Measuring sextupole and octupole resonance driving terms using turn-by-turn BPM data Andrea Franchi on behalf of the ASD Beam Dynamics Group Low Emittance Rings 2014 Workshop Frascati, 17th -19th September 2014

the physics behind the analysis from TbT BPM data to magnet strengths Contents the physics behind the analysis from TbT BPM data to magnet strengths Some precautions results for sextupole magnets measuring octupolar fields correction of resonance driving terms & lifetime Andrea Franchi Resonance Driving Terms

the physics behind the analysis from TbT BPM data to magnet strengths Contents the physics behind the analysis from TbT BPM data to magnet strengths Some precautions results for sextupole magnets measuring octupolar fields correction of resonance driving terms & lifetime Andrea Franchi Resonance Driving Terms

the physics behind the analysis Caesar’s cipher jdoold hvw rpqlv glylvd lq sduwhv wuhv, txduxp xqdp lqfroxqw ehojdh, doldp dtxlwdql, whuwldp, txl lsvruxp olqjxd, fhowdh, qrvwud jdool dsshoodqwxu. kl rpqhv olqjxd, lqvwlwxwlv, ohjlexv lqwhu vh gliihuxqw. jdoorv de dtxlwdqlv jduxpqd ioxphq, d ehojlv pdwurqd hw vhtxdqd glylglw. kruxp rpqlxp iruwlvvlpl vxqw ehojdh, surswhuhd txrg d fxowx dwtxh kxpdqlwdwh surylqfldh orqjlvvlph devxqw, plqlphtxh dg hrv phufdwruhv vdhsh frpphdqw, dwtxh hd txdh dg hiihplqdqgrv dqlprv shuwlqhqw lpsruwdqw ; suralpltxh vxqw jhupdqlv txl wudqv ukhqxp lqfroxqw, txlexvfxp frqwlqhqwhu ehooxp jhuxqw. txd gh fdxvd, khoyhwll txrtxh uholtxrv jdoorv yluwxwh sudhfhgxqw, txrg ihuh txrwlgldqlv surhollv fxp jhupdqlv frqwhqgxqw, fxp dxw vxlv ilqlexv hrv surklehqw, dxw lsvl lq hruxp ilqlexv ehooxp jhuxqw. gh ehoor jdoolfr Andrea Franchi Resonance Driving Terms

the physics behind the analysis Caesar’s cipher gallia est omnis divisa in partes tres, quarum unam incolunt belgae, aliam aquitani, tertiam, qui ipsorum lingua, celtae, nostra galli appellantur. hi omnes lingua, institutis, legibus inter se differunt. gallos ab aquitanis garumna flumen, a belgis matrona et sequana dividit. horum omnium fortissimi sunt belgae, propterea quod a cultu atque humanitate provinciae longissime absunt, minimeque ad eos mercatores saepe commeant, atque ea quae ad effeminandos animos pertinent important ; proximique sunt germanis qui trans rhenum incolunt, quibuscum continenter bellum gerunt. qua de causa, helvetii quoque reliquos gallos virtute praecedunt, quod fere quotidianis proeliis cum germanis contendunt, cum aut suis finibus eos prohibent, aut ipsi in eorum finibus bellum gerunt. de bello gallico ns=3 ! shift value il=len_trim(word_en) do k=1,il ic=ichar(word_en(k:k)) word_de(k:k)=char(mod(ic-ns-96,26)+96) enddo Andrea Franchi Resonance Driving Terms

