lecture 4: emittance compensation - pbpl home...
TRANSCRIPT
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?
• RF emittance
– Small beam dimensions
– Small acceleration field? Maybe not…
• Space charge emittance
– K.J.Kim treatment is very discouraging
n,th
1
2
h W
m0c2 x 5 10 4
x m( )
n,RF kRF RF kRF z( )2
x2
n,sc
mec2
(2 )2eE0
10I
I0(1+ 35 A)
A = x
z
, I0 =ec
re
n,sc 5 mm - mrad (I =100 A, E0 =100 MV/m)
Space-charge emittance control?
• Kim model indicatesmonotonic emittance growthdue to space-charge
• Multiparticle simulations atLLNL (Carlsten) showemittance oscillations,minimization possible:Emittance compensation
• Work extended by UCLA,INFN scientists to giveanalytical approach
• New high gradient designdeveloped and understood
• Many new doors opened
0.0
1.0
2.0
3.0
0 50 100 150 200 250 300 350 400
rms beam size (uniform beam)
rms emittance (uniform beam)
z (mm
), (m
m-m
rad)
z (cm)
PWT linacs
Focusing solenoidHigh gradient RF photocathode gun
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 focusingsolenoids)
• Very large self-fields
– Longitudinal debunching (charge limit)
– Radial oscillations (single component plasma)
• Optimization of beam handling with large parameter space andcollective effects. Multiparticle simulations are invaluable aid, buttime-consuming
• Understanding of non-equilibrium transport approached using rmsenvelope equations…
Transverse dynamics model
• After initial acceleration, space-charge field is mainlytransverse (beam is long in rest frame).
• Force scales as -2 (cancellation of electric defocusing
with magnetic focusing)
• Force dependent almost exclusively on local value ofcurrent density I / 2 (electric field simply from Gauss’law)
• Linear component of self-force most important. Weinitially 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, non-accelerating, space-charge dominated beam are described by a nonlineardifferential equation
• Separate DE for each slice (tagged by ),
• Each slice has different current
• External focusing measured by betatron wave-number
• In solenoid, beam is rotating, so envelope coordinates are in rotatingLarmor frame with same wave-number
• Rigid rotator equilibrium (Brillouin flow) depends on local value ofcurrent (line-charge density). “Pressure” forces negligible
x ,z( ) + k 2 r ,z( ) =re ( )
2 3x ,z( )
+ n,x2
2 x3 ,z( )
re ( ) = I( ) /I0
= z vbt
k = eBz /2p0
eq ( ) =1
k
re ( )2 3
I( ) = ( )v
Equilibrium distributions and space
charge dominated beams• Maxwell-Vlasov equilibria have
simple asymptotic forms,
dependent on parameter
• Emittance dominated gaussian
• Space-charge dominated uniform
• Uniform beam approximation very
useful
R I /2 2k nI0R << 1
R >>1
0
0.5
1
1.5
2
0 1 2 3 4 5
f(x)
x/
x=(
n/ k )
1/2
0
0.2
0.4
0.6
0.8
1
1.2
0 1 2 3 4 5
f(x)
x/
x=(I/2
3I0k
2)1/2
0
0.2
0.4
0.6
0.8
1
1.2
0 1 2 3 4 5
f(x)
x/
D~(
n/ k )
1/2
• Nominally uniform has Debye sheath• High brightness photoinjector beams have
R >1, 250!
R 5
The trace space model
• Each -slice component of the
beam is a line in trace space.
