Recent measurements of linear nonlinear optics at the

Recent measurements of linear & nonlinear optics at the ESRF storage ring Andrea Franchi (ESRF, Grenoble) on behalf of the Beam Dynamics & Diagnostics groups TW-DULER 2018, DIAMOND, 19 th -20 th April 2018

Outlines • Where we were in 2016 • Fast measurement and analysis of the orbit response matrix (ORM) • Measuring ultra-low coupling via turn-by-turn (Tb. T) BPM data • Calibrating sextupole magnets via chromatic functions and off-momentum ORM • Accuracy studies • Extra: dealing with “multiple beams” • Measuring momentum compaction: see Laura Torino’s talk Andrea Franchi Optics Measurements @ ESRF

Outlines • Where we were in 2016 • Fast measurement and analysis of the orbit response matrix (ORM) • Measuring ultra-low coupling via turn-by-turn (Tb. T) BPM data • Calibrating sextupole magnets via chromatic functions and off-momentum ORM • Accuracy studies • Extra: dealing with “multiple beams” • Measuring momentum compaction: see Laura Torino’s talk Andrea Franchi Optics Measurements @ ESRF

Where we were in 2016 • Linear optics measured & corrected weekly via ORM (10’+15’ , no need of switch BPM in Tb. T mode, works for any sextupolar optics, i. e. filling mode) => rms β-beat ~4 -5% 32 correctors ultra-low coupling εy/εx~1‰ 64 correctors PRSTAB 14, 034002 (2011) Andrea Franchi Optics Measurements @ ESRF

Where we were in 2016 • Linear optics measured & corrected weekly via ORM (10’+15’ , no need of switch BPM in Tb. T mode, works for any sextupolar optics, i. e. filling mode) => rms β-beat ~4 -5% ultra-low coupling εy/εx~1‰ • Measuring ultra-low coupling with Tb. T BPM data with kicked beam unsuccessful because of the low signal/noise of the coupling line in the Tb. T spectrum Andrea Franchi Optics Measurements @ ESRF

Where we were in 2016 • low kick • Linear optics measured & corrected weekly 1 mm via @ORM βx=35 m • nonlinearities (10’+15’ , no need of switch BPM in Tb. T mode, avoided works for any sextupolar optics, i. e. filling mode) =>line • coupling ~ background rms β-beat ~4 -5% ultra-low coupling noise εy/εx~1‰ • Measuring ultra-low coupling with Tb. T BPM data with kicked beam unsuccessful because of the low signal/noise of the coupling line in the Tb. T spectrum • large kick 3 mm@βx=35 m • nonlinearities pollute lines • coupling line >> background noise arxiv. org: 1603. 00281 Andrea Franchi Optics Measurements @ ESRF

Where we were in 2016 • Linear optics measured & corrected weekly via ORM (10’+15’ , no need of switch BPM in Tb. T mode, works for any sextupolar optics, i. e. filling mode) => rms β-beat ~4 -5% ultra-low coupling εy/εx~1‰ • Measuring ultra-low coupling with Tb. T BPM data with kicked beam unsuccessful because of the low signal/noise of the coupling line in the Tb. T spectrum • nonlinear lattice model and sextupole calibration from harmonic analysis of Tb. T BPM data (~2’, but need BPM in Tb. T mode with MAF filter ~15’x 2, works for a special sextupole optics only, tedious orbit control when calibrating sextupoles) PRSTAB 17 074001 (2014) Andrea Franchi Optics Measurements @ ESRF

Where we were in 2016 from norm. sext. lines to sextupole calibration • Linear optics measured & corrected weekly via ORM (10’+15’ , no need of switch BPM in Tb. T mode, works for any sextupolar optics, i. e. filling mode) => rms β-beat ~4 -5% ultra-low coupling εy/εx~1‰ • Measuring ultra-low coupling with Tb. T BPM data with kicked beam unsuccessful because of the low signal/noise of the coupling line in the Tb. T spectrum • nonlinear lattice model and sextupole calibration from harmonic analysis of Tb. T BPM data (~2’, but need BPM in Tb. T mode with MAF filter ~15’x 2, works for a special sextupole optics only, tedious orbit control when calibrating sextupoles) PRSTAB 17 074001 (2014) Andrea Franchi Optics Measurements @ ESRF

