High Harmonic Generation for the XUV seeding experiment at FLASH

for the

XUV seeding experiment

at FLASH

Dissertation

zur Erlangung des Doktorgrades

des Fa hberei hs Physik

der Universität Hamburg

vorgelegtvon

Manuel Mittenzwey

Hamburg

Prof. Dr. FlorianGrüner

Guta hter der Disputation : Prof. Dr. Markus Dres her

Dr. Jens Osterho

The Free ele tron LASer in Hamburg (FLASH), operating inthe Self-Amplied

Sponta-neous Emission (SASE) mode, is urrently the most intense femtose ond light sour e in

the eXtreme-UltraViolet (XUV) regime. However, the statisti al nature of SASE leads

to intensity u tuations of the Free Ele tron Laser (FEL) pulses. Moreover, the

ele -tron a elerationpro essintrodu esarrival-timeu tuationsofthe ele tron bun hatthe

undulator entran e, whi h leads to a temporal jitter with respe t to the

syn hroniza-tion system. This jitter limitsthe resolution of orresponding pump-probeexperiments.

In order to redu e these pulse-to-pulse u tuations, a seeding se tion for the ele tron

bun h has been installed in 2009/2010. Here, XUV seed pulses from a High-Harmoni

Generation (HHG) sour e are overlapped in spa e and time with the ele tron bun hes

in new variable-gap undulators installed upstream the existing SASE undulator. Behind

the undulator se tion, the seeded FEL radiation will be dee ted by a mirror into an

experimental ontainerfor beam hara terization. Afra tion ofthe opti allaser, used as

the drivinglaser forHHG,isalsotransported intotheexperimental ontainer,thus,

tem-poraldiagnosti slike ross- orrelationsbetween thesyn hronized seededFEL andopti al

pulses arepossible. Emphasisof this workisthe onstru tionand hara terization ofthe

HHG sour e.

Referat

Der Freie-Elektronen LASer in Hamburg (FLASH) arbeit zur Zeit in einem Modus, der

die spontane Undulatorstrahlung selbst verstärkt (SASE). Die resultierende Strahlung

liegt im extremultra-violetten (XUV) Spektralberei h und istdie stärkste Li htquelle in

diesemWellenlängenberei hüberhaupt. DasVerstärkungsprinzipSASEisteineMethode,

die das Anfangsraus henverstärktund dadur hintrinsis hstatistis henFluktuationenin

der Intensität und im Spektrum unterliegt. Zusätzli h werden dur h den

Bes hleuni-gungsprozess der Elektronen Ankunftzeituktuationen relativ zu einem

Syn hronisation-slaseramUndulatoreingangeingeführt,diesi hdur hAnkunftzeituktuationender

resul-tierendenLi htpulsebemerkbarma hen. DieseS husszuS hussFluktuationenreduzieren

die zeitli he Auösung von so genannten Pump-Probe Experimenten. Um diese

Fluktu-ationen zu reduzieren, wurde 2009/2010 eine seeding-Sektion in FLASH (sFLASH)

im-plemeniert. Hierbeiwerden Li htpulse einerHohe-Harmonis hen Quelle(HHG)in Raum

und Zeit mit den Elektronenpaketen überlagert und ans hlieÿend in den sFLASH

Un-dulatoren verstärkt. Na h der Undulatorstre ke wird die verstärkte Strahlung aus dem

FLASH-Tunnel in einen Experimentier ontainer reektiert. Zusätzli h wird ein Teil des

für die HHG Quelle benötigten Fundamentallasers direkt in den Experimentier ontainer

geführt, so dass zeitaufgelöste Kreuzkorrelationsexperimentemitdem Fundamental-und

der verstärkten XUV- Laserstrahlung dur hgeführt werden können. S hwerpunkt dieser

DieseArbeitistvonmirselbstständigverfasst wordenund i hhabekeineanderen alsdie

angegebenen Quellen und Hilfsmittelverwendet.

1 Introdu tion 1

1.1 Highharmoni generation . . . 1

1.2 Self-amplied spontaneous emission . . . 2

1.3 HHG vs. SASE . . . 3

1.4 Seeded FELs . . . 3

1.5 Thesis stru ture . . . 4

2 Fundamentals 5 2.1 Des riptionof laser pulses . . . 5

2.1.1 Geometri al opti s . . . 5

2.1.2 Gauss opti s . . . 6

2.2 XUV opti s . . . 8

2.3 HighHarmoni Generation . . . 9

2.3.1 Three-step model . . . 9

2.3.2 Quantum me hani al des ription . . . 12

2.3.3 Phase mat hing . . . 14

2.3.4 Properties of the harmoni s . . . 16

2.4 Free Ele tron Laser . . . 17

2.4.1 Ele tron sour eand a eleration . . . 18

2.4.2 Undulator radiation. . . 18

2.4.3 High-gain free-ele tron laser . . . 20

2.4.4 FEL operationmodes. . . 21

3 Experimental environment 25 3.1 seeded Free ele tron LASer inHamburg (sFLASH) . . . 25

3.1.1 FLASH fa ility . . . 26

3.1.2 sFLASH . . . 28

3.2 HighHarmoni Generation . . . 32

3.2.1 The laser system . . . 32

3.2.2 High harmoni generation sour e . . . 33

3.3 Diagnosti s . . . 39

3.3.1 Chara terization of the driving laser . . . 39

3.3.2 XUV opti s . . . 39

3.3.3 Separation of XUV radiation. . . 42

3.3.4 Chara terization of XUV radiation . . . 43

4.1 Fundamentallaser . . . 51

4.1.1 Laser pulse duration . . . 51

4.1.2 Laser beam proleand intensity . . . 51

4.1.3 Polarization . . . 55

4.1.4 Wavefront . . . 56

4.2 Calibration of the spe trometer . . . 56

4.2.1 Online alibration . . . 56

4.2.2 Quasi online alibration . . . 56

4.2.3 Absolute alibration . . . 57

4.3 HighHarmoni Generation . . . 59

4.3.1 XUV beam propagation . . . 60

4.3.2 Harmoni energy and onversion e ien y . . . 64

4.3.3 Harmoni spe trum- plateauand uto . . . 75

4.3.4 Wavefront . . . 79

4.3.5 Stability . . . 79

4.3.6 Quasi phase mat hing . . . 79

4.3.7 Two olor high harmoni generation . . . 81

4.4 Seeding FLASH . . . 83 5 Con lusion 85 5.1 Outlook . . . 88 5.1.1 Seed power . . . 88 5.1.2 Inje tion beamline . . . 89 5.1.3 Alternativeseeding . . . 90 A Abbreviations I

