Development of Carrier-Envelope-Phase-Stabilized, mJ-Class Laser Sources Generating Intense, Few-Cycle Laser Pulses

Development of

Carrier-Envelope-Phase-Stabilized,

mJ-Class Laser Sources Generating

Intense, Few-Cycle Laser Pulses

Mikayel M

USHEGHYAN

Institut f¨ur Physik, Fachbereich 10 - Mathematik &

Naturwissenschaften

Universit¨at Kassel

A thesis submitted for the degree of

Philosophiae Doctor (Ph.D.)

Doktor der Naturwissenschaften (Dr. rer. nat.)

Supervised by

Prof. Dr. Thomas B

PD. Dr. Andreas A

Die vorliegende Arbeit wurde vom Fachbereich Mathematik und Naturwissenschaften der Universit¨at Kassel als Dissertation zur Erlangung des akademischen Grades eines Doktors der Naturwissenschaften (Dr. rer. nat.) angenommen.

Erster Gutachter: Privat-Dozent Dr. Andreas Assion. Zweiter Gutachter: Professor Dr. Thomas Baumert.

Weitere Mitglieder der Pr¨ufungskommission: Prof. Dr. Martin E. Garcia, Prof. Dr. Thomas Giesen.

Affidavit

English

I hereby declare, that the doctoral thesis handed in is written autonomously without illicit means or instruments other than the ones mentioned in the text. Text that is verbatim or implicitly taken from other published or unpublished sources is made visibly. No part of this thesis was previously submitted.

Deutsch

Hiermit versichere ich, dass ich die vorliegende Dissertation selbstst¨andig, ohne unerlaubte Hilfe Dritter angefertigt und andere als die in der Dissertation angegebenen Hilfsmittel nicht benutzt habe. Alle Stellen, die w¨ortlich oder sinngem¨aß aus ver¨offentlichten oder unver¨offentlichten Schriften entnommen sind, habe ich als solche kenntlich gemacht. Dritte waren an der inhaltlich-materiellen Erstellung der Dissertation nicht beteiligt; insbesondere habe ich hierf¨ur nicht die Hilfe eines Promotionsberaters in Anspruch genommen. Kein Teil dieser Arbeit ist in einem anderen Promotions- oder Habilitationsverfahren verwendet worden.

Date:

Place:

Abstract

Over a decade ago the frontiers of ultrashort physics and nonlinear optics have reached the at-tosecond timescale (1 as = 10−18s). Currently, in order to generate isolated attosecond pulses, an intense femtosecond pulse is focused in a jet of a noble gas. Given the sufficient intensity, this leads to high harmonic generation (HHG), which can be filtered (gated) to create a single iso-lated attosecond pulse. The laser source for the driver pulses has to fulfill two requirements: the pulses need to be limited to the duration of only few optical cycles (<6 fs in case of the central wavelength of 800 nm) and the electric waveform needs to be stable. The latter in practice trans-lates to the stability of the phase between the carrier wave and the pulse envelope. This quantity is known as the carrier-envelope phase (CEP). As the field of attosecond physics expands, more accent is put on the reliability and robustness of the few-cycle, CEP-stable laser sources. This work addresses two distinct challenges in this area.

Currently, in order to generate mJ-level, few-cycle pulses the following approach is usually undertaken. First, nJ-level femtosecond pulses are generated in a laser oscillator. The generated oscillator pulses are stretched to tens or hundreds of picoseconds and injected into a laser ampli-fier. The pulses are then amplified to the mJ-level. This amplification method is known under the name of chirped pulse amplification. When the gain medium in the amplifier is titanium-sapphire (Ti:Sa) crystal, the pulse duration after the re-compression is on the order of 30 fs. As of this moment, it is not possible to achieve direct few-cycle output due to the effect known as the gain narrowing. This phenomenon stems from the Gaussian/Lorentzian profile of the gain curve, leading to stronger amplification of the spectral components near the gain maximum. Therefore, during the amplification process the spectrum of the pulses becomes narrower and the duration of the compressed output pulses increases. The gain narrowing presents the reason why the am-plifier output duration is limited to the 30-fs range. Hence, for achieving mJ-level, few-cycle pulses, these 30-fs amplifier pulses must be spectrally broadened using a nonlinear scheme and re-compressed afterwards, achieving the few-cycle regime. This leads to more complicated, less stable setups and ultimately sets a limit on the achievable energy of the few-cycle pulses.

There is also a trend in attosecond physics to perform experiments with more energetic XUV photons. This can be done by utilizing driver sources in the longer wavelength region. The rea-son for this is the scaling of the high harmonic cut-off frequency (which gives the most energetic photons) with λ2 of the driver laser wavelength. However, the reverse scaling of λ−5.5 of the HHG efficiency makes the wavelengths in the far-infrared region less useful. In practice, this trade-off leads to the necessity of driver sources in the mid-infrared region (3-5 µm). Since there is currently no laser medium with a sufficiently broadband (for supporting few-cycle duration)

emission curve in this wavelength region, the efforts have been concentrated on the development of parametric sources. When it comes to parametric sources in the mid-infrared, multitude of pump/seed sources and nonlinear crystals present viable options. As it stands, there is no universally accepted or “standard” scheme for generating fewcycle pulses in this wavelength region -various setups have their own up- and downsides.

In the scope of this work, two unique laser systems with a stable CEP are developed to tackle the described challenges.

The first laser system is an ultrabroadband, CEP-stable Ti:Sa amplifier. The compensation of the gain narrowing using custom spectral filters leads to a compact and robust amplification stage with a combination of output pulse parameters that has never before been demonstrated in the scientific literature. The direct output is CEP-stable with pulse duration of sub-13 fs and energy of 3.2 mJ. This system presents a major step forward towards direct generation of few-cycle, mJ-level pulses.

The second laser system is a CEP-stable, potentially few-cycle mid-infrared parametric am-plifier. It is pumped/seeded by a 30-fs Ti:Sa amam-plifier. The presented source explores the viability of using ultrashort laser pulses as pump/seed source for the system. The undertaken approach results in compressed, CEP-stable output with 300 µJ energy per pulse at the 3.4 µm central wave-length. The spectrum of the generated pulses supports few-cycle duration. Such a result has not been demonstrated before for Ti:Sa-pumped parametric amplifiers. This presents an important step towards simpler and more robust parametric sources in this wavelength region.

In this work, the connection between these two systems is also outlined. Namely, the use of the presented ultrabroadband Ti:Sa amplifier as a prospective pump/seed source for the developed mid-infrared parametric amplifier. At the end of the thesis, results of two-color HHG experiments with the mid-infrared parametric amplifier are presented.

Zusammenfassung

Vor ¨uber einem Jahrzehnt konnte mit Ultrakurzzeitpulslaser in Verbindung mit nichtlinearen Prozessen Attosekunden (1 as = 10−18s) Lichtpulsz¨uge generiert werden. Um isolierte Attosekunden-Pulse zu erzeugen, wird derzeit ein intensiver Femtosekunden Laserpuls in einem Edelgasstrahl fokussiert: Bei ausreichender Intensit¨at f¨uhrt dies zu einer Hohen Harmonischen Erzeugung (HHG), die gefiltert werden kann, um einen einzelnen isolierten Attosekundenpuls zu erzeu-gen. Die Laserquelle f¨ur die Treiberpulse muss zwei Anforderungen erf¨ullen: Die Pulse m¨ussen auf die Dauer von nur wenigen optischen Zyklen (<6 fs bei 800-nm Zentralwellenl¨ange) be-grenzt sein und das elektrische Feld muss stabil sein. Letzteres ¨ubersetzt sich in der Praxis in die Stabilit¨at der Phase zwischen der Tr¨agerwelle und der Einh¨ullenden des Laserpulses. Diese Gr¨oße wird als Carrier-Envelope Phase (CEP) bezeichnet. Mit zunehmender Verbreitung der Attosekundenphysik wird die Zuverl¨assigkeit und Robustheit der CEP-stabilen Laserquellen mit wenigen Zyklen immer wichtiger. Die vorliegende Arbeit befasst sich mit zwei unterschiedlichen Herausforderungen in diesem Bereich.

