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High Brightness Electron Sources, D. Dowell, Slac

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The Physics of High-Brightness Sources SSSEPB 2015 The Physics of High Brightness Sources D. H. Dowell David H. Dowell SLAC 1 Boeing thermionic gun w/ subharmonic bunchers PITZ gun ca. 1980: after 1985: Sheffield et al. Carlsten et al. S.V. Benson, J. Schultz, B.A, Hooper, R. Crane and J.M.J. Madey, NIM A272(1988)22-28 The Physics of High Brightness Sources D. H. Dowell < 0.5 meter 2 The Photocathode RF Gun System •Introduction – will concentrate on the RF gun •Field equations and eqns. of motion for photocathode guns •Intrinsic emittance and QE •Space charge limited emission •Simple optical model & RF emittance •Emittance compensation and matching •Solenoid aberrations •“Beam blowout” dynamics & 3rd order space charge The Physics of High Brightness Sources D. H. Dowell Basic components of the photocathode RF gun system 3 Maxwell’s Equations After 150 years, Maxwell’s equations still describe all the physics of photoinjectors! Faraday’s Law (1831) 𝛻×𝐸 =− 𝜕𝐵 𝜕𝑡 Ampere’s Law (1826) 𝛻 × 𝐵 = 𝜇0 𝐽 + Gauss’s Laws for electric and magnetic fields 𝛻∙𝐸 = 𝜌 𝜖0 𝛻∙𝐵 =0 1 𝜕𝐸 𝑐 2 𝜕𝑡 The current and charge densities obey the continuity equation: 𝛻∙𝐽+ 𝜕𝜌 =0 𝜕𝑡 where 𝐽 is the current density The Physics of High Brightness Sources D. H. Dowell 150 Years of Maxwell’s equations, Science, 10 July 2015, Vol 349, pp.136-137 4 Special Properties of Maxwell’s Equations There are useful observations to be made about the fields satisfying Maxwell’s Eqns. Gauss’ law (in differential form and cylindrical coordinates ) in a charge free-region becomes 1 𝜕 𝑟𝐸𝑟 1 𝜕𝐸𝜃 𝜕𝐸𝑧 + + =0 𝑟 𝜕𝑟 𝑟 𝜕𝜃 𝜕𝑧 𝛻 ∙ 𝐸 =0 With cylindrical symmetry, 𝜕𝐸𝜃 𝜕𝜃 = 0, and 𝜕𝐸𝑟 𝐸𝑟 𝜕𝐸𝑧 + =− 𝜕𝑟 𝑟 𝜕𝑧 r 𝑑 𝜕𝐸𝑧 𝛾𝑚𝑟 = 𝑒𝐸𝑟 = −𝑒 𝑟 𝑑𝑡 𝜕𝑧 𝑟=− 𝑒 𝜕𝐸𝑧 𝑟 𝛾𝑚 𝜕𝑧 ∴The fringe of the longitudinal field leads to a radial fringe field. 𝜕𝐸𝑧 radial acceleration ∝ 𝐸𝑧 -divergence The linear part of the radial 𝜕𝐸 field will be focusing if 𝑒 > 0 and electrons get focused to the axis. Electrons are defocused when 𝑒 𝑧 < 0. 𝜕𝑧 𝜕𝑧 When the field is accelerating the electron, 𝑒𝐸𝑧 > 0 making the z-derivative of 𝑒𝐸𝑧 is positive for an electron entering from a zero field region. Hence the electron is focused when entering an accelerating field. Correspondingly, the sign of the derivative reverses when the electron exits from high field to zero field. Which defocuses the electrons. This relation between the derivative of the longitudinal field and the radial field is expressed more generally by the Panofsky-Wenzel theorem (Panofsky and Wenzel, Rev. Sci. Instrum., 27, 976(1956)). See also Wangler, RF Linear Accelerators, pp. 166-167. This theorem has several important implications concerning the photocathode injector. For example, in pure TE-modes, Ez and its derivatives are zero and therefore produces no transverse momentum kick. The Physics of High Brightness Sources D. H. Dowell 𝜕𝐸 Assume that the field is uniform across the aperture, then 𝑟 = 0 𝜕𝑟 and the Lorentz eqn. can be written just as the z-partial derivative of Ez : 5 Lorentz Force: 𝐹 = 𝑒 𝐸 + 𝑐𝛽 × 𝐵 The Lorentz force and Maxwell’s equations together give the relativistic equation of motion (sort of a relativistic Newton’s eqn.): 𝐹= 𝑑 𝛾𝑚𝑐 𝛽 = 𝑒 𝐸 + 𝑐 𝛽 × 𝐵 𝑑𝑡 𝒅 𝜸𝒎𝒙 = 𝜸𝒎𝒙 + 𝜸𝒎𝒙 = 𝒆 𝑬𝒙 + 𝒚𝑩𝒛 − 𝒛𝑩𝒚 𝒅𝒕 𝒅 𝜸𝒎𝒚 = 𝜸𝒎𝒚 + 𝜸𝒎𝒚 = 𝒆 𝑬𝒚 + 𝒛𝑩𝒙 − 𝒙𝑩𝒛 𝒅𝒕 𝛾= 𝛾= 1 1 − 𝛽2 𝑑𝛾 𝑑𝛾 = 𝑧 = 𝛾 ′𝑧 𝑑𝑡 𝑑𝑧 𝐸 1853-1928 1902 Nobel Prize Longitudinal Dynamics • • • Extra force due to relativistic acceleration, 𝛾 > 0, and force is proportional to velocity. Transverse Dynamics For the transverse dynamics, the force decreases as 1 𝛾 vs. 1 𝛾3 for the longitudinal dynamics. The longitudinal acceleration couples the transverse and longitudinal velocities => transverse focusing during acceleration. 