http://loa.ensta.fr/UMR 7639

Laser-plasma acceleration

Physics and Applications

Jérôme FaureLaboratoire d’Optique Appliquée

Ecole Polytechnique, France

Outline of the class

• First class

• General introduction: what, why and how ?

• Theory of plasma wave driven by a laser pulse

• Second class

• Acceleration/injection of particles in plasma waves

• Applications of laser-plasma accelerators

Electron accelerators in our life / science

X-ray tube

E< 1 MeV

E< 300 keV

Electron microscope

E< 20 MeV

radiotherapy

E> 100 GeV

High-energy physics

E > 1 GeV

Synchrotron light source

Industrial Market for Accelerators

Application

Total

systems

(2007)

approx.

System

sold/yr

Sales/yr

(M$)

System

price (M$)

Cancer Therapy 9100 500 1800 2.0 - 5.0

Ion Implantation 9500 500 1400 1.5 - 2.5

Electron cutting and welding 4500 100 150 0.5 - 2.5

Electron beam and X rays

irradiators2000 75 130 0.2 - 8.0

Radio-isotope production (incl. PET) 550 50 70 1.0 - 30

Non destructive testing (incl.

Security)650 100 70 0.3 - 2.0

Ion beam analysis (incl. AMS) 200 25 30 0.4 - 1.5

Neutron generators (incl. sealed

tubes)1000 50 30 0.1 - 3.0

Total 27500 1400 3680

Total accelerators sales increasing more than 10% per year

The development of state of the art accelerators for HEP has lead to :

research in other field of science (light source, spallation neutron sources…)

industrial accelerators (cancer therapy, ion implant., electron cutting&welding...)

Accelerators : One century of exploration of the infinitively small

Quarks

Atom

Nucleus

Why alternative techniques ?

• Accelerating field in cavities limited to 50 MV/m by breakdown

LHC at CERN is 27 km circumference

ILC (electron-positron collider at several 100 GeVs), will be a LINAC, about 30 km long

• Use plasmas with much higher electric fields: > 100 GV/m

RF cavity: 1 m Plasma wave over 1 mm

Ez = 10-100 MV/m Ez = 10-100 GV/m

Motivation for alternative techniques

A plasma: free electrons and ions: already ionized

mGVnE ez /300 (for electron density ne=1019 cm-3)

z

dn/n0

Ez

z

With a plasma wave, EZ is 104 greater than in a RF cavity

compact accelerators possible

2/1

12

ep

pn

c

vp c

Plasma wave

Why plasmas ?

F ~ -dIlaser

F

Champ E

Laser vg ~ c

• Ponderomotive force pushes electrons:

• In a plasma: creates a plasma wave (wakefield)

A laser pulse to drive the plasma wave

Gas jet

laser

100 MeV

electrons

Experimental principle

plasma

1 mm

Ultra-intense laser

Energy: 1 J

Duration: 30 fs

Focal spot: 10 µm

Accelerating electrons in wakefields

PROS

• Extreme fields

1 MeV / 10 µm

Compact accelerator

Mitigates space charge

• fs bunches

< p/4

• no jitter

Ez

Er

Electron density

Pulsep≈1-10 µm

accelerating

focusing

Tajima & Dawson, PRL 1979

Surfing the wake !!

Injecting electrons in the wakefield

Fluid electrons trapped electrons

Electrons need a kick to catch the plasma wave

A simple solution: wavebreaking

We drive the wakefield so strongly that it breaks,

traps and accelerated the plasma electrons

Resonant condition for wakefield

excitation: ct ≤ p and w0 ≤ p

Bubble formation

Pukhov & Meyer-ter-Vehn, Appl. Phys. B 2002

Self-injection in the bubble regime

Example of self-injection in a simulation

Gas jet

laser

100 MeV

electrons

Experimental principle

plasma

1 mm

Ultra-intense laser

Energy: 1 J

Duration: 30 fs

Focal spot: 10 µm

5-pass amplifier :

200 mJ

8-pass amplifier : 2 mJ

Oscillator : 2 nJ, 15 fs

stretcher : 500 pJ, 400 ps

On target:

1 J, 30 fs, 0.8 mm,

10 Hz, 10 -92 m

Nd:YAG : 10 J

4-pass amplifier:

