Superconducting Electron Linacs

Transcription

1 Superconducting Electron Linacs Nick Walker DESY CAS Zeuthen Sept. 3 What s in Store Brief history of superconducting RF Choice of frequency (SCRF for pedestrians) RF Cavity Basics (efficiency issues) Wakefields and Beam Dynamics Emittance preservation in electron linacs Will generally consider only high-power highgradient linacs sc e + e - linear collider TESLA technology sc X-Ray FEL

2 Status 199: Before start of TESLA R&D (and 3 years after the start) MV L Lilje S.C. RF Livingston Plot courtesy Hasan Padamsee, Cornell

3 TESLA R&D 3 9 cell EP ities TESLA R&D

4 E z The Linear Accelerator (LINAC) ct λ c c z standing wave ity: bunch sees field: E z E sin(ωt+φ )sin(kz) E sin(kz+φ )sin(kz) For electrons, life is easy since We will only consider relativistic electrons (v c) we assume they have accelerated from the source by somebody else! Thus there is no longitudinal dynamics (e ± do not move long. relative to the other electrons) No space charge effects SC RF Unlike the DC case (superconducting magnets), the surface resistance of a superconducting RF ity is not zero: f RBCS exp 1.76 Tc / T T ( ) Two important parameters: residual resistivity thermal conductivity

5 SC RF Unlike the DC case (superconducting magnets), the surface resistance of a superconducting RF ity is not zero: f RBCS exp 1.76 Tc / T T ( ) Two important parameters: residual resistivity R res thermal conductivity losses surface area f -1 R s f when R BCS > R res f TESLA 1.3 GHz f > 3 GHz SC RF Unlike the DC case (superconducting magnets), the surface resistance of a superconducting RF ity is not zero: f RBCS exp 1.76 Tc / T T ( ) Two important parameters: residual resistivity thermal conductivity Higher the better! LiHe Nb I skin depth ρ / ωµ heat flow RRR Residual Resistivity Ratio

6 RF Cavity Basics Figures of Merit E P R z RF s RF power P Shunt impedance r s V rp s Quality factor Q : R-over-Q Q s r / Q stored energy ωu π energy lost per cycle P V ω U r s /Q is a constant for a given ity geometry independent of surface resistance Frequency Scaling r s f + 1/ f 1 normal superconducting Q f f 1/ normal superconducting r s f Q f normal superconducting

7 RF Cavity Basics Fill Time From definition of Q Allow ringing ity to decay (stored energy dissipated in walls) Combining gives eq. for U Assuming exponential solution (and that Q and ω are constant) Since U V U () t U () e V () t V () e Q P du dt ωu P du dt ω U Q ω t Q ω t Q RF Cavity Basics Fill Time Characteristic charging time: Q τ ω time required to (dis)charge ity voltage to 1/e of peak value. Often referred to as the ity fill time. True fill time for a pulsed linac is defined slightly differently as we will see.

8 f RF 1.3 GHz Q R/Q R RF Cavity Basics Some Numbers S.C. Nb (K) Ω 1 kω Cu Ω P (5 MV) cw! 5 W 1.5 MW P (5 MV) cw! 15 W 31 MW τ fill 1. s 5 µs f RF 1.3 GHz Q R/Q R RF Cavity Basics Some Numbers S.C. Nb (K) Ω 1 kω Cu Ω P (5 MV) P (5 MV) τ fill cw! cw! 5 W 15 W 1. s Very 1.5 high MW Q :: the the great 31 advantage MW of of s.c. RF 5 RF µs

9 RF Cavity Basics Some Numbers very f RF small 1.3 GHz power loss in in ity S.C. Nb walls (K) all all supplied power goes into accelerating the the beam Q very high R/Q RF-to-beam transfer efficiency 1 kω for for AC power, must include cooling power Ω R Cu Ω P (5 MV) cw! 5 W 1.5 MW P (5 MV) cw! 15 W 31 MW τ fill 1. s 5 µs RF Cavity Basics Some Numbers f RF 1.3 GHz Q R/Q R P (5 MV) for for high-energy higher S.C. Nb (K) gradient linacs Cu (X-FEL, LC), cw cwoperation not an an option due to to load on on cryogenics pulsed operation generally required numbers now represent 1 peak kω power P P pk pk duty cycle Ω 1 7 Ω (Cu linacs generally use very short pulses!) cw! 5 W 1.5 MW P (5 MV) cw! 15 W 31 MW τ fill 1. s 5 µs

