Emittance Compensation J.B. Rosenzweig ERL Workshop, Jefferson Lab 3//5
Emittance minimization in the RF photoinjector Thermal emittance limit Small transverse beam size Avoid metal cathodes? " n,th # 1 h$ %W m c & x # 5 '1 %4 & x m ( ) RF emittance " n,rf # k RF $ RF ( k RF % z ) % x Small beam dimensions Small acceleration field? Maybe not Space charge emittance Original K.J.Kim treatment discouraging " n,sc # m e c 1I ($) ee I (1+ 3 A) 5 A = " x " z, I = ec r e " n,sc # 5 mm - mrad (I =1 A, E =1 MV/m)
Space-charge emittance control: Emittance compensation Space-charge emittance evolution not monotonic in time Multiparticle simulations at LLNL (Carlsten) show emittance oscillations, minimization possible: Emittance compensation Analytical approach Scaling laws Prescriptions for design LCLS (Ferrario WP) Superconducting version?! z (mm), " # (mm-mrad) 3.. 1. High gradient RF photocathode gun Focusing solenoid rms beam size (uniform beam) rms emittance (uniform beam). 5 1 15 5 3 35 4 z (cm) PWT linacs Multiparticle simulations (UCLA PARMELA) Showing emittance oscillations and minimization
Transverse dynamics model After initial acceleration, space-charge field is mainly transverse (beam is long in rest frame). Force dependent ~ exclusively on local value of current density I /σ (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, λ rf
The rms envelope equation The rms envelope equation for a cylindrically symmetric, nonaccelerating, space-charge dominated beam r e & $ #" x ( $,z) + k % # r ( $,z) = ' 3 # x $,z ( ) ( ) + ( n,x '# 3 x ( $,z) Separate DE for each slice (tagged by ζ), " = z # v b t Each slice has different current I (" ) = #(" )v External focusing measured by betatron wave-number k " = eb z /p Envelope coordinates are in Larmor frame Rigid rotator equilibrium (Brillouin flow) depends on I(ζ). Pressure forces negligible " eq # ( ) = 1 r e %(#) k $ & 3 r e "(#) = I (# ) /I
! Equilibrium distributions and space charge dominated beams Maxwell-Vlasov equilibria have simple asymptotic forms, dependent on parameter f(x) 1.5 1! x =(" n /#k $ ) 1/ R " I /# k $ % n I Emittance dominated gaussian R << 1 Space-charge dominated uniform R >>1 Uniform beam approximation very useful 1. f(x).5 1 3 4 5 x/! 1. 1.8.6.4! x =(I/" 3 I k # ) 1/ f(x) 1 R " 5.8.6.4.! 1 3 4 5 x/!. 1 3 4 5 x/! " ~(# /$k ) 1/ D n % Nominally uniform has Debye sheath High brightness photoinjector beams have R >1, " # 5!
The trace space model Each ζ-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 9% ellipse x exp (#1/ ) contour Contrast with thermal trace space and nonlinear slice trace space
Envelope oscillations about equilibria Beam envelope is non-equilibrium problem Linearizing the rms envelope equation about equilibrium gives " $ # x (%,z) + k & $ x (%,z) = Dependent on betatron wave-number, not local beam size or current Small amplitude envelope oscillations proceed at 1/ times the betatron frequency or assuming uniform beam distribution k env = k " = 4#r en b,eq $ 3 = k p This is the matched relativistic plasma frequency
Phase space picture: coherent oscillations SC beam envelope oscillations proceed about different equilibria, with different amplitude but at the same frequency Behavior leads to emittance oscillations but not damping (yet) 1st compensation, after gun, before linac!/! eq, "/(K r ) 1/! eq " #! r ' x 1..8.6.4. " " " 1 3! eq1! eq! eq3 " 1 < " <" 3 "! rx Assume that beam is launched at minimum (e.g. at cathode) 1! "..4.6.8 1 (K r ) 1/ z/ # Small amplitude oscillation model
