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A Photon Collider at RelativisticIntensity

Dissertation zur Erlangung des akademischen Grades

Doctor rerum naturalium (Dr. rer. nat.)

vorgelegt dem Rat der Physikalisch-Astronomischen Fakultätder Friedrich-Schiller-Universität Jena

von M. Sc. Ben Liesfeld,

geboren am 19.03.1977 in Frankfurt am Main

Gutachter1. Prof. Dr. Roland Sauerbrey

Institut für Optik und QuantenelektronikFriedrich-Schiller-Universität, Jena, Germany

2. Prof. Dr. Francois AmiranoffLaboratoire pour l’Utilisation des Lasers IntensesEcole Polytechnique, Palaiseau, France

3. Prof. Dr. Dino A. JaroszynskiTerahertz to Optical Pulse SourceUniversity of Strathclyde, Glasgow, United Kingdom

Tag der letzten Rigorosumsprüfung:Tag der öffentlichen Verteidigung:

Abstract

In this work a novel, powerful - but compact and versatile - experimental scheme,coined “photon collider”, is presented. It provides counter-propagating, focused,ultra-short laser pulses and the diagnostic means to accurately achieve spatial andtemporal overlap of these pulses. The functionality and usability of the photoncollider was demonstrated experimentally. For the first time an autocorrelation mea-surement was carried out at full relativistic intensity exactly in the laser focus andtherefore under realistic experimental conditions. In a second experiment soft x-rayswere generated through Thomson backscattering from laser-accelerated electrons.Our results represent the first observation of Thomson backscattered photons in anall-optical setup. The backscattered radiation was used to obtain time-resolved spec-tra of the electrons in the plasma during the acceleration process. To our knowledgethese are the first time-resolved spectra of laser-accelerated electrons ever recorded.The photon collider, as a novel diagnostics tool, will shed light on such revolutionaryplasma acceleration concepts like bubble acceleration.

Its potential applications, however, have not yet been exploited with the presentedexperiments. Work is under way which will transform the photon collider to a truecollider - of electrons. The creation of a laser-based positron source is envisionedconsidering the new high-intensity lasers currently under construction. Future ex-periments may also explore the realm of non-linear quantum electrodynamics in theoptical regime.

Contents

1 Introduction 4

2 Laser-matter interaction 72.1 Non-linear Thomson scattering . . . . . . . . . . . . . . . . . . . . . 7

2.1.1 Perturbative approximation of the scattering process . . . . . 92.1.2 Rigorous treatment of the scattering process . . . . . . . . . . 14

2.2 Light propagation in a plasma . . . . . . . . . . . . . . . . . . . . . . 232.2.1 Linear propagation . . . . . . . . . . . . . . . . . . . . . . . . 242.2.2 Nonlinear propagation . . . . . . . . . . . . . . . . . . . . . . 25

2.3 Electron acceleration . . . . . . . . . . . . . . . . . . . . . . . . . . . 282.3.1 Laser wakefield acceleration . . . . . . . . . . . . . . . . . . . 282.3.2 Self-modulated laser wakefield acceleration . . . . . . . . . . . 292.3.3 Direct laser acceleration . . . . . . . . . . . . . . . . . . . . . 292.3.4 Bubble acceleration . . . . . . . . . . . . . . . . . . . . . . . . 31

2.4 Thomson backscattering in the linear regime . . . . . . . . . . . . . . 322.4.1 Scattering from a single electron . . . . . . . . . . . . . . . . . 332.4.2 Scattering from a laser-accelerated electron bunch . . . . . . . 36

3 Single-shot autocorrelation at relativistic intensity 413.1 The Jena Ti:Sa laser system (JETI) . . . . . . . . . . . . . . . . . . . 423.2 Protection of the laser system . . . . . . . . . . . . . . . . . . . . . . 433.3 Experimental setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . 443.4 Alignment of the photon collider . . . . . . . . . . . . . . . . . . . . . 473.5 Exemplary scenarios . . . . . . . . . . . . . . . . . . . . . . . . . . . 513.6 Experimental results . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

4 Thomson backscattering from laser-accelerated electrons 574.1 Experimental setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . 584.2 X-ray diagnostics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 634.3 Background and competing processes . . . . . . . . . . . . . . . . . . 684.4 Temporal change of total backscattered radiation . . . . . . . . . . . 714.5 Electron spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

5 Future prospects of the photon collider 815.1 Thomson backscattering as x-ray source . . . . . . . . . . . . . . . . 815.2 Electron collider and positron production . . . . . . . . . . . . . . . . 875.3 Non-linear QED . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

Bibliography 90

1 Introduction

A light intensity of I ≈ 1018 W/cm2 represents a relativistic threshold in the near-infrared wavelength regime. Electrons that are subjected to light intensities exceedingthis value oscillate with relativistic velocities and a number of fascinating phenomenaoccur which are due to the relativistic nature of the electron motion. In fact, theterm relativistic optics is used to describe this intensity regime [1]. On the basisof relativistic phenomena ultra-high laser intensities have opened up new excitingprospects in areas such as laser particle acceleration [2–6], laser-generated high energyphotons [7–9] or laser induced nuclear reactions [8, 10–14].

All these past experiments were based on the use of a single, ultra-short and ultra-intense laser pulse interacting with matter. In the course of this work experiments willbe presented introducing a new concept into relativistic optics: the use of counter-propagating laser pulses. The analogy to conventional facilities is so prominent thatwe coined this concept the “photon collider”.

The photon collider was developed by the high-intensity laser group at the Institutfür Optik und Quantenelektronik, Jena, to be used with the Jena titanium:sapphireterawatt laser system (JETI). It represents one of the compact terawatt lasers op-erating at near-infrared wavelengths which have become popular in the past decadeand which are capable of producing relativistic intensities. They are also referredto as “table-top” laser systems which means that they can still be installed onindustry-standard optical tables and that they are small enough to be operated bymedium-sized academic institutions. In 2002 intensities exceeding 1020 W/cm2 weredemonstrated with JETI [15]. To have some kind of reference to the “real world”considering this huge number: One would obtain such ultra-high intensity focusingall the sun light incident on earth onto a spot of 0.2 mm2.

A simplified diagram of the photon collider is shown in Fig. 1.1. In this schemean ultra-short laser pulse is divided into two laser pulses by a beam-splitter. Thetwo laser pulses are focused each by an off-axis parabolic mirror in a symmetric

4

1 Introduction

Figure 1.1: The photon collider setup: A high-intensity laser pulse is divided into two laserpulses by a beam-splitter. Both pulses are focused by 45 off-axis parabolic mirrors ontothe same point in space so that they are exactly counter-propagating. The time delay ∆τbetween the pulses may be adjusted by moving the beam-splitter. The counter-propagatingpulses constitute a standing wave in the focus (electric field indicated in blue, magneticfield indicated in green).

configuration where the plane of the beam-splitter is the symmetry plane. Why docounter-propagating laser pulses offer the means for novel laser plasma experiments?What are the powerful new aspects of the photon collider?

It is, actually, quite obvious: Instead of a single ultra-intense laser pulse the scien-tist now has control over two. Since ultra-short laser pulses incident on matter aresuch a rich source of many kinds of radiation and accelerated particles (see e. g. [16]),the photon collider may combine any two of these powerful sources or a particularsource and a laser pulse or just two laser pulses (and nothing in between - this willbriefly be discussed in Ch. 5). A necessary prerequisite to make this work practicallyis that the location of the two laser pulses in space and in time can be controlledvery accurately. The spatial accuracy must be as good as the focal radius (∼ 2 µm)and the temporal accuracy better than the pulse duration (∼ 80 fs).

The first experiment presented here is dedicated to the task to demonstrate thatthe required accuracy can be achieved in the photon collider setup. To this aim asingle-shot autocorrelation at relativistic intensity was carried out. Beyond the proofof principle of accurate control this represents the first pulse duration measurement atrelativistic intensity in the laser focus [17]. The pulse duration, in turn, is important

5

1 Introduction

to determine the laser intensity. The autocorrelation measurement is presented indetail in Ch. 3.

The second, even more challenging experiment which was carried out with the pho-ton collider is Thomson backscattering from relativistic laser-accelerated electrons - acombination of laser-based particle accelerator and counter-propagating ultra-intenselaser pulse. This is, again, the first measurement of its kind in an all-optical setup[18]. The photon collider in the Thomson backscattering configuration is, however,not only a source of backscattered radiation - which may be quite powerful by itself- but also an on-line diagnostics tool for the energy distribution of laser-acceleratedelectrons. The first in situ and time-resolved electron spectra are presented in Ch. 4.

During the short period of time that the photon collider has been operational (onlylittle more than about a year) only a small fraction (exactly these two previouslymentioned experiments) of the possible applications could be carried out. Ch. 5 willlist three more examples and describe what lies ahead of us. Finally the now followingchapter covers, subdivided into several short sections, some theoretical backgroundnecessary for the analysis of the experiments.

6

2 Laser-matter interaction

The mechanisms that are involved in the interaction of an ultrashort laser pulse withmatter, in our particular case with a gaseous target, are numerous and complex.In the context of this work we will focus on topics which are most important forthe analysis of the experiments described later. Still, this section may not at all becalled a complete description of the underlying physical principles. It will rather be asummary of well-established facts or elaborate on examples which are closely relatedto the experiments.

First, the theory of non-linear Thomson scattering will be discussed, which isthe basis for the single-shot autocorrelation experiment (Sec. 3), followed by shortoverviews over the propagation of a laser pulse in a plasma and electron accelerationmechanisms. At the end of this chapter I will return to a specific case of Thomsonscattering: Thomson backscattering in the linear regime in an all-optical setup. Thissection will be the basis for the analysis of the backscattering experiment in Sec. 4and makes use of almost all phenomena described up to then.

2.1 Non-linear Thomson scatteringThomson scattering is a classical description of scattering of photons from electronsin the regime where the scattered photon energy is small with respect to the electronenergy. For electrons at rest this corresponds to photon energies of less than 511 keV.If photon energies above this threshold occur, one usually refers to this process asCompton scattering. In very intense laser fields the classical description of Thomsonscattering is no longer valid since the electron motion becomes relativistic and theinfluence of the magnetic field on the electron motion can no longer be neglected.The motion of electrons in intense fields is non-linear and therefore radiation atfrequencies different from the fundamental laser frequency is emitted. This is thenon-linear Thomson scattering regime.

7


A parameter useful to determine the nature of the electron motion is the dimen-sionless laser field strength parameter

a0 =eE

meω0c,

where e is the electron charge, E the amplitude of the electric field, m the electronrest mass, ω0 the laser frequency and c the speed of light. In practical units a0 isgiven by

a0 = 0.85λ

1 µm

√I

1018 W/cm2,

where λ is the laser wavelength and I the laser intensity. Physically, a0 representsthe electron quiver momentum in an oscillating field in units of mec. For a0 1 theelectron motion can be considered classical.

An extensive relativistic theory of non-linear Thomson scattering has been madeavailable by Esarey et al. [19] and Lau et al. [20]. In the moderately relativistic regime(a0 < 1), however, the equations of motion of an electron in an electromagnetic wavemay be solved by perturbation theory. It can then be shown that the non-linearThomson scattered light at the second harmonic frequency is a non-linear effect ofsecond order. This allows to revert to well-known autocorrelation techniques forthe interpretation of our experimental results. I will therefore start with the ratherintuitive perturbative treatment of the scattering process and subsequently discussthe rigorous scattering theory.

Since this section only provides the background for the analysis of the experimentin Sec. 3 and 4 it seems appropriate to provide some reference values from the ex-periment for the following theoretical considerations. Simplifications arising fromexperimental conditions will be indicated as well. The laser light used in the exper-iment has a central wavelength of λ0 = 2πc/ω0 = 795 nm. This is located in thenear-infrared, invisible for the human eye. At this wavelength a laser pulse of 85 fs

duration contains about 30 oscillations of the electric field. Here, the pulse durationis typically determined as the full width at half maximum (FWHM) of a Gaussianenvelope. One photon of this wavelength has the energy of 1.56 eV. The laser pulsesmay be described with reasonable accuracy with Gaussian beams. Please note thatwhen focusing Gaussian beams the curvature of the wavefronts becomes negligiblein the focal region and the wave may be approximated by a plane wave for many

8


Figure 2.1: Observation of the Thomson scattered light: The electromagnetic wave propa-gates along the z-axis, the electric field is aligned with the x-axis and the electron oscil-lates in the x-z-plane. The electron orbit is shown in the reference frame of the electron(co-moving observer). The second-harmonic Thomson scattered light is observed alongthe x-axis.

applications.The second harmonic of the fundamental laser frequency ω0 has a wavelength of

λ2ω = 398 nm which is visible blue light. The laser light is linearly polarized andwe will from now on assume that the electric field vector E of any electromagneticwave discussed in this section is aligned with the x-axis. The propagation of thelaser light is set to be along the z-axis which lets the magnetic field vector B bealigned with the y-axis. In our experiments we chose to observe the interactionregion along the x-axis, the detector being placed in the x-z-plane. Fig. 2.1 showsthe orientation of the axes and the position of the observer. The trajectory of anelectron undergoing oscillations is indicated by a bold line (here, shown in the movingframe of the observer). This electron orbit, the characteristic “figure 8” movement,will be described in more detail in Sec. 2.1.2.

2.1.1 Perturbative approximation of the scattering process

The perturbative treatment starts with the formulation of the equations of motion ofthe electron. The scaling of the radiation emitted by the electron will be evaluatedwithout specifying if the exciting electromagnetic wave is a standing or travelingwave. The validity of the approximation will be verified computing higher orderterms and an example of two counter-propagating Gaussian pulses will be discussedwhich will be directly applicable to the experiment in Sec. 3.

9


Equations of motionThe relativistic Lorentz equation for a single electron in an electromagnetic fieldreads:

d

dt(γmr) = −e(E + r×B), (2.1)

where γ is the Lorentz factor, m the electron rest mass, r the electron velocity ande the electron charge. In the weakly relativistic regime the influence of the magneticfield on the motion of the electron will be small. Therefore its velocity along thez-axis will be small compared to its velocity along the x-axis. One may thereforewrite the Lorentz factor of the electron as

γ =1√

1− (r/c)2≈ 1√

1− (〈x2〉 /c2)= const., (2.2)

where the brackets in 〈x2〉 indicate the average over one laser period. Since theelectric field is aligned with the x-axis, E = (Ex, 0, 0), it follows that r × B =

(−zBy, 0, xBy) = (−zEx/c, 0, xEx/c)). The equations of motion can now be sim-plified to

γmx = −eEx + ez

cEx, (2.3)

γmy = 0, (2.4)

γmz = −ex

cEx. (2.5)

Neglecting the second term in Eq. 2.3 and using the initial conditions x(−∞) = 0

and z(−∞) = 0 and the electron being born in a maximum of the electric field, onecan integrate Eq. 2.3:

x(0) =e

γm

t∫−∞

E(t′) dt′. (2.6)

Please note that the Lorentz factor γ was set to be constant in Eq. 2.2.

Inserting Eq. 2.6 into Eq. 2.5 one obtains the acceleration of the electron along thez-axis

z(1) =e2

γ2m2cE(t)

t∫−∞

E(t′) dt′. (2.7)

10


The explicit form of E(t) has not yet been specified in any way which means thatthis formula is valid for traveling as well as for standing waves.

Emitted radiationTo calculate the radiation emitted by an electron at the second-harmonic (2ω0) fre-quency one may use the classic relation (see Jackson [21, p. 665])

|ET(ρ, θ, t)| = ez(t− ρ/c) sin θ

4πε0c2ρ, (2.8)

where ρ is the distance of the observer to the source, θ the angle of observation withrespect to the z-axis and ε0 the vacuum permittivity. The experimental conditionsfor the detection of the Thomson scattered light as described in Sec. 3.3 are as follows:

1. The radiation is detected in the x-z-plane under an angle of θ = 90 with respectto the z-axis while the distance ρ to the source is constant (see Fig. 2.1).

2. The envelope of the electric wave E(t) is varying slowly compared to the carrierfrequency and goes to zero for t → −∞.

3. The exposure of the detector (a CCD camera) is much longer than the time-scale of the interaction.

Taking condition (1) into account and inserting Eq. 2.7 into Eq. 2.8 the emittedintensity at 2ω0 may be written as

I2 =1

2ε0c |ET|2 =

e6

(4π)2ε0c5γ4m4eρ

2

∣∣∣∣∣∣E(t)

t∫−∞

E(t′) dt′

∣∣∣∣∣∣2

. (2.9)

Condition (2) allows to simplify

t∫−∞

E(t′) dt′ ≈ 1

iω0

E(t). (2.10)

11


Condition (3) means that one has to integrate the emitted radiation over time andthen arrives at the following relation for the detected Signal S2:

S2 ∼∞∫

−∞

∣∣E2(t)∣∣2 dt. (2.11)

Eq. 2.11 states that in the weakly relativistic regime the intensity of the non-linearThomson signal, S2, scales with the square of the laser intensity, I2. The 2ω-emissionof the laser plasma can therefore be used as a second-order autocorrelation signal.

Validity of the approximation

An estimate of the next higher order terms in the expansion of z (Eq. 2.5) will nowbe made. In order to carry out the time integrals it will be assumed that the electricfield has the form E(t) = E0 cos ω0t, which is a valid assumption for traveling as wellas for standing waves. The integration will be carried out from t = 0 using the initialconditions x(0) = 0, x(0) = 0, z(0) = 0, z(0) = 0. Starting again with Eq. 2.3

x(0) = − e

γmE0 cos ω0t = −ωc

γa0 cos ω0t,

where the dimensionless parameter a0 was inserted, integration yields

x(0) = − c

γa0 sin ω0t.

