High brilliance X-gamma ray sources based on LASER-matter interaction at high intensities

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Università di Pisa

Facoltà di Scienze Matematiche, Fisiche e Naturali

Corso di Laurea Magistrale in Fisica

HIGH BRILLIANCE X-γ RAY SOURCES

BASED ON LASER-MATTER

INTERACTION AT HIGH INTENSITIES

Relatori:

Prof. Danilo Giulietti Dott. Riccardo De Angelis

Laureando Alessandro Curcio Matricola: 498255

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Acknowledgements

I wish to thank my parents and my family for giving me the opportunity to get this goal. I warmly thank Prof. Danilo Giulietti for believing in me and for having allowed me to professionally grow during this year. I would like to say a big thank you to the Dr. Riccardo De Angelis for the opportunity to work at the ABC facility, and for introducing me to its wonderful team: Fabrizio, Pierluigi, Giuseppe, Mattia, Francesco, I'm very proud to have worked with you. I care a lot to say to you all, starting from the Prof. Giulietti to Dr. De Angelis and everyone of the ABC team, that I consider you as friends. I say thank you to all my friends from Salerno, my beautiful city. In particular Luca, Sibbone, Frank, Franzis, Fabrizio, Anti, Lisa, Dylan, Valentina: I consider you all part of my life, part of my family. I want to say thank you to my fellow musicians Francesca and Marco. I wish a successful career to you both: you are chasing a dream also for me. I say thank you to all my everlasting friends: my dear Kiara, Pigi, Alex, Ped, Peppe, Raaele, Giammy, Oscar and everyone who grew with me, unfortunately I cannot nominate everyone. I warmly want to say thanks to my friends from Pisa, I love you all. In particular Fabio (you are a brother for me), Angela, Mary, Azzurra, Silvia, Piero, Grace, Giovanni and Luigi.

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tellin' myself

it's not as hard as it seems... (Robert Plant, James Patrick)

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Introduction

We refer to synchrotron radiation as the electromagnetic radiation emitted when high energy charged particles are radially accelerated. It's well-known, since the late 1960s, that electrons in accelerator rings emit electromagnetic waves over a spectrum ranging from infrared to X-rays. This phenomenon was at rst considered as a problem, insofar the synchrotron emission can actually be an energy-loss channel in such elementary particle experiments. The point of view can drastically change if one thinks to use or even to improve emission mechanisms of this radiation, to get, for all purposes, a light source. On the other hand, the developments in LASER technology with regard to infrared and visible light have been a rather fast and fairly easy process, for the availability of technologies capable of producing coher-ent, brilliant light in those ranges from the very beginning. Not the same fate befell ultraviolet and X-ray light technologies that, before the advent of synchrotron facilities, had been stuck in their initial stage, not so much progressed since the days of their discovery, due to Röntgen towards the end of the nineteenth century. Help to make synchrotron light very interest-ing the spectacular properties of brilliance, directionality, monochromaticity, tunability and coherence, which are all together interconnected and make these kind of sources the only capable of working in the X-ray range with so high eciency and versatility. Behind the process of radiation emission that's realized in the storage rings, there's the presence of magnetic devices, namely bending magnets, undulators and wigglers, which interact with ul-trarelativistic electron bunches, accelerating them in radial direction and stimulating energetic and collimated electromagnetic radiation beams. We don't describe exactly what synchrotron facilities actually are, nor we do an overview of the possibilities they oer to their users, but we want to talk about several new X-rays sources, whose mechanisms are in close analogy to those of synchrotrons. The synchrotrons teach us that in order to obtain high quality radiation, with really useful properties, we need a radial accel-eration eld and relativistic electron bunches which interact with it. One could think: Is it really necessary to put up large buildings, or even use macroscopic magnetic devices to achieve radial acceleration of relativistic charges ?. We surely know that the to date developed technologies work well, but new trends in acceleration eld are taking hold, actually bringing

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in plasmas.

Thomson Scattering is the interaction of energetic electrons with an elec-tromagnetic wave. Thomson Scattering based sources work by coupling and synchronizing a LINAC with a LASER or by splitting a LASER beam so as to produce a LASER-plasma accelerated electron bunch with a rst branch and leaving the second interact with them. It's rather clear that Thomson Scat-tering has all the ingredients for the synchrotron radiation recipe, namely ultrarelativistic travelling electrons trasversally accelerated by a counter-propagating electromagnetic eld. Considering the LASER-LINAC congu-ration, radiation sources can be realized, very similar in properties to those of synchrotron facilities, but with a really notable saving in cost and size. On the other hand, riding the wave of new particle acceleration experiences, in which high intensity LASER-plasma interactions are involved, we can easily imagine, in a not too distant future, to get rid of the LINACs, giving birth to new X-rays Thomson Scattering sources that work with the only LASER device.

Betatron radiation in LASER-plasma acceleration experiments is the radia-tion emitted by longitudinally accelerated electron bunches which interact, in the so-called bubble regime, with an electrostatic ionic radial eld, gener-ated by positive charges, much heavier than the electrons, in a region deeply depleted of electrons by the high intensity LASER ponderomotive forces. Trapped electrons in this region, accelerated by plasma waves longitudinal elds, experience a radial eld which forces them to make the typical beta-tron oscillations. The process of longitudinal acceleration is called "LASER Wake Field Acceleration", or simply LWFA, because strong LASER pulses propagating in plasmas generate charge separation, responsible of huge ac-celerating eld gradients, through the excitation of electron plasma waves. The achievement of extended stable wakeelds is the main concern of plasma accelerators together with the controlled synchronized injection of charged particles into the wave buckets, and the generation of mono-energetic beams. The longitudinal elds can rapidly accelerate electrons to very high energies, as surfers gain velocity riding an ocean wave. The most appealing feature of LWFA is the possibility to achieve a particle energy gain unthinkable for the conventional accelerators which, limited by electrical breakdown phe-nomena, can work with 50 MV/m maximum elds. In a LASER-plasma experiment, where a 1018_/cm3 _{plasma is used and a centimetric acceleration}

path is maintained without signicative dephasing eects, GeV particles can be generated. The comparison between the two philosophies will leads us, in the next future, to lean toward the LASER-plasma acceleration in the aim of overcoming the intrinsic limits of the actual accelerators, nevertheless we can not actually say that the advances in this new technique are such to already oer a better-working alternative in respect to the conventional

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parable to those achieved via the actual techniques. LWFA is not strictly our subject so not too many other words we'll spend about in the course of this work, but the radiation emitted in such interactions is our concern. The rst chapter is an overview on the physics related to the LASER-plasma interaction, that is the frame in which every other topic of this work is col-located. Next chapters are organized in such a way to describe at rst the radiation emitted by relativistic charged particles, then the Thomson Scat-tering process in both LINAC-LASER and All-Optical congurations and the Betatron oscillations in plasmas at some extent. Finally we report on a LASER-plasma experiment at ENEA Research Center in Frascati, in which the main concern is on X-ray diagnostics in view of their utilization in the forthcoming experiments dedicated to the above mentioned innovative X-ray sources.

The present work of thesis has been carried out between the Physics De-partment of Pisa University and the ENEA Research Center in Frascati. The main ideas about innovative radiation sources based on LASERs and plasmas were showed during the lectures on Quantum Optics given by Prof. Danilo Giulietti. The experimental activity on LASER-plasma interaction and related X-ray diagnostics has been developed in the ABC ENEA facil-ity, led by Dr. Riccardo De Angelis. The most important theoretical part of the work has been developed with the fundamental collaboration of Dr. Giuseppe Dattoli.

