Self-injection in a moderately relativistic LWFA regime

Dottorato di Ricerca in Fisica

UNIVERSITÀ DI PISA

GRADUATE COURSE IN PHYSICS

UNIVERSITY OF PISA

PhD Thesis:

"Self-injection in a moderately relativistic

LWFA regime"

Candidate:

Supervisor:

Daniele Palla

Dr. Leonida A. Gizzi

Co-Supervisor:

Dr. Pasquale Londrillo

CONTENTS

Contents

1 Introduction iv

I Laser Wakeeld Electron Acceleration

vii

2 Laser Wakeeld Acceleration 2

2.1 Original Tajima and Dawson's Idea . . . 2

2.2 Single Particle Dynamics . . . 8

2.2.1 1-D Hamiltonian in a Wakeeld . . . 8

2.2.2 Trapping and Acceleration . . . 12

2.3 Injection Mechanisms: Overview . . . 14

2.4 Ionization Injection . . . 15

2.5 Downramp and Shock . . . 19

2.6 Bubble Regime . . . 20

II Overview of the Experiments

24

3.2 Interaction Chamber Setup . . . 27

4 Gas Jet Prole Characterization 30 4.1 Experimental Setup . . . 30

4.2 Molecular Density Proles . . . 34

4.2.1 Perturbed Density Proles . . . 36

5 Beam Properties Analysis 39 5.1 Lanex Scintillator Screen . . . 39

5.2 Energy Spectrum . . . 42

5.2.1 Analytical Approach . . . 42

5.2.2 A more Realistic Model . . . 47

5.2.3 Radia Software Approach . . . 50

6 Beam Emittance 54 6.1 Denition of Emittance . . . 54

CONTENTS

6.1.2 Emittance and rms Emittance . . . 56

6.1.3 Normalized Emittance . . . 60

6.2 Emittance Measurement with Pepperpot Technique . . . 62

7 Experimental Results 66 7.1 Plasma Density-Backing Pressure Conversion . . . 66

7.2 N2and He Injection Threshold: Experimental Results . . . 68

7.3 Emittance Vs. Laser Polarization . . . 71

7.4 Electron Bunch Spectra . . . 76

7.5 Downramp Test . . . 80

III Numerical Investigation of LWFA Regime and Discussions

82

8.2 Wakeeld Structure . . . 89

9 Identication of Self-Injection Mechanisms 94 9.1 Ionization . . . 94

9.2 Injection . . . 98

10 Acceleration, Emittance and Phase-Space Properties of Injected Beams 102 10.1 Acceleration and Dephasing Scale . . . 102

10.2 Emittance and Phase-Space . . . 106

IV Conclusion and Perspectives

110

V Appendix

113

11.2 Electron in a Plane Wave . . . 119

11.3 Pulse Propagation . . . 124

11.3.1 Ponderomotive force . . . 124

11.3.2 Laser Group Velocity . . . 127

11.3.3 Pulse Self-Focusing . . . 131

CONTENTS

11.4.1 Linear Waves . . . 135 11.4.2 Nonlinear Waves . . . 139

1 Introduction

Since their invention, in the 1930s, particle accelerators have increased their importance as fundamental tools in many elds, and there is no need to remark the constant gain in eciency and in performance during the rst years until now. Today, almost 90 years later, over 30000 particle accelerators currently operate in the world. Most of them are used in radiation therapy, ion implantation, nondestructive inspection, etc, while only ∼ 1% are used in high-energy physics as collider or synchrotron light/radiation source. Although several schemes and sizes for accelerators exist, a common phenomena, named electrical breakdown, limits the maximum achievable particle energy. The maximum accelerating eld generated by the radiofrequency (RF) cavities is limited[1] to ∼ 100MV/m; this implies that in order to increase the particles energy, the size of the accelerator must increase too, until tens of kilometer for the modern high energy accelerators. A strong reduction of dimensions, maintaining the same performance level, would be desirable for any type of particle accelerator. One possibility to overcome this limitation is to use plasmas as accelerating medium. Plasmas can support longitudinal electrostatic waves characterized by a strong electric eld (up to 103 _{times larger}

than RF) and a phase velocity close to the speed of light. These are the conditions required to accelerate charged particles. One way to excite plasma waves is based on the use of intense lasers.

In 1972, Rosenbluth and Liu[2], studied the excitation of a plasma wave by beating of two laser beams. Later, in 1979, Tajima and Dawson proposed[3] a new accelerator scheme based on laser-plasma interaction known as laser wakeeld acceleration (LWFA). Large amplitude plasma waves excited by an ultraintense and ultrashort laser can support an electric eld of the order of ∼ 100GV/m which, in principle, may provide the solution to reduce the acceleration length by a factor ∼ 103 _{with respect to the standard RF accelerators.}

After many years, high-power laser pulses with a suitable femtosecond duration have been develop and laser wakeeld acceleration has been explored with impressive results obtained word-wide[4, 5, 6]. The increase in the maximum beam energy in LWFA experiments has been rapid; from 0.2 GeV obtained in 2002[7] to the current record of 4.2 GeV obtained in 2014[8] from the group at the Lawrence Berkeley National Laboratory. Actually, more recent application requiring high quality electron bunches, with low energy spread and high stability are being considered, the most attractive being the X-ray free electron laser (XFEL).

In principle, higher-power lasers would be capable of driving even higher-energy electron beams[9]. How-ever, energy gain was not achieved by simply increasing the laser power; key parameters include also the plasma density, the length of the accelerator and the laser intensity. The maximum energy that can be gained by an electron increases by decreasing the plasma density; in fact, a lower plasma density allows a longer acceleration length to be achieved before reaching the so-called dephasing limit, which occurs when electrons travel faster than the plasma wave. In fact, at lower plasma density, the wave phase velocity increases as well as the plasma wavelength, allowing for a greater acceleration length.

Tightly focused laser pulses quickly diract in a vacuum, remaining focused and intense over a distance called Rayleigh length zR. Acceleration lengths used in laser wakeeld experiments are typically much

longer than zR and some sort of `guiding' is therefore needed to keep the laser intensity suciently high to

drive a wake throughout the structure. An example of guiding is the so-called self-guiding, which can be obtained due to relativistic self focusing by increasing the plasma density above a critical values nsf. Since it

requires a minimum plasma density, the maximum energy gain is intrinsically limited. In order to overcome this limitation, a plasma wave-guide[10] can be used, which may be obtained, for example, with a plasma channel[11] or a capillary discharge waveguide[6].

Another critical issue in LWFA is the control of the injection of electrons into the accelerating phase of the plasma wave. In order to improve the beam quality (energy distribution, emittance, charge, etc) it is crucial to reduce the phase space volume of the injected beam, and the injection plays a crucial role here. Self-injection is the basic injection scheme and occurs spontaneously when the amplitude of the plasma wave approaches the wavebreaking limit[12]. Other schemes involve multiple pulses[13, 14] or tailored/shock plasma density proles[15, 16, 17] to control injection and create tunable wakeeld accelerators. More recently, quality and stability of LWFA accelerated electron has been improved by using the so-called ionization injection, which is based on the injection of inner shell electrons of high-Z atoms ionized only on the laser peak[18, 19, 20, 21]. Development of all these physical processes is still in progress and many laboratories world-wide are engaged in the endeavor to make laser-plasma acceleration a viable solution for future generations of radiation sources. Motivated by these perspectives, entirely new infrastructures are being designed and constructed, like ELI, the Extreme Light Infrastructure, the most comprehensive European eort in applications of intense lasers currently under implementation in the Czech Republic, Hungary and Romania, or the EuPRAXIA project, a new design study for a Compact European Plasma Accelerator with Superior Beam Quality, that involves Germany, France, Italy, UK and Portugal and aims at demonstrating for the rst time free-electron laser operation with a plasma-based accelerator. These pan-European initiatives also motivated many well established intense laser laboratories across EU, including Italy, to upgrade their experimental and modelling capabilities to be able to access new domains of laser-plasma acceleration and to contribute to the design and construction of these infrastructures. Indeed, Italy is increasingly involved in these initiatives with major laser laboratory developments in Pisa (ILIL at INO-CNR) and Frascati (SPARC_LAB at LNF-INFN), and involving many institutions and a growing community. These circumstances make laser-plasma acceleration an attractive eld of research for talented young scientists and a promising area of development for many multi and cross-disciplinary applications in biology and medicine, cultural heritage, space and material sciences.

