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Argonne National Laboratory is managed by The University of Chicago for the U.S. Department of EnergyKJK, COOL05, 9/18-23/05
Transverse-Longitudinal Phase Space Manipulations
Kwang-Je KimArgonne National Laboratory
and The University of Chicago
COOL05International Workshop on Beam Cooling and
Related TopicsSeptember 18-23, 2005
Eagle Ridge, Galena, Illinois
2KJK, COOL05, 9/18-23/05KJK, COOL05, 9/18-23/05
Phase Space ManipulationAccelerators are for manipulation of beams in 6D phase spacePhase space manipulation in accelerators has been mainly in one 2D subspaces (z, δ) or (x, x’) and in Hamiltonian systemBeam cooling is an advanced phase space manipulation in weakly non-Hamiltonian systemManipulation involving 6-D phase space of Hamiltonian system can greatly enhance accelerator performanceThis and the next talk are about Hamiltonian phase space manipulation involving more than 2D
3KJK, COOL05, 9/18-23/05KJK, COOL05, 9/18-23/05
ContentsSome properties of Hamiltonian Transport
Transverse-longitudinal switching for x-ray FELs
Flat beam technique
Applications of flat beam– Smith-Purcell FEL– X-ray pulse compression
4KJK, COOL05, 9/18-23/05KJK, COOL05, 9/18-23/05
Beam Transport and Manipulation6D phase space: (x, x´, y, y´, z, δ)We will use 4D for notational simplicity
X
Beam matrix:
Transfer matrix: MX = M Xo, Σ = M Σo
⎟⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜⎜
⎝
⎛
′
′=
yyxx
∑⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢
⎣
⎡
′
′′′
==
21
2
y..yx...xy...xxyxxyxxx
x~x
M~
5KJK, COOL05, 9/18-23/05KJK, COOL05, 9/18-23/05
Hamiltonian TransportUnit symplectic matrix
M is symplectic:
Det M = 1
There are six constraints on 2D submatrices of M
⎥⎦
⎤⎢⎣
⎡−
=⎥⎦
⎤⎢⎣
⎡=
0110
00
22
2D
D
D J,J
JJ
JMJM~ =
6KJK, COOL05, 9/18-23/05KJK, COOL05, 9/18-23/05
Conserved Emittances2D-case (x, x´):
4D-case (x, x´, y, y´):–
– For uncoupled case, can find a symplectic transformation
–
2
2222
2 xxxxdetDD ′−′==ε ∑D~D
D D∑ ⎥⎦
⎤⎢⎣
⎡β
βε=
2 2 100
∑=ε conservedis24 detD
⎥⎦
⎤⎢⎣
⎡ε
ε=→∑ ∑
bb
aas
TT0
0
conservedis222 baI ε+ε=
7KJK, COOL05, 9/18-23/05KJK, COOL05, 9/18-23/05
Emittance Switching Theorem (E. Courant, …)
εa and εb are uniquely determined up to switching.
(εa, εb) → (εa, εb) or (εb, εa)
Proof:
QEDbaba
baba
∴=
+=+2
22
22
12
1
22
22
21
21
εεεε
εεεε
8KJK, COOL05, 9/18-23/05KJK, COOL05, 9/18-23/05
Projected Emittances for Coupled Cases
Projected emittances are not conservedSome properties: (K.L. Brown & R.V. Servanckx, SLAC-PUB 4679 (1989) )
– Start from uncoupled emittances εxo and εyo
– If εxo = εyo ⇒ εx = εy for all sεx ≥ εxo
– If εxo ≠ εyo
εx + εy ≥ εxo + εyo
Useful applications appear difficult, but …(A. Sessler’s talk)
(coupled)0≠⎟⎟⎠
⎞⎜⎜⎝
⎛Σ
Σ=∑ C,
C~C
y
x
emittancesprojected:det,det 22yyxx Σ=Σ= εε
9KJK, COOL05, 9/18-23/05KJK, COOL05, 9/18-23/05
Emittance Switching for X-Ray FELs
Enter presentation date [Your Presentation Title] 10
SASE FEL for 30 keV
• LCLS reference parameters:λ = 8 keV, λu = 3 cm, K = 3.7, Ip = 3.5 kA, Ee = 15 GeV,∆E/E = 10-4 (2x10-6 possible) εn = 1.2 mm-mrad, Lsat = 100 m