10 Cryogenic Power Requirements Basic Thermodynamics: Carnot Efficiency (T.K) η T. c.7% T T 3. room System efficiency typically.-.3 (latter for large systems) Thus total cooling efficiency is.14-.% P 5W /..5kW cooling Note: this represents dynamic load, and depends on Q and V Static load must also be included (i.e. load at V ). RF Cavity Basics Power Coupling calculated fill time was 1. seconds! this is time needed for field to decay to V/e for a closed ity (i.e. only power loss to s.c. walls). P Vt ( ) V()exp( ω t/ Q) Vt () t

11 RF Cavity Basics Power Coupling calculated fill time was 1. seconds! this is time needed for field to decay to V/e for a closed ity (i.e. only power loss to s.c. walls). however, we need a hole (coupler) in the ity to get the power in, and P in P Vt ( ) V()exp( ω t/ Q) Vt () t RF Cavity Basics Power Coupling calculated fill time was 1. seconds! this is time needed for field to decay to V/e for a closed ity (i.e. only power loss to s.c. walls). however, we need a hole (coupler) in the ity to get the power in, and this hole allows the energy in the ity to leak out! P out P Vt V t Q L () ()exp( ω / ) Vt () t

12 RF Cavity Basics circulator coupler ity Generator (Klystron) Z Z Z matched load Z characteristic impedance of transmission line (waveguide) RF Cavity Basics P for impedance mismatch Generator (Klystron) P ref Z Klystron power P for sees matched impedance Z Reflected power P ref from coupler/ity is dumped in load Conservation of energy: Pfor Pref + P

13 Equivalent Circuit Z Z Generator (Klystron) Z coupler ity Z L R C 1:n Equivalent Circuit Only consider on resonance: ω C note : Q R L 1 LC Z R V 1:n

14 Equivalent Circuit Only consider on resonance: ω 1 LC We can transform the matched load impedance Z into the ity circuit. V / n Zext R / β nz R V 1:n Equivalent Circuit Q Q load ext R ω L V1 / n1 + R Z ω L define external Q: Q ext Q Q Q load Zext ω L ext 1 1:n Zext R r s / V nz R / β coupling constant: P R Q Q ext β P Rext Qext Q load 1 + β

15 Reflected and Transmitted RF Power reflection coefficient (seen from generator): from energy conservation: 1 β ref for for P Γ P P 1+ β R Zext β 1 Γ R+ Zext β + 1 P Pfor Pref P for 1 Γ β 1 Pfor 1 β 1 + 4β P P ( 1+ β ) for V steady state ity voltage: Transient Behaviour Vˆ 1/ β 1 + β think in terms of (travelling) microwaves: for ( PforZ ) V ref ( ΓVfor) ( ) V n V + V for ref P r s P r for s β 1 Γ β + 1 from before: remember: ω ω ˆ β n V n( Vfor + Vref ) n( 1+Γ ) Vfor V steady-state for 1 + β result! P 4β P (1 + β ) for

16 Reflected Transient Power ( ω ) V ˆ () t 1 exp t/ql V β n 1 exp ( ω t/q ) V β L 1 + for V () t Vref () t Vfor n Vref () t β 1 exp ( ωt/ql ) 1 V for 1 + β Γ() t time-dependent reflection coefficient Reflected Transient Power Pref () t β 1 exp ( ωt/ql ) 1 P for 1 + β after RF turned off Vfor toff QL / ω Vˆ Vref exp ( ω t / QL ) n β Vfor exp t / Q 1+ β ( ω ) L t t t off P P ref for 4β exp / ( 1+ β ) ( ω t Q ) L