the physics behind the analysis Substitution cipher LIVITCSWPIYVEWHEVSRIQMXLEYVEOIEWHRXEXIPFEMVEWHKVSTYLXZIXLIKIIXPIJVSZEYPERRGERIM WQLMGLMXQERIWGPSRIHMXQEREKIETXMJTPRGEVEKEITREWHEXXLEXXMZITWAWSQWXSWEXTVEPMRXRSJ GSTVRIEYVIEXCVMUIMWERGMIWXMJMGCSMWXSJOMIQXLIVIQIVIXQSVSTWHKPEGARCSXRWIEVSWIIBXV IZMXFSJXLIKEGAEWHEPSWYSWIWIEVXLISXLIVXLIRGEPIRQIVIIBGIIHMWYPFLEVHEWHYPSRRFQMXLE PPXLIECCIEVEWGISJKTVWMRLIHYSPHXLIQIMYLXSJXLIMWRIGXQEROIVFVIZEVAEKPIEWHXEAMWYEPP XLMWYRMWXSGSWRMHIVEXMSWMGSTPHLEVHPFKPEZINTCMXIVJSVLMRSCMWMSWVIRCIGXMWYMX Plaintext alphabet: ABCDEFGHIJKLMNOPQRSTUVWXYZ Ciphertext alphabet: ZEBRASCDFGHIJKLMNOPQTUVWXY no longer a pure shift, but a masking, unsolvable unless the ciphertext is known …. until 9th century Al-Kindi’s Manuscript on Deciphering Cryptographic Messages Andrea Franchi Resonance Driving Terms

the physics behind the analysis Substitution cipher LIVITCSWPIYVEWHEVSRIQMXLEYVEOIEWHRXEXIPFEMVEWHKVSTYLXZIXLIKIIXPIJVSZEYPERRGERIM WQLMGLMXQERIWGPSRIHMXQEREKIETXMJTPRGEVEKEITREWHEXXLEXXMZITWAWSQWXSWEXTVEPMRXRSJ GSTVRIEYVIEXCVMUIMWERGMIWXMJMGCSMWXSJOMIQXLIVIQIVIXQSVSTWHKPEGARCSXRWIEVSWIIBXV IZMXFSJXLIKEGAEWHEPSWYSWIWIEVXLISXLIVXLIRGEPIRQIVIIBGIIHMWYPFLEVHEWHYPSRRFQMXLE PPXLIECCIEVEWGISJKTVWMRLIHYSPHXLIQIMYLXSJXLIMWRIGXQEROIVFVIZEVAEKPIEWHXEAMWYEPP XLMWYRMWXSGSWRMHIVEXMSWMGSTPHLEVHPFKPEZINTCMXIVJSVLMRSCMWMSWVIRCIGXMWYMX Andrea Franchi Resonance Driving Terms

the physics behind the analysis Substitution cipher LIVITCSWPIYVEWHEVSRIQMXLEYVEOIEWHRXEXIPFEMVEWHKVSTYLXZIXLIKIIXPIJVSZEYPERRGERIM WQLMGLMXQERIWGPSRIHMXQEREKIETXMJTPRGEVEKEITREWHEXXLEXXMZITWAWSQWXSWEXTVEPMRXRSJ GSTVRIEYVIEXCVMUIMWERGMIWXMJMGCSMWXSJOMIQXLIVIQIVIXQSVSTWHKPEGARCSXRWIEVSWIIBXV IZMXFSJXLIKEGAEWHEPSWYSWIWIEVXLISXLIVXLIRGEPIRQIVIIBGIIHMWYPFLEVHEWHYPSRRFQMXLE PPXLIECCIEVEWGISJKTVWMRLIHYSPHXLIQIMYLXSJXLIMWRIGXQEROIVFVIZEVAEKPIEWHXEAMWYEPP XLMWYRMWXSGSWRMHIVEXMSWMGSTPHLEVHPFKPEZINTCMXIVJSVLMRSCMWMSWVIRCIGXMWYMX Andrea Franchi Resonance Driving Terms

the physics behind the analysis Substitution cipher LIVITCSWPIYVEWHEVSRIQMXLEYVEOIEWHRXEXIPFEMVEWHKVSTYLXZIXLIKIIXPIJVSZEYPERRGERIM WQLMGLMXQERIWGPSRIHMXQEREKIETXMJTPRGEVEKEITREWHEXXLEXXMZITWAWSQWXSWEXTVEPMRXRSJ GSTVRIEYVIEXCVMUIMWERGMIWXMJMGCSMWXSJOMIQXLIVIQIVIXQSVSTWHKPEGARCSXRWIEVSWIIBXV IZMXFSJXLIKEGAEWHEPSWYSWIWIEVXLISXLIVXLIRGEPIRQIVIIBGIIHMWYPFLEVHEWHYPSRRFQMXLE PPXLIECCIEVEWGISJKTVWMRLIHYSPHXLIQIMYLXSJXLIMWRIGXQEROIVFVIZEVAEKPIEWHXEAMWYEPP XLMWYRMWXSGSWRMHIVEXMSWMGSTPHLEVHPFKPEZINTCMXIVJSVLMRSCMWMSWVIRCIGXMWYMX Andrea Franchi Resonance Driving Terms