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LASER-matter interaction at

high intensities

1.1 LASER-produced plasmas

When an high intensity LASER is sent on a solid target, a plasma on its surface is generated, and it is usually referred to as LASER-produced plasma. An high intensity LASER is generally a pulsed LASER; energies obtained via more than one amplication stages and high focalizations, up to the diraction limit, can play an important role on the intensity value, but the shortness of the pulse is more important than these parameters, determining impressive energy uxes per unit time and surface. It's given known that a pulsed LASER is a LASER device on which Q-switch or mode Locking techniques are applied, eventually together with a nal stage in which the pulse, stretched by a rst couple of gratings, is subsequently amplied and nally compressed in vacuum by a second pair of gratings (Chirped Pulse Amplication technique or, simply, CPA1_{) . A LASER-produced plasma}

is a matter state, in which partial or even total ionization is induced by LASER irradiation. It's a uid state composed by electrons and ions; plasma as a whole, is neutral, but for it's made up by charges, it's easy to guess its high reactiveness to electromagnetic stimulations. Figure (1.1) shows typical proles of density, temperature and expansion velocity of the plasma generated by irradiating a solid target with a powerful, 1 µm wavelength, nanosecond laser pulse.

Plasmas have a proper resonance, the so called plasma frequency in S.I. units:

ωp =

s nee2

mε0 (1.1)

where we denote as nethe number of electrons per unit volume, e the

elemen-tary charge, m the electronic mass and ε0 the dielectric constant of vacuum.

Many eects in plasmas can be related just to this resonance. Electrons are 11

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Figure 1.1: 2_{Typical temperature, density and expansion velocity proles of}

a plasma generated by irradiation of a solid target by a nanosecond LASER pulse focused on target at an irradiance of 1014_W/cm2_.

the major responsible of the plasma thermalization for their light mass, and of energy transfer's mechanisms, purely internal or even toward the external environment. A LASER-produced plasma is quite never a medium in ther-modynamical equilibrium, but in an average way an electronic temperature and a ionic one, generally quite lower, are dened; that's why it's usually referred to plasma temperature as the electronic one. Plasma temperature depends on the properties of electron heat conduction, and by the absorbed energy from the LASER intensity:

Te∼ 3 × 107(Iabs[W/cm

2_]

f ne[cm−3]

)2/3 eV (1.2) Where "f" is the ux limiter with typical value ∼ 0.1, it takes into account that not a simple classical description can completely describe the real elec-tronic heat conduction3_{. Typical temperature values for LASER-produced}

plasmas stand near hundreds electronvolt. Another very important parame-ter in the LASER-plasma inparame-teraction is the so called critical density; i.e. the plasma density at which the plasma frequency equals the LASER frequency.

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The LASER can not penetrate higher densities, because the refractive index becomes imaginary, so that the surface in which the plasma density is the critical one strongly reects the LASER radiation:

nc=

mω2₀ε0

e2 (1.3)

where ω0 is the LASER pulsation.

A fundamental topic about LASER-produced plasmas is the so-called hot electrons. These particles somehow don't obey to the statistical Maxwellian distribution of the velocities. When we say "somehow", of course we know that there are precise acceleration mechanisms, which origin by the absorp-tion of an important part of the LASER energy, giving birth to a big amount of energetic electrons, in respect to what provided by a distribution with center on a certain temperature. These mechanisms involve electron plasma waves, about which we're going to quite diusely talk soon.

1.2 Absorption mechanisms

Among the principal mechanisms of energy absorption we cite the Inverse Bremsstrahlung Absorption, the Ion Turbulence Absorption, Resonance Ab-sorption,Brunel eect.

1.2.1 Inverse Bremsstrahlung Absorption

Electrons, interacting with LASER elds, can transfer their acquired oscilla-tion's energy to the ions via collisions. The linear absorption coecient for this process is related to the imaginary part of the laser refraction index, i.e. it contains information about absorption eects:

kib= 3.1 × 107Zn2eln Λω −2 0 [1 − ( ωp ω0 )2]−1/2(Te[eV ])−3/2 cm−1 (1.4)

ln Λis the Coulomb logarithm for electron-ion collisions ∼ 24 − ln n1/2e

Te[eV ] for

Tim_me_i < 10Z2eV < Te. This absorption mechanism is favored in regions with

low temperatures and high densities. Low temperatures allow electrons and ions to interact strongly, due to the dependence of the Coulomb cross-section on the electron velocity as 1/v4

e; high Z increases the electron-ion Coulomb

interaction; high electron density raises the average number of the collisions. IBA is really eective near the critical surface, provided that the critical region variation length scale is not too short, so reducing the absorption spatial length, and on the contrary it should be not too long, because of the risk for LASER energy to be substantially dissipated before reaching the critical region. The electron velocity under the electric eld action is dened as quivering velocity:

vq=

eE0

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One can dene an eective velocity, which takes into account the thermal motion together with the LASER-induced quivering:

vef f =

q

v_th2 + v2

q (1.6)

For high intensity LASERS, i.e. very important quivering motion, the IBA process becomes less eective, for what we have already said about the Coulomb collisional cross-section, which decreases for increasing electrons velocities. Furthermore, if a big amount of hot electrons is produced at high LASER irradiance, the number of low-velocity electrons, the major responsi-ble for this type of absorption mechanism, is dramatically reduced, according to the fact that the rate of the energy gain from the LASER intensity can reach values much higher than the rate of electrons mutual energy-sharing, in such a way that a lack of low-energy electrons is produced in the Maxwellian distribution.

1.2.2 Ion Turbulence Absorption

The coecient in (1.4) is obtained by assuming random motion of the ions. It can be shown that a collective correlated motion can strongly enhance the Inverse Bremsstrahlung Absorption mechanism, this phenomenon is referred to as Ion Turbulence Absorption4_.

1.2.3 Resonance Absorption

When a high intensity LASER impacts obliquely on a solid target, with an electric eld component in the plane of incidence, i.e. −→EL ·

−−→

∇n_e 6= 0, an ecient longitudinal electron plasma oscillation at the critical surface can be excited. The damping of the excited electron waves leads to conversion of electromagnetic LASER energy into thermal energy. As we'll show later, a plasma wave can drive charges, in particular plasma electrons, leading them to very high energies. The turning point for a light wave impinging on a plasma density gradient at an incident angle θ, occurs a a density ntp

e given

by the classical theory2_:

ntp_e (θ) = nccos2θ (1.7)

If the electric eld is completely on the density gradient's direction θ = π/2, it is reected much before (at ne << nc) it eciently drives an electron

plasma wave at the critical density; on the other hand, for θ = 0, it has no component on the gradient's direction, and it cannot drive plasma waves. The angle at which a maximum eciency of the process is reached is ∼ arcsin 0.8(ω0Lne/c)

−1/3_{, in which L}

ne is the scale of the density gradient.

Now we show a simple way to relate the resonance absorption to the LASER polarisation and the turning point position. The real part of the dielectric

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function in a plasma can be expressed as: ε = 1 − ω 2 p ω2₀ = 1 − ne nc (1.8)

We get from the Poisson equation:

− → ∇ ·−→E = − − → ∇ε ·−→E ε = − → ∇ne· − → E nc− ne = −4πeδne (1.9) an expression for δne: δne= − → ∇n_e·−→E 4πe(ne− nc) (1.10)

which explains that the "s" polarisation can't drive a Langmuir wave because no longitudinal charge displacement can occur when−→EL·

−−→

∇ne= 0, while the

"p" polarisation can very eciently drive a plasma wave in proximity of the critical surface, for the charge displacement is not zero (−→EL·

−−→

∇ne6= 0) and

it increases when the critical surface is approached.

1.2.4 Brunel eect

Very intense LASER radiation, impinging obliquely on a metallic surface or on a sharply bounded overdense plasma, is capable to extract electrons and push out them into the vacuum; the kinetic energy this way acquired, is dissipated in several ways when the particles re-enter the plasma5_{, which}

is neutral. This eect is more ecient than the resonance absorption when the quivering velocities with which the electrons return into the plasma, are much bigger than the product ω0Lne. This eect becomes important in

regimes of electron relativistic quivering motion, namely for ultra short laser pulses.

1.3 Parametric instabilities

When LASER light propagates over a long region of underdense plasma with Lne >> λ0, many physical processes can take place, which may produce

elec-tron and ion plasma waves and scatter LASER light. The damping of these waves produces plasma heating, while the scattering reduces laser absorption in the plasma. In addition, self-focusing and lamentation may give rise to local enhancement of the laser intensity, making it easier to achieve insta-bility threshold conditions. A parametric instainsta-bility is characterized by the following relations, which express the conservation of energy and momentum:

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ω0 = ω1+ ω2 − → k0 = − → k1+ − → k2 (1.11) where ω0 and − →

k0 represent the pulsation and the wave vector of the

imping-ing LASER eld respectively. By excitimping-ing electronic and ionic waves or by scattering, the LASER eld gives birth to two kind of waves, which can be longitudinal electrostatic waves when electronic or ionic waves are produced, or transversal electromagnetic waves when light is scattered by the plasma medium. In the following we briey report the most important parametric instabilities, relative to the LASER-plasma interaction.