2.5 J, 400 ps

“Salle Jaune” laser

compressor

Vacuum

chamber

f=1m

w0=18 µm

30 fs

Typical experimental set-up

Gas jet

Si diodes

magnet

ICT

LANEX screen

Probe beamImage of the plasma

laser

Divergence = 6 mrad

6.0 x 1018cm-37.5 x 1018cm-31.0 x 1019cm-3

2.0 x 1019cm-35.0 x 1019cm-3 3.0 x 1019cm-3

7°

Discovery of the bubble regime

We adjust the plasma density to reach the resonant condition

ct ~ p

Energy distribution

ne=7.5×1018 cm-3, maxwellian distribution

ne=6×1018 cm-3, monoenergetic distribution

20 50 100 200Energy (MeV)

Div

erg

ence(°

)

1

0

-1

Div

erg

ence(°

)

20 50 100 200

Energy (MeV)

Div

erg

ence(°

)

1

0

-1

• charge about 100 pC

• dE limited by spectrometer

resolution

First quasi monoenergetic e-beams

Mangles et al, Imperial College:

70 MeV beam

Geddes et al, LBNL:

85 MeV beam

Faure et al, LOA:

170 MeV beam

5-pass amplifier :

200 mJ

8-pass amplifier : 2 mJ

Oscillator : 2 nJ, 15 fs

stretcher : 500 pJ, 400 ps

On target:

1 J, 30 fs, 0.8 mm,

10 Hz, 10 -92 m

Nd:YAG : 10 J

4-pass amplifier:

2.5 J, 400 ps

“Salle Jaune” laser

First conclusions

plasma

1 mm

In the past 10 years, a new technology for particle accelerators has emerged

Laser-plasma acceleration (laser-wakefield acceleration) is based on

- ultra-intense femtosecond laser

- laser-plasma interaction

- Highly nonlinear physics

In 2017:

- Proof-of-principle experiments have been demonstrated

- Physics well understood

- 100 MeV to few GeV beams driven by 100 TW to PW-class lasers

- Low repetition rate: 1 shot per second

- World record: 4 GeV using PW laser @ Berkeley National Lab. (USA)

.. ... .

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......

..

..

.

...

. . ..

..

...

..

...

...

. ......

..

Ultra-intense laser-plasma interaction

in the world > 50 groups in the world

Berkeley

California

1 PW

Commercial system

by THALES

South Korea

First multi-PW laser

Saclay, France

First 10 PW

in preparation

Outline of the class

• First class

• General introduction

• Theory of plasma wave driven by a laser pulse

Very basic plasma physics

• Natural plasma oscillation: plasma frequency :

• Plasma wavelength: p=2c/p. p=10 µm for ne=5×1018 cm-3

• EM Wave propagation possible only if > p

: the plasma is underdense

(n < 1021 cm-3 for λ= 1 µm). In most experiments, ne=1018-1019 cm-3

Basics: definition of laser intensity

• Consider a EM field as a plane wave

• Peak intensity is defined as

• For a Gaussian pulse (at focus)

• Example: E=1 J, w0=20 µm, t0=30 fs I0=5×1018 W/cm2

0

2

0

0

2

tw

EI

Relativistic regime of laser-plasma interaction

• Electron in laser field:

• Weakly relativistic case v/c <<1(magnetic component isneglected)

t

A

m

e

m

Ee

dt

dv

ee

Losc

ac

vosc

Relativistic regime is entered when a ~ 1 (I0 ~ 1018 W/cm2)

a: the normalized vector potential

• Laser E field linked to potential vector a by

• Normalized vector potential

• In practical units

• Example: I0=2×1018 W/cm2 (=1µm) a=1.2

t

AEL

cm

eAa

e

0

00

cm

eEa

e

]/[][105.8 22/1

0

10

0 cmWIµma

Fluid model: hypothesis (1)

Fluid model: hypothesis (1)

• Laser driver

– envelope

Definition of the laser pulse

with

Rewrite as: With

Density perturbation caused

by the laser pulse

Poisson equation

Physical meaning of scalar potential

Fluid equations

Some algebra

Motion of electrons in plasma after

averaging over fast oscillations

Ponderomotive force

Some algebra

Plasma wakefield equation

• Write equation on potential using:

Plasma wakefield equation

Plasma wakefield equation becomes:

Moving window

We follow the laser pulse:

Use new variables:

• Neglect derivatives in t compared to derivatives in z

• Physical meaning: the plasma responds adiabatically to slow changes of the driver

laser

1/zR

1/L0

1/p

Quasi-static approximation

• Solution behind the pulse (gaussian shape)