Gegenw¨artig wird zur Erzeugung von Laserpulsen in mJ-Bereich und wenigen Zyklen ¨ublicher-weise der folgende Ansatz gew¨ahlt. Zun¨achst werden in einem Laseroszillator Femtosekunden-pulse auf nJ-Niveau erzeugt. Die erzeugten OszillatorFemtosekunden-pulse werden dann auf einige zehn oder hundert Pikosekunden gedehnt und in einen Laserverst¨arker eingekoppelt und in den mJ-Bereich verst¨arkt. Diese Verst¨arkungsmethode ist unter der Bezeichnung Chirped Pulse Amplification bekannt. Wenn das Verst¨arkungsmedium ein Titan-Saphir-Kristall (Ti:Sa) ist, liegt die typische Pulsdauer der Laserpulsen in der Gr¨oßenordnung von 30 fs. Die Limitierung in der Pulsdauer hin zu k¨urzeren Pulsdauern ergibt sich aufgrund des Gain Narrowing Prozesses im Verst¨arker. Gain Narrowing bedeutet, dass es beim Verst¨arkungsprozess zu einer starken Einschn¨urung der spektralen Breite der verst¨arkten Laserpulse kommt, was letztendlich die Pulsdauer vergr¨oßert. Dieses Ph¨anomen beruht auf dem Gaußschen / Lorentzschen Profil der Verst¨arkungskurve, die bewirkt, dass die spektralen Komponenten in der N¨ahe des Verst¨arkungsmaximums eine viel h¨ohere Verst¨arkung erfahren. Zur Erzielung von Laserpulsen im mJ-Bereich und wenigen Zyklen muss daher, unter Verwendung eines nichtlinearen Schemas eine zus¨atzliche spektrale Verbrei-terung der Pulse erfolgen. Dies f¨uhrt zu komplizierteren, weniger stabilen Aufbauten und be-grenzt letztendlich die erreichbare Energie der Laserpulse mit wenigen Zyklen.

Eine weitere Anforderung an die Laserquellen, die sich aus der Attosekunden Spektroskopie ableitet ist, dass Attosekunden Lichtpulse im XUV Bereich f¨ur die Experimente ben¨otigt wer-den. Dies kann durch Verwendung von Treiberquellen im infraroten Wellenl¨angenbereich erfol-gen. Der Grund daf¨ur ist, dass die h¨ochst erzielbare Photonenenergie der Hohen Harmonischen

mit dem Quadrat der Treiberlaserwellenl¨ange λ2skaliert. Allerdings skaliert die HHG Effizienz umgekehrte mit λ−5.5, was bedeutet, dass die Laserquelle im fernen Infrarotbereich sehr inten-sive Laserpulse erzeugen muss. In der Praxis hat sich gezeigt, dass Laserquellen im Infrarot-bereich von 3-5 µm geeignet sind, um eine hohe Photonenenergie der Hohen Harmonischen bei gleichzeitig ausreichender Effizienz zu erreichen. Da es in diesem Wellenl¨angenbereich derzeit kein Lasermedium mit ausreichend breitbandiger Emissionskurve (zur Unterst¨utzung der Dauer von wenigen Zyklen) gibt, wurden die Anstrengungen auf die Entwicklung parametrischer Quellen konzentriert. Wenn es um parametrische Quellen im mittleren Infrarot geht, bieten sich eine Vielzahl von Pump-/Seed-Laserquellen und nichtlinearen Kristallen an. Derzeit gibt es in diesem Wellenl¨angenbereich kein allgemein akzeptiertes “Standard”-Schema. Verschiedene Aufbauten haben ihre eigenen Vor- und Nachteile.

Im Rahmen dieser Arbeit wird die Entwicklung von zwei einzigartigen Lasersystemen, die in der Lage sind CEP stabile Laserpulse zu generieren dargelegt.

Das erste Lasersystem ist ein ultrabreitband CEP-stabiler Ti:Sa-Verst¨arker. Die Kompensa-tion des Gain Narrowing unter Verwendung von speziell designten Spektralfiltern f¨uhrt zu einer kompakten und robusten Verst¨arkungsstufe. Es konnten Laserparameter erzielt werden, die in der wissenschaftlichen Literatur noch nie gezeigt wurden. Das Lasersystem erzeugt CEP-stabile Laserpulse mit einer Pulsdauer von unter 13 fs und einer Energie von 3.2 mJ. Dieses System ist ein großer Fortschritt auf dem Weg zur direkten Erzeugung von Laserpulsen in mJ-Bereich mit wenigen optischen Zyklen.

Das zweite Lasersystem ist ein CEP-stabiler parametrischer Mittelinfrarot-Verst¨arker. Der parametrische Verst¨arker wird mit einem 30-fs-Ti:Sa-Verst¨arker optisch gepumpt und geseedet. Mit diesem Ansatz ist es m¨oglich CEP stabile Laserpulse im Infrarotbereich von 3.4 µm bei einer Pulsenergie von 300 µJ zu erzeugen. Erste Messungen zeigen, dass mit diesen System Laser Pulse mit wenigen optischen Zyklen generiert werden k¨onnen. Dieses Ergebnis wurden bisher mit Ti:Sa-gepumpte parametrische Verst¨arker nicht erreicht.

Beide Systeme sind ein wichtiger Schritt in Richtung einfacherer und robusterer parametrische Laserquellen f¨ur die Attosekunden Physik.

In dieser Arbeit wird auch der Zusammenhang zwischen den beiden Lasersystemen skizziert. Insbesondere die große spektrale Breite, die mit dem Ti:Sa-Verst¨arker erzielt wird, er¨offnet voll-kommen neue M¨oglichkeiten bei der Entwicklung von parametrischen Verst¨arkern im mittleren Infrarotbereich.

Relevant Publications

Journal Articles

• M. Musheghyan, F. L¨ucking, Z. Cheng, H. Frei, and A. Assion. “0.24 TW ultrabroadband, CEP-stable multipass Ti:Sa amplifier”. In: Optics Letters 44.6 (2019), p. 1464. M. M. performed all the experiments, analyzed the data and wrote the manuscript.

• M. Musheghyan, P. Prasannan Geetha, D. Faccial`a, A. Pusala, G. Crippa, A. G. Ciriolo, M. Devetta, A. Assion, C. Manzoni, C. Vozzi and S. Stagira. “Tunable, Few-cycle, CEP-stable, Mid-IR Optical Parametric Amplifier for Strong-Field Applications”. (Manuscipt is currently in the process of submission). M. M. performed the overwhelming majority of the experiments, analyzed the majority of the data and wrote most of the manuscript.

Conference Contributions. Talks.

• M. Musheghyan, T. Le, Z. Cheng, P. Roth, F. L¨ucking and A. Assion. “Generation of CEP-Stabilized sub-15 fs, 0.1 TW pulses from a Compact Ti:Sapphire Amplifier”. In ATTO-2017, Xi’an, People’s Republic of China.

• M. Musheghyan, Z. Cheng, P. Roth, F. L¨ucking and A. Assion. “FWHM > 120 nm, 6 mJ, CEP-Stable Ti:Sapphire Multipass Amplifier”. In LPHYS’17, Kazan, Russian Federation. • M. Musheghyan, Z. Cheng, P. Roth, F. L¨ucking and A. Assion. “FWHM > 120 nm, 6 mJ, CEP-Stable Ti:Sapphire Multipass Amplifier”. In ASSL-2017 Session Code: AW1A.7, Nagoya, Japan.

• M. Musheghyan, F. L¨ucking, Z. Cheng, H. Frei and A. Assion. “Towards a sub-9 fs, 3 mJ, CEP-stable Multipass Ti:Sapphire Amplifier”. In CLEO-2019, Session Code: CF-9, Munich, Federal Republic of Germany.

• M. Musheghyan, F. L¨ucking, Z. Cheng, P. Maroju, H. Frei and A. Assion. “Sub-10 fs, Ultrabroadband, CEP-stable Multipass Ti:Sa Amplifier”. In ASSL-2019, Session Code: AM4A.4, Vienna, Republic of Austria.

• M. Musheghyan, P. Prasannan Geetha, D. Faccial`a, A. Pusala, Gabriele Crippa, A. Ciriolo, M. Devetta, A. Assion, C. Manzoni, C. Vozzi, S. Stagira. “Tunable, Few-Cycle, CEP-Stable Mid-IR Optical Parametric Amplifier for Strong-Field Applications”. In ASSL-2019, Session Code: AM2A.4, Vienna, Republic of Austria.

Conference Contributions. Posters.

• M. Musheghyan, F. L¨ucking, Z. Cheng and A. Assion. “Ultrabroadband (FWHM> 100nm) Ti:Sapphire Multipass Amplifier”. In Summer School ”Ultrafast dynamics with intense ra-diation sources”, Crete, Hellenic Republic.

