Lecture 4: Emittance Compensation J.B. Rosenzweig USPAS, UW-Madision 6/30/04
Emittance minimization in the RF photoinjector Thermal emittance limit Small transverse beam size Avoid metal cathodes? n,th 1 2 h W m 0 c 2 x 5 10 4 x m ( ) RF emittance n,rf k RF RF ( k RF z ) 2 2 x Small beam dimensions Small acceleration field? Maybe not Space charge emittance K.J.Kim treatment is very discouraging m n,sc e c 2 10I (2) 2 ee 0 I 0 (1 + 3 A) A = x, I 0 = ec 5 z r e n,sc 5 mm - mrad (I =100 A, E 0 =100 MV/m)
Space-charge emittance control? Kim model indicates monotonic emittance growth due to space-charge Multiparticle simulations at LLNL (Carlsten) show emittance oscillations, minimization possible: Emittance compensation Work extended by UCLA, INFN scientists to give analytical approach New high gradient design developed and understood Many new doors opened z (mm), (mm-mrad) 3.0 2.0 1.0 High gradient RF photocathode gun Focusing solenoid rms beam size (uniform beam) rms emittance (uniform beam) 0.0 0 50 100 150 200 250 300 350 400 z (cm) PWT linacs Multiparticle simulations (UCLA PARMELA) Showing emittance oscillations and minimization
Intense beam dynamics in photoinjector: a demanding problem Extremely large applied fields Violent RF acceleration (0 to ~3E8 m/s in < 100 ps) Large, possibly time-dependent external forces (rf and focusing solenoids) Very large self-fields Longitudinal debunching (charge limit) Radial oscillations (single component plasma) Optimization of beam handling with large parameter space and collective effects. Multiparticle simulations are invaluable aid, but time-consuming Understanding of non-equilibrium transport approached using rms envelope equations
Transverse dynamics model After initial acceleration, space-charge field is mainly transverse (beam is long in rest frame). Force scales as -2 (cancellation of electric defocusing with magnetic focusing) Force dependent almost exclusively on local value of current density I / 2 (electric field simply from Gauss law) Linear component of self-force most important. We initially assume that the beam is nearly uniform in r. The linear slice model Extend linear model to include nonlinearities within slices Scaling of design physics with respect to charge, RF
The rms envelope equation The rms envelope dynamics for a cylindrically symmetric, nonaccelerating, space-charge dominated beam are described by a nonlinear differential equation 2 r x (,z) + k 2 r (,z) = e ( ),z 2 3 x,z 2 3 x Separate DE for each slice (tagged by ), = z v b t Each slice has different current I( )= ( )v External focusing measured by betatron wave-number k = eb z /2p 0 ( ) + n,x ( ) In solenoid, beam is rotating, so envelope coordinates are in rotating Larmor frame with same wave-number Rigid rotator equilibrium (Brillouin flow) depends on local value of current (line-charge density). Pressure forces negligible eq ( ) ( ) = 1 r e k 2 3 r e ( ) = I( ) /I 0
Equilibrium distributions and space charge dominated beams Maxwell-Vlasov equilibria have simple asymptotic forms, dependent on parameter f(x) 2 1.5 1 x =( n /k ) 1/2 R I /2 2 k n I 0 Emittance dominated gaussian R << 1 Space-charge dominated uniform R >>1 Uniform beam approximation very useful 1.2 f(x) 0.5 0 0 1 2 3 4 5 x/ 1.2 1 0.8 0.6 0.4 x =(I/2 3 I 0 k 2 ) 1/2 f(x) 1 0.8 R 5 0.6 0.4 ~( /k ) 1/2 D n 0.2 0 0 1 2 3 4 5 x/ 0.2 0 0 1 2 3 4 5 x/ Nominally uniform has Debye sheath High brightness photoinjector beams have R >1, 250!