Summary of the Thesis and Contribution of the Author

My thesis work is centered on the study of the injection properties of LWFA using gas target with dierent ionization properties including He and N2. The study is centered on experimental work performed at the

Intense Laser Irradiation Laboratory (ILIL) in Pisa where application of LWFA including radiobiology and secondary radiation sources studies are being developed. For the thesis, a moderately relativistic LWFA regime was considered, characterized by a maximum intensity on target of I0 ' 2 × 1018W/cm2 with a

normalized vector potential a0= eEL/meωL' 0.96and a pulse duration of τL= 40fs. In these experimental

conditions self-focusing allows the main pulse to remain focused along the entire acceleration length without external guiding. In addition to that, self-focusing also leads to a signicant increase of the laser intensity enabling ultrarelativist regime. Background plasma is created via tunneling ionization during the interaction between the laser pulse leading front and the gas targets. The dierent ionization properties of nitrogen and helium, combined with the strong dependence of laser pulse intensity upon plasma density, produce substantial dierences in the accelerated electron bunch properties. Selected experiments designed to identify the role of ionization in the injection mechanism will be described. In addition, a preliminary experimental study of the role of tailored density prole will also be presented.

This thesis is structured in 4 main parts plus an appendix. The rst part, named Laser Wakeeld Electron Acceleration, gives a brief introduction on LWFA, from the original concept of Tajima and Dawson to the strongly non-linear bubble regime, including an overview of the most eective injection mechanisms and their pros and cons. The role of plasma and laser parameters in LWFA will also be discussed and the main limits explained. The second part, Overview of the Experiments, consists of ve chapters and gives a presentation of the basic experimental laser and target parameters, a description of the main electron bunch diagnostics used in the experiments, with a discussion of electron beam emittance and relevant experimental detection techniques. The second part also includes a presentation of the main experimental results obtained on gas ionization and plasma formation where experimental measurements on the injection thresholds in He and N2are presented. Also, evidence of the role of laser polarization on the beam emittance is provided and

electron energy spectra analysis for the two gas targets are shown. Preliminary results on the exploration of density prole tailoring are nally summarized. The third part, Numerical Investigation of LWFA Regime and Discussions consists of three chapters and is dedicated to the interpretation of experimental results via comparison with dedicated numerical simulations carried out to unfold the role of self-focusing and ionization in the injection process of the two gases and to study the eect of laser polarization on the transverse momentum of injected electrons. Finally, a fourth part is dedicated to conclusion and prospectives.

The author was fully involved in the experimental measurements, and was primarily in charge in the electron diagnostics including beam transverse prole and energy spectra, from design and construction to data analysis. The author was also directly engaged in the numerical simulations, from the denition of input parameters to the analysis and interpretation of numerical results. The author was also involved in the

Laser Wakeeld Electron Acceleration characterization of the gas-jet target with interferometric measurements of the molecular density proles, and was in charge of the design and construction of the preliminary experimental set up for the modication, via shock generation, of the density prole. The participation to all the aspects of the research involved in this thesis, including design, construction, data acquisition and analysis, numerical simulations and interpretation of experimental results, has given the author the opportunity to become fully aware of the complexity of the current approach to laser-plasma acceleration.

Laser Wakeeld Electron Acceleration

Part I

Laser Wakeeld Electron Acceleration

Figure 1: A schematic picture of Laser Wakeeld Acceleration (LWFA).

2 Laser Wakeeld Acceleration

2.1 Original Tajima and Dawson's Idea

In their original article[3], Tajima and Dawson proposed an innovative particle accelerator scheme in which the particles are accelerated in an underdense plasma medium. The main advantage in using a plasma instead of radio-frequency cavities in vacuum is the possibility to overcome the "electrical breakdown" limitation. The extremely high electric eld values up to ∼ 100 GV/m available in plasma[22], compared with the ∼ 100 MV/mbreakdown, limit give an idea of the impressive dierence. The acceleration scheme proposed by Tajima and Dawson, named laser wakeeld accelerator (LWFA), is based on electron plasma waves exited by an intense laser pulse. LWFA, nowadays, has became a branch of plasma physics that includes the study of several sub-topics as the laser driver propagation, the generation of plasma wave and injection mechanisms. In this section we will present the fundamental aspects of LWFA scheme assuming that the reader is familiar with the basic concepts of laser-plasma interaction. An appendix is also included with a detailed description of relevant laser-plasma interaction physics.

The LWFA mechanism is schematically presented in Fig.(1) and it can be summarized as follows: 1. A laser pulse, with a duration τL close to the resonance1 condition τL = π/ωp = (4e2n0/πme)−1/2,

where ωp is the plasma frequency, n0 is the background electron density and methe electron mass, is

used to excite, through the ponderomotive force, a plasma wave also called wakeeld.

2. The axial electric eld Ez generated by the plasma wave is used to accelerate a charged particle along

the laser driver direction.

1_{Alternatively, instead of a single resonant pulse is possible to use two long laser pulse of frequencies ω1and ω2 to excite a} plasma wave with the condition: ∆ω = ω1−ω2= ωp[2]. This mechanism lead to the plasma beat wave accelerator (P BW A)[23].

2.1 Original Tajima and Dawson's Idea Laser Wakeeld Electron Acceleration

Roughly speaking, the laser pulse is not directly involved in the acceleration process, but is the plasma wave itself that takes the place of the resonant cavity of a standard accelerator. At high laser intensity, the formal description of the entire acceleration process is not trivial due to the variety of nonlinear phenomena involved in the laser-plasma interaction. For this reason, Tajima and Dawson, considered a linear plasma wake in order to obtain some representative orders of magnitude of the acceleration parameters. Following their article, a plasma wake is most eectively generated if the length of the electromagnetic wave packet is half of the wavelength of the plasma waves in the wake. The reason is that a laser pulse pushes the plasma electrons, through the ponderomotive force, in opposite direction on rising and falling fronts. Thus, such a double "kick", should be in phase with plasma oscillations (see for instance Sec.(11.4)). Typical plasma densities of ∼ 1018_cm−3 _{require a resonant pulse duration τ}

Lof tens of femtoseconds. An alternative way of exciting

a plasma wake is to use two co-propagating beating laser pulses with angular wavenumber and frequencies of k1/2 and ω1/2 respectively to producing a modulated laser amplitude of the form cos(1/2(∆kz − ∆ωt)),

where ∆ω = ω1− ω2 and ∆k = k1− k2 . This is done by appropriately adjusting the laser frequencies

and plasma density to satisfy the resonance condition ∆ω = ω1− ω2 = ωp. When this is satised, large

amplitude plasma waves can be generated. Resonant excitation of a plasma wave using two laser pulses had been previously analyzed by Rosenbluth and Liu[2] for plasma heating applications.

In the case of a plasma wake excited by two beating long laser pulse, the acceleration scheme is named plasma beat wave accelerator (PBWA). Beat-wave excitation of plasma waves was rst demonstrated by Clayton et al. in 1985[25] using a CO2laser with a pulse duration of 2ns. Then, the acceleration of electrons

in a PBWA was demonstrated in 1993[26], again, by Clayton et al. by using a CO2 laser. During the last

decade of the 20th, until the advent and the development of the chirped pulse amplication[24] (CPA), which allowed to a suitable laser pulse duration of tens of femtoseconds, experimental investigation of PBWA was intensively pursued[23, 26, 27]. Unfortunately, limitations due to relativistic detuning for high amplitude plasma waves prevented further program of this scheme.

Independently of the acceleration scheme (LWFA of PBWA), Tajima and Dawson considered a linear plasma wake, for which the longitudinal electric eld can be written as:

Ez= E cos (kpz − ωpt) , (2.1)

where and E ≤ E0is the eld (wave) amplitude, E0≡ mecωp/eis the cold nonrelativistic wavebreaking eld

and ω2

p = 4πe2n0/meis the plasma frequency, where e, meand n0represent the absolute electron charge, the

electron rest mass and the plasma density respectively. The wake phase velocity approximation βph' 1has

2.1 Original Tajima and Dawson's Idea Laser Wakeeld Electron Acceleration βph' 1 along ˆz and the associated gamma factor is γph= ω/ωp 1, where ω is the laser driver frequency.