Phase space picture: coherent oscillations Emittance (area in phase space) is maximized at k p z = " /,3" / Emittance is locally minimized at k p z =,"," - 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.. 1. High gradient RF photocathode gun k z= "/ p k pz=3 " / k p z=, " Focusing solenoid # 1 # # 3 # 1 < # < # 3 rms beam size (uniform beam) rms emittance (uniform beam). 5 1 15 5 3 35 4 z (cm) PWT linacs Damping of e...! 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) * Particular solution - the invariant envelope (generalized Brillouin flow), slowly damping fixed point -analogue " inv (#,z) = 1 r e &(# ) % $ ( + ')%( z) (% )1/ Angle in phase space is independent of current # x ( $,z) = r e -( $ ) ( ) 3 # x ( $,z) Corresponds exactly to entrance/exit kick (matching is naturally at waists) " x # RF = $ 1 # % % x Matching beam to invariant envelope yields stable linear emittance compensation! % z " # 1 (rf or solenoid focusing) #" = ee /m c (accel. "wavenumber") " = $ # inv = % 1 &# $ inv &
Envelope oscillations near invariant envelope, with acceleration Linearized envelope equation & %#) " $ # x + ( + " $ # x + 1+, & %#) ( + "$ x = ' % * 4 ' % * Homogenous solution (independent of current) ' "# x = [# x $# inv ]cos) ( Normalized, projected phase space area oscillates, secularly damps as offset phase space (conserved!) moves in " $ # x 1+ % ' ln & ( z )** ),, ( & + + " n ~ " offset # inv ~ $ %1/ damping oscillation " x Oscillation (generalized matched plasma) frequency damps with energy k p = d $ & dz % 1+ " ( ) ln #( z) ' ) = 1+ " #* ( # offset phase space " $ # x = % 1 $ 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
Scaling in photoinjector design The envelope equation approach gives rise to powerful scaling laws RF acceleration also amenable to scaling Scale designs with respect to: Charge (k p ) RF wavelength (k RF ) Change from low charge (FEL) to high charge (LC, wakefield driver) design Change RF frequency from one laboratory to another (e.g. SLAC X-band, SC L-band)
Charge scaling Keep all accelerator/focusing parameters identical Density and aspect ratio of the bunch must be preserved Contributions to emittance scale with powers of beam size Space-charge emittance RF/chromatic aberration emittance Thermal emittance! i " Q 1/ 3 " x,sc # k p $ x #Q / 3 " x,th #$ x #Q 1/ 3 " x,rf #$ z $ x #Q 4 / 3 Beam is SC dominated, and emittancs do not affect the beam envelope evolution; compensation is preserved.
Wavelength scaling First, make acceleration dynamics scale: " and E " # $1 RF # E $ = constant Focusing (betatron) wavenumbers must also scale (RF is naturally scaled, " #,rf $ E ). Solenoid field scales as B! " #1. Correct scaling of beam size, and plasma frequency:! i " # Q " # All emittances scale rigorously as! n " #
Brightness, choice of charge and wavelength Charge and pulse length scale together as λ Brightness scales strongly with λ, This implies low charge for high brightness What if you want to stay at a certain charge (e.g. FEL energy/pulse) Mixed scaling: $ Q( nc) ( ) = # rf (m) a 1 & " n mm - mrad ' ) % # rf (m)( B e = I /" n # $ % For Ferrario scenario, constants from simulation: / 3 a 1 =1.5 a =.81 a 3 =.5 thermal charge RF/chromatic ( ) $ Q nc ' + a & ) % # rf (m)( 4 / 3 ( ) $ Q nc ' + a 3 & ) % # rf (m)( 8 / 3
Example: SC gun sigma_x_[mm] ε n [mm-mrad] 6 5 4 3 Scale Ferrario scenario to L-band, SC 6 MV/m peak (3 average) gun field! HOMDYN HBUNCH.OUT Simulation Q =1 nc R =1.69 mm L =19.8 ps ε th =.45 mm-mrad E peak = 6 MV/m (Gun) E acc = 13 MV/m (Cryo1) B = 3 kg (Solenoid) sigma_x_[mm] enx_[um] I = 5 A E = 1 MeV ε n =.6 mm-mrad 1 6 MeV 5 1 15 3.3 m z [m] z_[m]
Nonlinear Emittance Growth Nonuniform beams lead to nonlinear fields and emitance growth 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.
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 f x n b x ( ), f ( ) = 1 Under laminar flow, charge inside of a given electron is conserved; one may mark trajectories from initial offset x. Equation of motion x " = k p F( x ), F x x # x d ( ) = f ( ) Note, with normalization F ( x ) " x x = const.
