Orbit Measurement and Correction

1:1 steering,HERA-E/P

orbit feedback,HERA-E/P

global BBA inarcs, HERA-E(for polarization)

dispersionfree steering,HERA-E

overview of HERA (2-ring, 920 GeV proton on 27.5 GeV lepton collider, L=6.3 km)

overview of the SLC (50 GeV single linac, double-arc electron-positron collider, L=3 km)

BBAin IP

BBAin IP

globalBBA

1:1, SVD steering

1:1, SVD, DFS steering

1:1, SVD steering

BBAin IP

+ s,e,c, & m – orbit feedback

overview of RHIC (2-ring, 250 * 250 GeV proton-proton collider; 100 GeV ion-ion collider, L=6.8 km)

1:1 and SVD steering

BBA in IP

BBA in IP

types of relative alignment errors and their effect on the beam:

multipole field expansion n

nnnxz iz))(xia(biBB

Lorentz force

x

z

“normal” (e.g. up-right) component

“skew” (e.g. rotatedby 45° component)

n=1: Bz+iBx=(b1+ia1)(x+iz) Bz=b1x Bx=b1z

Fx = -qvBz = -qvb1x Fz = qvBx = qvb1z

the off-axis beam sees the next-lower order field (“feeddown effect”)

bn 0, an= 0:

example: (normal) quadrupole

for x0, the field experienced is that of a dipole

B

F

Introduction

bn=0, an 0:

p. 65 ,below Eq. 2.106

Bz=b0 Re(x+iz)0=b0

Bx=b0 Im(x+iz)0=0

Bz=b1 Re(x+iz)=b1xBx=b1 Im(x+iz)=b1z

Bz=a0 Im(x+iz)0=0Bx=a0 Re(x+iz)0=a0

normal dipole

“skew” dipole

Bz/b0

x

normal quad

“skew” quad Bz=a1 Im(x+iz)=a1zBx=a1 Re(x+iz)=a1x

x

Bz/b1

normal sextupole

skew sextupole

Bz=b2 (x2-z2)Bx=2b2xz

Bz=2a2xz Bx=a2(x2-z2) x

Bz/b2

(z=0)

x = xco + xβ + x dispersive orbit

defined by magnetic focussing

closed orbit

n

nnnxz iz))(xia(biBB

z=0, x=0: Bz/b2=(xco2+2xxco+x

2xco2)example:

off-axis beam sees sextupole, quadrupole, and dipole fields

off-axis in quadrupole dipole field:

leads to dispersive emittance growth when ≠ 0 since

x =η and x ~ <x2>1/2

(here x=transverse position offset)

leads to beam depolarization when magnetic moment of particle μ experiences a transverse deflecting field B

off-axis in sextupole quadrupole and dipole fields:

introduces additionally β-beats (gradient errors), vertical dispersion, betatron coupling, …

off-axis in accelerating structures wakefield kicks:

dilution of projected beam emittance since different particles within the bunch follow different trajectories

alignment (by surveying) beam-based alignmentsteeringfeedback

generates dispersion D~1/ρ

causes sampling ofnonlinear fields, contributesto emittancegrowth

solutions:

In addition to perturbing the design optics,

focus of this lecture

famous question: “where is zero?”

xco = xd + xb + xm

measured beamposition

beam centroid position with respect to thereference axis

quadrupolemisalignment

BPM electronic and/ormisalignment offset

the “answer” depends on application chosen:

BBA of single elements and global BBA

one-to-one steering

SVD steering

dispersion-free steering

absolute position of beam wrt magnet center - best solution (but invasive, time consuming, and for global BBA, complicated)BPM values zeroed, corrector values (and ) nonzero - beam may be still off-axisBPM readings and corrector values minimized using empirical weighting of the unknown offsetsBPM reading and corrector values minimized with additional constraints given by dispersive orbits

orbit correction method consequence(s)

Beam-based alignment of single elements: quadrupoles

typical residual alignment errors after surveying: ~ (100-200) μm

typical quadrupole alignment tolerances in storage rings (i.e. synchrotron light sources): ~ 10 μm circular collider IPs (i.e. B-factories, HERA, LHC) ~ 5 μm linear collider IPs (i.e. NLC, TESLA) < 1 μm

Beam-based alignment determines the relative offset between magnet centersand nearby BPMs. If the offsets are sufficiently stable, then simple orbit cor-rection (steering) can be used to maintain a well-centered orbit.