Where we were in 2016 from norm. sext. lines to sextupole calibration • Linear optics measured & corrected weekly via ORM (10’+15’ , no need of switch BPM in Tb. T mode, works for any sextupolar optics, i. e. filling mode) => rms β-beat ~4 -5% ultra-low coupling εy/εx~1‰ • Measuring ultra-low coupling with Tb. T BPM data with kicked beam unsuccessful because of the low signal/noise of the coupling line in the Tb. T spectrum • nonlinear lattice model and sextupole calibration from harmonic analysis of Tb. T BPM data (~2’, but need BPM in Tb. T mode with MAF filter ~15’x 2, works for a special sextupole optics only, tedious orbit control when calibrating sextupoles) PRSTAB 17 074001 (2014) Andrea Franchi Optics Measurements @ ESRF

Where we were in 2016 from norm. lines to&sextupole gradient model • Linear opticssext. measured corrected weeklyerror via ORM (10’+15’ , no need of switch BPM in Tb. T mode, works for any sextupolar optics, i. e. filling mode) => rms β-beat ~4 -5% ultra-low coupling εy/εx~1‰ • Measuring ultra-low coupling with Tb. T BPM data with kicked beam unsuccessful because of the low signal/noise of the coupling line in the Tb. T spectrum • nonlinear lattice model and sextupole calibration from harmonic analysis of Tb. T BPM data (~2’, but need BPM in Tb. T mode with MAF filter ~15’x 2, works for a special sextupole optics only, tedious orbit control when calibrating sextupoles) PRSTAB 17 074001 (2014) Andrea Franchi Optics Measurements @ ESRF

Where we were in 2016 fromoptics skew sext. lines to&sextupole model via ORM • Linear measured correctedtiltweekly (10’+15’ , no need of switch BPM in Tb. T mode, works for any sextupolar optics, i. e. filling mode) => rms β-beat ~4 -5% ultra-low coupling εy/εx~1‰ • Measuring ultra-low coupling with Tb. T BPM data with kicked beam unsuccessful because of the low signal/noise of the coupling line in the Tb. T spectrum • nonlinear lattice model and sextupole calibration from harmonic analysis of Tb. T BPM data (~2’, but need BPM in Tb. T mode with MAF filter ~15’x 2, works for a special sextupole optics only, tedious orbit control when calibrating sextupoles) PRSTAB 17 074001 (2014) Andrea Franchi Optics Measurements @ ESRF

Where we were in 2016 fromoptics norm. measured oct. lines to&octupole field model via (in quads) • Linear corrected weekly ORM (10’+15’ , no need of switch BPM in Tb. T mode, works for any sextupolar optics, i. e. filling mode) => rms β-beat ~4 -5% ultra-low coupling εy/εx~1‰ • Measuring ultra-low coupling with Tb. T BPM data with kicked beam unsuccessful because of the low signal/noise of the coupling line in the Tb. T spectrum • nonlinear lattice model and sextupole calibration from harmonic analysis of Tb. T BPM data (~2’, but need BPM in Tb. T mode with MAF filter ~15’x 2, works for a special sextupole optics only, tedious orbit control when calibrating sextupoles) PRSTAB 17 074001 (2014) Andrea Franchi Optics Measurements @ ESRF

Where we were in 2016 • Linear optics measured & corrected weekly via ORM (10’+15’ , no need of switch BPM in Tb. T mode, ! is s ly a n a ) y il a d ( r e works for any sextupolar quick i. e. filling mode) => we want aoptics, rms β-beat ~4 -5% ultra-low coupling εy/εx~1‰ • Measuring ultra-low coupling with Tb. T BPM data with kicked beam unsuccessful because of the low o it! d o t e k li ’d e w signal/noise of the coupling line in the Tb. T spectrum • nonlinear lattice model and sextupole calibration from harmonic analysis of Tb. T BPM data (~2’, but ! is s ly a n a le ib x le f re filter ~15’x 2, o m & r ie s a e need w BPM in Tb. T mode with MAF n a t e wan works for a special sextupole optics only, tedious orbit control when calibrating sextupoles) Andrea Franchi Optics Measurements @ ESRF

Outlines • Where we were in 2016 • Fast measurement and analysis of the orbit response matrix (ORM) • Measuring ultra-low coupling via turn-by-turn (Tb. T) BPM data • Calibrating sextupole magnets via chromatic functions and off-momentum ORM • Accuracy studies • Extra: dealing with “multiple beams” • Measuring momentum compaction: see Laura Torino’s talk Andrea Franchi Optics Measurements @ ESRF

