Two-dimensional Effecs on he CS Ineacion Foces fo an Enegy-Chiped Bunch ui Li, J. Bisognano,. Legg, and. Bosch
Ouline 1. Inoducion 2. Pevious 1D and 2D esuls fo Effecive CS Foce 3. Bunch Disibuion Vaiaion in a Chicane 4. eadaion fo 2D Ineacion 5. CS Foces fo a 2D-Bunch Self Ineacion 6. Summay
1. Inoducion In JLAB FEL, when he bunch is fully compessed by he chicane, beam fagmenaion is obseved in he second ac which has no been epoduced by 1D CS model. measuemen done by D. Douglas, P. Evushenko Thee ae some ohe unexplained obsevaions, such as one emiance measuemen in CTFII which shows song dependence of emiance gowh on ansvese bea funcion inside he chicane. These moivae us o sudy he 2D CS effec, as well as o invesigae ohe possible causes.
Quesions In which paamee egime is he 1D model a good appoximaion? How does bunch ansvese size influence he CS ineacion foces? Goal: full undesanding of he CS effec in bunch compession chicane so as o pedic and conol is advese effec. I is hoped ha a sysemaic analysis fo a simple Gaussian case can povide some insigh of he 2D effecs and seve as benchmak fo he 2D simulaion. fee space, unpeubed Gaussian bunch, cicula obi
Oveview of Basic Fomulas fo he CS Ineacion on a Cuved Obi ',, ' ' / ' ',, ' ' / ' ',, 3 3 d c J A d c A B c A E,,, eaded Poenial Loenz gauge: Elecomagneic Fields: c ' ' Loenz Foce On each elecon: B c v E e F assuming coninuous chage disibuion
Illusaion of CS ineacion fo a 2D bunch ',,
2. Pevious 1D and 2D esuls of Effecive CS Foces The CS ineacion influences boh he longiudinal and ansvese paicle dynamics via effecive CS foces Debenev, 1996 A Elecomagneic Fields: E,, B, A c d mv v Loenz Foce e E B In dynamic eqn: d c ae of kineic 2 d mc v d A enegy loss ee e e cd c cd c c eff F s Hee dynamics 2 d mc e eff Fs cd o dh d H eff F s plays an impoan ole in influencing paicle ansvese mc 2 e mc 2 e F eff s ' cd' Effecive longiudinal CS foce
1D Assumpion and esul Debenev, 1996 A D C B Chaaceisic disance: AB 2 1/3 2 24 z, DB 29 z 1D assumpion: 1/3 O x DB x 1, o 2 1/ 3 z 1 Effecive Foce in Seady-sae: F eff s 2 2Ne dz z 1/3 2/3 1/ 3 z z' 3 z z' z' fz:fom faco F eff s z 2 2N 3 1/3 p e 2 4/3 z 2/3 f z Gaussian bunch Feaue: no dependence on paicles x coodinaes
Pevious 2D esuls fo a Gaussian Bunch Dohlus,22 x 2 1/3 z As he deflecion ges bigge, he CS foce is smalle in ampliude as compaed o he igid-bunch esul wih he same pojeced bunch lengh
3.Bunch Disibuion Vaiaion in a Chicane Dawing fom P. Kejcik, SLAC Bunch compession is caied ou via Iniial z coelaion x coelaion in dispesive egions This gives x-z coelaion=> bunch deflecion in dispesive egions
Linea Opic Tanspo fom s o s x s z s 11 51 s x s x 12 52 s x s x z 16 s 56 s z wih iniial enegy chip u z Zeo uncoelaed spead x x un x s s uz z s z 16 56 s u z x s 1 u u un 56 s z Compession faco s 1 u 55 56 s s slope of 16 s deflecion z s 1 u s u= fo bunch wih no deflecion. 56 x z
Example of he Benchmak Chicane 1 u56 s
Example of he Benchmak Chicane assuming zeo iniial emiance and enegy spead
Sudies using 1D CS ineacion model is semi-selfconsisen fo CS foce calculaion: pojecing 2D bunch on design obi and assuming igid-bunch a eaded ime 1/3 2 1/3 k fo W z z z o Z k ia A 1.63i 2 1/3 2/ 3 3 z.94 Need o include evoluion of bunch x-z disibuion in he CS foce calculaion
4. eadaion fo 2D Ineacion The CS foce is influenced by he bunch x-z deflecion mainly hough eadaion idenify he souce paicles a eaded ime 2D bunch:, ', ' / c d ' 2 ' ', / c dd ' / c 1 1, ' / c 2 2 Fo each, he souce paicles ae on a coss-secion wih x-s dependence
1D line bunch ',, eadaion fo a 2D bunch pojeced on he design obi, assuming longiudinal disibuion fozen in he pas : ' / c 1 1, ' / c 2 2
3D eadaion, ', ' ' illusaion no o scale / c d 2 ' ', / c 2 dsin d d ' / c, Inesecion of he sphee ligh cone fo 4D space-ime wih he 3D bunch Because of dispesion, ansvese size is ofen much bigge han he veical size, so 2D is ofen a good appoximaion.
Pefec iniial Gaussian disibuion in ansvese and longiudinal phase space Iniial linea enegy chip Tanspo by linea opics fom exenal field, bunch is unpeubed by he CS self-ineacion. All CS ineacions ake place inside bend 5. CS Foces fo a 2D Bunch Self Ineacion Assumpions used in he analyical sudy of 2D CS foce: ' ' / ' ',, ' ' / ' ',, 3 3 d c J A d c
Analyical appoach: Fo CS foce on each es paicle in he bunch, solve 2D eadaion elaion analyically Idenify he souce paicle which emied field a a souce locaion. Find he phase space disibuion of he souce paicle by popagaing iniial Gaussian disibuion via linea opics o he souce locaion Fo CS foce on a es paicle, inegae conibuion fom all souce paicles in he iniial disibuion. Thin bunch analyical esuls Geneal cases numeical inegaion of inegands which is an analyical expession involving eadaion, linea opics and iniial Gaussian disibuion
Analyical esul of Effecive Longiudinal CS Foce fo a hin Gaussian bunch x 2 1/3 z agee wih Dohlus Tafic4 esul
Iniial beam paamees un un s z s u x s x s uz s z x s x s s z 1 56 56 52 51 56 52 51
egula bunch nx H 1 m 1 Vey hin bunch nx 4 H Vey hick bunch.1 m 1 5 nx 1 m H 1 3
Illusaion of CS ineacion fo a 2D bunch ',,
6. Summay of Key esuls Delayed esponse of CS foce fom bunch lengh vaiaion Dependence of CS foce on paicles ansvese disibuion aound full compession The high fequency field signal geneaed aound full compession and eaches he bunch sholy afe Dohlus pevious esuls can be explained as pa of he delayed esponse