– Potential

– Electric field

– Longitudinal

– Transverse

– E0 wavebreaking field2/1

0 ne

cmE

pe

Solutions

f, Ez/E0, dnz/n0 are normalized quantities and have the same amplitude

Resonance condition

0/ 0 Lf

Maximum amplitude for: 20 Lk p

In practical units:

0 1 2 3 4 50

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

kpL

0

F / a

02

Accelerating and focusing fields

a=0.5

Electron density

Pulse

Defocusing

Ez

Er

Focusing

Accelerating

Decelerating

1D Model: Plasma wave

• Low intensity limit (a2<<1), the potential is solution of

• One can use a 1D nonlinear fluid theory which works for a > 1

Co-moving coordinate

Plasma wave vector (p plasma frequency)

Integrate numerically this equation for a gaussian pulse

Example of 1D nonlinear plasma wave

a=0.3

f

dn/n0

Ez/E0

a=2

f

dn/n0

Ez/E0

Ez

Er

3D nonlinear wakefields

Focusing

Defocusing

Accelerating

Decelerating

a0=2

Electron density

Pulse

relativistic shift of p

Requires more complex models (fluid or kinetic)

+ computer simulations

Summary

• Intensity and a:

• Wakefield amplitude: proportional to laser intensity

• Wakefield max at resonance

]/[][105.8 22/1

0

10

0 cmWIµma

20 Lk p

http://loa.ensta.fr/UMR 7639

Laser-plasma acceleration

Physics and Applications

2/2

Jérôme FaureLaboratoire d’Optique Appliquée

Ecole Polytechnique, France

Outline of the class

• First class

• General introduction: what, why and how ?

• Theory of plasma wave driven by a laser pulse

• Second class

• Acceleration/injection of particles in plasma waves

• Applications of laser-plasma accelerators

How do we inject electrons into wakefields ?

How are electrons accelerated ?

Electron density

Pulselp≈1-10 µm

Bubble formationFirst class: wavebreaking

Injecting electrons in the wakefield

Fluid electrons trapped electrons

Electrons need a kick to catch the plasma wave

Summary / reminder of notation

• Intensity and normalized

vector potential a (linear pol.)

• Wakefield amplitude: comes from charge separation define scalarpotential for the plasma wave and its normalized counterpart

• Laser group velocity vg, plasma phase velocity vp

• Define Lorentz factor

laser

Plasma wakefield

laser

Plasma wakefield

We use a 1D fluid model for the plasma wave

Electron motion:Transverse longitudinal

z

1D Model: Plasma wave

• Low intensity limit (a2<<1), the potential is solution of

• One can use a 1D nonlinear fluid theory which works for a > 1

Co-moving coordinate

Plasma wave vector (wp plasma frequency)

Integrate numerically this equation for a gaussian pulse

Outline

• 1 D Hamiltonian model for electrons interacting

with laser field and plasma wave

References:E. Esarey and M. Pilloff, Phys. Plasmas 2, 1432 (1995)

E. Esarey et al., IEEE Trans. Plasm. Sci. 24, 252 (1996)

Hamiltonian of electron in laser and

plasma wave with potential F

kinetic energy potential energy

Let’s normalize the Hamiltonian

• H depends on time but in a a particular manner (z-vgt)

– Eliminate time using a canonical transformation

– With generating function

• New Hamiltonian:

Hamiltonian’s basic properties

• New Hamiltonian is then:

• Define the momentum conjugate to the position (or canonical

momentum)

• In our case, q=-qe and A is a transverse laser field. This translates in

normalized units into

• Hamiltonian expressed in terms of canonical momentum

Hamiltonian’s basic properties

• From Hamilton’s equations, one finds that in 1D the transverse

canonical momentum is conserved (constant of motion)

• For electrons initially at rest in front of the laser pulse, Cste=0 and

Trajectories in the wakefield

• The Hamiltonian does not depend on time constant of motion H0

We want to find the electron trajectories in phase space: uz(z)

• Solving the Hamiltonian for uz, one finds

• 2nd degree polynomial equation with solution

• If a(z) and f(z) and H0 are known then the initial conditions are

known the trajectory in phase space uz(z) is known

(Exercise)

Fluid trajectories: electrons

initially at rest in front of the

laser

Fluid electrons

The separatrix: limit of trapped

trajectories

Trapped electrons=Paddling surfer

Trapped orbits

Trapped electrons haveinitial kinetic energy

(Exercise)

Calculate and obtain minimum energy for trapping

We start from

Electrons in front of the laser pulse and on the separatrix

We can easily calculate

Injection threshold

(Exercise)

Injection thresholds

Larger wake amplitudes Slower wakes

Injection threshold is lower for large amplitude and slow wakefields

Wavebreaking ?