17 P P ref for 1.8 RF On Pref () t β 1 exp ( ωt/ql ) 1 P for 1 + β β.1 note: No beam!.6.4 β 1 β ω t Q / L Reflected Power in Pulsed Operation Example of square RF pulse with t 1 Q / ω P P ref for critically coupled under coupled over coupled β 1 β β ω t Q / L

18 Accelerating Electrons I t b Q t b b t π h/ ω b Qb t Assume bunches are very short σ z / c λrf model current as a series of δ functions: I () t Q δ ( t n t ) b b b n Fourier component at ω is I assume on-crest acceleration (i.e. I b is in-phase with V ) consider first steady state Accelerating Electrons I g I R ext R R β V acc V I I acc ( g ) 1 R + β what s I g?

19 Accelerating Electrons Consider power in ity load R with I b : P 4β (1 + β ) P for steady state! From equivalent circuit model (with I b ): V P I g R 1 + β V R I g R 1 ( + β ) I g P for R β NB: I g is actually twice the true generator current substituting for I g : Accelerating Electrons V I I V acc acc ( g ) 1 R + β P β I for R 1+ R β 1 R I P R for Pforβ 1+ β β R introducing K I P for beam loading parameter V acc K PforRβ 1 β 1+ β

20 Accelerating Electrons Now let s calculate the RF beam efficiency 4K β K power fed to beam: Pbeam IVacc Pfor 1 1+ β β hence: η RF beam P 4K β beam K 1 Pfor 1+ β β V Pref Pfor Pbeam P Pfor R reflected power: ( 1 η) β 1 K β Pfor 1+ β acc Note that if beam is off (K) Accelerating Electrons P ref β 1 Pfor β + 1 For zero-beam loading case, we needed β 1 for maximum power transfer (i.e. P ref ) Now we require β 1 K β β 1 K β Hence for a fixed coupler (β), zero reflection only achieved at one specific beam current. previous result

21 A Useful Expression for β efficiency: η RF beam P 4K β beam K 1 Pfor 1+ β β voltage: 1 r β s Vacc 1 I Pforrs Pforβ 1 + β optimum 1 K β β η V opt acc β 1 β rp s β for can show rs β opt 1+ r where r V / I beam acc beam beam current: 1 Nb 1 Qb 3.nC tb 337ns I 9.5mA For optimal efficiency, P ref : From previous results: K 3.8 β 384 Example: TESLA ity parameters (at T K): f 1.3 GHz rs / Q 1kΩ 1 Q 1 13 r 1 Ω V acc 5MV s η Pfor Pbeam + P Pbeam IVacc 37.5kW Vacc P 6.5 W rs 99.97% RF beam cw!

22 Unloaded Voltage V loaded P r for s β matched condition: 1 K β β V unloaded β 1 + β P r for s hence: V V unloaded loaded β 1 + β for K, β 1 From previous discussions: Pulsed Operation ( t / ) V () t V 1 e τ V beam Q L Q τ acc ω ω( β + 1) t t () t ( 1 ( t tfill V )/ τ acc e ) t > t fill fill Allow ity to charge for t fill such that V ( t ) V t ln() τ fill acc fill For TESLA example: τ 645µ s 447µ s t fill

23 After t fill, beam is introduced exponentials cancel and beam sees constant accelerating voltage V acc 5 MV Power is reflected before and after pulse Pulse Operation generator voltage ity voltage beam induced voltage RF on beam on reflected power: P / P ref for β Pref () t 1 exp ( ω t/ql ) 1 Pfor 1 + β t/µs Pulsed Efficiency total efficiency must include t fill : for TESLA η pulse t beam η tfill t + beam 95µ s 99.97% 446 µ s+95 µ s 68%