Fast ORM measurement and analysis • ORM analysis (dip. & quad. errors & tilts) needs to evaluate responses N & S Andrea Franchi Optics Measurements @ ESRF

Fast ORM measurement and analysis • ORM analysis (dip. & quad. errors & tilts) needs to evaluate responses N & S • today: done numerically (compute ORM for each magnet error & tilt) : num@ESRF’ 17 ~2’ x 2 (iterations), 64+256 magnets (existing storage ring) num@ESRF’ 19 ~4’ x 2 (iterations), 128+514 magnets ( new storage ring) Andrea Franchi Optics Measurements @ ESRF

Fast ORM measurement and analysis • ORM analysis (dip. & quad. errors & tilts) needs to evaluate responses N & S • today: done numerically (compute ORM for each magnet error & tilt) : num@ESRF’ 17 ~2’ x 2 (iterations), 64+256 magnets (existing storage ring) num@ESRF’ 19 ~4’ x 2 (iterations), 128+514 magnets ( new storage ring) • Analytic formulas for N & S were derived speeding up their computation analyt@ESRF’ 17 ~2” x 2 (iterations) collaboration with Z. Martí of ALBA analyt@ESRF’ 19 ~4” x 2 (iterations) arxiv. org: 1711. 06589 Andrea Franchi Optics Measurements @ ESRF

Fast ORM measurement and analysis • ORM analysis (dip. & quad. errors & tilts) needs to evaluate responses N & S • today: done numerically (compute ORM for each magnet error & tilt) : num@ESRF’ 17 ~2’ x 2 (iterations), 64+256 magnets (existing storage ring) num@ESRF’ 19 ~4’ x 2 (iterations), 128+514 magnets ( new storage ring) • Analytic formulas for N & S were derived speeding up their computation analyt@ESRF’ 17 ~2” x 2 (iterations) collaboration with Z. Martí of ALBA analyt@ESRF’ 19 ~4” x 2 (iterations) ORM diagonal block line (foc. err. ) ORM off-diagonal block line (coupl. ) Error: ~1 -2% rms No measur. difference in fit & correction of real data arxiv. org: 1711. 06589 Andrea Franchi Optics Measurements @ ESRF

Fast ORM measurement and analysis • Today’s ORM measurement: vary one by one the DC component of 16 x 2 steerers, store de corresponding orbit and infer the ORM => ~10’ (The complete ORM with all 96 x 2 steerers requires ~50’) ~300 μm rms orbit distor. Andrea Franchi Optics Measurements @ ESRF

Fast ORM measurement and analysis • Today’s ORM measurement: vary one by one the DC component of 16 x 2 steerers, store de corresponding orbit and infer the ORM => ~10’ (The complete ORM with all 96 x 2 steerers requires ~50’) ~300 μm rms orbit distor. • As of 2010 @ Diamond Light Source (*) the fast orbit feedback electronics is used to drive simultaneously the AC steerers with a programmable amplitude and frequency and to retrieve the ORM via harmonic analysis => ~43’’ for 172 x 2 steerers (15’ in DC mode) & ~17 μm rms orbit distortion (~170 μm in DC mode) (*) G. Rehm et al. , MOCNB 01@BIW 10, p. 44, TUPRI 083@IPAC 14, … Andrea Franchi Optics Measurements @ ESRF

Fast ORM measurement and analysis • Today’s ORM measurement: vary one by one the DC component of 16 x 2 steerers, store de corresponding orbit and infer the ORM => ~10’ (The complete ORM with all 96 x 2 steerers requires ~50’) ~300 μm rms orbit distor. • As of 2010 @ Diamond Light Source (*) the fast orbit feedback electronics is used to drive simultaneously the AC steerers with a programmable amplitude and frequency and to retrieve the ORM via harmonic analysis => ~43’’ for 172 x 2 steerers (15’ in DC mode) & ~17 μm rms orbit distortion (~170 μm in DC mode) • Since then, AC ORM measurements implemented in other labs (^ nonexhaustive list) (^) X. Yang et al. PRAB 20 054001 (2017), Z. Marti et al. MOPAB 102@IPAC 17, … Andrea Franchi Optics Measurements @ ESRF

