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Fourth International Accelerator School for Linear Colliders Beijing, China 7-18 September 2009 Olivier Napoly CEA-Saclay, France Preliminary Version 17 September 2009 Lecture A3 Beam Delivery System and Beam-Beam Effects Part 2 Slide 2 2 Beam Delivery System Collider Parameters, Physical Constants and Notations E, energyand p, momentum B , magnetic rigidity ( B = p/e ) L, luminosity Q, bunch charge N, number of particles in the bunch ( N = Q/e ) n b, number of bunches in the train f rep, pulse repetition rate x and y, horizontal and vertical emittances x * and y *, rms horizontal and vertical beam sizes at the IP z, bunch length , , , Twiss parameters , tune , phase advance, c = 299 792 458 m/s e = 1.602 177 33 10 -19 C m e c 2 = 510 999 e V 0 = 4 10 -7 N A -2 permeability 0 = 1/ 0 c 2 permittivity r e = 2.818 10 -15 m with Slide 3 3 Beam Delivery System ILC BDS Beam Delivery System The Beam delivery system is the final part of the linear collider which transports the high energy beam from the high energy linac to the collision point (Interaction Point = IP). Most Important Functions: Final Focus: Focus the beams at the interaction point to achieve very small beam sizes. Collimation: Remove any large amplitude particles (beam halo). Tuning: Ensure that the small beams collide optimally at the IP. Matching: Precise beam emittance measurement and coupling correction. Diagnostics: Measure the key physics parameters such as energy and polarization. Extraction: Safely extract the beams after collision to the high-power beam dumps. Slide 4 4 Beam Delivery System 14mr IR Final Focus E-collimator -collimator Diagnostics Tune-up dump Beam Switch Yard Sacrificial collimators Extraction grid: 100m*1m Main dump Muon wall Tune-up & Emergency extraction Linac Exit ILC Beam Delivery System Layout (RDR) Slide 5 5 Beam Delivery System Contents: Four (+ Two) Outstanding Questions ? Part 1 Q1: do we need a Final Focus System ? Luminosity, Emittance, Q2: do we need High Field Quadrupole Magnets ? Quadrupole Magnets, Multipoles, Superconducting Quadrupoles Part 2 Q3: do we need Flat Beams ? Beam-beam forces, Beamstrahlung, e+e- Pairs, Fast Feedback Q4: do we need Corrections Systems ? Beam Optics, Achromat, Emittance Growth Slide 6 6 Beam Delivery System Contents: Fundamentals in Beam Physics Part 1 Luminosity Emittance Magnetism (continued) Part 2 Beam-Beam Effects Beam Optics Synchrotron Radiation Slide 7 7 Beam Delivery System Magnetic Measurements Magnet fields are measured to checks: 1.the field integral of the main component ( Bdl for a dipole, Gdl for a quadrupole) 2.the field harmonics content (~ field homogeneity, or quality) These measurement are usually performed by a magnetic probe located at a Reference Radius R ref along the magnet bore. Hence, magnet experts use the following expansion: (sometimes with a +) so that Slide 8 8 Beam Delivery System Quadrupole Magnets: the Engineer Standpoint Given a Central Axis Oz which coincides with the Reference Trajectory, we introduce an Engineering Symmetry R S ( ) which allows categorize the Magnets into Engineering Multipoles. Notations: R ( ) is the rotation with angle around Oz. S is the symmetry with respect to the Oxy plane. is the combination of the two. Note that : 1) For a transverse B field (pseudo-vector) 2) For a longitudinal electrical current J 3) x y O x y B J x y B J S R( )R( ) O O Slide 9 9 Beam Delivery System Symmetric Current Distributions Consider a distribution of longitudinal electrical current J(r, ). If a current distribution is invariant with respect to R S then: 1) the integrated current through the plane is zero, since 2) 2 0 is a sub-multiple of 2 , since x y O J(r, ) zero x y O J(r, ) + Slide 10 10 Beam Delivery System Multipolar Current Distributions We introduce the following categories of current distributions J(r, ) : Monopole ( n =0):, azimuthally independent distributions Dipole ( n =1), such that: Quadrupole ( n =2): Sextupole ( n =3): Octupole ( n =4): Decapole ( n =5): . 