Tutorial on (Generating) High Brightness Beams

Transcription

1 Tutorial on (Generating) High Brightness Beams David H. Dowell SLAC & LBNL PAC Tutorial April,

2 Outline of Tutorial Intrinsic emittance and Q Space charge limited emission Simple optical model & R emittance mittance compensation and matching Solenoid aberrations Beam blowout dynamics & 3 rd order space charge Dowell - PAC Tutorial

3 The Photocathode R Gun System Basic components of the photocathode R gun system R requencies 44MHz, 433MHz.3 GHz:L-band.8 GHz: S-band 7GHz: X-band R Klystron Master Oscillator Drive Laser Drive Lasers: Nd:YL, Nd:glass, TiS, iber, Bucking Coil Gun -6 MeV e-beam Linac Section Solenoid Photocathodes Semi-Conductor: CsKSb, CsTe, Metal: Cu, Mg, +match to st linac section 3 Dowell - PAC Tutorial

4 Q and thermal/intrinsic emittance Space charge limited emission Simple optical model & R emittance mittance compensation and matching Solenoid aberrations Beam blowout dynamics & 3 rd order space charge Dowell - PAC Tutorial 4

5 mittance - xpt. nergy Quantum fficiency Photo-lectric mission and the 3-Step Model Photoelectric emission from a metal given by Spicer s 3-step model:. Photon absorption by the electron Comparison of theory & expt. lectron transport to the surface x - 3. scape through the barrier x -3 x -4 Metal Vacuum x -5 x -6 ermi nergy occupied valence states )Photon absorbed )lectrons 3)lectrons move to surface escape to vacuum e- e- e- Optical depth photon Potential barrier due to spillout electrons Direction normal to surface x -7 x Wavelength (nm) Theory at MV/m Theory at 5 MV/m xperiment. y =.8569x mittance - Theory *D. H. Dowell, K.K. King, R. Kirby and J.. Schmerge, PRST-AB 9, 635 (6) 5 Dowell - PAC Tutorial

6 Q and emittance depend upon electronic structure of the cathode Sum of electron spectrum yield gives Q: Q W DOS ( ) d Width of electron spectrum gives intrinsic emittance n eff Both are dependent upon the density of occupied states near the ermi level Simple electron gas model of a metal at kt = State Occupation # eff nergy spectrum of emitted electrons Bound electrons eff xcess energy eff ermi nergy, Vacuum nergy, = + eff 6 Dowell - PAC Tutorial

7 Refraction of electrons at the cathode-vacuum boundary Snell s Law for electrons Conservation of transverse momentum at the cathode-vacuum boundary p p p in out x x in out total sinin ptotal sinout p m in total out p m total W To escape electron longitudinal momentum needs to be greater than barrier height: in p m z eff in m eff m cos Refraction law for electrons: sinout n sin n in eff out cos Outside angle is 9 deg at in max which is typically ~ deg. Maximum internal angle for electron with energy which can escape: max in eff in 7 Dowell - PAC Tutorial

8 Q for a metal is the electron energy is the ermi nergy eff is the effective work function eff W Schottky Step : Optical Reflectivity ~4% for metals ~% for semi-conductors Optical Absorption Depth ~ angstroms raction ~.6 to.9 Q( ) R( ) Step : Transport to Surface e-e scattering (esp. for metals) ~3 angstroms for Cu e-phonon scattering (semiconductors) raction ~. ee d(cos ) eff eff Step 3: scape over the barrier d d d(cos ) Sum over the fraction of occupied states which are excited with enough energy to escape, raction ~.4 d d Azimuthally isotropic emission raction = raction of electrons within max internal angle for escape, raction ~. 8 Dowell - PAC Tutorial