The space frame dened by γph is the rest frame S

0

of the wake. According to Lorentz transformations, in the rest frame S0

we have: E_z0 = Ez, k 0 p= γph(kp− βphωp/c) , ω 0 p= γph(ωp− vphkp) , (2.2)

where we used that Kµ _{= (ω}

p/c, kp) is a 4-vector and Ez is invariant. The phase velocity of the wake

is dened by vph = ωp/k ' ωp/kp which, replaced in the ω

0

p expression, yield to ω

0

p = 0 as expected; in

fact, the electric eld is static in the rest frame S0

. Performing the same substitution in k0

p we get

k0_p= ω/ωp(ωp/vph− vphωp/c2) = ωp/(γphvph). At the end, the electric eld E

0

z and the potential eld Φ

0

in S0are given respectively by:

E_z0 = E cos  _ω p vphγph z0  , Φ0 = −Evphγph ωp sin  _ω p vphγph z0  . (2.3)

In such static potential, the maximum energy gain of an electron is given by:

ε0= e (Φ0_max− Φ0min) = 2eEvphγph ωp ≤2eE0cγph ωp = 2mec2γph, (2.4)

where the linear wavebreaking limit E0 denition has been used. The condition for an electron to gain the

maximum energy ε0

is to start at rest from the maximum of the potential Φ0

max and leave the wave at the

minimum Φ0

min. This acceleration mechanism implies that some electrons can be trapped by the wakeeld.

Those trapped electron are called injected electrons and the mechanism which enables trapping of electrons is called injection. Injection mechanisms will be discussed in detail in next chapter; for the moment we can imagine that a fraction of the background electron can be self-injected without perturbing the wake structure.

Transforming the energy (2.4) dened in the wave frame S0

back to the laboratory frame we nd:

ε = γph  ε0+ vphp 0 ' γε0+ cε0/c= 2γphε 0 (2.5)

2.1 Original Tajima and Dawson's Idea Laser Wakeeld Electron Acceleration

Figure 2: Variation of reported electron beam energy with laser power from various experiments.[9] and, nally: εmax' 4mec2γ2ph, γmax' 4  ω ωp 2 , (2.6)

where γph= ω/ωp has been used to dene the normalized electron energy γmax in term of laser and plasma

parameters only. The energy gained by a particle of charge e in an accelerating structure is simply propor-tional to the product of the averaged electric eld and the length of the accelerator: εmax' e hEzi Ld. In

our case hEzi = 2/πE ≤ 2/πE0 thus, the acceleration length needed to the reach the maximum energy (2.6)

is given by:

Ld'

2πc ωp

γ_ph2 = λpγph2 , (2.7)

where λp = 2πc/ωp is the plasma wavelength. In this model, the only plasma and laser parameters which

appear in εmax and La are ωp and ω respectively. The driver pulse intensity and duration are implicit

parameter; in fact, the laser pulse is assumed to be strong and resonant enough to excite a plasma wave with amplitude E0. It is convenient to spend a few words on this point.

In term of dimensionless quantities, we know that in a linear wakeeld regime φmin/max' ±βphEˆ, where

φ = eΦ/mec2 and ˆE ≡ Ez/E0 is the normalized eld. Since ˆE ∼ 1 used in our calculation implies also

φmin/max ' ±1, we nd that the normalized laser vector potential a = eA/mec2 should be2 a > 1. Not

surprisingly, considering a typical Gaussian driver pulse temporal prole, we nd a remarkable laser intensity

2.1 Original Tajima and Dawson's Idea Laser Wakeeld Electron Acceleration of3 _{I > 10}18_W/cm2 _{(using a typical λ ∼ 1 µm laser wavelength). This means that the driver pulse must}

be superintense.

This became possible with the invention of the chirped pulse amplication[24] that enabled the devel-opment of ultrashort, ultraintense lasers. Since then, a dramatic develdevel-opment of LWFA occurred also from an experimental point of view as summarized by Mangles in his overview of recent progress in LWFA[9], by examining a set of over 50 experiments. This work clearly shows the growing in electron beam energy with laser power (See Fig.(2)).

We consider now two examples in which electrons are accelerated up to εmax' 20 GeVand 20 MeV.

Assuming that the laser is strong enough to excite a wakeeld with the required amplitude E0 we nd,

according to Eq.(2.6), that γph = ω/ωp ≡ 100 and 3.2 respectively. Assuming an infrared laser pulse with

wavelength λ = 800 nm, we nd ω ' 2355 THz, which, combined with

ωp[THz] ' 55

p

n0[1018cm−3], (2.8)

gives the required plasma densities of n0' 2 × 1017cm−3and 1.8 × 1020cm−3respectively. The relative

wakeeld intensities are E0' 0.04 TV/mand 1.3 TV/m. In the same way, from Eq.(2.7), we can estimate

the acceleration lengths of Ld ' 76 mmand 25 µm. As expected from the magnitude of the electric eld,

the acceleration length is extremely small compared to the standard RF accelerators size. For example, the Stanford Linear Accelerator (SLAC), is able to accelerate an electron bunch up to ∼ 50 GeV in 3200 meters. It is interesting to note how lower density plasmas, despite the lower supported electric elds, produce higher energy electron bunches. The reason is the acceleration length Ld ∼ n

−3/2

0 , which is a dominant quantity

with respect to E0∼ n 1/2

0 . This is also clear in Fig.(3), where the same set of experimental data presented

in Fig.(1) is now reported in terms of plasma density. In fact, following Mangles and rewriting Eq.(2.6) as εmax∼ ncr/n0(ncr≡ ω2me/4πe2), nd a good agreement with the entire dataset.

Clearly, the possibility of miniaturizing a particle accelerator make LWFA very attractive, but several issues (not considered in the above model) must be taken into account. The main limits of the LWFA scheme can be summarized as follows:

ˆ LASER DIFFRACTION: Acceleration length is limited by the Rayleigh length zR = πw02/λ,

which represent the distance beyond which the pulse will be diracted. Increasing the waist w0 (i.e.

the minimum of pulse radius in vacuum propagation) at will, clearly, does not represent the solution because the pulse must be relativistic while intensity decrease as ∼ P0/w02, where P0is the maximum

3_{In 1979 (time of the publication of the Tajima and Dawson article), the maximum available pulse intensity was I ∼} 1015_W/cm2_.

2.1 Original Tajima and Dawson's Idea Laser Wakeeld Electron Acceleration

Figure 3: Variation of reported electron beam energy with plasma density from various experiments. The blue line shows the relation εmax= ncr/n0.[9]

power of the pulse. Generally, at a given P0, the optimization of the optical system is not trivial.

ˆ ELECTRONS DEPHASING: The acceleration length reported in Eq.(2.7) represents an intrinsic limit of LFWA. In fact Ld is noting but the distance traveled by the electrons before their phase slips

by 1/2 of a period with respect to the plasma wave. Along a distance of Ld the electron will remain

in the accelerating part of the wave. The linear acceleration length, or more correctly the dephasing length is dened by Ld(βz− βph) = λp/2 → Ld ' λpγph2 , where βz' 1 and (1 + βph) ' 2has been

used. We nd again the result (2.7), conrming the meaning of this limit.

ˆ ELECTRONS INJECTION: The creation, via laser-plasma interaction, of a suitable acceleration eld is just the rst step in the acceleration process. In fact electrons should be injected and nally accelerated. Injection can be distinguished in external and background injection. In the external injection scheme, injection is achieved by using an external bunch produced, for example, by a Linac. This is technically complex and is not considered here. In the background injection the injected electrons come directly from the background plasma. The latter process is hard to be controlled. The study and the comprehension of the various background injection mechanisms are fundamental in order to overcome this limitation.

ˆ BEAM QUALITY: The nonlinear nature of processes involved in injection and acceleration lead to electron bunches characterized by a lack of beam quality (large energy spread, emittance, stability, etc) compared to standard accelerators. Quality of the bunches strongly depends on the initial evolution of the phase space of the injected electrons. Ultimately, the control of beam properties is closely related to the comprehension of the injection process.

2.2 Single Particle Dynamics Laser Wakeeld Electron Acceleration In this thesis we are more concerned with the injection and beam quality problems but, as we will shows, also the guiding and dephasing must be necessarily taken into account.

2.2 Single Particle Dynamics

2.2.1 1-D Hamiltonian in a Wakeeld

In this section we will expand the basic model presented in Sec.(2.1) including, this time, the nonlinear wakeeld interaction and the analytic description of the injection process. The approach presented here is a summary of the work by E. Esarey and C.B. Schroeder [28, 29].

