C. Kamperidis- Table-top laser plasma wakefield electron accelerators

8/3/2019 C. Kamperidis- Table-top laser plasma wakefield electron accelerators

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Table-top laser

plasma wakefield

electron accelerators

C. KAMPERIDIS

Plasma Physics GroupImperial College London, UK

OLA - CRETE 2008


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Outline

Motivation for table-top/compact particle accelerators

Physics of laser driven wakefield electron accelerators(LWFA)

Ionisation by an intense laser pulse

Particle motion under a laser field / Ponderomotive Force

Plasma waves

Generation of non linear plasma waves

Relativistic plasma wave breaking limit (self-injection)

Energy gain

Energy losses in plasma wave generation

1st Session


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Outline

2nd Session

Propagation of ultra-short, ultra-intense laser pulses in underdense

plasmasNon linearities in plasmasPropagation beyond the Rayleigh length

Schemes of laser driven plasma based electron acceleration

LWFASelf Modulated - LWFA

Experiments performed in multi-TW table-top laser systems

Recent experimental landmarksGeV mono-energetic electron beams from 2D simulations

Conclusions/Applications/The Future


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1st Session Motivation

Review of major state-of-the-art traditional accelerator facilities

Conseil Europen pour la Recherche Nuclaire: www.cern.ch

Large Hadron Collider (LHC) - Multi-TeV proton collision energies

27 km


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DESY synchrotron facility in Hamburg

http://pr.desy.de/e113/indexeng.html

Review of major state-of-the-art traditional accelerator facilities


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1. RF technology is not getting cheaper2010 - proposed ILC (International Linear Collider) will be ready giving500 GeV at an estimated cost of 5 G$

2. RF technology needs a lot of spaceMaximum possible accelerating gradient in RF accelerators (limited by

material electric breakdown): ~ 100 MeV/mILC -> 31.5 MeV/m -> ~15 km

3. RF technology can not deliver ultrashort (~ fsec)electron bunchesILC will deliver e- bunches of 1 psec min

Applications of ~ fsec science : Ultrafast biological probing for example



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Conclusion: Current acceleration techniques havetwo main limits :

1.an upper limit on the accelerating gradient

2.a lower limit the pulse length.

Any new method that can improve any of thesefeatures will open up new possibilities in both highenergy, biomedical physics and many otherapplications.


Possible solution for the future:

THE LASER PLASMA WAKEFIELD

ACCELARATOR (LWFA)


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1st Session The LWFA

The Laser WakefieldThe Laser Wakefield

AcceleratorAccelerator1979 Tajima and Dawson

Same way a boat excites sea wakes


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1st Session Basic EM theory equations *

1

o

S E H E B

= =

ur ur uur ur ur

The Poynting vector represents the energy flux of an EM wave

( ) ( )2

20 0

1 / 377 / peak values

o

I S E E V cm I W cmc

= = =

ur ur

and its amplitude the intensity (W/cm2) of the wave

* Most of the basic theory of laser plasma interactions can be found in E. Clarks talk


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1st Session Physics of the LWFA :

Ionisation by an intense laser pulse

Keld > 1 multiphoton ion.

3 distinct processes :depending on the strength of the external laser field,

for a given atomic state and governed by the Keldysh parameter

Keld < 1 tunneling ion. Keld


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Particle motion under a laser field

Laser field equations

Lorentz force

But first, again some basic equations :

For a single electron, the electric and magnetic fieldcan be expressed in terms of the vector potential as


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By replacing in the Lorentz equation we obtain the

equations of motion of our test particle (electron)


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where

and is the normalised vector potential

The first term in the longitudinal equation of motion denotes a

gradual, linear drift with time

in useful experimental units,0 in m, I in 1018W/cm2


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Thus, if we transform in frame

moving with velocity

we obtain the famousfigure of 8 motion



2

0

4

a c

Laboratory frame

2 oscillation


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Ponderomotive force

Total energy of a relativistic particle

Kinetic energy gained

Averaged kinetic energy (Ponderomotive potential of the field)

of a relativistic particle over a field oscillation period 2/


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Ponderomotive force

And since there is there is net energy gain when

the laser is off, the Ponderomotive potential is related to the Ponderomotive force by

As we will see immediately, the Ponderomotive

force is the responsible force behind the creation ofthe plasma waves, which possess electric fields

with ideal accelerating properties.