9 Q for a metal is the electron energy is the ermi nergy eff is the effective work function eff W Schottky Step : Optical Reflectivity ~4% for metals ~% for semi-conductors Optical Absorption Depth ~ angstroms raction ~.6 to.9 Q( ) R( ) Step : Transport to Surface e-e scattering (esp. for metals) ~3 angstroms for Cu e-phonon scattering (semiconductors) raction ~. Q ~.5*.*.4*.* = 4x -5 ee d(cos ) eff eff Step 3: scape over the barrier d d d(cos ) Sum over the fraction of occupied states which are excited with enough energy to escape, raction ~.4 d d Azimuthally isotropic emission raction = raction of electrons within max internal angle for escape, raction ~. 9 Dowell - PAC Tutorial

10 Derivation of Photo-lectric Intrinsic mittance y x p x p total sin cos p total m z p p sin cos m sin cos x total n x p x mc / metal vacuum p sin d(cos ) d cos d eff eff x m Intrinsic emittance for photoemission from a metal d d(cos ) d n x 3mc eff Dowell - PAC Tutorial

11 Q & mittance are related via the excess energy Intrinsic emittance (microns/mm-rms) Q(copper) Define the excess energy as: excess eff ee eff R excess ( R( )) ( ) ( ( )) Q opt opt 8 eff ( eff ) n x 3mc 3mc eff excess ee mittance Q xcess nergy (ev) Dowell - PAC Tutorial

12 Intrinsic emittance and Q Space charge limited emission Simple optical model & R emittance mittance compensation and matching Solenoid aberrations Beam blowout dynamics & 3 rd order space charge Dowell - PAC Tutorial

13 Space Charge Limit (SCL) is different for DC diode and short pulse photo-emission Cathode Anode Space Charge ield Across a Diode, Child-Langmuir law: J CL 4 ev 9 m d 3/ V electron flow, d Space Charge ield Across a Short lectron Bunch from a Laser-driven Photocathode, parallel plate (capacitor) model: SCL applied Drawing by A. Vetter 3 Dowell - PAC Tutorial

14 Surface current density (A/cm) Comparison of space charge limits for Child-Langmuir and Short Pulse Geometries/Conditions 5 4 J SCL applied V d laser laser LCLS 3 J CL 4 ev 9 m d 3/ Voltage gain across gap (MV) Child-Langmuir limit Short pulse limit LCLS at nc LCLS at 5 pc LCLS typically operates at approximately half the space charge limit for short pulse emission and a factor of 4 to 5 higher than the space charge limit given by the Child-Langmuir law. 4 Dowell - PAC Tutorial

15 Transverse lectron Beam Shape: The beam core is clipped at the SCL e laser Q r elaser rm rf sin rf Q e m r Produces a flat, uniform transverse distribution in the beam core. lattens hot spots. Space Charge Limit of Gaussian peak Q Limited mission radial distribution follows laser & Q Space Charge Limited mission radial distribution saturates at the applied field Dowell - PAC Tutorial 3 3

16 Derivation of Schottky Scan unction: mitted charge vs. launch phase Begin with the Q for a metal cathode: R Q opt ee eff where the effective work function is eff W e e rf 4 sin rf Putting this into the Q formula gives, R Q opt ee W e e rf sin verything is known except for material work function, W, and the field enhancement factor,. it Schottky scan data to find them. rf /(4 ) 6 Dowell - PAC Tutorial

17 Derivation of Schottky Scan unction: mitted charge vs. launch phase W 4.83 ev;. it to LCLS data by Dao Xiang, SLAC Begin with the Q for a metal cathode: R Q opt ee eff where the effective work function is eff W e e rf 4 sin rf Putting this into the Q formula gives, R Q opt ee W e e rf sin verything is known except for material work function, W, and the field enhancement factor,. it Schottky scan data to find them. rf /(4 ) 7 Dowell - PAC Tutorial

18 Intrinsic emittance and Q Space charge limited emission Simple optical model & R emittance mittance compensation and matching Solenoid aberrations Beam blowout dynamics & 3 rd order space charge Dowell - PAC Tutorial 8