The 1-D dynamics of a single test electron in the presence of a plasma wave and a laser pulse can be described in terms of the normalized potential

φ (ζ) = eΨ (ζ) mec2

, (2.9)

where Ψ (ζ) is assumed to depend only on the comoving coordinates ζ = x − vphtand τ = t. In addition to

this, the driver pulse is assumed nonevolving (wake's phase velocity vph and amplitude are constant along

the entire interaction). Here we consider the pseudopotential Ψ (ζ) dened in the laboratory space frame, dened in terms of the vector potential A and the scalar potential Φ as follows:

Ψ (ζ) = Φ (ζ) −vph

c Ax(ζ) → Ex(ζ) = −∂ζΨ (ζ) , (2.10) where the relations ∂x = ∂ζ, ∂t = −vph∂ζ + ∂τ has been used. The potential (2.9) can be considered a

periodically oscillating function between the values φmin ≤ φ ≤ φmax with a nonlinear plasma wavelength

λp period. The explicit forms of φmin and φmax, in the approximation of A · ˆx ' 0 (i.e. Ψ = Φ), are given

in detail in section (11.4) (see Eq. (11.104)).

We are interested in the momentum transfer and the phase slippage between the electron and the wake; for this reason is convenient to use p and ζ as canonical coordinates instead of p and x. The 1-D equations of motion for an electron in a electrostatic potential is given by:

d˜p (ζ)

2.2 Single Particle Dynamics Laser Wakeeld Electron Acceleration where ˜p = p/mecis the normalized electron momentum. In a similar way we nd

dζ

dct= β − βph= ˜ p

γ− βph, (2.12)

where βph= vph/c. Equations (2.11) and (2.12) can also be obtained as canonical equation ∂ζH = −dζ/dct,

∂p˜H = dζ/dctfrom the following Hamiltonian[28]:

H (˜p, ζ) = 1 +p_e2
12 _{− β}

php − φ (ζ) .˜ (2.13)

The Hamiltonian (2.13) contains an extra term −βphp˜ with respect to the standard H (˜p, x) function of

an electron in an electromagnetic eld. This term comes from the generating function F1 = ˜pζ of the

canonical transformation dened by ˜p = ∂ζF1. In fact, the relation between these two Hamiltonians is

H (˜p, x) = H (˜p, ζ) − βph∂ζF1.

Since the electrons move predominantly in the forward direction (x direction), the paraxial approssimation can be used. This mean that the 2-D orthogonal motion on y − z plane can be simply decoupled from the longitudinal one. Assuming a linearly polarized laser eld and averaging over the fast oscillation we nd (see section (11.2)) ˜p⊥ = 1/

√

2a as transverse normalized momentum. Thus, the 3-D paraxial Hamiltonian can be dened as:

H (˜p, ζ) = γ_⊥2 +p_e2
12 _{− β}

php − φ (ζ) .˜ (2.14)

where γ2

⊥ = 1+1/2a2+ ˜p2⊥0and ˜p⊥0 1is the secular transverse momentum. The Hamiltonian (2.14) is time

independent and represents a constant of motion of the electron; in other word H (˜p, ζ) can be considered the normalized particle energy in the comoving wave frame. Eq.(2.14) can be solved as a quadratic function in ˜p and yield to the following solutions[29]:

˜ p = βphγph2 (H + φ) ± γph  γ_ph2 (H + φ)2− γ2 ⊥ 12 , (2.15)

where γph= (1 − βph2 )−1/2 is the Lorentz factor associated with the plasma wave phase velocity. At a given

2.2 Single Particle Dynamics Laser Wakeeld Electron Acceleration

Figure 4: Example of single electron orbits (dotted curves) in (˜p, ζ) phase space for γph = 10, φmax = 0.5

and γ⊥= 1. The solid curve is the separatrix, and the dashed curve is the cold uid orbit. The head of the

driver is at ζ = 0.[29]

of orbits is presented in Fig.(4). As we can see, closed orbit means that the electrons remain conned in a certain portion of spatial region ζ. Electrons with closed orbit are called injected electrons or trapped electrons. In the opposite case (i.e. open orbits), electrons simply passing through the wakeeld. Analysis of closed and open orbits enable us to distinguish between background electrons and injected electrons. The separatrix orbit denes the boundary between closed (injection) and open (non injection) orbit as showed in Fig.(4) (solid curve). The separatrix is dened by the stationary point of H (˜p, ζ). Now, a description is given on how to determinate the separatrix orbit.

The Hamiltonian H (˜p, ζ) has stable and unstable xed point dened by: ∂p˜H = ˜ p (γ2 ⊥+ ˜p2) − βph ≡ 0 → p˜f ix= γ⊥γphβph (2.16) and ∂ζH = ∂ζφ (ζ) ≡ 0 → ζmax+ jλp, ζmin+ jλp j ∈ Z, (2.17)

where ζmin/max are implicitly dened by φ(ζmin) ≡ φmin and φ(ζmax) ≡ φmax. The stable points are

(˜pf ix, ζmax+ jλp)while the unstable one are (˜pf ix, ζmin+ jλp). After replacing the unstable point inside

2.2 Single Particle Dynamics Laser Wakeeld Electron Acceleration

Figure 5: Initial electron momentum ˜ptrequired to be trapped by a plasma wake with eld amplitude Em/E0

and phase velocity. γp= 10, 4.5, 3.1and 2.6.

Hs(˜pf ix, ζmin) =

γ_⊥(ζmin)

γph

− φmin, (2.18)

where the relation (1 + γ2

phβph2 )1/2 = γph has been used. The relation (2.18) denes the minimum energy

needed by an electron to be injected.

We consider now a trapped plasma electron, with initial longitudinal momentum of ˜pt= γtβt, before

the passage of the laser and the excitation of the plasma wave. We want to determine the minimum initial momentum ˜pt to get self-injection. Since the Hamiltonian Ht(˜pt, ζ) is a constant of motion, it can be

determined using the following conditions: γ2

⊥ = 1 + ˜p2⊥0 ≈ 1, ˜pt and φ = 0. Using the general denition

(2.14) we obtain:

Ht(˜pt, ζ) = 1 +_ep2t

12 _{− β}

php˜t. (2.19)

Trapping of the electron will occur when the orbit dened by Ht(˜pt, ζ)coincides with a trapped orbit dened

by the separatrix orbit (Eq.(2.18)). The condition for trapping an electron is then dened by the following relation: Ht(˜p, ζ) ≤ Hs(˜pf ix, ζmin). In the opposite case of Ht(˜p, ζ) > Hs(˜pf ix, ζmin), as we know, the

electron will be on an open orbit (i.e. a background electron). Solving Ht(˜pt, ζ) = Hs(˜pf ix, ζmin) yield

to the minimum4 _{initial normalized momentum ˜pt for trapping in the plasma wave. The self-injection}

4_{Note that Ht( ˜pt, ζ) = Hs p˜}

2.2 Single Particle Dynamics Laser Wakeeld Electron Acceleration momentum threshold is then dened by:

˜ pt= γphβph(γ⊥− γphφmin) − γph  (γ⊥− γphφmin)2− 1 12 . (2.20)

Some typical values of ˜pt, expressed as function of φmin(Em/E0), where Em is the maximum electric eld

(see Eq.(11.104)), are showed in Fig.(5). For simplicity the electron quiver velocity is supposed to be zero: γ⊥ = 1. As we can see from Fig.(5), within an explored ranges of 0 ≤ ˆEm ≤ 2 and 2.6 ≤ γph ≤ 10, the

threshold momentum required for trapping varies strongly. The threshold momentum ˜ptdecrease for a larger

plasma wave amplitude and for lower plasma phase velocity γph. Moreover, observing that that ˜pt≥ 0 we

deduce that only electrons with momenta in the direction of the plasma phase velocity can be trapped. The best injection condition can be obtain by minimizing the Eq. (2.20) as function of φm(Em/E0) and

γph. Since ˜pt ≥ 0 and noting that the equation Ht(0, ζ) = Hs(˜pf ix, ζmin) has a continuous solutions, we

deduce that the minimum of Eq. (2.20) simply corresponds to the trapping threshold of an initially cold plasma electron, dened by ˜pt= 0 and Ht(0, ζ) = Hcold= 1. Finally, by solving Ht(0, ζ) = Hs(˜pf ix, ζmin)

we can easily obtain the cold electron injection condition in term of the elds:

φmin = γ⊥ γp − 1 → Eˆm= ˆEb= [2γ⊥(γph− 1)] 1 2_, (2.21)

where the equation (11.98) has been used. The eld amplitude (2.21) corresponds to the relativistic break-down amplitude Eb/E0for a cold plasma (see Eq.(11.105)). Trapping of ˜pt= 0electron implies a strong wave

particle resonance (wavebreaking) which is not described by the cold uid theory. In any case, since most of plasma electrons can be considered at rest before injection, we deduce that the wavebreaking injection represents the most ecient injection mechanism in terms of number of injected particles.

