study of transverse effects in a back-scattering coherent

Study of transverse effects in a back-scatteringcoherent Thomson source of X-rays

A.Bacci, M.Ferrario*,C. Maroli, V.Petrillo, A.Rossi, L.Serafini

Università e Sezione I.N.F.N. di Milano (Italy)*LNF, Frascati (Italy)

SPARC-PLASMONX Erice 9-14/10/05

Thomson back-scattering

X rays

Laser pulse

Electron beam

-10 -5 0 5 10

-10

-5

0

5

10

The incoherent linear and non linear radiation at ω=4γ2ωL is usuallyevaluated by calculatingthe emitted intensity by each single electron and then summing allcontributions at the collector.


If the laser pulse is long enough, collective effects can develop.The system electron beam + laser pulse behaves like a free-electronlaser with an electromagnetic wiggler.

In particular, if the time duration of the laser pulse ΔTL is larger than a fewgain lengths, i.e. if

ΔTL> (10) Lg/c

the electron of the beam can bunch and the f.e.l. instability can develop.

The coherent radiation is expected to have a spectrum bandwidth verymuch narrower than the incoherent radiation, a less broad angular distributionand (if the saturation is reached) a larger intensity.


J. Gea-Banacloche, G. T. Moore, R.R. Schlicher, M. O. Scully, H. Walther, IEEE Journal of Quantum Electronics, QE-23, 1558(1987).B.G.Danly, G.Bekefi, R.C.Davidson, R.J.Temkin,T.M.Tran,J.S.Wurtele, IEEE Journ. of Quantum Electronics, QE-23,103(1987).Gallardo, J.C., Fernow, R.C., Palmer, R., C. Pellegrini, IEEE Journal of Quantum Electronics 24, 1557-66 1988.

To evaluate the collective effects:

The field or potential (instead of the intensities) must be calculated and summed at the collector, taking into account possible interferences

In the trajectories of the electrons, the collective fields must be taken into account


)/(ˆ)()( )(T

tkzi LOccexyztAxyzt λω ++= − eA

3-d equations

single mode treatment

Slowly Varying Envelope Approximation

Averaged on radiation and laser wavelengths

Space charge effects neglected

Electrons modelled with macroparticles

Relativistic equations in the lab frame


Laser system:

Laser pulse characteristics:

wavelength λ=0,8 µm, power 1TW, timeduration T=5 psCircular polarization, focal spot diameterw0 >20 micron

z0 = πw02/4λL > 2,5 mm Rayleigh length

)()ˆ),((2

),(0

)(0

wOccetg

at LtzkiL

LLL

λω ++= +− errA

+

+−

+

+−

+

+

+Φ=)(

4

)1(

4exp

1

1

)(),(0

0

20

22

20

220

22

20

20

z

z

z

zw

yxi

z

zw

yx

z

z

z

zi

ctztrg

Guided pulse: g(r,t) step functionGaussian pulse:

)(

)()(

t

tt

td

d

j

jj γ

ρP

r =

Laser p

onderomotive

forces

[ ] ....)(Re2

||1

2)(

)(

220

20

+−

∂

∂−=

=∗

=

ti

j

j

Ljz

j

j

eAgal

gz

atP

td

d

θ

γ

γργ

jrx

rx

[ ]

[ ] ....))((Im1

1

4

||1

2)(

)(

220

20

+∇+

−

∇−=

=∗

⊥

=⊥⊥

ti

jL

j

Lj

j

j

eAg

kk

ga

ttd

d

θ

γη

γργ

jrx

rxP

bAk

kitA

ztL =∇−

∂

∂+

∂

∂⊥2),()( ρx

Electron equatio

ns

Radiation equatio

nColle

ctive

ponderomotive

effects


∑ −=s

ti

s

s

s

set

ttg

Nb )(

)(

)),((1 θ

γ

r

t2t Lρω=

xx Lk2ρ=Ai

mc

eA

R

b

−=

ω

γρω

22

3

1

2L

L20L

2b

0 16

)k

k1(a1

+

=ω

ω

γρ

.

20

20

1

4

L

L

a+≈

ωγω

.

Bunching factor

Normalization

Resonancecondition

)t)1k

k()t(z)

k

k1((

k2

k)t( L

jL

Lj −++=

ρθ

...)|(|1 )(22

0222

02 +++= = tLjzj j

gaP rxργγ

0jj / γγγ = Pj = pj/γ0ρ 0L20L amc

ea =


Laser pulse: time duration up to 5 psec, power 1-3 TW, varying w0, λ=0,8-1 micron

Electron beam counterpropagating respect the laser pulseQ=1nC, Lb=100-300micron, radius σ0=10-20 micron, I=1-2,5 KA Energy=15 MeV (γ=30) , transverse norm emittance up to 3 mm mrad, δγ/γ=10-4.