Free-expansion of slab beam Solution for electron positions: ( x( x ) = x + k z p ) F( x ) Distribution becomes more linear in density with expansion! Example case! f ( x( x )) = ( ) f x ( ) 1+ k pz f x Wavebreaking will occur when! final x is independent of initial x, In free-expanding slab, there is no wave-breaking for any profile dx ( )! = 1+ k pz dx f ( x ) " ( k p z) ( a ) ( ) = 1" x f ( x ) >1 > x'/k dx = dx f(x) 1.8.6.4..5 1.5.5 f(x_) f(x) 1 3 4 5 6 7 x/a 3 1 Initially parabolic profile becomes more uniform at k p z = 4 kz=4 kz=1!..4.6.8 1 1. 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, Solution with x eq x x x ( ) = k p x " + k # x = k p F( x ). [ ] cos( k # z) ( ) = x eq ( x ) + x " x eq ( x ) ( ) k " F x Wavebreaking occurs in this case for f x For physically meaningful distributions, smoothly, and wavebreaking occurs when ( ) = " k # For matched beam, k p = k " half of the beam wave-breaks. Stay away from equilibrium! When k p >> k! there is little wavebreaking, and irreversible emittance growth avoided. k p f ( x ) " k " z > # / ( ) cos k # z $ sin k # z' & ) % (
Extension to cylindrical symmetry: 1D simulations Sigma r [mm] Emittance n,rms [mm mrad]! r [mm], " n,rms [mm mrad] 7. 6. 5. 4. 3.. 1. Evolution of a Parabolic Beam in an RMS Matching Solenoid. 1 3 4 1 3 6 1 3 8 1 3 1 1 4 8. 6. 4.. Z [mm] Evolution of a Parabolic Beam through one Lens 1 Matched parabolic beam shows irreversible emittance growth after single betatron period Grossly mismatched single thin lens show excellent nonlinear compensation Explains robustness of first compensation. 1 1 3 1 3 3 1 3 4 1 3 5 1 3 6 1 3 7 1 3 z [mm] RMS beam size in red, emittance in blue
Emittance growth and entropy 7 Irreversible emittance growth is accompanied by entropy increase Far-from-equilibrium thinlens case shows ~uniform beam at maximum Near perfect reconstruction of initial profile Small wave-breaking region in beam edge Density [Macroparticles/mm ] (a) Density [Macroparticles/mm ] (b) Density [Macroparticles/mm ] 6 5 4 3 1...4.6.8 1. 1. r [mm] 16 14 1 1 8 6 4. 1.. 3. 4. 5. 6. 7. 8. r [mm] 7 6 5 4 3 1 Beam profile at launch Beam profile at maximum Beam profile, return to min. (c)...4.6.8 1. 1. r [mm]
Multiparticle simulation picture: LCLS case (Ferrario scenario).15.1.1.5.5 x (cm). x (cm). -.5 -.5 -.1 x' (rad) -.15 -. -.15 -.1 -.5.5.1.15.!z (cm) Spatial (x-z) distribution.3..1. -.1 -. -.3 -.15 -.1 -.5.5.1.15 x (cm) Trace-space distribution -.1 -.1 -.5.5.1 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.3 5. 4. rms beam size (nonuniform beam) rms emittance (nonuniform beam)..1! x (mm), " n (mm-mrad) 3.. 1. High gradient RF photocathode gun Focusing solenoid PWT linacs. 5 1 15 5 3 35 4 z (cm) Larger emittance obtained x (cm) -.1 -. -.3 -. -.15 -.1 -.5.5.1.15. z (cm) Spatial (x-z) distribution x' (rad).6.4.. -. -.4 -.6 -. -.15 -.1 -.5.5.1.15. x (cm) Case II: Gaussian beam Most emittance growth due to nonlinearity Large halo formation Trace-space distribution
Have we chosen the right beam shape? The first beer can Beer-can beam suffers from Edge erosion, non-uniform distr. Nonlinear fields at edges Practical difficulties with laser Luiten (Serafini) proposal: Use any ultra-short pulse Longitudinal expansion of radial parabolic profile I( r) = I ( 1" ( r / a) ) Uniform ellipsoidal beam created! Linear space-charge fields (3D)
New direction: marry Luiten proposal to emittance compensation! x (mm).5 1.5 1.5 4 6 8 z (cm) Beam envelope evolution Initial (not-too-optimized) PARMELA study Standard LCLS injector conditions Q=.33 nc, initial long. Gaussian σ t =33 fs (cutoff at 3 σ) trans. Gaussian with σ x =.77 mm (cutoff at σ). Final bunch length 1.43 mm (full), 14 A. 4 3.5! n (mm-mrad) 3.5 1.5 1.5 4 6 8 z (cm) Emittance evolution Beam distribution showing ellipsoidal boundary (1.5 MeV)
Challenges and advantages Laser very forgiving Shorter pulses possible? Cathode image charges drive incorrect final state, not but Excessive energy spread during compensation Charge fluctuations Experiment at LLNL, ORION or SPARC