Please note that it is still assumed that Eq. 2.2 is a valid approximation. This resultin turn is inserted into Eq. 2.5 which yields

z(1) =ω0c

2γ2a2

0 sin 2ω0t (2.12)

and thusz(1) =

c

4γ2a2

0(1− cos 2ω0t).

Returning to Eq. 2.3 and using the last result one obtains

x(2) =ω0c

8γ3a0

[(a2

0 − 8γ2) cos ω0t− a20 cos 3ω0t

]

12


which leads through integration to

x(2) =c

8γ3a0

[(a2

0 − 8γ2) sin ω0t− a20 sin 3ω0t

].

One can then calculate following Eq. 2.5

z(3) =ω0c

48γ4a2

0

[(24γ2 − 2a2

0) sin 2ω0t− a20 sin 4ω0t

].

Comparing this result with Eq. 2.12 one sees that the corrections to the amplitude ofthe second-harmonic signal arising from higher-order expansion terms are of the orderof a2

0/12γ2. Corrections to the observed power of the second-harmonic signal will beof the order of (a2

0/12γ2)2. Please note that this estimate of accuracy reflects theaccuracy of the perturbative approximation for which the basic assumption γ = const

was used. In Sec. 2.1.2 the perturbative approximation will be compared to the exactsolution.

Example: focused counter-propagating Gaussian laser pulses

The non-linear Thomson signal generated by two focused and counter-propagatingGaussian laser pulses will be examined. The notation that is used here is the mostcommonly used definition of Gaussian pulses (see [22, 23]). The fields of the laserpulses incident from the left and from the right will be called EL and ER, respectively.The laser pulse incident from the right reaches its focus at t = ∆τ and its focus isnot necessarily located at z = 0 but at some arbitrary position z = z1:

EL(r, z, t) = E0Lw0

w(z)exp

[− r2

w2(z)

]exp

[−i

kr2

2R(z)

]×

exp

[−

(t− z/c

τL

)2]

exp [i(ω0t− kz + ϕ(z)] ,

ER(r, z − z1, t−∆τ) = E0Rw0

w(z − z1)exp

[− r2

w2(z − z1)

]exp

[−i

kr2

2R(z − z1)

]×

exp

[−

(t−∆τ + (z − z1)/c

τL

)2]×

exp [i(ω0(t−∆τ) +−k(z − z1) + ϕ(z − z1)] .

13


Here, E0L and E0R are the field amplitudes, w0 is the waist of the Gaussian beams,w(z) = w0

√1 + (z/z0)2 the beam diameter, z0 the Rayleigh length, R(z) = z(1 +

(z0/z)2) the wavefront curvature, τL the laser pulse length and ϕ(z) = arctan(z/z0).τL is related to the laser intensity pulse duration at FWHM by τ = τL

√2 ln 2.

The integrand of the autocorrelation signal according to Eq. 2.11 is now calculated:

∣∣E2(t)∣∣2 =

∣∣E2R + E2

L + 2EREL

∣∣2=

∣∣E2R

∣∣2 +∣∣E2

L

∣∣2 + 4 |ER|2 |EL|2

+E2RE2∗

L + E2∗R E2

L + 2E2RE∗

RE∗L + 2E∗

RE∗LE2

L

+2ERELE2∗R + 2ERELE2∗

L . (2.13)

This result can be simplified taking into account the limited spatial resolution of theapplied optical imaging system (see Sec. 3.3). The autocorrelation obtained in theexperiment is not interferometric, i. e. spatially fast oscillating terms in Eq. 2.13 canbe neglected. Spatially fast oscillating terms are those in which terms like exp[−ikz]

and exp[ikz] do not cancel out. Therefore the only remaining terms are:

∣∣E2(t)∣∣2 =

∣∣E2R

∣∣2 +∣∣E2

L

∣∣2 + 4 |ER|2 |EL|2 .

In the experiment the signal is integrated over r which does not change the z-depen-dence of the expression. Carrying out the integration over time one finds

S2ω ∼ I20L

1 +(

zz0

)2 +I20R

1 +(

z−z1

z0

)2

+I0LI0R

1 +(

z−z1/2z0

)2

+z21

4z0

exp

[− 4

τ 2L

(z − z1/2

c− ∆τ

2

)2]

. (2.14)

This function and its application and some special cases will be discussed in Sec. 3.5.

2.1.2 Rigorous treatment of the scattering process

The non-linear Thomson scattering theory [19, 20, 24, 25] allows to compute the radi-ation spectrum of a single electron (with known trajectory r(t)) in an electromagneticwave decomposed into harmonic frequencies. In the following, a short summary of

14


this theory will be given. The energy radiated by a moving charge per unit solidangle Ω and per unit frequency ω is given by Jackson [21, p. 676]:

d2

dωdΩW =

e2ω2

16π3ε0c|n× (n× F(ω))|2 , (2.15)

F(ω) =

∞∫−∞

β(t) exp [iω(t− n · r(t)/c] dt, (2.16)

where n the unit vector pointing in the direction of observation and β(t) the velocityof the electron normalized to the speed of light. With the help of Eq. 2.15 one maycalculate the radiation spectrum of any given electron motion, if the trajectory andthen β(t) is known.

If the motion of the electron is a periodic function of time with period T and ifa net displacement r0 of the electron is associated with each period of motion, onemay write

β(t + mT ) = β(t), (2.17)

r(t + mT ) = mr0 + r(t), (2.18)

m being an integer number. Then, F(ω) can be written using the periodicity of themotion [20]:

F(ω) =∞∑

m=−∞

Fmδ(ω −mω1) (2.19)

ω1 =2π

T − n · r0/c, (2.20)

Fm =ω1

2π

T∫0

β(t) exp [imω1(t− n · r(t)/c] dt. (2.21)

Here, ω1 is the fundamental frequency of the emitted radiation. From Eq. 2.20 onecan see that this frequency is not necessarily equal to the laser frequency. The latteris a valid approximation only in the limit of weakly relativistic interaction, as will beshown for the case of the traveling wave.

The movement of a single electron in a strong plane electromagnetic wave has

15


been discussed in the literature at great length [19, 20, 24, 26–28]. I will give a shortsummary of the results since this is one of the few cases where a closed form solutioncan be obtained.

In the following, convenient normalizations are introduced: time is normalized by1/ω0 and distance by c/ω0. Normalized coordinates are indicated by a hat. For alinearly polarized plane wave E = E0ex cos(t−z), with E0 the electric field amplitude,Eq. 2.1 can be written in the following form:

d

dt(γβx) = a0(1− βz) cos(t− z), (2.22)

d

dt(γβy) = 0, (2.23)

d

dt(γβz) = a0βx cos(t− z). (2.24)

The electric field strength is expressed in terms of the dimensionless parameter a0 =

eE0/mω0c.

The electron trajectory has a closed form solution for the initial conditions:

x(t = 0) = 0, y(t = 0) = 0, z(t = 0) = zin

βx(t = 0) = βx0, βy(t = 0) = βy0, βz(t = 0) = βz0,

if the trajectory is expressed as a function of the phase parameter θ = t − z. Theproblem can be further simplified assuming the initial conditions:

βx0 = βy0 = 0,

since then βy = 0 for all times. The initial phase under which the electron is bornin the electromagnetic field is θin = zin. The trajectory of the electron is given by[20, 26]

γ = γ0 +(a2

0 sin θ − sin θin)2

2γ0(1− βz0),

γβz = γ − γ0(1− βz0),

γβx =1

γ0(1− βz0)a0(sin θ − sin θin),

16


x = a0(cos θin − cos θ)− (θ − θin) sin θin

γ0(1− βz0),

t =(θ − θin)

1− βz0

[1 +

a20(1 + βz0)

2

(1

2+ sin2 θin

)]+

a20(1 + βz0)

2(1− βz0)

[−sin 2θ

4+ 2 cos θ sin θin −

3 sin 2θin

4

].

The velocity components βx and βz are periodic functions of θ with a period of 2π.The time period of the electron motion T can be calculated by subtracting t(θ + 2π)

from t(θ):

T =2π

1− βz0

[1 +

a20(1 + βz0)

2

(1

2+ sin2 θin

)]. (2.25)

Fig. 2.2 shows sample trajectories of an electron in a plane electromagnetic wavefor various θin and a0. In Fig. 2.2a) and c) the trajectory in the laboratory frame isshown while in Fig. 2.2b) a reference frame is introduced (primed coordinates) whichmoves with the electron along the z-axis. The moving frame representation makesthe “figure-8” movement of the electron visible. For an initially resting electron(βz0 = 0, γ0 = 1) and θin = 0 the trajectory simplifies to

x = a0(1− cos θ),

z =a2

0

4(θ − 1

2sin 2θ).

The fact that the amplitude of the motion scales with a0 along the x-axis but scaleswith a2

0 along the z-axis is illustrated by Fig. 2.2a) and b). For increasing a0 theacceleration along the z-axis, the component induced by the magnetic field, becomesdominant.

The comparison of the exact solution of the equations of motion to the perturbativesolution from the previous section is shown in Fig. 2.3. For a0 = 0.1 the perturbativeand the exact solution for βz are almost identical (Fig. 2.3a)). This is no longerthe case when a0 approaches unity. Inserting the Lorentz factor from the exactsolution averaged over one period 〈γ〉 into Eq. 2.12, the amplitude of the oscillatingacceleration are approximately equal but the frequencies contained in the oscillationare not (Fig. 2.3b)). The total radiated power may therefore be estimated correctlyby the perturbative solution but the spectral distribution will differ.

The spectral properties of the emitted radiation may be derived from the exact

17


Figure 2.2: a) Trajectories of a free electron in a traveling electromagnetic wave for differ-ent laser field strengths a0 = 1 and a0 = 2 calculated for the initial conditions θin = 0and βz0 = 0 in the laboratory frame. b) The trajectories in the moving frame of the elec-tron (indicated by the primed coordinate) exhibit the characteristic “figure-8” movement.Laser field strength and initial conditions corresponding to a) were used. c) Electrontrajectories in the laboratory frame for a0 = 1, βz0 = 0 and θin = −0.1, −0.2 and −0.3.

18


Figure 2.3: Comparison of perturbative (solid line) and exact solution (dashed line) ofβz(t) for θin = 0. a) a0 = 0.1. Perturbative and exact solution are almost identical.b) a0 = 0.5. Perturbative and exact solution yield approximately the same amplitude ofβz which means that the calculated total radiated power will be similar. The frequenciescontained in the oscillation differ, however.

solution. Using the relation dθ/dt = γ0(1− βz0)/γ one can transform Eq. 2.21 into

Fm =ω1

ω0

θin+2π∫θin

γβ(θ) exp

[2πim

t(θ)− n · r(θ)T − n · r0

]dθ, (2.26)

which is more suitable for calculations using the parametrized equations. For a giveninitial phase θin of the electron one may now calculate the power of each of theharmonics emitted into a certain direction. This can generally be done solving theequations of motion numerically and applying Eq. 2.15 to the numerical solution,but since the exact solution is known, the radiated power may be calculated withmuch less effort through Eq. 2.26.

For simplicity, I will confine myself to the configuration of the experiment describedin the beginning of Sec. 2.1, i. e. observing under an angle of 90 with respectto the z-axis and along the axis of the electric field (x-axis). The dependency ofthe frequency of the emitted radiation on the initial phase and on the laser fieldstrength a0 according to Eq. 2.20 is shown in Fig. 2.4a). Here, the second-harmonicfrequency is given in units of the laser frequency ω0. The frequency of the secondharmonic drops with increasing laser field strength - a circumstance which is notquite intuitively comprehensible. It is due to the fact that the period of the electronmotion T becomes elongated for large a0 (cf. Eq. 2.25). The frequency of the emittedradiation also strongly depends on the initial phase θin of the electron.

19


Figure 2.4: Dependency of the frequency ω = mω1/ω0 of the emitted radiation on thelaser field strength a0. a) Magnitude of the second-harmonic frequency (m = 2) emittedby the electron over laser field strength a0 and initial phase θin. b) Magnitude of thefundamental (red) and the first two harmonic frequencies (green and blue). The limitinglines of the areas correspond to an initial phase θin = π/2 (lower line) and θin = 3π/2(upper line). The grey bar represents the frequency band of the interference filter used inthe experiment.

In Fig. 2.4b) the frequency bands of the fundamental and the first two harmonicfrequencies over a0 are shown. A single electron born at a certain phase in theelectromagnetic wave still emits radiation at discrete frequencies but a number ofelectrons born each at different initial conditions lead to a broader emission bandstructure. At higher intensities the frequency bands of the harmonics overlap. Inthe limit of large a0 this finally leads to an almost continuous emission. In ourexperiment the non-linear emission of the laser plasma is observed through a narrow-bandwidth interference filter with a fixed central wavelength of 2ω0. This narrowband of detection is indicated with a grey bar in Fig. 2.4b).

In order to estimate the dependency of the detected radiation on a0 the initial phaseof the electrons emitting at the fixed frequency 2ω0 and the power of the emittedradiation were determined. The result is displayed in Fig. 2.5. The contributionsof the first three harmonics were calculated (dotted lines) as well as the sum of allcontributions (solid line). For small a0 the total radiated power at 2ω0 is proportionalto a4

0 as was expected from the perturbative solution. For a0 approaching unity theradiated power increases faster, proportional to a5

0, but beyond a0 = 1 a maximumis reached and the radiated power then decreases. This may have an impact on the

20


0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.00

2

4

6

8

10

12

14

16

18

m = 4

m = 3

~a40

Rad

iate

d po

wer

/ ar

b. u.

a0

~a50

m = 2

Figure 2.5: Dependency of the power emitted per unit solid angle at the constant frequency2ω0 on the laser field strength a0. The contributions from the harmonic frequencies mω1

for m = 2, 3, 4 are displayed as dotted lines. The solid line represents the sum of thesecontributions. For a0 < 0.5 the radiated power is proportional to a4

0, for a0 approachingunity this changes to a5

0. Above a0 = 1 a maximum is reached and the radiated powerdecreases.

measured autocorrelation signal: The autocorrelation signal of laser pulses with apeak field strength of about a0 = 1 may lead to shorter pulse lengths than they arein reality. Laser pulses exceeding a0 = 1.3 on the other hand will appear broader.

Until now, the movement of an electron in an intense traveling electromagneticwave was discussed. However, counter-propagating laser pulses form a standing wavein the focal region. The description of the electron motion in a standing wave turnsout to be more complicated, and appears to be erratic [29]. The solution cannot begiven in closed form except for very particular initial conditions. The equations ofmotion for a standing wave are:

d

dt(γβx) = 2a0(cos t cos z − βz sin t sin z),

d

dt(γβy) = 0,

d

dt(γβz) = 2a0βx sin t sin z.

These equations can no longer be parametrized introducing a universal phase θ

21


Figure 2.6: a) Traveling wave: Electrons (indicated by circles) born at different moments intime (denoted by “A”, “B”, “C”) and at different positions along the z-axis experiencethe same initial conditions in the electromagnetic field since only the phase relative tothe wave is relevant. b) Standing wave: The initial conditions of electrons born in astanding wave cannot be reduced to a single parameter.

and a systematic analytical analysis is impossible. Fig. 2.6 illustrates the qualitativedifference between the traveling and the standing wave problem: Fig. 2.6a) showselectrons (circles) which are born in the electromagnetic wave at different positionsz0. One may find an appropriate time t0 for each of the electrons such that theyexperience the same initial conditions and therefore one may attribute to these elec-trons the universal initial phase θin. This is no longer possible for a standing wave(see Fig. 2.6b)) because the maximum amplitude of the field depends on z and on z

only.

SummaryThe rigorous non-linear Thomson scattering theory yields some important resultswhich cannot be obtained from perturbation theory:

1. The phase at which the electron is born in the electromagnetic wave stronglydetermines the electron trajectory and therefore the radiation spectrum.

2. The fundamental frequency of the emitted radiation depends on the laser fieldstrength a0 and on the initial conditions.

3. The radiated power is shifted to higher harmonics for larger a0. In the limita0 1 an almost continuous spectrum is obtained.

22


Especially result (1) makes the use of particle-in-cell simulations (PIC) necessary.The temporal step size used in these simulations must be small enough to calcu-late the higher harmonics emission. This clearly means a large increase in requiredcomputation time and memory size.

The effect of an infinitely extended plane and intense electromagnetic wave ona single electron has been discussed. The results permit to describe the radiationspectrum of a single electron and to estimate the emitted radiation of a populationof plasma electrons. I will now turn to the more complex interaction of a focusedlaser pulse with a gas-jet.

2.2 Light propagation in a plasmaA laser plasma can be treated with the so-called two-fluid theory, one fluid beingthe light plasma electrons, the other fluid being the heavy ions. The light electronswill exhibit rather fast oscillations compared to the slower (acoustic) oscillationssupported by the ions. In our case, where the time scale of laser-matter interactionis determined by the laser pulse length which is much smaller than 1 ps, one mayneglect the slow movement of the ions and consider the ion density as constant [30].

The oscillations which an electron plasma can support are characterized by theplasma frequency ωp = (e2ne/ε0m)−1/2, where ne is the electron density in theplasma, and ε0 the vacuum permittivity. The dispersion relation of the linear waveequation in a uniform plasma is given by

ω2 = ω2p + k2c2 (2.27)

The wave number k becomes imaginary for ω < ωp which means that the wave isabsorbed since the electrons are responding to the light wave in a characteristic timeω−1

p . This condition defines a critical electron density above which propagation isimpossible for a light wave of given frequency. The critical density for laser lightof λ = 795 nm, as was used in the experiments described below, is nc = 1.1 ×1021/λ2[µm2] cm−3 = 1.7×1021 cm−3. Target material is commonly called underdense(overdense) if its density is below (above) the critical density.