the physics behind the analysis Substitution cipher LIVITCSWPIYVEWHEVSRIQMXLEYVEOIEWHRXEXIPFEMVEWHKVSTYLXZIXLIKIIXPIJVSZEYPERRGERIM WQLMGLMXQERIWGPSRIHMXQEREKIETXMJTPRGEVEKEITREWHEXXLEXXMZITWAWSQWXSWEXTVEPMRXRSJ GSTVRIEYVIEXCVMUIMWERGMIWXMJMGCSMWXSJOMIQXLIVIQIVIXQSVSTWHKPEGARCSXRWIEVSWIIBXV IZMXFSJXLIKEGAEWHEPSWYSWIWIEVXLISXLIVXLIRGEPIRQIVIIBGIIHMWYPFLEVHEWHYPSRRFQMXLE PPXLIECCIEVEWGISJKTVWMRLIHYSPHXLIQIMYLXSJXLIMWRIGXQEROIVFVIZEVAEKPIEWHXEAMWYEPP XLMWYRMWXSGSWRMHIVEXMSWMGSTPHLEVHPFKPEZINTCMXIVJSVLMRSCMWMSWVIRCIGXMWYMX Andrea Franchi Resonance Driving Terms

the physics behind the analysis Substitution cipher LIVITCSWPIYVEWHEVSRIQMXLEYVEOIEWHRXEXIPFEMVEWHKVSTYLXZIXLIKIIXPIJVSZEYPERRGERIM WQLMGLMXQERIWGPSRIHMXQEREKIETXMJTPRGEVEKEITREWHEXXLEXXMZITWAWSQWXSWEXTVEPMRXRSJ GSTVRIEYVIEXCVMUIMWERGMIWXMJMGCSMWXSJOMIQXLIVIQIVIXQSVSTWHKPEGARCSXRWIEVSWIIBXV IZMXFSJXLIKEGAEWHEPSWYSWIWIEVXLISXLIVXLIRGEPIRQIVIIBGIIHMWYPFLEVHEWHYPSRRFQMXLE PPXLIECCIEVEWGISJKTVWMRLIHYSPHXLIQIMYLXSJXLIMWRIGXQEROIVFVIZEVAEKPIEWHXEAMWYEPP XLMWYRMWXSGSWRMHIVEXMSWMGSTPHLEVHPFKPEZINTCMXIVJSVLMRSCMWMSWVIRCIGXMWYMX Andrea Franchi Resonance Driving Terms

the physics behind the analysis Substitution cipher LIVITCSWPIYVEWHEVSRIQMXLEYVEOIEWHRXEXIPFEMVEWHKVSTYLXZIXLIKIIXPIJVSZEYPERRGERIM WQLMGLMXQERIWGPSRIHMXQEREKIETXMJTPRGEVEKEITREWHEXXLEXXMZITWAWSQWXSWEXTVEPMRXRSJ GSTVRIEYVIEXCVMUIMWERGMIWXMJMGCSMWXSJOMIQXLIVIQIVIXQSVSTWHKPEGARCSXRWIEVSWIIBXV IZMXFSJXLIKEGAEWHEPSWYSWIWIEVXLISXLIVXLIRGEPIRQIVIIBGIIHMWYPFLEVHEWHYPSRRFQMXLE PPXLIECCIEVEWGISJKTVWMRLIHYSPHXLIQIMYLXSJXLIMWRIGXQEROIVFVIZEVAEKPIEWHXEAMWYEPP XLMWYRMWXSGSWRMHIVEXMSWMGSTPHLEVHPFKPEZINTCMXIVJSVLMRSCMWMSWVIRCIGXMWYMX Andrea Franchi Resonance Driving Terms