• No thermal effects
• No nonlinearities (lines are
straight!)
x'
x
rms ellipse
Trace space distribution
90% ellipse
"5 times" rms ellipse
x
x’
x
x’
“Slice” phase spaces
x
x’
Projected phase space
x
x'
f(x,x')
x
x'x
x
exp 1/ 2( ) contour
The model
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
Dependent on betatron wave-number, not local beam size or current
• Small amplitude envelope oscillations proceed at 21/2 times thebetatron frequency or assuming uniform beam distribution
This is the matched relativistic plasma frequency
x ,z( ) + 2k 2 x ,z( ) = 0
kenv = 2k =4 renb,eq
3 = kp
Phase space picture:
coherent oscillations
• All oscillations of space-charge beam envelopeproceed about
– different equilibria,
– with different amplitude
– but at the same frequency
• Behavior leads to emittanceoscillations…but not damping(yet)
• Qualitative explanation of“1st compensation”, after gun,before linac…
r
r
1 2 3
eq1
eq2eq3
'
21< < 3
x
x
Assume that beam is launchedat minimum (e.g. at cathode)
0
0.2
0.4
0.6
0.8
1
1.2
0 0.2 0.4 0.6 0.8 1
/eq
, /(
Kr)1
/2eq
2
(2Kr)1/2z/2
Small amplitude oscillation model
Phase space picture:
coherent oscillations
• Emittance (area in phasespace) is maximized at
• Emittance is locallyminimized at
- the beam extrema!
• Fairly good agreement ofsimple model with muchmore complex beamline
• What about acceleration?– In the rf gun, in booster linacs…
r
r'
k z=3 /2p
k z=0,2 p
1
2
3
1< <
2
3
k z= /2p
x
x
kpz = / 2,3 / 2
kpz = 0, ,2
0.0
1.0
2.0
3.0
0 50 100 150 200 250 300 350 400
rms beam size (uniform beam)
rms emittance (uniform beam)
z (m
m),
(m
m-m
rad)
z (cm)
PWT linacs
Focusing solenoidHigh gradient RF photocathode gun
Damping of ...
Emittance damping: Beam envelope
dynamics under acceleration• Envelope equation (w/o emittance), with acceleration, RF focusing
• New particular solution - the “invariant envelope” (generalized
Brillouin flow), slowly damping “fixed point”
• Angle in phase space is independent of current
• Corresponds exactly to entrance/exit kick (matching is naturally at
waists)
• Matching beam to invariant envelope yields stable linear emittance
compensation!
x ,z( ) +
z( )
x ,z( ) +
8
z( )
2
x ,z( ) =re ( )
2 z( )3
x ,z( )1 (rf or solenoid focusing)
= eE0 /m0c2 (accel. "wavenumber")
inv ,z( ) =1
re ( )2 +( ) z( )
1/ 2
= inv
inv
=1
2
x RF =1
2
x
Envelope oscillations near invariant
envelope, with acceleration• Linearized envelope equation
• Homogenous solution (independent of current)
• Normalized, projected phase space area oscillates, secularydamps as offset phase space (conserved!) moves in…
x +
x +
1+
4
2
x = 0
x = x0 inv[ ]cos1+
2ln
z( )
0
x
x
x =1
2 x
damping
oscillation
“offset phase space”
Oscillation (matched plasma)
frequency damps with energy
kp =d
dz
1+
2ln z( )( )
=
1+
2
n ~ offset inv ~1/ 2
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 thatpropagation of non-uniform distributions in equilibrium leads toirreversible emittance growth (wave-breaking in phase space).
r
r'
Fixed point off-axis
"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.
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)
• Under laminar flow, the charge inside of a given electron
is conserved, and one may mark trajectories from initial
offset x0. Equation of motion
nb x0( ) = n0 f x0( ), f 0( ) = 1
x = kp2F x0( ) , F x0( ) = f ˜ x 0( )
0
x0
d ˜ x 0 = const.
