Frequency Map Analysis at CesrTA

Transcription

1 Frequency Map Analyss at CesrTA J. Shanks. FREQUENCY MAP ANALYSS A. Overvew The premse behnd Frequency Map Analyss (FMA) s relatvely straghtforward. By samplng turn-by-turn (TBT) data (typcally 2048 turns) across a grd n poston-space, one can map the coordnates (x, y) to a pont (Q x, Q y ) n frequency-space. Ths mappng s done by an nterpolated FFT on the frst 1024 turns of the TBT data. For stable orbts, the tune should be well-establshed after 1024 turns. One can consder the tune shft between the frst and last 1024 turns of TBT data as a comment on the stablty of the orbt. f the tune shft s relatvely large ( 0.1), the partcle s moton s chaotc and represents an unstable orbt. The large amount of nformaton from FMA can be analyzed n two ways. Frst, one can plot the data as a 2D hstogram n (x, y)-space, wth a color scale to ndcate the magntude of tune shft between the frst and last half of TBT data. Ths amounts to a dynamc aperture (DA) plot, where the boundary of the stable regon ndcates an estmate of the dynamc aperture of the machne. Second, one can use the frst half of TBT data for the ntal tunes (Q (1) x, Q (1) y ) and use these values as the axes n frequency-space. The data can then be plotted as a scatter plot, wth a color scale ndcatng the tune shft between the frst and second half of TBT data. Both representatons of the FMA are Poncaré surfaces, taken at an arbtrary startng poston s=0 n the rng. These two nterpretatons are equvalent, and one of the prmary goals of frequency mappng s to determne how one surface maps onto the other. When plottng the frequency map and dynamc aperture, we use a dffuson ndex D, defned as: D = ( ) 2 ( ) 2 Q (2) x Q (1) x + Q (2) y Q (1) y (1) n other words, D s the RMS tune shft between the frst and second half of the TBT data. B. Features n Frequency Space 1. Non-Defnte Torson and Drectons of Fast Escape Many of the domnant features n frequency maps result from propertes of the torsonal matrx M. Laskar goes through the detals n hs paper, but we wll brefly dscuss t here. Defne M as: ( 2 ) H 0 () M 2 (2) where s the ampltude (normalzed?), and H 0 s the ntegrable Hamltonan. (See Laskar s formalsm [1] for further detal) Ths 2 2 Jacoban matrx s equvalent to a generalzaton of the ampltude-dependent tune shft (neglectng synchrotron oscllatons). f the torson matrx M s a matrx descrbng a defnte quadratc form, solutons exst wth fnte tme stablty that do not exst f M descrbes a non-defnte torson. f the torson s non-defnte, then we wll see drectons of fast ( escape ) n our frequency map. a c f we let M 1 = and V = (x, y), then the vector V s an sotropc drecton (leadng to a drecton of c b fast escape) f V T M 1 V = ax 2 + 2cxy + by 2 = 0. The punchlne s ths: f det(m) > 0, the quadratc form s defnte and no sotropc drectons exst that s, no drectons of fast escape exst and the moton remans bounded. f det(m) < 0, two sotropc drectons emerge whch act as asymptotes for frequency dffuson, and fast escape may occur.