2n- pole ( n ): Note that, for the Monopole : Slide 11 11 Beam Delivery System The space J of longitudinal current distributions can be decomposed over the sum of symmetric subspaces as follows. 1.Since, one can decompose any current distribution into symmetric ( n=1 ) and antisymmetric components w.r.t. R S ( ) : 2.Since, one can decompose the current distrib- ution into symmetric ( n=2 ) and antisymmetric components w.r.t. R S ( /2) : etc, etc, leading to : dipole + quadrupole + octupole + 16-pole + . + monopole Decomposition of Current Distributions where is invariant under the rotation Slide 12 12 Beam Delivery System Question: Where are the Sextupoles, and Decapoles, ? Answer: Sextupoles ( n=3 ) and Decapoles ( n=5 ) are Dipole ( n=2 0 ) Natural Harmonics: Dodecapoles ( n=6 ) and 20-poles ( n=10 ) are Quadrupole ( n=2 1 ) Natural Harmonics: In general, 2n -poles with n =(2m+1)2 p are 2.2 p -pole ( n=2 p ) Natural Harmonics:. In other words, using the example of Dipoles, the ensemble of Dipole ( R S ( )- invariant) current distributions includes that of Sextupole ( R S ( /3) -invariant) ones. As a consequence, Dipole magnets built from a Dipole invariant current distribution most likely contains Sextupole, Decapole, etc components. Other harmonics are excluded, unless induced by non-invariant fabrication errors. Natural Harmonics Slide 13 13 Beam Delivery System A Quadrupole magnet can be designed as follows: 1.Define a current distribution J(r, ) over a 90 wide sector : 2.Rotate it by the symmetry R S ( /2) over the 2 nd sector 3.Repeat the R S ( /2) operation to the 3 rd sector 4.Repeat the R S ( /2) operation to the 4 th sector Quadrupoles Have Four Poles x y J (1) (r, ) + x y + J (2) (r, ) R S ( /2) J (1) (r, ) + x y J (2) (r, ) J (1) (r, ) + + J (3) (r, ) J (4) (r, ) + + Slide 14 14 Beam Delivery System The simplest Monopole magnet is given by longitudinal current line transporting a total current I z. By rotation symmetry :. From Ampres law :. The complex magnetic field is then given by The Simplest Monopole Magnet Nota Bene 1: The magnetic field inside of the conductor is an ideal focusing field ! Nota Bene 2: Point-like Magnetic Monopoles do not exist on earth ! ^ I B x y Slide 15 15 Beam Delivery System A Dipole Magnet The simplest Dipole Field configuration is given by a pair of linear conductors: x y +I+I II R0R0 This expression can be expanded for : Comparing to the mathematician expressions (page 71) one gets : Dipole : ; Sextupole: ; Decapole : (2m+1)-pole : B Nota Bene: This Dipole Magnet still has zero curvature radius ! Slide 16 16 Beam Delivery System A Quadrupole Magnet The simplest Quadrupole Field configuration is given by a quadruplet of conductors: x y +I+I +I+I R0R0 Comparing to the mathematician expressions (page 71) one gets : Quadrupole: Dodecapole: II II Slide 17 17 Beam Delivery System A Better Quadrupole Magnet An improved Quadrupole Field configuration is given by an octuplet of conductors: x y +I+I +I+I R0R0 Comparing to the mathematician expressions (page 71) one gets : Quadrupole: Dodecapole: cancels for II II +I+I II II +I+I The calculation proceeds like the preceding one leading to: Slide 18 18 Beam Delivery System A Sector Quadrupole Magnet This quadrupole is realised with 4 finite sectors of current density J. One can integrate elementary currents by using the preceding result : Quadrupole : Dodecapole:, cancels for Slide 19 19 Beam Delivery System An Ideal Quadrupole Magnet: Cos(2 ) distribution The engineer meets the mathematician when he tries to build current lines following a cos(2 ) angular distribution : with. The complex magnetic field is given by the following integral: This is a Pure Quadrupole, in the mathematician sense: Quadrupole Slide 