lund Feb 03, 2015 4- Mismatched beam resonances Analytical expression of the even and odd « eigenfrequencies » + = even mode - = odd mode Analytical expression of the mismatched beam envelope oscillations  a i and  b i = envelope initial mismatches R e and R o = coefficients function of [ , ,  ]

29. Page 29 J-M Lagniel ESS-Lund Feb 03, 2015 4- Mismatched beam resonances The analytical expression of the mismatched beam envelope oscillations ( obtained linearizing the nonlinear envelope differential equations ) gives incredible good results ! (up to 30-40% mismatches) Numerical integration of the nonlinear envelope equations of motion vs analytical expression

30. Page 30 J-M Lagniel ESS-Lund Feb 03, 2015 4- Mismatched beam resonances To study the mismatched beam resonances we have to consider the « Mismatched tunes » The mismatched tunes are ratios of terms which are  to  0t  Mismatched tunes function of [ , ,  ] [ , ,  ] parameters [ , ,  ] charts = ideal tools for mismatched beam resonance studies 2-D

31. Page 31 J-M Lagniel ESS-Lund Feb 03, 2015 4- Mismatched beam resonances Mismatched tune diagram for weak space charge [ ,  = 0.9,  = 1.0] Round beam  = 1 = particular case Evolution f (  =  y /  x )  Half-integer resonance always present in both even and odd modes i.e. cannot be avoided  The integer resonance can be present in the odd mode ( can be avoided )

32. Page 32 J-M Lagniel ESS-Lund Feb 03, 2015 4- Mismatched beam resonances Unequal emitances Higher space charge (  = 0.6 )  larger mismatched-tune footprints

33. Page 33 J-M Lagniel ESS-Lund Feb 03, 2015 4- Mismatched beam resonances Range of resonances (x, y) f (  )  = 1.02  = 1.02 (~ round beam)  = 1.5  = 1.0  = 2.2  = 1.0 Even mode Odd mode

34. Page 34 J-M Lagniel ESS-Lund Feb 03, 2015 4- Mismatched beam resonances [ , ,  = 1.0] charts Mismatched tunes x e x o y e y o

35. Page 35 J-M Lagniel ESS-Lund Feb 03, 2015 4- Mismatched beam resonances [ , ,  ] charts Integer resonance d = 1 (red)  = 1 (core) d = 0 (blue)  0 = 1 (halo)  = 1.0  = 2.0

36. Page 36 J-M Lagniel ESS-Lund Feb 03, 2015 4- Mismatched beam resonances [ , ,  ] charts Half-integer resonance (always present !) d  1 (red)  = 1 (core) d = 0 (blue)  0 = 1 (halo) Even  = 1.0Odd  = 1.0 Even  = 2.0Odd  = 2.0

37. Page 37 J-M Lagniel ESS-Lund Feb 03, 2015 4- Mismatched beam resonances [ , ,  ] charts with mismatched beam (sum) coupling resonances as well Odd  = 1.0 Even  = 1.0 Sum coupling resonances leading to instabilities present, even at low space charge ?? Mismatched beam dynamics dominated by the half-integer (and integer) resonances ??

38. Page 38 J-M Lagniel ESS-Lund Feb 03, 2015 4- Mismatched beam resonances Example of beam dynamics with mismatch (uncoupled particles)  0x = 60°/m [  = 2.8,  = 0.5,  = 1.0] yo xo ye xe 1/4 1/2 1/1

39. Page 39 J-M Lagniel ESS-Lund Feb 03, 2015 4- Mismatched beam resonances Phase-space portraits  0x = 60°/m [  = 2.8,  = 0.5,  = 1.0] (x,x’) even (y,y’) even (x,x’) odd (y,y’) odd

40. Page 40 J-M Lagniel ESS-Lund Feb 03, 2015 5- Summary Resonances in matched beams Resonances in mismatched beams @ New knowledge - Tune diagrams should be systematically used for linac resonance studies ( = circular machines) (TraceWin ?) - [ , ,  ] charts useful to locate the beam position with respect to the difference coupling resonances (Use them understanding their limits !) - The  0* < 90° rule must be respected (take care smooth approximation and synchrotron motion) - Think « mismatched beam tune » for resonance studies - [ , ,  ] charts very useful to locate the resonance positions (core / halo, working point choice) - Mismatched beam dynamics dominated by the half-integer (and integer) resonances (?) - Best (less worth) working point ? (less sensitive to mismatches) More core-particle and multiparticle simulations