19 Optical Model of the R Gun: Solenoid ffects Drive Laser f rf mc e sin e -6 MeV lectron Beam f rf ~-5 cm for MV/m which needs to be compensated for with solenoid Defocusing and ocusing R Lenses ocusing of Gun Solenoid Gun Solenoid Solenoid cancels defocusing of gun R and performs emittance compensation and matching to booster linac Solenoid has focal length of ~5 cm but is ~ cm long=> thick lens => aberrations The principal solenoid aberrations can be classified as : Chromatic Geometric Anomalous fields Misalignment (not discussed here) 9 Dowell - PAC Tutorial

20 Optical Properties of the Gun s R ields (MV/m) BNL/SLAC/UCLA style PC gun cavity, S-band, beadpull template file = MHz 7 6 phase shift or a phase shift between cells: Longitudinal and transverse electric fields along a 5mm radius e z: Superfish sin e sine 8 r: Superfish f mc fe mc e z: *cos(kz) 6 r: d(*cos(kz))/dx 4 e sin e sin e sin f mc mc mc sin sin dz/dz of Superfish z and with ~ and () ~ () -4 thus there is no net focusing across the iris between two adjacent -mode cells: e -6 sin e sin f f mc mc -8 Hence the optical strength of the gun is dominantly due to the defocus at the exit of the last cell, - e sin 5 e 5 (cm) fe mc e Dowell - PAC Tutorial

21 Normalized emittance (microns) x' (mr) f rf mc e sin e R mittance n x x xx x x' x x e cos e mc d d x e f rf x e e cose rf, for x e 9deg mc This is the linear part of the emittance, the non-linear part due to the R curvature is nd order in the phase spread of the bunch, e sine rf, for x e 9deg mc e cos e rf, total cos sin x e e mc 5 5 e = deg e =9 deg x (mm) xit phase (deg) st + nd st-order rf emittance nd-order rf emittance scales as the beam size squared >> a common feature of many emittance sources Dowell - PAC Tutorial

22 Intrinsic emittance and Q Space charge limited emission Simple optical model & R emittance mittance compensation and matching Solenoid aberrations Beam blowout dynamics & 3 rd order space charge Dowell - PAC Tutorial

23 Projected mittance Compensation* The radial envelope equation for each slice position, : xternal focusing by magnetic and R fields emittance acts like a defocusing pressure K( ) n ( ) r ( ) r ( ) k ( ) r r 3 3 r( ) r( ) acceleration changes magnification of the divergence Space charge defocusing, K is the generalized perveance Generalized perveance (Lawson, p. 7) K( ) I( ) 3 3 I I is the peak current of slice I is 7 amps *Serafini & Rosenzweig, Phys. Rev. 55, 997, p Dowell - PAC Tutorial

24 Projected mittance Compensation: How to align the slices? Beam nvelope quation: Assume no acceleration, zero slice emittance K r krr Consider solutions for small perturbations from equilibrium radius: r Substituting into the envelope eqn. and expanding in a Taylor series we get K K kr e kr e e r e constant terms, set sum to zero and solve for e : e K k r Dowell - PAC Tutorial 4 I 3 3 I k r K I+ I I- I nvelope eqn. for small amplitude radial perturbations: e e k e ke sin kez cos kez This is known as balanced or Brillouin flow, when the outward space charge force is countered by external focusing, usually a magnetic solenoid. This establishes the invariant envelope of Serafini & Rosenzweig. e sin kz cos kz

25 mittance (microns/mm-rms/a-rms) nvelope eqn. for small amplitude radial perturbations K e e k e ke sin kez cos kez e This is the wave equation with oscillating solutions: k e sin kz cos kz with the equilibrium wave number defined as: K kr To simplify the emittance calculation let s make the crude approx. that the focusing channel can be replaced by a thin lens such that kesin kez e 4 ke sin kez ; ke f e Derivation of projected emittance for a current spread in the slices 3 3 e I I The emittance due to lens strength dependence upon beam current: d di fe n, sccomp e I Using the expressions for /f e and the equilibrium wavenumber in this eqn. gives the projected emittance for a beam with a I rms spread in slice current: e I n, sin k cos I ez kez kez II I = A I = A e = mm = Distance along beamline (cm) 5 Dowell - PAC Tutorial