BBA using quadrupole excitation

s

(xb and xq are unknown)

xb=0

xq≠0

dipole kick

s

xb=0

xq=0

s

xb≠0

xq=0

ideal case nominal case(after steering “flat”)

ideal case(with misalignments)

single-pass measurement (transport line or linac):

Θ=K·xq

dipole kick (for horizontally focussing quadrupole)

change in integratedfield B(I), where B(I) depends on magnet properties

quadrupolemisalignment

Icor (A)

x (

mm

)

please replace quad with cor in figure caption 3.6, p. 74

example from the TESLA test facility (courtesy P. Castro, 2000)

s

triplet

corrector

BPM

total quadcurrentvariationof ~ 5%

minimumdisplacement

K (set fieldto zero)

ρ(eVs/m)Vs/m3

(in 1/m)

m

quad center

slope ~ R12

since x~ (Icor)

independentof absoluteBPM position (and electronicoffsets)

multiple-pass measurement (circular accelerator):

Θ≈K·xq- K x + O (K· x) s

BPM #1(as before) change in the closed orbit

offset at the quadrupole

solve iteratively using expression for the c.o.d. at the location of the kick (Eq. 2.34):

solving for x:

and inserting back into first eq. above:

example from SPEAR (courtesy J. Corbett, 1998)

quadrupole shunt circuitry (allowing variation of single quad given multiple magnets per power supply)

BPM #2

xq

BPM #1

BPM #2

quad center

xq=orbit offset at quadrupole(varied using local bump)

warning: new analyses…

x=

diff

ere

nce

orb

itm

easu

red a

t B

PM

quadrupole gradient modulation (LEP)

example from LEP (courtesy I. Reichel, 1998)

modulation of single quadrupoles with (non-resonant) discrete frequencies

response of beam to modulationof four different quadrupoles

FFT of turn-by-turn BPM measurements

~ 1

μm

FS

natural orbitdrift andcorrection ata single quad(time scale, 6 hrs)

orbit variationmeasured at BPMclose to modula-ted quad usingposition jitter(time scaleshort)

minimum givesrelative BPM offset (~200 μm)

orbitcorrection

relativeoffset

Beam-based alignment of single elements: sextupoles

Possible approaches (using sampled quadrupole field):

1. use additional quadrupole trim windings, mounted on the sextupoles, quadrupole-based BBA (KEK) – (assumes coinciding magnetic centers of trim windings and sextupole)

2. measure shift in betatron tune vs sextupole strength (PEP-II, HERA,…):

θ x~ ( xβ + xco )2 - ( zβ + zco )2 ~ … + xβxco + … zβzco + …

Θx~Bz=b2 (x2-z2)Θz~Bx=2b2xz

recall (normal sextupole):

off-axis in x (xco≠0) θx~xβ (quadrupole field); similarly in z

3. measure tune separation near the difference resonance vs sextupole strength (NLC): θz ~ … + xβzco + …

off-axis in z (zco≠0) θz~xβ (skew quadrupole field)

need foreach sextu-pole an indi-vidual power supply

1. using local orbit bumps (KEK,DESY):

horizontal sextupole alignment: θx = -0.5 Ks (xbump-xs)2

θz = -0.5 Ks (xbump-xs)zs

vertical sextupole alignment:

θz = Ks (x0-xs)(zbump-zs)

independent of the number of sextupolesdriven by a single power supply

2. using induced orbit kick vs sextupole position (KEK B-factory, SLAC FFTB)

example from the FFTB (courtesy P. Tenenbaum, 1998)sextupoles mounted on precision movers,

measure quadratic change in closed orbit(transfer line or linac)

center atdxb/dxm=0

change strength of all sextupoles by Ks measure induced orbit changerepeat for different bump amplitudes

x0 is a preferably large horizontal orbit

offset (for enhanced sensitivity)

xs and zs are the sextupole offsets

Possible approaches (using sampled dipole field):

Beam-based alignment of accelerating structures

s

x couple out power at natural resonantfrequencies of the structure (higher-ordermodes) if x,y≠0 using output coupler

example from the ASSET experiment at SLAC (courtesy M. Seidel and C. Adolphsen, 1998)

signal amplitudeat fc = 16 GHz ~ 7.5 MHz(after mixing)

ph

ase

(deg)

