USPAS Accelerator Physics 207 University of California, Davis Lattice Extras: Linear Errors, Doglegs, Chicanes, Achromatic Conditions, Emittance Exchange Todd Satogata (Jefferson Lab) / satogata@jlab.org Cedric Hernalsteens (IBA) / cedric.hernalsteens@gmail.com Randika Gamage (ODU) / bgama002@odu.edu http://www.toddsatogata.net/207-uspas
x x 0 s 0 +C = betatron phase advance Review Hill 0 s equation x 00 + K(s)x =0 quasi periodic ansatz solution x(s) =A p (s) cos[ (s)+ 0 ] (s) = (s + C) (s) 2 0 (s) (s) + (s)2 (s) Z ds (s) = (s) cos µ + (0) sin µ (0) sin µ x (0) sin µ cos µ (0) sin µ x 0 µ = M = I cos µ + J sin µ I = Z s0 +C s 0 0 0 ds (s) J(s 0 ) Tr M = 2 cos µ s 0 (0) (0) (0) (0) J 2 = I ) M = e J(s)µ 2
General Non-Periodic Transport Matrix We can parameterize a general non-periodic transport matrix from s to s 2 using lattice parameters and = (s 2 ) (s ) 0 q (s2 ) (s M s!s 2 = @ ) [cos + (s p )sin ] (s ) (s 2 )sin q A (s ) (s 2 ) [cos (s 2)sin ] [ (s 2 ) (s )] cos +[+ (s ) (s 2 )] sin p (s ) (s 2 ) This does not have a pretty form like the periodic matrix However both can be expressed as C S M = C 0 S 0 where the C and S terms are cosine-like and sine-like; the second row is the s-derivative of the first row! A common use of this matrix is the m 2 term: (C&M Eqn 5.52) x(s 2 )= p (s ) (s 2 )sin( ) x 0 (s ) Effect of angle kick on downstream position General phase advance, NOT phase advance/cell µ 3
Design Orbit Perturbations design trajectory Magnets RF Cavity Sometimes need a local change x(s) to the design orbit But we really only get changes in angle x 0 from magnets e.g. small dipole corrector : x 0 = B corrector L corrector /(B ) Changes to/corrections of design orbit from dipole correctors Linear errors add up via linear superposition x(s2 ) C S 0 x 0 = (s 2 ) C 0 S 0 x 0 (s ) x(s 2 )= x 0 (s ) p (s ) (s 2 )sin s x 0 (s 2 )= x 0 (s ) (s ) (s 2 ) [cos (s 2)sin ] 4
Checking Optics with Difference Orbits Horizontal Vertical Steering changes through linear elements are linear Compare calculated optical transport of Δx to measured Can localize focusing errors with enough BPMs Above picture is for single-pass AGS to RHIC transfer line Strong nonlinearities introduce dependence on initial orbits Physics of the AGS to RHIC Transfer Line Commissioning, T. Satogata, W.W. MacKay, et al, EPAC 96 5
Checking Optics with Difference Orbits Line: Predicted FODO orbit Dots: Measured orbit (with errors at s=700m) PEP-II vertical orbit during commissioning Projected vertical betatron oscillation in near-fodo lattice Discrepancy starts at s~700m Later found two quadrupole pairs had ~0.% errors Can clearly discriminate systematic optics errors from random BPM errors Y. Cai, 998, from M. Minty and F. Zimmermann, Beam Techniques: Beam Control and Manipulation 6
Control: Two-Bump x(s 2 )= x 0 (s ) p (s ) (s 2 )sin s x 0 (s 2 )= x 0 (s ) (s ) (s 2 ) [cos (s 2)sin ] x 0 (s ) x 0 (s 2 ) A single orbit error changes all later positions and angles Add another dipole corrector at a location where = k At this point the distortion from the original dipole corrector is all x that we can cancel with the second dipole corrector. s x 0 (s 2 )= x 0 (s ) (s ) (s 2 ) + angle from s 2 dipole Called a two-bump: localized orbit distortion from two correctors But requires = k between correctors 7
