Multiple Bunch Longitudinal Dynamics Measurements at the Cornell Electron-Positron Storage Ring

Transcription

1 Multiple Bunch Longitudinal Dynamic Meauement at the Conell Electon-Poiton Stoage Ring R. Holtzapple, M. Billing, and D. Hatill Laboatoy of Nuclea Studie, Conell Univeity, Ithaca, NY Abtact The Conell Electon-Poiton Stoage Ring (CESR) ha a longitudinal dipole-coupled-bunch intability that limit the total amount of cuent that can be toed in the ing without feedback. A a eult, it i one of the majo limitation fo highe toed cuent and luminoity. Thi pape epot the meauement of multiple bunch longitudinal dynamic done on CESR with a teak camea. The camea wa ued to meaue the dependence of the bunch ditibution on cuent and acceleating RF voltage, fo multiple bunche toed in CESR, a well a the effect of the longitudinal intability on the bunch ditibution. Meauement of the beam longitudinal bunch ditibution with multiple bunche peent in the ing, help give an undetanding of the intability, how it affect the bunch ditibution, and poibly give inight into a cue of the intability. 1. Multiple Bunch Opeation at CESR Multiple bunche efe to moe than one bunch of electon o poiton being peent in CESR at one time. Multiple bunche in CESR ae neceay to achieve high luminoity. Unde peent high enegy phyic colliding beam condition, thee ae up to nine nealy evenly paced tain of bunche, with up to five bunche in each tain, fo a maximum of 45 electon and 45 poiton bunche. The bunche ae epaated at coing point, othe than the inteaction point, by vetical and hoizontal electotatic epaato. The hoizontal epaato ae ued to make a petzeled obit to inue epaation between bunch coing at location othe than the inteaction point. The CESR acceleato complex i depicted in Fig. 1. In CESR, poiton and electon typically collide at the cente of ma enegie between GeV. The enegy of each beam can be between the enegy of GeV but nomally CESR i eithe at GeV o GeV. The multiple bunch meauement wee pefomed at both enegie; one of the conequence of the enegie being cloe togethe i that the diffeence in bunch length fo thee two diffeent enegie i negligible. In CESR thee ae two wiggle magnet, located nea the inteaction egion, which poduce ynchoton adiation fo the Conell High Enegy Synchoton Souce (CHESS). Opening and cloing the wiggle magnet affect the longitudinal phae pace of the beam a will be dicued late. Thee wee fou nomal conducting RF acceleating ection peent in CESR when the meauement wee pefomed. Since the meauement have been pefomed, they have been eplaced with upeconducting RF ection. Wok uppoted by the National Science Foundation. 1

2 The meauement wee pefomed on both the electon and poiton bunche when the oppotunity fo beam time wa available. Some of the paamete of CESR ae hown in table I. Enegy (on/off eonance) Cicumfeence Revolution Peiod RF Fequency 5.289/5.269 GeV m µec Hoizontal Tune Qx 1.53 Vetical Tune Qy 9.61 Longitudinal Tune Q MHz Hamonic numbe 1281 TABLE I. Paamete of the CESR toage ing. Electotatic Sepaato Fome Noth IR-L3 Coppe Mio Electotatic Sepaato Expeimental Aea Stoage Ring E+ Tanfe Line Synchoton E- Tanfe Line Electon Petzel Poiton Petzel Gun 15 KeV Linac Convete E+ 15 MeV E- 3 MeV RF Station CLEO RF Station Electotatic Sepaato CHESS Wet CHESS Eat Electotatic Sepaato 2

3 FIG. 1. The layout of the Conell Acceleato Complex. The teak camea expeimental aea and the mio ued to eflect the light out of the CESR vacuum chambe i located nea the fome Noth inteaction egion (IR). Duing high-enegy colliding beam phyic at CESR, thee ae many diffeent poible bunch pacing configuation. The 45 bunche ae plit into nine tain with each tain allowed to have a maximum of five bucket filled. Diffeent pacing of bunche in CESR eult in a facto of two vaiation in the intability thehold (fig. 2 -(c)). At the time of the meauement, the highet total cuent achieved i with nine tain with two bunche epaated by 42 n (fig. 2). Tain #1 Tain #2 Bunch #1 #2 #3 #4 #5 Bunch #1 28n 28n 28 o 294 n FIG. 2. Thee ae nine tain of bunche in CESR. Each tain i epaated by 28 o 294 n and can have up to five bunche pe tain. The pacing between bunche i 14 n. The highet luminoity in CESR, at the time of the teak camea meauement, wa achieved with bunche #2 and #5 occupied in each tain. Fo a ingle bunch in CESR the motion i decibed by intenal electomagnetic field fom the bunch and the extenal electomagnetic field fom the magnet and the RF ytem. When the bunch intenity i low, magnet and RF cavity field detemine paticle motion. At highe cuent beam geneated field do not ignificantly modify the dynamic o ingle bunch behavio i table in CESR [1]. Thi i not the cae fo multiple bunche in CESR. With multiple bunche a dipole-coupled-bunch longitudinal intability i diven by the inteaction of the multiple bunche with thei vacuum chambe envionment. Duing colliding beam thee ae many high intenity paticle bunche. Thee high intenity bunche contain a elatively lage chage and act a a ouce of electomagnetic field, called wakefield, which act back on the beam. A a eult the wakefield can caue a change in the ynchoton ocillation fequency, a change in the bunch ditibution, o intability. Thee collective effect can caue collective motion of the many paticle in the beam. A detailed deciption of collective effect i outide of the cope of thi pape, a numbe of pape which cove collective effect ae available [2,3]. An altenative deciption beam longitudinal phae pace can be detemined fom the beamgeneated pecta and compaed to the teak camea meauement. 2. Theoy We begin with the longitudinal phae pace denity,ρ( τ,φ), which can be witten in action-angle coodinate in tem of the ynchoton ocillation amplitude τ and phae φ. The longitudinal phae 3