𝑒𝐸 Matched beam for 𝑥 = 0 => 𝑚𝑥 − 𝛾 ′ 𝑥 𝑧 = 0, esp. if 𝐸𝑥 is the defocusing space charge force The Physics of High Brightness Sources D. H. Dowell 𝒅 𝜸𝒎𝒛 = 𝜸𝒎𝒛 + 𝜸𝒎𝒛 = 𝒆 𝑬𝒛 + 𝒙𝑩𝒚 − 𝒚𝑩𝒙 𝒅𝒕 where 𝛾 ′ = 𝑚𝑐𝑧2 6 𝑒𝐸 Estimate of the transverse focusing due to longitudinal acceleration only, 𝛾 ′ = 𝑚𝑐𝑧2 , without any transverse fields, 𝐸𝑥 = 0 : 1𝑑 2 𝑥 = −𝛾′𝑧𝑥 2 2 𝑑𝑡 multiply both sizes by 𝑥 to get Re-arranging terms gives: 1 𝑑 𝑥2 2 𝑥2 Integrating and solving for 𝑥 𝑧 : 𝑥 𝑑 𝑥 = −𝛾 ′ 𝑥0 𝑥 = −𝛾 ′ 𝑑𝑧 𝑥 𝑧 = 𝑥0 𝑒 −𝛾 𝑧 𝑑𝑧 0 ′𝑧 The relativistic 𝑥 𝑧-term can provide significant focusing near a cathode at high accelerating field. The figure compares focusing at cathode fields of 20 MV/m and 50 MV/m. This 𝛾′focusing helps to counteract the space charge defocusing forces during initial stages of the electron bunch’s formation by the laser and its acceleration from rest. 𝛾 ′ = 40 (~20 MV/m) 𝑥 𝑥0 𝛾 ′ = 100 (~50 MV/m) The Physics of High Brightness Sources D. H. Dowell 𝑥 = −𝛾′𝑥 𝑧, 7 z (mm) This is one reason why a high cathode field allows higher charge/peak current operation. Basic Cavity Shapes of Transverse Mode RF Guns 𝑅𝑐𝑎𝑣𝑖𝑡𝑦 𝒍𝒄𝒂𝒗𝒊𝒕𝒚 ~8 cm Most RF guns use the TM011 mode whose non-zero field components are (see Wangler, p30): 𝐸𝑧 = 𝐸0 𝐽0 𝑘𝑟 𝑟 cos 𝑘𝑧 𝑧 exp 𝑖 𝜔𝑡 + 𝜙0 𝑘𝑧 𝐸𝑟 = − 𝐸0 𝐽′ 0 𝑘𝑟 𝑟 sin 𝑘𝑧 𝑧 exp 𝑖 𝜔𝑡 + 𝜙0 𝑘𝑟 𝑖𝑘𝑧 𝐵𝜃 = − 𝐸0 𝐽′ 0 𝑘𝑟 𝑟 cos 𝑘𝑧 𝑧 exp 𝑖 𝜔𝑡 + 𝜙0 𝑘𝑟 𝑐 where 𝑘𝑟 = 2.405 𝑅𝑐𝑎𝑣𝑖𝑡𝑦 and 𝑘𝑧 = 𝜋 The Physics of High Brightness Sources D. H. Dowell Field labels for Pillbox Cavity Tmnp m: rotational asymmetry => quadrupole fields n: Radial dependence of field => r3 - nonlinearities p: Longitudinal mode of cavity => RF emittance 𝑙𝑐𝑎𝑣𝑖𝑡𝑦 𝑘 2 = 𝑘𝑟 2 + 𝑘𝑧 2 and 𝑘 = 𝜔 𝑐 Comparison of E&M equations with Superfish is pretty good (to first order)! 8 Synchronous phase and bunch compression* Energy gain in first half cell Bunch compression factor for 𝛼 = 1.8 𝛼= 𝑒𝐸𝑝𝑒𝑎𝑘 𝜆𝑟𝑓 4𝜋𝑚𝑐 2 The Physics of High Brightness Sources D. H. Dowell The beam asymptotically approaches the synchronous phase, 𝜙𝑠𝑦𝑛𝑐 , as it is accelerated from rest to c from the cathode. This is the phase slip between the bunch and the light-velocity rf fields while the bunch is non-relativistic. 9 *See K. Floettmann, RF-induced beam dynamics in rf guns and accelerator cavities, Phys. Rev . ST Accel. Beams 18, 064801(2015). Optical properties of the gun’s RF field Maxwell’s eqns. relate the momentum kicks of the radial electric field to the z- and t-derivatives of the longitudinal electric field: 𝐸𝑟 = − 𝑟 𝜕 2 𝜕𝑧 𝐸𝑧 and 𝑐𝐵𝜃 = 𝑟 𝜕 2𝑐 𝜕𝑡 𝐸𝑧 See K-J. Kim, NIM A275(1989)201-218 and references therein The radial momentum kick is then ∆𝑝𝑟 = 𝑒 𝐸𝑟 𝑑𝑧 𝑒 =− 𝛽𝑐 2 𝑟 𝜕𝐸𝑧 𝑑𝑧 𝛽𝑐 𝜕𝑧 𝐸𝑧 = 𝜃 𝑧𝑓 − 𝑧 𝐸0 sin 𝜙 𝐸𝑧 (𝑧) zf z The derivative of the Heaviside step function is a delta function: 𝜕𝐸 𝑧 = −δ 𝑧𝑓 − 𝑧 𝐸0 sin 𝜙 𝜕𝑧 ∆𝑝𝑟 𝑒 𝑟 𝑒𝐸0 sin 𝜙𝑒 𝑟 =− δ 𝑧𝑓 − 𝑧 𝐸0 sin 𝜙 𝑑𝑧 = 𝑟 = − 𝛾𝑚𝑐 2 𝛽𝑐 2𝛽𝛾𝑚𝑐 2 𝑓𝑟𝑓 The gun’s optical strength is dominantly due to the defocus at the exit of the last cell, or at the exit of any strong electric field, including DC. The defocusing strength is strong in a high-field gun. E.g., at 𝑒𝐸0 sin 𝜙𝑒 = 100MV/m the focal length is only -12 cm! ∴ 1 𝑒𝐸0 sin 𝜙𝑒 =− 𝑓𝑟𝑓 2𝛽𝛾𝑚𝑐 2 The Physics of High Brightness Sources D. H. Dowell Assume the RF field vs. z is a constant step function over the gun’s length. This looks like: 10 Implication of the gun’s defocus: RF Emittance* Phase-dependent focal strength 10 d x    d e  1   x e f   rf  x' (mR) 5 eE cos   x  0 2 e  x  2mc Phasedependent divergence  e = 0 deg 0 5 definition of normalized emittance:  e =90 deg 10 𝜖𝑛 = 𝛽𝛾 𝑥 2 𝑥′2 − 𝑥𝑥′ 𝜖𝑟𝑓,1 = 2 ≅ 𝛽𝛾𝜎𝑥 𝜎𝑥′ 1.5 1 0.5 0 x (mm) 0.5 1 1.5 𝑒𝐸0 cos 𝜙𝑒 𝜎𝑥 2 𝜎𝜙 2 2𝑚𝑐 This is the linear part of the emittance. The non-linear part due to the RF curvature is 2nd order in the phase spread of the bunch, 𝜖𝑟𝑓,2 = 𝑒𝐸0 sin 𝜙𝑒 2 2𝑚𝑐 2 2 𝜎𝑥 𝜎𝜙 2 High order emittance scales as the bunch length squared >> a common feature of many emittance sources. Normalized emittance (microns) 10 1st + 2nd 1st-order rf emittance 1 2nd-order rf emittance 0.1 The total rf emittance is the square-root of sum of squares: 𝜖𝑟𝑓,1+2 𝑒𝐸0 𝜙𝑒 2 2 2 = 𝜎𝑥 𝜎𝜙 𝑐𝑜𝑠 𝜙𝑒 + 𝑠𝑖𝑛2 𝜙𝑒 2 2𝑚𝑐 2 The Physics of High Brightness Sources D. H. Dowell 2 mc 2 f rf   eE0 sin e *See K-J. Kim, NIM A275(1989)201-218 0 20 40 60 80 Exit phase (deg) 100 120 RF emittance can be eliminated in a 2-frequency gun. See D.H. Dowell et al., PAC0? 