Estimate of Free-Electron Laser Gain Length in the ...accelconf.web.cern.ch/AccelConf/FEL2015/talks/moc03_talk.pdf · FEL Conf., Daejeon, 08/2015 [email protected] 1 Estimate

FEL Conf., Daejeon, 08/2015 1 [email protected]

Estimate of FEL Gain Length in the Presence of

e-beam Collective Effects

S. Di Mitri1, S. Spampinati2

1Elettra Sincrotrone Trieste 2Univ. of Liverpool, Cockroft Inst.

FEL Conf., Daejeon, 08/2015 [email protected] 2

Problem

Collective effects (Coherent Synchrotron Radiation, Geometric Transverse Wakefield)

“misalign” bunch slices in the transverse phase space: proj is increased

albeit slice may be not, whereas Lslice Lcoop << Lbunch.

Picture courtesy D.Douglas [1] Guetg et al. PRSTAB 18, 2015


Problem




This is the picture we have in mind:



Problem





iproj ,


iprojfproj ,,


Problem





iproj ,


Can we analytically

relate LG to proj ?

Simulations show an impact on SASE FEL output[1]:

iprojfproj ,,


Single Kick Error (Dipole-like)

Consider a bunch subjected to dipole-like kicks in the undulator[2],

e.g. due to steering magnets or misaligned quadrupoles.

[2] Tanaka et al. NIMA 2004 [3] Chae et al., FEL 2004

Microbunched e-beam

Bunching wavefront

After kick, SKE

1. Lack of photons/electrons overlap: the undulator spontaneous radiation does not

sustain efficiently the coherent emission,

2. Smearing of microbunching: the kick enhances the arrival time difference

of electrons belonging to the same

wavefront, so spoiling the phase coherence,

22, 1 cSKE

GSKEG

LL

,1 22,

cSKE

GSKEG

LL

Gc L






Microbunched e-beam

Bunching wavefront

After kick, SKE






22, 1 cSKE

GSKEG

LL

,1 22,

cSKE

GSKEG

LL

Gc L

The effect of angular divergence (“de-bunching”) is far more important than the lack of transverse overlap (“sustainment”).






Microbunched e-beam

Bunching wavefront

After kick, SKE






22, 1 cSKE

GSKEG

LL

,1 22,

cSKE

GSKEG

LL

Gc L

The effect of angular divergence (“de-bunching”) is far more important than the lack of transverse overlap (“sustainment”).

Experiment at LEUTL[3]:

t-dependent simulation

theory


We now consider error kicks that affect individual slices, e.g. from

CSR in a dipole, and from GTW in an RF cavity.

0,

2

0,0,

20,1,

'1'det

x

xx

xxx

xxxx

xx

Projected Emittance Growth

The “-matrix” provides an RMS estimate of proj induced by those

perturbations. For a single angular error (x’):

Consider the uncorrelated sum of m-consecutive CSR kicks in magnetic

compressors and GTW kicks in the linac. The resultant projected

emittance, e.g. at the undulator, turns out to be:

Twiss functions are at the location of the perturbation

...11

1,0,,

m

iin

infn P


We now consider error kicks that affect individual slices, e.g. from

CSR in a dipole, and from GTW in an RF cavity.

0,

2

0,0,

20,1,

'1'det

x

xx

xxx

xxxx

xx

RMS kick averaged over all the slices.

Projected Emittance Growth

The “-matrix” provides an RMS estimate of proj induced by those

perturbations. For a single angular error (x’):

Consider the uncorrelated sum of m-consecutive CSR kicks in magnetic

compressors and GTW kicks in the linac. The resultant projected

emittance, e.g. at the undulator, turns out to be:

Twiss functions are at the location of the perturbation

...11

1,0,,

m

iin

infn P

We will evaluate

this later.

...11

1,,,

m

iin

iinfn P


Collective Angle

...11

1,,,

m

iin

iinfn P


Collective Angle

in

colluin

,

2

, 1

...11

1,,,

m

iin

iinfn P


Collective Angle

This is solely determined by the linac dynamics.

in

colluin

,

2

, 1

This is the resultant angular spread of the bunch slices’ centroids.

This is <> in the undulator.

...11

1,,,

m

iin

iinfn P


Collective Angle

02 coll

In the UNDULATOR, if the beam is matched to a SMALL u, the slices are largely dispersed in angle:

02 coll

If the beam is matched to a LARGE u, the slices overlap in angle:

AFTER the kick(s), the slices follow different trajectories in phase space, but the projected emittance is preserved:

ctecoll 2

We expect a BIG impact on FEL gain

We expect a SMALL impact on FEL gain

This is solely determined by the linac dynamics.

In the LINAC, two slices are displaced along the direction of the kick.

The slice emittance is unperturbed, while the projected is enlarged.

in

colluin

,

2

, 1

This is the resultant angular spread of the bunch slices’ centroids.

This is <> in the undulator.