26 mittance Compensation Summary The beam at the cathode begins with all slices of nearly equal peak current propagating with equilibrium radii in a balanced flow. The beam envelope equation is linearized and solved for small perturbations about this equilibrium. The solution obtained shows all slice radii and emittances oscillate with the same frequency (determined by the invariant envelope), independent of amplitude. Assuming the slices are all born aligned, they will re-align at multiple locations as the beam propagates, with the projected emittance being a local minimum at each alignment. The beam size will oscillate with the same frequency, but shifted in phase by /. Dowell - PAC Tutorial 6

27 Transverse mittance (microns) Beam Radius at xit, rms (mm) mittance compensation includes matching the low energy beam to the booster linac to damp the emittance oscillations In addition to compensating for the emittance from the gun, it is necessary to carefully match the beam into a high-gradient booster accelerator to damp the emittance oscillations. The required matching condition is referred to as the errario working point and was initially formulated for the LCLS injector*. The working point matching condition requires the emittance to be a local maximum and the envelope to be a waist at the entrance to the booster. The waist size is related to the strength of the R fields and the peak current. R focusing aligns the slices and acceleration damps the emittance oscillations..5 5 cm S-Band TW Section.5 Solenoid = 8 G. =6 MV/m, = 5 deg 3 =6 MV/m, =.7 deg linac = 9 MV/m beam =6 MeV.5 Transverse mittance rms Radius z (cm) *M. errario et al., HOMDYN study for the LCLS R photoinjector, SLAC-PUB-84, LCLS-TN--4, LN-/4(P). 7 Dowell - PAC Tutorial

28 Transverse mittance (microns) Beam Radius at xit, rms (mm) Assume the R-lens at the entrance to the booster is similar to that at the gun exit with an injection phase at crest for maximum acceleration, e =/, so the angular kick is, e mc Taking the derivative gives the rf term needed for the envelope equation, The errario Working Point e mc with the accelerating field of the booster. or a matched beam we want the focusing strength of the accelerating field to balance the space charge defocusing force, i.e. no radial acceleration: match I since A I 3 match e mc.5 5 cm S-Band TW Section Solving gives the matched beam size:.5 Solenoid = 8 G. =6 MV/m, = 5 deg 3 =6 MV/m, =.7 deg linac = 9 MV/m beam =6 MeV matched I I A Transverse mittance rms Radius z (cm) matched is the waist size at injection to the accelerator. The matched beam emittance decreases along the accelerator due the initial focus at the entrance and Landau damping. 8 Dowell - PAC Tutorial

29 Intrinsic emittance and Q Space charge limited emission Simple optical model & R emittance mittance compensation and matching Solenoid aberrations Beam blowout dynamics & 3 rd order space charge Dowell - PAC Tutorial 9

30 935, Chromatic Aberration in the mittance Compensation Solenoid mittance due to the momentum dependence of the solenoid s focal length: d dp f n, chromatic x p x is the rms beam size at the solenoid f is the focal length of the solenoid p is the beam momentum is the rms momentum spread p This is a general expression for the emittance produced by a thin lens when the focal length is varied..g., it was used earlier to describe emittance compensation. In the rotating frame of the beam the solenoid lens focal strength is given by B( ) eb() K sin KL, K B() is the solenoid field fsol Bρ p L is the solenoid effective length B is the beam magnetic rigidity Chromatic emittance (microns/mm-rms / MeV)., 6. typical solenoid field, kg. 5 5 Solenoid Kxfield, B().6 (kg) p B pgev / ckg m e K p n, chromatic x sin KL KL cos KL p The solenoid is achromatic when tan KL KL 3 Dowell - PAC Tutorial