Note, with normalization F x0( ) x
Free-expansion of slab beam
• Solution for electron positions:
• Distribution becomes more linear in
density with expansion
• Example case
• Wavebreaking will occur when final
x is independent of initial x0,
• In free-expanding slab, we have no
wave-breaking for any profile
0
0.2
0.4
0.6
0.8
1
0 1 2 3 4 5 6 7
f(x_0)f(x)
f(x)
x/a
0
0.5
1
1.5
2
2.5
3
0 0.2 0.4 0.6 0.8 1 1.2 1.4
kz=4kz=1
x'/k
x
x x0( ) = x0 +kpz( )
2
2F x0( )
f x x0( )( ) =f x0( )
1+kpz( )
2
2f x0( )
2
kpz( )2
f x0( ) = 1 x0a( )
2
Phase space profile becomes more linear for kpz >>1
Initially parabolic profile becomesmore uniform at kpz = 4
dx
dx0= 1+
kpz( )2
2f x0( ) >1 > 0
dx
dx0= 0
Slab-beam in a focusing channel
• Add uniform focusing to equation of motion,
• Solution
with
• Wavebreaking occurs in this case for
• For physically meaningful distributions,smoothly, and wavebreaking occurs when
• For matched beam, half of the beam wave-breaks!
• Stay away from equilibrium! When there is littlewavebreaking, and irreversible emittance growth avoided.
x + k2x = kp2F x0( ).
x x0( ) = xeq x0( ) + x0 xeq x0( )[ ] cos k z( )xeq x0( ) =
kp2
k2F x0( )
f x0( ) =k 2
kp2
cos k z( )
2sin2k z
2
f x0( ) 0
k z > / 2
kp2
= k2
kp2>> k 2
Extension to cylindrical
symmetry: 1D simulations• 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
1.0
2.0
3.0
4.0
5.0
6.0
7.0
0 2 103 4 103 6 103 8 103 1 104
Evolution of a Parabolic Beam
in an RMS Matching Solenoid
Sig
ma r [m
m] E
mitt
ance
n,r
ms [m
m m
rad]
Z [mm]
0.0
2.0
4.0
6.0
8.0
10
0 1 103
2 103
3 103
4 103
5 103
6 103
7 103
Evolution of a Parabolic Beam through one Lens
r [m
m],
n,r
ms [
mm
mra
d]
z [mm]
RMS beam size in red, emittance in blue
Emittance growth and entropy
• Irreversible emittancegrowth is accompanied byentropy increase
• Far-from-equilibriumthin-lens case shows largedistortion at beammaximum, near perfectreconstruction of initialprofile
• Small wave-breakingregion in beam edge
0
100
200
300
400
500
600
700
0.0 0.20 0.40 0.60 0.80 1.0 1.2
Densit
y
[Macro
part
icle
s/m
m2]
r [mm](a)
0
2
4
6
8
1 0
1 2
1 4
1 6
0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0
Densit
y
[Macro
part
icle
s/m
m2]
r [mm](b)
0
100
200
300
400
500
600
700
0.0 0.20 0.40 0.60 0.80 1.0 1.2
Densit
y
[Macro
part
icle
s/m
m2]
r [mm](c)
Beam profile at launch
Beam profile at maximum
Beam profile, return to min.
Trace space picture
0.0
0.020
0.040
0.060
0.080
0.10
0.0 1.0 2.0 3.0 4.0 5.0
r' [r
adia
ns]
r [mm](a)
0.0
0.030
0.060
0.090
0.12
0.15
0.0 5.0 1 0 1 5
r' [r
adia
ns]
r [mm](b)
• Wave-breaking occurs
near beam edge at
emittance maximum
• Fortuitous folding in
trace space near “fixed”
point minimizes final
emittance
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
-0.050
0.0
0.050
0.100
0.15
-0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2
x (c
m)
z (cm)
-0.030
-0.020
-0.0100
0.0
0.010
0.020
0.030
-0.15 -0.1 -0.05 0 0.05 0.1 0.15
x' (
rad)
x (cm)
-0.10
-0.050
0.0
0.050
0.10
-0.1 -0.05 0 0.05 0.1
x (c
m)
y (cm)
• Case I: initially uniform beam (in rand t)• Spatial uniformity reproduced aftercompensation• High quality phase space• Most emittance is in beamlongitudinal tails (end effect)
Spatial (x-z) distribution Spatial (x-y) distribution
Trace-space distribution
Multiparticle simulation picture:
Nonuniform beam
• Case II: Gaussian beam• Most emittance growth due tononlinearity• Large halo formation
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
-0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2
x (c
m)
z (cm)
-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' (
rad)
x (cm)
Trace-space distribution
Spatial (x-z) distributionLarger emittance obtained
0.0
1.0
2.0
3.0
4.0
5.0
0 50 100 150 200 250 300 350 400
rms beam size (nonuniform beam)
rms emittance (nonuniform beam)
x (mm
), n (
mm
-mra
d)
z (cm)
PWT linacs
Focusing solenoidHigh gradient RF photocathode gun
The big picture: scaling of design
parameters in photoinjectors
• The “beam-plasma” picture based on envelopesgives 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 toanother (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 ofthe bunch must be preserved
• The contributions to the emittance scale with varyingpowers of the beam size
• Space-charge emittance
• RF/chromatic aberration emittance
• Thermal emittance
• Fortuitously, beam is SC dominated, and these emittancsdo not affect the beam envelope evolution; compensationis preserved.
