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Acceleration and Focusing by superstronglaser fields and their applications
to a laser micro-collider
Kazuhisa Nakajima
2nd ORION WORKSHOPSLAC, Feb. 18-20, 2003
KEK/JAERI
Outline
Laser acceleration and focusing mechanisms
Electron-positron micro-collider concept
Which is better for high energy acceleration,plasma wakefields or vacuum laser fields?
Necessity for high energy
acceleration mechanism
Effective constituent collision energy ofColliders (Physics Today, Jan. 2001)
1970 19801990 2000 20100
1
10
100
1,000
10,000
Colli
sion
Ene
rgy
(GeV
)Prin-Stan, VEPPII, ACO
ADONESPEAR, DORIS, VEPPIII
SPEARIIVEPPIVCESR
PETRA, PEP
TRISTANSLC, LEP
LEPIILC500
ISR
SppSTevatron I
LHC
e+e- colliders
Hadron colliders
RHIC
Energy frontier Accelerators
“The realization that we seem to be near the end of the road forconventional accelerators
has generated renewed interest in thepossibility of accelerators with
super-high accelerating fields.”
The Laser Acceleration of Particles Workshop, 1982, Los Alamos
(AIP Conference Proceedings No. 91, 1982)
To keep enregy frontier trend
Accelerating field limits
100cm 10cm 1cm 1mm 100m 10m 1m 0.1m1 MV/m
10 MV/m
100 MV/m
1 GV/m
10 GV/m
100 GV/m
1 TV/m
200 MHzproton linac
SLAC (SLC)
NLC
CLIC
3GHz 300 GHz30GHz300MHz 3THz 30THz 300THz 3PHz
W-band linac
LWFA
SM-LWFA
DLASWA
Wavelength
Frequency
Surface heating limit
Break
down
limit
Plasma growth limit
Plasma w
ave
breakin
g
Ele
ctri
c fi
eld
gra
die
nt
RF accelerators
Laser-Plasmaaccelerators
R.B. Palmer, AIP Conf. Proc. 91, 179-189 (1982)
1 EV/m Vacuum breakdown
Is plasma wakefield a high energyacceleration mechanism?
• The plasma wave-breaking causes background electron trapping. • The accelerating field should be less than the plasma wave-breaking limit.• The energy gain is limited by dephasing even ifthe optical guiding is accomplished.• Beam-plasma instabilities breaks high qualitybeam acceleration.• Field control is difficult.• High energy background particle and photon radiations
Many disadvantages kill a role of high energy accelerators.
The maximum accelerating energy
max w
1 p
eEmax
mc2 4 p2 eEmax
mc2kw
The maximum energy gain of particles trapped in the electricfield Emax of which phase velocity vp is given by
where w is the wavelength of the electric field,
kw 2 w p vp c p 1
1 p2
Assuming the maximum electrostatic potential eEmax
mc2kw
1
max 4 p2
max 1
eEmax
mc2 Lacc for the acceleration length Lacc cw
c vp
Super-high energy particle acceleration in plasma
The phase velocity of the electron plasma wave is given by
p vp
c 1
ne
nc
p nc
ne
1 2
where nc is the critical plasma density,nc cm 3
re0
2 1.11 10210 2 m
re e2
mc2 2.818 10 13 cm with the classical electron radius
The maximum energy gain is approximately given by
max 4 p2 4
nc
ne
4 1021
02 m ne cm 3
The acceleration length is Lacc p
1 p
2 p2p
p 2kp
reneWith the plasma wavelength
Lacc cm 2nc
ne
rene
2
02
rene
3 2
7.4 1027
02 m ne
3 2 cm 3
1015 1016 1017 1018 1019 1020 1021
106
102
104
10 m
11 cm
1 m
100 m
10 km
Ene
rgy
gain
Acce
lera
tion
leng
th L
acc
A single stage energy gain
Plasma density ne (cm-3)
max
Lacc
A naive estimate of 2.5 TeV electron linac
Consider a laser driver of
5 106
For 2.5 TeV electron acceleration,
Necessary plasma density is given by ne 8 1014 cm 3
Necessary acceleration length becomes Lacc ~ 3.3 km
the required energy gain is
0 1m
The required accelerating gradient is 0.76 GeV/m
The accelerator length will be the order of 1 km even if the multi-staging can be accomplished.