24 cw efficiency for s.c. ity: efficiency for pulsed linac: fill time: t fill ln() τ Quick Summary L τ η cw η Pbeam 1 P η for beam cw tfill + t beam t 1 Q Q r V + L s acc L ω ω(1 β) ω Q I Increase efficiency (reduce fill time): go to high I for given V acc longer bunch trains (t beam ) some other constraints: cyrogenic load modulator/klystron V f t acc rep pulse Lorentz-Force Detuning In high gradient structures, E detuning of ity and B fields exert stress on the V acc ity, causing it to deform. V acc Re As a result: 1 + iql ity off resonance by relative Vacc amount δω/ω 1 + 4QL equivalent circuit is now complex require voltage phase shift wrt generator V (and beam) by ϕ tan ( Q L ) Vacc 4Q power is reflected few Hz for TESLA For TESLA 9 cell at 5 MV, f ~ 9 Hz!! (loaded BW ~5Hz) [note: causes transient behaviour during RF pulse] 3 L

25 Lorentz Force Detuning cont. f k E k 1Hz/(MV/m) recent tests on TESLA high-gradient ity Three fixes: Lorentz Force Detuning cont. mechanically stiffen ity feed-forwarded (increase RF power during pulse) fast piezo tuners + feedback

26 Three fixes: Lorentz Force Detuning cont. mechanically stiffen ity feed-forwarded (increase RF power during pulse) fast piezo tuners + feedback reduces effect by ~1/ stiffening ring Three fixes: Lorentz Force Detuning cont. mechanically stiffen ity feed-forwarded (increase RF power during pulse) fast piezo tuners + feedback Low Level RF (LLRF) compensates. Mostly feedforward (behaviour is repetitive) For TESLA, 1 klystron drives 36 ities, thus vector sum is corrected.

27 Three fixes: Lorentz Force Detuning cont. mechanically stiffen ity feed-forwarded (increase RF power during pulse) fast piezo tuners + feedback Three fixes: Lorentz Force Detuning cont. mechanically stiffen ity feed-forwarded (increase RF power during pulse) fast piezo tuners + feedback recent tests on TESLA high-gradient ity

28 Wakefields and Beam Dynamics bunches traversing ities generate many RF modes. Excitation of fundamental (ω ) mode we have already discussed (beam loading) higher-order (higher-frequency) modes (HOMs) can act back on the beam and adversely affect it. Separate into two time (frequency) domains: long-range, bunch-to-bunch short-range, single bunch effects (head-tail effects) Wakefields: the (other) SC RF Advantage the strength of the wakefield potential (W) is a strong function of the iris aperture a. Shunt impedance (r s ) is also a function of a. To increase efficiency, Cu ities tend to move towards smaller irises (higher r s ). For S.C. ities, since r s is extremely high anyway, we can make a larger without loosing efficiency. W a 3 significantly smaller wakefields a

29 Long Range Wakefields t b V( ω, t) I( ω, t) Z( ω, t) Bunch current generates wake that decelerates trailing bunches. Bunch current generates transverse deflecting modes when bunches are not on ity axis Fields build up resonantly: latter bunches are kicked transversely multi- and single-bunch beam break-up (MBBU, SBBU) wakefield is the time-domain description of impedance Transverse HOMs wake is sum over modes: kc W t e ω t n ωnt/qn () sin( n ) n ωn k n is the loss parameter (units V/pC/m ) for the n th mode Transverse kick of j th bunch after traversing one ity: yq kc y e i t j 1 i i i ωni t/qn j sin i b i 1 Ei ωn ( ω ) where y i, q i, and E i, are the offset wrt the ity axis, the charge and the energy of the i th bunch respectively.

30 HOMs cane be randomly detuned by a small amount. Over several ities, wake decoheres. Effect of random.1% detuning (averaged over 36 ities). Still require HOM dampers Detuning abs. wake (V/pC/m) abs. wake (V/pC/m) next bunch no detuning with detuning time (ns) Effect of Emittance vertical beam offset along bunch train (n b 9) Multibunch emittance growth for ities with 5µm RMS misalignment

31 Single Bunch Effects Completely analogous to low-range wakes wake over a single bunch causality (relativistic bunch): head of bunch affects the tail Again must consider longitudinal: effects energy spread along bunch transverse: the emittance killer! For short-range wakes, tend to consider wake potentials (Greens functions) rather than modes Longitudinal Wake Consider the TESLA wake potential W ( z ct) V s W ( z) exp pc m [m] wake over bunch given by convolution: W, bunch ( z) W ( z z) ρ( z ) dz z z average energy loss: E qb W, bunch( z) ρ( z) dz For TESLA LC: E 46 kv/m V/pC/m (ρ(z) long. charge dist.) σ 3µm z tail -4-4 z/σ z head