2.2.2 Trapping and Acceleration

The maximum electron energy gain occurs for electrons which are trapped in close orbits just inside the separatrix boundary dened by Hs(˜pf ix, ζmin)(2.18). The Hamiltonian (2.14) is a constant of motion, thus,

for such injected electrons, the maximum and minimum energy gain is dened by the following condition: Ht p˜max/min, ζmax
= Hs(˜pf ix, ζmin). The solution of this equation is obviously similar to the result (2.20)

and gives:

2.2 Single Particle Dynamics Laser Wakeeld Electron Acceleration ˜ pmax/min= γphβph(γ⊥+ γph∆φ) ± γph  (γ⊥+ γph∆φ) 2 − γ2 ⊥ 1₂ , (2.22)

where ∆φ = φmax− φmin and ± give ˜pmax and ˜pmin respectively. Eq.(2.22) generalizes the Tajima and

Dowson's results (2.6). In the limit of strong eld γph∆φ  γ⊥ the equation (2.22) becomes:

˜ pmax' γph2 (1 + βph) ∆φ, p˜min' − ∆φ 1 + βph + γ 2 ⊥ 2∆φ. (2.23) Quite surprisingly, the minimum momentum ˜pmin is a negative quantity and corresponds to a deceleration

of the initial injected electron (remember that ˜pt ≥ 0). Formally, approaching the ωp → ω limits, which

represents a critical laser-plasma interaction, we obtain a perfectly symmetric LWFA scheme, characterized by ∆φ  γ⊥ and γph ∼ 1. In this case the wakeeld propagation is so slow that injected electrons can

be accelerated in the same way both in backwards and forward direction. We must take into account, however, that our equations are not fully valid in the ω & ωp limits because laser-plasma interaction with

overdense/dense targets involves dierent acceleration mechanism not discussed here.

In this thesis, all discussed target are low density target (gases), then the acceleration mechanisms is mainly in the forward direction and the condition ˜pmax/ |˜pmin| ' γph2 (1 + βph)2 1will be always satised.

On the other side, in the weak eld approximation γph∆φ  γ⊥ we obtain:

˜ pmax/min= γph  βphγ⊥±p2γphγ⊥∆φ  , (2.24)

which can be both positive quantities or ˜pmin< 0and ˜pmax> 0in the case of βph2  1. Again, the backwards

acceleration occurring in the overdense limits will not be considered here.

The nonlinear potential dierence ∆φ = φmax− φmin can be obtained in terms of wave amplitude ˆEmfrom

Eq.(11.104), and yield to:

∆φ = 2βph   γ⊥+ ˆ E2 m 2 !2 − γ2 ⊥   1 2 . (2.25)

In the linear regime (i.e. ˆEm2  2γ⊥ ) we obtain ∆φ ' 2βph

√

γ⊥Eˆm while, in the nonlinear case (i.e.

2γ⊥  ˆEm2 ≤ ˆEb2), the wake potential dierence can be written as ∆φ ' βphEˆ2m+ 2βphγ⊥. The upper

limit for the eld is the nonlinear wavebreaking eld ˆE2

2.3 Injection Mechanisms: Overview Laser Wakeeld Electron Acceleration model is no longer valid. Inserting the wavebreaking eld ∆φ = βphEˆb2+ 2βphγ⊥ (the linear term of Taylor

expansions is needed because ˆE2

b also has a linear term) into the ˜pmax expression (2.23) gives:

˜

pmax= γph2 (1 + βph) (2βphγ⊥(γph− 1) + 2βphγ⊥) ' 4γph3 , (2.26)

which is the maximum momentum (or energy ˜pmax' γmax) of a trapped electron at the wavebreaking limit.

The approximations βph = γ⊥ = 1 has been used in the expression (2.26). In the Tajima and Dawson's

articles, a linear wavebreaking limit ( ˆEm= 1), and a linear regime ∆φ = 2βph

√

γ⊥Eˆm' 2βphare taken into

account. By inserting ∆φ ' 2βph ' 2in the expression of ˜pmax in the case of strong eld (Eq.(2.23)) we

obtain ˜pmax' γmax= 4γp2= 4(ω/ωp)2, which coincide with the Tajima and Dawson's result (2.6).

As in the case of linear interaction, the energy gained by an accelerated electron is simply proportional to the product of the averaged electric eld and the length of the accelerator: εmax ' eLdephhExi, where

hExi ∝ Em. Assuming a maximum electric eld of Em= (2γ⊥(γph− 1))1/2E0, which corresponding to the

nonlinear wavebreaking limit, we nd Em = Eb ∼ n 1/4

0 . On the other side, according with Eq.(2.26), we

know that energies globally grows as γmax∼ n −3/2

0 . Combining these two results we obtain the dephasing

length dependence upon plasma density in the case on a nonlinear eld:

Ldeph∼ n−7/40 non − linear. (2.27)

The result (2.27) is slightly dierent from the linear case discussed by Tajima and Dawson's, in which Emax= E0∼ n

1/2

0 , γmax∼ n−10 and Ldeph∼ n−3/20 . However, in both cases the key point is the same: the

maximum electron energy can be reached at lower plasma densities as possible.

2.3 Injection Mechanisms: Overview

LWFA requires two fundamental step, the rst being the electron injection and the second the acceleration. One of the key challenges that has driven progress in the eld of laser wakeeld acceleration is the maximum achievable beam energy; this aspect mainly regards the optimization of acceleration. As we have discussed, the maximum achievable beam energy strongly depends upon plasma density n0 according to n

−3/2 0 or n

−1 0

for the non-linear or the linear interaction, respectively. At the same time, lower plasma densities require the laser to stay focused for the entire dephasing length which is of order of several centimeters in the case of n0∼ 1017cm−3. The problem to obtain a proper wakeeld along the entire dephasing length is central in

2.4 Ionization Injection Laser Wakeeld Electron Acceleration challenge in LWFA is the beam quality (such as transverse emittance, energy spread, etc) and the way to control it. In this context injection mechanisms plays a key role in the comprehension and in the improvement of the beam quality.

This thesis is mainly dealing with injection mechanisms. For this reason, the acceleration mechanism will always be treated as a consequence of injection properties at xed optical and laser congurations. Once claried this point, we can start to analyze the fundamental properties of electron injection. Using the simple 1-D model presented in Sec.(2.2.1) (see for instance Fig.(5)), we can resume the ways to control electrons trapping with a single laser pulse as follows:

ˆ Increasing ˜p: Using gas target, which is the standard in LWFA, plasma is created via tunneling ionization. An analogous eect of an increment of the initial longitudinal electron momentum ˜p can be obtained, at xed pulse properties, by exploiting ionization properties of the target.

ˆ Increasing ˆEm = Em/E0: The ratio ˆEm increases with plasma density and driver pulse intensity.

The limit is the wavebreaking eld ˆEb for which even the particles at rest can be injected.

ˆ Decreasing γph= ω/ωp: In order to reduce the plasma phase velocity we can operate in several ways.

The rst one is simply based on an increment of the plasma density n0. Alternatively it is possible to

locally reduce the phase velocity with a local plasma density downramp transition. Wavebreaking can also be obtained in this way.

Despite 3-D eects can be very important in to describe wake evolution and injection dynamics, the points described above, and based on a 1-D model, remain valid. The dierent injection schemes suggested by the three previous points have been experimentally investigated and demonstrated. They include the self-injection which may be transverse[7] (bubble/blow-out regime) or longitudinal (longitudinal self-injection[30]), density gradient injection[15, 16], shock injection[17, 31] and ionization injection[18, 19, 20, 21].

Since background plasma electrons considered here are obtained exclusively via laser tunneling ionization, ionization assumes a particular importance in this work, regardless of the injection mechanism. In fact, the main laser pulse used to create the wakeeld is the same that ionizes the gas target. For this reason tunneling ionization will be involved to explain our observations, including the injection mechanism. In the next section we will describe the basics of tunneling ionization, the role of ionization in trapping and the dynamics of the so-called ionization injection. Further injection mechanisms will be discussed later, in Sec.(2.5) and (2.6).