• M. Musheghyan, D. Faccial`a, P. Prasannan Geetha, A. Pusala, Gabriele Crippa, A. Ciri-olo, M. Devetta, C. Vozzi, S. Stagira. “High energy Ultrafast Mid IR Optical Parametric Amplifier for Strong field Science”. In ATTO-2019, Szeged, Hungary.

• M. Musheghyan, D. Faccial`a, P. Prasannan Geetha, A. Pusala, Gabriele Crippa, A. Ciri-olo, M. Devetta, C. Vozzi, S. Stagira. “High energy Ultrafast Mid IR Optical Parametric Amplifier for Strong field Science”. In ICPEAC-2019, Deauville, French Republic.

Contents

1 Introduction 1

1.1 High Harmonic Generation and Attosecond Pulses . . . 2

1.1.1 Reaching the Attosecond Regime via High Harmonic Generation . . . . 2

1.1.2 Requirements for the Driver Lasers . . . 4

1.2 Solid-State Lasers as Driver Sources for Attosecond Science . . . 5

1.2.1 Few-Cycle Laser Oscillators . . . 5

1.2.2 Reaching the mJ-level Energies and Few-Cycle Duration with Ti:Sa Laser Amplifiers and Nonlinear Compression Schemes . . . 5

1.2.3 Stabilizing the CEP of the Laser Pulses . . . 6

1.3 Optical Parametric Amplifiers . . . 7

1.3.1 General Information . . . 7

1.3.2 OPAs for Applications in Attosecond Science . . . 7

1.4 Thesis Outline . . . 8

1.4.1 The Main Challenges . . . 8

1.4.2 Chapter Roadmap . . . 8

2 Theoretical Background 11 2.1 Gain in Solid-State Laser Amplifiers . . . 11

2.1.1 Multipass and Regenerative Amplifiers . . . 11

2.1.2 Gain Curve Line shape: Gain Narrowing . . . 12

2.1.3 Gain Regimes . . . 14

2.1.4 Conclusions . . . 17

2.2 Carrier-Envelope Phase . . . 17

2.2.1 Definition . . . 17

2.2.2 The Carrier-Envelope Offset Frequency . . . 18

2.2.3 Measuring the fCEO: An f-to-2f Interferometer . . . 19

2.2.4 Stabilizing the CEP Slip in Oscillators: Feed-Back and Feed-Forward Schemes . . . 20

2.2.5 CEP Stabilization of Ti:Sa Amplifiers . . . 23

2.2.6 Quantifying the CEP Noise: Power Spectral Density and Integrated Phase Noise . . . 24

2.2.7 Conclusions . . . 25

2.3.1 Second-Order Nonlinear Processes and (N)OPA Design . . . 25

2.3.2 Broadband Amplification. Group-Velocity Mismatch . . . 28

2.3.3 CEP Stability in OPAs . . . 30

2.3.4 Conclusions . . . 31

3 Development of an Ultrabroadband Ti:Sa Laser Amplifier 33 3.1 Overview of the Ultrabroadband Amplifier Schemes . . . 34

3.2 The Design of the Amplification Plate . . . 35

3.2.1 Gain Narrowing Compensation: Gaussian Filters . . . 35

3.2.2 Amplification Stage: The Schematic and Overall Comments . . . 36

3.2.3 The First 6 Passes . . . 37

3.2.4 Pulse Picking and the Dazzler . . . 40

3.2.5 The Subsequent 8 Passes . . . 42

3.2.6 Beam Profiles and Ti:Sa Fluorescence . . . 44

3.3 The Full Experimental Schematic . . . 45

3.4 Results . . . 48

3.4.1 The Spectrum and Pulse Duration of the Ultrabroadband Output . . . 48

3.4.2 Power Stability . . . 50

3.5 Self-Phase Modulation Stage . . . 50

3.5.1 Experimental Setup . . . 50

3.5.2 Output Beam Profile . . . 52

3.5.3 The Spectrum and Pulse Duration of the SPM Stage . . . 52

3.6 The CEP Stabilization of the Ultrabroadband Amplifier . . . 54

3.6.1 Stabilization of the Oscillator fCEO . . . 54

3.6.2 Ultrabroadband Amplifier CEP Stability . . . 55

3.7 Outlook . . . 56

4 Development of a Few-Cycle CEP-stable Mid-IR OPA 59 4.1 Mid-IR OPA Design Considerations . . . 59

4.1.1 Literature Overview . . . 59

4.1.2 Pumping/Seeding with a Ti:Sa Amplifier . . . 61

4.2 System Description . . . 63

4.3 System Characterization . . . 66

4.3.1 HCF and OPA Output Spectra . . . 66

4.3.2 M2and Power Stability Measurements of the Mid-IR OPA . . . 69

4.3.3 Pulse Duration Measurements of the Mid-IR OPA . . . 70

4.3.4 Simulations of the Pulse Duration and Bandwidth . . . 73

4.3.5 CEP Measurements of the Mid-IR OPA . . . 75

CONTENTS xiii

5 Two-Color HHG Using the Developed Mid-IR OPA 81

5.1 The Experimental Principles and the Setup . . . 81

5.1.1 Two-Color Experiments . . . 81

5.1.2 The Experimental Setup and Higher-Order Nonlinear Effects in Air . . . 82

5.2 HHG in Atomic and Molecular Gases . . . 84

5.2.1 HHG in Krypton . . . 84

5.2.2 HHG in Molecular Gases: CO2and Methane . . . 87

5.3 Summary & Outlook . . . 89

6 Conclusions and Outlook 91 6.1 Overall Thesis Summary . . . 91

6.2 Key Results . . . 91

6.3 Outlook . . . 92

Appendix 95

A Additional Experimental Details Relevant for the Mid-IR OPA Development 95

B Data Analysis 97 Acronyms 99 List of Figures 103 List of Tables 105 Bibliography 107 Acknowledgments 123

Chapter 1

Introduction

Over half a century ago, the first laser based on the ruby crystal was developed by Theodore Maiman [Mai60]. This event gave birth to laser physics and opened unprecedented horizons for the field of optics. In the span of the last 60 years this relatively young field has already played a pivotal role in the recent crucial discoveries in physics. Apart from its profound effects on fundamental science [Raa87; Abr92; Ima99], the laser technology has thoroughly reshaped certain fields of chemistry [Dan03; Bla12], biology/medicine [Str72; Hor79; Tro83; Fuj86] and had a major effect on modern-day industry [Rea97; Sug14]. Contemporary laser physics is composed of myriads of fields - each requiring its own unique tools, systems and approaches.

The topic of this work is centered around one such field, namely, ultrafast physics. The foundation of ultrafast experimental physics relies on the utilization of laser pulses with short temporal duration. Consequently, the investigation of schemes for the generation of ever shorter laser pulses (with higher peak intensities) plays a crucial role for the further research in the field of ultrafast physics.

The goal of the presented thesis is the development of novel ultrafast laser sources, providing a stepping-stone for the future research in the areas that require intense electric fields. The present work can also be viewed as a logical continuation of the endeavors pursued by various groups across the years to make the frontiers of ultrafast laser technology more accessible and mature, ensuring the further development of the branch.

The purpose of this chapter is to provide a general overview of the history of the related topics, as well as to present the general structure of the thesis. As such, the chapter is divided into the following sections:

• Section 1.1 describes one of the current frontiers of ultrafast science - attosecond physics. • Section 1.2 reviews the history and the current state of the solid-state laser technology that

is necessary for reaching the attosecond regime.

• Section 1.3 introduces the field of parametric amplifiers.

• Section 1.4 draws a connection between the major points of the previous sections and presents the goals of this work.

1.1

High Harmonic Generation and Attosecond Pulses

1.1.1

Reaching the Attosecond Regime via High Harmonic Generation

The demonstration of the Q-switching technique [McC62] and the discovery of the mode-locking mechanism [Lam64; Har64] have allowed to confine the output energy of the laser in short tem-poral intervals, i.e. pulses. This has phenomenally increased the achievable peak intensities and enabled precise temporal measurements on the ultrashort timescale, giving start to what nowadays is colloquially known as ultrafast physics. As the field matured, the pulse durations decreased, becoming as short as several femtoseconds (10−15 seconds). These advances gave numerous breakthroughs in both fundamental and applied sciences: the ability to observe and control chemical reactions [Zew88; Zew00; Ass98]; a possible pathway towards nuclear fu-sion [Dit99]; advancements in the field of machine cutting and drilling [Liu97; Anc08] and a multitude of advancements in the field of medicine [Chi96; Sch01; Gat08].