1.3.1 Stimulated Brillouin Scattering (SBS)

When the LASER eld acts on a low frequency density perturbation, typi-cally of the ionic waves, it produces a current, then a reected wave which can make interference with the initial one: this positive feed-back loop can let an instability grow.

ω0 = ωB+ ωi − → k0 = −→ kB+ − → ki (1.12) Usual experimental situations are ω0 >> ωi and

− → k0 k − −→ kB, in such a way that: − → ki ∼ 2 − → k0 −→ kB ∼ − − → k0 ωB ∼ ω0(1 − 2 vs c) (1.13) where vs = q ZkTe+3kTi

mi is the velocity of sound in the plasma (mi the ionic

mass and Ti the ionic temperature). The scattered LASER light is

red-shifted by the acquiring of momentum by ions. This process tends to reject radiation out from the plasma, reducing its heating. The LASER intensity threshold for SBS results:

ISBS & 7 × 1012 Te[eV ] Lne[µm]λ0[µm] nc ne W/cm2 (1.14)

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1.3.2 Stimulated Raman Scattering (SRS)

When the LASER eld acts on a high frequency density perturbation, typ-ically of the electronic waves, it produces a current, than a reected wave which can make interference with the initial one: this positive feed-back loop can let an instability grow, just as in the case of SBS.

ω0 = ωR+ ωe − → k0 = −→ kR+ − → ke (1.15) It's easy to verify that this process is possible only for region ne< nc/4.We

report the red shift of incoming radiation:

ωR= ω0(1 −

r n_e nc

(1 + 3λ2

Dke2)) (1.16)

where the Debye length λD is dened as

q

ε0kTe

nee2 . The threshold intensity

for the activation of SRS results: ISRS & 4 × 107

1 Lne[µm]λ0[µm]

W/cm2 (1.17) For long scalelengths of density variation, up to 10% of the initial LASER energy can be converted by SRS into hot electrons, because of the production of plasma waves that allow some electrons to gain momentum, i.e. energy.

1.3.3 Two plasmon decay (TPD)

At the surface characterized by an electron density ∼ nc/4a LASER photon

can decay into plasmons (plasma waves). ω0 = ωB+ ωR − → k0 = −→ kB+ −→ kR (1.18) Plasma waves can undergo damping eects when for very high temperatures, the plasma wavelength become shorter than the Debye length which takes into account the shielding of electrical elds in the plasma medium. For small damping (k2

B+ k2R)λ2D . 0.1, the density at which TPD can occur is:

ne= nc 4 [1 − 3 2(k 2 B+ kR2)λ2D] (1.19)

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The range of densities for which this process is allowed is given by6_: 0.19 < ne nc < 1 4[1 − 3 2k 2 0λ2D] (1.20)

The two plasmons frequencies come out to be: ωB,R= ω0 2 [1 ± 3 4(k 2 B+ kR2)λ2D] (1.21)

These plasmons can couple with LASER eld, giving rise to the emission of 3/2ω0 radiation. We nally report the intensity threshold for TPD:

IT P D = 5 × 1012

Te[eV ]

Lnc/4[µm]λ0[µm]

W/cm2 (1.22) When the rst harmonic of a short pulse LASER decays in two plasmons, for intensities higher than the threshold, radiation at 3/2ω0 can be detected.

The 3/2ω0emission is characterized by a spectral structure due to the slightly

dierent frequencies of the red and blue plasmons. When this structure is re-solved, it provides information about the plasma temperature. Anyway, the presence of this parametric instability is the signature that electron plasma waves have been generated.

1.4 Filamentation and self focusing

The occurrence of lamentation and/or self-focusing of a LASER beam prop-agating in a plasma causes substantial modications of the plasma condi-tions, producing local density depression and an increase in the electron temperature. An increase in the local electron temperature aects the emis-sion properties of the plasma. In particular, the spectral distribution and intensity of the X-ray radiation emitted from the plasma region involved in the lamentation processes will result modied. All the emission processes arising will reect the modications occurring in the plasma, with a time re-sponse typically of the order of a picosecond [7, 8]. Filamentation of LASER light in a plasma can occur when a small perturbation in the transverse intensity prole of the incident laser beam induces a perturbation in the electron density. This perturbation can be generated either directly, via the ponderomotive force, or indirectly as a consequence of localized collisional heating and subsequent plasma expansion. We report the expression of the ponderomotive force: −→ Fp= − − → ∇I0 2cnc (1.23)

Once the perturbation has been generated, refraction of the laser light in the electron density perturbation enhances the intensity perturbation providing the positive feed-back for the instability. The most important theoretical

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results concerning the lamentation instability have been derived by using a simple model in which a sinusoidal intensity perturbation is imposed on a plane light wave interacting with a plasma. Recently, the eect of non-local heat transport has also been included9 _{to account for deviations from}

the classical Spitzer-Harm conductivity occurring when the electron temper-ature scale- length is shorter than the electron mean free path. In these conditions thermal transport must be described solving the electron Fokker-Planck equation. This modication to the electron heat conductivity leads to the following expression for the spatial growth rate of the lamentation instability: K = k⊥ 2√ε[2 ne nc (γp+ γT κSH κF P k2₀ k2 ⊥ ) − k 2 0 k2 ⊥ ]1/2 (1.24) where ε = 1 − ne/nc is the plasma dielectric function, κSH and κF P are

the Spitzer-Harm conductivity and the eective Fokker-Planck conductivity that accounts for non-local transport eects10 _{and k}

⊥is the wave-number of

the sinusoidal spatial modulation. We can basically notice three terms. In the rst term γp = (1/4)(Z/(Z + 1))(v2q/v2th) accounts for ponderomotive

eects, vq and vth being the quiver and thermal velocity of the electrons

respectively, and Z the charge state of the plasma. In the second term γT = c2S/ω02κSHkBTe accounts for thermal eects, S being the background

inverse bremsstrahlung heating rate and Te the electron temperature. The

third term is due to diraction and gives negative feedback as it tends to de-focus the beam. The important consequence of non-local electron transport in the theory of lamentation instability is that the threshold of the insta-bility is substantially reduced since, generally speaking, the growth rate is increased. In addition, an optimum perturbation wavelength is found which maximizes the growth rate, in contrast with the theory based on the classical electron transport, which predicts a constant growth rate over a wide range of perturbation wavelengths ([11] and references therein).

Finally we spend some words about the relativistic self focusing; the implicit expression of the relativistic ponderomotive force is given by:

−−→

F_prel= −mc2−→∇γ(I0) (1.25)

The spatial variation of the refraction index is given at relativistic intensi-ties by a spatial variation of the electronic masses, bigger where higher is the LASER intensity. The reason of the instability is due now just to the relativistic increasing of the electrons masses.

1.5 X-ray emission from LASER-produced plasmas

In the phase of plasma thermal relaxing, light is emitted by plasmas in basically three ways, which don't depend directly on the interaction with

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the LASER eld; they are the bremsstrahlung radiation, the electron-atom recombination radiation, and the atomic de-excitation transition lines. Elec-tron moving in the ionic environment with a sucient thermal velocity, don't return in a bound state, but they experience a friction force in such a way that by losing energy (decelerating), they are forced to irradiate. If the ther-mal energy is very high (hundreds electronvolt), the broad bremsstrahlung spectrum of radiation will extend up to the X-ray region. The recombina-tion radiarecombina-tion is due to those electrons which recombine in the lower atomic states of the free ions; when they undergo this process without having a sig-nicative kinetic energy, typical spectral edges become visible. Finally, the emission lines are due to transitions of electrons from upper to lower bound states.