2.4 Ionization Injection

Gas targets can be roughly divided in two categories: low-Z gases and high-Z gas. When using low-Z gases as hydrogen and helium, ionization generally occurs well before the main pulse arrives, leading to a cold

2.4 Ionization Injection Laser Wakeeld Electron Acceleration background plasma; for such cases, the 1-D injection mechanism has already been discussed in Sec.(2.2.1). For arbitrary targets, the probability W of tunneling ionization based on the AmmosovDeloneKrainov (ADK) model[32] for an electron with a nal energy ε is given by[33]

W (ε) ∼ exp  −2 3 λ λc a3γ_k3Ea |E|− γ_k3ε }ω  , (2.28)

where E ∼ cos(ωt) is the eld of the laser, Ea ≡ mec2k0/e, γk = (αf/a)(Ui/Uh)1/2is the Keldysh parameter,

with Uh = 13.6 eV the ionization potential of Hydrogen, Ui is the ionization potential, αf = 1/137 is the

ne structure constant, λc = 2.4263 × 10−10cmis the Compton wavelength and a is the normalized vector

potential amplitude. The ionization rate(2.28) integrated over a laser pulse gives the cumulative probability of extracting an electron with an ionization potential Ui and a nal energy ε. As we know, in the case of

LWFA experiments, relativistically intense (I & 1018_W/cm2_{, a & 1) laser pulses are used; then it becomes}

interesting to distinguish between outer atomic orbitals for which ionization processes occur before the peak of the pulse, and inner atomic orbitals for which ionization occurs on the peak. In general, by using gases with large gap between successive ionization potential Un+1/Un& 3 with Un. 100 eV it is always possible to

obtain the desired eects of separating in time the ionization on peak and o peak. Typical examples are nitrogen (U6/U5 = 5.6, U5= 98eV), oxygen (U7/U6= 5.7, U6= 138eV) or argon (U9/U8= 2.9, U8= 143

eV). In the case of ionization injection we are specically interested in such higher Z-gases, in which the inner shell electrons can be ionized only at the peak of the pulse.

From a mathematical point of view, ionization injection is not dierent injection mechanism from already discussed in Sec.(2.2.1), and only involves the further description of the initial state of ionized electrons. In order to explain the ionization injection we will follow the approach used by M. Chen in his original article[34]. As discussed in the appendix5_{, the transverse equation of motion for an electron in a plane wave is xed by}

˜

p_⊥= a⊥ (momentum conservation). As it appears clear, the higher is the eld a⊥ in correspondence of the

ionizing starting point, the greater will be the nal electron transverse momentum. In fact, if we take into account the ionization, for an electron ionized at a generic wake phase ζi we obtain:

˜

p_⊥(ζ) = a⊥(ζ) − a⊥(ζi) , (2.29)

where ∂ζ(˜p⊥(ζ) − a⊥(ζ)) = 0 and ˜p⊥(ζi) = 0 has been used (electron is born at rest). Assuming a

laser intensity well tuned with the target ionization potentials, since the ionization rate is maximum at

2.4 Ionization Injection Laser Wakeeld Electron Acceleration

Figure 6: Normalized laser eld ˆE (red lines) and normalized potentialφ (blue lines). The dashed lines show the ionization starting point of an inner shell electron. In this case ∆φi > 1 allowing to ionization

injection.[35]

the maximum of laser intensity (look at Eq.(2.28)), most of ionization will occur at the peak (inner shells electrons) or before the peak (outer shell electrons) of the laser pulse. We stress that if the electrons are ionized exactly at the peak of the laser electric eld, then a⊥(ζi) = 0 and the transverse momentum will

became ˜p⊥≈ 0far from the laser. However, some electrons will be ionized o-peak with nite (and generally

not small) a⊥(ζi), than a non null transverse momentum ˜p⊥≈ −a⊥(ζi)is expected to be conserved far from

the pulse. Therefore, since contribution to beam emittance is primarily due to ˜p⊥, in order to reduce the

beam emittance, the optimal situation occurs if electrons are ionized exactly at laser peak or far before the peak. For this reason, gas with suitable large gap between successive ionization potential or gas mixtures can be used to concentrate electrons ionization starting points on and far from peak. Clearly, the best gas (or gas mixtures) choices, strongly depend on pulse properties.

The Hamiltonian of a single electron ionized in a wakeeld and born at rest (˜p ' 0) can be obtained from the general expression (2.14), and yields:

Hi(ζi) ' 1 − φ (ζi) , (2.30)

2.4 Ionization Injection Laser Wakeeld Electron Acceleration section and valid for an already ionized medium. In particular, it appear analogous6 _{to the rst order}

expansion of H(˜p, ζ) ' 1 − βphp˜, which describes a plasma electron with an initial longitudinal momentum

˜

p. Considering that φ(ζi)plays the same role as ˜p, we already know that injection threshold will decrease

by increasing φ(ζi). The only way to increase φ(ζi)is to ionize electrons around the pulse peak, as shown

in Fig.(6). In fact, the wake potential starts growing in correspondence of laser pulse front, where the ponderomotive force which creates the wake is maximum. In the case of inner shell electrons we expect also that injection will start well before the wavebreaking limit, as in the case of relativistic trapping threshold ˜

pt.

The trapping condition is not modied by ionization, and is given by the usual relation previously determined: Hi(ζi) ≤ Hs(˜pf ix, ζmin). Finally, using the relations (2.18) and (2.30) we obtain:

1 − φ (ζi) ≤ γ⊥(ζmin) γph − φmin= 1 + a2_⊥(ζi)
1/2 γph − φmin, (2.31) where Eq.(2.29), γ2

⊥(ζmin) = 1 + ˜p2⊥(ζmin)and a⊥(ζmin) = 0 has been used. The trapping threshold (2.31)

is usually presented as:

∆φi≥ 1 − 1 + a2 ⊥(ζi)
1/2 γph = −φb, (2.32)

where the potential well ∆φi= φ(ζi) − φminis a positive quantity, while φb is the nonlinear breakdown eld.

As anticipated above, once considering the ionization, the injection threshold (2.32) shows that relativistic wavebreaking is not required to trap cold electrons because ∆φ ≥ −φb does not implies φmin = φb, but the

less restrictive condition of: φ(ζi) ≥ φb− φmin. Fig.(6) shows a LWFA conguration in which the injection

condition ∆φi& 1 is satised thanks to the large ionization potential φ(ζi). However, despite wavebreaking

is not required to get ionization injection, we observe that the condition ∆φi& 1 implies however relativistic

elds (elds are normalized). According to [34], the normalized vector potential a0 must be a0≥ 1.7.

The injection threshold (2.32) generalizes the result (2.21) obtained in the case of initially cold plasma. Ionization injection can be dened thanks to the Eq.(2.32) as a self-injection in which injected electrons are only those ionized around the pulse peak. In comparison to self-injection, tunneling ionization injection requires a lower laser intensity to trap electrons, and represents a useful mechanism for injecting electrons into lower density wakeelds[19] or to produce higher electron beam energies[9].

Anyway, since continuous injection in a gas target is usually inevitable, the energy spread control in this

6_{With φ(ζi)instead of β} php˜.

2.5 Downramp and Shock Laser Wakeeld Electron Acceleration mechanism remains a challenge. The best way to overcome this limitation consists in reducing the portion of the acceleration length in which the injection condition ∆φi& 1 is satised. Two dierent schemes using

this feature have been proposed. In the rst one, the target portion which contains the high atomic number gas contaminant is spatially limited[34]. The second one[21, 35] is based on the nonlinear evolution of the wakeeld, that under suitable conditions, ensures that the threshold ∆φi& 1 can be reached only in a limited

portion of the plasma target since φmin increases with time (until ∆φi < 1is obtained). This technique is

called self-truncated ionization injection.

2.5 Downramp and Shock

Injection schemes based on downward plasma density ramps were rst proposed by Bulanov, S., N. Naumova, F. Pegoraro and J. Sakai in they original article in 1998[12]. Downramp injection relies on the slowing down of the plasma wave velocity at the plasma density ramp.

Before we describe this method, it may be useful to recall the fundamental role of the plasma density n0

in the trapping process. As we know, according to the 1-D model presented in Chap.(2.2.1), a lower wake phase velocity γph is able to enhance the electrons trapping. This is evident in Fig.(5), where the minimum

of the injection threshold (which corresponds to a wavebreaking) decreases with decreasing γph. In a linear

approximation γph = ω/ωp ∼ n −1/2

0 , thus the simplest way to decrease γph is to increase the background

plasma density. The increment of the plasma density can be considered the fundamental tool to reach self-injection in every LWFA experiment; basically, it represents the simplest way to trigger self-injection. However, according with Eq.(2.26) (and also with experimental results showed in Fig.(3)), the increase of the plasma density has a negative eect on particles energy since ˜pmax ∼ n−3/20 or ˜pmax ∼ n−10 for the

nonlinear o linear regime respectively. In general, self-injection and acceleration require opposite conditions and separate plasmas must be setup[36].

