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Introduction to Stochastic Cooling
Hans Stockhorst 2nd September 2015 Forschungszentrum Jülich GmbH, Institut für Kernphysik/COSY BND School 2015 Landal Resort Eifeler Hof, Heimbach August 31 th – September 11 th , 2015 Aim of Beam Cooling
° Reduce transverse (horizontal and vertical) beam dimensions ° Reduce momentum spread in the beam ° No beam losses: phase space density increase
• Beam accumulation of (rare) particles: Cooling to make space available so that more particles can be stacked in the storage ring • Internal Target Experiments: Compensate growth in beam size, compensate mean energy loss, keep good beam-target overlap, keep momentum resolution
2nd September 2015 H. Stockhorst Folie 2 Early History of Stochastic Cooling at CERN
1968 Idea of stochastic cooling by van der Meer (CERN) 1972 First observation of Schottky noise at ISR. 1972 Theory of transverse stochastic cooling. 1975 First experimental demonstration of transverse cooling at ISR. 1975 pbar accumulation schemes for ISR and SPS. 1975 Filter method for momentum cooling. Palmer (BNL), Thorndahl (CERN) 1977 & 1978 Refinement of theory and detailed experimental verification at ICE. 1981 & 1982 Accumulation of several 10 11 pbars in AA from batches of several 10 6 pbars. 1986 & 1987 Construction of AC. 1983 Observation of W and Z bosons (carriers of the electro-weak force). 1995 Observation of top-quark at Fermilab.
S. van der Meer (1925 – 2011): Nobel prize in physics 1984 D. Möhl (1936 -2012): pioneer in stochastic cooling theory and beam physics
F. Caspers and D. Möhl, “History of stochastic beam cooling and its application in many different projects”, Eur.Phys.J. H36 (2012) 601-632
2nd September 2015 H. Stockhorst Folie 3 Description of Stochastic Cooling
° Time Domain: Sampling Picture • Statistical approach • Yields cooling rate equation • Mixing, electronic and particle noise introduced heuristically
° Frequency Domain: • Yields precise desciption of cooling with Mixing • Description of Betatron Cooling • Description of Momentum Cooling: ‹Filter Method, Palmer Cooling, Time of Flight Cooling • Hardware can be included in description (PU and KI response, amplifiers, Filters, power etc.)
References: D. Möhl, CERN Accelerator School CAS, CERN 87-03, 1987 D. Möhl, Stochastic Cooling of Particle Beams, Lecture Notes in Physics 866, Springer 2013
2nd September 2015 H. Stockhorst Folie 4 Time Domain Description of Betatron Cooling
Basic Setup: ° Due to quadrupole magnets in the ring and random initial phases: particles execute betatron oscillations around the nominal orbit ° Cooling goal: reduce betatron oscillations
• Pickup: measure position • Kicker: deflector
Betatron oscillation of single particle Nominal orbit
Electronic delay adjusted to particle travelling time Courtesy by D. Möhl, CERN from PU to Ki for nominal particle ( ∆p/p = 0 )
2nd September 2015 H. Stockhorst Folie 5 Normalized Phase Space Betatron Oscillations at position s in the ring: 4 Smooth sinusoidal approximation β ≈ R/Q 2 betatron oscillation x(s): a D ψ
cm 0 x( s )= a ⋅ cos( ϕψ ( s ) + ) @ y R dx( s ) y(s)= =−⋅ asin((s)ψ ϕ + ) -2 Q ds