4 pace denity can depend on τ, but fo paticle to be independent, it can not depend on φ. The longitudinal pectum of a beam ha a beam geneated ignal given by [4] ( ) I( ω)= ω δω kω nω k = n= 2π ( ) τdτ dφρ τ,φ e ikφ J k ( ωτ) (1) whee ω i the evolution fequency, ω i the ynchoton fequency, ρ( τ,φ) i the phae pace denity, and J k i an odinay Beel function of ode k. In geneal, the longitudinal bunch denity at time t can be witten a a Fouie expanion in φ ρτ,φ ( )= 1 ρ m ( τ)e imφ (2) 2π m= whee τ i the amplitude and φ i the phae, and the phae pace otate at fequency ω. At low cuent, the bunch motion i detemined by electomagnetic field of the acceleato and the beam geneated field ae negligible. A the cuent inceae, the phae pace tuctue i a eult of the beam inteaction with it own field. The beam-induced field depend on the amount of chage in the bunch and the bunch ditibution. It can be concluded fom equation 1 that azimuthal tuctue in longitudinal phae pace i peent when ynchoton ideband ae obeved [4]. Thee ynchoton ideband imply azimuthal phae pace. Following ae example of beam elated ignal which elate to the obevation of the dipole-coupled-bunch longitudinal intability in CESR. a) No Petubation (m=): Suppoe the beam i not petubed (m=) and the phae pace denity i Gauian in τ with an m bunch length σ τ and unifom in phae φ. The phae pace i then given ρτ,φ ( )= Q 2πσ exp τ τ 2σ τ which i ubtituted into equation 1 to give the cuent pectum of I( ω)= Qω exp ω 2 2 σ τ 2 ( ) δω nω. The pectum obeved fo the m= cae i a eie of line located at ω = nω, with a Gauian envelope with a m width of 1 σ. It hould be noted that ignal at the ynchoton ideband ae τ not peent in thi tate when the beam ha not been excited. b) Dipole Petubation (m= ± 1): Now if the beam ha a dipole petubation, the phae pace denity i given by and the eulting beam pectum i n= ρτ,φ ( )= Q 2πσ exp τ τ 2σ τ 1 + P τ 1 coφ (3) σ τ 4

5 I( ω)= Qω exp ω 2 2 σ τ 2 n= ωσ δω ( nω )+ P τ 1 2 n= δω ( nω + ω )+ δω ( nω ω ) n= The pectum conit of two pat. The fit pat i identical to the unpetubed beam, and the econd pat i the ignal fom the dipole petubation. The dipole ignal give ideband at ±ω. The phae pace of the dipole petubation i plotted in fig. 3 and. Signal at the fit ynchoton ideband have been detected at CESR with a pectum analyze and the eult of the meauement will be peented late in thi pape. 1 ρ 1 (τ,φ) φ(ad) τ/σ τ ρ 1 (τ) τ/σ τ ρτφπσ, FIG. 3. A phae pace plot of the dipole petubation (equation 3) whee ρ τ φ τ 1, Q and the facto P 1 = 2 fo plotting pupoe. A two dimenional pictue of the bunch ditibution in fig. whee ρ 1 ( τ)= ρ 1 ( τ,φ = ). ( )= ( ) c) Dipole and Quadupole petubation (m= ± 1, ± 2): Suppoe the beam ha a dipole and quadupole petubation, the phae pace denity i given by ρτ,φ ( )= Q 2πσ exp τ τ 2σ τ 1 + P τ τ 2 1 coφ + P 2 co ( 2φ ) 2 (4) σ τ σ τ and the eulting beam pectum i I( ω)= Qω exp ω 2 2 σ τ 2 ω 2 2 σ +P τ 2 2 n= n= ( ) δω nω n= + P 1 ωσ τ 2 n= δω ( nω + 2ω )+ δω ( nω 2ω ) δω ( nω + ω )+ δω ( nω ω ) The pectum conit of the unpetubed pat, the dipole petubation at ideband at ±ω, and the quadupole petubation at ideband of ±2ω. The phae pace, envelope of the dipole, and quadupole petubation, i plotted in fig. 4 and. The econd ynchoton ideband ha been detected alo with a pectum analyze at CESR, but it i only peent when the fit ynchoton n= 5

6 ideband i peent, o it i not a pue quadupole mode but a quadupole mode that ha been induced by a dipole mode ρ 2 (τ,φ) -1 φ(ad) ρ τ/σ τ ρ 2 (τ) τ/σ τ FIG. 4. A phae pace plot of the dipole and quadupole petubation (equation 4) whee ( τ,φ)= 2ρτ,φ ( )πσ 2 τ Q and the facto P 1 = 2 and P 2 = 1. A two-dimenional pictue fo the bunch ditibution in fig. when ρ 2 ( τ)= ρ 2 ( τ,φ = ). Highe ode tem (above the quadupole mode) have been obeved at CESR but they have been eithe induced o diven fom the dipole mode o they ae not the pimay mode of inteet. 3. Steak Camea Data Analyi Meauement of the CESR longitudinal bunch ditibution wee made with a 5-femtoecond eolution Hamamatu teak camea. A detailed deciption of the camea, expeimental et-up, and data analyi technique ae decibed in detail elewhee [1]. A bief dicuion will be povided hee. To detemine the bunch length, the longitudinal pofile of the beam ditibution ae fit to an aymmetic Gauian function with a contant backgound given by I(z) = I + I 1 exp 1 (z z) 2 2 (1 + gn(z z)a)σ whee I i the backgound pedetal, and I 1 i the peak of the aymmetic Gauian. The tem gn(z z)a i the aymmety facto that paameteized the hape of the aymmetic Gauian. The longitudinal pofile of the beam ditibution ae χ 2 minimized uing the minimization package Minuit [4]. A χ 2 minimization wa pefomed on each teak camea pictue χ 2 = n i [ Iz ( i ; A, I, I 1,z,σ) x i ] 2 x i 6