When an ultra-short laser pulse of an intensity I > 1018 W/cm2 impinges on aHe gas-jet those parts of the pulse exceeding 1014 W/cm2 ionize the gas through

23


multi-photon and tunnel ionization. This means the rising edge of the laser pulsewill already have fully ionized the He gas before the main part of the pulse interactswith the gas-jet. The plasma and variations of its electron density in turn change thepropagation of the laser pulse. One can attribute a refractive index η to the plasmaand using Eq. 2.27 one can write

η = ck/ω =√

1− ω2p/ω

2 =√

1− ne/nc. (2.28)

2.2.1 Linear propagation

If the plasma density is sufficiently low, the propagation of an intense laser pulsethrough a gaseous medium may be considered linear under certain conditions. Thecriterion for linear propagation will be the phase shift ∆φ that the laser pulse un-dergoes due to ionization. This regime will become important for experiments whereeffects of propagation must be excluded and will therefore be discussed in the follow-ing.

In the focal region of a Gaussian beam one may approximate the beam with aplane wave E(z, t) = E0 exp iφ, where φ = ω0t − k0η(t)z is the phase of the wave,k0 the wavenumber in vacuum and η(t) the time-dependent index of refraction ofthe plasma. One has to assume that the refractive index depends on time since itdepends on the ionization state of the gas.

For ω ωp and therefore ne nc one may approximate

η(t) ≈ 1− 1

2

ne(t)

nc

.

The rate of change of the electron density may be written as dne/dt = R(t)(ni−ne/Z),where R is a function of the pulse envelope, ni the neutral gas density and Z theionization state of the gas atoms (Z = 2 for He gas). Since the early rising edge ofa relativistic laser pulse fully ionizes the He gas one may assume ne = 2ni and thusdne/dt = 0 for the main interaction. The phase difference between the part of thewave propagating through vacuum and the part propagating through the plasma isthen

∆φ = k0ni

nc

∆z, (2.29)

where ∆z is the extension of the plasma along pulse propagation. While this is valid

24


Figure 2.7: a) Ponderomotive potential of a Gaussian laser pulse perpendicular to the axisof propagation (z-axis). b) Corresponding ponderomotive force vector field. The axesare normalized to the radius (half width at half maximum) of the laser pulse. In b) theextension of the laser pulse along the z-axis was set equal to its extension perpendicularto the z-axis which is not true for the laser parameters used in the experiments (focalradius of the order of few microns, pulse length of the order of 100 fs corresponding to30 µm).

for a homogeneous plasma, in the case of a gas-jet with Gaussian profile as used inthe experiment ni∆z must be replaced by the integration

∫ni(z)dz for an accurate

estimate.

For the propagation to be considered unaltered the phase shift must hold thecondition ∆φ < 1. For a homogeneous plasma of 1 cm length and laser parametersas used in our experiment (nc = 1.7 × 1021 cm−3), the gas density must be ni <

2 × 1016 cm−3 to ensure that the refractive index of the plasma does not alter thepropagation of the laser pulse significantly.

2.2.2 Nonlinear propagation

At high intensities a laser pulse may have a large impact on the laser plasma densitydistribution. The plasma index of refraction is then dependent on the laser intensitywhich gives rise to non-linear effects.

25


The Ponderomotive force

The laser pulses generating relativistic intensities are tightly focused - quite in con-trast to the conditions in the earlier simplified discussions of electrons in plane waves.Averaging the equation of motion of an electron in such an inhomogeneous field overthe fast laser oscillations one arrives in the case of low laser field strength a0 1 atthe description of a force acting on the electron [30]

Fp = − e2

4mω2∇|E|2. (2.30)

This force is called ponderomotive force and is directed along the gradient of thelaser intensity. For a Gaussian beam profile this means that electrons are expelledfrom the optical axis. The ponderomotive vector field and the corresponding po-tential φ, which can be obtained from Fp = −∇φ, are visualized in Fig. 2.7. Theconcept of the ponderomotive force remains valid even for higher intensities as wasshown in [29, 31]. In the fully relativistic description an additional factor of 1/ 〈γ〉,where γ is averaged over the fast laser oscillations, is introduced:

Fp = − e2

4 〈γ〉mω2∇|E|2.

In a quasi-static picture the electrons are driven away from the beam axis by theponderomotive force until the resulting electric field between electrons and (immo-bile) ions counteracts this effect. At high laser intensities the ponderomotive forcecan lead to complete cavitation [32]. In the light of these findings one has to re-formulate Eq. 2.28 since the electron density now depends on the distance from theoptical axis r:

η =√

1− ne(r)/nc.

A radial gradient of the electron density ∂ne/∂r < 0 as generated by the pondero-motive force implies a gradient ∂η/∂r < 0 and this in turn means that the phasevelocity vph = cη on axis is smaller than farther outside. The wavefronts are bentand the plasma acts like a positive lens: The laser pulse is focused. This effect iscalled ponderomotive self-focusing.

26


Relativistic self-focusing

The electrons in a tightly focused laser pulse are oscillating with relativistic velocities(as shown for intense plane waves). Their velocity will depend on the local laser fieldstrength and therefore the Lorentz factor will exhibit a spacial dependency γ = γ(r).The properties of the electrons oscillating in the field are changed and therefore theplasma frequency ωp is altered. Replacing ωp with ωp/γ(r) the refractive index ofthe plasma now reads

η =√

1− ω2p/(γ(r)ω2).

For moderate intensities (a0 < 1) the electrons are mainly oscillating along theelectric field vector (see Sec. 2.1.1) and γ =

√1 + (p/mc)2 ≈

√1 + a2/2. Therefore

a gradient in intensity ∂a2/∂r < 0 again leads to a refractive index profile ∂η/∂r < 0

which leads to self-focusing. Summarizing all these contributions to the refractiveindex in the limit ω2

p ω2 one may write

η(r) ≈ 1−ω2

p

2ω2

(1− a2

2+

δn

n0

),

where the a2/2 term represents relativistic self-focusing and the δn/n0 term pondero-motive self-focusing. Here, δn represents a perturbation of the homogeneous plasmadensity n0 induced by the ponderomotive force such that the resulting plasma densityne may be written as ne = n0 + δn.

Relativistic self-focusing sets in as soon as the laser power exceeds a certain criticalpower Pc. This critical power depends on the ratio of laser frequency to plasmafrequency only [33]:

Pc ≥ 16.2ω2

ω2p

[GW] = 16.2nc

ne

[GW]. (2.31)

This critical power is not very high compared to the power delivered by today’s multi-terawatt table-top lasers. For a given laser power increasing the plasma densityresults in lowering the threshold for relativistic self-focusing. This process has, ofcourse, its limit in the critical density. For a laser pulse with a pulse energy of600 mJ and a pulse duration of 80 fs focused into a He gas-jet with a particle densityof 3×1019 cm−3 (being fully ionized by the rising edge of the pulse) the critical powerPc ≈ 0.6 TW is exceeded by a factor of 25.

27


z- ct

I

z- ct

ne,Ia) b)

Figure 2.8: a) LWFA: A short laser pulse (cτ ≤ λp) shown as a dashed line drives a plasmawave. b) SM-LWFA: An initially long laser pulse (dashed line) breaks up into a train ofshorter pulses which match the condition of LWFA and resonantly drives a plasma wave(after [41]).

2.3 Electron accelerationSince many years ago laser plasmas have been seen as an ideal medium for high-field,but compact-sized (millimeter range) accelerators [34, 35]. The plasma as an ionizedmedium may sustain much higher fields than is possible to generate with conventionalaccelerator technology where material breakdown imposes a limit at about 55 MV/m

[36]. Electric fields in the GV/m range were inferred to be present in laser plasmasobserving the acceleration of injected electrons [37–40].

In many experiments during the past two decades a variety of acceleration mech-anisms was identified. A selection of these mechanisms will briefly be described inthe following with the focus on schemes where only a single laser pulse is needed toaccelerate initially resting electrons to relativistic energies. All these schemes have incommon that a laser pulse is focused into a gas-jet generating an underdense plasma.

2.3.1 Laser wakefield acceleration

An ultra-short laser pulse impinging on a plasma (the previously mentioned gas-jetis quickly ionized) drives a plasma wave with frequency ωp. The plasma wave followsthe driving laser pulse with a phase velocity determined by the laser pulse groupvelocity vplasma

ph = vg = cη with η as defined in Eq. 2.28. The electric fields associatedwith the plasma wave are now longitudinal. An electron can “ride” on the plasmawave and be accelerated to relativistic energies in the direction of laser propagation.

28


This process is most efficient when the laser pulse length cτ , where τ is the pulseduration, is shorter than the plasma wavelength λp = 2πc/ωp. This condition isdepicted in Fig. 2.8a). When a large number of electrons acquire a velocity close tothe phase velocity of the plasma wave, wave-breaking occurs [41]. The fast electronsare “surfing” on the wake of the plasma wave with velocity vplasma

ph . The name for thisprocess coined by Tajima and Dawson [34] is laser wakefield acceleration (LWFA).

2.3.2 Self-modulated laser wakefield acceleration

If the laser pulse length is longer than the plasma wavelength cτ > λp, the laser pulseundergoes a self-modulation instability. The leading edge of the laser pulse drives aplasma wave. The electron density modulation of the plasma wave in turn representsa periodic modulation of the refractive index. It acts on the long laser pulse such thatthe pulse is self-modulated and breaks up into a train of short pulses (see Fig. 2.8b)).These shorter pulses now match the conditions for LWFA and can resonantly drive aplasma wave. The self-modulated laser wakefield acceleration (SM-LWFA) is not asefficient as the pure LWFA, but still high-energy, even quasi-monoenergetic electronbeams can be generated.

2.3.3 Direct laser acceleration

Another acceleration process which is quite different in nature to the wakefield ac-celeration is direct laser acceleration (DLA). It was first proposed by Gahn et al. [42]and is closely related to the formation of a relativistic channel. The ponderomotiveforce expels electrons from the laser beam axis and generates a radial quasi-staticelectric field. Electrons which are accelerated along the laser propagation generate -being a current of charges - an azimuthal magnetic field. Both these fields combinedresult in an effective potential well for relativistic electrons. Electrons trapped inthis well will oscillate at the frequency ωβ = ωp/(2

√γ), the betatron frequency [43].

If the trapped electron is moving fast enough along the laser propagation, the laseroscillations may be in phase with the betatron oscillations in the frame of the elec-tron. In this case, an efficient energy coupling is possible. The energy gained by theelectron in this process directly results from the laser field and therefore the namedirect laser acceleration is appropriate.

29


0 5 10 15 20 25 30 35 40 45 50 55 600

5

10

15

20

25

30

35

40

0 5 10 15 20 25 30 35 40 45 50 55 600

5

10

15

20

25

30

35

40 b)a)

dN/d

E / 105 m

sr/M

eV

Energy / MeV

dN/d

E / 105 m

sr/M

eV

Energy / MeV

T = 7 MeV

Figure 2.9: a) Typical exponential electron spectrum measured with a high-resolution mag-net spectrometer. b) Electron spectrum recorded under similar experimental conditionsas a) exhibiting prominent quasi-monoenergetic peaks. The laser pulse parameters inthis experiment were: intensity I = 5 × 1019 W/cm2, pulse energy E = 660 mJ, pulseduration τ = 80 fs, peak plasma density ne = 8× 1019 cm−3 [44].

SM-LWFA and direct laser acceleration cannot be discriminated experimentally.Both lead to similar electron energy distributions. Only in PIC simulations thevarious contributions may be separated. A typical exponential electron spectrumgenerated by SM-LWFA and direct laser acceleration is shown in Fig. 2.9a) [44]. Thelaser parameters in this experiment match the conditions of SM-LWFA. Fig. 2.9b)shows an electron spectrum with a significant monoenergetic peak. Monoenergeticfeatures appear at a fraction of the laser shots and are not stable in terms of energyor intensity. Both spectra shown here were recorded under the same experimentalconditions with a high-resolution dipole magnet spectrometer.

The energy resolution of this kind of spectrometer relies on a small aperture whichimplies a small acceptance angle thus cutting out a small fraction of the total ac-celerated electron bunch. Small directional fluctuations of the electron beam createlarge changes in the recorded spectra. Conventional electron spectrometers as theone used here are basically offline diagnostics. After recording the electron spectraof a sequence of ten shots onto an imaging plate the target chamber must be ventedand the information on the image plates must be processed further.

30


2.3.4 Bubble acceleration

A novel regime of laser wakefield acceleration, which was coined “bubble accelera-tion”, was proposed by Pukhov and Meyer-ter-Vehn [45] in 2002 on the basis of PICsimulations. Short (τ < 7 fs) and intense (a0 > 1) laser pulses were predicted toproduce quasi-monoenergetic electrons at energies exceeding 50 MeV. Since then,experiments in a transition regime between LWFA and bubble acceleration (alsocalled forced laser wakefield regime, FLWF) demonstrated electron spectra with asignificant, non-exponential shape [36], and, recently, quasi-monoenergetic electronbeams in the range of 50 . . . 170 MeV with an energy spread of a few percent, limitedby the resolution of the detectors [3, 5, 46, 47].

Extensive numerical studies of the bubble regime were carried out which led to“rule of thumb” limits for the bubble regime as well as to scaling laws for the producedelectron beams [48]. According to these studies the optimum configuration for a laserpulse in the bubble regime is defined by

kpR ≈√

a0, τ ≤ R

c,

where kp = ωp/c is the plasma wave number and R the laser focus radius. Theenergy of the monoenergetic electrons Ee is scalable according to the relation

Ee ≈ 0.65mc2

√P

Prel

cτ

λ0

,

where P is the laser pulse power, Prel ≈ 8.5 GW the relativistic power unit and λ0

the laser wavelength.The number of electrons contained in the monoenergetic peak may also be deter-

mined by

Ne ≈1.8

k0re

√P

Prel

,

where k0 = 2π/λ0 is the laser wave number and re = e2/4πε0mc2 is the classicalelectron radius. Remarkably, the conversion efficiency of the laser energy into elec-trons is constant and about 20%. A conversion efficiency of 10% has already beendemonstrated in the transition regime [46].

Putting numbers into the “rule of thumb” scaling laws one obtains very promisingresults: With a pulse duration of about 5 fs (which is the projected pulse duration

31


Pump pulseand electrons

Probe pulse

Scattered x-rays

x, y

z

Figure 2.10: Thomson backscattering of a weak, non-relativistic laser pulse (incident fromthe right) from laser-accelerated electrons (incident from the left). The angle ϑ is thescattering angle of the photons with respect to the z-axis which will also be called angleof observation. dΩ is the differential solid angle covered by the detector.

e. g. for the new laser facility at the Max-Planck Institute of Quantum Optics,Garching, Germany) the expected electron energies are: 40 MeV for a pulse energyof 50 mJ, 150 MeV for 500 mJ and 250 MeV for 5 J. The prospects and applicationsof these ultra-short electron bunches in conjunction with the photon collider arediscussed in Sec. 5.

2.4 Thomson backscattering in the linear regime

A rather complex interaction scheme will be discussed in the following which makesuse of several phenomena presented above. A relativistic electron bunch that wasaccelerated by one or several of the various acceleration mechanisms is subjectedto a weak, non-relativistic counter-propagating electromagnetic wave. This processis commonly referred to as relativistic Thomson backscattering or inverse Comptonscattering. With the photon collider in mind we imagine having one strong laserpulse accelerating a bunch of electrons and another, weak and non-relativistic laserpulse counter-propagating and scattering from the electrons (see Fig. 2.10).

This situation is actually already included in the thorough treatment of the non-linear Thomson scattering (Sec. 2.1.2) in the limit a0 1, but there I confinedmyself to initially resting electrons and radiation scattered under an angle of 90. Itis very useful for the interpretation of the experiment in Sec. 4, however, to look atthe problem of backscattered radiation in more detail.

32


There is an analogous experimental scenario from accelerator physics which hasbeen worked on extensively in the past and which was introduced into text bookslong ago: undulator radiation. It is generated in synchrotrons by sending relativisticelectrons through static but spatially alternating magnetic fields. The differencesto the laser-accelerated electrons interacting with counter-propagating laser photonsare: The energy of the electrons is higher (synchrotrons are operating with GeVelectrons) and the wavelength of the undulator field that the electrons experience islarger (of the order of cm in the laboratory frame). A fully laser-based scheme mayoffer certain advantages over large synchrotron facilities and are discussed in Sec. 5.

It seems useful to start with reviewing a few well known facts from undulatortheory [49, pp. 135ff.]. First, the emission of a single electron will be discussed andthen the emission of a population of electrons of a given energy distribution.

2.4.1 Scattering from a single electron

The steps to the calculation of the emitted radiation will be to determine the fieldsthat the electron experiences in its frame of reference, to calculate the radiationpattern it emits in its frame of reference and then to transform this back to thelaboratory frame. Many properties of undulator radiation can probably most easilybe understood considering the classical Doppler effect: An electron which is at rest ormoving slowly compared to the speed of light and undergoing an oscillation (inducedby external fields) will emit radiation in a dipole pattern and the frequency of theradiation will not strongly depend on the position of the observer. If the electronis moving, however, the wavefronts are compressed along the propagation of theelectrons. The Doppler effect is strongly angle dependent. In the relativistic casethe observer will detect short wavelength x-ray emission along the propagation ofthe electrons. Towards larger angles the emission will have larger wavelengths up toinfrared and radio waves depending on the velocity of the electron.