It is not so attractive compared to the conventional scheme.
Laser acceleration fieldThe laser fields are written as E
A
ct, B A
in terms of the vector potential A of the laser field.
For a linearly polarized laser field, A A0 cos kz t e
Defining the normalized vector potential as a eA
mec2
the laser strength parameter is given by
where I is the laser peak intensity, and
is the laser wavelength with frequency
a0 2e20
2 I
me2c5
1 2
0.855 10 9 I1 2 W/cm2 0 m
0 2c 0 0
The amplitude of the transverse electric field of linealy polarized laser is
EL TV m mec2k
ea0 3.21
a0
m 2.7 10 9 I1 2 W cm 2
e.g. I =1×1018 W/cm2 gives EL =2.7 TV/m
Lawson-Woodward TheoremThe net energy gain of a relativistic electron interacting with an electromagnetic field in vacuum is zero.
The theorem assumes that (i) the laser field is in vacuum with no walls or boundaries present, (ii) the electron is highly relativistic (v ≈ c) along the acceleration path,(iii) no static electric or magnetic fields are present,(iv) the region of interaction is infinite,(v) ponderomotive effects (nonlinear forces, e.g. v x B force) are neglected.
(J.D. Lawson, IEEE Trans. Nucl. Sci. NS-26, 4217, 1979)
Acceleration mechanism must violate the Lawson-Woodward theorem.Usually it is not difficult!
Acceleration by Higher-Order Gaussian ModesThe x-polarized Hermite-Gaussian TEM10 mode is given by
Ex 4E0w0 x
2w 2 z exp
r2
w2 z
sin
Ez 4E0w0
2kw2 z exp
r2
w 2 z
1
2x2
w2 z
cos
2zx 2
Z Rw2 z sin
-3 -2 -1 1 2 3
-1
-0.5
0.5
1 Ex
Ez
x w0
TEM10
w0 z 0
Ex,z
E0
kz t zr 2
Z Rw2 z 2tan 1 z
ZR
0where w z w0 1
z2
ZR2
1 2
E0 2
w0
P
cP
c
4E0
2w02
On the z-axis Ex=0 and the axial field accelerating an electron with vz ≈ cEz
4E0w0
2kw2 z cos z
z kdz
2z2 2 tan 1 z
ZR
0 z 1
vz2
c 2
1 2where
The phase velocity of the accelerating field is
v ph c 1 d2 1 z 2 1
d 2 kZR 1 2 w0where is the divergence angle.
Near the focal point z Z R v ph 1d2 11 2c
2 c 2d2 1 2
The slippage distance is given by Zs ZR / 2
1c2 z
2
For z c Zs ZR
2The particle injection energy for a finite interaction length is given by
Einj m0c2 m0 c2
4eEz0 k
eEz0
kTinj m0c 2 m0c 2
4 d2 Z0 P 2
1
d
2 Z0 P
2
Z0 377where
The maximum energy gain is given by
W MeV eEz0 ZR 2e 2 P c 2 Z0 P TW 15.5 P TW
1m,w0 10m, P 10TWe.g.
Tinj 2.1MeV
For
W 49MeV
Vacuum Beat Wave Accelerator
(E. Esarey, PRE 52, 5443,1995)
Two laser beams of different frequencies cause an axial acceleration from the beat term in the v×B force.
dp
dt e E v B / c E A
c tB A
Using u p mc a eA / mc2
1 u2 1 2 1 2 1 2
dudt
a t
cu a d
dt u
a t
u a
0.5 1 1.5 2 2.5 3
-1
-0.5
0.5
1Beat-Wave
t
A A1 A2
FzFz Fz
The axial component of the nonlinear ponderomotive force is
Fz e
cv B z
mc 2
2z
a a
The total laser field is A z,r, t A 1 A 2 A1, A2; the vector potential of laser 1 and 2.
a2 a a a12 a2
2 2a1a2 cos 2 1 az2 ai a0iw0 i / wi exp r2 / wi
2
The energy equation along the axis isddz
a01a02
z
ksin 02 01
Along the axis r=0 and near the focus of the two lasers
v ph c 1 1 Z R1 ZR2 / kZ R1
v ph c for ZR2 ZR1
In an infinite interaction region zI , zF
F2 I
2 2a01a02kZR
-3 -2 -1 1 2 3
-3
-2
-1
1
2
3
z ZR1
w
w0w1
w2
1 2 2
The accelerated electron energy for the injection energy and WIP1 P2
WF MeV WI MeV 2 750 1 2 1 P1 TW 1 2
e.g.WI 20MeV
1 2 2 1 m
P 10TW
w01 1
WF 89MeV
W 70MeV
Energy gain
Effect of Radiative Losses
An attainable maximum energy is limited by radiative losses due to transverse quiver oscillations.