7 whee x i i the digitized ignal fom the ith pixel ( i=1 to 512) of the teak camea pofile and z i i detemined fom the fit to the aymmetic Gauian function. The fit will etun the mean z, aymmety facto A, backgound level I, peak of the aymmetic Gauian I 1, and width σ of the ditibution. Figue 6 i an example of a teak camea pofile fit to the aymmetic Gauian ditibution. The petinent infomation etieved fom the aymmetic Gauian ditibution i the m width σ z = m width = ( z z ) 2 12 = σ π A2, the mean of the ditibution z = mean = z π Aσ and the aymmety facto A. Thee quantitie ae the quoted eult fom the meauement. Many expeiment wee pefomed on CESR unde diffeent machine paamete. Duing each expeiment, a et of data, which conit of at leat ten teak camea pictue, wa taken unde the ame acceleato condition. The mean m width, σ z, and aymmety facto, A, fom each expeiment ae ued to examine the pule to pule fluctuation in the beam ditibution. Each teak camea ditibution i fit to the mean m width and aymmety facto whee the aea and mean poition of the aymmetic Gauian ditibution vay. The diffeence, o eidual, between the fit and the data will how vaiation between an aveage ditibution and individual pictue. The eidual, R ji i the ith pixel fom the expeiment with j teak camea pofile. The eidual i detemined fom the expeion Rji = xi I( zi; A, I, I1, z, σ) ( i ;,, 1,, σ) i detemined by doing a χ 2 fit to the aymmetic Gauian ditibution whee Iz AI I z uing the mean igma ( σ ) and mean aymmety facto ( A ) fom a given expeiment, and x i i the data fom the ith pixel. The mean ( z ), pedetal I, and peak ( I 1 ) of the expeion Iz ( i ; AI,, I1, z, σ) ae the fit vaiable. Figue 7 i an example of a ditibution, fit to the mean ditibution, and the eidual wa computed fom the fit to the data. The eidual fom each expeiment can alo be ummed to give the mean eidual. The mean eidual fom an expeimental data et i detemined by umming all the individual eidual in the data et. Becaue each pofile ha a diffeent mean poition, the eidual fom each pofile i hifted o the mean of the aymmetic Gauian i at the oigin. The eidual pixel intenity fom each pofile i then ummed and aveaged in two pec bin to emove the ganulaity of the calibation cuve. The mean eidual fo the jth two p bin i given by R m j ji = = 1 ( xi I( zi; A, I, I1, z, σ) ) 7 m

8 ( 1 σ) i the aymmetic Gauian fit to the data when the mean m width and whee Izi ; AI,, I, z, aymmety value i held contant. M i the numbe of pictue taken duing the expeiment that fit in the two pec bin. Figue 14 i an example of the mean eidual fom an expeiment. The following et of multiple bunch meauement wee made on CESR with the teak camea: 1) the bunch ditibution a a function of cuent, 2) the bunch length a a function of RF acceleating voltage, 3) the bunch ditibution duing High Enegy Phyic colliion, 4) the bunch length with wiggle magnet open and cloed, 5) the bunch ditibution when modulating the RF voltage, and 6) the bunch ditibution with the longitudinal dipole-coupled-bunch intability peent unde diffeent bunch pacing. Thee expeiment will be peented with comment about the eult of the expeiment will be dicued in each ection. 4. CESR Multiple Bunch Longitudinal Dynamic Meauement (I) Bunch Ditibution a a function of Cuent The electon bunch ditibution a a function of cuent wa meaued with nine tain of two bunche pe tain peent in CESR. Fo thi expeiment the bunch length of the fit bunch in the tain wa meaued and the epaation between the two bunche wa 42 n. Duing the meauement the wiggle magnet wee cloed and the bunch had a petzel obit. The total beam cuent wa vaied fom appoximately 18 ma to 216 ma and the RF acceleating voltage wa et at 6.54 ±.1MV. Ten teak camea pictue at each cuent and fitting the ditibution to an aymmetic Gauian function. Plotted in fig. 5 and i the mean and the oot mean eo of the bunch width and aymmety facto at each cuent etting RMS Bunch Length σ z (mm) Aymmety Facto Bunch Cuent (ma) Bunch Cuent (ma) 8

9 FIG. 5. The electon bunch length a a function of cuent with 18 bunche in the CESR ing. The aymmety facto a a function of cuent. A ingle pictue of the longitudinal bunch ditibution fo thee diffeent cuent etting i plotted in fig. 6 though (c) Intenity (abitay unit) Head Tail Intenity (abitay unit) (c) Intenity (abitay unit) FIG. 6. A ingle data acquiition of the electon longitudinal bunch ditibution at the cuent etting of: 1.3 ma pe bunch, 6. ma pe bunch, and (c) 12.5 ma pe bunch. A denoted in figue, the head of the ditibution i to the left of the cente and the tail of the ditibution i to the ight of the cente of the ditibution. Seveal obevation can be made fom the eult: 9

10 1) Between the cuent of 1 to 13 ma pe bunch, the electon bunch length inceae by appoximately 6.%. Thi teady inceae i not accompanied by an inceae in the aymmetic Gauian aymmety facto. Unlike the ingle bunch cae, the aymmety facto doe not change a a function of cuent in the multiple bunch cae [1]. 2) Fo a ingle bunch, the bunch length at low cuent i expected to be σ z = 17.mm 3 a calculated fo the paamete of CESR. It ha been meaued at low cuent to be σ z = ±. 35mm which i conitent with the multiple bunch meauement at low cuent of σ z = ±. 38mm [1]. 3) The lack of light intenity fo the teak camea wa a poblem at low cuent. It wa neceay to open up the teak camea lit apetue to 3µm to allow enough light fo the meauement. The 3µm apetue eolution coection wa made on thee meauement. 4) Duing thee meauement the longitudinal intability wa not detected by the teak camea o in the beam pecta. At thi bunch pacing the thehold cuent of the dipole-coupled-bunch longitudinal intability i appoximately 24 ma fo a ingle beam. Figue 7 though (c) ae plot of the ditibution fit to the mean ditibution with the width and aymmety facto held contant. Above the ditibution, the eidual fom the fit to the data i plotted. It i evident that thee i no geate than a 4% diffeence between the mean and actual ditibution. The longitudinal intability will be dicued late. 5 Head Tail 1 Reidual Reidual Intenity (abitay unit) -5 6 Head Tail Intenity (abitay unit)

11 1 Reidual Intenity (abitay unit) (c) FIG. 7. A ingle data acquiition of the electon longitudinal bunch ditibution fit to the mean ditibution with the m width and aymmety facto kept contant. The eidual fom the ditibution i plotted above the ditibution fo the cuent fo the etting of: 1.3 ma pe bunch, 6. ma pe bunch, and (c) 12.5 ma pe bunch. (II) Bunch Length a a function of the RF Acceleating Voltage The electon bunch length wa meaued a a function of the RF acceleating voltage. At each RF voltage etting ten teak camea pictue wee taken with a contant cuent in CESR. A meauement wa made at high (9mA pe bunch), and low (3.2 ma pe bunch) cuent, with nine tain of two bunche epaated by 42 n. Plotted in fig. 8 and i the mean bunch width and oot mean eo at each RF voltage etting. RMS Bunch Length σ z (mm) σ z (mm) = m1*v RF (MV) m2 m1 m2 Value Eo RMS Bunch Length σ z (mm) σ z (mm)= m1*v RF (MV) m2 m1 m2 Value Eo RF Voltage V RF (MV) RF Voltage V RF (MV) FIG. 8. The electon bunch length a a function of RF acceleating voltage with 18 bunche in the CESR Ring at a cuent of 9. ma pe bunch and 3.2 ma pe bunch.