Figure 1.1: The three-step model of the HHG depicted in four steps. The figure is taken from [Cor07]. First (a) the potential well is tilted due to the presence of an external electric field, which allows the bound electron to undergo tunnel ionization, then (b) & (c) the electron is accelerated in the driving laser’s electric field and finally (d) the electron recombines with the parent ion emitting an XUV photon.

As it came to pass, the technology matured to the point where terawatt-class table-top sources became broadly available. As a result, the intensities became sufficiently high to observe high-order nonlinear phenomena, namely, the high harmonic generation (HHG) in noble gases [Kra92; Cha97]. This process is described via the (semi-classical) three-step model [Cor93; Lew94]:

1. A “driver” laser pulse with (close to) a few-cycle duration tilts the potential well of the va-lence electron. The electron undergoes tunneling ionization. Corresponds to Figure 1.1 (a). 2. The ionized electron is accelerated in the electric field of the driver pulse. Corresponds to

Figure 1.1 (b) & (c).

3. During the next half-cycle of the driver pulse, the direction of the electric field reverses. The accelerated electron recombines with the parent ion and the gained kinetic energy is emitted in form of an extreme ultraviolet (XUV)/soft X-ray photon. Corresponds to Figure 1.1 (d).

1.1 High Harmonic Generation and Attosecond Pulses 3

Figure 1.2: The evolution of pulse duration across the years. Adapted from [Cor07].

It was soon realized that high-order harmonics generated by an atom in an intense laser field form trains of ultrashort laser pulses that have attosecond (10−18seconds) duration [Ant96; Pau01]. Understandably, after this realization the efforts were undertaken to isolate a single pulse from this attosecond pulse train, i.e. to generate an isolated attosecond pulse (IAP). The way to do this was to somehow filter out other elements of the attosecond pulse train - to “gate” the necessary region of the high-order harmonics and thus to generate the IAP. Since then, different gating techniques have been proposed and demonstrated experimentally [Cor94; Hen01; Tch03; Kie04; Sol06; San06]. These discoveries, as well as the continuous improvements of the tech-niques [Bal03; LM05] have ushered in the era of attosecond physics. The graph of how the pulse duration changed with the development of laser technology can be seen in Figure 1.2.

The attosecond regime has allowed the study of numerous novel physical effects [Gou04; Uib07; Gou10; Sch10; Sch12], signifying a new period in ultrafast physics. The different phe-nomena on ultrafast timescale are visualized in Figure 1.3. As it is visible, such physical effects as structural changes inside the molecules, spin-crossover/ligand-exchange dynamics and pho-toinduced reactions occur on relatively longer timescales (nanoseconds to hundreds of femtosec-onds). Lifetimes of highly excited states and other processes, such as coherent electron dynamics and delays in photoemission occur on timescales ranging from few femtoseconds to only tens of attoseconds [Kra18]. An in-depth review of many techniques and developments in this field can be found in [Kra09; Pop10; Cal16; Kra18].

Figure 1.3: The timescales of various physical phenomena. Adapted from [Kra18].

1.1.2

Requirements for the Driver Lasers

Since its inception, the field of attosecond science has seen a rapid development. Nowadays the attosecond pulses are generated utilizing the HHG process in gaseous media. For this method a femtosecond driver laser is necessary. This driver laser should satisfy 3 important parameters:

1. Few-cycle pulse duration. Most of the schemes of the IAP generation require the electric field of the driver pulse to have ideally a single maximum (corresponding to a single-cycle duration). That way, by filtering the harmonics emitted near this maximum, an IAP may be generated.

2. High intensity. In order for the high-order nonlinear process of HHG to occur, the peak intensity on the noble gas target needs to exceed 1013−1014_W/cm2_{. In this case energies}

of mJ-level are desirable1.

3. The stability of the carrier-envelope phase (CEP) is necessary for the efficient production of IAPs [Cal16]. Since the driver pulses have few-cycle duration and the HHG is a highly nonlinear process, even small changes in the CEP have a significant effect [Boh98; Nis03]. For the majority of the experiments the CEP root-mean squared (R.M.S) noise has to re-main in the range of hundreds of milliradians.

Independent from these requirements, it should be noted that the central wavelength of the driver pulse λd is crucial for the duration and efficiency of the generated IAP. This is due to the fact

that the high-order harmonic cut-off frequency scales with λ_d2, while the efficiency scales with λ_d−5.5[Col08].

1_{This energy level holds for driver sources at the central wavelength of 800 nm. For longer wavelengths higher} energies are necessary due to the reduced efficiency of the HHG process [Col08].

1.2 Solid-State Lasers as Driver Sources for Attosecond Science 5

1.2

Solid-State Lasers as Driver Sources for Attosecond

Sci-ence

1.2.1

Few-Cycle Laser Oscillators

The discovery of the titanium-doped sapphire (Ti:Sa) crystal [Mou86] has played a pivotal role in the development of ultrafast laser systems. Thanks to the broad emission curve, the Ti:Sa gain medium is able to support pulses with few-cycle duration - the single cycle being 2.67 fs at the Ti:Sa gain maximum near the central wavelength of 800 nm.

The development of various mode-locking techniques, specifically the Kerr-lens mode-locking [Spe91; Bra92] and the semiconductor saturable absorber mirror (SESAM) [Kel96] mode-locking has allowed the generation of pulses from laser oscillators with sub-10 fs duration. Improve-ments in this technology, namely the creation of chirped-mirrors [Szi94] have further improved the Ti:Sa oscillator technology, allowing for the generation of sub-two-cycle pulses directly from an oscillator [Mor99]. Nowadays commercial, sealed Ti:Sa oscillators are readily available with pulse duration of sub-7 fs and energy of a few nJ per pulse at 80 MHz repetition rate [SP].

1.2.2

Reaching the mJ-level Energies and Few-Cycle Duration with Ti:Sa

Laser Amplifiers and Nonlinear Compression Schemes

The attosecond pulse generation requires laser drivers with pulse energies on the mJ-scale. Since scaling up the output energy of a laser oscillator presents insurmountable technical difficul-ties [Sie86], laser amplifiers are used for this purpose. A “seed” pulse generated by an oscillator is sent through a “pumped” gain medium2 in which the amplification process occurs. However, direct amplification of ultrashort oscillator pulses is problematic. This is due to the high peak intensities, which cause damage in the amplifier gain medium. This is especially true for the case of Ti:Sa crystals, since the few-cycle duration reaches the laser-induced damage threshold (LIDT) rather quickly [Ute07].

To circumvent this problem, the method of chirped pulse amplification (CPA) was pro-posed [Str85]. The name3of the method originates from a similar technique used in radar tech-nology [Blo73]. The implementation in the optical case is rather different from the radar - the input seed pulse is stretched using some dispersive element (e.g., bulk material or gratings) and compressed after the amplification using an element with the opposite sign of dispersion. This method helps to avoid the limit imposed by the LIDT and amplify pulses to the target energy level.

Although the Ti:Sa crystal has a broadband emission curve [Mou86], achieving sub-30 fs pulses at the amplifier output presents a twofold problem.

2_{In the case of Ti:Sa the pumping is done by nanosecond diode-pumped solid-state lasers. However, direct} diode-pumping of Ti:Sa lasers is also possible [Rot09].

3_{To the author’s knowledge, the specific term “chirp” originates from an internal Bell Laboratories memorandum} titled “Not with a bang, but a chirp”. This, in its turn, is very likely a reference to the ending of the poem “The Hollow Men” by T.S. Eliot - “This is the way the world ends, Not with a bang but a whimper”.

• The first issue is the necessity of the management of the higher-order dispersion - the third-order dispersion (TOD) and the fourth-order dispersion (FOD). This problem can be nowadays considered solved - a few methods of addressing it have already been devel-oped. These include: Shaping the femtosecond pulses by a liquid crystal phase modula-tor [Wei92]; employing TOD mirrors [Len95] and utilizing such tools as an acousto-optic programmable dispersive filter (AOPDF) Dazzler (from company Fastlite [Kap02]).

• The second issue is the gain narrowing phenomenon [Sie86], which is a fundamental issue physically stemming from the atomic response to the external electric fields. This phe-nomenon is discussed in Chapter 3 and is one of the main challenges addressed in this work.