B Propagation of un ertainties III

C Te hni al details V

C.1 Software environment . . . V

C.2 Hardware . . . VII

D Code of pra ti e XI

D.1 The laser system . . . XI

D.2 Highharmoni generation sour e . . . XII

1.1 Peak brillian eof dierentlight sour es . . . 2

2.1 Cal ulated beam width . . . 7

2.2 Simple man'smodel. . . 10

2.3 Typi alHHG spe trum . . . 16

2.4 S hemati representation of anundulator . . . 19

2.5 GENISIS simulationof the mi robun hing pro ess . . . 21

2.6 Exponentialgrowth of the FEL pulse energy . . . 22

3.1 Free Ele tron LASer Hamburg at DESY . . . 25

3.2 S hemati representation of FLASH . . . 26

3.3 Overview of sFLASH . . . 28

3.4 Cross se tion of the inje tion beamline . . . 30

3.5 S hemati sket h of the laser system . . . 32

3.6 CAD model of the pit . . . 34

3.7 CAD model of the HHGsour e . . . 35

3.8 CAD model of the HHGtarget hamber . . . 36

3.9 AlternativeCAD model ofthe HHG target hamber. . . 38

3.10 Ree tivity of B

4

C for p-and s- polarized light. . . 40

3.11 CAD model of the tripletmirror system . . . 41

3.12 Ree tivity of S Si multilayer . . . 42

3.13 CAD model of the omplete HHG setup . . . 43

3.14 Transmission of the used metal lters . . . 44

3.15 CAD model of the diagnosti bran h. . . 46

3.16 Simulationof dierentspe trometer ongurations . . . 47

3.17 Simulationof dierentslits infrontof the spe trometer . . . 48

4.1 Measured spe trumand pulse length of the driving laser. . . 52

4.2 Measured fareld beam proleof the driving laser . . . 52

4.3 Beamsize and energy of the driving laser . . . 54

4.4 Absolute alibrationof the

21

st

harmoni . . . 57

4.5 Spe tra for two dierent spe tral positions of the harmoni omb. . . 58

4.6 Calibrated spe trum . . . 59

4.7 H

21

near eld beam prole (

f

NIR

= 1500

mm). . . 60

4.8 H

21

near eld beam prole (

f

NIR

= 3000

mm). . . 61

4.9 Fo us s an of H

21

. . . 62

4.10 H

21

yielddependent on the position of the gas target.. . . 65

4.12 H

21

yielddependent on the driving laser intensity . . . 67

4.13 Yieldof dierent targetgeometries . . . 68

4.14 Measured targetdensity of the gas valve . . . 69

4.15 Imaged damageof the rst tripletmirror.. . . 70

4.16 HHG signal dete ted with the HHmeter. . . 73

4.17 Correlation between argon and helium for the HHmeter signal . . . 75

4.18 Singleshot HHG spe trum . . . 76

4.19 HHG spe tra generated with a basi gas nozzle. . . 77

4.20 Spe tra for dierent pulse lengthsof the driving laser . . . 77

4.21 Optimized HHG spe trumfor

f

NIR

= 1500

mmand

f

NIR

= 3000

mm . . . . 78

4.22 S hemati representation of QPM. . . 80

4.23 Ee t of hydrogenon quasiphase mat hed HHG. . . 81

4.24 S hemati illustrationof two olor HHG . . . 81

4.25 Two olor HHG spe tra. . . 82

4.26 HHG tunnel spe trum . . . 83

C.1 Javadoo sdata display . . . VI

2.1 Harmoni orders . . . 9

3.1 Performan e of FLASH . . . 27

3.2 Partial ross se tion of argon. . . 45

4.1 Divergen e of H

21

. . . 63

4.2 Energy of H

21

. . . 76

5.1 Published spe i ations of HHG in argon . . . 86

A.1 Abbreviations . . . I

A.2 Physi al onstants . . . II

C.1 Te hni al data Basler slA1390-17fm . . . VII

C.2 Te hni al data Beta BariumbOrate . . . VIII

C.3 Te hni al data multi- hannelplate . . . VIII

C.4 Te hni al data GrenouilleFROG . . . VIII

C.5 Te hni al data HHmeter . . . VIII

C.6 Te hni al data XUV CCD . . . IX

C.7 Te hni al data wavefront sensor . . . IX

C.8 Te hni al data XUV dira tion grating . . . X

The free-ele tron laser in Hamburg (FLASH) is a user fa ility providing highly brilliant

ultrashort eXtreme UltraViolet (XUV) radiation to users. The established operating

mode SASE (Self-Amplied Spontaneous Emission) shows shot-to-shot u tuations in

the spe trum, the pulse prole, and the arrival time of the radiation. In ontrast, high

harmoni radiation is omparably stable, but with pulse energies orders of magnitude

lower than SASE. Seeding FLASH (sFLASH) with a high harmoni of an external laser

is a method to ombine the advantages of high harmoni generation (HHG) and FELs

in order to improve the performan e of the fa ility. In a rst stage of sFLASH, the

21

st

harmoni of a titanium-sapphire (Ti:Sa) laser is used for seeding. Between 2009 and 2010 the user operation of FLASH was stopped to implement the new 50 m long

sFLASH se tion. Here,the ele tronbun hes are overlappedwith theXUVhighharmoni

radiation in order to initiate the ampli ation pro ess. The amplied radiation enables

thenthepossibilityfortime-dependentexperimentswithoutthela koferrati u tuations

in the time and spe tral domainand without arrival timeu tuations between the XUV

radiation and the external laser. This hapter introdu es the main features of HHG and

the ommon Free-Ele tron Laser (FEL) operation mode SASE. It further des ribes the

ideahowto ombinetheadvantagesofea hte hnique. Finally,thestru ture ofthis thesis

is presented.

1.1 High harmoni generation

HHG is a method to generate XUV and x-ray radiation. Therefore, a femtose ond laser

pulse is fo used into a noble gas target. The orresponding laser eld bends the atomi

potentialof thenoblegasatoms. Thisdeformationleadstoanin reasedpossibilityofthe

valen e ele trons to tunnel through the atomi barrier. The subsequent motion is then

determined by the laser eld. The ele trons are a elerated by the laser eld and have a

small possibility to re ombine with their parent ion. The gained energy is then emitted

in terms of XUV orx-ray photons [Winterfeldt et al., 2008; Changand Corkum, 2010℄.

HHG is a oherent pro ess where a fra tion of the driving laser is onverted into odd

multiples of the fundamental frequen y. The onversion e ien y (CE) of this pro ess

limitsthepulseenergyofthe harmoni stothenanojouleorsub-mir ojouleregime. Inthe

timedomainthehighharmoni radiation onsistsofaburstofattose ondpulses

[Papado-giannisetal.,1999;Dres heretal.,2001;Hents heletal.,2001;Agostini,2004℄andenables

real-time observation of atomi -s ale ele tron dynami s [Dres her et al., 2002; Corkum

and Krausz, 2007℄. The driving laser and the XUV radiation are syn hronized

peak br

illianc

e (phot

ons/(s mr

ad mm 0.1% b

w))

2

2

energy (eV)

Figure1.1: Peak brillian eof dierent lightsour es. Courtesyof S. S hreiber, DESY.

limitedbytheirpulseduration. Duetothespe tralandtemporalstabilityhighharmoni s

arethetoolof hoi eformonitoringatomi -s aledynami s[T.Pfeifer,2006℄. Pump-probe

experiments [Uphues et al., 2008; Allison, 2010℄ and dira tive imaging [Ravasio et al.,

2009;Sandberg etal.,2007℄for instan e are some appli ationsof HHG.

1.2 Self-amplied spontaneous emission

The most ommon operation mode of an FEL is Self-Amplied Spontaneous Emission

(SASE), produ ing highly-brilliant radiation down to the XUV and x-ray regime. In

ontrast tosyn hrotrons the light pulsesof anFEL have a mu hhigher brillian e(gure

1.1) and are oherent. The pulse duration is in the femtose ond regime and is mu h

shorter than the pi ose ond pulses delivered by syn hrotrons. The pulse energy ex eeds

those of HHG by up toveorders of magnitude.

The radiation is suited for oherent x-ray dira tion imaging [Chapman et al., 2006;

Boganetal.,2008℄,inlineholography[Rosenhahn etal.,2009℄,Fourier-transform-

holog-raphy [Boutet etal., 2008; Man uso etal., 2010; Mar hesini et al.,2008℄, oherent x-ray

experi-ments [Krikunova etal., 2011;Chapman et al.,2007; Barty et al.,2008℄.

Unfortunately,thegenerationofthelightpulsesisbasedonastatisti alpro ess,leading

to errati u tuations of the pulse energy and the spe tral and temporal domain

[A k-ermann et al.,2007℄. Moreover, the ele tron a eleration pro ess introdu es arrivaltime

u tuations of the ele tron bun h at the undulator entran e, whi h leads to a temporal

jitterwithrespe t toexternallaser pulses. Thus, theresolutionof time-dependent

pump-probe experiments is redu ed [Dres her et al., 2010; Maltezopoulos et al., 2008; Azima

et al., 2007; Rad lie etal., 2007℄.

1.3 HHG vs. SASE

Both,HHG andSASE arete hniques togenerate XUV radiationfortime-resolved

exper-iments. HHG is able to provide oherent radiation in the nanojoule regime. In ontrast

the pulse energy of FLASH is up to

300 µ

J [S hreiber, 2011b℄. Due to the sto hasti natureof SASEthe radiationdiers initsspe trumand intensity onashot-to-shotbasis.

These jittersour es are absent inHHG.

In HHG the radiation onsists of odd multiples of the driving laser frequen y. The

lowestprodu edwavelengthdependsonthefundamentalwavelengthandthetargetwhi h

usually isanoblegas. Itisfurtherpossibletoshiftand/orbroaden thespe trumslightly

at the expense of the pulse energy. In an FEL the wavelength depends on the ele tron

energy and onthe

K

-parameterof the used undulators and an be hanged ontinuously. FLASH for instan e an produ e radiation with

4.1

nm

< λ < 47

nm [S hreiber et al., 2011;Tiedtke etal.,2009℄.