1.5.1 Bremsstrahlung

We just report the theoretical spectral distribution of the bremsstrahlung radiation produced by electron-ion collisions in plasmas:

W_Bν = 6.8 × 10−38Zn2_eT_e−1/2[oK]e−kTehν _ergs−1_cm−3_Hz−1 (1.26)

Furthermore we stress on the relation between the maximum of such a radi-ation spectrum and the corresponding electronic temperature:

Te[eV ] ∼ 500Emax[keV ] (1.27) 1.5.2 Recombination

About the peculiar feature of a recombination spectrum to have edges at some characteristics frequencies, we have already said something, but here below we see it in formulas:

hν = 1 2v

2

th+ EZn (1.28)

where a photon energy is equated to the thermal energy of an electron which recombines in an atomic level n losing an energy En

Z. In (1.29) the

recombi-nation radiation spectral distribution is reported: W_rν = 2.2 × 10−32W_BνZ 3_n2 e Te3/2 ∞ X n=1 1 n3[ 13.6eV × Z2 n2_kT e − hν kTe ] ergs−1cm−3Hz−1 (1.29) Finally we make the following comparison:

Wr

WB ∼ 2.4

13.6eV × Z2

kTe (1.30)

where an integration on the whole spectra is understood. From the equation above we deduce that for high temperatures, the bremsstrahlung processes dominates over the unfavored atomic recombination. But the ionic Coulomb eld is nevertheless important for plasmas composed by heavy elements.

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1.5.3 Atomic lines

The spectral lines due to bound to bound transitions in atoms are very important to characterize atomic species but especially in order to have in-formation about the plasma temperature and density. For hot plasmas the Doppler eect produces a gaussian broadening of the atomic lines from which the plasma temperature can be inferred:

∆νD = 6.65 × 10

−9

r Te[eV ]

A (1.31)

with A the mass number. For high density plasmas, the pressure broadening due to collisions between particle or to the interaction with high electrical elds (Stark eect) produces a Lorentzian line-shape, with FWHM:

∆νP =

σvrelnp

2π (1.32)

where np is the density of perturbers, σ the total cross section of the specic

collisional process, and vrel the relative velocity between particles. The

nat-ural broadening of these lines is usually negligible with respect to the above mentioned ones.

1.6 Plasma acceleration

The maximum acceleration electric elds in conventional accelerators are of the order of 50MV/m, for the air breakdown phenomena that occur even in the deepest achievable vacuum. To accelerate electrons up to 1T eV , one would need an accelerator ∼ 20Km long. The conventional accelerators are linearly (LINACs) or circularly shaped; the larger LINAC devices currently existing are few kilometers long, while the acceleration rings suer the prob-lem of breemsstrahlung losses whose trend is∝ E/(m R), where E/m is the energy-mass ratio of accelerated particles and R is the ring's radius. As a con-sequence, to achieve high energy without signicant losses R must tend to ∞. Plasma environment instead can oer very high acceleration elds on milli-metric scale without breakdown problems, being an already ionized medium. The plasma acceleration mechanism nds its reason in plasma waves excita-tion. Electron plasma waves are electronic charge density oscillations, which produce very high longitudinal electric elds up to (∼ 100GeV/m). At least three ways to stimulate electron plasma waves exist: one uses two LASERs beats such that ω1− ω2 = ωp[12, 13, 14], where ω1, ω2 are the LASER

fre-quencies and ωp the plasma wave frequency; coulombian excitation realized

by sending an electron bunch of an half plasma period, works on the direct interaction an ekectron bunch and the plasma electron density; nally, by using an ultrashort LASER pulse of an half plasma period time duration, one can produce a quasi-resonant electronic charge density displacement making

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possible a plasma wave growth. The plasma acceleration exploiting the gen-eration of plasma waves on the wake of a short LASER pulse is often referred to as LWFA, i.e. Laser Wake Field Acceleration.

1.6.1 Linear LWFA

When the the electronic charge variation is much less than the initial equi-librium density, a low amplitude plasma wave is produced. In this regime, the acceleration of the electrons driven by the plasma waves is called Linear Laser Wake Field Acceleration.

Consider the following scheme, which shows how the acceleration is possible. The pink-shadowed areas in gures (1.2) and (1.3), point out the acceleration region: electric eld in that region has the right sign to drive and accelerate forward electrons.

-+

-0.0 0.5 1.0 1.5 2.0 2.5 3.0 x 0.995 1.000 1.005 nen0

Figure 1.2: Electron charge density distribution ne along the x direction in

arbitrary units, divided by the unperturbed plasma electron density n0.

Plasma wave is a travelling longitudinal perturbation, therefore dephasing eects between accelerated electrons and plasma waves have to be controlled in order to not achieve the contrary purpose of decelerating electrons. Fur-thermore we point out that not the whole electron charge distribution is aected by the travelling perturbation, but only a small fraction of it as shown in gure (1.2). Now we estimate the longitudinal electric eld ampli-tude, starting from classical electrodynamics:

− → ∇ ·−→E = ρ ε = −e∆ne ε (1.33)

where ∆neis the eective variation of charge density involved in the plasma

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0.5 1.0 1.5 2.0 2.5 3.0 x -2 -1 1 2 Ex

Figure 1.3: Longitudinal electric eld along the x direction in arbitrary units, related to the excited plasma wave.

form ∆ne= ∆ne0sin(kpx−ωpt), we have an associated longitudinal electrical

eld: Ex= e∆ne0cos(kp x − ωp t) kpε (1.34) of maximum amplitude: E_x(max)= e∆ne0 kpε = ∆ne0 n0 n0c √ m ωpε ∼ ∆ne0 n0 p n0[cm−3] [ V cm] (1.35) so that we can see that bigger is the plasma wave amplitude, namely the variation of the electronic charge density, more intense results the longitu-dinal accelerating electric eld. The Bohm-Gross dispersion relation which describes the plasma oscillation modes ωpl at a certain temperature is:

ω2_pl= n0e 2 εm + 3kBT m k 2 p = ω2p+ 3kBT m k 2 p = n0e2 εm (1 + λ 2 Dk2p) (1.36)

For typical values of a LASER-produced plasma parameters, such as T ∼ 100 eV /kb and n0 ∼ 1018/cm3, one has λD ∼ 0.7µm and λp ∼ 33µm, so that the waves are only weakly aected by the pressure term λ2

Dk2p and a

. 60f s LASER pulse can eciently excite them. In order to be accelerated, electrons must enter with the right phase and the suitable velocity in the plasma wave in such a way to be eciently driven during the acceleration path. The wake phase velocity is imposed by the group velocity of the LASER pulse, in formulas:

v_glaser = c s 1 −ω 2 p ω2 = v wake φ (1.37)

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The Lorentz factor corresponding to the wake eld results: γp = 1 q 1 − (v wake φ c )2 = _q 1 1 − β2 p = ω ωp (1.38)

that's coherently a decreasing function of plasma density. To easy calculate the dephasing length, we impose that the electrons enter the decelerating region, after having traveled a λp/2long path at the speed of light c, i.e. in

the laboratory frame:

Ldeph= ctdeph =

cλp

2(c − vwake

φ )

≈ λpγp2 (1.39)

At this point we can estimate the maximum energy gained by the electrons15_:

E_maxelectrons= eE_x(max)Ldeph= 2π

∆ne0

n0

γ_p2mc2 (1.40) The (1.40) shows that the maximum energy gained scales as the inverse of the plasma density. Therefore, one could think to create long low density plasmas to maximize the acceleration elds over longer acceleration paths, but it's a pretty dicult goal, for the limited focalization lengths experimentally achievable. The most common experimental techniques exploited to enhance the acceleration lengths, consist in the use of gas capillary guides, in which an high power LASER beam can be focused over centimetric distances16_,

and the exploitment of the relativistic self-focusing, which allows the pulse to maintain high intensity values over several Rayleigh lengths, despite to the diraction.