11 1  1 L2   1     0 1  f sol 𝑓𝑒𝑓𝑓 1 0  1 0  1 L  1 L2   1  1   1 1  0 1    0 1  1      f eff  f rf  = 1 1 + 𝑓𝑟𝑓 𝑓𝑠𝑜𝑙 0 1 L  1 1  0 1     Image of electron emission from Cu cathode in LCLS-gun 𝑟𝑣𝑠 𝑟′𝑣𝑠 = = L  1  2 f eff   1   f eff L1 L2   f eff   L 1 1  f eff  L1  L2  Point-to-point imaging when zero 𝐿2 𝑓𝑒𝑓𝑓 1 − 𝑓𝑒𝑓𝑓 1− 1 𝑓𝑒𝑓𝑓 = 𝐿1 + 𝐿2 − 1− 𝐿1 𝐿2 𝑓𝑒𝑓𝑓 𝐿1 𝑓𝑒𝑓𝑓 𝑟𝑐 𝑟′𝑐 1 1 𝐿1 + 𝐿2 + = 𝑓𝑟𝑓 𝑓𝑠𝑜𝑙 𝐿1 𝐿2 1 𝐿1 + 𝐿2 𝑒𝐸0 sin 𝜙𝑒 0.1 + 1 1 1 = + ~ + ≈ 11 + 8 = 𝑓𝑠𝑜𝑙 𝐿1 𝐿2 2𝛽𝛾𝑚𝑐 2 0.1 ∗ 1 0.12 0.05𝑚 • • The solenoid focal length to image the cathode is only ~5 cm with RF on! Demonstrates beam quality The Physics of High Brightness Sources D. H. Dowell 1 Using the RF gun like a PEEM! 12 Optical Model of a RF Gun Drive Laser 2mc2 f rf   eE0 sin e frf ~-15 cm for 100 MV/m which needs to be compensated for with solenoid Gun Solenoid Defocusing and Focusing RF Lenses Focusing of Gun Solenoid Solenoid cancels defocusing of gun RF and performs emittance compensation and matching to booster linac. Solenoid has focal length of ~15 cm but is ~20 cm long=> thick lens => aberrations The principal solenoid aberrations can be classified as : Chromatic Geometric Anomalous fields Misalignment (not discussed here) The Physics of High Brightness Sources D. H. Dowell 2-6 MeV Electron Beam 13 The Physics of High Brightness Sources D. H. Dowell Quotes taken from P. Musumeci, SSSEPB, SLAC, July 2013 14 See Hommelhof et al.,… for ultra fast emission from needles and 2010 USPAS class notes, Electron Injectors for 4th Generation Light Sources by D. H. Dowell • High QE (QE>5%) at long wavelengths • fast time response: depends upon RF frequency + bunch length, typ. < 1ps • Uniform emission 𝑄𝑏𝑢𝑛𝑐ℎ 𝐸𝑙𝑎𝑠𝑒𝑟 𝑝𝑢𝑙𝑠𝑒 =𝑒 𝑄𝐸 ℏ𝜔 D. Dowell et al., Cathode R&D for future light sources, NIM A 622:13(2010) The Physics of High Brightness Sources D. H. Dowell QE and the Drive Laser 15 Emittances Near the Cathode Intrinsic (aka Thermal) Emittance, e intrinsic : Cathode’s material properties (EF ,w , EG ,EA , m* ,…) Cathode temperature, phonon spectrum Laser photon energy, angle of incidence and polarization Large scale space charge forces across diameter and length of bunch Image charge (cathode complex dielectric constant) effects space charge limit Emittance compensation Bunch shaping (beer-can, ellipsoid) to give linear sc-forces Rough Surface Emittance: Electron and electric field boundary conditions important Surface angles washout the exit cone Coherent surface modulations enhances surface plasmons Three principle emittance effects: Surface tilt washes out intrinsic transverse momentum > escape angle increases Applied field near surface has transverse component due to surface tilt Space charge from charge density modulation due to Ex surface modulation The Physics of High Brightness Sources D. H. Dowell Bunch Space Charge Emittance: 16 Photo-Electric Emission and the 3-Step Model 𝑄𝑒−𝑏𝑢𝑛𝑐ℎ = 𝑒 𝑄𝐸 𝑁𝛾 = 𝑒 𝑄𝐸 Metal Energy 1)Photon absorbed Excess energy: 𝐸𝑒𝑥𝑐𝑒𝑠𝑠 = ℏ𝜔 − 𝜙𝑤 e- 3)Electrons escape to vacuum e- Vacuum level, E=0 Potential barrier due to spillout electrons Fermi Energy occupied valence states energy Vacuum 2)Electrons move to surface e- 𝐸𝑙𝑎𝑠𝑒𝑟 𝑝𝑢𝑙𝑠𝑒 ℏ𝜔 •QE and emittance depend upon electronic structure of the cathode •Sum of electron spectrum yield gives QE: •Width of electron spectrum gives intrinsic emittance •Both are dependent upon the density of occupied states near the Fermi level QE  Optical depth photon number of emitted electrons EF   EF  W EDOS ( E )dE   n  Direction normal to surface *D. H. Dowell, K.K. King, R.E Kirby and J.F. Schmerge, PRST-AB 9, 063502 (2006) The Physics of High Brightness Sources D. H. Dowell Photoelectric emission from a metal given by Spicer’s 3-step model: 1. Photon absorption by the electron 2. Electron transport to the surface 3. Escape through the barrier   eff 17 Refraction of electrons at the cathode-vacuum boundary: Snell’s law for electrons Conservation of transverse momentum at the cathode-vacuum boundary: pxin  pxout in ptotal  2m  E    out ptotal  2m  E    EF  W  Refraction law for electrons: To escape electron longitudinal momentum needs to be greater than barrier height: pzin  2m  EF  eff  2m  E    cosin  2m  EF  eff  Outside angle is 90 deg at inmax which is typically ~10 deg. sin out  sin in n E   in E    EF  eff nout Maximum internal angle for electron with energy E which can escape: The Physics of High Brightness Sources D. H. Dowell in out ptotal sin in  ptotal sin out 18 cos inmax  E   EF  eff E  Elements of the