12 Seveal comment can be made about thi expeiment: 1) Fitting the data to the equation σ( V f )= AV m f the electon bunch length dependence on RF acceleating voltage wa detemined unde the aumption that the voltage fo all cavitie ae in phae. In the low cuent cae, the fit give m =.47 ±.3, in the high cuent cae, the fit give m =.39 ±.3. Fom ingle bunch meauement low cuent behavio of the bunch length i not peciely inveely popotional to the quae oot of the RF acceleating voltage and at high cuent, the bunch length dependence on RF voltage deviate fom the low cuent cae lightly due to collective effect [1]. Simila behavio i obeved in multiple bunch meauement. 2) A expected, the bunch length i hotened a the RF acceleation voltage inceae. At the highet RF voltage of 6.6 MV, the diffeence in bunch length fo the low and high cuent cae i appoximately 3%. 3) The dipole-coupled-bunch longitudinal intability wa not obeved in the bunch ditibution o the beam pecta duing thee meauement. (III) Bunch Ditibution duing High Enegy Phyic Colliion The bunch ditibution wa meaued fo both electon and poiton a a function of cuent duing high-enegy colliion. The meauement wee made with nine tain of two bunche with 42 n pacing. Steak camea pictue wee taken of the fit bunch in the tain when the beam wee colliding. Simultaneou detection of the bunch ditibution fo electon and poition i not poible, o no coelation between the beam can be infeed. The bunch length and aymmety facto i plotted a a function of the cuent in fig. 9 and fo electon and fig. 1 and fo poiton. Seveal obevation can be made fom the meauement: 1) The bunch length, fo both the electon and poiton, inceae a a function of cuent. At the time of thee meauement, the total peak cuent collided duing high-enegy phyic wa 17 ma pe beam. Each fill of CESR would lat 7 minute, o until the ingle beam cuent wa 12 ma, then CESR wa efilled. 2) Fo both electon and poiton the magnitude of the aymmety facto inceae a a function of cuent. The aymmety behavio fo a ingle bunch i alo obeved with multiple bunche. 3) The meauement of electon and poiton bunch length wee made on diffeent fill, o it i had to make any diect compaion. Fom the linea fit to the data, it i evident that the bunch length at the beginning, and end of the fill i vitually the ame fo both beam. Oveall, the electon and poiton exhibit the ame behavio in CESR. 4) At a bunch pacing of 42 n the electon bunch length fo the fit and econd bunch in the tain wa meaued duing colliding beam. The eult ae plotted in fig. 11 and. Each data point i the mean and m eo of 2 teak camea pictue. The bunch length fo the fit bunch in the tain i conitently lightly longe a a function of cuent than the econd bunch in the tain. The 12

13 bunch length inceae a a function of cuent. The aymmety facto of the two bunche cloely match each othe, and they both inceae with cuent. 5) A dipole-coupled-bunch longitudinal intability wa not obeved in the bunch ditibution o the beam pecta duing thee meauement. RMS Length σ z (mm) σ z (mm) = M + M1*I(mA) M M R Aymmety Facto Ay Fac=M+M1*I(mA) M M R Total Cuent (ma) Total Cuent (ma) FIG. 9. The electon bunch length and aymmety facto a a function of cuent duing highenegy colliion in CESR. The fit to the data i linea and point out the low deceae in the bunch length and aymmety facto a a function of cuent. The RF acceleating voltage duing the meauement wa 6.49 ±.1MV σ z (mm)= M + M1*I(mA) M M R Ay fac= M + M1*I(mA) M M R 2.16 RMS Length σ z (mm) Aymmety Facto Total Cuent (ma) Total Cuent (ma) FIG. 1. The poiton bunch length and aymmety facto a a function of cuent duing highenegy colliion in CESR. The bunch length and aymmety facto i fit to a line. The RF acceleating voltage duing the meauement wa 6.42 ±.1MV. 13

14 -.6 RMS Bunch Length σ z (mm) σ z (mm) Bunch #1 σ z (mm) Bunch #2 Aymmety Facto Aymmety Facto Bunch #1 Aymmety Facto Bunch # Total Bunch Cuent (ma) Total Cuent (ma) FIG. 11. The electon bunch length and aymmety facto fo bunche one and two a a function of cuent duing high-enegy colliding beam. Thee ae 42 n of pacing between each bunch duing the meauement. (IV) Bunch length meauement with vaying petzel obit and wiggle magnet The dependence of the longitudinal bunch ditibution on the wiggle magnet and petzel obit amplitude wa meaued with multiple bunche peent in CESR. The poiton bunch ditibution wa meaued with nine tain with two bunche in each tain epaated by 42 n. The RF acceleating voltage wa V RF = 6.89 ±.1MV duing the meauement. The meaued bunch length and aymmety facto a a function of petzel amplitude and wiggle magnet i lited in table II. Figue 12 i a plot of the bunch length a a function of petzel amplitude. Hoizontal Petzel Wiggle Open/Cloed RMS Bunch Length Aymmety Facto Amplitude (CU) σ z (mm) open ±.9.1 ±.1 cloed ± ±.1 15 open ±.8.1 ±.1 15 cloed ±.1.1 ±.1 25 open 17.2 ±.7.1 ±.1 25 cloed ±.7.9 ±.1 28 open 17.1 ±.7.11 ±.1 Table II. The poiton bunch length and aymmety facto of the CESR bunch ditibution a a function of petzel amplitude with wiggle magnet open and cloed. The hoizontal petzel amplitude unit ae compute unit and the conveion to epaation i appoximately 1 CU=1m fo one half of the hoizontal coing angle at the inteaction point. 14