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Plasma waves Plasma frequency

When many electrons are present, collectivebehaviour is exhibited

x

x

Static positive (ion)background -ALWAYS

Mobile electrons

Surface charge density

Capacitor-like E-field

Equation of motion

Simple harmonic

oscillation

Oscillation freq.


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Plasma waves Excitation of linear plasma waves

When a plane EM wave travels in free space is characterised by the

dispersion relation =ck, where c=1/(00)1/2

and k=2/For a medium/plasma with and , however the dispersion relation is altered

Assume an infinite and low intensity (i.e. collisions and magnetic effects are neglected) EM wave

The equation of motion of a test free electron within the plasma will be

with

By performing some mathematical manipulations,i.e. integrate the equation of motion over time and solve for velocity,

we can generate expression for current Je


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Plasma waves Excitation of linear plasma waves

1. Propagation of the mode isperpendicular to the initial field E-

oscillation,mode is EM in nature

2. Propagation of the mode is parallel

to the initial field E-oscillation,mode is electrostatic in nature

We can identify two oscillatory excited modesfrom the propagation of the EM wave within the medium/plasma

Exercise : Derive the current Je, use the curl of Faradays law (for EM and E-stat), use

Ampere-Maxwell equation (for EM only) to derive the dispersion relations of the new modes


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Plasma waves Generation of non-linear pw

To estimate how much and in what way a particular EM pulse(having finite durationrather than an infinite wave as before)ofarbitrary strengthwill affect the behaviour of the plasma wave

we will have to express some fundamental quantities such as the

plasma density variation(i.e. oscillation) neandthe electrostatic potential created by this variationin terms of our EM pulse primary quantity,

which is the normalized vector potential = eA/mc2.


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Lets assume an EM pulse (of finite duration),

linearly polarised, described by the vector potentialy : oscillation axis

x : propagation axis

Medium : cold, collisionless plasma

Density perturbations n=ne-n0 Electrostatic potential

New expressions for the electric and magnetic fieldsto be used in the Lorentz equation


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Equations of motion*

*These equations contain the momentum components px, py,which explicitly introduces the relativistic factor

Normalisations :

Coupling factor

Coulomb gauge,

assuming charge plasma fluid motionaffects instantly


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Vector field wave equation,

by replacing Eand Bin Amperes law

Continuity equation(since we are treating

our medium as fluid)

Poisson equation,only in terms of,

due to Coulomb gauge

The longitudinal eq. of motionRelates a fluid quantities (, )

with a field quantity


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The previous four equations provide a system where the importantparameters of the plasma wakefield are coupled with the strength

parameter of the EM pulse, albeit in a non easily solvable form

For simplification, we transform in

a frame moving at the groupvelocity of the EM pusle

New coordinatesin comoving frame

Also, assume that EM pulseevolution is negligible compared toplasma frequency, i.e. Quasi-Static

Approximation (QSA)

Thus, any time derivativeis neglected


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The four coupled equations can berewritten in the new coordinate

system

The second equation has a very important implication for plasmadensity evolution : when the velocity of the fluid reaches the groupvelocity of the EM pulse (i.e. ->g)

then the plasma density reaches infinity (ne->),black hole formation - impossible


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To cut a very long story short, we rearrange the above relations,

and we associate the plasma fluid velocity , and density ntothe electrostatic potential and vector potential , which whenreplaced in Poissons equation give in the new coordinate frame*

*All quantities are still normalised

which is a 2nd

order nonlinear ordinary differential equation for theelectrostatic potential , created by the density variation (n = ne/n0),created by the EM pulse =0()in the QSA and

can be solved numerically.


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Physical 1D system : EM pulse, =1m, pulse=100 fsec,

Plasma, n0=1018 cm-3 (p=30m) (i.e. cpulse ~ p)


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Important notes : in the Non Linear case, we observe plasma wavelength increase due to relativistic

effects (p->p1/2)

plasma density profile steepening

plasma wake electric field sawtoothing


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Plasma waves Relativistic pw wavebreaking (wb)

Lets remember the second of the final coupled equations

What happens when ->g???