Thepulse lengthofthe FEL radiationdepends ontheele tron bun h anditispossible

to hange the pulse length of FLASH between

10

fs

< ∆t < 200

fs (rms) [S hreiber, 2011b℄. In ontrast, the pulse length of the HHG related radiationis determined by the

pulse lengthof the driving laser. Furthermore, the driving laser pulse length has a large

impa t on the CE. Thus, the pulse length of the harmoni s is more or less xed. On

the other hand, the arrival time jitter of the radiation is only determined by the jitter

of the driving laser. In ontrast to an FEL, where the a eleration pro ess introdu es

arrival time u tuations in the order of

FWHM

≈ 80

fs (FullWidth at Half Maximum) [S hreiber,2011b℄. Thus, thetemporaljitterofanFELismu hlarger omparedtoHHG.

1.4 Seeded FELs

Seeding anFELis analternativete hnique toSASE, wherethelasing pro ess isinitiated

by an external laser and not from spontaneous emission. The external laser introdu es

a density modulation of the ele tron bun h leading to oherent radiation. In this ase,

only a fra tion of the ele tron bun h is being modulated, thus the pulse length of the

orresponding radiationdepends on the seed pulse length. In addition,the ampli ation

pro esstakespla eonlywithinthepulselength. Hen e,thearrivaltimeu tuationsofthe

jitter issmaller than the width of the ele tron bun h.

The wavelength of the radiationis determined by the seed, thus, itis an odd multiple

of the HHG driving laser. The pulse energy of a seeded FEL should be omparable to

SASE radiationinthe mi rojouleregime [Milt hevetal., 2008℄.

In summary, a seeded FEL ombines the advantages of the stability of HHG and the

highpowerofSASE.Therefore,oneisabletoimprovethetimeresolutionof orresponding

experimentsand to enhan e the energy resolution of wavelength-dependent experiments.

So far FLASH is operated in the SASE mode. This thesis des ribes the joint venture

between the University of Hamburg and DESY, where the FLASH fa ility is upgraded

to a seeded FEL (sFLASH). It is meant to get further insights of the pe uliarities of a

seeded operationmode and is arst step towards a seeded user fa ility(FLASH II).

1.5 Thesis stru ture

This thesis is separated into ve hapters. The following hapter des ribes the

funda-mental physi al on epts of HHG and FEL theory ne essary to understand this thesis.

In hapter 3 the experimental environment, i.e. FLASH, sFLASH, and the onstru ted

harmoni sour e are des ribed. Chapter 4 is the main part of this thesis and des ribes

the hara terization of the harmoni sour e and the a hievements of the seeding

experi-ment at FLASH. Chapter 5 on ludes and presents an outlook of sFLASH. Finally, the

This hapter is addressed to the fundamentals of this thesis. Primarily the theoreti al

on ept of HHG and FEL theory willbe presented.

2.1 Des ription of laser pulses

The behavior of laser pulses is based on opti s and has been investigated thoroughly in

the past. A omprehensive des ription an be found in [Saleh and Tei h, 1991; He ht

and Zaja , 1997; Frank L. Pedrotti, 1987; L.Bergmann et al., 2004℄. This hapter only

a ounts for the fundamentals and summarizes the ne essary issues to understand the

following hapters.

2.1.1 Geometri al opti s

The simplest way to des ribe the behavior of lightis interms of geometri al opti s. The

theory is based on a few postulates as des ribed for instan e in [Saleh and Tei h, 1991℄.

It is assumed that light travels in form of rays in a medium whi h is des ribed by the

refra tive index. The path of the rays is determined by Fermat's prin iple: "Light rays

travel along the path of least time". Based on these three prin iples the propagation of

light rays is determined.

Forrayshavingasmall angle

ϑ

tothe opti alaxisofanopti alelement,their behavior is des ribed by the paraxial approximation: sin

(ϑ)

≈ ϑ

. One onsequen e is the image equation for athin lens ora spheri al mirror:

1

b

+

1

g

=

1

f

,

(2.1)

where

g

isthe distan e fromthe obje ttothe lens,

b

is the distan e fromthe lens tothe image, and

f

is the fo al length of the lens. The equation an be employed to al ulate the position of an image depending on the distan e of the obje t to the fo using opti .

Theparaxialapproximationla ksonthefa tthataberrationsasforinstan e astigmatism

annot be des ribed. Astigmatismis indu ed, whenever the lightrays have a large angle

ϑ

tothe opti alaxis of the fo usingelement. In this ase, two linefo iappear instead of one pointfo us. Thelinefo iinthesagittaland meridionalplaneare separated inspa e:

f

m

= f

·

sin

(ϑ)

f

s

= f

·

1

sin

(ϑ)

where

f

m

, and

f

s

are the ee tive fo al lengths of the meridional and sagittal plane, respe tively.

In general, geometri al opti s is an idealized model and implies that

λ → 0

. It is not possible to des ribe ee ts like interferen e or dira tion. To a ount for the wave

nature of light the ele tromagneti eld theory has to be used. Some major results will

bepresented withinthe next se tion.

2.1.2 Gauss opti s

Light an be des ribed with the Maxwell equations and it is possible to dedu e simple

statements by employingsome assumptions. Firstly, the wave equation

∇

2

~E =

_c

1

2

∂

2

~E

∂t

2

,

(2.3)

with

~E

theele tri eld,

c

thespeed oflight,and

t

the timeisadire t onsequen e ofthe Maxwell equations. Foranele tromagneti wave

~E(~r, t) = E(~r)T(t)

equation 2.3 an beseparated in spa e and time and an be rewritten as:

∇

2

_{+ k}

2

_E(~r)

_{= 0,}

(2.4)



d

2

dt

2

+ ω

2



T

(t)

= 0,

(2.5) with

ω = kc,

and

k =

2π

λ

,

where

ω

istheangularfrequen y,

k

isthewaveve tor,and

λ

isthewavelength. Equation 2.4 is known as the Helmholtz equation. Assuming an ele tromagneti wave traveling

alongthez-dire tionwithaslowlyvaryingenvelope(paraxialapproximation),onesolution

of equation 2.4 orresponds toa Gaussianbeam [Ja kson, 1999℄ with

w(z) = w

0

s

1 +

 z

z

R



2

(2.6)

R(z) = z



1 +



z

R

z



2



(2.7)

ζ(z)

=

tan

−1

 z

z

R



(2.8)

w

0

=

r

λz

R

π

,

(2.9)

where

w(z)

isthe beamwidth depending onthe spatial oordinate

z

and

z

R

the Rayleigh length.

R(z)

orresponds to the wavefront radius of urvature,

ζ(z)

represents the phase

−200

−100

0

100

200

0

20

40

60

80

100

120

spatial coordinate z (mm)

beam width w (µm)

_{2 z}

R

w

0

4 z

R

Figure 2.1: Cal ulatedbeamwidth for aGaussianbeam with

λ = 38.1

nm,

w

0

= 40 µ

m, and

z

R

= 132

mm.

retardation of the beam relative to a uniform plane wave, and

w

0

is the minimal beam widthatitswaistdependingonthe wavelength

λ

andontheRayleighlength

z

R

. Ingure 2.1 the al ulated beam width

w(z)

is shown for a Gaussian beam with

λ = 38.1

nm

and

w

0

= 40 µ

m. In this ase the beam width

w

at position

z

is dened in terms of

the intensity: the Gaussianbeam has its peakintensity onaxis forevery position

z

. The beamradius

w

, however, is dened asthe radialdistan e tothe opti alaxisatwhi hthe intensityisredu ed to

1/e

2

ofitspeakintensity. AlsotheRayleighlengthdepends onthe

beam waistand refers tothe distan e

z

atwhi h the fo alspot area istwi e asbig asat its waist

w(z = 0) = w

0

,i.e. the beam radius is

√

2w

0

. The intensity at that positionis

I(z =

_±z

R

) = 0.5

· I(z = 0).

Dependingonthe ontextdierentdenitionsofthebeamwaistareused. Asexplained

before

w

orresponds tothe radius,where the intensity de reases by a fa tor of

1/e

2

. In

ontrast, the

FWHM

value of a beam is dened as the diameter of the beam at whi h the intensity dropsdown by afa tor oftwo. The rms width

σ

of abeam isdened asthe mean of the photon-distribution. In ase of a Gaussianbeam the rms-width (root mean

square) ishalf asbigasthe

w

-radius(

w = 2σ

). The dierent denitionsof the Gaussian beam widthsare onne tedas follows:

FWHM = 1.177w = 2.35σ

Besides, gure 2.1 indi ates that the beam width in reases linearly in the near eld.

For

z

≫ z

R

equation 2.6 an berewritten as

w(z) =

w

0

z

R

· z

Thedivergen e,i.e. halfopeningangleofthe one anbeexpressedby ombiningequation 2.9 and 2.10.