T (structure central axisto beam position inferedfrom BPMs)

relative positionoffsett

please fix scale on bottomplot of Fig. 3.10, p. 79

90

-90

Lattice diagnostics and R-matrix reconstruction

simple FODO lattice

corrector dipole quadrupolewith BPM

point-to-point transfer matrix:

plot x=R11x+R12x’max

measure R12=dx/dx’

method I “jitter data” plus careful 2-dimensional error propagation, parasitic

method II 2-point fit x(x’=0), x(x’≠0) or, for better measurement (smaller error), x vs x’

example: linac-to-ring transfer line (SLC)

to probe all elements in the beamline, need a second corrector dipole separated in betatron phase by π/2

angular deflection (“kick”) due to dipole:

since then

slope:

Bρ=magnetic rigidity =E(GeV)/0.3 [T-m] =p (MeV)/300 [T-m]

measurementat a single locationas the strengthof an upstreamcorrector isvaried

example: linac-to-ring transfer line (SLC), continued: “fitted” R12

then minimizeconcept: acquire data using all available BPMs while one upstream corrector is varied

fitting for the best incoming orbit given by x1, x1’

x(s) y(s)

data (solid line): measured trajectory fitted (dashed line): x1 + R1jxj’ (using R1j measurements and x1 fitted)

60 m, 20 BPMs 60 m, 20 BPMs

Comparison allows for localization of optics errors and BPM errors (these BPMs may then be excluded in subsequent steering procedures)

calc

Suppose there is poor agreement between expectation and measurement. Possiblecauses include: optical errors (including knowledge of B-fields) incomplete model corrector and BPM gain/polarity (and location) errors

example: lattice diagnostics in the main SLC linac (courtesy T. Himel and K. Thompson, 1999)

energy dependence of the point-to-point transfer matrix obtainedusing a Taylor series expansion to first order in the energy

expanded R-matrix:

with Nk(=240)=total number of klystrons, Ek=energy gain per klystronwith Nk(=30)=total number of sectors, Ek=energy gain per sector

1500 m, 300 BPMs

data (solid line): measured trajectory fitted (dashed line) as before with linear scaling of beam energy

evidence of 180° phaseshift in center of linac

chromatic phase advance in a FODO cell:

with

if not taken into account, expect a 90° cumulative error with =2.5%, =30sectors●2π/sector

(90° FODO cell)

Possible causes for discrepancy

unknown energy errors

procedure used for further analysis: 1. measure xm(s) versus the applied deflection 2. select a set of Ek’s 3. compute

4. iterate for minimum 2

BPMs BPM

c2

m2

N

1xxχ 2

measured

computed

result

before theenergy fit

after theenergy fit

(bad BPMs)

model still lacksall physics input(i.e. energy spreadvariation alongthe linac not takeninto account)

the model offersreduced complexity(30 fit parameters, which are well con-strained given the~1200 R-matrix elements)

the estimatedenergy error ~30%,greatly exceeding the estimateduncertainty

Nevertheless, the predictive power of the model is very useful; e.g. for localization of BPM and klystron phase errors

data (solid line) fitted (dashed line)

data (solid line) fitted (dashed line)

Beam Steering Algorithms

x = xco + xβ + x dispersive orbit

orbit defined by magnetic focussing

closed orbit

review: so far we have discussed beam-based alignment of single elements which, while accurate, is time consuming and so used most often during early stages of accelerator commissioning and in interaction regions, where single-element alignment tolerances are most critical lattice diagnostics which is often needed prior to being able to implement automated steering algorithms which may be model-dependent and are significantly less prone to errors if “bad BPMs” are excluded from the data samples until now (in the discussion of BBA and lattice diagnostics) only the coherent motion of the bunch centroid (center of charge) has been considered

Beam steering is applied over long regions of the accelerator and is significantly moretime-efficient than BBA of single elements. Note that while the center of charge is measured by the BPMs, the best beam steering algorithms ideally minimize the projected beam emittance.

Again (with x = horizontal or vertical),

σx = <x2>1/2

= <(xco+ xβ + x)2>1/2

which contains for example a dispersive term σ=<x2>1/2=2<2>1/2

The emittance εx ~ σx2/β, where

“One-to-One” steering (linear least-squares algorithm)Beam is steered to zero the transverse displacements measured by the BPMs. The BPMs are typically mounted inside the quadrupoles (where the β-functions are largest).

closed bump that would minimizethe BPM readings (but would alsogenerate dispersion)

Beam position measured at a downstream BPM (denoted by subscript, j):

indices idenote eachcorrector kick

In matrix form

Solving the matrix equation:

with

or x is the vector containing the BPM measurements

Θ is the vector containing the unknown kick angles

m is the total numberof BPM measurements

n is the total numbercorrectors

( (

i-j )

Again , the solution is given by

Here, θ is an (n1) matrix [n=number of correctors] M is an (nm) matrix x is an (m1) matrix [m=number of BPM readings (per plane of interest)]

If

m>n, the matrix M is overdetermined m=n, the matrix M is square and the solution is simply θ=M-1x

m<n, the matrix M is underdetermined and more measurements are needed (to ensure that the number of unknowns is at least equal to the number of measurements). To constrain the solution in this case, some parameter (e.g. the beam energy) needs to be varied and the measurement repeated (however, the measurement fails to be noninvasive)

minimization procedure:

The solution to the matrix equation can equivalently be found byminimizing

fitting function with unknowns Θi (i = number of unknown kicks)

measurements (j = number of BPM measurements)

i.e.