Same thing to a surfer riding a sea wave

Smooth/slow wave steepening,

good ride

Harsh/fast wave steepening,

bad ride


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The mathematical formulation for wavebreaking is rathercomplicated but it suffices to give the final expression of

the natural limiton the electrostatic fields supported by aplasma wave before the wave structure collapses

Cold RelativisticWaveBreaking field

Cold Non-RelativisticWaveBreaking field

Practical units Emax,limit~0.96(ne)1/2,

i.e. ne~1020 cm-3 ->

Emax,limit~10 GV/cm

Laser field factor,equal to 1 when laser is off


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Plasma waves Relativistic pw wb Self-injection

1D Plasma density profile steepening

Adapted from Nature, 431, 515 (2004)

The natural process of driving a non-linear plasma wave towavebreaking acts as automatic electron injector in the

accelerating parts of the plasma wake electric fields


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3D Plasma density profile steepening trajectory crossing /3D Wavebreaking Bubble regime

(identified in Pukhov, A. & Meyer-ter-Vehn, J. Laser wake field acceleration: the highlynon-linear broken-wave regime. Appl. Phys. B 74, 355361 (2002))

Electron density void /

Ion surplass / Plasma bubbleElectron trajectories


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Plasma waves Energy gain of accelerated e-

Natural limitation on acceleration distance *remember the plasma wake electric field points at

the centre of the plasma period/plasma bubble

The dephasing length : the maximum length thatan electron injected at the back of the plasma wake period

will travel before it starts to decelerate due to theelectrostatic fields of the front half of the plasma wake

In the comoving frame (ug),is equal to a length of half

the plasma wavelength p/2

In the laboratory frame,


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Plasma waves Energy gain of accelerated e-

Energy gain in the comoving frame (the wave frame)

By performing a Lorentz transformationon the electron momentum four-vector

Energy gain in the laboratory frame


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Plasma waves Energy losses in Wake generation

Simplistic model : EM energy contained in laser pulseequals E-static energy contained in the plasma wake

Laser EM energy Plasma wake E-static energy

Optimum length for asquare pulse to maximiseplasma wakes amplitude

Valid approximation for allinitial driver laser pulseshape (i.e. sine, square,

guassian)


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Depletion length(for a square pulse)

Dephasing length

Small laser amplitudes

The interaction isdephasing limited

Large laser amplitudes

The interaction isdepletion limited


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Third physical limitation length on the Laser Plasma Interaction

Natural defocusing - Rayleigh length :defined as the length were the cross section of the laser

pulse is doubled (or in simpler terms stays in focus)

Physical example :a laser pulse of=1 m, focused at a

focal spot of w0=20 m, with an achieved0=1, in a plasma with ne=5*10

18 cm-3

ZR= 1.2 mmLd = 2.98 mmLdp = 3.03 mm

Conclusion : WE NEED TO OVERCOME NATURALDEFOCUSING AND EXTEND THE INTERACTION LENGTH

2nd

SESSION


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1st Session Summary of LWFA

1. The ponderomotive force of a laser pulse excites plasma waves

2. Depending on the strength of the laser pulse, a bubble structurecan form withperfect Efield geometry for electron acceleration,i.e. both acceleratingand focusingkeeping a low energy spreadand emittance. This non-linear structure can break, leading to

self-injection of electrons in th`e accelerating parts of thewake

3. Beam loading end of injection, where the electrostatic field ofthe injected electrons cancels the accelerating field of the wake.

4. Dephasing the injected electron beam enters thedecelerating phase of the wake.

LWFA is a 4 step process


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Plasma waves Relativistic pw wb Self-injection

1D Plasma density profile steepening

Adapted from Nature, 431, 515 (2004)

The natural process of driving a non-linear plasma wave towavebreaking acts as automatic electron injector in the

accelerating parts of the plasma wake electric fields


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1st Session Bibliography

E. Esarey, P. Sprangle, J. Krall, and A. Ting,Overview of plasma-based accelerator concepts

IEEE Transactions on Plasma Science 24, 252 (1996)

Paul Gibbon,Short Pulse Laser Interaction with Matter

Imperial College Press, 2005

W. B. Mori,The physics of the nonlinear optics of plasmas atrelativistic intensities for short-pulse lasersIEEE Journal of Quantum Electronics 33, 1942 (1997)