PR2
3remc3 2a0
2 k 2 1
2z2
1
kZR
2
ree2 / mc 2; the classical electron radius
The electron energy equation with radiative loss is
ddz
a01
a02
k 2
3re
2 a012
ZR12
a 02
2
ZR22
Assuming k k1, a01 a02 , ZR1 Z R2
the maximum electron energy is given by
Wmax mc2 w01 1 3w01 / re 1 3
Wmax GeV 1.3 w014 3 m 1 m
e.g.
Wmax 1.3GeV for w01 1 22 1m
50 100 150 200
500
1000
1500
2000
2500
z cm
Electron Energy vs Distance
WI 100MeV
a0 1
w01 1 22 1m
Ponderomotive acceleration
E ˆ x E0 sin, B ˆ y E0
csin t t
z t c
d
dt x
eE0
mc1 z sin
d
dt z
eE0
mcx sin
d
dt y 0
d
dt
eE0
mcx sin
y const 0 1 z const 0 1 0
0 1 a02 cos 1 2 1 0
2
x 2a0 cos 1
0 2 a02 1 0 cos 1 2
z 1 2 1 0
0 2 a02 10 cos 1 2
a0 eE0
mc
eE02mc 2
(F.V. Hartemann et al.,PR E,51,4833,1995)
Energy-momentum equations of electrons in a linearly polarized plane waveddt
e
mc2 vE
where v
c,
1
1 2, p mc
dp
dt e E v B
z ca0
1 0 0 3
4a0
2 1 0
a0
2 10 sin28
sin
x ca0 sin 0 1 0
The maximum energy at
max 0 1 2a02 10 0 1
e2 I2
20 m2c5 1 0
E0
2 I
0c
2n 1
The transverse and longitudinal positions are
Electron energy Electron trajectory
0.5 1 1.5 2 2.5 3
1000
2000
3000
4000
2
a0 10 0 10
20000 40000 60000 80000
-600
-500
-400
-300
-200
-100
z
x
a0 10 0 10
Energy and trajectory of an electron
with
100 200 300 400 500
2.5
5
7.5
10
12.5
15
17.5
20
0 1
0 2
0 5 0 10
Scattering angle
•The quiver amplitude becomes comparable to the beam waist in the high field.•The electron can be scattered away with a high energy as the interaction is terminated within a wavelength.
Threshold power given by
Pmax 5
02
0 m2c5
e2
w0
4
0.21
02
w0
4
TW
Scattering angle
x
z
2
2 1
z 2 1 1 z 0 1 0
1 2 1 x2 z
2
arctan
2
1 0
0
1
1 2
0 1 0
x
arctan x
zz
0
arctan2
1 0 10 0
Ponderomotive scattering
x
0a0 w0
a0th w0
0
0At 0 No acceleration
Is vacuum laser acceleration possible to accelerate particles up to super high energy?
2 2a0w0
0
Consider energy gain of laser acceleration in vacuum.
Higher Order Gaussian Mode Acceleration
Laser Beat Wave Acceleration
2 3 2a0w0
0
assuming 0 1 22
max w0
0
3w0
re
1 3
Radiation loss limit
Poderomotive scattering for 0 ~1
40a02
with a0 w0
001 10 100 1000
103
105
107
101
a0
RadiationLoss limit
HGMA
LBWA
PMSA
Energy gain as a function oflaser intensity
(HGMA)
(LBWA)
(PMSA)
w0 0 10
0 100
Direct laser-particle interaction
A
A
a eA
mc2p e
cA
P ~ mcaa 1
a 1Pz ~ mca2
~1
a2
Non-relativistic interaction
Relativistic interaction
is the normalized vector potential.