2 2 2. Frequency Map Foldng A fold n the frequency map occurs f det(m) changes sgn n the regon of the tune plane spanned by the frequency map. The fold occurs along the lne det(m) = 0, and the frequency map wll have very dfferent characterstcs before and after the fold. Before the fold, the torson should be postve n order to ensure stablty of the beam. After the fold, the torson wll become negatve, leadng to drectons of fast escape. t s mportant to ensure that under normal operatng condtons (tunes and ampltudes), one does not enter any regons where drectons of fast escape may exst.. SUMMARY: FMA AT OTHER FACLTES Frequency mappng has been performed at many other facltes, as summarzed by Nadolsk and Laskar n PRST:AB [2]. The frequency maps for every machne wll be drastcally dfferent, and depend strongly on the sextupole dstrbuton. A small change n sextupole strengths can radcally change the map. n order to gan some sense of scale or perspectve, we can consder the footprnt of other machnes frequency maps n (Q x, Q y )-space. Footprnts can range anywhere from [ ] (SOLEL, after optmzaton) to [ ] (ESRF). t s not uncommon to see folds and drectons of fast escape n frequency maps that Nadolsk and Laskar deem acceptable, as long as they occur outsde the normal operatng regme. The authors also note that the dynamc aperture s overestmated by ths technque, n the sense that resonance lnes wll lmt the DA at much smaller ampltudes than ndcated by the plots. Ths wll be dscussed n more detal n a later secton.. FMA AT CESRTA: CTA 2085MEV For the frst part of ths study we wll focus on the cta 2085mev NORM optcs for CesrTA. We wll only consder the smplest case of an deal lattce wth a flat orbt for now. n each scenaro, coordnate-space s sampled n constant steps of 80µm over a regon large enough to span the entre dynamc aperture (typcally 20mm 20mm) turns are tracked. An nterpolated FFT wth a Hannng wndow s used to determne the tunes from the frst and last 1024 turns. Each FMA job s slced nto roughly 500 parallel jobs, each job consstng of a sngle y-coordnate and all desred x-coordnates. The resultng outputs are then combned for analyss n a Python scrpt. Addtonally, an nteractve Python scrpt was used n determnng how the FM maps back to coordnate space. Two sextupole dstrbutons are consdered. The frst s a smple 2-famly (2fam) dstrbuton, only optmzed to acheve the desred chromatctes. The second dstrbuton was optmzed usng Bengtsson s prescrpton [] to mnmze resonances, ampltude-dependent tune shfts, and maxmze dynamc aperture. Tune scans for the two sextupole dstrbutons are shown n fgure (1). FG. 1: Tune scans for the two sextupole dstrbutons beng analyzed. Left: two-famly (2fam). Rght: optmzed dstrbuton usng Bengtsson s formalsm.

3 The desgn workng pont for the lattce s (0.571, 0.628). Ths wll be taken as the workng pont for ntal analyss. A. 2-Famly Sextupoles Fgure (2) shows the dynamc aperture and frequency map for the 2-famly sextupoles wth the orgnal tunes. FG. 2: FMA for 2-famly sextupole dstrbuton, orgnal workng pont of (0.571, 0.628). The color scales are proportonal between the two plots. Frst, some global remarks about ths frequency map. The footprnt of ths frequency map s roughly [ ], whch s well wthn the range seen n Nadolsk and Laskar s studes. f anythng, ths footprnt s relatvely small compared to other unoptmzed sextupole dstrbutons. We do not see any folds n ths map, but there appear to be paths of fast escape along several resonances. t s nterestng to note that startng coordnates near a resonance tend to be ether attracted toward or repelled from the resonance. Ths phenomenon s dscussed by Laskar [1]. Next, we can dentfy what resonances are present n the frequency map and determne how they map back to coordnate-space. See fgure (). t s nterestng to note that the nearest matches for resonances are all offset from the smulated data by the same amount, ( , ). t s not clear at ths tme what caused ths shft. The resonances n fgure () can be mapped back nto coordnate space by observng sgnfcant features. To help wth ths process, an nteractve Python scrpt was developed. Ths allows the user to select ponts n frequency space, and the correspondng locaton n coordnate space s hghlghted. See fgure (4). The node at (0.5875, 0.622) les along the x-axs far from the workng pont, and therefore corresponds to the pont (1mm, 0mm) n the dynamc aperture plot. The Q x + 2Q y = resonance s therefore the leftmost curve n ths juncton, branchng up and to the left toward (0mm, 8mm). The node at (0.574, 0.644) n the tune plane maps back to (5mm, 7.5mm) n coordnate space. The Q y = 2 resonance maps to a horzontal lne at roughly y = 11mm. t s lkely that the dynamc aperture s not nearly as large as ths smulaton would have one thnk. The Q x +2Q y = resonance s lkely strong enough to lmt the dynamc aperture to approxmately 1mm 8mm, rather than 15mm 1mm. B. Optmzed Sextupoles Next, consder the optmzed sextupole dstrbuton. DA and FM plots are shown n fgure (5). Agan, a few global remarks frst. The frst mmedately apparent feature n the FM s the ntroducton of a fold along the rght edge of the map. Many lnes of fast escape are evdent after the fold, n the det(m) < 0 regon. The optmzed dstrbuton has a footprnt of about [ ]. Vertcally ths s nearly dentcal to the 2fam dstrbuton, however horzontally ths s a factor of three smaller. Several features can be mapped back to coordnate space. See fgure (6). The fold occurs along a lne stretchng vertcally from (11mm, 0mm) to (14mm, 12mm), arcng to the left. Ths lne s not vsble n the DA plot. The