For a static, weak undulator field the electron traveling at relativistic speed willexperience a Lorentz-contracted field of wavelength λ′ = λu/γ, where λu is the wave-length of the undulating magnetic field and γ the Lorentz factor of the electron. Thewavelength λ of the observed radiation is given by the undulator equation in the

33


relativistic limit (β ≈ 1):

λ = λu(1− β cos ϑ) ≈ λu

2γ2(1 + γ2ϑ2), (2.32)

where ϑ is the angle of observation with respect to the axis along which the electronspropagate (see Fig. 2.10). The approximation in Eq. 2.32 is valid for ϑ 1. Eq. 2.32represents a simplified version of the complete undulator equation (see [49]) and isshown in Fig. 2.11a) for a value of γ = 10.

The undulator equation can easily be adapted to the situation where a counter-propagating electromagnetic wave (a laser pulse) substitutes the static undulatorfield. The frequency of the electromagnetic wave in the electron frame is then givenby

ω′ = ω0

√1 + β

1− β= (1 + β)γω0 ≈ 2γω0 or

λ′ ≈ λ0

2γ,

where ω0 and λ0 are the frequency and the wavelength of the electromagnetic wave,respectively. Eq. 2.32 will therefore only be changed by a factor of two. Consideringonly radiation emitted along the z-axis (ϑ = 0) one arrives at the equation for theenergy of the emitted photons

~ωx = 2γ2~ωu

or, in the case of an electromagnetic wave,

~ωx = 4γ2~ω0. (2.33)

Since as far as our experiment is concerned the emitted photons have energies in thex-ray range, all quantities related to them will be labeled with a subscript “x”.

From Eq. 2.33 one can easily understand the great advantage of a Thomson scatter-ing scheme over conventional undulators: The laser wavelength is typically a factorof 104 shorter than the period of commonly used undulators. This allows for the useof 100 times less energetic electrons to generate x-rays of a given energy.

In the frame of the electron (the primed coordinate system) the emitted radiationwill have a natural bandwidth ∆λ′/λ′ ≈ 1/N0, where N0 is the number of oscillations

34


that the electron undergoes, corresponding to the number of periods of the undulatingfield. In the laboratory frame this bandwidth will be emitted in a central radiationcone of half angle:

ϑcen ≈1

γ√

N0

. (2.34)

This central cone will become narrower the more energy the electron has. For a5 MeV electron (γ ≈ 10) submitted to 30 laser oscillations, corresponding to an 80 fs

probe pulse, the expected x-ray photon energy in forward direction is 620 eV and theopening half angle of the central emission cone is ϑcen = 0.02 rad.

The energy radiated by the electron per unit solid angle Ω and per unit frequencyω can be calculated integrating the Liénard-Wiechert potentials and is given e. g. by[16, 19]. For our purposes it is sufficient to consider the limit of weak electromagneticwaves (a0 1), highly relativistic electrons (γ 1) and small observation angle(ϑ 1):

d2I

dωxdΩ= remcγ2N2

0 a20

(ω

4γ2ω0

)2

R(ωx, ω0),

where re is the classical electron radius. The resonance function R(ω, ω0) has theform

R(ωx, ω0) =

(sin(k′L/2)

k′L/2

)2

, (2.35)

where k′ = kx(1 + γ2ϑ2)/(4γ2) − k0, kx = ωx/c, k0 = ω0/c and L = N0λ0 theinteraction length. The resonance function is strongly peaked at

ωr = 4γ2 ω0

1 + γ2ϑ2, (2.36)

or written in terms of a resonant electron energy

γ2r =

ωx

4ω0

/(1− ωxϑ2

4ω0

). (2.37)

The relation between γr and ωx is displayed in Fig. 2.11b). Clearly, the angle ofobservation has a large impact on this relation. This will become important whenthe dependency of radiated spectra on ϑ is discussed later in this section or in theanalysis of the experiment in Sec. 4.5.

35


x / ë

z / ë

Energy / eV

ã

a) b)

- 2.5 - 1.5 - 1 - 0.5 0.5 1

- 1

- 0.5

0.5

1

Figure 2.11: a) Relativistic angular dependent Doppler shift: This graph displays the wave-length of the radiation emitted by an electron moving relativistically (γ = 10) along thez-axis according to Eq. 2.32. The radiation is observed in the laboratory frame. b) TheLorentz factor γ of the electron as a function of the observed backscattered photon energyfor various observation angles ϑ. Lower curve: ϑ = 0, middle curve: ϑ = 0.04 rad, uppercurve: ϑ = 0.06 rad.

2.4.2 Scattering from a laser-accelerated electron bunch

The total spectra radiated by a bunch of electrons of a given energy distribution f(γ)

may be obtained by integrating the spectrum of a single electron multiplied by f(γ)

over γ:

d2IT

dωxdΩ=

∫f(γ)

d2I

dωxdΩdγ. (2.38)

Here the electron distribution function is given as the normalized function

f(γ) =1

Nb

dNe(γ)

dγ,

where Nb is the number of electrons in the bunch.For a large number of periods N0 of the undulating field the resonance function

(Eq. 2.35) may be approximated by a delta function:

R → ∆ωrδ(ωx − ωr) = ∆γrδ(γ − γr),

where ∆ωr = ωr/N0 and ∆γr = 2γ3r ω0/(N0ωr). This approximation is valid for wide

36


energy distributions like exponential distributions which were measured in the SM-LWFA regime (see Fig. 2.9). The line broadening inherent to the single-electronspectrum is narrow compared to the contribution of the wide energy distribution.Carrying out the integration in Eq. 2.38 using the delta function approximation onearrives at

d2IT

dωxdΩ=

1

2remecN0a

20

γ3

1 + γ2ϑ2f(γ). (2.39)

The intensity distribution of the Thomson backscattered radiation can be assumedflat over the solid angle ∆Ω 1. The number of photons ∆Nx per frequency inter-val ∆ωx backscattered into this collection angle can then be calculated by dividingEq. 2.39 by the photon energy Ex = ~ωx and multiplying by the number of electronsin the bunch Nb:

∆Nx

∆ωx∆Ω=

Nb

~ωx

d2IT

dωxdΩ=

α

8ωL

NbN0a20γf(γ), (2.40)

where the fine-structure constant α = e2/(4πε0~c) was inserted. The number ofphotons observed in the corresponding energy interval reads

∆Nx

∆Ex∆Ω=

α

8EL

NbN0a20γf(γ), (2.41)

where E0 = ~ω0 is the energy of a laser photon.

The Thomson backscattered photon spectrum is typically recorded with a constantdetector solid angle ∆Ω with a constant energy resolution ∆Ex. From Eq. 2.41 theelectron spectrum may therefore be calculated by

Nbf(γ) =dNe(γ)

dγ=

∆Nx

∆Ex

8E0

αN0a20γ∆Ω

,

or in terms of electron energy expressed in MeV using Ee = mc2(γ − 1)

dNe(Ee)

dEe

=∆Nx

∆Ex

8E0

αN0a20γ∆Ω

dγ

dEe

(2.42)

Please note that γ as shown in Eq. 2.37 is dependent on the observation angle ϑ (seealso Fig. 2.11). In order to accurately calculate the electron spectrum according toEq. 2.42 the angle ϑ has to be determined experimentally.

37


Assuming an exponential energy distribution f(γ) of Nb electrons with

f(γ) = f0 exp[−γ/Te],

where Te is the electron temperature given in units of γ and f0 a normalization factor,Eq. 2.40 may be rewritten in terms of the scattered photon energy Ex:

dNx(Ex)

dEx

=α

8E0

∆Ωa20NbN0f0

√Ex

4E0(1− Exϑ2

4E0)exp

[−

√Ex

4E0(1− Exϑ2

4E0)/Te

].

(2.43)The maximum of the photon spectra is located at

Emax =4 T 2

e E0

1 + T 2e ϑ2

.

The backscattered photon spectrum for a given electron distribution function ofTe = 10 was calculated and is shown in Fig. 2.12a). The shape of the spectrumis strongly dependent on the observation angle ϑ. For larger ϑ the spectrum fallsoff steeper towards higher energies, the maximum of the spectrum shifts to lowerenergies. The maximum of the spectrum for ϑ = 0 is located at 624 eV. In Fig. 2.12b)backscattered spectra are displayed for various electron temperatures of Te = 4 . . . 12.An increase in temperature shifts the maximum of the photon emission towardshigher photon energies. Please note that the spectra shown in Figs. 2.12a) andb) were calculated using an absolute spectral bandwidth dEx and not a relativebandwidth of 0.1% as common in the accelerator and synchrotron community (seee. g. [50]).

Eq. 2.41 leads to several important conclusions:

1. The scenario in which a weak (non-relativistic) laser pulse and a laser-accelera-ted relativistic electron bunch are counter-propagating may be used to generateultra-short and well collimated x-ray pulses. The spectra of these x-ray pulseswill be broad if the energy distribution function of the electrons is broad. Inturn, quasi mono-energetic electron bunches will produce peaked x-ray spectra.

2. Eq. 2.34 shows that the divergence angle of the emitted x-rays decreases withincreasing energy. The higher the energy of the electrons, the more power is

38


Figure 2.12: a) The backscattered Thomson spectra strongly depend on the observationangle ϑ (cf. Eq. 2.39). This graph shows backscattered photon spectra generated fromelectrons with an exponential energy distribution (Te = 10 in units of γ) for observationangles in the range ϑ = 0 . . . 0.06 rad. b) Backscattered x-ray spectra calculated fromEq. 2.43 for ϑ = 0 assuming an exponential electron energy distribution with electrontemperatures of Te = 4 . . . 12.

39


radiated in forward direction.

3. The measurement of the Thomson backscattered radiation immediately re-veals the energy distribution function of the laser-accelerated electrons usingEq. 2.40. Thomson backscattering can therefore be used for in-situ time-resolved diagnostics of the electron acceleration process [51]. I would like topoint out that no assumptions about the electron spectrum must be madewhich is, however, the case for other methods using e. g. nuclear reactions[14].

40

3 Single-shot autocorrelation atrelativistic intensity

The accurate measurement of relativistic laser intensities is a difficult task. Thecommonly used method to determine laser intensities is the simultaneous, carefulmeasurement of pulse energy E, pulse duration τ and focus area A, and the calcu-lation of laser intensity according to I ≈ E/(τA). The experimental determinationof each of the parameters entering this equation is problematic at high intensities.For example classical autocorrelation techniques or phase resolved methods such asSpider or Frog for the determination of the pulse duration require the use of non-linear optical materials. Even at intensities many orders of magnitude below therelativistic regime these materials are destroyed by the laser pulse. Therefore thepulse duration is usually measured at low intensities, frequently using only part ofthe laser beam, and then extrapolated to the experimentally relevant intensity range.

Another method to directly measure maximum intensities employs field ionizationof dilute gases by the laser pulse. The maximum charge state measured in a massspectrometer is related to the maximum intensity by the Ammosov-Delone-Krainov(ADK) scaling law, which is well-established for non-relativistic intensities [52]. Theexperimental difficulty of this approach lies in the clear distinction of highly chargedstates of atoms generated by field ionization as opposed to other processes such as col-lisional ionization or recombination. Ions generated in the high-intensity focal regionmay also be shielded by ions generated in lower fields and never reach the detector[53]. It is under investigation whether the classical scaling law for field ionizationused for the intensity measurement remains valid in the relativistic intensity regime[54]. Relativistic intensities have also been estimated through the ponderomotivescaling of electron temperature [55–60] or laser-induced nuclear reactions [14, 61].

While these latter methods yield reasonable lower bounds for the laser intensitythey are, however, indirect and typically average over many thousand shots. It is

41

3 Single-shot autocorrelation at relativistic intensity

Figure 3.1: A schematic representation of the Jena laser system in which key componentsintroduced to prevent damage from back-reflected laser pulses during the photon colliderexperiments are highlighted in bold print: (1) Faraday isolator between regenerative am-plifier and 4-pass amplifier, (2) diagnostics unit consisting of beam-profiler and energymeter for monitoring of the back-reflected pulse which is coupled out at the isolator, (3)spatial filter.

therefore of interest to utilize a genuinely relativistic effect that is well understoodtheoretically for the determination of laser pulse parameters at relativistic intensities.Such an effect is non-linear Thomson scattering of light from free electrons (seeSec. 2.1.1).

With the help of the photon collider and a He gas-jet as target we carried out thefirst single-shot autocorrelation measurement at relativistic intensity [17], which willbe described below. I will start the discussion of the experiment with the descriptionof the Jena laser system and its adaption to this experiment and continue with theexperimental setup and the alignment procedure in detail. Various autocorrelationsignals will be discussed both in theory and on the basis of experimental data.

3.1 The Jena Ti:Sa laser system (JETI)

The Jena Ti:Sa laser system (JETI) is based on the principle of chirped pulse am-plification (CPA) [62, 63]. A schematic setup is shown in Fig. 3.1. A commercialoscillator generates pulses of 10 nJ, 45 fs at a central wavelength of 795 µm with a

42


repetition rate of 80 MHz. A pulse picker reduces the repetition rate to 10 Hz. Thepulses are lengthened to a duration of 150 ps in a grating stretcher and amplified inthree stages to a pulse energy of 1 . . . 1.6 J corresponding to an amplification of 108.In all these three stages frequency-doubled Nd:YAG lasers are used as pump lasers.

The first stage consists of a regenerative amplifier yielding an output energy of2 mJ. The second and third stage are multi-pass amplifiers. The 4-pass amplifier,pumped by 2×500 mJ, yields an output energy of 300 . . . 400 mJ, the 3-pass amplifier,pumped by 2 × 2.5 J, finally provides up to 1.6 J pulse energy. After amplificationthe beam profile is enlarged to a diameter of 50 mm and the pulses are shortenedagain in a 4-grating vacuum compressor down to 80 fs. Starting from the compressorthe beam must be guided in vacuum to the target chamber.

Several diagnostics are installed for regular use to characterize the laser pulses(naming the ones relevant for this experiment): A third harmonic generation (THG)autocorrelator measuring the pulse length with a dynamic range of up to 105 in a timeinterval of a few picoseconds around the pulse maximum and an interferometric pulsefront tilt diagnostic system. The pulse front tilt detection is necessary to achieve highintensity since a slight tilt of a laser pulse of 50 mm diameter and ∼ 30 µm lengthinduces a substantial increase of the effective pulse duration in the laser focus.

3.2 Protection of the laser system

The symmetric design of the photon collider on the one hand facilitates the alignmenttremendously but on the other hand creates a great risk: Assuming there is no matterin the focal region partially absorbing the laser pulses, theoretically, if a 50/50 beam-splitter is used, 50% of the original pulse energy are reflected back from the vacuumchamber towards the laser system. 60% of that amount will pass the compressor sothat 30% of the original energy on target will reach the last amplifier.

Two measures are in place to protect the laser system:

1. A spatial filter was introduced between the 3-pass amplifier and the 4-passamplifier (see Fig. 3.1). The filter consists of a 1:1.5 mirror telescope with aquartz pinhole (diameter 300 . . . 400 µm). At this point returning pulses thatare not perfectly aligned with the beam path will be suppressed.

43


2. A Faraday isolator was installed between 4-pass and regenerative amplifier(see Fig. 3.1). This is a crucial point in the laser setup since here a back-reflected pulse will be scaled down in diameter by the 9:1 lens telescope whichcorresponds to an increase in intensity of a factor 92.

The Faraday isolator consists of a Faraday rotator, two polarizing beam-splitters anda λ/2-waveplate. Looking from the regenerative amplifier along laser pulse propaga-tion: The first (“input”) polarizer is aligned with the laser polarization. The Faradayrotator then turns the polarization of the pulse by -45. The waveplate then turnsthe polarization back to 0 (rotating +45). The second (“output”) polarizer is ori-ented along this polarization such that the forward propagating pulse is transmittedunimpeded.

A pulse, back-reflected from the target chamber, will be polarized along the ori-entation of the output polarizer. It will be rotated by the λ/2-waveplate by -45

and rotated another -45 by the Faraday rotator since the rotation is insensitive todirection of propagation. The polarization of the back-reflected pulse will now bevertically aligned to the input polarizer. According to the contrast of the input po-larizer (5 × 10−6), most of the back-reflected light will be coupled out of the beampath onto a CCD camera and a power meter for monitoring.

With these measures in place there are still some risks remaining:

• The 3-pass amplifier is unprotected from back-reflected pulses.

• Back-reflected pulses are amplified in the 3-pass amplifier as well as - if noteliminated by the spatial filter - in the 4-pass amplifier.

• The input polarizer of the Faraday isolator plays the most important role forthe laser protection and will have to stand very high strain. In the experimentit turned out that this optical element will be damaged first.

3.3 Experimental setupThe setup of the photon collider for the autocorrelation measurement is as follows(see Fig. 3.2): The main laser pulse, the polarization of which is oriented along thex-axis, is divided into two pulses of equal energy by a 50/50 beam-splitter of 2.5 mm

44


Beamsplitter

Off-axisparabolicmirror

Off-axisparabolicmirror

Laser pulse

He-gas jet2 Interference filterù

z

x

2ù probe beam

Äô

z1

CC

D

a)

b)

1

2

3

Figure 3.2: a) Schematic of the experimental setup for two colliding laser pulses: Themain laser pulse is divided into two pulses of equal energy by a beam-splitter which maybe moved along the z-direction. Each pulse is focused into a He-gas jet by a 45 off-axisparabolic mirror. The 2ω self-emission of the plasma and the shadowgrams generatedby the 2ω probe pulse are observed perpendicular to the z-axis. b) Color image of theHe gas-jet during the experiment: (1) gas nozzle, (2) target mount, (3) parabolic mirrorwith quartz wafer debris shielding (removed in autocorrelation measurement). The brightoverexposed region in the center of the image is due to plasma emissions.