Three-Step Photoemission Model Fermi-Dirac distribution at 300degK f FD ( E )  1 1  e ( E  E F ) / k BT Electrons lose energy by scattering, assume e-e scattering dominates, Fe-e is the probability the electron makes it to the surface without scattering h h eff heff 1 Bound electrons ptotal  2m( E  )  p pnormal  2m( E  ) cos Emitted electrons .5 cos  max  E 0 0 5 EF EF+eff-h Energy (eV) E+h EF  eff p  ptotal E   10E + F eff EF+h  QE ( )  (1  R( )) 2 pnormal  EF  eff 2m Escape criterion: eff     schottky .5 Step 3: Escape over barrier Step 2: Transport to surface EF    eff dE N ( E   )(1  f FD ( E   )) N ( E ) f FD ( E )     dE EF  d (cos  )F   e e cos  2 1 max (E) ( E ,  ,  )  d The Physics of High Brightness Sources D. H. Dowell Step 1: Absorption of photon 0 1 2 1 0 N ( E   )(1  f FD ( E   )) N ( E ) f FD ( E )  d (cos  )  d 19 QE for a good metal, like Cu Step 3: Escape over the barrier and integrate up to max escape angle 1 R is the reflectivity eff is the effective work function eff  W  Schottky EF QE ( )  1  R( )  Fe e   Step 1: Optical Reflectivity ~40% for metals ~10% for semi-conductors Optical Absorption Depth ~120 angstroms Fraction ~ 0.6 to 0.9 Step 2: Transport to Surface e-e scattering (esp. for metals) ~30 angstroms for Cu e-phonon scattering (semiconductors) Fraction ~ 0.2 QE ~ 0.5*0.2*0.04*0.01*1 = 4x10-5   EF  eff EF dE     dE EF   d (cos  ) EF eff 2  d E  1 0 2 1 0  d (cos  )  d  •Azimuthally isotropic emission Fraction =1 •Fraction of electrons within max internal angle for escape, Fraction ~0.01 •Sum over the fraction of occupied states which are excited with enough energy to escape, Fraction ~0.04 D. H. Dowell et al., PRST‐AB 9, 063502 (2006) The Physics of High Brightness Sources D. H. Dowell E is the electron energy EF is the Fermi Energy 20 Performing the integrals gives the QE: e e 2 2  Eexcess (1  R(  ))     8eff ( EF  eff )  1  opt Vacuum Level e e ℏ𝜔 𝐸𝑒𝑥𝑒𝑠𝑠 𝑬𝒆𝒙𝒄𝒆𝒔𝒔 𝟐 Intrinsic emittance can be lowered from 0.5 to <0.35 microns/mm-rms, if excess energy is reduced from 0.4 to <0.2 eV. Unfortunately the QE goes to zero faster than the emittance! The Physics of High Brightness Sources D. H. Dowell EF  eff (1  R(  )) ( EF   )  QE  1   2   EF   1  opt 21 22 P. Musumeci, 2013 SSSEPB The Physics of High Brightness Sources D. H. Dowell Vacuum -e q’ e xd x Potentials Near the Surface eschottky  e eE0  0.0379 E MV / meV 40 eschottky  0.379eV @100MV / m The Physics of High Brightness Sources D. H. Dowell Cathode 23 RF processing and machining processes can produce rough surfaces which increase field emission and emittance very close to the cathode Field enhancement factor for various geometries E  E0 The Physics of High Brightness Sources D. H. Dowell Cathode damage during RF processing and beam operation 24 “High Voltage Vacuum Insulation, Basic Concepts and Technological Practice,” Ed. Rod Latham, Academic Press 1995 Emittance Due to a Tilted Surface z z px ,out ~ p x ,out T x pout  2mE    EF  eff  out Vacuum in Continuity of transverse momentum at surface: ~ p x ,in  ~ p x ,out Metal pin  2mE    p x ,in sin  out p  in  sin  in pout E   E    E F  eff p x ,in Max angle of incidence: p x ,out  p out 2 ~ sin  out cos T  cos  out sin T  pout cos  out sin T 2 sin  max  E    EF  eff The Physics of High Brightness Sources D. H. Dowell x E   Surface momentum proportional to intrinsic momentum, pout cosout 25 26 The Physics of High Brightness Sources D. H. Dowell Surface Emittance Due to an Applied Electric Field + … The transverse momentum due to an applied electric field is given by the integral of the acceleration along the electron’s trajectory, (x(t),z(t)). [D. J. Bradley et al., J. Phys. D: Appl. Phys., Vol. 10, 1977, pp. 111-125.], t p x, field  eE0 an k n  sin k n x(t )e kn z (t ) dt At large distances from the surface, z > a few n , the transverse field vanishes and transverse momentum gained in the field becomes constant. The variance of this transverse momentum, px,field , gives the emittance due to the applied field [See D. Xiang et al., Proceedings of PAC07, pp. 1049-1051],  field  2 an2eE0  x 2n mc2 n  2 kn This is the emittance only due to the transverse component of the applied field. Include the cross terms between the