• Vary K, εn, and Ee (Z.R. Huang)
• It pays to strive for an ultralow emittance e-beam
600.1121
400.1303.7
1300.5303.7
3001.2303.7
L sat(m)
εn(mm-mrad)
Ee(GeV)
K
shorter undulatorshorter undulator
shorter undulatorand shorter linac
11KJK, COOL05, 9/18-23/05KJK, COOL05, 9/18-23/05
An Emittance Switching Scheme for Improved X-Ray FEL Performance
(P. Emma, Z. Huang, P. Piot, and KJK)
Flat beam technique (units in m-rad)
Use short electron beam σz = 20 µ
Q = 15 pC, I = 90 A
Switch (x ↔ z)
( ) 7526 101010: −−− ⊗→γε⊗γε yx
53 1010520 −− =×⊗== µσσγε γzz
( )( )
GeV15@10,5007.22090
radm107.2,107.3:
10,10,10
4
56
577
−
−−
−−−
==×=⇒
−==×=
→⊗⊗
γδγ
γεδγσ
γεγεγε
AI
Partition zz
zyx
Enter presentation date [Your Presentation Title] 12
Emittance Exchange: Transverse to LongitudinalEmittance Exchange: Transverse to Longitudinal
Electric and magnetic fieldsElectric and magnetic fieldsηηkk
εx0εz0
εx0εz0
εx ≈ εz0εz ≈ εx0
εx ≈ εz0εz ≈ εx0
ηk = 1ηk = 1
transverse RF in a chicane (P.Emma)transverse RF in a chicane (P.Emma)
γεx0 =10 µmγεx0 =10 µm γεz0
=0.1 µmγεz0 =0.1 µm
γεx ≈0.1 µmγεx ≈0.1 µm γεz ≈10 µmγεz ≈10 µm
system also compresses bunch length
system also compresses bunch length
CrossCross--term term and 2and 2ndnd--order order dispersion dispersion limit limit exchangeexchange
SLACSLAC--PUBPUB--9225, May 20029225, May 2002
13KJK, COOL05, 9/18-23/05KJK, COOL05, 9/18-23/05
Flat Beam TechniqueTheory
– Y. Derbenev (1998)– R. Brinkmann, Ya Derbenev, and K. Floettmann(2001) – A. Burov, S. Nagaitsev, Ya Derbenev ( circular basis)– KJK
Experiment at FNAL A0– D. Edwards, H. Edwards, Ph. Piot,…– Yin-e Sun ( U of C thesis, May 2005)
14KJK, COOL05, 9/18-23/05KJK, COOL05, 9/18-23/05
Schematics of Flat Beam Experiment at FNPL
x = A cos κz
y = A sin κz
A cos κz
A sin (κz+π/2)
flat beamvortex beam
15KJK, COOL05, 9/18-23/05KJK, COOL05, 9/18-23/05
Fringe of Solenoidal Field ProducingKinetic Angular Momentum
16KJK, COOL05, 9/18-23/05KJK, COOL05, 9/18-23/05
Cathode Immersed in Solenoidal FieldMotion in solenoidal field is most conveniently described in a rotating (Larmor) frame
Particle coordinates right after cathode plane
( )sPsqB
dsd
2=
θ
⎟⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜⎜
⎝
⎛
κ+′
κ−′=
⎟⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜⎜
⎝
⎛
′
′
+=yx
yxy
x
yyxx
o
o
s 0
(x´,y´): thermal angular spread
mcqB
o 2=κ
⎥⎦
⎤⎢⎣
⎡β
β=⎥
⎦
⎤⎢⎣
⎡ε
ε=∑ 10
0o
oeff
oeff T,T,JJ,T
L-L
2
222
2 co
coc
c
xyyx σκ=′−′=
σκ+σ′
σ=β
L
17KJK, COOL05, 9/18-23/05KJK, COOL05, 9/18-23/05
After triplet transformation:
ε± = εeff ± L
⎥⎦
⎤⎢⎣
⎡β
β=⎥
⎦
⎤⎢⎣
⎡ε
ε→
±
±±
−−
++∑ 100
00
TT
T
( )222240∑ −ε=ε=+= LeffDsdet
424 thD ε=ε (even if transform before → after cathode surface
is not symplectic)22
theff ε+=ε∴ L
ththeff
eff
th
εεε
εεε
εεε
>>⎟⎟⎠
⎞⎜⎜⎝
⎛≈
−+
=
=∴
−
+
−+
LL
LL
for,22
2
18KJK, COOL05, 9/18-23/05KJK, COOL05, 9/18-23/05
X (mm)
Y (
mm
)
−5 0 5
−5
0
5−5 0 5
−5
0
5−5 0 5
−5
0
5
−5 0 5
−5
0
5−3 0 3
−3
0
3−10 0 10
−10
0
10
X3 X4
X8X7X6
X5−5 0 5
−5
0
5
X (mm)
Y (
mm
)−5 0 5
−5
0
5−5 0 5
−5
0