1.6.2 Blowout regime

At very high intensities (& 1018_W/cm2_{) the so-called blowout regime}

(cav-itation) can be achieved. The LASER ponderomotive forces, that in the relativistic case (a0 = 8.5 × 10−10pI0λ20 > 1) become very important,

gener-ate an electrons-cleared region, spherically shaped at a rst approximation17

whence also the name bubble regime. The relation which relates the LASER intensity to the dimension of the bubble for a certain plasma density is reported18_:

ωp

c R ∼ √

2a0 (1.41)

where R is approximately the radius of the bubble. In this case we don't talk properly about electron plasma waves, while the structure of LWFA is realized in these quasi spherical regions with positive charge redundance, where electrons are self injected from the same background plasma electrons and/or by the restoring forces provided by the ionic eld. The electrons injected inside the bubble bunch while accelerating forward by the bubble

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longitudinal electric eld (g. 1.4). The electric eld inside the bubble is determined by considering a uniform and static (for their high inertia) dis-tribution of ions spherically shaped. The transversal action of this eld is responsible of the oscillations we are going to study in a diuse way, called Betatron oscillations. While, for the bubble is propagating on the wake of the LASER pulse, the longitudinal oscillations result into an acceleration path in the laboratory frame. It's shown in [19] that an ultra short LASER front which impacts on a plasma, undergoes an erosion (said also pump de-pletion), nally etching back with a velocity vetch ∼ cω

2 p

ω2. This phenomenon

is typical of ultra-short pulses propagating through a plasma. The leading fronts of such pulses tend to be steepened during their propagation because they release energy in exciting electron plasma waves. Nevertheless a steep prole is very eective in exciting an electron charge density perturbation, that's why a feed-back loop is established, resulting in a localized (at the front) depletion of energy and in a futher steepening of the leading edge prole, namely in an erosion. Numerical simulations show that the number of photons is conserved in the erosion process, in such a way that we can talk about of photons deceleration19_{. The front of the LASER that excites}

the wake moves backward as the pulse etches back with vetch. The phase

velocity of the wake, at which accelerating electrons have to be matched, will be now

vφ= vglaser− vetch ∼ c(1 − 3ω2

p

2ω2) (1.42)

The laser energy will be depleted over a length Lpd ∼ ω

2 p

ω2 c τF W HM, where

τF W HM is the Full Width Half Maximum LASER pulse duration. The

de-phasing length, evaluated by imposing that at the dede-phasing point the elec-trons have traveled over a distance of the order ∼ R, results:

Ldeph=

2ω2 3ω2 p

R (1.43)

It's the distance after which the electrons begin to decelerate. The expression of the electron maximum energy gain in the blowout regime is:

E_maxelectrons= a0mc2

2ω2 3ω2 p

(1.44) One of the most interesting aspect of the blowout regime, is that the electrons injected in the bubble experience an oscillatory motion while accelerating for-ward, in such a way that they become ultrarelativistic synchrotron radiation sources. The calculations on what is known in literature as Betatron radi-ation in plasmas will be the aim of the chapter 3, where we will study the emission spectra in the blowout regime both in the non resonant and in the resonant case, i.e. when the electrons don't interact with the LASER eld, and when their oscillatory motion is forced by the LASER electromagnetic eld.

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Figure 1.4: (Color) A sequence of 2-dimensional slices (x-z) reveals the evo-lution of the accelerating structure (electron density, blue) and the laser pulse (orange). Each plot is a rectangular of size z= 101.7µm (longitudinal direction, z) and x = 129.3µm (transverse direction, x). A broken white circle is superimposed on each plot to show the shape of the blown-out re-gion. When the front of the laser has propagated a distance (a) z = 0.3 mm, the matched laser pulse has clearly excited a wakeeld. Apart from some local modication due to beam loading eects, as seen in (b) this wakeeld remains robust even as the laser beam propagates though the plasma a dis-tance of 7.5mm [as seen in (c) and (d)] or 5 Rayleigh lengths. After the laser beam has propagated 2mm [as seen in (b)] into the plasma, one can clearly see self-trapped electrons in the rst accelerating bucket. The radial and longitudinal localization of the self-trapped bunch is evident in part (c). After 7.5mm the acceleration process terminates as the depleted laser pulse starts diracting18_.

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Thomson Backscattering

The interaction of an electromagnetic wave with a counterpropagating elec-tron bunch is referred to as Thomson Backscattering. Head-on scattering of photons by relativistic electrons really oers particular and spectacular possibilities, which don't belong to other electron-photon interaction con-gurations. First of all, the relativistic Doppler eect experienced by elec-trons who see a radiation wave travelling toward themselves, leads to an important frequency-shift in the oscillatory dynamic of electrons under the eld. If one considers that an accelerated relativistic charge, seen as an electromagnetic source, produces nely collimated beams, great geometric properties of directionality can be achieved for the emitted radiation. A fur-ther relativistic Doppler eect, due to the motion of the electrons towards the observer in lab reference system, is the cause of a further frequency-shift, which characterizes the detected radiation: the combination of these two Doppler-shifts is the real cause for which infrared LASER light can be transformed in X-rays. By using low emittance electron bunches (we will dene later the emittance parameter, which takes into account the angular dispersion of a bunch), produced by LINAC or by LASER-plasma acceler-ation processes, one generally obtains powerful properties of brilliance. For what concerns the monochromaticity of the Thomson Backscattering pro-duced X-ray radiation, we must focus our attention, as we'll discuss later, on the angle-energy correlation. The re-emitting travelling electrons after the interaction, emit photons of a certain energy only at a well-established and in turn correlated angles: the most energetic photons are those perfectly backscattered. One can x a certain degree of monochromaticity inserting a diaphragm between the interaction region and the detectors with such an aperture to select only the desired bands from the emitted spectrum. An observation must be done about the fact that not perfectly monochromatic electron bunches can of course aect the monochromaticity degree of the radiation that one could expect, in fact electrons with dierent energies are responsible of dierent Doppler frequency-shifts, i.e. a spectral enlargement

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of the emitted radiation. Coherence in the radiation, with following radi-ation gain and amplicradi-ation as those in the X-FELs (X-ray Free Electron LASERs), can be achieved when a coherent microbunching occurs due to the fact that the already Doppler-shifted radiation emitted by the electrons in the bunch tail can interact with the electrons ahead. It can be showed that the average electron-electron distance in such a conguration becomes just the wavelength of the emitted radiation, so that the single electron emissions can add in phase. In this work we don't consider these collective phenomena inducing coherent summation of the single particle emissions, analyzing just the cases of incoherent superposition, more appropriate for the experimental settings we're interested to.

2.1 Dynamics of relativistic electrons interacting

with an arbitrary electromagnetic plane wave

packet

In this section, we calculate the electron trajectories from rst principles of relativistic mechanics when they interact with an electromagnetic plane wave, not necessarily monochromatic. The formalism Hamilton-Jacobi will be adopted here, as in[20]. The four-dimensional form of Hamilton-Jacobi equation for the interaction of our interest is:

gik(∂S ∂xi − e cAi)( ∂S ∂xk − e cAk) = m 2_c2 _(2.1)

where gik _{is the Minkowski tensor with (1, 3) signature, S the Hamilton}

function, xi _{the radius four-vector, e the modulus of the elementary charge,}

m the electronic mass and c the light speed in vacuum. We make explicit the planar symmetry in the argument of the four-vector potential by setting Ai ≡ Ai(ξ), with ξ = kixi = ω0(t + z_c), where the sign + in the phase

expression stands for the head-on interaction on the z-direction; then we impose the Lorentz gauge condition:

∂Ai ∂xi =

∂Ai

∂ξ ki= 0 (2.2) that is equivalent to Ai_k

i = 0. In order to nd the Hamilton principal

function, we look for a solution of the kind:

S = −fixi+ F (ξ) (2.3)

where fi is the four-vector who satises the condition fifi = m2c2, namely

the electron generalized four-momentum under the action of the electromag-netic eld, and F (ξ) is a unknown function to determine. Substituting (2.3) into (2.1) yields: 2η∂F ∂ξ + 2 e cfiA i₋e2 c2AiA i _{= 0} _(2.4)

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where η = fiki. From the above we can infer the expression for F and therefore for S: S = −fixi− efi η Z Aidξ + e 2 2η Z AiAidξ (2.5) Being ki_{= (}ω0 c , 0, 0, − ω0 c), we obtain for η: η = ω0 c (f0+ f3) (2.6) Expanding the square of fi, we have:

f₀2− f₃2− κ2 = m2c2 (2.7) where we have denoted by κ the modulus of the generalized transversal four-moment: −→κ is a constant of motion, and it will be naturally interpreted as the electron initial transverse moment just before impacting on the photons. It's straightforward to deduce at this point the following equation, by using (2.6),(2.7) and some algebra:

f0− f3 = (

ω0

c )