20 20 Beam Delivery System This is a Pure Quadrupole, in the mathematician sense: Quadrupole The Cos(2 ) Quadrupole Magnet Assuming a uniform radial distribution of currents : the complex magnetic field is given by the following expression: Slide 21 21 Beam Delivery System Superconducting Quadrupoles Slide 22 22 Beam Delivery System The LHC Cos( ) Dipole Magnet LHC Superconducting Dipoles have been wound from the cos( ) model, using two layers of cables and anglular wedge to fit the cos( ) distribution. Internal Diameter : = 56 mm Current Density: J = 500 A.mm -2 Dipole field : B = 8.33 T Current Lines: NI = ??? A.t angular wedges pole n1 Slide 23 23 Beam Delivery System LHC Arc Quadrupole Magnet LHC Superconducting Arc Quadrupoles have been wound from the sector model. Note that current sectors encompass the [0, 30] azimuth interval to eliminate the b 6 dodecapole harmonics. Their main parameters are: Internal Diameter : = 56 mm Current Density: J = 500 A.mm -2 Quadrupole gradient : G = 223 T/m Current Lines: NI =??? A.t pole n1 Slide 24 24 Beam Delivery System LHC Low- Quadrupole Magnets LHC Superconducting Low- Quadrupoles are realized from the cos(2 ) model. Fermilab Low- Quadrupole KEK Low- Quadrupole pole n1 pole n2 pole n4 pole n3 Internal Diameter : = 70 mm Current Density: J = 545-700 A.mm -2 Quadrupole gradient : G = 215 T/m Current Lines: NI = ??? A.t Slide 25 25 Beam Delivery System Copper at room temperature: 1.6 10 -8 .m at 10 K, 0 T (RRR =100): 1.6 10 -10 .m at 10 K, 7 T (RRR = 100): 4.3 10 -10 .m NbTi in normal state : 5-6 10 -7 .m in superconducting state : NbTi Superconducting Cables NbTi ingot (136 kg) mono-filament rod in Cu can (10 km) multi-filaments cable in Cu can multi-wire Rutherford cable Electric Resistivities: (Courtesy Oremet Wah Chang) High performance mono-filament rods achieve J C of 3000 A/mm 2 at 4.2 K and 5 T. Copper is necessary to flow the electric current in case of a magnet quench : J = 500 A/mm 2 at 2 K on average in the conductor. (Courtesy Furukawa Electric Co.) mono-filament hex. rods multi-filament cable Rutherford type cable ~1mm NbTi Wire (Courtesy Alstom/MSA) Slide 26 26 Beam Delivery System B vs. J C Limitations of SC Conductors Nb3Sn conductors allow for increase of field and gradient, but: 1.Cables are more expensive 5-10 and production volume 100, 2.Fabrication of accelerator magnets still at the R&D stage. MaterialT C (K) Nb9.2 NbTi9.5 Nb 3 Sn18 Slide 27 27 Beam Delivery System Normal Conducting Magnets Internal Diameter : =300 mm Current Density: J = 4 A/mm 2 Quadrupole gradient : G = 0.39 T/m Current Lines: NI =13000 A.t Normal Conducting are generally built with water cooled Cu conductors. In Air magnets, the magnetic field is generated by the current lines through the Maxwell-Ampres equation: Air magnets are very ineffective, as illustrated by this example: AirConductors Slide 28 28 Beam Delivery System Quadrupole gradient : G = 5.7 T/m Normal Conducting Magnets: the Role of the Iron Yoke The Iron Yoke increases the magnetic field in the Iron thanks to its much higher Permeability with, It is shaped to optimize the Magnetic Circuit through the Air internal chamber, and to reduce the Stray Fields at large radii. Iron YokeConductors Slide 29 29 Beam Delivery System Magnetic Saturation Introducing the auxiliary magnetic field H such that, one gets where r is the relative permeability. First magnetization curve, then hysteresis takes over. Saturation occurs around 2 T 10% for high quality electrical steel or permalloy (nickel-iron). rr 2 T The relative permeability can reach max. values from 5000 to 10000. In the saturation regime, B vs. J linearity is lost and increasing the electric current is not effective any more. As a consequence: 1. Normal Conducting Magnets are limited to ~2 T fields 2. In Superconducting Magnets, the Iron Yoke is mostly used to contain the transverse stray fields. H Slide 30 30 Beam Delivery System ILC-RDR Final Focus Systems To avoid parasitic collisions, the ILC FFS is based on a large 14 mrad crossing angle, with two alternate designs for zero (head-on) or 2 mrad crossing-angles. 