5
−5 0 5
−5
0
5−3 0 3
−3
0
3−10 0 10
−10
0
10
X3 X4 X5
X6 X7 X8
Removal of angular momentum and generating a flat beam
experiment simulation
X3 X4 X5 X6 X7 X8
N1 S2 S3 S4 N6 N7
350 351
S5
502
cavitybooster
L1 L2 L3
UV laserrf−gun
3770
1854
19KJK, COOL05, 9/18-23/05KJK, COOL05, 9/18-23/05
Applications of Flat BeamsSmith-Purcell FELAlso, image charge undulator (Ya Derbenev)
Compression of x-rays to pico-femtosecondpulses
20KJK, COOL05, 9/18-23/05KJK, COOL05, 9/18-23/05
Smith-Purcell RadiationAn electron beam traveling close and parallel to a metallic reflection grating
)cos1( θββλ
λ −= g m667SP µλ <
*S. J. Smith and E. M. Purcell, Phys. Rev. 92, 1069 (1953)
m667m4942/m3210
µπµπµ
λθ
λg = 173 µm, β = 0.35 (35 keV)
d = 100 µm, w = 62 µm,
b = 10 µm, L = 12.7 mm
Dartmouth parameters**
**J. Urata et al., Phys. Rev. Lett., 80, 516 (1998)
Image charge
Electron
Grating
θx
y z e-
e+
(Assume translational symmetry in y-direction)
21KJK, COOL05, 9/18-23/05KJK, COOL05, 9/18-23/05
Non-Linear Behavior in Smith-Purcell Radiation ? (Dartmouth, PRL 80 (1998) 516-519)
22KJK, COOL05, 9/18-23/05KJK, COOL05, 9/18-23/05
Smith-Purcell Experiment using SEM at U of C, Copying the Dartmouth Set-Up (O. Kap, A. Crew, KJK)
23KJK, COOL05, 9/18-23/05KJK, COOL05, 9/18-23/05
Heated Specimen Stage and Possible Black Body radiation background
24KJK, COOL05, 9/18-23/05KJK, COOL05, 9/18-23/05
Smith-Purcell FEL Theory(V. Kumar and KJK, 2005)
Interaction of e-beam with the surface mode(freely propagating EM mode with phase velocity βc and frequency ω(kz) = c)
< 0 : Group velocity in the opposite direction (C. Brau)
Thus SPFEL is a Backward Wave Oscillator (BWO)Optical energy accumulates exponentially to saturation without feedback mirrors
Start current condition:
ckz
β=ω
zdkdω
( )( )
bA
sat oeL
I.dy
dIdydI Γ
χπλβγ
=≥32
4
277
25KJK, COOL05, 9/18-23/05KJK, COOL05, 9/18-23/05
Simulation Results
0 1 2 3 4 5 6Time (ns)
10−8
10−6
10−4
10−2
100
102
P/∆
y (m
W/µ
m)
(a)
0 4 8 12z (mm)
0
5
10
15
P/∆
y (
mW
/µm
)
(b)
For I/∆y = 50 A/m, at saturation, P/∆y = 13.7 mW/µm
Power e-folding time = 0.2 ns (simulation)
0.17 ns (analytic formula)
Lasing wavelength = 694.5 µm (simulation)
694 µm (analytic formula)
I/∆y = 50 A/m
I/∆y = 36 A/m
After saturation
@ z = 0 Energy conversion efficiency = 0.8%
26KJK, COOL05, 9/18-23/05KJK, COOL05, 9/18-23/05
Smith-Purcell FEL for Terahertz Radiation Requires Flat Beams (KJK & V. Kumar)
Beam distance to grating surface should be
Beam width should be similar to diffraction limit
A set of beam parameters has been worked out satisfying the transverse profile, start current limit, and the requirement that the space charge effect in beam transport
Miniature Terahertz source
πβλ< 4~
µ20~<b
µπ
βλ<∆ 5004
~L~y
27KJK, COOL05, 9/18-23/05KJK, COOL05, 9/18-23/05
Pulse Compression via Transverse-Longitudinal Correlation (A. Zholents,..)
t
yPulse can be slicedor compressed withasymmetric cutcrystal
RF deflecting cavity RF deflecting cavity
(Adapted from A. Zholents' August 30, 2004 presentation at APS Strategic Planning Meeting.)
