Reflections on Rotators, Or, How to Turn the FEL Upgrade 3F Skew Quad Rotator Into a Skew Quad Rotator

Transcription

1 JLAB-TN August 24 Reflectons on Rotators, Or, How to Turn the FEL Upgrade 3F Skew Quad Rotator nto a Skew Quad Rotator D. Douglas ntroducton A prevous note [] descrbes a smple skew quad system that can be used to exchange horzontal and vertcal moton n the R Upgrade recrculator and thereby nfluence BBU thresholds. Ths system was devsed and nstalled wth the assumpton that the relevant HOs drvng the nstablty would stmulate ether pure horzontal or pure vertcal deflectons, and could thus be counteracted smply by out-couplng the kck to the other transverse plane. After successful operaton of ths system wth an ncrease of the observed BBU threshold from around 2 ma to over 8 ma t has been noted that t s n fact not a true skew quad rotator but rather s a reflecton. t has further been noted that suppresson of BBU drven by HOs of arbtrary polarzaton s desrable [2]. A thorough quanttatve nvestgaton of the relatonshp between reflectons and rotatons s underway [3]; here, we gve only a smple method for convertng the specfc reflecton nstalled n the R Upgrade nto a true rotaton. Reflectons on Rotators The 3F rotator descrbed n Reference [] smply nterchanges horzontal and vertcal phase spaces, magng each dentcally nto the other. Ths s characterzed by the followng transfer matrx, wheren the 2 x 2 off-dagonal sub-blocks are dentcal and n ths case betatron stable, magng an uprght ellpse (α) characterzed by beam envelope back to tself n the other transverse plane. wth sn sn Ths wll map a horzontal kck nto the vertcal plane, and a vertcal kck to the horzontal, and thus should suppress BBU drven by modes wth these specfc polarzatons. However, t wll equally clearly map a dagonal dsplacement and kck (x,x y,y ) back nto a smlar dagonal dsplacement and kck, thus potentally falng to suppress BBU drven va an HO of arbtrary polarzaton. We were however gently of 5

2 JLAB-TN August 24 2 of 5 remnded by Todd Smth [4] that earler work [5] shows a true rotaton wth a matrx nterchangng both planes wth equal ampltudes but dfferng rotatonal phasng wll provde the desred suppresson. The task at hand s thus to convert the above matrx to one wth the followng form usng components nstalled n the 3F regon of the Upgrade Drver ± µ ntroducng the desred sgn dfferences mmedately suggests use of the exstng rotator led (and followed) by upstream (and downstream) decoupled transport of ratonal phase advance quarter wave, half wave, or full wave, to produce an overall system wth the desred rotatonal phase propertes. We note that that half- or full-wave transport s descrbed by or, and quarter-wave can be descrbed by the followng matrx, where the leadng notaton s delberately chosen to suggest a potental approach. nspecton of the stuaton then suggests a soluton: Lead the rotator wth a matrx that s negatve dentty n one plane (half-wavelength) and a quarter wavelength n the other, tral t wth a smlar module wth the planes exchanged. The net transfer matrx s then ether or The product of the 2 x 2 sub-matrces s evaluated under the assumpton that the n s the betatron stable matched envelope characterstc of the exstng egenmode exchange module. sn sn sn sn The resultng matrces thus have the desred form of a true rotaton:

3 JLAB-TN August 24 µ ± We note that full wave () and three-quarter wave (-) wll work equally well, should that be otherwse called for. mplementaton Ths secton s called mplementaton rather than Practcal mplementaton for specfc reasons. See the Comments below. To convert the 3F regon to a pure rotaton, we frst recall that the preferred polarty s as shown n Fgure, and the nstalled system s confgured (durng reflector operaton) as n Fgure 2. We can lead wth n ether plane and a quarter-wave n the other, for the purposes of ths example we chose to start wth the half wave n the horzontal. After a bt of fddlng, we found that an addtonal par of quads was useful n makng the upstream and downstream telescopes symmetrcal and to get a robust match. Gven ths confguraton (Fgure 3) t was trval to obtan the desred characterstcs and get an overall transfer matrx that was a pure rotaton (Table ). Fgure : Natural 3F confguraton Fgure 2: 3F wth reflecton actvated Fgure 3: 3F wth rotaton actvated; nonexstent elements n blue. 3 of 5