Fast ORM measurement and analysis • Today’s ORM measurement: vary one by one the DC component of 16 x 2 steerers, store de corresponding orbit and infer the ORM => ~10’ (The complete ORM with all 96 x 2 steerers requires ~50’) ~300 μm rms orbit distor. • As of 2010 @ Diamond Light Source (*) the fast orbit feedback electronics is used to drive simultaneously the AC steerers with a programmable amplitude and frequency and to retrieve the ORM via harmonic analysis => ~43’’ for 172 x 2 steerers (15’ in DC mode) & ~17 μm rms orbit distortion (~170 μm in DC mode) • Since then, AC ORM measurements implemented in other labs (^ nonexhaustive list) • After first tests in 2012, AC ORM measurements have been resumed in 2017: • measurement: 34’’ for 96 x 2 steerers (7’’x 2+20’’ DS overhead) • orbit distortion: 250 μm rms (H), 25 μm rms (V), to be optimized • 8 steerers in parallel at steps of 2 Hz within 114 Hz & 130 Hz, 0. 5’’ to be repeated 14 times • data analysis duration: 12’ ( 3’x 4 ORM blocks, to be optimized) Andrea Franchi Optics Measurements @ ESRF

Fast ORM measurement and analysis • Today’s ORM measurement: vary one by one the DC component of 16 x 2 steerers, store de corresponding orbit and infer the ORM => ~10’ (The complete ORM with all 96 x 2 steerers requires ~50’) ~300 μm rms orbit distor. • As of 2010 @ Diamond Light Source (*) the fast orbit feedback electronics is used to drive simultaneously the AC steerers with a programmable amplitude and frequency and to retrieve the ORM via harmonic analysis => ~43’’ for 172 x 2 steerers (15’ in DC mode) & ~17 μm rms orbit distortion implementation plan (~170 μm in DC mode) perform it daily after special • Since then, AC ORM measurements implemented in other labs (^ top-up nonbunch cleaning (long exhaustive list) sequence @9 am): prefer • After first tests in 2012, AC ORM measurements have been resumed in short duration and accept 2017: orbit. DS distortion while IDs • measurement: 34’’ for 96 x 2 steerers large (7’’x 2+20’’ overhead) do rms not move • orbit distortion: 250 μm rms (H), 25 μm (V), to be optimized • 8 steerers in parallel at steps of 2 Hz within 114 Hz & 130 Hz, 0. 5’’ to be repeated 14 times • data analysis duration: 12’ ( 3’x 4 ORM blocks, to be optimized) Andrea Franchi Optics Measurements @ ESRF

Fast ORM measurement and analysis • Today’s ORM measurement: vary one by one the DC component of 16 x 2 steerers, store de corresponding inferoff-diagonal the ORM => ~10’ ORM diagonal block line AC Vs DCorbit and. ORM block line(The AC Vs DC complete ORM with all 96 x 2 steerers requires ~50’) ~300 μm rms orbit distor. • As of 2010 @ Diamond Light Source (*) the fast orbit feedback electronics is used to drive simultaneously the AC steerers with a programmable amplitude and frequency and to retrieve the ORM via harmonic analysis => ~43’’ for 172 x 2 steerers (15’ in DC mode) & ~17 μm rms orbit distortion (~170 μm in DC mode) • Since then, AC ORM measurements implemented in other labs (^ nonexhaustive list) • After first tests in 2012, AC ORM measurements have been resumed in 2017: • measurement: 34’’ for 96 x 2 steerers (7’’x 2+20’’ DS overhead) • orbit distortion: 250 μm rms (H), 25 μm rms (V), to be optimized • 8 steerers in parallel at steps of 2 Hz within 114 Hz & 130 Hz, 0. 5’’ to be repeated 14 times • data analysis duration: 12’ ( 3’x 4 ORM blocks, to be optimized) Andrea Franchi Optics Measurements @ ESRF

Outlines • Where we were in 2016 • Fast measurement and analysis of the orbit response matrix (ORM) • Measuring ultra-low coupling via turn-by-turn (Tb. T) BPM data • Calibrating sextupole magnets via chromatic functions and off-momentum ORM • Accuracy studies • Extra: dealing with “multiple beams” • Measuring momentum compaction: see Laura Torino’s talk Andrea Franchi Optics Measurements @ ESRF