The trace space model Each -slice component of the beam is a line in trace space. x x x x No thermal effects No nonlinearities (lines are straight!) Slice phase spaces Projected phase space x x The model f(x,x') x' "5 times" rms ellipse Trace space distribution x' rms ellipse x' x x x x 90% ellipse x exp( 1/ 2) contour Contrast with thermal trace space and nonlinear slice trace space
Envelope oscillations about equilibria Beam envelope is non-equilibrium problem, however Linearizing the rms envelope equation about its equilibria gives (,z) + 2k 2 x,z x ( ) = 0 Dependent on betatron wave-number, not local beam size or current Small amplitude envelope oscillations proceed at 2 1/2 times the betatron frequency or assuming uniform beam distribution k env = 2k = 4r e n b,eq 3 = k p This is the matched relativistic plasma frequency
Phase space picture: coherent oscillations r ' x All oscillations of spacecharge beam envelope proceed about different equilibria, with different amplitude but at the same frequency Behavior leads to emittance oscillations but not damping (yet) Qualitative explanation of 1st compensation, after gun, before linac / eq, /(K r ) 1/2 eq 2 1.2 0.8 0.6 0.4 0.2 1 2 3 eq1 eq2 eq3 1 < 2 < 3 rx Assume that beam is launched at minimum (e.g. at cathode) 1 0 0 0.2 0.4 0.6 0.8 1 (2K r ) 1/2 z/2 Small amplitude oscillation model
Phase space picture: coherent oscillations Emittance (area in phase space) is maximized at k p z = /2,3 /2 Emittance is locally minimized at k p z = 0,,2 - the beam extrema! Fairly good agreement of simple model with much more complex beamline What about acceleration? In the rf gun, in booster linacs z (mm), (mm-mrad) x r ' 3.0 2.0 1.0 High gradient RF photocathode gun k z= /2 p k pz=3 /2 k pz=0,2 Focusing solenoid 1 2 3 1 < 2 < 3 rms beam size (uniform beam) rms emittance (uniform beam) 0.0 0 50 100 150 200 250 300 350 400 z (cm) PWT linacs Damping of... r x
Emittance damping: Beam envelope dynamics under acceleration Envelope equation (w/o emittance), with acceleration, RF focusing x (,z)+ z () x (,z)+ 8 z () New particular solution - the invariant envelope (generalized Brillouin flow), slowly damping fixed point inv (,z)= 1 r e ( ) 1/2 ( 2 + )() z Angle in phase space is independent of current 2 x (,z)= ( ) ( ) r e 2() z 3 x,z 1 (rf or solenoid focusing) = ee 0 /m 0 c 2 (accel. "wavenumber") = inv = 1 inv 2 Corresponds exactly to entrance/exit kick (matching is naturally at waists) x RF = 1 2 x Matching beam to invariant envelope yields stable linear emittance compensation!
Envelope oscillations near invariant envelope, with acceleration Linearized envelope equation 1+ x + x + x = 0 4 Homogenous solution (independent of current) x = [ x 0 inv ]cos Normalized, projected phase space area oscillates, seculary damps as offset phase space (conserved!) moves in x 2 1+ 2 ln () z 0 n ~ offset inv ~ 1/ 2 x Oscillation (matched plasma) frequency damps with energy damping oscillation k p = d dz 1+ 2 ( ) ln () z = 1+ 2 offset phase space x = 1 2 x
Validation of linear emittance compensation theory Theory successfully describes linear emittance oscillations Slice code (HOMDYN) developed that reproduce multiparticle simulations. Much faster! Ferrario will lecture on this.. LCLS photoinjector working point found with HOMDYN Dash: HOMDYN Solid: PARMELA
Nonlinear Emittance Growth Nonuniform beams lead to nonlinear fields and emitance growth It is well known from the heavy ion fusion community that propagation of non-uniform distributions in equilibrium leads to irreversible emittance growth (wave-breaking in phase space). r' "Wave-breaking" occurs in phase space when slope of (r,r') distribution is infinite. This example is past wave-breaking, and irreversible emittance growth has occurred. r Fixed point off-axis Fixed point is where space-charge force cancels applied (solenoid) force. It is in the middle of the Debye sheath region.