4 JLAB-TN August 24 Table : DAD output for 3F transfer matrx n rotator (rather than reflector) mode. The elements 4, 23, 32, 4 match to only the 5- or 6-dgt level because that s the resoluton used to set the beta value n fttng for the quarter-wave transform. ****************************** * TRANSFORATON ATRX * ****************************** FRST ORDER ATRX E-5.679E E E+.E+.E E E E E+.E+.E E+.336E E E-4.E+.E E E E E-5.E+.E+ -.E+.E+.E+.E+.E+.E+ -.E+.E+.E+.E+.E+.E+ Comments The only skew elements are the embedded reflecton Ths s an exercse only; wouldn t necessarly recommend runnng off and nstallng the two extra quads for four reasons. Frstly, they mght not absolutely be necessary; the computaton was done usng telescopes wth two trplets smply because we knew we could get the answer quckly. We haven t checked to be sure we can t do t wth a doublet/trplet confguraton. Secondly, ths demonstraton doesn t actually solve the rght problem: we really want a rotaton from an unstable cavty back to tself, not just across the backleg. The approprate unstable cavty hasn t, however, yet been dentfed, and, even f t were, knowledge of the polarzaton of the offendng mode would be needed and could n fact be such that a reflecton, or even just a smple shft n phase advance, would provde adequate operatonal control of the nstablty. Thrdly, even f the matrx s rght, the nternal detals of the transport could preclude successful machne operaton. Though the nstablty mght be suppressed by the approprate choce of matrx, the envelope msmatch nternal to the system could potentally be severe enough that performance could be loss-lmted, not nstablty lmted. Fnally, addtonal quads may well not be requred to meet the desred condtons. Wth 3 other man quadrupoles avalable n the lnac and recrculator, t s vald to queston whether mposng the constrant that 3F be a pure rotaton (wth potental nasty nternal msmatch) s the rght operatonal approach. One may do better smply by runnng the rotator as we do now usng the upstream and downstream quads to match nto and out of t, and usng the 2F and 5F telescopes to massage the overall transfer matrx nto the proper form. We note that operatonal practce often gnored BBU-mposed constrants altogether. Were we nstablty lmted (to, say or 2 ma) we frequently just tuned up the transport system to be better matched and thereby adequately cured the nstablty lmt as well (that s, we pushed t out to over 5 ma). 4 of 5

5 JLAB-TN August 24 Acknowledgments Thanks to Geoff Krafft for tryng to get us to pay attenton to ths, and to Todd Smth for succeedng n gettng us to pay attenton to ths. Be sure to study Eduard Pozdeyev s treatment of the reflecton/rotaton; the ntal note looks great and t wll be done rgorously, generally, and n detal. Thanks to Chrs Tennant for causng the ruckus to begn wth by documentng BBU suppresson n the Upgrade. References [] D. Douglas, A Skew-Quad Egenmode Exchange odule (SQEE) for the FEL Upgrade Drver Backleg Transport, JLAB-TN-4-6, 2 ay 24. [2] G. Krafft ntally attempted to nstruct us of ths. As always, he was gnored. T. Smth, however, s NOT to be gnored, and wasn t, when he brought t up. T. Smth, prvate communcaton; E. Pozdeyev, prvate communcaton and unpublshed note; C. Tennant et al., to be presented at the 24 FEL Conference (Treste, Aug. 24) [3] E. Pozdeyev, work n progress. [4] Op. ct. [5] Rand, R.E. and T.. Smth, Beam Optcal Control of Beam Breakup n A Recrculatng Electron Accelerator, Partcle Accelerators, Vol., pp.-3, of 5