Wavebreaking as an injection mechanism

fluid

separatix

Zoom on the fluid trajectory

• As the wake amplitude increases (|fmin| increases), the fluid

trajectory gets closer to the separatrix

• 1D wavebreaking occurs when fluid and separatrix overlap

• All plasma electrons are then injected and accelerated

Wavebreaking / self-injection

Experimentally, wavebreaking is hard to control unstable beamsResearchers are finding ways to control the injection of electrons

Ponderomotive force in the beatwave: Fp ~ 2a0a1/λ0

The beatwave pre-accelerates electrons locally and injects them

INJECTION is local and short (30 fs) monoenergetic beams

Plasma wave

Pump pulse a0

Injection pulse a1

electrons

Principle of colliding pulse injection

Colliding pulse injectionZoom on the fluid trajectory

fluid

separatix

beatwave

The beatwave provides a bridge from the fluid orbit to the trapped orbit

There is no need for wavebreaking

wavebreaking Controlled injection

Injection beam130 mJ, 30 fs ffwhm=28× 23 µmI ~ 4×1017 W/cm2

Pump beam670 mJ, 30 fs, ffwhm=21×18 µmI ~ 4×1018 W/cm2

What it looks like in reality

Statistics (30 shots):

E = 206 +/- 11 MeV

charge = 13+/- 4 pC

dE = 14 +/- 3 MeV

dE/E = 6%

Very little electrons at low energy, dE/E=5% limited by spectrometer

3 mm gas jet

Stable monoenergetic beams

pump injection

pump injection

Jet exit

pump injection

Middle of jet

Zinj=225 μm

Zinj=125 μm

Zinj=25 μm

Zinj=-75 μm

Zinj=-175 μm

Zinj=-275 μm

Zinj=-375 μm

50 100 200 300 400Energy (MeV)

Beginning of jet

Tuning the beam energy

pulse

1 2 3

Effective slow down

1

2

3

Gradient scale length

In the density gradient, lp increases• causes the plasma wave to elongate• effective slow down of the back of the plasma wave• effective decrease of the phase velocity Facilitates trapping Decreases the threshold for self-injection

Other methods for controlling injectionInjection in a density gradient

Ionized electrons are born in the laser and in the wake itself • They have different initial

conditions compared to fluidelectrons

• “Dropping” them at the right phase so they can be trapped

Electrons from N+ to N5+

Electrons from N6+

Main idea:If we drop an electron at rest at this phase, it will be on a trapped orbit

Other methods for controlling injectionInjection using ionization

Outline of the class

• First class

• General introduction: what, why and how ?

• Theory of plasma wave driven by a laser pulse

• Second class

• Acceleration/injection of particles in plasma waves

• Applications of laser-plasma accelerators

• Very compact acceleration

• Source is very small and short:

– Few micron diameter

– Few femtosecond duration

Application using these

unique qualities

Unique properties of laser-plasma source

High resolution radiography of dense object with a low divergence, point-like electron source

Application exploiting small source size

Radiography of dense objects

Y. Glinec et al., PRL 94, 025003 (2005)

50 μm γ source size2010Cut of the object in 3D

Spherical hollow object in tungsten with sinusoidal structures etched on the inner part.

A. Ben-Ismail et al., App. Phys. Lett. 98, 264101 (2011)A. Ben-Ismail et al., Nucl. Instr. and Meth. A 629 (2010)

400 μm γ source size2005

Radiography of dense objects: experiments

Quicker Access

to Physics and

Applications

Laser Plasma Technologies

The market of non-destructive inspection

Airplane partsnuclear plants

X-rays produced by relativistic electrons

β

β.