ϑ =

w

0

z

R

=

λ

πw

0

,

(2.11) with tan

(ϑ)

≈ ϑ.

The phase retardation

ζ(z)

of a Gaussian beam (equation 2.8) hanges its sign at the position of the waist [Gouy, 1890℄. This ee t is known as the Guoy phase-shift

∆ζ = π

between the near eld on ea hside ofthe fo us.

Stri tlyspeaking, the aboveequationsare onlyvalidfora GaussianTEM

0

inx- and

y-dire tion, respe tively. [Siegman, 1993℄introdu ed the beam qualityfa tor

M

2

toextend

the above equations. The beam quality- or beam propagation- fa tor is dened as the

times dira tion limited (TDL) value for an arbitrary real beam ompared to a TEM

0

Gaussian beam". It is further shown that the above equation have to be modied in a

way that

λ → M

2

_λ

, where

M

2

≥ 1

in order to des ribe modes deviating from TEM

0

mode in x- and y- dire tion. During this work the modied beam width, as well as the

modied Rayleighlength is being used. The equations 2.6 and 2.9 an berewritten as

W(z) = W

0

s

1 +

 z

z

R



2

(2.12) with

z

R

=

πW

2

0

M

2

_λ

,

(2.13)

where

W

0

is the modied beam waist. During this work laser pulses are assumed to be Gaussian in the spatial and spe tral domain, unless it is expli itlynoted to be dierent.

Adetaileddedu tionofGaussianbeamproperties anbefoundin[Siegman,1986℄. Laser

related fundamentalsare further dis ussed in[Svelto, 1982;Rulliere,1998℄.

2.2 XUV opti s

The XUV regime is in the range of

10

nm

≤ λ ≤ 120

nm. Unfortunately, there are no existing materialshaving a high ree tivity within this regime. Most of the radiation is

beingabsorbedwithinapenetrationdepthoflessthan

1 µ

m,leadingtotypi alree tivity of

R

≤ 10

−4

at normal in iden e [Jaeglé, 2006℄. A solution to over ome this pe uliarity

are grazing in iden e mirrors, as well as normal in iden e multilayer mirrors. Grazing

in iden e mirrors have a large bandwidth ompared to multilayer mirrors. As explained

in hapter 2.1.1a fo usingopti employed ata grazingin iden e angle leads toan

astig-matism,whi hisusuallynotwanted. Thus, grazingin iden emirrorsare ommonlyused

for simple (non-imaging)ree tions. In ontrast, multilayer mirrors an be employed at

anglesaroundzero degree. Thosemirrors onsistof aperiodi thin layerstru tureontop

of the substrate, leadingto onstru tiveinterferen e of the ree ted XUV radiation. The

a hieved bandwidth is small ompared tothe grazing in iden e mirrors, but the fo used

found in [Jaeglé, 2006; Attwood, 2000℄. [Lawren e Berkeley National Laboratory, 2011℄

providesaweb-interfa eto al ulatetheree tivityandtransmissionofvariousmaterials.

A report ondira tiongratingsinthe XUVregimewhi hareused inthis thesishas been

givenby[S hroedter,2009℄andbasi gratingtheory anbefoundin[PalmerandLoewen,

2005℄.

2.3 High Harmoni Generation

HHG is a oherent nonlinear intera tion of light and matter [Shen, 1984; Menzel, 2001℄:

if a high intensity laser is fo used into a gas target, the atoms emit light pulses of odd

multiplesofthe in identfrequen y [W.F.Drake,2006;Jaeglé,2006℄. Therstobservation

of opti al harmoni s (se ond harmoni generation) has been des ribed in 1961 [Franken

etal.,1961℄oneyear aftertheinventionofthe laser[Maiman,1960℄. Firstexperimentson

low-orderharmoni generationingashavebeenpublished in[New andWard,1967;Ward

and New, 1969℄, whereas the rst high-order harmoni s have been observed in the late

1980s[M Phersonetal.,1987;M.Ferrayetal.,1988℄. Earlypublishedresultsarereviewed

in [Huillieret al., 1991;Gavrila, 1992℄. More re entreviews an befound in [Winterfeldt

et al., 2008; Brabe and Krausz, 2000℄. In table 2.1 a sele tion of harmoni s generated

withan

800

nmTi:Salaserare shown,whereH

21

willdrawanin reasedattention during the following work.

Table 2.1: Sele ted harmoni orders indierent units.

harmoni order

λ

(nm)

E

photon

(eV)

H

01

800

1.55

H

09

88.9

13.95

H

11

72.7

17.05

H

13

61.5

20.15

H

19

42.1

29.45

H

21

38.1

32.55

H

23

34.8

35.65

H

61

13.1

94.54

2.3.1 Three-step model

A simple empiri model has been developed to des ribe the pro ess of HHG [Corkum,

1993;Kulanderetal.,1993℄. Itiswidely knownasthesimpleman'smodel,thethree-step

model, or in the 1990s asthe two-step quasi- lassi al approa h [L'Huillier etal., 1993℄.

The model des ribeshowthe bound ele tronsof the gasatoms tunnelthrough itsatomi

barrier. In the se ond step the ele trons move lassi ally and may return tothe nu leus.

Thoseele tronsthatre ombinetothegroundstateemitharmoni sasillustratedingure

(a)

(b)

(c)

Figure 2.2: S hemati representation of the three-step model: (a) represents the tunnel

ionization, (b) the a eleration inthe ontinuum, and ( ) the re ombination.

First step: ionization

Wheneveralaserisfo useddown tointensitiesinthe orderof

10

15

W/ m

2

the amplitude

ofthe orrespondingele tri eldapproa hes

10

9

V/ m. Theseeldstrengthsex eedthat

of anatomi Coulombeld, leadingto adeformationof itsatomi potential. Hen e, this

deformation in reases the possibility for a bound ele tron to tunnel through its atomi

barrier. Tunnel ionization has rstly been des ribed for hydrogen [Keldysh, 1665℄ and

was extended toa generalizedtheory forarbitrary atoms [Ammosov et al., 1986℄.

Insummary,thetheoryintrodu estheKeldyshparameter[MiyazakiandTakada,1995℄:

γ =

s

I

p

2U

p

,

where

I

p

is the ionization potential of the gas and

U

p

is the ponderomotive potential. The ponderomotive potential orresponds tothe averagedenergy of the ele tron's quiver

motion in an ele tromagneti eld and isdire tly proportional to the wavelength square

of the driving laser. It is dened as

U

p

=

e

2

2m

e

ǫ

0

cω

2

I

_{∝ Iλ}

2

,

where

e

is the elementary parti le harge,

m

e

the ele tron mass,

ǫ

0

the va uum permit-tivity,

c

the speed of light,

ω

the angular frequen y,

I

the laser intensity, and

λ

is the wavelength of the laser. However, for

γ > 1

the ionizationtakes pla e interms of multi-photon ionization [Lin, 2006℄ and for

γ < 1

through tunneling. Multi-photon ionization is a simultaneous absorption of several photons, leading to the ionization of the atom

[Mainfray and Manus, 1991℄.

The numberof ele trons

N

e

an be dedu ed asin [Miyazakiand Takada, 1995℄ and is des ribed by the rate equation

where

N(t)

isthetimedependentneutralatomdensity and

W(t, E)

the tunnelionization rate. The ionizationrate depends onthe eldstrength

E

and leadstoanele tron density [Ammosov etal.,1986℄

N

e

(t) = N

0



1 −

exp



−

Z

t

−∞

W(t, E)dt



,

where

N

0

is the neutral atom density at

t = 0

.

Besides, ionizationof the gaseous medium hanges the dira tive index and may lead

to self-fo usingof the driving laser whi hitself has inuen e on the HHG pro ess. HHG

in a rapid ionizing mediumhas been des ribed thoroughly in [Raeet al., 1994; Raeand

Burnett, 1993a,b℄. However, laser elds in the order of the atomi Coulomb eld of

10

8

_{− 10}

9

V/ m, ne essary to drive the HHG pro ess, are known as strong elds where

the lassi alperturbation theory is not valid[Brabe , 2008℄.