κ2+ m2c2

η (2.8)

Now, by developing the algebra below:

f3x3− f0_x0 ₌ _(f3_{+ f}0_)(x3_{− x}0_{) + x}0_f3_{− x}3_f0₌ = (f3+ f0)(x3− x0) + (f3− f0)(x3+ x0) − f3x3+ f0x0 = = (f 3_{+ f}0_)(x3_{− x}0₎ 2 + (f3− f0_)(x3_{+ x}0₎ 2 (2.9) and by using preceding equations, one gets the following expression for S:

S = −→κ · −r→⊥− cη 2ω0 (ct − z) −κ 2_{+ m}2_c2 2η ξ − efi η Z Aidξ + e 2 2η Z AiAidξ (2.10)

By equating the S derivatives with respect to the −→κ components and to the η parameter to some constants, eventually zeroing each one for a suitable choice of reference frame, we achieve the useful results for the electron trajectories:

−→ r⊥ = − →_κ η ξ + e η Z ₋_→ A dξ z = ( c 2ω0 − κ 2_{+ m}2_c2 2η2 ( ω0 c ))ξ − ω0e cη2 − →_{κ ·}Z −→_{A dξ +} + ω0e 2 2cη2 Z AiAi (2.11)

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We can carry out the time derivative of S, corresponding to the opposite of the electron energy E = γ(t)mc2_{, in order to get an expression for the}

Lorentz invariant η, evaluated at the origin of times.

η = (ω0

c )(γ0mc + pz0) (2.12) By performing the integrals in (2.11), the electron trajectories can be estab-lished, for any plane temporally-shaped wave packet.

2.2 Radiation emitted by relativistic electrons

trav-elling against a gaussian temporally-shaped LASER

pulse

The aim now is to calculate the number of photons emitted in a Thomson Backscattering process (g. 2.1), starting from the electron trajectories. We're going to use the well-known formula21 _{for the number of scattered}

photons per unit of solid angle and pulsation: d2N dΩdω = α ω (2π)2 | Z dt(−→n × −→n ×−→β )eiω(t− − →_{n ·}−_r(t)−→ c ) |2 (2.13)

where α is the ne structure constant,−→β is the normalized (to c) electron velocity, and −→n is the unit vector which select the direction line from radia-tion is detected. One can, in principle, think to Fourier-expand the potential

Figure 2.1: Geometrical conguration for the scattering between the LASER pulse and the electron bunch22_.

vector of an arbitrary temporally shaped LASER plane pulse, in such a way that: − → A = n=+∞ X n=−∞ −→ anein(ω0t−k0z)= n=+∞ X n=−∞ −→ aneinξ (2.14)

Then, by inserting (2.14) into (2.11), one obtains trajectories expressed as sum of harmonic terms of the fundamental LASER frequency ω0. Here below,

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we report the Jacobi-Anger identities eiµ sin θ = n=+∞ X n=−∞ Jn(µ)einθ eiµ cos θ = n=+∞ X n=−∞ inJn(µ)einθ (2.15)

where Jn(x) is the Bessel function of rst kind; by using the generalized

identities for the Bessel functions of many variables, one should have an expression like this:

d2N dΩdω = α ω (2π)2 | Z dξ(−→n × −→n ×−→β )eiω(t− − →_{n ·−}_rn→Pn=+∞ n=−∞einξ c )|2 (2.16)

to be processed in this way:

d2_N dΩdω = α ω (2π)2 | n=+∞ X n=−∞ Jn({ ω−→n · −→rn c } n=+∞ n=−∞) Z dξ(−→n × −→n × f−→β )ei(ρ0_ω0ω−n)ξ |2 (2.17) for some ρ0 that later will be more clear; we have set f

− → β = ω0 c d−→r dξ as in

the derivation22 _{and J} n({ω−

→_{n ·−}→_r

n

c }

n=+∞

n=−∞) is the generalized Bessel function of

rst kind23_{. A gaussian temporal prole is from now on considered for the}

LASER pulse, polarized along y-direction. By solving the equations (2.11) respect to a normalized a potential vector−→a = _by a0 e

− ξ2

2ω2

0τ 2 cos ξ (a₀ is

also said relativistic parameter, already encountered in chapter 1), one has trajectories in the form:

x = x0ξ y = y0ξ + y1 a0 e − ξ2 2ω2₀τ 2 _{sin ξ} z = z0ξ + z1 a0 e − ξ2 2ω2₀τ 2 _{sin ξ +} z2 2 a 2 0 e − ξ2 ω2₀τ 2_{(ξ −}1 2sin 2ξ) (2.18)

where we have performed the integrations in (2.11) considering the gaus-sian prole static with respect to the oscillations, according to the condition

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ω0τ >> 1. Here the list of parameters appearing in (2.18): a0 = eA mc x0 = px0 η y0 = py0 η y1 = mc η z0 = ( c 2ω0 −(p 2 x0+ p2y0) + m2c2 2η2 ( ω0 c )) z1 = − ω0mpy0 η2 z2 = = ω0m2c 2η2 (2.19)

Now we can determine f−→β:

f βx = x0ω0 c f βy = ω0(y0+ a0 y1 e − ξ2 2ω2₀τ 2 _{cos ξ)} c f βz = ω0(z0+ a0 z1 e − ξ2 2ω2₀τ 2 _{cos ξ +}a20 2 z2 e − ξ2 ω2₀τ 2_{(1 − cos 2ξ))} c (2.20)

where we've considered again ω0τ >> 1. Being −→n = (cos φ sin θ, sin φ cos θ, cos θ),

we can move to develop an expression for (2.13). For simplicity, we'll work in spherical coordinates, in such a way that the three-dimensional vector under the integral operator in (2.13) becomes a two-dimensional vector, due to its orthogonality to the radial directionbn. In formulas:

e

βθ = (−→n × −→n × f

− →

β )θ =

= −fβxcos θ cos φ − fβycos θ sin φ + fβzsin θ

f βφ = (−→n × −→n × f − → β )φ= = fβ_xsin φ − fβ_ycos φ (2.21) With a straightforward calculation one gets:

d2N dΩdω = α ω (2πω0)2 | Z dξ( eβθ, fβφ)ei ω ω0(ρ0ξ−ρ1sinξ−ρ2sin2ξ)_|2 (2.22)

which, by using the generalized Bessel functions of two variables formulas23_,

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d2_N dΩdω = α ω (2πω0)2 | n,l=+∞ X n,l=−∞ Z dξJn(ρ1 ω ω0 )Jn−2l(ρ2 ω ω0 )( eβθ, fβφ)e i(ρ0_ω0ω−n)ξ |2 (2.23) where we have dened:

ρ0 = 1 − sin θ ω0 c (x0cos φ + y0sin φ) − ω0 c (1 + cos θ)(z0+ a2₀ 2 z2) ρ1(ξ) = ω0 c a0e − ξ2 2ω2₀τ 2_(y

1sin φ sin θ + z1(1 + cos θ))

ρ2(ξ) = − ω0 c a2₀e− ξ2 ω2₀τ 2 4 z2(1 + cos θ) (2.24) Now we have to perform the above integration over the interval [−ω0T

2 ,

ω0T

2 ],

where T is the time duration of the interaction, about which we'll discuss later. We specialize the result to the case of low intensities-scattering, the so called linear regime, when just one harmonic appears in the emission spectrum, according to the physical expectation that only the rst oscillation mode is excited in a physical system when the external force eld is not too high. In the linear case, the relativistic parameter is a0 << 1, therefore the

Bessel functions in (2.23) and the function ρ2 in (2.24) become:

ρ2 ∼ 0 Jl(ρ2 ω ω0 ) ∼ δl,0 Jn(ρ1 ω ω0 ) ∼ δn,0+ 1 2ρ1 ω ω0 (δn,1− δn,−1) (2.25)