i Q1/ 3
x,sc kp2
x2 Q2 / 3
x,RF z2
x2 Q4 / 3
x,th x Q1/ 3
Wavelength scaling
• First, must make acceleration dynamics
scale: and
• Focusing (betatron) wavenumbers must
also scale (RF is naturally scaled, ).
Solenoid field scales as
• Correct scaling of beam size, and plasma
frequency:
• All emittances scale rigorously as
E01
RF E0 = constant
B01.
,RF E0
iQ
n
Scaling studies: envelope
• 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.
0.000
0.100
0.200
0.300
0.400
0.500
0.600
0.700
x (
cm)
0.00 20.00 40.00 60.00 80.00 100.00 120.00
z (cm)
Q=0.36 nC
Q=2 nC
Q=6.36 nC
Q=11.7 nC
0.000
0.005
0.010
0.015
0.020
0.025
0.030
0.035
0.040
x/
0.00 2.00 4.00 6.00 8.00
z/
f=3.5 GHz
f=6.75 GHz
f=9.25 GHz
f=14 GHz
10.00 12.00
Beam size evolution, different charges
Beam size evolution, different RF
Scaling studies: emittance
0.10
1.00
10.00
100.00n (
mm
-mra
d)
0.1 1.0 10.0 100.0
Q (nC)
n ~ Q2/3
n ~ Q4/3
n = [(aQ2/3)2+(bQ4/3)2]0.5
0.00
2.00
4.00
6.00
8.00
n/
(x 1
03)
10.00
0.0 2.0 4.0
z/6.0 8.0 10.0 12.0
f=3.5 GHz
f=6.75 GHz
f=9.25 GHz
f=14 GHz
• Simulation studies
verify exact scaling ofemittance with
• Charge scan of
simulations gives
information about
“family” of designs
• Use to mix scaling
laws…
Scaling of emittance with charge (no thermalemittance), fit assumes addition in squares.
Evolution of emittance, normalized to
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:
• For Ferrario scenario, constants from simulation:
Be = 2I / n2 2
n mm-mrad( ) = rf (m) a1Q nC( )
rf (m)
2 / 3
+ a2Q nC( )
rf (m)
4 / 3
+ a3Q nC( )
rf (m)
8 / 3
a1 =1.5 a2 = 0.81 a3 = 0.052
charge RF/chromaticthermal
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 dimensions scale as .Current density scales as
B 1
Jsol2
Exercises
Problem 5: Assume the LCLS photoinjector has gradient of20 MV/m, and is run on the invariant envelope with =1,achieving a normalized emittance of 0.9 mm-mrad at 100A current. At the final energy of 150 MeV, what is theratio of the space-charge term to the emittance term in theenvelope equation?
Problem 6: (a) For the parameters of the LCLS designfamily (Ferrario scenario), if one desires to run at 1 nC,what is the optimum RF wavelength to choose tominimize the emittance? (b) If you operate at an RFwavelength of 10.5 cm, what choice of charge maximizesthe brightness?