45


thickness. These are focused by two F/2.5 45 off-axis parabolic mirrors into a pulsedHe gas-jet and their foci are overlapped in space. The time delay between the pulsesis adjusted so that they arrive at the focus at the same time by moving the beam-splitter. The electrons generated early in the laser pulse by ionization of the He gasare driven by both the electric and the magnetic field of the electromagnetic wave insuch a way that Thomson scattered light close to the second harmonic of the laserfrequency is generated. The self-emission of the laser plasma is imaged by a quartzlens onto a CCD camera under an angle of 84.

In the weakly relativistic regime the second-harmonic Thomson signal observedapproximately perpendicular to the z-axis is spectrally close to the second harmonicof the laser frequency and will be called second harmonic (axes are indicated inFig. 3.2a)). This can be seen from Eq. 2.20 in Sec. 2.1.2. An interference filter of10 nm bandwidth is introduced into the imaging setup to insure that only second-harmonic plasma emission is recorded. It was verified that the polarization directionof the second harmonic was in z-direction which is expected for second harmoniclight generated by non-linear Thomson scattering. The signal on the CCD camerais integrated along the y-axis (columns) and is spatially resolved along the laserpropagation (z-axis, rows) and it represents an autocorrelation trace of the laserpulse of second order at full intensity as was derived in Sec. 2.1.1.

With the same imaging setup shadowgrams of the laser plasma can be acquiredusing a frequency-doubled ultra-short probe pulse of variable delay which is back-lighting the interaction region. For the autocorrelation measurement the gas-jetmust be so dilute that propagation effects through ionization can be neglected (seeSec. 2.2.1). The particle density of the gas-jet which exhibits a Gaussian profile alonglaser propagation and an exponential profile along the nozzle axis was determined bytime-resolved interferometry with nitrogen gas (nitrogen having a much larger indexof refraction than helium). The results are shown in Fig. 3.3. In our autocorrelationmeasurements the peak density of the gas-jet did not exceed 1017 cm−3. We estimatethe upper limit of the integral phase shift to ∆φ < 0.5 (cf. Eq. 2.29) [64].

The laser was operated at 100 mJ pulse energy on target, i. e. 50 mJ in each ofthe counter-propagating pulses. The intensity of the laser pulses was determined to3× 1018 W/cm2. This corresponds to a laser field strength parameter a0 ≈ 1.2.

46


Figure 3.3: The gas-density n in front of the orifice of the subsonic, cylindrical nozzle(nozzle center at y = z = 0µm) at two different timings: a) opening time of the valveof 1 ms, b) opening time 170 µs. Sharp peaks in the low gas-density measurement areartifacts due to noise from the interferometric measurements. The gas-density at thefocal position in the autocorrelation measurement was determined through extrapolation.

3.4 Alignment of the photon collider

The experimental setup was designed in such a way that the foci of the laser pulsesare located in the center of the octagonal vacuum chamber. The parabolic mirrorscan be tilted along the x- and y-axes and translated in all three dimensions but theycannot be rotated around the incident beam axis. This degree of freedom has to beadjusted in the very beginning of the experiment when the mirrors are fixed in theirmounts. With the help of a HeNe laser the foci are adjusted to be in the center ofthe chamber.

The later fine-adjustment of the foci through tilting the mirrors changes the po-sition of the foci which can be corrected translating the parabolic mirrors in x, y, z

but in principle this first fixation defines the angle under which the two laser pulsescounter-propagate. This may be most easily understood if one remembers that theparabolic mirror is a circular cut-out of a (virtual) larger, axially symmetric parabo-loid. The position of the focus with respect to the surface is fixed. The rotation ofour parabolic mirror is therefore a degree of freedom necessary for the alignment.

47


Figure 3.4: Intensity distribution in the laser focus a) of the laser pulse propagating throughthe beam-splitter (6.5 µm2 FWHM) and b) of the laser pulse reflected from the beam-splitter (5.9 µm2 FWHM) .

Focus optimizationThe foci are optimized in vacuum by tilting the mirrors along x and y while imagingthe focal spot onto a CCD camera with a 40×microscope objective. The optimizationis carried out at reduced laser power introducing an attenuator into the beam pathbetween the last amplifier and the compressor, achieving a variable attenuation ofup to 105. Multiple images of the focal spots are taken using different sets of neutraldensity filters and are recomposed with the use of a computer program. This allowsfor a focus analysis with a dynamic range of ∼ 2000 using a common (and affordable)8bit CCD camera. The intensity distribution in the laser foci measured before theexperiment is shown in Fig. 3.4.

From the resulting images the FWHM-area A of the focal spot and the ratio q

of pulse energy inside the peak over energy outside the peak are determined. Theintensity I is then calculated with

I =E · qA · τ

,

where E is the pulse energy, and τ is the pulse duration (both parameters measuredseparately). Due to the limited field of view of the CCD camera the value of q

measured in such a way represents an upper limit of the real value (see Fig. 3.4).

48


Overlap in spaceIn the next step the laser foci in low-density He gas are examined. The plasmaself-emission of each of the beams is imaged under 90 onto an untriggered CMOScamera located in the vacuum chamber through two 8× microscope objectives (seeFigs. 3.5a) and b)). These microscope objectives are aligned each with the x- andthe y-axis. A filter for 800 nm prevents the camera from saturation by stray light.

With this technique the laser foci can be overlapped within an accuracy of ∼ 10 µm

given by the resolution of the imaging setup. Since the CMOS camera cannot beoperated in triggered mode the resulting standard video signal has to be de-interlacedto obtain meaningful images. The images shown in Fig. 3.5c)-f) are averaged over 10laser shots. The overall brightness of the images varies depending on the observationaxis (the emission patterns of the non-linear harmonics are non-homogeneous) andexperimental conditions (stray light).

Overlap in timeNow that the laser foci are overlapped within a few microns the time delay betweenthe ultra-short laser pulses has to be adjusted. To this aim the interaction region ofthe laser pulse with the He gas-jet is back-lighted with a frequency-doubled probepulse. Shadowgrams are recorded with the same imaging setup as used for thedetection of plasma self-emission. The probe pulse is approximately as short as themain pulse and its delay with respect to the main pulse can be adjusted with aresolution of about 100 fs, i. e. changes of the ionization front propagating throughthe plasma can be consistently traced with 100 fs resolution.

The delay adjustment takes place in two steps: First, the delay of the probe pulseis adjusted to the particular moment in time when the ionization front of the laserpulse incident from the right (propagating through the beam-splitter) reaches theposition of the vacuum focus. We have now marked this certain moment in timewith the arrival of the probe pulse and can use it as a reference. Secondly, the timedelay of the laser pulse incident from the left is adjusted by moving the beam-splittersuch that it reaches the focus at the same time delay of the probe pulse.

The fringes that are visible in each picture of Fig. 3.6 are due to diffraction ofthe probe pulse on a plasma cavity generated by each of the laser pulses. Thisphenomenon has been thoroughly examined under a wide range of experimentalconditions. In fact, it enables us to monitor the evolution of relativistic plasma

49


CMOSCamera

Adapter to x-y-ztranslationstage

Microscopeobjective(y-axis)

Focal point

Microscopeobjective (x-axis)

Beamsplitter

Mirror

Mirror

x

y

c) d)

e) f)

a) b)

200 µm

Figure 3.5: a) Picture of the diagnostics unit with four microscope objectives. Two 40×objectives are mounted along the z-axis (perpendicular to the plane of the image) tomonitor the focus size. Two smaller 8× objectives are mounted along the x- and y-axes to image the side-scattered light from the focus onto a CMOS camera. The fourobjectives are highlighted by a white circle. The camera is mounted on top of the unitand can be operated in vacuum. b) Drawing of the diagnostics unit (the z-axis objectiveswere omitted for clarity). c)-f) Images of the self-emission of a laser plasma generatedby one laser pulse and observed through the x- and y-objectives. c) Laser focus side view(along x-axis) of the pulse incident from the left, d) side view of the pulse incident fromright, e) top view (along y-axis) of the pulse incident from left, f) top view of the pulseincident from right.

50


b)

d)

200 µm

a)

c)

Figure 3.6: Shadowgrams of the interaction region: In a first step the delay of the back-lighting probe pulse was adjusted such that it passed the interaction region when the pulseincident from the right reached its focus. a) The laser pulse incident from the left is stilldelayed by ∆τ = −360 fs to the pulse incident from the right. Its ionization front isvisible about 100 µm left of the focus. b)-d) The delay of the pulse incident from the leftis gradually adjusted by 120 fs moving the beam-splitter such that both pulses pass thefocus at the same moment in time.

channels with 100 fs resolution and will be discussed elsewhere [65].

3.5 Exemplary scenariosIf the foci of the two parabolic mirrors are on the same optical axis, different situationsmay occur. Let us re-examine Eq. 2.14:

S(z, z1, ∆τ) ∼ 1

1 +(

zz0

)2 +R2

1 +(

z−z1

z0

)2

+4R

1 +(

z−z1/2z0

)2

+z21

4z20

× exp

[− 4

τ 2L

(z − z1/2

c− ∆τ

2

)2]

, (3.1)

where we divided by I0L and inserted the ratio of the intensities R = I0R/I0L. τL isthe laser pulse duration, ∆τ the time delay between the pulses and z0 the distancebetween the foci. τL is related to the laser intensity pulse duration FWHM τ byτ = τL

√2 ln 2.

The first two summands are Lorentzian peaks at the position of the foci (z = 0

and z = z1). The third summand is a product of a Lorentzian and a Gaussian, the

51


z

S

z

x

a) b)

c) d)

Figure 3.7: Illustrations of exemplary scenarios: The upper graph in each of the illustrationsshows a snapshot of the two counter-propagating Gaussian laser pulses indicating theposition of the foci and the time delay between the pulses. In each of the lower graphsthe expected autocorrelation signal is plotted (grey: interferometric autocorrelation, black:spatially averaged signal). a) perfect overlap in space and time (z1 = 0 and ∆τ =−z1/(2c)), b) overlap in space but a time delay remaining (z1 = 0 and ∆τ > 0), c) fociseparated in space and no time delay (z1 > 0 and ∆τ = 0), d) foci separated in spaceand time delay (z1 > 0 and ∆τ = 0). In all these examples the ratio of the intensities Rwas set to one.

52


center of which is located in the middle between the foci at z = z1/2. Far awayfrom this center the Gaussian dominates the behaviour of the term since it decaysexponentially. Close to the center, where the Gaussian does not vary much, theLorentzian peak shape dominates. Please note that the information about the timeduration of the pulse is contained in the Gaussian. For plane waves (z0 → ∞), acommon focus of the two beams (z1 = 0) and equal intensities from the left and theright (R = 1) Eq. 3.1 reduces to the well known intensity autocorrelation signal ofsecond order [66].

The resulting signal may exhibit a variety of shapes. A single peak structure, twoor even three distinct peaks are possible. We will discuss four exemplary situations:

Overlapping foci. If the foci are at the same position z = 0, the two pulses propa-gating in opposite direction will cross each other at a position determined bythe position of the beam-splitter. For zero delay ∆τ between the two pulsesthey will cross exactly in the focus generating the maximum intensity in thestanding wave (Fig. 3.7a). For ∆τ 6= 0 the overlap will occur outside the focusleading to a reduced signal with a slightly asymmetric shape (Fig. 3.7b).

Separate foci. If the foci are at different positions z = 0 and z = z1 on the opticalaxis, two approximately equal signals are expected due to each individual laserpulse if the delay ∆τ between the pulses satisfies the condition ∆τ z1/c.If both pulses overlap at or between the two foci, a more complex spatialsignal appears that may consist of one (Fig. 3.7c), two (Fig. 3.7d) or even threepeaks (not shown). The height of the observed peaks (in the case of a two-peakstructure) is not necessarily equal, even if the laser pulses are of equal intensity.

3.6 Experimental results

Fig. 3.8b) shows experimental data obtained for z1 = 0 compared with calculatedcurves based on Eq. (3.1). The data is integrated over 10 laser shots to increase theaccuracy of the non-linear curve-fit. Excellent agreement between the experimentallydetermined signals and the fitted curves of Eq. 3.1 are obtained for both the signalwith zero delay (∆τ = 0) as well as the two signals with a delay of ∆τ = ±66 fs.The data taken at non-zero delay between the pulses exhibits an asymmetric shape

53


a)

b)

c)

d)

Figure 3.8: a) Counter-propagating laser pulses with identical foci. b) Non-linear Thomsonsignal obtained for two counter-propagating laser pulses with identical foci. The experi-mental data represented by filled squares was measured for zero delay (∆τ = 0), while thedata represented by open squares and circles correspond to a delay of ∆τ = ±66 fs. Thesolid lines were obtained by carrying out a non-linear curve-fit to the experimental dataaccording to Eq. 3.1. For better visibility the data at non-zero delay (open symbols) wasscaled by a factor of two (left scale) with respect to the data at zero delay (right scale).c) Counter-propagating laser pulses with foci separated by a distance z1. d) Non-linearThomson signal obtained for z1 = 45 µm with the pulses intersecting each other in theleft focus (filled squares) and for z1 = 22µm (open squares).

54


Figure 3.9: a) Autocorrelation obtained from a THG autocorrelator using a small part ofthe beam cross-section. The dashed line indicates a Gaussian curve-fit with τ = 85 fsFWHM. b) Impact of the pulse front tilt on the effective pulse length in the laser focus.The intensity cross-section of two Gaussian pulses along the x-z-Axis is shown. Thepulse front of the left pulse is tilted with respect to the x-axis by an angle of 45 whichis greatly exaggerated. This leads to an increase of the effective pulse length τeff in thelaser focus as is indicated by the line graphs of the temporal pulse shapes.

as expected from the exemplary scenarios (Fig. 3.7b)). The curves are not perfectlysymmetric to z = 0 due to the delicate optical setup: Small movements of the cameraor any other optical element in the imaging setup had a large impact on the positionof the image on the CCD chip.

In Fig. 3.8d) autocorrelation results are shown for the case of non-identical fociof the two parabolic mirrors. The black curve shows an experimental non-linearThomson scattering signal obtained for a focus separation of z1 = 45 µm when thedelay ∆τ was adjusted such that the counter-propagating pulses intersected at theleft focus at z = 0, as shown in Fig. 3.7d). When the focus position is then moved toz = 22 µm a symmetric signal (red curve) is obtained. The data in Fig. 3.8d) does notcontain enough information about the Gaussian component of the autocorrelationfunction in Eq. 3.1 to carry out a non-linear curve-fit. Please note that only theGaussian component is determined by the laser pulse length.

The pulse duration τ and the Rayleigh length z0 were obtained for the fittedcurves in Fig. 3.8b). For the pulse duration of the full energy laser pulse at the focusposition we obtain τ = (112± 11) fs and for z0 = (9± 1) µm. This value of the pulse

55


duration is larger than the value of τ = (84 ± 5) fs measured with the classic THGautocorrelator (see Fig. 3.9a)). The THG operates at low intensity and takes only a(2 × 2) mm2 central cut-out of the full beam (Ø = 50 mm). The difference betweenthe two measurements may be explained by a slight tilt of the pulsefront. At theentry of the compressor a single laser pulse resembles a disk which is very large indiameter (50 mm), but which has a thickness of only 30 µm. A very small tilt in thepulse front thus has a large impact on the effective pulse length (see Fig. 3.9b)). Ourwavefront interferometer has a detection limit of 1’ of arc. The relative increase ofthe pulse duration τ by a wavefront tilt is given by [67]

τeff

τ=

√1 +

dFWHM sin α

cτ. (3.2)

The increase that is not detectable with the interferometer can therefore be as largeas 13%.

Another possible reason for the larger pulse duration determined by the photoncollider setup is related to the dynamics of the electrons in the laser focus. InSec. 2.1.1 it was assumed for simplicity that the electrons are oscillating locally. Inan experiment at relativistic intensities, however, the electrons may be acceleratedover a large distance and for instance be scattered out of the focal region. This mayhave a blurring effect on the autocorrelation signal and lead to a seemingly largerpulse length. PIC simulations are under way to quantitatively assess this effect.

The pulse duration measurement with the photon collider presented in this chap-ter represents the first single-shot autocorrelation measurement that has ever beencarried out at relativistic intensity. Since it takes place in the laser focus where laterthe actual experiment is carried out (e. g. laser-acceleration) it is a method which ismuch closer to the true experimental laser parameters than conventional methods.

56

4 Thomson backscattering fromlaser-accelerated electrons

The process of laser pulses scattered from accelerated electrons is closely related tothe generation of x-rays in undulator assemblies of synchrotrons. The close analogywas used in Sec. 2.10 to deduce some relations useful for the analysis and interpre-tation of the laser-generated radiation. So far, in Thomson scattering experimentsthe relativistic electron beams have been generated separately by conventional highenergy accelerators [68–72]. Apart from serving as a bright x-ray source, the backscat-tered radiation was also used for electron beam characterization [73, 74] and funda-mental studies of non-linear Compton scattering [75] and positron production fromphoton-photon interactions [7].

Recently, much progress has been made in laser-based accelerators producing rel-ativistic electrons with low divergence and a small energy spread [3–5]. These ad-vances have greatly increased the interest in the observation of Thomson scatteringwith an “all-optical interaction scheme”, where a laser-based accelerator is used inconjunction with a synchronized scattering laser pulse [50, 51, 76, 77].