0
a2
where W
mc2 for the energy gain W.
Reflection of electromagnetic wave on an electron plasma
2 k 2c2 p2
p 4nee
2
mp
Photon
ne
p
The reflection condition of the electromagnetic wave (photon) is given by
Electromagnetic wave
m ph
p
c2
Photon-’Electron Plasma’ Interaction
Electron Plasma
The dispersion relation of the electromagnetic wave in plasmawith electron density ne is
This means that a photon inside plasma has an effective mass:
where
a
Electron
de Broglie wave
ph ,p
a mec2
a
1 a 2 1a0
2
2
1 2
Reflection of an electron on an intense photon field
Electron-’Photon Plasma’ Interaction
Photon plasma
In an analogy to reflection of electromagnetic wave fromthe electron plasma, an electron in an intense laser field has a mass shift
Considering a laser pulse as a ‘plasma’ of photons, the photon plasma frequency is
me2 a 2me
2 1
2a0
2me2
An electron de Broglie wave will be reflected for its frequency
e
mec2
ph, p
a mec2
i.e.
An electrom with momentum is expelled in the condition:
Particle acceleration in super-strong laser fields in plasma
Laser field and electron space charge field can be given by the vector potential A and the scaler potential
E 1
c
A
t B A
The equation of a particle motion is written as
dp
dt eE
v
cB
e
1
c
A
t
v
c A d
dt
t
v
The equation of energy isd
dtmc2 ev
1
c
A
t
with
Basic equations
The Maxwell’s equations are1
c2 2
t2 2
A
4c
j 1
c
t
A
1
c
A
t
4
is the charge density, en
j envwhere j is the current density,
The continuity equation of plasma density n isn
tnv 0
assuming the plasma is an ensemble of cold fluids.
1D field-electron interaction in plasma
A Axe x Aye yA Axe x
for circular polarization,
for linear polarization.
In a 1D laser field according to the Coulomb gauge Az 0
A 0letting
the vector potential is expressed as
Consider the laser pulse frame propagating at a group velocity in plasma
vg k
c 1 p
2
02 p
4nee2
mwhere
is the plasma frequency for the ambient electron density ne.
Transforming the momentum equation to a new variable,
z vgt
d
dp
e
cA
0
p e
cA
This gives conservation of canonical transverse momentum.
The electron quiver velocity is given by
a eA
mc2
v pm
e
mcA
a
From the transverse component of the equation,
Defining the normalized vector potential
1 2 1 2with the Lorentz factor
d
dpz
mc2
vg
e
vg
0
g vg
c
emc2
assuming that prior to laser interactiona 0 0 , z 0
gz const. 0 1 g0 where
The longitudinal component of the energy equation becomes
This gives
the solutions on and are given by
z g
2 H0 H02 1 g
2 2 1 2
H02 g
22
a
g2 H0 gg g
2H02
2 1 2
with H0 0 1 g0
1 a2Using 1 u
2 uz2 z where z
1
1 z2
H02 g
22
H0 g H02 1 g
2 2 1 2
The space-charge potential is obtained from
(Continuity equation)1
c
n
t n 0
2 kp2 n ne 1 (Poisson's equation)
where n is the electron density, k p p c
In an initial equilibrium (prior to the laser pulse), the space-charge potential is negligible, i.e. =0
Space-charge potential
g 1For
d2d 2
ˆ k p2
2
1 a2
1 2 1
0 1 0 ˆ k p
kp
03 2 1 0
letting
(Nonlinear wake equation)
Assuming n n d
dn g z 0 n ne
g 0
g z
In the short-pulse limit L ckp 1
where L is a laser pulse duration.
1
In the long-pulse limit L ckp 1
1 a2 s
1 2 1
1 a2 s
1 2 1 a0
2 2 1 2
is nearly constant.