Control: Three-Bump (see computer lab) A general local orbit distortion from three dipole correctors Constraint is that net orbit change from sum of all three kicks must be zero Bump amplitude x b = S x 0 Only three-bump requirement is that S, S 2 0 8
Steering Error in Synchrotron Ring Short steering error x 0 in a ring with periodic matrix M Solve for new periodic solution or design orbit (x 0,x 0 ) M x0 x 0 0 0 + x 0 = x0 Note that (x 0 =0,x 0 =0) is not the periodic solution any more! x0 x 0 0 x 0 0 =(I M) 0 x 0 0 (I M) =(I e (2 Q)J ) = [e QJ e New closed orbit x 0 = 0 x 0 0 2 QJ e QJ = (2J sin( Q)) e QJ = (J cos( Q)+I sin( Q)) 2 sin( Q) tan( Q) x 0 0 = x0 0 2 [ 0 cot( Q)] ) if Q = k integer resonances 9
Steering Error in Synchrotron Ring We can use the general propagation matrix to find the new closed orbit displacement at all locations around the synchrotron p x(s) = x0 0 0 (s) cos[ Q] 2 sin( Q) This displacement of the closed orbit changes its path length If the revolution (RF) frequency is constant, then the beam energy changes, and there is an extra (small) term in the closed orbit displacement p x(s) = x0 0 0 (s) cos[ Q]+ x 0 0 (s) 0 2 sin( Q) pc dl where p L /dp p is the momentum compaction and C is the accelerator circumference. 0
RHIC Vertical Difference Orbit Where is the cusp? Predicted difference orbit Measured difference orbit Ring difference orbits also measure optics Compare against modeled/design orbit Integer part of tune readily visible: Q~30 Can easily find reversed, broken BPMs Can also find (or eliminate) focusing errors
RHIC Vertical Difference Orbit Where is the cusp? Predicted difference orbit Measured difference orbit Ring difference orbits also measure optics Compare against modeled/design orbit Integer part of tune readily visible: Q~30 Can easily find reversed, broken BPMs Can also find (or eliminate) focusing errors 2
LOCO and SVD Statistical Methods NSLS VUV synchrotron: before optics correction NSLS VUV synchrotron: after optics correction Measuring many difference orbits in a ring gives a ton of information about optics, BPM and corrector errors Singular value decomposition (SVD) optimization of error fits Has become standard method for correction of synchrotron lattices, particularly synchrotron light sources, led to other methods e.g. model independent analysis (MIA), independent component analysis (ICA) J. Safranek, Experimental Determination of Storage Ring Optics using Orbit Response Measurements, 997 3
Focusing Error in Synchrotron Ring Short focusing error in a ring with periodic matrix M Now solve for Tr M to find effects on tune Q Can be used for a simple measurement of M new = M at the quadrupole 0 f 2 Tr M = cos(2 Q new) = cos(2 Q 0 ) 2 0 f sin(2 Q 0) For small errors Q new = Q 0 + Q we can expand to find Q 0 4 f 0 Quadrupole errors also cause resonances when Q = k/2 : half-integer resonances 4
Natural chromaticity Chromaticity Correction N Q Q / p 0 p = 4 Q I K(s) (s) ds How can we control chromaticity in our synchrotron ring? We need a way to connect momentum offset to focusing Dispersion (momentum-dependent position) and sextupoles (nonlinear focusing depending on position) come to rescue x(s) =x betatron (s)+ x (s) Sextupole B field B y = b 2 x 2 B y (sext) = b 2 [x betatron (s)+ x (s) ] 2 b 2 x 2 betatron +2b 2 x betatron (s) x (s) Nonlinear! Total chromaticity from all sources is then I = [K(s) b 2 (s) x (s)]ds 4 Q like a quadrupole K(s)! Strong focusing (large K) requires large sextupoles, nonlinearity! 5