Non-equilibrium, nonlinear slice dynamics Matching of envelope to invariant envelope guarantees that we have linear emittance compensation; is it courting nonlinear emittance growth? Understanding obtained as before by: Heuristic analysis Computational models
Heuristic slab-model of non-equilibrium laminar flow Laminar flow=no trajectory crossing, no wavebreaking in phase space Consider first free expansion of slab (infinite in y, z) beam (very non-equilibrium) ( ) = n 0 f x 0 n b x 0 ( ), f 0 ()= 1 Under laminar flow, the charge inside of a given electron is conserved, and one may mark trajectories from initial offset x 0. Equation of motion x = k 2 p Fx ( 0 ), Fx 0 x 0 ( ) = f ( ) 0 x 0 d x 0 = const. Note, with normalization F ( x0 ) x
Free-expansion of slab beam Solution for electron positions: ( xx ( 0 ) = x 0 + k z p ) 2 Fx ( 0 ) 2 Distribution becomes more linear in density with expansion f( x( x 0 )) = Example case ( ) f x 0 ( ) 2 1+ k pz 2 f x 0 f( x 0 ) 2 ( k p z) 2 ( a ) 2 ( ) = 1 x 0 f(x) 1 0.8 0.6 0.4 0.2 2.5 f(x_0) f(x) 0 0 1 2 3 4 5 6 7 x/a 3 Initially parabolic profile becomes more uniform at k p z = 4 Wavebreaking will occur when final dx x is independent of initial x 0, = 0 dx 0 In free-expanding slab, we have no wave-breaking for any profile dx ( ) 2 = 1+ k z p dx 0 2 f( x 0 ) >1 > 0 x'/k 2 1.5 1 0.5 kz=4 kz=1 0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 x Phase space profile becomes more linear for k p z >>1
Slab-beam in a focusing channel Add uniform focusing to equation of motion, x + k 2 x = k 2 p Fx ( 0 ). [ ]cos k z Solution xx ( 0 ) = x eq ( x 0 )+ x 0 x eq ( x 0 ) with x eq ( x 0 ) = k 2 p k Fx 2 ( 0) Wavebreaking occurs in this case for ( ) For physically meaningful distributions, smoothly, and wavebreaking occurs when f( x 0 )= k 2 For matched beam, k 2 2 p = k half of the beam wave-breaks! Stay away from equilibrium! When k 2 2 p >> k there is little wavebreaking, and irreversible emittance growth avoided. k p 2 f( x 0 ) 0 k z > /2 ( ) cos k z 2sin k 2 z 2
Extension to cylindrical symmetry: 1D simulations Sigma r [mm] Emittance n,rms [mm mrad] r [mm], n,rms [mm mrad] 7.0 6.0 5.0 4.0 3.0 2.0 1.0 Evolution of a Parabolic Beam in an RMS Matching Solenoid 0.0 0 2 10 3 4 10 3 6 10 3 8 10 3 1 10 4 8.0 6.0 4.0 2.0 Z [mm] Evolution of a Parabolic Beam through one Lens 10 Matched parabolic beam shows irreversible emittance growth after single betatron period Grossly mismatched single thin lens show excellent nonlinear compensation Explanation for robustness of first compensation behavior 0.0 0 1 10 3 2 10 3 3 10 3 4 10 3 5 10 3 6 10 3 7 10 3 z [mm] RMS beam size in red, emittance in blue
Emittance growth and entropy 700 Irreversible emittance growth is accompanied by entropy increase Far-from-equilibrium thin-lens case shows large distortion at beam maximum, near perfect reconstruction of initial profile Small wave-breaking region in beam edge Density [Macroparticles/mm 2 ] (a) Density [Macroparticles/mm 2 ] (b) Density [Macroparticles/mm 2 ] 600 500 400 300 200 100 0 0.0 0.20 0.40 0.60 0.80 1.0 1.2 r [mm] 16 14 12 10 8 6 4 2 0 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 r [mm] 700 600 500 400 300 200 100 Beam profile at launch Beam profile at maximum Beam profile, return to min. (c) 0 0.0 0.20 0.40 0.60 0.80 1.0 1.2 r [mm]
Trace space picture r' [radians] (a) r' [radians] 0.10 0.080 0.060 0.040 0.020 0.0 0.0 1.0 2.0 3.0 4.0 5.0 r [mm] 0.15 0.12 0.090 0.060 0.030 Wave-breaking occurs near beam edge at emittance maximum Fortuitous folding in trace space near fixed point minimizes final emittance (b) 0.0 0.0 5.0 10 15 r [mm] Trace space plots of a freely expanding, initially Gaussian beam at the initial emittance (a) maximum and (b) minimum.
Multiparticle simulation picture: LCLS case (Ferrario scenario) 0.15 0.10 0.100 0.050 0.050 x (cm) 0.0 x (cm) 0.0-0.050-0.050-0.10 x' (rad) -0.15-0.2-0.15-0.1-0.05 0 0.05 0.1 0.15 0.2 z (cm) Spatial (x-z) distribution 0.030 0.020 0.010 0.0-0.0100-0.020-0.030-0.15-0.1-0.05 0 0.05 0.1 0.15 x (cm) Trace-space distribution -0.10-0.1-0.05 0 0.05 0.1 y (cm) Spatial (x-y) distribution Case I: initially uniform beam (in r and t) Spatial uniformity reproduced after compensation High quality phase space Most emittance is in beam longitudinal tails (end effect)
Multiparticle simulation picture: Nonuniform beam 0.3 5.0 4.0 rms beam size (nonuniform beam) rms emittance (nonuniform beam) 0.2 0.1 x (mm), n (mm-mrad) 3.0 2.0 1.0 High gradient RF photocathode gun Focusing solenoid PWT linacs 0.0 0 50 100 150 200 250 300 350 400 z (cm) Larger emittance obtained x (cm) 0-0.1-0.2-0.3-0.2-0.15-0.1-0.05 0 0.05 0.1 0.15 0.2 z (cm) Spatial (x-z) distribution x' (rad) 0.060 0.040 0.020 0.0-0.020-0.040-0.060-0.2-0.15-0.1-0.05 0 0.05 0.1 0.15 0.2 x (cm) Case II: Gaussian beam Most emittance growth due to nonlinearity Large halo formation Trace-space distribution
The big picture: scaling of design parameters in photoinjectors The beam-plasma picture based on envelopes gives rise to powerful scaling laws RF acceleration also amenable to scaling Scale designs with respect to: Charge RF wavelength Change from low charge (FEL) to high charge (HEP, wakefield accelerator) design Change RF frequency from one laboratory to another (e.g. SLAC X-band, TESLA L-band)
Charge scaling Keep all accelerator/focusing parameters identical To keep plasma the same, the density and aspect ratio of the bunch must be preserved i Q 1/3 The contributions to the emittance scale with varying powers of the beam size Space-charge emittance x,sc k 2 p 2 x Q 2/3 RF/chromatic aberration emittance x,rf 2 z 2 x Q 4/3 Thermal emittance x,th x Q 1/3 Fortuitously, beam is SC dominated, and these emittancs do not affect the beam envelope evolution; compensation is preserved.