4 4 Qx+ 2Qy= FG. : Labelng the resonances found n the FM from fgure (2). Qy = 2 resonance lne s obvous n the FM, and maps back to a dstorted ellpse at y = 11mm and returnng at y = 12.5mm. The groupng of resonances around (0.571,0.646) correspond to several of the same resonances seen n the 2fam frequency map, though agan the closest ratonal resonances are offset from the frequency map by roughly n Qx. V. PROPOSED WORKNG PONT To further understand propertes of the frequency map, a second workng pont should be explored. The workng pont (0.578, 0.542) looks to be clear of resonances and may be a good choce for a new workng pont. We now repeat the analyss for ths new pont. Explct mappng between DA and FM has not been analyzed yet. A. 2-Famly Sextupoles The dynamc aperture and frequency map for the 2fam sextupole dstrbuton usng the new workng pont s shown n fgure (7). The horzontal span n frequency-space has ncreased from 0.02 to 0.04, but the vertcal span has remaned roughly the same. Several resonances are now evdent that aren t attractors or repellers, and only show up n a color plot. The effectve dynamc aperture has drastcally changed from [11mm 7.5mm] to [5mm 10mm]. B. Optmzed Sextupoles The dynamc aperture and frequency map for the optmzed sextupole dstrbuton usng the new workng pont s shown n fgure (8). Agan, the dynamc aperture has drastcally changed from [20mm 7.5mm] to [10mm 10mm]. The orgnal workng pont has a much larger regon wth D < 6, ndcatng a larger regon of stablty. However, the stuaton s reversed when nterpretng the frequency map. Although the fold s stll present, the new torson-postve regon s larger than for the orgnal workng pont. By changng the workng pont we ve decreased the DA slghtly, whle ncreasng the stable regon n the frequency map.

5 5 Q y y (m) δq vs. (Q x, Q y ) Q x δq vs. (x, y) x (m) FG. 4: nteractve Python utlty for vsualzng how ponts n frequency space map back to coordnate space. Here, ponts along the Q x + 2Q y = resonance n the 2fam frequency map are selected, and ther correspondng locatons n coordnate space are shown. FG. 5: FMA for optmzed sextupole dstrbuton, orgnal workng pont of (0.571, 0.628). V. SUMMARY The technques and analyss tools have been developed for understandng and nterpretng frequency maps. t s now possble to determne what resonances n frequency space are restrctng the dynamc aperture n coordnate space. Tolerances for paths of fast escape and folds n frequency space are not yet well understood. Further nvestgatons are requred to understand why smulated frequency map data s offset by a constant from theoretcal resonance lnes. [1] J. Laskar, Frequency Map Analyss and Partcle Accelerators, Proceedngs of the 200 Partcle Accelerator Conference

6 6 Qy=2 Qy=2 Qx+2 Qy= Fol d e onanc s e ngr t m l DAng) t m l.dat r Ve ( Qy=1 4Qx-2 ng t m l A.D z r Ho FG. 6: Labelng the resonances found n the FM from fgure (5). The mappng of regons (1) and (2) are shown. The fold s not apparent on the dynamc aperture before labelng. Areas and are on the det(m ) > 0 sde of the fold, as s the workng pont. FG. 7: FMA for 2-famly sextupole dstrbuton, wth new workng pont of (0.578, 0.542). Vertcal grd spacng s dentcal to fgure (2), however the horzontal span s now larger. [2] L. Nadolsk, J. Laskar, Revew of Sngle Partcle Dynamcs for Thrd Generaton Lght Sources through Frequency Map Analyss [] J. Bengtsson, The Sextupole Scheme for the Swss Lght Source (SLS): An Analytc Approach, SLS Note 9/97

7 FG. 8: FMA for optmzed sextupole dstrbuton, wth new workng pont of (0.578, 0.542). Horzontal and vertcal grds are at the same spacng as n fgure (5). The dynamc aperture for the orgnal workng pont has a larger regon of δq < 10 6, therefore the new workng pont has a smaller stable regon n coordnate space. On the other hand, the torson-postve sde of the FM surface s sgnfcantly larger now, ndcatng a larger non-chaotc regon n frequency space. 7