As it stands, the direct output of Ti:Sa-based standard amplifiers is in the range of 30 fs. In or-der to reach the few-cycle durations necessary for the attosecond science applications, nonlinear compression schemes are used. These generally include either spectral broadening and subse-quent compression in a hollow-core fiber (HCF) [Nis96; Nis97; Sud05] or spectral broadening in bulk media and subsequent re-compression [Lu14; Sei17]. This allows to reach the few-cycle regime on a mJ-level, albeit putting a limit on the achievable output energy. In the cases when an HCF is used, the beam pointing instabilities also lead to less robust/stable experimental setups.

Another major issue in Ti:Sa amplifiers concerns scaling up the output pulse energy for higher repetition rates (>10 kHz). The Ti:Sa amplifiers can easily deliver mJ-level pulses at repetition rates of up to only several kHz. The limiting factor in this case is the large quantum defect of the Ti:Sa crystal, necessitating extensive cooling schemes (e.g., cryo-cooling). Although in principle possible, this presents a major technical difficulty that won’t be effectively resolved in the foreseeable future. Often, if higher repetition rates (in the range of 100 kHz) are necessary, other sources may be more suitable.

1.2.3

Stabilizing the CEP of the Laser Pulses

Another important parameter for the driver sources for the attosecond science is the CEP stability. This is an extensive topic, which requires in-detail explanation of the concepts. Hence, all the necessary information is presented in Chapter 2. In general, it can be said that the stabilization of the CEP for Ti:Sa sources is well-developed. Multiple methods exist, achieving CEP stability in the region of 150 mrad [L¨u14b].

Qualitative improvements in this technology may very well be possible, however they are not within the scope of this work and all the data presented in Chapter 3 and Chapter 4 may be classified only as incremental/quantitative upgrades.

1.3 Optical Parametric Amplifiers 7

1.3

Optical Parametric Amplifiers

1.3.1

General Information

As it came to pass, the laser technology became sufficiently mature to induce nonlinear phe-nomena in materials on a routine basis. As the field of nonlinear optics developed, it became clear that second-order nonlinear processes can be utilized for the operation of a new kind of laser amplifier - the optical parametric amplifier (OPA) [Akh65; Bau79; Gal98; Cer98]. Such an amplifier has several crucial differences from the standard laser amplifiers. Although these differences will be explained later in Section 2.3, they are mentioned here shortly:

1. Since no energy absorption is involved in the OPAs, there is no need for cooling of the gain medium to remove the excess heat4.

2. The central wavelength of the OPA output pulses is not strictly fixed and can often be varied to a great degree. Additionally, the gain curve may be far broader than in the case of standard gain media, since it is limited not by the linewidth of a (vibrational/electronic) transition, but by the phase matching conditions.

3. In suitable configurations, the CEP of the OPA pulses may be inherently stable [Bal02].

1.3.2

OPAs for Applications in Attosecond Science

As far as the applications in attosecond physics are concerned, the following points hold:

1. Since cooling is not required, OPAs with high average power and high pulse energy can be developed - an area for which Ti:Sa sources are problematic, as mentioned in Subsec-tion 1.2.2.

2. As it was mentioned in Subsection 1.1.2, the attosecond pulse duration (or the cut-off frequency of the HHG) depends on the central wavelength of the driver pulse. Since the overall efficiency decreases rapidly for longer wavelengths, the driver sources in the region of 3-5 µm are a good compromise. Currently there are no known suitable standard gain media for this task and OPAs can serve as an alternative. Additionally, the broader gain profile of the OPAs makes the direct generation of high-energy, few-cycle pulses a simpler task.

3. Since the CEP of the output OPA pulses can be inherently stable [Bal02], active stabiliza-tion schemes become less crucial or unnecessary, making the setups of the driver sources more reliable and robust.

In the cases when the LIDT becomes a problem, it is possible to implement the CPA approach for the OPA systems. If such a scheme is employed, the system bears the name of an optical parametric chirped-pulse amplifier (OPCPA) [Fle86; Dub92]. In comparison to the OPAs, the 4_{Some thermal management may be necessary, however this is done to achieve better phase-matching conditions.}

OPCPA systems suffer from the requirement of picosecond laser sources as pumps. As it stands, picosecond lasers are less developed than, for example, the nanosecond lasers necessary for Ti:Sa CPAs or even the femtosecond lasers, which can be used for pumping/seeding OPAs.

When it comes to the mid-infrared (mid-IR) region of 3-5 µm, there is effectively no “stan-dard” OPA scheme and multiple options exist. In total, this area is still novel and promising, prompting extensive investigations by multiple ultrafast physics groups. The development of an ultrafast OPA for this wavelength region can be beneficial for various fields of ultrafast laser physics. Not only has the mid-IR region proven crucial for the HHG in gases [Pop12] and in solids [Ghi18], but also for a multitude of applications ranging from pump-probe spec-troscopy [Wou97] to nonlinear optics [Dor15], atmospheric sensing [Mal15], bio-sciences [Her02a], as well as experiments involving laser-induced electron diffraction in molecules [Wol16].

1.4

Thesis Outline

1.4.1

The Main Challenges

During the discussion in the previous sections, points where the technology of driver sources for attosecond science can be improved were noted. They are summarized as follows:

1. Ti:Sa amplifiers are currently unable to directly generate pulses that come even close to the few-cycle duration. This makes additional nonlinear compression schemes necessary, resulting in setups that are bulkier and more prone to failure. Reaching direct few-cycle output is desirable. Such an amplifier would possess a much broader spectrum of the output pulses compared to the current systems and can be deemed as “ultrabroadband”. 2. There are various cases in which an OPA in the mid-IR would be more desirable. There

is no universally accepted “standard” scheme for this wavelength range. Depending on various parameters, different solutions are viable. As such, development of a few-cycle, CEP-stable mid-IR OPA possesses great potential.

These two seemingly disconnected topics are the main foci of the thesis. Later, in Chapter 2 it will be demonstrated that there is a direct connection between these two goals - an ultrabroadband Ti:Sa amplifier can serve as a pump/seed source for the mid-IR OPA.

1.4.2

Chapter Roadmap

The thesis is structured in the following manner:

• Chapter 2 summarizes the relevant theoretical background necessary for ultrabroadband amplifier and mid-IR OPA design. Associated problems are discussed. The connection between these two topics is outlined.

• Chapter 3 concerns the construction of an ultrabroadband Ti:Sa amplifier. The chapter starts with a summary of progress in this sphere. Afterwards the developed experimental setup is discussed, and the achieved results are examined and compared.

1.4 Thesis Outline 9

• Chapter 4 is dedicated to the progress in the development of the mid-IR OPA. This chapter is structured as the previous one. First, some general information on the existing mid-IR OPA designs is reviewed. This is followed by the presentation of the constructed OPA. Discussion and comparison with other parametric amplifiers is performed at the end of the chapter.

• Chapter 5 revolves around the applications of the developed mid-IR OPA. Data acquired from the two-color HHG experiments is presented.

• Chapter 6 concludes the thesis and summarizes all the results. A discussion regarding future possible improvements is offered.

As an ending note to this chapter, all the major concepts and their connections to the work performed in this thesis are visualized in Figure 1.4.

Driver Sources for Attosecond Physics

Few-Cycle Duration CEP Stability

Nonlinear

Compression GenerationDirect

Active Passive Few-Cycle OP(CP)As Standard Ti:Sa Laser Amplifiers High Intensity Low Power Ultrabroadband Ti:Sa Laser Amplifiers Gain-Narrowing Compensation Thermal Management Nanosecond

Pump PicosecondPump

No Quantum Defect Wavelength Tunability Fixed Wavelength Direct Intra-Pulse DFG _Ti:Sa-Pumped Few-Cycle Mid-IR OP(CP)A Shorter Path in Nonlinear Medium Pump/Seed Close to a Few-Cycle

Duration Energy ScalabilityStraightforward ImplementationSimple

Chapter 2: Theory Chapter 3: Sub-13 fs Amplifier Chapter 4: Mid-IR OPA Chapter 5: Applications High Power

Figure 1.4: The thesis structure.

Chapter 2

Theoretical Background

This thesis concerns the most recent advancements in Ti:Sa amplifier architecture, CEP stabi-lization schemes and mid-IR OPA design. Hence, a short summary of these topics is necessary beforehand. Since the discussion of each segment in detail would be overly extensive, only key results relevant to the thesis are summarized. The reader is kindly directed to the mentioned sources for all the derivations. This chapter is structured as follows:

• Section 2.1 presents the general information regarding solid-state laser amplifiers and laser gain. The types of amplifiers are presented, after which the gain narrowing phenomenon and different gain regimes are described.