14 mrad : the final doublet specialized to the incoming beam, another doublet is used for the outgoing beam compact quadrupoles with small R o outer radius. head-on : the final doublet is shared by the incoming and outgoing beams large aperture quadrupoles with large R i inner radius. 2 mrad : only the final D-quadrupole is shared the incoming and outgoing beams. 14 mrad Xing G =144 T/m, R i =10 mm Head-on, LHC Arc G =250 T/m, R i =28 mm 2 mrad Xing, LHC low- G =250 T/m, R i =35 mm Slide 31 31 Beam Delivery System ILC : 14 mrad Final Focus System l* = 3.5 m Combined sextupole -octupole Quadrupole QD0 NbTi cable : serpentine winding technology Slide 32 32 Beam Delivery System ILC : Head-on Final Focus System Electrostatic separators (26 kV/cm, Ti, 7 4 m) needed to separate e+ and e- beams. A dipole (~30 mT) is used to keep the incoming trajectory straight. SPS ZX separator l* = 4 m IP 1 st 2 nd etc... parasitic collisions = n51 m (= nc340 ns/2) Slide 33 33 Beam Delivery System ILC : 2 mrad Final Focus System Slide 34 34 Beam Delivery System Reasons for the Strong Quadrupoles Strong Quadrupoles allow for a short focal distance f ~ l*, needed for 3 reasons : 1. Chromaticity: it is kept under control by keeping l* < 10 m. This is a soft constraint, especially since the chromaticity is optically corrected by sextupoles. 2. Spent beam : in the Head-on and 2 mrad cases, the main constraint is to clear the outgoing beam and beamstrahlung after the collision: 3. Synchrotron radiation: in all cases, the need to clear the synchrotron radiation generated by the final doublet from the inner detector region (with a vertex detector at low radius, and masks) is ensured by tight collimation of the transverse beam tails. The collimation requirements are growing with l* Slide 35 35 Beam Delivery System Question n3: Do we need a Flat Beams ? Slide 36 36 Beam Delivery System Beam Beam Effects Slide 37 37 Beam Delivery System Beam-Beam Interactions (in short) The beam-beam interaction in a linear collider differs from the circular collider: High charge densities at the interaction point lead to very intense fields and to very strong interaction within single collision Strong mutual focusing of beams (pinch) gives rise to luminosity enhancement by a factor H D. As electrons/positrons pass through intense field of opposite beam, they radiate hard photons beamstrahlung, and lose energy in a random fashion. Interaction of beamstrahlung photons with intense field causes e + e pair production, which causes background and provides a luminosity signal. The beam-beam interaction is reduced with flat beams. Slide 38 38 Beam Delivery System Focusing Effect of a Round Current Line A simple circular current line with uniform density J of radius R generates: an external field which attracts same sign current ( qv ): with an internal focusing field for same sign current: i.e. the perfect magnetic focusing lens. B J x y Nota Bene: principle of a Lithium lens used for proton and muon focusing. Slide 39 39 Beam Delivery System Focusing Effect of a Round Line Charge Distribution A simple circular charge distribution with uniform density of radius R generates: an external field which repels same-sign charge q : with an internal defocusing field for same-sign charge q : E x y Slide 40 40 Beam Delivery System Focusing Effect of a Round Positron Beam A circular Positron beam (travelling towards us) of radius R is generating both charge and current J distribution with the relation: Effect on a Positron travelling with the beam, at r < R : the electric and magnetic forces cancel, yielding a repelling force vanishing for. Effect on an Electron travelling against the