Electron

mm plasma wigglers

Equipe A. Rousse et K. Ta Phuoc (LOA)

synchrotrons

free electron lasers

A. Rousse, K. Ta Phuoc et al, Phys. Rev. Lett. 2004

20 mrad

E > 3 keV

Characteristics of the source:- 105 photons/shot/0.1% BW @ 1 keV

- divergence: 10’s mrad

- Duration: 10’s fs

- Spectrum: 1-10 keV

- Source size: 1- 2 microns

Perspectives:- Increase radiation energy by controlling electron trajectories

- Use PW lasers

Radiation produced in a laser wakefield accelerator

Betatron radiation: fs X-ray source

Characteristics of the source:- 105 photons/shot/0.1% BW @ 1 keV

- divergence: 10’s mrad

- Duration: 10’s fs

- Spectrum: 10-1000 keV

- Source size: 1- 2 microns

Perspectives:- Produce a tunable and monochromatic source

- Use PW lasers

Radiation produced at the collision between a laser pulse and a relativistic electron

Compton scatteringfs X-ray source

Ta Phuoc et al., Nat. Phot. 2012

Using laser-plasma accelerators for probing matter ?

Time resolved electron diffraction: watching structural dynamicsin real time

Laser plasma accelerators:

- Femtosecond bunches- No jitter

But should be:

Lower energy: 5 MeVkHz (statistics, averaging)

Toward kHz laser-plasma acceleratorswith mJ-class kHz lasers

Laser pulse has to be resonant

with plasma wave:

R≈lp/2, ct≈lp/2

Pulse

R

lp

Lu et al., PRSTAB 10, 0613001 (2007)

Laser energy scaling Electron energy gain

30 fs 1 J 100 MeV-1 GeV3 fs mJ 1-10 MeV

Laser pulses of 5 fs, few mJ possible @ kHz !

Beaurepaire et al., NJP 16, 023023 (2014)

−8 −4 0 4 8 120

20

40

60

80

100

ElectronHspectraa6 b6

c6 ChirpH=H−4fs2 ChirpH=H0fs2 ChirpH=H8fs2

2 4 6 80

0.2

0.4

0.6

0.8

1

EnergyH5MeV6

dN

el/d

EH8fs2

H4fs2

H0fs2

H−4fs2

1

2

1 2

ElectricHfield

Envelope

H5a

.u.6

FWHMHpulseHdurationH5fs6

Cha

rgeH5

fC/s

hot6

3

3

−20HHHHHHH−10HHHHHHHHHHH0HHHHHHHHHHH10HHHHHHHHHH20HHHHH

TimeH5fs6

HHH HHHHHHHHHHHHH−10HHHHHHHHHHH0HHHHHHHHHHH10HHHHHHHHHH20HHHHH

TimeH5fs6

HHHHHHHHHHHHHH−10HHHHHHHHHHH0HHHHHHHHHHH10HHHHHHHHHH20HHHHH

TimeH5fs6

Am

plit

ude

H5a

.u.6

1

0.5

0

−0.5

−1

ChirpH5fs26

6.7 4.3 3.4 4.0 12.1 16.3

Laser:

3.5 fs – 3 mJE-beam

100 fC-20 pC, 5 MeV

Time-resolved diffraction experimentby streaking the Bragg peaks

30 nm Si nano membrane, S. Scott, M. Lagally, Univ. Wisconsin

Diffraction on single cristal nano-membranes

020

220

200

Gold, 20 nm Silicon, 30 nm

Exposure time 500 msHigh quality diffraction patterns

Z. He et al., Appl. Phys. Lett. 102, 064104 (2013)

Demonstration of time-resolved

electron diffraction

sample

CCD

kHz laser

Diffraction

patternpump pulseI1/I0-1(220) Data set I Data set II I1/I0-1(220)

0-order

unpumped

pumped

1mm

fit

data with DC gun

normalized

a

b

c

d

e

f

normalized

3.1 ± 0.8 ps 15.5 ± 0.8 ps

3.0 ± 0.2 ps 7.1 ± 0.4 ps

fit

data with DC gun

Silicon, 30 nmDynamics of (220) peak

220

400

ps resolution obtainedfor fs resolution, relativistic MeV beams are necessary

From engineering to fundamental science

and back

R & D in laser technology

New generation of ultra-intense lasers

Fundamental researchin

Laser-plasma interaction

New generation of particle beams

SOCIETAL APPLICATIONS

SOCIETAL APPLICATIONS

New regimesfor research

?

Thank you

• If you are interested: contact me !

jerome.faure@ensta-paristech.fr

http://loa.ensta-paristech.fr/appli/

• PhD positions available…