ultra-low coupling via (Tb. T) BPM data Idea: replace pulsed excitation with continuous AC excitation close to the betatron tune, d=Q-QAC (RHIC 1998 [*], Tevatron/RHIC 2008 - [^], LHC 2009 -[&], …. ) • thousands of Tb. T with no decoherence high spectral resolution • efficient data cleaning • but some precautions & corrections to interpret data (theory not completed yet) [*] S. Peggs, C. Tang, RHIC/AP/159, 1998; M. Bai et al. , PRL 80, 4673 (1998) [^] R. Miyamoto, Ph. D thesis, Univ. of Texas, Austin 2008; BNL C-A/AP/#410, 2010; PRSTAB 11 084002 (2008), X. Shen et al. , PRSTAB 16 111001 (2013), … [&] R. Tomás et al. , PRSTAB 5 054001 (2002), 8 024401 (2005) … 15, 091001 (2012), 16 -81003 (2013) … 19, 054001 (2016), … Andrea Franchi Optics Measurements @ ESRF

ultra-low coupling via (Tb. T) BPM data Idea: replace pulsed excitation with continuous AC excitation close to the betatron tune, d=Q-QAC (RHIC 1998 [*], Tevatron/RHIC 2008 - [^], LHC 2009 -[&], …. ) • thousands of Tb. T with no decoherence high spectral resolution • efficient data cleaning • but some precautions & corrections to interpret data (theory not completed yet) Very successful on hadron machines (beating, coupling, nonlinearities). Can it work in lepton rings with radiation damping & diffusion (and high chroma @ ESRF)? [*] S. Peggs, C. Tang, RHIC/AP/159, 1998; M. Bai et al. , PRL 80, 4673 (1998) [^] R. Miyamoto, Ph. D thesis, Univ. of Texas, Austin 2008; BNL C-A/AP/#410, 2010; PRSTAB 11 084002 (2008), X. Shen et al. , PRSTAB 16 111001 (2013), … [&] R. Tomás et al. , PRSTAB 5 054001 (2002), 8 024401 (2005) … 15, 091001 (2012), 16 -81003 (2013) … 19, 054001 (2016), … Andrea Franchi Optics Measurements @ ESRF

ultra-low coupling via (Tb. T) BPM data Betatron coupling described by two CRDTs Fxy & Fyx (*) Measurement with low chroma & detuning sextupole optics compare (εy/εx~1‰) ORM model with Tb. T harmonic analysis AMPLITUDE PHASE (*) PRSTAB 17 074001 (2014) Andrea Franchi Optics Measurements @ ESRF

ultra-low coupling via (Tb. T) BPM data Betatron coupling described by two CRDTs Fxy & Fyx (*) Measurement with low chroma & detuning sextupole optics compare (εy/εx~1‰) ORM model with Tb. T harmonic analysis AMPLITUDE PHASE (*) PRSTAB 17 074001 (2014) Andrea Franchi Optics Measurements @ ESRF

ultra-low coupling via (Tb. T) BPM data Betatron coupling described by two CRDTs Fxy & Fyx (*) Measurement with large chroma operational sextupole optics compare (εy/εx~1‰) ORM model with Tb. T harmonic analysis AMPLITUDE PHASE (*) PRSTAB 17 074001 (2014) Andrea Franchi Optics Measurements @ ESRF

ultra-low coupling via (Tb. T) BPM data Betatron coupling described by two CRDTs Fxy & Fyx (*) Measurement with large chroma operational sextupole optics compare (εy/εx~1‰) ORM model with Tb. T harmonic analysis AMPLITUDE PHASE synchrotron radiation+diffusion & high chroma => a dilemma: • small distance d=Q-QAC for coupling • large distance d=Q-QAC for β-beating (extra slides) • in both case accuracy is limited, until new theory including them is developed AC dipole & data cleaning OK for low-chroma lepton rings (*) PRSTAB 17 074001 (2014) Andrea Franchi Optics Measurements @ ESRF

Outlines • Where we were in 2016 • Fast measurement and analysis of the orbit response matrix (ORM) • Measuring ultra-low coupling via turn-by-turn (Tb. T) BPM data • Calibrating sextupole magnets via chromatic functions and off-momentum ORM • Accuracy studies • Extra: dealing with “multiple beams” • Measuring momentum compaction: see Laura Torino’s talk Andrea Franchi Optics Measurements @ ESRF

sextupoles Vs chromatic functions & ORM Motivation • Until Tb. T BPM data are quickly available and harmonic analysis is proved effective on any sextupole setting (high chroma & detuning, …) and bunch filling pattern, we seek a way to extend the linear analysis via ORM to obtain & correct sextupole models Andrea Franchi Optics Measurements @ ESRF