(algebra)

(reorganizing terms)

This equation, expressed in matrix form, is equivalent to the result from before:

i.e. vary (possibly numerically)the unknowns θi given the BPMmeasurements xj and the assumed transfer elementsMji so that the relation holds(as closely as possible)

Global beam-based steeringTake into account now that the electrical center of the BPMs ≠ center of quadrupole

xm

xq

beam position wrt quad center

beam position wrt reference axis

beam position wrt reference axis transported between quad (j) and quad (j+1)

beam position wrt quad center transportedbetween quads (j) and (j+1)

= beam position at quad k

sum over allupstream quadrupoles

xbpm

rearrange terms

again

function to be minimized:

measurements fitting function with unknowns xq, xbpm, and(x0,x0’)

Since the number of measurements is about ½ the number of unknowns, an independent set of measurements is required. One possibility is to scale the lattice (all quadrupole and corrector strengths) and repeat the measurement.

sum over upstream quadrupoles

Singular valued decomposition (SVD)Global orbit correction using the SVD algorithm offers the advantage of reducing not onlythe rms of the BPM measurements, but also the rms of the applied corrector strengths.Correctors compensating (or “fighting”) one another generate unwanted dispersion. This powerful algorithm may also be used in application to orbit feedback, dispersion-free steering, and in the computation of multiknobs. [caveat: “reversibility” in practicalapplications]

Matrix equation to be solved:

measured orbitdisplacements,

(linear) orbitresponse matrix,

(m1) matrix

m=number of BPM readingsn=number of correctors

matrix of corrector kickangles, (n1) matrix

If m>n, the matrix A may be decomposed (easilyusing matlab, for example)

(mn) matrix

U is an (mn) matrixV is an (nn) matrix

The column vectors of U and V are orthonormal satisfying (In is the (nn) identity matrix)

Again,

m>n: SVD solution will agree with the result of the least-square fit

m=n: A is square matrix and the solution for the correctors is

If none of the wi are zero, the solution is unique (example to follow). If one or more of the wi are zero (degenerate case), there may not be an exact solution. In this case, replace 1/wi by zero which gives still the solution in the least squares sense; i.e |A●θ-x| and |θ|2 are minimized.

m<n: Add rows with zeroes to the vectors and matrices in x=Aθ until the A matrix is square and apply the SVD algorithm as just described (again a degenerate case)

A is an (mn) matrixm=number of BPM readingsn=number of correctors

Numerical example of the SVD algorithm

correctors

BPMs

s

c1 - c2 = π

1 - 2 = π

x

using x=R12θ andplus a convenient normalization

1 - c2 = π/2

fifi12 sin ββR Matrix equation to be solved

Given the above geometry, there is no exact solution for placing the beam at arbitrary desired positions at both BPMs. However, there is an SVD-solution. Suppose we want to steer the beam to xt1=1 and xt2=0.Then

which may be decomposed as

with solution Θ=A-1 x

The positions at the BPMs with this solution are x1=1/2 and x2=-1/2. With this solution, the quadraticdistance to the target values ∑i(xti-xi)2 and the corrector strengths ∑jθj

2 are minimized.

Dispersion-free steering (Raubenheimer, Ruth)

one-to-one steering – a first step in orbit correction but imperfect since dispersive errors may be generatedglobal beam-based alignment of quadrupoles – works well at low beam currents where there is no wakefield-generated dispersionwakefield bumps – more local but highly sensitive to small perturbationsdispersion-free steering – even more local correction of dispersive errors including dispersive errors arising from transverse wakefields

absolute orbitt:

including now theenergy-dependence(adiabatic damping)

To constrain the system (taking into account the additional unknown energies), the beam energy may be changed (not so easy) or the lattice (quadrupoles and correctors) may be scaled to mimic a change in beam energy. Then

differenceorbit:

factor missing in text, Eq. 3.41

1K

ΔKκ

K is the quadrupolestrength

again, i=1,2,…,n, n = no. correctors j=1,2…,m m = no. BPMS

tmeasured center-of-charge.please replace “for a deflection applied in” with “between” before Eq. 3.41, p. 92