The improvement in laser-acceleration of electrons apparently involved complexelectron acceleration mechanisms. To date, they can only be understood through PICsimulations which clearly manifests the need for in situ and time-resolved diagnos-tics. However, conventional diagnostic methods like electron magnetic spectrometersonly allow for the spectral analysis of a small cone of electrons at a distance fromthe interaction region outside the plasma after the acceleration process is complete.Catravas et al. [50] and Tomassini et al. [51] pointed out that - similarly to diagnosingthe accelerator electron beam - the backscattered radiation in a laser-based setup canbe used to infer energy- and time-resolved information of the electron accelerationprocess in the laser plasma.

In the photon collider setup we can generate a strong and a weak (non-relativistic)

57

4 Thomson backscattering from laser-accelerated electrons

laser pulse using an appropriate 90/10 beam-splitter. The strong laser pulse is fo-cused into a He gas-jet and accelerates electrons in the laser plasma to relativisticenergies. The counter-propagating weak pulse is then scattered from those electrons.It was shown that each measured backscattered photon spectrum is directly relatedto the electron energy distribution at the specific moment in time when the elec-tron bunch and the counter-propagating laser pulse interact (see Sec. 2.10). Sincethe photon collider offers accurate control over both spacial and temporal overlapof pump and probe pulse it can be used to directly monitor the acceleration of theelectrons. This direct access to the processes in the laser plasma has not been avail-able to date. Our results represent the first observation of Thomson scattering fromlaser-accelerated electrons.

4.1 Experimental setup

The schematics of the experimental setup are shown in Fig. 4.1. The main laser pulseis divided by a 90/10 beam-splitter into a pump pulse and a probe pulse, the strongerpump pulse being reflected from the beam-splitter. Each of the pulses is focused byan F/2.5 45 off-axis parabolic mirror into a He gas-jet. The time delay between thelaser pulses may be adjusted by moving the beam-splitter as indicated in Fig. 4.1.To achieve an optimum focal spot the parabolic mirrors may be tilted around twoaxes and to achieve spatial overlap both mirrors may be moved in three dimensions.The setup is located in a vacuum chamber and is entirely computer controlled. Theadjustment of the exact spatial and temporal overlap of the two laser pulses wasdescribed earlier (see Sec. 3.4 and [17]).

JETI was operated at the following parameters: The pulse length of 85 fs of themain laser pulse was determined by a THG autocorrelation. The energy of theundivided laser pulse was 370 mJ focused to an intensity of 2 × 1019 W/cm2. Theprobe pulse was focused to an intensity of 1×1018 W/cm2. These nominal intensitiescorrespond to laser field strengths of a0 = 3 and a0 = 0.8, respectively. Non-lineareffects occurring in the beam-splitter substrate are negligible since at that point thebeam diameter is about 50 mm and the energy of the probe pulse passing throughthe beam-splitter is only 10% of the main pulse energy. This was confirmed bycalculations carried out with Lab2 [78].

58


Figure 4.1: a) Experimental setup of the Thomson backscattering experiment: The mainlaser pulse is divided into two pulses by a 90/10 beam-splitter both of which are focusedby off-axis parabolic mirrors into a He gas-jet. The backscattered radiation is observedwith an x-ray CCD camera through a hole in the mirror of the probe beam. The time-delay between pump and probe may be varied adjusting the position of the beam-splitter.A frequency-doubled laser pulse is used for shadowgraphy of the interaction area (angleof incidence exaggerated for better visibility). b) Setup of the x-ray diagnostics. Thedimensions of the respective element and its distance to the source are given in mm. A:laser focus, B: pinhole in probe beam parabolic mirror, C: lead pinhole and metallic filter,D: CCD chip.

59


Figure 4.2: a) Position of the relativistic channel generated by the pump pulse in the gas-jet (indicated by a red line). The pump pulse is incident from the left. The position ofthe vacuum focus (z = 0) was determined from second-harmonic emission at low gasdensity. b) Side image of the relativistic channel at high gas density in false colors. Theline graphs (white) were obtained binning the image along the y- and z-axis, respectively.A white arrow indicates the position of the vacuum focus (x = z = 0).

A cylindrical subsonic gas nozzle was used which created a pulsed He gas-jet witha density profile of Gaussian shape along the laser axis and with a peak gas densityof 6× 1019 cm−3 (see Fig. 4.2). The gas density was determined using time-resolvedinterferometry with an error of 20%. When focused into the rising edge of the Hegas-jet the pump pulse undergoes relativistic self-focusing and generates a plasmachannel. In this configuration electrons are accelerated efficiently in forward directionto relativistic energies by the pump pulse. The spectrum of the high-energy electronsis dominated by an exponential shape and was determined in earlier experiments (seeFig. 2.9 in Sec. 2.3.1 and [14, 44]). The temperature of the electron spectrum usuallyranges from 3 to 7 MeV. The probe pulse is scattered from these high-energy electronsand the backscattered x-ray photons are observed.

The position of the laser focus and of the relativistic channel with respect to thegas-jet as well as the dimensions of the channel were determined from side images.At low gas density, similar to the condition in the autocorrelation experiment, theposition of the vacuum focus is indicated by faint but highly peaked second-harmonicemission. At higher gas densities a bright and more than 100 µm long relativisticchannel is generated by the pump pulse. The gas density profile along the opticalaxis and the extension of the relativistic channel are shown in Fig. 4.2a). A typical

60


side view image of second-harmonic light emitted from the relativistic channel isdisplayed in Fig. 4.2b). A white arrow indicates the position of the pump pulsevacuum focus. In all side view images the pump pulse is incident from the left.

In Fig. 4.2b) it is clearly visible that channeling of the pump pulse (and thus thenon-linear emission) sets in before the pump pulse reaches its vacuum focus. On theother hand, in the case of overlapping foci, the probe pulse (incident from the right)must propagate through most of the gas-jet to reach the nominally highest intensityin its focus. Under these conditions the probe pulse does not undergo relativistic self-focusing and is not forming a single, stable plasma channel. Contrarily, filamentationreduces the effective intensity of the pulse which has to be taken into account inthe analysis of backscattered radiation. This was confirmed by side-imaging of theinteraction region: The probe pulse did not generate second-harmonic emissionsdetectable with our imaging setup.

An ultra-short, frequency-doubled probe pulse with variable delay propagating atan angle of approx. 90 through the interaction region was deployed to record shadowimages of the laser plasma with a time resolution of 100 fs. Monitoring the ionizationfronts of the laser pulses in the shadow images the spatial and temporal point ofinteraction of the laser pulses was reliably established. Fig. 4.3a)-d) shows a typicalset of shadowgrams of the interaction region where the strong pump pulse is incidentfrom the left generating the relativistic channel and the probe pulse is incident fromthe right. The structures that are visible on these images are plasma regions ionizedby the propagating laser pulses. The bright area at the very center of the images isdue to the non-linear self-emission (cf. Fig. 4.2b)) of the laser plasma and indicatesthe position of the relativistic plasma channel generated by the pump pulse. Theself-emission is visible here since the exposure time of the camera is long comparedto the duration of the interaction.

In the experimental situation shown in Fig. 4.3 the probe pulse has already under-gone filamentation which can be seen from the irregular filamented structure of theprobe pulse in contrast to the clear ionization front of the pump pulse. An upperlimit of 45 µm for the probe pulse focal diameter may be estimated from the shadowimages (nominal diameter 3 µm).

A hole of 3 mm diameter was drilled into the parabolic mirror of the probe beam insuch a way that it is aligned with the axis of the focused beams (z-axis in Fig. 4.1a)).An x-ray CCD camera (Andor DO420 BN) was placed on this axis for energy-resolved

61


Figure 4.3: a)-d) Sequence of shadow images of two counter-propagating laser pulses withidentical foci and both pulses interacting in the focus. The pump pulse is incident fromthe left, the probe pulse from the right. The white region in the center of the imagesis due to the self-emission of the relativistic channel of the pump pulse. e) Invertedimage of the scintillating screen (Kodak KF x-ray intensifier screen) which was used formonitoring the energy of the electrons passing through the hole in the parabolic mirror(see Fig. 4.1). The dotted lines indicate the limits of the gap between the dipole magnets.

62


detection of Thomson backscattered photons. A pair of dipole magnets was used toprevent electrons from reaching the CCD chip while the camera was additionallyshielded with a lead pinhole from background bremsstrahlung. A filter set was in-troduced between CCD and laser focus to block irradiation by laser light.

In conjunction with an x-ray intensifier screen (Konica KF) the dipole magnetswere also used to infer the energy of the electrons passing through the hole in theprobe beam parabolic mirror in a range of 5 . . . 12 MeV. Electron energies in thisrange were expected from earlier measurements with high-resolution electron spec-trometers (see Fig. 2.9). A typical image taken from the fluorescent backside of thescreen is shown in Fig. 4.3e). The plane of the magnets was not perfectly alignedto the axis of the electron beam which led to helical electron trajectories. Towardslower electron energies the electron trace is limited by the edge of the upper dipolemagnet.

The spectrum of the electrons passing through the hole reflects the energy distri-bution of only a small fraction of accelerated plasma electrons. At large distancesthe energy distribution of the electrons has been found strongly anisotropic and an-gle dependent [36, 42, 60, 79]. The information from the scintillating screen wastherefore primarily used as an on-line diagnostic to optimize the laser-accelerationprocess.

4.2 X-ray diagnostics

The x-ray CCD camera was operated in single-photon counting mode. A single x-rayphoton incident on the CCD chip creates electron-hole pairs in an amount which isproportional to the photon energy, on average 29 eV/count in the analyzed spectralrange. The charge cloud created by a single photon may diffuse over several pixels(blooming). For an accurate, energy-resolved measurement it is therefore necessaryto keep the number of photons incident on the chip so low that the charge clouds donot overlap in order to distinguish single events. For the analysis of the x-ray imagesan algorithm must be applied that recognizes the patterns of pixels which are knownto be created by a single photon. The rejection of all pseudo-events which do notmatch any of the known patterns ensures the accuracy of the spectral information.

Fig. 4.4a) shows a typical image (1024× 256 pixels) recorded by the camera which

63


a)

b)

Figure 4.4: Analysis of the x-ray CCD images: a) CCD image of the x-rays generated byone laser-shot. b) Enlargement of the image region indicated in a). Single photon eventsare distinguishable.

was exposed to a single laser shot. The electronic dark current was subtracted. Themagnified part of the image shown in Fig. 4.4b) clarifies that single photon eventsare discriminable. The position of the CCD camera relative to the laser focus isdisplayed in Fig. 4.1b). Two pinholes ensure that only radiation originating fromthe laser focus is detected by the CCD chip. The first pinhole is constituted bythe hole in the probe pulse parabolic mirror of 3 mm diameter which is located ata distance of 150 mm from the focus. A second lead pinhole of 10 mm diameterand 50 mm thickness combined with a thin metallic filter at a distance of 320 mm

ensures that bremsstrahlung radiation and stray light are blocked. The CCD chip of26.6 mm× 6.7 mm is positioned at a distance of 1500 mm from the laser focus.

The general functionality of the analysis algorithm is described in Fig. 4.5a)-d).The algorithm recognizes the single- and multi-pixel events listed in Fig. 4.5d). Thesquares represent pixels on the x-ray CCD chip. Two different thresholds were ap-plied: In order to be recognized as a valid pattern, the center pixel (marked red)must contain more counts than threshold No. 1 (T1). Concurrently, adjacent pixels(marked blue) must be higher than background (threshold No. 2, T2, where T1 > T2)and smaller than the center pixel. All neighboring pixels not belonging to the pattern(empty squares) are required to be on background level (≤ T2), otherwise the eventis rejected. We found that the optimum thresholds for our application were T1 = 7

64


Figure 4.5: Details of the analysis algorithm: a) Histograms of the various pattern classes ofa typical x-ray image as indicated in the graph and the sum of all contributions (dashedline). b) Event counts broken down into pattern classes. For this particular image,most of the events are contributed by class 2 patterns. c) Comparison of a raw imagehistogram (grey bars) and the resulting analyzed data (identical to the dashed line in a)).d) List of patterns recognized by the algorithm. Two different thresholds (T1 and T2,where T1 > T2) are used: The red center pixel is required to be higher than T1. Bluepixels must be higher than background (T2) but lower than the center pixel. White pixelsmust be on background level (≤ T2). Parts of the x-ray image which do not match thesepatterns are disregarded.

65


b)

0 500 1000 1500 2000 2500 30000,01

0,1

1Tran

smission

, Efficien

cy

Energy / eV0 500 1000 1500 2000 2500 3000

0,01

0,1

1

Tran

smission

, Efficien

cy

Energy / eV

a)

Figure 4.6: Quantum efficiency of the x-ray CCD camera and transmission of the filtersused in the experiment: The Quantum efficiency of the x-ray CCD camera given by themanufacturer is shown as a dotted line in both graphs. At 1740 eV the Kα-edge of Si isvisible. Both graphs also display the filter transmission (dashed line) and the resultingefficiency curve according to Eq. 4.1 (solid line). a) Mylar-aluminum filter consistingof 6 µm Mylar foil vaporized with 300 nm aluminum. b) 300 nm Ni filter consisting of3 × 100 nm Ni foil and 3 Ni support grids of 5 µm thickness. Filter transmission datataken from Henke et al. [80].

and T2 = 3.Each class of patterns contributes to a different part of the spectrum which is

illustrated by Fig. 4.5a). This is due to the fact that the probability of generatingmany-pixel events rises with increasing photon energy. Additionally, the energythreshold for the detection of many-pixel events is higher than for few-pixel events.We therefore take only into account the part of the histogram where all patternclasses are considered, i. e. in particular we disregard all channels below channel 16.For highly exposed images this algorithm may only recognize and analyze a fractionof all events registered by the CCD chip (down to 30%). This is compensated byscaling the resulting spectrum with the total number of counts in the image. Theaccuracy of this procedure was verified in experiments with characteristic x-ray lineemissions of variable intensity using a conventional x-ray tube.

In order to obtain a photon spectrum from the image data the histogram of theanalyzed data (like the one shown in Fig. 4.5c)) must be divided by the energydependent quantum efficiency of the camera and by the transmission of the filter set.The efficiency of the CCD chip and the transmission of filters used in this experimentare displayed in Fig. 4.6a) and b). The quantum efficiency of the chip is shown in

66


both graphs as a dotted line and the transmission curves of the filters as dashed lines.

The electronic noise of the camera as well as the analysis algorithm led to a de-creased energy resolution of the obtained spectra. In a separate experiment with aconventional x-ray tube line spectra at energies of 452 eV (Ti Lα), 930 eV (Cu Lα)and 1486 eV (Al Kα) were recorded. The energy broadening was estimated to be 10channels (∼ 290 eV) from the width of the emission lines and the energy calibrationwas determined to an average of 29 eV/count. It is known from [81] that at lower en-ergies the energy resolution of an x-ray CCD is no longer linear. This was confirmedin our experiments, but in the observed energy range the deviation of the energycalibration was < 2 eV. If the photon spectrum R(E) generated in the laser-plasmainteraction is slowly varying with energy E compared to the broadening functiong(E), one may write the photon distribution S(E) measured with the CCD as

S(E) =

∫T (E ′) q(E ′) R(E ′) g(E − E ′) dE ′

≈ R(E)

∫T (E ′) q(E ′) g(E − E ′) dE ′

= R(E) T (E), (4.1)

where T (E) is the transmission of the filter and q(E) the efficiency of the camera. Thebroadening function was assumed to be of Gaussian shape with a width of 290 eV.T (E) represents the blurred response function of the filter-detector assembly and isshown as solid black line in Figs. 4.6a) and b).

The CCD is sensitive to photons of up to 10 keV of energy. The number of photonsincident on the chip may be adjusted by decreasing the space angle covered by theCCD (increasing the distance to the source) or by the use of filters of various materialsand thickness. The advantages of a CCD in single-photon counting mode comparedto imaging soft x-ray spectrometers are the easy alignment of the camera with acollimated cw-laser running along the z-axis, through the laser focus and throughthe hole in the parabolic mirror and that optimization of the signal strength is easilyachieved. The x-ray CCD was therefore the detector of choice in the Thomsonbackscattering experiment.

67


4.3 Background and competing processes

In order to obtain the Thomson backscattered signal from the recorded spectra, it hadto be separated from background. Background is caused e. g. by accelerated plasmaelectrons generating bremsstrahlung radiation in the vacuum vessel or characteristicx-ray emission from material of the nozzle body (steel). From [82, 83] it is also knownthat a single relativistic laser pulse incident on a gas-jet may produce x-rays in therange of a few keV through betatron oscillations of the electrons in the relativisticchannel. This phenomenon is also called “plasma wiggler” and was first observedwith high energy electrons (GeV range) propagating through a preformed plasma[84, 85]. X-rays are generated when relativistic electrons propagate through an ionchannel and undergo betatron oscillations with the betatron fundamental frequencyωb = ckb = ωp/

√2γ (see also Sec. 2.3.3).

The properties of the plasma wiggler radiation are very similar to the radiation gen-erated in a conventional synchrotron wiggler (see [49]).The plasma wiggler strengthis given by

K = γkbr0,

where r0 is the excursion of the electrons in the ion channel. For K 1 the radiatedspectrum becomes continuous. The critical (maximum) frequency ωc is determinedby

~ωc[MeV] = 5× 10−21γ2ne[cm−3] r0[µm],

and the maximum of the radiated spectrum is located at about 0.29 ωc. The halfangle of the central emission cone is θcen = K/γ [48, 86]. The betatron radiation maytherefore be highly collimated to an emission angle as low as 50 mrad [82] which isof the same order as the expected half angle of emission of Thomson backscatteredradiation.