• Section 2.2 revolves around the physical quantity of the CEP. The notion of the CEP is introduced: the quantity is defined, after which an explanation regarding measurement technique, CEP noise sources and stabilization techniques is given.

• Section 2.3 reviews the field of parametric amplifiers. The main principles behind paramet-ric amplification are summarized: the difference to standard laser gain media is outlined, followed by specific design considerations and properties of CEP in OPAs.

2.1

Gain in Solid-State Laser Amplifiers

2.1.1

Multipass and Regenerative Amplifiers

As mentioned in Subsection 1.2.2, scaling up the energy of laser oscillators presents insurmount-able technical difficulties. Hence, the general practice is to generate a low-energy “seed” pulse via an oscillator and to amplify this seed separately in a laser amplifier. Laser amplifier, is, broadly speaking, an arrangement of path through a gain medium pumped by some energy source. For the case that we are interested in here, i.e. Ti:Sa technology, the gain medium is a Ti:Sa crystal pumped by a diode-pumped solid-state laser (DPSSL). Nowadays Ti:Sa ampli-fiers take use of the CPA technology [Str85] mentioned in Subsection 1.2.2. According to how exactly the passes through the gain medium are achieved, amplifiers are categorized in 2 types, which are visualized in Figure 2.1.

• First type is the multipass amplification scheme. In this design the passes through the gain medium are geometrically separated and their amount is either strictly fixed or not trivial to modify. This approach makes it possible to adjust the characteristics of the beam (e.g., focusing and spectral shaping) for each individual pass. In general, multipass amplifiers can operate with relatively modest stretching factors (in the range of tens of picoseconds). This allows to stretch the seed pulses in bulk material.

• Second type is the regenerative amplification scheme. In this case a resonator cavity is constructed in which an optical switch controls the number of passes through the gain medium. The passes become geometrically identical and their total number may be varied by adjusting the timing of the optical switch. Therefore, greater efficiency (compared to the multipass case) can be achieved in this type of amplifiers. However, this design requires high stretching factors for the input seed pulses (hundreds of picoseconds) to avoid the catastrophic beam collapse inside the amplifying medium (B integral, which is a measure of the nonlinear phase shift of light, is used in these cases as a descriptive quantity [Sie86]). Such stretching factors are, as a rule, achieved with a grating stretcher design.

Pockels Cell _Faraday Isolator Thin-Film Polarizer Thin-Film Polarizer λ/2 λ/4 Pump

Pump Ti:Sa Ti:Sa

Input Output Input

Output

Multipass Amplifier

Regenerative Amplifier

Figure 2.1: Multipass and regenerative amplifier schemes. Adapted from [Pasa; Pasb].

In a multipass amplifier, the number of passes (in this case two) is strictly fixed, whereas in a regenerative amplifier, this number can be varied by adjusting the timing of the Pockels Cell.

2.1.2

Gain Curve Line shape: Gain Narrowing

Both multipass and regenerative amplifiers are used to deliver mJ-level, CEP-stabilized pulses. The pulses directly from a laser amplifier are, however, not few-cycle and require a nonlinear compression stage (be it an HCF [Nis96; Nis97; Sud05] or spectral broadening in bulk media and subsequent re-compression [Lu14; Sei17]). The reason for this is the gain narrowing phe-nomenon [Sie86]. Assuming the simplest case without saturation effects or effects caused by the

2.1 Gain in Solid-State Laser Amplifiers 13

short pulse duration, the power gain line shape is the same as the atomic line shape [Hen00]: G(ω) = exp  σ NL 1 + [2(ω − ωa)/∆ωa]2  (2.1) Where:

σ - the emission cross section (for Ti:Sa it is 3 × 10−19cm2[Hen00]). N - the population inversion.

L- the path length of the amplification medium. ω - the angular frequency.

ωa- the atomic transition angular frequency.

∆ωa - the atomic linewidth (full width at half maximum (FWHM)) of the atomic resonance

an-gular frequency.

The equation is written for media with homogeneous linewidth broadening, i.e. for the Lorentzian line shape. In the case of inhomogeneous linewidth broadening, the expression would include a Gaussian instead of a Lorentzian [Sie86; Sve10].

0.5

1.0

_(a)

600

650

700

750

800

850

900

950

1000

Wavelength [nm]

0.0

0.5

1.0

Intensity [normalized]

(b)

FWHM = 209 nm

FWHM = 32 nm

Figure 2.2: Ti:Sa fluorescence and gain narrowing in a multipass amplifier.

(a). Fluorescence spectrum of a Ti:Sa crystal taken via Ocean Optics USB2000+ spectrometer. Spectrum was taken from an operational multipass amplifier with the crystal temperature at 185 K.

(b). Gain narrowing in a multipass amplifier. Every second pass is shown. Spectra taken via Ocean Optics USB2000+ spectrometer. The broadband seed (the violet curve) narrows down and the central wavelength slightly shifts (as evident from the green curve).

The described Lorentzian (or Gaussian) line shape of the gain curve leads to a higher gain near the peak central wavelength, ultimately shrinking the spectral bandwidth of the pulse. This phenomenon is know as the gain narrowing and puts a rather fundamental obstacle when trying to preserve the spectral bandwidth of the amplified pulses. An ample example of the severity of the gain narrowing effect is presented in Figure 2.2 (b), with a measured Ti:Sa fluorescence

curve in (a) to serve as a reference. These measurements were performed with an OceanOptics USB2000+ spectrometer during the development of the ultrabroadband multipass amplifier1, with the spectra in (b) corresponding to the seed input (violet curve), the spectrum after the second pass (blue curve) and the spectrum after the fourth pass (green curve) inside the multipass amplifier. It is clear that the FWHM shrinks by approximately 7 times in the given example.

2.1.3

Gain Regimes

As the seed becomes more energetic with each pass through the amplification medium, the satu-ration effects become noticeable and the gain per pass decreases. In general, 3 gain regimes are differentiated:

1. Small-signal gain: In this regime no saturation effects are present, the amplification factor is close to constant. This regime is sometimes known as the linear gain regime and can be expressed as2[Hen00]:

Glin= exp [σ NL] (2.2)

Where the σ , N and L parameters are the same as in Equation 2.1.

2. Intermediate-signal gain: Saturation effects become noticeable, the amplification factor de-creases with each subsequent pass.

3. Saturated/large-signal gain: the amplifier is in the saturation regime - the amplification factor becomes increasingly low (usually in the region of a factor of 2), significant imprint on the amplification process is noticeable. The equation governing this regime may be expressed as [Hen00]:

Gsat= Glin exp

 −Eout− Ein F_sat A  (2.3) Where:

E_out& Ein- the output and input pulse energies.

F_sat= ¯hω_σ - the saturation fluence. A- the illuminated area.

A simulation based on Equation 2.2 and Equation 2.3 gives a visual representation of the ampli-fication process. The linear gain coefficient Glin and the input energy Einare taken to be close to

the experimental values in Chapter 3. The Glin = 10 for 8 mm propagation in Ti:Sa, while the Ein

is 1 nJ. The saturation energy is set to 3 mJ.

Figure 2.3 shows how the amplification factor evolves when the output energy approaches the saturation energy (a) and how the output energy increases when increasing the number of passes through the Ti:Sa gain medium (b). From Figure 2.3 (b) the three gain regimes described previously can be visually differentiated. The region where the energy increases exponentially (where the red and violet curves overlap) is the small-signal gain regime. Once the two curves

1_{These plots are part of the results presented in Figure 3.4 and Figure 3.11 with a detailed discussion.} 2_{Assuming that gain does not depend on the frequency.}

2.1 Gain in Solid-State Laser Amplifiers 15

start to diverge, the intermediate-signal gain regime starts, followed by the saturated/large-signal gain regime. From the zoomed-in graph in (c) it is visible that the point at which the intermediate-signal gain starts is around 150 µJ.

0.0 0.5 1.0 1.5 2.0 2.5 3.0

Output Energy [mJ]

2

6

10

Amplification Factor

(a)

E

= 1 nJ

G

= 10

40

60

80

Ti:Sa Total Length [mm]

0

1

2

3

Output Energy [mJ]

(b)

E

= 1 nJ

E

= 3 mJ

G

38

41

44

0.1

0.2

0.3

(c)

Figure 2.3: The simulation of the gain factor and the amplification process in a Ti:Sa amplifier.

(a). The gain factor evolution for different output energies assuming Glin = 10 and Ein = 1 nJ. The

saturation energy is set to 3 mJ.