L0 cL p
2
L0 cL p
2
The averaging over a laser wavelength,
In the short-pulse limit 0 g 1let for vacuum
0 11 0
2a2
s
0 11 0
4a0
2
1
20a0
2
for0 1
z 0
1 0
2a2
11 0
2a2
0
1 0
4a0
2
11 0
4a0
2
a0 1
1
In the short-pulse limit ≈ 0
g20 1 g0 gg g
202 1 g0 2
2 1 2
g20 1 g0 gg g
202 g 0 2
a2 1 2
the maximum and minimum energies are
g0 1 g0 For
max g20 1 g0 gg g
202 g 0 2
a02 1 2
min 0
max g20 g 0 2
0 2
min 0
g0 1 g0 For
Scattering by ponderomotive potential
a0th
a0th g0 g 0
Scattering conditiong0 1 g0 Lth
Lth 1 a0th2 cos2 k 1 2
Electron trajectory in phase space
a0 a0th g2 1Scattering
-1 -0.5 0.5 1
50
100
150
200
k
g 10,0 1,0 0
a0 5 a0thTrapping
Simulations of ponderomotive acceleration
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Group velocity
g vg
c0.8
g 1.67
a0 5Below threshold
Above threshold a0 10
Ponderomotive acceleration and focusing
High energy booster acceleration of a pair-beam can be accomplished by the relativistic ponderomotive acceleration with focusing in vacuum ortenuous plasma.
The final energy is obtained approximately as
f a02
for a particle initially at rest.
E f GeV 0.37 10 21 I W/cm2 02 m
I 1022 W/cm2 E f 2.4GeVE.g.
I 1023W/cm2 E f 24GeV
I 1024 W/cm2 E f 240GeV
I 1025W/cm2 E f 2.4TeV
Acceleration
0 0.8mAt
El,m Re E0l,mel ,me
iti l ,m r,z Hl
x
2
Hm
y
2
0
exp
r2
4 2
z ct 2
4 z2
The electric field of Hermite-Gaussian modes is given by
0is the rms beam size. 0 1 z2 ZR
2
ZR is the Rayleigh length. z is the beam pulse length.
Consider the lowest mode of the laser beam with l=m=0, and the amplitude electric field E0
a0 e2mc2 E0 c
-3 -2 -1 1 2 3
0.2
0.4
0.6
0.8
1
1.2
1.4
U0U1
U=U0 +U1
r 0
The ponderomotive potential is
U0 r, z, t a02 0
2
2 exp
r2
22
z ct 2
2 z2
Focusing
The amplitude of the quiver oscillations is estimated to be
x
0
a00
2ZR z 1 2
ZR2 1 2 8 z
Off axis particles in a Gaussian laser beam are accelerated by the radial ponderomotive force and quickly expelled from the beam in the radial direction.To avoid transverse instability of motion, superposition of a Gaussian mode with l=m=0 and higher order modes, e.g. a linear combination of l=1, m=0 and l=0, m=1 with the same polarization vectors e1,0 =e0,1, and E1,0 =iE0,1 creates a potential well in the radial direction:
U1 r, z, t a12 r20
2
4 exp
r2 02
24
z ct 2
2 z2
where a1 is the dimensionless vector potential.
The total ponderomotive potential is given by
U U0 U1
The focusing force is given by
Fr
mc 2 U
r 2a1
2 a02 r0
2
4 a1
2 r3 02
6
exp r2
22
z ct 2
2 2
Focusing strength at r=0, and z-ct=0
KF Ft
mc2r
2a12 a0
2
02
The beam envelope equation on the rms beam radius rb isd2 rb
dz2 KF rb reN
2 23 zb rb
b
2
rb3 0
where N is the number of electrons in the bunch, zb is the rms bunch length, b is the geometric emittance, n the normalized emittance re is the classical electron radius.
b n
Space charge force Thermal emittance
The equilibrium radius is obtained fromd2 rb
dz2 0
Estimate of a focused beam size
1) The space-charge-force dependent beam size
rb reN
2 1 4 KF1 23 2 zb
1 2 reN
2 1 4 2a12 a0
2 1 4 zb
1 2
0
a1 a0Assuming 0 r0 2 zb 0
rb 2 1 4
2
r0
a05 2
reN
0
rb pm 2 1024 N
I5 4 W/cm2 r0 m 0
3 m
e.g. For 0 0.8mr0 10m
Np 1 1010
I 1.06 1025 W/cm2
rb 0.2 pm
Assuming a1 a0
rb n0
a0
1
2a02
0r0
b n n a02 a0
4 0 r0 2
rb pm 4 1023
I W/cm2 r0
0
rb b
KF1 4
1 4 b0
2a12 a0
2 1 4
2) The emittance dependent beam size
e.g. For 0 0.8mr0 10m
Np 1 1010
I 1.06 1025 W/cm2
rb 0.13 pm
If an electron-positron pair beam is focused, the space chrage forcewill be neglected.