sextupoles Vs chromatic functions & ORM Motivation • Until Tb. T BPM data are quickly available and harmonic analysis is proved effective on any sextupole setting (high chroma & detuning, …) and bunch filling pattern, we seek a way to extend the linear analysis via ORM to obtain & correct sextupole models • Measurement & correction of sextupoles RDTs @ Touschekdominated ESRF did not result in improved beam lifetime (unlike @ Diamond[^] ). Simulations indicate ü RDTs <-> DA, injection efficiency ü Chromatic functions <-> momentum acceptance, Touschek lifetime [^] R. Bartolini et al. PRAB 11 104002 (2008) Andrea Franchi Optics Measurements @ ESRF

sextupoles Vs chromatic functions & ORM Motivation • Until Tb. T BPM data are quickly available and harmonic analysis is proved effective on any sextupole setting (high chroma & detuning, …) and bunch filling pattern, we seek a way to extend the linear analysis via ORM to obtain & correct sextupole models • Measurement & correction of sextupoles RDTs @ Touschekdominated ESRF did not result in improved beam lifetime (unlike @ Diamond[^] ). Simulations indicate ü RDTs <-> DA, injection efficiency ü Chromatic functions <-> momentum acceptance, Touschek lifetime • The idea is to measure & fit the off-energy ORM & 2 nd-order dispersion • Chromatic functions ineffective for harmonic sextupoles, but new ESRF ring will have chromatic sextupoles only => OK [^] R. Bartolini et al. PRAB 11 104002 (2008) Andrea Franchi Optics Measurements @ ESRF

sextupoles Vs chromatic functions & ORM Chromatic functions D’y is a bit more complicated expression from meas. & fit of standard on-energy ORM from meas. & fit of 1 or 2 off-energy ORM the dispersive off-axis orbit across sextupoles introduces additional focusing (dβ/dδ) and dispersion (D’). arxiv. org: 1711. 06589 Andrea Franchi Optics Measurements @ ESRF

sextupoles Vs chromatic functions & ORM Chromatic functions from off-energy ORM example ORM at +- 100 Hz (delta=0. 16%) agreement meas. Vs AT model better in V than H (not understood) Andrea Franchi Optics Measurements @ ESRF

sextupoles Vs chromatic functions & ORM Chromatic functions from off-energy ORM Varying a sextupole (corrector) strength (i. e. current [-2, +2] A) and measuring variation of chromatic beating w. r. t. sextupole OFF ? ? ? good response Andrea Franchi Optics Measurements @ ESRF

sextupoles Vs chromatic functions & ORM Chromatic functions from off-energy orbit horizontal closed orbit at all BPMs for different RF frequencies from [ -400, 400] Hz in 10 Hz steps & fit a third-order polynomial Varying a sextupole (corrector) strength (i. e. current [-2, +2] A) BPM @ large lin. dispersion BPM @ low lin. dispersion Andrea Franchi Optics Measurements @ ESRF

sextupoles Vs chromatic functions & ORM Sextupole Calibration from Chromatic functions removing horizontal chromatic beating The calibration factor from magnetic measurements is 0. 1569 m-2 A-1 Andrea Franchi Optics Measurements @ ESRF

nonlinear magnetic model from … ORM analysis • observables: chromatic terms • better for lifetime (tbc experimentally) • linear system to be solved • requires at least 3 measurements at δ=0 & δ=±ε • works with BPMs in normal orbit mode • resolution independent upon sextupole setting • for octupoles & higher-order multipoles you need several measurements at large δ Tb. T analysis • observables: resonant driving terms • better for calibration of nonlinear magnets & DA (tbc experimentally) • linear system to be solved • requires 1 measurement at δ=0 • requires BPMs switch to Tb. T (MAF) mode • resolution dependent upon sextupole setting (high chroma => low accuracy) • you may characterize octupoles & higher-order multipoles with a single measurement Andrea Franchi Resonance Driving Terms

Outlines • Where we were in 2016 • Fast measurement and analysis of the orbit response matrix (ORM) • Measuring ultra-low coupling via turn-by-turn (Tb. T) BPM data • Calibrating sextupole magnets via chromatic functions and off-momentum ORM • Accuracy studies • Extra: dealing with “multiple beams” • Measuring momentum compaction: see Laura Torino’s talk Andrea Franchi Optics Measurements @ ESRF

Accuracy studies: ORM Vs Tb. T analysis systematic errors statistical errors from L. Malina’s talk, LER workshop 2018 @ CERN Andrea Franchi Resonance Driving Terms