(b). The output energy depending on the number of passes through the Ti:Sa crystal (expressed as the total Ti:Sa length). Ein= 1 nJ.

(c). Zoomed-in version of (b) in the region where the intermediate-signal gain regime starts and the simulated curves begin to show different dependence.

Red curve- the amplification taking into consideration the saturation behavior. Violet curve- the amplification assuming only linear gain.

Orange line- the saturation energy.

Since the gain in the small-signal regime is exponential and decreases in the subsequent regimes, the majority of the gain narrowing occurs precisely in this region (which, for the cases that we discuss, persists until 150-200 µJ). This is visualized in Figure 2.4. The single-pass gain curve is depicted in (a), while the effect of 1.5 · 105(up to 150 µJ, assuming input energy of 1 nJ) amplification on an experimentally measured seed spectrum (from Chapter 3) is shown in (b). The curve depicted in (a) was generated by using Equation 2.1. In this equation the ∆ωaatomic

linewidth parameter value was set to better match the experimental results observed in Chapter 3. The amplified spectrum is calculated by multiplying the input spectrum by the single-pass gain presented in (a) until the energy reaches 150 µJ.

Figure 2.5 shows the energy evolution in the presence of fluctuations in the seed energy (a) and in the linear gain coefficient3(b).

600

700

800

900

Wavlength [nm]

0

5

10

Amplification Factor

(a)

600

700

800

900

Wavelength [nm]

0.0

0.5

1.0

Intensity [normalized]

(b)

G

=

1.5 10

Figure 2.4: The simulation of the gain in a Ti:Sa amplifier excluding the gain saturation effects. (a). The wavelength-dependent gain curve. Based on Equation 2.1.

(b). Green curve - the input seed spectrum.

Blue curve - the output spectrum after 1.5 · 105 linear amplification. The spectrum is normalized for convenience. The gain process was simulated according to Equation 2.1.

0

20

40

60

80

Ti:Sa Total Length [mm]

0

1

2

3

Output Energy [mJ]

(a)

E

= 1.0 nJ

Ein = 0.5 nJ

E

= 1.5 nJ

G

= 10

20

40

60

80

(b)

G

= 10

Glin = 8

G

= 12

E

= 1.0 nJ

Figure 2.5: The simulation of the fluctuations in a Ti:Sa amplifier.

(a). The fluctuations in the output energy with the seed fluctuations equal to ±50%. Glinis kept constant

at 10.

(b). The fluctuations in the output energy with the Glinfluctuations equal to ±20%. Ein input energy is

2.2 Carrier-Envelope Phase 17

As it is visible, in both cases, despite the large fluctuations, the amplifier has stable energy output when operating near the saturation regime. This is the reason that laser amplifiers usually operate close to saturation.

Since the gain factor decreases dramatically, the gain narrowing compensation is crucial for the passes in which the seed is effectively in small-signal gain regime. The need for the com-pensation decreases as the seed becomes more energetic. It is straightforward to utilize this fact when dealing with multipass systems, since the passes in such systems are geometrically distinct. For a regenerative amplifier such an approach is rather impossible.

2.1.4

Conclusions

Taking into consideration the discussions in this section, the following conclusions can be drawn for Ti:Sa amplifiers:

1. The amplifier should operate in the saturation regime to improve the system stability. 2. The majority of the gain narrowing occurs in the small-signal gain regime4. By ensuring

(via the spectral filtering) a broad spectrum in the small-signal gain regime, an ultrabroad-band output from the amplifier can be guaranteed.

A deeper insight into the theory of laser amplification is provided in [Sie86; Sve10].

2.2

Carrier-Envelope Phase

2.2.1

Definition

The generated ultrashort laser pulses can be described both in the spectral and temporal domains. The discussion in the previous section was concentrated on the spectral domain and the associ-ated gain narrowing effect. In order to understand the notion of the CEP, it is helpful to remember the mathematical representation of the electric field of a Gaussian pulse in the temporal domain:

E(t) = Re {E₀exp[i(ω0t− φ (t))]} (2.4)

Where E(t) corresponds to the electric field, ω0is the carrier angular frequency and φ (t) is

the temporal phase. In this representation, the temporal phase can be expanded into Taylor series around the t = 0 point:

φ (t) = φ (0) + φ0(0) · t +1 2φ

00_{(0) · t}2₊1

6φ

000_{(0) · t}3_{+ ... (2.5)}

In such a case, the zero-order term in the series that does not depend on time will yield the absolute phase between the carrier-wave and the pulse envelope. This is precisely the CEP. It 4_{In the case of our future experimental setup presented in Chapter 3, the small-signal gain regime persists until} 150-200 µJ, as in the simulations presented in this section.

is evident from the physical nature of this quantity, that a shift in the CEP is equivalent to the difference between the phase and group velocities of the pulse. In general, the phase shift is directly connected to dispersion-inducing phenomena.

2.2.2

The Carrier-Envelope Offset Frequency

In order to gain a deeper understanding of the CEP of oscillator pulses, it is instructive to first switch back to the frequency domain representation. It can be mathematically shown that the pulse train emitted from a mode-locked oscillator can be represented in the spectral domain as an equidistant frequency comb with a separation between the frequency modes equal to the free spectral range of the cavity _2Lvg, where vg is the group velocity of the speed of light and L is the

optical length of the cavity [Rei99].

Figure 2.6: Spectral and temporal representations of an oscillator pulse train. Adapted from [L¨u14a]. The CEP of the first pulse in the temporal domain is 0. The phase slip ∆Φ is π

2 between each subsequent

pulse. CEP stabilization of the oscillator means that the fCEO is stabilized (in the case of this figure

fCEO= frep/4) and the ∆Φ phase slip is constant.

2.2 Carrier-Envelope Phase 19

(by convoluting the electrical field of a single pulse with a Dirac delta function) and Fourier trans-form the expression to the frequency domain, the following equation for the frequency modes fm

of the comb can be found [L¨u14a]:

fm= m · frep+ fCEO (2.6)

Where frep is the repetition rate of the oscillator and fCEOis called the carrier-envelope offset

(CEO) frequency: fCEO≡ 1 2π  δ Φ δ t mod 2π  = 1 2π  ∆Φ trt mod 2π  (2.7) Equation 2.6 makes it clear that the equidistant frequency comb produced by a mode-locked oscillator is shifted by the fCEO amount from the zero frequency. This is graphically depicted

in Figure 2.6. The fCEO describes the rate of change of the CEP (Φ) after one round trip in

the oscillator cavity. The ∆Φ quantity is the CEP slip per round trip. In Figure 2.6 the CEP of the first pulse is 0, while the ∆Φ phase slip is π

2 between every neighboring pulse. The phase

slip is affected by either dispersion changes inside the oscillator or the change in the nonlinear refractive index, which leads to the fluctuations in the self-phase modulation (SPM) process. From Equation 2.7:

∆Φ = 2πfCEO frep

(2.8) From this equation it follows that if the fCEO is stabilized to the N-th fraction of the repetition

rate, then every N-th pulse will have the same CEP value. As an example, fCEO = frep/4 in

Figure 2.6. Consequently, if the fCEOis set at zero, then the CEP slip will be zero and each pulse

in the pulse train will have the identical CEP.

2.2.3

Measuring the f

: An f-to-2f Interferometer

In order to determine the fCEO, self-referencing technique with an f-to-2f interferometer can be

used [Tel99]. The measurement is based on the fact that the pulse resulting from the second harmonic generation (SHG) process has double the amount of fCEO compared to that of the

fundamental pulse [L¨u14a].

If the fundamental spectrum spans an octave, then the lower frequencies of the resulting second harmonic beam will interfere with the same frequency components, which are present in the high-frequency range of the fundamental. The graphical representation of this idea is shown in Figure 2.7. This interference will result into a beat signal, which can be measured with a photodiode. From this signal the fCEOcan be calculated. In the case when the oscillator spectrum

is narrower than one octave, the SPM process in either bulk material or single-mode fibers is employed to broaden the spectrum, making it sufficiently for such a measurement [Apo00].

Figure 2.7: Visual representation of the frequency combs in an f-to-2f interferometer. The highlighted fre-quency component is present both in the fundamental and second harmonic combs. Adapted from [L¨u14a].

2.2.4

Stabilizing the CEP Slip in Oscillators: Feed-Back and Feed-Forward

Schemes

Figure 2.8 presents a generalized schematic of a chirped mirror oscillator. The parts that may in-fluence the fCEOare highlighted. In the case of the oscillators, the CEP slip needs to be stabilized.