31 Chromatic miitance (microns) Chromatic Aberration: Comparison with simulation & expt. n, chromatic x p d dp f sol n, chromatic x K sin KL f sol K sin KL, p KL cos KL p K B() B Comparison of qn. with Simulation. Projected Projected nergy Spread measured at 5 pc, 6 MeV. Slice KL = K = 5/m nergy Spread (KeV-rms) Assumes mm rms beam size at solenoid Solenoid chromatic aberration is a significant contributor to the projected emittance. But with a slice energy spread of KeV, the slice chromatic emittance is only ~. microns 3 Dowell - PAC Tutorial

32 Geometric Aberration of the mittance Compensation Solenoid In order to numerically isolate the geometrical aberration from other effects, a simulation was performed with only the solenoid followed by a simple drift. Maxwell s equations were used to extrapolate the measured axial magnetic field, B z (z), and obtain the radial fields [see GPT: General Particle Tracer, Version.8, Pulsar Physics, solenoid entrance preceding focus post focus pincushion shape time=e- time=.3995e-9 time=.4e-9 Transverse beam distributions: y(mm) GPT x(mm) y(mm) 4e-4 e-4 e-4 -e-4-4e-4-5e-4 e-4 5e-4 GPT x(mm) y(mm) GPT x(mm) 3 Dowell - PAC Tutorial

33 quadrupole phase angle (deg) Solenoid Axial ield (KG) Computing the emittance due to anomalous quadrupole fields of the solenoid quadrupole field (gauss) quad solenoid R sol R quad cos KL K sin KL cos KL sin KL cos KL K sin KL sin KL cos KL K cos KL sin KL K sin KL cos KL sin KL cos KL K sin cos K sin KL cos KL KL KL sin KL K sin KL cos KL fq sin KL cos KL K cos KL f q.5.5 Measured magnetic fields B z (z) x, qs R R () R R T ( ) sol quad sol quad () () det x det () () x, qs x, sol y, sol sin KL f q field z(cm) phase 5 4 normal skew z(cm) z(m) Comparison with simulation 33 Dowell - PAC Tutorial

34 The emittance due to anomalous quadrupole field at the solenoid entrance The previous expression is for the case of a quadrupole plus solenoid system where the quadrupole itself isn t rotated. When the quadrupole is rotated about the beam axis by angle,, with respect to a normal quadrupole orientation, then total rotation angle becomes the sum of the quadrupole rotation plus the beam rotation in the solenoid and the emittance becomes x, qs x, sol y, sol sin KL f q quad solenoid Comparison with simulation for a 5 meter focal length quadrupole followed by a strong solenoid (focal length of ~5 cm). The emittance becomes zero when KL n Adding a normal/skew quadrupole pair allows recovery of the emittance caused by this x-y correlation. 34 Dowell - PAC Tutorial

35 mittance (microns) x-rms beam size Summary of mittance Contributions from the Solenoid mittance due to chromatic aberration: p n, chromatic( x) x Ksin KL KL cos KL p mittance due to anomalous quad field: ( ) n, quad sol x, sol x, sol, ( ).46 4 n spherical x x sin KL f Spherical aberration emittance: q rms beam size from gun to linac solenoid pc, Rc=.6 mm pc, Rc=.3mm pc, Rc=.6mm pc, Rc=.mm pc, Rc=.3mm pc, Rc=.mm pc, Rc=.mm Distance from cathode (cm). LCLS 5 pc, slice n, spherical n, chromatic n, quad sol n, spherical n, chromatic. n,spherical rms beam size (microns) Chromatic emittance assumes KeV-rms energy spread 35 Dowell - PAC Tutorial

36 Intrinsic emittance and Q Space charge limited emission Simple optical model & R emittance mittance compensation and matching Solenoid aberrations Beam blowout dynamics & 3 rd order space charge Dowell - PAC Tutorial 36

