• V. Balandin, W. Decking, N. Golubeva
  DESY, Hamburg

The prob­lem of er­rors, aris­ing due to fi­nite BPM res­o­lu­tion in the re­con­struct­ed orbit pa­ram­e­ters, is one of the stan­dard prob­lems of the ac­cel­er­a­tor physics. Even so for the case of un­cou­pled mo­tion the co­vari­ance ma­trix of re­con­struc­tion er­rors can be cal­cu­lat­ed "by hand", the usage of the ob­tained so­lu­tion, as a tool for de­sign­ing of a "good mea­sure­ment sys­tem", is not straight­for­ward. A bet­ter un­der­stand­ing of this prob­lem is still de­sir­able. We make a step in this di­rec­tion by in­tro­duc­ing dy­nam­ics into this prob­lem, which seems to be stat­ic. We con­sid­er a vir­tu­al beam ob­tained as a re­sult of the ap­pli­ca­tion of a re­con­struc­tion pro­ce­dure to ‘‘all pos­si­ble val­ues'' of BPM read­ing er­rors. This beam prop­a­gates along the beam line ac­cord­ing to the same rules as any real beam and has all beam dy­nam­i­cal char­ac­ter­is­tics, such as emit­tances, dis­per­sions, be­ta­tron func­tions, and all these val­ues de­scribe the prop­er­ties of the BPM mea­sure­ment sys­tem. As an ap­pli­ca­tion we for­mu­late re­quire­ments for the BPM mea­sure­ment sys­tem of high-en­er­gy in­tra-bunch-train feed­back sys­tem of the Eu­ro­pean XFEL Fa­cil­i­ty in terms of the in­tro­duced con­cepts of error emit­tance and error Twiss pa­ram­e­ters.

• B. Faatz, N. Baboi, V. Balandin, W. Decking, S. Düsterer, J. Feldhaus, N. Golubeva, T. Laarmann, T. Limberg, D. Nölle, E. Plönjes, H. Schlarb, S. Schreiber, F. Tavella, K.I. Tiedtke, R. Treusch
  DESY, Hamburg
• J. Bahrdt, R. Follath, M. Gensch, K. Holldack, A. Meseck, R. Mitzner
  Helmholtz-Zentrum Berlin für Materialien und Energie GmbH, Elektronen-Speicherring BESSY II, Berlin
• M. Drescher, V. Miltchev, J. Roßbach
  Uni HH, Hamburg

FLASH has been a user fa­cil­i­ty since 2005, de­liv­er­ing ra­di­a­tion in the wave­length range be­tween 7 and 47 nm using the SASE prin­ci­ple. In order to in­crease user beam time and im­prove the ra­di­a­tion prop­er­ties de­liv­ered to users, a major ex­ten­sion of the user fa­cil­i­ty called FLASH II has been pro­posed by DESY in col­lab­o­ra­tion with the HZB, which is a seed­ed FEL over the pa­ram­e­ter range of FLASH. As log­i­cal con­tin­u­a­tion, the HHG de­vel­op­ment pro­gram start­ed with sFLASH, will re­sult in di­rect seed­ing. Be­cause in the fore­see­able fu­ture there will prob­a­bly not be HHG seed lasers avail­able at high rep­e­ti­tion rates down to wave­lengths of 4 nm, a cas­cad­ed HGHG scheme will be used to pro­duce short wave­lengths. After a first de­sign re­port, the pro­ject now en­ters its prepa­ra­tion phase until the de­ci­sion for fund­ing will be taken. Dur­ing this time, the FLASH beam pa­ram­e­ters after the pre­sent up­grade 2009/2010 will be char­ac­ter­ized and the pre­sent de­sign will be re-eval­u­at­ed and ad­just­ed. In ad­di­tion, com­plete start-to-end sim­u­la­tions will com­plete the sim­u­la­tions which have been per­formed so far, in­clud­ing a com­plete de­sign of the ex­trac­tion area.