This can be achieved in 2 ways:

1. Modifying the nonlinear refractive index or changing the dispersion inside the resonator cavity of the oscillator, hence affecting the CEP.

2. Changing the CEP of the pulses after the resonator cavity.

The fluctuations in fCEOinside the cavities of ultrafast oscillators are caused by several factors.

The quantitative values for these are as follows [Hel02]:

1. Crystal temperature and air pressure fluctuations, causing change in the linear refractive index. The measured δ fCEO= 1.4 MHz/K and 20 kHz/Pa respectively.

2. Amplitude-to-phase noise due to intensity-dependent beam pointing, resulting in δ fCEO=

5· 10−4Hz cm2/W for oscillators with prism-based compressors and δ fCEO= 10−4Hz cm2/W

for prismless systems.

There are a number of parameters one can tweak in order to induce the desired change of the CEP inside a laser cavity, such as: modulating pump laser intensity (changing the nonlinear refractive index) [Xu96]; changing the Ti:Sa crystal temperature (changing the linear refractive index) [Yun09]; changing the amount of path in glass inside the cavity (chirped mirror based oscillator cavities employ wedges for fine-control of the dispersion) [Xu96].

2.2 Carrier-Envelope Phase 21

Ti:Sa

CM 1

OC

Pump

Changing the linear refractive index by changing the temperature Changing the nonlinear refractive index

by changing the pump intensity

Changing the CEP (slowly) by adjusting the dispersion

Wedges CM 2 CM 4 CM 3 CM 5 CM 6 FM 2 FM 1

Figure 2.8: Parameters affecting the phase slip in a chirped mirror oscillator cavity.

A generic scheme of a Ti:Sa chirped mirror oscillator highlighting the parts that influence the phase slip. These parts are both a source of noise and parameters that can be tweaked to stabilize the fCEO.

FM 1, FM 2 - focusing mirrors. CM 1-4 - intracavity chirped mirrors. OC - output coupler.

CM 5, CM 6 - external chirped mirrors.

Figure 2.9: The schematic of a feed-back fCEO stabilization module. Adapted from [L¨u14a]. The pump

All these are visualized in Figure 2.8. In order to tweak these parameters and stabilize the fCEO, an f-to-2f interferometer is employed to measure the fCEO followed by a PI controller

that applies the correction to the pump intensity/crystal temperature/amount of dispersion. Due to its nature, this method is known under the name of feed-back fCEO stabilization. A rough

sketch is presented in Figure 2.9. However, all the mentioned parameters can correct noise up to relatively low frequencies. The “fastest” in this case is the pump intensity modulation. This method can correct noise theoretically up to 100 kHz [L¨u14a] (while in practice it is closer to 10 kHz). Changing the Ti:Sa temperature and the amount of the introduced dispersion are useful only when correcting long-term drifts occurring on the order of minutes and longer. Other than that, an important problem is the presence of both low and high frequency noise, meaning that the PI controller cannot efficiently cope with the situation. This leads to the dilemma of exchanging the long-term stability for robustness of the stabilization scheme [Moo14]. In order to avoid this, the method of stabilizing the fCEOafter the oscillator cavity should be employed.

f

= n∙f

+ f

α

= α

= α

f

= 0

x

y

z

-1

_order

λ

~ f

_(t)

Acoustic Wave

α

α

f

= n∙f

+ f

Figure 2.10: The acousto-optic frequency shifter in the feed-forward scheme. Adapted from [Kok10]. The featured acousto-optic frequency shifter (AOFS) is driven at the radio frequency of fRF= frep+ fCEO

for higher efficiency. The device is optimized for the Bragg’s angle of the -1stdiffraction order.

In order to stabilize the phase slip after the resonator, an AOFS can be used. A radio fre-quency (RF) wave applied to the AOFS crystal causes photon-phonon interaction and effectively generates a transient grating inside the AOFS - subtracting the fCEOin the -1st order of the

grat-2.2 Carrier-Envelope Phase 23

ing. This sets the phase slip to zero and results in an oscillator pulse train in which each pulse has the identical CEP. The method is known under the name of feed-forward5stabilization technique and can correct noise up to 300 kHz [L¨u14a]. A rough sketch of the idea behind the feed-forward method is presented in Figure 2.10. The AOFS is driven by an RF wave with the frequency of fRF= frep+ fCEO. In that case the AOFS can achieve high diffraction efficiency - nearly 70% of

the energy is in the f_CEO-subtracted -1storder. The major benefit of this approach is the fact that fCEOis directly subtracted and no feedback mechanism based on a PI controller is necessary. A

schematic implementation of this scheme for an oscillator is presented in Figure 2.11.

The feed-forward method completely avoids the dilemma of robustness vs. stability that is present in conventional feed-back schemes. Among the possible issues, the angular chirp induced by the AOFS may be mentioned. This can be corrected via prism, resulting in a small amount of residual spatial chirp [L¨u14a].

Figure 2.11: The schematic of a feed-forward fCEO stabilization of oscillator pulses. Adapted

from [L¨u14a].

2.2.5

CEP Stabilization of Ti:Sa Amplifiers

The limits of the oscillator R.M.S noise are on the order of 100-140 mrad for the feed-back [Ver12; Ver14] and sub-50 mrad for the feed-forward [L¨u12] schemes. However, the CEP R.M.S noise of the pulses after an amplifier has been, for the most cases6, around 250-400 mrad [Rau06; Li08;

5_{Since there is no feedback to the cavity.}

6_{These exclude the relatively recent results achieved with a feed-forward scheme in [L¨u14b]. This will be} touched upon later in this subsection.

Her11].

In the case of the amplifiers, a significant source of the CEP noise is due to the grating stretcher, which primarily used in regenerative amplifier schemes. Due to the presence of cavities in regenerative amplifiers, large stretching factors on the order of hundreds of picoseconds are necessary not to damage the gain medium due to the self-focusing effect. Although it is possible to stabilize the CEP of a CPA system with a grating stretcher, this is rather problematic [Kak04]. Even minuscule changes in the distance of the gratings in the stretcher induce a substantial CEP noise - a change on the order of 1 µm induces a CEP change of 3.7±1.2 rad [Li06].

Multipass amplifiers, on the other hand, can completely avoid this issue by using bulk stretch-ers to stretch the pulse duration to only several tens of picoseconds. Apart from the obvious benefit of not inducing any noise from the bulk stretcher, this leads to the possibility of a com-pact grating compressor design, significantly reducing the CEP noise from the latter. Besides the stretcher, the rest of the CEP fluctuations in an amplifier can be classified as “low-frequency”, and are (relatively) easy to correct [L¨u14a].

In the case of the amplifier output pulses, an f-to-2f interferometer is used in combination with a spectrometer in order to resolve the spectral modulation. From this spectrogram the CEP can be calculated by means of a fast Fourier-transform (FFT).

The best reported results for a CEP-stable amplifier, involving a feed-back fCEO

stabiliza-tion scheme for the seed oscillator, are in the range of 190 mrad R.M.S noise [And11a]. The results of the feed-forward scheme have shown to be superior, achieving sub-150 mrad R.M.S noise [L¨u14a; L¨u14b].

For compensating the amplifier-induced CEP drift, several solutions can be employed (e.g., changing the dispersion amount in the stretcher by varying the bulk material length). However, when employing the feed-forward stabilization scheme for the oscillator, an elegant solution is to utilize the phase of the AOFS RF wave, since after the oscillator CEP stabilization it remains as a free parameter [L¨u14b]. This last method will be demonstrated in Chapter 3.

2.2.6

Quantifying the CEP Noise: Power Spectral Density and Integrated

Phase Noise

When measuring the CEP noise, usually the R.M.S value is calculated. For the purpose of analyzing and quantifying the measured noise, it is convenient to introduce the power spectral density (PSD) and integrated phase noise (IPN) quantities:

The PSD visualizes the amplitude of each frequency component of the noise in the frequency domain independent of the measurement’s frequency resolution [L¨u14a]. Let us assume that Φ(tn) is the measured CEP at equidistant tn= n · ∆t (n = 0, 1, ..., N) time intervals. By taking the

discrete Fourier-transform of this quantity ˜Φ( fk) ( fk= k ∆ f = _N·∆tk ), the PSD can be defined as

follows: PSD( fk) =    | ˜Φ( f0)| N√∆ f for k = 0 √ 2| ˜Φ( fk)| N√∆ f for k = 1, 2, ..., N/2 (2.9)