Since important quantities like the excursion r0 of the electrons is not accessible inan experiment, the betatron spectrum cannot be predicted from the laser parametersalone unless PIC simulations are carried out. Comparing our laser parameters tothose of Rousse et al. [82] it is apparent that photons of much lower energy and oflarger angular spread are to be expected. Its contribution to the background shouldtherefore be small.

Background spectra were recorded with only the pump pulse incident on the He

68


Figure 4.7: a) Signal and background detected by the x-ray CCD camera. Enlarged regionof 400 . . . 2000 eV where significant Thomson signal was measured. b) X-ray spectrum upto 7 keV showing characteristic lines from the nozzle material. The data shown here wasaveraged over 10 laser shots and corresponds to the data point #13 at the maximum inFig. 4.10b). Please note that a) and b) are shown with a logarithmic scale. c) Single shotspectra (linear scale) obtained by subtracting background. Blue and red curves: ten shotsrecorded sequentially under constant experimental conditions. The hindmost spectrumclearly stands out from the others and represents a remarkably bright shot. Black curve:average over these ten shots which was used for further analysis.

69


gas-jet. The Thomson signal was then identified as the difference between spectrarecorded with both pulses incident on the target and these background spectra. Ear-lier experiments showed that gradual destruction of the nozzle body by acceleratedions leads to a continuously decreasing number of accelerated electrons. Due to thedeformation of the nozzle orifice constant optimum conditions for laser-accelerationcannot be provided over several hundred laser shots. It was therefore necessary torecord background spectra in regular intervals. Two consecutive background spectrawere averaged and subtracted from spectra containing both background and Thom-son signal.

Fig. 4.7a)-b) shows a sample x-ray signal (including the background) and the cor-responding background spectrum. The x-ray spectra typically extend beyond 7 keV

but significant Thomson backscattering signal was measured in the range up to 2 keV

only with our experimental configuration. In the range of 6 . . . 8 keV the Kα-linesfrom the nozzle material (mostly Fe, Cr) are visible. The Thomson backscatteringsignal as determined by subtraction amounts to about 30% background.

The pure Thomson backscattering signal obtained from this measurement by sub-traction is shown in Fig. 4.7c). The red and blue curves represent 10 single-shotspectra which were recorded under constant experimental conditions. The blackcurve is the average spectrum calculated from these ten shots. From the shot-to-shotfluctuations in these spectra a standard error for each channel may be calculated.The resulting error bars are displayed e. g. in Figs. 4.12 and 4.13.

An important property of Thomson backscattered x-rays in an all-optical setupis the small source size which is determined by the laser-generated electron beamprofile. In our experiment the Thomson backscattering source size was not directlyaccessible since we were not able to discriminate between Thomson scattered andbackground photons. However, the overall x-ray source size was determined whichrepresents an upper limit of the Thomson backscattering source size.

The shadow of the filter support grids becomes visible when adding all recordedlow-exposure images (see Fig. 4.8a)). In a simple geometric optics model the sourcesize S1 may be estimated from the softness of the shadow edges:

S1 =L1

L2

S2,

where L1 and L2 are the distance of the source and of the detector from the grid,

70


a) b)

Figure 4.8: Determining the source size from the filter grid. a) Adding all x-ray imagesrecorded during the experiment yields an image of the shadow of the filter support grids.b) Cross-section around a grid shadow. The shadow edge extends over a distance of about3 pixels corresponding to 78 µm on the CCD chip.

respectively, and S2 the extension of the shadow edge. The setup of the x-ray CCD isshown in Fig. 4.1b) and S2 was determined from an image cross-section as shown inFig. 4.8b). The source size S1 was calculated to 30× 30 µm2 which is in accordancewith results published in [82, 83] and with the diameter of the plasma channel (20 µm

FWHM) as shown in Fig. 4.2.

4.4 Temporal change of total backscattered radiation

Another independent indication that the recorded signal in fact arises from Thomsonbackscattering is the circumstance that the total number of backscattered photonsis dependent on the delay between pump and probe pulse. Let us first consider aconfiguration where the pump and probe pulse vacuum foci are identical. The delaybetween pump and probe may be adjusted by moving the 45 beam-splitter by adistance ∆s, where ∆s is measured normal to the beam-splitter surface. The changeof the pump pulse optical path amounts to ∆σ1 =

√2∆s (cf. Fig. 4.9b)).

We now define the delay between pump and probe pulse as the difference betweenthe moment in time when the pump pulse passes its vacuum focus and the moment intime when pump and probe pass each other. The choice of the pump pulse passing itsfocus as reference point in time is arbitrary. This choice turns out to be reasonable,however. We will apply this frame of reference even to situations where the pump

71


Figure 4.9: Enlarged sections of the experimental setup shown in Fig. 4.1a) illustrating theimpact of moving the parabolic mirror of the probe beam a) and of moving the beam-splitter on the time delay between pump and probe pulse b).

and probe vacuum foci are not identical.The change of the optical path ∆σ1 of the pump pulse shifts the point of interaction

by ∆σ1/2 or - in other words - changes the delay between pump and probe by∆τ = ∆σ1/2c. The situation is slightly more complicated, if one also considersconfigurations where the probe pulse focus is moved away from the pump pulse focusby a distance ∆z by moving the probe parabolic mirror. The change of the probepulse optical path can be estimated using geometric optics (see Fig. 4.9a)) to be∆σ2 = (1 + 1/

√2)∆z. The corresponding time delay ∆τ = ∆σ2/2c inflicted on

the moment of interaction must be considered when comparing experiments withdifferent focal configurations.

Two different experiments were carried out: In a first experiment A an Al-Mylarfilter was used in front of the CCD (transmission shown in Fig. 4.6). However,for this experiment the observation angle ϑ is unknown (see Fig. 2.10). Its resultsmay therefore serve for qualitative analysis of the total backscattered photon yieldonly and will not be considered later in Sec. 4.5 for spectral analysis. In a secondexperiment B a 300 nm Ni filter was deployed and the number of photons detectedin the range of 430 . . . 2030 eV could be increased fourfold. The observation angle ϑ

was determined experimentally. The signal to background ratio remained the samein both experiments (up to 0.3).

In Fig. 4.10 the number of photons integrated over the interval 430 . . . 2030 eV and

72


-200 0 200 400 600 800 1000-0.5

0.0

0.5

1.0

1.5

2.0

11

12

13

16

1819

Num

ber o

f pho

tons

/ 10

4

Delay / fs

-200 0 200 400 600 800 10000

1

2

3

4

5

Num

ber o

f pho

tons

/ 10

3

Delay / fs

a) b)

Figure 4.10: Integral number of photons in the range of 430 . . . 2030 eV. a) Data recordedin experiment A: The data indicated by squares was recorded with pump and probe focusbeing identical. Subsequently the focus of the probe pulse was moved away from the pumpfocus by 200 µm (data indicated by circles). A delay of 0 fs corresponds to the laser pulsesinteracting in the pump pulse focus. b) Data recorded in experiment B. Filled squaresindicate a first sequence of measurements, empty squares a second. Both delay curves a)and b) show good agreement. One error bar in each graph indicates the standard errorof the measurement deduced from shot-to-shot fluctuations of ten laser shots.

averaged over 10 shots is shown with respect to the pump-probe delay. Fig. 4.10a)shows data acquired in experiment A. The photon yield in the configuration of over-lapping vacuum foci (indicated by squares) shows a strong dependence on the pump-probe delay. It is close to zero at a delay of −200 fs and rises to a maximum at 300 fs,then slowly decreases again.

In order to determine if the dependence on the delay is due to the spacial conver-gence of the probe pulse only (opening angle ≈ 30) the probe pulse vacuum focuswas moved by ∆z = 200 µm from the pump pulse vacuum focus towards the incidentprobe pulse. If the dependence obtained from the earlier measurement was due onlyto the properties of the probe pulse, the maximum of the delay curve should haveshifted by ∆τ = ∆z/c ≈ 700 fs towards positive delay. The measured data indicatedby circles in Fig. 4.10a), however, fits nicely into the previously determined curve.Please note that the timescale of this additional data was adjusted according to theconsiderations at the beginning of this section. The increase in backscattered photonsmust therefore be related to an increase in high-energy electrons in the laser plasma.This seems reasonable considering that the probe pulse undergoes filamentation inthe gas-jet and is consequently not strongly focused.

73


Figure 4.11: a) The normalized photon yield in the interval 430 . . . 2030 eV averaged over10 laser shots is plotted over the delay between the laser pulses. In this way data fromdifferent experiments (indicated by different symbols) can be compared. At delays whereseveral data points are available an average is carried out and indicated as a solid line.b) Sketches of three scenarios of different pump-probe delay where the pump pulse isincident from the left generating a relativistic channel in which electrons are accelerated.The region where pump and probe pulse overlap is indicated with a hatched box.

The photon yield obtained from experiment B exhibits the same dependency, butwith an improved signal level (see Fig. 4.10b)). The maximum of the delay curve ismore pronounced. After recording a first data set (filled squares) the measurementwas repeated (empty squares) to ensure that the assumed Thomson signal did notdecrease continuously with an increasing number of shots. Due to the deformationof the nozzle body the Thomson signal drops overall, as does the background (notshown).

The results of the time-resolved photon yield measurements are summarized inFig. 4.11. All data acquired so far is displayed in Fig. 4.11a). The number of photonsdetected in the energy interval 430 . . . 2030 eV was normalized for each experiment tofacilitate comparison. At those delays, where more than one data point was available,an average was calculated and is shown as solid line. The underlying processes maybe visualized schematically as shown in Fig. 4.11b). Three different scenarios areselected and indicated accordingly in Fig. 4.11a):

1. A delay of −200 fs: The probe pulse passes the pump pulse vacuum focusbefore the pump pulse arrives. Both pulses meet before the pump pulse has

74


undergone self-focusing. At this moment in time effectively no electrons havebeen accelerated yet.

2. A delay of 200 fs: The probe pulse interacts with the laser-accelerated electronstowards the end of the relativistic channel. At this moment in time the numberof accelerated electrons reaches its maximum.

3. A delay of 600 fs: The probe pulse interacts with the electrons outside therelativistic channel and the electron beam is now diverging. At later delays thedivergence of the probe pulse may become relevant as well.

The timescale of the changes in the photon yield agrees well with our observationsof the non-linear Thomson scattering (Fig. 4.2). Typical side view images of theinteraction region at the second-harmonic frequency show bright emission over adistance of 100 . . . 200 µm along the z-axis. A laser pulse would need 300 . . . 600 fs

to propagate through this region corresponding to the rising edge of the graph inFig. 4.11a).

4.5 Electron spectra

In Sec. 2.4 the principles of Thomson backscattering and their application to the all-optical photon collider setup were discussed. It was shown that the detected x-rayspectra contain direct information about the electron energy at the location of in-teraction and at the certain moment in time of interaction. Thomson backscatteringmay therefore serve for on-line time-resolved electron spectroscopy.

We will briefly recall the most important interrelations from Sec. 2.4. The x-rayphoton energy Ex = ~ωx may be calculated from the electron energy using

Ex =4γ2E0

1 + γ2ϑ2,

where γ is the Lorentz factor of the electron, E0 = ~ω0 the laser photon energy andϑ the angle of observation with respect to the direction of electron propagation (seeFig. 2.10 for the definition of ϑ). Conversely, one may calculate the energy of the

75


electron (in terms of γ), which scattered the x-ray photon, by (cf. Eq. 2.36 and 2.37)

γ =

√Ex

4E0(1− Exϑ2

4E0). (4.2)

Having measured the backscattered photon spectrum ∆Nx/∆Ex and having deter-mined experimentally the observation angle ϑ we may now calculate the underlyingelectron spectrum ∆Ne/∆Ee using

∆Ne

∆Ee

=∆Nx

∆Ex

· 8E0

αN0a20mc2∆Ω

· 1

γ, (4.3)

where everything except γ, which may be calculated from Eq. 4.2, is experimentaldata or known constants.

In experiment B, where the Ni filter was deployed, the constants in Eq. 2.42 weredetermined as follows: The number of oscillations N0 contained in an 85 fs pulse ofλ0 = 795 nm is about 30. The solid angle of detection ∆Ω = 80 µsr in our experimentwas limited solely by the size of the CCD chip and not by pinholes. The angle ofobservation was experimentally determined to ϑ = 60 mrad by introducing a cross-hair into the beam path of the collimated laser marking the center of the beamprofile. Behind the laser focus the shadow of the cross-hair indicated the laser axisand therefore the axis of electron propagation and was used to directly measure theangle of observation ϑ on the surface of the parabolic mirror.

It can easily be seen from Eq. 4.2 that a ϑ as large as 60 mrad will have a sizeableimpact on the calculated electron spectrum. In fact, this ϑ is larger than the openinghalf angle of the central radiation cone for electrons of γ > 3. Therefore the detectedbackscattered Thomson signal is much weaker than the backscattered radiation di-rectly in forward direction where most of the radiated power is concentrated. Onthe other hand, observation under a non-zero angle ϑ may offer certain advantagesdue to the γ2 dependency of the x-ray energy Ex: If one observes the scattered ra-diation exactly in forward direction and analyzes an x-ray spectrum covering twooctaves, the obtained electron spectrum extends over only

√2 octaves which may be

a quite limited part of the electron spectrum. Observation under a non-zero angleϑ, however, increases the covered electron energy range.

After discussing the dependency of the total backscattering photon yield on the

76


400 600 800 1000 1200 1400 1600 1800 2000 22001

10

100

1000

Data set #13: 200 fs

dNx/d

Ex /

(29

eV)

Energy / eV0 5 10 15 20 25 30 35 40 45

0.1

1

10

100

1000

Te = 6.1 MeV

dNe/d

Ee /

(104 M

eV-1)

Energy / MeV

Te = 1.3 MeV

0 5 10 15 20 25 30 35 40 450.1

1

10

100

1000

dNe/d

Ee /

(104 M

eV-1)

Energy / MeV

6.5 MeV

1.4 MeV

400 600 800 1000 1200 1400 1600 1800 2000 22001

10

100

1000


dNx/d

Ex /

(29

eV)

Energy / eV

0 5 10 15 20 25 30 35 40 450.1

1

10

100

1000

dNe/d

Ee /

(104 M

eV-1)

Energy / MeV

3.3 MeV

1.5 MeV

400 600 800 1000 1200 1400 1600 1800 2000 22001

10

100

1000

dNx/d

Ex /

(29

eV)

Energy / eV

Data set #11: 0 fs

Figure 4.12: Thomson backscattering spectra recorded with the x-ray CCD camera and av-eraged over 10 shots (graphs on the left) and the electron spectra calculated from theseaveraged photon spectra (graphs on the right). The data set numbers and delays corre-spond to those given in Fig. 4.10.

77


pump-probe delay in the previous section it is of special interest to examine thechange in the electron spectra with respect to the delay. Electron spectra were cal-culated from selected backscattered x-ray spectra obtained from experiment B. Thecorresponding total photon yields are indicated by data set numbers in Fig. 4.10.The x-ray spectrum and the calculated electron spectrum are shown in Figs. 4.12and 4.13 arranged in rows. The error bars indicated in the x-ray spectra are stan-dard errors derived from shot-to-shot fluctuations of ten laser shots. At the timingcondition when the highest photon yield was generated the statistically most reliablespectra were obtained (here: set #13).

The electron spectra are composed of two distinct exponential energy distributionswith temperatures of about 1 MeV and 6 MeV, respectively. The temperatures areindicated by dashed lines in the electron spectra. The hotter temperature agrees wellwith our previous measurements using a conventional spectrometer. The surprisingresult is that the electron temperatures do not change significantly in the displayeddelay range of 600 fs. From early delays on the two populations exist. At delaytimes far from optimum delay, however, little data is available to reliably carry outnon-linear curve fits.

The total number of electrons Nb contained in the spectrum may be calculatedfrom Eq. 4.3. It was pointed out earlier that the laser field strength a0 of the probepulse was reduced due to filamentation and the effective value was estimated fromshadow images to be in the range of 0.05 . . . 0.8 (0.8 being the nominal field strength).Inserting the parameters from the non-linear curve fits from set #13 yields an electronbunch charge of 0.07 nC . . . 18 nC. Other groups reported total bunch charges of0.5 . . . 8 nC [36, 42, 79, 87, 88]. This indicates that our effective probe pulse laserfield strength is about 0.1 and thus significantly smaller than the nominal value.Future experiments will improve this value using specifically tailored, supersonicgas-jets.

We interpret our results from the time-resolved electron spectroscopy as follows:The laser acceleration occurs on a very short timescale of the order of the laser pulseduration. During this short time two electron populations (one of low temperature,about 1 MeV, one of higher temperature, about 6 MeV) are formed. The idea that theacceleration of a particular set of electrons would take place over the whole length ofthe relativistic channel is therefore wrong. The acceleration length is much shorter.While the pump laser pulse propagates through the channel, the total number of

78


400 600 800 1000 1200 1400 1600 1800 2000 22001

10

100

1000


dNx/d

Ex /

(29

eV)

Energy / eV

0 5 10 15 20 25 30 35 40 450.1

1

10

100

1000

dNe/d

Ee /

(104 M

eV-1)

Energy / MeV

7.4 MeV

1.0 MeV

0 5 10 15 20 25 30 35 40 450.1

1

10

100

1000

dNe/d

Ee /

(104 M

eV-1)

Energy / MeV

6.8 MeV

1.1 MeV

400 600 800 1000 1200 1400 1600 1800 2000 22001

10

100

1000


dNx/d

Ex /

(29

eV)

Energy / eV

0 5 10 15 20 25 30 35 40 450.1

1

10

100

1000

dNe/d

Ee /

(104 M

eV-1)

Energy / MeV

5.5 MeV

1.3 MeV

400 600 800 1000 1200 1400 1600 1800 2000 22001

10

100

1000


dNx/d

Ex /

(29

eV)

Energy / eV

Figure 4.13: Thomson backscattering spectra recorded with the x-ray CCD camera and av-eraged over 10 shots (graphs on the left) and the electron spectra calculated from theseaveraged photon spectra (graphs on the right). The data set numbers and delays corre-spond to those given in Fig. 4.10.