intrinsic, tilt and applied field momenta and compute the variance of the transverse momentum to get the total intrinsic, tilted-surface and high field emittance,  intrinsic tilt  field   x D. H. Dowell -- P3 Workshop     a k         2 eff 3mc 2 n n 2   eff 3mc 2 eE0   2k n mc 2  The Physics of High Brightness Sources D. H. Dowell 0 Comparison of these three emittances for emission from a diamond turned surface AFM of LCLS cathode sample LCLS gun & cathode parameters: an  17nm n  10microns k n  0.628 / micron an k n  0.011 ~35 nm Ea  57.5MV / m eff ( E0 )  W  Schottky Schottky  e  intrinsic tilt  field  x     1  a k  2 3mc eff 2    intrinsic  0.48microns / mm  rms x  tilt  3.7 10 3 microns / mm  rms x  field  0.13microns / mm  rms x n n 2 eE0  0.29eV 40  eE0 2   a k  n n 2  4k n mc  intrinsic tilt  field  0.49 microns / mm  rms x ∴ The intrinsic emittance is by far the largest contributor! See D. H. Dowell, 2012 P3 Workshop The Physics of High Brightness Sources D. H. Dowell ~10 microns   4.86eV W  4.8eV 28 Charge Density Modulations produced by Surface Roughness can increase the emittance due to space-charge forces Electrons are focused and can go through a crossover a few mm from the cathode for emission from a sinusoidal surface beam cross section ~6.5 mm from cathode 120 100 80 60 7000 40 6000 20 0 0 5000 2 4 6 8 10 8 10 of x-y x Mapping (microns) 6538 4000 6537.5 3000 6537 2000 1000 6536.5 6536 6535.5 0 -15 -10 -5 transverse position, x (microns) 0 5 6535 6534.5 2 4 6 xMapping (microns) of x-y 62.7 62.65 62.6 62.55 y (microns) Due to this crossover, space charge forces and other effects such as Boersch-scattering should be investigated. Since this ‘surface lens’ is non-linear it can also produce geometric aberrations and increase the emittance. Plus the time dependence of the RF field which changes the focus with time. Rich area of study! 0 62.5 29 62.45 62.4 62.35 62.3 62.25 See D. H. Dowell, 2012 P3 Workshop at Cornell for space-charge effects The Physics of High Brightness Sources D. H. Dowell 8000 y (microns) longitudinal position, y (microns) modulation amplitude = 0.02 microns; spatial wavelength = 10 microns; emittance = 0.16412 microns 62.2 0 2 4 6 x (microns) 8 10 Space Charge Limit (SCL) is different for DC diode, short pulse and long bunch photo-emission. Space-charge-limited current vs. space-charge-limited field Space charge limited current across a DC diode, Child-Langmuir law: 4 2e V 3/ 2 J CL   0 2 m d Space Charge Limited Field of a Short Electron Bunch from a Laser-driven Photocathode. Parallel plate (capacitor) model:  SCL   0 Eapplied Space charge limited field of an ellipsoid near the cathode (water bag model): Potential energy and longitudinal electric field of a uniformly charged ellipsoid (aka. water bag) next to the cathode The ellipsoid edge semi-axes of a and zm, corresponding to the edge radius and half the bunch length, respectively. [See Reiser, page 406], 𝜕𝜙𝑓𝑠 𝜌0 𝑎 𝐸𝑓𝑠,𝑧 𝜁 = − = 𝜁 for −𝑧𝑚 ≤ 𝜁 ≤ 𝑧𝑚 𝜕𝜁 6𝜖0 𝑧𝑚 𝜌0 = 𝐸𝑓𝑠,𝑧 3 𝑄𝑏𝑢𝑛𝑐ℎ 4 𝜋𝑎2 𝑧𝑚 𝜌0 4𝜋𝑎𝑧𝑚 ℎ𝑒𝑎𝑑 = 𝑎 => 𝑄𝑠𝑐𝑙−𝑒𝑙𝑙𝑖𝑝𝑠𝑜𝑖𝑑 = 𝐸𝑎𝑝𝑝𝑙𝑖𝑒𝑑 6𝜖0 3 The Physics of High Brightness Sources D. H. Dowell 9 30 z Transverse Electron Beam Shape: The beam core is clipped at the SCL eElaser  QE •SCL truncates the beam core to a flat, uniform transverse distribution. •SCL also flattens hot spots. below the SCL: 𝑄𝑏𝑢𝑛𝑐ℎ 𝑒𝐸𝑙𝑎𝑠𝑒𝑟 = 𝑄𝐸 ℏ𝜔  rm2 0 Erf sin rf  QE eElaser   e rm2 2 r2 Space Charge Limit of Gaussian peak QE Limited Emission Space Charge Limited Emission radial distribution follows laser & QE radial distribution saturates at the applied field 1 1 0.8 0.8 0.6 0.6 0.4 0.4 0.2 0.2 0 3 2 1 0 1 2 3 0 The Physics of High Brightness Sources D. H. Dowell QE scan: Emitted charge vs Laser pulse energy 31 3 2 1 0 1 2 3 J. Rosenzweig et al., NIM A 341(1994) 379-385 The Schottky scan: Emitted Charge vs. Laser Phase The Schottky scan can be divided into two regions. At low rf phases the emission is space charge limited while at higher field phases the emission becomes QE-limited. In the Schottky scan, these two phenomena are equal at the phase m . 