15 19 Wiggle Magnet Open Wiggle Magnet Cloed 18.5 RMS Width σ z (mm) Hoizontal Petzel Amplitude (CU) FIG. 12. The poiton bunch length with the wiggle magnet open and cloed a a function of the hoizontal petzel obit amplitude. It can be concluded that the petzel amplitude doe not ignificantly change the bunch length o aymmety facto appeciably. The wiggle magnet change the bunch length by appoximately 6% with zeo hoizontal epaation and with full petzel amplitude (25 CU). (V) Modulation of the RF Acceleating Voltage Modulating the phae o amplitude of the RF voltage with a ingle bunch poduce longitudinal collective motion of the bunch in CESR. The RF acceleating voltage wa modulated with a ignal geneato to obeve the editibution of paticle in longitudinal phae pace. A tacking ignal geneato fom a pectum analyze having vaiable amplitude and fequency wa ued to excite the deied mode of ocillation. When the induced mode of longitudinal ocillation wa peent, teak camea pictue of the longitudinal bunch ditibution wee taken. In conjunction a pectum analyze ecoded the amplitude and fequency of the induced ocillation by obeving a beam poition ignal at a location with non-zeo dipeion. Initially unde table condition, twenty teak camea pictue wee taken, to compae the change in the longitudinal ditibution with the intability peent. The peence of a mode of ocillation i obeved a a ignal at ynchoton fequency ( f ) ideband of the otation fequency ( f ) which i f ± mf in the beam pecta. The mode of ocillation induced wee the dipole (m=1), quadupole (m=2), extupole (m=3), and extupole mode (m=4). Duing thee meauement the RF acceleating voltage wa 6.58 ±.1 MV. 15

16 a) Dipole Mode (m=1) The dipole mode of ocillation wa induced in CESR in a ingle bunch of electon by modulating the phae of the RF voltage at khz when the electon bunch cuent wa appoximately 7 ma. To induce the dipole mode the bunch ditibution wa modulated with ignal geneato voltage amplitude of 2 dbm and 1 dbm. Figue 13 and (c) ae epeentative ingle data acquiition of the longitudinal pofile of the bunch ditibution at thee two diffeent voltage amplitude. The ditibution wee fit to the mean table ditibution and the eidual fom the fit i plotted in fig. 13 and (d). 1 1 Reidual 9 Reidual (d) Intneity (abitay unit) Data Mean Backgound Intenity (abitay unit) (c) Data Mean Backgound FIG. 13. The CESR electon bunch ditibution when the RF phae i modulated at the fequency of khz and ignal geneato voltage amplitude of -2 dbm and +1 dbm. The eidual fom the mean ditibution ae plotted above the figue fo the ignal geneato voltage of (c) -2 dbm and (d) +1 dbm. The fequency and amplitude of the beam pecta meaued by the pectum analyze while the beam wa modulated at khz i lited in table III. The dipole mode ignal caled in diect popotion with the modulation level with a quadupole mode ignal appeaing at the highe excitation level. f -2 dbm Sinewave +1 dbm Sinewave f (39 khz) -18 db -19 db f (369.9 khz, m=1) -53 db -23 db f 2 f (349.9 khz, m=2) db TABLE III. The beam pecta amplitude meaued by the pectum analyze when modulating the RF voltage at a fequency of khz. 16

17 Table IV i the eult of the bunch length and aymmety facto fom thee meauement. Thee wa no evidence that the longitudinal bunch ditibution changed in hape o ize when only the f ± f ignal wa peent in the beam pecta. No Dive -2 dbm Sinewave +1 dbm Sinewave RMS Width σ ±.11mm ±.1mm ±. 45mm Aymmety Facto.7 ±.1.5 ±.1.3 ±.6 TABLE IV. The electon bunch length and aymmety facto when RF phae wa modulated at a fequency of khz. The eo ba fo the data i the oot mean eo of the meauement. The bunch length with no dive and aymmety value i ued in fitting the data to detemine the pule to pule eidual and mean eidual fom the fit to the data. Seveal comment about the induced dipole mode bunch ditibution: 1) An ocillation in the aival time of the bunch ditibution i expected with the dipole mode peent. The time ocillation ha been meaued by an ocillocope to be a few picoecond in ize. A few picoecond time ocillation i in the noie of the teak camea tigge and cannot be meaued with the peent tigge ytem. 2) By inceaing the modulation voltage amplitude, the ignal at f ± 2 f, o the quadupole mode (m= ± 2) i induced. With the quadupole mode peent the change in the bunch ditibution ae lage and meauable with the teak camea. The longitudinal hape of the bunch with the quadupole mode peent can be compaed to the theoetical phae pace denity plotted in fig. 4. 3) The mean eidual fo the two diffeent voltage amplitude i plotted in fig. 14. Thee i little evidence of the quadupole mode tuctue in the -2 dbm cae, and in the +1 dbm cae, the quadupole mode i clealy viible Head Tail 2 Mean Reidual 5 Mean Reidual

18 FIG. 14. The mean eidual when the phae of the RF voltage wa modulated at a voltage amplitude of -2 dbm and +1 dbm and fequency of khz. The head of the bunch i to the left of the cente (zeo on the plot) and the tail of the bunch i to the ight of the cente. 4) An effot wa made to coelate the ignal geneato inewave phae with the teak camea. By locking the teak camea tigge with the ignal geneato, the phae of the ignal geneato could be changed, and the phae of the intability would follow the change in the phae. Thi would enable the teak camea to meaue the phae of the induced bunch ditibution change. Figue 15 diplay the bunch length a a function of teak camea pictue when the teak camea tigge wa uppoedly phae locked with the ignal geneato pule. Thi effot wa unucceful, and i evident fom the bunch length, due to the lack of coelation between the teak camea and the ignal geneato RMS Width σ z (mm) Steak Camea Pictue Numbe FIG. 15. The bunch length a a function of pictue numbe taken by the teak camea when modulating the RF voltage at a voltage amplitude of +1 dbm and fequency of khz. b) Quadupole Mode (m=2) The quadupole mode of ocillation wa induced in CESR in a ingle bunch of electon at a cuent of appoximately 8 ma. The bunch ditibution wa excited by diving the RF amplitude with a ignal geneato at khz fo thee diffeent voltage amplitude. Figue 16 ae typical ingle data acquiition of longitudinal bunch ditibution when the ignal geneato voltage wa dbm, (c) +5 dbm, and (e) +1 dbm. The ditibution wee fit to the mean table ditibution and the eidual fom the fit i plotted in fig

19 1 1 (d) Reidual Reidual -1-1 Intenity (abitay unit) 8 Data Mean Backgound Intenity (abitay unit) (c) Data 7 Mean Backgound Reidual 1-1 (f) Intenity (abitay unit) (e) Data Mean Backgound FIG. 16. A ingle data acquiition of the longitudinal bunch ditibution when the amplitude of the RF voltage i modulated at a fequency of khz and ignal geneato voltage amplitude of dbm, (c) +5 dbm, and (e) +1 dbm. The eidual fom the mean table ditibution fo the above ignal voltage of dbm, (d) +5 dbm, and (f) +1 dbm. Seveal comment can be made about the eult: 1) When diving the quadupole mode, the dipole mode i alway peent (table V). The dipole mode amplitude i not a lage a the quadupole mode amplitude, but it i alway peent. 2) Fom the mean eidual in fig. 17, a noticeable change in the bunch ditibution i peent at all thee voltage etting. The quadupole mode appea in the mean eidual at all thee voltage etting. 19