79


relativistic electrons changes (increases), but not their energy distribution.These measurements represent the first energy- and time-resolved diagnostics of

laser-accelerated electrons. So far, electrons accelerated in the SM-LWFA regimewhich exhibited an exponential energy distribution were examined. It will be mostexciting to apply this technique to the regime where quasi-monoenergetic electronsare generated which has been shown to be possible also at the Jena laser facility[44]. It is believed that in the transition regime towards bubble acceleration theprocess of acceleration takes place over a range in the order of millimeters [46]. Thetime-dependent evolution of this process may be monitored with the photon collider.

80

5 Future prospects of the photoncollider

In the previous section the first experimental observation of Thomson backscatteredphotons from laser-accelerated electrons in an all-optical setup was presented. There,the focus was set on the identification of the Thomson backscattered signal, its sep-aration from background and its application as an electron diagnostics tool. Theprospects of the photon collider setup, however, reach far beyond that. It may serveas a short-pulsed x-ray source with unique properties or it may actually provide col-liding particle beams - measuring up to its name. The photon collider may also be thebasis of a visionary experiment demonstrating non-linear quantum electrodynamics(QED). These applications will be discussed in the following.

5.1 Thomson backscattering as x-ray source

In order to assess the unique x-ray source properties of the photon collider appro-priately, I will first briefly review the requirements which are made on short-pulsedx-ray sources in modern experiments and then list sources which are available todayor which will be made available in a few years’ time. Finally, the capabilities of thephoton collider will be presented and compared to those of other sources.

One important application of short x-ray pulses are pump-probe experiments. Thegoal of these experiments is to achieve a better understanding of processes governedby atomic motion studying excited systems prior to vibrational relaxation, far fromequilibrium. In such an experiment a pump pulse (typically realized by an ultra-shortoptical laser pulse) induces e. g. a conformational change in the sample which is thenprobed by an other pulse after a variable delay. If the probe pulse is an x-ray pulse,direct observation of real-time changes in the sample by means of x-ray diffraction

81

5 Future prospects of the photon collider

Source Principle / Method Energy range/ keV

Source size/ µm

Bpk

ALS 5.0 Synchrotron 0.23 . . . 0.62 310× 16 6× 1020

APS Type A Synchrotron 3.5 . . . 12.0 360× 34 3× 1022

Bessy II U49 Synchrotron 140 . . . 500 65× 45 6× 1021

PLEIADES Thomsonbackscattering

up to 78 ∼ 50 ∼ 1018

Dense laser plasmas Characteristicx-ray emission

1 . . . 10 ∼ 10 ∼ 1018

Plasma wiggler / LOA Betatron radiation 1 . . . 10 20 2× 1019

XFEL FEL 4.0 . . . 12.4 110 ∼ 1033

Photon collider Thomsonbackscattering

0.4 . . . 2.0 ∼ 30 6× 1015

Photon collider (projected) Thomsonbackscattering

1 . . . 100 ∼ 30 ∼ 1020

Table 5.1: Characteristic parameters of selected x-ray sources: Undulators operating atthe fundamental frequency of Advanced Light Source (ALS, Berkeley), Advanced PhotonSource (APS, Chicago), Berliner Elektronenspeicherring-Gesellschaft für Synchrotron-strahlung (Bessy, Berlin). Further sources are Picosecond Laser-Electron InterActionfor the Dynamic Evaluation of Structures (PLEIADES, Livermore), the plasma wig-gler at the Laboratoire de l’optique appliquée (LOA, Palaiseau) and the European X-ray Laser project (XFEL, Hamburg). The peak spectral brightness Bpk is given inphotons/s/mm2/mrad2/0.1%BW, values are taken from literature given in the text.

is possible (in reciprocal space).

The timescale of interest for these experiments is in the range of a few 100 fs whichis the order of vibrational periods. The x-ray probe pulse must therefore be ultra-short and very intense since the diffraction efficiency of the sample may be very low.A small x-ray source size is advantageous since small sources can be focused ontosmall areas (yielding higher photon flux) using x-ray focusing optics and allow forbetter imaging quality (if the application involves imaging). Additionally, the delaybetween pump and probe must be well-known - a priori or a posteriori.

Tab. 5.1 lists examples of brilliant x-ray sources which are available for opticalpump x-ray probe experiments. As characteristic parameters the energy range, thesource size and the peak spectral brightness were selected which do not represent allimportant source properties but which may be a guideline for comparison.

82


Third-generation synchrotron facilitiesModern existing synchrotron facilities typically consist of a linear accelerator pro-viding electron bunches in the GeV range which are then injected into storage rings.They provide short-pulsed x-rays in the energy range of a few hundreds of eV upto tens of keV sending the GeV electrons through bending magnets, undulator orwiggler structures in the storage ring. The electrons are kept circulating for as longas possible to maximize the usable beam-time. Synchrotrons are therefore inflexibleconcerning the timing of the electron bunches [89].

The x-ray pulse duration is determined by the electron bunch length which is typ-ically ∼ 100 ps. In order to achieve better time-resolution, time-resolving detectors,streak cameras, are deployed which are limited in resolution to ∼ 1 ps [90, 91]. Theuse of a streak camera in turn lowers the effective number of photons available for themeasurement since the “long” x-ray pulse is distributed over many “slices” registeredby the streak camera.

Novel electron bunch slicing techniques and ultrafast x-ray mirrors have been de-veloped to generate truly femtosecond synchrotron radiation [92, 93]. However, thesetechniques reduce the x-ray intensity to the same degree as slicing by a streak cam-era. Generally, control over the pump-probe delay must be achieved by externalsynchronization which is very difficult at these time-scales. A jitter of 3 ps RMSbetween the pump laser pulse and the x-ray source may typically not be overcome[94].

Laser-plasma based x-ray sourcesWhen an ultra-short and ultra-intense laser pulse impinges onto a thin foil a micro-plasma is produced which is a source of many kinds of radiation and acceleratedparticles. In particular, a large number of accelerated, supra-thermal electrons gen-erate bremsstrahlung and characteristic x-ray line emission within the foil target.The line emission may be considered monochromatic (0.01% BW) and is isotropic.The source size is small, i. e. of the order of the laser focus (a few tens of microns).The generated x-ray pulse duration is of the order of the laser pulse duration [95–98]. A fraction of the isotropic emission may be collected and refocused using x-rayoptics. 1012 photons from a source size of 50 × 50 µm2 within 100 fs emitted into asolid angle of 4π are reported in [99].

Recently, two novel laser-plasma x-ray sources have been demonstrated: Betatron

83


radiation from relativistic electrons inside a plasma channel (the plasma wiggler,[82]) which was briefly discussed in Sec. 4.3 and bremsstrahlung from relativistic,laser-accelerated electrons [9, 14, 100]. The plasma wiggler represents a very bright,collimated and ultra-short (laser pulse-length limited) soft x-ray source with a sizeof down to 20 µm. Its peak spectral brilliance amounts to Bpk = 2 × 1019∗ (this isdifferent from the figure given in [82] which was miscalculated). The bremsstrahlungsource produces hard x-rays and gamma-rays with an opening angle of ∼ 3, a sourcesize of∼ 400 µm and a pulse duration of a multiple of the laser pulse length dependingon the converter target.

Hybrids: Thomson scattering with linear accelerators (LINACs)This scheme, where a laser pulse is scattered from accelerated electrons, was pro-posed in the early 60s [101, 102] but did not achieve competitive brightness untilthe development of high-intensity lasers. It has been demonstrated in many facili-ties [68–71, 75, 103, 104] and maximum photon energies up to the GeV range havebeen observed [7]. The achieved intensities were, however, only moderate. The x-raypulse duration is either determined by the laser pulse duration in a 90 scatteringgeometry (short-pulsed but low brightness x-rays) or by the electron bunch length incollinear geometry (long-pulsed and high brightness). Expensive and highly complexmechanisms must be installed to obtain control over the time jitter between laserand electron bunch which is ∼ 1.5 ps at best [69, 71, 72]. The only way to overcomethis shortcoming is to carry out sophisticated a posteriori delay measurements withlimited time-resolution [105] leading to a “Monte-Carlo-like” experimental strategy[94].

X-ray free electron lasers (X-FELs)They are the novel tool which will revolutionize the field - whereas “tool” is a shortword for an installation of more than 3 km length. An X-FEL consists of a linearaccelerator followed by an undulator of exceptional length (number of periods ∼ 103).The radiation is produced based on the same principles as in a synchrotron undulatorwith a few additional effects which are caused by a phenomenon widely known asthe FEL collective instability [106].

Electrons propagating through the undulator interact with the electromagnetic∗Spectral brightness is given in units of photons/mm2/mrad2/s/0.1%BW which are commonly

used in the synchrotron community. The units are omitted in the text for brevity.

84


field generated by other electrons. The interaction changes their energy and thechange is modulated at the x-ray wavelength. Additionally, due to the electron en-ergy spread, electrons will bunch together within a wavelength. Spatially coherentradiation is generated, the single electron fields superimpose and the bunching mech-anism grows even stronger. The gain is then proportional to the number of electronssquared which is huge. The laser-like properties of the radiation and the exceptionalrequirements on the electrons necessary for the FEL process lead to a bright x-raysource surpassing today’s existing sources by at least eight orders of magnitude.

FELs generating long-wavelength radiation may be compact-sized [107], but inorder to reach the hard x-ray regime electrons in the GeV range must be used whichcalls for large accelerator facilities. Currently, X-FELs at two locations are underconstruction: The Linac Coherent Light Source (LCLS) at Stanford, USA [108], andthe European X-ray Laser Project (XFEL) at Hamburg, Germany [109]. The formerwill be operational in 2009 at up to 8 keV, the latter in 2012 at up to 12 keV. Thepeak spectral brightness Bpk of XFEL is projected to be 1033∗ .

But even though X-FEL facilities will provide radiation of unprecedented qual-ity it will be difficult to conduct pump-probe experiments since - as it is with allaccelerator-based sources - sub-picosecond synchronization is hard to achieve. Aposteriori delay measurements will have to be developed to fully take advantage ofthe ultra-short X-FEL radiation [94, 105].

The photon colliderHow does the photon collider compare to the aforementioned sources? Since it isan all-optical scheme, it has intrinsical, built-in, synchronization at highest level asdo have all other purely laser-based sources. It is, in this sense, predestined forpump-probe experiments.

In our experiment we detected a maximum of 2 × 104 Thomson backscatteredphotons at optimum delay. This corresponds to an average spectral brightness ofBav = 5 × 103∗ and a peak spectral brightness Bpk = 6 × 1015∗ assuming a pulseduration of 85 fs. However, the observation angle was not optimized for maximumphoton yield but rather large, larger than the central emission cone of the detectedradiation. In forward direction the peak spectral brightness is calculated to be Bpk =

1.3× 1017∗. This is still not the best that the photon collider at our laser parameterscould do: The probe beam was filamented before it interacted with the electron

85


bunch thus decreasing the effective probe pulse intensity.Calculations have been carried out by Catravas et al. [50] assuming perfect spa-

tial overlap between the laser-accelerated electron bunch and the probe laser focus.Catravas et al. obtained Bav = 9 × 107∗ and Bpk ∼ 1020∗ for a typical, LWFAgenerated electron bunch of 5 nC and exponential energy distribution. The spectralbrightness using narrower, quasi-monoenergetic laser-accelerated electron bunchesmay be even higher. The photon collider has therefore the potential to advance to aregime which was until today reserved for large synchrotron facilities.

Rousse et al. used only a single laser pulse to obtain a brightness of Bpk ∼ 1019∗ -why would one bother to set up colliding laser pulses? There are important propertiesof Thomson backscattered photons which make the photon collider superior to thebetatron source. Considering e. g. the scalability: How may higher photon energiesbe achieved in these schemes? Generally, in both cases the photon energy scales withγ2. However, in the case of the plasma wiggler, the process of laser-acceleration isclosely linked to the wiggler motion and cannot be separated. The conditions foroptimum laser-acceleration may not coincide with optimum wiggler conditions. Theplasma density, e. g., must be low (< 1019 cm−3) to obtain quasi-monoenergeticelectron bunches [46], whereas the maximum photon yield from the plasma wiggleris obtained at maximum plasma density: The gain in radiated power is proportionalto the plasma density squared [48].

Rousse et al. carried out the plasma wiggler experiments producing electron beamsof 20 MeV temperature. If laser photons would be scattered from these electrons ina photon collider geometry, the maximum of the resulting photon spectrum wouldbe located at 10 keV (see Sec. 2.4) compared to an exponential photon spectrumfrom the plasma wiggler in the range of 1 . . . 10 keV. For 40 MeV which are reportedin [9] the maximum would be shifted to 40 keV. Another advantage of the photoncollider is that it conserves polarization, i. e. the emitted radiation has exactlythe polarization of the scattering laser pulse. Polarized intense x-rays are of greatinterest for probing magnetic materials, helical structures and other samples withpolarization dependent properties [49, Ch. 5 and references therein].

With the photon collider the parameters for laser-acceleration can be optimizedindependently from the x-ray generation. In consideration of the recent advancesin laser electron acceleration [3–5, 44] the photon collider thus offers the meansto generate unparalleled bright, polarized and short-pulsed x-rays in energy ranges

86


Figure 5.1: The “mini-orange” is part of a positron detector specially designed for exper-iments with the photon collider. This image was created by K. Haupt in a ray-tracingsimulation. Green, dark blue, cyan and red lines indicate positron trajectories of differ-ent energy generated in the laser focus. The positron trajectories are bent towards thetop by the magnetic field where they can be guided to a detector by quadrupole magnets.

exceeding 100 keV.

5.2 Electron collider and positron production

Ultrashort laser pulses have proven to efficiently accelerate electrons and even gen-erate quasi-monoenergetic electron beams. The next generation of high-power lasersystems aims at generating shorter pulses at higher intensities. These relativisticfew-cycle pulses are predicted to produce monoenergetic electron pulses of up toGeV energy. The scaling laws governing the underlying processes have been studiedthoroughly in PIC simulations and are thought to be reliable since the simulationsalso explain the recent results in the transition regime between LWFA and bubbleacceleration (see Sec. 2.3).

Having these ultra-short pulsed particle sources at hand it seems logical to conductelectron collider experiments. When relativistic electrons collide, electron-positronpairs may be generated. The photon collider may therefore serve as a positron

87


source. In collaboration with D. Habs and his group from Ludwig-Maximilians-Universität, Munich, a dipole magnet assembly was designed, a so called “mini-orange”, a common tool in high-energy physics. A 3D-view of the mini-orange isshown in Fig. 5.1. It enables us to guide positrons generated in the laser focus ontoa detector which may be placed at an arbitrary distance from the interaction region.

According to calculations of D. Habs and his group the number of positrons gen-erated in an electron collider experiment provides a measure of the luminosity L ofthe electron bunches. The luminosity of two electron bunches with electron numbersN1 and N2 and bunch radii r1 and r2 is given by

L =N1N2

r1r2

frep,

where frep is the laser repetition rate. The luminosity is a quantity that is notexperimentally accessible otherwise. Preliminary experiments will be carried out atthe Jena laser facility in order to test the detector assembly and to demonstratethe production of positrons. For our laser parameters the generation of about onepositron per laser shot is predicted.

5.3 Non-linear QEDThe electric fields which can be generated with ultra-high intensity lasers are thehighest fields achievable in a laboratory to date. Intensities in the laser focus ofup to 1021 W/cm2 correspond to electric field amplitudes of 1012 V/cm. The rapidprogress towards these extreme magnitudes (which may cautiously be extrapolatedinto the future) incited numerous discussions about phenomena of non-linear QEDwhich may come into reach using high-intensity laser technology.

A prominent effect of non-linear QED is the electron-positron pair productionin vacuum by a strong electric field which was first considered by Sauter [110].Schwinger was the first to derive the formula for the probability of pair creationfor the case of a static electric field [111]. According to Schwinger the probabilityfor pair creation reaches its optimum value approaching the critical electric fieldEcrit = 1.3 × 1016 V/cm which in turn would correspond to a laser intensity of5× 1029 W/cm2. This is a parameter regime which even the most optimistic augursof high-intensity laser physics do not envision [1]. Until now experiments involving

88


non-linear QED effects have been carried out using high-energy Compton backscat-tered photons from GeV electron beams to demonstrate vacuum pair creation andnon-linear Compton scattering [7, 75]. Other experiments targeting the low energyphoton-photon scattering in the optical domain yielded upper limits for the value ofthis cross-section [112, 113].

The extraordinary properties of counter-propagating, focused intense laser pulsesforming a standing wave have attracted attention from many theoreticians to care-fully evaluate pair creation in time-dependent electric fields [114–118]. The resultsare - to say the least - diverse, and this is a motivation to pursue the realization ofan experiment exploring pair creation in vacuum with optical photons. The photoncollider is the ideal tool to create standing, highly focused and intense electromag-netic waves. Other technological issues remain to be solved, e. g. the particle densityin the interaction region must be low, so low that not a single atom is found in thefocal region during the time of interaction. Non-linear QED with the photon collideris clearly not a short term project and probably not a project leading to a positiveresult at that, but the technology developed and tested with nowadays “moderate”laser intensities may be used successfully with tomorrow’s new laser generation.

89

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