• V. Balandin, R. Brinkmann, W. Decking, N. Golubeva
  DESY, Hamburg

A straight drift-quadrupole sys­tem, though not being an achro­mat, can trans­port cer­tain in­com­ing beam el­lipses with­out in­tro­duc­ing first-or­der chro­mat­ic dis­tor­tions. Sev­er­al ex­am­ples of such apoc­hro­mat­ic beam trans­port are avail­able in the lit­er­a­ture. In this paper we show that the pos­si­bil­i­ty of apoc­hro­mat­ic fo­cus­ing is a gen­er­al prop­er­ty: For every drift-quadrupole sys­tem there exist an unique set of Twiss pa­ram­e­ters (apoc­hro­mat­ic Twiss pa­ram­e­ters), which will be trans­port­ed through that sys­tem with­out first order chro­mat­ic dis­tor­tions. More­over, we prove that at the same time the apoc­hro­mat­ic Twiss pa­ram­e­ters bring the sec­ond order ef­fect of the be­ta­tron os­cil­la­tions on the shift of the av­er­age bunch path length to the min­i­mal pos­si­ble value and also min­i­mize the ef­fect of be­ta­tron os­cil­la­tions on bunch length­en­ing for Gaus­sian beam. As an ex­am­ple we con­sid­er the ap­pli­ca­tion of the apoc­hro­mat­ic fo­cus­ing con­cept to the de­sign of match­ing sec­tions and phase shifter of the post-linac col­li­ma­tion sec­tion of the Eu­ro­pean XFEL Fa­cil­i­ty.

• V. Balandin, R. Brinkmann, W. Decking, N. Golubeva
  DESY, Hamburg

An achro­mat is a fo­cus­ing sys­tem, in which as large a num­ber of high­er order aber­ra­tions as pos­si­ble is can­celed by sym­me­tries of the lin­ear op­tics and the rest is cor­rect­ed by the usage of third and high­er order mul­ti­poles. The first achro­mats ever con­sid­ered were repet­i­tive achro­mats, in which the can­cel­la­tion of high­er order aber­ra­tions re­lies on ap­pro­pri­ate se­lec­tion of cell tunes. Later on achro­mats, em­ploy­ing mir­ror sym­me­try, were also de­vel­oped. In this paper we re­move one su­per­flu­ous con­straint on the lin­ear op­tics in the the­o­ry of four cell mir­ror sym­met­ric achro­mats, make an ac­cu­rate con­sid­er­a­tion of two cell repet­i­tive achro­mats, and com­pare the num­ber of mul­ti­poles re­quired for each of those achro­mats. More­over, we con­tribute a point of view, from which both ap­proach­es to the achro­mat de­sign be­come iden­ti­cal. As a prac­ti­cal ap­pli­ca­tion we con­sid­er the de­sign of the arcs of the post-linac col­li­ma­tion sec­tion of the Eu­ro­pean XFEL Fa­cil­i­ty.

• N. Golubeva, V. Balandin, W. Decking
  DESY, Hamburg

We con­sid­er a beam line, in which pure be­ta­tron os­cil­la­tions are trans­verse­ly un­cou­pled, but which has nonze­ro hor­i­zon­tal and ver­ti­cal dis­per­sions si­mul­ta­ne­ous­ly. We show that trans­verse os­cil­la­tions in such a beam line could be chro­mat­i­cal­ly cou­pled if the hor­i­zon­tal dis­per­sion is nonze­ro in the ver­ti­cal bend­ing mag­nets and vice versa. We also show that the abil­i­ty of sex­tupoles to gen­er­ate chro­mat­ic cou­pling terms de­pends on the re­la­tion be­tween sex­tupole tilt an­gles and the di­rec­tion of the dis­per­sion vec­tor at the sex­tupole lo­ca­tions. We dis­cuss dif­fer­ent ap­proach­es to the setup of sex­tupole tilt an­gles de­pend­ing on chro­mat­ic aber­ra­tions taken for cor­rec­tion. As a prac­ti­cal ap­pli­ca­tion we con­sid­er the usage of tilt­ed sex­tupoles in the de­sign of the beam switch­yard at the Eu­ro­pean XFEL Fa­cil­i­ty.