Laser micro colliderTwo counter propagating laser-accelerated beams make a micro collider.
The space charge limited luminosity
L Np
2 frep
4 rb2
a050Np frep
2 3 2rer02
L cm 2s 1 2 10 30 I 5 2 W/cm2 06 m r0
2 m N p frep
0 0.8m
r0 10m
Np 1 1010
I 1.06 1025 W/cm2 EC.M. 5TeVe.g.
L 2 1040 frep cm 2s 1
P = 17 EW EL > 2×4 kJ
Required peak power and pulse energy
e+e- pair-beam micro-collider
Two counter-propagating laser-accelerated pair beams will createa new e+e-, e-e-, e+e+ micr-size collider without beam disruption at collision.
L Np
2 frep
4 rb2
a04 Np
2 frep
0r0
The emittance-limited luminosity is
where Np is the number of accelerated e+e- pairs and frep is the repetition rate of laser pulses.
L cm 2s 1 5.3 10 27 I2 W/cm2 03 m r0 1 m Np
2 frep[Hz]
E.g. For
0 0.8m
r0 10m
Np 1 1010
L 3 1042 frep cm 2s 1
I 1.06 1025 W/cm2
Pair-beam micro-collider concept
EC.M.[GeV]
1
I [W/cm2]
2.11021
P [PW/pulse]
3.3
EL [J]
1.6
L [cm-2s-1] at 1 Hz
1.2 1035
4 8.4 1021 13 6.4 2 1036
10 2.11022 33 16 1.2 1037
500 1.1 1024 1700 800 3 1040
1000 2.110243300 1600 1.2 1041
5000 1.1 102517000 8000 3 1042
Vacuum or tenuous plasma
factory
J/ factory
B factory
Luminosity of laser micro-colliders
1042
1039
1036
1033
1030
1027
Lum
inos
ity a
t 1 H
z [c
m-2s-1
]
100
110
C.M
. Ene
rgy
[GeV
]
100010000
1020 1021 1022 1023 1024 1025
Laser Intensity [W/cm2]
C.M. energy
Emittance-limitedluminosity
Space charge-limitedluminosity
0 0.8m r0 10m Np 1 1010
Summary
The proper shaped superstrong laser fields are useful for acceleratingand focusing electron and/or positron beams.
Two counter-propagating electron and positron beams result ina micro-collider with extremely high luminosity.
The plasma acceleration concept has difficulties in high energy gainand high beam quality.
A single stage energy gain < 1 GeV
I 1025W/cm2 E f 2.4TeV
rb 0.13 pm
L 3 1042 frep cm 2s 1
I 1025W/cm2
Np 1 1010
Np 1 1010
L 2 1040 frep cm 2s 1
The beam injector,The e+e- pair beam generator
Advanced Accelerator Research & Development at KEK
There are great interests in advanced accelerator researches, growing in the national and industry institutes, andthe universities around Japan.
The C.O.E. on A.A. R&D is conceived in KEKto organize developments of the future high energy acceleratortechnologies, holding long term visions of the post-LC, or the post-B factory.
We are eager to make collaboration and a word-wide network for A.A. R&D.
The 1st Workshop on Advanced Accelerator R&DApril 24, 25, 2003 at KEK, Tsukuba, Japan
The 1st Workshop on Advanced Accelerator R&DApril 24, 25, 2003 at KEK, Tsukuba, Japan
KEKJAERICRIEPAIST・・・
U. TokyoTokyo I. Tech.Tokyo M. U.Waseda U.U. UtsunomiyaU. KyotoU. OsakaU. Hiroshima・・・・・
KoreaKAER
I
ChinaIHEP
Fudan U.
Japan
Taiwan
India
U.S.A.SLAC ORION
BNLLBLNRLANL
US Univ.
EuropeCERN
RAL (UK)LULI, LOA
(France)GSI (Germany)
CNR (Italy)IGP (Russia)
Advanced Accelerator R&D NETWORK