the physics behind the analysis Substitution cipher LIVITCSWPIYVEWHEVSRIQMXLEYVEOIEWHRXEXIPFEMVEWHKVSTYLXZIXLIKIIXPIJVSZEYPERRGERIM WQLMGLMXQERIWGPSRIHMXQEREKIETXMJTPRGEVEKEITREWHEXXLEXXMZITWAWSQWXSWEXTVEPMRXRSJ GSTVRIEYVIEXCVMUIMWERGMIWXMJMGCSMWXSJOMIQXLIVIQIVIXQSVSTWHKPEGARCSXRWIEVSWIIBXV IZMXFSJXLIKEGAEWHEPSWYSWIWIEVXLISXLIVXLIRGEPIRQIVIIBGIIHMWYPFLEVHEWHYPSRRFQMXLE PPXLIECCIEVEWGISJKTVWMRLIHYSPHXLIQIMYLXSJXLIMWRIGXQEROIVFVIZEVAEKPIEWHXEAMWYEPP XLMWYRMWXSGSWRMHIVEXMSWMGSTPHLEVHPFKPEZINTCMXIVJSVLMRSCMWMSWVIRCIGXMWYMX I->e X->t L->h I->e E->a R->s M->I Z->m V->r + some extrapolations reveal Edgar Allan Poe’s novel “The Gold-Bug” Hereupon Legrand arose, with a grave and stately air, and brought me the beetle from a glass case in which it was enclosed. It was a beautiful scarabaeus, and, at that time, unknown to naturalists—of course a great prize in a scientific point of view. There were two round black spots near one extremity of the back, and a long one near the other. The scales were exceedingly hard and glossy, with all the appearance of burnished gold. The weight of the insect was very remarkable, and, taking all things into consideration, I could hardly blame Jupiter for his opinion respecting it. Andrea Franchi Resonance Driving Terms

the physics behind the analysis Frequency analysis & local variations Andrea Franchi Resonance Driving Terms

the physics behind the analysis Frequency analysis & local variations Andrea Franchi Resonance Driving Terms

the physics behind the analysis Frequency analysis & local variations Andrea Franchi Resonance Driving Terms

the physics behind the analysis Frequency analysis & local variations Andrea Franchi Resonance Driving Terms

the physics behind the analysis Frequency analysis & local variations Andrea Franchi Resonance Driving Terms

the physics behind the analysis Frequency analysis & local variations Andrea Franchi Resonance Driving Terms

the physics behind the analysis Frequency analysis & local variations Andrea Franchi Resonance Driving Terms

the physics behind the analysis human voice Storage Ring’s voice Andrea Franchi Resonance Driving Terms

the physics behind the analysis language spectrum: Storage Ring’s transverse spectra Andrea Franchi Resonance Driving Terms

the physics behind the analysis language spectrum: Storage Ring’s transverse spectra Andrea Franchi Resonance Driving Terms

the physics behind the analysis language spectrum: Storage Ring’s transverse spectra Andrea Franchi Resonance Driving Terms

the physics behind the analysis language spectrum: Storage Ring’s transverse spectra Andrea Franchi Resonance Driving Terms

the physics behind the analysis from TbT BPM data to magnet strengths Contents the physics behind the analysis from TbT BPM data to magnet strengths Some precautions results for sextupole magnets measuring octupolar fields correction of resonance driving terms & lifetime Andrea Franchi Resonance Driving Terms

from TbT BPM data to magnet strengths X(1,…,N)BPM =x(1,…,N)BPM /βx1/2,BPM Y(1,…,N)BPM =y(1,…,N)BPM /βy1/2,BPM Andrea Franchi Resonance Driving Terms

from TbT BPM data to magnet strengths X(1,…,N)BPM =x(1,…,N)BPM /βx1/2,BPM Y(1,…,N)BPM =y(1,…,N)BPM /βy1/2,BPM from TbT BPM data Andrea Franchi Resonance Driving Terms

from TbT BPM data to magnet strengths X(1,…,N)BPM =x(1,…,N)BPM /βx1/2,BPM Y(1,…,N)BPM =y(1,…,N)BPM /βy1/2,BPM from linear lattice model from TbT BPM data Andrea Franchi Resonance Driving Terms