𝑄𝑠𝑐−𝑙𝑖𝑚𝑖𝑡 = 𝜋𝜖0 𝑅2 𝐸𝑝𝑒𝑎𝑘 𝑠𝑖𝑛𝜙 𝐸𝑙𝑎𝑠𝑒𝑟−𝑝𝑢𝑙𝑠𝑒 ℏ𝜔  EF  W  e eErf sin rf /( 40 ) 1  R EF    QE  1   2  EF   1  opt  e e Measured Schottky scans agree with expected shapes due to SC- and QE-limits     2 The Physics of High Brightness Sources D. H. Dowell 𝑄𝑄𝐸−𝑙𝑖𝑚𝑖𝑡 = 𝑒𝑁𝛾 QE = eQE 32 data courtesy of M. Krasilnikov A virtual cathode forms and the electrons oscillate in a time-dependent potential well just in front of the cathode. The Physics of High Brightness Sources D. H. Dowell What happens when you go way above the SCL? 33 See: An Introduction to the Physics of Intense Charged Particle Beams by B. Miller, Plenum Press, New York, March 1985. The relativistic, paraxial ray and envelope equations Begin with the radial equation of motion (Lorentz eqn.): 𝑚 𝑑 𝛾𝑟 − 𝑚𝛾𝑟𝜃 = 𝑒 𝐸𝑟 − 𝛽𝑐𝐵𝜃 𝑑𝑡 And transform from time to z as the independent variable to get the paraxial ray equation. Reiser (p. 210) gives a modified version of the relativistic paraxial ray equation: 2 The accelerating damping term, discussed in slides 6-10 1 𝐾 − =0 𝑟 3 𝑟𝑚 2 external focusing, e.g. solenoid 𝑟𝑚 is the radius of the beam envelope K is Lawson’s generalized perveance. Gives the space-charge force. angular momentum term and x-y mixing entrance/exit radial kick given by the 𝜕 accelerating field, E z. 𝛾 ′′ ∝ 𝐸𝑧 ∝ 𝐸𝑟 𝜕𝑧 small near the cathode, unless curved normalized emittance term, 𝜖𝑛 = 4𝛽𝛾𝜖𝑟𝑚𝑠 And he writes the relativistic envelope equation as: 𝑟𝑚′′ 1 𝑝𝜃 + 2 𝛾 ′ 𝑟𝑚′ + 𝛾 ′′ 𝑟 + 𝑘0 2 𝑟𝑚 − 𝛽 𝛾 𝛽𝛾𝑚𝑐 The relativistic, generalized perveance is defined as 𝐾 ≡ 2 1 𝜖𝑛 2 1 𝐾 − − =0 𝑟𝑚 3 𝛽 2 𝛾 2 𝑟𝑚 3 𝑟𝑚 2 𝐼 𝛽 3 𝛾 3 𝐼0 where 𝐼0 = 17𝑘𝐴, characteristic current for electrons In the non-relativistic limit the perveance gives the same 𝑉 3/2 voltage-dependence as the Childs-Langmuir law: 𝐼 1 𝐾𝑁𝑅 = 3/2 𝑉 4𝜋𝜖0 2𝑒/𝑚 1/2 See Reiser, p. 197 The Physics of High Brightness Sources D. H. Dowell 𝛾′ 𝛾′′ 𝑝𝜃 𝑟 + 2 𝑟 ′ + 2 𝑟 + 𝑘0 2 𝑟 − 𝛽 𝛾 2𝛽 𝛾 𝛽𝛾𝑚𝑐 ′′ 34 Brillouin flow, confined flow and other flows Brillouin flow (1948) maintains constant beam size of 𝑟𝑚 by balancing the outward space-charge force against the focusing of a axial magnetic field. The conditions for BF are zero emittance, zero angular momentum and no acceleration. The ray equation for no radial acceleration then For magnetic focusing: 𝑘0 The magnetic field needed for Brillouin flow is 2 𝐾 𝑟𝑚 => 𝑟𝑚 = 𝑒𝐵 = 2𝛽𝛾𝑚𝑐 𝑒𝑐𝐵𝐵𝐹 𝐾 𝑘0 2 and with 𝐾= 2 𝐼 𝛽 3 𝛾 3 𝐼0 2𝑚𝑐 2 2 𝐼 = 𝑟𝑚 𝛽𝛾 𝐼0 ∴ to confine a 100 ampere beam inside a radius of 1mm with 𝛽𝛾 = 2 (~800 KeV) requires a magnetic field of The Physics of High Brightness Sources D. H. Dowell 𝑟 ′′ = 0 = 𝑘0 2 − c𝐵𝐵𝐹 = 78 𝑀𝑉/𝑚 or 𝐵𝐵𝐹 = 0.26𝑇 = 2.6 𝑘𝐺 35 The Concept of Slices and Emittance Compensation slice width is determined by the beam energy due to Lorentz contraction of the longitudinal fields v=0 2 𝛾 ~ r v~c r 1 𝛾 field lines inside a bunch Electric field lines of a point charge 2𝑧𝑚 Emittance Compensation •The beam at the cathode begins with all slices of nearly equal peak current propagating with equilibrium radii in Brillouin flow. •The beam envelope equation is linearized about the Brillouin equilibrium and solved for small perturbations about this equilibrium point. •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 /2. The Physics of High Brightness Sources D. H. Dowell 𝜁, 𝑧, 𝛾 36 Projected Emittance Compensation* The radial envelope equation for each slice position, zm <  < zm 0      r ( )   r ( )  2   External focusing by magnetic and RF fields   n2 ( ) K ( )  2 2 3 0   kr r ( )   r ( )    r ( )  Space charge defocusing, K is the generalized perveance acceleration changes magnification of the divergence Generalized perveance for each slice at  (Lawson, p. 117) emittance acts like a defocusing pressure I ( ) K ( )  3 3   I0 2 *Serafini & Rosenzweig, Phys. Rev. E55, 1997, p.7565 I is the peak current of slice  I0 is 17000 amps The Physics of High Brightness Sources D. H. Dowell  slice 37 Projected Emittance Compensation: How to align the slices? Beam Envelope Equation: Assume no acceleration, zero slice emittance  r  kr r  K r I+I I-I 0  r   e   Substituting into the envelope eqn. and expanding in a Taylor series we get    kr e   K  k     0  e  r  e2  K constant terms, set sum to zero and solve for e : e  Envelope eqn. for small amplitude radial perturbations: K 2 I 1  kr  3 3 I 0 kr    2 K  2 e   0 𝑘𝑒 = 2𝐾 𝜎𝑒 sin ke z     1   cos ke z    0  k  e        1   k sin k