s: position along the ring -4 -4 -2 0 2 4 R: ring radius ϕ: initial phase x @cmD Betatron phase: ψ(s) = Q ⋅s/R Beam emittance ε ∼ a2 • Q: number of betatron oscillations per turn • Q is no rational number (therefore: neither an integer nor an half integer)
2nd September 2015 H. Stockhorst Folie 6 Betatron Motion (artist’s view)
beam pipe
cooled beam
• Goal for betatron cooling: reduce amplitude of betatron oscillation
2nd September 2015 H. Stockhorst Folie 7 Optimal phase advance ∆ψ from PU to KI: 90 degrees plus (even) multiples of 180 degrees 4 PU KI Pickup (PU)
2 Particle 1: optimal reduction 1 50 100 150 200 250 300 350 2 90 0 -2 4 3 - 4 Particle 2: partial reduction PU2 KI
-100 100 200 300 400 500 600 Kicker (KI) ú: center of gravity -2 4 PU-4 KI Particle 3: no reduction, but Q is not an integer 32 1 2
- 100 100 200 300 400 500 600 Green dot : particle at KI before correction -2 Blue: after correction
2nd September 2015 H. Stockhorst Folie 8 -4 Finite Cooling Bandwidth W
° Single particle can not be resolved ° Finite bandwidth W of the cooling system: • A single particle produces a pulse at the pickup which is broadened to T S due to the finite bandwidth
*) 1 Sample time: T = S 2W PU KI
° A particle at time t 0 at the pickup is not only kicked due to its own error signal (coherent
effect ) but also due to other particles in the time interval t 0 –TS/2 ≤ t ≤ t0+T S/2: • the finite bandwidth results in HEATING (incoherent effect) due to the other
particles in the sample of length T S ° single test particle picture ➨ sampling picture of cooling
2nd September 2015 H. Stockhorst *) Courtesy by D. Möhl, CERN Folie 9 Beam Samples
° Finite bandwidth W of the cooling loop: ° The cooling system resolves samples of the beam
In a continuous coasting beam (DC beam) of length T (revolution period) and particle number N: • Number of equally spaced samples:
T Example: ℓ = = 2WT 10 S T N = 10 , T = 1 µs, W = 2 GHz ••• ••• S TS = 250 ps
TS x = 4000 Sample S 6 NS = 2.5 ⋅ 10 N N • Number of particles per sample: N S = = (sample size) ℓ S 2WT
2nd September 2015 H. Stockhorst Folie 10 Take a beam with N particles Beam Sample with mean E[x] = 0 and variance σ2 = E[x 2] ∫ 0
Beam Sample x Beam Sample Distribution
NS << N
±σ / N S
° Sample averages fluctuate around zero
° Standard deviation ±σ / N S
2nd September 2015 H. Stockhorst Folie 11 Sample Statistics
Take a beam with N particles with mean E[x] = 0 and variance σ2 = E[x 2] ∫ 0
0 x ° The sample averages 〈x〉S will fluctuate around zero: ±σ / N S Sample Relations: Average over the samples:
Mean: E x = E[x] = 0 S
2 2 2 Variance: E x = σ σ / N S S σ 2 E(x ) 2 = Squared mean: S N S Courtesy by D. Möhl, CERN Ψ µ =E[x] = ∫ x (x)dx Ψ µ σ 2=E[x] =∫ ( − x) 2 (x)dx 1= ∫Ψ ( x )dx
2nd September 2015 H. Stockhorst Folie 12 Example:
• Discrete distribution with values x = -1,0,1 having equal probability : µ = E[x] = 0 and σ2 = E[x 2] = 1/3 ((-1) 2 + 0 2 +1 2) = 2/3
• Form all possible samples with size N S = 2
Before Correction: 1 NS Sequence: Sample
2nd September 2015 H. Stockhorst Folie 13 Towards the Cooling Equation
The test particle receives a correction proportional to its error x: ∆x = λ⋅ x
The error x will change from x to x C: xC = x − λ x
Finite bandwidth ⇔ finite sample size:
Others: xxxC= −λ − ∑ x i all particles in the sample others except the test particle
Coherent Incoherent, heating cooling
1 x= x(N) − λ x ⇒ Or: C S∑ i xC = xgx − ⋅ S NS sample