X(1,…,N)BPM =x(1,…,N)BPM /βx1/2,BPM from TbT BPM data to magnet strengths X(1,…,N)BPM =x(1,…,N)BPM /βx1/2,BPM Y(1,…,N)BPM =y(1,…,N)BPM /βy1/2,BPM from linear lattice model Courant-Snyder TbT data for FFT from TbT BPM data Andrea Franchi Resonance Driving Terms

|V(-1,-1)|=(2Ix2Iy)½|FNS1| |V( 1,-1)|=(2Ix2Iy)½|FNS0| from TbT BPM data to magnet strengths |H(1,0)|=½(2Ix)1/2 |V(0,1)|=½(2Iy)1/2 |H(-2,0)|=(2Ix) |FNS3| |H(0,-2)|=(2Iy) |FNS2| |V(-1,-1)|=(2Ix2Iy)½|FNS1| |V( 1,-1)|=(2Ix2Iy)½|FNS0| X(1,…,N)BPM =x(1,…,N)BPM /βx1/2,BPM Y(1,…,N)BPM =y(1,…,N)BPM /βy1/2,BPM Courant-Snyder TbT data for FFT s-dep. s-dep. s-indep. kick-dep. kick-indep. Andrea Franchi Resonance Driving Terms

|V(-1,-1)|=(2Ix2Iy)½|FNS1| |V( 1,-1)|=(2Ix2Iy)½|FNS0| from TbT BPM data to magnet strengths |FNS3| ≈ M3(β,φ) × K2(sext) |FNS2| ≈ M2(β,φ) × K2(sext) |FNS1| ≈ M1(β,φ) × K2(sext) |FNS0| ≈ M0(β,φ) × K2(sext) from linear lattice model from magnet calibration |H(1,0)|=½(2Ix)1/2 |V(0,1)|=½(2Iy)1/2 |H(-2,0)|=(2Ix) |FNS3| |H(0,-2)|=(2Iy) |FNS2| |V(-1,-1)|=(2Ix2Iy)½|FNS1| |V( 1,-1)|=(2Ix2Iy)½|FNS0| from the model from measurement Andrea Franchi Resonance Driving Terms

|V(-1,-1)|=(2Ix2Iy)½|FNS1| |V( 1,-1)|=(2Ix2Iy)½|FNS0| from TbT BPM data to magnet strengths |FNS3| ≈ M3(β,φ) × K2(sext) |FNS2| ≈ M2(β,φ) × K2(sext) |FNS1| ≈ M1(β,φ) × K2(sext) |FNS0| ≈ M0(β,φ) × K2(sext) from linear lattice model from magnet calibration |H(1,0)|=½(2Ix)1/2 |V(0,1)|=½(2Iy)1/2 |H(-2,0)|=(2Ix) |FNS3| |H(0,-2)|=(2Iy) |FNS2| |V(-1,-1)|=(2Ix2Iy)½|FNS1| |V( 1,-1)|=(2Ix2Iy)½|FNS0| from the model from measurement Andrea Franchi Resonance Driving Terms

the physics behind the analysis from TbT BPM data to magnet strengths Contents the physics behind the analysis from TbT BPM data to magnet strengths Some precautions results for sextupole magnets measuring octupolar fields correction of resonance driving terms & lifetime Andrea Franchi Resonance Driving Terms

Some precautions Enough turns with suitable BPM data are needed to increase FFT resolution Vs. radiation damping (check FFT background noise) Andrea Franchi Resonance Driving Terms

Some precautions Enough turns with suitable BPM data are needed to increase FFT resolution Vs. radiation damping (check FFT background noise) Large kicker strength to increase FFT resolution Vs. second & higher-order contribution to RDTs (checks are possible, simulations and data analysis) unwanted modulation induced by 2nd order effects of sextupoles Andrea Franchi Resonance Driving Terms

Some precautions Enough turns with suitable BPM data are needed to increase FFT resolution Vs. radiation damping (check FFT background noise) Large kicker strength to increase FFT resolution Vs. second & higher-order contribution to RDTs (checks are possible, simulations and data analysis) linear model (β,φ) needs to be robust otherwise model & measured RDTs get corrupted (check BPM phase advance and invariance of tune line amplitude) Andrea Franchi Resonance Driving Terms