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2nd Session In brief

We will show how we can extend the propagation of alaser pulse beyond the naturally limiting length ofZR , by

exploiting various non-linear phenomena that inevitablyoccur

We will show results from recent experiments + somesimulation movies, visualising in real time what is reallyhappening


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2nd Session Propagation in underdense plasmas :

Non Linearities Refractive index modulations

A plasma is a medium with electric permittivity

and magnetic permeability , different thanthe vacuum equivalents 0, 0

Index of refraction is defined as

The phase velocity of a propagating EM wave is*

*Remember from dispersion relation


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With appropriate substitutions

Group velocityof EM wavePhase velocityof EM wave

Dependence ofon plasma density (ne), laserfrequency (0), laser intensity (0-as relativisticeffects effectively increase the critical density ncr)


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Thus can be expanded to first order accordingly

Modulations in these three parameters (ne, 0, 0) lead

to modulation of the index of refraction which in turnleads to modulation ofphand grof the EM pulse


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Physical system : A laser pulse (e.g. gaussian)

enters a homogeneous density plasma region ->Transverse intensity profile induces ponderomotive

radial expulsion of electrons ->

Transverse index of refraction modulationOr respectively transverse modulation in ph



Induced phenomenon : Self-focusing

Self-focusingacceleration


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Physical system : A laser pulse (e.g. gaussian) has

longitudinal intensity variation (0) ->Longitudinal variation plasma density will inducelongitudinal variation in index of refraction ->

Longitudinal variation in gr



Induced phenomenon : Self-compression

Self-compression rate (incomoving frame , )

d


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Induced phenomenon : Self-compression

Image courtesy of A.G.R. Thomas

2 d S i P i i d d l


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Physical system : A laser pulse (e.g. gaussian) has

longitudinal intensity variation (0) ->Longitudinal variation plasma density will inducelongitudinal variation in index of refraction ->

Longitudinal variation in ph

Induced phenomenon : Photon Acceleration

Frequency change rate (incomoving frame , )

2nd S i P ti i d d l


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Conclusion: Refractive index gradientsare found in almost all cases of laser

propagation in underdense plasmas, so it is

almost unavoidable not to experience

Self-focusingSelf-compression

Photon acceleration/deceleration

2nd S i P ti i d d l


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Propagation beyond the Natural Limit (ZR)

If the natural defocusing rate of a laser beam matches the inducedfocusing from any non-linear phenomena (i.e. self-focusing fromtransverse density gradients) then the laser pulse fronts will stay

focused and propagate for an extended length

Natural Defocusing Limit = Rayleigh range ZR=w02/

Adapted from http://upload.wikimedia.org/wikipedia/commons/9/94/Gaussianbeam.png

Typical experimental valuesw0=10 m, =1m ->

ZR=0.3mm

w0=1/e2 intensity radius



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If the natural defocusing rate of a laser beam matches the inducedfocusing from any non-linear phenomena (i.e. self-focusing from

transverse density gradients) then the laser pulse fronts will stayfocused and propagate for an extended length





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Two ways to induce a transverse density gradient(i.e. transverse index of refraction gradient)

I) Externally, for example with the use of the so-called gas filled discharge discharge capillaries

II) Internally, i.e. by allowing the laser pulse to self-evolve, see Self-focusing section, due to theunavoidable electron expulsion from axis due to the

radial ponderomotive force of the laser pulse



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Propagation beyond ZR External guiding

Transverse parabolicplasma profile

Gas filled discharge discharge capillaries =Hollow tubes of 100s m diameter, filled with gas (i.e. hydrogen)

where an electrical pulse of 100s A and 10s kVpasses, ionises and hydrodynamically expands the plasma

Transverse inverse parabolicindex of refraction

Maximum achieved guidingup to 50 mm



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Propagation beyond ZR External guiding

By matching the natural

defocusing rate

By matching the self-focusingacceleration (see Self-focusing

section)

We can derive a matched spotsize, where diffraction is balancedby self-focusing effects due to the

density (refractive index)

transverse gradients



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2 d Session Propagation in underdense plasmas :

Propagation beyond ZR Relativistic guiding

Respectively, the relativistically corrected index ofrefraction (due to relativistic mass increase) will be

Assume a laser pulse in the mildly relativisticregime, i.e. 0

2


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And since

2 Session Propagation in underdense plasmas :


Index of refraction profile similar to externally guiding case

Thus by replacing this index of refraction expression in the self-focusing acceleration equation and equate with the natural

diffraction rate (as we did before) we can derive the matched

spot size for relativistic self-guiding



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By replacing 0 with

We can derive a critical value for the power of the laserpulse, so that relativistic self-focusing is induced

(!!!) (!!!)