37 Space Charge Shaping*, aka Beam Blowout! Here we derive the radial force on an electron confined to a thin disk of charge. The surface charge density is assumed to have a radial quadratic surface charge density. The quadratic factor can be adjusted to cancel the 3 rd order space charge force of the disk s distribution. The mathematical technique* is Step : Compute the electrical potential energy on the axis of symmetry. Step : xpand this potential into a power series and multiply each term by the same order of Legendre polynomial to obtain the potential at any point on space. Step 3: Take the divergence of the potential to get the radial electric field. *J.D. Jackson, Classical lectrodynamics, 3 rd ed., p.-4 Step : Compute the axial potential for a parabolic charge distribution, ( r) r dr R r ds () r rdr 4 V( z) z r z r V() z z P R R 3 rdr r dr z r z r V z z R z z R z z R z V z V z / 3/ / 3 ( ) ( ) ( ) 3 3 uniform distribution parabolic distribution *Serafini, AIP Conf Proc 79, p645 99; Luiten et al., PRL, 93, 4,p.948 R 37 Dowell - PAC Tutorial

38 Step : xpand V in powers of r and multiply each term by the same order of Legendre polynomial to get the potential for a uniformly charged disk at any point P, 4 r r V (, r) RP (cos ) rp (cos ) P (cos ) P 3 4(cos ) R 8 R z z where r z cos r z P In the plane of the disk, z z = => cos = and r =, r so the Legendre polynomials are evaluated at zero, i.e. P n (). Then the potential in the plane of a uniformly charged disk is R V 3 4 ( ) R 4 R 8 3 R ollowing the same procedure for the parabolic term gives the potential, V V ( ) R R R The total potential in the plane of the disk is given by the sum of these potentials V( ) V ( ) V ( ) 38 Dowell - PAC Tutorial

39 Step 3: Take derivative of potential to get radial field. Total potential inside the disk: V( ) V ( ) V ( ) Summing uniform and parabolic potentials and collecting powers of r gives R 3 V R R O 4 6 ( ) R 8 R R The radial space charge field is then: ( ) V 3 R 6 R R 3 5 ( ) R 3 O P R is the radial space charge field at point P in the plane of the disk for <R Disk with surface charge density: ( ) f( rx) ( rx).5.5 Linear force No emittance ( ) 3R.5 R.5 Non-linear force mittance growth If we make: 3R Then there s no 3 rd order force and no space charge emittance during expansion of beam from cathode! 39 Dowell - PAC Tutorial

40 How large are these radial fields? x).5 ( ) 3R.5.5 R.5 rx 3 3 (, ) R 3 R 6 R R, 3R 3R 3 or or 5 pc and R=.6 mm (LCLS-like parameters): 36 MV / m This is nearly as large as the R field(!!!) which is peak cos rf = 57.5 MV/m R However, even with these strong fields they are now linear and the space charge emittance is greatly reduced by parabolic shaping of the transverse shape. Luiten et al., PRL, 93, 4,p Dowell - PAC Tutorial

41 Summary lectron emission physics Quantum efficiency and intrinsic emittance computed with the 3-step model Q agrees but expt. emittance is x-larger than theory Space charge limited (SCL) emission Child-Langmuir Law vs. short pulse emission short pulse emission has higher space charge limit than C-L law Analysis of space charge limited emission data SCL flattens the transverse profile R emittance A simple optical model, R defocusing at exit of gun canceled by solenoid Time-dependent emittance: st and nd order Projected emittance compensation Balanced flow and plasma oscillations Matching to first linac section, errario matching condition Solenoid aberrations Chromatic, projected vs. slice energy spread Geometric, scales with beam size to 4 th power Anomalous quadrupole fields, recoverable emittance with normal/skew quad correctors Beam blowout dynamics Transverse shaping to eliminate space charge emittance due to Serafini and Luiten Begin with very short bunch, a single slice which can be modeled as a disk of charge Parabolic transverse shaping eliminates 3 rd order space charge force 4 Dowell - PAC Tutorial

42 Thanks for your attention! Dowell - PAC Tutorial 4