Se ond step: propagation

Assoonasthe ele tronisionizeditistreatedasafree-ele tronand itssubsequenttra

je -toryisdeterminedbythe lasereld. Dependingonthephase ofthelasereld atthetime

of the ionizationthe ele trons gain dierent energies before the ele tri eld hanges its

algebrai sign. The sign hange auses the ele tron to de elerate and then to a elerate

ba k tothe parention. The averaged kineti energyanele tron angain inanos illatory

laser eld is [Gallagher,1988; Krauseet al.,1992℄:

 1

2

m

e

v

2



= U

p

1 + 2

os

2

(ωt

0

)
,

where

v

is the velo ity and

t

0

is the ionization time. This equation indi ates that the energy an ele tron an gain is between

U

p

and

3U

p

. Classi al simulations of the prop-agation have been published in [Corkum, 1993℄ and reveal a slight modi ation of the

previously dedu ed result. It is shown that the maximal energy an ele tron an gain is

3.17U

p

. Itisalsoshownthatanele tron,whi hgainedtheenergy of

3.17U

p

hastunneled at

ωt

0

= 17

◦

or

197

◦

of the driving laser. Thus, the maximal energy anele tron an gain

o urs twi e a laser y le. Here it is perspi uous that for an ellipti al polarization of

the driving laser the ele tron never returns to the parent ion. Thus, the CE of HHG is

strongly dependent onthe ellipti ityof the driving laser polarization[Budil et al., 1993;

Dietri h etal., 1994;S hulze etal., 1998℄.

Third step: re ombination

Some of the free-ele trons may re ombine with their parent ionand emit aphoton with

a maximal energy of

E

max

= 3.17U

p

+ I

p

,

(2.14)

where

I

p

is the ionization potential of the orresponding gas atoms. Equation 2.14 is known as the uto law and des ribes the uto behavior of the harmoni spe trum

[L'Huillier et al., 1993℄. The smallest possible wavelength in the harmoni spe trum

depends on

•

the wavelength of the driving laser,

•

the intensity of the driving laser,

•

and the ionizationpotential of the gaseousmedium.

The larger the wavelength, intensity, and ionizationpotential,the smallergets the

wave-length of the highest harmoni order.

In summary,the three-step model is a quasi- lassi al model whi h des ribes the HHG

pro esswithinthreedistin tsteps. Aself- ontainedfullyquantumme hani aldes ription

of the HHG pro ess is given in [Lewenstein et al., 1994℄ and will be des ribed in the

followingse tion.

2.3.2 Quantum me hani al des ription

The quantum me hani al des ription of the HHG pro ess is based on the single a tive

ele tron (SAE) approximation[Kulanderand Res igno,1991℄ andis known asthe strong

eld approximation (SFA) or as the Lewenstein model [Lewenstein et al., 1994℄. In the

SAE approximation it is assumed that the bound state wave fun tion evolves into a

ontinuum wave pa ket. One part of the wave pa ket willnever return, whereat another

part will return to the ioni ore. Here dierent possibilities a probable. Either the

ele tron s atters o, or hanges the dire tion to gain energy, or overlaps with the bound

state, whi h leads to a time dependent dipole moment and photo emission. The system

an bedes ribed with the orresponding time-dependent S hrödinger(TDSE) equation

i

h

∂

∂t

|Ψ(

~r, t)

i =



−

h

2

2m

e

^

∇

2

+ ^

V(

~r) − ~E(t)^x



|Ψ(

~r, t)

i,

(2.15)

where

V(

^

~r)

is the atomi potential operator,

~E(t)

is the time-dependent laser eld, and

^

x

is the dipole operator. However, the solution of equation 2.15 is a superposition of bound,os illating,and outgoing wave pa kets. In orderto solve the TDSE the following

approximationshave tobe made:

•

the system evolution depends only onthe ground state

|0

i

. All other states donot ontribute and an be negle ted.

•

depletion of the ground state an be negle ted.

•

the ele tron in the ontinuum moves asa free-ele tron in an ele tri eld, thus the atomi potentialhas noee t and an be negle ted.

The Fourier transform of the indu ed dipolemoment

ex(t) =

_{hΨ(x, t)|ex|Ψ(x, t)i}

(2.16)

leads dire tly to the harmoni spe trum.

The ansatz

|Ψ(x, t)

_{i = e}

iI

p

t/

h



a(t)|0

_{i +}

Z

d

3

_{~q b(~q, t)|~qi}



,

(2.17)

where

a(t)

is the ground state amplitude and

b(

~q, t)

is the amplitude of the ontinuum states, leads to the solution of the TDSE. Be ause the depletion of the ground state is

negle ted, the amplitude of the ground state is set to

a(t)

≈ 0

,

a(t)

≈ 0

. Equation 2.16 an be evaluated by further hanging the variables to the anoni al momentum

~p = ~q + e~

A(t)/c

:

ex(t) = i

Z

t

0

dτ

 2m

e

hπ

ν + iτ



3/2

× d

∗

x

(

~p

s

− e~

A(t)/c)

· e

−iS(~p

s

,t,τ)

× E(t − τ)

os

(ω(t − τ))d

x

(

~p

s

− e~

A(t − τ)/c) + c.c.

(2.18)

with

~p

s

=

~p

s

(t, τ) =

Z

t

t−τ

dt

′

e~

A(t

′

)/cτ

and

S(

~p

s

, t, τ) =

Z

t

τ

dt

′′

(

~p

s

− e~

A(t

′′

_)/c

2

₎

2

2m

e

+ I

p

!

,

where

d

x

(

~q) =

h~q|ex|0i

is the omponent of the dipole matrix element parallel to the polarizationfor bound-free transitions,

d

∗

x

its omplex onjugate,

S(

~p

s

, t, τ)

is the quasi- lassi al a tion,

~p

s

the momentum at the stationary point of the quasi- lassi al

momen-tum, and

τ = t − t

′

is the time the ele tron is travelling in the ontinuum. Equation

2.18 an beinterpreted interms ofthe simpleman's model: thebottomline ontains the

matrix element forthe transitionbetween the bound stateand the ontinuum,

S(

p

s

, t, τ)

orresponds to the propagation operator in the ontinuum, and

d

∗

x

orresponds to the transitionfrom the ontinuum ba k intoa bound state.

A Fourier transformation of equation 2.18 al ulates the harmoni spe trum. It an

further be shown [Salieres et al., 2001℄ that the k

th

Fourier omponent orresponds to a

sum over the quantum orbits

ex

k

=

X

n

a

n

exp

(−iS(

p

n

, t

n

, t

τ

)),

i.e. the k

th

harmoni depends oninterfering ontributions of dierent traje tories

(Feyn-man's path integral). Finally, it should be pointed out that this model is also able to

reprodu e the uto law of the simple man's model [Lewenstein et al., 1994℄. Equation

2.14 has tobe modied:

where the fa tor

F(U

p

/I

p

)

is a orre tion term due to quantum tunneling and quantum diusion.

F(U

p

/I

p

)

is equal to

1.3

for

I

p

≪ U

p

and de reases to

1

as

I

p

grows.

HHGisaperiodi pro esswhi ho urseveryhalflaser y le. Moreover,one anobserve

that two ele tron traje tories ionized at dierent phases of the laser eld will result in

identi al momentaatthe timeof re- ollision[Lewensteinet al.,1995;Bal ouetal.,1999;

Milo²evi¢ and Be ker, 2002; Liu et al., 2009℄. These two traje tories are known as the

short and the long traje tory orresponding tothe time the ele tron is traveling through

the ontinuum. Sin e the pro ess takes pla e every half period of the laser y le

T/2

the Fourier transformisa dis rete fun tionwith a separation orresponding to

1/(T/2) = 2f

. That is why the spe trum onsists only of odd multiple of the frequen y of the driving

laser [Winterfeldt et al., 2008℄. This symmetry an be avoided by adding the se ond

harmoni of the driving laser tothe HHG pro ess [Mauritsson etal.,2006℄.

2.3.3 Phase mat hing

So far only a single atom has been onsidered. However, HHG is a multi-parti le

pro- ess and the olle tive ee ts have to be taken into a ount. For e ient HHG the

phase- onditions of the atoms have tobe set that the orresponding radiationadds

on-stru tively. The wave-ve tor mismat hof the

q

th

harmoni order an bewritten as

∆k = qk(ω

f

) − k(qω

f

),

(2.20)

where

k

represents the wave-ve tor,

q

is the harmoni order, and

ω

f

is the frequen y of the driving laser. Equation 2.20 and the followingis mainlytaken from[T.Pfeifer, 2006℄.

The wave-ve tor

k = k(ω)

depends on itsfrequen y. It an bewritten as

k(ω) = k

vac

(ω) + k

disp

(ω) + k

plasma

(ω) + k

geom

(ω),

(2.21)

where

k

vac

(ω) = 2πω/c

is the wave-ve tor in free spa e,

k

disp

(ω)

is the ontribution due todispersion in a neutral medium,

k

plasma

(ω)

is due to dispersion in a plasma,and

k

geom

(ω)

is due to geometri dispersion. Constru tive interferen e o urs, if the phase-mismat hvanishes:

∆k = 0.