Major Limitation : For relativistic self-guiding to occurthe laser pulse must cover many plasma periods (pe) so

that there is time for self-focusing to evolve



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Longestself-guiding channels observed so far: 10.1 mm

Laser pulse : 25J, 500fsec, 50TW, w0=25mm -> Z

R=1.8mm,

0~ 2



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Summary

So far we have seen that any laser pulse propagating through ahomogeneous plasma density region will be affect by non-linear

phenomena such asself-focusing, self-compression, photon acceleration

Simply put, a pulse of initial strength 0will self-evolve during propagation in

0>

0

Given that the laser pulse is long enough (i.e. claser>>plasma)

Relativistic increase of electron mass leads to transverse density profile,ideal for self-focusing and hence self-guiding over many ZR

2nd Session Schemes of laser driven plasma based


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p

accelerators - LWFA

Typical Experimental values

Laser : 40fsec, 500mJ

w0 : 40m 5 m

I : 3*1017 3*1019 W/cm2

0 : 0.4 3.5

Plasma Density : ne ~ 5*1018 cm-3

Plasma length : Lplasma ~ 3 mm

Single pulse excites plasma waves, which when driven to highamplitudes, wave break, inject electronsin the accelerating phaseson the wake, thus leading toMono-Energeticelectron beams



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p

accelerators Self Modulated LWFABy utilising the Forward Raman instability, we create a beat pattern that

resonantly can drive a plasma wake to breaking conditions.From that point on it behaves as a usual LWFA



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p

accelerators SM LWFA Parametric instabilities

The plasma waves grow at the expense of the incoming EMwave, which is called the pump

Parametric instabilities occur only when their growth rate is fasterthan the characteristic evolution of the system, usually determinedby the duration of the EM pulse, i.e. short (


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1. The pump EM field causes the electrons of the plasma to oscillate at

pe

2. Pump photons are scattered from these density oscillations, atfrequencies equal to the sum and difference of the pump frequency

and plasma frequency (pmpe)

3. The interference between the pump and scattered beam causes avariation at the beat frequency (pe) in the overall EM pressure

4. These variations resonantly excite electron density fluctuations, whichin return generate more scattered photons at (pmpe), which beatwith pump photons and so on

Feedback Loop




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The Stimulated Raman Scattering (SRS)is the decay of an

incoming photon (pump laser) into another photon (eitherblueshifted or downshifted) and a plasmon

The emitted photon can be either:

co-propagating with the pump beam (Forward Raman Scattering)

counter-propagating (Raman Back Scattering)

can be scattered in any in between angle (Raman Side Scattering)




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Matching conditions for the feedback loop to grow



2nd Session Experiments performed Multi-TW


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table top laser systems Recent historical landmarks

First high energy SM-LWFA experiment

Until 2002 all the experimentalresults on electron accelerationsuffered from a large energy

spread, an inherent feature of theSM-LWFA mechanism as it involves

randomised wavebreaking of manyplasma periods

The bubble regime has been identified in 2002 and was madeaccessible only recently with the T3 laser systems that could inducewavebreaking conditions with laser pulses of a few plasma periods

long (i.e. claser

~ plasma

)

Modena, Nature, 377, (1995)

2nd Session Experiments performed Multi-TWbl l


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First Mono-Energetic LWFA experiments

But in 2004 Nature, 431,3 seminal papers werepublished

The set-up was similar in allexperiments

2nd Session Experiments performed Multi-TWt bl t l t


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Mangles et al,Imperial College, UK:

70 MeV beam

Geddes et al,Lawrence Berkeley, USA:

85 MeV beam

Faure et al,LOA, France:

170 MeV beam

All images taken from Nature, 431


First Mono-Energetic LWFA experiments

2nd Session Experiments performed Multi-TWt bl t l t


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First Mono-Energetic GeV experiment

Leemans et al,

Lawrence Berkeley, USA:

1000 MeV beam

Image taken from Leemans et al., Nature Physics, 2 (2006)

Long interaction length, i.e.