° The test particle receives a correction proportional to the sample average x S
2nd September 2015 H. Stockhorst Folie 14 Application of Sample Relations
1 NS 1 NS Sample the beam: meanx= x variance x2= x 2 S ∑ i S ∑ i N S i= 1 N S i= 1
1. First correction: Each particle in the sample is corrected as xk→ x k − gx S N new 1 S 2 2.a Calculate new sample variance: x2 = xgx − S ∑()k S N S k= 1
NS new 1 2.b Calculate new sample mean: x= xgx − =− (1g)x S∑()k S S ! N S k= 1
3. Repeat 1. & 2. for all samples and average over all samples: NS 2 1 2 σ C=E∑() xgx k − S 221 22 NS k= 1 Apply sample relations: σ σ σ C =−()2g −⋅ g N S
2221 22 Change of beam variance in one turn σ: ∆σ σ σ =−=−C ()2g − g ⋅ N S
2nd September 2015 H. Stockhorst Folie 15 Cooling Rate Equation (simple form)
1 N 1 Since T = ⇒ number of particles in a sample: N = S 2W S T 2W
g2 1 1 d∆σσ 2 1 2 2W =− =− =()2g − g 2 2 2 rate 2g σ σ τ 2 dt T N σ heating cooling ° Use large bandwidth W ° Large particle number N ⇒ cooling rate decreases g =1 2 ° Choose optimum gain g = 1 opt ° Optimal cooling rate 1 2W = τ 2 N σ opt
2nd September 2015 H. Stockhorst Folie 16 Example: Simulation of One-Turn Correction
• Discrete distribution with values x = -1,0,1 having equal probability : µ = E[x] = 0 and σ2 = E[x 2] = 1/3 ((-1) 2 + 0 2 +1 2) = 2/3
• Form all possible samples with N S = 2 g = 1
Before Correction: After Correction: 1 NS Sequence: Sample Sequence: Sample
2nd September 2015 H. Stockhorst Folie 17 Example: Mixing and second correction
Before Correction: After Correction: after Mixing Sequence: Sample Sequence: Sample average variance average variance
mean: 0 mean: 0 variance: 1/3 variance: 1/6 (average over 9 samples)
• After mixing: First turn: Second turn: sample average reappears! σ 2 = 2/ 3 →σ 2 = 1/ 3 → σ 2 = 1/6 • Cooling!
2nd September 2015 H. Stockhorst Folie 18 Mixing ° In the model up to now: After one turn sample averages are zero ⇒ error signals vanish: cooling will stop!?
° Fortunately: Particles in the beam have different momenta: ∆p ∫ 0 ⇒ spread in revolution time ∆T ⇒ particles will mix: sample error reappears ⇒ cooling proceeds until all particles have zero error
1 1L D( s ) ∆∆ T p αη = − α = ds For one revolution: = − η 2 ∫ T p γ L0 ρ (s) 2 γ = E/m 0c D(s): dispersion ρ: radius of curvature L: ring length
Mixing factor M: number of turns a particle T 1 M =S = with ∆T needs to migrate by one sample length T S: η∆∆ T 2WT p/p
2nd September 2015 H. Stockhorst Folie 19 The Mixing Dilemma
Delay for signal from PU to KI set to nominal particle Kicker to Pickup:
∆ ∆ T p 1 1 D( s ) KP = − η αη = − α = ds TKP p KPγ 2 KP KP s∫ ρ (s) KP KP sKP
Pickup to Kicker: 1 D( s ) ∆ ∆ TPK p 1 = − η αη = − α PK = ds PK PK2 PK s∫ ρ (s) TPK p γ PK sPK
T / 2 Mixing factor PU to KI: M * = S Courtesy by D. Möhl, CERN ∆TPK
° M should be small, best value M = 1 (good mixing) ° M* should be large (no mixing)
2nd September 2015 H. Stockhorst Folie 20 Unwanted Mixing from PU to KI: - Correction of coherent term -
Kicker pulse: 1 2 ∆TPK 1 K(T)1∆ PK =− =− 1 *2 TS / 2 M ∆TPK
-TS/20 TS/2 T /2 too fast too slow M * = S Time at kicker: T PK = Tref + ∆TPK ∆TPK
• Nominal particle ( ∆p/p = 0) receives maximum correction K = 1. • Particles which are too slow or too fast receives a reduced correction K < 1.