Magnet calibration Vs cycling at different currents Some precautions Enough turns with suitable BPM data are needed to increase FFT resolution Vs. radiation damping (check FFT background noise) Large kicker strength to increase FFT resolution Vs. second & higher-order contribution to RDTs (checks are possible, simulations and data analysis) linear model (β,φ) needs to be robust otherwise model & measured RDTs get corrupted (check BPM phase advance and invariance of tune line amplitude) Magnet calibration Vs cycling at different currents Andrea Franchi Resonance Driving Terms

the physics behind the analysis from TbT BPM data to magnet strengths Contents the physics behind the analysis from TbT BPM data to magnet strengths Some precautions results for sextupole magnets measuring octupolar fields correction of resonance driving terms & lifetime Andrea Franchi Resonance Driving Terms

results for sextupole magnets Andrea Franchi Resonance Driving Terms

results for sextupole magnets Andrea Franchi Resonance Driving Terms

results for sextupole magnets more than 50% of the RDT modulation stems from focusing errors (beta beating)!! model with perfect sextupoles Andrea Franchi Resonance Driving Terms

results for sextupole magnets model with perfect sextupoles Andrea Franchi Resonance Driving Terms

results for sextupole magnets model with perfect sextupoles Andrea Franchi Resonance Driving Terms

results for sextupole magnets model with perfect sextupoles Andrea Franchi Resonance Driving Terms

results for sextupole magnets model with fitted sextupole errors Andrea Franchi Resonance Driving Terms

results for sextupole magnets fit of 224 sextupole errors S4C15 calibration S4C15-16 S4C29-30 calibration of sextupole correctors Andrea Franchi Resonance Driving Terms

results for sextupole magnets Andrea Franchi Resonance Driving Terms

results for sextupole magnets fit of 224 sextupole “effective rotations” (better call them “skew-sextupole fields”) Andrea Franchi Resonance Driving Terms

the physics behind the analysis from TbT BPM data to magnet strengths Contents the physics behind the analysis from TbT BPM data to magnet strengths Some precautions results for sextupole magnets measuring octupolar fields correction of resonance driving terms & lifetime Andrea Franchi Resonance Driving Terms

beam-based nonlinear magnet calibration Andrea Franchi Resonance Driving Terms

beam-based nonlinear magnet calibration Andrea Franchi Resonance Driving Terms

the physics behind the analysis from TbT BPM data to magnet strengths Contents the physics behind the analysis from TbT BPM data to magnet strengths Some precautions results for sextupole magnets measuring octupolar fields correction of resonance driving terms & lifetime Andrea Franchi Resonance Driving Terms

correction of resonance driving terms & lifetime sextupole correctors (12 at the ESRF SR) may be used to retrieve the ideal RDTs: Andrea Franchi Resonance Driving Terms

correction of resonance driving terms & lifetime sextupole correctors (12 at the ESRF SR) may be used to retrieve the ideal RDTs: inj. septum Andrea Franchi Resonance Driving Terms

correction of resonance driving terms & lifetime sextupole correctors (12 at the ESRF SR) may be used to retrieve the ideal RDTs: hypothesis: the more “matched” the RDTs, the larger the dynamic aperture, and (hopefully) the longer the lifetime Andrea Franchi Resonance Driving Terms

correction of resonance driving terms & lifetime sextupole correctors (12 at the ESRF SR) may be used to retrieve the ideal RDTs: hypothesis: the more “matched” the RDTs, the larger the dynamic aperture, and (hopefully) the longer the lifetime filling pattern P= 6.03 GeV/c Ey Ex=4nm lifetime correctors OFF manual correction RDT correction multibunch (1 mA/bunch) low chroma 7 pm 16.2 h 24.2 h 22.4 h 16 bunch (~6 mA/bunch) large chroma 5 pm 2.5 h 3.2 h 2.0 h Andrea Franchi Resonance Driving Terms

RDTs are a powerful tool to check linear and nonlinear lattice models Conclusion the harmonic analysis of turn-by-turn BPM data provides a direct measurement of the resonant driving terms (RDTs, i.e. the β-functions of the nonlinear lattice) RDTs are a powerful tool to check linear and nonlinear lattice models RDTs are robust observables for beam-based calibration of nonlinear magnets Correcting RDTs (alone) seems not really effective in improving beam lifetime, specially for Touschek- dominated beams: off-momentum RDTs needed? Andrea Franchi Resonance Driving Terms