Study of transverse effects in a back-scattering coherent ...
Transcript of 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.
SPARC-PLASMONX Erice 9-14/10/05
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.
SPARC-PLASMONX Erice 9-14/10/05
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
SPARC-PLASMONX Erice 9-14/10/05
)/(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
SPARC-PLASMONX Erice 9-14/10/05
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
SPARC-PLASMONX Erice 9-14/10/05
∑ −=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 =
SPARC-PLASMONX Erice 9-14/10/05
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
SPARC-PLASMONX Erice 9-14/10/05
3-d code
Fourth order RKG for the particles
Explicit finite differences scheme forthe Schroedinger equation
SPARC-PLASMONX Erice 9-14/10/05
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
SPARC-PLASMONX Erice 9-14/10/05
-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
SPARC-PLASMONX Erice 9-14/10/05
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
SPARC-PLASMONX Erice 9-14/10/05
-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
SPARC-PLASMONX Erice 9-14/10/05
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
SPARC-PLASMONX Erice 9-14/10/05
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.