2nd September 2015 H. Stockhorst Folie 21 Improved Cooling Rate for betatron motion x(s), σ = x rms
1 1 dσ 2 2W 1 =− =⋅⋅2g sin( ∆µ )1 ⋅− −g 2 () MU + 2 *2 στ 2 dt N M σ
° Unwanted mixing ° Wanted mixing from PU to KI: from KI to PU:
M* should be large: M ≥ 1 should be small, no mixing from PU to Ki best value M = 1 ° For Betatron cooling: ° Electronic noise: ∆µ : phase advance from increase in heating, PU to KI additional term g 2 U • optimal: ∆µ = π/2 mod π ° U ∼ 1/N σ2: noise to signal ratio
2nd September 2015 H. Stockhorst Folie 22 Improved Optimal Betatron Cooling Rate
With optimal gain: + M+ 1 gopt = withM = 1 − *2 MUM+ for ∆µ = π/2
2 + + 2 1 1dx 2 2W (M ) 1 1dε W (M ) =−rms = ⋅ rms 2 =− =⋅ τ 2 x dt N M+ U ε τ dt N M+ U σ opt rms ε opt rms
2 2 Betatron motion σ = xrms Emittance ε
2nd September 2015 H. Stockhorst Folie 23 Asymptotic Behavior of Cooling Time τ
N 1 τ ∼ ()M+ U with U ≈ W N
τ
N N Mixing limit ⋅ M ⋅U W W
Amplifier noise limit: independent of particle number N
2nd September 2015 H. Stockhorst Folie 24 Noise-To-Signal Ratio for Betatron Cooling (1)
From frequency domain approach of cooling:
electronic noise power k(T+ T )G2 W U = = A R A Schottky particle powe r 2 2ZP′ 2 N( Z e ) f0 G A W ε ⋅ β P ZC
k: Boltzmann constant (= 1.38 x 10 -24 J/K) N: particle number
TA: equivalent amplifier noise temperature Ze: particle charge ε: beam emittance TR: pickup noise temperature f0: revolution frequency GA: amplifier gain
Z´P: pickup impedance [ Ω/m]
ZC: line impedance (50 Ω)
βP: betatron function at PU
2nd September 2015 H. Stockhorst Folie 25 Noise-To-Signal Ratio for Betatron Cooling (2)
To reduce the noise-to-signal ratio U
2 k(TA+ T R )G A W U = 2 2ZP′ 2 N (Z e ) f0G A W ε ⋅ β P ZC
• Cool pickup electrodes and use low noise amplifiers
• Use PU structures with large coupling impedance Z´P • Use large betatron function at PU • During cooling the emittance ε decreases and U increases ° If possible, move electrodes during cooling closer to the beam to
increase the coupling impedance Z´P • If N and Z large: cooling becomes independent of charge number
2nd September 2015 H. Stockhorst Folie 26 Noise-To-Signal Ratio: Example
W = 1 GHz N = 10 9 protons
TA: 20 K T R: 20 K f0 = 1 MHz Z’ = 3 k Ω/m Z : 50 Ω pl C ε = 5 mm mrad βPU : 8 m
GA: 1 • Thermal noise power in 1 GHz: 0.06 pW (at room temperature ≈ 1 pW) • Schottky signal noise power in 1 GHz: 0.2 pW
⇒ U = 0.3 ( = 5 at room temperature 300 K!)
⇒ Cryogenic cooling of pickup electrodes essential!