Cavity frequencyis harmonic h ofring rf frequency
Ideally, second cavityexactly cancels effectof first if phase advanceis n*180 degrees
Radiation fromhead electronsUndulator
Radiation fromtail electrons
28KJK, COOL05, 9/18-23/05KJK, COOL05, 9/18-23/05
Preliminary Optics Concept for 10 keV
29KJK, COOL05, 9/18-23/05KJK, COOL05, 9/18-23/05
X-Ray Pulse Compression at the APSERL with flat beam can achieve femtosecond x-rays (LUX @ LBL)
A modest but significant compression of the x-ray pulses from 100 ps → 1 ps can be achieved at the current APS setting by installing deflection a pair of cavity
Together with the advantage of operating user facility→ Enthusiastic support from APS users
30KJK, COOL05, 9/18-23/05KJK, COOL05, 9/18-23/05
The End
31KJK, COOL05, 9/18-23/05KJK, COOL05, 9/18-23/05
Smith-Purcell BWOe-beam:
EM wave:
phaseenergy
Energy
Bun
chin
gEnergy
Bun
chin
g
• At 690 µm, owing to the negative group velocity of the surface mode, there exists a feedback mechanism even without external resonator.
Backward Wave Oscillator (BWO).
• Hence, the e-beam has strongest interaction around 690 µm.
32KJK, COOL05, 9/18-23/05KJK, COOL05, 9/18-23/05
Squashed SCRF Cavities for the APS
10.6 cm10.6 cmλ
4-cell1-cell
1.8
665 mT
102 W
3 x 109
53 Ω/m
5.3 cm
4 MV
2.81 GHz
1.8Cell aspect ratio
4 MVVT
21.3 cmActive Cavity Length
230 Ω/mRT/Q
3 x 109Q
25 WPL
230 mTBMAX
2.81GHzFrequency
Squashed-cell shape was used to remove TM110 degeneracy.Modeled after KEK design (cell aspect ratio ~ 1.8). Pi-mode chosen for 4-cell cavity to minimize number of cells. Other modes have better frequency separation. 4-cell cavity has 230 mT maximum magnetic field. To ensure BMAX < 100 mT, three 4-cell cavities would be required.1-cell cavity has 665 mT maximum magnetic field and would require seven cavities.4-cell cavity has better RT/Q and will be much more compact than 1-cell cavities.
J. Waldschmidt
21.3 cm
33KJK, COOL05, 9/18-23/05KJK, COOL05, 9/18-23/05
Multiple Temporal Scales in Chemical Sciences
10-16 secLasers
Synchrotron X-rayXFEL
10-1510-1410-1310-1210-1110-1010-910-810-710-610-5Time
Photochemical Isomerization
Energy transfer in photosynthesis
Photoionization
Photodissociation
Electron transfer in photosynthesis
Proton transfer
Torsional dynamics of DNA
Protein internal motion
Electronic relaxation
Vibrational relaxation
Vibrational motion
Electronic dephasing
Solvent relaxation
Molecular rotation
G. R. FlemingChemical Applications of Ultrafast Spectroscopy1986
Probing Transient Molecular Structures in Photochemical Processes Using Pulsed X-rays
Synchrotron X-ray
34KJK, COOL05, 9/18-23/05KJK, COOL05, 9/18-23/05
Compression Results for 10 keV, UAM. Borland
Optics losses not included.Expect >30% optics transmission1
1S. Shastri
6 MV deflection
35KJK, COOL05, 9/18-23/05KJK, COOL05, 9/18-23/05
Minimum Achievable Pulse Length
Electron beamenergy
Deflectingrf voltage &frequency
Unchirped e-beamdivergence (typ.2~3 µ-rad)
Divergence dueto undulator (typ.~5 µ-rad)
Normal APS bunch is 40 ps rms
For 6 MV, 2800MHz (h=8) deflecting system, get ~0.4 ps!
(M. Borland)
2,
2,, radyey
axrayt Vh
E′′ += σσ
ωσ
36KJK, COOL05, 9/18-23/05KJK, COOL05, 9/18-23/05
Emittance Requirements for X-Ray FELsRequirements for x-ray FELs
γ εx, γ εy 0.1 × 10-6 m-rad
However, current state-of-the-art: γ εx ~ 1 × 10-6 m-rad δ γ/γ 10-6 (mc2 δγ = 2.5 keV)
Can phase space areas be exchanged?
~<
410−<γ
δγ~
~<
( ) ( ) ?10mradm1010mradm10 4276262 −−−− ⊗−→⊗−=γ
δγ⊗εεγ yx
37KJK, COOL05, 9/18-23/05KJK, COOL05, 9/18-23/05
20 ceBL σ=
22
21 reBmrL z+= φγ &
On the photocathode:
Production of Angular Momentum Dominated Beam
1 1/2 cellRF gun
buckingsolenoid
prim arysolenoid
RF coupler
secondarysolenoid
cathodeBPM
FNPL 1.625-cell RF gun, 1.3 GHz0 20 40 60 80 100
0
500
1000
1500
Z (cm)
Bz (G
auss
)