beam: the electric and magnetic forces add, yielding a non-vanishing Beam-Beam attracting force for. ( , J) x y (E, B) Slide 41 41 Beam Delivery System Coherent Beam-Beam Force The Coherent beam-beam force is the force acting on the electron bunch, i.e. the average of the forces over the electron distribution : Beam-Beam Effect : Round Case Incoherent Beam-Beam Force (assuming ) This beam-beam force can be easily extended to the case of a round Positron beam with a non-uniform circular charge distribution. The force acting on an individual Electron is given by: It is called the Incoherent beam-beam force. ( (r), J (r)) x y (E, B) However, two ingredients are still missing ! 1. realistic positron and electron bunches are not infinite lines : i.e. 2. colliding positron and electron bunches are not round: Slide 42 42 Beam Delivery System v = 0 v = c v < c ErEr Electrostatic field in the rest frame Lorentz Boost B 1/c 2 v E = 0 = Shock Wave in the lab frame F = q(E + v B) (1 vv/c 2 ) / r No Intra-beam Forces for = Strong Beam-Beam Forces No Wake: the Front and Rear parts of the bunch do not contribute to the beam-beam forces ! 1/ v = c The Electromagnetic Field of a Relativistic Particle This will be worked out during the Home Work session. Slide 43 43 Beam Delivery System Incoherent Beam-Beam Kick The incoherent beam-beam kick is given by integrating the effect of the force over the bunch collision: assuming that the position r stays constant over the collision ! Beam-Beam Effect : Round Gaussian Case Incoherent Beam-Beam Force This incoherent beam-beam force generated by a round Gaussian Positron beam, with on an individual Electron is given by: Hence the following expression of the weak incoherent beam-beam kick : Slide 44 44 Beam Delivery System The Disruption Parameter In the linear regime,, the incoherent beam-beam kick can be linearized as a thin focusing lens: Coherent Beam-Beam Kick Without demonstration, the expression of the weak coherent beam-beam kick is: Beam-Beam Effect : Round Gaussian Case Coulombic Interaction: In the far distance regime,, both the individual electron (incoherent kick) and the electron bunch (coherent kick) reach the point-like Coulombic interaction. Slide 45 45 Beam Delivery System Incoherent Beam-Beam Effect : Gaussian Flat Case Incoherent Beam-Beam Kick For flat Gaussian beams the incoherent beam-beam kick is given by: where is the complex error function. Using the following expansions: and. the round beam case can be retrieved and the linearized form, projected on each x and y plane, is given by:, with the disruption parameters : Coherent Beam-Beam Kick The coherent beam-beam kick is not amenable to easy analytic treatment. Slide 46 46 Beam Delivery System ++++++++++++++++++++++++++++++++++++++++++++++++++ Focusing Effect of a Sheet (ultra-flat) Beam The ultra-flat limit of a flat positron bunch is given by an infinite surface density of charge whose motion is generating surface current J S = c. B x y The force acting on an individual electron is given by for. The important result is that the e.m. fields are constant over space, and the resulting forces have an infinite vertical range ! In the realistic case where, the vertical range of the beam- beam forces is set by the smallest infinite size, i.e. by. E B E The electric and magnetic fields are given by the well known expressions: Slide 47 47 Beam Delivery System The preceding expressions assume that the beam-beam effect is weak and that the bunch particle trajectory are not perturbed during the collision. In Linear Colliders, this assumption is correct for the horizontal motion. On the contrary, the vertical beam-beam forces are strong because y * is very small, (a few nanometres) and the vertical charge densities are very large. Hence the particle trajectories are bent towards the other beam. There are 3 main consequences: 1. the Pinch Effect: the bunch are not rigid but they are pinched during the collision. This induces an enhancement of the luminosity by a factor H D ~1.4 2, 2. the Breamstrahlung effect : the individual electrons and positrons are emitting Synchrotron Radiation hard Photons while their trajectories are curved. This generates energy loss (~2-3 %), energy spread and secondary background. 