2nd September 2015 H. Stockhorst Folie 27 Time Evolution of Beam Emittance
dε W W =−()2gM+ − gM2 ε ⋅+ε g 2 (U) where Uε is independent from ε dt N N
−t / τ • First order differential equation for εε: ε ε ε (t)=( (0) −∞) e + ∞
1 W g =()2gM+ − g2 M Equilibrium emittance: ε ε = ⋅ (U ) τ N ∞ 2M+ − gM
1.0 Optimal cooling rate: 10 0.8 2 + 8 1 W (M )
0.6 L
= ⋅ e
N 6 U W H
τ N M ê
opt t 1 • 0.4 e 4 M+ = M = 1 M+ = M = 1 1 k(TR+ T) A 0.2 ε = ⋅ 2 ∞ M 2 2 Z P′ N() Ze f 0⋅ β P 0.0 0 0.0 0.5 1.0 1.5 2.0 0.0 0.5 1.0 1.5 2.0 ZC g g 2nd September 2015 H. Stockhorst Folie 28 Stochastic Cooling Systems at COSY (present status)
Stochastic Cooling in COSY: ° Horizontal system: 1.8 – 3.0 GHz (band II) ° Vertical system: 1.0 – 3.0 GHz (band I + II) ° Longitudinal system: 1.0 – 1.8 GHz (band I) ° Proton momentum range: 1.5 GeV/c to 3.6 GeV/c (β = v/c > 0.85) ° Systems can be independently adjusted
Longitudinal cooling with optical notch filter or Time-Off-Flight cooling technique
2nd September 2015 H. Stockhorst Folie 29 Example COSY
° Diagonal signal paths:
sPK = sKP = L/2 (L: ring length = 174 m) ° Dispersion zero on straights:
ηPK = η ⇒ M* = M
W = 2 GHz η = 0.1 5 10 -4 N = 10 protons T = 1 µs ∆p/p = 2 ⋅ 10 4 PU structures cooled: U ≈ 0 3 T 1 ⇒ M=S = = 13 ⇒ M* = 13 η∆∆ T 2WT p / p 2
emittance [mm mrad] emittance [mm 1
1 W 1 −1 = ⋅ ≈ 0.015 s or (τ ε )opt ≈ 65 s 0 0 50 100 150 200 250 300 350 400 τε opt N M time [s]
2nd September 2015 H. Stockhorst Folie 30 First Successful Betatron Cooling at COSY (1)
• First observation of vertical stochastic cooling at COSY eighteen years ago. • Vertical beam profiles were measured with EDDA. • Cooling with only band I (1 -1.8) GHz • Gain and phase not From logbook optimized at that time • No beam losses
2nd September 2015 H. Stockhorst Folie 31 First Successful Betatron Cooling at COSY (2)
Decrease of the Vertical Beam Width during Cooling • The vertical beam width 3.5 Fit = A + B x exp(-t/ τ) shrinks exponentially 3
towards an equilibrium. τ = 376 sec A = 0.63 mm Stockhorst, 1997 2.5 • The FWHM of the beam is gain of cooling system not optimized, band I only reduced by a factor of five 2 after about 1500 s. 1.5 FWHM [mm] FWHM • The system was not 1 optimized. 0.5 Profilmessungen.kaldat(ProfilDecrease_1.plot), H. 0 0 500 1000 1500 2000 t [sec]
2nd September 2015 H. Stockhorst Folie 32 Summary Betatron Cooling
° The cooling rate is proportional to bandwidth W and inversely proportional particle number N ° For stochastic cooling a DC beam is favorable ° Betatron cooling: • Phase advance ∆µ = π/2 plus multiple of π • Betatron tune Q not an integer or half integer © ° Signal delay adjustment ° Mixing is necessary ( η ∫ 0) and is produced by the dispersion in the magnetic lattice ° Mixing between KI and PU should be large, between PU and KI it should be low! ° Mixing is always a compromise, except when the lattice allows
to change the optics to make ηPK = 0 and ηKP ∫0
2nd September 2015 H. Stockhorst Folie 33 Stochastic Momentum Cooling
• Approximately: Similar rate equation for the rms relative
momentum spread σ ∆ = ( p/p) rms :
2 1 1 dσ 2W 1 2 =− =⋅⋅−2g 1 −g () M + U στ 2 dt N M *2 σ 2
• But: Momentum distribution Ψ(σ,t) changes during cooling • Therefore: For a complete description of momentum cooling a Fokker-Planck Equation for the time evolution of the beam momentum distribution Ψ(σ,t) must be solved:
∂ ∂ ∂ Ψ σ Ψ σ Ψ σ ( ,t ) = −F(σ , t ) (,) t − D(σ , t) (, t) ∂t ∂σ ∂ σ
Drift: coherent cooling Diffusion: incoherent heating