Doglegs +ˆx bend + bend Drift L or other optics Displaces beam transversely without changing direction What is effect on 6D optics? M dipole (, ) = 0 B @ cos sin 0 0 0 ( cos ) sin cos 0 0 0 sin 0 0 0 0 0 0 0 0 0 C sin ( cos ) 0 0 L/( 2 2 ) ( sin ) A 0 0 0 0 0 Be careful about the coordinate system and signs!! If ρ,θ>0, positive displacement points out from dipole curvature Be careful about order of matrix multiplication! (C&M 3.02) 6
Reverse Bend Dipole Transport What is the correct 6x6 transport matrix of a reverse bend dipole? It turns out to be achieved by reversing both ρ and θ ρθ=l (which stays positive) so both must change sign M dipole (, ) = 0 B @ cos sin 0 0 0 ( cos ) sin cos 0 0 0 sin 0 0 0 0 0 0 0 0 0 C sin ( cos ) 0 0 L/( 2 2 ) ( sin ) A 0 0 0 0 0 M drift = 0 L 0 0 0 0 0 0 0 0 0 0 0 L 0 0 B0 0 0 0 0 C @ 0 0 0 0 L/( 2 2 ) A 0 0 0 0 0 7
Aside: Longitudinal Phase Space Drift Wait, what was that M 56 term with the relativistic effects? Recall longitudinal coordinates are (z, δ) This extra term is called ballistic drift : not in all codes! Important at low to modest energies and for bunch compression Relativistic terms enter converting momentum p to velocity v Positive δ move forward in bunch δ (e-3) Negative δ move backward in bunch relative z position [mm] 8
Weak Dogleg +ˆx M dipole (, ) Drift L M dipole (, ) M dogleg = M dipole (, )M drift M dipole (, ) 0 M weak dogleg = @ 2 0 0 L 0 2 0 A @ 0 0A @ 2 2 0 A 0 0 0 0 0 0 0 L +2 L (, 0 ) in =0 = @ 0 0 A ) (, 0 ) out =( L, 0) 0 0 Strong dogleg can also be derived: D = L cos sin D 0 = L sin2 9
Dogleg Dispersion For weak dogleg (, 0 ) in =0 ) (, 0 ) out =( L, 0) L = 4 m =0.08 rad ) L =.2 m Does this make sense? x 0 ( )= BL (B ) = q p( + ) [BL] q p [BL]( )=( ) x0 ( = 0) 20
Achromatic Dogleg How can we make an achromatic dogleg? (, 0 ) in =(0m, 0) ) (, 0 ) out =(0m, 0) Use an I insertion (e.g. four consecutive π/2 insertions) M /2 = = J (Recall J 4 = I) M achromatic dogleg = 0 cos(2 ) sin(2 ) 0 @ sin(2 ) cos(2 ) 0A 0 0 achromatic! Any transport with net phase advance of 2nπ will be achromatic (nπ if all dipoles bend in same direction) common trick for matching dispersive bending arcs to nondispersive straight sections. 2
Achromatic Dogleg 22
Achromatic Dogleg: Steffen CERN School Notes K. Steffen, CERN-85-9-V-, 985, p. 55 23
First-Order Achromat Theorem A lattice of n repetitive cells is achromatic (to first order, or in the linear approximation) iff M n = I or each cell is achromatic Proof: Consider R M d 0 For n cells : R n = where 0 x @ x 0 A So the lattice is achromatic only if d =0or 2 0 x = R @ x 0 A M6 d = M 26 M n (M n + M n 2 +...+ I) d 0 but (M n + M n 2 +...+ I) =(M n I)(M I) So for n cells : R n = M n (M n I)(M I) d 0 M n = I cos µ tot + J sin µ tot ) µ tot =2 k M n = I S.Y. Lee, Accelerator Physics 24
Chicane +ˆx bend + bend + bend bend Divert beam around an obstruction e.g. vertical bypass chicane in Fermilab Main Ring e.g. horizontal injection chicane in CEBAF recirculating linac Essentially a design orbit 4-bump (4 dipoles) Usually need some focusing, optics between dipoles Usually design optics to be achromatic Operationally null orbit motion at end of chicane vs changes in input beam energy Naively expect M 56 <0 (bunch lengthening or decompression) Higher energy particles (+δ) have shorter path lengths But can compress bunches with introduction of longitudinal correlation 25