3. the Kink Instability: in the high disruption regime,, individual trajectories are kinked, i.e. oscillate through the opposite bunch leading to unstable regime. Strong Beam-Beam Forces Slide 48 48 Beam Delivery System ILC parameters: D y ~12 Luminosity enhancement H D ~ 1.4 Not much of an instability. The Pinch Effect Animations produced by A. Seryi using the GUINEAPIG beam-beam simulation code (D. Schulte). Slide 49 49 Beam Delivery System N 2 D y ~ 24 Beam-beam instability is clearly pronounced. Luminosity enhancement is compromised by higher sensitivity to initial offsets. Beam-Beam Instability Effects Animations produced by A. Seryi using the GUINEAPIG beam-beam simulation code (D. Schulte). Slide 50 50 Beam Delivery System Beam-Beam Simulations D y ~ 12 N x2 D y ~ 24 Animations produced by A. Seryi using the GUINEAPIG beam-beam simulation code (D. Schulte). Slide 51 51 Beam Delivery System Beamstrahlung is traditionnally characterized by the parameter: with the electron Compton wavelength. The average magnetic field of a Gaussian bunch is given by: Introducing Schwingers critical field: Beamstrahlung Electrons radiate against the coherent field of the positron bunch synchrotron radiation called Beamstrahlung. NeNe ee ee ee ee B = 0.32 T @ LEP, 60 T @ SLC, 440 T @ ILC ILC 500 0.05 nn 1.3 BS 2 % Slide 52 52 Beam Delivery System Since the luminosity is given by : one would like x y small to maximise L, AND ( x + y ) large to reduce BS. This is achieved by using Flat Beams : x >> y in such a way that does not depend on y. Thus, one can make y as small as possible to achieve high luminosity, while BS is fixed by x. The Beamstrahlung Parameter Beamstrahlung causes a spread in the e e center of mass energy. This effect is characterized by the Beamstrahlung parameter BS : Slide 53 53 Beam Delivery System Beamstrahlung Numbers The average value for Gaussian beams : BS and n (and hence ) directly relate to luminosity spectrum and backgrounds. Photons per electron : Average energy loss : ILC 500 0.05 nn 1.3 BS [%] 2 Slide 54 54 Beam Delivery System Beam-Beam Deflection Colliding with offset, e + and e - beams deflect each other. Tiny offsets (~nm) at IP cause large angular kick (~100 rad) and therefore large offsets of the beam a few meter downstream. Slide 55 55 Beam Delivery System Beam-Beam Deflection Animations produced by A. Seryi using the GUINEAPIG beam-beam simulation code (D. Schulte). Slide 56 56 Beam Delivery System Coherent Beam-Beam Deflection Curves The outstanding question of IP collisions is: How can one bring two 5 nm-wide spots to collide at the same point when the mechanical motions and vibrations are of the order of micrometers, or more ? The beam-beam deflection brings two fondamental ingredients of the answer: 1.A high sensitivity to small offsets: 2.Long range forces: the range of vertical beam-beam is set by x * ~ 1 m Horizontal beam-beam curveVertical beam-beam curve Slide 57 57 Beam Delivery System Collision Orbit Feedback Two additional ingredients permit the fast feedback system to correct the IP offset: the long pulse: the ILC pulse with 2820 bunches separated by 340 ns (~100m) is suitable to bunch- to-bunch correction system (analogy with a machine- gun shooting a moving target, and the eyes motion), beam position monitors with 1 m resolution can be used to measure and zero the B-B deflections. As a result, no active mechanical stabilization of final doublet is required up to ~ 100 nm jitter tolerance ( f < 3 GHz). Head-on Collisions14 mrad Collisions Slide 58 58 