2nd September 2015 H. Stockhorst Folie 34 Momentum Cooling at COSY Simulation: Measurement: Red : initial -30 dB Green : 40 s Magenta : 90s distribution final Blue : equilibrium
initial
time
5.5 min
Gain 126 dB Simulation: ° Protons p = 2.425 GeV/c γ = 2.771 ° Number of protons N = (6 – 8) ⋅ 10 8
° Lattice: γtr = 2.343 measured η = - 0.05 - 15 dB ° Dispersion D = D’ = 0 in telescopes - 20 dB
° Cooling system: band II (1.8 – 3) GHz ∆∆ f p - 30 dB ° Filter momentum cooling =η ⋅ f0 p 0
2nd September 2015 H. Stockhorst Folie 35 Stochastic Cooling in the COSY-11 Experiment (September 1998)
• Transverse and Longitudinal Cooling switched ON • Highest momentum 3.3 GeV/c • Cycle length one hour
• Counting rate becomes nearly constant when cooling is ON
2nd September 2015 H. Stockhorst Folie 36 Summary (1) • Stochastic particle beam cooling has been discussed in time domain. • The finite bandwidth of the signal processing system is too small to resolve a single particle. Instead samples of the continuous coasting beam (DC beam) with millions of particles are resolved. • This led to the sampling picture. • Considering a “test” particle in the sample it receives not only a correction at the kicker due to its own error but also “kicks” from the other sample particles. • This led heuristically to the concept of mixing and heating. • Unwanted mixing from pickup to kicker that diminishes the coherent cooling effect has been introduced to account for the finite pulse length at the kicker.
2nd September 2015 H. Stockhorst Folie 37 Summary (2) • The application of the sample relations to the beam samples led to the rate equation for stochastic cooling. • The equation has been improved by unwanted mixing and wanted mixing. Electronic noise that increases the heating term has been introduced ad hoc. For betatron cooling the coherent term is modified so as to include the case of a non-optimal betatron phase advance from pickup to kicker. • The cooling rate is proportional to the system bandwidth W and inversely proportional to the number of particles N in the beam. • Schottky particle and electronic noise lead to an equilibrium value in either momentum or betatron cooling. Cryogenic cooling the pickup electrodes and low noise amplifier will reduce the achievable equilibrium value. In addition, the noise-to-signal ratio U can be reduced with high sensitive pickups.
2nd September 2015 H. Stockhorst Folie 38 Exercises
° Proof the sample relations. ° Verify the steps in the application of the sample relations to cooling. ° Continue the cooling sampling tables: ° Show that the “beam” variance decreases exponentially. ° Verify the formula for the optimal cooling rate. Determine the optimal gain. ° Discuss why the optimal mixing factor is M = 1. (Note: from the analysis in frequency domain it follows that always M ≥ 1) ° Discuss why the mixing factor M > 1 increases the heating term.
° Discuss the unwanted mixing from PU to KI. 1 f+ cos( 2 f T )df Compare the parabolic kicker pulse with the expression ∫ π ∆ PK W f derived in frequency domain (W = f + - f-). − ° Verify the asymptotic behavior of the cooling time. ° The cooling rate was derived under the assumption of a continuous coasting beam (DC beam). Discuss the effect on the cooling rate if the beam gets bunched by an RF cavity. Hint: the bunch length is less than the ring length.
2nd September 2015 H. Stockhorst Folie 39 Thanks to the audience for listening
2nd September 2015 H. Stockhorst Folie 40