The phase-mismat hin free spa e

∆k

vac

= q

ω

f

c

−

qω

f

c

= 0

(2.22)

is zero, thus, only the dispersion due to neutral atoms, plasma, and geometri al ee ts

havetobe onsidered. Asthetherefra tiveindexofamediumdependsonthewavelength

the phase-mismat hdue to dispersionin aneutral medium an be expressed as

∆k

disp

(ω

f

) = (n(ω

f

) − n(qω

f

))

qω

f

c

,

(2.23)

where

n

isthe refra tiveindex. This ontributiontothephase-mismat hdependsonlyon the dieren eof the refra tive indi es. In ase of the

800

nm driving laser the refra tive

indexislargerthanoneandsmallerthanonefortheXUVradiation,thus,thewave-ve tor

mismat his apositivevalue.

As explainedin the previous se tion onlya fra tion of the ionized ele trons re ombine

and emit harmoni radiation. Themajority ofthe ele tronsremain ionizedand lead toa

hangeof the refra tive index:

n

plasma

(ω) =

r

1 −



ω

p

ω



2

with

ω

p

=

s

e

2

_N

e

ǫ

0

m

e

,

where

ω

p

isthe plasmafrequen y. The orresponding plasma ontributionto the phase-mismat h an be al ulated:

∆k

plasma

(ω) = qk

plasma

(ω

f

) − k

plasma

(qω

f

) =

ω

2

p

(1 − q

2

)

2qcω

f

(2.24) and is

∆k

plasma

(ω) < 0

for

q > 1

.

Thelast ontributioninequation2.21dependsonthefo usingpropertiesofthedriving

laser. As explained in hapter 2.1.2 the fo used Gaussian beam ontains an additional

phase ontributionalongthez-dire tionwhi hisknown astheGuoyphase(equation2.8).

The wave-ve tor an be approximated for

z << z

R

to

k

geom

(z) =

dζ(z)

dz

≈

1

z

R

.

Thus, the phase-mismat his:

∆k

geom

= qk

geom

(ω

f

) − k

geom

(qω

f

) =

q − 1

z

R

.

(2.25)

The total phase-mismat h an be al ulated by ombiningequation 2.20 -2.25:

∆k

tot

= (n(ω

f

) − n(qω

f

))

qω

f

c

|

{z

}

>0

+

ω

2

p

(1 − q

2

)

2qcω

f

|

{z

}

<0

+

q − 1

z

R

| {z }

>0

.

(2.26)

In order toa hieve onstru tiveinterferen e the phase-mismat hintrodu ed by free

ele -trons have to an el out the mismat h due to dispersion and geometri al onsiderations.

During this work only loose fo using geometries are used and therefore the geometri al

part of the phase-mismat h an be negle ted. Thus, the phase-mismat h is only due to

free-ele tronsand dispersioninthe neutralmedium,i.e. argon. NotethatHHG inhollow

waveguides introdu es an additional term, whi h is addressed to waveguide dispersion

[Durfee etal.,1999℄. This term isabsentduring this workan annot beused tooptimize

2.3.4 Properties of the harmoni s

In gure 2.3 a typi al HHG spe trum is plotted s hemati ally. The spe trum an be

divided in three se tions: the perturbative regime of the lower harmoni s whi h an

betreated with the lowest-order perturbation theory [Gavrila, 1992℄, the mid-harmoni s

whi hformaplateau,and thehighestharmoni s inthe uto. Thehigh-orderharmoni s

of the plateauandthe uto regimeare des ribed intermsof the SFA,whi hisnot valid

for lowerorders. Here the atomi potentialhas an inuen e onthe ele tron's movement,

thus the approximationsmade inthe SFA are not fullled anymore ( hapter 2.3.2).

1

3

5

7 9

11

13

15

17

19

21

23

25

27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61

plateau - SFA regime

cutoff

perturbative

regime

Figure 2.3: S hemati representation of a typi al HHG spe trum (odd orders shown on

top).

Asexplainedbefore, theharmoni s are separatedby

2ω

f

, where

ω

f

is thefrequen y of the driving laser. The spe tral bandwidth of ea h harmoni depends onthe intensity of

the driving laser [He et al., 2009℄ and is larger, if the long ele tron traje tory is favored,

thus for higherlaser intensities. Moreover, destru tive interferen e between the long and

the short traje torymay o ur at high intensities[Xu et al., 2008; Brunetti et al., 2008;

Zaïr et al., 2008℄. It is further possible to slightly shift the harmoni s towards lower

wavelengths. As explainedin hapter2.3.1 the probability toionize anatom depends on

the intensity of the driving laser. Thus, for high intensities the number of free-ele trons

in reases and the refra tiveindex hanges (se tion2.3.3). This variationof the refra tive

index leads to a spe tral blueshift of the driving laser and therefore to a blueshift of

the harmoni s [Wahlströmet al., 1993℄. Detailed al ulations an be found in [L'Huillier

et al., 1992; Raeand Burnett,1993a; Raeet al., 1994; Kan etal., 1995℄.

The harmoni sforma trainof attose ond pulsesinthe time domain[Paulet al., 2001;

Antoineetal.,1996℄. Inordertomeasuretheattose ondpulsestru tureoftheharmoni s,

the phase dieren ebetween two onse utive harmoni s is measured. This measurement

isbasedona ross- orrelationte hniquebasedonatwophotontransition. The te hnique

RABITT(Re onstru tionofAttose ondharmoni BeatingByInterferen eofTwo-photon

Transitions)uses the fa tthat the drivinglaser frequen y isexa tlyhalfof the frequen y

laser intoa gas, the spe trum onsists of peaks orrespondingto the harmoni orders. If

in additionthe driving laser overlaps in spa eand time, sidebands appear whi h depend

ontherelativephase ofthe onse utiveharmoni sandthetime-delayofthedrivinglaser.

In this way, the phase dieren e an be dedu ed. In ombinationwiththe harmoni

am-plitudethetemporalprole an bere onstru ted[Hents heletal.,2001;Pauletal.,2001;

Kienberger et al., 2002℄. Moreover, the time-dependent frequen y an be measured and

depends linearly onthe harmoni orderand on the hirp of the drivinglaser [Mauritsson

et al., 2004; Norinet al., 2002℄.

The oheren e time has to be onsidered separately for the long and for the short

traje tory. The short traje tory has a phase that does not vary mu h with intensity,

thus the emitted radiation has a long oheren e time and is well ollimated[Lewenstein

et al., 1995℄. In ontrast the phase of the long traje tory hanges rapidly with the laser

intensity [Gaarde et al., 1999℄, therefore, a shorter oheren e time and a more divergent

emission appears [Salières et al., 1995℄. The spatial oheren e has been investigated by

a series of a two-slit experiments. It has been shown that for moderate laser intensities

the harmoni s have a high degree of spatial oheren e. For high intensities the spatial

oheren e isredu ed, due tothe time-dependent index of refra tion[Ditmireet al., 1996;

Bartels etal.,2002℄.

2.4 Free Ele tron Laser

In 1971 the prin iple of an FEL has been proposed by [Madey, 1971℄. After the rst

observationofmagneti bremsstrahlungat

10.6 µ

m[Eliasetal.,1976℄the rstFEL os il-lator [Dea on etal.,1977℄went into operation. Today, almost35years later, a variety of

dierent FEL fa ilitieshavebeen onstru ted inorder to providehigh-brightness,

oher-ent, ultrashort, and short wavelengthradiationto the s ienti ommunity. For example

LCLS[Emma,2009;BoutetandWilliams,2010℄,XFELatSpring-8[Shintake,2010;Hara

etal.,2010℄andtheEuropeanXFEL[S hneidmillerandYurkov,2010℄areabletoprodu e

FEL-radiationwithawavelengthintheorderof

0.1

nm, whileotherfa ilitieslikeFLASH ( hapter3.1),SCSSatSPring-8[Shintake,2010℄,FERMIatElettra [Allariaetal.,2010℄,

and the SwissFEL [Garvey, 2010; Patterson et al., 2010℄ are operating in the VUV and

the softX-ray regime.