By always keeping in mind that ω0τ >> 1 and the formulas:

Jn(µe − ξ2 2ω2₀τ 2_)einξ ₌ k=∞ X k=0 µn(−1)ke− (n+2k)ξ2 2ω2 0τ 2 einξ 2n+2k _{k! Γ(k + n + 1)} Z dqJn(f (q)) cos wq = Jn(f (q)) sin wq − Z dqdJn(f (q)) df (q) df (q) dq sin wq ∼ ∼ Jn(f (q)) sin wq when df (q) dq << 1 (2.26)

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where f(q) = e−

q2 2ω2

0τ 2 and its derivative is ∝ 1/ω2

0τ2, we organize the result

in the following feature: d2N dΩdω = α ω T2 (2π)2 (V 2 θ + Vφ2) (2.27) in which: Vθ = −2

ω0(−z0sin θ + cos θ(x0cos φ + y0sin φ)) sinT ρ₂0ω

c T ρ0 ω + +( 2ω0e −T 2 8τ 2 c T (ω2 0− ρ20ω2)

(z1sin θ − y1cos θ sin φ) ×

×(−ρ₀ω cosT ω0 2 sin T ρ0ω 2 + ω0cos T ρ0ω 2 sin T ω0 2 ) − − ω0ωe −T 2 8τ 2 c2 _{T (ρ} 0ω − ω0)

(x0cos θ cos φ − z0sin θ + y0cos θ sin φ) ×

×(z1+ z1cos θ + y1sin θ sin φ) sin

T (ρ0ω − ω0) 2 + + ω0ωe −T 2 8τ 2 c2 _{T (ρ} 0ω + ω0)

(x0cos θ cos φ − z0sin θ + y0cos θ sin φ) ×

×(z₁+ z1cos θ + y1sin θ sin φ) sin

T (ρ0ω + ω0)

2 ) a0

(2.28) and

Vφ= −2

y0ω0cos φ sinT ρ₂0ω − x0ω0sin φ sinT ρ₂0ω

c T ρ0 ω + +(−2ω0e −T 2 8τ 2 y₁cos φ c T (ω2₀− ρ2 0ω2) (−ρ0ω cos T ω0 2 sin T ρ0ω 2 + ω0cos T ρ0ω 2 sin T ω0 2 ) + + ω0ωe −T 2 8τ 2 c2 _{T (ρ} 0ω − ω0)

(−y0cos φ + x0sin φ) ×

×(z1+ z1cos θ + y1sin θ sin φ) sin

T (ρ0ω − ω0) 2 − − ω0ωe −T 2 8τ 2 c2 _{T (ρ} 0ω − ω0)

(−y0cos φ + x0sin φ) ×

×(z₁+ z1cos θ + y1sin θ sin φ) sin

T (ρ0ω + ω0)

2 ) a0 (2.29) The cumbersome expressions above do not say anything too interesting at a rst sight. Nevertheless there are some features that have to be brought to

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light. From now on we'll consider an eective relativistic parameter aef f

0 ≡

a0e

−T 2

8τ 2 , which takes into account the actual pulse shape over the interaction

region (usually determined by the LASER focalization volume), noticing that it's nothing but a renormalization of the eld amplitude. The parameters (2.19)become for a0 << 1: x0 = tan(θe)c cos φe 2ω0 y0 = tan(θe)c sin φe 2ω0 y1 = c 2ω0γ0 z0 = c 2ω0 z1 = − c 4ω0γ02 (2.30) where we have introduced the information about the initial electron cong-uration, remembering that (easily provable with geometrical arguments) the angle θ is related to the initial electron polar θe and azimuthal φe angles,

and to the angle χ between θ and θe by the formula22:

χ =pθ2_{+ θ}2

e− 2θeθcos(φ − φe) (2.31)

γ0 is the initial Lorentz factor related to the initial energy-momentum of the

electron. By taking into account that the electrons we're thinking about are ultrarelativistic, we can expand all the expressions just for small angles. So we do an expansion for small θ0

esand small θ0s, stopping at the second order.

The expressions of ρ0 and ρ1 become:

ρ0 = 1 + γ₀2χ2 4γ₀2 ρ1 = = a0e − T 2 8ω2₀τ 2(γ0θsinφ − γ0θesinφe) 2γ2 0 (2.32) Here below the more evocative result for the number of photons per unit solid angle and pulsation:

d2N dΩdω = α ω T2 64γ2 0π2 a2₀e−T 24τ 2 (1−4(γ0θsinφ − γ0θesinφe) 2 (1 + (γ0χ)2)2 )sinc2(T 2(ρ0ω −ω0)) (2.33) in perfect agreement with [22], with the novelty of a T -dependent gaussian term, already discussed. Is really important to note that such a radiation spectrum consists of a single fundamental harmonic:

ωf =

4γ₀2ω0

1 + γ2 0χ2

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from which we infer a correlation between the emitted photon energy and the angle with respect to the drift direction from where it's radiated. We can think to integrate (2.33) with respect to Ω in the limit of very long interaction times (T >> 2π

ω0), namely after approximating the cardinal sine

with a Dirac delta: d2N dΩdω = α 2π ω a 2 0 e −T 2 4τ 2 γ 2 0 (1 + (γ0χ)2)2 ω0T × × (1 − 4(γ0θ sin φ − γ0θesin φe) 2 (1 + (γ0χ)2)2 )δ(ω − 4γ 2 0ω0 1 + (γ0χ)2 ) (2.35)

By a change in variables one can integrate the above expression, obtaining the number of photons per unit pulsation:

dN dω = α 64γ6 0 T a2₀ e −T 2 4τ 2 ω 2 ω2 0 (−8γ₀4ω 2 0 ω2 + 4γ 2 0 ω0 ω − 1 + 1 2γ 2 0θe2sin2φe) (2.36)

which is the same result in [22] generalized to a non perfectly on-axis elec-tron. What is nevertheless more interesting, is the total number of photons irradiated in a Thomson Scattering process like that so far studied, obtained with a straightforward integration of (2.36) over the interval [4γ2

0ω0/(1 + γ₀2θ2_max), 4γ₀2ω0] Nph= 1 2 α ω0 T a 2 0e −T 2 4τ 2 ψ2 1 + ψ2+2₃ψ4−1 2γ 2 0θ2esin2φe(1 + ψ2+1₃ψ4) (1 + ψ2₎3 (2.37) which is a common expression in the elds of the free electron LASERs and undulators, but with ψ = γ0θmax taking place of the usual strength

parame-ter, which in those devices depends on the applied magnetic eld. θmaxis the

maximum polar angle of detection. The importance of the exponential term e

−T 2

4τ 2 is limited to the cases when T & τ, for example, if T is determined

by the length of the focal zone, one would have T ∼ 10ps for a Rayleigh length zR ∼ 3mm. So 1/e corrections to the total number of photons be-come important for τ ∼ 10ps. It's fundamental to notice that the gaussian correction is not in contradiction with the energy conservation law: if one sends a LASER pulse at a certain energy against the electrons, the informa-tion of its energy is contained in the term a0, while the gaussian term only

takes into account the optimization of the energy transfer over the interac-tion region. Remember that the analysis carried out holds for one electron only, while in a real experimental setting, a whole electron bunch is made to collide against a LASER beam. Nevertheless in the limit of low-emittance and low energy spread bunches, all the results so far discussed change little, as we're going to see.