The scheme proposed by Bulanov et al. is based on the local reduction of γph in the correspondence of

a plasma density transition to overcome the limitation arising from increasing n0. Moreover, since injection

is local, in principle it allows well collimated, short and almost monochromatic electron bunches to be produced[37].

The phase velocity of the wake during a density transition can be calculated by considering the local phase ψ of the wake, which is given by:

ψ ≡ k [z − vpht] , k = ∂zψ, ωp= vphk = −∂tψ, (2.33)

where k is the plasma wavevector while ωp is the plasma frequency. Since we consider only underdense

2.6 Bubble Regime Laser Wakeeld Electron Acceleration density transition can be calculated once replacing k and ωp with their respective expression in terms of the

local phase. Thus we obtain

vph= ωp k = − ∂tψ ∂zψ ' ck (k + ζ∂zk) , (2.34)

where ζ = z − ct is the comoving coordinates and vph' chas been used. For a small variation of ∂zk(z)we

nd: βph' 1 1 + ζ_k∂zk (z) ' 1 −ζ k∂zk (z) = 1 − ζ 2n0 ∂zn0 (2.35) where k ∼ n1/2

0 has been used. Since ζ < 0 behind the driving pulse, the wake phase velocity will decrease

for decreasing density ∂zn0< 0. Injection triggering can be explained using the usual 1-D theory as a local

decreasing of the injection threshold ˜pt(see Eq.(2.20)). In the case of sharp transition (i.e. (ζ/k)∂zk(k)  1)

wavebreaking can be locally reached enabling the injection of cold electrons. Clearly, since plasma electrons are generally ionized via tunneling ionization, the corrected cold electron injection threshold (2.32) (which does not require wavebreaking) can be used also in the presence of downramp density transition.

An intuitive description of the downramp injection arises from the following observation: the plasma wave-length λp ' 2πcω−1p ∼ n

−1/2

0 locally increases with a decrease in density, thus some of the background

electrons returning to the optical axis will fall into the acceleration bucket of the wakeeld getting trapped. From theoretical[12, 37, 38] and experimental[15, 16, 17, 31] points of view, it is now established that downramp/shock enables a very eective way of controlling injection. However, a sharp density tran-sition, which is the best to enhance the trapping, is not easy to be implemented experimentally as we will discuss in detail is Section (7.5).

2.6 Bubble Regime

Plasma wavelength λp = 2πvph/ωp ∼ n −1/2

0 increases not only at lower plasma densities but also at higher

laser intensity. This nonlinear eect can be explained in term of the nonlinear correction of the wake phase velocity; for example γph becomes7 γph ' ω/ωp 1 + a2/8

at the rst order in a  1, where a is the amplitude of the normalized laser vector potential a. More rigorously, the nonlinear plasma wave increases as the wake amplitude increases (which depends on laser intensity). In fact, the nonlinear plasma wavelength can be expressed as[39]

2.6 Bubble Regime Laser Wakeeld Electron Acceleration

Figure 7: Scheme of principle of the bubble regime. Electron density wake is presented. The laser pulse propagates from left to right.

λp' 2 π  2πc ωp   ˆ_{Em+ ˆE}−1 m  ,  ˆEm 1  , (2.36)

where ˆEm= Em/E0is the normalized longitudinal electric eld. As we can see from Eq.(2.36), the nonlinear

increment of λpis not negligible in the case of strong eld. Such an increase in the plasma wavelength with

increasing wake (and laser) amplitude has an important eect on the geometry of plasma waves. In fact, since the laser intensity peak is generally maximum on axis (Gaussian radial prole), also the plasma wave amplitude is maximum on axis. Then, according to Eq.(2.36), the plasma wavelength on axis is larger than o axis. This causes the wave fronts of the plasma wave to become curved and take on a horse-shoe shape, as showed in Fig.(7). The greater the distance behind the driver, the more severe the curvature becomes. Moreover, since the nonlinear plasma wavelength λp= λp(r)becomes a function of the distance r from the

laser axis, the curvature increase also with the ratio λp/w0, where w0 is the waist of the pulse. In fact, the

transverse derivative of λp can be approximated as: ∂rλp ' (λp(0) − λp(w0))/w0. At the end we can say

that the wake front curvature becomes relevant if the following condition is satised: w0

λp . 1, (2.37)

where as λpwe consider the linear limit λp' 2πc/ωp. Another eect of nonlinear 3-D interaction is that the

laser intensity can be suciently high so as to completely expel all plasma electrons from the vicinity of the axis, thus forming a positively charged cavity (ions) surrounded by a dense region of electrons, immediately behind the laser pulse. Such a geometry, reached for a > 1 and w0. λp denes the so-called blow-out or

2.6 Bubble Regime Laser Wakeeld Electron Acceleration bubble regime[40]. The reason of such a name simply arises from the spherical geometry of the region from where the electrons are expelled. This is clear from Fig.(7), where two empty regions behind the pulse are showed. Moreover, it is important to observe that in the bubble regime the presence of electron cavitation behind the pulse implies that electrons must necessarily move also in transverse direction. Thus we cannot use the 1-D injection model to explain here the injection mechanism.

The plasma density prole is determined by balancing the laser radial ponderomotive force with the space charge force. For a Gaussian pulse prole, the complete blowout of plasma electrons occurs for a laser intensity satisfying:[41] a2 γ⊥ & 1 4(kpw0) 2 , (2.38)

which becomes a & (kpw0)2/4 in the high-intensity limit a2  1. The blow-out region of the wake is

characterized[42] by an accelerating eld of Ez ' (kpζ/2)E0 that is constant as a function of radius and

varies linearly as a function of distance behind the driver, and a focusing eld of Er ' (kpr/4)E0 that is

linear as a function of radius. Thanks to the linear focusing forces, in this regime, the normalized emittance of an accelerated electron bunch will be preserved during the acceleration.

As discussed in Ref.[43], despite its relevance, a conclusive theory of particle self-injection and trapping in the 3-D nonlinear bubble regime, which is drastically dierent from the nonlinear 1-D regime presented in Sec.(2.2), is still missing. Of the several analytic models proposed to describe the trapping mechanism inside the bubble, it is useful to report the results obtained by Thomas[44] (and further discussed by S. Corde[45]), since they showed the best agreement with both the experimental and numerical results[43]. Thomas, assuming that the orbits of the electrons reaching the back of the bubble (i.e., the ones that most likely will be self- injected) are elliptical, found the following injection threshold:

kpRb> 2 r ln2γ2 ph  − 1, (2.39)

where Rb is the radius of the bubble. Injection threshold reported in Eq.(2.39), in addition to a weak

dependence on the wake phase velocity, is based on a geometrical property of the bubble: the normalized radius R ≡ kpRb. Since threshold depends only on the plasma density and bubble size, we can determine

the minimum pulse properties required to reach the threshold. In fact, according to Lu[46], the radius of the bubble is related to the pulse energy and duration through

2.6 Bubble Regime Laser Wakeeld Electron Acceleration

Figure 8: (Left); Scaling of normalized bubble radii R⊥ (red line) and Rk (blue line) with normalized laser

eld strength a obtained via PIC simulation. The black dashed line is the quantity R = 2√awhich is the theoretical bubble size proposed Eq.(2.40).[43] (Right); Amount of self-injected charge in the bubble wake for dierent values of the wake velocity γph, and normalized laser eld strength a. Numerical results are

compared with analytical injection thresholds: (a) R > 4[46], (b) a > γ2

ph/2[42], (c) R > 2(ln(2γph2 )−1)1/2[44].

kpRp' kpw0' 2

√

a, (2.40)

where a ≥ 2 and a perfectly spherical bubble are assumed. Considering, for example, a phase velocity of γph = 15 and replacing Eq.(2.40) into Eq.(2.39), we nd an injection threshold of ath ' 5, which is in

agreement (order of magnitude) with experimental observations.