ρ=5 10-4 gain length Lg= 100-150 micron

Radiation λ=3,5 Ang ZR=1-4m

ρbar=2 no quantum effects


3-d code

Fourth order RKG for the particles

Explicit finite differences scheme forthe Schroedinger equation


Start from noise

As usually for three-d codes, it is time-consuming

0 1 2 3 4 5 6

1E-4

0,01

1

|A|2

t(psec)

0 1 2 3 4 5 6

1E-4

0,01

|b|

t(psec)

εn=0.45, guided pulse εn=1.13, w0=500micr

0 2 4 6

10-6

10-3

1

|A|2

t(psec)

0 2 4 6

10-3

10-1

<|b|>

|A|2sat=1.4 in 10 Lg (5psec) |A|2sat=0,12 in 5psec


-4 -3 -2 -1 0 1 2 3 426

27

28

29

30

31

32

33

-4 -3 -2 -1 0 1 2 3 429,974

29,976

29,978

29,980

29,982

29,984

-4 -3 -2 -1 0 1 2 3 4

29,976

29,977

29,978

29,979

29,980

29,981

29,982

-4 -3 -2 -1 0 1 2 3 4

29,97

29,98

29,99

-4 -3 -2 -1 0 1 2 3 4

29,94

29,95

29,96

29,97

29,98

29,99

30,00

-4 -3 -2 -1 0 1 2 3 4

29,93

29,94

29,95

29,96

29,97

29,98

29,99

30,00

X Axis Title

F

X Axis Title

F

X Axis Title

F

X Axis Title

F

F F

-3 0 329,970

29,972

29,974

29,976

29,978

29,980

29,982

29,984

-3 0 329,970

29,975

29,980

29,985

-3 0 329,970

29,975

29,980

29,985

-3 0 329,970

29,975

29,980

29,985

-3 0 3

29,94

29,95

29,96

29,97

29,98

29,99

30,00

t=1,2 ps 2,4 ps

3,6 ps 4,8 ps

Pz

theta

5 ps

-3 0 3

29,94

29,95

29,96

29,97

29,98

29,99

30,00

6,2 ps

Phase space pz versus the phase angle theta

Ideal case More realistic case


Lethargy phase

Exponential growth90 degree rotation inphase\space

SaturationPost saturationphase

-0,002 0,000 0,002 0,0040,0

0,5

1,0

1,5

ε=1.11

ε=0

|A|2max

δω/ω

Radiation spectrum w0=1000


-40

-20

0

20

40

(a)

1E-5

5,1E-4

0,001010

0,001510

0,002010

0,002510

0,003010

0,003510

0,004000

y

(b)

5E-5

0,005050

0,01005

0,01505

0,02005

0,02505

0,03005

0,03505

0,04000

-40 -20 0 20 40-40

-20

0

20

40

(c)

1E-4

0,02760

0,05510

0,08260

0,1101

0,1376

0,1651

0,1926

0,2200

x

y

-40 -20 0 20 40

(d)

1E-4

0,05010

0,1001

0,1501

0,2001

0,2501

0,3001

0,3501

0,4000

x

t=1 ps t=2psec

t=3ps t=4ps

Transverse radiation intensityfor emittance=1.8

Initially more chaotic, then smoother


W0=1000, emitt=1,11 δγ/γ=10-4

24

68

10

0,000

0,001

0,002

0,003

0,004

0,005

0,006

0,007

0,008

2

4

6

8

10

Z A

xis

Y Axis

X Axis

24

68

10

0,00

0,02

0,04

0,06

0,08

0,10

0,12

0,14

0,16

2

4

6

8

10

Z A

xis

Y Axis

X Axis

24

68

10

0,0

0,2

0,4

0,6

0,8

1,0

2

4

6

8

10

Z A

xis

Y Axis

X Axis

0 1 20,0

0,5

1,0

w0=200

w0=4000 µm

w0=400 µm

|A|2sat

εn (mm mrad)

Saturation intensity value (averaged on the transverse section) versusthe transverse normalized emittance for different w0

We have considerable emission also inviolation of the Pellegrini criterion for astatic wiggler. In fact, the emittancesconsidered largely exceed the valueγλ/4π, that in this case is 8,5 10-4 micron.On the other hand, on the basis of thefact that Lg/ZR=1.2 10-4, the criterion ofPellegrini can be rewritten in ageneralized form for both static andoptical undulators as

where α=

and gives εn<0.25

πγλαε 4// RgRN LZ≤

)/(ωρωd


Conclusions

The growth of collective effects in the back scattering Thomson process is possible provided that:

A low-energy , high-brigthness electron beam is availablewith short gain length

The optical laser pulse is long enought to permit the bunching and theinstauration of the instability.

In the interaction region the laser transverse and longitudinal profiles areflat.

study of transverse effects in a back-scattering coherent

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