z cos k z    0  e e   e 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. The Physics of High Brightness Sources D. H. Dowell Consider solutions for small perturbations from equilibrium radius: 38 Derivation of projected emittance for a slice current spread The emittance due to a sc-lensing strength dependence upon beam current is: Envelope eqn. for small amplitude radial perturbations K  e2   0 𝑘𝑒 = 2𝐾 𝜎𝑒  n, sc comp   e2 I This is the wave equation with oscillating solutions:  sin ke z     ke   0      0   ke sin ke z cos ke z  Using the expressions for 1/fe and the equilibrium wave number in this eqn. gives the projected emittance for a beam with a I rms spread in slice current:   1   cos ke z       1   with the equilibrium wave number defined as: K I  e2 1 0   0   1          1   ke sin ke z 1   0  ; ke2   e I sin ke z  ke z cos ke z  I 0 I 5 To simplify the emittance calculation let’s make the crude approx. that the solenoid focusing-sc defocusing channel can be replaced by a thin lens such that 1  ke sin ke z fe  n,  4 I  3 3 e2 I 0 emittance (microns) k  2kr  2 2 e d 1   dI  f e  The Physics of High Brightness Sources D. H. Dowell    2 Locations where the slices align 4 3 2 39 1 0 0 0.5 1 z (m) 1.5 2 The Ferrario Working Point: matching the low energy beam to the booster linac 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 Ferrario working point* and was initially formulated for the LCLS injector and based upon the theory of L. Serafini and J. Rosenzweig, see Phys. Rev. E55,1997, p.7565. 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 RF fields and the peak current. RF focusing aligns the slices and acceleration damps the emittance oscillations. Assume the RF-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=/2, so the angular kick is, eE0 Taking the derivative gives the rf term needed for the envelope equation, eE0  2     2 2     2 2 mc 2 2mc 2 since  eE 0 mc 2 with E0 the accelerating field of the booster. For a matched beam we want the focusing strength of the accelerating field to balance the space charge defocusing force, i.e. no radial acceleration: 2     match 2.5 Transverse Emittance (microns) Beam Radius at Exit, rms (mm) 125 cm S-Band TW Section Elinac = 19 MV/m 2 Solenoid = 2080 G. 1.5 Solving gives the matched beam size: E beam =62 MeV E0=106 MV/m, 0 = 50 deg E3=16 MV/m,  = 1.7 deg  I  0 2 3  2 I A  match  matched  1 Transverse Emittance rms Radius 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 damping during acceleration. 0.5 0 0 100 200 300 z (cm) 400 1 I   2 I A 500 *M. Ferrario et al., “HOMDYN study for the LCLS RF photoinjector”, SLAC-PUB-8400, LCLS-TN-00-04, LNF-00/004(P). The Physics of High Brightness Sources D. H. Dowell   40 Chromatic Aberration in the Emittance Compensation Solenoid Emittance due to the momentum dependence of the solenoid’s focal length: 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. E.g., it was used earlier to describe emittance compensation. In the rotating frame of the beam the solenoid lens focal strength is given by 1  K sin KL, f sol Chromatic emittance (microns/mm-rms /20 keV @6 MeV) 100 1935  1  0.020   6  K typical 2 solenoid field, 2 kG 10 B( 0 ) eB(0)  2 Bρ0 2p B(0) is the solenoid field L is the solenoid effective length Br0 is the beam magnetic rigidity B r0  1 p  33.356 p  GeV / c  kG  m e  n,chromatic   x2 K sin KL  KL cos KL 0.1 p p The solenoid is a first order achromat when 0.01 0 5 10 15  2  33.356  0.006 SolenoidKxfield, B(0) (kG) 20 tan KL  KL The Physics of High Brightness Sources D. H. Dowell  n,chromatic d 1    p dp  f  2 x  x is the rms beam size at the solenoid 41 Chromatic Aberration: Comparison with simulation & expt.  n,chromatic   x2 p d  1    dp  f sol  1 B(0)  K sin KL, K  f sol 2Br 0  n,chromatic   K sin KL  KL cos KL 2 x p p Chromatic Emiitance (microns) 1 Projected Energy Spread measured at 250 pC, 6 MeV Projected Assumes 1 mm rms beam size at solenoid 0.1 KL = 1 K = 5/m Slice 0.01 0.001 0 5 10 15 20 25 30 The Physics of High Brightness Sources D. H. Dowell Comparison of Eqn. with Simulation Energy Spread (KeV-rms) Solenoid chromatic aberration is a significant contributor to the projected emittance. But with a slice energy spread of 1 