Accuracy studies: ORM Vs Tb. T analysis artificial β-beating from Tb. T data from L. Malina’s talk, LER workshop 2018 @ CERN Andrea Franchi Resonance Driving Terms

Accuracy studies: ORM Vs Tb. T analysis artificial β-beating from Tb. T data ~10 μm/√Hz (Tb. T, 353 k. Hz) Vs ~10 nm/√Hz (ORM, 10 Hz aqn) from L. Malina’s talk, LER workshop 2018 @ CERN Andrea Franchi Resonance Driving Terms

Accuracy studies: ORM Vs Tb. T analysis artificial β-beating from Tb. T data The measured BPM phase advance is no longer the betatron BPM phase advance from L. Malina’s talk, LER workshop 2018 @ CERN Andrea Franchi Resonance Driving Terms

Accuracy studies: ORM Vs Tb. T analysis artificial β-beating from Tb. T data ~1/N 2 (Tb. T) Vs no dependence for ORM from L. Malina’s talk, LER workshop 2018 @ CERN Andrea Franchi Resonance Driving Terms

Accuracy studies: ORM Vs Tb. T analysis Error contribution to rms β-beating (in ‰) • Statistical errors (precision) the most significant (machine vibrations, orbit drifts [@ ESRF 15 μm rms => 5‰], …) • Systematic (accuracy): SVD on ORM: 3‰ (simulations over ten sets, w/wo 10 nm BPM noise) • Reproducibility (precision): 5‰ (H) & 2‰(V) over 5 consecutive ORM measurements (orbit corrected within 2μm rms) • Lattice non-linearities polluting Tb. T tune line (from simulations): 1 -2‰ accuracy @ lowest kick amplitude • BPM noise and harmonic analysis of Tb. T data: depends on methods Mean error βx-beating precision [‰] βy-beating precision [‰] @ ESRF 4 4 ORM @ ESRF 6 4 Method Tb. T See L. Malina LER workshop 2018 @ CERN & PRAB 20, 082802 (2017) Andrea Franchi Resonance Driving Terms

Summary • Fast measurement and analysis of the orbit response matrix (ORM): analysis OK, operational implementation ongoing • Measuring ultra-low coupling via turn-by-turn (Tb. T) BPM data: OK for low-chroma rings, not yet for highchroma lepton machines • Calibrating sextupole magnets via chromatic functions and off-momentum ORM: calibration OK, some open questions, sextupole model not yet • Linear optics: @ ESRF accuracy of ORM Vs Tb. T ~same (with precautions) Andrea Franchi Optics Measurements @ ESRF

Outlines • Where we were in 2016 • Fast measurement and analysis of the orbit response matrix (ORM) • Measuring ultra-low coupling via turn-by-turn (Tb. T) BPM data • Calibrating sextupole magnets via chromatic functions and off-momentum ORM • Accuracy studies • Extra: dealing with “multiple beams” • Measuring momentum compaction: see Laura Torino’s talk Andrea Franchi Optics Measurements @ ESRF

Dealing with “multiple beams” from P. Goslawski “TRIBs at BESSY II / MLS”, 21/9 NOCE 2017 Andrea Franchi Optics Measurements @ ESRF

Dealing with “multiple beams” beamlets into islands orbit meas. & correct. One or few RF buckets are filled with 1. only 1 beamlet in 1 island Andrea Franchi Optics Measurements @ ESRF

Dealing with “multiple beams” beamlets into islands orbit meas. & correct. One or few RF buckets are filled with 1. only 1 beamlet in 1 island 2. N beamlets in N islands (N=3 here) ~ CASE 1. Andrea Franchi Optics Measurements @ ESRF

Dealing with “multiple beams” beamlets into islands orbit meas. & correct. One or few RF buckets are filled with 1. only 1 beamlet in 1 island 2. N beamlets in N islands (N=3 here) ~ CASE 1. 3. N beamlets in N islands + beamlet at the centre Andrea Franchi Optics Measurements @ ESRF

Dealing with “multiple beams” beamlets into islands orbit meas. & correct. One or few RF buckets are filled with • centre of gravity of 3 islands is not x=0 (~300μm in this plot) and varies along the ring • how to distinguish island’s orbit (in some RF buckets) from the one of the beam on axis (in most of the RF buckets)? • How to correct those two orbits separately? Andrea Franchi Optics Measurements @ ESRF