32 accelerating field along bunch: RMS Energy Spread E( z) qw ( z) + E cos( π z/ λ + φ) b, bunch RF Minimum energy spread along bunch achieved when bunch rides ahead of crest on RF. Negative slope of RF compensates wakefield. For TESLA LC, minimum at about φ ~ +6º E/E (ppm) rms E/E (ppm) wake+rf z/σ z RF φ (deg) RMS Energy Spread

33 Transverse Single-Bunch Wakes When bunch is offset wrt ity axis, transverse (dipole) wake is excited. kick along bunch: V/pC/m 8 qb y ( z) W ( z' z) ρ( z ) y( s; z ) dz Ez ( ) z z Note: y(s; z) describes a free betatron oscillation along linac (FODO) lattice (as a function of s) 6 4 tail head z/σ z particle model Effect of coherent betatron oscillation - head resonantly drives the tail tail head eom (Hill s equation): solution: y + 1 kβ y1 ( ) y () s aβ()sin s ϕ() s + ϕ 1 head tail eom: q W ' σ z y + k y y1 E beam resonantly driven oscillator

34 BNS Damping If both macroparticles have an initial offset y then particle 1 undergoes a sinusoidal oscillation, y 1 y cos(k β s). What happens to particle? y y cos k s + s k s ( β ) sin ( β ) W ' qσ z kβ Ebeam Qualitatively: an additional oscillation out-of-phase with the betatron term which grows monotonically with s. How do we beat it? Higher beam energy, stronger focusing, lower charge, shorter bunches, or a damping technique recommended by Balakin, Novokhatski, and Smirnov (BNS Damping) curtesy: P. Tenenbaum (SLAC) BNS Damping Imagine that the two macroparticles have different betatron frequencies, represented by different focusing constants k β1 and k β The second particle now acts like an undamped oscillator driven off its resonant frequency by the wakefield of the first. The difference in trajectory between the two macroparticles is given by: W ' qσ 1 z y y1 y 1 cos ( kβs) cos( kβ1s) Ebeam kβ k β1 curtesy: P. Tenenbaum (SLAC)

35 BNS Damping The wakefield can be locally cancelled (ie, cancelled at all points down the linac) if: W ' qσ 1 E k k z beam β β1 This condition is often known as autophasing. It can be achieved by introducing an energy difference between the head and tail of the bunch. When the requirements of discrete focusing (ie, FODO lattices) are included, the autophasing RMS energy spread is given by: σe 1 W ' qσ z Lcell Ebeam 16 Ebeam sin β 1 ( πν ) curtesy: P. Tenenbaum (SLAC) Wakefields (alignment tolerances) bunch ities tail performs oscillation tail head y accelerator axis tail head head tail km 5 km 1 km ε ε NW ε E Ei f β 1 y c

36 Cavity Misalignments Wakefields and Beam Dynamics The preservation of (RMS) Emittance! control of Wakefields Dispersion Transverse Longitudinal Single-bunch beam loading E/E Quadrupole Alignment Structure Alignment

37 Emittance tuning in the Linac Consider linear collider parameters: DR produces tiny vertical emittances (γε y ~ nm) LINAC must preserve this emittance! strong wakefields (structure misalignment) dispersion effects (quadrupole misalignment) Tolerances too tight to be achieved by surveyor during installation Need beam-based alignment mma! Basics (linear optics) thin-lens quad approximation: y KY g ij g ij yi y j y j R (, ) 34 i j Y i Y j K i j y g KY Y i ij i j i i 1 linear system: just superimpose oscillations caused by quad kicks. y j