20 The ingle data acquiition of the bunch ditibution (fig. 16 (c) and (e)) exhibit a damatic change in the bunch ditibution fo the +5 dbm and +1 dbm voltage etting. dbm Sinewave +5 dbm Sinewave +1 dbm Sinewave f (39 khz) -17 db -16 db -15 db f f (369.9 khz, m=1) -61 db -56 db -52 db f ± 2 f (349.9 khz, m=2) -54 db -47 db -46 db TABLE V. The beam pecta amplitude fom the pectum analyze when modulating the amplitude of the RF voltage at khz Mean Reidual -5 Mean Reidual (c) 5 Mean Reidual FIG. 17. The mean eidual when the amplitude of the RF voltage i modulated at voltage amplitude of dbm, +5 dbm, and (c) +1 dbm and fequency of khz. 2

21 3) The bunch length, bunch width, and aymmety facto fo the thee diffeent voltage amplitude i lited in table VI. Bunch lengthening i obeved with an inceae in ignal geneato voltage. The bunch width bette decibe the bunch length when the quadupole o highe ode mode i peent. The bunch width i detemined by calculating Width = N i=1 ( z i z) 2 Iz i whee N i the numbe of CCD pixel within ±3σ of the mean z, z i i the location of the pixel, and Iz ( i ) i the pojection height. The bunch length and width in thi cicumtance i diplayed in table VI. N i=1 Iz ( i ) ( ) No Dive dbm Sinewave +5 dbm Sinewave +1 dbm Sinewave RMS Width ±.11mm ±. 3mm ±. 6mm ±. 4mm σ z Mean Width ±.12mm 2. 4 ±. 3mm ±. 6mm ±. 3mm Aymmety Facto.7 ±.1.9 ±.1.8 ±.2.1 ±.4 TABLE VI. The bunch length and aymmety facto when the amplitude of the RF voltage i modulated at a fequency of khz. c) Sextupole Mode (m=3) The extupole mode of ocillation wa induced in CESR on a ingle bunch of electon at a cuent of appoximately 8 ma. The bunch ditibution wa diven by modulating the phae of the RF voltage with the ignal geneato at the fequency of khz and voltage amplitude of +1 dbm. Figue 18 i a typical ingle data acquiition when the ignal geneato wa et to +1 dbm and the modulated longitudinal bunch ditibution wa fit to a table ditibution. The eidual fom the fit i plotted in fig. 18. The bunch length and aymmety facto fo thee meauement ae σ z = ±.8 mm and Ay =.7 ±.1. Thee wa no evidence fo the quadupole mode being peent when the extupole mode wa diven. The bunch ditibution exhibit a mall change and the deviation fom a table ditibution i le viible in the eidual than in the quadupole cae. The head of the longitudinal ditibution change hape lightly and thi i obeved in a ingle data acquiition. A bump i viible in the eidual (fig. 18 ) and in the mean eidual (fig. 18 (c)). The deviation fom the table ditibution i tonge in the head of the mean eidual than in the tail of the eidual. Table VII lit the beam pecta amplitude duing the meauement. 21

22 Reidual (c) Intenity (abitay unit) Data Fit Backgound Mean Reidual FIG. 18. A ingle data acquiition of the longitudinal bunch ditibution fit to the mean table ditibution when the phae of the RF voltage i modulated at +1 dbm. The eidual between the ingle data acquiition and the mean table ditibution. The mean eidual fo the data et i plotted in (c). The aow in and depict the onet of the extupole mode ocillation in the bunch ditibution. f f +1 dbm Sinewave f (39 khz, m=) -17 db f (369.9 khz, m=1) -62 db 2 f (349.9 khz, m=2) - f 3 f (329.8 khz, m=3) -52 db TABLE VII. The beam pecta amplitude fom the pectum analyze when modulating the phae of the RF voltage at khz. d) Octupole Mode (m=4) The octupole mode of ocillation wa excited in CESR on a ingle bunch of electon at a cuent of appoximately 1 ma. The bunch ditibution wa excited by diving the RF amplitude with a ignal geneato at a fequency of 39.8 khz at a voltage amplitude of +1 dbm. Figue 19 i a typical ingle data acquiition of the modulated longitudinal bunch ditibution. The bunch length and aymmety facto fo thee meauement ae σ z = ±.7mm and Ay =.8 ±.1. It i inteeting to note that with a 1 dbm ignal, the ame tength that poduced majo change in the bunch ditibution at khz (quadupole mode), little evidence of change in the bunch ditibution i noticed at khz (extupole mode) and 39.8 khz (octupole mode). 22

23 Reidual (c) Intenity (abitay unit) Data Fit Backgound Mean Reidual FIG. 19. A ingle data acquiition of the longitudinal bunch ditibution fit to the mean table ditibution when the ignal geneato voltage i +1 dbm and fequency i 39.9 khz. The eidual fom the mean ditibution when the ignal geneato voltage i +1 dbm. The mean eidual fo the octupole mode i plotted in (c). Fom the mean eidual, the bunch ditibution exhibit a light deviation fom a table ditibution. The quadupole and extupole mode wee not peent when the octupole mode wa excited (table VIII). +1 dbm Sinewave f (39kHz) -18 db f f (369.9 khz, m=1) -59 db f 2 f (349.9 khz, m=2) - f 3 f (329.9 khz, m=3) - f 4 f (39.9 khz, m=4) -52 db TABLE VIII. The beam pecta amplitude fom the pectum analyze when modulating the amplitude of the RF voltage at 39.9 khz. (VI) The Dipole-Coupled-Bunch Mode Intability in CESR a) Intability Popetie The CESR longitudinal intability i a dipole-coupled bunch intability, which i only obeved with multiple bunche pe tain. The chaacteitic of the intability ae decibed in detail elewhee [6]. The thehold of the intability depend upon the mode of opeation. We obeved that: 1) at the intability thehold cuent, a ignal at nf ± f appea in the beam pecta, which indicate a pedominately dipole motion in phae pace. 2) A the cuent i inceaed, well above the intability 23