Beam Delivery System Beamstrahlung photons and beam particles interact, and create e + e pairs. Three processes are important for incoherent pair production ( 0.6 ): Breit-Wheeler process ( e + e - ) Bethe-Heitler process (e e e + e - ) Landau-Lifshitz process (e + e - e + e - e + e - ) e e Pair Production Spectrum of pairs Nota Bene: the number of Real Photons in the process equals by the number of r in the Physicists names. Pairs are low energy particles affected by the beam (focused or defocused), hence a source of background. But they are curled by the field of the detector solenoid: B-H and L-L pairs provide a powerful signal proportional to the luminosity because they are produced by overlapping the e + and e beam distributions. Luminosity Monitor Calorimeter Slide 59 59 Beam Delivery System ILC Beam Parameters at Collision ParameterUnitILC nominal E0E0 GeV250 N10 2 x*x* nm639 y*y* nm5.7 zz mm 300 nbnb 2820 f rep Hz5 tt ns340 L 10 34 cm -2 s -1 2.0 BS %2 nn 1.3 Slide 60 60 Beam Delivery System Question n4: Do we need Correction Systems? Slide 61 61 Beam Delivery System Beam Linear Optics Slide 62 62 Beam Delivery System ILC BDS Optical Functions Slide 63 63 Beam Delivery System Horizontal Planar Symmetry Most accelerators are designed with simplifying assumptions : 1. a planar horizontal reference trajectory 2. an electromagnetic circuit symmetric with respect to the horizontal plane: Under these assumptions, the 6D Transfer Map satisfies: The Linearized Transfer Map (a matrix R ) commute with the matrix S y with Slide 64 64 Beam Delivery System General Form of Transfer Matrix The planar symmetric 6D transfer matrix simplifies to the following 2x2 blocks: in such a way that the vertical motion is decoupled from the two other planes. We make two more assumptions: 1.The BDS is a static system with no time dependence: hence trajectories do not depend on the z coordinate; 2.The energy is constant : no acceleration, energy loss due to synchrotron radiation is neglected at the first order is constant. Under these assumptions, the 6D transfer matrix further simplifies to Slide 65 65 Beam Delivery System Symplecticity The Transfer Matrix R is symplectic provided that : Note that, for a 2x2 matrix,, so that The symplecticity condition is fulfilled if : and Notice that is free. Slide 66 66 Beam Delivery System Example : Transport Matrix Through a Quadrupole We consider a straight line reference trajectory with passing through a quadrupole of length l Q with a field : which is null along the reference trajectory (no curvature, ). The equations of motion can be simplified since: 1. the energy is constant (magnetic fields do accelerate, but do not work); 2. the reference frame is constant; 3. and ; 4. ; 5. Slide 67 67 Beam Delivery System Example : Transport Matrix Through a Quadrupole At the first order in the variables the equations of motion lead to: Transverse motion: Longitudinal motion: The energy is constant : The path length difference is given by: Slide 68 68 Beam Delivery System Example : Transport Matrix Through a Quadrupole This equation can be easily solved to obtain the following transfer matrix: with assuming G > 0. For G < 0, the matrices and are interchanged. Slide 69 69 Beam Delivery System General Form of Beam Matrix with Planar Symmetry The general form of the Beam Matrix (cf. Part 1, page 51-53) which fulfils a) planar symmetry, b) no time correlations, and c) invariance of , simplifies to because the correlation coefficients by symmetry S y. Nota Bene: these assumptions are useful to design the Beam Delivery System at the 0 th order (cf. page 62). They are not adapted to investigate the effects of misalignments and errors in the beam line, hence the implementation of correction systems e.g. y -orbit correction. Slide 70 70 Beam Delivery System 2D Beam-Matrix Transport Assuming, for convenience, that the average beam offsets are