38 Introduce matrix notation Original Equation Defining Response Matrix Q: j y g KY Y i ij i j i i 1 Q G diag(k) + I Hence beam offset becomes G is lower diagonal: y Q Y g1 G g31 g3 g41 g4 g43 Dispersive Emittance Growth Consider effects of finite energy spread in beam δ RMS chromatic response matrix: K Q( δ) G( δ) diag + I 1+ δ G G( δ ) G() + δ R ( δ ) R () + T δ δ lattice chromaticity dispersive kicks y( δ ) ηy Q Q Y δ dispersive orbit: [ ( δ ) ()]

39 What do we measure? BPM readings contain additional errors: b offset static offsets of monitors wrt quad centres b noise one-shot measurement noise (resolution σ RES ) y Q Y b b R y y y BPM + offset + noise + y fixed from shot to shot random (can be averaged to zero) launch condition In principle: all BBA CAS algorithms Zeuthen deal with b offset Scenario 1: Quad offsets, but BPMs aligned BPM Assuming: - a BPM adjacent to each quad - a steerer at each quad simply apply one to one steering to orbit steerer quad mover dipole corrector

40 Scenario : Quads aligned, BPMs offset BPM 1--1 corrected orbit one-to-one correction BAD! Resulting orbit not Dispersion Free emittance growth Need to find a steering algorithm which effectively puts BPMs on (some) reference line real world scenario: some mix of scenarios 1 and BBA Dispersion Free Steering (DFS) Find a set of steerer settings which minimise the dispersive orbit in practise, find solution that minimises difference orbit when energy is changed Energy change: true energy change (adjust linac phase) scale quadrupole strengths Ballistic Alignment Turn off accelerator components in a given section, and use ballistic beam to define reference line measured BPM orbit immediately gives b offset wrt to this line

41 DFS Problem: E E y Q( ) Q() E Y E E M Y E Note: taking difference orbit y removes b offset Solution (trivial): 1 Y M y Unfortunately, not that easy because of noise sources: y M Y+ bnoise + R y 3µm random quadrupole errors % E/E DFS example µm No BPM noise No beam jitter µm

42 Simple solve DFS example 1 Y M y In the absence of errors, works exactly Resulting orbit is flat Dispersion Free (perfect BBA) original quad errors fitter quad errors Now add 1µm random BPM noise to measured difference orbit Simple solve DFS example 1 Y M y Fit is ill-conditioned! original quad errors fitter quad errors

43 Solution is still Dispersion Free but several mm off axis! DFS example µm µm DFS: Problems Fit is ill-conditioned with BPM noise DF orbits have very large unrealistic amplitudes. Need to constrain the absolute orbit minimise T T y y y y + σ σ + σ res res offset Sensitive to initial launch conditions (steering, beam jitter) need to be fitted out or averaged away R y

44 Minimise DFS example T T y y y y + σ σ + σ res res offset absolute orbit now constrained remember σ res 1µm σ offset 3µm original quad errors fitter quad errors Solutions much better behaved! DFS example µm! Wakefields! Orbit not quite Dispersion Free, but very close µm

45 DFS practicalities Need to align linac in sections (bins), generally overlapping. Changing energy by % quad scaling: only measures dispersive kicks from quads. Other sources ignored (not measured) Changing energy upstream of section using RF better, but beware of RF steering (see initial launch) dealing with energy mismatched beam may cause problems in practise (apertures) Initial launch conditions still a problem coherent β-oscillation looks like dispersion to algorithm. can be random jitter, or RF steering when energy is changed. need good resolution BPMs to fit out the initial conditions. Sensitive to model errors (M) Ballistic Alignment Turn of all components in section to be aligned [magnets, and RF] use ballistic beam to define straight reference line (BPM offsets) y y + s y + b + b BPM, i i offset, i noise, i Linearly adjust BPM readings to arbitrarily zero last BPM restore components, steer beam to adjusted ballistic line 6

46 Ballistic Alignment angle α i quads effectively aligned to ballistic reference b i qi ref. line L b with BPM noise 6 Ballistic Alignment: Problems Controlling the downstream beam during the ballistic measurement large beta-beat large coherent oscillation Need to maintain energy match scale downstream lattice while RF in ballistic section is off use feedback to keep downstream orbit under control

47 TESLA LLRF