Using a numerical particle in cell (PIC) approach, Benedetti et al.[43], showed that for a > 5 a signicant deviations from the spherical bubble shape can be observed. For this reason, the wake shape/size can be more correctly characterized by the two normalized radii; Rk ≡ kpRbk (longitudinal) and R⊥ ≡ kpRb⊥

(transverse), which are measured starting from the center of the bubble, as presented in Fig.(8). In general, the geometrical properties of the bubble are weakly dependent on γph, so the bubble shape can be simply

characterized by a, as in the analytic model. According with Ref.[43] we nd:

kpRbk≡ Rk' 2.9 + 0.305 · a, P IC (2.41)

which depends linearly on a, in contrast with Eq.(2.40). In any case, as shown in Fig.(8), the analytic model(2.40) is really close to the numerical one in correspondence of a ' 3.5 (spherical bubble), and

rep-2.6 Bubble Regime Overview of the Experiments resents a good approximation in the entire 2 ≤ a ≤ 7 range once considered the representative radius R as the smaller radius between R⊥ and Rk. As we can see from Fig.(8), the analytic threshold (2.39) is in a

qualitative agreement with the numerical injection threshold, which is found to be[43]:

ath(γpf) ' 2.75  1 +γph 22 21/2 . (2.42)

As we can see from Eq.(2.42), their numerical results suggest a stronger dependence of the injection threshold on γphcompared to the Thomas's model. Considering, again, γph= 15as example, we nd ath' 3.3, which

is in a better qualitative agreement with experimental results. A further (slightly) decrease of injection threshold, may be arise also from the evolution of the bubble, which is not taken into account in the previous model. In fact, Kalmykov et al.[47] showed a that self-injection can be induced by a slow temporal expansion of the bubble. The existence of a minimum expansion rate which ensures trapping in a spherical bubble is also discussed.

Overview of the Experiments

Part II

Overview of the Experiments

3 Experimental Conguration

During my Ph.D I had the opportunity to join several experiments dedicated to LWFA of electrons. Exper-iments were performed at the Intense Laser Irradiation Laboratory (ILIL), CNR Pisa (Italy), at Rutherford Appleton Laboratory (RAL), Didcot (UK) and at FLAME facility in the Laboratori Nazionali di Frascati, Frascati (Italy) from 2014 to 2016. All experiments described in this thesis were performed at the Intense Laser Irradiation Laboratory (ILIL) of the Istituto Nazionale di Ottica (INO) of CNR in Pisa using the 10 TW class Ti:Sa8_{laser system.}

3.1 Laser Pulse Properties

The infrared (λ = 800 µm) laser pulse is characterized by a beam quality factor of M2

. 1.5 and an ASE (amplied spontaneous emission) contrast of > 109_{, which enables almost pre-plasma free interaction[48].}

In other words, the ASE is suciently low to ensure that no premature plasma formation occurs prior to the arrival of the main femtosecond pulse. The laser has a 10 Hz repetition rate and delivers a τL . 40 fs

(FWHM) pulses with a total energy on target of

εL'

r π

4 ln 2P0τL' 380 mJ (3.1) focused down to a 20 µm diameter focal spot (FWHM) using a f/10 o-axis parabolic mirror. According to the general properties of single mode T EM00 (i.e. Gaussian) laser beam (see appendix), the maximum

intensity in vacuum is:

I0W/cm2 ' 6.36 · 1019 P0[TW] (w0[µm])2 ' 2.0 × 1018W/cm2, (3.2) where w0 ' F W HM/ √

2 ln 2 ' 16.9 µmis the waist of the beam. In a similar way we nd the maximum amplitude of the normalized vector potential:

a ' 0.85λ [µm]pI0[1018W/cm2] ' 0.96. (3.3)

3.2 Interaction Chamber Setup Overview of the Experiments Both quantities I0 and a dene a moderately relativistic regime. At the xed laser power P0, the pulse

amplitude scale as a ∼ w−1

0 while the focusing length scales as zR ∼ w02. The optical setup, based on f/10

OAP parabolic mirror, was chosen in such way to maximize the Rayleigh length ensuring, at the same time, an acceptable moderately relativistic intensity of a ' 1. In fact, we obtain a Rayleigh length of

zR=

πw20

λ ' 1.1 mm, (3.4)

which is matched to the length of the plasma target used in our experiments. Another important parameter which characterizes the laser-plasma interaction is the critical density:

nc1018cm−3 '

1089

(λ [µm])2 = 1.74 × 10

21_cm−3_{, (3.5)}

which distinguishes between underdense (n0 < ncr) and overdense (n0 > ncr) plasmas. As we will see,

in our case, densities are well underdense, in the range between 5 − 10 × 1018_cm−3_{. Finally, it is also}

interesting to consider the self-focusing threshold calculated according with the usual[49, 50] denition of Pcr[GW ] ' 17.5 · ncr/n0of the critical self-focusing power. Using P0= 9 TWas critical power we nd the

minimum plasma density required for self-focusing to occur:

9 [TW] ' 0.0175 · ncr nsf

[TW] → nsf = 3.4 × 1018cm−3. (3.6)

3.2 Interaction Chamber Setup

Laser-plasma interaction takes place inside a vacuum chamber, as schematically shown in Fig.(9). In our case we use a spherical chamber of ∼ 65 cm diameter. The main beam, with a diameter of ∼ 5.5 cm, is reected using a metallic mirror onto the f/10 o-axis parabolic mirror (OAP). The OAP focuses the beam on a gas-jet produced by a rectangular supersonic Laval nozzle, connected to a high pressure gas line and mounded on a remotely controlled 3-D motorized translation stage. A fast switching valve allows the jet to be operated with with pulsed operation of a few ms. Nitrogen (N2) or helium (He) gas-jets were used. The

target can be positioned with 10 µm resolution in all directions, with a maximum range of 4 mm.

The vacuum pressure in the chamber is kept below 10−4_{mbars. A dual stage rotary pump is used to go}

3.2 Interaction Chamber Setup Overview of the Experiments

Figure 9: Schematic layout of the experimental setup used for the experiment. Laser pulse is focused onto the gas target then, after diraction, is shielded by the aluminized output windows. Electrons (with typical energies of ≥ 10 MeV) are not shielded by 86 µm thick of thermoplastic media. Then, electron properties can be measured outside the chamber.

After a gas-jet cycle (tens of msec of duration), ∼ 3sec are normally needed to restore the desired pressure of . 10−4_{mbars. Laser pointing optimization is performed in vacuum using two remotely controlled dielectric}

turning mirror9_.

The laser-gas interaction is primarily characterized using Thomson scattering and shadowgraphy imaging. Thomson scattering is based on the elastic scattering of laser light from plasma background electrons. Radiation is emitted perpendicular to the polarization plane of the incident laser radiation, collected through an optical system and detected using a standard reex camera. The Thomson imaging diagnostic system (not shown in Fig.(9)) is basically used to follow propagation of the main laser pulse in the plasma.

Shadowgraphy of the plasma is realized with an auxiliary, frequency doubled (λ = 400nm) sub-picosecond probe pulse propagating along the axis perpendicular to both the main laser pulse propagation direction and the Thomson line of sight. Shadowgraphy is used to detect density gradients in a rapidly evolving plasma. By varying the time delay between main pulse and probe pulse we can follow the evolution of interaction (results are always integrated along the probe duration) and plasma size along the probe pulse propagation

9_{Two remotely controlled dielectric turning mirror are used to transport the main pulse from the compressor exit to the} interaction vacuum chamber. On the other side, the metallic mirror place inside the chamber and used for beam optimization is manually tunable.

3.2 Interaction Chamber Setup Overview of the Experiments line. Thomson and shadowgraphy images are used as a guide to optimize the propagation of the laser through the gas-jet. We will not present any result based of such diagnostics which are shown elsewhere[51, 52].

The transverse spatial prole of the accelerated electron bunch is monitored using a regular Lanex scintillating screen mounted on the bunch propagation direction outside the interaction chamber as shown in Fig.(9). A 70 µm thick Mylar windows with a diameter of 7 cm is used to extract electrons from the vacuum chamber. In addition, the exit windows is covered with a 20 µm thick Mylar aluminized layer lm in order to stop laser light. Lanex images, taken with a CCD camera (Andor) mounted as shown in Fig.(9), are used to measure bunch divergence, geometry, pointing stability and charge. A calibrated integrating current transformer (ICT) is also used inside the chamber, mounded along the beam trajectory, to measure the bunch charge.

Electron energy spectra are measured using a magnetic dipole inserted on the electron propagation line before the scintillating screen, as shown in Fig.(9). Lanex screen position is xed; swapping from beam spatial prole measurement to beam energy measurement (deection) only requires to set the magnetic dipole in position.

A detailed description of gas-jet characterization, Lanex scintillating screen, magnetic spectrometer and pepperpot techniques is given below in Sections (4) ,(5) and (6).