zero,, the beam matrix is transported from s 1 to s 2 as follows: 2D Vertical Transport: Clearly, the emittance is conserved The vertical beam Beta-functions, also called Twiss Parameters, transform as follows: Slide 71 71 Beam Delivery System Normalized Variables Conversely, given the initial and final beam matrices, the transfer matrix can be determined up to a phase. Using the Normalized Variables (cf. Part1, page 52) such that, one gets: is a 2D orthogonal matrix. The rotation matrix is the transfer matrix of the normalized variables: Slide 72 72 Beam Delivery System The Phase Advance As a result: The parameter such that : is called the phase advance. It allows to parameterize the transfer matrix as follows: One can check that when Slide 73 73 Beam Delivery System Ideal beam transport of a polychromatic bunch assumes that: 1) the centroids of all energy slices are aligned (no banana shape) 2) the emittance and Twiss parameters of all energy slices are identical (no butterfly shape) 3) the beam is matched to the ideal beam-line (requires a precise definition for single pass beam lines). Chromatic Filamentation is an emittance growth effect which occurs as soon as one of these conditions is broken, allowing an accumulation of phase errors along very long beam lines like the Main Linac or the BDS. 2D Chromatic Filamentation u v We consider the 3 rd source of filamentation, pictured as shown in the normalized phase space of the matched beam at the position s 1 where the mismatch shows up. Slide 74 74 Beam Delivery System The outgoing beam matrix at the position s 2 can be calculated as follows: After filamentation, the emittance is given by 2D Chromatic Filamentation u v Slide 75 75 Beam Delivery System The Perturbative Approach to Higher Order Optics In a single pass, based on TRANSPORT notations, the change of the co- ordinate vector is given by All terms with one subscript equal to 6 are referred to as chromatic terms, since the effect depends on the momentum deviation of the particle. All terms without subscript equal to 6 are referred as geometric terms, since the effect depends only on the central momentum. {j,k,n=1,6} Slide 76 76 Beam Delivery System s Energy spread chromatic aberrations The Final Doublet develops the highest magnetic field, and hence the biggest chromaticity Chromatic Aberrations at the IP Slide 77 77 Beam Delivery System K 1 (+ K 1 (- s x (- x(+ Sextupole Magnet in the doublet Local Chromatic Correction Local Chromatic Correction Final Quadrupole: x(+ x (- for Sextupole: Dispersion Slide 78 78 Beam Delivery System Local Chromaticity Correction Scheme Local chromatic corrections introduced in the NLC FFS, and now adopted by all LC designs. A single sextupole cures the chromaticity but introduces dramatic geometric aberrations, unless they are paired with -phase advance. Slide 79 79 Beam Delivery System End of Part 2 Slide 80 80 Beam Delivery System Conclusions The Beam Delivery System of a Superconducting Linear Collider is the scene of many accelerator physics outstanding issues: Non linear optics and high order correction systems High field magnets and superconducting technology Collimation and wakefields Synchrotron radiation and emittance growth Fast feedback system Luminosity optimisation and monitoring Strong beam-beam interaction Novel beam-physics concepts... These topics can only be exhausted with longer lectures and more thorough lecturer. Thank you for your attention. Slide 81 81 Beam Delivery System Acknowledgements I would like to thank: Weiren Chou for extending this invitation, Deepa Angal-Kalinin for the permission to re-use her lecture from 2008, My colleagues Antoine Chanc, Olivier Delferrire, Jacques Payet, Pierre Vdrine for providing expertise, material, corrections and proof-reading, My young colleague Reine Versteegen for all the above, plus considerable help in assembling the material presented.