24 thehold, the ignal at nf ± 2 f appea. 3) The thehold cuent fo the intability i dependent on the bunch pacing in the tain. 4) The intability degade the CESR luminoity. Figue 2 -(c) i a plot of the dipole-coupled-bunch intability cuent thehold fo the vaiou bunch pacing at the time of the teak camea meauement. (c) FIG. 2. The cuent thehold fo the dipole-coupled-bunch intability with nine tain of two bunche pe tain, thee bunche pe tain, and (c) fou bunche pe tain. Thee meauement wee made with the fou coppe RF acceleato ection in CESR and duing the ame peiod when the teak camea meauement wee made. 24

25 A pectum analyze wa ued in conjunction with the teak camea to meaue the amplitude and fequency of the beam pecta. An example of the CESR beam pectum fom a beam poition monito electode i diplayed in fig. 21 and. Figue 21 i the beam pecta below the intability thehold when nine tain of two bunche, epaated by 56 n, with a total beam cuent of 75 ma, i peent in CESR. Figue 21 i the beam pecta above the intability thehold when the bunch pacing i 56 n apat when a total cuent of 135 ma i peent in CESR. The ynchoton ideband aociated with the dipole-coupled-bunch longitudinal intability ae clealy viible at f = nf ± f (n=5644 in thi cae) nf nf -5-5 Amplitude (db) -6-7 Amplitude(db) -6-7 nf -f nf +f Fequency (GHz) Fequency (GHz) FIG. 21. The pectum analyze beam pecta centeed on a otation hamonic at GHz without the intability and with the intability peent. The ynchoton fequency fo CESR i f ~2kHz. Meauement of the bunch ditibution at eveal diffeent bunch pacing wee made with the dipole-coupled-bunch longitudinal intability peent. The longitudinal feedback ytem, which at the time of the meauement, conited of a hoizontal kicke magnet that damp the dipole longitudinal ocillation uing local dipeion, wa not fully opeational duing thee meauement[7]. At the time of the meauement, nomal high-enegy phyic configuation wa nine tain and 42 n pacing. Thi bunch pacing mode wa choen fo it high cuent thehold (fig. 2 ). The cuent thehold at thi bunch pacing wa not obtainable duing the teak camea meauement due to heating of the vacuum chambe. Othe bunch pacing configuation wee choen to obeve the dipole-coupled-bunch longitudinal intability. b) Nine tain with two bunche epaated by 14 n. The longitudinal coupled bunch intability ha a low cuent thehold in CESR with nine tain of two bunche with 14 n pacing. With thi bunch pacing the ingle beam intability cuent 25

26 thehold wa appoximately 11 ma. Fo thee meauement the RF acceleating voltage wa 6.31 ±.1 MV and the wiggle magnet wee cloed. Jut above the intability thehold, meauement of the electon bunch ditibution wee taken with the teak camea with the feedback on (no intability peent), and with the longitudinal feedback off (intability peent). Figue 22 i a typical ingle data acquiition of the longitudinal bunch ditibution taken with the teak camea with the feedback off and the dipole-coupled-bunch intability peent. The intability wa noted by the appeaance of a ignal at f f (369.9 khz) in the beam pecta. The ditibution wa fit to the mean table ditibution and the eidual i plotted above the ditibution. Figue 22 i a typical ingle data acquiition of the longitudinal bunch ditibution with the feedback on and declaed table in the beam pecta. The table bunch ditibution wa alo fit to the mean table bunch ditibution to exhibit the pule-to-pule fluctuation of the ditibution and teak camea. Reidual 5-5 Reidual 1-1 Intenity (abitay unit) Data Mean Backgound Intenity (abitay unit) Data Fit Backgound FIG. 22. A ingle data acquiition of the electon longitudinal bunch ditibution meaued by the teak camea when the longitudinal feedback i off and the longitudinal intability peent and the longitudinal feedback i on and the longitudinal intability not peent. The eidual when the intability wa peent (c), and not peent (d). The mean bunch length and aymmety facto fom the fit to the data i lited in table IX. Feedback On Feedback Off RMS Width σ z ±.13mm ±.9mm Aymmety Facto.7 ±.1.7 ±.1 TABLE IX. The electon bunch length and aymmety facto when the intability i peent (feedback off) and not peent (feedback on). The featue of the longitudinal intability fom the meauement ae the following: 26

27 1) A in the cae when the dipole ocillato i excited by the RF ytem the dipole mode i not detectable by the teak camea when the intability wa peent jut above the intability thehold. The bunch length and aymmety facto i in good ageement with each othe with and without the intability peent. 2) No tuctue i detected in the ingle data acquiition eidual o the mean eidual. Figue 23 i the mean eidual with and without the intability peent. Thee appea to be moe tuctue in the eidual when the intability wa not peent Mean Reidual Mean Reidual FIG. 23. The mean eidual with the intability peent and not peent. Exploing the intability behavio futhe, the cuent wa inceaed to appoximately 21 ma. Inceaing the cuent caued eveal inteeting featue to appea: 1) The longitudinal feedback ytem wa not able to damp the beam above 13 ma of total cuent. The f ± f ignal appeaed in the beam pecta egadle of the longitudinal feedback. 2) Figue 24 (c) i the bunch length a a function total cuent between 17 and 21 ma. The bunch length inceae damatically with cuent, and it ocillate in ize. Thi i a ignatue of the quadupole-coupled-bunch intability o a dipole ocillato diven to lage amplitude. 3) A the cuent inceae, the f ± 2 f ignal appea in the beam pecta. The f ± 2 f ignal gowth in tength i evident in fig. 24 (c). Thee i a damatic jump in the f when the longitudinal ditibution change damatically. 4) With thi inceae in cuent, imila behavio in the longitudinal bunch ditibution i obeved a with inceaing the RF modulation voltage amplitude in peviou expeiment. A the cuent inceae, the wake voltage in the RF cavitie become tonge, and excite the quadupole mode, and change the bunch ditibution damatically. The bunch ditibution ocillate violently above 195 ma and that i evident in fig. 24 and. The bunch ditibution ae fit to the table bunch 27 ± 2 f ignal amplitude