KeV, the slice chromatic emittance is only ~0.02 microns 42 Geometric Aberration of the Emittance Compensation Solenoid 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, Bz(z), and obtain the radial fields [see GPT: General Particle Tracer, Version 2.82, Pulsar Physics, http://www.pulsar.nl/gpt/]. post focus “pincushion” shape preceding focus solenoid entrance time=2.401e-009 time=2.3995e-009 time=1e-012 0 -1 0.004 2e-4 0.002 y(mm) y(mm) y(mm) Transverse beam distributions: 1 4e-4 0e-4 0.000 -2e-4 -0.002 -4e-4 -0.004 -2 -2 GPT -1 0 x(mm) 1 -5e-4 2 GPT 0e-4 x(mm) -0.005 5e-4 GPT 0.000 x(mm) 0.005 The Physics of High Brightness Sources D. H. Dowell 2 43 The emittance due to anomalous quadrupole field at the solenoid entrance 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 sin 2  KL    fq quad solenoid Comparison with simulation for a 50 meter focal length quadrupole followed by a strong solenoid (focal length of ~15 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. The Physics of High Brightness Sources D. H. Dowell  x ,qs   x ,sol y ,sol 44 Summary of Emittance Contributions from the Solenoid rms beam size from gun to linac Emittance due to chromatic aberration: 2.0 p 1.6 Emittance due to anomalous quad field:  n ,quad  sol ( x ,sol )   x2,sol sin 2 KL fq solenoid 1.8 200 pC, Rc=0.6 mm 1.4 100 pC, Rc=0.3mm 1.2 100 pC, Rc=0.6mm 1.0 10 pC, Rc=0.1mm 0.8 10 pC, Rc=0.3mm 0.6 1 pC, Rc=0.01mm 0.4 Spherical aberration emittance: 1 pC, Rc=0.10mm 0.2 0.0  n,spherical ( x )  0.0046 x4 0 20 40 60 80 100 Distance from cathode (cm) LCLS at 250 pC, slice emittance Emittance (microns) 1  n,spherical   n,chromatic   n,quad  sol  n , spherical   n ,chromatic 0.1  n, spherical The Physics of High Brightness Sources D. H. Dowell p x-rms beam size  n,chromatic( x )   K sin KL  KL cos KL 2 x 0.01 1 10 Chromatic emittance assumes 1 KeV-rms energy spread 3 0 0.5 1 1.5 2 rms beam size (microns) 2.5 3 45 Space Charge Shaping*, sometimes known as “Beam Blowout!” Step 1: Compute the axial potential for a Here we derive the radial force on an electron confined parabolic charge distribution, to a thin disk of charge. The surface charge density is assumed to have a radial quadratic surface charge  (r )   0 1   2 r 2 density. The quadratic factor can be adjusted to cancel the 3rd order space charge force of the disk’s distribution.  *J.D. Jackson, Classical Electrodynamics, 3rd ed., p.101-104 V ( z)  r dr P z R 4 0V ( z )    ds z2  r2 R  2   (r )rdr 0 R  0  R rdr r 3dr  V ( z)  2    2 2 2 0  0 z 2  r 2 z  r 0   0  2 2 1/2  1 2 2 3/2 2 2 2 1/2 2 3   z  R  z    z  z  R   z    V0 ( z )  V2 ( z )   2  z  R  2 0  3 3   uniform distribution z2  r2 parabolic distribution *Serafini, AIP Conf Proc 279, p645 1992; Luiten et al., PRL, 93, 2004,p.94802 The Physics of High Brightness Sources D. H. Dowell The mathematical technique* is Step 1: Compute the electrical potential energy on the axis of symmetry. Step 2: Expand 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.  46 Step 2: Expand V0 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,  V0 ( , r )  0 2 0 r   1 r2 1 r4 RP (cos  )  rP (cos  )  P (cos  )  P (cos  )  0 1 2 4   2 R 8 R3   z z 2 2 2 cos    r  r  z where r r 2  z2  z r In the plane of the disk: z = 0 => cos q = 0 and r = r, so the Legendre polynomials are evaluated at zero, i.e. Pn (0). Then the potential in the plane of a uniformly charged disk is  V0 ( r )  0 2 0 R  1 r2 3 r4  2 3 R  4 R 8 R     Following the same procedure for the parabolic term gives the potential, V2  V2 ( r )  0 2 2 0 2 4 1 3 1 3 r 2   R  Rr    3 4 8 R       The total potential in the plane of the disk is given by the sum of these potentials V ( r )  V0 ( r )  V2 ( r ) The Physics of High Brightness Sources D. H. Dowell P 47 Step 3: Take derivative of potential to get radial field. V ( r )  V0 ( r )  V2 ( r ) Total potential inside the disk: Summing uniform and parabolic potentials and collecting powers of r gives 4   0    2 R2  1  1  2 3 1 r 6 V (r )   O  r   R 1      2  2  R r  2  2  3 2  2 0   3  4 R  8 R R   V Er ( r )  r 0 1  1  3 1  r3 5  Er ( r )     2  2  Rr   3 2  2   O  r  2 0  2  R  16  R  R  Er r Er is the radial space charge field at point P in the plane of the disk for r