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

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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

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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

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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

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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)

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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

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Fringe of Solenoidal Field ProducingKinetic Angular Momentum

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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

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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

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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)

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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)

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Heated Specimen Stage and Possible Black Body radiation background

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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

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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%

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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

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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

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Preliminary Optics Concept for 10 keV

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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

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The End

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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.

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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

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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

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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

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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

)