33 mm, via guiding through a

Hydrogen filled, discharge

capillary

Note : Maximum electron

acceleration ~ 100 GeV in

km long linear accelerators

2nd Session Experiments performed Multi-TWtable top laser systems Simulations


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table top laser systems Simulations

The code used for the simulations is a 2D Particle-In-Cell, called OSIRIS

Physical system :

Laser - 1J, 30fsec, =800nm, w0=10m, 0=1.79

Plasma -ne=9*1018

cm-3

, Lplasma=10 mm

1st Simulation : Propagation of ultrashort (claser~plasma)

laser pulse in homogeneous underdense plasma - LWFA

Main results :

1. slow evolution(i.e. self-focusingand self-compressionto higher laseramplitudes)

2. stable >250MeV electron beamthat outruns the laser pulse

3. maintaining its Mono-Energetic/low energy spreadfeature

2nd Session Experiments performed Multi-TWtable top laser systems Simulations


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table top laser systems Simulations

Physical system :Laser 100J, 400fsec, =1054nm, w0=25m, 0=4.3

Plasma -ne=2*1018 cm-3, Lplasma=20 mm

Main results :

1. slow evolution(i.e. self-focusingand self-compressionto higher laseramplitudes) leads to complete pulse break

2. laser pulse is slowly matched to an ideal LWF Accelerator

3. Emax>GeV, >plasma)laser pulse in homogeneous underdense plasma SM-LWFA

2nd Session Conclusions


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The need for new particle acceleration techniques is obligatory,if we want to explore new realms of physics.

However, as more is not always the best, new particleacceleration techniques on a lower energy range (100-1000 MeV)can downsize and reduce the cost of an acceleration machine for

commercial purposes as well

Laser Wakefield Accleration,since 2004 when it wasfirst demonstrated, has drawn the attention (and the

funding) of 100s of research groups all over the world.

However, the reproducibility, stability, emittanceandchargeof the beams produced from LWFA are still one

order of magnitude worse than conventional accelerators.

2nd Session Conclusions : Exotic new research


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Control of electron self-injection Multiple laser beams


The ponderomotive force of a second laser beam (co-,counter- or cross-propagating) can dephase wake electrons and inject them in the wake.

2nd Session Conclusions : Exotic new research


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Control of electron self-injection External Impurities

Al5+->Al6+

IBSI =1.4e17 W/cm2

O5+->O6+

IBSI =4e16 W/cm2

Al9+->Al10+

IBSI =1e18 W/cm2

2nd Session Conclusions : Exotic applications


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Towards a table-top free-electron laserConventional bright x-ray source

180m

Downsized novel x-ray source

How x-rays are produced ???

The applications of suchsynchronised electron/x-ray beams

are numerous. From ultrafastbiomedical (molecular/atomic)imaging, to lithography for the

computer and automotive industry

2nd Session Conclusions : Proton/Ionacceleration applications


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acceleration applications

Main acceleration of protons/ions comes from electron sheathspropagating through and away from the surface of a thin metallictarget hit by a laser pulses similar to those described in this talk

Main applications ofthese protons/ionsCancertreatm

ent

FastIgnitor

(Tatarakis

talkonHiP

ER)


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Thank you for your timeand presence !!!

CollaboratorsRCUK - Basic Technology Grant AlphaX


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A.G.R. Thomas, S.P.D. Mangles, L. Willingale, C. Bellei, S. Kneip, S. Nagel,K. Krushelnick, Z. Najmudin

Plasma Physics Group, Imperial College London, UK

C.D. Murphy, K. Lancaster, P.S. Foster, C.J. Hooker, E.J. Divall,O. Cheklov, P. Norreys, J. Collier

CCLRC Rutherford Appleton Laboratory, UK

J.G. Gallacher, E. Brunetti, M. Wiggins, F. Bode, D.A. Jaroszynski

University of Strathclyde, UK

T.P. Rowlands-Rees, Tom Ibbotson, A.J. Gonslaves, S.M. Hooker

Clarendon Laboratory, University of Oxford, UK

RCUK Basic Technology Grant AlphaX

C. Kamperidis- Table-top laser plasma wakefield electron accelerators

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Transcript of C. Kamperidis- Table-top laser plasma wakefield electron accelerators