In ontrast to the syn hrotron light sour es, the FELs generate photon pulses with a

mu hhigherdegreeof oheren e anda pulselength down tothe femtose ond range. The

HHG sour es, on the other hand, are also apable to emit ultra-short oherent

XUV-pulses, but with anenergy in the nanojoules regime,while FELs typi allydeliver

mi ro-or millijoulepulses. FLASH for instan e is able to deliver pulses up to

300 µ

J [Treus h and Feldhaus, 2010℄,whereas LCLS has observed apulse energy of

3

mJ [LCLS, 2011℄.

Due to the phase and amplitude jitter of the a elerating ele tri eld, the ele tron

bun hes arriveatdierenttimesatthe undulatorentran e omparedtoanexternallaser

(arrival-time jitter). In the SASE operation mode, the FEL amplies the spontaneous

radiation produ ed in the beginning of the undulator se tion. The sto hasti nature of

on a shot-to-shot basis (spe tral and intensity jitter). Furthermore, the pulse shape and

the pulselength diers from shot toshot.

This hapterisontheonehanddire tedtothefundamentalsasso iatedwiththeorigins

of shot-to-shotu tuations, and onthe other hand to alternative operationmodes of an

FEL. DetailsonFEL related pro esses an be found in[S hreiber, 2010;S hmüser etal.,

2008;Huang and Kim,2007; Saldin etal.,1999b℄.

2.4.1 Ele tron sour e and a eleration

The ele trons are generated by a laser driven photo athode and are subsequently

a - elerated in a RF avity generating ele tron bun hes with a length in the pi ose ond

regime. The te hni al solutionas well as adetailed hara terization is given in[Stephan

et al., 2010; Fraser et al., 1986℄. However, the light pulses of the driving laser have a

temporalshot-to-shotjitterrelativetothesyn hronizationsystem whi histransferred to

the ele tron bun hes. Additionally, the ele trons experien etemporalu tuations of the

RF phase leading to a further arrival time jitter of the ele tron bun hes relative to the

syn hronization system.

In order to produ e ultra-relativisti ele tron bun hes with a high peak urrent the

ele trons are a elerated downstream the super ondu ting RF a elerating modules and

are longitudinally ompressed using the bun h ompressors realized as four dipole

mag-neti hi anes. As in ase of the RF gun, depending on their longitudinal position the

ele tronsexperien eadierentRFphaseduringthea elerationintheRF-modules. This

has a twofold ee t - it indu es a head-tail energy variation along the bun h and it also

hangesthemeanenergy and orrespondinglythebun harrivaltime. Espe iallytherst

a elerating modulewhere the bun hes are a elerated toalmost the speed oflighthas a

dire t inuen e onthe arrivaltime relative to the syn hronization system. In the bun h

ompressor,dependingonthemomentumthe ele tronstraveldierentpathways: thetail

ele trons with a lower momentum travel a shorter and the head ele trons with a higher

momentum travel a longer way. This leads to a longitudinal ompressionof the ele tron

bun h.

Therefore, the arrival time u tuations of the ele tron bun hes are mainly introdu ed

bythe rst a elerationmodule aswellas the bun h ompressor. The timing u tuation

aused by the ele tron gun is partly ompensated by the bun h ompressor. A detailed

des ription of the a eleration pro ess an be found in [Wiedemann, 2007℄. The

orre-sponding arrival time u tuation of the ele tron bun hes aswellas te hniques to redu e

those errati u tuationsis des ribed in[Löhl,2009℄.

2.4.2 Undulator radiation

The undulator is a periodi magneti stru ture onsisting of dipole magnets with

alter-nating polarity as illustrated in gure 2.4. The following does only summarize the key

aspe ts of undulators. Details an be found in [Hofmann, 2004℄ and [Duke, 2000℄.

x

y

z

u

Figure 2.4: S hemati representationof anundulator: yellow isthe sine-liketraje tory of

the ele tron and blue isthe emitted radiation.

an be hara terizedwith the undulator parameter

K =

eB

0

λ

u

2πm

e

c

.

(2.27)

An ele tron whi htravelsthrough the undulator is due tothe Lorentzfor emoving ona

sine-like traje tory and emits syn hrotron radiation. The radiation emitted at dierent

positionsinthe undulatorinterferesandis on entratedinanarrow onewithanopening

angle of

1/γ

, where

γ =

E

m

e

c

2

is the Lorentzian fa tor. The fundamental wavelength of

the undulator

λ

ph

=

λ

u

2γ

2



1 +

K

2

2

+ γ

2

_θ

2



(2.28)

dependsonthethe undulatorperiod

λ

u

,the energyofthe ele tron,the undulator param-eter, and the emission angle

θ

with respe t to the beam axis. The on axis wavelength (

θ = 0

) an be hanged by tuning the ele tron energy or the magneti eld of the un-dulator. The ele trons emit radiation independent of ea h other, thus, the radiation is

in oherentforwavelengths smallerthanthe bun hlengthanditspowerisproportionalto

the number of ele trons. The polarizationof the radiation is perpendi ular to the

mag-neti eld and the spe tral width is inversely proportional to the number of undulator

2.4.3 High-gain free-ele tron laser

In the high-gain FELs, the initial urrent density in the ele tron bun h is periodi ally

modulated at the radiation wavelength, thus reating the so- alled mi robun hes. This

mi robun h stru ture leads to oherent radiation. As explained inse tion 2.4.2 the

ele -tronsmoveonasine-liketraje tory. Assumedthey aresuperimposedwithan

ele tromag-neti wavethey angainorlooseenergydependingonthephaseoftheele tron os illation

relative to the phase of the ele tromagneti eld. The energy transfer an be des ribed

with the following equation:

dW

dt

=

~v

· ~F

= −ev

x

(t)E

x

(t),

∝

os

((k

l

+ k

u

)z − ω

l

t + ψ

0

) +

os

((k

l

− k

u

)z − ω

l

t + ψ

0

)

(2.29)

=

os

(ψ) +

os

(χ)

(2.30)

where

dW

is the transferred energy,

t

the time,

~F

the Lorentzian for e,

~v

the velo ity of the ele tron (see gure 2.4) and

E

x

is the magnitude of the ele tri eld in x dire tion.

k

l

and

k

u

are the wave number of the light wave and the undulator, respe tively,

z

the positioninthe undulator,

ω

l

the angularfrequen y of the lightwave, and

ψ

in equation 2.30 is known as the ponderomotive phase. It has been shown [S hmüser et al., 2008℄

that a ontinuous energytransfer fromtheele tron tothe lightwavealong the undulator

is present, if the ponderomotive phase is onstant. Furthermore, the ondition for a

sustained energytransfertothe ele tromagneti waveyieldsthe samelightwavelengthas

the onaxis wavelength of anundulator (equation 2.28 onaxis, i.e.

θ = 0

). The phase of the ele tromagneti waveisthen shiftedby

π

everyhalf y leof theos illatorymovement of the ele trons. The energy transfer from the ele trons to the light leads to aredu tion

ofthe ele tronmomentum. Thesmallertheele tronmomentum,thesmalleristhe radius

of the traje tory (

p = eB

0

r

). This leads to a urrent density modulationof the ele tron bun h. [S hmüser et al., 2008℄ showed also that this pro ess is self-sustained and leads

toasubstru ture of the ele tronbun h. This substru ture are longitudinalmi robun hes

with a distan e of

λ

ph

to ea h other. The ele trons in a mi robun h emit oherent light and itspowerisproportionalto

N

2

e

. Adetaileddes riptionof theself-modulationpro ess is published in [Kondratenko and Saldin, 1980℄.

However, the prin ipletogenerate oherent radiationworks only if the ele tron bun h

getsdensitymodulated,whi hisusually realizedwithanele tromagneti wave. Assumed

a seed ispresent, the phase spa e motionof the ele trons withinan ele tromagneti eld

an be des ribed by the pendulum equations, i.e the time derivative of an ele tron's

ponderomotive phase and the hange of the relative energy deviation. In addition, the

inhomogeneous wave equation for the ele tri eld of the lightwave and the evolution of

a mi robun h stru ture oupled with longitudinal spa e harge for es an be des ribed

mathemati ally. These equations an be solved analyti ally only for some simple ases

e.g. assuming a homogeneous harge density and very long ele tron bun hes as well as

negle ting the betatronos illationsand the dira tionof the EM-wave. Howeverin most

dierent tools like GENISIS [Rei he, 1999℄, GINGER [Fawley, 2004℄, and FAST [Saldin

et al., 1999a℄ are available. In gure 2.5 a GENISIS simulation of the mi robun hing

pro ess performedby Sven Rei he is shown.