Again,

absolute orbit difference orbits

minimize the absolute orbit and the difference orbit simultaneously:

(2M1)(2MN)

(N1)

SLC experience: initially, solution overconstrained by scaling lattice in steps (K/K=0%, -10%,-20%, -30%, +5%, 0%); later, the problem of hysteresis was overcome (Raimondi) by measuring the orbits of the e+ and e- beams (whichis equivalent to K/K=200%)

simulated orbit and dispersive emittance with single 100 μm quadrupole offset (courtesy R. Assmann, 2000)

simulated orbit and dispersive emittance with single 100 μm structure offset (courtesy R. Assmann, 2000)

(single particle model)

~ 200σβ

~ (3-4)σβ

simulated orbit and dispersive emittance with quadrupole and structure offsets afterapplication of one-to-one steering (courtesy R. Assmann, 2000)

solid line: quadrupole offsetdashed line: structure offset

~ (3-4)σβ

measured dispersive orbits

=0.9

=0.7=0.8

=1.0

beforecorrection

after(3 intera-tions)of DFSsteering

Remarks:

1) after DFS, the maximum orbit difference (neglecting the bad BPM) was reduced from ~ 1.5 mm to < 200μm

2) after DFS, the rms of the absolute orbit was larger – suggesting that significant BPM and/or quadrupole misalignments were still present

measured absolute orbits after two-beam dispersion free steering

errors

typical error sources and their rms contributions:

σ(xj) BPM resolution <10 μm σBPM BPM misalignments ~100 μm σsys systematic errors (jitter and drift) ~20 μm

to propagate errors, introduce a weighting functionm denotesthe differenterror sources

sum overj BPMmeasurements

function to be minimized:

one-to-one steering

beam-based alignment

dispersion-free steering (recall x is a vectorwith positions and orbit differences, the Mijare transfer matrix elements with and with-out the energy-equivalent scaling)

Error evaluation with dispersion-free steering (Assmann)function to be minimized:

goodness-of-fit parameter defined with emphasis on the leading error sources:

systematic errors contribute less thanthe alignment errors in measurements of the absolute orbits

BPM misalignments cancel in the measure-ments of the difference orbits

weightingby disper-sion cor-rection

weightingby orbit correction

in absence of “deterministic fits”(number of unknowns = number ofmeasurements with no degener-acies), skill and experience are required in determining optimum weighting factors:

rms of vertical beam positions:

Summary:

Using the multipole field expansion, we discussed relative alignment errors and (some of) their effects on the beam (e.g. “feed-down effect”)

Typical magnet-to-beam alignment tolerances,

n

nnnxz iz))(xia(biBB

storage rings (i.e. synchrotron light sources): ~ 10 μm circular collider IPs (i.e. B-factories, HERA, LHC) ~ 5 μm linear collider IPs (i.e. NLC, TESLA) < 1 μm

are considerably tighter than what can be achieved by precision alignment (surveying) which is on the order of (100-200) m

To overcome these difficulties, beam-based alignment and beam steering are used

We reviewed various methods of beam-based alignment of single magnetic elements including various methods for aligning quadrupoles and sextupoles.

As a precursor to beam steering methods, “lattice diagnostics” was presented, whichis yet another method for localizing optics and BPM errors (the “bad BPMs” would then be excluded from steering algorithms)

An example was presented, whereby an incomplete model with known shortcomings (asaften the case) could nonetheless provided information on optical and BPM errors

The steering method of choice (incidentally: unavoidable in large accelerators) depends on many factors including availability and reliability of BPMs relative importance of well-controlled orbits versus investment of effort relative importance versus investment of beam time prevailing conditions (i.e. presence of wakefields at high beam currents)

Common methods of beam steering were described and algorithms were given for numerical analysis

The methods presented included: one-to-one steering global beam-based alignment SVD steering dispersion free steering

Aspects of the mathematics were illustrated using simple examples 2-corrector / 2-BPM 2 minimization and SVD solution

Completely overdetermined systems (as nowadays typical) require additional constraints(represented by the “weighting factors”, which implies non-deterministic solutions whichrequire experience and skill to optimize

Once optimal orbits have been determined, orbit feedback (not yet presented) may beused to maintain the orbits (provided that they don’t drift with time)

Tutorial on SVD and Least-Squares including “bad BPMs” and RHIC data available on request