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2.2.1 Radiated photons by an electron bunch

For taking into account at an elementary level, but physically relevant, of the bunch properties, a gaussian transversal prole is considered. We're thinking to detect the radiation in the so-called far-eld region, in which (2.13)strictly holds, so we completely neglect the transversal spatial prole eects for what concerns the radiation spectrum, looking at them just to estimate the number of electrons who actually take part to the interaction, being irradiated by a LASER in its focal region. We satisfy this request introducing the lling factor z, which takes into account the transversal overlapping of the electronic bunch with the LASER beam. In formulas, we have a spatial radial prole:

R(re) ≡ ree −r2_e 2w2e 2πw2 e (2.38)

where re is the initial radial coordinate of one electron, we the radial

exten-sion of the electron bunch analogous to the beam spot radius at the waist for a LASER. So we dene the lling factor as:

z ≡ R dreR(re)e −2r2_e w2₀ R dreR(re) = w 2 0 w2₀+ 4w2 e (2.39)

where w0 is the radius of the LASER beam waist. With the help of the lling

factor, one can better estimate the number of photons emitted by an electron bunch. In order to account for the number of photons emitted per unit of solid angle and pulsation for a whole electronic bunch, we should sum over the degrees of freedom appearing in the one electron formulas, namely θe, φe

andγ0, by using the correct bunch distribution functions. If we consider a

bunch with a gaussian transversal distribution of velocities, a very common experimental feature, automatically we have a gaussian distribution of the electrons initial polar angles θe. The distribution function of the azimuthal

angles φe, for a circular electrons spot LINAC or LASER-produced, will

clearly be uniform over [0, 2π]. If we consider an electron bunch with energy spread which doesn't exceed 1%, we notice from (2.34) that the maximum energy spread in the radiated spectrum is of the order of 2%, which enlarges the spectrum in respect to the ideal condition when all the electrons have the same energy γ0mc2. At a rst approach we can consider the electron

bunches nearly monochromatic, with a Dirac delta distribution over the ini-tial energies. In the next section we'll show several spectra according to dierent experimental situations of our interest. It's worth noting at the end of this section that if the total number of electrons in a bunch is Ne, one has

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emitted: N_phtot= z Ne 1 2 α ω0T a 2 0e −T 2 4τ 2 ψ21 + ψ 2₊ 2 3ψ4− 1 4γ02(θemax)2(1 + ψ2+13ψ4) (1 + ψ2₎3 (2.40) where ψ = γ0θmax takes into account the detector cone semi-aperture θmax,

as we have already seen, and θmax

e is the electron beam semi-divergence.

It can be noticed from (2.34) that δω/ω ∼ ψ2_{: it's really important for}

the xing of monochromaticity of the exploited radiation, as it will be soon clear. Furthermore for ψ << 1 one has Ntot

ph ∼ ψ2, i.e. Nphtot ∼ δω/ω.The

above equation tells us in fact that if a well-collimated electron bunch in-teracts on-axis with a focused high power LASER, the number of scattered photons strongly depends on the maximum angle of detection θmax (which

in this case is comparable to χ ∼ θ). In fact, to the detriment of photons number received, one can x a certain degree of monochromaticity because of the energy-angle correlation (2.34). A Thomson source would oer great performances, insofar one considers the properties of high collimation, high brilliance of the produced radiation, in addition to the possibility of tuning the source by selecting a detection angular aperture, and of getting a good monochromaticity degree with an however high radiation ux.

My work has been inserted in the vivid italian framework of the physical research activities in Frascati (Rome), in particular at the ENEA center. An experiment on the Thomson scattering is actually in progress at the Frascati national labs (LNF) of INFN, and is in phase of designing at the ENEA ABC facility. The experimental conguration at least for now, in both cases, is that of LINAC-LASER, but nobody excludes a future development in the direction of the LASER-only conguration, by using electron bunches accel-erated via LASER-plasma interaction. Both LASERs and LINACS of the two labs are really dierent. In the next sections, we start by showing the properties of the radiation expected from a possible source based on the ABC LASER and a 5−10 MeV LINAC at the ENEA center: A feasibility study is carried out to some extent, in perspective to experimental campaigns which should be designed in the next future. Then we move to another congura-tion based on the FLAME LASER coupled with the SPARC LINAC, both of the LNF: some experiments are already in place and the results which have to be expected are shown. Finally an overview on a All Optical Thomson Backscattering X-γ source is carried out, based only on an ultra-short pulse LASER, utilized both to accelerate electrons and to interact with them.

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2.3 ABC LASER coupled with a 5 - 10 MeV LINAC

ABC LASER is an important LASER available in the ENEA labs of Frascati. It's an high power LASER whose characteristics will be reported in the chapter 4. In this section we do a feasiblity study24 _{for a Thomson source}

based on ABC coupled with a 5 − 10 MeV LINAC. We choose a set of parameters that are available for this LASER, not the only possible, but those which we'll be possibly used in the designing and the realization of the TS X-ray source. The normalized transversal emittance of a LINAC is dened22 _as

n = γ0weθmaxe /2: from now on we consider LINACs with

n∼ 2mm mrad. As we have already said, LINACs available at the ENEA facility can generate 1 − 10µs electron macrobunches, corresponding to a structure of ∼ 15ps microbunches, at 3GHz. The carried current by each microbunch is ∼ 6A corresponding to a number of electrons Ne ∼ 5 × 108. The data shown below allow us to predict the emission spectra and the spatial distribution of the X-rays beam generated after the interaction. Here is a summary of the experimental setting:

ω0 = 2πc λ = 2πc 1.054µm ∼ 1.79 × 10 15_rad/s τ = 3ns w0 ∼ 100µm EL ∼ 20J I0 = 2.12 × 1013W/cm2 −→ a0∼ 0.004

n ∼ 2mm mrad −→ we∼ 2mm , θemax< 1mrad

Ie ∼ 6A −→ Ne∼ 5 × 108 τe = 15ps

γ0 = 10, 20 (2.41)

We point out that for τ >> zR/c, and for τe ∼ zR/c, the interaction time

duration T will be determined by τe. We will study just a single microbunch

interaction; of course, the total number of emitted photons will depend on the total number of microbunches interacting with the LASER pulse. The starting formula we use here is:

d2N dΩdω = zNeα 4π2 ωa 2 0e −T 2 4τ 2 T2 Z dθe dφe dγe Θ(θe)Φ(φe)Γ(γe){ (1 − 4(γeθsinφ − γeθesinφe) 2 (1 + (γeχ)2)2 )sinc(T 2(ρ0ω − ω0))} (2.42)

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where Θ(θe), Φ(φe), and Γ(γ0) are normalized distribution functions

repre-sentative of the bunch, in particular:

Θ(θe) = e− θ2_e 2(θmaxe )2 p2π(θmax e )2 Φ(φe) = 1 2π if φe [0, 2π] , 0 otherwise Γ(γ0) = δ(γe− γ0) (2.43) By integrating over the solid angle Ω one gets the number of photons per unit of pulsation N(ω) ≡ dN/dω, while by integrating over the pulsa-tions, one can get the angular distribution in space of the radiation emitted N (Ω) ≡ dN/dΩ. Therefore evaluations are possible about the radiation spot which one would expect for example on a Imaging Plates (chap. 4 and 5), located at the right position to detect the emission. It's worth noting that in such conguration the lling factor z ∼ 1/1600 is really unfavorable, so one could think to better focalize the electron beam, or, if possible, to defocus the LASER maintaining an high intensity level. Our results are shown in g. (2.2), (2.3) and (2.4): 150 200 250 300 350 400 450 E@eVD 6 8 10 dNdE@eVD

Figure 2.2: 24_{Thomson spectrum obtained by xing the maximum}

accep-tance semi-aperture θmax ∼ 1/γ0 (γ0 = 10). The number of particles

ran-domly generated for the calculation is ∼ 10000.

The total number of photons emitted is straightforwardly obtained by nu-merical integration of the above function; a rough estimate gives: NT ot

photons ∼

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the LASER infrared light has been converted into soft X rays. 420 430 440 E@eVD 6 7 8 9 10 11 12 dNdE@eVD

Figure 2.3: 24_{Thomson spectrum, relative to the case with γ}

0 = 10, obtained

by xing the acceptance semi-aperture θmax ∼ 1/(10 √

10) in such a way to detect just radiation near 0.45 keV (on-axis) with a 10% monochromaticity degree. The number of particles randomly generated for the calculation is ∼ 10000.

By reducing the detection area, it's possible to narrow down the spectrum of the collected radiation, namely, as already discussed, to the detriment of the photons number arriving to the detectors (that however can remain very high) one can x a monochromaticity degree by reducing their acceptance cone aperture. Here we chose a 10% monochromaticity degree.

Figure 2.4: 24_{Thomson radiation spot on a screen at 1m, relative to the}

case with γ0 = 10; red stands for ∼ 1.4 × 102 photons/cm2. The number of