Wavelength scaling First, must make acceleration dynamics scale: and E 0 1 RF E 0 = constant Focusing (betatron) wavenumbers must also scale (RF is naturally scaled,,rf E 0 ). Solenoid field scales as B 0 1. Correct scaling of beam size, and plasma frequency: Q i All emittances scale rigorously as n
Scaling studies: envelope 0.700 PARMELA simulations used to explore scaling Charge scaling (non-optimized case) is only approximate. At large beam charges (beam sizes), beam is not negligibly small compared to RF wavelength. Wavelength scaling is exact, as expected. x (cm) x / 0.600 0.500 0.400 0.300 0.200 0.100 0.000 0.040 0.035 0.030 0.025 0.020 0.015 0.010 Q=0.36 nc Q=2 nc Q=6.36 nc Q=11.7 nc 0.00 20.00 40.00 60.00 80.00 100.00 120.00 z (cm) Beam size evolution, different charges f=3.5 GHz f=6.75 GHz f=9.25 GHz f=14 GHz 0.005 0.000 0.00 2.00 4.00 6.00 8.00 10.00 12.00 z/ Beam size evolution, different RF
Scaling studies: emittance n (mm-mrad) 100.00 10.00 1.00 0.10 n = [(aq 2/3 ) 2 +(bq 4/3 ) 2 ] 0.5 n ~ Q 2/3 n ~ Q 4/3 0.1 1.0 10.0 100.0 Q (nc) Scaling of emittance with charge (no thermal emittance), fit assumes addition in squares. n / (x 10 3 ) 10.00 8.00 6.00 4.00 2.00 0.00 0.0 2.0 4.0 f=3.5 GHz f=6.75 GHz f=9.25 GHz f=14 GHz 6.0 8.0 10.0 12.0 z/ Evolution of emittance, normalized to Simulation studies verify exact scaling of emittance with Charge scan of simulations gives information about family of designs Use to mix scaling laws
Brightness, choice of charge and wavelength Charge and pulse length scale together as Brightness scales strongly with, This implies low charge for high brightness What if you want to stay at a certain charge (e.g. FEL energy/pulse) Mixed scaling: n ( ) Q nc ( mm - mrad)= rf (m) a 1 rf (m) B e = 2I / n 2 2 For Ferrario scenario, constants from simulation: 2/3 a 1 =1.5 a 2 = 0.81 a 3 = 0.052 thermal charge RF/chromatic ( ) Q nc + a 2 rf (m) 4/3 ( ) Q nc + a 3 rf (m) 8/3
Some practical limits on scaling Scaling of beam size laser pulse length and jitter difficult at small emittance measurements difficult at small Scaling of external forces Electric field is natural - high-gradient implies short because of breakdown limits,. RF limitations may arise in power considerations Focusing solenoid B 1 dimensions scale as. Current density scales as J sol 2
Exercises Problem 5: Assume the LCLS photoinjector has gradient of 20 MV/m, and is run on the invariant envelope with =1, achieving a normalized emittance of 0.9 mm-mrad at 100 A current. At the final energy of 150 MeV, what is the ratio of the space-charge term to the emittance term in the envelope equation? Problem 6: (a) For the parameters of the LCLS design family (Ferrario scenario), if one desires to run at 1 nc, what is the optimum RF wavelength to choose to minimize the emittance? (b) If you operate at an RF wavelength of 10.5 cm, what choice of charge maximizes the brightness?