EXTRA SLIDES Andrea Franchi Optics Measurements @ ESRF

large sextupole “tilts” & octupole fields in quads PRSTAB 17 074001 (2014) Andrea Franchi Optics Measurements @ ESRF

sextupoles Vs chromatic functions & ORM Sextupole Calibration from Chromatic functions removing horizontal chromatic beating The calibration factor from magnetic measurements is 0. 1569 m-2 A-1 Andrea Franchi Optics Measurements @ ESRF

sextupoles Vs chromatic functions & ORM Sextupole Calibration from Chromatic functions with all 3 chromatic functions The calibration factor from magnetic measurements is 0. 1569 m-2 A-1 Andrea Franchi Optics Measurements @ ESRF

ultra-low coupling via (Tb. T) BPM data • AC dipole @ ESRF: magnetic kicker (shaker) producing H & V magnetic field with 2 sets of 6 coils inside a ferrite enclosure and outside a ceramic vacuum chamber (internally coated with a thin metallic layer to lower HF impedance). The shaker is driven by a 700 W (H) & 100 W (V) amplifiers. • store 6000 turns and perform an SVD cleaning prior harmonic analysis • harmonic analysis over 1024 or 2048 only on real signals x & y , not the complex x+ipx & y+ipy ! • betatron coupling described by two CRDTs Fxy & Fyx (*) PRSTAB 17 074001 (2014) Andrea Franchi Optics Measurements @ ESRF

ultra-low coupling via (Tb. T) BPM data low Vs high chroma sextupole optics Artificial β-beating from multiparticle & leptonic nature of the beam Low chroma & detuning Andrea Franchi Optics Measurements @ ESRF

ultra-low coupling via (Tb. T) BPM data low Vs high chroma sextupole optics Artificial β-beating from multiparticle & leptonic nature of the beam High chroma & detuning Andrea Franchi Optics Measurements @ ESRF

ultra-low coupling via (Tb. T) BPM data low Vs high chroma sextupole optics Artificial β-beating from multiparticle & leptonic nature of the beam High chroma & detuning Andrea Franchi Optics Measurements @ ESRF

ultra-low coupling via (Tb. T) BPM data low Vs high chroma sextupole optics Artificial β-beating from multiparticle & leptonic nature of the beam High chroma & detuning synchrotron radiation and diffusion & high chroma seem to limit the possibility of accurate measurement (& correction) of β-beating via AC dipole excitation … until new theory including them is developed Andrea Franchi Optics Measurements @ ESRF

nonlinear magnetic model from orbit BPM data • off energy additional focusing is provided by sextupoles • by measuring the ORM off energy information on sextupoles can be extracted on momentum δ=0: approach Nr. 1 off momentum δ≠ 0 including linear error model, to be pseudo-inverted:

nonlinear magnetic model from orbit BPM data • off energy additional focusing is provided by sextupoles • by measuring the ORM off energy information on sextupoles can be extracted approach Nr. 2 on momentum δ=0:

nonlinear magnetic model from orbit BPM data • off energy additional focusing is provided by sextupoles • by measuring the ORM off energy information on sextupoles can be extracted approach Nr. 2 on momentum δ=0: off momentum δ≠ 0 :

nonlinear magnetic model from orbit BPM data • off energy additional focusing is provided by sextupoles • by measuring the ORM off energy information on sextupoles can be extracted approach Nr. 2 on momentum δ=0: being tested @ ESRF off momentum δ≠ 0 : to be pseudo-inverted chromatic terms

Measuring momentum compaction • A quicker technique consists in measuring the variation of the x-ray intensity onto a monitor Vs Df. RF • Variation of δ => Variation of synchr. rad. intensity be DISR/I 0≅T δ => δ≅(DISR/I 0)/T a value (10 Infer δ from DISR after Df. RF c nd in g m ag . sources of systematic error: • 1. 6% • measured unsaturated soft end Vs CCDhard part) measured saturated main 1% => 0. 8% (~30 mm Cu) Crotch absorber absolute energy (~6 mm Al/Fe) Extraction windowerror 1% => (~6 mm Cd. W) Scintillator dipole field uncertainty (not. Objective -4) Ideal model 1. 7795 Model with errors 1. 827 ± 0. 004 (*) ID 20 1. 76 ± 0. 14 ID 21 1. 87 ± 0. 11 hard x-ray monitor 1. 867 ± 0. 004(^) (*) from 11 ORM models last 2017 run (^) statistical error, systematic to be found Andrea Franchi Optics Measurements @ ESRF