0

1

2

3

4

s/d

-100

-50

0

50

100

x [

µ

m]

z / λ

tra

n

s

ve

rs

e

b

e

a

m

s

ize

(

µm)

(a)

0

1

2

3

4

s/ı

-100

-50

0

50

100

x [

µ

m]

z / λ

(b)

0

1

2

3

4

s/

-100

-50

0

50

100

x [

µ

m]

z / λ

( )

Figure 2.5: GENISISsimulationofthe mi robun hingpro ess. The plotsshowthe

devel-opmentof the mi robun h stru ture withrespe t tothe transverse and

longi-tudinal oordinates in the bun h. The distan e of the mi robun hes is equal

tothe radiatedwavelength. 2.5(a): initialdistributionof the ele tron bun h.

2.5(b): developing mi robun h stru ture. 2.5( ): fully developed mi robun h

stru ture atthe end of the undulator. (Courtesy of Sven Rei he, PSI)

Inordertosolvethependulumequationsanalyti allythefollowingapproximationshave

to be made: the dependen e of the bun h harge density as well as the ele tromagneti

eld onthe transverse plane isnegle ted (one-dimensionalmodel). Theele tron bun h is

assumedtobeverylong,i.e. theee tsofthetailandheadofthebun h anbenegle ted.

Further,the hargedistributionisassumedtobehomogeneousand themodulationofthe

urrentdensityisassumedtobesmall. Theseassumptionsleadtotheso- alledthird-order

equation and an besolved analyti ally.

Itisfurtherpossibletodedu eanumberofproperties,des ribingtheFELpro ess. The

typi al emitted FEL radiation as a fun tion of the undulator length is shown in gure

2.6. During the rst few meters the ele tron bun h has an initial urrent density. As

soon as the ele tron bun h be omes modulated and the mi robun h stru ture develops

the emitted FEL pulse energy in reases exponentially. The power of the light wave an

bedes ribed with the followingequation:

P(z)

_∝

exp

(z/L

g

)

with

L

g

∝

λ

u

K

2

1/3

,

where

P(z)

isthepowerofthelightwaveand

L

g

isthegainlengthwhi hin reasesforlower wavelengths [S hmüser et al., 2008℄. After approximately

12

m themi robun hstru ture isfullydevelopedandsaturationsetsin. Nofurtherampli ationoftheemittedradiation

is possible.

2.4.4 FEL operation modes

In the previous se tion the FEL-ampli ation of an ele tromagneti wave has been

Figure 2.6: The exponential growth of the FEL pulse energy as a fun tion of the length

traveled in the undulator. After a few gain lengths the exponential growth

andthe mi robun hingstarts. The ir les orrespondtomeasurementsofthe

TESLA Test Fa ility[Ayvazyan etal., 2002℄. The progressing mi robun hing

is indi ated s hemati ally. The graphi is taken from[S hmüser et al., 2008℄

with kindpermissionof Springer S ien e and Business Media.

FEL-operationmodes, whi hwill bedis ussed inthis hapter.

Self-amplied spontaneous emission (SASE)

The most ommonoperationmode of anFEL isSASE, whi h has been observed for the

rsttimeintheXUVregimeattheTESLAtestfa ilityinHamburgatDESY[Andruszkow

et al., 2000℄. This operation mode does not need an external laser seed, it rather uses

the spontaneous undulator radiation as a seed to initiate the ampli ation pro ess: the

ele tron bun h onsists of randomly distributed ele trons. This distribution leads to a

whitenoisespe trumwithintherstpartoftheundulator. Thespe tral omponentwhi h

is within the FEL bandwidth willa t asthe seed for the following ampli ationpro ess.

However, SASE is based ona sto hasti pro ess, whi h leads tou tuating properties of

the FEL radiationona shot-to-shotbasis. The spe trum and the pulse prole issubje t

to errati u tuations. Moreover, the pulse energy and the pulse lengthvaries fromshot

toshot. Nevertheless, itispossible todes ribe theresultingradiationinastatisti alway.

In [Saldin et al., 1998℄ a detailed study of statisti al properties of the radiation from a

SASE FEL is given.

Seeding s hemes

Dierentseedings hemeshavebeenproposed,inordertoredu etheseerrati u tuations

of the FEL radiation. The following is primarily addressed to dire t seeding with an

external laser although a brief summary of other seeding s hemes is given. A general

Dire t seeding with an external laser

Seeding an FEL with an external laser has been prosed in [Garzella et al., 2004℄. The

idea is toinitiate the ampli ation pro ess of a high-gain FEL with anexternal seed, in

order tosuppress the errati u tuations of the SASE radiation. Asexplainedin hapter

2.3 HHG is a feasible method to generate VUV toXUV radiation with a high degree of

temporaland spatial oheren e and istherefore aneligible andidate.

Dierent simulations have been published [Milt hev et al.,2008; M Neilet al., 2007a℄

showingthat seedingwithanexternallaserleadstoFELradiationwithimproved

tempo-ral oheren eandspe tralbrightness. Theshot-to-shotstabilitymay learlybeenhan ed

and the pulse width isdetermined by the pulse width of the seed. The simulationsshow

further that the ampli ation pro ess washes out the attose ond pulse-train stru ture

of the harmoni s and sele tively amplies a spe i harmoni , whi h is determined by

the FEL bandwidth. It is ne essary to ensure that the seed overlaps with the ele tron

bun hinspa e,time,wavelength, andpolarizationinordertoinitiatethe seedingpro ess

before the SASE signal develops. Therefore, the seed energy threshold depends on the

wavelength [Togashiet al., 2011℄and is assumed tobeinthe nanojoule rangefor photon

energies smaller than

100

eV. Moreover, the ampli ationpro ess takespla e within the pulse length of the seed, hen e the arrival time jitter of the ele trons aused by the

a - eleration pro ess has noinuen e on the yielded FEL radiationas long as the temporal

jitter doesnot ex eed the ele tron bun h length.

So far dierent seeding experiments have been performed. The rst observation of a

seeded FEL has been reported atthe Deep Ultra-Violet FEL, where the third harmoni

from rystals

λ

seed

= 266

nm ledtoalargeampli ation[DiMauroetal.,2003℄. In Japan the SPring-8Compa t SASESour e(SCSS)test a eleratorhasbeen seeded witha

high-harmoni . Here su essful seeding has been reported for

λ

seed

= 160

nm and

λ

seed

=

61.2

nm [Lambert et al.,2008; Togashi et al., 2011℄. At SPARC a seedingexperiment is ongoing,where rst results have been a hieved at

λ

seed

= 260

nm [Giannessietal., 2008; T herbako et al.,2006℄and Max Labis preparing aseeding experiment as published in

[ƒuti¢ etal., 2010; Thorinet al., 2007℄.

It has been shown [Lambert et al.,2008;Togashi et al., 2011℄ that the intensity of the

seeded FEL radiationis enhan ed ompared to the SASE operation mode, although the

ratiobetweentheSASEandseedingasso iatedsignaldropsdownforsmallerwavelengths.

In additionit has been shown that the spiky stru ture of the spe trum in ase of SASE

is drasti allyredu ed by seeding the FEL.

Alternative seeding s hemes

High-gainharmoni generation(HGHG)isadierents hemeofseedinganFEL[Doyuran

et al., 2004; Yu et al., 2000; M Neil et al., 2007b℄. In this ase, a seed laser modulates

the ele tron bun h inarst undulator (modulator). Afollowingdispersivese tion

trans-fers the energy modulationintoa density modulationand ase ond undulator (radiator)

radiates higher harmoni s of the seed laser. In a se ond step the radiation passes a

a se ond modulator. The ele tron bun h travels then through another dispersive se tion

andanotherradiator,whi histunedtohigherharmoni s. In[Yuetal.,2003;Wangetal.,

2006;ƒuti¢etal.,2011℄rstresultshavebeenpublished. Thiss heme analsobeusedto

establish a temporal overlap between the ele tron bun h and the seed laser aspublished

in [Tarkeshian et al., 2010℄.

Re ently, e ho-enabled harmoni generation has been proposed [Stupakov, 2010;

Xi-ang et al., 2010℄. Here a long wavelength laser modulates the ele tron bun h to a short

wavelength mi robun hing: the ele tron bun h ismodulatedby alaserwith awave

num-ber

k

1

inarst modulator. Ittravelsthentroughadispersivese tionand issubsequently modulated by a se ond laser with

k

2

in a se ond modulator. After passinga se ond dis-persivese tion the e hosignal o urs atthe wavenumber

k

E

= nk

1

+ mk

2

, where

n

and