28 ditibution and the eidual i then computed and plotted above the ditibution. Figue 24 and ae two typical ingle data acquiition of the longitudinal bunch ditibution at a total cuent of 2 ma. In figue 24, thee i tong evidence of the quadupole mode, and in fig. 24, thee i little evidence of the quadupole mode. 2 1 Reidual Intenity (abitay unit) Data Mean Backgound (c) Reidual Intenity (abitay unit) RMS bunch length 2f Signal (dbm) Data Mean Backgound RMS Width σ z (mm) khz (dbm) 2f Total Cuent (ma) FIG A ingle data acquiition of the longitudinal bunch ditibution fit to the mean table bunch ditibution when the beam i untable. The beam pecta ignal of f ± f and f ± 2 f ae peent and beam cuent i 2 ma duing the meauement. Thee ae two conecutive pictue of the bunch ditibution and the cuent diffeence between the pictue i not moe than one ma. The 28

29 eidual ae plotted above the ditibution. (c) The bunch length and 2 f pecta amplitude a a function of cuent. 5) The tength of the intability i evident in the mean eidual. The mean eidual i plotted in fig. 25 fo thee diffeent cuent in CESR. The amplitude of the mean eidual exhibit the tength of the quadupole-coupled-bunch intability and it leen a the cuent deceae. Mean Reidual (c) Mean Reidual Mean Reidual FIG. 25. The mean eidual when the total cuent wa between ma, ma, and (c) ma. c) Nine tain with 3 bunche pe tain with bunch pacing of 28 n and 14 n. Highe luminoity i achieved by colliding highe total beam cuent. Highe beam cuent can be achieved in eveal way. The method choen fo CESR i to add moe bunche in a tain. Thi pead the total cuent out ove many bunche and educe the intabunch tanvee and longitudinal wakefield by educing the total ingle bunch cuent. One of the goal now at CESR i to inceae the numbe of bunche pe tain by having many bunche with low cuent athe than fewe bunche with high cuent. With thi in mind the dipole-coupled-bunch longitudinal intability wa meaued with the teak camea with nine tain of bunche, that conit of thee bunche in the tain. The bunch pacing between bunch #1 and #2 wa 28 and between bunch #2 and #3 wa 14 n. With thi bunch pacing

30 the dipole-coupled-bunch longitudinal intability thehold i appoximately 125 ma. Fo thee meauement the RF acceleating voltage duing thee meauement wa 6.51 ±.1MV, and the wiggle magnet wee cloed. Jut above the intability thehold the electon bunch ditibution wa meaued with the teak camea with the feedback on and no intability peent, and with the feedback off with the intability peent. Figue 26 i a typical ingle data acquiition of the longitudinal bunch ditibution taken with the feedback off and the dipole-coupled-bunch intability peent. Once again, the intability peence wa detemined by the peence of the f ± f ignal in the beam pecta. Figue 26 i a typical ingle data acquiition of the longitudinal bunch ditibution with the feedback on and the intability i not detected in the beam pecta with the longitudinal feedback on. Reidual Intenity (abitay Unit) (c) Data Fit Backgound Reidual Intenity (abitay unit) (d) Data Fit Backgound FIG. 26. A ingle data acquiition of the electon longitudinal bunch ditibution meaued by the teak camea when the longitudinal feedback i off and the longitudinal intability i peent and the longitudinal feedback i on and the longitudinal intability i not peent. The eidual fom the fit i plotted when the intability wa peent (c) and not peent (d) The meaued bunch length and aymmety facto fom the fit to the data i lited in table X. Feedback On 3 Feedback Off RMS Width σ z 18.4 ±.24 mm 18.2 ±.27mm Aymmety Facto.7 ±.1.7 ±.3 TABLE X. The electon bunch length and aymmety facto when the intability i peent (feedback off) and not peent (feedback on). Much like the two bunche pe tain cae, with thee bunche pe tain: 1) The dipole mode i not detectable by the teak camea. The bunch length and aymmety facto i in good ageement with each othe with and without the dipole-coupled-bunch intability peent. 2) No tuctue wa

31 detected in the eidual o the mean eidual. Figue 27 i the mean eidual with and without the intability peent. The backgound in the teak camea image have a lage noie component with the thee bunche peent pe tain. Thi inceae in noie i due to an exta bunch pe tain. A eduction in noie fo futue expeiment can be achieved by uing a light choppe Mean Reidual 1-1 Mean Reidual FIG. 27. The mean eidual with the intability peent and not peent. The cuent wa inceaed and eveal inteeting featue appeaed: 1) Identical to the two bunche pe tain cae, the longitudinal feedback ytem wa not able to damp the beam above 13 ma of total cuent. The intability wa too tong, the f ± f ignal would appea in the beam pecta independent of the feedback ytem. 2) Thee i a teady inceae in the bunch length a a function of cuent. A the cuent inceae, the f ± f and f ± 2 f ignal appea and gow in tength in the beam pecta a a function of cuent. In fig. 28 (c) the dipole ( f ± f ) and quadupole ( f ± 2 f ) ignal amplitude ae plotted, a well a the bunch length. 3) A the cuent i inceaed even highe (fig. 28 (d)), the bunch ditibution ocillate with a imila violate behavio a een in the two bunch pe tain cae. Figue 28 i a typical ingle data acquiition of the bunch ditibution at 17 ma of total cuent in CESR. The bunch ditibution i fit to the table bunch ditibution and the eidual i plotted above the ditibution. The eidual in fig. 28 doe not exhibit the quadupole mode. At highe cuent the quadupole mode i evident, a in Fig. 28, and it look imila in tuctue to the theoetical ditibution in fig

32 Reidual Reidual Intenity (abitay unit) Data Fit Backgound Intenity (abitay unit) Data Fit Backgound RMS Width σ z (mm) (c) RMS Width σ z f Synchoton Sideband 2f Synchoton Sideband Total Cuent (ma) Synchoton Sideband Signal Amplitude (dbm) RMS Width σ z (mm) (d) Total Cuent (ma) FIG. 28. A ingle data acquiition of the longitudinal bunch ditibution fit to the mean table bunch ditibution. The beam i declaed untable due to the peence of the f ± f and f ± 2 f ignal in the beam pecta. The cuent i 17 ma total. The eidual between the ditibution and the fit i plotted above the ditibution. (c) The bunch length and ynchoton ideband amplitude fo f ± f and f ± 2 f ignal between the cuent of ma. (d) The bunch length between the cuent of ma. The amplitude of the beam pecta wa not meaued duing thi expeiment but the f ± f and f ± 2 f ignal wee peent duing the meauement. 4) The mean eidual i plotted in fig. 29 and fo the two diffeent cuent ange. In both cae the quadupole-coupled-bunch intability i evident. 32