AGS to RHIC Transfer Line (vertical dogleg) The ATR vertical dogleg is not strictly a dogleg since the planes of the AGS and RHIC accelerators are not parallel 26
CEBAF Spreader Injection chicane Recombiner 27
CEBAF Spreaders/Recombiners Problem: Separate different energy beams for transport into arcs of CEBAF, and recombine before next linac Achromats: arcs are FODO-like, linacs are dispersion-free I insertion: betatron wavelength between dipoles Single dogleg: unacceptably high beta functions Two consecutive staircase doglegs with same total phase advance was solution Still quite a challenge in physical layout of real magnets! D. Douglas, R.C. York, J. Kewisch, Optical Design of the CEBAF Beam Transport System, 989 28
CEBAF Spreaders/Recombiners Dispersion One step recombiner Unacceptably large vertical beta function/beam size (550 m) Staircase two-step recombiner Acceptable beta functions and beam sizes in both planes (00 m) Dispersion 29
Mobius Insertion Fully coupled equal-emittance optics for e + e - CESR collisions Symmetrically exchange horizontal/vertical motion in insertion Horizontal/vertical motion are coupled Only one transverse tune degree of freedom! Q x,y : unrotated tunes Q,2 = Q x + Q y ± Q Q 2 = 2 4 2 Match insertion to points where β x =β y and α x =α y with phase advances that differ by π between planes Normal insertion: M erect = T 0 0 T Rotated by 45 degrees around s axis: M mobius = 0 T T 0 A purely transverse example of an emittance exchanger 30
Transverse/Longitudinal Emittance Exchange J.C.T. Thangaraj, Experimental Studies on an Emittance Exchange Beamline at the A0 Photoinjector, 202 3
Fermilab A0 Emittance Exchanger Dogleg Dogleg : Bending angle : dogleg dispersion L : dogleg length L c : RF cell length M = 0 B @ L c 4 (4L+L c ) 4 0 4L+L c 4 (4L+L c ) 4 + L c 4 (4L+L c ) 4 2 L c 4 L c 4 + L 2 c 4 C A J.C.T. Thangaraj, Experimental Studies on an Emittance Exchange Beamline at the A0 Photoinjector, 202 32
(xkcd interlude) http://www.xkcd.org/964/ This is known in accelerator lattice design language as a double bend achromat 33
Double Bend Achromat (approximate) You calculated constraints for the double bend achromat in your homework last night Keep lowest-order terms in θ, including θ 2 in upper right term since ρθ=l 34
Double Bend Achromat (approximate) M DBA = M DBA = 0 C S D @ C 0 S 0 D 0 A 0 0 35
Double Bend Achromat (approximate) The periodic solutions for dispersion for the general M matrix were derived in class and the text (periodic) = [ S0 ]D + SD 0 2( cos µ) 0 (periodic) = [ C]D0 + C 0 D 2( cos µ) It turns out that the equation is satisfied automatically! This is a consequence of the mirror symmetry of the system 0 The equation is satisfied if D=0: D = (L + l)(4f L 2l) =0 2f ) 4f L 2l =0 ) f = L +2l 4 =0 =0 ˆ = (L +2l) 2 =2f 36
Double Bend Achromat ˆ =2f =0.03 m L = l =2m =0.0 rad f = L +2l 4 Exact DBA : f = l 2 + 2 =.5 m (KL quad )=0.667 m tan( /2) ˆ = ( cos )+l sin DBA is also known as a Chasman-Green lattice Used in early third-generation light sources (e.g. NSLS at BNL) More after we discuss synchrotron radiation, H functions 37
Triple Bend Achromat Cell (ALS at LBL) Good reasons to minimize integral of dispersion (next week) Extra quadrupoles help with matching at ends of cell, α x,y =0 L. Yang et al, Global Optimization of an Accelerator Lattice Using Multiobjective Genetic Algorithms, 2009 38