3-D Gain Length

in

colluinfn

,

2

,, 1

22, 1 cSKE

GSKEG

LL

)],(1[ ,3, yxGDG LL

DGth L 3, ,1 22

3,,

thcoll

DGcollG

LL

We propose[7]:

[7] Di Mitri, Spampinati, PRSTAB 17, 2014

Single-slice dynamics

De-bunching

Projected dynamics


3-D Gain Length

in

colluinfn

,

2

,, 1

22, 1 cSKE

GSKEG

LL

)],(1[ ,3, yxGDG LL

DGth L 3, ,1 22

3,,

thcoll

DGcollG

LL

We propose[7]:



De-bunching

Projected dynamics

This depicts the SLICE dynamics

This depicts the PROJECTED dynamics


3-D Gain Length

in

colluinfn

,

2

,, 1

22, 1 cSKE

GSKEG

LL

)],(1[ ,3, yxGDG LL

DGth L 3, ,1 22

3,,

thcoll

DGcollG

LL

We propose[7]:



De-bunching

Projected dynamics

Theory vs. GENESIS simulations: 1) --- n,proj = n,slice = 0.5 m

2) --- n,proj = n,slice = 2.3 m

3) --- n,proj = 2.3 m > n,slice = 0.5 m

Intuitively, we expect LG of 3)

in between that of 1) and 2);

confirmed by simulations.

u := (<x><y>)1/2; the scan spans

different scenarios of radiation

diffraction.

E = 1.8 GeV

I= 3.0 kA

u = 2 cm

K=√2 = 0.1%

FEL = 1.6 nm

This depicts the SLICE dynamics

This depicts the PROJECTED dynamics


Comparison with M.Xie-LG

FERMI-like linac (S-band, 1.4 GeV, 0.5 kA, 10 nm), one-stage compression. C

hoi

ce o

f th

e

Lin

ac w

orki

ng p

oint


Low gain due to low I

proj due to CSR

proj due to GTW E C

proj limits Psat before E starts doing it




hoi

ce o

f th

e

Lin

ac w

orki

ng p

oint



proj due to CSR

proj due to GTW E C





hoi

ce o

f th

e

Lin

ac w

orki

ng p

oint


CSR strength

GTW

str

en

gth

FE

L S

ens

itiv

ity

to L

inac

sett

ing

CSR strength

GTW

str

en

gth

CSR strength

GTW

str

en

gth

Bn,6D

LG,coll Psat,coll


proj due to CSR

proj due to GTW E C





hoi

ce o

f th

e

Lin

ac w

orki

ng p

oint


CSR strength

GTW

str

en

gth

FE

L S

ens

itiv

ity

to L

inac

sett

ing

CSR strength

GTW

str

en

gth

CSR strength

GTW

str

en

gth

Bn,6D

LG,coll Psat,coll


proj due to CSR

proj due to GTW E C





hoi

ce o

f th

e

Lin

ac w

orki

ng p

oint


5-D BRIGHTNESS vs. COMPRESSION FACTOR


Collective Effects

tail head

CSR in a 4-Dipole Compressor[4]: GTW in RF cavities[5]:

yx

Din

fnyfnx

Dfn

CBIB

5,

,,

5,

[4] Dohlus, Emma, Limberg ICFA 38, 2005 [5] Raubenheimer PRSTAB 3, 2000 [6] Di Mitri, Cornacchia Phys. Rep. 539, 2014

head

tail

The effect of CSR and GTW on proj has opposite dependence on z.

The beam brightness[6] may show a local maximum as a function of the final z.

m

j

jGTW

n

i

iCSRin

BCCSR

LGTWinfn PPPP

11,

11,, 1...11

212

0

222

0,

221 zfx

totFODOzexGTWx

LLWQceZr

34

2,

2

,11zx

CSRxCSRx

Din

fnyfnx

Dfn CBIB 5

,,,

5,


Conclusions

LG,coll aims to include the beam projected dynamics. A deviation 10%

was found vs. 3-D time-dependent simulations, over a wide range of u.

When proj > slice, a considerable deviation from M.Xie-LG appears.

This suggests a larger u for optimum FEL performance.

The model can be used either for tuning of the accelerator in order

to maximize the FEL performance, or for specifying the FEL tolerance

on the beam projected emittance, vs. the undulator optics.

Further numerical studies will assess the limits of the proposed model:

<2coll> neglects correlations between consecutive CSR and GTW kicks.

the “-factor” in LG,coll might actually depend on e-beam parameters.

Psat and Lsat were re-scaled to the 1-D model.


Acknowledgements

A special thank to my collaborator, S. Spampinati

W. Fawley and E. Allaria are acknowledged for instructive

discussions and suggestions.

This work was funded by the FERMI project of Elettra Sincrotrone

Trieste.

Thank You for Your attention

Documents

Estimate of Free-Electron Laser Gain Length in the ...accelconf.web.cern.ch/AccelConf/FEL2015/talks/moc03_talk.pdf · FEL Conf., Daejeon, 08/2015 [email protected] 1 Estimate